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UBC Theses and Dissertations

Pressure broadening and coherence transients effects: a kinetic theory approach Coombe, Dennis Allan 1976

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PRESSURE BROADENING AND COHERENCE TRANSIENTS EFFECTS - A KINETIC THEORY APPROACH DENNIS ALLAN COOMBE B.Sc. (Hons.) U n i v e r s i t y o f C a l g a r y , 1970 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e D e p a r t m e n t o f CHEMISTRY We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA F e b r u a r y , 1976 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree that p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f d ke. t*t(s f r Lj The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date t^grcU 2 d, Il<(7b ABSTRACT The r e s p o n s e o f a p o l y a t o m i c g a s to m i c r o w a v e r a d i a t i o n i n c l u d i n g b o t h s t e a d y s t a t e ( p r e s s u r e b r o a d e n i n g ) and time d e p e n d e n t ( c o h e r e n c e t r a n s i e n t s ) e f f e c t s - i s d e s c r i b e d . t h e o r e t i c a l l y . The t r e a t m e n t i s b a s e d on s o l u t i o n s o f a quantum m e c h a n i c a l B o l t z m a h n e q u a t i o n and employs k i n e t i c t h e o r y methods w h i c h have p r e v i o u s l y been u s e d i n t h e e x p l a n a t i o n o f the f i e l d d e p e n d e n c e o f t r a n s p o r t phenomena ( S e n f t l e b e n - B e e n a k k e r e f f e c t s ) . Much o f t h e r e c e n t t h e o r e t i c a l work o f p r e s s u r e b r o a d e n i n g and c o h e r e n c e t r a n s i e n t phenomena i s b a s e d on a two ( e n e r g y ) s t a t e model f o r t h e g a s m o l e c u l e s . T h i s m o d e l , when d e v e l o p e d f r o m a d e n s i t y o p e r a t o r p o i n t o f v i e w , r e s u l t s i n a c o u p l e d s e t o f t h r e e e q u a t i o n s w h i c h a r e m a t h e m a t i c a l l y e q u i v a l e n t to the B l o c h e q u a t i o n s o f NMR. The p r e s e n t work r e e x a m i n e s t h i s d e s c r i p t i o n , and r e p l a c e s i t w i t h a two l e v e l m o d e l f o r t h e g a s s y s t e m . H e r e , t h e term " l e v e l " i m p l i e s e x p l i c i t c o n s i d e r a t i o n o f t h e r o t a t i o n -a l ( m a g n e t i c ) d e g e n e r a c y a s s o c i a t e d w i t h e a c h e n e r g y s t a t e . T h i s model g i v e s a more a p p r o p r i a t e r e p r e s e n t a t i o n o f the i n t e r a c t i o n o f m i c r o w a v e r a d i a t i o n w i t h a r e a l m o l e c u l a r s y s t e m . I n p a r t i c u l a r , a more c o m p l e t e s e t o f c o u p l e d e q u a t i o n s r e s u l t f r o m t h i s d e s c r i p t i o n and i n v o l v e q u a n t i t i e s i n a d d i t i o n to the t h r e e moments u s e d i n a two s t a t e a p p r o a c h . The most i m p o r t a n t o f t h e s e l a t t e r e f f e c t s a r e r e p r e s e n t e d by s p h e r i c a l i n the a n g u l a r momentum J o f the r e l e v a n t e n e r g y l e v e l s . An a n a l o g o u s t r e a t m e n t o f r o t a t i o n a l e f f e c t s has p r e v i o u s l y been u s e d i n S e n f t l e b e n - B e e n a k k e r s t u d i e s . S p e c i f i c m o l e c u l a r t y p e s o f i n t e r e s t i n m i c r o w a v e s p e c t r o s c o p y - d i a m a g n e t i c d i a t o n i c s and l i n e a r p o l y a t o m i c s , s y m m e t r i c t o p s , and i n v e r t i n g s y m m e t r i c t o p s - a r e t r e a t e d s e p a r a t e l y by t h i s two l e v e l a p p r o a c h . The v e c t o r (and t e n s o r ) n a t u r e o f the m o t i o n s a r e e m p h a s i z e d t h r o u g h o u t . The number o f r o t a t i o n a l p o l a r i z a t i o n s t h a t a r i s e i n t h e g e n e r a l two l e v e l c a s e i s o f t e n q u i t e l a r g e . The s i m p l e s t example o f a two l e v e l s y s t e m i s the j=0 to j = l t r a n s i t i o n o f a d i a m a g n e t i c d i a t o m i c . T h i s i s s t u d i e d i n the o n l y r o t a t i o n a l p o l a r i z a t i o n a f f e c t e d by l i n e a r l y p o l a r i z e d r a d i a t i o n i n the u s u a l e x p e r i m e n t s . The e f f e c t o f t h i s q u a n t i t y on b o t h s t e a d y s t a t e and t r a n s i e n t phenomena i s d e s c r i b e d , and a new " c o m b i n a t i o n " e x p e r i m e n t i s s u g g e s t e d as the b e s t way t o d e t e c t the p r e s e n c e o f t h i s a d d i t i o n a l p o l a r i z a t i o n . The D o p p l e r e f f e c t i s t r e a t e d by a p p r o p r i a t e l y i n c l u d -i n g the e f f e c t s o f t r a n s l a t i o n a l m o t i o n i n t h e quantum B o l t z m a n n e q u a t i o n . A more g e n e r a l s e t o f c o u p l e d moment e q u a t i o n s t h e n r e s u l t s , and the manner i n w h i c h the macro-s c o p i c v e l o c i t y p o l a r i z a t i o n s a r i s e i s t h e r e b y e s t a b l i s h e d . A m odel method s o l u t i o n o f the quantum B o l t z m a n n e q u a t i o n , e m p h a s i z i n g the p a r i t y i n v a r i a n c e o f the c o l l i s i o n s u p e r -o p e r a t o r , i s g i v e n f o r a s t e a d y s t a t e a b s o r p t i o n e x p e r i m e n t i n the a b s e n c e o f s a t u r a t i o n b u t i n c l u d i n g D o p p l e r e f f e c t s . some d e t a i l . H e r e , t h e i s T h r o u g h o u t t h i s t h e s i s , the r e l a x a t i o n r a t e s , a r e r e l a t e d to k i n e t i c t h e o r y c o l l i s i o n c r o s s s e c t i o n s by s o l v i n g the quantum B o l t z m a n n e q u a t i o n . E x t e n s i v e use i s made o f r o t a t i o n a l i n v a r i a n c e t o r e d u c e t h e number o f i n d e p e n d e n t c o l l i s i o n i n t e g r a l s , and t h e i r a p p r o x i m a t e e v a l u a t i o n i s a c c o n p l i s h e d w i t h i n the c o n t e x t o f the d i s t o r t e d wave B o r n a p p r o x i m a t i o n . A l l c o l l i s i o n i n t e g r a l s f o r the p u r e i n t e r n a l s t a t e p o l a r i z a t i o n s a r e f o u n d t o - b e e x p r e s s i b l e i n terms o f one t r a n s l a t i o n a l f a c t o r , w h i c h i s i t s e l f f u r t h e r a p p r o x i -mated by a m o d i f i e d B o r n a p p r o x i m a t i o n . C o r r e s p o n d i n g l y , the t r a n s l a t i o n a l f a c t o r w h i c h a r i s e s i n t h e r e l a x a t i o n o f m a c r o s c o p i c v e l o c i t y p o l a r i z a t i o n s i s c o m p l e t e l y s p e c i f i e d by (Z s) r e l a t i n g i t to the "' i n t e g r a l s o f t r a d i t i o n a l k i n e t i c t h e o r y . TABLE OF CONTENTS Page A b s t r a c t . i i L i s t o f F i g u r e s i x Acknowledgements x C h a p t e r I : I n t r o d u c t i o n 1 (a) T h e s i s I n t r o d u c t i o n 2 (b) F u n d a m e n t a l C o n c e p t s i n L i n e B r o a d e n i n g . 4 (c) A s p e c t s o f P r e s s u r e B r o a d e n i n g . . . . 8 (d) S a t u r a t i o n E f f e c t s 12 (e) D o p p l e r B r o a d e n i n g . 15 ( f ) R e l a t e d A r e a s o f S p e c t r o s c o p y . . . . 17 (g) Summary 22 C h a p t e r I I : The Two S t a t e S ystem 2 3 (a) I n t r o d u c t i o n . . 24 (b) K i n e m a t i c s . 26 (c) Dynamics and the R o t a t i n g Wave A p p r o x i m a t i o n 30 (d) The R e l a x a t i o n M a t r i x 36 (e) S t e a d y S t a t e A b s o r p t i o n and the G e n e r a l T r a n s i e n t E x p e r i m e n t . . . . . 41 ( f ) D e t e c t i o n o f R a d i a t i o n 45 (g) T r a n s i e n t A b s o r p t i o n 51 (h) T r a n s i e n t E m i s s i o n 55 ( i ) Summary 61 v i i Page C h a p t e r I I I : S e n f t l e b e n - B e e n a k k e r E f f e c t s and the L i n e a r i z e d W aldmann-Snider C o l l i s i o n S u p e r o p e r a t o r ^ 2 (a) I n t r o d u c t i o n . 6 3 (b) The L i n e a r i z e d Waldmann-Snider E q u a t i o n ^5 (c) V a r i o u s C h o i c e s o f B a s e s 7 2 (d) Moment E q u a t i o n s - the S h e a r ." -V i s c o s i t y C o e f f i c i e n t f o r N,,. . . . . 79 (e) C o l l i s i o n a l E x p r e s s i o n s . . . . . . . 86 ( f ) The D i s t o r t e d Wave B o r n A p p r o x i -., m a t i o n (DWBA) 105 (g) E v a l u a t i o n o f the T r a n s l a t i o n a l C o l l i s i o n I n t e g r a l s - the M o d i f i e d B o r n A p p r o x i m a t i o n . 118 (h) Summary 127 C h a p t e r IV: A K i n e t i c E q u a t i o n f o r P r e s s u r e B r o a d e n -r-- i n g and the L i n e a r i z e d C o l l i s i o n O p e r a t o r . . . . . . . . . 128 (a) I n t r o d u c t i o n . . 1 2 9 (b) G e n e r a l E q u a t i o n o f M o t i o n - t h e Waldmann-Snider form 130 (c) A p p r o p r i a t e C h o i c e o f B a s i s and R e s u l t i n g Moment E q u a t i o n s 136 (d) G e n e r a l C o l l i s i o n E x p r e s s i o n s . . . . 140 (e) The D i s t o r t e d Wave B o r n and the M o d i f i e d B o r n A p p r o x i m a t i o n 159 ( f ) Some C o n n e c t i o n s w i t h o t h e r work. . . 170 C h a p t e r V: Two L e v e l Systems 175 (a) I n t r o d u c t i o n . . . . . . . . . . . . . 176 (b) DDLP C a s e - D i m e n s i o n a l C o n s i d e r a -t i o n s and C o n n e c t i o n s w i t h C h a p t e r IV 180 (c) DDLP Case - Moment Methods and C o n n e c t i o n w i t h F l y g a r e 193 v i i Page (d) R e s o n a n t T r a n s i t i o n s i n Symmetric Top M o l e c u l e s 211 (e) I n v e r t i n g Symmetric Tops and the I n v e r s i o n S p e c t r a o f NH 220 ( f ) Summary . . "' . 228 C h a p t e r V I : The j = 0—• j = l Case f o r DDLP 229 (a) I n t r o d u c t i o n 230 (b) E q u a t i o n s o f M o t i o n - V e c t o r and S c a l a r Forms 232 (c) C o l l i s i o n I n t e g r a l s 242 (d) S t e a d y S t a t e A b s o r p t i o n and t h e G e n e r a l T r a n s i e n t E x p e r i m e n t . . . 253 (e) T r a n s i e n t A b s o r p t i o n and T r a n s i e n t E m m i s s i o n f o r a Two L e v e l S y s t e m . . . 262 ( f ) Summary o f the Two L e v e l P o i n t o f View . 268 C h a p t e r V I I : V e l o c i t y E f f e c t s i n P r e s s u r e B r o a d e n i n g . . 274 (a) I n t r o d u c t i o n 275 (b) A K i n e t i c E q u a t i o n w h i c h i n c l u d e s V e l o c i t y E f f e c t s 279 (c) The Moment Method a p p l i e d t o a D i s c u s s i o n o f V e l o c i t y E f f e c t s . . . . 286 (d) C o l l i s i o n a l D e s c r i p t i o n o f V e l o c i t y E f f e c t s 292 (e) Model Methods a p p l i e d to V e l o c i t y E f f e c t s 306 ( f ) T h e s i s C o n c l u s i o n 316 R e f e r e n c e s 318 A p p e n d i x A: The G e n e r a l i z e d B o l t z m a n n E q u a t i o n o f S n i d e r and S a n c t u a r y 327 A p p e n d i x B: I r r e d u c i b l e T e n s o r s o f SO(3) 333 v i i i L I S T OF FIGURES Page 1. A b l o c k d i a g r a m o f the i d e a l i z e d e x p e r i m e n t . . . . 5 2. P l o t s o f P ( t) f o r v a r i o u s v a l u e s o f 54 c 3. D i a g r a m showing t r a n s i e n t a b s o r p t i o n and t r a n s i e n t e m i s s i o n . . . . . . . . . . . . . . . . . . . . . . 58 4. E v e n and odd v i s c o s i t y c o e f f i c i e n t s f o r N^. . . . . 85 5. C o u p l i n g d i a g r a m £ . . . . . . 95 6. C o u p l i n g d i a g r a m / 9 . . . . . . . . . . . 154 7. R o t a t i o n a l c h a r a c t e r i s t i c s . . . . . . . . . . . . . 178 8. Some p o s s i b l e s h a p e s o f A(u) v e r s u s u c u r v e s . . . . 261 9. R o t a t i o n a l m o t i o n s p r o d u c e d by l i n e a r l y p o l a r i z e d r a d i a t i o n 270 10. A c o m b i n a t i o n m i c r o w a v e a b s o r p t i o n - o p t i c a l l i g h t l i g h t s c a t t e r i n g e x p e r i m e n t . . . . . . . . . . . . 273 ACKNOWLEDGEME NTS The a u t h o r s i n c e r e l y w i s h e s to thank Dr. R. F. S n i d e r f o r h i s e n c o u r a g e m e n t , i n s i g h t , and e f f o r t i n the p r o d u c -t i o n o f t h i s t h e s i s . He has the a b i l i t y to g e t t h e b e s t p o s s i b l e o u t o f t h o s e who work w i t h him. Thanks a r e a l s o due t o D r . J.A.R. Coope, whose d e p t h o f u n d e r s t a n d i n g has s e r v e d as a model t o s t r i v e t o w a r d s . I am i n d e b t e d to Ruth Hasirian and J u d y W r i g h t , who p e r s e v e r e d t h r o u g h the t y p i n g o f t h i s e x t r e m e l y d i f f i c u l t m a n u s c r i p t . The f i n a n c i a l s u p p o r t o f t h e N a t i o n a l R e s e a r c h C o u n c i l t h e H.R. M a c M i l l i a n F o u n d a t i o n , and t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a i s g r a t e f u l l y a c k n o w l e d g e d . F i n a l l y , a p p r e c i a t i o n i s e x p r e s s e d to t h o s e who c o n t r i b u t e d e m o t i o n a l s u p p o r t - t o my p a r e n t s , who have been " k e e p i n g the f a i t h " f o r y e a r s , and t o J e a n e t t e , f o r h e r p a t i e n c e and u n d e r s t a n d i n g . A l l q u o t a t i o n s a r e from L e w i s C a r r o l l ' s " A l i c e ' s A d v e n t u r e s i n W o n d e r l a n d " and " T h r o u g h the L o o k i n g G l a s s . " CHAPTER I I n t r o d u c t i o n "Mine i s a l o n g sad t a l e " s a i d t h e Mouse t u r n i n g t o A l i c e and s i g h i n g . " I t i s a l o n g t a i l , c e r t a i n l y " s a i d A l i c e , l o o k i n g down w i t h wonder a t the Mouse's t a i l , " b u t why do you c a l l i t s a d ? " (a) T h e s i s I n t r o d u c t i o n T h i s t h e s i s r e p r e s e n t s an a t t e m p t a t a u n i f i e d f o r m u l a t i o n o f s e v e r a l r e l a t e d m i c r o w a v e e x p e r i m e n t s - p r e s s u r e b r o a d e n i n g , s a t u r a t i o n b r o a d e n i n g and D o p p l e r b r o a d e n i n g o f s t e a d y s t a t e l i n e s h a p e s as w e l l as the c o r r e s p o n d i n g t i m e domain t r a n s i e n t e m i s s i o n and a b s o r p t i o n e f f e c t s - on a d i l u t e gas o f p o l y -a t o m i c m o l e c u l e s . The a p p r o a c h i s b a s e d on the s o l u t i o n o f a quantum m e c h a n i c a l B o l t z m a n n e q u a t i o n and i s p r e s e n t e d i n a manner t h a t i s c o n s i s t e n t w i t h e a r l i e r e x p l a n a t i o n s o f the f i e l d d e p e n d e n c e o f t r a n s p o r t phenomena ( S e n f t l e b e n - B e e n a k k e r e f f e c t s ) . I t i s hoped t h a t t h i s s y s t e m a t i c d e v e l o p m e n t w i l l p r o v i d e a framework f o r t h e d e s c r i p t i o n o f m o s t gas p h a s e phenomena i n t h e b i n a r y c o l l i s i o n r e g i m e and t h u s a l l o w a g r e a t e r u n d e r s t a n d i n g o f b o t h e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s . The t h e s i s i s d i v i d e d i n t o s e v e n c h a p t e r s . The f i r s t p r o v i d e s a q u a l i t a t i v e o v e r v i e w o f t h e g e n e r a l f i e l d o f p r e s s u r e b r o a d e n i n g and r e l a t e d phenomena. The more s p e c i f i c a r e a s o f c o n c e r n a r e t h e n p r e s e n t e d i n a q u a n t i t a t i v e f a s h i o n i n c h a p t e r I I , b a s e d on a c o n s i s t e n t , b u t somewhat e l e m e n t a r y , a p p r o a c h o f o t h e r w o r k e r s . C h a p t e r I I I summarizes t h e t h e o r e t i c a l methods employed i n t h e s t u d y o f S e n f t l e b e n -B e e n a k k e r e f f e c t s . T h e s e same k i n e t i c t h e o r y methods t h e n a l l o w a r e - e x a m i n a t i o n , i n c h a p t e r s IV, V, and V I , o f t h e i n t e r n a l s t a t e e f f e c t s d e s c r i b e d i n c h a p t e r I I . I n d e e d , C h a p t e r IV d i s c u s s e s t h e t r e a t m e n t q u i t e g e n e r a l l y , w i t h p a r t i c u l a r e m p h a s i s on c o l l i s i o n a l a s p e c t s . The a p p r o a c h i s 3 p u r s u e d f u r t h e r i n c h a p t e r V and c o n n e c t i o n s a r e made w i t h t h e more e l e m e n t a r y t h e o r y o f c h a p t e r I I , f o r s p e c i f i c m o l e c u l a r t y p e s . C h a p t e r VI i s i n c l u d e d as an i l l u s t r a t i v e example o f the new e f f e c t s w h i c h a r e p r e d i c t e d from t h i s k i n e t i c t h e o r y t r e a t m e n t . C h a p t e r V I I p o i n t s o u t t h a t t h e methods o f c h a p t e r I I I a r e a l s o u s e f u l i n d i s c u s s i n g t h e c o m p l i c a t i o n s i n t r o d u c e d by t r a n s l a t i o n a l ( v e l o c i t y ) e f f e c t s . C o n c e p t u a l l y t h e n , c h a p t e r s IV t h r o u g h V I I c a n be v i e w e d a s a s y n t h e s i s o f c h a p t e r s I I and I I I . (b) F u n d a m e n t a l C o n c e p t s i n L i n e B r o a d e n i n g B e c a u s e t h e n a t u r e o f t h i s t h e s i s i s t h a t o f c o n n e c t i o n and c o m p a r i s o n , and b e c a u s e t h e huge volume o f m a t e r i a l a l r e a d y p u b l i s h e d on p r e s s u r e b r o a d e n i n g and r e l a t e d phenomena demands a c e r t a i n amount o f e d i t i n g , i t was f e l t t h a t a q u a l i t a t i v e o v e r v i e w o f t h e s u b j e c t s h o u l d f i r s t be p r e s e n t e d . T h i s c h a p t e r , t h e n , i s i n t e n d e d a s a g e n e r a l p e r s p e c t i v e on t h e " s t a t e o f a f f a i r s " w i t h o u t i n v o k i n g t h e d e t a i l e d mathe-m a t i c a l a r g u m e n t s o f l a t e r c h a p t e r s . I n c l u d e d a r e d e f i n i t i o n s o f t e r m s u s u a l l y employed i n t h e e x p l a n a t i o n o f t h e s e e f f e c t s a s w e l l as a d e s c r i p t i o n o f how t h e s e m i c r o w a v e e x p e r i m e n t s c a n be v i e w e d i n r e l a t i o n to a n a l o g o u s e x p e r i m e n t s i n o t h e r r e g i o n s o f t h e e l e c t r o m a g n e t i c s p e c t r u m . An i d e a l i z e d e x p e r i m e n t a l s e t up i s shown i n f i g u r e 1. I t c o n s i s t s o f t h r e e components - a s o u r c e o f a d i s t u r b a n c e , a s y s tern t o w h i c h t h e d i s t u r b a n c e i s a p p l i e d , and a de t e c t o r w h i c h m e a s u r e s the r e s p o n s e o f t h e s y s t e m to t h e d i s t u r b a n c e . I n p a r t i c u l a r , t h e s o u r c e may be c o n t i n u o u s wave (c.w.) m i c r o -wave r a d i a t i o n , t h e d e t e c t o r a s t e a d y s t a t e f r e q u e n c y d e t e c t -i n g s y s t e m , and t h e s y s t e m a c o l l e c t i o n o f c o m p l e t e l y i s o l a t e d , n o n t r a n s l a t i n g atoms ( o r m o l e c u l e s ) e a c h h a v i n g a number o f d i s c r e t e i n t e r n a l s t a t e e n e r g y l e v e l s . A t r a n s i t i o n ( s a y a b s o r p t i o n ) between two o f t h e s e d e f i n i t e e n e r g y s t a t e s s h o u l d r e s u l t i n a s h a r p l i n e i n t h e d e t e c t e d s p e c t r u m . Y e t e x p e r i -m e n t a l l y , t h e s e l i n e s have a f i n i t e w i d t h . What f a c t o r s a r e i n v o l v e d i n c r e a t i n g t h e o b s e r v e d shape? S u c c e e d i n g p a r a -g r a p h s w i l l d i s c u s s the v a r i o u s c o n t r i b u t i o n s , i n t u r n . 5 source gas system detector y=0 y=L A z A F i g u r e 1: A b l o c k d i a g r a m o f the i d e a l i z e d e x p e r i m e n t . The most f u n d a m e n t a l ( u n i v e r s a l ) c o n t r i b u t i o n t o an 1 2 o b s e r v e d l i n e shape i s the n a t u r a l l i n e shape ' w h i c h r e s u l t s from an i n t e r r u p t i o n o f t h e r a d i a t i o n p r o c e s s by a t o t a l l y quantum m e c h a n i c a l phenomena — s p o n t a n e o u s e m i s s i o n - ( a l t h o u g h c l a s s i c a l e l e c t r o m a g n e t i c t h e o r y i n t e r p r e t s n a t u r a l b r o a d e n i n g i n t e r m s o f r a d i a t i o n d a m p i n g ) . The n a t u r a l l i n e shape c a n be shown to have a L o r e n t z i a n form T , , M 1 I ( w ) - w 72—p ( O J - W ) + -j-o 4 w i t h t h e h a l f w i d t h y c o r r e c t l y i d e n t i f i e d a s the t o t a l s p o n t a n e o u s t r a n s i t i o n p r o b a b i l i t y p e r u n i t t i m e . Spontcineous e m i s s i o n i s c a u s e d by t h e i n t e r a c t i o n o f t h e quantum, m e c h a n i c a l s y s t e m w i t h a p h o t o n even when t h e e x p e c t a -t i o n v a l u e o f t h e e l e c t r i c f i e l d i s s t r i c t l y zero,. T h a t t h e r e e x i s t s a n o n - z e r o p r o b a b i l i t y o f h a v i n g a p h o t o n p r e s e n t when t h e r e i s no e l e c t r i c f i e l d i s a quantum m e c h a n i c a l a s p e c t o f l i g h t - namely t h a t E, t h e o p e r a t o r f o r t h e e l e c t r i c f i e l d , and n, t h e o p e r a t o r f o r t h e number o f p h o t o n s , do n o t commute [E,n] = E and hence obey an u n c e r t a i n t y r e l a t i o n o f t h e form AE&n £ j EI . . The s p o n t a n e o u s t r a n s i t i o n p r o b a b i l i t y p e r u n i t t i m e , and t h u s the n a t u r a l l i n e w i d t h y , v a r i e s as the f r e q u e n c y c u b e d , w 3. I n t h e m i c r o w a v e r e g i o n o f t h e s p e c t r u m , t h i s . o c o n t r i b u t i o n i s many o r d e r s o f m a g n i t u d e s m a l l e r t h a n o t h e r b r o a d e n i n g mechanisms ( t o be d i s c u s s e d below) and hence c a n be s a f e l y n e g l e c t e d . However, s i n c e t h e n a t u r a l w i d t h i n c r e a s e s m a r k e d l y w i t h f r e q u e n c y , a n e g l e c t o f t h e s p o n t a n e o u s e m i s s i o n mechanism i n o p t i c a l r e g i o n s o f t h e s p e c t r u m c a n n o t be so e a s i l y r a t i o n a l i z e d . From t h e s e c o n s i d e r a t i o n s , i t i s f e l t t h a t a c l a s s i c a l d e s c r i p t i o n o f t h e r a d i a t i o n f i e l d i s s u f f i c i e n t i n r e g i o n s where t h e n a t u r a l l i n e shape c a n be n e g l e c t e d , b u t a QM t r e a t m e n t o f l i g h t i s m a n d a t o r y when-e v e r s p o n t a n e o u s e m i s s i o n i s a s i g n i f i c a n t b r o a d e n i n g mechanism. The r e a l i z a t i o n t h a t t h e i n d i v i d u a l m o l e c u l e s i n a gas a r e n o t i s o l a t e d b u t i n c o n s t a n t , m u t u a l i n t e r a c t i o n b r i n g s one t o t h e much d i s c u s s e d phenomena o f p r e s s u r e b r o a d e n i n g . The t e r m " p r e s s u r e b r o a d e n i n g " i s d e r i v e d from t h e u s u a l e x p e r i m e n t a l o b s e r v a t i o n t h a t an i n c r e a s e i n t h e p r e s s u r e Ln t h e gas c e l l ( i . e , an i n c r e a s e i n t h e number o f i n t e r a c t i n g m o l e c u l e s ) l e a d s t o an i n c r e a s e d b r o a d e n i n g o f t h e l i n e u n d e r s t u d y . V a r i o u s q u a l i t a t i v e a s p e c t s o f p r e s s u r e b r o a d e n -i n g - d e f i n i t i o n s , n a t u r a l d i v e r s i o n s and s u b - c l a s s i f i c a t i o n s -a r e o u t l i n e d i n t h e n e x t few p a r a g r a p h s . The more d e t a i l e d m a t h e m a t i c a l d e s c r i p t i o n s b e g i n i n c h a p t e r I I . ( c ) Aspects of Pressure Broadening There are two l i m i t i n g cases of pressure broadening -termed the s t a t i s t i c a l ' ^ ' ^ and i m p a c t 5 ' ^ l i m i t s . In the former, the p e r t u r b e r s are assumed to move past the r a d i a t i n g molecules i n f i n i t e l y slowly.. T h i s c r e a t e s a s t a t i c but random p e r t u r b a -t i o n i n the energy l e v e l s of the r a d i a t i n g molecule. The impact theory, on the other hand, c o n s i d e r s sharp, i m p u l s i v e c o l l i s i o n s which c r e a t e an incoherence i n the emitted r a d i a t i o n . The r e g i o n s of v a l i d i t y of each of these approaches can be s t a t e d i n terms of three r e l e v a n t parameters t w-to^, the d e v i a t i o n from l i n e c e n t r e , T • the d u r a t i o n of a c o l l i s i o n , • c and the mean time between c o l l i s i o n s . Then, as d i s c u s s e d 7 8 by S p i t z e r and H o l s t e i r i , i n the b i n a r y c o l l i s i o n regime (X < 'C _) the impact theory is. v a l i d near the l i n e c e n t r e c i [ ( O J - W ) T < 1] while the s t a t i s t i c a l theory i s v a l i d i n the o c 9 wings [ (OJ-w )T > 1] . ' As the pr e s s u r e i s i n c r e a s e d p a s t the o c d i l u t e qas reaion, (T > T„) , the concept of an i s o l a t e d c a: c o l l i s i o n becomes meaningless and ci s t a t i s t i c a l theory must ap p l y . A u n i f i e d treatment of the two t h e o r i e s has been presented by Anderson,^ 0 and l a t e r by F u t r e l l e . 3 2 Margenau and Lewis" have reviewed the s t r u c t u r e of plasma s p e c t r a l l i n e s s t a r t i n g with the c l a s s i c work of 13 Holtsmark. Here, i n a d d i t i o n to the presence of n e u t r a l p e r t u r b e r s , c o n s i d e r a t i o n must be g i v e n to the e f f e c t s of ions and e l e c t r o n s . For the f a s t moving e l e c t r o n e f f e c t s , a form of the impact theory i s u s u a l l y s u i t a b l e , while f o r the he a v i e r (and hence slower moving) i o n s , a s t a t i s t i c a l theory i s deemed necessary. The e f f e c t s of charged p e r t u r b e r s i s known g e n e r a l l y as S t a r k b r o a d e n i n g . The i m p a c t t h e o r y o f b r o a d e n i n g by n e u t r a l m o l e c u l e s i s now d i s c u s s e d i n g r e a t e r d e t a i l . H e r e , a L o r e n t z i a n l i n e s hape i s g e n e r a l l y e x p e c t e d f o r an i s o l a t e d l i n e , w i t h a 1 14 w i d t h rjr~.. The p i o n e e r i n g work o f A n d e r s o n u s h e r e d xn t h e 1 2 modern e r a o f i m p a c t b r o a d e n i n g t r e a t m e n t s i n t h a t t h e r e l a -t i o n s h i p between t h e i n t e r m o l e c u l a r f o r c e s and c o l l i s i o n s was e x p l i c i t l y d i s p l a y e d . A n d e r s o n ' s t h e o r y , i n one s e n s e , r e p r e s e n t s a g e n e r a l i z a t i o n o f t h e work o f F o l e y ^ t o i n c l u d e i n e l a s t i c ( d i a b a t i c ) c o l l i s i o n a l e f f e c t s . T h a t i s , ™ — = ~ x 2 1 1 ™— e l a s t i c + i n e l a s t i c . T h i s g e n e r a l i z a t i o n a l l o w e d l 2 1 2 A n d e r s o n ' s t r e a t m e n t t o be a p p l i c a b l e t o l i n e s i n the m i c r o -wave r e g i o n where i n e l a s t i c e f f e c t s a r e e x p e c t e d t o o c c u r q u i t e r e a d i l y „ To see t h i s , c o n s i d e r t h e q u a n t i t y ^J°_ where kT "(0 " i s t h e f r e q u e n c y d i f f e r e n c e between th e two l e v e l s u n d e r c o n s i d e r a t i o n and "kT" r e p r e s e n t s an a v e r a g e a v a i l a b l e k i n e t i c e n e r g y . I n t h e o p t i c a l r e g i o n a t n o r m a l t e m p e r a t u r e s "Pr— >> 1 and t h e r e i s n o t s u f f i c i e n t t r a n s l a t i o n a l energy a v a i l a b l e on t h e a v e r a g e t o cause, i n e l a s t i c " o p t i c a l " t r a n s -- i t i o n s . Thus e l a s t i c : ( a l s o c a l l e d p h a ^ e - s h i f t i n g o r a d i a b a t i c ) c o l l i s i o n s dominate,, y^™ =•• e l a s t i c . I n t h e m i c r o w a v e r e g i o n , however, "^ V° << 1 and by s i m i l a r r e a s o n i n g , i n e l a s t i c e f f e c t s must be c o n s i d e r e d . . I n h i s t r e a t m e n t , A n d e r s o n - e m p l o y e d a s e m i - c l a s s i c a l d e s c r i p t i o n w h e r e i n t h e t r a n s l a t i o n a l d e g r e e s o f f r e e d o m a r e t r e a t e d c l a s s i c a l l y . He f u r t h e r assumed t h a t t h i s t r a n s l a t i o n a l m o t i o n c o u l d be a p p r o x i m a t e d by s t r a i g h t l i n e t r a j e c t o r i e s - i n d i c a t i n g t h a t d i s t a n t c o l l i s i o n s ( l a r g e i m p a c t p a r a m e t e r s ) s h o u l d 10 be d o m i n a n t . A n d e r s o n ' s t r e a t m e n t i s d i s c u s s e d i n d e t a i l (and some f u r t h e r s i m p l i f i c a t i o n s p o i n t e d o u t ) i n a p a p e r by 15 Ts a o and C u r n u t t e . I n summary, t h e n , A n d e r s o n ' s work on i s o l a t e d l i n e s g i v e s a L o r e n t z i a n l i n e shape w i t h an e x p r e s -s i o n f o r t h e w i d t h i n t e r m s o f d e t a i l e d ( e l a s t i c and 2 i n e l a s t i c ) c o l l i s i o n a l e f f e c t s and t h e i n t e r m o l e c u l a r p o t e n t i a l . The e a s e w i t h w h i c h i n e l a s t i c c o l l i s i o n s o c c u r i n t h e m i c r o w a v e r e g i o n a l s o n e c e s s i t a t e s c o n s i d e r a t i o n o f c o l l i s i o n -a l c o u p l i n g s between v a r i o u s l i n e s . I n d e e d , t h e o v e r l a p p i n g  o f s p e c t r a l l i n e s i n t h e m i c r o w a v e p o r t i o n o f t h e s p e c t r u m i s a l m o s t t h e r u l e r a t h e r t h a n t h e e x c e p t i o n . K o l b and G r i e m 1 6 17 and B a r a n g e r , i n a s e r i e s o f p a p e r s , i n d e p e n d e n t l y e x t e n d e d A n d e r s o n ' s work t o i n c l u d e o v e r l a p p i n g l i n e s . I n e s s e n c e , t h e one r e l a x a t i o n r a t e (•jjj—) d e s c r i p t i o n o f an i s o l a t e d l i n e i s 2 1 r e p l a c e d by a m a t r i x o f r e l a x a t i o n r a t e s (——) - t h e o f f z2 d i a g o n a l e l e m e n t s o f w h i c h a r e u s u a l l y d e s i g n a t e d a s f r e q u e n c y c o u p l i n g t e r m s . I t was B a r a n g e r , w i t h h i s d e f i n i -t i o n o f " l i n e s p a c e " ( l a t e r t e r m e d L i o u v i l l e s p a c e ) who p r o b a b l y f i r s t r e a l i z e d t h a t t h e p r o b l e m o f p r e s s u r e b r o a d e n i n g was n a t u r a l l y a q u e s t i o n o f o p e r a t e r s ( o b s e r v a b l e s) and how t h e y e v o l v e and r e l a x . T h i s b a s i c c hange i n a t t i t u d e i s 18 most e l e g a n t l y p r e s e n t e d i n a p a p e r by F a n o and i t i s t h i s a p p r o a c h w h i c h I f e e l i s n e c e s s a r y f o r t h e c l e a r e s t u n d e r -s t a n d i n g o f t h e v a r i e t y o f e f f e c t s i n v o l v e d . N e e d l e s s t o s a y , t h e " o p e r a t o r " p o i n t o f v i e w w i l l be e m p h a s i z e d t h r o u g h -o u t t h i s t h e s i s . The e f f e c t s on l i n e s h a p e s o f d e g e n e r a t e m a g n e t i c s t a t e s ( r o t a t i o n a l i n v a r i a n c e ) and v a r i o u s i n v a r i a n c e p r o p e r t i e s o f -11 c o l l i s i o n p r o c e s s e s have been d i s c u s s e d i n a c o m p r e h e n s i v e 19 f a s h i o n by Ben-Reuven . He t h e n a p p l i e d t h e s e i d e a s to a t r e a t m e n t o f o v e r l a p p i n g l i n e s , and t h e t r a n s i t i o n f r o m r e s o n e n t t o n o n - r e s o n e n t (Debye; z e r o f r e q u e n c y ) l i n e s h a p e s 2 0 i n m i c r o w a v e s p e c t r a . By t h e s e c o n s i d e r a t i o n s , he d e r i v e d d e t a i l e d c o l l i s i o n a l s e q u e n c e s f o r the p h e n o m e n o l o g i c a 1 21 p a r a m e t e r s i n t h e Van V l e c k - W e i s k o p f l i n e shape and showed how i t c o u l d be g e n e r a l i z e d . I n c o n n e c t i o n w i t h Debye r e l a x a t i o n , some f u r t h e r comments 2 2 seem a p p r o p r i a t e . B i r n b a u m t r e a t e d t h e p r o b l e m o f non-r e s o n a n t a b s o r p t i o n s t a r t i n g f r o m a quantum m e c h a n i c a l k i n e t i c e q u a t i o n a p p r o a c h . However, t h e k i n e t i c e q u a t i o n c h o s e n by 23 B i r n b a u m (Wang-Chang U h l e n b e c k e q u a t i o n ) i s i n a p p r o p r i a t e i n t h a t i t does n o t t r e a t t h e d e g e n e r a t e m a g n e t i c s t a t e s p r o p e r l y - y e t n o n - r e s o n a n t a b s o r p t i o n i s f u n d s i n e n t a l l y c o n -c e r n e d w i t h j u s t t h e s e s t a t e s , . An a l t e r n a t e t r e a t m e n t has 2 4 been g i v e n by T i p and H c C o u r t , w h i c h i s b a s e d on t h e a p p r o p r i a t e e q u a t i o n f o r t h e s e " z e r o f r e q u e n c y " e f f e c t s -2 5 2 G th e Waldraann-Snider e q u a t i o n . ' T h i s t h e s i s g e n e r a l i z e s the work o f T i p and M c C o u r t t o " n o n - z e r o f r e q u e n c y " e f f e c t s ( r e s o n a n t a b s o r p t i o n ) . As d i s c u s s e d i n c h a p t e r IV, t h e a p p r o p r i a t e k i n e t i c e q u a t i o n has t h e Waldmann-Snider f o r m , 12 (d) S a t u r a t i o n E f f e c t s The d e s c r i p t i o n s o u t l i n e d a b o v e have i m p l i c i t l y assumed low power i n p u t by t h e s o u r c e . An a d d i t i o n a l b r o a d e n i n g e f f e c t , termed s a t u r a t i o n (or power) b r o a d e n i n g , o c c u r s when t h e s o u r c e o f r a d i a t i o n i s e e r y s t r o n g . T h i s has been 27 t r e a t e d by Townes and Schawlow and, u s i n g a d e n s i t y m a t r i x • 28 p o i n t o f v i e w , by K a r p l u s and S c h w i n g e r . S a t u r a t i o n w i l l be d e s c r i b e d i n some d e t a i l i n t h e n e x t c h a p t e r . The p r e s e n t d i s c u s s i o n i s l i m i t e d t o a few q u a l i t a t i v e r e m a r k s . B a s i c a l l y , f i r s t o r d e r e f f e c t s i n t h e s o u r c e - s y s t e m c o u p l i n g ]£*E_0 c r e a t e f o r c e d o s c i l l a t i o n s between t h e two l e v e l s r e s o n a n t w i t h t h e r a d i a t i o n , b u t do n o t change t h e e q u i l i b r i u m p o p u l a t i o n s o f t h e s e two s t a t e s ( i . e . no s a t u r a t i o n ) . I t i s t h i s s a t u r a t i o n t h a t has b e e n d i s c u s s e d i n t h e p r e c e e d i n g s u b s e c t i o n where i t has been p o i n t e d o u t t h a t t h i s f o r c e d o s c i l l a t i o n d e c a y s a t a r a t e — — ( a s s u m i n g an i s o l a t e d l i n e ) . 2 However, a t h i g h e r p o w ers, s e c o n d o r d e r t e r m s i n t h e c o u p l i n g i i ' E ^ became s i g n i f i c a n t as w e l l , and t h e s e t e r m s a r e r e s p o n -s i b l e f o r c r e a t i n g a n o n - e q u i l i b r i u m p o p u l a t i o n d i f f e r e n c e w h i c h c a n r e l a x t o i t s e q u i l i b r i u m v a l u e a t a r e l a x a t i o n r a t e — - . The r e c o g n i t i o n t h a t ( a t l e a s t ) two d i s t i n c t r e l a x a -t i o n p r o c e s s e s a r e o c c u r i n g h e r e , means t h a t t h e two r e l a x a -t i o n t i m e s a r e n o t n e c e s s a r i l y e q u a l . I n d e e d , one r e c o g n i z e s t h a t t h e r e l a x a t i o n o f p o p u l a t i o n d i f f e r e n c e s g o v e r n e d by ^ — , c a n o n l y o c c u r t h r o u g h i n e l a s t i c c o l l i s i o n s w h i l e ^ — 1 2 p r o c e s s e s i n c l u d e c o n t r i b u t i o n s f r o m b o t h e l a s t i c and i n -e l a s t i c e f f e c t s . From t h e s e a r g u m e n t s , i t i s e x p e c t e d t h a t 13 —— > ——, d e p e n d i n g on how e f f e c t i v e t h e e l a s t i c c o l l i s i o n 2 1 1 c o n t r i b u t i o n i s to — — . I f , as has been s u g g e s t e d , t h e i n -T 2 e l a s t i c c o l l i s i o n a l e f f e c t s d o m i n a t e r e l a x a t i o n p r o c e s s e s i n t h e m i c r o w a v e r e g i o n , t h e n *v< . T h i s i s known as t h e a 2 T l s t r o n g c o l l i s i o n m o d e l . The q u a l i t a t i v e d i s c u s s i o n j u s t g i v e n w i l l be p u t i n q u a n t i t a t i v e t e r m s i n c h a p t e r I I and d e v e l o p e d f u r t h e r i n l a t e r p a r t s o f t h e t h e s i s . I n f a c t , c o n s i d e r a b l e e m p h a s i s w i l l be p l a c e d on t h i s a s p e c t o f p r e s s u r e b r o a d e n i n g . B e f o r e l e a v i n g t h e g e n e r a l a r e a o f p r e s s u r e b r o a d e n i n g , i t i s f e l t t h a t some s e l e c t e d c o n t r i b u t i o n s o f R.G. G ordon 29 s h o u l d be m e n t i o n e d . He has d e v e l o p e d a t r e a t m e n t o f l i n e w i d t h s and s h i f t s t h a t d o e s n o t make t h e a p p r o x i m a t i o n s o f p e r t u r b a t i o n t h e o r y and s t r a i g h t l i n e t r a j e c t o r i e s , i n c o n -t r a s t t o A n d e r s o n . B e c a u s e G o r d o n ' s a p p r o a c h h e r e i s o f a more c l a s s i c a l n a t u r e , i t i s e s p e c i a l l y u s e f u l i n " p i c t u r i n g " v a r i o u s c o l l i s i o n a l e f f e c t s . Some comments on how p o p u l a t i o n s r e l a x have been g i v e n i n a n o t h e r p a p e r ^ " by G o r d o n . A d d i t i o n a l 31 32 a p p r o a c h e s i n v o l v i n g s e m i c l a s s i c a l and c o m p l e t e l y quantum c a l c u l a t i o n s o f l i n e s h a p e s , and o t h e r r e l a x a t i o n phenomena, have a l s o b e en d e s c r i b e d . However, i t i s G o r d o n ' s e f f o r t s t o c o r r e l a t e w i d e l y v a r y i n g g a s p h a s e c o l l i s i o n phenomena 3 3 3 4 t h a t has been most a p p r e c i a t e d . Two a r t i c l e s ' i n p a r t i c u l a r i l l u s t r a t e t h i s p h i l o s o p h y . (The work p r e s e n t e d i n t h i s t h e s i s has a l s o been m o t i v a t e d by t h e same u n d e r l y i n g i d e a . Namely, t h a t a f u l l e r u n d e r s t a n d i n g o f i n t e r m o l e c u l a r f o r c e s c a n be g a i n e d by a c o m p a r i s o n o f r e s u l t s w i t h i n a u n i f i e d framework.) 14 The r e c e n t r e v i e w a r t i c l e by H. R a b i t z ^ " * p r o v i d e s f u r t h e r r e f e r e n c e s to more d e t a i l e d a s p e c t s o f p r e s s u r e b r o a d e n i n g o f s p e c t r a l l i n e s i n t h e m i c r o w a v e r e g i o n . 15 (e) D o p p l e r B r o a d e n i n g 27 The D o p p l e r c o n t r i b u t i o n to t h e l i n e shape i s t h e r e s u l t o f t h e t r a n s l a t i o n a l r e a c t i o n o f t h e m o l e c u l e s . I n d e e d , i f a s t a t i o n a r y m o l e c u l e e m i t s l i g h t o f f r e q u e n c y OJ , t h e n t h i s o m o l e c u l e , when moving w i t h v e l o c i t y v, w i l l e m i t l i g h t whose f r e q u e n c y i s s h i f t e d by an amount k « v . S i n c e a v e l o c i t y v o c c u r s w i t h a p r o b a b i l i t y p r o p o r t i o n a l t o exp[-mv2/2k T ] , t h e r e s u l t a n t D o p p l e r l i n e shape c a n be r e g a r d e d a s an i n f i n i t e sum ( i . e . i n t e g r a l ) o v e r t h e v a r i o u s v e l o c i t y com-p o n e n t s , w e i g h t e d w i t h a B o l t z m a n n d i s t r i b u t i o n : - namely X D < » ) - - / / /dv(^S- 7 ) 3 / 2 ox P[-gL] « ( k . v - A c ) B B , mc t l / 2 mc 2 ,Aw 2 . ( > e x P [ - — (—) ] . 2TTk TOO B o B o C o n s e q u e n t l y t h e r e s u l t a n t p r o f i l e i s G a u s s i a n . I t t h e n f o l l o w s t h a t t h e h a l f w i d t h a t h a l f maximum i s 2kT 1/2 W o AOJ = ( I n 2) — . From t h e e a r l i e r d i s c u s s i o n o f D m c p r e s s u r e b r o a d e n i n g , e a c h v e l o c i t y component s h o u l d p o s s e s s a f i n i t e w i d t h o f i t s own, due t o m o l e c u l e - m o l e c u l e i n t e r -a c t i o n s a f f e c t i n g t h e i n t e r v a l s t a t e s . The o b s e r v e d l i n e shape s h o u l d be due t o a c o m p o s i t e o f p r e s s u r e b r o a d e n i n g and D o p p l e r e f f e c t s . I f t h e e f f e c t s were i n d e p e n d e n t , t h e n t h e l i n e shape would be t h e c o n v o l u t i o n o f t h e D o p p l e r p r o f i l e I ^ ( O J ) and t h e L o r e n t z i a n p r o f i l e I (u) , namely D i-i 1(d)) = /doj' I ( O J ' ) I ( O J - O J ' ) . T h i s r e s u l t i n g l i n e s h a p e i s D L 3 6 known a s a V o i g t p r o f i l e and i s a s u f f i c i e n t d e s c r i p t i o n f o r many q u a l i t a t i v e f e a t u r e s o f t h e l i n e s h a p e s . However, 16 i t d o e s n o t r e p r e s e n t t h e whole s t o r y . The e f f e c t s o f v e l o c i t y -37 c h a n g i n g c o l l i s i o n s ( D i c k i e n a r r o w i n g ) a r e n o t i n c l u d e d -t h i s a s p e c t i s t r e a t e d i n d e t a i l i n c h a p t e r V I I . F o r t h e p u r -p o s e o f t h e p r e s e n t d i s c u s s i o n , however, t h e D o p p l e r h a l f w i d t h Acdp i s s m a l l i n t h e m i c r o w a v e r e g i o n o f t h e s p e c t r u m , and t h u s ( e x c e p t f o r q u i t e low p r e s s u r e s i n t h e gas c e l l where p r e s s u r e b r o a d e n i n g i s i n s i g n i f i c a n t ) t h e D o p p l e r e f f e c t on l i n e s h a p e s c a n be n e g l e c t e d . 17 ( f ) R e l a t e d A r e a s o f S p e c t r o s c o p y A l t h o u g h t h i s t h e s i s i s c o n c e r n e d w i t h l i n e s h a p e s and r e l a t e d phenomena i n t h e m i c r o w a v e r e g i o n o f t h e s p e c t r u m , a g r e a t d e a l o f a d d i t i o n a l u n d e r s t a n d i n g c a n be o b t a i n e d by • c o m p a r i n g r e l a t i v e m a g n i t u d e s o f t h e e f f e c t s and v a r i o u s a p p r o a c h e s t o t h e g e n e r a l p r o b l e m i n t h e p a r t s o f t h e e l e c t r o -m a g n e t i c s p e c t r u m w h i c h s u r r o u n d t h e microwave r e g i o n . To t h i s end, some comments on a n a l o g o u s gas phase phenomena i n t h e r a d i o f r e q u e n c y and optice.1 f r e q u e n c y r e g i o n s a r e now p r e s e n t e d . D e s p i t e i t s r e l a t i v e l y l a t e d e v e l o p m e n t , n u c l e a r m a g n e t i c 3 8 r e s o n a n c e now r a n k s as p r o b a b l y the most h i g h l y s t u d i e d , most h i g h l y d e v e l o p e d form o f s p e c t r o s c o p y , A s e t o f t h r e e c o u p l e d e v o l u t i o n - r e l a x a t i o n e q u a t i o n s f o r a s p i n 1/2 cystem ( e i t h e r i n t h e f o r m o f B l o c h o r R e d f i e l d e q u a t i o n s ) i s an i d e a l way t o c o n c i s e l y d e s c r i b e t h e l i n e s h a p e , s a t u r a t i o n , and r e l a t e d p h e n o m e n a « ( I n f a c t , , one would be w e l l on t h e way t o u n d e r s t a n d i n g t h e c o r r e s p o n d i n g e f f e c t s i n t h e m i c r o w a v e r e g i o n i f an a n a l o g o u s s e t o f c o u p l e d e q u a t i o n s c o u l d be d e r i v e d and t h e i r l i m i t s o f v a l i d i t y e s t a b l i s h e d , . ) I n t h e B l o c h e q u a t i o n s , t h e T_ r e l a x a t i o n o f M and M i s r e l a t e d 2. + — t o t h e r e l a x a t i o n o f t h e c o h e r e n c e s !"><—™| and | ' ~ ^ > < " j l > w h i l e t h e T^ r e l a x a t i o n o f M i s p r o p o r t i o n a 1 t o t h e r e l a x a -t i o n o f a p o p u l a t i o n d i f f e r e n c e . Thus t h e T , , T 2 n o m e n c l a t u r e employed i n e a r l i e r p a r a g r a p h s i n t h e d i s c u s s i o n o f m i c r o w a v e r e l a x a t i o n s i s s e e n t o be c o n s i s t e n t w i t h the NMR d e f i n i t i o n s . F u r t h e r , p h e n o m e n o l o g i c a l l i n e shape e x p r e s s i o n s i n v o l v i n g and a r e o f the same form*"^ a s t h e m i c r o w a v e case. I t s h o u l d be s t r e s s e d , however, t h a t t h e a c t u a l mechanism f o r r e l a x a t i o n i n gas ph a s e NMR c a n d i f f e r g r e a t l y f r o m t h e m i c r o wave s i t u a t i o n . I n d e e d , as d i s c u s s e d above, c o l l i s i o n s a r e d i r e c t l y r e s p o n s i b l e f o r r e l a x a t i o n i n t h e m i c r o w a v e r e g i o n . T h i s i s a l s o t r u e f o r n u c l e a r s p i n r e l a x a t i o n i n monatomic g a s e s . H e r e r e l a x a t i o n t i m e s a r e e x t r e m e l y l o n g b e c a u s e d i r e c t c o l l i s i o n a l e f f e c t s on n u c l e a r s p i n s t a t e s a r e v e r y weak. I n p o l y a t o m i c g a s e s , t h i s same "weakness o f c o l l i s i o n s means t h a t o t h e r r e l a x a t i o n mechanisms a r e p r e f e r r e d , when a v a i l a b l e . T h u s , f o r example, n u c l e a r s t a t e s i n p o l y a t o m i c s a r e r e l a x e d p r e d o m i n a n t l y by i n t r a m o l e c u l a r c o u p l i n g ( s p i n -r o t a t i o n ) t o r o t a t i o n a l s t a t e s w h i c h a r e i n t u r n r e l a x e d by c o l l i s i o n s . E v e n when t h i s mechanism d o m i n a t e s , however, t h e r e l a x a t i o n t i m e s a r e f o u n d t o be much l a r g e r t h a n m i c r o w a v e r e l a x a t i o n t i m e s . A k i n e t i c t h e o r y a p p r o a c h t o b o t h t h e m o n a t o m i c ^ " 3 and p o l y a t o m i c ^ * 5 c a s e s have been d i s c u s s e d by Chen and S n i d e r . D o p p l e r e f f e c t s p l a y a n e g l i g i b l e r o l e i n NMR s p e c t r a l l i n e s h a p e s s i n c e t h e w a v e l e n g t h s i n v o l v e d a r e much l a r g e r t h a n t h e l e n g t h o f t h e c e l l and hence no s p a t i a l i n h o m o g e n i e -t i e s r e s u l t i n g f r o m t h e r . f . f i e l d o c c u r . ( I n f a c t , any s p a t i a l d e p e n d e n c e i s more r e a d i l y a t t r i b u t a b l e t o f i e l d 41 i n h o m o g e n i e t i e s i n t h e magnet and t o d i f f u s i o n e f f e c t s . F i n a l l y , i t s h o u l d be p o i n t e d o u t t h a t i n NMR no m e n t i o n i s e v e r made o f n a t u r a l l i n e shape e f f e c t s and t h a t a c l a s s i c a l d e s c r i p t i o n o f t h e r a d i a t i o n f i e l d i s a l w a y s e m p l o y e d . 19 The c o h e r e n t s o u r c e means t h a t m a c r o s c o p i c o s c i l l a t i o n s a r e i n d u c e d i n t h e s y s t e m w h i l e t h e c o h e r e n t d e t e c t o r means t h a t b o t h i n - p h a s e and o u t - o f - p h a s e components o f t h e s e o s c i l l a t i o n s c a n be d e t e r m i n e d . I t i s t h i s c o h e r e n t n a t u r e o f t h e f i e l d -s y s t e m i n t e r a c t i o n t h a t a l l o w s t i m e - d e p e n d e n t ( F o u r i e r t r a n s -4 2 form) s t u d i e s to be u n d e r t a k e n . F o u r i e r t r a n s f o r m t e c h n i q u e s g i v e t h e same i n f o r m a t i o n a s t h e s t e a d y s t a t e e x p e r i m e n t s b u t 43 have t h e marked a d v a n t a g e o f i m p r o v e d s i g n a l t o n o i s e r a t i o . An a d d i t i o n a l t e c h n i c a l a d v a n t a g e o f d e t e c t i o n i n t h e NMR r e g i o n i s t h a t one i s d e a l i n g w i t h low f r e q u e n c y and l o n g l a s t i n g s i g n a l s , so t h a t many c o h e r e n t d a t a p o i n t s c a n e a s i l y be m e a s u r e d . T h i s e x p l a i n s why c o h e r e n t methods have been p i o n e e r e d i n t h e NMR r e g i o n . S p e c t r o s c o p y i n t h e o p t i c a l r e g i o n o f t h e s p e c t r u m s h o u l d be compared and c o n t r a s t e d w i t h t h e r a d i o f r e q u e n c y (NMR) and m i c r o w a v e r e s u l t s . F o r an i s o l a t e d l i n e i n t h i s r e g i o n o f t h e s p e c t r u m , D o p p l e r b r o a d e n i n g i s t h e d o m i n a n t e f f e c t (remember A O J D ^ W q ) . The p r e s s u r e b r o a d e n i n g w i d t h s a r e much s m a l l e r t h a n i n t h e m i c r o w a v e r e g i o n s i n c e o n l y e l a s t i c c o l l i s i o n a l e f f e c t s s h o u l d c o n t r i b u t e t o ^ — . (Thus t h e 2 V o i g t p r o f i l e has been a t r a d i t i o n a l d e s c r i p t i o n ! ) A l s o , t h e 3 n a t u r a l l i n e w i d t h c o n t r i b u t i o n ( Y ^ w ) has i n c r e a s e d t o o t h e p o i n t where i t c a n become c o m p a r a b l e w i t h p r e s s u r e b r o a d e n -i n g e f f e c t s , a t l e a s t a t l o w e r p r e s s u r e s . T h e s e c o n s i d e r a -t i o n s , p l u s the a d v e n t o f i n t e n s e , m o n o c h r o m a t i c c o h e r e n t ( l a s e r ) l i g h t s o u r c e s have c o n t r i b u t e d t o a new f o r m o f h i g h r e s o l u t i o n s t e a d y s t a t e s p e c t r o s c o p y c a l l e d L amb-dip s p e c t r o s -44 c o p y . H e r e , o n l y a p a r t i c u l a r v e l o c i t y g r o u p i s r e s o n a n t 20 (li) = di -f k'v) w i t h t h e e x t r e m e l y ' m o n o c h r o m a t i c l a s e r s o u r c e o. — — and a narrow l i n e i s o b s e r v e d . I n t h i s way, t h e D o p p l e r s p r e a d has been e l i m i n a t e d and o n l y the homogeneous l i n e b r o a d e n i n g e f f e c t s ( n a t u r a l a n d / o r p r e s s u r e w i d t h s ) r e m a i n . The e x i s t e n c e o f i n t e n s e c o h e r e n t l a s e r l i g h t s o u r c e s has opened up t h e p o s s i b i l i t y o f time d e p e n d e n t ( F o u r i e r A t r a n s f o r m ) s p e c t r o s c o p y i n t h e o p t i c a l r e g i o n o f t h e s p e c t r u m where a n a l o g u e s o f NMR n u t a t i o n and f r e e i n d u c t i o n d e c a y e x p e r i m e n t s a r e c a r r i e d o u t . (The g e n e r a l term c o h e r e n c e t r a n s i e n t s w i l l be u s e d t o d e n o t e t h e s e e f f e c t s . ) H e r e , a S t a r k s w i t c h i n g method o f c r e a t i n g p u l s e s i s e m p l o y e d . A c o n t i n u o u s wave l a s e r l i g h t s o u r c e i s e f f e c t i v e l y s w i t c h e d i n and o u t o f r e s o n a n c e w i t h t h e m o l e c u l a r l e v e l s by t h e b r i e f a p p l i c a t i o n , o f a S t a r k f i e l d w h i c h c a u s e s t h e e n e r g y s e p a r a -t i o n between m o l e c u l a r l e v e l s t o change m o m e n t a r i l y . The method o f c o h e r e n t d e t e c t i o n e m p l o y s , by n e c e s s i t y , a h e t e r o d y n e t e c h n i q u e so t h a t i t i s t h e d i f f e r e n c e f r e q u e n c y w h i c h i s a c t u a l l y p i c k e d up„ The b a s i c t h e o r y f o r t h e s e 4 6 o p t i c a l e f f e c t s d a t e s b a c k t o t h e work o f Feynman and V e r n o n who were t h e f i r s t t o r e c o g n i z e t h a t e l e c t r i c d i p o l e i n t e r -a c t i o n s w i t h a two s t a t e s y s t e m c o u l d be e x p r e s s e d i n t h e same form (i«e„ t h r e e c o u p l e d e q u a t i o n s ) as NMR i n t e r a c t i o n g b I n d e e d , w i t h t h e a d d i t i o n o f r e l a x a t i o n t i m e s , t h i s c o u p l e d s e t o f e q u a t i o n s l o o k s l i k e t h e B l o c h e q u a t i o n s - hence t h e o b v i o u s NMR a n a l o g i e s . S e v e r a l a t t e m p t s a t o b t a i n i n g quantum m e c h a n i c a l B o l t z m a n n e q u a t i o n s , v a l i d f o r t h e o p t i c a l r e g i o n , s h o u l d be 21 m e n t i o n e d . I n d e e d , t h e s e p r o v i d e a m o l e c u l a r b a s i s f o r t h e above m e n t i o n e d " o p t i c a l B l o c h e q u a t i o n s . " One a p p r o a c h i s 47 r e p r e s e n t e d by the work o f Berman and Lamb who a l s o r e c o g n i z e , a t l e a s t p h e n o m e n o l o g i c a l l y , t h e e x i s t e n c e o f s p o n t a n e o u s e m i s s i o n e f f e c t s ( n a t u r a l l i n e w i d t h s ) i n t h e i r e q u a t i o n s . A s e c o n d method has been p r o d u c e d by t h e J I L A 48 g r o u p a t B o u l d e r , C o l o r a d o . More d e t a i l e d c o m p a r i s o n s between t h e s e a p p r o a c h e s and t h e p o i n t o f v i e w t a k e n i n t h i s t h e s i s w i l l be l e f t t o l a t e r c h a p t e r s . I n f a c t , many o f t h e r e m a r k s made i n t h i s t h e s i s , a l t h o u g h s t r i c t l y i n t e n d e d f o r a p p l i c a t i o n i n t h e m i c r o w a v e r e g i o n , c o u l d a l s o be a p p l i e d t o t h e o p t i c a l r e g i o n . 2 2 (g) Summary C h a p t e r I has d i s c u s s e d q u a l i t a t i v e l y , the v a r i o u s f a c t o r s a f f e c t i n g a m i c r o w a v e l i n e shape and c o n t r a s t e d t h i s s i t u a t i o n w i t h t h o s e f o u n d i n NMR and t h e o p t i c a l r e g i o n s . F u r t h e r , . i t has been p o i n t e d o u t t h a t t h e s e l a t t e r two r e g i o n s c a n a l s o be s t u d i e d by time d e p e n d e n t s p e c t r o s c o p i c t e c h n i q u e s . Thus i t s h o u l d n o t be s u p r i s i n g t h a t t h e microwave r e g i o n e x h i b i t s o b s e r v a b l e t r a n s i e n t phenomena as w e l l . T h e s e e f f e c t s s h a l l be d i s c u s s e d a t g r e a t l e n g t h i n t h e n e x t c h a p t e r , b a s e d on the q u a n t i t a t i v e a p p r o a c h o f F l y g a r e and 49 c o w o r k e r s . ( S t e a d y s t a t e l r n e s h a p e s c a n a l s o be d i s c u s s e d n a t u r a l l y w i t h i n t h i s framework,,) A l t h o u g h v a r i o u s o t h e r g r o u p s have o b s e r v e d m i c r o w a v e t r a n s i e n t behaviour,- t h i s a u t h o r f e e l s t h a t F l y g a r e * s papex s show t h e g r e a t e s t a p p r e c i -a t i o n o f t h e o b s e r v e d e f f e c t s and h i s a p p r o a c h w i l l be f o 1 lowed CHAPTER I I The Two S t a t e System "Would you t e l l me, p l e a s e , w h i c h way I o u g h t t o go from h e r e ? " " T h a t d e p e n d s a good d e a l on where you want to g e t t o , " s a i d the C a t "I d o n ' t much c a r e where -" s a i d A l i c e "Then i t d o e s n ' t m a t t e r w h i c h way you go,"_ s a i d the C a t . 