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The microwave spectrum of chlorine dioxide

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HOUSTON.
TEXAS
THE RICE INSTITUTE
THE MICROWAVE SPECTRUM OF CHLORINE DIOXIDE
by
James L loyd K i n s e y
A THESIS
SUBMITTED TO THE FACULTY
IN PARTIAL FUIFILUVtENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY IN CHEMISTRY
H ouston, Texas
A p r i l , 1959
'
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Number: 3079758
____
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ACKNOWLEDGEMENT
I w o u l d l i k e t o e x p r e s s my s i n c e r e a p p r e c i a t i o n t o
D r . G e o r g e R, B i r d ,
research,
who d i r e c t e d t h e e a r l y s t a g e s o f t h i s
f o r h i s p a t i e n t g u id an c e and p e rs o n a l i n t e r e s t
a n d t o D r . R o b e r t P. C u r l f o r h i s e n t h u s i a s m ,
ad v ice,
frie n d s h ip d u rin g h is d i r e c t i o n o f the l a t t e r p a r t .
would a l s o l i k e
and
I
t o t h a n * J a m e s B a i r d f o r a l l o w i n g me a c c e s s
t o h i s e x p e r i m e n t a l d a t a a n d f o r many s t i m u l a t i n g d i s c u s ­
sio n s,
some o f w h i c h w e r e a b o u t c h l o r i n e d i o x i d e .
I am a l s o i n d e b t e d t o t h e R o b e r t A. W elch F o u n d a t i o n
a n d t o t h e S o u t h e r n F e l l o w s h i p s Fu nd f o r f e l l o w s h i p s
d u r i n g t h e summer o f 1 9 5 8 a n d t h e a c a d e m i c y e a r 1 9 5 8 - 5 9
resp ec tiv e ly .
I w ould l i k e
t o t h a n k D r . L. C. B i e d e n h a r n f o r t h e
l o a n o f a copy o f a n u n p u b l i s h e d p a p e r by Eugene W igner.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T a b le o f C o n t e n t s
ACKNOWLEDGEMENT
PART I
:
INTRODUCTION..........................................................
PART I I
:
THE THEORY OF HYPERFINE STRUCTURE IN
1
NON-LINEAR MOLECULES......................................
10
The M e th o d o f Van VTeck a n d L i n .......................
12
The T e n s o r C o u p l i n g M e t h o d ....................................
15
A B r i e f Summary o f t h e N o t a t i o n .......................
25
Reduced M a t r ix E le m e n ts f o r t h e E l e c t r i c
D i p o l e M o m e n t.....................................................................
27
M a t r i x E l e m e n t s f o r t h e Zeeman E f f e c t . . . .
34
PART I I I :
EXPERIMENTAL RESULTS......................................
37
G e n e r a l T e c h n i q u e s a n d P r o c e d u r e s ..................
37
R e s u l t s o f P r e v i o u s I n v e s t i g a t i o n s ................
33
The 43 t o 45 K i l o m e g a c y c l e R e g i o n ..................
42
Summary a n d D i s c u s s i o n ..............................................
58
REFERENCES...........................................................................................
67
permission of the copyright owner. Further reproduction prohibited without permission.
PART I
INTRODUCTION
Because th e r e a c tio n s
an im p o rtan t r o l e
t o be a b l e
of f r e e r a d i c a l s p la y such
in ch em ical systerns,
to s tu d y t h e i r p ro p e r tie s
In our c a se ,
it
is
one w o u l d l i k e
i n some d e t a i l .
the unusual h v p e rfin e s tru c tu re * in
t h e m icrow ave s p e c t r u m c a u se d b y th e p r e s e n c e
electro n s
th at
is
w i l l be c e n t e r e d
of prim ary i n t e r e s t .
of un p aired
Our c o n s i d e r a t i o n
on t h e m i c r o w a v e s p e c t r u m o f t h e r a d i c a l
ch lo rin e d io x id e .
Some r e m a r k s c o n c e r n i n g t h e s p e c t r a
free rad icals
in g e n e r a l w i l l p re c e d e d i s c u s s i o n of the
p ecu lia rities
o f t h e C102 s p e c t r u m .
U n fo rtu n ately ,
rad icals
one o f t h e c h a r a c t e r i s t i c s
is ex trem e chem ical r e a c t i v i t y so t h a t
of
of free
it
is
g e n erally not p o ssib le
t o o b t a i n them f o r a p p r e c i a b l e
l e n g t h s of tim e e x c e p t
i n v e ry m inute c o n c e n t r a t i o n s .
T his does n o t n e c e s s a r i l y p re c lu d e th e p o s s i b i l i t y of ob­
s e r v a tio n of s p e c t r a
of r e a c t i v e r a d i c a l s as
is evidenced
b y t h e w o r k o f D o u s m a n i s , S a n d e r s , a n d Townes on t h e OH
rad ical
(l) .
The l a b o r i o u s n e s s
of t h e i r ex perim ental
* F o r m o le c u la r s p e c t r a , th e d i s t i n c t i o n betw een " f i n e "
and " h y p e r f i n e " s t r u c t u r e i s n o t c l e a r .
The t e r m " h y p e r f i n e " w i l l be used th r o u g h o u t t h i s t h e s i s to a p p ly t o a l l
such s t r u c t u r e .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-2 w o r k on t h i s
in d icates
r e la tiv e ly lo n g -liv ed
rad ical,
however,
t h a t f u r t h e r e x p e r i m e n t s on u n s t a b l e s p e c i e s
w i l l be c h a l l e n g i n g a t b e s t .
T h e m o s t d e s i r a b l e s y s t e m o f t h i s n a t u r e w o u l d be
a m o l e c u l e h a v i n g one o r m o re u n p a i r e d e l e c t r o n s b u t
s t i l l p o s s e s s i n g some c h e m i c a l s t a b i l i t y .
C hlorine d i ­
o x i d e h a s 33 e l e c t r o n s a n d , w h i l e n o t among t h o s e s u b ­
stan ces
o r d in a r ily considered
prepared
ch em ically s ta b le ,
i n a p u r e s t a t e and i s
a t low p r e s s u r e s
c a n be
s u ffic ie n tly tractab le
in the vapor phase
to p e r m i t m icrowave
m easurem ents.
O th er sim ple m o lecules s a t i s f y i n g
a r e few i n d e e d .
A nalyses
n itric
(15 e l e c t r o n s )
oxide
(2)
o f t h e microwave s p e c t r a
m ag n e tic m olecule in s p i t e
have b een c o m p le te d .
these c o n d itio n s
a n d o x yg e n (3)
of
(a p a r a ­
o f a n e v e n number o f e l e c t r o n s )
N itro g en d io x id e
(4 ,5 )
rep resen ts a
much more c o m p l i c a t e d c a s e b e c a u s e o f a s y m m e t r y o f t h e
m o lecu le.
P aram eters f i t t i n g
t h e NO2 s p e c t r u m t o w i t h i n
o n e m e g a c y c l e h a v e now b e e n o b t a i n e d b y B a i r d a n d B i r d
C hlo rine
trio x id e
in the vapor phase
(7) .
(4 1 e l e c t r o n s )
is re p o rte d
(6).
to exist
An a t t e m p t was made b y t h e a u t h o r
t o p r e p a r e t h i s compound a n d t o s e a r c h f o r i t s
in a b r a s s w aveguide sy s te m ,
spectrum
b u t no l i n e s were o b s e r v e d .
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-3 I t was s u p p o s e d a t
present
t h e time t h a t a n y c h l o r i n e
i n t h e s a m p l e was r e a c t i n g a l m o s t i n s t a n t a n e o u s ­
l y w ith the m e ta l g u id e .
B rack ett
trio x id e
A s i m i l a r s t u d y was made b y
( 8 ) who s u c c e e d e d
in p r e p a r i n g sam ples and m a in ­
t a i n i n g th em i n a q u a r t z w a v e g u i d e s y s t e m l o n g e n o u g h t o
make a r e l i a b l e
th is
search fo r l i n e s .
None w e r e f o u n d
in
c a s e e i t h e r a n d i t was c o n c l u d e d t h a t CIO^ i s
probably a p la n a r or n e a r p la n a r s tr u c tu r e
and h e n c e i t s
m icrow ave s p e c t r u m e i t h e r f o r b i d d e n o r q u i t e w eak .
The o n l y r e m a i n i n g p o s s i b i l i t i e s ,
besides c h lo rin e
d i o x i d e , a r e a few c om pounds o f t h e e l e m e n t s o f t h e t r a n s ­
i tio n group.
M ag n etic s u s c e p t i b i l i t y m easurem ents
in d icate
t h a t many s u c h com p ou n ds h a v e p a r a m a g n e t i c p r o p e r t i e s ,
b u t alm ost a l l
o f them e i t h e r have no v a p o r p r e s s u r e
or
such a hig h d e g re e
o f symm etry t h a t no e l e c t r i c d i p o l e
moment e x i s t s .
is p o ssib le
It
m olecules in t h i s
group,
th at
t h e r e may b e some u s a b l e
h o w e v e r , p a r t i c u l a r l y some o f
t h e more u n s t a b l e s p e c i e s .
The h y p e r f i n e s t r u c t u r e a r i s i n g f r o m i n t e r a c t i o n s
i n v o l v i n g th e s p i n o f an u n p a ir e d e l e c t r o n can i n t r o d u c e
fo rm id ab le c o m p le x itie s
i n t o t h e m icrowave s p e c t r u m ,
e s p e c ia lly i f n u c le a r spins are a lso p resen t .
of a n a ly s is ,
I n term s
i t m ea n s t h a t t h e n u m b e r o f e n e r g y l e v e l s
and hence l i n e s
is m u ltip lie d s e v e ra lfo ld .
In a d d itio n ,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-4 tb e p ro p e r choice of a b a s is f o r s o lu tio n of a given
p r o b l e m may n o t be c l e a r c u t .
There a re
th ree p o s s ib i l i ti e s
f o r the c o u p lin g of
t h e t h r e e a n g u l a r momenta, N ( r o t a t i o n ) ,
spin),
and S ( e l e c t r o n s p i n )
I
(n u clear
to g iv e an in te r m e d i a te
a n d a n o v e r a l l a n g u l a r momentum, F .
They a re as
fo llo w s:
(a)
J
scheme N + S = J ;
J + I = F
(b)
G scheme I + f f = G ;
G + N = "F
(c)
E
E+S = F
scheme N -t-T =E ;
A l l t h r e e schem es c a n o c c u r n a t u r a l l y a s good a p p r o x i ­
m a t i o n s , n a m e l y when t h e d o m i n a t i n g t e r m i n t h e
fin e
in te rac tio n
actio n ,
h am ilto n ian is
(b) F e r m i c o u p l i n g ,
Largeness
of the
(a) s p i n - r o t a t i o n
i n t e r a c t i o n betw een th e m a g n e tic d i p o l e
(d ip o le -d ip o le
in te r­
does n o t correspond to any of th ese coupling
schem es b u t r a t h e r t o a n i n t e r m e d i a t e s i t u a t i o n .
is,
in te r­
and (c ) q u a d r u p o l e c o u p l i n g .
of th e n u c le u s w ith t h a t o f the e l e c t r o n
actio n )
hyper-
That
t h e i n t e r m e d i a t e quantum number i n t h a t c a s e d o es
n o t c o r r e s p o n d t o a n y a n g u l a r momentum.
There
i s n o a. p r i o r i r e a s o n f o r e x p e c t i n g a n y g i v e n
r e l a t i o n s h i p betw een th e s i z e s
o f th e v a r i o u s term s
the h a m ilto n ia n f o r a p a r t i c u l a r problem .
In fa c t,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
in
a
-5 m o l e c u l e c o u l d c o n c e i v a b l y a p p r o x i m a t e one c o u p l i n g
sc hem e f o r some e n e r g y l e v e l s ,
ferent lev els,
and r e q u i r e
describe s t i l l
others .
a n o t h e r schem e f o r d i f ­
in te rm e d ia te cou p lin g to
T r a n s i t i o n s betw een l e v e l s
b e l o n g i n g t o d i f f e r e n t s c h e m e s may be a l l o w e d .
Because of th e e x is te n c e
o f a number o f i n v a r i a n t s
o f t h e e n e r g y m a t r i x u n d e r u n i t a r y t r a n s f o r m a t i o n s and
c ertain fac to riz atio n s
of the s e c u l a r eq u atio n
e n t of th e c o u p lin g scheme,
independ­
t h i s a m b i g u i t y n e e d n o t be
an in su rm o u n ta b le h andicap i f a r o t a t i o n a l a ssig n m en t
can be made.
if
F o r d i a t o m i c a nd s y m m e t r i c t o p m o l e c u l e s ,
the b y p e rfin e s p l i t t i n g
i s s m a l l c o m p a red t o t h e
r o t a t i o n a l c o n s t a n t B*, th e r o t a t i o n a l assig n m en t w i l l
s t i l l be a p p a r e n t b e c a u s e o f t h e r e g u l a r i t y o f t h e r i g i d
r o t o r spectrum of such a m o le cu le .
F o r asym m etric tops
The a b s e n c e o f a n a x i s
the s i t u a t i o n
is q u ite d i f f e r e n t .
o f sym m etry removes th e d e g e n e r a c y
due t o t h e quantum num ber K c h a r a c t e r i s t i c
of sym m etric
*
B = 7r 9 , w h e r e . XB i s t h e moment o f i n e r t i a a b o u t t h e
i n t e r m e d i a t e a x i s . F o r m o le c u le s w ith an a x i s o f sym m etry,
th e r o t a t i o n a l sp ectru m is a s e r i e s of ev en ly spaced l i n e s ,
2B a p a r t .
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-6 tops .
The s p e c t r u m becomes im m ensely e n r i c h e d and
i r r e g u l a r l y spaced.
E xam ination of the g ro ss a p p e a r ­
ance of th e sp e c tru m no lo n g e r s u f f i c e s f o r a s s ig n m e n t,
e v e n when t h e r e
i s no b y p e r f i n e s t r u c t u r e a t a l l .
u su a l procedure
in t h i s
im ate r i g i d
case is
to c a l c u l a t e an a p p ro x ­
r o t o r s p e c t r u m f r o m moments o f i n e r t i a
b a s e d on a r e a s o n a b l e m o d e l s t r u c t u r e
f o r the m o le c u le .
S u c h a m o d e l may be o b t a i n e d f r o m X - r a y ,
frac tio n ,
electro n d i f ­
or o th e r s p e c tro s c o p ic d ata or perhaps guessed
f r o m t h e known s t r u c t u r e s
rate,
The
o f s i m i l a r compounds.
At a n y
th e d a ta a v a i l a b l e a r e seldom r e l i a b l e enough f o r
an a c c u r a t e p r e d i c t i o n o f t h e m icrowave s p e c t r u m b e c a u s e
of th e extrem e s e n s i t i v i t y
lev els
is
to sm all changes
obtainable
is an id ea
of the r o t a t i o n a l e n erg y
i n t h e moments o f i n e r t i a .
