THE EFFECT OF COLLISIONS IN PRECISION SPECTROSCOPY

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THE EFFECT OF COLLISIONS IN PRECISION SPECTROSCOPY

V. Alekseev

To cite this version:

V. Alekseev. THE EFFECT OF COLLISIONS IN PRECISION SPECTROSCOPY. Journal de

Physique Colloques, 1981, 42 (C8), pp.C8-29-C8-35. �10.1051/jphyscol:1981804�. �jpa-00221699�

JOURNAL DE PHYSIQUE

ColZoque C8, suppldrnent au n o 12, Tome 42, ddcembre 1981 page C8-29

T H E E F F E C T OF C O L L I S I O N S I N P R E C I S I O N SPECTROSCOPY

V.A. Alekseev

P.N. Lebedev Physical I n s t Z t u t e , Moscow, U.S.S.R.

I n g e n e r a l , t h e i n f l u e n c e o f c o l l i s i o n s on t h e a c c u r a c y o f s p e c t r o s c o p i c measurements i n g a s e s i s e r y s i m p l e . In t h e r e g i o n o f r e l a t i v e l y low d e n s i t i e s t h e s h i f t 61 and w i d t h of t h e l i n e a r e d i r e c t l y r o p o r t i o n a l t o t h e d e n s i t y N of p e r t u r b i n g p a r t i c l e s

( s e e @.a. ^/1/P. T h i s s t a t e m e n t r e m a i n s v a l i d f o r t h e v a s t maf o r i t y o f expeEiments b o t h i n l i n e a r and n o n l i n e a r s p e c t r o s c o p y . l n t h i s Y a r t i c l e we s h o u l d l i k e t o d i s c u s s some d e v i a t i o n s from t h a t r e g u l a - r i t y .

The f i r s t p a r t of t h e p a p e r i s d e v o t e d t o t h e d i s c u s s i o n o f a s p e c i f i c i n t e r a c t i o n o f t h e c o l l i s i o n a l b r o a d e n i n g o f t h e l i n e w i t h t h e b r o a d e n i n g due t o s a t u r a t i o n . T h i s i n t e r a c t i o n d e t e r m i n e s

t h e dependence of c o l l i s i o n a l s h i f t of t h e l i n e 8 o n t h e l i g h t i n - t e n s i t y and macroscopic p a r a m e t e r s o f t h e e x p e r i m e n t a l s e t u p , a n d must be t a k e n i n t o a c c o u n t i n e s t i m a t i n g t h e a c c u r a c y of s p e c t r o s c o - p i c measurements o r t h e f r e q u e n c y r e p r o d u c i b i l i t y of t h e o p t i c a l s t a n d a r d .

In t h e second p a r t of t h e p a p e r we c o n s i d e r t h e s h i f t o f t h e l i n e a r i s i n g i n t h e c o l l i s i o n a l n a r r o w i n g phenomena (Dicke e f f e c t ) . T h i s s h i f t may p l a y a n i m p o r t a n t r o l e i n microwave f r e q u e n c y s t a n - d a r d s w i t h o p t i c a l pumping a n d , as f a r as we know, i t has n o t been d i s c u s s e d b e f o r e .

1. The S h i f t of t h e L i n e i n t h e O p t i c a l Frequency S t a n d a r d due t o C o l l i s i o n s

P o r t h e d i s c u s s e d e f f e c t i t i s v e r y e s s e n t i a l t h a t i n t h e r e g i o n o f r e l a t i v e l y low p r e s s u r e , i . e . w i t h i n t h e framework o f t h e impact a p p r o x i m a t i o n , t h e s h i f t and t h e w i d t h of t h e n o n l i n e a r r e - s o n a n c e o f a b s o r p t i o n become n o n l i n e a r f u n c t i o n s o f p r e s s u r e /2-5/.

T h i s can be e x p l a i n e d by t h e f o l l o w i n s i m p l e c o n s i d e r a t i o n .

