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Microwave and laser spectroscopy of high-L Rydberg states of H(2) and its isotopes

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D IS S E R T A T IO N
MICROWAVE AND LASER SPECTROSCOPY OF HIGH-L
RYDBERG STATES OF H2 AND ITS ISOTOPES
S u b m itte d by
P h illip L ee Ja co b so n
D e p a rtm e n t o f P h y sics
In p a rtia l fu lfillm e n t o f the re q u ire m e n ts
fo r th e D e g ree o f D o c to r o f P h ilo so p h y
C o lo ra d o S ta te U n iv e rs ity
F o rt C o llin s , C o lo rad o
S p rin g 1998
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Number: 9835004
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Copyright 1998, by UMI Company. All rights reserved.
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UMI
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C O L O R A D O STATE U N IV E R SIT Y
M arch 26. 1998
WE
H EREBY
PREPARED
UNDER
RECO M M EN D
OUR
THAT
SU PE R V ISIO N
THE
BY
D IS SE R T A T IO N
P H IL L IP
JA C O B SO N
E N T IT L E D M IC R O W A V E AND LA SER S P E C T R O SC O PY OF H IG H -L
R Y D B E R G STA TES O F H2 AND ITS ISO T O PE S BE A CC EPTED AS
F U L F IL L IN G
IN
PA R T
R E Q U IR E M E N T S
FOR
THE
D O C T O R O F PH IL O SO P H Y .
C o m m ittee on G rad u ate W ork
A d v iser
D e p artm en t H ead
ii
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DEG REE
OF
A B S T R A C T OF D IS S E R T A T IO N
M IC R O W A V E AND L A S E R SPE C T R O SC O PY O F H IG H -L R Y D B E R G
ST A T E S OF H 2 AND ITS IS O T O P E S
T he h ig h est-L (^ 4 ), n = 9 ,1 0 R ydberg le v e ls o f H 2, H D , and D 2 w ere
stu d ie d in a fast m o le c u la r beam , u sin g both o p tic a l an d m icro w av e
sp e c tro sc o p y . T his is th e f irs t sy ste m a tic stu d y o f h ig h -L R y d b erg sta te s
in all th re e iso to p es. B o th sp e c tro sc o p ic m eth o d s w e re based on S ta rk
io n iz a tio n o f R ydberg le v e ls re s o n a n tly e x c ite d by a D o p p le r-tu n e d C 0 2
la s e r. T he laser e x c ita tio n sp e c tru m gave o p tic a l s p e c tra w ith lin e w id th s
o f ~ 300 MHz (0.01 cm*1). M ic ro w av e sp e ctra w ere o b ta in e d u sin g th e
s e le c tiv e laser e x c ita tio n fo r d e te c tio n o f m ic ro w a v e -in d u c e d tra n s itio n s ,
w h ich gave lin e w id th s o f a b o u t 1 M H z.
T he R ydberg fin e s tru c tu re re v e a le d in th e o p tic a l sp e c tro sc o p y w as
a lm o st id en tica l in a ll th re e iso to p e s . Some d iffe re n c e s in the s tru c tu re ,
due to d iffe re n c e s in th e q u a d ru p o le m om ents o f th e th re e ions, w ere
c le a rly o b serv ed , e s p e c ia lly fo r th e h e te ro n u c le a r H D + io n . H ow ever, th e
m o st sig n ific a n t v a ria tio n s in th e sp e c tra , o b se rv ed fo r th e th ree iso to p e s ,
w ere in the re la tiv e lin e in te n s itie s . T he P auli e x c lu s io n p rin c ip le fa v o re d
odd R ro ta tio n a l sta te s in H 2, fa v o re d even R ro ta tio n a l sta te s in D 2, and
had no e ffe c t on the a sy m m e tric HD m o lecu le.
iii
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M icro w av e s p e c tro s c o p y w as used to stu d y h ig h -L R y d b erg fine
stru c tu re o f th e H 2 an d D 2 n = 9 ,1 0 R ydberg le v e ls b o u n d to the ground
(v = 0 ,R = 0 ) s ta te s o f H 2+ a n d D 2+. The m e a su re d fin e s tru c tu re in te rv a ls
w ere an aly ze d w ith in th e p o la riz a tio n m o d el, and y ie ld e d p re c ise v alu es
fo r th e s c a la r a d ia b a tic d ip o le p o la riz a b ilitie s o f b o th io n g round sta te s.
T he m easu red p o la r iz a b ilitie s w ere, fo r (0 ,0 )H 2+, a s= 3 .1680(7) aG3 an d ,
fo r (0 ,0 )D 2+, a s= 3 .0 7 1 6 (4 ) a03. T hese m e a su re m e n ts h av e a lre a d y
m o tiv a te d im p ro v e m e n ts to e x is tin g th e o re tic a l tre a tm e n ts o f th is sim p le st
m o lecu le.
P h illip L ee Ja c o b so n
D e p a rtm e n t o f P h y sics
C o lo ra d o S ta te U n iv e rsity
F o rt C o llin s , CO 80523
S p rin g 1998
iv
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Table o f C o n ten ts
Chapter 1
1
Rydberg states o f m olecular hydrogen
1.1 Introduction and historical background................................................................. I
1.2 Theoretical description of H2 high-L Rydberg states...........................................7
1.3 Motivation and overview of thesis structure
28
Chapter 2 ________________________________________________________________ 30
Experimental technique and details o f equipment
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Introduction to the laser RESIS technique..........................................................30
Population and preparation o f Rydberg states.................................................... 33
Laser resonant excitation between Rydberg states............................................44
Stark ionization and detection o f Rydberg states...............................................48
Example o f observed H2 Rydberg state transitions........................................... 55
Description o f microwave RESIS technique...................................................... 59
Details o f the microwave spectroscopy apparatus.............................................63
Chapter 3
69
Laser spectroscopy o f n—9 ,10 high-L Rydberg states o f H 2, HD, and D 2
3.1 Introduction o f the laser spectroscopy experiment
69
3.2 Expected theoretical transition structure............................................................. 71
3.3 Measured optical spectra for
HD, and D2.................................................... 77
3.4 Discussion o f the measured spectra................................................................. 112
3.5 Conclusions o f the laser spectroscopy experiment.......................................... 139
Chapter 4 ________________________________________________________________142
M icrowave spectroscopy o f R=0, n=9,10 high-L Rydberg states o f H 2 and D 2
4.1
4.2
4.3
4.4
4.5
Theoretical fine structure of H2/D2 high-L Rydberg states
143
Measured microwave RESIS transitions...........................................................150
EFS intervals extracted from measured transitions..........................................177
EFS related to Veff and fitted for dipole polarizabilities.................................. 190
Conclusions o f the microwave spectroscopy
205
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Appendix A
202
Ro-vibrational energies and core parameters
Appendix B
205
Description o f Ef2) contributions
Appendix C _____________________________________________________________ 208
Spin structure calculations fo r (0,0) H2 and D 2 states
Appendix D
218
Calibration procedure fo r optical spectra
References
221
vi
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C hapter 1
R ydberg states o f m olecular hydrogen
1.1 In tro d u ction and h isto r ic a l b a ck grou n d
R ydberg a to m s a n d m o le c u le s are d e fin e d as th o s e w ith one e le c tro n
in a h ig h ly e x c ite d s ta te , w h o se o rb it is m uch la r g e r th a n the siz e o f th e
u n d e rly in g ion c o re . S u ch sy ste m s have b een in v e s tig a te d in a to m ic
sp e c tro sc o p y fo r a lo n g tim e 1. T he large siz e o f R y d b e rg atom s a llo w s th e
stu d y o f a w ide ra n g e o f p h e n o m e n a , in c lu d in g th e s tro n g in te ra c tio n o f
R y d b e rg sy stem s w ith e x te rn a l su rro u n d in g s, s p e c if ic a lly , th e ir s e n s itiv ity
to e x te rn al e le c tric o r m a g n e tic fie ld s. R e se a rc h e rs h a v e lea rn ed to
m a n ip u late th e se R y d b e rg sy ste m s in w ays th a t o ffe r a n e w to o l to g ain
in s ig h t into v a rio u s to p ic s in a to m ic p h y sic s. T h e in itia l e x p e rim e n ts
w ere im p o rtan t, b u t it w as n o t u n til the in tro d u c tio n o f tu n a b le la se r
te c h n iq u e s th a t e ff ic ie n t p o p u la tio n o f s p e c ific R y d b e rg sta te s w as m ad e
p o s s ib le , a llo w in g a v a rie ty o f e x p erim e n ts to be c o n s id e re d 12.
A p a rtic u la r c la s s o f R y d b e rg atom s is m a d e up o f th o se w h ere th e
h ig h ly ex cite d e le c tro n a lso has h ig h o rb ita l a n g u la r m o m en tu m (L ). W ith
su ffic ie n tly h ig h L, th e e le c tro n o rb it a p p ro a c h e s a c ir c u la r p a th and d o es
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n o t o v e rla p th e io n . F ro m the e le ctro n v ie w p o in t, th e io n core can be
a p p ro x im a te d by a “ p o in t-c h a rg e ” . T h is is a lso tru e fo r m o le c u la r
R ydberg sta te s. F o r la rg e enough L, th e s p a tia l e x te n t o f ev en a
m o le c u la r ion can be n e g le c te d . T he d e fin itio n o f h ig h L is b est
u n d e rsto o d by c o n s id e rin g th e C oulom b in te ra c tio n b e tw e e n the c o re and
the R ydberg e le c tro n . T he c e n trifu g a l p o te n tia l b a rr ie r 3, p ro p o rtio n a l to
L ( L + l) /r 2, d e te rm in e s th e p o in t o f c lo s e s t a p p ro a c h b e tw e e n the c la ss ic a l
o rb it o f a R y d b erg e le c tro n and the ion c o re , d e fin e d as th e in n e r c la s s ic a l
tu rn in g p o in t o f th e ra d ia l m otion. F or e le c tro n s o f a n g u la r m om entum L,
the in n e r tu rn in g p o in t is g re a te r th a n L ( L + l) /2 (a 0: B o h r ra d iu s) fo r all
v a lu e s o f the p rin c ip a l q u a n tu m n um ber, n. I f th is m in im u m se p a ra tio n
from the core is g re a te r th a n the sp a tia l e x te n t o f th e c o re w a v efu n c tio n ,
th e re is n e g lig ib le p e n e tra tio n . F or H 2 R y d b e rg s ta te s , th is re q u ire m e n t is
m et by sta te s w ith L > 4.
M o le c u la r R y d b e rg system s have th e a d d itio n a l c o m p lic a tio n o f th e
in te rn a l core m o tio n . E v en th e sim p le st d ia to m ic m o le c u la r ions have
b o th v ib ra tio n and r o ta tio n . In a d d itio n , fo r lo w -L R y d b e rg sta te s, th e re
is a stro n g c o u p lin g b e tw e e n the o rie n ta tio n o f th e c o re io n and the o rb it
o f th e R ydberg e le c tro n . T he e ffe c t o f th is c o u p lin g is th a t, along w ith
th e to ta l a n g u la r m o m e n tu m , only the p ro je c tio n o f th e o rb ita l an g u la r
m om entum onto th e c o re a x is is a c o n se rv e d q u a n tity , la b e le d w ith the
qu antum n u m b er, A. F o r h ig h -L sta te s, th e R y d b e rg e le c tro n is far
e n o u g h from th e c o re th a t b o th the ro ta tio n a l a n g u la r m o m en tu m o f th e
2
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c o re (R ) and the R y d b e rg e le c tro n o rb ita l a n g u la r m om entum (L ) are
c o n se rv e d . In th is “ L -u n c o u p le d ” or “H u n d ’s c a s e -d ” sy ste m 4, the ion
c o re ro ta te s fre e ly w ith in th e o rb it o f the R y d b e rg e le ctro n . In th e se tw o
ex tre m e s, low -L v s. h ig h -L R ydberg sta te s, v ery d iffe re n t co u p lin g
sch em es are u sed , b u t fo r L>4 the la tte r “c a s e -d ” lim it is a p p ro p ria te in
H 2.
For e x tre m e ly h ig h o rb ita l a n g u la r m o m en tu m (L ^6 ) R ydberg s ta te s ,
th e se p a ra tio n o f th e R y d b erg e le c tro n and th e H 2+ co re m akes it p o ss ib le
to reg a rd th e e le c tro n as a se n sitiv e , n o n -in tru s iv e p ro b e o f the io n . T he
R y d b erg e le c tro n i t s e l f h as little e ffe c t on th e c o re ; how ever, long ran g e
in te ra c tio n s e x h ib it d is tin c tiv e sp littin g s o f th e e le c tro n energy le v e ls,
w h ich are re la te d to th e io n c o re stru c tu re . S tu d ie s o f th is R ydberg
e n erg y level fine s tru c tu re e ffe c tiv e ly use th e e le c tro n as a p ro b e o f th e
m any p ro p e rtie s o f th e H2+ ion. The lo n g -ra n g e in te ra c tio n s d ire c tly
re la te the R y d b erg e le c tro n en erg y lev els to fu n d a m e n ta l p ro p e rtie s o f th e
io n , such as th e q u a d ru p o le m om ent, d ip o le p o la riz a b ilitie s , and h y p e rfin e
c o n sta n ts. T hese in te ra c tio n s w ere used in d e sc rib in g the en erg y le v e ls in
a h ig h -p re c is io n stu d y o f th e n=10 in te rv a ls o f H 2 R ydberg s ta te s 5-6. A
v e ry p rec ise (0 .0 2 % ) v a lu e fo r th e (v = 0 ,R = l; th e v ib ra tio n a l and
ro ta tio n a l q u an tu m n u m b ers re sp e c tiv e ly ) q u a d ru p o le m om ent w as fo u n d ,
w h ich could only be a c c o u n te d fo r a fte r th e th e o ry o f th e free H2+ sy ste m
w as c o n sid e ra b ly im p ro v e d . The sam e e x p e rim e n t a lso d ete rm in e d the
s c a la r and te n s o r p o la riz a b ilitie s ( a s, a t) o f th e sam e ion lev el. In o th e r
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s im ila r m easu rem en ts, H 2 n= 27 R y d b e rg fin e stru c tu re in te rv a ls w ere
m e a su re d to e x tra c t the H 2 + (0 ,1 ) h y p e rfin e c o n sta n ts7, a n d H 2 ( 0 ,l) n = 2 8
to (0 ,3 )n = 1 6 in te rv a ls w e re m ea su re d to e x tra c t the H 2 + ( 0 ,l ) - ( 0 ,3 )
ro ta tio n a l splitting*. E ac h o f th e se m e a su re m e n ts, a lth o u g h n e a rly
id e n tic a l in th e ir m e th o d s, te s ts a d iffe re n t p ro p e rty o f th e io n c o re . T h is
se rv e s to illu stra te th e fle x ib ility o f th is e x p e rim e n ta l te c h n iq u e , u sin g
R y d b e rg sta te s to g a in in s ig h t in to th e fu n d am e n ta l io n c o re .
T he m easu rem en t o f su ch io n co re p ro p e rtie s has b e e n an im p o rta n t
stim u lu s for im p ro v e m e n ts in th e th e o re tic a l d e sc rip tio n o f H 2 +- A t f ir s t
one m ig h t think th a t th e H 2 * io n is a c o m p le te ly u n d e rs to o d sy ste m in
m o le c u la r p h y sics, bu t c o n tra ry to th is b e lie f, even th is s im p le s t o f
m o le c u le s is u n d e rsto o d fa r le ss p re c is e ly th a n the o n e -e le c tro n a to m . In
a to m ic system s, th e ro le o f th e h y d ro g e n ato m was c ru c ia l to the
d e v e lo p m e n t o f fu n d a m e n ta l, q u a n tu m m ec h an ica l th e o rie s . T he sim p le
h y d ro g e n atom w as, and s till is, an im p o rta n t te stin g g ro u n d fo r new
p h y s ic a l effe cts, in c lu d in g re la tiv is tic an d QED e ffe c ts. T h e a b ility to
u n d e rs ta n d th is a to m ic sy s te m w as p rim a rily b ecau se th e n o n - r e la tiv is tic
H a m ilto n ia n could be so lv e d c o m p le te ly . In H 2 th e n o n - re la tiv is tic
p ro b lem is m uch m ore c o m p le x . In fa c t, o n ly in th e o n e -e le c tro n
m o le c u la r ion, H 2 *, d o es a c o m p le te s o lu tio n o f the n o n - r e la tiv is tic ,
C o u lo m b H a m ilto n ian se em a re a so n a b le g o a l.
T he th e o re tic a l d e s c rip tio n o f H 2 + is p o ssib le at v a rio u s le v e ls o f
a p p ro x im a tio n , d e p e n d in g on th e d e s ire d p re c isio n . T he s ta n d a rd
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te c h n iq u e s w ere o u tlin e d n ic e ly in a re v ie w by C a rrin g to n , M cN ab, an d
M o n tg o m erie 9. T he s im p le s t ap p ro ach u ses th e B o rn -O p p e n h e im e r
a p p ro x im a tio n w ith a “ c la m p e d -n u c le u s” , w h ic h ig n o re s th e n u c le a r
m o tio n and any e ffe c ts it m ig h t have on th e p o te n tia l or the e le c tro n ic
s ta te s . T he w a v e fu n c tio n is ap p ro x im ate d by th e p ro d u c t o f a sin g le
e le c tro n ic w a v e fu n c tio n and a n u c le ar w a v e fu n c tio n . T his tre a tm e n t le a d s
to a re a so n a b le s o lu tio n o f the H 2 + sy ste m ; h o w e v e r, the p re c is io n o f th e
re s u lts is lim ite d by th e ra tio o f the e le c tro n to n u c le i m ass (m e/m n).
The n e x t le v e l o f p re c is io n can be re a c h e d w ith the a d ia b a tic
m eth o d o f tre a tin g H 2 *, w h ic h in clu d es the d ia g o n a l c o u p lin g s o f th e
n u c le a r m o tio n and e le c tro n ic w a v e fu n c tio n s. T h is tre a tm e n t, w h ile an
im p ro v e m e n t o v e r th e B o rn -O p p e n h e im e r a p p ro x im a tio n , re ta in s th e b a sic
a ssu m p tio n th a t th e Hz* w a v e fu n c tio n can be w ritte n w ith a sin g le
e le c tro n ic sta te w a v e fu n c tio n as a fa c to r. T h e a c c u ra c y o f a ll su c h
“ a d ia b a tic ” c a lc u la tio n s is g e n erally lim ite d to a b o u t (m e/m „) fo r m o st io n
p ro p e rtie s.
N o n -a d ia b a tic c a lc u la tio n s , w h ich a v o id th e “ a d ia b a tic ” a s s u m p tio n
th a t the w a v e fu n c tio n h a s a sin g le e le c tro n ic s ta te w a v e fu n c tio n as a
fa c to r, yield th e h ig h e s t p re c isio n ; h o w e v e r, th e s e a re ty p ic a lly m uch
m o re d iffic u lt. T h ere is no w idely a c c e p te d , s in g le a p p ro ach to su ch
c a lc u la tio n s. A n u m b e r o f a lte rn a tiv e m e th o d s h a v e been c o m p a red by
B abb and S h e rtz e r10. T h e m eth o d o f B ab b , a s d e sc rib e d in th is re fe re n c e ,
d e v e lo p s th e n o n -a d ia b a tic p o rtio n o f the w a v e fu n c tio n in a fo rm al
5
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p e rtu rb a tiv e th e o ry .
A n o th e r a p p ro a c h is the fin ite e le m e n t m eth o d
(FE M : J. S h e rtz e r) ", w h ic h e s s e n tia lly breaks up th e in te g ra tio n o f th e
S c h ro d in g e r e q u a tio n in to “ e le m e n ts ” th a t are e v a lu a te d in d iv id u a lly , by
e x p a n d in g th e w a v e fu n c tio n in to lo c a l basis sets. A th ird a p p ro a c h , a
v a ria tio n a l c a lc u la tio n ,12 d e p e n d s on th e a d ju stm en t o f a tria l
w a v e fu n c tio n . T h ese la st tw o te c h n iq u e s are lim ite d so fa r to c a lc u la tio n s
o f th e g round sta te o f th e io n . O n th e o th er hand, th e p e rtu rb a tiv e
te c h n iq u e , d e v e lo p e d by B ab b , is a m ore g eneral a p p ro a c h , w h ich h o ld s
p ro m ise for b e in g a b le to tre a t a v a rie ty o f sta te s o f H 2 +. D e v e lo p m e n t o f
a d d itio n a l im p ro v e m e n ts to th e a d ia b a tic m ethods is u n d e r w a y 13, b u t can
o n ly be te ste d by m e a su re m e n ts o f p re c isio n g re a te r th a n m e/m„.
In c o n tra st to th e a to m ic o n e -e le c tro n sy stem , th e re has been v ery
little e x p e rim e n ta l w o rk on H 2 +. T he rev iew by C a rrin g to n et a l.s a lso
c o v ers th is w e ll. T he lac k o f e x p e rim e n ts hin d ers th e d iff ic u lt th e o re tic a l
im p ro v e m e n ts th a t are n e c e ssa ry to co m p lete the n o n -re la tiv is tic tre a tm e n t
o f th e ion and m ove b e y o n d it. W ith o u t m easu rem en ts to co m p are w ith ,
d isc re p a n c ie s b e tw e e n d iffe re n t th e o re tic a l te c h n iq u e s a re d iffic u lt to
re s o lv e . In the p a st, d ire c t s tu d ie s o f H D + w ere p io n e e re d by W ing e t
a l . M, w h ile o th e r su c c e ss fu l e x p e rim e n ts m easured th e h y p e rfin e s tru c tu re ,
b o th in H 2 + by J e f f e r ts 15 and m o re re c e n tly in H D + by C a rrin g to n et a l . 16
H o w ev er, d ire c t sp e c tro sc o p ic m e a su re m e n ts o f H 2 + are se rio u sly lim ite d
d u e to e x p e rim e n ta l d iff ic u ltie s . O n th e other hand, in d ire c t stu d ie s u sin g
R y d b e rg lev e ls to p ro b e th e io n h ave been very re w a rd in g , as m e n tio n e d
6
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ab o ve. U sin g th is in d ire c t ap p ro a ch , the e x p e rim e n t rep o rted in C h a p te r 4
o f th is th e s is re s u lts in im p o rta n t new in fo rm a tio n about th e a b so lu te
ground sta te o f the m o le c u la r io n s, H 2 * and D 2 +. T he m easu red d ip o le
p o la riz a b ilitie s are p re c ise e n o u g h to c h a lle n g e c u rre n t th e o re tic a l
m o d els, and p e rh a p s e v en p re c ise enough to re q u ire the a n tic ip a te d
tre a tm e n t o f r e la tiv is tic c o n trib u tio n s to the H am ilto n ian .
1.2 T h e o retica l d escr ip tio n of H2 h igh-L R ydberg states
T his s e c tio n w ill g iv e a d e sc rip tio n o f th e stru c tu re in an H 2 h ig h -L
R ydberg sy ste m . S e v e ra l ste p s w ill be c o v e re d in a p e rtu rb a tiv e a p p ro a c h
to the th e o ry , as d e fin e d by S tu rru s 6. The firs t w ill be the z e ro th -o rd e r
stru c tu re , in c lu d in g b o th the ion core and h y d ro g e n ic e le c tro n e n erg y
lev e ls. T his w ill be fo llo w e d by d isc u ssio n o f th e h ig h e r o rd e r e le c tric
in te ra c tio n s b e tw e e n th e e le c tro n and core, as re fle c te d in th e fine
stru c tu re s p littin g o f th e h y d ro g e n ic e le c tro n le v e ls . This e le c tric fin e
stru c tu re w ill be re fe rre d to as E FS. F in a lly , as a sm all p e rtu rb a tio n to
each o f th e se le v e ls , th e m a g n e tic in te ra c tio n s o f th e n u c le a r and e le c tro n
sp in s w ill be in c lu d e d . T h is fin a l c o n trib u tio n , th e m ag n etic sp in
stru c tu re (M F S ), is o n ly e x p e rim e n ta lly e v id e n t w hen c o n sid e rin g th e h ig h
re s o lu tio n , m ic ro w a v e sp e c tro sc o p y e x p e rim e n t in C h ap ter 4.
1.2-1
T he free ion c o re and R y d b erg e le c tro n tre a te d se p a ra te ly .
H igh-L R y d b erg sy ste m s are in te re stin g in th e ir sim ila rity to th e
fa m ilia r h y d ro g e n ato m . As illu stra te d by F ig . 1-1, i f the R y d b erg e le c tro n
7
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is fa r e n o u g h aw ay, th e s p a tia l e x te n t o f th e ion c o re is n e g lig ib le . T he
o rb ita l ra d iu s o f a R y d b e rg e le c tro n is p ro p o rtio n a l to n2, w ith th e siz e o f
a n n= 10 R ydberg sta te b ein g ~ 1 0 0 a o as co m p ared to th e 2 a 0 siz e o f th e
io n . In a d d itio n , fo r L >4, th e in n e r tu rn in g p o in t o f the e le c tr o n ’s ra d ia l
m o tio n is > 1 0 a o. U n d e r th e se c o n d itio n s , th e m o le c u la r io n is c o n sid e re d
a fre e ly ro ta tin g body and tre a te d in d e p e n d e n t o f th e h ig h -L R y d b e rg
e le c tro n . T he m otion o f th e e le c tro n is th a t o f an e le c tro n b o u n d to a
s in g ly c h a rg e d p a rtic le ( o f m ass tw o ), i.e . th e h y d ro g en a to m . T he w eak
in te ra c tio n o f the core ro ta tio n a l a n g u la r m om entum and th e o rb ita l
a n g u la r m om entum o f the R y d b e rg e le c tro n w ill be in c lu d e d la te r as a
Ion core
Rydberg
electron
(-)
Fig. 1-1: Molecular Rydberg systems. This figure shows the “separated” nature o f a
Rydberg system. The ion core is in its ground electronic state and, with high enough L,
the Rydberg electron will not penetrate the ion core.
8
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p e rtu rb a tio n . T he re s u ltin g sta te s are b e s t la b e le d by the q u an tu m
n u m b ers
(v,R )nL N ,
w h ere v is the ion v ib ra tio n a l q u an tu m n u m b er, R is
the io n ro ta tio n a l a n g u la r m om entum , n is th e R y d b e rg e le c tro n p rin c ip a l
q u a n tu m n u m b er, L is the R ydberg e le c tro n o rb ita l a n g u la r m o m en tu m ,
and N = R + L is th e v e c to r c o u p le d to ta l a n g u la r m om entum .
T he c o m p le te n o n -re la tiv is tic , C o u lo m b H a m ilto n ia n can be w ritte n
fo r th e R y d b e rg m o le c u le , w hich c o n sists o f tw o n u c le i and tw o e le c tro n s.
T his e q u a tio n is g iv en as
The la b e ls c o rre s p o n d to the n u c le i (n) an d th e e le c tro n s (e) w ith th e
c o rre sp o n d in g n u m e ric a l la b e ls. T he m a sse s a re d e fin e d
as the n u c le a r
m ass (M x) and e le c tro n m ass (m ). T he p o s itio n v e c to rs are re la tiv e to th e
lab fram e .
T he H a m ilto n ia n is m ore u se fu l w h en tra n s fo rm e d into re la tiv e
Ja co b i c o o rd in a te s , w h ere each new c o o rd in a te is d e fin e d re la tiv e to the
c e n te r o f m ass o f th e p rev io u s q u a n titie s . T h ese re la tiv e c o o rd in a te s are
P - fn2 ~ rnl
% = *ei ~ ( M £ , + M 2rn2)/(M , + M2) = reI - ^ :cm
h = l i -((M , + M 2)rn;cm+ mreI)/(M , + M 2 + m) = ?e2 - rI;cni,
= ((M, + M 2 + m)fI;cm+ mre2) / ( m , + M 2 + 2m)
9
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Eq u 2
and illu stra te d in F ig. 1-2. W ith th e se d e f i n i t i o n s , th e k in e tic en erg y
term s (firs t b rack eted term in Eq. 1-1) can be rew ritten in th e form
H ke =
i- i2
IU 1
2M total
i- i2
|Pi|_ iPzl
2 n n + 2p, + 2 p 2
Eq. 1-3
w here each su b scrip t re fe rs to the su b sc rip t o f the d e fin e d re la tiv e
c o o rd in a te s, and the m o m en ta re p re se n t d e riv a tiv e s w ith re s p e c t to each
o f th o se c o o rd in a te s. T he red u c ed m asses are defined as
e=
m
M, + M 2 + m
M,M2
~ M, + M,
m(M. + M ,)
M, + Mj + m
m(M, + M2 + m)
M, + M 2 + 2m
Eq. 1-4
.
.
m
(l + e)
U sing th e d e fin itio n s o f E q. 1-2, the C oulom b in te ra c tio n s o f Eq. 1-1 are
d e riv e d in term s o f the new c o o rd in a te s. T he com plete H a m ilto n ia n can
be w ritte n in a “ se p a ra te d ” form ,
H = H 0(rcm) + HJJore(r,,p ) + Hoyd(f2) + V (r,,r2,p)
10
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Eq. 1-5
Pp
0
2M total
2
|Pi|
e2
e2
e2
2n n + 2 P. + | p f _
r. +
1
m
m
,+
2
m
.
'o
2
rI
M,
.
M, + M 2 P
1 _ e
2^ 2
Eq. 1-6
|r.
V=
tel te-(»-«)?!I
r, + er. +
2
M, + M ,
m
M,
r2 + Er. “ M, + M 3
— >
m,
M i
Fig. 1-2: Coordinates defined for a molecular Rvdbere system. The Rydberg electron is
much further from the nuclei than the core electron. T2 > n .
11
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T he first b ra c k e te d te rm in Eq. 1-6 is the H a m ilto n ia n fo r th e free ion.
T he second b ra c k e te d te rm , by in clu d in g th e a d d itio n a l C o u lo m b fa c to r in
th e p o te n tia l, is the H a m ilto n ia n for the h y d ro g e n ato m and th e last te rm
is th e rem a in in g C o u lo m b p o te n tia l. Ig n o rin g th e c e n te r o f m ass m o tio n ,
th e tw o p a rts o f th e R y d b e rg m o lecu le are c o n s id e re d “ s e p a ra te ly ” , and
th e z e ro th -o rd e r e n e rg ie s are
Eq. 1-7
In itia lly , th e en erg y le v e ls o f a m o le c u la r R y d b e rg sy ste m a re g iv en by
th e sum o f th e s e tw o in d e p e n d e n t p arts, th e ro -v ib ra tio n a l e n e rg y o f th e
free ion and th e h y d ro g e n ic e n e rg y o f th e R y d b e rg e le c tro n . T he se co n d
te rm in Eq. 1-7 is the e n e rg y fo r a h y d ro g en ic e le c tro n w h e re SRred is th e
R y d b erg c o n sta n t c o rre c te d fo r th e red u ced m ass.
91. = —
"i
Eq. 1-8
1+ 8
SR. = 10973731534 c m '1
In the a d ia b a tic a p p ro x im a tio n for H 2 *, th e v e c to r-c o u p le d
e ig e n sta te s o f to ta l a n g u la r m om entum (N ) c an be w ritte n as
12
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V oAvRnLNM ( P » *j » *2 ) ~
£ (R, m ,, L, m2|R, L, N, M) v ^ vRnii (p, r, )y ^Lnij (r2)
HII.IBj
Eq. 1-9
(P.f,) =
" D*
(p)f„(r,';p)
T he w a v e fu n c tio n s are lin e a r c o m b in a tio n s o f the p ro d u c ts o f the H 2 +
e ig e n s ta te s and th e h y d ro g e n ic R ydberg e le c tro n e ig e n s ta te s , as given in
Eq. l- 7 b . The fu n c tio n , fa A (r'i ;p), is th e e le c tro n ic w ave fu n c tio n o f th e
c o re e le c tro n , w ith r 'i re fe rre d to the in te rn u c le a r a x is, and g (p ) is th e ro v ib ra tio n a l w a v e fu n c tio n '7.
T he energy le v e ls are c a lc u la te d u sin g th e se e ig e n s ta te s . The io n
c o re e n e rg ie s have been c a lc u la te d and ta b u la te d fo r the lo w e st ro ta tio n a l
an d v ib ra tio n a l levels'*. A few o f th ese e n erg ies a re g iv en in A ppen d ix A.
T he E (0) stru c tu re fo r H 2 R y d b e rg sta te s is show n in F ig. 1-3, to illu stra te
h ow e ac h ro -v ib ra tio n a l le v e l h a s an e n tire m a n ifo ld o f a llo w e d n lev e ls
b o u n d to it, w here n is th e p rin c ip a l q u an tu m n u m b e r fo r th e h y d ro g en ic
R y d b e rg ele ctro n .
1.2-2
E ffe c tiv e p o te n tia l m odel o f lo n e -ra n e e in te ra c tio n s
T he rem ain in g te rm in th e C oulom b H a m ilto n ia n , g iv en in Eq. 1-6,
re p re s e n ts the lo n g -ra n g e in te ra c tio n s b etw een the R y d b e rg e le c tro n and
th e io n co re. T his p o te n tia l can be ex p an d e d , u n d e r th e a ssu m p tio n s th a t
a) the R ydberg e le c tro n is d istin g u is h a b le fro m th e o th e r e le c tro n
b) the R ydberg e le c tro n is n o n -p e n e tra tin g so r 2 > n , r 2 > p.
13
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All H2energies are in cm*1
Splittings are not to scale.
R=2
v = li R=!
R=0
▲
2191.126 cm*
174.24
R=2
58.23
7
30-----20 ------
-1097.1
-1354.4
10------
n=9'
Fig. 1-3 : Lowest rotational and vibrational levels for H2*. Each level has an entire
manifold of bound Rydberg levels (n). The binding energies are independent o f the core
quantum numbers.
14
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A s d e riv e d by S tu rru s e t a l . '7 fo r H 2 , n e g le c tin g th e te rm s o f o rd e r e, th is
g iv e s th e re su lt
Eq. 1-10
cos02, = ?,-?,
cos02p = r2 -p
T he p e rtu rb a tio n V c a n be show n to r e s u lt in an “ e ffe c tiv e
p o te n tia l” from w hich th e R y d b e rg fine s tru c tu re e n e rg ie s a re
c o n v e n ie n tly c a lc u la te d . E s s e n tia lly , th e f ir s t- o rd e r p e rtu rb a tio n due to V
re s u lts in a m u ltip o le e x p a n s io n c o n ta in in g a ll th e p e rm a n e n t e le c tric
m o m en ts o f the io n c o re . T h e se c o n d -o rd e r p e rtu r b a tio n re s u lts in
a d d itio n a l term s in th e e ff e c tiv e p o te n tia l, w h ic h re p r e s e n t th e e ffe c ts o f
in d u c e d e le c tric m o m e n ts, s u c h as the d ip o le p o la r iz a tio n e n e rg y .
T he re s u ltin g e ff e c tiv e p o te n tia l no lo n g e r d e p e n d s e x p lic itly on the
c o re e le c tro n c o o rd in a te , r i , b u t in stead d e p e n d s on a fe w e le c tric
p ro p e rtie s o f the ion c o re th a t a p p ear as c o e ffic ie n ts in th e d e fin e d
p o te n tia l. The form o f th e e ffe c tiv e p o te n tia l is
Eq. 1-11
d (p ) = d ip o le m o m e n t [d (p)= 0 fo r H 2 + and D 2 +]
Q (p) = q u a d ru p o le m om ent
ccx(p ) = a d ia b a tic d ip o le p o la r iz a b ilitie s ( s c a la r an d ten so r)
15
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T he m a trix e le m e n ts o f th is e ffe c tiv e p o te n tia l a re to b e e v a lu a te d w ith in
th e red u ced su b sp ac e o f R y d b e rg s ta te s , re p re se n te d by th e b a sis
fu n c tio n s,
V vRnLNM
T he p e rtu rb e d R y d b e rg e n e rg ie s are c a lc u la te d w ith in th is su b sp a c e
by u sin g Veff in s ta n d a rd p e rtu rb a tio n th eo ry . T he m a jo r c o n trib u tio n to
th e en erg y lev e ls co m es fro m th e d ia g o n a l m atrix e le m e n ts o f the
p o te n tia l, i.e. the firs t o rd e r p e rtu rb a tio n e n e rg ie s. T h e s e fin e s tru c tu re
s p littin g s (E FS) are g iv en by
v X u , = |(v,R ),(n,L > ,N )
E (l* —< V | / R y d
I v ^ l
Eq. 1-13
v Ryd)
| e(Q)[ iiL||r"3|nL] + ^ - ( a t )[nL|r‘4flnL]V r L n JP2(cosB 2p)| RLN)
T he core p a ra m eters a re c a lc u la te d by a v erag in g o v e r th e n u c le a r
w a v e fu n c tio n , e.g.
Eq. 1-14
E v en fo r HD, th e firs t term in E q. 1-11, p ro p o rtio n a l to th e d ip o le
m o m en t, d(p ), d o es n o t a p p e a r in Eq. 1-13, sin ce o n ly th e o ff-d ia g o n a l
16
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m a trix elem en ts are n o n -z e ro . T he hydro g en ic ra d ia l m a trix elem en ts are
g iv en b y 19
(r-3\
J
z
Y
V U UJ
i
_____________ _____________ f a V
n3L(L
+l/2VL +l ) ^ m J
, x4
(r-<\ J jL ) __________ 3n2 —L(L +1)__________
KT ' nL U J 2ns(L - 1 /2 ) l( l + 1/2)(L + l)(L + 3/2)v m /
Eq. 1-15
w h ere the red u c ed m ass c o rre c tio n s have been in c lu d e d e x p lic itly . In
a d d itio n ,
/ D IV |_ I „
2S(S- 1 ) -4 L (L + 1)R(R +1)
^RLN|p , ( cos0 2(,) RLN^ - 2(2 R _ J)(2 R + 3)(2L _ ])(2L + 3)
where
Eq' , ' 16
S = R(R + l) + L(L + l)-N (N + l)
A few o f the c a lc u la te d m a trix elem en ts o f th e c o re p a ra m e te rs are
ta b u la te d in A p p en d ix A . A s an exam ple o f the e le c tr ic fin e stru c tu re ,
c o n sid e r an H 2 R y d b erg sta te w ith n=10 and L > 4. T he c a lc u la te d fine
stru c tu re e n e rg ie s fo r c o re
sta te s w ith (v= 0,R = 0 and 1) are lis te d in T able
1-1 and show n in F ig. 1-4.
T hese en erg ies scale ro u g h ly as n*3 and
co n v erg e as L in c re a se s.
1.2-3
R vdberg se rie s m ixing: o ff-d ia g o n a l e le m e n ts o f V -ff
A fter the d ia g o n a l e n e rg ie s are d e te rm in e d , th e s e c o n d -o rd e r
c o rre c tio n s are c a lc u la te d u sin g the o ff-d ia g o n a l m a trix e le m e n ts. In th is
m an n er, the E (2) c o rre c tio n s are g iven by
17
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Table 1-1: Electric fine structure energies for H2 , n—10. The following E(I) energies are
calculated using Eq. 1-13 and the core parameters given in Appendix A. Each state is
labeled by (R)LN, an abbreviation for (0,R)iiLn. The last column gives the series mixing
energies, E , as described in Appendix B and * means the value is less than 0.0001 cm '1.
H 2 11=10 electric fine structure (E FS)
Label
E(1) (cm'1)
Ea) (cm '1)
Label
E(I) (cm*1)
Ea) (cm '1)
(0)Hs
-0.0971
0.0016
(0)L8
-0.0091
*
(O H 4
-0.3995
-0.0038
(1)L7
-0.0807
-0.0001
(DHs
+0.3557
-0.0091
(1)L8
+0.1102
-0.0003
(UH*
-0.2717
-0.0077
(1)1*
-0.0593
-0.0003
(0)I6
-0.0398
0.0005
(0)M9
-0.0048
*
(DIs
-0.2121
-0.0013
(1)M8
-0.0548
*
d )l6
+0.2308
-0.0030
(1)M9
+0.0803
*
(1)17
-0.1482
-0.0024
(l)Mio
-0.0413
*
(0)K7
-0.0183
0.0001
(1)K«
-0.1258
-0.0005
(1)K7
+0.1563
-0.0011
(DKs
-0.0902
-0.0009
w h ere th e su m m atio n in c lu d e s a ll d is c re te an d c o n tin u u m s ta te s th a t w ill
m ix w ith e ac h level o f in te re s t. T his is a ra th e r ted io u s c a lc u la tio n and is
d e sc rib e d in m ore d e ta il in A p p en d ix B. T he E (2) v a lu e s are
18
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(H)
(I)
(K)
L=5
L=6
L=7
(M)
(L)
L=8
L=9
N=5
E
(1)
0.1 cm"1
8
E ( 0 ) -----------------------------------------------------
- - - -,-g
9
10
8
9
7
............. 8
6
R =0 core
R=1 core
Fig. 1-4: H2 fine structure energies. This figure plots the n=10, L>4 Rydberg structure
for an H 2 ion core with either R=0 or 1. E -Ecore - 1097.074 cm '1 based on Eq. 1-7.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
re p re s e n ta tiv e o f th e d e v ia tio n from th e p u re “c a s e -d ” co u p lin g a n d , as L
in c re a s e s, th e s e c o n trib u tio n s b ecom e a lm o st n e g lig ib le . C o m p a riso n
w ith the E (,) v a lu e s , as given in T ab le 1-1, sh o w s th a t the n e x t
p e rtu rb a tiv e c o n trib u tio n w ould be n e g lig ib le a t th is level.
1.2-4
E ffe c ts due to in c lu d in g n u c le a r an d e le c tro n sp in s .
This se c tio n w ill in tro d u ce th e p re v io u s ly n e g le c te d sp in s o f th e
R y d b erg sy ste m . T he re su ltin g e n erg y c o n trib u tio n s w ill o n ly be
c o n sid e re d fo r th e h ig h re so lu tio n , m ic ro w a v e sp e c tro sc o p y e x p e rim e n t
d isc u sse d la te r in C h a p te r 4, sin ce th e e ffe c ts a re n e g lig ib le in th e
o b se rv e d la s e r tr a n s itio n s . The tre a tm e n t u se d h e re is the sam e as th a t
p re s e n te d by S tu rru s e t a l.5-6 and others*-7 in m e a su rin g H 2 R y d b erg s ta te s .
A lth o u g h n o t a rig o ro u s tre a tm e n t, th e a g re e m e n t o f the sp in H a m ilto n ia n
w ith e x p e rim e n t is a d e q u a te at th e n e c e s s a ry le v e l o f p re c isio n . T h e sp in
H a m ilto n ia n is tre a te d in two p a rts. F irs t, th e h y p e rfin e stru c tu re o f th e
fre e H 2 + ion is c o n sid e re d . T his stru c tu re g iv e s th e larg e st s p littin g s .
S e c o n d , th e m a g n e tic fin e stru c tu re due to th e R y d b erg e le c tro n s p in is
in c lu d e d . T h is tre a tm e n t is only a p p ro x im a te and is based on the c lo s e ly
a n a lo g o u s h e liu m a to m . The sum o f th e s e tw o p a rts gives th e to ta l sp in
H a m ilto n ia n . F in a lly , th e p o ssib le e ffe c ts o f e x c h a n g e are c o n s id e re d .
O nce the a p p ro p ria te b a sis is c h o sen and d e fin e d , th e spin stru c tu re is
c a lc u la te d fo r th e s ta te s to be m ea su re d in C h a p te r 4.
The h y p e rfin e H a m ilto n ia n fo r th e fre e H 2 + io n is giv en by
D a lg a rn o 20 as
20
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Eq. 1-18
H „ , s = b l • S, + c ( I • p) • (Sc ■p) + dR • Sc.
T h e to ta l n u c le a r sp in (I), the c o re e le c tr o n spin (Sc), and the ro ta tio n a l
a n g u la r m o m e n tu m (R) are the d e fin e d q u a n tu m num bers fo r th e sy ste m .
T he in te r n u c le a r se p ara tio n o f th e tw o p ro to n s is p. The h y p e rf in e
c o u p lin g c o n sta n ts (b, c, and d) c o n ta in the in te rac tio n o f the e le c tr o n and
p ro to n m a g n e tic d ip o le m om ents. T h ese c o n sta n ts are d e p e n d e n t on the
v ib ra tio n a l and ro tatio n a l q u a n tu m n u m b e rs o f the ion core an d are
c a lc u la te d 21 to be
H 2+ (
v
= 0 ,
R =0)
b = 8 8 1.547 MHz
c= 1 2 8 .7 3 1 M H z
d=42.526 M H z
c = 1 9 .9 9 M H z
d = 2 1.52 M H z
D 2+ 0 = 0 , R =0)
b = 1 3 5 .8 8 5 MHz
As d e s c rib e d by S tu rru s 6, Eq. 1-18 c a n be w ritte n in a m ore u s e fu l
m an ner. By re w ritin g the se co n d te rm as a p ro d u c t o f ir re d u c ib le te n s o rs ,
the H a m ilto n ia n ta k e s the form
H „K = [b + | ] i S I +cT I:|(I .S t ) T |2|(p,p) + d R S t .
Eq. 1-19
W here T [21(x ,y ) are se co n d -ra n k irr e d u c ib le te n s o r o p e ra to rs w h ic h are
te n s o r p ro d u c ts o f the v ector o p e ra to r s x and y, as d efined in E d m o n d s 22.
T he e ffe c ts o f the R y d b e rg e le c tr o n sp in are c o n s id e ra b ly s m a lle r,
b u t n e c e s s a ry to in clu d e. The h ig h -L , R =0 m o le c u la r R y d b e rg sta te s
21
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c o n s id e re d here a re v e ry s im ila r to p re v io u s ly s tu d ie d h e liu m R ydberg
s ta te s. The m a g n e tic fin e s tru c tu r e (M FS) fo r h e liu m , as d e fin e d in th e se
p re v io u s e x p e rim e n ts 23*24, is g iv e n by
Eq. 1-20
a = fine s tr u c tu r e c o n s ta n t
SRoo= R ydberg c o n s ta n t fo r infin ite n u c le a r m a s s
S
r
= R ydb erg e le c t r o n spin
V* = e x ch a n g e e n e r g y p a ra m e te r
B o th e le c tro n m a g n e tic d ip o le s o f the sy ste m c o n tr ib u te to the spin
s tru c tu re by in te r a c tin g w ith th e m otional m a g n e tic H eld a n d w ith each
o th er, g iv in g the first t h r e e te r m s . This d e s c r ip tio n o f th e h eliu m
str u c tu r e is a d e q u a te to d e s c r ib e the a n a lo g o u s m o le c u l a r R ydb erg states.
T he final te rm in th e a b o v e eq uation is an a d - h o c a tte m p t to a c c o u n t
fo r the e ffects due to e x c h a n g e o f the tw o e le c tr o n s . In h e liu m , w hen the
to ta l w a v e fu n c tio n is p r o p e r l y sy m m etrize d , th e s y m m e tr ic and a n ti­
sy m m e tric states are fo u n d to be split by an “ e x c h a n g e e n e r g y ” . The size
o f th is effect, w h ic h is d e p e n d e n t on the sp a tia l e x te n t o f the ground sta te
e le c tro n w a v e fu n c tio n a n d its overlap w ith th e R y d b e rg e le c tro n
w a v e fu n c tio n , is p a r a m e te r iz e d by the c o e ff ic ie n t V x. A ny p e n e tra tio n o f
th e c o re by the R y d b e rg e le c tr o n will give a s p littin g o f th e energy lev e ls,
p ro p o rtio n a l to V x. I f a ll o th e r in te ra c tio n s in E qs. 1-19 a n d 1-20 w ere
22
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a b se n t, the e ig e n s ta te s o f th e p ro b le m would be e ig e n s ta te s o f to tal
e le c tro n spin, the “ s i n g l e t ” and “ t r ip l e t ” states. H o w e v e r, fo r th e s e n o n ­
p e n e tra tin g R y d b e rg le v e ls , the e x c h a n g e e n erg ies are sm a ll e n o u g h to be
a lm o s t c o m p le te ly n e g lig ib le , and c e rta in ly sm all e n o u g h th a t th ey d o n ’t
d e te rm in e the e ig e n s ta te s .
