UBC Theses and Dissertations

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UBC Theses and Dissertations

Some aspects of nuclear spin relaxation for dilute polyatomic gases Sanctuary, Bryan Clifford 1971

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U 2 M SOME ASPECTS OP NUCLEAR SPIN RELAXATION FOR DILUTE POLYATOMIC GASES by BRYAN CLIFFORD SANCTUARY B. Sc. (Hons.), University of B r i t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of CHEMISTRY We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1971 i v In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l l m e n t of the requirements f o r an advanced degree at the U n i v e r i s t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y 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 study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or 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 allowed without my w r i t t e n p e r m i s s i o n . Department of CHS* > STrtY  The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada Date hJov-t^Jiox, I > /17 / •  i i A B S T R A C T T h e d e n s i t y d e p e n d e n c e o f n u c l e a r s p i n r e l a x a t i o n i n p o l y a t o m i c g a s e s i s s t u d i e d t h e o r e t i c a l l y . I n p a r t i c u l a r , t h e e x p e r i m e n t a l l y o b s e r v e d n o n - l i n e a r d e p e n d e n c e o f o n d e n s i t y a t h i g h d e n s i t y i s e x p l a i n e d . A q u a n t u m 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 i s u s e d t o f o r m u l a t e t h e t h e o r y . T h e p a r t i c u l a r B o l t z m a n n e q u a t i o n u s e d d e s c r i b e s t h e t i m e e v o l u t i o n o f 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 n d t h i s a f f o r d s a n a d e q u a t e d e s c r i p t i o n o f a d i l u t e g a s . N o r e s t r i c t i o n o n t h e i n t e r n a l s t a t e s o f t h e p o l y a t o m i c m o l e c u l e s i s r e q u i r e d . A n e x p r e s s i o n f o r T ^ i s o b t a i n e d i n t e r m s o f a r e -l a x a t i o n m a t r i x , t h e e l e m e n t s o f w h i c h a r e c o l l i s i o n I n -t e g r a l s b e t w e e n t h e v a r i o u s i n t r a m o l e c u l a r t r a n s i t i o n p r o -c e s s e s w h i c h c o n t r i b u t e t o t h e r e l a x a t i o n r a t e . T h e d i a g o n a l t e r m s o f t h i s m a t r i x d e s c r i b e t h e r e l a x a t i o n r a t e a n d f r e -q u e n c y s h i f t s f o r e a c h i n t e r n a l s t a t e t r a n s i t i o n f r e q u e n c y w h i l e t h e o f f - d i a g o n a l t e r m s a c c o u n t f o r t h e c o l l i s i o n a l o v e r l a p p i n g b e t w e e n t h e v a r i o u s i n t e r n a l s t a t e f r e q u e n c i e s . T h e c o l l i s i o n i n t e g r a l s a r e p a r t i a l l y e v a l u a t e d b y a d i s -t o r t e d w a v e B o r n a p p r o x i m a t i o n a n d a r e r e d u c e d t o t r a c e s o v e r i n t e r n a l s t a t e s a n d i n t e g r a l s o v e r t h e r e l a t i v e v e l o c i t y i i i of the c o l l i d i n g p a i r . The i n t e r n a l s t a t e t r a c e s are eva lua ted e x a c t l y w h i l e the r e l a t i v e v e l o c i t y i n t e g r a l s are l e f t as s c a l i n g parameters . The t h e o r e t i c a l e x p r e s s i o n i s a p p l i e d to the sym-me t r i c top molecules CH^F and CF^H to account f o r the e x p e r i m e n t a l l y observed "step e f f e c t " . On the b a s i s of exper imen ta l evidence an i n t r a m o l e c u l a r d i p o l a r mechanism i s used and i t i s shown that t h i s g ives adequate agree-ment w i t h the exper imen ta l d a t a . The steps a r i s e as a r e s u l t o f the phase r andomiza t ion of the low l y i n g r o t a -t i o n a l K l e v e l s . V TABLE OP CONTENTS PAGE A b s t r a c t i i L i s t o f Tables v i i L i s t of F i g u r e s i x Acknowledgement x i INTRODUCTION 1 PART I SURVEY OF THE KINETIC THEORY OF GASES BASED ON THE BOLTZMANN EQUATION.' 4 Chapter 1 I n t r o d u c t i o n 5 Chapter 2 A B a s i s of T r e a t i n g D i l u t e Gases. 11 Chapter 3 A G e n e r a l i z e d Boltzmann E q u a t i o n f o r Molecules w i t h I n t e r n a l S t a t e s 18 3.1 D e r i v a t i o n 1 8 3.2 Formal S c a t t e r i n g Theory.... 24 3 - 3 E x p l i c i t Forms of the C o l l i s i o n Superoperator and L i n e a r i z a t i o n 35 Chapter 4 P r o p e r t i e s of }^ 4 l PART I I NUCLEAR SPIN RELAXATION OF GASES.. 56 Chapter 1 I n t r o d u c t i o n 57 Chapter 2 Boltzmann E q u a t i o n Approach t o Nuclear Spin R e l a x a t i o n 64 Chapter 3 C o r r e l a t i o n F u n c t i o n Approach.... 8 l Chapter 4 R e l a t i o n Between C o r r e l a t i o n and Boltzmann Approach 90 v i PART I I I PARTIAL EVALUATION OF THE x'S 106 Chapter 1 I n t r o d u c t i o n 1 0 7 Chapter 2 The D i s t o r t e d Wave Born Approximation (DWBA) 1 1 0 Chapter 3 The I n t e r m o l e c u l a r P o t e n t i a l 1 1 3 Chapter 4 E x p r e s s i o n f o r the x's 1 1 6 4 . 1 I n t e g r a t i o n Over the Center of Mass 1 2 2 4 . 2 S e p a r a t i o n of V e l o c i t y Terms from I n t e r n a l S t a t e Traces f o r B 1 124 4 . 3 S e p a r a t i o n of V e l o c i t y Terms from I n t e r n a l S t a t e Traces f o r B 2 1 2 7 4 . 4 General E x p r e s s i o n f o r T j ^ C i f i ' f ' ) 1 2 9 Chapter 5 D i s c u s s i o n of R e s u l t s 1 3 1 PART IV APPLICATION TO SYMMETRIC TOPS 1 3 6 Chapter 1 I n t r o d u c t i o n 1 3 7 Chapter 2 Symmetry P r o p e r t i e s o f Symmetric Top M o l e c u l e s , CX3Y 146 2 . 1 The R o t a t i o n a l F u n c t i o n s 1 5 1 2 . 2 S p i n F u n c t i o n s 1 5 4 2 . 3 T o t a l Symmetry 1 5 8 Chapter 3 D e s c r i p t i v e Example of High Frequency - E f f e c t s 1 6 6 Chapter 4 The E f f e c t of High F r e q u e n c i e s On the Nuclear S p i n R e l a x a t i o n of C F 3 H 1 7 9 D i p o l e - d i p o l e I n t r a m o l e c u l a r C o u p l i n g H a m i l t o n i a n f o r CF 3H Molecules 1 8 6 4 . 2 E v a l u a t i o n of the d_(2q) V e c t o r s f o r AK^O and AJ=0 1 9 4 4 . 3 Approximation of the R e l a x a t i o n M a t r i x 2 0 5 Chapter 5 F i t t i n g of Experiments to C F 3 H at 2 9 7°K 2 2 1 5 . 1 The E f f e c t o f V a r y i n g y 2 2 1 5 . 2 The E f f e c t of V a r y i n g 6 '. 224 5 . 3 The E f f e c t o f V a r y i n g c 2 3 0 5 . 4 F i t to E x p e r i m e n t a l Data 2 3 3 v i i DISCUSSION ,. 241 BIBLIOGRAPHY 247 APPENDIX A 254 APPENDIX B 266 APPENDIX C 273 APPENDIX D 281 LIST OF TABLES TABLE PAGE V a l i d i t y C o n d i t i o n s of Quantum Mechanical Boltzmann Equations. 5 5 C h a r a c t e r Tables f o r and C_ and C 3 M u l t i p l i c a t i o n ^Table . J v 148 3 S p i n F u n c t i o n s ( o f f - a x i s s p i n s o n l y ) . 1 5 5 4 M o l e c u l a r Constants f o r CFoH and CH 3F. 0 1 7 3 5 Values of Parameters f o r CHoF From Leas t Squares F i t . 1 7 8 6 Proton D i p o l a r C o u p l i n g Tensor C o e f f i c i e n t s f o r CF 3H . 1 9 3 7 Values of T(KK') f o r CF H at 2 9 7°K. • 5 204 8 Values of A l K K ' K»V- 2 1 5 9 Values of uvv.T- f o r CF_H. 2 1 9 KK d j i x LIST OF FIGURES FIGURE PAGE Different Ways of Graphing the Density Dependence of Spectral Lines. 6 l 2 Example of T ] L Minimum. 138 3 P l o t of T - , / p versus p f o r CHF~. The Value of T / p f o r the F l u o r i d e Spins? i s 3.5 msec/Amagat. 140 4 P l o t of T 1 / p versus p f o r CH"3F. l 4 l 5 Comparison of Proton R e l a x a t i o n Data f o r CF3H. 143 6 M o l e c u l a r Geometry of a Symmetric Top Molecule CXgY. 147 7 Permutation of N u c l e i 2 and 3 of CX^Y Mo l e c u l e . 160 8 T h e o r e t i c a l F i t to R. Dong's 1 Proton T j R e l a x a t i o n Data. , 174 9 P l o t o f G(k) versus D e n s i t y f o r k Equal Zero to Seven. 175 10 Overhauser E f f e c t Measurements f o r CF3H at Room Temperature. 180 11 K T r a n s i t i o n s Allowed i n Each Submatrix o f the R e l a x a t i o n M a t r i x 213 12 The R e l a x a t i o n M a t r i x gA for. Small Values of P o s i t i v e and Negative k . 220 13 T-j/n vs n f o r CP H at 297°K. The E f f e c t of V a r y i n g y w i t h 6=0 and e=0. 223 14 T-j/n vs n f o r CF^H at 297°K. The E f f e c t of V a r y i n g 6 with y=.02 and £=0. The Arrows I n d i c a t e K T r a n s i t i o n s . 226 LIST OF FIGURES (cont inued) FIGURE 15 T-j/n vs n f o r CFoH at 297°K. The E f f e c t of v a r y i n g y w i t h 6 = . 4 and £=0. The Arrows I n d i c a t e K T r a n s i t i o n s . 16 3(coo i,) vs n f o r CFoH at 297°K. The E f f e c t of ' V a r y i n g 6 w i t h y=.02 and C=0. 17 G(co3 4) vs n f o r C F o H at 297°K. The E f f e c t of ' V a r y i n g y w i t h 6 = . 4 and C=0. 18 T ] / n vs n fo r CF3H at 297°K. The E f f e c t o f V a r y i n g r, w i t h Y = - ° 5 and 6 = . 4 . 19 Dens i ty Dependence o f D iagona l E l e m e n t(3, 4 ) of the R e l a x a t i o n M a t r i x . The E f f e c t o f V a r y i n g £ w i t h y=.05 and S = . 4 . 20 T h e o r e t i c a l T x / n vs n f o r C F ^ H at 297°K w i t h Y=.10, 6=.50 and 5 = - .Ho. 21 T h e o r e t i c a l P l o t o f T ] / n vs n f o r Y=.10, . 4 0 and <5=.50 w i t h the Expe r imen ta l Da ta . 22 Dens i ty Dependence of the D iagona l Elements of the R e l a x a t i o n M a t r i x w i t h y=.10, £ = - . 4 0 and 5=.50. 23 I n t e r m o l e c u l a r Sepa ra t i on ACKNOWLEDGEMENT My most s i n c e r e thanks and a p p r e c i a t i o n go to P r o -f e s so r R. P . Sn ide r who supe rv i sed my graduate s t u d i e s . By h i s i n s t r u c t i o n and h e l p , wi thout which t h i s t h e s i s would not have been p o s s i b l e , he has r e v e a l e d to me not only the beauty and s i m p l i c i t y of s c i e n c e , but a l s o the cha rac t e r of a great man. I t i s w i t h p l easu re tha t I acknowledge h i s s u p e r v i s i o n . I am a l s o indebted to P ro fes so r s J . A . R. Coope and M. Bloom f o r t h e i r i n t e r e s t and h e l p f u l i n s i g h t i n t o many aspects o f t h i s p r o j e c t . My w i f e , Mingy , deserves s p e c i a l acknowledgement f o r her encouragement, pa t i ence and unders tanding o f a non-academic na tu re . INTRODUCTION The m o t i v a t i o n f o r the work pre s e n t e d i n t h i s t h e s i s i s to e x p l a i n the s o - c a l l e d " s t e p - e f f e c t " which has been ob-served f o r p r o t o n n u c l e a r s p i n r e l a x a t i o n of CH^F and CF^H. 1 2 Since i t was f i r s t r e p o r t e d ' , doubt has been cast on the e x p e r i m e n t a l r e s u l t s , and attempts to reproduce the data have so f a r not always been s u c c e s s f u l - " 5 . There are c o n s i d e r a b l e e x p e r i m e n t a l d i f f i c u l t i e s to overcome i n p e r f o r m i n g n u c l e a r s p i n r e l a x a t i o n experiments i n gases, such as weak s i g n a l s and long r e l a x a t i o n times as w e l l as the e l i m i n a t i o n of the e f f e c t s of i m p u r i t i e s and w a l l s . The author understands t h a t these shortcomings w i l l soon be overcome, and t h a t the e x p e r i m e n t a l r e s u l t s w i l l be agreed upon, although the e x i s t -i n g data i s shown i n the course of t h i s t h e s i s not to be unreasonable. Prom a t h e o r e t i c a l p o i n t of view, i t i s necessary to r e -t u r n to the fundamentals of s t a t i s t i c a l mechanics, and to b u i l d a theory which takes i n t o account a l l the i n t e r n a l s t a t e s of gaseous polyatomic m o l e c u l e s , and t r e a t them i n 'a c o n s i s t e n t manner. Up to now, only a r e s t r i c t e d treatment of i n t e r n a l s t a t e s has been p r o p e r l y f o r m u l a t e d ^ ' . The n e c e s s i t y of t a k i n g i n t o account , a l l aspects of the i n t e r n a l s t a t e s to e x p l a i n the 2 "s teps" i s a good excuse to extend the t h e o r y . A n a t u r a l way of t r e a t i n g gases i s by some Boltzmann equa t ion and i t i s from t h i s approach tha t the problem i s fo rmula t ed . S ince the g e n e r a l i z e d Boltzmann equa t ion used here^ must be v a l i d to f a i r l y h igh d e n s i t i e s , u s u a l l y con-s i d e r e d to be ou t s ide the d i l u t e gas reg ime, care i s g i v e n to e x p l a i n i n g the range of i t s a p p l i c a b i l i t y . Al though at these h i g h d e n s i t i e s the range o f v a l i d i t y i s s t r e t c h e d some-what, i t i s b e l i e v e d tha t the concepts have a f a r g r ea t e r a p p l i c a b i l i t y than to gases on ly and can be used w i t h suc -cess i n deve lop ing t h e o r i e s which i n v o l v e molecu la r i n t e r -n a l s t a t e s i n g e n e r a l . Of p a r t i c u l a r i n t e r e s t and use i n the e x t e n s i o n o f the Boltzmann equa t ion to i n c l u d e non-degenerate i n t e r n a l s t a t e s i s tha t i t can be a p p l i e d t o pressure broadening problems of spec t roscopy . The c o l l i s i o n term takes i n t o account the e f -f e c t s of p ressure broadening and through a n a l y s i s of the c o l -l i s i o n dynamics i t should be p o s s i b l e to g l ean important i n -fo rmat ion on the form of the i n t e r m o l e c u l a r p o t e n t i a l . H i t h e r t o , a p p l i c a t i o n to such phenomena was beyond the scope of the Boltzmann e q u a t i o n . In Pa r t I some b a s i c ideas of po lya tomic molecules are expounded and the g e n e r a l i z e d Boltzmann equa t ion i s b r i e f l y r ev i ewed . F o l l o w i n g t h i s , i n Par t I I , the nuc l ea r s p i n r e l a x -3 a t i o n i n gases i s t r e a t e d . The n o v e l t y of t h i s p a r t i s the n a t u r a l i n c l u s i o n of r o t a t i o n a l f r e q u e n c i e s which are u s u a l l y i g n o r e d i n oth e r treatments. Since the more common method of f o r m u l a t i n g n u c l e a r s p i n r e l a x a t i o n i s v i a time c o r r e l a -te t i o n theory , some e f f o r t i s made to show the e q u i v a l e n c e of the two approaches. In p a r t i c u l a r the c o l l i s i o n i n t e g r a l s , or the c o r r e l a t i o n times, must be the same and t h i s i s shown to be the case. In Part I I I these c o l l i s i o n i n t e g r a l s are i n v e s -t i g a t e d i n more d e t a i l . F i n a l l y i n Part IV, the concepts and t h e o r i e s of the f i r s t t h r e e p a r t s are a p p l i e d to symmetric top molecules and as s p e c i a l c a s e s , to the h i g h d e n s i t y de-pendence of the p r o t o n n u c l e a r s p i n r e l a x a t i o n o f CF3H and CHoF. PART I SURVEY OP THE KINETIC THEORY OP GASES BASED ON THE BOLTZMANN EQUATION 5 CHAPTER 1 INTRODUCTION A d i l u t e gas composed o f m o l e c u l e s w h i c h obey c l a s s i c a l mechanics i s assumed t o be d e s c r i b e d a t a g i v e n t i m e , t , by a phase space d i s t r i b u t i o n f u n c t i o n , f ( R , P, t ) . Here R and P_ a r e r e s p e c t i v e l y the p o s i t i o n and momentum. T h i s f u n c t i o n i s assumed t o s a t i s f y a n o n - l i n e a r i n t e g r o - d i f f e r e n t i a l equa-t i o n f i r s t d e r i v e d , on p h y s i c a l arguments o n l y , by B o l t z m a n n ^ , and w h i c h b e a r s h i s name. I f t h i s Boltzmann e q u a t i o n c o r -r e c t l y d e s c r i b e s the time e v o l u t 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 , t h e n t h i s w i l l be r e f l e c t e d by agreement between c a l c u -l a t e d and o b s e r v e d m a c r o s c o p i c p r o p e r t i e s o f t h e gas a t a l l t i m e , i n c l u d i n g i t s o v e r a l l approach t o e q u i l i b r i u m . S i n c e m i c r o s c o p i c m o l e c u l a r p r o c e s s e s a r e r e v e r s i b l e , t h e r e a r e m a t h e m a t i c a l problems i n o b t a i n i n g a decay t o e q u i l i b r i u m . T h i s i n v o l v e s making an a s s u m p t i o n about the d i r e c t i o n o f t i m e , and one such a s s u m p t i o n , c a l l e d m o l e c u l a r chaos 1*" 1, s t a t e s : b e f o r e a c o l l i s i o n , t he m o l e c u l e s a r e u n c o r r e l a t e d . The s o l u t i o n o f the Boltzmann e q u a t i o n , o r e q u i v a l e n t l y the c a l c u l a t i o n o f f , i s not s i m p l e and h a l f a c e n t u r y 6 p a s s e d b e f o r e Chapman and Enskog i n d e p e n d e n t l y p r e s e n t e d a method f o r o b t a i n i n g a p p r o x i m a t e s o l u t i o n s . T h e i r method assumes t h a t the 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 and t h e r e t u r n t o " l o c a l " e q u i l i b r i u m o c c u r s l i n e a r l y . The Chapman-Enskog method and o t h e r a p p r o x i m a t i o n s a r e d i s c u s s e d a t l e n g t h i n s e v e r a l b o o k s 1 0 . Other a s p e c t s o f t h i s B o l tzmann e q u a t i o n b e s i d e s c a l c u l a t i n g t h e d i s t r i b u t i o n f u n c t i o n a r e a l s o o f i n t e r e s t . I n p a r t i c u l a r , H. G r a d ^ has examined i t from a m a t h e m a t i c a l p o i n t o f view i n an attempt t o put k i n e t i c t h e o r y on a f i r m m a t h e m a t i c a l b a s i s . A l s o a g r e a t d e a l o f work i s c u r r e n t l y underway w i t h mock c o l l i s i o n terms 1** w h i c h , i t i s hoped, s i m p l i f y t h e mathematics n e c e s s a r y t o s o l v e t h e t r u e Boltzmann e q u a t i o n , w h i l e r e t a i n i n g t h e g e n e r a l p h y s i c a l f e a t u r e s . N o t a b l e among t h e s e i s t h e BGK 1^ model w h i c h has been used i n s o l v i n g boundary v a l u e and i n i t i a l v a l u e problems a n a l y t i c a l l y . The c l a s s i c a l Boltzmann e q u a t i o n , however, i g n o r e s the many phenomena which i n v o l v e the m o l e c u l e s ' i n t e r n a l s t a t e s . I n p a r t i c u l a r , by quantum m e c h a n i c s , t h e i n t e r n a l s t a t e energy s p e c t r u m i s d i s c r e t e . The f i r s t B oltzmann c o l l i s i o n t erm t h a t was i n t r o d u c e d t o account f o r t h i s was due t o Wang Chang and U h l e n b e c k 1 ^ . E s s e n t i a l l y , t h e i r o n l y change from the B oltzmann e q u a t i o n f o r monatomic gases i s t h e i n t r o d u c -t i o n ' o f a s e t o f c o l l i s i o n c r o s s - s e c t i o n s d e s c r i b i n g the p r o b a -b i l i s t i c changes i n i n t e r n a l s t a t e s and t o r e p l a c e t h e d i s -7 t r i b u t i o n f u n c t i o n f ( R , P, t ) by a s e t o f d i s t r i b u t i o n f u n c t i o n s f-^CR, P, t ) , one f o r each i n t e r n a l s t a t e . T h i s B o l tzmann e q u a t i o n i s s u f f i c i e n t f o r d e s c r i b i n g t r a n s p o r t p r o c e s s e s when c o l l i s i o n a l changes o f i n t e r n a l energy a r e i m p o r t a n t , but f a i l s t o have enough f l e x i b i l i t y t o acco u n t f o r phenomena w h i c h depend on the degeneracy o f t h e i n t e r n a l s t a t e s energy spectrum. To account f o r t h i s , a quantum Boltzmann e q u a t i o n was " d e r i v e d " i n d e p e n d e n t l y by Waldmann^ and S n i d e r ^ and has s u b s e q u e n t l y been used t o d e s c r i b e a v a r i e t y o f t r a n s p o r t and r e l a x a t i o n p r o c e s s e s . Even t h i s i s not s u f f i c i e n t l y g e n e r a l f o r t h e d e s c r i p t i o n o f c e r t a i n one p a r t i c l e phenomena. I n p a r t i c u l a r , t h e e q u a t i o n does not c o n s e r v e a n g u l a r momentum and t h e f a u l t l i e s i n t h e way the c o l l i s i o n o p e r a t o r has been l o c a l i z e d i n p o s i t i o n . Re-c e n t l y , i t has been shown-^ how a more g e n e r a l l o c a l i z a t i o n p r o c e d u r e can l e a d t o c o n s i s t e n c y w i t h a l l c o n s e r v a t i o n l a w s , namely o f mass, l i n e a r momentum, a n g u l a r momentum and energy. A n o t h e r s h o r t c o m i n g o f t h e Waldmann-Snider c o l l i s i o n term i s t h a t i t i s a p p l i c a b l e o n l y t o t h o s e ( s i n g l e t ) den-s i t y m a t r i c e s " d i a g o n a l " i n i n t e r n a l energy. T h i s r e s t r i c -t i o n has been s t r e t c h e d i n the a p p l i c a t i o n o f the Boltzmann 1 8 e q u a t i o n s i n c e t h e Zeeman s p l i t t i n g i s s t a n d a r d l y I g n o r e d i n t h e c o l l i s i o n o p e r a t o r but i t i s f u l l y a c c o u n t e d f o r d u r i n g the f r e e p a r t i c l e motions o f the m o l e c u l e s between c o l l i s i o n s . T h i s i s presumably a v e r y good a p p r o x i m a t i o n 8 f o r s m a l l magnetic f i e l d s , but not when i t i s n e c e s s a r y t o have d e n s i t y m a t r i c e s " n o n - d i a g o n a l " i n i n t e r n a l energy as i s needed t o account f o r h i g h f r e q u e n c y e f f e c t s i n n u c l e a r 19 s p i n r e l a x a t i o n and t o d e s c r i b e 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 ^ ? . S i n c e t h e s e phenomena m o t i v a t e t h i s t h e s i s , a Boltzmann e q u a t i o n g e n e r a l enough t o be used i n a l l t he cases mentioned above i s r e q u i r e d . Such an e q u a t i o n has r e c e n t l y been " d e r i v e d " ? and f o l l o w s v e r y c l o s e l y t h e o r i g i n a l d e r i v a t i o n by S n i d e r except t h a t o p e r a t o r t e c h n i q u e s a r e used e x c l u s i v e l y , and t h e a s s u m p t i o n (used a t l a t e r s t a g e s i n r e f e r e n c e 6) t h a t the d e n s i t y o p e r a t o r s be d i a -g o n a l i n i n t e r n a l energy i s not made. T h i s more g e n e r a l B oltzmann e q u a t i o n i s d i s c u s s e d i n some d e t a i l i n t h e f o l l o w -i n g s e c t i o n s . The s t a r t i n g p o i n t o f the d e r i v a t i o n i s based on t h e N - p a r t i c l e Quantum L i o u v i l l e o r von Neumann Equ a t i o n * 1 , Here i t i s assumed t h a t an N - p a r t i c l e system can be c o m p l e t e l y d e s c r i b e d by an N - p a r t i c l e d e n s i t y o p e r a t o r , j u s t as i n c l a s s i c a l mechanics i t i s assumed t h a t a d i s t r i b u t i o n f u n c -t i o n g i v e s a complete d e s c r i p t i o n o f a c l a s s i c a l N - p a r t i c l e (1) 9 system. The name Quantum L i o u v i l l e e q u a t i o n i s t h u s c l e a r , i t b e i n g t h e analogue o f t h e f a m i l i a r L i o u v i l l e e q u a t i o n f o r c l a s s i c a l systems. I f however th e system i s a d i l u t e gas i n which most o f t h e t i m e t h e m o l e c u l e s a r e f r e e , a g r e a t simp-l i f i c a t i o n i s e x p e c t e d . Then i t can be assumed t h a t a s u f f i -c i e n t d e s c r i p t i o n i s a f f o r d e d by the one p a r t i c l e ( s i n g l e t ) d e n s i t y m a t r i x , p , r a t h e r t h a n by p^ N^. F u r t h e r , f o r an i d e a l gas w h i c h i s one i n which the m o l e c u l e s e v o l v e i n d e -p e n d e n t l y o f the o t h e r s , the changes w i t h time r e s u l t o n l y from the one p a r t i c l e h a m i l t o n i a n // . Thus an " i d e a l " gas would obey the o n e - p a r t i c l e L i o u v i l l e e q u a t i o n f o r a l l t i m e s , I f t h e m o l e c u l e s i n t e r a c t w i t h o t h e r s , t h e n Eq.( 2 ) w i l l be v a l i d f o r o n l y t h o s e t i m e s d u r i n g which an e n c o u n t e r w i t h o t h e r m o l e c u l e s i s not t a k i n g p l a c e . The 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 s t h u s r e s p o n s i b l e f o r d e s t r o y i n g t h e v a l i d i t y o f Eq.( 2 ) , w h i c h can be thought o f as an e q u a t i o n v a l i d i n t h e l i m i t t h a t V+0. To q u a l i t a t i v e l y a ccount f o r t h e s e c o l l i s i o n s , a gas i s p i c t u r e d as h a v i n g m o l e c u l e s w h i c h are d e s c r i b e d by a one p a r t i c l e d e n s i t y o p e r a t o r whose time e v o l u t i o n i s i n p a r t Eq . C 2) w i t h an added c o l l i s i o n term t o account f o r the c o l l i s i o n s o f one m o l e c u l e w i t h a n o t h e r . F o r d e n s e r gases t h i s d e s c r i p t i o n o b v i o u s l y b r e a k s down and two p a r t i c l e , and h i g h e r , p^n\ d e n s i t y o p e r a t o r s a r e r e q u i r e d a l o n g w i t h a l l o w a n c e f o r m u l t i p l e c o l l i s i o n s . However, s i n c e o n l y d i l u t e gases are c o n s i d e r e d i n t h i s t h e s i s , t h e s e c o m p l i c a t i o n s a r e not i n c l u d e d . 11 CHAPTER 2 A BASIS OF TREATING DILUTE GASES I f T i s t h e average time between c o l l i s i o n s and T £ N { . i s t h e average time f o r t h e d u r a t i o n o f an i n t e r a c t i o n , t h e n i t i s e x p l i c i t l y assumed t h a t ^ C 3 ) T_ > > T. . f i n t . T h i s i s e s s e n t i a l l y a statement w h i c h i m p l i e s t h a t 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 i s o f s h o r t range and 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 . C o l l i s i o n s a r e t h e r e f o r e " c o m p l e t e " i n t h e sense t h a t b e f o r e t h e next i n t e r a c t i o n , a f t e r an a v e r -age time o f x 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 o f t h e p r e v i o u s c o l l i s i o n i s n e g l i g i b l e and can be i g n o r e d . The s h o r t n e s s o f t h e i n t e r a c t i o n t i m e a l s o a v o i d s anomalous c o l l i s i o n a l phenomena such as o r b i t i n g . To a i d i n t r e a t i n g a gas i t i s c o n v e n i e n t t o t r a n s f o r m the d e n s i t y o p e r a t o r by means o f a 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 ^ i n t h e t r a n s l a t i o n a l s t a t e s w h i l e r e t a i n i n g the e x a c t d e n s i t y o p e r a t o r f o r t h e i n t e r n a l 12 s t a t e s . The a p p r o p r i a t e t r a n s f o r m a t i o n i s g i v e n b y 2 2 where now the Wigner 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 m a t r i x i s a f u n c t i o n o f p o s i t i o n , R, momentum, P, as w e l l as b e i n g an o p e r a t o r i n i n t e r n a l s t a t e space. At e q u i l i b r i u m , t h i s f u n c t i o n - o p e r a t o r t a k e s t h e f o r m , (5) where n i s t h e number d e n s i t y , m the mass o f t h e mole-c u l e , \i-]nx , 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 and Q t h e p a r t i t i o n f u n c t i o n f o r i n t e r n a l s t a t e s , (6) 13 I t i s u s e f u l t o d e f i n e the r e d u c e d ' p e c u l i a r v e l o c i t y W=P/(2mkT) , and t h e l o c a l e q u i l i b r i u m i n t e r n a l s t a t e d e n s i t y o p e r a t o r p ^ ^ e x p f ~Hln1/4T]/Q so t h a t 2 3 w i t h b o t h "fw = W i*p[-v_* and p ( o ) b e i n g u n c o r -r e l a t e d and i n d e p e n d e n t l y n o r m a l i z e d t o 1 , tr I t i s o f paramount i m p o r t a n c e , as emphasized i n t h e p r e v i o u s s e c t i o n , t h a t t h e m o l e c u l e s ' i n t e r n a l s t a t e s be p r o p -e r l y t r e a t e d . F o r t h i s p u r p o s e , the i n t e r n a l s t a t e s a s s o c i -a t e d w i t h the i n t e r n a l h a m i l t o n i a n a r e c o n s i d e r e d as two d i f f e r e n t t y p e s : 1) Those s t a t e s f o r w h i c h t h e energy s e p a r a t i o n between l e v e l s depends on an e x t e r n a l f i e l d , ( i . e . Zeeman o r S t a r k s p l i t t i n g s , l a b e l l e d by m a )• 2) Those s t a t e s f o r which the energy s e p a r a t i o n i s i n d e -14 pendent o f e x t e r n a l f i e l d s ( I . e . r o t a t i o n a l and v i b r a -t i o n a l l e v e l s l a b e l l e d by a ). I f t h e i n t e r n a l s t a t e s a re denoted by |a,ma> t h e n ( 9 ) Kint i = i^, where E i s t h e energy a s s o c i a t e d w i t h t h e s t a t e |a,m >. I n z e r o f i e l d o r t h e l i m i t t h a t t h e f i e l d v a n i s h e s , t h e E a w ^ e n e r g i e s become independent o f and t h e r e i s a degen-e r a c y o f t h e ma s t a t e s . The energy s e p a r a t i o n s , o r f r e q u e n c i e s , between v a r i o u s l e v e l s i n t h e p r e s e n c e o f an e x t e r n a l f i e l d must be d i s t i n g u i s h e d i n o r d e r t o a v o i d c o n f u s i o n when t r e a t -i n g b o t h t y p e s o f f r e q u e n c i e s . I t i s c o n v e n i e n t t o d i s c u s s the i n t e r n a l s t a t e f r e q u e n -c i e s i n co m p a r i s o n t o t h e average time between c o l l i s i o n s , T f . That i s , t h e q u a n t i t y C O T ^ i s a u s e f u l d e l i m i t e r . S i n c e T f i s i n v e r s e l y p r o p o r t i o n a l t o t h e d e n s i t y o f g a s, the two t y p e s o f f r e q u e n c i e s a r e p a r a m e t e r i z e d and d i s t i n g u i s h e d by (10) 15 and ( I D C J ^ ^ Tf <^  H/^ where H i s the e x t e r n a l f i e l d and n i s t h e d e n s i t y . I t i s assumed t h a t a l l m s t a t e s w i t h i n an a a l e v e l have th e same Boltzmann w e i g h t , w h i l e d i f f e r e n t B oltzmann w e i g h t i n g can be i m p o r t a n t t o t h e p o p u l a t i o n o f t h e a l e v e l s . T h i s can be e x p r e s s e d as ci2) I co^^- | << AT/* where co , = ( E - E , )/Ti S i n c e v e r y h i g h f i e l d s would be r e q u i r e d t o l i f t t h e degen-e r a c y t o the o r d e r o f kT, t h i s i s presumably a v e r y good a p p r o x i m a t i o n . T h i s a s s u m p t i o n , however, does not u n e q u i -v o c a l l y d i s t i n g u i s h between co , and co / f r e q u e n c i e s . The advantage o f u s i n g C O T ^ l i e s i n an a b i l i t y t o d i s c u s s s i m u l t a n e o u s l y t h e e f f e c t s o f f i e l d , d e n s i t y and energy s e p a r a t i o n s . I t o c c u r s n a t u r a l l y i n the d e s c r i p t i o n 16 o f such phenomena as s p i n r e l a x a t i o n , p r e s s u r e b r o a d e n -i n g 2 ^ ' 2 ^ a n c i S e n f t l e b e n e f f e c t s 2 ^ , a l l o f which a r e due t o th e phase r a n d o m i z a t i o n o f o f f - d i a g o n a l elements o f t h e 27 d e n s i t y m a t r i x . I n essence i t can be argued t h a t f o r toff >> 1 and w i t h Eq.( 2) s a t i s f i e d , t h e m o l e c u l e i s f r e e t o o s c i l l a t e w i t h f r e q u e n c y & many t i m e s be-tween c o l l i s i o n s . As a r e s u l t i t a c c u m u l a t e s a l a r g e amount o f phase, w h i c h , when averaged over a l l t h e m o l e c u l e s i n t h e system would d e s t r u c t i v e l y i n t e r f e r e , t h u s e l i m i n i n -a t i n g any c o n t r i b u t i o n t o t h e o f f - d i a g o n a l elements o f t h e d e n s i t y m a t r i x which depends on t h i s f r e q u e n c y . On t h e o t h e r hand, i f U>T.£>< 1 an average o v e r a l l t h e m o l e c u l e s would f i n d them i n a r e s t r i c t e d phase o f t h e i r o s c i l l a t i o n and would t h u s be more l i k e l y t o c o n s t r u c t i v e l y i n t e r f e r e r e s u l t i n g i n a f i n i t e c o n t r i b u t i o n t o the o f f - d i a g o n a l e l e -ments o f t h e d e n s i t y m a t r i x . C l e a r l y , f o r t h e case t h a t C13) C J ^ - Zf » 1 t t h e d e n s i t y m a t r i x becomes d i a g o n a l i n a and t h e o f f -d i a g o n a l d e n s i t y m a t r i x elements a r i s e o n l y between t h e m-, s t a t e s f o r a p a r t i c u l a r a l e v e l . A l t h o u g h t h i s 17 a s s u m p t i o n l e a d s t o s i m p l i f i c a t i o n s and t o c o r r e c t d e s c r i p -t i o n s o f t h e S e n f t l e b e n e f f e c t s 2 3 5 2 ^ , i t s h o u l d not be assumed t o h o l d g e n e r a l l y . One has o n l y t o i n c r e a s e t h e d e n s i t y s u f f i c i e n t l y u n t i l co yx - 1 and t h e n E q . ( 1 3 ) i s otoc X v i o l a t e d . T h i s , i n f a c t , i s t h e essence of t h e o b s e r v a t i o n o f t h e " s t e p - e f f e c t " w h i c h i s d i s c u s s e d i n d e t a i l i n P a r t IV of t h i s t h e s i s . S i m i l a r l y t h e S e n f t l e b e n e f f e c t s r e s u l t from i n c r e a s i n g H/n i n com m ' Tf . Other phenomena dependent on a change o v e r from cox^ , >> 1 t o cox^  << 1 a r e found i n microwave s p e c t r o s c o p y and i n p a r t i c u l a r i n t h e i n v e r s i o n s p ectrum o f Ammonia 2^. I n t h e l a t t e r , a t low d e n s i t i e s t h e p o s i t i v e and n e g a t i v e i n v e r s i o n f r e q u e n c i e s a r e s e p a r a t e d and uncoupled', but a t h i g h e r d e n s i t i e s they merge and o v e r -l a p , r e s u l t i n g i n the o b s e r v e d f r e q u e n c y b e i n g c e n t e r e d a t 30 z e r o f r e q u e n c y . U s i n g Fano's t h e o r y o f p r e s s u r e b r o a d e n i n g , Ben-Reuven was a b l e t o o b t a i n e x c e l l e n t agreement w i t h t h e e x p e r i m e n t a l d e n s i t y dependence on the l i n e shape u s i n g j u s t t h e d e s c r i p t i o n o u t l i n e above. I n o r d e r t o t r e a t such e f f e c t s , i . e . phase r a n d o m i z a t i o n o f the o f f - d i a g o n a l elements o f t h e d e n s i t y m a t r i x , an a p p r o -p r i a t e e q u a t i o n must be found f o r the c o r r e c t e v o l u t i o n o f t h e s i n g l e t d e n s i t y o p e r a t o r . I t must be v a l i d f o r b o t h l i m i t s o f cox^  and t h e o n l y r e s t r i c t i o n i s t h a t the gas be s u f -f i c i e n t l y d i l u t e so t h a t 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 . CHAPTER 3 A GENERALIZED BOLTZMANN EQUATION FOR MOLECULES WITH INTERNAL STATES 3 . 