T h i s i s w e l l known ( s e e e.g: / t h a t t h e overwhelming con- t r i b u t i o n t o t h e s a t u r a t e d absorption r e s o n a n c e i s made by a small g r o u p of m o l e c u l e s whose v e l o c i t y p r o j e c t i o n s V on 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 l i g h t a r e c o n c e n t r a t e d i n a narrow r e g i o n ~d n e a r z e r o v a l u e s o f 'If d e t e r m i n e d from t h e c o n d i t i o n ~ l 4 l t l d $ ,

where I( i s t h e f i e l d wave-vector and (I i s t h e homogeneous l i n e w i d t h . A f t e r t h e p r o c e s s of c o l l i s i o n a molecule c h a n g e s i t s v e l o c i - t y . Let t h e a v e r a g e s c a t t e r i n g a n g l e be l i m i t e d 8<< ^1. I n t h e p r o c e s s o f e l a s t i c s c a t t e r i n g t h e changing o f v e l o c i t y i s e q u a l t o

0 8 . ^{If 8} i s s o small t h a t ${{Ku@ ^% dub# Ok)a b e i n g t h e Doppler l i n e w i d t h , t h e s c a t t e r i n g t h r o u g h a n g l e d t a k e s t h e molecule o u t o f t h e r e g i o n Klatrl 6 8 , t h e e l a s t i c s c a t t e r i n g i n t e r r u p t s t h e i n t e r a c t i o n of molecule w i t h t h e f i l e d a n d p l a y s t h e r o l e of a b r o a d e n i n g f a c t o r . I n t h e o t h e r l i m i t i n g c a s e ,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981804

C8-30 JOURNAL DE PHYSIQUE

X ) ) A % ~ , t h e change of t h e molecule v e l o c i t y does n o t t a k e t h e mo- l e c u l e o u t s i d e t h e r e g i o n of resonant i n t e r a c t i o n w i t h t h e f i e l d , and t h e v e l o c i t y changing does not c o n t r i b u t e t o t h e s h i f t and w i d t h of t h e l i n e . A s a r e s u l t , i n two l i m i t i n g c a s e s t h e formulae d e s c r i b i n g t h e width and s h i f t of t h e l i n e a r e d i f f e r e n t ;

where r ^{and A} a r e t h e width and s h i f t of t h e l i n e g i v e n by t h e u s u a l broadening t h e o r y , and t h e complex parameter V = v'+ ^it''

d e s c r i b e s t h e v e l o c i t y changing. I n p a p e r s / 3 , 5 / t h e p r a c t i c a l l y e x a c t formulae a r e g i v e n d e s c r i b i n g t h e behaviour of 'd and i n t h e wide range of d e n s i t i e s and l e a d i n g t o Eq. ⁽¹⁾ i n two l i m i - t i n g c a s e s .

Values r ^, dJ and 9 a r e p r o p o r t i o n a l t o t h e g a s e n s i t y E/ ,

s o t h a t i n t h e r e g i o n of d e n s i t i e a where t h e e q u a l i t y $ ^{~ ~ a ~ a}

i s v a l i d , t h e s l o p e of c u r v e s 8 ) and $(M) changes. Such a v a r i a t i o n of t h e s l o p e of c u r v e r ( N ) has been observed f o r d i f - f e r e n t g a s e s i n /2,7-1 0 / . The measurement of t h e s l o p e of curve

Flu) i n t h e r e g i o n of low d e n s i t i e s i s more complicated, and h a s been, so f a r , made o n l y f o r t h e i n f r a r e d t r a n s i t i o n i n CH

( A = 3.39 prn) /?/. The q u a l i t a t i v e dependence b f d obtaindd i n t h i s experiment i s shown i n f i g u r e b t h e s o l i d l i n e .

$he prono nced growth of t h e s l o p e o f curve $(HI w i t h d e n s i t y i n - c r e a s i n g shows t a t v a l u e s of para- m e t e r s A and $' a r e very c l o s e and have t h e o p p o s i t e s i g n s .

Let us assume t h a t t h e r e e x i s t s a n a d d i t i o n a l r e a s o n of broadening b e s i d e s t h e c o l l i s i o n s , e.g. t h e f i e l d broadening. ? Y i t h t h e f i e l d being i n c r e a s e d , t h e width of resonance i s i n c r e a s i n g t o o w h i l s t t h e d e n s i - t y o f p a r t i c l e s 'M remains c o n s t a n t . This means t h a t w i t h t h e growth of f i e l d i n t e n s i t y t h e c o l l i s i o n a l broa- dening changes i t s dependence on den- s i t y , and i s moving from t h e r e g i o n ( l a ) t o t h e r e g i o n ( l b ) . I n o t h e r words, w i t h t h e growth of t h e f i e l d

C

Af i n t e n s i t y t h e r e o c c u r s a t r a n s i t i o n from curve 1 t o c u r v e 2 ( s e e t h e T i - g u r e ) . I n c a s e of CH l i n e i t means t h a t w i t h t h e i n c r e a s i n g of f i e l d i n t e n s i t y t h e a h f f t of t h e l i n e i s i n c r e a s i n g too. Thus, t h e j o i n t a c t i o n of c o l l i s i o n s and t h e f i e l d broadening l e a d s t o t h e dependence of t h e l i n e s h i f t on t h e f i e l d i n t e n s i t y , w h i l s t t h e f i e l d broadening i t s e l f d o a n o t pro- duce any s h i f t .