Both h y p e rf in e a n d m a g n e tic sp in c o n trib u tio n s are c o m b in e d to
g iv e the total sp in H a m ilto n ia n ,
Eq. 1-21
= [b + | ] I S c + c T |!|( I , S j - T |!|(p,p) + d R S c
T h is H a m ilto n ia n is n e a r ly d ia g o n a l in the chosen b a s is 6, illu s tr a te d in
Fig. 1-5. T he n u c le a r sp in s ( I i, I 2) c o u p le to form to ta l n u c le a r spin I.
The p o ssib le v a lu e s o f I are r e s tric te d by the Pauli e x c lu s io n p rin c ip le , as
d isc u sse d b elo w . T h e n u c le a r sp in (I) co uples w ith th e sp in o f the co re
e le c tro n (Sc) to fo rm a to ta l c o re spin ( F c). This sp in ( F c) is th e n c o u p le d
to the total a n g u la r m o m e n tu m (N ) to form an in te r m e d ia te a n g u la r
m om entum ( J i ) . F in a lly , the R y d b e rg e le ctro n sp in
(Sr)
is a d d e d to form
th e to ta l a n g u la r m o m e n tu m J . In sum m ary,
N=R+ L
(I = I, + I2; Fc = I + Sc)
J, = N + Fc
J = Jl + Sr
23
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Eigenstates : \|/ = | { (vR,nL; N ),(I,S C; Fc); J! } ,S R; J nij >
T h e s p e c ific states o f in te r e s t in c h a p te r 4 are the R=0 sta te s o f H 2 +
and D 2 +. T h ese diatom ic ion c o re s c o n s is t o f e ith e r tw o p ro to n s o r two
d e u te ro n s b oun d by a core e le c tro n in its ground state. T h ese sy ste m s are
qu ite d if f e r e n t in th e ir spin c o u p lin g , due to the Pauli e x c lu s io n p rin cip le .
As d e s c r ib e d by L ib o ff25, the to ta l n u c le a r w a v efu n c tio n s m u s t be p ro p erly
I
R
Fig. 1-5: Vector coupling of angular momenta and spins. The order o f addition is shown
graphically to aid in understanding the coupling.
sy m m e triz e d w ith respect to in te rc h a n g e o f the two id e n tic a l n u c le i. To
be gin w ith , Hz* c o n sists o f tw o fe rm io n s (p ro to n s, each o f n u c le a r spin
1=1/2) a n d , to obey the Pauli p r in c ip le , the total w a v e fu n c tio n m u st be
a n tis y m m e tric . The p o ssib ilities in c lu d e th ree sy m m etric (1=1) sp in states
and one a n tisy m m e tric (1=0) state. T he sp a tia l w a v e fu n c tio n s d e p e n d on
the r o ta tio n a l m o tio n (sym m etric fo r e v e n R and a n tis y m m e tric for odd
R). C o m b in in g th ese to form an a n tis y m m e tric w a v e fu n c tio n re q u ire s that
24
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even R sta te s h a v e o n ly the sing le 1=0 sta te an d odd R sta te s h a v e the
three 1=1 sta te s. T h is m u ltip lic ity in the s p in sta te s will fa v o r th e o d d R
levels by 3:1.
D 2 + is m ad e o f tw o b o so n s (d e u te ro n s , e a c h o f n u c le a r s p in 1=1).
These to ta l w a v e fu n c tio n s m ust be sy m m e tric u n d e r e x ch a n g e o f th e tw o
id en tical p a rtic le s . T h e a llo w e d spin sta te s d if f e r from th o se in H 2 by
having six s y m m e tric sta te s (one w ith 1=0 a n d fiv e with 1=2) a lo n g w ith
the three a n tis y m m e tr ic (1=1) spin sta te s. W h e n c o m b in ed to fo rm a
sy m m etric w a v e fu n c tio n , the even R le v e ls h a v e the six a llo w e d (1=0,2)
spin sta te s and th e o d d R lev e ls have o n ly th e th re e 1=1 sta te s. In D 2 *, th e
even R le v e ls a re fa v o re d o v e r the o dd R le v e ls b y 2:1. T he e x p e c te d
ratio o f “ fa v o r in g s t a te s ” is d isc u sse d la te r w h e n c o m p a rin g th e d if f e r e n t
o b se rv ed s p e c tra . As e x p e c te d , and n o te d in th e sp ectra, HD* d o e s no t
follow an y n u c le a r sp in s ta tis tic s due to h a v in g n o n -id e n tic a l n u c le i.
As an a d d itio n a l n o te, the h y p e rfin e s tr u c tu r e is s ig n if ic a n tly
d iffe re n t b e tw e e n the tw o iso to p es. F o r th e R = 0 cores, th e to ta l c o re sp in
has only one a llo w e d v a lu e for H 2 +, w h ile D 2 + h a s three d if f e r e n t v a lu e s
o f core spin . T h is d if f e r e n c e leads to m ore c o m p lic a te d s tr u c tu re in the
ob serv ed D 2 R y d b e rg tra n s itio n s.
As m e n tio n e d a b o v e , the spin H a m ilto n ia n is nearly d ia g o n a l in the
chosen b a s is and th e e n e rg ie s can be c a lc u la te d u sin g the fo rm u la s g iv e n
in A pp en d ix C. A ll m a trix e lem en ts w e re c a lc u la te d for a g iv e n (R =0)nL N
level. T he r e s u ltin g su b -m a tric e s w ere th e n d ia g o n a liz e d to in c lu d e any
25
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m ix in g s that c o n trib u te to the s p littin g s . F o r an e x a m p le o f t h e sp in
s tru c tu r e , c o n sid er the R y d b e r g le v e ls w ith n = 1 0 , N = L = 5 , an d b o u n d to an
R=0 c o re . The sp littin g o f th e le v e ls is illu stra te d in F ig. 1-6, a n d c an be
fo llo w e d as the c o u p lin g is d e s c rib e d . The e le c tro n sp in s are b o th 1/2 and
the n u c le a r spins are 11= 1 2 = 1/2 fo r H 2 . T hese sp in s a re c o u p le d to g ive
1=0, w ith sym m etry a llo w in g o n ly th e even v a lu e s o f I fo r e v e n R le v e ls.
T h en coupling I to the c o re e le c tr o n spin gives the h y p e rf in e le v e l w ith
F c= l / 2 . F c and the a n g u la r m o m e n tu m (N) are c o u p le d to form J 1 ,
re s u ltin g in each F C,L le v e l b e in g sp lit into (2 F C+ 1) d if f e r e n t le v e ls .
F in a lly , the R y dberg e le c t r o n s p in is coupled w ith J i to sp lit e a c h o f these
le v e ls into two d iffe re n t le v e ls . N o te that th e s y m m e tric R=0 c o re g iv es
“ h e liu m -lik e ” H 2 spin s ta te s , e x h ib itin g the sam e 4 - f o l d stru c tu re ! T he
a n a lo g o u s stru ctu re in D 2 is illu s tr a te d , alo ng w ith th e o th e r F c sta te s ,
w h ic h are the re s u lt o f h a v in g th e a llo w ed 1=2 n u c le a r spin . F o r th e n=10,
L=5 le v e ls, the spin s tr u c tu r e is c a lc u la te d for e a c h n o n -z e ro te r m . T hese
d ia g o n a l en erg ies are g iv e n in T a b le C -l and i llu s tr a te d in Fig. 1-6.
The spin s tru c tu re d e s c r ib e d h ere is o nly im p o r ta n t in th e
m ic ro w a v e sp e c tro sc o p y e x p e r im e n t presen te d in c h a p te r 4. T h e se le c tio n
ru le s th a t govern the o b s e r v e d e le c tric dipole m ic r o w a v e tr a n s itio n s
re q u ir e AFc=0 and fa v o r A L =A J=+ 1. Since Fc d o e s n o t c h a n g e , th e larg e
h y p e rfin e sp littin g b e tw e e n d if f e r e n t Fc lev els d oes n o t a p p e a r in the
tr a n s itio n energies. In a d d itio n , sin c e the sp in s tr u c tu r e fo r a d ja c e n t L
26
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H2
J1
(0)H5
4.5
5
E
•
J-'spin
4
0.5
10 Mhz
5.5
D
rc j r
2.5
3
4
5
2
2.5
3.5
4.5
5.5
6.5
7.5
142.221
Mhz
10 Mhz
85
6
i
7
5
4.5
4
Fc I
0.5
5.5
213.332
Mhz
6
_
Fc
t_ L 5 _
5.5
5
4.5
3.5
4
5
4
3
Fig. 1-6 : Spin structure diagrams for H 2 and D 2 . The n=10, (0)Hs state for each isotope
27
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levels is very sim ila r, th e tra n s itio n sp in stru cture is q u ite sm a ll (ty p ic a lly
< 5 M H z). This will be sh ow n later, in C hapter 4, for all o b se rv e d
tra n s itio n s.
1.3 M otivation and overview o f thesis stru ctu re
As m en tion ed a b o v e , several ex perim ents have a lre a d y b e e n c arried
out in H 2 , using a fast b e am , re so n a n t ex cita tio n m e th o d 5'*. B y c o m p a rin g
m ea su re d R ydberg fine stru c tu re in te rv a ls with the p o la r iz a tio n m od el
p re se n te d above, th e se e x p e rim e n ts have su c ce ssfu lly m e a s u re d a v a rie ty
o f ion core p a ra m eters, b u t w ere lim ite d by signal to n o ise (S /N ). F o r the
study p resen te d here, se v e ra l m ajo r im pro vem en ts o f the a p p a r a tu s and
te c h n iq u e have been m ad e , re s u ltin g in a su b stan tial in c re a s e in S/N.
T hese im p ro v em en ts to the e q u ip m en t, as discu ssed in d e ta il in C h a p te r 2,
a llo w e d aH L levels ( ^ 4 ), w ith n=9 and 10, to be o b se rv e d in H 2 , H D , and
D 2 . The im proved S/N m ade p o ssib le the o b se rv atio n o f th e p r e v io u s ly
u n seen R ydberg sta te s b o u n d to R=0 core levels, w h ic h in tu r n m ade
p o ss ib le the m ic ro w a v e re s o n a n c e e x p erim e n t d e sc rib e d in C h a p te r 4.
T hese R=0 states, w h ic h are bound to th e absolute g ro u n d s ta te o f the
m o le c u la r ion, e x h ib it th e sim p le st R y d b erg fine stru c tu re a n d lead to a
d e te rm in a tio n o f the d ip o le p o la riz a b ilitie s o f the ion c o re s. T he stu d y
d e sc rib e d in C h ap ter 4 d e te rm in e s these p ro perties w ith s u f f ic ie n t
p re c isio n to ch allen ge e v e n the m ost p re c ise th e o re tic a l d e s c r ip tio n o f
th ese sim p le m o le c u la r ions.
28
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T he r e m a in d e r o f th is th e s is is d iv id e d in to th re e p a rts . C h a p te r 2
de scrib e s th e e x p e r im e n ta l a p p aratu s and te c h n iq u e u se d in b o th the la s e r
and m ic ro w a v e r e s o n a n c e m ea su re m e n ts. T h is in c lu d e s e s p e c ia lly the
details o f the im p ro v e d a p p a ra tu s , w hich m a d e th e s e e x p e rim e n ts p ossib le.
N ext, C h a p te r 3 p r e s e n ts the la s e r sp e c tro s c o p y o f h ig h -L R y d b e rg states
o f H 2 , HD, and D 2 , a lo n g w ith th e d etails e x tr a c t e d fro m the o b se rv ed
spectra. C h a p te r 4 c o n c lu d e s by p resen tin g th e m a in e x p e r im e n t, the
m icrow av e re s o n a n c e m e a su re m e n ts o f R =0, H 2 a n d D 2 R y d b e r g states,
and the r e s u ltin g d e te r m in a tio n o f the H 2 "’ a n d D 2 + d ip o le p o la riz a b ilitie s .
29
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C h ap ter 2
E xp erim en tal tech n iq u e and details o f eq u ip m en t
2.1 In trod u ctio n to the laser R E S IS technique
The re s o n a n t e x c ita tio n , JJtark io n iz a tio n s p e c tro s c o p y (R E S IS )
tec h n iq u e , d e v e lo p e d by P a lfre y 26 fo r stu d y in g h e liu m R y d b e rg s ta te s , has
sin c e been a p p lie d to se v e ra l stu d ie s o f H 2 R y dberg s ta te s 5' * 17-27. T he
R E S IS te c h n iq u e , as sh o w n s c h e m a tic a lly in Fig. 2-1, c o n s is ts o f th re e
m a in co m p o n e n ts, a c h a rg e -e x c h a n g e r e g io n , a laser e x c ita tio n r e g io n , and
a S ta rk io n iz a tio n d e te c to r. A fast io n b eam is m ass s e le c te d an d ste ere d
in to a Cs v a p o r c e ll w h e re R ydberg le v e ls are p o p u la te d th r o u g h c h a rg e
e x c h a n g e b e tw e e n th e ions and Cs v a p o r. A ny re s id u a l ions a re th en
s te e re d out o f th e b e a m w ith a h ig h f ie ld p re -io n iz e r, w h ich a ls o se rv e s to
e m p ty the u p p e r (n > 1 6 ) R ydberg le v e ls . A n u m b er o f the n e u tr a l p a rtic le s
a re in R ydberg le v e ls w ith n=9 or n = 1 0 , w hich can be e x c ite d by th e laser
in to h ig her n' le v e ls . T h ese tr a n s itio n s o c c u r in the la s e r in te r a c t io n
re g io n (LIR) w h e re a s in g le -fre q u e n c y , C O 2 laser is D o p p le r tu n e d into
re so n a n c e w ith th e f a s t n eu tral m o le c u le s . Any m o le c u le s e x c i te d to the
u p p e r n' states are su b s e q u e n tly S ta rk io n iz e d in the R y d b e rg d e te c t o r and
30
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RESIS : Laser spectroscopy apparatus
Positive ions
Neutral Molecules
Signal Molecules/Ions
ion
source
switching
magnet
pre-ionizer
and deflector
LIR: laser
interaction
region
Rydberg
detector
A"
2 meters
Cs cell
8 mm dia.
aperture
3 mm dia.
aperture
pre-ionizer
empties upper
n' states
ionization
threshold
27K
27K
captured
Rydberg state
populations
->
resonant laser
excitation to
upper n' state
27K
co2
detector is set to
ionize the target
upper state n -27
laser
• ••
101
101
101
laser excited electrons
captured electrons
Fig. 2-1: Laser spectroscopy experiment and RESIS technique. This illustrates the order
of equipment and the method used to study n=9 and 10 Rydberg states in H 2 , HD, and D2 .
31
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th e r e s u ltin g ions are m e a s u re d in a c h an n e l e le c tro n m u lt i p li e r (C E M ).
T h e fast (—11 keV ) ion b eam is pro duced in a s ta n d a rd ,
d u o p la s m a tr o n ion so u rc e . T h e a p p ro p ria te u ltra -h ig h p u r ity g a s b o ttle ,
e ith e r H 2 , D 2 , or a 50 /5 0 m ix o f both (fo r HD)], is a tta c h e d to th e s o u rc e .
T he d e s ire d beam o f io n s is m ass se le c te d with a s w itc h in g m a g n e t and
fo c u s e d d o w n the b e a m lin e , into the C s vapor c ell. T he io n s c a p tu r e
e le c tr o n s , from the Cs ta r g e t, to form fast neutral p a rtic le s . M a n y
R y d b e rg sta te s are fo rm ed in th is p ro c e ss , with all n le v e ls p o p u l a t e d in
som e u n k n o w n d is trib u tio n . F o llo w in g the Cs c e ll, a p r e - i o n i z a ti o n
re g io n is u sed to em pty h ig h - n ' le v e ls and d e fle c t any r e s id u a l io n s o u t o f
the b e am . T his is a p r e p a r a to r y step b e fo re the ty p ic a l la s e r e x c i ta t io n
from n = 1 0 to n'=27 (o r n = 9 to n '= 17,20). Any p o p u la tio n in th e u p p e r
R y d b e rg sta te w ould la te r be io n iz e d a n d d e te cted , fo rm in g a b a c k g r o u n d
sig n a l p re s e n t even in th e a b s e n c e o f la s e r e x cita tio n . By e li m in a ti n g th is
b a c k g ro u n d , the p r e -io n iz e r is im p o r ta n t for o b ta in in g o p tim u m s ig n a l- to no ise.
T h e beam th e n e n te rs th e la s e r in te ra c tio n re g io n (L IR ), w h e re it is
i n te r s e c te d by a C O 2 laser. T his la s e r is d isc retely tu n e d by a d if f r a c t i o n
g ra tin g to one o f the - 5 0 d is c r e te (9-11 pm ) lines on w h ic h th e s ta n d a rd
C O 2 la s e r can operate. F u r th e r f in e -tu n in g is a c c o m p lis h e d t h r o u g h the
D o p p le r e ffe c t, by c h a n g in g th e in te r s e c tio n a n g le b e tw e e n th e m o le c u la r
beam a n d the laser. W hen th e an g le is scanned th e a p p a r e n t f r e q u e n c y , as
seen by th e R ydberg e le c tr o n , is tu n e d c o n tin u o u sly . W h en th e D o p p le r
32
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
shifted fre q u e n c y c o m e s into reso n an c e, the R y d b e r g e le c tro n can a b s o rb
a photon and be e x c ite d to an u p p e r n' state. A s m en tio n e d b e fo re , ty p ic a l
tran sitio n s are fro m n=9 le v e ls to n'= 17,20 o r fro m n=10 lev e ls to
n '= 2 6 ,2 7 ,2 9 ,3 0 . T h e se tr a n s itio n s are all n e a r e n o u g h to som e d is c r e te ,
single C O 2 fr e q u e n c y th a t th e D opp ler tu n in g is s u ffic ie n t to c o v e r the
rem aining d iff e re n c e .
The d e te c tio n re g io n is set to Stark io n iz e the laser e x c ite d s ta te s .
The resu ltin g io n s are th e n fo c u se d and s te e re d into a ch an nel e le c tro n
m u ltip lie r (C E M ) w h e re th e c u rre n t is m o n ito r e d w ith a lo ck -in a m p lifie r .
The laser is a m p litu d e m o d u la te d at a fixed f r e q u e n c y and the C E M
current, m e a s u re d s y n c h ro n o u s ly w ith the la s e r m o d u la tio n f re q u e n c y , is
defined as a s ig n a l. T his sig n a l can be m e a s u r e d as a fu n ctio n o f the
D oppler in te r s e c tio n a n g le (0) b etw een the la s e r and n eutral b e am , w h e re
0 is d efined to b e zero w h e n th e tw o are c o u n te r p r o p a g a tin g .
2.2 P op u latio n and p rep aration o f R y d b e r g states
2.2-1
D u o p la s m a tr o n io n source a n d b e a m optics
The ion s o u rc e u se d in th e se e x p e rim e n ts w as a High V o ltag e
E n gineering m o d e l D P -2. A d ia g ra m o f the s o u r c e is show n in Fig. 2-2 .
Two in d e p e n d e n t g as fee d lin e s, along w ith tw o m e d iu m c u rre n t
feed thro ug hs, w e re s u p p o rte d on a filam ent f la n g e . A tta c h e d to one o f the
gas lines was a K u rt J. L e s k e r m od el K J L - 5 3 1 1 th e rm o c o u p le g a u g e u se d
to m onitor the s o u rc e p re s s u r e , w h ich was k e p t a ro u n d 60 m T o rr d u rin g
33
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ion Source
tungsten filament
potential
(300 V)
potential
(lOkv)
acceleration
potential
(llkV ) ^
(not to scale)
probe electrode
ceramic insulator
anode pinhole (100 pm)
high voltage insulator
magnet coils
lens electrode
side view o f filament post
1/4“ stainless
steel rod
spot
welds
medium current
threaded vacuum
feedthroughs
0.040" Tungsten
wire filament
stainless steel
butt couplers
w/set screws
93 cm
Fig. 2-2: Duoplasmatron ion source. The ion source and the extraction einzel lens are
shown above, along with a schematic of the applied potentials. An expanded view o f the
filament is shown at the bottom.
34
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o p e ra tio n . The c u rre n t fee d th ro u g h s w ere 1/4" s ta in le s s steel rods and
co n n ec te d to a tu n g s te n w ire fila m e n t (0.040" d ia m e te r, 3N8 p u rity ,
E le ctro n ic Space P ro d u c ts In te rn a tio n a l) using c o u p le r s and sta in le s s steel
p o sts, as show n. T he tu n g ste n w ire w as bent u n d e r t o r c h h e a tin g and th en
sp o t-w eld ed onto th e fila m e n t p o sts. T hese p osts w e re re u s e a b le (3-4
tim e s), by sim p ly g rin d in g o f f the old tu ngsten and r e -w e ld in g a new
fila m e n t onto them . T he source and e le ctro n ics w ere d e s c rib e d in m ore
d e ta il by S tu rru s6.
The sou rce re g io n w as s e p a ra te d from the re s t o f th e b e am lin e by
the anode a p e rtu re o f 100 pm . T h is d iffe re n tia l a p e r tu r e a llo w e d the
b e am lin e to be k e p t a t 10*6 T orr w ith a d iffu sio n p u m p . The e n tire so u rce
w as held at 11 kV a n d the hot fila m e n t (new fila m e n ts ty p ic a lly ru n n in g at
30 A /3 V ) p ro d u ce d a p la s m a at the anode. The so u rc e h a d to be co o le d by
c irc u la tin g d e io n iz e d w a te r, k e p t in a h e a t-e x c h a n g in g re s e rv o ir. Ions
w ere acc elera te d by th e g ro u n d ed e x tra c tio n e le c tro d e o f an e in z e l lens
and co llim a te d . T he io n beam w as m ass-se lec te d by a sw itc h in g m ag n e t
field and ste ere d th r o u g h the a p p ro p ria te port, d ow n th e b e a m lin e . A
se co n d lens was u se d to focus th e beam through a re m a in in g d iff e re n tia l
a p e rtu re and into th e c h a rg e ex ch a n g e region. The io n b eam s w ere
ty p ic a lly a few p A a n d on the o rd e r o f 1/4" in d ia m e te r.
2.2-2
Cs c e ll ch arg e e x c h a n g e region
The ion beam w as fo cu sed into a charge c a p tu re re g io n w here
R y d b erg levels w ere c o llis io n a lly p o p u lated , p r o p o r tio n a l to n '3.2*
35
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C a p tu re into the v a rio u s nL R ydb erg le v e ls is a n o n -re s o n a n t p ro c e ss , so
p o p u la tio n o f th e lo w e r n = 9 ,1 0 levels is d e te r m i n e d p u rely by the v e lo c ity
o f th e m o le c u la r b e a m an d the species o f th e c o ll i s i o n a l target. P re v io u s
stu d ie s have u se d g as ta r g e t c e lls , w hich a re m a d e w ith v ariab le leaks to
c h an g e the ta r g e t d e n s ity an d sp ecies. T h e s e c e lls u se d d iffe re n t gases
su c h as He, X e, K r, A r, an d N 2 28 and all w e re s u f f i c i e n t to give
re a so n a b le c h a rg e e x c h a n g e ; h o w e v er, th e o u te r e le c tr o n binding e n erg y
fo r each o f th e s e s p e c ie s is q u ite large in s o m e c a s e s . Since c h arg e
c ap tu re is e x p e c te d to d e p e n d on this b in d in g e n e r g y , and the o u te r Cs
e le c tro n is b o u n d by o n ly 3.89 eV, a n e w c h a r g e e x c h a n g e cell was
d e sig n e d to c o n ta in C s v a p o r. T his s m a lle r b i n d in g e n erg y w as e x p e c te d
to give m ore p o p u l a t i o n in the n=10 R y d b e rg le v e ls w hich are b oun d by
o n ly 0.136 eV.
The Cs c ell w a s sim p le in design an d s h o w n in Fig. 2-3. A
c o m b in a tio n o f v a rio u s p u b lis h e d c o n f ig u r a tio n s w a s u sed in the fin al
d e s ig n 29. T he m a in c o n s id e r a tio n was s im p lic ity a n d lo n g -te rm o p e ra tio n .
T he c o ncept w as to h a v e a h e a te d pipe th a t w o u ld m a in ta in a Cs v a p o r
a lo n g the ion b eam p a th . E ac h end o f the “ h o t p i p e ” had a c ooled e n d c a p ,
w h ic h c o n d e n se d th e C s to c o n ta in it w ith in th e p ip e and re c irc u la te it
back into the ho t r e g io n . A re s e rv o ir, a tta c h e d to th e h ot region, k e p t the
d e n sity c o n sta n t, an d w a s n e c e s s a ry sin ce s o m e C s d o e s leak o u t in to the
su rro u n d in g v a c u u m .
36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
O p e r a t i o n a l Det ail s
5 g Cesium
4 0 0 hrs. r u n t i m e
T
1 - 40 C
T . - 100 T
?>
T, - 150 C
P =- 10 I o r r
c o o l i n g ICO ~ 20 m l / m i n
-*
- C o o l i n g lines
C ooled end caps with
8 m m a p e r t u r e s ; T,
H o t r e g i o n ; s tai nl es s s teel
m e sh wick recirculates Cs;
C
C s r e s e r v o i r k ep t hot to
p u m p into cell a b o v e ; T
Fig. 2-3 : Cesium charge exchange region. A Cs vapor cell was designed for efficient
population of high-L Rydberg states. The operational characteristics are given along with
a picture of the region.
37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The cell w as c o n s tru c te d using tw o s ta in le s s steel tu b e s (5/8" d ia .,
5.5" long) so ld e re d to g e th e r to form a T sh a p e, w ith a larger Cu rod (1.5 "
dia., 2.5" long) su rro u n d in g th e jo in t. The v a p o r region w as m a in ta in e d
in the h o rizo n tal tu b e, w h ich had copper “e n d c a p s ” a ttac h ed at each end.
A 304 sta in less ste el “ f a b r ic ” m esh wick (2 5 0 x 250 th re a d s/in .) was
rolled and p lac ed in sid e th is tu b e to aid in re c ir c u la tin g the Cs. The e n d
caps, w rapped w ith 1/8" c o p p e r tubing ( s o ld e r a tta c h e d ), had 6m m dia.
apertu res, a llo w in g th e ion b e am to pass th ro u g h . Tap w ater c o oling
flow ed th ro u g h the tu b in g at a rate o f —50 m l/h r. The v ertical sta in le s s
steel pipe w as s o ld e re d to the top o f a 2 -p ie c e th re a d e d re se rv o ir. The
cen ter co p p er se c tio n w as w ra p p e d several tim e s (R~10£2) w ith n ic h ro m e
h e a te r wire (O m e g a N i8 0 /C r2 0 ), insulated by 1/16" N extel 312 fib e r
sleeve (O m ega X C - 1 16-25). T he re se rv o ir w as a lso w rapped w ith the sam e
h e a te r w ire /in su la tio n . (F o r sch em atics and in itia l o p e ra tio n see la b b o o k
PLJ7.)
The e n tire Cs cell hung on an a d ju s ta b le m o u n t a ttac h ed to a 6"
D ependex v a cu u m flan g e. F o u r th erm o c o u p le s, fed in thro u g h an O m e g a
M F T -116-5 v acu u m fe e d th ro u g h , were a tta c h e d to the cell to m o n ito r th e
tem peratures o f th e tw o end c a p s, the c h a m b e r, and th e reserv o ir. The
w a ter lines p a sse d into the v acu u m through C a jo n th read ed fe e d th ro u g h s
and the c u rre n t cam e in th r o u g h a Pave th re a d e d , e le ctrica l fe e d th ro u g h .
The entire cell w as p la c e d into a 6" vacuum c h a m b e r, w hich had
differen tial p u m p in g a p e rtu re s (6m m dia. on th e so u rc e side and 3m m d ia.
38
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o n th e d e te c to r sid e ). T he c h a m b e r, w ith a d if f u s io n pum p and w a te r
c o o le d trap, w as k e p t at 10‘6 T o r r d u rin g o p e ra tio n .
The Cs cell r e s e r v o i r w as lo a d e d und er a c h e m ic a l fume h o o d a n d
d ry -arg o n a tm o s p h e re , sin c e C s re a c ts read ily w ith w a te r vapor and
fla s h e s easily in h u m id su rro u n d in g s . Due to the v io le n t sparks th at o c c u r
w h ile loading (o r c le a n in g ) th e c e ll, flam m ab le s u b s ta n c e s should be
a v o id e d , such as p a p e r to w e ls a n d lo o se c lo th in g . In a d d itio n , p ro p e r
c h em ic al g lo v es s h o u ld be w o rn. [W hen cle an in g th e c ell, ethanol is
som etim es p re fe rre d o v e r w a ter; h o w ev er, this p r a c tic e is m ore d a n g e ro u s
b e ca u se o f the e x p lo s iv e h a z a rd .] To load the c e ll, th e bottom o f the
r e s e rv o ir was u n s c re w e d . T he Cs, w hen hand w a r m e d abov e its m e ltin g
tem p e ra tu re , w as p o u r e d o u t o f the bro k en g lass a m p o u le into the
rese rv o ir, w h ic h w as th e n sc re w e d o n to the re s t o f th e a p p aratu s. W h en in
th e beam line and e v a c u a te d , th e Cs w as “ lo a d e d ” in to the m ain c h a m b e r
by heating o n ly the r e s e r v o ir up to 150°C for an h o u r . T his served to
p um p Cs up on to the ste e l m e sh w ic k where it c o n d e n s e d . D uring n o rm a l
o p e ra tio n , the m ain h o riz o n ta l se c tio n was h eated a n d m ain ta in e d at
~105°C (± 5°) and the r e s e r v o ir at ~150°C (± 10°) w ith c u rre n ts o f ~ 1A .
T his kept an o p e ra tin g te m p e ra tu re o f ~40°C on th e e n d c a p s to c o n d e n s e
th e Cs vapor b e fo re e s c a p in g th ro u g h the beam a p e r tu r e s . U nder th e s e
o p e ra tin g c o n d itio n s , a 5g Cs lo ad c o u ld last up to 3 0 0 hou rs o f ru n tim e .
H ow ever, d u rin g m o s t e x p e rim e n ts , th e b eam lin e h a d to be c han ged m o re
freq u e n tly , w hich m e a n t b rin g in g the Cs ch am ber up to a tm o sp h e re u s in g
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
dry a rg o n to b a c k fill a n d le ft flow ing w h ile o pen. Even w ith th e s e
p re c a u tio n s , d e g ra d a tio n o f th e Cs w as o b s e r v e d every tim e th e v a c u u m
c h a m b e r w as o p e n ed .
2.2-3
P re -io n iz a tio n regio n : “ d o u b le q u e n c h e r”
The n e u tra l “ c h a r g e -c a p tu re d ” b e a m p a sse d into the p r e - io n iz e r
reg io n im m e d ia te ly fo llo w in g the Cs cell c h a m b e r. A high e le c tr ic fie ld
io n iz ed any R y d b e rg sta te s w ith n>16, w h ic h serv ed the p u rp o se o f
em p ty in g a n y sta te s th a t w o u ld later be io n iz e d in the d e te c to r. T h is
greatly r e d u c e d the b a c k g ro u n d c u rre n t s e e n in the d etecto r. In a d d itio n
to the h i g h - f ie ld reg io n , a se t o f d e f le c tio n p la te s was used to s te e r any
re sid u a l io n s , o u t o f th e beam path, into a F a ra d a y cup.
P re v io u s s tu d ie s 30 have show n th a t m o le c u la r R ydberg s y s te m s a re
able to r e p o p u la te th e s e “e m p tie d ” u p p e r s ta te s . R ydberg e le c tro n s b o u n d
to v i b r a tio n a lly e x c ite d c o re s (e.g. v = l , n = 7 ) g ain energy from the io n
core and a re e x c ite d into a d ja ce n t, h ig h e r n sta te s, bound to a g ro u n d
v ib ra tio n a l c o re (su c h as v= 0, n=27). F o r an illu s tra tio n o f w h ic h
v ib r a tio n a lly e x c ite d sta te s should be c o n s id e r e d , see Fig. 1-3. T h is
r e g e n e ra tio n h a p p e n s on a v e ry sho rt ( 1 0 0 n s ) tim e sca le, so it w as a s s u m e d
that o n c e it o c c u rre d , an a d d itio n a l field r e g io n would a g ain e m p ty th e
v=0 R y d b e rg le v e ls , fu r th e r red u c in g th e a v a ila b le backgro und .
A m u ltip le p r e - io n iz e r was d e s ig n e d , sh o w n sc h e m a tic a lly in F ig 24. S e v e ra l s ta in le s s ste el p la te s , each w ith a 4m m a p ertu re, w ere a rr a n g e d
in a “ s t a c k e d ” fo rm a tio n , c o a x ia l w ith the b e am . Each sm all p la te c o u ld
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Pre-ionization Region
Deflection
region
Ionizing regions
i
C-
5.5 cm - )
neutral beam
and residual
ion beam
0.45 cm
gaps
fl
V .
Faraday “cup”
deflected
ions
il
fl
H HT
■■
.W
'D
threaded steel rod
support posts
insulating nylon screws
The deflection region is 2.54 cm long
and the plate separation is I . I cm.
Fig. 2-4: Pre-ionization region. High electric fields are used to ionize Rydberg states to em pty n>16.
neutral beam
(n < !6 )
be held at a p o ten tial (V ) w h ile the la rg e r p late was k e p t g ro u n d e d . E ach
e le ctro d e was c o n n e c te d to a 15 kV feedthrou gh; h o w e v e r, th e d e sig n
lim ite d the h ig h est p o te n tia l to 10.5 kV b ecau se o f a rc in g .
R e g en e ratio n o f th e u p p e r R yd berg states was o b se rv e d in H 2 .
W hile in d iv id u a lly v a ry in g the n u m ber o f regions th a t w ere “ o n ” , th e
b a ck g ro u n d CEM c u rre n t w as m o n itored in the d e te cto r. P lo ttin g this
b a ck g ro u n d c u rre n t vs. d e te c to r d e fle c tio n voltage r e s o lv e d the
back g ro u n d R ydberg p e a k , as show n in Fig 2*5. In th is fig u re , tw o g rap h s
are show n, one fo r an a to m ic beam (H) and an oth er fo r a m o le c u la r beam
(H 2 ). C learly , w hen th e first ele ctro d e w as turned on, th e b a c k g ro u n d w as
s ig n ific a n tly d e crea se d . T his sh ow ed the in itial “e m p ty in g ” o f th e u p p e r
R yd berg levels, lea v in g v e ry little for the d e te c to r to io n iz e. N o tic e , in
the m o le c u la r beam th e r e w as s till a sm all level o f r e s id u a l c u rre n t,
a lth o u g h alm ost n o th in g rem a in e d in the ato m ic sy stem . T h is re p r e s e n ts
“re g e n e r a tio n ” o f the o n c e em p tied R ydb erg levels. W hen th e se c o n d
e le ctro d e was tu rn ed on, the p eak in the m o le c u la r sy ste m still d e c re a s e d
w hile n oth in g c h an g e d in th e ato m ic system . Since the s e c o n d e le c tro d e
was only 50 nsec d o w n s tre a m , th is show s th a t the re p o p u la tio n o f th e
up per R ydberg sta te s o c c u rs ra p id ly . S u b seq u e n tly , the th ir d re g io n
d e crea se d the b a c k g ro u n d even m ore.
U pon fu rth er te s tin g , a p e c u lia rity w as o bserved. W hen the th ird
e le c tro d e was tu rn ed on, th e back g ro u n d w as low ered, b u t i f th e m id d le
e le c tro d e was tu rn ed off, th e b ack g ro u n d w en t dow n even f u r t h e r 31, by
42
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CEMDC vs. preionizer settings
-15
V,=V =V3=0
V,=3 kV
V,=V2=3 kV
V ,=V2=V3=3 kV
-10
-5
0
^£
w
i
U
-16 r -14 -12 _
12
-1 0
-
I
I
T
T
T
v,=v2=v3= 0
V =3 kV
V =V3=3 kV
- -
-2 -
5
6
7
Vy deflection voltage (times +/- 500 V)
Fig. 2-5: Preionizer data. This measurement shows the background CEM current at
different settings o f the preionizer. The more field regions used, the lower the
background was, but only in molecular states. See labbook PLJ12 p. 9-13.
about 17%. In o th e r w o rd s , havin g on ly th e f irs t and th ird r e g io n s on
gave the lo w e s t b a c k g ro u n d . A p o ssib le e x p la n a tio n is th a t o p tim u m
r e g e n e ra tio n o c c u r r e d w ith a zero field d r i f t re g io n . W hen a ll th re e
e le c tro d e s a re on, th e re is no zero f ie ld . B y g ro u n d in g the m id d le
e le c tro d e , a 50 nse c re g io n is created w h e re th e re is no field p r e s e n t. F o r
the m e a s u re m e n ts m ad e h ere, th e m id d le e le c tr o d e was g ro u n d e d and o n ly
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
th e first and th ird w e re o p e ra te d at V=10.5 kV. U n d e r th e s e c o n d itio n s,
th e typical H 2 o p tic a l sig n a l (F ig . 2 -1 2) was a b o u t h a l f th e siz e o f the
re s id u a l b a c k g ro u n d (s ig n a l/C E M D C - 1/2).
2.3 Laser r e so n a n t e x c ita tio n between R yd b erg sta te s
The n e u tra l b e a m e n te re d th e la se r in te ra c tio n re g io n (L IR )
im m e d ia te ly a fte r th e p r e - io n iz e r . An Adkin M IR L -5 0 c w C O 2 la s e r was
b ro u g h t into the v a c u u m c h a m b e r th ro u g h a 2" Z n S e w in d o w . T h e laser
w a s d isc retely tu n a b le u sin g a d iff ra c tio n g ratin g at one end o f th e laser
c a v ity . Inside the L IR v a c u u m c h a m b e r, illu stra te d in F ig. 2-6, Cu
m irro rs , m o u n ted o n a r o ta tio n a l p la tfo rm , ste e re d th e la s e r into the
m o le c u la r beam . T h e la s e r an d m o le c u la r beam s w ere p a r a l le l to and ~2
cm above the p la tf o r m . The a n g le betw een the tw o b e a m s w as d e fin e d as
0, w here 0 = 0° fo r c o u n te r p r o p a g a tin g beams and 0 = ± 1 8 0 ° fo r
co p ro p a g a tin g . T h e p la tfo rm w as a tta c h e d to a sh a ft, m o u n te d on a
fe rro flu id ic ro ta ry f e e d th ro u g h . E x te rn a l to the v a c u u m th e fe e d th ro u g h
w a s attached to a N e w p o r t m o d e l 4 9 6 , com p uter c o n tr o lle d ro ta r y stage.
T he entire p la tfo rm c o u ld be ro ta te d 360°, w hich a llo w e d a s im ila r range
in the in te rse c tio n a n g le ; h o w e v e r, th e ro tation w as not a llo w e d w ith in
± 3 0° o f p arallel to k e e p the m o le c u la r beam from h ittin g th e la s e r m irror
or p o st. M agnet c o ils m o u n te d on the outside o f the c h a m b e r a llo w e d the
m a g n e tic field to be z e ro e d at th e in te rse c tio n o f the la s e r an d m o le c u la r
b e am s.
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Laser Interaction Region (LIR)
ZnSe
_________j
Cu mirrors
CO, laser
-4*—
i
r'
Vacuum Region
Iferrofluidic rotary
vacuum feedthrough
'
i
?!
/
Newport 496
---- J
rotation stage .________________
Fig. 2-6: Laser interaction region. Schematic for the LIR used in measuring the optical
scans presented in Chapter 3. A computer controlled stage allowed higher resolution and
automatic scanning.
F in e -tu n in g o f th e laser fre q u e n c y w as ach iev ed by D o p p le r
sh iftin g . R otating th e in te r s e c tio n ang le D opp ler tu n ed the a p p a re n t
fre q u e n c y (v ’) o f the C O 2 laser, as “ s e e n ” by the m o v in g R y d b e rg sy stem .
T his tu n in g o f the la s e r fre q u e n c y ( v L) w as g ov ern ed by the D o p p le r
fo rm u la
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
w h e re P is th e beam v e lo c ity (v /c ). W hen tu n ed in to r e s o n a n c e the
R y d b e rg e le c tro n was e x c ite d fro m a lo w er (n = 9 ,1 0 ) le v e l into an u p p e r
( n '= 1 7 ,2 7 ), io n izab le level. T h e d isc re te la s e r fr e q u e n c y w a s cho sen c lo se
to th e d e s ir e d tra n s itio n fr e q u e n c y . This is illu s tr a te d in F ig . 2-7, w here
the C O 2 la s e r lines are sh o w n n e x t to the a llo w e d t r a n s it i o n s fo r n=9,10
and 11 R y d b e rg states. F o r m o le c u la r b e am s w ith la rg e e n o u g h v e lo c ity
(P > 0 .0 0 1 ), the con tin u o u s D o p p le r tu n in g r a n g e w as s u f f i c i e n t to fill in
the g ap s b e tw e e n these d is c re te la s e r lines and a llo w c o n tin u o u s tu nin g.
E le c tr ic dipole t r a n s itio n s , in d u ce d by th e s in u s o id a l fie ld o f the
la se r, w e re o b se rv ed w h e n th e D o p p le r tu n e d fre q u e n c y ( v 1) a n d the
tr a n s itio n fre q u e n c y (v 0) f o r th e tw o d if fe re n t le v e ls w e re th e sam e. Such
tr a n s itio n s w ere governed by th e s tre n g th o f the a p p lie d fie ld (laser
p o w e r) and d iffe re n t s o u rc e s o f lin e b ro a d e n in g ( m o s tly D o p p le r
b ro a d e n in g ) . The o b se rv ed lin e w id th s w ere on th e o r d e r o f ~0.01 c m '1
(3 0 0 M H z).
For each m e a s u r e d sp ectru m , a r e p r e s e n ta tiv e lin e w id th is
g iv e n in th e c o rre sp o n d in g t a b le (T ables 3-3 to 3 -1 1 ). In th e o b se rv ed
s p e c tra , fo r all three is o to p e s , th e n= 9-20 tr a n s itio n s w e re sy s te m a tic a lly
b ro a d e r ( - 0 . 0 1 2 c m '1) th a n th e o th e r two tr a n s itio n s ( —0 .0 0 8 cm*1). The
m o st p r o b a b le e x p la n atio n f o r th is d iffe re n c e lies in th e fa c t th a t the n= 920 t r a n s itio n s were all m e a s u r e d w ith a d if fe r e n t la s e r th a n th e others.
T h is la s e r w as passed th ro u g h a cu rv ed m irro r, w h ic h g a v e a sm a ller w a ist
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6 0
n = 11
500
8 0
Energy separation of
hydrogenic Rydberg levels
11155
(many states not shown)
22
2 3
2 4
2 5
2 6
2 7
2 8
2 9
3 0
31
3 2
n=10
n -16
n -17
n=9
Discrete CO, laser lines
00'1-100 band
i, i
P(50)
vL(c m ')
L ...
8 7 0
.
. .
1
8 9 0
I
1 . . 1 -
I
9 1 0
t
.
i
i -
- i —
9 3 0
i -
—
P(I6)
P(2) R(0)
i
i .
.
1 .
9 5 0
. i
R(20)
. i . - . J .
9 7 0
i
i
R(30)
i
I
9 9 0
Fig. 2-7: CO 2 laser lines. The transition energies for a few Rydberg levels are compared to the allowed discrete laser frequencies.
and ; th e re fo re , a la rg e r a n g u la r w idth for those tra n s itio n s .
Several d iffe re n t e ffe c ts c o n trib u te to the o v e ra ll ob serv ed
lin e w id th s. The tra n s it tim e for the intersection o f th e la s e r (w aist ~ 3m m )
w ith the m olecular beam (P = 0 .00344), gave a w id th o f 0 .0 0 4 c m '1 (12 9
M H z). How ever, the m ain p o rtio n o f the linew idth cam e from D o p p le r
b ro a d e n in g . The m o le c u la r beam was restric te d by th e p re -io n iz e r (4 mm
d ia. a p e rtu re ) and the d e te c to r en trance (6 mm dia. a p e rtu re ). These
p o in ts w ere separated by ~2 m eters, giving a full d iv e rg e n c e angle o f 3
m rad and a D oppler w id th o f - 0 .0 1 c m '1 (300 M H z). T ra n sit and D o p p le r
b ro a d e n in g are enough to e x p la in the o bserved w id th s; how ever, b oth
s a tu r a tio n effects and the n a tu ra l linew idths are e x p e c te d to c o n trib u te .
A ll m e a su re d spectra w ere tak e n w ith less than 15 W and, because
p re v io u s m ea su re m e n ts6 sh o w e d th a t these tra n s itio n s sa tu ra te at a b o u t 20
W o f la s e r power, any re la te d e ffe c ts on the lin e w id th s w e re ignored. In
a d d itio n , the lifetim es o f th e s e states are ty p ic a lly > l p s e c . , which g iv e s a
n e g lig ib le natural lin e w id th o f ~5*10*6 cm*1 (0.16 M H z).
2.4 S tark ionization and d etection o f R yd b erg states
T he Stark io n iz a tio n d e te c to r provides a h ig h fie ld reg io n to S ta rk
io n iz e h ig h -n ' R ydberg s ta te s, w h ile also b o ostin g t h e ir en erg y to
d is tin g u is h them from c o llis io n a lly produced ions c re a te d along the
b e a m lin e . The resu ltin g io n s w ere deflected by a lo w field , energy
a n a ly z e r and co llected in a c h a n n e l e lectro n m u ltip lie r (C E M , G alileo
m o del 4 7 0 0 ). To e n ab le lo c k in a m p lifie r detectio n , th e la s e r was
48
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
am p litu d e m o d u la te d w ith a 50/50 ro ta tin g c h o p p e r w heel (a llo w in g th e
la s e r to pass th ro u g h 50% o f the tim e. The C E M c u rr e n t was m o n ito r e d
sy n c h ro n o u sly w ith the la s e r m o d u la tio n ( r e f e r e n c e ) freq u en cy . A n y
R ydberg states e x c ite d by the laser w ere io n iz e d and the re su ltin g io n
cu rre n t, sy n c h ro n o u s w ith the refe ren c e f re q u e n c y , w as c o llec ted in th e
C E M as “ laser s i g n a l ” (and defined as CEM A C ). T h is general d e te c tio n
sch em e was u se d f o r m any e x p erim e n ts, in c lu d in g th o se re fe re n c e d a t the
beg inn ing o f S e c tio n 2 .1; ho w ever, th re e m a jo r im p ro v e m e n ts in th e
d e te c to r m ade th e s e n e w m ea su re m e n ts p o s s ib le . A p ic tu re o f the n e w
U H V R ydberg d e te c t o r 32 in te rio r is show n in F ig . 2-8.
2.4-1
Im p ro v e m e n ts to the R ydb erg d e te c to r .
