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Nuclear magnetic resonance studies on clathrate hydrates Raghunathan, Parthasarathy 1966

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NUCLEAR MAGNETIC RESONANCE STUDIES ON CLATHHATE HYDRATES by PARTHASARATHY RAGHUNATHAN M.A., M.Sc., U n i v e r s i t y o f M a d r a s , I n d i a , 1959 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department of CHEMISTRY We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA May, 1966 V I n 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 m e n t o f t h e r e q u i r e -ments f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e 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 s t u d y . I f u r t h e r a g r e e 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 c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of C U A M S T ^ , The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date Ayl, 1766 The U n i v e r s i t y o f B r i t i s h C o l u m b i a FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY OF PARTHASARATHY RAGHUNATHAN M s S c „ s U n i v e r s i t y o f M a d r a s / 1959 FRIDAY s MAY 27S 1966 a t 3:30 P.M. IN ROOM 261. CHEMISTRY BUILDING COMMITTEE IN CHARGE Chairman: E. T e g h t s o o n i a n N. B a r t l e t t Z. A. M e l z a k B. A. D u n e l l R. S t e w a r t C A. McDowell J = . T r o t t e r E x t e r n a l E x a m i n e r ; R„ E, R i c h a r d s , F.R„S= P h y s i c a l C h e m i s t r y L a b o r a t o r i e s S o u t h P a r k s Road O x f o r d , E n g l a n d R e s e a r c h S u p e r v i s o r : C. A, McDowell, NUCLEAR MAGNETIC RESONANCE STUDIES ON CLATHRATE HYDRATES ABSTRACT W i t h a v i e w t o o b t a i n i n g i n f o r m a t i o n on t h e n a t u r e and e x t e n t o f m o l e c u l a r m o t i o n o f e n c l a t h r a t e d " g u e s t s " and t h e i r i n t e r a c t i o n w i t h the " h o s t " l a t t i c e s , n u c l e a r m a g n e t i c r e s o n a n c e a b s o r p t i o n o f t e n gu e s t s p e c i e s i n c l u d e d i n the c l a t h r a t i o n v o i d s o f f u l l y d e u t e r a t e d h y d r a t e s has been s t u d i e d from a t e m p e r a t u r e o f 77°K upwards. The F ^ re s o n a n c e l i n e shapes f o r CF^ and SFg and the H"'' resonan c e l i n e shape f o r e t h y l e n e o x i d e ( C 2 H 4 O ) i n t h e i r r e s p e c t i v e c l a t h r a t e h y d r a t e s i n d i c a t e t h a t t h e s e m o l e c u l e s a r e but l i t t l e r e s t r i c t e d by t h e w a l l s o f t h e c l a t h r a t e c a v i t i e s , and r e o r i e n t f r e e l y about chosen axes o f symmetry a t low t e m p e r a t u r e s and a t random a t h i g h e r t e m p e r a t u r e s . Above 150°K, a l i m i t e d i s o t r o p i c t r a n s l a t i o n , o r " r a t t l i n g " , o f an SFg g u e s t m o l e c u l e up t o a d i s t a n c e o f 0.5 K f r o m t h e c e n t r e o f the c l a t h r a t i o n volume has been d e m o n s t r a t e d . P r o t o n r e s o n a n c e has been s t u d i e d f o r propane i n two specimens o f the c l a t h r a t e h y d r a t e , one o f w h i c h was r i c h e r i n gu e s t c o n t e n t t h a n t h e o t h e r . F o r t h e s e two specimens i t has been s u g g e s t e d t h a t , below 160°K, propane assumes a s t a g g e r e d C2 c o n f i g u r a t i o n i n s i d e t h e c l a t h r a t e c a v i t y . M o l e c u l a r C2 - a x i s r e o r i e n t a t i o n s s u p e r p o s e d on m e t h y l r e o r i e n t a t i o n s have been p r o p o s e d , and a t h e r m a l a c t i v a t i o n energy b a r r i e r o f 1 . 7 0 + 0 . 0 8 k c a l / m o l e has been c a l c u l a t e d f o r the above m o t i o n from 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 measurements i n the 77 K -110 K range. C l o s e r t o the m e l t i n g p o i n t o f the two specimens, d i f f u s i o n o f propane t h r o u g h the h o s t l a t t i c e has been i n d i c a t e d , and d i f f u s i o n a l a c t i v a t i o n e n e r g i e s o f 1.40 + G.02 k c a l / m o l e and 0.75 + 0.05 k c a l / m o l e have been o b t a i n e d f o r t h e g u e s t - r " i c h and g u e s t - p o o r s p e c i m e n s , r e s p e c t i v e l y . I n s h a r p c o n t r a s t t o the above r e s u l t s , t h e low t e m p e r a t u r e p r o t o n r e s o n a n c e o f t h r e e h a l o m e t h a n e s s C H 3 X (X = C l , B r , I ) , i n s i d e h y d r a t e h o s t c a v i t i e s has r e v e a l e d d e f i n i t e c o n s t r a i n t s t o r e o r i e n t a t i o n a l and t r a n s l a t i o n a l m o t i o n , the second moment d a t a i n d i c a t i n g o n l y l o w - a m p l i t u d e o s c i l l a t o r y m o t i o n s o f t h e CH3 groups I i n t h e s e c l a t h r a t e s . A t r i p l e t l i n e shape has been o b s e r v e d f o r the CH^Br c l a t h r a t e a t 77°K. At h i g h e r t e m p e r a t u r e s , e x p a n s i o n o f the h y d r a t e l a t t i c e s has been p r o p o s e d , w h i c h p e r m i t s f r e e C - j - r e o r i e n t a t i o n s o f t h e CH3 groups o f the t h r e e g u e s t m o l e c u l e s . From the a s s o c i a t e d l i n e w i d t h t r a n s i t i o n s 3 a c t i v a t i o n e n e r g i e s o f 2.48 f 0.32, 9.30 + 0.25,. and 6.80 + 0 . 5 0 k c a l / m o l e have been c a l c u l a t e d f o r t h e p o t e n t i a l b a r r i e r h i n d e r i n g m e t h y l r e o r i e n t a t i o n i n the CH3CI-, CR/jBr-,, and CH3I-h y d r a t e s , r e s p e c t i v e l y . The m o t i o n a l model p r o p o s e d f o r t h i s t e m p e r a t u r e range i s a d e q u a t e l y s u p p o r t e d by measurements o f 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 s . F o r the d i c h l o r o m e t h a n e c l a t h r a t e h y d r a t e , , a c l e a r l y r e s o l v e d d o u b l e t c h a r a c t e r i s t i c o f r i g i d p r o t o n p a i r s has been o b t a i n e d a t 77°K. The p o s s i b l e e x i s t e n c e a t low t e m p e r a t u r e s o f an a l i g n e d g u e s t m o l e c u l e i n a s u i t a b l y -s i z e d c a v i t y i s t h e r e b y i n d i c a t e d . A . l i n e shape a n a l y s i s o f t h i s d o u b l e t s p e r f o r m e d on an. IBM 7040 computer, y i e l d e d an a c c u r a t e H-H i n t e r a t o m i c d i s t a n c e o f 1.73 A f o r t h e " g u e s t " d i c h l o r o m e t h a n e m o l e c u l e . T h i s v a l u e has been d i s c u s s e d i n the l i g h t o f r e s u l t s f r o m e a r l i e r microwave s t u d i e s o f d i c h l o r o m e t h a n e . The p r o t o n r e s o n a n c e l i n e w i d t h and second moment r e s u l t s between 77°K and 286°K f o r i - a m y l groups i n c l u d e d as g u e s t s i n the c l a t h r a t e h y d r a t e o f ( i - C ^ H ^ ^ ) ^ N F have been i n t e r p r e t e d i n terms of s i m p l e m o t i o n a l models of t h e s e g u e s t m o i e t i e s . The r e s u l t s complement the r e p o r t e d c r y s t a l s t r u c t u r e o f t h i s c l a t h r a t e . F o r the anal o g o u s h y d r a t e c o n t a i n i n g " g u e s t " n - b u t y l g r o u p s , p r o t o n second moments i n t h e same t e m p e r a t u r e range have s u p p o r t e d t h e d i s o r d e r e d g u e s t s t r u c t u r e r e p o r t e d from a p r e v i o u s X - r a y d i f f r a c t i o n s t u d y . I n a d d i t i o n , h y s t e r e s i s has been d e m o n s t r a t e d i n the second moment cu r v e o f t h i s c l a t h r a t e beyond 248°K, and t h i s has been a s c r i b e d t o a phase t r a n s i t i o n . GRADUATE STUDIES F i e l d o f Study: P h y s i c a l C h e m i s t r y T o p i c s i n P h y s i c a l C h e m i s t r y Seminar i n C h e m i s t r y T o p i c s i n C h e m i c a l P h y s i c s T o p i c s i n I n o r g a n i c C h e m i s t r y D i f f e r e n t i a l E q u a t i o n s Quantum C h e m i s t r y S p e c t r o s c o p y and M o l e c u l a r S t r u c t u r e Dr. A. V. Bree Dr. A. R. Coope Dr. W. A. B r y c e Dr. B. A. D u n e l l Dr. C.'A; McDowell Dr. N. B a r t l e t t Dr. H. C. C l a r k Dr. J . T, Kwon Dr. W. R= C u l l e n Dr. S. A , J e n n i n g s Dr. J . A. R, Coope Dr, K. B. Harvey Dr. L. If. Reeves Dr. C. R e i d Dr. Eo J . W e l l s Dr. L. G. H a r r i s o n Dr. S. M e l z a k Dr. J . T r o t t e r D i g i t a l Computer Programming Dr. J„ R. H.Dempster S o l i d S t a t e C h e m i s t r y C r y s t a l S t r u c t u r e A B S T R A C T W i t h a v i e w t o o b t a i n i n g i n f o r m a t i o n on t h e n a t u r e and e x t e n t o f m o l e c u l a r m o t i o n o f e n c l a t h r a t e d " g u e s t s " and t h e i r i n t e r a c t i o n w i t h t h e " h o s t " l a t t i c e s , n u c l e a r m a g n e t i c r e s o n a n c e a b s o r p t i o n o f t e n g u e s t s p e c i e s i n c l u d e d i n t h e c l a t h r a t i o n v o i d s o f f u l l y d e u t e r a t e d h y d r a t e s has been s t u d i e d f r o m a t e m p e r a t u r e o f 77°K upwards. 19 1 The F r e s o n a n c e l i n e shapes f o r CF^ and SFg and t h e H r e s o n a n c e l i n e shape f o r e t h y l e n e o x i d e (CgH^O) i n t h e i r r e s p e c t i v e c l a t h r a t e h y d r a t e s i n d i c a t e t h a t t h e s e m o l e c u l e s a r e b u t l i t t l e r e s t r i c t e d b y t h e w a l l s o f the c l a t h r a t e c a v i t i e s , and r e o r i e n t f r e e l y about chosen axes o f symmetry a t low t e m p e r a t u r e s and a t random a t h i g h e r t e m p e r a t u r e s . Above 150°K, a l i m i t e d i s o t r o p i c t r a n s l a t i o n , o r " r a t t l i n g " , o f an SF^ g u e s t m o l e c u l e u p t o a o o d i s t a n c e o f 0.5A fr o m t h e c e n t r e o f t h e c l a t h r a t i o n volume has been d e m o n s t r a t e d . P r o t o n r e s o n a n c e has been s t u d i e d f o r propane i n two specimens o f th e c l a t h r a t e h y d r a t e , one of w h i c h was r i c h e r i n g u e s t c o n t e n t t h a n t h e o t h e r . F o r t h e s e two specimens i t has been s u g g e s t e d t h a t , b e l o w 160°K, propane assumes a s t a g g e r e d c o n f i g u r a t i o n i n s i d e the c l a t h r a t e c a v i t y . M o l e c u l a r - a x i s r e o r i e n t a t i o n s s u p e r p o s e d on m e t h y l r e o r i e n t a t i o n s have been p r o p o s e d , and a t h e r m a l a c t i v a t i o n e nergy b a r r i e r o f 1.70 £ 0.08 k c a l / mole has been c a l c u l a t e d f o r t h e above m o t i o n fa?om 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 measurements i n t h e 77 °K - 110°K r a n g e . C l o s e r t o t h e m e l t i n g p o i n t o f t h e two sp e c i m e n s , d i f f u s i o n o f propane t h r o u g h t h e h o s t l a t t i c e has been i n d i c a t e d , and d i f f u s i o n a l a c t i v a t i o n e n e r g i e s o f 1.40 + 0.02 k c a l / m o l e and 0.75 + 0.05 k c a l / m o l e have been o b t a i n e d f o r t h e g u e s t - r i c h and g u e s t -poor s p e c i m e n s , r e s p e c t i v e l y . I n s harp c o n t r a s t t o t h e above r e s u l t s , t h e low t e m p e r a t u r e p r o t o n r e s o n a n c e o f t h r e e h a l o m e t h a n e s , CHgX (X = C I , B r , I ) , i n s i d e h y d r a t e h o s t c a v i t i e s has r e v e a l e d d e f i n i t e c o n s t r a i n t s t o r e o r i e n t a t i o n a l and t r a n s l a -t i o n a l m o t i o n , the s e c o n d moment d a t a i n d i c a t i n g o n l y l o w - a m p l i t u d e o s c i l l a t o r y m o t i o n s of t h e CHg g r o u p s i n t h e s e c l a t h r a t e s . A t r i p l e t l i n e shape has been o b s e r v e d f o r t h e CHgBr c l a t h r a t e a t 77°K. A t h i g h e r tempera-t u r e s , e x p a n s i o n o f t h e h y d r a t e l a t t i c e s has b e e n p r o p o s e d , w h i c h p e r m i t s f r e e C g - r e o r i e n t a t i o n s o f the CHg groups o f t h e t h r e e g u e s t m o l e c u l e s . From t h e a s s o c i a t e d l i n e w i d t h t r a n s i t i o n s , a c t i v a t i o n e n e r g i e s o f 2.48 + 0.32, 9.30 + 0.25, and 6.80 + 0.50 k c a l / m o l e have been c a l c u l a t e d f o r the p o t e n t i a l b a r r i e r h i n d e r i n g m e t h y l r e o r i e n t a t i o n i n t h e C H ^ C l - , CHgBr-, and C H g l - h y d r a t e s , r e s p e c t i v e l y . The m o t i o n a l model p