The q u a n t i t a t i v e i n v e s t i g a t i o n of t h i s e f f e c t has been c a r -

r i e d out i n / I I / . I n t h i s paper we p r e s e n t formulae only f o r t h e

l i m i t i n g e a s e J ' + e << ¹

He r e I =(4 7 4 )Z,&r+ ~'1' ^{i s} t h e s a t u r a t i o n p a r a m e t e r ; fp , t h e r a d i a t i v e l i n e w i d t h ; d , t h e d i p o l e momentum o f t r a n s i t i o n , a n d E , t h e f i e l d a m p l i t u d e . The f u n c t i o n $1) depends o n macro- s c o p i c p a r a m e t e r s o f t h e e x p e r i m e n t , f o r e x a n p l e i t depends on t h e r a t i o between t r a n s v e r s e dimensions o f t h e o p p o s i t e t r a v e l l i n g wa- v e s . But i n a l l c a s e s i t v a r i e s between u n i t y and 1.5, when t h e s a t u r a t i o n p a r a m e t e r changes from z e r o t o u n i t y .

We e s t i m a t e now t h e s h i f t of t h e most i n v e s t i a t e d F:' ^corn-

ponent o f t h e v i b r a t i o n a l - r o t a t i o n a l t r a n s i t i o n P o f t h e band

$a o f methane ( A = 3!39 p m ) , which i s of p a r t i c u l a r i n t e r e s t f o r t h e f r e q u e n c y s t a b i l l z a t i o n . From /2/ i t f o lows t h a t f o r t h e

7 4

s e l f b r o a d e n i n g of t h i s l i n e we have df'/dd = 10 3 Hz/Torr, dg'/'~

= 2.10 Hz/Torr, d A / d ~ = 1 0 Hz/Torr, ~ ( A ~ Y ) / A ~ I 05 Hz/Torr, a n d 4dB8 --" 2.105 Hz. The r a d i a t i v e l i n e w i d t h of t h i s t r a n s i t i o n i s v e r y small and may be n e g l e c t e d . Using t h e s e d a t a we c a n f i n d t h a t a t p r e s s u r e N = 3 mTorr t h e r e s o n a n c e s h i f t s by a p p r o x i m a t e l y 100 Hz, when t h e s a t u r a t i o n p a r a m e t e r c h a n g e s from z e r o t o u n i t y . A t N = 1 mTorr we g e t 6(1=i) - S ( I x o ) 5 1 0 Hz. I n t h e c a s e u n d e r c o n s i d e r a t i o n t h i s s h i f t depends on t h e s q u a r e o f r e s s u r e , s o i t d e c r e a s e s r a p i d l y when N d e c r e a s e s . A t hf$ 10-8 T o r r t h e s h i f t becomes v e r y small, and may be n e g l e c t e d . I t ' s d i f f i c u l t t o g i v e e s t i m a t e s f o r o t h e r t r a n s i t i o n s u s e d f o r t h e s t a b i l i z a t i o n due t o t h e l a c k o f i n f o r m a t i o n a b o u t t h e b r o a d e n i n g o f t h e s e l i n e s . I n t h i s c o n n e c t i o n we s h o u l d l i k e t o n o t e t h a t a g e n e r a l s h a p e o f ( 2 ) d o e s n o t a l l o w one t o p r e d i c t even t h e s i g n o f t h e s h i f t . N e v e r t h e l e s s , i t seems u s e f u l t o d i s c u s s t h e o r d e r o f magnitude of such k i n d o f a s h i f t f o r t h e t r a n s i t i o n .used f o r s t a b i l i z a t i o n .