24 (a) I n t r o d u c t i o n T h i s c h a p t e r d e s c r i b e s the s t e a d y s t a t e and t r a n s i e n t r e s p o n s e s o f a s y s t e m o f g a s e o u s m o l e c u l e s to c o h e r e n t r a d i a -t i o n , e m p l o y i n g the b a s i c a p p r o a c h o f F l y g a r e and h i s c o -w o r k e r s . ^ As s u c h , i t i s p r e s e n t e d w i t h i n the c o n t e x t o f a two s t a t e model f o r the i n d i v i d u a l m o l e c u l e s . The term "two s t a t e " , as employed h e r e , i m p l i e s two d i s t i n c t n o n - d e g e n e r a t e e n e r g y e i g e n s t a t e s o f the f r e e one m o l e c u l e i n t e r n a l s t a t e H a m i l t o n i a n . I m p l i c i t i n t h i s model i s the a s s u m p t i o n t h a t the n a t u r a l r e s o n a n c e f r e q u e n c y between t h e s e two s t a t e s i s s u f f i c i e n t l y d i f f e r e n t f r o m any o t h e r n a t u r a l f r e q u e n c y s e p a r a t i o n between o t h e r e n e r g y s t a t e s . T h i s i m p l i e s t h a t the c o h e r e n t r a d i a t i o n c a n i n t e r a c t r e s o n a n t l y w i t h o n l y one p a i r o f s t a t e s a t a t i m e , v / h i l e i t e f f e c t i v e l y i g n o r e s the p r e s e n c e o f the r e m a i n i n g e n e r g y s t a t e s . G e n e r a l i z a t i o n s o f the two s t a t e model c a n r e a d i l y be v i s u a l i z e d i n s e v e r a l d i r e c t i o n s . I n d e e d , the c o n s i d e r a t i o n o f t h e p r e s e n c e o f m a g n e t i c quantum numbers ( r o t a t i o n a l d e g e n e r a c y ) l e a d s to the c o n c e p t o f a two l e v e l model w h i c h i s t r e a t e d i n d e t a i l i n c h a p t e r s IV, V and V I . T h i s g e n e r a l i z a t i o n to c o r r e c t l y i n c l u d e p r o p e r t i e s o f the r o t a -t i o n a l m o t i o n - the d i s t i n c t i o n between the two s t a t e and two l e v e l m o d e l s - i s a m a j o r e m p h a s i s o f t h i s t h e s i s . A f u r t h e r e x t e n s i o n t o d e s c r i b e t r a n s l a t i o n a l m o t i o n s and t h e D o p p l e r e f f e c t i s g i v e n i n c h a p t e r V I I . By r e s t r i c t i n g the p r e s e n t d i s c u s s i o n to t h e two s t a t e c a s e , however, much o f the b a s i c phenomena ( b o t h f r e e m o t i o n and c o l l i s i o n a l a s p e c t s ) a p p e a r i n a r e l a t i v e l y c l e a r and 25 s i m p l e manner. I n f a c t , many o f the q u a l i t a t i v e r e m a r k s o u t -l i n e d i n c h a p t e r I a r e p u t on a q u a n t i t a t i v e b a s i s i n t h i s c h a p t e r . I n terms o f the b l o c k d i a g r a m o f f i g u r e 1, t h e s o u r c e i s a m o n o c h r o m a t i c , c o h e r e n t beam o f mi c r o w a v e r a d i a -t i o n and i s s p e c i f i e d as 2 c o s ( W t - k y ) . T h i s s o u r c e a c t s on the gas sample and d r i v e s i t o u t o f e q u i l i b r i u m . D i f f e r e n t t y p e s o f e x p e r i m e n t s c a n be p e r f o r m e d and a r e d e s c r i b e d i n some d e t a i l . B e c a u s e t h e two s t a t e model i s employed f o r the s y s t e m m o t i o n , t h e r e s p o n s e p o r t r a y e d by the s y s t e m i s e x p e c t e d , i n e a c h c a s e , t o be as s i m p l e as p o s s i b l e . (b) K i n e m a t i c s The two s t a t e s o f the "two s t a t e " s y s t e m a r e t h e e n e r g y e i g e n s t a t e s |a> and |b> o f t h e f i e l d f r e e H a m i l t o n i a n ^ w i t h a s s o c i a t e d e i g e n v a l u e s E and E , (E < E , ) . T h i s a l l o w s a b a b /2/ to be w r i t t e n as o ( 2 . 1 ) = E |a><a + E !b><b| . o a b 1 The n a t u r a l r e s o n a n c e f r e q u e n c y o f t h i s s y s t e m i s E — E ( 2 . 2 ) U) = b a , o -ft w h i l e t h e a p p l i e d r a d i a t i o n o f f r e q u e n c y to d e v i a t e s f r o m r e s o n a n c e by an amount ( 2 . 3 ) Aw = 0) - w . o The e l e c t r i c d i p o l e moment o p e r a t o r f o r t h e two s t a t e s y s t e m i s w r i t t e n a s ( 2 . 4 e e ) u = u d + y where ( 2 . 4.b) y d = |a> u a a<a| + ! b>yfab<b> | v = ' a > y a b < b | + ' b > y b a < a | * 27 In many i n s t a n c e s , t h e d i a g o n a l p a r t o f the d i p o l e moment o p e r a t o r [ i ^ i s i d e n t i c a l l y z e r o . C o n s i d e r a t i o n o f e q u a t i o n s (2.1) and (2.4) shows t h a t t h e f o u r o p e r a t o r s |a><a|, |b><b[, |a><b|, and |b><a| form an o p e r a t o r b a s i s f o r t h e two s t a t e s y s t e m . T h a t i s , any o p e r a t o r c an be e x p r e s s e d i n terms o f t h e s e f o u r b a s i s o p e r a t o r s . Indeed,, t h e s t a t e o f the s y s t e m i t s e l f , , r e p r e s e n t -ed by the d e n s i t y o p e r a t o r p, has a r e p r e s e n t a t i o n i n terms o f t h i s b a s i s s e t (2.5) p = p |a><a| + p w I a>< b | + p. |b><a| + p.. |b><b| aa 1 1 ab' 1 ba' 1 bb 1 1 Here p , p , a r e t h e p o p u l a t i o n s o f s t a t e s |a> and |b>, a a bb r e s p e c t i v e l y , w h i l e p . and p. a r e r e f e r r e d to as c o h e r e n c e s - - J • ar> 0a and measure t h e amount o f s u p e r p o s i t i o n between t h e s e two states. E q u a t i o n (2,5) i s n o t t h e o n l y p o s s i b l e r e p r e s e n t a t i o n o f t h e s t a t e o f t h e s y s t e m , however. As p o i n t e d o u t by 2 Fano, t h e d e n s i t y o p e r a t o r may be c o n v e n i e n t l y e x p r e s s e d i n terms o f any b a s i s set. o f o p e r a t o r s . F o r t h e two s t a t e s y s t e m i n t e r a c t i n g w i t h r a d i a t i o n v i a i t s e l e c t r i c d i p o l e , a d e s c r i p t i o n i n v o l v i n g the d i p o l e o p e r a t o r i t s e l f m i g h t be a p p r o p r i a t e . I n d e e d , as s h a l l be s e e n s h o r t l y , a r e p r e s e n t a -t i o n i n terms o f t h e f o u r o p e r a t o r s 1, A N e p /U , and p i s v e r y u s e f u l . T h e s e f o u r o p e r a t o r s a r e d e f i n e d as (2.6) 1 =» |a><aj + |b><b| AN0f,= N([a><a| - |b><b|) Vi « y . Ia><bl + v i . Ib><al ab1 ' ba1 ' fj « i y - b a | b><a| - i u a b | a > < b | E q u a t i o n s (2.6) can be e q u a l l y v i e w e d as a t r a n s f o r m a t i o n between b a s i s s e t s o f o p e r a t o r s , w i t h the i n v e r s e t r a n s f o r m a -t i o n g i v e n as (2.7) |a><a| - -| 1 + | ^ » b X b J . i i - I M.p 2 2 N . |a><b| =•- ~ - (y + i y ) ab | b><a| = (y - i y ) ba In t h e s e c o n d c h o i c e o f b a s i s , t h e d e n s i t y o p e r a t o r p i s w r i t t e n as ( 2 . 8 , p » I X ,. AN ^ + P 4 " 2N 2 y"N U N v;ith t h e c o e f f i c i e n t s o f t h e two e x p r e s s i o n s ' r e l a t e d by (2.9) AN - N ( p a a - p b b ) P r " ! ( l J b a y a b + W b a * p. = ^h-(y, v w ~ i 1 ui J v,.) i 2r ba ab ab ba T h e s e r e l a t i o n s a r e e a s i l y o b t a i n e d by a p p l y i n g t h e t r a n s -f o r m a t i o n (2.7) to e q u a t i o n ( 2 . 5 ) . The AN, P and p n o t a -t i o n i n e q u a t i o n ( 2 C 8 ) i s due t o F l y g a r e - t h e s e numbers a r e r e l a t e d to t h e e x p e c t a t i o n v a l u e s o f AN ef.rU and u , r e s p e c t i v e l y , see e q u a t i o n ( 2 . 2 3 ) . B e f o r e c l o s i n g t h i s s e c t i o n , i t s h o u l d be s t r e s s e d a g a i n t h a t t h e r e p r e s e n t a t i o n s i n terms o f the two b a s i s s e t s (2.6) and (2.7) a r e e x a c t l y e q u i v a l e n t - t h e y p r e s e n t t h e same e f f e c t s f r o m a d i f f e r e n t p o i n t o f v i e w b u t c o n t a i n t h e same amount o f i n f o r m a t i o n . (c) Dynamics and the R o t a t i n g Wave A p p r o x i m a t i o n A s y s t e m o f gas m o l e c u l e s , d e s c r i b e d by a o n e - p a r t i c l e d e n s i t y o p e r a t o r a , e v o l v e s i n t i m e due t o b o t h f r e e m o t i o n and c o l l i s i o n a l e f f e c t s . The f r e e m o t i o n , i n t h e p r e s e n c e o f an o s c i l l a t i n g e l e c t i v e f i e l d , i s g o v e r n e d by the t i m e d e p e n d e n t H a m i l t o n i a n (2.10) 2^=?/ - 2]iE c o s ( w t - k y ) , ' o o w h i l e t h e c o l l i s i o n a l e f f e c t s a r e d e s c r i b e d by a g e n e r a l r e l a x a t i o n m a t r i x , w h i c h i s to be d i s c u s s e d l a t e r . The e q u a t i o n o f m o t i o n t h a t O s a t i s f i e s i s t h e n (2.11) i j^o = l?6fo] - i / ? a The method o f s o l u t i o n o f t h i s e q u a t i o n f o l l o w s the a p p r o a c h t a k e n i n t h e c o r r e s p o n d i n g NMR e q u a t i o n s and i s d e s c r i b e d i n t h e s u c c e e d i n g p a r a g r a p h s . I n s h o r t , i t i s b a s e d on a t r a n s f o r m a t i o n . t o a " r o t a t i n g f r a m e " , i n - p h a s e w i t h t h e a p p l i e d e l e c t r i c f i e l d , and a s u b s e q u e n t " r o t a t i n g wave a p p r o x i m a t i o n " w h e r e i n t h e h i g h l y o s c i l l a t o r y p a r t s o f the e q u a t i o n a r e d r o p p e d . The r e s u l t i n g e q u a t i o n i s g o v e r n e d by an e f f e c t i v e t i m e i n d e p e n d e n t H a m i l t o n i a n . The t r a n s f o r m a t i o n o f the d e n s i t y o p e r a t o r t o t h e frame r o t a t i n g a t f r e q u e n c y "w" i s a c c o m p l i s h e d by (2.12) p = e x p S ( t - y / c ) ] O e x p [ S ( t - y / c ) ] 31 where S i s a time i n d e p e n d e n t o p e r a t o r w h i c h i s d e f i n e d to s a t i s f y the r e l a t i o n s (2.13) ,s] = o [s,y ] = - win (s,y ) » wiy o A p p l y i n g the t r a n s f o r m a t i o n (2.12) to t h e e q u a t i o n o f m o t i o n (2.11), t h e e x a c t r e s u l t (2.14) = L % - S - 2R s E q c o s ( W t - k y ) ,p] - i / ? g ( t ) p . i s o b t a i n e d . H e r e , ^ ( t ) i s t h e t r a n s f o r m e d c o l l i s i o n m a t r i x ( b e i n g e x p l i c i t l y time d e p e n d e n t ) , w h i l e y g i s t h e t r a n s f o r m e d d i p o l e moment o p e r a t o r (2.15) y ••-= e x p [ £ S ( t - y / c ) J y e x p [ ~ | S ( t - y / c ) ] S )r, v\ C o n s i d e r a t i o n o f (2.13), (2.15) and t h e i d e n t i t y (2.16) c o s ( w t - k y ) = -|{ exp [ i ( Wt-ky) ] + e x p [ ~ i ( wt-ky) ] } i m p l i e s t h a t t h e r a d i a t i o n - s y s t e m c o u p l i n g i n t h e r o t a t i n g f r a m e c a n be s p l i t i n t o a t i m e - i n d e p e n d e n t p a r t and a r a p i d l y o s c i l l a t i n g p a r t . I n d e e d , (2.17) -2y E cos(oot-ky) = -E y + r a p i d l y o s c i l l a t i n g terms s o o 3 2 The n e g l e c t o f t h e s e r a p i d l y o s c i l l a t i n g terms i n e q u a t i o n c o n s t i t u t e s t h e r o t a t i n g wave a p p r o x i m a t i o n and i s t h e method by w h i c h the r a d i a t i o n - s y s t e m c o u p l i n g w i l l be t r e a t e d i n t h i s c h a p t e r ( i n d e e d , i n t h i s t h e s i s ) . C o r r e c t i o n s to t h i s r o t a t i n g wave a p p r o x i m a t i o n v i a p e r t u r b a t i o n t h e o r y 3 have been t r e a t e d e l s e w h e r e . W i t h i n t h i s a p p r o x i m a t i o n , e q u a t i o n (2.14) c a n be w r i t t e n as E q u a t i o n (2.18) i s c o n s i d e r e d as t h e f u n d a m e n t a l e q u a t i o n g o v e r n i n g the e v o l u t i o n o f the gas m o l e c u l e s . W h i l e t h e d i s c u s s i o n i n t h i s s e c t i o n has been q u i t e g e n e r a l so f a r , the r e m a i n d e r e x p l i c i t l y assumes t h e two s t a t e model f o r t h e m o l e c u l e s . The d e n s i t y o p e r a t o r P i n e q u a t i o n (2.18) i s a f u n c t i o n o f t i m e . T h i s i m p l i e s t h e e x p a n s i o n c o e f f i c i e n t s i n any b a s i s e x p a n s i o n o f p w i l l be t i m e d e p e n d e n t and s a t i s f y moment e q u a t i o n s d e r i v a b l e from e q u a t i o n (2.18). I n p a r t i c u -l a r , t h e e x p a n s i o n s (2.5) and (2.8) a r e now c o n s i d e r e d . I n t h e f i r s t b a s i s , t h e c o u p l e d s e t o f e q u a t i o n s a r e ( 2 . 1 7 ) , as w e l l as the t i m e d e p e n d e n t terms ( 2 . 1 8 ) **n " [K -5 - y v p l + ^ s p (2.19) i^-|pab = Vab(Paa-pbb> ^ A W p a b - i R . , P . ab;ab ab 33 i h — p = E U ( P - P ) + "h Atop, - i R, . p. 1 3 t b a . o M b a ybh M a a n ^ba b a ; b a ' b a i*j | D aa = " E o y a b P b a + V b a P a b " 1 R a a ; a a f P a a ^ i a ^ ' 1 R a a , b b , P b b - p b b ^ ^ S T ^ b = " E o y b a P a b + V a b P b a " 1 R b b ; b b { p b b " P b b ^ - 1 R b b ; a J P a a " P i a ) | . S i n c e o n l y t h o s e p a r t s o f t h e c o l l i s i o n m a t r i x . ( t ) have been r e t a i n e d w h i c h a r e t i m e i n d e p e n d e n t , i t f o l l o w s t h a t //^  ' o n l y c o n n e c t s e l e m e n t s o f p h a v i n g the same f r e q u e n c y . ( T h i s p r o p e r t y i s a l s o c o n s i s t e n t w i t h the c o n s e r v a t i o n o f e n e r g y i n t h e d e t a i l e d c o l l i s i o n p r o c e s s e s c o n t r i b u t i n g t o IP , as d i s c u s s e d i n t h e n e x t s e c t i o n . ) The f a c t t h a t / ^ ">> s s p r e s e r v e s f r e q u e n c y i m p l i e s t h a t m a t r i x e l e m e n t s o f a r e m ost n a t u r a l l y c o n s i d e r e d i n t h e " f r e q u e n c y " b a s i s ( 2 . 5 ) . E q u a t i o n s o f m o t i o n i n t h e a l t e r n a t e b a s i s (2.8) a r e s t i l l d e s i r a b l e , however, i n v i e w o f t h e m a c r o s c o p i c i n t e r -p r e t a t i o n t h a t t h i s d e s c r i p t i o n y i e l d s . The moment e q u a t i o n s a r e t h e n P 34 (2.20, £L p r + A03P. + ^ - 0 4-P. - AWP + K 2 E ( ^ ) + - 0 d t l r o 4 T • ^ ^AH - E P . + i - ( ^ M - " i ^ S a „ Q d t 4 o 1 4 4 2 U 2 where IC = — — and t h e t r a n s f o r m a t i o n (2.6) has been u s e d 4 ^ 2 I n e q u a t i o n (2.20) t h e i d e n t i f i c a t i o n (2.21) — = R + R, , - R - R,, V^.^-J./ T ^ a a ; a a bb;bb aa; bb bb;aa T^ ab;ab ba;ba has been made. E q u a t i o n (2.20) a l s o assumes t h a t t h e r e l a x a -t i o n m a t r i x i s r e a l and t h a t <AN> i s c o l l i s i o n a l l y u n c o u p l e d from <1> . E q u a t i o n s (2.19) and (2.20) a r e s e e n to be t h e m i c r o -wave a n a l o g u e s ( f o r a t w o - s t a t e system) o f the R e d f i e l d and B l o c h e q u a t i o n s o f NMR. A l t e r n a t i v e l y , one m i g h t s a y t h a t e q u a t i o n (2.20) a l l o w s a c l a s s i c a l m e c h a n i c a l i n t e r p r e t a t i o n w h i l e (2.19) p o i n t s o u t t h e quantum m e c h a n i c a l m o t i o n s . How-e v e r , i t i s e m p h a s i z e d o n c e more, t h a t (2.19) and (2.20) a r e c o m p l e t e l y e q u i v a l e n t d e s c r i p t i o n s o f the same p h y s i c a l phenome na. The e f f e c t o f the t r a n s f o r m a t i o n (2.12) on the d i p o l e moment o p e r a t o r (2.4) c a n be shown to be (2.22) P ( t ) = N t r { 0 ( t ) u } {p ( t ) - + i P i ( t) } e x p [ i ( w t - k y ) ] + { P r ( t ) ' - i P ^ t ) } e x p l - i ( w t - k y ) ) + N(u + U,,) + ( a a o b b ) A N ( t) aa bb 2 where (2.23) P r ( t ) = f t r ( p ( t ) V i > P j L ( t ) = ~ t r ( p ( t ) y } AN (t) = t r {p ( t ) AN} E q u a t i o n s (2.22) and (2.23) p o i n t o u t most c l e a r l y the d e s c r i p t i o n s o f t h e m o t i o n s i n t h e " l a b f r a m e " v e r s u s th e " r o t a t i n g f r a m e " . F u r t h e r , f o r t h e two s t a t e s y s t e m , t h e s e e q u a t i o n s show t h a t t h e t i m e d e p e n d e n c e o f t h e d i a g o n a l p a r t s o f t h e d i p o l e moment a r e d e t e r m i n e d c o m p l e t e l y by t h e e v o l u t i o n o f t h e p o p u l a t i o n d i f f e r e n c e A N ( t ) . 3 6 (d) The R e l a x a t i o n M a t r i x The p u r p o s e o f the p r e s e n t d i s c u s s i o n i s to g i v e a p h y s i c a l p i c t u r e o f t h o s e t y p e s o f d e t a i l e d c o l l i s i o n p r o -c e s s e s w h i c h c a n c o n t r i b u t e t o any p a r t i c u l a r m a t r i x e l e m e n t o f « T h i s s e c t i o n i s p r o d u c e d w i t h i n t h e c o n t e x t o f a two s t a t e s y s t e m where t h e c o m p l i c a t i o n s o f r o t a t i o n a l d e g e n -e r a c y a r e n o t p r e s e n t t o d i s t r a c t f r o m t h e b a s i c e f f e c t s . T h e s e s h o r t c o m i n g s a r e c o r r e c t e d i n c h a p t e r IV where a r i g o u r o u s p r e s e n t a t i o n o f c o l l i s i o n a l e f f e c t s i s g i v e n . In e s s e n c e , one would l i k e t o v i s u a l i z e how t h e m i c r o -s c o p i c v i e w o f c o l l i s i o n s between two m o l e c u l e s i s r e l a t e d to the o v e r a l l m e asured r e l a x a t i o n r a t e s p e c i f i e d i n terms 4 o f R.. ... The e l e m e n t a r y p r o c e s s e s a r e g i v e n i n terms o f w e l l d e f i n e d d i r e c t i o n a l l y a v e r a g e d c r o s s s e c t i o n s C d e s c r i b i n g the c o l l i s i o n between a m o l e c u l e o f i n t e r e s t and a p e r t u r b e r m o l e c u l e , w i t h t h e p a i r h a v i n g a d e f i n i t e i n i t i a l r e l a t i v e t r a n s l a t i o n a l e n e r g y e r „ The d i r e c t i o n a l l y a v e r a g e d t r c r o s s s e c t i o n s a r e i n d e p e n d e n t o f t h e c e n t r e o f mass v e l o c i t i e s b e f o r e and a f t e r c o l l i s i o n . T h e s e e l e m e n t a r y p r o c e s s e s a r e t h e n a v e r a g e d o v e r an e q u i l i b r i u m d i s t r i b u t i o n o f i n i t i a l p e r t u r b e r i n t e r n a 1 s t a t e s e a v e r a g e d o v e r an e q u i l i b r i u m d i s t r i b u t i o n o f i n c r e a s i n g r e l a t i v e t r a n s l a t i o n a l e n e r g y , and summed o v e r f i n a l s t a t e s o f the p e r t u r b e r . I t i s t h e s e k i n e t i c a l l y a v e r a g e d r e s u l t s w h i c h a r e i d e n t i f i e d w i t h the r e l a x a t i o n m a t r i x e l e m e n t s . C o n s i d e r R. . .... w i t h i ^ i ' , j ^ j ' . T h i s m a t r i x e l e m e n t r e p r e s e n t s the t o t a l r a t e t h a t the one m o l e c u l e s t a t e | i , > < : j ' | c a n be • c o n v e r t e d to t h e s t a t e | i > < : j i by c o l l i s i o n s . I n v i e w • A ' , -» o f t h e p r e v i o u s r e m a r k s , the i d e n t i f i c a t i o n 8kT 1 / 2 ( 2 ' 2 4 ) R . . . , . , = n ( — > ^ t r / d £ ; r e x P [ - e ; r ] e ; r i / i ' D * j ' E I exp f-e , ] 6 ( e ' - e ) c c ' C 2 2 2 ^ c ( | i ' > < j « | , |c^><c'|, e ' t r + | i > < j | , | c 2 > < c 2 | f s h o u l d be r e a s o n a b l e . Here 6(E'-E) e x p r e s s e s t h e f a c t t h a t t h e t o t a l r e l a t i v e e n e r g y i s c o n s e r v e d i n t h e c o l l i s i o n p r o c e s s e s , s e e c h a p t e r IV f o r some t e c h n i c a l a s p e c t s o f e n e r g y c o n s e r v a t i o n . I n e q u a t i o n ( 2 . 2 4 ) , b e c a u s e t h e c o h e r e n t s t a t e s '{:i><j| and | i ' x j ' | a r e i n v o l v e d , e n e r g y c o n s e r v a t i o n r e q u i r e s two e q u a l i t i e s (2.25) e t r + + e. - e t r , + e c , + e L , t r c 2 j t r ' c 2 j ' S u b t r a c t i n g one c o n s e r v a t i o n law f r o m t h e o t h e r i m p l i e s t h e f r e q u e n c y c o n s e r v a t i o n p r o p e r t y . (2.26) e. - e. = e., - e i ^ i ' r j ^ j * i j i D The m a t r i x e l e m e n t s R , . and R = = o f the l a s t s e c t i o n aa jbb DD;aa o b v i o u s l y have t h e f o r m o f e q u a t i o n ( 2 . 2 4 ) , w i t h z e r o f r e q u e n c y 38 The r e l a x a t i o n m a t r i x e l e m e n t s R. . .. a r e s l i g h t l y more i J ! J- D c o m p l i c a t e d i n t h a t t h e y r e p r e s e n t a l l p o s s i b l e r o u t e s by w h i c h the c o h e r e n c e |i><j| c a n r e l a x . As s u c h , the form o f R. . . . i s a-D ; JO • 8kT 1/2 ' (2.27) R. . . . = n ( — - ) / d e ^ / d £ ^ , e x p t - E ^ , 1 e. , ' i D J J O y t r t r * c t r * t r 1 Z Z e x p [ - e c , ] C 2 C 2 ' 2 [ C ( | i > < j | ; | c 2><c 2| , e t r , ^ U > < j | , | c 2><c 2| , e f c r ) 6 ( e ' - e ) + | 2 c ( | i > < j | ; | c 2 > < c 2 | , e t r , - > | k > < k | , | c 2 > < c 2 l , e t r ) k 6 ( e ' - e ) + | £ C ( | j > < j | ; | c 2 > < c ' | , e t r f - > | k > < k | , | c 2 > < c 2 | , e t r ) k E q u a t i o n (2.27) i n c l u d e s t h e m a t r i x e l e m e n t s R , R., . , , ^ a a ; a a bb;bb R ,. ' , and R, , o f t h e l a s t s e c t i o n as s p e c i a l c a s e s . ab;ab ba; ba Com p a r i n g the f o r m s (2.24) and (2.27) w i t h ( 2 . 2 1 ) , e x p r e s s i o n s f o r T^ and T 2 i n t e r m s o f t h e c o n t r i b u t i n g m i c r o s c o p i c c o l l i s i o n e v e n t s c a n be e s t a b l i s h e d . I n d e e d , (2.28) ~ - = h v — ; t r ' " t r | J t r ' c , c 1 c ? °o Z-1/2 1 = n(-2£l) /ae / d E f . e x p [ - e ] c E E e x p [ - e . ] 2 2 + [ E 6(e'-e) c (|i><i|,|c•><c'|,e |k><k|,|c ><c2J e t r> k i^ + E 6 (e ' -£) c(| j><j| , Ic-xc-|,etrl-|k><k}Jc2><c2r,etr)] k^ j and (2.29) 1/2 i _ . n ( ^ } / d E f c r / d E t r | e x p [ - E t r , ] £ t r , E Z exp[-e c,] +[6<e'-e> c(|i><j |Jc 2 ><c'| fe t r I*|i><j| f |c2><c2|, e t r) + '6(e'-e) c(|j><i| f |c2><c'|fetrI-*-|j><-i|,|c2><c2|fetr) + E $<e'-e) c( |i><i|,|c 2><c 2| ,e t r ,-^IkXkl, |c 2><c 2| ,e t r) E 6(e'-e> c ( | j>< j | , I c '><c • | , e tr-,| k><k | , | c 2><c 2 | , e ^ ) ] k a r e t h e r e s u l t s . T h e s e e q u a t i o n s d e m o n s t r a t e t h a t o n l y i n e l a s t i c c o l l i s i o n s c o n t r i b u t e t o ( T ^ 1 w h i l e ( T 2 ) 1 has b o t h e l a s t i c and i n e l a s t i c e f f e c t s . T hus, i t i s e x p e c t e d t h a t and t h a t t h e s t r o n g c o l l i s i o n m o d e l , ( e q u a l i t y o f and T 2 ^ ' would be v a l i d o n l y when e l a s t i c c o n t r i b u t i o n s to (T ) ^ a r e s m a l l . I n c l o s i n g t h i s s e c t i o n , i t s h o u l d be e m p h a s i z e d t h a t the above i d e n t i f i c a t i o n s , e s p e c i a l l y (2.24) and ( 2 . 2 7 ) , a r e n o t p r e s e n t e d a s d e r i v e d r e s u l t s . R a t h e r , t h e y r e p r e s e n t an a t t e m p t t o a i d i n v i s u a l i s i n g the c o l l i s i o n a l a s p e c t s o f t h e two s t a t e p r o b l e m . A r i g o r o u s t r e a t m e n t o f c o l l i s i o n s i s p r e s e n t e d i n c h a p t e r I V . 41 (e) S t e a d y S t a t e A b s o r p t i o n and the G e n e r a l T r a n s i e n t E x p e r i m e n t The s i m p l e s t e x p e r i m e n t s w h i c h c a n be c a r r i e d o u t a r e t h e t r a d i t i o n a l s t e a d y s t a t e a b s o r p t i o n e x p e r i m e n t s . Here t h e gas i s i n i t i a l l y assumed to be i n e q u i l i b r i u m , w h i c h i m p l i e s (2.31) P r ( o ) = 0 P.(o) = 0 l AN(O) = AN eq and t h e n i s s u b j e c t e d t o a c o n t i n u o u s wave (cw) m i c r o w a v e s o u r c e . The d e s i r e d s o l u t i o n i s t h e s t e a d y s t a t e r e p r e s e n t i n g the l o n g time b e h a v i o u r o f the s y s t e m a f t e r a l l t r a n s i e n t s have d i e d away. A t t h i s s t a g e , the o n l y o s c i l l a t i o n s a r e a t the f r e q u e n c y o f the a p p l i e d f i e l d . T h i s s o l u t i o n i s m o s t e a s i l y o b t a i n e d by s e t t i n g a l l time d e r i v a t i v e e q u a l t o z e r o i n t h e r o t a t i n g f r a m e . Thus e q u a t i o n s (2.20) become P (2.32) A U P . + — = 0 1 T 2 . 2 >AN % i -AoJP + K E (—•—) + — = 0 r o 4 T - E P . + VN:ANe^ .- o . o l 4 The s o l u t i o n to t h e s e e q u a t i o n s , s u b j e c t t o t h e i n i t i a l c o n d i t i o n (2.31) a r e t h e n e a s i l y f o u n d to be (2.33) 2 p -4- E Aw r 4 o N E Q " _1 + ( A U ) 2 + (1±)K2E 2 T 2 T 2 2 1 P . - — — - E x 4 o 1 2 A N T e q -A- + ( A w ) 2 +• ( - i ) < 2 E 2 T 2 T 2 ° + ( A w ) 2 A N = T 2  A N . _ T eq 1 , . , 2 1 2 2 H 5- + (AW) + — K E T„ 2 E q u a t i o n s (2.33) s h o u l d be v i e w e d w i t h i n the q u a l i t a t i v e comments o f c h a p t e r I . The e x p r e s s i o n f o r g i v e s the l i n e -shape f o r a m i c r o w a v e l i n e , when b o t h D o p p l e r e f f e c t s and n a t u r a l l i n e w i d t h s a r e assumed n e g l i g i b l e and when no o v e r -l a p p i n g w i t h o t h e r s p e c t r a l l i n e s i s s i g n i f i c a n t . The above e q u a t i o n d o e s a c c o u n t f o r s a t u r a t i o n b r o a d e n i n g , however, and shows how i t becomes n e g l i g i b l e i n t h e low ( i n p u t ) power 2 2 c a s e when < E <<1, t h a t i s , when A N = A N . In t h i s l i m i t , o eq i t i s s e e n t h a t T^ p l a y s no r o l e i n d e t e r m i n i n g t h e s t e a d y s t a t e ( 2 . 3 3 ) . Thus the f u n d a m e n t a l l i n e w i d t h p a r a m e t e r i s • i , t h e r e l a x a t i o n time a s s o c i a t e d w i t h the r e l a x a t i o n o f the V c o h e r e n c e between the two s t a t e s . 43 A s e l e c t i v e d i s c u s s i o n o f m i c r o w a v e t r a n s i e n t e x p e r i m e n t s b a s e d on the work o f McGurk e t a l * i s g i v e n i n the r e m a i n d e r o f t h i s c h a p t e r . A d d i t i o n a l m a t e r i a l , e x p e r i m e n t a l d e t a i l s and r e s u l t s , may be f o u n d i n a s e r i e s o f p a p e r s by t h i s 5 6 7 8 g r o u p . ' ' ' E a c h e x p e r i m e n t i s r e p r e s e n t e d by a p a r t i c u l a r time d e p e n d e n t s o l u t i o n to e q u a t i o n s ( 2 . 2 0 ) . As f i r s t i n d i c a t e d 9 by T o r r e y , f o r p r o b l e m s o f t h i s t y p e , t h e s e t i m e d e p e n d e n t s o l u t i o n s t a k e t h e f o r m — a t — b t r p " * 1 ^ (2.34) f ( t ) = Ae + Be c o s Qt + ^— s i n Qt + D, where f ( t ) r e p r e s e n t s any o f •Pi ( t) P ( t ) , and A N ( t ) . Here t h e t i m e c o n s t a n t s a, b + i f l , b-i£2 a r e t h e t h r e e r o o t s o f t h e s e c u l a r d e t e r m i n a n t (2.35) 2 A(u) E ( U + ^ - ) ( U + -r^—)2 + I | f i J l j ( u + ^ - ) o l o 2 < E o l o 1 + (u + p — — — ) = 0 KE T ' o 2 w h i c h j u s t depend on t h e p r o p e r t i e s o f t h e s y s t e m . The c o n s t a n t s A,B,C and D, on t h e o t h e r hand, depend on the p a r t i c u l a r e x p e r i m e n t . S i n c e t h e t i m e i n d e p e n d e n t p a r t D r e p r e s e n t s the a p p r o p r i a t e s t e a d y s t a t e s o l u t i o n [ e q u a t i o n s ( 2 . 3 3 0 ] , e q u a t i o n (2.34) s p e c i f i e s t h e manner i n w h i c h t h e s t e a d y s t a t e v a l u e s a r e a p p r o a c h e d . The s e c u l a r e q u a t i o n (2.35) c a n o n l y be s o l v e d i n c l o s e d form f o r c e r t a i n c a s e s see s e c t i o n s (g) and ( h ) . The c o n s t a n t s A,B,C and D a r e a l s o e v a l u a t e d f o r s p e c i f i c e x p e r i m e n t s i n t h e s e s e c t i o n s . ( f ) D e t e c t i o n o f R a d i a t i o n B e f o r e p r o c e e d i n g w i t h the d e t a i l e d d i s c u s s i o n o f t h e t r a n s i e n t e x p e r i m e n t s , some g e n e r a l comments on the d e t e c t e d r a d i a t i o n a r e p r e s e n t e d - the t h i r d s t a g e t h a t i s i l l u s t r a t e d i n f i g u r e 1. T h e s e r e m a r k s a r e , f o r the most p a r t , i n d e p e n -d e n t o f the p a r t i c u l a r m i c r o w a v e e x p e r i m e n t t h a t i s b e i n g c a r r i e d o u t on the g a s sample and a r e t h e r e f o r e c o l l e c t e d i n one s e c t i o n . F u r t h e r , t h i s s e p a r a t i o n o f t h e d e t e c t e d r a d i a t i o n f r o m the u n d e r l y i n g s y s t e m m o t i o n i m p l i e s t h a t t h e s e r e m a r k s a r e a l s o r e l a t i v e l y i n d e p e n d e n t o f t h e two s t a t e model u s e d i n t h i s c h a p t e r and so may be r e a p p l i e d t o the m o d i f i e d s y s t e m m o d e l s d i s c u s s e d i n l a t e r c h a p t e r s . The r e s u l t a n t e l e c t r i c f i e l d i s c a l c u l a t e d by s o l v i n g M a x w e l l ' s e q u a t i o n s i n t h e p r e s e n c e o f a d i e l e c t r i c ( p o l a r i z a b l e ) medium. T h e s e e q u a t i o n s a r e o f t h e f o r m ^ (2.36) V * E = 4irp p o l V • B = 0 1 8E T 4TT J c S t c ~ p o l = VxB 1 9_B c a t = -VxE where (2.37) B p o l = -V»P and J ~po 1 F o r a gas i n the a b s e n c e o f e x t e r n a l s t a t i c f i e l d s (2.38) V'P = 0 , 4 6 and t h e wave e q u a t i o n c t c 3 t i s obtained"'"''' as the e q u a t i o n o f m o t i o n o f the e m i t t e d f i e l d . I n t h e e x p e r i m e n t s t h a t a r e t o be d e s c r i b e d , t h e p o l a r i z a -t i o n P ( t ) a r i s e s i n r e s p o n s e t o an a p p l i e d e l e c t r i c f i e l d E = E q c o s ( w t - k y ) . By i s o t r o p y , t h e p o l a r i z a t i o n P ( t ) i s a l s o a l o n q t h e E d i r e c t i o n (where E j„ y) . Thus e q u a t i o n o o (2.39) r e d u c e s to t h e one d i m e n s i o n a l wave e q u a t i o n . (2.40) sr— b - - - — - „ 9 y c 2 3 t 2 c whore t 1" 1^ "cL""'"i z ? t i o n a 3 ono the. E d i r e c t i o n i s e x p r e s s e d j. - -- -• _ o " • as [compare (2.22).] (2.41) P ( t ) ~ 2P ( t ) c o s (to t ~ k y ) - 2P_L ( t ) s i n ((trt-ky) y -y, , ... H ) A N ( T ) + ( u + y )N «c a a D o N e g l e c t i n g the t i m e d e r i v a t i v e s o f P (t)., P ± ( t ) and AN ( t ) , the s o u r c e term i n e q u a t i o n (2..40) i s e v a l u a t e d a p p r o x i m a t e -l y a s 2 (2.42) - — - -0J 2{2P ( t ) c o s ( w t - k y ) - 2 P . ( t ) s i n ( w t - k y ) } 3 t 2 1 The wave e q u a t i o n (2.40, w i t h e q u a t i o n (2.42) as the s o u r c e term c a n be a p p r o x i m a t e l y s o l v e d i n the form (2.43) E = 2[e^ c o s ( w t - k y ) + e r s i n ( t o t - k y ) ] where t h e s l o w l y v a r y i n g q u a n t i t i e s e ^ y t ) and e r ( y t ) s a t i s f y (2 . 4 4 , ! l i " - la P . pL.*™V 3 y c i 3y c r . T h i s i n v o l v e s n e g l e c t i n g t h e f o l l o w i n g q u a n t i t i e s : 3 2 e , 3 2 e . 3e 3e". 3 2 e 3 2 e . 1 i r l r i 3 t 3 t 3 y 3 y As shown i n f i g u r e 1, t h e i n c i d e n t r a d i a t i o n 2 E Q c o s ( u t - k y ) e n t e r s t h e sample a t y=0 and the r e s u l t i n g r a d i a t i o n i n t h e form o f e q u a t i o n (2.43) e x i t s a t y=l. The a p p r o p r i a t e b o u n d a r y c o n d i t i o n s f o r t h e d i f f e r e n t i a l e q u a t i o n s (2.44) a r e t h e n (2.45) e . = E a t z=0 l o e = 0 , r so t h a t t h e c o r r e s p o n d i n g s o l u t i o n s a r e g i v e n i m m e d i a t e l y (2.46) e . ( y t ) = E + • 2 - ^ P . ( t ) 1 o c 1 40 e ( y t ) - P ( t ) . c r The f i e l d w h i c h the d e t e c t o r o b s e r v e s a t y=& i s (2.47) E U t ) = 2E ' c o s ( w t - k i ) + 4 7 r U ) £ [P (t) s i n ( w t - k J l ) O C 3C + P i ( t ) c o s ( w t - k £ ) ] w h i c h i s o b t a i n e d f r o m a c o m b i n a t i o n o f e q u a t i o n s (2.43) and ( 2 . 4 6 ) . F o r a " s q u a r e law" d e t e c t o r , t h e measured s i g n a l 2 S i s p r o p o r t i o n a l t o t h e t i m e a v e r a g e E ( t ) o f t h e s q u a r e o f the e l e c t r i c f i e l d e n t e r i n g the d e t e c t o r , namely (2.48) s - B E 2 (t) $ b e i n g the p r o p o r t i o n a l i t y c o n s t a n t f o r the p a r t i c u l a r d e t e c t o r . The change i n s i g n a l As due t o the p r e s e n c e o f t h e gas m o l e c u l e s i s f r o m e q u a t i o n s (2.47) and (2.48) (2.49) AS = S - 3E 2 o 1 ^ v . ( t ) + i l W p [ 2 ( F C ) + 2 ( t ) J c " 4 9 In t h i s e q u a t i o n , a l l r a p i d l y o s c i l l a t i n g terms have v a n i s h e d due t o t h e time a v e r a g i n g . When t h e E q f i e l d i s p r e s e n t , a b s o r p t i o n o f r a d i a t i o n o c c u r s , see s e c t i o n ( g ) . On t h e o t h e r hand, i f E q i s a b s e n t , e m i s s i o n c a n o c c u r , see s e c t i o n ( h ) . S i n c e A s(=S) i s i n t h i s c a s e q u a d r a t i c i n b o t h P and P., r 1 and hence p r o p o r t i o n a l t o t h e s q u a r e o f the number o f m o l e -c u l e s , i t i s c l e a r t h a t t h i s i s c o h e r e n t s t i m u l a t e d e m i s s i o n and n o t s p o n t a n e o u s e m i s s i o n (in s p o n t a n e o u s e m i s s i o n , t h e power e m i t t e d i s d i r e c t l y p r o p o r t i o n a l t o the number o f m o l e c u l e s p r e s e n t ) . A m o d i f i e d d e t e c t i o n scheme commonly u s e d t o d e t e c t t r a n s i e n t e m i s s i o n employs a r e f e r e n c e f i e l d E ( t ) , g i v e n as (2.50) E r ( t ) = 2 E r c o s ( 0 J R t-k&) , w h i c h o s c i l l a t e s a t a f i x e d f r e q u e n c y tor w h i l e t h e s y s t e m i t s e l f i s o s c i l l a t i n g a t f r e q u e n c y to and p r o d u c i n g an e m i t t e d f i e l d E ( t ) . The two f i e l d s i n t e r f e r e t o g i v e a E s i g n a l (2.51) AS =3 E f ( t ) E E ( t ) a t t h e d e t e c t o r . T h i s i n v o l v e s t h e b e a t f r e q u e n c i e s to +to r o and |w -w I. The p r e c i s e f o r m f o r E _ ( t ) i n terms o f P ( t ) r o E r i s d i s c u s s e d i n s e c t i o n ( h ) . E q u a t i o n (2.49) [ o r (2.51)] g i v e s the e x p r e s s i o n f o r t h e d e t e c t e d s i g n a l . H e r e , P^ ( t) and P ^ t ) a r e s o l u t i o n s o f t h e c o u p l e d s e t o f e q u a t i o n s (2.20) s u b j e c t t o a c e r t a i n s e t o f i n i t i a l c o n d i t i o n s . E a c h p a r t i c u l a r s e t o f i n i t i a l c o n d i -t i o n s c o r r e s p o n d s to a p a r t i c u l a r e x p e r i m e n t w h i c h c a n be c a r r i e d o u t on the two s t a t e gas s a m p l e . The f o l l o w i n g s e c t i o n s d i s c u s s the two main t y p e s o f t r a n s i e n t e x p e r i m e n t s t r a n s i e n t a b s o r p t i o n ( b o t h o n - r e s o n a n c e and o f f - r e s o n a n c e c a s e s ) and t r a n s i e n t e m i s s i o n ( p r o d u c e d by c o n t i n u o u s wave or p u l s e i r r a d i a t i o n ) . T h e s e e x p e r i m e n t s a r e t h e m i c r o w a v e a n a l o g s o f NMR r o t a t i o n and f r e e i n d u c t i o n d e c a y e x p e r i m e n t s , r e s p e c t i v e l y . A d d i t i o n a l d i s c u s s i o n s and r e f e r e n c e s c a n be f o u n d i n t h e p a p e r by McGurk e t a l . * (h) T r a n s i e n t A b s o r p t i o n T r a n s i e n t a b s o r p t i o n a r i s e s when r a d i a t i o n i s b r o u g h t i n t o ( o r n e a r ) r e s o n a n c e w i t h a two s t a t e s y s t e m i n a time s h o r t r e l a t i v e to the r e l a x a t i o n t i m e . T e c h n i c a l l y , t h i s i s a c h i e v e d t h r o u g h the method o f S t a r k s w i t c h i n g . H e r e , t h e two s t a t e s y s t e m , w h i c h i s o r i g i n a l l y f a r o f f - r e s o n a n c e w i t h a c o n t i n u o u s wave s o u r c e o f m i c r o w a v e r a d i a t i o n , i s s u d d e n l y b r o u g h t i n t o r e s o n a n c e by the a p p l i c a t i o n o f a S t a r k f i e l d . T h e r e a r e two c a s e s o f s p e c i a l i n t e r e s t where t h e s o l u t i o n s to t h e s e c u l a r e q u a t i o n ( 2 . 3 5 ) , and t h u s to ( 2 . 2 0 ) , c a n be w r i t t e n i n c l o s e d f o r m - the o n - r e s o n a n c e c a s e , Aw=0, w h i c h c a n be u s e d t o s t u d y the r e l a t i o n s h i p between T^ and T^, and t h e s t r o n g c o l l i s i o n model c a s e T ^ = T 2 » w h i c h c a n be u s e d to s t u d y t h e o f f - r e s o n a n t t r a n s i e n t s i g n a l . The o n - r e s o n a n c e c a s e g i v e s t h e t r a n s i e n t r e s p o n s e as (2.52) 2 . +K A N E - t / T n . . e q , o f f e c o s S2t-1, p ( t ) = 1 I — J i 4 T T T 2 T 2 1 . 1 1 T*T + T - t / T + f 1 2 _ m , e s i n flt } + i _ T ] / 1 1 2 2 ^ - A - + —— K E T „ 2 when t h e i n i t i a l c o n d i t i o n s a r e g i v e n by e q u a t i o n ( 2 . 3 1 ) . H e r e , t h e o s c i l l a t i o n f r e q u e n c y ft and the d e c a y time T a r e g i v e n by 2 2 1 1 1 2 1 / 2 (2.53) ft = [ < 2 E / - ±.<±- - = - > ] • 2 1 T = 1 1 T T 1 2 E q u a t i o n (2.52) i s a p a r t i c u l a r example o f t h e g e n e r a l form ( 2 . 3 4 ) , w h e r e i n a = 0, b=^- and ft i s g i v e n by ( 2 . 5 3 ) . As c a n be s e e n from e q u a t i o n s (2.52) and (2.53) s t u d i e s o f t h i s t y p e c a n be u s e d t o d e t e r m i n e t h e r e l a t i v e s i z e s o f and T^. I n d e e d , a s an example, t h e r o t a t i o n a l t r a n s i t i o n J=0 to J = l i n OCS was f o u n d t o obey T ^ = T 2 V'J-t^J-" an a c c u r a c y o f 15%.^ T h i s r e p r e s e n t s an e x p e r i m e n t a l j u s t i f i c a t i o n o f t h e s t r o n g c o l l i s i o n m o d e l . A l t e r n a t i v e l y , w i t h t h e s t r o n g c o l l i s i o n model as a w o r k i n g h y p o t h e s i s , o f f - r e s o n a n c e t r a n s i e n t a b s o r p t i o n e x p e r i m e n t s c a n be c a r r i e d o u t by v a r y i n g t h e d e g r e e o f o f f r e s o n a n c e c h a r a c t e r Aw. The s o l u t i o n to (2.20) i n t h i s c a s e ( a g a i n w i t h t h e i n i t i a l c o n d i t i o n s ( 2 . 3 1 ) ) , i s g i v e n a K2AN E (2.54) P, = ^ "o r [e t ^ T c o s ftt-l]m — { i 4 1 1 . 2 ~~2 ft T 1 T 2 . . - t / T s i n ft t t + lj - 1] e J J — } — - + ft T^ wher e 2 ? ? 1 / 2 (2.55) » [ ic E + (Aw) •] o F i g u r e 2 i l l u s t r a t e s the shape o f P ^ ( t ) f o r v a r i o u s v a l u e o f « 54 0 2 5 7 10 12 15 17 20 22 25 0 2 5 7 10 12 15 17 20 22 25 F i g u r e 2: P l o t s o f P ^ ( t ) f o r v a r i o u s v a l u e s o f ft, w i t h time i n m i c r o s e c o n d s . A r e l a x a t i o n time o f 5 usee i s assumed. (From McGurk e t a l . , Adv. Chem. P h y s . XXV, 1, (1975).) 55 (h) T r a n s i e n t E m i s s i o n The phenomenon o f t r a n s i e n t e m i s s i o n must be c o n s i d e r e d i n two p a r t s . I n t h e f i r s t s t a g e , t h e s y s t e m i s p r e p a r e d i n some n o n e q u i l i b r i u m s t a t e by i n t e r a c t i o n w i t h e x t e r n a l r a d i a -t i o n . The s e c o n d s t a g e o c c u r s when t h e e x t e r n a l r a d i a t i o n i s s h u t o f f and t h e s y s t e m i s a l l o w e d t o d e c a y back t o i t s e q u i l i b r i u m s t a t e . I t i s i n t h i s s e c o n d s t a g e t h a t t h e a c t u a l e m i s s i o n p r o c e s s e s o c c u r . I n t h e d e s c r i p t i o n g i v e n h e r e , t h e b e h a v i o u r o f t h e s y s t e m i n t h e e m i t t i n g s t a g e i s f i r s t c o n s i d e r e d f o r an a r b i t r a r y n o n e q u i l i b r i u m p r e p a r a t i o n ( i . e . a r b i t r a r y v a l u e s o f p r ( t 1 ) » P i * f c l * a n d ^ N ( t 1 > a t t i m e t ^ ) . Then t h e t h r e e u s u a l methods o f p r e p a r i n g t h e s y s t e m a r e d i s c u s s e d - namely, c o n t i n u o u s wave i r r a d i a t i o n and two p u l s e t e c h n i q u e s . I n t h e a b s e n c e o f any r a d i a t i n g f i e l d ( E Q = 0 ) , t h e sy s t e m e q u a t i o n s ( 2 . 2 0 ) c a n be e a s i l y s o l v e d i n t e r m s o f t h e a r b i t r a r y i n i t i a l c o n d i t i o n s a t t i m e t o g i v e , f o r t > t ^ f ( 2 . 5 6 ) P ( t ) = e x p [ - ( t - t . ) / T 0 ] {P ( t , ) c o s [Aw( t - t . ) ] - P. ( t . ) s i n t A w ( t - t . ) ]} P j,(t) = e x p [ - ( t - t 1 ) / T 2 ] { P i ( t 1 ) c o s [ A a ) ( t - t 1 ) ] + P r ( t x ) s i n [Ao)( t - t 1 ) ] } N ( t ) = AN + e x p [ - ( t - t , / T . ) ( A N ( t . )-AN ) eq X X i eq The e m i t t e d f i e l d E ( t ) i s t h e n c a l c u l a t e d by c o m b i n i n g ( 2 . 5 6 ) E w i t h ( 2 . 4 7 ) , when E = 0 . The r e s u l t i s 56 (2.57) E E ( t ) - 4 TTU)£ e x p [ - ( t - t . ) / T ] { p . ( t c o s [w ( t - t )-k£+tot ] c 4 TTOjfc C + P r ( t 1 ) c o s ( A 0 3 t 1 ) } e x p [ - t / T 2 ] sin(« ot-kil) ] T h i s e q u a t i o n s t a t e s t h a t t h e s y s t e m e m i t s l i g h t a t i t s n a t u r a l f r e q u e n c y o f o s c i l l a t i o n to , w h i l e i t r e l a x e s a t a As i n d i c a t e d i n s e c t i o n ( f ) , t h e r e a r e two methods u s e d t o d e t e c t t h i s s i g n a l . I n t h e f i r s t f d i r e c t ) method, t h e f i e l d i t s e l f i s t i m e a v e r a g e d by t h e d e t e c t o r . The r e s u l t e q u a t i o n ( 2 . 5 6 ) . The s e c o n d , more u s u a l , method o f d e t e c t i o n i n v o l v e s t h e b e a t i n g o f t h e p u r e e m i s s i o n f i e l d (2.57) w i t h some r e f e r e n c e f i e l d ( 2 * 5 0 ) . The d e t e c t e d s i g n a l i s t h e n ( 2 . 5 1 ) . An o s c i l l a t o r y d e c a y t o a s t e a d y v a l u e i s o b s e r v e d w h i c h i s s i m i l a r i n a p p e a r a n c e t o t h e t r a n s i e n t a b s o r p t i o n e x p e r i m e n t . B o t h t r a n s i e n t a b s o r p t i o n and t r a n s i e n t e m i s s i o n c a n be o b s e r v e d i n t u r n , by e m p l o y i n g t h e method o f S t a r k s w i t c h i n g . W i t h t h e S t a r k f i e l d o f f , t h e a p p l i e d r a d i a t i o n r a t e — . 2 i s e q u a t i o n (2.49) w i t h E =0 and P ( t ) and P . ( t ) g i v e n by -1- o r x o f f r e q u e n c y w^ i s r e s o n a n t w i t h t h e m o l e c u l a r l e v e l s (w - u J ) and t r a n s i e n t a b s o r p t i o n c a n be m e a s u r e d . Then r o s u d d e n l y , t h e S t a r k f i e l d i s t u r n e d on, s h i f t i n g t h e m o l e c u l a r e n e r g y l e v e l s to a new f r e q u e n c y d i f f e r e n c e w * and the m o l e c u l e s e m i t a t t h i s f r e q u e n c y . The e m i t t e d r a d i a t i o n m i x e s w i t h t h e a p p l i e d r a d i a t i o n ( o f f r e q u e n c y w^) t o c r e a t e a b e a t s i g n a l a t t h e d e t e c t o r . The o v e r a l l scheme i s i n d i c a t e d i n f i g u r e 3. To c o m p l e t e t h e d i s c u s s i o n o f t h e t r a n s i e n t e m i s s i o n e x p e r i m e n t , e x p r e s s i o n s f o r p r ( a n d ^ ^ ^ ^ J must be o b t a i n e d t o i n s e r t i n t o e q u a t i o n ( 2 . 5 6 ) . T h r e e s u c h methods o f p r e p a r a t i o n a r e o u t l i n e d . The f i r s t method c o n s i s t s o f i r r a d i a t i n g t h e s y s t e m w i t h c o n t i n u o u s wave r a d i a t i o n u n t i l a s t e a d y s t a t e has been a c h i e v e d . The s o l u t i o n f o r t h i s c a s e has a l r e a d y been d i s c u s s e d ( s e e e q u a t i o n ( 2 . 3 3 ) ) and c o n s e q u e n t l y , t h e v a l u e s K 2 E Aw o (2.58) P /AN r G q 1 2 T l 2 2 +.(Aw) J + < - ± ) K ^ E T„ 2 ° K 2 E o 1 4 T p. /AN 1 G q 1 2 T l 2 2 * + (Awr + (—-) K E T „ 2 T 2 c a n be s u b s t i t u t e d i n t o e q u a t i o n ( 2 . 5 6 ) . 58 Stark 0 Emission A bsorption Emission 10 /xsec F i g u r e 3: D i a g r a m showing t r a n s i e n t a b s o r p t i o n ( f i e l d o f f ) and t r a n s i e n t e m i s s i o n ( f i e l d on) as o b t a i n e d from S t a r k s w i t c h i n g . (From McGurk e t a l . , Adv. Chem. P h y s . XXV, 1, (1975).) Two o t h e r methods o f p r e p a r a t i o n i n v o l v e p u l s e t e c h n i q u e s where t h e p u l s e t i m e s a r e assumed s h o r t i n com-p a r i s o n t o t h e r e l a x a t i o n t i m e s . T h u s , d u r i n g t h e e v o l u t i o n o f t h e p u l s e , t h e s y s t e m e q u a t i o n s (2.20) c a n be s i m p l i f i e d t o r e a d dP ( 2 . 5 9 ) -_E + A W P . - 0 f j ( *M) - E o P . = 0 ^ ± - A 0 3 P + K 2 E - 0 d t r o 4 The s o l u t i o n t o t h e s e e q u a t i o n s ( a g a i n e m p l o y i n g t h e method o f L a p l a c e t r a n s f o r m s and t h e i n i t i a l c o n d i t i o n s ( 2 . 3 1 ) ) , c a n be w r i t t e n as (2.60) P ( t , ) ^ 0 — ; AN s i n ( K E t j r 1 i i 4 • eq o l AN ( t , ) ' V A N c o s ( K . E . t . ) f o r K.E >>AOJ 1 eq o 1 o The two p u l s e t e c h n i q u e s t h e n c a n be e x p r e s s e d i n a n a l o g y t o NMR as < 2' 6 1> * i / 2 = " 2 I C E " P i U i r / 2 } = A N e q o P r ( t f / 2 ) = 0 N ( t v / 2 ) = 0 and 60 (2.62) t = IT K E W - 0 P ( t ) » 0 r i r AN(t ) = -AN IT eq Thus the e m i t t e d f i e l d i s a maximum f o l l o w i n g a ir/2 p u l s e and i s z e r o f o l l o w i n g a TT p u l s e . 61 ( i ) Summary I n c o n c l u s i o n , i t s h o u l d be m e n t i o n e d t h a t o t h e r e x p e r i m e n t a l s i t u a t i o n s have a l s o been s t u d i e d . F o r example, a d d i t i o n a l ( m u l t i p l e ) p u l s e e x p e r i m e n t s have been c a r r i e d 1 1 2 o u t ' i n a n a l o g y w i t h NMR. F u r t h e r , t h e t r e a t m e n t employed by F l y g a r e and c o w o r k e r s has n o t c o m p l e t e l y n e g l e c t e d t h e q u e s t i o n s o f m a g n e t i c quantum number (m) d e p e n d e n c e o r v e l o c i t y c o n t r i b u t i o n s . R a t h e r , we f e e l t h a t t h e i r t r e a t m e n t i n t h e s e a r e a s i s i n c o m p l e t e and i t i s t h e aim o f t h i s t h e s i s t o r e - e x a m i n e t h e s e e f f e c t s . CHAPTER I I I Senftleben-Beenakker E f f e c t s and the L i n e a r i z e d Waldmann-Snider C o l l i s i o n Superoperator "I c o u l d have done i t i n a much more complicated way" s a i d the Queen, immensely proud. \ 63 (a) I n t r o d u c t i o n C h a p t e r I I I makes a s e e m i n g l y a b r u p t d e p a r t u r e from the s p e c t r o s c o p i c d i s c u s s i o n s o f the p r e v i o u s two c h a p t e r s i n t h a t t h e s u b j e c t u n d e r d i s c u s s i o n h e r e i s t r a n s p o r t phenomena, where n o n - e q u i l i b r i u r a e f f e c t s a r e c a u s e d by v e l o c i t y and t e m p e r a t u r e g r a d i e n t s , i n s t e a d o f a p p l i e d r a d i a t i o n f i e l d s . . However, the g a s e o u s s y s t e m under s t u d y i s t h e same and the mechanism by w h i c h t h e s y s t e m r e t u r n s to e q u i l i b r i u m ( c o l l i s i o n s ) i s c o n s e q u e n t l y the same. T h i s c h a p t e r - i n d e e d , t h i s t h e s i s - shows t h a t much c a n be g a i n e d from a c l o s e r i n s p e c t i o n o f b o t h phenomena from a u n i f i e d p o i n t o f v i e w . Over the p a s t d e c a d e , i m p r o v e d e x p e r i m e n t a l t e c h n i q u e s and more d e t a i l e d o b s e r v a t i o n s have shown t h a t t r a n s p o r t phenomena and p r e s s u r e b r o a d e n i n g a r e more c l o s e l y a s s o c i a t e d 1 2 3 t h a n o r i g i n a l l y i m a g i n e d . S e n f t l e b e n - B e e n a k k e r s t u d i e s , ' ' w h e r e i n e x t e r n a l s t a t i c e l e c t r i c and m a g n e t i c f i e l d s m o d i f y t r a n s p o r t c o e f f i c i e n t s , have been p a r t i c u l a r l y w e l l d e v e l o p e d i n t h i s time s p a n . T h e s e show t h a t i n t e r n a l s t a t e s do a f f e c t t h e t ; i a n s l a t i o n a l t r a n s p o r t c o e f f i c i e n t s , , E q u a l l y w e l l , D o p p l e r e f f e c t s and v e l o c i t y r e l a x a t i o n s must be t a k e n i n t o a c c o u n t f o r a c o m p l e t e d e s c r i p t i o n o f s p e c t r o s c o p i c phenomena. Thus, i n e a c h i n s t a n c e , t h e r e a r e s e c o n d a r y mechanisms w h i c h c o u p l e t r a n s l a t i o n a l and i n t e r n a l d e g r e e s o f f r e e d o m . U s u a l k i n e t i c t h e o r y a p p r o a c h e s to t r a n s p o r t phenomena and S e n f t l e b e n - B e e n a k k e r e f f e c t s a r e e x t r e m e l y d e t a i l e d i n t h e i r t r e a t m e n t o f r o t a t i o n a l a s p e c t s and t r a n s l a t i o n a l e f f e c t s . T h i s i s i n d i r e c t c o n t r a s t to the s i m p l i f i e d 04 t r e a t m e n t o f the s y s t e m m o t i o n i n the s p e c t r o s c o p i c c a s e , d e s c r i b e d i n C h a p t e r I I . C h a p t e r I I I w i l l t h e r e f o r e o u t -l i n e t h e methods employed i n t h e d e s c r i p t i o n o f S e n f t l e b e n -B e e n a k k e r e f f e c t s w i t h an eye t o a p p l y i n g t h e s e c o n c e p t s t o the m i c r o w a v e phenomena i n s u b s e q u e n t c h a p t e r s . I n f a c t , t h i s c h a p t e r s e r v e s the d u a l p u r p o s e o f i n t r o d u c i n g n o t a t i o n and d e f i n i n g q u a n t i t i e s t h a t w i l l be u s e d a g a i n , as w e l l as p r e s e n t i n g S-B f o r m u l a e w h i c h c a n be d i r e c t l y compared w i t h " s p e c t r o s c o p i c " r e s u l t s . The p r e s e n t a t i o n o f S e n f t l e b e n - B e e n a k k e r e f f e c t s g i v e n h e r e i s a d m i t t e d l y somewhat l i m i t e d , i n t h a t the e m p h a s i s w i l l be on m a t h e m a t i c a l t e c h n i q u e s r a t h e r t h a n on the com-p l e t e d e s c r i p t i o n o f t h e a c t u a l p h y s i c a l phenomena. A f u l l e r a c c o u n t o f t h e t h e o r y and e x p e r i m e n t s c a n be f o u n d i n 1 . , . 1 , 2 , 3 , 4 s e v e r a l e x c e l l e n t r e v i e w s . 65 (b) The L i n e a r i z e d W a l d m a n n - S n i d e r E q u a t i o n An e q u a t i o n o f m o t i o n , a p p r o p r i a t e t o the s t u d y o f S e n f t l e b e n - B e e n a k k e r e f f e c t s , must f i r s t be e s t a b l i s h e d . The s t a r t i n g p o i n t i s the g e n e r a l i z e d quantum B o l t z m a n n e q u a t i o n o f S n i d e r and S a n c t u a r y ^ v a l i d f o r m o l e c u l e s w i t h i n t e r n a l s t a t e s t r u c t u r e , T h i s r e p r e s e n t s t h e most g e n e r a l e q u a t i o n o b t a i n e d w i t h i n t h e p h i l o s o p h y o f t h e o r i g i n a l B o l t z m a n n a p p r o a c h and i t s " d e r i v a t i o n " i s g i v e n i n a p p e n d i x A. The s u p e r o p e r a t o r n o t a t i o n employed i n e q u a t i o n (3.1) i s a l s o d i s c u s s e d i n t h i s a p p e n d i x . E q u a t i o n (3.1) i s now s p e c i a l i z e d t o o b t a i n t h e w o r k i n g e q u a t i o n f o r t h i s c h a p t e r - an e q u a t i o n u s u a l l y known a s 6 7 t h e W aldmann-Snider e q u a t i o n . ' To t h i s end, t h e d e s c r i p -t i o n o f t h e t r a n s l a t i o n a l d e g r e e s o f f r e e d o m i s c l a r i f i e d by 8 the i n t r o d u c t i o n o f t h e W i g n e r d i s t r i b u t i o n f u n c t i o n B e c a u s e o f t h e d e f i n i t i o n ( 3 . 2 ) , f ( r p _ t ) i s s t i l l an o p e r a -t o r i n i n t e r n a l s t a t e s p a c e . A l s o , i t s h o u l d be e m p h a s i z e d t h a t t h e t r a n s f o r m a t i o n (3.2) i s o n e - t o - o n e ( i . e . has a u n i q u e i n v e r s e ) and no i n f o r m a t i o n i s l o s t . I t i s an 9 example o f t h e Weyl c o r r e s p o n d e n c e between t r a n s l a t i o n a l (3.1) h o p e r a t o r s and ph a s e s p a c e f u n c t i o n s . The r e p r e s e n t a t i o n i n terras o f f ( r p _ t ) i s p r e f e r r e d t o one i n v o l v i n g p, i n t h a t a c l a s s i c a l i n t e r p r e t a t i o n c a n be a s s o c i a t e d w i t h f e v e n t h o u g h i t i s q u a n t a l i n n a t u r e . I n d e e d , the f r e e m o t i o n s u p e r o p e r a t o r /C / when s u b j e c t e d to t h i s t r a n s f o r m a t i o n , g i v e s r i s e t o b o t h a d r i f t term from the t r a n s l a t i o n a l s t a t e s and an i n t e r n a l s t a t e L i o u v i l l e o p e r a t o r oC . . The t r a n s -l n t f o r m e d c o l l i s i o n term i s much more c o m p l i c a t e d . I t i s most c o n v e n i e n t l y e x p r e s s e d u s i n g a p a r a m e t r i z a t i o n due t o B a e r w i n k l e and G r o s s m a n n 1 0 a c c o r d i n g t o t h e f o r m ^ 1 <3<3> Tl + £ ' hf ^ i n t f " J ( ^ t } W JL t I*i 6 3 (3.4) J ( r p t ) = - - ~ - (h t r / e x p [ - r ( x . k - ^'Z^JiP& f 1 (r+-|(y-x) » P+B-Jt) f 2 (r+|-[y+x] , p+B+k) dBdqdkdKdxdy Her e, {BfjkK) i s a s u p e r a t o r a c t i n g on i n t e r n a l s t a t e opera t o r s o n 1 y ,• na in e 1 y (3.5) J7(£3--)A 0 8 <&+%\ t l l c ' i " £ > A < ~ " ^ l H + l ^ " " 3 . > -<B+q|fi|k+K>A<k-K|t +|3-q> 67 T h i s i s a momentum r e p r e s e n t a t i o n of^ 7 . The c o l l i s i o n e x p r e s s -i o n (3.4) i s non l o c a l s i n c e t h e c o l l i d i n g m o l e c u l e s a r e n o t a t t h e same p o s i t i o n , nor a t t h e p o s i t i o n r o f the d i s t r i b u -t i o n f u n c t i o n i n t h e d r i f t terms on t h e l e f t hand s i d e o f ( 3 . 3 ) . F o r the d e s c r i p t i o n o f d i l u t e gas t r a n s p o r t phenomena, i t i s u s e f u l to employ a l o c a l i z a t i o n a p p r o x i m a t i o n by n e g l e c t i n g the jx and y p o s i t i o n d e p e n d e n c i e s i n f ^ and f ^ . T h a t i s , f i s assumed t o be a s l o w l y v a r y i n g f u n c t i o n o f p o s i t i o n . The n e t r e s u l t i s a l o c a l c o l l i s i o n o p e r a t o r o f t h e f o r m (3.6) J ( r g t ) •-*• J Q ( r p t ) = -6 4 T T 3 i 1 \ 2 t r 2 f^f(Boko) f 1 ( r , J'+B.-JO £ 2 ( r g + B + M d3dk 4a 2 , ~ / £ 2 ~ p £ + £ 2 - i ( 2 i r ) % z t r 2 / J - | ) , o , - j ^ - f ^ r p ^ ' t ) f 2 ( - ^ £ + £ 2 ' £ i t } dSi ' d £ 2 F u r t h e r , i t i s assumed t h a t t h e d i s t r i b u t i o n f u n c t i o n o f i n t e r e s t i s d i a g o n a l i n i n t e r n a l e n e r g y . T h i s a s s u m p t i o n l i b i s b a s e d on t h e c o n c e p t o f "phase r a n d o m i z a t i o n " w h i c h s t a t e s t h a t a l l o b s e r v a b l e i n t e r n a l s t a t e e f f e c t s c a n be d e s c r i b e d i n terms o f a d e n s i t y o p e r a t o r w h i c h a l m o s t commutes w i t h th e i n t e r n a l s t a t e h a m i l t o n i a n . The r e m a i n -i n g h i g h f r e q u e n c y p o r t i o n s o f t h e d e n s i t y o p e r a t o r , i f p r o d u c e d , v a n i s h so q u i c k l y due t o t h e i r r a p i d l y o s c i l l a t i n g f r e e m o t i o n , t h a t t h e y a r e n o r m a l l y n o t o b s e r v e d and hence c a n be d i s r e g a r d e d . D i a g o n a l i t y o f f ( r p t ) i n e n e r g y a l s o i m p l i e s t h a t t h e t r a n s i t i o n o p e r a t o r s i n e q u a t i o n (3.5) a r e on t h e e n e r g y s h e l l . C o u p l i n g t h i s s t a t e m e n t w i t h t h e L i p p m a n n - S c h w i n g e r i n t e g r a l e q u a t i o n (3.7) p|E> = {1 + £ i m + E * i £ t}|E>, £-*0 t h e W a l dmann-Snider c o l l i s i o n o p e r a t o r (3.8) 4+2 ( r D t ) = ( o JW S ( r p t   2 l T ) H 2 t r /dp. x [ / d p i . < n | _ ^ | - P 1 , > f 1 ( r £ 1 ' ) f 2 ( r p 4 p 2 - p 1 ' ) <=-f± - P 1 ' l < 5 ( E ) t + | ^ > , P,-P, ,Pp"P + ( 2TTi) 1{<=^f-\t|^|-^->f1(rp)f2(rp2) f x ( r p ) f 2 ( r p 2 ) < p 2 - p | t + | p 2 - p > } ] i s o b t a i n e d . The 6(E) f a c t o r e x p r e s s e s c o n s e r v a t i o n o f e n e r g y a c r o s s the t o p e r a t o r . F i n a l l y , i n t r a n s p o r t t h e o r y , i t i s u s u a l l y assumed t h a t t h e gas i s c l o s e t o l o c a l e q u i l i b r i u m , so t h a t t h e l i n e a r i z a t i o n 69 (3.9) f ( r p t ) = f ( 0 ) (1 + <|>) can b e • p e r f o r m e d . H e r e , ,2 , % n t . f ( o ) e - W e x P [ - — ] m l / 2 ( v . v , ( 3 * 1 0 ) ~ = , T l 3 / 2 ' Q ' " = ( I k T ) ( 2 TTm k T) ' a r e t h e l o c a l e q u i l i b r i u m W i g n e r d i s t r i b u t i o n f u n c t i o n d e n s i t y o p e r a t o r and t h e r e d u c e d p e c u l i a r v e l o c i t y , r e s p e c t i v e l y . The l o c a l d e n s i t y n, t h e mass o f t h e p a r t i c l e m, and t h e l o c a l t e m p e r a t u r e T a r e u s e d i n t h e above d e f i n i t i o n s . Note t h a t i n e q u a t i o n ( 3 . 