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Laboratory measurements of microwave absorptivity and refractivity spectra of gas mixtures applicable to giant planet atmospheres

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L a b o ra to ry m easu rem en ts o f m icrow ave a b so rp tiv ity an d
re fra c tiv ity sp e c tra of gas m ix tu re s applicable to giant p la n e t
a tm o sp h e re s
Spilker, Thom as R ichard, Ph.D .
Stanford University, 1990
C o p y r ig h t © 1 9 9 0 b y S p ilk e r , T h o m a s R ic h a r d . A ll r ig h ts r e se rv ed .
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LABORATORY M EA SU REM EN TS OF
M IC R O W A V E A B S O R P T IV IT Y A N D R E F R A C T IV IT Y SPE C T R A O F
G A S M IX TU R ES A P P L IC A B L E T O G IA N T P L A N E T A TM O SPH ER E S
A D IS S E R T A T IO N
S U B M IT TE D T O T H E D E P A R T M E N T O F E L E C T R IC A L E N G IN EE R IN G
A N D T H E C O M M IT T E E O N G R A D U A T E STU D IES
O F S T A N F O R D U N IV E R S IT Y
IN P A R T IA L F U L F IL L M E N T O F T H E R E Q U IR E M E N T S
FOR TH E D EG REE OF
D O C T O R O F P H IL O S O P H Y
By
T hom as R ichard Spilker
Ju n e 1990
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© C op y rig h t b y T hom as R. S pilker 1990
A ll R ights R eserved
ii
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I certify that I have read this dissertation and that in m y opinion it is fully adequate,
in scope and quality, as a dissertation for the degree o f D o cto r o f Philosophy.
Prof. Von R . E shlem an (Principal A dvisor)
I certify that I have read this dissertation and that in m y opinion it is fully adequate,
in scope and quality, as a dissertation for the degree o f D o cto r o f P hilosophy.
/
Prof. G. L eonard T yler
I certify that I have read this dissertation and that in m y opinion it is fully adequate,
in scope and quality, as a dissertation for the degree o f D o cto r o f Philosophy.
Prof. R obert M . G ray
A pproved for the U niversity C om m ittee on G raduate Studies:
N
Dean o f G raduate Studies
iii
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Abstract
A ccurate fo rm u lae fo r p red ictin g m icrow ave ab so rp tiv ities and refractiv ities o f gas m ix tu res o f
hydrogen, helium , m ethane, and am m onia are vital fo r interp retin g results fro m rad io occu ltatio n and
ra d io a stro n o m ic a l in v estig atio n s o f the atm o sp h ere s o f th e g ia n t p lan ets.
In th o se atm o sp h eres
am m o n ia is the source o f essen tially all m icro w av e o p ac ity at levels accessible by rad io m ethods.
U nfortunately , accurate m ethods fo r calculating m icro w av e ab sorptivities o f gas m ix tu res con tain in g
am m onia under specified conditions have eluded spectroscopists fo r m ore than h a lf a century, partly due
to a lack o f adequate laboratory data.
A m icro w av e spectrom eter, based o n a cav ity reso n ato r, w as constructed to m easu re refractiv ity
spectra o f the tran sp aren t gases hydrogen, h elium , an d m ethane, an d to m easu re both refractiv ity and
absorptivity spectra o n gas m ixtures containing am m o n ia. G as conditions in clu d ed tem peratures from
about 2 1 0 to 320 K , pressures from 1 to 8.2 atm ., and th e sp ectra covered m icrow ave frequencies from 9
to 18 G H z.
D a ta on the transparent gases confirm that th e ir den sity -n o rm alized refractivities are inv arian t w ith
tem perature, pressure, and frequency w ithin m easu rem en t accuracies. R efractivities are co n sisten t w ith
accepted values a t optical frequencies: 135 X l O
6
fo r hydrogen, 35 X 10~6 fo r helium , and 4 4 0 x 10 ‘ 6
for m ethane, un d er standard conditions. R efractivities o f m ixtures are linear in the partial p ressures o f
constituent gases.
A bso rp tiv ity d ata on m ixtures containing am m o n ia show serious d isagreem ents w ith c u rren t p re­
diction form alism s. V an V leck -W eissk o p f theory, and B en -R eu v en theory as m o d ified by B erge and
G ulkis (1976), both fail to accurately pred ict observed tem perature, pressure, and frequency dependences.
O ptim ization techniques th at fit a param eterized B erg e an d G ulkis form alism to the data pro d u ced a new
form alism that m ore accurately predicts m icrow ave absorption by am m onia.
iv
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T h e new fo rm alism p red icts th a t at th e low tem peratures ex ta n t in the atm o sp h eres o f the g ian t
p lanets, am m o n ia w ill be less opaque to m icrow aves than prev io u sly thou g h t. T his w ill req u ire revis­
ing upw ard the am m onia abundances inferred from radio m ethods, by a few percent to as m uch as 50% .
v
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Acknowledgements
F irst and forem ost, I w ould lik e to th an k m y research advisor, V .R . E sh lem an , an d m y co-advisor,
G .L. T yler, fo r their guidance in this research. T heir scientific insight an d unquestio n ab le integrity w ere
an inspiration in the p u rsu it o f excellence. R .M . G ray serv ed as th e third reader. I also th an k L en T yler
for the opportunity to w ork w ith the V oyager pro ject and exp erien ce planetary ex ploration at its finest.
I am grateful to the num erous persons w ho helped w ith the m echanics o f this research. B ill C rosby
and B laine B olich taught m e the m achine shop skills needed to b u ild com ponents o f the spectrom eter. M y
good friend T ony Straw a o f the A eronautical and A stronautical E n gineering D ep artm en t in troduced me to
gas system technology, and m achinist A ldo R ossi assisted w ith critical m achining o f the sealing surfaces
on the sim ulation cham ber. T hanks go to G eneral Sealants C orp o ratio n fo r p ro viding their G S -1 0 0 seal­
ant that enab led operations at 213 K . T ony Fraser-S m ith an d Paul M cG ill p ro v id ed the tim e base o scilla­
tor and digital m ultim eter used in frequency counter and therm om eter calibrations; D on B ag an o ff provided
the pressure g au g e calibration apparatus. M aintenance o f the laboratory site, S ite 522 a t the 150-foot dish
antenna, w as handled by Bill C rosby and B ill Trabucco, w ho are both equally adept at scientific inquiry and
plum bing. D ick Sim pson an d Paul R osen k e p t the D ata G eneral M V /1 0 0 0 0 co m p u ter o perating and gave
valuable advice about its use. C alvin T eague m anaged the o th er system , the S T A R V A X cluster, th at did
jo b s the M V /10000 w as not con fig u red to d o . D iscussions o f the topic o f this research w ith R .L. Poynter
o f JP L provided valuable insight into its role in planetary science and spectroscopy.
I appreciate the m any scintillating discussions w ith the faculty, sta ff an d g raduate students of Stanford
STA R L ab and the Stanford C enter for R ad ar A stronom y, and the abundant sup p o rt from w hat m ust be one
o f the best groups o f secretaries in the w orld.
F inally, I th an k m y parents, M r. and M rs. W .G . Spilker, for their u n failin g sup po rt and encourage­
m ent throughout this seem ingly endless p roject.
T his research w as supported by the P lan etary A tm ospheres P ro g ram o f the N atio n al A eronautics and
Space A dm inistration.
vi
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Table of Contents
A b s tr a c t
iv
A c k n o w le d g e m e n t s
vi
T a b le o f C o n te n ts
v ii
L ist o f T a b le s
ix
L is t o f F ig u r e s
xi
1 I n t r o d u c tio n
2
3
4
5
1
1.1
Background and M o t i v a t i o n ...............................................................................................................
1
1.2
O rganization and S c o p e .......................................................................................................................
5
1.3
C o n t r i b u t i o n s ........................................................................................................................................
6
R a d io T e c h n iq u e s F o r
th e S tu d y o f G as G ia n t P la n e ts
2.1
Introduction and G eneral D iscussion
2 .2
2.3
8
..............................................................................................
8
R adio A stronom y
...............................................................................................................................
10
R adio O ccultation
...............................................................................................................................
16
M ic r o w a v e A b so r p tio n a n d th e A m m o n ia M o le c u le
23
3.1
M icrow ave Propagation in an Isotropic M e d i u m ........................................................................
23
3.2
Structure o f the A m m onia M o l e c u l e .............................................................................................
27
3.3
M icrow ave A bsorption by A m m onia Inversion
29
3.4
Pressure B roadening o f the A m m onia Inversion L ines
........................................................................
...........................................................
E x p e r im e n t S tra teg y , A p p a r a tu s, an d P r o c e d u r e s
4.1
T echniques for M easuring A bsorptivity and R efractivity
4 .2
E xperim ent A pparatus
4.3
34
38
.......................................................
38
.......................................................................................................................
43
L aboratory Procedures, D ata R eduction, and U n c e r ta in tie s .......................................................
51
L a b o r a to r y M e a su r e m e n ts o f M ic r o w a v e R e fr a c tiv ity S p e c tr a on
G a se o u s M e th a n e , H y d r o g e n , an d H e liu m
59
5.1
D ata on Pure M ethane and M ethane M ixed W ith H ydrogen
..................................................
59
5 .2
D ata on Pure H y d r o g e n ......................................................................................................................
67
5.3
D ata on P ure H e l i u m ..........................................................................................................................
72
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6
7
8
L a b o r a to r y M e a su r e m e n ts o f M ic r o w a v e A b so r p tiv ity and R e fr a c tiv ity
S p e c tr a on G a s M ix tu r e s C o n ta in in g A m m o n ia
................................................................................................................................
6.1
D ata O rganization
6 .2
D ata o n P ure G aseous A m m onia
6.3
D ata o n G as M ixtures C ontaining A m m onia
..................................................................................................
.............................................................................
O p tim iz a tio n o f a P a r a m e te r iz e d B e n -R e u v e n F o r m a lism to F it
th e A m m o n ia D a ta
80
80
82
89
110
7.1
The B en-R euven Form alism as M odified by B erge and G u l k i s ................................................... 110
7.2
Param eterization o f the B erge and G ulkis F o r m a l i s m ....................................................................117
7.3
Procedures For O ptim ization o f the P aram eterized F o r m a l i s m ...................................................122
7.4
A nalysis and R e s u l t s ............................................................................................................................... 127
C o n c lu s i o n
156
8.1
S u m m a r y .................................................................................................................................................... 156
8.2
P rincipal C onclusions
8.3
S uggestions F o r F uture R esearch
...........................................................................................................................158
......................................................................................................159
A p p e n d ix A
D esig n o f th e R e so n a to r S ig n a l P r o b e s
A p p e n d ix B
A M e th o d o f C o m p e n sa tin g F o r S p ectru m
In te r m e d ia te F r e q u e n c y E rr o r s
A p p e n d ix C
162
A n a ly z e r
165
L a b o ra to ry P r o c e d u r e s
173
C .l
C a l i b r a t i o n ................................................................................................................................................173
C .2
Screen Af and C ounter Af M ethods
C .3
Standard P r o c e d u r e ...................................................................................................................................180
A p p e n d ix D
................................................................................................. 176
D a ta R ed u c tio n P ro c ed u r e s an d U n c e r ta in ty
A n a ly sis
189
D .l
Screen Af M e t h o d s ....................................................................................................................................190
D .2
C o u n ter Af M e t h o d s ...............................................................................................................................196
D .3
Standard P r o c e d u r e .................................................................................................................................. 202
R e fe r e n c e s
224
viii
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List o f Tables
1 .1
C onstitu en t abundances in the tropospheres o f th e g ian t p l a n e t s ........................................................
4.1
R esonator center frequencies, m ode num bers, and Q
5.1
M easured refractivities o f pure m ethane using the screen Afm ethod
.................................................
62
5.2
M easured refractivities o f pure m ethane using the co u n ter Af m e t h o d .................................................
65
5.3
M easured refractivities o f a m ixture o f m ethane and h y d r o g e n ............................................................
66
5.4
M easured refractivities o f pure hydrogen n ear 213 K
..............................................................................
68
5.5
M easured refractivities o f pure hydrogen n ear 273 K
..............................................................................
70
5.6
M easured refractivities o f pure helium n ear 213 K
...................................................................................
73
5.7
M easured refractivities o f pure helium near 273 K
...................................................................................
75
5.8
M easured refractivities o f pure helium n ear 213 K
...................................................................................
78
6
.1
6.2
6.3
..............................................................................
2
44
M easured absorptivities o f pure am m onia at room tem perature (from Bleaney
and L oubser, 1 9 5 0 ) .............................................................................................................................................
85
M easured refractivities and absorptivities o f pure a m m o n i a ................................................................
88
M easured refractivities and absorptivities o f 0.82% am m onia in hydrogen, 213 K
.......................
93
6.4 M easured refractivities and absorptivities o f 0.82% am m onia in hydrogen, 273 K
.......................
95
.........................
97
......................
98
99
6.5
M easured refractivities and absorptivities o f 6.7% am m onia in hydrogen, 273 K
6 .6
M easured refractivities and absorptivities o f 0.82% am m onia in hydrogen, 323 K
6.7
M easured refractivities and absorptivities o f 0.82% am m onia in helium , 213 K
...........................
6 .8
M easured refractivities and absorptivities o f 0.82% am m onia in helium , 273 K
................................ 101
6.9
M easured refractivities and absorptivities o f 0.82% am m onia in helium , 313 K
................................ 103
6.10 M easured refractivities and absorptivities o f 6.7% am m onia in helium , 313 K ................................... 105
ix
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6.11 M easured refractivities and absorptivities o f 0.82% am m onia in 89.2% hydrogen
and 9.95% helium , 213 K ..................................................................................................................................... 106
6
.12 M easured refractivities and absorptivities o f 0.82% am m onia in 89.2% hydrogen
and 9.95% helium , 273 K ..................................................................................................................................... 108
x
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List of Figures
2.1
N atural planetary rad io e m is s io n s ..............................................................................................................
11
2 .2
R ad io occultation ex p erim en t i l l u s t r a t i o n .............................................................................................
17
2.3
R adio occultation exp erim en t g e o m e t r y .................................................................................................
20
3.1
P hysical in terpretation o f p ropagation constants a and P
...............................................................
25
3 .2
G eom etric m odel o f an am m onia m olecule
........................................................................................
28
3.3
L ow pressure am m onia absorption s p e c tr u m ...................................................................................
3.4
H igh pressure am m onia absorption spectrum
4.1
C avity resonator d e s i g n .........................................................................................................................
4 .2
A tm ospheric sim ulation ch am b e r design
4.3
Schem atic diagram o f the integrated m icrow ave sp e c tro m e te r......................................................
7.1
O ptim ization softw are design
7 .2
T rajectories o f optim al G|-|2-Z|-|2 pairs, param etric in C ................................................................ 133
7.3
O ptim al G j-)2 values v s hydrogen partial p r e s s u r e ............................................................................ 136
7.4
O ptim al Z h 2 values v r hydrogen partial p r e s s u r e .............................................................................142
7.5
O ptim al Z h 2 values v s optim al G h 2 values, w ith best-fit quartic p o l y n o m i a l ..................... 143
7 .6
Illustration o f tem perature and pressure ranges o f laboratory data and atm ospheric conditions . 149
7.7
C om parison o f laboratory d a ta from this w ork and predictions o f various form alism s . . . .
7.8
C om parison o f laboratory data from other w ork and predictions o f various form alism s
7.9
C om parison o f radio astronom ical data and results o f radiative transfer m odel predictions
A .l
D esign o f the new reso n ato r signal p r o b e s ......................................................................................... 164
33
.....................................................................................
35
45
..............................................................................................
47
48
...................................................................................................................... 125
152
. . . 153
. . 155
xi
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Chapter 1
Introduction
T his chapter provides an introduction to the research reported in this m anuscript. Section
1 .1
places
the w ork in the co n tex t o f m icro w av e spectroscopy and planetary science, discusses the m otivation for
pursuing this research topic, and states the goals o f th e w ork. The scope o f this w o rk and organization
o f th e m an u scrip t are o u tlin e d in S ectio n 1.2. R ead ers m ay fin d this sectio n useful, along w ith the
T able o f C ontents, in d eciding w h ich parts m ay be safely skipped and w hich m u st be read d iligently.
T he final section sum m arizes contributions to know ledge m ade by the au th o r as p art o f this research.
1.1
Background and Motivation
T o an ex traterrestrial visitor approaching o u r so lar sy stem fo r the first tim e, E arth an d all the o th er
"solid" bo d ies o f the p la n e ta ry sy stem w o u ld ap p ear to b e relativ ely in sig n ifica n t m em b ers o f th e
fam ily. F o u r o f the n o n -so lar objects, accounting fo r ab o u t 99.5% o f the p lan etary sy stem m ass, w ould
com m and the visitor's attention. T hese fo u r objects are k now n to us individually as the planets Jupiter,
Saturn, U ranus, an d N ep tu n e, an d co llectiv ely as the gia n t p la n e ts. U nlik e the o th er p lanets, w hose
atm ospheres are b u t tiny fractions o f th eir m asses, the g ian t planets have atm ospheres th at constitute a
significant fraction o f th e ir m asses, suggesting the com m on variant on th eir co llectiv e label: the g a s
g iant p la n e ts. W ith the re c e n t en co u n ter o f the N ep tu n e system b y V o y ag er 2 all fo u r g ian t planets
have been v isited by ro b o t sp a c e cra ft th at p ro v id ed a w ealth o f new in fo rm atio n a b o u t them . O ne
im portant categ o ry o f th at info rm atio n concerns the constituents o f the p lanets, i.e. the ch em ical e le ­
m ents and m olecular species th at are p resen t in them , and th e abundances o f those constituents.
C onstituents o f the g ian t p lanets are th ought to be m uch m ore rep resen tativ e o f the p reso lar neb u la
that condensed to form the p resen t so lar sy stem than those o f the o th er planets. W hen piecing together
1
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Ju p ite r
Saturn
Uranus
N eptune
Hydrogen (H 2 )
0.89
0.94
0.85
0.85
H elium (He)
0.11
0.06
0.15
0.15
M ethane (CH 4 )
0.002
0.0045
A m m onia (NH 3 )
V ariable w ith
altitude; less
than 2 5 0 ppm ?
V ariable w ith
V ariable with
altitude; 0.023
o r less
altitude; upper
lim it unknow n
V ariable with
V ariable w ith
V ariable with
altitude; less
than 100 ppm ?
altitude; u p p er
lim it unknow n
altitude; upper
lim it unknow n
T able 1.1: A p p ro x im ate n u m b e r m ixing ratio s o f the fo u r m ost ab u n d an t c o n stitu en ts in the upper
tropospheres o f the g ia n t planets. See text below .
the evolu tio n o f the p reso lar n eb u la to the cu rren t state o f the so la r system , k n o w led g e o f con stitu en t
abundances in all the planets is an im portant p iece o f the puzzle. H y d ro g en and helium , th e m ajo r con­
stitu en ts o f th e S u n , also d o m in ate the atm ospheres o f the g ian t p la n e ts w ith sim ilar m ix in g ratios.
M an y o th er species are presen t, b u t com p ared to hydrogen m o st h av e tiny co n cen tratio n s. T h e four
principle atm ospheric co n stitu en ts o f the g ian t plan ets are listed in T able 1.1 above. M ixing ratios for
hydrogen and h eliu m are from T y le r et al. (1989) for N eptune, and L indal et al. (1987) fo r the others.
U n certain ties in th o se figures ra n g e from about 3 to 5% . T h e Jo v ian m eth an e m ixing ratio is from
H anel et al. (1979). Interpretations o f V oyager spacecraft infrared d ata suggest th at at Saturn the mixing
ratio o f hydrogen m ay b e as larg e as 0.97 and th a t o f h elium as sm all as 0.03 (C onrath et al., 1984);
the stated m ethane m ix in g ratio a t Saturn is fro m C ourtin et al. (1 984). C o n d en satio n cau ses m ixing
ratios o f som e less ab u n d an t co n stitu en ts to vary considerably w ith altitude, w ith upper lim its d ictated
by the global abundances o f each species o r by atm ospheric ch em ical reactions. T h e u p p er lim it fo r the
m ethane m ixing ratio o f U ranus is derived from m odeling to yield a best fit to V oy ag er 2 rad io occu lta­
tion data (L indal e t a l., 1987). A m m onia m ixing ratios at Jupiter and Saturn w ere m easured by V oyager
2
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radio occu ltatio n ex p erim en ts (L in d al et al., 1981; L ind al e t al., 1985). A t Ju p ite r those data extend
d ow nw ard below the bottom s o f the am m onia clo u d s so the stated lim it is n o t affected by saturation.
B ut if am m onium hyd ro su lfid e clo u d s e x ist at lo w er levels th e u p p e r lim it m ay b e h ig h er below those
clouds. D ata d id n o t ex ten d b elow saturation-lim ited levels a t S atu rn , b u t there is reason to believe the
67 p p m m easured at the deep est datum is n e a r the unsaturated abundance b elow the am m onia cloud base.
A lthough am m onia is a relatively sm all co m p o n en t o f these atm ospheres, it is condensible at som e
levels and can fo rm c lo u d s; thus it is im portant in atm ospheric d y n am ics. W a te r vapor w ith in Earth's
atm osph ere is a g o o d ex am p le o f the im portance o f such a c o n stitu en t. A lth o u g h w ater v a p o r is a
m in o r com ponent o f E arth 's atm osphere com pared to n itrogen and o x y g en , its ability to tran sp o rt large
am ounts o f energy through phase changes has a trem endous im p act o n atm ospheric dynam ics. Species
w ith relativ ely low abundances can thus b e im portant fo r studying a p lan et's w eath er and w eath er phe­
nom ena in general. O ne d ifference betw een am m onia and w ater is th a t the v ap o r pressure o f am m onia at
a g iven tem perature is m uch h ig h er than th at o f w ater. F o r equal p artial p ressu res o f the tw o species
m uch low er tem peratures are req u ired to condense am m onia. T his p o in ts to a n o th er characteristic com ­
m on am ong the atm ospheres o f the g ia n t planets: at p ressu re levels u p to a fe w bars they are consider­
ably cold er than E arth's atm osphere. W here tropopause tem peratures a t E arth are about 220 K, a t Jupiter
they are ab o u t 110 K (L in d al e t al., 1981), and at U ran u s an d N ep tu n e they approach a frigid 5 0 K
(L in d al et a l., 1987; T y le r et al., 1989). Such low tem p eratu res c a n co n d en se am m o n ia and create
clouds even w hen its p artial pressure is q u ite sm all.
A m m o n ia is e sp ecially im p o rtan t to research ers using radio m ethods to stu d y the atm ospheres of
giant planets. R adio astronom ical and rad io occultation experim ents y ield vertical profiles o f m icrow ave
absorptivity alm ost entirely due to the strongly absorbing am m onia com p o n en t. H ydrogen, helium , and
m ethane are m ore a b u n d an t than am m onia, b u t at p ressures less than a few tens o f bars they are essen­
tially tran sp aren t to m icrow ave radiation. T he researchers then rely o n form ulae th at calculate absorp­
tivity d ue to am m onia, g iv en e n v iro n m en tal conditions, to e x tra c t am m o n ia ab u n d an ce pro files from
absorptivity p rofiles. H o w ev er, those abundance p rofiles can be n o m o re accu rate than th e am m onia
absorptivity form ulae used in the calculations. T hese topics are d iscu ssed in m ore d etail in C hapter 2.
3
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U nfortunately th e m icrow ave absorption spectrum o f am m onia h as b een a theorist's nightm are since
first o b serv e d m o re th an h a lf a century ago (C leeton and W illiam s, 1934). T h eoretical analysis o f the
absorption o f p ro p ag atin g electrom agnetic energy by m olecules re p re sen te d as sim ple harm onic o scil­
lators (V an V leck a n d W eissk o p f, 1945) yielded absorptivity pred ictio n s th a t fo r m any species agreed
quite w ell w ith lab o rato ry m easurem ents, y et for am m onia erred by facto rs o f tw o or m ore, especially at
h ig h er gas p ressu res. A s th e m ag n itu d e o f the p roblem b ecam e e v id e n t in the late 1940's and early
1950's m any researchers m easu red am m onia absorptivities in the laboratory under conditions dictated by
the lim itations o f th e ir e q u ip m e n t E ach new se t o f d ata p ro d u ced em p irical m odifications to the eq u a­
tions from the theo retical approach, b u t each new m odification late r p ro v ed to have significant trouble in
unsam pled areas o f the variable space, w hich consists o f the conditions th at affect m icrow ave absorption
by am m onia. A q u an tu m m echanical analysis o f the problem (B en-R euven, 1966) im proved agreem ent
betw een theory an d o bservations, especially for p u re am m onia at hig h er pressures, but it to o had trouble
w hen the am m onia w as m ixed w ith large am ounts o f other gas species such as hydrogen an d helium .
A no th er series o f laboratory m easurem ents and subsequent em pirical m odifications ensued. The last
m ajor m odification to equations from B en-R euven theory w as p u b lish ed b y B erge and G ulkis in 1976.
T h at form alism , w h ich pred icts th e m icrow ave absorptivity o f am m onia u n d e r conditions extan t in the
atm o sp h ere s o f th e g ia n t p lan ets, has been in co m m o n u se by p la n e ta ry scien tists fo r m ore than a
decade, but it also suffers inaccuracy in certain significant p ortions o f the variab le space. T hese errors
can be m uch larg er than the accuracy o f radio astronom ical and especially rad io occultation data already
(and at g re a t e x p en se!) acq u ired . S ignificant inform ation co n tain ed in th o se data is being m issed or,
w orse yet, m isinterpreted, due to o u r inaccurate know ledge o f the b eh av io r o f m icrow ave energy propa­
gating through gas m ix tu res like the atm ospheres o f the g ia n t planets.
T hese shortfalls in o u r k n ow ledge about gas m ixtures containing am m o n ia w ere the initial m otiva­
tion fo r this w ork. It w as recognized th a t a coherent program o f lab oratory m easurem ents w as required,
one th at w o u ld allo w ex am in atio n o f th e dependences o f am m o n ia ab so rp tiv ity o n all th e significant
m acroscopic co n d itio n s affectin g it in those atm ospheres.
S p ectro sco p ists as w ell as planetary scien­
tists w ould find such la W a to ry data valuable. O nce laboratory data w ere acquired, know ledge contained
4
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in the data m ust be in corporated into a more accurate se t o f form ulae (h ereafter called a fo rm a lism ) for
calculatin g m icro w av e ab so rp tio n due to am m onia u n d er g iv en co n d itio n s. R esearchers using radio
astronom ical and rad io o ccultation m ethods c o u ld then apply the new fo rm alism to yield m ore accurate
am m onia abundance calculations. T hese needs defined the goals o f the research presented here.
Som etim es in the p u rsu it o f these goals o th e r d ata w ere acquired, eith er o u t o f procedural necessity
or because the apparatus re q u ired to m eet the goals m ade acquisition o f th e d ata a sim ple m atter. T he
m ethod chosen for m easuring absorptivities o f gas m ixtures c o u ld also m ake m easurem ents o f refractivi­
ties w ith little additio n al effort; thus for each m easured absorptivity, a refractiv ity was also m easured.
E nsuring the accuracy o f absorptivity m easurem ents on gas m ixtures o f am m o n ia in hydrogen o r helium
required oth er m easurem ents on pure hydrogen and helium w ith o u t the am m o n ia com ponent, resulting
in a considerable body o f data on the m icrow ave refractivities o f p u re hydrogen and helium u n d er various
conditions. T esting the m icrow ave sp ectro m eter and e x p erim en t p ro ced u res dev elo p ed fo r this w ork
could be done w ith a w ide range o f test gases, so the o th er c o n stitu en t prev io u sly m entioned, m ethane,
w as included. L ike hydrogen and helium , m ethane is essen tially tran sp aren t at m icrow ave frequencies,
so only refractivity data w ere acquired on i t T hese subordinate data are reported here also.
1.2
Organization and Scope
T he m ajority o f this w ork m ay be org an ized into four areas: m otivation fo r the w o rk in the per­
spective o f p lan etary scien ce; theoretical aspects o f electro m ag n etic w av e p ropagation in absorbing
m edia and o f absorption by am m onia; laboratory m easurem ents; and ap p licatio n o f those m easure­
m ents to im p ro v e o u r kno w led g e concerning m icrow ave p ro p ag atio n in th e atm ospheres o f the g iant
planets. C h ap ter 2 is a d escrip tio n o f the rad io rem ote sensing techniques av ailab le fo r p ro b in g these
atm ospheres. It points o u t the need for spectroscopic d ata in red u cin g raw d a ta fro m those experim ents
to useful know ledge about the atm ospheres studied.
T heoretical aspects are treated in C hapter 3, w hich begins w ith a rev iew o f electrom agnetic w ave
propagation in a refractive, absorbing m edium . F ollow ing this is a d iscu ssio n o f the m ech a n ism o f
5
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m icrow ave abso rp tio n by am m onia and aspects o f its b eh av io r u n d er c ertain enviro n m en tal conditions.
A thorough understanding o f this phenom enon requires a full q u an tu m m echanical approach to th e p ro b ­
lem , b u t su ch an in vestigation is b eyond the scope o f this w ork. In the lig h t o f disagreem ents b etw een
observatio n s an d theory p o in ted o u t later in this w ork, it appears th at p h y sicists have y et to ach iev e a
fully adequate analysis o f the interactions betw een m icrow ave photons and am m onia m olecules. C hap­
ter 3 includes im p o rtan t v ocabulary that w ill be u sed th roughout the rem ain d er o f the text.
B uilding on the back g ro u n d p rovided by C h ap ter 3, laboratory m easurem ents o f m icrow ave refrac­
tivity and absorptivity spectra, the co re o f this w ork, are describ ed in the n e x t three chapters. C h ap ter 4
begins w ith the theory o f th e cavity reso n ato r m ethod an d the reasons fo r choosing that technique, and
continues w ith a d escription o f the apparatus, a m icrow ave spectrom eter, co n stru cte d b y th e au th o r to
im plem ent th e m easurem ents. H ardw are an d p rocedural innovations arising from the construction p ro ­
ject, o f interest to m icrow ave engineers and spectroscopists, are considered augm entations to the prim ary
topics o f this w ork and are included in A ppendices A and B. T his is also true o f the ex a c t ex perim ental
procedures u sed and uncertainty analyses for the m easurem ents, w hich are discussed in A ppendices C and
D , respectively. C h ap ter 5 presents refractivity d ata acq u ired o n the tran sp aren t gases m ethane, h y d ro ­
gen, and helium , and C h ap ter
6
presents absorptivity and refractivity d ata o n p u re am m onia and various
m ixtures o f hydrogen, helium , and am m onia. Included in C hapter
6
are d ata on p u re am m onia p ublished
by B leaney and L o u b ser in 1950 b u t n o t previously available in tab u lar form .
T he data o f C h ap ter
6
fin d application in C h ap ter 7, w here the B erge an d G ulkis (1976) m o d ifica­
tion to the B en-R eu v en form alism for predicting m icrow ave absorption is e x ten d e d to reflect the new
know ledge co ntained in those data. T his extension is em pirical in nature and is aim ed at providing m ore
accurate predictions o f am m o n ia absorptivity under given conditions. E x am p les o f the im provem ent in
absorptivity predictions are given, and som e im plications fo r planetary science are discussed.
A sum m ary o f the p rin cip al findings from this w ork appears in the co n clu d in g C h a p te r
8,
w hich
also contains several suggestions for future research.
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1.3
Contributions
In the course o f this w ork the au th o r has m ade the follow ing contributions to planetary atm ospheric
science and m icrow ave spectroscopy:
1
. Perform ed the first program o f m easurem ents addressing the dependences o f all know n significant
factors affectin g m icrow ave p ropagation in gas m ix tu res sim ilar to those fo u n d in the tro p o s­
pheres o f the g ian t p lanets: Jupiter, Saturn, U ranus, and N eptune;
2 . V erified th a t th e re fractiv ities o f hyd rog en , h elium , an d m eth an e are d ire c tly pro p o rtio n al to
n u m b er d en sity , add lin early in m ixtures, an d are in d ep en d e n t o f freq u en cy o v er the ran g e o f
conditions and frequencies observed;
3 . E stab lish ed th a t the B erge & G ulkis (1976) m o dification o f th e B en-R euven form alism p red ic­
ting m icro w av e a b so rp tio n by gaseous am m o n ia does n o t accu rately p re d ic t the tem perature
d ep en den ce o f th at p h enom enon, an d th at its pred ictio n o f ab sorption line w idths th at d ep en d
linearly on the partial pressures o f broadening gases does n o t appear to be correct;
4 . D ev ised a param eterized version o f the B erge & G ulkis am m onia form alism , and by fitting it to
data from this w o rk and from previous researchers, d eveloped a new fo rm alism that is consider­
ably m ore accurate than its predecessors o v er a w ide range o f gas conditions an d frequencies;
5 . Invented a new signal p ro b e fo r use w ith m icrow ave cavity resonators, th a t com bines the ch arac­
teristics o f a m agnetic field pro b e an d a tu n ed circu it, y ielding c o n sid erab ly hig h er SN R than
probes p rev io u sly u se d w ith co ax ia lly fed reso n ato rs, an d th at featu res a m ech a n ical d esig n
enabling o ptim ization o f prob e-to -reso n ato r coupling fo r the requirem ents o f a particu lar ex p eri­
m ent; and
6
. D eveloped new techniques fo r m easuring resonance center frequencies, using an analog spectrum
analyzer an d a h igh reso lu tio n frequency counter, th at virtually elim inate uncertainty introduced
by d rift o f the spectrum analyzer's interm ediate frequency (IF), reducing those uncertainties from
ab o u t ±500 kH z to as little as ±1 kH z in th e experim ents o f this w ork.
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Chapter 2
Radio Techniques for the Study of Gas Giant Planets
T his ch ap ter discusses the prim ary m ethods o f using electrom agnetic radiation at radio frequencies to
p ro b e the atm ospheres o f gas g ian t planets, and points o u t the necessity o f reliable laboratory d ata for
th e p ro p e r in terp re tatio n o f results from these m ethods. S ection 2.1 g iv e s a b rie f in tro d u ctio n to the
three radio m ethods, and explains the inability o f th e rad ar astronom y m eth o d to p ro b e th e atm ospheres
o f the gian t p lan ets. T he tw o m ethods th at are useful fo r studying gas g ian t p lanets, rad io astronom y
and radio occultation, are covered in m uch greater detail in Sections 2.2 and 2.3, respectively.
2.1
Introduction and General Review
A great d ea l of our know ledge about plan ets o th er th an E arth h as co m e fro m the rem ote sensing o f
electrom agnetic radiation propagating from those plan ets. R ad io w aves m ake up an im portant p ortion
o f th at radiation. The ability o f rad io w aves to propagate through m ed ia th at are essen tially opaque to
shorter w avelengths has yielded inform ation that w ould n o t be available from optical or infrared observa­
tions. R adio m ethods are especially im portant in studies o f planets th at are shrouded by g lobal clouds,
such as V enus, S aturn's larg est m oon Titan, and all the gas g ian t p lan ets. F o r exam ple, rad io ex p eri­
m ents have m easured tem peratures and pressures in the atm ospheres o f all the gian t plan ets (L indal et
al., 1981, 1985, a n d 1987; T y ler et al., 1989), and have recen tly laid to re st speculation that the surface
o f Titan m ig h t b e covered w ith a global ocean (M uhlem an et al., 1989).
T here are th ree general types o f planetary m easurem ents using radio w aves. O ne, radio astronom y,
is a passive m ethod, g athering inform ation about a p lan et b y receiving the its natu ral electrom agnetic
em issions. The v a st m ajority o f radio astronom y experim ents are Earth based, bu t o ccasionally they can
8
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be space based, as in the P lanetary R adio A stronom y experim ents aboard the tw o V o y ag er spacecraft.
T he other tw o are active m ethods, requiring an artificial source o f radio energy that som ehow interacts
w ith the subject p lan et and thus carries inform ation about the p la n e t to a receiver; by definition both are
form s o f radar. R a d a r astronom y refers to the m onostatic ra d ar m ethod, w here a radio signal is transm it­
ted and reflections received from essentially th e sam e location w ith respect to the target. A s w ith rad io
astronom y, such experim ents m ay b e co n d u cted entirely fro m an E arth based station. R adio occultation
is a bistatic ra d a r m ethod, w here the tran sm ittin g an d receiv in g antennae are w idely sep arated , an d
requires placin g a sp acecraft in the vicinity o f th e targ et p lan et. It relies on recep tio n o f a refracted
signal rather than a reflected signal.
R adar astronom y is critically dependent o n th e strength o f reflections from the target body. A trans­
m itter connected to a highly directional antenna is used to send a narrow -band, coherent signal to a target
planet, w hile a receiv er collects the p art o f th e signal reflected from the target body back to the antenna.
N atural em issions o f the target, even those at th e frequency o f th e rad ar signal, are considered noise, and
m ust be elim inated as m uch as possible from the received signal. D u e to the p resence o f n o ise and the
g reat distances involved, the tran sm itter m u st be extrem ely p ow erful, on th e o rd er o f a m egaw att, and
the target bodies m ust have abrupt ph ase c h an g e surfaces providing strong reflections. T his m ethod has
been used w ith g reat success to probe the in n e r planets, Jupiter's G alilean m oons, and S aturn's rings and
its m oon T itan. T hese bodies share a co m m o n characteristic: they possess distinct, w ell-d efin ed solid
o r liquid surfaces that provide the large contrast in dielectric constants needed for a substantial reflection.
U nfortunately the g ian t planets them selves p o ssess no such interfaces w ithin the regions accessib le to
externally generated radio w aves, so radar astronom ical m ethods cannot be used on those planets.
R adio astronom ical and radio occultation m ethods have been used quite successfully to study atm os­
pheres o f all the g ian t planets. To the delig h t o f planetary scientists large am ounts o f data have been
collected, providing inform ation not available b y other m eans. D irectly m easured data fro m eith e r radio
m ethod, how ever, are o f little interest to those scientists. T o th em the m ost useful inform ation, such as
tem peratures, pressures, and abundances o f atm ospheric constituents, are the resu lt o f in terpretation o f
vertical absorptivity o r refractivity profiles p ro d u ced by reduction o f the raw data. Interpretation requires
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know ledge o f the radio propagation characteristics o f the significant atm ospheric constituents to translate
directly o r indirectly m easured quantities in to useful know ledge about those con stitu en ts in the sub ject
atm osphere. A lthough in m any cases theory can provide reasonable predictions o f th e rad io b ehavior o f
various atm ospheric species, theory m u st b e reg ard ed w ith suspicion unless it is b ack ed by appropriate,
reliable laboratory data. F o r m any species there is g o o d agreem ent betw een theoretical predictions and
laboratory' m easurem ents. H o w ev er fo r so m e species there is sig n ifican t d isagreem ent betw een lab o ra­
tory d ata and predictions o f current theories. A m m onia is o n e o f these p roblem species.
K now ledge o f the rad io propagation beh av io r o f am m onia v ap o r is particularly im p o rtan t to studies
o f the gian t planets. A m m onia is reco g n ized as the m ajor source o f rad io absorption in the atm ospheres
o f those planets, and also p lays an im p o rtan t role in atm ospheric dynam ics sin ce it is a cloud-form ing
constituent. If the radio behavior o f am m onia is accurately know n, radio astronom ical an d radio occulta­
tion d ata can be interpreted to yield accu rate inform ation about its ab undance and d istrib u tio n in g ian t
p lan et atm ospheres. P lan etary scientists c a n use th a t in fo rm atio n to u n d erstan d b e tte r th e d y nam ic
processes in those atm ospheres. L a b o ra to ry m easu rem en ts rep o rted later in th is w o rk in d icate th at
current theories d o not predict the radio absorptivity o f am m onia w ith sufficien t accuracy to fully use the
inform ation contained in d ata already acquired by radio m ethods. Specifically, tem perature dependences
appear to be inaccurate a t cu rren t levels o f interest, and thus using those theories to ex trap o late d o w n ­
w ard fro m laboratory data, at tem peratures w ithin ab o u t 5 0 K o f ro o m tem p eratu re, to tem peratures
observed in the atm ospheres o f the g ian t planets, is pro b ab ly unw arranted.
A bundance inform ation on any co n stitu en t is useful also for constraining theories about th e form a­
tion o f the so lar system .
A m m o n ia is im p o rtan t is this re sp e c t sin ce it is th o u g h t to be th e m ajo r
nitrogen-containing m olecule o f the prim ordial solar nebula.
2.2 Radio Astronomy
R a d io astronom y is a p assiv e m ethod, u sin g a p lan et's n atu ral in co h eren t rad io em issio n s as the
source o f th e inform ation obtained. T hese em issions arise fro m m echanism s such as co n tin u u m therm al
10
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Reproduced
with permission
of the copyright owner.
Further reproduction
Planet
Earth
prohibited without p e r m is s io n .
Atmosphere
F igure 2.1: N atural radio em issions propagate from the atm osphere o f a p lanet tow ard Earth, w here they can be received by a radio
telescope. E nergy is radiated in all directions, bu t only th at escaping from the atm osphere in the precise direction o f the antenna
can be received. R adiation o f any w avelength can originate fro m any level in the atm osphere and the surface, if a distinct surface
exists, but increasing tem peratures and atm ospheric absorption w ith depth result in w avelength-dependent zones o f effective em is­
sion th at m ove deeper into the atm osphere as w avelength increases.
em ission from the planet's atm osphere, as illu strated in F ig u re 2 .1 , surface (if a w ell-defined surface
exists), rings, etc., and syn ch ro tro n radiation from ch arg ed particles traveling w ithin the m agnetosphere
o f the target p la n e t A n E arth-based rad io telescope collects a sam p le o f th e em issions, and that sam ple
is analyzed to in fer inform ation ab o u t the p la n e t A rad io telescope consists o f tw o m ain subsystem s: a
sensitive radio receiver an d a highly directional antenna. T h e receiv er selects a range o f radio frequencies
to be m onitored an d m easures the strength o f the signals in that ra n g e su p p lied by the antenna, w hose
directional capabilities are used to select the target body.
Since w avelength an d size are m ajo r determ ining factors o f th e reso lu tio n o f a telescope, an optical
telescope has a reso lu tio n advantage o v e r a single radio telescope. A o n e m eter o p tical telescope has a
diam eter-to-w avelength ratio,
, 0 f approxim ately
2
x 1 0 6 , y ield in g an ideal reso lu tio n about a tenth
o f an arcsecond. T h e e ffectiv e reso lu tio n o f E arth -b ased o p tical telesco p es is lim ited m uch m ore b y
atm ospheric e ffects ("seein g "), to ab o u t h a lf an arcseco n d at b e st.
(T h is is th e m otivation fo r the
H ubble Space T elescope, w hich w ill n o t experience the atm ospheric disto rtio n s.) O n the other hand, a
radio telescope o p eratin g at a w av elen g th o f 5 cm w ou ld n e ed a n a n ten n a aperture nearly 100 k m in
diam eter to achieve a tenth o f an arcsecond ideal resolution. Practical steerable antennae have diam eters
up to about 100 m eters, w h ich w ou ld y ield an ideal reso lu tio n n e a r 100 arcseconds at the 5 cm w ave­
length. T his is la rg e r than the larg est ap p aren t d isk sizes o f the p lan e ts as seen fro m E arth. A radio
antenna usually receiv es sig n als fro m the en tire d isk sim u ltan eo u sly , so the m easurem ents m ade are
spatially unresolved disk averages.
Interfero m etric m ethods circ u m v e n t these reso lu tio n lim its by u sin g signals fro m two o r m ore
w idely separated antennae to synthesize an oriented virtual aperture as larg e as the distan ce betw een the
antennae. An exam ple is the V ery L arg e A rray n ear Socorro, N ew M ex ico . B y m ak in g m easurem ents
at tw o closely sp aced frequencies the in h eren t am biguity o f an in terfe ro m eter can be resolved. U sing
these m ethods rad io astronom ical reso lu tio n s n e a r o n e arcseco n d h ave b een ach iev ed (N ap ier et al.,
1983). T his is su fficien t to reso lv e sm all parts o f the d isk s o f Ju p ite r an d S aturn (see, for exam ple,
G rossm an et al., 1989), providing m easurem ents that are no t lim ited to full-disk averages.
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W h eth e r the m easurem ents are d isk averages o r spatially reso lv ed po rtio n s o f the disk, plan etary
electrom agnetic em issions are very broad-band. T here is m easurable p o w e r at w avelengths from g reater
than o n e m e ter to less th an a m icron. R ad io astronom ical ex p erim en ts determ in e th e intensity o f these
em issions o v e r a relativ ely narrow band o f w avelengths. Intensities are usually expressed in term s o f an
effective b rig h tn ess tem perature (o r sim ply b rightness tem perature), T g , w hich is defined as the physical
tem perature o f an ideal black body radiator that produces the sam e p o w e r spectral density as th e targ et at
the observed w avelength. Planets do n o t behave as ideal black bodies; th eir brightness tem peratures vary
considerably w ith w avelength. Som e em issions are non-therm al, such as th e synchrotron radiation at the
longer w avelengths from planetary m agnetospheres, and these can cause elev a ted brightness tem peratures
at those w avelengths, b u t even w ithout the effects o f non-therm al em issions there are sig n ifican t v aria­
tions o f brightness tem peratures w ith w avelength.
T his v ariability is o b serv ed in all so lar system planets w ith sig n ifican t atm ospheres, w h eth er o r no t
they have d istin c t o b serv ab le so lid or liq u id surfaces. T h e gas g ian t plan ets h ave n o such su rfaces and
thus have no ab ru p t lo w er lim it to the radio em ission zone, n o r an ab ru p t up p er lim it to the tem perature
o f the observ ab le m aterial. M anifestations o f this differen ce distin g u ish the rad io sp ectra o f gas g ian t
planets from those o f th e in n er planets. Spectra o f the in n e r planets show ch aracteristic lim iting b rig h t­
ness tem peratures as w avelength increases to the longer, highly penetrating w avelengths traveling essen ­
tially unim p ed ed fro m th e surfaces o f those planets. Surface tem peratures are th e h ighest o bservable, so
once the o b serv ed w avelength is large enough to see the surface further increases in w avelength w ill not
see m aterial at h ig h er tem peratures, and the brightness tem perature sp ectru m flattens. G as g ian t planets
do n o t have this constraint. T heir observable therm al rad io em issions originate from w ithin the atm os­
phere itself, n o t from a radiating (and reflecting) surface as w ell. L o n g er w avelengths are free to p en e­
trate fro m d eep er levels at higher tem peratures, and thus the o b serv ed b rightness tem peratures steadily
increase w ith w avelength, w ith no lim iting value.
O bserved brightness tem peratures are functions o f the properties and physical conditions o f the source
m aterials. T h e atm o sp h eric vertical p ro file o f tem perature, T (z ) w h ere z is an altitude variable, is a
critical factor. A n o th er is the radio absorptivity (opacity) o f m aterial betw een th e source and observer.
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A bsorptivity is generally a function o f tem perature and pressure (and thus z) as w ell as observed frequen­
cy f, so it is w ritten a ( f ,z ) . In teg ratin g ab so rp tiv ity fro m a source a t h eig h t z to the o b serv er, at a
h eig h t that is essentially infinite, yields the in teg ra ted norm al opacity, x(f,z):
(2 . 1)
R adiative tran sfer theory show s th at the brightness tem perature o f a sm all p ortion o f a gas g ian t p lan e­
tary disk is given by
( 2 .2 )
w here |i is the direction cosine o f the lin e-o f-sig h t direction to the o bserver relativ e to th e local norm al
and x is the x(f,z) given by E quation 2.1. T h eo retical brightness tem peratures fo r fu ll-d isk m easu re­
m ents are o b tain ed by averaging o v er p. fro m zero to unity. T he integral o v er all x is eq u iv alen t to an
integral over altitude from the observer, w here z is essentially infinite an d x is zero, to the cen ter o f the
planet, w here z = 0 and x is effectively infinite. T he resulting brightness tem perature represents a balance
betw een tem peratures that increase w ith depth producing therm al em issions w ith h ig h er p o w e r spectral
densities, and opacities th a t also increase w ith depth and tend to absorb the energy p ropagating upw ard
from the hig h er tem perature zones below . A t hig h er altitudes the o p acity can be q u ite low fo r a given
radio w avelength, bu t tem peratures are also low so little energy is radiated. A t lo w er altitudes m uch
m ore energy is radiated at that w avelength, but high integrated opacity prevents the radiation from escap­
ing. T he precise definitions o f "high" and "low " altitudes fo r a specific atm osphere depend on th e w ave­
length in question.
A w eighting fu n c tio n for a specific atm osphere and w avelength, derived from the arg u m en t o f the
integral in E quation 2.2, gives the relativ e co n tribution to the observable pow er spectral d en sity at that
w avelength from the m aterial at a given altitude. E xam ples o f w eighting functions at 2 an d
6
c m w ave­
lengths in the atm osphere o f S aturn are given by G rossm an et al. (1989), sh ow ing th e ch aracteristic
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peak w ithin a relativ ely narrow altitude ran g e (usually given in term s o f total pressure) w h ere there are
significant contrib u tio n s to th e escaping radio energy. T hese exam ples p o in t o u t the general b ehavior o f
the w eighting functions as w avelength increases: the altitude o f the p eak d ecreases, and the w idth o f the
altitude ran g e yield in g sig n ifican t co ntributions increases. This is the reaso n lo n g e r w av elen g th s see
deeper levels in an atm osphere.
R adio telescopes d o no t m easure brightness tem peratures directly, but in stead m easure the po w er o f
the radio em issions co llected by th e antenna from a target planet, w ithin a frequency range determ ined by
the receiver. T he received po w er m easured by the telescope, P r , is
P r = k B ( T R + TA) ,
(2.3)
where k is B oltzm ann's constant, B is the receiv er bandw idth, T r is the receiv er n o ise tem perature, and
T a is the effective tem p eratu re o f the antenna. T a is m uch m ore th an sim ply the p h y sical tem perature
o f the antenna; it represents all en erg y collected from all the radio sources radiating tow ard th e antenna,
including the target p lan e t. T herm al radiation by the antenna structure itself is a tin y p a rt o f the pow er
arriving at the receiver sin ce it is n o t coherently focused on th e receiving elem e n t o f the antenna. T a is
the p ro d u ct o f the an ten n a resp o n se pattern w ith the source pattern surrounding the antenna. Sources
outside the antenna m ain lo b e u su ally yield n eg lig ib le contrib u tio n s to T a , an d m o st antenna m ain
lobes are narrow enough that they contain only one significant source at a tim e. V ery accurate m easure­
m ents o f T r are p o ssib le, to w ithin about 0.1 K, leaving T a the o n ly unknow n in E quation 2.3. T he
brightness tem perature o f the source is then obtained via deconvolution from th e know n antenna pattern.
In both the m easu rem en t step in th e field and the subsequent deconvolution, calib ratio n o f the antenna
pattern is critical to the accuracy o f the reduced data. C alibration errors are m a jo r co n trib u to rs to the
uncertainties o f radio astronom ical m easurem ents.
E quatio n s 2.1 and 2 .2 p ro v id e a m eans o f calculating a theoretical b rig h tn ess tem p eratu re given
inform ation ab o u t the tem p eratu re and absorptivity profiles. H o w ev er, it is n o t p o ssib le to in v ert the
equations and unam biguously d eriv e tem perature and absorptivity profiles g iv en a m easu red brightness
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tem perature. A ssum ptions that defin e a tem p eratu re pro file, such as w e t o r dry adiabatic lapse rates,
allow a solu tio n to b e o btained, and the re su lt is a vertical a b so ip tio n p ro file that is d ep en d en t on the
initial assum ptions. I f it is know n th at a single co n stitu e n t o f th e o b serv ed atm osphere is responsible
fo r the radio opacity, and if all physical condition dependences o f radio opacity fo r that species are accu­
rately know n o v er the ran g e o f sam pled conditions, a vertical ab undance p rofile fo r that co n stitu en t can
be extracted fro m the absorptivity profile by a sim ple pro ced u re. F o r each absorptivity datum , the asso­
ciated condition values {i.e. tem perature, partial pressures o f significant non-absorbing constituents, etc.)
are su b stitu ted into the absorptivity form alism . T his leaves the p artial p ressure o f the absorbing co n ­
stituent as th e on ly unknow n in the form alism , w hich is eq u ate d to the o bserved absorptivity. N u m eri­
cal inversion, som etim es as sim ple as a sin gle division, then yields the estim ated partial p ressu re o f the
absorbing co nstituent. F orm alism s fo r calculating opacities o f absorbing constituents are often theo ret­
ical predictions unsupported by appropriate laboratory m easurem ents, and this leads to large uncertainties
in the derived abundance profiles even if the assum ed tem perature profiles are c o rre c t
2.3 Radio Occultation
U nlike rad io astronom y, radio occultation is an active m eth o d requiring a radio transm itter as w ell
as a receiver, w ith highly directional antennae fo r both. A n o th er rath er expensive requirem ent is that a
spacecraft c arry in g o n e o f those subsystem s m u st be flow n n e a r an d b ehind th e p lan et to be studied, as
show n in F ig u re 2.2. D ue to the physical size and m ass o f the co m p u ter equipm ent needed at the receiv ­
ing end o f th e system , ra d io occu ltatio n ex p erim en ts h av e h isto rically p laced the tran sm itter on the
spacecraft. T h is lim its the transm itted p o w er to a few w atts, a feeb le w h isp er co m p ared to the din o f
radio noise n atu rally p re se n t th ro ughout the so lar system . E x p erim en ts using this m ethod are called
dow nlink rad io occultation experim ents. T he transm itter-p lan et-receiv er system is a reciprocal system :
if the transm itter w ere to b e m oved fro m th e antenna on the sp acecraft to the antenna at E arth, and the
receiver m o v ed to the antenna on the spacecraft, the signal p o w er receiv ed w ould be exactly the sam e as
in the original configuration. N ew technologies in signal processing and com puter m iniaturization m ay
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PLA N ET
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T o Earth
SENSIBLE
A T M O SPH E R E
S P A C E C R A F T T R A JE C T O R Y
Figure 2.2: R adio occultation geom etry, show ing the progressively greater signal refraction during im m ersion, w hen the spacecraft passes behind the planet.
D ata also are obtained during em ersion, as the spacecraft em erges from behind the planet, at a different location in the atm osphere.
allow u p lin k rad io occu ltatio n ex perim ents, placing the re c e iv e r o n th e sp a c e cra ft an d using an E arth
based tran sm itter w ith a b o u t ten tho u sa n d tim es the o u tp u t p o w e r o f the sp ace cra ft b ased transm itters
(T yler, 1987). T his pro d u ces an in crease in the system S N R equal to the relativ e in crease in transm it­
ted pow er (but decreased by higher receiver noise tem peratures), prom ising m uch m ore accurate m easure­
m ents over a larger ran g e o f depths in the atm ospheres o f the g ian t planets.
T h e electrom agnetic signal transm itted from a rad io o ccultation sy stem is a sinusoidal w ave w hose
precisely know n frequency is derived from the output o f an ex trem ely stable o scillator. T his signal is o f
exceptional purity in both frequency an d am plitude. V ery n arrow b andw idth is necessary to allow tight
bandpass filtering th at rem oves nearly all the broad-band rad io noise. A t the receiving station very accu­
rate m easurem ents o f signal po w er (and thus am plitude), frequency, an d p h ase are m ade before, during,
and after th e signal is im peded by th e intervening planet's atm osphere. It is com m on practice fo r the
radio occultation ex p erim en t system on a single sp acecraft to tran sm it tw o differen t frequencies, co h er­
ently d eriv ed from th e sam e o scillato r signal, w ith in d ep en d e n t m easu rem en ts on the tw o signals at
E arth. M easu rem en ts before an d after the atm ospheric o ccu ltatio n estab lish the sig n al's "free-space"
pow er. Signal pow er relative to this free-space pow er, n o t absolute po w er, is im p o rtan t to the results o f
the experim ent, so u n lik e radio astronom ical experim ents ab so lu te c alib ratio n o f th e receiv ed signal
pow er is n o t critical. M easurem ent o f relativ e pow er to w ithin a few p e rc e n t is a sim ple m atter.
T hese raw d ata are o f no direct use to planetary scientists, bu t inform ation derived fro m them are o f
great interest. T h e records o f th e signal frequency, in conjunction w ith inform ation ab o u t the trajectory
o f the spacecraft during the occultation, yield vertical profiles o f atm ospheric refractivity. T hose profiles
can be used to deduce m ixing ratios o f the m ajor atm ospheric constituents an d generate vertical profiles
o f tem peratures and pressures. Signal am plitude data can then yield vertical profiles o f radio absorptiv­
ity. If there is only o n e co n stitu en t responsible for essentially all radio ab sorption by the atm osphere,
and if the rad io b ehavior o f that species is accurately know n under the o b serv ed atm ospheric conditions,
the absorptivity profiles can be translated into abundance profiles for th a t c o n stitu en t by th e sam e p ro ­
cedure used w ith ra d io astronom ical d a ta (see p. 16). T h is is a p p licab le to the g ia n t p lanets, w here
am m onia vap o r is b eliev ed to b e resp o n sib le fo r nearly all absorption at the freq u en cies u sed fo r radio
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occultation ex perim ents. A m ore detailed discussion o f the red u ctio n o f raw radio occultation d a ta fo l­
low s here.
A lthough the radio signal transm itted from the spacecraft m ay have a very accurately determ ined fre­
quency, various m echanism s generally cause the frequency receiv ed at the E arth based station to b e quite
d ifferen t from the transm itted frequency. T he larg est d ev iatio n s are D o p p le r shifts d eterm in ed by the
rad ial co m p o n en t o f the spacecraft's velocity relativ e to the receiv in g station. C alcu latin g this radial
velocity involves m any co m plex relative m otions, such as E arth 's rotation and m o v em en t aro u n d the
barycenter o f th e E arth-M oon system , that barycenter’s orbital m otion aro u n d the sun, an d various tidal
forces exerted by o th er m assiv e bodies in th e so lar system , as w ell as the com plex m otion o f the sp ace­
cra ft itself. A ll th ese m otions, and thus th e d ire c t D o p p ler sh ift, can be v ery accurately d eterm ined.
W hen the sp acecraft is in clo se proxim ity to a very m assive p la n e t th ere is also a gravitational (general
relativistic) re d sh ift due to the signal clim bing o u t o f th e p lan et's d eep potential w ell. T h is frequency
shift can also be determ ined quite accurately, bu t usually it is n eg lig ib ly sm all.
A noth er freq u en cy pertu rb atio n closely relate d to D o p p le r shifting occurs w hen the signal p ath is
refracted o r is otherw ise d eviated from a vacuum g eodesic by a n intervening body, such as a p lanetary
atm osphere. D o p p le r sh ift o f a signal is determ in ed by the ra te o f ch an g e o f the ray path length from
the spacecraft to the receiving station — d u e to the curved sig n al path this m ay be quite differen t from
the rate o f c h an g e o f the d ire c t distance betw een the tw o. T h is causes the o bserved D o p p ler sh ift to
deviate from th at expected w hen considering only the spacecraft-to-station relative velocity. T he precise
deviation is d eterm in ed by characteristics o f the intervening body, in this c ase its atm o sp h ere, and it is
this deviation th at is crucial to the radio occultation experim ent.
T h e geo m etry o f the rad io occultation ex p erim en t is diag ram m ed in F ig u re 2.3, after E sh lem an et
al. (1980). C ertain sim plifying assum ptions m u st be m ade p ertain in g to the shape and h om ogeneity o f
the atm osp h ere to allow in v ersio n o f the d ata. F o r a slow ly ro tatin g in n e r p lan e t such as V enus an
assum ption o f spherical sym m etry is sufficient, b u t this is inadequate for the g ian t planets w here o b late­
ness m ust be considered. A signal transm itted fro m the sp acecraft travels along ray asym ptote R 2 until
it reaches the planet's atm osphere, w here refraction bends its p a th tow ard the p la n e t U pon reem erging
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R ay A sym ptotes
R2
SPA C E
CRA FT
g~P
R es
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EARTH
STATION
Re
PL A N E T
CEN T ER
F igure 2.3: Schem atic d iagram o f the geom etry o f an atm ospheric radio occultation experim ent. See text for explanation.
from the atm osphere it has tu rned by the refraction angle, a , and th ereafter travels along ra y asym ptote
R1 to a receiving station a t E arth. In th e spherically sym m etrical case b o th ray asym ptotes are offset
from the planet's c en te r by a, the im p a ct param eter. T he clo se st approach o f such a ray to the planet's
center is called the ra y p eria p sis, show n a t poin t P, w hich g enerally m oves d eep er into the atm osphere
as a increases. S o m e o f the distances a n d angles in the figure m ay be calc u lated from E arth and planet
ephem erides, sp acecraft trajectory data, an d know ledge o f th e location o f th e receiving station on Earth's
surface: distances R e ,R e s , R s, D, and r, and angle y. A dditional in form ation need ed to com plete the
characterization o f th e ray p ath is available from differences betw een the observed D o p p ler sh ift o f the
refracted signal and the expected D o p p ler sh ift had the signal propagated through the interplanetaiy m ed­
ium o v er the en tire d istan ce fro m sp acecraft to E arth, via the "straight-line" g eodesic R e s. F o r a given
instant this residual D o p p ler shift, calc u lated from the receiv ed signal frequency, along w ith trajectory
inform ation that p rovides accu rate d eterm ination o f the sp acecraft's velo city at the tim e th e signal was
transm itted, can be used to co m p u te angle a . O nce this angle is know n all th e o th er quantities show n
in F igure 2.3 quick ly follow . T im e series tables o f com puted values o f a an d a can b e used to calculate
instantaneous approxim ations o f ^ a /d a > and subsequent integral inversion o f those tim e series can then
yield vertical refractivity profiles.
I f an atm ospheric gas m ixture consists alm ost entirely o f n on-polar species, its refractivity provides
a m easure o f the ratio o f its m ean m olecular m ass to its tem perature. V ertical refractivity pro files estab­
lish this ratio o v er a range o f altitudes. If eith er q uantity is know n th e o th e r follow s, b u t there is no
m ethod o f unam biguously d eterm ining both from refractivity profiles alone. F ortunately th ere are other
m ethods, such as infrared spectrom etry, th at can m ake tem perature m easurem ents calibrating the profiles
obtained from radio occultation data. A n instrum ent such as the Infrared R adiom eter Interferom eter and
Spectrom eter (IR IS) aboard th e V oyager sp acecraft can m easure the tem perature o f an atm osphere's tropopause. A pplying this to th e o ccu ltatio n refractivity m easurem ents estab lish es the m ean m olecular
m ass a t the tropopause. I f the m ixing ratios o f the m ajor atm ospheric constitu en ts are co n stan t o v e r the
range o f altitudes sam p led by the occultation experim ent, then th at m ean m o lecu lar m ass can be used
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throughout, producing an accurate vertical p rofile o f tem perature. V ariable m ixing ratios co m p licate the
procedure, an d inform ation from other sources m ay b e needed to extract tem perature profiles.
O n ce th e signal frequency d ata have b een analy zed it is po ssib le to ob tain additional in fo rm atio n
from the signal am plitude data. D ifferential refraction w ith altitude by the p lanet’s atm osphere generally
causes defocusing o f the initially narrow radio beam , decreasing its intensity upon arrival a t E arth . T he
vertical refractivity profiles obtained from reduction o f the D o p p ler data allow accurate calculation o f the
loss in signal p o w er due to this e ffe c t P ow er losses b eyond those due to defocusing m ay be assu m ed
due to absorption w ithin the atm osphere. A t any p a rticu lar tim e this excess loss is the in teg rated loss
o ver the en tire signal p ath through the atm osphere. L osses a t tw o adjacent tim e steps provide a m easure
o f the chang e in absorption w ith a , w hich are in v erted to produce a vertical p rofile o f radio absorptivity.
I f a sin gle atm ospheric constituent is know n to b e responsible for the o b serv ed rad io absorption it
is straightfo rw ard to tran slate the absorptivity p ro file into a v ertical abundance p ro file fo r th at co n stit­
uent. H o w ev er this hin g es on an accurate fo rm alism fo r p red ictin g th e ab so rp tiv e b e h av io r o f th at
constituent un d er th e conditions observed in the atm osphere. T hus the problem o f predicting th e b eh av ­
ior o f am m onia m ay b e a significant source o f trouble in translating rad io occultation d a ta fro m g ian t
planets into useful inform ation concerning am m onia abundances in their atm ospheres.
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Chapter 3
Microwave Absorption and the Ammonia Molecule
T h e m ajo rity o f this w ork involves q u an titativ e discu ssio n s o f the ab so rp tio n an d refractio n o f
m icrow ave signals propagating in a gaseous m edium containing am m onia. T his ch ap te r provides back­
g round inform ation to facilitate understanding o f subsequent chapters. S ectio n 3.1 is a sum m ary o f the
basic con cep ts an d vocabulary o f the th eory o f electrom agnetic w ave p ro p ag atio n in isotropic m edia.