What
o f th e g e n e r a l a p p e a r a n c e and
d e n s i t y o f t h e s p e c tr u m and th e a p p ro x im a te l o c a t i o n o f
certain
in te restin g
tran sitio n s
(e .g .,
those
invo lv in g
low v a l u e s o f t h e r o t a t i o n a l q u a n t u m n u m b e r . )
The c o n d i t i o n o f t h e s m a l l n e s s o f b y p e r f i n e s p l i t t i n g
i s much m ore s e v e r e f o r t h e a s y m m e t r i c r o t o r s
sym m etric o n e s .
than f o r
S in c e th e a v erag e sp a c in g s betw een r i g i d
ro to r lin e s are c o n sid era b ly sm aller in th is
case,
the
h y p e r f i n e s e p a r a t i o n c o u ld e a s i l y be l a r g e enough t o
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-7 oause a c o n f u s in g o v e rla p of m u l t i p l e t s w h ile being
q u ite sm all
in com parison to th e e n e r g y d i f f e r e n c e s
betw een r o t a t i o n a l l e v e l s .
rig id
r o t o r spectrum ,
For a s u f f i c i e n t l y sparse
t h i s problem is n o t e n c o u n te re d ,
a lth o u g h a com plete a ssig n m en t w ith in a m u l t i p l e t
still
l i k e l y t o be t r o u b l e s o m e .
O ther,
more d i r e c t m e t h o d s e x i s t f o r d e t e r m i n a t i o n
or c o n firm a tio n o f an a ssig n m e n t.
o r Z eeman e f f e c t c a n b e r e s o l v e d ,
N o tab ly ,
t r a n s i t i o n w i l l be i n d i c a t e d b y t h e
resolved
com ponents .
if
the S t a r k
one o r b o t h o f t h e
v a l u e s o f t h e o v e r a l l a n g u l a r momentum (F)
the
is
involved
in
t o t a l number of
D e te rm in a tio n o f F v a lu e s does n o t
n e c e s s a r i l y y i e l d an u n e q u iv o c a l a ssig n m e n t s in c e
there
i s u s u a l l y more t h a n one c h o i c e f o r t h e r o t a t i o n a l and
i n t e r m e d i a t e quantum number f o r a g i v e n v a lu e o f F .
o f the p o s s i b i l i t i e s
can u s u a l l y be e l i m i n a t e d by con­
s i d e r a t i o n of the p re d ic te d
A f u r t h e r aid
rig id
sitio n s
r o t o r spectrum .
to assignm ent is
a number of b e a t f r e q u e n c i e s ,
by B a ird
Some
(5)» be r e p e a t e d
called
the requirem ent th a t
"com bination r u le s "
in the sp ectru m .
When t r a n ­
from a g iv e n e n e rg y l e v e l to b o th of a p a i r of
energy le v e ls a re allow ed,
b e t w e e n t h e two l i n e s
the d if f e r e n c e
in frequency
corresponds to the energy se p a ra tio n
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-8 o f the p a i r .
I f there
makes t r a n s i t i o n s
i s a n o t h e r e n e r g y l e v e l which
to each of the l e v e l s
of the p a i r ,
th e n t h a t f r e q u e n c y d i f f e r e n c e w i l l be r e p e a t e d by th e
s e c o n d two l i n e s .
C onsider,
f o r exam ple,
the case of
a m o le c u le f o r w h ic h t h e J scheme i s a p p r o p r i a t e * .
of the t r a n s i t i o n s
±1;A J = 0, ± 1 ;
None
o b e y i n g t h e s e l e c t i o n r u l e s AN = 0 ,
AF = 0 ,± 1 a r e s t r i c t l y f o r b i d d e n .
For
h i g h N, o n l y t h e AN = A J ~ A F l i n e s a r e a p p r e c i a b l y i n ­
ten se,
and n o c o m b i n a t i o n r u l e s c a n o c c u r w i t h i n t h e
h y p erfin e m u lt ip le t b e lo n g in g to any r o t a t i o n a l
itio n .
At low N, h o w e v e r ,
tran s­
t h e o t h e r s become more a l l o w e d
a n d one may e x p e c t t o s e e m ore l i n e s p e r m u l t i p l e t w i t h
a c h arac teristic p a ttern
sev eral features
of com bination r u le s
having
in d ep e n d en t o f the c o u p lin g scheme.
R e t u r n i n g t o c o n s i d e r a t i o n o f c h l o r i n e d i o x i d e , we
n o t e t h a t t h e c a l c u l a t e d moments o f i n e r t i a
require a
r i c h l y p o p u l a t e d m icrow ave s p e c t r u m and t h a t 1 = 3 / 2 f o r
both C l ^
and 0 1 ^ 7 .
The l a t t e r
in fo rm atio n leads
to the
♦ S i m i l a r rem arks r e g a r d i n g th e s e l e c t i o n r u l e s a p p ly to
b o t h t h e G and E schem es . F o r i n t e r m e d i a t e c o u p l i n g t h e r e
may be n o a n a l o g o u s s t r i c t s e l e c t i o n r u l e on t h e i n t e r ­
m e d i a t e quantum num ber s i n c e i t i s n o t r e q u i r e d b y sym m etry
in th is c a s e .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-9 p red ic tio n
of a spectrum c o n s i s t in g of a p a i r of q u a r te ts
f o r e v e r y bigb N r o t a t i o n a l t r a n s i t i o n
ap p lies.
The G s c h e m e p r e d i c t i o n
if
i s o ne q u i n t e t and one
t r i p l e t p e r high N t r a n s i t i o n w h ile t h a t
is a s e t of fo u r d o u b le ts .
sidered
t h e J sc hem e
o f t h e E scheme
The s p e c t r u m w i l l be c o n ­
i n d e t a i l b e lo w , b u t i t m ig h t be m en tio n ed a t
th is point
th a t the c h a r a c t e r i s t i c
the J schem e.
appearance
is
th a t of
The s p e c t r u m c o n s i s t s a l m o s t e n t i r e l y o f
many c l o s e l y s p a c e d q u a r t e t s w i t h o n l y o c c a s i o n a l t r i p l e t s
(perhaps
in com pletely resolved q u a rte ts )
iso lated sin g le lin e s .
tere stin g
J_ _ _ _
regions
However,
and a few
i n some e s p e c i a l l y i n ­
t h i s p a t t e r n b r e a k s down.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
10 -
PART I I
THE THEORY OP HYPERPINE STRUCTURE IN
NON-LINEAR MOLECULES
H yperfine s t r u c t u r e i n m o le c u le s,
is
as i n atom s,
a p p ro a c h e d t h e o r e t i c a l l y from t h e s t a n d p o i n t o f
p e rtu rb a tio n theory.
The t h e o r e t i c i a n ' s
task,
once
th e c o rr e c t h am ilto n ian f o r th e p e rtu r b a tio n has been
chosen,
is
to c a l c u l a t e i t s m a t r i x e le m e n ts and s o lv e
a s e c u la r e q u atio n .
S im p lify in g assum ptions can
u s u a l l y b e made t o f a c t o r t h e s e c u l a r e q u a t i o n i n t o
fairly
s m a l l b l o c x s so t h a t i n p r a c t i c e
the o n ly non­
t r i v i a l problem i s g e n e r a l l y t h e c a l c u l a t i o n o f th e
m a trix elem en ts.
In th is
s e c t i o n a d i s c u s s i o n i s g iv e n o f th e m ethods
p re v io u sly ap p lied in a tta c k in g the th eo ry of h y p erfin e
stru ctu re
i n n o n -lin e a r m o lecu les* .
A d ifferen t tech ­
nique f o r c a l c u l a t i o n of th e m a tr ix elem ents o f th e h y p e r­
f i n e I n t e r a c t i o n h a m i l t o n i a n i n a p u r e c o u p l i n g sc h em e
is
i n t r o d u c e d and i t s m e r i t s p o i n t e d o u t .
T h is m ethod,
* L i n e a r m o l e c u l e s may h a v e n o n - v a n i s h i n g c o m p o n e n t s o f
o r b i t a l a n g u l a r momentum a l o n g t h e i n t e r a t o m i c a x i s .
We
a r e n o t i n t e r e s t e d i n e f f e c t s a r i s i n g from t h i s p o s s i ­
b ility .
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-1 1 w h i c h we c a l l t h e t e n s o r c o u p l i n g m e t h o d ,
i s based
on th e use o f v e c t o r r e c o u p l in g c o e f f i c i e n t s
to sim p lify
m a t r ix e le m e n ts o f o p e r a t o r s form ed by th e c o u p lin g o f
irred u c ib le ten so rs.
I t w i l l be u sed t o c a l c u l a t e
the
m a t r i x e l e m e n t s f o r t h e S t a r k a n d Zeeman e f f e c t s .
When no e x t e r n a l f i e l d s a r e p r e s e n t ,
the t o t a l
h a m ilto n ia n f o r t h e r o t a t i o n a l problem i s :
(I I -1)
H * Hq + Hn s + HI S + Hp 4* Hq .
Hq r e p r e s e n t s t h e e n e r g y o f r o t a t i o n o f t h e r i g i d f r a m e
o f th e m o lecu le.
The e n e r g y e x p r e s s i o n s a n d e i g e n ­
f u n c t i o n s f o r t h i s t e r m a r e w e l l known ( 9 ) a n d w i l l n o t
be t r e a t e d h e r e .
I t arises
HNS i s
the s p i n - r o t a t i o n i n t e r a c t i o n .
f r o m two s o u r c e s
— the d ir e c t in te r a c tio n
b e tw e e n t h e m a g n e t i c d i p o l e s due t o t h e e l e c t r o n s p i n and
to the o v e r a ll
ro tatio n ,
and from t h e s p i n - o r b i t i n t e r ­
a c t i o n o f t h e u n p a i r e d e l e c t r o n v i a a Van V l e c k t r a n s ­
fo rm ation (1 0 ).
re la tiv istic
HjS i s
t h e term d e s c r i b i n g t h e n o n -
i n t e r a c t i o n b e t w e e n t h e m a g n e t i c d i p o l e s due
to th e n u c le a r and e l e c t r o n sp in s
to as th e d ip o le - d ip o le t e r n .
is
a rela tiv istic
(11).
It is
referred
The F e n n i c o u p l i n g ,
c o r r e c t i o n to H jg.
f o r e l e c t r o n s w i t h some "s'* c h a r a c t e r .
Hp ( 1 2 ) ,
I t i s im p o rta n t only
The Hq t e r m comes
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-1 2 from t h e e l e c t r o s t a t i c
I n t e r a c t i o n betw een th e n u c l e a r
q u a d r u p o l e moment a n d t h e g r a d i e n t o f t h e e l e c t r i c
j
ij
j
i.
o f th e m olecule a t th e n u c le u s
(13) -
field
These l a s t f o u r
term s com prise th e h y p e r f in e i n t e r a c t i o n h a m ilto n ia n .
!;
!;
!
j
The M etho d o f Van VLectc and L i n
' '
Methods f o r e v a l u a t i o n o f t h e e f f e c t s o f th e h y p e r -
j
fin e
i n t e r a c t i o n h a v e b e e n d i s c u s s e d b y a number o f
!
au th o rs
(^> 1 0 ,
14,
15).
The w o r k o f L i n ( 1 4 ) o n t h e
th e o r y o f th e n i t r o g e n d io x id e spectrum i s
sig n ifican ce.
The h a m i l t o n i a n i s
of esp ecial
w ritte n in a coordinate
system f i x e d i n th e m o le c u le f o r h i s t r e a t m e n t .
T his
c h o i c e i n t r o d u c e s a s l i g h t d i f f i c u l t y a r i s i n g from t h e
f a c t t h a t t h e c o m p o n e n t s o f t h e t o t a l a n g u l a r momentum
I n t h e m o l e c u l a r fram e s a t i s f y com m utation r e l a t i o n s h i p s
w ith th e sig n o f i
r e v e r s e d from w hat w ould a p p e a r i f
c o m p o n en ts were m e a su re d r e l a t i v e
of axes.
(I,
S,
the
to a space fix e d s e t
The c o m p o n e n t s o f t h e i n t e r n a l a n g u l a r m om enta
G) on t h e o t h e r h a n d ,
o bey com m utation r u l e s w it h
th e norm al s ig n o f i i n e i t h e r fram e.
Van V l e c k ( 1 0 ) h a s c o n s i d e r e d t h i s a n o m a l y .
p o in ts out th a t i f
(I,
S,
He
r e v e r s e d i n t e r n a l a n g u l a r m om enta
G, w h e r e I x = e t c . )
a re in tro d u ced ,
th en the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-1 3 s i g n o f i i n t h e com m utation r u l e s i n th e m o le c u l a r fram e
is
c o n s i s t e n t f o r a l l a n g u l a r momenta.
R elatio n sh ip s
b e t w e e n t h e v a r i o u s a n g u l a r m o m e n ta h a v e t o b e r e w r i t t e n
in th is
system .
have N = J-t-S.
For exam ple,
i n s t e a d o f J = N + S, we
J and S can th e n be th o u g h t o f a s p r e -
c e s s i n g a ro u n d N w i t h o u t n u t a t i o n so t h a t t h e m ethod o f
p ro jectio n
(1 6 ) can be u s e d t o o b t a i n t h e m a t r i x e l e m e n t s
d i a g o n a l i n N.
Van V l e c k n o t e s t h a t many o f t h e s e w i l l
t u r n o u t t o be o f t h e same f o r m a s m a t r i x e l e m e n t s o c c u r ­
r i n g i n a t o m i c s p e c t r o s c o p y p r o b l e m s a nd c a n be f o u n d i n
s t a n d a r d w o r k s on t h a t s u b j e c t
(17).
M atrix e lem en ts f o r
t h e s p i n - r o t a t i o n i n w hat c o r r e s p o n d s t o o u r J scheme
are given in h is paper.
L i n u s e d t h e r e v e r s e d a n g u l a r momentum n o t a t i o n t o
fa c ilitate
c a lc u la tio n o f the
rem aining m a tr ix e le m e n ts i n
b o t h J and G schemes f o r t h e sy m m etric to p c a s e .
Because
o f th e s m a lln e s s o f th e c o u p lin g term s i n com parison to
th e r o t a t i o n a l energy,
he d e c id e d t h a t m a tr ix e le m e n ts
c o n n e c t i n g d i f f e r e n t r o t a t i o n a l l e v e l s c o u ld be i g n o r e d
to a good ap p ro x im atio n .
S — ^ and t h a t F i s
In th is
case,
the f a c ts
th at
d i a g o n a l i n a n y c o u p l i n g sc h e m e l e a d
to th e f a c t o r i z a t i o n o f th e s e c u la r d eterm in an t in to
i
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
a
-1 4 n u m b e r o f 2X2 o r 1X1 b l o c k s a l o n g t h e m a i n d i a g o n a l .
The c o m p l e t e l y d i a g o n a l m a t r i x e l e m e n t s w e r e
o b t a i n e d by t h e m e t h o d o f p r o j e c t i o n .