It i s w e l l known t h a t I n c a s e o f t h e t r a n s i t i o n o f methane

A ⁼ 3.39 p m t h e s h i f t o f t h e l i n e due t o c o l l i s i o n s i s v e r y small, s o t h a t t h e p a r a m e t e r s A a n d V u a r e v e r y small t o o , more t h a n hundred t i m e s s m a l l e r t h a n t h e l i n e w i d t h . T h i s i s a t y p i c a l s i t u a - t i o n f o r a l l t h e r o t a t i o n a l - v i b r a t i o n a l t r a n s i t i o n s i n s p h e r i c a l - - t o p t y p e m o l e c u l e s , among which ( b e s i d e s CH4) SF6 a n d Os04 a r e u s e d f o r t h e f r e q u e n c y s t a b i l i z a t i o n .

The s h i f t o f t h e l i n e a r i s e s due t o t h e d i f f e r e n c e i n ampli- t u d e s o f s c a t t e r i n g f a n d f i n t h e i n i t i a l and f i n a l s t a t e s o f t h e molecule u n d e r i l v e s t i g g t i o n . I f v a l u e s f a n d f c o i n c i d e , t h e s h i f t of t h e l i n e would be e q u a l t o z e r o , w h i l s t $he l i n e w i d t h i s e q u a l t o t h e r a t e o f i n e l a s t i c t r a n s i t i o n s . There a r e two r e a - s o n s l e a d i n g t o t h e d i f f e r e n c e of v a l u e s f and f2: l ) t h e d i f f e r e n c e i n a s c a l a r p a r t o f t h e i n t e r a c t i o n of a m a l e c u l e i n s t a t e s 1 and 2 w i t h t h e p e r t u r b e r s , which l e a d s t o d i f f e r e n t s h i f t s o f l e v e l s 1 and 2 d u r i n g t h e p r o c e s s o f c o l l i s i o n ; 2 ) t h e d i f f e r e n c e i n t h e a n g u l a r monrentum r e o r i e n t a t i o n c r o s s - s e c t i o n s i n s t a t e s 1 a n d 2 , which i s i n a c c o r d a n c e w i t h o p t i c a l theorem l e a d i n g t o t h e d i f f e r e n c e i n v a l u e s f a n d f2.

The f i r s t r e a s o n p l a y s a n i m p o r t a n t r o l e i n c a s e of t r a n s i - t i o n s s i t u a t e d i n t h e v i s i b l e and u l t r a - v i o l e t r e g i o n s of s p e c t r a , Because i n t h i s c a s e t h e s t a t e s 1 and 2 a r e t h e l e v e l s w i t h d i f f e - r e n t e l e c t r o n i c s t a t e s , s o t h a t t h e i r i n t e r a c t i o n w i t h t h e p e r t u r - b e r s i s q u i t e d i f f e r e n t . For t h i s r e a s o n t h e s h i f t o f s u c h l i n e s , a s a r u l e , h a s t h e o r d e r of magnitude as t h e l i n e w i d t h . I n c a s e of v i b r a t i o n a l - r o t a t i o n a l t r a n s i t i o n s t h e f i r s t r e a s o n i s n o t o p e r a - t i n g b e c a u s e t h e l e v e l s 1 a n d 2 have t h e same e l e c t r o n i c s t a t e s . R e v e r t h e l e s s , t h e s e c o n d r e a s o n c a n l e a d t o r a t h e r l a r g e v a l u e s o f t h e s h i f t , i f s t a t e s 1 and 2 have d i f f e r e n t a n g u l a r momenta a n d t h e r e o r i e n t a t i o n o f a n g u l a r momenta i n t h e p r o c e s s o f c o l l i s i o n com- p a r a b l e w i t h e l a s t i c s c a t t e r i n g .

The p o t e n t i a l o f i n t e r a c t i o n o f a m o l e c u l e w i t h t h e p e r -

t u r b i n g p a r t i c l e c a n be w r i t t e n i n t h e form:

JOURNAL DE PHYSIQUE

wher_e VS and Vv a r e t h e s c a l a r and v e c t o r p a r t s o f i n t e r a c t i o n ;