r o p o s e d f o r t h i s t e m p e r a t u r e range i s a d e q u a t e l y s u p p o r t e d by measurements o f H^ 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 s . F o r the d i c h l o r o m e t h a n e c l a t h r a t e h y d r a t e , a c l e a r l y r e s o l v e d d o u b l e t c h a r a c t e r i s t i c o f r i g i d p r o t o n p a i r s has been o b t a i n e d a t 77°K. The p o s s i b l e e x i s t e n c e a t low t e m p e r a t u r e s o f an a l i g n e d g u e s t m o l e c u l e i n a s u i t a b l y - s i z e d c a v i t y i s t h e r e b y i n d i c a t e d . A l i n e shape a n a l y s i s o f t h i s d o u b l e t , p e r f o r m e d on an IBM 7040 computer, y i e l d e d an a c c u r a t e H-H o i n t e r a t o m i c d i s t a n c e o f 1.73A f o r t h e ' g u e s t * d i c h l o r o m e t h a n e molecule,, T h i s v a l u e has been d i s c u s s e d i n the l i g h t o f r e s u l t s f r o m e a r l i e r m i c r o -wave s t u d i e s o f d i c h l o r o m e t h a n e . The p r o t o n r e s o n a n c e l i n e w i d t h and second moment r e s u l t s between 77 °K and 286 °K f or i - a m y l groups i n c l u d e d as g u e s t s i n the c l a t h r a t e h y d r a t e i v o f ( i - C g H j j ) ^ N F have been i n t e r p r e t e d i n terms of s i m p l e m o t i o n a l models o f t h e s e g u e s t m o i e t i e s . The r e s u l t s complement the r e p o r t e d c r y s t a l s t r u c -t u r e o f t h i s c l a t h r a t e . F o r t h e a n a l o g o u s h y d r a t e c o n t a i n i n g ' g u e s t ' n - b u t y l g r o u p s , p r o t o n second moments i n the same t e m p e r a t u r e range have s u p p o r t e d t h e d i s o r d e r e d g u e s t s t r u c t u r e r e p o r t e d f r o m a p r e v i o u s x - r a y d i f f r a c t i o n s t u d y . I n a d d i t i o n , h y s t e r e s i s has been d e m o n s t r a t e d i n t h e second moment c u r v e o f t h i s c l a t h r a t e beyond 248°K, and t h i s has been a s c r i b e d t o a phase t r a n s i t i o n . v i TABLE OF CONTENTS Page A b s t r a c t i i L i s t o f T a b l e s v i i i L i s t o f F i g u r e s x Acknowledgments X 1 1 D e d i c a t i o n x i i i CHAPTER I INTRODUCTION 1 CHAPTER I I STRUCTURES OF CLATHRATE HYDRATES 12 2.1 O u t l i n e ; 12 2.2 The B a s i c P o l y h e d r a l U n i t 12 2.3 The H o s t L a t t i c e s o f Gas H y d r a t e s 13 2.4 C o m p a t i b l e 'Guests' 15 2.5 The P e r a l k y l S a l t H y d r a t e C l a t h c a t e s 20 2.6 T e t r a i -Amyl Ammonium F l u o r i d e H y d r a t e 21 2.7 T e t r a n - B u t y l Ammonium F l u o r i d e H y d r a t e » 23 CHAPTER I I I THEORY OF THE INFLUENCE OF MOLECULAR MOTION ON NUCLEAR RESONANCE LINE SHAPES 27 3.1 The L i n e Shape F u n c t i o n ...... 27 3.2 N u c l e a r D i p o l e - D i p o l e I n t e r a c t i o n 28 3.3 Second Moment o f the Resonance L i n e 31 3.4 M o t i o n a l E f f e c t s 33 3.5 R i g i d R o t o r . • . . . • . © « • © . . © . © . . . . . o . . . . . . . . . • • • • © • 36 3.6 G e n e r a l R e o r i e n t a t i o n ........©©... 37 3.7 M o l e c u l a r M o t i o n and C o r r e l a t i o n F u n c t i o n s ...... 40 3.8 S a t u r a t i o n , C h o i c e o f H-^  F i e l d s , and F a s t P assage • 45 CHAPTER IV APPARATUS 47 4.1 The S t a i n l e s s S t e e l P r e s s u r e C e l l 47 4.2 The NMR S p e c t r o m e t e r 48 4.3 Low Temperature Equipment 51 CHAPTER V MATERIALS AND METHODS 54 5.1 M a t e r i a l s • 54 5.2 P r e p a r a t i o n and E s t i m a t i o n 55 ( a ) The P r e s s u r e C e l l Method 55 (b) D i r e c t F o r m a t i o n i n Sample Tubes 57 v i i ( c ) T e t r a i - A m y l Ammonium F l u o r i d e H y d r a t e . ( d ) T e t r a n - B u t y l Ammonium F l u o r i d e H y d r a t e 5.3 S p e c t r o m e t e r C a l i b r a t i o n s ( a ) C a l i b r a t i o n o f t h e S c a n n i n g F i e l d (b) C a l i b r a t i o n o f H m , t h e Sweep A m p l i t u d e s ..... 5.4 Measurements .. a) L i n e w i d t h s ...................... b) Second Moments c) A c t i v a t i o n E n e r g i e s f r o m L i n e w i d t h D a t a ,d) T^Measurements by S i g n a l S a t u r a t i o n ,e) T^Measurements b y A d i a b a t i c F a s t P a s s a g e .... CHAPTER V I THE HYDRATES OF CARBON TETRAFLUORTDE, SULPHUR HEXAFLUORIDE, ETHYLENE OXIDE, AND PROPANE ....... 6.1 O u t l i n e 6.2 R e s u l t s and C o n c l u s i o n s ( a ) C a r b o n T e t r a f l u o r i d e i n t h e 'Type I * H o s t L a t t i c e ( b) S u l p h u r H e x a f l u o r i d e i n t h e 'Type I I 1 H o s t L a t t i c e ' ( c ) E t h y l ene O x i d e i n the 'Type I * H o s t L a t t i c e . . ( d) Propane i n t h e 'Type I I ' H o s t L a t t i c e ....... CHAPTER V I I THE HALOMETHANE HYDRATES 7.1 O u t l i n e 7.2 R e s u l t s and C o n c l u s i o n s ( a ) The Monohalomethane H y d r a t e s (b) The D i c h l o r o m e t h a n e H y d r a t e CHAPTER V I I I THE TETRA-ALKYL AMMONIUM "SALT" HYDRATE CLATH-RATES 8.1 O u t l i n e 8.2 R e s u l t s and C o n c l u s i o n s . ( a ) ( i - C H .,) NF~38D 20 (b) ( n - C ^ H y 4 N F ^ l . S D 2 0 CHAPTER I X SUMMARY AND FURTHER PROSPECTS BIBLIOGRAPHY APPENDIX I APPENDIX I I • APPENDIX I I I Page 57 58 59 59 59 59 60 60 61 61 63 66 66 67 67 73 82 84 95 95 95 95 105 113 113 113 113 122 135 140 148 152 155 v i i i L I ST OF TABLES Page 1. Thermodynamic F u n c t i o n s f o r " H y d r a t e " I c e S t r u c t u r e s 4 2. B a r r i e r s t o flotation o f G u e s t s i n Q u i n o l C l a t h r a t e s ........ 6 3. C r y s t a l S t r u c t u r e D a t a on Gas H y d r a t e s 16 4. L i s t o f Known Gas H y d r a t e s 17 5. P h y s i c o - C h e m i c a l D a t a on Some Type I Gas H y d r a t e s 18 6. P h y s i c o - C h e m i c a l D a t a on Some Type I I Gas H y d r a t e s 19 7. Some T y p i c a l E x p e r i m e n t a l V a l u e s o f C l a t h r a t e C o m p o s i t i o n .. 56 8. E x p e r i m e n t a l Second Moments f o r CF^ C l a t h r a t e H y d r a t e s 68 19 9. T h e o r e t i c a l F Second Moments f o r CF^ i n t h e D e u t e r a t e d H y d r a t e L a t t i c e 69 19 10. T h e o r e t i c a l F Second Moments f o r SFg i n the D e u t e r a t e d H y d r a t e L a t t i c e 77 11 . E x p e r i m e n t a l L i n e w i d t h s and Second Moments f o r E t h y l e n e O x i d e i n t h e D e u t e r a t e d H y d r a t e L a t t i c e 83 12. T h e o r e t i c a l H"^  Second Moments f o r E t h y l e n e Oxide i n t h e D e u t e r a t e d H y d r a t e L a t t i c e 83 13. C a r t e s i a n Hydrogen C o o r d i n a t e s G e n e r a t e d f o r the n-Propane C h a i n - 8 5 14. T h e o r e t i c a l H* Second Moments f o r P r o p a n e i n the D e u t e r a t e d H y d r a t e L a t t i c e P . . . . 8 7 15. D a t a on t h e Monohalomethane H y d r a t e s 96 16. T h e o r e t i c a l Second Moments f o r M e t h y l "Guest" i n D e u t e r a t e d H y d r a t e s • ••• 98 17. E x p e r i m e n t a l Second Moments f o r D i c h l o r o m e t h a n e i n t h e D e u t e r a t e d H y d r a t e L a t t i c e 108 18. Computed Hydrogen Atom C o o r d i n a t e s 118 19. P o s s i b l e i - A m y l C h a i n M o t i o n s and C o r r e s p o n d i n g R e s i d u a l S e c o n d Moments * 120 i x LIST OF TABLES (CONTD) Page 20. Computed H y d r o g e n Atom C o o r d i n a t e s 125 21. T h e o r e t i c a l H 1 Second Moments: The D e u t e r a t e d H y d r a t e o f T e t r a n - B u t y l Ammonium F l u o r i d e 129 LIST OF FIGURES Page 1. 14 2. 14 3. 22 4. 22 5. A Guest C a t i o n i n t h e T e t r a i - A m y l Ammonium F l u o r i d e H y d r a t e .. 24 6.( a) One o f t h e F o u r E q u i v a l e n t C a t i o n s i n T e t r a n - B u t y l Ammonium 25 6(b) The F i f t h C a t i o n i n T e t r a n - B u t y l Ammonium F l u o r i d e H y d r a t e , 26 7. 3 J 30 8. 34 9. 34 10. 38 11. B l o c k D i a g r a m o f 30 M c / s e c . Wide L i n e NMR S p e c t r o m e t e r 49 12. 53 13. 74 14. 76 15. 81 16. ( C g H g ) ~ D e u t e r a t e d H y d r a t e : Second Moment v e r s u s Temperature .. 86 17. Temperature Dependence of L i n e w i d t h s 91 18. 93 19. T y p i c a l H* Resonance S p e c t r a o f t h e Bromomethane C l a t h r a t e .... 97 20. P r o l a n Linewitlthx'ChangaH.fDri t h e H a l o m e t h a n e ' C l a i h i a t e s 104 21. 104 22. 107 x i LIST OF FIGURES (CONTP) Page 23. C a l c u l a t e d F i t t o E x p e r i m e n t a l L i n e Shape ( C H 2 C 1 2 ) ~ d e u t e r a t e d h y d r a t e 109 24. S u g g e s t e d A l i g n m e n t o f D i c h l o r o m e t h a n e i n the "Type I " H j t d r a t e Cage a t 77°K 112 25. L i n e w i d t h and S e c o n d Moment D a t a , ( i - C 5 H i ; i ) 4 N F ~ DgO C l a t h r a t e 115 26. G u e s t C h a i n C a r b o n Atoms and T h e i r I s o t r o p i c Temperature F a c t o r s .. 116 27. Second Moment v e r s u s Temperature: ( n - C . H Q ) . N F ~ D p0 C l a t h r a t e . 124 x i i ACKNOWLEDGMENTS To P r o f e s s o r C A . M c D o w e l l , mentor and g u i d e , who i n t r o d u c e d me t o m a g n e t i c r e s o n a n c e , and t o whom I am p r o f o u n d l y i n d e b t e d f o r h i s h e l p , encouragement, and s u p p o r t , t o P r o f e s s o r B.A. D u n e l l , f o r s e v e r a l v a l u a b l e s u g g e s t i o n s t h a t have u r g e d t h i s work f o r w a r d , t o D r . D.F.R, G i l s o n , f o r p r o v i d i n g c o n t e x t s of s t i m u l u s and f r i e n d s h i p a t a c r u c i a l h o u r and f o r some v e r y r e w a r d i n g s u g g e s t i o n s , t o M e s s r s . T. C y r and B. S a r k a r , f r i e n d s of i n e s t i m a b l e c h e e r , who g e n e r -o u s l y gave o f t h e i r t i m e and e f f o r t s i n p r e p a r i n g many o f t h e d i a g r a m s i n t h i s t h e s i s , t o D r . G.W. S m i t h of G e n e r a l M o t o r s C o r p o r a t i o n , M i c h i g a n , U.S.A., and t o the s t a f f o f t h e Computing C e n t r e , U n i v e r s i t y of B r i t i s h C o l u m b i a , f o r some e d i f y i n g h i n t s w h i c h i n s p i r e d t h e F o r t r a n IV programmes i n t h i s t h e s i s , t o Mr. W. G r i b a f o r v a l u a b l e h e l p w i t h some of t h e i l l u s t r a t i o n s and t o M r s . 0. Thompson f o r t h e promptness she showed i n r e n d e r i n g i n t o n e a t t y p e s c r i p t t h e h i e r o g l y p h i c s c a l l e d . m y h a n d w r i t i n g . TO THE MEMORY o f MY BELOVED FATHER 1 CHAPTER I INTRODUCTION The i n c l u s i o n o f one compound w i t h i n t h e c r y s t a l l a t t i c e o f a n o t h e r , w i t h o u t any f o r m a l c h e m i c a l f o r c e s e x i s t i n g between them, was 1 2 e a r l y r e c o g n i z e d ' when h y d r o q u i n o n e formed u n u s u a l l y l a r g e c r y s t a l s i n th e p r e s e n c e o f m o l e c u l e s l i k e s u l p h u r d i o x i d e and c a r b o n monoxide. T h a t t h e s e s t r u c t u r e s were i n d e e d a p a r t i c u l a r c l a s s o f m o l e c u l a r compounds, i n w h i c h one component formed a "cage" t y p e o f l a t t i c e e n c l o s i n g t h e o t h e r as a " g u e s t " , was shown by t h e d e t a i l e d X - r a y s t u d i e s o f P a l i n and 3 4 P o w e l l . ' The c l a s s i c a l e tymology, " c l a t h r a t e compounds", was t h e n i n t r o d u c e d (Greek: KLETHRA s o c a g e * o r ' t r a p ' ) , and t h e a p p r o p r i a t e n e s s o f t h i s naming i s o b v i o u s i n v i e w o f t h e c r y s t a l s t r u c t u r e o f t h e s e com-pounds. T y p i c a l c l a t h r a t e s o f q u i r i o l i n d i c a t e d t h a t e x t e n s i v e h y d r o g e n -b o n d i n g between q u i n o l m o l e c u l e s l e d t o two i n t e r p e n e t r a t i n g t h r e e -2 d i m e n s i o n a l n e t w o r k s e n c l o s i n g p s e u d o - s p h e r i c a l c a v i t i e s w i t h i n w h i c h g u e s t m o l e c u l e s were h e l d b y weak van d e r W a als and London d i s p e r s i o n f o r c e s . 