To o b t a i n t h e
o f f - d i a g o n a l e l e m e n t s i n t h e 2X2 b l o c k s ,
he em ployed an
i n g e n io u s t r l c x w hich u se d o n ly th e i n v a r i a n c e o f th e
t r a c e o f a m a t r i x u n d e r s i m i l a r i t y t r a n s f o r m a t i o n s and
the f a c t th a t th e s p in - r o ta tio n in te r a c tio n i s
d ia g o n a l in th e J scheme.
co m pletely
I f we w r i t e down t h e e q u a t i o n s
e x p re ssin g th ese c o n d itio n s f o r the tra n s fo rm a tio n J
sc hem e t o G sc hem e o f
( a ) t h e 2X2 m a t r i c e s o f a n y o n e o f
t h e c o u p lin g term s and o f th e s p i n - r o t a t i o n i n t e r a c t i o n
( b ) the squares of th e se m a tric e s ,
and ( c ) t h e s q u a r e o f
t h e i r sum, e n o u g h r e l a t i o n s h i p s w i l l e x i s t t o p e r m i t t h e
o f f - d i a g o n a l e le m e n ts i n b o th o f th e c o u p lin g schemes to
be e x p re s s e d i n term s o f th e d ia g o n a l o n e s.
L in,
A ccording to
t h i s i s an e x t e n s i o n o f t h e Goudsmit i n s p e c t i o n
m ethod (1 8 ).
These r e s u l t s were e x te n d e d by B a ir d
(5) to th e case
o f t h e asym m etric r o t o r by c o n s i d e r a t i o n o f th e e x p a n s io n
o f t h e a s y m m e t r i c t o p wave f u n c t i o n s i n t e r m s o f Wang
co m b in atio n s o f th e sym m etric to p f u n c t i o n s .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-1 5 The T e n s o r C o u p l i n g M e th o d
The a l g e b r a o f i r r e d u c i b l e t n e s o r s i n q u a n t u m m e c h a n i c s
i s p rom inent i n
t h e g e n e r a l t h e o r y o f a n g u l a r momenta.
The o r i g i n a l d e v e l o p m e n t s o f t h e t h e o r y o f t e n s o r s w e r e
made b y R a c a h ( 1 9 ) a n d i n d e p e n d e n t l y b y W i g n e r ( 2 0 ) .
Synopses o f t h e s e and su bseq uen t tr e a tm e n ts a p p e a r i n a
number o f e x c e l l e n t t e x t s r e c e n t l y p u b l i s h e d
(21,
22,
2 3 ).
I t s h o u l d be e m p h a s i z e d t h a t o u r d i s c u s s i o n b e l o w i s
n o t t o b e c o n s i d e r e d a new c o n t r i b u t i o n t o t h e t h e o r y i n
th at a ll
th e fundam ental r e l a t i o n s h i p s g iv en a re e i t h e r
c o n t a i n e d i n t h e o r i g i n a l works o r a r e o b t a i n a b l e from
them by m a k i n g m i n o r e x t e n s i o n s .
first
T his i s ,
however,
the
tim e t o o u r knowledge t h a t t h e s e m ethods have b e en
a p p lie d s y s t e m a t i c a l l y to th e problem s o f h y p e rf in e i n t e r ­
a c tio n in m o lecu les.
The n o t a t i o n u s e d i n t h i s s e c t i o n w i l l b e m a i n l y
t h a t o f W igner ( 2 0 ) .
av ailab le
Since h i s p a p e r i s la m e n ta b ly u n ­
to th e g en eral p u b lic ,
we r e f e r t h e r e a d e r t o
a b r i e f su m m ary o f h i s n o t a t i o n a t t h e e n d o f t h i s s e c t i o n
and a l s o t o Edmonds' book (22) w hich i s
w ritten in th is
n o t a t i o n and c o n ta in s an e x p la n a tio n o f i t .
A lthough
Edmonds may n o t b e c o n s i d e r e d a p r i m a r y s o u r c e f o r t h e
eq u atio n s g iv en ,
it
w i l l be c i t e d a s a r e f e r e n c e whenever
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 16p o s s i b l e i n o r d e r to m inim ize th e in c o n v e n ie n c e o f
check in g.
R e f e r e n c e t o e q u a t i o n s a p p e a r i n g i n E dm o n d s '
b o o k w i l l b e made b y e n c l o s u r e i n p a r e n t h e s e s o f a
c a p i t a l E f o l l o w e d by h i s n u m b erin g o f t h e e q u a t i o n .
L e t u s b e g i n b y c o n s i d e r i n g t h e t h r e e co m m u tin g
a n g u l a r momenta
o p erato rs T ^ ) ,
J2
and
T is
Jy
j 2 , and j
and th e th r e e te n s o r
U ( k 2 ), a n d V ( k 3 ) .
U w ith
and
J
, V w ith
L e t T commute w i t h
and
J2.
r e l a t e d t o t h e same d e g r e e o f f r e e d o m a s
That i s ,
e tc.
S u p p o s e t h a t T, U, a n d V c o u p l e i n some way t o f o r m
a t e n s o r X o f ramc K t h a t i s
momentum J .
related
to the t o t a l a n g u la r
We c h o o s e t h e c a s e i n w h i c h T a n d U c o u p l e
t o g i v e a n i n t e r m e d i a t e w h i c h c o u p l e s t o V t o g i v e X.
T his i s
w r itte n sy m b o lically :
(II-2)
(t C ^ ) ,
U(k2 ) )
W(kl 2 ) , V ( k 3 ), X( KQ) .
i n a n a lo g y to th e n o t a t i o n f o r th e c o u p lin g o f the a n g u la r
momentum v e c t o r s .
We c h o o s e t h e r e p r e s e n t a t i o n i n w h i c h
t h e a n g u l a r m o m e n t a c o u p l e i n t h e same way a s t h e t e n s o r s :
1( J i , J2 )
A*1 e q u a t i o n f o r t h e r e d u c e d
m a t r i x e l e m e n t s o f X(K) i n t h i s
by su c ce ssiv e a p p lic a tio n s o f
re p re s e n ta tio n i s ob tain ed
(E 7 . 1 . 5 ) :
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-1 7 (^"3)
( * ( ^ J H ) j l2_;
,J
IIX(K)^yC
jfKz I
hi
IIW Cfe\2)lU11i'll >
I
(H -4)
h'
^3
^3 f
J'
T
<
(*"H
HVCk3) l u j 3>.
< * i . ' . i p i.z - J P J ' I i x o o i m },, j p h z • h -J >
=
[ ( . 2 ^ l ) C
'
2 X
ill
ii
I j'
,4 - 0 ( 2 1 C + l ) ( a i t t ^
O
t t j li i - l ) ( 2
hz.
(if
ii
h
T
•U
h
K 1
■ in
t u
v
0 } ^
^2.
h
i" p
7
<X“ i3\tVCte3 ) U i 3> ] .
The c o m p l e t e m a t r i x e l e m e n t s c a n t h e n be o b t a i n e d b y t h e
use o f th e d e f i n i t i o n o f th e reduced elem ents
(see
s e c t i o n below on t h e n o t a t i o n ) .
The o b j e c t i o n may b e r a i s e d a t t h i s p o i n t t h a t we
h a v e n o t t r e a t e d t h e m o s t g e n e r a l c a s e b e c a u s e t h e r e may
b e ways i n w h i c h T,
U, a n d V c a n c o u p l e t o g i v e a
r e s u l t a n t o f r a n k K t h a t do n o t c o r r e s p o n d t o t h e c o u p l i n g
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 18sc h e m e f o r t h e r e p r e s e n t a t i o n we w i s h t o u s e .
exam ple, i n a d d i t i o n t o
For
(II-2 ) the fo llo w in g p o s s ib i l i ty
e x ists:
(II-5)
TC IO ,
Y 0 ; , ) , X '(K Q ) .
H o w ev e r , a r e c o u p l i n g t r a n s f o r m a t i o n c a n b e a p p l i e d
t a x e X' i n t o
( i i - 6)
t h e form d e s i r e d
X'
to
(E 6 . 1 . 5 ) :
ICQ) = 2
U ls^+ oj*
ba
K
.X
j bi
f e >-
\ t,
K
\
S
KQ ) .
so t h a t we c a n a l w a y s g e t t o a f o r m s u i t a b l e f o r u s e i n
(II-U ).
I n t h e c a s e where any o f
equal to z ero ,
to a s in g le
(II-7)
the
term ,
j 0
IO
or K is
sum i n t h e t r a n s f o r m a t i o n w i l l r e d u c e
and s in c e
^
C
the tran sfo rm a tio n i s
(E 6 . 3 . 2 ) :
( - l ) a " k+C
b 1
triv ia l.
[ ( I b + O U c t O l ' ,/‘
1;
We o b t a i n t h e r e l a t i o n s h i p
in th is case:
( I I -8)
KQ) =
X ((.L > ,) KQ).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-1 9 As i t
tu rn s out,
are of th is
a l l t h e o p e r a t o r s we a r e i n t e r e s t e d i n
type.
Among t h e o p e r a t o r s we w i l l h e c o n s i d e r i n g ,
ten so rs
depending on th e n u c le a r and e le ctro n ic s p a t i a l c o ­
o r d i n a t e s p la y an im p o rta n t r o l e .
elem ents o f a t y p i c a l
The r e d u c e d m a t r i x
t e n s o r TN( k ) o f t h i s
t y p e may be
o b t a i n e d m ost c o n v e n i e n t l y by e v a l u a t i n g t h e m a t r i x elem en t
o f th e component T ^ ( k ,0) and th e n u s in g th e d e f i n i t i o n
o f the reduced e le m e n t:
(11-10)
OfN HT»•%( k ) l \ Y* N - > -
< V N M | TN a , 0 ) \ r ' N ‘M>
M-N'
E v a lu a tio n o f th e m a trix elem ent i t s e l f i s n o t d i f f i c u l t
b ecau se i n th e m o le c u la r c o o r d i n a te system ,
t h e compo-
n e n t s j ( k , q ) o f t h e t e n s o r m ay b e c o n s i d e r e d a s c o n s t a n t s .
T h is comes a b o u t t h r o u g h t h e B o rn -O p p e n h e im e r a p p r o x i ­
m ation
(2 4 ) w h ic h a l l o w s t h e s e o p e r a t o r s t o be r e p l a c e d
by t h e i r e x p e c ta ti o n v a lu e s i n th e v i b r a t i o n a l and
e le c tro n ic s ta te of i n te r e s t.
The c o m p o n e n t T ^ ( k , 0 ) i s
t h e n o b t a i n e d f r o m th e m t h r o u g h t h e t r a n s f o r m a t i o n :
j _______________
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
w h e re
Y are
th e E u le r ia n a n g le s l o c a t i n g th e m o le c u la r
fram e w ith r e s p e c t to th e sp a c e a x e s , and
is
(d ,(3 , y )
qU
*
t h e qO e l e m e n t o f t h e k - d i m e n s i o n a l i r r e d u c i b l e
r e p r e s e n t a t i o n o f t h e r o t a t i o n g r o u p f o r t h e r o t a t i o n by
T hus:
(11-1 2 )
<YNM|TN( k , 0 )
1 3U k,< 3)
S-
< y n m | £ ^ 0 (* , ( ^ Y O W ’N 'M ) .
S i n c e <S1<:
is
e s s e n t i a l l y a s p h e r i c a l h a rm o n ic
(E 4 . 1 . 2 5 ) ,
i t s m a tr ix e le m e n ts a r e r e a d i l y o b ta in e d i n e i t h e r th e
sy m m etric to p o r a s y m m e tric to p r e p r e s e n t a t i o n .
n a tio n o f
C om bi­
( I I - I O ) and (1 1 -1 2 ) th e n le a d s to th e red u ced
e le m e n ts.
We t u r n now t o t h e v a r i o u s t e r m s i n t h e h y p e r f i n e
i n te r a c tio n h a m ilto n ia n .
T hat each o f th e se i s
re p re se n t­
a b l e a t l e a s t b y a sum o f t e n s o r s o f t h e f o r m X ( 0 ) i s
o b v io u s from t h e f a c t s
th a t,
s c a l a r q u a n t i t i e s and t h a t ,
a c tio n s ,
as e n e rg ie s,
th e y re p re se n t
b e in g p u r e ly i n t e r n a l i n t e r ­
th e y a re in d e p e n d e n t o f th e o r i e n t a t i o n o f th e
c o o r d i n a t e s y s te m to w h ich t h e o p e r a t o r s a n d s t a t e s a r e
referred .
F or th e sake o f c l a r i t y ,
h o w e v e r , we w i l l show
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-2 1 th e c o u p lin g i n d e t a i l .
The c o m p l e t e s p i n - r o t a t i o n i n t e r a c t i o n h a m i l t o n i a n
h a s b e e n d e r i v e d b y Van V l e c k
(1 0 ).
We w r i t e i t
i n th e
f o l l o w i n g w ay:
(11-14)
h n s = t n -"s
+ UN(1):C!S C2)
.
w h e re
(1 1 -1 5 )
Tn =
a0N + W N( 2 ) - N
.
The m a t r i x e l e m e n t s c a n be f u r t h e r s i m p l i f i e d b y m a k in g
u se o f th e r e l a t i o n s h i p
(II-16 )
f" m«
(E 7 . 1 . 1 ) :
<V m'\\WnU ) - N I U M> =
^N
N '
(_i)N'+N+l
M*' I
N
The r e d u c e d m a t r i x e l e m e n t o f N i s g i v e n b y (E 5 . 4 . 3 ) :
( 11 - 1 7 )
< t ,N, llMl\*u N“ > = [ ( 2 N ' + \ ) ( N ' v i ) N 1] ^
5* a" •
w h ic h g i v e s
(ii-is)
<m , * ,\iwNu ) - * u \ M * y = - { 5 1 \ [
[U N 1v \ ) ( N ‘ v O N ' 1 ^
]>
1M
2
N'
1
M’
HW n U ) \ U N > .
\
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-2 2 -
i
The s e c o n d t e r m i n
(II-1 4 ) is
q u a d ru p o le " i n t e r a c t i o n .
That i s ,
c a lle d a
"pseudo-
a lth o u g h n o t a r i s i n g
fro m t h e p r e s e n c e o f a n y a c t u a l q u a d r u p o l e m om ent, i t
i s o f t h e same f o r m .
A ll m a tr ix e le m e n ts f o r t h i s p a r t
w i l l v a n i s h i d e n t i c a l l y u n l e s s S —1 .
The H jS t e r m i s
o f t h e same fo r m a s t h e c l a s s i c a l
e x p r e s s i o n f o r t h e i n t e r a c t i o n o f two d i p o l e s :
(11-19)
H1S =
w h e re g g , g N, j j l q ,
[ ( I 'S ) - 3 ( I - f ) ( S - r ) / r z ] .
a re ,
r e s p e c tiv e ly , th e e le c t r o n i c
and n u c l e a r g y ro m a g n e tic r a t i o s and th e B ohr and n u c l e a r
m a g n e to n s .