%,PI n , G a r e t h e v i b r a t i o n a l quantum numbers and t h e d i r e c t i o n o f a x e s o f t h e i n v e s t i g a t e d a n d p e r t u r b i n m o l e c u l e s . F o r t h e above r e a s o n , t h e dependence on e ^{and p i n Eq. 7 3 )} i s v e r y weak a n d may be n e g l e c t e d . A f t e r t h a t o n l y t h e v e c t o r p a r t o f t h e p o t e n t i a l ( 3 ) c a n provoke t h e s h i f t o f t h e l i n e , f o r i t h a s n o n v a n i s h i n g m a t r i x e l e m e n t s between s t a t e s w i t h d i f f e r e n t p r o j e c t i o n s o f a n g u l a r momen- tum. I n c a s e o f a s y m m e t r i c a l m o l e c u l e s t h e v e c t o r p a r t o f t h e i n t e - r a c t i o n h a s t h e same o r d e r o f magnitude as t h e s c a l a r one, and c a n produce a r e l a t i v e l y l a r g e s h i f t . But i n c a s e o f s p h e r i c a l - t o p t y p e m o l e c u l e s i t i s e v i d e n t t h a t t h e v e c t o r p a r t of i n t e r a c t i o n i s much s m a l l e r t h a n t h e s c a l a r p a r t , t h e r e f o r e t h e s h i f t o f t h e l i n e is much s m a l l e r t h a n t h e width.

The s i t u a t i o n i s d i f f e r e n t i n t h e c a s e o f e l e c t r o n i c t r a n s i - t i o n i n 1 2 m o l e c u l e , A = 0.63 p m , which i s u s e d f o r s t a b i l i z a - t i o n i n t h e v i s i b l e r e g i o n o f s p e c t r a /12/. I n t h i s c a s e t h e s h i f t o f t h e l i n e i s t e n f o l d l a r g e r t h a n f o r t h e above mentioned l i n e o f -

met bane.

The e x p e r i m e n t a l measurement o f t h e dependencel(~)/f2/does n o t r e r e a l t h e ghanging-of a s ope of t h i s c u r v e , s o we c a n assume t h a t

~ q , e * p 3 . 1 0 Hz ( 8 I 0 - j ) . We a l s o have no i n f o r m a t i o n a b o u t va- l u e s A a n d 9 ' . b u t f o r t h e e s t i m a t i o n we have c h o s e n d A / d d

3 d L l % ~ lo6 Xz/Torr. S u b s t i t u t i n g t h e s e v a l u e s i n Eq. ( 2 ) a t p r e s s u r e N = 100 r#Torr we g e t 8(1=i) - X(I= ⁰⁾ = 104 HZ.

It i s n o t e w o r t h y t h a t a t N = l o 0 m T o r r t h e c o l l i s i o n a l w i d t h o f t h e l i n e i s commensurable w i t h t h e r a d i a t i v e l i n e w i d t h , s o t h a t w i t h a f u r t h e r d e c r e a s i n g of p r e s s u r e t h e f u l l w i d t h o f t h e l i n e r e m a i n s s t a b l e . N e v e r t h e l e s s , f r o & t h e v i e w p o i n t o f t h e s h i f t depen- d i n g on t h e s a t u r a t i o y i t i s v e r y u s e f u l t o d e c r e a s e t h e p r e s s u r e . I n t h i s c a s e rp> ^{fttl ,} s o t h a t t h e s h i f t o f t h e l i n e provoked by t h e s a t u r a t i o n d e c r e a s e s p r o p o r t i o n a l l y t o hf. For t h e o t h e r v i - s i b l e t r a n s i t i o n s t h e s i t u a t i o n must be v e r y similar w i t h t h e d i a - c u s s e d t r a n s i t i o n i n 12 .

2. The S h i f t o f t h e Line o f Radiofrequency S t a n d a r d w i t h t h e O p t i c a l Pumping

The i n h e r e n t f e a t u r e of t h e f r e q u e n c y s t a n d a r d w i t h o p t i c a l pumping i s a r e l a t i v e l y h i g h p r e s s u r e o f t h e b u f f e r g a s i n a c e l l , which i s u s e d f o r t h e n a r r o w i n g of' t h e l i n e due t o Dicke e f f e c t /13/. I n t h i s p a p e r we s h o u l d l i k e t o draw a t t e n t i o n t o t h e f a c t t h a t i n t h e c o u r s e o f s u c h n a r r o w i n g t h e r e a p p e a r s a r e l a t i v e l y l a r g e s h i f t of t h e l i n e , which i s i n d e p e n d e n t of t h e b u f f e r g a s den- s i t y .

A s a r u l e , f o r a t h e o r e t i c a l i n v e s t i g a t i o n o f t h e Dicke n a r - rowing phenomena t h e model d e s c r i p t i o n i s u s e d /14-16/. I n s u c h a d e s c r i p t i o n t h e s h i f t o f t h e l i n e mentioned above does n o t a p p e a r . Por t h i s r e a s o n , we s h a l l s h o r t l y r e l a t e t h e r e s u l t s o f t h e e x a c t t r e a t m e n t o f t ; ~ i s phenomenon d e v e l o p e d i n /17/.