1 ) D e r i v a t i o n Two as s u m p t i o n s a r e i n h e r e n t i n t h e " d e r i v a t i o n " o f any Boltzmann e q u a t i o n 1 0 . F i r s t , 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, se c o n d , t h a t time i s g i v e n a d e f i n i t e d i r e c t i o n . The f i r s t a s s u m p t i o n e f f e c t i v e l y r e s t r i c t s t h e range o f a p p l i c a b i l i t y o f t h e Boltzmann e q u a t i o n t o a den-s i t y range f o r wh i c h t r i p l e and h i g h e r c o l l i s i o n s a r e n e g l i g i b l e and the second l e a d s t o decay t o e q u i l i b r i u m . c\ , CN) The N - p a r t i c l e H a m i l t o n i a n , /x , c o n s i s t e n t w i t h a l l o w i n g o n l y b i n a r y c o l l i s i o n s i s t a k e n as (14) where rlji, i s t h e one-molecule H a m i l t o n i a n and \ ^ i s a two m o l e c u l e p o t e n t i a l energy o p e r a t o r . U s i n g E q . ( l 4 ) , i s p o s s i b l e t o d e r i v e a h i e r a r c h y o f e q u a t i o n s f o r t h e r e -19 duced d e n s i t y o p e r a t o r s . (15) f = - f c o Zl*/c*-«)\ • T h i s was f i r s t done by B o g o l u b o v ^ 1 , Born and G r e e n ^ 2 , K i r k w o o d ^ ^ and Y v o n ^ and t h e f i r s t two e q u a t i o n s i n the BBGKY h i e r a r c h y a r e ( N-1 - N -2 - N ) ( 1 6 ) and where ( 1 8 ) and ( 1 9 ) sit b^/Zt = X, + Vtdk ( 1 7 ) At h<£*lzt =• X T x \ \ l + - W ( V „ + V a s^ 20 The s u b s c r i p t s l a b e l m o l e c u l e s , i t b e i n g assumed t h a t and p ^ n ^ a r e symmetric t o p a r t i c l e i n t e r c h a n g e . I f t h e r e were no c o l l i s i o n s , \ f j j = 0 , and the h i e r a r c h y would reduce t o the i d e a l gas case mentioned b e f o r e . I n t h i s case t h e f o r m a l s o l u t i o n t o Eqs.(2) and (16) i s (20) When 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 t h e l a s t term i n Eq.(l6) does make a c o n t r i b u t i o n and t h e s o l u t i o n , Eq.(20), i s no l o n g e r v a l i d . T h i s , however, happens o n l y w h i l e mole-c u l e s one and two a r e w i t h i n t h e range o f t h e p o t e n t i a l ^X2 On t h e o t h e r hand, w h i l e does r e s u l t i n a non-zero con-t r i b u t i o n , 1/aj and would be e x p e c t e d t o be n e g l i g i b l e u n l e s s t h e b i n a r y c o l l i s i o n a s s u m p t i o n were v i o l a t e d . Hence d u r i n g t h e time i n t e r v a l t h a t m o l e c u l e s one and two a r e c o l -l i d i n g , t he s o l u t i o n t o t h e second BBGKY e q u a t i o n i s (2D ,o r . V*). . . ^1 "I These " r e a s o n a b l e " c o n j e c t u r e s about t r e a t i n g t h e BBGKY h i e r a r c h y f o r a d i l u t e gas a l o n g w i t h t h e m o l e c u l a r chaos a s s u m p t i o n a re s u f f i c i e n t f o r o b t a i n i n g a c l o s e d e q u a t i o n 2 1 f o r t h e s i n g l e t d e n s i t y o p e r a t o r . These assumptions a r e e x p l i c i t l y : 1) D u r i n g t h e t i m e t h a t two m o l e c u l e s a re i n t e r a c t i n g , the s o l u t i o n o f t h e second BBGKY E q u a t i o n i s E q . ( 2 1 ) . 2 ) There a r e t i m e s , " l o n g " b e f o r e t h e c o l l i s i o n o f mole-c u l e s one and two, d u r i n g w h i c h t h e e f f e c t o f i s n e g l i g i b l e and the m o l e c u l e s a r e f r e e . At t h e s e t i m e s i t i s assumed 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 f a c t o r s i n t o a p r o d u c t o f two s i n g l e t d e n s i t y o p e r a t o r s , one f o r each m o l e c u l e , s i n c e b e f o r e i n t e r a c t i n g , no c o r -r e l a t i o n s h o u l d e x i s t between m o l e c u l e s . T h i s assump-t i o n can be e x p r e s s e d as f o l l o w s : ( 2 2 ) gl*UA = P ? , ( t ^ t o ~ > " ° ° -Here the t i m e " l o n g " b e f o r e a c o l l i s i o n means a time l o n g compared t o the d u r a t i o n o f a c o l l i s i o n but s h o r t compared t o the time between c o l l i s i o n s . Thus from a c o l l i s i o n p o i n t o f v i e w , the l i m i t t0-»- - 0 0 can be t a k e n w h i l e m a c r o s c o p i c a l l y t must be s m a l l . T h i s s e p a r a t i o n o f time s c a l e s i s i m p l i c i t l y assumed when t h e f o r m a l l i m i t tQ-> - <» i s t a k e n . At such t i m e s i t i s assumed t h a t t h e m o l e c u l e s a re s u f f i c i e n t l y s e p a r a t e d ( i n p a r t i c u l a r no bound p a i r s t a t e s o c c u r ) , so t h a t t h e 22 p a r t i c l e s a r e n o n - i n t e r a c t i n g and the p r o d u c t c o n d i -t i o n , Eq.(22) i s j u s t i f i e d . 3) F i n a l l y , s i n c e t h e m o l e c u l e s a r e not i n t e r a c t i n g a t ti m e s t Q - 0 0 , the time dependence o f ( t Q ) and p 2 ^ 0 ) s n o u l d be governed o n l y by the one p a r t i c l e L i o u v i l l e e q u a t i o n Eq.(2) w i t h s o l u t i o n Eq. ( 2 0 ) . Assumptions 1, 2, and 3 can be combined t o r e l a t e P^^ t o p^(t) p^(t) as f o l l o w s ( 2 3 ) The s u p e r o p e r a t o r i s d e f i n e d when the l i m i t i n e q u a t i o n ( 3 ) e x i s t s . I n p a r t i c u l a r t h e l i m i t e x i s t s f o r c e r t a i n po-t e n t i a l s a n d i t i s f o r t h i s c l a s s o f p o t e n t i a l s t h a t t h e Boltzmann c o l l i s i o n s u p e r o p e r a t o r can be d e f i n e d . I f , i n p a r t i c u l a r , t h e M i l l e r wave o p e r a t o r , (24) -XL = JU^JL J^A*A.HC%0/t]s^pirA, Kta/-€ t e - > - o o 7 e x i s t s , where K=H, + ^ ( J L , t h e n i t can be shown t h a t i t , e x i s t s 2 3 and can be written as (25) _ J C L ^ ft =• R _ o l -By analogy of Eqs. ( 2 3 ) and (24), ft.* i s c a l l e d here the 35 M i l l e r superoperator and i t has been shown that . ft,* can be written i n the form, E q . ( 2 5 ) , even when ft i s not given by Eq.(24). ( 2 ) By su b s t i t u t i n g P,\ from Eq. ( 2 3 ) into the f i r s t BBGKY equation, the Boltzmann equation i s obtained, ( 2 6 ) ^"St" " + Xntt- ^ ? ' ? x where the t r a n s i t i o n superoperator i s defined as (27) H = V - O . * and only s i n g l e t density operators appear. This Boltzmann equation has a d r i f t term , and a binary c o l l i s i o n term, sfcasts. °^ //^ PiP2 ' T h e a s s u m P t i o n that P,^ 2^ factors before a c o l l i s i o n introduces a time d i r e c t i o n and t h i s "Boltzmann property"" 1" 0 accounts for the eventual decay to equilibrium. In "deriving" t h i s equation the only assumptions about the 24 time s c a l e s are E q . ( 3 ) and assumption 2 . In p a r t i c u l a r the i n t e r n a l s t a t e energ ies are t r e a t e d e x a c t l y and no assump-t i o n s about the t ime s c a l e s o f t h e i r i n t e r n a l motions are r e q u i r e d . In the remainder of t h i s s e c t i o n more e x p l i c i t forms of the Boltzmann e q u a t i o n , E q . ( 2 6 ) , are p r e sen t ed . 3 . 2 ) Formal S c a t t e r i n g Theory A p h y s i c a l p i c t u r e a s s o c i a t e d w i t h a s c a t t e r i n g experiment i s a stream of f ree p a r t i c l e s , e . g . a molecule beam, i m p i n g -i n g on a t a r g e t ; i s s c a t t e r e d ; and at a d i s t a n c e f a r from the s c a t t e r i n g cen te r the r a d i a l d i s t r i b u t i o n of p a r t i c l e s i s measured. The number o f p a r t i c l e s p a s s i n g through a g i v e n s o l i d angle per u n i t of t ime de f ines the d i f f e r e n t i a l c ross s e c t i o n . A process such as t h i s shou ld have the incoming beam, composed of a s t a t i s t i c a l mix tu re o f s t a t e s , d e s c r i b e d by a d e n s i t y o p e r a t o r . F u r t h e r , s i n c e c o l l i s i o n s w i t h i n a beam are n e g l i g i b l e , a one p a r t i c l e d e n s i t y ope ra to r d e s c r i p -t i o n i s s u f f i c i e n t . Thus the p r ev ious d i s c u s s i o n of c o l l i s i o n s w i t h i n a d i l u t e gas has many s i m i l a r i t i e s to a beam e x p e r i -ment. However, u s u a l s c a t t e r i n g theo ry -^ ± s developed f o r s ta te vec to rs which are envisaged as wave packets im p in g in g upon a s c a t t e r i n g c e n t e r . For purposes h e r e , t h i s theory nust be adapted f o r d e n s i t y o p e r a t o r s , and before d i s c u s s i n g 25 t h i s i t i s u s e f u l t o summarize the r e s u l t s o f t h e u s u a l s c a t t e r i n g t h e o r y so the p a r a l l e l development f o r d e n s i t y o p e r a t o r s can be more e a s i l y seen. T h i s can be done i n two e q u i v a l e n t ways; by a time dependent or t i m e i n d e p e n -dent a p p r o a c h ; the l a t t e r method u s u a l l y b e i n g more u s e f u l i n t r e a t i n g m o n oenergetic p r o c e s s e s . A l t h o u g h t h i s i s not a r e s t r i c t i o n h e r e , the t i m e i n d e p e n d e n t t h e o r y w i l l be used s i n c e t h e r e s u l t i n g o p e r a t o r e q u a t i o n s are f o r m a l l y more e a s i l y h a n d l e d , even though t h e time-dependent approach i s p h y s i c a l l y more t r a n s p a r e n t . The M i l l e r wave o p e r a t o r , E q . ( 2 4 ) , can be w r i t t e n as ( 2 8 ) ^ M^U LL( + To) where ( 2 9 ) u(to) = yC\/t]juxP[-jL Ku/t] i s a u n i t a r y o p e r a t o r . T h i s can be c a s t i n t o an i n t e g r a l e q u a t i o n 3 ^ (30) a (t 0) = 1 - _ _ J **f>U HUt/tl V-i Mtpl-X kt/£j Jt 2 6 so t h a t i f 9, o p e r a t e s on a s t a t e v e c t o r ^ 0 ( E ) wh i c h i s an e i g e n f u n c t i o n o f the f r e e , t w o - p a r t i c l e h a m i l t o n i a n K = -( 3 D K V 0 ( E ) = B VQCE) t h e n ( 3 2 ) H The i n t e g r a l , however, i s not w e l l d e f i n e d i n t h e l i m i t t h a t tQ-»- - 0 0 s i n c e i n t h i s l i m i t i t may o s c i l l a t e w i d e l y . To remedy t h i s a convergence f a c t o r e + e t , e>o i s i n t r o d u c e d - ^ and e v e n t u a l l y the l i m i t e -* 0 i s t a k e n . I n t e g r a t i o n t h e n g i v e s , C 3 3 ) _ac£) V»<e) - ('J.+ S C E ) \ O V . C e ) where G(E) i s the t o t a l ( H ) p a i r p a r t i c l e Green's f u n c t i o n ( 3 4 ) &(e) = JU^UL ( E - M ^ + x.e)" 1 . 6 -> C7+ 27 The M i l l e r wave o p e r a t o r t h u s t a k e s a f r e e s t a t e i n t o a t o t a l s c a t t e r i n g s t a t e , ip(E) , The words " f r e e " and " t o t a l " a r e used i n t h e same c o n t e x t as the y a r e f o r a " f r e e p a i r p a r t i c l e o p e r a t o r " (which c o n t a i n s t h e " f r e e " p a i r p a r t i c l e h a m i l t o n i a n , K ) and a " t o t a l " p a i r p a r t i c l e o p e r a t o r ( wh i c h c o n t a i n s t h e " t o t a l " h a m i l t o n i a n , Hf = K + V,j_ ) # Hence ty(E) , w h i c h i s p r e p a r e d i n t h e p a s t (say as tQ-»- -°° ) , i s l a b e l l e d by t h o s e quantum numbers wh i c h commute w i t h the f r e e p a i r h a m i l t o n i a n , K , and t h e s e a r e changed i n the s c a t t e r i n g p r o c e s s by V-^ • Thus ip(E) , the t o t a l s t a t e , i s an e i g e n f u n c t i o n o f *H.(i' and i n t h e p a s t was e q u a l t o (E) . The o p e r a t o r e q u a t i o n t h a t n(E) s a t i s f i e s i s one form o f the Lippmann-Schwinger e q u a t i o n - ^ , w hich can a l s o be w r i t t e n i n terms o f the f r e e p a i r p a r t i c l e Green's f u n c t i o n C 3 5 ) . ( 3 6 ) _ T L ( E ) = 1 + GCE)V, ( 3 7 ) aoCE) = ( £ - K W e ) 28 by (38) -ILCE) = 1 + Cr0CE)Vi£L(E) . The q u a n t i t y , V-j_2^(E) , d e f i n e s the t r a n s i t i o n o p e r a t o r , t(E) = V^SLCB) (39) = Vi+ V, G,C£)tCE). I f i s not an e i g e n f u n c t i o n o f K i t i s not p o s s i -o b l e t o e x p l i c i t l y i n t e g r a t e Eq.(30) f o r a g i v e n energy as was done f o r ^ Q ( E ) . However i t i s p o s s i b l e t o p r o c e e d by s p e c t r a l l y decomposing U w i t h r e s p e c t t o K . Thus i f i s t h e s p e c t r a l measure o f K , t h a t i s K=/EdPg t h e n t h e M i l l e r wave o p e r a t o r and t h e t r a n s i t i o n o p e r a t o r can be w r i t t e n as _TL = M^jt UL (to) (40) t 0 - " > - 0 0 J 29 and (41) t - J t(E) <SPE = t^-O- . I f 9 o r t o p e r a t e on an e i g e n s t a t e o f K , t h e n o n l y t h e Ji(E] o r tCE) component w i l l s u r v i v e and the s i t u a t i o n r e -v e r t s t o the p r e v i o u s d i s c u s s i o n . I f n o t , and 9 and t o p e r a t e on a s t a t e o f mixed e n e r g i e s , t h e n Pjr w i l l s e l e c t out t h e energy components o f t h i s s t a t e . F o r m a l l y , analogous Lippmann-Schwinger e q u a t i o n s can be f o u n d , C42) =. 1 + GQ V i a t J O . C43) where 6,8 - fG.OB dPe C44) Here "VC i s t h e s u p e r o p e r a t o r d e f i n e d i n Eq.(19) and B 30 i s an o p e r a t o r ( i . e . Q o r t a b o v e ) , upon w h i c h G Q a c t s . (Hence G Q i t s e l f i s a s u p e r o p e r a t o r ) . The second e q u a l i t y i n E q . ( 4 4 ) f o l l o w s by dou b l y s p e c t r a l l y decomposing B w i t h r e s p e c t t o K . The l a s t e q u a l i t y uses th e s p e c t r a l r e p r e -s e n t a t i o n o f the s u p e r o p e r a t o r , K B = ff (z-s')<jp£5c)pe> ( 4 5 ) = t f ua d(P„oB , o r ^ X ~- tf^od<P„o where i s the s p e c t r a l measure o f . T h i s l a s t r e s u l t i s i m p o r t a n t s i n c e t h e f r e q u e n c y , o r d i f f e r e n c e between energy l e v e l s , o c c u r s . Even though i t i s customary t o use the h a m i l t o n i a n and i t s energy e i g e n v a l u e s t o d e s c r i b e phenomena, i t i s more n a t u r a l t o use a s u p e r o p e r -a t o r w h i c h p i c k s out energy d i f f e r e n c e s and hence i s c l o s e r t o what i s e x p e r i m e n t a l l y o b s e r v e d , as w e l l as b e i n g i n d e p e n -dent o f an a r b i t r a r y " z e r o " r e f e r e n c e energy. The p o i n t o f the p r e v i o u s b r i e f d i s c u s s i o n o f the common 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 s t h a t i t n a t u r a l l y l e a d s t o energy d i f f e r e n c e s once t r a n s i t i o n s a r e i n c l u d e d . I n o r d e r t o c o n t i n u e and d i s -31 cuss s c a t t e r i n g t h e o r y f o r d e n s i t y o p e r a t o r s , i t i s n e c e s s a r y t o o b t a i n g o v e r n i n g e q u a t i o n s f o r the s u p e r o p e r a t o r s w h i c h a r e analogous t o t h o s e f o r t h e o p e r a t o r s . I n o t h e r words, what a r e t h e s u p e r o p e r a t o r e q u i v a l e n t s o f t ( E ) , 9(E) , t , 9 , and t h e v a r i o u s Green's f u n c t i o n s ? The answers have a l r e a d y been i m p l i e d i n E q s . ( 2 5 ) and ( 2 7 ) and t h i s w i l l now be more e x p l i c i t l y s t a t e d . Comparisons o f Eqs.(41) and ( 2 7 ) and Eqs.(23) and (24) suggest t h a t _/ and o,^  a re t h e s u p e r o p e r a t o r e q u i v a l e n t s o f t and o, . T h i s i s i n d e e d t r u e as can be seen by w r i -t i n g o,2> as t h e l i m i t o f the a n a l o g o u s u n i t a r y e v o l u t i o n s u p e r o p e r a t o r , ta--> -<° ir where (48) U^Q - & x . p [ i j f ' J ^ p l ' X X W i ] . P r o c e e d i n g as b e f o r e , i s c a s t i n t o an i n t e g r a l equa-t i o n which can be s o l v e d f o r one f r e q u e n c y component i f i t o p e r a t e s on an e i g e n o p e r a t o r A o f "K* , i . e . , from (49) SLxA^X^xUitjf) ^ 32 i t f o l l o w s t h a t ( 5 0 ) JixiLjjB - (i + - # r w t f ) t y * ) f t o n l y i f (5D Yi f\ - £ co* R . The q u a n t i t y u>) i s the s u p e r o p e r a t o r , -/ ( 5 2 ) MCuS) = ( t c o - X C \ A 6/ . The s u p e r o p e r a t o r e q u i v a l e n t s o f the Lippmann-Schwinger equa-t i o n s are'' t h e n ( 5 3 ) _ X L ^ (uS) = 1 -t M(co)Vlx and ( 5 4 ) -CLXCLJ) - I + M.0(u) V,AJXx(uS) where t h e f r e e p a r t i c l e Green's s u p e r o p e r a t o r i s ( 5 5 ) M.cu) = (tcj - X + a ) " ' . 33 Comparison w i t h E q . ( 4 4 ) r e v e a l s t h a t (so 6 = JtJo) and f u r t h e r i t can be shown t h a t ( 5 7 ) Cr - M. (o). By a n a l o g y t o t ( E ) , the t r a n s i t i o n s u p e r o p e r a t o r f o r f r e -quency i s d e f i n e d as 7 M = V,a _a ( 5 8 ) The a n a l o g y can be t a k e n f u r t h e r by c o n s i d e r i n g 9,^ o p e r a t i n g on an o p e r a t o r w h i c h i s not an e i g e n o p e r a t o r o f X Then and ft^fcu) become s p e c t r a l components o f t h e more g e n e r a l s u p e r o p e r a t o r s 0 and , ( 5 9 ) and (60) 34 From E q s .(58) and (60), can be i d e n t i f i e d as the Boltzmann c o l l i s i o n supe ropera to r , E q.(27). The Lippmann-Schwinger equat ions f o r and "3 f o l l o w (6D = 1 + and 7 - V + VM.CV] (62) = V + VJS. cv] where the square b racke t s mean tha t the super - superopera to r ac t s on the enc losed superopera to r . These are found to be jtcai = [je*) d u a (61) 6 -> Ot 35 I t i s o f c o u r s e p o s s i b l e t o go f u r t h e r i n t h e d e v e l o p -ment o f ( s u p e r ) n - o p e r a t o r s , a l t h o u g h m a t h e m a t i c a l l y the d i s t i n c t i o n between o p e r a t o r s and s u p e r o p e r a t o r s , i s r a r e l y made. However, p h y s i c a l l y , i t i s not n e c e s s a r y t o p r o c e e d f u r t h e r . The i m p l i c a t i o n s o f t h e n a t u r a l o c c u r r e n c e o f *J and -H.^ w i l l be d i s c u s s e d i n C h a p t e r 4 . B e f o r e p r o c e e d i n g however, v a r i o u s more e x p l i c i t forms of the Boltzmann 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 g i v e n which a r e u s e f u l i n l a t e r d i s c u s s i o n s and a p p l i c a t i o n s . 3 . 3 ) E x p l i c i t Forms 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  and L i n e a r i z a t i o n From Eqs. ( 2 7 ) , (25) and ( H i ) , the Boltzmann c o l l i s i o n s u p e r o p e r a t o r can be w r i t t e n ? (65) * J f l = t f l - fltt+ t f l t ^ J - O u t f i t * . I f A, which i s a two p a r t i c l e o p e r a t o r , i s d i a g o n a l i n energy so t h a t ( 6 6 ) ye A - on - o t h e n t h e l a s t two ( g a i n terms) can be combined t o g i v e 36 (67) S u b s t i t u t i o n o f t h e s e two forms i n t o the Boltzmann equa-t i o n g i v e s two e q u a t i o n s ; one v a l i d f o r d e n s i t y o p e r a t o r s d i a g o n a l i n t h e i n t e r n a l s t a t e e n e r g y , (68) and t h e o t h e r more g e n e r a l l y v a l i d when Eq. ( 6 6 ) i s not r e -q u i r e d t o h o l d . (69) The f i r s t e x p r e s s i o n has many s i m p l i f y i n g a s p e c t s and i t i s s u f f i c i e n t f o r d e s c r i b i n g t h e e v o l u t i o n o f when the o f f - d i a g o n a l elements o f p depend o n l y on the deg e n e r a t e ( o r n e a r l y d e g e n e r a t e ) i n t e r n a l s t a t e s . T h i s e q u a t i o n c o n t a i n s o n l y the z e r o f r e q u e n c y component, "^j(o) o f °J and i s t h e Waldmann-Snider (W-S) e q u a t i o n ' . The second 37 e q u a t i o n i s more c o m p l i c a t e d because t h e p r i n c i p a l v a l u e terms i n t h e Green's f u n c t i o n s do not c a n c e l and c o m p l i -7 . 4 0 c a t e d f r e q u e n c y terms a r i s e . These a r e d i s c u s s e d i n the next c h a p t e r . As s t a t e d e a r l i e r , p r a c t i c a l d e s c r i p t i o n s o f a gas a r e a i d e d by making use o f the 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 , E q . ( 4 ) . F u r t h e r i t i s u s u a l l y assumed t h a t the n o n - e q u i l i b r i u m gas i s o n l y l i n e a r l y d i s p l a c e d from 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(°) } E q . ( 5 ) 1 0 . Hence f may be w r i t t e n a p p r o x i m a t e l y as where <t> i s a l i n e a r h e r m i t i a n p e r t u r b a t i o n and the a n t i -commutator i s r e q u i r e d s i n c e <t> does not i n g e n e r a l com-( 7 0 ) mute w i t h f (o) and f must be h e r m i t i a n The l i n e a r -i z e d B oltzmann e q u a t i o n f o r $> i s t h e n ( 7 D -4- A X, £ and the l i n e a r 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 61 i 42 i s 2 > -<K*f <10"pdj,x+ ( w f x t * The s u b s c r i p t s , 1 and 2, r e f e r t o m o l e c u l e s one and two r e s p e c t i v e l y , u i s the r e d u c e d mass o f t h e p a i r , g' and g_ ar e the r e l a t i v e v e l o c i t i e s b e f o r e and a f t e r a c o l l i s i o n , w h i l e C73) t\ = < A 5- / t \A £') 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 space. I f t h e compos i t e quantum numbers a,m„ i n t r o d u c e d b e f o r e l a b e l t h e e i g e n v a l u e s E^,-^^ and s i n c e t h e e i g e n -v e c t o r s form a complete s e t o f b a s i s v e c t o r s (see Eq.(9) t h e n $> can be expanded, (7*0 > < « ' ^ / (fi 39 where E q u i v a l e n t l y t h i s can be w r i t t e n i n terms of p r o j e c t i o n o p e r a t o r s , 2^ **_ = I «*• ^  <*><<*- " " I * / > (76) where (77) Z F *< 'Hi. For the case t h a t ai^'t f-» 1 d i s c u s s e d above, no o f f - d i a g o n a l elements of <j) between d i f f e r e n t a l e v e l s are important and a s u f f i c i e n t approximation t o <j> would b e 2 3 40 ( 7 8 > 0 = 2 - P 0 P However when t h e v a r i o u s a l e v e l s do i n t e r f e r e , t h e more g e n e r a l E q . ( 7 6 ) must be used. if 4 l CHAPTER 4 PROPERTIES OF D The W-S e q u a t i o n , E q . ( 6 8 ) , was o r i g i n a l l y d e r i v e d t o a c c o u n t f o r phenomena i n which t h e d e g e n e r a t e or n e a r l y d e g e n e r a t e i n t e r n a l s t a t e s a r e t r e a t e d e x a c t l y . F o r cases t h a t t h e d e g e n e r a t e s t a t e s a r e not i n c l u d e d , the W-S e q u a t i o n r e d u c e s t o the Wang-Chang Uhlenbeck (WCU) e q u a t i o n 1 ^ , and c o n s e q u e n t l y i t i s c o n s i s t e n t w i t h r e s u l t s o b t a i n e d i n e a r l i e r work w i t h the WCU e q u a t i o n and i t s p r e d e c e s s o r s . Treatments o f s p e c i f i c problems w i t h t h e W-S e q u a t i o n (which cannot be d e s c r i b e d by the WCU e q u a t i o n ) have r e s u l t e d i n s u c c e s s f u l 26 d e s c r i p t i o n s o f S e n f t l e b e n e f f e c t s and n u c l e a r s p i n r e l a x a -19 4? t i o n , and p a r t i a l s u c c e s s i n t r e a t i n g t h e thermomagnetic 44 t o r q u e . I n t r e a t i n g t h e s e e f f e c t s , the degeneracy o f t h e l e v e l s i s o f t e n l i f t e d by t h e Zeeman s p l i t t i n g . J u s t how de-g e n e r a t e t h e l e v e l s have t o be i n o r d e r t o f a l l w i t h i n t h e v a l -i d i t y range o f t h e W-S e q u a t i o n needs c a r e f u l e x a m i n a t i o n , and i t i s not f u l l y a g r e e d upon as t o what t h i s c r i t e r i o n i s . I t has been s t a t e d ^ t h a t o n l y f o r w a a' T^>>1 i s t h e W-S e q u a t i o n a p p l i c a b l e , but t h e r e a r e most p r o b a b l y low l y i n g 4 2 o r a c c i d e n t a l l e v e l s such t h a t to ,<to Hence t h i s c r i t e r i o n b r e a k s down s i n c e t h e W-S e q u a t i o n i s c e r t a i n l y v a l i d f o r s m a l l f r e q u e n c i e s . I n o r d e r t o s p e c i f y j u s t how s m a l l they must be, i t i s n e c e s s a r y t o r e t u r n t o t h e Boltzmann e q u a t i o n t o see e x a c t l y what the e f f e c t i s o f a s -suming d i a g o n a l i t y o f the d e n s i t y m a t r i x i n i n t e r n a l energy. I n t h e l a s t s e c t i o n i t was shown t h a t the z e r o f r e q u e n c y com-ponent o f the g e n e r a l Boltzmann c o l l i s i o n s u p e r o p e r a t o r , °J(o) , c o r r e s p o n d s t o t h e W-S c o l l i s i o n term. E x a m i n a t i o n o f E q . ( 4 9 ) f o r 0,% i m p l i e s t h a t f o r °J(o) and t h i s c o n d i t i o n can be met a p p r o x i m a t e l y p r o v i d e d t h a t the f r e q u e n c y i s " s m a l l " . S m a l l n e s s here means t h a t t h e f r e q u e n -c i e s must s a t i s f y s i n c e t h e i n t e g r a t i o n i s c o r r e c t l y o n l y o f t h e o r d e r o f an i n t e r a c t i o n t ime and not from minus i n f i n i t y . The consequence f o r t h e to , f r e q u e n c i e s i s t h a t an e x t e r n a l l y a p p l i e d f i e l d must be s m a l l so as t o s a t i s f y E q . ( 8 0 ) . I f to , f r e q u e n c i e s ( 7 9 ) ( 8 0 ) « 1 43 o c c u r i n E q . ( 7 9 ) > t h e n i t might be t h a t w a c / r i n t ^ 1 * ' by E q . ( 3 ) j i t f o l l o w s t h a t w / T f,>>l and by phase randomi-z a t i o n 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 s a re d i a g o n a l i n a . Thus the W-S e q u a t i o n a d e q u a t e l y d e s c r i b e s t h e s e e f f e c t s . F o r t h e m^ l e v e l s , i t i s assumed t h a t OJ ,x. <<1 i n * * m^ m^ ' i n t which case E q . ( 7 9 ) i s a g a i n a p p r o x i m a t e l y s a t i s f i e d and t h e W-S e q u a t i o n a p p l i c a b l e . I t i s thus seen t h a t the W-S equa-t i o n i s v a l i d f o r t h e s e d i f f e r e n t s e t s o f l e v e l s f o r d i f f e r -ent r e a s o n s : 1) t h e to . f r e q u e n c i e s a re a l l phase r a n d o m i z e d , w ,x~>>l, a a ' aa f ' 2) when phase r a n d o m i z a t i o n o f the co t f r e q u e n c i e s does m^niot not t a k e p l a c e to , x. ^<<1 so t h a t the a p p r o x i m a t i o n , E q . ( 7 9 ) i s s u f f i c i e n t . The c o n d i t i o n f o r v a l i d i t y o f t h e W-S e q u a t i o n i s t h e r e f o r e u ™ ™ i T - a n d a) ,X^>>1. m^mj^ i n t aa 1 r By i g n o r i n g the f r e e p a r t i c l e change i n the 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 s d u r i n g an i n t e r a c t i o n , the p o s s i b l i t y o f c o l l i s i o n a l t r a n s i e n t e f f e c t s cannot a r i s e . As a r e s u l t Eq. (80) i s t h e c o n d i t i o n f o r t h e v a l i d i t y o f the M a r k o v i a n Approx-i m a t i o n , and i s i n f a c t the same one wh i c h i s o f t e n employed i n the d e r i v a t i o n o f t h e master e q u a t i o n . The memory t e r m i n a g e n e r a l i z e d master e q u a t i o n t a k e s t h e form o f a c o n v o l u t i o n ^ T (81) 44 where M ( x ) i s a k e r n e l o p e r a t o r and p ( t - x ) i s the a p p r o -p r i a t e d e n s i t y o p e r a t o r f o r t h e problem. P r o v i d e d t h e memory t i m e , x o f M ( T ) i s s h o r t i n comparison t o t h e time s c a l e m o f change o f p , t h a t i s , p ( t - x m ) ~ p ( t ) , an i n t e g r a t i o n o v e r T g i v e s ( 8 2 ) n e ft) -I n t r e a t i n g a d i l u t e gas from t h e Boltzmann e q u a t i o n p o i n t o f v i e w , t h i s memory time c o r r e s p o n d s t o t h e i n t e r a c t i o n t i me o f two m o l e c u l e s T±nt • T n e c o n d i t i o n t h a t p ( t - x ) i s s l o w l y v a r y i n g f o r t h i s t i me i n t e r v a l t h e n f o l l o w s from E q . ( 8 0 ) . Memory between c o l l i s i o n s i s r e f l e c t e d i n the p r e s e n c e o f o f f - d i a g o n a l elements o f the d e n s i t y o p e r a t o r . That i s memory i s r e t a i n e d when cox^<l and d e s t r o y e d (phase random-i z e d ) when wxf>>l . I t i s t h e n m e a n i n g l e s s t o d i s c u s s t h e e f f e c t o f W T i n t ~ 1 f o r a Boltzmann e q u a t i o n s i n c e t o x f > > u ) x i n t ~ l and by phase r a n d o m i z a t i o n the component o f p w i t h t h i s f r e -quency v a n i s h e s . F o r example i n the WCU e q u a t i o n , cox f>>l and the p o s s i b i l i t y o f memory ( o f f - d i a g o n a l i t y ) n ever a r i s e s . F u r t h e r , as d i s c u s s e d above, t h e W-S e q u a t i o n can have e i t h e r l i m i t f o r to x~ but co ,x. ,<<1 . That i s , a l t h o u g h t h e p o s s i b i l i t y o f memory i s p r e s e n t , c o l l i s i o n a l t r a n s i e n t e f f e c t s 45 cannot a r i s e . On the o t h e r hand, i n d e r i v i n g t h e g e n e r a l i z e d Boltzmann e q u a t i o n i n Ch a p t e r 2 . 1 , E q . ( 8 0 ) was not r e q u i r e d t o h o l d , and as a r e s u l t the ,K. f r e q u e n c i e s i n E q . ( 7 9 ) a r i s e n a t u r a l l y . I n c o n t r a s t t o t h e W-S e q u a t i o n i t i s c l e a r t h a t i n c l u d i n g yC f r e q u e n c i e s i s o n l y i m p o r t a n t when o ) i ^ n t i s not s m a l l i n the sense o f E q . ( 7 9 ) . However u r i n t must not be so l a r g e t h a t t h e component o f p w i l l phase randomize f o r c e r t a i n l y C 0 T j _ n t > > - ' - i m p l i e s I O T ^ > > 1 and the WCU e q u a t i o n t h e n f o l l o w s . C o n s e q u e n t l y i t i s f o r t h e i n t e r m e d i a t e range o f toT^n1_ t h a t the g e n e r a l i z e d Boltzmann e q u a t i o n i s p a r t i c u l a r l y u s e f u l . More e x a c t l y , c o x m u s t be l a r g e enough so t h a t co f r e q u e n c i e s a r e i n c l u d e d , but s m a l l enough so t h a t cox f<l. Then t h e p o s s i b i l i t y o f memory e f f e c t s (cox^-l) and c o l l i s i o n a l t r a n s i e n t s ( n o n - n e g l i g i b l e C U T ) can b o t h be r e a l i z e d . Such a s i t u a t i o n i s l i k e l y t o a r i s e f o r dense gases when t h e mean f r e e t ime d e c r e a s e s so t h a t i t a t l e a s t becomes comparable t o T i n t ' T a b l e ( 1 ) summarizes t h e v a l i d i t y c o n d i t i o n s and v a r i o u s cases t h a t can a r i s e . A b e t t e r p r o c e d u r e t h a n making t h e M a r k o v i a n a p p r o x i m a t i o n p ( t - x ) - p ( t ) i s t o assume ( 8 3 ) 46 t h i s keeps th e f r e e p a r t i c l e t i m e e v o l u t i o n o f p d u r i n g the c o l l i s i o n . S i n c e f o r a master e q u a t i o n i s not r e -s t r i c t e d t o b e i n g a two p a r t i c l e o p e r a t o r , E q . ( 8 3 ) s h o u l d be a p p r o p r i a t e f o r many s i t u a t i o n s where f r e e p a r t i c l e m o t ions need be c o n s i d e r e d . I n h i s s t u d y o f p r e s s u r e b r o a d e n i n g , Pano30 ^as o b t a i n e d from Zwanzig's g e n e r a l i z e d master equa-46 t i o n , a c o l l i s i o n o p e r a t o r , m(w) , which i s o f s i m i l a r form t o t h e Boltzmann c o l l i s i o n o p e r a t o r , E q . ( 5 8 ) . E x p l i c i t l y , c o m p a r i s o n o f E q . ( 5 8 ) and Pano's E q . ( 4 6 ) w i t h the i d e n t i -f i c a t i o n t h a t tyf" A and X~ X0 shows t h a t C 8 4 ) i U X ^ ( w + ie) - "3 Cui) . However a c t i n g on an a r b i t r a r y o p e r a t o r A, I t f o l l o w s t h a t C85) JL^X ^ ( W 4 ^ d ) f f = *3c^)Rj: 7 / 9 , e —> o+ u n l e s s A i s an e i g e n o p e r a t o r o f H- w i t h s p e c t r a l f r e q u e n c y a) . I n e x p e r i m e n t s , a broadened and s h i f t e d f r e q u e n c y i s always o b s e r v e d , and t h i s i s what b o t h m(to) or^7(a>) a r e d e s -c r i b i n g . That i s , c o l l i s i o n a l e f f e c t s r e s u l t i n p r e s s u r e 47 b r o a d e n i n g o f t h e f r e e p a r t i c l e f r e q u e n c i e s by t a k i n g a s p e c t r a l l i n e a t co and m o d i f y i n g i t t h r o u g h b i n a r y i n t e r a c t i o n s . F a n o 3 0 has shown t h a t t h e a n t i h e r m i t i a n p a r t o f m(co) i s p o s i t i v e - s e m i d e f i n i t e which i s a n e c e s s a r y c o n d i t i o n t h a t t h e master e q u a t i o n have an H-theorem. The same can be p r o v e n i or and t h u s , i n a s p e c i a l c a s e , f o r t h e W-S term. Hence, i n t h i s c a s e , decay t o e q u i l i b r i u m o c c u r s . C o n s i d e r now t h e case when more t h a n one n a t u r a l f r e q u e n c y a r i s e s i n a problem. Then, as was seen i n t h e l a s t s e c t i o n , i t i s not p o s s i b l e t o r e p l a c e by ftco , but r a t h e r t h e s p e c t r a l r e p r e s e n t a t i o n o f must be used. When th e b r o a d e n i n g due t o t h e t/f«o) ' s f o r each co i n the spectrum o f K-i s s m a l l so t h a t t h e o b s e r v e d l i n e s a r e w e l l s e p a r a t e d , t h e n the c o l l i s i o n s u p e r o p e r a t o r g i v e s r i s e t o a sum o f c o n t r i b u -t i o n s f o r t h e o b s e r v e d s p e c t r u m , each s p e c t r a l component e v o l -v i n g and d e c a y i n g i n d e p e n d e n t l y o f t h e o t h e r components. How-47 e v e r , as i s w e l l known i n microwave s p e c t r o s c o p y , f r e q u e n c y components do o v e r l a p ( e . g . t h e i n v e r s i o n s p ectrum o f N H 3 c i t e d e a r l i e r ) 2 ^ c o n s e q u e n t l y g i v i n g r i s e t o i n t e r f e r e n c e terms between d i f f e r e n t *7 c't°) components. I n e f f e c t t h e coup-l i n g c o n t r i b u t i o n s a l l o w f o r t h e p o s s i b i l i t y o f coherences o r t r a n s i e n t s w h i c h may b u i l d up w i t h i n a gas between f r e q u e n c y components. I t s h o u l d not be t o o s u r p r i s i n g t h e n , t h a t a g e n e r a l H-theorem f o r *3 i s not p o s s i b l e . I t has been argued HS e l s e w h e r e ' t h a t t h i s i n d e e d seems t o be t h e c a s e . T h i s can be r a t i o n a l i z e d by r e a l i z i n g t h a t t h e s e c o h e r e n c e s s h o u l d be o f a t r a n s i e n t n a t u r e and w i l l e v e n t u a l l y d i e o u t . Then each f r e q u e n c y component, l e f t t o i t s e l f , w i l l a l s o decay, and an o v e r a l l approach t o e q u i l i b r i u m f o l l o w s . To u n d e r s t a n d t h e o r i g i n o f t h e s e i n t e r f e r e n c e t e r m s , c o n s i d e r a square b r a c k e t o r c o l l i s i o n i n t e g r a l 1 0 , C n,nl = « R ' W J I H ^ Both A and B a r e two p a r t i c l e o p e r a t o r s . F o c u s i n g a t t e n t i o n on one f r e q u e n c y component o f A upon w h i c h "3 o p e r a t e s , c o n t r i -b u t i o n s t o t h e square b r a c k e t a r e a sum o f terms o f t h e form, (86) where (87) 49 where (89) R - T fljf X 0 ^ ' = /9 and KA =E A . , -fito. ,'=E -E , and A, , , K=E , A. , , k k r k k k " kk k k' k k 1 k' kk' T r e a t i n g o n l y t h e g a i n t e r m , each c o n t r i b u t i o n t o t h e sum i n Eq.