9 ) , t h e r e i s no w o r r y a b o u t c o m m u t a t i o n p r o b l e m s between f a n d <j) s i n c e f i s assumed t o be d i a g o n a l i n i n t e r n a l e n e r g y . (A d i s c u s s i o n o f c o m m u t a t i o n t r o u b l e s t h a t c a n o c c u r upon l i n e a r i z a t i o n whenever o f f d i a g p n a l i t i e s i n i n t e r n a l e n e r g y do o c c u r i s g i v e n by 12 S n i d e r . ) The l i n e a r i z e d W aldmann-Snider c o l l i s i o n o p e r a t o r i s d e f i n e d as (3.11) J W S ( r p t ) -»• - f ( 0 ) (rpt)^?<f) w i t h , i n t h e u s u a l n o t a t i o n , (3.12) $<j> = - ( 2 T r ) 4 - f c 2 t r 2 / f 2 0 ) d p 2 { / t g , « j ) 1 ' + ( | > 2 ' ) 6 ( K ) t ^ d ( u q ' ) + + (2TTi)" 1[t g(<j) 1+<()_) - (* 1+4>,)t^ I> g J. z X z , g Here g and g* a r e t h e r e l a t i v e v e l o c i t i e s a f t e r and b e f o r e q c o l l i s i o n w h i l e t ^ , r e m a i n s an o p e r a t o r on i n t e r n a l s t a t e s , (3.13) < a |t g,|b> = <ayg|t|bug'> . Thus the l i n e a r i z e d W a l d mann-Snider e q u a t i o n t a k e s the form <*.!«) ( « + i / f i / i n t , * . . - T i T ( | ? . £ . | F } f < ° » S u b s t i t u t i o n o f t h e c o n s e r v a t i o n l a w s i n t o (3.14) g i v e s (3.15) i / R ^ I N T ) < J ) = X where X i s the s o u r c e t e r m f o r t h e non e q u i l i b r i u m s i t u a t i o n . T h i s i s c a l c u l a t e d to be (3.16) X . 2 [ W ] ( 2 ) : ( - V v ] ( 2 ) - ( (| - — ) ( W 2 - 3 / 2 ) -" ~ — o 3 c v 0 / . - < ? / > „ i n t m t i n J . ,„2 i n t i n t , ^ ]l'Z0 + z[l" ~ 3 / 2 ) + — c ~ ^ — ] V V • 1-VlnT) E q u a t i o n (3.15) r e p r e s e n t s t h e e q u a t i o n o f m o t i o n f o r t h e gas u n d e r t h e i n f l u e n c e o f e x t e r n a l g r a d i e n t s and r e p r e s e n t s t h e w o r k i n g e q u a t i o n o f c h a p t e r I I I . I t c o n t a i n s a w e l l d e f i n e d , t h o u g h a p p r o x i m a t e , c o l l i s i o n s u p e r o p e r a t o r $ , whose p r o -p e r t i e s a r e d i s c u s s e d f u r t h e r i n s e c t i o n s (d) and ( e ) . The a p p r o x i m a t i o n s i n v o l v e d i n a r r i v i n g a t ^ - namely l i n e a r i z a -t i o n , l o c a l i z a t i o n , and e n e r g y c o n s e r v a t i o n - have been p r e -s e n t e d i n some d e t a i l h e r e t o f a c i l i t a t e c o m p a r i s o n s w i t h t h e form o f t h e c o l l i s i o n s u p e r o p e r a t o r d i s c u s s e d i n c h a p t e r IV and V I I . (c) The V a r i o u s C h o i c e s o f B a s e s A c h o i c e o f b a s i s f o r t h e d e s c r i p t i o n o f t r a n s p o r t and S e n f t l e b e n - B e e n a k k e r phenomena i s now d i s c u s s e d . T h i s s e c -t i o n i s p r e s e n t e d n o t o n l y as a p r e l i m i n a r y to t h e r e m a i n i n g s e c t i o n s o f c h a p t e r I I I b u t a l s o as a summary o f the u n d e r -s t a n d i n g t h a t has d e v e l o p e d o v e r t h e p a s t d e c a d e r e g a r d i n g t h e a p p r o p r i a t e n e s s o f e a c h t y p e o f b a s i s . The " i n t u i t i o n " t h a t has thu s been g a i n e d f r o m a s t u d y o f the S e n f t l e b e n -B e e n a k k e r e f f e c t s , c a n t h e n be u s e d ( b e g i n n i n g i n c h a p t e r IV) f o r c h o o s i n g an a p p r o p r i a t e b a s i s f o r t h e d e s c r i p t i o n o f t h e s p e c t r o s c o p i c phenomena. F o r c o m p u t a t i o n a l p u r p o s e s , i t i s c o n v e n i e n t t o u s e a b a s i s t h a t i s o r t h o n o r m a l w i t h r e s p e c t t o t h e i n n e r p r o d u c t o f an a p p r o p r i a t e H i l b e r t s p a c e . I n t h i s c h a p t e r , a H i l b e r t s p a c e o f t e n s o r o p e r a t o r s A(W) i s i n v o l v e d - t h e s e o p e r a t o r s *>«» a r e b o t h t e n s o r v a l u e d f u n c t i o n s o f t h e r e d u c e d p e c u l i a r v e l o c i t y W and t e n s o r v a l u e d o p e r a t o r s , d i a g o n a l i n i n t e r n a l e n e r g y , o v e r t h e s p a c e o f i n t e r n a l s t a t e s o f one m o l e c u l e . The i n n e r p r o d u c t between any two s u c h t e n s o r o p e r a t o r s A. (W) and A „ ( W ) i s d e f i n e d as 1 ~ 2 ~ (0) 1/2 . (0) 1/2 (3.17) <<A. A >> = t r / d p ( - ) A. (W) ( ) A.(W) 1 2 ~ n 1 ~ n 2 ~ t t where i s t h e o p e r a t o r a d j o i n t and i s t h e t e n s o r t r a n s p o s e . b a s i s s e t s w h i c h a r e b o t h o r t h o g o n a l and n o r m a l i z e d i n t h i s i n n e r p r o d u c t a r e now d e f i n e d . The t r a n s l a t i o n a l m o t i o n i s d e s c r i b e d by a s e t o f t e n s o r v a l u e d f u n c t i o n s I J p S(W) o f W w h i c h a r e d e f i n e d a s 1 3 (3.18) •PS,„i _ C t t 1 / 2 r ( s + l ) < 2 p M i 2 p + l l } 1 / 2 L P + 1 / 2 ( W 2 ) [ W ] ( P ) r ( s + p + 3 / 2 ) 2 P + 1 ( p ! ) 2 S { ^ / 2 n s + l ) } 1 / 2 w P L P + l / 2 ^ ^ ( P ) ^ , l 2 P ( s + p + 3/2) s " 4 ^ ^ s p ( W ) exp[|^]V(P)(W) Here t h e v e l o c i t y m a g n i t u d e i s e x p r e s s e d i n terms o f combina-t i o n s o f a s s o c i a t e d L a g u e r r e p o l y n o m i a l s L P + 1 / / 2 ( w 2 ) o r e q u i v a l e n t l y i n t e r m s o f n o r m a l i z e d 3 - d i m e n s i o n a I h a r m o n i c o s c i l l a t o r wavef unc tions-j? (W) . The a n g u l a r d e s c r i p t i o n i s /' sp 14 g i v e n i n terms o f t h e i r r e d u c i b l e C a r t i s i a n t e n s o r s 1j ^ P^(W) w i t h components i n a s p h e r i c a l b a s i s ( 3 . 1 9 ) ^ ( P ) M ( W ) E e ( p ) m ( . ) P y < P ) (W) - 4 T T 1 / 2 i P Y P M ( W ) . He r e Y (W) a r e t h e s p h e r i c a l h a r m o n i c s o f Edmonds and t h e pm e.<p) m f o r m t n e c a r t e s i a n b a s i s f o r s p h e r i c a l t e n s o r s , a s P s d i s c u s s e d i n a p p e n d i x B. The L (w) b a s i s i s t h e o n l y one w h i c h has been employed i n gas k i n e t i c t h e o r y f o r t h e d e s c r i p t i o n o f the t r a n s l a t i o n a l d e g r e e s o f f r e e d o m . I n c o n t r a s t , t h r e e s e t s o f t e n s o r o p e r a t o r b a s i s have p r o v e n u s e f u l i n t h e d e s c r i p t i o n o f t h e i n t e r n a l s t a t e m o t i o n , 74 The f i r s t s u c h b a s i s r e c o g n i z e s the i n t e r n a l l e v e l s t r u c t u r e and i s the q r a n k t e n s o r (3.20) ( P j v ) ' 1 / 2 y ( q ) <J)<^J(vv), w i t h (vv) b e i n g the o p e r a t o r (3.21) (Pi(vv) = E|jmv><jmv|, composed o f p a r t i c u l a r k e t - b i a c o m b i n a t i o n s o f e i g e n s t a t e s o f t h e i n t e r n a l s t a t e H a m i l t o n i a n . H e r e j g i v e s t h e t o t a l a n g u l a r momentum m a g n i t u d e [ j ( j + l ) J , m g i v e s t h e com-p o n e n t m"fc o f a n g u l a r momentum i n t h e s p a c e f i x e d d i r e c t i o n , and v i s a c o m p r e h e n s i v e l a b e l f o r t h e o t h e r i n t e r n a l d e g r e e s o f f r e e d o m . C o n s i s t e n t w i t h the p i c t u r e o f phase r a n d o m i z a t i o n g i v e n e a r l i e r , t h e o p e r a t o r s |jmv><jmv| a r e c o n s i d e r e d t o be d i a g o n a l i n i n t e r n a l s t a t e e n e r g y , i n p a r t i c u l a r v and v l a b e l d e g e n e r a t e s t a t e s . A l s o i n e q u a t i o n (3.20)^ t h e B o l t z m a n n w e i g h t - E ( j m v ) / k T (3.22) P j v = - - (2j + l ) and t h e i r r e d u c i b l e C a r t e s i a n t e n s o r o p e r a t o r s ^ ' ^ ' ( j) have t i l been u s e d . T h e s e l a t t e r q u a n t i t i e s a r e q r a n k c o m b i n a t i o n s o f t h e a n g u l a r momentum v e c t o r o p e r a t o r J w h i c h p o s s e s s t h e c o n v e n i e n t n o r m a l i z a t i o n 75 (3.23) t r # j ( v v ) y ( q ) ( J ) ^ ( q , ) ( J ) } - ( 2 j + l ) 6 q q , E l q > where E ^ q ' i s t h e i d e m p o t e n t i d e n t i t y t e n s o r f o r symmetric t r a c e l e s s t e n s o r s o f r a n k q. The i d e n t i f i c a t i o n (3.24) ^ ( q l V ( J ) ^ J ( v v . ) ^ e ( q ) V ( . ) q y ( q , ( J ) ^ j ( v v ) = i q ( ( 2 j + l ) ( 2 q + l ) ] 1 / 2 J, (-1) j - m ( _l q I,) I j m v x j m ' v l c o m p l e t e s the d e f i n i t i o n o f t h e s e t e n s o r o p e r a t o r s . F u r t h e r p r o p e r t i e s a r e d i s c u s s e d i n a p p e n d i x B. (A g e n e r a l i z a t i o n o f th e t e n s o r o p e r a t o r s (3.24) w i l l p r o v e u s e f u l i n d e s c r i b i n g the s p e c t r o s c o p i c e f f e c t s a s d e v e l o p e d i n c h a p t e r IV.) A c o m b i n a t i o n o f (3.18) and (3.20) g i v e s t h e c o m p o s i t e t e n s o r o p e r a t o r b a s i s (3.25) A .- = L P S ( W ) (p. ) " 1 / 2 Z / ( q ) ( J $ j < v v ) p q s ] w ~ J v J w h i c h i s c a p a b l e o f d e s c r i b i n g s i m u l t a n e o u s t r a n s l a t i o n a l and i n t e r n a l s t a t e e f f e c t s . T h e s e t e n s o r s a r e o r t h o n o r m a l i n t h e i n n e r p r o d u c t ( 3 . 1 7 ) , namely (3.26) <<A .- IA , . , . ,-, ,>> p q s ] W p'q's'D? v * v ' = 6 < p q s j v v p ' q ' s ' j ' v ' V ) E ( Q ) ( . ) Q ( [ [) ( Q ) E ( P > ( ] ] ) ( Q ) T h i s b a s i s i s used i n t h e d i s c u s s i o n o f c o l l i s i o n s g i v e n i n s e c t i o n s (e) and ( f ) . A s e c o n d c h o i c e f o r t h e i n t e r n a l s t a t e d e s c r i p t i o n has been f o u n d u s e f u l i n t h e d i s c u s s i o n o f t h e s i m p l e r S e n f t l e b e n -B e e n a k k e r e f f e c t s . T h i s b a s i s i s s i m i l a r i n s t r u c t u r e t o t h e t r a n s l a t i o n a l b a s i s g i v e n i n e q u a t i o n ( 3 . 1 8 ) , h a v i n g t h e f o r m (3.27) R ( q ) Q ^ i n t ) [J] (<*> T h i s i s s e e n t o c o n s i s t o f i r r e d u c i b l e C a r t e s i a n t e n s o r s ( rt \ [J] and n o r m a l i z e d Wang-Chang-Uhlenbeck p o l y n o m i a l s R t ^ q * ^int^kT^ * T h e l a t t e r q u a n t i t i e s a r e o b t a i n e d f o r e a c h q by Gram-Schmidt o r t h o n o r m a l i z a t i o n o f t h e s e r i e s o f powers o f ^ £ n t / k T a c c o r d i n g t o (3.28) ( 2 q + l ) " 1 Q " 1 t r [ J J ( q ) ( - ) q [ J ] { q ) e " ^ i h t / k T R 1 ^ l^Bl)R^\^l) I n t e r m s o f ( 3 . 2 7 ) , t h e s e c o n d c o m p o s i t e b a s i s has t h e form (3.29) B S L p S ( W ) [ J ] ( q ) R< q ) <# / k T ) p q s t ~ t i n t = . E ( p . ) 1 / 2 A * l q ) (e . /kT) |jv * ] V p q s j v v t 3V and i s a l s o o r t h o n o r m a 1 i n t h e i n n e r p r o d u c t ( 3 . 1 7 ) . I n d e e d , (3.30) < <B |B , , u l » = 6 - ( p q s t | p ' q ' s ' f ) E ( q ) ( O q ( [ [ ) ( q ) E ( p ) ( ] ] ) q p q s t 1 p ' q ' s ' t * i s i m m e d i a t e l y v e r i f i e d . As the s e c o n d form o f (3.29) i l l u s -t r a t e s , the B ^ b a s i s i s e s s e n t i a l l y a p a r t i c u l a r a v e r a g e p q s t o f t h e A b a s i s . F o r t h i s r e a s o n , th e b a s i s (3.25) i s pqs j the more f u n d a m e n t a l one. The B ^ b a s i s has been u s e d t o p q s t d e s c r i b e t r a n s p o r t phenomena when the i n t e r n a l s t a t e e f f e c t s a r e n o t t o o c o m p l i c a t e d , see t h e n e x t s e c t i o n . W h i l e b o t h (3.25) and (3.29) have been employed i n t h e d i s c u s s i o n s o f low s t a t i c f i e l d S e n f t l e b e n - B e e n a k k e r e f f e c t s , h i g h s t a t i c f i e l d measurements have n e c e s s i t a t e d t h e i n t r o -d u c t i o n o f a t h i r d b a s i s s e t f o r a s a t i s f a c t o r y d e s c r i p t i o n o f t h e o b s e r v e d p h e n o m e n a . ^ I n t h i s i n s t a n c e , t h e o p e r a t o r s (3.31) -1/2 f f (vv) I n P j v ' 1 | jmv>*jihv|- ( 2 j + l ) J ( w ) f i - m , where t h e l e v e l p o p u l a t i o n (vv) has been e x p l i c i t l y c o n -s i d e r e d a s one b a s i s e l e m e n t . Note t h a t i n c o n t r a s t t o t h e i n t e r n a l s t a t e b a s e s (3.20) and ( 3 . 2 7 ) , t h i s b a s i s s e t d o e s n o t p o s s e s s the p r o p e r t y o f r o t a t i o n a l i n v a r i a n c e - one r e a s o n why t h i s b a s i s i s n o t employed u n l e s s a b s o l u t e l y n e c e s s a r y . I n d e e d , t h i s b a s i s i s n o t c o n s i d e r e d f u r t h e r i n c h a p t e r I I I , a l t h o u g h some comments on the a p p l i c a b i l i t y o f an a n a l o g o u s b a s i s f o r t h e d e s c r i p t i o n o f s p e c t r o s c o p i c e f f e c t s a r e g i v e n i n c h a p t e r s IV and V. I n summary, a l l o f t h e b a s e s ( 3 . 2 0 ) , (3.27) and (3.31) a r e e q u i v a l e n t i n t h a t t h e y a r e c o n n e c t e d by w e l l d e f i n e d t r a n s f o r m a t i o n s . However, the most a p p r o p r i a t e c h o i c e -t h e one r e q u i r i n g the f e w e s t terms to g i v e an a d e q u a t e r e p r e s e n t a t i o n o f t h e s y s t e m - d e p e n d s v e r y much on the s i t u a t i o n to be d e s c r i b e d . G e n e r a l l y , t h e b a s i s w h i c h most c l o s e l y r e s e m b l e s the f r e e m o t i o n e i g e n o p e r a t o r s i s c o n -s i d e r e d t h e most a p p r o p r i a t e , w i t h c o l l i s i o n a l c o n s i d e r a -t i o n s p l a y i n g a s e c o n d a r y r o l e . ^ ' ^ ' ^ A g a i n , n o t e t h a t a l l t h r e e b a s i s s e t s a r e d i a g o n a l i n j quantum numbers, c o n s i s t e n t w i t h the c o n c e p t o f p h a s e r a n d o m i z a t i o n m e n t i o n e d e a r l i e r . (d) Moment E q u a t i o n s - t h e She a r V i s c o s i t y C o e f f i c i e n t f o r N The t h i r d s t a g e i n t h e t r e a t m e n t o f the S e n f t l e b e n -B e e n a k k e r e f f e c t s i o the e s t a b l i s h m e n t o f the e q u a t i o n s o f m o t i o n f o r t h e e x p e c t a t i o n v a l u e s o f the r e l e v a n t b a s i s e l e m e n t s , s t a r t i n g f r o m e q u a t i o n (3.-15). S i n c e t h e p u r p o s e o f t h i s c h a p t e r i s to i n t r o d u c e a few u s e f u l a s p e c t s i n v o l v e d i n t h e t r e a t m e n t o f S-B e f f e c t s , one s i m p l e example w i l l -s u f f i c e . To t h i s end, t h e f i e l d d e p e n d e n c e o f the s h e a r v i s c o s i t y c o e f f i c i e n t f o r N^ i n a s m a l l m a g n e t i c f i e l d i s 3 18 c o n s i d e r e d . W h i l e t h i s p r o b l e m has been t r e a t e d b e f o r e , ' the p r e s e n t d e v e l o p m e n t i s s l i g h t l y d i f f e r e n t . T h i s p a r t o f c h a p t e r I I I i s p r o v i d e d f o r c o m p l e t e n e s s t o s t r e s s the p a r a l l e l i s m i n the t r e a t m e n t o f moment e q u a t i o n s f o r t r a n s -p o r t p r o p e r t i e s and f o r spec t r o s c o p i c ph e.nomena * The s y m m e t r i c t r a c e l e s s p a r t o f t h e v i s c o u s p r e s s u r e ( 2) t e n s o r I I ' i s raacroscopically. r e l a t e d t o the s y m m e t r i c ( 2 ) t r a c e l e s s p a r t o f t h e v e l o c i t y g r a d i e n t [V v ] by the p h e n o m e n o l o g i c a l e q u a t i o n (3.32) H ( 2 ) « -2Qi[Vvo] where JQ i s the p h e n o m e n o l o g i c a l s h e a r v i s c o s i t y c o e f f i c i e n t t e n s o r . The k i n e t i c t h e o r y o f g a s e s i d e n t i f i e s the p r e s s u r e t e n s o r as a m i c r o s c o p i c o r s t a t i s t i c a l a v e r a g e o f the symmetric ( 2) t r a c e l e s s p a r t o f t h e one m o l e c u l e momentum f l u x m[W] , namely 80 (3.33) n ( 2 ) = — T t r / m [ W ] ( 2 ) f dp ~ m ~ — 2nkT r r r 1 (2) f ( Q ) . ' = ^ 2 t r / / 2 [W] — <f> dp Her e , $ r e p r e s e n t s t h e s o l u t i o n t o t h a t p a r t o f e q u a t i o n (3.15) w h i c h i s r e s p o n s i b l e f o r t h e momentum f l u x . The e q u a t i o n t o be s o l v e d i s t h e r e f o r e (3.34) 2 [ W ] ( 2 ) : t - V v ] = ( $ +\ I . ) «J> . ~ ——o v >> i n t A c o m p a r i s o n o f t h e p h e n o m e n o l o g i c a l and s t a t i s t i c a l d e f i n i -(2) t i o n s o f II a l l o w s a k i n e t i c t h e o r y e v a l u a t i o n o f t h e p h e n o m e n o l o g i c a l JQ i n terms o f t h e s o l u t i o n o f e q u a t i o n (3.34) . The s o l u t i o n i s o b t a i n e d by a moment method w h e r e i n an a d e q u a t e d e s c r i p t i o n e m p l o y i n g t h e s m a l l e s t number o f moments i s d e s i r e d . F o r t h e c a s e o f N 2 i n a s m a l l m a g n e t i c f i e l d , t h e u s e o f two t e n s o r moments has g i v e n good a g r e e m e n t w i t h e x p e r i m e n t . I n te r m s o f t h e o r t h o n o r m a l b a s i s s e t ( 3 . 2 9 ) , a two moment a n s a t z expands t h e d i s t r i b u t i o n f u n c t i o n <j> i n t h e fo r m (3.35) * = < B 2 0 0 0 > : B 2 0 0 0 + < B Q 2 0 0 > : B Q 2 0 0 , where t h e e x p e c t a t i o n v a l u e s <A> a r e d e f i n e d a s (3.36) <A> = <<<|>|A>> f o r any operator A. In p a r t i c u l a r , note that equation (3.33 can be r e w r i t t e n i n the form ( 3. 37> n<2) - 2|kT < B 2 o Q o > S u b s t i t u t i o n of equation (3.35) i n t o the equation of motion (3.34) and employing the s p h e r i c a l b a s i s t e n s o r s e f ^ ^ allows the set of coupled moment equations to be w r i t t e n i n component form as (3.38) < B 2 0 0 0 > U + * < B 0 2 0 0 > y " - / 2 ^ ^ o j ( 2 ) y <B 0 2 0 0>^ + r i|-<B 2 0 0 0^ - i y . T i n < B 0 2 0 0 > ^ - 0 In equation (3.38), the r e l a x a t i o n parameters < 3 ' 3 9 > < < B 2 0 0 0 ^ B 2 0 0 0 > > = ? <<B \P\B >> 1 „(2) 0 2 00"^' 0200 T. i n t «Bo2ooWB2ooo» - I E<2> have been employed. F u r t h e r , the p a r t i c u l a r form of ^ i n t f o r N 2 i n a weak magnetic f i e l d has allowed the e v a l u a t i o n (3.40) /. 8 jV = -YH'[ J , B y o ] K i n t 0200 z o2oo " - y W % 0 2 0 0 to be p e r f o r m e d . The c o u p l e d e q u a t i o n s (3.38) have t h e s o l u t i o n (3.41). < B 2 0 0 0 > = 2/2 j [ - l ) L a ) ( 2 ) U x i - i y « T i n t ' - /2 T [_Vv ] ( 2 ) U + / ^ T i n t / 2 [ - V v ] ( 2 ) U f o r w o 1-iywx. . ""o i n t The a p p r o x i m a t i o n employed i n e q u a t i o n (3.41) seems t o be v e r i f i e d by e x p e r i m e n t and b a s i c a l l y s t a t e s t h a t t h e c o l l i s i o n a l c o u p l i n g between v e l o c i t y and i n t e r n a l a n g u l a r momentum d i r e c t i o n s i s weak. S u b s t i t u t i o n o f (3.41) i n t o (3.37) g i v e s , f o r t h e two moment a n s a t z , t h e s h e a r p r e s s u r e t e n s o r (3.42) I I ( 2 ) - Z e ( 2 ) - y ( - l ) V 2 ) U - 2nkT T I e ( 2 ) - V i - l ) e ( 2 ) ^ ( • ) 2 [ - V V Q ] ( 2 ) ,„ _ ( 2 ) - y , ,.y T i n t _ ( 2 ) y + * > 2 n k T J e - ' ^ - i > ^ _ ^ O l - V v , i n t C o m p a r i s o n o f (3.42) and (3.32) i m p l i e s t h a t t h e s h e a r v i s c o s i t y c o e f f i c i e n t t e n s o r f o r N i n t h e p r e s e n c e o f a 8 3 weak magnetic f i e l d has the form ( 3 . 4 3 ) J3 = T ~ E ( 2 ) + A n , where ( 3 . 4 4 ) n = nkTT and ( 3 . 4 5 ) A n - ***nkT g . e ( 2 ) - ^ - l ) V T - ^ - e ( 2 ) ^ « i n t Ex p r e s s i o n s ( 3 . 4 3 ) , ( 3 . 4 4 ) and ( 3 . 4 5 ) show t h a t i n the presence of a weak magnetic f i e l d , there are f i v e indepen-dent v i s c o s i t y c o e f f i c i e n t s which are produced by the c o l l i s i o n a l c o u p l i n g of v e l o c i t y and i n t e r n a l angular momentum d i r e c t i o n s . F u r t h e r , these complex c o e f f i c i e n t s A n ^ can be d i v i d e d i n t o t h e i r r e a l and imaginary p a r t s a c c o r d i n g to An, An.+ A n -y V i i ri ( 3 . 4 6 ) — J l + i — J i where ( 3 4 7 ) K . i ! i T 2 { j } - T T i n t \ 2 2 2S A n " •>! „ y w x . ^ - T i n t , 2 2 2 n 1+y OJ T . i n t T h e s e c o e f f i c i e n t s a r e p l o t t e d i n f i g u r e 4 as a f u n c t i o n o f UWT. and a r e s e e n t o behave as L o r e n t z a b s o r p t i o n and d i s I n t p e r s i o n s h a p e s , r e s p e c t i v e l y . They a r e i n v e r y good a g r e e -ment w i t h t h e e x p e r i m e n t a l c u r v e s f o r N . F i g u r e 4: E v e n and odd v i s c o s i t y c o e f f i c i e n t s f o r 36 (e) C o l l i s i o n a l E x p r e s s i o n s I n t h e d e s c r i p t i o n o f t r a n s p o r t phenomena, m a t r i x e l e m e n t s o f t h e l i n e a r i z e d W a ldmann-Snider c o l l i s i o n s u p e r o p e r a t o r a r i s e n a t u r a l l y . I n p a r t i c u l a r , f o r t h e e v a l u a t i o n o f t h e v i s c o s i t y c o e f f i c i e n t s o f t h e t h r e e c o l l i s i o n a l p a r a -m e t e r s T, T. ^, and \b a r e i n v o l v e d . Thus the c a l c u l a t i o n o f i n t c o l l i s i o n m a t r i x e l e m e n t s i s r e q u i r e d and, i n d e e d , r e p r e s e n t s an i m p o r t a n t a s p e c t o f k i n e t i c t h e o r y . T h i s has been t h e s u b j e c t o f d e t a i l e d i n v e s t i g a t i o n s - f i r s t f o r d i a m a g n e t i c 13 19 d i a t o m i c s and l a t e r e x t e n d e d t o g e n e r a l p o l y a t o m i c s . I n t h e l a s t two s e c t i o n s o f c h a p t e r I I I , some o f t h e s e c o n s i d e r a -t i o n s a r e p r e s e n t e d w i t h t h e i d e a o f r e a p p l y i n g t h i s t e c h n i c a l e x p e r t i s e i n l a t e r c h a p t e r s t o t h e c o l l i s i o n i n t e g r a l s o f p r e s s u r e b r o a d e n i n g and r e l a t e d phenomena. The l i n e a r i z e d W a l d mann-Snider c o l l i s i o n s u p e r o p e r a t o r has s e v e r a l g e n e r a l symmetry p r o p e r t i e s ^ ' ^ ' 2 ^ t h a t r e d u c e t h e number o f i n d e p e n d e n t m a t r i x e l e m e n t s . The u s u a l c a s e where $ c a n be assumed i n d e p e n d e n t o f t h e e x t e r n a l s t a t i c e l e c t r i c and m a g n e t i c f i e l d s i s t r e a t e d h e r e . The p a r i t y s u p e r o p e r a t o r i s d e f i n e d s u c h t h a t (3.48) // A(W) £ -A A(-W) where 7\ i s the p a r i t y o p e r a t o r a c t i n g on quantum m e c h a n i c a l 5 i n t e r n a l s t a t e s . I t c a n t h e n be shown t h a t (3.49) A ?f- 71 ft. 87 and h e n c e ^ does n o t mix o p e r a t o r s o f d i f f e r e n t p a r i t y . The t i m e r e v e r s a l s u p e r o p e r a t o r $ , d e f i n e d as (3.50) Q A(W) = O A f - W j e " 1 where 9 i s t h e i n t e r n a l s t a t e a n t i l i n e a r t i m e r e v e r s a l o p e r -a t o r , d o e s n o t commute w i t h (jf , however. I n d e e d , g i m i l a r c o n s i d e r a t i o n s ^ e s t a b l i s h ( 3 . 5 1 ) 0 ^ = 6 ^ / 9 where ^ ^ i s t h e s u p e r o p e r a t o r a d j o i n t o f ^ . E q u a t i o n (3.51) i m p l i e s t h a t ^ c a n mix o p e r a t o r s o f d i f f e r e n t t i m e r e v e r s a l symmetry. The d e f i n i t i o n o f t h e r o t a t i o n s u p e r o p e r a t o r R i s g i v e n by (3.52) R A(W) = r A ( R - W ) r - 1 i n t e r m s o f a 3x3 C a r t e s i a n r o t a t i o n m a t r i x R and t h e i n t e r n a l s t a t e r o t a t i o n o p e r a t o r r . Then t h e r o t a t i o n a l p r o p e r t i e s o f $ a r e s t a t e d s u c c i n c t l y a s 1 3 (3 .53) R (H = <K R and i s an i s o t r o p i c s u p e r o p e r a t o r w h i c h c a n c o u p l e o n l y t e n s o r o p e r a t o r s o f t h e same r o t a t i o n a l symmetry. More d e t a i l e d a s p e c t s o f r o t a t i o n a l i n v a r i a n c e o f ^ a r e e x p l o r e d m o m e n t a r i l y . C o l l i s i o n i n t e g r a l s < < A | ^ | A'>> have t h e u n i t s o f i n v e r s e t i m e and t h u s c a n be c o n s i d e r e d a s c o l l i s i o n r a t e s o r f r e q u e n c i e s . I t i s t r a d i t i o n a l t o e x p r e s s t h e s e m a t r i x e l e m e n t s as k i n e t i c c r o s s s e c t i o n s (3.54) £ D (pqsjvv' I p'q's'j*v"v«") = n 1 ( f k T ) 1 / 2 < < A p q s j v v ' l ^ l A p ' q " s ' j ' v " v ' " > > 8 kT 1/2 Here (~—) i s the a v e r a g e r e l a t i v e v e l o c i t y , n i s t h e d e n s i t y , and t h e b a s i s c h o s e n i s d e f i n e d i n e q u a t i o n (3.25) M a t r i x e l e m e n t s i n t h e s e c o n d b a s i s (3.29) a r e j u s t a p p r o -p r i a t e l y a v e r a g e d v e r s i o n s o f ( 3 . 5 4 ) . • The e f f e c t s o o f t h e r o t a t i o n a l i n v a r i a n c e p r o p e r t y o f on t h e k i n e t i c c r o s s s e c t i o n s i s now p r e s e n t e d i n d e t a i l . The t e n s o r c r o s s s e c t i o n s o f e q u a t i o n (3.54) c a n be e x p r e s s e d i n terms o f a se t £"> ( | - -) o f s c a l a r c r o s s s e c t i o n s by t h e r e l a t i o n (3.55) £ ( p q s j v v [p'q's* j ' v ' v ' ) = | ( - 1 ) k + q + q ' f l ( k q q ' ) 1 / 2 fi(kpp') 1 / 2 V(qkq') ( . ) q ' + k n ( q , ) V ( k p p ' ) ] ] ( q , ) { ^ ( p q s j v v I p'q's'j'vW ) k where V(kpp') a r e C a r t e s i a n 3 - j t e n s o r s and fi(kpp') a r e t h e i r 13 14 n o r m a l i z a t i o n f a c t o r s . ' I n e q u a t i o n (3.55) t h e c o u p l i n g scheme c h o s e n i s one w h i c h f i r s t c o u p l e s the v e l o c i t y t e n s o r s 89 t o g e t h e r and l i k e w i s e f o r the a n g u l a r momentum t e n s o r s , and t h e n c o n s i d e r s the d i r e c t i o n a l c o r r e l a t i o n between v e l o c i t y and a n g u l a r momentum s p a c e s . The c h o i c e o f c o u p l i n g scheme i s p r e d i c a t e d on the i d e a t h a t t h e e f f e c t s o f t h e non-s p h e r i c a l p a r t o f t h e i n t e r m o l e c u l a r p o t e n t i a l a r e weak. I n d e e d , i f t h e i n t e r m o l e c u l a r p o t e n t i a l were i s o t r o p i c , t h e n o n l y the k=0 component would r e s u l t . Thus k^O c o n -t r i b u t i o n s measure t h e c o l l i s i o n a l c o u p l i n g between v e l o c i t y and a n g u l a r momentum d i r e c t i o n s and a r e e x p e c t e d to be s m a l l e r t h a n the k=0 t e r m s . I n f a c t , t h e a p p r o x i m a t i o n i|/'*ty<<l u s e d i n t h e p r e v i o u s s e c t i o n i s a p a r t i c u l a r example o f t h i s t y p e o f r e a s o n i n g . The s c a l a r c o e f f i c i e n t s i n (3.55) a r e g i v e n by (3.56) G ( p q s j v v l p ' q ' s ' j ' v ' v ' ) = ( - 1 ) k + P + P ' ( 2 k + l ) f t ( k q « q ) ' 1 / 2 ft(kp'P)1/2V(qIkq) ( - ) k + q [ [ ( q ) V ( k p ' p ) ] ] ( q ) ( • ) P + p , + q + q ' S ( p q s j v v | p « q ' s ' j ' v ' v ' ) S u b s t i t u t i o n o f (3.25) i n t o (3.54) and t h e n i n t o (3.56) a l l o w s the i n t e g r a l o v e r t h e c e n t r e o f mass momentum t o be e v a l u a t e d . The t r a n s f o r m a t i o n s u s e d a r e (3.57) W ' + W ' = W. + W .«= /2 if W2 - V1 = / 2 y W 2 ' - V? 1' '= /2-v' , m_. 1/2 and t h e r e s u l t ( f o l l o w i n g t h e methods o f Chen, M o r a a l and S n i d e r ) i s (3.58) ( p q s j v v | p ' q ' s ' j 1 v * v ) O w n E f « U ^ ' > > 1 / 2 T ( k ) O l q n j v v l J t ' q ' n ' j ' v ' v ' ) , + (-1) ^  " ( £qn j v v | I ' q • n ' j * v ' v « ) } - ( ? U 1 1 Z fft(k&&') } 1 / 2 (k) { j $ j l v . < P j v P j t v , ) 1 / 2 y l ' ( ^ n j v v j v 2 q U ' n ' j ' v ' v ' J 2 v 2 q ' ) k J 2 2 J 2 2 J 2 2 J 2 2 + (-1)*' E (p P . , v , ) 1 / 2 r " ( ^ n j v v j v q|£'n'j'v'j'v'v ' q ' ) j 2 v 2 =>2 2 3 l x * i V i The s e c o n d form r e p r e s e n t s t h e u s u a l p r e s e n t a t i o n w h i l e t h e f i r s t f o r m i s t h a t p r e f e r r e d by t h e p r e s e n t a u t h o r . In (k) e q u a t i o n ( 3 . 5 8 ) , the t r a n s f o r m a t i o n I c o n n e c t s t h e d e s -c r i p t i o n i n terms o f t h e i n i t i a l and f i n a l moments o f one m o l e c u l e ( r e p r e s e n t e d by the i n d i c e s p s p ' s ' ) , w i t h t h e d e s -c r i p t i o n s ^ ^ r i n terms o f t h e i n i t i a l and f i n a l r e l a t i v e moments o f t h e c o l l i d i n g p a i r ( r e p r e s e n t e d by t h e i n d i c e s ZnH'n'). T h i s i s a n e c e s s a r y c o m p l i c a t i n g f a c t o r due t o t h e p r e s e n c e o f v e l o c i t y p o l a r i z a t i o n s and t h e i r r o t a t i o n a l p r o p e r t i e s - i t r e d u c e s to t h e i d e n t i t y t r a n s f o r m a t i o n when 91 (k) p=s=p ,=s'=0. R e f e r e n c e 13, a p p e n d i x B d e f i n e s I and g i v e s a f o r m u l a f o r i t s e v a l u a t i o n . The two e f f e c t i v e s c a l a r c r o s s s e c t i o n s i n t r o d u c e d i n e q u a t i o n (3.58) a r e g i v e n by (3.59) ( f c q n j ^ v j A ' q ' n ' j J v J v J ) 1/2 - ( - l ) : ( - l ) q + q , ( 2 i r , V ( ^ ) -hf** • ) - 1 / 2 f l ( k ^ . ) - 1 / 2 if P j 2 V 2 P j 2 V 2 , 1 / 2 V ( q ' q k ) ( . ) k V ( k £ ' i t ) + 1 { ( 2 j 2 + l ) X ( 2 j ' + l ) } j 2 v 2 i ' v • J 2 2 {/d Y / d ( y g ' ) . e x p [ - i ( Y 2 + Y ( 2 ) ) ] L * " ( Y ) L * ' n ' ( Y ' ) Z/ ( Q ) ( J ) t r 1 t r 2 C ^ 1 ( v 1 v 1 ) ^ ^ 2 ( v 2 v 2 ) ^ ^ 1 / 2 < y g 1 11 Ug • > (2 j ^ l ) ^ ^ K v , ' / i2iv'v'y^- (",1,)<V9' | 6(K) t + | y g > ] 1 v v i v i ' " 2 1 2 2' 1/2 J. 1 1 Z 2 2 ( 2 j £ + 1 ) - L / ' 4 2 + ( - l ) q + q ' ( 2 T T i ) * 1 6 ( J 1 J 2 | j { j 2 ) 6 ( v 2 | v « ) fdy L £ n (y)L 1 ' n ' (y) t r . t r ^ T 7 2 { < v g 111 y g x ^ 9 ( v • v )fl2 (v v ) 1 2 ( 2 j 1 + l ) 1 / 2 "~ - . 1 2 < y g | t + | y g > 6 ( v 1 ! v ^ ) } ] } 9 2 and (3.60) " ( f c q n j j V j v J A ' q ' n ' j ^ v ' v ' ) 1/2 - <-l>(-l ) * + ^ 2 i r)V<j^>'.. - 3 ) 2 " ( k q q ' ) - 1 / 2 « ( k ^ ' ) - 1 / 2 P J 2 v 2 > V 1 / 2 V ( q ' q k ) ( . ) k V ( k £ a ) + ? { ( 2 j + l ) X ( 2 j ' + l ) } -•52 2 3 1 1 {fay fd(\ig') exp [ - | ( y 2+Y ' 2 ) ]L % n (Y)L * ' n ' (Y ') 14 ( Q ) (J ) t r 1 t r 2 [ ^ ( v 1 - 1 ^ 2 2 ( V 2 V 2 > ; ^ l / a ^ g l t l ^ ^ (2j 1+l) P" 5!, , , , i 0 j 2 / - , ..^/ ( ~ 2 } <Ug' |6(K) t + [ u g > ] 1 1 1 2 2 2 ( 2 j 2 + l ) 1 / 2 2 + ( - l ) q + q ' ( 2 T T i ) - 1 6 ( J 1 j j j ' j l ) / d Y e~ V..L* r i<Y>L A , n , ' m t r l t r 2 t ^ 7 ^ 2 < U l l t l V 2 > l ! C , l 1 ( , ' i ; ; i > ^ 2 2 < ; ' 2 v 2 ' 2 2 <yg| t + | y g > 6 ( v 1 v 2 | v ^ v 2 ) }) } 93 P h y s i c a l l y , the c r o s s s e c t i o n ^ ' d e s c r i b e s e f f e c t s where t h e p o l a r i z a t i o n s , b o t h b e f o r e and a f t e r t h e c o l l i s i o n , r e f e r t o t h e same m o l e c u l e . C o l l i s i o n p r o c e s s e s w h e r e i n t h e p o l a r i z a -t i o n s a r e p a s s e d from one m o l e c u l e t o t h e o t h e r a r e a c c o u n t e d r e s o n a n t p r o c e s s e s f a l l i n t o t h i s l a t t e r c a t e g o r y . As has been s e e n , t h i s s u b d i v i s i o n o f c o l l i s i o n a l e f f e c t s i s a n a t u r a l one w h i c h f o l l o w s d i r e c t l y f r o m t h e l i n e a r i z a t i o n o f t h e a p p r o p r i a t e quantum m e c h a n i c a l B o l t z m a n n e q u a t i o n . The e v a l u a t i o n o f t h e a n g u l a r i n t e g r a l s and t h e i n t e r n a l s t a t e t r a c e s i n e q u a t i o n s (3.59) and (3.60) p r o d u c e s t r a n s i -t i o n o p e r a t o r m a t r i x e l e m e n t s i n an a n g u l a r momentum r e p r e s e n -t a t i o n . More p a r t i c u l a r l y , t h e u s e o f where Y, (p) a r e t h e s p h e r i c a l h a r m o n i c s o f Edmonds and f o r i n t h e " t r a n s f e r c r o s s sec t i o n s " , 5~" In p a r t i c u l a r , (3.61) |ug> = I |pXs><Xs|p> = £ |pXs> i " Y* (p) AS A S AS 21 (3.62) < 3 i m i v 1 3 2 m 2 V 2 | A | j i m i V l j 2 m 2 V 2 > g i v e s e x p r e s s i o n s forZ>', a n d j > " i n t e r m s o f t m a t r i x e l e m e n t s of the form (3.63) < J 1 m 1 v 1 J 2 m 2 v 2 p X s | t | j ^ m ^ v ^ j ^ m ^ v ^ p U ' s ^ 94 B u t t h e t r a n s i t i o n o p e r a t o r i t s e l f i s a l s o a r o t a t i o n a l l y 2 2 i n v a r i a n t q u a n t i t y . I t f o l l o w s t h a t t h e m a t r i x e l e m e n t s (3.63) c a n be e x p r e s s e d i n t e r m s o f r e d u c e d ( s c a l a r ) quan-19 2 2 t i t i e s . S e v e r a l c o u p l i n g schemes have been d e s c r i b e d ' f o r t h e i r d e c o m p o s i t i o n . I n ^ t h i s c h a p t e r , o n l y th e t r a n s l a -t i o n a l - i n t e r n a l c o u p l i n g scheme i s u s e d s i n c e i t i s t h e scheme most s u i t a b l e f o r t h e d i s t o r t e d wave B o r n a p p r o x i -m a t i o n ( g i v e n b e l o w ) . T h u s , the m a t r i x e l e m e n t (3.63) i n t h e t r a n s l a t i o n a l - i n t e r n a l c o u p l i n g scheme i s w r i t t e n a s (3.64) < J 1 m 1 v 1 J 2 m 2 v 2 p X s | t | j j m ^ v ^ J 2 m 2 v 2 p ' X ' s ^ I O . .1 , -1/2, . , j l + j 2 + X+ n + j2 + A' • „ m i + m 2 + S = (27TiU) (pp') ( - 1 ) (-1) [ ( 2 j ' + 1 ) ( 2 J 1 + 1 ) ( 2 X « + l ) ] 1 / 2 Z [(2< + 1 ) ( 2 / + 1 ( 2 / + 1 ] 1 / 2 L \ L 2 L > A Tin-* v ' i ' v 1 X ' ; / / Z; i v i v X) l— * J 1 1 J2'2 1 2 3 1 1 J2 2 ' where / - " i s the Y u t s i s , L e v i n s o n , and V a n a g a s 2 3 t y p e d i a g r a m , see f i g u r e 5. I t s h o u l d be n o t e d t h a t t h e i n d i c e s oh t h e r i g h t hand s i d e o f e q u a t i o n (3.64) a r e i n t h e r e v e r s e o r d e r t o t h o s e on t h e l e f t hand s i d e - t h i s s h o u l d be k e p t i n mind when i n t e r p r e t i n g f o r m u l a e i n v o l v i n g t h e r e d u c e d T m a t r i x e l e m e n t s . S u b s t i t u t i o n o f (3.64) i n t o (3.59) and (3.60) a l l o w s a l l m a g n e t i c summations t o be p e r f o r m e d e x p l i c i t l y , t o 19 y i e l d t h e t o t a l l y ( r o t a t i o n a l ) i n v a r i a n t e x p r e s s i o n s . F i g u r e 5: C o u p l i n e d i a g r a m 96 65) It ' (Sqnj.jVjvJfc'q'n' j j v j v j ) - ( - l ) ( - l ) k + q + q , f t ( k q q ' ) - 1 / 2 i q - q , [ ( 2 q + l ) ( 2 q ' + l ) ] 1 / 2 t (2j.+1) ( 2 j ' + 1 ) ] 1 / 2 2 [(2/.+1) (2/'+l) (2/+1) ( 2 Z ' + 1 ) ] 1 / 2  1 1 I ll2 L 1 1 E P V 2 P ^ 2 V 2 1 / 2 '< i • v • J 2 2 q* q * j i j i J i J i A l ft(k2,&•)'1/2(-i);!' + £ , [<2£ + l ) ( 2 £ ' + l ) ] 1 / 2 { ( 2 j 2 + l ) ( 2 j 2 + l ) } 1 / 2 ( 2 3 ^ l ) ( 2 j 2 + l ) 1 / 2 ( 2 J . + 1 ) ( 2 j _ + l ) y ( 8 k T } / d Y Y V d g ' ( g V ) - J L _ ^ £ «Y) ^ . £ (Y ' ) 6 (B +B +HS E . gg ' 1 1 2 1 JL 1 ,2 -E. X X . X " X - » • H2X+1) (2X " + !)] ""(2A-+1) <2A«<t-l) (-1) 1/2 X+X1 j „v„ 2 J 2 2 X • X •" SL • X X" I ( ) ( ) 0 0 0 0 0 0 X' l x I' k SL X'"/ • X" T ^ v ^ ^ X - / ^ / ; ^ ^ P i v + £ J 2 2 6 | j ( j , ) 6 ( v | v . , n ( k . j - 1 / 2 q+q' j , v , ( 2 j +1 V J 1 J 2 1 J 1 J 2 ; v 2 1 2' 1 i H ' J 2 2 . J 2 J 2 2 3 1 q 3 1 X 2J.+1 1/2 [ < 2 q + l > ( 2 q ' + 1 , ] 1 / 2 ^ ^ ( ^ l , X , 7 2 J 2 + l , l 6 ( v i | v i , 2* ( | ^ ) 1 / 2 fi(kur1/'2(-i)W,[(2Ui„2£,i„^ ( k 1 *'> y 8 k T 0 0 0 Y2 1/? X" k X X " / d Y h ^ n £ ( Y ) ^ n ' £ ' ( Y ) Z „ C 2 X + l ) X / ' i ( 2 X » + l ) (-1) ( ) M g n X" n * XX 0 0 0 T( J 1 v ^ J 2 v 2 X " ; k o k ; J 1 v 1 J 2 v 2 X ) + 6 ( V ; L |v^) ( - l ) k + q + q ' i l j 2 ( ^ H ) 1 / 2 fi(k?^')"1/2(+i)£+£'t(2£+i) ( 2 £ ' + l ) ] 1 / 2 ( k ) y 8kT 0 0 0 Y2 E 1/2 X" k X X " / d Y f e ^ n £ ( Y ) ^ n ' i f ( Y ) X X « ( 2 X + 1 ) (2X" + D ( - 1 ) ( q Q q > ™ j l V l j 2 V 2 X " ; k 0 k ; j l V l j 2 V 2 X ) ] and (3.66) B " ( J l q n j 1 v 1 v 1 I J l ' q ' n ' J ^ ^ ^ k = ( - 1 ) ( - l ) q + q , n ( k q q ' ) " 1 / 2 i q ' " q [ ( 2 q + l ) ( 2 q ' + l ) ] 1 / 2 [ ( 2 j . + 1 ) ( 2 j 1 + 1 ) ] 1 / 2 Z . [ ( 2 / +1) ( 2 / +1) (2/ + 1)] 1 / 2 x • • ^ 1^2 ^ X i -^2 X* 1/2 [ ( 2 ^ + 1) ( 2 / 2 + l ) ( 2 £ ' + l ) ] X / Z (-1) 3 2 V 2 P 1/2 *2 V2 j 2 v 2 { - ( 2 J 2 + l) ( 2 j i + l ) j { v j > (-D j l + j 2 + j i + j 2 D 2 / 2 J 2 q / f ^ 1)} 1/2 * 2 q ' l 2 j / i q -1/2 £ + £' ' 1/2 ( 2 j ' + l) (2J-+1) 1/2 ft(k££') ^ ( - i ) [(2£ + l) (2A'+1) ] 1 / 2 [ — ! — — £ _ ] (23^ + 1) ( 2 3 2 + l) ( ^ , 1 / 2 ^ Y 2 / a g . ( g . ) 2 - ^ ^ ( Y i ^ . ^ i y , 5 i v i " 3 2 V 2 ' 2 V l j 2 V 2 2 AA X A ) , E „ ,„[ (2A + 1) (2A- + 1) ] 1 (2A'+l) (2A"' + l) (-l)X+A'" ( A • A •*' E • A A • I ) ( ) O O O 0 0 0 A ' / A 1 U ' k ^ A / ' A " P j . v . 1/2 E { 3 2 V 2 J l " 1 ' j 2 V 2 ( 2 j 2 + 1 ) ( 2 3 ' + D > <5( J x J 2 | j ^ j 2 ) f i ( k q q ' ) ,,-1/2 .q+q' J l 1 ( 2 k + l ) " 1 / 2 [ ( 2 J 2 + l) ( 2 j ^ 4 - l ) ] 1 / 2 [ ( - D k + q + q , 6 ( v 1 v 2 | v j v ' ) " ( k i i £ , ) " 1 / 2 ( - i ) [(2X + 1) ( 2 £ ' + l ) ] 1 / 2 (k ^ ^ ) 0 0 0 < f ? ) 1 / 2 ^ ^ 7 ? n l < Y ) < , , , . ( y ) x E x . . < 2 X + i , 1 / 2 < 2 X " + i , ( - i ) ^ k A A 0 < > T( J 1 v ^ J 2 V 2 A " ; q q ' k ; J l V l J 2 v 2 A ) + 6 ( v ^ | vjv!, ) Q ( m . , - 1 / 2 I W , [ ( 2 U l ) ( n . + l ) ] 1 / 2 ( k 1 T4~ J o o o v 8 k T / d Y h ^ n l i y ) ' ( Y ) A£« (2A + 1 ) 1 / 2 ( 2 X » + 1) (-1) X" ) pg n)6 n x, AA 0 0 0 T * ( j l V l j 2 V 2 X " ; q q ' k ; j l V l j 2 V 2 X ) ] * E q u a t i o n s (3.65) and (3.66) a r e exac t and may be v i e w e d a s the r e s u l t o f c o m b i n i n g t h e r o t a t i o n a l i n v a r i a n c e p r o p e r t i e s o f t w i t h the r o t a t i o n a l i n v a r i a n c e p r o p e r t i e s o f ^ a s a w h o l e . They a r e c o m p l i c a t e d b e c a u s e t h e y c o n t a i n b o t h i n t e r n a l s t a t e and v e l o c i t y p o l a r i z a t i o n e f f e c t s . W h i l e e q u a t i o n s (3.65) and (3.66) a r e t h e f o r m s f o r ' ar.d r~ " t h a t a r e u s u a l l y p r e s e n t e d , a l t e r n a t e ( b u t c o m o l e t e l v e q u i v a l e n t ) f o r m s c a n be o b t a i n e d . One s u c h f o r m whose f o r m a t i s c l o s e r t o e x p r e s s i o n s commonly u s e d i n p r e s s u r e b r o a d e n i n g s t u d i e s , i s o b t a i n e d by e m p l o y i n g t h e t r a n s f o r m a -t i o n (3.67) T ( £ v £, v, A ; / v % v X) a a b b ^ a-" b a a b b = 6(1 v x".v .X"|jl v £ v A ) S ( / Z./|000) a a b b ' a a b b a b 1 a a b b * a^ b^ a a b b a l o n g w i t h the i d e n t i f i c a t i o n s " 77T^ / E d E ^ 4 (kT) y g and < 3^> < F ^ 1 / 2 ' ^ Y Z ^ - 2 d g ^ n A ( Y , ^ . ( Y ' ) /EdE /dE' £ ( Y ) ^ n . £ . < T ' ) "1T~2 e x p [ - | ( Y 2 + Y ' 2 ) 1 • y g Here (3.70) ^ n £ ( Y ) .« e " Y / 2 < a ( Y ) . The r e s u l t s a r e t h e n 1 (kT) 2 ( 3 . 7 i ) £ > 1 ( f c q n j ^ v j & ' q ' n ' j ' v ' j v j ^ (kT) 2 2 2 / E d E / d E ' ^ n J i ( Y ) jCn«Jl« (Y') e x p t - J ^ ] P j v P j ! v ' 1/2 £ { 2 2 t _ _ 2 - _ £ _ } fi(E. + E , +E-E . , ,-E., ,-E') j 2 v 2 { 2 3 2 + 1 ) ( 2 ^ 2 + 1 ) 31 V1 32 V2 3 i V i 32 V2 C ( E ; ^ q j 1 v 1 v 1 j 2 v 2 | £ ' q ' j ^ v ' v - j ' v ' ) fc 101 and (3 .72) ( f c q n j ^ v j f c ' q ' n - j > v ' v £ ) k = Y ' 2 + Y 2 — r EdE I"" d E » ^ " p (Y) ' 2. * (y ' ) e x P [ _ 2 3 ( k T ) 2 ° P • P • i i 1 / 2 1 2 V 2 2 J l J l 1 J 2 2 J l 1 E i 'v' ~ E i ' v ' ~ E , ) J l 1 J 2 2 C » ( E / U q J i v i v l J 2 v 2 U . q . J i v i J 2 v 2 y 2 ) k The q u a n t i t i e s C'(E) and C" (E) a r e e f f e c t i v e l y r o t a t i o n a l l y i n v a r i a n t f o r m s o f e n e r g y d e p e n d e n t t o t a l c r o s s s e c t i o n s . E q u a t i o n s (3.61) and (3.62) d e m o n s t r a t e t h a t t h e k i n e t i c c r o s s s e c t i o n s ' , r e p r e s e n t p a r t i c u l a r a v e r a g e s ( o v e r ic ic t r a n s l a t i o n a l e n e r g y and s e c o n d m o l e c u l e i n t e r n a l s t a t e s ) o f t h e s e t o t a l c r o s s s e c t i o n s . The p a r t i c u l a r k i n d o f a v e r a g -i n g employed - e q u a l B o l t z m a n n w e i g h t s f o r i n c o m i n g and o u t g o i n g s t a t e s - has been c h o s e n so as to y i e l d t h e most r~ r- 24 s y m m e t r i c f o r m s p o s s i b l e f o r u n d e r t i m e r e v e r s a l . k k To c o m p l e t e t h e p r o c e s s w h i c h began w i t h e q u a t i o n ( 3 . 6 7 ) , t h e p r e c i s e f orms f o r C ' ( E ) k and C"(E) a r e g i v e n . Namely, 102 ( 3 . 7 3 , C » ( E , A q J 1 v 1 y 1 J 2 v 2 | £ ' q J i v i v i J ' v ' ) k TT-ft2 [ (2q+l) ( 2 q ' + l ) ( 2 ^ + l ) ( 2 £ ' + l ) ] 1 / 2 . q-q •+£ y 2 g 2 fi(k^Jl-)1/2n(kqq')1/2 [ ( 2 j + 1 ) ( 2 J 1 + 1 ) ] 1 / 2 , £ ,^, (-1) 2 2 ^ 1 * 2 * XX,,, / 1 + / 2 + x -[ ( 2 / 1 + l ) (2-1+1) (2/ 1/2 / l ^ ^2 } [ ( 2 X + i , ( 2 A ' + i ) (2X-+D ( 2 A ' « f i , ] 1 / 2 (_ 1 )k+q+q ,+ s-+ j 1 . (-1) A+A"' [ (2X . » + i , ( 2 X " H , ) 1 / 2 ( ) . ( ) 0 0 0 0 0 0 A « A ••' I' A A " I ( 2 J J + 1 ) ( 2 j ' + l ) ( 2 j 1 + l ) ( 2 j ' + l ) ( 2 j 1 + l ) ( 2 j 2 + l ) 1/2 q' q k j i j l ' l iX' ^ X k £ X ••' ^  « X " 3 t S U i v ' j - v ' A . ; / ^ , , J i v 1 J 2 v 2 A ) S M j ' v i J 2 V 2 X . . . ; / . / 2 Z . ; J i v 1 J 2 v 2 X . . ) U x L 2 L l l L ' I 0 0 0 0 ° ) 6 ( 3 j [ v p 2 v 2 l J 1 v 1 v 1 J 2 v 2 ) 6(X-X»'|XX", ] and (3.74) C " ( E ' £ q J 1 V 1 V 1 J 2 V 2 | £ 1 q ' j ^ v j J ' V ' V ' ) TT^2 [ ( 2 q+l) ( 2 q'+l) (2&+1) ( 2 & '+1) ] 1 / 2 .q'-q+.fc+£' U 2 g 2 ft(k££«)1/2iT(kqq')1/2 ( 2 j + 1 , ( 2 j ' + l ) ] 1 / 2 I , , A X z . ( - D [ ( 2 / 1 + l ) ( 2 Z 2 + D (2^ + 1) (.2/J + l ) (2/^ + 1) ( 2 / ' + l ) ] 1/2 K 2 X + D ( 2 X - + D (.2X-+D ( 2 X -+D ] 1 / 2 ( - D X + X - 1/2 X ' X ' " £ ' X X " 1 (-1) [ (2X-+1) (2X" + 1) ] ^ ( ) ( ) 0 0 0 0 0 0 ( 2 j 1 + l ) ( 2 j 2 + l ) ( 2 j ' + l ) <2fj£ + l ) { ( 2 j 1 + l ) (2 J 2 + D 1/2 } (-1) j l + j 2 + j i + j 2 2 ^ 2 -^ 2 {{ L 2 q ' Z 2 ' l 31 3 1 k / 1 * I u X ' A x / £' k £ X I ' X " [ S ( j ' v i J 2 V 2 X . ; / : i Z 2 / ; J l V l J 2 V 2 X ) S M j ' V i J ' V 2 X » . ; , : 1 / 2 / . ; J l V l J 2 V 2 X » , - 6 ^ x / 2 ^ l ^ ' l O 0 0 0 O 0 ) < S ( A , X " , l X X " ) < S < 3 i 3 ^ | j : L D 2 ) S f v ' v ' v ' v ' v V V V )1 1 1 1 2 2 1 1 1 2 2' ' 104 The S m a t r i x v e r s i o n s (3.73) and (3.74) may a p p e a r more f a m i l i a r to some t h a n the T m a t r i x forms (3.65) and (3.66) g i v e n e a r l i e r . They a r e s t i l l c o m p l i c a t e d , however, m a i n l y due t o t h e p r e s e n c e o f v e l o c i t y p o l a r i z a t i o n s . I n d e e d , t h e c o m p l i c a t e d o r i e n t a t i o n a l a v e r a g i n g ( 6 j and 9 j symbols i n v o l v i n g £ and £') r e q u i r e d t o g e t r o t a t i o n a l l y i n v a r i a n t c r o s s s e c t i o n s , and t h e nonequ i l i b r ium O / ^ ^ (Y) , ^ , ^ , ( Y ' ) ) a v e r a g i n g o v e r t r a n s l a t i o n a l e n e r g y i n e q u a t i o n s (3.71) and ( 3 . 7 2 ) , a r e a d i r e c t c o n s e q u e n c e o f t h e s e t r a n s l a t i o n a l p o l a r i z a t i o n s . I t s h o u l d be e m p h a s i z e d t h a t e q u a t i o n s (3.73) and (3.74) a r e e x a c t and a r e c o m p l e t e l y e q u i v a l e n t to the T m a t r i x e x p r e s s i o n s . B o t h r e p r e s e n t r o t a t i o n a l l y i n v a r i a n t e x p r e s s i o n s i n the t r a n s l a t i o n a l - i n t e r n a l c o u p l i n g scheme. S and T m a t r i x e x p r e s s i o n s i n o t h e r c o u p l i n g schemes c a n a l s o be v i s u a l i z e d ( b u t w i l l n o t be p r e s e n t e d e x p l i c i t l y ) . I n d e e d , a m u l t i t u d e o f e q u i v a l e n t exac t e x p r e s s i o n s - S v e r s u s T, one c o u p l i n g scheme v e r s u s a n o t h e r - c a n be o b t a i n e d . Which i s p r e f e r a b l e ? The c h o i c e i s d e t e r m i n e d by the a p p r o x i m a t e scheme t o be u s e d i n e v a l u a t i n g t h e s e m a t r i x e l e m e n t s . T h r o u g h o u t t h i s t h e s i s , the d i s t o r t e d 13 25 wave. B o r n a p p r o x i m a t i o n (DWBA) ' w i l l be e m p l o y e d . The a p p r o p r i a t e s t a r t i n g p o i n t i s t h e n a T m a t r i x r e p r e s e n t a t i o n i n the t r a n s l a t i o n a l - i n t e r n a l c o u p l i n g scheme. I n c h a p t e r I I I , t h i s i s g i v e n by e q u a t i o n s (3.65) and ( 3 . 6 6 ) . 10! ( f ) The D i s t o r t e d Wave B o r n A p p r o x i m a t i o n (DWBA) The k i n e t i c t h e o r y c o l l i s i o n i n t e g r a l s d e s c r i b e d i n t h i s c h a p t e r i n v o l v e bo th t r a n s l a t i o n a l and i n t e r n a l s t a t e e f f e c t s . The d i s t o r t e d wave B o r n a p p r o x i m a t i o n i s a method o f a p p r o x -i m a t e l y e v a l u a t i n g t h e s e m a t r i x e l e m e n t s , i n w h i c h b o t h -t y p e s o f m o t i o n s a r e c o n s i d e r e d . I t i s ba s e d on a d i v i s i o n o f t h e i n t e r m o l e c u l a r p o t e n t i a l V i n t o a s p h e r i c a l ( i n t e r n a l s t a t e i n d e p e n d e n t p a r t V q and a r e m a i n i n g non s p h e r i c a l ( a n i s o t r o p i c ) p a r t V^. Thus (3 .7 5 ) V = V + V o 1 where i s w r i t t e n a s A . w i t h R = R R as the p o s i t i o n o f m o l e c u l e 2 r e l a t i v e t o rtui} n u 2 ] m o l e c u l e 1 and^J^ ' CL/2 a S t l l G i n * - e r n a " - s t a t e o p e r a t o r s f o r m o l e c u l e s 1 and 2, r e s p e c t i v e l y . The r a d i a l (R) d e p e n -d e n c e i s s p e c i f i e d by the s c a l e r c o e f f i c i e n t h, ( R ) . The * 1 ^ 2 ^ p o t e n t i a l (3.75) i s q u i t e g e n e r a l , a l t h o u g h i t d o e s n o t i n c l u d e momentum d e p e n d e n t p o t e n t i a l s s u c h a s the s p i n -r o t a t i o n t y p e (3.77) v = ( J . + J,) • R x P Z/ (R) , s r ~1 ~2 - — / w h i c h c a n be s i g n i f i c a n t f o r m o l e c u l e s w i t h s p i n 1/2 n u c l e i . I n e q u a t i o n ( 3 . 7 5 ) , the n o n s p h e r i c a l p a r t V i s assumed much " s m a l l e r " t h a n V , so t h a t a p e r t u r b a t i o n e x p a n s i o n i n o c a n be p e r f o r m e d . Thus (3.78) t = Vfl = t Q + t + t 2 + ..., where (3.79) t = V fl 0 o o and / (3.8 0) t . = V. + V G ( t . ) + V , G ( t ) 1 1 o 1 1 o = fl v , n o 1 o Here fl i s t h e t r a n s p o s e ( o r A. c o n j u g a t e ) o f the s p h e r i c a l o M i l l e r wave o p e r a t o r flQ m e n t i o n e d i n a p p e n d i x A. i n t h e n o t a t i o n o f ( 3 . 1 3 ) , e q u a t i o n (3.78) i s w r i t t e n as (3.81) t g g | = < y g | t 1 | y g * > A 1 / 2 < 1 . * - 1 / X A 2 ~ ~ w i t h the t r a n s l a t i o n a l d e p e n d e n c e e n t i r e l y f a c t o r e d i n t o t h e q u a n t i t y (3.02) A ^ <gg') = < Mg | V 1jU ] ( R) b / y l 2 < ( R) flQ | Mg ' > A ( o ) ( g g 1 ) =<Mg|v ft |yg'> oo ~ ~ ~ o o ~ E q u a t i o n (3.81) s t a t e s t h a t t h e t r a n s i t i o n o p e r a t o r , i n f i r s t o r d e r , i s a sum o f p r o d u c t s o f t r a n s l a t i o n a l and i n t e r n a l s t a t e c o n t r i b u t i o n s . F o r t r a n s i t i o n m a t r i x e l e m e n t s o f the f o r m ( 3 . 6 3 ) , one has t ^ e x p r e s s i b l e a s , ( 3 . 8 3 ) . ^ m ^ j j n i j V j X s l ^ | j j m j v • j J m j v - X • s • > ^ 1 l 2 ; j i v i l l ^ l ) | | j i v i > < J ^ l l ^ 2 U 2 , | l i 2 " 2 > ... J , + J 9 + A-m -m -s z r <X \\A {/\ (gg')|iX->(-l) 1 2 1 2 £ , where <|| ||> i s a r e d u c e d m a t r i x e l e m e n t a c c o r d i n g t o 2 1 Edmonds c o n v e n t i o n . The i n v e r s e r e l a t i o n s h i p t o ( 3 . 6 4 ) , when combined w i t h t h e d e f i n i t i o n ( 3 . 8 3 ) , i m p l i e s the r e l a t i o n (3.84) T . 1 ( j - v i J 2 v 2 X ' ; / 1 Z 2 / ; J 1 v 1 J 2 v 2 X ) = 2 T T y i ( p p ' ) - L / 2 < A | | A / l / 2 ( g g ' ) || X ' x j ^ J I ^ ||j{v'> <• II 0 U 2 ) „ . . J l + J 2 + X - 3 I - 3 2 - X , ( - l ) ^ 1 + Z 2 + < 3 2 v 2 | | j ; 2 ||3»v 2> [ ( 2 j ' + l ) ( 2 j ' + l ) ( 2 A'+l) ( 2 ^ + 1 ) ( 2 / : 2 + l ) ( 2 / + 1 ) ] " 1 / 2 108 between r e d u c e d q u a n t i t i e s . The t r a n s i t i o n o p e r a t o r t c a n be d i v i d e d i n t o i t s r e a l and i m a g i n a r y p a r t s (3.85) t g . i ( t g + t g + j + I ( t g - t g + ) = g 2 g g 2 g g ( t 9 ) . - 2 T T i . 7 t , g 6 ( E ) t , g + d ( u g V ) g h g ' g ' ~ where the optical theorem has been used in the second line of equation ( 3 . 8 5 ) . T n e reduced T matrix expression analogous to (3.8 5) is (3.86) T i j i v ' j ' v ^ M / ^ j / l ' j ^ j ^ X ) -T h ( j i V i j 2 V 2 X , ' / l ^ ^ l V 1 ^ 2 V 2 X ) 1 1/2 V V X + j i + j 2 + X ' + - [ ( 2 j 1 + l) (2j 2+l) (2X+D ) ' E (-1) A * X 5 l V 5 2 * 2 E [ (24'+l) <2£'+l) (2/-+1) ( 2 2 ,+D (24_+l(2/+l) ] 1 / 2 ^ 1 ^ 2 ^ D l D l 3 1 D 2 3 1 3 2 X X ' X I ^ 1 ^2 ^ T f j ' v ' j ' v ' X ; / ^ ^ ; J l V ; L J 2 v 2 X ) 109 6(E ,+E-r - +E- - -E -E . -E . ) t r ^1 V1 3 2 V 2 t r 3 1 V 1 D 2 V 2 wi t h (3.87, ^ ( ^ v ^ . v - X ' ^ ^ j ^ j ^ X ) -i ^ < 3 i j i j 2 V 2 X , ^ l ^ ' V l j 2 V 2 X ) " l ™ J l V l J 2 V 2 A ^ l Z 2 ^ ^ V x j 2 V 2 A , ) ( - l ) D l + J l + ^ . ^ 2 ^ ^ , „ X + X ' ^ ( 2 J 1 - M ) ( 2 j 2 + 1 ( 2 A ^ 1 ) , 1 / 2 * ' ( " l ( 2 j ^ + l ) ( 2 j 2 + l ) (2X C o n s i d e r now t h e q u a d r a t i c i n T p a r t o f e q u a t i o n ( 3 . 6 5 ) , d e n o t e d by ^ ^(--) . I n s e r t i o n o f e q u a t i o n (3.84) i n t o t h i s e x p r e s s i o n y i e l d s d i r e c t l y (3.38) [ ^ ; ( £ q n j 1 v 1 v 1 | £ ' q , n ' j j v | v ^ ) k = P, v P j , v , 1/2 z { 3 2 2 3 2 2 } Z 3 2 2 ' 4 1 l 2 < D 2 v 2 (-1) 1 2 j q ' - q [ ( 2 q + l ) ( 2 g ' + l ) ) 1 / 2 fl(kqq')1/2 (2Z 2+1) { 1 1 } 3 1 3 1 q 3 i j l ' l < J 2 v 2 H 2 U 2 ) H J 2 v 2 > 110 [ £ j 1 } - ^ n ^ ^ , n , ^ ^ 2 l E J i + E J 2 v 2 - E J l v 1 - E j 2 v 2 > k The l i n e a r i n T p a r t s o f ( 3 . 6 5 ) , i n d i c a t e d a s ^ ( — ) ^ , r e q u i r e more i n v o l v e d m a n i p u l a t i o n s , i n t h a t the t r a n s f o r -m a t i o n (3.86) must f i r s t be e m p l o y e d . S u b s e q u e n t use- o f (3.84) t h e n y i e l d s (3.89) ] ^ l ( £ q n j 1 v i v 1 i £ ' q ' n ' j | v j v ^ ) k = Z { 3 2 V 2 J 2 2 J l 1 J 2 2 . 2 2 P V v . 1/2 P , „ v „ P-;»v.. 2 2 } { 1 1 2 2 } 1/2 P • P • ~i V T V J l 1 J 2 2 [ ( 2 J i + l ) ( 2 j 2 + l ) ( 2 j £ + l ) ( 2 j ^ + l ) 1/2 i . . . 6 . . . 6 . -r E 3 l 3 i 3 2 3 2 V 2 V 2 2 Z l / 2 Z i 1 fk ^ 1 * 1 , .q+q' f ( 2 q + l ) ( 2 q ' + l ) , 1 / 2 l2 V I l < D 2 V 2 l l i / 2 ) l l ^ V 2 > | 2 , * q q' } T 2 7 7 I ) j l 3 1 3 1 3 i 3 1 31 I l l ,<-i> £ p 1 ) < » - " / ' C i f 2 l ' ' , " V v ; ^ l H : l „ v „ « j „ v „ - E j 1 v 1 -E j , ^ J ^ I l r f'l ' l l i i v J X J J v j I l ^ i ' l l j . T . * « ^ ^ }C •*• X X X -sZJi •(-l)k+c^+^, <J1v'||^ zi)i|j»v»><j»v»Hj?</i,|| jiyi>] P • P • i i I / 2 j. r 2 2 2 2 } 6 6 6 + j v ( 2 ^ 2 + 1 ) ( 2 j 2 + 1 ) j l j i j 2 j 2 V 2 V 2 J 2 2 J 2 2 r , o - , • . ,,,1/2 ( - D q + q ' r (2q + l ) (2q'+l) , 1 / 2 k ^ ^ ' M /0\ 3 1 3 1 3 1 ^ V i " ^ " ^ ^ - C - l ) - q + q ' < J l v 1 l l i i k ) | | 3 1 v i > 6 V i V , ] . E q u a t i o n s ( 3 . 88) <a«d (3.89) r e p r e s e n t t h e DWBA f o r £ ' . They c o n s i s t o f p r o d u c t s o f i n t e r n a l s t a t e f a c t o r s w i t h t h r e e t r a n s l a t i o n a l i n t e g r a l s E p " ^ > I S ^ ^ and £>n^"' • A g a i n , i t i s e m p h a s i z e d t h a t t h e DWBA a l l o w s a s e p a r a t i o n o f i n t e r n a l s t a t e and t r a n s l a t i o n a l m o t i o n s . T h i s s e p a r a t i o n i s i n c o m -p l e t e , however, i n t h a t t h e t r a n s l a t i o n a 1 i n t e g r a l s s t i l l d e pend e x p l i c i t l y on the e n e r g y i n e l a s t i c i t y (3 . 9 0 ) 1 ' V T V ^ l 1 J 2 2 3 1 V 1 - E 3 2 V 2 T h i s i s the amount o f e n e r g y t r a n s f e r r e d between t r a n s l a -t i o n a l and i n t e r n a l s t a t e s d u r i n g a c o l l i s i o n . E x p l i c i t l y , the t h r e e t r a n s l a t i o n a 1 i n t e g r a l s a r e g i v e n as < 2 * > H 2 ( f ^ ) 1 / 2 Q ( k £ £ ' ) - . 1 / 2 u - 3 / 2 // e x p I z i ! | L l ! 1 I l * n ( ) L i ' n . ( I ) {m)l+l V(£'£k) ( - ) k v ( k / V ) ( - ) Z + / ' A / - ( g g ' ) A * l *(gg'> *1^2 ~~ 1^2 <5 ( f u g ' 2 - | u g 2 + x) dug' dY E ' E L2 / d E _ / dE- exp[-i(-S + e : ( k T ) 2 ° t r o t r 2 k T k T t r u 2 g ' 2 5(E« - E ' + X ) n ( k £ £ ' ) _ i / ^ E [2TTVr(gg') J"/' ] t r t r XX-M _ 1 / 2 X r ^ T T l ! 2 f o r , M 1 / 2 1 2 X ' X" < X H A 4 ^ 2 ( I g , ) | , X , > ^ n £ ( Y ) ^ „ . & . ( Y ) - < X , , l l A ^ ! / . . ( 9 g , , ) | | A ' » [ ( 2 U 1 ) <2&'+l) (2X +1) (2X» + 1 ) ] 1 / 2 i ^ ^ * * " £ £ ' k X X ' k X X » k ( - 1 ) X + X + ^ ( ) ( ) { > 0 0 0 0 0 o 9 2 ) ( £ ( D r-n) 113 ( 2 1 T ) V ( 8 S ) 1 / 2 « ( ^ ' ) - 1 / 2 T r - 3 / 2 / / e x p [ ^ 2 ^ , 2 ] L * " ( Y ) L £ , n ' ( Y ' ) ( . ) £' + i L V ( £ ' £ k ) ( . ) k v ( k ^ ' / ) A / ( g g M A ^ ' *(gg')<S(iyg' 2 - I y g 2 + K ) d ( ] i g ' ) d y *1*2 ^1*2 ~ ~ 2 ~ = + r0 0 r°° 1 E t r E t - r -, / dE„ / d E ' e X p f _ ± ( - ± £ + _h£)j 2 p t r o t r 2 kT kT (kT) : t r 4^ 2 6 ( E ^ r - E t r + * > « U * * ' > 1 / 2 * y g' X X • X ' - X 1 " [ 2 T T y 2 ( g g ' ) 1 / 2 ] 2 [ ( 2 £ + l ) (2*'+l) (2X+1) (2X-+1) (2A"+1) ( 2X-+1) ] 1 / 2 , , V £ + £ ' + X + X » + X " + X " » , (-k) { £ X X " £ • X ' X ' " > < ) (-1) 0 0 0 0 0 0 I + 4'+k X ' / X £ • k £ X " ' X " J < ^ I ! A/. (gg') | | X ' > < X " | | A / ( G G - ) | | X " t > * ^ 2 ~ ~ ^ n £ ( Y , ^ n ' £ ' ( Y ) (3.93, C ' 1 / 2 ) , k q q , -(2TT) H 2 iry 1/2 1 , - 2 r £ n , t _ %' n 1 £ + £ ' v U ' £ k ) ( • ) k ^ e [ A ( k ) , ( g g ) l qq ~ ~ 2TT ,°° Tr t i 2 ~ - _ v 2 — 2 I V E t r l l ^ ' ^ n r ( ^ Y r y r E i ( ) ( ) fl(k££') 7 A X * 0 0 0 0 0 0 1(21+1) (2£'+l) 2X+1) ( 2 X ' + 1 ) ] 1 / 2 [2 iry 2 g] \ { < M k ^ ! ( g g ) | | A ' > + < A | | A ^ ! ( g g ' ) | | X ' > * > qq — qq • In p a r t i c u l a r , (3.94) £ < 1 / 2 ) a n l ' n ' ) k k o 5 I ^ 1 ' 2 > ( £ n £ • n • ) fc. In p r i n c i p l e , j j v m e a s u r e s v e l o c i t y c h a n g e s a n d j ^ * 1 ' g o v e r n s the r e l a x a t i o n r a t e s o f i n t e r n a l s t a t e p o l a r i z a t i o n s . In the l a n g u a g e o f c h a p t e r I , ! ^ 1 ^ d e s c r i b e s " v e l o c i t y c h a n g i n g " r- (1) c o l l i s i o n s w h i l e / ^ has t o do w i t h a k i n d o f "phase p c h a n g i n g " c o l l i s i o n a l e f f e c t . J~>^^2^ i s l i n e a r i n and c o n t r i b u t e s to f r e q u e n c y s h i f t s . A d i s t o r t e d wave B o r n a p p r o x i m a t i o n c a n a l s o be d e r i v e d f o r j ^ ^ . I n d e e d , s u b s t i t u t i o n o f (3.84) and (3.86) i n t o e q u a t i o n (3.66) y i e l d s an e x p r e s s i o n f o r / T " t h a t i s a n a l o g o u : to e q u a t i o n s (3.00) and ( 3 . 