6—12 S u b s e q u e n t l y , v on S t a c k e l b e r g and c o - w o r k e r s c o n c l u d e d t h a t t h e gas 13 h y d r a t e s , w h i c h had been known f o r many y e a r s , were a l s o t y p i c a l l y c l a t h -r a t e s t r u c t u r e s , a l l r e f e r a b l e t o two c u b i c c r y s t a l i o g r a p h i c t y p e s ( i and I i ) o f r a t h e r h o l l o w hydrogen-bonded l a t t i c e s . T h e i r c o n c l u s i o n s , b a s e d on 14 c r y s t a l s t r u c t u r a l a n a l y s e s , were c o r r o b o r a t e d by C l a u s s e n and P a u l i n g and 15 M a r s h . A n o v e l s e r i e s o f p o l y h e d r a l c l a t h r a t e h y d r a t e s was soon d i s -16—23 C o v e r e d , f r o m X - r a y c r y s t a l s t u d i e s o f s e v e r a l p e r a l k y l ammonium, su l p h o n i u m , and phpsphonium s a l t h y d r a t e s , i n w h i c h a l k y 1 c h a i n s b e l o n g i n g t o a t e t r a h e d r a l l y d i s p o s e d c e n t r a l c a t i o n were shown t o l i e as g u e s t s i n l a r g e p o l y h e d r a l c a v i t i e s formed by a hydrogen-bonded, c r y s t a l l i n e w a t e r l a t t i c e . I n r e c e n t y e a r s , a m u l t i p l i c i t y o f such c l a t h r a t e s t r u c t u r e s has 24—26 27 been s t u d i e d . A number o f d e s c r i p t i v e r e v i e w s and a book on c l a t h — r a t e s a r e a v a i l a b l e . The f o r m a t i o n and s t a b i l i t y o f c l a t h r a t e s a r e d i c t a t e d by v a r i o u s f a c t o r s . The c r y s t a l s t r u c t u r e o f t h e " h o s t " ( e . g . , q u i n o l ) l a t t i c e i s an open arrangement; o f t e n t h i s s t r u c t u r e i s n o t the normal "cv-structure" c h a r a c t e r i s t i c o f t h e c a g e - f o r m i n g m o l e c u l e when i t c r y s t a l l i z e s a l o n e , b u t i s a m e t a s t a b l e " g - s t r u c t u r e " , w h i c h r e q u i r e s t he i n c l u s i o n o f a g u e s t com-ponent f o r i t s s t a b i l i t y . S e c o n d l y , t h o u g h l i m i t i n g c o m p o s i t i o n f o r m u l a e a r e p o s s i b l e f o r a l l t h e s e s t r u c t u r e s c o r r e s p o n d i n g t o maximum f i l l i n g o f the h o s t - l a t t i c e v a c a n c i e s by a g u e s t m o l e c u l e each, t h e a c t u a l c o m p o s i t i o n r e s u l t i n g f r o m a c r y s t a l l i z a t i o n o f t h e h o s t and g u e s t t o g e t h e r may v a r y o v e r a w i d e r a n g e , and f o r t h i s r e a s o n c l a t h r a t e s a r e n o t s t o i c h i o m e t r i c 28—31 compounds. R e c e n t phase s t u d i e s on gas h y d r a t e s a r e n o t e w o r t h y i n t h i s r e s e c t . T h i r d l y , t e r a r e s t r i n g e n t r e q u i r e m e n t s on t h e s i z e o f t h e 3 g u e s t m o l e c u l e s t h a t may e n t e r a c e r t a i n h o s t - l a t t i c e c a v i t y w i t h o u t s e r i o u s l y d i s t o r t i n g i t . I n t h e n e x t c h a p t e r , t h e s e p o i n t s w i l l be i l l u s -t r a t e d w i t h p a r t i c u l a r r e f e r e n c e t o the v a r i o u s t y p e s o f c r y s t a l s t r u c t u r e s known f o r t h e c l a t h r a t e h y d r a t e s s t u d i e d i n t h i s t h e s i s . The thermodynamic a s p e c t s i n v o l v e d i n c l a t h r a t e f o r m a t i o n have been 32 33 34 t h o r o u g h l y examined by v a n d e r Waals and P l a t t e e t e w , B a r r e r e t a l . , ' 35 • and, more r e c e n t l y , by C h i l d , J r . , .?These t r e a t m e n t s a r e b a s e d on t h e c o n s i d e r a t i o n t h a t t h e c l a t h r a t e s a p p r o x i m a t e t o i d e a l s o l u t i o n s o f gas m o l e c u l e s i n t h e c e l l s o f a s o l i d l a t t i c e , r a t h e r l i k e the " c e l l s " c o n -36 s i d e r e d by L e n n a r d - J o n e s and D e v o n s h i r e . A s s u m i n g a L e n n a r d - J o n e s (12-6) p o t e n t i a l f u n c t i o n f o r t h e i n t e r a c t i o n energy of a g u e s t m o l e c u l e w i t h an atom i n t h e w a l l o f i t s c e l l , one c a n a p p l y s t a t i s t i c a l m e c h a n i c a l methods and, f r o m the p a r t i t i o n f u n c t i o n f o r t h e e n c l a t h r a t e d g a s , e s t i m a t e t h e f r e e - e n e r g y , e n t h a l p y , and d i s s o c i a t i o n p r e s s u r e o f t h e c l a t h r a t e . I f c l a t h r a t e s t a b i l i t y i n t h e absence o f any c h e m i c a l b i n d i n g between h o s t and g u e s t may be e x p l a i n e d i n terms p a r t l y o f the v a n d e r Waals-London i n t e r -a c t i o n between t h e g u e s t m o l e c u l e s and t h e w a t e r m o l e c u l e s o f t h e framework 37 and p a r t l y o f t h e energy o f the h o s t - l a t t i c e h y d r o g e n bonds , t h e n t h e f o r m a t i o n o f c l a t h r a t e s c a n be e x p l a i n e d i n terms o f a d e f i n i t e t h e rmo-dynamic " d r i v i n g f o r c e " made up o f e n t h a l p y and e n t r o p y f a c t o r s . More e x p l i c i t l y , t h e f r e e - e n e r g y d i f f e r e n c e , ( F - Fcv) o r ^ F between a P. P> CV; w a t e r m o l e c u l e o f t h e empty m e t a s t a b l e ( f i - ) h y d r a t e framework and t h a t o f o r d i n a r y i c e (cv - ) , may be e x p e c t e d t o be r e p r e s e n t e d by where i s t h e d i f f e r e n c e i n e n t h a l p y between t h e h y d r a t e framework and p > o/ i c e , and i s t h e d i f f e r e n c e between t h e r e s p e c t i v e e n t r o p y v a l u e s . 8 , 0 / However, r e s u l t s f r o m e a r l i e r s t a t i s t i c a l m e c h a n i c a l m e t h o d s 3 3 ' ^ f o r a g r e e d r e m a r k a b l y w e l l f o r b o t h Type I and Type I I h y d r a t e l a t t i c e s w i t h t h e c o r r e s p o n d i n g /H v a l u e s c a l c u l a t e d f r o m v a n d e r Waals and London 37 e l e c t r o n i c i n t e r a c t i o n terms by P a u l i n g . The o b v i o u s c o n c l u s i o n , t h e n , i s t h a t t h e r e i s h a r d l y any e n t r o p y d i f f e r e n c e between the two empty f r a m e -works and o r d i n a r y a - i c e , a f a c t w h i c h i n d i c a t e s a g r e a t s i m i l a r i t y i n t h e i n t e r m o l e c u l a r f o r c e s t h a t d e t e r m i n e t h e v i b r a t i o n s o f t h e s e hydrogen-bonded s t r u c t u r e s . More r e c e n t l y ' , a r e a p p r a i s a l o f t h e ^ F ^ ^ v a l u e s f o r t h e Type I and Type I I h y d r a t e l a t t i c e s has been made, and t h e e x i s t e n c e of s m a l l b u t d e f i n i t e v a l u e s o f A£L d e m o n s t r a t e d . These v a l u e s a r e q u o t e d 8,a . i n t h e f o u r t h column of T a b l e I , w h i c h summarizes a l l t h e thermodynamic f u n c t i o n s p r o p o s e d f o r t h e m e t a s t a b l e h y d r a t e s t r u c t u r e s t o - d a t e . TABLE I Thermodynamic F u n c t i o n s f o r ' H y d r a t e * I c e S t r u c t u r e s R e l a t i v e t o cv - I c e a t 273°K. Form of H y d r a t e I c e AF„ /H • A5„ R e f e r e n c e B,cv . B,cv *~B,cv c a l / m o l e c a l / m o l e e.u./mole P2atte«uw and/ ( 3 8 1 Type I 167 van d e r W a a l s v 1 (33) ~167 B a r r e r and S t u a r t 160 P a u l i n g v ' 300 180 -0.42+0.23 C h i l d , J r . ^ 3 5 ^ Type I I 196 v a n der Waals/„ 2\ and P l a t t e e u w * ' 88-128 B a r r e r and R u z i c k a ( 3 4 ) 200 P a u l i n g ^ 3 7 ) ( 3 5 ) 190 200 0.05+0.08 C h i l d , J r . v ' T h e r e a r e two b a s i c a s s u m p t i o n s i n v o l v e d i n the c e l l model o f van d e r Waals and P l a t t e e u w : i ) t h e c o n t r i b u t i o n by t h e c a g e - f o r m i n g m o l e c u l e t o t h e p a r t i t i o n f u n c t i b m i s n o t a f f e c t e d by the p r e s e n c e o f g u e s t m o l e c u l e s , and i i ) t h e f o r c e - f i e l d i n a cage has s p h e r i c a l symmetry and a g u e s t m o l e c u l e can r o t a t e f r e e l y i n i t . U s e f u l as t h e s e a s s u m p t i o n s a r e f o r a g a a e r a l u n d e r s t a n d i n g of the p r o p e r t i e s of c l a t h r a t e s , t h e y a r e n o t v a l i d f o r c e r t a i n c l a t h r a t e s , p a r t i c u l a r l y a t (39) 40 41 low t e m p e r a t u r e s , S t a v e l e y and c o - w o r k e r s ' f i n d t h a t t h e m o l a r h e a t c a p a c i t i e s o f t h e q u i n o l c l a t h r a t e s o f a r g o n , k r y p t o n , and methane a r e l i n e a r f u n c t i t t f i s o f t h e p e r c e n t a g e f i l l i n g o f t h e c a v i t i e s , so t h a t e x t r a -p o l a t i o n t o z e r o f i l l i n g s h o u l d y i e l d a c o n s t a n t v a l u e c o r r e s p o n d i n g t o t h e m o l a r h e a t c a p a c i t y o f t h e empty g - q u i n o l l a t t i c e . However, a c o n s t a n t v a l u e i s n o t o b t a i n e d , e s p e c i a l l y a t low t e m p e r a t u r e s . F u r t h e r m o r e , the o e n t r o p y o f the g u e s t S^, o b t a i n e d a t 298 K f r o m i t s o b s e r v e d l o w - t e m p e r a t u r e h e a t c a p a c i t y C , by i n t e g r a t i o n , v i z . , S = 298 ~ dT , ( 2 ) s T i s s m a l l e r t h a n t h e v a l u e c a l c u l a t e d f r o m t h e t h e o r y o f van d e r W a a l s and P l a t t e e u w by 2-3 e n t r o p y u n i t s . T h i s s u g g e s t t h a t , c o n t r a r y t o t h e assump-t i o n o f t h e c e l l model, the l o w - f r e q u e n c y v i b r a t i o n s of t h e q u i n o l l a t t i c e a r e s u f f i c i e n t l y p e r t u r b e d by t h e g u e s t m o l e c u l e f o r t h e i r c o n t r i b u t i o n t o th e t o t a l e n t r o p y t o be s i g n i f i c a n t l y r e d u c e d . S i m i l a r low t e m p e r a t u r e s t u d i e s on c l a t h r a t e h y d r a t e s , w h i c h may d i s c l o s e e v i d e n c e o f d i s t o r t i o n o f th e h o s t l a t t i c e s t r u c t u r e s , have n o t so f a r been r e p o r t e d , t h o u g h such 42 e v i d e n c e s h o u l d be s u r p r i s i n g because ( i ) McKoy and S i n a n o g l u have p o i n t e d o u t t h a t even t h e l o w e s t f r e q u e n c i e s i n the v i b r a t i o n a l s p e c t r u m of t h e empty h y d r a t e l a t t i c e a r e r a t h e r h i g h i n t h e thermodynamic s c a l e t o be s u f f i c i e n t l y p e r t u r b e d by t h e " r a t t l i n g " g u e s t m o l e c u l e s i n t h e i r c a v i t i e s 12 and ( i i ) X - r a y d i f f r a c t i o n s t u d i e s have n o t r e v e a l e d any d i s t o r t i o n o f t h e h o s t l a t t i c e f o r t h e Type I and Type I I c l a t h r a t e h y d r a t e s . The second a s s u m p t i o n of t h e c e l l m o d el, namely, t h e a s s u m p t i o n o f f r e e r o t a t i o n o f g u e s t m o l e c u l e s i n s p h e r i c a l l y - s y m m e t r i c f o r c e - f i e l d s , 43 26 has been i n v a l i d a t e d by a v a r i e t y ' of e x p e r i m e n t a l d a t a . Among t h e c l a t h r a t e s o f t h e q u i n o l s e r i e s , s u l p h u r d i o x i d e has been shown d i r e c t l y by (4) X - r a y F o u r i e r a n a l y s i s v 7 t o r e o r i e n t about one a x i s i n i t s q u i n o l cage. (44) I n f r a - r e d s t u d i e s o f t h e c a r b o n d i o x i d e - q u i n o l c l a t h r a t e show t h a t r o t a t i o n of t h e g u e s t m o l e c u l e i s l i k e w i s e h i n d e r e d . From a d e t a i l e d i n f r a -r e d s t u d y of the c a r b o n m o n o x i d e - q u i n o l c l a t h r a t e o v e r a wide t e m p e r a t u r e 45 r a n g e , B a l l and McKean have s u g g e s t e d a c o m b i n a t i o n o f l i b r a t i o n a l and r a t t l i n g m o t i o n s of t h e g u e s t m o l e c u l e . H e a t c a p a c i t y measurements on t h e 46 same c l a t h r a t e by S t e p a k o f f and C o u l t e r have borne o u t s u b s t a n t i a l l y . . 