(88) g i v e s (90) 1tjl't,»" > O b v i o u s l y f o r k=j and k'=j' t h e two r e s o l v e n t s can be combined t o g i v e i 2 7 r S ( o ) . Hence f o r t h i s case t h e energy l o s t from one m o l e c u l e i s g a i n e d by the o t h e r . I n t h e case t h a t k ^ j and/or k V j ' t h e s i t u a t i o n i s not so s i m p l e , and the d i f f e r e n t f r e q u e n c i e s g i v e r i s e t o p r i n c i p a l v a l u e t e r m s . F a n o 3 0 f i r s t i n c l u d e d such terms i n h i s m(to) o p e r a t o r , and p o i n t e d out t h a t t h e y m a n i f e s t t h e m s e l v e s p h y s i -c a l l y as f r e q u e n c y s h i f t s . T h i s i s not t h e o n l y s o u r c e f o r 50 f r e q u e n c y s h i f t s from t7 , s i n c e the l o s s term a l s o c o n t r i -b u t e s t o them. That i s , i n p e r f o r m i n g p r a c t i c a l c a l c u l a t i o n s t h e i n a d e q u a t e knowledge 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 and i g n o r a n c e o f t h e c o l l i s i o n dynamics t h w a r t s a t t e m p t s t o c a l -c u l a t e t h e s e f r e q u e n c y s h i f t s . Only w i t h t h i s i n f o r m a t i o n a v a i l a b l e i s i t p o s s i b l e t o d i s c o v e r t h e r e l a t i v e i m p o r t a n c e o f such terms and t o j u s t i f y a p p r o x i m a t i o n s w h i c h a r e i n v a r i a -b l y made i n o r d e r t o s i m p l i f y the c a l c u l a t i o n s . One u s e f u l a p p r o x i m a t i o n w h i c h e l i m i n a t e s c o l l i s i o n a l t r a n s i e n t s i s t o assume t h a t the v a l u e s o b t a i n e d f o r t h e v a r i o u s m a t r i x e l e -ments wh i c h c o n t r i b u t e t o QA,A] a r e i n s e n s i t i v e t o a l l f r e q u e n c i e s . T h i s can be e x p r e s s e d as and i s commonly c a l l e d t h e impact a p p r o x i m a t i o n ^ . W i t h i t , the p r i n c i p a l v a l u e terms and hence t r a n s i e n t e f f e c t s v a n i s h , and the f o r m a l c o l l i s i o n o p e r a t o r i s o f t h e W-S form. The meaning o f t h e impact a p p r o x i m a t i o n i s t h a t t h e c o l l i s i o n a l t r a n s i e n t s a r e v e r y s h o r t l i v e d and d i e out b e f o r e t h e c o l -l i s i o n i s o v e r . I t i s a l s o p o s s i b l e t o s p e c i f y an i n t e r m o l e c u l a r p o t e n t i a l , and t h e n e v a l u a t e t h e square b r a c k e t i n t e g r a l s . T h i s i n f a c t 5 1 As p a r t i a l l y u n d e r t a k e n i n P a r t I I I . One advantage o f t h i s i s t h a t t h e p r i n c i p a l v a l u e terms can be d i r e c t l y s t u d i e d r a t h e r t h a n e l i m i n a t e d as i s done i n an impact a p p r o x i m a t i o n . S i n c e t h e c o n d i t i o n f o r t r a n s i e n t e f f e c t s r e q u i r e s b o t h tox f<l and C 0 T j _ n ^ < l D U t not n e g l i g i b l e , i t might be e x p e c t e d t h a t ( J k ) T i n - t : be r e s t r i c t e d t o a s m a l l f r e q u e n c y r a n g e . T h i s s u g g e s t s a r e a s o n a b l e a p p r o x i m a t i o n might be t o assume a l l e f f e c t s from t r a n s i e n t s a re e q u a l and l e a d s t o as u s e f u l a s i m p l i f i c a t i o n i n t r e a t i n g the c o l l i s i o n b r a c k e t s as the impact a p p r o x i m a t i o n . E f f e c t i v e l y t h i s i s <<A\i,'J | A>>-<<A 14,*3b*)\A>> . I n p a r t i c u l a r f o r a l l e f f e c t s w h i c h a r i s e due t o f r e q u e n c i e s o n l y t h o s e w h i c h a r e not phase r a n d o m i z e d , i . e . u )T^<l, need t o be c o n s i d e r e d . Of t h e s e f r e q u e n c i e s , the a s s u m p t i o n s t a t e s t h a t t h e c o l l i s i o n a l i n t e g r a l s a r e i n s e n s i t i v e t o v a r i a t i o n s i n to and hence, i f t o x i n t < l and not n e g l i g i b l e , a l l such c o n t -r i b u t i o n s from a l l f r e q u e n c i e s a r e t h e same. I n g e n e r a l E q . ( 8 6 ) has r e a l and i m a g i n a r y p a r t s and t h e s e c o n t r i b u t e r e s p e c t i v e l y t o the l i n e w i d t h 1/x.. and f r e q u e n c y s h i f t 6 . . ( 9 2 ) 52 Although simple formulae f o r the h e r m i t i a n , and a n t i -hermit i a n , p a r t s of "3 cannot g e n e r a l l y be found, they can be f o r m a l l y r e l a t e d by* a. a* and (94) 7 r ^ + Z X . Consequently Eq.(92) can be w r i t t e n as (95) ^ W C ^ - = - XWLlkd" + Hence 1/x =-<<A- I 3 « |A >>(A) i s r e a l w h i le i j i j i j i6^j=i<<A^,j I |A.j.j>>f*} i s pure imaginary. Furthermore i f * The s u p e r o p e r a t o r a d j o i n t 1s i s d e f i n e d f o r a s u p e r o p e r a t o r a by « f l i a i a » = < : < a * f l i G » . 53 = - « / i ftH T and 6 v a n i s h e s . The f i r s t e q u a l i t y f o l l o w s from t h e h e r m i t i c i t y o f , t h e second from the f a c t t h a t t7 p r e s e r v e s a d j o i n t n e s s and t h e t h i r d f r o m A. •= At. . A l s o f o r one s p e c t r a l component o f *3 , t h e a n t i h e r m i t i a n p a r t o f i s n e g a t i v e s e m i d e f i n i t e and the l i n e w i d t h s obey ( 9 7 ) Comparison o f E q . ( 8 6 ) w i t h E q . ( 9 2 ) i m p l i e s t h a t 54 where T h i s j u s t r e f l e c t s t h e f a c t t h a t even though th e r a t e o f decay o f t h e whole system i s a sum o f t h e i n d i v i d u a l r a t e s , t h e t i m e f o r t h e whole system t o decay i s t h e r e c i p r o c a l sum o f t h e i n d i v i d u a l r a t e s . F o r t h e o f f - d i a g o n a l c o n t r i b u t i o n s ,<<A. ,\X. 'O I A . >>t? t h e r e i j 1 ' k j a r e no s i m p l e r e l a t i o n s h i p s nor p o s i t i v e d e f i n i t n e s s . I n t h e impact a p p r o x i m a t i o n where *3(o) i s used t h e r e a r e c o n s i d e r a b l e s i m p l i f i c a t i o n s and symmetry p r o p e r t i e s a r i s i n g i n t h i s case liq 41 have been d i s c u s s e d by Ben-Reuven y and by S n i d e r 55 TABLE 1 V a l i d i t y C o n d i t i o n s o f Quantum M e c h a n i c a l Boltzmann E q u a t i o n s E q u a t i o n V a l i d i t y C o n d i t i o n s WCU Only B i n a r y C o l l . x >x. ^ COX^>>1 f i n t f w-s » » 0 > a a , T f » i G e n e r a l i z e d Boltzmann E q u a t i o n ( G B E ) w .x. <_<<1 mm' m t S p e c i a l L i m i t i n g Cases C o n d i t i o n s S i m p l e s t E q u a t i o n 1- t o a a ' T f > > 1 / ' W C U mm' i -' 2. co „ / X->>lJ aa i ' and "mm'V^J a ) " m m ^ i n t ^ 1 G B E b ) ° W T i n t < < : L W ~ S F o r a or m f r e q u e n c i e s tox*.=l JL a ) a j T i n t " 1 G B E b) a ) T i n t < < 1 w _ s PART I I NUCLEAR SPIN RELAXATION OF GASES 57 CHAPTER 1 INTRODUCTION I n P a r t I some f u n d a m e n t a l i d e a s c o n c e r n i n g t h e m o t i o n of gases a r e p r e s e n t e d . Now t h e s e c o n c e p t s a r e a p p l i e d t o d e s c r i b i n g t h e l o s s o f n u c l e a r magnetism w h i c h may have b u i l t up w i t h i n a gas. T h i s magnetism i s due t o the n u c l e a r s p i n s w h i c h c e r t a i n a tomic n u c l e i p o s s e s s . P o l y a t o m i c m o l e c u l e s may have s e v e r a l d i f f e r e n t s uch n u c l e i , as w e l l . a s o t h e r degrees of freedom and a p p l i c a t i o n o f an e x t e r n a l magnetic f i e l d c auses the t o t a l n u c l e a r s p i n s t a t e t o s p l i t i n t o a m u l t i p l e t o f nondegenerate d i r e c t i o n a l l y dependent components. The ones most n e a r l y " l i n e d up" w i t h the e x t e r n a l f i e l d a r e the most e n e r g e t i c a l l y f a v o u r a b l e , and the m o l e c u l e s attempt t o ob-t a i n a Boltzmann d i s t r i b u t i o n among the l e v e l s . I n d o i n g s o , a t r a n s f e r o f energy t o the o t h e r modes o f m o t i o n w i t h i n t h e system (known as t h e l a t t i c e ) may o c c u r . I n o r d e r t o t r a n s f e r energy between n u c l e a r s p i n s t a t e s and l a t t i c e s t a t e s , t h e r e must be a mechanism by which t h e two a r e c o u p l e d . F o r s o l i d s and l i q u i d s t h e most e f f i c i e n t t r a n s f e r a r i s e s from t h e e f f e c t s o f n e i g h b o u r i n g m o l e c u l e s ( i n t e r m o l e c u l a r p r o c e s s e s ) but f o r 58 g a s e s , where most o f t h e time the m o l e c u l e s a r e f r e e , t h e t r a n s f e r o c c u r s w i t h i n t h a t same m o l e c u l e t o o t h e r molecu-l a r degrees o f freedom and t h e s e i n t u r n t r a n s f e r energy d u r i n g t h e b r i e f i n t e r m o l e c u l a r p r o c e s s t o the r e s t o f t h e system ( i n t r a m o l e c u l a r p r o c e s s e s ) . I n t h i s way t h e n u c l e a r s p i n energy i s d i s t r i b u t e d t h r o u g h o u t t h e gas and s p i n r e -l a x a t i o n o c c u r s . The r a t e o f change i n t h e m a g n e t i z a t i o n M ( t ) i s o f t e n s u f f i c i e n t l y w e l l a p p r o x i m a t e d by t h e p h e n o m e n o l o g i c a l B l o c h e q u a t i o n s , On t h e b a s i s o f symmetry, the r e l a x a t i o n t e n s o r f o r a gas can be w r i t t e n CD d rjct) d t C2) T 77 where Tj_ and a r e t h e l o n g i t u d i n a l and t r a n s v e r s e r e l a x a t i o n t i m e s and a i s t h e i n t e r m o l e c u l a r c h e m i c a l s h i f t . 5 9 T h i s means t h a t f o r many s p i n s y s t e m s , the time f o r energy t o be t r a n s f e r r e d between the s p i n s and the l a t t i c e can be c h a r a c t e r i z e d by one t i m e c o n s t a n t c a l l e d the s p i n l a t t i c e r e l a x a t i o n t i m e T-^ . The most p o p u l a r way o f i n t e r p r e t i n g the s p i n l a t t i c e r e l a x a t i o n r a t e , T-j_ ^ i s i n terms o f t h e o s p e c t r a l d e n s i t i e s w h i c h a r i s e f o r t h e v a r i o u s i n t r a m o l e -c u l a r p r o c e s s e s . Such s p e c t r a l d e n s i t i e s c o r r e c t l y i n c l u d e t h e whole m o l e c u l a r spectrum and depending on the mechanism and the m o l e c u l a r geometry they r e f l e c t t h e v a r i o u s a l l o w e d t r a n s i t i o n s between the m o l e c u l a r energy l e v e l s . However t h e i n t e n s i t y due t o most of t h e s e t r a n s i t i o n s c o n t r i b u t e s n e g l i g i b l y t o the r e l a x a t i o n r a t e . Only the s m a l l e s t r o t a -t i o n a l and n u c l e a r Zeeman energy d i f f e r e n c e s ( f r e q u e n c i e s ) need be r e t a i n e d i n most c a s e s . T h i s i s b e c a u s e , a t t h e d e n s i t i e s where most e x p e r i m e n t s a r e p e r f o r m e d , t h e h i g h e r f r e q u e n c i e s a r e phase randomized.. However the b e h a v i o u r o f t h e s m a l l e r f r e q u e n c i e s i s t y p i c a l o f what happens t o a l l f r e q u e n c y components. That i s , as x f i n c r e a s e s t h e c o n t r i -b u t i o n t o the s p e c t r a l d e n s i t y from t h i s f r e q u e n c y d e c r e a s e s due t o phase r a n d o m i z a t i o n . Two u s e f u l ways o f g r a p h i n g t h i s b e h a v i o u r a r e o f t e n i n v o k e d . F i r s t t h e f r e q u e n c y spectrum i s p l o t t e d f o r v a r i o u s d e n s i t i e s , and t h e s p e c t r a l l i n e s broaden as a r e s u l t o f i n c r e a s e d p r e s s u r e . A l t e r n a t i v e l y , i n s t e a d o f 60 c o n s i d e r i n g the whole f r e q u e n c y spectrum f o r v a r i o u s d e n s i t i e s , a component o f t h e s p e c t r a l d e n s i t y a t t h e a p p r o p r i a t e i n t e r -n a l t r a n s i t i o n f r e q u e n c y t o 0 can be p l o t t e d a g a i n s t t h e d e n s i t y . T h i s g i v e s a maximum c o n t r i b u t i o n a t a d e n s i t y such t h a t o o ^ x ^ l . B e t t e r y e t i s t o p l o t n t i m e s t h e s p e c t r a l d e n s i t y a g a i n s t d e n s i t y and i n t h i s p l o t , each c o n t r i b u t i o n i n c r e a s e s f r om z e r o a t z e r o d e n s i t y t o a f i n i t e v a l u e at l a r g e d e n s i t i e s . T h i s i s i n f a c t t h e c o n t r i b u t i on t o n/T^ from t h i s f r e q u e n c y component. A l t h o u g h the concept o f o v e r l a p p i n g s p e c t r a l l i n e s i s u s e f u l i n s p e c t r o s c o p i c s t u d i e s where f r e q u e n c y i s p l o t t e d d i r e c t l y , f o r n u c l e a r r e l a x a t i o n e x p e r i m e n t s where t h e d e n s i t y dependence o f T^ i s s t u d i e d , t h e n/T^ v e r s u s n p l o t i s p a r t i -c u l a r l y i l l u m i n a t i n g and i s t h e method w h i c h b e s t i l l u s t r a t e s phase r a n d o m i z a t i o n . These t h r e e t y p e s o f g r a p h i n g a r e i l l u s -t r a t e d i n f i g . C D . At h i g h e r d e n s i t i e s , the t o t a l s p e c t r a l d e n s i t y w i l l have c o n t r i b u t i o n s from o t h e r (as w e l l as t h e Zeeman) f r e q u e n c y components s i n c e i n t h e f r e q u e n c y spectrum p i c t u r e t h e s e com-ponents have broadened so much t h a t t h e s p e c t r a l d e n s i t y i s n o n - n e g l i g i b l e a t the low f r e q u e n c i e s where t h e e x p e r i m e n t s a r e u s u a l l y p e r f o r m e d . These h i g h e r f r e q u e n c i e s c o r r e s p o n d t o the r o t a t i o n a l l i n e s , and t h e i n t e r p r e t a t i o n o f r e c e n t e x p e r i -ments w h i c h a r e d i s c u s s e d i n P a r t IV r e q u i r e s t h a t they must be i n c l u d e d i n the e x p r e s s i o n f o r T" 1. P r e s s u r e b r o a d e n i n g "3 «JO\ ) n t < n 2 < n 3 n F i g u r e 1. D i f f e r e n t Ways of G r a p h i n g the D e n s i t y Dependence o f S p e c t r a l H L i n e s . 62 o f t h e s e l i n e s i s e s s e n t i a l t o a c o r r e c t d e s c r i p t i o n o f t h e p r o c e s s e s . Moreover d i f f e r e n t r e l a x a t i o n t i m e s must be c o n -s i d e r e d f o r d i f f e r e n t components and t h e p r o p e r c o l l i s i o n a l s h i f t s o f t h e l i n e s must be i n c l u d e d . I t i s the purpose h e r e t o d e r i v e f o r m u l a e f o r w h i c h account f o r such f r e q u e n -c i e s i n c l u d i n g t h e e f f 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 s h i f t -i n g . I t i s assumed t h r o u g h o u t t h a t t h e d e s c r i p t i o n o f a gas p r e s e n t e d i n P a r t I can be used f o r t h e s p i n systems h e r e . I n p a r t i c u l a r , t h e t i m e s c a l e s and f r e q u e n c i e s obey the same l i m i t s and bounds as d i s c u s s e d e a r l i e r , t h e s t a t e o f t h e gas i s a d e q u a t e l y d e s c r i b e d by a one p a r t i c l e d e n s i t y o p e r a t o r , and the g e n e r a l i z e d Boltzmann e q u a t i o n can be employed. F o r n u c l e a r s p i n s y s t e m s , i t i s e x p e c t e d t h a t the range o f a p p l i -c a b i l i t y o f t h e B o l tzmann e q u a t i o n i s v a l i d up t o f a i r l y h i g h d e n s i t i e s even though the system i s r e s t r i c t e d t o b e i n g a d i l u t e gas. However, as d i s c u s s e d i n P a r t I , a t t h e s e h i g h e r d e n s i t i e s d e c r e a s e s . There a r e two consequences o f t h i s . F i r s t , as s t a t e d above, v a r i o u s f r e q u e n c i e s w h i c h a r e u s u a l l y phase randomized i n a r e l a x a t i o n e xperiment become i m p o r t a n t and must be i n c l u d e d . Second, as d e c r e a s e s w i t h i n c r e a s -i n g d e n s i t y i t approaches T j _ n t , and c o l l i s i o n a l t r a n s i e n t e f f e c t s become i m p o r t a n t when the f r e q u e n c i e s a r e l a r g e enough 63 so t h a t x i s n o n - n e g l i g i b l e , but s t i l l ux f,<l. I t i s i n t 1 f o r t h e s e r e a s o n s t h a t t h e g e n e r a l i z e d Boltzmann e q u a t i o n must be used. The o n l y p r e v i o u s d e t a i l e d t h e o r i e s w h i c h e x i s t f o r gases a r e t h o s e by Oppenheim and Bloom50 5 51 where gases and l i q u i d s a r e s t u d i e d from a t i m e c o r r e l a t i o n p o i n t o f v i e w , and by Chen and S n i d e r 1 9 > ^ ^ W h e r e gases a r e t r e a t e d from a Boltzmann e q u a t i o n p o i n t o f v i e w . The d i s c u s s i o n i n t h i s t h e s i s f o l l o w s c l o s e l y the l a t t e r a p p r o a c h . The r e l a t i o n s h i p between r e s u l t s o b t a i n e d by the Boltzmann approach and by t i m e c o r r e l a t i o n t h e o r y are compared. I t i s shown t h a t f o r s p i n systems t h e two methods a r e e q u i v a l e n t . 6 4 CHAPTER 2 BOLTZMANN EQUATION APPROACH TO NUCLEAR SPIN RELAXATION I t has a l r e a d y been argued t h a t f o r a d i l u t e g a s , an 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 i s t h e g e n e r a l i z e d Boltzmann e q u a t i o n . F u r t h e r , s i n c e t h e n o n - e q u i l i b r i u m m a g n e t i z a t i o n i s assumed t o be o n l y l i n e a r l y d i s p l a c e d from the e q u i l i b r i u m m a g n e t i z a t i o n , M ^ j i t i s s u f f i c i e n t t o use t h e l i n e a r i z e d B oltzmann e q u a t i o n , E q . ( I - 7 D , where t h e d e n s i t y o p e r a t o r ( o r t h e c o r r e s p o n d i n g Wigner-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 ) f i s (3) at t ( 4 ) f f o r one p a r t i c l e . The e q u i l i b r i u m i n t e r n a l s t a t e one p a r t i c l e 65 d e n s i t y o p e r a t o r i s ( 5 ) where ini i s t h e i n t e r n a l s t a t e one p a r t i c l e h a m i l t o n i a n . I t i s assumed t h a t t h e m o l e c u l e s a r e i n t h e i r e l e c t r o n i c and v i b r a t i o n a l ground s t a t e s , and t h a t t h e s e s t a t e s have no d e g e n e r a c i e s a s s o c i a t e d w i t h them. F u r t h e r , t h e r o t a t i o n a l and n u c l e a r s p i n s t a t e s a r e assumed t o be o n l y weakly c o u p l e d so t h a t J and I a r e good quantum numbers, and i n d e p e n d e n t l y p r e c e s s about t h e e x t e r n a l f i e l d . I n o t h e r words t h e e x t e r n a l f i e l d i s l a r g e enough so t h a t J and J _ are f a r more s t r o n g l y c o u p l e d t o t h e f i e l d d i r e c t i o n t h a n t o each o t h e r . T h i s h i g h f i e l d l i m i t i s v a l i d whenever t h e c o u p l i n g c o n s t a n t c between J and I_ and t h e Zeeman f r e q u e n c i e s w ^ obey t h e f o l l o w i n g i n e q u a l i t y - * 2 T h i s i s t r u e under u s u a l e x p e r i m e n t a l c o n d i t i o n s f o r n u c l e i i n a f i e l d o f s e v e r a l thousand gauss. C < 1 ( 6 ) • 66 S i n c e t h e r o t a t i o n a l and s p i n quantum numbers J and I a r e good, the p r o j e c t i o n o p e r a t o r s i n t h e p e r t u r b a t i o n , E q . ( I - 7 6 ) , can be c o n s i d e r e d as p r o j e c t i n g onto t h e s e s t a t e s . S i n c e symmetry p r o p e r t i e s may r e s t r i c t r o t a t i o n and s p i n s t a t e s ( e . g . o r t h o - p a r a h y d r o g e n ) , a p r o j e c t i o n onto a r o t a t i o n a l s t a t e may r e s t r i c t t h e s p i n s t a t e . Hence the p r o j e c t i o n o p e r a t o r s a r e t a k e n as C7> = T..P* where "T^ o p e r a t e s on s p i n s o n l y and & on r o t a t i o n s o n l y . The one p a r t i c l e h a m i l t o n i a n f o r t h e i n t e r n a l motions o n l y C n o ' 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 t a k e n a s , C 8 ) w h i c h i n c l u d e s t h e n u c l e a r and r o t a t i o n a l Zeeman h a m i l t o n i a n s , H_T and % and the r o t a t i o n a l h a m i l t o n i a n % r e s p e c t i v e l y . The moment o f i n e r t i a t e n s o r , "L^ , i s c o n s t a n t i n the body f i x e d frame o f t h e m o l e c u l e , and t h e c o n t r i b u t i o n , M-c > i s t h e i n t r a m o l e c u l a r c o u p l i n g h a m i l t o n i a n . T h i s can be w r i t t e n i n terms o f " l a t t i c e " , F 0„ , and s p i n A„ o p e r a t o r s , 6 7 A c t u a l l y A and F are t h e q t h s p h e r i c a l components o f t h e £-th rank i r r e d u c i b l e t e n s o r s , A , and F (see XJ XJ a p p e n d i x (A)) and t h e c o u p l i n g c o n s t a n t f o r the q t h compon-e n t s i s C^q. I n t h i s way, b o t h F ^ q and A £ q s a t i s f y the commutation r e l a t i o n s , where I and J a r e t h e z-components o f the s p i n and Z Z 1+ and J J + a r e t h e r a i s i n g and l o w e r i n g s p i n and r o t a t i o n o p e r a t o r s , e.g. I ± =/2 I e%\ , A l s o and F ^ q form o r t h o g o n a l s e t s o f o p e r a t o r s namely 68 The c o r r e s p o n d i n g L i o u v i l l e s u p e r o p e r a t o r i s a n a l o g o u s l y . (12) X = X L x + X z t X + X c . F i n a l l y t h e e q u i l i b r i u m d e n s i t y o p e r a t o r , E q . ( 5 ) , i s a p p r o x i m a t e d i n t h e h i g h t e m p e r a t u r e l i m i t by, as l o n g as the c o n d i t i o n s a r e a l l s a t i s f i e d . I t r e mains now t o s p e c i f y t h e p e r t u r b a t i o n which a c c o u n t s f o r t h e n o n - e q u i l i b r i u m p a r t o f t h e d e n s i t y o p e r a t o r . I n o r d e r t o account f o r t h e n o n - e q u i l i b r i u m m a g n e t i z a t i o n , t h e 5 l i n e a r e x p a n s i o n must have a term p r o p g r t i o n t a l t o J_, t h a t i s (13) 0 = b i t ) - I 69 where now I_ i s t h e n u c l e a r s p i n f o r one m o l e c u l e . b_(t) measures the d e v i a t i o n o f t h e s p i n p o l a r i z a t i o n from e q u i l i -b r i u m a t t i m e t . However a p e r t u r b a t i o n o f t h e form o f E q . ( 1 5 ) i s g r o s s l y i n a d e q u a t e i n a c c o u n t i n g f o r s p i n r e l a x a -t i o n i n p o l y a t o m i c s because c o l l i s i o n s a r e v i r t u a l l y i n -e f f e c t i v e a t c h a n g i n g n u c l e a r s p i n s t a t e s - ' . T h i s a p p r o x i -m a t i o n i s w r i t t e n w i t h t h e r e s u l t , see Eq.( 3 ) , t h a t no s p i n r e l a x a t i o n o c c u r s , v i a an i n t r a m o l e c u l a r mechanism. T h i s means t h a t t h e l a t t i c e must be p o l a r i z e d by c o u p l i n g w i t h t h e n u c l e a r s p i n system and t h i s l a t t i c e p o l a r i z a t i o n i s i n t u r n l o s t by t h o s e c o l -l i s i o n s w h i c h a r e e f f e c t i v e at c h a n g i n g th e l a t t i c e s t a t e s . C o n s e q u e n t l y i t i s n a t u r a l t o c o n s i d e r t h e s i m u l t a n e o u s p o l a r i z a t i o n o f b o t h th e l a t t i c e and t h e n u c l e a r s p i n s . Such terms s h o u l d appear i n t h e n o n - e q u i l i b r i u m p a r t o f t h e d e n s i t y o p e r a t o r and f o r s p i n systems an adequate form f o r <{> i s (17) 7 0 i s t h e p r o j e c t i o n o p e r a t o r , E q . ( 7 ) . G^'^ has a s i m i l a r meaning t o b_(t) and measures the v a r i o u s components o f the s i m u l t a n e o u s p o l a r i z a t i o n s o f t h e s p i n s and l a t t i c e w i t h i n a m o l e c u l e . By s u b s t i t u t i o n o f t h e p e r t u r b a t i o n , E q . ( 1 7 ) and t h e i n t e r n a l h a m i l t o n i a n E q . ( 8 ) i n t o t h e l i n e a r i z e d Boltzmann e q u a t i o n , moment e q u a t i o n s can be found by m u l t i p l y i n g by, « l | and by < < ^ A £ , q " F t f f f i I • T h e f l r s t o f t h e s e i s Here use has been made o f the a p p r o x i m a t i o n t h a t c o l l i s i o n s a r e i n e f f e c t i v e a t c h a n g i n g n u c l e a r s p i n s and hence the r i g h t hand s i d e i s s e t e q u a l t o z e r o (see Eq. 1 6 ) . E q . ( l 8 ) i s i n t h e form o f an inhomogeneous B l o c h e q u a t i o n w i t h i n h o m o g e n e i t y G w h i c h , t o reduce t o t h e B l o c h e q u a t i o n , must be e x p r e s s e d i n terms o f b ( t ) . I f t h e magnetic f i e l d i s t a k e n i n t h e z d i r e c t i o n , H =H z, t h e n i t i s c l e a r * —o o from E q . ( l ) , t h a t t h e z component o f b w i l l r e s u l t i n t h e B l o c h e q u a t i o n f o r w h i l e t h e x and y components 71 l e a d t o an e x p r e s s i o n f o r . Thus t o c a l c u l a t e T-^  , E q . ( l 8 ) may be s i m p l i f i e d t o i (t) = _XJ ***** fr « **W«l»^%?AtA^ The e q u a t i o n s l e a d i n g t o T 2 a r e ( 2 0 ) Ci-.^ll SfrS I A*.. Gl»»h) and a s i m i l a r e q u a t i o n f o r b . The moment e q u a t i o n f o r GH»-q i s a a1 a t (21) 2 _ J r -72 w h e r e t h e f r e q u e n c i e s a r e to=u)T - to , a n d C22) C J ^ = 11 : ( z i W > I ? : f e ) ) ^ > w h i l e t h e c h a r a c t e r i s t i c t i m e f o r d e c a y o f o n e " c o m p o n e n t " i s C23). I n o b t a i n i n g E q . ( 2 1 ) , a t e r m h a s b e e n n e g l e c t e d w h i c h i s p r o p o r t i o n a l t o C„ G ^ ' , " ^ • T h i s t e r m w o u l d i n v o l v e s e c o n d ^ ^ Zq aa o r d e r c o u p l i n g o f t h e l a t t i c e t o t h e s p i n s a n d i s a s s u m e d n e g l i g i b l e . E x a m i n a t i o n o f E q.(23) r e v e a l s t h a t T ' i s u n s y m m e t r i -c a l l y n o r m a l i z e d b e t w e e n i n d i c e s a a ' a n d 86' . I t i s 55 b o t h p h y s i c a l l y a n d m a t h e m a t i c a l l y c o n v e n i e n t t o h a v e s y m m e t r i c a l l y n o r m a l i z e d T'S , a n d o n e w a y t h i s c a n b e a c c o m p l i s h e d i s b y d e f i n i n g d " 8 > o<oi ' C2H) % c—'pf) " j«f>) VtL~"'et'J 73 where t h e square o f t h e norm i s C25) f J '2}T = « **-t ?>V V- I fi't » T h i s n a t u r a l l y l e a d s t o a r e n o r m a l i z a t i o n o f b o t h E q s. ( 1 9 ) and ( 2 1 ) . S u b s t i t u t i o n o f x _ 1 i n t o E q . ( 2 1 ) g i v e s ^ "t C26) and f o r t h e b z e q u a t i o n C 2 7 } * b , a ) ^ _ x\\ g - ° * c ' f r f r <*~' where ( 2 8 ) ^e<ot' = C 74 F u r t h e r , t h e q u a n t i t i e s d ^ l a n d j" 1^, c a n b e t r e a t e d a s v e c t o r s .djM s p a n n i n g t h e i n d e x s p a c e ( a a ' ) w h i l e ( 2 9 ) Mococ'pp' - PP') i s a c o m p o n e n t o f a r e l a x a t i o n m a t r i x R ^ q i n t h i s s p a c e . C o n s e q u e n t l y E q s . ( 2 6 ) a n d ( 2 7 ) c a n b e w r i t t e n a s ( 3 0 ) cJ t I ( I - H ) C « t » ) a n d ( 3 D 7 5 where the dot p r o d u c t f o r ( aa') space i s ( 3 2 ) etc*.' S i n c e t h e r a t e o f decay o f b depends on t h e s i z e o f C , i t can be seen t h a t b w i l l r e l a x more s l o w l y t h a n the &q -"J* components w h i c h i n t u r n r e l a x t h r o u g h c o l l i s i o n s . As p. a r e s u l t o f t h i s d i f f e r e n c e s i n t ime s c a l e s , & can be c o n s i d e r e d t o be i n a s t e a d y s t a t e t h a t i s r o t a t i n g i n t h e frame o f b _ 7 J That i s , t h e o n l y components o f r w h i c h do not r a p i d l y decay out by c o l l i s i o n s a r e t h o s e w h i c h a r e i n phase w i t h t h e f r e e l y r o t a t i n g b . T h i s can be e x p r e s s e d as ( 3 3 ) at W i t h t h e c o n s t a n t r o t a t i n g frame a p p r o x i m a t i o n , E q . ( 3 1 ) can be f o r m a l l y s o l v e d , t h a t i s 7 6 C3*U ? - X - ^ f-0 p r o v i d e d t h e i n v e r s e o f t h e m a t r i x R^l e x i s t s . S u b s t i t u t i o n o f t h i s i n t o Eq.(30), g i v e s a r e s u l t i d e n t i c a l t o t h e B l o c h e q u a t i o n , f rom which. T^ can be i d e n t i f i e d as T h i s e q u a t i o n can be used t o t r e a t t h e e f f e c t s o f h i g h f r e q u e n c i e s <^aa' s i n c e a l l t h e s e appear n a t u r a l l y i n t h e r e l a x a t i o n m a t r i x . F u r t h e r m o r e , t h e x's depend on t h e p a r t i c u l a r s t a t e s i n t o w h i c h t h e y a r e p r o j e c t e d , and a gener-a l i z e d c o l l i s i o n c r o s s s e c t i o n f o r t h e s e s t a t e s can be d e f i n e d as G6") (j ( W / P P ' ) = 77 where t h e average r e l a t i v e speed f o r m o l e c u l e s o f mass y i s <v> = 78kT/yiT When h i g h f r e q u e n c y terms a r e phase randomized o r i g n o r e d , t h e e x p r e s s i o n f o r T s i m p l i f i e s s i n c e t r a n s i t i o n between d i f f e r e n t a l e v e l s a r e u n i m p o r t a n t . Thus the o n l y c o l l i s i o n t i m e s t h a t e n t e r a r e t h e E x " 1 ( a a | 8 8 ) ^ x T 1 The d i s t i n c t i o n i s u s u a l l y n e v e r made between t h e a t h components o f x and i t i s a good a p p r o x i m a t i o n t o assume a l l X T 1 independent o f a . I n f a c t i t i s found e x p e r i -£q m e n t a l l y t h a t x 0 i s even independent o f t h e t e n s o r compon-x-q ent q , and w i t h t h i s a s s u m p t i o n , S i m i l a r t o Eq.C36). the e f f e c t i v e c o l l i s i o n c r o s s s e c t i o n can be i d e n t i f i e d as - T C371 78 In t h i s way the e x p r e s s i o n f o r reduces to the form u s u a l l y encountered i n NMR. By s u b s t i t u t i n g d ^ ^ w r i t t e n w i t h the s p i n and l a t t i c e pa r t s separa ted , and us ing the completeness o f the p r o j e c t i o n o p e r a t o r , £P^=1, Eq.(35) f o r T becomes, The s p e c t r a l d e n s i t y i s de f ined by , T W«V8uS) = - a. 4« « hf I ( */t,t + « ) " I & , » (39) The e x p r e s s i o n f o r T , E q.(35), i s c o n s i s t e n t w i t h r e s u l t s ob ta ined from other approaches and can be used to eva lua te T^ i n s p e c i a l cases . T h i s i s demonstrated to some exten t i n chapter 4. In order to o b t a i n a va lue f o r T-^  i t i s necessary to know: 1) The c o u p l i n g cons tants C which can sometimes be c a l c u l a t e d from f i r s t p r i n c i p l e s , or when t h i s i s not 79 p o s s i b l e , are t h e o b j e c t o f i n t e n s e e x p e r i m e n t a l p u r s u i t . 2} M o l e c u l a r c o n s t a n t s ; such as r o t a t i o n a l c o n s t a n t s , s p a t i a l arrangement of t h e atoms w i t h i n a m o l e c u l e , n u c l e a r s p i n v a l u e , yT , y , e t c . As a r u l e t h e s e can be found ex-cept f o r complex m o l e c u l e s . 3) The i n t e r m o l e c u l a r p o t e n t i a l : n e c e s s a r y t o c a l c u l a t e T . I n p r i n c i p l e t h i s can be c a l c u l a t e d , but because o f t h e c o m p l e x i t y o f the problem i t has been done o n l y f o r t h e s i m p l e s t o f m o l e c u l e s . I n most c a s e s , t h i s i s an unknown q u a n t i t y which i n h i b i t s a c t u a l e v a l u -a t i o n o f the T ' S and h a s , f o r the most p a r t , escaped e x p e r i m e n t a l p r o b e s . I n summary, the above c a l c u l a t i o n o f T-^  f o l l o w s t h e sim-p l e p r o c e d u r e o f t a k i n g a p p r o p r i a t e moments of the l i n e a r i z e d B o ltzmann e q u a t i o n a f t e r s p e c i f y i n g an a p p r o ximate form f o r the p e r t u r b a t i o n , <J> , o f the n o n - e q u i l i b r i u m p a r t o f t h e d e n s i t y o p e r a t o r . The c h o i c e o f <j> i s c r i t i c a l , but the c h o i c e here f o l l o w s c l o s e l y from the p h y s i c s and can be m o d i f i e d t o t a k e I n t o account any l i n e a r phenomena t h a t a r i s e . C o n s t r u c t i n g t h e moment e q u a t i o n s by use o f s t a n d a r d commutation r e l a t i o n s and t e n s o r a n a l y s i s g i v e s e x a c t f o r m u l a e f o r the change of t h e v a r i o u s c o e f f i c i e n t s w hich appear i n <f> . T h i s r e s u l t s i n a c o u p l e d s e t o f f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n s whose g e n e r a l a n a l y t i c a l s o l u t i o n can be a d i f f i c u l t t a s k . O f t e n , however, 80 i t i s p o s s i b l e t o approximate t h e s e t as was done w i t h the c o n s t a n t r o t a t i n g frame a s s u m p t i o n , Eq.(33). T h i s p a r t i -c u l a r a p p r o x i m a t i o n i n s e r t s o n l y the dominant f r e q u e n c y I n t o t h e s p e c t r a l d e n s i t y by i g n o r i n g t h o s e c o n t r i b u t i o n s w h i c h a r e out o f phase w i t h u>-^ and assumed s m a l l . S u b s t i t u t i o n o f t h i s s o l u t i o n i n t o t h e f i r s t moment e q u a t i o n g i v e s a l i n e a r e q u a t i o n f o r b . S i n c e b i s p r o p o r t i o n a l t o t h e non-e q u i l i b r i u m m a g n e t i z a t i o n , t h i s e q u a t i o n has t h e same form as t h e B l o c h e q u a t i o n , and T^ can be i d e n t i f i e d i n terms o f m o l e c u l a r p a r a m e t e r s and c o l l i s i o n f r e q u e n c i e s . 81 CHAPTER 3 CORRELATION FUNCTION APPROACH At z e r o time i t i s assumed t h a t t h e system i s i n a s t a t i c magnetic f i e l d , H Q + AH . At t h i s t i m e , t h e AH f i e l d i s t u r n e d o f f , and t h e m a g n e t i z a t i o n b e g i n s t o decay from M(0) t o t h e e q u i l i b r i u m m a g n e t i z a t i o n which i s d i s -p l a y e d i n t h e s t a t i c magnetic f i e l d H^. I t i s assumed t h a t t o a good a p p r o x i m a t i o n , t h e m a g n e t i z a t i o n i s a l i n e a r f u n c -t i o n o f t h e magnetic f i e l d and t h a t t h e decay from M(0) t o o c c u r s l i n e a r l y . The o b j e c t i s t h e n t o account f o r the l i n e a r decay o f the n o n - e q u i l i b r i u m m a g n e t i z a t i o n , a f t e r AH i s removed, i n terms o f t h e d y n a m i c a l p r o p e r t i e s o f the system. F o r m a l l y t h e m a g n e t i z a t i o n M ( t ) i s g i v e n by d o ) M(t) = JA. ^ ( t ) ^ (N) (N) where y i s t h e magnetic moment o p e r a t o r and p i s the d e n s i t y o p e r a t o r f o r the N - p a r t i c l e system. At e q u i l i b r i u m t h e d i s t r i b u t i o n w hich t h e system a t t a i n s i s t a k e n t o be 82 B o l t z m a n n ( o r t h e c a n o n i c a l e n s e m b l e ) , (41) (si) , i s composed o f a n u c l e a r Zeeman term w h i c h g i v e s r i s e t o t h e m a g n e t i z a t i o n , and a t e r m , l e f t q u i t e g e n e r a l f o r now, w h i c h s p e c i f i e s the " l a t t i c e " and t h e coup-l i n g s between t h e s p i n s and the l a t t i c e , H'u) = - ^ - H 0 + H ? A n a l o g o u s l y , 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 i s (43) JT - X 2 j + X u K f A / ) c o n t a i n s b o t h t h e i n t e r - and i n t r a - m o l e c u l a r c o n t r i -b u t i o n s . F o r i d e n t i c a l n u c l e i , t h e magnetic moment o p e r a t o r 83 CM) ^ = £ ^ 1 where i s t h e t o t a l n u c l e a r s p i n f o r t h e whole N - p a r t i c l e system. From t h e s e e q u a t i o n s t h e e q u i l i b r i u m m a g n e t i z a t i o n i s g i v e n by, On t h e b a s i s t h a t t h e magnetic f i e l d , H , i s weak enough —o so t h a t t h e Zeeman s p l i t t i n g s a r e s m a l l i n comparison t o kT, t h e n t h e Zeeman p a r t o f t h e e x p o n e n t i a l i n Eq . ( 4 6 ) can be expanded, C16) In p e r f o r m i n g t h i s e x p a n s i o n , c a r e must be t a k e n f o r i n W r w ) 57 g e n e r a l I and do not commute and the Kubo t r a n s f o r m ^ 1 I_ , o f I_ a r i s e s , C18) j = ^ C P ^ J - 1 j The e q u i l i b r i u m m a g n e t i z a t i o n i s t h e n g i v e n by M0 * p ^ ' " ' ' i l - t i . - X 0 • jj0 and t h i s d e f i n e s t h e e q u i l i b r i u m s u s c e p t i b i l i t y Y . One —o s p e c i a l form f o r Y^ i s fo u n d by expa n d i n g the r e m a i n i n g 8 5 e x p o n e n t i a l s and r e t a i n i n g o n l y the f i r s t t e r ms. Then t h e Kubo t r a n s f o r m becomes n e g l i g i b l e , and M a T - 1 , wh i c h i s C u r i e ' s law -^ o 5 8 . The n o n - e q u i l i b r i u m M p a r t o f M can be w r i t t e n f o r NE t i m e s t^O as I t depends on t h e i n i t i a l c o n d i t i o n s when t h e p e r t u r b i n g f i e l d was removed, i . e . C 5 2 5 CN) and t h e time e v o l u t i o n o f p ( t ) .. The f o r m a l s o l u t i o n o f 86 t h e Quantum L i o u v i l l e E q u a t i o n , Eq. (1-1), f o r p ^ N \ t ) i s C535 p(W\t) = **p[-4. r H \ / + ] (>"%) . To c o n t i n u e , p^~N^Co) must be s p e c i f i e d , and t h e p h y s i c a l b a s i s o f c h o o s i n g t h i s i s t h e f o l l o w i n g . A f t e r t h e p e r t u r b i n g f i e l d AH has been removed, t h e decay t o th e o r i g i n a l magne-t i z a t i o n , M^, o c c u r s . The d e n s i t y o p e r a t o r f o r t h e system a t t h e b e g i n n i n g o f t h i s decay i s A g a i n , by th e 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 , the e x p o n e n t i a l s can be expanded as was done i n Eq.