We s h a l l n e g l e c t t h e degeneracy of s t a t e s , and c o n s i d e r t h e

l i n e a r p r o b l e ~ ~ w i t h r e s p e c t t o t h e f i e l d , The a b s o r p t i o n power

( o r t h e l i n e i n t e n s i t y ) P(d) i s c o n n e c t e d w i t h d e n s i t y m a t r i x

element o f t h e t r a c s i t i o n Ftz(*) , v b e i n g t h e molecule ve1.o-

c i t y , i n t h e f o l l o r ~ i n g manner

where d i s t h e f r e q u e n c y of t h e e l e c t r o m a g n e t i c wave and WL i s t h e m a t r i x element of t h e magfietic d i p o l e momentum o p e r a t o r . For t h e d e n s i t y m a t r i x element w e have :

[ip, + a ~ ) + + ] L ~ F ) - J ~ ( i i , & ) ~ n 7 ~ ~ ' ⁼ rnW&)> ⁽⁵⁾

where d Q = t 3 - d ~ ~ ; dr;z i s t h e f r e uency of t r a n s i t i o n ; K - ,

t h e wave v e c t o r of the f i e l d , and ~ ( 3 ) , t h e ilaxrrellian d i s t r i b u t i o n f u n c t i o n . The parameter V f i i ) an6 t h e k e r n e l of t h e c o l l i s i o n i n - t e g r a l c a n be e x p r e s s e d i n terms of ~ o l e c u l e and p e r t u r b i n g p a r t i c l e s c a t t e r i n g a m p l i t u d e s f i and f i i n s t a t e s 1 and 2 191. For t h e

I!

f o l l o w i n g i t i s v e r y i m p o r t a n t t h a t 3 depends on t e molecule ve- l o c i t y 9 ( s e e e.g. /19/). The e x i s t e n c e of a Dicke narrowing i n d i c a t e s t h a t t h e s e a m p l i t u d e s c o i n c i d e w i t h g r e a t a c c u r a c y , and t h a t i n e l a s t i c s c a t t e r i n g i s v e r y weak. Pie c a n , t h e r e f o r e , r e p r e s e n t t h e s e a m p l i t u d e s i n t h e form:

where v a l u e 4fL d e s c r i b e s a s m a l l d i f f e r e n c e between a m p l i t u d e s f and h and a p o s s i b l e e x i s t e n c e of i n e l a s t i c s c a t t e r i n g ;

R e s p e c t i v e l y , we can e x p r e s s v a l u e s VC5) and A(5,iF') i n t h e form:

where $.(f), A. /&I); d l , d A {.A-!;+A!~ {: + d ! : ~ f t ;

a n d , e v i d e n t l y , ( o S I < < I J ~ ~ ; ~ A A I & I A ~ I .

Combining t h e terms w i t h V 0 and & i n t h e i n t e g r a l o p e r a t o r

6 ^{,and nJ and} i n t h e o p e r a t o r o b we g e t :

Values 3 and A , and t h e r e f o r e , t h e o p e r a t o r s U and A a r e p r o p o r t i o n a l t o t h e d e n s i t y of p e r t u r b i n g p a r t i c l e s N. T h e r e f o r e , a t t h e s u f f i c i e n t l y l a r g e d e n s i t y , valuel/U//becomes much l a r g e r as compared t o /%$I . S i n c e i n e q u a l i t y ,~AuII<<UUI( i s always f u l f i l l e d , t h e s h a r e ( i X p + " U I f i z ~ ~ ) i n Eq. ( 8 ) c a n be con- s i d e r e d a s p e r t u r b a t i o n .