1 „ , . , 4 0 , 4 7 , 48 ,. + - 1 . - T + 4 9 s i m i l a r r e s u l t s . H e a t c a p a c i t y ' , m a g n e t i c s u s c e p t i b i l i t y , p a r a m a g n e t i c r e s o n a n c e " ^ , n u c l e a r q u a d r u p o l e r e s o n a n c e ^ , and d i e l e c t r i c 52 l o s s e x p e r i m e n t s on v a r i o u s q u i n o l c l a t h r a t e s have e s t a b l i s h e d d e f i n i t e e n e r g y b a r r i e r s t o i n t e r n a l r o t a t i o n o f t h e g u e s t m o l e c u l e i n i t s c a v i t y . The r e s u l t s o f t h e s e e x p e r i m e n t s a r e p r e s e n t e d i n T a b l e I I . TABLE I I B a r r i e r s t o R o t a t i o n o f G u e s t s i n Q u i n o l C l a t h r a t e s G u e s t M o l e c u l e Temperature B a r r i e r E x p e r i m e n t R e f e r e n c e Range, K H e i g h t , c a l / m o l e 0 2 0.25-4.0 128 M a g n e t i c Meyer e t . a l . , v ' S u s c e p t i b i l i t y 7 TABLE I I C o n t d . B a r r i e r s t o R o t a t i o n o f G u e s t s i n Q u i n o l C l a t h r a t e s G u e s t M o l e c u l e T mperature Range, K B a r r i e r H e i g h t c a l / m o l e E x p e r i m e n t R e f e r e n c e °2 1.5-4.2 -250 P a r a m a g n e t i c Resonance '•6 1 ( 5 0 ) F o n e r e t . a l . , ' 100-300 -200 H e a t C a p a c i t y G r e y and S t a v e l e y ^ ^ . N 2 1.5-25 940 (51) N u c l e a r Q u a d r u p o l e Meyer and S c o t t v Resonance 120-300 -1100 H e a t C a p a c i t y G r e y and S t a v e l e y ^ ^ ^ 15-100 511 H e a t C a p a c i t y „ ,_L , 1 (47) C w u l t e r e t . a l . . CO 100-300 -1200 H e a t C a p a c i t y G r e y and S t a v e l e y ^ ^ 40 298 745) 2867) H e a t C a p a c i t y S t e p a k o f f and C o u l t e r ( 4 6 ) C H 4 150-300 J^O Heat C a p a c i t y P a r s o n a g e and S t a v e l e y ^ 4 0 ) 15-50 193 H e a t C a p a c i t y C o u l t e r e t . a l . , ^ ^ CHgOH 133-292 2300 D i e l e c t r i c L o s s n A ( 5 2 ) D r y d e n ' CHgCN 133-292 -18000 -N u c l e a r m a g n e t i c r e s o n a n c e s t u d i e s ' ^ on t h e q u i n o l c l a t h r a t e s o f methane, f l u o r o f o r m , m e t h y l a l c o h o l , and m e t h y l c y a n i d e have i n d i c a t e d t h a t t h e r e i s i s o t r o p i c r o t a t i o n o f t h e e n c l a t h r a t e d m o l e c u l e s even a t 77°K. H a r r i s e t . a l . on t h e o t h e r hand, have r e c e n t l y r e p o r t e d a s m a l l a n i s o -19 t r o p y i n t h e F c h e m i c a l s h i f t t e n s o r o f t h e q u i n o l c l a t h r a t e o f f l u o r o -form. These a u t h o r s have a l s o shown t h a t t h e a n i s o t r o p y f o r NFg s i m i l a r l y t r a p p e d i n t h e q u i n o l l a t t i c e i s much l a r g e r . A n a l o g o u s s t u d i e s on o r i e n t a t i o n e f f e c t s o f g u e s t m o l e c u l e s i n 42 c l a t h r a t e h y d r a t e s have been v e r y s p a r s e . McKoy and S i n a n o g l u have arg u e d a g a i n s t t h e e x i s t e n c e o f a h i g h b a r r i e r t o r o t a t i o n o f the g u e s t m o l e c u l e s i n t h e Type I h y d r a t e s , b u t a s i m i l a r g e n e r a l i z a t i o n has n o t y e t a p p e a r e d 55 f o r t h e Type I I c l a t h r a t e s of w a t e r . W i l s o n and D a v i d s o n have p r e s e n t e d d i e l e c t r i c e v i d e n c e f o r f r e e l y r e o r i e n t i n g (>500kc/sec.,) a c e t o n e m o l e c u l e s i n h y d r a t e c a v i t i e s a t 200°K. S i m i l a r l y , from t h e pronounced d i e l e c t r i c a b s o r p t i o n i n the v h f . (~6Gc/sec.,) r e g i o n o f t h e h y d r a t e s o f e t h y l e n e o x i d e and t e t r a h y d r o f u r a n , D a v i d s o n e t a l . f ^ , c o n c l u d e t h a t e n t i r e encaged m o l e -c u l e s r o t a t e i n t h e i r c a v i t i e s even down t o l i q u i d a i r t e m p e r a t u r e . A c c o r d -i n g l y , t h e r e l a x a t i o n t i m e s a s s o c i a t e d w i t h t h e r e o r i e n t i n g ' g u e s t ' d i p o l e s a r e v e r y s h o r t , ~10 ^ s e c , and t h e i r a c t i v a t i o n e n e r g i e s a r e o n l y of t h e o r d e r o f 0 . 5 k c a l / m o l e . S u c h a b e h a v i o u r i s s u r p r i s i n g f o r p o l a r ( o x y g e n a t e d ) m o l e c u l e s i n n e a r l y s p h e r i c a l c a v i t i e s , a l t h o u g h t h e e f f e c t i v e van d e r Waals d i a m e t e r s of e t h y l e n e o x i d e and t e t r a h y d r o f u r a n , 4.6A and 5.6.A., r e s p e c t i v e l y , deduced f r o m s c a l e d m o dels, would show t h a t i n f a c t b o t h m o l e c u l e s e n j o y c o n s i d e r a b l e f r e e space i n t h e i r c a v i t i e s f o r u n r e s t r i c t e d r o t a t i o n . V e r y 57 r e c e n t X - r a y a n a l y s i s o f the f o r m e r h y d r a t e shows i t t o be a h i n d e r e d a x i a l r o t o r i n i t s c a v i t y , w h i l e f o r t h e t e t r a h y d r o f u r a n h y d r a t e s t a b i l i z e d 58 by i n c l u s i o n o f HgS as t h e ' h e l p g a s ' , f r e e r o t a t i o n i s i n d i c a t e d . 15 From X - r a y s t u d i e s of c h l o r i n e h y d r a t e , P a u l i n g and M a r s h found t h a t t h e c h l o r i n e m o l e c u l e s were r e s t r i c t e d t o s t e r i c a l l y f a v o u r a b l e p o s i t i o n s i n n o n - s p h e r i c a l c a v i t i e s . On t h e o t h e r hand, X - r a y s t u d i e s o f 12 v o n S t a c k e l b e r g and M u l l e r have l e d them t o p r o p o s e freedom of r o t a t i o n f o r s u l p h u r d i o x i d e m o l e c u l e s i n t h e SOg h y d r a t e . F o r t h e same c l a t h r a t e , 59 H a r v e y e t a l . , . i n f e r r e d f r o m i n f r a - r e d e v i d e n c e t h a t t h e m o t i o n o f t h e SOg m o l e c u l e i s l i m i t e d - b e i n g o f a h i n d e r e d - r o t a r y and h i n d e r e d - t r a n s l a -9 t o r y n a t u r e . Such d i v e r s i f i e d r e s u l t s s u g g e s t t h a t m o l e c u l a r shape and s i z e o f g u e s t m o l e c u l e s a r e q u i t e i m p o r t a n t ' f a c t o r s t h a t d e c i d e t h e p o t e n t i a l f i e l d i n s i d e c l a t h r a t e c a v i t i e s and t h e b a r r i e r s t o r e o r i e n t a t i o n . The L e n n a r d -J o n e s (12-6) p o t e n t i a l f u n c t i o n , w h i c h E a s ±he g e n e r a l two-parameter f o r m ^ = e o [ ( r o / r ^ 1 2 - 2 ( r o / r ) 6 ] <3> i s e s s e n t i a l l y a f u n c t i o n o f t h e p o t e n t i a l minimum ( e Q ) and i t s p o s i t i o n ( r ) i n a s p h e r i c a l c a v i t y o f r a d i u s r . I t i g n o r e s t h e shape and s i z e o f the g u e s t m o l e c u l e i n t e r a c t i n g w i t h t h e w a l l s o f t h e c a v i t y . A more r e a l i s -t i c p o t e n t i a l s h o u l d t a k e i n t o a c c o i i n t t h e shape-dependent e f f e c t i v e s i z e o f t h e g u e s t m o l e c u l e i n t e r a c t i n g w i t h the c a v i t y w a l l s . K i h a r a ' s (12-6) 60 p o t e n t i a l i s one suc h , w h i c h a s s i g n s a c o r e , p, t o each g u e s t m o l e c u l e . A c c o r d i n g l y , t h e f u n c t i o n w i l l be o f t h e f o r m «p> - •.[<v*>12 - * v / } ( 4 ) w i t h e m the p o t e n t i a l minimum and p m i t s domain. I t i s t h i s l a t t e r p o t e n -t i a l w h i c h has been shown t o p r e d i c t more r e l i a b l e d i s s o c i a t i o n p r e s s u r e s 42 f o r r o d l i k e m o l e c u l e s c o n f i n e d i n h y d r a t e c a v i t i e s The o v e r a l l p i c t u r e emerging f r o m t h e above s u r v e y , t h e n , i s t h a t a gaseous ' g u e s t ' m o l e c u l e s u f f e r s a l a r g e and n e g a t i v e e n t r o p y change when b e i n g i n c l u d e d i n a c l a t h r a t e l a t t i c e . A s f i r s t s u g g e s t e d by Evans and 61 R i c h a r d s f r o m e x t e n s i v e t h e r m o c h e m i c a l measurements on q u i n o l c l a t h r a t e s , t h e g u e s t m o l e c u l e l o s e s most or a l l o f i t s gas-phase t r a n s l a t i o n a l e n t r o p y , and, i f p o l y a t o m i c , p o s s i b l y some o r a l l o f i t s r o t a t i o n a l e n t r o p y . I t i s , however, e x p e c t e d t o g a i n v i b r a t i o n a l e n t r o p y as a r e s u l t o f i t s o s c i l l a -t i o n s i n t h e h o s t c age. I n o t h e r w o r d s , i t i s i n s i d e a p o t e n t i a l w e l l , and has t o a c q u i r e an a c t i v a t i o n e nergy t o surmount t h e w e l l . F u r t h e r m o r e , when 10 a l a r g e s i z e and a n o n - s p h e r i c a l shape of t h e g u e s t m o l e c u l e a r e i n v o l v e d , 26 t h i s p o t e n t i a l w e l l w i l l have a n a r r o w b o t t o m w i t h s t e e p s i d e s . A l i g n m e n t of an o p t i m u m - s i z e d g u e s t m o l e c u l e i n an o p t i m u m - s i z e d c a v i t y s h o u l d be t h e o r e t i c a l l y p o s s i b l e a t l e a s t a t l o w t e m p e r a t u r e s . I t s h o u l d t h e r e f o r e be of g r e a t i n t e r e s t t o s t u d y e x p e r i m e n t a l l y t h e n a t u r e o f a l i g n m e n t and m o t i o n a l b e h a v i o u r o f g u e s t m o l e c u l e s , p a r t i c u -l a r l y among t h e l e s s i n v e s t i g a t e d gas h y d r a t e s . The f r e q u e n c y o f o s c i l l a -t i o n , v g, o f g u e s t m o l e c u l e s i n h y d r a t e c a v i t i e s ( r e g a r d e d as ' c a v i t y 42 4 5 o s c i l l a t o r s ' ) has been shown r o u g h l y t o l i e i n t h e range 10 - 10 -1 s e c . , s i n c e vg = l / 2 n ( % ) * ( 5 ) where m i s t h e o s c i l l a t o r - m a s s and 3C i s the f o r c e - c o n s t a n t . T h i s f r e q u e n c y range i s i d e a l f o r s t e a d y - s t a t e n u c l e a r m a g n e t i c r e s o n a n c e (NMR) t e c h n i q u e s t o be m e a n i n g f u l l y a p p l i e d . A range o f g u e s t m o l e c u l e s o f d i f f e r e n t shapes may be chosen, and t h i s range may be ext e n d e d t o c o v e r n o t o n l y t h e Type I and Type I I h y d r a t e - f o r m e r s , b u t a l s o some of t h e q u a t e r n a r y ammonium c a t i o n s w h i c h f o r m p o l y h e d r a l w a t e r c l a t h r a t e s . I n s e v e r a l o f t h e s e p e r a l k y l a t e d 16—23 c a t i o n h y d r a t e s t r u c t u r e s t h e o b s e r v e d n o n - c u b i c symmetries o f t h e h o s t - l a t t i c e s i n d i c a t e t h a t d i s t o r t i o n due t o t h e g u e s t c a t i o n s i s p o s s i b l e . F u r t h e r , t h e r e i s some measure o f p o s i t i o n a l and o r i e n t a t i o n a l d i s o r d e r i n t h e d i s t r i b u t i o n o f c a t i o n s w i t h i n t h e v o i d s o f t h e s e h o s t - l a t t i c e s , and unambiguous i d e n t i f i c a t i o n o f e l e c t r o n d e n s i t y peaks i n v o l v i n g t h e d e t a i l s of c a t i o n o r i e n t a t i o n s Lhas> t h e r e f o r e been f o u n d t o be d i f f i c u l t . A l s o , t h e p o s i t i o n s o f t h e p r o t o n s b e l o n g i n g t o t h e p e r a l k y l g u e s t c a t i o n s have n o t been deduced f r o m X - r a y d i f f r a c t i o n d a t a , and p r o t o n m a g n e t i c r e s o n a n c e m i g h t p r o f i t a b l y be employed as a complement t o t h e s e d i f f r a c t i o n s t u d i e s . 11 As w i l l be seen i n t h e n e x t c h a p t e r , c l a t h r a t e s t r u c t u r e s have t h e un i q u e advantage o f b e i n g a b l e t o p r o v i d e a f a i r degree o f s e l e c t i v e i s o -l a t i o n , o f a g u e s t m o l e c u l e or a s i n g l e g u e s t c a t i o n i n a h o s t c a v i t y . N e i g h b o u r i n g c a v i t i e s i n a c l a t h r a t e a r e f a r enough a p a r t f o r . ; m a g n e t i c d i p o l a r i n t e r a c t i o n s between two g u e s t m o l e c u l e s t o be n e g l i g i b l e ; o w i n g t o t h e i r dependence on the i n v e r s e cube of t h e d i s t a n c e , and r e l i a b l e e s t i m a t e s o f t h e l i n e w i d t h , second moment, and 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 o f a g u e s t m o l e c u l e a r e t h e r e f o r e p o s s i b l e . One ma j o r p r o b l e m i n a p p l y i n g the NMR t e c h n i q u e i s t h a t b o t h t h e g u e s t m o l e c u l e and t h e h o s t - l a t t i c e may c o n t a i n n u c l e a r s p i n s of the same s p e c i e s , p a r t i c u l a r l y p r o t o n s , and c o m p u t a t i o n s o f t h e second moments f o r such s t r u c t u r e s w o u l d be d i s t u r b i n g l y complex. A f l u o r o c a r b o n g u e s t m o l e -19 c u l e i n t h e h y d r a t e l a t t i c e i s an o b v i o u s f i r s t c h o i c e s i n c e the F r e s o n -ance may be s t u d i e d a p a r t f r o m t h a t o f the l a t t i c e p r o t o n s . I n o t h e r c a s e s , t h i s p r o b l e m may be c i r c u m v e n t e d by f o r m i n g t h e c l a t h r a t e h y d r a t e s f r o m pure DgO and the g u e s t component. D i p o l a r b r o a d e n i n g due t o t h e l a t t i c e n u c l e i i s t h e r e b y e l i m i n a t e d , t h e c o n t r i b u t i o n t o t h e second moment of t h e 1 2 H -D i n t e r a c t i o n s b e i n g n e g l i g i b l e , p a r t i c u l a r l y i n such open s t r u c t u r e s . 