The first im p ro v e m e n t was d e sig n in g th e e n tir e d e te c to r in s id e o f an
u ltra -h ig h v acuum (U H V ) 10" C o n flat Tee, w h ic h w as pum ped o u t by a 6"
d iffu sio n pum p, to p p e d w ith a LN 2 trap . The p r e s s u r e was m a in ta in e d at
~ 5 x l 0 '9 Torr, a p p ro x im a te ly 20 tim e s low er th a n p re v io u s d e sig n s. A n y
back g ro u n d gas in th e io n iz atio n reg io n in c r e a s e s th e b a ck g ro u n d C E M
c u rre n t due to c o llis io n a l io n iz atio n . By d e c r e a s in g the p ressu re , th e
back g ro u n d level w a s s ig n ific a n tly red u c ed .
The seco n d im p ro v e m e n t was c o n s tr u c tin g th e d e te cto r in a
lo n g itu d in al c o n f ig u r a tio n w ith the io n iz in g fie ld s p a ra lle l to th e b e a m
d ire c tio n . Fig. 2-8 is m ea n t to illu stra te the e f f e c t o f hav in g lo n g itu d in a l
fie ld s. As the m o le c u la r beam e nters th e d e te c to r , th e m o le cu le s
e x p erien c e a slo w ly ra m p e d p o ten tial (a low fie ld ). T his lo w -field r e g io n
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6 mm dia. apertures
2.4 cm
Molecular Beam
1
V
3
2V
3~
V
1
Deflection Plates
Lens
V = 0 to l5kV
Channeltron
Detector
Background ions
Signal molecules/ions
• ►
The solid line above represents the potential as a function of the
position along the detector. The background ions are slowed down
as a result o f the initial low electric field. At the peak, the field
becomes high enough to ionize the desired Rydberg states and the
resulting signal ions are accelerated and, therefore, tagged with a known
energy boost. This boost allows the signal to be separated from the
background as shown by the solid vs. dashed lines.
Beam
Viewing
System
Fig. 2-8 : Schematic o f the UHV Rydberg detector. The detector operation is illustrated
underneath a photograph o f the apparatus. The potential is plotted vs. position in the
detector, and the expected behavior o f both the ions and the molecules is illustrated. The
velocity at each position is represented by the magnitude o f the corresponding arrow.
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is u se d to retard any re s id u a l ions in the beam . W h en th e beam reach es
the m ain e le c tro d e , a h ig h field is e x p e rie n c e d d ue to th e rapidly
d e c re a sin g p o te n tia l. R y d b e rg sta te s in the a p p ro p r ia te levels w ill be
io n iz e d and the re s u ltin g ions w ill be “ta g g e d " w ith an energy boo st
e q u iv a le n t to th e a p p lie d p o te n tia l. Any b a c k g ro u n d ions will sim ply
re c e iv e a boost e q u iv a le n t to the p rev io u s r e ta r d a tio n , giving them no net
e n e rg y change. In th is w ay, the sig n a l ions are d is tin g u is h e d from the
b a c k g ro u n d ions by t h e i r re s u lta n t en erg y d if fe r e n c e . A p air o f d e fle c tio n
p la te s are used to e n e rg y a n a ly z e the ions and s te e r th e m both
h o riz o n ta lly and v e r tic a lly . A len s is used to fo cu s th e signal ions into
the CEM .
F o r v e rs a tility , a p a ir o f s im ila r io n izatio n r e g io n s were in sta lle d
sim u lta n e o u s ly , one fo r h ig h fie ld io n iz a tio n (n '= 1 7 ) a n d the other for low
field io n iz atio n (n '= 2 7 ). E ach v o lta g e fe e d th ro u g h w as rated for 15kV.
The seco n d reg io n w as d e sig n e d w ith o n e -th ird th e e le c tro d e spacing o f
the fir s t region. C o n se q u e n tly , the “ s h o rt-g a p ” io n iz e r has three tim es
h ig h e r field for th e sam e a p p lie d v o lta g e . T h is a d d itio n a l io nizer a llo w s
n=9 R ydberg sta te s to be stu d ie d w ith the sam e d e te c to r as the n=10
R y d b e rg states, s im p ly by c h a n g in g w hich e le c tr o d e s have voltage.
T he final s ig n i f ic a n t im p ro v e m e n t was th e a d d itio n o f a C o lu tro n
BVS-1 beam v ie w in g s y s te m . T he CEM is p o s itio n e d the same d ista n c e
o ff-a x is as a m u lti-c h a n n e l p late and p h o sp h o r sc re e n system . A vacuum
w in d o w and CCD c a m e r a a llo w v ie w in g o f the s ig n a l beam on the
51
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ph o sp h o r. T h e s ig n a l is focused onto th is b eam v ie w in g sy ste m and the
p o la rity o f th e d e fle c tio n field is then re v e rse d to s te e r the sig n a l into the
sy m m e tric a lly p la c e d CEM . T h is view ing c a p a b ility a llo w s optim um
se ttin g o f the d e te c to r , in c lu d in g focusing o f the sig n a l beam . The
c o lle c tio n e ff ic ie n c y is im p ro v ed s ig n ific a n tly by h a v in g the view er for
the sim ple r e a s o n th a t th e o p e ra to r can now be sure th e sign al beam is
fo cu sed p ro p e rly .
2.4-2
O p e ra tio n o f the R ydberg d e te c to r.
The R y d b e rg d e te c to r p o te n tia ls are set to io n iz e the s p e c ific up per
sta te being d r iv e n by th e laser e x cita tio n an d to s te e r th e re s u ltin g ions
in to the CEM . R y d b e rg sta te s are exp ected to io n ize (d ia b a tic a lly ) at
field s ran g in g fro m l / 9 n 4 to l / 4 n 4 a.u. (e.g. a to m ic h y d ro g e n n= 27 levels
ionize b e tw e e n 1000 and 2000 V /c m )33. The io n iz a tio n o f R y d b erg states
is g e n erally in d e p e n d e n t o f the ion core sp e cie s. In a d d itio n , the d iffe re n t
a n g u la r m o m e n tu m sta te s are ex p ec te d to io n iz e at the sam e field . The
d e te c to r is u s u a lly tu n e d up on a large a to m ic la se r s ig n a l, sin c e they are
e a sie r to see th a n the m o le c u la r signals and th e re is ty p ic a lly no
d iffe re n c e in io n iz a tio n b eh av io r.
To o p tim iz e th e d e te c to r setting s, a la s e r sig n a l w as m ea su re d as a
fu n ctio n o f th e io n iz a tio n (strip p in g ) p o te n tia l
V s,
the d e fle c tio n
(V x
and
V y)
and lens
(Viens)
( V s) .
A t each se ttin g o f
v o lta g e s w ere set to
m axim ize th e s ig n a l. As the io n iz in g field w as in c re a s e d , it re a ch e d a
th re sh o ld w h e re th e s p e c ific , la s e r excited R y d b e rg s ta te s b e g a n to be
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
io n iz ed and o b s e r v e d as signal. W hen th e strip p in g field r e a c h e d
a p p ro x im a te ly th r e e tim e s the c ritic a l t h r e s h o ld , R yd berg s ta te s b e g a n to
be ionized in the p r e lim in a r y reg io n (3 tim e s lo n g er g a p ) a n d n o t ta g g e d
w ith the d e s ire d e n e r g y boost. T his c a u s e d the signal to d e c r e a s e w ith
in cre asin g V s. T h e re s u ltin g io n iz a tio n y ie ld cu rves are sh o w n in F ig . 2-9
fo r both a to m ic h y d r o g e n and ato m ic d e u te r iu m signals. T h e s e
m ea su re m e n ts are fo r n= 10-27 t r a n s itio n s a n d show th a t th e d e te c to r
se ttin g s are in d e p e n d e n t o f mass.
12
r-
XI
^ .
atomic H
_-
------ atomic D
—
e3
03
>-
r
c
00
l/9n 4
• /
—
ka
u
V)
CB
•
l/4n 4
•
•
.
V
0
500
/1000
1500 2000 2500 3000 3500 4000 4500 5000
Detector Ionization Field (V/cm)
Fig. 2-9: Ionization yield curves. Laser signals for n= 10-27 transitions were measured as
a function o f the applied ionization field. There was no noticeable difference between
hydrogen and deuterium ionization. [PLJ 15 p. 121 and PLJ16 p.23]
53
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The R ydb erg d e te c to r w as set d iffe re n tly fo r e a c h ch o sen la s e r
sp e ctru m . The io n iz a tio n y ie ld c u rv e s are show n in F ig. 2-10 for e ach o f
th e th re e m easured tr a n s itio n s , n = 1 0 -2 7 , n=9-20 a n d n = 9 -1 7 . The
io n iz a tio n field is b a s e d on the m e a su re d p o te n tia l a n d th e know n d e te c to r
gap size as seen in F ig . 2-8. T he lon g gap (2 .4 cm ) w as u sed for n = 2 7 a n d
th e sh o rt gap (0 .8cm ) w as u sed fo r n= 17 and n= 2 0. T h e se yield c u rv e s
w e re used to set the s tr ip p in g p o te n tia l to the o p tim u m v a lu e fo r eac h
la s e r scan. E ach y ie ld c u rv e b e g in s to rise at a fie ld o f a b o u t l / 9 n 4 a .u .,
w h ic h is the lo w est d ia b a tic io n iz a tio n field for s ta te s o f p rin cip al
q u a n tu m num ber n.
T h e m ax im u m o f each c u rv e o c c u r s at about tw ice
6
5 -
2 -
"
♦
0 —•
0
■----■ n=20
♦— ♦ n=17
•
--------------------------------------------------
'
' 5000
•
•
10000
15000
20000
•
Detector Ionization Field (V/cm)
Fig. 2-10: Ionization yield curves for three different upper states, n=17, 20 and 27. The
field depends on the applied voltage (Vs) and the specific gap sizes. The long (2.4 cm)
gap was used for n=27 and the short (0.8 cm) gap was used for n=17 and 20. The arrows
denote the expected ionization threshold, l/9n a.u. [PLJ15 p.l 18, 121; PLJ14 p.79]
54
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
th is value, w here all S ta rk lev els are e x p ec te d to io n iz e 33. T he m o le c u la r
states w ere e x p e c te d to io n iz e w ith the same b e h a v io r and any o b s e r v e d
d e viatio ns from th e s e y ield curves will be d isc u sse d in the n e x t c h a p te r.
For laser tra n s itio n s from n=10 to n=27, the R y d b erg d e te c to r w a s set
w ith a field o f a p p ro x im a te ly 2000 V/cm. T he long-gap io n iz e r w as used ,
w ith an a pp lied v o lta g e o f Vs=5000 V. As m en tio n ed above, the sig n a l
ions were b oo sted by the a d d itio n a l 5kV, g iving them a to tal e n e rg y o f
~16 keV. The la s e r sig n a ls w ere energy a n a ly z e d by the lens a n d v e rtic a l
d e flection v o lta g es. The CEM cu rrent was m e a su re d as a fu n c tio n o f the
vertical d e fle c tio n v o lta g e (V y) to see the re s o lv e d signal and to m a x im iz e
th e settings. T h ese sc an s are show n in Fig. 2-11 fo r two d if f e r e n t se ttin g s
o f the d e te cto r. T he first scan was for the n= 27 setting at V s= 5 k V a n d the
second scan was fo r the n=17 setting o f 10.5kV. The s e p a ra tio n o f th e
signal peak from th e b a c k g ro u n d peak w as tw ic e as large for the s e c o n d
scan since the b o o stin g p o te n tia l was tw ice as large.
2.5 Exam ple o f ob served H 2 Rydberg state tran sitio n s
As an e x a m p le o f the RESIS technique and the e q u ip m en t d e s c rib e d
here, co n sid er the n = 10 to n'= 27 laser tra n s itio n s m easu red in H 2 R y d b e rg
states. A b o ttle o f h y d ro g e n w as attach ed to the ion source. A n H 2 +
beam, a c c e le ra te d to 11.1 keV , was steered into the Cs v ap o r c e ll, w h e re
R ydberg states w ere c o llis io n a lly populated. T he beam e n te re d th e p r e ­
ionizer reg io n w h ere all sta te s w ith n>16 w ere io n ized , thus e m p ty in g the
55
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
t/1
20
400
•
—
600
background ions
5 kV boosted signal ions (n*=27)
10.5 kV boosted signal ions (n“ 17)
800 1000 1200 1400 1600 1800 2000 2200 2400 2600
Vy (volts; +/- deflection voltage)
Fig. 2-11: Resolved signal peaks. The signal peaks are shown for two different settings
of the ionization potential. The deflection voltage serves to energy analyze the different
components of the beam allowing the resolution o f the signal peak from the background
ions. [PLJ15p.ll5, 122]
u p p e r sta te s and d e fle c tin g the re s id u a l ions o u t o f th e b e am . T h e n e u tra l
R y d b e rg beam p a sse d in to th e la s e r re g io n w ith a v e lo c ity o f
P = v /c —0.00344. To e x c ite th e R y d b erg tr a n s itio n fro m n = 1 0 -2 7 , th e C O 2
P (1 6 ) lin e (v = 94 7.74 2 c m '1) w as c h o sen , b a sed on Fig. 2-7. T h is la s e r
fre q u e n c y was ste ere d in to th e L IR and the in te rse c tio n a n g le w a s ste p p e d
by 0.03°, scanning from 80° to 130°.
The R ydberg beam e n te r e d the U H V d e te cto r, w h e re th e lo w field
(lo n g -g a p ) ionizer was u se d to io n iz e n'= 27 R yd berg s ta te s in th e H 2
beam . T he d etector was o p tim iz e d fo r m ax im u m io n iz a tio n y ie ld by
m e a su rin g the signal as a fu n c tio n o f field (V s). The p e a k fo r n '= 2 7 w as
fo u n d to be with V s=5kV , g iv in g a field o f - 2 0 8 0 V /cm . T he s ig n a l beam
was fo c u se d and d e fle c te d in to th e b e a m v ie w e r to o p tim iz e th e d e te c to r
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
settin g s and th e n d e f le c te d into the C E M fo r m e a su re m e n t. T he la s e r w as
am p litu d e m o d u la te d at 203 Hz and the C E M c u rre n t, c o n v e rte d (an d
a m plified by 108) to a v o lta g e sig n a l, w as m e a s u re d w ith a lo c k -in
a m p lifie r (E G & G P r in c e to n A pplied R e s e a r c h M odel 5110). T y p ic a l
settin g s fo r the lo n g - g a p d e te cto r (n '= 2 7 ) w e re V s= 5kV , VX= 4 9 0 V ,
Vy=±3560V , V lens= 3 4 0 0 V , C E M H V = -8 0 0 V .)
The lo c k -in s ig n a l was m easu red as a f u n c tio n o f the la s e r
in te rse ctio n an g le. T h e neutral beam w as m o n ito re d on a F a ra d a y cup at
the end o f the d e te c to r , to m aintain a c o n s ta n t beam c u rren t.
T he
resu ltin g scan is sh o w n in Fig. 2-12 w h e re e a c h p o in t re p re se n ts a fiv e second average o f th e sig n a l. The m e a s u re d s tru c tu re gives a lm o s t a
direc t map o f th e n = 1 0 fin e structu re. F u r th e r d e ta ils o f line
id e n tific a tio n s w ill be d isc u sse d in C h a p te r 3. U n its o f the sig n a l a x is are
arb itrary and th e h o r iz o n ta l axis sho w s th e c a lib r a te d e n erg ies a lo n g w ith
a few c o rre sp o n d in g a n g le s . A n g le -to -e n e rg y c a lib ra tio n s are d e s c r ib e d
fu rth er in C h a p te r 3 a n d A ppendix D. T y p ic a l m e a su re m e n ts h a v e a 60
nA neutral b eam an d sig n a l/b a c k g ro u n d - 1 / 2 . [In p re v io u s m e a s u r e m e n ts ,
th is S/B was re p o rte d by Sturrus to be - 1 / 3 0 0 . ]
57
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Typical laser scan of H2 n=10-27 transitions
P (I6 )C O j laser line
P = .00344
946.0
947.0
948.0
[122.42°!
(103.24°)
[85.57°)
Transition Energy ( c m 1)
[intersection angle (degrees)]
Fig. 2-12: Laser scan o f H 2 n= 10-27 transitions. The observed structure is directly related to the n=10 Rydberg EFS. The strong line
going ofTscale is the n= 10-27 transition o f atomic deuterium, which exists in the mass two ion beam. Each point represents a five
second average o f the signal. A typical signal corresponds to an ion count rate o f about 10s s '1.
2.6 D escrip tion o f m icrow ave RESIS te ch n iq u e
The m ic ro w a v e re s o n a n c e e x p erim e n ts ( C h a p te r 4) u se d a v a ria tio n
o f the RESIS a p p a ra tu s d e sc rib e d in the a b o v e se c tio n s. By a d ding a
se c o n d la se r re g io n an d a m icrow ave re g io n , h ig h e r p r e c is io n
m e a su re m e n ts o f R y d b e rg fine structu re can be a c h ie v e d by d ire c tly
indu cin g m ic ro w a v e tra n s itio n s betw een sta te s o f the sam e n. The
sc h em a tic sh o w n in F ig. 2-13 illu stra te s the p o s itio n in g o f th is new
equ ip m en t. T he c a rto o n at the bottom o f the fig u r e show s a
r e p re s e n ta tio n o f the lev e l pop ulatio n s at d if f e r e n t sta g e s o f th e beam line.
As d e sc rib e d a b o v e , the Cs cell an d p r e i o n i z e r p o p u la te d the n
s ta te s and p re - io n iz e d th e u p p e r n' states, r e s p e c tiv e ly . S im ila rly , the
f ir s t laser p re p a re d th e lo w e r n states by d e -p o p u la tin g a s p e c ific fine
stru c tu re lev e l (e.g . n = 1 0 , L=6; 101). T h is c r e a te d a p o p u la tio n d ifferen c e
b e tw e e n n e ig h b o rin g fin e stru c tu re levels, su c h as the 101 and 10K states.
T he second la s e r w as tu n e d to the same tr a n s itio n as the firs t and, with
the d e te c to r set to io n iz e and d e te c t the la s e r e x c ite d u p p e r sta te , this
la s e r was u sed to m o n ito r any change in the lo w e r n sta te (101).
An a m p litu d e m o d u la te d (@ 2.85kH z) tr a v e lin g m ic ro w a v e field
e n te re d the m ic ro w a v e re g io n and p ro p ag a ted p a r a lle l to the m o le cu la r
b eam , e ith e r in th e sam e d ire c tio n or o p p o site d ire c tio n , d e p e n d in g on the
d e sire d D o p p le r sh ift. T he m icrow ave fre q u e n c y w as sc a n n e d and the
CEM curren t, s y n c h ro n o u s w ith the 2.85 k H z m o d u la tio n freq u e n c y , was
m easu red on a lo c k -in a m p lifie r. When the m ic ro w a v e fre q u e n c y reached
59
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ion source
Microwave
region
pre-ionization
and deflection
UHV
detector
\
\
LIR #1
LIR #2
Cs cell
Positive ions
Neutral Molecules
Signal Molecules/Ions
27K
27K.
■C
*■
27K
O-
10K
QO■
-------
I OK
•••••♦
10K
•»««
»•
••
101
101
••••
■■••
101
Fig. 2-13: Microwave spectroscopy apparatus. An additional LIR and a microwave
region were added to the beamline. Below the schematic, populations o f various
Rydberg levels are shown to illustrate the effects o f the LIRs and microwave region. The
first LIR depletes the 101 population. The microwave field drives a resonant transition
from the 10K to 101 levels. The second LIR, tuned to the same angle as the first,
monitors the 101 population. This laser transition will increase in size when the
microwave transition is induced.
the r e s o n a n t freq uen cy , a tr a n s it i o n w as d r iv e n (from 10K to 101) an d a
sig n al w as m easu red. By c h o o s in g d if f e r e n t laser t r a n s it i o n s an d s c a n n in g
o v e r th e ex p ec te d f re q u e n c y ra n g e s, th e e n tire fine s t r u c tu r e w as m a p p e d .
The in te ra c tio n o f th e R y d b e rg e le c tr o n w ith the o s c i ll a t o r y e le c tr ic
field can be m odeled u s in g s ta n d a r d tim e d e p e n d e n t p e r t u r b a ti o n th e o ry .
60
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T he m o le c u la r beam , tr a v e lin g w ith ve lo c ity v=(Jc, s p e n d s a tim e T = L /p c
in the field, w here L is the le n g th o f the m icro w av e in te r a c tio n re g io n .
T he tra n s itio n p ro b a b ility is d e fin e d by
x
w h e re th e am p litu d e (A) is d e p e n d e n t on the d ip o le m a trix e le m e n t
b e tw e e n the two sta te s and th e a p p lie d m icro w ave p o w e r, to is th e a p p lie d
fre q u e n c y , and too is the “ D o p p le r f r e e ” re so n a n t fre q u e n c y . The
c o e f f ic ie n t o f to, in x(to), g iv e s th e re la tiv is tic D o p p le r s h if t, d e p e n d e n t
on th e d ire c tio n o f p r o p a g a tio n r e la tiv e to the m o le c u la r beam .
S everal a p p ro x im a tio n s w e re m ad e in a rriv in g at E q. 2 -2 . T h e first
w as th e rotatin g w ave a p p ro x im a tio n , w hich a llo w s th e n o n - r e s o n a n t
te rm s to be n e g le cted . S e c o n d , any re fle c te d w ave fro m th e te r m in a tio n
end o f th e m icrow ave re g io n w ill be tra v e lin g in the o p p o s ite d ir e c tio n to
th e m ain field. T his r e f le c tio n , w ith c o e ffic ie n t T, c a n a lso d riv e
tr a n s itio n s , but the f re q u e n c y w ill be D o p p le r s h if te d d if f e r e n tly th a n the
d e s ire d tra n sitio n . I f the m ic r o w a v e reg io n is d e s ig n e d w e ll e n o u g h ,
m in im iz in g the re fle c tio n c o e f f ic ie n t, these c o n tr ib u tio n s can be
n e g le c te d , as done in Eq. 2 -2 . T h is w ill be d is c u s s e d l a t e r in S e c tio n 4.32.
As su ggested in Fig. 2 -1 3 , th e fine stru c tu re in te rv a l fo r the (0)1 OU
an d (0)1 OK7 R y dberg le v e ls w as m e a su re d for H 2 . A p lo t o f th e C E M
sig n a l as a fu n ctio n o f the s c a n n e d m ic ro w a v e f r e q u e n c y is g iv e n in Fig.
61
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2-14 . The tr a n s itio n was m ea su re d w ith a 50Q tra n s m iss io n line region,
one m eter in le n g th (L). W ith a b eam v e lo c ity , p= 0.0 0 3 4 0 4 , the
in te rac tio n tim e w as T=0.98 |xsec g iv in g a linew idth o f 1.02 M H z. With
the ex p ec te d tr a n s itio n frequ ency o f 6 3 0 .7 9 MHz, the D o p p le r s h ift was
2.15 MHz. F or a m icro w ave field c o p ro p a g a tin g with the H 2 b eam , the
m easu red lin e c e n te r was at 6 32 .9 4 M H z. The resolved s p in s tru c tu re is
e x p lained in S e c tio n 4.1. E ach d a ta p o in t rep re se n ts the a v e ra g e o f three
(0)10I6 - (0)101^ microwave transition
0.10
—
0.08 -
CA
‘5
•Tip*.
%•. ' *
/
r.
♦i
0.04 -
%
•
&
op
0.02 -
S
u
CJ
0.00
-
.^ *T
•'tUiiif
—
0.02
628
'*
629
*
630
i'i*'
'•
,
631
632
633
634
635
636
637
638
Microwave Frequency (MHz)
Fig. 2-14: H2 microwave RESIS signal. The CEM signal is plotted as a function o f
microwave frequency. The microwave field is copropagating with the neutral H 2 beam.
Each point represents an average o f 3 scans, each at 10 sec./pt.
62
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
sc a n s, rep eated one a fte r th e o th er, eac h a v erag in g for 10 se c /p t. T his
a llo w s each scan to be m e a s u re d a cro ss th e en tire lin e s h a p e w ith m in im al
flu c tu a tio n in beam c u rr e n t o r laser p o w e r.
2.7 D etails o f the m icro w a v e sp ectr o sco p y a p p a r a tu s
The laser reg io n s u se d in the m ic ro w a v e e x p e rim e n t w e re o f a m uch
sim p le r d esig n than the o n e used in th e laser e x p erim e n t. I llu s tr a te d in
Fig. 2 -1 5 , this d esig n had a s in g le - m irr o r g eo m etry , w h e re th e a c tu a l
p o in t o f laser in te rs e c tio n w ith the b eam m ov ed along the b e a m p a th as a
fu n c tio n o f the angle. T h e b e n e fit o f u sin g these re g io n s w a s th e in te rn a l
m u -m e ta l shielding , w h ic h a llo w e d an a lm o s t co n tin u o u s m a g n e tic sh ie ld
fro m the beginning o f L IR # 1 to the e n d o f L IR #2. T his s h ie ld in g re d u c e d
th e S ta rk m ixing o f R y d b e rg lev els due to any m otion al e le c t r i c fie ld s.
B etw een the tw o L IR s, fo ur d if f e r e n t m ic ro w a v e r e g io n s w e re used
d u rin g th is e x p erim e n t to re a c h d if fe re n t freq u e n cy ra n g e s. T w o o f th e se
w e re the same as th o se u s e d by S tu rru s6. In a d d itio n , tw o n e w m ic ro w a v e
re g io n s w ere used for the e x p e rim e n ts p re s e n te d in C h a p te r 4. D e ta ils o f
eac h m icro w ave se ctio n w ill be d isc u sse d here.
The m ain reg io n u se d in these e x p e rim e n ts w as o f a n o ff - a x is , 50Q
tra n s m is s io n line g e o m e try . A tra n s m is s io n line has b o th i n n e r and o u te r
c o n d u c to rs. By lo w e rin g th e in n er c o n d u c to r and c h a n g in g its siz e to
m a in ta in the im pedance m a tc h , a la rg e r c ro s s -s e c tio n w as c re a te d fo r the
m o le c u la r beam to pass th r o u g h . Such an a rra n g e m e n t is s h o w n in F ig . 216, g iv in g a sch em atic o f R e g io n A [d e s ig n a te d the “ A l f o r d ” re g io n , a fte r
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Laser Interaction Region
R otation Stage
beam
C 0 2 Laser
Mu-metal shield
Fits inside 6 Dependex can with
4" Dependex side ports.
"
Mirror
(attached to rotation stage)
^ ---------------
1 ---------------
H2 beam
>•
CO2 laser
ZnSe vacuum
w indow
Fig. 2-15: Different LIR designs used during the microwave experiment.
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the m a n u fa c tu re r]. T h is m ic ro w a v e re g io n was used for the h ig h e s t-L
tr a n s itio n s , w hich a re the m o st se n s itiv e to stray e le c tric fie ld s. Both
c o n d u c to rs are c o p p e r, g o ld -c o a te d to o b ta in high c o n d u c tiv ity . The
T y p e-N fe e d th ro u g h s are h ig h - p r e c is io n m atched a d a p te rs , c o n s tru c te d to
tr a n s m it the m ic ro w a v e p o w e r with m in im a l re fle c tio n due to g e o m e tric a l
m is m a tc h . The fre q u e n c y ran g e o f th is re g io n was lim ite d to < 1 800 M H z;
h o w e v e r, the m e a su re d r e fle c tio n c o e f f ic ie n t over th a t range w as <0.10
and sy m m e tric b e tw e e n th e tw o ends. T he entire re g io n w as s h ie ld e d w ith
w e ld ed (A m u n ea l) m u -m e ta l s h ie ld in g , w hich reduced th e m a g n e tic field s
to B < 20 m G. The m e a s u re d in te rn al e le c tr ic Helds w ere - 5 m V /cm ,
c o n s is te n t w ith the e x p e c te d size o f m o tio n a l ele ctric fie ld s in m a g n e tic
H elds o f th is size, as d is c u s s e d later in C h a p te r 4.
T he o th e r new re g io n (R e g io n B ) w as also a tra n s m is s io n line, but
o f a c o a x ia l g e o m e try , sh o w n in Fig. 2 -1 7 . This re g io n left v e ry little
sp a ce fo r the neutral beam to pass th ro u g h , resu ltin g in s ig n if ic a n t
a c c u m u la tio n o f stra y c h a rg e on the s u rfa c e s . M easu red in te rn a l H elds
w ere fo u n d to be - 1 5 0 m V /cm . This r e g io n does seem to go h ig h e r in
fre q u e n c y , but only to - 2 0 0 0 MHz. T h e m u-m etal sh ie ld in g in th is re g io n
is in te rn a l to the v a c u u m d ue to the s lig h tly m ag netic sta in le s s ste e l
vacu u m c h a m b e r th a t s u rro u n d s the re g io n . Also, th e T y pe-N
f e e d th ro u g h s are lo w p r e c is io n and h a v e larg e re fle c tio n s at h ig h e r
f re q u e n c ie s , m ain ly due to the c o u p lin g in o f the m ic ro w a v e H eld . In
su m m a ry , th is reg io n w as u se d for a fe w m e a su re m e n ts , b ut fu tu re use
65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Microwave Transmission Line Region
2 " B ra ss D e p e n d e x
-T -
T y p e-N
R F fe e d th ru s
W e ld e d m u -m e ta l
s h ie ld in g
L = 1 m e te r
0.050 in. thick
mu-metal
C o p p e r c o n d u c to rs
(gold coated)
0 .6 5 c m
M u - m etal s h ie ld in g
Region A
Fig. 2-16: Primary microwave region. An off-axis transmission line region,
manufactured by Alford with high-precision vacuum feedthroughs, is shown above.
66
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Alternate Transmission Lines Used
T ypeN
feedthrus
110 cm
-:r -
T.
-1
Solid copper
center conductor (2.26 cm dia.)
4" stainless
D ependex
Region B
Copper outer conductor
(5.20 cm dia.)
8 mm
aperture
2" Stainless Dependex
Type-N ----RF feedthrus
W rapped m u-m etal
shielding
30 cm
2.92 cm-
Region C
E
u
oo
in
0.36 cm x 0.94 cm
0.51 cm
Fig. 2-17: Alternate transmission line regions. The two regions shown were used for
various transitions. Region C was useful for higher frequencies (> 2 GHz) and higher
beam current due to the shorter length.
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
sh o u ld be a v o id e d sin c e the o th e r r e g io n s are fa r su p erio r.
The p r e v io u s ly used tr a n s m is s io n lin e 6-34 is a lso an o ff-a x is
g e o m e te ry . R e g io n C uses a r e c ta n g u la r c o n fig u ra tio n w ith a w a v e g u id e
as an o u te r c o n d u c to r. This reg io n e x te n d s b e y o n d 2 GHz in its fre q u e n c y
ran g e ; how ever, th e s h o r t length lim its the r e s o lu tio n o f th ese
m e a su re m e n ts. In a d d itio n , the 2" D e p e n d e x a d a p te rs are s ta in le s s ste e l
and h ave s ig n ific a n t m ag n e tic fie ld s at the w eld ed jo in ts . T h is s e r io u s ly
lim its the use for h ig h -L tra n s itio n s t h a t are se n s itiv e to m o tio n a l e le c tric
fie ld s, since it is n o t p o ssib le to e n tir e ly sh ie ld fro m this e ffe c t.
The final r e g io n used was a G -b a n d w a v e g u id e (no in n e r
c o n d u c to r), m o d ifie d in len gth from th e p re v io u s u s e 6, m ak in g th e
in te ra c tio n len g th 80 cm (in stea d o f 30 cm ). D im en sio n s and ty p ic a l
c h a r a c te r is tic s are lis te d in T able 2- 1 fo r the m icro w av e re g io n s u s e d in
th is ex p erim en t.
r
R egion
(fr. ran g e GHz)
A : T rlin e
DC - 1.7
D,
(cm )
10.0
d2
(cm )
1.25
L ength
(cm )
100
E -fie ld
(m V /cm )
5
(av g .)
0.10
R efere n ce
(b o o k ,p g .)
PL J20
p.9
B : C oax.
DC - 2.0
5.20
2.26
80
150
0.15
PLJ19
p .62
C : T r-lin e
DC - 2.5
5 .8 4 x
2 .9 2
0.94 x
0.36
30
40
0.30
PLJ20
p .2 7 ,4 2
D : G -band
3.2 - 6.0
4 .7 5 5
2.215
80
36
0.10
P L J19
p.9
Table 2-1: Microwave regions. The first column gives the type of region and the
frequency range in GHz. The next two columns give the cross-sectional dimensions. Di,
the larger dimension, is the size of the outer conductor or the larger side o f the
waveguide. D 2 , the smaller dimension, is the diameter o f the inner conductor or the small
side o f the waveguide. L is the interaction length o f the region. E and T are typical
values for stray electric fields and the reflection coefficients as measured in the reference.
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C hapter 3
L aser sp ectroscopy o f n=9, 10 high-L R ydberg states o f
H2, HD, and D2
3.1 Introduction o f the laser sp ectrosco p y ex p erim en t
L aser and m icrow ave stu d ies o f high-L Rydberg sta te s o f H 2 h av e
been w id e ly pursued for th e last 15 y e a r s 3-7 1 17-27-35. S everal o f th e se
m e a su re m e n ts have y ield ed h igh p r e c is io n m easurem ents o f p ro p e rtie s o f
the H 2 + ( v = 0 ,R = l) ion core. In c o n tra s t, stud ies o f high-L s ta te s o f the
o th e r sta b le H 2 isotopes have been v ir tu a lly absent, e x c e p t fo r a sin g le
stu d y o f nF states in HD 36. The m e a su re m e n ts presented in th is c h a p te r,
in c lu d in g H 2 , HD, and D 2 , re p re se n t th e first system atic stu d y o f h ig h -L
R y d b e rg sta te s for all three iso to p es. C om p ariso n o f the th ree iso to p e s
p ro v id e s an in te restin g new p e rs p e c tiv e on the physics o f h ig h -L H 2
R y d b erg sta te s.
T he g eneral p ro p erties o f h ig h -L m o le cu la r R ydberg s ta te s w ere
d isc u sse d in C hapter 1. The s im ila rity o f the m olecu lar ions, H 2 +, H D +
and D 2 +, su g g e sts that there w ould be little d ifference b e tw e e n the th re e
sy ste m s, o th e r than the m ass v a ria tio n s and the small e ffe c ts o f th e H D +
d ip o le m om ent. This e x p ec te d s im ila rity o f the ions leads to the
69
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c o n c lu sio n th a t sim ila r h ig h -L Rydberg sta te s s h o u ld e x is t for all th r e e
sy ste m s. T hese sta te s c o u ld be used to study th e la tte r two m o le c u la r io n s
in th e same way they h av e b e e n used fo r H 2 +.
All three m o le c u la r sy ste m s sho uld e x h ib it a lm o st id entical
R y d b erg fine stru c tu re . T h e p o la riz a tio n m odel p re s e n te d in C h a p te r 1 is
e x p e c te d to p re d ic t th is s tr u c tu r e , but som e s u b tle d iffe re n c e s b e tw e e n the
iso to p e s m ust be c o n s id e re d . A ctu ally, the b ig g e s t d iffe re n c e seen
b e tw e e n the sp e ctra o f th e d iffe re n t R y dberg m o le c u le s was in the lin e
stre n g th s , not the stru c tu re . T he laser e x p e rim e n ts r e p o rte d in th is
c h a p te r give rough m aps o f th e R ydberg fine s tr u c tu re , show ing
sig n ific a n t d e ta ils th a t m u st be c o n sid ere d in p la n n in g any high p r e c i s i o n
m ea su re m e n ts. The s p e c tr a p re se n te d in th is c h a p te r w ere used to p la n
a nd im p lem ent a m e a s u re m e n t o f the Hz+ and D 2 + (v = 0 ,R = 0 ) d ip o le
p o la riz a b ilitie s , w hich is p re s e n te d in C h a p te r 4.
The fo llo w in g d is c u s s io n is broken into th r e e d iffe re n t s e c tio n s .
T he first rein tro d u ce s the th e o ry d isc u sse d in C h a p te r 1 and re la te s th e
le v e l stru c tu re to th e e x p e c te d stru c tu re o f o p tic a l tr a n s itio n sp e c tra . T h is
s e c tio n illu stra te s the stro n g s im ila ritie s b e tw e e n th e th re e iso to p es.
D iffe re n c e s b etw een the is o to p e s are m a in ly due to th e sm all v a ria tio n in
c o re p a ra m eters (e.g. the q u a d ru p o le m o m e n ts) a n d th e sm all e n e rg y s h if ts
du e to the d ip ole m o m en t in H D , which are left to th e d isc u ssio n in
S e c tio n 3.4. The secon d s e c tio n is a sim p le c a ta lo g o f the m easured
sp e c tra . This p art b e g in s w ith a d e sc rip tio n o f th e d a ta a c q u isitio n a n d
70
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calib ra tio n u sed f o r eac h m easurem en t. E ach s p e c tru m is p re s e n te d and
follow ed by a c o rr e s p o n d in g tab le. The ta b le s c o n ta in ob serv ed and
calcu late d v a lu e s f o r the energy p o sitio n s a n d s tre n g th s o f the id e n tifie d
tran sitio n s. T h e fin a l section d isc u sse s the lin e id e n tific a tio n s and o th e r
d e ta ils o f the s p e c tr a .
3.2 Expected th eo r etica l stru cture fo r R yd b erg sp ectra
The m o st b a s ic th e o re tic a l tre a tm e n t o f th e h ig h -L R y dberg fine
structu re c o n s id e rs th e m o le c u la r ion core a n d th e R ydberg e le c tro n as
in d ep en d en t s y s te m s . A t this cru de level, th e sy s te m takes on the
p ro p erties o f a h y d ro g e n atom , w here the p r o to n is re p la c e d by the io n
c o re . Each r o - v ib r a tio n a l level o f the ion h a s an e n tire m an ifo ld o f b o u n d
hy d ro g en ic le v e ls . A few o f th ese R y dberg le v e ls , for each o f th e lo w e s t
(v ,R ) ion le v e ls , a re sh o w n in Fig. 3-1. T he v ib r a tio n a l and ro ta tio n a l
e n erg ies have b e e n c a lc u la te d " .
E lectric d ip o le tra n s itio n s b e tw ee n h ig h -L R y d b e rg lev e ls o b e y th e
se lec tio n ru le , A v=A R =0. The h y d ro g en ic b in d in g en erg y is in d e p e n d e n t
o f (v,R ), so the d if f e r e n c e b etw een any p a ir o f R y d b e rg levels, w ith
com m on (v ,R ), w ill be th e same for all r o - v ib r a tio n a l levels. T h is p o in t is
show n in Fig. 3-1 a n d is im p o rtan t w hen c o n s id e rin g th e o b se rv ed s p e c tra .
In the absence o f fin e stru c tu re , all n —»n' tr a n s itio n s w ill be at the sa m e
tra n s itio n e n e rg y g iv e n by AE(0) = E(0)[n-] - E (0)[„j. T he only d iff e re n c e s
fro m this e n e rg y w ill be due to the e ffe c ts o f th e R y d b e rg fine s tr u c tu re .
71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
^ 2
R=2
V=1
R=1
R=0
v=l, n=9 and n=10
levels are unstable due
to autoionization
2191.126 cm*1
9
174.24 cm -i
R=2
v=
0 ,
r
=
_ _
i
27
R=0 ---- —
ZL
.2 0 .
2 0
_ !Z ,
27
20
17
17
Laser
induced
transitions
'
n2
n=6
\
; io
10
-1097.1
cm
1 0 ——
All three transitions shown would be
the same energy
9
o
J0)
9
- i 354.4
AE
n = 9 ------
(0)
- E [n=9]
=-379.6 - (-1354.4)
= 974.8 cm
= E
(0)
[ n -17]
Fig. 3-1: Hydrogenic structure o f molecular Ha Rydberg levels. Transitions between any
pair o f Rydberg levels are at the same energy.
72
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T he lo n g -ran g e in te ra c tio n s betw een the R y d b e rg e le ctro n and the
m o le c u la r ion core a re m o d eled by the e ffe c tiv e p o te n tia l given in Eq. 111. T he c alcu lated fin e stru c tu re energies are also g iv e n in S ection 1.2-2.
T he e ffe c tiv e p o te n tia l is n e a rly diagonal in the c h o se n (v,R)nLN basis.
T h e re fo re , firs t-o rd e r c o n trib u tio n s to the en erg y a re th e dom inant
c o n trib u tio n to the fin e stru c tu re that sp lits eac h n le v e l. The a d d itional
s e c o n d -o rd e r e n e rg ie s , as d e fin e d in Eq. 1-17, are in c lu d e d for fu rth er
a c c u ra c y in d e sc rib in g the R ydb erg levels. T h ese c o n tr ib u tio n s are
d o m in a te d by the q u a d ru p o le m ixing o f d iffe re n t R y d b e rg series bou nd to
(v ',R ') levels, w here AR = 0 ,± 2. This term , alo n g w ith th e two
p o la riz a tio n term s, is d e n o te d
E q (2).
in c lu d e the strong d ip o le m ix in g s,
In HD, the 2nd- o r d e r en erg ies also
E d (2 ),
due to the b ra c k e te d term in Eq.
1-11. (N ote th at th is te rm d o e s not c o n trib u te to E (1) sin c e it is pu rely
o ff-d ia g o n a l in th is b a s is a n d even then this term is o n ly p resent in
R y d b e rg levels o f H D .) T he sum o f these e ffe c ts, E (l) +
E q (2) + E d (2\
g ives the total fine s tr u c tu r e splittin g and is p re s e n te d in Fig. 3-2 fo r the
n=9, L=5 levels b o u n d to R=1 H 2 + ion cores. T his f ig u r e show s the
re la tiv e c o n trib u tio n o f eac h fine structure sh ift and illu s tra te s th at all
th re e iso to p es have s im ila r fine structure. The z e r o th - o r d e r en ergies are
g iv en to show the sm a ll e ffe c ts o f the red u c ed m ass R y d b e rg co nstant
O R red).
73
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Fine structure energies for each isotope
(R = l) n=9, L=5
H 2___________ H D________________D 2
c CI)
C.
E(0)=
c (2)
—d )
C .Q
t
-•544
N=4
-002
~'370
004
-1354.41
491
N=5
_ (2 )
_ (2 )
C ,D
t,Q
-(1 )
H
*-564
023
0.00
—
.531
_ _
-.001
'3.8.°
.021
0.001
-360
-.002
—
-1354.54
-1354.60
-.028
003
-0.001
.481
526
All splittings are to scale and
energies are given in cm*'.
Fig. 3-2: Fine structure energies. The structure for all three isotopes is given.
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-.003
As m entioned a b o v e , the tr a n s itio n s b e tw ee n h y d r o g e n ic levels
b o u n d to various R le v e ls o n ly d iffe rs by the fin e s tr u c tu r e p resen t. T h e se
e n e rg y sp littin g s are a p p ro x im a te ly p ro p o rtio n a l to n ' 3, so th e fine
stru c tu r e fo r the u p p e r s ta te o f th e tra n s itio n ( n f) is a lm o s t n e g lig ib le .
S in ce all tra n sitio n s o c c u r n e a r th e sam e z e ro th - o rd e r e n e rg y and the
u p p e r sta te fine stru c tu re is “n e g lig ib le ” , a sp e c tru m o f R y d b erg
tr a n s itio n s is exp ected to b e a lm o s t a d ire c t m ap o f th e fin e stru c tu re for
th e lo w e r state. All R le v e ls w ill be p re s e n t, d is tr ib u te d aro u n d AE(0).
T he o b se rv e d tra n s itio n e n e r g ie s (AE) are re la te d to th e d iag o n a l, lo w er
sta te fin e structu re c o n tr ib u tio n s E ( I ) („*=9 or n«io) by
AE = AE(0) + E g , - E g - E g .
Eq. 3-1
F o r s im p lic ity in c o m p a rin g the d iffe re n t is o to p e s , a fin e stru c tu re
p a ra m e te r A, is d e fin e d to r e p r e s e n t th e c o rre c tio n s to th e en erg y le v e ls
due to th e p e rtu rb in g p o te n tia l. The d e fin itio n
A = A E -A E (0)
Eq- 3-2
is illu s tr a te d in Fig. 3-3 a n d u se d la te r fo r p lo ttin g th e d iff e re n t sp e ctra .
T he tra n sitio n s tre n g th s a re g o v e rn e d by e le c tr ic - d ip o le s e le c tio n
ru le s and m atrix e le m e n ts. T he s tro n g e st a llo w e d lin e s are fo r those w ith
A L=A N =+1. O ther t r a n s it i o n s are allo w ed w ith (A L = + 1, AN=0) and
(A L =A N =-1), but are e x p e c te d to be an o rd e r o f m a g n itu d e w eak er in m ost
c a se s. As already n o ted , o n ly tr a n s itio n s o b e y in g A v=A R =0 are a llow ed .
T r a n s itio n strengths w ill a ls o be a ffe c te d by th e p o p u la tio n s o f the
d if f e r e n t lev els and w ill be d is c u s s e d in m ore d e ta il in S e c tio n 3.4.
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
n —2.1
, cn c cm ',
-150.5
(Example energies are not to scale.)
k
- - ________ -
Transition energy AE is given by:
aF
A E <0)
\
E(,)_______
n=10
A E
= AE(0) + E(I)
(n'=27)- E(1)
E’(n=10) - EC)
c (n=l0)
A = AE - AE(0)
£a)
-1097 cm*1
Fig. 3-3: Transition energies. The expected transition energies are calculated and shown
in the figure. This can also be used to extract E(1).
76
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3.3 M easured o p tic a l spectra for H2, H D , and D2
3.3-1
D e ta ils o f the m ea su re m e n t: d a ta a c q u is itio n /c a lib r a tio n
This c o m p a ra tiv e study o f m o le c u la r R y d b e rg lev e ls in c lu d e d th r e e
d iffe re n t tra n s itio n s fo r each isotop e, n = 1 0 -2 7 , n = 9 -2 0 , and n = 9-1 7.
T hese h y d ro g en ic tr a n s it i o n energies w ere s e le c te d be ca u se th ey are all
near strong C 0 2 la s e r lin e s. Table 3-1 lists th e s p e c if ic AE(0) e n e rg ie s a n d
freq u e n cie s u se d fo r e a c h spectrum . W ith th e la s e r tu n e d to the
a p p ro p ria te lin e an d th e d e te cto r set as d e s c r ib e d in S ection 2 .4 -2 , th e
CEM signal w as m e a s u r e d as a fu n c tio n o f th e D o p p p le r shift v ia the
ro ta tio n stage a n g le . A t each in cre m e n t o f th e a n g le , the signal was
a v e ra g e d to im p ro v e th e sig n a l-to -n o is e ra tio . T h e sta g e was c o m p u te r
c o n tro lle d and m o v e d in the same d ire c tio n t h r o u g h o u t any g iv e n scan .
T his w as to e lim in a te a n y error due to n o n - r e p e a t a b il i ty (or b a c k la s h ) in
the stage p o sitio n .
Table 3-1: Hydrogenic transition energies rAE(0)1. The hydrogenic transition energies are
given for each isotope and for each transition. The reduced mass Rydberg constant is
also given in column 2. In addition, the chosen laser line is identified and the frequency
is given. The * marking the D2 n=9-17 transition notes that the 10R(18) laser line was
used for this scan instead o f the 10R(20) used for the others. All energies are in cm '1.