8 9 ) . E x p l i c i t l y , the r e s u l t s a r e (3.94) r ^ ; ( a q n J 1 v 1 v 1 U ' q ' n ' j 2 v 2 v ' ) k = P i v P i , v i 1 / 2 , + / . T. { 2 2 i i - M E (-1) 2 ^ 2 i q ' - q ( 2 j +1) (2j'+1 K ' 3 2 V 2 2 1 i i ^ 2 i q' q k j (2q-fl) ( 2 q '+l) 1/2 , q ^ 1 ^ 1 , , q ' ^ 2 ^ 2 } I . . , . / 2 ' l I J 3 1 3 1 3 1 3 2 3 2 3 2 < 1 l v l H 9 { / l , | | j i v i > < J i v i l l j ? « Z f > | | J l v l > J 2 2 and ( 3 . 9 5 ) £ : a q n J 1 v l V l | A . q . n ' J 2 v . 2 v 2 ) k 11G P • P • , i 1/2 s f * 1 } 3 i V i • I { i - i 2-1} - i»v" p i v p i v J l 1 J l 1 J2 2 i " v " J 2 2 ( 2 j +1) ( 2 j _ + l ) 1 / 2 r £ £ i r ( 2 j £ + l ) ( 2 j 2 ' + l ) (-i) q+q' 3232 ( 2 q + l ) ( 2 q ' + l ) 1/2 fl(kqq1) J S / 1^2 i q ^ / { q ' l2 C2  { . „ • } { . „ . • } 3 1 31 31 32 32 32 q ' q \ I 2 <i ^2 / l / J <D 1 V 1 I|J?{ /1 )|| J i v r < j » v » | | ^ ^ ) | | J l v 1 > < J 2 v 2 | L 5 ^ 2 ) | | j u v p < j ,v,|IO(^"2,|| j 'v'> i { 6 ,6 . + 6- .6 - ) J 2 2M'i'2 1 1 J 2 2 2 V v , v ' v„ v i v ' v ' v „ v . ' ' 1 1 2 2 1 1 2 2 Cp 1 }an/V 2 U ' n ' / 7 I / 2 U j l l v „ + E , - E - E ) ^ D l 1 D 2 2 3 1 1 3 2 2 P • P • . i 1/2 + S " { 2 2 1 1 3 2 v 2 ' ( 2 3 2 + 1 ) ( 2 3 i + 1 ) 3W q+q* j i j i j 2 j 2 2 1 T i « > 2 1/2 < 2 k + l ) [ ( 2 q + l ) ( 2 q ' + l ) ( k q q ' ) ] ; k q q ' V l v ' v 2 v - V ; L v ' V ^ ' A g a i n the r e s u l t s a r e p r o d u c t s o f e x p l i c i t i n t e r n a l s t a t e f a c t o r s and the t r a n s l a t i o n a 1 i n t e g r a 1 s E>~ ^ ^  ^  , FZ^^ andfZ}^''' v p ^ h As i s d i s c u s s e d i n r e f e r e n c e ( 2 5 ) , i t i s g e n e r a l l y e x p e c t e d tha t £ k >> r ^ J ^ , u n l e s s r e s o n a n t p r o c e s s e s p l a y a m a j o r r o l e ( H y d r o g e n i s o t o p e c o l l i s i o n s p r o v i d e an example o f t h i s 2 6 l a t e r e f f e c t . ) The n e g l e c t o f c o n t r i b u t i o n s e n t i r e l y c a n t h e r e f o r e p r o v i d e a f u r t h e r u s e f u l a p p r o x i m a t i o n i n many c a s e s . (g) E v a l u a t i o n o f the T r a n s l a t i o n a l C o l l i s i o n I n t e g r a l s -The M o d i f i e d B o r n A p p r o x i m a t i o n W i t h i n t h e d i s t o r t e d wave B o r n a p p r o x i m a t i o n , t h e r e s t i l l r e m a i n s the p r o b l e m o f e v a l u a t i n g the t r a n s l a t i o n a l f ac t o r s ]_y^1 ^ , /T ^ 1 ^  and J~^^ ^ / 2 ^ T h i s has been i n v e s t i g a t e d v p ^ n i n some d e t a i l , see r e f e r e n c e s (13) and ( 2 5 ) . The p r e s e n t t r e a t m e n t i s l i m i t e d t o a d e s c r i p t i o n o f t h e t r a n s l a t i o n a l i n t e g r a l s w h i c h a r i s e i n c o n n e c t i o n the s h e a r v i s c o s i t y c o e f f i c i e n t f o r N^• T h i s i s s u f f i c i e n t t o i l l u s t r a t e t h e s a l i e n t f e a t u r e s o f t h e a p p r o a c h t a k e n by S n i d e r and c o -w o r k e r s . I n a d d i t i o n , the r e m a r k s p r e s e n t e d h e r e a r e u l t i -m a t e l y a p p l i c a b l e t o t h e c r o s s s e c t i o n s e n c o u n t e r e d i n s p e c t r o s c o p i c s t u d i e s - see c h a p t e r s IV and V I I f o r f u r t h e r d i s c u s s i o n s o f t h i s p o i n t . The u n d e r l y i n g f e a t u r e s t o S n i d e r ' s a p p r o a c h a r e as f o l l o w s . F i r s t , t h e e n e r g y i n e l a s t i c i t y (3.96) i s n e g l e c t e d whenever p o s s i b l e , o r i f t h i s g i v e s a v a n i s h i n g r e s u l t , A i s c o n s i d e r e d t o f i r s t o r d e r o n l y . S e c o n d , the t r a n s i t i o n o p e r a t o r f o r t h e s p h e r i c a l p a r t o f t h e p o t e n t i a l , A ^ ° ^ ( g g ' ) , i s t r e a t e d e x a c t l y whenever i t a r i s e s . C o n v e r s e l y , oo — ~ f o r A \^ ^ , ( g g 1 ) , a c r u d e a p p r o x i m a t i o n i s u s e d w h i c h i s b a s e d on a p p r o x i m a t i n g V q as a r i g i d s p h e r i c a l p o t e n t i a l f o r m o l e c u l e s o f d i a m e t e r d . The e f f e c t o f t h e M i l l e r o p e r a t o r fiQ ( w h i c h d e s c r i b e s t h e c o l l i s i o n d y n a m i c s o f t h e s p h e r i c a l p o t e n t i a l ) i s t h e n a p p r o x i m a t e d as e x c l u d i n g a r i g i d c a r e o f 119 r a d i u s d from t h e r a d i a l i n t e g r a l . From the d e f i n i t i o n ( 3 . 0 2 ) , t h e a p p r o x i m a t e form f o r (gg*) i s t h e r e f o r e (3.97) h W { g g . ) = h " 3 / e ^ V ^ U J b . ' ( R) d R / l ^ 2 ~~ R>d J t x L 2 L where K i s d e f i n e d as the change i n wavenumber d u r i n g t h e c o l l i s i o n (3.?8) K = y ( g ' - g ) A I n o b t a i n i n g e q u a t i o n ( 3 . 9 7 ) , t h e t r a n s l a t i o n a l k e t | y g ' > has been e x p r e s s e d i n p o s i t i o n r e p r e s e n t a t i o n . T h i s 27 a p p r o x i m a t i o n was f i r s t u s e d by Chen and S n i d e r , and 28 25 . 29 t h e n s u b s e q u e n t l y by M o r a a l and S n i d e r . S n i z g a x has termed i t t h e " m o d i f i e d B o r n a p p r o x i m a t i o n " t o t h e t r a n s i -t i o n o p e r a t o r s i n c e , i n e f f e c t , e q u a t i o n (3.97) i s j u s t a m o d i f i e d F o u r i e r t r a n s f o r m o f t h e n o n s p h e r i c a l p o t e n t i a l . By r o t a t i o n a l i n v a r i a n c e , (gg') must be p r o p o r t i o n a l t o 'tj (K) i n t h i s a p p r o x i m a t i o n , and i t s e x p l i c i t f o r m i s g i v e n as 0 0 (3.99) A U ) (gg') = 4 T T i / h " 3 / j (KR) b (R) R 2dR Zf ^ < K) , *1^2 R>d *-lA2A \J \frhere j^(KR) i s the s p h e r i c a l B e s s e l f u n c t i o n . F o r t h e s p h e r i c a l c a s e o f a m u l t i p o l e - t y p e p o t e n t i a l , b ^ ^ / t 1 * ) = k ^ l ^ 2 ^ R ^ ^' a n < ^ t * i e r a d i a l i n t e g r a l c a n be e v a l u a t e d i n c l o s e d f o r m w i t h t h e r e s u l t (3.100) A J ^ (gq') = 4 T T i / b / / l / 2 Z ( K h 3 d ^ " 1 ) " 1 (Kd,yU) (K) (2-rrhy)" 1 ( 2 l l l 2 2 J L ) <K) (The s c a l a r ^ ^ 2 ^ a s t''ie dimensions of l e n g t h and i t s _ square can be i n t e r p r e t e d as a kind of d i f f e r e n t i a l c r o s s s e c t i o n . ) The essence of the mod i f i e d Born approximation i s seen from equations (3.99) or (3.100)- the t r a n s i t i o n opera-tor A ^ ' (gg') does not depend s e p a r a t e l y on the d i r e c t i o n of g and g' but only on the momentum t r a n s f e r K. With these comments as a background, the three c o l l i s i o n para-meters T, T. ^ and i> are now d i s c u s s e d , i n t The f i r s t c o l l i s i o n parameter to be co n s i d e r e d i s T. From equation (3.39), i t i s seen t h a t t h i s d e s c r i b e s the r e l a x a t i o n of a pure v e l o c i t y p o l a r i z a t i o n . Since the s p h e r i c a l p a r t of the p o t e n t i a l / =/^ = J T 2 = 0 can c o n t r i b u t e to v e l o c i t y r e l a x a t i o n and, w i t h i n the co n t e x t of the DWBA, i t i s assumed to dominate, the r e l a x a t i o n of a pure v e l o c i t y p o l a r i z a t i o n i s approximated by employing the e f f e c t of the s p h e r i c a l p o t e n t i a l o n l y . T h i s i m p l i e s t h a t the t r a n s l a -t i o n a l f a c t o r s to be c o n s i d e r e d are of the form J - l ( 1 ) (£n000/Jt' n ' 000/0) , f l ( 1 * (JlnOOO/in • 000/0) , and '— p o '—p o J~L, ( 1 /' 2 ) (Inln' ) ; where i=l' = 2 and n=n'=0 f o r the p a r t i c u l a r n ooo v e l o c i t y p o l a r i z a t i o n t r e a t e d i n T. F u r t h e r , as d i s c u s s e d i n r e f e r e n c e (13), the i n t e r n a l s t a t e f a c t o r s are such that the c o n t r i b u t i o n s of iTp '^ a n d L>h^^^ s e p a r a t e l y v a n i s h , with the r e s u l t t h a t the r e l a x a t i o n of a pure v e l o c i t y 121 p o l a r i z a t i o n , i n t h e s p h e r i c a l p o t e n t i a l a p p r o x i m a t i o n , i s d e s c r i b e d i n terms o f the one t r a ns l a t i o na 1 i n t e g r a l (1) ( J l n O O o l i i ' n ' O O o l 0 ) o , From e q u a t i o n (3.91) and ( 3 . 9 2 ) , t h e e x p l i c i t f o r m f o r t h i s t r a n s l a t i o n a l i n t e g r a l i s ( 3 . 1 0 1 ) / Z ( 1 ) U n 0 0 0 Z ' n ' O O O l o ) . *—' v o 2 = 6 M , (2£ + l ) ~ 1 ( 2 7 r ) " 1 / / e ~ Y L £ n ( y) ( -) 1 [ L £ n ' ( y) - LZn\y')) Y ( 2 T r ) 4 * V A ^ g g ' J A ^ ' d g ' d y T h i s c a n be e x p r e s s e d i n terms o f the ft i n t e g r a l s o f Chapman 3 0 and C o w l i n g , (3.102) n < A ' 8 ) 5 ( ^ ) 1 / 2 //e-Y 2 Y 2 S + 3 ( 1 - c o s £ X ) a ( g X ) d cos X dY where (3.103) x = J . J i 0(gX) - < 2 i r ) 4 * V A^,(gg')A^°,*(gg') In g e n e r a l , t h e i d e n t i f i c a t i o n i s somewhat c o m p l i c a t e d , i n v o l v i n g s e v e r a l w e i g h t e d sums o v e r fi i n t e g r a l s . The g e n e r a l f o r m u l a e a r e g i v e n i n r e f e r e n c e ( 1 3 ) . F o r the p a r t i c u l a r > • " 122 t r a n s l a t i o n a l i n t e g r a l u s e d i n t h e e v a l u a t i o n o f T, the s i m p l e r e s u l t 1/2 (3.104) / T ( 1 ) ( 2 0 0 0 | 2 0 0 0 | 0 ) - ^ C - 2 ^ ) « ( 2 ' 2 ) '—' V O D K I i s o b t a i n e d . The r e s u l t i n g e x p r e s s i o n f o r T i s a l s o d e r i v e d i n r e f e r e n c e (13) and i s g i v e n a s M Vo^  i f*™. 1 / 2 r ( 0 ) (2 0 0 0 (3.105) - - n ( — ) h> (2 0 0 0 | n ft(2'2) 5 I n summary, the r e l a x a t i o n o f a p u r e t r a n s l a t i o n a l p o l a r i z a -t i o n c a n u l t i m a t e l y be e x p r e s s e d , i n a p p r o x i m a t e f o r m , i n terms o f the Q i n t e g r a l s o f Chapman and C o w l i n g . E q u a t i o n (3.105) i s a p a r t i c u l a r example o f t h i s r e s u l t . The T - n t p a r a m e t e r , o b t a i n e d i n s e c t i o n ( d ) , i s an example o f t h e r e l a x a t i o n o f a p u r e i n t e r n a l s t a t e p o l a r i z a - t i o n - see e q u a t i o n (3.39) f o r the d e f i n i t i o n o f T ^ n t « A s s u c h , the t r a n s l a t i o n a l f a c t o r s w h i c h a r e i n v o l v e d i n i t s c o l l i s i o n a l d e s c r i p t i o n a r e o f t h e form £ ( 1 * ( 0 0 / / / |00/'/'/• |0) < 0 0 / Z 1 ^ 2 l 0 0 ^ ^ i ^ 2 l 0 ) o ' a n d - ^ h 1 / 2 ) ( 0 0 0 0 ) o q q * I n a m a n n e r s i m i l a r t o t h e p u r e v e l o c i t y p o l a r i z a t i o n c a r e , t h e c o n t r i b u -t—(1/2) . t r o n from J^,^ c a n c e l s . A t t h e same t i m e , t h e J J v term v a n i s h e s i d e n t i c a l l y s i n c e t h e s p h e r i c a l p o t e n t i a l c a n n o t a f f e c t i n t e r n a l s t a t e c h a n g e s . Thus, the r e l a x a t i o n o f a p u r e i n t e r n a l s t a t e 123 p o l a r i z a t i o n i n v o l v e s o n l y the t r a n s l a t i o n a l f a c t o r 1 ' ( 0 0 / / , / n | 00 / 11 'I' I0) . T h i s i n t e g r a l , w h i c h i n v o l v e s p ^ /" 1*- 2 ^ 1 *• 2 o the n o n s p h e r i c a l p a r t o f the i n t e r m o l e c u l a r p o t e n t i a l , i s e s t i m a t e d by means o f the m o d i f i e d B o r n a p p r o x i m a t i o n . I n d e e d , the s u b s t i t u t i o n o f e q u a t i o n (3.100) i n t o (3.91) f o r the p a r t i c u l a r c a s e when '=n=n'=0 l e a d s d i r e c t l y to (3.106) ( 0 0 / / ] y 2 | 0 0 / 7 1 / 2 | 0 ) 1 r r r 1 , 2 , 2 X . . .1 2 1 ,2. — • / / e x p [ . - - ( Y +Y ) ] 6 (-yg --yg ' ) Try (8ykT) /? / ] * dy d ( y g ' ) V ( O ^ ) (')2^lfl) (K) V U ) (K) E m p l o y i n g t h e r e l a t i o n (3.107) V ( 0 / < > ( - ) 2 ^ U ) (K )2y U ) (K) - ( 2 / + l ) 1 / 2 and t h e t r a n s f o r m a t i o n s 1 9 1 ? 2 1/2 (3.108) / / 6 ( i y g ^ - i - yg •"*) d (y g ') dy = 8 TT y ( 2 y k T ) ' co cc / / dTdy o y 1x2 „ 2 y = ( 8 y k T ) ~ ft K r 1 , 2 .2, T = -jCY + Y' ) a l l o w s e q u a t i o n (3.10G) to be r e w r i t t e n i n the form (3.109) £p ( 0 0 / / 1 / 2 | 0 0 / / 1 / 2 | 0 ) o 2 i r 3 u b b r» kT d ^ W / ( 4 - 1 ) where t h e a s y m p t o t i c form o f the s p h e r i c a l B e s s e l f u n c t i o n u s e d i n d e f i n i t i o n (3.100) has been u s e d i n a p p r o x i m a t i n g t h e y i n t e g r a l . The m a n i p u l a t i o n s i n v o l v e d i n e q u a t i o n s (3.106) t h r o u g h (3.109) have been p r e s e n t e d more g e n e r a l l y i n r e f e r e n c e ( 2 5 ) . E q u a t i o n (3.109) r e p r e s e n t s the " m o d i f i e d B o r n a p p r o x i m a t i o n " f o r the t r a n s l a t i o n a l i n t e g r a l C p 1 5 ( 0 ' 0 / i 1 i 2 | 0 0 ^ ^ 2 | 0 ) Q . An e x p r e s s i o n f o r T ± n t i n t e r m s r—(1) o f JLjp , w h i c h i s q u i t e a n a l o g o u s to t h e e x p r e s s i o n f o r T /—(1) i n terms o f ^  , c a n t h e n be o b t a i n e d , a l t h o u g h t h i s has n o t been done e x p l i c i t l y . The f i n a l r e l a x a t i o n p a r a m e t e r w h i c h o c c u r s i n t h e s t u d y o f the s h e a r v i s c o s i t y o f N 2 i s an example o f a p r o - d u c t i o n c o l l i s i o n i n t e g r a l . H e r e , an i n t e r n a l s t a t e p o l a r i z a t i o n i s p r o d u c e d from a t r a n s l a t i o n a l p o l a r i z a t i o n by c o l l i s i o n . G e n e r a l l y , t h e s e p r o d u c t i o n i n t e g r a l s a r e one o r more o r d e r s o f m a g n i t u d e s m a l l e r t h a n the r e l a x a t i o n c o l l i s i o n i n t e g r a l s . F u r t h e r , i n many c a s e s , the p r e v i o u s l y employed a p p r o x i m a t i o n o f n e g l e c t i n g the e n e r g y i n e l a s t i c i t y i n the t r a n s l a t i o n a l f a c t o r s l e a d s to a v a n i s h i n g r e s u l t . The p a r a m e t e r Tij', d e f i n e d by e q u a t i o n ( 3 . 3 9 ) , i s a p a r t i c u l a example o f t h i s . H e r e , the o n l y t r a n s l a t i o n a l i n t e g r a l w h i c h c o n t r i b u t e s i s j ^ , 1 * ( 00/ / Z 2 | 2 0 / 'I J L2 \ x ) 2 and an a p p r o x ma t i o n w h i c h i s f i r s t o r d e r i n A = x/kT ( 3 . 110) J ^ I V 1 ) ( 0 0 / / 1 / 2 | 2 0 / 1 / 'Z 2|x) = : A C v 1 ) " ( 0 0 Z / 1 / 2 | 2 0 / « Z ^ 2 | 0 ) 2 where t h e p r i m e h e r e d e n o t e s the d e r i v a t i v e w i t h r e s p e c t t o A. The e v a l u a t i o n o f Ti/i i s d i s c u s s e d i n some d e t a i l i n r e f e r e n c e ( 1 8 ) . The r e m a r k s o f t h i s s e c t i o n a r e n o t i n t e n d e d as a com-p l e t e a c c o u n t o f t h e t r e a t m e n t o f c o l l i s i o n i n t e g r a l s b u t r a t h e r a r e p r e s e n t e d as a p e r s p e c t i v e i n w h i c h to v i e w t h e c o l l i s i o n a l c a l c u l a t i o n s employed i n t h e s u c c e e d i n g c h a p t e r s C h a p t e r s IV t h r o u g h VI d e a l c o m p l e t e l y w i t h i n t e r n a l s t a t e p o l a r i z a t i o n s and so i t s h o u l d n o t be s u r p r i s i n g t h a t t h e o n l y t r a n s l a t i o n a 1 i n t e g r a l w h i c h a r i s e s i s j ^ ^ 1 5 ( 0 0 / / 1 / 2 | 0 0 y / | / 2 | x ) q . I n c h a p t e r V I I , t r a n s l a t i o n a l p o l a r i z a t i o n s o c c u r and w i t h them, c o n s i d e r a t i o n o f L>^1) (&n000 | In' 000 | 0) q i s n e c e s s i t a t e d . F i n a l l y , i t s h o u l d be m e n t i o n e d t h a t the a l g e b r a i c method employed i n t h i s t h e s i s f o r the e v a l u a t i o n o f t h e t r a n s l a t i o n a l f a c t o r s i s , o f c o u r s e , n o t the o n l y a p p r o a c h w h i c h c a n be t a k e n . L e s s c r u d e a l g e b r a i c m o d e l s , s e m i -126 c l a s s i c a l WKB methods, or exact quantum s c a t t e r i n g techniques co u l d a l l be a p p l i e d to o b t a i n b e t t e r estimates f o r the 29 c o l l i s i o n parameters. Indeed, S h i z g a l has shown t h a t the m o d i f i e d Born approximation tends to underestimate the c r o s s s e c t i o n and has i n v e s t i g a t e d v a r i o u s other techniques 31 f o r e v a l u a t i n g the t r a n s l a t i o n a l f a c t o r s . 127 (h) Summary T h i s c o n c l u d e s a r e v i e w o f k i n e t i c t h e o r y as i t i s a p p l i e d to the s t u d y o f the S e n f t l e b e n - B e e n a k k e r e f f e c t s . O b v i o u s l y , the t r e a t m e n t i s v e r y i n v o l v e d when the d e t a i l s a r e t o be u n d e r s t o o d . B u t from t h i s s t u d y , a g r e a t d e a l has been l e a r n e d a b o u t the k i n e t i c b e h a v i o u r o f gas s y s t e m s when i n t e r n a l s t a t e s p l a y an i m p o r t a n t r o l e . The aim o f the r e s t o f t h i s t h e s i s i s to p r e s e n t the t o p i c s o f p r e s s u r e b r o a d e n i n g and c o h e r e n c e t r a n s i e n t s to the same d e p t h o f u n d e r s t a n d i n g . In p a r t i c u l a r , the t r e a t m e n t o f c o l l i s i o n s i s g e n e r a l i z e d , see c h a p t e r s IV and V I I . The moment method i s a l s o a p p l i e d , i n c h a p t e r s V, VI and V I I , to d e s c r i b e t h e s y s t e m m o t i o n s . Thus, t h e s t u d y o f the S e n f t l e b e n -B e e n a k k e r e f f e c t s has g r e a t l y i n f l u e n c e d the method o f t r e a t m e n t o f p r e s s u r e b r o a d e n i n g and c o h e r e n c e t r a n s i e n t phenomena, as p r e s e n t e d i n t h i s t h e s i s . I n d e e d , t h e u s e o f a u n i f i e d n o t a t i o n and p a r a l l e l d e v e l o p m e n t p u t s the s u b -j e c t s o f t r a n s p o r t p r o p e r t i e s and s p e c t r o s c o p i c phenomena o f g a s e s on the same f o o t i n g f o r , i t i s hoped, a more c o m p l e t e u n d e r s t a n d i n g o f b o t h . CHAPTER IV A K i n e t i c E q u a t i o n f o r P r e s s u r e the L i n e a r i z e d C o l l i s i o n B r o a d e n i n g O p e r a t o r and "Have you g u e s s e d the r i d d l e y e t ? " t h e H a t t e r s a i d , t u r n i n g t o A l i c e a g a i n . "No, I g i v e i t up," A l i c e r e p l i e d . "What's the ansv/er ? " "I h a v e n ' t t h e s l i g h t e s t i d e a , " s a i d the H a t t e r . "Nor I , " s a i d t h e March H a r e . 129 (a) I n t r o d u c t i o n The a p p r o a c h o u t l i n e d i n c h a p t e r I I had t h r e e u n s a t i s -f a c t o r y a s p e c t s - the a b s e n c e o f a w e l l d e f i n e d c o l l i s i o n m a t r i x , no o b v i o u s g e n e r a l i z a t i o n to i n c l u d e d e g e n e r a t e ( m a g n e t i c ) quantum numbers, and no s a t i s f a c t o r y d i s c u s s i o n o f p o s s i b l e t r a n s l a t i o n a l e f f e c t s . F o r t u n a t e l y , the a p p r o a c h to S-B e f f e c t s o u t l i n e d i n c h a p t e r I I I has c o n s i d e r e d p r e c i s e l y t h e s e t h r e e a s p e c t s i n g r e a t d e t a i l . Thus a s y n t h e s i s o f c h a p t e r s I I and I I I seems a p p r o p r i a t e t o p r o -d u c e a k i n e t i c t h e o r y t r e a t m e n t o f p r e s s u r e b r o a d e n i n g and c o h e r e n c e t r a n s i e n t s . C h a p t e r s IV, V, and VI a p p l y to methods o f c h a p t e r I I I t o t h e f i r s t two a s p e c t s . The f i n a l c h a p t e r o u t l i n e s a k i n e t i c t h e o r y d i s c u s s i o n o f t h e t r a n s -l a t i o n a l e f f e c t s . The f o r m a t o f c h a p t e r IV p a r a l l e l s t h a t o f c h a p t e r s I I and I I I . Namely, the a p p r o p r i a t e e q u a t i o n o f m o t i o n i s f i r s t o b t a i n e d and a u s e f u l c h o i c e o f b a s i s i s d e f i n e d . W i t h t h e s e b a s i s e l e m e n t s , t h e most g e n e r a l moment e q u a t i o n s a r e t h e n d e r i v e d . A t h o r o u g h d i s c u s s i o n o f a g e n e r a l c o l l i s i o n m a t r i x e l e m e n t i s t h e n p r e s e n t e d i n c l u d i n g a d i s -t o r t e d wave B o r n a p p r o x i m a t i o n to the e x a c t form and a m o d i f i e d B o r n a p p r o x i m a t i o n f o r the t r a n s l a t i o n a l f a c t o r . Some comments on the r e l a t i o n s h i p o f t h e s e c o l l i s i o n a l c o n s i d e r a t i o n s to the work o f o t h e r s t h e n c o m p l e t e t h i s c h a p t e r . (b) G e n e r a l E q u a t i o n o f M o t i o n - The Waldmann S n i d e r Form As i n c h a p t e r I I I , the s t a r t i n g p o i n t i s the g e n e r a l i z e d B o l t z m a n n e q u a t i o n o f S n i d e r .and S a n c t u a r y . 1 In p r i n c i p l e , t h e c o l l i s i o n s u p e r o p e r a t o r .^f c a n c o u p l e any f r e q u e n c y component | E > < E ' | o f the d e n s i t y o p e r a t o r f o r the p a i r o f c o l l i d i n g m o l e c u l e s to any o t h e r f r e q u e n c y component | E " > < E " ' I . As a c o n s e q u e n c e o f t h e d e f i n i t i o n o f 2/ g i v e n i n a p p e n d i x A, t h i s f r e q u e n c y c o u p l i n g c a n be w r i t t e n a s (4.1) Tr{ ( | E"><E " i ) + J | E X E • | } = < E » | t | E > 6 E , E l I , - S E E „ < E ' | t + | E " » -<E" | G ( t) j E><E 1 | t + | E'"> - <E" | 11 E><E ' | G ( t + ) | E'"> = <E"|t|E>6 E, E„,_- < S E E „ < E ' | t +|E"»> - < ^ " l t l E > < E - l t + l E " > { E _ E ; + i £ + E . , U E > + i £ ] (Here " T r " d e n o t e s the t r a c e o v e r t h e s t a t e s o f the p a i r o f c o l l i d i n g m o l e c u l e s . ) Thus f r e q u e n c y c o u p l i n g s r e q u i r e o f f - t h e - e n e r g y s h e l l t m a t r i x e l e m e n t s . O n - t h e - e n e r g y - s h e l l t m a t r i x e l e m e n t s ( d e n o t e d by t^) a r e a s s o c i a t e d w i t h com-p l e t e d c o l l i s i o n s w h i l e o f f - t h e - e n e r g y - s h e l l t m a t r i x 2 e l e m e n t s a r e a s s o c i a t e d w i t h d u r a t i o n o f c o l l i s i o n e v e n t s . A u s e f u l a p p r o x i m a t i o n i s to i g n o r e t h e o f f - e n e r g y - s h e l 1 m a t r i x e l e m e n t s i n eqn. (4.1) . T h i s l e a d s to a c o m p l e t e d c o l l i s i o n ( i m p a c t ) a p p r o x i m a t i o n t o . T h i s a p p r o x i m a t i o n i s d e n o t e d as 3^/ f o r i t was H e s s 3 who f i r s t p o i n t e d o u t t h a t f r e q u e n c y c o n s e r v i n g c o l l i s i o n s a r e e x p r e s s i b l e i n terms o f o n - t h e - e n e r g y - s h e l 1 t m a t r i x e l e m e n t s o n l y , ( 4 l 2) Tr{ ( | E"><E"'|)+ 2/ H | E><E « | } + 2 n i <E" I t„ I E X E ' I t * I E"' >} ' d 1 ' d 1 The a p p r o x i m a t i o n -*• / a c t u a l l y d e c o u p l e s ( c o l l i s i o n a l l y ) the d i f f e r e n t f r e q u e n c y components o f the d e n s i t y o p e r a t o r . Thus the w i d t h s o f i s o l a t e d l i n e s , c o r r e s p o n d i n g to t h e r e l a x a t i o n o f a p a r t i c u l a r f r e q u e n c y component, a r e e x p r e s s i b l e e x a c t l y i n terms o f w h i l e t h e o v e r l a p p i n g o f s p e c t r a l l i n e s r e q u i r e s the f u l l e x p r e s s i o n ( 4 . 1 ) . The d i s c u s s i o n s p r e s e n t e d i n t h i s t h e s i s a r e r e s t r i c t e d t o p r e s s u r e b r o a d e n i n g and c o h e r e n c e t r a n s i e n t s o f i s o l a t e d l i n e s , so t h a t 7^ i s t h e a p p r o p r i a t e c o l l i s i o n s u p e r -o p e r a t o r . W i t h the r e p l a c e m e n t o f t h e e q u a t i o n o f n m o t i o n f o r the one p a r t i c l e d e n s i t y o p e r a t o r becomes (4.3) i|£ = / P + t r 2 - ^ H P P 2 * 132 P r o c e e d i n g w i t h the m a n i p u l a t i o n s o f the t r a n s l a t i o n a 1 s t a t e s as o u t l i n e d i n c h a p t e r I I I , the B o l t z m a n n e q u a t i o n f o r the Wigner d i s t r i b u t i o n f u n c t i o n i s g i v e n a s ( 4 . 4 ) M + £ . i f . - i / t X f - f<°Vl«i>) 31 m 9 r t r w h e r e a t i s a l i n e a r i z e d l o c a l i z e d c o l l i s i o n o p e r a t o r o f Waldmann-Snider fo r m, namely ,2 "3/2 J " ~ 2 '~Z2 (o) -W' (4.5) K « j > ) = -(2 T T)H [ - r 7 7 ] ~ n t r 0 /dp. (27TmkT) f ( o ) t r f [/dp'dp 2<ug| t d|yg'>{ - — <})(W') _ (o) t r , . 9 f ' ' f + — 4>(W^))<Uy'|6(E-E*)t' ji-ig: n n z ~ - ~ 6 ( p + p 2 - P ' - p 2 ) + X2TTi) " 1 [<yg| t d | y g > f ( o ) t r f<°> f < ° ) t r f ( o ) { f _ i _ ^ w ) + - i - — * 2 ( w 2 ) } / \ 4 (o) (o) t r , . f ( o ) t r f 2 f 2 f ( 0 ) i t i _[£ _ J ^(w) + _£ ± <J>,(W_>]<pg t ' yg>]] n n ~ n n 2.-2. ~ (The a c t u a l form o f the l i n e a r i z a t i o n employed i n e q u a t i o n (4.4) i s d i s c u s s e d i n more d e t a i l m o m e n t a r i l y . ) In e q u a t i o n (4.5) , i t i s i m p l i c i t l y i m p l i e d t h a t ^ ? c o l l i s i o n a l l y c o u p l e s one p a r t i c l e i n t e r n a l s t a t e f r e q u e n c y components to and to'' o n l y i f to = to' - a l t e r n a t e l y , i f to and to' d i f f e r g r e a t l y i n f r e q u e n c y , t h e n t h e i r c o r r e s p o n d i n g f r e q u e n c y 133 components a r e n o t c o u p l e d , c o 1 1 i s i o n a 1 1 y . Note t h a t t r a n s -l a t i o n a l d e g r e e s o f f r e e d o m a r e c l a s s i f i e d as zero, f r e q u e n c y , b e c a u s e o f p o s i t i o n l o c a l i z a t i o n . F u r t h e r , when <f i s r e s t r i c t e d to z e r o f r e q u e n c y components, e q u a t i o n (4.5) i s e q u i v a l e n t to the l i n e a r i z e d W aldmann-Snider c o l l i s i o n . o p e r a t o r . Thus e q u a t i o n (4.4) r e p r e s e n t s a c o l l i s i o n a 1 1 y d e c o u p l e d s e t o f e q u a t i o n s - one f o r e a c h f r e q u e n c y compor n e n t , and i n p a r t i c u l a r , the z e r o f r e q u e n c y e q u a t i o n i s the l i n e a r i z e d W aldmann-Snider e q u a t i o n employed i n the 4 d e s c r i p t i o n o f S e n f t l e b e n - B e e n a k k e r e f f e c t s . T i p has a l s o r e a c h e d t h e s e c o n c l u s i o n s v i a a d i f f e r e n t a p p r o a c h and o b t a i n e d e q u a t i o n (4.4) as the l o n g time l i m i t o f the g e n -e r a l i z e d B o l t z m a n n e q u a t i o n . The s p e c i f i c f o r m o f l i n e a r i z a t i o n u s e d i n t h i s c h a p t e r d i f f e r s f r o m t h a t o f the p r e v i o u s c h a p t e r . I n e q u a t i o n ( 3 . 9 ) , the e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n r e p r e s e n t s a l o c a l e q u i l i b r i u m s t a t e and <$> d e s c r i b e s the d e v i a t i o n f r o m t h i s s t a t e . No p r o b l e m s o f c o m m u t a t i o n between f^°^ and (j> a r i s e b e c a u s e ( J ) i s a l w a y s d i a g o n a l i n e n e r g y . F o r t h i s and s u b s e q u e n t c h a p t e r s , t h e a l t e r n a t e l i n e a r i z a t i o n (4.6) f = f + f = f + f (p i s c h o s e n . Here f^°^ i s t h e a b s o l u t e M a x w e l l i a n w h i l e § i s no l o n g e r r e s t r i c t e d to b e i n g d i a g o n a l i n e n e r g y . T h i s " n o n - d i a g o n a l i t y " would l e a d to c o m m u t a t i o n p r o b l e m s i f a form a n a l o g o u s to (3.9) were c h o s e n ( i e . e i f f ^  ^  = £*°'<j>) 5 as d i s c u s s e d by S n i d e r . The form ( 4 . 6 ) , on t h e o t h e r hand, c o m p l e t e l y a v o i d s t h i s p r o b l e m . I n s t e a d , s l i g h t l y more c o m p l i c a t e d c o l l i s i o n m a t r i x e l e m e n t s , ( w i t h a d d i t i o n a l i n t e r n a l s t a t e B o l t z m a n n f a c t o r s ) , a r e o b t a i n e d , as d i s -c u s s e d f u r t h e r i n s e c t i o n ( d ) . A l s o , t h e form (4.6) l e a d s to c o l l i s i o n a l e x p r e s s i o n s w h i c h c a n most e a s i l y be r e l a t e d t o t h o s e o f o t h e r w o r k e r s i n p r e s s u r e b r o a d e n i n g phenomena. The g e n e r a l r e s u l t (4.4) i s now m o d i f i e d t o o b t a i n an e q u a t i o n s u i t a b l e f o r the d e s c r i p t i o n o f p r e s s u r e b r o a d e n -i n g and c o h e r e n c e t r a n s i e n t phenomena. To t h i s end, t h e L i o u v i l l e s u p e r o p e r a t o r , d e s c r i b i n g t h e m o t i o n o f t h e s y s t e m under t h e i n f l u e n c e o f a time d e p e n d e n t e l e c t r i c f i e l d , i s s p e c i f i e d as (4.7) / A = / A + [-2U-E c o s ( w t - k « r ) , A ] , where A i s an a r b i t r a r y one m o l e c u l e o p e r a t o r . The f l o w p 9 term ~ • i s n e g l e c t e d i n t h e p r e s s u r e b r o a d e n i n g l i m i t . (The e f f e c t s o f the f l o w term a r e d i s c u s s e d i n c h a p t e r V I I . ) As a r e s u l t , t h e B o l t z m a n n e q u a t i o n becomes (4.8) | | = - i / f c ^ f -±/^ [-2y-E Q c o s ( w t - k . r ) , f ] - f [ ° } ^ ((J>) . T h i s i s e q u i v a l e n t to e q u a t i o n ( 2 . 4 ) , b u t w i t h a w e l l d e f i n e d c o l l i s i o n o p e r a t o r . A g a i n , t h e t r a n s f o r m a t i o n to 13 5 a " r o t a t i n g f r a m e " i s u s e f u l to a p p r o x i m a t e l y e l i m i n a t e the e x p l i c i t t i m e d e p e n d e n c e i n t h e i n t e r a c t i o n . I n t h i s frame t h e t r u n c a t e d e q u a t i o n i s (4.9) | | = - i / f c [ # 0 - S , f ] by t h i s t r a n s f o r m a t i o n f r e q u e n c y . The c o l l i s i o n o p e r a t o r i s u n a f f e c t e d b e c a u s e (R. p r e s e r v e s ( i n t e r n a l s t a t e ) (c) A p p r o p r i a t e C h o i c e o f B a s i s and R e s u l t i n g Moment E q u a t i o n The q u e s t i o n o f an a p p r o p r i a t e b a s i s i s now t a c k l e d . C o n s i d e r a t i o n o f two p r o p e r t i e s o f {P\. - namely f r e q u e n c y c o n s e r v a t i o n and r o t a t i o n a l i n v a r i a n c e - l e a d s a l m o s t 2 i m m e d i a t e l y to a b a s i s o f the form (4.10) [ j i v i > < j f v - f ] ( q ) V EE i q Z. ( 2 q + l ) 1 / 2 ( - I j W m, m _ f 3 i q 3 f < ) I . 3 . m. v. ><i „m -mi V m f I J i i i J f f f - e ( q ) V ( . ) q [ J i v . x j f v f ] ( q ) T h e s e t e n s o r o p e r a t o r s f o r m an 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 S O ( 3 ) , and a r e s i m u l t a n e o u s l y e i g e n -f u n c t i o n s o f the f i e l d f r e e i n t e r n a l s t a t e L i o u v i l l e o p e r a t o r / = [ % ,] w i t h f r e q u e n c y w „ . „ =4r(e . „ J l 1 f f J l 1 £ . ) . O p e r a t o r s o f t h i s t y p e were f i r s t i n t r o d u c e d by Hess and K o h l e r . and c a n be c o n s i d e r e d a s an e x t e n s i o n o f t h e ^ ^ q . ' ( J ) P _ . v v , o p e r a t o r s c o n s i d e r e d i n t h e l a s t c h a p t e r to o f f - d i a g o n a l i t i e s i n " j " . I n d e e d , t h e r e l a t i o n (4.11, I J v 1 > < J v f , « " » . P J i f ( 2 j + l ) ' i s e a s i l y v e r i f i e d by a c o m p a r i s o n o f (4.10) w i t h ( 3 . 2 4 ) . As s u c h , t h e b a s i s (4.10) c a n be c o n s i d e r e d as a s y n t h e s i s 137 o f the b a s i s (2.15) o f c h a p t e r I I and the b a s i s (3.24) o f c h a p t e r I I I . The p r o p e r t i e s o f the [ j v i> < j f v f q * a r e d i s c u s s e d f u r t h e r i n A p p e n d i x B . The moment e q u a t i o n s f o r t h i s r a t h e r a b s t r a c t (or a t l e a s t v e r y "quantum m e c h a n i c a l " ) b a s i s a r e now d i s c u s s e d . I n v o l v e d i s the i n n e r p r o d u c t t t ' - W ' (4.12) < <A | B > > = t r / d p A ( W ) B ( W ) (2TrmkT) ' t •*• t r { A B > when b o t h A , B a r e i n d e p e n d e n t o f the v e l o c i t y . T h i s i n n e r p r o d u c t r e f l e c t s the form o f l i n e a r i z a t i o n c h o s e n , e q u a t i o n ( 4 . 6 ) . The moment e q u a t i o n s , i n t h e i r m o s t g e n e r a l form a r e w r i t t e n a s (4 .13) i * | ^ < [ j v X j ' v ' ] ( q ) > ^ f t t r U j v X j ' v ' ] ( q ) f > -t^Aoj . „ , , v , < t j v > < j ' v ' ] { q ) > 3 V 3 - I « ( [ jv>< j 'v'] ( q ) ) + | $ | ( [ j v X j 'v'] +>> ( • ) q (<[ j v x j ' v ' ] ( q ) > - <[ j v x j ' v ' ] ( q ) > e q 138 q + q ? 1/2 q q ? q i -i-H" + 2 Z i : [ ( 2 q + l ( 2 q } ' i ^ ( - l ) 3 3 j " v " q x q 2 j " j ' j q i + q 2 ( q i ) ( q l } ( q ? } V ( q q 2 q l ) ( ' ) ^ < J , V , | M |i j"v">< [ jv>< j " v , R J > q + q 2 1/2 q q i q ? - i ' + V Z Z i 2 [ ( 2 q + l ) (2q + l ) 1 / 2 { 1 ^ ( - l ) 3 * 3 j " v " q x q 2 j " j ' j V ( q q i q 2 ) ( « ) < [j"V"><j ' v ' ] li j " v " M where t h e i d e n t i f i c a t i o n (4.14) [ # -S, [ j v x j ' v * ] ( q ) ] = **iAw . ., , [ j v x j «v'] ( q ) O ] V ] V =1i(a) . . . , , -OJ) [ j v x j ' v ' ] ( q ) has been u s e d , a s s u m i n g the i n t e r n a l s t a t e e n e r g i e s a r e i n d e p e n d e n t o f the m a g n e t i c i n d i c e s . I n e q u a t i o n ( 4 . 1 3 ) , a c o m p l e t e l y g e n e r a l form o f the e x t e r n a l f i e l d - s y s t e m i n t e r a c t i o n has been employed, (4.15) = Z M %C m t "v. q 1 V 1 1 ( q i ) V l ( q l } where M i s the o p e r a t o r p r o p e r t y and ^  i s the V l 139 e x t e r n a l f i e l d p r o p e r t y . W i t h the form ( 4 . 1 5 ) , i t i s s e e n t h a t the c o l l i s i o n i n d e p e n d e n t p a r t o f (4.13) i s the g e n e r a l ! -7 za t i o n o f the p r e c e s s i o n e q u a t i o n s o f Fano to i n c l u d e o f f d i a g o n a l i n " j " c o u p l i n g s . T h a t i s , w h i l e Fano d e s c r i b e s p o l a r i z a t i o n s [ j v ^ x j v ] w i t h i n e a c h " j " s h e l l , ( c a u s e d by the p r e s e n c e o f e x t e r n a l s t a t e f i e l d s ) e q u a t i o n (4.13) a l l o w s n o n - d i a g o n a l i n " j " c o h e r e n c e s [ j i ^ v ^ > < : J £ V £ J • T h e s e a r i s e b e c a u s e o f r e s o n a n c e i n t e r a c t i o n ( i n the r o t a t i n g f r a m e ) . E q u a t i o n (4.13) r e p r e s e n t s the f u n d a m e n t a l r e s u l t o f t h i s c h a p t e r . I t b e a r s l i t t l e r e s e m b l a n c e t o e q u a t i o n s (2.18) o r (2.19) o f c h a p t e r I I . However, w i t h the s p e c i f i c f orm o f the r a d i a t i o n - s y s t e m i n t e r a c t i o n g i v e n as (4.16) — -y * E_ i n t ~ ~o c o n n e c t i o n to c h a p t e r I I c a n be made. C h a p t e r V i s d e v o t e d t o j u s t t h i s t a s k . The r e m a i n i n g p o r t i o n o f t h i s c h a p t e r d e a l s w i t h the c o l l i s i o n a l p a r t o f e q u a t i o n ( 4 . 1 3 ) . 14 0 (d) General C o l l i s i o n Expressions The e v a l u a t i o n of the general c o l l i s i o n matrix element << [ j i v i > < j £ v f ] ^  | $ | j [ v [ > < : J £ v £ ] ^  >> i s now discussed. E x p l i c i t l y , a combination of (4.12) and (4.5) y i e l d s (4.17) « [ J i v i > < j f v f ] . C q ) \ (R | j !v!><j £v£] ( q )>> - ( 2 T r ) 4 R 2 n t r 1 ) 2 / dp dp 2 ( [ j . v . x j f v £ ] C q ) ) + , , , r r C 0 ) t r (0) R , V L > < I I V I ] C q ^ + f 2 C ° ^ Y ° ) [ j i V i > < J f V f ] ( ^ ) } < u g « |t t 6 C E-E' ) |yg> S ( p + p 2 - p ' - P 2 ' ) . i , , r , ( 0 ) t r . (0) ..., , , , v , n (q) ( 2 i r i ) [ < v g | t j v g > { f JL 2 l : ) i v i 3 £ f J  a ~ n n t I ( 0 ) t I l 2 ( 0 ) [ J l v i » < J i v i ] ^ ' * £ 2 C ° J » f ( 0 ) [ J . , i > < J . v . ] ( O J < P g | t J | y g > ] ] where [i!v!><i Iv~] denotes a coherence a s s o c i a t e d with L J i i J f f J ( 2 ) molecule 2. This form represents a g e n e r a l i z a t i o n of equation (3.25) i n the sense that operators that are o f f - d i a g o n a l i n 14 1 " j " a r e c o n s i d e r e d . In a n o t h e r s e n s e , however, i t i s l e s s c o m p l i c a t e d s i n c e no v e l o c i t y p o l a r i z a t i o n s a r e p r e s e n t . I n d e e d , t h e r e m a i n d e r o f t h i s c h a p t e r e m p h a s i z e s t h e s i m i l a r i t i e s i n t h e c o l l i s i o n a l a s p e c t s o f b o t h p r o b l e m s . Thus, w h i l e c o n s i d e r i n g c o m p l e t e l y d i f f e r e n t phenomena h e r e a d d i t i o n a l i n s i g h t i n t o t h e methods o f the l a s t c h a p t e r may be g a i n e d . The remarks made i n c h a p t e r I I I c o n c e r n i n g t h e i n v a r i a n c e p r o p e r t i e s o f t h e c o l l i s i o n s u p e r o p e r a t o r a r e e q u a l l y v a l i d h e r e , s i n c e b o t h u l t i m a t e l y r e s u l t f r o m t h e p r o p e r t i e s o f C/H. In p a r t i c u l a r , ^ p r e s e r v e s p a r i t y , i s r o t a t i o n a l y i n v a r i a n t , y e t can c o n n e c t o p e r a t o r s o f d i f f e r e n t t i m e r e v e r s a l symmetry. The m a t r i x e l e m e n t s (4.17) r e f l e c t t h e s e p r o p e r t i e s . Two u s e f u l i d e n t i t i e s can be e s t a b l i s h e d a l m o s t i m m e d i a t e l y - namely, (4.18) « [ j i v i > < j f v f ] ( q ) \ j ! v ! x j -v£] ( q ) > > * (-D [ J f V f X J i V . l ^ ^ l ^ l j ^ v ^ x j ' v ! ] ^ 3 > > and (4.19) < < J i V i > < J f V f ] ( q ) 1^1 [ J i V i > < J £ V f ] C q ) > : Pj (2J.+1) ( 2 j ! + l ) J Pj ( 2 j f + l ) * J f < < • U j v - x j . v ^ ^ l ^ l [ j i v i x j f v f ] ( ^ > > . 142 Here i s t h e " t i m e r e v e r s e d " o p e r a t o r (4.20) . A T = 0 A as The f i r s t r e s u l t , e q u a t i o n ( 4 . 1 8 ) , f o l l o w s from (CT^A) 1^ -( s e e a p p e n d i x A) and t h e a d j o i n t p r o p e r t i e s o f t h e b a s e s ( 4 . 1 0 ) , as d i s c u s s e d i n a p p e n d i x I I I . E q u a t i o n (4.19) i s a c o n s e q u e n c e o f t h e i d e n t i f i c a t i o n o f t h e s u p e r o p e r a t o r a d j o i n t o f C/^ '> t h e t i m e r e v e r s e d s u p e r o p e r a t o r f o r "C/ - i n o t h e r words see a p p e n d i x A. When e q u a t i o n (4.19) i s r e s t r i c t e d t o t h e " d i a g o n a l i n j " c a s e , i t becomes (4.22) « [ j i v . x j f v f ] ( q ) |<& | [ j ! v ' x j ' v ' ] ( q ) > > = t_ _ 3 — •/ ?JL 1 « [ J , v ! > < j ' v ^ ] ^ ) | ^ | [ j v . x j v £ ] ^ ( 2 j + l ) ( 2 j ' + l ) The e x t r a B o l t z m a n n f a c t o r s i n t h i s e q u a t i o n a r e a d i r e c t c o n s e q u e n c e o f t h e form o f l i n e a r i z a t i o n c h o s e n i n t h i s c h a p t e r , e q u a t i o n ( 4 . 6 ) . W i t h t h e method o f l i n e a r i z a t i o n c h o s e n i n c h a p t e r I I I , t h e s e e x t r a n e o u s f a c t o r s d i s a p p e a r so t h a t t h e m a t r i x e l e m e n t s d e f i n e d i n t h e l a s t c h a p t e r a r e more s y m m e t r i c a l t h a n t h e ones d e a l t w i t h h e r e . F o l l o w i n g t h e p r a c t i c e employed i n t h e s t u d y o f S e n f t l e b e n -B e e n a k k e r e f f e c t s , m a t r i x e l e m e n t s o f (J^ a r e e x p r e s s e d as 14 3 k i n e t i c t h e o r y c r o s s s e c t i o n s (4.23) £> C [ j i v . x j f v £ ] ( q ) | [ j ! v ! x j f v f ] ( q , ) ) 5 n'^-iw~^ « [ J i v i > < J f v £ ] C q ) | ^ | [ i i v ! x j f v f ] t ^ » = <5 , E ( q ) / ^ ( [J • V-><j -v.] ( q ) | [j .' v ! x j ' v i ] ( q ) ) qq i ^ U J i i •> f f J 1 u i i J f f J o I t i s t h e r o t a t i o n a l i n v a r i a n c e o f t h a t i m p l i e s t h a t t h e t e n s o r c r o s s s e c t i o n ^ can be e x p r e s s e d i n terms o f one s c a l a r c r o s s s e c t i o n . The l a s t l i n e o f e q u a t i o n (4.23) i s t o be compared w i t h e q u a t i o n (3.55) o f t h e p r e v i o u s c h a p t e r . T h i s shows t h a t t h e a p p l i c a t i o n o f t h e p r i n c i p l e s o f r o t a t i o n a l i n v a r i a n c e i n t h e p r e s e n t c h a p t e r i s much s i m p l e r t h a n i n c h a p t e r I I I . In p a r t i c u l a r , t h e r e i s o n l y one s c a l a r c r o s s s e c t i o n f o r each t e n s o r c r o s s s e c t i o n . From (4.17) and ( 4 . 2 3 ) , t h e s c a l a r c r o s s s e c t i o n i s g i v e n as (4.24) ^ ( [ j i v . x j f v f ] ( q ) | [ j ! v ! > < j ^ ] ( q ) ) o ° ( - l ) ( 2 7 T ) 4 ^ ( - ^ ^ ) ^ ( 2 q + l ) - 1 F / q ) ( 0 2 q t r 1 ) 2 - 4 7 T - / d Y UJivi*<Jfvf]Cq))+ 144 (2) i n t /d ( u g»)e' r e > { Q kT [ j ! v : > < j 1 1 J 1 v ' f V f (q) 76\ (2) i n t e +Q kT [ j > < f v f ] ( q ) } < y g ' | (2) t + <5(E- E') yg> 2 / . < 2 ) i n t + ( 2 i r i ) [<yg| t r e y ! > { Q kT [ j ! v.' <> j i v ! . ] L J I I J f f J (q) m t kT e +75 [ j ! v ! x j ' v ' ] ( q ) } ^ (2) k T k T I ^  J <wg|.t Iyg>]] To o b t a i n e q u a t i o n ( 4 . 2 4 ) , t h e t r a n s f o r m a t i o n ( 4 . 2 5 ) tj = _ i _ ( w + W _) Y = — ( W ? - W ) j] 2 + y 2 = W2 + W2 (2 1 FmkT) " 3 / d p d p 2 = - i - f d y d i TT 145 C2TrmkT)" 3/dpdp 2dp ,dp. 26(p + p 2 - p , - p 2 ) = / d ( y g ' ) d 5 d Y W-/dir e x p [ - <]  2 ] = 1 3/2 Tf t o r e l a t i v e and c e n t r e o f mass v a r i a b l e s has been employed. The ( t r i v i a l ) c e n t r e o f mass i n t e g r a t i o n has a l s o been p e r f o r m e d . I n d e e d , t h e ease w i t h w h i c h the c e n t r e o f mass i n t e g r a t i o n can be c a r r i e d out r e p r e s e n t s a s e c o n d s i m p l i f y i n g f e a t u r e o b t a i n e d by n e g l e c t i n g t h e p r e s e n c e o f v e l o c i t y p o l a r i z a t i o n s . In t h e l a n g u a g e o f t h e l a s t c h a p t e r , I , , „ „. . -»- 6„„6 6 „ l o 6 t n  b 6 1 ' p s p ' s ' ; £ n £ 1 n ' £0 no £*0 n'O I t i s c o n v e n i e n t , at t h i s s t a g e , t o d i v i d e t h e c o l l i s i o n s e c t i o n i n t o two p a r t s ; namely (4.26) C ( [ j i v i > < j f v f ] ( q ) | [ j ! v ! x j ^ ] ^ ) o = G , C [ J i v i > < J f v f ] C q ) | [ j ! v ! x j ^ v ' ] ( q : ) ) o + G ' , , ( [ ^ v i > < j f v f ] ( q ) l ; i i v i > < J f v f ] C q ' : ) 3 o - E . , . , exp [ __J 23 2 ] j2y2 Q ^ i i t 1 i 2 v 2 [j !v ! x j 'vL] ( q ) / P . , ,) L J l I J f f J v i ' v i ; o J 2 2 exp•[" E j 'jV'J + A . s i — G " ( t J i v i > < j f v f ] ( q V j v , i 14 6 /P- i . [j - v . x j 'v' ] ( q ) ) U Vj' LJ 1 i J f f 1 J o T h i s d i v i s i o n i s b a s e d on w h e t h e r t h e i n t e r n a l s t a t e c o h e r e n c e s [j ± v ± > < j £ v £ ] and [j !><j £ v £ ] ^  r e f e r t o t h e same m o l e c u l e ( £ ' ) or t o d i f f e r e n t m o l e c u l e s ( £ " ) . E x p l i c i t l y , t h e s e c r o s s s e c t i o n s have t h e f o r m (4.27) G , ^ 3 i V i > < 3 f V ( q V J 2 V J J i v i > < i ^ v £ ] ( q V j , v , ) 0 = ( - l ) ( 2 , ) 4 h 2 ( ^ T ) ^ ( 2 q + l ) - 1 E ^ q ) ( . ) 2 q t r 1 > 2 - 4 7 I / d Y ( [ J i v i > < j £ v £ ] ( q ^ ) V J 2 V 2 x [ / d ( y g ' ) e x p [ - Y , 2 ; < y g | t | i i g ' > [ j | v ! > < j £ v £ ] ( q ) / 7 . , , ~ . . ~ J 2 V 2 < y g ' | t f 6 ( E - E ' ) | y g > (2iri)~ 1exp{-Y 2}[<ug|t|yg> [ j j v l x j ' v ' ] ( q ) /O. , }2V2 - [ J i v i > < j ^ v £ ] t q V j I v I < y g | t + | y g > ] ] 2 V2 and 147 (4.28) J 2 2 J I I IT d y C t ^ v . x j . v , ] ^ ) ) ^ ^2 2 x [ / d ( y g ' ) e x p - Y ' 2 < U g | t | y g » > ^ > . , v , [ j ! v ! ><j Lv'.] ( q ) J l V l (2) <yg|t + 6 ( E - E « ) |yg> + ( 2 T r i ) - 1 e x p - Y 2 [ < U g | t | p g > ( ^ , [ j ! v ! x j 'v'] ( q ) • - ~ 3 i \ 1 1 1 r (2) - , v , [j ! v ! x j 'v'] C q ) < y g l t + l y g > ] ] . J l 1 1 1 (2) ~ T h e s e e q u a t i o n s a r e a n a l o g o u s t o e q u a t i o n s (3.59) and ( 3 . 6 0 ) , r e s p e c t i v e l y . R o t a t i o n a l i n v a r i a n c e o f t h e t r a n s i t i o n o p e r a t o r i t s e l f , as e x p r e s s e d by e q u a t i o n (3.64) f o r t h e t r a n s l a t i o n a l - i n t e r n a 1 c o u p l i n g scheme, a l l o w s a r e p r e s e n t a t i o n o f t h e s c a l a r c r o s s s e c t i o n s (4.27) and (4.28) e n t i r e l y i n terms o f r e d u c e d ( i n v a r i a n t ) q u a n t i t i e s . The e v e n t u a l a n a l o g s o f (3.65) a r e f o u n d t o be (4.29) G , ( J i v i > < J f v f ] ( q ) I ^ i n ^ J ^ f ] 1 ^ ^ (q) 148 i .+j c i ! + i I , . J i J f , . J 1 J f 1 ( - l ) i (-i) v ' * "E t r ~Li2v2 2 / E t r d E t r e x p F f ~ e x P — n ( k T ) / j 2 v 2 t r t r K 1 Q Tl Fl ( y g 1 — o 5! /dE. 6 ( E . +E. +E„ -E., ,-E., ,-E .) • ) 2 j 2 v 2 t r J f V f J 2 V 2 t r ^ r V f ^ 2 V 2 t r'-I [ ( 2 j ! + l K 2 j 2 j 2 + l ) (2A' + 1) (-1) XX 1 T ( j !v! J 2 v ' X ' ; i 1 l 2 J L ; j .v. j 2 v 2 A ) T* ( j f v • j 2 v ' A ' ; lxl2L\ j £ v f J 2 v 2 X ) •E, -E . V I / E ^ d E ^ e x p " ^ exp[ ^ f l (kT ) J 2 V 2 ( y g ) — ( 2 j 2 + l ) (2X + 1) « Cj ± J f I J i J f ) [ T ( j ? v ! J 2 v 2 X ; 0 0 0 ; j . v i J 2 v 2 X ) 6 ( v f | v p + T ( j f v f J 2 v 2 X ; 0 0 0 ; j f v f J 2 v 2 X ) 6 ( v . | v ! ) ] and ( 4 . 3 0 ) 14 9 C - D i 1 f ( - i ) 1 1 — I /E dE exp[ F ^ ] e x p [ i i l g — ] CkT") -j ' v' K 1 ^ • l l •E •E . , , kT I fdE+ <5(E. +E. +E -E., ,-E., ,-E* ) I I [ ( 2 j ! + l ) ( 2 j ' + l ) ] J 5 ( 2 j ! + l ) ( 2 X ' + l ) ^ Z 2 A A ' ^ ^ i • ^ 2 ' ( - 1 ) 1 ( -1 ) r q A 2 r 2 l i -i ' i l l J 2 J £ ? i / 2 Z 2 q { z i z x / } [ ( 2 / 1 + 1 ) ( 2 i 2 + l ) ( 2 / ^ + 1 ) ( 2 Z 2 + l ) ] : T C J J v ^ v I X ' ; ^ / 2 Z J J i v i J 2 v 2 X ) T * ( j ^ v > j f v £ X ' ; / - z^2 / ; j f v f J 2 v 2 X ) + l (-i) i ( 2 q + l ) -% 1 - E 2 ^ t r t r ( k T K X 1 Z Z T X./E^dE^expI. ^] - E . r 3 i V i , exp[ ] 150 2 ( 2 A ' + 1 ) [ ( 2 j +1) ( 2 j U l ) ] ^ ( - l ) * 1 T * ( J i V . j£vp« ;qqO; j £ y f j ?v!X') i _ p + i ^ - i ) t ( 2 q + l ) - ' £ — i - _ J /E dE exp[ ^ j e x p f — ^ ] (kT) -X' r t r K 1 — ! i i 7 ( 2 X ' + 1 ) [ (2j • + 1) (2j + 1 ) ] ' 2 ( - 1 ) (yg') T ( j f v f j ! v ! X ' ; q q 0 ; j i v i j £ v £ x « ) In e q u a t i o n s (4.29) and ( 4 . 3 0 ) . t h e i d e n t i f i c a t i o n s (3.68) and (3.69) have been employed. In a d d i t i o n , the d e f i n i t i o n (4.26) i n c l u d e s a l l t h e B o l t z m a n n f a c t o r s . As s u c h , t h e forms (4.29) and (4.30) c l e a r l y i l l u s t r a t e t h e u n e q u a l B o l t z m a n n a v e r a g i n g o f t h e i n i t i a l and f i n a l s t a t e s - a d i r e c t c o n s e q u e n c e o f t h e f o rm o f l i n e a r i z a t i o n ( 4 . 6 ) . T h i s i s c o n t r a s t e d w i t h t h e s y m m e t r i c a l a v e r a g i n g o f i n i t i a l and f i n a l s t a t e s employed i n e q u a t i o n s (3.65) and ( 3 . 6 6 ) . The g r e a t e r r o t a t i o n a l s i m p l i c i t y o f t h e forms (4.29) and ( 4 . 3 0 ) , due t o t h e a b s e n c e o f v e l o c i t y p o l a r i z a t i o n s , i s a l s o r e a d i l y a p p a r e n t . The r e s u l t s (4.29) and (4.30) a r e e x a c t and r e p r e s e n t t h e s t a r t i n g p o i n t f o r t h e a p p r o x i m a t e c a l c u l a t i o n s employed i n the n e x t s e c t i o n . But f i r s t , o t h e r forms o f t h e s e e x a c t r e l a t i o n s a r e o b t a i n e d i n o r d e r t o compare w i t h q u a n t i t i e s c a l c u l a t e d b y o t h e r r e s e a r c h e r s . A s m e n t i o n e d i n c h a p t e r I I I , t h e s e a d d i t i o n a l f o r m s c a n r e p r e s e n t p r e f e r r e d s t a r t i n g p o i n t s f o r a l t e r n a t e a p p r o x i m a t i o n s c h e m e s , u s e d t o e v a l u a t e t h e g e n e r a l i z e d c r o s s s e c t i o n s . T h e t r a n s f o r m a t i o n ( 3 . 6 7 ) c a n b e a p p l i e d t o o b t a i n a n S m a t r i x r e p r e s e n t a t i o n f o r t h e c r o s s s e c t i o n s , s t i l l w i t h i n a t r a n s l a t i o n a l - i n t e r n a l c o u p l i n g s c h e m e . I n d e e d , t h e r e s u l t s ( 4 . 3 1 ) 5 ' C [ J i v i > < j f v f ] ( q ) | [ j ! v ! > < j f v f ] C q ) ) o (-D j , ? , / E t r d E t r e x P [ k T ~ ] e x P £ " T r ™ C V T 1 2 2 0 =• I / d E + 6 ( E . + E . + E . - E * - E . , , - E . , , ) ( y g ' ) 2 j 2 v 2 t r t r 3 2 v 2 3 f V f t r 3 2 V 2 3 f v f I [ ( 2 j !+l) ( 2 j f + l ) ] J s ( 2 j f + l ) ] ! s ( 2 J 2 i - l ) ( 2 X + 1 ) L A X ' x [ ' S ( j i v i | j ! v ! ) 6 ( j f V f | j £ v f ) 6 ( J 2 v 2 X | J 2 v 2 X ' ) 6 ( Z x L 2 L | 0 0 0 ) - S U W I j ' v . X ' ; / 1 / 2 ^ J i v . J 2 v 2 X ) 152 s * ( j f v f j 2 v 2 x - ; L l L 2 l ; j f v f j 2 v 2 x ) ] and ( 4 . 3 2 ) r 3 > " ( [ j i v i > < j f v f ] ( q ) | [ j ! v ! > < j f v f ] ( q ) ) o = i C-i) ~~2 I /E dE exp[ j ^ ] e x p [ ] ( k T ) ^ j ' v - t r t r k r k ^ + — — o I J " d E + 6 CE . + E . E ^ - E . , , - E . , , - E ' ) C v g ' ) 2 J 2 V 2 J f V f J 2 V 2 + 3 l V i ^ * • I I [C2j •'•M)(2j ' + l ) ] % ( 2 j ' + 1 ) ( 2 X ' + 1 ) L 1 l , 2 l XX' 1  Al k2 [C2 / ^ l) (2 /2+l) (2/. | + 1) ( 2 / 2 + l)]% (-1) ^ + L 2+ 1 1 + 1 2 + Z + q C-D j2+Ji+Ji+Jf Mi^i V ^ i ' l ^ X [ 6 ( ^ 1 / > 2 / ' / 2 / |00000) 6 C X | x ' ) 6 ( j . v . | j ' v ! ) 6 ( j £ v f | j ' v ' ) 6 C J 2 v 2 | j f v p 153 - S U ' v 1 ; j ! v ! A - ; ^ v . j ^ A ) S * ( j [ v ' j f v f A ' ; L [ L 2 L 5 j f v f j 2 v 2 A) ]' a r e a l m o s t i m m e d i a t e . T h e s e e q u a t i o n s a r e the a n a l o g s o f (3.71) and ( 3 . 7 2 ) , r e s p e c t i v e l y , and have been p r e v i o u s l y r e p o r t e d . 2 A t o t a l a n g u l a r momentum ( t o t a l J ) c o u p l i n g scheme has been q u i t e p o p u l a r i n p r e s s u r e b r o a d e n i n g s t u d i e s , and r e p r e s e n t s an a l t e r n a t e method o f c o u p l i n g t h e a n g u l a r momentum q u a n t i t i e s i n v o l v e d i n (3.63) t o f o r m a s c a l a r ( r e d u c e d ) m a t r i x e l e m e n t . 8 In t h e n o t a t i o n o f C u r t i s s and c o w o r k e r s , (4.33) < J 1 m 1 v 1 J 2 m 2 v 2 p A s | t | j ^ m j v | j ^m^v^p'A 1s'> ro • . ^ r • , j 1 + 3 2 + X + 3 i H J 2 + X \ ™ l m 2 + s = ( 2 m y ) (pp') ( - i ) (-1) I [ ( 2 j + l ) ( 2 j + l ) ] % ( 2 J + l ) /Q T J ( j i v ' J 2 v » j A ' | i 1 v 1 J 2 v 2 J A ) , where /Q i s the Y u t s i s c o u p l i n g d i a g r a m shown i n f i g u r e 6. A g a i n , n o t e t h e r e v e r s e d o r d e r o f i n d i c e s employed on o p p o s i t e s i d e s o f t h e e q u a l s i g n . A c o m b i n a t i o n o f (4.33) w i t h t h e F i g u r e 6: C o u p l i n g d i a g r a m /? 155 t h r e e e q u a t i o n s ( 4 . 2 6 ) t h r o u g h ( 4 . 2 8 ) , a n d t h e s u b s e q u e n t d e f i n i t i o n o f a r e d u c e d S m a t r i x f o r t h i s c o u p l i n g s c h e m e , ( 4 . 3 4 ) = < S ( j ^ v ' J 2 v 2 J A ' | J 1 v 1 J 2 v 2 J A ) - T J ( j ^ v ^ J 2 v 2 J X ' | j 1 v 1 J 2 v 2 J X ) , l e a d s e v e n t u a l l y t o t h e i n v a r i a n t e x p r e s s i o n s ( 4 . 3 5 ) I" 1 ( [ j . V . x i - V -1 ^ ! f i ! v ! > < i l v ' J ^ ^ C_=> ^ 1 i i " ± £ J | L J i i " t i - 1 J h + h j i + j f i r . . . - E t r " E V V ? ( - l ) i * ( - ! ) l - j I / E d E e x p [ r H ] e x p [ ] ( k - r r j 2 v 2 t r t r k l — i i i — -, i r R y /dE«. S(E«. + E . + E . - E ' -E], , - E . , , ) ( y g ' ) 2 J 2 V 2 t r t r J 2 V 2 ^ f V f t r ^ 2 V 2 ^ f V f . I [ ( 2 j + l ) ( 2 j ' + ( 2 j + l ) ( 2 j • + 1 ) ] ' S ( 2 J + 1) ( 2 J + 1) X X ' j j ' J J ' J J { q j i - f i i q i j ~ q -} i 5 ' q -i 156 ( - i ) j i + j f + j l + j ^ ( - i ) J 2 + J 2 • ( . D J - J ^ - J ' ^ - ^ ' .J ,. [S C J I v l j - v - j - X - J i V i J 2 v 2 J X ) S J ( j f v f J 2 v ' j ' A ' | j f v f J 2 v 2 J X ) 6 ( j i V i : i 2 v 2 ^ A : i f v f j h i v i 3 2 v 2 ^ ' X ^ f v f J ' )] and (4.36) G " ( [ J i v i > < j £ v f ] C c l ) | [ j ! v J > < j f v ' ] ( q ) ) = (-10 i (-i) I _ / E t i . d E ^ e x p [ j ^ j e x p t ] ( k T ) Z j ' v ' t r t r kT nil (pg ^ 4 / d E « 6 t v / E ^ 2 * E « - v r E ^ - E « ' I [ 2 j+l) (2 j •• + ! ) ( 2 j + l ) (23 ] } 5 ( 2 J + 1 ) ( 2 J + 1) X X ' j j ' {q 3 M f ] J q i ' i 3 q -} C- i ) C- i ) j 2 ( - D - j - 5 + x * - x 157 [ S J ( j ' v | j ! v ! j ' A ' | j . v . J 2 v 2 J X ) S J * ( j W ' j f v f j ' X ' | j f v f J 2 v 2 J A ) - <5(Xjj |X'JJ • ) 6 ( j . v . j 2 v 2 j f v f j 2 v 2 | j - v ^ l v l j - v ' j ' v p ] . E q u a t i o n s (4.35) and (4.36) a r e e x a c t l y e q u i v a l e n t t o (4.31) and ( 4 . 3 2 ) . They can a l s o be o b t a i n e d d i r e c t l y from the f i r s t 9 s e t by t h e t r a n s f o r m a t i o n (4.37) ' S J ( £ v J L v . i x U v £,v,jX) v a a b b ^ 1 a a b b ^ = ( _ l ) X + J + 3 { ( 2 l +1)(2£ +1) (2X + 1) (25*1) (2J + 1)]** I (-1/ [ ( 2 / a f i ) ( 2 v m 2 M ) r t i ^ I U , * a v A b^ ^ 11 ,£b / b] t h e i n v e r s e o f whi c h i s g i v e n i n r e f e r e n c e ( 8 ) , e q u a t i o n ( 7 9 ) . A c t u a l l y , a t l e a s t f o r t h e c a s e o f no v e l o c i t y p o l a r i z a t i o n s , t h i s s e c o n d method o f o b t a i n i n g e q u a t i o n s (4.35) and (4.36) i s much more cumbersome t h a n t h e f i r s t and i n d e e d has be e n m e n t i o n e d o n l y t o emp h a s i z e t h a t t he t r a n s f o r m a t i o n i s known and t h a t t h e methods employed h e r e a r e s e 1 f - c o n s i s t e n t . The r e s u l t s (4.35) and (4.36) a r e a l s o p r e s e n t e d i n r e f e r e n c e Some a d d i t i o n a l comments r e g a r d i n g t h e s e two e q u a t i o n s a r e g i v e n i n the l a s t s e c t i o n o f t h i s c h a p t e r . (e) The D i s t o r t e d Wave Born and M o d i f i e d Born A p p r o x i m a t i o n s The e x a c t c o l l i s i o n e x p r e s s i o n s o f t h e l a s t s e c t i o n , i n t h e forms (4.29) and (4.'30), a r e now e v a l u a t e d a p p r o x i m a t e l y , , u s i n g t h e method employed i n t h e p r e v i o u s c h a p t e r . T h i s method a l l o w s an a p p r o x i m a t e s e p a r a t i o n o f i n t e r n a l s t a t e and i t r a n s l a t i o n a l m o t i o n s w h i c h , i n t u r n , p r o d u c e s a u s e f u l p h y s i c a l i n t e r p r e t a t i o n o f t h e v a r i o u s c o l l i s i o n a l e f f e c t s . I n d e e d , t h e c a t e g o r i z a t i o n i s e s p e c i a l l y u s e f u l i n v i s u a l i z i n g t h e r o l e s o f t h e t r a n s l a t i o n a l s t a t e s - t h e i r s e c o n d a r y n a t u r e i n t h e r e l a x a t i o n o f p u r e i n t e r n a l s t a t e p o l a r i z a t i o n s (as d i s c u s s e d i n t h i s c h a p t e r ) v e r s u s t h e i r more a c t i v e r o l e i n t h e r e l a x a t i o n v e l o c i t y p o l a r i z a t i o n s (as d e t a i l e d i n c h a p t e r V I I ) . Thus t h e a p p r o a c h u s e d h e r e has t h e a d v a n t a g e o f ' f u r t h e r a p p l i c a b i l i t y t o t r a n s l a t i o n a l r e l a x a t i o n and t h e p r e s e n t t r e a t m e n t o f t h e t r a n s l a t i o n a l m o t i o n s s h o u l d be c o n t r a s t e d w i t h the comments g i v e n i n t h e f i n a l c h a p t e r . The l i n e a r i n T t e r m s i n e q u a t i o n s (4.29) and (4.30) a r e f i r s t t r a n s f o r m e d u s i n g t h e o p t i c a l t h e o r e m , e q u a t i o n ( 3 . 