12 CHAPTER I I STRUCTURES OF CLATHRATE HYDRATES 2.1 O u t l i n e I n t h i s c h a p t e r an a c c o u n t w i l l be g i v e n o f the g e n e r a l s t r u c t u r a l a s p e c t s o f the 'Type I s and 'Type I I ' ! V gas h y d r a t e s . We s h a l l compare t h e s e and t h e s t r u c t u r e s f o u n d i n s e v e r a l p e r a l k y l a t e d ammonium s a l t h y d r a t e s . Two r e p r e s e n t a t i v e s t r u c t u r e s o f t h e - l a t t e r k i n d , f o r w h i c h c r y s t a l d a t a a r e a v a i l a b l e i n c o m p l e t e d e t a i l f o r t h e h o s t as w e l l as t h e g u e s t m o i e t i e s , w i l l be d e s c r i b e d , The two s t r u c t u r e s c h o s e n a r e o f p a r t i c u l a r i n t e r e s t f o r t h e p r e s e n t work. 2.2 The B a s i c P o l y h e d r a l U n i t 7—12 15 X - r a y i n v e s t i g a t i o n s * and a s t u d y o f a p p r o p r i a t e p h y s i c a l 14 models have r e v e a l e d t h a t i n t h e c l a t h r a t e h y d r a t e c r y s t a l s , w a t e r forms t h r e e - d i m e n s i o n a l h y d r o g e n bonds t o g i v e r i s e t o r e p e a t i n g p a t t e r n s o f 13 c e r t a i n h o l l o w p o l y h e d r a . The most f u n d a m e n t a l u n i t o f t h e s e p o l y h e d r a i s the p e n t a g o n a l dodecahedron, O^Q , formed f r o m t w e n t y w a t e r m o l e c u l e s ( F i g . l a ) . The t w e n t y oxygens c o n s t i t u t e i t s t w e n t y a p i c e s . E a c h oxygen i s hydrogen-bonded t o f o u r o t h e r s , t h r e e o f w h i c h a r e i n t h e same u n i t and th e f o u r t h b e l o n g s t o a n e i g h b o u r i n g dodecahedron. The h y d r o g e n bonds are. d i s t r i b u t e d n e a r l y t e t r a h e d r a l l y about any oxygen atom, and t h e t h i r t y edges o f t h e p e n t a g o n a l dodecahedron a r e t h u s h y d r o g e n bonds about 2.8A l o n g . The 3 volume of t h i s p o l y h d r o n i s about 169A, i . e . , i t w o u l d e n c l o s e a s p h e r e o f a p p r o x i m a t e r a d i u s 3.4A. Owing t o i t s f i v e f o l d a x i a l symmetry, t h e p e n t a -g o n a l dodecahedron c a n n o t be c l o s e - p a c k e d w i t h i t s e l f t o f i l l space c o m p l e t e l y . C r y s t a l s t r u c t u r e s b u i l t up by s t a c k i n g t h e s e dodecahedra must t h e r e f o r e c o n t a i n n o n - r e g u l a r " v o i d s " w h i c h a r e t h e m s e l v e s p o l y h e d r a . I n t h e h y d r a t e s t r u c t u r e s so f a r s t u d i e d , t h e s e a d d i t i o n a l p o l y h e d r a a r e a l l l a r g e r t h a n the p e n t a g o n a l dodecahedron. They a r e t h e t e t r a k a i d e c a h e d r o n , F i g . l b ; t h e p e n t a k a i d e c a h e d r o n , and t h e h e x a k a i d e c a h e d r o n , F i g . l c . The volumes of t h e s e hydrogen-bonded cages a r e , r e s p e c t i v e l y , about 230, 250 and 260A i . e . , c o r r e s p o n d i n g a p p r o x i m a t e l y t o e n c l o s a b l e sphex.esof r a d i i 3.7A - 4 .0A. 2.3 'The H o s t L a t t i c e s o f Gas H y d r a t e s The two o r d e r e d t y p e s , I and I I , o f s i m p l e gas h y d r a t e s r e p r e s e n -t e d i n F i g . 2 may now be d e s c r i b e d . I n a Type I h y d r a t e , o f c u b i c u n i t c e l l w i t h edge ~12A, d o d e c a h e d r a a r e c e n t r e d a t the c o r n e r s and b o d y - c e n t r e o f t h e cube. Those a t t h e c o r n e r s a r e j o i n e d t o a d j a c e n t c o r n e r dodecahed-r a by two a d d i t i o n a l h y d r o g e n bonds t o f o r m h e x a g o n a l r i n g s o f w a t e r m o l e c u l e s , b e l o n g i n g t o t e t r a k a i d e c a h e d r a . The b o d y - c e n t r e d dodecahedron i s a t t a c h e d t o a l l t he c o r n e r dodecahedra by s i n g l e h y d r o g e n bonds and (a) FIG 1 The P o l y h e d r a l Cages f o u n d i n t h e C l a t h r a t e H y d r a t e s : !a) The Dodecahedron bj The t e t r a k a i d e c a h e d r o n c) The H e x a k a i d e c a h e d r o n F I G 2 The C u b i c Gas H y d r a t e Frameworks ( a ) Type I ( b ) Type I I 15 a l s o t o two o f t h e added 'hexagon' w a t e r m o l e c u l e s . The u n i t c e l l o f 46 w a t e r m o l e c u l e s c o m p r i s e s two d o d e c a h e d r a l and s i x t e t r a k a i d e c a h e d r a l v o i d s . I f a g u e s t m o l e c u l e , G, o c c u p i e d e v e r y v o i d , t h e l i m i t i n g c o m p o s i t i o n w o u l d be G.SlHgO; o n l y "the l a r g e r c a v i t i e s a r e o c c u p i e d , t h i s c o m p o s i t i o n w o u l d be G.T^g H^O' I t s h o u l d be n o t e d t h a t t h e s t a c k i n g o f t h e s e p o l y h e d r a i s su c h t h a t f a c e s , r a t h e r t h a n edges o r a p i c e s , a r e s h a r e d between n e a r e s t n e i g h b o u r s . The s t r u c t u r e o f Type I I h y d r a t e s may be r a t i o n a l i z e d i n t h e 14 f o l l o w i n g way . TJie dodecahedron i s h e r e deformed s l i g h t l y so t h a t t h e a n g l e s a r o u n d two o p p o s i t e m o l e c u l e s a r e e x a c t l y t e t r a h e d r a l . These two t e t r a h e d r a l p o i n t s a r e t h e n s u p e r p o s e d on what w o u l d be t h e s i t e s of p a i r s o f c a r b o n atoms i n a diamond l a t t i c e . T h i s l e a d s t o t h e f o r m a t i o n o f h e x -k a i d e c a h e d r a l v o i d s , w h i l e t h e h y d r o g e n bonds a r e s t i l l o f l e n g t h ~2.8A. T h e r e a r e t h e n 136 w a t e r m o l e c u l e s p e r c u b i c u n i t c e l l (edge ~17A), w h i c h f o r m s i x t e e n d o d e c a h e d r a l and e i g h t h e x a k a i d e c a h e d r a l v o i d s . When o n l y t h e l a r g e r v o i d s a r e o c c u p i e d , a l i m i t i n g c o m p o s i t i o n of G.lTHgO i s o b t a i n e d . I f a Type I I h y d r a t e i s formed i n t h e p r e s e n c e o f G and v e r y s m a l l ' h e l p gas' m o l e c u l e s , e.g., H^S, t h e n d o u b l e h y d r a t e s a p p r o a c h i n g t h e c o m p o s i t i o n G.aHgS.lTHgO a r e o b t a i n e d . By f i l l i n g b o t h t y p e s o f v o i d s i n 62 t h i s manner, one s t a b i l i z e s t he open w a t e r l a t t i c e c o n s i d e r a b l y T a b l e I I I summarizes t h e e s s e n t i a l f e a t u r e s o f t h e c r y s t a l s t r u c t u r e o f t h e two t y p e s o f h y d r a t e s , and i n T a b l e IV i s f o u n d an e x t e n s i v e l i s t i n g o f gas h y d r a t e s now known. 2.4. C o m p a t i b l e 'Guests' M o l e c u l e s w h i c h a r e h y d r o p h o b i c , and n o t l i k e l y t o i n t e r a c t s t r o n g l y w i t h w a t e r , f o r m c l a t h r a t e h y d r a t e s . There i s a l i m i t e d number o f TABLE I I I CRYSTAL STRUCTURE DATA ON GAS HYDRATES H y d r a t e S t r u c t u r e Type eg c i 2 -H y d r a t e R e f : 15 Type I I eg: H y d r a t e s o f C H C 1 Q C H 2 C 1 2 , C 2 E Z E 1 *5 Ref: 1 1 Space Group Symmetry P 3 m n d m U n i t C e l l Edge A ~12A ~17A No. of Water M o l e c u l e s i n U n i t C e l l 46 136 D o d e c a h e d r a l „ V o i d s ( V o l . leoA1*) T o t a l Number 16 C e n t r e d a t (21 1)(3 3 I) v 8 8 8 A 8 8 8 ' , 3 1 3 w l 3 3 N V 8 8 8 A 8 8 8'' / 5 5 ^8 8 l u l l I) 8 M 8 8 8 ' / 7 5 3 w 5 7 JJ\ K8 8 8 M 8 8 8 ' / 5 1 5 w 7 3 5x K8 8 8 ^ 8 8 8 ' (111)^11) v 8 8 8 A 8 8 8 ' / 3 5 ^8 8 I) (I I I 8 M 8 8 8 , 1 5 5 w 3 7 5x ^8 8 8 A 8 8 8 ' T e t r a k a i d e c a h e d r a l H e x a k a i d e c a h e d r a l V o i d s ( V o l . 2 3 0 A 3 ) V o i d s ( V o l . 2 6 0 A 3 ) T o t a l Number C e n t r e d a t None (| P 0 f ) T o t a l Numbe: • None C e n t r e d a t (1 1 Iw 3 3 3, ^2 2 2 A 4 4 4 J (o o i ) ( ^ | ) ( o i o ) ( | | i ) (±o o)(|^i) C J TABLE IV LIST OF KNOWN GAS HYDRATES C o m p o s i t i o n and S t r u c t u r e G u e s t M o l e c u l e , G 6G,46H 20 C u b i c , P 3 ' m n a = 11.9 - 12.2k 0 A r ,Kr ,Xe ,CH 4 ,H2S ,CF 4 ,PH 3 ,H2S e ,AsH 3, CHFg^HgCl^Clg.COg.CHgBr.BrCl.SOg, N 2 0 ,CH 2C1 2, c y c l o - d g H g , 0 ^ j C ^ . C H g C H F g , CgHgF ,CH 3SH ,CH 2: CHF ,C0S , 0 ^ 0 8G.136H 20 C u b i c , F„3 d m a = 17.0-17.51 o C H g l ^ F g ^ F C l g j C H C l g ^ F g C l g j C H C l g F , C H 2 C l 2 , C H g C H C l 2 , C H g C F 2 C l ,(CHg) ^,0^1, C ^ B r ,CgHg, (CHg) gCH ,CHC 1:CHC1,n-C g H ^ r 8G.16H 2S.136H 20 C u b i c , F,3 d m a = 17.2 - 17.61 o CHgBr,CHgI ,CHgCHF 2,CHF:CF 2,SF 6 ,C0S, C H g C l g j C H C l g j C C ^ . C C l g F ^ C C l g B r . C S g j C H g C C l F g , C 2 H 5 C l , C 2 H 5 B r , C 3 H 8 , ( C H 3 ) 2 S i C t l 3 N P 2 , C 6 H 6 , CH 2C1CH 2C1, n-C g H ? B r ,C 4H g0 20G.172 H 2 0 T e t r a g o n a l , 4/mmm ; Br, a = 23. c& , c = 12.21 o ' o R e f e r e n c e s : 6-12, 14, 15, 57, 58, 62, 64, 65. 18 TABLE V ^HYSICO-CHEMICAL DATA ON SOME TYPE I GAS HYDRATES M o l e c u l a r Diameter (A ) f r o m models o r * c a l c u l a t e d G u e s t M o l e c u l e G M o l e c u l a r Volume of L i q u i d G (cm 3) D i s s o c i a t i o n P r e s s u r e o f \ H y d r a t e a t 0°C(atm) D e c o m p o s i t i o n t e m p e r a t u r e o f H y d r a t e a t l a t m (°c) 3.08* 4.06 A r C H 4 29 39 1 05 26 -42.8 -29.0 3.38* 4.70 K r co 2 43 29*;./ 3 / 14.5 12.3 -27.8 -24.0 4.95 N 20 10.0 -19.3 '• - • C2?2 42 5.7 -15.4 5.50 C2H4 49 5.5 -13.4 5.43 C2H6 55 5.2 -15,8 '- CHgF - 2.1 -4,40* P H 3 46 1.6 -6.4 4.40 4.30* Xe C F 4 43 1.5 ~1.0 -3.4 4.10 H 2S 36 0.731 +0.35 4.50 A S H 3 55 0.613 +1.80 5.65 C2H5F 60 CO. 530 +3.70 4.40 H 2 S e 38 0.346 +8.0 5.06 CHgCl 53 0.311 +7.5 5.00 so2 44 0.297 +7.0 5.17 c i 2 45 0.252 +9.6 5.50 CH gSH 54 0.239 +10.0 5.33 CHgBr 55 0.187 +11.1 5.68 51 0.045 + A s s o l i d COg ++ R e c e n t X - r a y s t u d i e s o f A l l e n and J e f f r e y (Ref .64) have shown t h a t B r H y d r a t e has a t e t r a g o n a l c r y s t a l s t r u c t u r e r a t h e r t h a n t h e c u b i c s t r u c t u r e o f Type I h y d r a t e s . 19 TABLE V I PHYSICO-CHEMICAL DATA ON SOME TYPE I I GAS HYDRATES M o l e c u l a r d i a m e t e r , A , f r o m models o r • c a l c u l a t e d G u e s t M o l e c u l e G M o l e c u l a r Volume o f l i g u i d G, cm D i s s o c i a t i o n P r e s s u r e o f H y d r a t e a t 0°c,iatm. D e c o m p o s i t i o n temp, o f H v d r a t e a t 1 atm. c 6.28 C 3 H 8 75 1.74 5.7 5.76* S F 6 - - 1 . 0 6.06 ( C H 3 ) 2 0 - - -6.50 ( C H 3 ) 3 C H - - -6.20 W 1 70 0.201 4.8 6.08 C H 2 C 1 2 63 0.116 ' 1.7 5.70 C H 3 I 62 0.074 4.3 6.00 CH 3CHC1 2 84 0.055 -6.44 CHClg 80 0.050 1.6 20 62 9 g u e s t s , e.g., a c e t o n e , e t h y l e n e o x i d e and t e t r a h y d r o f u r a n , w h i c h d i s s o l v e i n w a t e r and y e t f o r m h y d r a t e s . The h o s t l a t t i c e shows r e m a r k a b l e s e l e c t -i v i t y i n e n c a g i n g c e r t a i n m o l e c u l e s b u t n o t o t h e r s . I n f a c t , w h e t h e r Type I o r Type I I h y d r a t e i s formed a t a l l depends upon t h e d i m e n s i o n s o f t h e g u e s t m o l e c u l e s . Hydrogen and h e l i u m a r e t h u s t o o s m a l l t o be e n c l o s e d s u c c e s s -f u l l y by any o f t h e i c e c a v i t i e s . The l a r g e s t c a v i t y , t h e h e x a k a i d e c a h e d r o n o f t h e Type I I s t r u c t u r e , s e t s an up p e r l i m i t t o the s i z e o f t h e e n t e r i n g g u e s t m o l e c u l e . The f r e e d i a m e t e r o f t h e l a r g e r c a v i t y o f t h e Type I s t r u c t u r e i s ~5.9A. The up p e r l i m i t o f s i z e c o m p a t i b l e w i t h h y d r a t e f o r m a t i o n i s r e p r e s e n t e d by a m o l e c u l a r volume o f about 85 m l . i n t h e l i q u i d s t a t e ( s e e T a b l e s V and V l ) . M o l e c u l e s l a r g e r t h a n n - p r o p y l bromide a r e t o t a l l y e x c l u d e d f r o m gas h y d r a t e f o r m a t i o n . L i k e w i s e , 1 , 2 - d i c h l o r o e t h a n e , w h i c h has an awkward m o l e c u l a r shape, does n o t g i v e a h y d r a t e , a l t h o u g h i t s 6 3 more compact i s o m e r , 1 , 1 - d i c h l o r o e t h a n e , does so. Hammerschmidt has u t i l i z e d t h i s s e l e c t i v i t y o f h y d r a t e f o r m a t i o n f o r f r a c t i o n a t i o n o f n a t u r a l g a s e s . 