W ith that fo undation estab lish ed , S ections 3.2, 3.3, an d 3.4 d escrib e th e stru c tu re o f th e am m onia
m olecule and the m anifestation o f that structure in the absorption o f m icrow aves p ropagating in a gas
m ixture containing am m onia, alo n g w ith historical d ev elo p m en t o f th e u n d erstan d in g o f this ph en o m ­
enon. M o st o f these three sections sum m arizes the relev an t inform ation contained in the landm ark 1955
book by Tow nes and Schaw low , "M icrow ave Spectroscopy," w hich serves as a general reference. Speci­
fic contributions by oth er researchers are cited individually.
3.1
Microwave Propagation in an Isotropic Medium
In a h o m ogeneous, isotropic, lo ssless dielectric m edium , un ifo rm p lan e elec tro m ag n etic w aves
propagate w ith unvarying am plitude. T he p h aso r expressions for the fields o f such w aves traveling in
the x direction are:
E (x ) = E 0 e ' i P x ;
H(x) = H 0 e ' i P x .
(3.1)
T h e field am plitude coefficients E o and H o in these expressions are n o t independent, b u t a re related by
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the in trin sic im pedance, T|, o f th e m edium : T] = E o/H o- N o te th a t a sin g le param eter, (5, is needed to
characterize the E an d H fields in the en tire space. C alled th e p h a se constant, f} is closely related to the
frequency and p ro p ag atio n velocity o f th e w aves. It defines the w av elen g th via the relad o n X = 2 it/p .
W hen m easuring the characteristics o f a m edium in w hich absorption o f the propagating electrom agnetic
w aves is insignificant, m easu rem en t o f any sin g le quan tity is su fficie n t to ch aracterize the w ave fields,
as long as th a t quan tity can b e u se d to calcu late (J. I f th e m ed iu m is an im p erfe ct dielectric, com m only
called a lo ssy dielectric, th e am plitude o f the w aves decreases as they p ro p ag ate in the x direction, w ith
phasor expressions:
E (x) = E 0 e - “ x e ' j P x ;
H(x) = H 0 e ' “ x e ' j P x .
(3.2)
In this case tw o param eters, called the pro p a g a tio n constants a an d p, are necessary. O ne o f the p aram ­
eters, p , is identical to th at in the lossless case discu ssed above. T h e o th e r para m e ter is the attenuation
constant, a , also called th e abso rp tio n coefficient or a b so rp tivity. It gives the in v erse scale length fo r
the exponential d ecay o f th e field am plitudes w ith increasing x. If th e m ech a n ism fo r th e loss o f energy
from the in cid en t w ave is in d eed absorption by the m ed iu m an d n o t scattering, th e energy lo st from the
w ave becom es "therm alized," o r therm al energy o f the m edium , thus raisin g its tem perature. T h e energy
is not, in general, reradiated later a t the sam e frequency. T his is the c ase w ith ab sorption o f m icrow aves
by am m onia in the atm ospheres o f the g ian t planets. F ig u re 3.1 illu strates the p h y sical interpretation
o f the propagation constants.
T he propagation constants are determ ined by tw o quantities characteristic o f the m edium , perm ittiv­
ity e and p erm eab ility p..
F o r gases at m o derate pressures the p e rm ittiv ity is in g en eral com plex and
can be expressed as real and im aginary parts:
e = e ’ - je " .
(3.3)
T he com m o n ly used refractiv e index n is clo sely tied to perm ittiv ity : n 2 = e'. F o r such gases the d if­
ference betw een p and Po, the p erm eability o f free space, is neg lig ib le. E x cep t in cases w here extrem e
24
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X
F igure 3.1: T h e phy sical interpretation o f propagation constants a and ji fo r a plane electrom agnetic w ave
traveling in th e + z d irectio n . A bsorptivity a is the scale length fo r the ex p o n en tial decay o f th e field
am plitudes. P hase con stan t p is ph ase change p e r unit length, w hich determ ines the w avelength X.
accuracy is im portant, p. m ay be assum ed to b e en tirely re al an d have a value o f |io- F o r real p , the
propagation constants at a frequency © are given by (R am o, W hinnery, and V anD uzer, 1965):
(3.4)
By form ing the quo tien t a /p the ex plicit dependences o n co and p are elim inated:
(3.5)
T he ratio e / e ' in E quation 3.5 is called the loss ta n g en t o f the m edium . Its inverse is th e q u a lity fa c to r,
Q , o f the m edium : Q = £'/E" . An approxim ation o f E q u atio n 3.5 that is linear in the loss tan g en t and
estim ates a /p to w ithin 0.5% w hen the loss tangent is less than 1 0 '2, is u sed in C h ap ter 4 to d eriv e an
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expression fo r a in term s o f m easurable quantities. A loss tangent o f 10-2 corresponds to absorptivities
g reater than 104 d B /k m fo r m icro w av es, and loss tangents o f gaseous m ed ia are a lm o st alw ays m uch
less th an that. In gas m ix tu res sim ilar to th e atm ospheres o f giant p lan ets such h u g e ab sorptivities
occur only at th e extrem ely high p ressu res existing at atm ospheric levels fa r below th e lo w est accessible
by radio occultation and radio astronom ical methods.
I t is im portant to note th at in th e preceding equations the real and im aginary p arts o f e, £' and e",
are not independent o f each other. T h eir m utual dependences are given by the K ronig-K ram ers relations
(see, e.g., R am o, W hinnery, and V an D uzer, 1965):
(3.6)
(3.7)
w here co and to' are radian frequencies. A fully accurate calculation o f e' from e" requires k n o w led g e o f
the value o f e" o v er all frequencies fro m zero to infinity, and likew ise a fully accurate calculation o f e"
from e' requires know ledge o f the value o f £' o v er all frequencies. F o r exam ple, even i f e" is m easured
to great accuracy o v er a lim ited range o f frequencies, the integral in E quation 3.6 w ill be incom plete and
subsequent calculations o f £' in the frequency range o f the data w ill suffer truncation errors. U nexpected
behavior o f e" o utside the frequency ran g e o f the m easurem ents could fu rth er com pound the problem .
B ut these integrals tend to em p h asize the b ehavior o f the "known" quantities (e" in E q u atio n 3.6, o r e'
in E quation 3.7) at frequencies n e a r the argum ent frequency to. Practical calculations using the K ronigK ram ers relations generally use all av ailab le inform ation ab o u t the know n quantity w h ile appropriate
interpolations fill any gaps, bu t m u st rely o n assum ptions about its behavior in the extrem es o f frequen­
cy, especially for very high frequencies, w here no data are available. If the frequency range o f the desired
result is nested w ithin a m uch broader frequency range w here reliable data are available, even relatively
large errors in the assum ed values at the frequency extrem es can yield sm all errors in the result.
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3.2
Structure o f the Ammonia M olecule
T h e "norm al" am m onia m o lecu le is a relativ ely sim p le sy ste m con tain in g o nly fo u r atom s, one
nitrogen and three h ydrogen. T h ere are several variations th a t o c c u r naturally, such as am m onia w ith
one, tw o, o r all three h y d ro g en ato m s rep laced by d eu teriu m atom s, b u t this discussion w ill focus on
"norm al" am m onia, w hich is by far the m o st abundant version, w ith o n e N l4 atom an d three H i atom s.
N eutral nitrogen has fiv e o u ter shell electrons, two in th e filled s o rb ital and one in each o f the three p
orbitals. N eutral hy d ro g en has a sin g le electro n in th e h alf-filled s o rb ital. A co v alen t bond is form ed
when a hydrogen atom and o n e o f th e n itrogen atom 's h a lf-filled p o rb itals sh are th eir electrons. T he
three p orbitals o f n itro g en allow th ree hydrogen atom s to bo n d to a single nitrogen atom and hence
establish the atom ic ratio o f the m olecule. T he nitrogen atom ’s b o n d in g orbitals are gen erally tho u g h t
to be h y b rid rp 3 o rbitals (S orriso, 1982), but there is reaso n to q u estio n this conclusion. A m m onia's
bond angle, the angle betw een lines connecting the nitrogen nucleus w ith tw o o f th e hydrogen nuclei, is
less than 107 deg rees.
T h e e x p ec te d angle fo r s p 3 h y b rid o rb itals is g rea te r than 109 deg rees. If
am m onia's bon d in g orb itals w ere tru ly r p 3 hybrids, m utual re p u lsio n o f the h ydrogen atom s should
push the b o n d angles to g re ate r values, as is o bserved in the w ater m olecule (S ienko an d P lane, 1966),
not to sm aller values. P erhaps future explanation o f this unex p ected deviation w ill contribute to a better
understanding o f am m onia's m icrow ave absorption characteristics.
T h e m olecule m ay be pictured as p art o f a tetrahedron. T he three hydrogen atom s are the basal v er­
tices, th e nitrogen atom is at the center, and the filled, u nbonded sp3 orbital points tow ard the apex. An
axis o f radial sym m etry passes through the apex, the nitrogen nucleus, an d the centroid o f the base. The
net resu lt o f rotating the m olecule 120 degress is a view o f the m o lecu le that is indistinguishable from
the original.
A m olecule w ith this type o f sym m etry is know n in m o lecu lar spectroscopy as a "sy m ­
m etric-top" m olecule.
A neutral am m onia m olecule has asym m etries that contrib u te to its properties. T he nitrogen atom
is displaced m ore than a third o f a bon d length from the plane o f the hydrogens. It has a higher electron
affinity than th e hy d ro g en atom s, and thus the electrons th at w ere asso ciated w ith the hydrogen atom s
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Hydrogen
+
Nitrogen
Molecular
Axis
+
+
T otal A ngular
M o m en tu m
F ig u re 3.2: A m odel o f an am m onia m olecule. T he charge separation that produces am m o n ia's p o w er­
ful dipole m om ent is ev id en t fro m the positive charges o n the hydrogen atom s and a negative charge on
the nitrogen atom . T h e m ag n itu d e o f the total m o lecu lar an g u lar m o m en tu m v ecto r is q u a n tiz ed w ith
the rotational q u an tu m n u m b er J . T he projectio n o f th e total an g u lar m om entum v ecto r o nto the m ole­
cular sym m etry axis is also q uantized w ith the rotatio n al qu an tu m num ber K.
spend m ost o f their tim e n ear the n itrogen atom . T he hy d ro g en atom s acquire a strong p o sitiv e charge
w hile the triply red u ced nitrogen ato m acquires a very strong negative ch arg e from the th ree electrons
p rovided by the hydrogens, as show n in F igure 3.2. T h is separation o f charge produces a p o w erfu l elec­
trostatic d ip o le m om ent. A t a d istan ce o f one n an o m e te r fro m the m o lecu lar cen ter th e e lectric field
strength produced b y the dipole m easures several hundred thousand volts p er centim eter (B leaney, 1946).
T h e am m onia m olecule is far fro m being a static entity. It can vibrate m u ch as i f the atom s w ere
m asses and the bonds w ere springs connecting th e hydrogens to the nitrogen. T he en tire m o lecu le can
also rotate. T h e en erg y o f the v ib ratio n al states and th e a n g u lar m om enta o f the rotatio n al states are
quantized, so they c an only take o n discrete values. T h e total angular m om entum v ector o f the m olecule
is quantized, an d the uncertainty p rinciple requires that it cannot have zero m agnitude. Its m agnitude is
restricted to be a p ositive integral m ultiple o f the total g ro u n d state an gular m om entum , an d th a t integer
num b er is the ro tatio n al quantum n u m b er J . T he p rojection o f the total angular m om entum v ecto r onto
the m o lecu lar sy m m etry axis is also quantized in units o f th e g ro u n d state an g u lar m om en tu m . T he
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m agnitude o f th e pro jected angular m o m en tu m v ecto r is g iv en by the in te g e r quantum n u m b er K tim es
th e total ground state angular m om entum . F ig u re 3.2 sh o w s the relatio n sh ip betw een these v ectors is.
B oth num bers J and K m ust be sp ecified to u n am b ig u o u sly d efin e the ro tatio n state o f th e m olecule.
L ik e J , K can n o t be zero, and b y definition J m u st alw ays b e greater than o r eq u al to K. I f J = K, as in
the ground state w here J = K = 1, the total m o m en tu m v e c to r is aligned w ith the m o lecu lar sym m etry
axis an d the m olecule spins around th at axis.
T h e pow erful dipole m om ent o f the am m o n ia m olecule allow s strong coupling o f electrom agnetic
radiation to these vibrational an d ro tatio n al states, cau sin g transitions w hich p ro d u ce ab sorption and
em ission phenom ena. H o w ev er all such transitions for am m o n ia are a t infrared an d hig h er frequencies
and at m oderate pressures contribute insignificantly to absorption at m icrow ave frequencies. T he m echa­
n ism responsible for the observed m icro w av e absorption spectrum o f am m onia is d iscu ssed in the fol­
low ing section.
3.3
Microwave Absorption by Ammonia Inversion
A s show n in F igure 3.2 an am m onia m olecule has a n itrogen atom w hich is considerably o ffse t fro m
the plane co ntaining the nuclei o f th e hyd ro g en atom s. If, in F ig u re 3.2, a force w ere ap p lied to the
nitrogen atom pushing it to the left, and a force o f equal m agnitude w ere d istributed am ong the hydrogen
atom s, pushing th em to the right, it w o u ld b e po ssib le to b rin g the n itro g en atom to th e p lan e o f the
hydrogen nuclei. If the nitro g en nucleus w ere to be p u sh ed slightly past th at p lane, it w ou ld suddenly
rearrange its bonding orbitals such th at the n itro g en atom w o u ld snap to a stable position on th e left o f
the hydrogen nuclei, producing a m olecule th at is a m irror im age o f the o rig in al m olecule. S in ce there is
no "handedness" asym m etry to am m onia m olecules with th ree norm al h ydrogen atom s this m irro r im age
m olecule is no differen t from the original m olecule; it m erely "points" in the o pposite d irection, w ithout
having rotated. T his sudden p enetration o f th e nitrogen a to m through the p lan e o f the hyd ro g en nuclei,
som etim es described as the m olecule "turning itse lf inside out," is called inversion.
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It is po ssib le that the forces needed to p u sh th e nitrogen nucleus through the hyd ro g en plane accord­
ing to the law s o f classical p hysics co u ld arise from very energetic vibration o f the m olecule. A t ro o m
tem perature, how ever, the fraction o f am m o n ia m olecules th at could have this large am o u n t o f vibrational
energy is truly in sig n ific an t U n d er the law s o f q uantum m echanics the pred ictio n fo r th e significance o f
inversion is quite different from the classical prediction. T h e m oving (from vibration) nitrogen atom sees
the plane o f the hydrogens as the peak o f a potential barrier. In an ex am p le o f th e qu an tu m m echanical
"tunneling" effect, the w ave function o f th e nitrogen nucleus has a finite am plitude on th e o th er side o f
the barrier, an d this is m an ifested as a fin ite p ro b ab ility th a t the n itro g en w ill p e n e tra te the p o tential
barrier despite its lack o f sufficient energy to penetrate it classically. E xperim ents d em onstrate that o cca­
sionally this does indeed o ccu r to m o lecules o f room tem perature am m onia gas, a n d in th is m icroscopic
dom ain "occasionally" is approxim ately 2 0 to 30 billion tim es p er second!
T he average rate a t w hich an am m onia m olecule inverts is a function o f the stren g th o f the potential
barrier at the hydrogen plane, and that strength is closely tied to the rotation state o f the m olecule. W hen
an am m onia m olecule is spinning aro u n d an axis roughly p arallel to its sym m etry axis, w hich is the case
w hen J and K are nearly equal, cen trifu g al forces pull the hydrogen atom s farth er apart, decreasing the
strength o f the potential b arrier an d thus allow ing inversions to occur m ore frequently. I f the m olecule is
spinning around an axis that is nearly p eip e n d ic u la r to the sym m etry axis, w hich occurs w hen J is m uch
larger than K, the centrifugal forces pu ll th e nitrogen aw ay from the hydrogens, increasing the strength o f
the potential barrier and causing inversions to occur less frequently.
Inversion results in a splitting o f the energies o f a m olecule's ro tatio n al and rotatio n al-v ib ratio n al
state transitions (D ennison and U hlenbeck, 1932) that produce absorption o r em ission in infrared regions
o f the electrom agnetic spectrum . This allow s relativ ely low energy transitions to o ccu r betw een the sublevels o f the sp lit transitions. T hese low en erg y transitions g iv e rise to the m icrow ave ab sorption spec­
trum o f am m onia. An im p o rtan t aspect o f such transitions is that w hen they resu lt in th e absorption o f
m icrow ave photons, it is not necessary fo r the absorbing m olecules to reradiate photons o f the sam e fre ­
quency to return to the lo w er energy levels. T h e added energy m ay be converted to translational k inetic
energy o f the m olecules in the gas through collisions w ith those m olecules, therm alizing th e energy th a t
30
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was once p a rt o f th e m icrow ave rad iatio n p ro p ag atin g th rough the g a s; thus tru e abso rp tio n occurs.
B ecause the m icrow ave a b so rp tio n sp ectru m o f am m o n ia arises from th e in v ersio n p h en o m en o n , it is
often called the am m onia in version spectrum . Initial predictions o f the ex istence o f the am m o n ia inver­
sion spectrum w ere based on theoretical predictions and experim ental observations o f sp lit rotational and
rotational-v ib ratio n al transitions a t in frared frequencies (W right and R an d all, 1933; B lean ey , 1946).
T he first d irect observations o f m icrow ave absorption by am m onia w ere m ade by C leeto n and W illiam s
(1934) using a crude form o f absorption cell constructed from rubberized cloth and wood!
T h e m agnitude o f the splitting o f a rotational o r rotational-vibrational energy level is related to the
potential b arrier heig h t o f the m olecule, and thus its rotational state. T h e m ag n itu d e o f th e low energy
transition betw een sublevels d eterm in es the en erg y an d thus the freq u en cy o f the m icro w av e photons
absorbed. A n isolated m olecule in a co n sta n t rotatio n al state, n o t sub jected to frequent, co llisio n s w ith
other m olecules {i.e., the m ean d uration o f collisions is m uch sm aller th an the m ean tim e betw een co lli­
sions, as in a gas at m icro b ar pressures), can absorb only an extrem ely n arrow range o f frequencies.
N ote the size o f this range is finite, n o t zero. T he uncertainty principle allow s m o lecules to absorb
a photon o f a freq u en cy w hich is slig h tly d ifferen t fro m the optim um abso rp tio n freq u en cy , o r cen ter
fr e q u e n c y , o f the m olecule. In a c o llectio n o f a larg e n u m b er o f m olecules, still e ssen tially collisio n less, D opp ler shifts cau sed by the therm al m otion o f the m olecules can also sh ift the cen ter frequencies
o f individual m olecules aw ay from the cen te r frequency o f a stationary m olecule. A t tem peratures above
a fraction o f a K elvin the m agnitude o f the D o p p ler shifting is far g reater than the range in tro d u ced by
the uncertainty p rinciple, and still is m any orders o f m agnitude sm aller than the cen ter frequency. The
frequency shifts p ro d u ced by therm al m otion are statistical in nature, d eterm in ed by the distrib u tio n o f
energies in a gas at a given m acroscopic tem perature and th e orientations o f the m o lecu lar trajectories
w ith respect to the propagation direction o f the m icrow aves. T he observed m icrow ave ab sorption sp ec­
trum o f the collection o f m olecules is the sum o f all the very narrow (in frequency) co ntributions o f the
individual m olecules, each exhibiting its p articular D o p p ler sh ift o f the cen ter frequency. O v erlap o f the
individual contributions causes the sum m ed spectrum to appear as a continuum m uch b ro ad er in frequen­
cy than the narrow range th at w ould be observed in the absence o f D oppler effects.
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A lin ear p lo t o f ab sorptivity again st frequency fo r the co llectio n o f m olecules in a p articu lar ro ta ­
tional state yields the b ell-shaped cu rv e fam iliar to spectroscopists. W h en the frequency axis o f the p lo t
includes zero frequency, th e o rig in o f the term absorption line is q u ite evident: th e b ell-shaped featu re
occupies such a narrow range o f frequencies com pared to th e m agnitude o f its cen ter frequency it appears
to be a vertical line. T he shape o f a curve ob tain ed by p lotting abso rp tiv ity due to a sin g le line ag ain st
frequency is called the line shape. A p aram eter frequently used to d escrib e an im p o rtan t aspect o f a line
shape is the line w idth. T he usual definition o f the w idth o f a line is the difference betw een the freq u en ­
cy o f m axim um absorptivity (w hich is also the center frequency o f the line) and the frequencies at w hich
the absorptiv ity is h a lf this m ax im u m value; this defines th e h a lf w id th at h a lf m a xim u m , o r H W H M ,
o f the line. U sually there are tw o frequencies a t w hich the abso rp tiv ity is h alf th e m ax im u m value, o n e
below and one above the center frequency. T he difference o f these tw o frequencies is an alternate d efini­
tion o f line w idth called th e f u l l w id th at h a lf m axim um , o r F W H M . A t low pressures, lin e shapes are
very nearly sym m etrical about th e cen te r frequency, so the FW H M is sim ply tw ice the H W H M . H o w ­
ever, un d er som e co nditions experim entally o bserved line shapes becom e asym m etrical, so the F W H M
is n o t tw ice the H W H M ; in ex trem e cases the asym m etry is so p ro n o u n ced th a t th e co n cep t o f a line
w idth has n o o b v io u s p h y sical connection. T h e subjects o f lin e sh ap es and lin e w idths h ave receiv ed
considerable attention o v er a span o f m any years, and w ill b e d iscu ssed in m ore d etail later.
S ince the p o ten tial b arrier h e ig h t o f an am m o n ia m o lecu le is tied to its ro tatio n al state, the ro ta ­
tional state d eterm ines the cen ter frequency o f the ran g e o f m icro w av e photons the stationary m olecule
can absorb. It is com m on p ractice to refer to an am m onia ab sorption line by the q u an tu m num bers o f
the rotation al state associated w ith it, such as the "(3, 1) line," w hich refers to th e line arising from
m olecules in the rotatio n al state w ith J = 3 an d K = 1. A m m onia m o lecu les in th e ir various rotational
states can absorb m icrow aves in a large num ber o f w idely spaced freq u en cy ranges. T here are literally
hundreds o f lines w ith frequencies betw een 15 and 40 GHz.
T he m ax im u m ab sorptivity o b serv ed in a given line is a m easu re o f its lin e stre n g th , w hich fo r
d ifferent lines can v ary o v e r m any orders o f m agnitude. T h e strength o f a g iv en line is d u e in p art to
intrinsic properties d eterm ined b y the qu an tu m num bers o f the ro tational state associated w ith it, bu t the
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22
23
Frequency, GHz
Figure 3.3: A p o rtion o f the am m onia inversion sp ectru m as it w o u ld ap p ear fo r extrem ely lo w pres­
sures. A ctual line frequencies and relative strengths are show n; all lines strong enough to show o n this
display are included. T his frequency ran g e includes the stro n g est line o f the am m onia inversion spec­
trum : th e (3,3) line at 23.87 G H z.
m ajor factor in the huge range o f line strengths o bserved is the equally disparate populations o f the rota­
tional states that g iv e rise to the d ifferent lines. A t ro o m tem perature m ost am m onia m olecules are in
low energy ro tatio n al states w ith sm all values o f J and K. T h e strengths o f the lines associated with
those states are m uch larg e r than lines associated w ith hig h energy, sparsely populated rotation states.
Figure 3.3 illustrates a sm all portion o f the am m onia in version spectrum as it w ould appear at ex trem e­
ly low pressu res, w here D op p ler broadening is a m ajo r co n trib u to r to the line w idths. A lthough large
differences in line strengths are ev id en t in this figure th ere are also m any lines w ithin this frequency
range that are m uch too w eak to show on this display. A n absorption spectrum w ith easily identifiable
individual lines is called a line spectrum-, m icrow ave lin e spectra are characteristic o f absorbing gases
under very low pressures.
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3.4
Pressure Broadening of the Ammonia Inversion Lines
A t higher pressures the relative tim e th e m o lecules sp en d in collisions w ith oth er m olecules o f the
gas increases, an d the effects o f these collisions becom e im portant. W hen an am m onia m olecule un d er­
goes a collision th e geom etry o f the m olecule is tem porarily altered. T his changes the h eig h t and shape
o f the poten tial barrier seen by the n itro g en atom , and thus changes the c en te r frequency o f the sm all
range o f m icrow ave photons th e m olecule can absorb. T he d irectio n and m agnitude o f the frequency
shift d epen d on the details o f the collision an d the p recise tim ing o f the absorption during the collision.
C ollision details and absorption tim ing are statistical in nature, w ith sm aller shifts in frequency being
m ore com m on than larger shifts. In a large co llectio n o f am m onia m olecules w here collisions o ccu r
frequently, a significant fraction o f the m olecules absorbing m icrow ave photons do so w hile undergoing
collisions. T h is m eans th a t m any o f the m o lecu les in a p a rticu lar ro tatio n al state w ill absorb photons
at frequencies shifted aw ay from the narrow line characteristic o f isolated m olecules in that rotation state.
A ny m acroscopic sam ple o f gas will contain a huge nu m b er o f m olecules in any o f the rotational states
w ith m oderate values o f J and K and thus the sum o f the co ntributions o f all m olecules in a p articu lar
state w ill again be m anifested as a bell-shaped, co ntinuous b an d o f absorption, bu t it w ill span a ran g e
o f frequencies th a t is considerably broader than the ran g e o f th e collisionless, D oppler-broadened case.
T he increase in absorption line widths due to m olecular collisions is called pressure broadening. T his is
th e d o m in a n t line-broadening m echanism in the tro p o sp h eres and stratospheres o f the g ia n t planets.
E ven at m illibar pressures it overw helm s D op p ler broadening, and at pressures as low as a few bars o f a
Jovian m ixture the line w idths are com parable to their cen ter frequencies.
P ressure broadening transform s am m onia's inversion spectrum from a line spectrum into an absorp­
tion continuum . A s line w idths increase the individual lines begin to overlap each other. W hen average
line w idths are com parable to the average frequency spacing betw een lines the characters o f the individual
lines are lost. T he sum o f the absorptivity contrib u tio n s o f the superposed pressure b ro ad en ed lines is
an absorption continuum o v er a w ide b an d o f frequencies, as show n in Figure 3.4.
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800
T otal A bsorptivity
(3, 3)
(7 ,3 )
0
10
20
30
40
50
Frequency, GHz
F ig u re 3.4: A m m onia inversion spectrum p ressu re b ro ad en ed by approxim ately 2 0 0 m illibars o f pure
am m onia o r 2 bars o f a Jovian gas m ixture. T h e lo w er tw o curves sh o w the absorption spectra o f tw o
o f the hundreds o f individual b roadened lines, th e (3, 3) lin e at 23.87 G H z and the (7, 3) lin e a t 18.02
G H z. T he h ig h est curve is the absorption c o n tin u u m resu ltin g from the sum o f th e co n trib u tio n s from
all the individual lines. N ote th at the freq uen cy ran g e o f F igure 3.3 is a tenth o f the ran g e here; the
individual ch aracters o f the lines visible there h a v e been com pletely lo st in the continuum .
W hen laboratory m easurem ents o f ab sorptivity are m ade on gases at very low pressures it is p o ssi­
ble to study the shapes and broadening b ehavior o f individual lines. Thus there is a large body o f know ­
ledge concerning th e low pressure characteristics o f absorption lines, including theoretically deriv ed line
shapes that closely m atch o bserved shapes. T he theory o f V an V leck and W eissk o p f (1945) accurately
predicts line shapes and strengths for am m onia an d a w ide variety o f o th er gases at low pressu res. This
theory is a q u an tu m m echanical treatm ent o f absorbing m olecules th at are harm onic o scillators in a collisional environm ent, im m ersed in a bath o f electrom agnetic radiation. E ach harm onic o scillato r repre­
sents a m olecule w ith a w ell-defined resonant frequency, so in applications to the beh av io r o f am m onia
vap o r one o scillato r represents o n e m olecule o f am m onia in a co n sta n t rotational state. T h e form alism
p ro d u ced by this analysis fo r p redicting absorptivities due to g aseo u s am m onia is a sum m ation o f the
absorptivity contributions o f each individual ab sorption line at the frequency o f interest. F o r each line,
35
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absorptivity is g iv en b y the p ro d u c t o f a lin e in ten sity factor, g iv in g th e absolute m ag n itu d e o f the
line's absorptivity at lin e center, and a line shape facto r that gives th e frequency dependence o f ab so rp ­
tivity due to th at line:
“ ( U = X X A (J, K ,m ) F(J, K, m ,fo) ,
J
(3.8)
K
w here a ( f 0 ) is th e absorptivity at o b serv ed freq u en cy f0 , J and K are the m o lecular rotatio n al q u an tu m
num bers, an d m is a vector o f th e p ertin en t m acroscopic conditions o f the gas, such as tem p eratu re an d
partial pressures o f the gas species present. E ach com bination o f J an d K values corresponds to a unique
absorption line arising from in version o f am m o n ia m olecules in the (J, K) rotatio n al state. F u n ctio n
A (J, K, m) is the line intensity factor. F unction F (J , K, m , f) is the line shape factor, and in E quation
3.4 it is evalu ated at th e argum ent frequency fo- It is the sum o f tw o sim pler line shapes at p o sitiv e an d
negative values o f the center frequency, having the functional form
F(J, K , m , f )
fc(JK)
Af
Af
(fc(J.K)-f) +Af2
(fc ( J , K ) + f) + A f ‘
(3.9)
w here fc (J, K) is the cen ter frequency o f the (J, K) lin e and Af is the line w idth, given as the H W H M o f
the line. T he line w idth is a function o f the m acroscopic conditions m o f the gas.
V an V leck-W eisskopf theory successfully predicts the asym m etry observed in individual absorption
lines as pressure increases: at equal displacem ents fro m the line cen ter frequency the ab sorptivity in the
line's high frequency tail is greater than that o f the lo w frequency tail, and in general does no t approach
zero even at very high frequencies. B u t the analysis th at produces the Van V leck-W eisskopf form alism
hinges on the assum ption that th e relativ e tim e m o lecu les o f the gas spend betw een co llisio n s is m uch
greater than the tim e sp e n t in collisions. T h is is a valid assum ption fo r gases at lo w pressures, b u t it
fails at high er pressures. F or Jovian gas m ixtures th e agreem ent betw een V an V leck -W eissk o p f theory
and laboratory m easurem ents deteriorates rapidly at pressures greater than about h alf a bar; the p redicted
absorptivity o f the low frequency w ings is too sm all, and in the hig h frequency w ings it is too large.
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U nfortunately laboratory d ata at h ig h e r pressures d o little to reveal lin e shape d etails and behavior,
d u e to the u n reso lv ab ility o f in dividual lin es. T h eo ries d ealin g w ith ab sorption line sh ap es at h ig h er
p ressures a re thus inheren tly less w ell estab lish ed . A m o re com plex qu an tu m m ech an ical an aly sis by
B en-R euven (1966) attem pted to p red ict h ig h p ressu re b ehavior. B en-R euven retain ed th e "in freq u en t
collisions" assum ption o f Van V leck an d W eissk o p f fo r g ases at low pressures, bu t ad d ed other assum p­
tions for m o lecu les in a nearly co n stan t sta te o f co llisio n , ap p ro p riate fo r high p ressu re s. H e also
included the effects o f coupling betw een ad ja c e n t lines, w h ich becom es im p o rtan t as th e lines o verlap.
In the low p ressu re lim it the resulting fo rm alism ap p ro ach es th e V an V leck -W eissk o p f form alism . A t
high pressures it p redicts ab sorptivities in th e lo w freq u en cy tails th at are ab o u t tw ice those o f V an
V leck W eissk o p f theory, and in the high frequency tails they are correspondingly low er. In this pressure
regim e the B en-R euven form alism is accu rate to w ithin 5 -10% for p u re am m o n ia gas a n d Jovian m ix­
tures. B ut fo r Jo v ian m ixtures a t m oderate pressures, w here neith er the "infrequent collisions" n o r the
"frequent collisions" assum ptions are appropriate, neith er form alism adequately describes existing lab o ra­
tory data. Surprisingly, at extrem ely high p ressu res g reater than 100 bars o f a Jovian m ixture, the B enR euven form alism again show s sy stem atic d isa g ree m en t w ith lab o rato ry data (M orris and P arso n s,
1970; B erge and G uilds, 1976).
A fter the high pressure laboratory d ata o f M orris and P arsons becam e available em pirical m odifica­
tions m ade to the B en-R euven form alism co rrected the o b serv ed inaccuracies at extrem ely high pressures
(B erge and G ulkis, 1976). T hose d ata w ere taken a t a single tem perature, and the correctness o f the tem ­
perature dependences in the m odification as w ell as the orig in al form alism is in doubt (B im b au m , 1987;
P oynter, 1987). T he sparseness o f reliable d a ta on Jovian g as m ixtures at pressures betw een one and ten
bars prevented useful analysis o f the prediction errors at m oderate pressures, and these rem ained trouble­
som e. This lack o f data w as the m otivation fo r the laboratory m easurem ents perform ed as p art o f this
w ork. A m ore detailed and quantitative presen tatio n o f the B erge and G ulkis m odification to the B enR euven form alism is contained in C hapter 7.
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Chapter 4
Experiment Strategy, Apparatus, and Procedures
T he purpose o f this w o rk is to p ro v id e a m eans fo r accurately d escribing the prop ag atio n o f m icro­
w aves in th e atm osphere o f a g ian t p lan et such as Ju p ite r o r Saturn. T h e g o al o f the experim ental p o r­
tion o f the p ro je c t is to m easure the param eters needed to attain that end. S in ce a w ide frequency range
is used in the probing o f atm ospheres w ith rad io w aves, the frequency dependence o f those param eters is
an im p o rtan t aspect o f the experim ental program . P lan etary atm ospheres also ex h ib it a w ide range o f
m acroscop ic gas co n d itio n s th at m ay affect rad io p ro p ag atio n . S u ch co n d itio n s in clu d e tem perature,
pressure, q uantity o f th e absorbing gas(es) present, an d th e quan tity and ch em ical species o f the other
(foreign) gases present. T he ex p erim ent p rogram m u st address these variab les as w ell. T his chapter
begins by describing the first step in the program : identifying the m easurable q uantities w hich can p ro ­
vide the inform ation n ecessary to attain the stated g o al, and d evising a stra te g y fo r m easu rin g those
quantities. Section 4.2 describes the m icrow ave sp ectro m eter d esigned an d co n stru cted fo r perform ing
the m easurem ents w hile varying all the appropriate m acroscopic conditions. A su m m ary o f th e proce­
dures used in the experim ents, reduction o f the raw data, and uncertainties follow s in S ection 4.3.
4.1
Techniques For Measuring Absorptivity and Refractivity
T h e problem o f m easuring the electrom agnetic p ropagation characteristics o f a given m ed iu m is by
no m eans new . M any varied m ethods have been used to m ake such m easurem ents o n a w ide ran g e o f
m aterials, w ith a considerable range o f accuracies. T here are tw o d om inant techniques for m easuring the
characteristics o f gaseous m edia at m icrow ave frequencies: absorption cell m ethods, and cavity resonator
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m ethods. E ach technique has its ow n advantages a n d disadvantages, an d the requirem ents o f a specific
ex p erim en t w ill usually d eterm in e w hich m eth o d w ill be used. Since the tw o m ethods m easu re the
sam e prop erties o f a gaseous m ed iu m there m ay be som e overlap, so in som e cases eith er m e th o d could
be used to obtain adequate results.
A bsorption cells are usually one o f tw o types, w aveguide cells o r coaxial cells. T h e th eory o f elec­
trom agnetic w av e propagation in w aveguides an d coaxial system s is w ell un d ersto o d (see, e.g ., R am o,
W hinnery, and V an D uzer, 1965), an d instrum ents based on these devices have p ro v id ed valuable data
(see, e.g., T ow nes and Schaw low , 1955). A n absorption cell is a length o f w aveguide o r (usually rigid)
coaxial conductors closed a t its ends so the m edium under test can b e contained w ith in the sp ace inside
the w aveg u id e o r betw een the conductors o f the coaxial system . If electro m ag n etic w aves o f know n
am plitude en ter one end o f the cell, their am plitude decreases as they propagate th ro u g h the c e ll due to
en erg y lo st to the absorbing gas, an d their w avelength varies w ith the index o f refractio n o f the gas.
T here are various m eans o f exam ining the o u tp u t o f the cell to deduce the characteristics o f the m edium
inside the cell. O ne such m eans is a direct m easurem ent o f the ch an g e in am plitude and p h a se o f the
ou tp u t as the test gas is intro d u ced into the cell. A nother involves totally reflectin g th e w av es at the
end o f the cell so they propagate back through the cell, thus doubling its effective length, and exam ining
the standing w ave pattern p roduced w hen the reflected signal interferes w ith the in p u t signal. A sig n ifi­
cant advantage o f the absorption cell m ethods is that the w aveguide an d coaxial system s m ay b e used
o v er a continuous range o f frequencies, so very high frequency resolution is possible. T his is often v al­
uable w hen m easuring the characteristics o f gases at very low pressures, w here closely spaced individual
absorption lines m ay need to be resolved.
T h ere are tw o m ajor disadvantages to absorption cell system s. If the test gas is a w eak absorber,
the length o f the cell needed to produce easily m easurable effects in the signal m ay becom e im practically
large. T his is th e case w ith gas m ixtures sim ilar to those in the atm ospheres o f the g ian t planets, w here
am m onia is a sm all fraction o f a gas m ixture in w hich the oth er significant constituents are essen tially
tra n sp a re n t A bsorption cells also su ffe r an inh eren t p roblem w ith reflections g en erated at the en d s o f
the cell. Som e type o f so lid m aterial, called a "w indow ," is necessary at the ends to contain the test gas
39
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w ithin the cell. T he reflections fro m the w indow s interfere w ith the m easurem ent o f the d ire ct effects o f
the test gas. It can be d ifficult to accurately p re d ic t the beh av io r o f these reflections given the changing
intrinsic im pedance environm ent in the cell, an d this results in larger errors.
A cavity resonator is w ell su ited to m easu rem en t o f propagation constants o f a gaseous m edium due
to its sensitivity to the characteristics o f its contents. T h e electrom agnetic theory o f cav ity resonators is
w ell understood (see, e.g., R am o, W hinnery, an d V an D uzer, 1965), an d th eir u tility in m icrow ave spec­
troscopy is w ell established (see, e.g., Tow nes and Schaw low , 1955). C ircu lar cylindrical resonators are
p articularly useful since they are relativ ely e asy to m achine fro m co m m o n ly availab le stock, high Q
values can be attained w ith m oderate care in construction, an d their com pactness p rom otes tem perature
uniform ity. Such resonators h ave m any d ifferen t resonances o f tw o basic types. T ra n sverse electric
m odes, o r T E m odes, h ave field configurations in w h ich the electric fields are alw ays p erp en d icu lar to
the sym m etry axis o f the cylinder. T ra n sverse m a g n etic m odes, o r T M m odes, h av e m ag n etic fields
p erpendicu lar to the sym m etry axis. T h eo retically there are an infinite n u m b e r o f both types o f re so ­
nances in any resonator, but w ithin a bounded ran g e o f frequencies the n u m b er and frequency spacing o f
resonances is determ ined by the specific geo m etry and size o f the resonator. A disadvantage o f a cavity
reso n ato r system is that its frequency resolution is lim ited to the frequencies o f its usable resonances. In
m easurem ents on gases at very low pressures this can be a fatal flaw . H o w ev er fo r gases at m o derate to
high pressures, w here the m icrow ave absorption lines h ave broadened to th e ex ten t that th e ab sorption
spectrum appears as a slow ly-varying (in frequency) continuum , this presents no m ajo r problem . Since
the lim ited resolution o f a resonator system w o u ld n o t b e a pro b lem w ith m easurem ents su ch as those
needed for this w ork, a cavity resonator system w as chosen fo r these experim ents.
E ach resonance o f a cavity resonator has tw o d escriptive param eters that are easily m easured w ith a
spectrum analyzer. The center freq u en cy, fc , is the frequency at w hich field am plitudes inside the cavity
are m axim ized fo r a constant input pow er. T he half-pow er bandw idth, b , is the d ifference o f the tw o fre­
quencies, one above and one below fc , w here th e field am plitudes decrease to
tim es those at fc (the
stored energy decreases by a facto r of */2). A related param eter is the resonance q uality fa c to r, o r fig u re
o f m erit, Q . It should n o t b e co n fu sed w ith th e sim ilar param eter Q o f a gaseous m ed iu m . T o avoid
40
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ambiguity, the quality factor associated with gaseous media w ill be written Qg, w h ile that associated
with a resonance w ill be Q c. Q c is defined as the ratio o f the energy stored in the fields o f a resonance
at its fc, divided by the average energy lost per radian change in phase. If Q c is much greater than unity
it may be closely approximated by a sim ple function o f the measurable resonance characteristics:
I f a resonator is filled w ith a hom ogeneous m edium o f refractive index n an d quality factor Qg, both
the center frequency and bandw idth o f each resonance are affected. F o r a T E m ode resonance w ith mode
orders j, k, and m , in a circu lar cylin d rical reso n ato r w ith cy lin d er rad iu s a an d length d , th e cen ter fre­
quency o f that resonance is given by:
(4.2)
w here c is the p lane-w ave propagation sp eed in a vacuum , and p'jk is th e value o f the argum ent r at the
k*b zero o f Jj’(r). the first derivative o f the jth order B essel function o f the first kind. For a fixed cavity
geom etry the quantity u n d er the rad ical is a constant, so cen te r frequency is inversely p ro p ortional to the
refractive index o f the contained m edium . If QCv is the quality facto r o f the reso n an ce w h ile the cavity
is evacuated, then Qc|, the quality factor o f the resonance w hile the cavity is filled w ith the test m edium
("loaded"), is given by (R am o, W hinnery, and V anD uzer, 1965):
1
Q cl
1 , 1
Q cv
Qg
u
..
^ ^
T h e situation w ith a T M m ode resonance is analogous for both fc and Q.
If the bandw idth bv an d cen ter frequency fcv o f a given resonance are m easured w ith the cavity ev ac­
uated, and altered values b| and fc | o f th e sam e resonance are m easured w hile the cavity is loaded w ith the
test m edium , the refractive index and absorptivity o f the m edium m ay be calculated. Inside the resonator
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the values fo r X and thus (3 a re im posed by cavity geom etry and d o n o t v ary w ith the reso n ato r contents;
thus they carry n o in fo rm atio n about the m edium . T he sh ift in fc c arries in fo rm atio n , specifically the
propagation sp eed in th e m edium and thus its refractive index. K now ledge o f the refractive index allow s
calculation o f P in the g en eral case, given the electrom agnetic w ave's frequency.
S ince the refractive
index o f a vacuum is unity an d fc| is inversely proportional to the refractiv e index o f the test m edium ,
n = ^
'Cl
.
T he real p art o f the relative p erm ittivity can also be calculated:
(4.4)
e' = n 2 . M o st atm o sp h eric gas m ix ­
tures have n values very near unity, so use o f the refractive index can b e so m ew h at aw kw ard. R efractivity v , defined by v = n-1, is often used for gases. In terms o f the m easu red frequencies o f the resonance,
If the loss tangent o f the m edium is m uch less than unity, E quation 3.5 m ay b e approxim ated by:
a _
p
2
£ '
’
(4.6)
so that
(4.7)
T he loss tangent e"/e' is VQg, w hich from E quation 4.3 is VQC| - "*/Qcv. S u b stitu tin g m easured esti­
m ators for the Q values yields:
If v is sm all enough that fcv = fcl> this sim plifies to
a = | ( b , - b v) .
(4.9)
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It is im portant to note th at the units on a as expressed h ere are nepers p e r u n it length. P lanetary astron­
om ers usually express absorption in optical depths. O ne n e p e r o f attenuation reduces the fi e ld strengths
o f a w ave to H e o f the incid en t value; one optical depth o f attenuation reduces the intensity o f the w ave
to H e o f the in cid en t value. T he pow er transported by an electrom agnetic w ave is proportional to the
square o f its field strengths, an d thus one neper o f attenuation is equivalent to tw o optical depths. T hese
units are rare ly w ritten explicitly, so in either case absorptivities are ex p ressed as length- 1. Potential fo r
am biguity m akes it advantageous to express absorptivities in dB p e r un it length. T he conversion factors
are:
1 neper = 2 optical depths = 20 lo g io £ (approx. 8.686) dB .
(4.10)
T he preceeding analysis establishes the m ethod by w hich m easurem ents o f resonance center frequen­
cies and bandw idths o f a cavity resonator th at contains an iso tro p ic gaseous m edium can be used to cal­
culate the refractiv ity v and absorptivity a o f the m edium . R efractiv ity can be used to calcu late the
phase co n stan t [3 o f a w ave trav elin g through the m edium . T h u s the reso n ato r m easurem ents supply
sufficien t inform ation to characterize electrom agnetic w av e p ropagation in the m edium a t the available
resonance frequencies.
4.2
Experiment Apparatus
T he apparatus used in this program o f experim ents is a m icrow ave sp ectro m eter consisting o f five
m ajor subsystem s: a cavity resonator, a sim ulation cham ber, a gas m an ifo ld to supply the sim ulation
cham ber, m icrow ave electronics to support the resonator, an d a therm al control and m onitoring system .
T he author fabricated the first three subsystem s, w hile m o st com ponents o f the la tte r tw o are co m m er­
cially available instrum ents. F igure 4.3 show s a schem atic d iag ram o f th e integ rated spectrom eter.
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C enter Frequency. G Hz
TE M ode O rder
A pprox, Q c
9.17
012
19,010
11.24
013
21,010
12.74
221
21,050
13.62
014
22,870
13.70
222
19,540
15.64
023
24,810
16.18
015
22,910
17.01
224
12,410
17.43
024
25,550
T a b le 4.1: C enter frequencies, m ode n um bers, an d ap p ro x im ate Q c at 273 K fo r th e observed usable
resonances o f the cavity resonator constructed for this w ork.
T h e co re o f the spectrom eter is the m icrow ave cav ity resonator, th e sensor w hich m akes m e a su re­
m ents o f re fractiv ity an d ab sorptivity po ssib le. T he g e n e ra l design o f the re so n a to r is illu stra te d in
Figure 4.1. It is a rig h t circu lar cy lindrical cav ity re so n a to r having three m ain structural parts an d tw o
signal prob es. T he stru ctu ral parts are the cy lin d er and tw o en d plates, th e base and th e head. A ll w ere
m achined from brass stock. T able 4.1 g ives the m ode num bers, cen te r frequencies, a n d m axim um fig­
ures o f m erit (Qc) fo r each o f the nine usable resonances w ithin the 9 to 18 G H z range.
T he en d o f the cy lin d er w hich jo in s to the h ead p late has tw o m ode-supression slits. C ertain T E
resonance m odes have center frequencies w hich are the sam e as those o f corresponding T M m odes, a p h e ­
nom enon called degeneracy. T hese TE m odes h av e high Q c , w hile those o f the d egenerate T M m odes
usually are considerably low er. The m ode supression slits p revent cu rren t flow betw een the cylinder and
end plate, disrupting the T M m odes and preventing their interference w ith the desirable T E m odes.
T w o signal probes th at couple m icro w av e en erg y in an d out o f th e reso n ato r are m ounted in the
head plate. T h e probes are the unique asp ect o f this reso n ato r, and greatly enhance the resonator's ap p li­
cability to these m easurem ents. T heir design, fabrication, a n d testing are described in A ppendix A.
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Signal
Probe
Ports
Head Plate
Top View
2.58 cm R (ID)
CD
Modesupresslon
Slit
CD
LO
Cylinder
Side View
Base Plate
F igure 4.1: D esign o f the cavity reso n ato r c o n stru cte d fo r this w ork. A ll m ajo r parts are brass. M o d e
supression slits elim inate undesirable T M m odes th a t w ou ld interfere w ith the h ig h er-Q c T E m odes.
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T h e resonator is m ounted inside a sim ulation cham ber, a v acuum /pressure ch a m b e r w hich contains
the gas m ixture under test. T h e design o f the cham ber is illustrated in F ig u re 4 .2 . A ll m ajo r structural
parts excep t the base plate are m ad e w ith stan d ard 6 " Schedule 4 0 steel pipe an d fittings. T he base plate
w as m ach in ed fro m 1/2" ste e l sto ck and w elded to the pipe. P orts have been m ach in ed into the lid to
accom odate tw o gas fittings, tw o herm etically sealed SM A feed-through jac k s, an d a th erm om eter probe
fitting. A lthough the ch am b e r is rated to ten atm ospheres, w ith a m uch h ig h er u ltim ate strength to give
a larg e safety factor, oth er considerations lim it th e experim ents to e ig h t atm ospheres absolute pressure.
A gas m anifold controls th e contents o f the sim ulation cham ber. T he m an ifo ld has tw o branches
th at co n n ect to the tw o gas fittings in the lid o f the cham ber. A ll p ipes o f both b ran ch es are 1/4" OD
stainless steel w ith S w agelok™ connectors, an d all valves are N u p ro m o d el S S -4 P 4 T stainless steel.
O ne branch carries the p ressure gauges an d serves as an exhaust p o rt w hen the ch am b er pressure is above
am bient. T he oth er branch connects a vacuum p u m p and up to fo u r bottles o f gas to th e cham ber.
T w o B ourdon tube pressure gauges are co nnected to the first branch. O ne, a vacuum /pressure gauge
that can m easure vacuum to 8 0 0 to rr below or p ressure to 760 torr above am bient, is isolated by a valve
from the rest o f the system w h en ch am b er pressures are above its range. T h e d ial o f this g au g e is m o d i­
fied fo r greater accuracy, w hich for m ost p ressures is ±5 torr. N ear am bient p ressu re, accuracies o f ±3
torr m ay be achieved. The o th e r g au g e m easures high pressures fro m 10 to 105 p sig , accurate to ± 0.2
psi. C alib ratio n charts are g en e ra ted for both gauges by p lacin g th em in p ara lle l w ith v ery accurate
H eise gauges. F o r absolute p ressu re m easurem ents both gauges d epend on accurate k n ow ledge o f am bi­
ent pressure. T his in form ation is p ro v id ed by a B endix m odel N .A E R 0 .1 9 3 6 .U S N an ero id aviation
barom eter that reads am bient p ressu res in m illibars, w ith a p recision o f 0.1 m illib ar. E ach p ressure
datum is reco rd ed in the units o f th e gauge in v o lved in the m easurem ent. D u rin g d a ta red u ctio n all
pressures are converted to atm ospheres w ith these conversion constants:
1 atm = 14.6956 psi = 76 0 torr = 1013.25 m illibar.
(4.11)
If cham b er pressure is at o r below am bient, p assiv e venting is im possible. A v acu u m p u m p c o n ­
nected to the second branch o f th e m anifold is used to produce low er pressures o r ev acu ate the cham ber.
46
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Reproduced
with permission
Chamber Cross-section
Top View of Chamber Lid
S h o w n with r e s o n a t o r m o u n te d
of the copyright owner.
Microwave
Feed-through
Jack Ports
Lid
O-rlng
Groove
Resonator
Further reproduction
-
Thermometer
Probe Port
Gas
Fitting
Ports
j
prohibited without p e r m is s io n .
Coaxial
Cables
15.42 cm Dia.
Microwave
Feed-through
Jack Ports
Figure 4.2: Design o f the atmospheric simulation chamber constructed for this work. Scale is 1/3 actual size.
Reproduced
with permission
RTD
Thermometer
Frequency
Counter
Exhaust
of the copyright owner.
□
Sweep
Generator
Thermal Chamber
Spectrum
Analyzer
Further reproduction
zzflz
Vacuum
Pump
\T e m p .
Probe
R esonator
prohibited without p e r m is s io n .
E xhaust
Gas
Supply
Tanks
Atmospheric
Simulation
C ham ber
Figure 4.3: Schem atic diagram o f the integrated microw ave spectrom eter constructed for this work.
It can also ev acu ate all gas supply lines w ith o u t depressurizing the cham ber, elim inating o n e source o f
system atic e rro r w hen generating gas m ixtures. T he pu m p is a W elch M fg . C o. D uo-Seal™ o il-sealed
rotary pum p capable o f pressures less than
0 .1
torr.
Supply lines o riginate at each gas tan k an d connect through a system o f valves to the seco n d branch
o f the m anifold. E ach tank is fitted w ith a standard reg u lato r w ith back-up valves to p rev en t c ro ss-c o n ­
tam ination. T he gas supply lines an d valve sy ste m p erm it m ixing o f up to fo u r d ifferen t g ases in the
cham ber. T h e fo u r gases used w ere o b tain ed from th e L iq u id C arbonic C orporation an d assay ed as:
helium , 9 9.9% p u rity ; h y d ro g en , 9 9 .9 9 % p u rity ; am m onia, 99.99% p u rity ;
and m eth an e, 9 9.5%
purity, w ith eth an e as the m ajor im purity.
T em p eratu res in the sim u latio n c h am b e r are c o n tro lled by p lacing th e en tire c h am b e r assem b ly
inside a M issim ers m odel F T 5 V -7 4 X 1 7 0 therm al unit w ith an internal fan to circulate air for tem p era­
ture uniform ity. T his provides th erm o statically m aintained tem peratures fro m approxim ately 2 1 0 K to
485 K . A lthough the therm ostat allow s tem p eratu re cycling ±1 K around the se t point, th e cham ber's
long therm al tim e constant keeps fluctuations in sid e the ch am b er sm aller than the 0.1 C (o r K ) reso lu ­
tion o f the in d ependent th erm om eter d escrib ed below . Stainless steel ports in the M issim ers u n it allow
m icrow ave cables, gas m anifold lines, an d therm om eter cables to connect w ith external instrum ents.
C h am b er tem peratures are m onitored w ith a therm om eter system that is independent o f the therm al
co n tro l unit. T he in stru m en t is an O m eg a E n g in eerin g m odel H H -72R T D th erm o m eter w ith a 100Q
platinum R T D probe, w ith 0.1 C reso lu tio n and an advertised accuracy o f ± 0.4 C o r better. It w as cali­
b rated before and after each ex p erim en t using a w ater ice bath and a high reso lu tio n digital m ultim eter;
adjustm ents w ere n ev er m ore than 0.2 C. T he pro b e sensor is in the tip o f a stainless steel sheath nine
inches long an d 1/8 inch in diam eter. It is inserted through a fitting in th e lid o f th e ch am b er, an d is
locked in place and sealed by a sw aging n u t. W h en inserted, the probe tip is fiv e m illim eters fro m the
head plate o f the resonator.
D uring experim ents tem perature m easurem ents w ere m ade w ith each pressure m easurem ent to allow
corrections fo r the effects o f slightly differin g tem peratures. A s stated above the uncertainty in absolute
tem perature m easurem ents m ade w ith the R T D th erm om eter w as ± 0.4 C, bu t the g reat m ajority o f that
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was system atic error, m ostly e rro r intro d u ced by the instrum ent's piecew ise lin ear approxim ation o f the
platinum p ro b e resistan ce v s. tem p eratu re cu rv e. C alib ratio n o f th e in stru m e n t w as a m uch sm aller
source o f system atic error, sin ce the zero p o in t co u ld be se t w ithin ± 0 .0 5 C an d the scale size set at 500
C w ithin ±0.1 C. M ost o f th e rem aining e rro r w as d u e to the 0.1 C reso lu tio n o f the therm o m eter d is­
play. T he m ajo r source o f ran d o m e rro r w as electrical noise in the therm om eter. U nlike resolution error
this noise w as tim e-v arian t an d w as visib le as toggling o f the la s t d igit o f the tem perature display as the
indicated tem perature ch an g ed o n e fro m reso lv ab le value to the next. T ests w ith a slow ly bu t steadily
drifting ch am b e r tem perature sh o w ed that the display sp en t approxim ately o n e fifth o f its tim e toggling
and four fifths on a steady d isp lay o f one value. T his suggested that noise m agnitudes co rresponded to
about ±0.01 C in the display; i f the in d icated tem perature w as w ithin 0.01 C o f the b oundary betw een
adjacent reso lv ab le values, d isplay toggling w ould occur. In fact this effect w as used in the recording o f
tem perature d ata. W hen a tem perature m easurem ent w as m ade w hile the display w as toggling, the tem ­
perature was recorded as m idw ay betw een the adjacent display values. If a m easured system tem perature
at one tim e w as T , and later the m easured sy stem tem perature w as T+A T w ith o u t intervening calibration
adjustm ents, th e d ifference betw een system tem peratures at the tw o tim es w as reliably AT ±0.1 C.
T he re so n ato r is en erg ized an d m o nitored w ith a m icrow ave sw eep gen erato r, a coaxial cable feed
system , a m icrow ave spectrum analyzer, and a high-resolution frequency counter. T he cable system was
assem bled from basic p arts, an d th e oth er com ponents are com m ercially available instrum ents.
T he sw eep gen erato r is a H ew lett-P ack ard m odel 8690B , w ith plug-in oscillato rs fo r the frequency
bands 7.0 to 12.4 G H z, 12.4 to 18.0 G H z, 18.0 to 2 6 .0 G H z, and 2 6 .0 to 4 0 .0 G H z. O u tp u t po w er
varies w ith frequency, from as m uch as 40 m W in the low est band to as little as 5 m W in the highest.
M o st o f that p o w e r is reflec te d by th e in p u t signal pro b e on the reso n ato r. M uch o f the rem aining
pow er is lost to the w alls and o th e r parts o f the h ead p late ports. T he actual p o w e r enterin g th e resonat­
o r is a sm all fraction o f the p o w er p roduced by the generator, so saturation effects are no t im portant.
C oaxial cables carry m icrow ave energy from the generator to the reso n ato r and from the resonator to
the spectrum analyzer. T he cab le s are b u ilt from 5 0Q , 0.085" O D sem irig id ca b le stock w ith a tin­
plated copper o u te r conductor, a silver-plated copper inner conductor, and a P T F E (Teflon™ ) dielectric.
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Soldered to the cab le stock are gold-plated O m ni-S pectra type SM A connectors. Tests show ed th e cables
perfo rm ed w ell a t frequencies u p to 24 G H z. D u e to the p ro fo u n d im p ed ance m ism atch betw een the
reso n ato r an d the ex tern al electronics, tw o T eled y n e M icro w av e m odel T -7S 83U -40 isolators are vital
com ponents o f th e feed system . O ne is placed betw een the sw eep g en erato r a n d th e resonator, th e other
betw een the resonator and the spectrum analyzer.
T h e spectrum an aly zer consists o f a H ew lett-P ack ard m odel 141T fram e and disp lay unit, w ith a
m odel 8555A R F u n it and a m odel 8552B IF u n it Its internal m ix er accepts frequencies up to 18 G Hz.
T h e R F unit has an o u tp u t ja c k fo r its local o scillato r (LO ), w hich controls the tuning o f the analyzer.
T his is connected to a H ew lett-Packard m odel 5245L high-resolution frequency counter. T he counter has
a m odel 5255A plug-in frequency converter w h ich allow s m easurem ent o f th e spectrum analyzer L O fre ­
q u en cy from 2.6 G H z to its u p p e r lim it o f 4 G H z. N ine d ig it resolution is possib le in this range, bu t
the L O is rarely stab le enough to afford m ore than seven or e ig h t d ig it accuracy. The co u n ter is calib rat­
ed w ith a 1 M H z signal accurate to one p art in 109. T here are problem s w ith drift o f the spectrum an a­
lyzer's interm ediate frequency, especially w ith tem perature changes, and these can affect the accuracy o f
frequency m easurem ents m ade w ith this system . A ppendix B describes this problem and its solution.
4.3
Laboratory Procedures, Data Reduction, and Uncertainties
T h e laboratory an d d ata reduction procedures sum m arized here w ere m eans o f im plem enting strate­
gies develo p ed in Section 4.1 on the apparatus describ ed in S ection 4.2. A detailed discussion o f the
procedures and the uncertainties arising from th em is quite lengthy an d m ight distract the read er from the
m ain line o f reasoning, so rather than present them here details are covered in tw o appendices. N otably,
d etails on corrections applied to the raw d ata are co n tain ed there. A ppendix C covers the laboratory
procedures used to collect the necessary raw data. A ppendix D presents the data reduction procedures and
form ulae for the d ifferen t laboratory p rocedures, including u ncertainty analyses for each. T his section
briefly sum m arizes those procedures and p resen ts som e high-level results o f the uncertainty analyses to
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precede th e presentation o f the d a ta in C h ap ters 5 an d
6
. T h ere w ere five m ain categories o f procedures
directly involved w ith these lab o rato ry m easurem ents: eq u ip m en t calibration p ro ced u res, gas m ixture
generation procedures, d ata collection procedures m easuring only refractivities, d ata collection procedures
m easuring refractivities and absorptivities, and data reduction procedures. D ata reduction procedures will
be discussed with their associated laboratory procedures.
Six separate instrum ents req u ired calibration. T h ree o f those w ere p ressu re gauges: th e barom eter,
the vacuum and low pressure gauge, and the high p ressu re gauge. A ll three w ere calib rated by sim u lta­
neous m easurem ents in p arallel w ith referen ce instrum ents o f know n high accuracy. T h e prim ary te m ­
perature m easurem ent device, the platin u m R T D therm om eter, was calibrated by a com bination o f an ice
bath for a zero-point reference an d a resistan ce calib rated w ith a high-resolution dig ital m u ltim eter fo r
scale size adjustm ent. A one M H z cry stal-controlled tim e base oscillator provided the calibration signal
for the frequency counter. The freq u en cy counter, alo n g w ith a calibrated attenuator, w as then used for
calibration o f the various frequency and po w er m easuring functions o f the m icrow ave spectrum analyzer.
For m ore detailed discussions o f all th ese procedures se e Section C .l o f A ppendix C . T h e procedures
presented in A ppendix B fo r circum venting spectrum analyzer interm ediate frequency d rift could also be
view ed as calibration techniques.
D espite considerable e ffo rt in p ro p e r calibrations o f the pressure gauges, those gau g es h ad fu n d a­
m ental resolution lim itations th a t co u ld n o t be im p ro v ed by practical m eans. Such lim itatio n s co u ld
present serio u s problem s in accu rately d eterm in in g th e m ixing ratios o f gas sp ecies th a t w ere m in o r
com ponents in m ixtures. T his is analogous to a p ro b lem com m only experien ced in ch em istry la b o r­
atories: dissolving o r o therw ise m ixing a tiny but accu rately know n quan tity o f one co m p o u n d into a
large q u antity o f another. T he solution to this p roblem , p aralleling the so lu tio n devised by the c h e m ­
ists, em ploy ed a series o f progressive dilutions to arrive at m ixtures with a sm all but accu rately know n
am m onia com ponent. An analysis o f the procedure used to generate such m ixtures w as sufficiently g e n ­
eral that it could be applied to m ixtures featuring m ore b alanced m ixing ratios. T h at analysis p roduced
first-order estim ates o f the m ixing ratio uncertainties o f each com ponent o f a g as m ixture. T h e ex p e ri­
m ent involving m ethane and hydrogen is an ex am p le o f application o f the general an aly sis to a m ore
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balanced m ixture. R esu lts o f th at application w ere calcu lated n u m b er m ixing ratios an d uncertainties o f
0.375 ± 0.0 0 4 fo r the m eth an e co m p o n en t an d 0.625 ± 0 .0 0 4 fo r hy d rog en .
M ixtures containing am m o n ia w ere usually targeted a t one o f tw o am m o n ia n u m b er m ixing ratios:
0.0672 o r 0.00820. T h e first w as achieved w ith a single dilution by the broadening gas, eith er hydrogen
o r helium , an d w as accu ra te to ± 0 .0 0 1 0 , o r 1.3%. T he sm aller ratio w as o f p rim a ry in terest in this
w ork since it w as m ore rep resen tativ e o f co nditions on am m o n ia in th e atm ospheres o f gian t planets.
T hose m ixtures required a seco n d dilution step after partial venting o f w e ll m ixed ch am b er contents, and
their am m o n ia m ixing ratio u n certain ties w ere ab o u t ± 0 .0 0 0 1 7 , o r 2 .0 % . If a m ix tu re w ith that targ et
m ixing ratio h ad been g en erated directly ffo m a single m ixing step, using the sam e pressure gauges, the
uncertainty w o u ld h a v e b een a factor o f five larger. In so m e ex p erim en ts the broadening gas w as no t
solely hydrogen or heliu m b u t w as a Jo v ian m ixture o f the tw o, about 90 % hydrogen and 10% helium
by num ber. T a rg et m ix in g ratio s and th eir uncertainties fo r those m ix tu res w ere: hy d ro g en , 0.8937
± 0.0037; heliu m , 0 .0 9 8 2 ± 0 .0 0 3 9 . A ctual m ixing ratio s ach ie v ed in th e ex perim ents deviate slightly
from those targ et values, bu t the uncertainties are the sam e. The target n u m b e r m ixing ratio an d uncer­
tainty fo r the am m onia c o m p o n en t o f three-gas m ixtures w as identical to that o f tw o-gas m ixtures.