We c a n r e w r i t e t h i s e x p r e s s i o n a s :
(1 1 -2 0 )
Rn ( 2 ) - S = Y
WIS .
The F e rm i c o u p l i n g i s a l r e a d y i n t h e d e s i r e d f o r m ,
b e i n g t h e s c a l a r p r o d u c t o f tw o v e c t o r s :
(11- 21)
w here ^ ( C ) i s
HF r
CX-S')
t h e v a l u e o f t h e wave f u n c t i o n o f t h e o d d
e l e c t r o n e v a l u a t e d a t t h e n u c l e u s w hose s p i n i s
The n u c l e a r q u a d r u p o l e c o u p l i n g i s
d e s ir e d form .
It is
I.
lik e w is e in th e
t h e s c a l a r p r o d u c t o f two s e c o n d r a n k
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-2 3 t e n s o r s — t h e n u c l e a r e l e c t r i c q u a d r u p o l e m om ent a n d
th e m o le c u la r
9,
* * quad ru pole c o u p l i n g c o n s t a n t "
(referen ce
C h a p te r 6 ):
(1 1 - 2 2 )
Hq
Vk (2 ) '.0 ^ ( 2 )
=
.
The h a m i l t o n i a n s f o r t h e S t a r k a n d Zeem an e f f e c t s
b o th have th e
same f o r m :
H' = T- Z
(II-2 3 )
w here T = ^ = > 1 ^ (1 ),
th e e l e c t r i c
d ip o le
m o l e c u l e a n d Z - E,
th e e l e c t r i c
fie ld
S tarK e f f e c t ;
m om ent
in te n sity
o f th e
f o r th e
T= - g 0 )*eS a n d Z ■=B, t h e m a g n e t i c f i e l d
i n t e n s i t y f o r t h e Z eem an e f f e c t .
A lth o u g h (1 1 -2 3 ) i s
two v e c t o r s ,
w ritte n
th e s i t u a t i o n h ere i s
f ro m t h a t o f t h e F e r m i c o u p l i n g .
as th e
sc a la r
p ro d u c to f
fu n d a m e n ta lly d i f f e r e n t
The r e a s o n i s
th a t E
and B a r e n o t o p e r a t o r s and t h e r e f o r e can be ta x e n o u ts id e
th e m a tr ix e le m e n ts i n t a c t .
T h is m akes t h e a c t u a l m a t r i x
e le m e n ts to be c a l c u l a t e d th o s e o f a f i r s t
raiu c t e n s o r
r a t h e r th a n o f an i n v a r i a n t :
(1 1 -2 4 )
F M 'l H 'l- F M )
= 2.
FV
\Tt \ -
=
( - F ' M ' l T i F K )
t-O *
F N \ ) - Z
2 .0
*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-2 4 S i n c e t h e r e i s no i n t e r n a l c o u p l i n g p r o b l e m f o r e i t h e r
o f th e se e f f e c ts ,
p a rtic u la rly
th e m a tr ix e le m e n ts w i l l reduce to a
sim p le fo rm .
T h ese a r e c a l c u l a t e d b e lo w .
I t i s p e r t i n e n t a t t h i s p o i n t t o r e f l e c t on w h e t h e r
th e i n t r o d u c t i o n o f th e e l a b o r a t e fo rm a lism o f te n s o r
a lg e b r a r e a l l y le a d s to any a d v a n ta g e s i n th e c a l c u l a t i o n
o f m a trix e le m e n ts.
A lth o u g h i t m u st be a d m itte d t h a t
t h e a c t u a l am ount o f l a b o r r e q u i r e d t o g e t m o st m a t r i x
e le m e n ts by hand c a l c u l a t i o n i s
t h i s m eth o d a s i t
is
p ro b a b ly a s g r e a t u sin g
u sin g o th e r s ,
t h e m e th o d h a s some
u n q u e stio n a b le v i r t u e s .
Once t h e t e c h n i q u e h a s b e e n m a s t e r e d ,
th e c a lc u la tio n
o f m a tr ix e le m e n ts o f any o p e r a t o r o f th e ty p e d is c u s s e d
becom es a r a t h e r r o u t i n e p r o c e d u r e .
In f a c t,
th e answ er
c a n b e w r i t t e n down a t o n c e i n t e r m s o f t h e 9 - j s y m b o ls
and a l l
t h e work: com es i n e v a l u a t i n g t h e s e .
For t h is
r e a s o n , o n e i s much l e s s l i k e l y t o m ake e r r o r s u s i n g t h i s
m e th o d t h a n when u s i n g ,
w h e re a r t i f i c i a l
fo r in sta n c e ,
Van V T e c k 's m e th o d
c o n c e p t s s u c h a s r e v e r s e d a n g u l a r m om enta
a r e r e q u i r e d and p h y s i c a l r e a s o n in g i s n e c e s s a r y a t
s e v e r a l p o i n ts d u rin g th e c a l c u l a t i o n .
The r o u t i n e n e s s
o f t h i s m e t h o d s u g g e s t s t h e p o s s i b i l i t y o f p r o g ra m m in g i t
f o r c o m p u t e r c a l c u l a t i o n w h i c h w o u ld b e q u i t e d i f f i c u l t f o r
Van V l e c k ' s m e t h o d .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T here i s a ls o th e p o i n t t h a t a l l m a tr ix e le m e n ts
are ,
fo rm a lly a t l e a s t ,
la tio n .
hand,
on a n e q u a l b a s i s i n t h i s
fo rm u ­
I n Van V l e c k ' s a n d L i n ' s w o rk , o n t h e o t h e r
u s e o f t h e m e th o d o f p r o j e c t i o n m ax e s t h e o f f -
d ia g o n a l e le m e n ts p a r t i c u l a r l y d i f f i c u l t to c a l c u l a t e .
A B r i e f Summary o f t h e N o t a t i o n
(a)
The W ig n e r 3 - j s y m b o l i s
a sy m m etrize d C ie b sc h -G o rd a n
c o e ffic ie n t:
(b )
An i r r e d u c i b l e t e n s o r o f r a n x x i s
q u a n ti t ie s T (x ,a ) s a t i s f y i n g
a set of
(2 x 4 -1 )
th e tra n s fo rm a tio n
p ro p e rtie s:
w h e r e R i s a r o t a t i o n o p e r a t o r a n d t B ^ I(^ ( R ) i s
th e
q *q e l e m e n t o f t h e c o r r e s p o n d i n g x - d i m e n s i o n a l i r ­
r e d u c i b l e r e p r e s e n t a t i o n o f t h e p r o p e r t h r e e d im e n
s io n a l r o ta tio n gro u p .
with permission of the copyright owner. Further reproduction prohibited without permission.
- 26(c)
Reduced m a t r i x e le m e n ts a r e a s t a te m e n t o f th e
W ig n e r-E c k a rt th eo rem
(2 5 ,2 6 ).
T hey a r e d e f i n e d
by:
(11-27)
<t' j ' m ' l T U v ^ U j m > =
*
V-m
(d)
The W ig n e r 6 - j
sy m b o l i s
c^.
<x '*
>
M
m I.
r e l a te d to th e re c o u p lin g
o f t h r e e a n g u l a r m o m e n ta :
(11.28)
U
^
U *
f ( 2 > 12* 0 ( 2 j 2S+O ] ' *
J
^
fa v is v J
^ i v > i*-) jrv2 i
I t is
i, , (
.
e s s e n t i a l l y t h e sam e a s t h e R a c a h c o e f f i c i e n t W:
(11 -2 9)
^
\&
J
fa
\ -
C - \)
i zV ^ v3
J
iv fa f a \ f a
(e)
J ^
The W ig n e r 9 - j sy m b o l i s
•
re la te d
to th e r e c o u p lin g
o f f o u r a n g u l a r m o m e n ta :
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
-2 7 -
(11-3 0 )
I
<h
i2
<j iz
is
i4
*34
\
r
[(2
-
i 3 4 v 1^
7
( t ju & O
^34 »^\ ( £ l , j O
i2-4 3 > .
Reduced M a tr ix E le m e n ts f o r th e E l e c t r i c
D i p o l e Moment
T a b u l a t e d b e lo w a r e e x p r e s s i o n s f o r a l l t h e n o n - z e r o
re d u c e d m a tr ix e le m e n ts o f th e e l e c t r i c
d i p o l e moment f o r
J sc h em e c o u p l i n g .
The c o m p l e t e e l e m e n t s c a n be o b t a i n e d
fro m th em b y u s e o f
(1 1 -2 7 ).
The E schem e r e d u c e d
e l e m e n t s c a n a l s o b e o b t a i n e d fro m t h e t a b l e s b y r e p l a c i n g
J ev ery w h ere by E and in te r c h a n g in g th e r o l e s o f I and S.
The s i m p l e s t way t o a r r i v e a t t h e G schem e e l e m e n t s ,
th e o th e r hand,
is
on
to s t a r t a t t h e b e g in n in g and c a l c u l a t e
them w i t h o u t u s i n g t h e t a b l e s .
The G sc h em e e l e m e n t s w i l l
be p a r t i c u l a r l y
sim p le to c a l c u l a t e b e c a u s e
used in s te a d o f
( I I -4 ).
( I I - 3 ) can be
The Y a p p e a r i n g i n m a t r i x e l e m e n t s h a s p r e v i o u s l y
r e p re s e n te d th e e ig e n v a lu e s o f an u n s p e c ifie d s e t o f
o p e r a to r s r e q u ir e d i n a d d itio n to th e
j 's
a n d m 's t o f o r a
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-2 8 a c o m p lete s e t .
I t now t a k e s o n t h e s i g n i f i c a n c e o f t ,
t h e a s y m m e t r i c t o p q u a n tu m n u m b e r .
To make o u r e q u a t i o n
( I I -4 ) c o rre sp o n d to th e J
s c h e m e , we make t h e f o l l o w i n g i d e n t i f i c a t i o n s f o r t h e
a n g u l a r m o m e n ta :
(II-3 1 )
^
; J -?
.
and f o r th e o p e r a to r s :
T ^ ) = JZ
(11-32)
UCb 2 ) = v ( t e 3 ) * \
feV2= K = \
E q u a tio n
( I I
-4 ),
th e n ,
(H -3 3 )
o
.
read s:
i.F W jU C tN * ,s )J ' , I , F ' >
=
3
<
[ ( 2 F * \ ) ( 2 F V 0 ( 2 1 + 0 (.2 1 ’ + \ ) ] ^
Y
The tw o 9 - J s y m b o l s ,
red u ce to a 6 - j
N
W
< S\\ V WS > < I \ I U I > .
sin c e th e y c o n ta in z e ro s,
sy m b o l.
w ill each
When t h e s e a r e p u t i n a l o n g w i t h
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
t h e v a l u e s o f t h e r e d u c e d m a t r i x e l e m e n t s o f 1 , we g e t :
(II-3 4 )
<
\\ t t ' N 1j S ) J ' j I j F 1')
=
[ ( Z F M ) ( 2P ‘v O
( 2 J > 0 U l 'v
\ \> U X rN '>
3"
j
n
r r
F
F'
Xf
I
n
j
N‘ s l
E x a m i n a t i o n o f t h e 6 - j s y m b o l s w i l l show t h a t o n l y t h e
m a t r i x e l e m e n t s t h a t do n o t v a n i s h w i l l be t h o s e f o r w h i c h
M' = N , M-t i
(1 1-3 5)
J' = T , T t \
F' = F , F t i
T hus, i n g e n e r a l ,
a n y v a l u e o f N , J , F may b e c o n ­
n e c t e d t o 27 o t h e r s so t h a t t h e r e s h o u l d b e 27 d i f f e r e n t
p o s s ib le e x p re s s io n s f o r th e red u ced m a trix e le m e n ts.
E ach o f t h e s e e le m e n ts ,
w h ich h a s a l l
how ever, h a s a u n iq u e c o u n t e r p a r t
th e changes i n N ,J ,F e x a c tly re v e r s e d .
T h e s e w i l l h a v e t h e sam e f u n c t i o n a l f o r m b u t d i f f e r e n t
a rg u e m en ts.
F o r e x a m p le ,
t h e m a t r i x e l e m e n t f o r N, J , F - *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
!
-3 0 N + 1 ,J ,F -1
c a n b e o b t a i n e d fro m t h a t f o r N, J , F - s - N - l, J ,
F -t-1 , b y r e p l a c i n g N w i t h N+l a n d F w i t h F - l i n t h e f i r s t
e x p re ssio n .
T h is m eans t h a t o n l y 14 e x p r e s s io n s hav e to
be t a b u l a t e d .
The e x p r e s s i o n s f o r t h e 6 - j
s y m b o ls u s e d i n t h i s
a n d t h e n e x t A p p e n d ix w e r e t a k e n fro m T a b l e 5 o f Edmonds 1
b o o k a f t e r b e i n g c h e c k e d a g a i n s t t h e R acah c o e f f i c i e n t s
t a b u l a t e d by B ie d e n h a rn , B l a t t ,
q u a n titie s
j^NTJ ltyU\N 1* 1> [ '
a n d R ose ( 2 7 ) .
The
a r e t h e sam e a s t h e l i n e s
s t r e n g t h s g i v e n i n t h e t a b l e s o f Schw endem an a n d L a u r i e
(2 8 ) e x c e p t f o r a f a c t o r e q u a l to th e sq u a re o f th e
comDonent of the diDole along the molecular a x i s involved
6
i n th e t r a n s i t i o n and a f a c t o r o f 10 .
In o rd e r to p re s e rv e sp a c e ,
we s u p p r e s s t h e s y m b o ls
S and I i n th e n o t a t i o n f o r th e s t a t e s
change.
s i n c e t h e y do n o t
We a l s o i n t r o d u c e t h e a b b r e v i a t i o n s a = I + J + F ,
b = S+J+N.
J _______
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-3 1 -
T
n
rW
4:-n
rt
+•
.0
n
-t-
rii
y—z
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-3 2
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-3 4 M a t r i x E l e m e n t s f o r t h e Zeeman E f f e c t
We g i v e h e r e t h e J schem e m a t r i x e l e m e n t s f o r t h e
Zeeman e f f e c t i n t h e c a s e w h e re t h e d i r e c t i o n o f t h e f i e l d
may b e t a k e n a s t h e a x i s o f q u a n t i z a t i o n .
The c a s e o f
s i m u l t a n e o u s S t a r k a n d Zeeman e f f e c t s w i l l n o t be c o n ­
s id e re d .
The G sc h em e e l e m e n t s c a n b e o b t a i n e d fro m t h e
t a b l e s by r e p l a c i n g J w ith G and in te r c h a n g i n g th e r o l e s
of N and I .
I n t h e Zeeman e f f e c t t h e E schem e e l e m e n t s
a re p a r t i c u l a r l y sim p le
u sin g
( I I - 3 ).