2.5 The P e r a l k y l S a l t H y d r a t e C l a t h r a t e s C r y s t a l s o f a number o f h y d r a t e d p e r a l k y l ammonium, s u l p h o n i u m , and 20 phosphonium s a l t s a r e r e l a t e d s t r u c t u r a l l y t o the gas h y d r a t e s . I n t h e s e s t r u c t u r e s a g a i n , the b a s i c framework c o n s i s t s o f hydrogen-bonded p e n t a g o n a l d o d e c a h e d r a , as have been p o s t u l a t e d to e x i s t i n t h e gas h y d r a t e s . B u t t h e r e i s now a s i g n i f i c a n t d i f f e r e n c e i n t h e s t e r e o c h e m i c a l mode o f a s s o c i a -t i o n o f t h e s e d o d e c a h e d r a , and much l a r g e r u n i t c e l l s a r e o b t a i n e d . Thus i n t h e o r t h o r h o m b i c c r y s t a l l i n e s e r i e s , , o f some o f t h e s e s a l t h y d r a t e s , F i g . 3, dode c a h e d r a s h a r e f a c e s t w o - d i m e n s i o n a l l y t o f o r m i n f i n i t e columns i n t h e 'a' d i r e c t i o n ; w h i l e i n t h e t e t r a g o n a l c r y s t a l l i n e s e r i e s , F i g . 4, d o d e c a h e d r a s h a r e f a c e s i n ^ H ^ Q O ^ Q ) u n i t s w h i c h a r e c o r n e r - l i n k e d . I n b o t h t h e s e c a s e s , t e t r a k a i - and p e n t a k a i d e c a h e d r a l v o i d s a r e formed. These s t r u c t u r e s show a f u r t h e r d i f f e r e n c e f r o m t h e gas h y d r a t e s i n t h a t t h e a n i o n s o f t h e q u a t e r n a r y s a l t s a r e hydrogen-bonded t o t h e w a t e r framework and a r e a p a r t of t h e p o l y h e d r a l ' h o s t 1 . The ' g u e s t s ' , t h e n , w o u l d be t h e c a t i o n i c a l k y l c h a i n s w h i c h p r o j e c t i n d i v i d u a l l y i n t o t e t r a k a i - and p e n t a -k a i d e c a h e d r a . T h e r e f o r e , t h e i n t e r a c t i o n between t h e h o s t and g u e s t s t r u c t u r e s i n t h e s e c l a t h r a t e s cannot be s o l e l y t h a t of van d e r W a als and London d i s p e r s i o n f o r c e s , b u t must i n c l u d e a p o l a r i n t e r a c t i o n . A c o n d i t i o n f o r t h e f o r m a t i o n o f t h e s e s a l t h y d r a t e s i s t h a t t h e a n i o n s hydrogen-bond s t r o n g l y w i t h t h e w a t e r s t r u c t u r e . Oxygen and f l u o r i n e b e i n g p a r t i c u l a r l y s t r o n g l y h y d r o g e n - b o n d i n g , s u c h s t r u c t u r e s a r e c r y s t a l l i z e d e a s i l y f r o m aqueous s o l u t i o n s of a c e t a t e s , n i t r a t e s , c a r b o n a t e s , o x a l a t e s , t u n g s t a t e s , b e n z o a t e s , and f l u o r i d e s of q u a t e r n a r y a l k y l c a t i o n s . The c a t i o n s them-s e l v e s , on t h e o t h e r hand, must be non-hydrogen-bonding, s i n c e t h e y p l a y t h e same r o l e as t h e gas m o l e c u l e s i n t h e gas h y d r a t e s , i . e . , of f i l l i n g t h e 22 l a r g e r p o l y h e d r a . I n s e v e r a l o f t h e s e s t r u c t u r e s , t h e r e i s d i s o r d e r i n t h e d i s t r i b u t i o n o f t h e c a t i o n s w i t h i n t h e v o i d s o f t h e h o s t l a t t i c e . E v i d e n c e has been found a l s o f o r f r e e w a t e r m o l e c u l e s o c c u p y i n g some of t h e p e n t a g o n a l d o d e c a h e d r a ^ ' ^ ' 2.6 T e t r a i - A m y l Ammonium F l u o r i d e H y d r a t e , ( i - C 5 H i ; L ) 4 N + F ~ . 38H 20 The c l a t h r a t e s t r u c t u r e o f t h i s compound has been shown by a 17 d e t a i l e d t h r e e - d i m e n s i o n a l s i n g l e c r y s t a l X - r a y a n a l y s i s . The c r y s t a l s a r e o r t h o r h o m b i c , w i t h a = 12.08A, b = 21.6LA, and.•c,c=s. 12.82A. There a r e two f o r m u l a u n i t s i n the c e l l ; w a t e r m o l e c u l e s and t h e a n i o n s t o g e t h e r c o n s t i t u t e a p o l y h e d r a l hydrogen-bonded framework. The n i t r o g e n F I G . 3 Dodecahedra s h a r i n g f a c e s t w o - d i m e n s i o n a l l y , f o r m i n g t e t r a k a i d e c a h e d r a l and p e n t a k a i -d e c a h e d r a l v o i d s . F I G . 4 Arrangement of f i v e f a c e - s h a r i n g dodecahedra ( l e f t ) , c o r n e r - l i n k e d t o one t e t r a k a i d e c a h e d r o n ( c e n t r e ) and two p e n t a k a i d e c a h e d r a ( r i g h t ) . o f t h e c a t i o n i s a t t h e common v e r t e x o f two t e t r a k a i - and two p e n t a k a i -d e c a h e d r a . The a l k y l c h a i n s e x t e n d i n t o t h e s e v o i d s , F i g . 5, and a r e t e t r a -h e d r a l l y d i s p o s e d w i t h r e s p e c t t o t h e N + s i t e . T h e r e i s an optimum c o r r e s -pondence between the d i m e n s i o n s o f the c a v i t y and o f t h e g u e s t i o n s i n t h i s s t r u c t u r e , w h i c h c r y s t a l l i z e s v e r y r e a d i l y and has a m e l t i n g p o i n t o f 31 C. Temperature f a c t o r s a s s o c i a t e d w i t h t h e f r a c t i o n a l a t o m i c c o o r d i n a t e s f o r th e a l k y l c a r b o n s s u g g e s t t h a t t h e r m a l m o t i o n o f an a l k y l c h a i n i n the p e n t a k a i d e c a h e d r a l c a v i t y i s s i g n i f i c a n t l y l a r g e r t h a n t h a t of a c h a i n i n th e s m a l l e r c a v i t y . No d i s o r d e r has been r e p o r t e d f o r t h i s s t r u c t u r e . 2.7 T e t r a n - B u t y l Ammonium F l u o r i d e H y d r a t e , ( n - C 4 H g ) 4 N + F ~ . 3 2 . 8 H 2 0 21 T h i s c l a t h r a t e i s t e t r a g o n a l w i t h u n i t c e l l d i m e n s i o n s a = 23.52A and c = 12.30A. The h o s t s t r u c t u r e i n c l u d i n g t h e F~ a n i o n c o n s i s t s of f i v e f a c e - s h a r i n g p e n t a g o n a l d o d e c a h e d r a r e l a t e d b y a 4 2 screw a x i s w i t h t h e i r c e n t r e s l o c a t e d a t t h e (0,0,^) and (i , 2",0) p o s i t i o n s . The i n t e r s t i c e s a r e t e t r a k a i - and p e n t a k a i d e c a h e d r a . There a r e t w e n t y o f t h e s e l a r g e r p o l y -h e d r a l v o i d s p e r u n i t c e l l , s i x t e e n o f them t e t r a k a i d e c a h e d r a and t h e r e m a i n i n g f o u r p e n t a k a i d e c a h e d r a . S i n c e t h e r e a r e f i v e m o l e c u l e s o f ammonium s a l t p e r u n i t c e l l , t h e s e t w e n t y v o i d s a r e c o m p l e t e l y f i l l e d b y t h e tw e n t y n - b u t y l c h a i n s . F o u r o f t h e s e c a t i o n s , w h i c h a r e e q u i v a l e n t , e x h i b i t o r i e n t a t i o n a l d i s o r d e r about one m i r r o r p l a n e as shown i n F i g . 6 a , w h i l e t h e f i f t h d i s p l a y s b o t h o r i e n t a t i o n a l and p o s i t i o n a l d i s o r d e r a b o u t two n o n - c r y s t a l l o g r a p h i c m i r r o r p l a n e s , F i g . 6b. Some o f t h e p e n t a g o n a l 21 do d e c a h e d r a w h i c h a re d i s t o r t e d b y t h e c a t i o n s a r e presumed t o be o c c u p i e d b y f r e e w a t e r m o l e c u l e s . 1 F I G . 6a. One o f the F o u r E q u i v a l e n t C a t i o n s i n T e t r a n - B u t y l Ammonium F l u o r i d e H y d r a t e , v i e w e d down t h e c - A x i s . ( X = V e r t i c e s w i t h Maximum D i s t o r t i o n ; a l k y l c h a i n s numbered as i n r e f . 2 l ) . M i r r o r P l a n e ( O r i e n t a t i o n a l D i s o r d e r ) 26 •Y' D i s o r d e r ) . F I G . 6b. The F i f t h C a t i o n i n T e t r a n - B u t y l Ammonium F l u o r i d e H y d r a t e , v i e w e d p e r p e n d i c u l a r t o t h e c - A x i s . ( X = V e r t i c e s w i t h Maximum D i s t o r t i o n ; a l k y l c h a i n s numbered as i n r e f . 2 l ) . 27 CHAPTER I I I THEORY OF THE INFLUENCE OF MOLECULAR MOTION ON NUCLEAR RESONANCE LINE SHAPES 3.1 The L i n e Shape F u n c t i o n 6 6 — 7 2 Fundamental n u c l e a r m a g n e t i c r e s o n a n c e (NMR) t h e o r y s t a t e s t h a t when an assembly o f n u c l e i w i t h m a g n e t i c moments i s immersed i n a c o n s t a n t m a g n e t i c f i e l d § o , energy i s a b s o r b e d f r o m a r a d i o f r e q u e n c y f i e l d —» a p p l i e d p e r p e n d i c u l a r t o H q and of f r e q u e n c y n e a r t h e n u c l e a r Larmor p r e c e s s i o n f r e q u e n c y m s u c h t h a t uu = 2TTV = Y^ 0 » where y i s a c h a r a c t e r i s t i c c o n s t a n t , t h e n u c l e a r g y r o m a g n e t i c r a t i o . The r a t e a t w h i c h power i s a b s o r b e d by the n u c l e i i s p r o p o r t i o n a l t o t h e i m a g i n a r y component o f t h e n u c l e a r m a g n e t i c s u s c e p t i b i l i t y x = x' ~ ix"» The dependence o f t h i s component x" o n f r e q u e n c y i s g i v e n by a shape f u n c t i o n g(v ) and i s d e f i n e d .(6) 28 as X."(v) = r r x ovg(v) where g ( v ) i s a " n o r m a l i z e e l " f u n c t i o n ^ i . e . , g ( v ) d v = 1 O "I I n ( 6 ) , X 0 = NM- (3kT) ( l + l ) / ( l ) i s t h e s t a t i c n u c l e a r s u s c e p t i -b i l i t y o f N, a u n i t volume o f t h e n u c l e i . The n u c l e i have s p i n quantum number I and m a g n e t i c moment (j, = g B l , g b e i n g t h e " n u c l e a r g - f a c t o r " and B t h e n u c l e a r magneton. The s t u d y o f the l i n e shape f u n c t i o n , g ( v ) , and o f t h e p a r a m e t e r c a l l e d " l i n a w i d t h " , 1 / , where * 2 - * t « i > i » • <7> i s t h e e s s e n c e o f w i d e - l i n e NMR s p e c t r o s c o p y . S e v e r a l i n t e r a c t i o n s may c o n t r i b u t e t o the b r o a d e n i n g o f t h i s shape f u n c t i o n i n s o l i d s , c h i e f among w h i c h a r e ( a ) m a g n e t i c i n t e r a c t i o n between n u c l e a r d i p o l e s and (b) c o u p l i n g v i a t h e n u c l e a r q u a d r u p o l e . Owing t o t h e i r l ow s p i n number ( l 4 v , H and F ± y - t h e n u c l e i o f i n t e r e s t i n t h i s s t u d y -p o s s e s s a v a n i s h i n g e l e c t r i c a l q u a d r u p o l e moment and, c o n s e q u e n t l y , t h e second form o f i n t e r a c t i o n s m e n t i o n e d above i s a b s e n t . T h i s s u r v e y w i l l t h e r e f o r e c o v e r o n l y i n t e r a c t i o n s o f t h e f i r s t t y p e . I n h o m o g e n e i t i e s o f t h e a p p l i e d m a g n e t i c f i e l d s (SQ) o v e r t h e sample volume a r e , o f c o u r s e , a n o t h e r 73 o b v i o u s s o u r c e o f l i n e b r o a d e n i n g , b u t r e c e n t advances i n t h e d e s i g n o f e l e c t r o m a g n e t s have l e d t o co m p l e t e e l i m i n a t i o n o f t h e s e e f f e c t s . 3.2 N u c l e a r D i p o l e - D i p o l e I n t e r a c t i o n . B e s i d e s t h e e x t e r n a l l y a p p l i e d m a g n e t i c f i e l d , t h e f i e l d a c t i n g on a c e r t a i n n u c l e u s i i s i n most c r y s t a l s e n t i r e l y due t o t h e i n t e r a c t i o n 29 between i t and t h e o t h e r n u c l e i j . I n o r d e r t o c a l c u l a t e t h e c o n t r i b u t i o n o f n u c l e u s j t o t he f i e l d a c t i n g on i and p a r a l l e l t o 3q, we c a l l r^r t h e i n t e r n u c l e a r d i s t a n c e and Y-• t h e a n g l e between the v e c t o r r . . and 3 ( s e e f i g u r e 7 ) . The c o n t r i b u t i o n , t h e n , i s where n • d e n o t e s t h e a b s o l u t e v a l u e of the m a g n e t i c moment of j . The - s i g n J a r i s e s f r o m t h e two p o s s i b l e o r i e n t a t i o n s o f t h e d i p o l e i n t h e m a g n e t i c f i e l d . However, t h e i n t e r a c t i o n between e q u i v a l e n t n u c l e i i n v o l v e s n o t o n l y a c o n t r i b u t i o n o f the s t a t i c f i e l d due t o t h e m a g n e t i c d i p o l e , b u t a l s o a dynamic t e r m . A n a l o g o u s t o t h e m e c h a n i c a l c a s e o f two w e a k l y c o u p l e d pendulums, the e q u i v a l e n t n u c l e i may exchange energy when the two o s c i l l a -t i o n s a r e i n phase. F o r t h i s c a s e one has t o c a l c u l a t e t he en e r g y l e v e l s w i t h t h e a i d o f quantum m e c h a n i c a l p e r t u r b a t i o n t h e o r y . T h i s has been done 74 b y P a k e f o r two p r o t o n s , and i t t u r n s o ut t h a t , because o f t h e dynamic t e r m and t h e s p i n exchange, t h e s t a t i c component o f t h e d i p o l a r m a g n e t i c 3 f i e l d between two e q u i v a l e n t n u c l e i i s —\ t i m e s as l a r g e as t h a t f o r two non-e q u i v a l e n t n u c l e i . T r a n s i t i o n s , t h e n , between a d j a c e n t n u c l e a r energy l e v e l s l e a d to two r e s o n a n c e l i n e s g i v e n b y hv = 2(JL H ± | iir~ 3(3CosVl ) T - (9) o 2 F o r a c o l l e c t i o n o f i s o l a t e d " p a i r s " i n a c r y s t a l , t h e l i n e shape t h a t r e s u l t s i s a s u p e r p o s i t i o n o f the d o u b l e t s f o r a l l t h e c r y s t a l l i t e s , t h e i n t e n s i t y o f each w e i g h t e d a c c o r d i n g t o t h e p r o b a b i l i t y o f t h e c o r r e s p o n d i n g o r i e n t a t i o n . F o r an i s o t r o p i c powder, t h e p r o b a b i l i t y of a d i p o l e b e i n g i n an o r i e n t a t i o n between Y and Y+dY i s : 30 dp = i S i n W ; d P = - i dCosY .