T here is a cav eat to these m ixing ratio uncertainties. M icrow ave spectroscopists are usually fam il­
ia r w ith am m o n ia’s noto rio u s adsorption behavior. C areless use o f am m o n ia in an apparatus can resu lt
in the unw anted p resence o f trace am ounts o f am m onia fouling the results o f su b seq u en t very low p res­
sure experim en ts m o n th s a fte r the last inten tio n al in tro d u ctio n o f am m o n ia in to a te s t cell (Poynter,
1987). A m m o n ia m o lecu les have a p a rticu larly h igh affinity fo r c e rta in m etallic su rfaces an d w ill
adsorb o n to the surfaces o f a te s t cell an d p ressu re vessel, a t the expense o f the am m onia content o f the
contained gas. L ater, w hen th e pressure inside the vessel is significantly reduced, som e o f the adsorbed
am m onia d eso rb s from the su rfaces and becom es re m n a n t am m onia g a s w ithin the te s t cell. Since
am m onia is such a strong m icrow ave absorber w ith a rich se t o f absorption lines over a w ide frequency
range this re m n a n t gas can w reak havoc on experim ents w ith m ore w eak ly absorbing species. It can
also affect the actual concentration o f am m onia in su b seq u en t experim ents involving am m onia. F ortu­
nately the experim ents o f this w ork are n o t low -pressure experim ents; spectroscopists usually deal w ith
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am m o n ia partial p ressures that are fractions o f a to rr in stead o f the tens o f to rr involved h ere. F lu ctu a­
tions o f the am m o n ia m ixing ratio d u e to ad so rp tio n an d d e so rp tio n effects sh o u ld be fa irly sm all.
H o w ev er, these effects are n o t y et fully u n d ersto o d an d the m ag n itu d es in v o lv ed are little m o re than
gu essw o rk (P oynter, 1987), so the stated m ixing ratio s sh o u ld b e view ed w ith som e caution.
A larg e su b set o f th e m easurem ents rep o rted h ere d o n o t involve m ixtures co n tain in g am m onia.
P u re gases hydrogen, helium , and m ethane, an d any m ixtures o f those gases, are essen tially transparent
to m icrow ave radiation un d er the tem peratures an d pressures o f these experim ents, so only m easurem ents
o f refractiv ity w ere attem pted. M o st refractiv ity m easurem ents on hydrogen and helium w ere done in
experim ents that w ere adjuncts to m easurem ents on m ixtures containing am m onia, m easuring effects on
the resonances due to the dielectric constants o f th e transparent gases, so procedures used in th o se experi­
m ents w ere the sam e as those for m easurem ents involving am m onia. E arlier m easurem ents o n m ethane
and a m ixture o f m ethane and hydrogen used different procedures o f tw o m ain categories, c a lle d screen Af
m ethods and counter Af m ethods. B o th cate g o rie s in v o lv ed m easuring the sh ift in the freq u e n c y o f a
resonance, called Af, as gas is added to or rem o v ed from the cham ber.
T h e re w ere tw o v arian ts o f the screen Af m ethods, th e full screen Af m ethod and the ab b rev iated
screen Af m ethod. In these m ethods frequency shifts w ere m easu red direcd y fro m the sp ectru m analyzer
display screen. T he full screen Af m ethod used tw o frequency sh ift m easurem ents, o n e d u rin g addition
o f a gas to the cham ber and one during rem o v al o f th at gas. R efractivity o f th e reso n ato r co n ten ts was
then calcu lated by dividing the average o f the m easu red frequency shifts by th e cen ter freq u en cy o f the
resonance under loaded conditions. A veraging o f the tw o shifts elim inated the effects o f lin e a r drift in
the sp ectru m analyzer IF and cham ber tem perature, reducing errors. M axim um un certain d es fro m this
m ethod varied w ith the m agnitudes o f the m easu red refracd v ities an d th e particu lar settings o f the sp ec­
trum analyzer, w ith ty pical calc u lated values o f ± 2 -3 % , b u t after n orm alization by num ber d e n sity the
g reat m ajority o f the m easured refracdvides w ere w ithin about 1% o f the average value. T he abbreviated
screen Af m ethod, used in ju s t one experim ent on a m ixture o f m ethane and hydrogen, m easu red frequen­
cy shifts o n ly during rem oval o f gas fro m th e cham ber. T his prevented the lin ear drift can cellatio n fea­
ture o f the full m ethod, bu t allow ed a series o f m easu rem en ts on a gas sam ple w ith absolutely con stan t
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m ixing ratios. M axim um relativ e errors w ith this m eth o d w ere larg er than those o f the full screen Af
m ethod, fro m ± 10% at the lo w est p ressu re to ± 3% at th e highest, b u t th e consistency o f d ata from th at
experim en t indicates the actual errors w ere m uch sm aller than the calculated m axim um uncertainties.
C o u n ter Af m ethods elim in ated d irect u se o f th e sp ectru m analyzer screen in m easu rin g frequency
shifts ex cep t as an aid in tuning to the precise cen ter frequency o f a resonance. L ike the screen Af m eth ­
ods there w ere tw o variants, a full co u n ter Af m eth od an d an abbreviated co u n ter Af m ethod. The m ajor
difference betw een the co u n ter Af m ethods and the screen Af m ethods w as th e m ethod o f m easuring fre­
quency shifts. In the counter Af m ethods resonance center frequencies w ere m easured w ith the frequency
co u n ter (v ia the spectrum analyzer) ju s t before an d ju s t a fte r adding gas to o r rem o v in g gas from the
cham ber. T he difference betw een the tw o m easured center frequencies w as the frequency shift. T w o such
shifts, one upon adding gas and o n e upon rem oving it, w ere averaged in the full m ethod, an d dividing
that averaged shift by the m easured center frequency under loaded conditions yielded the calculated refrac­
tivity o f th e resonator contents. A veraging p erfo rm ed th e sam e drift can cellatio n role as th a t in th e full
screen Af m ethod. This m ethod reduced calcu lated m axim um uncertainties to ± 0.8-1% , an d after density
norm alization all values obtained w ere w ithin 0.4% o f the average. U ncertainties w ere less dependent on
the refractivity v alue than w ith the full screen screen Af m ethod so higher refractivities p ro d u ced sm aller
relativ e uncertainties. O nly three refractiv ity m easurem ents on pure m ethane u se d th e fu ll co u n ter Af
m ethod; inform ation gained as a re su lt o f these m easurem ents verified th at better tem p eratu re and fre­
qu en cy m easurem ents w ere necessary, m o tiv atin g upg rad es in the ap p aratu s and d ev elo p m en t o f the
procedures o f Appendix B.
T h e abbreviated co u n ter Af m ethod w as u sed in o n e experim ent, sim ultaneous w ith th e previously
described u se o f the abbreviated screen Af m ethod. A gain, the linear drift can cellation featu re o f the full
m eth o d w as no t p resen t in the ab b rev iated m eth o d sin ce freq u en cy shifts w ere only m easured upon
rem oval o f gas from the cham ber. M axim um relative errors im proved w ith refractivity fro m about ± 8 %
at the lo w est pressure to ± 1.7% at the h ig h e s t
W ith th e apparatus in its final configuration and th e m ethods o f A ppendix B allow ing repeatable fre­
quency m easurem ents o f high accuracy, a new set o f procedures, referred to as "standard procedure," was
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devised to p erfo rm the needed m easurem ents on gas m ixtures containing am m onia. Standard procedure
defined a series o f steps th at acquired absorptivity and refractivity d a ta o n a m ixture o f am m onia and one
or tw o broadening gases u n d er varying pressures and frequencies w hile tem perature and m ixing ratios
w ere held constant w ithin the capabilities o f the apparatus. H ardw are considerations helped organize the
collection o f th e data.
F o r a gas m ix tu re w ith co n stan t m ixing ratios there w ere three m acroscopic co nditions un d er the
control o f the experim enter that affected m icrow ave absorptivity an d refractivity o f the m ixture: tem per­
ature, total p ressu re, and freq u en cy o f the m icrow ave radiation. T em perature w as the m o st difficu lt o f
these to vary d u e to the long therm al tim e con stan t o f the apparatus. T otal pressure co u ld be red u ced by
venting o f the ch am b e r contents, bu t constant m ixing ratios could only be g u aran teed by venting. Thus
a reduction in total p ressu re co u ld be perform ed only w hen all possible m easurem ents at the higher p res­
sure w ere com plete. M icrow ave frequency w as quite sim ply controlled, b u t o f course w as lim ited to the
usable resonance frequencies o f the cavity resonator. The fundam ental inform ation unit in this program
o f m easurem ents, the datum , w as defined as all necessary m easurem ents o f resonance center frequency,
bandw idth, tem peratures, pressure, an d any o th er quantities req u ired to sp ecify the characteristics o f a
single resonance an d conditions inside the resonator, w hile those conditions w ere being held co n stan t to
w ithin the capabilities o f th e apparatus. The ease o f frequency changes helped defin e the next o rg an iza­
tional level, the series. O ne series o f d ata consisted o f datu m m easurem ents for each o f the usable reso ­
nances w ithin the 9 to 18 G H z frequency range, w hile the conditions inside the resonator w ere held co n ­
stant to w ithin th e capabilities o f the apparatus; thus a series represents a collection o f discrete sam ples
o f absorptivity an d refractiv ity sp ectra o f the gas m ixture. T h e first series m easured on a gas m ixture
with an am m onia m ixing ratio o f 0.0 0 8 2 was usually a t a total p ressu re o f
com pleted, total pressure w as d ecreased by venting to
6
8
atm . W hen that series was
atm . and another series w as recorded, follow ed
by a series at 4 atm ., a series at 2 atm ., an d finally one at 1 atm . T his p ro g ressio n o f series defined one
sequence o f data, so nam ed since the five total pressures w ere available in sequence by venting. T em ­
peratures w ere h eld co n sta n t th ro u g h o u t this procedure. A t the in term ed iate m ix in g step w here the
am m onia m ixing ratio w as 0.0 6 7 2 a single series w as som etim es reco rd ed at a total pressure o f 8.2 atm .
56
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T he proced u re that collected all these d ata on a m ixture o f specific gases at a single targ et tem perature, at
all m ixing ratios, total pressures, an d frequencies desired, w as called o n e experim ent. F o r exam ple, the
set o f proced u res co llectin g all d ata at all m ixing ratios, to tal pressures, an d frequencies, on m ixtures o f
am m onia in heliu m a t 313 K, is referred to as "the 313 K am m onia in helium experim ent." T his o rg a n ­
izational vocabulary is also used in describing the reduced d ata p resented in C hapter 6 .
R eso n an ce ban d w id th changes arising fro m the d ielectric co n stan t o f the gas in sid e th e resonator,
com m only labeled "dielectric loading" (although true dielectric loading is only one o f the effects causing
the b andw id th changes in response to d ielectric con stan t ch an g es), w ere a p o tential source o f erro r in
absorptivity m easurem ents. R em oving those bandw idth changes from the raw d ata req u ired experim ents
on pure hydrogen an d helium to characterize them . R efractivities w ere the yardstick b y w hich d ielectric
loading effects on the p u re gases w ere interpolated, an d to a m inor ex ten t ex trapolated, to the m ixtures
containing am m onia, so dielectric loading experim ents necessarily in clu d ed refractivity m easurem ents.
T h e p ro ced u re u sed fo r th o se experim ents w as exactly the sam e as th a t u sed fo r th e m easu rem en ts on
m ixtures w ith am m onia m ixing ratios o f 0.0082.
A few m easurem ents w ere perform ed on p u re gaseous am m onia, and these d ev iated so m ew h at from
standard procedure. A t each tem perature only o n e o r tw o series w ere recorded. T h ese m easurem ents
p u sh ed the apparatus n ear its practical lim its o f absorption m easurability. Som e reso n an ces w ere ren ­
dered unusab le by absorptivities as high as 2500 dB /km .
R eduction o f the raw data from an experim ent required d ata on the characteristics o f each resonance
w ith the reso n ato r ev acu ated and at the target tem perature o f th at e x p e rim e n t A series m easuring such
characteristics was called a baseline series. W ith few exceptions, each experim ent in clu d ed one baseline
series before sequences o f loaded resonator data, and another baseline series afterw ard. In a procedure that
included the effects o f sm all tem perature differences the baseline data w ere corrected and averaged to yield
b aseline values fo r reso n an ce cen ter frequency an d bandw idth, fCv an d b v , for each resonance. E ach
datum from the loaded series provided a center frequency and bandw idth that w ere corrected to yield values
fc | and b | . R efractivity o f the loaded resonator contents w as reduced from these values using the reduc­
tion form ula deriv ed in Section 4.1, E quation 4.5 . A bsorptivity w as red u c ed w ith E q u atio n 4.8.
57
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U n d er g o o d co n d itio n s o f S N R an d reso n an ce sym m etry the u n certain ty in refractiv ity m easure­
m ents did n o t depend on the m agnitude o f the refractivity being m easured, an d m ax im u m uncertainties as
sm all as ± 2 .5 x lO 6 co u ld b e attained w ith refractivities varying from a few h u n d red to a few thousand
tim es lO"6. R esonance asy m m etry an d decreases in SN R induced by high absorp tiv ity a t tim es caused
uncertainties to be m u ch larger. E ach reso n an ce h a d its ow n "p ersonality" th a t v a rie d as conditions
varied, so separate uncertainties h a d to be calculated fo r each datum . T h is w as also true o f absorptivity
uncertainties. U n d er g o o d c o n d itio n s m e asu red ab sorptivities c o u ld b e accu rate to ± 0 .9 dB /km , b u t
decreases in SN R and the p resen ce o f induced resonance asym m etries could degrade th at by a factor o f ten
or m ore. In experim ents on p u re am m o n ia the extrem ely hig h ab sorptivities c o u ld d eg rad e th at figure
by a factor o f a hundred and even render a resonance unusable.
G lobal absorptivity an d refractivity uncertainties treated here an d in A ppendix C rep resen t the propa­
gation o f errors from o nly those fundam ental m easurem ents directly affecting cav ity reso n ato r m ethods,
m ainly frequency o r tem perature m easurem ent uncertainties. If the system tem perature can be held fairly
constant the cavity resonator m eth o d w ill yield faithful m easurem ents o f th e abso rp tiv ity and refractivity
o f the resonator contents, regardless o f th at system tem perature. T he investigator m ust d ecid e the appro­
p riate tem perature an d u n certain ty to be associated w ith the m easured ab so rp tiv ity an d refractivity and
th eir uncertainties. F o r exam ple, refractivity uncertainties include the effects o f tem p eratu re uncertainty
on the m easured center frequency o f a resonance since therm al expansion o r contraction has a direct affect
o n those freq u en cies, e v e n if the re so n ato r contains only a vacuum . O n th e o th e r h a n d v ariatio n o f
absorptivity or refractiv ity o v er th e tem perature, pressure, and m ixing ratio u n certain ty ranges is thus
excluded from those q uantities' expressed uncertaintainties. Propagation o f u n certainties in fundam ental
conditions to global uncertainties assum es know ledge o f the dependences o f ab sorptivity and refractivity
o n those fundam ental conditions. D ata from this w ork casts doubt on the accuracy o f c u rren t know ledge
a b o u t those dependences, so condition uncertainties have been left w ith those condition m easurem ents.
T his section has been a b rief sum m ary o f procedures used in the laboratory m easurem ents and is not
intended to be a rigorous d ev elo p m en t o f the m ethods. R eaders desiring m o re detail ab o u t any o f these
procedures and uncertainties are strongly urged to read A ppendices B, C, and D.
58
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Chapter 5
Laboratory Measurements of Microwave Refractivity
Spectra on Gaseous Methane, Hydrogen, and Helium
T his ch ap ter is d ev o ted to presentations and discussion o f refractiv ity d a ta co llected on gas species
that exhibit essen tially n o m icrow ave absorption under the co nditions o f the experim ents. The behavior
o f these gases is quite p redictable; the m easurem ents rep resen t sim ple verification o f theory. Since the
d ata w ere co lle c te d in th e co u rse o f th is w ork, h o w ev er, it w as th o u g h t appropriate to re p o rt them .
M easured refractivities o f pure m ethane and m ethane m ixed w ith h ydrogen are presen ted in Section 5.1.
T his is follow ed by the d a ta on pure h y d ro g en in Section 5.2 . F in ally , Section 5.3 p resen ts d ata on
pure helium . In each section, tables o f da ta are preceded by a discussion o f the d ata and, w here appro­
priate, interpretation o f the results.
5.1
Data on Pure Methane and Methane M ixed With Hydrogen
T his section presents all d ata collected on the refractivity o f p u re m ethane and a m ixture o f m ethane
and h y drog en w ith n u m b e r m ixing ratio s o f 0.375 ± 0 .0 0 4 an d 0.6 2 5 ± 0 .0 0 4 , resp ectiv ely .
F our
different m ethods from tw o categories, screen Af m ethods and co u n ter Af m ethods, w ere used to acquire
these data. T h e m ethods are discussed briefly in Section 4.3 and are d etailed in A ppendix C, with data
reduction procedures and uncertainty analyses in A ppendix D.
U nlike am m onia, m ethane m olecules under m oderate pressures h ave no perm an en t electric dipole
m om ent so m icrow ave radiation does n o t easily couple to them . A bsorptivities fo r m ethane are m any
orders o f m agnitude less th an those o f am m onia, far below th e sensitivity o f the apparatus u sed in this
w ork. M eth an e rem ains essen tially tran sp aren t until n ear o p tical freq u en cies. T he K ronig-K ram ers
59
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relations, E quations 3.6 and 3.7, pred ict th at if a m ediu m is transparent ov er a w ide range o f frequencies
its dielectric co n stan t and thus its refractivity sh o u ld be very nearly co n stan t w ithin th at range. M o d er­
ate tem peratu res should also n o t affect m ethane's transparency as long as they do n o t cau se significant
ionization. T h e refractiv ity o f m ethane u n d e r the ra n g e o f conditions sam pled in this w o rk sh o u ld be
directly proportional to its n u m b er density.
H ydro g en is very sim ilar to m ethane in these respects. A lthough it is expected th at the refractivity
o f hydrogen w ou ld b e d ifferen t fro m th at o f m ethane at the sam e n u m b er density, it sh o u ld also be
d irectly p roportional to the hyd ro g en n u m b er density. T here are n o anticipated in teractio n s betw een
h ydrogen and m ethane that w ould affect their refractivities in m ixtures o f the tw o gases, so th e refractiv­
ity o f a m ixture should be given by adding the refractivity contributions o f the individual com ponents.
V erificatio n o f th e linearity o f these refractivities is accom plished by norm alizing m easu red refrac­
tivities by n u m b er density to th a t o f an ideal gas at ST P. This allow s direct com parison o f sam ples at
w idely differing tem peratures and pressures. T h e norm alization procedure, based on ideal gas law s, is
detailed in A ppendix D , page 222.
T he ran g e o f conditions sam pled in the m easurem ents on pure m ethane spanned p ressures from 1 to
3.5 atm ., tem peratures from ab o u t 21 0 to 373 K , and the resonator frequencies betw een 9 an d 18 G H z.
S im ilar con d itio n ranges in m easurem ents on am m o n ia w ould im m ediately reveal th at th e no rm alized
refractivity o f am m onia is a strong function o f tem perature, pressure, an d frequency. M easurem ents on
th e m ixture o f m ethane and hydrogen w ere p erfo rm ed w ith a single resonance, the T E 0 2 4 m o d e at about
17.4 G H z, at a single tem perature, 273 K , and a t total pressures from 1 to about
8
atm .
U ncertainties in th e m easurem ents depended on the d ata collection m ethod and various oth er factors
detailed in A ppendix D . S eparate uncertainties w ere calcu lated fo r each d atu m and are p resen ted in the
d ata tables w ith the m easured values. U ncertainties in tem perature m easurem ents are ±3 K since these
d a ta w ere co llected before the upgrade to the p latin u m R T D therm om eter. T otal pressure u ncertainty is
± 0 .0 1 4 atm . e x cep t fo r p ressures n ear 1 atm . w here the uncertainty is ± 0.004 atm . T em p eratu re and
pressure uncertainties co u ld a d d as m uch as ± 1 .8 % to the refractivity uncertainties arising directly fro m
th e cavity reso n ato r m ethod.
60
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R esults o f the p u re m ethane m easurem ents are consistent w ith reffactivity being directly pro p o rtio n ­
al to num ber density an d independent o f tem perature, pressure, or frequency. T he norm alized reffactivity
value o b tain ed by av erag in g all m easu rem en ts using the screen Af m ethod is 4 3 0 X 1 0 '6. A sim ilar
average o f the three m easurem ents u sin g the co u n ter Af m ethod, all m ade w ithin a sh o rt tim e span o f
each other, is 440 X 10-6. W hen the effects o f tem perature and pressure uncertainties are co m b in ed with
d irect uncertainties in the m easured refractiv ities these tw o values are seen to b e co n sisten t. A lhough
refractiv ity valu es m e a su re d w ith th e c o u n te r Af m eth o d are in h eren tly m o re accu ra te th an tho se
m easured w ith the screen Af m ethod, tem perature an d p ressure m easu rem en ts are e q u a lly uncertain.
R efractiv ities m e asu red w ith the sc reen Af m eth o d are listed in T ab le 5.1, fo llo w e d by listings o f
refractivities m easured w ith the co u n ter Af m ethod in T able 5.2.
M easured refractivities on the m ixture o f m ethane and hydrogen w ere quite self-co n sisten t and w ere
also consistent w ith th e m easurem ents o n pure m ethane. A s w ould be expected, actual errors appear to
be m uch sm aller than the calcu lated u ncertainties. A greem ent betw een the ab b rev iated screen Af and
counter Af m ethods w as very close. D ifferen ces betw een norm alized refractivity v alu es as m easu red by
the tw o m ethods w ere less than 2 % o f th e values fo r all total pressures ex cep t the low est,
1
atm ., w here
the difference w as 3.7% . R esults w ere quite co n siste n t w ith refractiv ity o f th e m ix tu re b ein g directly
proportional to n u m b er density. T h ey w ere also co n sisten t w ith refractiv ity o f the m ix tu re being the
lin ear sum o f the refractiv ities o f the in d iv id u al com p o n en ts. U sing 4 3 0 x
10*6
fo r the n o rm alized
reffactivity o f m ethane and 135 x 10"6 fo r the norm alized refractivity o f hydrogen, the pred icted n orm al­
ized refractivity o f the m ixture is 246 x 10~6. U sing 4 4 0 x 10 - 6 for m ethane predicts 24 9 x 10‘6 for the
m ixture. Both predictions are co n sisten t w ith the m ixture results, bu t the 44 0 X 10 - 6 figure agrees m ost
closely. M easured d ata on th e m ixture are listed in T ab le 5.3.
61
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Measured Refractivities
Gas species: Methane (CH4)
M easurem ent m ethod: Screen Af
M easured
Tem perature. K
2 1 1 .8
T o tal Pressure,
Atm.
Frequency,
GHz
3.00
9.16
11.23
13.59
15.61
16.99
16.99
17.42
17.42
1652
1673
1675
1661
1673
1664
1686
1681
±42
±40
± 38
± 37
± 36
± 36
± 36
±36
42 7
432
430
427
429
429
432
43 4
±
±
±
±
±
±
±
±
3.00
15.61
16.99
17.42
17.44
1616
1621
1641
530
±36
± 36
±36
± 22
422
42 4
429
416
± 12
± 12
± 12
±20
2 1 1 .6
2 1 0 .6
2 1 0 .8
2 1 0 .2
2 1 1 .2
2 1 0 .0
211.4
214.2
1 .0 0
R efractivity (v ) x 10 6
R efractiv ity X
N o rm alized to
14
13
13
12
12
12
12
12
223.2
3.00
9.17
10.26
11.23
12.71
13.59
15.62
17.41
1561
1567
1587
1579
1571
1564
1589
±41
±40
± 39
± 38
±37
±36
± 35
425
427
432
430
428
426
43 3
±
±
±
±
±
±
±
248.2
3.00
9.17
10.27
11.23
11.44
12.71
13.59
13.68
15.15
15.61
16.98
17.40
1424
1420
1430
1428
1440
1417
1414
1424
1410
1420
1431
± 40
± 38
± 37
± 37
± 36
±35
±35
± 34
±34
± 33
±33
431
430
433
433
436
42 9
428
431
427
430
433
± 15
± 14
± 14
± 14
± 14
± 13
± 13
± 13
±13
±13
±13
14
14
13
13
13
12
12
T able 5.1: M easured refractivities o f pure gaseous m ethane at various p ressu res from 1 to 3.5 atm and
tem peratures from ab o ut 210 to 373 K . D ensity norm alized refractiv ities in the last co lu m n show that
the data are co n siste n t w ith m eth an e refractivity being directly pro p o rtio n al to n u m b er d en sity . This
table continues on the fo llow ing tw o pages.
62
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Measured Refractivities
(continued from previous page)
Gas species: M ethane (CH4)
M easurem ent m ethod: Screen Af
M easured
T em perature. K
T o tal Pressure,
Atm.
248.2
3.00
9.17
12.71
13.59
13.68
15.15
15.61
16.98
17.40
1410
1432
1412
1403
1409
1404
1417
1419
272.6
3.00
17.41
1306 ± 32
4 3 4 ± 13
272.6
3.00
9.16
10.27
11.23
1271 ± 38
1286 ± 37
1287 ± 36
42 3 ± 15
42 8 ± 15
428 ± 15
273.4
3.00
11.44
12.70
13.58
13.67
15.14
15.60
16.97
17.40
1276
1294
1275
1272
1261
1271
1276
1276
426
432
426
42 4
421
42 4
426
42 6
273.6
3.50
3.00
17.40
17.41
1549 ± 35
1319 ± 3 2
443 ± 12
441 ± 13
294.2
3.00
9.16
10.26
11.23
11.47
12.70
13.58
1187
1198
1205
1195
1207
1201
±29
±29
± 28
±28
± 28
± 27
42 6
43 0
433
429
433
431
± 13
± 13
± 13
± 13
±13
± 12
294.2
3.00
9.16
10.26
11.23
11.47
12.70
13.58
1168
1194
1198
1185
1204
1192
±37
±36
±35
±34
±34
±33
419
428
430
42 5
43 2
428
± 16
± 15
±15
± 15
± 15
± 14
Frequency,
GHz
R efractivity (v) x 10 6
±40
± 36
±35
± 35
± 34
±34
±33
±33
± 35
± 35
± 34
± 34
± 33
±32
± 32
± 32
R efractivity x
N orm alized to
42 7
43 4
427
425
427
42 5
42 9
430
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
15
14
13
13
13
13
13
13
14
14
14
14
13
13
13
13
63
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M easured Refractivities
(continued from previous page)
Gas species: Methane (CH4)
M easurem ent m ethod: Screen Af
M easured
Tem oerature. K
T otal Pressure,
A tm .
294.6
3.00
10.26
1200 ± 29
4 3 2 ± 13
295.2
3.00
10.26
11.23
11.44
11.44
1197 ± 3 6
1203 ± 2 8
1186 ± 3 4
1 2 0 0 ± 28
431
435
42 7
432
295.2
3.00
17.40
1184 ± 3 1
427 ± 14
296.2
3.00
9.16
9.16
11.23
11.23
12.70
12.70
13.58
13.58
15.60
1188 ± 2 9
1184 ± 3 7
1212 ±28
1203 ± 35
1213 ± 2 8
1190 ± 3 3
1199 ± 2 7
1188 ± 3 3
1178 ± 3 1
429
428
438
435
439
430
433
429
42 6
±
±
±
±
±
±
±
±
±
13
16
13
15
13
15
12
14
14
373.2
3.00
9.13
10.23
944
951
95 4
94 9
958
945
94 6
945
94 3
946
953
430 ±
433 ±
43 4 ±
432 ±
436 ±
43 0 ±
431 ±
430 ±
429 ±
431 ±
434 ±
18
18
17
17
17
16
16
16
16
15
15
Frequency,
GHz
1 1 .2 0
11.41
12.69
13.57
13.65
15.12
15.58
16.95
17.37
R efractiv ity (v) x 10 6
± 35
± 33
± 32
± 32
± 31
± 30
± 30
± 29
± 29
± 28
± 28
R efractivity x
N orm alized to
±15
±13
± 15
± 13
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Measured Refractivities
Gas species: Methane (CH4)
M easurem ent m ethod: Counter Af
M easured
T em perature. K
T otal Pressure,
A tm .
Frequency,
GHz
R efractivity (v) x 106
272.6
273.6
273.6
3.00
3.50
3.00
17.4082
17.4037
17.4075
1326 ± 13
1534 ± 13
1311 ± 13
R efractiv ity x 10 6
N o rm alized to STP
441 ±
439 ±
438 ±
T able 5.2: M easured refractivities o f p u re gaseous m ethane at tem peratures n e a r 273 K an d p ressures of
3 and 3.5 atm . using the co u n ter Af m ethod.
65
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M easured Refractivities
Gas m ixture, by number mixing ratio:
m ethane, 0.375; and hydrogen, 0.625 .
M easurem ent methods: Abbreviated Screen Af and Counter Af (simultaneous
measurements).
M easured
Tem perature. K
273.0
273.0
273.0
273.0
273.0
273.0
273.0
273.0
273.0
273.0
T o tal P ressure,
Atm.
Frequency,
GHz
1 .0 0
17.427
17.423
17.414
17.406
17.397
242
499
1001
1489
1981
±24
± 30
± 39
±48
±56
24 2
24 6
24 9
248
248
±
±
±
±
±
17.427
17.423
17.414
17.406
17.397
251
505
1010
1502
1996
±21
±24
±27
± 30
± 33
251
248
251
25 0
250
± 23
± 14
± 9
± 6
± 5
2.03
4.01
6 .0 1
7.98
1 .0 0
2.03
4.01
6 .0 1
7.98
R efractivity (v) X 10 6
R efractiv ity x 10 6
N o rm alized to S T P
26
17
11
9
8
T able 5.3: M easured refractiv ities o f a m ixture o f gaseous m ethane an d hydrogen a t various p ressures
from 1 to 8 atm . and a tem perature o f 273 K. T h e first group w as m easured using the abbreviated screen
Af m ethod and the seco n d , m easu red sim u ltan eo u sly w ith the first, used the a b b re v ia ted c o u n te r Af
method.
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5.2
Data on Pure Gaseous Hydrogen
T his section p resen ts d a ta on th e refractiv ity o f pure hydrogen. A ll th ese d ata w ere collected w ith
the standard procedure discussed in Section 4.3 and A ppendix C. D ata reduction procedures and an uncer­
tainty analysis are in A ppendix D . T he ran g e o f conditions in v estig ated include tem peratures n ear 213
and 273 K , pressures n ear 1, 2, 4,
6
, and
8
atm ., at the frequencies o f the cavity resonator's nine usable
resonances betw een 9 and 18 G H z. D ata on the gas at 213 K are p resen ted in T ab le 5.4, an d those at
273 K are in T able 5.5.
H ydrogen, like m eth an e, is essen tially transparent to m icro w av e rad iatio n a t the frequencies and
un d er the conditions stu d ied in this w ork. L ike m ethane, its refractiv ity sh o u ld also be directly p ro p o r­
tional to num ber density. R efractivity m easurem ents have been norm alized b y num ber density to that o f
the gas at ST P, as detailed in A ppendix D . It is expected th at th e no rm alized values should be constant
to w ithin the uncertainties o f the m easurem ents.
T he results agree w ith those expectations. D ata values are gen erally co n sisten t w ith a norm alized
refractivity o f p u re h y d ro g en b etw een 133.5 x 10-6 and about 134 x
1 0 -6 .
T he spread o f the d ata is
som ew hat larg er than anticipated, m o stly d u e to the sm aller values ob tain ed in the final ex p erim en t at
273 K . T h a t ex p erim en t w as done so m e tim e after the last hig h p ressu re g au g e calib ratio n , so those
m easured pressures are m ore lik ely to be in erro r than the others. It is p o ssib le that system atic pressure
errors larger than th e stated uncertainty o f the gauge m ay have influenced the lo w m easured refractivities.
W ithin the m easurem ent accuracies n o system atic variation o f the n orm alized refractivity w ith frequen­
cy, tem perature, o r pressure is indicated. A t th e low est pressures, n ear 1 atm., tem perature an d pressure
uncertainties could add as m uch as 0 .8 % to refractivity uncertainties arising directly from the cavity reso ­
n ato r m eth o d ; at h ig h e r p ressu res they are less im portant, as little as 0.4 % at
8
atm . T h e ran g e o f
internally co n sisten t values g iv en above is centered about 1 % lo w er than th e generally accepted value at
optical frequencies (so d iu m D line), ab o u t 135 x 10_6. If resu lts o f th e last ex p erim e n t a t 273 K are
om itted the average o f the rem ain in g d ata is 134.85 X 10-6, w ithin 0.2% o f th a t o p tical value.
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities
Gas species: Hydrogen (H2)
T arget tem p eratu re: 213.15 K
M easured
Tem perature. K
T otal P ressure,
Atm.
Frequency,
GHz
213.70
213.65
213.35
213.25
213.25
213.25
213.25
213.25
213.15
0.999
0.999
0.997
0.997
0.997
0.997
0.997
0.997
0.996
9.17397
11.25375
12.75804
13.64027
13.71823
15.65447
16.19839
17.02348
17.44676
172.7
170.2
169.3
169.8
171.0
172.5
173.8
174.1
173.5
+ 3 .2
± 3 .1
+ 3 .2
+ 3 .0
± 3 .0
± 3 .2
± 3 .4
± 3 .5
± 3 .4
135.2
133.3
132.6
132.9
133.9
135.1
136.1
136.4
135.9
± 3.3
± 3 .2
± 3.3
± 3.2
± 3.2
± 3 .3
± 3 .5
± 3.6
± 3.5
213.15
213.10
213.10
213.15
213.15
213.20
213.20
213.20
213.15
1.974
1.974
1.974
1.974
1.974
1.975
1.975
1.975
1.974
9.17783
11.25184
12.75372
13.63565
13.71358
15.65184
16.19287
17.02062
17.44383
340.1
339.9
338.7
338.6
339.2
340.3
340.8
341.9
341.4
± 3 .2
± 3 .1
± 3 .2
± 3.0
± 3 .0
± 3 .2
± 3 .1
± 3 .2
± 3 .1
134.4
134.3
133.8
133.8
134.0
134.6
134.7
135.2
134.9
± 2 .7
± 2 .6
± 2 .7
± 2 .6
± 2 .6
± 2.7
± 2 .6
± 2 .7
± 2 .6
212.95
213.00
213.10
213.10
213.15
213.15
213.15
213.15
213.20
3.971
3.972
3.974
3.974
3.975
3.975
3.975
3.975
3.976
9.17464
11.24794
12.74930
13.63091
13.70882
15.64641
16.18726
17.01473
17.43778
688.2
686.7
685.4
686.5
686.2
687.7
687.6
688.2
688.2
± 3 .2
± 3 .1
± 3 .2
± 3 .0
± 3.0
± 3 .0
± 3 .0
± 3 .1
± 2 .9
135.1
134.8
134.5
134.7
134.7
135.0
135.0
135.1
135.1
± 1.8
± 1.7
± 1.8
± 1.7
± 1.7
± 1.7
± 1.7
± 1 .7
± 1 .7
212.95
213.05
212.95
213.00
212.95
213.10
213.15
213.05
213.15
5.991
5.994
5.991
5.993
5.991
5.996
5.997
5.994
5.997
9.17149
11.24408
12.74491
13.62623
13.70409
15.64105
16.18172
17.00887
17.43179
1032.2
1029.9
1030.2
1030.4
1031.8
1030.6
1030.1
1033.0
1032.0
± 3.3
± 3 .2
± 3.3
± 3 .1
± 3 .1
± 3 .1
± 3 .1
± 3.2
± 3.0
134.3
134.0
134.0
134.1
134.2
134.1
134.1
134.4
134.3
± 1.5
± 1.4
± 1.5
± 1.4
± 1.4
± 1.4
± 1 .4
± 1.4
± 1.4
R efractivity (v ) x 10 6
R efractivity x 10 6
N orm alized to STP
T able 5.4: M easu red refractivities o f p ure gaseous hydrogen at tem peratures n ear 213 K and pressures
n ear 1, 2, 4, 6 , an d 8 atm . T his table continues on the n ex t page.
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities
(continued from previous page)
G as species: Hydrogen (H2)
T arget tem p eratu re: 213.15 K
M easured
T em perature. K
212.80
212.90
213.00
213.05
213.00
212.90
212.95
212.95
213.00
T o tal Pressure,
Atm.
8.157
8.161
8.165
8.167
8.165
8.161
8.163
8.163
8.165
R efractiv ity (v) x 106
Frequency,
GHz
9.16806
11.23987
12.74015
13.62116
13.69899
15.63517
16.17566
17.00254
17.42532
1407.1
1404.8
1404.2
1402.2
1404.5
1407.1
1405.2
1405.7
1403.7
± 3 .4
± 3 .3
± 3 .4
± 3 .2
± 3 .2
± 3 .2
± 3 .2
± 3 .3
± 3 .2
R efractiv ity x 106
N orm alized to STP
134.4
134.2
134.1
133.9
134.1
134.4
134.2
134.2
134.1
±
±
±
±
±
±
±
±
±
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
69
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities
G as species: Hydrogen (H2)
T arget tem p eratu re: 273.15 K
M easured
Tem perature. K
T o tal P ressure,
A tm .
Frequency,
GHz
R efractivity (v) x 106
___________________
R efractiv ity x 106
N orm alized to STP
272.85
272.85
272.85
272.90
272.85
272.85
272.85
272.85
272.85
0.998
0 .9 9 8
0.998
0.998
0.998
0.998
0.998
0.998
0.998
9.16984
11.24205
12.74266
13.62378
13.70168
15.63820
16.17879
17.00580
17.42865
134.1
134.8
136.2
135.9
135.8
136.7
136.2
136.7
135.8
± 3 .3
± 3 .2
± 3 .3
± 3 .1
± 3 .1
± 3 .1
± 3 .1
± 3 .3
± 3 .2
134.3
134.9
136.3
136.1
136.0
136.9
136.3
136.8
136.0
± 4 .0
± 3 .9
± 4 .0
± 3 .9
± 3 .9
± 3 .8
± 3 .8
± 4 .1
± 4 .0
273.15
273.15
273.15
273.15
273.15
273.15
273.15
273.15
273.15
1.006
1.006
1.006
1.006
1.006
1.006
1.006
1.006
1.006
9.16980
11.24200
12.74259
13.62371
13.70162
15.63814
16.17872
17.00568
17.42855
138.8
139.1
137.6
137.2
135.9
136.0
135.9
136.6
137.2
± 3 .6
± 3 .9
± 3 .9
± 3 .3
± 3 .7
± 3.6
± 3 .6
± 3 .3
± 3 .5
138.0
138.2
136.8
136.3
135.0
135.1
135.0
135.7
136.4
± 4 .4
± 4 .6
± 4 .6
± 4 .0
± 4 .4
± 4 .3
± 4 .3
± 4 .1
± 4 .2
273.55
273.35
273.35
273.35
273.35
273.40
273.40
273.35
273.30
1.975
1.974
1.974
1.974
1.974
1.974
1.974
1.974
1.974
9.16865
11.24059
12.74100
13.62201
13.69990
15.63616
16.17670
17.00355
17.42636
264.5
264.9
262.6
262.2
261.3
262.1
260.9
262.0
262.7
± 3 .6
± 3 .9
± 3 .9
± 3 .3
± 3 .7
± 3 .6
± 3 .6
± 3 .3
± 3 .5
134.1
134.3
133.2
132.9
132.5
132.9
132.3
132.8
133.2
± 3 .2
± 3 .3
± 3 .3
± 3 .0
± 3 .2
± 3 .2
± 3 .1
± 3 .0
± 3 .1
272.40
272.40
272.40
272.40
272.40
272.40
272.40
272.40
272.40
3.967
3.967
3.967
3.967
3.967
3.967
3.967
3.967
3.967
9.16612
11.23749
12.73750
13.61825
13.69614
15.63189
16.17226
16.99890
17.42161
540.0
540.6
541.2
541.6
540.4
540.6
539.9
542.5
540.0
± 3 .3
± 3 .2
± 3 .3
± 3 .1
± 3 .1
± 3 .3
± 3 .2
± 3 .3
± 3 .2
135.8
135.9
136.0
136.1
135.9
135.9
135.7
136.4
135.7
± 2 .0
± 2 .0
± 2 .0
± 1 .9
± 2 .0
± 1.9
± 1 .9
± 1.9
± 1 .9
T able 5.5: M easu red refractiv ities o f p u re gaseous hydrogen at tem peratures near 273 K and pressures
n ear 1, 2 ,4 , 6 , a n d 8 atm . R esu lts o f tw o experim ents p erfo rm ed ab o u t six m onths apart are com bined
in this table. T h e table continues on the next page.
70
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities
(continued from previous page)
Gas species: Hydrogen (H2)
T arget tem p eratu re: 273.15 K
M easured
T em perature. K
T o tal P ressure,
Atm .
Frequency,
GHz
R efractivity (v) x
R efractivity x 106
N orm alized to ST P
273.10
273.15
273.15
273.15
273.15
273.15
273.15
273.15
273.15
4 .0 5 2
4.053
4.053
4.053
4.053
4.053
4.053
4.053
4.053
9.16614
11.23751
12.73750
13.61826
13.69613
15.63187
16.17225
16.99888
17.42136
538.2
539.0
537.6
537.5
536.4
536.7
536.0
536.9
549.8
± 3 .6
± 3 .9
± 3 .9
± 3 .3
± 3 .7
± 3 .6
± 3 .6
± 3 .4
± 3 .5
132.8
133.0
132.7
132.6
132.4
132.4
132.3
132.5
135.7
± 2 .1
± 2 .1
± 2 .1
± 2.0
± 2 .1
± 2 .0
± 2 .0
± 2 .0
± 2 .0
273.55
273.55
273.40
273.30
273.30
273.30
273.35
273.30
273.20
6 .0 1 1
6.008
6.006
6.006
6.006
6.007
6.006
6.004
9.16374
11.23458
12.73417
13.61472
13.69258
15.62783
16.16808
16.99448
17.41708
801.0
799.5
799.4
797.7
796.0
796.0
794.2
796.1
795.8
± 3 .6
± 3 .9
± 3 .9
± 3 .3
± 3 .7
± 3 .6
± 3 .6
± 3 .4
± 3 .5
133.4
133.2
133.2
132.9
132.6
132.6
132.3
132.6
132.6
±
±
±
±
±
±
±
±
±
1.6
1.7
1.7
1.6
1.6
1.6
1.6
1.6
1.6
272.70
272.75
272.80
272.90
273.00
273.15
273.15
273.15
273.25
7.974
7.9 7 6
7.977
7.9 8 0
7.983
7.988
7.988
7.988
7.9 9 0
9.16125
11.23152
12.73069
13.61098
13.68882
15.62354
16.16361
16.98977
17.41226
1071.8
1072.0
1076.9
1076.1
1075.3
1075.3
1075.3
1080.1
1077.4
± 3 .3
± 3 .2
± 3 .3
± 3 .1
± 3 .1
± 3 .3
± 3 .2
± 3 .3
± 3 .2
134.2
134.2
134.8
134.7
134.6
134.6
134.6
135.2
134.9
±
±
±
±
±
±
±
±
±
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
273.20
273.15
273.15
273.15
273.15
273.15
273.15
273.15
273.20
8.254
8.253
8.253
8.253
8.253
8.253
8.253
8.253
8.254
9.16104
11.23126
12.73004
13.61072
13.68854
15.62322
16.16339
16.98944
17.41190
1094.8
1096.0
1092.7
1091.8
1091.4
1090.7
1090.7
1092.4
1093.5
± 3 .6
± 3 .9
± 3 .9
± 3 .3
± 3 .7
± 3 .6
± 3 .6
± 3 .4
± 3 .5
132.7
132.8
132.4
132.3
132.2
132.2
132.2
132.4
132.5
±
±
±
±
±
±
±
±
±
1.3
1.3
1.3
1.2
1.3
1.2
1.2
1.2
1.2
6 .0 1 1
71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.3
Data on Pure Gaseous Helium
T his section presents d a ta on the refractivity o f pure helium . A ll these d ata w ere o b tained using the
standard procedure discussed in Section 4.3 and A ppendix C. D ata reduction procedures and an uncertain­
ty analysis are in A ppendix D . T he range o f conditions investigated include tem peratures n ear 2 1 3 ,2 7 3 ,
and 313 K, p ressu res n e a r 1 , 2 , 4 ,
6
, and
8
atm ., at the frequencies o f the cav ity resonator's nine usable
resonances betw een 9 and 18 G H z. D ata on the gas at 213 K are p resen ted in T ab le 5.6, those at 273 K
are in T able 5.7, and those at 313 K are in T able 5.8.
H elium , lik e m ethane an d hydrogen, is transparent to m icrow ave rad iatio n at the frequencies and
under the co n d itio n s stu d ie d in this w ork. T hus, like th e oth er tw o gases already discu ssed , its refrac­
tivity sh o u ld also be d irectly p roportional to n u m b er density. It is the o n ly m onatom ic species studied
in this w ork. W here hydrogen m olecules have a w eak m icrow ave absorption m ech an ism (m any orders
o f m agnitude less than am m onia, even at the lin e c en te r n ear 1.4 G H z) arisin g from the w ell know n
nu clear spin p airin g tran sitio n , heliu m has on ly its electro n ic tran sitio n s a t o p tic a l freq u en cies an d
higher. T h u s the expected m icrow ave absorptivity o f helium is far sm a lle r still than hydrogen. As in
previous presentations o f d ata, refractivity m easurem ents h ave b een n o rm a liz e d by n u m b er density to
that o f the gas at ST P as detailed in A ppendix D . N orm alized refractivities sh o u ld be con stan t to w ithin
the uncertainties o f the m easurem ents.
T he results support those predictions. D ata values are consistent w ith a no rm alized refractivity o f
pure helium betw een 33.8 x 10 - 6 and 34.2 x 10"6. This sm all v alue causes larger relative uncertainties,
±3% at best, than in the m easurem ents on hydrogen. A t pressures n ear 1 atm . tem perature and pressure
uncertainties co u ld add up to
0 .8
% to refractivity uncertainties arising directly fro m the cavity resonator
m ethod; at h ig h er p ressu res they are less im portant, as little as 0.4% at
8
atm . O nce again the spread
o f the d ata is slightly larger than anticipated, bu t is still w ithin the uncertainties determ in ed in A ppendix
D . W ithin the accuracy o f th e m easurem ents n o system atic variation o f the n o rm alized refractivity w ith
frequency, tem perature, or pressure is indicated. The range o f internally co n sisten t values given above is
centered 2.9% lo w er than the accepted value at optical frequencies (sodium D line), about 35 x 10-6.
72
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities
Gas species: Helium (He)
T arget tem p eratu re: 213.15 K
M easured
Tem perature. K
T o tal Pressure,
Atm.
Frequency,
G Hz
R efractiv ity (v ) X 106
213.05
213.05
213.15
213.10
213.15
213.15
213.15
213.20
213.15
1.004
1.004
1.004
1.004
1.004
1.004
1.004
1.004
1.004
9.18055
11.25518
12.75752
13.63970
13.71766
15.65650
16.19767
17.02567
17.44899
45.5
4 6 .0
4 5.3
44.7
4 4 .0
45.1
4 5 .6
44.3
45 .0
213.15
213.15
213.15
213.15
213.15
213.15
213.15
213.15
213.15
1.979
1.979
1.979
1.979
1.979
1.979
1.979
1.979
1.979
9.18019
11.25472
12.75699
13.63913
13.71708
15.65584
16.19701
17.02497
17.44829
85.6
87.0
87.2
86.2
86.5
213.10
213.05
212.95
212.90
213.05
213.15
213.25
213.35
213.55
3.993
3.992
3.990
3.989
3.992
3.994
3.995
3.997
4.001
9.17937
11.25372
12.75585
13.63789
13.71584
15.65442
16.19555
17.02342
17.44667
174.7
175.5
176.6
177.2
176.6
177.7
176.4
176.6
177.7
213.20
213.25
213.35
213.15
213.15
213.05
213.05
213.05
213.10
6.006
6.007
9.17857
11.25275
12.75476
13.63674
13.71469
15.65310
16.19420
17.02201
17.44527
261.4
262.1
261.8
261.9
260.5
261.9
259.7
259 .2
258.1
6 .0 1 0
6.005
6.005
6 .0 0 2
6 .0 0 2
6 .0 0 2
6.003
R efractivity x 106
N orm alized to STP
± 3 .2
± 3 .1
± 3 .2
± 3 .0
± 3 .0
± 3 .2
± 3 .1
± 3 .2
± 3 .4
35.4
35.8
3 5 .2
34.8
34.2
35.0
35.4
34.4
35 .0
± 2 .7
± 2 .6
± 2 .7
± 2 .6
± 2 .5
± 2.7
± 2 .6
± 2.7
± 2.8
± 3 .2
± 3 .1
± 3 .2
± 3 .0
± 3 .0
8 6 . 8 ± 3 .2
86.1 ± 3 .1
85.2 ± 3.2
85.1 ± 3 .4
33.8
34.3
34.4
3 4.0
34.1
3 4 .2
3 4 .0
33 .6
33 .6
±
±
±
±
±
±
±
±
±
1.8
1.7
1.8
1.7
1.7
1.8
1.7
1.8
1.8
± 3 .2
± 3 .1
± 3 .2
± 3 .0
± 3 .0
± 3 .2
± 3 .1
± 3 .2
± 3 .1
34.1
34.3
34.5
34 .6
34.5
34.7
34.5
34.5
34.7
±
±
±
±
±
±
±
±
±
1.3
1.3
1.3
1.2
1.2
1.3
1.2
1.3
1.2
± 3 .2
± 3 .1
± 3 .2
± 3 .0
± 3 .0
± 3 .2
± 3 .1
± 3 .2
± 3 .1
34.0
34.1
3 4.0
34.0
33.9
34.0
33.7
33.7
33.5
±
±
±
±
±
±
±
±
±
1.1
1.1
1.1
1.0
1.0
1.1
1.1
1.1
1.0
T able 5.6: M easured refractivities o f pure gaseous helium at tem peratures n ear 213 K and pressures near
1, 2 ,4 , 6 , and 8 atm . T h is table continues on the next page.
73
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities
(continued from previous page)
Gas species: Helium (He)
T arget tem p eratu re: 213.15 K
M easured
Tem perature. K
T otal Pressure,
Atm.
Frequency,
G Hz
213.15
213.15
213.15
213.15
213.20
213.35
213.50
213.45
213.50
8.017
8.017
8.017
8.017
8.019
8.024
8.030
8.028
8.030
9.17777
11.25177
12.75363
13.63555
13.71348
15.65176
16.19280
17.02053
17.44376
R efractivity (v) x 10^
349.1
349.4
350.2
348.7
348.7
347.8
346.1
346.2
344.8
± 3 .2
± 3 .1
± 3 .2
± 3 .0
± 3 .0
± 3 .2
± 3 .1
± 3 .2
± 3 .1
R efractivity x 106
N orm alized to STP
34.0
34.0
34.1
33.9
34.0
33.9
33.7
33.7
33.6
± 0.9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities
Gas species: H elium (He)
T arget tem p eratu re: 273.15 K
M easured
TemDerature. K
T otal P ressure,
Atm.
Frequency,
GHz
273.15
273.15
273.15
273.15
273.15
273.15
273.15
273.20
273.15
0.999
0.999
0.999
0.999
0.999
0.999
0.999
0.999
0.999
9.17079
11.24321
12.74396
13.62516
13.70308
15.63980
16.18047
17.00760
17.43047
36.1
36.2
36.6
36.3
36.4
36.6
37.0
36.5
36.7
+ 3.3
± 3 .2
± 3 .3
± 3 .1
± 3 .1
± 3 .1
± 3 .1
± 3 .2
± 3 .0
36.1
36.2
36.7
36.4
36.5
36.7
37.1
36.6
36.8
± 3 .5
± 3 .4
± 3 .5
± 3 .3
± 3 .3
± 3 .3
± 3 .3
± 3 .4
± 3 .2
273.15
273.25
273.25
273.30
273.25
273.25
273.25
273.25
273.20
1.000
1.000
9.17075
11.24315
12.74390
13.62511
13.70301
15.63974
16.18039
17.00745
17.43036
34.9
35.5
35.5
34.8
35.2
34.8
34.2
34.2
34.2
± 3 .3
± 3 .5
± 3 .4
± 3 .1
± 3 .3
± 3 .3
± 3 .2
± 3 .2
± 3 .2
34.9
35.5
35.5
34.8
35.2
34.8
34 .2
34.2
34.2
± 3 .5
± 3 .6
± 3 .6
± 3 .3
± 3 .5
± 3 .4
± 3 .4
± 3 .4
± 3 .4
273.15
273.15
273.15
273.15
273.15
273.15
273.20
273.20
273.25
1.999
1.999
1.999
1.999
1.999
1.999
1.999
1.999
1.999
9.17050
11.24284
12.74355
13.62471
13.70262
15.63928
16.17993
17.00702
17.42989
67.6 ± 3 .3
6 8 . 2 ± 3 .2
69.4 ± 3 .3
69.4 ± 3 .1
69.8 ± 3 .1
70.1 ± 3 .1
70.7 ± 3 .1
7 0.6 ± 3 .2
69.5 ± 3 .0
33.8
34.1
34.7
34.7
3 5 .0
35.1
35.4
35.3
34.8
±
±
±
±
±
±
±
±
±
273.20
273.20
273.20
273.15
273.15
273.20
273.25
273.40
273.55
1.923
1.923
1.923
1.923
1.923
1.923
1.924
1.925
1.926
9.17047
11.24281
12.74352
13.62471
13.70260
15.63925
16.17987
17.00690
17.42980
65.2
65.9
65.0
64.5
65.2
65.9
33.9
34.3
33.8
33.6
33.9
34.3
34.4
3 4 .6
34.5
± 2 .3
± 2 .5
± 2 .4
± 2 .2
± 2 .3
± 2 .3
± 2 .3
± 2 .2
± 2 .3
1 .0 0 1
1 .0 0 1
1 .0 0 1
1 .0 0 1
1 .0 0 1
1 .0 0 1
1 .0 0 1
R efractivity (v) x 106
______________
± 3 .6
± 3 .9
± 3 .8
± 3 .3
± 3 .7
± 3 .6
6 6 . 1 ± 3 .6
66.4 ± 3 .3
66.4 ± 3 .5
R effactivity x 106
Normalized .to STP
2 .1
2 .1
2 .1
2 .0
2 .0
2 .0
2 .0
2 .1
2 .0
T able 5.7: M easu red refractivites o f pure gaseous helium at tem peratures n ear 273 K and pressures n ear
1, 2, 4, 6 , an d 8 atm . R esults fro m tw o experim ents p erform ed ab o u t four m onths apart are co m b in ed
in this table. T he table continues on the follow ing two pages.
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities
(continued firom previous page)
Gas species: Helium (He)
T arget tem p eratu re: 273.15 K
M easured
Tem perature, K
T otal P ressure,
Atm.
Frequency,
GHz
273.10
273.1 0
273.15
273.15
273.15
273.20
273.25
273.25
273.20
4.007
4.007
4.008
4.008
4.008
4.008
4.009
4.009
4.008
9.16987
11.24208
12.74270
13.62381
13.70170
15.63823
16.17884
17.00586
17.42870
136.4
136.1
135.7
135.6
136.8
137.4
137.6
139.1
137.9
± 3 .6
± 3 .2
± 3 .3
± 3 .1
± 3 .1
± 3 .1
± 3 .1
± 3 .2
± 3 .0
34.0
34.0
33.9
33.8
34.1
34.3
34.4
34.7
34.4
± 1.6
± 1.5
± 1.5
± 1.4
± 1 .5
± 1.4
± 1.4
± 1.5
± 1.5
273.00
273.05
273.10
273.10
273.15
273.15
273.15
273.15
273.15
4.008
4.008
4.009
4.009
4.010
4.010
4.010
4.010
4.010
9.16985
11.24207
12.74266
13.62378
13.70168
15.63820
16.17879
17.00576
17.42863
132.4
131.7
132.6
132.7
132.4
133.1
132.9
133.3
133.5
± 3 .6
± 3 .9
± 3 .8
± 3 .3
± 3 .7
± 3 .6
± 3 .6
± 3 .3
± 3 .5
33.0
32.8
3 3.0
33.1
3 3.0
33.2
33.1
33.2
33.3
± 1.6
± 1.7
± 1.7
± 1 .5
± 1.6
± 1.6
± 1 .6
± 1.5
± 1.6
273.15
272.95
272.75
272.75
272.65
273.70
272.75
272.80
272.90
6.005
6.001
5.996
5.996
5.994
5.995
5.996
5.997
5.999
9.16923
11.24130
12.74181
13.62289
13.70077
15.63719
16.17776
17.00470
17.42754
205.9
205.9
205.4
203.2
204.7
204.1
204.8
207.0
204.7
± 3 .3
± 3 .2
± 3 .3
± 3 .1
± 3 .1
± 3 .1
± 3 .1
± 3 .2
± 3 .2
34.3
34.3
34.2
33.8
34.1
34.0
34.1
34.5
34.1
±
±
±
±
±
±
±
±
±
1.2
1.2
1.2
1.1
1.1
1.1
1.1
1.2
1.2
273.15
273.15
273.20
273.20
273.15
273.15
273.15
273.15
273.15
6.050
6.050
6.051
6.051
6.050
6.050
6.050
6.050
6.050
9.16918
11.24123
12.74174
13.62282
13.70071
15.63711
16.17767
17.00459
17.42744
206.0
206.5
204.8
203.1
202.9
202.5
202.2
202.3
201.6
± 3.6
± 3 .9
± 3 .8
± 3 .3
± 3 .7
± 3 .6
± 3 .6
± 3 .3
± 3 .5
34.0
34.1
33.8
33.6
33.5
33.5
33.4
33.4
33.3
±
±
±
±
±
±
±
±
±
1.3
1.4
1.3
1.2
1.3
1.3
1.3
1.2
1.2
R efractivity (v) x 106
R efractivity X 106
N orm alized to ST P
76
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities
(continued from previous page)
Gas species: H elium (He)
T arget tem perature: 273.15 K
M easured
Tem perature. K
T otal Pressure,
Atm.
Frequency,
GHz
273.15
273.15
273.15
273.15
273.15
273.15
273.15
273.15
273.15
8.012
8.012
8.012
8.012
8.012
8.012
8.012
8.012
8.012
9.16863
11.24056
12.74097
13.62197
13.69986
15.63613
16.17667
17.00356
17.42637
271.8
271.4
271.9
271.0
271.5
271.7
272.2
274.3
271.4
± 3 .3
± 3 .2
± 3 .3
± 3 .1
± 3 .1
± 3 .1
± 3 .1
± 3 .2
± 3 .2
3 3 .9
3 3 .9
3 3 .9
3 3 .8
3 3 .9
3 3 .9
3 4 .0
3 4 .2
3 3 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0.9
± 0 .9
± 0 .9
273.15
273.15
273.15
273.15
273.15
273.15
273.20
273.20
273.20
8.255
8.255
8.255
8.255
8.255
8.255
8.256
8.256
8.256
9 .16854
11.24045
12.74083
13.62182
13.69970
15.63596
16.17647
17.00331
17.42613
275.3
276.9
276.4
276.7
276.9
276.5
276.3
277.3
276.8
± 3 .6
± 3 .9
± 3 .8
± 3 .3
± 3 .7
± 3 .6
± 3 .6
± 3 .3
± 3 .5
3 3 .4
3 3 .5
3 3 .5
3 3 .5
3 3 .6
3 3 .5
3 3 .5
3 3 .6
3 3 .5
± 1.0
± 1.0
± 1.0
± 0 .9
± 1.0
± 1 .0
± 1.0
± 0 .9
± 1 .0
R efractivity (v) X 106
R efractiv ity x 106
N o rm alized to ST P
77
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M easured Refractivities
G as species: Helium (He)
T arget tem perature: 313.15 K
M easured
T em perature. K
T otal Pressure,
A tm .
Frequency,
GHz
R efractivity (v) x 106
R efractiv ity X 106
N o rm alized to ST P
312.95
312.95
313.00
313.05
313.05
313.10
313.10
313.15
313.15
1.013
1.013
1.013
1.013
1.013
1.013
1.013
1.013
1.013
9.16400
11.23488
12.73456
13.61509
13.69296
15.62824
16.16824
16.99487
17.41755
27.5
28.9
28.4
29.4
29.6
30.3
30.1
29.7
30.2
± 3 .7
± 4 .0
± 3 .8
± 3 .4
± 3.8
± 3 .6
± 3 .6
± 3 .4
± 3 .6
31.1
32.7
32.1
33.3
33.5
34.3
34 .0
33.6
3 4 .2
± 4 .3
± 4 .7
± 4 .5
± 4 .0
± 4 .4
± 4 .3
± 4.3
± 4 .0
± 4.2
313.00
313.00
313.00
312.95
312.95
313.00
313.05
313.05
313.05
1.935
1.935
1.935
1.935
1.935
1.935
1.935
1.935
1.935
9.16375
11.23457
12.73420
13.61472
13.69258
15.62780
16.16805
16.99438
17.41706
54.7
56.7
56.3
57.1
57.4
58.2
57.2
58.2
58.4
± 3 .7
± 4 .0
± 3 .8
± 3 .4
± 3 .8
± 3 .6
± 3 .6
± 3 .4
± 3 .6
32.4
33 .6
33.3
33.8
34 .0
34.4
33.9
34.5
3 4 .2
± 2 .8
± 2 .9
± 2 .9
± 2 .7
± 2 .8
± 2 .7
± 2 .7
± 2 .6
± 2 .7
313.20
313.25
313.25
313.30
313.35
313.35
313.35
313.30
313.25
4.057
4.057
4.057
4.058
4.059
4.059
4.059
4.058
4.057
9.16318
11.23388
12.73341
13.61387
13.69173
15.62684
16.16703
16.99332
17.41598
117.2
118.5
118.7
119.5
119.2
119.9
120.1
121.1
120.1
± 3 .7
± 4 .0
± 3 .8
± 3 .4
± 3 .8
± 3 .6
± 3 .6
± 3 .4
± 3 .6
33.1
33.5
3 3.6
33.8
33.7
33 .9
34 .0
34 .2
34 .0
± 1 .7
± 1.8
± 1.7
± 1.6
± 1.7
± 1.7
± 1.7
± 1.6
± 1.7
313.20
313.15
313.15
313.15
313.10
313.10
313.15
313.15
313.15
6.059
6.058
6.058
6.058
6.057
6.057
6.058
6.058
6.058
9.16264
11.23320
12.73267
13.61309
13.69094
15.62593
16.16611
16.99236
17.41499
176.3
178.7
176.5
177.0
177.1
178.1
177.0
177.6
177.2
± 3 .7
± 4 .0
± 3 .8
± 3.4
± 3 .8
± 3 .6
± 3.6
± 3.4
± 3.6
3 3.4
33.8
33.4
33.5
33.5
33.7
33.5
3 3 .6
33.5
±
±
±
±
±
±
±
±
±
1.2
1.3
1.3
1.2
1.3
1.2
1.2
1.2
1.2
T able 5.8: M easured refractivities o f p u re gaseous helium at tem peratures n ear 313 K and pressures near
1, 2 , 4, 6, and 8 atm . T h is table continues on the n e x t page.
78
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Measured Refractivities
(continued from previous page)
G as species: H elium (He)
T arget tem p eratu re: 313.15 K
M easured
Tem perature. K
313.15
313.35
313.35
313.35
313.35
313.35
313.30
313.25
313.25
T o tal P ressure,
A tm .
8.245
8.2 5 0
8.250
8.2 5 0
8.250
8.2 5 0
8.249
8.248
8.248
Frequency,
GHz
R efractivity (v) x 106
9.16207
11.23251
12.73185
13.61220
13.69005
15.62492
16.16504
16.99124
17.41386
238.3
240.1
240.9
242.1
242.2
242.8
242.9
243.5
242.1
± 3 .7
± 4 .0
± 3 .8
± 3 .4
± 3 .8
± 3 .6
± 3 .6
± 3 .4
± 3 .6
R efractivity x 106
N o rm alized to STP
33.1
33.4
33.5
33.7
33.7
33.8
33.8
33.9
33.7
±
±
±
±
±
±
±
±
±
1.1
1.1
1.1
1.0
1.1
1.0
1.0
1.0
1.0
79
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Chapter 6
Laboratory Measurements of Microwave Absorptivity
and Refractivity Spectra on Gas Mixtures
Containing Ammonia
T his ch ap te r lists, in tab u lar form , all the d a ta co llected as p a rt o f this w ork concerning m icrow ave
absorption and refraction by pure gaseous am m onia and gas m ixtures containing am m onia, and includes
data on pure am m onia by o th er researchers (B leaney and L oubser, 1950). It begins w ith a discussion o f
data organization in S ection 6.1. Section 6.2 presents the d ata o n p u re gaseous am m onia; m o st o f the
data w ere collected by B leaney and L oubser, and are in cluded fo r com pleteness and to m ake them avail­
able in tab u la r form . D a ta on gas m ixtures co n tain in g sm all am o u n ts o f am m o n ia are p re se n te d in
Section 6.3. T hey include am m onia in otherw ise pure m o lecu lar hydrogen (H 2 ), am m onia in o therw ise
pu re atom ic heliu m (H e), and am m onia in a Jo v ian m ixture o f m o le c u lar h ydrogen an d atom ic helium ,
w hich is 90% hydrogen and 10% helium by num ber.