8 6 ) . As t h i s t r a n s f o r m a t i o n i s e x a c t , t h e r e s u l t i n g e q u a t i o n s a r e a l s o e x a c t and t h e s e a r e c o l l e c t e d as " f o u r e f f e c t i v e c r o s s s e c t i o n s c o r r e s p o n d i n g t o f o u r s e p a r a t e c o l l i s i o n a l p r o c e s s e s . E x p l i c i t l y , t h e s e a r e (4.58) £ , + . C [ J i v i x j f v f ] ( c l ) | [ j ! v ! > < j ^ ] ^ 160 f 1 . . - i ' J f r . , J i + : i f 1 r ' r " E t r , r j 2 V 2 , (kr) 3 2v> 0 * R y /dE. 6 ( E . +E. +E„ -E., ,-E., ,-E* ) ( y g ' ) 2 j 2 V 2 " J £ V f J 2 V 2 " j * v * J 2 V 2 t r I [ ( 2 j !+l) ( 2 j f + l ) ] l s ( 2 j 2 + l ) (2X'+1) ( - l ) q J i J f ^ T C j j v J j ' v ' X V ; / ^ / ^ . v . j ^ X ) T * ( j ^ v f J 2 . v 2 X ' ; / 1 / 2 / ; j f v £ J 2 v 2 X ) and (4.39) r ^ ' ^ ( [ j i v i > < j f v f ] ( q ) | [j ! v ! > < j f v f ] C q ) ) o 1 r v v ' ' . r C t r , r 3 2 V 2 2 • i , , i -\ i 4 , , ^ t r t r ( k T ) - j ' v ' X - j v l l l 2 l J 2 v 2 X n f i 2 .1 — 2 2 5 C J i J f h i i p / d E t r 6 ( E t r - E t r + x ) ( 2 J 2 + l ) ( 2 X ' + l ) 161 x [ T * ( j | v ! J 2 v * A ' ; I lL 2l \ jv j 2 v 2 A) T (j ! v ! j \ v 2 A ' 5 / ! / 2 A; j v j 2 v 2 A) 6 ( v f | v p + T U f V p ' v ' A ' * L 1 L 2 L ; J v J 2 v 2 X ) T * ( j ^ v p 2 v 2 A ' ; / ^ 2 / ; j v j 2 v 2 A) 6 C v . | v p ] - E . , , T • "XT' " CkT) j 2 v 2 (yg-) 2 ( 2 j 2 + l ) (2A-+1) x [ « ( v f | v p T h ( j ! _ v | j . 2 v 2 A « ; 000 ; j i v i j 2 v 2 x « ) + 6(v. | v p T * ( j f v p 2 v 2 A ' ;000; j £ v f J 2 v 2 A ' ) ] from equation ( 4 . 2 9 ) , and (4.40) £ V [ Ji Vi > < Jf Vf] C q ) | [ J i ^ > < J ^ v £ ] C ^ ) o (kT) j - v ! + 2 " —o I fdE+6(E. +E. + E, - E . , , - E . , , - E * ) • ) 2 j 2 v 2 t r 3 f V f J 2 V 2 t r J i V i ^ f V f t r ( y g I I [ ( 2 j ! + l ) ( Z j U D J ^ C Z j ; + ( 2 X ' + 1 ) x Z 2 XXV 2 Z . + Z 9+ / • + L ' + Z +q C- i ) ( - D j 2 + J i + j i + J f q Z 2 Z 2 , { 1 q * i / i { j i ji j£} I 2 ^2 q { L i Z. ^ } [ ( 2 Z. 1+l ) ( 2 / 2 + l ) ( 2 / - + 1 ) ( 2 ^ 2 + 1 ) ] T ' ( J i v ' j ! v ! X ' ; / X L 2 L ; J i v i J 2 v 2 X ) T * ( j i v i J f v f A ' \L x l 2 l ; j f v f j 2 v 2 X ) and (4.41) V * f f n J i + if 1 v y , F ' • -Etrn "^ i v i . IT Fl ( M g ' ) — / d E t r 6 ( E t r - E t r + x ) ( 2 X ' + 1) [ ( 2 J . + 1 ) ( 2 j £ + ! ) ] ' [ ( 2 j £ + l ) ( 2 j ! + l ) ] ! s 1G3 I ( -D j l + J2 + : ii + j f q + J f + J i "2-^ 2 (-1) 1 1 (-1) ' 1 [(2 + (2 /. 2 + l) ( 2 / • • M)(2 / l 2 + l ) ] : i -i -i i ' r^-2 ^ 2 I -i ' i < J 9 r ^ 2 ^ i / , T ( j f v f j ! v ! x - ; / 1 L 2 L ; J 1 v 1 J 2 v 2 x ) J . V . exp [ R* r] + exp[ - E . JfVf kT ]} (-i) (2qH-l)Js " 2 X 'E r r d E t r e XP [ ~ £ ] - I L ^ - 2 - ( 2 X ' + l ) ( k T ) " X (yg') x [exp[ j ^ j T — ] [ ( 2 J . + 1 ) (2j f + l ) ]^(- l ) 1 1 J l l T h ( j i v i j f v f x ' 'qq°^f v f j i v i x ' ) " E j £ v f i q + j F + J 4 + eXp[ -j^ f-i- ] [ ( 2 j ! + l ) ( 2 j f + l ) ] 2 ( - l ) 4 1 164 T h ( j f v f j ! v ! X ' ; q q O ; J i v i J f v f X ' ) ] from e q u a t i o n ( 4 . 3 0 ) . In e q u a t i o n s (4.39) and ( 4 . 4 1 ) , i s d e f i n e d by e q u a t i o n ( 3 . 8 7 ) . I g n o r i n g f o r t h e moment th e l i n e a r i n ( s h i f t ) c o n t r i b u t i o n s , i t i s s e e n t h a t t h e g e n e r a l i z e d c r o s s s e c t i o n 0 i s a sum o f f o u r c o l l i s i o n a l c o n t r i b u t i o n s L> Q = +0 + ^ 3 - 0+ L> +0+ - 0, each o f w h i c h c o n s i s t s o f a r o t a t i o n a l l y i n v a r i a n t , e n e r g y d e p e n d e n t c r o s s s e c t i o n ( i . e . q u a d r a t i c i n T ) t h a t has been B o l t z m a n n a v e r a g e d o v e r b o t h t h e r e l a t i v e t r a n s l a t i o n a l e n e r g y and t h e e n e r g y s t a t e s o f t h e s e c o n d m o l e c u l e . The p a r t i c u l a r c o l l s i o n p r o c e s s w h i c h each o f t h e s e e f f e c t i v e c r o s s s e c t i o n s c o n s i d e r s , i s f o u n d by a n a l y z i n g t h e arguments o f t h e v a r i o u s r e d u c e d t r a n s i t i o n o p e r a t o r s . F o r example, i n a " d i a g o n a l " ( j . v . = j ! v ! , j - V = j L v L ) c r o s s s e c t i o n , j- 1 1 1 I I i t jG +0 d e s c r i b e s c o l l i s i o n a l e v e n t s where t h e s t a t e o f t h e m o l e c u l e o f i n t e r e s t i s n o t changed ( p h a s e sh i f t i n g ' ' c o l 1 i s i o n s ) w h i l e JZ, -0 a l l o w s f o r i n e l a s t i c c o n t r i b u t i o n s t o t h e r e l a x a t i o n . F i n a l l y , £_> +0 and £_> -0 c o n s i d e r " t r a n s f e r p r o c e s s e s " , where p o l a r i z a t i o n s a r e p a i r e d f r o m one m o l e c u l e t o a n o t h e r by c o l l i s i o n , and t h e i r c o n s e q u e n t e f f e c t on r e l a x a t i o n . T h e s e l a t t e r c o l l i s i o n e v e n t s a r e u s u a l l y i g n o r e d . I n d e e d , t h e sum o f t h e f i r s t two e f f e c t i v e c r o s s s e c t i o n s , ZT +0 and -0, can be v i e w e d as t h e r o t a t i o n a l l y c o r r e c t e d c o u n t e r p a r t s t o e q u a t i o n s (2.24) o r (2.27) o f c h a p t e r I I . T h e s e a r e a l s o t h e o n l y c r o s s s e c t i o n s t h a t a r i s e i n f o r e i g n gas b r o a d e n i n g . 165 The d i s t o r t e d wave Born approximations to these four e f f e c t i v e cross s e c t i o n s can r e a d i l y be obtained. S u b s t i t u i o n of equation (3.84), and equation (3.87) where necessary, leads d i r e c t l y t o . ' (4.42) C+([J iv i><j £v f] ( q ) | [j ! v ! > < j f v « ] ( q ) ) o >2V2 Q /,lJL l L J f J£ 1 ^2 V2 ( / J ( Z, ) <ViH '^l- 1 I l i i v J x j ' v j r l l ^ 1 l|jfv f v f U 2 ) < J 2 V 2 M J ) 2 M i 2 v 2 > | exp[-A/2] ( 2 / + 1) -*5 (2 >C 9 + l) E i1 D (oo i L./ Joo/. I . L 9 | E . + E . - E . , , - E . , , ) P 1 2 J f V f J 2 V 2 J f V f ^2 V2 ° and (4.43) G ' - ( [ J i v i x j f v f ] ( q ) | [ j ! v ! x j f V f ] ( q ) ) o 1G6 - E . i ' v ' J 2 2 I , I e x p [ - L ± ] I H J i J f U i J f ) j . v - j v 3 2 v 2 l < j 2v 2 l l ^ 2 U 2 ) I I J ^ ^ I 2 e x p [ - A / 2 ] ( 2 l + \ V h (2 /., + !) x 1 { < J V | ^ l ^ ^ 1 1 J j v i > * < J v l 1 S i ^ ^  I I J j V ^ « C v f | v f ) 2 ( 2 j !+l) (2/. 1 + + < J V I IcOt ( ^ ' l l3fvf><jv| 1 1 J f V ^ i ^ J v ! ) } ( 2 j f + l ) ( 2 / . 1 + l ) The terms l i n e a r i n T ^ v a n i s h i d e n t i c a l l y . A n a l o g o u s e x p r e s s i o n a r e f o u n d f o r , namely (4.44) f , + ([j.v.><jfvf] ( q ) | [j !v!><jfvf] ( q ) ) o - E = (-D I exp[ 3 2 v 2 i ' v 1 3 1 1 i 1 v ' 3 1 1 kT 1) /+q+j[+jf+j2+j{ 167 (-D I ,+ l 0 * 4 i * I x < j i v i ' I $ i I IJ i v i > < J i v i l \ D i I l j f v f > i v i M ^ 2 2 l h 2 v 2 > * <j f 'v f | |^ 2 " 2 - || 3 2v 2> ( 2 . X + l ) ! s e x p [ - A / 2 ] £ ; ( 1 ) C 0 0 / / L 2 ! 00 / / W ' | E + E f vf J 2 V 2 E j i v i " E J fVf" 1 0 and "(4.45) -G " - ( [ J i V i > < J f V f ] ( q ) | [j!v!><j£v£] - E - E I ( e x p [ - J i - i ] +exp[ ]) J I (-1) f ^ ^ Q Q J 2 V2 ^-1^2 r * 2 * 2 q , J i J 2 r 4 2 4 i 4. / 2 q ] (2 / + 1 ) ~ ^ exp[-A/2] 160 - - , , n { L l 3 , , , , n ( 1 V G ^ C O O ^ i. , ^ 2|00 / ^ . ^ 2 j = J 1 V E J 2 v 2 - E J i v i - E ^ v ^ o -E . -E. r 3 i V i J f v f rM {exp[ — ^ ]-exp[ ] } £ ^ ( 0 0 0 0 ) oqq In t h i s i n s t a n c e , the c o n t r i b u t i o n from the s h i f t does not a u t o m a t i c a l l y v a n i s h . Equations (4.42) through (4.45) are analogous to equations (3.88), (3.89), (3.94) and (3.95) of the l a s t chapter. The t r a n s l a t i o n a l i n t e g r a l s ^ and L?^ are d e f i n e d by equations (3.91) and (3.93), r e s p e c t i v e l y . These t r a n s l a t i o n a l f a c t o r s are now approximated. As i n the p r e v i o u s chapter, the chosen method of e v a l u a t i o n i s the modified Born approximation. Since t h i s , i n i t s e l f , i s a "high temperature" approximation, f i "\ E • ~* E . the terms i n v o l v i n g -Oh a r e s m a 1 1 i n t n e l i m i t J i V i J f Vf<<1, kT see equation (4.45), and i n consequence t h i s t r a n s l a t i o n a l i n t e g r a l w i l l not be c o n s i d e r e d f u r t h e r . Thus, w i t h i n t h i s approximation, a l l matrix elements of ^ are r e a l and r e l a x a t i o n of any i n t e r n a l s t a t e e f f e c t s i s d e s c r i b e d i n terms of only one t r a n s l a t i o n a l 169 (1) P i n c h a p t e r I I I , w i t h t h e f i n a l r e s u l t b e i n g e q u a t i o n (3.109) £ ^ ( 0 0 / / 1 Z 2 I 0 0 ^ ^ i ^ 2 I X J o " I t S e v a l u a t i o n h a s b e e n d e s c r i b e d ( f ) Some C o n n e c t i o n s w i t h O t h e r Work In c o n c l u d i n g c h a p t e r IV, a few q u a l i t a t i v e r e m a r k s a r e p r e s e n t e d r e l a t i n g the c o l l i s i o n a l t r e a t m e n t employed i n t h i s t h e s i s to a p p r o a c h e s t a k e n by o t h e r w o r k e r s . h more q u a n t i -t a t i v e c o m p a r i s o n between a l t e r n a t e methods o f e v a l u a t i n g the e f f e c t i v e c r o s s s e c t i o n s would p r o v e e x t r e m e l y u s e f u l , b u t t h i s i s n o t the aim o f t h i s t h e s i s . I n d e e d , s u c h a d e t a i l e d i n v e s t i g a t i o n c a n be c o n s i d e r e d o n l y as a c o n s e q u e n c e o f the p r e s e n t work, i n w h i c h a s i n g l e framework i s d e v e l o p e d t o encompass v a r i o u s g a s - p h a s e r e l a x a t i o n e f f e c t s . C o m p a r i s o n w i t h t h e e f f e c t i v e c r o s s s e c t i o n s o f t r a n s -p o r t phenomena and S e n f t l e b e n - B e e n a k k e r e f f e c t s has a l r e a d y b e e n . g i v e n . As p r e v i o u s l y n o t e d , the o n l y change i s i n t h e c h o s e n method o f l i n e a r i z a t i o n , w h i c h r e s u l t s i n d i f f e r e n t B o l t z m a n n f a c t o r s . F u r t h e r c o n n e c t i o n s a r e d e v e l o p e d i n c h a p t e r V I I where t h e e f f e c t s o f v e l o c i t y r e l a x a t i o n on l i n e s h a p e s a r e c o n s i d e r e d . The t r a n s l a t i o n a l -i n t e r n a l c o u p l i n g scheme i s employed t h r o u g h o u t , a s the one m ost n a t u r a l f o r a DWBA method o f e v a l u a t i o n . G o r d o n , K l e m p e r e r , and S t e i n f e l d ^ have d e f i n e d an e f f e c t i v e c r o s s s e c t i o n i n terms o f r e d u c e d S m a t r i x e l e m e n t s i n a t o t a l J c o u p l i n g scheme t h a t i s a n a l o g o u s to t h e * e x p r e s s i o n , e q u a t i o n ( 4 . 3 5 ) , d e r i v e d i n t h i s c h a p t e r . An a c t u a l d e r i v a t i o n o f the GKS e x p r e s s i o n o n l y seems to have a p p e a r e d i n p r i n t much l a t e r , as p a r t o f a p a p e r by F i t z and M a r c u s . ^ The r e l a t i o n s h i p between the GKS e x p r e s s i o n and ' i s e s t a b l i s h e d i n r e f e r e n c e (2) - the t r a n s f e r c r o s s s e c t i o n J^" vanishes f o r f o r e i g n gas broadening. With the GKS e x p r e s s i o n as a s t a r t i n g p o i n t , many approximate methods of e v a l u a t i o n have been s t u d i e d , i n c l u d i n g e n t i r e l y quantum mechanical c a l c u l a t i o n s and a l s o s e m i - c l a s s i c a l 12 c a l c u l a t i o n s . The review a r t i c l e by R a b i t z g i v e s some r e f e r e n c e s to t h i s work. The approximate scheme employed i n t h i s t h e s i s may g i v e a cruder estimate of these p u r e l y i n t e r n a l s t a t e c r o s s s e c t i o n s than those methods j u s t mentioned, but has the advantage of being d i r e c t l y a p p l i c a b l e to the study of t r a n s l a t i o n a l r e l a x a t i o n e f f e c t s as w e l l (see chapter V I I ) . , 13 The approximate method f i r s t employed by Anderson, and l a t e r extended by Tsao and C u r n u t t e , ^ and F i u t a k and 15 Van Krancadonk i s jr.ent3.or.ed because of i t s widespread acceptance and p o p u l a r i t y . I t i s one example of a semi-c l a s s i c a l -approach f o r c a l c u l a t i n g g e n e r a l i z e d c r o s s s e c t i o n s - these are u s u a l l y viewed (see Gordon et a l 1 0 ) as o b t a i n a b l e from the f u l l y quantum e x p r e s s i o n f o r ' i n the t o t a l J c o u p l i n g scheme by f o r m a l l y a l l o w i n g the o r b i t a l angular momenta to approach i n f i n i t y , . We suggest t h a t the Anderson s e m i - c l a s s i c a l f o r m u l a t i o n , i n p a r t i c u l a r , i s more compactly viewed as r e s u l t i n g from the same l i m i t i n g procedure a p p l i e d to the t r a n s l a t i o n a l - i n t e r n a l c o u p l i n g scheme r e p r e s e n t a t i o n f o r £^'. Indeed the p a r a l l e l develop-ment of the two approximate schemes (Anderson's method and the p r e s e n t work) i s q u i t e s t r i k i n g . Both r e p r e s e n t a p e r t u r b a t i o n approach and r e s u l t i n products of i n t e r n a l s t a t e f a c t o r s w i t h t r a n s l a t i o n a l f a c t o r s . The t r a n s l a t i o n a l f a c t o r s a r e e v a l u a t e d a l g e b r a i c a l l y i n e ach c a s e , w i t h v e r y c r u d e a s s u m p t i o n s a p p l i e d to the t r a n s l a t i o n a l m o t i o n . The main d i f f e r e n c e s i n the two methods a r e t h e a c t u a l form o f t h e p e r t u r b a t i o n and the p r e c i s e t r e a t m e n t o f the t r a n s l a -t i o n a l s t a t e s . The p a r a l l e l i s m r e m a i n s , however, and be-c a u s e t h e d e v e l o p m e n t s a r e so a n a l o g o u s , a s t e p - b y - s t e p c o m p a r i s o n seems f e a s i b l e , and c o u l d l e a d to a more c o m p l e t e u n d e r s t a n d i n g o f b o t h t r e a t m e n t s . 16 P i c k e t t has r e c e n t l y p r o d u c e d a r e l a x a t i o n m a t r i x w h i c h i s s i m i l a r t o t h e ' o f t h i s t h e s i s , and d i s c u s s e s the m e r i t s o f b o t h the t o t a l J and t r a n s l a t i o n a l - i n t e r n a l c o u p l i n g schemes. An a l t e r n a t e b a s i s f o r h i s r e l a x a t i o n m a t r i x i s a l s o p r e s e n t e d , w h i c h i s n o t r o t a t i o n a l l y i n v a r i a n T h i s b a s i s i s u s e f u l , however, when s t r o n g e x t e r n a l s t a t i c e l e c t r i c o r m a g n e t i c f i e l d s a r e p r e s e n t , and may be v i e w e d as a g e n e r a l i z a t i o n o f the b a s i s (3.31) to t h e " o f f -d i a g o n a l i n j " c a s e . The p r e s e n t work has n o t c o n s i d e r e d s u c h a b a s i s a l t h o u g h the comments made i n c h a p t e r I I I r e g a r d i n g the b a s i s (3.31) must a l s o be c o n s i d e r e d r e l e v a n t to the s p e c t r o s c o p i c c a s e - i n p a r t i c u l a r , i t was f o u n d n e c e s s a r y to r u t h l e s s l y a p p r o x i m a t e t h e " m" d e p e n d e n c e o f the r e l a x a t i o n m a t r i x i n t h i s b a s i s , b e f o r e any t r a c t a b l e t r e a t m e n t c o u l d be d e v i s e d . P i c k e t t t r e a t s the t r a n s l a -t i o n a l d e g r e e s o f f r e e d o m c l a s s i c a l l y and d o e s n o t c o n s i d e r t h e p o s s i b i l i t y o f any t r a n s l a t i o n a l r e l a x a t i o n e f f e c t s . He has p r o d u c e d a r e l a x a t i o n m a t r i x to d e s c r i b e t h e d e c a y 173 o f t h e m i c r o w a v e t r a n s i e n t e f f e c t s - one o f the g o a l s o f the p r e s e n t work as w e l l . B u t i n c o n t r a s t to P i c k e t t ' s q u a l i t a t i v e r e m a r k s , t h i s l a t t e r a s p e c t i s t r e a t e d i n m a t h e m a t i c a l d e t a i l i n the n e x t two c h a p t e r s o f t h i s t h e s i s . 17 An e x c e l l e n t p a p e r by L i n and M a r c u s c a n be v i e w e d as a more c o m p l e t e d e v e l o p m e n t o f P i c k e t t ' s a p p r o a c h . A g a i n , a r e l a x a t i o n m a t r i x i s d e r i v e d w h i c h i s a n a l o g o u s to £ ^ ' i n a t o t a l J c o u p l i n g scheme, w h i l e no a t t e m p t i s made t o i n c l u d e t r a n s l a t i o n a l r e l a x a t i o n e f f e c t s . E q u a t i o n s o f m o t i o n , c o m p l e t e w i t h c o l l i s i o n a l s h i f t s , a r e d e t a i l e d to d e s c r i b e t r a n s i e n t b e h a v i o u r , and t h e e f f e c t s o f s t a t i c f i e l d s a r e a l s o d i s c u s s e d . Much o f the p a p e r i s r e s t r i c t e d to d i a m a g n e t i c d i a t o m i c s ( o r l i n e a r p o l y a t o m i c s ) . A s e m i -c l a s s i c a l t r e a t m e n t f o r c a l c u l a t i n g t h e a p p r o p r i a t e r e l a x a -18 t i o n r a t e s f o r OCS i s p r e s e n t e d i n a s u b s e q u e n t p u b l i c a t i o n . The work o f L i n and M a r c u s r e p r e s e n t s a p o i n t o f v i e w a l t e r n a t e t o t h e k i n e t i c t h e o r y a p p r o a c h o f t h i s t h e s i s . 19 2 0 F i n a l l y , the a p p r o a c h o f T u r n e r and S n i d e r ' s h o u l d be m e n t i o n e d . T h e s e w o r k e r s a r e i n the p r o c e s s o f d e v e l o p -i n g methods o f a p p r o x i m a t i n g c r o s s s e c t i o n s from an o p e r a t o r  v i e w p o i n t . T h i s i d e a i s r a d i c a l l y d i f f e r e n t f r o m any o f t h o s e m e n t i o n e d a b o v e , b u t f i l l s a l o g i c a l gap i n the g e n e r a l t r e a t m e n t o f gas p h a s e r e l a x a t i o n p r o c e s s e s . I n d e e d , as 21 e m p h a s i z e d by Fano, p r e s s u r e b r o a d e n i n g s t u d i e s a r e f u n d a m e n t a l l y c o n c e r n e d w i t h the p r o d u c t i o n and r e l a x a t i o n o f o p e r a t o r q u a n t i t i e s ( c o h e r e n c e s ) - e f f e c t s more g e n e r a l t h a n p o p u l a t i o n s . The e f f e c t i v e c r o s s s e c t i o n s a r e 174 i n t e r p r e t a b l e as a p p r o p r i a t e a v e r a g e s o v e r o p e r a t o r d i f f e r -2 2 2 e n t i a l c r o s s s e c t i o n s . ' (The d e n s i t y o p e r a t o r v i e w -p o i n t i s n e c e s s a r i l y employed t h r o u g h o u t t h i s t h e s i s , as w e l l . ) Y e t a l l o f the methods m e n t i o n e d i n t h i s s e c t i o n a r e c o n c e r n e d w i t h a p p r o x i m a t i n g S o r T m a t r i x e l e m e n t s , r a t h e r t h a n the c r o s s s e c t i o n s t h e m s e l v e s . I t i s f e l t t h a t the a p p r o a c h o f T u r n e r and S n i d e r g i v e s a more n a t u r a l d e s c r i p t i o n o f the p h y s i c a l p r o c e s s e s i n v o l v e d , and may l e a d to new a p p r o x i m a t e methods o f e v a l u a t i n g c r o s s s e c t i o n s w h i c h a r e u n a t t a i n a b l e f r o m t h e more t r a d i t i o n a l p o i n t o f v i e w . CHAPTER V Two L e v e l S y s t e m s " C o n s i d e r y o u r v e r d i c t , " t h e K i n g s a i d to the j u r y "Not y e t , n o t y e t ! " t h e R a b b i t h a s t i l y i n t e r r u p t e d " T h e r e ' s a g r e a t d e a l to come b e f o r e t h a t ! " 17G ( a ) I n t r o d u c t i o n T h i s chapter a p p l i e s the two l e v e l approximation to the more gen e r a l c o n s i d e r a t i o n s presented i n chapter IV. The approximate moment equations f o r s p e c i f i c molecular types-diamagnetic d i a t o m i c s and l i n e a r p o l y a t o m i c s , symmetric top molecules, and i n v e r t i n g symmetric tops - are i n v e s t i g a t e d w i t h i n the c o n t e x t of t h i s approximation. In p a r t i c u l a r , the m o d i f i c a t i o n s of the simple ( two state) theory o u t l i n e d i n chapter I I due to a r i g o u r o u s c o n s i d e r a t i o n of the mag-n e t i c s u b l e v e l s are demonstrated. The diamagnetic d i a t o m i c and l i n e a r polyatomic (DDLP) case i s t r e a t e d i n c o n s i d e r a b l e d e t a i l i n s e c t i o n s (b) and (c) . The remaining tv/o s e c t i o n s t r e a t the symmetric top and i n v e r t i n g symmetric top s i t u a -t i o n s i n a more c o n c i s e manner, emphasizing the s i m i l a r i t i e s and d i f f e r e n c e s with DDLP case. The v e c t o r i a l ( a c t u a l l y , the t e n s o r i a l ) a s p e c t s of the problem are emphasized i n the method of p r e s e n t a t i o n , s i n c e t h i s r e p r e s e n t s the u n d e r l y i n g d i f f e r e n c e between the two s t a t e and two l e v e l p o i n t s of view. T h i s t e n s o r i a l approach has the f u r t h e r advantage of more c l e a r l y demonstrating the connections with the methods employed i n the study of the Senftleben-Beenakker e f f e c t s . A theory d e a l i n g with e l e c t r i c d i p o l e induced t r a n s i -t i o n s between ro ta t i o na1 l e v e l s or s u b l e v e l s must, of n e c e s s i t y , be concerned with the e l e c t r i c d i p o l e moment operator U, the r o t a t i o n a l angular momentum operator j , and the r e l a t i o n between them. The o p e r a t o r J i s a l w a y s d i a g o n a l i n " j " quantum numbers. F o r d i a m a g n e t i c d i a t o m i c s and l i n e a r p b l y a t o r a i c s , p* J=0 (see f i g u r e 7a) and c o n s e q u e n t l y the e l e c t r i c d i p o l e moment o p e r a t o r c o n t a i n s no d i a g o n a l i n " j " m a t r i x e l e m e n t s . F o r s y m m e t r i c top m o l e c u l e s , the v e c t o r s J and }i a r e n o t n e c e s s a r i l y o r t h o g o n a l ( s e e f i g u r e 7b, where t h e quantum number k d e s c r i b e s the p r o j e c t i o n o f J on ]i) , and hence t h e e l e c t r i c d i p o l e o p e r a t o r c a n c o n t a i n a d i a g o n a l i n " j " p a r t . F i n a l l y i n an i n v e r t i n g s y m m e t r i c top l i k e NH^, t h a t p a r t o f y w h i c h i s d i a g o n a l i n " j , | k | " c a n be f u r t h e r s u b d i v i d e d i n t o i n v e r s i o n m o t i o n s - ( s e e f i g u r e 7c) - and ]i i s e n t i r e l y o f f - d i a g o n a l i n the i n v e r s i o n quantum number. T h e s e q u a l i t a t i v e r e m a r k s must be k e p t i n mind when s p e c i f y i n g the two l e v e l s y s t e m to be c o n s i d e r e d . I n d e e d , the g e n e r a l e x p r e s s i o n f o r the e l e c t r i c d i p o l e moment o p e r a t o r i s (5.1) c t 3 i v i > < 3 f v f 5 (1) J l J f 1 f w h i l e t h e two l e v e l e x p r e s s i o n s f o r the d i p o l e moment o p e r a t o r i n e a c h o f the t h r e e c a s e s o f f i g u r e 7 a r e t r u n c a t i o n s o f t h i s e x p a n s i o n . E x p l i c i t l y , t h e s e a r e g i v e n a s (5.2a) u = C M x j - l ] (1) + c [ j - i x j ] (1) A3 (2 j+1 ) - fo ld degenerate ^ ( o ) Q ) P ^ ( 2 j ) Q)F? (2 j -1) - fo ld degenerate ^ ( 0 ) 0 ) R - < y ( 2 j " 2 ) ( 0 )P jk (2 j + 1 ) - fold degenerate 15 ( o )Q)fJ k' - ^ ( 2 j ) ( 0 ) P , jk j i k ^ = ( 2 j - l)-fold degenerate ^ ( o ) ( 0 ) R _ | k 2 5 ( 2 J " 2 ) Q ) P _ j-ik jkl+> •I jk 1-5 (2 j+1) - fo ld degenerate ( 2 j +1 Mold degenerate " ^ ( 2 J ) Q ) P , jk »I+x+l Sl -x - l R o t a t i o n a l c h a r a c t e r i s t i c s p o l y a t o m i c s , (b) symmetric o f (a) d i a m a g n e t i c d i a t o m i c s and l i n e a r t o p s , and (c) i n v e r t i n g s y m m e t r i c t o n s . 179 (5.2b) u = c [ jk><j - lk) (1) + C [ j - l k > < j k ] (1) + C . [jk>< j k ] (1) + C [ j - l k > < j - l k ] (1) j - l (5.2c) M = C + [ j | k | + > < j | k | - ] (1) + C [j|k|-><j|k|+] (1) where t h e p r e c i s e e x p r e s s i o n s f o r the s c a l a r c o e f f i c i e n t s a r e n o t o f c o n c e r n a t t h i s moment. In t h e t r u n c a t i o n s employed i n e q u a t i o n s ( 5 . 2 ) , t h e r e i s the i m p l i c i t assump-t i o n t h a t the o t h e r i n t e r n a l s t a t e d e g r e e s o f f r e e d o m ( d e s c r i b i n g e l e c t r o n i c and v i b r a t i o n a l m o t i o n s ) c a n be removed from d i r e c t c o n s i d e r a t i o n . S i n c e th e s p a c i n g between t h e s e s t a t e s i s much l a r g e r t h a n t h a t o f t h e r o t a -t i o n a l s t a t e s , the d r o p p i n g o f t h e s e terms i s c o n s i s t e n t w i t h the i d e a s o f p h a s e r a n d o m i z a t i o n as o u t l i n e d i n c h a p t e r I I I . Two l e v e l a p p r o x i m a t i o n s to o t h e r o p e r a t o r s o f p h y s i c a l i n t e r e s t , s u c h as ]i and A N , c a n a l s o be e x p r e s s e d i n terms o f the a b s t r a c t b a s i s [ j ^ v . x j f v f ] f o r e a c h o f the t h r e e p h y s i c a l s i t u a t i o n s o f f i g u r e 7. T h i s w i l l be done i n the a p p r o p r i a t e s e c t i o n s w h i c h f o l l o w . I n t h e s e s e c t i o n s , however, the e m p h a s i s i s s t i l l on the more a b s t r a c t b a s i s o p e r a t o r s [ > < j f v ] * ^ , as i t i s t h i s d e s c r i p t i o n w h i c h i s the e a s i e s t to m a n i p u l a t e and to g e n e r a l i z e . (b) DDLP Case - D i m e n s i o n a l C o n s i d e r a t i o n s and C o n n e c t i o n s w i t h C h a p t e r IV C o n s i d e r the two l e v e l s j and j - 1 i n f i g u r e ( 7 a ) . The u n p e r t u r b e d j l e v e l i s ( 2 j + l ) f o l d d e g e n e r a t e ( t h a t i s , t h e r e a r e 2 j + l p o s s i b l e " m.. " v a l u e s ) w h i l e the j - 1 l e v e l i s ( 2 j - l ) f o l d d e g e n e r a t e . The t o t a l d i m e n s i o n a l i t y o f the two l e v e l " s t a t e s p a c e " i s t h e r e f o r e 4 j w h i l e t h e d i m e n s i o n a l i t y i n the o p e r a t o r s p a c e i s the s q u a r e o f t h i s , 2 • "' •' • 16 j . Thus the - o p e r a t o r , space,; i s c o m p l e t e l y spanned by • 2 16 j i n d e p e n d e n t o p e r a t o r s . N o t e ; t h a t the t e n s o r o p e r a t o r [ j ' X j " ] ^ q^ has 2q+l i n d e p e n d e n t o p e r a t o r s [ j ' X j " ] f o r V =0,+1,...+ . In p a r t i c u l a r , the o p e r a t o r s [ j > < j ] ^ ° ^ , [ J > < j ] ^ ^ , . . . [ j X j ] ^ - ' J c a n be formed f o r e a c h l e v e l j , 2 p r o d u c i n g ( 2 j + l ) i n d e p e n d e n t o p e r a t o r s . S i m i l a r l y the 2 j-1 l e v e l c a n be d e s c r i b e d i n terms o f ( 2 j - l ) o p e r a t o r s o f the form [ j-l><j-1 ] ( ° } , [ j-1>< j - 1 ] ( 1 } , . . . [ j-l>< j - 1 ] ( 2 j " 2 5 . 2 0 2 2 The r e m a i n i n g 1 6 j - ( 2 j + 1 ) " - ( 2 j - 1 ) = 2 ( 4 j -1) o p e r a t o r s a r e formed w i t h the o f f d i a g o n a l p a i r s [ j > < j - l ] ( q ) , t j - l > < j ] ( q ) r u n n i n g from [ j><j-1] ^^' , . . . [ j > < j - 1 ] ^ 2 3 . As an example o f t h e s e c o n s i d e r a t i o n s , [see a l s o c h a p t e r VI] the j = l c a s e o f f i g u r e 7a i s spanned by the o p e r a t o r b a s i s [ 0 > < : 0 ] ^ 0 ^ , [ j > < j ] ( o ) , [1><1] [1><1] ( 2 ) p l u s [1><0] [ 0 > < 1 ] ( 1 ) . I t i s e a s i l y v e r i f i e d t h a t t h i s l i s t c o n t a i n s 16 i n d e p e n d e n t o p e r a t o r s , a s i t s h o u l d . A s s o c i a t e d w i t h t h i s b a s i s o f the o p e r a t o r s p a c e f o r the two l e v e l s y s t e m , t h e r e a r e r a t e e q u a t i o n s f o r the f o u r t y p e s o f t e n s o r o p e r a t o r s [ j x j ] ^ q ' , [ j - l > < j - 1 ] * q * , [ j><j-1) ' q ' f and [ j - l > < j ] ^ q ' . T h e s e e q u a t i o n s a r e o b t a i n e d by s p e c i a l i z i n g e q u a t i o n s (4.13) and (4.16) to the two l e v e l s i t u a t i o n . T hese r e s u l t s a r e (5.3) g | <[]><]] ( q > > = - i V < [ j > < j ] ( q ) | ^ | [ j > < j ] ( q ) » 0 ( < [ i > < j ] ( q ) > - < [ j > < j ] ( q ) > ) - i ^ « t j > < j ] ( q ) | ^ | [ J - l > < 3 - l ] ( q ) » 0 6 ( J (< [ j - i x j - l ] ( q ) > - < [ j-l>< 3-1] ( q > > p o ) till - E i q + q 2 ( 2 q + l ) 1 / 2 ( 2 q ? + l ) 1 / 2 < j | M b - l > { q ^ 1 q 2 j - 1 j j V ( q q 2 l ) C ) 1 + q 2 E q < [ j - 1 > < j ] ( q 2 > > ( - 1 ) q + q 2 + 1 + E i q + q 2 ( 2 q + l ) 1 / 2 ( 2 q 0 + l ) 1 / 2 <j-l||y||j>* {* ^ q 2 2 j - 1 j j V ( q l q „ ) ( O q 2 + 1 < [ j > < M ] ( Q 2 ) > E ( - l ) q + q 2 + 1 and o 182 (5.4) ± + i y | < ( j-l>< j-1] ( q ) > = - i h < < [ j - l > < j - l ] ( q ) | ^ l f j - i x j - i ] ( q )>>. (< [ j - i x j - i ] ( q ) > - < [ j - l X j-1] < q ) > ) - i ^ « [ j - l > < j - i ] ( q ) $ | [j><j] ( q l » j < [ j > < j ] ( q ) > -< [ j > < i l l q ) > > - I i q + q 2 ( 2 q + l ) 1 / 2 (2q + 1 ) 1 / 2 ^4 2 < j - l | | y | | j > * ( '.} V ( q q l ) ( . ) q 2 E j j-1 j-1 2 < f j > < j - l ] ( q 2 ) > ( - l ) q + q 2 + 1 + Z i q + q 2 ( 2 q + l ) 1 / 2 q2 (2q + l ) 1 / 2 < j | | y | | j - l > * { q 1 q 2 } V ( q l q 0 ) ( - ) 1 + q 2 j j-1 j-1 2 - , ( a o ) . , . , . Ct+Oo + l ~o f o r the " d i a g o n a l " t e n s o r o p e r a t o r s w h i l e 18 3 (5.5) i"h — <{ j>< j-1] ( q ) > = -fcAoj< [ j X j - 1 ] ( q ) > - ils«tj><j-ll ( q ) |^| [ j X j - 1 ] ( q )>> * < [ j > < j - l ] ( q ) > + Z i q + q 2 ( 2 q + l ) 1 / 2 ( 2 q 0 + l ) 1 / 2 < j | | y ! | j - i > * { q 1 * 2 ) " q 2 ' j - i j - i j V ( q q 2 D ( . ) 1 + q 2 E o < [ j - 1 >< j-1 ]. ( q 2 } > (-1 ) q + q 2 + J -- Z i q + q 2 . ( 2 q + l ) 1 / 2 ( 2 q 2 + l ) 1 / 2 < j | | M | | j - l > * { q ^ "} * 2 3 3-1 J V ( q l q , ) ( . ) q 2 + 1 < [ j X j ] ( q 2 ) > E ( - l ) q + q 2 + 1 Z ~ o and (5.6) i-K y | < [ j - l x j ] ( q ) > = +tiAu < I j - l x j ] ( q ) > - i K « t j - l x j ] ( q ) |^| [ j - l x j ] ( q ) » * < [ j - l x j ] ( q ) > + Z i q + q 2 ( 2 q + l ) 1 / 2 ( 2 q 2 + l ) 1 / 2 < j - l | | y | | j > * C q 2 S q 2 j J-1 D V ( q q 2 l ) ( . ) q 2 + 1 E < [ j X j ] ( q 2 ) > ( - 1 ) q + q 2 + 1 - Z i q + q 2 ( 2 q + l ) 1 / 2 ( 2 q 2 + l ) 1 / 2 < j - l | | M | | j > * {* 1 q 2 > q 2 j - i j - i j V ( q l q 2 ) ( . ) q 2 + < [ j - l > < j - l ] ( q 2 ) > E Q ( - l ) q + q 2 + 1 184 r e p r e s e n t t h e moment e q u a t i o n s f o r the " o f f d i a g o n a l " p a r t s . The above e q u a t i o n s a r e t h e e x a c t g e n e r a l i z a t i o n o f e q u a t i o n (2.19) f o r r o t a t i o n a l t r a n s i t i o n s i n d i a m a g n e t i c d i a t o m i c s or l i n e a r p o l y a t o m i c s . T h e s e e q u a t i o n s d e s c r i b e t h e e v o l u -t i o n o f t h e two 1 e v e l ( b u t 4 j s t a t e s , b e c a u s e o f the m a g n e t i c s u b l e v e l s ) s y s t e m compared to the two s t a te s y s t e m d e s c r i b e d i n c h a p t e r I I . F o r an a r b i t r a r y j , t h e y a r e s t i l l too c o m p l i c a t e d and some f u r t h e r a p p r o x i m a t i o n must be contem-p l a t e d . (The j = l c a s e i s a t l e a s t t r a c t a b l e and w i l l be d i s c u s s e d i n c h a p t e r VI.) The e q u a t i o n s f o r t h e f i r s t f o u r moments [ j x j ] ^ ° ^ f [ j-l>< j - 1 ] ( o ) , [ j X j - 1 ] ( 1 ) and [ j - i x j ] ( 1 ) c a n be o b t a i n e d d i r e c t l y f r o m the above e q u a t i o n s u s i n g the p r o p e r t i e s o f the 3 - j t e n s o r s V f q l q ^ ) ( s e e a p p e n d i x B) and the e x p r e s s i o n s 2 f o r the 6-j s y m b o l s . I n d e e d , (5.7) < [ j > < j ] J ( 0 ) > = - i f c « [ j x j ] ( 0 ) |$| [ j x j ] ( o ) > > o « [ j > < j ] ( o ) > - < [ j x j ] ( o ) > ) e q - i * « [ j x j ] ( o ) [ j - i x j - i ] < o ) » o ( < [ j - l > < j - l ] ( o ) > .1/2 - < [ j - l > < j - D ( 0 ) > ) ^ r ? r i V ( 0 1 1 ) ( . ) 2 e q ( 2 j + l ) 1 / 2 E < [ j - l X j ] U ) > + _ i v ( 0 1 1 ) ( •) 2< [ j > < j - l ] U ) >E and (5.8) i-fc ^ | < [ j-l>< ( 0 ) > = - i V < [ j - l > < j - l ] ( o ) \(H\ [ j - i x j - l ] ( o ) » ( < [ j - l > < j - l ] ( o ) > - < [ j - l > < j - l ] ( o ) > ) eq - i 1 \ « [ j - i > < j - i ] ( o ) | ^ | [ J > < J ] ( o ) » .1/2 < [j><j] ( o )>-< [j><j] ( ° } > > ^ T~7o 1 V ( 0 U ) ( . ) 2 e q ( 2 j - l ) 1 / 2 .1/2 E < [ j>< j-1] ( 1 ) >+ ^ ~ i V(011) ( •) 2< [ j - l > < j ] U ) > E (2J-5-1) ' " ° a r e t h e moment e q u a t i o n s f o r the two d i a g o n a l p o l a r i z a t i o n s w h i l e t h e l o w e s t moments w h i c h a r e o f f d i a g o n a l i n " j " a r e g i v e n a s (5.9) ±h |^<( j > < j - l ] U ) > = - 1 i A w < [ j > < j - l ] ( 1 ) > - i ^ « [ j x j - l ] ( 1 ) | ^ | [ j x j - 1 ] ( 1 ) > > * < [ j x j - l ] ( 1 ) > . .1/2 . . .1/2 + - - i J i a - 7 7 - r V ( 1 0 1 ) . E < [ j - l x j - l ] ( o ) > ^ - T J X ( 2 j - l ) 1 / 2 ~° ( 2 j + l } 1 / 2 v ( i i o ) ( . ) < [ j > < j ] ( o ) > E o - [ 2 j ^ . ^ T I 1 / 2 y j 1 / 2 v ( i i i ) ( • ) 2 K < [ j - l > < j - l ] ^ > > + [ ¥ l J j ± i l T . ] 1 / 2 y j l / 2 v { 1 1 1 ) ( . ) 2 13 6 3 V(112) ( • ) 3< [ j><j] ( 2 ) > E Q and (5.10) i i i j | < [ j - l > < j ] ( 1 ) > = +tiAco < [ j - l > < j ] ( 1 ) > - ih«[j-l><j] [ j - i x j ] ( 1 ) » * < [ j - l > < j ] ( 1 ) > • i nn 1 / 2 i \ -,, -1/2 + 3 1/2 V ( 1 0 1 ) ( O B < [ J > < 3 I ( 0 ) > U j + l ) 1 ^ ~° ( 2 j - l ) 1 / 2 V ( I I O ) ( . , < [ M > < M ] ( O ) > E O - h V J ^ y ' 2 ^ i / 2 ~o 23(23+1) J V ( l l l ) (-) 2E < [ j > < j ] U ) > + [ 0 3 ! j " 1 ! J 1 / 2 U J 1 7 2 V ( l l l ) ( . ) ~~ 23 v Z3-1) M j - l X j - l ] ( 1 ) > E - i [ ( 2 j + 3 ) ( p + l ) 1/2 1/2 U 3 J 2o 1 l ( 2 j + l ) (2j) ( 2 j - l ) J ^ vu2i,«.,v»><'ic">* i • , 2 i i i , ? : a ; , ( n j i i ) i l / 2 *"/2 V(112) ( *) 3< [ j - l > < ( 2 ) > E ~o 187 In e q u a t i o n s (5.7) t h r o u g h ( 5 . 1 0 ) , the r e l a t i o n (5.11) <j||u||j-l> = u < j | | r | | j - l > = y j 1 / 2 has been u s e d w h i c h i s v a l i d f o r d i a m a g n e t i c d i a t o m i c s and l i n e a r p o l y a t o m i c s . The f i r s t t h i n g to n o t i c e a b o u t e q u a t i o n s (5.7) t h r o u g h (5.10) i s t h a t t h e y a r e n o t a c l o s e d s e t . T h a t i s , t h e y i n v o l v e t h e a d d i t i o n a l f o u r moments [ j > < j ] ^ ^ , [ j><j] ( 2 ) , [ j - l > < j - l ] ( 1 ) and [ j - l > < j - 1 ] ( 2 ) . The e q u a t i o n s o f m o t i o n f o r t h e s e l a t t e r f o u r moments c a n a l s o be o b t a i n e d from the g e n e r a l f o r m s (5.3) and (5.4) b u t f u r t h e r unknown moments w i l l be i n t r o d u c e d a t t h i s s t a g e as w e l l . I n d e e d , the e x a c t c l o s u r e o f t h e s y s t e m o f e q u a t i o n s w i l l n o t be c o m p l e t e u n t i l a l l o f the moments w h i c h span the s p a c e 2 ( h a v i n g a t o t a l o f 1 6 j i n d e p e n d e n t components) a r e i n t r o -d u c e d . E x c e p t f o r the s i m p l e s t c a s e j = l , t r e a t e d i n c h a p t e r V I , the number o f moments i n t h e e x a c t p r o b l e m r a p i d l y r i s e s to unmanageable p r o p o r t i o n s as the quantum number j i n c r e a s e s , Thus, some a p p r o x i m a t e scheme must be e m p l o y e d . T r a d i t i o n a l a p p r o x i m a t e methods a r e e q u i v a l e n t , i n the n o t a t i o n o f t h i s s e c t i o n , to n e g l e c t i n g the e f f e c t o f the o p e r a t o r s [ j > < j ] ( 1 ) , [ j > < j ] ( 2 ) , [ j - l > < j - l ] ( 1 ) and (2) [ j - l X j - 1 ] i n t h e s e t o f e q u a t i o n s (5.7) t h r o u g h ( 5 . 1 0 ) . T h i s i s d i s c u s s e d f u r t h e r i n the n e x t s e c t i o n . However, t h e r e a p p e a r s to be no r i g o r o u s j u s t i f i c a t i o n f o r t h i s r a t h e r ad-hoc p r o c e d u r e f o r c l o s i n g the s y s t e m o f e q u a t i o n s . 108 Rather, i t seems that they are ne g l e c t e d because no p h y s i c a l i n t e r p r e t a t i o n has been made f o r these e f f e c t s . The p r e s e n t work i n d i c a t e s t h a t , c o n t r a r y to t h i s o p i n i o n , these e f f e c t s can be w e l l c h a r a c t e r i z e d (both m athematically and p h y s i c a l l y ) and should be i n c l u d e d f o r a b e t t e r d e s c r i p t i o n (and under-standing) of the two l e v e l problem. The most obvious method of d e s c r i b i n g the e f f e c t of the four o p e r a t o r s [ j x j ] ( 1 ) , [ j X j ] ( 2 ) , [ j-1><j-1) ( 1 ) , and ( 2) [ j - l X j - 1 ] i n an approximate manner i s to extend the ad-hoc procedure one step f u r t h e r . Namely, i n c l u d e the equations of motion f o r these four l e v e l p o l a r i z a t i o n s and trun c a t e the system of equations a t t h i s p o i n t . Thus from the g e n e r a l two l e v e l equations (5.3) and ( 5 . 4 ) , the trun c a t e d equations (5.12) i1h j|< [ j>< j] ( 1 ) > = - ik<< [ j>< j] ( 1 ) |^| [j>< j] U ) > > 0 < [ j > < j ] ( 1 ) > - i1 i«[j><j] U ) \(H\ [ j - l X j - 1 ] ( 1 ) > Q < » - i > < J - i ) ( 1 ) > + l 3 f ^ l l / 2 M l 1 / 2 v U i i ) ( . ) J ; o < [ j > < j - u u > > 5 0 a n d 10 9 (5.13) 3 - | < [ j > < j ] ( 2 ) > = - i 1 i « [ ' j > < j ] ( 2 ) ^ | t j > < j ] ( 2 l » o < [ j > < j ] ( 2 ) > - ^«[j><j](2)|(S|[j-l><j-l](2)» < [ j - 1 x j - 1 ] ( 2 ) > + i r ( 2 ^ ; { ^ > ) 1 ^ y 3 ^ v ( 2 i i , ( . ) 2 E < [ j - i > < j ] ( 1 ) > - i t t i i ± i L L i ± i i ] i / 2 u j i / 2 K ' ~o l J 3 1 1 ( 2 j + 2j ( 2 j - 1 ) 1 M D V(211) ( •) 2< [ j>< j-1] U ) >E d e s c r i b e s the p o s s i b l e p o l a r i z a t i o n s o f t h e up p e r l e v e l w h i l e 3 ( 1 ^ (5.14) i l l —-< [ j - l > < j-1] v-' > = -it\«[ j - i x j - l ] ( 1 ) |$| [ j - i x j - l ] ( } » < [ j - l > < j - l ] ( 1 ) > [ j - i x j - l ] ( 1 ) |#| [ j><j] U ) > > <[ j><j] ( 1 ) > + ^ l l l ] 1 - ! ) ^ 2 V I 1 ' " V d l D O 2 E o < [ j X j - l , < 1 > > - t 2 - l 2 - - [ ) ] 1 / 2 M j 1 / 2 V ( l l l ) ( . ) 2 < [ j - l > < j ] ( 1 ) > E o and 3 (2) (.5.15) 1 ^  a~t < t j-i>< j - i ] - i t i < < [ j - i x j - l ] ( 2 ) \(H\ [ j - i x j - l ] ( 2 ) > > < [ j - i x j - l ) ( 2 ) > -i-fi<< [ j - l > < j - l ] ( 2 ) | $ | t j x j ] ( 2 ) » < ( j > < j ] . ( 2 ) > • * ! 7 * f J i H & ^ 1 / a « 1 / a v ( 2 i i ) ( . ) 2 S o < t j > < j - i ] ( 1 ) > - j [ ( 2 j ~ 3 ) ( j - l ) 1/2 1/2 v ( 2 1 1 ) ( . ) 2 < [ j . 1 > < j ] ( D > E 1 (2 j + 1) 2j (2 j-1) J M 3 V ^ J - J - ^ ; U J J t o g i v e t h e a n a l o g o u s m o t i o n s a s s o c i a t e d w i t h the l o w e r l e v e l . E q u a t i o n s (5.7) t h r o u g h ( 5 . 1 0 ) , and (5.12) t h r o u g h ( 5 . 1 5 ) , form a c l o s e d s e t o f e q u a t i o n s d e s c r i b i n g t h e r e s p o n s e o f a two l e v e l s y s t e m to an a p p l i e d r a d i a t i o n f i e l d . T h e r e i s s o m e t h i n g to be s a i d f o r c l o s i n g the e q u a t i o n a t t h i s p o i n t . I n d e e d , t h e e x a c t e q u a t i o n s p o i n t o u t t h a t a t h i g h i n c i d e n t power ( l a r g e E ) t h e p o p u l a t i o n d i f f e r e n c e ( d e s c r i b e d i n terms o f t j X j ] and [ j - l X j - 1 ] i s changed f r o m i t s e q u i l i b r i u m v a l u e and s i m u l t a n e o u s l y , p o l a r i z a t i o n s w i t h i n t h e two r o t a t i o n a l l e v e l s a r e p r o d u c e d . I n c l u s i o n o f t h e above f o u r moments w o u l d , a t l e a s t a p p r o x i -m a t e l y , t a k e t h e s e e f f e c t s i n t o a c c o u n t . T h e s e e f f e c t s s h o u l d d e f i n i t e l y be p r o d u c e d (and e x p e r i m e n t a l l y o b s e r v a b l e ! ) u nder s t e a d y s t a t e i r r a d i a t i o n a t h i g h i n c i d e n t power. A l s o , the moments w h i c h a r e i g n o r e d by t h i s l a t t e r ad hoc p r o c e d u r e , namely [ j > < j - l ] ( 2 ) , f j > < j - l ] ( 3 ) , [ j - l > < j ] ( 2 ) , [ j - i x j ) ( 3 ) , b e g i n to l o s e p h y s i c a l s i g n i f i c a n c e f o r the e l e c t r i c d i p o l e p r o b l e m . One f u r t h e r a p p r o x i m a t e scheme has been i n v e s t i g a t e d -c a l l e d the l a r g e " j " a p p r o x i m a t i o n . T h a t i s , does the d e g r e e o f c o u p l i n g o f t h e s e h i g h e r moments change as the p a i r o f r o t a t i o n a l l e v e l s c o n s i d e r e d r e a c h h i g h e r (more c l a s s i c a l ) v a l u e s o f " j " ? One m i g h t e x p e c t so, on p h y s i c a l g r o u n d s , s i n c e l a r g e r v a l u e s o f " j " mean a much l a r g e r r a n g e o f p o s s i b l e "m." v a l u e s and p o s s i b l e a more even d i s t r i b u -t i o n ( i . e . no [j><j] , [ j — 1> < j —1] p o l a r i z a t i o n s ) o v e r t h e s e v a l u e s . U n f o r t u n a t e l y , t h i s d o e s n o t seem t o be t h e c a s e as a s h o r t c o n s i d e r a t i o n o f ( 5 . 9 ) , (5.10) o r (5.14) shows. I n d e e d , as the l a r g e " j " l i m i t i s t a k e n i n t h e s e e q u a t i o n s , the c o u p l i n g d o e s change s l i g h t l y b u t u l t i m a t e l y comes to a " j " i n d e p e n d e n t v a l u e . Thus t h e r e i s no p o s s i b i l i t y t h a t t h i s c o u p l i n g v a n i s h e s f o r s u f f i c i e n t l y h i g h " j " v a l u e s . A n a l o g o u s r e s u l t s h o l d f o r the h i g h e r moment e q u a t i o n s f o r [ j > < c j ] [ j><j] ^ 2 ^ , [ j - l > < j - l ] (2) and [ j - l X j - 1 ] a s w e l l - the c o u p l i n g to a d d i t i o n a l moments becomes ° j " i n d e p e n d e n t a t s u f f i c i e n t l y h i g h " j " . A g a i n , t h e s e e q u a t i o n s a r e n o t c l o s e d , even i n the l a r g e " j " l i m i t , as the a d d i t i o n a l moments [ j><j-1] ^ 2 ' , [ 3 > < j —1] ^ [ j - l > < j ] ( 2 ) and [ j - l > < j ] ( 3 ) s t i l l come i n t o p l a y . Thus a t t h i s t i m e , t h e r e seems no r e a l hope o f sa t i s f a c t o r i l y 2 c l o s i n g the s y s t e m o f e q u a t i o n s u n t i l a l l o f the 1 6 j moments a r e i n c l u d e d . The r e m a i n i n g s e c t i o n s o f t h i s a r e t h e r e f o r e based on the e x t e n d e d ad-hoc method o a p p r o x i m a t e l y r e p r e s e n t i n g the m o t i o n o f the s y s t e m (c) DDLP Case - Moment Methods and C o n n e c t i o n s w i t h F l y g a r e The ad hoc methods d e s c r i b e d i n the l a s t s e c t i o n a r e , i n r e a l i t y , moment methods a p p l i e d to the s p e c t r o s c o p i c p r o b l e m . I n d e e d , an e x p a n s i o n o f the d i s t r i b u t i o n f u n c t i o n f ( v t ) i n t h e form -w2 (5.16) f ( v t ) = r — { [ j>< j ] i 0'< [ j>< j ] K O ) > (2-rrmkT) ' + [ j > < j ] U ) • 3 < [ j > < j ) ( 1 ) > + [ j>< j ] ( 2 ) :5< [ j>< j ] ( 2 ) > + l j - l x j - 1 ] ( o ) < [ j - l x j - l ] ( o ) > + [ j - l X j - 1 ] ( 1 ) - 3< [ j - l X j - 1 ] U ) > •+ [ j - l > < j - 1 ] ^ :5< [ j - l X j - 1 ] '> + tj><j-1] U ) -3< ( [ j><j-1] U ) ) + > + [ j-l>< j ] ( 1 ) -3< ( [ j-l>< j ] U ) ) + > ) and s u b s e q u e n t s u b s t i t u t i o n i n t o e q u a t i o n (4.13) l e a d s to the c o u p l e d s e t o f moment e q u a t i o n s (5.7) t h r o u g h ( 5 . 1 0 ) , p l u s (5.12) t h r o u g h ( 5 . 1 5 ) . The i d e n t i t i e s (1) t (1) (5.17) ( [ j x j - 1 ] 1 ) = - [ j - l x j l «[]><]'] ( q ) I [j'xj'"] (q,)>> = 6 q q l E ^ ^ . . , ^ . , .„, which are d i s c u s s e d i n appendix B, are a l s o u s e f u l i n these m a n i p u l a t i o n s . I f i n s t e a d , a f u r t h e r t r u n c a t i o n of the d i s t r i b u t i o n f u n c t i o n f ( v t ) i n the form -W2 (5.18) f ( v t ) = - — • { [ j>< j] ( 0 ) < [ j>< j] ( 0 ) > ( 2 TTmkT) ' + [ j - i x j - l ] ( o )< [ j - i x . j - i ] ( o )> (1) (1) t + [ j > < j - l ) U ' - 3 < ( [ j X j - l ] U ' ) > + [ j-l>< j] ( 1 ) -3< ( [ j-l>< j] U ) ) +> i s s u b s t i t u t e d i n t o (4.13), an a b b r e v i a t e d v e r s i o n of the coupled s e t of moment equations (5.7) through (5.10) r e s u l t , Namely, the terras i n v o l v i n g the moments of [ j X j ] ^ ^', [ j - l > < j - 1 ] ( 1 ) , [ j > < j ] ( 2 ) , and [ j - l > < j - l ] ( 2 ) are no longer p r e s e n t . For both expansions (5.16) and (5.18), the appro-p r i a t e e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n i s of the form . . -W2 -2£/kT -w2 (5.19) f ( 0 ) ( v ) ~ 6 - 6 (27rmkT) 3 / 2 Q (27rmkT 3 / 2 tt:><D] < [ D > < 3 ] > Si + [ j-l>< j - l ] ( o )< [ j - i x j - l ] ( o ) > e q The e x p a n s i o n s o f the d i s t r i b u t i o n f u n c t i o n (5.16) o r (5.18) i n terms o f t h e b a s i s (4.10) i s a n a l o g o u s to t h e e x p a n s i o n o f t h e d i s t r i b u t i o n f u n c t i o n i n the s t u d y o f the S e n f t l e b e n - B e e n a k k e r e f f e c t s u s i n g the b a s i s (3.24 ) . I n b o t h c a s e s , e x p l i c i t d e t a i l s o f i n t e r n a l s t a t e l e v e l -s t r u c t u r e a r e e m p h a s i z e d . These b a s i s t y p e s a r e e s p e c i a l l y u s e f u l f o r any d e t a i l e d c o l l i s i o n a l c a l c u l a t i o n s ( s e e c h a p t e r s I I I and IV) and a r e m a n d a t o r y whenever the f r e e m o t i o n a s p e c t s a r e s u f f i c i e n t l y c o m p l i c a t e d . I n t h e s t u d y o f S e n f t l e b e n - B e e n a k k e r e f f e c t s , an a l t e r n a t e r e p r e s e n t a t i o n o f i n t e r n a l s t a t e m o t i o n s ( s e e e q u a t i o n [ 3 . 2 7 ] ) was f o u n d u s e f u l i n i n t e r p r e t i n g r e s u l t s f o r c a s e s i n w h i c h t h e i n t e r n a l s t a t e m o t i o n s were n o t t o o c o m p l i c a t e d . I n p a r t i c u l a r , the d i s c u s s i o n o f t h e s h e a r v i s c o s i t y c o e f f i c i e n t f o r N 2 g i v e n i n s e c t i o n I I I - ( d ) , employed t h i s d e s c r i p t i o n . T h i s b a s i s t y p e e m p h a s i z e d the m o l e c u l a r , o b s e r v a b l e s , f o r example t h e r o t a t i o n a l a n g u l a r momentum o p e r a t o r f o r the m o l e c u l e J . As s u c h , a b a s i s o f t h i s type a l l o w s an a l t e r n a t e , more p h y s i c a l ( o r a t l e a s t , more c l a s s i c a l ) i n t e r p r e t a t i o n o f the i n t e r n a l s t a t e m o t i o n s The r a n g e o f a p p l i c a b i l i t y i s however more l i m i t e d . S i m i l a r comments h o l d f o r the s p e c t r o s c o p i c p r o b l e m . The d i s c u s s i o n o f a two l e v e l s y s t e m w i t h the e x p l i c i t n e g l e c t o f t h e more c o m p l i c a t e d i n t e r n a l s t a t e m o t i o n s (2) (2) 13) i n v o l v i n g [ j > < j - l ] , [ j - l > < j ] , [ j > < j - l ] and (3) [ j - l > < j ] a g a i n a l l o w s the p o s s i b i l i t y o f a more c l a s s i c a l " p i c t u r e " o f the phenomena. T h i s i s a c h i e v e d t h r o u g h the o p e r a t o r t r a n s f o r m a t i o n .1/2 (5.20) Vi = - i M J / 9 - { [j>< U ) + [ j - l > < j ] ( 1 ) > 3 ' 2 " ^ ( [ 3 > < i - l ] U ) ~ U - l > < i ] ( 1 ) } 1 = ( 2 j + l ) 1 / 2 [ j > < j ] ( 0 ) + ( 2 j - l ) 1 / 2 [ j - l > < j - l ] ( 0 ) AN - ±llzl ( I l ^ < 3 - 1 ] ( 0 ) „ [ j > < j ] ( 0 ) } 23 ( 2 J - D V 2 ( 2 j + 1 ) l / 2 ( J ) P . = ( 2 j + l ) 1 / 2 [ j > < j ] ( 1 ) ^ 1 ) { i ) P j - l = ( 2 j - l ) 1 / 2 [ j - l > < j - l ] ( 1 ) ^ 2 > C £ ) P j - ( 2 j + i ) 1 / 2 [ j > < j ] ( 2 ) ^ ( 2 ) ( ^ P j _ l = ( 2 j - l ) 1 / 2 [ j - l > < j - H ( 2 ) w h i c h c a n be t h o u g h t o f a s a g e n e r a l i z a t i o n o f (2.6) t o t h e two l e v e l c a s e . T h e s e o p e r a t o r s a r e h e r m i t i a n and m u t u a l l y o r t h o g o n a l i n t h e i n n e r p r o d u c t ( 4 > 1 2 ) , t h a t i s , <<l|AN>> = 0, <<y|y>> = 0 e t c . , and have t h e n o r m a l i z a t i o n s 4 • 2 (5.21) « l | l > > = 4 j <<AN|AN>> = _ l _ = _ i <<U = <<y |0>> = 2 U 2 j < < ^ ( 1 ) ( J j P j | ( - ) ^ ( 1 ) ( J ) P j > > = 3 ( 2 j + l ) « V ( 2 ) U ) P , ( •) 2fV ( 2 ) (J ) P S » = 5 ( 2 j + l ) < < ^ U ) ( ~ ) P j - l ' ( * ) ^ U ) ( ~ ) P J - 1 > > = 3 < 2 J - D <<y ( 2^J ) P j „ 1 | ( O 2 7 ^ ( 2 ) (J ) P j _ 1 > > = 5 ( 2 j - l ) Note t h a t a l l n o r m a l i z a t i o n s have t h e same a s y m p t o t i c b e h a v i o u r i n " j " , namely b e i n g l i n e a r i n j f o r l a r g e j . i s c o n v e n i e n t i f one i s i n t e r e s t e d i n a t t e m p t i n g a l a r g " j " a p p r o x i m a t i o n . The d i s t r i b u t i o n f u n c t i o n f ( v t ) c a n a l t e r n a t e l y be expanded i n t h i s b a s i s . The r e s u l t _w2 (5.22) f ( v t ) = £ r7=. {1 |£ + M<M> + Ui3<Ji> ( 2 7 , m k T ) J / 2 q j ( i l l z i , 2V 3 * C ^ < ^ l , , - J , ' j < f e ^ V 2 , , - , , ' * ! U j - l ) " 0 ( 2 j + l) " ( 2 i - l ) ^ 198 i s e q u i v a l e n t t o e q u a t i o n (5.16) and i n d e e d c a n be o b t a i n e d from (5.16) u s i n g e q u a t i o n s (5.20) and ( 5 . 2 1 ) . A f u r t h e r t r u n c a t e d moment e x p a n s i o n f o r f ( v t ) w h i c h i s e q u i v a l e n t to e q u a t i o n (5.18) t a k e s the form , e ~ w 2 „ <1> , A N < A N > , r 3 < P (5.23) f ( v t ) = j-^ U — + — + — (2mnkT) '  J ( 4 j - 1 ) 2y j j 13 • 3 < )3 > + ~ — } 2U j The e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n f ( v ) has the moment e x p a n s i o n -W 2 - % / k T -W2 <1> '(5.24) f ( o } - — 2 — e ° - * — {1 — ( 2 T r m k T ; 3 / < J Q (27rmkT) J /^ 4 ~J A N < A N > + — 3 > 4-i -1 C J . ") D and r e p r e s e n t s the a p p r o p r i a t e e q u i l i b r i u m f o r m f o r e i t h e r (5.22) o r ( 5 . 2 3 ) . I n terms o f t h e p h y s i c a l q u a n t i t i e s (5.20), the moment e q u a t i o n s t a k e t h e form A « I | £ | A N > > (5.25) -rx <1> = = (<AN> - < A N > ) 91 . . 2 eq « l | ^ l 1 » 0 —. ( < 1 > _ <!> ) 4j eq 199 9 2 1/2 7 « A N | 0 e | l » ^ < A N > - - | 3 1 / 2 V ( 0 1 1 ) ( . ) - E o <£> ~ °- « 1 > - < l > e q ) <<AN|^.|AN>> 2. ( < A N > ~ <AN> ) 4 j 2 - l G q D f^<y> „ + A w < £ > + | ( ^ - ) 1 / 2 - T ^ I V ( l l l ) ( . ) 2 ^ ^ ' ( ^ P . ^ 3<<y |$U>>> ~ ^ - — £ <u> 2 y 2 j ^<P> 2 ? i 2 = - Aw<y>+ (-—if—) y j V ( 1 0 1).E <AN> 3 t ~ 3 l / 2 4 j 2 _ 1 ~o + 2 lI2JLtlLLi±lI11/2 P 2 j V ( 1 2 1 ) C ) 3 E <7y { 2 ) ( J ) P > * 1 6 j ( 2 j - l ) J ( 2 j + l ) ' « y | $ | y » . _ 3 —=-= — <y> 2 y ^ j 200 f o r t he f i r s t f o u r moments, n e g l e c t i n g any c o l l i s i o n a l s h i f t s . The r e m a i n i n g f o u r moment e q u a t i o n s a r e (5.26) 1/2 ~ <lj{1) (^)Pj> = + | ( — ) V C l l l ; ) (•) 2 K Q < U > 2 3 JI r r ^ i ) 2 — — — <V*i> ( 2 3 - l f - - ^ - ^ < f ] ( i ) P ^ > |-<,Z/1) ( J ) P • ,> = + :! ( ~ ) V ( l l l ) (•) 2 E <u> 3 t y ~ 3-1 n ^3 ~o ~ ; ^ 1 ) ^ > P j - l l < g l ^ - ) ^ > P j - l > > o , ^ ^ £JH v — J *. . 1 L/C I / >7 I ~" J J. . . * " ' / T \ 9 < (2) > „ _ ' l r.3(j + l ) ( 2 j + 3) 1 / 2 V ( 2 1 1 ) C ) 2 E <M> Tl L$ {JJl j -fc 1 2 j ( 2 j - l ) J ( 2 j + l ) y «o}2) ( J ) P J / ? I / J 2 ) ( J ) P >> J — 2 < Z / ( J )Pj> ^ 2 > « J - " j K ^ 2 ' ' J - » f J - l > > . < ^ 2 , C j > g - , t r ( 2 j - l ) . ^ W.' Fj-1 ( 2 j - l ) ^ . • < y 2 ,i » v i i e i ' 2 , ' J - 1 P 3 > > o V 2 > 1 J ) P . > (2j-f-l) <? " 3 W i t h i n t h e c o n t e x t o f t h e two l e v e l a p p r o x i m a t i o n as d i s -c u s s e d i n the p r e v i o u s s e c t i o n , e q u a t i o n (5.25) a r e e x a c t w h i l e e q u a t i o n s (5.26) a r e a p p r o x i m a t e ( t r u n c a t e d ) v e r s i o n s . I f , i n s t e a d , t h e a b b r e v i a t e d form (5.23) i s c h o s e n f o r f ( v t ) , t h e n e q u a t i o n s (5.25) w i t h o u t t h e terms i n v o l v i n g < | / 1 ) ( J L ) p j > ' < ^ 2 ) ( , l ) p j > '  <z/1) {Vpj-i>r a n d  K^ 2) ( , l ) p i - i > ' a r e n e c e s s a r y to d e s c r i b e t h e d i a g o n a l i n " j " p o l a r i z a t i o n s o f the p a i r o f l e v e l s , a s a c l o s e e x a m i n a t i o n o f the f r e e m o t i o n terms i n (5.25) and (5.26) i n d i c a t e s . T h e s e q u a n t i t i e s are a g a i n more "molecular" i n nature, r a t h e r than e x p l i c i t l y i n t e r n a l l e v e l dependent p r o p e r t i e s . The c o l l i s i o n a l e f f e c t s couple the varioxis moments i n a l t e r n a t e ways. I t i s p o s s i b l e to d e r i v e the i d e n t i t i e s (5.28) o v 2j ' ^  ( 2 j - 1 ) < < A N | ^ H > > - ( i l ^ i ) ( « [ 3 - i > < 3 - H ^ ^ l [ J - i > < J - Ht o J » n « t j > < j ] ( 0 , l ^ l [ j - l > < j - l ] l 0 , > 0 ( 4 J 2 - ! ) 1 / 2 «[j-l><j-l] ( 0 )K|[j><j] l 0 ,» 0 ( 4 j 2 - l ) 1 / 2 + « [ J X J 1 ( P ) 1^ 1 tj><J] ( Q ) » (2j+l) «lWl'l»0 = (2j + l ) « [ j><j] C O ) |#| [ j X j ] ( 0 )>> o + ( 4 j 2 - l ) 1 / 2 « [ j X j ] (°> |^| [ j - l X j - 1 ] ( o ) » ( + ( 4 - j 2 - l ) 1 / 2 « [ j - l X j - l ] C o ) |^| [ j X j ] < ° > » + (2j-l)<< ( j - l X j - 1 ] ( 0 ) |<£| [ j - l X j - 1 ] ( ° } > > 0 203 2 1/2 « I | # | A N » o - < ± i ^ > { < § 2 ± ± ) « [ j > < j ] ( 0 )|£l t j - i > < j - i ] ( 0 , » ( - « [ j > < j i ( 0 )|^l.[j><j] ( 0 ) » •+ « [ ] - i > < j - i ] ( 0 ) | ^ | [ j - i > < j - i ) ( 0 ) » o 1/2 ' ( f j T T ! « f 3 - i > < . i - i ] ( o ) l ^ h j > < j ] ( o ) » 0 } 2 1/2 « A N W | I » q - (±Lzi,{• ( | i ± i , « [ j - i > < j - i ] ( 0 , | ^ | [ j > < j ] ( 0 ) » < + < < [ j - i x j . i i ( o ) i^ei [ j - i x j - i i ( o ) o - « t j > < j 3 ( 0 ) i ^ i l j > < j ] ( 0 ) > > 1/2 < f ^ > « [ j > < j ] ( 0 ) ! ^ l t j - i > < j - i ] ( o ) » o } and « u k | u > > = <<v>l<#|y>>, - 4 i t « I J > < - J - U U ) l#l [ J X j - 1 ] U ) » 0 + « [ j - l > < j ] U ) |#| I j - i x j ] ( 1 ) » Q > . The p r o p e r t i e s o f r o t a t i o n a l i n v a r i a n c e , p a r i t y , and f r e q u e n c y c o n s e r v a t i o n have been u s e d t o a c h i e v e t h e s e i d e n t i f i c a t i o n s , as d i s c u s s e d i n c h a p t e r I V . To s i m p l i f y the d e s c r i p t i o n f u r t h e r , some o f t h e s e c o l l i s i o n a l . terms must be a p p r o x i -mated - i t i s u s u a l 3 ' 4 t o s e t <<AN [/£ | 1>> and <<l|^|AtI>> e q u a l t o z e r o on the b a s i s t h a t the r a t e o f s c a t t e r i n g o u t o f the u p p e r l e v e l a p p r o x i m a t e l y e q u a l s the r a t e o f s c a t t e r -i n g o u t o f t h e l o w e r l e v e l . T h i s a p p r o x i m a t i o n has the e f f e c t o f c o m p l e t e l y d e c o u p l i n g t h e m o t i o n o f <1> f r o m t h e o t h e r e q u a t i o n s . F i n a l l y , i n e q u a t i o n s ( 5 . 