@ow c o n s i d e r t h e o p e r a t o r 4 . The o p t i c a l theorem and t h e d e t a i l e d b a l a n c e p r i n c i p l e y i e l d t h e e q u a l i t i e s :

By making use of t h e s e r e l a t i o n s , i t i s v e r y e a s y t o s e e ' t h a t t h e Kaxwellian d i s t r i b u t i o n f u n c t i o n i s t h e e i g e n f u n c t i o n of t h e ope- r a t o r U c o r r e s p o n d i n g t o z e r o e i g e n v a l u e d ~ @ ) = 0. It may be shown t h a t eigenfunc t i o n s of t h i s o p e r a t o r Y)n(e) are o r t h o g o n a l

J/$(G)]-' %(+) dd = Pnm , and a l l o t h e r e i g e n v a l u e s A n ,

e x c e p t a= ^0, a r e t h e p o s i t i v e numbers, and we may assume t h a t t h e y do n o t d i f f e r from 3 by t h e order-o_f magnitude. Thus,

s i m u l t a n e o u s l y w i t h inequality~l~u>)lb~ll,Il<vl f o r nfO, t h e inequa- l i t y An >> l l ~ u l l , ladl i s a l s o f u l f i l l e d .

Now we c a n seek t h e s o l u t i o n of Eq. ( 3 ) i n t h e form o f ex- p a n s i o n i n s e r i e s ?,2(9) = % am(P4(*) . Making use o f -the p e r t u r b a - t i o n t h e o r y , w i t h r e s p e c t t o parameters IlAUll /An , I K * I / ~ , ^{n # O ,}

we c a n determine v a l u e s Qm. According t o Eq. ( 4 ) o n l y one o f

t h e s e v a l u e s i s connected w i t h t h e i n t e n s i t y of t h e l i n e

C8-34 JOURNAL DE PHYSIQUE

F o l l o w i n g t h e p e r t u r b a t i o n t h e o r y p r o c e d u r e f o r c a l c u l a t i n g qo ,

we g e t /17/:

2 -1

P ( a ) ^{= 2 w IE%} lmi2 Re id^ ^{+ Gee} - f-1 ,

where

G,,= { % ~ ~ ) = ~ ( ~ j J i ~ i + a i c ( ~ ~ ~ @ } = i j t i j ~ ( ~ ) d ~ +

+ j ~ ~ c s ) Y,,~,/~)dii + JA~(G,C') (j',,[~?ci~' ^r i[KC)oh+ d , , , + i & n , ^{( 9 )}

F ~ ~ = z A ~ ' G ~ , , G ~ ~ = Z ~i'[<~+);n - ~ ( ~ ~ l e n J ' o ~ + ^{( 1 0 )}

n +o "3%

' +ip~~a)..Y~, + i2 ~ o H r * n J .

+ ^f:m - ^JOB

braking u s e o f i q s . ( 9 ) a n d (1 0 ) , and n e g l e c t i n g t h e t e r m s fiA ,

8% , J ~ / A we g e t

Y,%t r'

9

( ~ w r ' , + r ' ) ~ + ( i l ~ ~ + r ' ) ~ ( 1 1 )

Thus, t h e l i n e i n t e n s i t y ( 1 1 ) h a s t h e form o f a L o r e n t z i a n c o n t o u r w i t h w i d t h d,, + r' and s h i f t r o o + f" . ^{Values 80, and P e e} r e p r e - s e n t w i d t h and s h i f t o f t h e l i n e o f t h e u s u a l impact b r o a d e n i n g t h e o r y . Value f' c o n s i s t s o f two terms. The f i r s t term-.&v]a/!~~&/~

d e c r e a s e s w i t h t h e i n c r e a s i n g o f d e n s i t y , a n d d e s c r i b e s t h e pheno- mena o f Dicke narrowing. The second o n e , as w e l l a s t h e q u a n t i t y

r N ^{, i s} i n d e p e n d e n t o f d e n s i t y . These d e n s i t y - i n d e e n d e n t t e r m s produce t h e q u a l i t a t i v e d i f f e r e n c e between (1 1 ) , ^{(1 27 a n d} similar f o r m u l a e produced by t h e model c a l c u l a t i o n . The e x i s t e n c e o f s u c h d e n s i t y - i n d e p e n d e n t t e r m s i n t h e w i d t h o f t h e l i n e i s v e r i f i e d i n t h e e x p e r i m e n t of T u r r a y and J a v a n /20/ ( s e e a l s o d i s c u s s i o n i n /17/). It i s v e r y i m p o r t a n t t o n o t i c e t h a t t h e a p p e a r a n c e o f s u c h amendments i n t h e w i d t h a n d s h i f t o f t h e l i n e i s c l o s e l y - c o n n e c - t e d w i t h t h e dependence of p a r a ~ e t e r on t h e v e l o c i t y y , a n d w i t h t h e e x i s t e n c e o f t h e i n t e g r a l o p e r a t o r b U .