6.1
Data Organization
In the atm ospheres o f the gian t planets there are five in d ep en d en t m acroscopic conditions that sig ­
n ificantly affect am m o n ia absorptivity: tem perature; the p a rtia l p ressu res o f hydrogen, helium , and
am m onia; an d the frequency o f the m icrow ave radiation. T his five dim ensional variable space presents
a m in o r challenge in the organization o f the d ata for presentation. P ractical considerations in the d ata
collection procedures, discussed in C hapter 4 and A ppendix C, offered an initial fram ew ork.
T h e basic unit o f data, one datum , consists o f m easured refractiv ity and absorptivity values and all
the in form atio n n eed ed to specify th e m acroscopic co nditions o f th e su b ject gas m ixture. F o r a Jovian
80
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m ix tu re o f gases this involves a m inim um o f sev en num bers. In these experim ents it w as n o t possible
to m easure th e d ifferent partial pressures independently since B ourdon gauges m easure o n ly total pres­
sure. P artial pressures m u st b e calcu lated from k n ow ledge o f m ixing ratios and the total p ressu re, thus
using four n um bers in stead o f three to specify the three partial p ressu res. O ne o f the m ix in g ratios is
alw ays red u n d an t since their sum m u st b e unity. E ach datum from this w ork is defin ed b y eig h t n um ­
bers: the m ix in g ratios o f hydrogen, helium , and am m o n ia (som e o f w hich m ay be zero); total p res­
sure; tem perature; frequency; and the m easu red refractivity an d absorptivity values. S in ce o n ly one
gas w as inv o lv ed in m easurem ents on p ure am m o n ia no m ixing ratio inform ation is needed, an d each
datum can b e defined by five non-zero num bers. B leaney and L oubser d id n o t m easure refractivities, so
their data require only four num bers each.
T hese d a ta are organized into series, referring to the laboratory procedure o f that nam e (one p ro ce­
dural series generates one d ata series). O ne series o f d ata includes all absorptivity m easurem ents, o v er
all available frequencies, o n a gas m ixture w hose rem aining m acroscopic conditions are h eld constant
w ithin the capabilities o f the apparatus. A series o f d ata represents d iscrete sam ples o f th e absorption
spectrum o f a single specific gas m ixture. T he d ata p resen ted in this ch ap ter are g ro u p e d by series.
M o st o f the series collected as part o f this w o rk c o n sist o f nine data, reflecting the n in e u sab le reso ­
nances availab le from the cavity resonator. T h e six series o f data on p u re am m onia fro m B lean ey and
L oubser vary in size from 9 to 24 points as ex p lain ed in Section 6.2.
D ata series are grouped into seq u en ces (again referrin g to the laboratory pro ced u re o f th at nam e),
w hich are v ery useful fo r exam ining p ressure dependences. A sequence o f d ata is th e collection o f all
series a t all available total pressures, on a gas m ixture w hose tem p eratu re and m ixing ratio s are held
constant. In this chapter a sequence o f data, g ro u p ed by series, is p resented in a single table. F o r ex am ­
ple, the en tire collection o f B leaney and L o u b ser am m o n ia data co n stitu tes o n e sequence, and is all
p resented in T able 6.1 o f Section 6.2. H ow ever, ju s t because som e series o f d ata are g ro u p ed into one
table does n o t im ply that those series constitute a single sequence; despite their being listed in one table
the data o f T able 6.2 are no t a single sequence because the series are at w idely differing tem peratures.
81
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6.2
Data on Pure Gaseous Ammonia
T o date the m o st co m p reh en siv e se t o f laboratory m easurem ents o f co n tin u u m m icrow ave absorp­
tion by pure g aseous am m onia is the high quality d ata o f B leaney an d L o u b se r (1950), w ho fabricated
m ultiple cavity reso n ato rs to m ake m easurem ents o v er a ran g e o f d iscrete freq u en cies an d pressures.
T heir frequency ran g e spans 3.75 to 36 G H z, w hich brackets the p eak o f th e am m o n ia inversion spec­
trum n ear 24 G H z. M easu rem en ts w ere m ade at pressures o f 100, 300, an d 9 0 0 torr, an d 2, 4 , and 6
atm ospheres, a ran g e th at dem onstrates the transition as p ressure increases fro m a distin ctly resonant
absorption spectrum to w ard a no n reso n an t D ebye spectrum . A b so rp tiv ities w ere sm all enough at the
100 and 300 to rr p ressu res th at 24 resonances w ere usable, providing abso rp tiv ity m easurem ents at 24
discrete frequencies w ithin the stated frequency range. H igher am m onia p ressures show ed larger absorp­
tivities that rendered som e resonances unusable. A t six atm ospheres p ressu re the absorptivities becam e
so large (over 22,000 dB p er km , o r 0.051 o p tical depths p er cm !) th a t o n ly n in e reso n an ces w ere still
usable, but this is sufficien t to reaso n ab ly constrain the shape o f the spectrum .
In contrast w ith th e larg e ran g es o f frequencies an d pressures in clu d ed in th eir pro g ram o f m easure­
m ents, all w ere m ade a t o n e tem perature, "room tem perature," ab o u t 297 K. C o n seq u en tly these data
provide no inform ation ab o u t tem perature dependence. N or were refractivity d ata collected.
T he form o f presentation B leaney and L o u b ser chose for their d ata m ay h ave b een convenient for
spectroscopists o f th at era, but ex tractin g absorptivity values from it is a tedious an d tim e-consum ing
task. A n effo rt w as m ade by R .L . P o y n ter to locate listings o f the o riginal ab so rp tiv ity m easurem ents
but he found that no num erical listings w ere available in any form (P oynter, 1987); it seem s that the
original m easurem ents m u st be considered lost. T he d ata are p resented in g raphical form only, and they
are all com pressed into a single grap h . T h at graph is linear in freq u en cy v on the ab scissa and in a
derived value, cc/v2, on th e ordinate. B ecause the m easured absorptivity values sp an n ed fo u r orders o f
m agnitude the sm aller valu es o ccu p y a very narrow zone at the bottom o f the graph. T his results in a
serious truncation o f the accuracy o f those original m easurem ents. B leaney and L o u b ser q u o te the accu ­
racy o f their m easurem ents as ± 3-4% , b u t for the sm allest m easured absorp tiv ity , 2 d B /k m at 100 ton-
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p ressu re an d 3.7 5 G H z, the d isp lay form at lim its th e accuracy to app ro x im ately ± 50% . T h e greatest
losses o f accuracy a re seen w here the absorptivities are sm allest, in th e low an d high frequency tails o f
the sp e c tra at th e lo w e st pressures, 100 and 300 torr. A lth o u g h the sm allest absorptivities o bserved in
the high freq u en cy tails are considerably higher than those in th e low frequency tails, the display form at
squanders the p o ten tial im provem ent in accuracy there b y dividing the absorptivities by v 2.
A bso rp tiv ity values w ere extracted from the grap h by hand. First, a 200% photocopy en largem ent
o f the o rig in al g rap h w as m ade. T his w as thoroughly c h eck ed fo r system atic nonlinearities in the grid
due to the original prin tin g or photocopying processes. T he o n ly m easurable d istortions w ere in an area
of the g rap h c o n tain in g no d ata. A m icrom eter accu rate to 0.001 inch w as used to m easure the position
o f each p o in t P oints on the grap h are represented as circles, w hich after the 200% m agnification w ere
o f considerable size. R ather than attem p t locating th eir centers b y eye, displacem ents w ere m easu red to
both ex trem a o f the circles and those figures w ere averaged. D isplacem ents w ere converted to values o f
v and a / v 2 , an d v alu es fo r a w ere c alcu lated fro m them .
T h ere m ay now be sim p ler m ethods o f
extract- in g the in fo rm atio n fro m the graph using a co m p u ter-co n tro lled elec tro n ic scanning device.
T his w o u ld be an a lm o st trivial b u t certain ly w orthw hile p ro je c t fo r any in stitu tio n possessing such
capability.
T ab le 6.1 p resen ts results o f the extraction p ro cess. If the la s t sig n ifican t d ig it o f a listed ab so rp ­
tivity v a lu e is a zero it is underlined. S ignificant digits in the absorptivity values represent the accuracy
o f the ex tractio n m easu rem en ts only, n o t th e 3-4% accu racy o f the ex p erim en tal m ethods q u o ted by
B leaney a n d L oubser. U ncertainties in pressures, tem peratures, a n d frequencies w ere n o t stated in their
paper. T h e table constitutes one sequence o f data, w ith series at each o f the six d ifferent pressures.
Som e d ata on p u re am m onia w ere taken as p a rt o f this w ork, alth o u g h the o riginal p u rp o se fo r
those m easu rem en ts w as to te st the p ractical limits o f th e apparatus, n o t to use them in the analysis.
T hese d a ta include b o th absorptivities and refractivities. T hey a re p resen ted in T ab le 6.2, bu t as n o ted
before the fo u r series a t three w idely d ifferent tem peratures d o no t constitute a single sequence. U n cer­
tainties in pressu re m easurem ents are ±0.007 atm osphere except fo r those n ear 1 atm osphere, w here the
u ncertainties are ± 0 .0 0 4 atm osphere. T em perature uncertain ties are ±0 .4 K . F requency m easurem ents
83
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w ere extrem ely accurate; all freq u en cies a re accu rate to at le ast six sig n ifican t digits, and so m e are
accurate to eight. T h e effects o f these uncertainties have n o t been carried over to the expressed u n certain­
ties in the ab so rp tiv ity and re fractiv ity m easu rem ents. T h e u n certainties show n on th e m easurem ents
reflect only those d irecd y associated w ith the cav ity reso n ato r m ethod, such as the accuracy o f th e b an d ­
w idth and cen ter frequency m easurem ents, SN R , uncertainties in dielectric loading, coupling effects, etc.,
as d iscu ssed in Section 4.3 and A ppendix D.
R efractivity d ata from the m easurem ents o n p u re am m onia im m ediately dem onstrate the differences
betw een am m onia an d the transparent gases o f C h ap ter 5. M agnitudes o f the am m onia refractiv ities are
m uch larger than those o f the transparent gases, an d their b ehavior is decidedly nonlinear in all the condi­
tions involved: freq u en cy , tem p eratu re, a n d p ressu re. A t S T P a sam ple o f hydrogen, fo r exam ple,
w ou ld have a refractivity n ear 135 x 10-6, w hile a sam ple o f am m onia at S T P w o u ld have a refractivity
at 10 G H z n ear 3000 x 10_6, a factor o f 22 larger. F o r am m onia a frequency m ust b e specified since its
refractivity is frequency d e p e n d e n t A t the lo w est p ressure m easured, 0.192 atm ., the ab sorption spec­
trum o f am m onia is still resonant in character, w ith a peak near 24 G H z. R efractivities increase sig n ifi­
can tly w ith freq u en cy o ver th e 9 to 18 G H z ra n g e o f th a t series. A t h ig h er p ressures th e ab so rp tio n
sp ectru m is in transition to a m o re n o n -reso n an t D e b y e shape. T he m easu red refractiv ity sp ectru m o f
the series at 0.5 atm . is approxim ately fla t o v e r th a t frequency range, w hile those o f th e series at 1 atm .
d ecrease w ith frequency. T hese results agree qualitativ ely w ith predictions o f the K ronig-K ram ers rela­
tions. A q uantitative application o f those equatio n s to these results w as no t attem pted, p artly due to the
lack o f data o v er a sufficiently w ide range o f freq u en cies to p erform the calculations w ithout involving
sig n ifican t assum ptions.
M easu re d absorptivities on p u re am m onia, w h eth er fro m B leaney and L o u b ser o r from this w ork,
g en erally sup p o rt th e original fo rm alism o f B en -R eu v en (1966). T h o se o f this w ork m easu red at 0.5
and 1 atm . p ressure, 300 K , are co n sisten t w ith the d ata o f B leaney and L oubser. B en-R euven theory
agrees w ith d a ta on p ure am m onia at the 5 -10% lev el, d epending on pressure. It is m ixtures o f sm all
am ounts o f am m onia in other gases, discu ssed in th e n ex t section, that have been sources o f d isag ree­
m ent betw een laboratory m easurem ents and theoretical predictions.
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M easured Absorptivities
G as m ixture: pure ammonia (NH3)
T em perature: "Room temperature" (approx. 297 K) for all measurements
N o te : T h ese d ata fro m B lean ey and L o u b se r (1950) d o no t rep resen t new m easurem ents. T h ey are
included for com pleteness, and to make them generally available in tabular form .
T otal P ressure,
to rr
Frequency,
GHz
A bsorptivity ( a )
dB /km
2
3.75
7.795
8.96
9.86
16.79
17.99
19.91
20.0
20.4
20.9
21.5
22.1
22.8
23.7
24.55
25.4
26.0
26.6
26.95
27.5
30.0
31.2
32.9
36.0
32
286
425
689
876
1020
1260
1545
1730
1950
2120
21Q0
1840
1530
1390
1305
990
630
510
260
75
3.75
7.795
9.08
10.25
17.09
17.63
18.05
18.86
19.61
20.3
20.9
112
152
212
1220
1415
1605
1790
2060
2275
2410
12
18
18
T able 6.1: M easured absorptivities o f p u re gaseous am m onia, by B leaney and L o u b ser (1950). This
table continues o n the fo llo w in g two pages.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Absorptivities
(continued from previous page)
G as m ixture: pure ammonia (NH 3 )
T em perature: "Room temperature" (approx. 297 K) for all measurements
N o te : T h ese d a ta fro m B lean ey and L o u bser (1950) d o n o t represent n ew m easurem ents. T hey are
included fo r com pleteness, and to m ake them generally available in tab u lar form .
T otal Pressure,
torr
Frequency,
GHz
A bsorptivity ( a )
dB /km
300
21.8
22.7
24.0
25.4
26.95
28.1
2 9.0
30.0
30.3
32.0
32.9
35.85
36.15
2650
2810
2930
29Q0
21Q0
2490
23Q0
2160
2080
1730
1505
1150
980
900
3.75
7.885
9.29
10.58
17.15
17.36
18.05
19.31
21.2
22.1
23.4
24.0
25.8
27.0
28.15
30.0
32.95
34.4
36.0
143
625
865
1112
2660
2690
2915
3161
3570
3690
3850
3840
4020
4080
4060
4150
3990
4140
3970
3.75
3.87
9.92
10.07
21.4
256
277
1540
1630
45Q0
1520
(2 atm .)
86
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M easured Absorptivities
(continued from previous page)
G as m ixture: pure ammonia (NH 3 )
T em perature: "Room temperature" (approx. 297 K) for all measurements
N o te : T hese d ata fro m B lean ey and L o u b ser (1 9 5 0 ) do n o t rep resen t n ew m easu rem en ts. T h ey are
included fo r com pleteness, and to m ake them generally available in tabular form.
T otal Pressure,
ton-
Frequency,
GHz
A bsorptivity ( a )
dB /km
1520
(2 atm .)
22.9
23.3
24.4
25.4
33.45
33.75
33.9
35.7
36.7
4735
46 3 0
5040
5135
6220
60Q0
6250
5930
6320
3040
(4 atm.)
3.75
9.77
10.25
21.3
22.5 5
23.7
24.5
25.7
33.3
33.9
35.55
36.3
412
2480
2730
8290
8890
9050
9360
12490
12540
12980
13420
3.75
10.07
10.19
22.55
23.7
24.55
33.5
33.95
36.4
433
2930
3040
11710
12390
13320
19840
20090
22160
4560
(6 atm .)
10020
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Measured Refractivities and Absorptivities
Gas m ixture: pure ammonia (NH 3 )
M easured
Tem perature. K
T o tal P ressure,
A tm .
Frequency,
GHz
R efractivity (v ) x
212.95
212.95
212.90
212.90
212.95
212.95
213.05
213.05
213.15
0.192
0.1 9 2
0.192
0.1 9 2
0.192
0.192
0.192
0.1 9 2
0.192
9.17011
11.24200
12.74205
13.62303
13.70114
15.63721
16.17762
17.00332
17.42732
1183
1218
1258
1268
1249
1278
1284
1359*
1289
±15
±12
±21
±20
± 8
±18
±30
±11
±13
96
218
255
386
380
813
808
1003
1328
± 10
±21
± 37
±20
±23
±75
±93
±46
±95
300.10
300.05
300.05
300.00
299.95
299.95
299.95
0.50 0
0 .5 0 0
0.5 0 0
0.5 0 0
0.5 0 0
0.5 0 0
0.5 0 0
9.15235
11.22050
13.59674
13.67507
15.60994
16.97409
17.53655**
1547
1557
1624
1582
1446
1497
1487
±32
±27
±57
±27
±88
±28
±28
225
412
728
741
1153
1476
1598
± 7
± 10
±28
±29
±36
± 23
±37
300.00
300.00
299.95
299.95
300.00
300.00
1.000
1.000
1.000
1.000
1.000
1.000
9.13957
11.20680
13.57899
13.65808
16.95451
17.51707**
2948 ± 2 9
2781 ± 113
2933 ± 7 6
2827 ± 3 1
2654 ± 3 4
2601 ± 2 8
538
1088
1502
1497
2388
2432
± 56
± 93
±93
± 95
± 55
± 47
313.15
313.25
313.65
313.60
313.45
1.001
1.001
1.002
1.002
1.001
9.1 3 9 2 0
11.20587
13.57784
13.65698
16.95354
2740 ± 3 7
2616 ± 113
2771 ± 7 2
2662 ± 2 8
2466 ± 3 4
485
895
1332
1323
2195
± 50
± 59
±77
± 54
±64
*
A bsorptivity (a )
d M ffl ___
T his value is con sid ered unreliable. It is thought th at an error w as m ade in read in g o r transcribing
the frequency co unter display.
** T he TE 0 2 4 resonance w as rendered unusable by the high absorptivities encountered in these exper­
im ents. A stro n g er reso n an ce o f lo w er intrinsic Q and slightly hig h er frequency w as used.
Table 6.2: M easu red absorptivities and refractivities o f pure gaseous am m onia.
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6.3
Data on Gas Mixtures Containing Ammonia
U n lik e the d ata p resen ted in the previous section, the d ata presen ted in this section w ere all obtained
as p art o f this w ork, an d all m easurem ents include refractivities as w ell as ab sorptivities. T h e gas m ix­
tures stu d ied co n sist o f a sm all am o u n t o f g aseo u s am m o n ia in o n e o r m ore essen tia lly tran sp aren t
broadenin g gases: am m o n ia in o th erw ise p u re m o lecu lar hydrogen (H 2 ), am m o n ia in o th erw ise pure
atom ic h eliu m (H e), an d am m onia in a Jo v ian m ixture o f 90% hydrogen and 10% h eliu m by num ber.
A n im p o rtan t asp e c t o f this p ro g ram o f m easurem ents is th a t all p e rtin e n t m acro sco p ic co n d itio n s
know n to significantly affect m icrow ave absorption by am m o n ia w ere varied to p ro v id e m ore detailed
inform ation about those dependences. Strategy and m ethods are described in C hapter 4 an d A ppendix C.
D ata presented in this section are organized by m acroscopic conditions. T he h ighest level is defined
by the species o f the broadening gas o r gases, separating the d ata into three broad categories. A ll data on
am m onia in o therw ise p u re hydrogen are presen ted first, in T ables 6.3 through 6.6. N ex t are all data on
am m onia in otherw ise pure helium , T ables 6.7 through 6.10. F in ally d ata o n am m o n ia in the Jovian
m ixture o f hydrogen and helium are in T ables 6.11 an d 6.12. W ith in each broadening gas categ o ry the
n ext level o f o rg an izatio n is by increasing targ et tem perature. F o r exam ple, in the h eliu m -b ro ad en ed
category data at 213 K a p p ear first, follow ed by d ata at 273 K, then d ata a t 313 K. In m o st cases each
tem perature category w ill be represented by one table. T he exceptions are the 273 K hydrogen-broadened
group and the 313 K h eliu m -b ro ad en ed g roup, w h ere there are d ata at tw o d ifferen t am m o n ia m ixing
ratios, abo u t 0.0082 and 0.067 by num ber. T he data in those groups are separated into tw o tables.
E ach table contains one sequence o f data, grouped by series, at a fixed target tem perature. A single
sequence m aintains con stan t m ixing ratios o f all gases, and thus all partial pressures in v o lv ed are direct­
ly co n tro lled b y total p ressure. M o st o f the sequences contain five series representing to tal pressures
n ear 1, 2, 4 , 6, an d 8 atm ospheres, in that order. S ince d ata w ere taken at o n ly one p ressu re in som e
experim ents, T ables 6 .6 and 6.1 0 contain o n ly one series each. T able 6.5 con tain s tw o series, bu t the
tw o represen t repeated m easurem ents since they are under essentially identical conditions. Finally, each
series is ordered by the m icrow ave frequency o f the m easurem ents, from low est to highest.
89
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U ncertainties in basic co ndition m easurem ents, i.e. tem perature, total p ressure, and frequency, are
straightforw ard and sim ilar to th o se given in Section 6.2. T em peratures are w ithin ± 0.4 K and frequen­
cies are accurate to at least six sig n ifican t dig its. Pressure m easurem ents n ear o n e atm o sp h ere u sed the
sam e low -p ressu re g au g e u sed in th e p u re am m o n ia m easurem ents, so they a re accu rate to ± 0.004
atm osphere. H ig h er pressures, m easured w ith a different gauge, are accurate to ± 0.014 atm osphere.
M ixing ratio uncertainties are m o re inv o lv ed . T he m ixing ratio at any step in an ex p erim en t m ay
be the resu lt o f m ultiple additions o r dilutions o f a gas in a m ixture, each carrying its ow n uncertainties
th at contrib ute to the u n certain ty o f the m ix in g ratio in question. A ppendix D c o v ers this in som e
detail. T h e tw o sequences taken on m ix tu res w ith an am m onia m ixing ratio n ear 0.067, T ab les 6.6 and
6.10, are relatively sim ple: those m ixtures are the resu lt o f a single m ixing o f am m o n ia w ith o nly one
broaden ing gas, eith er hydrogen o r heliu m . F o r those m ixtures uncertain ty in th e am m o n ia m ixing
ratio is ± 0.0010, o r ±1.3% . Since th e sum o f m ixing ratios m u st be unity th e m ixing ra tio uncertainty
fo r the broadening gas is also ± 0 .0 0 1 0 , w h ich is a m ere ± 0.1% due to its m uch la rg e r m ix in g ratio.
M o st o f th e d a ta p resen ted in this sectio n w ere taken w ith an am m onia m ixing ra tio n e a r 0.0082.
A chieving this required a dilution step, increasing the relative uncertainties. T he am m onia m ixing ratio
uncertainty for such m easurem ents is ± 0 .0 0 0 1 7 , o r ± 2.0% . If the b roadening gas is p u re hyd ro g en or
pure helium , as is the case in T ables 6.3, 6.4 , 6.6, 6.7, 6.8, and 6.9, then the u ncertainty in its m ixing
ratio m ust also be ± 0.00017, w hich is n egligibly sm all. I f the broadening gas is the Jovian m ixture o f
hydrogen and helium , as in T ables 6.11 an d 6.12, the am m onia m ixing ratio uncertainty is n o t affected,
but the m ixing ratio u n certainties o f the h ydrogen and helium ju m p to ab o u t ± 0.004. T h is is ± 0 .4 %
for the hydrogen com ponent and ±4% fo r the helium .
A gain, the u n certain ties in pressu res, tem p eratu res, freq u en cies, and now m ix in g ratio s, are n o t
carried forw ard to the uncertainties ex p ressed for the m easured refractivities and ab so rp tiv ities. The
global uncertainties in the m easured d ata reflect only those individual uncertainties d ire c tly associated
w ith the cavity reso n ato r m ethod, as d iscu ssed in S ection 4.3. T he tw o series o f d a ta in T a b le 6.5 are
w orth noting since they are essen tially rep eated m easurem ents. T he differences betw een th e tw o are
considerably sm aller than the sizes o f th e erro r bars.
90
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R esults o f the refractivity m easurem ents confirm that am m onia is distin ctly d ifferen t from the gases
o f C hapter 5. R efractivities e x h ib it n o n lin ea r b ehavior w ith all co nditions varied: frequency, tem pera­
ture, and pressures. A t the lo w er p ressures w here the absorption sp ectru m is still reso n an t in character
w ith a peak n ear 24 G H z, the refractivity o f a fixed m ixture increases significantly w ith frequency in the
9 to 18 G H z range. H ig h er pressures w ith absorption spectra approaching a D eb y e shape produce refrac­
tivities th at m ay d ecrease w ith frequency in th at range. T his is in q u alitativ e ag reem en t w ith the p red ic­
tions o f the K ronig-K ram ers relations. It is possible to extract refractiv ities o f th e am m onia com ponent
o f a m ixture by subtracting th e calcu lated refractivities o f the bro ad en in g gases, assu m in g they are still
lin ear in the presen ce o f sm all am ounts o f am m onia, although the u n certainties invo lv ed are large due to
the sm all am m onia m ixing ratio and uncertainties in the refractivities o f the broadening gases. T ypical
uncertainty in such an o p eratio n is ± 20-30% . A m m onia refractiv ities c alc u la te d b y this m ethod are
extrem ely large co m p a re d to the gases o f C h ap ter 5, in a g reem en t w ith the m easu rem en ts on pure
am m onia p re se n ted in S ectio n 6.2. U n d er conditions w here the tran sp aren t g ases h ad refractivities o f
tens to a few hun d red tim es 1 0 '6, calcu lated am m onia refractivities system atically v aried fro m three- to
ten-thousan d tim es 1CH>. T his is evidence o f stro n g er coupling o f the d ip o la r am m o n ia m olecules w ith
m icrow ave rad iatio n . D esp ite these larg e m agnitudes, calc u latio n s o f bulk refrac tiv ities w ithin the
atm ospheres o f the gian t p lan ets involve am m onia m ixing ratios so sm all th at its refractiv ity co n trib u ­
tion can be neglected (A tkinson an d Spilker, 1990). T he refractivity results rep o rte d h ere are m ostly o f
spectroscopic interest.
A bsorp tiv ity m easu rem en ts rep o rted h ere have m uch m ore im p o rtan t im p licatio n s fo r studies o f
giant planet atm ospheres. D ata indicate th at although predictions o f the m odified B en-R euven form al­
ism o f B erge and G ulkis are in m any cases m ore accurate than those o f o ld e r V an V leck -W eissk o p f
form alism s, neither prediction schem e adequately represents the laboratory m easurem ents presented here.
W ithin a single series, even if th e p redicted absorptivity m agnitudes are reaso n ab ly close to the data (but
not necessarily w ithin error bars) the predicted frequency dependences disagree w ith the d ata. W ithin one
data sequence the relativ e d ifferen ces betw een predictions and d ata v ary sy stem atically w ith pressure,
suggesting problem s w ith p ressu re dependences o f the prediction form alism s. E x p erim en ts at different
91
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tem peratures also sh o w clear evidence o f system atic v ariation in the differences betw een pred ictio n s and
data, suggestin g problem s w ith tem perature depen d en ces. Specifically, although predictions o f the B erg e
and G ulkis fo rm alism are m o st accurate n ear ro o m tem perature, w here m o st previous laboratory m easu re­
m ents have clu stered , d ata at the low est tem perature, 213 K , indicate the form alism o verstates am m o n ia
absorption at th at tem p eratu re. I f this tren d co n tin u es, ab so rp tiv ities at the m uch lo w e r tem p eratu res
encountered in the atm ospheres o f g ian t planets m ay be seriously overestim ated by the B erge an d G ulkis
form alism . T he in tensity fac to r o f V an V leck -W eissk o p f form alism s is id en tical to th a t o f B en -R euven
form alism s, so its tem p eratu re d ep en d en ce m ay also be erro n eo u s. B oth types o f form alism s are co m ­
m only used in red u ctio n and interpretation o f ra d io astronom ical an d radio occultation d ata fro m the g ian t
planets. As d iscu ssed in C h apter 2, accurate k n o w led g e o f absorptivities is critical to co rrect interpretation
o f rad io astronom ical and radio occultation data. S ignificant overestim ates o f the absorptivity o f am m o n ia
can lead to significant underestim ates o f the am m onia abundance in an atm osphere under study.
92
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Measured Refractivities and Absorptivities
Gas m ixture, by number mixing ratio:
am m onia, 0.00818 ; hydrogen, 0.99182 .
T arget tem perature: 213.15 K
M easured
Tem perature, K
T otal Pressure,
Atm.
Frequency,
GHz
1.001
2 23.6
226.9
229.9
232.1
232.2
235.1
237.0
240.9
240.3
± 3 .2
+ 3 .1
± 3 .2
± 3 .4
± 3 .0
± 3 .5
± 3.4
± 4 .7
± 3 .5
1.7
6.6
9.6
11.6
12.9
23.7
29.3
4 1 .2
4 7 .0
± 0 .9
± 0 .9
± 1.4
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 2 .7
± 1.4
R efractivity (v ) x 106
A bsorptivity ( a )
d B /k m
213.15
213.05
213.00
213.00
213.05
213.10
213.10
213.15
213.15
1.001
1.001
9.17891
11.25315
12.75517
13.63717
13.71512
15.65355
16.19459
17.02233
17.44563
212.85
212.85
212.90
212.95
212.90
213.00
213.05
213.05
213.10
1.964
1.964
1.964
1.964
1.964
1.965
1.965
1.965
1.965
9.17701
11.25081
12.75252
13.63434
13.71227
15.65034
16.19124
17.01896
17.44212
430.9
434 .6
438.3
4 3 9 .6
44 0 .4
440.4
444.3
44 5 .2
441.8
± 4 .1
± 3.8
± 3 .8
± 3.6
± 3 .7
± 3 .5
± 4 .1
± 4 .8
± 4 .0
9 .0
18.7
26 .9
36.1
37.7
62.2
72 .5
91.2
101.5
± 0 .9
± 0 .9
± 1.4
± 0 .9
± 1.4
± 1.4
± 1.4
± 3 .6
± 1.4
212.90
212.90
212.85
213.00
213.00
213.00
213.00
213.05
213.15
3.970
3.971
3.972
3.975
3.976
3.978
3.979
3.981
3.984
9.17309
11.24606
12.74713
13.62865
13.70656
15.64395
16.18467
17.01188
17.43512
857.6
857.6
861.4
857.0
857.3
848.9
850.2
848.1
843.7
± 3 .6
± 3 .4
± 3 .5
± 3.3
± 3 .7
± 5 .8
± 4 .8
± 4 .8
± 4 .7
2 9.5
58.1
7 6 .0
92.5
93 .0
131.5
145.5
171.2
175.9
± 1 .4
± 1 .4
± 5 .0
± 1.4
± 1 .4
± 2 .3
± 2 .3
± 3 .6
± 2 .3
212.85
212.90
213.00
212.95
213.00
213.05
213.10
213.10
213.15
5.984
5.984
5.985
5.984
5.984
5.985
5.985
5.984
5.985
9 .16924
11.24141
12.74190
13.62307
13.70099
15.63771
16.17824
17.00518
17.42818
1278.2
1271.2
1272.2
1267.7
1264.1
1248.3
1248.0
1249.6
1241.7
± 7 .0
± 3 .4
± 6 .0
± 3 .4
± 3 .7
± 10.6
± 6 .0
± 7 .2
± 5 .8
54 .6
9 6 .6
118.1
136.5
133.6
179.5
191.7
210.3
2 1 8 .0
± 1.4
± 1.4
± 5 .0
± 2 .3
± 2 .3
± 2 .3
± 2 .3
± 3 .6
± 2 .3
1.000
1.000
1.000
1.000
1.000
1.000
Table 6.3: M easured absorptivities and refractivities o f 0.818% am m onia in hydrogen, at approxim ately
213 K. T his table continues on the next page.
93
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Measured Refractivities and Absorptivities
(continued from previous page)
Gas m ixture, by number mixing ratio:
am m onia, 0.00818 ; hydrogen, 0.99182 .
T arget tem perature: 213.15 K
M easured
Tem perature. K
T otal P ressure,
A tm .
Frequency,
GHz
212.85
212.85
212.85
212.75
212.85
212.85
212.85
213.05
212.95
8.160
8.160
8.160
8.157
8.161
8.161
8.162
8.170
8.166
9.16506
11.23634
12.73612
13.61699
13.69492
15.63077
16.17101
16.99763
17.42028
R efractivity (v) x 106
1734.8
1723.0
1726.6
1714.0
1707.9
1692.8
1695.6
1694.4
1694.9
± 1 1 .6
± 4 .9
± 8 .1
± 3 .8
± 4 .8
± 11 .0
± 9 .2
± 9 .0
± 8 .8
A bsorptivity (
dB /km
N .U .
129.0 ± 1.4
147.4 ± 9 .6
174.2 ± 5 .0
176.8 ± 3 .2
222.1 ± 5 .0
234.6 ± 5 .0
250 .6 ± 6 .4
263.0 ± 3 .2
94
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Measured Refractivities and Absorptivities
Gas m ixture, by number mixing ratio:
am m onia, 0.00817 ; hydrogen, 0.99183 .
T arget tem p eratu re: 273.15 K
M easured
T em perature. K
T otal Pressure,
Atm.
Frequency,
GHz
273.05
273.05
273.15
273.15
273.15
273.15
273.15
273.15
273.15
1.002
1.002
1.002
1.002
1.002
1.002
1.002
1.002
1.002
9.16952
11.24164
12.74216
13.62323
13.70113
15.63753
16.17807
17.00498
17.42781
164.3
166.4
169.0
170.9
171.7
175.1
177.1
179.3
180.5
± 3 .3
± 3 .2
± 3 .1
± 3 .1
± 3 .1
± 3 .6
± 3 .6
± 3 .9
± 3 .9
2.7
2.2
6.9
8.6
6.0
14.4
19.2
2 1 .2
28.5
± 0 .9
± 0 .9
± 1.2
± 0 .9
± 0 .9
± 1.9
± 0 .9
± 1.8
± 4 .0
273.15
273.15
273.15
273.15
273.15
273.15
273.15
273.15
273.20
1.935
1.935
1.935
1.935
1.935
1.935
1.935
1.935
1.936
9.16816
11.23996
12.74026
13.62120
13.69909
15.63521
16.17568
17.00247
17.42529
313.6
316.1
318.1
320.5
320.7
323.5
324.7
326.6
325.1
± 3 .3
± 3 .2
± 3 .2
± 3 .7
± 3 .3
± 4 .9
± 4 .8
± 4 .3
± 4 .8
6.9
10.4
17.0
21.8
21.6
38.1
46.4
54.5
65.4
± 0 .9
± 0 .9
± 1.6
± 0 .9
± 0 .9
± 2 .4
± 1 .4
± 2 .7
± 3 .1
273.05
273.10
273.15
273.15
273.15
273.15
273.15
273.15
273.15
3.980
3.981
3.982
3.982
3.982
3.982
3.982
3.982
3.982
9.16513
11.23626
12.73608
13.61674
13.69460
15.63018
16.17051
16.99707
17.41977
643.6
645.1
646.8
6 47.5
6 48.5
644.9
64 4 .4
64 4 .2
641.8
± 3 .3
± 3 .2
± 3 .3
± 3.2
± 3.8
± 5 .6
± 5 .5
± 5.4
± 5 .3
29.3
34.9
49.9
61.9
62.1
92.2
105.0
117.2
128.3
± 0 .9
± 0 .9
± 2 .6
± 1.4
± 2 .3
± 3 .3
± 1.4
± 1.8
± 3 .1
273.05
273.05
273.15
273.15
273.15
273.15
273.15
273.25
273.25
5.988
5.988
5.990
5.990
5.990
5.990
5.990
5.993
5.993
9.16221
11.23269
12.73208
13.61244
13.69031
15.62533
16.16547
16.99180
17.41438
962 .5
963.3
961 .4
96 3 .8
9 6 2 .0
956.1
95 6 .4
9 5 5 .2
9 5 1 .5
± 3 .7
± 3 .6
± 4 .0
± 3 .8
± 3 .8
± 6 .3
± 6 .2
± 6 .1
± 5 .9
37.8
60.9
82.5
97.9
97.7
128.2
143.6
156.6
166.0
± 1 .2
± 1 .4
± 4 .4
± 2 .3
± 2 .3
± 4 .2
± 2 .3
± 2 .7
± 3 .1
R efractiv ity (v) x 106
A bsorptivity (a )
dB /km
T able 6.4: M easu red absorptivities and refractivities o f 0.817% am m onia in hydrogen, at approxim ately
273 K . T h is table continues on the n ex t page.
95
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M easured Refractivities and Absorptivities
(continued from previous page)
Gas m ixture, by number mixing ratio:
am m onia, 0.00817; hydrogen, 0.99183 .
T arg et tem p eratu re: 273.15 K
M easured
T em perature. K
T otal Pressure,
Atm.
Frequency,
G Hz
273.15
273.15
273.15
273.1 5
273.15
273.15
273.15
273.15
273.15
8.026
8.026
8.026
8.026
8.026
8.026
8.026
8.026
8.026
9.15927
11.22912
12.72812
13.60817
13.68604
15.62051
16.16050
16.98657
17.40906
R efractiv ity (v) x 106
____________________
1283.9
1282.4
1272.8
1277.9
1274.3
1265.0
1264.2
1263.4
1257.4
± 5 .4
± 4 .9
± 4 .8
± 4 .6
± 4 .6
± 6 .3
± 6 .2
± 6 .1
± 5 .9
A bsorptivity ( a )
dB /km
53.9
84.3
107.5
124.1
123.7
158.6
172.3
184.5
193.0
± 1.2
± 1.4
+ 4.4
2.3
± 3.2
+ 3.3
+ 3.2
Hr 2.7
± 3.1
96
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities and Absorptivities
Gas m ixture, by number mixing ratio:
am m onia, 0.0670 ; hydrogen, 0.9330 .
T arget tem perature: 273.15 K
M easured
T em perature. K
T otal Pressure,
Atm.
Frequency,
GHz
273.15
273.15
273.15
273.15
273.15
273.15
8.175
8.175
8.175
8.175
8.175
8.175
9.14498
11.21272
13.6659
15.6004
16.9637
17.3860
2849
2746
2750
2556
2613
2586
±71
± 67
± 64
± 88
±28
±51
400
652
889
1175
1226
1369
± 38
±51
±81
±95
± 80
±96
273.20
273.15
273.15
273.15
273.15
273.15
273.15
8.206
8.204
8.204
8.204
8.204
8.204
8.204
9.14490
11.21272
13.58744
13.66566
15.60016
16.96356
17.38591
2858
2747
2806
2768
2571
2620
2590
±49
± 67
±78
±48
±87
±28
±50
391
634
899
907
1157
1252
1335
±29
±59
± 62
± 63
±94
±81
±94
R efractivity (v) x 106
A bsorptivity
dB /km
T ab le 6.5: M easured absorptivities an d refractivities o f 6.70% am m onia in hydrogen, a t approxim ately
273 K.
97
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Measured Refractivities and Absorptivities
G as m ixture, by number mixing ratio:
am m onia, 0.00816; hydrogen, 0.99184 .
T arget tem perature: 323.15 K
M easured
Tem oerature. K
T otal Pressure,
Atm.
Frequency,
G Hz
323.65
323.60
323.40
323.40
323.40
323.35
323.35
323.25
323.25
0.996
0.996
0.995
0.995
0.995
0.995
0.995
0.995
0.995
9.16125
11.23148
12.73072
13.61098
13.68885
15.62348
16.16356
16.98961
17.41221
R efractivity (v) x 106
____________________
142.9
144.6
144.3
144.6
144.6
147.5
148.2
151.2
151.3
± 4 .9
± 4 .8
± 4 .9
± 4 .7
± 4 .7
± 4 .7
± 4 .7
± 4 .8
± 4.6
A bsorptivity (a )
dB /km ____
0.8
2.0
3.5
6.1
4.0
9.2
11.5
15.6
19.8
± 0 .9
± 0 .9
± 1 .8
± 1.4
± 0 .9
± 1.4
± 2 .3
± 1 .8
± 1 .4
T able 6.6: M easured absorptivities and refractivities o f 0.816% am m onia in hydrogen, at approxim ately
323 K .
98
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Measured Refractivities and Absorptivities
Gas m ixture, by number mixing ratio:
am m onia. 0.00822 : helium. 0.99178 .
T arget tem perature: 213.15 K
M easured
Tem perature. K
T otal Pressure,
A tm .
Frequency,
GHz
213.15
213.10
213.15
213.15
213.15
213.15
213.10
213.15
213.25
1.001
1.001
1.001
1.001
1.001
1.001
1.001
1.001
1.002
9.18004
11.25453
12.75674
13.63886
13.71682
15.65548
16.19661
17.02450
17.44778
100.7
102.4
105.4
106.3
104.8
108.8
110.5
112.7
114.2
± 3 .2
± 3 .1
± 3 .0
± 3 .0
± 3 .0
± 3 .5
± 3 .4
± 3 .5
± 3 .4
0.9
1.4
3.7
4.2
4 .2
8.6
10.6
16.2
18.8
± 0 .9
± 0 .9
± 1.4
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 2 .7
± 0 .9
213.20
213.25
213.15
213.15
213.10
213.00
213.05
213.00
213.05
1.969
1.970
1.969
1.969
1.968
1.967
1.968
1.967
1.968
9.17927
11.25357
12.75564
13.63765
13.71559
15.65402
16.19507
17.02280
17.44606
184.7
188.4
192.1
194.5
194.9
20 2 .6
205.1
212.1
212.7
± 3 .2
± 3 .1
± 3 .0
± 3 .0
± 3 .0
± 3 .5
± 3 .4
± 3 .6
± 3 .4
2.3
5.0
9.2
12.4
14.3
27.8
33.9
47.8
58.2
± 0 .9
± 0 .9
± 1.4
± 0 .9
± 0 .9
± 0 .9
± 1.4
± 2 .7
± 2 .3
213.05
213.15
213.15
213.15
213.15
213.15
213.15
213.15
213.20
3.995
3.997
3.997
3.997
3.997
3.997
3.997
3.997
3.998
9.17765
11.25154
12.75329
13.63514
13.71309
15.65111
16.19201
17.01958
17.44284
361.6
368.1
376.1
378.9
376.6
388.3
394.3
401.8
397.6
± 3 .2
± 3 .6
± 3 .4
± 3 .4
± 3 .4
± 7 .2
± 5 .6
± 5 .5
± 5 .3
11.0
2 2.9
3 5 .0
47 .5
4 9 .6
87.9
105.0
134.9
159.7
± 1.4
± 0 .9
± 2 .3
± 2 .3
± 2 .3
± 4 .1
± 4 .1
± 2 .7
± 2 .3
213.10
213.15
213.05
213.15
213.10
213.05
213.10
213.05
213.05
5.998
6.000
5.997
6.000
5.998
5.997
5.998
5.997
5.997
9.17609
11.24962
12.75105
13.63279
13.71072
15.64853
16.18933
17.01677
17.44003
531.7
539.7
551.6
551.5
549.4
553 .2
559.9
566.9
558.8
± 4 .2
± 3 .9
± 4 .0
± 4 .7
± 4 .8
± 7 .7
± 4 .3
± 4 .4
± 5 .6
23.9
4 6 .2
70.4
89.0
96.5
157.1
181.3
224.5
262.9
± 1.4
± 2 .3
± 4 .1
± 2 .3
± 4 .1
± 5 .0
± 4 .1
± 4 .1
± 4 .1
R efractivity (v) x 106
A bsorptivity (a )
_________________________________ d B /km
T ab le 6.7: M easured absorptivities and refractivities o f 0.822% am m onia in helium , at approxim ately
213 K . T his table continues o n the n ex t page.
99
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities and Absorptivities
(continued from previous page)
G as m ixture, by number mixing ratio:
am m onia, 0.00822 ; helium, 0.99178 .
T arget tem p eratu re: 213.15 K
M easured
Tem oerature, K
T o tal P ressure,
A tm .
Frequency,
G Hz
213.15
213.10
213.05
213.05
213.10
213.00
213.15
213.10
213.15
8.015
8.013
8.011
8.011
8.013
8.009
8.015
8.013
8.015
9.17448
11.24768
12.74886
13.63042
13.70839
15.64595
16.18670
17.01403
17.43730
R efractiv ity (v) x 106
707.0
712.9
7 2 4 .0
725.1
72 0 .2
718.4
722.8
7 2 8 .2
715.1
± 5 .9
± 3 .4
± 3 .4
± 3 .5
± 3 .5
±10.3
± 4 .4
± 4 .6
± 7 .4
A bsorptivity (a )
dB /km
39.5
76.7
108.9
137.6
145.0
228.8
2 64.7
3 0 9 .6
357.1
± 1.4
± 2 .3
± 4 .1
± 4 .1
± 4 .1
± 9 .6
± 4 .1
± 4 .1
± 4 .1
100
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities and Absorptivities
Gas m ixture, by number mixing ratio:
am m onia, 0.00818 : helium. 0.99182 .
T arget tem p eratu re: 273.15 K
M easured
Tem perature. K
T otal Pressure,
A tm .
Frequency,
G Hz
273.15
273.15
273.20
273.15
273.15
273.15
273.15
273.15
273.15
1.001
1.001
1.001
1.001
1.001
1.001
1.001
1.001
1.001
9.17042
11.24272
12.74340
13.62455
13.70246
15.63905
16.17965
17.00663
17.42951
71.6
74.7
75.9
77.7
78.1
81.8
83.6
86.4
86.9
± 3 .3
± 3 .2
± 3 .1
± 3 .1
± 3 .3
± 3 .1
± 3.2
± 3 .6
± 3.5
0.5
1.6
1.9
2.9
2.4
6.8
9.6
13.9
17.2
± 0 .9
± 0 .9
± 1.4
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 2 .7
± 0 .9
273.05
273.05
273.15
273.15
273.15
273.15
273.15
273.15
273.15
1.931
1.931
1.931
1.931
1.931
1.931
1.931
1.931
1.931
9.16990
11.24208
12.74264
13.62373
13.70164
15.63808
16.17863
17.00553
17.42839
128.2
131.9
134.9
138.0
137.9
143.8
146.4
150.6
151.0
± 1.2
± 1 .2
± 1.2
± 1.3
± 1.1
± 1.1
± 1.2
± 1.6
± 1.5
1.8
4.1
6.5
8.8
7.4
17.9
23.4
31.0
40.4
± 0 .9
± 0 .9
± 1.4
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 2 .7
± 2 .3
273.15
273.15
273.15
273.15
273.15
273.10
273.05
273.00
272.95
4 .0 0 0
4 .0 0 0
4 .0 0 0
4 .0 0 0
4 .0 0 0
3 .9 9 9
3.998
3.997
3.997
9.16867
11.24055
12.74089
13.62185
13.69975
15.63591
16.17641
17.00309
17.42597
261.8
267.8
272.5
276.1
2 7 5 .6
28 2 .6
2 8 3 .6
2 94 .4
2 90 .0
± 3 .3
± 3 .2
± 3 .4
± 3 .2
± 3.3
± 3.6
± 4 .2
± 6 .1
± 4 .7
6.9
14.4
23.5
29.0
30.7
58.7
72.0
90.4
110.6
± 0 .9
± 0 .9
± 1.8
± 0 .9
± 2 .3
± 2 .3
± 2 .3
± 2 .7
± 2 .3
273.15
273.15
273.10
273.05
273.05
273.05
272.95
272.95
272.95
6.001
6.001
5.999
5.998
5.998
5.998
5.996
5.996
5.996
9.16745
11.23907
12.73923
13.62006
13.69797
15.63384
16.17424
17.00088
17.42380
394.7
399.8
4 0 3 .0
407.3
4 0 6 .0
4 1 5 .0
41 7 .5
423 .9
414 .5
± 3 .6
± 3 .5
± 3 .5
± 3 .4
± 3 .4
± 6 .2
± 6 .2
± 7 .7
± 6 .5
15.2
29.2
44.8
60.3
60.4
107.3
126.0
158.5
186.0
± 0 .9
± 0 .9
± 2 .7
± 2 .3
± 2 .3
± 2 .3
± 2 .3
± 2 .7
± 2 .3
R efractivity (v) x 106
____________________
A bsorptivity (a )
dB /km ____
T able 6.8: M easu re d absorptivities and refractivities o f 0 .818% am m onia in helium , a t approxim ately
273 K. T his table continues on the n ex t page.
101
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
M easured Refractivities and Absorptivities
(continued from previous page)
Gas m ixture, by number mixing ratio:
am m onia, 0.00818 ; helium, 0.99182 .
T arget tem p eratu re: 273.15 K
M easured
T enm erature. K
T otal Pressure,
Atm.
Frequency,
GHz
273.15
273.15
273.15
273.15
273.15
273.15
273.15
273.15
273.10
8.013
8.013
8.013
8.013
8.013
8.013
8.013
8.013
8.012
9.16633
11.23767
12.73759
13.61834
13.69623
15.63192
16.17234
16.99887
17.42168
R efractivity (v) x 106
517.1
524.5
531.6
534.2
533.0
537.8
535.4
542.4
536.3
± 3 .7
± 3 .5
± 3 .6
± 3 .4
± 3 .4
± 7 .0
± 6 .8
± 6 .8
± 6 .7
A bsorptivity ( a )
dB /km
26.7
49.6
71.6
93 .6
92.8
154.0
178.1
221.3
243.3
± 0 .9
± 1.4
± 3 .6
± 2 .3
± 2 .3
± 2 .3
± 2 .3
± 2 .7
± 4 .1
102
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities and Absorptivities
Gas m ixture, by number mixing ratio:
am m onia, 0.00819 : helium. 0.99181.
T arget tem perature: 313.15 K
M easured
T em oerature. K
T otal Pressure,
Atm.
Frequency,
G Hz
313.15
313.15
313.15
313.15
313.15
313.15
313.15
313.15
313.15
1.002
1.002
1.002
1.002
1.002
1.002
1.002
1.002
1.002
9.16374
11.23455
12.73414
13.61465
13.69253
15.62770
16.16791
16.99423
17.41689
55.1
56.9
58.4
59.9
59.4
63.1
63.7
65.6
67.4
± 3 .4
± 3 .3
± 3 .2
± 3 .2
± 3 .2
± 3 .2
± 3 .2
± 3 .2
± 3 .2
0.9
1.3
1.8
3.1
2.6
5.9
8.5
10.4
12.8
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 1.8
± 0 .9
313.10
313.05
313.05
313.05
313.05
313.05
313.05
313.05
313.05
1.916
1.916
1.916
1.916
1.916
1.916
1.916
1.916
1.916
9.16330
11.23400
12.73351
13.61397
13.69183
15.62688
16.16705
16.99331
17.41594
103.0
105.8
108.0
110.3
110.1
115.4
117.0
119.8
121.9
± 3 .4
± 3 .3
± 3 .2
± 3 .2
± 3 .2
± 3 .4
± 3 .4
± 3 .4
± 3 .4
2.1
3.2
5.1
7.4
7.2
15.1
19.5
26.6
32.0
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 2 .7
± 1.2
313.15
313.15
313.05
313.10
313.15
313.15
313.15
313.15
313.15
3.999
3.999
3.998
3.999
3.999
3.999
3.999
3.999
3.999
9.16228
11.23273
12.73206
13.61241
13.69027
15.62508
16.16518
16.99133
17.41393
213.7
218.2
222.2
225.0
224.1
230.8
232.5
236.1
237.8
± 3 .4
± 3 .3
± 3 .2
± 3 .2
± 3 .2
± 3 .5
± 3 .5
± 3 .5
± 3 .4
5.8
11.5
18.4
22.4
24.1
45.1
55.8
71.5
86.0
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 1 .2
± 1 .2
± 2 .7
± 1 .2
313.05
313.10
313.15
313.15
313.25
313.25
313.30
313.25
313.10
5.993
5.994
5.995
5.995
5.997
5.997
5.998
5.997
5.994
9.16132
11.23155
12.73071
13.61094
13.68880
15.62342
16.16349
16.98961
17.41219
318.4
323.4
328.0
332.5
331.9
337.2
337.4
337.3
337.7
± 3 .4
± 3 .3
± 3 .4
± 3 .2
± 3 .3
± 3 .7
± 3 .7
± 3 .8
± 3 .7
12.6
21.8
33.9
4 4 .6
46.4
81.1
95.4
120.8
146.2
± 0 .9
± 0 .9
± 2 .8
± 1.2
± 1.2
± 1 .2
± 2 .1
± 2 .7
± 2 .1
R efractivity (v) x 106
____________________
A bsorptivity ( a )
d B /k m
T able 6.9: M easu red absorptivities and refractivities o f 0.819% am m o n ia in helium , a t approxim ately
313 K. T his table continues on the n ex t page.
103
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities and Absorptivities
(continued from previous page)
Gas m ixture, by number mixing ratio:
am m onia, 0.00819 ; helium, 0.99181.
T arget tem p eratu re: 313.15 K
M easured
T em perature. K
T otal Pressure,
Atm.
Frequency,
GHz
313.60
313.65
313.45
313.25
313.15
313.15
313.05
313.00
313.00
8.023
8.024
8.019
8.014
8.011
8.011
8.009
8.008
8.008
9.16037
11.23036
12.72932
13.60948
13.68735
15.62181
16.16183
16.98786
17.41041
R efractivity (v) x 106
422.4
429.3
437.1
440.4
43 7 .6
439.9
44 0 .2
440.8
439.7
± 3 .5
± 3 .3
± 3 .4
± 3 .2
± 3 .3
± 3 .6
± 3 .6
± 3 .6
± 3 .6
A bsorptivity (oc)
d B /km
19.8
37.0
54.6
70.8
70.8
117.1
136.7
169.2
189.2
± 0 .9
± 0 .9
± 2 .8
± 1 .2
± 1 .2
± 2 .1
± 2 .1
± 2 .7
± 2 .1
104
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities and Absorptivities
G as m ixture, by number mixing ratio:
am m onia, 0.0671 : helium. 0.9329 .
T arget tem p eratu re: 313.15 K
M easured
T em perature, K
T o tal Pressure,
Atm.
Frequency,
GHz
313.00
313.05
313.05
313.05
313.05
313.15
313.15
8.213
8.214
8.214
8.214
8.214
8.217
8.217
9.14863
11.21621
13.59186
13.67006
15.60448
16.96840
17.39054
R efractivity (v) x 106
____________________
1706
1691
1735
1701
1548
1586
1581
± 34
±59
±57
±38
±63
±34
±33
A bso rp tiv ity (a )
dB /km
267
447
732
740
1057
1247
1340
± 22
±41
± 55
± 55
±71
±81
±85
T able 6.10: M easu red absorptivities and refractivities o f 6.71% am m o n ia in helium , at approxim ately
313 K .
105
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities and Absorptivities
Gas m ixture, by number mixing ratio:
am m onia, 0.00816 ; helium, 0.09954; hydrogen, 0.8923 .
T arget tem p eratu re: 213.15 K
M easured
T em oerature. K
T otal Pressure,
Atm.
Frequency,
GHz
1 .0 0 1
1 .0 0 0
9.17893
11.25318
12.75520
13.63722
13.71515
15.65359
16.19466
17.02243
17.44571
211.6
213.5
216.6
217.1
218.2
220.5
222.0
225.4
224.6
± 3 .2
± 3 .1
± 3 .3
± 3 .0
± 3 .2
± 3 .5
± 3.5
± 4 .7
± 4 .0
2.3
5.6
9.3
11.3
11.9
22.3
26.1
36.5
4 2 .9
± 0 .9
± 0 .9
± 1.8
± 0 .9
± 0 .9
± 0 .9
± 0 .9
± 2 .7
± 1.4
213.20
213.15
213.15
213.15
213.15
213.15
213.15
213.15
213.15
1.934
1.934
1.934
1.934
1.934
1.934
1.934
1.934
1.934
9.17724
11.25109
12.75280
13.63468
13.71261
15.65065
16.19162
17.01922
17.44246
395.8
399.2
40 5 .0
403.7
403.3
408.5
4 1 0 .0
414.1
411 .0
± 3 .2
± 3 .1
± 4 .8
± 3 .0
± 3 .1
± 7 .4
± 6 .0
± 5 .9
± 5 .7
8.2
17.4
2 5.8
32.7
34.0
57.4
66.5
84.6
98.3
± 1.4
± 0 .9
± 1 .8
± 0 .9
± 1.4
± 1.4
± 1.4
± 2 .7
± 1 .4
213.30
213.25
213.20
213.20
213.25
213.20
213.15
213.15
213.05
4.002
4.001
4.000
4.000
4.001
4.000
3.999
3.999
3.998
9.17348
11.24651
12.74756
13.62917
13.70710
15.64456
16.18530
17.01258
17.43574
805.7
806.6
815.8
807.8
806.2
797.7
800.7
804.7
796.6
± 3 .5
± 3 .2
±11.3
± 3 .1
± 3 .2
± 8 .7
± 6 .0
± 8 .9
± 6 .4
30.0
56.5
81.1
92 .2
95.1
130.8
145.7
168.7
180.0
± 2 .3
± 1.4
± 5 .5
± 2 .3
± 2 .3
± 2 .3
± 2 .3
± 2 .7
± 2 .3
5.995
9.16989
11.24219
12.74264
13.62400
13.70192
15.63870
16.17933
17.00623
17.42935
1197.4
1191.0
1202.3
1187.6
1183.8
1172.7
1170.0
1178.4
1163.6
± 6 .7
± 3 .5
±11.9
± 3 .4
± 3 .4
± 9 .4
± 6 .1
± 8 .9
± 8 .7
56.9
94.1
122.3
135.7
138.8
183.0
198.3
218.1
228.3
± 4 .1
± 1 .4
± 1 0 .0
± 2 .3
± 2 .3
± 2 .3
± 2 .3
± 2 .7
± 2 .3
213.25
213.25
213.25
213.25
213.15
213.10
213.10
213.10
213.10
212.95
213.15
213.15
213.15
213.20
213.20
213.20
213.20
213.15
1 .0 0 1
1 .0 0 1
1 .0 0 1
1 .0 0 0
1 .0 0 0
1 .0 0 0
1 .0 0 0
6 .0 0 0
6 .0 0 0
6 .0 0 0
6 .0 0 2
6 .0 0 2
6 .0 0 2
6 .0 0 2
6 .0 0 0
R efractivity (v) x 10 6
A bso rp tiv ity ( a )
________________
dM m ___
T able 6.11: M easured absorptivities and refractivities o f 0.816% am m onia in a m ix tu re o f 90% h y dro­
gen and 10% helium , at approxim ately 213 K. T his table continues on the n ex t page.
106
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Measured Refractivities and Absorptivities
(continued from previous page)
G as m ixture, by number mixing ratio:
am m onia, 0.00816 ; helium, 0.09954 ; hydrogen, 0.8923 .
T arget tem perature: 213.15 K
M easured
T em perature. K
T otal Pressure,
Atm.
Frequency,
G Hz
213.05
213.05
213.15
213.15
213.15
213.05
213.00
213.05
213.00
8.007
8.007
9.16629
11.23781
12.73766
13.61878
13.69668
15.63282
16.17310
16.99985
17.42276
8 .0 1 1
8 .0 1 1
8 .0 1 1
8.007
8.005
8.007
8.005
R efractivity (v) x 106
1590.6
1581.3
1593.7
1571.4
1566.8
1549.3
1555.7
1554.1
1542.2
± 7 .2
± 3 .5
±11.9
± 3 .4
± 4 .6
±12.7
±11.7
± 8 .9
± 8 .8
A bsorptivity ( a )
dB /km
76.9
129.2
164.9
176.0
180.9
225.5
240.5
254.1
264.2
± 4 .1
± 2 .3
±10.0
± 2 .3
± 5 .0
± 5 .0
± 2 .3
± 2 .7
± 5 .0
107
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
M easured Refractivities and Absorptivities
Gas m ixture, by number mixing ratio:
am m onia,
T arget tem perature:
0 .0 0 8 1 7 ;
helium,
0 .0 9 9 2 1 ;
hydrogen,
0 .8 9 2 6 .
2 7 3 .1 5 K
M easured
Tem oerature. K
T otal Pressure,
Atm.
Frequency,
GHz
273.15
273.15
273.15
273.20
273.20
273.20
273.20
273.20
273.15
0.999
0.999
0.999
0.999
0.999
0.999
0.999
0.999
0.999
9.16962
11.24174
12.74226
13.62333
13.70123
15.63766
16.17821
17.00512
17.42798
154.6
158.1
161.8
163.8
164.4
166.8
168.1
169.5
169.9
± 3 .3
± 3 .2
± 3 .3
± 3 .1
± 3 .1
± 3 .3
± 3 .2
± 3 .3
± 3 .3
± 0 .9
± 0 .9
± 1 .4
± 0 .9
± 0 .9
± 0 .9
± 0 .9
2 2 . 1 ± 1 .8
26.7 ± 1 .4
273.15
273.15
273.25
273.35
273.45
273.45
273.45
273.45
273.45
1.942
1.942
1.942
1.943
1.944
1.944
1.944
1.944
1.944
9.16831
11.24015
12.74045
13.62140
13.69929
15.63544
16.17591
17.00271
17.42552
296.8
299.9
303.8
305.4
305.9
308.8
310.3
311.6
311.3
± 3 .3
± 3.2
± 3 .3
± 3 .1
± 3 .1
± 3 .3
± 3 .3
± 3 .3
± 3 .5
4 .6
10.1
16.6
21.7
20.8
37.3
4 3 .6
54.5
65.4
± 0 .9
± 0 .9
± 1 .8
± 0 .9
± 0 .9
± 1 .4
± 1 .4
± 1.8
± 1 .4
273.15
273.15
273.15
273.05
273.05
273.10
273.10
273.10
273.10
3.984
3.984
3.984
3.982
3.982
3.983
3.983
3.983
3.983
9.16549
11.23670
12.73657
13.61728
13.69516
15.63080
16.17114
16.99772
17.42045
604.3
606.9
608.4
608.4
607.8
605.8
605.3
605.4
602.2
± 3 .4
± 3 .2
± 3 .4
± 3 .3
± 3 .3
± 3 .4
± 3 .7
± 3 .8
± 4 .2
18.3
34.0
47 .9
58.3
58 .2
89.5
103.1
120.0
129.2
± 0 .9
± 0 .9
± 1.8
± 1.4
± 1.4
± 2 .3
± 1 .4
± 2 .7
± 2 .3
273.15
273.15
273.15
273.05
273.05
273.00
273.00
273.00
272.95
5.996
5.996
5.996
5.994
5.994
5.993
5.993
5.993
5.992
9.16275
11.23334
12.73280
13.61325
13.69112
15.62626
16.16646
16.99284
17.41547
904.2
905.3
904.8
904.3
902.8
896.6
895.1
892.7
888.3
± 3 .5
± 3 .4
± 3 .7
± 3 .3
± 3 .3
± 3.7
± 3 .7
± 4 .3
± 4 .2
35.3
59.2
7 8.4
94.4
94.2
129.1
143.4
159.4
169.7
± 0 .9
± 1.2
± 2 .7
± 1.4
± 1.4
± 1 .4
± 1.4
± 1.8
± 1 .4
R efractivity (v) x 10^
A b so ip tiv ity (a )
dB /km
1.7
3.5
6.0
6.9
7.1
14.4
16.9
T able 6.12: M easured absorptivities an d refractiv ities o f 0.817% am m onia in a m ixture o f 90% hydro­
g en and 10% heliu m , at approxim ately 273 K . T his table continues on th e n e x t p age.
108
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Measured Refractivities and Absorptivities
(continued from previous page)
Gas m ixture, by number mixing ratio:
am m onia, 0.00817 ; helium, 0.09921; hydrogen, 0.8926.
T arget tem perature: 273.15 K
M easured
Tem perature. K
T otal P ressure,
Atm.
Frequency,
GHz
273.00
273.15
273.10
273.05
273.00
273.05
273.05
273.05
273.05
8 .0 1 0
9 .1 5 9 9 6
11.22996
12.72901
13.60921
13.68708
15.62171
16.16176
16.98789
17.41042
8.014
8.013
8 .0 1 2
8 .0 1 0
8 .0 1 2
8 .0 1 2
8 .0 1 2
8 .0 1 2
R efractiv ity (v) x 10 6
1208.5
1206.6
1202.0
1202.1
1198.2
1188.1
1186.1
1184.4
1178.6
+ 4 .4
± 4 .0
± 4 .2
± 3 .9
± 3 .9
± 4 .4
± 4 .3
± 4 .3
± 4 .2
A bsorptivity (a )
d B /km
51.7
83.4
107.0
123.2
121.1
162.3
173.9
192.6
200.1
± 1.2
± 1.2
± 5.5
± 2 .3
± 2 .3
± 1.4
± 1.4
± 1.8
± 1 .4
109
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Chapter 7
Optimization o f a Parameterized Ben-Reuven Formalism
to Fit the Ammonia Data
If the d ata presen ted in C hapter
6
are to be useful to researchers studying the g ia n t planets, it is vital
that there be an accurate m eans o f interpolation and extrapolation to g as conditions and frequencies other
than those u sed in th e p ro g ram o f experim ents. T h is ch ap te r describ es the pro ced u re b y w hich such a
m eans w as devised. T h e m odified B en-R euven fo rm alism p u b lish ed by B erge an d G ulkis (1976) for
predicting the absorptivity o f gas m ixtures consisting o f hydrogen, helium , and am m onia, is detailed in
Section 7.1. Section 7.1 then describes the substitution o f free param eters fo r constants in that form al­
ism so an o p tim ization routine can adjust the form alism to fit th e available data. Section 7.3 covers the
optim izatio n p ro ced u re an d S ectio n 7.4 p resen ts the resu ltin g new form alism , w hich is show n to be
considerably m ore accu rate at in term ediate pressures than that o f B erge and G ulkis. Its ap plicability is
dem onstrated o v er a frequency ran g e from 2 G Hz (and possibly m uch low er) to at least 40 G H z.