(M7), and L551 87 By u se o f t h e s e r e s u l t s , E q . ( 5 1 ) becomes C 5 5 J £ t , « ( t ) = p Git) - aa where a l l the time dependence i s I n the c o r r e l a t i o n f u n c t i o n , GCt) , d e f i n e d by Grit) s 4 . C ^ t i ^ p l - - c X ' " ) t / i ] U i C571 = « M i / M nt)>"! The i n n e r p r o d u c t , w h i c h i s a c o r r e l a t i o n f u n c t i o n f o r t h e N - p a r t i c l e s y s t e m , i s d e f i n e d by the above e q u a t i o n , and i C t ] i s C581 l i t ) = **p[-JL*€"klt) I . The h a l f range F o u r i e r t r a n s f o r m d e f i n e s t h e r e s p o n s e f u n c t i o n o f t h e c o r r e l a t i o n f u n c t i o n 88 (59) r i i i r i « « j ' ^ ^ 6 r i u i f i n w h i c h t h e r e s o l v e n t o f the L i o u v i l l e o p e r a t o r (to+X^^+ie) 1 a p p e a r s . Eq. (59) can be r e l a t e d t o T^, T 2 and a . F o r example, by Eq.(56), and the B l o c h e q u a t i o n , E q . ( l ) , t h e G ( t ) component o f G ( t ) must obey the r e l a t i o n zz ( 6°> - t/T, The h a l f range F o u r i e r t r a n s f o r m g i v e s (61, T, = * y N T f d " I t f o l l o w s from Eq.(6l) t h a t i n o r d e r t o c a l c u l a t e T i t i s n e c e s s a r y t o e v a l u a t e J ( o ) . A master e q u a t i o n approach i s 89 50 59 one g e n e r a l method o f t a c k l i n g t h i s p r o b lem ' . However f o r gaseous systems e i t h e r the low d e n s i t y l i m i t o f the master e q u a t i o n o r a Boltzmann e q u a t i o n can be used. The e q u i v a l e n c e o f t h e s e two approaches i s g i v e n i n the next c h a p t e r . 90 CHAPTER 4 RELATION BETWEEN CORRELATION AND BOLTZMANN APPROACH E q u a t i o n ( 3 8 ) f o r T" 1 i s c o n s i s t e n t i n form w i t h r e s u l t s o b t a i n e d by t i m e c o r r e l a t i o n t h e o r y . F o r example, the s c a l a r c o u p l i n g f o r s p i n r o t a t i o n i n t e r a c t i o n can be w r i t t e n , ( c i s t h e c o u p l i n g c o n s t a n t assumed indep e n d e n t o f q ) ( 6 2 ) K . = tc )_ Lit Lit c-)* where [^T)^1^ = e see appendix ( A ) ) . T h i s l e a d s t o ^ 8 agreement w i t h p r e v i o u s r e s u l t s ( 6 3 ) T 3 i -f Co*T* s i n c e no i n t r a m o l e c u l a r t r a n s i t i o n s between J s t a t e s o c c u r . S i m i l a r l y , f o r d i p o l e - d i p o l e i n t e r a c t i o n s , t h e h a m i l t o n i a n 91 f o r a d i a t o m i c m o l e c u l e can be w r i t t e n ^ 0 , (64) Here 1=1+1 and n i s t h e u n i t v e c t o r w h i c h s p e c i f i e s 1-2 the o r i e n t a t i o n o f t h e m o l e c u l e i n t h e l a b o r a t o r y . J~Lji can be r e w r i t t e n as (see ap p e n d i x ( A ) ) (65) T F o l l o w i n g Chen and S n i d e r J , i t i s found t h a t f o r low t e m p e r a t u r e s when o n l y one J s t a t e i s o c c u p i e d , and low d e n s i t i e s so t h a t i n t r a m o l e c u l a r t r a n s i t i o n s between J s t a t e s a r e phase r a n d o m i z e d , (66) f 7i<r)J =• - _ I l l f l 92 A l s o f o r a f i x e d I , t h e re p l a c e m e n t 6 1 can be made where a i s d e f i n e d by Chen and S n i d e r I S u b s t i t u t i o n o f t h e s e i n t o E q . ( 3 8 ) , r e s u l t s i n (C^ =-6yH") C681 f i _ -f 1 o r f o r 1 = 1 , 1^=12=^, a-^Jg and (£91 5 ( * J - ' ) ( * T + 3 ) [ J + a 4- T i "| T h i s i s t h e w e l l known f o r m u l a used f o r gas However t h e s e r e s u l t s can be compared t o t h e c o r r e l a t i o n f u n c t i o n approach o n l y i f t h e c o r r e l a t i o n t i m e s , x ' s, w h i c h 93 a r i s e i n t i me c o r r e l a t i o n t h e o r y a r e c o n s i s t e n t w i t h t h e x's d e r i v e d h e r e . E v a l u a t i o n o f t h e s e v i a t h e B oltzmann e q u a t i o n f o l l o w s i n p r i n c i p l e once 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 s p e c i f i e d . C e r t a i n a s p e c t s o f such a c a l c u l a t i o n a r e c o n s i d e r e d i n P a r t I I I . I n t h e t i me c o r r e l a t i o n a p p r o a c h , the c o r r e l a t i o n t i m e s a r e u s u a l l y l e f t as p h e n o m e n o l o g i c a l c o e f f i c i e n t s w h i c h a r e f i t t e d t o e x p e r i m e n t a l r e s u l t s . Then w i t h E q . ( 3 7 ) i t i s p o s s i b l e t o o b t a i n an e m p i r i c a l v a l u e f o r t h e e f f e c t i v e c o l l i s i o n c r o s s s e c t i o n , o ^  . The t h e o r y o f 50 Oppenheim and B l o o n r goes f u r t h e r and r e l a t e s t h e c o r r e l a t i o n f u n c t i o n s t o 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 . I f t h e two methods a r e t o a g r e e , t h e n i t must be shown t h a t th e x's o r , b e t t e r , t h e s p e c t r a l d e n s i t i e s o b t a i n e d by e i t h e r methods ar e e q u i v a l -e n t . I n t h e r e m a i n d e r o f t h i s s e c t i o n i t i s shown t h a t f o r s p i n systems t o f i r s t o r d e r i n gas d e n s i t y and f o r l i n e a r d e v i a t i o n from e q u i l i b r i u m t h e two r e s u l t s agree . E q u i v a l e n c e between the Boltzmann and t ime c o r r e l a t i o n approach has been 63 c o n s i d e r e d by s e v e r a l a u t h o r s and agreement f o r v a r i o u s t r a n s p o r t c o e f f i c i e n t s has been f o u n d . However, a g e n e r a l e q u i v a l e n c e between the two approaches i n the low d e n s i t y gas r e g i o n i s not a v a i l a b l e and the r e a s o n f o r t h i s i s i n d i c a t e d . The r e s u l t s a r e based on Zwanzig's g e n e r a l i z e d master e q u a t i o n ° o f w h i c h the master e q u a t i o n used by Bloom and 9 4 Oppenheim^O i s a s p e c i a l case. Zwanzig's equation li <J0] where p ^ ^ C t ) i s the projected, "relevant" part of the system, an C] 5 6V The p r o j e c t i o n operator (1 - ) picks out the remaining " i r r e l e v a n t " part of the system so that In Eq .C70) , i t has been assumed that a l l the i n i t i a l conditions it*) are i n the relevant part of p 'and as a r e s u l t , Eq . (70) i s (N) a closed equation for p . Also the zero l i m i t of i n t e g r a t i o n 95 i s t h e o r i g i n a l z e r o o f t i m e . ^ S i n c e a p r o j e c t i o n o p e r a t o r t e c h n i q u e i s u s e d , o n l y a (N) l i n e a r p a r t o f p i s " r e l e v a n t " by Eq . ( 7 0 ) and hence a t most, o n l y a l i n e a r i z e d B oltzmann e q u a t i o n can be o b t a i n e d . T h i s i s a l i m i t a t i o n on t h e master e q u a t i o n a p p r o a c h and e x c l u d e s such phenomena a s , f o r example, n o n - l i n e a r Knudsen e f f e c t s . I n g e n e r a l , a t r a n s p o r t c o e f f i c i e n t , o r a s p e c t r a l d e n s i t y , can be w r i t t e n as t h e one s i d e d F o u r i e r t r a n s f o r m 65 o f a c o r r e l a t i o n f u n c t i o n ( 7 3 , = *~ 4 rV" " i 6 J UP, , u<*lt)f* Here th e N - p a r t i c l e i n n e r p r o d u c t i s d e f i n e d , C74) « ft I 6 » < N ) H Jh. fl'^g f o r N - p a r t i c l e o p e r a t o r s A and B and t h e time dependence o f (N) U i s g i v e n by UfM)(t) = *s»p X^-t/tJ U C A > ) 96 (N) I n t h i s t h e s i s , U ( t ) i s t h e 5-th r a n k t e n s o r o p e r a t o r a r i s i n g from t h e n o n - e q u i l i b r i u m p a r t o f t h e l i n e a r i z e d d e n s i t y o p e r a t o r , v i z ( 7 5 ) ^ rt where A(o) i s a c o n s t a n t £-rank t e n s o r and t h e t i l d e d enotes t h e Kubo t r a n s f o r m s . E q . ( 7 3 ) can now be i n t e g r a t e d and t h e r e s u l t i s i n terms o f t h e r e s o l v e n t o f ( 7 6 ) J(co) = U L U ' w ) / C ^ - f Xc%jLeJx\ LL1*)) . Note, t h i s i s o f t h e same form as E q . ( 5 9 ) . However, i t i s v i r t u a l l y i m p o s s i b l e t o s o l v e the N - p a r t i c l e p r o b l e m and hence some a p p r o x i m a t e s o l u t i o n i s r e q u i r e d . To o b t a i n an e x p r e s s i o n from Zwanzig's M a s t e r E q . ( 7 0 ) w h i c h i s e q u i v a l e n t t o t h a t from t h e Boltzmann e q u a t i o n , a b i n a r y gas e x p a n s i o n must be used. However c a r e must be t a k e n i n s p e c i f y i n g t h e p r o j e c t i o n o p e r a t o r w h i c h w i l l r e s u l t i n t h e N - p a r t i c l e system b e i n g r e d u c e d t o any i t h p a r t i c l e 9 7 s i n g l e t d e n s i t y o p e r a t o r . The r e a s o n l i e s i n a f u n d a m e n t a l d i f f e r e n c e between the two t r e a t m e n t s . The maste r e q u a t i o n a p p r o a c h t r e a t s t h e r e l e v a n t p a r t o f t h e system and t h i s i s u s u a l l y assumed t o be t h e i t h p a r t i c l e , w i t h t h e r e m a i n i n g i r r e l e v a n t p a r t b e i n g a l l t h e o t h e r m o l e c u l e s . I n c o n t r a s t , t h e Boltzmann a p p r o a c h t r e a t s a l l t h e m o l e c u l e s s y m m e t r i c a l l y . As y e t i t has not been p o s s i b l e t o e x a c t l y c o n s t r u c t a p r o j e c t i o n o p e r a t o r t o do t h i s . An a p p r o p r i a t e p r o j e c t i o n o p e r a t o r w h i c h can be used t o show ( a p p r o x i m a t e l y ) t h e e q u i v a l e n c e i s now d e f i n e d . Pour a p p r o x i m a t i o n s a r e : 1 ) The L i o u v i l l e s u p e r o p e r a t o r i s o f t h e form E q . ( I - 1 9 ) , so t h a t o n l y b i n a r y i n t e r a c t i o n s a r e c o n s i d e r e d . 2) F o r a l o w - d e n s i t y gas 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 can be n e g l e c t e d i n the N - p a r t i c l e e q u i l i b r i u m d e n s i t y o p e r a t o r , C77) where (see E q . ( I - l 4 ) ) . As a r e s u l t C78) . IT e • 9 8 where p o i i s t h e e q u i l i b r i u m d e n s i t y o p e r a t o r f o r t h e i t h m o l e c u l e . 3) To f i r s t o r d e r i n gas d e n s i t y i t i s assumed t h a t where <J> i s t h e p e r t u r b a t i o n f o r t h e i t h p a r t i c l e . 4) I t i s assumed t h a t t h e c o l l i d i n g p a r t n e r i s a t e q u i l i -b r i u m . T h i s a s s u m p t i o n need not be made, but i s v a l i d f o r s p i n systems and r e s u l t s i n a s i m p l e r p r o o f . I n 30 f a c t , i t i s t h e same a s s u m p t i o n made by Fano i n t r e a t i n g p r e s s u r e b r o a d e n i n g , ( i . e . one m o l e c u l e i s out o f e q u i l i b r i u m and i n t e r a c t s w i t h an e q u i l i b r i u m b a t h o f p e r t u r b e r s ) . By Eq.(75) and u s i n g a more c o n v e n i e n t n o r m a l i z a t i o n f o r t h e N p a r t i c l e s y s t e m ^ , t r M p ^ ^ = 1 , i t i s seen t h a t and t h i s must a l s o h o l d f o r any t r a c e o v e r any p a r t i c l e ( 8 0 ) 99 C811 p 0. - O A. W i t h t h e s e p r o p e r t i e s i t i s p o s s i b l e t o d e f i n e a p r o j e c t i o n o p e r a t o r onto t h e i t h p a r t i c l e by C 8 2 ] A. where t h e t r a c e i s o v e r a l l p a r t i c l e s but t h e i t h . I t f o l l o w s t h a t ess, e r & c 9" = er *x and hence 6 i^, i s i d e m p o t e n t , 100 However, 6^  o p e r a t e s o n l y on t h e n o n - e q u i l i b r i u m p a r t o f (N) , x t h e d e n s i t y o p e r a t o r and not on t h e t o t a l ^ i n E q . ( 7 5 ) , i . e . C 8 5 ) < ? r e x efN) * & r> s r i . T h i s i s because o f t h e r e q u i r e m e n t s , E q s . ( 8 0 ) and (83) CN) W i t h t h i s p r o j e c t i o n o p e r a t o r , U v becomes C86T) where TJ^ i s a s i n g l e t o p e r a t o r . E q u a t i o n (73) can be w r i t t e n C871 J(u>) = 4 _ 1 J _ « u A / |g. £ U + ' W i » ' U ) 101 where t h e p r o j e c t i o n o p e r a t o r i s t h e same on b o t h s i d e s by a s s u m p t i o n 4 . F a n o J has d i s c u s s e d s o l u t i o n s o f Zwanzig's g e n e r a l i z e d m a s t e r e q u a t i o n when o n l y one p a r t i c l e i s out o f e q u i l i b r i u m and c o n s i d e r s t h e d i l u t e gas l i m i t d e f i n e d above. I n f a c t t h e p r o j e c t i o n o p e r a t o r , 6^  , i s j u s t t h e one he used and h i s r e s u l t s f o r t h e b i n a r y gas e x p a n s i o n can be w r i t t e n C88) i 1 /p where eCj^ i s d e f i n e d by E q . ( I - l Q ) . The m(to) o p e r a t o r has been shown t o be t h e same as t h e Boltzmann c o l l i s i o n s u p e r o p e r a t o r ^ , but s i n c e E q . ( 8 7 ) i s l i n e a r , o n l y t h e l i n e a r i z e d B o l tzmann c o l l i s i o n s u p e r o p e r a t o r , E q . ( I - 7 1 ) , a p p e a r s . Thus by p e r f o r m i n g t h e t r a c e o v e r a l l but t h e i t h p a r t i c l e , E q . ( 8 7 ) becomes T M = ( 8 9 ) isz 1 € u+1 ( u + - * ^  ^ e H w-<>> AHA, Q £ '/ 102 where t h e s c a l a r p r o d u c t f o r one p a r t i c l e i s d e f i n e d by <<A|B>>=trA +p J B . The CJ component o f t h e l i n e a r i z e d o l B o ltzmann c o l l i s i o n s u p e r o p e r a t o r i s (J^(UT) and a c c o u n t s f o r c o l l i s i o n s between p a r t i c l e i and a l l t h e r e m a i n i n g p a r t i -c l e s . F u r t h e r , i f a l l t h e m o l e c u l e s a r e i d e n t i c a l , ( 9 0 ) J XTtA e->o s i n c e the l a b e l i can be dropped. The e x p r e s s i o n , E q . ( 9 0 ) , i s a r e s u l t w hich i s d e r i v e d by E q . ( I - 8 4 ) i n terms o f t h e l i n e a r i z e d B oltzmann c o l l i s i o n s u p e r o p e r a t o r . By use o f E q . ( 5 9 ) , t a k i n g AH i n the z t h d i r e c t i o n AH=AHz , and assuming a l l the N s p i n s a r e i d e n t i c a l ( E q . ( 4 5 ) ) , T i s C 9 D T h i s same r e s u l t can be o b t a i n e d from the l i n e a r i z e d 1 0 3 Boltzmann e q u a t i o n , E q . ( 3 ) where <)>(t) must g e n e r a l l y s a t i s f y , (92) 4>(t) = -e 0(0) and f o r t h e s p i n system c o n s i d e r e d h e r e , ( 9 3 ) 0(o) = ^ r / X z AH £ Use o f t h e B l o c h e q u a t i o n f o r and t h e d e f i n i t i o n o f M N E ( t ) , g i v e s A W ^ z . = ^ T / T ' MNEC°\ ( 9 4 ) = tf«<Yz-#I* / I n t e g r a t i o n o ver t from z e r o t o i n f i n i t y g i v e s an i d e n t i c a l r e s u l t t o E q . ( 9 1 ) . • 104 T h i s completes t h e e q u i v a l e n c e o f t h e two approaches f o r s p i n s y s t e m s , and, i n p a r t i c u l a r , f o r d e r i v i n g T^ . I t i s p o s s i b l e t o e x t e n d t h e p r o o f t o i n c l u d e T 2 , and more g e n e r a l l y t o abandon a s s u m p t i o n 4 and o b t a i n e q u i v a l -63 ence f o r more g e n e r a l systems . However w i t h o u t assump-t i o n 2 i t does not seem p o s s i b l e t o c o n s t r u c t a p r o j e c t i o n o p e r a t o r w h i c h i s a p p r o p r i a t e f o r the above p r o o f . PART I I I PARTIAL EVALUATION OP THE x»S 107 CHAPTER 1 INTRODUCTION P a r t i a l e v a l u a t i o n o f t h e c o l l i s i o n b r a c k e t s i s under-t a k e n i n o r d e r t o f i n d t h e i r r e l a t i v e magnitudes. T h i s i s p a r t i c u l a r l y u s e f u l when t r e a t i n g a r e l a x a t i o n m a t r i x such as E q . ( 1 1 - 2 9 ) , w h i c h c o n t a i n s many unknown x ' s . I f t h e s e reduce t o o n l y a few parameters t h e n t h e y can be more e a s i l y and m e a n i n g f u l l y f i t t e d . These few r e m a i n i n g c o n s t a n t s r e f l e c t i g n o r a n c e 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 and the c o l l i s i o n dynamics. I n t h e t r e a t m e n t i n t h i s t h e s i s , the d i s t o r t e d wave Bo r n a p p r o x i m a t i o n ( D W B A ) ^ ' ^ , the impact a p p r o x i m a t i o n ^ and more g e n e r a l methods are d i s c u s s e d . The impact a p p r o x i -49 m a t i o n has been used w i t h g r e a t s u c c e s s by Ben-Reuven . A l t h o u g h many o f t h e c o n c e p t s between h i s approach (based on Fano's m(to) o p e r a t o r ) and t h e Boltzmann approach used here a r e r e a d i l y i n t e r c h a n g e a b l e , t h e r e I s an i m p o r t a n t d i f f e r e n c e i n t r e a t m e n t . Ben-Reuven u t i l i z e s symmetry p r o p e r t i e s t o c o n s t r u c t h i s r e l a x a t i o n m a t r i x i n terms o f q u a n t i t i e s d e r i v e d from s t a t i s t i c a l m echanics. However, r a t h e r t h a n s p e c i f y an i n t e r m o l e c u l a r p o t e n t i a l , and e v a l u a t e t h e s e q u a n t i t i e s , a b e s t f i t t o experiment i s foun d . I n t h i s work an attempt i s made t o use not o n l y symmetry p r o p e r t i e s , but a l s o t o e x p r e s s 108 the c o l l i s i o n times i n terms of simpler q u a n t i t i e s , namely in t e g r a l s over r e l a t i v e v e l o c i t i e s . In Ben-Reuven's case for NH^ i n v e r s i o n 2 ^ where only a few frequency components are used, the relaxa t i o n matrix i s small and r e l a t i v e l y e a s i l y handled. However, as mentioned e a r l i e r , there are many terms i n the relaxa t i o n matrices i n general, and i t i s expedient to p a r t i a l l y evaluate these terms as much as possible by performing traces over i n t e r n a l states i n order to reduce the unknowns to as few as possible. In doing t h i s more d e t a i l e d evaluation, i t i s assumed that dispersion and induction forces are neglected. This i s an undesirable r o assumption but w i l l be used for s i m p l i c i t y . For diatomic molecules, considerable progress has been made i n evaluating c o l l i s i o n i n t e g r a l s and studying the 70 c o l l i s i o n problems . In p a r t i c u l a r , useful r e l a t i o n s h i p s have been found between the c o l l i s i o n brackets for the W-S 71 72 c o l l i s i o n superoperator ' . These l a t t e r studies have been made almost e n t i r e l y f or the Senftleben e f f e c t s , and include expansion tensors i n v e l o c i t y W as well as angular momentum, J . In nuclear spin r e l a x a t i o n , the expansion tensors are independent of W . This leads to great s i m p l i -f i c a t i o n of the v e l o c i t y i n t e g r a l s , i n p a r t i c u l a r , the i n t e -gration over the center of mass, which can always be done, i s es p e c i a l l y simple. However the r e l a t i v e v e l o c i t y terms present 109 more problems t h a n u s u a l because now the g e n e r a l i z e d Boltzmann c o l l i s i o n s u p e r o p e r a t o r i s used w h i c h depends on the i n t e r n a l f r e q u e n c i e s i n a c o m p l i c a t e d way. These terms s i m p l i f y g r e a t l y i n the Impact a p p r o x i m a t i o n . I n s p i t e o f t h e s e d i f f e r e n c e s , t h e work on d i a t o m i c s forms t h e f o u n d a t i o n s f o r t h e t r e a t m e n t w h i c h i s p r e s e n t e d f o r p o l y a t o m i c s . I n 72 p a r t i c u l a r the work o f Chen, M o r a a l and S n i d e r i s c l o s e l y f o l l o w e d . 110 CHAPTER 2 THE DISTORTED WAVE BORN APPROXIMATION (DWBA) I t I s n e c e s s a r y t o know what form t o t a k e f o r t h e t r a n s i t i o n o p e r a t o r , t , w h i c h appears i n the c o l l i s i o n s u p e r o p e r a t o r , E q . ( I - 7 2 ) . Here i t i s chosen t o use the DWBA and t h i s i s now b r i e f l y o u t l i n e d . The essence o f t h e a p p r o x i m a t i o n i s t h a t 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 can be w r i t t e n i n two p a r t s , ( i ) v = + \/, where V q i s t h e s p h e r i c a l p a r t and p l a y s no r o l e i n r e -o r i e n t a t i n g t h e m o l e c u l e s , but can change t h e i r r e l a t i v e v e l o c i t i e s w h i l e t h e n o n s p h e r i c a l p a r t , V , a c c o u n t s f o r m o l e c u l a r r e o r i e n t a t i o n and i s c o n s i d e r e d t o be s m a l l i n com p a r i s o n t o V . T h i s s u g g e s t s a p e r t u r b a t i o n e x p a n s i o n o f o r t i n powers o f the n o n s p h e r i c i t y ( 2 ) I l l where t n i s the n t h o r d e r i n V . S u b s t i t u t i o n o f E q . ( 2 ) and Eq.Cl) i n t o the Lippmann-Shwinger e q u a t i o n ( 1 - 4 3 ) and e q u a t i n g powers o f g i v e s ca) t . = V. + V0 G0t0 = \ ^ l . and ( 4 ) t, ~ SL0 V , _C1 where t h e t r a n s p o s e o f Si i s o (5) -n~o = ' + Z o & 0 . I f t=t + t 1 t h e n t 1 i s t h e f i r s t DWBA t o t . From E q . ( 4 ) , t h i s i s i n terms o f and the M i l l e r wave 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 the p o t e n t i a l . Hence a m a t r i x element o f t ^ between an i n i t i a l < i | and f i n a l |f> s t a t e can be w r i t t e n <*-l*,lf> (6) = <U/_fi,,v;_£iJO 112 and t h e s t a t e s a r e " d i s t o r t e d " by ttQ . The second DWBA i s found by e q u a t i n g terms i n Eq.(I - 4 3 ) w h i c h a r e second o r d e r i n . T h i s g i v e s (7) tx - t. ~ X, &a \/0 &o A w h i c h has been w r i t t e n i n a s y m m e t r i c a l form. 113 CHAPTER 3 THE INTERMOLECULAR POTENTIAL The i n t e r m o l e c u l a r p o t e n t i a l depends on t h e r e l a t i v e p o s i t i o n o f the two c o l l i d i n g m o l e c u l e s , R , and t h e o r i e n t a t i o n o f the two m o l e c u l e s . F o r d i a t o m i c m o l e c u l e s i t i s s u f f i c i e n t t o r e p r e s e n t t h e o r i e n t a t i o n by t h e m o l e c u l a r a x i s a l o n e , but f o r p o l y a t o m i c s i t i s n e c e s s a r y t o i n c l u d e v a r i o u s a n g l e s t o f u l l y f i x the m o l e c u l e s i n space. F o r m a l l y t h i s o r i e n t a t i o n i s s p e c i f i e d by n-j_ and r ^ f o r p a r t i c l e one and two r e s p e c t i v e l y . I t i s assumed t h a t 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 i s indepen d e n t o f t h e magnetic f i e l d . A m u l t i p o l e e x p a n s i o n can be p e r f o r m e d and g i v e s -t • • • . 114 T h i s i s o b t a i n e d from a two c e n t e r e x p a n s i o n (see a p p e n d i x D)) (P) f o r uncharged m o l e c u l e s . The v a r i o u s m u l t i p o l e s , M , a r e where p Cn^  ) i s the charge d i s t r i b u t i o n w i t h i n t h e i t h m o l e c u l e , and d i p o l e q u a d r u p o l e o c t a p o l e — i M ( 2 ) M (3) e t c I n a more compact n o t a t i o n , t h e m u l t i p o l e e x p a n s i o n can 72 be w r i t t e n C i o ) 115 where a l l t h e t e n s o r s a r e symmetric t r a c e l e s s , and t h e s t r e n g t h o f each c o n t r i b u t i o n i s g i v e n by b(l^,l^) . The term b(o,o) c o r r e s p o n d s t o the s p h e r i c a l p a r t , V " 0 , o f V , i . e . b(o,o)/R = V Q . 1 1 6 CHAPTER 4 EXPRESSION FOR THE x'S Wi t h 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 p e c i f i e d , and t h e DWBA, the t r a n s i t i o n o p e r a t o r can be s u b s t i t u e d i n t o 51. » and the e x p r e s s i o n f o r T c a l c u l a t e d . T h i s t a s k i s s i m p l i -f i e d by s e v e r a l f a c t o r s and a l r e a d y mentioned i s t h e v e l o c i t y independence o f t h e e x p a n s i o n t e n s o r s . A l s o , s i n c e m o l e c u l e two i s i n e q u i l i b r i u m , i t does not appear i n t h e n o n - e q u i l i -b r i u m p a r t . Hence i t i s e x p e c t e d t o c o n t r i b u t e o n l y a con-s t a n t f a c t o r t o T . S i n c e 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 cannot change s p i n s t a t e s , t h e s p i n o p e r a t o r s commute w i t h (R„ , and t h e n o r m a l i z e d x's , E q . ( 1 1 - 2 4 ) , become, ex c e p t f o r s p i n symmetry r e s t r i c t i o n , i n dependent o f t h e s p i n t r a c e s . F i n a l l y t he x's a r e s c a l a r s , and s i n c e t h e r e l a t i v e v e l o -c i t y p a r t must be s c a l a r , i t f o l l o w s , from t e n s o r i a l a r g u -ments, t h a t m u l t i p o l e c o n t r i b u t i o n s w h i c h a r i s e i n the t r a n s i -t i o n o p e r a t o r s , t , must be o f the same r a n k . I n o t h e r words, as p o i n t e d out by Ben-Reuven, c o l l i s i o n s do not mix d i f f e r e n t 4 9 m u l t i p o l e s p e c i e s 117 I f t h e 21-th rank n o r m a l i z e d o p e r a t o r f o r s p i n and l a t t i c e a r e d e f i n e d (ID " ?f Fx ft. t h e n t h e e x p r e s s i o n f o r x 1 can be w r i t t e n < 1 2 ) % u u r ) L L '»* L a 1 . 1 8 The Boltzmann w e i g h t s i n E q . ( 1 2 ) , f o l l o w from t h e o r i g i n a l _ i (I) d e f i n i t i o n o f T £ x(ifi'f«) , E q . ( I - 2 4 ) i n w h i c h t h e o V f ' s app e a r , w h i l e i n the d e f i n i t i o n o f |i><f| above, t h e Boltzmann w e i g h t s have not been i n c l u d e d . The i n t e g r a l o p e r a t o r , z, , can be i d e n t i f i e d from E q . ( I - 7 2 ) as (see E q . ( I - 7 ) ) , The t r a c e i n E q . ( l 2 ) i s ov e r a l l t h e i n t e r n a l s t a t e s o f b o t h m o l e c u l e one and two. There a r e f o u r a d d i t i v e c o n t r i b u t i o n s t o E q . ( 1 2 ) , two of w h i c h a r e e x p l i c i t l y i n d e p e n d e n t o f G Q , and two e x p l i -c i t l y l i n e a r i n G . The e n e r g i e s w h i c h appear i n t h e s e o f r e e p a r t i c l e Green's f u n c t i o n s have been d i s c u s s e d i n P a r t I , C h a p t e r ( 3 . 2 ) . F o r example, from E q . ( I - 3 7 ) , t h e second term has t h e forms 119 The t r a n s i t i o n o p e r a t o r a c c o u n t s f o r changes i n t h e i n t e r n a l energy o f b o t h p a r t i c l e s w h i l e t h e Green's f u n c t i o n i n v o l v e s t a k i n g t h e t o t a l energy d i f f e r e n c e a c r o s s t h e t r a n s i t i o n o p e r a t o r , ( s e e E q . ( I - 4 4 ) ) . Thus th e i n t e r n a l energy d i f f e r e n c e o v e r t*= f o r p a r t i c l e one i s -fito.,. = E. ,-E. where g» ^ 1 'x 1 ' i H. l i x i ' l .= E l i x i ' l , and t h e change i n t h e r e l a t i v e x n t 1 1 x ' ? 2 v e l o c i t y i n fiio , = h\x Cg - g ) . To show th e energy g g d i f f e r e n c e f o r p a r t i c l e 2 e x p l i c i t l y , i d e n t i t i e s must be i n t r o d u c e d , C15I such, t h a t E q . C l 4 ) i s CIS) The Green's f u n c t i o n i n E q . ( l 6 ) i s t h e n o f t h e form 120 ( 1 7 ) G0U'iilJ>xW)= J 2 Z ' Prom an analogous t r e a t m e n t o f t h e e n e r g i e s , t h e G + terms o i n E q . ( 1 2 ) i s G Q(f'ff£f 2g'g) w i t h t h e a p p r o p r i a t e i d e n t i t i e s i n t r o d u c e d as sums over "ftf. and 7^ ' F o r t h e terms i n E q . ( 1 2 ) e x p l i c i t l y i ndependent o f G ' s , i t f o l l o w s from t h e c y c l i c p r o p e r t y o f t h e t r a c e s t h a t i = i ' f o r the f i r s t , and f = f f o r t h e second. U t i l i z i n g t h e s e p r o p e r t i e s , E q . ( 1 2 ) becomes ( 1 8 ) f£ 121 where ( t h e s u b s c r i p t I r e f e r s t o t h e t e n s o r c o u p l i n g rank.), and (20) " l 1 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 i s r e p l a c e d by E n t n and . o n l y t h o s e terms i n t o second o r d e r i n t h e n o n s p h e r i c a l i n t e r m o l e c u l a r p o t e n t i a l V-^  , a r e r e t a i n e d . To i n d i c a t e t h i s , t h e t i l d e ' s on the B's a r e dropped, i . e ( 2 D B > S. Note t h a t when t h e f r e q u e n c i e s a r e t h e same i n the two Green's f u n c t i o n s o f E q . ( l 8 ) t h e y combine t o a d e l t a f u n c t i o n o f e n e rgy. 122 The e q u a t i o n f o r t h e x*s i s now i n a c o n v e n i e n t form f o r t h e subsequent e v a l u a t i o n and t h e sequence f o r t h i s p r o c e d u r e i s as f o l l o w s : 4.1 i n t e g r a t i o n o v er t h e c e n t e r o f mass. 4.2 s e p a r a t i o n o f r e l a t i v e v e l o c i t y term from t h e i n t e r n a l s t a t e t r a c e s f o r . 4.3 s e p a r a t i o n o f r e l a t i v e v e l o c i t y terms from t h e i n t e r n a l s t a t e t r a c e s f o r • 4.4 g e n e r a l e x p r e s s i o n s f o r T~"*"(if i ? f •) . I n p e r f o r m i n g s t e p s 1 t o 4, no new a p p r o x i m a t i o n s a r e made. 4.1) I n t e g r a t i o n Over t h e C e n t e r o f Mass. c Co) S u b s t i t u t i o n o f j w and W , whi c h a r e g i v e n f o l l o w i n g E q . ( I - 7 ) i n t o £ g i v e s where t h e momentum, p , has been changed t o t h e r e d u c e d v e l o c i t y , s i n c e o n l y one t y p e o f m o l e c u l e i s t r e a t e d . The re d u c e d v e l o c i t i e s can be t r a n s f o r m e d i n t o r e l a t i v e c o o r d i n a t e s (22) 123 by where t h e v e l o c i t y o f t h e c e n t e r o f mass i s ^ and t h e r e l a t i v e v e l o c i t y y_ i s r e l a t e d t o g_ by S i n c e t h e J a c o b i a n o f t h e t r a n s f o r m a t i o n o f dW-^ dW,, t o dSdy i s one, ? becomes, <25> 5= ( ^ f ^ f ^ p ( - 5 V d s ] [ f f ^ ( - ^ ) ^ E x a m i n a t i o n o f E q . ( l 8 ) r e v e a l s t h a t the q u a n t i t y on w h i c h ^ o p e r a t e s depends on g_ and g_' , but i s independent o f 124 Hence t h e I n t e g r a t i o n o v e r t h e c e n t e r o f mass i s C26~) CWgio^, and c i s now C27) 5 ^ ^ i ^ ^ ' " ^ * " - ' 0 ' 4.2) S e p a r a t i o n o f the V e l o c i t y Terms  from I n t e r n a l S t a t e T r a c e s f o r B S i n c e £2 a f f e c t s o n l y 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 o f i r s t DWBA t r a n s i t i o n o p e r a t o r can be w r i t t e n 1 2 5 C 2 8 i ~ l_ Cn,] C-tfJ 0 S i n c e t h e o p e r a t o r s , |i?><f | , o p e r a t e o n l y on t h e i n t e r n a l s t a t e s p a c e , t h e t r a c e s o ver m o l e c u l e s one and two s e p a r a t e from t h e terms w h i c h depend on t h e r e l a t i v e v e l o c i t i e s . S u b s t i t u t i o n o f t ^ J , i n t o E q . ( 1 9 ) , and s e p a r a t i n g t h e s e t e r m s , becomes 8, ( i f x ' - f U ^ -C29) I X (if HO TT(A^;) a/*#.) e > « * 126 where the t r a c e o v e r the I n t e r n a l s t a t e s o f m o l e c u l e 1 i s (30) >{ [ j }j. f) r ^ / W & ^ f ^ ^ ^ and ( 3 D Ob The t r a c e o v e r m o l e c u l e 2 g i v e s 1 2 7 t o r s must be o f the same t e n s o r r a n k . 4 . 3 ) S e p a r a t i o n o f V e l o c i t y Terms from I n t e r n a l S t a t e  T r a c e s f o r The e x p r e s s i o n f o r B^ , E q . ( 2 0 ) o c c u r " l i n e a r l y " i n t . C o n s e q u e n t l y i f t i s r e p l a c e d by t-^ i n t h e f i r s t DWBA, w i l l v a n i s h by th e argument t h a t f o r a s c a l a r r e l a t i v e v e l o c i t y term + %^ = 0 . ' I t i s n e c e s s a r y t h e r e f o r e , t o use the second DWBA and r e p l a c e t by t^t E q . ( 7 ) . By s u b s t i t u t i o n o f E q . ( 7 ) , B„ becomes The Green's f u n c t i o n s i n E q . ( 3 3 ) i n v o l v e energy d i f f e r e n c e s a c r o s s t he t r a n s i t i o n o p e r a t o r s as d e s c r i b e d i n E q s . ( l 4 ) t o ( 1 7 ) . E x a c t l y t he same p r o c e d u r e can be c a r r i e d out f o r B^, and a p p r o p r i a t e i d e n t i t i e s o v er i n t e r n a l s t a t e s o f m o l e c u l e 2 , ( 3 3 ) 128 Eq.(15), and over the r e l a t i v e v e l o c i t y must be inserted. By separating the traces of molecule one, molecule two, and the r e l a t i v e v e l o c i t i e s may be written I n^fj;) f[*u$ii J " \ r r & v/; <?„ amu-A* where (35) V.*,' r ^ f " I V J A f > and 1 2 9 ( 3 6 ) . (_! -oil 4 . 