2 6 ) , t h e c o l l i s i o n a l i d e n t i f i c a t i o n s (5.29) « 7 ^ 1 ) P J I ^ I Z ^ 1 5 ( J j ) P J » 0 = ( 2 j + l ) « [ j > < j ] ( 1 ) -l/oi [j><j] ( 1 ) > > o « t j > < j J ( 1 ) | ^ | [ J - l > < J - l ] ( 1 ) » p « [ j - l > < j - l ] ( 1 ) | / ? | [ j x j ] ( 1 , » o « y ( 1 ) ( J J P J . J ^ I ^ 1 5 ^ ) p j _ 1 » 0 - ( 2 j - i ) « [ j - l > < j - l ] U > |#| [ j - i x j - l ] o (2) p l u s s i m i l a r r e s u l t s f o r ^ / ( j ) have been u s e d . I n t h e l a r g e " j " l i m i t where o n l y t h e moments o f A* 1^ and A* 2* a r e p r e s e n t due t o t h e f r e e m o t i o n , i t i s c o n v e n i e n t t o a p p r o x i m a t e t h e c o l l i s i o n a l e f f e c t s by e m p l o y i n g r e a s o n i n g a n a l o g o u s to t h a t used t o d e c o u p l e < 1 > „ In t h i s l i m i t t h e n , an a p p r o p r i a t e s y s t e m o f moment e q u a t i o n s u s e f u l i n d e s c r i b i n g the b e h a v i o u r o f a two l e v e l s y s t e m i s (5.30) 2 - < A N > = - J- 3 1 / 2 V ( 0 1 1 ) ( - ) 2 E '<y> - i— ( < A N > - < A N > ) d t ~U ~o ~ T eq 3 • 2 35— <U>= +A6)<y> + - ~ r - V ( l l l ) ( < - ) 2 E < A ( 1 ) > ~ ~<y> °t ~ ~ X/A+^ ~0 £ ~ * 2 2 "4T-= - A0J<U> + ^ V ( 1 0 1 ) ' E <AN> + y V ( 1 2 1 ) ( « ) 3 E < A dt - 3 1 / 2-K ~o 6!/ 2-fc ~° 1- <y> T 2 3< A ( D > 3 2 1 . ( 1 ) — a * = + i m V ( l l l ) (•) E <y> - - — - < A L X ; > d t 2 1 / 2 ^ ~o ~ T (1) 9 < A ( 2 ) > 1/2 ^ - f r = — T 7 n V ( 2 1 D ( - r E <y> - - 7 7 7 < A L ^ > 3 t 2 1 / 2 £ ~ ° T { 2 ) * In e q u a t i o n ( 5 . 3 0 ) , the d e f i n i t i o n s 206 (5.31) 1 < < A N | ^ | A N > > 4 j 1 3<<y|^|y>> 3<<y |^|IJ>>O 2 2 2]i j 2P j 1 « A ( 1 ) | ^ | A ( 1 > > > ~ o (1) 4 j 1 < < A ( 2 ) | ^ | A ( 2 ) o (2) 4 j have been e m p l o y e d . The d e s c r i p t i o n t o t h i s p o i n t has e m p h a s i z e d the v e c t o r and t e n s o r n a t u r e o f the q u a n t i t i e s u n d e r c o n s i d e r a t i o n -s o m e t h i n g w h i c h i s e s p e c i a l l y u s e f u l when t r e a t i n g t h e c o l l i s i o n a l a s p e c t s o f the p r o b l e m . F u r t h e r , t h i s t e n s o r i a l c h a r a c t e r p l a y s an i m p o r t a n t r o l e whenever a d d i t i o n a l s t a t i c e l e c t r i c o r m a g n e t i c f i e l d s a r e p r e s e n t . ( A c t u a l l y , a s s t u d i e s o f S e n f t l e b e n - B e e n a k k e r e f f e c t s have shown, the t r e a t m e n t p r e s e n t e d h e r e i s more a p p l i c a b l e t o t h e "weak" s t a t i c f i e l d c a s e . A d e s c r i p t i o n o f t h e h i g h s t a t i c f i e l d e f f e c t s on p r e s s u r e b r o a d e n i n g and c o h e r e n c e t r a n s i e n t s , on the o t h e r hand, s h o u l d r e q u i r e an e x p a n s i o n o f the d i s t r i -b u t i o n f u n c t i o n i n terms o f an a l t e r n a t e b a s i s s e t , r e p r e s e n t -i n g an a p p r o p r i a t e g e n e r a l i z a t i o n o f the b a s i s ( 3 . 3 1 ) . F o r the p r e s e n t p u r p o s e , however, i t i s s u f f i c i e n t to i l l u s t r a t e how t h e s e e q u a t i o n s d e s c r i b e t h e r e s p o n s e o f a two l e v e l s y s t e m to a l i n e a r i l y p o l a r i z e d r a d i a t i o n f i e l d . I n d e e d , f o r t h i s g e n e r a l c l a s s o f e x p e r i m e n t s , each t e n s o r q u a n t i t y i n (5.30) h a s , a t most, o n l y one component t h a t i s a f f e c t e d by the r a d i a t i o n . Thus the t e n s o r e q u a t i o n s (5.30) c a n be r e d u c e d to a c o u p l e d s e t o f s c a l a r , e q u a t i o n s , as i s now shown. The r e s t r i c t i o n to l i n e a r l y p o l a r i z e d l i g h t i m p l i e s t h a t t h e e l e c t r i c f i e l d v e c t o r E c a n be w r i t t e n as ~o (5.32) E = e ( 1 ) E° ~o ~o o ( D o where e i s one component o f t h e s p h e r i c a l b a s i s t e n s o r s e ( q ) v d e f i n e d i n a p p e n d i x B. When e q u a t i o n (5.32) i s s u b -s t i t u t e d i n t o ( 5 . 3 0 ) , d e c o u p l e d s e t s o f e q u a t i o n s r e s u l t . I n p a r t i c u l a r , the s e t c o n t a i n i n g <AN>, o f i n t e r e s t h e r e , i s e v a l u a t e d u s i n g t h e r e l e v a n t 3 - j symbols (5.33) 1 1 0 ( ) 0 0 0 (-1) ,1/2 1 1 1 ( ) = 0 0 0 0 1 1 2 ( > = (15) 0 0 0 •1/2 and i s w r i t t e n e x p l i c i t l y as 2 08 (5.34) ! — < A N > = + T E ° <\i>° ~ i ~ ( < A N > - < A N > ) dt o ~ T± eq § T < y > ° = + A o j < y > ° - ^ - < ] i > ° A p > ° - - A « < y > ° + ^ E ° < A N > E ° < A ( 2 ) > ° 8t - ~ 3^ o 3 ( 1 0 ) 1 / 2 ^ ° = < y > ° T 2 a< A<2>>o . 1 E o < f j > o . 1 < A ( 2 ) > o at ~ d o ) 1 / 2 ° - T{22) * The s e t o f e q u a t i o n s (5.34) i s termed " t h e s c a l a r e q u i v a l e n t " o f e q u a t i o n s (5.30) f o r l i n e a r l y p o l a r i z e d l i g h t . Note t h a t no component o f <A^"^> i s a f f e c t e d by t h e r a d i a t i o n i n t h i s s e t o f e q u a t i o n s . The o t h e r s e t s o f e q u a t i o n s do n o t i n v o l v e <AN> and c o n s e q u e n t l y r e q u i r e o t h e r methods o f e x c i t a t i o n b e f o r e t h e y c a n be o b s e r v e d . T h e s e a r e n o t c o n s i d e r e d i n t h i s t h e s i s . I n o r d e r t o make c o n n e c t i o n w i t h t h e work o f McGurk e t 4 a l . , two a d d i t i o n a l t r a n s f o r m a t i o n s must be p e r f o r m e d . The f i r s t i n v o l v e s the t r a n s f o r m a t i o n o f b a s i s t e n s o r s e °-»- [ z] as d i s c u s s e d i n a p p e n d i x B. I n d e e d , w i t h t h e c h o i c e o f o b a s i s t e n s o r s employed up t o t h i s p o i n t , t h e q u a n t i t i e s E Q , < y > ° and < y > ° a r e c o m p l e x . The [z] ( q ) b a s i s r e c t i f i e s t h i s i n c o n v e n i e n c e , and t h e e x a c t r e l a t i o n s h i p s a r e (5.3-5) E « E °e ( 1 ) :'= - i E ° 2 = E 2 -.0 0 0 o z y = y°e ( 1 ) - - i y ° fi - Vz * — o ^ y = y ° e o ( 1 ) = - i y ° a - y z a ~ o ^ • ^(2) ^ ( 2 ) 0 ^ ( 2 ) . ^ ( 2 ) 0 ^ , ( 2 ^ ^ ( 2 ) ^ ( 2 ) ^ ^ F i n a l l y , t h e n o r m a l i z a t i o n o f McGurk e t a l . i s employed f o r the d i p o l e moment and i t s c o n j u g a t e , namely (5.36) P r = \ y z P i = ~ 1 * z A p p l y i n g (5.35) and (5.36) t o t h e s e t o f e q u a t i o n s (5.34) g i v e s t h e g e n e r a l i z a t i o n o f t h e two s t a t e p r o b l e m to t h e two l e v e l c a s e ( 5 . 3 7 ) = E P . d t 4 z i 2 — P . = A O J P - —!=— E <•—-> d t i r . 1.2 z 4 3 i \ ^ — P = - AO J P . - ^ — P d t r x T 2 r d - < * ( I , 1 / 2 A ( 2 ) > t a _ _ J _ ^ ( 5 1 / 2 A ( 2 ) > d t K2' 2 z i (2) l2 2 z T z T h i s c o m p l e t e s t h e d i s c u s s i o n o f t h e two l e v e l a p p r o x i m a -t i o n f o r t h e d i a m a g n e t i c d i a t o m i c o r l i n e a r p o l y a t o m i c t y p e o f m o l e c u l e , p r e s e n t e d f r o m a moment method a p p r o a c h . In s u c c e e d i n g s e c t i o n s , a s i m i l a r d e s c r i p t i o n o f t h e two l e v e l a p p r o x i m a t i o n f o r s y m m e t r i c t o p s and i n v e r t i n g s y m m e t r i c t o p m o l e c u l e s i s g i v e n . The a c t u a l p r e s e n t a t i o n i s s h o r t e r , and an a t t e m p t i s made t o p o i n t o u t t h e d i f f e r e n c e s i n t h e v a r i o u s d e s c r i p t i o n s . 1_ T. (<• A N A M . 210 eq _ i H ! . E < ^ { 5 ) l / 2 f l ( 2 ) > 151? 2 2 Z 2 211 (d) R e s o n a n t T r a n s i t i o n s i n Symmetric Top M o l e c u l e s The e l e c t r i c d i p o l e i n t e r a c t i o n o f r o t a t i o n a l l e v e l s o f sy m m e t r i c top m o l e c u l e s w i t h r a d i e i t i o n c a n be d i v i d e d i n t o two p a r t s . T h i s d i v i s i o n c o r r e s p o n d s to the s e p a r a t i o n o f the e l e c t r i c d i p o l e moment i t s e l f i n t o d i a g o n a l and o f f -d i a g o n a l terms i n the r o t a t i o n a l quantum number,, In p a r -t i c u l a r , f o r two r o t a t i o n a l l e v e l s " j " and " j - 1 " t h e e l e c t r i c d i p o l e o p e r a t o r c a n be w r i t t e n as e q u a t i o n (5.26) and e l e c t r o m a g n e t i c r a d i a t i o n c a n c a u s e r e o r i e n t a t i o n w i t h i n t h e " j " l e v e l , r e o r i e n t a t i o n w i t h i n " j - 1 " l e v e l , o r t r a n s i -t i o n s between t h e s e two l e v e l s J The f i r s t two " r e o r i e n t a -t i o n " e f f e c t s o c c u r when l i g h t o f v e r y low ( a p p r o x i m a t e l y zero) f r e q u e n c y i s vised t o p e r t u r b the g a s e o u s s y s t e m o f sym m e t r i c t o p m o l e c u l e s and comes u n d e r the g e n e r a l h e a d i n g o f Debye r e l a x a t i o n . A k i n e t i c t h e o r y a p p r o a c h t o t h i s 5 p r o b l e m has been g i v e n by T i p and M c C o u r t . T r a n s i t i o n s b etween two r o t a t i o n a l l e v e l s c a n o c c u r when l i g h t t h a t i s a p p r o x i m a t e l y r e s o n a n t w i t h the e n e r g y d i f f e r e n c e o f the two r o t a t i o n a l l e v e l s u n d e r c o n s i d e r a t i o n . T h i s " r e s o n a n t " t r a n s i t i o n p r o c e s s i s now d i s c u s s e d , i n d e t a i l , , As shown i n f i g u r e ( 7 b ) , t h e two l e v e l s y s t e m u n d e r c o n s i d e r a t i o n i s d i a g o n a l i n "k" quantum number, s i n c e t h e j j r o j e c t i o n o f J on U i s u n a l t e r e d by e l e c t r i c d i p o l e t r a n s i t i o n s , and c o n s i s t s o f the two r o t a t i o n a l l e v e l s " j " and " j - 1 " . Thus the d i m e n s i o n a l i t y o f t h e o p e r a t o r s p a c e 2 i s a g a i n 1 6 j and i s spanned by the f o u r t y p e s o f o p e r a t o r s [ j k > < j k ] ( q ) , [ j - l k > < j - l k ) ( q ) , . [ j k > < j - l k ] { q ) ' a n d [ j - l k > < j k ] ( q > . B e f o r e p r o c e e d i n g to t h e e q u a t i o n s o f m o t i o n f o r a s e l e c t e d number o f t h e s e o p e r a t o r s , a few comments on the s y s t e m - r a d i 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 s h o u l d be made. The o r i g i n a l t i m e - d e p e n d e n t i n t e r a c t i o n i s o f the form where t h e e l e c t r i c d i p o l e moment o p e r a t o r has been s p l i t up i n t o a p a r t y ^ w h i c h i s p a r a l l e l t o the a n g u l a r momentum o p e r a t o r J and a pai"t w h i c h i s p e r p e n d i c u l a r to J . As d i s c u s s e d i n t h e i n t r o d u c t i o n , t h i s i s e q u i v a l e n t , i n t h e two l e v e l a p p r o x i m a t i o n ( s e e a l s o e q u a t i o n ( 5 . 2 6 ) ) , t o (5.38) vV i n t = - 2 y . E Q c o s ( W t - k « r ) c o s ( w t - k « r ) - 2p.«E c o s ( w t - k »r) (5.39) - 1 {<jk||y||jk>[ j k x j k [ (1) + < j - l k | | y j | j - l k > [ j-l k > < j - l k ] (1) } and (5.40) - 1 {< jk||y ||j-lk> [ j k x j - l k ] (1) 3 1/2 + < j-lk||y ||jk> [ j-lk>< jk] (1) } where, f o r s y m m e t r i c t o p s , 213 (5.41) < jk||u ||j'k'> = M 6 k k , i j + j ' + 1 [ (2 j + 1) (2 j ••»•!) ] 1 / 2 k j 1 j ' (-1) ( ) ~k 0 k Then, a s d i s c u s s e d i n c h a p t e r I I , t h e r o t a t i n g wave a p p r o x i -m a t i o n c a u s e s the d i a g o n a l p a r t o f U to be d r o p p e d . The e f f e c t i v e t i m e - i n d e p e n d e n t i n t e r a c t i o n i n t h e r o t a t i n g frame i s t h e r e f o r e g i v e n a s (5.42) . = - y X - E • ^  l n t ~ •* W i t h e q u a t i o n (5.41) a s t h e i n t e r a c t i o n H a m i l t o n i a n , the g e n e r a l e q u a t i o n s o f m o t i o n c a n be f o u n d from (4.13), and i n c l u d e (5.43) it?a!<[jk><jk] { o ) > = - ift«ijk><jk] ( o ) | # | [ j k x j k ] ( o ) > > ( (<[jk><jk] ( o )>-<[ jk>< jk] ( 0 ) > e q ) - [ j k > < j k ] ( o ) |(^| [ j - l k x j - l k ] ( 0 ) > > 0 ( < i j - i k x j - i k ] ( 0 ) > - < [ j - i k x j - i k ] ( 0 ) > „ ) - <3fcllvll3- i;>* i e q (2J+1, 1/ 2 v ( o i i ) ( - ) 2 E <[j-ik><jk] ( 1 ) > + <^Hv\\i*>* i ( 2 j - l ) 1 / 2 V(011) ( • ) 2 <[ jk>< j - l k ] ( 1 ) > E Q 214 and (5.44) i ^ ~ < [ j-lk>< j - l k ] ( o ) -ifc<< [ j - l k > < j - l k ] ( o ) |^| [ j-l k > < j - l k ] ( o )>> (<[j-lk><j-lk] ( o ) > - < [ j - l k > < j - l k ] ( o ) > ) e q M< l j-ik>< j - i k ] ( o ) |^| [jk><jk] ( o )>> o « [ j k > < j k ] ( 0 ) > - < [ j k > < j k ] ( 0 ) > „ ) - < j - i H M , h * > * i e q ( 2 j - l ) 1 / 2 v ( 0 l l ) ( O 2 E <[jk><j-lk] ( 1 ) > + <3K|MI J - i k > * i . ~° ( 2 j + l ) 1 / 2 V(011) (•) 2< t j - l k > < j k ] ( 1 ) > f o r the lowest order d i a g o n a l p o l a r i z a t i o n s while (5.45) i t i - | < [ j k x j - l k ] ( 1 ) > a t - i t \ « [ j k > < j - l k j ( 1 ) |^| [ j k x j - l k ] ( 1 )>>* < [ j k > < j - l k ] ( 1 ) > - h A w < t j k > < j - l k ] { 1 ) > + . < j k l | M | l j - l k > * v ( 1 0 1 ) ( O E < [ j . l k > < j . l k ] ( o ) > ( 2 j - l ) 1 / 2 ~ ° _ i U - l k l H l j k * V ( 1 1 0 ) (•)<[ j ' k X j k ] ( 0 ) > E ( 2 j - l ) X / ^ ~ ° - [ 3 1 / 2 2j < 2 j - l ) ] ' <Jk||u|| j - l k > * V ( l l l ) ( . ) 2 E <( j - l k > < j - l k j U ) > 2 j ( 2 j + l ) ] <Dk||jj|| j - l k > * V ( l l l ) ( - ) 2 < [ j k > < j k ] ( 2 ) > E ~o - i r. J 2 j-3) (j-1) 1/2 •• ., , l T 2 j T l ) 2j ( 2 j - l j ' 1 <Dk||u|h-lk>* V ( 1 2 1 ) ( . ) E ~o <[ j - l k > < j - k ] ( 2 ) > + i r ( 2 j + 3 ) ( j + l ) 1/2 ,, J 1 l ( 2 j + l ) 2 j ( 2 j - l ) - ] <3k||Ll||j-lk>* V(112) ( • ) 3 < [ j k > < j k ] ( 2 ) > and ( 5 . 4 6 ) iP\~ < I j - l k X j k ] ( 1 ) > = Au< [ j - l k x j k ) ( 1 ) > -i f c « [ j - l > < k j ( 1 ) y{ \ [ j - l x k j ( - 1 > » o * < [ j - l k x j k ] ( 1 ) > +. L — <j-lk||y j|jk>* V ( 1 0 1 ) ( - ) E < [ j k x j k ] ( o ) > ( 2 j + l ) - L / 2 ~° —"77172 < J - l k | | j j | | j k > * V(110) ( . ) < [ j - l k > < j - l k ] ( 0 ) > E (2 ] - l ) „o [ 2 ^ 2 ^ T i } ] 1 / 2 < 3 - l k | | a | | j k > V ( l l l ) (.) 2 E <[ j k X j k ] ( 1 ) > f'2j ( 2 j " l ) 1 V 2 < J - l k | | y | | j k > V ( l l l ) ( * ) 2 < [ j - l k X j - l k ] ( 1 ) > E Q  i [ ( 2 j + i ? 2 J ( 2 j - l ] ] 1 / 2 <3-13c||yHjK>V(121)(.) 3 E < [ j k X j k ] ( 2 ) > 1 [ T 2 W ( 2 j i 2 J - l ) l l / 2 <J-l*IMl5k>V<112) ( . ) 3 < [ j - l k > < j - l k ) ( 2 ) > E r e p r e s e n t t h e l o w e s t o r d e r o f f - d i a g o n a l e f f e c t s . The r e s u l t (5.43) t h r o u g h (5.46) a r e c o m p l e t e l y a n a l o g o u s to the DDLP c a s e , e q u a t i o n s (5.7) to ( 5 . 1 0 ) , i n the r o t a t i n g frame e x c e p t t h a t the r e d u c e d m a t r i x e l e m e n t o f \i u s e d i n t h i s s e c t i o n i s g i v e n by e q u a t i o n ( 5 . 4 0 ) . A g a i n , t h e e q u a t i o n s a r e n o t c l o s e d . The p o s s i b l e a p p r o x i m a t e schemes f o r c l o s i n g t h e s e e q u a t i o n s , g i v e n i n c o n n e c t i o n w i t h the DDLP c a s e , c a n a l s o be a p p l i e d h e r e . T h i s i s n o t done e x p l i c i t l y , however, s i n c e i t f o l l o w s d i r e c t l y f r o m the p r e v i o u s s e c t i o n ' s work. R a t h e r , f u r t h e r c o n s i d e r a t i o n s a r e c o n f i n e d to t h e above n o n - c l o s e d s e t o f f o u r moment e q u a t i o n s , i n o r d e r t o most c o n c i s e l y p r e s e n t the s i m i l a r i t i e s and d i f f e r e n c e s between the sy m m e t r i c t o p and t h e p r e v i o u s l y t r e a t e d c a s e s . I n d e e d , t h e above s e t o f e q u a t i o n s , (5.43) t h r o u g h ( 5 . 4 5 ) , g i v e s one r e p r e s e n t a t i o n o f t h e p r o b l e m . An a l t e r n a t e r e p r e s e n t a t i o n c a n be f o u n d i n terms o f a s e c o n d , p e r h a p s more " p h y s i c a l " , b a s i s . The t r a n s f o r m a t i o n i s g i v e n (5.47) 1 = ( 2 j + l ) 1 / 2 [ j k X j k ] ( 0 ) + (.2 j - l ) 1 ' [ j - l k > < j - l k ] a s: (o) AN } [ j k x j - l k ] (1) - i < j - l k | | u ||jk> [ j - l k x j k ] (1) 217 ^ . iJJ i lL^^> t j k >< j . l k ] (i) . <3 - ikIy[ | iJ i> [ j . l k > < j k 3 (i) 3 3 y 1 ) ( J ) P . , = (2 j+l) 1 / 2 [ jk>< jk] U ) ~ J K ( 2 ) ( J ) P . V = (2j+l ) 1 / 2 [ jk><jk] ( 2 ) ( ~ ) P j - lk = (2j - l ) 1 / 2 [ j - lk><j- lk] ( 1 ) y ' ( 2 ) < £ > p j _ l k = ( 2 j - D 1 / 2 [ j - l k > < j - l k ] ( 2 ) I n t h i s s e c o n d b a s i s , t h e f o u r moment e q u a t i o n s (5.43). t h r o u g h (5.46) become 3, v « l | ^ | A u » o (5.43) T r — <1> - : (<AK> - <AN> ) dt . 2 eq ( i Y 1 ) 4 j ( < 1 > " < 1 > e q > - | 3 1 / 2 V ( 0 1 1 ) ( . ) 2 E O « 5 > < < A N |<£| i>> ° (<l> - <i> ) 4 j eq < < A N | ^ | A N > > x ( < A N > - < A N > ) 4 j 2 - l G q 2 2 (1) , < < l i ' ^ ^ > > o V ( l l l ) ( . ) E < 2/ . (J ) P . > - 3 | — ~° «/ ~ 3 l < j k M j - l k > = - Ato<y> + — 1 — ( - 2 4 — ) l<jkj'u||j-lk>| 2 v(101) (.) - 3 l / 2 ^ 4 j 2 _ 1 r <AM> 2 r ( 2 j ~ 3 ) 1/2 l<jkj| y B j - l k > S 0 < A N > " * [ 6 j (2 j + l ) ] ( I F I T V ( 1 2 1 ) ( . ) 3 E o Y 2 ) ( J ) P . , 1 > + | ^ 2 y ^ ^ ^ < j k | y | l j - i k > | 2 V ( 1 2 1 ) ( > ) 3 E n<2/ ( 2 ) (j)Pn:> ( 2 ] + l ) ~o J 3 « y | # | y > > < jk||y !|j-lk> . 3 - Q • <y> E q u a t i o n (5.48) i s o b v i o u s l y t h e a n a l o g o f e q u a t i o n ( 5 . 2 5 ) , I n o r d e r to compare t h e s y m m e t r i c t o p c a s e w i t h t h e two s t a t e s y s t e m o f c h a p t e r I I and the DDLP c a s e , a c l o s e r l o o k a t e q u a t i o n s (5.48) i s i n o r d e r . As d i s c u s s e d i n s e c t i o n ( c ) , o n l y one component o f y (and y) i s e x c i t e d by l i n e a r l y p o l a r i z e d l i g h t . T h i s a l l o w s the v e c t o r n a t u r e o f t h e p r o b l e m to be e l i m i n a t e d . I n p a r t i c u l a r , e q u a t i o n (5.48) s t a t e s t h a t t h e p o l a r i z a t i o n s ^ J ^ P j k a n < ^ ' (J)P_. c o m p l e t e l y d e c o u p l e from the r e s t o f the e q u a t i o n s and so a r e u na f f ec t e d by the a p p l i c a t i o n o f the ( l i n e a r l y p o l a r i z e d ) r e s o n a n t r a d i a t i o n . T h i s i s n o t so s u r p r i s i n g i n i t s e l f , s i n c e t h e same e f f e c t was f o u n d i n the DDLP c a s e , ( I n d e e d , b e c a u s e t h e r o t a t i n g wave a p p r o x i m a t i o n e f f e c t i v e l y d r o p s the y t/ term f r o m the i n t e r a c t i o n , see e q u a t i o n (5.42) , the s y m m e t r i c t o p and DDLP p r o b l e m s become v e r y s i m i l a r i n m o s t r e s p e c t s . ) However, r e c a l l i n g e q u a t i o n (5.39) w h i c h e f f e c t i v e l y d e s c r i b e s y „ i n terms o f t h e two o p e r a t o r s Z / ( 1 ) ( ^ ) P j k a n d ' ^ / ' ( 1 ) ( J ) j - l k ' ° r m o r e e x p l i c i t l y (5.49) y = •i { <jk|[y[l jk>? (l) j < j-iLkHy ll3-»ik> ~ " ^ 1 / 2 ( 2 j + l ) 1 / 2 # ~ ^ ( 2 j - l ) 1 / 2 ? . v ( 1 > ( J ) P e q u a t i o n s (5.48) s t a t e t h a t y ^ i s u n a f f e c t e d by t h e r a d i a t i o n no m a t t e r how i n t e n s e t h e s o u r c e ! T h i s l a t t e r r e s u l t i s i n d i r e c t cofi t r a d i c t i o n w i t h t h e two s t a t e p r o b l e m , see e q u a t i o n ( 2 . 2 2 ) , s i n c e i n t h a t c a s e t h e d i a g o n a l p a r t s o f the d i p o l e moment o p e r a t o r depend on the p o p u l a t i o n s . The p o p u l a t i o n s , o f c o u r s e , d_o change a t h i g h i n c i d e n t power when s a t u r a t i o n s e t s i n . B e c a u s e o f t h i s a s s o c i a t i o n , t h e two s t a t e model s h o u l d be employed w i t h c a u t i o n - even t o p r o b l e m s f o r w h i c h the model i s s u p p o s e d l y a p p l i c a b l e . No i n c o n s i s t e n c i e s o f t h i s t y p e a r i s e i n the DDLP c a s e s i n c e t h e d i p o l e moment o p e r a t o r i s c o m p l e t e l y o f f - d i a g o n a l i n the " j " quantum number. (e) I n v e r t i n g Symmetric Tops and t h e I n v e r s i o n S p e c t r u m o f NH 3 T h i s s e c t i o n l o o k s a t a t h i r d p o s s i b l e r e l a t i o n s h i p between U and J , and a t h i r d t y p e o f two l e v e l s y s t e m . The c a s e u n d e r s t u d y h e r e i s a v e r y common one i n m i c r o -6,7 wave s p e c t r o s c o p y - the i n v e r s i o n s p e c t r u m o f ammonia. The i n v e r t i n g m o t i o n o f t h e ammonia m o l e c u l e i s a t t r i b u t a b l e t o t h e quantum m e c h a n i c a l t u n n e l l i n g e f f e c t w h e r e i n t h e n i t r o g e n a torn has the a b i l i t y to t u n n e l t h r o u g h the p l a n e o f t h e t h r e e h y d r o g e n s and t h u s ean e x i s t on e i t h e r s i d e o f t h e p l a n e i n s u c c e s s i v e i n s t a n t s o f t i m e . The s m a l l p o t e n t i a l b a r r i e r s l o w s down t h i s ( e s s e n t i a l l y v i b r a t i o n a l ) m o t i o n t o t h e e x t e n t t h a t i t f a l l s i n t h e m i c r o w a v e r e g i o n o f t h e s p e c t r u m . I n most sy m m e t r i c t o p s l i k e C H 3 F o r C F 3 H , on t h e o t h e r hand, t h e p o t e n t i a l b a r r i e r to t h i s m o t i o n i s so h i g h t h a t t h e m o l e c u l e s e x i s t i n one form o r t h e o t h e r f o r an i n f i n i t e l y l o n g t i m e . The f r e e ; m o t i o n e i g e n s t a t e s f o r t h e i n v e r t i n g m o l e c u l e s c a n t h e n be c o n s i d e r e d as l i n e a r c o m b i n a t i o n s o f t h e f r e e m o t i o n e i g e n s t a t e s o f t h e n o n - i n v e r t i n g m o l e c u l e s . I n p a r t i c u l a r , the i d e n t i f i c a t i o n s ( 5 . 5 0 ) | j{k|m+>= | j |k|m> - ( - l ) j |j-|k|m>}J+> |j|k|m->= | j | k|m>+(-l) j | j - | k|m>} |-> c a n be e s t a b l i s h e d . 0 E q u a t i o n s (5.50) s t a t e s t h a t , b e c a u s e o f the i n v e r t i n g m o t i o n , o n l y the m a g n i t u d e o f k ( t h e p r o -j e c t i o n o f J o n t o p) r e m a i n s as a good quantum number. From e q u a t i o n s (5.41) and ( 5 . 5 0 ) , the r e d u c e d m a t r i x 9 e l e m e n t f o r t h i s c a s e i s g i v e n as •i k j 1 j (5.51) < j | k|+||y ||j | k|-> = yi ( - 1 ) J ( 2 j + l ) ( - l ) ( - Ik| 0 Ik where y = <+|y|->, and c o n s i d e r a t i o n i s r e s t r i c t e d to a s i n g l e " j " m a n i f o l d ( i . e . p u r e i n v e r s i o n s p e c t r a ) . F i n a l l y , i t i s n o t e d t h a t b e c a u s e o f p a r i t y , y i s c o m p l e t e l y o f f -d i a g o n a l i n the i n v e r s i o n quantum number. The two l e v e l s y s t e m to be t r e a t e d i n t h i s s e c t i o n i s t h e r e f o r e t h a t shown i n f i g u r e ( 7 c ) . I n t h i s i n s t a n c e t h e 2 d i m e n s i o n a l i t y o f the o p e r a t o r s p a c e i s (4j=2) and i s spanned by the f o u r t y p e s o f b a s i s o p e r a t o r s [ j | k | + > < j | k | + ] ( q ) , [ j | k | + > < j | k | - ] ( q ) , [ j | k | - > < j | k | - ] ( q ) and Cj|k|-><j|k|-] ^ q^ f o r e v e r y v a l u e o f q=0,...., 2 j . A g a i n the e q u a t i o n s o f m o t i o n f o r a l l o f t h e s e o p e r a -t o r s c a n be d e t e r m i n e d f r o m the g e n e r a l e q u a t i o n s ( 4 . 1 3 ) . In p a r t i c u l a r , e q u a t i o n s f o r [j|k|+><j|k|+] [ j | k | - X j | k | - l ( 0 ) , [ j | k | + X j | k | - ] U ) , and [ j | k | - > < j | k | - ] ( 1 ) c a n be e s t a b l i s h e d and a r e g i v e n as 222 (5.52) i f c ~ < [ j | k | + > < j | k | + ] < 0 ) > = - i * « t j | k | + > . < j | k l + ] ( 0 ) [ j | k | + > < j | k | + ] ( 0 ) » o (< [ j | k | + > < j | k | + ] ( o ) > - < [ j | k | + > < j | k | + ] ( o ) > ) e q - i t , « t j i k i + x j | k i + ] ( o ) i ^ e i [ j i k i - > < j i k i - . ] ( o ) » o « [ j | k | - > < j | k | - ] ( ° > > . < [ j | k | - > < j | k | . 1 ( ° > > ) e q + 1 . / 0 < j i k | + | i p | | j | k | - > * V(011) ( - ) 2 E {2^1)X//L ~° < [ j | k | - > < j | k | + i ( q )> i — < j | k | - | | n | | j | k | + > * ( 2 j + l ) ± / ' ' V ( O l l ) C ) <[j|k|.+><j-|k|-] U > > E ~u and (5.55) i f c ^ l < [ j | k | - X j | k | + l ( 0 ) > = - U « [ j | k | - > < j | k | - ] < o ) |^| [ j | k | - x j | k | - ] ( o ) » o ( < [ j | k | - > < j | k | - ] ( ° > > - < [ j ] k | - > < j | k | - ] ( o ) > ) - i ^ « [ j | k | - > < j | k | . ] ( o ) | « | [ j | k | + ] ( o ) » o « f j | k | + > < j | k | + ] ( o ) > - < [ j | k | + x j | k | + ] ( o ) > _ ) 223 + < j | k | - | | y | | j | k | + > * V ( 0 1 1 ) ( . ) 2 E ( 2 j + l ) X / ~ <t:i|k|+><j|k|-] ( 1 > > - i - — • < j | k | + | | n | | j [ k | - > * ( 2 j + l ) J - / -V ( 0 1 1 ) ( . ) 2 <[jIk|-><jIk|+](1)> E Q for the moments of rank " 0 " while (5.54) ifc | ^ •<[ j |k | + ><j |k |-] ( 1 ) > = - i * » « [ j | k l > x j | k | - ] U ) |^ | [ j |k | + x j | k | - ] U ) » o <[ 3 |k | + ><j |k I-] ( 1 ) >-% Aw<[ j |k | + ><j |k |-3 ( 1 ) > + i - T 7 7 <j |k| + l)y||j |k|->* V ( 1 0 1 ) ( . ) E (2J+.1) •L/-'S ~° <{j |kl-xj |k - ] ( o ) > i ^ < j | k | _ | | y | | j | k | + > * ( 2 j + l ) X / ' i v ( i i o ) ( . ) <E J | k | + ><j|k| + [ (°)> E O + - ^ ~ - 2 t^jTBTiT3 <3 |k| + ||u||j |k|->* V ( l l l ) ( . ) 2 E Q <[j | k | - x j jk|-] ( 1 ) > " ( 2 j + ^ i / 2 [ T J T J + T T I 1 / 2 <*M-|p!|iM+>* V ( l l l ) ( . ) 2 < [ 3 | k | + x j | k | + 3 ( ° ) > E O + i _ c ( 2 3 + 3 M 2 j - l ) 3 l / 2 ~° ( 2 j + l ) i / 2 2Dt3+D <j |k|+||y ||j |k|->* V ( 1 2 1 ) ( . ) 3 E q < [ j |k |-xj |k|-] ( 2 )> 224 — S T ? [ ( 2 i ( i ! i 2 j ~ ] 1 / 2 <3UI - l lMi i j|M +>* v d i 2 ) ( 2 j + l ) J / " 3 (2) (.) .<[j|k|+><j.|k|+] v ; > E Q and (.5.55) i t l < [ j | k | - X j |k|+[ ( 1 ) > = - i ^ « [ j | k | - x j | k | + ] ( 1 ) |^| [ j | k | - X j | k | + ] U ) > > <[ j | k | - x j |k|+] ( 1 ) > + ^ Aw<[j | k | - x j |k| + ] ( 1 ) > + 1 i / ? <j lk|.-|y ||j |k|+>* v ( i o i ) (•) E ( 2 j + l ) ~ "° < [ j | k | + x j | k | + ] ( 0 ) > ^ 7 I <j|k| +||y||j|k|->* ( 2 j + l ) X / ^ V ( H O ) ( . ) < [ j | k | - > < J | k | - J ^ > E Q + ( 2 J ^ ) 1 / 2 t 2 T T W ] 1 / 2 < j | k |-||y ||j | k |+>* V ( l l l ) ( . ) 2 E q < [ j | k |+>< j | k |+] ( 1 ) > - ( 2 j H , ' , i / z ' • 2 T ( W L L / 2 <3IHH-||PIIJIM->" v c i i i x - ) <j |k|-||u||j |k|+>* v ( 1 2 1 ) ( . ) 3 E q <[j|k|+><j|k|-] ( 2 ) > V(112) ( • ) 3 <[j|k|-><j|k - ] ( 2 ) > E ~o 2 are the equations f o r the next two moments of i n t e r e s t . The s et of equations (5.52) through (5.55) are o b v i o u s l y analogous to the se t s (5.7) to (5.10) and (5.43) through (5.46) obtained i n the pr e v i o u s s e c t i o n s . The remarks made there are a l s o p e r t i n e n t to t h i s c a se. In p a r t i c u l a r , a tra n s f o r m a t i o n of o p e r a t o r s (5.56) 1 = ( 2 j + l ) 1 / 2 { [ j | k | ~ X j | k i - ] ( 0 > + [ j | k | + X j | k U ] ( 0 ) } AN - (2 j + l ) 1 / 2 { [ j | k | - x j | k | _ ] (o) . [ j l k j + X j I k i-f] ( o ) } + y [ j | k | - x j |k|+] H = l K I [ 3 7 T J T i T ^ 1 / 2 i {MU|H+><DIM-](1) - y [ j | k | - > < j | k | + ] e tc . can be made to r e c a s t the above set of equations i n t o an a l t e r n a t e form, although t h i s i s not done here e x p l i c i t l y . Before l e a v i n g the d i s c u s s i o n of the ammonia i n v e r s i o n s p e c t r a , a t t e n t i o n i s drawn to a f e a t u r e p e c u l i a r to t h i s s y s t e m . The i n v e r s i o n t r a n s i t i o n i n NH i s u n i q u e i n the se n s e t h a t m o t i o n s i n v o l v e d i n t h e e l e c t r i c d i p o l e t r a n s i t i o n between the s t a t e s |+> and |-> i s c o m p l e t e l y s e p a r a t e from t h e r o t a t i o n a l e f f e c t s due t o t h e p r e s e n c e o f m a g n e t i c s u b -l e v e l s . A l l o t h e r c a s e s d i s c u s s e d i n t h i s c h a p t e r i n v o l v e e l e c t r i c d i p o l e t r a n s i t i o n s between r o t a t i o n a l l e v e l s and a s a c o n s e q u e n c e t h e two e f f e c t s a r e i n t i m a t e l y m i x e d . The s e p a r a t i o n o f e f f e c t s i n the ammonia p r o b l e m a l l o w s the f o u r t y p e s o f o p e r a t o r s [ j | k | + > < j | k | + ) ( q ) , ( j | k | + > < j | k | - ] ( q ) , [ j | k | - > < j | k | + ] ( q ) and [ j | k j - > < j | k | - ] ( q ) t o be c o m p a c t l y r e p r e s e n t e d a s a m a t r i x o f o p e r a t o r s J U i + l ) 1 " l | ->< + | l - x - l / f o r e a c h v a l u e o f q=0,1,...,2j . Here ® d e n o t e s a d i r e c t p r o d u c t . Then a l l p o s s i b l e e f f e c t s f o r t h i s two l e v e l s y s t e m a r e d e s c r i b e d i n terms o f a d i r e c t p r o d u c t o f p o s s i b l e r o t a t i o n a l e f f e c t s w i t h p o s s i b l e two s t a t e e f f e c t s . In p a r t i c u l a r , t h e s e t o f e q u a t i o n s (5.52) to (5.55) show t h a t the p a r t i c u l a r o p e r a t o r s ( 2 j + l ) 1 / ^ (2j+D W . ( 2 j + l ) 1 / 2 t/0) {VP • I k| and ^ —r4^—'®|-><•{• | a r e o f d i r e c t c o n c e r n i n t h e two ( 2 j + 1 ) L / Z l e v e l p r o b l e m b u t o t h e r o p e r a t o r s s u c h a s 14 ( J ) P • Ikl iJ { J ) J- ' 'g>| + ><-| and \f- ~<SJ+><-| a r e n o t a f f e c t e d by ( 2 j + l ) J ' / - ( 2 j + l ) i / i ! " l i n e a r l y p o l a r i z e d r a d i a t i o n . From t h i s p o i n t o f v i e w , one m i g h t say t h a t the two s t a t e a p p r o x i m a t i o n o f c h a p t e r I I i n c o r r e c t l y c o n s i d e r s t h a t the f o u r o p e r a t o r s Vt ( J ) p i l k i / l + > < + ! l + > < -( 5 . 5 8 ) 7 I L - ^ L U L L ® ^ ( 2 j + l ) 1 / 2 1 | - X + | |-><-d e s c r i b e t h e e v o l u t i o n o f t h e s y s t e m . The d i r e c t p r o d u c t a p p r o a c h f o r t h e d e s c r i p t i o n o f u m i g h t a l s o p r o v e u s e f u l i n s e p a r a t i n g the e f f e c t s o f r e o r i e n t i n g v e r s u s phase c h a n g i n g c o l l i s i o n s i n e x p r e s s i o n s f o r —— . 2 228 ( f ) Summary T h i s c h a p t e r l i a s shown t h a t a r i g o u r o u s a p p r o a c h to t h e two l e v e l p r o b l e m r e q u i r e s c o n s i d e r a t i o n o f o r i e - n t a t i d n a l p o l a r i z a t i o n s o f e a c h o f the l e v e l s a t h i g h i n c i d e n t powers, i n a d d i t i o n to the o v e r a l l p o p u l a t i o n change ( s a t u r a t i o n e f f e c t s ) o f the p a i r o f l e v e l s . T h i s i s a g e n e r a l c o n c l u -s i o n , v a l i d f o r e a c h o f the t h r e e m o l e c u l a r s y s t e m s d i s -c u s s e d , and some e x p e r i m e n t a l m a n i f e s t a t i o n s o f t h e s e o r i e n t a t i o n a l phenomena sh o u I d be o b s e r v a b l e , i f l o o k e d 3 f o r . L i u and Marcus a p p e a r to be the o n l y o t h e r w o r k e r s who have r e c o g n i z e d t h a t a d d i t i o n a l e f f e c t s a r e p o s s i b l e due t o t h e p r e s e n c e o f m a g n e t i c d e g e n e r a c y , a l t h o u g h t h e i r p r e l i m i n a r y e s t i m a t e s seem t o i n d i c a t e t h a t s u c h e f f e c t s a r e s m a l l . The p r e s e n t c h a p t e r has gone f u r t h e r by i d e n t i f y i n g the e x a c t n a t u r e o f t h e s e added e f f e c t s and i n d e e d , has shown t h a t t h e y a r e a n a l o g o u s to i n t e r n a l s t a t e p o l a r i z a -t i o n s f o u n d i n the s t u d y o f S e n f t l e b e n - B e e n a k k e r e f f e c t s . I n t h e n e x t c h a p t e r , f u r t h e r c o n s i d e r a t i o n i s g i v e n to t h e s e o r i e n t a t i o n a l p o l a r i z a t i o n s i n the p a r t i c u l a r c a s e o f the j = 0<-*jal t r a n s i t i o n s o f DDLP m o l e c u l e s , where an " e x a c t " t r e a t m e n t i s p o s s i b l e . E s t i m a t e s o f r e l a x a t i o n t i m e s i n d i c a t e , f o r t h i s c a s e a t l e a s t , t h a t t h e s e e f f e c t s a r e s i g n i f i c a n t . 229 CHAPTER VI. The j=0*-«j=l Case f o r DDLP "I d o n ' t s e e , " s a i d the C a t e r p i l l e r . "I'm a f r a i d I c a n ' t p u t i t more c l e a r l y , " A l i c e r e p l i e d v e r y p o l i t e l y , " f o r I c a n ' t u n d e r s t a n d i t m y s e l f , to b e g i n 'with." 230 (a) I n t r o d u c t i o n I n c o n t r a s t to the t h r e e c a s e s t r e a t e d i n the l a s t c h a p t e r , the j = 0*-+ j - 1 m i c r o w a v e t r a n s i t i o n f o r d i a t o m i c s and l i n e a r p o l y a t o m i c s r e p r e s e n t s a s i t u a t i o n f o r w h i c h no a p p r o x i m a t e d e c o u p l i n g scheme i s n e c e s s a r y b e c a u s e o f the s m a l l d i m e n s i o n a l i t y o f the p r o b l e m w i t h i n the two l e v e l a p p r o x i m a t i o n . T h a t i s , a c o m p l e t e d e s c r i p t i o n o f t h i s s y s t e m c a n be g i v e n i n terms o f the moments [ 0 > < 0 ] ^ ° ^ , [ 1 X 1 ] ( o ) , [ 1 X 1 ] [ 1 X 1 ] ( 2 ) , [ 1 X 0 ] ( o ) , and [ 0 X 1 ] ( T e c h n i c a l l y , a s i m p l i f y i n g a s s u m p t i o n r e g a r d i n g t h e c o l l i s -i'onal m o t i o n o f [ 0 X 0 ] ( o ) + i [ i > < i ] i s a c t u a l l y employed i n t h i s c h a p t e r , see equa t i o n (6 .18) ) . The j=0*-* j = l p r o b l e m i s t r e a t e d i n some d e t a i l w i t h an e m p h a s i s p l a c e d on the p h y s i c a l i n t e r p r e t a t i o n o f the v a r i o u s phenomena i n v o l v e d and i n p a r t i c u l a r , how the r e o r i e n t a t i o n e f f e c t s a r e i n -c l u d e d i n t h e r e s p o n s e o f the s y s t e m . A t h o r o u g h u n d e r -s t a n d i n g o f t h i s example s h o u l d a l s o p r o v i d e a g r e a t e r a p p r e c i a t i o n o f the g e n e r a l two l e v e l p r o b l e m . F i n a l l y , the f o r m a t o f t h i s c h a p t e r p a r a l l e l s t h a t o f c h a p t e r I I to a l a r g e e x t e n t , i n o r d e r t h a t the two l e v e l and two s t a t e a p p r o a c h e s c a n be m o s t e a s i l y c o n t r a s t e d . The c o n t e n t s o f t h i s c h a p t e r a r e as f o l l o w s . F i r s t , the v e c t o r and s c a l a r f o r m s o f the e q u a t i o n s o f m o t i o n a r e p r e s e n t e d as a p a r t i c u l a r c a s e o f the r e s u l t s o f the l a s t c h a p t e r , a l t h o u g h an a l t e r n a t e method i s employed f o r t h e i r d e r i v a t i o n . The g e n e r a l r e m a r k s o f c h a p t e r IV on c o l l i s i o n s a r e s p e c i a l i z e d to a l l o w a DWBA c a l c u l a t i o n o f the r e l a x a -t i o n r a t e s i n v o l v e d , f o r s e v e r a l f o r m s ' o f the i n t e r m o l e c u l a r p o t e n t i a l . S t e a d y s t a t e and t r a n s i e n t s o l u t i o n s to the two l e v e l e q u a t i o n s o f m o t i o n a r e t h e n e s t a b l i s h e d , i n o r d e r to compare the r e s u l t s w i t h the s i m p l e r two s t a t e t h e o r y o f c h a p t e r I I . A d i s c u s s i o n o f the p h y s i c a l i n t e r p r e t a t i o n s a l l o w e d by the two l e v e l model c o m p l e t e s t h i s c h a p t e r . (b) E q u a t i o n s o f M o t i o n - V e c t o r and S c a l a r Forms The g e n e r a l s i t u a t i o n f o r r o t a t i o n a l t r a n s i t i o n s j - 1 — * i n d i a m a g n e t i c d i a t o m i c s and l i n e a r p o l y a t o m i c s has been p r e s e n t e d i n c h a p t e r V, s e c t i o n s (b) and ( c ) . T h e r e t h e 2 d i m e n s i o n a l i t y o f the o p e r a t o r s p a c e was shown to be 16j . O b v i o u s l y , the s i m p l e s t c a s e to be t r e a t e d i s j - 1 , w h e r e i n the t e n s o r o p e r a t o r s [ 0 > < 0 ] ( o ) , [ 1 > < 1 ] ( o ) , [ 1 > < 1 ] ( 1 ) , [1><1] ( 2 ) , [1><0] ( 1 ) and [0><1] ( 1 ) a r e s u f f i c i e n t to span the c o m p l e t e . s p a c e . The e q u a t i o n s o f m o t i o n f o r t h e s e o p e r a t o r s a r e t h e n s p e c i a l c a s e s o f e q u a t i o n s (5.7) t h r o u g h ( 5 . 1 0 ) , p l u s e q u a t i o n s (5.12) t h r o u g h ( 5 . 1 5 ) . I n f a c t , the t r e a t m e n t and comments employed i n s e c t i o n s (b) and (c) a r e a g a i n a p p l i c a b l e , i n c l u d i n g the t r a n s f o r m a t i o n (5.20) f r o m one b a s i s s e t o f o p e r a t o r s to a s e c o n d s e t and the e n s u i n g moment e q u a t i o n s f o r t h e s e o p e r a t o r s . ( O b v i o u s l y , however, the l a r g e " j " l i m i t d i s c u s s e d i n c h a p t e r V has no r e l e v a n c e h e r e . ) B u t r a t h e r t h a n p u r s u i n g t h i s a p p r o a c h , an a l t e r -n a t e , more c o n c i s e d e r i v a t i o n o f the a p p r o p r i a t e moment e q u a t i o n s i s g i v e n i n t h i s s e c t i o n . By so d o i n g , the v e c t o r n a t u r e o f the p r o b l e m i s e m p h a s i z e d f u r t h e r . The e q u a t i o n s o b t a i n e d by e i t h e r method a r e i d e n t i c a l . W i t h i n the c o n t e x t o f a two l e v e l ( j = 0 «-» j = l ) a p p r o x i -m a t i o n , the d i s t r i b u t i o n f u n c t i o n f i s expanded i n a c o m p l e t e o r t h o g o n a l s e t o f o p e r a t o r s a s - w 2 , r r C r 1<1> AN<AlJ> II • 3 < ^  > (6.1) f = — I - T — + + . — (2TTmkT) 7 1 J 2U y-3<u> y U ) (J) • y { 1 ) (jj y ( 2 ) ( j ) + "17" + ~ 3 •+ where (6. 2) : 1 = P + P. o 1 A N = 2 ( P - I P ) 2 o 3 1 P =' P P P + P . U P ~ o ~ o 1- o y .= i { p . y p - p U P , ~ 1 - o o ~ 1 y { 1 ) ( f 5 = y{2]'<£> = t}2)<vpi and (G.3) <<l-|l>> = 4 < < A N | A N > > = 3 <<y |u>> = 2)i * * 2 <<U VI>> = 2]S «Zj{1) ( j ) . ^ x ) (j)>>= 9 <<J( 2 ) (J) : Z^2) (J) >>= 15 Here P q and a r e t h e p r o j e c t i o n o p e r a t o r s f o r the j=0 and j = l l e v e l s , r e s p e c t i v e l y . The e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n i s g i v e n i n t h i s b a s i s a s -W (6.4) f(°} = - ~ — r l < l > + A N < A N > } ( 2 T T M K T ) 3 / 2 1 E Q E Q As d e s c r i b e d by e q u a t i o n ( 4 . 9 ) , t h e d i s t r i b u t i o n f u n c t i o n s a t i s f i e s , i n t h e r o t a t i n g f r a me, an e q u a t i o n o f the form (6.5) !*§£- - i t f ^ n t S V ' l M 1 " , ! where, f o r t h i s m o l e c u l a r s y s t e m , (6.6) 7[ ^ = // - S -e f f r L o \1 *E - ~o 23 5 = t A t o P 1 - (P pp. + P.yp )«E 1 o ~ 1 l ~ o ~c In o r d e r to o b t a i n t h e moment e q u a t i o n s from (6.1) and ( 6 . 5 ) , v a r i o u s commutators must be e v a l u a t e d . I n p a r t i c u l a r , i t i s t r i v i a l to show t h a t (6.7) [P y ] = - i u [P, Vi] = iy iPjA]' = 0 f o r A f y , y [ y , A N ] = - 2 i y [ y , l ] = o The r e m a i n i n g c o m m u t a t o r s r e q u i r e a more d e t a i l e d a n a l y s i s , F o r t h i s p u r p o s e , t h e u s e f u l i d e n t i t y (6.8) P i P f P = Z ( - i ) q ( 2 q + l ) 1 / 2 V ( l l q ) ( • ) 3 3 q=0,l,2 Z/(q) <J>Pi j 1 j ' j ' 1 j 1 1 q J ~T72 ( 2 j + l ) ( 2 j ' + l ) ( ) ( j { } (2j+1) 0 0 0 0 0 0 j j j ' p l a y s an i m p o r t a n t r o l e . E q u a t i o n (6.8) c a n then be s p e c i a l i z e d to g i v e y 2 (6.9) P U P U P = ' V (11 0 ) P o ~ l ~ o -,1/2 o n ' r i . . y_ (1) p n y p U P , = v ( i i o ) — + y i v ( i i i ) ( - ) l J - o = l ,1/2 3 ' - - v V J . ^ / v , ^ 3 From e q u a t i o n ( 6 . 9 ) , t h e two c o m m u t a t o r s (6.10) [y,y] = i [ p yp,yp - p,yp yp, + ( P yp,yp ) t o~ 1~ o 1~ o~ 1 o~ 1- o - (p.yp y p . ) t ] 1~ o~ 1 o 4y i A N 2 5 1/2 f~- V(110) — + 2 i y ^ ( | ) - L / ^ V ( 1 1 2 ) ( [ y , y ] = p yp yp - (p yp up )t + p up yp, o ~ 1~ o o~ 1~ o 1~ o~ 1 - ( P , y p y p , ) t 1 ~ o ~ 1 1) ,J 2 " ( > 2y i v ( i i i ) • f o l l o w d i r e c t l y . To e v a l u a t e the r e m a i n i n g two c o m m u t a t o r s , the c o e f f i c i e n t s a and 6 i n q q (6.11) • [ U . , ^ q ) = a q V ( l q l ) ( ')V + B q V ( l q l ) . ji f o r q=l,2 must be e s t a b l i s h e d . E q u a t i o n (6.9) c a n be u s e d to t h i s end as w e l l . I n d e e d , cc^ i s e v a l u a t e d from e q u a t i o n (6.11) by m u l t i p l y i n g by U and t a k i n g the t r a c e (6.12) t r {y [ y , ^ q ) (J) ] } = t r { ( y , y ] ^ q ) ( J) ) V ( i q l) • t r { y y > q w h i c h i m p l i e s (6.13) a = 3 i 6 q qr 1 A n a l o g o u s l y , y t i m e s e q u a t i o n (.6.11) and s u b s e q u e n t t r a c i n g y i e l d s (6.14) t r { y [ y , ^ ( q ) ( J>]> = t r ( [y,y] y { q ) ( J) ]} = 3 v ( l q i ) • t r ( y y > q and the e v e n t u a l i d e n t i f i c a t i o n (6.15) 3 = - i / 1 5 6 „ q q t 2 r e s u l t s . The o r t h o g o n a l i t y and n o r m a l i z a t i o n s ( e q u a t i o n ( 6 . 3 ) ) . o f the v a r i o u s o p e r a t o r s were a l s o u s e d i n a r r i v i n g a t e q u a t i o n s ( 6 . 1 3 ) and ( 6 . 1 - 5 ) . The r e m a i n i n g commutators o f i n t e r e s t a r e t h e r e f o r e (6.16) [p , ,Zy < 1 ) (J) ] = 3 i V(lll)»y ,(2) lU,lj <J>] = - / I 5 i V ( 1 2 1 ) ' y The moment e q u a t i o n s a r e new r e a d i l y o b t a i n e d by s u b -s t i t u t i o n o f the form ( 6 . 1 ) i n t o e q u a t i o n ( 6 . 5 ) u s i n g t h e c o m m u t a t i o n r e l a t i o n s ( 6 . 7 ) , ( 6 . 1 0 ) and ( 6 . 1 6 ) . The r e s u l t i s a s e t o f f i v e c o u p l e d e q u a t i o n s A 2 < < A N | $ | A N > > ( 6 . 1 7 ) — <AN> + TTE -<y> + - ( < A N > - < A N > ) = 0 d t -ft ~o ~ 4 eq a t?\ / i s . « U { 2 ) ( J) l ^ | t / 2 ) ( J > > > 1 ~ <"7i 2 ) ( J)> = 0 4 3 , ( 1 ) 3 «tf1)<V Wr} 1 ) (£)» < TZ<U ' U)> - T. E .V(111)-<U> + J J- = i < (J)> = 0 . 2 •j|<ii> - A O J < J J > - E O - V ( I I I ) '<ZJ1) ( J ) > + < < u . l ^ h i > > 0 < H > = 0 2 y 2 2 + A w < H > ~ I T E o < A N > " ! V ( ^ ) 1 / 2 S o ' v ( 1 1 2 ) (*>, < 2 / 2 ) ( J ) > + - ~ «yK|M»<vi> = o J ~ 2U H e r e , th e s i x t h moment e q u a t i o n i n v o l v i n g <1> has been d e c o u p l e d and i g n o r e d , by e m p l o y i n g the c o l l i s i o n a l a p p r o x i -m a t i o n ( 6 . 1 8 ) < < l | ( ^ | A N > > = < < A N | ^ ? | I > > = 0 . E q u a t i o n s ( 6 . 1 7 ) a r e t h e c o u n t e r p a r t s to e q u a t i o n s (5.30) o b t a i n e d i n t h e l a s t c h a p t e r . In p a r t i c u l a r , t h e same t y p e s o f e f f e c t s t h a t were n o t e d i n t h e l a s t c h a p t e r o c c u r h e r e as w e l l - o n l y the r e l a t i v e m a g n i t u d e s o f the c o u p l i n g s have c h a n g e d , s i n c e t h e y a r e a c t u a l l y " j " d e p e n d e n t . The methods p r e s e n t e d h e r e c a n a l s o be g e n e r a l i z e d to p r o v i d e an a l t e r n a t e , more d i r e c t d e r i v a t i o n o f the moment e q u a t i o n s o b t a i n e d i n t h e l a s t c h a p t e r . The i d e n t i t y ( 6 . 8 ) i s p a r t i c u l a r l y u s e f u l i n t h i s r e g a r d . However, d i m e n s i o n a l c o n s i d e r a t i o n s and the use o f the b a s i s []><]'] (q) a l l o w a more c o m p l e t e v i e w o f the g e n e r a l p r o b l e m , e s p e c i a l l y i n r e g a r d s to what i n f o r m a t i o n i s l o s t by a t r u n c a t e d e x p a n s i o n and i n t h e t r e a t m e n t o f t h e c o l l i s i o n a s p e c t s . In summary t h e n , b o t h d e r i v a t i o n s have t h e i r own m e r i t s a so b o t h have been p r e s e n t e d i n t h i s t h e s i s . The s c a l a r f orms o f e q u a t i o n (6.17) f o r l i n e a r l y p o l a r i z e d i n c i d e n t r a d i a t i o n f o l l o w from the comments p r e s e n t e d i n c h a p t e r V, s e c t i o n ( c ) . I n d e e d , e m p l o y i n g e q u a t i o n s ( 5 . 3 3 ) , (5.35) and ( 5 . 3 6 ) , the s c a l a r e q u a t i o n s o f i n t e r e s t become (6.19) 81 4 E P . + z 1 i ,,>AN > <2L AJI> T, 4 " 4 ) = 0 eq 0 P t r •P + Awp . = 0 - AOJP ~ 2 T 2 1 where t h e r e l a x a t i o n r a t e s have been i d e n t i f i e d as 1 3<<MI$|M>>6 3<<IJ o 2)1 2 1_ T t ) 2 ) ( J ) i ^ t / 2 ) ( J ) >> o 3 P E q u a t i o n s (6.19) and (6.20) a r e t h e f u n d a m e n t a l e q u a t i o n s g o v e r n i n g the J = O*-H> j=*l t r a n s i t i o n i n d i a m a g n e t i c d i a t o m i c s and l i n e a r p o l y a t o m i c s . In a d d i t i o n to the t h r e e e x p e c t a -t i o n v a l u e s u s u a l l y employed i n the d e s c r i p t i o n o f the two s ta te s y s t e m ( s e e c h a p t e r I I ) . a f o u r t h s c a l a r moment has a p p e a r e d w h i c h c a n be d i r e c t l y t r a c e d to the c o n s i d e r a t i o n o f the m a g n e t i c s t a t e s o f t h e u p p e r 1 e v e l . T h e s e e q u a t i o n s r e p r e s e n t the s i m p l e s t example o f the d i f f e r e n c e between the two s t a t e and two l e v e l a p p r o a c h e s . The c o l l i s i o n a l a s p e c t s w i l l be e x p l o r e d f u r t h e r i n s e c t i o n (c) w h i l e s u b s e q u e n t s e c t i o n s w i l l d e a l w i t h v a r i o u s s o l u t i o n s o f t h e s e e q u a t i o n s . (c) C o l l i s i o n I n t e g r a l s T h i s s e c t i o n i l l u s t r a t e s how t h e DWBA e x p r e s s i o n s , d e r i v e d i n c h a p t e r IV f o r a g e n e r a l i n t e r n c i l s t a t e c o l l i s i o n a l i n t e g r a l , c a n be s p e c i a l i z e d to the e v a l u a t i o n o f the c o l l i s i o n i n t e g r a l s o f i n t e r e s t h e r e , e q u a t i o n s ( 6 . 2 0 ) . I n so d o i n g , e s t i m a t e s o f t h e s e c o l l i s i o n i n t e g r a l s a r e o b t a i n e d f o r s e v e r a l assumed f o r m s o f the i n t e r m o l e c u l a r p o t e n t i a l s . The s t a r t i n g p o i n t i s e q u a t i o n s (4.42) and (4.43) o f c h a p t e r IV w h i c h , f o r DDLP c a n be r e w r i t t e n as (6.21) £ ( [ j . > < j f ) ( q ) | [ j ! X j ' ] ( q ) ) 0 / £ " ( 1 ) (2^ + l ) - 1 / 2 i -/ • J - ! j < ^ = - -ex P[> i ( e -e >] ( - i , ^ l + 3 f + 3 i {." ~> 1 3 f 3 1 • j f j'f q <3 illc9 1 U l )« Y f ^ ^ l r ^ + D T ~ V J i J f 1 J i J f 3 e x P [ - i(£,-e, ) ] =• 1 1 2 j 3 ± 2 ( 2 j ± + l ) ( 2 * +1) + 6 ( 3 i 3 f l ) 3 3 x l < J l l l j z i ? b f > l 2 2 (2j£+l) ( 2 ^ + 1) Here 243 r ~ ( i ) • i s an i n t e g r a l i n v o l v i n g o n l y the t r a n s l a t i o n a l m o t i o n , see e q u a t i o n ( 3 . 1 0 9 ) , w h i l e I', i s the a s s o c i a t e d sum o v e r the i n t e r n a l s t a t e s o f the c o l l i s i o n p a r t n e r , namely ( 6 • 2 2 ) 1 ; = T< — 1 3 2 „ , „ , , t / 2 , Q ( 2 / +1) F o r the s p e c i a l c a s e o f p o p u l a t i o n s , (6.21) r e d u c e s to (6.23) £ ( [ j > < j ] ( 0 ) | [ j - X j ' ] ^V/Tp15 (2L+D~1/2 1 ^ I< A l ) II ^ . I 2 11^' 6... | e x p [ . - ( £ r e j , ) ] ^ . , + 1 ) ( 2 ^ + 1 ) - e x p [ - - ( £ . - e . , ) ] 2 -i n ' 1 / 2 1 / 2 J J ( 2 j + l ) " / ( 2 j ' + l ) J - / ' " ( 2 / . 1 + l ) w h i c h shows t h a t o n l y i n e l a s t i c e f f e c t s c o n t r i b u t e . In the above e x p r e s s i o n s , the form 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 f o r DDLP, namely, , ' (6.24) < j ' \\JL' }|| j"> = i D ' + ^ '+J" [ ( 2 j ' + l ) (2/.'+l) ( 2 j " + l ) ] 1 / 2 j * C j " ( ) 0 0 0 i s to be e m p l o y e d . A c o m b i n a t i o n o f t h e s e e q u a t i o n s w i t h the e q u a t i o n s (5.28) y i e l d s ( 6 . 2 5 ) 1 .skT, 1/ 2,— (1) ( 2 / + 1) - 1 / 2 4 j 244 r j *1 j _ 1 2 {( 1 0 0 0 4 j 3-1 I, J. 2 ( ) 0 0 0 ^ ^ ^ - f V 1 2TJT + 3 / x 3-1 _ 2 ( ) 2 j * j , j - l 0 0 0 expr- I<e_-e._ 1)] ( ^ S4±i>• + . * (' A ' ) 2 j ^ j , j - l 0 0 0 1 /2 [ ( 2 j + A ) ( 2J + / . + 1 ) ( 2 j - / ) ( 2 J - A - 1 ) ] j / 1 j { ± L _ 1 ( ) ^3 0 0 0 e x p l - ^ i e . - e ^ ) ) + e x p [ - | < e . . ^ - e ] + E e x P [ - | ( e . - e j _ 1 ) ] 3 -^i 3 " 1 o _ 1 3 ll 3 2 ( A )^ ( 2 j + l ) + E e x p [ - r ( E , - e . ) l ( ) 0 0 0 j ^ 3 3 0 .0 0 < •„ * ( 2 j + l ) } • .. ' ... • -245 (6. 27, : i - " ( a f . 1 ' 2 / ; ' 1 ' ^ ! , - 1 ' 2 x; ( t U l V p H p *2 0 0 0 (4 j + 3) 2j ( j + (j + 2 ) - 3 [ 2 j ( j + l ) - / 1 ( / 1 + l ) J [(2 j - 1 ) ( j + 1 ) ( Z +1) ] 2j (j + 1) (2j + 3) ' 1 - j 1 2 + £ exp[-|{e,-e )] (2j+l) ( A ) } 5Vj 3 d 0 0 0 The e x p l i c i t fo rmulas f o r the 6- j symbols i n e q u a t i o n ( 6 . 2 1 ) , t (2j+^ 1 ) (2j + / 1 + l ) ( 2 j - / 1 ) (2j-/.1-l))1/2 I , j j L { 1 >= ( - D 1 2 j j 2{3[2j ( j + l ) - / 1 U 1 + l ) ] (2 j ( j + 1 ) - ^ (/.1 + l ) - l ] - 4 j 2 ( j + 1 ) 2 } ( 2 j - 1 ) 2 j ( 2 j + l ) ( 2 j + 2 ) ( 2 j + 3 ) have been used i n a r r i v i n g a t the above f o r m u l a e . For the j= l case o f i n t e r e s t f o r t h i s c h a p t e r , the forms f o r the r e l a x a t i o n r a t e s \ ~ t \~ a n < * h~ s i m p l i f y to 1 2 p X 2 f 1 X X 0 2 1 4 ° ^1 1 2 0 0 0 Z J . o 0 0 0 1 j A 0 2 1 e x p [ - i ( e -e.-) + £ { ( ) e x p [ - i ( e _ - E )] 2 0 1 j ^ 0 , l 0 0 0 ^ j ° (2j + l ) + ( j ^ 1 1 ) 2 e x p t - i - C e . - e , ) ] - 2 - ± t i } } 0 0 0 2 j 1 , e . 3 o , I ^ B ( ^ £ m ( 2 / + 1 , - i / 2 l . 2 ^ 2 1 j A ° 2 1 {£ e X p [ - i ( e -e )] ( ) (2j+l) + Z e x P t - r ( E _ - e )] j 2 j ° 0 0 0 j ^ J X 1 2 - , ( 1 (2j + l )> 0 0 0 a n d #6 3 1 ) i_ - " ^Js i , 1 / 2 r ( 1 ) ( 2 M ) " 1 / 2 i ' {-— c1 ^ V (6.31) - ( ^) S p <2^> 2 0 ( 0 Q 0> j JL. 1 2 ( ^ ? + 2 / ? - 5 ^ 2 - 6 / . - 2 0 ) + Z e x P [ - i ( e _ - e )] (2j + l ) ( ) } 1 1 1 1 j £ L 2 j 1 0 0 0 E q u a t i o n s ( 6 . 2 9 ) , (6.30) and (6.31) a r e t h e DWBA e x p r e s s i o n s f o r the r e l a x a t i o n r a t e s o f i n t e r e s t i n t h i s c h a p t e r . The 3 - j symbols d e t e r m i n e w h i c h r o t a t i o n a l l e v e l s a r e c o l l i s i o n -a l l y c o u p l e d . As t h e s e e q u a t i o n s i n d i c a t e , i f o n l y one a n i s o t r o p i c p o t e n t i a l > d o m i n a t e s a l l t h r e e r a t e s a r e p r o p o r t i o n a l to the same t r a n s l a t i o n a l and p e r t u r b e r m o l e c u l e f a c t o r s , namely VZ} X ^ ( 2 / + 1) X / ^ 2 and I ' , r e s p e c t i v e l y . P l2 The t h r e e r a t e s d i f f e r o n l y b e c a u s e o f the e x p l i c i t e x p r e s s -i o n s f o r the i n t e r n a l s t a t e f a c t o r s o f the " f i r s t " m o l e c u l e . The above t h r e e e q u a t i o n s a r e nov; e v a l u a t e d f o r f o u r d i f f e r e n t m u l t i p l e p o t e n t i a l s - the d i p o l e - d i p o l e (/iL ~ JL 2 ~ 1 ' ^ = t h e d i P o l e ~ < l u a d r u P o l e = 1 t <t 2 = 2, L - 3 ) ; the q u a d r u p o l e - d i p o l e (/ = 2 , ^ 2 = 1 , / = 3 ) ; and t h e quadrupo l e - q u a d r u p o l e (/.^  = 2, j^^ - 2, £ - 4 ) . The summations and i n t e g r a t i o n s i n v o l v e d a r e p e r f o r m e d e x p l i c i t l y w i t h the a i d o f one a d d i t i o n a l a p p r o x i m a t i o n - the h i g h t e m p e r a t u r e a p p r o x i m a t i o n . F o r t h e " f i r s t m o l e c u l e " p a r t s o f e q u a t i o n s ( 6 . 2 9 ) , (6.30) and ( 6 . 3 1 ) , t h i s amounts to n e g l e c t i n g the d i f f e r e n c e s i n B o l t z m a n n w e i g h t s . F o r t h e " p e r t u r b e r " m o l e c u l e e x p r e s s i o n , I* , the h i g h t e m p e r a t u r e a p p r o x i m a t i o n a g a i n a l l o w s the d i f f e r e n c e s i n B o l t z m a n n f a c t o r s to be n e g l e c t e d . I n d e e d , f r o m e q u a t i o n s (6.22) and ( 6 . 2 4 ) , the h i g h t e m p e r a t u r e a p p r o x i m a t i o n f o r any m u l t i p l e l 2 g i v e s 248 (6.32) I ' = I (2 j +1) (2 + — ~ ± - i k2 j j ' y  J2 J2 2^ ^2 2^ '2 0 0 0 B e x p [ - — • j (j +1) j / j ' = Z (2j +1) i Z (2j'+l) (• ;^ j 2 -J 2» 2 o o o B e x p [ - — j ( j +1) ] = I' (2J.+1) k T ^ '2 Q J 2 = 1 where B i s the r o t a t i o n a l c o n s t a n t . F i n a l l y , a s d i s c u s s e d i n c h a p t e r IV, t h e n e g l e c t o f e n e r g y i n e l a s t i c i t y (as s u g g e s t e d by a h i g h t e m p e r a t u r e a p p r o x i m a t i o n ) c o u p l e d w i t h the a s s u m p t i o n t h a t t h e o n l y v e c t o r i a l d e p e n d e n c e o f the t r a n s i -t i o n o p e r a t o r m a t r i x e l e m e n t i s a s s o c i a t e d w i t h the d i r e c t i o n o f t h e momentum t r a n s f e r v e c t o r , y i e l d s a m o d i f i e d B o r n a p p r o x i m a t i o n to the t r a n s l a t i o n a l c o l l i s i o n i n t e g r a l , e q u a t i o n (3.109). F o r a m u l t i p o l e form o f i n t e r a c t i o n p o t e n t i a l , the c o e f f i c i e n t s h. a r e g i v e n a s * 1* 2 k My M / r , _ , .,^2 (2/ + 1) 1 .1/2 1 4 2 ( 6 - 3 3 ) [ ( 2 , 1 + 1 ) , ( 2 4 2 + 1 ) ^ ( 2 / + 1 ) l / 2 249 the where My and M, are the m u l t i p o l e moments and y i s *1 ^2 reduced mass. A combination of (3.109) and (6.33) g i v e s the m o d i f i e d Born approximation f o r the t r a n s l a t i o n a l i n t e g r a l s (6 . 3 4 ) C ( 1 , ( 0 0 2 1 l | 0 0 2 1 l | 0 ) = 5 1 / 2 ^ p ° h^kT 3d r j ( 1 ) ( 0 0 3 1 2 | 0 0 3 1 2 | 0 ) - 7 1 / 2 * I ! E ^ p ° h kT 6d r—(1) I I 1/2 2TT3y Q 2D 2 £ ^ V ' ( 0 0 3 2l!0032l|0) = 7 ' -x—-P ° li kT 6d ^ ( 1 ) ( 0 0 4 2 2 | 0 0 4 2 2 | 0 ) n = 9 1 / 2 ^ I S ^ ! P ° h 2kT 3 0d 6 where D i s the d i p o l e moment and Q i s the quadrupole moment. S u b s t i t u t i o n of the high temperature approximations (6.32) and (6.34) i n t o equations (6.29) through (6.31) g i v e s estimates of the three r e l a x a t i o n r a t e s f o r the f i r s t few m u l t i p o l e i n t e r a c t i o n s . Indeed, i n terms of the convenient q u a n t i t y 3 ,r . , r » A - 8kT 1/2 2TT y (6.35) A = n l-zrr-) (-3 ) V V h kT the r e l a x a t i o n r a t e : 2 2 lT 'DD 3 2 2 a 2 2 j = A_ D D T DD 3 ^2 p d i a s s u m i n g the d i p o l e - d i p o l e p o t e n t i a l d o m i n a t e s , w h i l e (6.37) (—) - - —f-1 d 2 2 T2 °e ^ d 4 2 2 = A D Q T DO 34 P d when the d i p o l e - q u a d r u p o l e i n t e r a c t i o n i s assumed. S i m i l a r l y , a q u a d r u p o l e - d i p o l e i n t e r a c t i o n l e a d s to r e l a x a t i o n r a t e s 2 2 1 x _ 3A Q D ( 6 ' 3 8 ) ^ Q D 20 d4 2 2 1 . _ A Q D ( T 2 } Q D 6 d 4 251 and a q u a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n y i e l d ; ( 6.39) <!-) - H A s f a i T.'QQ 100 ,6 1 * • d 1 = 7A Q 2 0 2 (T 2'QQ 30 d 6 2 2 1_ = 7A Q O ( T QQ 30 ^6 p d f o r the t h r e e r e l a x a t i o n r a t e s . The above c a l c u l a t i o n s d e m o n s t r a t e two p o i n t s . F i r s t , the g e n e r a l DWBA r e s u l t s d e r i v e d i n c h a p t e r IV a r e s p e c i a l i z e d t o o b t a i n e x p r e s s i o n s f o r the d e s i r e d i n t e r n a l s t a t e r e l a x a -t i o n r a t e s by m a n i p u l a t i n g o n l y the i n t e r n a l s t a t e f a c t o r s . Any o f the r e l a x a t i o n r a t e s i n t r o d u c e d i n c h a p t e r V c a n a l s o ba s t u d i e d i n t h i s manner. S e c o n d l y , i t i s shown t h a t f o r the j=0 t o j = l c a s e a t l e a s t , the t h r e e r e l a x a t i o n r a t e s a r e o f T 1 1 1 2 s i m i l a r s i z e s , 'V 'v ^ — . I n p a r t i c u l a r , t h e r a t i o = 1.5 1 2 p 1 has been e s t a b l i s h e d i n t h e f i r s t two e q u a t i o n s , (6.36) and ( 6 . 