7.1
The Ben-Reuven Formalism as M odified by Berge and Gulkis
T h e original B en-R euven form alism fo r calculating the m icrow ave absorptivity o f a gas is the result
o f a quantu m m echanical treatm en t o f the problem no t sp ecifically aim ed at absoiption by am m onia
(B en-R euven, 1966). A lthough its application to o th er absorbing gas species has been considered less
than successfu l (Poynter, 1987; B im b au m , 1987), w hen applied to the theoretically troublesom e cases
o f gaseous am m onia and gas m ixtures containing am m o n ia it provided better agreem ent w ith laboratory
data than previous theoretical approaches. T he general fo rm o f the B en-R euven form alism is identical to
that o f the old er V an V leck-W eisskopf form alism (V an V leck and W eisskopf, 1945):
110
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“ ( U = X X A ( J ,K ,m ) F (J ,K , m . f j
J
,
(7.1)
K
w here a (f0 ) is absorptivity a t observed frequency f0 , J and K are th e m olecular rotational q uantum n u m ­
b ers, and m is a v ecto r o f th e p ertin en t m acroscopic co nditions o f th e gas, such as tem perature an d
p artial pressures o f th e gas species present. E ach com bination o f J and K values corresponds to a unique
a b sorption line arisin g from inversion o f am m onia m olecules in the (J , K) rotational state. T h e fu n c ­
tion A (J, K, m ) g iv es the in ten sity o f the (J, K) line u n d er conditions sp ecified by m . T he function
F (J, K, m , f), called the line shape fa cto r, determ ines the frequency dependence o f absorptivity due to the
(J, K) line un d er conditions m , and is evaluated at the o bserved frequency f0 . T he absorptivity at f 0 d u e
to the (J , K) lin e is the p ro d u ct o f A (J, K, m ) and F (J , K, m , f0). T otal absorptivity is then given by
the sum m ation o v e r all individual absorption lines (i.e., all possible com binations o f J and K values) o f
the contribution o f each line a t frequency f0 .
T he differences betw een the V an V leck-W eisskopf an d B en-R euven form alism s lie in the line shape
factors F (J, K, m , f); intensity factors A (J, K, m ) o f the tw o form alism s are identical. T he V an V leckW eissk o p f line sh ap e facto r is the sum o f tw o sim pler lin e shapes a t p ositive and n egative values o f th e
line center frequency (T ow nes and Schaw low , 1955). It has a sin g le internal param eter, the line w idth,
w hich is a fu n ctio n o f the m acroscopic conditions o f th e gas. B en -R eu v en ’s lin e shape facto r is m ore
com plex (com pare w ith E quation 3.9, Section 3.4), and has two additional internal param eters:
'U J 'K),
{(' - | U J .K) + 8 f - YW } + 4 , V
w here fc (J, K) is the c e n te r frequency o f the (J, K) line, y is the line w id th param eter corresponding to
the line w idth in th e V an V leck -W eissk o p f line shape factor, ^ is a line-to-line coupling param eter, an d
8
is a frequency sh ift param eter. A ll three param eters are functions o f the m acroscopic conditions o f the
gas. T h e line intensity factor A (J, K, m ) is given by:
87t 2ni
A (J, K, m ) =
2
| M-ij|
.
(7.3)
111
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w here T is absolute tem perature; p.jj is the dipole m om ent m atrix elem ent fo r the transition from state i
to state j w h ich gives rise to the (J, K) line; and n (- is the concentration o f ab so rb in g m olecules in the i
(low est energy) state. W hen this expression is ev alu ated in term s o f J , K, the specific ch aracteristics o f
the am m onia m olecule, an d the m acroscopic conditions o f the gas, it becom es (B erge and G uilds, 1976):
A(J, K, m) = 1 2 3 0
( 2 J + 1) K 2 fc(J,K )
j(j+i)
y
(7.4)
w h ere S(K ) is a degeneracy facto r equal to 1.5 unless K is an integral m ultiple o f 3, w hen S (K ) is 3;
P n h 3 is the partial pressure o f am m onia in atm ospheres; and f(J, K, T) is given by:
(7.5)
T h e param eter y in the expression fo r A (J, K, m ) is the sam e line w idth p aram eter y that appears in the
expression fo r F (J, K, m , f).
U sing the ex perim ental results o f Fren kel el al. (1966) an d assistance fro m A. A . M aryott, W rix o n
e t al. (1971) pro d u ced a version o f the B en-R euven form alism in w hich theoretically d erived expressions
fo r the internal param eters resulting fro m the o riginal qu an tu m m echanical analysis w ere rep laced w ith
em pirical relations. It was recognized th at the B en-R euven approach w as m uch m ore accurate at hig h er
p ressu res than the o ld e r V an V leck -W eissk o p f th eo ry . M o rris and Parsons (1970) m ade lab o rato ry
m easurem en ts o f the absorptivity o f tw o-gas m ixtures in w hich am m onia w as a relativ ely m in o r c o n ­
stitu en t. T he broadening gases in clu d ed hydrogen an d helium , appropriate to the study o f the a tm o s­
p h eres o f the g ia n t planets. T hese m easu rem en ts w ere m ad e o v er a large ran g e o f p ressu res, fro m
a p proxim ately 10 to 700 b ars, w h ich is ap p licab le to regions far below the tro p o p au ses o f th e g ian t
p lanets. A t p ressu res above 100 bars th ere w ere surprising system atic deviations o f the d a ta from the
predictions o f th e B en-R euven form alism . B erge and G ulkis (1976) published a m o d ified B en -R euven
form alism w hich fit the M orris and P arsons data, and presen ted new em pirical relatio n s fo r th e internal
param eters. T he m odified form alism has a slightly d ifferent form:
112
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a (fd =
C X l A ( J , K , m ) F (J, K, m , f 0) .
J
(7.6)
K
T he correctio n factor C , w h ich forces a fit to the M orris an d P arsons d ata, is e m p iric a l and is the
m ajor contribution o f the B erge an d G ulkis m odification:
C = 1. 0075 + 0.0308 + 0.0552 ■
i
T
(7.7)
w here T is tem perature in K elvins and PH 2 is the partial pressure o f m o lecu lar h y d ro g en in atm ospheres.
U nfortunately the M orris and Parsons d ata are all at "room tem perature," so the tem p eratu re dependence
used in the expression fo r C can n o t be verified by that data.
T he new em pirical relations B erge and G ulkis p u b lish ed fo r th e internal p aram eters y, £, and
8
are
based on results o f the original q u an tu m m echanical treatm ent o f the problem : total lin e w id th y, c o u p ­
ling £, and frequency shift 5 sh o u ld be lin ear in the partial pressures o f th e g ases p re se n t in a m ixture.
T he form o f the expression fo r y is:
y( J, K, m) = £ G, y ((J,K,T,P;) ,
(7.8)
w here subscrip t i specifies a gas species in the m ixture and P /i s the partial p ressure o f th at species. In
this expression the total line w idth o f an am m onia absorption line is g iv en by the su m o f contributions
o f all gas species i to th e bro ad en in g o f th e line. T he contribution by a single gas sp ecies is th e product
o f the tem perature-pressure su b function y/ (J, K, T, P /) fo r that species and a sc alin g co efficien t G /.
T he role o f the subfunction is to describe any tem perature o r pressure dependence. S in ce theory indicates
y should be lin ear in the partial pressu res, all y/ subfunctions in the B erge and G u lk is fo rm alism have
P / as a sim ple lin ear m u ltip lier;
the tem p eratu re depen d en ces are so m ew h at m o re co m p lex .
The
am m onia self-broadening term has another factor p resen t in the y; subfunction: th e self-b ro ad en ed line
w idth p aram ete r y p (J, K), w h ich is a strong function o f the rotational state. S in c e all tem perature,
pressure, and rotational state d ependences should be covered by the y/ subfunctions, th e coefficients G ;
sh o u ld be constants, and this is th e case in the B erge and G ulkis form alism . T h e ir ex p ressio n fo r y
113
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inclu d ed o n ly the th ree gas sp ecies w h ich have sig n ifica n t co n trib u tio n s to th e total b roadening o f
am m onia lines in a Jovian atm osphere: m o lecu lar h ydrogen, ato m ic h elium , and am m onia. T hus, in
E quation 7 .8 , the su b scrip t / can b e H 2 , H e , o r NH 3 . T he fu ll ex p ressio n is:
Y (J , K, m ) = G h 2( ^ ) 3 P h 2+ G HaP f )
3
p He + G Nh 3( ^ ) Y 0 (J, K) P NHa G Hz,
(7.9)
w here T is tem perature in K elvins, an d the partial pressures are in atm ospheres. B erge and G ulkis' fit to
the M orris an d Parsons d ata yielded G h 2= 2.318, G|_)e = 0.79, and GNH 3 = 0.75.
T he fo rm o f th e expression for the coupling p aram eter C, is identical to th at fo r y :
C (J, K, m ) = X Z (- £,-(J, K ,T , P j ) ,
(7.10)
/
and the
su b function fo r a given species is identical to the y,- sub fu n ctio n fo r th at species. T hus the
only differences in the expressions fo r y and C are the coefficients G ; and Z /:
C ( J , K ,m ) = Z H2( ^ )
3
P H 2 + Z He ( ^ )
3
P H e + Z NH3( ^ ) y
0
( J . K ) P NH3 G Hz.
(7.11)
B erge and G ulkis' fit to th e data y ielded Z h 2 = 1-92, Z ^ g = 0.3, an d ZNH 3 = 0-49.
In a Jovian gas m ixture the frequency sh ift param eter
8
does n o t depend on tem perature o r the partial
pressures o f th e no n -p o lar species, an d thus is a function on ly o f th e am m onia partial pressure:
5 = D P Nh3 GHz,
(7.12)
w ith P NH3 again in atm ospheres. B erge and G ulkis' fit y ielded a value o f -0.45 for D.
W hen co n sid erin g th e B erge and G ulkis form alism in th e co n tex t o f a Jo v ian atm osphere at total
pressures less than ten atm ospheres it is seen that som e term s in the fo rm alism are m ore critical than
others to
the u ltim ate accuracy
o f the calculations. T he expression fo r correction factor C , E quation 7.7,
involves
the hyd ro g en p artial p ressu re d iv id ed by the tem p eratu re. A t the o n e b ar lev el in Ju p iter's
114
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atm osphere, w h ere the hydrogen partial pressure w ould be about 0 . 8 8 atm ospheres, tem peratures are near
165 K (L indal e t al., 1981). The pressure-dependent term o f the correction factor is thus far sm aller than
the (nearly u nity) co n stan t term . F o r total pressures less th an about a h u n d red bars both the p ressured ep en d en t term and the deviation o f th e co n stan t term fro m unity are far sm aller than the accuracies o f
o u r m easu rem en ts o f absorptivities in ou ter p la n e t atm ospheres o r the accuracies o f any lab o rato ry
m easurem ents to date. T hus in this c o n tex t it is appropriate to use a sim ple constant, unity, fo r C .
In th e sam e context it is fo u n d th a t som e term s in expressions fo r y an d
E quations 7 .9 and 7.11,
are m u ch m ore critical than others. B ecau se th e abundances o f am m onia in the atm ospheres o f Ju p iter
and S aturn are so sm all, up to about 25 0 ppm a t Ju p iter (L indal et al., 1981) a n d less at S atu rn (L indal
e t al., 1985), the contributions o f the am m onia self-action term s in y and C, are far less than one p ercen t
o f the totals, so those term s are alm o st negligible. M o d erately large errors in the values o f the G n h 3
and ZfvjHa coefficients w ill have little effec t on the accu racy o f calcu latio n s o n Jo v ian gas m ixtures.
S ince helium abundances o f all ou ter p lan et atm ospheres are m uch higher than those o f am m onia, ran g ­
ing fro m 5 o r
6
p ercen t by n u m b er at S aturn (L in d al et a l., 1985) up to ab o u t 15 p ercen t a t U ranus
(C onrath e t al., 1987), the contributions o f the helium term s w ould be expected to be m uch larger than
the am m on ia self-broadening term s. T h is is partially true, b u t the heliu m term s are sm aller than their
relativ e abundances w o u ld suggest. F o r instance, the relativ e abundance o f h elium in Ju p iter's tro p o s­
phere is about 11 percent by nu m b er (L indal et al., 1981), b u t the helium broadening term in the ex p res­
sion fo r y is o nly 3.5 p ercen t o f the total broadening. H eliu m is less efficien t as a line b roadening gas
than h y dro g en . G iv en equal partial p ressures o f m olecular hydrogen and helium , th e broadening due to
the hydrogen w ill be approxim ately three tim es as larg e as th e broadening due to helium (T ow nes and
Schaw low , 1955; B erge and G ulkis, 1976). As w ith the coefficients for th e am m onia term s, there w ill
be fairly w ide tolerances fo r erro r in the values o f the h eliu m term co efficients. The hyd ro g en terms
dom in ate the ex p ressio n s for y and
constitu tin g m ore than 95 p ercent o f the totals. T olerances fo r
error in the hydrogen term coefficients w ill closely parallel the overall tolerance for error in the results o f
calculations using the form alism , and thus it is critical to accurately determ ine those coefficients.
T he freq u en cy sh ift p aram eter
8
is sig n ifican t only w hen the partial p ressure o f am m onia is about
115
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0.5 atm osphere o r m ore. In a Jo v ian atm ospheric context, S is negligible w hen the total p ressure is less
than a few hundred bars. Indeed, in th eir application o f the B en-R euven form alism to radio astronom ical
studies o f Jupiter's atm osphere, W rixon e t al. (1971) om itted
8
entirely.
T he only qu an tu m m ech an ical req u irem ents discu ssed fo r J and K are th at th ey m u st be integers
greater than zero, so it m ig h t seem th a t a fully accurate com putation using this fo rm alism w ou ld require
com puting the co ntributions o f an in fin ite n u m b er o f individual lines, thus requiring infinite tim e. B ut
there are physical lim its on the size o f J (and thus K): if J is too large, the an g u lar m om entum is large
enough that centrifugal forces rip th e m olecule apart. M ore co nvenient from the standpoint o f reducing
the com putational w orkload are the relativ e p o p ulations o f am m onia m olecules in the h ig h er rotational
states. A t the tem peratures en co u n tered in p lanetary atm ospheres the rotatio n al states w h ich are m o st
densely populated have sm all J values, n e a r J= 3 (T ow nes and Schaw low , 1955). W ith the increasing J
values associated w ith hig h er rotational energies, populations o f the states d ecrease rapidly, and thus the
observed intensities o f the lines due to m olecules in those states also decrease rapidly. P redictions o f the
form alism 's intensity factor, in general agreem ent w ith the laboratory data o f P o y n ter and K ak ar(1 9 7 5 ),
indicate that all lines w ith J values g re a te r than 16 have such low intensities th a t n o noticeab le d eteri­
oration o f accuracy results i f they are ignored. In a p ractical com putation o f absorp tiv ity using a V an
V leck-W eissk o p f o r B en -R euven alg o rith m it is su fficien t to sum o v er J values fro m 1 to 16 an d K
values from 1 to J. This is no t a particu larly dem anding task fo r a co m p u ter o f even m odest speed.
7.2 Parameterization of the Ben-Reuven Formalism of Berge and
Gulkis
To optim ize the fit o f an equation to a data set it is necessary to h ave adjustable p aram eters, know n
as fr e e para m eters, w ithin th e equation. I f there is an analytic solution fo r the values o f the free p a ra m ­
eters, they are calculated directly from th e data. If there is no know n analytic solution, then it is th e jo b
o f an optim ization procedure to vary the values o f the free param eters, assigning q u ality -o f-fit values to
each com bination o f param eter values according to som e quantitative m easure, and find th at com bination
116
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o f p aram eter values w h ich yields the h ig h est quality o f fit. A sim p le exam ple is a fit o f the C artesian
equation o f a straig h t lin e, y = ax + b , to a set o f points (X, y ), w ith th e q u an titativ e m easure o f fit
being the square o f the y deviation o f a datum , and the ov erall q u ality -o f-fit in d icato r being the su m o f
these squ ares o v e r all d a ta points. T h e free p aram eters in this eq u atio n , a an d b, m ay be adju sted to
control th e slope and positio n o f the lin e it represents. In this c ase there is an analytic solution fo r the
o p tim al p a ra m e te r v a lu e s, b u t it is also p o ssib le to ch eck an o p tim izatio n ro u tin e by g iv in g it a
p ro b lem such as this an d com paring th e re su lt to the analytic solu tio n . It is im p o rtan t in an o p tim i­
zation task that the free param eters be independent; it m ust no t b e possible to o ffset exactly the effect o f
varying o n e p aram eter b y varying o th er param eters. If the p aram eters are not independent there w ill be
an infinite nu m b er o f d ifferent (but related) optim al solutions. In the exam ple above the param eters are
indeed independent. H ow ever, i f the fit is attem pted w ith an equation containing three free param eters,
such as y = a(bx + c), i t is quickly fo u n d that the param eters are n o t in d ependent and one of them (a in
this case) m ust b e elim in ated in o rder to find a unique optim al solution.
C hoices o f param eters to use in an optim ization o f th e B en-R euven form alism o f B erge and G ulkis
to a set o f laboratory d ata depend on the m acroscopic conditions u n d er w hich th e d ata w ere taken. If the
d ata are taken at very lo w pressures w here the total am o u n t o f line broadening is quite sm all it is p o ssi­
ble to reso lv e individual absorption lines, and thus use o f p aram eters w hich w ould vary from one lin e to
another m ay be possible. A t hig h er pressures w here line broadening is larger than the spacing betw een
lines the in dividual ch aracters o f the lines are lost. T he p aram eters in these cases m u st be sen sitiv e to
the shape o f the co n tin u u m absorption spectrum and no t depend on resolving individual lines. T he data
used in this pro ject are all o f the higher-pressure class, and this restricted the param eter choices available.
S ince the essen c e o f this w ork is to generate a form alism w hich accu rately predicts absorptivity due to
am m onia g iv en the re le v an t m acroscopic conditions, the p aram eters u sed in this m odification o f the
B en-R euven form alism m u st reflect the connection betw een m acroscopic conditions and absorptivity.
T he first step in param eterization o f the B en-R euven form alism w as to assum e th e line shape facto r
F (J, K, m , f) of E quation 7.1 is correct. T his equation is the result o f B en-R euven's original w ork, used
unchanged b y B erge and G ulkis. E rrors in predicting the frequency dependence o f am m onia's absorption
117
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sp ectru m w ere assum ed due to errors in determ in in g the co rrect values fo r the elem ents o f E quation 7.1,
n o t in the eq uation itself. In ex am in in g E q u atio n 7.1 it is seen th a t it alread y contains three internal
param eters w hich reflect the m acroscopic co n d itio n s:
7,
£, an d
8
. It w o u ld n o t be w ise to u se
7
an d £
as free param eters, how ever, sin ce it is already know n they are n o t sim ple valu es but are sum s o f terms
involving independent m acroscopic conditions. It w ould be technically p o ssib le to u se
7
an d £ as free
param eters, but further analysis w ould then b e necessary to isolate the effects o f an individual broadening
gas on the param eters o bserved in this o v ersim plified approach. P aram eterization necessarily involves
and £, but it m u st be at a m ore fundam ental lev el. D esp ite its sim plicity c o m p ared to
particu lar case there are also problem s w ith using
8
7
an d
7
in this
, the freq u en cy -sh ift param eter, as a free param eter.
F o r the d a ta u sed here the shifts in line cen ter freq u en cies are sm all en o u g h that th e ir e ffects on the
am m onia absorption spectrum are sm aller than th e accuracy o f the data, so no useful in form ation about
8
m a y be o b tain ed from them . T he em pirical equation fo r
8
p u b lish ed by B erge and G ulkis w ill be used
here, w ith no attem pt at m odification.
T he em p irical equations fo r
7
and t, p u b lish e d by B erge and G ulkis o ffe r o b vious ch o ices o f free
p aram eters. E ach term in the tw o eq u atio n s co n sists o f a su b fun ctio n w h o se ro le is to d escrib e the
d ep en d en ce o f th a t term on m acro sco p ic c o n d itio n s (also ro tatio n state, in the am m o n ia self-actio n
term s), and a scaling coefficient. If the subfunctions are indeed co rrect and the coefficients are m ade free
p aram eters, then fitting o f the lin e sh ap e fa c to r to d a ta taken u n d er various c o n d itio n s sh o u ld y ield
co efficient values that are co n stan t to w ithin the accu racy o f th e data. If the su bfunctions are incorrect,
fitting the lin e shape to those sam e d ata should y ield co efficien t values th at vary sy stem atically w ith the
conditions. B y adopting this approach six free param eters h ave b een introduced. In E quations 7.9 and
7.11 they are: G h 2, G|_|e , G n H 3 » Z h 2, Z\-\e> ^
^N H • C aution m ust be ex ercised in su b seq u en t steps
to a v o id introducing too m any param eters in to the form alism . A lthough it has sufficient degrees o f free­
d om that independence o f the free param eters is n o t in jeo p ard y , the finite size o f the d ata set precludes
the u se o f an arbitrarily large n u m b er o f param eters.
S ince th e num ber o f free param eters is o f concern, it is fortunate that the intensity facto r A (J, K, m )
has a sim p ler role than that o f th e line shape factor. W here F (J , K, m , f) m u st d escrib e the freq u en cy
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dependence o f a line's absorption spectrum , a role w hich involves m any degrees o f freedom in the result,
th e intensity fa c to r c a n b e red u c ed to a single n u m b er w hich g ives th e line's strength. A lthough the
equation used by B erg e and G u lk is fo r A (J, K, m ) , E quation 7.2 , is rich in constants and coefficients
th at co u ld be ch an g ed into free param eters, the use o f all the possib ilities w ould overtax the d ata set and
w o u ld m o st lik ely p ro d u ce m ean in g less resu lts.
T he lin e w id th p a ra m e te r y in th e d en o m in ato r o f
E quation 7.2 has already been p aram eterized due to its treatm ent as p a rt o f th e line shape factor. It was
d ecid ed to use a sin g le additional param eter, a leading coefficien t for th e en tire expression, and assum e
th a t the dependences on the m acro sco p ic co nditions and ro tatio n states w ere as given in E quation 7.2.
S ince individual abso rp tio n lines can n n o t be reso lv ed by the data, the v alue o f that param eter w ill no t
v ary w ith J or K. It m ay be tak en o u tsid e the sum m ations on J and K in E q u atio n 7.4 as a general
scaling param eter. If assum ing the correctness o f th e condition dependences expressed in E quation 7.2 is
n o t ju stifie d , th e values o f this scalin g param eter obtained by fitting the p aram eterized form alism to the
d ata w ill show sy stem atic v ariatio n s w ith the changing co n d itio n s. It has b een show n in Section 7.1
th at u n d er the ran g e o f conditions seen in the d ata the B erge an d G ulkis correction facto r C , E quation
7.7, can be treated sim ply as unity, effectively disappearing. S ince a general scaling param eter is needed
anyw ay, it was co n v en ien t to le t C assum e that role.
T he presence o f a scaling p a ra m e te r is im portant to insure th at the p aram eters w ithin the line shape
facto r are free to adjust the frequency dependence o f the absorption spectrum independent o f its general
m agnitude. T his prevents the o ptim ization routine from attem pting to fit the integrated m agnitude o f an
experim entally m easu red absorption spectrum by adjusting the p aram eters w ithin the line shape factor.
T his is indeed p o ssib le w ith th e B en -R euven line shape, especially if th e data cover a relatively sm all
range o f frequencies, so decoupling m agnitude from frequency dependence is vital.
T he general form o f the original B en-R euven form alism is show n in E quation 7.1. In this form al­
ism as applied to am m onia absorptivity, the m acroscopic condition v ecto r m w o u ld co n sist o f the tem ­
p erature and the partial pressures o f all gases p resen t in the m ixture in question, including am m onia. If
the entire right h an d side o f E quation 7.1 w ere to b e w ritten as a single fun ctio n called O B R follow ed
by its variable list, th a t function w o u ld be w ritten O B R (f0 , T , P n h 3. P a , P b . Pc> •••)• T he ex p licit
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variables o f the function are the frequency o f o b serv atio n an d the m acro sco p ic conditions. I f the sam e
treatm ent is applied to th e m odified B en-R euven form alism o f B erge an d G ulkis, E quation 7.6, the v aria­
ble list contains on ly the three p artial pressures sig n ifican t in a Jo v ian atm osphere. N am ing the single
function B G B R , it w ould b e w ritten B G B R (f0 , T, PNH 3 . Phfe- ^H e)- N o te that in both the original and
m odified v ersions o f the form alism the rotational states and th e specific characters o f am m onia and the
o th e r gases th at a ffect it are internal to the fu n ctio n .
A co m p u ter su b ro u tin e d esig n ed to p erfo rm
calcu latio n s o f am m onia absorptivity using the m o d ified fo rm alism w o u ld n eed to receiv e only five
num bers fro m the calling routine, and all rotational state an d gas species ch aracteristic considerations
w o u ld be com pletely transparent to the routine calling this subroutine.
In the process o f param eterizing the m odified B en-R euven form alism o f Berge and G ulkis seven free
param eters have been introduced, and these may be considered additional variables. Six o f these variables
appear in th e lin e shape factor, th ree appear in the intensity factor, an d one is a scaling co efficien t o v er
the entire equation. W ritten in the form o f E quation 7.6 this w ould ap p ear as:
^ » Gnh3i Ghj, Ghb) F(J»
K,
m, fo>
GfsjH3>Gh^ G ^g,
Z^h3, Z ^ ^
Z ^ g ) . (7.13)
T he sem icolons in the v ariab le lists separate tw o d istin ct types o f variables: those before the sem icolon
rep resen t conditions that affect absorption by am m onia lines, an d those a fte r the sem icolons are p aram ­
eters representing k n ow ledge about the behavior o f am m onia and its interactions w ith the other gases in
a Jovian m ixture. A representation o f the param eterized form alism as a single function P B G B R co u ld
be w ritten in the form:
a ( f 0) = P B G B R (f 0 , T, P nh3. P h2. P hbJ C , G nh3, G h2, G hb, Z nh3, Z h2, Z H8) •
(7.14)
A ro u tin e w hich calls a subroutine im plem entation o f P B G B R m u st su p p ly all 12 o f th ese n um bers
n eeded to p erform a calculation. N o te that the rotational q u an tu m nu m b ers J an d K are ab sen t fro m th e
variab le list. T hey are com pletely internal to P B G B R . T he variables p reced in g the sem icolon are the
m easured conditions that affect m icrow ave absorption by am m onia, and those after it are the form alism 's
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param eters. It is th e jo b o f the d ata to p rovide the values for the m easured conditions. A com plete set
o f m easured values for these conditions, coupled w ith the m easured absorptivity o f the gas m ixture under
those conditions, co nstitutes o n e datum . It is the jo b o f the o p tim izatio n ro u tin e to su p p ly values for
th e param eters, ju d g e their consistency w ith th e m easu red data, an d ev entually arrive at values fo r the
param eters th at reflect know ledge about am m o n ia and its interactions w ith the oth er gas species.
In sum m ary, the steps n ecessary to calcu late a p redicted ab sorptivity using this param eterized v er­
sion o f B erge and G ulkis' m odified B en-R euven form alism are:
1. D eterm ine the frequency f and m acroscopic conditions (tem perature and partial pressures) fo r the
calculation. In an optim ization task this inform ation is pro v id ed by the data.
. D eterm in e values fo r the param eters C , G h 2>G|-le> GNH 3 . Z h 2. Z He> and Z n h 3-In th e optim i­
zation task these values are p ro v id ed by the optim ization routine.
2
3. D eterm ine values fo r the rotatio n al q u an tu m num bers J and K. In a softw are im plem entation o f
the fo rm alism this is d o n e w ith n ested loops w hich sta rt w ith J = K = 1 and iterate through the
necessary values. T he ou ter lo o p iterates J fro m 1 to 16; th e in n er loop iterates K fro m 1 to J.
4 . O btain the values fo r fc (J, K) and Yq (J, K). P o y n ter and K akar (1975) have m e asu red m ost o f
the necessary values and provide m ethods o f calculating those not m easured.
5. U se th e valu es fro m steps 1 th ro u g h 4 to c a lc u late y, C, an d
7.12, respectively.
6
8
fro m E q u atio n s 7.9, 7.1 1 , and
. C alculate the values o f F (J , K, m , f) an d A (J, K, m ) from E quations 7.2 and 7.4, respectively.
7. M u ltip ly the p ro d u c t o f the valu es o f F a n d A fro m step 6 by th e v alue o f p aram e te r C. T he
resu lt is the p redicted absorptivity co n tribution by the (J , K) line at frequency f.
8
. R ep eat steps 3 through 7 for all values o f J and K specified in step 3.
9.
Sum the contributions o f all th e in dividual lines. T he resu lt is the p redicted total absorptivity at
frequency f d u e to the am m onia in the gas m ixture d escrib ed in step 1 , based on the param eter
values o f step 2 .
K now ledge o f the pro p er values for param eters C , G h 2, G |-ie’ GNH 3 . Z h 2, Z|_)e , and Z n h 3 is th e key to
obtaining accurate predictions w ith this m ethod. T his is the goal o f the n ex t tw o sections.
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7.3
Procedures for Optimization o f the Parameterized Formalism
In the previous section a param eterized version o f the m odified B en-R euven form alism o f Berge and
G ulkis w as d eveloped in o rd er to obtain a new version o f th e form alism o ptim ized to fit an expan d ed set
o f laboratory data. T he param eterized fo rm alism contains seven free param eters w hich an optim ization
routine m ay ad ju st to obtain th e best fit. T h is section discusses the steps in th at optim ization process.
T he data to be used in this p rocedure com e prim arily from tw o sources, an d fro m an im plicit third
source. T h e first source is th e set o f h ig h -quality lab oratory m easurem ents o f the abso rp tiv ity o f pure
am m onia gas m ade by B leaney and L oubser (1950), described in C h ap ter 6 . U nfortunately all their m ea­
surem ents w ere m ade at "room tem perature" (about 297 K), so no tem perature dependence inform ation is
available from them . T he m ajo r prim ary source is the set o f laboratory m easurem ents p erform ed as p art
o f this w ork, describ ed in C hapters 5 and
6
. T h e im p licit source is the set o f high p ressu re data taken
by M orris an d P arsons an d u sed by B erge and G ulkis in their analysis. T hese d ata a re not used directly
bu t instead estab lish values o f the optim izatio n p aram eters fo r very high pressu res, and this analysis
m ust accom odate those lim its.
T he design o f the d ata set reduces the com plexity o f the optim izatio n process by b reaking the w hole
problem into sm aller separable problem s. T his optim izatio n p roblem can be ch aracterized as a sevenparam eter fit o f a rather in v o lv ed equation to a lim ited set o f data. A seven-param eter fit is n o t an in h er­
ently intractable p ro b lem bu t is usually difficult, requiring considerable co m p u ter tim e; also, w ith only
a few hundred d ata points, there are n o t m any eq u iv alen t points covering each degree o f freedom . Since
the internal fo rm u lae in the p aram eterized fo rm alism do n o t involve cross products o f partial pressures,
effects o f term s stem m ing fro m a specific gas species m ay be easily isolated. W h en dealing w ith the
B leaney and L o u b ser d ata on p u re am m onia the p artial pressures o f hy d ro g en and helium are zero, so all
term s involv in g those partial p ressures are zero. T h is elim inates fro m consideration th e free param eters
G h 2.
Z H2> ar|d z He o f E quations 7.9 an d 7.11, leaving only th ree free p aram eters: G n h 3, z NH3>
and the g lo b a l m ag n itu d e p a ra m e te r C o f E q u atio n 7.6. S u b seq u en t w o rk w ith d a ta on m ix tu res o f
am m onia in fo reig n broad en in g gases w ill use th e values fo r G n h 3 an d Z n h 3 o b tain ed from the pure
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am m onia data, so th o se w ill no lo n g er b e free p aram eters. W h en using d ata on am m o n ia in hydrogen
from this w ork the partial pressure o f heliu m is zero, elim inating p aram eters G|_|e and Z|_|e . G N H 3 and
Z Nh 3 have been determ ined, leaving only GH 2 . Z h 2 >and C as free p aram eters. Sim ilarly, using data on
am m onia in heliu m fro m this w ork involves only G h 8 , Z ^ g , and C as free param eters. T h u s the data
se t d esign allow s separating the initial seven-param eter optim izatio n p ro b lem into three m u ch sim pler
three-param eter problem s, m ore than doubling the num ber o f equivalent d ata points p er free param eter.
T hese data are organized into series and sequences, as described in Section 6.1. D ata presented in the
tables o f C h ap ter
6
are g ro u p ed by series. A series represents d iscrete sam ples o f th e ab sorption sp ec­
trum o f a single specific gas m ixture and is the sm allest practical block o f data an o p tim izatio n routine
c o u ld u se. M o st o f th e series co llected as p a rt o f this w o rk c o n sist o f n in e p o in ts, reflectin g the nine
usable resonances available w ith the cavity resonator. T he six series from B leaney and L o u b se r vary in
size from 9 to 24 points as explained in Section 6.2. D ata series are gro u p ed into seq u en ces, w hich are
very useful fo r exam ining pressure dependences. A sequence o f d ata is the collection o f all series at all
available total pressures, on a gas m ixture w hose tem perature and m ixing ratios are held constant.
T hese optim ization tasks o f course require co m p u ter softw are b ased on an optim izatio n algorithm .
T he algorithm chosen is th e "A m oeba" algorithm from the very useful han d b o o k "N um erical R ecipes,"
by Press e t al. (1986), a dow n h ill sim plex m ethod th a t finds a single lo ca l m inim um o f sin g le-v alu ed
m ultivariate functions. In this application the v ariables o f the fun ctio n are the free p aram eters o f the
param eterized form alism developed in the previous section. S tarting th e pro ced u re req u ires defining a
n on-degenerate sim plex in the p aram eter space. T his is accom plished by specifying the p aram eter space
coordinates o f the sim plex vertices, o r by specifying a "seed point" vertex consisting o f initial p aram eter
values along w ith corresponding step sizes that determ ine th e oth er vertices. If the function has m u lti­
ple local m in im a the ch o sen initial sim plex w ill influence w hich m inim um the algorithm w ill find.
The single-valued m ultivariate function to be m inim ized is th e quality -o f-fit function fo r th e p aram ­
eterized form alism com pared to a collection o f data. T his m easure m ust b e defined such th at the value o f
the function decreases as th e quality o f fit im proves. F o r a sin g le datu m it is usually appro p riate to use
the squared deviation, the difference o f the p redicted and m easured absorption values. T he q u ality-of-fit
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function fo r the collection o f d ata w o u ld then be the sum o f the squared deviations o v e r all d ata points in
the collection. It is possible to use m easures other than th e squared deviations, and indeed attem pts w ere
m ade to use one o f these alternatives: instead o f sum m ing the squared deviations, the abso lu te values o f
the deviations w ere sum m ed. T his ap p ro ach h ad th e undesirable effect o f overem phasizing d ata points
w ith low absorptivities, w here th e relativ e errors w ere larger. In the d ata from this w ork the m easured
absorptivities uniform ly increased w ith frequency, so this alternate q uality-of-fit definition overem pha­
sized the data at low frequencies a t the ex p en se o f relatively m ore accurate d ata at h ig h er frequencies. It
is entirely possible that the squared-deviations approach m ight overem phasize the data at hig h er frequen­
cies, w here the abso lu te errors are g en erally larger. H ow ever, curves op tim ized using th a t m ethod,
w hich agreed w ell w ith the d ata at the higher frequencies, w ere alm ost alw ays w ithin the error bars o f the
data at low er frequencies; curves optim ized on the sum m ed absolute values o f the deviations agreed with
the data at low er frequencies but often lay considerably outside the error bars o f the higher frequency data.
F o r this reason the quality-of-fit function b ased on absolute values w as abandoned in favor o f that based
on squared deviations. T he procedure fo r calculating the quality-of-fit value fo r a set o f data points and a
set o f param eter values is then:
1. R ead a datum p oint, including the frequency o f observation, pertin en t m acroscopic conditions,
and m easured absorptivity;
2 . U sing the param eterized fo rm alism fro m Section 7.2 w ith the p aram eter values as giv en , calcu ­
late the predicted absorptivity for the conditions o f step 1 ;
3 . Subtract the predicted absorptivity value from the m easured value and square th at difference;
4 . R ep eat steps 1 through 3 fo r each d atu m point, then sum all the step 3 results. T his final sum
is the quality-of-fit value.
The optim ization routine adjusts the param eter values to find that com bination o f values m inim iz­
ing the q u ality -o f-fit value. T his n u m b e r is useful in calculating the stan d ard d eviation o f the d ata
points from the predictions o f the optim ized form alism . T he standard deviation should be sim ilar in size
to the o n e -o m easurem ent erro r bars o f th e d ata. L arge standard deviations m ay indicate problem s w ith
the optim ization, or that the form alism being fit to the data is sim ply in consistent w ith the d ata.
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A m m onia A bsorption
L ine D ata
D R IV IN G PR O G R A M
E xperim ent
D a ta F ile
A M OEBA
O u tp u t
D ata File
U ser C o n tro l
O ptions
Q U A LITY -O F-FIT FU N C T IO N
PBGBR
F igure 7.1: G eneral organization o f the optim ization softw are. T h e d riving p ro g ram read s all num erical
data and m akes it av ailab le to the quality -o f-fit subprogram , im p lem en ts op tio n s availab le to the user,
and initiates A m oeba. A m o eb a adjusts the values o f th e p aram eters to m in im ize th e q u ality -o f-fit
function, w hich u ses P B G B R to calc u late pred icted ab sorptivities b a se d on the p ara m e te r values and
m easured conditions. W hen finished, the driving program w rites th e results to an o u tp u t file.
A softw are p ack ag e to p erfo rm th e optim ization w as w ritten in F o rtra n 77 an d im p lem en ted on a
D ata G eneral M V /10000 com puter. T he pack ag e co n sisted o f th ree m ain parts: a d riving program , a
slightly m odified im p lem entation o f A m oeba that w as contained w ithin the driving p rogram , and a su b ­
program to calculate the value o f the quality-of-fit function. A b lo ck d ia g ra m o f the p ack ag e organiza­
tion is show n in F ig u re 7.1 above. T he driving p rogram h ad 5 prim ary tasks:
1. Reading fro m a file the d ata concerning the individual am m onia ab so ip tio n lines (center frequen­
cies and self-broadened line w idths) and making them available to the quality-of-fit subprogram ;
2 . R eading fro m an o th er file th e co llectio n o f experim ent d ata an d m aking them av ailab le to the
quality-of-fit subprogram ;
3. Reading and im plem enting user specified options for the procedure;
4 . Preparing th e initial sim plex from user input and starting A m oeba;
5 . W riting the final resu lts to an o u tp u t file.
D ata read in steps 1 and 2 are p assed to the q u ality-of-fit subprogram v ia a F o rtran C O M M O N block.
U ser options o f step 3 include using absolute deviations o r relative deviations (absolute deviation divided
by the datum value), fixing the values o f o n e o r m ore param eters w hile th e re st rem ain free, setting the
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v alu e o f the co n v erg en ce criterion, an d obtaining a "full listing" from the process, an option ex p lain ed
below . It w as fo u n d th at the "relativ e deviations" o p tion p ro d u ced results sim ilar to those o b tain ed by
using th e absolute valu es o f the deviations as the m easu re o f fit, so that option w as no t used in the final
results. In step 4 the u se r is p ro m p ted fo r an initial seed p o in t o f param eter values and co rresponding
initial step sizes. T h e starting sim plex fo r A m oeba is g en erated fro m these values.
A m oeba is w h o lly co n tain ed w ithin the d riving pro g ram . O n e m inor m odification to the A m oeba
alg o rith m given in "N u m erical R ecipes" allow s th e u ser to req u est a "full listing" o f the process. W ith
this o p tion en ab led (v ia user interaction w ith th e driving p rogram ) results o f each iteration are p rin ted to
the o u tp u t file, tracing th e path to convergence.
In the no rm al c o u rse o f its o p eratio n A m oeba m akes rep eated calls to a function su b p ro g ram th at
calcu lates and returns the value o f the function to be m inim ized. F o r this application th at function is
the quality -o f-fit fun ctio n for the in p u t d ata set and the cu rren t p aram eter values. Its value is calcu lated
as previously outlined, and thus the subprogram m u st calculate absorptivities p redicted by the p aram eter­
ized form alism using those param eter values. T h e rou tin e to calculate the predicted absorptivities (called
PB G B R in F igure 7.1) is w holly contained w ithin the q u ality -o f-fit function subprogram .
T his optim ization package can be run w ith any n u m b er o f entries in the input d ata set. If desired, it
co u ld operate on the en tire collection o f experim ental d ata at o nce. H ow ever, such an approach w ou ld
pro v id e only one v alue fo r each p aram eter, values rep resen tin g th e best possible fit assum ing th at the
p aram eter values do n o t change w ith tem p erature o r pressure. It is im portant in this analysis to look fo r
changes in the param eter values w ith tem perature an d pressures, so the global data se t approach is u n d e­
sirable except to d em onstrate the ultim ate success o r failure o f such an approach. R esults o f such o p ti­
m ization attem pts are described in the n ex t section. Instead, separate optim izations w ere perform ed w ith
each individual series o f d ata on p u re am m onia o r am m onia in a single foreign gas.
O ptim izations w ere no t attem pted w ith d ata series o n am m onia in a Jovian m ixture o f hydrogen and
helium ; those d ata w ere reserved as a test o f th e lin earity o f m ixing the broadening gases. If th e total
broadening and coupling o f a line is indeed the lin e a r sum o f term s as show n in E quations 7.9 and 7.11,
then the coefficients G H 2 and Z |~|2 as determ in ed from th e d ata on am m onia in hydrogen, and Gf_je and
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Z(_je as determ ined fro m the data on am m onia in h elium , should com bine to m ake accurate predictions
about am m o n ia in a m ixture o f hydrogen and helium . I f those predictions turned ou t to b e inaccurate it
w o u ld be evidence th at E quations 7.9 and 7.11 are fu n d am en tally incorrect, and th a t n o n -lin ear term s
should appear in those equations. In that case th e d ata on m ixtures o f all three gases could pro v id e infor­
m atio n about the non-linearities.
7.4
Analysis and Results
T h is section details use o f the tools an d p rocedures d escrib ed in S ection 7.3 fo r finding o ptim ized
values o f param eters introduced into the B erge an d G ulkis form alism . T his process w as n o t as straight­
fo rw ard as hoped. Several problem s arose, som e d u e to n oise in the d ata o r the b eh av io r o f the tools
used, b u t som e due to fundam ental disagreem ents b etw een the d ata and the basic form o f th e B erge and
G ulkis form alism . A few assum ptions w ere req u ired to arrive at a m odified form alism th at m ore accu­
rately predicts m icrow ave absorption by am m onia. N o n e o f th e assum ptions are based on the quantum
physics o f am m onia m olecules interacting w ith each o th e r an d w ith o th er m o lecules in the p resence o f
electrom agnetic radiation; explaining the physics o f su ch interactions is beyond the scope o f this work.
T he g o a l is to provide a form ula or series o f form ulae, in corporating the know ledge rep resen ted in the
new d ata, yielding m ore accurate predictions o f the m icrow ave absorptivity o f am m onia u n d er the range
o f conditions o f interest to researchers studying the atm ospheres o f the giant planets.
T h e first task in the analysis o f the am m onia absorp tiv ity d ata w as to d eterm ine the values o f the
p aram eters representing the am m onia self-action coefficients, G[\|H 3 and Z n h 3- T hree-param eter optim i­
zations, w ith the third param eter being the global m agnitude param eter C , w ere perform ed o n each o f the
six series o f B leaney an d L oubser d ata an d on the fo u r series o f p u re am m onia data from this w ork.
Since th e Bleaney and L oubser data w ere also used to determ in e the self-action coefficients in the B erge
and G u lk is form alism , no large disagreem ents w ith th eir values ( G n h 3 = 0-75 and Z |\jh 3 = 0 A 9 ) w ere
ex p ected . Som e differences are possible d u e to errors intro d u ced by hand extraction o f frequencies and
absorption values from the graphical displays.
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T he results w ith the B leaney a n d L o u b ser series at 100, 300, and 900 torr p ressures w ere consistent
to w ithin 3-4% , com m ensurate w ith the noise lim its o f th e data. The averaged values o f th e param eters
w ere: C = 0.94, G NH 3 = 0.74, and ZNH 3 = 0.50. A greem ent w ith the B erge and G ulkis values is quite
good. A lthough the low values o f C m ay in d icate problem s w ith the in ten sity facto r A (J, K, m ) o f
E q u atio n 7.4, it is also possible th a t there w ere problem s w ith the p ressure or tem p eratu re m easu re­
m ents o f Bleaney and Loubser.
Series at 2, 4, and
6
atm . p roduced less consistent results. T he series at 2 atm . y ielded C = 0.94 as
before, but w ith G N H 3 = 0-90 and Z |\j }-|3 = 0-62. T he 4 atm . series p roduced results m ore like those at
the lo w er pressures, w ith C = 0.96, G N H 3 = 0.76, and ZNH 3 = 0.53.
w ith the
6
B ut all attem pts at optim izatio n
atm . d ata diverged. F u ll listings o f the processes show ed the valu es o f G n h 3 and Z n h 3
becom ing quite large, in d ependent o f in itial valu es o r step sizes: at the sam e tim e C seem ed to be
asym ptotically approaching a value n ear 1.042.
F u rth er scrutiny o f the relativ e inconsistency o f the higher p ressure d ata rev ealed a p ro b le m that
w ould be experienced m any times in this analysis. M any optim izations w ere p erfo rm ed w ith th e value
o f C fixed so the optim ization pro ced u re w o u ld find the best-fit values for G n h 3 ar)d ZN H 3 given the
fixed value o f C . T hese sets o f tw o -p aram eter optim izations typically covered a ran g e o f C values from
0.8 to 1.1. F or a giv en series from th e low er p ressure data, 100 through 900 torr, fix in g C m ore than a
few p ercen t different from its three-param eter optim ized value caused the tw o-param eter optim ized value
o f the quality-of-fit function to be significantly larger. T his evinced the linear independence o f the three
param eters. H ow ever a t pressures greater than 900 torr the linear independence o f th e param eters becam e
m ore tenuous. F o r a given series, fixing a value o f C quite different from the th ree-p aram eter optim ized
value w ould cause the procedure to converge to different values o f G n h 3
ZNH 3 , but the quality-of-fit
value fo r the new re su lt w ould no t b e sig n ifican tly g re a ter than the three-p aram eter o p tim ized value.
T he hig h er the pressure, the less the variation in q u ality-of-fit values fo r d ifferent solutions. T his gave
the appearance o f an under-constrained system that w ould n o t have a well defined unique solution.
E ach data series w as plotted along w ith, fo r each set o f param eter values, the predictions o f the fo r­
m alism . This y ielded clues to the sources o f the problem , and those sources lay in the b eh av io r o f the
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form alism 's line shape factor and th e noise associated w ith the m easurem ents. T h e three param eters are
indeed linearly independent; effects o f variation o f one param eter cannot b e ex actly o ffset by variations
in the o th er tw o. H ow ever, d ifferen t sets o f param eter values can p ro d u ce p red icted absorption spectra
such that, although they are n o t exactly the sam e, th eir subtle differences are sm aller than the n o ise in
the data. The sim ilar roles o f y an d £ in th e n um erator o f the form alism 's lin e sh ap e factor contrib u te to
this behavior. A m o eb a is b lind to n oise and w ill faithfully try adjusting the p aram eters to fit even the
error-induced features o f the m easured spectra if at all possible. T he Bleaney and L o u b ser series at higher
pressures w ere m ore susceptible to this p roblem d u e to the sm all num ber o f points in those series.
Fortunately fo r this w o rk all laboratory data o n m ixtures u se am m onia p artial pressures o f 4 0 0 torr
o r less, w here the B leaney an d L o u b ser d a ta produce consistent results fo r p ressu res o f 900 to rr o r less,
so the values G n h 3 = 0.74 a n d Z n h 3 = 0.50 sh o u ld w ork w ell. In a p lan e ta ry context, an am m onia
partial pressure g reater than 9 0 0 to rr in a Jovian m ixture requires a total p ressu re g reater than 5000 bars,
w hich w ou ld o ccu r at depths far b elo w those observable w ith rad io experim ents. T hus these values for
the self-action coefficients sh o u ld also w o rk w ell at pressures fo u n d in the o bservable reg io n s o f giant
p lan et trop o sp h eres.
T h e g lo b al a ccu racy o f th e optim ized fo rm alism is fa irly in sen sitiv e to these
coefficients anyw ay, so any inaccuracy introduced by errors here w ould be relativ ely sm all. V alues for C
w hich are consistently less than u n ity are w orth noting, but at this p o in t attem pting to inco rp o rate that
know ledge into the form alism w as n o t practical.
Tire B leaney an d L o u b ser data w ere all taken at "room tem perature" an d thus carry no inform ation
about tem perature dependence. U nfo rtu n ately the d ata on pure am m onia from th is w ork, no t originally
intended to be p art o f it, did little to rectify this situation. W ith in the p ractical p ressu re ran g e o f the
apparatus pure am m onia is a v ery strong absorber, and m aking th e m easurem ents p ushed the capabilities
o f the resonator nearly to its lim its. S om e o f the resonances w ere absorbed into the n oise level, ren d er­
ing them unusable and decreasin g th e n u m b er o f d ata points in already sparse series; m easurem ents
m ade w ith the surviving resonances su ffered larger relative errors than m easurem ents on less opaque gas
m ixtures. These situations caused the am biguity problem s experienced w ith the h ig h er pressure B leaney
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and L o ubser d ata to be q u ite pronounced w ith these data, so for each series tw o-param eter optim izations
had to b e perform ed w ith a w ide range o f C values.
R esults o f op tim izatio n s on d ata taken at 300 K, essen tially the sam e as "room tem perature," w ere
consistent w ith those from the lo w er pressure B leaney and L o u b ser data. T w o series o f d ata w ere taken
at that tem perature, at p ressu res o f 0.5 an d 1.0 atm . F o r both series, fixing C betw een 0 .9 4 and 0.97
y ielded o p tim ized v alu es o f G N H 3
Zj\jH 3 w ith in 5% o f th e 0.7 4 an d 0 .5 0 , resp ectiv ely , fro m the
B leaney and L o u b ser data. Series at other tem peratures p roduced far less co n sisten t results. T he series at
313 K, 1.0 atm ., w ould be ex p ected to yield results n o t sig n ifican tly d ifferen t from the 300 K , 1.0 atm .
series; the d ifferen ce is o n ly 4% in tem perature. T he values o f C that p ro d u ce d results agreeing m o st
closely w ith th e p revious o n es w ere still in the ran g e 0.90 to 0.9 5 , b u t this y ield ed G n h 3 and ZN H 3
values that w ere b oth sm a lle r than the lo w er p ressure B leaney and L o u b ser values by ab o u t 15-20% .
T h e results o f a th ree-p ara m e te r fit on th a t series w ere: C = 0.9 2 , G N H 3 = 0-64, and Z N H 3 = 0-39.
R easonable values o f C ap p lied to the d ata at 213 K resulted in G n h 3 an d Zfgf-13 values that w ere again
significantly sm aller than th o se obtained from the room tem perature data.
R ather than risk draw in g unw arranted conclusions from a sparse data set, attem pts to fu rth er clarify
the tem perature dependence o f absorption by pure am m onia w ere abandoned. T h e low er pressure data o f
B leaney and L oubser, in agreem ent w ith the 300 K data from this w ork, verify the lin ear p ressu re d epen­
dence o f the self-actio n term s in E quations 7.9 an d 7.11 o v e r th e range o f am m o n ia p artia l pressures
observable in th e atm o sp h ere s o f th e g ian t planets. T he tem p eratu re d ep en d en ces e x p ressed in those
term s m ay or m ay n o t be correct, b u t even 50% errors w ould not significantly affect the g lo b al accuracy
o f the form alism as ap p lied to the g ian t planets. It w as d ecid ed to retain the tem p eratu re an d pressure
dependences as show n in E q u atio n s 7.9 and 7.11, and use th e co n stan t p aram ete r values determ in ed
above fo r the scaling coefficients: G N H 3 = 0.14, and Z n h 3 = 0.50.
W ith the self-action coefficients determ ined, the next task w as the m o st critical — determ ining the
appropriate values and dependences for tire hydrogen coefficients G h 2 and Z h 2 o f Equations 7 .9 and 7.11,
using the d ata o n am m o n ia in hydrogen p resen ted in C h ap ter
6
. It w as an ticip ated th a t to fit the d ata
best those coefficients m ight have to vary w ith tem perature or pressure, b u t the m agnitude and character
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o f the variations w ere surprising. Problem s w ith the lin ear independence o f the param eters w ere again
encountered, bu t a technique w as devised that allow ed the analysis to continue.
As a test o f correctness o f the p ressure d ep en d en ce o f the form alism w ith fixed param eter values,
three-param eter o ptim izations w ere p erform ed on en tire sequences o f data. F o r a single sequence this
w ould produce o n e value each fo r C , G h 2 . and Z h 2 » providing the best fit to the data assem blage that
constant param eter values could allow. T he results in d icated th at no single set o f param eter values was
co n sisten t w ith the data o v er the en tire ran g e o f p re ssu re s rep resen ted in one sequence.
U sing the
c o n stan t p ara m e te r values to calculate p red icted absorptivities m ight y ield agreem ent w ith d ata at an
interm ediate p ressu re like 4 atm ., bu t predictions fo r higher and low er pressures system atically d iverged
from the d ata at those pressures, often w ith o b v io u sly d ifferen t frequency dependences. O ne o r m ore
param eters w ould have to vary w ith pressure and possibly tem perature. Separate optim izadon and analy­
sis on each series from the collection o f sequences w as required.
T hree-param eter optim izations were perform ed separately on each series, each accom panied by m any
tw o -p aram eter o ptim izations featuring various but fix ed values o f C. It w as im m ediately obvious that
the linear independence problem experienced w ith som e o f the p u re am m onia data w as universally p re ­
sen t and even m ore pronounced w ith these data. O ften, a sequence o f tw o-param eter optim izations w ith
C ranging from
0 .6
to 1.3 yielded m inim um q u ality -o f-fit values w ithin 5% o f the value p roduced by a
fu ll th ree-p aram eter optim ization. R esults o f the th ree-p aram eter o p tim izations w ere alm o st useless.
T h ey yield ed a huge range o f param eter values: C v aried fro m 0.6 to 1.4, G h 2 from 0.25 to 2.5, and
Z |-|2 from -0.5 to 4.3 . T here w as no clea r pattern to the variations except that the extrem e values w ere
often asso ciate d w ith series at low pressures,
1
or
2
atm ., w here the relativ e errors in the data w ere
generally larger. T his suggested that the p articular param eter values determ ined by full three-param eter
optim izations m ight be strongly influenced by n oise in the data.
Individually considered, the set o f tw o-param eter o ptim izations on a single data series w as no t m ore
inform ative. S in ce the variations o f the quality -o f-fit function value w ere obviously in th e n oise fo r a
w ide range o f C , there w ere no criteria clearly indicating that one choice o f param eter values w as better
131
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than another. T he problem seem ed hopelessly in determ inate. H ow ever, a technique using m ultiple d ata
series w as developed that, if it d id n o t resolve th e indeterm inacy, at least circum vented it.
T he technique involved graphing the tw o-param eter optim ized values o f G h 2 and Z h 2 from a given
series, param etric in the fixed p aram eter C. T he grap h has G h 2 as the abscissa an d Z |- |2 as the ordinate.
E ach value o f C w as associated w ith a unique p air o f G -Z values. P lo ttin g the pairs fo r a n u m b e r o f
d ifferen t C values allow ed defining the trajectory o f the optim ized solutions in the G -Z p lane as the
value o f C changed. The resu ltin g trajectory w as nearly lin ear fo r reaso n ab le values o f C , though fo r
extrem e values it co u ld som etim es becom e quite curved. T he advantage o f this technique was d em o n ­
strated w hen the trajectories from series taken a t the sam e tem perature b u t at d ifferen t p ressu res w ere
plotted together. A n exam ple o f this is show n in F ig u re 7.2, w here trajectories o f three series at 27 3 K,
4,
6,
and
8
atm . total pressure are p lotted. Slopes o f the trajectories vary w ith pressure, a ch aracteristic
often observed in these graphs. T he three trajectories in tersect each o th er w ithin a sm all n eighborhood
around Gj-j2 = 2.04, Z h 2 = 1.26. W ithin that n eighborhood the C values associated w ith all three trajec­
tories vary less than ± 1 %, from 1.07 to 1.09!
T his su g g ests that fo r these series there m ight be a
p referred value for C n ear 1.08, w ith the associated valu es o f G h 2 and Z h 2. If it is assum ed th a t C is
independen t o f pressure as it seem s to be for the 4,
the preferred results for the
1
and
2
6
, an d
8
atm . series, then C = 1.08 sh o u ld also give
atm . series at th at tem perature.
T he assum ption that C is independent o f pressure is equivalent to assum ing correctness o f the p re s­
sure dependence expressed in the intensity factor A (J, K, m ) o f Van V leck-W eisskopf and B en-R euven
form alism s. This is supported by the d ata o f figure 7.2, and it w ill be seen later that h elium -broadened
data also support it. If C did vary w ith pressure this analysis w ould b e m uch m ore d ifficu lt and m ig h t
be im pracdcal with the data currently in hand. T here w ouid be no means o f resolving the linear in d ep en ­
dence problem , so the d eterm ination o f p ro p er p a ra m e te r values w ould be speculative at best. T his
analysis pro ceed s b ased on the assu m p d o n that C is in d ependent o f pressure, w ith the realizatio n that
finding a general solution to the p ro b lem th at is ap p licab le o v er the en tire range o f co n d id o n s re p re ­
sented by the data m ay require m odifying o r abandoning that assum pdon.
F or C = 1.08, param eter values o b tained for the 273 K series at 2 atm . are G h 2 = 1.86, Z h 2 = 1.30;
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2.4
A
4 Atm
2.2
2.0
0.9
0.8
0.95
0.9
1.0
0.95
CM
X
1.0
0.8
Nl
0.9
0.95
1.0
12
1.3
1.2
0.6
0.4
1.4
1.6
1
2.0
.8
2.2
2 .4
2.6
2.8
G H2
F igure 7.2: T rajectories o f the tw o -p aram eter optim ized values o f G (-)2 and Z (-|2 param etric in C, w hich
rem ained fixed during the optim izations, fo r series at 273 K and 4, 6 , an d 8 atm . total p ressure. N um bers
beside specific points are the values o f C producing th e G h 2 and Z h 2 values o f those points. P o in ts w ith
C = 1.05 and the 4 atm . p o in t w ith C = 1 .1 are unlabeled to p revent o bscuring the in tersection n eighbor­
hood. W ith C = 1.08 points from the three trajectories occupy the sm allest possib le neighborhood.
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fo r the 1 atm . series, they w ere G h 2 = 0-36, Z h 2 = 3.87. P aram eter values from the 2 atm . series that
differred from those o f the higher pressure series w ere no t particularly surprising, b u t the large deviations
o f the values o f the 1 atm . series fro m those o f the others w ere unexpected. It w as decided th at the data
from the sequence at 213 K sh o u ld b e processed and com pared w ith the 273 K results b efore attem pting
to draw any generalizations ab o u t the behavior o f the coefficients.
R esults from the m any tw o -p aram eter optim izations on the five series at 213 K sho w ed both sim i­
larities w ith and differences fro m those o f the 273 K series. The G -Z trajectories fro m the
6
and
8
atm .
series intersected n ear G h 2 = 2.09, Z h 2 = 1.31, w h ere both had C = 0.95. T he trajecto ry o f the 4 atm .
series was slightly rem o v ed fro m th at in tersection; w ith C = 0.95, th at series y ield ed G h 2 = 1-93 and
Z h 2 = 1.22. T his difference w as larger than the corresponding difference o f the 27 3 K series, b u t not
large enough to be a serious p ro b lem . W ith C = 0.95 fo r both, the 2 atm . series p ro d u ced G h 2 = 1-61,
Z (-)2 = 1.65, w hile the
1
atm. series y ielded G (-|2 = 0.14, Z h 2 = 4.83. T he trend to w ard sm aller values o f
G h 2 and larger values o f Z h 2 at the tw o lo w est pressures, especially the
1
atm . series, w as the sam e for
the series at 213 K and 273 K, lending sup p o rt to the unexpected results fro m the 273 K data.
Since only a single series w as taken at 323 K
(1
atm . total pressure) the G -Z trajecto ry intersection
technique could n o t be used. H o w ev er for reasonable values o f C the G h 2 and Z h 2 values o b tained lent
m o re supp o rt to the 1 atm . re su lts a t 213 K and 273 K . W ith C betw een 0.9 5 a n d 1.05, G h 2 w as
betw een 0.1 and 0.3 and Z h 2 w as betw een 3.9 and 4.3.
A ll the G -Z points fro m th e 213 K sequence w ith C = 0.95, and all those from th e 273 K sequence
w ith C = 1.08, w ere plotted on a sin g le graph, along with the G h 2 = 2.318 and Z h 2 = 1.92 p o in t deter­
m in ed for the B erge and G ulkis fo rm alism from the very high pressure data o f M o rris and P arsons. It
w as seen th at all eleven o f these loci w ere very close to a fairly sim ple curve in the G -Z plane. A p o ly ­
nom ial least-squares fit to the cu rv e y ielded a quartic polynom ial in G h 2 w ith very sm all scatter. A t the
tim e this relationship w as considered o f little practical value, but it proved useful in la te r analyses.
T hese results show large variatio n s in G h 2 and Z h 2, arousing suspicion about th e assum ption that
C is independent o f pressure. A n alternate assum ption w ould be that G h 2 is in d ep en d e n t o f pressure.
T h at assum ption requires that C and Z h 2 bo th vary w ith pressure, ju s t as fixing C re su lts in variations
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in both G r 2 an d Z(-|2. S ince th eory held th at G h 2 should be co n stan t the p o ssib ility th at G h 2 is in d e­
p en d en t o f pressu re w as eq u ally plausible an d deserved fu rth er exam ination. R esu lts o f analyses under
that prem ise are discussed late r in this section.
A nalysis o f the p ressu re dependence o f G h 2 w as begun by p lotting o p tim ized G h 2 values against
th eir associated hydrogen partial pressures, show n in F igure 7.3 (a) and (b). As pressures increase above
approxim ately
1
atm . G h 2 seem s to asym ptotically approach a lim iting v alue. T h is w ould no t con tra­
d ic t the findings o f B erge an d G u lk is assigning G h 2 a v alu e o f 2.318, in dependent o f pressure. T heir
analysis w as b ased on the M orris an d Parsons data, taken at pressures o f tens to hundreds o f atm ospheres
and a single frequency. The absence o f spectral inform ation precludes in d ependent determ ination o f G (-)2
at a single p ressure, req u irin g assu m p tio n s about its behavior. A t such h igh p ressu res an asym ptotic
approach w ith rates such as those su g g ested b y F igure 7.3 w ould be v ery clo se to the lim iting value, so
system atic v ariad o n o f G h 2 in that ran g e o f pressures w ou ld be in significant co m p ared to the m easure­
m en t errors in th e data. It is quite reaso n ab le to conclude fro m the M orris and P arsons d ata th a t a co n ­
stant value for G h 2 is appropriate.
Figure 7.3 indicates that the b ehavior o f G h 2 a t the tw o tem peratures d iffers slightly: the approach
scale length appears shorter a t 273 K than a t 213 K. Thus the pressure d ep en d en ce o f G h 2 w ill involve
a tem perature d ependence. It is obvious fro m Figure 7.3 th at a full characterizatio n o f the beh av io r o f
G h 2 for all pressures requires m ore d a ta at hydrogen partial pressures below
1
atm .; m ore d ata betw een
1 and 2 atm . w o u ld also be helpful. T h e trend in G h 2 as pressure decreases is certainly no t am enable to
extrap o lad o n b elo w the d ata at
1
atm . pressure, and m ay even be an a rd fa c t o f previous assum ptions,
su ch as assum ing that the B en -R eu v en line shape facto r is co rrect o r that C is in d ep en d en t o f pressure.
In the low p ressu re lim it the B en-R euven form alism s co llap se to a V an V leck -W e issk o p f form alism
(B en-R euven, 1966), and this w ill req u ire th a t G h 2 again take on a value g re a te r than 2 (B en-R euven,
1966; W rixon e t al., 1971). T he details o f how G h 2 w ould ch an g e from valu es in the ran g e o f 0.1 to
0.4 at 1 atm . to a value g re a ter than 2 at v ery low pressures are unkn o w n . R eso lv in g this question
w o u ld require m o re spectral d ata than currently exist at pressures in the ran g e fro m ab o u t 0.25 to 1 atm.