C h o o s i n g t h e Z a x i s t o be t h e d i r e c t i o n o f t h e f i e l d ,
we g e t t h e f o l l o w i n g h a m i l t o n i a n :
(11- 3 6 )
H2 =
S *
A p p lic a tio n o f e q u a tio n s
( I I - 4 ) and ( I I - 5 ) g iv e s f o r th e
J schem e:
( I I -37)
((W ,S )T ,I,F
=
k
1S«
3 [ ( 2 3 > l ) ( 2 T '- H ) ( 2F * 0 CZF't- 0 ] 2
F '
(-1
F
I j
U '
(IMIU
J*
1 J
<SUS1\S> < H U \ \ I > .
i
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-3 5 We n o t e t h a t t h i s
(1 1 -3 8 )
re q u ire s th a t:
N = M } NA'= M ) T = r • 1 = 1,1*1 ;
f
'- F, F t I .
T h u s t h e r e w i l l b e n i n e e l e m e n t s c o n n e c t i n g a n y J , N, P
in th is
c a s e , o n l y f i v e o f w h ic h m u st be t a b u l a t e d .
sim p lic ity ,
we s u p p r e s s a l l t h e n o t a t i o n f o r s t a t e s e x c e p t
J a n d P s i n c e t h e o t h e r s do n o t c h a n g e .
S u b s t i t u t i o n f o r t h e 9 - j s y m b o ls a n d t h e r e d u c e d
m a tr ix e le m e n ts y i e l d s :
We t a b u l a t e
For
J
J
F’
F
I
b e lo w t h e q u a n t i t i e s
\
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E le m e n ts
Matrix
J Scheme
II:
Table
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
-3 7 PART I I I
EXPERIMENTAL RESULTS
G e n era l T e c h n iq u e s and P ro c e d u re s
The c h l o r i n e d i o x i d e s a m p l e s u s e d i n t h i s
s t u d y w e re
p r e p a re d by h e a t i n g a m ix tu r e o f p o ta s siu m c h l o r a t e and
o x a l i c a c i d t o 8 0 °C , a t w h ic h p o i n t t h e y r e a c t a c c o r d i n g
to :
(III-1 )
2 KCIO3 + 2 H2 C2 C>4 -> 2C102 + K2 C2 04 + 2H2 0 +• 2C02 .
The e v o l v e d g a s was d r i e d b y p a s s a g e o v e r a n h y d r o u s
calciu m c h lo r i d e and p h o sp o ro u s p e n to x id e ,
a t liq u id n itro g e n te m p e ra tu re .
th e n tra p p e d
The t r a p was t h e n a l l o w e d
t o warm t o t h e t e m p e r a t u r e o f a d r y i c e - a c e t o n e b a t h an d
t h e i m p u r i t i e s C02 , 0 2 , a n d C l 2 pumped o f f ,
p u r e r e d - o r a n g e c r y s t a l s o f C102 .
u n d e r vacuum i n t o
le a v in g th e
The C102 was d i s t i l l e d
a s a m p le c o n t a i n e r a n d s t o r e d u n d e r i t s
own v a p o r p r e s s u r e a t d r y i c e t e m p e r a t u r e u n t i l u s e d .
The s a m p l e s e x h i b i t a s u r p r i s i n g
c o n d itio n s.
s t a b i l i t y under th e se
The m e l t i n g p o i n t o f a s a m p le w h ic h h a d b e e n
s t o r e d i n t h i s f a s h i o n f o r a w e ek w as m e a s u r e d a t - 6 0 ° ^ 1 C
a s c o m p a re d t o
-5 9 ° C r e p o r t e d i n t h e l i t e r a t u r e
The s p e c t r o m e t e r i s
(2 9 ).
o f th e c o n v e n tio n a l S ta rk
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-3 8 m o d u la tio n t y p e .
A d e ta ile d d e s c r ip tio n o f th e i n ­
stru m e n t a p p e a rs i n B a ir d 's t h e s i s
s o u r c e s w e re a v a i l a b l e
(5 ).
As no o s c i l l a t o r
in t h i s la b o r a to r y f o r th e f r e ­
quency re g io n s o f i n t e r e s t in t h i s
to use fre q u e n c y m u l t i p l i c a t i o n .
stu d y ,
it
was n e c e s s a r y
T h i s w as a c c o m p l i s h e d
b y d o u b l i n g t h e f r e q u e n c y f ro m a 2 K -3 3 o r QK-289 k l y s t r o n
u s in g a c o m m erc ial
(D e M o rn a y -B o n ard i) c r y s t a l h a rm o n ic s
g e n e ra to r.
The Zeem an e f f e c t s w e re s t u d i e d w i t h s t a t i c
g e n e r a t e d b y a s o l e n o i d w rapped a ro u n d t h e
fie ld s
s a m p le c e l l .
The c o n s t r u c t i o n o f t h e c o i l was s u c h t h a t a c u r r e n t o f
o n e a m p e re c o r r e s p o n d e d t o a n a x i a l f i e l d
9 .0 g au ss.
o f a p p ro x im a te ly
Lead s t o r a g e b a t t e r i e s p r o v i d e d t h e c u r r e n t
s o u r c e w h ich c o u ld be v a r i e d to g i v e a n y v a l u e o f t h e
fie ld
up t o a b o u t 1 0 0 g a u s s .
A l l t h e m e a s u r e m e n t s w e re m ade a t p r e s s u r e s b e tw e e n
5 and 40 m ic ro n s o f m e rc u ry and a t d r y i c e t e m p e r a t u r e .
R e s u lts o f P re v io u s I n v e s t i g a t i o n s
P r e v i o u s t o o u r i n v e s t i g a t i o n s o n t h e m ic r o w a v e
s p e c t r u m o f c h l o r i n e d i o x i d e a r e t h o s e o f D. P . S m i t h a n d
J a m e s B a i r d o n t h e sam e p r o b l e m .
S in c e o n ly th e g e n e r a l
f e a t u r e s o f t h e i r r e s u l t s w ill be o f i n t e r e s t i n o u r
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-3 9 d isc u ssio n ,
t h e f r e q u e n c i e s o f t h e i r l i n e s w i l l n o t be
g iv e n h e re .
A lis t
in v e s tig a to rs is
o f th e l i n e s m easured by b o th th e s e
a v a i l a b l e i n th e t h e s i s o f B a ird
On o b s e r v i n g t h e i r r e s u l t s ,
(5 ).
one n o te s t h a t th e m ost
s t r i k i n g f e a t u r e s o f th e b u lk o f th e c h lo r in e d io x id e
s p e c tr u m a r e t h e a m a z in g r e g u l a r i t y o f i t s
a p p earan ce
a n d t h e e n o rm o u s n u m b e r o f l i n e s p r e s e n t .
P ra c tic a lly
a n y w h e r e i n t h e a c c e s s i b l e m ic r o w a v e r e g i o n a n a l m o s t
m o n o to n o u s s e q u e n c e o f q u a r t e t s i s
is
seen.
Now, w h i l e t h e r e
no r e a s o n t h a t q u a r t e t s p e r s e s h o u l d b e c o n s i d e r e d
u n d e sira b le ,
th e a b sen c e o f In fo rm a tio n ab o u t th e s iz e s
o f th e v a r io u s i n t e r a c t i o n p a ra m e te r s and th e abundance
o f p r e d i c t e d a s w e l l a s o b s e r v e d l i n e s m akes i t
im p o s sib le
t o p a i r one q u a r t e t w ith th e o t h e r one b e lo n g in g to th e
sam e r o t a t i o n a l t r a n s i t i o n .
T hus,
th e l o c a t i o n s o f th e
q u a r t e t s g i v e no c l u e t o t h e i r a s s i g n m e n t b e c a u s e o n e h a s
no i d e a how f a r t h e y m ay b e d i s p l a c e d fro m t h e p r e d i c t e d
ro ta tio n a l tra n s itio n .
The f a c t t h a t so m any q u a r t e t s a r e
se e n i n d i c a t e s t h a t f o r m ost l e v e l s ,
t h e J schem e a p ­
p ro x im a tio n i s good.
im p lie s t h a t th e
s p in -ro ta tio n
T h is ,
sp littin g
m ent i s p ro b a b ly g r e a t .
in tu rn ,
i s l a r g e and hen ce th e d i s p l a c e ­
A d d itio n a lly ,
th e re i s a lre a d y
a f a i r l y la r g e u n c e r ta in ty in th e lo c a tio n o f th e p re ­
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-4 0 d ic te d r o t a t i o n a l t r a n s i t i o n s becau se o f th e u n r e l i a b i l i t y
o f th e s t r u c t u r a l d a ta .
The S t a r k e f f e c t i s
lin e s .
l i k e w i s e o f no h e l p f o r t h e s e
F o r n o n e o f t h e l i n e s o b s e r v e d b y e i t h e r S m ith
o r B a i r d was r e s o l u t i o n o f t h i s e f f e c t r e p o r t e d .
T h is
te n d s to c o n firm th e s u s p ic io n a lr e a d y p r e s e n t t h a t th e
l i n e s t h e y o b s e r v e d come m a i n l y fro m h i g h N t r a n s i t i o n s .
A n o th e r p e r p l e x i n g a s p e c t i s
t h e e x i s t e n c e o f two
n a tu r a lly abundant iso to p e s o f c h lo rin e
( 0 1 ^ 5 7 5 . 4# a n d
C i37 2 4 . 6 ^ ) h a v i n g t h e same s p i n a n d c o m p a r a b l e m a g n e t i c
a n d q u a d r u p o l e m o m e n ts .
O rd in a rily ,
g e t t i n g d a t a f r o m tw o i s o t o p i c
th e p o s s i b i l i t y o f
s p e c i e s w o u ld b e a w elcom e
c irc u m s ta n c e b e c a u s e o f th e a d d it i o n a l in f o r m a tio n g a in e d
fro m i t .
In o u r c a se ,
how ever,
th a n o u tw e ig h e d by th e i n a b i l i t y
l i n e s b e lo n g in g to
t h i s a d v a n t a g e i s m ore
t o d i s t i n g u i s h b e tw ee n
t h e tw o i s o t o p e s .
Some r e l a t i v e l y u s e f u l r e s u l t s a r e c o n t a i n e d i n t h e
w ork o f B e n n e t t ,
In g ram , an d S c h o n lan d (3 0 ) on t h e p a r a ­
m a g n e tic r e s o n a n c e s p e c t r a o f l i q u i d and s o l i d
of c h lo rin e d io x id e .
For th e d ilu te liq u id
s o lu tio n s
s o lu tio n s
t h e y o b s e r v e d a s p e c t r u m c o n s i s t i n g o f f o u r l i n e s w hose
m u t u a l s e p a r a t i o n w as i n d e p e n d e n t o f t h e f r e q u e n c y a n d
e q u a l t o 17 g a u s s .
W eissm a n ( 3 1 ) h a s show n t h a t i n
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
-4 1 liq u id
so lu tio n s ,
a l l th e i n t e r a c t i o n term s e x c e p t th e
F e r m i c o u p l i n g a r e a v e r a g e d o u t t o z e r o b y t h e random
tu m b lin g o f th e m o le c u le s i n th e s o l u t i o n .
th e o b s e rv e d sp e ctru m i n th e l i g h t o f t h i s
In te rp re tin g
re s u lt,
we
o b t a i n a v a l u e o f - 4 8 Me f o r t h e F e r m i c o n s t a n t ,
- ( l 6 V 3 ) s n^ e
•
In th e s o lid
s o lu tio n s t h is a v e ra g in g does n o t o c cu r
a n d t e r m s fro m t h e d i p o l e - d i p o l e i n t e r a c t i o n a n d t h e
s p i n - o r b i t i n t e r a c t i o n o f th e odd e l e c t r o n a ls o
c o n trib u te .
In t h e i r p ap er,
re s o lu tio n
o f th e s o l i d
of i t
B e n n e tt,
e t a l,
s o l u t i o n sp ectru m and g iv e an i n t e r p r e t a t i o n
i n te rm s o f a g e n e r a l f i n e
t h e fo rm H ' = Aa I a Sa
p re ta tio n is
c la im a p a r t i a l
s tr u c tu r e h a m llto n ia n o f
y I y S y +A z I z S z_ .
If
th e ir in te r-
a c c e p te d and th e r e a s o n a b le a ssu m p tio n o f a
n e g l i g i b l y s m a l l s p i n - o r b i t i n t e r a c t i o n i s m ad e , v a l u e s
can be c a l c u l a t e d f o r th e d i p o l e - d i p o l e i n t e r a c t i o n p a r a ­
m e te rs .
F o r a num ber o f r e a s o n s , h o w ev er,
th e s e v a lu e s c o u ld
be b e li e v e d o n ly w ith s tr o n g r e s e r v a t i o n s .
p la c e ,
a n a c c u r a c y o f o n l y 10# t o 30# i s
c o n s ta n ts th e y g iv e .
In th e f i r s t
c la im e d f o r th e
A l o o k a t t h e a c t u a l s p e c t r u m fro m
w h i c h t h e y w e re p r e s u m a b l y c a l c u l a t e d show s t h a t t h e r e ­
s o l u t i o n w as so d u b i o u s t h a t e v e n t h e s e l a r g e l i m i t s o f
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-4 2 e r r o r a re p ro b a b ly n o t s u f f i c i e n t .
A lso ,
th e a u th o r i s
c o n v in c e d t h a t t h e i r t h e o r e t i c a l tr e a tm e n t o f th e d a ta
is
not c o rre c t.
T h e ir n o ta tio n i s ,
how ever,
s u ffic ie n tly
o b s c u r e t o him t h a t t h i s m ay n o t b e t h e c a s e .
No
t h e o r e t i c a l r e e v a lu a tio n o f th e e x p e rim e n ta l c o n s ta n ts
was a t t e m p t e d b e c a u s e o f t h e p o o r n e s s o f t h e d a t a .
The 43 t o 45 K i l o m e g a c y c l e R e g io n
I t i s a p p a r e n t by t h i s
tim e t h a t an y i n s i g h t i n t o
t h e c h l o r i n e d i o x i d e p r o b l e m w i l l h a v e t o come f r o m some
p o r t i o n o f t h e s p e c t r u m w h e re t h e g e n e r a l a p p e a r a n c e d o e s
not p re v a il.
O ur e f f o r t s h a v e b e e n c o n c e n t r a t e d o n s u c h a
r e g i o n l o c a t e d b e t w e e n 43 a n d 45 k i l o m e g a c y c l e s .
in th is
In te re st
re g io n began w ith th e su g g e stio n o f B a ird t h a t
t h e r e m ay h a v e b e e n some r e s o l v a b l e S t a r tc e f f e c t s n e a r
44.6KMC.
When a n e x a m i n a t i o n o f h i s r e c o r d e r t r a c i n g s
le a d to th e d is c o v e ry o f a c l e a r l y re s o lv e d S ta r k lo b e i n
o n e o f th e m , e f f o r t was b e g u n t o l o o k i n t o
th e hope o f r e l o c a t i n g th e l i n e
th is
re g io n in
t o w h ic h t h i s l o b e b e l o n g e d .