Isotope
h2
*red
109707.45
n = I 0-27
946.584
n = 9 -2 0
1080.144
n = 9 - l7
974.802
HD
1 0 9717.40
9 4 6 .670
1080.242
974.891
d2
1 0 9722.38
946.713
1080.291
974.935*
L aser
—
10P(16)
9 R (2 4 )
10R (20)
947.7420
108 1 .0 8 7 4
975.9304
77
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The a p p aren t la s e r fr e q u e n c y was D o p p le r tu n e d into reso n an c e
w ith d if f e r e n t fine s tru c tu re tra n s itio n s by c h a n g in g the in te rs e c tio n a n g le
(6 ) b e tw e e n the laser and th e n e u tra l R yd berg b e am . T he tun ing is a
c o n tin u o u s function o f the a n g le , lim ited o n ly by th e a n g u la r in cre m e n t
siz e. The D opp ler tu n ed fr e q u e n c y is g iv en by
v' - -
1 + PCosG
Eq. 3-3
w h ic h w as intro d u ced in S e c tio n 2.3. The in te r s e c tio n a n g le (0) is a
d ire c t fu n c tio n o f the m e a s u re d stag e angle (0<>bs)- T he d iffe re n c e
b e tw e e n in te rse ctio n a n g le a n d sta g e angle, a lth o u g h m in o r, was first
in tro d u c e d by S tu rru s17 a n d d is c u s s e d in m o re d e ta il in A p p e n d ix D. The
m o d e le d re la tio n , as d e s c r ib e d in th e a p p en d ix , w as g iv e n by
Eq. 3-4
w h e re the c o e ffic ie n ts w e re f itte d a g ain st m e a s u re d a to m ic sign als. E ach
m e a su re d spectrum was c a lib r a te d in sim ila r fa s h io n a n d , using the fitte d
c o e f f ic ie n ts and beam v e lo c ity ((3), the m e a su re d s ta g e a n g le was
c o n v e rte d into an ap p aren t fr e q u e n c y (v '), w h ic h d e te r m in e d AE fo r eac h
6. A su m m ary o f the m e a s u r e m e n t details fo r th e m o le c u la r sp e c tra is
p re s e n te d in T able 3-2. T he c a lib r a tio n n u m b e r re f e r s to T able D-2.
The u n c ertain ty in m e a s u r e d line p o s itio n s w a s d e p e n d e n t on both
th e c a lib r a tio n error and th e flu c tu a tio n o f th e la s e r f r e q u e n c y during a
sc an . T he c alib ra tio n e rro rs due to the fitte d c o e f f ic ie n ts gave an
u n c e r ta in ty o f a p p ro x im a te ly ± 0 .0 0 2 cm*1 a n d e rro rs in the fitted
78
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ve lo c itie s c o n tr ib u te —0.003 cm*1 to the u n c e rta in ty . In a d d itio n ,
v a ria tio n in th e a b so lu te stage angle ( - 0 .0 2 ° ) , as ob serv ed by the stag e
c o n tro lle r, g a v e an a d d itio n al ± 0 .0 0 2 cm*1 e rro r. The laser, w h ic h w as not
freq uen cy s ta b iliz e d during these m e a su re m e n ts , was e x p e c te d to d r if t by
at m ost ± 0.001 cm*1. In sum m ary, the u n c e r ta in ty in the c a lib ra te d line
p ositio n s was e x p e c te d to be ± 0 .0 0 4 cm*1.
Table 3-2: Laser spectroscopy details. Measurement details for each isotope and for each
o f the three measured spectra. The transition label is given along with the angular range
o f the scan, the step size, and the averaging time per point. The calibration number refers
to Table D-2.
C al. #
0.03°
T im e/p t.
(sec)
5
2
R eferen ce
labbook
PL J15, p .4
80°-120°
0.05°
1
1
PLJ14, p . 109
n = 9 -l7
80°-130°
0.05°
5
2
P L J14, p . 132
n = 1 0 -2 7
90°-130°
0.05°
20
4
P L J16, p .32
n= 9-20
78°-125°
0.05°
5
1
PL J14, p .80
n = 9 - 17
75°-135°
0.05°
10
1
PL J14, p .74
n = 1 0 -2 7
88°-150°
0.05°
20
3
PL J15, p . 134
n = 9-20
70°-130°
0.05°
5
1
PL J14, p .l 14
n = 9 - 17
40°-102°
0.05°
20
4
PL J15, p . 144
n = 10-27
Range
(degs)
80o-130o
n= 9 -2 0
L abel
h
2
HD
d
2
Step
A c o n v e n ie n t check on the a ssig n e d m e a su re m e n t u n c e rta in ty e x ists
in the H 2 s p e c tra . W hen m easu ring the H 2 tr a n s itio n s , the sw itc h in g
m agnet is set to s e le c t p a rticle s w ith a to m ic m ass o f 2. T h is m ass tw o
beam c o n ta in e d a sm a ll am ount o f re s id u a l a to m ic d eu teriu m , w h ic h has a
stro ng tr a n s itio n at e ac h o f the sam e h y d ro g e n ic energy p o sitio n s (A=0) as
H 2 (w ith in 0 .0 0 0 2 cm*1). Since th e re w as no a p p a re n t d iffe re n c e b e tw e e n
the two sy s te m s in the d etecto r, the a to m ic sig n a l w as o b se rv e d d u rin g
79
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
eac h o f the th ree H 2 sp e c tra . T he e x pected a to m ic lin e p ositio n w as
c a lc u la te d p re c ise ly an d co m p ared to each o f th e o b se rv e d p o sitio n s . The
a g re e m e n t was on the level o f the ex pected ± 0 .0 0 4 c m '1 u n c ertain ty .
3.3-2
P re s e n ta tio n o f the data: s p e c tra an d tables
Each m e a su re d sp e c tru m is presen ted at th e top o f each fig u re
( F ig s .3-4 to 3 -12). T h e CEM sig n al is p lo tte d as a fu n ctio n o f the lin e
p o s itio n A. The tra n s itio n s are id e n tifie d by n u m b e rs , which re fe r to
T ab les 3-3 to 3-11, d is c u s s e d belo w . Some w e a k e r a llo w ed tr a n s itio n s are
id e n tifie d and la b e le d w ith an “ a ” , in d icatin g an “ a lte r n a te ” u p p e r s ta te ,
such as (0,1)10G 4-(0,1)27H 4. T he strong tr a n s it i o n w ould be w ith A N =+1.
O nly L>4 levels w ere in c lu d e d in the tables.
A sim u la tio n o f the sp ectru m is p re s e n te d b e n e a th each p lo t. T h e
sim u la tio n s, used to te s t the a g re em e n t o f th e lin e p o sitio n s, w ere
c a lc u la te d using the p o la riz a tio n m odel p re s e n te d in C hapter 1 w ith th e
th e o re tic a l core p a ra m e te r s in A p p en d ix A. T h is m o d e l was e x p e c te d to
a c c u ra te ly p re d ic t the p o sitio n s o f the L>4 le v e ls . O n the o th er h a n d ,
tra n s itio n s w ith L=4 sh o w a s ig n ific a n t d e p e n d e n c e on the n e g le c te d
h ig h e r ord er term s in th e e ffe c tiv e p o te n tia l5, w h ic h are p ro p o rtio n a l to r ' 6
and r*7. The o b s e rv e d sy s te m a tic d isc rep a n cy in th e nG state tr a n s itio n s
w as on the o rd er o f 0.01 c m '1. The rela tiv e line s tre n g th s were also
m o d e le d , inclu ding s e v e ra l e ffe c ts due to level p o p u la tio n s and io n iz a tio n
b e h a v io r. This w ill be d isc u sse d in the fo llo w in g se c tio n . C o n trib u tio n s
from each R level a re g iv e n se p a ra te ly , at the b o tto m o f each plot. T h is
80
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
illu s tr a te s th e o v e rla p p in g o f l in e s , w h ich re s u lts in th e to ta l s im u la tio n .
E a c h o f th e R sim u la tio n s a re p lo tte d on the sam e r e la tiv e s c a le w ith in a
g iv e n fig u re , except w h e n n o te d d iffe re n tly (e.g. R = 2,3 in F ig. 3-6).
E ac h spectrum is fo llo w e d by a table o f e n e r g ie s and s tr e n g th s . An
id e n tif ie d state label is g iv en in the firs t colum n, b a s e d on th e lo w e r sta te
o f th e tra n s itio n and la b e le d
(R )L
n
-
T his a b b re v ia tio n re fe rs to the
(0 ,R )n L N -(0 ,R )n '(L + l)(n + d t r a n s it i o n , w ith n and n' g iv e n by the s p e c tru m
o f in te re s t. The se co n d c o lu m n lis ts the peak # th a t c o rr e s p o n d s to th a t
s p e c ific lab eled tra n s itio n . F o r e x a m p le , in the H 2 n = 10-27 s p e c tru m o f
F ig . 3 -4 , p e ak # 1 is id e n tif ie d as th e ( 0 , l ) 1 0 G 4 - ( 0 , l ) 2 7 H s t r a n s itio n . In
the c o rre sp o n d in g T ab le 3-3 , t h e (1)G4 state label ( c o lu m n 1) is r e la te d to
p e a k #1 (co lu m n 2). T he n e x t c o lu m n s give the o b s e r v e d an d c a lc u la te d
e n e r g ie s , fo llow ed by the d i f f e r e n c e betw een the tw o . T he o b s e r v e d an d
c a lc u la te d stre n g th s a re g iv e n n e x t. An a ste risk on th e o b s e r v e d s tre n g th
r e p r e s e n ts an id en tifie d line t h a t c o n s is ts o f at le a s t tw o u n r e s o lv e d
t r a n s itio n s , so the re p o rte d s t r e n g t h is the to ta l o f e a c h . W h en a n a ly z in g
the Aobs-Acic, b lend in g o f th e lin e s m u st be c o n s id e re d , sin c e it e ff e c ts the
r e p o r te d A0bs- The last c o lu m n s in e a c h table give th e e n e rg y
c o n tr ib u tio n s that are u se d to c a l c u l a t e the to ta l tr a n s it i o n e n e rg y .
81
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H2 n = 10 - 27
ato m ic I) line
IOt I’
Possibly
(0)F,
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A (Ali""=946.584 cm ’)
Fir. 3-4: H, n= 10-27 laser spectrum. The CliM signal is plnlled as a function of calibrated A (cm
All numbered lines are identified in Table 3-3.
A simulation, based on calculated energies and estimated strengths, is shown below the data, lor R=0 and R=l levels only.
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Table 3-5: H2 n=9-17 tabulated energies and strengths. Observed transitions are labeled according to the lower state (R)LN quantum
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Some R=4 lines are identified in Table 3-9; however, there are no R=3 lines seen.
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3.4 D iscu ssion o f th e m easured sp ectr a
T his se c tio n w ill d isc u ss the m e a su re d s p e c tra w ith th e in te n tio n o f
le a v in g the re a d e r w ith c o n fid e n ce in th e lin e id e n tific a tio n s and a n
u n d e rsta n d in g o f th e d iffe re n c e s b e tw ee n th e s p e c tra o f the th re e d if f e r e n t
iso to p e s . Two g e n e ra l to p ic s w ill be c o n s id e re d . T he e x p ec te d , sm a ll
d iffe re n c e s in th e R v d b e re fine s tru c tu re w ill be d isc u sse d firs t. S e c o n d ,
the re la tiv e tra n s itio n stre n g th s w ill be d is c u s s e d , in c lu d in g e ffe c ts d u e
b o th to p o p u latio n o f d iffe re n t lev e ls an d to th e ir s ta b ility a g a in st
ra d ia tiv e and n o n -ra d ia tiv e decay. T h ese tw o e ffe c ts e x p la in a lm o s t all
th e d iffe re n c e s b e tw e e n th e o b serv ed s p e c tra . T he d isc u ssio n o f th e
re la tiv e stre n g th s w ill fo c u s m ainly on b u ild in g a m o d el to s im u la te th e
lin e in te n sitie s in th e m e a su re d sp e c tra .
3.4-1
Is o to n ic v a ria tio n s in th e e x p e c te d R v d b e re fin e s tru c tu re
The m ost o b v io u s d iffe re n c e b e tw e e n th e th re e iso to p es o f H 2 + is
th e w id e v a ria tio n in ro ta tio n a l and v ib ra tio n a l e n e rg ie s o f th e fre e io n .
A lth o u g h the in te rn u c le a r p o te n tia l is id e n tic a l fo r th e th ree io n s, th e
d iffe re n t n u c le ar m a sse s le a d to a d iffe re n t re d u c e d m ass (p.) fo r th e
n u c le a r m otion in e a c h iso to p e .
—
p
M,
h
E q . 3 -5
M2
^
T he ro ta tio n a l an d v ib ra tio n a l e n e rg ie s are o f th e com m on fo rm 37,
R~
ft2 R(R + l)
,
2W '
—BR(R +1)
Eq. 3-6
112
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and
w h ere (v ,R ) are the v ib ra tio n a l and ro ta tio n a l quantum n u m b e rs an d p is
th e in te rn u c le a r se p a ra tio n . A n illu s tra tio n o f the ro -v ib ra tio n a l s tru c tu re
is g iv en in Fig. 3-13, and v a lu e s for B and vo, for e ach o f th e th re e
iso to p e s, are given in T ab le 3 -1 2 . T he v a lu e s in th is ta b le co m e fro m th e
V(p)
v=2 —
v = lv= 0
R=2
x
R= 1
V *. 200x.............R=0
C 2B
Fig. 3-13: Rotational and vibrational energy levels for the core ion. The parameters B
and v0 are defined by Eq. 3-6. V(p) is the internuclear potential o f the ion.
ta b u la te d e n e rg ie s g iv en in A p p e n d ix A and Eq. 3-6. T he d e p e n d e n c e o f
th e se e n e rg ie s on th e re d u c e d m ass is sig n ific a n t. H o w ev er, sin c e th e
R y d b e rg sp e c tra p re se n te d in S e c tio n 3.3 a ll obey th e s e le c tio n ru le s
113
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Av = 0; AR = 0 ,
th e se core en erg y d iffe re n c e s do not lead to any d ire c t e ffe cts o n th e
o b se rv e d R ydberg s tru c tu re .
Table 3-12: Ion core dependence on mass. The ratio o f reduced mass to proton mass is
given in the first row. The other two rows give the rotational and vibrational constants
for each isotope. These values are based on Eq. 3-6 and the energy levels given in
Appendix A.
HDT
H r
d 2+
p /m p
0 .5000
0 .6 6 6 6
0.9995
B (c m '')
29 .1 2
21.93
14.70
v 0 (cm *‘)
2191.13
1912.99
1577.15
The d iffe re n c e s in to ta l c o re ion m ass do h av e two n o tic e a b le
e ffe c ts on th e sp e c tra . F irst, the to tal core m ass m o d ifies th e e le c tro n ic
e n e rg y lev els th ro u g h th e re q u ire d red u ced -m ass R ydberg c o n sta n t; th u s,
m o d ify in g the h y d ro g e n ic tra n s itio n e n erg ies, w h ic h vary by <
0 .2
cm ’ 1
b e tw e e n each iso to p e . T hese v alu es w ere g iv e n in T able 3-1. E a c h o f th e
m easu red sp e c tra w as p lo tte d as a fu n ctio n o f A, d e fin e d by Eq. 3-2,
w h ich a lread y has th e se c a lc u la te d values o f AE(0) su b tra c te d fro m th e
m ea su re d freq u en cy (v ') g iv en by Eq. 3-3. In th is w ay, the ta b u la te d A0bs
v a lu e s for th e R y d b erg tra n s itio n s have a lre a d y acc o u n te d for th e s e m ass
d iffe re n c e s.
The seco n d e ffe c t o f the co re ion m ass is to a lte r the ra d ia l sc a le o f
th e R ydberg e le c tro n w av e fu n c tio n . T his in tu rn a ffe c ts th e ra d ia l
114
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e x p e c ta tio n v alu es w hen c a lc u la tin g th e fine s tru c tu re . T he c o rre c t ra d ia l
e x p e c ta tio n v alu es fo r ion m a ss M | (and e le c tro n m a ss m e) are fo u n d from
the in fin ite n u c le a r m ass v a lu e s by th e su b s titu tio n
M ore su b tle d iffe re n c e s b e tw e e n the th re e is o to p e s re s u lt fro m th e
e ffe c ts o f n u clear m ass on th e v ib ra tio n a l w ave fu n c tio n s . T h is o c c u rs in
sp ite o f the fact th a t the in te rn u c le a r p o te n tia l is id e n tic a l fo r th e th re e
io n s. I f the in te rn u c le a r p o te n tia l w ere p u rely h a rm o n ic , th e n th e
d iffe re n c e w ould be so le ly in th e sp a tia l w id th o f th e v ib ra tio n a l
w av efu n c tio n , w h ich fo r a sim p le harm onic o s c illa to r is p ro p o rtio n a l to
p*1/4. T his w ould p re d ic t a 2 0 % in c re a se in th e w id th o f th e fu n c tio n fo r
H 2 + as com pared to D 2 +. T h e a c tu a l p o te n tia l, h o w e v e r, is q u ite
a n h arm o n ic and the a d d itio n a l re s u lt is an in c re a s e in th e a v e ra g e
in te rn u c le a r se p a ra tio n < p> . F o r ex am p le, p re c ise c a lc u la tio n s 3* o f <p>
fo r th e th ree ions in the R =0 s ta te g iv e the re s u lts in T a b le 3 -1 3 . T h en ,
sin ce th e core p ro p e rtie s Q, a s, and a t also d e p en d o n p, th e re is a
c o rre sp o n d in g ch an g e in e a c h o f th e s e ion p ro p e rtie s .
A special e ffe c t is o b s e rv e d in the q u a d ru p o le m o m e n t o f H D +. I f
<Q>oo w ere com puted a b o u t th e c e n te r o f th e in te r n u c le a r a x is , th e c e n te r
o f c h arg e (CC) o f th e ion, it w o u ld be 1.6143 e a 02, a b o u t m id w ay b e tw ee n
the n u c le i in H 2 + and D 2 +. H o w e v e r, th e a p p ro p ria te q u a d ru p o le m o m en t
is co m p u ted ab o u t th e c e n te r o f m ass (CM ) o f th e io n . It can be sho w n
1 15
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Table 3-13: Dependence o f core parameters on mass. The calculated values o f the (0,0)
ion core parameters show significant dependence on the ion mass. A more complete
tabulation o f these parameters is given in Appendix A.
(v = 0 ,R = 0 ) ion
param s.
<P> (a 0)
h 2+
HD+
2.0634
2 .0 5 4 4
2.0438
<Q>oo (e a Q2)
1.6390
1.7315*
1.6061
< a s>oo (a G3)
3.1675
3 .1 1 9 0
3.0712
< a t>oo (a c3)
3.995
3 .885
3.777
d
2+
* <Q>oo about the center o f charge is 1.6143, by Eq. 3-7.
th a t
Q cm = Qcc + e(Az)2,
Eq. 3-7
w h ere Az is the d is ta n c e b etw een CM a n d C C , o r a b o u t < p > / 6 .
C o n se q u e n tly , th e q u a d ru p o le m om ent o f H D + is a b o u t 7.3% la rg e r th a n
e ith e r H 2 + or D 2 +T hese su b tle d iffe re n c e s in th e io n c o re p a ra m e te rs a ffe c t the
s p e c tra enough to be e v id e n t in the o b s e rv e d s tru c tu re . T h is can be se en
by c o n sid e rin g th e o b se rv e d p o sitio n s o f s e v e ra l p a irs o f lin e s, each
sh o w in g large q u a d ru p o le sp littin g . T he fo llo w in g tra n s itio n s ,
a)
( l) n G 4 - ( l) n G s
b)
( l ) n H s - (l)n H fi
c)
(l)n l6 - ( l ) n l 7
116
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w e re u se d . From th e ta b u la te d lin e p o s itio n s (A 0 bs) and the c a lc u la te d
E n (1) and E„(2), we can e stim a te the fir s t o rd e r fine stru c tu re e n e rg y (E n(1))
u sin g T a b le s 3-3 to 3 -1 1 . T he a p p lie d c o n v e n tio n is d e fin e d by
E«>— Z L .+ E W - E ® .
Eq. 3-8
T h en , as in tro d u c ed in C h a p te r I, we e x p e c t
E (n0 = A 0 (nL) + A 2 (nL)(P2 (cos0)),
Eq. 3-9
so w e can estim ate th e v a lu e o f A 2 by c a lc u la tin g the a p p ro p ria te <P 2 >
c o e ffic ie n t fo r each term and su b tra c tin g th e values o f Eq. 3-8 fo r e ac h
p a ir o f in te rv a ls. T he A 2 c o e ffic ie n t is th e n com pared to th e “ e x a c t” form
A 2 (nL) = - e Q ( r '3)|iL “ ^ “a ,(r~l )nL.
Eq. 3-10
S c a lin g th e e stim ate d A 2 v a lu e by th e < r‘3> m atrix e le m e n t g iv e s a v a lu e
a p p ro x im a te ly equal to th e q u a d ru p o le m o m en t.
A 2 (nL)
e2
(r ~4 ) 1j.
c
, ,,
' ( > - i =eQ +3 “ ' { a -
'
T ab le 3-1 4 show s the e stim a te d v a lu e s o f —A 2 / < r '3> in fe rre d fro m th e n in e
s p e c tra o f S ectio n 3-3 fo r th e (R = 1 )G , H, and I states.
A d ire c t c o m p ariso n o f th e se sc a le d in te rv a ls in d ic a te s th a t th e
q u a d ru p o le m om ent o f D 2 + is s lig h tly sm a lle r th an for H 2 +, w h ile th a t o f
H D + is s u b s ta n tia lly la rg e r. T h is is e v en c le a re r if the a v e ra g e o f th e
th re e v a lu e s o f com m on L is p lo tte d a g a in s t the av erag e o f th e ra d ia l
m a trix e le m e n t ratio (th is p lo ts the a v e ra g e o f Eq. 3 -1 1 ), as sh o w n in F ig.
3 -1 4 . T he lin e a r slo p e s a p p e a r a p p ro x im a te ly equal, as e x p e c te d fo r th e
117
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Table 3-14: Scaled tensor structure factors (A 2 ) for each isotope and spectrum.
Transition
-A 2 (nL)/<r'3>nL (eao2)
G (L=4)
H (L=5)
I(L = 6)
n= 10-27
1.738
1.705
1.674
n=9-20
1.737
1.708
1.684
n=9-17
1.738
1.702
1 .6 8 8
n= 10-27
1.848
1.805
1.781
n=9-20
1.838
1.789
1.765
n=9-17
1.857
1.817
1.830
n= 10-27
1.705
1.667
1.643
n=9-20
1.697
1.694
1.645
n=9-17
1.700
1.667
1.620
h2
HD
d2
p o la riz a b ilitie s , and the e x tra p o la te d in te rc e p ts are c o n siste n t w ith th e
th e o re tic a l q u a d ru p o le m om ents (sh o w n by o p en c irc le s) fo r th e th re e
iso to p e s. The in cre ase in the H D + q u a d ru p o le m om ent, due to th e o ffs e t
b e tw ee n the c e n te r o f m ass and c e n te r o f c h a rg e is very c le a r from th e
d ata. T he p re d ic te d d iffe re n c e s in th e o th e r c o re p ro p e rtie s ( a s and a t)
are too sm all to be c o n clu siv ely re s o lv e d fro m the o p tic al sp e c tra .
The fin al d iffe re n c e b etw een th e iso to p e s is the sm all, se co n d o rd e r
en erg y sh ifts due to m ixings b e tw e e n d iffe re n t R ydberg le v e ls. T h ese
m ix in g s d ep en d on the p o sitio n s o f th e p e rtu rb in g lev e ls, w h ich in tu rn
d ep en d d ire c tly on the iso to p e -d e p e n d e n t, ro -v ib ra tio n a l e n e rg ie s . T h is
iso to p ic ch an g e in the re la tiv e p o s itio n o f p e rtu rb in g lev e ls w ill
118
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1.90
G(L=4)
1.85
H (L=5)
1.80
00
>
eo
A
HD
1.75
La
V
1-70
I
1.65
1.60
1.55
0
1
2
3
4
5
6
7
8
(<r*4> /< r'3> )avg * 1 0 '2
Fig. 3-14: Extrapolation o f the quadrupole moments for each isotope. The inferred
energy splittings are used to estimate A2 for each pair o f observed nL lines. These scaled
values are plotted as a function o f the matrix element ratio. The intercept gives the
quadrupole moment and the slope is 1/3 the tensor dipole polarizability. The open circles
represent the theoretical estimates o f <Q>oo as given in Table 3-13.
s ig n ific a n tly a ffe c t th e E (2) e n erg y c o n trib u tio n an d , in som e c a se s, e v e n
ch ange th e d ire c tio n o f th e p e rtu rb a tio n . T h is c a n be seen by r e fe re n c e to
F ig . 3-1. T he q u a d ru p o le m ix in g (A R =2), w ill m ix the (0 ,0 )n = 1 0 le v e l
w ith both th e (0 ,2 ) n = 9 an d n= 10 le v e ls. S in c e th e s e tw o le v e ls are
ev en ly sp a ce d a ro u n d th e ( 0 , 0 ) 1 0 level in H 2 , th e n e t c o n trib u tio n w ill be
a lm o st z ero . H o w e v e r, in D 2 th e ro ta tio n a l s p a c in g is sm a lle r, b rin g in g
th e (0 ,2 )1 0 le v e l c lo s e r so th e n e t p e rtu rb a tio n b e co m es n e g a tiv e . T h is is
re fle c te d in th e ta b u la te d E (2) valu es sh o w n in T a b le s 3-3 an d 3-9.
119
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A m o re s ig n ific a n t c o n trib u tio n to the R ydberg s e rie s p e rtu rb a tio n s
co m es from th e p e rm a n e n t d ip o le m om ent in H D . T he d ip o le m ixin g s
o b e y the se le c tio n ru le s A R =±1, A L =±1, and A N =±1. T h e se m ix in g s are
m u ch stro n g e r th a n th e q u a d ru p o le m ix in g s d e sc rib e d a b o v e , b ecau se th ey
a re due to a lo w e r m u ltip o le m o m en t. The to ta l e n erg y s h if t on a g iven
R y dberg lev e l is on th e o rd e r o f 0.02 cm*1. T hese p e rtu rb a tio n s are liste d
in th e ta b le s o f S e c tio n 3.3 . T he la rg e st e ffe c t o f th e d ip o le s h ifts is seen
in th e n=10 le v e ls (L = 4 ,5 ,6 ) w h ere a n ear d e g en e ra c y o c c u rs b etw een the
(0 ,0 )n = 1 0 and the ( l , l ) n = 6 R y d b e rg lev e ls. T he (0 ,0 ) - (1 ,1 ) en erg y
s p littin g is c a lc u la te d to be 1954.85 cm*1. T he h y d ro g e n ic n= 10 lev el is
b o u n d by -1 0 9 7 .1 7 c m * 1 and th e n =
6
level is bound by -3 0 4 7 .7 1 cm*1.
T he re s u lt is a d iffe re n c e in z e ro th -o rd e r e n e rg ie s o f o n ly 4 .3 2 cm*1. T h is
is c o m p a rab le to th e fin e s tru c tu re o f the (1 ,1 )6 L n le v e ls , w h ic h m u st be
in c lu d e d in th e e n e rg y d e n o m in a to r to give re lia b le e s tim a te s o f th e
m ix in g and lev e l s h ifts . S u ch a c a lc u la tio n g iv es a d ip o le s h ift o f -0.4
c m * 1 for th e (0)1 OHs lev e l in H D , w h ic h is c le a rly o b se rv e d in the
sp e ctru m o f F ig. 3 -7 . N o te th a t th e (0 ,0 )n = 1 0 le v e ls w ith L >7 a re not
a ffe c te d , sin c e th e re a re no (1 ,1 )6 L n lev els w h ich can m ix w ith th ese
sta te s .
Id e n tific a tio n o f th e (0 ,0 )n = 1 0 sp e c tra l lin e s in HD w as at first
q u e s tio n a b le , w hen b a se d s o le ly on th e line p o s itio n s . A t th e end o f
S e c tio n 3.4, th e io n iz a tio n b e h a v io r o f the (0 ,R ) n = 1 0 -2 7 tra n s itio n s in
HD w ill be sh o w n to d e p e n d on th e sp e c ific R, due to s tro n g d ip o le
120
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m ix in g s w ith n e ig h b o rin g R y d b erg le v e ls . This se n sitiv e io n iz a tio n
b e h a v io r gave im p o rtan t a d d itio n a l in fo rm a tio n and w as u se d to p o s itiv e ly
id e n tify th ese p e rtu rb e d R =0 le v e ls in HD.
The R ydberg fin e stru c tu re o b se rv e d in the th ree d iffe re n t iso to p e s
w as a cc u ra te ly d e scrib e d by th e p o la riz a tio n m odel d e fin e d in C h a p te r 1.
T he d iffe re n c e s d isc u sse d a b o v e, v a ria tio n s in the co re p a ra m e te rs and the
2
nd-o rd e r co rre ctio n s, w ere a lso in c lu d e d w hen c a lc u la tin g th e o re tic a l
p o sitio n s for each o f th e e x p e c te d tra n s itio n s . As liste d in th e ta b le s o f
th e p re v io u s sectio n , the o b se rv e d stru c tu re m atched w ell w ith th e
c a lc u la tio n s , w hich lends c re d ib ility to th e line id e n tific a tio n s . H o w ev e r,
th e re la tiv e stren g th s sh o u ld also be d e sc rib e d in o rd er to h a v e a d d itio n a l
in fo rm a tio n on w hich to b a se th e id e n tific a tio n s . T hese s tre n g th s w ill be
d isc u sse d next.
3 .4 -2
Iso to p ic d iffe re n c e s in th e re la tiv e tra n s itio n s tre n g th s
U pon com parison, th e m ea su re d sp e c tra are o b v io u sly d iff e re n t
b e tw e e n th e isotopes (e.g . th e n = 1 0 -2 7 tra n s itio n s, F ig s. 3 -4 ,7 ,1 0 ),
h o w e v er, the d iffe re n c e s are m ain ly du e to v a ria tio n s in th e re la tiv e
tra n s itio n stren g th s. F or in s ta n c e , th e stru c tu re in all th re e sy s te m s is
q u ite sim ila r, but th e s trik in g d iffe re n c e b etw een the H 2 an d D 2 s p e c tra is
th e n u m b er o f lines seen in th e la tte r. T h is w ill be e x p la in e d by
c o n sid e rin g both the re la tiv e p o p u la tio n s o f d iffe re n t co re ro ta tio n a l
le v e ls (e .g . R = 1 co m p ared w ith R = 2) an d th e sta b ility o f th e u p p e r and
lo w e r sta te s o f the tra n s itio n .
121
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M any d if f e r e n t v ib ra tio n a l and ro ta tio n a l le v e ls a re p o p u la te d in the
ion source. T he v ib ra tio n a l p o p u latio n is o f little in te r e s t sin c e only
R ydberg le v e ls b o u n d to th e low est v=0 le v e l w ere o b s e rv e d , as d isc u sse d
later. The ro ta tio n a l le v e l p o p u latio n s are g o v e rn e d by a B o ltz m an n
d istrib u tio n ,
Eq. 3-12
w here E (R ) is th e ro ta tio n a l en erg y and kT is ta k e n to be room
te m p era tu re ( - 1 /4 0 eV o r 2 0 1 .6 c m '1). T h is d is trib u tio n fa v o rs the g ro u n d
ro ta tio n a l le v e l (R = 0 ) and fa lls o f f e x p o n e n tia lly fo r in c re a s in g R. T he
ty p ic al (2 R + 1 ) m u ltip lic ity fa c to r is a c c o u n te d fo r by th e m u ltip lic ity o f
th e d iffe re n t N s ta te s .
As d is c u s s e d in C h a p te r 1, p o p u la tio n o f th e r o ta tio n a l lev els is also
g o v erned by th e P a u li e x c lu sio n p rin c ip le , w h ich d e te rm in e s th e a llo w e d
c o u p lin g o f the n u c le a r sp in s in rela tio n to th e sy m m e try o f th e core
w a v efu n c tio n . T h is is d isc u sse d in d e ta il by L ib o ff25. In H 2 , w ith tw o
ferm ions ( I p= l / 2 ) , th e e v en R, sp a tia lly sy m m etric s ta te s m u st have a n ti­
sym m etric sp in s ta te s (1=0), w h ile the odd R s ta te s m u st h a v e 1=1.
C o n se q u e n tly , in H 2 th e odd R states are fa v o re d by a f a c to r o f 3:1, due to
the m u ltip lic ity (21+1) o f a llo w e d spin s ta te s. In D 2 , w ith tw o bosons
(Id = l), th e ev en R, s p a tia lly sym m etric sta te s m u st h a v e sy m m etric sp in
sta te s (1= 0,2), w h ile th e odd R states m ust h av e 1=1. C o n se q u e n tly , in D 2
th e even R s ta te s a re fa v o re d by 6:3. T here is no su c h sy m m e try e ffe c t in
HD. T his c o n trib u tio n a lo n e d e scrib e s th e la rg e s t d is c re p a n c y betw een
122
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th e H 2 and D 2 s p e c tra . S in ce th e R = 0,2 le v e ls are fa v o re d in D 2 , but
d isfa v o re d in H 2 , th e D 2 s p e c tra w ill have a g re a te r n u m b e r
p ro m in e n t
(o n e+ fiv e ) o f
tra n s itio n s fo r e ach L th a n for H 2 (th re e fo r e a c h L ). The
n u c le a r sp in s ta tis tic s are m o d eled by a d isc re te v a lu e d fu n c tio n d efin ed
as:
H2
HD
D2
I(R ) = 1 / 3 fo r e v en R, 1 fo r odd R
I(R ) = 1 fo r all ro ta tio n a l lev els
I(R ) = 1 fo r ev en R, 1/2 fo r odd R.
The to ta l s tre n g th d e p e n d e n c e fo r d iffe re n t c o re ro ta tio n a l lev els is
g iv en by the p ro d u c t o f the tw o te rm s, I(R )* B (R ). A c o m p a ris o n o f th is
c a lc u la te d m o d el w ith a c tu a l m e a su re d stre n g th s fo r e a c h iso to p e is show n
in F ig. 3-15. E ac h c o n trib u tio n is c a lc u la te d and th e n n o rm a liz e d to th e
o b serv ed R=1 s tre n g th s fro m th e ta b le s . T he la b e le d tra n s itio n s and peak
n um bers re fe r to th e s p e c tra p re s e n te d in th e p re v io u s s e c tio n . T here is
rea so n a b le a g re e m e n t b e tw e e n th e R=1 and R=2 le v e ls , sin c e th o se ch o sen
lin e s are w ell re s o lv e d fe a tu re s in th e sp e ctra . O n th e o th e r h an d , the R=0
lin e s are u n re s o lv e d in a ll th re e iso to p e s, so th e c a lc u la te d stre n g th o f the
( 2 ) 1 4 line m u st be in c lu d e d . T his a d d itio n a l c o n trib u tio n is g iv en by the
lig h te r sh ad ed p o rtio n in F ig . 3 -15.
The im p o rta n c e o f th is in te n s ity d e p en d e n ce on R is seen w hen
co m p arin g th e H 2 a n d D 2 s p e c tra (F ig . 3-4 an d 3 -1 0 ). A s m en tio n ed
ab o v e, the D 2 sp e c tru m c le a rly has m ore lin e s. T h is is a d ire c t
c o n seq u e n ce o f th e fa c t th a t each L lev el is s p lit in to 2R+1 d iffe re n t
123
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
le v e ls , lab eled by N . S in ce th e even R sta te s d o m in a te the D 2 sp ectru m ,
th e re are 5 lines fo r e a c h R =2 and 1 lin e fo r e ach R = 0, fo r each L . T h is is
in c o n tra s t to the 3 lin e s fo r R=1 that d o m in a te in th e H 2 spectrum . It is
th e n c le a r th at the D 2 sp e c tru m is e x p ected to h av e tw ic e as many stro n g
lin e s as the c o m p arab le sc an in H 2 -
R=1
Measured strength ( S ^ )
Calculated strength: B(R)*I(R)
R=0
R=0
R=0
■eeo
Tc5
op
R=2
33
R=2
I
R=1
I
R=2
bn
HD
n-*-I7
peaks #16,1,21
n-9-17
peaks #30,1,5
R=1
d2
n=10-27
peaks #27.1.4
Fig. 3-15: Relative populations of ion core rotational levels. For each isotope, the
observed and calculated strengths are compared for R=0, 1, and 2 levels with L=N=4.
The discrepancy with R = 0 is explained by the blended (2)L» in all three observed lines.
This additional R=2 contribution is shown by the lighter shaded bar that adds to the R=0
calculated contribution.
The line in te n s itie s also depend on L and w e re m o d eled by fittin g a
w e ll-re so lv e d set o f R=1 lin e s in the H 2 n = 1 0 -2 7 sp e c tru m . These
tr a n s itio n s in clu d ed L = 4 -9 and had e s s e n tia lly no b le n d e d com ponen ts to
1 24
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
co m p lica te the m e a su re d stre n g th s. T he c h o se n lin e s are show n in F ig . 316 along w ith a lin e re p re se n tin g the fitte d sh a p e w h ich is g iven by
f(L) = -0 .0 6 4 (L -6 )2 + 0.857.
T h is p ara b o lic sh a p e is cen tere d aro u n d the la rg e s t p eak d eterm in ed to be
th e (1)1 Ole sta te (w ith L = 6 ). Once fitte d , th is m o d el w as co m p ared to th e
an alo g o u s peaks o b se rv e d in HD, F ig. 3-7, an d w as seen to a g re e q u ite
w e ll, so no v a ria tio n s w ere m ade fo r th e d iffe re n t iso to p es. The m odel
l.o
f(L) = -0.064(L-6)2 + 0.857
measured spectrum (Fig. 3-4)
(1)10HS
0
) 101 ,
0.8
(1)10G,
0.6
0.4
02
0.0
-0.7
-0.6
-0.5
-0.4
-0.3
-0 2
0.0
Energy (cm*1)
Fig. 3-16: Fitted relative strength dependence on L. A set o f H2 R=1 lines, selected from
the n= 10-27 spectrum in Fig. 3-4, were fitted for their strength dependence on L. This
model was seen to match all three isotopes.
assu m ed th at the lin e stre n g th , w ith in a set o f R y d b e rg levels w ith a
sp e c ific L, w as p ro p o rtio n a l to the s ta tis tic a l w e ig h t (2N + 1). In o th e r
w ords the m odel w as m u ltip lie d by th e n o rm a liz e d s ta tistic a l w eig h t,
125
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2 N + 1 )/(2 L + 1 ). T he c o m p le te m odel u sed to p re d ic t tr a n s itio n stre n g th s
w as
Eq. 3-13
= exp
T he c a lc u la te d s tre n g th s w ere th e n n o rm aliz e d to a s in g le , p ro m in e n t
featu re in th e d a ta , ty p ic a lly the (1 )I 6 lin e . O nly the s tro n g e s t tra n s itio n s
sa tisfy in g AN=AL=+1 w ere in clu d e d .
The fin a l c o n s id e ra tio n fo r the re la tiv e tr a n s itio n stre n g th s is th e
s ta b ility o f th e R y d b e rg le v e ls in v o lv e d . F o r th e m e a su re m e n ts m ade
here, th e life tim e s o f th e n = 9 ,1 0 h igh-L R y d b e rg le v e ls had to be long
eno ugh to su rv iv e b e tw e e n th e Cs cell and th e laser. T h e C s v ap o r c e ll,
w hich p o p u la te d th e n = 9 ,1 0 le v e ls o f in te re s t, w as a b o u t 4 0 cm from th e
laser re g io n . T he m o le c u la r beam passed fro m the C s c e ll to th e laser
reg io n in < lp .se c . T he ra d ia tiv e life tim e s o f th e n = 9 ,1 0 le v e ls w ith L>4
a re long e n o u g h th a t a s ig n ific a n t p o p u la tio n re m a in e d in th e s e ex cite d
lev els by th e tim e th e b eam e n te re d the L IR . A ny n = 9 ,1 0 R ydberg
e le c tro n s b o u n d to v ib ra tio n a lly e x c ite d c o re s, h o w e v e r, h av e a n o th e r
p o ssib le decay p a th , v ib ra tio n a l a u to io n iz a tio n (e.g . ( l , l ) 1 0 G s -*■ ( 0 ,l) e G s
w ith e= 0 .1 3 5 eV ). T he life tim e fo r this p ro c e ss is a b o u t 1 n se c.
C o n se q u e n tly , no v ib ra tio n a lly e x c ite d R y d b e rg le v e ls w e re se en in th is
ex p erim e n t.
126
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A seco n d c o n d itio n is th a t th e u p p e r s ta te o f th e la se r tr a n s itio n
(n '= 1 7 ,2 0 , o r 27) s u rv iv e s to the R y d b erg d e te c to r. R a d ia tiv e d e c a y is n o t
a p ro b lem h ere (s in c e th e life tim e is oc n 3), b u t n o w ro ta tio n a l
a u to io n iz a tio n p la y s an im p o rta n t ro le. I f an u p p e r R y d b erg s ta te [e.g .
(0 ,2 )2 7 H s] lie s a b o v e th e io n iz a tio n th re s h o ld o f a n a d ja c e n t ro ta tio n a l
lev el [(0 ,0 )eH s)], th e R y d b erg m o le cu le m ay a u to io n iz e . In o th e r w o rd s,
a ro ta tio n a lly e x c ite d io n co re, w ith a bound R y d b e rg e le c tro n , c an g iv e
up som e its ro ta tio n a l e n e rg y to th e e le c tro n . T h e e le c tro n is th e n e je c te d
as a free e le c tro n . F ig s. 3-17, 3 -1 8 , and 3-19 illu s tr a te the e n e rg y le v e l
stru c tu re for eac h is o to p e and show w hich R y d b e rg le v e ls are sta b le fo r
eac h ro ta tio n a l le v e l. In e sse n c e , th is e ffe c t is a n o th e r re su lt o f th e m ass
d iffe re n c e s in th e is o to p e sin c e th e ro ta tio n a l s p littin g s becom e s m a lle r
w ith in c re a sin g m ass. T he net re s u lt is th a t th e D 2 le v e ls w ill h a v e a
g re a te r nu m b er o f sta b le ro ta tio n a l co re le v e ls fo r any giv en u p p e r n'
sta te . For e x a m p le , n '= 2 7 sta te s are only s ta b le fo r R =0 and 1 in H 2 , b u t
are sta b le fo r R = 0 ,l,2 ,3 in D 2 . T h is is an a d d itio n a l re a so n w hy th e
sp e ctru m show n fo r H 2 in F ig. 3-4 has fe w er lin e s th a n the D 2 sp e c tru m
sh o w n in F ig. 3-10.
127
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H 2 rotational energy levels
(autoionization behavior)
Rotational
energy
575.47
347.10
174.24
58.23
binding
energy
cm *1
v= 0
27
-150.5 n -2 7
0 .0
cm*
20
TT
-274.3 n - 2 0
-379.6 n —17
TT
The arrow represents rapid autoionization
due to AR = -2 relaxation o f the ion core.
(quadrupole mixing)
Fig. 3-17: Rotational energy level structure for H2 . The Rydberg levels bound to
rotationally excited molecular ions can autoionize to an adjacent (AR=2) rotational level.
The ion core gives up rotational energy to the Rydberg electron, ejecting it as a free
electron. The arrow represents possible autoionization for the given Rydberg level.
Stable levels include R=0,1 for n —27; R=0,l,2 for n —20; R=0,l,2,3 for n —17.
128
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
HD rotational energy levels
(autoionization behavior)
Rotational
energy
434.71
261.86
131.32
v= 0
binding
energy
cm ' 1
-150.5
n —27*
-274.3
n -20*
-379.5
n —17*
27
43.86
-i
0 . 0 cm
T7
20
20
17
17
The arrow represents rapid autoionization
due to AR = -2 relaxation o f the ion core.
(quadrupole mixing)
Fig. 3-18: Rotational energy level structure for HD. The rotational splittings are smaller
than in H2 , allowing more Rydberg levels to remain stable to autoionization. Stable
levels include R=0,l,2 for n -2 7 ; R=0,l,2,3 for n -2 0 ; R=0-4 for n -1 7 .
129
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D 2 rotational energy levels
(autoionization behavior)
Rotational
energy
292.16
175.76
bubM
88.05
v= 0
binding
energy
cm '
n ,f = 4 2
-150.5
n -2 7
20
-274.3
n-20«
20
-379.7
n -1 7 -
77
77
29.39
-l
0 . 0 cm
17
<The arrow represents rapid autoionization
due to AR = -2 relaxation o f the ion core.
(quadrupole mixing)
Fig. 3-19: Rotational energy level structure for D 2 . The rotational splittings are the
smallest in this system, allowing stability for almost all the levels shown. Only the R=4,
n —27 state is unstable. The dashed line between n -2 7 and n"=42 represents allowed
quadrupole mixing.
130
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In su m m ary , th e sim u la tio n s p re se n te d a lo n g w ith th e m easured
sp e c tra in th e p re v io u s se c tio n w ere c a lc u la te d b a se d on th e three m ain
e ffe c ts d isc u sse d a b o v e . F irst, the s tru c tu re , c a lc u la te d u sin g the
p o la riz a tio n m odel d e fin e d in C h ap ter 1, v a rie d w ith eac h iso to p e due to
th e su b tle d iffe re n c e s in th e ion core p a ra m e te rs (e .g . q u ad ru p o le
m o m en ts). S eco n d , th e re la tiv e stre n g th s w ere m o d eled u sin g the fo rm u la
in Eq. 3-13, S (R ,L ,N ). F in a lly , only R le v e ls w ith tra n s itio n s to stab le
u p p e r n' R y d b erg le v e ls w ere included in th e s im u la tio n s . The total
sim u la tio n s, w hen c o m p a re d to the m ea su re d s p e c tra , a g re e d quite w ell in
m o st cases, e s p e c ia lly fo r H 2 and D 2 . F o r e x a m p le , c o n sid e r the H 2 n = 9 20 sp ectru m in F ig . 3 -5 . T he sim ila rity b e tw e e n th e d a ta an d the
sim u la tio n is re m a rk a b le , ev en for th e w e a k e r R —2 le v e ls w hich are sta b le
in n '= 2 0 , b u t are n o t sta b le in n'= 27.
In th e m e a su re d sp e c tra o f HD, th e a g re e m e n t w ith th e sim u la tio n s
is n o t q u ite so good. T he d isc re p a n c y c an be se e n th e m o st in the n = 1027 sp ectru m (F ig . 3 -7 ). T he p re d ic te d s tre n g th s a g re e w ell for peaks #511; h o w ev er, the r e la te d lin e s w ith p o sitiv e A w e re o b se rv e d w ith w eak er
in te n sity th a n e x p e c te d . F or exam ple, th e th re e R = l , L =4 lin e s (peak s
# 5 ,2 5 , and 27) e x h ib it th e stro n g e st e ffe c t. T h is b e h a v io r seem s to
d e c re a se w ith lo w er n ', sin ce th e sam e b e h a v io r is less p ro n o u n ced in the
n = 9 -2 0 and n = 9 -1 7 s p e c tra fo r HD. In th e se n= 9 s p e c tra , it seem s th a t
o n ly the ( 1 )G 3 sta te is m issin g and all o th e rs a g re e w ith th e sim u la tio n
q u ite w ell. T h is d e v ia tio n from the s im u la tio n is n o t y e t u n d ersto o d .