(10) F I G . 7 The P a r a l l e l F i e l d , h., due t o a m a g n e t i c moment p,. The measured f i e l d d i s p l a c e m e n t o f a d o u b l e t component f r o m t h e l i n e c e n t r e i s ^\h = ± | |ir~ 3(3Cos 2Y-l) = * cv(3CosV-l) ( l O a ) T h e r e f o r e dp = fa # dCosY d ^ i dCosY d ^ i and we g e t the p r o b a b i l i t y d i s t r i b u t i o n ( " l i n e shape") i p _ = _ i _ d ± ( i i ) d ^ 2o/3 V —J The l i m i t s 0 ^ Y < TT r e q u i r e - 2 ^ & 1. The powder l i n e shape ( e q u a t i o n a l l ) i s shown by the dashed l i n e i n F i g . 8, w h i c h i s t h e sum of t h e f u n c t i o n s h a v i n g t h e + and - s i g n s . I n g e n e r a l , however, t h e f i e l d s o f more remote n u c l e i have a l s o t o be i n c l u d e d . The u s u a l a s s u m p t i o n i s t h a t t h i s " e x t r a m o l e c u l a r " b r o a d e n i n g ca n be r e p r e s e n t e d by a G a u s s i a n t y p e f u n c t i o n . D e n o t i n g t h i s f u n c t i o n by i the sm e a r i n g o u t o f s h a r p f i n e - s t r u c t u r e components b y t h e G a u s s i a n s i s r e p r e s e n t e d b y 7 ^ +00 ^ = ( 1 2) —00 T h i s "smoothed" f u n c t i o n g ( ^ i ) i s t h e s o l i d c u r v e i n F i g . 8. F o r s p i n systems more c o m p l i c a t e d t h a n t h e p a i r , more complex a n g u l a r dependences o f 76 l i n e p o s i t i o n s on c r y s t a l l i n e o r i e n t a t i o n may be f o u n d , b u t t h e s e may be 77 t r e a t e d i n a s i m i l a r way t o t h e above example. F i g . 9 shows t h e " s h a r p " and "smoothed o u t " l i n e shapes f o r a " t r i a n g l e " o f n u c l e i . F o r g r o u p s o f more t h a n 4 i n t e r a c t i n g n u c l e i o f s p i n ^, t h e r e s o n a n c e l i n e shape w i l l of c o u r s e be so complex as t o d e f y a p r e c i s e c a l c u l a t i o n e x c e p t i n e x t r e m e l y symmetric c o n f i g u r a t i o n s . The l i n e i n such i n s t a n c e s i s o f t e n s t r u c t u r e l e s s , and a p p r o x i m a t e s t o a G a u s s i a n d i s t r i — b u t i on. 3.3 Second Moment o f t h e r e s o n a n c e l i n e 78 1 Van V l e c k has d e v e l o p e d a t e c h n i q u e f o r o b t a i n i n g v e r y u s e f u l s t r u c t u r a l i n f o r m a t i o n b y comp u t i n g a pa r a m e t e r c a l l e d t h e " s e c o n d moment", <^H^>, o f a g i v e n l i n e shape f u n c t i o n g(^Ji) I n g e n e r a l , t h e n t h moment of a f u n c t i o n i s d e f i n e d as i t s "mean n t h power w i d t h " : <tfn>= ]7^ )ng(^ )d^  --j+Q- f oon+1 ^  ... ( i 3 ) . 79 , . F o r o ur p u r p o s e , n=2. The second f o r m of e q u a t i o n (.13) i s i n c l u d e d 2 because i t i s the u s u a l b a s i s f o r c a l c u l a t i n g (^H ) f r o m e x p e r i m e n t a l c u r v e s , w h i c h a re u s u a l l y d i s p l a y e d i n t h e f o r m o f t h e d e r i v a t i v e s o f t h e l i n e shape. The e s s e n c e o f Van V l e c k ' s quantum m e c h a n i c a l c a l c u l a t i o n i s t h e f o l l o w i n g : t h e t o t a l H a m i l t o n i a n of t h e e n t i r e s p i n s y s t e m i n t h e main m a g n e t i c f i e l d w i l l be + %' ( I 4 ) i»o where HQ i s t h e l a r g e energy term due t o t h e i n t e r a c t i o n o f t h e n u c l e a r s p i n s w i t h t h e main m a g n e t i c f i e l d and $J, t h e d i p o l a r i n t e r a c t i o n e n e r g y t e r m , i s a p e r t u r b a t i o n . The moments, t h e n , may be e x p r e s s e d i n terms o f c e r t a i n a p p r o p r i a t e o p e r a t o r s i n v o l v e d i n e q u a t i o n (14) and t h e i r commuta-t o r s w i t h # , i t s e l f . The second moment, f o r i n s t a n c e , i s w r i t t e n as // 2\ T r ( # S - S &) 2 ' Y 2 * 2 T r ( S v 2 ) wheare S i s t h e x- component o f t h e t o t a l n u c l e a r s p i n I = J I . . The x j_ 1 f a c i l i t y o f e m p l o y i n g t h e t r a c e n o t a t i o n l i e s i n t h e wellknowm f a c t t h a t t h e t r a c e o f an o p e r a t o r , v i z . t h e d i a g o n a l sum o f t h e c o r r e s p o n d i n g m a t r i x , i s i n d e p e n d e n t of t h e manner i n w h i c h t h e b a s i s w a v e - f u n c t i o n s a r e c h o s e n . H e r e t h e t r a c e s may be e v a l u a t e d i n t h e r e l a t i v e l y s i m p l e r e p r e s e n t a t i o n a p p r o p r i a t e t o t h e u n c o u p l e d s p i n s , i n w h i c h t h e i n d i v i d u a l 1^ and m^ a r e "good" quantum numbers. A l l t h e odd moments v a n i s h f o r j^' « ^ ' ^ k T . The second moment r e a d s + • § N I l f i 2 g Y i V v i H ^ f i f - 1 ) 2 ^ ••••• ( 1 5 ) where t h e f i r s t term a r i s e s f r o m t h e N r e s o n a n t n u c l e i i n the sample and s the second comes f r o m t h e i r i n t e r a c t i o n w i t h t h e n o n - r e s o n a n t s p i n s 1^. I n t h e c a s e o f a p o l y c r y s t a l l i n e m a t e r i a l , w i t h random o r i e n t a t i o n s o f t h e i n t e r n u c l e a r v e c t o r s , f has t o be a v e r a g e d o v e r a s p h e r e , 2 2 4 and s i n c e ( 3Cos Y - l ) = -g , t h e second moment i s ( ^ 2 } = | I ( I + 1 ) N ^ Y V z E r"6 i j > i J + T 5 N I V • gY 2I f(lf+ D ' l S ( 1 8 ) F o r s i m p l i c i t y , t h e c o n t r i b u t i o n s t o t h e second moment i n p o l y -c r y s t a l s may be d i v i d e d i n t o two p a r t s : " i n t r a " - i n t e r a c t i o n s between n u c l e i i n t h e same m o l e c u l e , and " i n t e r " - i n t e r a c t i o n s between t h e n u c l e i o f n e i g h b o u r i n g m o l e c u l e s . Owing, however, t o t h e i n v e r s e s i x t h power o f i t h e r i j , B and t h e r ^ ' s , t h e sums i n the above e q u a t i o n r a p i d l y c o n v e r g e , 80 and one 'may u s u a l l y r e p l a c e t h e summations by an i n t e g r a l f o r i n t e r a c t i o n s 81 beyond r = 6A. K r o o n has d e r i v e d e x p r e s s i o n s ( l 5 ) and ( 1 6 ) above i n an i n t e r e s t i n g manner by s t a r t i n g o f f w i t h t h e s t a t i s t i c a l d e f i n i t i o n o f t h e second moment. I f t h e n u c l e a r p o s i t i o n s change i n a c r y s t a l l a t t i c e a t r a t e s comparable t o t h e Larmor f r e q u e n c y ( ~ 1 0 ^ c p s ) , t h e a v e r a g e d i p o l a r i n t e r a c t i o n s a r e r e d u c e d and t h e l i n e w i d t h n a r r o w s . 3.4 M o t i o n a l E f f e c t s The e f f e c t o f n u c l e a r m o t i o n on t h e l i n e w i d t h i s s i m p l y u n d e r s t o o d i n an i d e a l i z e d case of a n u c l e u s t h a t " s e e s " one n e i g h b o u r a t a t i m e , a l t h o u g h t h i s n e i g h b o u r i s exchanged f r o m t i m e t o t i m e . The n e i g h b o u r a d i s t a n c e r away p r o d u c e s a l o c a l f i e l d h, and as a r e s u l t o f t h i s the J. 0 c • p r e c e s s i o n a l m o t i o n of t h e n u c l e u s w i l l be changed by an amount Su) = ± Y ^ i o c ' Sou i s t h e n the l i n e w i d t h i n t h e absence o f m o t i o n . I f t h e n e i g h -b o u r s i t s f o r a t i m e j, an e x t r a p h a s e - a n g l e 61 ( b e s i d e s t h a t due t o t h e u n i f o r m p r e c e s s i o n ) i s a c c u m u l a t e d , 6$ = i 6cjjT • 34 FIG. 8. THEORETICAL LINE SHAPE FOR STATIC PAIR. FIG. 9. THEORETICAL LINE SHAPES FOR STATIC A. (a) without broadening, (b) broadened by neighbours. A f t e r n s u c h i n t e r v a l s , t h e mean square phase a n g l e i s g i v e n b y A $ 2 = n ( 6 $ ) 2 = n ( 6 w ) 2 T 2 I n a t i m e T, n i s g i v e n b y T n = — T L e t u s suppose t h a t i n i t i a l l y t h e n u c l e i s t a r t p r e c e s s i n g i n s t e p . B y t h e t i m e Tg. c o n v e n t i o n a l l y c a l l e d t h e " s p i n - s p i n " o r " t r a n s v e r s e " r e l a x a t i o n 2 t i m e , when A<£ = 1, t h e y w i l l be: c o n s i d e r a b l y out o f s t e p . T h i s t i m e o f d e p h a s i n g c o r r e s p o n d s t o an e f f e c t i v e s p r e a d i n p r e c e s s i o n a l f r e q u e n c e s o f A<u = ? ( 1 7 ) v2 That i s , t h e l i n e w i d t h i s , and t h e e q u a t i o n ( l 7 ) may be compared w i t h l2 e q u a t i o n ( 7 ) . S i n c e 1 - ( T 2 / t ) ( 6 O ; 2 ) ( T 2 ) , we have 2 Au = (6a>) T ( 1 8 ) Thus, t h e s m a l l e r T i s , t h e s m a l l e r i s the l i n e w i d t h , and we have d e s c r i b e d 82 n a r r o w i n g by m o t i o n . More c o m p l i c a t e d e x p r e s s i o n s have been d e r i v e d f o r t h i s phenomenon. We n o t e t h a t t h e l i n e w i d t h measures t h e m o t i o n d i r e c t l y . The above e x p r e s s i o n h o l d s , o f c o u r s e , o n l y i f T i s s m a l l enough f o r a n u c l e u s t o " f l i p " b e f o r e s i z e a b l e d e p h a s i n g o c c u r s . Thus, n a r r o w i n g i s o b s e r v e d o n l y i f du)T < 1 • S t a t e d d i f f e r e n t l y , a v e r y l a r g e T s i m p l y means a " f r o z e n " o r " r i g i d " l a t t i c e , f o r w h i c h Au = 6UJ. 36 /, -I t i s c h a r a c t e r i s t i c t h a t t h e l i n e n a r r o w s due t o m o l e c u l a r m o t i o n when l / has a v a l u e o f t h e o r d e r o f t h e " f r o z e n " l a t t i c e l i n e w i d t h ( e x p r e s s e d as an a n g u l a r f r e q u e n c y ) . T h i s c o r r e s p o n d s t o a r e o r i e n t a t i o n f r e q u e n c y o f t h e o r d e r o f ,5kc/sec, w h i c h i s e x t r e m e l y s m a l l compared t o the f r e q u e n c i e s o r d i n a r i l y a s s o c i a t e d w i t h a t o m i c and m o l e c u l a r r o t a t i o n s 12 —1 and v i b r a t i o n s ( a b o u t 10 sec ) . One must be c i r c u m s p e c t , t h e r e f o r e , when comparing l i n e n a r r o w i n g w i t h o t h e r p h y s i c a l e f f e c t s , such as d i e l e c t r i c phenomena. I t i s c o n c e i v a b l e t h a t v e r y d i f f e r e n t f r e q u e n c y s c a l e s a r e i n v o l v e d i n two d i f f e r e n t e x p e r i m e n t s , so t h a t a d i e l e c t r i c l o s s peak, f o r i n s t a n c e , may o c c u r a t a v e r y d i f f e r e n t t e m p e r a t u r e f r o m a l i n e n a r r o w i n g , though b o t h a r e c o n c e r n e d w i t h t h e same m o t i o n . 