W ith the d ata in h an d analysis w ith o u t unduly risky extrapolation is lim ited to p ressu res o f 1 atm . or
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(a) G h 2
vs
H ydrogen Partial P ressure, 213 K
2.5
2.0
0.5
0.0
0
2
6
4
8
10
8
10
H ydrogen Partial Pressure, A tm ospheres
(b) G h 2 vs H ydrogen P artial Pressure, 273 K
2.5
2.0
0.5
0.0
0
2
4
6
H ydrogen Partial Pressure, A tm ospheres
F igure 7.3: V ariation o f th e scaling co efficient G h 2 w ith hydrogen p artial pressure, at (a) 213 K, and
(b) 273 K . As pressures in crease G h 2 appears to asym ptotically approach a lim iting value. F rom the
data o f M orris and P arsons, B erge and G ulkis determ ined a constant value o f 2.318, appropriate for pres­
sures o f tens to hundreds o f atm ospheres. T h at value is theoretically in d ependent o f tem perature bu t this
has not been verified by laboratory data.
136
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g reater. Previous p ractical ap p lications o f B en-R eu v en -d eriv ed form alism s to g ian t p la n et atm ospheres
have relied on a V an V leck-W eisskopf fo rm alism for total p ressures less than 1 atm . (W rixon et al., 1971;
de P ater and M assie, 1985), and that w as o n e option available here.
Ignoring the beh av io r o f GH 2 at h ydrogen partial pressures less than ab o u t 1 atm . m ad e describing
the variation o f G h 2 w ith p ressure relatively sim ple. T here are com m only u se d m athem atical form s for
asym ptotic approaches like those o f F ig u re 7.3, and tw o o f th e m ost co m m o n are inverse exponential
approach and hyperbolic (o r sim ilar) approach. In an x-y p lane a sim ple inverse exponential approach to
a value y = k could be described w ith an equation y = k(1 - e 'x). This m ust b e shifted along the abscissa
to m atch the behavior o f the data, w hich suggests an in tercept o f that axis at a positive v alu e instead o f
x = 0. A pproach rate scale lengths m ust also be adjusted. T hese m odifications produce an equation with
three parameters:
(7.15)
w here the param eters are k, s, and x0 . T he abscissa intercept and the approach scale are controlled by x 0
and S respectively. Substituting the variables fro m the am m onia absorptivity p ro b lem and renam ing the
param eters w ith appropriate sym bols produces the equation:
(7.16)
w hich can be fit to the d ata o f F igure 7.3.
A sim ple hyperbolic approach is given by the rectilinear hyperbolic eq u atio n xy = c. A coordinate
transform ation is required to provide asym ptotic approach to y = k and an abscissa intercept a t a positive
value, and a param eter is needed for approach rate scaling. Since this m odified hyperbola has a vertical
asym ptote, one m ore param eter is needed to specify its location. T he resu lt o f this transform ation is the
four-param eter equation:
y = k- -
a
r
■
(7.17)
137
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w here x 0 sets the location o f the vertical asym ptote, a controls the abscissa in tercept, an d r controls the
approach rate. T h ere is interactio n am ong those three p aram eters, so a fit o f d a ta to th at equation is
m ore than m ere assigning values to them , one by one, according to various aspects o f the d ata; all m ust
be optim ized sim ultaneously. N o te that this equation has an infinite d isco n tin u ity a t x = x 0 , the verti­
cal asym ptote. R enam ing variables an d param eters yields the equation to be fit to th e data:
G h *= 9 " ^ ----------7
P H
- P
'
(7 ' 18)
or, equivalently (but w ith a d ifferent value o f a),
Gh, = 9
;pH2-p)r
(7.19)
A nother softw are pack ag e based on A m oeba, sim ilar in design to that used to fit the p aram eterized
B en-R euven form alism to th e basic data, w as used to fit E quations 7.16 and 7.19 to the d ata show n in
Figure 7.3 and a po in t representing the B erge and G ulkis high pressure value o f G(-)2 » 2.318, a t 500 atm.
T he inverse ex ponential equation, E q u atio n 7.16, pro v ed incapable o f ad eq u ately d escrib in g the data.
Scale length s fo r the asym ptotic approach w as n o t constant. It need ed to b e m uch sh o rter for hydrogen
p artial pressu res n ear 1 atm . th an fo r h ig h er pressu res. M od ify in g E q u atio n 7 .1 6 to h av e a variable
approach scale w ould be possible, bu t this w ou ld require adding at least on e m ore param eter an d m aking
the argum ent o f the exponential n o n lin ear in
This changing approach scale len g th is m o re ch arac­
teristic o f an hyperbolic approach than an inverse exponential approach.
As expected from results o f the fit to the inverse ex ponential equation, fitting th e m odified hy p er­
bolic equation, E quation 7.1 9 , to the d ata w as m uch m ore successful. D ata from th e sequence at 213 K
yielded param eter values g = 2.339, a = 0.3477, p = 0.7715, and r = 0.6741. T h o se from the sequence
at 273 K yielded g = 2.3 4 0 , a = 0.2080, p = 0.9881, and r = 0.3272. P aram eter g appears to be fairly
in sen sitiv e to tem perature, b u t the others have sig n ifican t tem p eratu re d ep en d en ces. W ith o nly tw o
points each controlling those tem perature dependences it w ou ld seem that lin ear descrip tio n s w o u ld be
appropriate, but oth er con sid eratio n s argue again st such a sim ple approach. A lin e a r fit to o p tim ized
138
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values o f param eter r w ould yield r less than zero fo r tem peratures g re a ter than about 330 K. N egative
values o f r d estro y th e asy m p to tic ch aracter o f E q u atio n 7.19 an d at high p ressures w ou ld v iolate the
approach o f G h 2 to values n ear the B erge and G ulkis high p ressure result. C ontinuity o f E quation 7.19
w ith tem perature req u ires th a t w h ile r m ay approach zero, it m u st n e v e r be less than zero. Sim ilarly, a
lin ear fit to the o p tim ized values o f p aram eter a w o u ld also y ie ld a less th an zero fo r tem peratures
g reater than 363 K . N e g a tiv e values o f a p ro d u ce an asy m p to tic ap p ro ach from above the lim iting
value, so at low pressures p redicted values o f G h 2 w ould be m u ch larg er than the high pressure value as
opposed to the m uch sm aller values uniform ly p resen t in the data at all tem peratures, including 323 K.
A gain, continuity requires th at w h ile a m ay approach zero, it m u st n e v e r be less than zero.
P aram eter p also has a lim iting value, bu t in this case it is n o t zero . T h e value o f p sets the loca­
tion o f the vertical asym ptote in E quation 7.19. P redictions o f th at eq u atio n are valid o nly fo r hydrogen
partial pressures above the infinite discontinuity produced by the v ertical asym ptote. H ow ever, fo r tem ­
peratures g reater than about 27 6 K , a linear fit to the optim ized results p laced the asym ptote at hydrogen
partial pressures larg er than th e sm allest o b serv ed in the d ata, a b o u t 0.9 9 1 8 atm ; at 323 K a lin ear fit
places the asym ptote at 1.159 atm . T his conflicts w ith the resu lts at 323 K th a t show th e value o f G h 2
is still quite sm all at
1
atm . total pressure, suggesting the b eh av io r o b serv ed in the lo w er tem perature
1
atm . data is continuous w ith th at o bserved a t 323 K. T he location o f th e asym ptote m ay approach that
m inim um pressure as tem p eratu re increases, but m u st n ev er ex ceed it. A n im portant side effect o f the
vertical asym ptote in this eq u atio n concerns gas m ixtures containing helium . F o r a given total pressure
the hydrogen partial pressure is decreased in such a m ixture, and care m ust be ex ercised to insure that it
is greater than the value o f p, o r calculation o f the value o f G h 2 m ay be w ildly in error.
All three cases discussed above require an approach to a lim iting value as tem perature increases, so
an asym ptotic form is required. In these applications an hyperbolic approach has the d isadvantage o f
requiring a vertical asym ptote, so use o f exponential approach form s w ou ld avoid infinite discontinuities
at tem peratures low er than those o f the data. Param eters r and a approach zero as tem perature increases,
so they can use the sim ple form
139
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(7.20)
Si
in the case o f param eter r, and
(7.21)
s.
in the case o f p aram eter a . C oefficients r0 and a 0 are initial values fo r the param eters a t zero tem pera­
ture, and T s is a tem peratu re scale length for the asym ptotic approach. O f course T s w ill h ave differen t
values in the expressions fo r the tw o param eters. Param eter p approaches a n o n-zero lim it and thus has
a slightly d ifferent form:
(7.22)
w here p a is the u p per lim it for the location o f the vertical asym ptote in E quation 7.19 and p 0 sets its
location at zero tem p eratu re. T he constant T s is again a tem perature scale length for the asym ptotic
approach, and w ill have a v alue different from those in the expressions fo r the o th er p aram eters. U nlike
the other p aram eters it is n o t unreasonable for p to take on n egative values. T he up p er lim it p a is an
artifact o f th e d ata set an d th e m ethod o f analysis, and has no physical significance w hatsoever. W hen
additional d ata at low er pressures becom e available that artifact can b e e lim in a te d F o r th at m atter, E qua­
tion 7.19 is only a seg m en t o f a piecew ise approxim ation to the true beh av io r o f G h 2, so existen ce o f
the vertical asym ptote is also an artifact. A dditional d ata in the p ressure range from 0.25 to 2 atm . w ill
allow dispen sin g w ith th e lim ited hyperbolic approxim ation and yield an approxim ation th at is co n tin ­
uous for all pressures. U n til then the hyperbolic approxim ation m ust be reg ard ed as a necessary evil.
W hen the param eters o f Equations 7.20, 7.21, and 7.22 are evalu ated and those expressions are sub­
stituted into E quation 7.19, the result is a form ula describing the p ressu re and tem perature dependences
o f G h 2 as represented by the data:
140
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•T/116.8
G Hj = 2.34
1
-
(7.23)
e ( 9.024 - T/20.3 ) _ 0.9 9 1 8 + P H
n
2,
where
r = 8.79 e ~ T/83
(7.24)
In these equations the gas tem perature is in K elvins and th e h ydrogen partial pressure, P h 2» is in atm.
E quation 7.23 is a critical elem ent o f the absorptivity p red ictio n form alism that is the goal o f this work.
A sim ilar characterization o f the pressure and tem perature d ependences o f Z (-)2 is also a critical ele­
m ent in the form alism . T h a t analysis began in the sam e m an n er as th at fo r G h 2» p lo ttin g th e values
obtained fo r Zj- |2 again st their associated hydrogen partial pressures. T he high p ressu re value used by
B erge and G ulkis, 1.92, w as included at 500 atm. and w as assum ed to apply at both tem peratures, 213
and 273 K. T h e beh av io r seen in those plots, show n in F ig u re 7.4, is m ore com plex than those o f Fig­
ure 7.3. B oth m ay have a m inim um som ew here in the ran g e o f 4 to
8
atm . pressure, increasing slow ly
to the high p ressure value as pressure increases. T he data o f this w ork do n o t clearly show th at increase;
in fact, data at pressures n e a r o r above 2 atm ., at 273 K , and n ear o r above 4 atm . at 213 K are all co n ­
sistent w ith a co n stan t v alue o f Z ^
2
near 1.30. It is the B erge and G ulkis v alue at hig h er pressures that
requires the rise no t seen in F igure 7.4. L ike the b e h av io r o f G h 2 the approach o f Z h 2 to the high
p ressure value sh o u ld be asym ptotic, but the data span an in su fficien t p ressu re range to eith er confirm
o r refute th a t ex pectation. O n the o th er side o f the m inim um the d ata in crease rap id ly below 2 atm.,
giving the appearance o f a vertical asym ptote som ew here below
1
atm .
T his g en eral b ehavior is sim ilar to the b ehavior o f functions d escribing the p o tential o f a neutral
hydrogen atom 's electron at a given radius from its p aren t nucleus. A ttem pts to fit such functions to the
d ata m et w ith failure. T hey req u ired m ore free p aram eters than those used in the analysis o f G (-|2 and
w ere thus extrem ely sensitive to noise in the data. O p tim ized functions resulting from this approach
generally d id a p o o r jo b o f estim ating Z h 2 values desp ite th e abundance o f param eters, indicating that
the functional form w as inappropriate for the data, so this approach w as abandoned.
141
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(a) ZH 2 vs H y d ro g en Partial Pressure, 213 K
0
2
4
6
H ydrogen P artial Pressure, A tm ospheres
(b)
Zh2
vs
8
10
8
10
H ydrogen P artial P ressure, 273 K
4
3
2
1
0
2
4
6
H ydrogen Partial Pressure, A tm ospheres
Figure 7.4: V ariation o f scaling co efficient Z h 2 w ith th e partial pressure o f hydrogen, a t (a) 213 K, and
(b) 273 K. F rom the data o f M orris and P arsons, B erge an d G ulkis determ ined a co n stan t valu e o f 1.92,
appropriate for pressures o f tens to hundreds o f atm ospheres. T h at value is theoretically in d ep en d en t o f
tem perature b u t this has n o t been verified by laboratory data.
142
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5 .0
4 .0
CM
X
N
2.0
1.0
0.0
0.5
1.0
1.5
2.0
2.5
GH2
F igure 7.5: G raph illustrating th e close relationship betw een G h 2 and Z h 2 fo r all tem peratures. C ro ss­
es m ark positions o f d ata from this w o rk an d the h ig h pressure datu m from B erg e and G ulkis, on the
extrem e rig h t o f the graph. T he solid line is the quartic equation fit to the data.
A second approach exploited the close relatio n sh ip betw een G h 2 and Z h 2 m entio n ed in the analysis o f
the b ehavio r o f G h 2.
A s
illustrated in F igure 7.5, points consisting o f G h 2 - Z h 2 pairs ob tain ed from
the data o f this w ork and the high p ressure p air from B erg e and G ulkis all lie very n ear a single-valued
quartic curve in the G -Z plane, regardless o f tem perature. E rror bars have been om itted fo r clarity and are
show n in F igures 7.3 and 7.4. S catter o f the points aro u n d the curve is uniform ly sm aller than the error
bars. W ithin the error lim its o f the data, points at a giv en tem perature are distributed along the curve by
pressure, w ith those at
1
atm . total p ressu re in the u p p er left o f th e graph, pro g ressin g to th e rig h t as
pressure increases tow ard the B erge and G ulkis poin t a t the extrem e right. T hen i f the value o f G (-)2 is
determ ined, the quartic equation allow s unam biguous determ ination o f the associated value o f Z h 2. This
approach gave quite satisfactory results. T he quartic eq uation for Z h 2 given a value o f G (-)2 is:
143
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Z h 2 = 5.7465 -
7 .7 6 4 4 G h 2 + 9.1931 G h 2 -
5 .6 8 1 6 G h 2 + 1.2307 G h 2 .
(7.25)
Figure 7.5 reiterates th e n eed fo r m ore data in the pressure range below 2 atm .
E quations 7.23 an d 7.25 fo r the critical hydrogen broadening an d coupling coefficients are based on
the assum ption,supported by d ata a t hig h er pressures, that the g lobal correction facto r C is independent
o f pressure. T heory o f course w o u ld have C constant (unity) un d er all c o n d itio n s. It also w ould have
G h 2 constant for all conditions, b u t the d ata show this is inco n sisten t w ith con stan t C . As m entioned
earlier, analysis w as also perfo rm ed un d er the assum ption that G h 2 w as co n stan t, requiring variation of
C w ith pressure. M an y tw o -p aram eter optim izations w ere p erfo rm ed on each series a t 213 an d 273 K,
w ith various bu t fixed values o f G h 2. Each optim ization produced th e values o f C and Z (-(2 needed to fit
the data b e s t L ik e the results w ith fix ed C , optim ized quality -o f-fit function values fo r d ifferent G h 2
varied only slighdy. A variation o f the trajectory intersection technique paired C and Z |- |2 values from a
single series on a C -Z plane, allow ing determ ination o f the trajectories o f the optim ized values param et­
ric in G |-|2. T rajectories from both sequences w ere p lotted on one graph to a llo w com parisons. There
w ere the expected sim ilarities to the G -Z trajectories: trajectories from the h ig h pressure d a ta o f one
sequence intersected each o th er w ithin a sm all neighborhood o f the C -Z plane, w ith sim ilar values o f
G h 2 : trajectories from the d ata at lo w e r pressures w ere considerably m ore rem oved fro m those neigh­
borhoods. T here w ere also v ery im p o rtan t differences. G -Z trajectories fro m th e tw o 1 atm . series at
213 and 273 K yield ed roughly sim ila r G h 2 and Z h 2 values from sim ilar C values, w here C -Z trajec­
tories from those series y ielded w idely differing values o f C and Z h 2 fo r sim ila r values o f G h 2- The
behavior o f the G -Z trajectories assum ing C w as independent o f p ressu re w ere m u ch m ore am enable to
general characterization than C -Z trajectories assum ing G h 2 w as in d ep en d en t o f p ressure, so the tech­
n ique involving C -Z trajectories w as n o t p ursued further. T his does n o t m ean th e assum ption that C is
co n stan t is correct and the assum ption that G h 2 is constant is not. H o w ev er, com pletion o f the optim i­
zation process co u ld be accom plished w ith the data in hand if C w ere assum ed in d ep en d en t o f pressure,
w here it appears likely th at com pletion under the alternate assum ption w ould req u ire more data.
T his com pletes analysis o f d ata on am m onia in otherw ise pure hydrogen, th e m ost critical part o f
the optim ization process. The n ex t task is sim ilar analysis o f data on am m onia in helium .
144
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A nalysis o f the effects o f h elium on am m onia is m uch less critical th an the analysis o f the effects
o f hydrogen. This is fortunate, since y et m ore problem s w ere encountered w ith the data on am m onia in
helium . C om parisons o f th e d ata w ith predictions o f the original B erge and G ulkis form alism confirm ed
errors in freq u en cy , tem perature, and p ressure dependences. A t th e tw o tem peratures n o t too rem oved
from room tem perature, 273 and 313 K , the predicted spectra often crossed the data, pointing to errors in
frequency dependences, and the relationships betw een predicted and observed spectra changed w ith pres­
sure, indicating problem s w ith p ressure dependences. A bsorptivities at 213 K w ere uniform ly sm aller
than predicted, an indication that the tem perature dependence m ay also be in error.
A nalysis began w ith three-param eter and m any tw o-param eter optim izations o f param eters C , G ^ e and Z ^ e fo r each series o f data from Tables 6 .7 , 6 .8 , and 6.9. L in ear independence problem s experienced
w ith the hyd ro g en data w ere also q u ite in evidence in the heliu m data. T h ree-p aram eter optim izations
w ere once again essentially useless, and quality-of-fit function valu es from tw o-param eter optim izations
varied only slightly o v er a significant range o f C values. T h e trajectory intersection technique from the
hydrogen d ata analysis w as applied.
Initial G -Z plane plots o f the pairs o f G(_ie and Z|_|e values obtained from each series revealed differ­
e n t behavior by the helium data. U nlike the hydrogen data, G -Z trajectories o f series at 4 ,
6
, and
8
atm .
w ere roughly parallel to each o th er at all th ree tem peratures, 213, 273, and 313 K . F ortunately the tra­
jecto ries w ere closely spaced, and in each case there w as a unique, very narrow ran g e o f C values such
that the associated optim al G ^ e and Zj_|e values for the series at those pressures occupied a sm all neigh­
borhood in th e G -Z plane. T his supported the observation m ade in the hydrogen d ata analysis that at the
higher pressures o f a sequence there appeared to be a preferred v alu e for C, and allow ed determ ining the
(apparently) preferred values o f C at each tem perature and the associated values o f G(_(e and Z ^ e for the
three highest pressures. A t 213 K the values w ere C = 0.92, G(_|e = 0.53, and Z ^ e = 0.16; at 273 K
they ch ang ed to C = 1.08, G(_|e = 0-55, andZ|_)e = 0.125; a t 313 K they w ere C =
1.1 1 ,
G(-je = 0.565,
and Z fqe = 0.1 0 . V alues o b tain ed for C a t 213 and 273 K by this m eth o d clo sely paralleled those
obtained from the hydrogen data, and its value at 313 K applied to th e series o f hydrogen d ata at 323 K
produced results consistent w ith those from hydrogen d ata at lo w er tem peratures. A t all tem peratures the
145
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values o f G ^ e and Z|_|e associated w ith those C values w ere sm aller than the high pressure values used
by B erge and G ulkis. T his indicates th at at these pressures effects o f helium on am m onia as co m p ared
to those o f hydrogen are even less significant than suggested by the B erge and G ulkis form alism .
F o r all tem peratures, behavior o f the G -Z trajectories at the tw o low est pressures prevented applying
the C values d eterm ined by the trajecto ry in tersec tio n technique. T raje cto ries at those pressures b o re
little resem blance to those o f the hig h er pressures. T hey w ere often rath er convoluted, and in one case,
th e series at 1 atm. and 273 K, all o ptim izations yield ed values o f Z ^ e less than zero. N o reaso n ed
cho ices fo r C applied to any o f the 1 and
2
atm . trajectories pro d u ced Gj-|e and Z|_je values w ith reco g ­
n izab le relationships to the others. B eh av io r a t d ifferen t tem peratures offered tantalizing hints that sy s­
tem atic variations w ere involved, bu t w ith o u t d ata at fin er p re ssu re resolution the relatio n sh ip s w ill
rem ain hidden. The best th a t co u ld be d o n e w ith the d ata availab le w as characterization b ased on the
b eh av io r at the higher pressures w ith the hope th at errors in troduced by applying those relatio n sh ip s to
lo w e r pressures w ould not be too large. F o r a g iv en tem perature, variations in the coefficients at the
h ig h e r pressures w ere w ithin the noise, and thus no m eaningful p ressu re dependences could be applied.
F orm ulae describing the coefficients could only express tem perature dependences.
T he coefficient values could be adequately d escribed w ith sim ple linear form ulae. N o problem s w ith
predicted co efficient values that are zero o r n egative ex ist below tem peratures n ear 50 0 K, w ell b eyond
th e range o f extrapolability. F or G[_)e the form ula is
(7.26)
an d fo r Z|_)e it is
(7.27)
T h is com pletes the analysis o f data on am m onia in helium .
T he final rem ain in g task in this analysis is ch aracterizatio n o f the tem perature dependence o f th e
global m agnitude param eter C. T he v alu e C = 1 is inco n sisten t w ith the data on am m onia in hydrogen
146
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o r helium (and the d ata on am m o n ia in a m ixture o f hydrogen and helium ), indicating th a t errors exist in
the tem perature dependence o f the intensity factor o f the B erge an d G ulkis fo rm alism and p ossibly in
other areas as w ell. T he tem perature dependence o f C provides no inform ation ab o u t the nature o f those
errors but perform s the applications-oriented task o f forcing a fit to the laboratory data.
V alues for C fro m d ata o n am m onia in hydrogen agreed qu ite w ell w ith those fro m da ta on am m on­
ia in helium . Since hy d ro g en is by far the m o st significant foreign c o m p o n en t in m ixtures containing
both hydrogen and helium , values from the hydrogen data at 213 and 273 K (w here com plete sequences
w ere available), 0.95 an d 1.08 respectively, w ere used. F o r a th ird co n tro l p o in t the value from the
helium data at 313 K, 1.11, w as used. A quadratic fit to those d ata yields the equation
C
0.337
+
n 0 4
70>600
.
(7.28)
P olynom ial fits are n otorious fo r th eir b ehavior outside the ran g e o f co ntrol so th e qu ad ratic equation
m ay not be the best fo r ex trapolations to higher o r low er tem peratures. U ncertainties in th e values o f C
w ere about ± 4-5% assum ing th at C is independent o f pressure; th at figure includes u ncertainty in the
am m onia m ix in g ratios. G iv en th o se u n certainties a range o f lin ear ap p ro x im atio n s is also co n sisten t
w ith the data, from
C = 0.87 + 0.0 0 0 6 T ,
(7.29)
C = 0.35 + 0.0 0 2 6 T .
(7.30)
to
H ow ever, the q u adratic fo rm u la
(Joiner
et
al.,
yields go o d agreem ent w ith d ata taken b y o th er researchers
1989), to b e d iscu ssed later. O ther extrapolation
at 203 K
schem es w ill also b e discussed. E q u a­
tions 7.28 through 7.30 are valid fo r pressures in the range covered by the d ata o f this w ork.
W ith E quation 7.28 the n ew version o f the B en-R euven-based fo rm alism is co m p lete. A ll seven
147
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param eters h ave been ch aracterized as com pletely as th e data allow . T he n e w form alism can b e su m m a­
rized as a series o f basic procedures th at start from specifications o f all p ertinent m acroscopic conditions
and arrive at a calculated absorptivity. F iv e condition values m u st be specified: tem perature T o f the
gas m ixture, p artia l pressures o f am m onia, hydrogen, an d h elium , P n h 3. P H 2 >
PHe> resp ectiv ely ,
and the frequency o f observation f. The procedure is then:
1. D eterm ine the values o f G|-)2 >
G|-le> ^He> anc* C , fro m E quations 7.23, 7.25, 7.26, 7.27,
and 7.28, respectively; also use G n h 3 = 0-74 an d Z fqn 3 = 0.50. N o te that in E quation 7.23 the
partial pressure o f hydrogen m ust be greater than the value o f param eter p as discussed below.
2. C alculate the B en-R euven param eter
8
from E quation 7.12 w ith D = -0.45.
3. F o r each am m onia absorption line from J= K = 1 to J= K = 1 6 , w ith K alw ays less than or equal to
J , perform the follow ing procedure:
a. O b tain values fo r fc (J , K) and Yq (J, K) fro m th e laboratory sp ectral data o f P o y n ter and
K akar (1975).
b. U se the necessary con d itio n values, coefficients fro m step
1,
an d y 0 (J, K) fro m step 3a to
calculate the B en-R euven param eters y and C, fro m E quations 7.9 and 7.11, respectively.
c. C alcu late the valu es o f F (J, K, m , f) and A (J, K, m ) from E q u atio n s 7.2 and 7.4, resp ec­
tively.
d. C alcu late the ab sorpdvity at frequency f due to the (J, K) line by m ultiplying th e pro d u ct o f
F and A from step 3c by the value o f p aram eter C from step 1. T his predicted ab sorpdvity is
in units o f optical depths p er cm.
4. Sum th e contributions o f all individual lines b y sum m ing the results o f each iteration o f step 3.
T he resu lt is the p red icted absorptivity o f the gas m ixture at freq u en cy f, in o ptical dep th s p e r
cm . If desired, co n v ersio n to units o f d B /k m is done by m u ltip ly in g this resu lt by a con stan t,
2 x 10 6 lo g 1 0 e, w hich is approxim ately 4.3429 x 105.
A b so rp iiv id es p red icted by this m eth o d agree w ell w ith all d ata u sed in its generation. U se o f the
m eth o d as an in terp o latio n sch em e should yield pred ictio n s accurate to w ithin 5-10% d e p en d in g on
frequency and pressure; higher pressures and frequencies are relatively m ore accurate than low er ones.
E xtrapolations p resen t problem s that researchers applying the new form alism m ust consider.
M ost applicau o n s o f the new form alism to studies o f gian t p lanet atm ospheres w ill require extrapoladon. R ad io occultation experim ents generally use frequencies from about 2 to 9 G H z, and rad io astro­
nom ical ob serv atio n s span frequencies from less than 1 G H z to the su b m illim eter range. F igure 7.6
illustrates the im portance o f extrap o latio n s in tem perature by show ing the tem peratures and pressures
148
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J
i
*
\
\
|#
\
\
v
\
|
i
\
\
i
|
i
|
—
j-■■ |
j ->p |
i■
|
|
I*- i1 |
| —i— i
i■ r
&
JO
A♦
A ♦
I 1
s2
10
50
100
150
200
Tem perature, K
250
300
350
F ig u re 7.6: A co m parison o f the tem perature and p ressure ran g es o f laboratory d a ta on m icro w av e
absorption by gas m ixtures applicable to th e atm ospheres o f the giant planets, and co nditions m easured
by the V oy ag er radio occultation experim ents at Ju p iter (circles), S aturn (squares), U ranus (diam onds),
and N ep tu n e (triangles). V alues fo r Jupiter, Saturn, and U ranus at the bottom o f the graph are ex trap o ­
lations based on w ork by L ew is and P rinn (1984). C rosses represent d ata from this w ork; d ata at 203 K
from Jo in er et al. (1989) and at 193 K from Steffes and Jenkins (1987) are m arked w ith an "x."
o f cu rren t laboratory d ata plotted w ith conditions observed at the giant planets by V o y ag er radio occulta­
tion experim en ts. N o te th ere is n o o v erlap betw een tem peratures at w hich lab oratory d ata e x ist and
those m easured by radio occultation experim ents. R adio astronom ical experim ents at large w avelengths
have pro b ed m ore deeply into the atm ospheres o f Ju p iter and Saturn, to levels th at are w ith in the tem ­
perature range o f laboratory m easurem ents, but those at shorter w avelengths still require extrapolation.
T he first problem area likely en co u n tered in applications involving extrapolation is the req u irem en t
that h y drog en partial pressures m u st be greater than the value o f param eter p o f E quation 7.19; v io lat­
ing this invalidates E quation 7.23. T h at param eter value m ay be calculated by
p = 0.9918 - e ( 9.024-T/20.3) _
(7 31)
149
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A t low tem peratures the v alu e o f p m ay be considerably less than 0.9918 atm . and even less than zero,
but it can n ev er b e g reater th an 0.9918 fo r any tem perature. U sing
1
atm . as th e lo w er lim it for P (-|2
w ill insure validity o f E quation 7.23 regardless o f tem perature. A t lo w er tem peratures, w here using the
new form alism to the low est p ressu res p o ssib le m ay be desirable, p m ust be calcu lated fro m E quation
7.31 and com pared w ith P |-|2 b efore proceeding w ith an absorptivity calculation. I f P h 2 is too sm all the
new form alism cannot be used and an alternate m ethod m ust be substituted.
C urrent absorptivity calcu latio n schem es often use a V an V leck -W eissk o p f form alism at total p re s­
sures below 1 atm . (W rixon et al., 1971; de P ater and M assie, 1985). T h is is a v iable alternative here,
although there m ay also be p ro b lem s w ith V an V leck -W eissk o p f fo rm alism s. D ata su g g est problem s
w ith the tem perature dep en d en ce o f the intensity factor o f the B erge an d G ulkis B en-R euven form alism ,
and the intensity factor in V an V leck -W eissk o p f form alism s is identical. It m ay be m ore accurate to use
a Van V leck-W eisskopf fo rm alism m odified w ith a correction factor C, the global m agnitude param eter
from this w ork. U nless so m e m atching m eth o d is u sed any approach using tw o d ifferen t form alism s
introduces a d iscontinuity at th e b oundary p o in t betw een the tw o form alism s. T h is is unfortunate, but
the data show th at devising a sin g le form alism valid at all pressures w ill req u ire additional laboratory
data on am m onia in hydrogen at pressures from about 0.5 to 2 atm., w ith n arro w er pressure resolution.
A ttem pts to extrapolate to v ery high freq u en cies at low pressures w ill also e n co u n te r problem s.
T he observed beh av io r o f G |- |2 an d Z h 2 w ith pressure causes the value o f Z h 2 to becom e m uch larger
than that o f G h 2 fo r pressures n ear
1
atm . A t frequencies m uch higher th an a line center frequency this
m ay cause the n um erator o f th e rig h tm o st quo tien t o f E quation 7.2 to be less than zero, yielding absorptivities less than zero. T his is n o t a problem a t m icrow ave frequencies, the intended range o f application
o f the new form alism , but extrapolations to frequencies in the subm illim eter range m ay be troublesom e.
The last p roblem area involves extrapolations in tem perature. M o st research ers studying the g ian t
planets are interested in tem peratures considerably below the low est fo r w hich there are data. Suggested
errors in the tem perature dependence o f the intensity factor poin t to absorptivity predictions by the B erge
and G ulkis form alism th at are too large, but uncertainties in the d ata m ak e it d ifficu lt to estim ate by
150
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
w h at fa c to r they are to o large. A t 125 K various extrap o latio n schem es for C consistent w ith the data
su g g est inten sities th at can b e from 50% to 95% o f th e B erg e a n d G u lk is values. U n certainties in tem ­
perature dependences internal to the line shape factor m ay add another few percent to that range.
T he quad ratic form ula, E quation 7.28, m ust break do w n so m ew h ere below 20 0 K . A t tem peratures
low er than 37 K E quation 7.28 yields values for C less than zero, w hich w ould resu lt in p redicted absorptivities less than zero. O ne technique th at prevents this p ro b lem uses a tangent to the quadratic equation at
a chosen b reak p o in t tem p eratu re T g .
A t tem p eratu res above this b reak p o in t E q u atio n 7.28 is used;
below it th e tangent line is used. W hen T g is chosen, the equation o f the tangent line is given by
C =
T2
\
I
T
B " 0.337 + - J _ - — A - I I
7 0 ,6 0 0
j
\ 110.4
35,300
.
(7 .3 2 )
T g can be set fro m w ell above 200 K to as low as 154 K, w here the equation o f th e tangent line w ould
yield C = 0 for T = 0. A version using a break point tem perature o f 180 K has been applied to experim en­
tal m odels o f V o y ag er 2 radio occultation d ata from N ep tu n e th a t p roduced p red icted signal b e h av io r in
good agreem ent w ith observed behavior.
In itial tests o f the new form alism com pared its predictions ag ain st the d ata on am m onia in Jovian
m ixtures o f hydrogen and helium , presen ted in T ables 6.11 an d 6.1 2 , th a t w ere n o t u sed in generating
the form alism . E ach series confirm ed that the new fo rm alism w as considerably m ore accurate than
either the B erg e and G ulkis B en-R euven o r V an V leck -W eissk o p f form alism s. F ig u re 7.7 show s tw o
exam ples from those d ata w ith the predictions o f Van V leck-W eisskopf, B erge and G ulkis B en-R euven,
and the n e w form alism s. D ata near 2 atm . total p ressure at 213 K in F ig u re 7.7(a) illustrate the general
overestim ation o f absorption by the B erge an d G ulkis form alism at th at tem perature. Its p redicted fre­
quency dependence is also incorrect, disagreeing with the data by about 10% at the highest frequency and
m ore than 30% at th e low est. D ata at 8 atm . total p ressure and 27 3 K , in F igure 7.7(b), better illustrate
frequency dependence errors. A lthough the predictions o f the B erge and G ulkis form alism are fairly accu­
rate o v er this freq u en cy range, w ithin about 10-15% , it is obvious th at the errors are system atic an d at
low er and h ig h er frequencies they w ill b e larger. In F ig u re 7.7 the im provem ent o b tain ed w ith the new
151
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(a) A b so rp tiv ity o f N H 3 + (H 2 + H e) at 213 K
120
N u m b er M ixing R atios: N H 3, 0.0 0 8 2 ; H e, 0.0992; H 2 ,0.893
T otal Pressure: 1.934 A tm
100
B erge & G u lk is B R
T h is W ork
vvw
9
10
11
12
13
14
Frequency, G H z
15
16
17
18
(b) A bso rp tiv ity o f N H 3 + (H 2 + H e) at 273 K
N u m b er M ixing R atios: N H 3, 0.0082; H e, 0.0992; H 2, 0.893
T otal Pressure: 8 A tm
200
150
B erge & G ulkis BR
T h is W ork
VVW
9
10
11
12
13
14
15
16
17
18
Frequency, GHz
F ig u re 7.7: E xam ples o f ab sorptivity d ata o n am m o n ia in a Jo v ian m ixture o f h ydrogen and heliu m
com pared to the predictions o f V an V leck -W eissk o p f (V V W ), B erge and G ulkis B en-R euven, and the
new B en-R euven-based form alism from this w ork. T he data, taken as part o f this w ork, are used only to
test the accuracy o f prediction m ethods and w ere n o t u sed in generating the new form alism . Im proved
accuracy w ith the new form alism is evident. A lso illustrated is the convergence o f V V W predictions and
the d a ta as p ressu re decreases, p o in tin g to the u sefulness o f a V V W form alism at p ressu res to o low to
allow use o f the n ew form alism .
152
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A bsorptivity o f N H 3 + (H 2 + H e) at 203 K
500
F rom Joiner, Steffes, and Jenkins (1989)
T otal P ressure: 2 A tm
VVW
400
B e rg e & G u lk is B R
1
300
< 200
T h is W o rk
100
32
33
34
35
36
Frequency, GHz
37
38
39
40
Figure 7.8: A bsorptivity d ata on 1.75% am m onia in a Jovian m ixture o f hyd ro g en and heliu m at 203
K, com pared w ith the predictions o f V an V leck-W eisskopf, B erge an d G ulkis B en-R euven, and the new
B en-R euven -b ased fo rm alism fro m this w ork. T w o independent sets o f data, in d icated by circles and
squares, are from Jo in er e t al. (1989), an d represent m in o r extrapolation in tem perature and significant
extrapolatio n in freq u en cy fro m the d a ta used to g en erate the n ew form alism . T h e g lo b al correction
factor C from this w ork w as extrap o lated using the quadratic form ula given in E quation 7.28.
form alism is quite ev id en t inboth sets o f data. A gain, data show n in F igure 7.7 are independent o f the
form alism and w ere n o t used in its generation.
R ecent laboratory d ata by Jo in er et al. (1989), show n in Figure 7.8, becam e availab le after the new
form alism w as com plete. T hose d ata w ere taken at a single pressure, 2 atm ., and a single tem perature,
203 K, at frequencies fro m 32 to ab o u t 4 0 G H z. T h is represents a sm all extrap o latio n in tem perature
from the data used to g en erate the new form alism , but the frequency range is a rath er large extrapolation;
it is on the o th er side o f the p eak o f th e am m onia inv ersio n spectrum n ear 24 G H z. F ig u re 7.8 also
show s predictions o f V an V leck-W eisskopf, B erge a n d G ulkis B en-R euven, and th e new form alism s.
T em perature extrapolation o f the g lo b al correction factor C was done with the best-fit quadratic form ula,
E quation 7.28. E xtrapolation upw ard in frequency to 4 0 G H z appears to cause n o problem s.
153
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P revio u s d a ta from the sam e group (Steffes an d Jen k in s, 1987) a t p ressu res o f 1, 2, 4, an d
6
atm .
and tem peratures as low as 193 K support extrapolations d ow nw ard in freq u en cy to 2 G H z and possibly
m uch low er. T h e uncertainties in those d ata m ake them c o n sisten t w ith bo th the B erge and G ulkis and
the n ew form alism s, bu t in general they clu ster m ore closely around the n ew form alism .
D ifferen ces betw een th e B erge and G ulkis an d the new form alism s h ave im plications for am m onia
abu nd ances a t th e g ia n t plan ets. In general, the new fo rm alism p re d ic ts sm aller ab so rp tiv ities by a
g iv en co n cen tratio n o f am m onia u n d er the low tem p eratu re co nditions o f those atm ospheres. A bsorp­
tivities m easu red b y radio occultation and radio astronom ical m ethods w ill thus require m o re am m onia
in those atm o sp h eres to pro v id e the observed absorption. F o r exam ple, th e V o y ag er rad io occultation
experim ents a t Saturn m easu red absorptivities dow n to a lev el w h ere th e total pressu re w as about 1.26
bars (L in d al e t al., 1985).
A t th at level the am m onia n u m b er m ix in g ratio n e e d e d to pro v id e the
o b serv ed absorptivity w as 66.9 ppm according to the B erge and G ulkis form alism . T h e new form alism
p red icts am m onia absorptivities that are 5 to 40% less th an th o se p red icted by the B erg e and G ulkis
form alism ; using the tan g en t line extrapolation schem e w ith a break p o in t tem perature o f 180 K that
figure is about 30% . This w ould require about 95 p p m am m onia to ex p lain the observed absorptivity.
T h e new fo rm alism w ill also im pact radiative tran sfer m odeling o f the atm ospheres o f the giant
planets. R esu lts o f these m odels depend critically o n the absorptivity p ro files used. M o d els are fre ­
quently used in attem pts to m atch predicted therm al radio em ission sp ectra to the rad io spectra observed
by radio astronom ical m ethods. D r. Im ke de Pater o f the U n iversity o f C alifo rn ia at B erkeley A strono­
m y D ep artm en t has used a version o f the new form alism in radiative tran sfer m odels o f Ju p iter, and has
k indly p ro v id ed displays o f som e o f the results obtained. F igure 7.9 is o n e o f those displays, show ing
the results o f m o d el calculations using Van V leck-W eisskopf, B erge an d G ulkis B en-R euven, and the
n ew fo rm alism s. T h is p articu lar m odel postulates an am m o n ia n u m b er m ixing ratio o f 30 0 p p m fo r
total pressu res g reater than 0.6 6 bar, decreasing at low er p ressu res to 1.5 p p b at 0.3 bar. In the rela­
tively "w arm " atm osphere o f Ju p iter there is not a g reat deal o f d ifference v isible betw een predictions o f
th e B erge and G u lk is and n ew form alism s, b u t the new fo rm alism ’s sm a lle r ab sorptivities do allow
radiation to penetrate from d eep er levels, raising pred icted brightness tem peratures ab o u t 10-20 K . A t
154
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450
BEN REUVEN PROFILE
VAN VLECK WEISKOPFF
400
SPILKER
NH3: 3 .0 E -4
350
/
d e c r e a s e NH3 a t 0.66 bar, dow n to 1.5E -9 a t 0.3 b a rs
£ 300
(U
o,
eQJ
K 250
<u
C
200
150
100
- .5
0
.5
lo g a r ith m w a v e le n g th (cm
1
1.5
F ig u re 7.9: F u ll-d isk rad io astro n o m ica l b rig h tn e ss tem p e ra tu re m easu rem en ts o f J u p ite r w ith the
resu lts o f radiative tran fer m o deling o f the Jo v ian atm osphere based on V an V le c k -W e issk o p f (V VW ),
B erg e and G ulkis B en-R euven, and the new B en -R eu v en -b ased form alism from this w ork, p ro v id ed by
D r. Im ke de P ater. A ll m odels u se identical (po stu lated ) vertical profiles o f am m onia's m ix in g ratio, so
differences reflect differences in predicted opacities from those form alism s.
th e low er tem p eratu res o f th e m o re d is ta n t p la n ets the d ifferen ce w ou ld b e m o re n o tice a b le . L onger
w avelengths on the rig h t side o f the d isplay p en etrate m ore deeply into the atm o sp h ere w here tem pera­
tu res and pressu res are higher. T h e "lo w p ressu re" assu m p tio n th at is the b asis o f th e V a n V leckW e issk o p f fo rm alism is in v alid at the h ig h e r p ressu res, so its predictions th ere d iffer m ark ed ly from
those o f the tw o B en-R euven-based form alism s.
155
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Chapter 8
Conclusion
T his ch ap ter concludes the p resentation o f this research effort. S ectio n 8.1 gives a b rie f sum m ary
o f the m ajor topics addressed by the research. Its organization parallels the organization o f the preceding
chapters o f this m an u scrip t. S ectio n 8.2 lists the prin cip al co n clu sio n s reach ed in the investigations.
T he final section, S ection 8.3, g iv es concluding rem arks and offers several avenues o f future research
suggested by the results o f this research.
8.1
Summary
This w ork covered a broad ran g e o f activities related to scientific inquiry. It began with recognition
o f the need fo r accurate m easurem ents o f a natural phenom enon, leading to the design an d fabrication of
an instrum ent, th e m icro w av e sp ectro m eter, to p erform the need ed m easu rem en ts. T h e m easurem ents
w ere made, and those data w ere analyzed to extract know ledge about the natural phenom enon and express
it in a quantitative form useful to researchers w ho could pro fit from that know ledge.
T he m icrow ave sp ectro m e ter constru cted fo r this research was based on a cavity resonator, exp lo it­
ing the sen siv ity o f such reso n ato rs to the physical properties o f th e ir contents. It p ro v id ed m easure­
m ents o f absorp tiv ity an d re fractiv ity spectra on gaseous m edia at n in e frequencies betw een 9 and 18
G H z, at tem peratures as low as 21 0 K, and at pressures u p to 8.2 atm . V arious d esig n and p rocedural
innovations, such as th e new k in d o f signal p ro b e fo r coupling m icrow ave energy in and out o f the re so ­
nator, resulted in sig n ifican t im provem ents in the sensitivity o f the in stru m en t an d y ielded absorptivity
m easurem ents that w ere m uch m ore accurate than previous m easurem ents on sim ilar gas m ixtures.
156
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T h e spectrom eter was used to m easure refractiv ity sp ectra on g ases know n to be essentially trans­
p arent to m icrow ave radiation. G ases investigated are m ajor com ponents o f the atm ospheres o f the giant
planets. M easurem ents w ere m ade on pure m ethane, m ethane m ixed w ith hydrogen, pure hydrogen, and
pure helium , o ver a significant range o f tem peratures and pressures.
T h e principal use o f the spectrom eter w as m easurem ents o f absorptivity and refractivity spectra o f
gas m ixtures containing am m onia. G ases and m ixtures investig ated in clu d ed p u re am m onia, am m onia
in hydrogen, am m onia in helium , an d am m o n ia in a Jo v ian m ixture o f hyd ro g en and h elium , approxi­
m ately 90% hydrogen and 10% helium by num ber. T h e conditions on the m ixtures, i.e. tem peratures
and pressures, w ere varied o ver the w id est practical ra n g e to p ro v id e as m uch inform ation as possible
about the condition dependences o f am m onia absorptivity. A t each tem perature, spectra w ere m easured
at total pressures o f 1 ,2 , 4,
6
, and
8
atm . D ata w ere tak en on m ixtures o f am m onia in hydrogen at 213
and 273 K, w ith a sin g le spectrum at 1 atm . and 323 K. M ix tu res o f am m onia in helium also included
a full sequence o f sp ectra at 313 K . M easurem ents on three-gas m ixtures o f am m onia in hydrogen and
helium w ere m easured a t 213 and 273 K.
T h e data on am m onia an d m ixtures containing am m onia, along w ith relevant d ata and results from
previous researchers, w ere used to derive a new form alism fo r predicting the m icrow ave absorptivities of
gas m ixtures sim ilar to the atm ospheres o f the g ia n t planets. T o accom plish this the B en-R euven-based
form alism given by B erg e and G ulkis (1976) w as m odified, introducing seven free param eters that could
be adjusted by an optim ization routine to fit the laboratory data. D espite significant problem s requiring
assum ptions, the optim ization techniques applied to the d ata y ield ed characterizations o f th e behavior o f
those param eters as gas conditions varied. In co ip o ratin g the equations describing param eter behavior
into the param eterized form alism p roduced a new am m o n ia absorption form alism that w as show n to be
m ore accurate than e ith e r the B erge and G ulkis o r V an V leck -W eissk o p f form alism s in the im portant
transition region w here n eith er the "low pressure" o r "h ig h pressure" approxim ations central to those
theories are valid. E xtrapolation schem es w ere suggested fo r application o f the new form alism to co n d i­
tions presen t in the atm ospheres o f the giant p lanets, and ex am p les o f such applications w ere given.
157
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8.2
Principal Conclusions
P rin cip al conclusions o f this research belong to three general categories: in stru m en t techniques,
transparent gases, and gas m ixtures containing am m onia. T hey w ill be discu ssed in th at order.
T echniques yielding the increased sensitivity and accuracy o f the m icrow ave spectrom eter provided
the substance o f this research. T h e state o f m icrow ave spectroscopy on am m onia, a field w ith m ore
than h a lf a century o f accum ulated study, requires that to be useful, new pro g ram s o f laboratory data
collection m u st expend considerable e ffo rt in m inim izing m easurem ent uncertain ties. C u rren t fo rm al­
ism s fo r predicting the m icrow ave absorptivity o f Jovian-like gas m ixtures h ave been h o n ed to the point
that, under m o st gas conditions, unless new d ata have uncertainties that are relatively sm all fractions o f
their values they w ill no t be able to d iscrim inate betw een com peting form alism s. T hree m ajor findings
of this w ork related to attaining sm aller uncertainties are:
1. T he new signal probe design detailed in A ppendix A contributed significantly to reducing uncer­
tainties in absorptivity m easurem ents. It provided higher efficiency in translating resonance field
energy into propagating m icrow ave energy in cables to the spectrum analyzer, y ielding sim ultan­
eous im provem ents in SN R and reso n ato r Q ; it provided adjustability o f reso n ato r coupling to
allow optim ization for the requirem ents o f p articu lar experim ents; and it sig n ifican tly reduced
the m ajor source o f "dielectric loading" bandw idth changes, the variation o f im pedance environ­
m ents in and around the p ro b e due to m ultiple reflection interfaces.
2. Procedures described in A ppendix B for elim inating the effects o f inaccurate and drifting spectrum
analyzer interm ediate frequencies greatly reduced the uncertainties o f refractivity m easurem ents.
3. A ccurate m easurem ents o f tem peratures and pressures w ere vital to attaining accuracies prom ised
by th e im provem ents d escrib ed above. C areful attention to calibration techniques p lay s a large
p art in realizing the potential accuracies o f therm om eters and pressure gauges. U nless tem pera­
ture and pressure uncertainties are k ep t sm all m easurem ents w ith a sensitive sp ectro m e ter are
likely no t w orth the e ffo rt
Im provem ents in refractivity m easurem ents w ere applied to the transparent gases discussed in the
previous section. T hose d ata revealed n o surprises and confirm ed theories o f the b ehavior o f such gases:
1. D ata w ere consistent with d ensity-norm alized refractivities invariant w ith tem perature, pressure,
or frequency; refractivities o f m ixtures w ere linear in the num ber densities o f th e gases.
2 . M easured values o f the d ensity-norm alized refractivities w ere consistent w ith accepted values at
optical frequencies.
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T he prim ary subject o f this research w as the m icrow ave absorptivity o f am m onia an d gas m ixtures
containing am m onia. Several conclusions could be m ade from the d ata acquired as p art o f this w ork:
1. N either the V an V leck -W eissk o p f n o r the B erge and G ulkis B en -R euven form alism s adequately
describe the b eh av io r o f am m o n ia in the transition pressure reg io n betw een the "low pressure"
and "high pressure" regim es used in th eir derivations. E rrors exist in the tem perature, pressure
and frequency dependences o f both formalism s.
2. A conclusion im p licit in th e form s o f the V an V leck-W eisskopf a n d the B erge an d G ulkis BenR euven form alism s, that ab so ip tio n line w idths and coupling are lin ear w ith the partial pressures
o f the gases in a m ix tu re co ntaining am m onia, hydrogen, and/or h elium , is no t co n sisten t w ith
the laboratory data o f this w ork.
3. A new form alism , d ev elo p ed as p art o f this research, allow ed system atic variation o f absorption
line broadening and coupling coefficients and yielded m ore accurate predictions o f the m icrow ave
absorptivity o f am m onia in g as m ixtures applicable to the atm ospheres o f the g ian t planets.
4. U sing the n ew form alism , red u ctio n o f new rad io astronom ical an d rad io occultation data and
rein terp retatio n o f ex istin g d ata fro m the g ia n t plan ets w ill in crease the e stim ated am m onia
abundances at those planets o v er abundances estim ated using the B erge an d G ulkis form alism .
T he am o u n t o f the in crease w ill v ary w ith conditions, esp ecially tem p eratu re, and m ay range
from a few percent to as m uch as 50% .
5. M ore laboratory d ata are n eed ed on the m icrow ave absorptivity o f am m onia in Jovian-like m ix­
tures at p ressures betw een about 0.25 and 2 atm ospheres, w ith sufficiently narrow resolution in
pressure to define the transition from m oderate and high pressure behavior to low pressure behav­
ior. D ata are also needed at lo w er tem peratures.
8.3
Suggestions for Future Research
T his research suggests several avenues fo r continued research concerning the m icrow ave sp ectro s­
copy o f am m onia an d its a p p licatio n to plan etary science. Possible investig atio n s in v o lv e laboratory
w ork, analysis o f laboratory data, application o f the results o f this research, and theoretical investigation
o f the phenom enon o f m icrow ave absoiption by am m onia and the effects o f foreign gases.
M ore laboratory absorptivity data are needed to resolve the transition from m oderate to high pressure
regim es exhib itin g B en -R e u v e n -lik e b eh av io r, to lo w er p ressu res resem b lin g V an V le ck -W eissk o p f
behavior. D ata should be taken a t p ressures fro m ab o u t 0.25 to 2 atm ., w ith a spacing o f 0.25 atm . or
less, at three o r m ore d ifferen t tem peratures. It w ould probably be advantageous to red u ce and analyze
the data as it is acquired to allow additional m easurem ents in any specific problem areas discovered.
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M ore accurate definition o f th e tem perature dependence o f absorption by am m onia w ill also require
m ore laboratory d ata a t lo w e r tem peratures. T he satu ratio n v a p o r p ressure o f am m onia is such th at it
can produce m easurable m icrow ave absorption in the atm ospheres o f the giant planets a t tem peratures as
low as 125 K . A t the p resen t tim e ex trapolations to th at tem p eratu re m u st span a gap that is nearly as
large as the full ran g e o f tem peratures fo r w hich d ata exist. U ncertainties in the d ata prev en t tight co n ­
straints on ex trapolations, so such larg e extrap o latio n s p la c e relativ ely loose constraints on ab so rp tiv i­
ties. D a ta at lo w er tem p eratu res w ould in crease th e ran g e o f co n tro l, p lacing tig h ter constraints on
tem perature d ependences, an d w ould reduce the req u ired span o f extrapolations. A ttem pts to m easure
am m onia ab sorptivities at lo w er tem peratures w ill be lim ited by th e com bination o f spectrom eter sen si­
tivity lim its and th e vapor p ressure o f am m onia, w hich decreases rap id ly w ith tem perature. Sensitivities
attained w ith the apparatus constructed fo r this research w o u ld enable m easurem ents at tem peratures as
low as 175 K , w here the saturation vap o r pressure o f am m onia is ab o u t 3 torr.
G reater sensitivities co u ld be attained by increasing the Q o f the cavity resonator. T his w ill depend
on new m aterials fo r the surfaces o f the cavity. T echniques applied in this research attained Q values as
high as 80% o f the theoretical values fo r brass, the m aterial used. G old o r silver plating w ould increase
the ex pected Q values, b u t only b y a few p e rc e n t M uch m ore sig n ifican t increases co u ld resu lt from
applying a lay er o f one o f the n ew h ig h -T c superconducting m aterials to the surfaces o f th e cavity. T his
w ould enab le m easurem ents in a w ide ran g e o f new areas. R esearch w ould be need ed to determ ine the
extent o f am m onia interaction w ith superconducting m aterials, through such phen o m en a as adsorption
or even chem ical reactions, before laboratory m easurem ents using such m aterials could be attem pted.
O nce n e w laboratory d ata are available it w ill be necessary to m odify current prediction form alism s
to re fle c t kn o w led g e c o n tain ed in the new data. N ew approaches to analysis o f the d ata m ight yield
better w ays to describe the beh av io r observed. O ne highly d esirable result from such analyses w ould be
the develop m en t o f a single absorptivity p rediction fo rm alism that is valid fo r all pressures, elim inating
the current n ee d for tw o different form alism s to handle different pressure ranges.
T h e new form alism o f this w ork provides im m ediate avenues o f research in applications to existing
radio astronom ical and rad io occultation data from the gian t planets, and to radiative transfer m odeling o f
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those atm ospheres. M any o f those d ata have b een interpreted w ith the B erge and G ulkis absorption fo r­
m alism , so rein terp retin g them in the lig h t o f the new fo rm alism w ill chan g e th eir in d icated am m o n ia
abundances. D ifferences betw een predictions o f the n ew form alism and the Berge and G ulkis form alism
depend on tem peratures and pressures, so reinterpretations w ill require point-by-point recalculations.
Finally, research is n eed ed to un d erstan d the ph y sics o f interactions o f am m onia m o lecu les w ith
m icrow ave rad iatio n , and th e effects o f fo reig n gases. R e su lts p resen ted in the p rev io u s ch ap te r are
entirely em p irical and represent no new k n ow ledge o f the underlying physics. T he failure o f previous
theoretical treatm ents o f the p ro b lem by V an V leck and W eissk o p f and B en-R euven d em onstrate that
physicists h av e n o t y et uncovered all p henom ena inv o lv ed in the interactions. W hen all such p h en o m ­
ena are w ell understood extrapolations from laboratory d a ta to the conditions existing in the atm ospheres
of the gian t planets w ill n o t be so troublesom e.
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Appendix A
Design o f the Resonator Signal Probes
T he signal probes used on the reso n a to r built fo r this p roject are a new design. H istorically, reso n a­
tors in the X and K m icrow ave frequency bands h ave b een fed b y w aveguide system s th a t suffered from
sm all useful bandw idths. A t low er frequencies, w here w id er bandw idth coaxial cables co u ld b e used, an
alternate m eth o d was to co u p le the signal to an d fro m the reso n ato r v ia m agnetic fie ld probes w hich
w ere sm all lo o p antennae. T he first p air o f signal p robes b uilt for this reso n ato r w ere m agnetic field
loops. W h ile using those probes pro b lem s w ere encountered w ith in su fficien t o u tp u t sig n al strength,
especially at the higher frequencies w here absorption by am m onia is stronger. T here w ere also problem s
with sensitivity o f the probe im pedances to the dielectric constant o f the cavity contents, due to refle c ­
tions generated by m ultiple im pedance contrasts w ithin the probes.
M y solution to these pro b lem s w as to design a new type o f probe, illustrated in F ig u re A .I. It
resem bles a m agnetic field loop, b u t the lo o p is o p en . T he loop is fashioned fro m the ex ten d ed center
conductor o f the 0.085" O D sem irigid coaxial cable th at feeds th e resonator. I f it w ere a true m agnetic
field probe th e loop w ould connect to the o u ter c o n d u cto r o f th e cable. H ow ever this lo o p has a sm all
gap betw een its end and the cable outer conductor. T he loop and gap behave as the inductance and capa­
citance o f a lo w -Q tuned circuit. T his circuit's cen te r frequency is slightly hig h er than the highest used
in the experim ents, so the probe response increases w ith frequency over the range o f frequencies used. In
the signal in jecto r probe, cu rren ts in the lo o p generate m agnetic fields n ear the p o rt iris th at propagate
into the cavity. C urrents in the detector p ro b e loop are driven by fields entering the p ort v ia the iris. A
brass bulkhead soldered to the end o f the cab le o u ter conductor grounds the ou ter c o n d u cto r to the port
w all. T his isolates the loop from the inactive portion o f th e port, betw een the cable o u ter c o n d u cto r and
the port w all, th at itself resem bles a coaxial cable and could cause interfering reflections. A thin lay er o f
indium m etal o n the port w alls in the vicin ity o f the bulkhead ensures a g o o d sliding connection.
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Both probes are mounted on the head plate with highly m odified SM A connectors that serve in a
mechanical capacity only. Fem ale SM A -to-solder-post jacks function as a mounting flanges. Each is
fixed to the head plate, over a probe port, with machine screws. The center post has been removed, and
the resulting hole in the PTFE dielectric is enlarged to allow a snug slip fit to the 0.085" sem irigid
cable. A male SM A cable connector ("plug") with its center pin and dielectric rem oved is soldered to the
outer conductor o f the cable. The plug has a threaded sleeve which mates to the threads o f the mounting
flange. The distance from the loop to the iris may be continuously adjusted by turning the threaded
sleev e o f the plug, allowing control of the coupling between the probe and the resonator. Strong coup­
ling yields a high signal-to-noise ratio (SN R ) but decreases the figures o f merit (Q c) o f all resonances.
This is desirable when measuring large absorptivities that can significantly decrease the SN R . For
sm all absorptivities the SNR is not as important, but high Q c values are advantageous;
coupling
should be weaker than that o f the high-absorptivity case. The adjustability o f the new probes permits
optimization o f coupling for the needs o f a particular experiment.
Tests o f the new probes compared their performance to the older magnetic field loops. With the
new probes adjusted to give resonance figures o f merit approximately equal to those obtained with the
old probes, the output signal strength o f the TE012 resonance at 9.17 GHz was only slightly higher than
with the old probes. At higher frequencies, however, the improvement in signal strength increased. The
signal from the TE024 resonance at 17.43 GHz was more than 10 dB stronger than with the magnetic
field probes. B y adjusting the new probes for stronger coupling it was possible to increase the output
signal strength an additional 5 dB or more, but at the expense o f up to 25 % decrease in Q c . Adjusting
them for much weaker coupling increased the observed Q c o f the TE024 resonance from approximately
21,000 (with the old probes) to 25,500. A s a final test, the gaps in the loops were shorted to the outer
conductor o f the coaxial cable to make closed loops; the probes were now magnetic field loops. The
output signal strengths o f all resonances fell drastically, especially those at higher frequencies. Perform­
ance returned to the original levels when the gap was restored.
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R esonator Signal Probe D esign
Scale: 1 cm = 0.25 cm
Coaxial Cable
Outer Conductor
S o ld e r
J o in t'
PTFE
Dielectric
M o d ifie d
SMA Plug
'/ r
/ —
?~T
T h read A dvance
0. 72 mm per turn
(show n fully engaged)
PTFE Spacer
Coaxial Cable
Inner Conductor
M o d ifie d _
SMA Jack
Head
Plate
Head
Plate'
S o ld e r
Joint'
S lid in g
Bulkhead
P r o b e ,
Loop '
C av ity
In te rio r
Figure A .l: D esign o f the n ew reso n ato r signal probes w ith adjustable coupling.
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Appendix B
A Method of Compensating For
Spectrum Analyzer Intermediate Frequency Errors
M easurem ents o f the refractivities o f gases w ith a m icrow ave cav ity resonator require very accurate
m easurem ents o f resonance center frequencies. F o r exam ple, in o rd er to m easure th e refractivity o f a gas
m ixture w hose refractiv ity is 150 x 10 - 6 to an accuracy o f o n e p erc e n t a t a frequency o f 15 G H z, the
c en ter freq u en cy o f th e resonance under both the ev acu ated and lo ad ed conditions m u st be m easured to
w ithin app ro x im ately 10 kH z. N orm ally this is n o t an u n reaso n ab le accuracy fo r a high resolution
spectrum analyzer coupled w ith a frequency counter.
T w o signal frequencies internal to a spectrum analyzer are critical to its accuracy. An input signal
is m ixed w ith a signal th a t is an harm onic o f the analyzer's lo ca l o scilla to r, o t L O , producing a signal
w hose frequency is the difference o f the tw o m ixed signals. A filter selects only those signals w ithin a
narrow ban d (the w idth o f w hich is usually adjustable) aro u n d a p articu lar w ell-determ ined frequency,
called the interm ediate fr e q u e n c y o r IF . T he user m anually tunes the L O to bring the frequency o f the
m ixed signal into the filter passband, and the m ixed sig n al then passes to th e electronics that d isplay its
strength on the an aly z er screen. W hen the frequency o f the m ix ed signal is in the c en te r o f the IF p ass­
band, the input signal frequency is equal to th e L O harm onic frequency plus or m inus the IF; the analy­
zer has p ro v isio n s allow ing the user to discern "plus" from "m inus."
A ccurate m easurem ent o f the
frequency o f the inp u t sig n al requires accurate k n o w led g e o f both the analyzer's internal IF an d th e LO
frequency need ed to bring the IF signal to the center o f the IF filter passband.
T h e sp ectru m analyzer used in the experim ents p erform ed as p a rt o f this w ork is described in S ec­
tion 4.2, pag e 51. It has m any ex cellen t features w hich m ade it easily adaptable to the task. A band
select sw itch allow s the user to select the harm onic o f the L O used in the internal m ixer, and sim ultan­
eously specify w hether the frequency being displayed is the L O harm o n ic frequency plus the IF, o r the
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L O harm onic frequency m inus the IF. A band is desig n ated by an in teg er num ber and e ith e r "+" o r
F o r exam ple, the b an d labeled "4+" displays the stren g th o f an in p u t signal w hose frequency is four
tim es that o f the L O , plus the IF: fjnput = 4f|_o + IF • F o r band "3-," fjnput = 3f|_o ■ IF . T he L O sig­
nal is available v ia a front-panel jac k , so connecting the L O to a frequency co u n ter fo r m easurem ent o f
the L O frequency to w ithin 1 kH z is a sim ple m atter. Ideally the IF o f this analyzer m odel is set a t p re ­
cisely 2.05 G H z, co m p en sated so it varies considerably less than 1 k H z fro m th at value. If this w ere
still the case, accurate frequency m easurem ents (and thus refractivity m easurem ents) w ith this particular
spectrum an aly zer w o u ld be sim ple an d straightforw ard: the L O frequency is m easured once, an d the
input signal frequency, accurate to w ithin a few kH z, is calculated using the equations o utlined above.
U nfortu n ately age has taken its toll on the 20-y ear-o ld m achine, an d the IF filte r is no longer as
stable as its design specifications state. In fact, in test m easurem ents o f a know n, precise frequency, the
IF w as o bserv ed to vary betw een 2.0 4 9 2 and 2.0502 G H z, o r 2.0497 G H z ± 5 0 0 kH z. T his variation
w as sensitive to am bient tem peratures b u t experim ents sho w ed th at tem perature alone, even the internal
tem perature o f the analyzer, was no t a good predictor o f the IF value. F ortunately the IF d rift rate w as
low , and at tim es the IF rem ained alm ost constant though considerably differen t from the nom inal value.