131
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On the o th e r h an d , th e re is stro n g e v id e n ce th a t the p re s e n c e o f th e
d ip o le m om ent in HD d o e s e ffe c t th e io n iz a tio n b e h a v io r o f th e u p p e r
R y d b erg states and c a u se s o th e r le v e ls to be ab sen t in th e m e a su re d
s p e c tra . For ex am p le, F ig . 3-18 sh o w s th a t th e R=2 lev el o f HD has an
n '= 2 0 R ydberg lev el th a t is sta b le to a u to io n iz a tio n , y et th e R =2 le v e ls
a p p e a r to be a b se n t fro m th e m ea su re d sp e c tra in F ig. 3-8. T he fo llo w in g
d isc u ssio n w ill d e sc rib e the d e ta ils o f th e d ip o le m ix in g s and w ill
c o n c lu d e by sho w in g v a rio u s s p e c tra th a t c le a rly illu s tra te th e e ffe c t o f
th e HD d ip o le m om ent on m easu red tra n s itio n s .
A diagram o f th e HD ro ta tio n a l s p littin g s is show n a g a in in F ig . 320. In th is fig u re , th e “ s ta b le ” R y d b e rg le v e ls w ith n'= 20 or n '= 2 7 are
c o n n e c te d to th e a d ja c e n t ro ta tio n a l le v e ls (AR=1) th ro u g h a d ip o le m ix ed
R y d b erg level o f h ig h e r n ", sh o w n by th e dash ed lin e . The n' R y d b e rg
sta te s are ex p ected to io n iz e at lo w er e le c tric H elds, p ro p o rtio n a l to
( n '/n " ) 4 i f the c o u p lin g b e tw ee n th e tw o se rie s is su ffic ie n tly s tro n g .
O nce th e n" lev e ls a re e m p tie d by S ta rk io n iz a tio n , th e n' p o p u la tio n w ill
re fill th e em pty n" le v e ls and io n iz e . T he n e t e ffe c t is so m e tim e s re fe rre d
to as “fo rced a u to io n iz a tio n ” 39. I f th is e ffe c t is to be s ig n ific a n t in th e
io n iz a tio n b eh av io r o f th e R y d b erg s ta te s in th is e x p e rim e n t, it m u st o c c u r
on th e tim e sc ale in w h ic h th e m o le c u le s p a ss th ro u g h th e io n iz e r,
ty p ic a lly
-2 0
nsec.
132
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
HD rotational energy levels
(dipole mixing behavior)
i
Rotational
energy
434.71
m:
261.86
131.32
binding
energy
cm ' 1
-150.5
v= 0
n"=42
43.86
0 . 0 cm"
„ --
n =32*
n —27
n"=22»
-274.3
n —20-
_LL
-379.5
n'=17Represents dipole mixing which
causes “faster” ionization o f
the main n - 2 0 , 27 levels
(mixing requires AR=1, AL=1)
Fig. 3-20: Rotational energy level structure for HD. Dipole mixing o f Rydberg levels is
allowed for series with (AR=1). The dashed lines show Rydberg levels that will mix with
the n -2 0 and n —27 Rydberg levels. The mixed n" level is numbered also.
133
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T he stro n g d ip o le m ixing in HD le a d s to very d iffe re n t b e h a v io r
b e tw ee n R y d b e rg sta te s bound to d if fe re n t ro ta tio n a l le v e ls o f th e io n
co re. T h e n = 9 -2 0 s p e c tra o f Fig. 3-21 w e re tak en at v a rio u s s e ttin g s o f
the R y d e rg d e te c to r io n iz a tio n fie ld (E s). A t th e h ig h e st fie ld a p p lie d , the
“a to m ic -lik e ” R=0 le v e ls are the m o st p ro m in e n t fe a tu re s. T he n e x t lo w e r
field w as u se d fo r th e co m p lete scan sh o w n in Fig. 3-8 and th e sp e c tru m
show n h e re is la b e le d c o rre sp o n d in g ly . In th is scan, m o st o f th e R = 0 and
R=1 le v e ls a re o b se rv e d a t the sam e fie ld se ttin g . T e n ta tiv e la b e ls (+ )
m ark th e R = 2 tra n s itio n s based on th e s p e c tra show n b elo w . T h e tw o
s p e c tra a re fo r lo w e r s e ttin g s o f th e d e te c to r field . A t th e lo w e s t s e ttin g ,
2063 V /cm , o n ly le v e ls co m p a rab le to n '= 2 7 are e x p ected to io n iz e . S in ce
o n ly R = 2, n '= 2 0 le v e ls a re o b se rv ed , th is seem s c o n siste n t w ith th e
e x p e c te d m ix in g o f (2 )n '= 2 0 lev els w ith ( l) n " = 2 4 le v e ls and th e
io n iz a tio n at a fie ld sig n ific a n tly lo w e r th a n the e x p ected fie ld f o r n '= 2 0 ,
7000 V /cm , b a sed on th e y ield c u rv e s sh o w n in Fig. 2-10.
S im ila r b e h a v io r w as o b se rv ed in th e n= 10-27 sp e c tru m o f H D .
Fig. 3-2 2 sh o w s th e d e te c to r io n iz a tio n y ie ld curves fo r th re e d if f e r e n t
tr a n s itio n s , e a c h w ith a d iffe re n t R. A s e x p e c te d , the R=0 le v e l is a lm o st
id e n tic a l w ith th e a to m ic n= 10-27 y ie ld c u rv e show n in F ig. 2 -9 ,1 0 . T he
R=1 an d R = 2 sig n a ls sh o w s ig n ific a n tly d iffe re n t b e h a v io r, b u t c o n s is te n t
w ith th e e x p e c te d “ fo rc e d a u to io n iz a tio n ” b e h av io r. T his is e x p la in e d by
the (2 )n '= 2 7 m ix in g w ith the (l) n " = 4 2 a n d th e ( l)n '= 2 7 m ix in g w ith th e
134
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
HD n = 9 - 20
i
i
|
i
i
r
r
0.5
R=0 lines are most prominent
Es = 10313 V/cm
0.4
0.3
0.2
+ denotes identifiable R-2 transitions
a
1C
6
0.1
i
u>
E. = 7500 V/cm
12
14
13
II
5
16
.5
.
+ il + A +
.§> oo
s
U4
<-> 0 .0
(ft
R=l lines are noticeable
-
E =4688 V/cm
0.2
-0.3
R=2 lines dominate
E =2063 V/cm
-0.4
I
-1.0
I
L
J
-0.5
L
J
0.0
I ..I
I
0.5
I
I
1.0
Transition energy (minus E<0-1080.242 cm'1)
Fig. 3-21: HP 9-20 laser spectra vs. ionization field (Es). The HD 9-20 transitions were measured at different ionization fields.
H D n = 1 0 - 2 7 y ie l d c u r v e s
1.0
;------------------------------
1
0— R=0 vield curve
R=1 '
c
R=2
e xpe ct ed ion izat io n field i n " )
0. 0
0
500
:
:
1000
1500
1
2000
:
2500
1-------------------------------- ^
3000
3500
4000
—
= -----------------
4500
5000
D e t e c t o r Ionization Field (V/cm)
Fig. 3-22: HD n’=27 detector io n izatio n vield curves. The ionization dependence is seen
to.vary dramatically with rotational levels in n’=27 Rydberg levels of HD. These n =27
levels will be mixed with other adjacent Rydberg levels, by the dipole moment. The
adjacent n” Rydberg levels are noted above each curve and the expected ionization field
is marked. The expected fields are based on Fig. 2-9 and the assumption that the peak
n=27 signal occurs at -2000 V/cm (roughly l/5n4). For these measurements, see PLJ16.
p.8,21,27.
(0 )n "= 3 2 . T he n = 1 0 -2 7 sp e ctru m fro m Fig. 3-7 is sh o w n a g a in in F ig . 323, w ith an a d d itio n a l scan im m e d ia te ly below , tak en w ith th e d e te c to r
io n iz a tio n field set at E s=0. T his m ig h t seem lik e a u to io n iz a tio n , ex cep t
th a t the d e te c to r s till has som e “h ig h -fie ld ’* re g io n s. As d e s c rib e d in
C h a p te r 2, the d e te c to r has b o th a len s and a v e rtic a l d e fle c to r set to steer
th e sig n al into a C E M . T he v e rtic a l d e fle c tio n field is a lw a y s at le a st
1000 V/cm . T his is h ig h en ough to e x p la in the io n iz a tio n o f le v e ls m ixed
136
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
HD n = 10 - 27
0.20
0.15
0.10
E. = 1875 V/cm|
0.05
0.00
a -0.05
1
i
u>
^4
-0.10
to -0.15
2
w
transitions observed
with E = 0 V/cm
-0.20
-0.25
R=0
-0.30
-0.35
R=2
-0.40
-0.45
-0.4
-
0.2
0.0
0.2
0.4
0.6
Transition energy (minus E(O-946.670 cm'1)
Fig. 3-23: HD n= 10-27 R=2 spectrum. A spectrum taken at Vs=0 V shows a majority o f R=2 lines that ionize at very low fields. The
labeled peaks correspond to Fig. 3-7, which is repeated in the top part o f this figure for comparison. See labbook PLJ16.
in to n"=42 R y d b e rg le v e ls , as se en in the lo w er scan o f F ig . 3 -2 3 . T his
scan aids in id e n tify in g th e R=2 lev els in th e n = 1 0-27 HD sp ectru m .
As m en tio n e d a b o v e , the R =0, n=10 R yd b erg le v e ls in HD are
n e a rly d e g e n e ra te w ith th e ( l , l ) n =
6
R ydberg le v e ls. T he la rg e
p e rtu rb a tio n s c a u se d by the stro n g dipole s h ifts w ere e n o u g h to cause
se rio u s d o u b ts as to th e lin e p o sitio n s o f th e R =0, n = 1 0 , L = 4 ,5 ,6
tra n s itio n s. H o w ev er, as d e sc rib e d here, the io n iz a tio n b e h a v io r in HD
v a rie d d is tin c tly w ith R. D ue to th is e ffe ct, y ie ld c u rv e s ta k e n on each o f
th e m easu red tra n s itio n s w ere su ffic ie n t to id en tify the “ m is s in g ” R=0
lin e s (at le a st th e L=5 and
6
lin e s). T hese are la b e le d in th e ta b le s o f th e
p re v io u s se c tio n th a t a cco m p an y the m easured sp e ctra .
We in v e s tig a te d w h e th e r the same “fo rc e d a u to io n iz a tio n ” pro cess
o c c u rs due to th e q u a d ru p o le m ix in g p resen t in R=2 R y d b e rg le v e ls o f D 2 .
As seen in F ig . 3 -1 9 , th e R =2, n '= 2 7 R ydberg lev el can m ix in to the R=0
m a n ifo ld w ith n "= 4 2 . S ince n'= 2 7 ionizes at a p p ro x im a te ly 2000 V/cm ,
n "= 4 2 w ould be e x p e c te d to io n iz e at ab o u t 340 V /cm . I f th e (2)27L n
sta te s io n ize by “ fo rc e d a u to io n iz a tio n ”, th ey should io n iz e a t m uch lo w er
fie ld s than (0 )2 7 L n sta te s. For a v a rie ty o f tra n s itio n s , th e la s e r signal
w as m easu red as a fu n c tio n o f th e d e te cto r io n iz a tio n fie ld . T he atom ic
tra n s itio n w as u sed fo r a co m p a riso n . T hese m easu red s ig n a ls a re p lo tte d
in F ig . 3-24. T he D 2 sig n a ls w ere chosen to o b se rv e th e b e h a v io r o f R =0,
1, an d 2 tra n s itio n s . T h ere w as no n o tic ea b le ch an g e, a lth o u g h the R=2
w o u ld be e x p e c te d to p e ak at a six tim es lo w er field i f th e q u a d ru p o le
138
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m ix in g was s u ffic ie n t to cau se “ fo rced a u to io n iz a tio n ” . We c o n clu d ed
th a t the q u ad ru p o le m ix in g is not stro n g en ough to cau se n o tic ea b le
c h an g e s in the io n iz a tio n b e h av io r o f any o f th e th re e iso to p es.
Ionization yield curves for n=9-20 transitions
CO
op
c
«
— — atomic yield
D2 R=0
D2 R=1
" —
D, R=2
j _
*
«e
BO
CO
.
J
tr*
2000
4000
6000
8000
10000
12000
14000
Detector Ionization Field (V/cm)
Fig. 3-24: Ionization yield curves vs. rotational level (R). The detector ionization o f
n - 2 0 Rydberg levels in molecular deuterium is independent o f the rotational angular
momentum (R). The ionization yield curves are the same as for the atomic deuterium
signal. This shows that there is no quadrupole mixing o f the (R=2, n -2 7 ) level with the
adjacent (R=0, n"=42) level. See PLJ14 p. 119-121. Transitions: (0)16, ( 1 ) 1-9 , and (2)Ig
3.5 C onclusions o f the laser sp ectr o sco p y exp erim en t
The high-L R y d b e rg tra n s itio n s, m easu red fo r n = 1 0-27, n = 9 -2 0 , and
n = 9 -1 7 in all th ree iso to p e s , H 2 , HD, and D 2 , re p re s e n t th e first
sy ste m a tic study o f h ig h -L R ydberg stu c tu re in th e th re e iso to p es. The
m easu red R ydberg fin e stru c tu re w as v ery s im ila r in all th ree iso to p e s.
139
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The sm all d iffe re n c e s d u e to th e m ass d e p en d e n ce o f th e io n core
p a ra m eters, in p a r tic u la r th e q u ad ru p o le m om ent, w e re o b se rv e d (F ig . 314) and a g re e d w e ll w ith th e th e o re tic a l p a ra m e te rs, a t th e lev el o f
p re c isio n a v a ila b le in th is e x p erim e n t. T he m ea su re d s p e c tra show ed
ex trem e d iffe re n c e s in th e lin e in te n sitie s. A fte r c o n s id e rin g both the
re la tiv e p o p u la tio n s o f th e d iffe re n t core ro ta tio n a l le v e ls an d the
a u to io n iz a tio n b e h a v io r o f th e u p p er sta te s in th e d if f e r e n t iso to p e s,
sim u la tio n s w e re c o n s tru c te d to m odel th e o b se rv e d s tr u c tu r e . In m ost
cases, th e se s im u la tio n s a g re e d w ell w ith th e m e a su re d s p e c tra .
T he m o st o b v io u s d iffe re n c e betw een th e s p e c tra w as e x p la in e d by
the ro ta tio n a l le v e ls in H 2 an d D 2 , w hich e x h ib ite d v e ry d iffe re n t re la tiv e
stre n g th s, due to th e P a u li e x c lu sio n p rin c ip le . T h is g a v e a fav o re d sta tu s
to the odd R le v e ls in H 2 a n d e v en R lev els in D 2 . T h e n e t re s u lt w as th a t
the D 2 sp e c tra w e re d o m in a te d by the n u m ero u s R =2 lin e s . A n a d d itio n a l
featu re o b se rv e d in th e s p e c tra w as the io n iz a tio n b e h a v io r o f HD. The
d ip o le m ix in g o f a d ja c e n t R y d b erg levels w as stro n g e n o u g h to m ark ed ly
e ffe ct th e S ta rk io n iz a tio n . T h is b eh av io r a id e d in th e id e n tific a tio n o f
the o b se rv ed lin e s .
In c o n c lu s io n , th e c o m p lic a te d R ydberg fin e s tr u c tu r e fo r H 2 , HD,
and D 2 w as m e a su re d . S im u la tio n s agree q u ite w ell w ith th e m easu red
line p o sitio n s a n d , in m o st c a se s, the re la tiv e in te n s itie s . T h ese
sim u la tio n s, as d e s c rib e d in th e p rev io u s se c tio n , w ere b a s e d on:
140
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a) th e s tru c tu re p re d ic te d by the p o la riz a tio n m o d e l, in clu d in g E (2)
c o n trib u tio n s and sm all d iffe re n c e s in th e c o re p a ra m e te rs
b) th e re la tiv e p o p u la tio n s o f d iffe re n t ro ta tio n a l le v e ls in the io n
c o re (a n d d iffe re n t L R ydberg le v e ls)
c) th e s ta b ility o f th e up p er n' R ydberg le v e ls .
T he good a g re e m e n t b e tw ee n the sim u la ted and m e a su re d sp e c tra show s
th a t all th re e sy s te m s (H 2 , HD, and D 2 ) c an be a c c u ra te ly d e sc rib e d by th e
lo n g -ran g e p o la r iz a tio n m odel and also g iv es c o n fid e n c e in th e line
id e n tific a tio n s. T h e se id e n tific a tio n s are im p o rta n t fo r p la n n in g fu tu re
m ea su re m e n ts w ith th e h ig h e r p re c isio n m ic ro w a v e te c h n iq u e d e sc rib e d in
C h ap ter 2. T h e se s p e c tra w ere used to p la n th e e x p e rim e n t p rese n te d in
th e next c h a p te r.
As one fin a l p o in t, the io n iz a tio n b e h a v io r o b s e rv e d in the HD
sp e c tra has sh o w n stro n g d ep en d en ce on the p e rm a n e n t d ip o le m om ent o f
H D +. T his b e h a v io r a id e d in the id e n tific a tio n o f th e (0 ,0 ) n=10
tra n s itio n s (F ig . 3 -7 ), w h ic h a llo w ed the p la n n in g o f a m ic ro w a v e
reso n an ce e x p e rim e n t to m ea su re th ese s tro n g ly p e rtu rb e d R ydberg
in te rv a ls. A h ig h p re c is io n m easu rem en t, s im ila r to th e o n e p re se n te d in
th e next c h a p te r, sh o u ld g iv e a p rec ise re s u lt fo r th e d ip o le m om ent;
h o w ev er, th is w as o n ly m ade p o ssib le by the c o n fid e n t id e n tific a tio n o f
th e o p tical lin e s p re s e n te d here.
141
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C hapter 4
M icrow ave sp ectroscop y o f R=0, n=9,10 high-L
R ydberg sta tes o f H2 and D2
The o p tic a l s p e c tra o f C h a p te r 3 are o f h ig h e n o u g h p re c is io n to see
iso to p ic d iffe re n c e s in th e ion c o re q u ad ru p o le m o m en ts; h o w e v e r, the
h ig h e st p re c isio n stu d ie s o f h ig h -L R y d b erg le v e ls u se th e m ic ro w a v e
reso n an c e (R E S IS ) tec h n iq u e , d e sc rib e d in S ectio n 2 .6 . T h is m etho d
d ire c tly m easures the energy d iffe re n c e s betw een R y d b e rg le v e ls o f th e
sam e n. The o p tic a l sp e c tra set th e sta g e for a n u m b e r o f m e a su re m e n ts
o f th is type. The e x p erim e n t c h o se n fo r th is th e s is p ro je c t fo c u s e d on the
H 2 and D 2 sy stem s, and s p e c ific a lly th e (v = 0 ,R = 0 ) h ig h e s t-L R y d b erg
sta te s w ith n=9 and 10 fo r each iso to p e . T hese sy ste m s e x h ib it th e sim p le
“h e liu m -lik e ” fine stru c tu re , illu s tr a te d in Fig. 4 -1 . T he d o m in a n t
c o n trib u tio n to the fin e stru c tu re in te rv a ls o f th e se is o tro p ic sy ste m s is
the d ip o le p o la riz a tio n energy
E s - y a s( r 4).
Eq. 4-1
S ince the h ig h er o rd e r c o n trib u tio n s v ary sy s te m a tic a lly w ith L, it m ig h t
be p o ssib le to e x tra c t p rec ise v a lu e s o f the d ipole p o la riz a b ility o f
H 2 +(0 ,0 ) and D 2 +( 0 , 0 ) from m e a su re m e n ts o f th is ty p e.
142
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T h ese sim p le st o f m o le c u la r io n s, a lo n g w ith th e ir sim p le s t fin e
s tru c tu re le v e ls , are the m o st p ro m is in g c a n d id a te s for c o m p a ris o n w ith
c u rre n t th e o re tic a l tre a tm e n ts. E x is tin g c a lc u la tio n s o f th e p o la r iz a b ility
are n o t e x p e c te d to be m ore p re c ise th a n a b o u t
0
. 1 %, and c a lc u la tio n s o f
h ig h e r p re c is io n pose a s u b s ta n tia l c h a lle n g e sin c e they m u st in c o rp o ra te
th e “n o n - a d ia b a tic ” c o rre c tio n s to th e H 2 + w a v e fu n c tio n . S tu d ie s in b o th
iso to p e s w ill be v alu ab le sin c e th e ra tio o f e le c tro n /n u c le a r m a ss is th e
b a sic p a ra m e te r d e te rm in in g th e siz e o f th e s e c o rre c tio n s. In a d d itio n ,
m e a su re m e n ts in the (v = 0 ,R = 0 ) g ro u n d s ta te o f th e ion m ay b e o f g re a te r
in te re s t th a n p rev io u s m e a su re m e n ts 5 in th e f ir s t e x cite d ro ta tio n a l sta te
(v = 0 ,R = l) . T h is in te re st w o u ld be due to th e w id e r v a rie ty o f th e o re tic a l
te c h n iq u e s th a t e x ist fo r tre a tin g th e m o le c u la r ground s ta te . M e a s u rin g
th e h ig h e s t-L lev e ls is im p o rta n t sin c e th e e ffe c tiv e p o te n tia l b e c o m e s
in c re a s in g ly m ore acc u ra te w ith h ig h e r L s ta te s . In a d d itio n , by
m e a su rin g b o th n=9 and 10, th e re d u n d a n c y p ro v id e s a c h ec k o f th e
c ritic a l c a lc u la tio n s o f
2
nd-o rd e r e n erg y c o n trib u tio n s , up o n w h ic h th e
a n a ly sis o f the m ea su re m e n ts d e p e n d s.
4.1 T h e o r e tic a l fine str u c tu r e o f H 2 /D 2 high-L R y d b e r g sta te s
4.1-1
E le c tric fine s tru c tu re (E F S ) tra n s itio n s
T he th e o re tic a l d e s c rip tio n o f h ig h -L R y d b e rg m o le c u le s,
in tro d u c e d in C h a p te r 1, w as b a se d on th e d e fin itio n o f an e ffe c tiv e
p o te n tia l. T he R ydberg e le c tro n fin e s tru c tu re w as c a lc u la te d u sin g th e
143
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red u c ed R y d b erg w a v e fu n c tio n (E q. 1-12) and c h a ra c te riz e d by the
q u an tu m n u m b ers, (v ,R ) fo r the io n , (n ,L ) fo r the R y d b e rg e le c tro n , and
N = R +L for th e c o m b in e d a n g u la r m om entum th ro u g h lo n g -ra n g e
in te ra c tio n . T h is lo n g -ra n g e in te ra c tio n w as re p re s e n te d by an e ffe c tiv e
p o te n tia l, fo rm u la te d as a m u ltip o le e x p a n sio n o f th e C o u lo m b
H a m ilto n ia n , and u sed in a sta n d ard p e rtu rb a tiv e tre a tm e n t.
For the p re c is io n re q u ire d h e re , a m ore c o m p le te re p re s e n ta tio n o f
the e ffe c tiv e p o te n tia l w as n e c e ssa ry 5, ra th e r th an th e s im p lifie d
e x p re ssio n in E q. 1-11. H ig h er o rd e r term s w ere in c lu d e d , up to the r ' 7
d ep en d en ce; h o w e v e r, th e lev e ls c o n sid e re d in th is e x p e rim e n t w ere bound
to an iso tro p ic R = 0 ion c o re and had no d ia g o n a l te n s o r c o n trib u tio n s . In
o th e r w ords, th e se R y d b e rg en erg y le v e ls w ere d e te rm in e d to fir s t o rd er
by the sim ple e q u a tio n
E< ' > = - ^ ( r J^
+ B ,( r1^
+ B ,(r2- ’ ) lL
a ,.
3e 4 a„ _ e 2 C„
2 -P.-'
^
10
B 7 oc
all higher order contributions
w h ere
a s = a d ia b a tic d ipole p o la riz a b ility
Co = a d ia b a tic q u a d ru p o le p o la riz a b ility
(3S = n o n a d ia b a tic d ip o le p o la riz a b ility
< r 2 *s>nL= h y d ro g en ic ra d ia l m a trix e le m e n t, c o rre c te d
fo r the re d u c e d m ass o f th e io n
144
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4 .2
The d ia g o n a l, ra d ia l m a trix elem en ts w ere c a lc u la te d w ith the fo rm u la s 40
Z V
3n 2 —L(L + 1)
2n K
/ -*\
( ZX 3 5 n 4 —n2[30L(L+ 1) —25] + 3(L —l)L(L + l)(L + 2 )
' r '«»■ ~ l b J
8 n 7 ( L - 3/2XL- 1)K(L + 2)(L + 5/2)
/ -7 \
( Z)
7
63n* ~ n2[?0L(L + *) ~ 1051+ 15(L ~ X)L(L + 1XL + 2) - 20L(L -h 1) +12
8 n 7 (L - 2)(L - 3/2XL- 1)K(L + 2 ) ( l + 5/2XL + 3)
K b (L - 1/2)L(L + 1/2XL + 1)(L + 3/2)
b = a„
as defined in Eq. 1-7.
VP-2'
T he c a lc u la te d fin e s tru c tu re e n erg ies, lis te d in T ab le 4-1 and illu s tra te d
in Fig. 4-1, a re m o stly du e to the d iag o n al c o n trib u tio n s o f th e e ffe c tiv e
Table 4-1: Electric fine structure energies. The EFS for R=0, n - 1 0 , L=5-9 levels in H 2
are presented. The diagonal contributions proportional to each power o f the radial
coordinate are given first, and then the total E ^ is given in column 5. The off-diagonal
EC) matrix elements o f Veffare next, along with the relativistic corrections. Theoretical
values were used for B4 = -1.585641, B 6 = 7.862, B?= 4.948 (in a.u.).
(A ll valu es are in M H z)
L abel
B4<r‘4>
B 6 < r 'V
(O)lOHs
-2 9 1 5 .1 5 1
4 8 .230
( 0 ) 1 0 I«
-1 1 9 3 .8 2 4
(0 )1 0 K 7
B 7 < r‘ />
E
Ew
Erel
-2864.81
47.76
-1 8 .7 1
8.675
0.243 -1184.91
14.62
-13.81
-5 4 7 .9 1 8
1.909
0.036
-5 4 5 .9 7
2.52
-10.22
(O)lOLg
-2 7 2 .4 6 2
0.470
0.062
-2 71.93
0.01
-7 .4 7
(0 )1 0 M 9
-143.401
0.119
0.001
-1.09
-5 .3 0
2.111
-143.281
145
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Fine structure energies
for H 2, (0)1 OL Rydberg states
.(0) ______________________________________________________________
9
EFS
500 Mhz
6
----------------1
AE
L 5
E<I>
E (2)+ E rel
Fig. 4-1: R -0 fine structure energies. The fine structure energies are illustrated for the
n=10, L=5-9 Rydberg levels o f H2 . The first level designates the diagonal contribution o f
the effective potential, E(I). The shifted line then shows the net contribution o f E(2) and
Erei, as given in Table 4-1. These small corrections are necessary when considering the
measured intervals (AE), reported in Section 4.3 and represented by a dashed arrow for
the (0)10H-I interval.
146
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p o te n tia l; h o w e v e r, th e sm all sh ifts show n g ra p h ic a lly a lso in c lu d e the
2 nd-o rd e r E (2) c o n trib u tio n s and re la tiv is tic c o rr e c tio n s 41,
r:
( ’Lr ) —
Erd\n
n
3
3
4n
1
L+ l / 2 '
Eq. 4-3
T h ese fine s tru c tu re e n e rg ie s are c h a ra c te riz e d by h a v in g o n ly a sin g le
s p littin g b e tw ee n e a c h p a ir o f a n g u la r m om entum le v e ls (L = N ). A ccu rate
m ea su re m e n t o f th e fin e stru c tu re in te rv a ls w ill y ie ld v a lu e s e a sily related
to the d ip o le p o la r iz a b ility , a s.
4.1-2
S p in stru c tu re tra n s itio n s
The m ic ro w a v e R E SIS te c h n iq u e is p re c ise e n o u g h to re so lv e some
o f th e p re v io u sly n e g le c te d sp in stru c tu re . A s d is c u s s e d in C h a p te r 1, the
tre a tm e n t o f the n u c le a r sp in s and e le c tro n sp in s is in c lu d e d
p e rtu rb a tiv e ly . T h e R = 0 lev el o f e ac h ion c o re is s p lit in to d iffe re n t Fc
sp in lev els. T h is la rg e h y p e rfin e s p littin g d o es n o t h a v e a d ire c t e ffe c t on
th e tra n s itio n s tru c tu re , due to a s tric t AFc=0 s e le c tio n ru le , so it can be
tre a te d as a c o n trib u tio n to the core en erg y le v e l. In th is a p p ro x im a tio n ,
e ac h o f the th re e a llo w e d D 2 + core h y p e rfin e le v e ls h as an e n tire m an ifo ld
o f R ydberg le v e ls b o u n d to it. E ach R ydberg le v e l e x h ib its the sam e R=0
E F S , but the a d d itio n a l sp in s tru c tu re w ill be v e ry d if f e r e n t, as show n in
F ig . 1-6. T he m a jo r c o n trib u tio n to th e o b se rv e d m ic ro w a v e tra n s itio n s
c o m es from the m a g n e tic in te ra c tio n s o f the e le c tro n s p in s (m a g n e tic fine
s tru c tu re : M F S).
147
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T he e x p e c te d tra n s itio n stru c tu re fo r e a c h iso to p e and AL tra n s itio n
is g iven in T a b le C -3 b and T able C -4 b . E ach le v e l is la b e le d by the
a sso c ia te d q u a n tu m n u m b ers,
Fc = I + Sc
J ,= N + Fc
J = J ,+ S R
O nly the fa v o re d tra n s itio n s w ith A L = A Ji= A J= + l a re g iv e n . W eaker
allo w e d tra n s itio n s are at le a st 60 tim e s s m a lle r in p re d ic te d stre n g th and
can be n e g le c te d . E ven i f th ey were p re s e n t, th e p o s itio n s w ould be w ell
aw ay from the m a in tra n s itio n freq u en cy .
(0)1 O
H s
T he sp in s tru c tu re for th e H 2
and (0)1016 is illu s tra te d in F ig . 4 -2 , a lo n g w ith the strong
tra n s itio n s th a t w ill o c c u r. T he spin s tru c tu re in D 2 is tre a te d in a sim ila r
fa sh io n , b u t is fa r m ore c o m p lica te d , h a v in g 24 sp in co m p o n e n ts fo r each
L level.
The m ic ro w a v e sp e c tro sc o p y p re s e n te d in th e fo llo w in g sectio n s
w ill m e a su re th e
A E s p jn
tra n s itio n s. E ach tr a n s itio n w ill be fitted w ith a
th e o re tic a l lin e s h a p e b ased on the c a lc u la te d sp in s tru c tu re p resen te d in
A p p en d ix C. T he fitte d lin e c e n te r w ill th en be th e v a lu e fo r
in fe rre d d ia g o n a l p o la riz a tio n energy,
A E (1),
su b tra c tin g o f f th e s m a lle r p e rtu rb a tio n s,
A E .
The
is e x tra c te d from AE by
A E (2)
an d
A E r e i-
p o la riz a b ility is d e te rm in e d by c o m p arin g th e m e a su re d
F in a lly , the
A E (1)
v a rie s.
148
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values as L
Spin structure energies for H 2 ,
(0)1 OH and (0)101 Rydberg states
A
Strongest allowed
transitions, AJ=+1
20MHz
^
T
(2 )
L=5
AF,
AE
spm
4.5
5
4
5.5
6
5
. ( 1)
Expected transition spin
splittings for (0)10H-I
AE spin
Fig. 4-2: Spin structure for R=0 states o f H 2 . The “helium-like” spin structure is shown
for the R=0 levels o f H2 having L=5 and 6 . The splittings are on the order o f 20 MHz;
however, the similarity o f the two adjacent L levels gives a smaller observed transition
splitting o f approximately 5 MHz. This is illustrated by the stick diagram at the bottom.
149
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4.2 M easu red m icrow ave R E SIS tra n sitio n s
T he re s o n a n t e x c ita tio n o f R y d b erg e le c tro n s by a tra v e lin g
m icro w av e fie ld , as d e sc rib e d in C h a p te r 2, is an e x te n s io n o f th e laser
R E SIS te c h n iq u e . The m o d ifie d te c h n iq u e is in tro d u c e d in S e c tio n 2.6
and fo llo w e d by the d e ta ils o f the a p p a ra tu s used in th is e x p e rim e n t. Tw o
la s e r re g io n s w ere used, one to d e p o p u la te a s p e c ific R y d b e rg level and
th e o th e r to m o n ito r the sam e le v e l fo r any su b s e q u e n t c h a n g e in its
p o p u la tio n . B etw een th e tw o la s e r reg io n s, a m ic ro w a v e re g io n ex cite d
re s o n a n t tr a n s itio n s b e tw ee n R y d b e rg lev els o f co m m o n n (p rin c ip a l
q u an tu m n u m b e r). Due to th e lo n g (~1 p se c ) in te ra c tio n tim e s, the
re s o lu tio n o f th is tec h n iq u e w as ~1 M Hz. T h is w as a p p ro x im a te ly 300
tim e s h ig h e r re so lu tio n th a n th e la s e r sp e c tro sc o p y .
The e x p e rim e n t c o n s is te d o f se ttin g e ac h la s e r to th e d e sire d
tra n s itio n , tu n in g the d e te c to r to m ax im ize th e sig n a l fro m the second
la se r, c h o o sin g an a p p ro p ria te m icro w av e p o w e r, th e n s c a n n in g the
a m p litu d e m o d u la te d m ic ro w a v e freq u e n cy . B y m o n ito rin g th e CEM
c u rre n t as a fu n c tio n o f th e fre q u e n c y , a sig n a l w as m e a s u re d fo r th at
p a rtic u la r d ire c tio n o f p ro p a g a tio n . The d ire c tio n w as th e n re v e rse d an d
th e sam e tra n s itio n was m e a su re d a t the D o p p le r s h ifte d (~2(3<oo)
fre q u e n c y . E ac h signal ty p ic a lly re q u ire d m any h o u rs o f a v e ra g in g to
o b ta in re a s o n a b le s ig n a l/n o ise . F o r each tra n s itio n , th e sa m e p ro c e d u re
w as fo llo w e d , tu n in g the la s e rs , s e ttin g the d e te c to r, c h o o s in g an
a p p ro p ria te m icro w av e p o w e r, and scan n in g th e fre q u e n c y .
150
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The m ic ro w a v e p o w e r w as ch o sen clo se to s a tu ra tio n , b u t low
e n o u g h to a v o id a n y u n w a n te d b ro ad en in g e ffe c ts o r s h if ts in the
lin e c e n te rs. T he tra n s itio n p ro b a b ility a m p litu d e is p ro p o rtio n a l to the
sq u a re o f th e a p p lie d e le c tric fie ld and the <z> m a trix e le m e n t. T he
re q u ire d p o w er, fo r th e n e c e ssa ry e le c tric fie ld , v a rie d w ith the g eo m etry
o f each reg io n , so a s a tu ra tio n c u rv e w as m ea su re d fo r e a c h , to d e te rm in e
th e a p p ro p ria te p o w e r. T h ese sa tu ra tio n cu rv es are sh o w n in F ig . 4-3,
w h ere the m ic ro w a v e sig n a l is p lo tte d as a fu n c tio n o f th e a v e ra g e
m icrow ave p o w er. T he re c o rd e d p o w er was m e a su re d w h ile the
m icro w av e field w as a m p litu d e m o d u lated at a 5 0 /5 0 d u ty c y c le and
arro w s d enote w h ic h p o w e r w as c h o sen . For a g iv en r e g io n , the a p p lied
p o w e r could be sc a le d in v e rs e ly p ro p o rtio n a l to th e s q u a re o f e ach
tra n s itio n m a trix e le m e n t. In o th e r w ords, the h ig h e r-L tra n s itio n s , w ith
sm a lle r m atrix e le m e n ts, w ould s a tu ra te at a h ig h e r p o w e r.
The m e a su re d tra n s itio n s , fo r th e R=0 H 2 an d D 2 R y d b e rg sta te s
w ith n= 9,10 and L = 4 -9 , are lis te d in T able 4-2 a lo n g w ith d e ta ils o f each
m ea su re m e n t. T he firs t c o lu m n la b e ls the tra n s itio n w ith th e c o n v e n tio n ,
n L (L + l). F or e x a m p le , th e tra n s itio n (O)lOHs - (0)1016 is d e n o te d 10HI.
T he second co lu m n lis ts th e re fe re n c e to the lab b o o k . T he th ird colu m n
lis ts the m ic ro w a v e re g io n u se d , refe ren c in g the n o ta tio n a s s ig n e d in
C h a p te r 2. The la s t c o lu m n s g iv e the freq u en cy s te p s iz e , th e to ta l
a v e ra g in g tim e p e r p o in t, and the a v erag e m ic ro w a v e p o w e r u sed.
151
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H2 (0)10L 8-H ,
I
I
>
I
I
|- T T
I— I— I-
eo
—
*
(= .’
Region A
t
-i I i i ■ L-j L i L
0
200
400
600
800
1000 1200
Power (jj.W)
_1— I
Helium 10G-H
1— i
' I
1 'I
*******
+
1
♦
es
s
QO
L/
O
0 .0
—
I
-j
L_i
I
0 .2
i
Region B
l i l .
0 .4
0 .6
0 .8
1.0
Power (fj.W)
H , (1)9K 6 -L7
u 1 I 1—
—i 1 j
ca
e
Ioo
I
-
♦ ♦
-L .l
0
2000
1 _i
Region C
I u I__ .
_l
i
4000
6000
8000
10000
Power (|iW)
H2(0)10G4-H5
T~T~_
ea
a
BO
Region D
? t.
0 .0
,
.
0 .2
1
0 .4
0.6
Power (fiW)
Fig. 4-3: Saturation curves for microwave power settings. A different saturation curve
was measured for each microwave region. These different saturation curves were used to
determine the power settings listed in Table 4-2. The arrows denote the chosen
microwave power for each region used.
152
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Table 4-2: Details o f measured microwave transitions. (* is discussed in section 4.3)
Reference
(PLJ #,p.#)
Label
iH2
Microwave
Region
Stepsize
(MHz)
Time/pt
Power
(pW)
t
j
sec/pt
0.05
10GH
18, p . 1 1 2
D
0 .2
120
10HI
19, p. 84
A
0 .1
80 sec/pt
100
10IK
19, p.82
A
0.05
60 sec/pt
100
10KL
17, p.17
A
0.05
5 min/pt
100
10LM
17, p.33
A
0.05
6
min/pt
400
10LM*
20, p.75
*A
0.05
25 sec/pt
400
9HI
20
, p.35
C
0.25
60 sec/pt
2000
9IK
17, p.63
A
0.05
60 sec/pt
200
9KL
17, p.53
A
0.05
100
10GH
19, p.33
D
0.5
1
10HI
11, p.139
B
0 .2
4 min/pt
0 .6
10IK
18, p.25
A
0 .1
5 min/pt
120
10KL
18, p.15
A
0 .1
5 min/pt
100
10LM
18, p.3
A
0 .1
7 min/pt
120
9IK
14, p.33
B
0 .2
2
min/pt
1 .0
9KL
18, p.43
A
0 .1
6
min/pt
150
|d 2
sec/pt
200
1
i
min/pt
5.0
To ex tra ct th e “ s p in le s s ” in te rv a l fro m e a c h m ea su re m e n t, th e
m icro w av e tra n s itio n s w ere fitted fo r a lin e c e n te r by in c o rp o ra tin g th e
e x p e c te d spin s tru c tu re , as defined in C h a p te r 1, in to a c o m p o site
lin e sh a p e . The sp in s tru c tu re for each tr a n s itio n w as c a lc u la te d and
p re se n te d in A p p e n d ix C. Each p a ir o f le v e ls w as e x p e c te d to have a
tra n s itio n p ro b a b ility am p litu d e re p re se n te d by E q. 2 -2 , w here the
lin e c e n te r, co0, is m o d ifie d to include th e sp in s tru c tu re , w hich v a rie d fo r
153
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e ac h tra n s itio n . As an e x am p le , c o n s id e r the H 2 10HI in te r v a l, w h ic h w as
e x p e c te d to have spin s tru c tu re th a t s p lit each R y d b erg le v e l in to fo u r
d if f e re n t le v e ls. T hese s p littin g s a re g iv e n in T ab le C -3 a fo r th e 10H and
101 le v e ls se p a ra te ly . The d iffe re n c e s , g iv en in T ab le C -3 b , a re d e fin e d
as th e tra n s itio n spin s tru c tu re . F o r th is in te rv al, th e th e o r e tic a l lin e sh a p e
c o n s is te d o f fo u r se p a ra te s in c 2 (x ) fu n c tio n s added to g e th e r, w ith fix e d
re la tiv e a m p litu d e s, a c c o rd in g to th e ir re sp e c tiv e s ta tis tic a l w e ig h ts
(2 J+ 1 ). T he c a lc u la te d sp in s tru c tu re w as added in to th e re s o n a n c e
fre q u e n c y e x p lic itly . The fo llo w in g lin e sh a p e w as u se d to f it th e 10H I
tr a n s itio n fo r a c o n sta n t o ffs e t (c ), a n in te ra c tio n tim e (T ), an a m p litu d e
(A ), a n d th e “ sp in le s s” tra n s itio n fre q u e n c y (v 0).
sin(it(v + 2380 - v 0) t )
rc(v + 2380 —v 0) t
Eq. 4-4
i(7 t(v -2.590- v 0)T)
t(v —2390 —vc) t
7
sin(7t(v —0377 -
vg) t
)
t(v —0377 —v 0) t
7
T he ad eq u acy o f such a f ittin g fu n c tio n w as d e te rm in e d u sin g th e
m e a su re d H 2 10HI and 9IK in te rv a ls . B o th o f th ese in te r v a ls , sh o w n in
F ig . 4 -4 b ,4 g , have w ell re so lv e d s p in stru c tu re . Tw o q u e s tio n s h a d to be
a n sw e re d in o rd e r to ju s tif y th e u se o f th e lin e sh a p e fu n c tio n g iv e n a b o v e.
F irst, by fo rc in g the re la tiv e a m p litu d e s , th e a ssu m p tio n w as m ad e th a t all
154
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le v e ls are p o p u lated u n ifo rm ly and any d iffe re n c e s in th e tra n s itio n
s tre n g th s are n e g lig ib le . S e c o n d , the sp in s tru c tu re w as in c lu d e d e x a c tly ,
n o t a llo w in g for any e rro r. A ny d e v ia tio n fro m e ith e r o f th e se
a ssu m p tio n s w ould in tro d u c e a d d itio n a l e rro r on th e fitte d lin e c e n te rs.
T h ese so u rc e s o f e rro r w e re c o n sid e re d by fittin g th e 10HI and 9IK
in te rv a ls w ith fu n ctio n s th a t allo w ed fo u r in d e p e n d e n t a m p litu d e s and
fo u r in d e p e n d e n t lin e c e n te rs . T hese fo u r in d e p e n d e n t m ea su re m e n ts (tw o
d ire c tio n s fo r each o f th e in te rv a ls ) gave fitte d p a ra m e te rs th a t were u sed
in e stim a tin g the u n c e rta in ty to assig n the o th e r m e a su re d in te rv a ls th a t
had u n re so lv e d stru c tu re .
In fittin g the tw o m e a su re d in te rv a ls, 10H I an d 9 IK , th e am p litu d e s
sh o w ed a t m ost a
10
% d e v ia tio n from the e x p e c te d re la tiv e stre n g th s.
T h is flu c tu a tio n in th e a m p litu d e w ill add so m e u n c e rta in ty in to the
m e a su re d lin e c e n te r; h o w e v e r, th a t u n c e rta in ty w ill be g o v e rn e d by b o th
the p o s itio n o f the lin e r e la tiv e to the o th e r sp in s tru c tu re and the w id th o f
th e lin e . T h is a d d itio n a l e rro r w as e stim a te d fo r e a c h in d iv id u a l in te rv a l.
A th e o re tic a l lin e sh a p e w as c re a te d for e ach in te rv a l a c c o rd in g to the
e x p e c te d sp in stru c tu re , b u t w ith a given lin e v a rie d by
1 0
% in its
a m p litu d e . Since m o st o f th e in te rv a ls have u n re s o lv e d stru c tu re , the
la rg e s t c o n trib u tio n o f th is e ffe c t cam e from th e e x tre m e sp in
c o m p o n e n ts, since th o se s h ifte d the sid es o f th e lin e sh a p e . A sy ste m atic
tre a tm e n t o f the m e a su re d in te rv a ls , sh iftin g e a c h c o m p o n e n t in d iv id u a lly
155
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and fittin g for the sh ifte d lin e c e n te r, g ave re a so n a b le e stim a te s o f th e
e rro r, d e fin e d by a ampi.
In a d d itio n , the te ste d fits o f th e 10HI and 9IK in te rv a ls gav e fo u r
in d e p e n d e n t lin e c e n te rs fo r e ac h m e a su re m e n t. T he e x p e c te d sp in
stru c tu re w as su b tra c te d from e a c h o f th e a p p ro p ria te lin e c e n te rs and th e
r e s u ltin g fre q u e n cie s w ere a v e ra g e d . T he a v erag e lin e c e n te r w as
co m p ared to the fitte d lin e c e n te r o b ta in e d w ith the lin e sh a p e d e fin e d
a cc o rd in g to Eq. 4 -4 . T hese v a lu e s w ere d iffe re n t, b u t w ell w ith in the
s ta tis tic a l fit e rro rs. H o w ev er, th e d iffe re n c e s seem ed to v a ry
sy s te m a tic a lly w ith the e x te n t o f th e sp in stru c tu re , s h iftin g
a p p ro x im a te ly 0.1% o f th e o v e ra ll sp in stru c tu re w id th . In o th e r w o rd s,
fo r th e 10HI in te rv a l, the sp in s tru c tu re w as spread o v e r 4 .9 7 0 M H z and
th e m e a su re d e rro r w as 0.0 0 5 M H z. T he m axim um d e v ia tio n w a s in c lu d e d
as an a d d itio n a l e rro r, d e fin e d as crSpin, fo r each o f the in te rv a ls , a lth o u g h
it w as u su a lly m uch sm a lle r th a n th e d o m in a n t s ta tis tic a l fit e rro rs (<Tm).
The e stim a te s for th e se a d d itio n a l sy s te m a tic e rro rs, due to a m p litu d e and
spin s tru c tu re u n c e rta in tie s, are su m m a riz e d in T ab le 4-3.
The o b serv ed re so n a n c e s fo r R =0 H 2 and D 2 are p re s e n te d below .
E ach m easu red in te rv a l w as fitte d w ith a fu n ctio n sim ila r to th e 10HI
fu n c tio n g iven by Eq. 4 -4 . T he a p p ro p ria te spin s tru c tu re w as fix e d into
the lin e sh a p e along w ith th e re la tiv e stre n g th s. The fitte d lin e c e n te r w as
re c o rd e d alon g w ith th e s ta tis tic a l e rro r a sso c ia te d w ith th e w e ig h te d fit.
W ith o u t th e p o ssib le sy ste m a tic e rro rs d isc u sse d ab o v e, th e se s ta tis tic a l
156
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Table 4-3: Additional uncertainties in the meaured intervals. (All values in MHz)
Label
O'spin
h2
0
0.009
10HI
0.001
0.005
10IK
0.009
0.003
10KL
0.020
0.002
10LM
0.019
0.001
10LM*
0.019
0.001
9HI
0
0.007
9IK
0.001
0.004
9KL
0.009
0.003
0
0.009
10HI
0.007
0.005
10IK
0.015
0.003
10KL
0.012
0.002
10LM
0.009
0.001
9IK
0.011
0.004
9KL
0.014
0.003
10GH
!D2
10GH
erro rs w o u ld a v e ra g e to a sm a lle r v a lu e w h e n th e a v erag e o f th e tw o
d ire c tio n s is ta k e n . H o w ev er, the a d d itio n a l e rro rs w ere in c lu d e d
se p a ra te ly to a rriv e a t th e to ta l e rro r fo r th e m easu red in te rv a ls . T he to ta l
e rro r is g iv e n by
w here A and B re p re s e n t th e tw o D o p p le r d ire c tio n s m ea su re d .