4 ) G e n e r a l E x p r e s s i o n f o r r ^ f i f i ' f M E q u a t i o n s ( 2 9 ) , and ( 3 4 ) can be s u b s t i t u t e d i n t o E q . ( l 8 ) t o g i v e , X.uur) W L (37) 130 where ( 3 8 ) l v e s t a l ) 6 0 C i ' H l i i i and (39) 1 3 1 CHAPTER 5 DISCUSSION OP RESULTS By assuming the DWBA and a g e n e r a l form f o r 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 , E q . ( l O ) , an e x p r e s s i o n has been o b t a i n e d f o r t h e x ' s , E q . ( 3 7 ) . P o r a p a r t i c u l a r case o f b a s i s s t a t e s , i . e . | i x f | , t h e i n t e r n a l s t a t e t r a c e s can be e v a l u a t e d and t h e dependence on t h e s e s t a t e s o b t a i n e d . W i t h t h i s know-l e d g e and some t r u n c a t i o n o f t h e i n t e r m o l e c u l a r m u l t i p o l e e x p a n s i o n , t h e x's a r e r e d u c e d t o unknown i n t e g r a l s o v e r the r e l a t i v e v e l o c i t y . However, E q . ( 3 7 ) i s v e r y complex and w i t h i n the a p p r o x i m a t i o n s , ( i . e . i g n o r i n g i n d u c t i o n s and d i s -p e r s i o n f o r c e s , and u s i n g t h e DWBA), t h i s e q u a t i o n a c c o u n t s f o r a l l p o s s i b l e exchanges o f i n t e r n a l energy and t r a n s l a -t i o n a l energy o f one p a r t i c l e out o f e q u i l i b r i u m w i t h a l l o t h e r s a t e q u i l i b r i u m . The d e t a i l e d d y n a m i c a l p r o c e s s e s a r e d i f f i c u l t t o t r e a t p r o p e r l y , and a l l t h e p o s s i b l e cases need t o be c a t e g o r i z e d and s e p a r a t e l y s t u d i e d . I t i s not t h e i n t e n -t i o n i n t h i s t h e s i s t o u n d e r t a k e such a program, but c e r t a i n a s p e c t s o f E q . ( 3 7 ) a r e now d i s c u s s e d . The g r e a t e s t d i f f i c u l t y l i e s i n the dependence o f t r a n s l a -t i o n a l v e l o c i t y i n t e g r a l s on t h e i n t e r n a l s t a t e s o f m o l e c u l e s 132 one and two. By c a r e f u l e x a m i n a t i o n o f the r e l a t e d Eqs. ( 3 8 ) and (39) f o r T £ 1 ( i f i ' f ) , i t can be seen t h a t t h i s c o m p l e x i t y v a n i s h e s i f a l l t h e Green's f u n c t i o n s have the same f r e q u e n c y dependence. I n t h i s case a l l t h e sums o v e r the second mole-c u l e ' s i n t e r n a l s t a t e s g i v e s and t h e sums o v e r £ f\[<i j f f'J can be per f o r m e d s i n c e t h e y now s e p a r a t e from the v e l o c i t y i n t e g r a l s . I n f a c t t h e whole e x p r e s s i o n , E q . ( 3 7 ) , r e d u c e s t o a r e a l q u a n t i t y , independent of any i n t e r n a l f r e q u e n c i e s . T h i s i s t h e impact a p p r o x i m a t i o n and t o g e t h e r w i t h t h e DWBA, no f r e q u e n c y s h i f t s a r i s e . I t can be c o n c l u d e d t h a t s i m u l t a n e o u s use o f b o t h t h e DWBA and t h e impact a p p r o x i m a t i o n i s t o o s e v e r e and d e s t r o y s t h e e f f e c t s o f c o l l i s i o n a l t r a n s i e n t s w h i c h a r e t h e o n l y s o u r c e o f f r e q u e n c y s h i f t s i n t h e f i r s t o r d e r DWBA. At low d e n s i t i e s t h e s e c o l l i -s i o n a l t r a n s i e n t s most l i k e l y p l a y a minor r o l e , and t r e a t m e n t s w h i c h make use o f the impact a p p r o x i m a t i o n s o n l y a r e adequate. However, f o r t h e h i g h e r d e n s i t y c a s e s o f i n t e r e s t here i t i s n e c e s s a r y t o i n c l u d e c o l l i s i o n a l t r a n s i e n t s s i n c e the l i m i t T ±nt~x *< i s approached. F o r t h i s case i t i s e x p e c t e d t h a t 133 n e g l e c t i n g f r e q e n c i e s i n the c o l l i s i o n a l p r o c e s s l e a d s t o err o n e o u s r e s u l t s . Of c o u r s e the r e l a t i v e i m p o r t a n c e o f e i t h e r a p p r o x i m a t i o n cannot be found w i t h o u t c a l c u l a t i o n s , s i n c e e m p i r i c a l f i t t i n g w i t h e i t h e r a p p r o x i m a t i o n can r e p r o d u c e t h e e x p e r i m e n t a l r e s u l t s . However, o n l y t h e q u a l i t a t i v e b e h a v i o u r o f t h e c o l l i s i o n p r o c e s s i s b e i n g p r e s e n t e d . I t i s , o f c o u r s e , p o s s i b l e t o mod i f y t h e impact a p p r o x i m a t i o n t o t a k e i n t o a c c o u n t t r a n s i e n t e f f e c t s as has been p o i n t e d out i n P a r t I , and ap p r o x i m a t e <<A|/tJ|A>> by «A\i3(w)\k» . When t h e e n e r g i e s i n t h e Green's f u n c t i o n s a r e not t h e same, p r i n c i p a l v a l u e terms a r i s e as w e l l as d e l t a f u n c t i o n s . The d e l t a f u n c t i o n s e x p r e s s the c o n s e r v a t i o n o f energy exchanged i n a c o l l i s i o n , i . e . Tito. . , + tlto. . , + ftto . = 0 ' i i ' X2X2 S S I n c o n t r a s t , t h e p r i n c i p a l v a l u e terms t a k e i n t o account p o s s i b l e r e s o n a n c e s , w h i c h may be s e t up d u r i n g a c o l l i s i o n . M a t h e m a t i c a l l y t h e s e a r e f r e q u e n c y s i n g u l a r i t i e s , and as such c o r r e s p o n d t o e x c i t e d o r m e t a s t a b l e s p e c i e s . These l o n g e r l i v i n g s p e c i e s r e s u l t i n a m o d i f i c a t i o n o f t h e d i s p e r s i o n t e r m f o r a p a r t i c u l a r f r e q u e n c y component, and may g i v e r i s e t o a f r e q u e n c y s h i f t . Prom a l e s s a b s t r a c t p o i n t o f v i e w , Eq.(37) f o r x ~ 1 ( i f i t f ' ) can be used t o e v a l u a t e s p e c i a l c ases o f c o l l i s i o n i n t e g r a l s , 1 3 4 ( o r c o r r e l a t i o n t i m e s ) . I t s h o u l d not be d i f f i c u l t t o f i n d t h e dependence on £ so t h a t t h e r e l a t i v e v a l u e s o f s p i n r o t a t i o n , £ = 1 , d i p o l a r , £ = 2 , e t c . , mechanisms can be found . A l s o , i t i s now p o s s i b l e t o e x p e r i m e n t a l l y s t u d y t h e i n t e r n a l s t a t e dependence o f t h e x's by m o l e c u l a r beam t e c h n i q u e s , and such t r e n d s s h o u l d f o l l o w from t h e i n t e r n a l s t a t e t r a c e s i n Eqs. ( 3 0 ) and ( 3 6 ) . These c o l l i s i o n t i m e s a r e s t u d i e d a t low enough d e n s i t i e s t h a t i t s h o u l d be a good a p p r o x i m a t i o n t o i g n o r e t h e p r i n c i p a l v a l u e terms and t h i s g r e a t l y s i m p l i f i e s t h e c a l c u l a t i o n . Then t h e i n t e r n a l s t a t e p a r t s c o n t a i n e d i n t h e X's , would s e p a r a t e from t h e r e l a t i v e v e l o c i t y i n t e g r a l s , and can the n be e x a c t l y e v a l u a t e d . Of p a r t i c u l a r i n t e r e s t a r e t h e t o t a l l y d i a g o n a l terms f o r the dominant mechanism a t low d e n s i t i e s , i . e . x^ " * " ( i i i i ) . The more r e s t r i c t e d p r o c e s s e s w h i c h i n v o l v e c o l l i s i o n a l t r a n -s i t i o n s between i n t e r n a l s t a t e s a r e e x p e c t e d t o add l i t t l e t o t h e r a t e o f decay and hence 135 o r (42) T h i s has a l r e a d y been i n d i c a t e d i n E q . ( I - 9 8 ) where t h e c r o s s s e c t i o n f o r one c h a n n e l i s s m a l l e r t h a n t h e t o t a l c r o s s s e c t i o n . T h i s l a r g e n e s s o f T ^ ( i f i * f ) i s r e q u i r e d t o e x p l a i n t h e s t e p e f f e c t i n P a r t I V , and i s i n f a c t e s s e n t i a l t o a c o r r e c t d e s c r i p t i o n . PART IV APPLICATION TO SYMMETRIC TOPS 137 CHAPTER 1. INTRODUCTION C o n s i d e r a b l e e x p e r i m e n t a l e v i d e n c e e x i s t s which con-f i r m s t h e form o f T T when h i g h e r f r e q u e n c y t e r m s , to , 1, ' a a 7 3 a r e n e g l i g i b l e . F o r example, e x a m i n a t i o n o f E q . ( I I - 6 3 ) , f o r s p i n - r o t a t i o n and Eq. ( 1 1 - 6 8 ) , f o r d i p o l a r i n t r a m o l e -c u l a r i n t e r a c t i o n s show t h a t as x v a r i e s minima can o c c u r when qtox=l . S i n c e x i s i n v e r s e l y p r o p o r t i o n a l t o d e n s i t y , i t i s seen t h a t as the d e n s i t y i n c r e a s e s , qtox may change from b e i n g g r e a t e r t h a n t o l e s s t h a n one and o b s e r v a t i o n o f t h i s change-over i s e x p e r i m e n t a l p r o o f o f t h e form o f T . From such an e x p e r i m e n t , the d e n s i t y dependence o f T_^  f o l l o w s a c u r v e s i m i l a r t o f i g . ( 2 ) . By f i t t i n g t h e minima t o qtox=l i t i s p o s s i b l e t o o b t a i n a v a l u e f o r x and t h u s the e f f e c -t i v e c r o s s s e c t i o n . F u r t h e r , by f i t t i n g t h e h i g h e r d e n s i t y dependence, t h e e f f e c t i v e c o u p l i n g c o n s t a n t can be f o u n d . A l l t h i s y i e l d s v a l u a b l e i n f o r m a t i o n on the m o l e c u l a r p r o p -e r t i e s and t h e v a l u e s o b t a i n e d a r e i n f a i r agreement w i t h t h o s e o b t a i n e d by o t h e r methods ( n o t a b l y m o l e c u l a r beam d a t a ) . Hence i t can be c o n c l u d e d t h a t t h e assumptions w h i c h l e a d t o the low d e n s i t y e x p r e s s i o n f o r T n a r e r e a s o n a b l e . DENSITY F i g u r e 2 . Example of T1 Minimum. H OO OO 139 In c o n t r a s t t o the c o n s i d e r a b l e volume of p r e c i s e e x p e r i -mental evidence f o r the low d e n s i t y r e l a x a t i o n d a t a , h i g h e r d e n s i t y i n f o r m a t i o n i s i n a c c u r a t e and exp e r i m e n t a l agreement i s poor between r e s e a r c h e r s . The behaviour p r e d i c t e d i n t h i s r e g i o n based on phase r a n d o m i z a t i o n of the Larmor frequency only i s t h a t T^ be a l i n e a r f u n c t i o n of the d e n s i t y when WT^<1 . To check t h a t r o t a t i o n a l s t a t e s do not c o n t r i b u t e t o the r e l a x a t i o n r a t e , ^ at t h i s d e n s i t y , the s p e c t r a l d e n s i t i e s f o r the Zeeman f r e q u e n c i e s are e v a l u a t e d with the Zeeman f r e q u e n c i e s r e p l a c e d by ti) f , i . e . J o (to t ) . On ^ aa 3 it v aa the b a s i s t h a t J 0(to , ) < < J 0(toq) such " r a p i d l y o s c i l l a t i n g " x aa *> 75 1 terms are ig n o r e d . U n t i l the experiments by Dong and 2 Bloom l i t t l e doubt was ca s t on t h i s . They r e p o r t e d s t a r t l i n g d e v i a t i o n s from the expected l i n e a r d e n s i t y dependence of prot o n T^ f o r the symmetric top molecules CF^H and CH^F. Por f l u o r o f o r m , a s e r i e s of breaks were observed which when p l o t t e d as T^/n versus n were i n t e r p r e t e d as a s e r i e s o f " s t e p s " (see f i g . 3)*. F o r methyl f l u o r i d e only one " s t e p " was 1 ft observed (see f i g . 4 ) . F o l l o w i n g t h i s , Speight and Armstrong pre s e n t e d evidence which i n d i c a t e d d e v i a t i o n from the expected l i n e a r d e n s i t y dependence of T-^  f o r ^  gas. These r e s u l t s show no sudden breaks, but r a t h e r a smooth d e c r e a s e - i n T^. Mot i v a t e d by Dong's r e s u l t s , Armstrong- 3 and Waugh independently checked these symmetric top molecules and although they *Dong used p f o r d e n s i t y w h i l e i n t h i s t h e s i s n i s used. 400 < < < cu (A) E 20( 100 0 Figure 3 T|/p V S ^3 CHF, 297°K 389 °K 297 "K (FLUORINE) (Vertical Seale=10:1) 20 40 SO p (AMAGATS) 6 0 Plot of Tx/-p versus p for CHF3. The Value of Ti/p for the Fluorine Spins Is 3.5 msec/Amagat. o F l u o r i n e 2 9 7 ° K C 1 0 2 0 3 0 4 0 p (AMAGATS) F i g u r e 4. P l o t o f T^/p v e r s u s p f o r CH 3F. 142 g e n e r a l l y agreed w i t h the CH^F d a t a , were i n complete d i s -agreement w i t h t h e f l u o r o f o r m d a t a . The r e s u l t s f o r CP^H a r e r e p r o d u c e d i n f i g . ( 5 ) . C o n s e q u e n t l y i t i s not c l e a r as t o what t h e c o r r e c t d e n s i t y dependence i s f o r t h i s mole-c u l e , but i t i s c l e a r t h a t the p r e v i o u s t h e o r y f a i l s t o account f o r the o b s e r v e d r e s u l t s . F u r t h e r , i t i s d i f f i c u l t t o j u s t i f y t h e sharp b r e a k s i n Dong's r e s u l t s from t h e d a t a he p r e s e n t s and t h e s o l i d l i n e s drawn t h r o u g h h i s p o i n t s a r e i d e a l i z a t i o n s w i t h no t h e o r e t i c a l f o u n d a t i o n s . I n t h e r e m a i n d e r o f t h i s t h e s i s , i t i s shown how t h e n o n - l i n e a r d e n s i t y dependence i s e x p l a i n e d f o r t h e s e mole-c u l e s u s i n g t h e t h e o r y o f P a r t s I , I I and I I I . That i s , c a l c u l a t i o n s a r e c a r r i e d out t o g i v e q u a n t i t a t i v e agreement f o r CH^F and CF-^H. F o r CH^F a p a r t i c u l a r l y s i m p l e model i s t a k e n which has l i t t l e p h y s i c a l j u s t i f i c a t i o n , and a l e a s t s q uares f i t i s o b t a i n e d . F o r CF^H, a more d e t a i l e d c a l c u l a -t i o n a t 297°K i s g i v e n . I t i s g e n e r a l l y f e l t t h a t such sharp 77 " s t e p s " cannot be e x p e r i m e n t a l l y i n t e r p r e t e d . However i t i s d e monstrated here t h a t such s t e p s can a r i s e t h e o r e t i c a l l y and moreover t h e s e e m i n g l y s p u r i o u s s c a t t e r o f e x p e r i m e n t a l r e -s u l t s may a l s o be s i g n i f i c a n t . A l t h o u g h i t was not f e a s i b l e t o p e r f o r m a l e a s t s quares f i t , r e a s o n a b l e agreement i s ob-t a i n e d by t r i a l and e r r o r . I t i s found t h a t f r e q u e n c y s h i f t s 1-1 .9-o *o T / n v s n fo r C E H a t 2 9 7 ° K 1 3 . 8 --+— • ' 0 o £ -6-U c \ A . 3 -o o o R. Armstrong' • 0. Waugh*" A R . Dong' o • • • • o • A A A A A A * A A A . 2 -A A 12 18 2 4 3 0 3 6 4 2 4 8 5 4 6 0 6 6 £ D e n s i t y n ( a m a g a t s ) F i g u r e 5 . Comparison o f P r o t o n R e l a x a t i o n Data f o r C F 3 H . 00 144 a r e e s s e n t i a l f o r a c o r r e c t d e s c r i p t i o n , a l t h o u g h o v e r l a p between f r e q u e n c y components can a l s o be i n c l u d e d . The r e l a x a t i o n t i m e s f o r the few low l y i n g f r e q u e n c y components must be l o n g e r t h a n t h e o r d i n a r y low d e n s i t y c o r r e l a t i o n t i m e s . By v a r y i n g the t h r e e q u a n t i t i e s , s h i f t , w i d t h and o v e r l a p , v a r i a t i o n i n the d e n s i t y dependence o f T-^/n can be o b t a i n e d and f u r t h e r m o r e t h i s v a r i a t i o n i s q u i t e s e n s i t i v e t o s m a l l changes i n t h e p a r a m e t e r s . 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 h i g h d e n s i t y depen-dence i s thus s i m p l y u n d e r s t o o d u s i n g t h e same phase r a n d o m i -z a t i o n concept w h i c h i s so s u c c e s s f u l i n a c c o u n t i n g f o r the low d e n s i t y T^ minima d a t a . I n e f f e c t , as t h e d e n s i t y i n -c r e a s e s , t h e r e a r e added c o n t r i b u t i o n s t o n/T^ from c e r t a i n h i g h e r m o l e c u l a r f r e q u e n c i e s . I n c h a p t e r 2 t h e symmetry i s s t u d i e d f o r CX^Y m o l e c u l e s and t h e n i n c h a p t e r 3 , a d e s c r i p t i v e example i s g i v e n w h i c h emphasizes t h e p r o c e d u r e and t h e 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 . The s i m p l e model p r e s e n t e d i s a p p l i e d t o CH^F, and a f i t t o expe r i m e n t i s o b t a i n e d . T h i s d e m o n s t r a t e s t h a t the f u n c t i o n a l form f o r i s c o r r e c t , and i l l u s t r a t e s how the f r e q u e n c y terms can e x p l a i n the e x p e r i m e n t s . However the numbers g i v e n have l i t t l e p h y s i c a l s i g n i f i c a n c e s i n c e many i m p o r t a n t symmetry f e a t u r e s ' o f the m o l e c u l e have not been i n c l u d e d . I n 1 4 5 c h a p t e r 4 , the symmetry f e a t u r e s are I n c l u d e d I n t h e s t u d y o f f o r CF^H. C h a p t e r 5 i s d e v o t e d t o f i t t i n g t h e p r o t o n T, e x p e r i m e n t s f o r CF^H. 146 C H A P T E R 2 S Y M M E T R Y P R O P E R T I E S O P S Y M M E T R I C T O P M O L E C U L E S , C X Y I n t h i s c h a p t e r , s y m m e t r i c t o p m o l e c u l e s a r e i n v e s t i g a t e d . T h e t w o m o l e c u l e s o f p a r t i c u l a r i n t e r e s t , C P ^ H a n d C H ^ F , a r e d i s c u s s e d s i m u l t a n e o u s l y a s C X ^ Y m o l e c u l e s w h e r e a t o m s X a n d Y e a c h h a v e n u c l e a r s p i n ^ . T h e X a t o m s w i l l b e r e f e r r e d t o a s t h e " o f f - a x i s " a t o m s w h i l e a t o m Y l i e s a l o n g t h e s y m m e t r y a x i s o f t h e m o l e c u l e , r , a n d w i l l b e r e f e r r e d t o a s t h e " o n - a x i s " a t o m . ( s e e f i g . 6 ) I n g e n e r a l a C X ^ Y m o l e c u l e b e l o n g s t o t h e p o i n t g r o u p C^v, 24 78 79 a n d c l a s s i f i c a t i o n u n d e r t h i s g r o u p i s w e l l k n o w n » 1 ^ * T h e r e a r e 5 s y m m e t r y o p e r a t i o n s b e s i d e s t h e i d e n t i t y ; t w o r o t a -t i o n s a b o u t t h e s y m m e t r y a x i s b y 2tv/3 ( C ^ ) , a n d b y - 2 T T / 3 , ( C * ) , a s w e l l a s t h r e e r e f l e c t i o n s (crv) i n p l a n e s p a r a l l e l t o t h e s y m m e t r y a x i s . T h e r e a r e 3 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 s , A-j_, A g a n d E , w h i l e f o r t h e s u b g o u p o f c 3 v > n a m e l y C ^ , t h e t w o d i m e n s i o n a l E r e p r e s e n t a t i o n s p l i t s i n t o t w o d e g e n e r a t e o n e d i m e n s i o n a l r e p r e s e n t a t i o n s E a n d E^ ( s e e t a b l e ( 2 ) ) . A r o t a -t i o n b y 2TT/3 i s e q u i v a l e n t t o t h e p e r m u t a t i o n o f t w o p a i r s o f i d e n t i c a l n u c l e i ^ s o t h a t f o r e i t h e r B o s e - E i n s t e i n s t a t i s t i c s F i g u r e 6. M o l e c u l a r Geometry of a Symmetric Top Mo lecu le CX ,Y . Note r x v and 6XY r e f e r to atom X and atom .Y , not to the c o o r d i n a t e system X > Y > Z • 148 TABLE 2 CHARACTER TABLES C 3 E ° 3 C l A 1 1 1 E l 1 e e 2 E2 1 e 2 e C 3 , E 2 C 3 V A l 1 1 1 A2 1 1 -1 E 2 -1 0 MULTIPLICATION TABLE C 3xC A E i E 2 A A E i E 2 E l E i E2 A E 2 E2 A E l e = exp 2 i r i / 3 149 or F e r m l - D i r a c s t a t i s t i c s , the t o t a l w a v e f u n c t i o n f o r t o t a s i n g l e m o l e c u l e must be i n v a r i a n t . T h i s means t h a t it. . to r t o t must t r a n s f o r m as the t o t a l l y s y m m e t r i c , o r A^, r e p r e s e n -t a t i o n . F u r t h e r m o r e , the r e f l e c t i o n s a permutes one p a i r o f i d e n t i c a l n u c l e i o n l y and t h u s $tot must be o f A^ symmetry f o r Bose and A^ f o r F e r m i s t a t i s t i c s . W i t h i n the a p p r o x i m a t i o n o f s e p a r a t i n g t h e d i f f e r e n t i n t e r n a l m o l e c u l a r m o t i o n s , can be w r i t t e n as a p r o d u c t o f e l e c t r o n i c \p v i b r a t i o n a l , , r o t a t i o n a l , IJJ and Q V 3T* n u c l e a r s p i n ^ wave f u n c t i o n , namely (1) * t o t = ^ v M l • I f t h e e l e c t r o n i c c o n t r i b u t i o n i s assumed t o be i n v a r i a n t t o and a v , t h e n the symmetry o f ty^ot depends o n l y on t h a t o f il) \\> IJJ v r I The CX Y m o l e c u l e s c o n s i d e r e d here a r e n o n - p l a n a r and 3 have two frameworks^^which cannot be r e l a t e d by a p r o p e r r o t a -t i o n . These a r e the two i n v e r t e d s p e c i e s . I n many m o l e c u l e s i n c l u d i n g CF^H and CH^F t h e b a r r i e r t o i n v e r s i o n between th e l e f t and r i g h t handed frameworks i s v e r y h i g h and the m o l e c u l e i s " t r a p p e d " i n one o f two c o n f i g u r a t i o n s . As t h i s b a r r i e r becomes l o w e r , i n v e r s i o n s can t a k e p l a c e more e a s i l y . An example o f t h i s i s NH^. The p o i n t i s t h a t even though i n v e r s i o n 150 may be very slow, i t i s s t i l l a f e a s i b l e process by quantum-mechanical tunnelling and should be accounted f o r . The consequence of t h i s i s not that the energetics are changed since only for NH^ i s the b a r r i e r low enough so the inver-sion doubling can be experimentally resolved, but that p a r i t y becomes a good quantum number rather than the projection of the t o t a l angular momentum on the symmetry axis, J.*r=k . The magnitude of k, K=|k| remains however a good quantum number. The only v i b r a t i o n a l motion that i s considered here Is the p o s s i b i l i t y of framework inversion. Otherwise, the v i b r a -t i o n a l states are assumed to be t h e i r (symmetric under C^y) ground states. Under the symmetry group the two d i f f e r e n t frameworks each have such a symmetric state. Under C^ v the two frameworks are mixed, and the states can be c l a s s i f i e d as either even or odd under framework inversion. Hence ty v i s written as |p> for the symmetric (p=0) or antisymmetric (p=l) states. The symmetry of ty^0t i s thus determined by the product ty ty ty^. p r T I An i n v e s t i g a t i o n of the symmetry of the r o t a t i o n and spin function under CU symmetry i s given f i r s t , and then ty, , i s constructed to be consistent with the symmetry of the group C^ v. Since operator methods have been used throughout t h i s t h e s i s , the appropriate projection operators are introduced 151 wh i c h p r o j e c t o n l y s t a t e s o f t h e p r o p e r s p i n and r o t a t i o n symmetry. 2.1) The R o t a t i o n a l F u n c t i o n s The magnetic f i e l d i n d e p e n d e n t energy l e v e l s a r e ob-t a i n e d from t h e r o t a t i o n a l h a m i l t o n i a n I n t h e body f i x e d c o o r d i n a t e system and f o r a symmetric t o p , t h e moment o f i n e r t i a t e n s o r can be w r i t t e n i n the form (3) ^ I ^ = J where B and C a r e r e s p e c t i v e l y t h e r o t a t i o n a l c o n s t a n t s about axes p e r p e n d i c u l a r and p a r a l l e l t o t h e d i r e c t i o n o f the symmetry a x i s , r and r e l a t e d t o t h e moment o f i n e r t i a t e n s o r s I R and I . by B=h / 8 i r 2I R , and C=h / 8 i r 2I r . The 152 energy l e v e l s E j ^ . a r e t h e n g i v e n by, (4) E J k = ^BJ(J+1) + -£(C-B)k2 I n a p p e n d i x ( A ) t h e E u l e r i a n a n g l e s , RE(a,$,y) , f o r a c t i v e r o t a t i o n s w i t h r e s p e c t t o the l a b o r a t o r y o r space f i x e d frame have been d e f i n e d and t h e symmetric t op wave f u n c t i o n d i s c u s s e d . These a r e (5) where t h e Mk m a t r i x element o f t h e ( 2 J + 1 ) - dimen-s i o n a l i r r e d u c i b l e r e p r e s e n t a t i o n o f the 3 - d i m e n s i o n a l 81 r o t a t i o n group , a c c o r d i n g t o t h e n o t a t i o n o f Edmonds A l t h o u g h t h e d i s t i n c t i o n between k and K=|k| i s not n e c e s s a r y f o r t h e energy, i . e . E J k = E J K i t i s i m p o r t a n t f o r t h e symmetry p r o p e r t y o f ip . I n p a r t i c u l a r , a r o t a -J JJCIYI t i o n by 2ir/3 about t h e body f i x e d symmetry a x i s i s e q u i v a l e n t t o c h a n g i n g y t o y + 2TT/3 w i t h the consequence t h a t 1 5 3 ( 6 ) Thus t h e symmetry o f t h e r o t a t i o n a l s t a t e s i s d e t e r m i n e d e n t i r e l y by t h e quantum number k . The symmetry r e s t r i c -t i o n s a r e t h u s s u f f i c i e n t l y w e l l d e s c r i b e d by p r o j e c t i o n o p e r a t o r s onto k su b s p a c e s . S i n c e t h e energy a l s o depends on J , i t i s u s e f u l t o a l s o s e p a r a t e d i f f e r e n t J s t a t e s . Such a p r o j e c t i o n o p e r a t o r onto t h e J , k subspace i s g i v e n e x p l i c i t l y by (7) An immediate consequence o f E q . ( 6 ) i s t h a t i n t h e absence o f any o t h e r degrees o f freedom, t h e symmetry o f ty ^ i s d e t e r m i n e d s o l e l y by ty , and t h e o n l y v a l u e s o f t o t JkM k w h i c h a r e a l l o w e d a r e m u l t i p l e s o f 3n. T h i s f o l l o w s from the r e q u i r e m e n t C0ty. . = ty, . The p r e s e n c e o f t h e 3 too t o t s p i n degrees o f freedom i s t o a l l o w a compensation f o r t h e l a c k o f symmetry w h i c h a r i s e s when k i s not a m u l t i p l e o f 3 . 154 2.2) S p i n F u n c t i o n s The th ree i d e n t i c a l i n - p l a n e s p i n s of CX^Y form a t o t a l s p i n of 1=3/2 or 1=1/2 w i t h coresponding sym-metry adapted s p i n f u n c t i o n s denoted by "V^,/ir and M T or yt/xfrtx ( t a b l e 3 ) . The p r o j e c t i o n opera to rs onto these s t a t e s can be c o n s t r u c t e d i n s e v e r a l ways and perhaps 82 the s i m p l e s t i s to use the group theory fo rmu la where % i s the d imension of the a t h i r r e d u c i b l e r e p r e -a s e n t a t i o n , h i s the order of the g roup , and xtg) i s the c h a r a c t e r of the group o p e r a t i o n f o r the a t h 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 . (see t a b l e 2) Here , i s to operate on s p i n s and thus must be e q u i v a l e n t to a pe rmutat ion of the s p i n s t a t e s s i n c e the opera to r r o t a t e s the molecu le about the symmetry a x i s by 2TT/3 . Such a r o t a t i o n i s a c t u a l l y e q u i v a l e n t to the c y c l i c p e r m u t a t i o n , Pia^ of the n u c l e i which can a l s o be g i v e n as ( 9 ) (C_) . 3 s p i n ^IA ^ 3 1 5 5 TABLE 3 S p i n F u n c t i o n s ( o f f - a x i s s p i n s o n l y ) Symmetry T o t a l I m I Symmetry Adapted S p i n F u n c t i o n s A 3/2 3/2 aaa 1/2 l/f3(a&a + a a 3 + 3 a a ) - 1 / 2 l / f 3 ( B a 3 + 3 3 a + a 3 3 ) - 3 / 2 3 3 3 E x 1/2 1/2 1 ^ 5 ( 3 a a + e a 3 a + e 2 a a 3 ) - 1 / 2 1 ^ 3 ( a 3 3 + e 3 a 3 + e 2 3 3 a ) E 2 1/2 1/2 1 ^ 3 ( 3 a a + e 2 a 3 a + e a a 3 ) - 1 / 2 l / ^ 3 ( a 3 3 + e 2 3 a 3 + e 3 3 a ) e = e x p ( 2 i r i ) 3 The o r d e r , e.g. a 3 3 , r e f e r s t o n u c l e u s 1,2 and 3 r e s p e c t i v e l y where a c o r r e s p o n d s t o m j = h and 3 c o r r e s p o n d s t o m = . 156 where P^ and Px$ a r e the i n t e r c h a n g e p e r m u t a t i o n s o f n u c l e i 1 and 2 , and 2 and 3 , r e s p e c t i v e l y . The p r o j e c -t i o n o p e r a t o r i s t h e n (10) where e = exp. ( i 2 i r / 3 ) and t h e c l a s s i f i c a t i o n a c c o r d i n g t o the i n t e g e r s i n t h a t s=3n c o r r e s p o n d s t o A symmetry w h i l e s=3n±l c o r r e s p o n d t o E-^  and E^ symmetries r e -s p e c t i v e l y . Because o f t h e r e l a t i o n (c^)Sp±nl^ = £S > t h e p r o j e c t e d s p i n s t a t e s if^^ave e i g e n f u n c t i o n s o f b o t h J% and ( C 3 ) S p i n namely (11) and Ss rI>Ml - rXtfir (12) but not an e i g e n f u n c t i o n f o r ^ L L , © - ^ 1 5 7 C13) P13 V j , n r -F o r s p i n \ n u c l e i , s c o m p l e t e l y d e t e r m i n e s t h e magni-tude o f t h e t o t a l n u c l e a r s p i n s t a t e s o f t h e o f f - a x i s n u c l e i . A l s o , because o f t h e s i m p l e r e p r e s e n t a t i o n o f i n terms o f the P a u l i s p i n o p e r a t o r s , namely t h e D i r a c exchange r e l a t i o n , an e x p l i c i t e x p r e s s i o n i n terms o f s p i n o p e r a t o r s i s e a s i l y o b t a i n e d f o r P,m (and f o r ) . I n p a r t i c u l a r P»a3 i s g i v e n i n terms o f I ^ , I_ 2 , and l g by ( 1 5 ) and becomes 158 2 . 3 ) T o t a l Symmetry S i n c e o n l y the n u c l e a r and r o t a t i o n a l f u n c t i o n s can change under , the e q u i v a l e n t o p e r a t o r f o r t h e t o t a l wave f u n c t i o n i s ( 1 7 ) c3 ~ P,» (cXo*. C o n s e q u e n t l y t h e t o t a l w a v e f u n c t i o n t r a n s f o r m s under C^ a c c o r d i n g t o S i n c e t h e t o t a l w a v e f u n c t i o n must be i n v a r i a n t t o C^, t h e s p i n and r o t a t i o n a l symmetries a r e r e l a t e d by the r e q u i r e m e n t t h a t (m an i n t e g e r ) ( 1 9 ) k + s = 3 m . Thus k d e t e r m i n e s b o t h t h e symmetry s p e c i e s and the t o t a l nu-c l e a r s p i n o f t h e o f f - a x i s n u c l e i . As mentioned above, ty must be a n t i s y m m e t r i c t o p e r -t o t m u t a t i o n o f one p a i r o f s p i n % n u c l e i . The e q u i v a l e n t 159 o p e r a t i o n f o r i s an i n v e r s i o n t i m e s a r o t a t i o n by T about an a p p r o p r i a t e a x i s . The i n v e r s i o n c o n s i d e r e d here i s the u m b r e l l a m o t i o n o f r e p l a c i n g z by -z ( i . e . r e f l e c t i o n i n the p l a n e p e r p e n d i c u l a r t o the symmetry a x i s , o"h ). T h i s , however i s not t h e same as p a r i t y w h i c h r e p l a c e s x, y, and z by - x , -y and -z . Under the p a r i t y o p e r a t i o n Pj becomes ±ty , The e f f e c t o f b o t h i n v e r s i o n and p a r i t y i s t o change t h e o r d e r o f t h e s p i n l a b e l s as i l l u s t r a t e d i n f i g . ( 7 ) . Thus i n v e r s i o n ( a ^ ) i s e q u i v a l e n t t o p a r i t y and a r o t a t i o n about the body f i x e d z a x i s by TT , P (20) (21) oj = R z (rr) P S i n c e z i s the body f i x e d a x i s , R (TT) changes y t o Y + 7 T Hence the e f f e c t o f i n v e r s i o n on ^ i s 85 JkMP (22) 160 161 F o r t h e o p e r a t i o n o f r e f l e c t i o n i n t h e xz p l a n e , see f i g . ( 7 ) , t h e e q u i v a l e n t o p e r a t i o n i s ( 2 3 ) (^)Xz ~ RxCrtoi = R y ( n ) P . S i n c e r o t a t i o n about ^y^71") e q u i v a l e n t t o t h e t r a n s -f o r m a t i o n , 0t Ct + TT 3 -*• IT - 3 Y -> ir - y and f o r R ( T T ) ^ a -*• a + ir 3 -*• TT - 3 Y -> _Y the e f f e c t o f ( a v ) on ^ J k M p i s X z (24) Here t h e t r a n s f o r m a t i o n p r o p e r t i e s 162 (25) o f t h e r o t a t i o n a l m a t r i c e s have been u s e d ^ . T h i s a v r e f l e c t i o n i n t e r c h a n g e s n u c l e i 2 and 3 w h i l e the o t h e r r e f l e c t i o n s , permute t h e o t h e r p a i r s 1 ,2 and 1 ,3 • S i n c e k changes s i g n under t h e a y o p e r a t i o n , i s not an e i g e n f u n c t i o n o f a y . I t i s p o s s i b l e t o c o n s t r u c t an a p p r o p r i a t e wave f u n c t i o n c o n s i s t e n t w i t h b o t h the o p e r a t i o n and , namely (26) where k i s no l o n g e r a good quantum number but r a t h e r K = |k| and p a r e . The c o e f f i c i e n t n can be found by 1 6 3 r e q u i r i n g On p e r f o r m i n g t h e a p p r o p r i a t e o p e r a t i o n s and u s i n g E qs. ( 1 3 ) and (24), n must be ( - l ) J + K + p , so t h a t f o r K ^ 0 When K = 0 , t h e a p p r o p r i a t e e x p r e s s i o n i s (29) K.n VT'H1 V„= C-f") s i n c e a d i f f e r e n t n o r m a l i z a t i o n i s r e q u i r e d , and J + p must be odd. Thus f o r K = 0 , o n l y one h a l f t h e number o f s t a t e s a r e p r e s e n t as t h e r e a r e f o r K ^ 0 . T h i s i s w e l l 164 known e x p e r i m e n t a l l y f o r NH^ . C o r r e c t l y when c o n s i d e r i n g K s t a t e s , K ^ 0 shou ld be t r e a t e d s e p a r a t e l y from K = 0 to a v o i d e r r o r s i n n o r m a l i z a t i o n . S ince the molecu les t r e a t e d here are assumed to i n v e r t s l o w l y , the energy l e v e l s are not s i g n i f i c a n t l y m o d i f i e d by i n c l u d i n g the i n v e r s i o n energy , i . e . , (30, (K*}L)VT.r= E „ V W where ^tiJ i s the i n v e r s i o n h a m i l t o n i a n . The a p p r o p r i a t e opera to r which p r o j e c t s on ly the JKMp s t a t e i s - C-0 l7&rt-sp)<7-% rl s pi ( 3 D I t i s not necessary to l a b e l (P w i t h an s s i n c e K determines the s p i n symmetry. Fu r thermore , M i s not 165 r e t a i n e d and a s t a t e i n E q . ( 3 1 ) , (32) jiih-Sp)- lHn}i-s))P> has been w r i t t e n f o r t h e p r o d u c t o f t h e r o t a t i o n |JkM> , s p i n |s> and p a r i t y , |p> s t a t e s . By use o f t h e p r o j e c t i o n o p e r a t o r d e f i n e d by Eqs. ( 7 ) and ( 1 0 ) , (P J K M p can be s i m p l i f i e d f o r K ? 0 t o M (33) = 3 f •<• - C-,rMn-k „><74s,l and f o r K = 0 (34) 166 CHAPTER 3 DESCRIPTIVE EXAMPLE OP HIGH FREQUENCY EFFECTS I n o r d e r t o I l l u s t r a t e the t y p e o f d e n s i t y dependence e x p e c t e d from the h i g h f r e q u e n c y c o n t r i b u t i o n s t o T 1> a q u a n t i t a t i v e l y c r u d e , but q u a l i t a t i v e l y i n f o r m a t i v e example i s d i s c u s s e d . The model i s as f o l l o w s : suppose t h a t t h e o n l y c o l l i s i o n a l c o u p l i n g w h i c h a r i s e s i s between + and -f r e q u e n c y components w h i c h a r i s e from th e d i p o l a r i n t r a -m o l e c u l a r mechanism. F o r s i m p l i c i t y o n l y one t y p e o f f r e q u e n c y i s c o n s i d e r e d , f o r example k f r e q u e n c i e s , and from above the non-zero c o u p l i n g i s assumed t o a r i s e o n l y between a f r e -quency component w and ui, n =-w. . ,. A l s o assume t h a t a l l k k t K K KK t h e T'S w h i c h can a r i s e a r e independent o f t h e r o t a t i o n a l quantum numbers and can be t r e a t e d i n the impact a p p r o x i m a t i o n . F u r t h e r m o r e l e t the d v e c t o r s w h i c h appear i n Eq.(11 -25) depend o n l y on t h e Boltzmann w e i g h t s and i g n o r e o t h e r depen-dence on t h e J's and k's . T h i s can be e x p r e s s e d as ( 3 5 ) d 167 where b c o n t a i n s t h e c o u p l i n g c o n s t a n t and a l l o t h e r m o l e c u l a r parameters, ( E^ . = k 2 ( C - B ) where C and B a r e r o t a -t i o n a l c o n s t a n t s ) . F i n a l l y i t i s assumed t h a t o n l y one t y p e o f f r e q u e n c y t r a n s i t i o n i s i m p o r t a n t , and i n t h i s example i t i s a r b i t r a r i l y chosen as Ak=+2. Hence t h e f r e q u e n c i e s , w, , i n c r e a s e l i n e a r l y w i t h k , These sweeping a p p r o x i m a t i o n s g r e a t l y s i m p l i f y t h e r e l a x a t i o n m a t r i x , w h i c h i n t h e impact a p p r o x i m a t i o n , can be w r i t t e n as an i n f i n i t e sum o v e r k o f 2 x 2 m a t r i c e s which d i f f e r o n l y i n t h e v a l u e o f the f r e q u e n c y component, (36) (37) R - J 1 6 8 Here use has been made of the symmetry p r o p e r t i e s f o r the m a t r i x elements of R which are w e l l known i n the impact a p p r o x i m a t i o n . The l i n e w i d t h , y , the s h i f t , 6 , and the c o l l i s i o n a l c o u p l i n g e lement , C are a l l independent of the J ' s and k ' s as assumed above. Note t h a t , y , 6 , and c are a l l d e f i n e d to be independent of d e n s i t y which has been accompl ished by m u l t i p l y i n g each element by x , the c o l l i s i o n (or c o r r e l a t i o n ) t ime of the low d e n s i t y r e l a x a t i o n mechanism, i . e . ( 3 8 ) ( 3 9 ) and (40) 169 . - 1 The d_ v e c t o r s i n the t o t a l e x p r e s s i o n f o r T^ become i n t h e above a p p r o x i m a t i o n , (41) d / Jf and so the t o t a l e x p r e s s i o n f o r T^ 1 i s oo r, 4-C42) - ? The m a t r i x can be e a s i l y i n v e r t e d , t h e m a t r i x - v e c t o r m u l t i -p l i c a t i o n t r i v i a l l y c a r r i e d out and t h e elements summed t o g i v e , 170 T, 8 4 G(k) i s p r o p o r t i o n a l to the l i n e shapes of the k, k+2 and k+2, k t r a n s i t i o n . In the above e q u a t i o n s , C 0 T 2 the low f requency c o n t r i b u t i o n to the r e l a x a t i o n r a t e . Th is va lue can e a s i l y be f i t t e d from e x p e r i m e n t a l data when W J C T 2 > > 1 and a l l the G(k) c o n t r i b u t i o n s are n e g l i g i b l e . The approx imat ions which l e a d to Eq . (43) are not as d r a s t i c as i t might appear , a l though no e f f o r t has been made to p h y s i c a l l y j u s t i f y them. However i t i s p a r a l l e l to the t reatment used f o r the i n v e r s i o n spectrum of ammonia. As a r e s u l t each 2 x 2 r e l a x a t i o n m a t r i x , , i s of the same form as o b t a i n e d by Ben-Reuven 2 ^>^9 m However, a d i f f e r e n c e l i e s i n the magnitude of the f requency components. For NH^ i n v e r s i o n there i s on ly one p o s i t i v e and one negat i ve f r e -quency to c o n s i d e r , w h i l e i n the case t r e a t e d h e r e , a l l the r o t a t i o n a l f r e q u e n c i e s , , shou ld appear . In t h i s 171 t r e a t m e n t , the d i f f e r e n t f requency components have weights a p p r o p r i a t e l y d e f i n e d from the Boltzmann d i s t r i b u t i o n w h i l e the r e l a x a t i o n m a t r i c e s , , d i f f e r from one to the next only i n the magnitude of t h a t appears t h e r e i n . Each c o n t r i b u t i o n , G(k) , i s uncoupled from the others -1 w i t h the t o t a l e x p r e s s i o n f o r be ing a s u p e r p o s i t i o n of independent l i n e shapes. The purpose i s now to demonstrate the e f f e c t of d e n s i t y on these te rms . Th is i s i l l u s t r a t e d u s i n g Dong's CH^F p ro ton r e l a x a t i o n d a t a . The va lue of ^2 = T 2 n ^ S r e P ° r t e c * by Dong f o r the CH^F s p i n r o t a t i o n c o r -r e l a t i o n t ime i s (9 .64 + 0 . 