New r e c o r d e r t r a c i n g s i n t h e n e i g h b o r h o o d o f B a i r d ' s
lin e
r e v e a l e d tw o l i n e s h e r e w i t h r e s o l v e d S t a r k e f f e c t s
r a t h e r th a n one a s e x p e c te d .
As f a c i l i t y i n t h e f r e q u e n c y
d o u b l i n g t e c h n i q u e d e v e l o p e d , m o re p o w e r a n d a w i d e r r a n g e
v
o f f r e q u e n c i e s b e ca m e a v a i l a b l e ,
a n d a w h o le m u l t i t u d e o f
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-4 3 l n t e r e s t i n g l i n e s w e r e f o u n d , I n c l u d i n g m any w i t h
r e s o l v e d S ta r k : e f f e c t s a n d some w hose S t a r k l o b e s w e re
re tro g ra d e
( t o lo w f r e q u e n c y r a t h e r t h a n t o h i g h a s
n o rm a lly ).
T ab le I I I i s a l i s t o f a l l t h e l i n e s m e a su re d i n
th is
re g io n .
stro n g ,
( w ),
A p p r o x i m a t e i n t e n s i t i e s — w ea k , m edium ,
v ery s tr o n g ,
(m ),
(s),
(v s),
e t c — a r e i n d i c a t e d b y t h e s y m b o ls
e tc .
N a rro w ly s p a c e d ,
in c o m p le te ly
r e s o l v e d m u l t i p l e t s a r e d e n o t e d b y "Q,!l a p p e a r i n g a f t e r
th e fre q u e n c y o f th e c e n te r o f th e m u l t i p l e t .
w e re a l s o o b s e r v e d b y B a i r d a r e u n d e r l i n e d .
q u e n c ie s g iv e n i n t h i s
The f r e ­
t a b l e r e p r e s e n t th e b e s t v a lu e s
o b t a i n e d fro m s e v e r a l r e p e a t e d m e a s u r e m e n t s .
b e l i e v e d t o be a c c u r a t e t o w i t h i n
The i n t e r e s t i n g
L in e s t h a t
T h ey a r e
O .IM c.
l i n e s a r e o f two t y p e s .
The l e t t e r s
o f th e a lp h a b e t a p p e a rin g i n th e ta b le b e fo re l i n e s d e n u m e ra te th o s e w ith r e t r o g r a d e S t a r k e f f e c t s .
w ith re s o lv e d S ta rk
o rd er.
in to
L in e s
( o r Z eem an) e f f e c t s a r e n u m b e r e d i n
I t w ill be n o tic e d th a t th e se l a t t e r l in e s f a l l
two w e l l s e p a r a t e d p a r t s ,
th e lo w e r g ro u p te r m in a ­
t i n g w i t h t h e l i n e a t 4 3 ,4 4 4 .1 M c a n d t h e u p p e r g r o u p
b e g i n n i n g w i t h t h e l i n e a t 4 4 , 2 4 2 . lM c .
T hese l i n e s a r e
show n i n F i g u r e s 1 a n d 2 .
i
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-4 4 T ab le I I I :
A b s o rp tio n L in e s o f C h lo r in e D io x id e
I n t h e 4 3 j 0 0 0 t o 4 5 , 0 0 0 M e g a c y c le R e g io n
4 2 ,4 5 3 .3
4 3 ,2 7 6 .7 0
7 9 2 .9 2
(1 2 )
2 8 5 .5
(s )
8 4 6 .1 2
(1 3 )
2 9 4 .8
(m )
8 5 1 .7 6
(a)
3 2 7 .3
8 5 5 .7 0
(b )
3 3 0 .3
8 5 7 .8 0
(14)
3 3 8 .1 6
(m)
9 3 6 .3
(1 5 )
3 4 8 .6 0
(v s)
937-5
(c)
3 5 5 .0 4
9 3 3 .2
(3 )
3 5 9 .8 0
(1 )
4 2 ,9 4 8 .5 0
(vw)
(2 )
^ 3 ,0 5 4 .2
(vw)
3 7 0 .4 6
(w)
(1 6 )
3 9 8 .0 8
(m)
(3 )
0 6 0 .2 4
(w)
(1 7 )
4 1 3 .6 0
(s)
(4 )
1 0 4 .1 8
(w)
(1 3 )
4 1 6 .1 0
(mw)
(3 )
1 1 2 .7 8
(w)
(1 9 )
4 2 3 .6 0
(mw)
(6 )
1 2 4 .9 5
(mw)
(2 0 )
4 2 8 .7 2
(mw)
(7 )
1 4 0 .3 0
(m s)
(2 1 )
4 3 5 .6 8
(mw)
(8 )
2 0 1 .5 4
(w)
(2 2 )
4 4 4 .1 0
(mw)
(9 )
2 1 5 . 8 4 (m)
(1 0 )
2 2 7 .2 6
(w)
(e)
6 4 8 .7
(1 1 )
2 7 4 .5 6
(w)
(f)
7 0 7 .8
4 4 7 .0 6
2 7 5 • 80
1
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-4 5 T a b le I I I C o n tin u e d
(23)
4 4 ,2 4 2 .0 8
(m s)
(J)
4 4 ,5 8 2 .5 8
(24)
3 1 0 . 3 0 (vw )
6 1 2 .0 6
(25)
3 7 6 .0 0 ( v s )
6 1 4 .8 0
(26 )
4 2 9 .7 4
6 1 5 .8 8
(g )
5 0 6 .4 0
6 1 9 .5 0
(27)
5 1 5 .2 8 ( s )
6 2 2 .6 8
(h )
5 2 6 .1 0
(30)
6 2 7 .2 0 ( s )
(28)
5 4 5 . 8 0 (w)
M
6 5 5 .8 4
(i)
5 5 5 .2 9
(29)
5 6 4 .1 0 ( s )
(s)
6 6 1 .8 Q
(31)
7 0 3 .0 0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(m)
I
-4 6 -
o
FIGURE
I:
LOWER
GROUP
OF
CHLORINE
DIOXIDE
L IN E S
SL
m
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-4 8 The r e s o l u t i o n o f t h e S t a r k e f f e c t o c c u r s i n a m o s t
b i z a r r e m anner in t h i s
re g io n .
The t h e o r y t e l l s
us t h a t
t h e e n e r g y , w h ic h i s i n d e p e n d e n t o f t h e q u a n tu m n u m b er
M when no f i e l d s a r e p r e s e n t ,
s p lits in to a s lig h tly
2
d i f f e r e n t v a l u e f o r e a c h v a l u e o f M when t h e r e i s a n
e le c tric
fie ld .
The s p l i t t i n g
second o r d e r p e r t u r b a t i o n .
c a n b e c a l c u l a t e d by
T h is l e a d s to an e x p r e s s io n
f o r th e f r e q u e n c y d i f f e r e n c e from th e z e r o f i e l d c a s e o f
2 2
t h e forro (A+-BM )E w h e re E i s t h e i n t e n s i t y o f t h e e l e c t r i c
fie ld .
S in c e th e c o n s t r u c t i o n o f th e a p p a r a tu s i s
t h a t th e e l e c t r i c
fie ld
i s p la c e d p a r a l l e l to
v e c t o r o f t h e m ic r o w a v e r a d i a t i o n ,
M i s AM = 0 .
T hus e a c h l i n e
is
such
th e e l e c t r i c
t h e s e l e c t i o n r u l e on
r e q u i r e d t o h a v e a n u m b er
o f S t a r k l o b e s e q u a l t o t h e l a r g e r o f t h e two v a l u e s
o f P i n v o l v e d i n t h e t r a n s i t i o n when P i s
th is
case
( f o r F h a l f - i n t e g r a l th e num ber i s
The r e s o l v e d S t a r k e f f e c t l i n e s
h o w ev er,
In te g ra l as in
in th is
seem t o b e i n c o n t r a d i c t i o n t o
p re d ic tio n .
For a l l
e x c e p t p e r h a p s tw o ,
F -t^ ).
re g io n ,
th e t h e o r e t i c a l
such l i n e s o f th e e n t i r e
re g io n
t h e r e i s o n ly one S t a r k l o b e ,
even
when t h e S t a r k e f f e c t h a s b e e n r e s o l v e d aw ay f ro m t h e
r e g u l a r l i n e b y m any m e g a c y c l e s .
o n l y re m a in s i n g l e ,
The S t a r k l o b e s n o t
b u t u s u a lly r e t a in t h e i r sharpness
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-4 9 e v e n up t o q u i t e l a r g e v a l u e s o f t h e f i e l d a s i s
i n F i g u r e 3 f o r two t y p i c a l l i n e s .
see,
show n
As we s h a l l p r e s e n t l y
t h e Zeeman e f f e c t s o f t h e l i n e s h e r e make i t
t h a t n o t a l l o f th e t r a n s i t i o n s
v a lu e o f F = l ,
c e rta in
se e n can have th e l a r g e r
so we a r e f a c e d w i t h t h e p r o b l e m o f m a k in g
t h e e x p e r i m e n t a l o b s e r v a t i o n b e c o n s i s t e n t w ith th e
th eo ry .
T h e r e a p p e a r s t o b e o n l y o n e p o s s i b l e way o u t o f t h i s
d i le m m a .
ta in in g
The p a r t o f t h e S t a r k e f f e c t e x p r e s s i o n c o n t h e Mc d e p e n d e n c e m u s t b e z e r o o r n e a r l y so f o r
a l l th e se lin e s .
T h i s e x p r e s s i o n a r i s e s fro m a l a r g e
num ber o f se co n d o r d e r p e r t u r b a t i o n te r m s ,
w h ic h c a n n o t b e c a l c u l a t e d b e f o r e
th e v a lu e s o f
th e sp e ctru m i s
an aly ze d
b ecau se th e y c o n ta in en erg y d i f f e r e n c e s as d e n o m in a to rs.
F o r e a c h i n d i v i d u a l te r m i t
2
o f th e c o e f f i c i e n t o f M i s
m a g n itu d e o f th e
can be s e e n t h a t th e m ag n itu d e
n o t g r e a t l y s m a lle r th a n th e
" c o n s ta n t" te rm .
The s i g n s ,
how ever,
a r e n o t a l w a y s t h e same a n d w h a t m u s t h a p p e n i s
"B" t e r m s c a n c e l o u t w h i l e t h e
"A" t e r m s a d d .
th a t th e
It
se e m s e x t r e m e l y c o i n c i d e n t a l t h a t t h i s w o u ld o c c u r f o r
a l l th e l in e s in t h i s
r e g i o n a n d t h e r e m ay b e some f u n d a ­
m e n ta l re a so n t h a t t h i s a v e ra g in g has to ta x e p la c e .
th is
is
so,
If
i t w o u ld u n d o u b t e d l y l e a d t o some v e r y i n t e r -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CM.
MMipni
Pin#*/'
£•1360
V/ CM
I
i
E - 2270
V/ CM
| '
m
FIGURE 3:
RESOLUTION
OF
TYPICAL
TWO
OF THE
CHLORINE
STARK
EFFECT
DIOXIDE LINES
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-5 0 e s t in g in f o r m a tio n a b o u t th e m o le c u le .
e ffe ct is ,
h ow ever,
The c a u s e o f t h i s
n o t a p p a re n t and w i l l p ro b a b ly have
t o be u n d e r s t o o d o n l y a f t e r t h e r e s t o f t h e s p e c t r u m .
The two p o s s i b l e e x c e p t i o n s t o t h i s
p a tte rn of
i r r e g u l a r S ta r k e f f e c t s a re th e l i n e s a t 4 3 ,3 4 8 .6 and
4 4 , 2 4 2 . 1 Me.
The f i r s t o f t h e s e a p p e a r s t o h a v e t h r e e
re so lv a b le S ta rk lo b e s,
a lth o u g h th e r e i s
some e v i d e n c e
t h a t t h i s a p p a r e n t r e s o l u t i o n may b e d u e t o t h e a p p e a r ­
a n c e o f w eak l i n e s a s t h e S t a r k v o l t a g e i s
in c re a se d .
The l o b e s a t 4 4 , 2 4 2 . 1 a r e n o t a t a l l w e l l r e s o l v e d b u t
th e shape o f th e
S ta r k lo b e d e f i n i t e l y
t h a n one c o m p o n e n t.
s u g g e s t s m o re
Two o r t h r e e s e e m s t o b e t h e b e s t
g u e s s f o r th e num ber o f lo b e s h e r e .
S in c e th e p e c u l i a r i t i e s o f th e S t a r * e f f e c t o f
c h lo rin e d io x id e
cause a la rg e p a r t o f th e in fo rm a tio n
o r d i n a r i l y s o u g h t f ro m t h a t s o u r c e t o b e w i t h h e l d ,
s t u d y o f t h e Zeem an e f f e c t was m ade a l s o .
se v era l p o ssib le
S u c h a m oment may a r i s e
u n p a ire d e l e c t r o n s ,
n u c le a r s p in s ,
a w h o le .
is
T here a r e
s o u r c e s o f a m a g n e t i c d i p o l e m om ent
w h ic h can i n t e r a c t w i t h a m a g n e tic f i e l d
th is e ffe c t.
a
to produce
from t h e s p i n o f
from t h e i r o r b i t a l m o tio n ,
fro m
o r fro m t h e r o t a t i o n o f t h e m o l e c u l e a s
In m o le c u le s l i k e
an u n p a ire d e le c t r o n ,
c h l o r i n e d i o x i d e w here t h e r e
th is is
th e o n ly so u rc e o f any
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-5 1 l m p o r t a n c e s i n c e t h e m a g n e t i c m oment o f t h e e l e c t r o n i s
m uch l a r g e r t h a n t h o s e fro m a n y o f t h e o t h e r s o u r c e s .
U n lik e th e e l e c t r i c
d ip o le ,
t h e m a g n e t i c moment c a n
have c o m p le te ly d ia g o n a l e le m e n ts ,
f i r s t o r d e r Zeeman e f f e c t .
is
i.e .,
th e re is a
The e n e r g y o f e a c h l e v e l
s p l i t by a m a g n e tic f i e l d
i n t o 2F+1 l e v e l s w h ic h t o
f i r s t o rd e r a re e v e n ly spaced and c e n te re d about th e o r i ­
g in a l one.
The m a g n e t i c f i e l d i n t h i s e x p e r i m e n t was
p a r a l l e l t o t h e d i r e c t i o n o f p r o p a g a t i o n o f t h e m ic r o w a v e s
a n d h e n c e t h e s e l e c t i o n r u l e o n M when t h e m a g n e t i c f i e l d
o n ly i s
p r e s e n t i s AM = ± 1 .
th e l i n e s p r o p o r t io n a l to
T h is c a u s e s a s p l i t t i n g o f
( M i l ) A '- M A w h e re A ' a n d A
a re c o n sta n ts c h a r a c te r is tic
sta te s,
re s p e c tiv e ly ,
o f th e f i n a l and i n i t i a l
a n d a r e c a l c u l a b l e fro m T a b l e I I .