157
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In the fig u re s p re s e n te d below , the fitte d Iin e s h a p e s a re in clu d ed as
th e so lid lin e . T he e rro r b a rs on each p o in t re p re s e n t th e s ta tis tic a l n o ise
in the a v e ra g e o v e r m u ltip le scans o f th e sam e tr a n s itio n . E ach tra n s itio n ,
as m en tio n ed in C h a p te r 2, w as m easu red se v e ra l tim e s , re p e a te d ly
scan n in g a c ro ss th e lin e to a v erag e out any ran d o m f lu c tu a tio n s in the
sig n al. T he re p o rte d sig n a l is an a v erag e o f N c o n s e c u tiv e sc a n s. The
e rro r b a r w as d e te rm in e d in th e fo llo w in g m an n er:
a) The sta n d a rd d e v ia tio n (SD ) w as c a lc u la te d fo r th e se t o f N
p o in ts a t e a c h fre q u e n c y .
b) The a rms o f th e s e SD v alu es w as c a lc u la te d , in c lu d in g all
fre q u e n c ie s , to a v e ra g e out an y ran d o m f lu c tu a tio n s .
c) T he a err w as c a lc u la te d by d iv id in g a rms by th e sq u a re ro o t o f N ,
and ± a err w as a ssig n e d as the e rro r b ar fo r e a c h p o in t in the
a v e ra g e d sc an .
E ach m e a su re m e n t w as m ade fo r both p ro p a g a tio n d ire c tio n s and both are
p re se n te d to g e th e r on th e sam e freq u e n cy sc ale fo r c o m p a ris o n , to
illu stra te the d e c re a s in g siz e o f the D o p p ler sh ift as th e fre q u e n c y
d e crea se s (h ig h e r L ). E ach m easu red tra n s itio n w as f itte d to e x tra c t th e
“ sp in le s s ” tr a n s itio n e n e rg y . The fitte d v a lu e s are g iv e n in T ab le 4 -4 a,b
fo r each d ire c tio n o f p ro p a g a tio n . The g e o m e tric m e a n o f th e v alu es,
g iv en in the la s t c o lu m n , re p re se n ts the D o p p le r-fre e in te rv a l fo r each
tra n s itio n . T he e rr o r b a rs re fle c t the co m b in ed u n c e r ta in tie s .
158
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H 2 1 0 G H (copropagating)
ss
-O
w
CO
5150
5160
5180
5170
5190
5200
Microwave Frequency (MHz)
H 2 10G H (counterpropagating)
— 1— 1— 1___ 1
5150
l
5160
1___ 1___ 1___ 1
l
1___ 1___ 1___ 1___ l___ 1___ 1___ 1___ 1___ l___ 1___ 1___ 1___ 1___
5170
5180
5190
5200
Microwave Frequency (MHz)
Fig. 4-4a: H2 (0)1 OG4 - Hs microwave transition. The four spin components are well
resolved. The structure exhibits a strong dependence on the exchange energy term
included in the spin structure, due to the 10G state.
159
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H 2 1O H I (copropagating)
C
3
JO
hm
ee
cso
00
K
2
UJ
u
1645
1650
1655
1660
1665
1670
Microwave Frequency (MHz)
10H I (counterpropagating)
c
3
x;
W
CO
CO
eop
K
2
w
u
I
1645
1 La
1650
I I
1
I
.
.
.
.
1655
I
1660
1665
Microwave Frequency (MHz)
Fig. 4-4b: H2 (0)1 OH5 - k microwave transition.
160
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1670
H 7 1O IK (copropagating)
c3
X!
i—
eo
"ra
cop
5>
2
UJ
U
i
625
i
630
635
Microwave Frequency (MHz)
H ? 10IK (counterpropagating)
s
xi
i—
3
W
*3
c
.2P
M
U
625
630
635
Microwave Frequency (MHz)
Fig. 4-4c: H2 (0)1016 - K 7 microwave transition.
161
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H 2 1 0 K L (copropagating)
e3
a
ac
00
cw
s
UJ
U
270
275
280
Microwave Frequency (MHz)
H 2 10K L (counterpropagating)
es
ee
e3op
2
ua
u
270
275
Microwave Frequency (MHz)
Fig. 4-4d: H2 (0)1 OK 7 -Lg microwave transition.
162
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
280
H 2 1 0 L M (copropagating)
s
3
es
3e
on
c»
s
U
U
125
130
135
Microwave Frequency (MHz)
H 2 10LM (counterpropagating)
2
'E
3
es
c
00
izi
2
u
u
125
130
Microwave Frequency (MHz)
Fig. 4-4e: H2 (0)1 OLg —M9 microwave transition.
163
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
135
9 H I (copropagating)
c3
JO
3
3C
BO
53
2
u
U
2280 2285 2290 2295 2300 2305 2310 2315 2320 2325 2330
Microwave Frequency (MHz)
H - 9H I (counterpropagating)
e
3
•E
3
3
c
op
K
s
u
U
2280 2285 2290 2295 2300 2305 2310 2315 2320 2325
Microwave Frequency (MHz)
Fig. 4-4f: H2 (0)9Hs -16 microwave transition.
164
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2330
H 7 9 IK (copropagating)
e3
JO
es
*3
c00
u
U
l
860
l
l
865
i
870
Microwave Frequency (MHz)
FL 9 IK (counterpropagating)
e3
JO
w
CO
es
eoo
X
S
w
u
860
865
870
Microwave Frequency (MHz)
Fig. 4-4g: H 2 (0)916 - K7 microwave transition.
165
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H ? 9 K L (copropagating)
3
JD
ka
ee
73
c
.2?
s
tu
u
i
1
365
L
370
375
Microwave Frequency (MHz)
H ? 9K L (counterpropagating)
c
3
XW
>
w
c
op
W
S
w
u
"3
365
370
375
Microwave Frequency (MHz)
Fig. 4-4h: H2 (0)9K? - Lg microwave transition.
166
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D 2 1O G H (copropagating)
I I l-l I I 'l I I I I I I I I I I I I I I 1 1 I I I I I I I I I I I I I I I I I I I 1 1 I I I I I I
25
.o
w
g
75
C
qo
35
|_
u
u
■ I I I I I ' ' I I ■ ' I I I ' I I I I ■ I I I I ' I I » I 11 1 1 I ' 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 ' 1
5300 5305 5310 5315 5320 5325 5330 5335 5340 5345 5350 5355
Microwave Frequency (MHz)
D 9 1OGH (counterpropagating)
e
s
a
C
B
sop
<11
o
s
U
U
5300 5305 5310 5315 5320 5325 5330 5335 5340 5345 5350 5355
Microwave Frequency (MHz)
Fig. 4-5a: D2 (0)1 O G 4 - Hs microwave transition.
167
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D 2 1 0 H I (copropagating)
1
xs
«
ecs
BO
K
2
u
u
1676 1678 1680 1682 1684 1686 1688 1690 1692 1694 1696 1698 1700
Microwave Frequency (MHz)
D~ 10H I (counterpropagating)
c
3
w
eo
JD
sop
K
s
u
u
1676 1678 1680 1682 1684 1686 1688 1690 1692 1694 1696 1698 1700
Microwave Frequency (MHz)
Fig. 4-5b: D 2 (0)1 O H s - 16 microwave transition.
168
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D-, 1 OIK (copropagating)
c
s
-C
i_
eo
To
c
00
<73
2
u
U
I
630
632
■
634
■
'
■
*
l
636
'
1
l
1
l
638
l
■
■
I
■
640
r
t
.
I
.
■
642
■
.
I
i
i
i
■
644
646
644
646
Microwave Frequency (MHz)
D~ 10IK (counterpropagating)
e
3
3
CO
C
00
c/5
2
u
w
11I 1' ' ' I ' ' ' ' I ' I 11I 1' 11I 11' ' I ' '
630
632
634
636
638
640
642
Microwave Frequency (MHz)
Fig. 4-5c: D 2 (0)1016 - K.7 microwave transition.
169
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D 2 1 0 K L (copropagating)
e3
JS
w
eo
"«
c
QO
35
S
u
u
266
268
270
272
274
276
278
280
278
280
Microwave Frequency (MHz)
D~ 10K L (counterpropagating)
M
'53
jw
i
eo
"eo
eoo
i»
S
u
u
I I I I ' ' I ■■I I I I I I I
266
268
270
272
274
276
Microwave Frequency (MHz)
Fig. 4-5d: D 2 (0)1 OK7 - Lg microwave transition.
170
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D 2 1OLM (copropagating)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11
3
JO
hm
eo
"3
c
00
aj
£
u
u
125
126
J_L '
127
'
» ■
I I I '
128
IIII
129
J_l
I L
130
131
132
133
134
135
134
135
Microwave Frequency (MHz)
D~ 10L M (counter-propagating)
e3
JO
CO
eco
oo
K
£
w
o
125
126
127
128
129
130
131
132
133
Microwave Frequency (MHz)
Fig. 4-5e: D 2 (0)1 OLg - Mg microwave transition.
171
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D~ 9 IK (copropagating)
n
'53
h-
co
To
c
BO
cw
s
CD
U
■1■1I ■i 1■I ' ■■■I 1■■■I ■■■■I 1■■1I ■■■■I ■
864
866
868
870
872
874
876
878
880
882
884
882
884
Microwave Frequency (MHz)
D~ 9 IK (counterpropagating)
s
3
-d
w
3
3
S
op
izj
S
u
u
864
866
868
870
872
874
876
878
880
Microwave Frequency (MHz)
Fig. 4-5f: D 2 (0)916 - K 7 microwave transition.
172
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D 2 9 K L (copropagating)
c
s
ea
c
BO
C»
2
LU
U
352
354
356
358
360
362
364
366
368
366
368
Microwave Frequency (MHz)
D 2 9K L (counterpropagating)
s
3
S3
C3
S
00
(»
UJ
u
352
354
356
358
360
362
364
Microwave Frequency (MHz)
Fig. 4-5g: D2 (0)9K7 - Lg microwave transition.
173
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 4-4a: Measured microwave transitions in H2 R=0 Rydberg states. The labeled transitions are related to the microwave region
by column 2, which are discussed in chapter 2. The following columns give the fitted linecenters for each direction o f propagation
followed by the reduced x2 and the degrees o f freedom, d. In parentheses, the statistical fit errors are given. Finally, the average
linecenter is given by the geometric mean o f the two fitted linecenters and the total error is defined by Eq. 4-5.
Label
Region
A(coprop.)
(MHz)
d, X2/d
B(ctrprop.)
(MHz)
d. X2/d
I0GH
D
5188.044(34)
80,1.27
5159.867(37)
80,1.45
5173.936(27)
10HI
A
1664.896(11)
86,1.96
1653.544(9)
116,1.29
1659.210(9)
10IK
A
632.945(8)
126,1.42
628.649(7)
115,1.22
630.793(11)
10KL
A
274.982(18)
78,0.75
273.167(19)
78,0.97
274.073(24)
I0LM
A
130.073(15)
86,1.14
129.275(13)
86,1.11
129.673(21)
♦10LM
A(rev)
130.128(15)
77,0.93
129.219(12)
96,0.74
129.673(21)
9H1
C
2311.11(10,31*)
101,1.76
2296.29(9,31*)
93,1.10
2303.686(230)
9IK
A
867.518(6)
156,1.16
861.621(6)
156,1.04
864.564(6)
9KL
A
371.962(9)
115,1.09
369.431(6)
115,0.95
370.694(11)
Average
Linecenter
represents a m easurem ent made w ith the m icrow ave region reversed (d iscu ssed in S ectio n 4 .3 )
d en otes an uncertainty.due to the un exp ected assym etry present in the m easured line (F ig . 4 - 4 0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 4-4b: Measured microwave transitions in P 2 R=0 Rydberg states.
Label
Region
A(coprop.)
(MHz)
d, x 2/d
B(ctrprop.)
(MHz)
d, x2/d
IOGH
D
5334.08(11)
57,0.95
5316.82(10)
58,1.51
5325.44(7)
10H1
B
1691.454(91)
61,1.13
1683.424(80)
56,1.10
1687.43(6)
10IK
A
639.576(8)
66,0.85
636.507(11)
71,1.23
638.040(17)
I0KL
A
273.698(11)
53,1.44
272.392(11)
52,1.47
273.044(15)
10LM
A
129.741(10)
40,1.07
129.132(9)
40,1.05
129.436(11)
9IK
B
875.389(45)
61,1.35
871.139(33)
61,1.10
873.261(30)
9KL
A
360.390(17)
67,0.66
358.646(19)
67,0.93
359.517(19)
Average
Linecenter
A q u ick e x a m in a tio n o f the m easured tra n sitio n s re v e a ls sev eral
fea tu res. F irst, th e D 2 tra n s itio n s do not e x h ib it th e sam e re so lv e d
stru c tu re as the H 2 tra n s itio n s . This w as e x p ected since each lin e sh a p e is
a c o m p o site o f 24 u n d e rly in g spin stru c tu re tra n s itio n s, as d isc u sse d
e a rlie r. In a d d itio n , se v era l o f the D 2 tra n s itio n s w ere m e a su re d w ith only
ad eq u ate S/N in o rd e r to get a reaso n ab le fit o f th e fre q u e n c y p o sitio n .
These tra n s itio n s w ere lim ite d by the n o ise level p re se n t. F o r the goal o f
th is e x p e rim e n t, th e h ig h est-L (KL and LM ) in te rv a ls w ere m o st
im p o rtan t; th e re fo re , m ore tim e was sp en t av erag in g th o se sig n a ls so they
ty p ic a lly h ave th e b e st S/N . O ne final fe a tu re , a lth o u g h s u b tle , show s up
in the H 2 10GH in te rv a l. The u n d erly in g sp in stru c tu re , due to the 10G
lev el, e x h ib its a stro n g d ep en d en ce on the ex ch an g e en erg y term in the
H am ilto n ian . T h is is seen by the g rea ter s p littin g o f the tw o , lo w er
freq u en cy p e ak s. T he fitte d lin esh ap e in c lu d e s a s ig n ific a n t e x ch an g e
c o n trib u tio n as d e fin e d by Eq. 1-21. T he ex ch an g e p a ra m e te r V x, has
been p re v io u sly re p o rte d as 0.3 0 (1 0 ) fo r the R = 1 10G s ta te , b a sed on
re so lu tio n o f th e m ore co m p lica te d R = 1 sp in s tru c tu re 5. T he R=0 in te rv a l
m easured h ere e x h ib its sim p le r spin stru c tu re , four p eak s fu lly re so lv e d .
A n a ly sis o f th e 10G exchange term req u ire d an ite ra tiv e a p p ro a ch to
in clu d e the o ff-d ia g o n a l c o n trib u tio n s. T he norm al sp in s tru c tu re w as
c alcu late d and in c lu d e d in the fittin g fu n ctio n . In a d d itio n , th e ex ch an g e
term w as a d d ed to the fittin g fu n ctio n in o rd e r to fit for d ia g o n a l
c o n trib u tio n s o f V x. T he fitte d p ara m eter w as th en used to c a lc u la te the
176
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
co m p lete d ia g o n a liz e d sp in stru c tu re . T he fittin g fu n ctio n w as m o d ifie d
to includ e (1) the o ff-d ia g o n a l c o n trib u tio n s o f th e to tal sp in H a m ilto n ia n ,
(2) the d ia g o n a l sp in s tru c tu re , n e g le ctin g th e e x ch an g e te rm , and (3 ) th e
p a ra m e te riz e d d ia g o n a l ex ch an g e c o n trib u tio n . F ittin g th is fu n c tio n g iv e s
a value o f V x= 1 .1 2 (2 0 ). A lth o u g h la rg e r th a n p re v io u sly re p o rte d , th e
assu m p tio n rem a in s v a lid th a t for h ig h er L le v e ls , th ere is no a p p re c ia b le
c o n trib u tio n o f the ex ch a n g e term . Vx=0 fo r all L >4. T his w as ju s tif ie d
by fittin g th e H i 10HI in te rv a l for the sam e p a ra m e te r, re s u ltin g in a v a lu e
c o n siste n t w ith z ero . T he exchange c o n trib u tio n should be s im ila r fo r
b o th sy stem s, in th e 10G le v e ls, so the sam e p a ra m e te r w as used to
d ete rm in e th e sp in s tru c tu re fo r the D 2 10G H in te rv a l. S ince th is in te rv a l
has poor S/N and no re s o lu tio n , this was a c c e p ta b le .
4.3 EFS in terv a ls extracted from m easu red tran sition s
T he m e a su re d lin e c e n te rs p resen te d in T a b le 4-4 had to be c o rre c te d
fo r p o ssib le sy s te m a tic freq u e n cy sh ifts. T h e tw o m ajor e ffe c ts
c o n sid e re d w ere S ta rk s h ifts due to any stra y (d c) e le c tric fie ld s and
lin e c e n te r s h ifts th a t o c cu r b ecau se o f u n w a n te d m icrow ave re fle c tio n s at
th e te rm in a tio n end o f th e m icrow ave re g io n . T he sh ifts due to e ac h o f
th e se e ffe c ts w ere e stim a te d and su b tra c te d fro m th e m easu red
lin e c e n te rs. The re s u lt w as a set o f m e a su re d e le c tric fine s tru c tu re (E F S )
in te rv a ls.
177
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.3-1
S ta rk s h ifts due to re s id u a l e le c tric fie ld s .
The S tark s h ift o f an en erg y lev e l is d ire c tly d e p e n d e n t on th e size
o f th e e le c tric fie ld p re s e n t. The e n erg y sh ift o f a g iv e n sta te co m es from
c o u p lin g to the n e ig h b o rin g lev els and is in v e rse ly p ro p o rtio n a l to the
se p a ra tio n from each p e rtu rb in g lev e l. T he lin e a r S ta rk s h ift for an (n L m )
lev el p e rtu rb e d by th e n e ig h b o rin g n ( L + l) level is g iv e n by
pL+l _
CS
(eE0(nLm|z|n(L + l)m))2
Eq. 4-6
~
L+l
w here
Eo
is the e le c tric fie ld and
A E
l ,l + i
is the fin e s tru c tu re s e p a ra tio n
o f th e tw o lev e ls. T he <z> m atrix e le m e n t can be w ritte n as
|(nLm|z|n(L + l)m)|2 = { ^ | (n2 - (L + 1)2)
(L + 1)2 - m2
(2L + 3)(2L + 1)
Eq. 4-7
T he to ta l sh ift o f a g iv e n nL level can be c a lc u la te d by ta k in g a w e ig h te d
a v erag e o v e r th e d iffe r e n t m sta te s. U sin g
Eq. 4-8
2 L T l | LmJ=i L(L+1)-
and in c lu d in g b oth u p p e r and low er p e rtu rb in g s ta te s , th e to ta l S ta rk s h ift
is re p re ste n te d by
( ? “ •)
Es
3(2L +1)
iV -L * )
(L + l)(n2 - ( L + l ) 2)
AE
l x +1
Eq. 4-9
w h ere the d e n o m in a to rs a re given by th e fin e s tru c tu re s e p a ra tio n o f th e
le v e ls. C lea rly , as th e e n e rg y se p a ra tio n s becom e s m a lle r, th e S ta rk s h ifts
w ill becom e larg e r. T h is b e h a v io r is u se fu l sin ce th e tra n s itio n s to be
178
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m easu red are in th e n = 9 ,1 0 R y d b e rg lev els. T he fin e s tru c tu re se p a ra tio n
sc a le s as n '3, so n = 2 7 fin e s tru c tu re levels w ill be m uch m ore se n sitiv e to
e le c tric fie ld s and c a n be u se d as d ia g n o stic lin e s.
A fu rth e r s im p lific a tio n to Eq. 4-9 can be m ade fo r “ h e liu m -lik e ”
R y d b erg le v e ls , su c h as th e R =0 sta te s c o n sid e re d h e re . T he fine
stru c tu re fo r th e se le v e ls , as g iv en in Eq. 4 -2 , is m o stly due to the first
te rm , w hich is p ro p o rtio n a l to h y d ro g en ic <r*4> m a trix e le m e n ts. T hese
can be w ritte n in an e x a c t form fo r each L lev el an d th e e n e rg y term can
be su b stitu te d fo r e a c h d e n o m in a to r. This s im p lifie s to a S ta rk sh ift g iv en
by
(L -l)(L -1 .5 )(n 2 —L2)
5n2 - ( L - l ) ( L + l)
E . - C f = l (L —0.5)(L)(L + l)(L +1.5)
2a.
Eq. 4-10
(L + 2)(L + 2J)(n 2 - (L + 1)2)
5n2 - (L)(L + 2)
w h ere C = 2.4883 x 10‘10 M H z /(V /c m )2, the c o n v e rsio n fro m ato m ic u n its.
H elium tr a n s itio n s w ere c h o sen as th e p rim a ry , e le c tric fie ld
d ia g n o stic lin e s fo r th is e x p e rim e n t fo r tw o im p o rta n t re a so n s . F irst, the
fin e stru c tu re in te rv a ls in H e liu m can be c a lc u la te d w ith v e ry h ig h
p re c isio n . T his is n e c e s s a ry fo r c a lc u la tin g b o th th e e x p e c te d lin e
p o sitio n s and th e S ta rk s h ift “r a te ” w hich d ep en d s on th e n e ig h b o rin g
in te rv a ls. S eco n d , th e a to m ic R E SIS sig n als h ave m uch h ig h e r S/N , w hich
a llo w s q uick m e a su re m e n ts o f th e d ia g n o stic lin e s. To u n d e rsta n d the
n e c e ssa ry H eliu m s tru c tu re , c o n s id e r the illu s tra tio n in F ig . 4 -6 . This
179
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
stru c tu re w as c a lc u la te d u sin g the m odel p re se n te d by D ra k e .42 T h is
m ethod, based on a q u a n tu m d e fe c t e x tra p o la tio n , has b een c o m p a re d to
e x p erim e n tal v a lu e s an d ag re ed on the ppb lev e l up to n = 20. T he
accu racy o f th is c a lc u la tio n fo r n=27 sh o u ld be a d eq u a te . T he n = 2 7 lev e ls
for H elium R y d b e rg s ta te s w ith L =2-4 are show n (o n ly the s in g le t D
state). T he p rim a ry d ia g n o s tic lin es (A and B in T ab le 4-5) are sh o w n
w ith a rro w s. In o rd e r to use th e H elium tra n s itio n s tw o q u e stio n s m u st be
an sw ered. F irst, th e c a lc u la te d stru c tu re has to be a c c u ra te in o rd e r to
p red ic t th e z e ro -fie ld in te rv a l. Second, the sh ift rate m u st be c o rre c t in
o rd er to d e te rm in e th e c o rre c t e le c tric H eld. T hese a re c o n sid e re d b elo w ,
by an aly zin g th e p rim a ry d ia g n o stic lin e s, th e 2 7 ,F - IG and th e 2 7 ID - IF,
d efin ed as A and B re s p e c tiv e ly .
The sp in s tru c tu re o f th e 27F sta te s in H eliu m , as show n in F ig . 4 -6 ,
is quite sm all. B e c a u se o f a llo w e d m ix in g , th e la b e lin g o f th e s e le v e ls is
not q u ite a c c u ra te . T he 2 7 ‘Fs and 273F3 are s u ffic ie n tly m ix ed to be
in d istin g u ish a b le to th e la s e r d riv in g the 10lD2-27F tr a n s itio n (s ). T h is
e ffe ct w as e v id e n t w h e n d riv in g the 2 7 , F - IG m icro w av e re so n a n c e . S in ce
both the F and G s ta te s hav e m ix tu res o f th e s in g le t and trip le t s ta te s ,
th ere are fo u r p o s s ib le m ic ro w a v e tra n s itio n s w hen th e la se r is d riv in g
into the “ s in g le t” s ta te . T he m easu red lin e sh a p e fo r th is d ia g n o s tic
tra n s itio n is show n in F ig . 4-7. The freq u e n cy ran g e c o v e re d in th is scan
w as only enough to o b se rv e th re e o f the e x p e c te d tra n s itio n s . T he fitte d
lin e sh a p e, g iv en by th e so lid lin e , had the re la tiv e sp in stru c tu re in c lu d e d
180
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
_
105.065 ■g ,
*■ ■A . 104.740 3g *
------------ 104.261
- y ~ y 103.736 3g 4
3G3
A
Ip _ LT----------------
▲
-1.507 3p
4 Mhz
_^345_>P;
-2.758 T,
▼
_T_T_
B
n=27 'D2 ______ r
-558.643 Mhz
Fig. 4-6: Calculated Helium n—27 Rydberg fine structure. The intervals shown by arrows
were used as primary diagnostic lines to determine any stray electric fields inside the
microwave regions. These diagnostics are labeled A and B in Table 4-5.
e x p lic itly . T he re a s o n a b ly good fit show s th a t th e s p in s tru c tu re for th ese
H elium R y d b erg s ta te s w as c a lc u la te d a c c u ra te ly . T h e e le c tr ic fin e
stru c tu re o f H e liu m R y d b e rg sta te s has been stu d ie d w ith s e v e ra l highp re c isio n sp e c tro sc o p y e x p e rim e n ts .43 T hese m e a s u re m e n ts sh o w ed very
go o d a g re em e n t, on th e le v e l o f 10kH z, b e tw e e n th e o b s e rv e d in te rv a ls
and c a lc u la te d s tru c tu re in n= 10 R ydberg le v e ls . T h is a c c u ra c y is
ex p ected to be m u ch h ig h e r in th e n=27 le v e ls .
181
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
cTo
0.00
102
103
104
105
106
107
108
Frequency (MHz)
Fig. 4-7: Helium 27lF -lG primary diagnostic line. The side features are due to alternate
transitions that occur due to the mixed singlet and triplet G states.
T he S tark sh ift “ r a te s ” a lso d ep en d ed on th e c a lc u la te d fine
s tru c tu re . To check b o th th e c a lc u la te d ra te s and fin e s tru c tu re , re d u n d a n t
m e a su re m e n ts o f th e p rim a ry d ia g n o stic lin e s w ere u se d in d e te rm in in g
th e e le c tric field . M e a su rin g b o th the 2 7 , F - IG a n d th e 2 7 * 0 - ^ tra n s itio n s
g av e tw o in d e p e n d e n t s h ifts an d ; th e re fo re , tw o in d e p e n d e n t v alu es fo r
th e e le c tric field . T h ese tra n s itio n s , as seen in F ig . 4 -6 , w ere at v ery
d iffe re n t fre q u e n c ie s. In a d d itio n , sin ce th e a d ja c e n t in te rv a ls w ere
la rg e r, th e sh ift rate fo r th e 2 7 1D - IF w as m uch s m a lle r th a n the o th e r
d ia g n o s tic lin e . T he fo llo w in g sum m ary d e sc rib e s th e re s u lts o f a sin g le
te s t o f th e s e d ia g n o s tic s . T he re su lts g av e a re a s o n a b le e stim a te fo r th e
e le c tric fie ld p re se n t d u rin g th e m e a su re m e n ts an d th e u n c e rta in ty o f
- l m V /c m , w hich w as sm a ll en o u g h to h av e n e g lig ib le e ffe c ts on any o f
th e m e a su re d m o le c u la r R y d b e rg in te rv a ls.
182
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Line
A
B
Shift “rate”
E-field
(MHz)
(MHz/(V/cm)2)
(mV/cm)
-0.204
-0.087
-7903.531
-1882.395
5.1
6.8
Vcaic
vobs
5v
(MHz)
(MHz)
105.065
558.643
104.861(3)
558.556(4)
The S tark s h ift ra te s and in te rv a ls w ere c a lc u la te d for each
d ia g n o stic tra n s itio n , as liste d in T able 4 -5 . A n u m b er o f d ia g n o stic s
w ere n e ce ssa ry fo r d iffe re n t reaso n s. T he p rim a ry lin e s, A and B, w ere
o n ly d ire c tly used tw ic e ; h ow ever, th e se tw o tra n s itio n s were used to
c a lib ra te the n= 27 m o le c u la r in te rv a ls (lin e s C an d D ), w hich w ere u sed
fre q u e n tly . T hese m o le c u la r in te rv a ls w ere u se d sin ce tuning up a H eliu m
b eam , w hile m e a su rin g th e H 2 or D 2 s ig n a ls , w as u n d esired
e x p e rim e n ta lly . S e v e ra l o th e r H elium tra n s itio n s w ere used fo r d iffe re n t
re a so n s. The G d ia g n o s tic line w as used fo r a re g io n having a
sig n ific a n tly h ig h e r e le c tric H eld, sin ce th is tra n s itio n had a sm a lle r s h ift
ra te . F in a lly , th e H lin e w as used in th e G -b a n d w aveguide reg io n , sin c e
th is reg io n does n o t p ro p a g a te fre q u e n c ie s b e lo w 3 GH z.
The d ia g n o s tic lin e s w ere in te rsp e rse d th ro u g h o u t the d a ta
a c q u isitio n fo r the d iffe re n t m o le cu la r R y d b e rg in te rv a ls. T able 4-6
re la te s each m ea su re d H 2 and D 2 tra n s itio n w ith the c o rre sp o n d in g
d ia g n o stic lin e . T he m ea su re d d ia g n o s tic fre q u e n c y is given in the th ird
c o lu m n and th e c a lc u la te d value (T ab le 4 -5 ) is su b tra c te d from it to g iv e
the d ia g n o stic lin e ’s S ta rk sh ift. T his d iffe re n c e w as assum ed to be
to ta lly due to the e x is tin g stray e le c tric fie ld . U sin g the c a lc u la te d s h ift
“ r a te ” o f the d ia g n o s tic lin e , an in fe rre d v a lu e o f the e le ctric fie ld w as
183
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c a lc u la te d and sh o w n in co lu m n 5 o f T ab le 4 -6 . By re fe re n c e to T ables 44 a ,b , it can be seen th a t th e la rg e s t stray fie ld s w ere fo u n d w ith
in te ra c tio n reg io n B (~ 8 0 V /cm ), th e sm a lle st w ith re g io n A (—10 m V /cm ),
and in te rm e d ia te v a lu e s w ith re g io n s C and D. T he S ta rk s h ifts for the
R=0 m o le cu la r le v e ls w ere c a lc u la te d using E q. 4 -1 0 . U sin g the m easu red
v a lu e o f th e e le c tric fie ld , th e re su ltin g S ta rk s h ift fo r e a c h n= 9,10
tra n s itio n w as d e te rm in e d . B oth th e c a lc u la te d “r a te ” a n d th e “ m ea su re d ”
sh ift are g iv en in th e la st tw o co lu m n s o f T ab le 4 -6 . T h e s e S ta rk sh ifts
w ere su b tra c te d o f f th e m ea su re d lin e c e n te rs lis te d in T a b le 4-4; h o w ev er,
in all cases th e se s h ifts w ere m uch sm a lle r th an th e e x p e rim e n ta l e rro rs.
T able 4-5: Electric field diagnostic lines. The diagnostic transitions are labeled along
with the description. The calculated “zero-field” frequencies are given, followed by the
calculated Stark shift “rate” based on Eq. 4-9. The frequencies labeled “a” are calibrated
positions using the A and B Helium diagnostic lines. The “b” denotes a transition
previously measured by Hessels34.
Label
Transition
Ecalc
Stark shift “rate”
(MHz)
M H z/(V /cm )2
A
27 ‘F ^G Helium
105.065
-7903.531
B
27 ‘D ^F Helium
558.643
-1882.395
C
H2 (1)27 H6-I7
[196.800(22)]*
-956.195
D
D2 (2)27 I8-K9
[112.699(6)]“
-945.255
E
27 3F3-3G4 Helium
106.493
-7602.137
F
17 3F3-3G4 Helium
424.049
-281.278
G
10 3G5-3H6 Helium
[491.967]b
-11.618
H
27 'P -lD Helium
4757.088
+1059.682
184
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T able 4-6: Calibrated electric field and calculated Stark shifts. The measured diagnostic
lines used for each transition are labeled in column 2 and the observed energy is given in
column 3. The differences between the measured and calculated values o f the diagnostic
transition are in column 4. Based on these shifts and the shift “rates” in Table 4-5, the
inferred electric field is given in column 5. This electric field, along with the calculated
shift rates for the molecular transitions (given in column 6), is used to determine the Stark
shifts for each measured molecular Rydberg transition.
Label
h
Diagn
Eobs
8E
E field
Shift rate
Label
(MHz)
(MHz)
(mV/cm)
MHz/(V/cm)2
Stark shift
(MHz)
2
10GH
10HI
10IK
10KL
10LM
♦10LM
9HI
9IK
9KL
H
C
C
E
E
A
B
F
F
4758.57(5)
196.745(2)
196.750(2)
105.873(5)
105.992(4)
104.874(2)
555.60(7)
423.969(4)
423.967(6)
1.412(5)
-0.049(20)
-0.044(20)
-0.620(5)
-0.501(4)
-0.191(2)
-3.04(7)
-0.080(4)
-0.082(4)
36.5
6.9
6.5
8.4
7.6
4.6
34.1
15.7
15.9
-1.01
-1.98
-0.36
2.05
10.75
10.75
-0.39
0.35
2.98
-0.0013
-0.0001
0
0.0001
0.0006
0.0002
-0.0005
0.0001
0.0008
H
G
D
D
D
G
D
4758.57(5)
491.88(2)
112.599(6)
112.595(10)
112.624(6)
491.91(1)
112.579(6)
1.412(5)
-0.087(20)
-0.084(6)
-0.088(10)
-0.059(6)
-0.06(1)
-0.104(6)
36.5
86.5
8.6
8.8
7.2
71.9
9.6
-1.03
-2.01
-0.37
2.08
10.93
0.36
3.03
-0.0014
-0.0150
0
0.0002
0.0006
0.0019
0.0003
d2
10GH
10HI
10IK
10KL
10LM
9IK
9KL
4.3-2
R e fle c te d m ic ro w a v e fie ld s and re s u ltin g lin e s h ifts .
E ffects due to a re f le c te d fie ld are m ost e v id e n t in th e H 2 9H I
tra n s itio n , show n in F ig . 4 -4 f. T he a ssy m etry in the lin e s h a p e is d u e to a
fe a tu re at the c e n te r fre q u e n c y , g>o==2 3 0 3 .6 9 M H z. T his f e a tu re c a n be
e x p la in e d by the la rg e r r e fle c tio n c o e ffic ie n t in th is re g io n , T ~ 0 .3 . E ach
185
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reg io n w ill h av e som e p o rtio n o f the tra v e lin g w av e re fle c te d back in the
o p p o site d ire c tio n . T he th e o re tic a l lin e sh a p e o f Eq. 2-2 d o es n o t c o n sid er
th e e ffe cts o f su c h a w av e.
A m ore a p p ro p ria te m o d el fo r the in te ra c tio n o f a R y d b e rg m olecule
w ith a tra v e lin g m ic ro w a v e fie ld in clu d es te rm s p ro p o rtio n a l to T and T2.
R ath er th an Eq. 2 -2 , fo r a c o p ro p a g a tin g m ic ro w a v e fie ld , the c o rre ct
tra n sitio n p ro b a b ility is g iv e n by
Sin(x) Sin(y)
Eq. 4-11
y2 _ e(z)E0T |2
in
w h ere the a m p litu d e is d e p e n d e n t on the <z> m a trix e le m e n t fo r the
tra n s itio n , the m ic ro w a v e e le c tric field , and th e in te ra c tio n tim e . The
firs t term is the m a in tr a n s itio n , w hich ap p ea rs a t a freq u e n c y co, D o p p ler
sh ifte d h ig h e r th a n th e re s o n a n t freq u en cy coo- T he o th e r tw o term s are
d ep en d e n t on th e r e fle c tio n c o e ffic ie n t. The se c o n d term g iv e s a
tra n s itio n fo r th e o th e r D o p p le r sh ifte d fre q u e n c y . H o w ev er, th e m ost
in te re s tin g is th e th ird te rm . B ein g lin e ar in T, th is term c o n trib u te s m ore
186
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to th e lin e sh a p e th a n th e T2 term . In a d d itio n , th is lin e is c en tere d at the
re so n a n t fre q u e n c y , so th e re is m ore overlap w ith the m a in tra n s itio n .
As an ex am p le, c o n s id e r a re s o n a n t freq u en cy o f o>o=130 M H z w ith a
b eam v e lo c ity o f p = 0 .0 0 3 4 4 . F o r sm all enough re fle c tio n s , as is the case
h e re , the seco n d te rm in the lin e sh a p e above can be n e g le c te d . W ith
r=0.1, the lin e sh a p e s fo r th e firs t and third term s o f E q. 4 -1 1 , along w ith
th e ir c o m b in atio n , a re show n in F ig. 4-8. T he lin e a r T term re s u lts in a
sm all -0.040 M H z s h if t in th e ex p ec te d lin e c e n te r. S in c e the H 2 (0 )1 0LM
tra n s itio n is n e a r th is fre q u e n c y and th e re fle c tio n c o e ffic ie n t o f the
A lfo rd reg io n is s im ila r, su ch a sh ift m ust be c o n sid e re d .
The “r e f le c tio n ” lin e s h ift can be estim ate d by th e c o n trib u tio n s o f
th e lin e a r term at th e h a lf-m ax im u m p o in ts o f th e m ain re so n a n c e .
Reflection Effects
— main transition
T term contribution
” ” combined terms
x>
es
.fi
2
eu
s
o
i
128
130
132
Microwave Frequency (MHz)
Fig. 4-8: Example resonance lineshape including reflections. The main Doppler shifted
resonance is shown with a solid line. The linear T term is the dashed line and the sum of
the two contributions is the dotted line. Clearly, the linear term has an effect o f shifting
the apparent linecenter.
187
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C o n sid e r the fre q u e n c y p o in ts w here x=±7t/2, w h ic h a re n e a rly th e h a lf­
m a x im a . At th e se fre q u e n c ie s , the lin e a r te rm c o n trib u te s
S m ( ± ^ ( l + 2p)-p< o0T j
A= = - V 2r,C os((J>-6)------- - --------------------------
”
Eq. 4-12
± ^(1 + 2 P )-P < 0 „T
T h is e x p re ssio n is fo r th e c o p ro p a g a tin g fie ld o n ly , w ith Ti b e in g the
re fle c tio n from “ end 1” . T ak in g th ese tw o a m p litu d e s h ifts , a lo n g w ith
th e slo p e at th o se p o in ts , a s h ift in the lin e c e n te r is g iv e n by
A .- A .
|slope|
Eq. 4-13
|slope| = —^-V 2
TC
8v= = - ^ : P
co0T C os ( P cd0t
) C os ( 4 » - 8 )
l+ 4 p - ( f P o . T )
I f th e p ro p a g a tin g fie ld is re v e rs e d , a s im ila r c o n trib u tio n is fo u n d , but
th e re fle c tio n w ill com e fro m th e o th er en d o f th e m ic ro w a v e re g io n and
is la b e le d T 2 - T his re fle c tio n s h ift is g iv en by
5v~ s ^ P ® o T C o s ( p © 0t ) c o s ( * - 8 )
1
1 - 4 P - ( | po>0t )
188
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Eq. 4-14
an d is the o p p o site s ig n s h ift. I f both re fle c tio n s are o f the sa m e
m ag n itu d e and p h a se , th e re fle c tio n sh ifts w ill can cel each o th e r. In o th er
w o rd s, by a v e ra g in g b o th d ire c tio n s o f p ro p a g a tio n , th is e ffe c t m ay can cel
o u t.
To d e te rm in e w h e th e r the re fle c tio n s a re th e sam e an d c a n c e l, or
w h e th e r th ere is a s y s te m a tic sh ift on th e lin e c e n te r, the H 2 (0)1 OLM
tra n s itio n w as re m e a s u re d w ith the m ic ro w a v e re g io n re v e rse d in
o rie n ta tio n . In th is w a y , Ti and T 2 w ill h a v e o p p o site e ffe c ts o n th e
lin e c e n te r. I f d iffe re n t, th e n e t re fle c tio n s h if t sh o u ld be o p p o s ite than
w ith the o rig in a l o r ie n ta tio n . T his m e a su re m e n t is liste d in T a b le 4 -4 a
an d lab eled w ith an a s te r is k . T he a v erag e lin e c e n te r is the sa m e fo r both
o rie n ta tio n s o f the m ic ro w a v e re g io n , illu s tra tin g th a t th is re g io n h as
w e ll-m a tc h e d fe e d th ro u g h s . N o t only are th e re fle c tio n s low , b u t th ey are
a lso alm ost id e n tic a l.
To c o n sid e r th e e ffe c ts on the o th e r tra n s itio n s , the fre q u e n c y
d e p en d e n ce o f Eq. 4 -1 4 w as d e te rm in e d . A s sh o w n in Fig. 4 -8 , th e
re fle c tio n sh ift co m es fro m th e fa c t th a t th e u n -D o p p le r s h ifte d te rm ,
lin e a r in T, lie s u n d e rn e a th th e m easu red tr a n s itio n . As the tr a n s itio n
freq u e n c y in c re a se s, th e D o p p le r sh ift also in c re a s e s and th is c o n trib u tio n
o f th e lin ear T term b e c o m e s n e g lig ib le . P lo ttin g Eq. 4-14 as a fu n c tio n
o f freq u en cy show ed a n o s c illa to ry , b u t d e c a y in g b e h av io r, as e x p e c te d .
F o r th e H 2 tra n s itio n s , w ith r = 0 . 1 , the re fle c tio n s h ifts w ere: 10H I 0 .0 0 2 , 10IK - 0 .0 0 5 , 10K L - 0 .0 1 3 , 10LM - 0.041 (all v a lu e s in M H z).
189
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For D 2 , the sh ifts w o u ld be even sm aller, sin c e P is sm a lle r. S ince the
10LM tra n sitio n is the m o st se n sitiv e to re fle c tio n s h if ts , and no effe ct
w as o b serv ed , no c o rre c tio n s w ere m ade to any o f th e m ea su re d
lin e ce n ters.
4.4 EFS related to V err and fitted for d ip o le p o la riza b ilities
The m easu red tra n s itio n lin e ce n ters, a fte r c o rre c tin g fo r the
sy stem atic S tark sh ifts, gav e a set o f m easured fin e s tru c tu re in te rv als.
T hese in te rv als w ere d e fin e d as AE, illu stra te d in F ig . 4 -1 . To re la te the
m easu red AE to the e ffe c tiv e p o te n tia l, the c o rre c tio n s m e n tio n ed in
S e ctio n 4.1-1 m ust be c o n sid e re d . The lead in g o rd e r, re la tiv is tic en erg y
sh ifts are d efin ed by Eq. 4 -3 . T hese com e from th e sta n d a rd “p 4”
c o n trib u tio n to th e H a m ilto n ia n . A m ore s ig n ific a n t c o rre c tio n com es
from the p e rtu rb a tiv e tre a tm e n t o f the e ffe c tiv e p o te n tia l. To e x tra c t the
d ip o le p o la riz a b ilitie s , th e m easu red in te rv a ls m u st be re la te d d ire c tly to
th e d iag o n al m a trix e le m e n ts o f Veff. H ow ever, th e 2 nd-o rd e r en erg y
c o rre c tio n s are n o t n e g lig ib le . T hese E(2) e n e rg ie s m u st be c a lc u la te d ,
and su b tra cted from th e m ea su re d e n erg ies, to a rriv e a t an in fe rre d value
o f th e diagonal EFS in te rv a l. Since these c o rre c tio n v a lu e s re q u ire a n o n ­
triv ia l c a lc u la tio n , the a sso c ia te d errors u ltim a te ly lim it th e a n a ly sis o f
the m easu rem en ts. B e ca u se o f th is lim ita tio n , a c a re fu l c o n sid e ra tio n o f
the so u rces o f e rro r w ill be p resen ted .
190
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C a lc u la tio n o f th e E (2) e n e rg y sh ifts w as b a se d on E q. 1-17. T h ree
lim ita tio n s e x iste d in p e rfo rm in g th is “in fin ite ” su m . T h ese lim itin g
fa c to rs arose as a re su lt of:
(a) tru n c a tin g the e ffe c tiv e p o te n tia l to th e le a d in g term s,
(b) d e te rm in in g a c c u ra te v a lu e s for th e o ff-d ia g o n a l m atrix
elem en ts o f the n e c e ssa ry co re p a ra m e te rs , and
(c) c a lc u la tin g the a p p ro p ria te energy d e n o m in a to rs for each
p e rtu rb in g lev el.
The sum was e v a lu ated o v e r b o th d isc re te and c o n tin u u m p e rtu rb in g
le v e ls; how ever, b o th p a rts had to be tru n c a te d . T h e d is c re te c o n trib u tio n
w as tru n c a te d at a high v a lu e o f n ( - 3 0 ) and th e re m a in d e r was
e x tra p o la te d to n —►<». T he c o n tin u u m lev els w ere e v a lu a te d out to 100
a.u . A ll p o ssib le R and L v a lu e s w ere in clu d e d , s in c e th e se are lim ite d by
th e g o v ern in g se le c tio n ru le s . T he v ib ra tio n a l le v e ls w ere in clu d ed up to
v= 2 . H igher v ib ra tio n a l le v e ls w e re not n e c e ssa ry , sin c e th e c o n trib u tio n s
b ecam e n e g lig ib le beyond v = 2 , due to the larg e e n e rg y se p a ra tio n b e tw ee n
th e le v e l o f in te re s t and an y p e rtu rb in g lev e ls .b ound to the (v>2) se rie s.
E ach o f the th re e lim ita tio n s lis te d ab o v e w ill be d is c u s s e d in m ore d e ta il.
The e ffe c tiv e p o te n tia l, as g iv en by Eq. 1-11, tru n c a te d a fte r th e
( r '4) term , has p re v io u sly b e e n d e fin e d as the “ lo w e s t-o rd e r p o la riz a tio n
m o d e l” (L O P M )17. T ru n c a tin g th e p o te n tia l to o n ly th e s e lo w e st th ree
term s w as show n in C h a p te r 3 to a c c u ra te ly m odel th e o p tic a l sp e c tra , at
th e n e ce ssa ry lev el o f p re c is io n . H ow ever, as m e n tio n e d b e fo re , the
191
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m icro w av e sp e c tro sc o p y is o f h ig h eno u g h p re c isio n to re q u ire m o re term s
in th e p o te n tia l, at le a s t fo r th e d ia g o n a l c o n trib u tio n s . O n th e o th e r
h a n d , only the LO PM w as u sed in c a lc u la tin g the E (2) s h ifts . T h is
tru n c a tio n o f the p o te n tia l g iv e s th e la rg e st sy ste m a tic so u rc e o f e rro r in
c a lc u la tin g E(2). C o n sid e r T a b le B - l , w hich show s a b re a k d o w n o f th e
c a lc u la te d sh ifts, fo r e ac h p e rtu rb in g (v ',R ')L ' se rie s an d e a c h o f th e th ree
le a d in g o rder term s [Q = q u a d ru p o le m atrix e le m e n t and P = d ip o le
p o la riz a b ility m a trix e le m e n t (b o th s c a la r and te n s o r)]. T he la rg e s t
c o n trib u tio n s com e fro m th e (0 ,2 )L '= L p e rtu rb in g s e rie s , an d th e d o m in an t
term is the QQ in te ra c tio n . (T h is QQ label re fe rs to the c o n trib u tio n
in v o lv in g the p ro d u c t o f tw o o ff-d ia g o n a l m atrix e le m e n ts o f th e
q u a d ru p o le m om ent: e .g . I < 0,2 | Q | 0 ,0 > | 2.) T h is term is e ff e c tiv e ly
p ro p o rtio n a l to r*6, w h ile th e n e x t tw o o rd ers are p ro p o rtio n a l to r '7 an d r '8
re s p e c tiv e ly . As can be se en in th is ta b le , the term s d e c re a s e ; h o w e v e r,
th e n e x t order, r*9 is n o t e x p e c te d to be n e g lig ib le (< 0.01 M H z). In
a d d itio n , tru n c a tin g th e p o te n tia l to th e LOPM n e g le c ts th e n e x t h ig h e r
term , the h e x ad e ca p o le m o m e n t (<j>), p ro p o rtio n a l to r*5. T he re s u ltin g Q<j>
term w ould also give an r '8 c o n trib u tio n to E(2). A re a s o n a b le e s tim a te o f
th e sy ste m atic e rro r re s u ltin g from th e tru n c a tio n w as ta k e n to be h a lf o f
th e sm a lle st c o n trib u tio n , d u e to th e PP term . T his is th e e rro r re p o rte d in
T ab le 4-7, along w ith th e to ta l E (2) and E rei c o rre c tio n s to e ac h (0 ,0 )n L
le v e l co n sid ere d in th is e x p e rim e n t. A s an e x am p le, fo r th e H 2 , (0 ,0 )1 0 H s
sta te th is gives E (2)= 4 7 .7 6 ± l .35 M H z.