1 ) x 1 0 " 1 1 sec -amagats . I t i s assumed t h a t the d i p o l a r c o r r e l a t i o n t ime i s the same. With t h i s v a l u e , and the r o t a t i o n a l cons tants f o r CH^F, the va lue of w, x~ i s A f t e r f i t t i n g ^QT2 ^° l ° w e r l i n e a r d e n s i t y r e g i o n , th ree parameters remain to be f i t t e d . These are b 2 ( y - £ ) j Y 2 - ? 2 and 6 . In order to o b t a i n a f i t to the e x p e r i m e n t a l p o i n t s , T-^/n i s p l o t t e d a g a i n s t the d e n s i t y n . From E q . ( 4 3 ) , T-^/n i s approximated i n the n u m e r i c a l c a l c u l a t i o n by t r u n c a t i n g the sum over k at k=7, k l 2 (44) 172 (45) where G(k) = nG(k) . The r e s u l t s of a l e a s t squares f i t r o u t i n e f o r t h i s approx imat ion to the CH^F p ro ton r e l a x a t i o n data at 297°K, 350°K and 433°K are g i v e n i n f i g . ( 8 ) , and the va lues f o r the parameters are g i ven i n t a b l e ( 5 ) . To f u r t h e r i l l u s t r a t e how the d e n s i t y dependent c o n t r i b u t i o n s are m a n i f e s t , the l i n e shapes , G(k) are p l o t t e d i n f i g . ( 9 ) f o r the 297°K f i t parameters . For example, the f i r s t c o n t r i b u t i o n , 'G(O), corresponds to k=0->-2 coupled to k=2->-0. Th i s p i c t o r i a l l y demonstrates the concept of phase r a n d o m i z a t i o n emphasized throughout t h i s t h e s i s . As seen from f i g . ( 9 ) , at low enough d e n s i t i e s there i s n e g l i g i b l e c o n t r i b u t i o n from any h i g h f requency component w .^ , but as the d e n s i t y i n c r e a s e s , each G(k) goes through a maximum va lue and then decreases to a constant v a l u e . The maxima occur at r e g u l a r i n t e r v a l s of 40 amagats because of the l i n e a r f requency i n c r e a s e , and from f i g . ( 8 ) the t h e o r e t i c a l curve i s a r e s u l t of on ly the f i r s t h igh f requency component. Th is c o i n c i d e s n i c e l y w i t h exper iment . 173 TABLE 4 MOLECULAR CONSTANTS CH3P 0 F 3 H r e f . B Mc/sec 24 , 5 3 6 . 1 2 10,348 .74 88,24 C Mc/sec 1 5 0 , 0 0 0 5 , 6 0 0 88,24 r HP X- 2 . 0 3 1 .977 90 r HH o A 1 . 8 0 90 9 HP 30 ° 5 1 ' 38 ° 5 7 ' 1 0HH 90° 1 C S R ( H ) ( K H z ) * 3 . 8 6 ± .39 .4 ± .05 2 CSR< P> (KH )2 z 2 5 . 6 2 ± 3 . 8 5 4 8 . 1 ± 2 . 1 2 Q r 297°K 2 7 , 5 6 0 . 3 5 , 3 7 8 . T 2 S T 1 sec-amagat ( . 9 6 4 + . 0 1 ) x 1 0 ~ 1 0 ( 2 . 5 6 ± . 1 1 ) x I O " 1 0 2 <J>297° = 12 - 3 3 < K > 2 9 7 ° = 4 -19 Y F = = 2 . 5 2 x I O4 e.m.u. Yu = n 2 . 6 7 x I O 4 e .m.u. 3 5 0 - i 120 _ j , , , IO 2 0 3 0 4 0 D E N S I T Y p ( A M A G A T S ) F i g u r e 8. T h e o r e t i c a l F i t to R. Dong'.s 1 P ro ton T-, R e l a x a t i o n D a t a . 3 . 5 n P ( A M A G A T S ) F i g u r e 9. P l o t of ^ ( k ) versus Dens i t y f o r k Equal Zero to Seven. 176 As e x p l a i n e d e a r l i e r , an a l t e r n a t i v e way o f v i e w i n g t h e s e r e s u l t s would be t o c o n s i d e r the f r e q u e n c y s p e c t r u m , and t h e n superimpose v a r i o u s d i f f e r e n t d e n s i t y c a l c u l a t i o n s . T h i s was o r i g i n a l l y done by Ben-Reuven f o r t h e NH^ i n v e r s i o n 29 s p ectrum , and has r e c e n t l y been d i s c u s s e d f o r CH^F and 84 CF^H by C l a r k . Both p i c t u r e s a r e e q u i v a l e n t and each demonstrates a d i f f e r e n t a s p e c t o f the e f f e c t o f i n c r e a s i n g d e n s i t y . I f the l i n e shape ( o r s p e c t r a l d e n s i t y t i m e s d e n s i t y ) i s p l o t t e d a g a i n s t t h e f r e q u e n c y i t i s found t h a t a t low enough d e n s i t i e s the v a r i o u s f r e q u e n c y components a r e independent and thus r e s o n a t e a t t h e i r own c h a r a c t e r i s t i c f r e q u e n c i e s . As t h e d e n s i t y i n c r e a s e s o v e r l a p p i n g and s h i f t i n g becomes i m p o r t a n t . E v e n t u a l l y , a t h i g h enough den-s i t i e s , v a r i o u s l i n e s a c t u a l l y merge and form b r o a d bands c e n t e r e d a t one o r s e v e r a l non-resonant f r e q u e n c i e s . Thus a sca n i n d e n s i t y has a c o n s i d e r a b l e e f f e c t on the f r e q u e n c y spectrum. I t i s not i m p l i e d t h a t t h i s t r e a t m e n t e x p l a i n s t h e CH^F d a t a , and a s u c c e s s f u l f i t s h o u l d be p o s s i b l e w i t h any o f the CH^F r o t a t i o n a l f r e q u e n c i e s . As can be seen from t h e e x p r e s s i o n f o r G ( k ) , the maximum o c c u r s when ^ " ^ ^ ^ n ' Hence t h e f r e q u e n c y s h i f t f i x e s t he p o s i t i o n o f t h e s t e p . The i n t e n s i t y o f each c o n t r i b u t i o n a t t h e maximum i s g i v e n 177 by 2x 2b 2/(y+C) so that as y + ? decreases, the i n t e n s i t y increases. In p a r t i c u l a r , from the d e f i n i t i o n of y as a r a t i o of the relaxation time for the dominant mechanism to the relax a t i o n time for one component, a small y i s expected, (see Eq. ( 1 - 9 8 ) ) . For CH^F, a better procedure than using Ak=+2 t r a n s i t i o n s would be to use AJ=+1 and +2 frequencies. The J t r a n s i t i o n s correspond to the lowest frequencies i n the r o t a t i o n a l spec-trum of CH^F, and are expected to come into phase sooner than the larger k frequencies. Also undesirable i n t h i s approximate treatment, i s the coupling, C , between p o s i t i v e ' and negative k frequencies. Such a coupling requires a change i n spin species during a c o l l i s i o n and i t has been assumed up to t h i s chapter that such c o l l i s i o n a l t r a n s i t i o n s are forbidden. Hence £ would be zero. Again, a better procedure would be to include p o s i t i v e and negative J f r e -quencies since no spin symmetry r e s t r i c t i o n s exist for sym-metric top molecules. However the treatment here c l e a r l y i l l u s t r a t e s both the physical process and the method of evaluating the r e l a x a t i o n matrix. In the following chapter T n i s evaluated i n more d e t a i l but for CF H, however, the 1 3 procedure and q u a l i t a t i v e picture remains the same. That i s , some p a r t i a l uncoupling approximation i s made and sym-metry properties are used to simplify the relaxation matrix. 178 TABLE 5 VALUES OP PARAMETERS FOR CH_F FROM LEAST SQUARES FIT 297°K 350°K 433°K C T p o 2 2 x 2 b 2 ( Y - C ) Y 2- ? 2 3-320 .356 .118 - 1 . 2 0 5 3-977 1.157 • 351 •2.023 5.076 2.583 .662 - 2 . 6 2 0 179 C H A P T E R 4 T H E E F F E C T O F H I G H F R E Q U E N C I E S ON T H E N U C L E A R S P I N R E L A X A T I O N O F C F 0 H I n h i s s t u d y o f C H ^ F a n d C F ^ H , D o n g r u l e d o u t t h e p o s s i b i l i t y t h a t h i s p r o t o n T-^ d a t a w e r e d u e t o s u r f a c e e f f e c t s , p a r a m a g n e t i c i m p u r i t i e s o r i n s t r u m e n t a l d e f e c t s . F u r t h e r h e i n v e s t i g a t e d t h e h i g h d e n s i t y r e g i o n o f t h e f l u o r i n e T-^ , a n d f o u n d t h a t n o d e v i a t i o n s f r o m t h e l i n e a r d e n s i t y d e p e n d e n c e w a s o b s e r v e d f o r e i t h e r m o l e c u l e , t h a t i s , o n l y t h e p r o t o n s g a v e a n o n - l i n e a r d e n s i t y d e p e n d e n c e . O v e r h a u s e r e x p e r i m e n t s w e r e a l s o p e r f o r m e d , a n d t h e r e s u l t s i n d i c a t e t h a t a t h i g h e r d e n s i t i e s i n t h e r e g i o n w h e r e d e n s i t y d e v i a t i o n s w e r e f i r s t o b s e r v e d , t h e O v e r h a u s e r g a v e p o s i t i v e e n h a n c e m e n t f o r b o t h m o l e c u l e s . T h i s i s e x p e r i m e n t a l e v i -d e n c e t h a t t h e i n t r a m o l e c u l a r d i p o l a r m e c h a n i s m c o n t r i b u t e s t o t h i s p h e n o m e n a . T h e O v e r h a u s e r m e a s u r e m e n t f o r C F ^ H i s r e p r o d u c e d i n f i g u r e ( 1 0 ) . T h e l o w e r d e n s i t y m e a s u r e m e n t s , h o w e v e r , i n d i c a t e t h a t t h e s p i n r o t a t i o n m e c h a n i s m d o m i n a t e s f o r b o t h t h e p r o t o n a n d f l u o r i n e s p i n s . A p a r t f r o m t h e i n d e -p e n d e n t e x p e r i m e n t s o f A r m s t r o n g ^ a n d W a u g h , t h i s c o n c l u d e s C H F 3 ft 10 2 0 3 0 4 0 ' 5 0 6 0 7 0 SO o Figure 10, Overhauser Eff e c t Measurements f o r CHFo at P (A''*AC"-C^S) Room Temperature. 181 the exper imenta l d a t a . I t i s p o s s i b l e to draw g e n e r a l obvious c o n c l u s i o n s from these measurements: 1. the low d e n s i t y i s dominated by s p i n - r o t a t i o n ; 2 . f l u o r i n e data i s dominated by s p i n - r o t a t i o n to d e n s i t i e s as h igh as s t u d i e d and show no n o n - l i n e a r d e n s i t y d e v i a -t i o n s ; 3. at h i g h e r d e n s i t i e s f o r p ro ton r e l a x a t i o n , d i p o l a r i n t e r -a c t i o n becomes i m p o r t a n t ; 4. the p ro ton r e l a x a t i o n r a t e i n c r e a s e s s i g n i f i c a n t l y and p o s s i b l y suddenly at h i g h e r d e n s i t i e s a l though the magnitude of the e f f e c t i s i n q u e s t i o n . The most probable reason why c o n c l u s i o n 2 i s observed l i e s i n the magnitude of the e f f e c t i v e s p i n r o t a t i o n c o u p l i n g constant f o r f l u o r i n e r e l a t i v e to the c o u p l i n g constant f o r the p r o t o n s . From t a b l e ( 4 ) , i t i s seen tha t the f l u o r i n e c o u p l i n g cons tants are much g r e a t e r than f o r the protons i n both CF^H and CH^F. Hence the protons should be more s e n s i -t i v e than the f l u o r i n e s to e x t r a c o n t r i b u t i o n s that may a r i s e from h i g h f requency s p i n r o t a t i o n or d i p o l a r te rms . For CF^H, on ly AJ and no AK t r a n s i t i o n s can a r i s e from the s p i n r o t a t i o n mechanism f o r the p ro ton s i n c e i t l i e s on the symmetry a x i s . The c o u p l i n g tensor consequent ly has components which do not mix K s t a t e s . Th is i s not t r u e 1 8 2 f o r CH^F. I t i s t h e r e f o r e not p o s s i b l e to r u l e out h i g h f requency e f f e c t s which a r i s e from s p i n r o t a t i o n , but i t i s not cons idered here because of the Overhauser ev idence and the r e s u l t of c o n c l u s i o n 2 . In order to d i s c u s s h igh frequency c o n t r i b u t i o n s to f o r the p ro ton r e l a x a t i o n of CF^H, the g e n e r a l e x p r e s -s i o n E q . ( 1 1 - 3 5 ) must be adapted to the case at hand. That - 1 * i s f o r T. , i - y c\ d"*\ fa'*)", j"* the d : v e c t o r s E q . ( 1 1 - 2 5 ) and the r e l a x a t i o n m a t r i x R M see E q . ( 1 1 - 2 9 ) must be e v a l u a t e d . S ince only the i n t r a -m o l e c u l a r d i p o l a r mechanism i s c o n s i d e r e d , £ i s 2 (second rank t e n s o r ) , and q v a r i e s from - 2 to 2 . F u r t h e r -more , the p ro ton s p i n , 1 ^ i s equa l to h and t h i s can be s u b s t i t u t e d i n E q . ( 4 6 ) . As a r e s u l t of the d i s c u s s i o n i n chapter 2 , i t f o l l o w s t h a t the a p p r o p r i a t e set of quantum numbers which l a b e l the The s p i n degeneracy ( 2 1 ^ + 1 ) i s i n c l u d e d i n the p a r t i t i o n f u n c t i o n r a t h e r than e x p l i c i t l y as has been done i n E q . ( I I - 3 5 ) 183 symmetric top mo lecu la r r o t a t i o n a l and s p i n s t a t e s are the angu lar momentum quantum numbers J , K=|k| , M , p a r i t y p , the s p i n symmetry number s and s p i n component m^ . . These are summed over i n the dot product n o t a t i o n ( c f . E q . ( 1 1 - 3 2 ) ) sub jec t to the symmetry r e s t r i c t i o n s of chapter 2 . The i n f i n i t e r e l a x a t i o n m a t r i x i n Eq.(46) must be reduced or approximated to a s i z e and form so t h a t the c a l c u l a t i o n i s p o s s i b l e . The d e t a i l s of the r e d u c t i o n are p resented i n s e c t i o n (3) but there are two n a t u r a l approx imat ions which e f f e c t i v e l y reduce i t s s i z e and which are d i s c u s s e d h e r e . F i r s t , the Boltzmann weights e v e n t u a l l y r e s u l t i n the c o n t r i -b u t i o n to T ^ be ing very s m a l l f o r l a r g e va lues of J and K . ( A c t u a l l y the Boltzmann weights are i n c l u d e d i n the d v e c t o r s , w h i l e the r e l a x a t i o n m a t r i x elements do not d i e o f f f o r l a r g e J and K v a l u e s . ) At 297° K the most p robab le va lue of J i s about 33 and f o r K about 19 f o r CF^H . Thus i t i s expected tha t t r u n c a t i n g the i n f i n i t e sum at J=60 and f o r k=-J to J shou ld be a good a p p r o x i m a t i o n . A second r e d u c t i o n i n the s i z e of the r e l a x a t i o n m a t r i x comes from the magnitude of the f requency components. I t has been s t r e s s e d tha t on ly the lowest l y i n g f r e q u e n c i e s are i m p o r t -a n t . Thus i t might be a good approx imat ion to i n c l u d e only the s m a l l e s t J and K l e v e l s . Numer ica l c a l c u l a t i o n s indeed 184 c o n f i r m t h i s , so that most of the h i g h e r f r e q u e n c i e s are complete ly phase randomized and need not be i n c l u d e d , (see e . g . f i g . (9) f o r CH3P) In p a r t i c u l a r , f o r CF^H, the K l e v e l s p l i t t i n g s are the lowest i n the r o t a t i o n a l spect rum. A c o n s i d e r a b l e number of K f r e q u e n c i e s l i e below the f i r s t few J t r a n -s i t i o n f r e q u e n c i e s . Therefore i t may be reasonab ly assumed t h a t J f r e q u e n c i e s do not c o n t r i b u t e , and the K 's are r e s p o n s i b l e f o r the e f f e c t . As a f u r t h e r argument to j u s t i f y t h i s a s s e r t i o n , i t i s expected tha t f o r s i m i l a r J and K f r e q u e n c i e s , the magnitude of the c o n t r i b u t i o n to T ^ from the K t r a n s i t i o n s would be l a r g e r than those from J . Th i s f o l l o w s by s t u d y i n g the r o t a t i o n a l energy l e v e l s f o r a symmetric top m o l e c u l e . There i s one s e p a r a t i o n of the same frequency f o r each J l e v e l b u t , except f o r a c c i d e n t a l d e g e n e r a c i e s , each J f requency a r i s e s only once. For C F 3 H t h i s means tha t about 60 equa l K f r e q u e n c i e s a r i s e and hence the i n t e n s i t i e s are enhanced. As a r e s u l t of these arguments, on ly the K f r e q u e n c i e s are t r e a t e d and these have the i n t r a m o l e c u l a r d i p o l a r s e l e c t i o n r u l e AK=0 , ±1 and ± 2 . Moreover , an average over J l e v e l s i s taken f o r each K . I f AJ t r a n s i t i o n s are i g n o r e d , t h i s J ave rag ing i s an exact p rocedure . In t h i s t reatment of CFoH, i t i s assumed 1 8 5 tha t AJ=0. The procedure f o r a d a p t i n g Eq . (46) f o r CF^H i s as f o l l o w s . F i r s t the i n t r a m o l e c u l a r d i p o l e - d i p o l e c o u p l i n g h a m i l t o n i a n i s d i s c u s s e d . Th i s e n t a i l s s p e c i f y i n g what the a p p r o p r i a t e l a t t i c e , F^ , and s p i n , , o p e r a t o r s a r e . With t h e s e , the d v e c t o r s can be e v a l u a t e d and from t h i s c a l c u l a t i o n , the s e l e c t i o n r u l e s are o b t a i n e d . F o l l o w i n g t h i s , the r e l a x a t i o n m a t r i x i s approximated and the e x p r e s -s i o n f o r TJ-1- put i n t o convenient form f o r f u r t h e r a n a l y s i s . Only h igh f requency e f f e c t s are t r e a t e d , and q u a n t i t i e s which a r i s e w i t h zero r o t a t i o n a l energy are i g n o r e d . That i s , i t i s assumed that the low d e n s i t y r e g i o n of n/T^ obeys the u s u a l theory which p r e d i c t s a constant d e n s i t y depend-dence , ( n i T l 'o where the constant A can a r i s e from other mechanisms b e s i d e s the d i p o l a r , such as s p i n r o t a t i o n i n t e r a c t i o n . S ince d i f f e r e n t c o r r e l a t i o n t imes are used f o r h i g h f r e -quency and low f requency e f f e c t s , t h i s i s q u i t e c o n s i s -t e n t . The va lues ob ta ined from h i g h f requency e f f e c t s are not a f f e c t e d by the va lue of A . A c a l c u l a t i o n by Dong'*' which e s t i m a t e s at the most an 1 1 % c o n t r i b u t i o n to A 186 from the d i p o l a r mechanism n o t e d , but t h i s i s n e i t h e r v e r i f i e d nor d i s p r o v e d i n the t reatment h e r e . There i s , however, disagreement w i t h the e x p e r i m e n t a l va lue of A w i t h Dong r e p o r t i n g A""*" = .42 sec./amagat w h i l e from Armst rong 's data ( f i g . 5 ) A - 1 - 1 sec ./amagat . R e c e n t l y 92 Beckmann has s t u d i e d CF^H and agrees w i t h an e x t r a p o -l a t e d va lue of Waugh's and Armst rong 's h i g h e r d e n s i t y va lue of about A -''" - .8 sec ./amagat , see f i g . ( 5 ) . Th is va lue i s used i n t h i s t r e a t m e n t . 4 .1) D i p o l e - d i p o l e I n t r a m o l e c u l a r C o u p l i n g H a m i l t o n i a n  f o r CF^H M o l e c u l e s . There are two types of c o n t r i b u t i o n s to the d i p o l e -d i p o l e h a m i l t o n i a n between n u c l e a r s p i n s w i t h i n a CX-^ Y m o l e c u l e . One term a r i s e s from o f f - a x i s , o f f - a x i s s p i n s (between XX atoms) and the o ther from o n - a x i s , o f f - a x i s s p i n s (between XY atoms) . Th is can be g e n e r a l l y w r i t t e n as CT) K j^ = I t h h - - il t l t T , 1+ : Qui. where the o f f - a x i s atoms are l a b e l l e d 1 to 3 , and the o n -a x i s atom i s l a b e l l e d 4 , (see f i g . 6 ) . 1^ i s the n u c l e a r s p i n opera to r f o r the i t h atom, and C . . i s the symmetric 1 8 7 t r a c e l e s s c o u p l i n g t e n s o r which couples the i t h and j t h sp ins through the m o l e c u l a r geometry. For the p ro ton coup-l i n g i n CF^H , on ly o f f - o n a x i s need be r e t a i n e d , and the second term can thus be dropped. I t i s p o s s i b l e to w r i t e y(cjj i n terms of symmetry adapted s p i n and l a t t i c e o p e r a -t o r s as f o l l o w s , ( e = e x p(2iTis/3) ), where, see E q . ( l O ) , (49) = p s i , = j C L + e " ' ? a * £ ' ? i ) and (50) The equ iva lence between the f i r s t term of Eq.(47) and Eq.(48) 188 can be checked by summing over s i n E q . (48) and u s i n g the d e f i n i t i o n s Eqs . (49) and ( 5 0 ) . I t i s worth n o t i n g tha t the o n - o f f a x i s c o n t r i b u t i o n i s i d e n t i c a l i n form to a s p i n r o t a t i o n c o u p l i n g h a m i l t o n i a n i f 1^ i s r e p l a c e d by J I t i s t h e r e f o r e not d i f f i c u l t to extend t h i s t reatment to i n c l u d e the s p i n - r o t a t i o n mech-an ism. For subsequent c a l c u l a t i o n s , (I_) I_ i s w r i t t e n The c o u p l i n g between d i f f e r e n t atoms depends on t h e i r i n t e r n u c l e a r s e p a r a t i o n , and the gyromagnetic r a t i o a c c o r d -e d c o u p l i n g t e n s o r must be ( i n the body f i x e d c o o r d i n a t e system) i n g to 8 3 ^ X . V r X Y From f i g u r e ( 6 ) , i t i s seen tha t the (5D 3 KKK 'XT ( 5 2 ) 189 By use of the r e l a t i o n s (zz)S r ZZf^ , (Z) 5- i cfs,o (.53) (x)' = - i - [ C l - C ^ ^ ) x - X B s u ~ * p Y ] , A, the symmetry adapted c o u p l i n g tensors are 3C i ^ - 0 C54) and O Y C55) 190 In terms of the b a s i s f u n c t i o n s e (see appendix ( A ) ) , q these become ( 5 6 ) _ 3 \^ (fx C^ Y p3 1 XY L /6 r X Y L J (57) 3 [ b?* eU + or i n g e n e r a l ( 5 8 ) 191 Here the constants b„ are (59) Th is h a m i l t o n i a n i s f o r o n - o f f a x i s c o u p l i n g and f o r t h i s case AJ and AK t r a n s i t i o n s of 0 , ±1 , and ±2 can a r i s e . I f o f f - o f f a x i s c o u p l i n g were t r e a t e d , AK c o u l d not be ±1 s i n c e 0 = TT/2 . Us ing the r o t a t i o n m a t r i c e s d e f i n e d i n appendix (A ) , the b a s i s v e c t o r s e2 which are i n the m o l e c u l a r frame can be expressed i n terms of the b a s i s v e c t o r s e^ of the l a b . frame E q . ( 5 8 ) then becomes, (60) (C)r=it b 4 € * fL.-if-B) + R corresponds to the E u l e r i a n ang les a B y ?ov the t r a n s -f o r m a t i o n from the l a b o r a t o r y to the m o l e c u l a r f rame. The h a m i l t o n i a n can now be w r i t t e n f o r the o n - o f f c o u p l i n g as H c n = Z Z t C?i£ (c)l C (61) - t t where the s p i n and l a t t i c e opera to rs are and Table 6 g i v e s the va lues of b 0 f o r CF^H. 1 9 3 TABLE 6 PROTON DIPOLAR COUPLING TENSOR COEFFICIENTS FOR CF^E 2 . 7 5 x 1 0 5 s e c " 1 . 1 6 6 4 + . 4 8 8 8 . 1 9 7 6 a = 3 h Y H Y F / r ^ p t , 0 ( 9 ) = b Q / a b ± 1 ( 9 ) = b ± 1 / a b ± 2 ( 6 ) = b ± 2 / a 9 E 9 H F 1 9 4 4.2) E v a l u a t i o n of the d ^ 2 q ^ V e c t o r s f o r K^O ,AJ=0. With the l a t t i c e , ( C ) s , and s p i n , ( I I , , ) S ope ra to rs = q ~~4 q s p e c i f i e d , the d ^ 2 c ^ v e c t o r s can be o b t a i n e d . One compon-ent i s , S >5 where ( j j ^ p p r o j e c t s onto a s t a t e of d e f i n i t e J , K (hence s p i n ) and p a r i t y , p , ( c f . E q . ( 3 3 ) ) . The c o u p l i n g c o n -s t a n t s b^ are now i n c l u d e d i n the l a t t i c e o p e r a t o r s , see E q . ( 6 3 ) , r a t h e r than separate as i n d i c a t e d i n Eq . ( 4 6 ) . Under p a r i t y , the opera to r (C) i s even and hence on ly s t a t e s of the same p a r i t y are connected , namely (65) + -*-*"+> -+->•-, + 4* - . The c o n v e n t i o n a l + and - i n d i c a t e s t a t e s of even (p=0) and odd (p=l) p a r i t y r e s p e c t i v e l y . I f the p r o j e c t i o n opera to rs 6j*f» are s u b s t i t u t e d from E q . ( 3 3 ) , s i x t e e n terms a r i s e . Of t h e s e , 12 cor respond to t r a n s i t i o n s from k=l to k ' = l . Other va lues of k and k' cannot be connected wi thout exceeding the s e l e c t i o n r u l e Ak = ±2 . Thus, when AJ ^ 0 c o n t r i b u t i o n s are i g n o r e d , 1 9 5 the 12 terms modify only the low d e n s i t y r e g i o n of T^ 1 . A l though such terms should be i n v e s t i g a t e d when c a l c u l a t i n g at low d e n s i t i e s , they are not g i v e n here because they do not a f f e c t the h igh f requency phenomena. The remain ing four terms account f o r a l l h igh f requency phenomena from k 7* 0 t r a n s i t i o n s , v i z , (K ,K ? ^ 0 ) < 6 6 > Each of the f o u r terms i s very s i m i l a r but d i f f e r i n the signs- of k and s . I t i s now shown tha t Eq . (66 ) i s the same d / 2 ^ component e v a l u a t e d under symmetry o n l y , i . e . C67) 81& ^ < ? . < r ^ < ? 4 < ? r ^ J . 1 9 6 The l a s t form of E q . ( 6 7 ) f o l l o w s from the d e f i n i t i o n of the i n n e r p r o d u c t , the commutation of s p i n and r o t a t i o n o p e r a t o r s , and the h i g h temperature approx imat ion of the e q u i l i b r i u m d e n s i t y o p e r a t o r , P q . The ob jec t i s to r e l a t e the f i r s t term i n E q . ( 6 6 ) to the l a s t th ree i n E q . ( 6 6 ) . Cons ider the o p e r a t i o n of permut ing two s p i n s , e . g . 2 and 3 i n f i g u r e ( 7 ) • From E q . ( 1 3 ) the e f f e c t on the s p i n f u n c t i o n only i s to change the s i g n of |s> to |-s> , hence , <-s/ CXI^'l-S') r ^ s / &P»(Ilj* l s ' > ( 6 9 ) The r o t a t i o n about the x a x i s by TT , R x (TT ) , i s e q u i v a -Q C l e n t to the t r a n s f o r m a t i o n , 6 -> TT - 9 <J> -> $ + TT 197 it The e f f e c t on ( C ) S of R (ir) I S then = q x " 4=-3L U (70) I f I Is r e p l a c e d by -I , I t Is found t h a t , ( 7 D R x ( i r ) Ccf = I k,C-'> ^ - . ^ ' c f ^ . ^ O where use has been made of the r e l a t i o n ( - 1 ) b ^ = b^. In u s e , I i s equa l to ± ( k - k ' ) s i n c e S) * operates between symmetric top wave f u n c t i o n s S)M^  a n d ^n'-ti ' C o n s e Q u e n t l y H k+k' the ( - 1 ) = ( - 1 ) can be f a c t o r e d out of the sum over I when o p e r a t i n g between such f u n c t i o n s . S ince the e f f e c t of the r o t a t i o n R x ( T T ) on | JkM> i s (72) R x ( T T ) I T & M > - c-if\T~k*>, 198 i t f o l l o w s t h a t With the r e l a t i o n s , E q s . ( 7 3 ) and ( 6 9 ) , the f o u r c o n t r i b u t i o n s i n E q . ( 6 6 ) are found to be equa l and the p roof i s complete . The argument g i ven above i s s t r i c t l y v a l i d on ly when K and K' ^ 0 . To i n c l u d e the case when K or K' = 0 the p r o j e c t i o n o p e r a t o r , E q . ( 3 * 0 must be used . A l though not e x -p l i c i t l y proved h e r e , the same va lue f o r dS^^ i s o b t a i n e d f o r the cases ( K or K1 = 0 ) , but w i t h on ly h a l f the i n t e n -s i t y as f o r K and K* ^ 0 . Prom the p o i n t of view of symmetry, t h i s f o l l o w s from the f a c t tha t on ly one h a l f o f the p a r i t y s t a t e s are present when K = 0 , ( c f . E q . ( 2 8 ) ) . Under symmetry however, k i s a good quantum number, and runs f o r g i v e n J from - J to J . S ince the energy depends only on k 2 , a +k and - k g i ve equal c o n t r i b u t i o n , except f o r k = 0 when only one term i s p r e s e n t . The r e s u l t of the above argument conf i rms a w e l l known 91 theorem which s t a t e s t h a t when an o p e r a t i o n i s unfeas ib le -^ ( 7 3 ) 199 ( e . g . I n v e r s i o n f o r CHF^) , the molecu les can be t r e a t e d as b e i n g i n two d i f f e r e n t independent frameworks ( i n t h i s case the l e f t and r i g h t handed forms of CHF^ )• The s p i n and r o t a t i o n t r a c e s have been c a l c u l a t e d i n Appendix ( B ) , and the combined r e s u l t i s [tUif'-lhfj; (74) Hi The l a s t f a c t o r i n E q.(74) i s the s p i n t r a c e , and i s v a l i d f o r ( s p i n h) o n - o f f a x i s c o u p l i n g . There i s no d i f f e r e n c e between the va lue of t h i s term f o r Ak/0 t r a n s i t i o n s , but f o r Ak=0 the weight v a r i e s from A to E symmetry by 1 0 : 1 . To d i g r e s s s l i g h t l y , i f o f f - a x i s o f f - a x i s c o u p l i n g were b e i n g t r e a t e d i t i s found that t r a n s i t i o n of the type E-,-«-> E (Co symmetry) cannot occur a l though A E-j_ , 2 A «-> Eg c a n . The f r a c t i o n a l p o p u l a t i o n of the J , k r o t a t i o n a l l e v e l P j k i s (75) PlA = 1 < 6\J 7A^> = * E U / h T . M Q. 2 0 0 The p a r t i t i o n f u n c t i o n , Q , must account f o r the s p i n degen-eracy } g ( I , k ) , which by E q . ( 1 9 ) depends on the p a r t i c u l a r k l e v e l , (76) Q - Z OJ+O ~a In the absence of s p i n degeneracy the r o t a t i o n a l p a r t i t i o n f u n c t i o n , , can be approximated at h igh temperatures by (77) Q„ - / T T ( k T ) = / TT ( T where B and C are the r o t a t i o n a l cons tants f o r symmetric t o p s . I f i t i s assumed t h a t 1/3 of the k l e v e l s (k=3n) have s p i n I = 3 / 2 f o r the o f f - a x i s s p i n s , and 2 / 3 of the k l e v e l s , k= 3 n ± l , have o f f - a x i s s p i n I = 1/2 , Q can be approximated a s , (78) whe re 1^ i s the s p i n of the on a x i s n u c l e u s , a l s o equa l 201 to 1/2, thus (79) Q = 16 Q 3 r For CF 3 H at 297°K t h i s has the va lue 1.8868 x 1 0 5 . For computa t iona l pu rposes , i t i s convenient to re -arrange E q.(74). The f o l l o w i n g d e f i n i t i o n s are u s e f u l ; ( 8 0 ) W - ~ ^ b * C & ^ = C bJLCt**> which separates the angle dependence of from the mag-n i t u d e of the c o u p l i n g constant c , and < 8 l ) /* r i J r n i s independent of the s i g n of k . Thus Eq.^46) becomes ( 1 ^ = 1/2 k^k ' and the sum over k ' s i s by symmetry t w i c e the sum over K 's ) 2 0 2 Note t h a t T(JKK) i s independent of q , and i n the a p p r o x i -mat ion tha t the r e l a x a t i o n m a t r i x i s a l s o independent of q , the sum over q g i v e s the va lue 1 0 . A l s o the summa-t i o n over J can be performed and w i t h ( 8 3 ) ''a where {KK' } i n d i c a t e s the l a r g e r of K and K* , E q . ( 8 2 ) i s - 80 c (84) T , -t T ( K , ) [ ( R T ' ] K K , T C K » K « . ) Mi K w i t h the a p p r o p r i a t e r e l a x a t i o n m a t r i x , ( c f . E q s . ( 1 1 - 2 4 ) and ( H - 2 9 ) ) , 2 0 3 S K - " T L T ( K K') V 1 /«,«' T ( H - K - ) / ' M"|K'" J ( 8 5 ) = 1 ' S The c o u p l i n g c o n s t a n t s , c 2 , b^(9) , and the p a r t i t i o n f u n c t i o n are known. The q u a n t i t i e s T (KK ' ) have been com-puted at 2 9 7 ° K by summing over the exact e x p r e s s i o n up to J. = 6 0 , These are g i v e n i n t a b l e ( 7 ) f o r s m a l l K and K* v a l u e s . E q u a t i o n (84) accounts f o r a l l the h i g h f requency e f f e c t s which a r i s e from the i n t r a m o l e c u l a r d i p o l a r mechanisms w i t h AJ = 0 , AK = ± 1 , ± 2 . I t does not account f o r the zero f r e -quency c o n t r i b u t i o n to T^- which i s not used i n the present t r e a t m e n t . Prom the low d e n s i t y form of (l/T-j_) = A/n , the t o t a l e x p r e s s i o n a p p r o p r i a t e f o r a l l d e n s i t i e s past the Larmor minima up to as f a r as the Boltzmann equat ion i s v a l i d , i s ( 8 6 ) 204 TABLE 7 VALUES OF T (KK ' ) FOR CF 3 H AT 2 9 7 ° K K K' T (KK 1 ) K K' T (KK' ) 0 1 . 2 5 1 7 3 5 0 2 . 9 2 8 7 3 6 1 2 . 8 1 9 2 0 2 1 3 1 . 2 8 8 2 7 5 2 3 1 . 1 8 1 4 6 2 2 4 1 . 2 5 7 2 6 6 3 4 1 . 4 7 7 0 3 0 3 5 1 . 2 2 2 8 0 1 4 5 1 . 7 2 2 3 9 9 4 6 1 . 1 8 6 3 4 6 5 6 1 . 9 2 7 8 0 7 5 7 1 . 1 4 8 8 5 8 6 7 2 . 1 0 0 3 8 4 6 8 1 . 1 1 0 9 9 1 7 8 2 . 2 4 5 4 4 0 7 9 1 . 0 7 3 2 1 0 8 9 2 . 3 6 7 1 1 7 8 1 0 1 . 0 3 5 8 3 2 9 1 0 2 . 4 6 8 7 4 0 9 1 1 . 9 9 9 0 9 0 1 0 1 1 2 . 5 5 3 0 2 9 1 0 1 2 . 9 6 3 1 4 8 1 1 1 2 2 . 6 2 2 2 6 0 1 1 1 3 . 9 2 8 1 1 8 1 2 1 3 2 . 6 7 8 3 5 4 1 2 14 . 8 9 4 0 7 7 1 3 14 2 . 7 2 2 9 5 0 1 3 1 5 . 8 6 1 0 7 1 14 1 5 2 . 7 5 7 4 4 5 14 1 6 . 8 2 9 1 2 4 1 5 1 6 2 . 7 8 3 0 6 2 1 5 1 7 . 7 9 8 2 5 0 1 6 1 7 2 . 8 0 0 8 5 4 1 6 1 8 . 7 6 8 4 4 0 1 7 1 8 2 . 8 1 1 7 4 3 1 7 1 9 . 7 3 9 6 9 4 1 8 1 9 2 . 8 1 6 5 4 1 1 8 2 0 . 7 1 1 9 8 0 1 9 2 0 2 . 8 1 5 9 4 8 1 9 2 1 . 6 8 5 2 7 8 2 0 5 I t remains to eva lua te the r e l a x a t i o n m a t r i x and t h i s i s approximated i n the f o l l o w i n g s e c t i o n . 4 . 