3 7 ) . H e r e , /i = 1 and the 3 - j symbol d e s c r i b i n g the e l a s t i c 1 i 1 L l l> c o n t r i b u t i o n s t o ^ — , ^ Q 0 , v a n i s h e s i d e n t i c a l l y . The 1 1 1 o r d e r i n g — — >_ — — has a l s o been f o u n d by L i u and M a r c u s i n 1 2 t h o s e c a s e s when e l a s t i c (and r e o r i e n t a t i o n ) c o n t r i b u t i o n s T 1 2 to — - a r e s m a l l . C o n v e r s e l y , the r a t i o — — = 0.9 has been 2 T l fo u n d f o r the r e m a i n i n g two c a s e s [see e q u a t i o n s (6.30) and (6.39)] where j£ = 2. I n t h i s i n s t a n c e , t h e e l a s t i c 252 c o n t r i b u t i o n s ^ ^) a r e e f f e c t i v e i n i n c r e a s i n g the ^7— 1 1 2 r e l a x a t i o n r a t e . T h i s l a t t e r o r d e r i n g , — — , i s r e m i n i s c e n t 1 2 o f the one u s u a l l y assumed i n q u a l i t a t i v e d i s c u s s i o n s , see e q u a t i o n ( 2 . 3 0 ) . I n d e e d , the q u a l i t a t i v e argument i s a l s o b a s e d on the i m p o r t a n c e o f e l a s t i c c o l l i s i o n s i n d e t e r m i n i n g the s i z e o f ^ — . 2 A " g e n e r a l i z e d s t r o n g c o l l i s i o n m o d e l " , i n w h i c h (6.40) p r o v i d e s a u s e f u l " i d e a l i z e d " s i t u a t i o n i n w h i c h the t r a n s i e n t r e s p o n s e o f a two l e v e l s y s t e m c a n be p r e s e n t e d i n c l o s e d form - see s e c t i o n ( e ) . The c a l c u l a t i o n s o f the p r e s e n t s e c t i o n i n d i c a t e t h a t s u c h a model i s r e a s o n a b l e . (d) s t e a d y S t a t e A b s o r p t i o n and t h e G e n e r a l T r a n s i e n t E x p e r i m e n t The e q u a t i o n s o f m o t i o n (6.19) d e s c r i b e the e v o l u t i o n o f a two l e v e l s y s t e m i n the r o t a t i n g f r a m e . In t h i s s e c t i o n some comments on the s o l u t i o n s to t h e s e e q u a t i o n s a r e g i v e n w h i c h r e q u i r e no a s s u m p t i o n s on the r e l a t i v e s i z e s o f T^, T 2 and T . T h i s s e c t i o n g e n e r a l i z e s the d i s c u s s i o n s g i v e n i n s e c t i o n (e) o f c h a p t e r I I t o t h e two l e v e l c a s e . As d i s c u s s e d i n c h a p t e r I I , t h e t r a d i t i o n a l s t e a d y s t a t e a b s o r p t i o n e x p e r i m e n t c o r r e s p o n d s to s e t t i n g a l l time d e r i v a -t i v e s e q u a l t o z e r o i n the e q u a t i o n s o f m o t i o n when w r i t t e n i n the r o t a t i n g f r a m e . Thus e q u a t i o n s (6.19) become (6,41) -E P. + < ^ > - i - < ^ > z x • 1^ 4 T^ 4 eq z i T 2 p v/ z AWP. + i — P = 0 x T 2 r > + E < ^ > + E r z „j.2 4 z „ + 2 u • ( 2 ) (J) 2 {i> p T h i s s e t o f f o u r s i m u l t a n e o u s e q u a t i o n s i n f o u r unknowns i s e a s i l y s o l v e d to g i v e 2 54 (6.42) ay p - E AOJ<-—— > 2 z 4 eq (Aw)2 + ( I - ) 2 + 2 T + — T ay j i 4 .2 z T £} 2 P . = 8_y_ i_<MH.> 9 i 2 Z T2 4 e * (ACO)2 + ( i - , + H - E 2 { T2 9*2 2 T + — T 1 4 Hi E 2 !B <Mli> g^2 z T 2 4 eq 2 T + — T (AOJ) 2 + ( i - ) 2 + ^ E 2 { 1 „ 4  T 2 9 ^ 2 T2 (Aw) + ( T X 2 -) + 2 T 2y 2 D 9* 2 2 T z ( A w ) " + (• - ) 2 + sy' K2 T + —- T E 2 { 1 4 P ) z T_ 4 eq E q u a t i o n s (2.33) a r e o b t a i n e d from e q u a t i o n s (6.42) i n t h e l i m i t >> —— , ^ — ( i . e . when t h e r e l a x a t i o n t o e q u a l p 1 2 p o p u l a t i o n s i n t h e t h r e e m a g n e t i c s t a t e s o f t h e j = l l e v e l i s v e r y much f a s t e r t h a n the r e l a x a t i o n o f t h e p o p u l a t i o n d i f f e r e n c e between t h e j=0 a n d , j = l l e v e l s and t h e r e l a x a t i o n 255 of the coherences between the j=0 and j=l l e v e l s ) . A l s o , 2l.i 2 2 i n the low i n p u t power case where 1 E <<1, equations 9h Z (G.42) show that there i s no s a t u r a t i o n , <~hAlL> - <ilAiI> 4 4 eq and there i s no p o l a r i z a t i o n o f the j = l l e v e l , -> 0. F i n a l l y , the o b s e r v e d a b s o r p t i o n , w h i c h i s p r o p o r t i o n a l t o P^ ( s e e e q u a t i o n ( 2 . 4 9 ) ) , " l o o k s " t h e same as the two s t a t e d e s c r i p t i o n f o r any i n p u t power. T h i s i s e f f ] see n from e q u a t i o n (6.42) by d e f i n i n g T^ = T^ + •- T^ i n the e x p r e s s i o n f o r P^. A g e n e r a l d i s c u s s i o n o f t h e t r a n s i e n t s o l u t i o n s a d m i t t e d by e q u a t i o n s (6.19) i s now g i v e n . S e c t i o n (e) p r e s e n t s more d e t a i l e d r e m a r k s on t r a n s i e n t a b s o r p t i o n and t r a n s i e n t e m i s s i o n e x p e r i m e n t s . F o r ea s e o f m a n i p u l a t i o n , t h e r e s u l t s a r e d e s c r i b e d i n terms o f t h e f o l l o w i n g q u a n t i t i e s , z 1 T = kE t $ 1 Z kE T z 2 M = <AN> Y 1 4 1 kE T z p M - ^ - < A N > 6 - A w o 4 eq kE z The r e s u l t i n g e q u a t i o n s a r e now o f the form / (6.44) + 3 P + r 6P . = 0 1 256 ci 1 - r — P . + 3 P . - 6 P + M + — Y = 0 d T i i r 4 dM — + OH - P . = CtM d T i o dY - r — + yY - P . = 0 d T ' l W r i t t e n i n t h i s manner, e q u a t i o n s (6.44) a r e s e e n to be a 2 g e n e r a l i z a t i o n o f the form t r e a t e d by T o r r e y and by McGurk 3 e t a l . i n t h e i r a p p e n d i x A l . The L a p l a c e t r a n s f o r m o f a f u n c t i o n f ( T ) i s d e f i n e d as (6.45) f ( u ) / f ( T ) e dT o and when a p p l i e d to e q u a t i o n s (6.44) y i e l d s the m a t r i x e q u a t i o n (6.46) / u + 3 6 -6 u + 3 0 -1 -1 0 \ u+a o u+y / ± - P M r o — P. M i o IT* O \ M ra u+a o Here r , i , m , and y a r e t h e i n i t i a l v a l u e s o f P /M , o o o o r o P./M , M/M , and Y/M , r e s p e c t i v e l y . The d e t e r m i n a n t o f 1 0 0 o 1 1 c o e f f i c i e n t s A(u) i n e q u a t i o n (6.46) i s e v a l u a t e d as (6.47) A(u) = (u + 3 ) 2(u+o) (u+y) + (u + 3) (u+y) +^(u + B)(u+a) 2 + 6 (u+y) (u+a) . From C r a m e r ' s r u l e , the s o l u t i o n ' s to e q u a t i o n s (6.46) c a n t h e n be w r i t t e n i n the f o r m _ M g (u) ( 6 ' 4 8 ) f ( u ) = "uATuT where f (u) i s any one o f P^, P.^ , M and Y and g(u) i s a q u a r t i c f o r m i n u w h i c h i s d i f f e r e n t f o r e a c h o f t h e f o u r q u a n t i t i e s . I f , a s p r o v e n below, i t i s f o u n d t h a t t h e e q u a t i o n (6.49) A(u) = 0 has a t l e a s t two r e a l n e g a t i v e r o o t s -a and -b, t h e n t h e q u a r t i c f o r m e q u a t i o n (6.47) c a n be f a c t o r e d as (6.50) A(u) = (u+a) (u + b) [ (u+c) 2 + J i 2 ] and e q u a t i o n (6.49) c a n be expanded i n p a r t i a l f r a c t i o n s to g i v e H e r e , t h e r e i s a d i f f e r e n t s e t o f c o e f f i c i e n t s A,B,C,D and E f o r e a c h o f the f o u r q u a n t i t i e s P., P., M and Y. The 1 1 i n v e r s e L a p l a c e t r a n s f o r m o f e q u a t i o n (6.51) i s (6.52) f ( T ) = A e " a T + B e ~ b T + C e " ° T COSQT + ^ e " c T s i n J ] T . T h i s r e p r e s e n t s the g e n e r a l form o f t h e t i m e d e p e n d e n t s o l u t i o n s t o t h e two l e v e l p r o b l e m . E q u a t i o n (6.52) i s the c o u n t e r p a r t o f e q u a t i o n ( 2 . 3 4 ) , w h i c h d e s c r i b e s the two s t a t e c a s e . When T r e l a x a t i o n i s v e r y f a s t , t h e P f a c t o r (u+y) d o m i n a t e s e q u a t i o n (6.47) and t h e a p p r o x i m a t e r e s u l t (6.53) A ( u ) = (U+Y) [ (u + 3) 2 ( u + a ) + (u + (3) + 6 2 ( u + a ) ] = ( u + Y ) [ e q u a t i o n ( 2 . 3 5 ) ] i s o b t a i n e d . E q u a t i o n (6.53) t h u s i l l u s t r a t e s how t h e two s t a t e c a s e c a n be r e c o v e r e d from the two l e v e l d e s c r i p t i o n . B e f o r e c l o s i n g t h i s s e c t i o n , i t must be e s t a b l i s h e d t h a t the q u a r t i c e q u a t i o n (6.49) a l w a y s p o s s e s s e s two r e a l n e g a t i v e r o o t s . T h i s i s n e c e s s a r y i n o r d e r t o have d e c a y to an e q u i l i b r i u m s t a t e . ( F o r t h e two s t a t e s y s t e m o f c h a p t e r I I , t h e c u b i c e q u a t i o n (2.35) must be shown to have one n e g a t i v e r e a l r o o t i n o r d e r t o d e s c r i b e d e c a y t o 259 e q u i l i b r i u m and t h i s i s t r i v i a l l y p r o v e n . ) F o r the q u a r t i c form, i t i s n o t so t r i v i a l t o show t h a t A ( u ) < 0 f o r some v a l u e s o f u<0, and must be a c c o m p l i s h e d i n s e v e r a l s t e p s . F i r s t , b e c a u s e a l l o f the c o e f f i c i e n t s i n e q u a t i o n (6.47) a r e p o s i t i v e , t h i s r e p r e s e n t s a q u a r t i c form w h i c h opens upward. S e c o n d l y s i n c e d ^ i s a c u b i c f o r m , a l s o w i t h a l l p o s i t i v e c o e f f i c i e n t s , t h e n t h e c u b i c e q u a t i o n ( 6 . 5 4 ) ^ 1 0 du has no p o s i t i v e r e a l r o o t s and t h u s a l l maxima and minima o f A(u) must o c c u r f o r u<0. I n o t h e r words, A ( u ) i s m o n o t o n i c a l -l y i n c r e a s i n g f o r u>0. To p r o c e e d f u r t h e r , a l l p o s s i b l e r e l a t i o n s h i p s between t h e r e l a x a t i o n r a t e s a , 3 , and y must be c o n s i d e r e d . F o r the c a s e o f a l l r e l a x a t i o n t i m e s e q u a l a = 3 = Y = s r (6.55) A ( u ) = ( u + s ) 2 [ ( u + s ) 2 + 6 2 + s/4] and e q u a t i o n (6.49) has two e q u a l n e g a t i v e r e a l r o o t s . F o r the c a s e a=3= s^y, e q u a t i o n (6.47) t a k e s the form 2 1 2 (6.56) A ( u ) = ( u + s ) [ ( u + s ) (u+f) + (u+y) + j ( u + s ) + 6 (u+y)] The r e m a i n i n g c u b i c p o r t i o n o f (6.56) must have a t l e a s t one n e g a t i v e r e a l r o o t , so t h a t a g a i n e q u a t i o n (6.49) has two n e g a t i v e r e a l ( b u t u n e q u a l ) r o o t s . F o r the c a s e s 260 a=Y = s/B and 3=Y=s/a, s i m i l a r r e a s o n i n g a g a i n shows t h a t (6.49) has two u n e q u a l n e g a t i v e r e a l r o o t s . F o r the r e m a i n i n g s i t u a t i o n s where the r e l a x a t i o n r a t e s a r e u n e q u a l , a d i f f e r e n t a p p r o a c h i s t a k e n . C o n s i d e r the s i t u a t i o n f o r w h i c h a<3<Y or y<B<a. Then, the e v a l u a t i o n (6.57) A ( u ) | u = _ 3 = 6 2 (Y-3) (a-B) <0 shows t h a t A(u) i s n e g a t i v e and hence t h e q u a r t i c f o r m must have c r o s s e d t h e A(u)=0 a x i s . T h i s i m p l i e s a t l e a s t two r e a l n e g a t i v e r o o t s o f e q u a t i o n ( 6 . 4 9 ) . I f the r e l a t i o n -s h i p s B<ct<Y or Y<c t<3 o c c u r , t h e n A(u) e v a l u a t e d a t u= - a , i s a g a i n n e g a t i v e . F i n a l l y , t h e e v a l u a t i o n o f A(u) a t u=-y e s t a b l i s h e s t h a t A(u) i s n e g a t i v e a t t h i s p o i n t i f a<Y<3 o r B<Y<A« Thus, f o r a l l p o s s i b l e r e l a t i o n s h i p s between the r e l a x a t i o n r a t e s , e q u a t i o n (6.49) p o s s e s s e s a t l e a s t two n e g a t i v e r e a l r o o t s . A v a r i e t y o f s h a p e s f o r the A(u) v e r s u s u p l o t s a r e s t i l l p o s s i b l e , a s e x e m p l i f i e d by f i g u r e 8. The comments p r e s e n t e d i n t h i s s e c t i o n a r e n o t a c t u a l l y c o n f i n e d t o the j=0 to j = l t r a n s i t i o n i n DDLP m o l e c u l e s b u t a r e e q u a l l y v a l i d whenever the tv/o l e v e l s y s t e m c a n be a p p r o x i m a t e l y d e s c r i b e d by f o u r c o u p l e d s c a l a r e q u a t i o n s -see f o r example, e q u a t i o n s ( 5 . 3 4 ) . I n su c h s i t u a t i o n s , the r e l a t i v e s i z e o f the c o u p l i n g o f the o r i e n t a t i o n a 1 moment may be e x p e c t e d to d i f f e r n u m e r i c a l l y from the v a l u e 1/4 use d i n t h i s c h a p t e r , b u t t h i s has no e f f e c t on the g e n e r a l f o r m o f the s o l u t i o n s . 261 A(u) A(u) / \ u \y u A(u) 1 / A(u) u u F i g u r e Some p o s s i b l e s h a p e s f o r A(u) v e r s u s u c u r v e s . 262 (e) T r a n s i e n t A b s o r p t i o n and T r a n s i e n t E m i s s i o n f o r a Two L e v e l System T r a n s i e n t a b s o r p t i o n and t r a n s i e n t e m i s s i o n e x p e r i -ments, as d i s c u s s e d i n s e c t i o n s (g) and (h) o f c h a p t e r I I , a r e now r e e x a m i n e d w i t h i n the c o n t e x t o f the two l e v e l s y s t e m . The t r e a t m e n t g i v e n h e r e i s more c o n c i s e t h a n t h a t o f c h a p t e r I I , and e m p h a s i z e s the m o d i f i c a t i o n s i n the s y s t e m m o t i o n due t o the p r e s e n c e o f a f o u r t h s c a l a r moment. As i n c h a p t e r I I , some a d d i t i o n a l a s s u m p t i o n s on the' r e l a t i v e s i z e s o f the p a r a m e t e r s i n v o l v e d i s r e q u i r e d i n o r d e r to p r e s e n t the t r a n s i e n t a b s o r p t i o n e x p e r i m e n t i n c l o s e d f o r m . C o n t r a r y to the s i t u a t i o n d i s c u s s e d i n c h a p t e r I I , however, an " o n - r e s o n a n c e " a s s u m p t i o n ( i . e . A«= 0) p r o d u c e s no u s e f u l s i m p l i f i c a t i o n * The o t h e r c a s e t r e a t e d i n s e c t i o n (g) o f c h a p t e r I I doe s have a w o r t h w i l e g e n e r a l i z a t i o n - the g e n e r a l i z e d s t r o n g c o l l i s i o n m o d e l, e q u a t i o n (6.40) - w h i c h a l l o w s the two l e v e l t r a n s i e n t a b s o r p t i o n e x p e r i m e n t to be s o l v e d i n c l o s e d f o r m . T h i s h y p o t h e s i s o f a g e n e r a l i z e d s t r o n g c o l l i s i o n model has been made a t l e a s t p l a u s i b l e by the c a l c u l a t i o n s p r e s e n t e d i n s e c t i o n (c) o f t h i s c h a p t e r . E m p l o y i n g e q u a t i o n (6.40) and a s s u m i n g t h a t the s y s t e m i s i n i t i a l l y a t e q u i l i b r i u m , e q u a t i o n s (6.46) a r e r e w r i t t e n a s 26 3 (6.50) O — 1 where s = (kE T,) . The d e t e r m i n e n t (6.47) i n t h i s n o t a t i o n o 1 t a k e s the f a c t o r i z e d f o r m (6.59) A ( u ) = (u+s) 2 [ (u+s) 2 + 6 2 + 5/4] As m e n t i o n e d i n s e c t i o n ( e ) , t h e s o l u t i o n s to e q u a t i o n (6.58) have t h e g e n e r a l form (6.48) and, i n p a r t i c u l a r , P^ i s e v a l u a t e d as (6.60) ;.' = - M ° 1 U ( ( u + s ) 2 + 6 2 + 5/4] C(u + s) + D E [ (u+s) 2 + 6 2 + 5/4] U where t h e c o e f f i c i e n t s C,D and E i n t h e p a r t i a l f r a c t i o n e x p a n s i o n s a t i s f y the s y s t e m o f t h r e e e q u a t i o n s (6.61) 0 = C + E -M = sC + D + 2sE o 2 2 -M = E ( 6 + 5/4 + S ) o The s o l u t i o n s o f e q u a t i o n s (6.61) a r e + M s (6.62) C = -E = ° s 2 + 6 2 + 5/4 -M ( 6 2 + 5 / 4 ) D = ° 2 2 s + 6 + 5/4 and t h e i n v e r s e L a p l a c e t r a n s f o r m o f e q u a t i o n (6.60) c a n t h e n be p e r f o r m e d to y i e l d — M S (6.63) P.(T) = ° [1 - e " S T c o s t ( 6 2 + 5 / 4 ) 1 / 2 T ] ) 1 s +6 +5/4 e - S ( 6 2 + 5 / 4 ) 1 / 2 M Q _ s i n [ (6 +5/4) T] s + 6 + 5/4 E q u a t i o n (6.63) i s a p a r t i c u l a r example o f the g e n e r a l tirae d e p e n d e n t s o l u t i o n (6.52) and the r e m a i n i n g moments P ( T ) , M ( T ) , and Y(T) c a n be e v a l u a t e d i n a s i m i l a r manner. F o r example M 2 6 5 (6.64) Y(T) = — - { e " S c o s [ (6 +5/4) '-T] - 1 6 +5/4+S - S T + s i n [ ( 6 2 + 5/4) 1 / 2 T ] (6 + 5/4) d e s c r i b e s the t r a n s i e n t b e h a v i o u r o f the o r i e n t a t i o n a 1 p o l a r i z a t i o n d u r i n g a b s o r p t i o n . U s i n g the t r a n s f o r m a t i o n s ( 6 . 4 3 ) , e q u a t i o n s (6.63) and (6.64) a r e r e w r i t t e n i n the o r i g i n a l n o t a t i o n as _ k 2 E < M N » c n r, / 4 - \ z 4 eq 1 1 - t / T n . (6.65) P ± ( t ) = - [- - — e c o s f i t —- + 9, • • • - T 2 + fie"t/T s i n f l t ] tfJViv K 2(E )2<4^ > 2 1 n 2 + U 1 - t / T s i n f i t , + T e - I T " 3 where (6.66) Q 2 = Aw 2 + 5 / 4 k 2 ( E ) 2 z T h e r e a r e two p o i n t s to n o t i c e a b o u t e q u a t i o n s ( 6 . 6 5 ) . F i r s t , t h e t r a n s i e n t s o l u t i o n s d e c a y to the p r e v i o u s l y e s t a b l i s h e d s t e a d y s t a t e v a l u e s ( s e e e q u a t i o n s ( 6 . 4 2 ) ) a s t-* 0 0. S e c o n d l y , the o b s e r v e d t r a n s i e n t a b s o r p t i o n , a s d e t e c t e d v i a e q u a t i o n ( 2 . 4 9 ) , i s o f the same form as t h a t g i v e n i n the tv/o s t a t e c a s e , see e q u a t i o n (2.54) . O n l y the p r e c i s e d e f i n i t i o n o f the f r e q u e n c y o f o s c i l l a t i o n 9, as g i v e n by e q u a t i o n (6.66) has been changed b e c a u s e o f p r e s e n c e o f the f o u r t h moment. The t r a n s i e n t e m i s s i o n e x p e r i m e n t r e q u i r e s no a d d i t i o n a l a p p r o x i m a t i o n , s u c h as the g e n e r a l i z e d s t r o n g c o l l i s i o n m o d e l , to a l l o w the s o l u t i o n to be p r e s e n t e d i n c l o s e d f o r m . I f the e x c i t i n g r a d i a t i o n i s t u r n e d o f f a t time t=0, t h e n from e q u a t i o n s ( 6 . 1 9 ) , the b e h a v i o u r o f the sy s t e m f o r t i m e s t>0 i s g o v e r n e d by the s e t o f e q u a t i o n s ( 6. .6 / ) 5 t < V + - % q } " ° d t ° 2 T <J2 P — p + Atop. + — P = 0 d t r i T 2 r P. - Awp + — P. = 0 d t i r T 2 i A s s u m i n g an a p p r o p r i a t e s e t o f i n i t i a l c o n d i t i o n s < A N > ( 0 ) , ( j )>< 0) , P r ( 0 ) , and P ^ ( 0 ) , the s o l u t i o n s to e q u a t i o n s (6.67) a r e i m m e d i a t e l y e s t a b l i s h e d as 267 (6.68) P ( t ) e P . ( t) = e~ ' 2 {P . (O) cosAoJt + P (O) sinAwt} <Au> ( t ) = <AN> + (<AN> (o) - £ A N > ) e - t / T i eq eq (J)> ( t) = < / 2 > ( J ) > ( 0 ) e t/Tp 2 z The s t e a d y s t a t e v a l u e s , e q u a t i o n s ( 6 . 4 2 ) , r e p r e s e n t one s u c h s e t o f a p p r o p r i a t e i n i t i a l c o n d i t i o n s . However, i n d e p e n d e n t o f t h e c h o i c e o f i n i t i a l c o n d i t i o n s , e q u a t i o n s the t r a n s i e n t e m i s s i o n p r o c e s s . T h i s c o n c l u d e s the d i s c u s s i o n o f t h e t r a n s i e n t b e h a v i o u r o f a two l e v e l s y s t e m . A l t h o u g h a d d i t i o n a l t r a n s i e n t e x p e r i -ments c a n be p e r f o r m e d ( s e e c h a p t e r I I ) , t h e examples p r e -s e n t e d i n t h i s s e c t i o n a r e s u f f i c i e n t t o i l l u s t r a t e t h e manner i n w h i c h a two l e v e l d e s c r i p t i o n o f the phenomena c a n b e o b t a i n e d . (6.68) d e m o n s t r a t e t h a t <AN> and d e c o u p l e d u r i n g 260 ( f ) Summary o f the Two L e v e l P o i n t o f View The two l e v e l p o i n t o f v i e w i s now summarized by e m p h a s i z i n g the p h y s i c a l i n t e r p r e t a t i o n o f the phenomena w h i c h the p r e s e n t t r e a t m e n t a l l o w s . Here, the " e x a c t " two l e v e l s y s t e m ( j = 0 to j - - l ) t r a n s i t i o n i n DDLP) i s c o n s i d e r e d e x p l i c i t l y , a l t h o u g h s i m i l a r a r g u m e n t s c a n be a p p l i e d t o the g e n e r a l two l e v e l s y s t e m s d i s c u s s e d i n c h a p t e r V. C o n s i d e r the j = l , m=0 s t a t e o f the upper l e v e l . T h i s i s the o n l y one o f the t h r e e m a g n e t i c s t a t e s t h a t i s a f f e c t e d by the l i n e a r l y p o l a r i z e d r a d i a t i o n . I n t h e a b s e n c e o f c o l l i s i o n s , t h e n , a d e s c r i p t i o n o f the u p p e r l e v e l i n terms o f the b a s i s o p e r a tors|lm><lm* | would seem the most a p p r o p r i a t e b e c a u s e o n l y one o f t h e s e o p e r a t o r s i s a f f e c t e d by the r a d i a t i o n . Thus the two l e v e l p r o b l e m i s r e d u c e d to t h e two s t a t e p r o b l e m - b u t o n l y i n t h e a b s e n c e o f c o l l i s i o n s . C o l l i s i o n a l c o u p l i n g between t h e d e g e n e r a t e m a g n e t i c s t a t e s r e q u i r e s t h a t the s t a t e s j - 1 , m=+l a l s o be c o n s i d e r e d . The p r e c i s e manner i n w h i c h t h e s e m a g n e t i c s t a t e s e n t e r the p r o b l e m f o l l o w s from the assumed r o t a t i o n a l i n v a r i a n c e o f the c o l l i s i o n s u p e r o p e r a t o r . I n d e e d , the b a s i s o p e r a t o r s |lm><lm'| must be decomposed i n t o r o t a t i o n -a l l y i r r e d u c i b l e p a r t s , y i e l d i n g an a l t e r n a t e d e s c r i p t i o n o f the j = l l e v e l . I n p a r t i c u l a r , (6.69) 1 0 X 1 0 269 w h i c h i s a s p e c i a l c a s e o f the i n v e r s e t o e q u a t i o n ( 3 . 2 4 ) . E q u a t i o n (6.69) s h o w s ' t h a t when r a d i a t i o n i n t e r a c t s w i t h the j = l , m = 0 s t a t e , two e f f e c t s a r e p r o d u c e d . F i r s t , the o v e r a l l p o p u l a t i o n o f the j = l l e v e l i s a l t e r e d . S e c o n d , the p o p u l a t i o n o f the m = 0 s t a t e r e l a t i v e t o the m = +_l s t a t e s i s s e l e c t i v e l y c h a n g e d , p r o d u c i n g a p o l a r i z a t i o n o f the j = l l e v e l , d e s c r i b e d by'^^^ . These two e f f e c t s have i n d e p e n d e n t c o l l i s i o n a l r e l a x a t i o n r a t e s . ( 2 ) In o r d e r to r a t i o n a l i z e why i t i s a *J^J (J) p o l a r i z a -t i o n t h a t must be formed i n s t e a d o f a /L^^ ( J) o r some o t h e r p o l a r i z a t i o n , a p i c t o r i a l a r g ument c a n be e s t a b l i s h e d as an a l t e r n a t i v e to the more m a t h e m a t i c a l d e r i v a t i o n s p r e s e n t e d e a r l i e r . T hese p h y s i c a l p i c t u r e s have t h e f u r t h e r a d v a n t a g e o f d e m o n s t r a t i n g the i n t e r r e l a t i o n s h i p between the s c a l a r and v e c t o r i a l ( o r t e n s o r i a l ) d e s c r i p t i o n s , a s w e l l a s e m p h a s i z i n g the d i f f e r e n c e s between the m i c r o s c o p i c and m a c r o s c o p i c m o t i o n s o f the s y s t e m . As i l l u s t r a t e d i n f i g u r e 9, the a p p l i c a t i o n o f l i n e a r l y p o l a r i z e d m i crowave r a d i a t i o n , w i t h i t s p o l a r i z a t i o n v e c t o r o s c i l l a t i n g i n t h e z d i r e c t i o n , p r o d u c e s a macro-•; scopes, o s c i l l a t i o n o f t h e g a s e o u s s y s t e m , a l s o i n the z d>js?c; n\ '"'' .'I n terms o f t h e i n d i v i d u a l m o l e c u l e s , however, . t h i s ' v f e c r o s ' c o p i c o s c i l l a t i o n i s e q u i v a l e n t t o r o t a t i o n s ' ' - - f a r o u n d ^ n * 1 t r a r y a x i s i n the x-y p l a n e , w i t h e q u a l • . . - , •-number s:-j£-f ro'61ecules r o t a t i n g i n the p o s i t i v e and n e g a t i v e •. ' dir.ee %\\<>i!g&.-. .Indeed, the r o t a t i o n a l m o t i o n i s n e c e s s a r y n o '^'?5duce• t h e m a c r o s c o p i c o s c i l l a t i o n i n a c o l l e c t i o n o f 27 0 R o t a t i o n a l m o t i o n s p r o d u c e d by l i n e a r l y p o l a r i z e d r a d i a t i o n . r i g i d r o t o r s because the p o s i t i v e and negative ends of the molecular d i p o l e don't tunnel through each other, i n g e n e r a l . Since there are no components of t h i s macroscopica l y o s c i l l a t i n g d i p o l e i n the x "or y d i r e c t i o n s , i t i s r e q u i r e d t h a t equal numbers of molecules must be r o t a t i n g i n the p o s i t i v e and negative sense.. The e f f e c t s of these motions on the o r i e n t a t i o n of J can be e a s i l y understood. Remembering that j i s perpen-d i c u l a r to \i f o r each i n d i v i d u a l molecule, the induced r o t a t i o n a l motion t h e r e f o r e produces an alignment of J v e c t o r s i n the x-y plane, a t the expense of J v e c t o r s i n the z d i r e c t i o n . T h i s alignment of i n d i v i d u a l J v e c t o r s i s expressed m a c r o s c o p i c a l l y as one component of the p o l a r i z a t i o n tensor 2j ^ 2 ^ (J ) . F i n a l l y , granted the e x i s t e n c e of t h i s o r i e n t a t i p n a l p o l a r i z a t i o n whenever a s u f f i c i e n t l y i n t e n s e microwave source i s a p p l i e d , what i s the best way to d e t e c t i t s presence? C o n s i d e r a t i o n s i n t h i s chapter of both steady s t a t e and t r a n s i e n t s o l u t i o n s i n d i c a t e t h a t the e f f e c t of the f o u r t h moment on P . ( t) - and hence, on the observed 1 a b s o r p t i o n - i s not d r a m a t i c . Indeed, i n each i n s t a n c e , the. g e n e r a l shajie of the s o l u t i o n remains u n a l t e r e d and • only the exact magnitude of the parameters i s changed . We f e e l t h a t a more d i r ec t measurement of the o r i e n t a t i o n a 1 e f f e c t i s p r e f e r r e d i n order to unambiguously e s t a b l i s h i t s e x i s t e n c e , e x p e r i m e n t a l l y . To t h i s end,, a new e x p e r i -ment i s suggested which i s a combination' of i n t e n s e microwave (cw) i r r a d i a t i o n and o p t i c a l , d e p o l a r i z e d R a y l e i g h (DPR) l i g h t s c a t t e r i n g . The i n t e n s e m i c r o w a v e r a d i a t i o n p o l a r i z e s the medium w h i l e t h e e x i s t e n c e o f a DPR s p e c t r u m i n d i c a t e s t h e p r e s e n c e o f a "^Y (J) p o l a r i z a -4 t i o n . H e r e , t h e l i n e w i d t h o f the DPR s p e c t r u m i s p r o p o r t i o n a l to the r e l a x a t i o n r a t e ^ — . E x p e r i m e n t a l l y , the P s e t - u p s h o u l d be somewhat a n a l o g o u s to t h a t used f o r m i c r o 5 w a v e - o p t i c a l d o u b l e r e s o n a n c e measurements - see a l s o f i g u r e 10. 273 DPR detector — F i g u r e 10: A c o m b i n a t i o n m i c r o w a v e a b s o r p t i o n -l i g h t s c a t t e r i n g e x p e r i m e n t . o p t i c a l 274 CHAPTER V I I V e l o c i t y E f f e c t s i n P r e s s u r e B r o a d e n i n g " I t ' s too l a t e t o c o r r e c t i t , " s a i d the Red Queen: "when you've once s a i d a t h i n g , t h a t f i x e s i t , and you must take t h e c o n s e q u e n c e s . " (a) I n t r o d u c t i o n T h i s f i n a l c h a p t e r d i s c u s s e s the i n f l u e n c e o f m o l e c u l a r v e l o c i t i e s on p r e s s u r e b r o a d e n i n g . I n d e e d , c h a p t e r V I I c a n be c o n s i d e r e d as a g e n e r a l i z a t i o n o f the tv/o s t a t e model o f c h a p t e r I I to i n c l u d e t r a n s l a t i o n a l m o t i o n ( i n terms o f W) i n a manner a n a l o g o u s to the way i n w h i c h the two s t a t e model was g e n e r a l i z e d i n c h a p t e r s IV, V and VI to a c c o u n t fox-r o t a t i o n a l m o t i o n ( i n terms o f J ) . The r e s u l t i n g model o f the m o l e c u l e a l l o w s f o r a c o m p l e t e d e s c r i p t i o n o f the m o l e c u l a r m o t i o n s - o s c i l l a t i o n (u,y), r o t a t i o n ( J) , and t r a n s l a t i o n (W) . F u r t h e r , t h i s c h a p t e r j u s t i f i e s the n e g l e c t o f t r a n s l a t i o n a l e f f e c t s i n e a r l i e r c h a p t e r s , by i n d i c a t i n g u n der wha t c o n d i t i o n s t h i s i s r e a s o n a b l e . R e l a t i o n s h i p s between S e n f t l e b e n - B e e n a k k e r e f f e c t s and t h o s e o f p r e s s u r e b r o a d e n i n g a r e m o s t r e a d i l y e s t a b l i s h e d w i t h i n the c o n t e x t o f the p r e s e n t c h a p t e r , s i n c e i n e a c h c a s e , bo t h i n t e r n a l s t a t e and t r a n s l a t i o n a l m o t i o n s a r e o f c o n c e r n . I n p a r t i c u -l a r , the p r e s e n t t r e a t m e n t o f c o l l i s i o n s i s v e r y s i m i l a r to t h a t i n c h a p t e r I I I . F i n a l l y , as e s t a b l i s h e d by the p r e s e n t p o i n t o f v i e w , c e r t a i n a r e a s where a d d i t i o n a l work i s r e q u i r e d a r e i n d i c a t e d . 27G The e f f e c t o f a d i s t r i b u t i o n o f m o l e c u l a r v e l o c i t i e s on the shape o f a s p e c t r a l l i n e was m e n t i o n e d i n c h a p t e r I . In p a r t i c u l a r , i t was p o i n t e d o u t t h a t i f the D o p p l e r s h i f t o f the n a t u r a l o s c i l l a t i o n f r e q u e n c y o f the m o l e c u l e s was i n c l u d e d , t h e n i n the a b s e n c e o f c o l l i s i o n s , the r e s u l t i n g l i n e shape e x p r e s s i o n i s G a u s s i a n w i t h a c h a r a c t e r i s t i c wid th Aw . o F u r t h e r , i f c o l l i s i o n s a r e assumed to a f f e c t o n l y the i n t e r n a l s t a t e s o f the m o l e c u l e s , then a V o i g t p r o f i l e i s o b t a i n e d - a c o n v o l u t i o n o f the G a u s s i a n w i t h a L o r e n t z l i n e s h a p e f o r the p u r e i n t e r n a l s t a t e e f f e c t s . To t h i s p o i n t t h e n , t h i s t h e s i s has n o t y e t d i s c u s s e d the p o s s i b i l i t y o f " v e l o c i t y c h a n g i n g " c o l l i s i o n s and t h e i r e f f e c t on l i n e s h a p e s . D i c k e 1 was the f i r s t to d e s c r i b e the e f f e c t s o f " v e l o c i t y c h a n g i n g " c o l l i s i o n s on the p u r e G a u s s i a n p r o f i l e and he f o u n d t h a t the l i n e s h a p e n a r r o w s - t h i s i s known as " D i c k e n a r r o w i n g " . The e f f e c t s o f bo th " v e l o c i t y c h a n g i n g " and " i n t e r n a l s t a t e c h a n g i n g " c o l l i s i o n s has been c o n s i d e r e d 2 by G a l e n t r y and r e s u l t e d i n a more i n v o l v e d p r e s s u r e d e -p e n d e n c e o f the l i n e s h a p e , a l t h o u g h the s o l u t i o n i s s t i l l r e p r e s e n t e d as a c o n v o l u t i o n o f t r a n s l a t i o n a l and i n t e r n a l 3 s t a t e f a c t o r s . N e l k i n and G h atak u s e d a method i n v o l v i n g a k i n e t i c e q u a t i o n to o b t a i n D i c k e ' s r e s u l t s w h i l e , i n an 4 e x c e l l e n t p a p e r by R a u t i a n and Sobelman, a k i n e t i c e q u a -t i o n f o r m u l a t i o n o f G a l a n t r y ' s "combined p r o b l e m " was f i r m l y e s t a b l i s h e d . I n a d d i t i o n , by p o i n t i n g o u t t h a t a s i n g l e c o l l i s i o n e v e n t c a n l e a d to b o t h v e l o c i t y c h a n g e s and i n t e r n a l s t a t e c h a n g e s , they showed t h a t a g e n e r a l l i n e s h a p e i s no t e x p r e s s i b l e as a c o n v o l u t i o n o f t r a n s l a -t i o n a l and i n t e r n a l s t a t e f a c t o r s . S i m i l a r c o n c l u s i o n s were o b t a i n e d i n d e p e n d e n t l y by G e r s t e n and F o l e y . ^ The p r e c e e d i n g t r e a t m e n t a r e a l l o f a c l a s s i c a l and p h e n o m e n o l o g i c a l n a t u r e , e s p e c i a l l y i n t h e i r h a n d l i n g o f the i n t e r n a l s t a t e p a r t s . A d e s c r i p t i o n o f t h e s e (no-s a t u r a t i o n ) l i n e s h a p e s , o b t a i n e d from a more f u n d a m e n t a l s t a r t i n g p o i n t , i s p r e s e n t e d i n a s e r i e s o f p a p e r s by a g r o u p o f r e s e a r c h e r s f r o m J I L A . I n d e e d , b e g i n n i n g w i t h a c o m p l e t e l y quantum m e c h a n i c a l t r e a t m e n t o f b o t h r a d i a t i o n and m a t t e r , S m i t h . e t a l o b t a i n e d a quantum m e c h a n i c a l k i n e t i c e q u a t i o n c a p a b l e o f d e s c r i b i n g the combined e f f e c t s o f D o p p l e r and p r e s s u r e b r o a d e n i n g . They the n s i m p l i f i e d 7 t h i s r e s u l t i n a s e c o n d p a p e r by e m p l o y i n g a c l a s s i c a l p a t h a p p r o x i m a t i o n f o r the t r a n s l a t i o n a l m o t i o n s . In so d o i n g , the f o r m a l e x p r e s s i o n s a r e p r e s e n t e d i n s u c h a way t h a t the p h y s i c a l p r o c e s s e s w h i c h i n f l u e n c e D o p p l e r and p r e s s u r e b r o a d e n i n g a r e s e p a r a t e d ( b u t n o t u n c o r r e l a ted) . The p o s s i b l e c o r r e l a t i o n e f f e c t s a r e e x p l a i n e d i n a s e p a r a t e Q p a p e r , u s i n g a "one i n t e r a c t i n g l e v e l " a p p r o x i m a t i o n w h i c h i s a p p r o p r i a t e to a d i s c u s s i o n o f o p t i c a l t r a n s i t i o n s . I n f a c t , b e c a u s e t h e s e a u t h o r s a r e c o m p l e t e l y c o n c e r n e d w i t h the e x p l a n a t i o n o f o p t i c a l l i n e s h a p e s , some d i s c r e t i o n s h o u l d be employed i n d i r e c t l y a p p l y i n g t h e i r work to the c o r r e s p o n d i n g m i c r o w a v e c a s e . 9 Berman has i n d e p e n d e n t l y d e r i v e d a v e r y s i m i l a r quantum m e c h a n i c a l k i n e t i c e q u a t i o n . In a d d i t i o n to the a p p l i c a t i o n o f t h i s e q u a t i o n to the s t u d y o f s p e c t r a l 1 i n e s h a p e s , 1'° ' 1 1 t h i s a u t h o r (and c o w o r k e r s ) have ernisloyed i t i n the e x p l a n a t i o n o f v e l o c i t y m o d i f i c a t i o n s to c o h e r e n c e 12 13 t r a n s i e n t e f f e c t s . ' A g a i n , however, the work o f Berman i s d i r e c t e d t o w a r d s o p t i c a l , t r a n s i t i o n s . 14 15 However, i t i s the work o f Hess ' t h a t m o s t c l o s e l y r e s e m b l e s the method o f a p p r o a c h e m p h a s i z e d t h r o u g h -o u t t h i s t h e s i s . A p a p e r d e s c r i b i n g v e l o c i t y c o r r e c t i o n s 14 to DPR s p e c t r a was l a t e r e x t e n d e d , i n a c o m p r e h e n s i v e p a p e r , 1 " * t o more g e n e r a l s p e c t r a l l i n e s h a p e s . In t h i s l a t t e r work, the c o m p l e t e t r a n s i t i o n (as the gas d e n s i t y i n c r e a s e s ) f r o m the D o p p l e r b r o a d e n e d r e g i o n to the p r e s s u r e b r o a d e n e d r e g i o n i s e s t a b l i s h e d . T h i s a l s o a l l o w e d c o n n e c t i o n s to be made between the p r e v i o u s l y m e n t i o n e d work on l i n e -s h a p e s and the k i n e t i c t h e o r y methods w h i c h a r e m o s t a p p r o -p r i a t e to the s t u d y o f m i c r o w a v e phenomena. In the f o l l o w i n g s e c t i o n s , a k i n e t i c t h e o r y a p p r o a c h to v e l o c i t y e f f e c t s i n m i c r o w a v e p r e s s u r e b r o a d e n i n g and c o h e r e n c e t r a n s i e n t s i s p r e s e n t e d w i t h i n the g e n e r a l f r a m e -work e s t a b l i s h e d i n t h i s t h e s i s . T h i s i n c l u d e s some c o n -s i d e r a t i o n s on the e v a l u a t i o n o f the a p p r o p r i a t e r e l a x a t i o n e x p r e s s i o n s . S p e c i f i c o b s e r v a t i o n s r e l a t i n g t h i s work to t h a t o f e a r l i e r w o r k e r s a r e g i v e n where a p p r o p r i a t e . (b) A K i n e t i c E q u a t i o n w h i c h i n c l u d e s V e l o c i t y E f f e c t s T h i s s h o r t s e c t i o n p r e s e n t s a k i n e t i c e q u a t i o n a p p r o -p r i a t e to the d i s c u s s i o n o f the e f f e c t s o f v e l o c i t y on s p e c t r o s c o p i c phenomena. I t s d e r i v a t i o n p o i n t s o u t the a s s u m p t i o n s and a p p r o x i m a t i o n s i n h e r e n t i n the p r e s e n t t r e a t m e n t , and a l l o w s a c o m p a r i s o n w i t h the k i n e t i c e q u a t i o n s (3.15) and (4.9) u s e d i n the d e s c r i p t i o n s o f S e n f t l e b e n -B e e n a k k e r e f f e c t s and p u r e l y i n t e r n a l s t a t e s p e c t r o s c o p i c e f f e c t s , r e s p e c t i v e l y . C o n n e c t i o n s w i t h the k i n e t i c e q u a t i o n s c i t e d i n the p r e v i o u s s e c t i o n c a n a l s o be more r e a d i l y e s t a b l i s h e d . As i n c h a p t e r s I I I and IV, the s t a r t i n g p o i n t i s the 16 g e n e r a l i z e d B o l t z m a n n e q u a t i o n o f S n i d e r and S a n c t u a r y , e q u a t i o n (3.1) - see a l s o a p p e n d i x A. F o r c o n v e n i e n c e , i t i s r e w r i t t e n h e r e as (7,1) ; i l X f ^ = / t r P + 0 +/. XP + t r ^ P P ^ where ( 7 * 2 ) ^ - t r P = to9 = ^'p3; L ,P = [-2[l'E c o s (w t - k - r ) , p] , 1 ~ ~o — — and t h e c o l l i s i o n s u p e r o p e r a t o r i s d e f i n e d by e q u a t i o n (A. 1 8 ) . A r e p r e s e n t a t i o n o f e q u a t i o n (7.1) i n terms o f the Wigner 280 d i s t r i b u t i o n i s a g a i n p r e f e r r e d , i n o r d e r to make the h a n d l i n g o f the t r a n s l a t i o n a l d e g r e e s o f f r e e d o m more m e a n i n g f u l . I n d e e d , e m p l o y i n g the e q u i v a l e n t forms o f the Wigner d i s t r i b u t i o n f u n c t i o n (7.3) f-(rgt) = / e 1 3 " ~ / t l < P + | 3 I P I E ~ l 2 > d 3 = ^ / e - ^ - g / ^ r + l n l p l r - ^ d q ; h e q u a t i o n s (7.2) i n t h i s new r e p r e s e n t a t i o n a r e g i v e n by (7.4) ±j / e ^ ' ^ ^ + i q l ^ p l g - l ^ d q = - i J . ^ f f (rpt) h ~ and i j / e l q * r / <p+|qUop|p-|q>dq = / Q f ( r p t ) h (7 . 5 ) ~ / e " 1 ! ! * ^ <r+^|/( 1p |r~^>d n= [ - 2 U - E q c o s ( w t - k - r ) , ± - / e - i D - f i ^ ^ l p l r - ^ J c o s ^  d n h +. [-2U-E s i n ( o ) t - k . r ) , ~ fe~^ < r + § | p | r -§> s i n ^ dr,] ~ ~ *" ~ h ~ A 1 201 F i n a l l y , the t r a n s f o r m e d c o l l i s i o n o p e r a t o r has the same form as e q u a t i o n ( 3 . 4 ) , e x c e p t i n t h i s i n s t a n c e , the W i g ner d i s t r i b u t i o n f u n c t i o n s on w h i c h i t a c t s a r e no l o n g e r r e s t r i c t e d to b e i n g d i a g o n a l i n i n t e r n a l s t a t e e n e r g i e s . The t r a n s f o r m a t i o n s r e p r e s e n t e d by e q u a t i o n s ( 7 . 4 ) , (7.5) and (3.4) a r e e x a c t - no i n f o r m a t i o n a b o u t the quantum n a t u r e o f the t r a n s l a t i o n a l d e g r e e s o f f r e e d o m has been l o s t . I t i s s u f f i c i e n t f o r the p r e s e n t p u r p o s e , however, to c o n s i d e r t h a t the d i s t r i b u t i o n f u n c t i o n s a r e o n l y s l o w l y v a r y i n g f u n c t i o n s o f p o s i t i o n . I n t h i s i n s t a n c e , e q u a t i o n (7.5) c a n be a p p r o x i m a t e l y e v a l u a t e d as ( 7. 6 ) 2_ /e-iD-E^<r+5|/1p|r-^>drj' [-2y*E o co s (0) t-k «r ) , f ( r p t ) ] i * - 3 ^ i a -, . „ + I " l T 7 ' a i £ ( E 2 t ) 3 + : J where [A,B] + i s the a n t i - c o m m u t a t o r o f A and B, and the i d e n t i f i c a t i o n s (7.7) c o s ^ 2 . ^ 1 s i n .^2- 'v^ 2- a - - -~~ t-2u«E co s (co t-k • r ) ] 3r dr Z ~o - — = k [ - 2 y * E s i n ( t o t - k - r ) ] have a l s o been e m p l o y e d . F u r t h e r , as d i s c u s s e d i n c h a p t e r I I I , t h i s a s s u m p t i o n o f s l o w l y - v a r y i n g p o s i t i o n d e p e n -d e n c e a l l o w s the c o l l i s i o n term to be s i m p l i f i e d a s w e l l . I n d e e d , the l o c a l i z e d c o l l i s i o n o o e r a t o r J ( r p t ) i s the - o ~ — r e s u l t , where J ( r p t ) has the same form as e q u a t i o n ( 3 . 8 ) , a s s u m i n g t h a t i n t e r n a l s t a t e f r e q u e n c i e s a r c c o n s e r v e d . T hus, w i t h the above t r e a t m e n t o f the t r a n s l a t i o n a l d e g r e e s o f f r e e d o m , the k i n e t i c e q u a t i o n (7.1) i s r e w r i t t e n a s (7.8) TTT + * -K— f - r k • Tr— J : U d t m d r 2 3 r 9 P + s = -L/^JC f - i/fc[-2u«E c o s ( u t - k «r) , f ] + J ( r g t ) F i n a l l y , t he l i n e a r i z e d v e r s i o n o f e q u a t i o n (7.8) i s o b t a i n e d , u s i n g the s p e c i f i c l i n e a r i z a t i o n , e q u a t i o n (4.6) . In p a r t i c u l a r , the a n t i c o m m u t a t o r terms i n e q u a t i o n (7.8) i s s m a l l e r t h a n the o t h e r o s c i l l a t o r y term by a f a c t o r o f < 7 ' 9 > 5 ' m^F " < l | ^ - 5 ) < < 1 and so i s n o t c o n s i d e r e d f u r t h e r . ( S m i t h e t a l . 6 have employed s i m i l a r r e a s o n i n g to a p p r o x i m a t e t h e f l o w term i n t h e i r k i n e t i c e q u a t i o n . ) The l i n e a r i z e d l o c a l i z e d c o l -l i s i o n s u p e r o p e r a t o r $ , d e f i n e d by e q u a t i o n (4 .17) , i s a g a i n o f t h e Waldmann-Snider f o r m . The l i n e a r i z e d v e r s i o n o f e q u a t i o n (7.8) i s then 283 (7.10) | 4 + £ ' f = ' i 4 ' C f - i / * [ - 2 U - E c o s ( o j t - k . r ) , f ] d t m or ^ o ~ ~o — — - f ^ u ^ - V f - f f 0 * ) ] t r v \ t r t r As w i t h a l l e q u a t i o n s u s e d to d e s c r i b e the i n t e r a c t i o n o f an o s c i l l a t i n g f i e l d w i t h a s y s t e m o f m o l e c u l e s , e q u a t i o n (7.10) c o n t a i n s a time d e p e n d e n t H a m i l t o n i a n . I t i s d e s i r a b l e t o a p p r o x i m a t e l y e l i m i n a t e t h i s time d e p e n d e n c e by a t r a n s f o r m a t i o n to a r o t a t i n g frame w i t h a s u b s e q u e n t n e g l e c t o f t h e h i g h l y o s c i l l a t o r y terms ( t h e r o t a t i n g wave a p p r o x i m a t i o n ) . I n t h e p r e s e n t c a s e , t h e t r a n s f o r m a t i o n used i n c h a p t e r IV i n terms o f the i n t e r n a l s t a t e o p e r a t o r S c a n a g a i n be e m p l o y e d . T h i s o p e r a t o r has no e f f e c t on the c o l l i s i o n term a s l o n g as (P\_ i s l o c a l and p r e s e r v e s f r e q u e n c y . The r o t a t i n g wave a p p r o x i m a t i o n to e q u a t i o n (7.10) i s I n e q u a t i o n ( 7 . 1 1 ) , i t i s u s u a l t o n e g l e c t the e f f e c t o f 3 the g-^ - term i n t h e r o t a t i n g frame - t h i s i s c o n s i s t e n t w i t h the i d e a o f a s l o w l y v a r y i n g p o s i t i o n d e p e n d e n c e . E q u a t i o n (7.11) i s the a p p r o p r i a t e k i n e t i c e q u a t i o n to be u s e d f o r a d e s c r i p t i o n o f the m o d i f i c a t i o n s o f 284 microwave e x p e r i m e n t s due to v e l o c i t y e f f e c t s , b o t h o f the s t e a d y s t a t e and t r a n s i e n t v a r i e t y . S i m i l a r v e r s i o n s o f t h i s e q u a t i o n have been o b t a i n e d i n d i f f e r e n t c o n t e x t s , as d i s c u s s e d i n the p r e v i o u s s e c t i o n . Many o f t h e s e a u t h o r s p r e f e r to s p l i t the c o l l i s i o n o p e r a t o r i n t o " v e l o c i t y c h a n g i n g " and "phase c h a n g i n g p a r t s - namely, (7.12) (R [ $ ] = $ V C [ ( J ) ] + $ p c [ * l where - w 2 (7.13) # [ « ] E - ( 2 T T ) V [ G 3 / 2 3 " 1 " t r 2 / d £ 2 ( 2 l T m k T ) / d p ' d p 2 6 ( p + P 2 - p ' - P 2 ) < y g | t | y g ' > ' ( o ) ' t r f < o ) ' .«>>• f < ° ) , t r { - — [4>(W) - <Mw>] + — — n n ~ n n [<J>2(w2) - <J> 2 (w 2 )]<<yg'|6(E-E') t + | y g > and 2 285 -W (7.14) $ [(M = -(2TT) 4- L 1 2 [ — ^ — • ] n t r /dp p C ( 2 T r m k T ) J / ^ [ / d p ' d p 2 6 ( p + P 2 - p ' - P 2 ) < P g | t | y g ' > f ( o ) t r - f < 0 ) ' f ( o ) . f 2 0 ) ' t r q / <MW) +1 ^ d> 2 (w 2 )} <Ug ' I 6 (E-E * ) t + I Mg> + ( 2 7 T i ) _ 1 [<]ig j fc | y g> » % • ^ ( o ) (o) t r ( o ) t r f ' (o) f { - — <J)(w) + - — <Mw„)> n n ~ n n ~ 2 ( o j t r f ( o ) f <j)(w) + - — 4> ( w „ ) J n n n n 2 ~2 < y g | t + | y g > ] ] . In p a r t i c u l a r , i t i s se e n t h a t t ^ i s the a p p r o p r i a t e pc c o l l i s i o n o p e r a t o r f o r p u r e i n t e r n a l s t a t e c o l l i s i o n e f f e c t s ( s e e c h a p t e r IV) s i n c e ^ c = 0 when (j) i s i n d e p e n -d e n t o f t h e v e l o c i t y . M o r e o v e r , b e c a u s e o f t h e o p t i c a l t h e o r e m , w v a n i s h e s when <$> i s i n d e p e n d e n t o f i n t e r n a l pc s t a t e s . T h i s d i v i s i o n o f d\ i s i n many ways s i m i l a r to 17 t h e d i v i s i o n employed by Chen e t a l . i n t h e i r e v a l u a t i o n o f c o l l i s i o n m a t r i x e l e m e n t s o b t a i n e d i n the s t u d y o f S e n f t l e b e n - B e e n a k k e r e f f e c t s . The s u c c e e d i n g s e c t i o n s e x p l o r e t h e i m p l i c a t i o n s o f equa t i o n ( 7 . 1 1 ) . (c) The Moment Method A p p l i e d to a ' D i s c u s s i o n of V e l o c i t y E f f e c t s The moment method :has been a p p l i e d i n chapter I I I to the d i s c u s s i o n of Senftleben-Beenakker e f f e c t s , and. i n chapte r s V and VI to the s o l u t i o n of the two l e v e l problem. In the pr e s e n t s e c t i o n , t h i s method i s a p p l i e d to equation (7.11) i n order to determine the e f f e c t s of the Doppler s h i f t on . s p e c t r o s c o p i c phenomena. For the purpose of i l l u s t r a t i o n , i t i s s u f f i c i e n t to c o n s i d e r as r e p r e s e n t a t i v e of the i n t e r n a l s t a t e n o t i o n s , the j=0 to j=l two l e v e l system f o r the DDLP case. T h i s system was c o n s i d e r e d i n chapter VI i n the absence of any v e l o c i t y e f f e c t s . A l t e r n a t e c h o i c e s of two l e v e l systems are p o s s i b l e , see chapter V, but the b a s i c d e s c r i p t i o n remains unchanged. The s i m p l e s t a n satz which al l o w s v e l o c i t y m o d i f i c a t i o n to be i n c l u d e d r e q u i r e s f to have the form 2 . • (7.15) f ( p t ) , €l ' { i < l > + M 1 M > + (2-nmkT) 3/ 2 ^ 2 )( J ) > n 2 ' a 3 10 10 3 10 l O , + L y : < L " y > + ~ L • p : < L g > } 2y. ~ ~ '" 2y " where (see a l s o equation (3.18)) 207 (7.16) L 1 0 ( W ) = 2 1 / 2W E q u a t i o n (7.15) c a n be compared d i r e c t l y w i t h e q u a t i o n (6.1) w h i c h r e p r e s e n t s the d e s c r i p t i o n o f the two l e v e l s y s t e m i n t h e a b s e n c e o f v e l o c i t y e f f e c t s . I n d e e d , e q u a t i o n ( 5 . 24) i s the r e l e v a n t e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n f o r b o t h s i t u a t i o n s . The a p p r o p r i a t e moment e q u a t i o n s a r e f o u n d by sub-s t i t u t i n g e q u a t i o n (7.15) i n t o the k i n e t i c e q u a t i o n ( 7 . 1 1 ) . (As m e n t i o n e d e a r l i e r , the — • term i n the r o t a t i n g frame has been d r o p p e d . ) The r e s u l t s a r e k T ( 7 . i 7 ) Jr<y> - Aco<vl> + { - 8 - ) 1 / 2 k - < L 1 0 u > d t ~ ~ m - ti 2 | " r - EO.V(III) -<ZJ1) (J)> + |-<y> = o + Au»<u> - ( ^ ) 1 / 2 k . < L 1 0 y > a t — m ~ ~ - ! % < ! > 1 / 2 5 o - v ( i i 2 , . < ^ w + i. <p> = o T 2 28 y l l A i i + 2 E . < y > + 1_ ( < A n>_ < A n> J = 0 d t t T ~ o ~ T eq < ^ 1 > ( J ) > " | E o - V ( l l l ) . < U > + -L-<^ ( 1 )(j)> d t d_ < / 7(2) d t (15) 1/2 Eo'V(121)- <y> + ~ < ^ 2 ) ' ( J ) > = 0 i - < L 1 0 y > _ A u ) < L 1 0 y > + '(_B-) 1/ 2k<y> + i - < L 1 0 M > = 0 d t ~ m ~ ~ T ^ ~ |-<L 1 0p> + A U < L 1 0 y > - ( 1^,1/2 <y> + :L_< L%> . 0 ~ ~ m » ~ T _ ~ d t In e q u a t i o n s (7.17), i t i s assumed t h a t <<AN|^|I>> = 0, w h i c h i m p l i e s t h a t <1> i s d e c o u p l e d f r o m the o t h e r moments. The r e l a x a t i o n t i m e s a r e d e f i n e d as (7.18) — 1 <<AN \(Jl | A N > > Q (J) T , P* i ) (J ) >> T «tj2) (j) \(R\2j2) (j ) >> i _ T. = 3 <<UJ^IE = 3 2y" 2y* i _ T « L 1 0 y | ^ 1 0 U » o « L 1 0 £ | ^ | L 1 0 a > = 3 2y' 2y' E q u a t i o n s (7.17) a r e t h e g e n e r a l i z a t i o n o f e q u a t i o n s (6.17) to i n c l u d e v e l o c i t y e f f e c t s . N o t i c e t h a t o s c i l l a -t i o n s 1J,M; r o t a t i o n s Z^' 1' ( J) , ^ ' 2 ' (J) , and t r a n s l a t i o n s L ° i J i L' L° )J a r e a l l c o n t a i n e d i n t h i s p i c t u r e . In p a r t i c u l a r the o r i g i n o f the v e l o c i t y e f f e c t s has been c l e a r l y i n d i -c a t e d - t h e y r e p r e s e n t a f l u x o f o x c i l l a t i n g d i p o l e s p , y i n the d i r e c t i o n o f p r o p o g a t i o n k p f the i n c i d e n t l i g h t beam More complex v e l o c i t y p o l a r i z a t i o n s c a n a l s o o c c u r . T h e s e i n c l u d e t h o s e moments t h a t have been n e g l e c t e d by 10, 01 01• the a n s a t z ( 7 . 1 5 ) ; f o r example <L A N > , <L y>, <L y>, 2 0 2 0 * <L y>, and <L y>. Th e s e h i g h e r o r d e r c o r r e c t i o n s to the v e l o c i t y d e s c r i p t i o n become s i g n i f i c a n t a t l o w e r d e n s i t i e s when t h e y a r e p r o d u c e d a t a r a t e no l o n g e r i n s i g n i f i c a n t compared t o t h e i r r e l a x a t i o n t i m e s . (The a p p r o p r i a t e o r d e r p a r a m e t e r i s the r a d i a t i o n w a v e l e n g t h d i v i d e d by the mean f r e e p a t h . ) A l s o , b e c a u s e i t i s the D o p p l e r s h i f t o f the o sc i l l a t i o n w h i c h b a s i c a l l y g i v e s r i s e t o t h e s e v e l o c i t y p o l a r i z a t i o n s , i t i s e x p e c t e d t h a t the p o l a r i z a -ps ps . t i o n s <L y>, <L y> w i l l o c c u r one o r d e r o f m a g n i t u d e P s s o o n e r t h a n the c o r r e s p o n d i n g p o l a r i z a t i o n <L*"AN>. F o r example, i n d e r i v i n g e q u a t i o n ( 7 . 1 7 ) , i t i s c o n s i s t e n t t o c o n s i d e r t h e p o l a r i z a t i o n s <Lx<"'y> and < L x 0 y > w h i l e i g n o r i n g the e f f e c t s o f <L 1 0AN>. W i t h the above d e s c r i p t i o n o f the v e l o c i t y e f f e c t s , the k i n e t i c t h e o r y t r e a t m e n t s o f S e n f t l e b e n - B e e n a k k e r e f f e c t s and the s p e c t r o s c o p i c e x p e r i m e n t s a r e seen t o have much i n common. In the f i r s t c a s e , v e l o c i t y p o l a r i z a t i o n s o f the gas sample a r e c r e a t e d by the p r e s e n c e o f v e l o c i t y o r t e m p e r a t u r e g r a d i e n t s . In the s e c o n d c a s e , i t i s the o s c i l l a t i n g microv/ave f i e l d w h i c h e s t a b l i s h e s t h e s e f l u x e s . In b o t h c a s e s , the d e s c r i p t i o n s o f the a c t u a l v e l o c i t y p o l a r i z a t i o n s p r e s e n t become more c o m p l i c a t e d as the number o f g a s e o u s m o l e c u l e s i n the c e l l i s d e c r e a s e d . The com-p a r i s o n s i n the i n t e r n a l s t a t e m o t i o n s p r o d u c e d by e a c h k i n d o f e x p e r i m e n t have been e m p h a s i z e d i n p r e v i o u s c h a p t e r s w h i l e t h e s i m i l a r i t i e s i n the t r e a t m e n t s o f c o l l i s i o n a l e x p r e s s i o n s i s e s t a b l i s h e d i n t h e n e x t s e c t i o n . V a r i o u s s o l u t i o n s to e q u a t i o n (7.17) c a n be e n v i s a g e d , c o r r e s p o n d i n g to d i f f e r e n t m i c r o w a v e e x p e r i m e n t s , w h i c h i l l u s t r a t e t h e e f f e c t s o f the v e l o c i t y p o l a r i z a t i o n . F o r example, a n o - s a t u r a t i o n , s t e a d y s t a t e e x p e r i m e n t l e a d s to a L o r e n t z i a n l i n e s h a p e w i t h a m o d i f i e d w i d t h 1 k B T 1 14 ;r— + ( ) T , , a s s u m i n g —->>Ato. (Hess has a l s o f o u n d t h i s T 2 m f T e f f e c t when c o n s i d e r i n g v e l o c i t y m o d i f i c a t i o n s to DPR l i n e s h a p e s by a moment method a p p r o a c h . ) However, s i n c e l i t t l e e x p e r i m e n t a l work has been done on v e l o c i t y m o d i f i c a -t i o n s to m i c r o w a v e e x p e r i m e n t s ( e s p e c i a l l y t r a n s i e n t e x p e r i m e n t s ) , e x p l i c i t s o l u t i o n s to e q u a t i o n (7.17) a r e n o t p u r s u e d h e r e . The l a c k o f e x p e r i m e n t a l o b s e r v a t i o n s p r o b a b l y stems from the f a c t t h a t u n d e r n o r m a l gas p r e s s u r e c o n d i t i o n s , v e l o c i t y e f f e c t s p l a y a m i n o r r o l e i n the m i c r o -wave r e g i o n o f the e l e c t r o m a g n e t i c s p e c t r u m . ( T h i s i s n o t the c a s e i n the o p t i c a l r e g i o n s . ) T h e s e f a c t s a l s o i m p l y , h o w e v e r , t h a t the d e s c r i p t i o n o f v e l o c i t y e f f e c t s i n the m i c r o w a v e r e g i o n s h o u l d be much s i m p l e r t h an i n the o p t i c a l r e g i o n s . I n the l a n g u a g e o f the p r e s e n t s e c t i o n , the m i c r o w a v e c a s e s h o u l d be d e s c r i b e d s u f f i c i e n t l y a c c u r a t e l y by the moments ^ X°u and L x ^ y r w h i l e t h e o p t i c a l r e g i o n s g e n e r a l l y r e q u i r e t h a t a d d i t i o n a l v e l o c i t y p o l a r i z a t i o n s s u c h as L^^AN and L 2 ^ y e t c . , be t a k e n i n t o a c c o u n t . Thus, the m i c r o w a v e r e g i o n has the a d v a n t a g e o f o f f e r i n g a d e f i n i t e v e l o c i t y p o l a r i z a t i o n t o be m e a s u r e d . (d) C o l l i s i o n a l Description of V e l o c i t y E f f e c t s The previous section has discussed an equivalence between transport properties and spectroscopic phenomena. In the present section, t h i s equivalence i s extended to the step by step treatment of the c o l l i s i o n a l aspects. Indeed, the c o l l i s i o n a l expressions obtained i n chapters III and IV are shown to be s p e c i a l cases of the general r e s u l t s presented here. Employing the natural basis set (7.19) Apqstj.v. | j f v f ] = L p S(W) [j . v. ><j f V f ] ( q ) and the inner product (4.12), matrix elements of are equivalently expressed as generalized c o l l i s i o n cross sections (7.20) £ ( p q s [ j . v . | j f v f ] | p ' q ' s » [ j ! v ! j j ^ ] ) E n~l(-^mr)h « A p q s [ 3 i v i | j £ v f ] | |Ap'q's'[J!v!