192
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The E (2) c a lc u la tio n w as d ire c tly d e p en d e n t on th e o re tic a l e stim ate s
o f th e core p a ra m e te rs , an d sp e c ific a lly on th e re q u is ite o ff-d ia g o n a l
m a trix ele m e n ts o f Q , a s, and a t. F o r H 2 +, th e n e c e ssa ry m a trix elem en ts
w ere o b ta in e d 44 from an e x p lic it c a lc u la tio n u sin g th e a d ia b a tic
w a v e fu n c tio n s fo r H 2 +. T hey are ex p ec te d to be a c c u ra te to a b o u t 0.05%
fo r Q and 0.2% fo r a s and a t. T his is p rec ise en o u g h to c o n trib u te
T able 4-7: Energy shifts for each Rydberg level. All values are in MHz.
EG>
Ere,
9H
91
9K
9L
10G
10H
101
10K
10L
10M
-35.48(1.77)
-13.48(22)
-9.34(4)
-5.53(1)
157.11(14.45)
47.76(1.35)
14.62(18)
2.52(3)
0.01(1)
-1.09(1)
-23.667
-16.945
-12.016
-8.246
-25.792
-18.713
-13.813
-10.219
-7.471
-5.302
91
9K
9L
10G
10H
101
10K
10L
10M
-53.26(21)
-15.12(3)
-11.691(7)
-266.10(13.2)
-69.43(1.25)
-24.31(18)
-10.13(3)
-5.555(6)
-3.031(2)
-16.945
-12.016
-8.246
-25.792
-18.713
-13.813
-10.219
-7.471
-5.302
Label (nL)
h2
1
I
J
i
D2
i
193
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
n e g lig ib le e rro r, co m p a red to the tru n c a tio n e rro r. The v alu es u sed are
g iv en in A p p en d ix B . T here is no a d d itio n a l c o n trib u tio n to the re p o rte d
e rro r in E(2) from an y u n c e rta in tie s in the c o re p a ra m e te rs.
F in a lly , th e en erg y d e n o m in ato r fo r e ach p e rtu rb in g lev el co m es
from th e o re tic a l e stim a te s. T h is energy c a lc u la tio n has tw o p o s s ib le
fa u lts. F irst, the ro -v ib ra tio n a l en erg ies o f th e ion core w ill be d iff e r e n t
fo r each p e rtu rb in g se rie s. T h ese values h av e b e e n c a lc u la te d fo r b o th H 2
and D 2 and ta b u la te d " . E x p erim en tal te s ts o f th e s e v alu es have p ro v e n
th e ir a ccu racy to <0.01 c m '1.* T his e rro r, w h en p ro p ag a ted th ro u g h th e
E (2> c a lc u la tio n s, w as n e g lig ib le . S econd, th e e n e rg y d e n o m in a to r can
b eco m e qu ite sm all in the c ase o f a n earby p e rtu rb e r (n e a r-d e g e n e ra c y ).
W hen th is o c c u rs, th e E (2) sh ift w ill be d o m in a te d by the c o n trib u tio n
fro m th is sin g le p e rtu rb in g le v e l. For th is re a so n , the d iag o n al EFS
e n e rg ie s are in c lu d e d in the en erg y d e n o m in a to r to im prove th e a c c u ra c y
o f the c a lc u la te d shift.* The b e st te st fo r any n e a r d e g e n e ra c ie s w as in
m easu rin g re d u n d a n t n=9 and n=10 R ydberg in te rv a ls . If, a fte r a ll th e
c o rre c tio n s w ere m ad e , the tw o lev e ls a g re ed in th e a n a ly sis, a n e a r
d eg en eracy e rro r w as ru led o u t, since such an e ffe c t w ould be very
d iffe re n t for the tw o R ydberg lev els.
Im p ro v em en ts to the c ritic a l e v a lu a tio n o f th e E (2) c o n trib u tio n s
h av e been c o n sid e re d . The lim itin g facto r, tru n c a tio n o f the p o te n tia l,
c o u ld be d e c rea se d by ad d in g h ig h e r o rd er te rm s; h o w ev er, the re q u ire d
o ff-d ia g o n a l m a trix e le m e n ts w ould have to be c a lc u la te d and w ere n o t
194
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
a v a ila b le a t th is tim e . C a re fu l e v a lu a tio n o f b o th th e co re p a ra m e te rs and
e n e rg y d e n o m in a to rs has a lre a d y s ig n if ic a n tly red u c ed a d d itio n a l e rro rs
th a t aro se b e c a u se o f the tw o c o n trib u tio n s d e s c rib e d ab o v e. O th e r
n u m e ric a l m eth o d s fo r c a lc u la tin g E(2), in v o lv in g very d iffe re n t
a p p ro a c h e s, are c u rre n tly u n d e r in v e s tig a tio n .
I 3 -45
The tw o c o rre c tio n e n e rg ie s , E (2) a n d E rei, w ere c a lc u la te d f o r e ach
le v e l. The c o n trib u tio n to e ach m e a su re d tr a n s itio n is d e te rm in e d by
su b tra c tin g E(l+ d - E( d - T h ese d iffe re n c e s a re lis te d in T ab le 4 -8 a lo n g
T able 4-8: Inferred polarization energies. The measured intervals are used to infer values
for the diagonal contributions o f the polarization potential. The calculated energy
corrections are subtracted from the observed energies to obtain the final values.
h
2
Label
AE(obs>
ae®
A E „I
10GH
10HI
10IK
10KL
5173.937(27)
1659.210(9)
630.793(11)
274.073(24)
129.673(21)
2303.687(230)
864.564(6)
370.693(11)
-109.35(14.5)
-33.14(1.36)
-12.10(18)
-2.51(3)
-1 .1 0 (1)
2 2 .0 ( 1 .8 )
4.14(22)
3.81(4)
7.079
4.900
3.594
2.748
2.170
6.722
4.929
3.770
5276.208(14.5)
1687.45(136)
63930(18)
273.835(38)
128.603(23)
2274.97(1.8)
855.495(220)
363.113(41)
5325.44(7)
1687.45(6)
638.040(17)
273.044(15)
129.435(11)
873.259(30)
359.517(19)
196.67(13.2)
45.12(1.26)
7.079
4.900
3.594
2.748
2.170
4.929
3.770
5121.69(13.2)
1637.43(13)
620.266(180)
265.726(34)
124.741(13)
830.19(21)
352317(36)
i
!
10LM
9HI
9IK
9KL
AE( ^inferred
d2
10GH
10HI
10IK
10KL
10LM
9IK
9KL
14.18(18)
4.57(3)
2.524(6)
38.14(21)
3.43(4)
195
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w ith the c o rre sp o n d in g o b se rv e d fin e stru c tu re in te rv a l, w h ic h h a s a lre a d y
been c o rre c te d fo r th e S tark e ffe c t. E ach en erg y s h ift is s u b tra c te d o f f th e
o b serv ed v a lu e o f c o lu m n
2
, a n d th e re s u ltin g en erg y is g iv e n in th e la s t
co lu m n . T hese in fe rre d e n e rg ie s , AE (1 ’inferred, can be d ire c tly r e la te d to
th e e ffe c tiv e p o te n tia l.
The fin e s tru c tu re e n e rg ie s g iv e n by Eq. 4-2, w ith th e firs t
c o e ffic ie n t re p la c e d by B 4 , w e re re la te d to the in fe rre d in te rv a ls b y
*p(') —p(')
~
n(L-t-l)
_ pd
nL
= B 4 A (r-4) + B 6 A(r-6) + B 7 A (r-7) '
Eq. 4-15
T he h y d ro g e n ic ra d ia l m atrix e le m e n ts w ere c a lc u la te d u sin g th e fo rm u la s
g iv en in S e c tio n 4 .1 -1 . The a n a ly s is w as sim p lifie d by d iv id in g th ro u g h
E q. 4-15 by A<r*4>. T he re s u lt is an e q u a tio n lin e a r in th e ra tio
A < r' 6 >/A <r‘4>, e x c e p t fo r the sm a ll fin a l term , p ro p o rtio n a l to A<r*7>.
T he in fe rre d e n e rg ie s (in a .u .), sc a le d by the c a lc u la te d v a lu e s o f th e
m a trix e le m e n ts, w ere p lo tte d as a fu n c tio n o f the ra tio , A<r*6> / A < r'4>.
T h is set o f v a lu e s w as fitte d fo r th e a p p ro p ria te c o e ffic ie n ts , w h e re B 4
gav e the in te rc e p t, B 6 g av e th e s lo p e , an d B 7 gave a s lig h t c u rv a tu re o f the
lin e . The ra tio s are g iv e n in T a b le 4-9 alo n g w ith th e fitte d v a lu e s and
th e d iffe re n c e s b e tw e e n th e f itte d and m ea su re d v a lu e s.
The H 2 fit w as w e ig h te d a c c o rd in g to th e e rro r b a rs sh o w n , a n d a ll
th re e c o e ffic ie n ts w e re a llo w e d to v a ry . F ig u re 4-9 sh o w s th e re p o rte d
d a ta and th e fitte d lin e (so lid ). In a d d itio n , a stra ig h t lin e (d o tte d ) is
g iv en , w h ich n e g le c ts th e r *7 d e p e n d e n c e . T his illu s tra te s th e c o n v e rg e n c e
196
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o f the p o la riz a tio n m odel as L in cre ases. T h e re w as little r*7 d e p en d en ce
fo r the h ig h est-L in te rv a ls and, th e re fo re , n e g le c t o f th e r*8 c o n trib u tio n
w as ju s tifie d . T h e fitte d re s u lts are g iv en in T a b le 4 -1 0 and th e e rro r bars
are the s ta tis tic a l fit e rro rs fo r each p a ra m e te r. S in c e th e B 7 c o e ffic ie n t
v a rie s n e g lig ib ly w ith p, the c o re in te rn u c le a r s e p a ra tio n , th e fitte d value
o f B 7=-19.7 (2 .0 ) a g reed fa irly w ell w ith th e v a lu e fo u n d in th e p rev io u s
m easu rem en t3 o f H 2 (v = 0 ,R = l) n=10 R y d b erg in te r v a ls , B 7= -21(6). In
a d d itio n , sin ce th is c o e ffic ie n t w as not e x p e c te d to v a ry s ig n ific a n tly , B 7
w as fixed fo r th e D 2 fit, show n in Fig. 4 -1 0 . V a ry in g B 7 by one stan d ard
d e v ia tio n and fittin g ag ain d id no t change th e o th e r v a lu e s s ig n ific a n tly .
Table 4-9: Analysis o f the inferred polarization energies. AE(,) is in a.u. The fitted
values, along with the difference from the measured values, are in the last two columns.
h2
d2
Label
A<r6>/A<r4>
A<r'7>/A<r'4>
AE(1)/A<r‘4>
Fit values
5fit(10*3)
10GH
1.19319e-2
1.56147e-3
-1.51936(418)
-1.51996
0.60(4.2)
10m
4.63457e-3
3.47834e-4
-1.55443(125)
-1.55404
-0.39(1.3)
10IK
2.11291e-3
1.02473e-4
-1.56942(44)
-1.56924
-0.18(44)
10KL
1.05345e-3
3.50175e-5
-1.57631(22)
-1.57633
0.02(22)
10LM
5.49152e-4
1.28101e-5
-1.58002(28)
-1.57990
-0.12(28)
9HI
4.41868e-3
3.24134e-4
-1.55518(123)
-1.55529
0.11(1.2)
9IK
1.95857e-3
9.13748e-5
-1.57013(40)
-1.57025
0.12(40)
9KL
9.34770e-4
2.90738e-5
-1.57711(18)
-1.57716
0.05(18)
10GH
1.19351e-2
1.5621 le-3
-1.47406(380)
-1.47692
2.9(3.8)
10HI
4.63583e-3
3.47976e-4
-1.50754(120)
-1.50783
0.29(1.2)
10IK
2.11349e-3
1.02515e-4
-1.52187(42)
-1.52194
0.07(42)
10KL
1.05373e-3
3.50318e-5
-1.52880(20)
-1.52857
-0.23(20)
10LM
5.49301e-4
1.28153e-5
-1.53173(16)
-1.53192
0.18(16)
9IK
1.95910e-3
9.14121e-5
-1.52286(39)
-1.52288
0.02(39)
9KL
9.35024e-4
2.90857e-5
-1.52939(16)
-1.52934
-0.05(16)
197
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-1 .5 1
fitted potential
without r'7 contribution
-1.52
10GH
-1.53
-1.54
10HI
9H I
<
-1.56
10IK
.57
9 IK
10IK
-1.57
9IK
10KL
9KL
-1.58
10KL
9KL
-/lOLM
-1.58
IOLM
-1.59
0.000
0.004
0.002
-1.60
0.000
0.002
0.004
0.006
0.008
0.010
0.012
A < r6> /A < r 4>
Fig. 4-9: Plot o f measured H 2 intervals as a function o f radial matrix elements. An
expanded view is shown for the highest-L intervals.
198
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.014
-1 .4 6
fitted potential
without r*7 contribution
-1.47
-1.48
10GH
- 1 .4 9
-1 .5 0
-1.52
10HI
W
<
10IK
9IK
-1.52
10IK
9IK
-1.53
9KL
10KL
10KL
-1.53
10LM
10LM
-1.54
0.000
0.002
-1.55
0.000
0.002
0.004
0.006
0.008
0.010
0.012
A<r"6> /A < r 4>
Fig. 4-10: Plot o f measured D2 intervals as a function of radial matrix elements. An
expanded view is shown for the highest-L intervals.
199
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0 .0 1 4
T able 4-10: Fitted coefficients for H 2 and D2 - The error bars on the fitted parameters are
assigned according to the variation in each necessary to increase x2 by one. In other
words, with a single parameter fixed to its best value the others were fitted to the data.
The resulting y} was recorded. The same parameter was then varied by a fixed amount
and the other parameters were then fitted again. The new x2 was calculated. When the
new x2 varied from the original by one, the difference in the parameter from the best
value was assigned as the error, (e.g. the B6 parameter had a best value o f 7.94 in H2 ;
however, when varied by 0.51 and the other two parameters were fitted, the x2 would
change by one)
Coefficient
h 2 (0 ,0 )
D2(0,0)
b4
-1.58401(36)
-1.53579(18)
b6
7.94(51)
7.51(16)
b7
-19.7(5.6)
Fixed (-19.7)
X2red
0.125
0.70
In te rp re tin g th e f itte d B 4 c o e ffic ie n t req u ires c o n sid e ra tio n o f the
o rig in a l d e riv a tio n , in c lu d in g the H a m ilto n ia n (E q. 1-6) and th e defined
p o la riz a tio n p o te n tia l (E q. 1-11). T he e m ass d e p e n d e n c e w as neg lected
w hen ex p an d in g th e p o te n tia l in th e m u ltip o le se rie s. W hen th ese term s
are k e p t, a d d itio n al fa c to rs are found fo r each term in th e p o te n tia l. As
d isc u sse d by S tu rru s5, th e d ip o le term in V is m u ltip lie d by (1+ e). T his
lead s to a facto r
( 1 + e )2
in the Veff term c o rre sp o n d in g to th e d ipole
p o la riz a tio n en erg y . S tu rru s et al. a rg u e d th at sin ce th is fa c to r arose in a
ch an g e o f v a ria b le s in v o lv in g th e R y d b erg e le ctro n , it w ould not be lik e ly
to a rise in a c a lc u la tio n o f the p o la riz a b ility o f th e fre e H 2 + io n . Indeed
the e x p re ssio n , e x c lu siv e o f th e
( 1 + e ) 2
fa c to r, could be show n to reduce to
the c o n v e n tio n a l e x p re ss io n fo r th e d ip o le p o la riz a b ility o f H 2 +. This led
S tu rru s et al. to d e fin e th e r ' 4 term in VCff by
200
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
R ecen tly , S h e rtz e r46 has show n th at, i f the p o la riz a b ility o f H 2 * is
c alcu late d w ith o u t c o n s tra in in g th e p rotons to be s ta tio n a ry , a facto r o f
e x a c tly th is form d o es in d e e d o c cu r in the c a lc u la tio n o f the free io n ’s
p o la riz a b ility . T his is m o st e a sily seen by w ritin g th e H am ilto n ia n fo r th e
free ion in an e le c tric fie ld .
Eq. 4-16
2M
2M
2m
+ ern2- e r eI)-E
By changing v a ria b le s,
P = r„2 ~ rnl
?1 =
f e. - f e ,
Eq. 4-17
+% a)/2
?„ = (M(rnI + f n2) + mrel )/(2M + m)
Eq. 4-18
+
r,+ p /2 |
|r ,- p /2 |
+ e(l + 8)r, -E
_1__ 2_
1
1
m
2M + m
201
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T he fac to r (1+e) m u ltip lie s the d ip o le i n te r a c tio n term , j u s t as in the
R y d b erg c a lc u la tio n .
One w ay to in te rp r e t th is term , a n d to see its c o n n e c tio n to th e
m o tio n o f the p ro to n s is to note th a t th e c e n te r o f m ass o f the ion w ill
a c c e le ra te in the e le c tric fie ld
eE
acra ~ (2 M + m)
T h is w ill give rise to a “ fic titio u s fo rc e ” , in th e fram e o f th e ion, and w ill
be seen by th e e le c tro n .
Ffict = -m a = -ee E
T he e le c tro n sees th is as an a d d itio n a l e le c tr ic fie ld
so th e net in te ra c tio n e n e rg y for the e le c tro n is
Eq. 4-19
= +e(l + e)E r,
In th is w ay, it is c le a r th a t p re c ise ly th is fa c to r w ill o c c u r in any p ro p e r
c a lc u la tio n o f the fre e ion p o la riz a b ility . It is e q u a lly c le a r th at th e
sp e c u la tio n o f S tu rru s et a l .5, th at such a fa c to r w o u ld be ab sen t from th a t
c a lc u la tio n , is c o m p le te ly fa lse . T h e re fo re , it se e m s m uch m ore c o rre c t to
in c lu d e th is fa c to r in th e d e fin itio n o f th e p o la r iz a b ility , re su ltin g in
Eq. 4-20
202
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
S in ce ( l + e ) 2= l . 000545 fo r H 2 + and 1.000272 fo r D 2 +, th is c o n v e n tio n
m o d ifie s th e v a lu e o f a s in fe rre d from th e o b s e rv a tio n s by 0.0 5 4 % and
0 .0 2 7% re s p e c tiv e ly . W ith th e c o n v en tio n d e fin e d by Eq. 4 -2 0 , the
re s u ltin g v a lu e s o f a s, o b ta in e d from the fitte d v a lu e s o f B 4 , are show n for
b o th ions in T ab le 4 -1 1 .
Table 4-11: Comparison o f measured and theoretical polarizabilities. The calculated
polarizabilities are based on several different techniques as mentioned in the text. The
percentage difference between the experimental (E) and theoretical (T) values are given
for each set, H2+ and D2+. The polarizabilities are in (ao3). The fitted values for B6 are
given in the last row and compared to theoretical values47.
Technique
h 2+
(E-T)/E %
d 2+
(E-T)/E %
Measured
3.1680(7)
—
3.0716(4)
—
Ad/cn4*
3.1713
-0.104(22)
3.0732
-0.052(13)
A d '3
3.1667
+0.042(22)
3.0708
+0.026(13)
Non-ad49
3.1682(4)
-0.006(25)
3.0714(4)
+0.007(18)
Non-adS0
3.1685
-0.016(22)
3.0719
-0.010(13)
Measured B6
7.94(51)
Calculated B6
7.82
7.51(16)
1.5(6.4)
7.25
.
3-5(2.1)
P rio r to th is e x p e rim e n t, th e o re tic a l c a lc u la tio n s o f a s w ere lim ite d
by tw o a p p ro x im a tio n s:
1) the a d ia b a tic H 2 + w a v e fu n c tio n s (a s su m in g a sin g le e le c tro n ic
sta te as a fa c to r)
2) the “ c la m p e d -n u c le u s ” a p p ro x im a tio n fo r a s, (c o m p u tin g a ± (p )
and ct||(p) an d a v e ra g in g each o v e r ro -v ib ra tio n a l fu n c tio n s,
203
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
w hich e ffe c tiv e ly n e g le c ts the in flu e n ce o f ro ta tio n a l e n erg ies on
the induced d ip o le m o m e n t.)31
W ith in th e se two a p p ro x im a tio n s, the b e st c a lc u la te d v a lu e s are show n in
the seco n d line o f T ab le 4 -1 1 .
The a v a ila b ility o f p re c ise m ea su re m e n ts has stim u la te d a good deal
o f re c e n t th e o re tic a l a c tiv ity . W illiam C lark o f CU B o u ld e r13, has
re c e n tly rep o rted c a lc u la tio n s o f a s fo r b o th ion g ro u n d sta te s, w hich
a v o id the second a p p ro x im a tio n , b u t s till use the firs t. T h ese re su lts,
show n on lin e 3 o f T ab le 4 -1 1 , c le a rly rem ove m ost o f th e d isc re p a n c y
w ith e x p erim en t. The re m a in in g d iffe re n c e s are now o n ly 0 .0 4 2 (2 2 )% in
H 2 + and 0.0 2 6 (1 3 )% in D 2 +. T his is a p p ro x im a te ly th e e x p e c te d (m e/m n)
lev e l for th e n o n -ad ia b atic e ffe c ts .
E ven m ore re c e n tly , tw o in d e p e n d e n t c a lc u la tio n s, w h ich avo id b o th
a p p ro x im a tio n s giv en a b o v e , have b een in fo rm a lly re p o rte d .49 30 An
in d e p e n d e n t third c a lc u la tio n is a p p a re n tly close to c o m p le tio n .44 T ab le
4-11 show s the tw o n o n -a d ia b a tic c a lc u la tio n s th a t h a v e b een rep o rte d .
T he re s u lts o f both S h e rtz e r and M oss a re in e x c e lle n t a g re e m e n t w ith
e x p e rim e n t for both ions.
A ll o f these new , n o n -a d ia b a tic c a lc u la tio n s use d iffe re n t m eth o d s.
T he c a lc u la tio n o f S h e rtz e r u ses the fin ite -e le m e n t m eth o d [FE M ], a
p u re ly num erical a p p ro a ch to d e a lin g w ith the th re e -b o d y p roblem . T h is
m eth o d w ould be d iffic u lt to ap p ly to any state o th e r th a n th e ion g ro u n d
sta te . T he c a lc u la tio n s o f M oss and th o se o f B abb ( s till in p ro g re ss) use
204
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m eth o d s th a t sh o u ld be e a s ily a p p lic a b le to o th e r s ta te s o f H 2 *. C learly ,
th is m easu rem en t has a lre a d y stim u la te d s ig n ific a n t im p ro v e m e n ts in the
th e o ry o f th is fu n d a m e n ta l ion.
4.5 C o n clu sio n s o f th e m icrow ave sp e c tr o sc o p y
M icrow ave s p e c tro s c o p y o f H 2 and D 2 R y d b e rg le v e ls b o u n d to the
g ro u n d sta te o f th e io n c o re h a v e y ie ld e d p re c ise v a lu e s fo r th e sc ala r
a d ia b a tic d ip o le p o la r iz a b ilitie s . T hese re s u lts sh o w e d a s ig n ific a n t
d isc re p a n c y w ith the b e st a v a ila b le th e o re tic a l c a lc u la tio n s , w h e n in itia lly
re p o rte d . T he d isc re p a n c y sc a le d as e x p ected w ith m a ss, d e c re a s in g for
th e D 2 system . T he H 2 + p o la r iz a b ility [ a s= 3 .1 6 8 0 (7 )] w as m e a su re d w ith
a p re c isio n o f 0 .0 2 2 % , w h ic h w as beyond the le v e l at w h ic h th e a d ia b atic
a p p ro x im a tio n , m ade in tr e a tin g the ion core, w as e x p e c te d to fa il. The
m ea su re d re s u lt has a lre a d y s tim u la te d a d d itio n a l th e o r e tic a l tre a tm e n ts o f
th e H 2 + ion. N ew er c a lc u la tio n s , sp e c ific to th e R =0 le v e l, a g re e q u ite
w e ll w ith th e m e a su re d v a lu e s ; h o w e v er, a m ore g e n e ra l tre a tm e n t o f the
io n is yet to be p re s e n te d an d w ill be a g re a te r c h a lle n g e . T he D 2 +
p o la riz a b ility [ a s= 3 .0 7 1 6 (4 )] h a s aid ed by d e te rm in in g th e m ass
d e p en d e n ce o f new th e o re tic a l c a lc u la tio n s.
205
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix A
R o-vib ration al energies and core p a ram eters
T h e o re tic a l p re d ic tio n s o f th e energy s p littin g s fo r ro ta tio n a l and
v ib ra tio n a l lev els o f the fre e H 2 +, H D +, and D 2 + io n s a re ta b u la te d in th e
lite r a tu r e .1* B elo w are ta b le s o f a few o f the lo w e st le v e ls (v< 3 a n d R <5),
w h ich are used to c a lc u la te th e E (0) en erg ies o f th e R y d b e rg sy ste m s.
T h ese sp littin g s d e te rm in e th e se p a ra tio n b e tw ee n n e ig h b o rin g R y d b erg
e le c tro n lev els, w hich b e c o m e s im p o rta n t w hen c o n s id e rin g th e m ix in g
b e tw e e n v a rio u s se rie s.
H
v \R
0
1
2
0
-21380.812
-19189.686
-17125.772
2
+ (a ll v a lu e s are in c m '1)
1
-21322.578
-19134.517
-17073.564
H D + (a ll
v \R
0
1
2
0
-21516.071
-19603.078
-17786.218
2
-21206.574
-19024.624
-16969.574
3
-21033.717
-18860.886
-16814.644
4
-20805.352
-18644.591
-16610.010
v a lu e s are in c m '1)
2
1
-21472.209
-19561.219
-17746.300
-21384.749
-19477.753
-17666.709
3
-21254.209
-19353.183
-17547.926
206
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
-21081.358
-19188.246
-17390.666
D 2 + (a ll v a lu e s are in c m '1)
0
v\R
0
1
2
1
-21711.47
-20134.32
-18621.85
-21682.08
-20106.02
-18594.63
2
-21623.42
-20049.55
-18540.29
4
3
-21535.71
-19965.12
-18459.06
-21419.31
-19853.08
-18351.26
E ach o f these free ions c an be d e sc rib e d by a few p a ra m e te rs
in c lu d in g a q u ad ru p o le m om ent (Q ) and a d ia b a tic d ip o le p o la riz a b ilitie s
( a s and a t). T hese v a lu e s have b e en c a lc u la te d and m e a su re d fo r the (0 ,1 )
lev el o f H 2
* 5
The c a lc u la te d v a lu e s for o th e r le v e ls w e re th e n sc ale d to
m atch th e (0 ,1 ) e x p erim e n tal r e s u lts , d e n o te d by an a s te ris k . An
a d d itio n a l s h ift in Q fo r HD* is in c lu d e d , as d isc u sse d in th e te x t. T hese
v a lu e s w ere used to c a lc u la te e n e rg ie s for th e laser e x p e rim e n t; h ow ev er,
som e s lig h t changes w ere m ade fo r th e m icrow ave e x p e rim e n t and are
d isc u sse d th e re .
< V=0,R | Q | V=0,R > (ea02)
R
0
1
2
3
4
"
hT
'
1.63902
*1.64295
1.65082
1.66262
1.67836
HD+
1.73147
1.73459
1.74082
1.75017
1.76264
d 2t
1.60608
1.60801
1.61186
1.61764
1.62535
207
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
< v=0,R
R
0
1
2
3
4
a s | v=0,R
h 2+
HD"
3.1675
*3.1770
3.1960
3.2245
3.2625
0
1
2
3
4
3.995
*4.015
4.054
4.113
4.192
A
04*
©
II
>
n r
3.1190
3.1260
3.1401
3.1613
3.1895
a
O
II
>
V
R
> (a 03)
HD"
3.885
3.899
3.928
3.971
4.029
d
2"
3.0712
3.0758
3.0850
3.0988
3.1173
(ao3)
D 2"
3.777
3.786
3.805
3.833
3.870
208
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A ppendix B
D escrip tion o f E(2) con trib u tio n s
The fo llo w in g is a b rie f d e s c rip tio n o f th e m eth o d used in
c a lc u la tin g th e s e c o n d -o rd e r e n erg y c o n trib u tio n s fo r H 2 and D 2 . T he sum
in Eq. 1-17 w as e v a lu a te d , for each s ta te o f in te re s t, u sin g the LO PM
p resen te d in E q. 1-11. To c a lc u la te th e e n e rg y s h ift fo r a given sta te
(0,R)nLN , th e sum w as c arried o u t o v e r a ll d is c re te and con tin u u m sta te s
th at w ere m ix e d w ith th a t p a rtic u la r le v e l. T h is in c lu d e d v'= 0 to 5, R '= 0
to 5, and L '= 0 ,L ± 2 .
C o n sid e r th e th re e term s o f th e e ff e c tiv e p o te n tia l used to c a lc u la te
the re q u ire d m a trix e le m e n ts. T h ese th re e te rm s can co m b in e to giv e
th ree d iffe re n t e ff e c tiv e pow ers o f r*s. T h e < Q x Q > te rm (r*6), th e
< Q x P > te rm (r*7), a n d th e < P x P > te rm (r*8). A s an ex am p le, T ab le B -l
show s a b re a k d o w n o f each o f th e se c o n tr ib u tio n s fo r eac h (v ',R ')L ' se rie s,
both d is c re te and c o n tin u u m . T he to ta l E (2) c o n trib u tio n is the sum o f all
th ese v a lu e s.
209
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T able B -l: The second-order fine structure energies for the H 2 and P 2 (0,0)10Hs states.
Each mixing series is listed in the first column. The energies are broken down into
separate contributions for each m ix in g term and for each type o f perturbing level,
whether discrete or continuum states. The sums are given in the last column, for each
series. The total E(2) contribution is given at the bottom o f each section. All values are in
MHz.
E(2) calcu lation s for (0,0)10H 5 state
Label
(vl,R,)L'
Q Q
Discrete
QP
PP
Sums
Continuum
Q Q
Q P
PP
h2
d2
(0,0)H
(0,2)F
(0,2)H
(0,2)K
(1,0)H
(U )F
(U )H
(U )K
(2,0)H
(2,2)F
(2,2)H
(2,2)K
42.63
65.87
-0.29
0.00
2.67
-2.04
0.00
0.00
0.04
-0.01
0.00
0.00
4.21
5.26
-0.01
0.00
0.69
-0.37
0.00
0.00
-0.01
0.00
0.00
0.04
0.18
0.11
0.00
-0.26
0.04
-0.02
0.00
0.00
0.00
0.00
0.00
0.00
-0.08
-27.70
-32.68
0.00
0.00
-0.57
-0.88
0.00
0.00
0.00
-0.01
0.00
-0.06
-3.02
-2.88
0.00
0.00
-0.14
-0.17
0.00
0.00
0.00
0.00
-2.48
-0.01
-0.09
-0.07
-0.11
0.00
-0.01
-0.01
0.00
0.00
0.00
0.00
-2.44
46.87
40.43
-35.93
-.37
3.40
-3.15
-1.06
0.00
0.03
-0.01
-0.01
47.76
(0,0)H
(0,2)F
(0,2)H
(0,2)K
(1,0)H
d ,2 )F
(L2)H
0 ,2 )K
(2,0)H
(2,2)F
(2,2)H
(2,2)K
0.00
39.63
-42.02
-0.32
0.00
1.53
-0.60
0.00
0.00
0.03
0.00
0.00
0.00
3.98
-2.99
-0.01
0.00
0.39
-0.09
0.00
0.00
-0.01
0.00
0.00
0.04
0.12
-0.05
0.00
-0.07
0.03
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-0.08
-27.55
-31.78
0.00
0.00
-0.48
-0.70
0.00
0.00
0.00
0.00
0.00
-0.06
-2.88
-2.69
0.00
0.00
-0.10
-0.13
0.00
0.00
0.00
0.00
-2.33
-0.01
-0.08
-0.06
-0.08
0.00
-0.01
-0.01
0.00
0.00
0.00
0.00
-2.29
43.58
-75.57
-34.86
-0.15
1.95
-1.28
-0.84
0.00
0.02
0.00
0.00
-69.43
0 .0 0
f
210
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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0.00
-005
-1.39
-1.11
TT
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-32.61
0.32
-i .17
0.64
;sE
lO'i-
0.00
149.28
310.55
-0.20
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-117.35
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ir ^ .p o
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Total G
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-168.83
-120.95
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SumG 302.15
-0.20
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(0,0)1( IH5
H2+ (in Ml te)
1
1
Mar 216 98 no cutoff
(nu'R')L
LHI 6X30.FOR
PH2CNT.FOR
QP
P2
Q2
Q2
QP
P2
(0,0)H
(0,2)F
(0t2)H
(0.2)K
0.00 0.00 0.04
0.00 0.00
42.63 4.21 0.18 -0.08 -0.06
65.87 5.26 0.11 -27.70 -3.02
-0.29 -0.01 0.00 -32.68 -2.88
-2.48
-0.01
-0.09
-0.07
(1,0)H
(1.2)F
(1,2)H
(1*2)K
0.00
2.67
-2.04
0.00
(2,0)H
<2,2)F
(2,2)H
(2,2)K
0.00
0.04
-0.01
0.00
Series Sums:
(Q,As,At)
(1.63856,3.17128,4.0078)
(1.64455,3.18567,4.0370)
(1.64455,3.18567,4.0370)
(1.64455,3.18567,4.0370)
-2.44
46.87
40.43
-35.93
48.82
K)
►
—
*
to
0.00 -0.26
0.69 0.04
-0.37 -0.02
0.00 0.00
0.00 0.00 -0.11
0.00 0.00 0.00
-0.57 -0.14 -0.01
-0.88 -0.17 -0.01
(.34266,.83930,1.73040)
(.31371,.78531,1.66558)
(.31371,.78531,1.66558)
(.31371,.78531,1.66558)
-0.37
3.400
-3.15
-1.06
-1.18
Sum H
Sum F
Sum K
Total H
Total F
Total K
0.00 0.00 0.00 0.00 0.00
-0.01 0.00
0.00 0.00 0.00
0.00 0.00
0.00 0.00 0.00
0.00 0.00 -0.01 0.00 0.00
0.09
-2.78
63.82 4.89 -0.13 -28.27 -3.16 -2.69
45.34 4.89 0.22 -0.08 -0.06 -0.01
-0.29 -0.01
0 -33.57 -3.05 -0.08
0.09
-2.78
34.46
50.30
-37.00
(-.026280,.035708,. 195633)
(-.028915,.026167,.167104)
(-.028915,.026167,.167104)
(-.028915,.026167,.167104)
0.00
0.03
-0.01
-0.01
0.01
Grand Total:
47.76
rA-8 terms -2.69
^ CD CD O
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(0,0)1 ( K7
H2+ (in Ml
LHRAX30.FOR
(nu'R')L
N>
q2
(0,0)K
(0,2)H
(0,2)K
(0.2)M
0.00
6.29
2.37
-0.78
Qp
000
0.25
0.11
0.03
(1,0)K
0.00
0.78
0.08
-aoi
0.00
-0.08
-0.01
0.00
(1.2)K
(1,2)M
(2,0)K
(2,2)H
(2,2)K
(2,2)M
M ari !8 98 no cutoff
PH2CNT.FOR
Q2 QP
P2
-0.03
0.00
0.00
0.00
(Q,As,At)
Series Sums:
-0.03
0.00
0.00
0.00
0.00
0.00
-0.89
-3.47
0.00
6;oo
0.00
aoo
0.00 0.00 0.00 (.34266,.83930,1.73040)
0.00 OOO o.oo (.31371,.78531,1.66558)
-0.02 o.oo 0.00 (.31371,78531,1.66558)
-0.07 -0.01 0.00 (.31371,.78531,1.66558) ;
SumK
SumH
SumM
0.00 0.00 0.00
000 0.00 0.00
OOO 0.00 0.00
0.00 0.00 0.00
0.03
2.29 0.10 -0.03
5.51 017 0.00
-0.79 -003 0.00
Total K
TotalH
Total M
1.36
5.68
-4.52
0.00
o.oo
0.06
0.15
(1.63856,3.17128,4.0078)
(1.64455,3.18567,4.0370)
(1.64455,3.18567,4.0370)
(1.64455,3.18567,4.0370)
-0.06
6.54
1.53
-4.43
3.58
-0.86
-0.11
-0.09
-1.06
000 0.00 0.00 (-.026286,.035708,.195633)
0.00 OOO 0.00 (-.028915,.026167,.167104)
0.00 0.00 0.00 (-.028915, 026167,.167104)
0.00 0.00 OOO (-.028915,.026167,.167104)
-0.03
-0.91 0.06 -0.03
0.00 0.00 0.00
-3.54 0.16 0.00
0.00
Grand Total:
2.52
rA-8 terms -0.06
0.00
0.00
0.00
0.00
0.00
s :
c
+
CM
X
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(0,0)1( L8
(nu'R')L Q2
LHIIX30.FOR
p2
QP
■ta)
Mar 2!8 98 Ino cutoff
PH2CNT.FOR
Q2
QP P2
(Q,As,At)
Series Sums:
(0.2)N
0.00
300
-1.01
-0.79
0.00 -001
0 08 0.00
-6.02 0.00
-0.02 0.00
0.00
0.00
0.13
-0.94
0.00
0.00
-6.01
-0.03
0.00
0.00
0.00
o.oo
(1.63856,3.17128,4.0078)
(1.64455,3.18567,4.0370)
(1.64455,3.18567,4.0370)
(1.64455,3.18567,4.0370)
( iM
(1,2)!
(1.?)L
(1.2)N
0.00 0.00 000
0.07 -0 01 0.00
0 02 0.00 0.00
0.01 0.00 0.00
0.00
000
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
000
0.00
(.34266,.83930,1.73040)
(.31371,78531,1.66558)
(.31371,.78531,1.66558)
(.31371,78531,1.66558)
(2.0)L
(2,2)1
(2,2)L
(2,2)N
0.00
0.00
0.00
0.00
0.00 0.00
0.00 o.oo
0.00 0.00
o.oo 0.00
(0,0)L
(0,2)1
-0.01
3.08
-1.17
-1.78
0.12
to
i—*
t/*
0.00
-0.08
-0.02
-0.01
-0.11
0.00
0.00
0.00
0.00
SumL
Sum I
SumN
0.00
0.00
0.00
000
-6.01
-1.03 -6.02 -6.01
2.93 0.07 o.oo
-0.80 -0.02 0.00
Total L
Total I
Total N
-1.2
3.00
-1.79
0.00
0.00
0.00
0.00
0.00
-0.13 -0.01 0.00
o.oo 0.00 0.00
-0.94 -0.03 0.00
(-.026280,.035708,. 195633)
(-.628915,.026167,.167104)
(-.028915,026167,.167104)
(-.028915,.026167,.167104)
0.00
0.00
0.00
0.00
0.00
Grand Total:
0.01
rA-8 terms -0.01
O
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216
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OlO,
C O lp s
i^io
—
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S m! co I
O .o'S!
ih>
Mar 2
flO
o>
H2+ (in MHz).
00
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(0,0) 9 H5
(nu'R')L
LHIIX30.FOR
PH2CNT.FOR
QP
Q2
Q
QP
P2
p2
(0,0)H
(0.2)F
(0,2)H
(0.2)K
0.00 0.00 -0.41
0.00 0.00
56.34 5.55 019 •0.09 0.07
-11.64 0.70 0.01 -30.28 -3.35
0.92 -0.05 0.00 -42.87 -3.69
-2.77
-0.01
0.10
-0.09
no cutoff
(Q,As,At)
Series Sums:
(1.63856,3.17128,4.0078)
(1.64455,3.18567,4.0370)
(1.64455,3.18567,4.0370)
(1.64455,3.18567,4.0370)
-3.18
61.91
-46.08
-47.62
-34.97
K>
■vl
(i,o)H
(1.2)F
(1,2)K
0.00 0.00 0.24
3.69 0.98 0.06
-2.27 0.40 0.02
-0.01 0.00 0.00
0.00 -0.12 (.34266, .83930,1.73040)
0.00 (.31371,.78531,1.66558)
0.66 -0.16 0.61 (.31371,78531,1.66558)
-1.15 -0.22 -0.01 (.31371,.78531,1.66558)
0.00
0.00
-0.36
4.73
-3.52
-1.39
0.00
-0.54
SumH
Sum F
SumK
0.00 000 0.00
0.00 0.00
0.00 0.00
0.07 0.02 0.00
-0.01 0.00 o^oo 0.00 0.00
0.00 0.00 0.00
0.01 0.00
0.43
-13.92 -1.10 0 68 -30.94 -3.51
60.10 6.51 0.25 -0.09 -0 07
-0.93 0.05 0.00 -4403 -3.91
Total H
total F
total K
-53.15
66.69
-49.02
(2,0)H
(l>2)F_
(1j2)H
(1.2)K
0.00 (-.026280,.035708,.195633)
0.00 (-028915,626167,167104)
0.00 (-.028915,.026167,.167104)
0.00 (-.028915,.026167,.167104)
-3.11
-3.00
0.01
0.10
0.00
0.05
-0.01
-0.01
0.03
Grand Total:
-35.48
rA-8 terms -3.54
H2+ (in Mlfc )
eo
o>
80
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(0,0)9 16
(nu'R')L
Mar 2
no cutoff
I
I
LHI •X30.FOR
PH2CNT.FOR
P2
Q2
QP
Q2
QP
P2
(Q,As,At)
(0,0)1
(0,2)G
(0,2)1
(02)L
0.00 0.00 -0.18 0.00
19.51 1.19 0.03 -0.01
-14.21 -0.69 -0.01 -4.82
-1.56 -0.07 0.00 -12.97
Series Sums:
0.00
-0.01
-0.39
-0.76
-0.24
0.00
-0.01
-0.01
(1.63856,3.17128,4.0078)
(1.64455,3.18567,4.0370)
(1.64455,3.18567,4.0370)
(1.64455,3.18567,4.0370)
-0.01
0.00
0.00
0.00
(.34266, .83930,1.73040)
(.31371,.78531,1.66558)
(.31371,.78531,1.66558)
(.31371,78531,1.66558)
-0.42
20.71
-20.13
-15.37
-15.21
K>
•—*
00
(1,0)1
(1,2)G
(1.2)1
(1.2)L
0.00
2.34
-0.39
-0.02
0.00
0.00
-0.10
-0.30
0.00
0.00
-0.02
-0.04
(2,0)1
(2,2)G
(2,2)1
(2,2)L
0.00
0.00
0.00
0.00
Sum 1
SumG
Sum L
0.00 0.00 0.00
0.00
-0.01 0.00 0.00
0.00
0.00 0.00 0.00
0.00
0.00 0.00 0.00
0.00
-0.17
-14.6 -0.74 -0.21 -4.92
21.84 1.53 0.04 -0.01
-1.58 -0.07 0.00 -13.27
Total 1
Total G
Total L
-21.14
23.39
-15.73
0.00 -0.02
0.34 0.01
-0.05 0.00
0.00 0.00
-0.03
2.69
-0.56
-0.36
1.74
0.00
0.00
0.00
0.00
-0.27
-0.41 -0.26
-0,01 0.00
-0.8 -0.01
(-.026280,.035708,. 195633)
(-.028915,.026167,.167104)
(-.028915,.026167,.167104)
(-.028915,.026167,.167104)
0.00
-0.01
0.00
0.00
-0.01
Grand Total:
-13.48
rA-8 terms -0.44
o cm •«- o
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(0,0)1CIG4
(nu’R')L Q2
(0,0)G
(0,2)D
(0,2)G
(0.2)1
n MHz )
LDMX30.FiDR
P2
QP
MAR31 98
PD2CNTJ OR
Q2
QP
P2
no cutoff
(Q,As,At)
Series Sums:
0.000 0.000 5.994
0.000
0.000 -30.652 (1.60503, 3.07315, 3.78540)
141.192 24.603 1.368
-0.620 -0.823 -0.295 (1.60791, 3.07988, 3.79903)
-79.333 -8.503 -0.226 -162.637 -25.420 -1.056 (1.60791, 3.07988, 3.79903)
-0.238 0.041 -0.005 -113.244 -15.231 -0.570 (1.60791, 3.07988, 3.79903)
-24.658
165.425
-277.175
-129.247
-265.655
to
to
(1|0)G
(1|2)D
(1,2)G
(1.2)1
0.000
4.531
1.463
-0.002
-0.052
1.757 0.240
0.454 0.034
0.000
0.000
0.000
0.000
-0.014
-3.047
-2.728
-0.037
-1.041
-0.785
0.000
-1.083
-0.026
-0.095
-0.063
(.28405, .67239,1.35869)
(.26722, .64095, 1.32136)
(.26722, .64095,1.32136)
(.26722, .64095,1.32136)
-1.135
6.451
-2.232
-3.578
-0.494
(2,0)G
(2.2)D
(2,2)G
(2.2)1
SumG
Sum D
Sum 1
0.000
0.000
0.028 -0.014
0.062 -0.021
0.001
0.003
0.002
0.000
0.000
0.000
0.000
0.000
0.000
-0.013
0.005
7.359
145.751 26.346 1.611
-0.634 -0.860
-77.808 -8.070 5.753 -165.684 -26.461
-0.240 0.041 -0.005 -115.985 -16.011
0.000
Total G 171.893
Total D -305.157
Total 1 -132.833
0.000
0.000
-0.001 (-.01802, .02209, .12744)
0.000 (-.01940, .01752, .11402)
0.000 (-.01940, .01752, .11402)
0.000 (-.01940, .01752, .11402)
0.052
-33.841
-0.321
-32.887
Grand Total:
-0.633
-266.097
rA-8 terms -26.482
,5rA-8 terms -13.241
0.000
0.017
0.043
-0.008
Is * *
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222
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in in-CM
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.o . o ' o
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
D2+ i[in MH;0
(0,0)1016
(nu'R)L
<0,0)1
(0,2)6
(0,2)1
(0.2)L
LDMX30.FOR
Q2
QP
P2
0.000 0.000 -0.085
MARS11 98
PD2CNT.FOR
Q2
QP
P2
n o cutoff
(Q,As,At)
Series Sums:
0.000 0.000 -0.230 (1.60503, 3.07315, 3.78540)
-0.315
14.462
-27.200
-11.615
13.593 0.871 0.017 -0.012 -0.006 -0.001 (1.60791, 3.07988, 3.79903)
-20.683 -1.021 -0013 -5.086 -0.389 -6.008 (1.60791, 3.07988, 3.79903)
-0.590 -0.025 -0.001 -10.391 -0.599 -0.009 (1.60791, 3.07988, 3.79903)
-24.668
to
to
u>
0.000 0.000 -0.016
0.000 0.000 -0.007 (.28405, .67239,1.35869)
0.000 6.666 0000 (.26722, .64095,1.32136)
(1,0)!