The t o t a l n u m b e r o f c o m p o n e n t s , p r o v i d e d A' i s d i f f e r e n t
fr o m A, i s
2 (F » 4 -F ).
For a ll
t h e n u m b e re d l i n e s ,
Zeeman s p l i t t i n g s
o c c u r r e d a t m a g n e tic f i e l d s o f 5 .0 g a u s s o r l e s s .
is
T h is
i n c o n t r a s t t o t h e o t h e r l i n e s i n t h e s p e c t r u m w h ic h
a r e e i t h e r u n a f f e c te d by su ch sm a ll f i e l d s o r a re o n ly
sm e a re d s l i g h t l y by th em .
An a m u s i n g ,
th o u g h n o t
p a r t i c u l a r l y i n f o r m a t i v e e f f e c t was n o t i c e d w i t h t h e l i n e
a t 4 - 3 ,2 8 5 .5 Mc , w h i c h s p l i t a t a n e x t r e m e l y lo w v a l u e o f
th e f i e l d .
i
The f i e l d was d e c r e a s e d a n d f i n a l l y t u r n e d
.
.
.
.
.
.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-5 2 o f f e n t i r e l y b u t t h e r e was some r e s i d u a l r e s o l u t i o n e v e n
th e n ,
a l t h o u g h a v e r y c a r e f u l e x a m i n a t i o n w as r e q u i r e d
to re v e a l i t .
P r e s u m a b l y t h i s was d u e t o a Zeeman e f f e c t
c a u se d by th e e a r t h ' s m a g n e tic f i e l d .
T h is i s
th e f i r s t
t i m e t o o u r k n o w le d g e t h a t s u c h a n e f f e c t h a s b e e n
o b s e r v e d i n m ic r o w a v e s p e c t r a .
I n m ost p l a c e s ,
th e c lo s e sp a cin g s o f th e l i n e s
p r e v e n t e d c o m p l e t e r e s o l u t i o n o f t h e i r Zeeman e f f e c t s
b e ca u se o v e rla p p in g to o k p la c e b e fo re th e f i e l d
c o u l d be
m ade l a r g e e n o u g h t o g i v e a n y c o n f i d e n c e t h a t a l l t h e
r e s o lu tio n p o s s ib le had ta k e n p la c e .
In f a c t ,
o n ly one
l i n e - - t h e one a t 4 4 , 2 4 2 . 1 - - w a s b o t h s t r o n g e n o u g h a n d
i s o l a t e d enough f o r a r e a l l y c o n v in c in g jo b t o be d o n e .
For th is lin e ,
f o u r Zeem an c o m p o n e n ts w e re r e s o l v e d a n d
f o l l o w e d up t o a f i e l d
o f a b o u t 40 g a u s s .
At t h i s
t h e s m a l l e s t s e p a r a t i o n b e tw e e n l i n e s i n th e
Zeeman
m u l t i p l e t was l a r g e e n o u g h ( a b o u t 2 Me) t h a t i t
is un­
l i k e l y t h a t t h e r e w e r e m o re r e m a i n i n g u n r e s o l v e d .
l i n e may be a s s i g n e d ,
th e n ,
as a F = 1 -* F = 1
F can have th e v a lu e 1 o n ly i f
N is
0 ,1 ,2 ,
fie ld
T h is
tra n s itio n .
o r 3.
O n ly
tw o t r a n s i t i o n s i n v o l v i n g no o t h e r v a l u e s o f N a r e p r e ­
d ic te d to l i e
an yw here n e a r t h i s r e g i o n .
2 q2 ^ 2 H 3 P r e d i c t e d a t 4 4 , 9 5 0 f o r C l ^ o
They a r e th e
a n d a t 4 3 ,7 0 0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-5 3 ryj
f o r Cl
0^ a n d t h e Oqq- *
* p r e d i c te d a t 6 0 ,0 0 0 f o r
C l^ O g and a t 5 ^ ,6 0 0 f o r C l^ O
.
I t se em s s u f f i c i e n t l y
u n l i k e l y t h a t t h e s p i n - r o t a t i o n i n t e r a c t i o n c o u l d be
l a r g e e n o u g h t o p l a c e t o O oo"^1 ! ! i n t h i s
re g io n th a t
i t may b e r e a s o n a b l y a s s u m e d t h a t t h i s l i n e i s
p a r t o f th e
2 ^ - ^ 2. - r o t a t i o n a l t r a n s i t i o n .
02
11
T h e r e a r e s e v e r a l o t h e r l i n e s f o r w h ic h t h e n u m b e r
o f Zeem an c o m p o n e n ts i s
fid e n c e .
Known w i t h a f a i r a m o u n t o f c o n ­
I n T a b l e IV , a s h o r t sum m ary o f t h e e x i s t i n g
i n f o r m a t i o n o n t h e Zeeman e f f e c t s o f a l l
th e l i n e s i s
g iv e n .
O t h e r i n f o r m a t i o n o f i m p o r t a n c e b e s i d e s t h e n u m b er
o f c o m p o n e n ts i s
sp littin g s
c o n t a i n e d i n t h e Zeeman e f f e c t s .
I f th e
o f w e ll re s o lv e d m u lt i p le t s a re m easured a t
a num ber o f f i e l d
may b e o b t a i n e d ,
stre n g th s ,
v a lu e s o f
\k '\
th o u g h n o t u n a m b ig u o u sly .
o f th e s e v a lu e s w ith th o s e c a l c u l a t e d
and
|A1
C o m p ariso n
(T a b le I I ) assu m in g
N —2 f o r p u r e c o u p l i n g sc h e m e s a l l o w s o n e t o d e t e r m i n e t h e
ty p e o f c o u p lin g a p p r o p r ia te f o r th e l e v e l s in v o lv e d i n
a n y t r a n s i t i o n f o r w h ich t h e s e c o n s t a n t s can b e o b t a i n e d .
A lso , b e c a u se o f th e i n v a r i a n c e o f th e t r a c e o f th e
Zeeman e f f e c t m a t r i x ,
a known v a l u e o f A f o r a n y l e v e l
w i l l p r e d i c t u n iq u e ly th e v a lu e o f t h a t c o n s ta n t f o r
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-5 4 T a b l e IV :
D e s c r i D t i o n o f Zeeman E f f e c t s
(1 )
S p l i t by f i e l d o f 2 .0 g a u s s .
d e t a i l e d m e a su re m e n t.
Too weax f o r m o re
(2 )
B r o a d e n e d b y v e r y lo w f i e l d .
by f i e l d o f 1 .1 g a u s s .
C o m p le te ly w ashed
(3)
I s s p l i t i n t o 6 c o m p o n e n ts a t a f i e l d o f 6 . 5 g a u s s .
B e g i n s t o w ash o u t a t 11 g a u s s .
(4)
T h r e e r e s o l v a b l e c o m p o n e n ts a t 1 . 1 g a u s s ,
g a u s s , w ashed o u t a t 4 . 0 g a u s s .
(5)
C le a r ly s p l i t a t 2 .2 g a u s s .
a t 6 .6 g a u s s .
(6)
P o u r o r s i x c o m p o n e n ts b e s t g u e s s .
(7 )
S i x c o m p o n e n ts r e s o l v e d a t 5 . 5 g a u s s .
p r o b a b l y t h e c o r r e c t n u m b e r.
(6)
B e g in s t o
(9 )
F a i r l y u n a f f e c t e d b y f i e l d s up t o 4 . 4 g a u s s .
B roaden s a t t h a t v a lu e and w ashes o u t a t h i g h e r f i e l d s .
s p lit a t 2 gauss.
S t i l l o n ly
W ashes o u t
out
f o u r a t 2 .2
two c o m p o n e n ts
T h is i s
q u ic k ly .
( 1 0 ) W ashes o u t a t 2 . 5 g a u s s .
( 1 1 ) At l e a s t f o u r Zeeman c o m p o n e n t s .
( 1 2 ) S p l i t s a t 0 . 5 5 g a u s s , s e v e n r e s o l v a b l e c o m p o n e n ts
from 0 .7 7 to 1 . 0 g a u s s .
P r o b a b ly h a s e i g h t c o m p o n e n ts.
( 1 3 ) A p p e a r s t o h a v e s i x c o m p o n e n ts a t 2 . 7 5 g a u s s .
( 1 4 ) A p p ears t o h av e a t l e a s t f o u r c o m p o n e n ts.
( 1 5 ) S p l i t s i n t o m any l o b e s .
i s te n .
B e s t g u e s s f o r n u m b e r o f them
( 1 6 ) P r o b a b l y h a s f o u r o r s i x Zeem an c o m p o n e n ts .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-5 5 T a b l e IV C o n t i n u e d
(1 7
S p l i t b y w eak f i e l d .
(18
S p l i t b y we a x f i e l d .
(1 9
S p l i t b y we a x f i e l d .
(20
S p l i t b y w eak f i e l d .
(21
S p l i t b y w eak f i e l d .
(22
S p l i t b y we a x f i e l d .
(2 3
D e f i n i t e l y h a s f o u r c o m p o n e n ts.
Zeem an s p l i t t i n g
c o e f f i c i e n t s m e a s u re d a s 0 .1 6 9 2 and 0 . 1 2 3 0 .
(2 4
Much t o o wea.< f o r Zeeman s t u d y .
(2 5
P r o b a b l y h a s s i x Zeeman c o m p o n e n ts .
(26
S i x r e s o l v a b l e c o m p o n e n ts a t 6 . 6 g a u s s , s e v e n a t
20 g a u s s .
May h a v e some i n t e r f e r e n c e f r o m o t h e r
lin e s .
S i x o r e i g h t p r o b a b l e Zeeman c o m p o n e n t s .
(2 7
A fa irly
(28
At l e a s t f o u r c o m p o n e n t s .
Too w eax a n d c ro w d e d
to a llo w a r e a s o n a b le e s t im a t e .
(2 9
B roadens a t 7 . 5 g a u s s .
S m e ars o u t c o m p l e t e l y a t
15 g a u s s .
P r o b a b l y m any c o m p o n e n ts .
(30
P ro b a b ly s i x c o m p o n e n ts .
(3 1
P r o b a b l y s i x o r e i g h t c o m p o n e n ts .
u n c e r t a i n g u e s s o f s i x c o m p o n e n ts .
Too c ro w d e d t o b e s u r e .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-5 6 t h e l e v e l d i f f e r i n g fro m i t o n l y i n t h e i n t e r m e d i a t e
q u a n tu m n u m b e r.
T h is in f o r m a tio n c o u ld p ro v e u s e f u l
i n c o n firm in g t e n t a t i v e
a ssig n m en ts o r in e lim in a tin g
a c c i d e n t a l c o m b in a tio n r u l e s i n th e sp e ctru m .
The m o s t c a r e f u l m e a s u r e m e n t s o f t h e s e c o n s t a n t s
w e re m ade o n t h e 4 - 4 ,2 4 2 .1 l i n e .
by t h e f i e l d
stre n g th .
The s p l i t t i n g
d iv id e d
s t r e n g t h was p l o t t e d a g a i n s t t h e f i e l d
T h i s g a v e f o u r s t r a i g h t l i n e s w h ic h w ere e x ­
t r a p o l a t e d to z ero f i e l d
s t r e n g t h to e lim in a te any
second o r h ig h e r o rd e r e f f e c t s .
V a lu e s o f 0 .1 6 9 2 and
0 .1 2 3 0 w e r e o b t a i n e d i n t h i s way f o r
not n e c e ssa rily re s p e c tiv e ly ).
JAM a n d
fA! ( b u t
E i t h e r c h o ic e o f s ig n
f o r 0 .1 6 9 2 i m p l i e s i n t e r m e d i a t e c o u p l i n g a s d o e s t h e
c h o ic e o f th e n e g a tiv e s i g n f o r 0 .1 2 3 0 .
p o sitiv e
C h o ic e o f t h e
s i g n f o r 0 .1 2 3 0 c o r r e s p o n d s f a i r l y
c lo s e l y to
t h e G sc h em e v a l u e .
The c o m b i n a t i o n r u l e s a r e a n a s p e c t o f t h e s p e c t r u m
t h a t re m a in s to be c o n s i d e r e d .
A co m p lete r o t a t i o n a l
t r a n s i t i o n c o n s i s t s o f 38 l i n e s f o r AN = 0 ,
In e i t h e r o f th e se c a se s,
33 f o r A N = i l .
th e re e x is ts a c h a ra c te ris tic
p a t t e r n o f r e p e a te d fre q u e n c y d i f f e r e n c e s in d e p e n d e n t
o f th e s i z e o f th e i n t e r a c t i o n p a ra m e te r s and o f th e
c o u p lin g schem e.
F i g u r e 4 show s s u c h a p a t t e r n f o r h a l f
a AN=0 t r a n s it i o n .
The o t h e r h a l f ,
w h ic h d i f f e r s o n l y
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RATTERNS
FOR
1
COMBINATION
RULES
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FIGURE
A .
4: THEORETICAL
_a
-5 8 i n t h e i n t e r m e d i a t e q u a n tu m n u m b e r f o r o n e o f t h e
ro ta tio n a l s ta te s ,
ance.
w i l l h a v e e x a c t l y t h e same a p p e a r ­
The d i f f e r e n c e s c a n b e o f a n y s i z e a n d t h e g r o u p s
c a n o v e r l a p i n a n y w ay, b u t t h e d i f f e r e n c e s h a v e t o be
r e p e a t e d i n t h e c o n n e c t e d way t h e y a r e sh o w n .
In th is
r e g i o n t h e r e a r e a n o v e r w h e l m in g n u m b e r o f
such re p e a te d fre q u e n c y d i f f e r e n c e s .
i n T a b l e s V, V I a n d V I I .
th e se
ta b le s
I t c a n be s e e n a t o n c e from
t h a t m any m ore c o m b i n a t i o n r u l e s e x i s t t h a n
a re r e q u ir e d by t h e o r y .
how ever,
T hese a r e l i s t e d
In s p ite o f t h is p ro fu sio n ,
t h e r e a r e no c o m b i n a t i o n r u l e s c o n f o r m i n g t o t h e
p r e d i c t e d p a t t e r n e v e n t h o u g h a n u m b e r o f them a r e
re p e a te d s e v e ra l tim e s .
way t o t e l l w h i c h a r e
B ecause o f t h i s ,
th e a c c id e n ta l o n es,
th ere i s
no
s o a l l o f them
h av e t o be k e p t i n m in d .