|j^]» . Equations (7.19) and (7.20) are generalizations of equations (3.54) and (3.55) to include "off-diagonal i n j " s i t u a t i o n s . As discussed i n chapter I I I , the r o t a t i o n a l invariance property of allows the tensor cross s e c t i o n to be expressed i n terms o f a set I — )v of s c a l a r cross sections, according to (7.21) ECpqstJ.v.lj^lp'q's'fJivllj'v^]) 293 I C- l ) k + q+qWq') ( O ^ t^'Vpp')]]^' 3 k The i n t e g r a t i o n over the centre of mass momentum i s then formally performed (k) by the transformation I„ . , , , so that the scalar cross sections of ' InZ'n';psp's' equation (7.21) are expressible i n terms of r e l a t i v e v e l o c i t y cross sections and £3 k • E x p l i c i t l y , t h i s transformation to r e l a t i v e v e l o c i t y cross sections i s (7.22) & P q s [ J i v i | j f v f ] | p ' q » s ' [ J J v ; | j ^ ] ) k f2k+n T \ "Cku') 1 2 (k) AnA'n' fitkpp') 1 X,lnJ>n';psp's• { J S ^ q n ^ v J j £ v £ ] | r q'n^j ! v ^ + (-1)*' £ U q n U ^ J j ^ l r q ' n ' t J l v l l j ^ v ' D j J . Equations (7.21) and (7.22) are generalizations of equations (3.55) and (3.58) of chapter I I I . Here, the r e l a t i v e v e l o c i t y s c a l a r cross section j~^ k i s given e x p l i c i t l y as (7.23) ( ^ q n [ i i v i | j f v f ] | r q ' n ' [ j ! v ! | j ^ v ' ] ) ] = ( - D i ^ ' - ^ ' i ' ^ ^ 1 " ^ ( - D ^ c k q q . ) ^ a c m - ) - ' 5 294 -E -E. , , - i ' i f I I t - 1 1 " " t r "J2V2 1 f h 2 T /E cnl exp[ ^ j e x p f — £ - = • ] — — T / d E (Y)-/? , (Y 1) .^2 t r t r * L k l J 1 L kT J , n 2 t r ^ l n £ v y v i n' ; (kT) j 2 v 2 i • v " J 2 V 2 6(E. +E. +E -E., ,-E.. ,-E* ) J f v f J 2 v 2 t r J f v ' j 2 v 2 - t r J J [(24+1) (2£'+l)(2q+l)(2q'+l)]' S [(2X+1) (2X'+1) (2X'+1)(2X" l + l ) ] ' S X X' X'X'1 1 [(2j ! + l ) ( 2 j 2 + l ) (2X'+1) (2j f+l) (2j 2+l) (2X' "+!)] I [ ( 2 / +l)(2 i£+l)(2/ + 1 ) ( 2 ^ + 1)]' T C j J v J j ' v ' X ' ; / ^ ^ ^ v . j ^ X ) T ( j f v f J 2 v 2 X " « ; / - L 2 l ; j f v f J 2 v 2 X « ) (-1) 1 ^ C-i) (-D f1 J f f X « X ' * f X X" £ l0 0 0 J '•O 0 0 ,£+£' 1 r 1 *" 2, I. X' /_ X ) {.!., k 2) Ivkz L ^ 1 J X'" /L'X" •E. iq 1 q k t r , _, 3 2 V 2 , wtr -7; + C " i } 7 ^ ' E t r d E t r e x P t k F^-Pt ^ T 2 ^ M ^ M n r M (kT) H g ( 2 j 2 + l ) (2k+l)' 29! I i [ ( 2 ^ 1 ) ( 2 £ « + l ) ( 2 q + l ) ( 2 q ' + l ) ] i s (J J' J) (* J' [ (2X +1) ( 2A ' +1) ] k XX { [ 2 j ^ l ) ( 2 X ' + l ) ] 1 2 6 ( j f v f | j ' v « ) ^ (-l)"* 1 A T ( j ! v ! j 0 v . , X l ; k O k ; j . v . j 0 v . , X ) q , k } J x xJ2 2 ' ' J x xJ 2 2 J l i . •) ! n j r J l l f + [ 2 j ' + l ) ( 2 X + l ) ] J s 6 ( j . v . | j ! v ! ) 2 f t - l ) 3 f X T * ( j ^ k 2 v 2 A ; k 0 k ; j £ v f J 2 v 2 X ' ) | q ' q k } ( - l ) k + q + q ' } , J f J £ J i w h i l e i s a n a l o g o u s l y e x p r e s s e d as ( 7 . 2 4 ) £ ^ " ( J J - q n L j^l j f v f ] jil'q'n' [ j !v ! | j £ v £ ] ) - (-1) i q + q ' + £ + £ ' n f k q q ' ) - ' * HXklVyh - U r I f E* d E ! ( k T ) ^ J 2 v 2 t r L r * i V i ' - E e x P [ H ^ l e x p L - p p - 3 ~ / d E t r / { n £ M A n ' * ' ^ n — ( y g ' ) <5(E. +E. + E + - E . , , - E . , , - E ' ) j f v f j 2 v 2 t r j ' v - t r J J [(2A+lK2A ,+l)C2q+l)(2q'tl)]^ [ (2 j - +1) ( 2 j ' + l ] * ( 2 j +1) X X ' A M X " ' [(2X+1) (2\l+l)Yi (2X'+1) (2X'"+1) 2 96 I [ ( 2 / +1)(2/ +1)(2 / +1)(2X + 1 ) ( 2 / A+1)(2/C :+1)]' 4 1( 2 /. T ( J i v i J 1 v ! A ' ^ l / ( 2 / ; J i v i J 2 v 2 X ) i i i X1 X1" V X X" £ r^' ^  2 ^ 2i / q ^ 1 ^ l i 1^' ^ * U U U U U U J2 J f J i J.1 3 i 3 f X'" X"J / 2 q ' / 2 + ( - i ) ^ " 1 " ^ iq+q'+^+^' nCkqq')"'8 ncm1)"^ — ^ £ / E ' ds' (kTT j 2 v 2 t r t r i 'v1  j l 1 r ~ E t r " E J i v i A 2 - -Cyg') 2 I ( ( j ! J ' 3 C0 0 0 } (2k+iyH [(2A+l)(2A'+l)(2X +l)C2X'+l)] ! s [ * * 2 J i S*£ SJil£ V i " V 1 3 ^ ! * 1 " 2 1 ' * 1 ' ] ' 297 T(j £v fj!v!A';qq'k;j.v.j'v«A) ( - 1 ) ^ ^  * ( - l ) k + q + q ' + 6 i i ' f i 5 . [(2j .+1) (2j >1) (2A+l)]' i J 2 J i v 2 v | j ^ j f v^v. J i J f T ( j ^ j ^ A j q q ' k j j ^ j i v J A 1 ) (-1) ] . The derivations of these expressions are not given here but can be obtained by a straightforward generalizations of the methods used to obtain equations (3.65) and (3.66). (These methods are outlined i n chapter I I I , section (e), 18 and presented i n greater d e t a i l by Hunter and Snider. ) The S matrix 19 versions of (7.23) and (7.24) have been quoted previously and are obtainable from these equations by employing the d e f i n i t i o n (3.67). F i n a l l y , f o r the case of no v e l o c i t y p o l a r i z a t i o n s 1=1'=n=n'=k=0, equations (7.23) and (7.24) reduce to equations (4.35) and (4.36), r e s p e c t i v e l y . Equations (7.23) and (7.24) are exact and express r o t a t i o n a l l y i n v a r i a n t generalized cross sections i n terms of r o t a t i o n a l l y i n v a r i a n t T (or S) matrix elements i n the t r a n s l a t i o n a l - i n t e r n a l coupling scheme. This representation i s p a r t i c u l a r l y appropriate to s i t u a t i o n s i n which the anisotropic part of the intermolecular p o t e n t i a l i s assumed small. With t h i s assumption, the reduced t r a n s i t i o n operator T( ^i^2^ ^ l S c o n v e n i e n t l y expressed as equation (3.84), and a d i s t o r t e d wave Born approximation to the generalized cross s e c t i o n s J ^ " ^ and J~>^ can be established. For convenience of wr i t i n g , these are expressed as sums, , = 2-> +L> and = l\ according to (7.25) £ , + (£qn[j.v i|j fv f]|£'q'n'[j!vJ|j'v^) k 298 3 2V2 0 i 1 v1 ]2 2 i i <^  2V2 I ! c0 2 ^  2^ I 1 ^  2V2> i 2 C 2 X2 + 1 ) _ 1 e x Pt^/2] d T ( 1 ) - l T ( 1 ) ) ( ^ n ^ / T /oU'n'/'Z ' /JE. +E. -E., ,-E., ,), ° v OP **1 2' 1 2' 3fvf J2V2 j ^ v ^ J2V2 (7. 26) £ ' - Uqn [j | j f v f ] | £' q' n' [j ! v! | j £v • ]) R -E, = i q + q ' I e x p t - ^ f 2 ] I fi(kqq')"15 [(2q+D(2q' + l ) ] J s J2V2 Q hVl h*2 I 2 (2Z 2+D" i(-D •'l J i •*! J i Ji Jf < j i V i l Ic9 1 ^  i ' 'Dl ^  I I J!v!>exp[-A/2] 2 9 9 V ( 1 ) UnV I , Z oU ' n 1 / ' X ' / J E . - +ET - - E . - E . ) , ^ A i^.2> ^ ^ 1 A 2 ' j 1 v 1 j 2 v 2 j i v i J 2 v 2 ' k i ' J f .. r ^  1 ^ 1 k i rq' q k , f J f h J f J i n ( 1 ) (^n/ / , / J r n 1 / ^ I / 0 | E T - +ET - - E . - E . ) . ] p i * A : A 2 I M ^ 2 ' J 1 v 1 j 2 v 2 j £ v f J 2 v 2 ' k J C-i) 2iri q+q1 - E . 3 2 v 2 I e x p t - ^ - ^ ]. fi(kqq')"2 [(2q+l) (2q ' + l ) ] 2(2j 2+l) (2k+l)' J 9 V 2 2 W f V f l l ^ k ) M w s c J i V i l J i v y - ( - l ) k + q + q ' ^ i v j l j l ^ l l j ^ «C j f v f | j f v f ) ] ; S M + ( A c l n tJ i v i I J £ v £ 3 IA' q' n' [j ! v! I j f v f ]) k - E . . . . . i q , _ q n(kqq>yh I exp[ ] [ ( 2 q + l ) ( 2 q ' + l ) ] J 5 (-1) 1 £ J 2 V? kT • 7 7 Q * 1 * 2 ^ t i i l2+l2 r q ' ^2 <2, r q t'l I ^ ^ k' t 1 2 A 1 *• V 2 ' 300 < j i V i H i ) i I I J f v £ > < j f V f l l c y 2 l | j 2 v 2 > exp[-A / 2 ] C U n ^ £ J l ' n ' L* L\L O | E . +E. - E . , , - E . , , ) , v>Dv -Dp ^ K - ^ l t'2 l *~ ^1^2' j £ v £ J 2 v 2 j j v j j f V J / k and (7.28) ^ " - ( ^ q n U ^ I j £ v f ] l £ 'q 'n ' [ J j j £ v f ]) R - E . , , . . = i q "" q ' fUkqq 1)^ I e x p [ - j ^ ] (-1) 1 f [(2q+l) (2q- +1) ] % *2 V2 — Q — y / ' / / 1 q q 1 k i v / / I • , £ J i J l - " i J f J2 t < J i v i l l i ) l ^ l 5 | l 5 l V < 3 l i l l l ^ l 1 l"lj fv f> exp[-4/2] < ; ^ | | J ! ^ 2 ) | | 5 2 v 2 > < 5 2 v 2 l l ^ ^ ; ) | U ; v : > X [ 6 ( j 2 v 2 | j f v p 6 ( j ! v ! | j f v f ) n^ 1^ ( W / i / o U ' n ' / ' / ! / 1 l E - r - + E - r - - E . , , - E . , , ) . i_>p 1 2 ' ^1*»2' j 1 v 1 j 2 v 2 3Jv» j j v j ^ k + 6 ( j„v 0 j !v!)6(j !v' i .v.) w 2 2 J x xJ ^Jx 1 Jx xJ 301 U ^ O n / I / \Vn' I ' / ! /' |E, - +E, - -E. , ,-E. P 1 2 1 2 1 j 1 v 1 j 2 v 2 . j'v^ j + 2iri (2k+l) -1 I e x P [ J2 V2 i 'v' J l 1 kT ] fi(kqq')"'2[(2q + l ) ( 2 q ' + l ) ] Q <j .v. J i I i l l j ? i q ) | | J £ V f > < J i v f | | c p ^ , ) | | j ! v J > ( - l ) C £ n £ ' n ' ) k q q 4 6 ( J 2 v 2 | j ! v p 6 ( j ' v j | j i v i ) - 6 U 2 v 2 | j f v ' ) 6 ( j ^ v ^ | j f v f ) ] . These DWBA expressions reduce to equations (4.42) through (4.45) f o r the case of no v e l o c i t y p o l a r i z a t i o n s (£=n=£'=n'=k=0), and to equations (3.88), (3.89), (3.94) and (3.95) when only "diagonal i n j " p o l a r i z a t i o n s are considered. (In t h i s l a t t e r case, care must be taken to observe the d i f f e r e n t l i n e a r i z a t i o n (3.9) employed i n chapter I I I , which leads to d i f f e r e n t Boltzmann weights i n these expressions, and the d e f i n i t i o n (4.11), which gives d i f f e r e n t (2j+l) degeneracy factors.) Equations (7.25) through (7.28) involve the t r a n s l a t i o n a l i n t e g r a l s [J.^ , I>p^ a n d ^ " h ^ w n :"- c n s t i H must be evaluated. In analogy with the approach taken i n chapter I I I , section (g) and chapter IV, section (e), these i n t e g r a l s are estimated by a (high temperature) modified Born approximation and only f o r a s p e c i f i c example - the r e l a x a t i o n of the dipole moment f l u x described by << L | E "^u>> . Other c o l l i s i o n a l rates such as density) corrections to the d e s c r i p t i o n provided i n the l a s t section, can be evaluated s i m i l a r l y . which a r i s e when making (low 302 importance. In p a r t i c u l a r then, the r e l a t i o n s h i p (7.29) (io ooo| io ooo|o)0 .= f t f ^ 2 « (1,1) can be e s t a b l i s h e d (see chapter I I I , equation (3.101) or reference (17)) when (H s) e n e r g e t i c a l l y i n e l a s t i c c o l l i s i o n s can be neglected. Here the Q ' i n t e g r a l s are those of Chapman and C o w l i n g 2 1 w h i l e , i n p a r t i c u l a r , fi^1'1^ i s r e l a t e d to the d i f f u s i o n constant. The t r a n s f o r m a t i o n back from the r e l a t i v e v e l o c i t y d e s c r i p t i o n t o the d e s c r i p t i o n i n terms of the v e l o c i t i e s of the two c o l l i d i n g molecules, as r e q u i r e d by equation (7.22), i s given i n t h i s case by Further, because of the s p h e r i c a l p o t e n t i a l i s the i d e n t i t y operator f o r i n t e r n a l s t a t e s , c o n t r i b u t i o n s from J3> u l t i m a t e l y produce an i n t e r n a l s t a t e (7.30) I (Q) 10 10;£n£n = 6 f a c t o r « u | ( , ) j J > > > while c o n t r i b u t i o n s from va n i s h i d e n t i c a l l y because t r l , 2 { i f ! f ( 2 ) } H 0 • Next, f o r terms i n v o l v i n g K , a modified Born approximation to t h i s c o l l i s i o n i n t e g r a l i s employed, namely (7.31) Ep V n ^ / ^ U n / ; / . ^ 2 | 0 ) 0 303 21 as discussed by Snider. In the absence of v e l o c i t y p o l a r i z a t i o n s , ?,=n=0 and equation (7.31) becomes equation (3.109). When v e l o c i t y p o l a r i z a t i o n s are present - of i n t e r e s t here - i t i s useful to further assume that the d i f f e r e n t i a l cross section \ CL I. L\ L 2^ l S independent of the v e l o c i t y so that equation (7.31) i s approximately (7.32) C y 3 ( ^ n Z Z XL 2 | £ n / Z ^ ^ O ) , & r - ( D Wnn b p (00/ L x / l 2 | 0 0 / L 1 / 2 | 0 ) Q The weight f a c t o r (7.33) Wnn = 2 / Y 3 d Y exp [-Y 2]-^n£(Y)A^(Y) i s the only part of equation (7.32) that depends on the presence of the v e l o c i t y p o l a r i z a t i o n s . Indeed, the transformation (7.22) a f f e c t s only 17 equation (7.33). Further, as discussed by Chen et a l , the approximate a e f f e c t of t h i s transformation i s to average Wnn to unity i n contributions and to cause contributions from yj> to vanish. This has the i n t e r e s t i n g consequence that equations (7.25) and (7.26) reduce to equations (4.42) and (4.43). (To see t h i s , consider equations (7.25) and (7.26) f o r k=0, I S y ^ contibutions ignored, and I S p ^ given by equations (7.32).) Thus, the re l a x a t i o n of the "composite p o l a r i z a t i o n " L ^ t W j u can be approximately s p l i t into two parts - the f i r s t representing the r e l a x a t i o n of LX^(W) under the influence of the sp h e r i c a l p o t e n t i a l and the second giving the pure i n t e r n a l state r e l a x a t i o n of y. E x p l i c i t l y , (7.34) 3 04 10 l / n , 10 Tf _ " « L M | # | L y » 0 10 10 « L u| : L y » B 3 < < :UhU» n f 8 k T ^ 1 ^ T J M - (1,1) J^l<#ly» 0 i 16n + (-^r—) 2 where the prime indicates that only type contributions are considered. 1 I Consistent with the DWBA approach, the term (——) i s expected to be smaller (1 1) 2 than the s p h e r i c a l part, 16n fi . The d i v i s i o n exhibited by equation (7.34) into " v e l o c i t y changing" and "phase changing" parts i s reminiscent of the d i v i s i o n of the c o l l i s i o n superoperator i t s e l f , equation (7.12). In the treatment of c o l l i s i o n s presented i n t h i s t h e s i s , a hierarchy of approximations has been described. This can be interrupted at any stage i f more exact expressions are desired. However, i f t h i s hierarchy i s followed to i t s most approximate end, a simple p h y s i c a l " p i c t u r e " of the r e l a x a t i o n process r e s u l t s . Consider the a r b i t r a r y v e l o c i t y and i n t e r n a l state p o l a r i z a t i o n s in f i q ) L (W) and Cs , r e s p e c t i v e l y . Then the r e l a x a t i o n of such a "composite p o l a r i z a t i o n s can be expressed as (7.35) £J(Aqn|jlqn) 0 * F ^ 1 - 1 (£n000 | £n000 | 0) + G^cooZ lxl2\ML 4l/2\0){ where F, G are c a l c u l a b l e factors and (/_ L ^/.2) represents the dominant anisotropic p o t e n t i a l . Equation (7.34), which describes the relaxation of 10 the composite p o l a r i z a t i o n L (W)p, i s an example of t h i s structure. The r—Cl") term i n v o l v i n g b> does not contribute to pure v e l o c i t y r e l a x a t i o n (q=0) -P 20 for example, see equation (3.105) describing the r e l a x a t i o n of L (W) . On r - ( l ) the other hand, the b v part of equation (7.35) vanishes i f only i n t e r n a l state r e l a x a t i o n processes are o f i n t e r e s t . In the spectroscopic case, the operators ^ are generally "off-diagonal i n j " while i n Senftleben-Beenakker studies, i s r e s t r i c t e d to "diagonal i n j " forms. The r e l a x a t i o n of y, s p e c i f i e d by ir} , i s an example of the f i r s t type and the r e l a x a t i o n of l 2 C 2) -1 ^ (*1) i described i n chapter III by T^ n t> i s an example of the second type of pure i n t e r n a l state decay. In both spectroscopic and S-B experiments, the production terms J^"(£qn|£'q ,n ,)k, kj^O, are expected to be small. However, while i n the former they can be s a f e l y neglected, i n the l a t t e r they are fundamental to a proper d e s c r i p t i o n of the phenomena and can not be ignored. Thus i n chapter I I I , a production term, \p, was a necessary part of the formalism. This completes the d e s c r i p t i o n of the c o l l i s i o n a l aspects of v e l o c i t y e f f e c t s on spectroscopic phenomena and t h e i r r e l a t i o n s h i p to Senftleben-Beenakker studies. (e) Model Methods a p p l i e d to V e l o c i t y E f f e c t s . The moment method lias been employed t h r o u g h o u t t h i s t h e s i s to s o l v e t h e v a r i o u s e q u a t i o n s o f m o t i o n and o b t a i n a d e s c r i p t i o n o f the m o t i o n s o f the gas s y s t e m . In f a c t , however, t h i s d e s c r i p t i o n i s o f l i m i t e d v a l i d i t y and a p p l i e s o n l y i n the " c o n t i n u u m r e g i o n " where the d e n s i t y o f gas m o l e c u l e s i s s u f f i c i e n t l y h i g h . In t h i s r e g i o n , the gas m o l e c u l e s c o l l i d e o f t e n enough to a l l o w any non-e q u i l i b r i u m c h a r a c t e r to be d e s c r i b e d i n terms o f a v e r a g e p o l a r i z a t i o n s o f the g a s . As the d e n s i t y i s l o w e r e d , the number o f p o l a r i z a t i o n s n e c e s s a r y to g i v e a good d e s c r i p t i o n o f the gas m o t i o n s i n c r e a s e s to unmanageable p r o p o r t i o n s . T h i s r e f l e c t s the p h y s i c a l s i t u a t i o n where c o l l i s i o n s a r e so i n f r e q u e n t t h a t e a c h m o l e c u l e has i t s i n d i v i d u a l v e l o c i t y c h a r a c t e r i s t i c s . To e f f e c t i v e l y d e s c r i b e t h e m o t i o n s o f the gas i n t h i s d e n s i t y r e g i o n , an a l t e r n a t e a p p r o a c h must be t a k e n . The model method i s s u c h an a l t e r n a t e a p p r o a c h . The model method r e t a i n s the e x a c t f r e e s t r e a m i n g p a r t s o f the e q u a t i o n o f m o t i o n b u t employs an a p p r o x i m a t e (model) c o l l i s i o n o p e r a t o r w h i c h p o s s e s s many q u a l i t a t i v e and a v e r a g e p r o p e r t i e s o f the t r u e c o l l i s i o n o p e r a t o r . T h i s method was o r i g i n a l l y a p p l i e d to the c l a s s i c a l 2 2 2 3 2 4 l i n e a r i z e d B o l t z m a n n e q u a t i o n ' ' and l a t e r e x t e n d e d to l i n e a r i z e d quantum B o l t z m a n n e q u a t i o n s , b o t h the Wang-2 5 2 6 27 Chang U h l e n b e c k ' and the Waldmann-Snider v e r s i o n s . 3 07 I n d e e d , model methods a p p l i e d to t h i s l a s t e q u a t i o n would a l l o w the d e s c r i p t i o n o f S e n f t l e b e n - B e e n a k k e r e f f e c t s i n the c o n t i n u u m r e g i o n ( c h a p t e r I I I , s e c t i o n ( d ) ) , t o be e x t e n d e d to the r a r e f i e d gas (or Knudsen) r e g i m e . The p r e s e n t s e c t i o n a p p l i e s the model method t o the s t e a d y s t a t e , non s a t u r a t i o n l i n e s h a p e p r o b l e m . As s u c h , the m o d e l l i n g o f the l i n e a r i z e d c o l l i s i o n o p e r a t o r , e q u a t i o n ( 7 . 1 2 ) , o f the Waldmann-Snider form i s c o n s i d e r e d . T h i s s h o u l d be s u f f i c i e n t to i n t r o d u c e t h e method to the d i s c u s s i o n o f s p e c t r o s c o p i c phenomena. T h i s s e c t i o n may be c o n s i d e r e d as a " j u s t i f i c a t i o n " o f the p r e v i o u s l y p r e s e n t e d moment method a p p r o a c h to s p e c t r o s c o p i c phenomena, i n the s e n s e t h a t the moment method r e p r e s e n t s the h i g h d e n s i t y ( c o n t i n -uum) l i m i t o f t h e model method. The r e s t r i c t i o n to n o n - s a t u r a t i o n , s t e a d y s t a t e e f f e c t s , i n t r o d u c e s a g r e a t s i m p l i f i c a t i o n i n t o the d e s c r i p t i o n . I n d e e d , as f a r as the i n t e r n a l s t a t e s a r e c o n c e r n e d , t h e r e a r e o n l y two o p e r a t o r s , U and U, w h i c h a r e p e r t u r b e d from e q u i l i b r i u m . E q u i v a 1 e n t l y , the two o p e r a t o r s (7.36) y + = y + iJJ t y_ = y - i y = ( y + ) can be u s e d , and a r e u s e d h e r e , to d e s c r i b e t h e i n t e r n a l s t a t e non e q u i l i b r i u m c h a r a c t e r . Thus, the d i s t r i b u t i o n 3 08 f u n c t i o n a p p r o p r i a t e t o t h i s p r o b l e m i s w r i t t e n as W 2 - # o / k T (7.37) f ( v ) = ~ { + $ ( v ) } f (2irmkT) ' U where the p e r t u r b a t i o n , <J>(v), has the form (7 .38) <J> (v) = (J) , (v) + <J> (v) Here, <t>+(v) and (v) a r e t h o s e p a r t s o f <{> (v) w h i c h a r e p r o p o r t i o n a l to y and y , r e s p e c t i v e l y . More p r e c i s e l y , ~ + — these r e l a t i o n s h i p s a r e (7.39) <j>+(v) = H + - V V ) * « y l . y » <{> (v) = y « A (v) 3y_-< ( J J _ ) + > <<y I -y >> where t h e l a s t f o r m s r e p r e s e n t t h e h i g h d e n s i t y l i m i t s . I n d e e d , t h i s s e c t i o n s p e c i f i e s the manner i n w h i c h the l i m i t s (7.39) o c c u r . E m p l o y i n g e q u a t i o n s (7.37) and (7.38) i n e q u a t i o n ( 7 . 1 1 ) , t h e e q u a t i o n s o f m o t i o n f o r <{>+(v) and cf> (v) a r e . . 3)1 «E (7.40) -i(AlO-k-v)<J> + + <^ [<J> + ] = iK <-^-> ~ + ~ ° eq <<y +I•y + > > a nd (7.41) i (Ato-k-v) 4»_ + r e s p e c t i v e l y . T hese e q u a t i o n s a r e c o l l i s i o n a l u n c o u p l e d on the a s s u m p t i o n t h a t c o l l i s i o n s c o n s e r v e f r e q u e n c y . As (7.40) and (7.41) a r e complex c o n j u g a t e s o f e a c h o t h e r , i t i s s u f f i c i e n t t o s t u d y j u s t one o f the p a i r . The r e m a i n d e r o f t h i s s e c t i o n w i l l c o n s i d e r t h e s o l u t i o n o f (7.14) f o r v a r i o u s m o d els o f the c o l l i s i o n s u p e r o p e r a t o r I n t h e p a s t , two m o d e l s have been t r a d i t i o n a l l y e mployed, and (7 .43) <#[<}>] = 3 y _ - < u + > 1 (4> -3U *<U > x <<u a ~ •u >> + <<U I •]! >> T b In o r d e r to l i m i t s , the make p r o p e r c o n n e c t i o n i d e n t i f i c a t i o ns w i t h t h e h i g h d e n s i t y 1 9 « L 1 0 y J ^ | L 1 0 M _ » o 1 T b « L 1 0 U : L 1 0 M » a r e made, v/here t h e l a s t e q u a l i t i e s f o l l o w , i g n o r i n g c o l l i s i o n a l s h i f t s . However, t h e s e " t r a d i t i o n a l " methods i g n o r e an i m p o r t a n t i n v a r i a n c e p r o p e r t y of. (f{, namely c o n s e r v a t i o n o f p a r i t y (7.45) (JI [-<()} = $el$ei + #0£<?>0] where <{>e and <f)° a r e t h e e v e n and odd ( i n v e l o c i t y ) p a r t s o f <j) , r e s p e c t i v e l y . E q u a t i o n (7.45) i m p l i e s t h a t (7.42) and (7.43) a r e s p e c i a l c a s e s o f the more g e n e r a l m o d e l l i n g ; (7.46) (R^°J = ^ - + T a T b and 32-'<}! + > . 1 . e - 3 ^ - " < ^ + > 1 1 ,o (7.47) fluj = — < < ; ~ | > ; + » + — (*! <<y-y./>>) + T~ * a — — — c «" — —• •" o In e q u a t i o n ( 7 . 4 7 ) , the i d e n t i f i c a t i o n (7.43) — x 1 5 « : J 2 0 J L i _ | ^ | L 2 % _ » 311 T . r 2 0 . i 2 0 c < < L )J lh y >-» 1 5 « L 2 0 U J ( ^ | L 2 % _ » = « L 2 % I ; L 2 0 ^ a g a i n f o l l o w s from the h i g h d e n s i t y l i m i t . T hus, i n o r d e r t o t r e a t a l l r e l e v a n t c a s e s , e q u a t i o n (7.41) i s s o l v e d u s i n g the model ( 7 . 4 7 ) , and t h e n s p e c i a l c a s e s o f t h i s model a r e c o n s i d e r e d . W i t h $ e x p r e s s e d as e q u a t i o n ( 7 . 4 7 ) , the p e r t u r b a t i o n <j>_ i s d i v i d e d i n t o 3y -<y > 3H_*<H+> (7.4 9) <b = 1 r- + * + (<*> - ~r- < rr-) ' f - < < y | • y > > - - <<.yj-y>^ and t h e c o u p l e d s e t o f i n t e g r a l e q u a t i o n s - ItC < > t-r 4 eq <<y_f *y_>> 3y -<y+> 3y «<y+> ( i A w + —)$°_ = i J S , Z f « y J . y _ » + <*- " « y ~ | - y _ » ) ] 3u -<y > , , -W 2 a r e o b t a i n e d f r o m e q u a t i o n ( 7 . 4 1 ) . T h e s o l u t i o n o f t h i s s e t o f e q u a t i o n s a l l o w s t h e m o m e n t < J J + > t o b e e x p r e s s e d a s (7.51) <y > = t r / a p 6 ^-pr (y ) «j> ~ + ( 2TTmkT) ~ " M 2 (iAw + i-) • 1K 2 <M«> E d P e * 4 e q ~ ° ( 2 T r m k T ) 3 / 2 [ ( iAw+^-) (iAw+i-) + ( k - v ) 2 ] T, T b c 2 (iAw + -W T I _ 1 }/ d p e b _ T c T a ( 2 T T m k T ) 3 / 2 [ (iAu+i-) (iAu+i-) + ( k - v ) 2 ] O f c o u r s e , t h e e x p r e s s i o n f o r t h e a b s o r p t i o n c o e f f i c i e n t i s p r o p o r t i o n a l t o < y > , w h i c h c a n b e o b t a i n e d f r o m < y + > b y t h e r e l a t i o n <y > - <y > (7.52) <y> =•—— 7 7 — — E q u a t i o n (7.51) g i v e s a t h r e e r e l a x a t i o n p a r a m e t e r d e s c r i p t i o n o f a m i c r o w a v e a b s o r p t i o n l i n e s h a p e , v a l i d o v e r a w i d e r a n g e o f g a s d e n s i t y . I n p a r t i c u l a r , i t g i v e s t h e 3 13 D o p p l e r l i n e s h a p e a t t h e l o w d e n s i t y e x t r e m e a n d a L o r e n z t i a n l i n e s h a p e a t t h e h i g h d e n s i t y l i m i t . E q u a t i o n (7.51) r e p r e s e n t s t h e m o s t c o m p l e t e d e s c r i p t i o n g i v e n i n t h i s t h e s i s o f t h e m a n n e r i n w h i c h t h e t r a n s i t i o n f r o m t h e D o p p l e r b r o a d e n e d r e g i m e t o t h e p r e s s u r e b r o a d e n e d r e g i m e i s a c h i e v e d . T h e s i m p l e r c o l l i s i o n m o d e l e q u a t i o n s (7.42), (7.43) a n d (7.46), l e a d t o s p e c i a l c a s e s o f t h e r e s u l t (7.51). I n p a r t i c u l a r , t h e V o i g t p r o f i l e i s o b t a i n e d w i t h a o n e r e l a x a t i o n t i m e m o d e l , ( T = T = T ) , c b a (7.53) <U > = - i K 2 < M l i > E / - * £ - £ -w 2 4 e q ~ ° ( 2 7 T m k T ) 3 / 2 1 . ( I A U ) + — ; a [ ( i A w + — ) 2 + ( k - v ) 2 ] Ta ~ I f t h e m o d e l , e q u a t i o n (7.43), i s e m p l o y e d (T = T^ ) , t h e r e s u l t i n g e x p r e s s i o n (7.54) 2 (iAo) + i ~ ) _ i K 2 < J 5 M > E f d p e b 4 e q ~ ° ( 2 T r m k T ) 3 / 2 [ ( i A a > + ~ ) 2 + ( J c - v ) 2 ] . . , -w 2 (iAto + ~ ) 1 - { — - — } / d p e T ^ T T 3 / 2 1 2 2 b a (2TTmkT) ' [ ( i A u + — ) + ( k * v ) ] T b " p r o d u c e s a l i n e s h a p e t h a t has a l s o been d i s c u s s e d p r e v i o u s -l y . 4 ' 1 5 F i n a l l y , i f T = T , the model (7.46) g i v e s C cl •w2 2 fcAN dp e (7.55) <U > = - i k < i 3 - r - > 1 3 77T ~ + 4 e q- ~° (2 TTmkT) / (iAw + •—•) _b 1 1 2 t ( i A ( J + — ) ( i A w + — ) + (k«v) ] T T, - ~ a b E q u a t i o n (7.55) r e s u l t s f r o m e m p h a s i z i n g p a r i t y c o n s e r v a t i o n I t s a s s o c i a t e d l i n e s h a p e has n o t been c o n s i d e r e d p r e v i o u s l y . E q u a t i o n (7.53) y i e l d s a l i n e w i d t h t h a t i n c r e a s e s mono t o n i c a l l y w i t h p r e s s u r e , even a t v e r y lev; gas d e n s i t i e s . T h i s i s i n c o n t r a d i c t i o n w i t h u s u a l e x p e r i m e n t a l o b s e r v a -t i o n . E q u a t i o n s (7.54) and (7.55) a l l o w f o r more g e n e r a l p r e s s u r e d e p e n d e n c e s o f the l i n e s h a p e . Of the two, the form (7.55) g i v e s a s i m p l e r d e s c r i p t i o n . More c a r e f u l e x p e r i m e n t a l measurements on the p r e s s u r e d e p e n d e n c e o f l i n e s h a p e s a r e r e q u i r e d i n o r d e r t o a s c e r t a i n w h i c h form w i l l g i v e a more f a i t h f u l r e p r e s e n t a t i o n o f the o b s e r v e d b e h a v i o u r . The more g e n e r a l p r o b l e m o f d e s c r i b i n g the e f f e c t s o f v e l o c i t y on s a t u r a t e d l i n e s h a p e s and c o h e r e n c e t r a n s i e n t e x p e r i m e n t s i n the microwave r e g i o n , have y e t t o be a t t e m p t e d from a model method a p p r o a c h . The r e s u l t , o f c o u r s e , would be a d e s c r i p t i o n v a l i d o v e r a whole r a n g e o f gas d e n s i t i e s and such t h a t e q u a t i o n s (7.17) a r e o b t a i n e d a the h i g h d e n s i t y l i m i t s . A t r e a t m e n t o f .these t y p e s o f e f f e c t s f o r the o p t i c a l r e g i o n has been g i v e n by.German e t a l . ^ b u t we f e e l t h a t f u r t h e r c o n s i d e r a t i o n i s n e c e s s a r b e f o r e t h e i r s i m p l e model c a n be a p p l i e d to t h e microwave c a s e . 316 ( f ) T h e s i s C o n c l u s i o n T h i s t h e s i s has p r e s e n t e d the s u b j e c t s o f p r e s s u r e b r o a d e n i n g and c o h e r e n c e t r a n s i e n t e f f e c t s i n the microwave r e g i o n o f the s p e c t r u m from a k i n e t i c t h e o r y p o i n t o f v i e w . The v a r i o u s f r e e m o t i o n a s p e c t s o f the m o l e c u l a r m o t i o n s -o s c i l l a t i o n s , r o t a t i o n s , and t r a n s l a t i o n s - have a l l been t r e a t e d i n a u n i f o r m manner. In p a r t i c u l a r , the c o n c e p t o f a two l e v e l s y s t e m has been d e v e l o p e d i n o r d e r to a n a l y z e t h e r o t a t i o n a l m o t i o n s i n a c o n s i s t e n t f a s h i o n . The c o l l i s i o n a l e f f e c t s on t h e s e m o t i o n s have a l s o been d i s c u s s e d i n d e t a i l - the e x a c t forms o f the r e l e v a n t m a t r i x e l e m e n t s o f the c o l l i s i o n s u p e r o p e r a t o r ^ ? have been s p e c i f i e d and t h e s e have been a p p r o x i m a t e l y e v a l u a t e d w i t h i n the c o n t e x t o f a d i s t o r t e d wave B o r n a p p r o x i m a t i o n . The v e c t o r (and t e n s o r ) n a t u r e o f the m o t i o n s have been e m p h a s i z e d t h r o u g h o u t t h e t h e s i s . T h i s g i v e s a c l e a r e r u n d e r s t a n d i n g o f t h e p h y s i c a l m o t i o n s t h a t a r e i n v o l v e d , i n d i c a t e s a d d i t i o n a l e f f e c t s w h i c h s h o u l d be o b s e r v a b l e e x p e r i m e n t a l l y , and a l l o w s the s m a l l e s t number o f i n d e p e n d e n t c o l l i s i o n m a t r i x e l e m e n t s to be s p e c i f i e d ( t h r o u g h use o f the r o t a t i o n a l i n v a r i a n c e p r o p e r t i e s o f ^ ) . The methods employed i n t h i s t h e s i s p a r a l l e l t h o s e u s e d t o d e s c r i b e t h e S e n f t l e b e n - B e e n a k k e r e f f e c t s . I n d e e d , c o n n e c t i o n s w i t h the t h e o r y o f the S-B e f f e c t s have been s t r e s s e d t h r o u g h o u t ( e s p e c i a l l y i n t h e c o l l i s i o n a l t r e a t -m e n t s ) . T h i s " u n i f i e d p o i n t o f v i e w " s h o u l d c o n t r i b u t e t o a more c o m p l e t e u n d e r s t a n d i n g o f b o t h s p e c t r o s c o p i c and t r a n s p o r t phenomena. As an example o f t h i s , c o n s i d e r the use d e x t e n s i v e l y i n t h e e x p l a n a t i o n o f S e n f t l e b e n - B e e n a k k e r e f f e c t s . The work o f t h i s t h e s i s has shown how t h e s e p o l a r i z a t i o n s a l s o a r i s e i n p r e s s u r e b r o a d e n i n g and c o h e r e n c e t r a n s i e n t e f f e c t s . As a f u r t h e r i n d i c a t i o n o f the u s e f u l n e s s o f s u c h a u n i f i e d a p p r o a c h , e x p e r i m e n t s i n v o l v i n g t h e e f f e c t s o f 28 o s c i l l a t i n g f i e l d s on t r a n s p o r t phenomena s h o u l d be p o i n t e d o u t . These e x p e r i m e n t s r e p r e s e n t m o d i f i c a t i o n s o f the u s u a l S e n f t l e b e n - B e e n a k k e r e f f e c t s s i n c e o s c i l l a t i n g f i e l d s a r e u s e d i n p l a c e o f , o r i n c o n j u n c t i o n w i t h , the s t a t i c e l e c t r i c or m a g n e t i c f i e l d s . As the f r e q u e n c y o f t h e a p p l i e d f i e l d i s swept t h r o u g h t h e n a t u r a l o s c i l l a t i o n f r e q u e n c i e s o f the m o l e c u l e s , r e s o n a n c e s o c c u r - j u s t a s i n t h e n o r m a l s p e c t r o s c o p i c e x p e r i m e n t s d i s c u s s e d i n t h i s t h e s i s . The e f f e c t s o f t h e s e r e s o n a n c e s on the t r a n s p o r t p r o p e r t i e s o f the gas a r e t h e n o b s e r v e d . The a p p l i c a t i o n o f t h e t h e o r y p r e s e n t e d i n t h i s t h e s i s to t h e (quantum m e c h a n i c a l ) d e s c r i p t i o n o f t h e s e phenomena, r e p r e s e n t s a n a t u r a l e x t e n s i o n o f the p r e s e n t work. r o t a t i o na 1 w h i c h have been BIBLIOGRAPHY C h a p t e r I 1. V/. He i t i e r , The Quantum Theory of R a d i a t i o n 3rd ed . , (Clarendon Press, Oxford, 1 9 5 4 ) . 2. W. H. Louise11, R a d i a t i o n and Koise i n Quantum  E l e c t r o n i c s , (McGraw-Hill, Hew York, 1 9 6 4 ) . 3 . H . Kuhn, P h i l . Mag. _18, 987 (1'934). 4 . H . Margenau, P h y s . Rev. 4_8 , 755 ( 1 9 3 4 ) . 5. H . A. L o r e n t z , P r o c . Amsterdam A c a d . 8, 591 ( 1 9 0 6 ) . 6. H . M. F o l e y , P h y s . Rev. 69_, 616 ( 1 9 4 6 ) . 7 . L. S p i t z e r , P h y s . Rev. J58, 348 ( 1 9 4 0 ) . 8 . T. H o l s t e i n , P h y s . Rev. 7_9' 7 4 4 ( 1 9 5 0 ) . 9. R. E. M. Hedges e t a l . , P h y s . Rev. A6, 1519 (1972) . 10. P . W. A n d e r s o n , P h y s . Rev. 8_6, 809 ( 1 9 5 2 ) . 11. R. P. F u t r e l l e , P h y s . Rev. A5, 2162 ( 1 9 7 2 ) . 12. H . Margenau and M. L e w i s , Rev. Mod. P h y s . 31 , 569 (1959) 13 . J . H o l t s m a r k , P h y s i k . Z. 2J3, 162 ( 1 9 1 9 ) . 14 . P . W. A n d e r s o n , P h y s . Rev. 76, 647 ( 1 9 4 9 ) . 15. C. J . Tsao and B. C u r n u t t e , J . Quant. S p e c t r y . Rad. T r a n s f e r 2, 41 ( 1 9 6 2 ) . 16. A. C. K o l b and H. Gri e m , P h y s . Rev. I l l , 514 ( 1 9 5 8 ) . 17 . M. 112 B a r a n g e r , P h y s . Rev. I l l , 481 ( 1 9 5 8 ) , 111 :, 855 (1958) . , 494 (1958) 18 . U . Fano, P h y s . Rev. 131, 259 ( 1 9 6 3 ) . 19. A . Ben-Reuven, P h y s . Rev. 141, 34 ( 1 9 6 5 ) . 20 . A. Ben-Reuven, P h y s . Rev. L e t t e r s 14, 349 ( 1 9 6 5 ) , P h y s . Rev. 145, 7 ( 1 9 6 6 ) . 21. J . 227 H. Van V l e c k and V. F. W e i s k o p f , Rev. Mod ( 1 9 4 5 ) . . P h y s . 17, 319 22. G. Birnbaum, P h y s . Rev. 150, 101 ( 1 9 6 6 ) . 23. C. S. Wang Chang and G. E. U h l e n b e c k , T r a n s p o r t  Phenomena i n P o l y a t o m i c M o l e c u l e s , ( U n i v e r s i t y o f M i c h i g a n P u b l i c a t i o n CM-681, Ann A r b o r , 1 9 5 1 ) . 24. A. T i p and F. R. M c C o u r t , P h y s i c a 52., 109 ( 1 9 7 1 ) . 25. L. Waldmann, Z. N a t u r f o r s c h 12a, 660 ( 1 9 5 7 ) . 26. R. F. S n i d e r , J . Chem. P h y s . 3_2, 1051 ( 1 9 6 0 ) . 27. C. H. Townes and A. L . Schawlow, Mi c r o w a v e S p e c t r o s c o p y , ( M c G r a w - H i l l , New Y o r k , 1 9 5 5 ) , p. 371. 28. R. K a r p l u s and J . S c h w i n g e r , P h y s . Rev. 7_3 , 1020 ( 1 9 4 8 ) . 2 29. R. G. Gordon, J . Chem. P h y s . 4_4_, 3083 ( 1 9 6 6 ) . 30. R. G. Gordon, J . Chem. P h y s . 4_6, 4399 ( 1 9 6 7 ) . 31. W. B. N e i l s o n and R. G. Gordon, J . Chem. P h y s . 58, 4131 ( 1 9 7 3 ) . 32. R. S h a f e r and R. G. Gordon, J . Chem. P h y s . 58, 5422 (1973) . 33. R. G. Gordon, Adv. Mag. Reso n . 3_' 1 ( 1 9 6 8 ) . 34. R. G. Gordon, W. K l e m p e r e r , and J . I . S t e i n f e l d , Ann. Rev. P h y s . Chem. X_9, 215 (1968) . 35. H. R a b i t z , Ann. Rev. P h y s . Chem. _25 , 155 ( 1 9 7 4 ) . 36. H. G r i e m , P l a s m a S p e c t r o s c o p y , ( M c G r a w - H i l l , New Y o r k , 1964) . 37. R. H. D i c k e , P h y s . Rev. 8_9, 472 (1953 ) . 38. See, f o r example, A. Abragam, The P r i n c i p l e s o f N u c l e a r  M a g n e t i s m , ( C l a r e n d o n P r e s s , O x f o r d , 1 9 6 1 ) . 39. A. C a r r i n g t o n and A. D. M c L a c h l a n , I n t r o d u c t i o n to  M a g n e t i c R e s o n a n c e , ( H a r p e r and Row, New Y o r k , 1 9 6 7 ) . 40. a) F. M. Chen and R. F. S n i d e r , J . Chem. P h y s . 4 6, 3937 ( 1 9 6 7 ) . b) F. M. Chen and R. F. S n i d e r , J . Chem. P h y s . 48, 3185 ( 1 9 6 8 ) . 41. R. K a i s e r , E . B a r t h o l d i , and R. R. E r n s t , J . Chem. P h y s . 60, 2966 (1974) . 42. T. C. F a r r a r and E. D. B e c k e r , P u l s e and F o u r i e r  T r a n s f o r m MWR, (Academic P r e s s , New Y o r k , 1 CJ71). 43. A. G. M a r s h a l l and M. B. Comisarow, A n a l y t i c a l C h e m i s t r y 47 , 491A (1975) . 44. R. G. Brewer, S c i e n c e 178 , 247 ( 1 9 7 2 ) . 45. R. G. Brewer and R. L. Shoemaker, P h y s . Rev. L e t t e r s 27, 631 ( 1 9 7 1 ) , P h y s . Rev. A6, 2001 ( 1 9 7 2 ) . 46. R. P. Feynman and F. L. V e r n o n , J . A p p l . P h y s . _2_8, 49 (1957) . 47. p. R. Berman and W. E. Lamb, P h y s . Rev. A_2, 2435 ( 1970), A4, 319 (197 1 ) , P. R. Berman, P h y s . Rev. A_5, 927 ( 1 9 7 2 ) . 48. W. R* C h a p p e l l e t a l . , J . S t a t . P h y s . 3_, 401 ( 1 9 7 1 ) . 49. J . C. McGurk e t a l , Adv. Chem. P h y s . XXVII, 1 ( 1 9 7 5 ) . 50. P. G l o r i e u x and B. Macke, Chem. P h y s . £, 120 ( 1 9 7 4 ) . 51. J . H. S. Wang e t a l . , Chem. P h y s . 1, 141 ( 1 9 7 3 ) . C h a p t e r I I "1. J . C. McGurk, T. G. S c h m a l z , and W. H. F l y g a r e , Adv. Chem. P h y s . XXVII, 1 ( 1 9 7 5 ) . 2. U. Fano, Rev. Mod. P h y s . 29^, 74 ( 1 9 5 7 ) . 3. See, f o r example, W. R. Sal z m a n , P h y s . Rev. A_5, 789 (1972) . 4. D. A. Coombe e t a l . , J . Chem. P h y s . £ 3 , 3015 ( 1 9 7 5 ) . 5. H. J e t t e r e t a l . , J . Chem. P h y s . 5_9, 1796 ( 1 9 7 3 ) . 6. J . C. McGurk e t a l . , J . Chem. P h y s . 6_0, 2923 ( 1 9 7 4 ) . 7. J . C. McGurk e t a l . , J . Chem. P h y s . 60, 4181 ( 1 9 7 4 ) . 8. J . C. McGurk e t a l . , J . Chem. P h y s . j61_, 3759 (1974 ) . 9. H. C. T o r r e y , P h y s . Rev. 7j3, 1059 ( 19 4 9 ) . 10. J . D. J a c k s o n , C l a s s i c a l E l e c t r o d y n a m i c s , ( J o h n W i l e y and Sons, New Y o r k , 1 9 6 2 ) . 3 21 11. R. P. Feynman et a l . , The Feynman Lec t u r e s on Ph y s i c s V o l . II (Addison-Wcsley, New York, 1 9 6 5 ) . 12. H. Mader et a l . , J . Chem. Phys. 62, 4300 ( 1 9 7 5 ) . C h a p t e r III.. 1. J . J . M. B e e n a k k e r and F. R. McCourt, Ann. Rev. P h y s . Chem. 21, 4 7 (197 0) . 2. J . J . M. B e e n a k k e r i n L e c t u r e N o t e s i n P h y s i c s , V o l . 21, T r a n s p o r t Phenomena, ( e d . G. K i r c z e n o w and J . H a r r o , S p r i n g e r - V e r l a g , B e r l i n , 1 9 7 4 ) , pp. 414-468. 3. R. F. S n i d e r i n L e c t u r e N o t e s i n P h y s i c s , V o l . 21, T r a n s p o r t Phenomena, ( e d . G. K i r c z e n o w and J . M a r r o , S p r i n g e r - V e r l a g , B e r l i n , 1 9 7 4 ) , pp. 469-517. 4. H. M o r a a l , P h y s . R e p o r t s 17_, 225 ( 1 9 7 5 ) . 5. R. F. S n i d e r and B. C. S a n c t u a r y , J . Chem. P h y s . 55, 1555 ( 1 9 7 1 ) . 6. L. Waldmann, Z. N a t u r f o r s c h . 12a, 660 ( 1 9 5 7 ) . 7. R. F. S n i d e r , J . Chem. P h y s . 3_2, 1051 ( 1 9 6 0 ) . 8. E. W i g n e r , P h y s . Rev. 4_0, 479 ( 1 9 3 2 ) . 9. H. Weyl, Z. P h y s . 4_6, 1 ( 1 9 2 7 ) . 10. K. B a e r w i n k l e and S. Grossmann, Z. P h y s i k . 198, 277 (1967 ) . 11. a) M. W. Thomas and R. F. S n i d e r , J . S t a t . P h y s . _2, 61 ( 1 9 7 0 ) . b) J . A. R. Coope and R. F. S n i d e r , J . Chem. P h y s . 57, 4266 (197 2) . 12. R. F. S n i d e r , J . Math. P h y s . J5, 1580 ( 1 9 6 4 ) . 13. F. M. Chen, H. M o r a a l , and R. F. S n i d e r , J . Chem. P h y s . 51, 542 (1972) . 14. J . A. R. Coope, R. F. S n i d e r , and F. R. M c C o u r t , J . Chem, P h y s . 4_3 , 2269 ( 1 9 6 5 ) , J . A. R. Coope and R. -F. S n i d e r , J . M ath. P h y s . 11., 1003 ( 1 9 7 0 ) , J . A. R. Coope, J..,Math. P h y s . 11, 1591 ( 1 9 7 0 ) . . •^y^* 15. J . A. R. Coopc and R. F. S n i d e r , J . Chem. P h y s . 5_6, 2049 ( 1 9 7 2 ) . 16. J . J . n. B e e n a k k e r , J . A. R. Coope, and R. F. S n i d e r , P h y s . Rev. A4_, 788 (1971) . 17. J . A. R. Coope e t a l . , P h y s i c a 79A, 129 ( 1 97 5 ) . 18. F. R. M c C o u r t and R. F. S n i d e r , J . Chem. P h y s . 47, 4117 ( 1 9 6 7 ) . 19. L. W. H u n t e r and R. F. S n i d e r , J . Chem. P h y s . 61, 1160 ( 1 9 7 4 ) . 20. A. C. L e v i and F. R. M c C o u r t , P h y s i c a 3_8, 415 ( 1 9 6 8 ) . 21. A. R. Edmonds, A n g u l a r Momentum i n Quantum M e c h a n i c s , ( P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , 1 9 7 4 ) . 22. L . W. H u n t e r and R. F. S n i d e r , J . Chem. P h y s . 6_1, 1151 (1974) . 23. A. P. Y u t s i s , I . B. L e v i n s o n , and V. V. Va n a g a s , T h e o r y o f A n g u l a r Momentum, ( I s r e a l Program f o r S c i e n t i f i c T r a n s l a t i o n s , J e r u s a l e m , 1 9 6 2 ) . 24. D. A. Coombe e t a l . , J . Chem. P h y s . £ 3 , 3015 ( 1 9 7 5 ) . 25. R. F. S n i d e r , P h y s i c a 7_8, 387 ( 1 9 7 4 ) . 26. R. A. J . K e i j s e r e t a l , , P h y s . L e t t . A45, 3 ( 1 9 7 3 ) . 27. F. M. Chen and R. F. S n i d e r , J . Chem. P h y s . 4 6, 3937 ( 1 9 6 7 ) . 28. H. M o r a a l , Z. N a t u r f o r s c h . 28a, 824 ( 1 9 7 3 ) . 29. B. S h i z g a l , J . chem. P h y s . _58 , 3424 ( 1 9 7 3 ) . 30. S. Chapman and T. G. C o w l i n g , The M a t h e m a t i c a l T h e o r y  o f N o n u n i f o r m G a s e s , (Cambridge U n i v e r s i t y P r e s s , C a m b r i d g e , 197 0) . 31. B. S h i z g a l , t o be p u b l i s h e d . C h a p t e r IV 1. R. F. S n i d e r and B. C. S a n c t u a r y , J . Chem. P h y s . 5 5, 1555 ( 1 9 7 1 ) . 3 23 2. D. A. Coombe e t a l . , J . Chem. P h y s . 5^3_, 3 015 ( 19 7 5 ) . 3. S. He s s , Z. N a t u r f o r s c h . 2 2a, 187 ( 1 9 6 7 ) . 4. A. T i p , P h y s i c a 5_2, 493 ( 1 9 7 1 ) . 5. R. F. S n i d e r , J . Math. P h y s . J3, 1580 ( 1 9 6 4 ) . 6. S. Hess and W . E . K o h l e r , Z. N a t u r f o r s c h 23a, 1903 (1968) . 7. U. Fano, P h y s . Rev. 133B, 8 28, ( 1 9 6 4 ) . 8. See L. W. H u n t e r and C. F c C u r t i s s , J . Chem. P h y s . 58 , 3897 ( 1 9 7 3 ) , and r e f e r e n c e s t h e r e i n . 9. L. w. H u n t e r and R. F. S n i d e r , J . Chem. P h y s . 61_, 1151 ( 1 9 7 4 ) . 10. R. G. Gordon e t a l . , Ann. Rev. P h y s . Chem. 1_9, 215 (1968) . 11. D. E. F i t z and R. A. M a r c u s , J . Chem. P h y s . 5_9, 4380 ( 1 9 7 3 ) . 12. H. K a b i t z , Ann. Rev. P h y s . Chem. 2 5, 155 ( 1 9 7 4 ) . 13. P. W . A n d e r s o n , P h y s . Rev. 7_6» 6 4 7 ( 1 9 4 9 ) . 14. C. J . T s a o and B. C u r n u t t e , J . Quant. S p e c t r y . Rad. T r a n s f e r _2, 41 (1962) .. 15. J . F i u t a k and J . Van Kranendonk, Can. J . P h y s . 40_t 1085 (1962) , 4_1, 21 (1963) . 16. H. M. P i c k e t t , J . Chem. P h y s . 6^ L, 1923 ( 1 9 7 4 ) . 17. W . L i u and R. A.' M a r c u s , J . Chem. P h y s . 63_, 272 ( 1 9 7 5 ) . 18. W. L i u and R. A. M a r c u s , J . Chem. P h y s . _63, 290 ( 1 9 7 5 ) . 19. R. E. T u r n e r and R. F. S n i d e r , to be p u b l i s h e d . 20. R. F. S n i d e r and R. E. T u r n e r , t o be p u b l i s h e d . 21. U. Fano, P h y s . Rev. 131, 259 ( 1 9 6 3 ) . 22. R. F. S n i d e r , J . Chem. P h y s . 63^, 3256 ( 1 9 7 5 ) . C h a p t e r V 1. A. C. L e v i , F. R. M c C o u r t , and A. T i p , P h y s i c a 39_, 165 (1968) . 2. A. R. Edmonds, A n g u l a r Momentum i n Quantum M e c h a n i c s , ( P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , 1 9 7 4 ) . 3. W. L i u and R. A. M a r c u s , J . Chem. P h y s . _63_, 27 3 ( 1 9 7 5 ) . 4. J . C. McGurk e t a l . , Adv. Chem. P h y s . X X V I I , 1 ( 1 9 7 5 ) . 5. A. T i p and F. R. M c C o u r t , P h y s i c a 5_2, 109 (1971) . 6. C. H. Townes and A. L . Schawlow, M i c r o w a v e S p e c t r o s c o p y , ( M c G r a w - H i l l , New Y o r k , 1 9 5 5 ) . 7. A. Ben-Reuvan, P h y s . Rev. 145, 7 ( 1 9 6 6 ) . 8. R. F. S n i d e r , p r i v a t e c o m m u n i c a t i o n . 9. D. P o l d e r , P h y s i c a 9, 908 ( 1 9 4 2 ) . C h a p t e r VI 1. W. L i u and R. A. M a r c u s , J . Chem. P h y s . 6_3 , 290 ( 1 9 7 5 ) . 2. H. C. T o r r e y , P h y s . Rev. 7_6, 1059 ( 1 9 4 9 ) . 3. J . C. McGurk e t a l . . Adv. Chem. P h y s . XXVII, 1 ( 1 9 7 5 ) . 4. S. H e s s , Z. N a t u r f o r s c h . 24a, 1675 ( 1 9 6 9 ) . 5. A. M. Ronn and D. R. L i d e , J . Chem. P h y s . 4_7 , 3669 ( 1 9 6 7 ) . C h a p t e r V I I 1. R. H. D i c k e , P h y s . Rev. 8_9, 472 ( 1 9 5 3 ) . 2. L. G a l a t r y , P h y s . Rev. 122, 1218 ( 1 9 6 1 ) . 3. M. N e l k i n and A. G h a t a k , P h y s . Rev. 13 5 A, 4 ( 1 9 6 4 ) . 4. S. G. R a u t i a n and J . I . Sobelmann, S o v i e t P h y s i c s -U s p e k h i 9, 701 (1967) . 325 5. J . I . G e r s t e n and H. M. F o l e y , J . O p t . S o c . Amer. 58, 933 (1968) . . 6. E. W. S m i t h e t a l . , J . Quant. S p e c t r o s c . Radia.t. T r a n s f e r 11, 1547 ( 1 9 7 1 ) . 7. E..W. S m i t h e t a l . , J . Quant. S p e c t r o s c . R a d i a t . T r a n s f e r _11, 1567 (1971) . 8. J . Ward e t a l . , J . Qu a n t . S p e c t r o s c . R a d i a t . T r a n s f e r 14 , 555 (1974) . 9. see P. R. Berman, P h y s . Rev. A5, 927 ( 1 9 7 2 ) , and r e f e r e n c e s t h e r e i n . 10. P. R. Berman, J . Q u a n t . S p e c t r o s c . R a d i a t . T r a n s f e r 12, 1331 (1972) . 11. P. R. Berman, A p p l i e d P h y s i c s 1_, 283 ( 1 9 7 5 ) . 12. J . S c h m i d t e t a l . , P h y s . Rev. L e t t . 3_1' 1103 (1973 ) . 13. P. R. Berman, P h y s . Rev. A l l , 1668 ( 1 9 7 5 ) . 14. S. He s s , Z. N a t u r f o r s c h . 25a, 350 ( 1 9 7 0 ) . 15. S. He s s , P h y s i c a &1_, 80 ( 1 9 7 2 ) . 16. R. F. S n i d e r and B. C. S a n c t u a r y , J . Chem. P h y s . 55, 1555 (1971) . 17. F. M. Chen, H. M o r a a l , and R. F. S n i d e r , J . Chem. P h y s . _51, 542 (1972) . 18. L. W. H u n t e r and R. F. S n i d e r , J . Chem. P h y s . 6_1, 1160 (1974) . 19. D. A. Coombe e t a l . , J . Chem. P h y s , £ 3 , 3015 ( 1 9 7 5 ) . 20. S. Chapman and T. G. C o w l i n g , The M a t h e m a t i c a l T h e o r y  o f N o n u n i f o r m G a s e s , (Cambridge U n i v e r s i t y P r e s s , C a m b r i d g e , 1 9 7 0 ) . 21. R. F. S n i d e r , P h y s i c a 78_, 387 ( 1 9 7 4 ) . 22. P. L . B h a t n a g e r e t a l . , P h y s . Rev. 94_, 511 ( 1 9 5 4 ) . 23. E. P. G r o s s and E . A. J a c k s o n , P h y s . F l u i d s 2_, 432 (1959) . 24. L. S i r o v i t c h , P h y s . F l u i d s 5_, 908 ( 1 9 6 2 ) . 3 26 25. F. B. Hanson and T. F. Morse, Phys.. F l u i d s 10, 345 (19 6 7 ) . 26. C. D. 3 o l e y et a l , Can. J . Phys. 5_0, 2158 ( 1 9 7 2 ) . 27. G, T o n t i and R„ C. Desai, to be p u b l i s h e d . 28. V. D. Berman et a l . , JETP L e t t e r s 5_, 85 (1967 ) . Append ix A •1. R. F. Snider and B. C. Sanctuary, J . Chem. Phys. 5_5, 1555 (197.1) -2. J . M. Jauch, B. M i s r a , and A. G. Gibson, Ilelva. Phys. Acta 4JL, 513 (1968 ) . Appendix B 1. F. M. Chen, H. Moraal, and R. F. Sn i d e r , J . Chem. Phys. 51 , 54 2 (197 2) . 3 27 APPENDIX A The G e n e r a l i z e d Boltzmann Equation of Snider and Sanctuary The s t a r t i n g p o i n t i s the N-molecule L i o u v i l l e equation where the H a r a i l t o n i a n ^ f " 1 " i s assumed to be a sum of one-and two-molecule terms (A.2) 2/ ( N ) = + Z V. . 1 l i<3 13 (N) and the N-molecule d e n s i t y p i s normalized a; (N) (A.3) t r p 1 ' = N ! n Equations f o r the reduced d e n s i t y o p e r a t o r s (A.4) p ( N ) t r ( N _ n ) p ( N ) / ( N - n ) 1 can be d e r i v e d from e q u a t i o n (A.l) by taking the a p p r o p r i a t e t r a c e s . In p a r t i c u l a r , the f i r s t two equations i n t h i s quantum BBGKY h i e r a r c h y are 9 o (1) ( A . 5 ) = A 1 P { - J + ^ 2 J / 1 2 P 1 2 } and 3 p ( 2 ) 328 where (A.7) ^ A = V A - AV. and 12 ' '12 ' " 1 2 T h i s s e t o f e q u a t i o n s i s n o t c l o s e d and some a p p r o x i m a t i o n scheme ( u s u a l l y by t r u n c a t i o n ) must be d e v e l o p e d to c l o s e i t . The t r u n c a t i o n scheme d e s c r i b e d h e r e i s w i t h i n the " p h i l o s o p h y " o f t h e B o l t z m a n n equation." 1" Namely, ( i ) o n l y b i n a r y c o l l i s i o n s a r e i m p o r t a n t and t h e s e a r e assumed to o c c u r on a time s c a l e s h o r t compared to the t i m e between c o l l i s i o n s T^; and ( i i ) t h e p a i r d e n s i t y o p e r a t o r f a c t o r s b e f o r e a c o l l i s i o n . Thus, a s s u m p t i o n ( i ) a l l o w s the term ( 3 ) i n v o l v i n g t o ^ e n e 9 " l e c t e < ^ x n e q u a t i o n ( A . 6 ) , w h i l e a s s u m p t i o n ( i i ) i m p l i e s t h a t t h e p a i r d e n s i t y o p e r a t o r s a t i s f i e s the a s y m p t o t i c c o n d i t i o n . (A.9) t r - l i m E p f ^ U ) - p { 1 * < t ) p * 1 } ( t o ) 3 •+ 0, O 329 Here " c o" i s a time l o n g compared to the d u r a t i o n o f a c o l l i s i o n b u t s h o r t compared to the time between c o l l i s i o n s . (3 ) E q u a t i o n ( A . 6 ) , w i t h the P 1 2 3 t e r m n e g l e c t e d , and e q u a t i o n (A.O), l e a d s to the d e f i n i t i o n o f the H o l l e r s u p e r o p e r a t o r as ( A . i o ) p ( 2 ) ( t ) = n / p ( 2 ) ( t ) p ( 1 ) (t) = t r - l i m e - ^ ^ ' ^ o e 1 * * ^ P ( 2 > (t) p ( 1 > ( t) t o F u r t h e r , i f the M o l l e r wave o p e r a t e r UQ e x i s t s , i t has been 2 p r o v e n t h a t (A. 11) = S u b s t i t u t i o n o f e q u a t i o n (A-.10) i n t o e q u a t i o n ( A . 5 ) , 1 r e s u l t s i n the g e n e r a l i z e d quantum B o l t z m a n e q u a t i o n (A.12) i*^£ P j L = / 1 P 1 + t r 2 ^ 1 2 P i P 2 where t h e s u p e r s c r i p t n o t a t i o n has been d r o p p e d and where t h e t r a n s i t i o n s u p e r o p e r a t e r i s d e f i n e d as (A. 13) = 2^ E q u a t i o n (A.12) i s the s u p e r o p e r a t o r a n a l o g o f the t r a n s i t i o n 330 o p e r a t o r (A.14) t = Vft t h a t a p p e a r s i n t h e u s u a l f o r m a l t h e o r y o f s c a t t e r i n g . I n d e e d , s u b s t i t u t i o n o f (A.11) and (A.14) i n t o (A.13) g i v e s the i d e n t i f i c a t i o n (A.15) ^/A = thtt+ - fiAt+ A form f o r a l t e r n a t e t o e q u a t i o n (A.15) i s o b t a i n e d by e m p l o y i n g t h e Lippmann-Schwanger i n t e g r a l e q u a t i o n f o r Most p r e c i s e l y , t h i s i n t e g r a l e q u a t i o n i s w r i t t e n as (A. 16) fi = 1 + G [VwQ] = 1 + G [ t] where G i s a " G r e e n ' s f u n c t i o n s u p e r o p e r a t o r " (A.17) G [ t ] = l i m (-?<{+ i e ) " 1 [ t ] £+0+ E q u a t i o n s (A.15) and (A.16) c a n t h e n be combined t o g i v e (A.18) A = tA - A t + - t A G [ t + ) - G [ t ] A t + The e x p r e s s i o n s f o r J*, e q u a t i o n s (A.15) o r ( A . 1 8 ) , g e n e r a l l y i n v o l v e " o f f - t h e - e n e r g y - s h e l l " t r a n s i t i o n o p e r a t o r s t . I n many a p p l i c a t i o n s , however, i t i s c o n v e n t i o n a l to a ssume t h a t o n l y " o n - t h e - e n e r g y - s h e l 1 " t r a n s i t i o n o p e r a t o r s t a r e i m p o r t a n t . In t h i s c a s e , e q u a t i o n (A.17) i s a p p r o x i -mated as (A.19) $A = J H A t,A - A t * + 2 TTi t , 6 ( / f ) t + d d d ' Here,J i s a " f r e q u e n c y c o n s e r v i n g " t r a n s i t i o n s u p e r o p e r a t o r In a d d i t i o n to t h i s p r o p e r t y o f " f r e q u e n c y c o n s e r v a t i o n , j f H p o s s e s s e s an i m p o r t a n t symmetry p r o p e r t y under tim e r e v e r s a l - namely (A.20, BXfr1 -X where r e p r e s e n t s t h e s u p e r o p e r a t o r a d j o i n t and 0 i s the t i m e - r e v e r s a l s u p e r o p e r a t o r (A.21) &>A = 6 A 8 - 1 , h e r e e x p r e s s e d i n terms o f the ( a n t i l i n e a r ) time r e v e r s a l o p e r a t o r 0. T h i s p r o p e r t y o f ^  u n d e r tim e r e v e r s a l f o l l o w s d i r e c t l y f r o m the b e h a v i o u r o f t ^ u n d e r tim e r e v e r s a l . I n d e e d , w i t h ( A . 1 4 ) , ( A . 1 6 ) , and t , r e s t r i c t e d t o d the e n e r g y E, t ^ c a n be c o n v e n i e n t l y e x p r e s s e d a s 332 (A.22) t , ( E ) = V + s t - l i m V ( E - P^ 2 ) + i E ) " 1 V c-»-o+ so t h a t u n d e r time r e v e r s a l , (A.23) @t (E) = 0 t d (E) 8 -1 V + s t - l i m V [ E _ ^ / ( 2 ) _ . e ] - l v t > ) . T h i s f o l l o w s f r o m t h e h e r m i t i a n p r o p e r t i e s o f V a n d ^ A c o m b i n a t i o n o f e q u a t i o n s (A.23) and (A.19) the n e s t a b l i s h e s e q u a t i o n ( A . 2 0 ) . APPENDIX B I r r e d u c i b l e Tensors of. S 0 ( 3 ) A s p h e r i c a l b a s i s f o r the 2q-f-l dimensional space of (q)V 1 m u symmetric t r a c e l e s s rank q t e n s o r s i s e . These are chosen to s a t i s f y the group o p e r a t i o n s (B.l) e ( ^ ) V - V e ( q ) V Qj[^ e ( q ) V - [ q ( q + l ) - ; V ( V ± 1 ) ] 1 / 2 e ( * ^ are the i n f i n i t e s i m a l r o t a t i o n o p e r a t o r s of the group act-ing on rank q t e n s o r s . E x p l i c i t l y , ^ i s g i v e n by (B.2) 0 ( q ) = - i ^ n 1 ® --10 z-Z®l ®1, ( n t h p o s i t i o n ) <y 2 n=l g where e i s the t h i r d rank completely antisymmetric, tensor, = . ( q ) v To complete the s p e c i f i c a t i o n of the b a s i s e , i t i s ( CT ) O necessary o n l y to s p e c i f y e as ( B . 3 ) e ( q ) o = ^[_A23lAAn- [ h 2q(cLi)Z Then the remaining e^V, V/0 are obtained from (B.3) by employing the raising or lowering operatorsJ)|q' . Thus, for q=l, the spherical basis tensors are 334 ( B . 4 ) e ( 1 ) ° - - ^ ( x + i y ) (1)0 e = i z ( 1) -1 i , ~ . ^  , e = (x - i y ) The a s s o c i a t e d c o v a r i a n t b a s i s e ^ q * i s d e f i n e d so t h a t (B.5) .««> - (-1) e l ' ) - V - <e«*>V The n a t u r a l p r o j e c t i o n o p e r a t o r E * q ^ i s t h e n r e p r e s e n t e d as ( B . 6 ) E ( q ) = E e ( q ) V e < q ) - E ( - l ) q _ V e ( q ) V e ( q ) _ V V w i t h t h i s d e f i n i t i o n o f the s p h e r i c a l b a s i s t e n s o r s , the s p h e r i c a l components o f any symmetric t r a c e l e s s t e n s o r c a n be e a s i l y o b t a i n e d . I n p a r t i c u l a r , the t e n s o r [ j ^ v ^ X j ^ v ^ ] ^ q^ has been c o n s i d e r e d i n t h e main body o f the t h e s i s . The s p h e r i c a l components o f t h i s t e n s o r a r e ( B . 7 ) [ J I V i > < W , ' , V E e ( " V ( . ) q l ) i v i > < j f v f ] , q l l / o 3 i ~ m • 1 f = i q Z ( 2 q + l ) 1 / 2 ( - l ) 1 ^ v m l l J i V i X J f - f ' f m ;m i f 1 f The i n v e r s e o f e q u a t i o n (U.7) i s n -m (D.8) | j .m .v .><j-m_v.| = I ( - 1 ) q ( 2 q + l ) ( - 1 ) 1 1  I J i i i J f f f ' q, v , j i q 3 f >. . ^ . , (q) v ( H 3 i v i > < 3 f v f ] -m . V m _ r f w h i c h shows t h a t the b a s i s [ j .v . > < : j ^ v ^ ] ^ q^ V i s c o m p l e t e . .i a f f F u r t h e r m o r e , t h e s e o p e r a t o r s p o s s e s s the a d j o i n t p r o p e r t y (B.9) ( [ j i v i > < j f v f ] ( q > ) + = (-1)3"1 3 f [ j f v f > < j ± v .] ( q ) and a r e o r t h o g o n a l i n t h e s e n s e t h a t (B.10) t r ( ( [ j i v i > < j f v f ] ( q ) V) T [ j ^ v \ _ > < j f v ^ ( q , ) V < } D -1 • " l ^ D i v.v*. v . v ' qq' V V J i J f ^ J i - i - f J f i i f f ^ -I n p a r t i c u l a r , when r e s t r i c t e d t o b e i n g " d i a g o n a l i n j " t h e o p e r a t o r i d e n t i f i c a t i o n /q) #TI/T»3 ( B . i D [ j v i > < j f ] ( q ) - J r — ~ \ / 2 1 f 1 4 7 ( 2 j + l ) ' i s u s e f u l . F i n a l l y , the r e d u c e d m a t r i x e l e m e n t (B.12) < 3 i v i j | [ j i v .><j fv £) ( q ) !|j fv f>- i q (2q + l ) 1 / 2 i s e s t a b l i s h e d from e q u a t i o n (B. 7) ,. ( a ) V Also using the s p h e r i c a l b a s i s tensors e ' , i t i s seen t h a t the components of the 3 - j tensors V(qq q^) are e x a c t l y the 3 j symbols ( B i , r f q q ' q \ - ( q 2 ) V 2 ( g i ) V l « > v < , q + q i + q 2 ( B . 13) ( ) e e e ( • ) v vx v 2 V ( q q ] q 2 ) ( q) V Indeed, the c h o i c e of phase f o r the e , as i n d i c a t e d by equation ( B . 3 ) , i s made p r e c i s e l y so that the component of the 3 - j tensors are r e a l . For other purposes, i t i s sometimes convenient to choose b a s i s tensors which are r e a l . Tha t i s , i n s t e a d of ( B „ 3 ) , the b a s i s tensors (B.14) £ «-i e ( 1 ) ° , ( 2 ) ,2,1/2 (2)0 [ z] == «• {--) e etc coul d be a l t e r n a t e l y employed. 

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