A n IF inaccuracy o f ± 500 kH z contributes directly to a frequency m easurem ent accuracy o f no better
than ± 500 k H z. F o r m ost gases such inaccuracy w ou ld ren d er attem p ted re fractiv ity m easurem ents
useless. S ince the $50-100,000 fo r a n ew spectrum analyzer w as no t av ailable, the alternative w as to
som ehow m easure the IF to significantly reduce that error. T h e IF cannot be m easured w ith a frequency
counter. A n IF signal is qu ite w eak co m p ared to the LO output, varies considerably in am plitude w ith
the in p u t sig n al strength, and has no pro v isio n fo r its av ailab ility ex tern al to the sp ectru m analyzer.
Thus the IF m ust be inferred rather than m easured directly, and the inferred value is then used to com pute
a m ore accurate value for the input signal frequency.
T he problem o f inferring the IF can be divided into cases based on the behavior o f the IF with time.
T he sim plest case, the "C onstant IF" case, involves an IF that has strayed from its nom inal value by an
unknow n am ount, b u t does n o t vary significantly from the pertu rb ed v alue w ith in the tim e necessary to
m ake the req u ired m easurem ents. A m o re com plex case, the "D rifting IF" case, occurs w hen the IF is
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not only displaced from its nom inal v alu e but also changes w ith tim e. If the d rift rate varies sufficiently
slow ly that it rem ains fairly co n stan t o v er the tim e req u ired fo r th e m easurem ents, lin ear d rift can be
assum ed. T h e case o f no n -lin ear, rap id ly vary in g d rift is a v ery d ifficu lt p ro b lem to ov erco m e, and
fortunately solving th a t p ro b lem w as no t necessary. T he d rift rates o b serv ed v aried slow ly; o v e r a
period o f five m inutes they w o u ld rem ain fairly constant, and thus a "L inearly D riftin g IF" case can be
applied w ithout sig n ifican t loss o f accuracy. W hen the IF rem ain ed nearly stab le the sim p ler C on stan t
IF case applied. T his c ase w ill b e treated first.
W hen the spectrum analyzer is perform ing w ithin its design specifications, finding an input sig n al’s
frequency is a m atter o f solving a single equation w ith one unknow n: the inp u t signal frequency. T he
m ajor source o f erro r lies in the accuracy o f the m anual L O tuning. In the case o f the aging analyzer the
EF v alu e becom es an unknow n also, and the sim ple reduction fo rm u la becom es a single eq uation w ith
two unknow ns. A n o th er equation, linearly independent o f the first, is need ed to so lv e for eith er o r both
o f the tw o unknow ns. O btaining tw o linearly independent equations is accom plished by reasoned use of
the spectrum analyzer's band se lec t capabilities.
O btaining the first independent equation is essentially identical to a norm al frequency m easurem ent.
The analyzer b an d select is adjusted so the band designation is o f t h e "+" type an d the in p u t signal falls
w ithin the tuning ran g e o f that band. T he L O frequency m ultiplication factor fo r this band w ill be given
the sym bol A, an d thus th e full designation o f the chosen band is "A+." W hen the L O is tuned to place
the p roper signal in the center o f the IF passband the L O frequency is m easured. In th e absence o f errors
the correct value this m easu rem en t should yield is represented by the sym bol f|_0,A- B ut L O frequency
m easurem ents are subject to ran d o m errors, such as errors in tuning the L O to the ex act frequency that
centers the d esired signal on the screen (i.e., places the signal resu ltin g fro m m ixing the inp u t signal
with the L O harm onic signal in the c en te r o f the IF passband) an d errors resulting from the frequency
instability o f the L O itself. T he su m o f all such errors for an L O frequency m easu rem en t m ade w hile
the spectrum analyzer is set for a band num ber o f A w ill be sym bolized by ea> T hus the m easured L O
frequency i*LO,A is sum o f the proper L O frequency plus the ran d o m erro r term : i*LO,A = fl_0,A + £A-
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T he relatio n sh ip betw een the true inp u t signal frequency, fjnput. and the p ro p er L O frequency and the IF
is given by
^input= ^ L O , A +
>
(B .l)
while the relationship betw een the indicated input signal frequency, f jnd. and the m easured L O frequency
and IF is
find= A fi_o,A + IF = A(f|_ 0 ,A+ e A) + IF
.
(B.2)
A seco n d linearly independent equation is o b tained by m easuring the freq u en cy o f the sam e inp u t
signal w hile th e spectrum analyzer is set fo r a b an d w ith harm onic m u ltip lier n u m b er B and a d e s i g ­
nation, band ” B-." T he relationship betw een the indicated in p u t signal frequency an d the m easured L O
frequency and IF is
find = B f Lo ,B - I F = B (fL0 ,B +£B ) - IF
(B.3)
Solving E q u atio n s B.2 and B.3 sim ultaneously yields the d esired result, a v alue fo r the indicated input
signal frequency th at is independent o f the IF value:
•
*
,
A f lo, a + B f lo , b
find = ------------2 --------------
■
(B 4 )
A lthough it ap p ears th at the IF has n o t b een determ ined, one w ay to look a t this re su lt is that the IF
value is calc u lated and applied internally in the solution o f the system o f equatio n s. T o verify th at the
m ethod yields the co rrect result, expressions for the m easured L O frequencies are substituted into E qua­
tion B.4 and th e IF is added and subtracted to give
[ A ( f LO.A+ E a ) + IF] + [ B j f L O i B + t B ) - IF]
find
=
2
A E a+ B E b
=
f
input +
2
T he indicated signal frequency obtained by this m ethod d iffers from th e true signal frequency by a ra n ­
d o m error term arising from m easurem ent errors in the L O frequencies. T hus th e true signal frequency is
reco v ered w ith o u t system atic bias. T he ran d o m error is ex p ected to be sim ilar in size to th a t o f a
m easurem ent using the norm al procedures fo r a know n IF.
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T he m eth o d o u d in ed above does no t w o rk if the IF d rift rate is large. A ssum e th at the IF is drifting
a t a co n stan t rate D , the first m easu rem en t is m ade a t tim e to w hen the IF value is IF q » an d the tim e
elapsed betw een th e first and second L O frequency m easurem ents is T. T h e first L O frequency w ould be,
in the absence o f random errors and with errors included,
(B.5)
V /hen the se co n d m easu rem en t is m ade the IF has d rifted an am o u n t D T . The second L O frequency
w ould be
fLO,B-B [T|nput+(IF0+DT)] ,
f LO.B = g [ f input+( IFq+DT)] + 6 b .
(B.6)
S ubstituting the m easured L O frequencies fro m E quations B .5 and B . 6 into th e reduction form ula, E q u a­
tion B.4, yields the indicated signal frequency:
DT
2
A ba+ B b b
+
2
(B.7)
T he indicated sig n al frequency is sy stem atically biased by h a lf the total IF drift that occurs betw een the
tw o m easurem ents. If D an d T are sm all en o u g h that the d rift-in d u ced term is sm aller than th e expected
random errors, then this resu lt reduces to the C o n stan t IF case. H ow ever i f D is large (since T, the tim e
required to p erfo rm one m easurem ent pro ced u re, is fairly co n stan t) the resu ltin g bias to the frequency
m easurem ents is significant and can bias subsequent refractivity calculations. Since drift rates this large
w ere som etim es observed during IF tests, a m ore pow erful procedure was needed to handle th at situation.
A lthough d rift rates could b e large they d id not vary sig n ifican d y o v er a period o f several m inutes,
so the IF value c o u ld be assum ed to be linear w ith tim e d u rin g a m easurem ent sequence; hence this case
is the "L inearly D riftin g IF" case. A ssum e again that the IF d rift rate is D; the first m easurem ent o f an
L O frequency is m ad e at a tim e w hen the IF v alu e is IF q ; an d the tim e req u ired betw een consecutive
m easurem en ts is T . This la st assum ption is th e least rig o ro u s. H ow ever, in tim ed tests o f the p ro ce­
dure being d ev elo p e d here the tim e betw een consecutive m easurem ents varied only ±10% o r less. It will
be show n th a t for m y purposes this variation introduced acceptably sm all erro rs in the results.
169
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In the L inearly D riftin g IF case tim e is a critical factor. T h e tim e variation o f the IF introduces a
third unknow n, the d rift rate D, in to the in dividual red u ctio n form ulae. I f a m easurem ent is m ad e at a
tim e N T after th e initial m easurem ent, the co rrect reduction fo rm u la w ould be o f the form
find = M f lo,m ± ( IF 0 + N DT)
(B.8 )
T h e u nk now ns are fjncj, IFo, and D. A th ird in d ep en d e n t equ atio n , an d thus a third m easurem ent, is
necessary to provide a com plete system o f equations. If it w ere po ssib le to m ake sim ultaneous m easure­
m ents on tw o differen t bands the d rift term in E q u atio n B . 8 co u ld b e elim inated, since T w ould be zero,
but sim ultan eo u s m easurem ents are im possible. B u t it is p o ssib le to sy n th esize a m easu rem en t at a
given tim e fro m tw o m easurem ents using the sam e b an d b u t sep arated in tim e. Specifically, i f p ro c e ­
dures fo r the C o n stan t IF case are u sed and the seco n d LO m easu rem en t (using band B-) is im m ediately
follow ed by another m easurem ent using band A +, the tw o m easurem ents w ith b an d A + m ay be used to
synthesize a virtual m easurem en t using b an d A + th a t is sim ultaneous w ith the m easurem ent using b an d
B-. F o r the three m easurem ents the ideal (free o f ran d o m error) and m easured L O frequencies are:
Ideal
Measured
^ ( f input- IFo)
[finput+ ( IF 0 + DT)]
- [ f in p u t-( I F 0 + 2 D T ) ]
The system o f individual reduction equations is:
find = A f l o , A 1 +
IFo
find =
B f * LO,B - ( I F o + D T )
find =
A f lo,A 2 +
( IF 0+ 2 D T )
(B.10)
.
Solving the sy stem o f equations B .1 0 for f jncj p ro d u ces th e d e sired result, a red u ctio n equation that
expresses the indicated input signal frequency in terms o f the three m easured LO frequencies:
(B. 11)
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T o verify th a t this m ethod produces th e co rre ct re su lt th e m easu red L O frequencies from E quations B .9
are substituted into E quation B .l 1, yielding
T hus th e m eth o d recovers the true inp u t signal freq u en cy w ith no system atic bias. T he size of the ran­
dom error term is expected to be the sam e o r sm aller than the random erro r term associated with a single
frequency m easurem ent
T h is m e th o d assum es the elapsed tim e betw een th e first and seco n d m easurem ents is the sam e as
that betw een the second an d third m easurem ents. T ests m entioned previously indicate tim es could vary
up to 10% fro m an average value. T h e e ffect o f such variability on the accuracy o f the re su lt sh o u ld be
exam ined. C all th e total tim e elap sed betw een the first an d third m easurem ents 2 T . A ssum e the tim e
betw een th e first and second m easurem ents is (1 - E )T , w h ere E is a relative error, so the tim e betw een
the second and third m easurem ents is (1 + E )T . T his m odifies the second line o f E quations B.9:
[ finput+ ( IF q+ D(1 "E)T)] + £ b .
(B.12)
T he reduction E quation B . l l rem ains unchanged since the m ethod assum es equal tim e spacing.
Substituting the new m easured LO frequency from E quations B .12 into Equation B. 11 yields
t
t
•ind ~ ■input
DEI
"
2
A .
"*"4
c
, B
A2 J g
0
B
A system atic bias has in d eed been introduced. A n u p p er bound for the m agnitude o f th at bias in this
application o f the m ethod can be estim ated fro m test d ata. T h e m axim um value for E is a b o u t 0.1, the
10% noted earlier; the largest IF drift observed over a p eriod o f three consecutive m easurem ents was less
than 50 kH z, so an upper bound fo r the d rift rate D w o u ld be 25 kH z p e r T. Substituting these values
into the drift term above, the up p er bound on the system atic erro r is 1.25 kH z. T his is sm aller than the
ex p ected size o f the ran d o m erro r term and w ill n o t sig n ifican tly im p act refractivity m easurem ents,
especially those on gases w ith large refractivities. T o m in im ize system atic errors w hen m easuring low
refractivities, m easurem ents should not be m ade during periods o f large IF drift rates.
171
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T h e L inearly D rifting IF m ethod g iv en above m ay be ex ten d ed by m aking o n e m o re m easu rem en t
o f the L O frequency on b an d B-, after the
th ree L O m easu rem en ts o f th e n orm al L in early D riftin g IF
m ethod. A reduction equation for the final three m easurem ents is deriv ed by the sam e m ethods u sed to
d eriv e E quation B . l l , yielding a very sim ila r resu lt. T h e n e t effec t is to sy n th esize an L O freq u en cy
m easu rem en t on band B - that is sim u ltan eo u s w ith the se c o n d m easu rem en t on b an d A + , p ro d u cin g
u nbiased m easurem ents o f IF and input frequencies th at are o n e m easurem ent period, T, later than those
pro duced by the first three L O frequency m easurem ents. This E x tended D rifting IF m ethod has potential
fo r indicating the size o f m easurem ent errors associated w ith the indicated frequencies.
R ed u ctio n o f the First set o f three L O m easurem ents is d o n e w ith E q u atio n B . l l , w ith the m in o r
alteration th at the first LO frequency m easurem ent on band B- is referred to by th e su b scrip t B1 instead
o f B, distinguishing it from the second m easu rem en t on b an d B-, referred to b y the su b sc rip t B 2. T he
second set o f three m easurem ents, subscripted B 1 , A 2, and B 2, is red u ced w ith the equation:
A *
B / *
*
\
find = Tj'f LO,A2 + 4 P LO, B1 + f LO.B2/
•
(B.12)
In this w ork the E xtended D rifting IF m ethod w as used on all baseline m easurem ents as discu ssed in
Section 4.3. O bserved differences betw een the tw o indicated resonance center frequencies for each set o f
m easurem en ts w ere alw ays less than 10 kH z, w ere usually 1-4 k H z, an d w ere often less than 1 kH z.
D ifferences larger than 4 kH z w ere alm o st alw ays associated w ith ch am b er tem perature d rift rates that
w ere la rg e r than usual. IF values w ere also calc u lated fo r each m easu rem en t set. It w as p o ssib le to
track IF ch an g es an d e stim ate long term d rift rates by e x a m in in g IF valu es fo r th e su ccessio n o f
m easurem en t sets. O bserv ed differences in the tw o in d icated IF values o f each set w ere com binations o f
IF drift and m easurem ent error. W hen the IF d rift rate as in d icated by values a t successive points w as
low the differences w ere usually in the 1-4 k H z range. F o r larg er long-term d rift rates the differences
w ere consistent with the drift rates.
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Appendix C
Laboratory Procedures
T h is appendix describes p rocedures u se d in collecting the d ata rep o rted in C h ap ters 5 an d
6
. Four
d ifferen t types o f experim ents on gases and gas m ixtures w ere perform ed: refractiv ity m easurem ents on
the transparent gases m ethane and m ethane m ixed w ith hydrogen; refractivity and dielectric loading m ea­
surem ents on hydrogen and helium , both transparent gases; refractivity an d absorp tiv ity m easurem ents
on pure am m onia; an d refractivity and absorptivity m easurem ents on m ixtures o f am m onia, hydrogen,
and helium . Each required variations in procedure, and som etim es procedures varied w ith th e tem perature
o f the g as in the ex p erim en t A nother im portant p rocedure category, eq u ip m en t calibration, is included.
C .l
Calibration
E quipm ent calibration is a critical asp ect o f any experim ental project. T h is is esp ecially true w hen,
as in this w ork, the goal o f the ex p erim en ts is to pro v id e m ore accurate k n o w led g e o f th e q uantities
being m easured. M any independent m easurem ents o f fundam ental quantities m ust be m ade to arrive at a
deriv ed m easurem ent o f gas absorptivity and refractivity. Inaccuracy in each fundam ental m easurem ent
contributes to uncertainty in the derived m easurem ents. M aintaining high accuracy in the re su lt requires
extrem ely high accuracies in the fundam ental m easurem ents, and this is no t po ssib le i f equipm ent used
to m ake the m easurem ents is not properly calibrated. T he accuracies attained in this w o rk w ere results
o f considerable forethought and extrem e diligence and care in the procedures, especially calibrations.
T he raw d ata from these experim ents inv o lv ed four fundam ental types o f m easurem ents: gas p re s­
sure, gas tem perature, m icrow ave signal frequency, and m icrow ave signal po w er. G as p ressures w ere
m easu red w ith o n e o f tw o B ourdon gau g es in co njunction w ith the aneroid b aro m eter. In m o st cases
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tem peratures w ere m easured w ith th e p latinum R TD therm om eter, although early experim ents relied on
the less accurate th erm om eter b u ilt into the therm al cham ber. F requencies w ere m easured by the fre­
quency co u n ter in con ju n ctio n w ith the spectrum analyzer, and relativ e p o w er w as m easu red w ith the
spectrum analyzer. E ach o f these dev ices required careful calib ratio n to insure the accuracy o f th eir
m easurem ents. Som e in strum ents n e ed ed calibration only a few tim es during th e course o f the ex p eri­
m ent program . O thers w ere calibrated before each experim ent and checked afterw ard.
C alibration o f the p ressu re gau g es w as the m ost difficu lt and tim e-co n su m in g p ro ced u re. It w as
d one three tim es during the ex p erim en t program . T he gauges h ad to be rem o v ed fro m the apparatus,
taken to an o th er laboratory for calib ratio n , an d reinstalled. F o rtu n ately the th ree calib ratio n s did not
reveal any large changes in the characteristics o f either gauge. C alibration o f a g au g e w as accom plished
by placing it in parallel w ith a g au g e o f sim ilar ran g e and kno w n high accuracy, th en m ak in g dual
m easurem ents at a series o f pressures. T h ese results w ere used to g en erate a calib ratio n ch art. T he
m aster gauges w ere m aintained by p erio d ic adjustm ents using d ead w eig h t calib ratio n system s. D ual
m easurem ents w ere m ade every
2
psi fo r the high pressure gauge, and every
20
to rr fo r the vacuum and
low -pressu re gauge. B oth gauges depen d ed on the aneroid b aro m eter fo r absolute p ressu re m easure­
m ents.
T h e b aro m eter w as c a lib rate d by m aking sim ultaneous m easu rem en ts o f p re ssu re w ith the
F ederal A viation A gency b aro m eter a t P alo A lto A irport, C alifornia, w h ile o u r b aro m eter w as resting
beside the airp o rt b arom eter. Its co n sisten cy w as such that after the initial ad ju stm en t su b seq u en t
calibrations required no further adjustm ents.
T he spectrum analyzer w as som ew hat easier to calibrate since it did not h ave to b e transported for
the procedure. T w o aspects o f the device required calibration: screen w idth, the ran g e o f frequencies
displayed; an d screen height, the signal pow er dim ension o f the screen. Screen w idth an d linearity w ere
m easured w ith the frequency counter, w hose calibration is described later. M easurem ents w ere m ade for
each o f the screen w idth settings used in the experim ents. Screen h e ig h t w as calib ra te d w ith a signal
from the sw eep generator th a t passed through a calibrated attenuator. B oth w idth and h eig h t calibrations
w ere done fo u r tim es during the ex p erim en t program , and w ere qu ite co n siste n t fro m c alib ratio n to
calibration.
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T h e other two instrum ents, the frequency counter an d the R T D therm om eter, w ere calibrated before
and after each ex p erim en t. F req u en cy co u n ter calib ratio n w as quite sim ple. A n adjacent laboratory
m aintained a one M H z tim e base o scillato r w ith a sig n al accurate to o n e p a rt in 109. T hat signal was
used as an inp u t signal to the frequency counter, an d the in d icated freq u en cy allow ed calculating the
frequency o f the counter's ow n one M H z internal tim e base oscillator. T his yielded a correction factor to
be applied to frequencies m easured by the counter. In every case, the frequency co u n ter m easured the
reference signal as an unvarying 999,999.5 Hz. T hus the correction factor w as 1.0000005.
C alibration o f th e platin u m R TD therm om eter involved tw o adjustm ents. T he therm om eter read in
C elsius units, and o n e ad ju stm en t set the zero poin t o f that scale. A n ice bath m ixture o f liquid w ater
and ice, both from d istilled w ater, w as used for the zero -p o in t reference. A styrofoam -insulated alum i­
num contain er held the m ixture. The zero po int was adjusted as the therm o m eter tem perature probe was
fully im m ersed in th e co n stan tly stirred bath. T his allo w ed settin g th e zero p o in t to w ithin ±0.05 C.
A nother adjustm ent set th e therm om eter's scale size. A calibrated resistance w as substituted for the tem ­
p erature probe. T he resistance value w as that o f the p ro b e at a tem perature o f 500 C. It w as adjusted
w ith a digital m ultim eter designed for four-w ire resistance m easurem ents, accurate to five digits. Effects
o f the tw o adjustm ent potentiom eters w ere n o t com pletely independent, so calibration w as a stochastic
process som etim es req u irin g several iterations o f the tw o -ad ju stm en t p rocedure to properly set both the
zero p o in t and scale. W hen a long p eriod o f tim e p assed betw een one calib ratio n and the n ex t adjust­
m ents w ere not alw ays required, but w hen required they w ere usually o n ly ±0.1 C and w ere never more
than ± 0 .2 C. C alibration checks after an experim ent alw ays ag reed w ith the calibration adjustm ents
m ade before the ex p erim en t
It w as observed that the therm om eter chassis tem perature slightly affected its zero-point setting and
thus the accuracy o f indicated tem peratures. T em perature o f the chassis, m ounted on the side o f the ther­
m al cham ber, varied w ith changes in laboratory room tem perature, so a T a y lo r probe-style bim etallic
coil therm o m eter c alib rated in the F ah ren h eit scale w as in serted into th e R T D therm o m eter case to
m easure chassis tem peratures. N orm al chassis tem perature w as 70 F, b u t this varied fro m
68
to 75 F
with ro om tem perature. C haracterization o f this effect involved m aking zero poin t m easurem ents in an
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ice b ath as the ch assis tem p eratu re w as varied. Z ero p o in t offsets as m uch as -0.1 and +0.2 C w ere
observed. O ffsets o f th e 500 C reference m easurem ents w ere also observed, an d they closely track ed the
zero p o in t offsets. A s a p art o f all tem perature m easurem ents in the experim ents using the R T D th er­
m om eter, th erm o m eter chassis tem peratures w ere recorded allow ing subsequent correction o f th e in d i­
cated tem peratures. T hus th e ± 0 .4 C absolute accu racy o f th e instrum ent w as m aintained. H ow ever,
due to the calibration m ethod and th e therm om eter's operating principles, m easu red tem peratures w ithin
a few C o f the zero p o in t w ere m uch m ore accurate.
R efractivity m easurem ents involving m ethane w ere done before the R TD therm om eter w as acquired,
so tem perature m easurem ents in those experim ents relied on th e therm om eter b uilt into the M issim ers
therm al cham ber. T ests show ed th a t the th erm om eter w as strongly affected by am bient tem peratures.
E fforts to co rrect in d icated tem peratures via k n o w led g e o f am b ien t tem peratures w ere o n ly p artially
successful, reducing the global uncertainty to ±3 C . If am b ien t tem peratures w ere stable the ch am b er
tem perature w as stab le to w ithin ± 0 .5 C, bu t the u ncertainty in the center o f that range w as still ±3 C.
O nly rudim entary room tem perature control was available. D iurnal effects w ere quite evident. A m bient
tem peratures w ere stab le fo r relativ ely sh o rt periods each day, an d this p u t tim ing lim its on the p ro c e ­
dures that could b e used for refractivity m easurem ents.
S om e o f the m eth an e m easurem ents w ere d o n e before the frequency co u n ter w as acquired, causing
m ore sensitivity to am b ien t co n d itio n s. It w as suspected that th e spectrum analyzer's interm ediate fre ­
quency (IF) drifted w ith am bient tem perature changes, a suspicion later verified by the frequency counter.
This w as additional m otiv atio n to restrict m easurem ents to periods o f stable am bient conditions. L ack
o f the frequency co u n te r also added system atic enrors as large as ±5% to frequency changes as m easured
on the spectrum an aly zer screen, since there w as no accurate w ay to calibrate its screen w idth. A cq u isi­
tion o f the frequency co u n ter allow ed calibration o f actual screen widths, so the d ata could be re-reduced
with corrected frequency shifts. This decreased the frequency shift uncertainties to ±1% o r better.
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C.2
Screen Af and Counter Af Methods
T h e refractivity m easurem ent procedure th a t em erg ed from this m ass o f drift p ro b lem s an d re stric ­
tions w ork ed as w ell as could b e expected fro m the equipm ent, b u t used fairly large quantities o f the test
gases. Frequency, pressure, and tem perature m easurem ents necessary to arrive at a refractivity m easu re­
m ent h a d to be m ade w ithin a fairly sh o rt tim e span so drifts in c h am b e r (and thus reso n ato r an d gas)
tem p eratu re an d sp ectru m an aly zer IF w ere m in im ized . F o r in stan ce, reso n an ce freq u en cies u n d er
vacuum conditions co u ld not be m easured one day a n d used w ith loaded frequency data fro m an other day
since the absolute ch am b er tem peratures on th e tw o days could b e as m uch as
6
C different. T herm al
expansion o r contraction o f the resonator m aterials caused the c en te r frequencies o f all resonances to vary
system atically with tem perature. T he variation c o u ld be as much as 320 kH z p e r degree C. Frequencies
u nder v acu u m and lo ad ed conditions had to be m easu red w ithin a few m inutes o f each o ther, a t a tim e
w hen am b ien t conditions w ere stable. This k e p t ch am b er tem peratures w ithin ± 0.5 C o f each o th er but
still u ncertain to ±3 C . Such sh o rt tim e spans w ere achieved by m easuring the chan g e in a single reso­
nance's cen te r frequency on the spectrum analyzer screen as the evacu ated cham ber w as loaded w ith the
test gas, an d then again w hen th e ch am b er w as su bsequently re-evacuated. W ith th e ch am b e r stab ilized
on a targ et tem perature, a given resonance w o u ld be displayed on th e screen, aligned w ith a p ro m in en t
fiducial m ark. A m easured pressure o f test gas w o u ld be in troduced into the cham ber, causing th e reso­
nance cen te r frequency to decrease from the refractive effect. T he relatively large m ass and high therm al
co n d u ctiv ity o f the sim ulation c h am b e r co n trib u ted to th e quick d e cay o f the p ressu rizatio n th erm al
transient. D rift due to the transient w as negligible afte r tw o m inutes. T he sh ift in the c en te r frequency
could then be m easured directly from the screen, w hich cou ld be d iv id ed into ab o u t 4 0 0 reso lu tio n ele­
m ents. T h is sh ift w as biased by any d rift o f th e sp ectru m an aly z er IF or ch am b e r tem perature. To
approxim ately co u n ter th e effects o f such drifts the c h am b e r w as th en evacuated, and after an identical
delay fo r decay o f the therm al transient the up w ard sh ift in the cen ter frequency w as m easured. A v erag ­
ing the tw o frequency shifts elim inated the effects o f lin ear drifts o n the frequency sh ift m easurem ent.
T he center frequency itself could b e read directly from the spectrum analyzer's tuning display, accurate to
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about 4 digits. R efractiv ity o f the test gas w as then calcu lated by divid in g th e freq u en cy shift by the
center frequency. T h is frequency shift w as called Af, and thus this p rocedure w as term ed th e screen A f
m ethod. M o st m easurem ents involving m ethane u sed this m ethod.
In the absence o f a suitable tracking generator, a visible display o f a resonance o n the spectrum ana­
ly zer screen w as accom plished by using the frequency sw eep features o f both th e spectrum analyzer and
the m icrow ave sw eep generator. T he cen ter frequency and screen w idth o f the sp ectru m analyzer w ere set
to co v er a frequency ran g e k n o w n to in clu d e th e d esired resonance. A n internally g e n erated saw tooth
w aveform sw eeps the local o scillato r frequency, co v erin g the screen range. F ro n t-p an el adjustm ents
provided one sw eep in tw o seconds. T he sw eep generator was adjusted to provide a frequency-sw ept sig­
nal w ith a sim ilar frequency range, bu t at a m uch h ig h er repetition rate, u p to 100 H z. As the spectrum
analyzer slow ly sw ept th e frequency range, the sw eep generator signal w ould ex cite th e resonator a t the
sam e frequency the spectrum analyzer w as receiving (w ithin the sp ectru m an aly zer tuning bandw idth),
once per g enerator sw eep. E ach such e v e n t w ould cau se the spectrum an aly zer to d isplay th e resonator
resp o n se to th e in p u t sig n al at th at frequency. A t o th er tim es the sp ectru m a n a ly z e r display show ed
essentially z ero received pow er. T h e appearance o f a synchronization e v e n t w as a v ery narrow clipped
spike on the screen, w here th e h eig h t o f the spike sh o w ed the relative p o w e r rece iv e d by th e spectrum
analyzer from the o u tp u t p ro b e o f th e resonator. T h ese events occu rred up to 2 0 0 tim es p e r spectrum
analyzer sw eep, producing a den se train o f spikes th at represented sam ples in freq u en cy o f the resonator
response. T he tips o f the spikes traced an outline o f the resonator frequency response curve. Theoretical
calculations and hardw are tests verified that the sw eep rate from the gen erato r w as slo w enough that the
resonator w as fully energized to steady state conditions by the input signal.
A fter the frequency co u n ter becam e available the screen A f m ethod w as m odified to use its ability to
m easure frequencies very accurately. T im ing was still im portant due to tem perature d rift uncertainty and
spectrum analyzer IF drift. U nfo rtu n ately the m ethods o f A ppendix B fo r overcom ing th e IF d rift p ro b ­
lem w ere no t form ulated until after all m ethane m easurem ents w ere fin ish ed an d the m ag n itu d e o f the
problem w as recognized. T hus the pressurization-evacuation technique w as retained. B u t the frequency
sh ift itse lf w as no t directly m easured. Instead, w h ile at vacuum , the cen te r frequency o f the resonance
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was m easu red by tuning the u n sw ep t spectrum analyzer to the c en te r frequency and m easuring the spec­
trum an aly z er local o scillato r (L O ) frequency w ith the frequency c o u n ter. A fte r ch am b er loading and
decay o f th e therm al tran sien t the spectrum analyzer w as retuned to the shifted cen ter frequency an d the
LO frequency m easured. T he cham ber w as then evacuated and the center frequency rem easured. A verag­
ing the tw o vacuum m easu rem en ts m inim ized lin ear drift effects. R efractiv ity w as then calc u lated by
dividing the calc u lated freq u en cy sh ift (vacuum cen ter frequency less lo ad ed c en te r frequency) by the
loaded center frequency. This procedure was term ed the counter A f m ethod. It shared one disadvantage o f
the screen Af m eth o d in that both req u ired venting o f the gas sam ple to th e atm osphere fo r each datum
p o in t taken. F o rtu n ately the volum e o f the ch am b er w as sm all enough th at th e total m ethane release
has n o t significantly affected global w arm ing, but subsequent m easurem ents requiring a large n u m b er of
data points w ould have been im practical w ith these m ethods.
O ne set o f refractivity m easurem ents on a m ixture o f m ethane and hydrogen used a procedure incor­
porating abbreviated versions o f both the screen Af and counter Af m ethods. A vailability o f the frequen­
cy counter allow ed calibrating the spectrum analyzer screen w idth, elim inating o n e source o f uncertainty
from the screen Af m easurem ents. W h en the cham ber had stabilized at the targ et tem perature o f 0 C the
center frequency o f the selected resonance, th e T E 0 2 4 m ode, w as m easured v ia m easurem ent o f the sp ec­
trum analyzer L O frequency. T he screen Af m ethod w as used to m easure the frequency shift as m ethane
w as in trod u ced into the c h a m b e r to a p ressure o f 3 atm . A n o th er m easu rem en t o f the sh ifted cen ter
frequency w as m ade w ith the frequency counter. M easurem ents up to this p o in t pro v id ed a refractivity
m easurem ent on the m ethane co m ponent o f the m ixture being generated. H ydrogen w as added to a total
pressure o f
8
atm ., and this w as allow ed to m ix for 12 hours. Screen Af m easurem ents during this step
w ould have b een m eaningless since th e gas m ixture w as inhom ogeneous. A fter m ixing w as com plete
the cham b er w as d ep ressu rized in steps to
6
, 4, 2, and 1 atm . total pressure, an d finally to a vacuum .
Each depressu rizatio n step w as p re c ed ed by a cen ter frequency m easurem ent, w as acco m p an ied by a
screen Af m easurem ent, an d w as follow ed by another cen ter frequency m easurem ent. S ubtracting the
center frequency before the pressure change from the frequency afterw ard yielded a frequency sh ift by the
counter Af m ethod. A fter correcting the frequency shifts m easured by the screen Af m ethod fo r screen
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w idth calibration, frequency shifts m easu red by the tw o m ethods agreed to w ithin 1%. A ll freq u en cy
shifts betw een a g iv en pressure and vacuum could b e added to give a total frequency sh ift fo r th at p re s­
sure, and th at sh ift could be used to calculate the refractivity o f the m ixture at th at pressure.
This p rocedure h ad various advantages and disadvantages. It featured the advantage o f m aintaining
an unvarying m ixing ratio at all total pressures, w here individually m ixed sam ples at each total p ressu re
w ould have had slightly differing m ixing ratios. O ne d isadvantage was that it allow ed m easurem ents o n
the gas m ixture on ly during d epressurization, prev en tin g the lin ear cham ber tem perature and sp ectru m
analyzer IF drift com pensation characteristic o f previous procedures.
It w as recognized that b etter tem perature m easurem ents w ere necessary, so the R T D th erm o m eter
system w as acquired an d integrated into the apparatus. T his reduced tem perature uncertainties by a facto r
o f about ten, ev en m ore for tem peratures n e a r 0 C. T h e therm o m eter u n it is a tw o channel device an d
w as supplied w ith tw o tem perature probes. Suspicion th a t the spectrum analyzer IF d rifted in resp o n se
to tem peratu re changes m otivated th e in stallation o f th e ex tra p ro b e into the sp ectru m an aly zer in a
location w here it w ou ld m easure tem peratures n ear the prim ary tuning filter. A ttem pts w ere m ad e to
predict the IF know ing the spectrum analyzer internal tem perature, bu t these m et w ith failure. A lth o u g h
changing tem perature w as a go o d predictor o f a drifting IF, tem perature alone w as n o t an adequate pred ic­
tor o f the absolute IF value.
C.3
Standard Procedure
R esults fro m m easurem ents w ith the counter Af m eth o d led to experim ents th at dem o n strated th e
m agnitude o f the spectrum analyzer IF drift problem . M eth an e is a relatively strongly refracting gas so
the observed frequency shifts w ere large enough that errors introduced by IF and tem perature drift did no t
m ake the m easurem ents excessively u ncertain. H y d ro g en and esp ecially helium , at a given n u m b er
density, are m uch w eaker refractors than m ethane, and sim ilar experim ents w ith those gases w o u ld h ave
been u selessly inaccurate. T his led to the fo rm u latio n o f the m ethods p resented in A pp en d ix B fo r
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m easuring and circum venting th e spectrum analyzer IF. A ll subsequent ex p erim en ts m ade use o f those
m ethods. M o n ito rin g o f the sp ectru m analyzer internal tem perature allo w ed o n -th e -sp o t d ecisions o f
w hich A ppendix B procedure w ould be appropriate.
W ith all the hardw are m odifications in place and new m ethods designed, the sensitivity and accuracy
o f the apparatus w as significantly im proved. Procedures taking advantage o f the upgrades w ere quite dif­
ferent from previous procedures. T em perature m easurem ents w ere m uch m ore accu rate and the m ethods
o f A ppendix B had bro u g h t ab o u t long term repeatability o f accurate frequency m easu rem en ts, so it was
n o t necessary to red u ce the tim e span betw een m easurem ents under vacuum an d lo ad ed conditions to a
m atter o f m inutes. A ll su b seq u en t experim ents involved m easuring reso n an ce b an d w id th s as w ell as
center frequencies, and this b ro ught new problem s and procedures to the collection o f the basic data.
O n e new p ro b lem in tro d u ced by abso rp tiv ity m easurem ents involved th e e ffects o f changing the
dielectric co n stan t o f the m ed iu m in sid e th e resonator and around th e signal p ro b es, a g ro u p o f effects
often lum p ed u n d er the sin g le term d ielec tric loading. C h anging the v alu e o f the dielectric constant
inside the resonator causes a chan g e in the electric field strengths and thus a change in the energy stored
in the fields, m uch the sam e w ay th at changing the dielectric con stan t o f the d ie le ctric in a capacitor
changes its capacitance. S in ce th e Q o f a reso n an ce depends on the en erg y sto re d in the fields, the
altered dielectric co n stan t changes the Q o f the resonance, even if the m edium is p erfectly transparent.
T his p articu lar e ffe c t is true dielectric loading. T he d ielectric constants o f the g ases inv o lv ed in this
w ork are only slightly d ifferen t fro m th a t o f a vacuum so the effect is sm all, b u t if ab sorptivities to be
m easured are also sm all dielectric loading can cause significant errors. D ielectric co n stan t changes also
affect the im pedance en v iro n m en t around the signal probes. Since the im pedance o f the coaxial cables
feeding the probes is a co n stan t 50 Q , probe im pedance changes alter the m atch at the p ro be-to-cable
interface and ch an g e th e am o u n t o f energy co u p led ou t o f th e reso n ato r, thus c h an g in g th e Q o f the
resonance. P robe im pedance is also a com plex function o f frequency and thus, u n lik e true dielectric
loading, the Q ch an g e beh av io r o f resonances at different frequencies can be m ark ed ly different. This
effect can be as large or larger than true dielectric loading, but it and other m inor effects are often lum ped
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un d er the g eneral heading "dielectric loading." Signal p robes d esigned as p a rt o f this w ork, described in
A ppendix A , w ere quite successful in m inim izing the im pedance change effect.
T he effects included in dielectric loading are additive w ith th e prim ary effect o f absorptivity on reso­
nan ce bandw idth (see the discussion in Joiner et al., 1989). If b andw idth m easurem ents can be m ade on
tran sp aren t gases to ch aracterize th e behavior o f each reso n an ce as the refractivity (i.e., dielectric co n ­
stant) o f its contents vary, th at inform ation can be used to su b tract b andw idth chan g e due to dielectric
loading fro m the total bandw idth change upon loading, leaving the bandw idth change due to absorption.
T his w as o n e m o tiv atio n fo r ex p erim en ts on p u re h y d ro g en an d h elium . In addition to refractiv ity
m easurem ents fro m those ex perim ents, d ielectric loading d ata fro m the bandw idth m easurem ents w ere
used to co rrect th e m easured bandw idth changes fro m experim ents on am m onia in hydrogen and helium .
A ll m easu rem en ts on hydrogen and h elium and all on am m o n ia in hydrogen an d heliu m used a
procedure that hereafter w ill be referred to as "standard procedure." It is a generally applicable procedure
for accurately m easuring resonance center frequencies and bandw idths under any conditions. T he raw data
produced are sufficient to calculate refractivity and eith er absorptivity o r dielectric loading effects o f the
sam ple. A se t o f m easurem ents on a single resonance, w ith a fixed gas sam p le u n d er fixed conditions,
defines o ne raw datum point. It m ust contain su fficient in fo rm atio n to com pletely characterize the co n ­
ditions on th e gas sam ple, and to determ ine the reso n an ce ce n te r frequency and bandw idth. F o r a fixed
sam ple u n d e r fixed conditions d a ta w ere taken (or attem pted) on each o f the nine usable resonances
available fro m the resonator, one by one, yielding up to nine data points. T he set o f all d ata points on
this fixed sam ple is called a series.
A series o f m easurem ents begins w ith verification o f equ ilib riu m conditions w ithin the sim ulation
cham ber. T h e total pressure m u st be acceptably n ear the d esired value, therm al transients from previous
pressure changes m ust have decayed, and the tem perature m u st b e stable w ithin a few tenths o f a K o f
the targ et tem perature. A full set o f condition m easurem ents is recorded: the d ate and tim e the series is
begun; barom etric pressure; indicated cham ber gauge pressure; indicated cham ber tem perature; spec­
trum analyzer internal tem perature; and therm om eter chassis tem perature. S om e o f these quantities vary
slow ly, so repeating a fu ll set o f conditions m easurem ents fo r each datum is unnecessary. Full sets are
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recorded o nly at the begin n in g o f each series, at c h an g eo u t o f the sw eep g enerato r p lu g -in (once p er
series), and at the en d o f each series. A fter the initial condition m easurem ents the first datum p o in t is
m easured an d recorded; then the spectrum analyzer and sw eep generator are retuned to the n ex t resonance
and the n ex t p o in t is recorded; and so on until all resonances accessible to the cu rren t sw eep er plug-in
have been m easured. P o w er to the sw eep g en erato r is turned off, the plug-in is replaced, an d p o w e r is
restored. W h ile the new p lug-in is w arm in g another full set o f condition m easurem ents are recorded.
The rem aining data points are recorded, a third full se t o f condition m easurem ents are recorded, and prep­
arations are begun fo r the n ex t series.
O ne datum as recorded using this procedure consists o f many separate m easurem ents o f fundam ental
quantities. T h o se q uantities an d com m ents concerning them are, in the o rd er they are m easu red and
recorded: initial indicated cham ber tem perature; initial spectrum analyzer internal tem perature; sp ec­
trum analyzer band designations and L O frequency m easurem ents (up to four each on tw o different bands)
necessary to accurately determ in e the reso n an ce cen te r frequency using the appropriate m eth o d from
A ppendix B, and any notes concerning the estim ated accuracy o f the frequency m easurem ents; m easured
resonance bandw idth, spectrum an aly zer screen w idth setting used, and estim ated accuracy; if the in d i­
cated cham ber tem perature o r spectrum analyzer tem perature has changed since the initial m easurem ents,
final values fo r them ; description o f any asym m etry observed in the resonance frequency response curve;
and any g en eral com m ents about the m easurem ents. In practice, com pletion o f this p ro ced u re req u ired
an average o f ab o u t ten m inutes, including initial tuning o f the spectrum analyzer and sw eep er to the
resonance.
Standard p rocedure calls fo r the first and last series o f any experim ent to be baseline series, those
recorded w ith the ch am b er evacuated and at the targ et tem perature fo r the experim ent. A ccuracy o f the
reduced data from an ex p erim en t is critically d ependent on good baseline m easurem ents. R eso n an ce
frequency d ata from baseline series at different tem peratures w ere used to characterize the center frequency
tem perature dependences o f all the usable resonances. T h at inform ation w as needed fo r reducing the raw
frequency d ata to refractiv ities. O ne featu re characteristic o f all b aseline series is that each b aseline
cen ter frequency m easurem ent used the E x ten d ed D rifting IF m ethod from A ppendix B, req u irin g four
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separate m easurem ents o f the spectrum analyzer L O frequency, regardless o f the tem perature stability o f
the spectrum analyzer.
H ardw are considerations w ere strong driv ers fo r the o rganization o f this p ro g ra m o f experim ents.
T he ease w ith w hich different conditions w ere controlled determ ined how often they w ere changed. Spec­
trum analyzer and sw eep g en erato r frequencies w ere the sim plest an d q u ick est to v ary , so that w as the
only condition intentionally altered betw een th e individual d ata in a series. S ince all oth er conditions
w ere held constant w ithin the capabilities o f th e apparatus, frequency w as th e on ly v ariab le in a series.
T he next easiest change to m ake w as a decrease in total pressure, by venting or active pum ping. W hen
a series w as finished, th e critical elem en t in the preparations fo r the n ex t series w as reducing the total
pressure o f the gas m ixture in th e cham ber. T his p rev en ted alteration o f th e m ix in g ratios from the
initial series. A change o f m ixing ratio s o f the test m ixture w as fairly sim ple to p erfo rm bu t w as tim e
consum ing. I t w as a sim p le m atter to add one gas to the m ixture, changing the m ixing ratios, b u t 12
hours was required to allow the gases to thoroughly rem ix. T h is p rocedure w as perfo rm ed only three or
four tim es in one experim ent. D ue to the long th erm al tim e constant o f the e q u ip m en t w ithin the ther­
m al cham ber, significant tem perature changes w ere the m o st d ifficu lt to m ake, requiring m ore than
12
hours for all therm al transients to decay. T hus in the course o f one experim ent no significant tem pera­
ture changes w ere attempted.
T hese levels o f difficulty helped establish an organizational vocabulary fo r the p rogram . V ariation
of the m ost accessible param eter, frequency, defin ed a series. L ikew ise, variation o f the n ex t m ost acces­
sible param eter, total p ressu re, d efin ed a seq u en c e . A sequence is the set o f all series taken o n a gas
sam ple w ith fix ed m ixing ratios and a co n stan t targ et tem perature. E ach series in a sequence is at a
unique total pressure; standard sequences contain series at 8 , 6 , 4 , 2, and 1 atm. total pressure. M eeting
the requirem ent o f fixed m ixing ratios w as a sim ple m atter. O nce the desired m ixture had been generated
at the highest total pressure, all low er pressures o f that precise m ixture w ere available in sequence (hence
the nam e applied to the org an izatio n al concept) by partial venting o f the cham ber. A pro ced u re that
produced data on m ixtures containing particu lar gas species at one or m ore m ixing ratio s, at a con stan t
target tem perature, was called one e xp erim en t. A n ex p erim en t could involve o n e o r m ore sequences.
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F o r exam ple, the en tire pro ced u re that p roduced d ata on the absorptivities and refractivities o f m ixtures
o f am m o n ia in helium a t 313 K , w ith am m onia nu m b er m ixing ratio s o f 0.067 an d 0.0 0 8 2 , w as referred
to as the "313 K am m onia in h eliu m experim ent."
E xperim ents on p u re hydrogen and helium required a single sequence since no m ixing w as involved.
Those experim ents produced refractivity and dielectric loading data that w ere used in the reduction o f data
from experim ents involving am m onia. T he procedure fo r these experim ents was:
1. P erform precalibration, including frequency counter and R T D therm om eter calibration;
2
. Stabilize the evacuated cham ber at the target tem perature;
3. P erform the initial baseline m easurem ents;
4. A d d the test gas to the cham ber to a total p ressu re o f
8
atm .;
5. Perform a standard sequence o f m easurem ents;
6.
E vacuate the cham ber an d perform the final baseline m easurem ents;
7. P erform postcalibration.
W hen experim ents calling for identical target tem peratures w ere perform ed in im m ediate succession, the
final baseline and postcalibration from one w ould also serve as the initial baseline and precalibration fo r
the next. E xperim ents o n pure gaseous am m onia u sed a pro ced u re very sim ilar to the one listed above,
w ith ch ang es to steps 4 an d 5. In those procedures the am m onia w as added to th e ch am b er to the targ et
pressure fo r the series to be m easured. A single series w as m easured instead o f the standard sequence. In
the 300 K p u re am m onia experim ent, after the series a t 1.0 atm . p ressu re w as finished the cham ber w as
pum ped dow n to 0.5 atm . and another series w as m easured.
T he b u lk o f the d a ta taken in this p rogram involved m ixtures o f gaseous am m onia, hydrogen, and
helium , at v ario u s m ix in g ratio s. M aking contro lled m ixtures o f these gases added an o th er level o f
com plexity to the p rocedures. T h ro u gh o u t the m ixture generation procedures it w as assu m ed that all
gases involv ed o b ey ed ideal gas law s. F o r the ran g e o f tem peratures and pressures invo lv ed the errors
that w ould arise fro m this assum ption w ere insignificant com pared to instrum ent uncertainties. C ontrol
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o v e r num b er m ixing ratios w as exercised by controlling partial pressures, an d this w as accom plished by
carefu l m easu rem en t o f the to tal p ressu re o f th e m ixture after each step in th e process. T h e standard
procedure fo r generating m ixtures containing am m o n ia had to b e m odified fo r the experim ents at 213 K
since they c alled for am m onia partial pressures up to 0.55 atm . The vap o r pressure o f am m onia at 213
K is considerably less than that.
Standard procedures for generating gas m ixtures w ere designed to m inim ize the uncertainties in the
am m onia p artial pressure at v ery low m ixing ratios. U ncertainties in pressure m easurem ents by the low
pressure g auge w ere ±5 torr, w hich is ± 10% fo r a 50 to rr pressure. This w as clearly u n acceptable. T he
m eth o d used to circum vent such larg e uncertain ties w as a series o f p ro g ressiv e dilu tio n s. A lthough
m easuring a 50 to rr pressure w ith the low p ressure g auge y ie ld ed large relativ e uncertainties, the sam e
± 5 torr uncertain ty applied to a 500 torr m easu rem en t is a m uch m ore acceptable ± 1% . T h u s standard
procedure began w ith rather large initial am m onia partial pressures that w ere then diluted by o th er gases.
T hroug ho u t the procedure pressure m easurem ents w ere m ade only after allow ing therm al transients from
pressure changes to decay, and sim ultaneous tem perature m easurem ents w ere m ade to k eep any tem pera­
tu re changes fro m biasing p recise calcu latio n s o f the resu ltin g m ixing ratios. A t tem peratures above
213 K the initial target load o f am m onia w as 4 1 9 torr, o r 0.551 atm . T he n e x t step, the first dilution,
w as accom plished by raising th e total ch am b er p ressu re to
8.2
atm . by addition o f hydrogen o r helium ,
y ielding an in itial m ixture w ith an am m onia n u m b er m ixing ratio a b o u t 0.0 6 7 2 . F o r m ixtu res that
w ould eventually contain both hydrogen and helium the first dilution w as done w ith hydrogen only; no
helium w as added in this step. T he m ixture w as allo w ed at least 12 hours to m ix thoroughly. T hen the
pressure w as red u ced to 1 atm . by venting. M ixing ratios w ere precisely m aintained b y this operation,
so the gas m ixture consisted o f 0.0672 atm . am m onia, with the rem ainder being the foreign gas species.
A nother dilu tio n step to 8.2 atm . total p ressu re b ro u g h t the am m onia m ixing ratio to 0.00820. If the
foreign gas w as to be a m ixture o f about 90% hydrogen and 10% helium , the dilution w as accom plished
in tw o steps. T he first gas added, hydrogen, b ro u g h t the ch am b er total p ressure to 7.387 atm . H elium
w as then added to a total p ressure o f 8.2 atm . I f desired, one m ore cycle o f venting to 1 atm . an d d ilu ­
tion to
8.2
atm . w o u ld have brought the am m onia m ixing ratio to
0 .0 0 1 0 0
, etc.
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A full sequence o f m easurem ents w as p erfo rm ed on the m ixtures w ith am m o n ia m ixing ratios o f
0.0082. O n ly single series o f m easurem ents w ere perfo rm ed on som e m ixtures w ith am m o n ia m ixing
ratios o f 0.067. T his w o rk was co n cern ed w ith m ixtures applicable to th e atm ospheres o f g ia n t planets,
w here am m onia m ixing ratios are sm all enough th a t broadening o f the in v ersio n lines is d o m in ated by
foreign gas broadening. In m ixtures w ith an am m onia m ixing ra tio o f 0.067 am m onia self-broadening
is respo nsib le fo r ab o u t 60% o f the to ta l broadening, so this is n o t co n sid ered represen tativ e o f those
atm ospheres. Series m easured on those m ixtures w ere fo r spectroscopic in te re s t T he full p rocedure for
these experim ents was:
1.
Perform precalibration, including frequency counter and R T D therm om eter calibration;
2.
Stabilize the evacuated ch am b er at the target tem perature;
3.
Perform the initial baseline m easurem ents;
4.
G enerate the initial m ixture w ith an am m onia m ixing ratio o f 0.067;
5.
Perform one series o f m easurem ents (optional);
6
.
7.
8
.
9.
D ilute to a m ixture w ith an am m onia m ixing ratio o f 0.0082;
Perform a standard sequence o f m easurem ents;
Evacuate the cham ber and p erform the final baseline m easurem ents;
Perform postcalibration.
A gain, if experim ents calling for identical target tem peratures w ere perform ed in im m ediate succession,
the final baseline and postcalibration fro m one w ould also serve as the initial b aseline and precalibration
for the n e x t
T he procedure given above could n o t be used fo r the experim ents on m ixtures at 213 K due to the
vapor pressure problem . I f the sam e g eneral pro ced u re w ere follo w ed the initial am m onia load into the
cham ber w ould be lim ited to less than
200
torr, significantly increasing relative uncertainties in am m o n ­
ia m ixing ratios in su b seq u en t m ixtures. Instead, the initial dilution w as perfo rm ed a t 273 K, beginning
w ith an initial am m onia lo a d o f 537 torr, or 0.707 atm . P ressurization w ith h ydrogen or helium to 8.2
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atm . and ven tin g to 1 atm . after m ixing w as com plete left 0 .0 8 6 2 atm . am m onia in the cham ber. The
cham ber w as then rep ressu rized w ith th e foreign gas to 8.2 atm . total p ressu re. C ham ber tem perature
w as then b ro u g h t to 213 K, w hich caused th e total p ressu re to d ecrease to 6.4 atm ., o f w hich 0.0672
atm . w as am m onia. M easured pressures after the cooling step agreed extrem ely w ell w ith pressures pre­
dicted by th e id eal gas law s fo r that tem perature change. A dding m ore o f th e foreign gas o r gases to a
total pressure o f 8.2 atm b ro ught the am m onia m ixing ratio to the desired v alue o f 0.0082. T his proce­
dure replaces steps 2 through
6
o f the standard procedure. U nfortunately this elim inated the initial base­
line m easurem ent, bu t this w as unavoidable since the ch am b er co u ld not be evacuated after bringing it to
the target tem perature. C onsistency o f the final baseline m easurem ents am ong the three experim ents at
213 K w as excellent, so it is th o u g h t the loss o f initial baselin e m easurem ents d id not im p act the accu­
racy o f the ex p erim en t results.
188
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Appendix D
Data Reduction Procedures and Uncertainty Analysis
T h e d ata collection p rocedures o f A ppendix C generated raw d a ta ab o u t the c o n d itio n s in sid e the
cavity resonator and the associated cen ter frequencies and bandw idths o f its resonances. T ech n iq u es out­
lined in Section 4.1 w ere used to d evise data red u ctio n procedures ex tracting know ledge ab o u t th e gas
m ixtures inside the reso n ato r from those raw data. T his appendix presents the d ata red u ctio n procedures
used, along w ith erro r analyses o f the m ethods. U ncertainties in fundam ental co ndition m easurem ents,
such as tem peratures and pressures, have no t been carried o ver to the ex p ressed uncertain ties in ab so rp ­
tivity o r refractiv ity m easurem ents. S u ch tran sfers o f uncertainties rely on k n o w led g e o f the co rre c t
dependences o f the derived m easurem ents on those fundam ental conditions. A lthough the current state o f
know ledge on this su b ject seem s to y ie ld d ep en d en ce predictions th at at least ap p ro x im ate o bserved
dependences, in som e cases the differences betw een predictions and observations w ere large enough that
it w as thou g h t best to leave uncertainties in fu n d am en tal condition m easurem ents w ith those m easure­
m ents, rath er than attem p t possibly erroneous p ropagation o f con d itio n uncertain ties to ab so rp tiv ity or
refractivity u ncertainties. T hus the e rro r analyses fo r absorptivities and refractiv ities treat only those
uncertainties that directly affect the cavity reso n ato r m ethod o f m easurem ent. T hese u n certainties may
indeed include fundam ental quantities w hen they directly affect reso n ato r ch aracteristics; fo r exam ple,
therm al expansion o r contraction o f the m aterials o f the resonator cau se its reso n an ce frequencies to be
quite sensitive to tem perature, even if it contains o nly a vacuum . H ow ever, effects such as the variation
o f p ro p ag atio n constants o v er the ran g e o f uncertain ties in tem p eratu res, p ressu res, o r m ixing ratios
have n o t been included in expressed uncertainties in absorptivities o r refractivities.
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D .l
Screen Af Methods
R eduction o f refractivity d a ta acquired by the screen Af m ethod w as a fairly sim ple procedure. O nly
three m easurem ents o f resonance characteristics w ere required: vacuum cen ter frequency, fov; cen ter fre­
quency shift upon pressu rizatio n o f the cham ber w ith the test gas, Af 1 ; an d center frequency shift upon
evacuation o f the cham ber, Af2 - A lthough it appears that the tw o freq u en cy shifts w ou ld be redundant,
both are affected by d rift o f the spectrum analyzer IF and cham ber tem perature. B oth o f these drifts w ere
slow ly varying q uantities o v er th e tim e span o f o n e refractiv ity m easurem ent, so both w ere do m in ated
by the linear co m ponent o f drift. A veraging the tw o shifts, one upw ard an d one do w n w ard in frequency,
cancelled the lin ear c o m p o n en t o f drift. The loaded cen ter frequency o f th e resonance is calculated by
subtracting the averaged frequency shift from the vacuum center frequency, though th e difference between
vacuum and loaded cen ter frequencies is essentially negligible fo r all th e m easurem ents m ade w ith this
m ethod. T hus th e d ata red u ctio n fo rm u la fo r a single m easurem ent by th e screen Af m ethod is
A f , + Afp
v = Af.yg = —---2--- i ---- .
f
'
1
(D.D
Af, + A f J
2 Ifcv "
A s described in A ppendix C o n e set o f m easurem ents w as m ade using an abbreviated version o f the
screen Af m ethod in w hich freq u en cy shifts w ere m easured only during d epressurizations. This ensured
constant m ixing ratios in th e gas sam p le at each step in the sequence o f p artial depressurizations, but a
side effect o f the m eth o d w as loss o f the lin ear d rift can cellation th e tw o-w ay m easurem ents provided.
T otal pressures are indexed w ith th e order reversed fro m the actual o rder o f m easurem ent; the last p res­
sure before ev acu atin g the ch am b e r,
1
atm ., is c alled P -|, an d the initial to tal p ressure,
8
atm ., is P 5 .
F requency shifts are w ritten Af; to m atch the to tal p ressure b efore the p re ssu re ch an g e, so Af-| is the
shift due to the dep ressu rizatio n from P \ to a v acu u m and A fs is the sh ift d u e to th e depressurization
from P 5 to P 4 ,
6
atm . A t the n th step in the seq u en ce the in d icated to tal freq u en cy sh ift d u e to a
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depressurization fro m th at total p ressu re to a vacuum is given by the sum o f frequency shifts
1
through
n. T he refractivity o f the gas m ixture at the n th step is given by the red u ctio n form ula
n
Z A f/
v n= —
-------- -
(D.2)
fcv-lA f,
/=1
S everal sources o f e rro r co n trib u te to the g lo b al u n certainties in refractiv ities m easu red w ith the
screen Af m ethod. T h ese sources include m easurem ents o f tem perature, pressure, cen ter frequency, and
frequency shifts. S in ce tem peratures fo r these experim ents w ere m e asu red w ith the built-in therm al
cham ber th erm om eter, absolute tem peratures w ere accurate on ly to ± 3 C . A bsolute tem perature errors
affect calculated refractivities through th e tem perature dependence o f the gas sam ple's refractivity, not
from d irect effects o n the reso n ato r m ethod, so tem perature u n certainties are left w ith th e tem perature
m easurem ents. R esonance center frequencies and frequency shifts do in fact vary w ith tem perature but in
such a m anner that they give an accurate m easurem ent o f the refractivity o f the resonator contents regard­
less o f absolute tem perature, as long as th at tem perature is stable. T h e reso n ato r m ethod w ill thus faith­
fully yield a reliab le m easu rem en t o f the refractivity o f w h atev er is inside th e resonator, u n d er w hatever
conditions ex ist;
it is then up to the in v estig ato r to d eterm in e the tem p eratu re o f th o se contents.
B ecause resonance frequencies are directly affected by tem perature variations the tem perature stability o f
the system during the m easurem ents is o f m uch g reater im portance; i.e., the system tem perature should
not drift significantly o v er the tim e span o f the m easurem ents. D espite the 3 C uncertainty in absolute
tem peratures the tem perature stab ility o f the system co u ld be kept to w ithin ± 0 .5 C o f an average value.
D ata on the tem perature dependences o f the resonance center frequencies show ed that they w ere quite
predictable, w ith relative frequency changes being identical fo r all resonances tested. This m ade it po ssi­
ble to use a single form ula to determ ine ^ c / g j for a given resonance by scaling the observed param eters
o f a reference resonance by center frequency. T he form ula resulting from this approach was referenced to
the TE 0 2 4 resonance, w hich had the highest center frequency o f those used:
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9f_C =
-fc
3T
1 7 .4 3 1 G H z
3 1 9 .2 ^
+ ( T - 2 7 3 .1 5
K)|o. 1 8 9 2 4
^ § j
(D.3)
T em p eratu re instab ility lim its c o u ld then be tran slated to cav ity frequency uncertainty, c alled A f ^ j »
using the fin ite form o f the E quation D .3:
Af
=
~CI
[3 1 9 .2 + 0 . 1 8 9 2 4 ( T - 2 7 3 .1 5 )]A T
1 7 .4 3 1 G H z
kH z,
(D.4)
w here fc | is in G H z and T is the system tem perature in K . T hese uncertainties directly affected the accu ­
racy o f refractivity m easurem ents. T h eir m agnitudes, calculated by E quation D .4, w ere usually betw een
30 and 300 k H z w hile the o bserved frequency shifts v aried from a few to a few tens o f M H z, but this
m ay be overly pessim istic. T he tem perature uncertain ty used w as the ± 0.5 C characteristic o f the lo n g ­
term tem perature stability o f the system , as long as am b ien t conditions w ere fairly stable. In the sh o rt
tim e required for a single screen Af m easurem ent it w as highly unlikely that tem peratures w ould drift the
full 1 C fro m lim it to lim it. A d rift o f 0.5 C has b e en assum ed, b u t ex perience w ith the equ ip m en t
suggests this m ay still be rath er p essim istic. S ig n ifican t red u ctio n s o f this uncertainty w ere po ssib le
from the lin ear drift cancellation characteristics o f the full screen Af m ethod. The im provem ent depended
on the am ount o f non-linear d rift present, a quantity th a t could not b e m easured w ith this apparatus, b u t
even in the w o rst cases the u ncom pensated uncertain ties w ere less than h a lf that indicated by E q u atio n
D .4. T he ab b rev iated v ersion did n o t perm it d rift cancellatio n so in th at m ethod these uncertainties are
as given in E q u atio n D .4.
T em p eratu re in stability also affected ce n te r frequency m easurem ents b u t
shifts o f a couple hundred kH z w ere insignificant com pared to cen ter frequencies o f several GHz.
Pressure m easurem ent errors had no direct effects on the resonance m easurem ents oth er than refrac­
tive effects. T hus the p ressure uncertainties are n o t included in the refractivity uncertainty calculations
here. P ressu re m easu rem en t uncertainties do in fluence u n certainties in the m ixing ratios o f gas m ix ­
tures, b u t those w ill be treated later in this appendix.
Errors associated w ith m easurem ents o f resonance center frequencies directly affected refractivity cal­
culations. M any o f th e m easurem ents w ere done w ith o u t the aid o f the frequency counter, lim iting their
192
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accuracies to the accuracy o f the spectrum analyzer tu ning display, about ± 0 .2 % , but it w ill b e seen that
uncertainties in th e frequency shift m easurem ents are far m ore sig n ific a n t E rrors associated w ith tuning
the spectrum analyzer precisely to the cen ter frequency o f a resonance w ere only a few k H z so w ere truly
insignificant in this m ethod. B ecause th e uncertainty in cen ter frequency m easurem ents is a direct result
o f spectrum analyzer lim itations it is called AfsA- Its m agnitude is given by AfsA = ^cv^500 >
a
range about 18 to 35 M H z.
E rrors affecting frequency sh ift m easurem ents w ere m o st im portant. The resonator frequency uncer­
tainty due to tem perature instability, d iscu ssed above, is one o f those errors. A n o th er is a fundam ental
lim itation on the accuracy o f the m easurem ents im posed b y the resolution o f the spectrum analyzer d is­
play screen. E ach m ajo r fiducial div isio n on th e screen consists o f about 40 reso lu tio n elem ents. A
front-panel screen w idth selector sets the frequency span o f one o f the m ajor divisions; the entire screen
is ten o f those spans. T hus the resolution lim it o f the screen, Afres , is given by dividing the frequency
span o f one division by the num ber o f resolution elem ents, 40, in one division:
.,
_
f8S
Frequency Span / D iv.
4 0 R es. E lem ents / D iv .
'
<P-5)
M easured frequency shifts w ere also affected by screen w idth calibration errors. Post-experim ent calibra­
tion o f screen w idths red u ced these errors to ± 1 % or better, so the uncertainty due to calib ratio n errors,
A f c a l , is ju s t A f a v g / 1 0 0 .
Including these uncertainties in the appropriate quantities o f the reduction equation fo r the full
screen Af m ethod yields
v±Av =
-------
^ a v 9 ± K a l + Af—
------------
fcv± AfSA+ AfAT - Af
+ A fAT
.