Summary a n d D i s c u s s i o n
O n ly o n e d e f i n i t e a s s i g n m e n t h a s b e e n p o s s i b l e i n
th e c h lo r i n e d i o x id e sp e ctru m and i t
That i s
i s a p a r t i a l one.
t h e a s s i g n m e n t o f t h e l i n e a t 4 4 ,2 4 2 .1 M c a s one
o f th e F » l- * 1
tra n s itio n .
t r a n s i t i o n s f o r t h e 2 o 2 ^ 2n
ro ta tio n a l
The v a l u e s a n d s i g n i f i c a n c e s o f t h e i n t e r ­
m e d i a t e q u a n tu m n u m b e r s a n d t h e i d e n t i t y o f t h e c h l o r i n e
i s o t o p e a r e unknown.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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- 61T ab le V II:
N u m e ric a l V a lu e s o f t h e C o m b in a tio n R u le s
A
8 .4 ,
8 .6
B
1 2 .2 ,
1 2 .1
C
1 5 .1 ,
1 5 .3 ,
D
1 8 . 0 , 1 8 .3
E
2 0 .2 ,
2 0 .5 ,
P
3 0 .5 ,
3 0 .5 ,
G
5 2 .6 ,
X
1 3 4 .3 ,
1 3 4 .4
Y
1 3 8 .9 ,
1 3 9 .0 ,
Z
1 4 1 .3 ,
1 4 1 .5
a
1 4 7 .1 ,
1 4 7 .3
2 0 .8
b
1 4 9 .0 ,
1 4 9 .3 ,
3 0 .6
c
1 5 5 .6 ,
1 5 5 .7
5 2 .7
d
1 6 1 . 6 , 1 6 1 .8
H
5 8 . 6 , 5 8 .8
e
1 6 9 .5 ,
169. 8 ,
I
6 3 .1 ,
6 3 .1
f
1 8 2 .0 ,
1 8 2 .3
J
6 4 .8 ,
6 5 .O
S
1 8 7 .7 ,
1 8 7 .7
K
6 7 .5 ,
6 7 .5 , 6 7 .6
h
1 9 6 .3 ,
1 9 6 .6
L
7 5 .4 , 7 5 .5 , 7 5 .8
i
1 9 7 .5 ,
1 9 7 .8 ,
M
8 0 . 1 , 8 0 .1
j
2 0 8 .3 ,
2 0 8 .4
N
8 5 .4 ,
8 5 .5
k
2 2 5 .3 ,
2 2 5 .4
P
8 7 .0 ,
8 7 .1
1
2 3 4 .0 ,
2 3 4 .2
Q
9 0 .5 ,
9 0 .8
m
2 3 5 .5 ,
2 3 5 .8
R
9 7 .4 ,
9 7 .5
n
2 7 3 .1 ,
2 7 3 .2 ,
2 7 3 .3
S
1 0 3 .0 , 1 0 3 . 3
P
2 8 8 .4 ,
2 8 8 .4 ,
2 8 8 .6
T
1 0 5 .7 , 1 0 5 .9
q
3 0 3 .7 ,
3 0 3 .7 ,
3 0 3 .8
U
1 1 1 .6 , 1 1 1 .7 ,
r
3 1 0 .7 ,
3 1 0 .8
V
1 2 1 .3 , 1 2 1 .3
s
3 1 9 .1 ,
3 1 9 .4
vf- 1 3 3 . 9 , 1 3 3 . 9
t
3 8 9 .7 ,
3 8 9 .9
1 5 .4 ,
1 5 .5
111 . 9
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1 3 9 .3
1 4 9 .6
1 6 9 .8
1 9 7 .9
2 7 3 .3
-62I n f o r m a tio n a b o u t th e re m a in in g l i n e s w ith r e s o lv e d
S t a r k o r Zeem an e f f e c t s ,
c e r ta in ly n o t a b se n t.
e ffe c ts
th o u g h l e s s
c o n c lu siv e ,
Prom t h e s i m p l e f a c t s
is
t h a t th e se
c a n b e r e s o l v e d some i m p o r t a n t r e s u l t s c a n b e
o b ta in e d f o r th e s e l i n e s w ith o u t even c o n sid e rin g
d e t a i l s o f th e r e s o lu tio n .
itiv e
One s u c h r e s u l t i s
th e
th e p o s­
i d e n t i f i c a t i o n o f th e l i n e s a s th o s e o f c h lo r in e
d io x id e .
T h is i s
a n o n - t r iv i a l r e s u l t because th e
f a ilu r e of th e se lin e s
to conform t o
th e g e n e ra l a p p e a r­
a n c e o f t h e C102 s p e c t r u m m i g h t h a v e s u g g e s t e d t h a t t h e y
w e re d u e t o i m p u r i t i e s .
how ever,
O n ly a p a r a m a g n e t i c s u b s t a n c e ,
c o u ld have any a p p r e c ia b le
Zeem an s p l i t t i n g s a t
s u c h lo w f i e l d s b e c a u s e o f t h e s m a l l n e s s o f m a g n e t i c
d i p o l e m o m en ts i n m o l e c u l e s h a v i n g no f r e e e l e c t r o n s p i n s
and c h lo r in e d io x id e i s
th e o n ly p a ra m a g n e tic m o le c u le
t h a t c o u l d c o n c e i v a b l y be p r e s e n t i n t h e g a s p h a s e i n
s u f f i c i e n t c o n c e n t r a t i o n to c o n t r i b u t e to th e sp e ctru m
u n d e r th e e x p e r im e n ta l c o n d it i o n s o f t h e s e m e a su re m e n ts.
A n o th e r i n f e r e n c e
b ility
t h a t c a n b e m ade f r o m t h e r e s o l v a ­
o f e i t h e r o f t h e two e f f e c t s i s
d u e t o f a i r l y lo w N t r a n s i t i o n .
t h a t th e l i n e s a re
T h is i s
im p lie d by th e
g e n e ra l ten d e n cy o f d e c re a s in g c o e f f i c i e n t s f o r b o th
e f f e c t s w i t h i n c r e a s i n g N.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-6 3 R e f e r e n c e t o T a b l e IV w i l l r e v e a l t h a t t h e lo w N
c o n d itio n is
a lso
s u g g e s te d r a t h e r e m p h a tic a lly by th e
s t r u c t u r e s o f t h e Zeeman e f f e c t s .
For a l l
th e l i n e s
e x c e p t 2 ,9 * 1 0 ,2 4 and 29, i n d i c a t i o n s a r e s tr o n g t h a t
no v a l u e s o f F h i g h e r t h a n f i v e o r s i x a t t h e m o s t a r e
in v o lv e d .
tw o ,
S i n c e N c a n d i f f e r f ro m F b y no m ore t h a n
th is lim its
c o n s id e ra b ly th e p o s s i b i l i t i e s
a s s ig n m e n ts f o r t h i s g ro u p o f l i n e s .
fo r N
B e s i d e s t h e two
a l r e a d y m e n t i o n e d t h e r e a r e o n l y f o u r lo w N t r a n s i t i o n s
p re d ic te d n e a r th e re g io n o b se rv ed :
4 . at
13
a t 4 5 ,3 0 0 and 4 1 ,9 0 0 ;
3 5 ,2 0 0 a n d 3 6 , 4 0 0 ; 4 1 3 ~> 3
3
51 5 - * 4 22 a t 4 4 , 8 0 0 a n d 4 1 , 3 0 0 ; a n d 4 q 4-^ 4J 3 a t 5 1 ,2 0 0
and 5 0 , 000.
The Zeeman e f f e c t s o f s e v e r a l o f t h e l i n e s a r e r e ­
s o l v e d w e l l e n o u g h t o a l l o w p r o b a b l e F a s s i g n m e n t s t o be
m ade w i t h some c o n f i d e n c e ,
th o u g h th e y a r e l e s s c e r t a i n
th a n th e a ssig n m e n t f o r l i n e
23.
T hese l i n e s a r e :
l i n e 7 - - F = l - » 2 o r 2 -*• 1 ; l i n e 1 2 — F - 2 - > 2 ; l i n e 1 3 - - F = 1 - ^
2 o r 2 -* l; l i n e
2 o r 2 -> l.
1 5 — F = 2 -*3 o r 3 - » 2 ; a n d l i n e
F o r m any o f t h e o t h e r l i n e s
2 5 — F = 1 -^
t h e n u m b er o f
Zeem an c o m p o n e n ts i s b e l i e v e d t o b e Known t o w i t h i n two
o r th re e c h o ic e s .
One w o u ld t e n d ,
fro m t h e b e s t e s t i m a t e s
f o r F a ssig n m e n ts,
to c o n c lu d e t h a t a l l
th e l i n e s i n t h i s
r e g i o n a r e p r o b a b l y p a r t o f t h e 2 Q2 - > 2 11 r o t a t i o n a l t r a n s ­
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-6 4 itio n ,
ed.
w ith b o th i s o t o p ic
T here i s ,
s p e c ie s p e rh a p s b e in g r e p r e s e n t ­
h ow ever, enough u n c e r t a i n t y i n th e e s t i m a t e s
t h a t th e o th e r p o s s i b i l i t i e s
c an n o t be e n t i r e l y d is c o u n te d .
The c o m b i n a t i o n r u l e s b e tw e e n t h e s e l i n e s a r e e x ­
t r e m e l y n u m e ro u s a s c a n be s e e n i n T a b l e s V a n d V I .
The
p r o b a b i l i t y t h a t some o f t h e s e a r e a c c i d e n t a l r e p e t i t i o n s
i s q u ite h ig h .
I t is
u n lik e ly ,
how ever,
a s m a l l n u m b e r o f th em c a n b e s o .
t h a t m o re t h a n
U n fo rtu n a te ly ,
in
s p i t e o f th e l a r g e num ber o f th e s e r e l a t i o n s h i p s and
th e c lo s e n e s s o f t h e i r a g re e m e n t,
by t h e o r y c a n n o t be f i t t e d .
th e p a t t e r n
re q u ire d
T h is may b e b e c a u s e some
o f t h e i m p o r t a n t l i n e s i n t h e p a t t e r n a r e t o o w eak t o be
o b s e r v e d o r a r e o u t s i d e t h e r e g i o n w h e re t h e m e a s u r e m e n t s
w e re m a d e .
At a n y r a t e ,
t h i s f a i l u r e m ak e s t h e i r i n t e r ­
p r e t a ti o n im p o ssib le a t t h i s
tim e .
The f a c t t h a t t h e r e
a r e s o m any r e p e a t e d d i f f e r e n c e s c o n n e c t i n g t h e l i n e s
in th is
r e g io n f a v o r s th e a ssu m p tio n t h a t m o st o f th e
l i n e s a r e due t o a s i n g l e
co u rse,
ro ta tio n a l tra n s itio n ,
and, o f
th e 2
2
i s th e forem ost c a n d id a te .
02
ii
The l i n e s w i t h r e t r o g r a d e S t a r k e f f e c t s a r e n o t
v ery in fo rm a tiv e .
th o s e h a v in g t h i s
T h at th e y a re n o t to be g ro u p e d w ith
e f f e c t reso lv e d i s
show n b y t h e com­
p l e t e a b s e n c e o f c o m b i n a t i o n r u l e s b e t w e e n t h e two s e t s .
R e t r o g r a d e S t a r k e f f e c t s o c c u r o n l y when o n e o f t h e l e v e l s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-6 5 in v o lv e d In th e t r a n s i t i o n i s
c lo s e ly ly in g l e v e l.
s t r o n g l y p e r t u r b e d by a
No s i t u a t i o n o f t h i s
so rt is p re­
d ic te d f o r any o f th e l in e s p re d ic te d n e a r t h i s
re g io n
up t o N = 1 2 .
T h is i s i n ag re em e n t w ith th e f a c t t h a t
th e l i n e s a re
r e l a t i v e l y u n a f f e c te d by m a g n e tic f i e l d s
e v e n up t o 2 0 g a u s s .
l i n e s may b e l o n g t o
I t may b e t h a t t h e
two g r o u p s o f
two h a l v e s o f t h e sam e r o t a t i o n a l
t r a n s i t i o n o r t h e y may b e l o n g t o e n t i r e l y d i f f e r e n t o n e s .
T h e r e a r e no i n d i c a t i o n s f r o m
c o m b i n a t i o n r u l e s w h ic h o f
th ese cases a p p lie s .
D e s p ite t h e num ber o f s i g n i f i c a n t c o n c l u s i o n s i t h a s
been p o s s ib le
t o d r a w fro m t h e r e s u l t s o f t h i s
o f m e a su re m e n ts ,
se rie s
t h e p ro b lem o f an e x p l a n a t i o n o f th e
m ic r o w a v e s p e c t r u m o f c h l o r i n e d i o x i d e r e m a i n s .
s a t i s f a c t o r y s o l u t i o n may s t i l l
A
be f a r aw ay, a lth o u g h
th e a u th o r b e li e v e s t h a t such a s o l u t i o n w i l l e v e n tu a lly
be p o s s i b l e .
One a p p r o a c h t o t h e p r o b l e m t h a t m i g h t p r o v e
f r u i t f u l i n a n y f u t u r e w o rk i s a d e t e r m i n e d e f f o r t t o
e x t r a c t m o re I n f o r m a t i o n fro m t h e Zeem an e f f e c t s o f t h e
l i n e s i n th e 4 3 ,0 0 0 t o 4 5 ,0 0 0 m e g a c y c le r e g i o n .
w o u ld be e x t r e m e l y t e d i o u s w o rk a s i t
T h is
w o u ld n e c e s s i t a t e
s w e e p in g t h r o u g h e a c h l i n e v e r y s l o w l y a t a n u m b e r o f
m a g n e tic f i e l d s an d o b s e rv in g th e r e s u l t s on a r e c o r d e r
tr a c in g r a t h e r th a n an o s c illo s c o p e .
I n t h i s way t h e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-6 6 r e s o l u t i o n and s e n s i t i v i t y m ig h t b e im p ro v e d en o u g h to
p e rm it d e f i n i te
P a s s i g n m e n t s t o b e m ade a n d a c c u r a t e
Zeem an c o e f f i c i e n t s
c a l c u l a t e d f o r enough o f th e l i n e s
t o a l l o w t h e c a l c u l a t i o n o f some o f t h e i n t e r a c t i o n p a r a ­
m e te rs .
A n o th e r p ro m is in g a p p ro a c h i s
t o go t o h i g h e r f r e ­
q u e n c i e s a n d s e a r c h f o r m o re i n t e r e s t i n g
in
lin e s,
e sp e c ia lly
t h e n e i g h b o r h o o d o f 6 0 , 0 0 0 Me w h e re t h e °oo~*’ ^11 i s
p re d ic te d .
T h i s may t u r n o u t t o b e i m p o s s i b l e i n t h i s
la b o ra to ry because i t
is
d i f f i c u l t w ith th e e x i s t i n g
e q u ip m e n t to o b t a i n f r e q u e n c i e s a s h ig h a s th o s e r e p o r t e d
h ere.
Though n e i t h e r o f t h e s e s u g g e s t i o n s i s p a r t i c u l a r l y
i n v i t i n g because o f t h e i r e x p e rim e n ta l d i f f i c u l t y ,
a u th o r f e e ls
w h ile .
th e
t h a t some e f f o r t a l o n g t h e s e l i n e s i s w o r t h ­
I t may be t h a t o n l y a s m a l l a m o u n t o f a d d i t i o n a l
i n f o r m a t i o n w i l l be r e q u i r e d to s o lv e t h i s
re c a lc itra n t
p ro b lem .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-6 7 -
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