It i s i n t e r e s t i n g t o e s t i m a t e v a l u e r* . ^{A s} s e e n from Eq. ( 1 2 ) , t h i s v a l u e i s p r o p o r t i o n a l t o t h e impact w i d t h o f t h e li e 8 a n d t o t h e r a t i o KW'A dAk),/A o f Doppler w i d t h a n d t h e r a t e o c o l i s i o n . The w i d t h o f t h e l i n e comes up t o i t s minimum, when t h e e q u a l i t y L ~:(iib)%n =Im, which d e t e r n i n e s t h e b u f f e r g a s d e n s i t y , i s v a l i d . AsS"u*ting t h a t % , , . - ' d c , , , we g e t rn-4(KlrJY2. ^{A s} a r u l e , t h e l i n e w i d t h comes up t o i t s minimurn I n the r e g i o n o f d e n s i t i e s

N - 4 1 T o r r . Assuming A t o be q u a 1 t o A = N I B = 3.10" cm-3 5 . 1 0 ~ cm/s 5.10* cm' = 7.5. 106 s+ and A d a n & w l o 5 H Z , we g e t a r e l a t i v e l y l a r g e v a l u e o f ?dl0 Hz. It should be n o t e d t h a t t h i s s h i f t c a n be f o u n d i n t h e e x p e r i m e n t o n l y due t o i t s de- pendence on t h e kind of t h e b u f f e r g a s , o r by t h e c o ~ p a r i s o n w i t h t h e m o l e c u l a r b e a r e - p e r i m e n t s .

R e f e r e n c e s

1 . SDBRLEIAZJ I.I., VAIXSHTCIN L.A., YUKOV E.A. E x c i t a t i o n o f Atoms a n d Broadening o f S p e c t r a l L i n e s . S p r i n g e r - V e r l a g , 1981.

2. BAGAEV C.N. BAKLANOV E.V., CHSBOTAEV V.P., JETP L e t t . 76 (1972) 9 ; 1 6 (19723 243.

3 . A L E ~ E E v V.A., AItDRZ3VA T. L., SOBBLNTi'l I. I. Sov. Phys. JETP 37

( 1 9 7 3 ) 413.

4. BAKLANOV E.V., Opt.Spectrosk., 38 ^{(1975) 24.}

5. KOCIUNOV V. P., RAUTIAW S. C., SHALAGIM A.m., Sov.Phys. JETP (3977) 723.

6. LATU;B lIV.E.,Jr., Phys.Rev.,l 4A (1964) 1429.

7. .

ZIA'PTICK A. T. . KUKNIT li.A.%VAB A. , Chem. Phys.Lett. 2 (1 976 ) 176.

8. ILEYElI T.W., RIIODES C.K., HAUS H.A. Phys.Rev. A12 (1975) 1993.

9. VASILENXO L, S., KOCIIANOV V. P., CHEBOTAEV V. P., Opt. Commun. 2

( 1 9 7 7 ) 409.

10. CAHUZAC Ph., Iv'IARIE E., ROBAUX O., VETTER R., B13m?ATT P.R. J.Phys.

B L (1978) 645.

ALSKSEZV V.A., YATSENBO L.P. JETP L e t t . 2 ( 1 9 7 9 ) 389; SOV.

Phys.JETP z ( 6 ) (1979) 1083.

SCHWEITZPUR \V. G. ,Jr., KESSLER E. G., Jr. , DESLATTES R, D. ,LAYER H.

VJHETSTONE J.R., Appl.Opt., 1 2 (1973) 2927.

DICKE R., Phys.Rev. ( 1 9 5 v 472.

RAUTIAN S. G., SOBELF5AR I. I. Sov.Pbys.Usp. 9 (1967) 701.

GALATRY L., Fhys.Rev. 122 (1961) 1218.

GERS'I'EN 9. I., FOLEY H.- JOSA 58 (1968) 933.

ALEKSEEV V.A., XALYUGIH A.V. Zb.Eksp.Teor.Piz. 8 0 (1981 ) 897.

ALERSEEV V. A. , ANl3R:SEVA T. L., SOBSLNAE I. I., S o T ~ h ~ s . JET?

( 1 9 7 2 ) 325.

ALEKSEXV Y. A. , BULYUGIE A.V. , S o v . P h y ~ . JETP, 4 (1 978) 4-77.

fiWRRAY J . R . , J A V A A . J o u r n . of Modol.Spectr.,& (1972) 1.