(1,2)0
(1,2)!
(1,2)L
0.941 0.149 0.006
0.367 -0.043 -0.001
-0.006 -0.001 OOOO
0.083 0.014 -6.661
-0.201 -0.025 0.001
(2,0)1
(2,2)6
(2,2)1
(2,2)L
0.000 0.000 0.000
0.041 0.007 0.000
0.001 0.660 0.000
0.000 0.000 0.000
0.000 0.000 0.000 (-.01802, .02209, .12744)
0.000 0.000 0.000 (-.01940, .01752, .11402)
0.000 oooo 0.000 (-.61940, .01752, .11402)
-0.001 oooo 6.000 (-.01940, .01752, .11402)
-0.023
1.096
-0.509
-0.234
(.26722, .64095,1.32136)
(.26722, .64095,1.32136)
0.330
-0.093
-0.257
Sum 6 14.575 1.013 0.023 -0.012 0.006 0.001
Sum! -21.051 -1.064 0.115 -5.169 -0.403 0.246
SumL -0.596 -0.026 -0.001 -10.593 0.624 -0.010
Total 6 15.592
Total 1 -28048
Total L -11.850
0.000
0.034
-0,001
-0.001
0.032
Grand Total:
-24.306
rA-8 terms -0.350
.5rA-8 terms -0,175
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(0,0)1 (IK7
(nu'R')L
D2-F (in MHz)
I■
LDMX30.FOR
Q2
P2
QP
0.000 0.000 0.031
(0,0)K
(0,2)H
5.412 0.233 0.603
(0,2)K -10.567 0.375 -6.003
(0.2)M -0.823 -0.025 0.000
MAR 31 9 8
1
PD2CNT.FOR
QP
Q2
P2
no cutoff
(Q,As,At)
Series Sums:
. .
0.000
0.002
•0.894
-3.436
0.000
-6.001
-0.053
-0.144
-0.025
0.660
-0.001
-0.002
(1.60503, 3.07315, 3.7854)
(1.60791,3.07988,3.79903)
(1.60791,3.07988,3.79903)
(1.60791, 3.07988, 3.79903)
-0.056
5.645
-11.893
-4.43
-10.734
(1,0)K
to
to
4*.
(1.2)K
(1,2)M
0.000
0.716
0.082
-6.008
0.000 -0.002
0.069 0.002
-6.007 0.000
-0.001 0.000
0.000
0.001
0.000
0.000
0.000
0.000
0000
0.000
0.000
ao o o
0.000
0.000
-6.031
6.127 0.302 0.005
SumH
SumK -10.649 -0.382 -0.036
Sum M 0.831 -6.026 0.000
(2,0)K
(2,2)H
(2,2)K
(2,2)M
total H 6.431
total K -12.057
Total M -4.503
0.000 0.000
0.000 0.000
0.014 -0.662
0.059 -0.005
0.001
0.000
0.000
0.000
(.28405, .67239,1.35869)
(.26722, .64095,1.32136)
(.26722, .64095, 1.32136)
(.26722, .64095,1.32136)
0.606
0.000 0.000 0.000 (-.01802, .02209, .12744)
0.000 6.600 0.000 (-.01940, .01752, .11402)
0.660 6.060 0.000 (-.01946, .01752, .11402)
0.000 6.600 0.000 (-.01940; .01752, .11402)
0.029
-0.001
-0.002 -0.001 0.000
-0.908 -0055 0.027
Grand Total:
-3.495 -0.149 0.002
-10.129
rA-8 terms -0.060
.5rA-8 terms -0.030
-0.003
0.787
-0.105
-0.073
0.000
-0.001
0.000
0.000
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(0,0)1( 1 8
(nu'R')L
(0,0)L
<0.2)1
(0.2)L
(0.2)N
D2+ (in MHj0
MAR! 1198
r '
LDIJIX30.FOR
PD2CNT.FOR
p2
Q2
Q2
p2
QP
QP
0.000 0.000 -0.009
—
. . . .
2.409 0.071 0.001
-5.727 -0.149 -0.001
-0.842 -0.020 0.000
no cutoff
(Q,As,At)
Series Sums:
0.000 0.000 -0.002 (1.60503, 3.07315, 3.78540)
0.000 0.000 0.000 (1.60791, 3.07988, 3.79903)
0.127 -0.006 0.000 (1.60791, 3.07988, 3.79903)
-0.949 -0.031 0.000 (1.60791,3.07988,3.79903)
-0.011
2.481
-6.010
-1.842
-5.382
N>
K>
(1.0)L
(1.2)1
(1.2)L
(1.2)N
0.000 0.000
0.000
0.115 •0.008 0.000
-0.023 -0 001 0.000
0.608 0.000 0.000
0.000 0.000 0.000
0.000 0.000 0.000
-6.002 0.000 oooo
-0.015 -6.001 o.ooo
0.000
(.28405, .67239,1.35869)
(.26722, .64095,1.32136)
(.26722, .64095,1.32136)
(.26722, .64095,1.32136)
-0.123
-0.026
-0,024
-0.173
0.000
o.ooo
0.000
0.000
(2,0)L
<2.2)1
(2.2)L
(2|2)N
0.000
0.000
0.000
0.000
Sum I
SumL
SumN
2.294 0.063
-5.750 -0.150
-0.850 -6.020
total 1
total L
total N
2.36
-6.05
-1.87
0.000
0.000
0.000
0.000
-0.009
0.000
0.000
0.000
o.ooo
0.000
0.000
0.000
0.000
0.000 (-.01802, .02209, .12744)
0.000 (-.01940, .01752, .11402)
0.000 (-.01940, .01752, .11402)
0.000 (-.01940, .01752, .11402)
-0.002
0.001
0.000 0.000 0.000
0.010
o.ooo
-0.129 -0.006 -0.002
0.964 0.032 oooo
0.000
0.000
0.000
0.000
0.000
Grand Total:
-5.555
rA-8 terms -0.011
.5rA-8 terms -0.006
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(0,0)10M9
D2+ (in MHz)
LDIMX30.FOR
MAR 31 98
1
PD2CNT.F0R
Q2
QP
■
-
•
no cutoff
■'
•
(nu'R')L Q2
QP
(0,0)M
(0,2)K
(o
(0.2)0
m
0.000
1.171
-3.321
-0.633
0.000
0.024
-0.064
0.011
0.003
0.000 0.000
O.OOO 0.000 aooo
0.000 -6.011 oooo
o.ooo -0.154 o.ooo
0.000
o.ooo
0.000
0.000
(1.60503, 3.07315, 3.78540)
(1.60791,3.07988, 3.79903)
(1.60791, 3.07988, 3.79903)
(1.60791, 3.07988, 3.79903)
(1,0)M
(1,?)K
(1,2)M
(1, 2)0
0.000
0.015
-0.006
-0.005
0.000
0001
0.000
o.ooo
0.000
o.ooo
0.000
0.000
0.000 o.ooo
0.000 0.000
0:000 0.000
-6.662 0.000
'
o.ooo
oooo
0:000
o.ooo
(.28405, .67239, 1.35869)
(.26722, .64095,1.32136)
(.26722, .64095,1.32136)
(.26722, .64095,1.32136)
(2,0)M
(2,2)K
(2,2)M
(2,2)0
0.000
0.000
0.000
0.000
P2
P2
(Q.As.At)
Series Sums:
-0.003
1.195
-3.396
-0.798
-3.002
K)
to
ON
oooo
0.000
oooo
o.ooo
-6.003
SumK
1.156 0.023 0:000
Sum M -3.327 -0^064 0.003
SumO -0.638 -6:0li 0.000
ToialK
1.179
ToiajM -3.405
Total 0 -0.805
0.000
oooo
aooo
0.000
o.ooo
o.ooo
oooo
oooo
oooo
0.000
oooo
6:000
0.000
0.000 o.ooo oooo
-0.011 oooo aooo
-ai5 6 0.000 0.000
0.600
0.000
o.ooo
oooo
0.000
-0.016
-0.006
-0.007
-0.029
(-.61802,
(-.01940,
(-.01940,
(-.01940,
.02209, .12744)
.01752, .11402)
.01752, .11402)
01752, .11402)
0.000
0.000
0.000
0.000
0.000
Grand Total:
-3.031
rA-8 terms -0.003
,5rA-8 terms -0.002
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(0,0)916
D2+ (in MHa0
1
j. .
LDMX30.FOR
Q.
P2
QP
.
(nu'R')L
..
MAR 31 98
no cutoff
..
PD2CNT.FOR
QP
Q2
P2
(Q,As,At)
Series Sums:
(0,0)1
0.000 0.000 -0.173
0.000 0.000 -0.224 (1.60503, 3.07315, 3.78540)
(0,2)G 17.739 1.089 0.019 -0.012 0.005 -0.001 (1.60791,3.07988, 3.79903)
(0,2)1 -51.567 -2.488 -0.030 -4.774 -0.373 0.008 (1.60791, 3.07988, 3.79903)
-1.629 -0.065 -0.001 -12.722 0.715 -0.011 (1.60791,3.07988, 3.79903)
(0-2)1-
-0.397
18.829
-59.240
-15.143
-55.951
(1,0)1
(1,2)G
(1.2)1
(1.2)L
0.000
3.024
0.429
0.018
0.000 -0.017
0.570 0.028
-0.048 0.001
- ° 6o2 0.000
0.000 0.000 -0.007
0.000 6.666 0.000
-0082 -0.014 -0.001
0.246 -0.030 -6.661
(.28405, .67239,1.35869)
(.26722, .64095,1.32136)
(.26722, .64095,1.32136)
(.26722, .64095,1.32136)
2.734
....................
0.000
-0.050
0.001
0.000
0.000
0.008
0.000
0.000
0.000
0.000
O.OOO 0.000
0.000
0.000
0.000 -6.001
0.175
Sum G 20.713 1.675 0.047
0.012
Suml -51.997 -2.536 -0.221 -4.856
SumL -1.647 -0.067 0.001 -12.969
(2.0)1
(2,2)G
(2.2)1
(2,2)L
Total G 22.417
Total 1 -60.237
Total L -15.441
-0.024
3.630
-0.575
-0.297
0.000
OOOO
0.000
o.ooo
0.000
0.000
0.000
0.000
0.253
-0.005 -O.661
-0.307 0.240
0.745 -6.612
(-.01802,
(-.01940,
(-.01946,
(-.01940,
.02209,
.01752,
.01752,
.01752,
.12744)
.11402)
.11402)
.11402)
0.000
-0.042
-0.044
Grand Total:
-53.261
rA-8 terms -0.428
.5rA-8 term s-0.214
-
0.001
-
0.001
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(0,0)91C7
(nu'R')L
D2+ (inMHi
r
LDMX30.FOR
P2
Q2
QP
(0,0)k
0.000 0.000 -0.048
6.922 0.278 0.003
(0,2)H
(0,2)lk -23.808 -0.807 -0.007
(0.2)M -1.936 0057 0.000
MAR 31 98
PD2CNT.FOR
QP
Q2
0.000
-0.001
-0.650
-3.497
0.000
O.OOO
-0.040
-0.144
0.000
0.000
-0.011
-0061
0.000
0.000
-6.001
-0.005
no cutoff
P2
(Q,As,At)
-0.019
0.000
-0.001
-6.002
(1.60503, 3.07315, 3.78540)
(1.60791, 3.07988, 3.79903)
(1.60791, 3.07988, 3.79903)
(1.60791, 3.07988, 3.79903)
Series Sums:
-0.067
7.202
-25.313
-5.636
-23.814
(1,0)K
(1,2)H
(1.?)K
0.000
8.121
-0.097
0026
0.000 -0.002
0.760 0.018
0.008 0.000
-0.001 O.OOO
(2,0)K
(2.2)H
(2,2)K
(2,2)M
0.000
-0.001
O.OOO
0.000
0.000
OOOO
0.000
0000
-6.001
oooo
oooo
o.ooo
(.28405, .67239,1.35869)
(.26722, .64095,1.32136)
{.26722, .64095, 1.32136)
(.26722, .64095,1.32136)
oooo
0.000
0.000
o.ooo
•0.023
-0.001 0.000 0.000
-0.661 -0.041 -0.021
-3.558 -0.149 -6 ’6o2
(-.01802, .02209, .12744)
(-.01940, .01752, .11402)
(-.61940, .01752, .11402)
(-.61940, .01752, .11402)
-0.003
8.899
-0.117
-0.087
8.692
O.OOO
O.OOO
0.000
0.000
-0.036
SumH 15.042 1.038 0.021
SumK -23^905 -0.815 0.057
SumM -1.956 -0!058 O.OOO
Total H 16.100
fotai k -25.500
Total M -5.723
0.000
O.OOO
0.000
0.000
0.000
0.000
oooo
0.000
-
0.000
0.001
0.000
0.000
-0.001
Grand Total:
-15.123
rA-8 terms -0.059
,5rA-8 terms -0.030
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
02+ (in MHao
(0,0)9L.8
(nu'R')L'
LDI4X30.FOR
P2
QP
oi
98
!MAR 31
I
i no
cutoff
1
PD2CNT.FOR !
P2
QP
(Q,As,At)
Series Sums:
q2
(0,0)1
0.000 0.000 -0.012
(0,2)1
3.003 0.079 0.001
(0,2)L -12.018 0.291 0.002
(0.2)N
-1.608 -0.035 0.000
0.000
0.000
-0.054
-0.596
0.000 -0.001 (1.60503, 3.07315, 3.78540)
0.000 0.000 (1.60791, 3.07988, 3.79903)
-0.003 0.000 (1.60791, 3.07988, 3.79903)
-0.019 0.000 (1.60791, 3.07988, 3.79903)
•0.013
3.083
-12.368
-2.258
-11.556
(1.0)L
(1.2)»
(1.?)L
(1,2)N
0.000
-0.077
0.024
-a015
0.000
0.005
-0.001
0.001
0.000
O.OOO
0.000
0.000
O.OOO
0.000
0.000
o.ooo
0.000
0.000
0.000
o.ooo
0.000 0.000 0.000
0.000 0.000 0.000
-o;ooi 0.000 0.000
o.oio -0.001 0.000
(.28405, .67239,1.35869)
(.26722, .64095, 1.32136)
(.26722, .64095, 1.32136)
(.26722, .64095, 1.32136)
0.000 0.000
0.000 o.ooo
0.000 o.ooo
oooo o.ooo
(-.01802, .02209, .12744)
(-.61940, .01752, .11402)
(-.01940, .01752, .11402)
(-.01940, .01752, .11402)
0.000
-0.082
-0.026
-0.027
-0.135
o.ooo
o.ooo
0.000
0.000
-0.013
Sum I
2.926 0.074 a o o i
SumL -12.042 -0.292 0.014
Sum N -1.623 -0.036 0.000
(2.0)L
(2.2)!
(2.2)L
(2,2)N
total I
3.001
total L -12.407
Total N -2.285
0.000
o.ooo
o.ooo
o.ooo
-0.001
0.000 0.000 0.000
-6.055 -6.003 -0.001
0.606 -6.020 0.000
I
I
0.000
0.000
0.000
0.000
0.000
Grand Total:
-11.691
rA-8 terms -0.014
.5rA-8 terms -0.007
To c a lc u la te th e se c o n d -o rd e r c o n trib u tio n s , th e o re tic a l e s tim a te s
for the ion c o re p a ra m e te rs are re q u ire d . T h e se v a lu e s are th e o ffd iag o n al m a trix e le m e n ts o f Q (q u ad ru p o le m o m e n t), a s (sc a la r d ip o le
p o la riz a b ility ), a n d a t (te n s o r d ip o le p o l.). T a b le B -2 gives th e v a lu e s
used to c a lc u la te th e E (2) c o n trib u tio n s used in C h a p te r 4.
Table B-2: Core parameters for (0,0) levels in H2 * and D2*- The <v,Rlx|v’,R'> values for
each x parameter are listed for each molecule. All the matrix elements have been
calculated directly using the adiabatic ion wavefimctions.44 The boxed values are actually
slightly different than the best theoretical values (3.18517, 0.92680, and 4.0364).
M a t r i x e l e m e n t s o f Q, a s, a n d a t u sed for E (2)
c a l c u l a t i o n s w ith ( v , R ) — (0,0)
(v',R')
Q
as
at
(0,0)
(0,2)
(1,0)
(1,2)
(2,0)
(2,2)
1.63856
1.64430 ;
0.34266 |
0.31371
-0.026280
-0.028915
3.17128
3.18567 ;
0.83930
0.78531
0.035708
0.026167
4.0078
4.0370 !
1.73040
1.66558
0.195633
0.167104
(0,0)
(0,2)
(1,0)
(1,2)
(2,0)
(2,2)
1.60503
1.60791
0.28405
0.26722
-0.01802
-0.01940
3.07315
3.07988
0.67239
0.64095
0.02209
0.01752
3.78540
3.79903
1.35869
1.32136
0.12744
0.11402
h2
d2
230
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A ppendix C
Spin s t r u c t u r e c a l c u a t i o n s fo r (0 ,0) H 2 a n d D 2 sta te s
The sp in s tru c tu re for R =0 R ydberg sta te s o f H 2 and D 2 was
d e sc rib e d in S e c tio n 1.2-4. T he spin H a m ilto n ia n w as d e fin e d by Eq. 121. U sing th e a n g u la r m om entum c o u p lin g schem e p re s e n te d th ere, th e
m atrix e le m e n ts fo r e a c h term o f the spin H a m ilto n ia n can be w ritte n in a
m o re u sefu l m an n er. E ach o f th e term s are show n b e lo w .
Eq. C-l
x ( - 1) j,+f> l[5I(I + 1)(2I + 1)(2FC+ lX2Fc' + lX2R + 1X2R' + 1X2N + lX2N' + 1)]1/2
,J1f2Fj+R.+L+l+l/2
—R(R + 1X2R + l)(2Fc + l)(2Fc' + l)(2N + lX2N' + 1)
231
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
,
_ . . fJ s R j ; H
<v -|L -s r |v >= { 1
^
fc
n
'
j;lfR
Nj | ,
']
LJ
l
n
N
1/2
—L(L + 1X2L+1X2N + 1X2N' + 0(2.1, + lX2J; + 1)
j,
^
f;
n 'U r
l
n 'H i
N
F. 11 1
N
L h 1 F.
^ J ,-^ 2 F j+ N + N V R + L ^ I+ l/2
;
X
(-
sc
f;'
S.
11/2
—L(L + 1X2L + 1X2N + 1X2N' + lX2Fc + lX2Fc' + 1)
a
|[ ( 2 J ,
:
?
a
s
;•
a
+ 1X2 J; + lX2F + 1X2F' +1)]
+R+L+L'+I+Sc+S|l ♦I
( v ' p , -[st - 3 f ( f ■St )JV) = ( - l ) w '*N'*,!
X
-[30(21, + lX2J| + 0(2FC+ lX2Fc' + lX2N + 0(2N ' + 1X2L + lX2L' +1)]
f L'
o
2
o
LM J
o ;ii
S
j
,
j;l[I
s, [i
Sc FC
'| ( R
Fc s c. 2
L'
N
N 'l
N'
f;
j;
N 2
fc i
j
, lj
(T h ere are a few m in o r d isc re p a n c ie s w ith a p re v io u s re p o rt o f these
fo rm u la s5, m o stly due to ty p o g ra p h ic a l erro rs in th e re fe re n c e ; how ev er,
th e se le c tio n ru le s in T a b le IX o f th at re p o rt are c o rre c t an d u se d h ere.)
T hese fo rm u la s w ere used to c a lcu late b o th d ia g o n a l and offd iag o n a l c o n trib u tio n s fo r a g iv en state. T able C -l p re s e n ts an exam ple
b reak d o w n o f the n o n -z e ro , d iag o n a l m atrix e le m e n ts. N e x t, T ab le C-2
g iv es an ex am p le o f th e o ff-d ia g o n a l c o n trib u tio n s . T h ese e n e rg ie s are
232
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table C -l: An example o f diagonal, R=0 spin structure. (0)1 OHs state for each isotope
H2
I
Fc
Ji
J
IS c
0
.5
.5
.5
.5
4.5
4.5
5.5
5.5
4
5
5
6
0
0
0
0
0
0
0
(n=10, N = L = 5 )
L-S r
SR-X[SC]
T jO )
fc
MFS
-3.183
2.604
-3.134
2.652
0.707
-0.579
-0.482
0.408
3.890
8.391
-8.921
-2.244
L-S r
SR-X[SC]
p (i)
fc
MFS
-3.184
2.605
-3.136
2.653
-3.184
2.476
-3.007
2.460
-2.991
2.531
-3.061
2.653
-3.184
2.274
-2.805
2.182
-2.712
2.219
-2.750
2.327
-2.857
2.476
-3.007
2.653
0.708
-0.579
-0.482
0.408
3.891
8.394
-8.921
-2.245
-0.425
0.330
-0.259
-220.761
-214.346
-218.508
-212.569
-215.931
-210.349
-212.927
-207.739
LSC
6.365
6.365
-5.304
-5.304
D 2 (n = 1 0 , N = L = 5 )
I
Fc
Ji
J
IS c
0
.5
.5
.5
.5
4.5
4.5
5.5
5.5
4
5
5
6
0
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
3.5
3.5
4.5
4.5
5.5
5.5
6.5
6.5
3
4
4
5
5
6
6
7
2.5
2.5
3.5
3.5
4.5
4.5
5.5
5.5
6.5
6.5
7.5
7.5
2
0
0
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
4
4
5
5
6
6
7
7
8
0
0
0
-213.332
-213.332
-213.332
-213.332
-213.332
-213.332
-213.332
-213.332
142.221
142.221
142.221
142.221
142.221
142.221
142.221
142.221
142.221
142.221
142.221
142.221
LSC
6.368
6.368
-5.306
-5.306
-3.821
-3.821
-1.910
-1.910
0.425
0.425
3.184
3.184
6.368
6.368
4.882
4.882
2.972
2.972
0.637
0.637
-2.123
-2.123
-5.306
-5.306
0 .2 1 2
-0.033
0.028
0.283
-0.245
0.708
-0.505
0.576
-0.448
0.432
-0.353
0.228
-0.193
-0.063
0.054
-0.463
0.408
233
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
146.113
150.358
144.874
148.837
142.912
147.059
140.336
144.992
137.178
142.629
133.445
139.976
c a lc u la te d and the s u b -m a tric e s a re d ia g o n a liz e d to d e te rm in e th e to ta l
sp in stru c tu re sp littin g s. In th is ta b le , only th e n o n -z e ro m a trix e le m e n ts
a re g iv en . T he su b -m a tric e s a re p re se n te d to g iv e a c o m p le te se t o f
n u m b ers. S tric t AI=AJ=0 s e le c tio n ru le s cause the I S C te rm s to be zero
fo r all o ff-d ia g o n a l m atrix e le m e n ts . Since th e o th e r tw o h y p e rfin e term s
are zero fo r all R=0 sta te s, o n ly th e th re e MFS term s g iv e n o n -z e ro m atrix
ele m e n ts.
The to ta l sp in s tru c tu re v a lu e s are su m m arized fo r th e R =0 sta te s in
T ab le C -3a fo r H 2 and T a b le C -4 a fo r D 2 . T he e x c h a n g e p a ra m e te r
(V x= 1 .1 2 ) is in clu d ed on th e 10G s ta te s c a lc u la te d h e re . T he “tra n s itio n
sp in s tru c tu re ” is p re se n te d (T a b le C -3b for H 2 an d T a b le C -4 b fo r D 2 ) fo r
e ac h o f the m ea su re d in te rv a ls to be d isc u sse d in c h a p te r 4. T h is
s tru c tu re is c a lc u la te d u sin g th e c o n v e n tio n E (n ,L ) - E ( n ,L - l) . A ll v alu es
a re in M H z and the sta te s a re la b e le d by (n ,L ). T he D 2 s tru c tu re
p re se n te d in T ab les C - l , C -2 , an d C -4 a depends on H FS c o n sta n ts
( b = l 3 5 .5 8 9 , c= 1 9 .8 9 7 , and d = 2 1 .3 9 5 ) th a t are d iffe re n t th a n th o se g iven
in th e tex t. H o w ev er, the tr a n s itio n stru c tu re g iv en in T a b le C -4 b is
u n a ffe c te d by th is sm all c h a n g e .
H 2 o f f - d i a g o n a l spin s t r u c t u r e
(1=0, R = 0 , N =L =5)
< F C=0.5, J i = 4 .5 , J = 5 | H spi„|0.5, 5.5, 5> = - 1 . 0 5 6 M H z
L S C te rm
=
0 MH z
L - S r te rm
=
-0.528 M H z
S r - X te rm = - 0 . 5 2 8 MH z
234
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table C-2: Off-diagonal matrix elements for R=0 D2 spin structure. This example o f the
o ff-d ia g o n a l spin structure is for n=10, L=N=5 : the (0,0)10Hs state. The first column is
the total a n g u la r momentum (J), which must be the same for both state 1 and state 2.
Columns 2 a n d 3 are the states being mixed by the spin Hamiltonian, labeled as (Fc, J i) .
C o lum n 4 is the total contribution, the sum o f the last three columns. Column 5, 6, and 7
are the first three terms in Eq. 1-20, the MFS Hamiltonian.
J
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
S tate 1
1.5 3.5
1.5 3.5
2.5 2.5
1.5 3.5
1.5 3.5
1.5 3.5
1.5 4.5
1.5 4.5
2.5 3.5
0.5 4.5
1.5 4.5
1.5 4.5
1.5 4.5
1.5 5.5
1.5 5.5
2.5 4.5
1.5 5.5
1.5 5.5
1.5 5.5
1.5 6.5
1.5 6.5
2.5 5.5
1.5 6.5
1.5 6.5
2.5 6.5
S ta te 2
2.5 2.5
2.5 3.5
2.5 3.5
1.5 4.5
2.5 3.5
2.5 4.5
2.5 3.5
2.5 4.5
2.5 4.5
0.5 5.5
1.5 5.5
2.5 4.5
2.5 5.5
2.5 4.5
2.5 5.5
2.5 5.5
1.5 6.5
2.5 5.5
2.5 6.5
2.5 5.5
2.5 6.5
2.5 6.5
2.5 6.5
2.5 7.5
2.5 7.5
T otal
0 .5 8 6
4 .1 6 9
-1 .3 4 2
-0 .9 3 4
3.6 7 5
0 .0 6 4
0.611
5.733
-1 .6 1 5
-1 .0 5 7
-0 .8 6 7
4 .9 5 0
0 .0 1 6
0.555
6 .3 7 4
-1 .7 1 2
-0 .5 9 2
5 .3 8 2
-0 .1 2 6
0.3 9 9
5.738
-1 .6 6 5
4 .7 3 3
-0 .3 7 7
-1 .3 8 5
L Sc
0
3.891
0
0
3.891
0
0
5.302
0
0
0
5.302
0
0
5.836
0
0
5.836
0
0
5.199
0
5.199
0
0
L-Sr
0
0
-1 .4 3 8
-1.001
0
0
0
0
-1.591
-0 .5 2 8
-1 .0 4 4
0
0
0
0
-1 .5 3 2
-0 .8 3 7
0
0
0
0
-1.341
0
0
-1.001
235
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
S r -X
0 .5 8 6
0 .2 7 8
0 .0 9 6
0 .0 6 7
-0 .2 1 6
0 .0 6 4
0.611
0.431
-0 .0 2 4
-0 .5 2 8
0 .1 7 7
-0 .3 5 2
0 .0 1 6
0 .5 5 5
0 .5 3 7
-0.181
0 .2 4 5
-0 .4 5 5
-0 .1 2 6
0 .3 9 9
0 .5 3 8
-0 .3 2 3
-0 .4 6 7
-0 .3 7 7
-0 .3 8 5
E x a m p l e o f D 2 ( 0 ) 1 0 H 5 s p in s t r u c t u r e
(s u b -m a tric e s to be d ia g o n a liz e d )
J = 3
J = 4
J = 5
J = 6
J = 7
-220.761
0386
. 4.169
0386
150358
4.169
-1 3 4 2
-1 3 4 2
144.874,
3.675
0.064
0
-0.934
-218308
0.611
5.733
0
3.675
0.611
148.837
-1.615
8394
-1.057
0
0
0
0
-1.057
-8.925
0
0
0
0
0
0
-212369
-0.867
4.950
0.016
0
0
-0 .867
-215.931
0355
6.374
-2 3 4 5
0
0
0
0
0
-2 1 0 3 4 9
-0 3 9 2
5382
-0.126
3.891
0
0
0
0
0
-2 1 4 3 4 6
-0.934
-207.739
4.733
. -0 .3 7 7
4.733
142.629
-1.385
0
-0 3 9 2
-212.926
0399
5.738
0
5382
0399
144.992
-1.665
0 >
0.064
5.733
-1.615
142.912,
0
0
4.950
0355
147.059
-1.712
0 '
0
0.016
6.374
-1.712
140.336,
0 >
-0.126
5.738
-1.665
137.178,
-0.377
-1 3 8 5
133.445,
Diagonalize these 5 matrices to get the 22 eigenvalues. Combined with the single J=2
and J=8 levels, this gives the 24 spin states for the (0)1 OHs D 2 level.
236
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table C-3a: The diagonalized spin structure o f all R=0 L >4 states o f H2. All values are
in MHz. States are labeled (n,L) across the top and the first column labels the possible
combinations for the (Ji,J) quantum numbers.
C o m p l e t e s p in s t r u c t u r e f o r R = 0 H 2
(n,L)
CJi.J)
(L-.5,L-1)
(L-.5,L)
(L+.5,L)
(L+.5,L+1)
(10,4)
(10,5)
(10,6)
(10,7)
(10,8)
(10,9)
5.265
12.831
-13.858
-4.043
3.890
8.455
-8.985
-2.244
2.652
6.075
-6.395
-1.667
1.924
4.575
-4.784
-1.287
1.459
3.570
-3.713
-1.024
1.144
2.863
-2.965
-.834
(9,5)
(9,6)
(9,7)
(9,8)
5.336
11.598
-12.325
-3.078
3.638
8.333
-8.773
-2.287
2.639
6.276
-6.562
-1.766
2.001
4.897
-5.093
-1.404
(n,L)
(Ji,J)
(L-.5X-1)
(L-.5,L)
(L+.5,L)
(L+.5,L+1)
Table C-3b: Transition spin structure for measured R=0 intervals o f H2. All values are
in MHz. Only the strongest transitions are considered, which include those with
AJi=AJ-+l.
T r a n s i t i o n s p in s t r u c t u r e
(n,L,L+l)
(Ji,J)
(L-.5,L-1)
(L-.5,L)
(L+.5,L)
(L+.5,L+1)
(10,4,5) (10,5,6) (10,6,7) (10,7,8) (10,8,9)
-1.375
-4.376
4.873
1.799
-1.238
-2.380
2.590
.577
-.729
-1.500
1.612
.380
-.465
-1.006
1.071
.264
-.315
-.707
.748
.190
(9,5,6)
(9,6,7)
(9,6,8)
-1.698
-3.265
3.553
.792
-1.000
-2.057
2.211
.521
-.638
-1.380
1.469
.361
237
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table C-4a: The diagonalized spin structure o f all R=0 L>4 states o f D 2 - All values are
in MHz. States are labeled (n,L) across the top and the first column labels the possible
combinations for the (Ji,J) quantum numbers.
C o m p l e t e s p in s t r u c t u r e for R = 0 D 2
Fc
.5
.5
.5
.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
(n,L)
(Ji, J)
(L--5,L-1)
(L-.5,L)
(L+.5,L)
(L+.5.L+1)
(L-1.5JL-2)
(L-1.5,L-1)
(L-.5,L-1)
(L-.5,L)
(L+.5,L)
(L+.5,L+1)
(L+1.5,L+1)
(L+1.5,L+2)
(L-2.5,L-3)
(L-2.5,L-2)
(L-1.5,L-2)
(L-1.5,L-1)
(L-.5,L-1)
(L-.5,L)
(L+.5,L)
(L+.5,L+1)
(L+1.5,L+1)
(L+1.5,L+2)
(L+2.5,L+2)
(L+2.5,L+3)
(10,4)
(10,5)
5.137
12.801
-13.776
-4.301
-225.156
-215.407
-222.685
-212.764
-218.994
-209.447
-214.234
-205.569
147.359
155.272
145.160
153.328
142.411
150.662
138.746
147.246
134.082
143.049
128.357
137.921
3.892
8.458
-8.989
-2.245
-220.809
-214.182
-218.801
-212.426
-216.257
-210.300
-213.152
-207.804
146.113
150.672
144.608
149.290
142.588
147.541
140.037
145.414
136.932
142.896
133.242
139.976
(10,7)
(10,8)
(10,9)
2.653
1.924
6.078
4.577
-6.3981
-4.786
-1.668
-1.288
-218.542 -217.170
-213.911 -213.750
-217.059 -216.034
-212.598 -212.732
-215.263 -214.699
-211.056 -211.563
-213.142 -213.160
-209.284 -210.242
144.874 144.145
148.294 146.796
143.691
143.201
147.212
145.930
142.203
142.062
145.903
144.914
140.403
140.725
144.360
143.745
138.282
139.185
142.579
142.419
135.828
137.438
140.553
140.933
1.459
3.571
-3.714
-1.024
-216.277
-213.348
-215.380
-212.836
-214.349
-211.919
-213.181
-210.897
143.680
145.791
142.913
145.085
142.014
144.274
140.982
143.357
139.815
142.332
138.509
141.197
1.145
2.864
-2.966
-.834
-215.663
-213.578
-214.938
-212.916
-214.118
-212.178
-213.202
-211.364
143.366
145.084
142.732
144.498
142.005
143.837
141.185
143.098
140.269
142.282
139.256
141.387
(10,6)
238
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table C-4a: (continued)
(n,L)
Fc (Ji, J)
.5
(L-.5,L-1)
.5
(L-.5X)
.5
(L+.5,L)
.5
(L+.5,L+1)
1.5 (L-1.5,L-2)
1.5 (L-1.5,L-1)
1.5 (L-.5,L-1)
1.5 (L-.5,L)
1.5 (L+.5,L)
1.5 (L+.5,L+1)
1.5 (L+1.5,L+1)
1.5 (L+1.5,L+2)
2.5 (L-2.5,L-3)
2.5 (L-2.5,L-2)
2.5 (L-1.5,L-2)
2.5 (L-1.5X-1)
2.5 (L-.5X-1)
2.5 (L--5,L)
2.5 (L+.5,L)
2.5 (L+.5,L+1)
2.5 a+ i.5x+ i)
2.5 (L+l .5,L+2)
2.5 (L+2.5,L+2)
2.5 (L+2.5,L+3)
(9,6)
(9,7)
(9,8)
3.640
8.337
-8.777
-2.288
-220.492
-214.137
-218.469
-212.343
-216.011
-210.231
-213.095
-207.797
145.861
150.553
144.250
149.079
142.219
147.290
139.755
145.177
136.841
142.728
133.452
139.933
2.640
6.279
-6.565
-1.766
-218.605
-213.912
-217.053
-212.520
-215.224
-210.918
-213.108
-209.104
144.861
148.497
143.572
147.316
142.016
145.926
140.184
144.323
138.069
142.502
135.660
140.455
2.002
4.899
-5.095
-1.405
-217.377
-213.769
-216.150
-212.658
-214.737
-211.401
-213.133
-209.997
144.223
147.118
143.175
146.154
141.946
145.044
140.531
143.787
138.927
142.379
137.128
140.816
239
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T able C-4b: Transition spin structure for measured R=0 intervals o f D2 . All values are
in MHz. Only the strongest transitions are considered, which include those with
AJi=AJ=+l. These values are insensitive to the HFS constants because of a strict AFc=0
selection rule.
T r a n s i t i o n s p in s t r u c t u r e
(n,L,L+l)
Fc
.5
.5
.5
.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
(Ji, J)
(L-.5,L-1)
(L-.5,L)
(L+.5,L)
(L+.5,L+1)
(L-1.5X-2)
(L-1.5,L-1)
(L-.5,L-1)
(L-.5,L)
(L+-5,L)
(L+.5,L+1)
(L+1.5,L+1)
(L+1.5,L+2)
(L-2.5,L-3)
(L-2.5,L-2)
(L-1.5,L-2)
(L-1.5,L-1)
(L-.5.L-1)
(L-.5,L)
(L+.5,L)
(L+.5,L+1)
(L+1.5,L+1)
(L+1.5,L+2)
(L+2.5,L+2)
(L+2.5,L+3)
(10,4,5) (10,5,6) (10,6,7) (10,7,8) (10,8,9)
-1.245 -1.238
-4.343 -2.381
4.787
2.591
.577
2.056
4.347
2.267
1.224
.271
1.741
3.884
-.171
.338
.994
2.737
-.756
-.853
1.082
.010
-2.235 -1.481
-1.247 -1.238
-4.600 -2.378
-.552
-.918
-4.038 -2.078
.178
-.386
-3.121 -1.639
1.291
.366
-1.832 -1.054
1.350
2.850
-.317
-.153
2.586
4.885
.577
2.055
-.729
-.465
-1.500 -1.006
1.071
1.613
.264
.380
1.372
.893
.161
.103
1.025
.654
-.135
-.104
.564
.350
-.508
-.356
-.017
-.022
-.958
-.655
-.729
-.465
-1.499 -1.005
-.490
-.288
-1.282
-.846
-.141
-.047
-.988
-.640
.322
.258
-.615
-.388
.903
.629
-.160
-.087
1.610
1.070
.380
.264
(9,6,7)
(9,7,8)
-.638
-.315 -1.000
-.707 -2.058 -1.380
1.470
.748 2.212
.521
.362
.190
1.228
1.887
.614
.224
.143
.069
1.417
.903
.443
-.177
-.138
-.080
.487
.787
.232
-.484
-.687
-.259
-.013
-.025
-.020
-.894
-.467 -1.306
-.638
-.315 -1.000
-.707 -2.056 -1.379
-.678
-.397
-.181
-.587 -1.763 -1.162
-.203
-.070
-.009
-.882
-.438 -1.364
.429
.347
.202
-.853
-.537
-.259
1.229
.858
.454
-.227
-.123
-.050
2.209
1.468
.747
.522
.362
.190
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A ppendix D
C a l i b r a t i o n p r o c e d u r e f o r o p t i c a l s p e c tr a
T he c a lib ra tio n te c h n iq u e used a to m ic h y d ro g e n and re lie d on the
a c c u ra te c a lc u la tio n o f th e tra n s itio n e n e rg ie s . A n e u tra l atom ic h y d ro g en
beam w as u sed to m e a su re th e R ydberg n= 10-27 tra n s itio n w hich w as
c a lc u la te d to be 9 4 6 .3 2 6 7 5 c m '1. T he beam v e lo c ity w as h ig h enough
(P —0.0 0 4 8 6 8 ) th a t s e v e ra l la s e r lin es c o u ld be D o p p le r tu n e d into
reso n an c e. T he C O 2 la s e r lin e s used ra n g e d fro m 10P (14) to 10P(20). For
each c h o sen lin e , th e re so n a n c e w as m e a su re d to o b ta in th e o b serv ed
lin e c e n te r ( 0 Obs)- S in c e th e stage co u ld ro ta te on b o th sid e s o f the beam ,
the tra n s itio n s w e re re c o rd e d for both p o s itiv e and n e g a tiv e angles.
Id ea lly , th e o b se rv e d a n g le w ould be the sam e as th e a c tu a l in te rse c tio n
an g le (0 ); h o w e v e r, d u e to any sm all m is a lig n m e n t, th e la s e r could “ w a lk ”
aro u n d on the m irro rs . T h is m otion w as m o d e le d as a sin u so id a l
d e v ia tio n . In a d d itio n , th e alig n m en t to a b s o lu te z ero w as very d iffic u lt,
and hard to m a in ta in b e tte r than 0.02°, so a c o n s ta n t o ffs e t had to be
d e te rm in e d . In a s im ila r, b u t som ew hat d iff e r e n t fo rm th a n the one used
by S tu rru s617, the o b s e rv e d angle was re la te d to th e in te rs e c tio n an g le by
Eq. D-l
241
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
S u b stitu tio n into Eq. 2-1 gave a form ula th at c o u ld be fitted a g a in st the
know n la se r fre q u e n c ie s. T h is fit used
vL=vHjEEL
i+pcose
Eq.D-2
w here Vh was the c a lc u la te d hydrogen tra n s itio n energy (n= 10-27) and v L
w ere the la se r fre q u e n c ie s. A sam ple c a lib ra tio n 52 is show n in T ab le D -l
w here the fo u r la s e r lin e s w ere used to m easu re the eight re so n a n c e s. T he
fitte d c alib ra tio n p a ra m e te rs w ere:
A =-0.45( 1)°
B = -0 .3 7 (3 )°
C = 0 .2 5 (l)°
P= 0.0 0 4 8 6 8 (2 )
The c a lib ra tio n p ro c e d u re was re p e ate d a fte r each m easu red
spectrum . T his gave a c o m p le te c alib ra tio n fo r each data set, in c lu d in g a
d ete rm in a tio n o f the b eam v e lo c ity . F or d iffe re n t m ass beam s, the
c a lib ra te d v e lo c ity w as c o rre c te d by the sq u a re ro o t o f the m ass ratio (e.g .
fo r HD, the m ass ratio is 1/2.9995524 and p = 0 .0 0 2 8 1 0 8 ). T he m e a su re d
Table D-l: Example calibration for laser spectroscopy experiments. The observed angles
are fitted to the known laser frequencies according to Eqs. D -l and D-2, to extract the
calibration factors A, B, C, and p. The fitted angles in column 4 correspond to the
observed angles in column 3 according to Eq. D -l. Eq. D-2 is inverted to calculate a
theoretical intersection angle, given in column 5. 50= 0fit - 0caic
Line
Laser Frequency (c m 1)
P(14)
P(16)
P(18)
P(20)
P(20)
P(18)
P(16)
P(14)
949.4793
947.7420
945.9802
944.1940
944.1940
945.9802
947.7420
949.4793
Oobs
Ofit
Ocalc
50
133.21°
108.10°
86.08°
62.89°
-61.69°
-85.09°
-107.41°
-132.85°
133.20°
108.00°
85.85°
62.49°
-62.54°
133.20°
108.01°
85.82°
62.51°
-62.51°
-85.82°
-107.99°
-133.23°
-85.82°
-108.01°
-133.20°
0.00°
-0.01°
0.03°
-0.02°
-0.03°
0.00°
0.02°
-0.03°
242
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
c a lib ra tio n s are su m m a riz e d in T ab le D-2 and n u m b e re d to be re fe re n c e d
in T able 3-2. T he d iffe re n c e in th e B and C c o e ff ic ie n ts b etw een
c a lib ra tio n s 2 and 3 is due to th e rea lig n m e n t o f th e L IR in te rn a lly . E very
tim e the b e am lin e w as o p e n e d , a co m p lete re a lig n m e n t w as n e ce ssa ry
in clu d in g rem o v a l o f th e in te rn a l m irro rs. T h is p ro c e d u re c au sed the
sin u so id a l e rro rs to c h a n g e s lig h tly . The rea so n fo r th is is th e ir
d ep en d en ce on a lig n in g th e in co m in g laser e x a c tly a lo n g th e c e n tra l axis
o f the ro ta tio n sta g e . A ny d e v ia tio n w ill cau se th e B and C e rro rs in the
a n g le.
Table D-2: Calibrations used for the laser scans. Fitted coefficients are presented, as
defined by Eq. D -l. Notice that between calibrations 2 and 3 the B and C factors
changed significantly. This is due to the beamline/LIR being opened and realigned.
C at. #
1
A
-0 .4 5 (1 )°
B
-0.37(3)*
C
+0.25(1)°
P = v/c
0 .0 0 4 8 6 8 (2 )
R eference
PLJ14 p . 111
2
-0 .4 4 (1 )°
-0 .3 6 (2 )°
+0.25(1)°
0 .0 0 4 8 6 5 (1 )
PLJ14 p . 149
3
-0 .4 2 (2 )°
-0 .1 6 (4 )°
+0.03(2)°
0 .0 0 4 8 7 2 (3 )
PLJ15 p . 137
4
-0 .4 8 (2 )°
-0 .1 7 (5 )°
+0.03(2)°
0 .0 0 4 8 6 8 (4 )
PLJ15 p . 149
243
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
R eferences
1T. F. Gallagher, Rydberg Atoms, Cambridge University Press, Cambridge, 1994, p. 4-8.
2 T. W. Hansch, Appl Opt. II, 895, (1972).
3 H. A. Bethe and E. E. Salpeter, Quantum Mechanics o f One- and Two-Electron Atoms, Academic Press
Inc., New York, 1957, p. 8 .
4 G. Herzberg, Molecular Spectra and Molecular Structure, Van Nostrand (New York), 1950; E. E. Eyler
and F. M. Pipkin, Phys. Rev. A 27,2462 (1983).
5 W. G. Sturms, E. A. Hessels, P. W. Arcuni, and S. R. Lundeen, Phys. Rev. A 44, 3032 (1991).
6 W. G. Sturms, Ph.D. dissertation, Department o f Physics, University of Notre Dame. 1988.
7 Z. W. Fu, E. A. Hesseis, and S. R. Lundeen, Phys. Rev. A 46, R5313 (1992).
8 P. W. Arcuni, Z. W. Fu, and S. R. Lundeen, Phys. Rev. A 42, R6950 (1990).
9 A. Carrington, I. R. McNab, and C. A. Montgomerie, J. Phys. B 22,3551 (1989).
10 J. F. Babb and J. Shertzer, Chem. Phys. Lett. 189, 287 (1992); J. F. Babb and A. Dalgamo, Phys. Rev.
Lett. 66, 880 (1991).
11 L. R. Ram-Mohan, S. Saigal, D. Dossa, and J. Shertzer, Comput. Phys. 4, 50 (1990); J. Shertzer and F. S.
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transition spin structure (Table C-4b) to these values is negligible, since they only enter in off-diagonal
contributions. A 10% variation in the HFS constant b varies the spin structure by at most four kHz.
244
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
“ A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1974, p. 72.
23 D. R. Cok and S. R. Lundeen, Phys. Rev. A 19, 1830 (1979); 24,3283 (9181).
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31 Labbook PLJ12, p. 81.
32 This detector was designed, constructed, and tested by C. Fehrenbach and D. Fischer.
33 T. F. Gallagher, Rydberg Atoms, Cambridge University Press, Cambridge, 1994, p. 85.
34 E. A. Hessels, Ph.D. dissertation, Department of Physics, University of Notre Dame, 1987.
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44 J. Babb, private communication.
45 W. G. Sturrus and B. Komara, private communication.
46 J. Shertzer, private communication.
245
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
47 D. M. Bishop and L. Cheung, J. Phys. B 11, 3133 (1978); 12, 3135 (1979).
48(H2~): D. M. Bishop and B. Lam, Mol. Phys. 65 , 679 (1988); (D2’’): J. Babb, private communication.
49J. Shertzer and J. DeGuzman, (in preparation).
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32 Labbook PLJ14, p. lll .
246
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