3 ) Approx imat ion of the R e l a x a t i o n M a t r i x The x ' s which appear i n the r e l a x a t i o n m a t r i x have been p a r t i a l l y e v a l u a t e d i n Par t I I I . In t h i s s e c t i o n the r e s u l t s of tha t t reatment are adapted to CF^H a l though the same p r o -cedure would be a p p r o p r i a t e f o r CH^F. In s e c t i o n 1 of t h i s c h a p t e r , a p a r t i a l c o l l i s i o n a l u n c o u p l i n g of the problem was suggested . I t i s based on the i n e f f i c i e n c y of c o l l i s i o n s to change the symmetry of the t o t a l n u c l e a r s p i n s t a t e , and hence on ly c o l l i s i o n a l t r a n s i -t i o n s of AK = ±3n are p o s s i b l e . The s i g n i f i c a n c e of t h i s can be understood by c o n s i d e r i n g an element of the c o l l i s i o n opera to r f o r one J - a v e r a g e d f requency component, ( 8 7 ) = « (R/ *> ' " » T~(K,K') ~T(K",K'") By the i n t r a m o l e c u l a r i n t e r a c t i o n f o r the d i p o l a r mechanisms i t f o l l o w s that K -K ' , and K"-K'" can e i t h e r be 0 , ±1 , or ±2 . However, the c o l l i s i o n a l s e l e c t i o n r u l e s imply tha t 206 K-K" = 3n and K ' -K" 1 = 3m where n and m are i n t e g e r s . These s e l e c t i o n r u l e s reduce the non -zero elements i n the r e l a x a t i o n m a t r i x s u b s t a n t i a l l y . I f such AK^O c o l l i s i o n a l t r a n s i t i o n s are present the n o n s p h e r i c a l p a r t of the i n t e r m o l e c u l a r p o t e n t i a l must be r e s p o n s i b l e . The f i r s t term which can cause a AK = ±3 t r a n s i t i o n i s the o c t a p o l e moment ft . In p a r t i c u l a r , the ft - 3ft component of t h i s t e n s o r can g i ve a torque xxx xyy * & about the symmetry a x i s r e s u l t i n g i n a AK = ±3 s e l e c t i o n r u l e . Such t r a n s i t i o n s have been observed by O k a ^ i n NH^- rare gas c o l l i s i o n s . Higher terms i n the i n t e r m o l e c u l a r m u l t i p o l e expansion than the o c t a p o l e w i l l be ignored i n t h i s t reatment f o r CF^H. Prom Par t I I I , the g e n e r a l e x p r e s s i o n s f o r a T i s g i v e n by E q . ( I I I - 3 7 ) . For the case h e r e , runs from 1 to 3 c o r -respond ing to d i p o l e , quadrupo le , and o c t a p o l e moments r e -s p e c t i v e l y f o r molecu le one, and £ g = 1 o n l y co r respond ing to the d i p o l e moment of molecu le 2 . Some i n d i c a t i o n has been g i ven concern ing the d i f f i c u l t i e s encountered i n e v a l -u a t i n g the H ' s . (See E q s . ( I I I - 3 8 ) and ( 1 1 1 - 3 9 ) . ) Here they are t r e a t e d as phenomenologlcal parameters . In o rder to reduce the number of pa ramete rs , i t i s assumed f o r mathe-m a t i c a l convenience tha t i s r e a l and t h a t 2 0 7 Fur thermore , i t i s assumed t h a t the va lues of the » 's do not vary much w i t h changing K ' s . I t i s convenient to d e f i n e U = 1 ) , ( 8 9 ) ~ \ S U / K K'IT K ' ) ( 9 0 ) and f o r K^K" , K y K " 1 , ( 9 1 ) J ( l ) * \ E U J * KK'K-K'") . Even though the va lues of the q u a n t i t i e s are u n a f f e c t e d by changing va lues of K , the symmetry r e l a t i o n s d e r i v e d e a r -l i e r s t i l l depend on the order i n which the k ' s appear . In p a r t i c u l a r 5 ( i ± ) •*• -8(1^ a s K K1 and K' K i n E q . ( 9 0 ) , and y(i-^) > 0 * These assumptions are made only to reduce the number of a d j u s t a b l e parameters . M a t r i x elements 208 of the r e l a x a t i o n m a t r i x are t h e r e f o r e (92) - r i f c f f / ) - * c f o ) + d k ) v- *LACOKH>] and (p i s the Boltzmann weight f o r the K r o t a t i o n a l s t a t e K J averaged) L-lZ^T Kit That on ly = 3 c o n t r i b u t e s to the 5 term f o l l o w s from the AK = ±3 s e l e c t i o n r u l e over the t r a n s i t i o n opera to r t o g e t h e r w i t h the requirement tha t only the o c t a p o l e moment can cause t h i s . S t i l l there are 6 unknown parameters i n R K K ' K K ' * Th is i s reduced to f o u r phenomenologica l parameters by adding the s h i f t s , 6(1) + 6(2) + 6(3) = 6 . Fu r the rmore , any combinat ion of the y ( ^ ) ' s may be used f o r a g i v e n component, but to reduce computa t ion , y = y ( l ) i s used w h i l e y (2) and y(3) are set equal to z e r o . Th is cho ice i s (93) T R K V " 209 complete l y a r b i t r a r y and o ther cho ices of the r e l a t i v e s i z e s of Y ( D J Y ( 2 ) , and y ( 3 ) cannot be d i s t i n g u i s h e d from t h i s one f o r low va lues of K. The J - a v e r a g e d , AJ=0 , XV \ K ,K ' , K",K'!!/ ' s have been e v a l u a t e d i n Appendix (D) to be The 6 a r i s e s from the AK = ±3 s e l e c t i o n r u l e . With K,K'±3 the va lues of the K'5 and the o ther m o l e c u l a r cons tants from Table ( 4 ) , the on ly unknowns i n E q.(84) are y , 6 and c, . The s i t u a t i o n i s s i m i l a r to the s imple case t r e a t e d i n the l a s t c h a p t e r , E q . ( 4 3 ) , a l though now the m a t r i x i s somewhat more c o m p l i c a t e d . However, v a r i o u s symmetry r e l a t i o n s s i m -* p l i f y R , and these are now d i s c u s s e d . Note f i r s t that R f a c t o r s i n t o 3 c o l l i s i o n a l l y uncou-p l e d m a t r i c e s , R ^ , E E ^ J and R G ^ which correspond r e s p e c t i v e l y to symmetries A , E^, and of the f i r s t K appear ing i n a m a t r i x element of R . Hence, i n p a r t i c u l a r , i f K = 3n, *Prom here on T „ R i s w r i t t e n R . 210 t h e n , R ., ,, belongs to R f l . Th is r e d u c t i o n means tha t KK K K i t i s p o s s i b l e to t r e a t R. m a t r i x s e p a r a t e l y from the R„ and Rg m a t r i c e s . That i s , f r e q u e n c i e s such as co^  ^ are c o l l i s i o n a l l y uncoupled from io,_ _ , s i n c e a t r a n s i t i o n of 5 , 3 AK = 5 -3 = 2 i s r e q u i r e d over the t r a n s i t i o n opera to r to couple these f r e q u e n c i e s and t h i s v i o l a t e s the AK = ±3 c o l -l i s i o n a l s e l e c t i o n r u l e . On the o ther hand to0 _ i s coupled 3 , 5 to 03^  2 s i n c e AK = 5-2 =3 • N o t i c e f u r t h e r t h a t the f r e -quency to„ n i s uncoupled from io 0 .. and hence R u n -3 , 5 3,1 — A couples even f u r t h e r . These o b s e r v a t i o n s are summarized below f o r symmetry, a = A , E.^  or Eg , and g i ves a l l the non - ze ro elements i n R : =a d i a g o n a l components; (n = ± 1 , ±2) o f f - d i a g o n a l components; (n = (~2 m = f l ) ( 9 6 ) 211 fix + Ptt;m. \ y ( * ^ ^ 3 The o f f - d i a g o n a l terms couple f r e q u e n c i e s , to to D K,K+n U K±3 ,K±m ' f r e c l u e n c l e s > U K , K ± n t 0 UK,K+m f ° r n = 1 m=2 or n=2 m=l. In p a r t i c u l a r , m a t r i x R has o f f - d i a g o n a l elements c o n n e c t i n g f r e q u e n c i e s w i t h AK = +2 to AK= - 1 , and elements c o n n e c t i n g AK = -2 to AK = +1 , but no e l e -ments c o u p l i n g AK = +2 to AK = +1 ,±2 . Hence R^ un^-couples i n t o two s m a l l e r m a t r i c e s , denoted by R* and R~ where an element of R + couples f r e q u e n c i e s AK = +2 t o —a AK = -1 , and R^ couples f r e q u e n c i e s of AK = -2 to AK = +1 . As a r e s u l t of t h i s decompos i t ion R f a c t o r s i n t o 6 s m a l l e r m a t r i c e s , R^ , R^ , Rg , a n c ^ ' However, the re are r e l a t i o n s between these m a t r i c e s . From E q . ( 9 4 ) f o r /) I »( K' K" K"'J t n e f o l l o w i n g symmetry r e l a t i o n s are found by permut ing the K ' s , (the K 's must of course obey the s e l e c t i o n r u l e s ) ( 9 8 ) K ' ' " K ' v/ / a. 4 A \ ^ I -K r K ' , - K " , - K u ' J 212 The m a t r i x elements of R* then obey (to K K = - w K , K to K , - K ( 9 9 ) - rt";-K"' Consequent ly I f (Jf1",,,,, v l l v J n runs from - K to K , i t a u t o -m a t i c a l l y accounts f o r (R~ K , K „ K m ) a so tha t changing the sums to i n c l u d e negat i ve K va lues e l i m i n a t e s the n e c e s s i t y of r e t a i n i n g both R + and R~ m a t r i c e s . The second e q u a l i t y i n E q . ( 9 9 ) r e l a t e s the R^ elements to Rg where K* belongs to symmetry 3 , Th is j u s t s t a t e s tha t a t r a n s i t i o n A E symmetry i s equa l to the o p p o s i t e t r a n s i t i o n f o r E -»• A. Th is i s i l l u s t r a t e d i n f i g . ( 1 1 ) , and hence t r a n s i t i o n s from A -«-»- E-j_ and A E t r a n s i t i o n s are accounted f o r i n S A on ly so t h a t 2 R A a c -counts f o r a l l A to E t r a n s i t i o n s , but not the E^ Eg t r a n s i t i o n s . These however are taken i n t o account by one R* m a t r i x (see f i g . ( 1 1 ) ) . - E 2 K \ r / r / / / \ / / \ 1 \ s \ / r \ / \ t \ / / \ f \ y f 2 3 4 5 7 RA R ~A R E 2 R + E , R E , R E 2 F i g u r e 11. K T r a n s i t i o n s A l lowed i n Each Submatr ix of the R e l a x a t i o n M a t r i x . 214 Prom these symmetry r e l a t i o n s between the elements of the r e l a x a t i o n m a t r i x , and u s i n g the f a c t tha t (101) T (KK' ) = T (K 'K) = T ( - K - K ' ) , Eq .(84) can be s u b s t i t u t e d i n t o Eq . (86) to g i v e , -2L =. fl + M (102) y ^ The dot products run over a l l the a l l o w e d va lues of K which f o l l o w from the s e l e c t i o n r u l e s f o r R* and Rg . The ma-t r i x R A i s g i v e n i n f i g . ( 1 2 ) and a s i m i l a r m a t r i x occurs f o r 2 ^ 2 " T h e ^ ' s a r e S i v e n in t a b l e (8) ( f o r s m a l l va lues of K) and have been obta ined by n u m e r i c a l l y summing up to J= 60 at 297°K. With the va lues of Q=1.8868 x 1 0 5 , T =2.65'x 1 0 ~ 1 0 sec-amagats and c 2 = ( 2 . 7 5 4 ) 2 x 1 0 ^ s e c - 2 , the a p p r o p r i a t e form f o r f i t t i n g to the 297°K T x /n CHP^ data i s .-/ (103) ^ = / " f l 4 -ooo^sj'SLj{^f'T + T . [ Z l j ' T j w i t h the only unknowns b e i n g y , 6 and £ T A B L E 8 VALUES OF K V i 1 2 3 0 1 1 . 0 0 0 0 0 .91582 1 . 0 0 0 0 0 1 2 .93615 .87325 .96752 2 3 !88200 .88827 .93739 3 4 .84373 .89599 .91706 4 5 .81237 .90242 •90179 5 6 .78471 .90791 . 8 9 0 8 3 6 7 .75939 .91228 . 8 8 3 3 3 7 8 .73575 .91541 . 8 7 8 5 3 8 9 .71342 .91726 .87580 9 10 .69217 .91786 .87467 10 11 .67187 .91727 .87475 11 12 .65240 .91556 .87573 12 13 .63369 .91281 .87737 13 14 .61569 .90911 .87945 14 15 . 5 9 8 3 3 .90452 .88180 15 16 .58159 .89913 .88428 16 17 .56542 . 8 9 3 0 1 .88678 17 18 .54980 . 8 8 6 2 3 .88919 18 19 .53470 .87885 .89143 19 20 .52010 .87095 .89346 TABLE 8 ( cont inued) 216 K K» 1 2 3 0 2 1 . 0 0 0 0 0 •77745 1 . 0 0 0 0 0 1 3 . 9 8 5 3 3 .79159 . 9 8 0 0 5 2 4 . 9 6 7 7 5 . 8 0 8 7 0 .95733 3 5 . 9 4 8 7 7 . 8 2 6 5 6 . 9 3 5 4 8 4 6 . 9 2 9 0 9 . 8 4 4 0 9 . 9 1 5 9 0 5 7 . 9 0 9 0 8 . 8 6 0 7 2 . 8 9 9 0 9 6 8 . 8 8 8 9 7 . 8 7 6 1 1 .88515 7 9 . 8 6 8 9 3 . 8 9 0 1 1 . 8 7 3 9 8 8 10 ' . 8 4 9 0 5 . 9 0 2 6 3 .86539 9 11 . 8 2 9 4 0 .91363 .85914 10 12 . 8 1 0 0 4 .91281 . 8 5 4 9 8 1 1 13 . 7 9 1 0 1 .93122 . 8 5 2 6 6 12 14 . 7 7 2 3 2 . 9 3 7 8 8 .85194 13 15 . 7 5 3 9 9 .94319 . 8 5 2 5 8 14 16 .73604 . 9 4 7 2 4 .85437 15 17 .71845 . 9 5 0 0 7 .85712 16 18 . 7 6 1 2 4 .95177 . 8 6 0 6 4 17 19 . 6 8 4 4 0 .95239 . 8 6 4 7 9 18 20 . 6 6 7 9 2 . 9 5 2 0 1 . 8 6 9 4 0 19 21 .65180 . 9 5 0 6 8 . 8 7 4 3 6 217 TABLE 8 ( cont inued) VALUES OP )^( ( K K ' K" K'") K K' K" K. K' K" 0 2 0 - 1 . 0 0 0 0 0 0 1 3 1 . 0 2 5 8 8 1 3 4 3 .10179 1 2 1 - 1 .27943 2 4 2 1 - . 1 3 2 7 5 1 2 4 2 .25530 2 4 5 4 .15128 2 3 2 0 .07295 3 5 3 2 .08204 3 4 6 4 . 0 9 8 2 1 4 6 7 6 .13567 4 5 7 5 .22014 5 7 8 7 . 12823 4 5 4 2 .26533 5 7 5 4 .15605 6 7 9 7 .11968 6 8 6 5 .11584 5 6 5 3 .12977 7 9 10 9 .13705 7 8 10 8 .18552 8 10 12 10 .10529 7 8 7 5 .23214 8 10 8 7 . 13632 9 10 12 10 .11952 9 11 9 8 .12105 8 9 8 6 . 13857 10 12 13 12 .12578 10 11 13 11 .15478 11 13 15 13 . 0 8 5 7 5 10 11 10 8 .19670 11 13 11 10 .11260 11 12 11 9 . 13035 12 14 12 11 .11362 12 13 15 13 . 10968 13 15 16 15 . 0 9 4 6 0 13 14 16 14 . 12842 14 16 18 16 .06966 13 14 13 11 .16454 14 16 14 13 .09186 14 15 14 12 .11519 15 17 15 14 . 10065 15 16 18 16 .09577 16 18 19 18 .09171 16 17 19 17 . 10645 17 19 17 16 .07467 16 17 16 14 . 13673 18 20 18 17 .08572 17 18 17 15 .09765 218 TABLE 8 ( cont inued) K K* K" K K' K" K'" 0 2 - 3 -1 .02765 0 1 3 4 .02765 1 3 4 6 .05855 1 2 - 2 -1 - . 1 2 4 1 3 2 4 5 7 .24400 1 2 4 5 .15480 3 5 6 2 .11925 2 3 -1 0 •11925 4 6 7 9 .10901 3 4 6 7 .05855 5 7 8 10 .20828 5 6 2 3 .13831 6 8 3 5 .13831 6 7 9 10 .10901 7 9 10 12 .12124 7 8 4 5 .14421 8 10 11 13 .17479 8 9 5 6 .13420 9 11 6 8 .13420 9 10 12 13 .12124 i o 12 13 15 .11696 10 11 7 8 .12027 11 13 14 16 .14551 11 12 9 8 .12067 12 14 9 11 .12067 12 13 15 16 . I I696 13 15 16 18 .10532 13 14 10 11 .09837 14 16 17 19 .12056 14 15 11 12 .10358 15 17 12 14 .10358 15 16 18 19 .10532 16" 18 19 21 .09077 16 17 13 14 .08002 7JL8 20 15 17 .08582 17 18 14 15 .09077 18 19 21 22 .09077 19 ' 20 16 17 .06499 / 3L \ \ y f x i . 3L It, V / K ( 11 ,.in 219 TABLE 9 VALUES OF w K K , x 2 FOR CF^H K K ' AK=1 0 1 1 .215742 1 2 3 . 6 4 7 2 2 7 2 3 6 .078714 3 4 8 . 5 1 0 2 0 0 4 5 1 0 . 9 4 1 6 8 1 5 6 1 3 . 3 7 3 1 6 9 6 7 1 5 . 8 0 4 6 5 5 7 8 1 8 . 2 3 6 1 3 0 8 9 20 .667618' 9 10 2 3 . 0 9 9 0 9 1 10 11 2 5 . 5 3 0 5 6 3 11 12 2 7 . 9 6 2 0 6 7 12 13 3 0 . 3 9 3 5 5 5 13 14 3 2 . 8 2 5 0 2 7 14 15 3 5 . 2 5 6 5 1 6 15 16 3 7 . 6 8 8 0 1 9 16 17 4 0 . 1 1 9 5 0 7 17 18 4 2 . 5 5 0 9 8 0 18 19 4 4 . 9 8 2 4 6 8 19 20 4 7 . 4 1 3 9 5 6 K K ' AK=2 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 4 . 8 6 2 9 7 0 9 . 7 2 5 9 4 3 1 4 . 5 8 8 9 1 2 1 9 . 4 5 1 8 7 4 2 4 . 3 1 4 8 5 0 2 9 . 1 7 7 7 9 5 3 4 . 0 4 0 7 8 7 3 8 . 9 0 3 7 4 8 4 3 . 7 6 6 7 2 4 4 8 . 6 2 9 7 0 0 5 3 . 4 9 2 6 6 1 5 8 . 3 5 5 6 3 7 6 3 . 2 1 8 6 1 3 6 8 . 0 8 1 5 8 9 72 . 9 4 4 5 6 5 7 7 . 8 0 7 5 2 6 8 2 . 6 7 0 5 0 2 8 7 . 5 3 3 4 7 8 9 2 . 3 9 6 4 5 4 9 7 . 2 5 9 4 3 0 J i.3 6,? J, 1 * 6>1 7 V - - - - - J I 3 ,A I 1 C i U . T« : 1 T i J ! 4 ^A"A"' _ j _ iL-r T - v t • 7 r : r • ^  • J-V*. ->-w| ^Tu>a • J I TL, + i— i r 1 ? 1 r • P. F i g u r e 1 2 . The R e l a x a t i o n M a t r i x R A f o r S m a l l Va lues of P o s i t i v e and Negat ive k . -%-to o 221 CHAPTER 5 FITTING OF EXPERIMENTS • TO C F 3 H AT 2 9 7°K In t h i s chapter the e x p e r i m e n t a l data i s f i t t e d by v a r y i n g y, 6 and C . To demonstrate how the d i f f e r e n t parameters a f f e c t the form of T]_ , each parameter i s v a r i e d i n d i v i d u a l l y . 5 -1) The E f f e c t of V a r y i n g y . When the frequency s h i f t , 6 , and the o v e r l a p C , are both set equa l to z e r o , the a n a l y t i c e x p r e s s i o n f o r n/T^ i s _£o 00 (104) n+ s o i l 't*'*iC^+^)j;>K..l<ri+c&.)' = A + Z &f«,...) . The l i n e w i d t h y i s g i v e n by the f o r m u l a , From t a b l e (8) I t i s seen t h a t X does not vary much w i t h K w h i l e i f Y ^ K ' ^ ' E q . ( 1 0 4 ) corresponds to a sum of s p e c t r a l d e n s i t y t imes d e n s i t y t e r m s . Th is i s the u s u a l e q u a t i o n g i -ven f o r n u c l e a r s p i n r e l a x a t i o n . V a r i o u s va lues of y have been s u b s t i t u e d i n t o Eq . . (104) , and T-^/n e v a l u a t e d , the va lue of A b e i n g taken as 1.25 amagat/sec. to g i ve a low d e n s i t y va lue of 0. 8sec ./amagat f o r T^/n-. The r e s u l t s of changing y w i t h 40 a l lowed f r e q u e n c i e s r e t a i n e d i n Eq . (104) (see t a b l e 9) are p resented i n f i g . ( 1 3 ) . I t i s found t h a t the re i s very l i t t l e d e v i a t i o n from the constant va lue of 0 . 8 s e c . / amagat over a l a r g e range of va lues of y . The l a c k of any s t r u c t u r e ( i . e . s teps ) and the s m a l l e f f e c t ( d e v i a t i o n from 0 . 8 sec ./amagat) of Y = 1 conf i rms a c o n c l u s i o n by Dong 1 tha t keep ing one (or indeed many) s p e c t r a l terms of the type ^2^ W KK'^ l s n ° k s u f f i c i e n t to account f o r the l a r g e e x p e r i -m e n t a l l y observed d e v i a t i o n from 0 .8 sec ./amagat . In f a c t , f o r a f requency taken a r b i t r a r i l y as co^  h , the c o n t r i b u t i o n to T " 1 i s ( c f . E q . ( 1 0 3 ) ) 3. y (106) S - f co j ") r .oooqis x a x[TC*,*^ ~i ~ — ^ where G(u) .) i s the l i n e shape. The l a r g e s t c o n t r i b u t i o n KK. L i £ . 7 -O cn o E .6-U (D 5. C \ . 4 -. 3 ^ 12 18 F i g u r e 1 3 . T n /n vs 2 4 3 0 . 3 6 4 2 4 8 D e n s i t y n ( a m a g a t s ) 5 4 6 0 7 2 n 6=0 and c=0. f o r CP 3 H at 297°K. The E f f e c t o f V a r y i n g y w i t h 224 to T " 1 from t h i s component occurs f o r l a r g e d e n s i t i e s when C O 3 << Y 3 i| n • For Y = l a n < i w i t h the va lues from t a b l e s (7) and (8) t h i s c o n t r i b u t i o n i s (107) G(coo h) = .004729 amagat/sec. 3 ' MAX which enhances the r e l a x a t i o n r a t e n e g l i g i b l y . For s m a l l e r va lues of y , the maximum va lue of G i s l a r g e r , but the maximum occurs on ly at h i g h e r and h i g h e r d e n s i t i e s , e . g . f o r Y = ' 0 2 , (108) G(to_ ,.) = .23645 amagat/sec. 3» 4 MAX Th is i s t y p i c a l of a l l f requency components. 5 .2) The E f f e c t of V a r y i n g 6 . I f a f requency s h i f t i s i n c l u d e d , the r e l a x a t i o n m a t r i x i s s t i l l d i a g o n a l (C=0) and the a n a l y t i c a l e x p r e s s i o n f o r G i s changed to (109) ^ 2 2 5 Th is form i s s i m i l a r to the case s t u d i e d i n chapter 3 , E q . ( 4 3 ) , and here as i n tha t c a s e , the maximum va lue occurs when l o ^ T T ^ / n = - 6 r a t h e r than at n = 0 0(6 = 0) . Conse-quent l y a va lue of 6 which i s o p p o s i t e i n s i g n to ( J °KK'^2 / / n i s expected to move the p o s i t i o n of the maximum t o lower va lues of d e n s i t y as |6| i n c r e a s e s . S ince CF^H i s an o b -l a t e top f o r K'<K , wj(]£! i s negat i ve and thus 6 must be p o s i t i v e . In i l l u s t r a t i n g the dependence of T-^/n on the f requency s h i f t , a s m a l l va lue of y i s used . Th is i s because y i s the l i n e w i d t h and a s m a l l va lue tends to separate the l i n e s . In f i g s . ( 1 4 ) and ( 1 5 ) , v a r i o u s va lues of 6 are p l o t t e d w i t h y=.02 and A=1 .25 amagat/sec. The va lue of T-j_/n i s c a l c u l a t e d at every 2 amagats, and t h e r e f o r e some s t r u c t u r e may be m i s s e d . Even s o , these p l o t s d r a m a t i c a l l y demonstrate the importance of a f requency s h i f t on the i n t e n -s i t y and the s t r u c t u r e of T-^/n. The exact e f f e c t of 6 upon a component i s to s h i f t i t i n such a manner tha t phase r a n -d o m i z a t i o n occurs at lower or h i g h e r d e n s i t i e s depending on the va lue of 6 . The low l y i n g f requency components are f i r s t a f f e c t e d and the s h i f t of these to lower d e n s i t i e s as <S i n c r e a s e s i s r e s p o n s i b l e f o r the r i p p l e d e f f e c t ev ident i n f i g u r e s (14) and ( 1 5 ) . In these f i g u r e s , the v a r i o u s d i p s have been l a b e l l e d by the f requency component which i s S = o ro D e n s i t y n ( a m a g a t s ) F i g u r e 14 T]_/n vs n f o r C F 3 K at 297°K. The E f f e c t o f V a r y i n g <S w i t h y=,02 and c=0. The Arrows I n d i c a t e K T r a n s i t i o n s . • 2 H 1 I I 1 1 1 1 1 1 1 w 12 18 2 4 3 0 3 6 4 2 4 8 5 4 6 0 6 6 ^ D e n s i t y n ( a m a g a t s ) F i g u r e 15. T-,/n vs n f o r C F 3 H at 297°K. The E f f e c t of V a r y i n g y w i t h 6 = . 4 and c=0. The Arrows I n d i c a t e K T r a n s i t i o n s . 228 r e s p o n s i b l e f o r the d i p . Comparison shows tha t as 6 i n c r e a s e s the r i p p l e s move to lower d e n s i t y . The g r e a t e s t i n t e n s i t i e s a r i s e from AK=±1 f r e q u e n c i e s , w h i l e the AK=±2 a r e . i n t e r s p a c e d between every second AK=±1 component (see t a b l e ( 9 ) f o r the va lues of " K K t t 2 ) • Th is i s r e s p o n s i b l e f o r the a l t e r n a t e i n t e n s i t y of the r i p p l e s . C o n s i d e r i n g a g a i n a t y p i c a l f requency component, i . e . 0 0 3 4 » ^ n e e x a c t va lue which corresponds to t h i s case i s , (see t a b l e s ( 7 ) , (8) and ( 9 ) ) (110) - • OO&HiS'X'tl ^ 7 7 0 3 )X ( < H 3 7 3 ) X (. Q0336g) *  F i g u r e (16) i s a p l o t of the d e n s i t y dependence of t h i s com-ponent f o r s e v e r a l d i f f e r e n t va lues of 6 w i t h y= . 0 2 . Th is c l e a r l y i l l u s t r a t e s the e f f e c t of 6 . I f the l i n e w i d t h i s i n c r e a s e d , the r i p p l e s which occur i n T ,/n are smoothed out .32-1 O CD CO \ O o £ jo 3 0 18 2 4 3 0 3 6 4 2 4 8 Dens i t y n ( a m a g a t s ) 5 4 6 0 F i g u r e 1 6 . G ( u v s n f o r C F 3 H at 297°K. The E f f e c t of V a r y i n g 6 w i t h y = . 0 2 J » and c=0. 230 as a r e s u l t of the broadening of the l i n e s . For <5 = . 4 t h i s i s p l o t t e d i n f i g . ( 1 5 ) w i t h y = .02, .05 and .1 . To f u r t h e r i l l u s t r a t e the e f f e c t of the l i n e w i d t h , G(w^ ^) i s p l o t t e d i n f i g . ( 1 7 ) w i t h 6 = .4 and y = .02, .05 and .1 . Al though not exact f i t s , these p l o t s do approximate the e x p e r i m e n t a l p o i n t s and i n f a c t l i e w i t h i n e x p e r i m e n t a l e r r o r . I t might be tha t these r i p p l e s are r e s p o n s i b l e f o r the seemingly spur ious s c a t t e r of e x p e r i m e n t a l r e s u l t s . 5.3) The E f f e c t of V a r y i n g g . When L, i s i n c l u d e d an exact a n a l y t i c a l e x p r e s s i o n i s i m p o s s i b l e to m e a n i n g f u l l y w r i t e down a f t e r the r e l a x a t i o n m a t r i x i s i n v e r t e d . The e f f e c t of z, i s to couple (or o v e r -l a p ) f requency components as e x p l a i n e d i n chapter 4.3. The e f f e c t of v a r y i n g £ f o r f i x e d y = .05 and 6 = .4 i s g i v e n i n f i g . ( l 8 ) . I t can be seen tha t o v e r l a p p i n g l i n e s can have a c o n s i d e r a b l e e f f e c t on the shape of the p l o t . In p a r t i -c u l a r , note i n f i g . ( l 8 ) t h a t the ? = -.3 p l o t g i ves a s t e p . I t shou ld be p o i n t e d out tha t the va lues of the a c t u a l o f f -d i a g o n a l i t i e s i n the u n i n v e r t e d r e l a x a t i o n m a t r i x are not as l a r g e as the va lue of K, • From t a b l e (8) the I n t e r n a l s t a t e t r a c e terms are found to be about o n e - t e n t h . Thus the a c t u a l va lues of the o f f - d i a g o n a l elements which appear i n the . 2 6 n 10 12 1 4 16 18 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 CO Dens i t y n ( a m a g a t s ) M F i g u r e 1 7 . G(toq-h) vs n f o r CFoH at 297°K. The E f f e c t o f V a r y i n g y w i t h 6 = . 4 D ' and c=0. F i g u r e 18. T ,/n vs n f o r CF^H at 297°K. The E f f e c t of V a r y i n g c w i t h y=-°5 J and 6=.4. 233 r e l a x a t i o n m a t r i x are about o n e - t e n t h of C . I t i s found n u m e r i c a l l y tha t the o f f - d i a g o n a l elements of the i n v e r t e d r e l a x a t i o n m a t r i x add l i t t l e to the t o t a l c o n t r i b u t i o n to n/T^. However the e f f e c t on the d i a g o n a l elements i s n o n - n e g l i g i b l e . Th is i s demonstrated by p l o t -t i n g the d i a g o n a l element of the i n v e r t e d r e l a x a t i o n m a t r i x which corresponds to the co^  jj f requency component w i t h Y= . 0 5 , 6 = .4 and r, = + . 4 , + . 3 , - . 3 ( f i g . ( 1 9 ) ) . The e f f e c t of the o f f - d i a g o n a l i t y i s to i n t r o d u c e i n t e r f e r e n c e terms which modify the i n t e n s i t y of the l i n e and i n t r o d u c e s a t e l l i t e s (see f i g . ( 1 9 ) and the bump i n the .3 l i n e ) . These s a t e l l i t e s are a r e s u l t of o v e r l a p p i n g of the f requency com-ponent w i t h o ther f requency components. In the p a r t i c u l a r example h e r e , to^ ^ o v e r l a p s w i t h to^  ^ , cog ^ , u o 1 J tog rj . A l l these ove r laps are taken i n t o account w h i l e a l l o thers are f o r b i d d e n by the AK=±3 s e l e c t i o n r u l e . I f £ becomes too l a r g e , the s a t e l l i t e s o s c i l l a t e between n e g a t i v e and p o s i t i v e v a l u e s , (note the £ = .4 va lue becomes negat i ve i n f i g . ( 1 9 ) ) . 5 . 4 F i t to E x p e r i m e n t a l D a t a . By changing the va lues of y , 6 and z, v a r i o u s shapes f o r the T-,/n vs n p l o t can be o b t a i n e d . With a low d e n s i t y o Q) CO 2 3 5 va lue of T-^/n = 0 . 8 sec ./amagat , the t h e o r e t i c a l curves can reproduce the e x p e r i m e n t a l data w i t h reasonab le va lues of Y , 5 and x, . The va lues of y , 6 and t, are reasonab le i f they s a t i s f y the f o l l o w i n g c r i t e r i a . The th ree parameters are a l l r a t i o s of h i g h f requency c o l l i s i o n i n t e g r a l s and the low d e n s i t y c o r r e l a t i o n t i m e , = 2 . 5 6 x 1 0 - 1 0 s e c . -amagat. (see E q s . ( 8 9 ) , (90) and (91)) Hence a va lue of Y = l would mean t h a t the h i g h f requency components r e l a x at the same r a t e as the Larmor f requency te rms . I t has been argued throughout t h i s t h e s i s tha t h igh f requency components r e l a x l e s s e f f e c t i v e l y and consequent ly would have longer " c o r r e l a t i o n " t i m e s . Thus y shou ld be l e s s than one and p o s i t i v e . For the o v e r l a p parameter , £ , i t i s r e a -sonable t h a t y >|£ , | . A rgu ing from the 2 x 2 r e l a x a -t i o n m a t r i x of chapter 3 , t h i s c o n d i t i o n i s necessary f o r a p o s i t i v e d e f i n i t e r e l a x a t i o n m a t r i x . There shou ld be no r e s t r i c t i o n s however on the s igns of £ or on 6 . By s t u d y i n g the t rends d i s c u s s e d above i t i s p o s s i b l e to guess va lues of y , 6 and r, which f i t the e x p e r i m e n t a l d a t a . One such p l o t i s g i ven i n f i g . (20) f o r T-j/n vs n w i t h Y= .10 , r, = - . 4 0 and 6 = . 5 0 , w h i l e f i g . (21) i s the same p l o t s l i g h t l y d i s p l a c e d so as to g i v e a b e t t e r f i t to the e x p e r i m e n t a l p o i n t s . The cho ice of va lues of parameters i s not easy s i n c e the shape of T n /n v a r i e s g r e a t l y w i t h s m a l l .1 1 , 1 1 1 1 1 I I I I 6 12 18 2 4 3 0 3 6 4 2 • 4 8 5 4 6 0 6 6 Dens i t y n ( a m a g a t s ) F i g u r e 20. T h e o r e t i c a l T i/n vs n f o r CFoH at 297°K w i t h Y = - 1 0 » 6 =.-50 a n d . £= - .40 , ro cr\ .84 £ -7-o u> o E - 6" o CD . 5 -co . 2 4 T / p vs p f o r C F H a t 297' K o R. Armstrong* • Waugh 4" A R. Dong ' A A A A A A * A A A ^ A 6 1 2 1 8 2 4 3 0 3 6 4 2 4 8 5 4 6 0 D e n s i t y /° ( a m a g a t s ) F i g u r e 2 1 . T h e o r e t i c a l P l o t of T^/n vs n f o r Y= '10, w i t h the Exper imenta l D a t a . C=-.i|0 arid 6=.50 IV) - 4 238 changes i n y , t, and 6 . Th is i s a consequence of the l a r g e number of f requency components which are i n v o l v e d as i n d i c a t e d i n f i g . ( 2 2 ) . Here a l l the d i a g o n a l elements of R - 1 are p l o t t e d f o r the same va lues of y , £ and 6 as are used i n f i g s . ( 2 0 ) and ( 2 1 ) . The o r i g i n of the step i s due to the e x t r a c o n t r i b u t i o n s of the AK = .2 terms to the envelope of the AK = 1 components. E v e n t u a l l y , the f r e -quency components d i e o f f due to the Boltzmann w e i g h t s , but t h i s happens at d e n s i t i e s o u t s i d e the range of exper iment . I t shou ld be s t r e s s e d tha t a l though f i t t i n g the d e t a i l s of the e x p e r i m e n t a l data and a t t e m p t i n g t o account f o r the apparent s t r u c t u r e i n the e x p e r i m e n t a l da ta i s not e a s y , the i n t e n s i t y of the p l o t does not vary much u n l e s s u n r e a -sonable va lues of y , ? and 6 are used . In p a r t i c u l a r , i t was not p o s s i b l e to account f o r Armst rong 's l a r g e s tep from the low d e n s i t y va lue of about one sec ./amagat , to about 0 .7 sec./amagat at 12 amagats. The va lue of 1 . sec./amagat i s t h e r e f o r e b e l i e v e d to be i n c o r r e c t . On the o ther hand i t i s not p o s s i b l e to r u l e out Dong's .42 sec./amagat va lue s i n c e the re i s . m o r e than ample i n t e n s i t y to f i t h i s d a t a . Th is was not undertaken s i n c e the data of Armstrong and Waugh at h i g h e r d e n s i t i e s are i n agreement. I t may be tha t w a l l e f f e c t s or the geometry of the apparatus i n t r o d u c e s an e x t r a mechanism. Th is would have the e f f e c t of s c a l i n g the D e n s i t y n ( a m a g a t s ) F i g u r e 22 . D e n s i t y Dependence of the D iagona l Elements of the R e l a x a t i o n M a t r i x w i th y=.10t c=-.40 and 6=.50 240 r e s u l t s which the e x p e r i m e n t a l r e s u l t s i n f i g . ( 2 1 ) suggest At present e x p e r i m e n t a l work i s underway to r e s o l v e t h i s d i s c r e p a n c y - ^ . 241 DISCUSSION In order to understand and e x p l a i n the step e f f e c t i t i s necessary to r e t u r n to the fundamentals of s t a t i s t i c a l mechanics and o b t a i n a k i n e t i c equat ion which takes i n t o account a l l p o s s i b l e i n t e r n a l s t a t e f r e q u e n c i e s . The g e n -e r a l i z e d Boltzmann equat ion d i s c u s s e d i n Par t I accompl ishes t h i s f o r a d i l u t e gas where the s i n g l e t d e n s i t y opera to r i s a s u f f i c i e n t d e s c r i p t i o n of the gaseous s t a t e . Th is equa -t i o n should be f l e x i b l e enough f o r d e s c r i b i n g a number of phenomena which i n v o l v e a l l the i n t e r n a l s t a t e s of gaseous molecu les and should be p a r t i c u l a r l y u s e f u l i n i n v e s t i g a -t i n g p ressure broadening prob lems. Th is has been i m p l i e d through the connec t ion of the Boltzmann c o l l i s i o n superoper -a t o r w i t h Fano' s m(w) operator of p ressure broadening t h e o r y . ^ i s more g e n e r a l than m(co) i n that the Boltzmann e q u a t i o n approach a l l o w s n o n - l i n e a r phenomena i n gases to be t r e a t e d w h i l e a master equat ion approach can on ly t r e a t l i n -ear d e v i a t i o n s . On the o ther hand, the master e q u a t i o n approach i s v a l i d f o r a l l o rders of d e n s i t y . I t would be of i n t e r e s t to apply the concepts of s e p a r a t i n g t ime s c a l e s and of t r e a t i n g i n t e r n a l s t a t e s , as d e s c r i b e d i n Par t I, to h i g h e r d e n s i t i e s v i a a g e n e r a l i z e d master a q u a t i o n . Up to t h i s t ime 242 many of these ideas appear to have been m i s s e d . The Boltzmann e q u a t i o n c o u l d a l s o be extended by i n c l u d i n g h i g h e r o rders of c o l l i s i o n s . Such ex tens ions would l e a d to k i n e t i c equat ions f o r t r e a t i n g chemica l r e a c t i o n s o ther than b i m o l e c u l a r . In Par t I I the Boltzmann equat ion approach i s used to d e r i v e a g e n e r a l e x p r e s s i o n f o r T""1 . A l though i t i s shown to be e q u i v a l e n t to other approaches i n the low d e n s i t y r e -g ime, the importance of t h i s s e c t i o n i s the i n c l u s i o n of h i g h e r f r e q u e n c i e s which are u s u a l l y c o n s i d e r e d to be phase randomized at d e n s i t i e s of e x p e r i m e n t a l i n t e r e s t . Th is theory i s an e x t e n s i o n of the work of Chen and S n i d e r - ^ who t r e a t e d d i a t o m i c s w i t h d i p o l a r i n t r a m o l e c u l a r mechanism. No such r e s t r i c t i o n s were i n c l u d e d i n t h i s t h e s i s on the type of molecu le or the mechanism i n v o l v e d . I t i s n e c e s s -ary however to s p e c i a l i z e E q . ( 1 1 - 3 5 ) f o r a p p l i c a t i o n . In Par t IV t h i s i s done f o r symmetric top m o l e c u l e s . Th is task i s s i m p l i f i e d somewhat i f the low f requency (Larmor) e x p r e s -s i o n i s s u f f i c i e n t , but to take i n t o account the h i g h e r f requency phenomena the complex r e l a x a t i o n m a t r i x must be approximated i n a reasonable way. An approx imat ion which s i m p l i f i e s the r e l a x a t i o n m a t r i x i s p resented i n Par t I I I . Th is i n v o l v e s the use of the D i s t o r t e d Wave Born A p p r o x i m a t i o n , a p a r t i c u l a r form f o r the i n t e r m o l e c u l a r p o t e n t i a l and a p a r t i a l e v a l u a t i o n of 243 the c o l l i s i o n i n t e g r a l s . I t has been shown how the i n t e r n a l s t a t e and r e l a t i v e v e l o c i t y terms can be separated and the i n t e g r a t i o n over the cente r of mass per formed. A g e n e r a l d i s c u s s i o n of t h i s procedure f o r d i a t o m i c s w i t h the W-S 72 c o l l i s i o n superoperator has been r e c e n t l y i n v e s t i g a t e d : The r e s u l t s there g i ve r e l a t i o n s h i p s between v a r i o u s c o l l i s i o n i n t e g r a l s and he lp i n r e l a t i n g v a r i o u s exper iments . Only the f o u n d a t i o n s of such a t reatment have been g i ven i n Par t I I I where p o l y a t o m i c s and the g e n e r a l i z e d c o l l i s i o n s u p e r -opera to r are used . The d i f f i c u l t y comes i n d e s c r i b i n g the c o m p l i c a t e d energy exchanges between two molecu les as they c o l l i d e . I t i s f i r s t necessary to c l a s s i f y the p o s s i b l e processes and t h e n , f o r d i f f e r e n t c l a s s e s of energy exchanges and t r a n s i e n t e f f e c t s , the a c t u a l c o l l i s i o n problem can be t a c k l e d . Some approx imat ion such as d i s c u s s e d i n Par t I may a i d i n t h i s , and i t shou ld be p o s s i b l e to f i n d r e l a t i o n s h i p s between low d e n s i t y c o r r e l a t i o n t imes and the h igh d e n s i t y w i d t h s , s h i f t s and ove r lap parameters . C e r t a i n l y the depen-dence of the low d e n s i t y c o r r e l a t i o n t imes on the i n t e r n a l s t a t e s can be e a s i l y found by s e p a r a t i n g out the i n t e r n a l s t a t e t r a c e s . In f a c t , the )r{,'s , E q . ( I I I - 3 0 ) , c o n t a i n t h i s i n f o r m a t i o n . As mentioned above such an a n a l y s i s would a l s o be u s e f u l i n p ressure broadening t h e o r y , and i t i s the o p i n -i o n of the author tha t s t u d y i n g the c o l l i s i o n problem w i l l 244 be f a r more s u c c e s s f u l than t r y i n g to f i t phenomenological parameters to exper iment . , In P a r t IV , the t h e o r i e s p resented i n the f i r s t th ree p a r t s are used to e x p l a i n the n o n - l i n e a r d e n s i t y dependence of CP^H and CH^P. The p h y s i c a l p i c t u r e of phase randomiza -t i o n of the r o t a t i o n a l l e v e l s adequate ly d e s c r i b e s the p r o -c e s s e s , and i t i s demonstrated tha t only the lowest l y i n g l e v e l s are r e s p o n s i b l e . I t i s shown however tha t merely adding s p e c t r a l d e n s i t y -terms of the type J ( waQ^) i s not s u f f i c i e n t to e x p l a i n the s teps but tha t the w i d t h , s h i f t and o v e r l a p must be i n c l u d e d . A g reat v a r i e t y of behav io r i s p o s s i b l e w i t h s m a l l changes i n these parameters and the e x p e r i m e n t a l data i s not p r e c i s e enough to d i s t i n g u i s h b e -tween d i f f e r e n t p o s s i b l e se ts of v a l u e s . In f a c t the u n c e r t -a i n t y of even the low d e n s i t y va lues f o r the c o r r e l a t i o n t ime i n h i b i t s the f i t t i n g . Prom a t h e o r e t i c a l p o i n t of view t h i s parameter i s needed i f the c o m p l i c a t e d i n t e g r a t i o n over r e l a t i v e v e l o c i t i e s i s to be a v o i d e d . I t should be p o s s i b l e to f i t the observed d e n s i t y dependence w i t h t h i s one p a r a -meter a f t e r more d e t a i l e d c a l c u l a t i o n of the c o l l i s i o n i n t e -g r a l s , as d i s c u s s e d above, has been completed . E x p e r i m e n t a l d e t e r m i n a t i o n of the low d e n s i t y c o r r e l a t i o n t i m e , however, i s not easy s i n c e the s i g n a l decreases w i t h d e n s i t y and w a l l (Knudsen) e f f e c t s may o c c u r . 