(D. 6 )
± Afcal + Afres + Af AT
U ncertainty AfsA *n the denom inator com pletely o v erw helm s o th er uncertainties there. Its m agnitude is
193
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a few ten s o f M H z, w h ile the sum o f all others is o n ly a few tenths o f a M H z. T h u s the u ncertainty in
the denom in ato r can be treated as ju s t AfgA- T he calcu lated glo b al uncertainty is then
_
av -
A f ca| + A f res + ^ A T
------- — -*cv
A favgA fsA
H
a 'a v g
(' f CV
+ in significant higher-order term s.
„
(D.7)
Af gyg '|
T h e first term o f this equation is the first order g lo b al uncertainty d u e to frequency sh ift uncertainties. It
is u sually an o rd er o f m agnitude o r m ore larg er th an th e second, w h ich is the first o rd e r glo b al u n cer­
tainty d u e to center frequency uncertainties.
As an exam ple o f the application o f the red u ctio n and uncertainty form ulae fo r the screen Af m ethod
o ne actual datum is red u ced here. T h e exam ple is a screen Af m easu rem en t o f the refractiv ity o f pure
m ethane a t an indicated tem perature o f +23 C (296 K ) and a total p ressu re o f 3 atm . T he T E 0 1 4 reso ­
nance w as used, w ith a m easured fcv o f 13.60 M H z. W ith the sp ectru m an aly zer screen w idth selector
adjusted fo r a w idth o f 2 M H z/D iv, the m easured pressu rizatio n an d depressu rizatio n freq u en cy shifts
w ere Af -| = 17.1 and Af2 = 17.0 M H z respectively. T h e screen w idth correction facto r fo r those m easure­
m ents is 0.955, so the corrected Afavg is given by
A f ^ = 0.9 5 5
(17.1 + 17.0 M H z !
= 16.28 M H z.
Frequency uncertainties are calculated w ith the appropriate formulae:
a.
_ 16.28 M H z
,
A 'cai “ ------^ ----- = 163 kH z;
Af
r9s
= ___ 2 MH ? / D.iv -___ = < o k H 7 4 0 R es. E lem . / D iv .
5
’
(13.60 GHz - 16.28 M H z ]
AfaT -
17.431 GHz
319.2 ^
+
K
0 .18924 ^
1
(2 3
k:
(0.5 K ) = 126 kHz;
K
_ 13.60 G Hz _
SA ~ ----- 500----- " 2 7 -2 M H z -
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T h e calculated refractiv ity and u ncertainty are then given by E quations D .l (w h ere fCv - Afa vg is used
fo r the fc | in the equation) and D.7:
v =j
1— M Hz
=
(1 3.60 G H z - 16.28 MHz)
Av =
(163 + 5 0 + 126 kHz)
(13.60 G H z - 16.28 MHz)
+
-6
(16.28 MHz) (27.2 MHz)
(13
6Q GHz _
= (25.0 X 1 0 '6) + (2.4 X 1 0 '6) = 27 .4 x 10 '
16
6
2 g M H z) 2
.
In this exam ple the first term in th e equation giving Av is slightly m o re than a fa cto r o f ten larger than
the second. T his dem onstrates the dom inance o f frequency shift uncertainties in this m ethod.
G lobal u n certainties in the ab b rev iated screen Af m ethod are c alc u lated in a sim ilar m an n er, but
som e o f the individual uncertainties m u st be h and led differently. T he in teg rated frequency sh ift at the
ndi step is a sum o f n m easured frequency shifts, each carrying its ow n u ncertainties, b u t in general the
uncertainties are n o t independent and cannot be sim ply added. S ince the lo ad ed ce n te r frequency in the
denom inato r o f E q u atio n D .2 is calcu lated from th e vacuum c en te r frequency, all in tegrated frequency
shifts are referen ced to the system tem perature at the tim e o f the vacuum m easu rem en t. As p reviously
discussed, system tem p eratu re stab ility w as such th a t tem peratures c o u ld vary o v e r a ± 0.5 C range.
T hus the m axim um tem perature d ifference that could be encountered betw een any tw o steps, the differ­
ence betw een the low an d high ends o f that range, is 1 C. A gain, drift rates varied slow ly enough that it
w as highly u nlikely th a t drift betw een m easurem ents at P i and vacuum (o r betw een any tw o adjacent
steps) w ould be the full 1 C, bu t w ith o u t m ore accurate inform ation that m axim um value m ust be used.
E quation D .4 is then u sed to translate th e tem perature uncertainty to frequency uncertainty.
U ncertainties due to spectrum an aly zer screen resolution add statistically. T h ese errors should be
random , so although the m axim um erro rs w ill add linearly, the expected relative error should decrease as
Vn.
All m easurem ents w ere m ade w ith a 2 M H z/D iv . screen w idth setting, so th e m ax im u m Afres at
the IIth step is
n tim es
th e m ax im u m e rro r contrib u ted by a single step, 50 kH z: Afre s = n x 50 kH z;
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but the ex p ected erro r is Vn tim es th e expected e rro r contribution fro m a sin g le step. T h e uncertainties
show n in the p resen tatio n o f these d ata in C hapter 5 use m ax im u m errors an d thus m ay be som ew hat
pessim istic.
S ince all freq u en cy shift m easurem ents w ere m ade w ith the sam e screen w idth setting, calibration
uncertainties Afc a | w ill add linearly, w ith no statistical dim in u tio n o f the ex p ected relativ e error. This
allow s calculating a to tal Afc a | fo r th e n [h step from the total freq u en cy sh ift w ith th e sam e form ula
used fo r the individual steps: AfCal = A ftot/joo , w here
n
Aftot = X Af/ •
(D.8)
<=1
T he expression fo r the global uncertainty o f the m ethod is very sim ilar to E quation D.7:
AV -
Af ca| + Af res
-------
^A T
,
—---------- +
^ to t ^ S A
,
.
.
,. ,
.
o + insig n ifica n t higher-order term s.
m
(D.9)
(fcv - Af,ot)
C enter frequency m easu rem en t uncertainty AfgA is unchanged fro m the usual screen Af m ethod. In the
abbreviated m ethod, how ever, it is even less significant com pared to the frequency shift uncertainties.
D.2
Counter Af Methods
In the counter Af m ethod frequency shifts w ere not m easured directly. Instead, the frequency counter
w as used fo r m easurem ents o f cen ter frequency un d er evacu ated a n d lo ad ed conditions, and frequency
shifts w ere subsequently calculated by the difference o f the m easu red cen te r frequencies. This m ethod
w as used before the techniques o f A ppendix B w ere devised and before installation o f the R TD therm om ­
eter. Spectrum an aly zer IF and ch am b er tem perature drifts affected the co u n ter Af m ethod ju s t as they
did the screen Af m ethod. Thus the tw o-w ay technique used in the screen Af m eth o d was retained fo r its
linear drift com pensation characteristics. The tw o frequency shifts w ere labeled Af 1 and Af2 , w here the
196
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subscript
1
refers to m easurem ents ju s t before and after pressurization, and
2
refers to m easurem ents ju st
before and after re-evacuation: Af; = fcv ,/- fc l,/• A s in the screen Af m eth od these two shifts w ere aver­
aged to produce AfaVg . T h e im provem ent in resolution o f cen ter frequency m easurem ents achieved by
use o f the freq u en cy c o u n ter m ade the effects o f spectrum an aly z er IF an d ch am b er tem perature drift
e v id en t in the tw o m easurem ents o f the loaded cen te r frequency. T h o se tw o m easurem ents w ere also
averaged to co m p en sate fo r the linear com ponents o f drift, b u t it w ill be seen later that frequency shift
uncertainties w ere still the m o st significant.
A s in equation D .l, the refractivity is calculated by dividing the frequency sh ift by the loaded center
frequency:
fc l, av g
2
Af 1 + Af 2
(fcl, 1 + fcl,2 )
fcl, 1 + fcl,2
(D. 10)
2
T his reduction form ula requires four independent c en te r frequency m easurem ents w ith th eir associated
uncertainties. L a te r exam ination o f those uncertainties w ill d em onstrate th e advantages o f this m ethod
over the screen Af m ethod.
T he one se t o f m easurem ents on a m ixture o f m ethane and hydrogen w ere done using the abbreviated
screen Af m ethod an d an abbreviated version o f th e counter Af m ethod, sim ultaneously. T here is much
sim ilarity betw een the abbreviated counter Af m ethod an d the abbreviated screen Af m ethod. T h e m ajor
difference is that in the abbreviated counter Af m ethod, a frequency shift w as calculated from m easu re­
m ents o f reso n an ce c en te r freq u en cy before and after a d epressurization step rath e r than m easuring it
directly fro m the spectrum an aly zer screen. O rganization o f this ex p erim en t is d escrib ed in the d iscu s­
sion o f its screen Af m eth o d aspects, p ag e 190 o f S ectio n D .l, an d n o m en clatu re is identical to that
described there. F o r the abbreviated counter Af m ethod, frequency shifts are given by
^f i
fcl,/
fcl,(1-1) •
( PM)
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w here fc |, / is the resonance cen ter frequency m easured w ith the ch am b er a t P /, ju s t before depressurizing
to the n ex t lo w er pressure, P ( / - 1 ) , and fcl, (/'- 1 ) is the cen ter frequency m easu red a t P ( / - 1 ) ju s t after the
depressurization. R efractiv ity o f the gas m ixture at th e n * step is then g iv en by the sam e form ula used
for the abbreviated screen Af m ethod, Equation D.2, w ith one m in o r change: the loaded center frequency
at any given step has been m easured direcdy, and becom es the den o m in ato r o f the reduction form ula:
Z A f/
Vn =
I7
■cl.n
.
( D .1 2 )
U ncertain ty p ropagation in the full counter Af m eth o d is rath er d ifferen t fro m th a t o f the screen Af
m ethod. T here are several uncertainty sources in the cen ter frequency m easurem ents th a t carry through
to the calcu lated refractivities, and a few o f sources that d o not. A ny e rro r source th at causes all fo u r
frequency m easurem ents to b e in error by a co n stan t facto r w ill can cel upon the div isio n in E q u atio n
D .10. Such sources include frequency counter absolute calibration error, absolute cham ber tem perature
error, and spectrum analyzer IF absolute error. IF errors on ly approxim ately cancel, b u t the second o rd e r
term s th at do n o t can cel are ex ceedingly sm all. A b so lu te ch am b e r tem p eratu re u n certain ties rem ain
associated with those tem perature m easurem ents.
E rro r sources th at d o affect calculated refractivities are errors in tuning the sp ectru m analyzer p re ­
cisely to a reso n an ce's c en ter frequency, and those a sso ciate d w ith in stru m en t stab ilities, in clu d in g
frequency co u n ter stability, ch am b er tem perature stability, and sp ectru m an aly zer IF stability. U n cer­
tainty d ue to tuning e rro r varies w ith the bandw idth and SN R o f th e reso n an ce being m easured. F o r the
refractivity m easurem ents on transparent species bandw idths rem ained narrow and SN Rs rem ained high,
yielding sm all tuning errors. Subsequent trials d em onstrated that u n d er go o d bandw ith an d SN R co n d i­
tions tuning errors w ere usually n ear
1
kH z, w ith the w orst errors (u su ally on the T E 2 2 1 resonance due
to its relatively low SN R ) reaching a m axim um o f 5 kH z. T his uncertain ty is called AftunT w o uncertainties associated with instrum ent stabilities, ch am b e r tem perature and spectrum analyzer
IF uncertain ties, w ere th e m ajo r sources o f e rro r in th e c o u n te r Af m eth o d .
C h a m b e r tem p eratu re
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variations w ere d iscu ssed in the analysis o f th e screen Af m eth o d , and th at source o f uncertainty is also
p resen t in this m ethod. T em p eratu re v ariatio n uncertain ties affe ct refractiv ity calculations prim arily
w hen the tem p eratu re o f a m easurem ent under loaded conditions differs from th a t o f the com plem entary
vacuum m easurem ent, so in th e num erator o f E quation D .10 the uncertainty is associated w ith each A f/,
n o t w ith each frequency m easurem ent. In the den o m in ato r it is associated w ith both m easurem ents o f
loaded center frequency. As in the screen Af m ethod this uncertainty is labeled A f ^ y , and is calculated
using E quation D .4. A fter the m ethods o f A ppendix B w ere dev ised it w as possible to characterize spec­
trum an aly z er IF b eh av io r an d assign realistic u n certainties b ased on th at behavior. T he n om inal IF
value is 2.05 G H z, bu t o ver a long p eriod o f tim e and u n d er a co n sid erab le ran g e o f am bient tem p era­
tures it w as fo u nd to vary u p to 20 0 k H z above and 800 k H z below that. T hus it w as treated as 2.0497
± 0.0005 G H z in the w orst case. In a sh o rt tim e span an d u n d er stab le am bient conditions IF drifts w ere
m uch sm aller. In the tim e span required fo r o n e d atum m e asu rem en t by the c o u n ter Af m eth o d the
observed d rifts w ere less than 50 kH z, usually m uch less. A v erag in g the up w ard and do w n w ard fre­
quency shifts can celled th e lin ear co m p o n en t o f IF drift, fu rth er red u cin g th e propagating u ncertainty.
Frequency uncertainty due to IF drift is labeled A fjp .
F requen cy c o u n ter stab ility is a source o f relativ ely sm all errors. S ince calibrations o f that in stru ­
m en t w ith the 1 M H z referen ce w ere alw ays co n sisten t to the last d ig it o f its sev en-digit display, in sta­
bilities m ust b e sm aller than 5 parts in 108. F o r a m easurem ent on a 10 G H z signal this w ould p roduce
an erro r o f 0.5 kH z o r less. T he actual m agnitude o f co u n te r in stabilities m ay be considerably sm aller
than this b u t w ith no m eans to m easure it this m axim um v alu e m u st be used.
U n certain ty d u e to
frequency co u n ter instability is called AfFC ■
O ne error source in the screen Af m ethod that is notably ab sen t here is AfsA • T he spectrum an aly ­
zer tuning d isplay is n o t inv o lv ed in these frequency m easu rem en ts, so its accuracy lim its do n o t apply.
T his reduces th e m ag n itu d e o f uncertainties in the d eno m in ato r o f the reduction form ula from tens o f
M H z to far less than one M H z. W here uncertainties in the d en o m in ato r w ere o f only m ild significance
before, they are even less sig n ifican t w ith the co u n ter Af m ethod.
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A pply in g th e uncertainties to the appropriate quantities yields the full reduction fo rm u la w ith global
uncertainty. In the num erator o f E quation D .10, each center frequency m easurem ent is affected by A fp c
and Aftun • E ach Af/ is affected by A f|p and A f ^ j as w ell. A ll fo u r uncertainties affect both o f the fre­
quency m easurem ents in the denom inator. This produces the full reduction form ula
(Af, -t-Af2 ) ± [4 A fFC + 4 A flun + 2 A f|F + 2 A fAT)
v ± A v = ---------------------- s—
-t(fcl.1 + U \ z ) ± 2 (A fpC + A f tun + Af I P + A f ATj
.
( D .l3)
E xtracting th e global uncertainty fro m this yields
4 A fFC + 4 A f ,un + 2A f |p + 2 A fAT
4A f ^
(Af Fc + Af tun + Af |F + Af ATJ
Av =
f c|,1 + f c|,2
(fc|1 + f . , J 2
cl21
+ in significant higher-order term s.
(D .l4)
Interpretatio n o f the term s is the sam e as E quation D .7. A no th er exam ple using real d a ta w ill dem o n ­
strate the relativ e m agnitudes o f term s in E quation D .14 and the individual uncertainties involved.
T his exam ple is a co unter Af refractivity m easu rem en t on a sam ple o f pure m ethane at +0.5 C and
3 atm . pressu re, using the T E 0 2 4 resonance. M easu red center frequencies w ere, in the o rd er m easured:
fc v .l = 17.430368 G H z; f d j = 17.407528 G H z; fc |,2 = 17.407536 G H z; fc v ,2 = 17.430348 G Hz.
M ax im u m values fo r all u n certainties w ere used in the calculation, so Aftun = 5 kH z, AfpQ = 0.5 kH z,
A f|p = 50 kH z, and A f ^ j = 159.5 k H z as calcu lated by E q u atio n D .4 w ith AT = 0.5 K . It is im m ediate­
ly obvious th a t the d rift-related uncertainties dom inate. F rom the cen ter frequencies the tw o frequency
shifts are obtained: Af-| = 22.840 M H z, and Af2 = 22.8 1 2 M H z. E quation D .10 gives the calcu lated
refractivity:
y =
22 .8 4 0 + 2 2 .8 1 2 M H z
=
1 7 .4 0 7 5 2 8 + 1 7 .4 0 7 5 3 6 GHz
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E qu atio n D .14 gives the global uncertainty in the calculated refractivity:
(2 + 2 0 + 1 0 0 + 319 kHz)
4 (22.826 MHz) (o.S + 5 + 50 + 159.5 kHz)
GHz)
(1 7 .4 0 7 5 2 8 + 17.407536 GHz ) 2
= (l2.7 X 10'6) + (o.Ol6 X 10'6) .
T he relativ e uncertainty in this exam ple is less than h a lf that o f the screen Af m ethod exam ple, m ainly a
resu lt o f elim inating the errors associated w ith m easuring freq u en cy shifts directly from the spectrum
analyzer screen. G lobal uncertainties arising from uncertainties in calcu lated frequency shifts are nearly
three orders o f m agnitude larger than those due to center frequency uncertainties, and 95% o f the frequen­
cy sh ift uncertain ties are due to instrum ent drift. F u rth e r im provem ents in refractiv ity m easu rem en t
accuracy w ould thus require reducing the uncertainties arising from ch am b er tem perature and spectrum
analyzer IF instability.
U ncertainty analysis for the abbreviated co u n ter Af m ethod is also sim ilar to that for the abbreviated
screen Af m ethod. U ncertainties due to cham ber tem p eratu re stability are identical and are referred to
w ith the label A f^y. A lthough this m ethod does n o t u se th e sp ectru m analyzer screen to m easu re fre­
quency shifts so uncertainty due to the resolution o f the screen is no t involved here, another uncertainty
o f equal m ag n itu d e takes its place: that due to sp ectru m an aly z er IF stability, A f|p s- A s p reviously
d iscussed, in th e tim e required for one step the IF co u ld d rift as m uch as 50 kH z, but u n d er norm al
conditions IF d rifts w ere m uch sm aller than that. If the d rift w ere constant at that m axim um rate the
resulting errors w o u ld add, producing a global erro r at the
step o f n x 50 kH z. As in the full counter
Af m ethod, uncertainties in absolute center frequency m easurem ents th at constitute the den o m in ato r o f
the reduction form ula are truly insignificant com pared to the frequency shift errors, so they are om itted
here. T hu s a first-o rd er approxim ation fo r the g lo b al un certain ty in the refractivity m easu red by the
abbreviated co u n te r Af m ethod at the n th step is
(D .l 5)
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T his is a m ax im u m u ncertainty th at assum es the IF d rifted u n ifo rm ly at its m ax im u m rate d u rin g the
entire experim ent, an d the cham ber tem perature stay ed co n stan t u n til ju s t before the final vacuum m ea­
su rem en t w hen it changed by a full 1 C. A ctual errors should b e m uch sm aller than this m axim um .
D.3
Standard Procedure
A fter the results o f experim ents using the co u n ter Af m ethod w ere analyzed the m ethods o f A p p en ­
dix B w ere devised, enabling accurate m easurem ent o f the sp ectru m analyzer IF and accurate m easu re­
m en t o f absolute resonance center frequencies. Installation o f the p latin u m R TD th erm om eter w ith the
probe inside the sim ulation cham ber beside the resonator reduced absolute tem perature uncertainties from
± 3 C to +0.4 C and allow ed m onitoring o f tem perature stability to th e instrum ent's resolution o f 0.1 C.
The ability to accurately and repeatably m easure absolute tem peratures and absolute frequencies e lim i­
nated the requirem ent th at resonance center frequency m easurem ents u n d er vacuum and loaded conditions
be m ade w ithin a sh o rt tim e o f each other. This led to the d ev elo p m en t o f a radically d ifferent approach
to the m easu rem en t o f refractivities, the "standard p rocedure" p resen ted in A ppendix C. O f co u rse the
new proced u res req u ired different d ata reduction p rocedures and d ifferen t global uncertainty red u ctio n
form ulae. U nlike the m ethods previously discussed in this appendix standard procedure involved m easur­
ing resonan ce bandw idths as w ell as cen ter frequencies, m aking p o ssib le calculation o f absorptivities o f
the resonato r contents a t resonance frequencies and adding another level o f com plexity to the experim ent
procedures, data reduction procedures, and uncertainty analyses.
In all facets o f standard procedure, condition m easurem ents also required reduction from the raw data
recorded from in strum ent displays to the desired absolute quantities. C ondition m easurem ents in cluded
tem perature, total pressure, m ixing ratios, and frequency. Frequency w as m easured repeatedly as p art o f
the resonance characteristics m easurem ents, but the rem aining condition m easurem ents w ere independent
o f resonato r m easurem ents. M ixing ratios w ere no t fundam ental m easurem ents but w ere calculated from
m easurem ents o f total p ressu res and tem peratures at each step o f the m ixing process. A discussion o f
m ixing ratio calculations and uncertainties appears later in this appendix.
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E ach d atu m in clu d ed at le ast o n e and p o ssib ly tw o m easu rem en ts o f ch am b er tem p eratu re. O ne
m easurem ent w as recorded before the frequency and bandw idth m easurem ents began, and if the indicated
cham b er tem perature had changed during the m easurem ent process a final tem perature m easurem ent was
recorded. T he first was called T j n ^ i , and if the seco n d was m ade it was called T jncj 2 . R eco rd ed chassis
tem peratu re o f th e R TD th erm om eter u n it w as u se d to rem ove th at so u rce o f erro r fro m th e indicated
tem perature, yielding a corrected cham ber tem perature called T q . If two tem peratures w ere recorded they
w ere corrected separately, then averaged to yield T 0 . Spectrum analyzer internal tem peratures w ere also
reco rd ed , but sin ce extrem e accu racy o f those m easu rem en ts w as no t critic al to the ex p erim en ts no
corrections w ere applied to them.
C h am b er total pressure m easurem ents w ere n o t m ade fo r each datum , bu t a t least three an d as m any
as fo u r m easurem ents w ere m ade p e r series, as d e scrib ed in A ppendix C: o n ce at the start o f the series,
o nce o r tw ice w h en the sw eep g e n e ra to r plug-in w as changed, and o nce at the e n d o f the series. B aro­
m etric p re ssu re an d ch am b er tem p eratu re m easu rem en ts w ere m ad e sim u ltan e o u sly w ith each total
p ressu re m easurem ent. T he total p ressu re m easurem ents w ere m ad e in the units o f the g au g e used and
w ere su b seq u e n tly co rrected w ith g au g e c alib ratio n d ata an d co n v erted to u n its o f atm . B aro m etric
p ressu re d ata w ere converted to units o f atm . an d ad d ed to the co rrec ted g au g e pressures to y ield the
necessary absolute pressures.
O ften p ressure m easurem ents w ithin a series w ere m ade at cham ber tem peratures d eviating slightly
from the target tem perature and from each other. A ccording to ideal gas law s the pressure o f a closed gas
sy stem o f co n stan t v olum e is directly pro p o rtio n al to its tem perature; thus m easured p ressures varied.
As a ch eck on th e accuracy o f the m easurem ents an d to m onitor for gas leaks, each m easured p ressu re o f
a series w as translated via ideal gas law s to the p ressu re that sam ple w ould have a t the ta rg e t tem p era­
ture. T hese translated pressures w ere com pared fo r consistency w ith the high pressure gauge accuracy o f
0 .0 1 4 atm (0.2 p si, in the gauge u n its) o r the low p ressu re g au g e accuracy, w h ich w as 0.0 0 4 atm at
p ressu res n e a r am b ien t and 0.007 atm o th erw ise (3 an d 5 torr, resp ectiv ely , in the g auge u n its). A
decrease in translated pressure g reater than th e g au g e accuracy fro m one m easu rem en t to the n e x t w as
203
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assum ed due to leakage. T his occu rred in one ex p erim en t early in the p ro g ram w hen the high pressure
gauge w as no t isolated from th e ch am b er betw een pressure m easurem ents. T here w ere n o cases in which
the indicated pressure increased b y m o re than the gauge accuracy, an event that w ould have c a st serious
doubt on the validity o f the stated gauge accuracy. M easurem ents consistent w ith no leakage w ere aver­
aged, and then for each datum total pressure w as "de-translated" to the cham ber tem perature o f the datum ,
again using ideal gas law s. In the series w here leaks w ere indicated, translated pressures w ere scaled for
each d atu m assum ing a co n stan t lea k rate and equal tim e req u ired for each d atum 's m easurem ents, and
those scaled pressures w ere de-translated.
R eduction o f baseline d ata m ade use o f the standard tem perature reductions, bu t since baseline data
by definition required an ev acu ated reso n a to r no p ressure reduction w as involved. O th e r baseline data
requiring reduction w ere those necessary fo r producing accurate m easurem ents o f the reso n an ce center
frequencies and bandw idths at the targ et tem perature, T targ, o f the experim ent. T hese included such data
as frequency counter calibration data, spectrum analyzer screen w idth calibration data, and others.
B aseline center frequency reductions w ere the m ost com plex, partly due to th e use o f the E xtended
D rifting IF m ethod o f A ppendix B fo r every datum . T his m ethod req u ired fo u r in d ep en d en t m easure­
m ents o f the spectrum analyzer L O frequency. T w o each o f both indicated center frequency and indicated
spectrum analyzer IF values, called: fCVi jnd,1 ! *cv, ind,
2
1 IF'md,1 ; and IFjncjr 2 , resp ectiv ely , w ere
ex tracted fro m the fo u r L O m easu rem en ts o f each d atum . T h e IF values fro m all d a ta w ere plo tted
against their associated spectrum analyzer internal tem peratures as a check o f accuracy and IF drift trends.
D ue to the absence o f absorbing m edia for baseline m easurem ents resonance SN R w ere m axim ized and
bandw idths w ere m inim ized, m aking these m easurem ents the m ost accurate o f an experim ent. O bserved
IF data w ere quite accurate, usually co n sisten t to w ithin 1-2 kH z and alw ays w ithin 5 kH z, and allow ed
close m onitoring o f IF d rift beh av io r d uring the baselin e series. T hese d ata p ro v ed v alu ab le in later
reductions o f loaded resonator data.
C enter frequencies extracted from the L O m easurem ents w ere not final reduced values. A fter check­
ing the tw o values for co n sisten cy they w ere av erag ed to p roduce fcv,ind,avg • T his v alue w as then
204
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corrected for frequency counter calib ratio n erro r to produce th e final co rrected center frequency for the
baseline series. T w o b a selin e series w ere reco rd ed for e a c h experim ent, o n e p a rt o f the precalibration
phase and the other p art o f the postcalibration phase. O rigin o f each final c en ter frequency w as identified
w ith a subscript, "pre" o r "post." T h u s fCv ,p re = fCv,ind,avg x 1.0000005, using fCv,ind,avg from the
precalibration series; an d fcv .p o st = fcv,ind,avg x 1.0000005, using fcv.ind.avg from the postcalibra­
tion series. N ote that these frequency m easurem ents are m ade with cham ber tem peratures that in general
are different, n either o f w hich is T ta rg - A v eraging the tw o center frequencies at this p o in t w ould have
been m eaningless. W ith E quation D.3 as a base it w as p o ssib le to translate those m easured center fre­
quencies to center frequencies that w ould have been m easured had the cham ber tem perature been Ttarg:
f c v .p r e .ta r g
f c v , p re +
at,
5T
T
(Ttarg
T c pr9J
,
(D.16)
'ta r g
w here the partial derivative is evaluated at T targ • T he translation equation fo r the postcalibration result
is identical, w ith "post" su b stitu ted fo r "pre." T he final, g lo b al vacuum c e n te r frequency o f the reso ­
nance fo r the experim ent, fcv.targ >
the average o f ^cv,pre,targ ^nd fcv.post,targ •
Bandw idth reduction was a much sim pler procedure. M easured bandw idths from both baseline series
w ere m ultiplied by a screen w idth correctio n factor, derived fro m m easurem ents o f actual screen w idths
using the frequency counter, to correct for screen w idth calibration errors. T he tw o corrected bandw idths
w ere averaged to yield b v , the final, global baseline resonance bandw idth for the experim ent. D ifferences
betw een the tw o m easured bandw idths w ere considered w hen assigning uncertainties to the absorptivities
calculated w ith that b aseline data. N o elaborate tem perature equalization p rocedure like th at used in the
center frequency red u ctio n w as necessary fo r th e bandw idth m easurem ents. R esonance bandw idths do
indeed vary w ith tem perature bu t on ly slightly. F o r exam ple, the bandw idth o f the T E 0 2 4 resonance,
norm ally about 70 0 kH z, changed only 15 kH z w ith a 60 K tem perature change. T em perature variations
during an experim ent, rarely exceeding ±0.5 K , w ould cause undetectable changes in bandw idths.
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R eduction o f d ata acquired with the loaded resonator w as th e n ex t task. T he procedure is quite sim i­
lar to th at fo r the baselin e data, w ith three m ain differences: p re ssu re reductions w ere n ecessary since
gases w ere involved, each d atum p o in t at a given set o f target conditions w as m easured only once, and
the E xtended D rifting IF m ethod was not used, requiring few er m easurem ent o f L O frequencies fo r each
datum. T em perature and total pressure data w ere reduced as previously described.
C enter frequency reductions w ere sim pler than those o f the baseline m easurem ents. Since there was
no duplication o f m easurem ents no tem perature equalization and averaging p rocedure w as necessary, and
the sim pler m ethods from A ppendix B (the C onstant IF an d the D rifting IF m ethods) p roduced on ly one
value each p e r d atu m p o in t fo r the resonance center frequency and spectrum analyzer IF. T he IF m ea­
surem ents assum ed a m o re im portant role in these m easurem ents. U n d er lo ad ed co nditions som e reso­
nances had a significantly lo w er SN R (e.g., the T E 2 2 1 m ode) o r w id er bandw idth (e.g., the TE 2 2 4 m ode)
than their nearest neighbors. B oth conditions degrade the accuracy o f tuning the spectrum an aly zer to
the resonance cen ter frequency, and this in turn degrades the accu racy o f th e cen te r frequency and IF
m easurem ents. U n d e r such conditions it can be m ore accu rate to interp o late an IF value from n eig h ­
boring m easurem ents, use that IF value to reduce the individual L O m easurem ents by the norm al, single
m easurem ent m ethod, and average the resulting indicated center frequencies. T o facilitate this procedure,
the IF value e x tracted fro m each d atum w as plotted against its associated sp ectru m an aly zer internal
tem perature m easu rem en t.
W hen ab sorptivities o f the reso n ato r co n ten ts w ere low , as in baselin e
m easurem ents o r m easurem ents at low pressures, such a p lo t yielded a sm ooth, fairly predictable curve.
A t high absorptivities one o r tw o o f the IF values w ould deviate from the sm ooth cu rv e characteristic o f
the other values, indicating th at those deviating w ere subject to larger errors. R efractivity data fro m the
affected m easurem ents w ere reduced using both center frequency values, that from the A ppendix B m eth­
ods and th at fro m the interpolated IF. The refractivities from th e in terpolated IF m ethod show ed m uch
better agreem ent w ith results from resonances with higher SN R and n arro w er bandw idths. R egardless o f
the m ethod used to generate it, the resonance center frequency carried forw ard to su b seq u en t steps is an
indicated frequency, fcl.in d >requiring a calibration correction.
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T h at co rrectio n is the frequency co u n ter calibration correction, w hich is identical to th at p erform ed
in the red u c tio n o f baselin e data: fcl = fcl.ind x 1.0000005. O n ly one such v alu e w as g en erated fo r each
resonance so no averaging w as necessary. Frequency fc | is the final value fo r the loaded cen ter frequency
un der the co n d itio n s in the resonator. N ote th at this value corresponds to the cham ber tem perature T c .
not TtargB andw idth reduction under loaded conditions w as identical to the baseline procedure. T he indicated
bandw idth w as m ultiplied by the appropriate screen w idth correction facto r yielding b ^ to t. the total co r­
rected bandw idth o f the resonance under the loaded conditions. This total bandw idth includes the effects
o f dielectric loading that are subtracted in the final reduction step. A gain, bandw idth tem perature d epen­
dence is so w eak it w as negligible.
W ith values in h an d for the resonance characteristics un d er vacuum conditions, fcv.targ an d b v , and
those u n d e r lo ad ed conditions, fc | and b | t o t » th e n e x t step w as reduction to m easu red refractiv ity and
absorptivity values. A m in er ad justm ent w as necessary before reducing the refractivity d ata. L o ad ed
center frequency fc | w as m easured at a cham ber tem perature T c , in general slightly d ifferent from T targ ,
the tem p eratu re asso ciate d w ith the baseline ce n te r freq u en cy fcv.targ ■ T o co rre c t this m ism atch the
baseline frequency w as translated to the tem perature o f the loaded frequency m easurem ent, T q , w ith an
equation very sim ilar to E quation D . l 6 :
f1cv - f1cv, targ + [1^g y ,
( T c - T ,a r g )
,
(D .17)
1targ
w here again, th e partial derivative is given by E quation D .3. N ow fCv and fc | co u ld be substituted direct­
ly into the refractivity reduction equation
v =
yielding the refractivity o f the gas m ixture inside the resonator under the loaded conditions.
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B andw idth data also required one m inor correction before the final reduction step. E ach total loaded
bandw idth, b |,to t»included th e effects o f dielectric loading that needed to be subtracted. D ielectric load­
ing m easurem ents w ere m ade sim ultaneously w ith refractivity m easurem ents during experim ents o n pure
hydrogen and helium a t the sam e tem peratures an d o v er the sam e ran g e o f pressures as experim ents on
m ixtures containing am m onia. C hanges in bandw idth observed with the transparent gases w ere assum ed
due to various effects com bined un d er th e dielectric loading label. R efractivities an d dielectric constants
are inseparab ly tied, so m easured refractiv ities o f the am m onia m ixtures w ere u sed to in terp o late or
extrapolate fro m the d ielectric loading d ata to the m agnitudes o f dielectric loading effects co n tain ed in
m easurem ents o f lo ad ed bandw idths. R efractivities o f m ixtures containing am m onia w ere som ew hat
higher than identical m ixtures w ith o u t am m onia at th e sam e total pressure, m aking necessary th e inter­
polations and m inor extrapolations o f th e d ielectric loading data. D ielectric loading bandw idth changes
w ere usually m uch sm aller than bandw idth changes d u e to absorption; they w ere often eith er zero or
near the lim it o f detectability o f the apparatus. T h e largest m easured bandw idth change due to dielectric
loading w as 5 0 kH z w h ile th e la rg e st d u e to ab so rp tio n in th e m ix tu res o f h y d ro g en , h eliu m , and
am m onia was about 4 M H z.
B andw idth change due to dielectric loading effects was calculated for each datum point and subtracted
from the total bandw idth to yield the loaded b andw idth due to absorptivity, b | . T his value, along with
the three other final bandw idth and center frequency values, was applied to the reduction equation
(D.19)
yielding the absorptivity o f the resonator contents. T he units on a as calculated w ith E quation D .19 are
N epers p e r unit length. C onversion to o th er units is describ ed in S ection 4.1, p age 43. F o r this w ork
absorptivities w ere converted to decibels p e r unit length by m ultiplying the results o f E quation D .19 by
20
log io e.
U ncertainty analysis for standard p rocedure is considerably m ore com plex than fo r procedures p re v i­
ously treated in this appendix. It is developed in the sam e order as the standard data reduction procedures:
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fundam ental con d itio n m easurem ents; b aseline m easurem ents; lo a d e d reso n a to r m easurem ents; and
final reduction.
C ondition m easu rem en t uncertainties w ere as discussed earlier. A b so lu te tem perature uncertainty
w ith the R T D th erm om eter w as ±0.4 C w ith a resolution o f 0.1 C e x cep t n e a r 0 C w here the uncertain­
ty w as n ear the reso lu tio n lim it. M easurem ents w ith the low pressure g au g e w ere accurate to ±0.007
atm. excep t n ear am bient pressure, w hen they were ± 0.004 atm . T h e high p ressu re gauge w as accurate
to ± 0.014 atm . (0.2 p si in th e gauge units). T hat gauge co u ld be re a d to a reso lu tio n o f 0.1 psi, but
tests show ed that tem perature effects, gauge hysteresis, and calibration e rro r degraded its uncertainty to
± 0 . 2 psi.
B aseline m easurem ent error sources included tem perature m easurem ent uncertainties, errors affecting
m easurem ents o f center frequencies, and errors affecting m easurem ents o f bandw idths. M ixing ratios and
pressures w ere no t involved.
T he th ree m ajor sources o f erro r in center frequency m easurem ents w ere frequency counter calibra­
tion error, tem perature error, an d ran d o m m easurem ent errors including sp ectru m an alyzer tuning erro r
and nonlin ear IF drift. F req u en cy counter calibration errors, discussed in the c o u n ter Af analysis, w ere
less than a b o u t 0.5 kH z. A b so lu te tem perature errors w ere n o t im portant to th e reso n ato r m ethod itself,
but relative tem perature differences betw een baseline and loaded m easurem ents w ere. Such tem perature
differences co u ld be co rrected to an e x ten t lim ited by the resolution o f the R T D therm om eter, 0.1 C.
D ifferences w ere thus u n certain to ±0.1 C, and the frequency uncertainty p ro d u ced is given by Equation
D .4. Spectrum analyzer tuning erro r and n on-linear IF drift w ere errors that co u ld n o t be experim entally
tested separately, but the E x tended D rifting IF m ethod used in the baseline m easurem ents provided much
data on the co m b in ed effects o f the tw o (and any other contributing ran d o m e rro r sources). The tw o
center frequencies pro d u ced b y that m ethod w ere usually w ithin 1-4 kH z o f each o th e r and w ere never
differen t by m ore than 10 kH z. T hese three erro r sources w ere id en tified by the sym bols A fF c for
frequency co u n ter calibration uncertainty, A f ^ j for tem perature d ifference u ncertainty, and Afrancj for
random tuning and non-linear IF drift errors.
209
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Final baseline center frequencies w ere averages o f tw o individual center frequency m easurem ents that
had been translated to the target tem perature. T he translation procedure w as quite accurate and introduced
negligible errors, but tem perature difference errors carried along undim inis'ned T ranslating both frequen­
cies to T targ involved a relative frequency uncertainty arising fro m 0.1 C o f tem perature difference uncer­
tainty.
U pon a v erag in g th is m ax im u m u n certain ty is halved. E x p ected relativ e e rro r w ill actually
decrease due to the statistical b ehavior o f sum m ed random quantities. M ax im u m errors due to the oth er
tw o sources add linearly but then are divided by tw o upon averaging. E xpected erro r from addition o f the
Afrand term s fro m th e tw o m easurem ents should n o t double, again due to statistical behavior. F requen­
cy c o u n te r c alib ratio n errors are sy stem atic so n o statistical relative e rro r reduction is ex p ected from
A fp c . T he net re su lt is that after averaging, the m ax im u m uncertainty in the final cen ter frequency is
d f AT
A fc v = A frand + A fFC + —
.
(D.20)
w here Afrancj an d A fp c are the uncertainties associated w ith an individual center frequency m easurem ent
as given above and A f ^ j is the frequency uncertainty produced by a tem perature difference uncertainty of
0.1 C. T he term a U t / 2 can be interpreted as the m axim um possible frequency d ifference betw een the
final m easu red v alu e fCv an d the actual cen ter freq u en cy at T targ as m e a su red b y the therm om eter.
U ncertainty Afcv was carried forw ard to the final reduction step.
B aseline bandw idth m easurem ents w ere su b ject to three prim ary sources o f uncertainty, errors due
to the finite resolution o f the spectrum analyzer display, both vertical an d horizontal; screen w id th c a li­
bration error; an d degradation o f th e displayed reso n ato r frequency response cu rv e due to finite SNR.
Spectrum analyzer screen resolution has been p reviously d iscu ssed and is approxim ately 4 0 resolution
elem ents p er m ajor fiducial division in both the horizontal (frequency) and vertical (signal pow er) dim en ­
sions. S lopes o f the resp o n se curve displays w h ere they intersected th e h alf p o w e r fiducial lin e w ere
approxim ately o n e vertical reso lu tio n unit p er h o rizontal resolution unit, so an erro r o f o n e resolution
210
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unit in locating the v ertical po sitio n o f the p e ak p o w e r poin t p ro p ag ated to an e rro r o f o n e horizontal
resolution u n it in the location o f each half-po w er intercept. This uncertainty adds to those d u e to finite
horizontal resolution, yielding a m ax im u m e rro r o f tw o resolution units a t each h alf-p o w er in te rc e p t A
bandw idth m easu rem en t requires tw o in tercep t lo catio n m easurem ents, so each in d iv id u al b andw idth
m easurem en t carries a m axim um e rro r d u e to finite screen resolution o f four tim es the m inim um fre­
quency resolution. H ow ever, bandw idths recorded fo r each datum point did no t rep resen t a single m ea­
su re m e n t R esponse curve displays w ere readjusted and bandw idths rem easured sev eral tim es p er datum
point, w ith each m easurem ent representing ten o r m ore d isplay sw eeps. T he recorded bandw idth w as an
average o f those m easurem ents. T rials in d icated that m easu red bandw idths on a reso n an ce un d er fixed
conditions d id indeed vary w ithin p lu s o r m inus fo u r reso lu tio n u nits, b u t as e x p e c te d th e averages o f
several such m easurem ents varied ab o u t h a lf th a t am o u n t o r less. F o r this analysis th e uncertainty due
to screen reso lu tio n , A b s R , is c o n sid e re d to b e tw ice the freq u en cy resolution fo r the screen w idth
setting used in the m easurem ent:
Ah
_ o
SR
Frequency Span / D iv.
4 0 R es. E lem en ts / D iv.
Screen w idth calibration errors w ere sm a lle r than those in the screen Af m ethod sin ce calibrations
w ere perform ed several tim es during the p ro g ram o f experim ents, n o t afterw ard. R e p eatab ility o f the
calibrations w as ±0.5% o r better, so screen w id th calibration uncertainty is given by Abca| = 0.005bv.
T he accuracy o f positioning the p eak p o w er poin t o f a resonance and picking the h alf-p o w er inter­
cepts degraded if the SN R w as relatively low . T he only resonance thus affected during baseline m easure­
m ents w as the w eaker T E 2 2 1 m ode, so th e uncertainty d u e to SN R degradation o f th e resp o n se curve,
AbsNR>w as essentially zero for o th e r reso n an ces. F o r th e T E 2 2 1 m ode an estim a ted u n certainty w as
recorded w ith the other data at the tim e o f the m easurem ent.
T he to tal uncertain ty in each b aselin e b andw idth m easu rem en t w as the sum o f the th ree u n cer­
tainties discussed. T w o b aseline m easurem ents w ere av erag ed to yield the final b aselin e bandw idth for
each resonance, bv. M axim um errors add lin early in th e averaging, bu t tw o e rro r so u rces, AbsR and
211
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AbSNR> w ere random quantities so the expected relative eirors due to those sources should decrease. The
uncertainty in a sin g le m easu rem en t is the sam e as the m ax im u m u n certain ty in the final bandw idth
after the averaging step, w hich is carried forw ard to the final reduction step:
Afc)v = A bcai + A b SR + A b s N R .
(D.22)
U ncertainties in m easurem ents m ade w ith the resonator lo ad ed w ith a te st gas m ixture w ere sim ilar
to those from b aseline m easu rem en ts, w ith so m e additions an d w ith the deletio n o f m ultiple m easure­
m ent averaging. S N R effects, lim ited to the T E 2 2 1 m ode in baseline m easurem ents, could and did affect
all o f the resonances as ab sorptivities in creased. H igh absorptivity also in d u ced asym m etry in som e o f
the resonances, affecting the accuracy o f bandw idth and especially center frequency m easurem ents.
L oaded center frequency m easurem ents w ere subject to the sam e frequency counter calibration uncer­
tainties and tem perature reso lu tio n uncertainties as individual b aseline m easurem ents, bu t also show ed
the effects o f larger spectrum analyzer tuning errors and uncertainty due to induced resonance asym metry.
T he random error term , w hich includes uncertainties due to tuning errors and non -lin ear IF drift, could be
m uch larger fo r lo ad ed m easu rem en ts as a re su lt o f in creased tuning errors from decreased SN R and
increased bandw idth. N otes reco rd ed w ith the m easurem ents g ave an e stim ated tuning uncertainty that
w as incorporated into the total random uncertainty Afrand- W hen absorptivities were extrem ely high, as
in the m easurem ents on p u re gaseous am m onia, this uncertainty could dom inate all oth er uncertainties
ex cept that due to induced resonance asym m etry.
C enter frequency uncertainty due to induced asym m etry, AfSym, w as d ifficu lt to quantify and could
be relatively large. F ortu n ately during m easurem ents on m ixtures o f am m onia in hydrogen or helium
this effect w as in sig n ifican t fo r m o st resonances; it was a m ore significant p ro b lem in m easurem ents
on pure am m onia. D uring the m easurem ents notes w ere reco rd ed describing any observed asym m etry,
and these w ere used to assign a v alue to AfSym- In pure am m o n ia m easurem ents AfSy m could b e as
large as several hundred kHz.
212
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T h e total uncertainty in the loaded center frequency is the su m o f the uncertainties discussed above:
A fci = A fp c + AfAT + A frand + A fsym .
(D.23)
H ere A fp o is 0.5 kH z as before; Af^T is given by E q u atio n D .4 w ith AT = 0.1 K; Afrancj is the sam e
as in baseline m easurem ents unless SN R o r bandw idth degradation occured, w hen th e estim ated accuracy
from record ed notes w as used; and AfSy m w as determ ined fro m recorded notes describing reso n an ce
asym m etry. T his final cen te r frequency uncertainty w as carried forw ard to the final reduction step.
B andw idth m easurem ents on the loaded resonator carried uncertainties arising from induced resonance
asym m etries and dielectric loading as well as all bandw idth uncertainties discussed fo r individual baseline
m easurem ents. As in c en te r frequency m easurem ents, it w as difficult to quantify the effect o f asy m m e­
tries on m easu red bandw idths. Fortunately the effect o f m oderate asym m etry on b andw idth w as n o t as
significant as it w as on cen te r frequency. T he recorded notes concerning asym m etry w ere used to assign
a value to the frequency uncertainty d u e to induced asym m etry, AbSy m . If the asym m etry w as severe,
as in som e o f the attem pted m easurem ents on pure am m onia, the resonance w as declared unusable.
A lthough bandw idth change due to dielectric loading effects, bQL, was n o t subtracted from the total
loaded bandw idth until the final reduction step, A boL. the uncertainty in the m easu red dielectric loading
bandw idth change, is included here. This allow s all uncertainties associated w ith the loaded bandw idth to
be included w ith the total loaded bandw idth uncertainty calculated here. T he sm all m agnitudes o f boi_,
coupled w ith the n eed to interpolate or extrapolate from the values m easured in dielectric loading ex p eri­
m ents, m ade uncertainties in boi_ a significant fraction o f the value. F o r interpolated b o L the u n certain­
ty w as Aboi_ = bD L/ 4 . If b o L w as extrapolated, the uncertainty w as A boL = ^ D L ^ F o r each resonance all these uncertainties w ere sum m ed to yield the total uncertainty A b |:
A b , = A b cai + A b SR + A bsN R + A b sym + A b DL .
(D.24)
The next step, reduction from m easurem ents to refractivities and absorptivities, uses this result.
213
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T h e fo u r m easured resonance ch aracteristics, fCv , f c l, t>v >^
b | , required to calc u la te the refrac-
tivity and absorptivity fo r one d atu m point, carried u n certainties Afcv , Afc | , A bv , and A b |, respec­
tively. E ach total uncertainty is the sum o f uncertainties arising from differen t erro r so u rces previously
discussed. G lobal uncertainties fo r the re fractiv ity an d absorptivity values w ere c a lc u la te d in term s o f
these total uncertainties, no t their ind iv id u al com ponents.
Substituting center frequencies w ith their expressed uncertainties into the refractivity reduction equa­
tion, E quation D .18, produces th e full reduction equation w ith the expressed global uncertainty:
E xtracting the global uncertainty from this equation yields
+ insignificant higher order term s,
(D.26)
w hich is the global uncertainty in the asso ciated refractivity m easurem ent d u e to all sig n ific a n t sources
o f erro r that directly affect the cav ity re so n ato r m ethod. In typical m easurem ents the se c o n d term o f
E quation D .26 was three orders o f m agnitude sm aller than the first, so it w as usually n eg lig ib le. W hen
the SN R was high for the loaded m easurem ents as w ell as baseline m easurem ents the g lo b al uncertainty
w as often less than 0.5% o f the m easu red refractivity.
C alculation o f the global absorptivity uncertain ty involves all four total uncertainties. T h e absorp­
tivity reduction equation, E quation D .19, is sim p lified slightly by d istributing the v acu u m cen ter fre ­
quency in the num erator o f the leading factor over the difference quantity:
(D.27)
S u b stitu tin g m easu red v alu es w ith th eir a sso c iate d to tal u n certain ties p ro d u ce s th e fu ll red u ctio n
equation with the expressed global uncertainty:
214
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a±Aa = i
(D.28)
E xtracting the global uncertainty yields
cv + fcv ^ c l
+ in significant higher order term s.
(D.29)
T his is the global uncertainty in the calcu lated ab sorptivity due to all sig n ifica n t e rro r sources directly
affecting the cavity reso n ato r m ethod. As in E quation D .1 9 the units a re N ep ers p e r unit length, w hich
are then c o n v erted to d ecibels p er unit length. T he first tw o term s in sid e th e brackets are first o rd er
global uncertainties arising from uncertainties in the m easured baseline and lo ad ed bandw idths. The third
term , global uncertainty d u e to u n certainties in the cen ter frequency m easu rem en ts, is three orders o f
m agnitude sm aller than th e first tw o. U n d er the b e st c o n d itio n s o b serv ed A by and A b| w ere ab o u t 5
kH z each, so the best accuracy th at could be attained w ith the apparatus w as a b o u t ± 0 .9 d B /^ m .
W ith the c h a rs c tc n a slio a o f all uncertainties pertaining to th e cav ity re so n ato r m eth o d and fu n d a­
m ental condition m easurem ents com plete, uncertainties in the d erived cond itio n m easurem ents, the m ix­
ing ratios o f gases in a g en erated m ixture, are now treated. T here w ere larg e variations in the com plexi­
ties o f the procedures u sed to generate m ixtures studied in this w ork. T h e sim p le st w ere the sam ples
consisting o f only one species th at do n o t actually q u alify as m ix tu res. N e x t w ere m ixtures o f two
gases gen erated in a sin g le m ixing step. T h e m o st com plex, an d thus the m o s t su scep tib le to error,
w ere the m ixtures o f three gases requiring the successive d ilution pro ced u re. F o r the uncertainty an a­
lysis m ost m ixture gen eratio n p rocedures can be treated as p arts o f a sin g le g en eral procedure, w ith
differences produced by stopping at different steps in the general procedure.
T h e general p rocedure consists o f various steps in w hich gases are added to the sim ulation cham ber
o r the its contents are partially vented, follow ed by m easurem ents o f total p ressu re and tem perature w ith
th eir accom panying u ncertainties. C om plete m ixing is assu m ed after each a d d itio n o f gas. It begins
w ith the addition o f gas A to a m easured pressure o f P i ± A P i at a m easured tem p eratu re o f T-| ± AT-).
215
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T h e p a rtia l p ressu re o f A and its u ncertainty a t this step, P a ,1 ± A P a ,1 , is still ju s t P t ± A P -|.
Ideal gas law allow s calculation o f the concentration o f A:
NA|1 _
V
P,±AP,
(D .30)
R ( T ,± A 1 ,)
w here R is the R y d b erg constant. N ow a d ifferent gas, B, is added to a total pressure o f P 2 ± A P 2 at
m easured tem perature T 2 ± AT 2 , w here T 2 is only slightly d ifferent from T 1 . A t the new tem perature
the concentration o f A rem ains constant but its partial pressure is changed:
= (P1 ± A P 1) 1 +
(t 2 ± a t 2) - (t 1 ± a t ,)
T 1 ± AT 1
(D .31)
W hen the tem peratures are differen ced in the num erator o f the b rack eted quantity any system atic errors
com m on to both m easurem ents o f absolute tem perature su b tract out, leaving only the e rro r asso ciated
w ith a tem p eratu re difference, ATd . Separate analysis show ed that w hen the tem perature d ifference
betw een pressu re m easurem ents is sm all it is the uncertainty in tem perature difference, no t the uncertain­
ty in absolute tem perature, th a t is th e m ajor co n trib u to r to tem p eratu re-in d u ced e rro r in m ixing ra tio
determ inations. As discussed in Section 4.2, w hen tem peratures v/ere m easured with the R T D therm om ­
eter the uncertain ty in tem perature differences w as a b o u t 0.1 C. I f the therm al cham ber th erm o m eter
w as used, as in the m e th an e w ith h ydrogen ex p erim en t, th at u n certain ty w as a fa c to r o f ten larger.
E xtracting the new partial pressure and its uncertainty yields
(D .32)
(D.33)
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T h e ellipses in E q u atio n D .33 re p re se n t in sig n ific a n t h ig h er-o rd er term s in the d iv isio n u n certain ty
expansions.
T h e n u m b er m ixing ratio o f gas A a fte r step
2
, R a ,2 . is given by the partial pressure o f A d ivided
by the total pressure:
P A, 2 - A P A, 2 _
R a ,2 - A R A, 2
p
2±
a p
2
P A, 2
“
AP a ,2
P A,2 A P 2
P -3 4 )
P2
T runcating all uncertain ty ex p an sio n s to first o rd e r and su bstituting the ex p ressio n fo r AP/\,2 from
E quation D .33 yields the uncertainty in th e m ixing ratio:
AR
.
A H A,2
~
AP i T * ,
P 2. TT .i
P
P i AT* |
P 2_ T
P
T,
P i I T 2 — T 1 1AT i
D_
_2
T i
+
P 1 A P2T2
D T
P , P ?2 T1 1
D
*
(D.35)
T he first term is first o rd er m ixing ratio uncertain ty d u e to uncertainty in th e initial lo ad o f gas A; the
second is due to tem perature d ifference uncertainty; the third is due to uncertainty in absolute tem p era­
ture m easurem ents- ~r.t
fom ui is d u e to uncertainty in the total pressure m easurem ent after adding
gas B. All oth er term s are higher o rd er in the sm all quantities and h ave been truncated. N u m b er m ixing
ratio s m ust sum to unity so th at o f B fo llo w s im m ediately:
R b ,2 ± A R b ,2 = 1 - ( R a ,2 ± A R a , 2 ) -
U n certainty A R b,2 is exactly the sam e as A R a ,2 •
Substituting representative values for param eters in Equation D .35 dem onstrates the relativ e im p o r­
tance o f the term s. U sing P-| = 0.55 atm ., AP-j = 0.007 atm., T-| = 273.0 K, AT^ = 0.4 K , P 2 = 8.2
atm ., A P 2 = 0.014 atm ., T 2 = 273.5 K, an d AT<j = 0 .1 K, the indicated m ixing ratio o f A is 0.0672.
T he first uncertainty term is 854 x 1 0 -6 ; the seco n d is 25 x 1 0 -6 ; the third is 0 .1 8 X 1 0 -6 ; and the
fourth is 115 x
1 0 -6
. M o st o f the m ixing ratio uncertainty is due to pressure m easu rem en t uncertainty
in the initial load. T erm s involving tem p eratu re uncertainties constitute less than 3% o f the total, and
the absolute tem perature uncertainty is tru ly n egligible. The second term is actually sm aller than the
u ncertainty in the first. Pressure m easu rem en t uncertainty A P-|, 5 torr, is app ro x im ated as 0.007 atm.
217
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To greater precision it is 0.00658 atm. U sing the more precise 0.00658 atm. to calculate the first term,
the sum o f the first and second terms is still smaller than the first term alone when it is calculated using
the 0.007 atm. approximated uncertainty. It is evident that with the temperature measurement capability
o f the R I D thermometer the effects of temperature measurement uncertainties are insignificant compared
to direct pressure measurement uncertainties.
The next step in the general procedure is venting o f the w ell mixed chamber contents to reduce the
concentration o f A. Total pressure is brought to P 3 ± A P 3 at measured temperature T3 ± AT3 , where
again T3 is only slightly different from
or T2 . Pressure and temperature at the start o f the venting
process are irrelevant since m ixing ratios (and in the absence o f leakage, concentrations) o f A and B are
exactly the same as for step 2. M ixing ratios also remain constant during venting so the partial pressure
o f A at T3 can be calculated afterward:
where second order term in the m ultiplicative uncertainty expansion is insignificant and has been trun­
cated from the expression on the right.
Dilution by the addition o f m ore gas B is the final step in this general procedure. Gas is added to a
total pressure o f P 4 ± A P4 at measured temperature T4 ± AT4 , where T4 is only slightly different from
the target temperature or any o f the previously measured temperatures. As before, the slight temperature
change from the previous measurement causes a slight change in the partial pressure o f A:
(D.37)
Extracting P /^4 and its uncertainty from this expression,
(D.38)
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w here ATd is the uncertainty in the temperature difference between T3 and T 4 . D ividing this by the
total pressure after step 4 yields the final number m ixing ratio o f A and its uncertainty:
(D.40)
so
(D.41)
(D.42)
The number m ixing ratio o f B is again calculated by R b ,4 ± AR[j,4 = 1 - (R a ,4 ± A R a ,4 ) •
Continuing the exam ple begun after step 2 , let P 3 = 1.000 atm., A P 3 = 0.004 atm., T3 = 273.3
K, A T 3 = 0.4 K, P 4 = 8.2 atm., A P 4 = 0.014 atm., and T4 = 272.9 K. After step 2 the m ixing ratio
o f A and its uncertainty were 0.0 6 7 2 and 0.0010, respectively. That uncertainty is about 1.3% o f the
value. Propagating it and the subsequent measurements through Equations D.41 and D .42 finds the final
m ixing ratio o f 0.00818 with an uncertainty o f 0.0 0 0 1 7 , or about 2.0% . This is far better than the
uncertainty would have been had the mixture been generated directly with an initial ammonia load near
0.067 atm., accurate to +0.007 atm.
I f the broadening gas (gas B in the general procedure above) is not a single species but is a mixture
o f tw o species, the results for the m ixing ratios o f gas A and their uncertainties at each step are the
same.
Additional analysis is required to calculate m ixing ratios o f the tw o foreign gases and their
uncertainties. D ividing the broadening gas B into com ponents B-| and B2 (representing hydrogen and
helium , respectively, in the experiments), all operations up to and including step 4 have included only
219
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gases A and B -|. The procedure for generating a three-gas mixture was identical to the procedure for
generating mixtures o f ammonia in hydrogen up to step 4, the second load o f B-|, and thus all partial
pressures and uncertainties remain unchanged to that p o in t
Step 4 is modified only slightly. The target pressure for P 4 is less than it w ould be for a two-gas
mixture, to leave room for B2 - This difference is visible only when values are substituted into Equa­
tions D .41 and D .42; the equations them selves are unchanged. The partial pressure o f B 1 may be
calculated:
P b i . 4 ± A P b i >4 = ( P 4 ± A P 4 ) - ( P Ai4 ± A P Ai4) = ( p 4 - p A4) ± ( A P 4 + A P Ai4) .
(D.43)
The uncertainty in P b i ,4 is the sum o f the uncertainties in P 4 and P a ,4 • In the experiments A P 4 was
considerably larger than A P a ,4 , so A P b i,4 was about the same size as A P4 .
The final operation, step 5, was an addition o f B 2 to a total pressure o f P 5 ± A P 5 at a temperature
o f T5 ± AT5 . Partial pressures o f A and B1 scale with temperature. Since the concentration o f A has
not changed after step 3 its partial pressure can be calculated from that reference:
t
5 ±
a t
5
t
3 ±
a t
3
P a .5 ± A P a ,5 = ( P a ,3 ± A P a ,3)
’
(D.44)
The final values for the partial pressure o f A and its uncertainty can be extracted from that equation,
yielding
(D.45)
3
(D .46)
The partial pressure o f B1 is scaled from its value after step 4:
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+ " a t4 ’
Pb i , 5 - A P b 1 , 5 - (Pb1, 4 ± AP b - m)
(D.47)
yielding the final values
B1,5 -
A P B15 = ^
14
(D .48)
* B1,4 ■p-
+ U§_
AP B1,4 + P B1,4
In practice this uncertain ty is n o t too d iffe re n t fro m APg-,
4
t 4|a t 4
(D.49)
, so it is only slightly larger than A P 4 .
T he sum o f the partial pressures o f A and B1 at step 4 is ju s t P 4 , w ith uncertainty A P4 , and this also
scales w ith tem perature to step 5:
;P a .s + P b i .s) ± A (P Ai5 + P b i .s) = ( P 4 ± A P 4) ^ s
(D .50)
E xtracting the v alu e and u n certain ty from this equation, su b tractin g the su m o f partial p ressures o f A
and B1 from th e total pressure a t step 5, and pro p ag atin g the uncertainties, produces the fin al value for
the partial pressure o f B2 an d its uncertainty (truncated to first order):
P B2,5 -
A P B2,5
Ts
P5 _ P4T-
I' 44
AP 5 + ^ A P 4 + P 4
14
ATd
P .5 1 )
'
IT5 - T 4 j AT 4
(D-52)
^
T42
U ncertainty in P b 2 given by E quation D .52 is dom inated by the first tw o term s, w hich are fundam ental
pressure m easu rem en t uncertainties that in practice w ere the sam e. T hus this uncertainty is about tw ice
the uncertainty in P g i , m aking P q 2 the m ost uncertain o f the partial pressures. M ix in g ratios fo r the
221
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
three gases are calculated by dividing their p artial pressures after step 5, given in Equations D .45, D .48,
a n d D .5 1 , by P 5 . U ncertainties in all these quantities pro p ag ate norm ally through the division.
A n ex am p le w ill dem onstrate the ty p ical values e n co u n te red in this w ork. F o r conv en ien ce the
exam ple p reviously developed fo r the general p rocedure w ill be used as a starting point, since until step
4 there are no differences in procedure. A ssum e fo r the altered an d additional values and uncertainties:
P 4 = 7.387 atm ., A P 4 = 0.014 atm ., T 4 = 272.9 K , A T 4 = 0.4 K , P 5 = 8.2 atm ., A P 5 = 0.014 atm .,
and T 5 = 273.2 K. A fter step 5 m ixing ratios and th eir uncertainties are: A, 0.00819 ±0.00017; B1,
0.8937 ± 0 .0 0 3 7 ; B 2 , 0.0982 ± 0.0039. N o te that th e m ix in g ra tio o f gas A, representing am m onia,
has in creased (b u t insignificantly) o ver the value o b tain ed in the tw o-gas exam ple, d u e to the slightly
h igher final tem perature. H ad the tem perature at step 5 been the sam e as that o f step 4 in the tw o-gas
exam ple, th at m ix in g ratio w o u ld have b een the sam e. Im p o rtan tly , the uncertainties are identical
betw een the tw o exam ples. T h e accuracy o f the ratio P B 1 /P B 2 . representing the hydrogen-to-helium
ratio in the experim ents, is ab o u t ± 4.4% , b u t the accuracy o f the m ixing ratio o f hydrogen, by far the
m ore critical o f the foreign broadening gases, is better than ± 0.5% .
M ixin g ratios and their uncertainties fro m the refractiv ity m easurem ents on a m ixture o f m ethane
and hydrogen can be calc u lated from steps 1 and 2 o f the g en eral p rocedure. T he larger m ixing ratio
u ncertainties o f that ex p erim en t reflect the m uch larg er tem p eratu re u n certain ties en co u n tered w hen
tem perature m easurem ents w ere m ade w ith the therm al cham ber’s built-in therm om eter.
S om e o f the refractivity m easurem ents h av e been norm alized to allow com parisons despite differing
conditions. This norm alization scales a m easured refractivity by density to the density o f a gas at STP,
yielding the ST P-norm alized refractivity v s t p - U sing ideal gas law the norm alization equation is
_
,,
STP
1 atm
T
P
2 7 3 .1 5 K
’
_ „.
(D ’53)
w here v is the m easured refractivity o f the gas sam ple un d er the total p ressu re an d tem perature condi­
tions existing in the ch am b er at the tim e o f the m easurem ent, and P and T are the m easured values of
222
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
those conditions in atm . an d K. T he m easu red refractiv ity carries uncertainty Av from the sources p re ­
viously d iscu ssed . O f co u rse pressure and tem perature m easurem ents carry u n certainties o f th e ir ow n,
A P and A T. Including those in E q u a d o n D .53 y ield s the g lo b al uncertain ty in the S T P -n o rm alized
refracdvity , Av st p - T runcated to first o rder this uncertainty is given by
»
Avstp -
1 atm
2 7 3 .1 5 K
IA v + v ^ r + v X |P
^
E quadons D .53 and D .54 co u ld also be u sed in n o rm alizing the calc u lated refracdvity o f one co m p o n e n t
o f a m ixture. In that case P m ust be the partial p ressure o f that co m p o n en t and A P is the g lo b al u n ce r­
tainty in th at partial pressure.
223
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