245 There i s no doubt that the n o n - l i n e a r d e n s i t y depen-dence can a r i s e t h e o r e t i c a l l y and f u r t h e r s i m i l a r phenomena should be observab le i n other po l ya tomic molecu les w i t h low 25 l y i n g r o t a t i o n a l l e v e l s . In f a c t C l a r k has observed d e v i a -t i o n s from the C l a u s i u s - M o s s o t t i equat ion f o r the d e n s i t y dependence of the d i e l e c t r i c r e l a x a t i o n i n CH^F a n d CF^H. I t shou ld be p o s s i b l e to e x p l a i n these r e s u l t s by adding l i n e shape terms of the type i n d i c a t e d i n t h i s t h e s i s . C l a r k ' s c o n c l u s i o n that h i g h e r s p e c t r a l d e n s i t y terms which i n c l u d e the f i r s t r o t a t i o n a l f r e q u e n c i e s cannot account f o r the d e -v i a t i o n i s erroneous and i s the same mistake made i n n u c l e a r s p i n r e l a x a t i o n . Frequency s h i f t s as w e l l as s m a l l e r va lues of the c o r r e l a t i o n t imes should be used . I t i s s t r e s s e d that i t i s not p o s s i b l e to r u l e out h igh f requency e f f e c t s by the smal lness of h i g h e r s p e c t r a l d e n s i t y terms wi thout c o n s i d e r i n g these m o d i f i c a t i o n s . I t i s suggested tha t a f t e r the e x p e r i m e n t a l r e s u l t s of CH^F and CF^H are agreed upon, f u r t h e r s t u d i e s be made on s i m i l a r m o l e c u l e s . High f requency e f f e c t s should a r i s e f o r i n t r a m o l e c u l a r s p i n r o t a t i o n and quadrupolar mechanisms and hence , i n p a r t i c u l a r , the o ther ha logenated methanes and the haloforms should be i n v e s t i g a t e d . S ince CH^X and CX-^ H (X= C l , B r , I ) have l a r g e moments of i n e r t i a the h i g h f requency e f f e c t s shou ld occur i n an e x p e r i m e n t a l l y a c c e s s i b l e r e g i o n . 246 No h igh f requency phenomena can a r i s e from an i n t e r m o l e c u l a r r e l a x a t i o n mechanism and hence paramagnetic molecu les are not expected to g i ve r i s e to s t e p s . Up to the present t ime most n u c l e a r s p i n r e l a x a t i o n exper iments i n the gas phase have been performed on r e l a t i v e l y s imple m o l e c u l e s . As more complex systems are s t u d i e d i t i s b e l i e v e d that d e v i a -t i o n s from the l i n e a r d e n s i t y dependence of w i l l become the r u l e r a t h e r than the e x c e p t i o n . 247 BIBLIOGRAPHY 1 R. Dong, P h . D . d i s s e r t a t i o n , U n i v e r s i t y of B. C . 1 9 6 7 . 2 R. Dong and M. Bloom, Phys . Rev. L e t t . 2_0, 9 8 1 ( 1 9 6 8 ) . 3 R. Armst rong , p r i v a t e communicat ion. 4 J . Waugh, p r i v a t e communicat ion. 5 L. Waldmann, Z. N a t u r f o r s c h , 1 2 9 , 6 6 0 ( 1 9 5 7 ) . 6 R. F. S n i d e r , J . Chem. Phys . 32_, 1 0 5 1 ( I 9 6 0 ) . 7 •R. F. Sn ider and B. C. S a n c t u a r y , J . Chem. Phys • 5 5 , 1 5 5 5 ( 1 9 7 1 ) . 8 See f o r example A . Abragam, The P r i n c i p l e s of Nuc lear  Magnet ism, Oxford U n i v e r s i t y P r e s s , O x f o r d , Eng land , ( 1 9 6 1 ) and C. P. S l i c h t e r , P r i n c i p l e s of Magnetic Reson -ance , Harper & Row, N. Y. ( 1 9 6 3 ) . 9 L. Bo l tzmann, Wien. B e r . 66_, 275 ( 1 8 7 2 ) . 10 See f o r example, J . 0 . H i r s c h f e l d e r , C. F. C u r t i s s and R. B. B i r d , M o l e c u l a r Theory of Gases and L i q u i d s , (John W i l e y , N . Y . , 1 9 5 8 ) and S . Chapman and T. G. C o w l i n g , The Mathemat ica l Theory of Non-Uni form Gases. (Cambridge U n i v e r s i t y P r e s s , London, 1 9 5 2 ) . 11 S . Chapman, P h i l . T rans . Roy. Soc . A216, 279 ( 1 9 1 6 ) . 12 D. Enskog, K i n e t i s c h e Theor ie der Vorgange i n m'assig  verdi innten Gasen. D i s s e r t a t i o n , U p p s a l a , A lmquis t and W i k s e l l , 1 9 1 7 . 13 H. G r a d , Phys . F l u i d s 6 , 147 ( 1 9 6 3 ) . 14 See f o r example, C. C e r c i g n a n i , Mathemat ica l Methods i n  K i n e t i c T h e o r y . , Plenum P r e s s , N . Y . , 1 9 6 9 . 15 P. F. Bhatnager , E. P. G r o s s , M. Krook , Phys . Rev. 9_4, 511 ( 1 9 5 4 ) . 248 16 C. S . Wang Chang and G. E. Uh lenbeck, "T ransport Phenomena i n Po lyatomic G a s e s " , U n i v e r s i t y of M i c h i g a n R e p o r t , CM-681 ( 1 9 5 D • 17 M. W. Thomas and R. F. S n i d e r , J . S t a t . Phys . 2 , 6 l ( 1 9 6 9 ) . 18 F. M, Chen and R. F. S n i d e r , J . Chem. Phys . 46, 3 9 3 7 ( 1 9 6 7 ) . ~ ~ 19 F. M. Chen and R. F. S n i d e r , J . Chem. Phys . 50 , 4 0 8 2 ( 1 9 6 9 ) . 20 J . von Neumann, Mathemat ica l Foundat ions of Quantum  Mechan ics , P r i n c e t o n U n i v e r s i t y P r e s s , ( 1 9 5 5 ) . 2 1 F. Wigner , Phys . Rev. 40_, 749 ( 1 9 3 2 ) . 22 For a d e t a i l e d account of t h i s p r o c e d u r e , see r e f e r -ence 1 7 . 23 J . A. R. Coope and R. F. S n i d e r , "A C o l l i s i o n a l l y Un -coupled Model f o r the C o n t r i b u t i o n of 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 to the T ranspor t P r o p e r t i e s of D i l u t e G a s e s . " (to be p u b l i s h e d ) . 24 See C. H. Townes and A. L Schawlow, Microwave S p e c t r o s -copy . M c G r a w - H i l l , I n c . , ( 1 9 5 5 ) , f o r a rev iew of p ressure broadening where t h i s p o i n t i s i l l u s t r a t e d . 25 R. B. C l a r k , M.Sc . D i s s e r t a t i o n , U n i v e r s i t y of B. C. ( 1 9 7 1 ) . 26 J . J . M. Beenakker and F. R. McCourt , Ann. Rev, of Phy -s i c a l Chemistry , 2 1 , 47 ( 1 9 7 0 ) . 27 J . A. R. Coope and R. F. S n i d e r , "Genera l F o r m u l a t i o n of S e n f t l e b e n E f f e c t s " (to be p u b l i s h e d ) . 28 J . A. R. Coope, R. F. S n i d e r , F. R. McCourt , J . Chem. P h y s . . 5 3 , 3 3 5 8 ( 1 9 7 0 ) , and J . J . M. Beenakker , J . A. R. Coope, R. F. S n i d e r , Phys . Rev. A , 4_, 7 8 8 ( 1 9 7 D -29 A. Ben-Reuven, Phys . Rev. L e t t . 1^, 349 ( 1 9 6 5 ) . 30 TJ. Fano , Phys . Rev. 1 3 1 , 259 ( 1 9 6 3 ) . 3 1 N. Bogolubov, J . Phys . ( U . S . S . R . ) 1 0 , 2 6 5 ( 1 9 4 6 ) . 249 32 M. Born and H. S . Green , A Genera l K i n e t i c Theory of  L i q u i d s , (Cambridge U n i v e r s i t y P r e s s , N . Y . , 1 9 4 9 ) . 33 J . G. K i rkwood , J . Chem. Phys . 15., 72 ( 1 9 4 7 ) . 34 J . Yvon, La Theor ie des F l u i d e s et de 1 ' e q u a t i o n  d 1 e t a t , ( P a r i s , Hermann et C i e , 1 9 3 5 ) . 35 J - M. J a u c h , B. M i s r a and A. G . G i b s o n , H e l v . P h y s . A c t a 4 1 , 513 ( 1 9 6 8 ) . 36 See f o r example R. Newton, S c a t t e r i n g Theory of Waves  and P a r t i c l e s , McGraw H i l l ( 1 9 6 6 ) , N. F. Mott and H. S . W. Massey, The Theory of Atomic C o l l i s i o n s , O x f o r d , F a i r Lawn, N. J . , 1 9 4 9 -37 A. T ip and F. R. McCourt , ( i n p r e s s , P h y s i c a ) exc lude t h i s p o i n t by t r e a t i n g p ressure broadening w i t h the W-S c o l l i s i o n superoperator o n l y . 38 B. D. Lippmann and J . Schwinger , Phys . Rev. 79., 469 ( 1 9 5 0 ) . 39 J . M. J a u c h , H e l v . Phys . A c t a . 3 1 , 127 ( 1 9 5 8 ) : ' 40 U. Fano ( r e f . 3 0 ) d i s c u s s e s p r i n c i p a l va lue terms which a r i s e from a master e q u a t i o n approach . These are the same as d i s c u s s e d h e r e . 41 R. F. S n i d e r , J . Math. Phys . 5 , 1 5 8 ( 1 9 6 4 ) . 42 See E q . ( 9 ) of Reference ( 1 9 ) . 43 F. M. Chen and R. F. S n i d e r , J . Chem. Phys . 48_, 3 1 8 5 ( 1 9 6 8 ) . 44 See f o r example, S . K. Hsu and R. F. S n i d e r , Phys . of F l u i d s 1 4 , 517 ( 1 9 7 0 ) ; A. L e v i , F. R. McCourt , P h y s i c a 42 , 3 6 3 T T 9 6 9 ) ; A. L e v i and J . J . M. Beenakker , Phys . L e t t . 2 5 A , 350 ( 1 9 6 7 ) ; H. M o r a a l , P h . D . D i s s e r t a t i o n , U n i v e r s i t y of W a t e r l o o , Canada. 45 L. Waldmann, S t a t i s t i c a l Mechanics of E q u i l i b r i u m and  N o n - E q u i l i b r i u m , Ed . J . Meixner (North H o l l a n d P u b l . Co. (Amsterdam, 1 9 6 5 ) p . l 7 7 and A. T i p , P h y s i c a 52., 4 9 3 ( 1 9 7 D . 46 See f o r example R. Zwanz ig . J . Chem. Phys . 33_, 1388, (1960) or Par t I I - c h a p t e r 4 . 250 47 The o r i g i n a l work was done by M. Baranger , Phys . Rev. I l l , 4 8 1 ; i l l , 4 9 4 ; 1 1 2 , 8 5 5 ( 1 9 5 8 ) , and extended by U. Pano ( r e f . 3 0 ) and A. Ben-Reuven ( r e f s . 29 and 4 9 ) . 48 The c l a s s i c a l impact approx imat ion was developed by H. A. L o r e n t z , P r o c . Acad . S c i . Amsterdam 8_, 5 9 1 ( 1 9 0 6 ) and quantum m e c h a n i c a l l y by J . H. Van V l e c k and V. F. We isskopf , Rev. Mod. Phys . 1 7 , 2 2 7 ( 1 9 4 5 ) . (See a l s o r e f s . 47 and 4 9 . ) 49 A. Ben-Reuven, Phys. ' Rev. l 4 l , 34 ( 1 9 6 6 ) , l 4 _ 5 , 7 ( 1 9 6 6 ) . 50 I. Oppenheim and M. Bloom, Can. J . Phys . 3 9 , 845 ( 1 9 6 1 ) . 51 I. Oppenheim and M. Bloom, I n t e r m o l e c u l a r F o r c e s , E d . J . 0 . H i r s c h f e l d e r , (John Wi ley & Sons , N . Y . , 1 9 6 7 ) . 52 J . A. R. Coope, M o l e c u l a r P h y s i c s , 2 1 , 2 7 1 ( 1 9 7 1 ) . 53 See Chapter I I - 3 , E q . ( 5 5 ) f o r the e q u i v a l e n t s tep i n t ime c o r r e l a t i o n approach. 54 R e c e n t l y B. S h i z g a l ( p r i v a t e communication) has c a l c u l a t e d the e f f e c t s of c o l l i s i o n s i n r e l a x i n g n u c l e a r s p i n s . U s i n g only a d i p o l e - d i p o l e i n t e r m o l e c u l a r i n t e r a c t i o n and the known va lues f o r the parameters f o r He3 he was a b l e to o b t a i n e x c e l l e n t agreement w i t h exper iment . In t h i s case T-^=33 minutes which i s so slow i n comparison to the observed T^ i n , f o r example, CH^F and CF-.H tha t such mechanisms can be i g n o r e d i n comparison t o o ther i n t r a m o l e c u l a r i n t e r a c t i o n s . 55 F. R. McCourt and H. M o r a a l , Chem. Phys . L e t t . 9_, 39 ( 1 9 7 1 ) (see a l s o r e f . 7 1 ) . 56 A. D. Buckingham, I n t e r m o l e c u l a r F o r c e s , Ed . J . 0 . H i r s h f e l d e r , (John Wi ley & Sons N. Y . , 1 9 6 7 ) . 57 R- Kubo, J . Phys . Soc . J a p a n , 1 2 , 570 ( 1 9 5 7 ) . 58 See f o r example J . M. Z iman, P r i n c i p l e s of the Theory of  S o l i d s , Cambridge U n i v e r s i t y Press , ( 1 9 6 5 ) . 59 R. Kubo and K. T o m i t a , J . Phys . Soc . Japan 9 , 8 8 8 ( 1 9 5 4 ) . 60 See page 316 of A. Abragam r e f . 8 . 2 5 1 61 See Eq.(39) of P. M. Chen and R. P. S n i d e r , J . Chem. Phys. 48_, 3185 (1968). 62 M. Bloom and I. Oppenheim, Can. J . Phys . 41, 1580 (1963). ~~ 63 See f o r example R. Zwanz ig ,Phys . Rev. 129, 486 ( 1 9 6 3 ) ; J . L. Jackson and P. Mazur , P h y s i c a 3_0, 2295 (1964); J . A. McLennan J r . and R. J . Swenson, J . Math. Phys . 4, 1 5 2 7 (1963); W. Kohn and J . M. L u t t i n g e r , Phys . Rev. 108, 590 (1957). 64 The o r i g i n a l zero t ime i s taken as the t ime when the i n i t i a l c o n d i t i o n s of the macroscopic n o n - e q u i l i b r i u m s t a t e b e g i n to decay. Th is i s i n c o n t r a s t to the t ime zero j u s t before a c o l l i s i o n which i s used i n the " d e -v i a t i o n " of the Boltzmann e q u a t i o n i n Par t I. 65 R. Zwanz ig , Annual Reviews of P h y s i c a l C h e m i s t r y , Ed . H. E y r i n g , V o l . 16, (1965). 66 In Par t I the n o r m a l i z a t i o n t r „ p ^ ^ = N! was u s e d . The n o r m a l i z a t i o n used here i s more convenient s i n c e i t avo ids N! c o e f f i c i e n t s which c o m p l i c a t e the subsequent e x p r e s s i o n s . 67 See r e f e r e n c e 7; T ip ( r e f . 45) has d i s c u s s e d t h i s e q u i v -a lence f o r the W-S e q u a t i o n . 68 See r e f . 71 where t h i s r e s t r i c t i o n has not been made i n the g e n e r a l f o r m u l a t i o n of the c o l l i s i o n processes f o r d i a t o m i c s w i t h the W-S c o l l i s i o n s u p e r o p e r a t o r . 69 K. S . N e i l s e n , M.Sc . d i s s e r t a t i o n , U n i v e r s i t y of B .C . (1969). 70 M. P. P a t t e n g i l l , C. F. C u r t i s s and R. B. B e r n s t e i n , J . Chem. Phys . 5_4, 2 1 9 7 (1971) and e a r l i e r r e f e r e n c e t h e r e i n . 71 H. Moraa l and R. F. S n i d e r , Chem. Phys . L e t t . 9,401 (1971). 72 F. M. Chen, H. Moraa l and R. F. Sn ider ( to be p u b l i s h e d ) . 73 M. Bloom (to be p u b l i s h e d ) a rev iew of recent d e v e l o p -ments i n NMR. 252 74 W. N. Hardy , Can. J . Phys . 4_4, 2 6 5 ( 1 9 6 6 ) . 75 R. Dong and M. Bloom, Can. J . Phys . 4_8, 7 9 3 ( 1 9 7 0 ) and a l s o r e f . (5 1 ) • 76 P. A. Speight and R. Armstrong Can. J . Phys_4_7, 1475 ( 1 9 6 9 ) . 77 P. Beckmann ( p r i v a t e communication) r e p o r t i n g on the g e n e r a l concensus at a recent symposium at W a t e r l o o , ( 1 9 7 D . 78 G. H e r z b e r g , I n f r a r e d and Raman Spet ra of Po l ya tomic  M o l e c u l e s , D. Van Nostrand Co. ( 1 9 4 5 ) . 79 D. M. Denn ison , Rev. Mod. Phys . 3 , 2 8 0 ( 1 9 3 D . 80 E. B. W i l s o n J r . , J . Chem. Phys . 3 . 276 ( 1 9 3 5 ) . 81 A. R. Edmonds, Angular Momentum i n Quantum Mechan ics , P r i n c e t o n U n i v e r s i t y P r e s s , (I960). 82 See f o r example M. T inkham, Group Theory and Quantum  Mechanics , M c G r a w - H i l l Book Co. , (1964) . 83 P. A. Kaempffer , Concepts i n Quantum Mechanics ? Academic P r e s s , ( 1 9 6 5 ) , page 245. 84 The l i n e shapes are i n f a c t , the s p e c t r a l d e n s i t i e s f o r g i v e n f r e q u e n c i e s . 85 J . T. Hougen, J . Chem. Phys . 3 7 , 1433 ( 1 9 6 2 ) . 86 Townes and Schawlow ( r e f . 24) i s not c o n s i s t e n t from t h e i r t r a n s f o r m a t i o n E q . ( 3 - 3 3 ) to the r e s u l t i n g phase i n E q . ( 3 - 3 4 ) . 88 G. B i rnbaum, J . Chem. Phys . 4j5, 2455 ( 1 9 6 7 ) . 89 The term r i g i d framework was i n t r o d u c e d by E. B. Wi l son ( r e f . 8 0 ) t o d e s c r i b e the two c o n f i g u r a t i o n s of methane which are e n e r g e t i c a l l y i s o l a t e d and which l e a d to a d e s c r i p t i o n of t h a t molecule under T symmetry r a t h e r than T d . 90 S . N Ghosh, R. T rambaru lo , W. Gordy, J . Chem. Phys . 20 , 6 0 5 ( 1 9 5 2 ) . 2 5 3 9 1 H. C. L o n g u e t - H i g g i n s , M o l . Phys . §_, 4 4 5 ( 1 9 6 3 ) . 92 P. Beckmann ( p r i v a t e communicat ion) . 93 T. Oka, J . Chem. P h y s . , 4 9 , 3 1 3 5 ( 1 9 6 8 ) . 94 M. E. Rose , Elementary Theory of Angular Momentum, J . W i l e y & Sons , I n c . , N . Y . , 1 9 5 7 -95 J . A. R. Coope, R. F. S n i d e r , F. R. McCourt , J . Chem. P h y s . , 4j_, 2 2 6 9 ( 1 9 6 5 ) ; J . A. R. Coope, R. F. S n i d e r , J . Math . P h y s . , 1 1 , 1 0 0 3 ( 1 9 7 0 ) ; J . A. R. Coope, J . Math . P h y s . , 1 1 , 1 5 9 1 ( 1 9 7 0 ) . 96 P. Y i , I. O z i e r and C H . Anderson , Phys . Rev. l 6 _ 5 , 92 ( 1 9 6 8 ) . 2 5 4 APPENDIX A ROTATIONS OP CARTESIAN AND SPHERICAL TENSORS The laws of p h y s i c s are u n a f f e c t e d by any c o o r d i n a t e change and hence a t r a n s f o r m a t i o n from one frame to another must leave the equat ions -unchanged . Examples of q u a n t i t i e s which are t ransformed under r o t a t i o n s are t e n s o r s of rank j which have 3 J i n d i c e s . For example one component of a f o u r t h rank t e n s o r T i s T w i t h respec t to a c o o r d i n -s xzyx ^ ate system x , y , z but i f t h i s t e n s o r i s to be w r i t t e n w i t h respec t to another c o o r d i n a t e sys tem, say X , Y , Z , the re are i n g e n e r a l 3 c o e f f i c i e n t s r e q u i r e d to r e l a t e the two. The approach of C a r t e s i a n t e n s o r theory i s to c o n s t r u c t C a r t e s i a n t e n s o r s which are i r r e d u c i b l e under such a group as S 0 ( 3 ) . In c o n t r a s t , s p h e r i c a l t e n s o r theory chooses i n -dependent t e n s o r components and c o n s t r u c t s them i n such a way t h a t they are e i g e n f u n c t i o n s of the r o t a t i o n generato r about some s p e c i a l a x i s . Both methods are e q u i v a l e n t , and the purpose of t h i s appendix i s to demonstrate how to pass from one to the o t h e r . Then the r o t a t i o n m a t r i c e s are d e -f i n e d and d i s c u s s e d w i t h respec t to a c t i v e and p a s s i v e 2 5 5 r o t a t i o n s , and f i n a l l y symmetric top wave f u n c t i o n s are c o n -s i d e r e d . Many books are a v a i l a b l e which t r e a t s p h e r i c a l 8 l 82 94 t e n s o r theory ' 5 , and a s e r i e s of th ree papers t r e a t 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 ^ . 1 . C a r t e s i a n and S p h e r i c a l Tensors . Any 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 can be expressed i n n a t u r a l form which means tha t i t can be w r i t t e n i n terms of • a symmetric t r a c e l e s s tensor of p o s s i b l y lower r a n k . The n a t u r a l form t e n s o r of symmetry j i s a symmetric t r a c e l e s s t e n s o r of rank j , and can be p r o j e c t e d out of the 3J* d i -mens iona l t e n s o r space by a 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 , . For example, the n a t u r a l t e n s o r of weight f o u r c o r r e -sponding to the f o u r t h rank t e n s o r T above i s the 4 t h rank ( 4 ) ? 4 ) 4 symmetric t r a c e l e s s t e n s o r T = E 0 T , or i n g e n e r a l , (1) [ a ] = E ( J ) ©J ( a ) j where (a)^ = a a a . . . ( j - f o l d t e n s o r p r o d u c t ) . Here 0 '^ i s the j - f o l d dot product between neares t i n d i c e s and s i n c e each dot c o n t r a c t s one p a i r of i n d i c e s , E^'^ must be of ( ") rank 2 j . S ince E i s a p r o j e c t i o n o p e r a t o r i t must be idempotent , E ^ ^ 0 J ' E^'^ = and i t s d imension i s 256 E U ) @ 2 J E ( , j ) _ 2 j + i . The 2 j + l independent components which are a s s o c i a t e d w i t h t h i s 2 j + l d i m e n s i o n a l space can be i d e n t i f i e d as the s p h e r i c a l t e n s o r components. F o l l o w i n g r e f . ( 9 5 ) , E^'^ can be w r i t t e n w i t h r e s p e c t to a b a s i s e^ such tha t (2) E(J> = 2 . e J ( - l ) m e ^ m=-j m - m which i s a s p e c t r a l r e s o l u t i o n of E ^ . The v e c t o r s e^ form the s p h e r i c a l b a s i s f o r the 2 j + l d i m e n s i o n a l space and are chosen to be e i g e n f u c t i o n s of the commuting opera to rs ^ and ^ , (3) fe^ = j ( J + l ) e J (4) i e l = meJ" y m m In a d d i t i o n i t i s p o s s i b l e to c o n s t r u c t (5) / ± e^ = f(j+m)(j±m+l) e J ill V / rr m i l 2 5 7 i and j . where }f  ^1 are the r a i s i n g and l o w e r i n g o p e r a t o r s . In s p h e r i c a l tensor theory the i d e n t i f i c a t i o n of the angu la r momentum magnitude as j and i t s component as m i n the l a b -o r a t o r y f i x e d frame (LFF) i s u s u a l l y made. To c o n s t r u c t the b a s i s i t i s necessary to s p e c i f y on ly one b a s i s element and t h i s i s taken a s , ( 6 ) eJ! = ( - 1 ) J / ( 2 ) J / 2 ( x+ iy ) ' and by use of E q . ( 5 ) , a l l the o thers can be c o n s t r u c t e d f o r a g i v e n j . Any symmetric t r a c e l e s s C a r t e s i a n t e n s o r can be w r i t t e n i n the s p h e r i c a l b a s i s as f o l l o w s , ( 7 ) f a ] ( J ) = £ e J ( - D m V (a) m m -m The Y ^ ( a ) ' s are s c a l a r components of ^ w i t h r e s p e c t to the b a s i s e^ and are e s s e n t i a l l y the s p h e r i c a l ha rmon ics , Yj^Bt})) but w i t h a d i f f e r e n t n o r m a l i z a t i o n . The n o r m a l i z a t i o n 9 5 can be found by use of the r e l a t i o n 258 ( 8 ) P.(b-a) .= ( ( 2 j ) ! / 2 J ( J ! ) ^ b ) J Q J E ( J ) Q J ( t ) J f o r the complete c o n t r a c t i o n of two I r r e d u c i b l e tensors C b"2^ and £3.7^ where P . ( b * a ) i s the u s u a l l y normal-i z e d Legendre p o l y n o m i a l . From E q . ( 7 ) t h i s i s equa l to ( 9 ) P . ( b ^ ) = [ ( 2 j ) ! / 2 J ( j : ) 2 ] ^ Y J ( b ) Y J m ( a ) ( - l ) m and from the a d d i t i o n theorem f o r s p h e r i c a l harmonics the r e l a t i o n s h i p between Y ' s i s , (20-The angles 0 and <j> d e s c r i b e the o r i e n t a t i o n of the u n i t v e c t o r a i n a convenient frame 259 2. R o t a t i o n s and R o t a t i o n M a t r i c e s . The E u l e r i a n angles (a3y) 3 - r e chosen to agree w i t h 8l Edmonds , page 7- In t h i s t reatment x , y and z denote the LPF w h i l e X, Y and Z i s the c o o r d i n a t e system which d i f f e r s from x , y and z by r o t a t i o n R(a3y) • Th is i s r e f e r r e d to as the body f i x e d f rame, BFP . By d e f i n i t i o n , R(a3y) i s where the pr imed c o o r d i n a t e system i s the i n t e r m e d i a t e one produced by r o t a t i n g f i r s t by Rz(<*) • R(a$y) can be w r i t t e n e n t i r e l y i n terms of XYZ or xyz by use of the f o l l o w i n g r e l a t i o n s , 0.3) R(a3 Y) = R Z ( Y ) R V,(3 ) R (a) R Z ( Y ) R z ( a ) R y ( B ) R z ( Y ) R y ( - 6 ) R z ( - a ) (12) R z ( a ) R z ( - y ) R (-B) R z ( a ) R Y(3 ) R z ( Y ) and (13) R z ( a ) R y ( 3 ) R z ( - a ) R Z ( - Y ) R Y ( B ) R Z ( Y ) 260 so tha t R ( Y ) R V , ( 3 ) R (a) = R z ( a ) R (3) R Z ( Y ) (14) Z y z z y = R z ( a ) R y ( 3 ) R Z ( Y ) . Th is d e f i n i t i o n of E u l e r i a n angles d e s c r i b e s the d i f f e r e n c e between the two c o o r d i n a t e systems. I f a v e c t o r i s o r i e n t e d i n one sys tem, then the same v e c t o r w i l l have d i f f e r e n t c o -o r d i n a t e s i n the r o t a t e d c o o r d i n a t e system. An a c t i v e r o t a -t i o n d e s c r i b e s the change i n the v e c t o r components, w h i l e a p a s s i v e r o t a t i o n leaves the components f i x e d and r o t a t e s the b a s i s v e c t o r s . E i t h e r cho ice g i v e s the same r e s u l t , but once one i s chosen i t i s necessary to be c o n s i s t e n t . A t e n s o r q u a n t i t y f can be d e f i n e d w i t h r e s p e c t to e i t h e r the LFF or the BFF, ( !5) f = I e£(B) C J ' ( B ) ( - l ) m = Ee j (L ) CJ" ( L ) ( - l ) n . m m ~™ n n - n I t i s p o s s i b l e to r e l a t e the c o e f f i c i e n t s C^ n (L ) to C^_m(B) or the b a s i s e J ( B ) to e J ( L ) where B r e f e r s to the BFF m n and L to the LFF . Of p a r t i c u l a r i n t e r e s t i n t h i s t h e s i s 261 i s to r e l a t e the c o e f f i c i e n t of the d i p o l a r c o u p l i n g t e n s o r , which are known i n the BFF to the e J ( L ) b a s i s . The a c t i v e n r o t a t i o n R ( a 3 y ) r o t a t e s the b a s i s as f o l l o w s , (16) e J ' (B) = R(a0Y )e«J(L) m m 81 and i n terms of the r o t a t i o n m a t r i c e s i n Edmonds , e^(B) = Z e ^ ( L ) ( - l ) n e J ( L ) ® J R ( a 3 Y ) e J ( B ) (17) = I e3AL) J )^( -a-e -Y) n n •W,T>I. = Z e j t L ) J D _ ( a 8 Y ) In p a r t i c u l a r , the s p h e r i c a l harmonics t r a n s f o r m a c c o r d i n g to - n - m The r o t a t i o n m a t r i c e s of Edmonds, ^ ) , d i f f e r from those of o thers 8 2 , 9 4 D , i n that the s igns of a , 3 and Y are changed. Th i s d i s t i n c t i o n produces the f o l l o w i n g change, 262 f) and a r i s e s as a r e s u l t of the o r i g i n a l d e f i n i t i o n of the r o t a t i o n s . Edmonds t r e a t s on ly p a s s i v e r o t a t i o n s and hence a f i n i t e r o t a t i o n about the £ a x i s i s ( 2 0 ) R P a s s l v e ( a ) = e " ^ 4 / f l where j ^ . i s the r o t a t i o n generator about E, . In c o n t r a s t , o ther t reatments d e f i n e a c t i v e r o t a t i o n s w i t h the r e s u l t , ( 2 1 ) R a c t ± V e ( a ) = e - ^ / * In the t reatment h e r e , a c t i v e r o t a t i o n s are used w i t h Edmonds r o t a t i o n m a t r i c e s and hence the minus s igns appear i n E q s . ( 1 7 ) and ( 1 8 ) . 3 . Symmetric Top Wave F u n c t i o n s . The a b s t r a c t s t a t e , |JkM> i s s p e c i f i e d by the t o t a l angu la r momentum, J ; the component i n the L F F , M ; and 263 and the component i n the BFF, k . I f the LFF and BFF c o -i n c i d e , k and M are the same. However, when the two frames are o r i e n t e d d i f f e r e n t l y , M and k must be d i s t i n g u i s h e d . The angu lar moment i n the BFF may be d e s c r i b e d i n terms of the k e t s |Jk> w h i l e i n the LFF by |JM> , where the C ' s are expansion c o e f f i c i e n t s . In g e n e r a l (22) kM and (23) JkM (a3y) i s ^JkMCaBY) = < R taSY)|JkM> = <R(agY) l|JM>c£ M = < l l R C a S Y ) - 1 |JM>c£ (24) = N ^ ' ^ k M ^ C Y P a ) = 5 < i i j N > c k M £ > M , ( a 3 Y ) ( - i ) N+M 2 6 4 where use has been made of ( 2 5 ) R ( a g Y ) " 1 = R z ( - Y ) R y ( - B ) R z ( - a ) , Edmonds *£)'s and symmetry r e l a t i o n s between the S^'s There i s a f u r t h e r requirement t h a t a r o t a t i o n by x about the m o l e c u l a r a x i s has the f o l l o w i n g r e s u l t , (26) R (x)IJkM> = e - l X k / h JkM> and from Eq . (24) t h i s i s t r u e only i f N=k , ( 2 7 ) * T M , ( ^ « = <l|Jk>C? M j E ) T . ( a B Y ) ( - l ) The c o e f f i c i e n t s are found by n o r m a l i z a t i o n to g i ve (28) * j k M ^ 0 J r ) = / ^ T ( a 3 Y ) ( _ 1 ) k+M Th is d e f i n i t i o n d i s a g r e e s w i t h those g i ven by Rose and Edmonds, but agrees w i t h Y i , O z i e r and A n d e r s o n 9 ^ i f Rose ' s d e f i n i t i o n 265 of the rotation matrices are used, (29) 7J4 f )7 c + e*) c-') T)J C-<(3 * ) C " ) ^ TT M - ft 4-TT P' (*<32f) 266 APPENDIX B EVALUATION OF From E q . ( l V - 6 7 ) i t i s s u f f i c i e n t to e v a l u a t e the f o l -lowing two q u a n t i t i e s s & * l > " ( < \ r * i l « \ 7 * 1 C 1 ) = ±[ hk * (cf ? (cf r j r c-,f ( 2 ) where ( i l ^ ^ , CO)^ , T-j^ and 7S are d e f i n e d by E q s . ( I V - 6 2 ) , ( I V - 6 3 ) ( I V -7 ) and ( I V - 1 0 ) r e s p e c t i v e l y . The ant icommutat ion i n E q . ( l ) has been e v a l u a t e d and g i ves the 2 6 7 f r a c t i o n a l p o p u l a t i o n of a g i v e n J , k r o t a t i o n a l l e v e l . These are d i s c u s s e d i n Par t IV , chapter 4 . 2 and because of the i n t i m a t e r e l a t i o n between s p i n and r o t a t i o n a l s t a t e s , the s p i n degeneracy i s i n c l u d e d i n the p a r t i t i o n f u n c t i o n Q. 1 . The R o t a t i o n Traces S u b s t i t u t i o n of E q s . ( I V - 5 ) and ( IV -7) i n t o the t r a c e over r o t a t i o n a l s t a t e s g i v e s (3) where a l l p o s s i b l e angles a$y of R are to be i n t e g r a t e d o v e r . By use of the symmetric top wave f u n c t i o n s , E q . ( I V - 5 ) j 8 1 and Edmonds E q . ( 4 . 2 . 7 ) t h i s becomes, 268 The i n t e g r a t i o n can be performed (Edmonds E q . ( 4 . 6 . 2 ) , and g i v e s , ( 5 ) . T X -j\ M + ^ g / j ^J\{J * j -H' n J \ M' ij- -M From the p r o p e r t i e s of the 3 - j c o e f f i c i e n t s , I = , i . J .« T, ,^ / n sk+k'+M+M'+q k-k' , and q = M-M' . Hence the ( - 1 ) M becomes ( - 1 ) and the sum over M and M' can be p e r -formed (Edmonds Eq . ( 3 . 7 - 8 ) ) 2 6 9 ( 6 ) JdWs-r^ i-*f£2-bzbt.(-<)1 f-i'.* i) J j - s The s u b s c r i p t q has been dropped s i n c e the r i g h t hand s i d e of E q . ( 6 ) i s independent of q . As a consequence of 6^ £, , s " = - s " . From the d e f i n i t i o n of the b^ 's i t i s seen t h a t b_ 1 = ( - 1 ) b and hence b ^ b^C-l) = | b ^ | 2 . With these o b s e r v a t i o n s , t h e t r a c e term i s , 2^  - ^ ^ 2*. 2 . The S p i n Trace As a r e s u l t of the r o t a t i o n a l t r a c e s i t i s s u f f i c i e n t to set s " = - s " . From the d e f i n i t i o n of 7^ E q . ( I V - l O ) , i t f o l l o w s tha t <8) % c n S = ( u f Z s „ 2 7 0 and s i n c e the p r o j e c t i o n opera to rs are idempotent and o r t h o g o n a l Eq . (2 ) becomes, (9) = f A (US": C l i f f , cfsyti- £<;v". Fur thermore , the symmetry number s " does not a f f e c t 1^ which i s s i t u a t e d on the symmetry a x i s . Hence the t r a c e over Ijj g i v e s (1^ = h) ( 1 0 ) :$» r 6 £ The n o r m a l i z a t i o n 5/3 and 1/3 comes from (11) E ^ 2 ) = 5/3 u and (12) Iz|Iz, : S = 1 / 3 I? = V 3 1^(1^+1) (2I 4 +1) 271 s " s " ( 2 ) s i n c e (II^) = (IT^) : E i s a second rank symme-t r i c tensor (Appendix A ) . By use of E q . ( I V - 4 9 ) , the tensor c o n t r a c t i o n s i m p l i -f i e s to h a v i n g used the r e l a t i o n I 2 = 9 / 4 + 2(1^- T_2 + 1 1 • I + 1 2 * > I = 1^  + 1 2 + I3 , and I^' I = 3 / 4 , f o r s p i n 1/2. With t h i s r e l a t i o n and the p r o j e c t i o n o p e r a t o r , E q . ( I V - l 6 ) , the s p i n t r a c e becomes ( 1 4 ) The l a s t term i s odd i n !_. and hence t r a c e s to z e r o . By 272 2 per fo rming the t r a c e w i t h I_ be ing 1 5 / 4 and 3 / 4 , the i s p i n t r a c e g i ves ( 1 5 ) 5'" The 4 i n the numerator comes from the s p i n degeneracy ( 2 1 + 1) f o r I = 3 / 2 and 2 ( 2 1 + 1 ) f o r 1 = 1 / 2 , and a g a i n the independence of q m e r i t s d ropp ing that s u b s c r i p t . The d v e c t o r components i n E q . ( I V - 6 7 ) f o l l o w from E q s . ( l ) , ( 7 ) and ( 1 5 ) . 2 7 3 APPENDIX C EVALUATION OF fl ( A k' A" 4"'/ From E q s . ( I I I - l l ) , ( I V - 3 3 , - 6 2 a n d - 6 3 ) the s p i n and r o t a t i o n a l opera to rs can be c o n s t r u c t e d . S u b s t i t u t i o n i n t o E q . ( I I I - 3 0 ) g i v e s X L 7, A,-*S 7 A r > i J A , - * ( l ) _ , - O-C 2 4, * For the case of i n t e r e s t h e r e , = 1 , 2 or 3 o n l y . The il-^th rank m u l t i p o l e o r i e n t a t i o n tensor of molecu le one i s L't/J , and w i t h r e s p e c t to the l a b o r a t o r y frame b a s i s elements t h i s i s r a n " 0 = 2 e ' * ' Z*' ( 2 ) 274 The body f i x e d m u l t i p o l e components are r e a l and have been taken as u n i t y . The a c t u a l m u l t i p o l e s t r e n g t h i s i n -c luded i n the r e l a t i v e v e l o c i t y i n t e g r a l s . S ince n u c l e a r s p i n s p e c i e s cannot change d u r i n g a c o l l i s i o n , the only r - 4*1 a l l o w e d k t r a n s i t i o n over L ^ J are due to those com-ponents of the o c t a p o l e moment, (& =3) hav ing n^=±3. I t f o l l o w s t h a t k» and k" , and k and k'" can d i f f e r on ly by ±3. Thus the e q u a l i t i e s s '=s" and s = s"' must h o l d . The consequence of t h i s i s the c a n c e l l a t i o n of the s p i n t r a c e s from the t o t a l e x p r e s s i o n f o r X , and the i n -s e r t i o n of the r e s t r i c t i o n s 6, n j _ , 6. , + 0 r r , where k ' , k"±3n k , K ±jm n , m = 0, ±1. The a p p r o p r i a t e b a s i s opera to rs become, o) nzxtrl = Z CQ5 %A'  C o n s i d e r i n g f i r s t the denominator of E q . ( 3 ) , i t i s found to be i d e n t i c a l to 3" ( Jkk ' s ^ " 1 )<S „ ,„ e v a l u a t e d i n appendix s " r s B, E q . ( B - 7 ) when J = J ' . Thus E q . ( l ) can be w r i t t e n as (no J changes are cons ide red ) 275 X ( ?X TA', J V , liO ~ ^ J* 14" W-'X'^S - 0 -j - y* [ j va ^ The term "K can be eva lua ted by s u b s t i t u t i o n of E q s . ( 3 ) and ( IV - 7) , • f~ ( 5 ) 3 -276 E v a l u a t i o n of { }^  g i v e s (6) w h i l e the i n t e g r a t i o n s i n { > 1 f a c t o r i n t o f o u r te rms . A f t e r s u b s t i t u t i o n of the symmetric top wave f u n c t i o n s , t h i s becomes n, ( 7 ) These are e v a l u a t e d i n e x a c t l y the same way as i n appendix B by use of Edmonds E q s . ( 4 . 2 . 7 ) and ( 4 . 6 . 2 ) to g i ve 277 From the p r o p e r t i e s of the 3 - j c o e f f i c i e n t s , and Edmonds E q . ( 6 . 2 . 3 ) the sum over a l l M's i s a 6 - j c o e f f i c i e n t , so t h a t , (9) r 3 yl c ^ y * < W « 5 Combining the r e s u l t s of E q s . ( 4 ) , ( 9 ) and (B -6 ) becomes 278 . / 7 * J ) fr A J ) fy 3. J ) f.J-C> J^f^^J? Th is e x p r e s s i o n can be reduced f u r t h e r by c a n c e l l a -t i o n of the 3 - j c o e f f i c i e n t s so tha t U D f  7 *  7 ) S i m i l a r l y the second 3 - j term g i v e s the s i g n ( -1 ) ( _ l ) n 2 + k ' W i t h the r e s u l t t h a t the t o t a l s i g n of E q . ( l O ) i s ( _ 1 ) £ l + n 2 + n 3 . T h e ( - 1 ) 5 , 1 can be embedded i n t o a 3 - j c o e f f i c i e n t by a p p r o p r i a t e permutat ion of the k ' s so 279 that E q . ( l O ) can be w r i t t e n , (12) where (13) D""-- M l and the r e l a t i o n f o r both b ' s and C ' s , (14) b = b ( - l ) n ^ has been used . Thus M. depends on ly on the s i g n of , *1 and C For the purpose of Par t IV , on ly the J - a v e r a g e d q u a n t i t y 280 i s r e q u i r e d . Th is case i s (15) where m and n can be ±1 or 0 o n l y . 2 8 1 APPENDIX D MULTIPOLE EXPANSION FOR TWO MOLECULES Cons ider f i r s t the one c e n t e r expansion of a charge d e n s i t y pCr) about a p o i n t at a d i s t a n c e R which i s out-s i d e p ( r ) . That i s , Laplace's e q u a t i o n CD V a V = o h o l d s . The p o t e n t i a l i s then C2) where (see f i g . ( 2 3 ) Cert. 0 + 1 i s the g e n e r a t i n g f u n c t i o n f o r Legendre p o l y n o m i a l s , 282 (4) From appendix A , i t f o l l o w s t h a t (see E q . ( A - 8 ) (5) P - P.(r-«)= -^Ur < < W E ' V and t h u s , ( 6 ) ^ a j . z ( £ ) - i ^ ( f r 0 - ^ - « ) where (7) T ^ C O = — T F L ~ J , 283 Consequently a one cente r expansion g i v e s , (8) and i f R>>r then i n t e g r a t i o n g i v e s the m u l t i p o l e expansion C9) v= 2 n f*V Xf/O and the nth m u l t i p o l e i s c i o -= J" ec> r r ] w c / r A n a l o g o u s l y , f o r a two cente r e x p a n s i o n , see f i g . ( 2 3 ) , the p o t e n t i a l i s | g + r, - 0. j where 284 (12) 5 + r- -fi I By the b i n o m i a l expansion of (r^ - r_ 2) n , t h i s can be w r i t t e n , (13) S e t t i n g s = n - t Eq . ( 13 ) becomes, (14) fl^-fj J . T r O ~ L s.' t! and by E q . ( 7 ) , (15) H^I' - a S/X-o * I > 285 so tha t the m u l t i p o l e expans ion f o r molecu le 1 and 2 i s C"> V = Z cst tf* e ^ A l t l where (17) <=» = I*±+*ti '• = cts <->)*• a 5 + c C s - e t ) ! S t t ! The c o e f f i c i e n t C i s used t o generate the n u m e r i c a l fac -S X/ t o r i n Eq . ( I I I-8). The m u l t i p o l e s are u s u a l l y denoted by M ( 0 ) = C, M ( 1 ) = M ( 2 ) E M ( 3 ) = Q (18) ~ - E = f o r the mono- , d i - , q u a d r a - and o c t a p o l e s r e s p e c t i v e l y 286 R R - r = / ( R - i ) • ( R - l ) F i g u r e I n t e r m o l e c u l a r S e p a r a t i o n . 

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