3.5 R i g i d R o t o r The t e r m " r i g i d r o t o r " i s u s e d t o den o t e a group o f n u c l e i r o t a t i n g about one o r more a x e s . The group i s r i g i d i n the sense t h a t we s h a l l assume t h e d i s t a n c e s and a n g l e s between t h e r a d i u s v e c t o r s o f t h e n u c l e i w i t h i n t h e group t o be c o n s t a n t . The p r o c e d u r e whereby t h e change i n second moment c a n be c a l c u l a t e d may now be d e m o n s t r a t e d f o r t h e s i m p l e case o f a p a i r o f p r o t o n s r o t a t i n g about a f i x e d a x i s . W i t h t h e a x i s o f r o t a t i o n p o i n t i n g i n t h e Z - d i r e c t i o n o f t h e c o o r d i n a t e s y s t e m , the ma g n e t i c f i e l d w i l l have p o l a r a n g l e s 9 and $ and t h e r o t a t i n g p a i r o f n u c l e i w i l l have a r a d i u s v e c t o r r and p o l a r a n g l e s B and a (Fig» 1 0 ) . The f i e l d o f t h e n u c l e u s P a t P' i s g i v e n b y e q u a t i o n ( 8 ) . We may e x p r e s s (3Cos Y - l ) i n a s e r i e s o f L e g e n d r e p o l y n o m i a l s and f i n d f o r the average f i e l d h = ± I nr ( 3 C o s V l ) = + I ^ r " 3 ( 3 C o s 2 e - l ) ( 3 C o s 2 8 - l ) . We e x p r e s s t h e second moment i n the same way as b e f o r e and f i n d 37 I n t h e case of a p o l y c r y s t a l l i n e sample, we have a g a i n t o a v e r a g e o v e r a l l p o s s i b l e o r i e n t a t i o n s of H o , and t h i s may be e x p r e s s e d as ^ r o t . , " TO K I ' D Y ^ O S V D 2 ^ (19) I f B = , i . e . , i f t h e r o t a t i o n a l a x i s i s normal t o t h e i n t e r -n u c l e a r v e c t o r , we see t h a t , from (16) , <32)rot - i (^ 2Wid 3.6 G e n e r a l R e o r i e n t a t i o n I n a few c a s e s t h e n a t u r e o f t h e r o t a t i o n , and the " t i l t " o f t h e 83 r o t a t i o n a l a x i s t o t h e i n t e r n u c l e a r v e c t o r , have been d e t e r m i n e d . The r o t a t i o n a l e f f e c t s of more c o m p l i c a t e d systems o f n u c l e i on t h e i r s e c o n d 84 moment have been t r e a t e d i n d e t a i l by Andrew and Eades and b y F r a t i e l l o 85 86 '* and D o u g l a s s . S m i t h has r e c e n t l y o b s e r v e d t h a t t h e s e two methods l e a d e s s e n t i a l l y t o t h e same ave r a g e v a l u e o f r^j» E x c e l l e n t r e v i e w s on m o t i o n a l e f f e c t s i n n u c l e a r m a g n e t i c r e s o n a n c e w i t h r e g a r d t o m i c r o s c o p i c 87 88 and m a c r o s c o p i c m o t i o n s have been g i v e n b y P o w l e s and Andrew . The e f f e c t s o f " i n - p l a n e " r o t a t i o n a l o s c i l l a t i o n s on t h e second moment have 89 b een examined b y Andrew . F o r t h e i n t e r a c t i o n o f a n u c l e u s r o t a t i n g u n i f o r m l y i n a c i r c l e w i t h a s t a t i o n a r y n u c l e u s l y i n g i n t h e p l a n e o f r o t a t i o n , the r e d u c t i o n f a c t o r P o f t h e second moment has been shown t o be 3 2 2 P = 1 - 7> cv S i n B , where and g a r e as d e f i n e d i n f i g u r e 10. More g e n e r a l i z e d e x p r e s s i o n s 90 91 have been d e r i v e d r e c e n t l y ' f o r two t y p e s o f r e o r i e n t i n g n u c l e i . I n 38 39 t h e f i r s t o f t h e s e , c o m p u t a t i o n s have been made f o r " o u t - o f - p l a n e " r o t a t i o n a l o s c i l l a t i o n s , i . e . , f o r t h e p l a n e of r o t a t i o n of the moving n u c l e u s o r i e n t e d a t a r b i t r a r y a n g l e s w i t h r e s p e c t t o t h e s t a t i o n a r y n u c l e u s . I n t h e second, the c a s e has been t r e a t e d o f a p a i r o f n u c l e i b o t h o f w h i c h r e o r i e n t i s o -t r o p i c a l l y about t h e i r f i x e d c e n t r e s o f mass. The " i n t r a " - m o l e c u l a r second moment f o r such a case w o u l d , o f c o u r s e , be z e r o and t h e s m a l l r e s i d u a l s e cond moment w h i c h p e r s i s t s w i l l be p u r e l y " i n t e r " - m o l e c u l a r i n n a t u r e . I n t h e e x t r e m e l y m o b i l e c a s e of t r a n s l a t i o n a l d i f f u s i o n t h r o u g h t h e l a t t i c e , when t h e c e n t r e s o f mass o f n u c l e i s t a r t t o s h i f t r e l a t i v e t o each 92 o t h e r , t h e o v e r a l l second moment i s a n n i h i l a t e d . Many h y d r o c a r b o n s a r e n o t e w o r t h y i n t h i s r e g a r d . A t y p e of m o t i o n t h a t i s i n a sense i n t e r m e d i a t e between r e -93 94 o r i e n t a t i o n and d i f f u s i o n has o f t e n been d i s t i n g u i s h e d i n p o l y m e r s ' ; t h i s c o m p r i s e s l a r g e s c a l e c h a i n m o t i o n and " c h a i n t w i s t i n g " . A n a l o g i e s t o t h e s e models may be f o u n d i n c e r t a i n r e s u l t s r e p o r t e d i n t h i s t h e s i s ( c h a p t e r V I I I ) . I t i s i m p o r t a n t t o m e n t i o n h e r e t h a t v i b r a t i o n s w i t h i n a s u i t a b l e t i m e s c a l e and o f s u i t a b l e a m p l i t u d e s w i l l a l s o a f f e c t t h e second moment 95—99 and l i n e w i d t h of a s p e c t r u m . 69 I t has been contended f r o m a t h e o r e t i c a l s t a n d p o i n t t h a t t h e second moment s h o u l d i n f a c t r e m a i n i n v a r i a n t d u r i n g m o l e c u l a r m o t i o n s i n c e t h e " m o t i o n " o f t h e r e o r i e n t i n g n u c l e i i s o n l y a f o r m of " f r e q u e n c y m o d u l -a t i o n " o f t h e n u c l e a r Larmor f r e q u e n c y and s i n c e t h e a n g u l a r v e l o c i t y o f m o t i o n , cu r, w o u l d t h u s o n l y endow t h e s p e c t r u m w i t h s i d e b a n d s a t f r e q u e n c i e s nuur on e i t h e r s i d e o f t h e c e n t r a l r e s o n a n c e peak, n b e i n g an i n t e g e r . However, l a t e r e x p e r i m e n t s * ^ ' have shown t h a t a t f r e q u e n c i e s of t h e o r d e r o f t h e s p e c t r a l l i n e w i d t h (~10 c/s) a t any c a t e , t h e s e s i d e b a n d s move o u t v e r y f a r w i t h t h e i r i n t e n s i t y f a l l i n g o f f as ^ t o a p o i n t where t h e y become u n o b s e r v a b l y weak and a r e i n f a c t l o s t i n t h e " n o i s e " o f any a c t u a l r e c o r d i n g s y stem. Thus, o n l y a n a r r o w e d c e n t r a l p o r t i o n o f t h e s p e c t r u m i s o b t a i n e d i n p r a c t i c e . When, t h e r e f o r e , second moments a r e p l o t t e d as a f u n c t i o n o f t e m p e r a t u r e , o n l y t h e h o r i z o n t a l p o r t i o n s * r e p r e s e n t i n g " f u l l y r i g i d " o r " f u l l y r e d u c e d " second moments a r e s i g n i f i c a n t ; t h e second moments i n t h e f a l l i n g p o r t i o n s ( i n w h i c h s p e c t r a l n a r r o w i n g i s a c t u a l l y o c c u r r i n g ) a r e dependent on how many s i d e b a n d s t h e r e c o r d i n g equipment i s a c t u a l l y a b l e to show up above the n o i s e . A t t e m p t s t o c a l c u l a t e t h e temperature-dpendence o f T > t h e " c o r r e l a t i o n i t i m e " f o r t h e m o t i o n , f r o m t h i s f a l l i n g p o r t i o n a r e c t h e r e f o r e m e a n i n g l e s s . 3.7 M o l e c u l a r M o t i o n and C o r r e l a t i o n F u n c t i o n s H a v i n g e s t a b l i s h e d ( s e c t i o n 3.4) t h a t linewdtdth t r a n s i t i o n s a r e cau s e d o n l y by m o l e c u l a r m o t i o n s w h i c h o c c u r w i t h 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 o f t h e o r d e r o f vh-, , we s h a l l now show t h a t i n t r a n s i t i o n r e g i o n s ' l o c , ' t h e l i n e w i d t h i s a s e n s i t i v e parameter,, and may be u t i l i z e d t o y i e l d a g r e a t d e a l o f i n f o r m a t i o n on the t e m p e r a t u r e dependent r a t e p r o c e s s e s s u c h as t h e r m a l r e o r i e n t a t i o n , " t u n n e l i n g " o f groups o f atoms, and d i f f u s i o n . 68 The f i r s t s i m p l e model employed by Bloembergen, P u r c e l l , and Poun d ( B P P ) D 102 t o d i s c u s s t h i s t h e o r y was l a t e r m o d i f i e d b y Kubo and T o m i t a , who gave a r i g o r o u s t h e o r y b a s e d on a d e n s i t y m a t r i x method. When t h e r e i s a r a p i d r e o r i e n t a t i o n o f m o l e c u l e s , the l o c a l f i e l d s e t up by t h e n u c l e a r magnets f l u c t u a t e s about some average v a l u e , and t h e Van V l e c k f o r m u l a (15) i s d e v e l o p e d i n t o a F o u r i e r s p e c t r u m as i n e q u a t i o n ( 2 l ) : ( # 2 > = ^ YVI(I+I) A r~° (t)[acoB a*.^t)-!] A v . t d +00 = |- Y 2 n 2 l ( l + l ) [ F° . (t)J =J Jo^dw (21) I n ( 2 l ) , t h e J o ( o ) ) ' s a r e the F o u r i e r i n t e n s i t i e s ( o r s p e c t r a l e n e r g y d e n s i t i e s ) , and the time-dependent r . . - f u n c t i o n , F.°(t), i s c o r r e l a t e d t o i t s v a l u e a t a ti m e (t+T-) i n such a way t h a t i t decays expo-n e n t i a l l y i n t h a t t i m e w i t h a t i m e - c o n s t a n t T C .1 To s o l v e t h e i n t e g r a l i n e q u a t i o n ( 2 l ) , BPP ad o p t e d a d e f i n i t e c u t - o f f f r e q u e n c y , u^, above w h i c h t h e F o u r i e r components o f Jq(UJ) a r e assumed n o t t o a f f e c t t h e re s o n a n c e e n v e l o p e . T h i s a s s u m p t i o n i s p e r h a p s i n a d e q u a t e f o r systems whose m o l e c u l a r m o t i o n s have a d i s t r i b u t i o n o f c o r r e l a t i o n t i m e s ' ^ 3 . wm> f r o m e a r l i e r d e f i n i t i o n s , w i l l be ^ Y<^ZP I n f a c t , u) m = Q^ Y •")> 2 > where a i s an u n d e f i n e d c o n s t a n t i n t r o d u c e d t o c o v e r t h e u n c e r t a i n t i e s i n the l i m i t s o f t h e i n t e g r a t i o n employed and a l s o t h e i n a c c u r a t e d e f i n i t i o n o f t h e l i n e w i d t h w i t h r e g a r d t o l i n e shape. ., 84 104 V a l u e s o f <* range f r o m 0.3 t o 0.9 i n s t u d i e s on v a r i o u s h y d r o c a r b o n s * . Kubo and Tomita'''^ 2 have shown t h a t a i s a c t u a l l y (81n2) . F o r i s o t r o p i c t u m b l i n g o f t h e r ^ j ' s a ^ t h e i r l a t t i c e s i t e s , 0°<t)2>= = r-° [ r ( 3 C o 8 2 f i r l ) 2 dn - I Z »:! , 42 and, f r o m ( 2 l ) and ( 2 2 ) , we o b t a i n 0 1+U) T C i 2 2 P u t t i n g a i m = c v Y < ^ H ^ , t h i s becomes < ^ > = "g; Y2*2I(I+D 5 r j 6 t a n - 1 ^ ^ 2 ^ " T c ] -...(23) From t h i s r e l a t i o n s h i p , one o b t a i n s T c f r o m second moment measure-105 ' ments i n t h e r e g i o n o f t h e l i n e w i d t h t r a n s i t i o n s . However, i n a b i d i n g by second moments, a p a r t f r o m t h e a l r e a d y m e n t i o n e d u n r e l i a b i l i t y o f second moments i n t h e t r a n s i t i o n r e g i o n , one i s g e n e r a l l y c o m p e l l e d t o o v e r l o o k t h e f a c t t h a t " i n t e r " - m o l e c u l a r c o n t r i b u t i o n s t o the second moment may v a r y w i t h t h e m o t i o n , and hence w i t h t h e t e m p e r a t u r e , i n a d i f f e r e n t way fr o m t h e " i n t r a " - m o l e c u l a r v a l u e . 106 Gutowsky and Pak e have r e d e f i n e d e'quation( 23) i n terms o f f r e q u e n c y , and have i n t r o d u c e d y , t h e c o r r e l a t i o n f r e q u e n c y , w h i c h i s t h e r a t e ^ v c = — - — J a t w h i c h m o l e c u l a r c o n f i g u r a t i o n s change a p p r e c i a b l y . T T T C F o r t r e a t i n g a t r a n s i t i o n f r o m one d e f i n i t e l i n e w i d t h , C g a u s s , t o a nar r o w e r , l i n e w i d t h , B g a u s s , c h a r a c t e r i s t i c o f a s p e c i a l i z e d m o t i o n , t h e i r 86 e q u a t i o n c a n be m o d i f i e d r a t h e r a c c u r a t e l y t o g i v e 2TTVc = cYY&H-Jtan [TT( 6 H 2 - B 2 ) / 2 ( C 2 - B 2 ) ] j 1 (24) where §H i s t h e l i n e w i d t h ( i n g a u s s ) i n t h e t r a n s i t i o n r e g i o n . When t h e r e o r i e n t i n g group of n u c l e i has o n l y a s m a l l moment of i n e r t i a , i t i s p o s s i b l e t h a t t h e r e o r i e n t a t i o n i s n o t w h o l l y t h e r m a l l y a c t i v a t e d , b u t i n c l u d e s a c o n t r i b u t i o n f r o m quantum m e c h a n i c a l " t u n n e l i n g " t h r o u g h the b a r r i e r h i n d e r i n g t he p e r i o d i c m o t i o n . O r d e r - o f - m a g n i t u d e e s t i m a t e s o f such e f f e c t s have been made^^ 1^9^ 43 From our p h y s i c a l p i c t u r e of t h e a c t i v a t e d n a t u r e o f r e o r i e n t a -t i o n a l p r o c e s s e s , we may e x p e c t v c t o obey an A r r h e n i u s r e l a t i o n s h i p , -E/HT , X v c = v<» e (25) 'so t h a t we can o b t a i n t h e a c t i v a t i o n energy E 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 ; v i s a c o n s t a n t " l i m i t i n g f r e q u e n c y " f o r t h e m o t i o n . CO The a s s u m p t i o n s o f t h e t h e o r y a r e t h u s j u s t i f i e d o n l y i n so f a r as s t r a i g h t - l i n e ' A r r h e n i u s ' p l o t s a r e o b t a i n e d f o r a c e r t a i n l i n e w i d t h t r a n s i t i o n r e g i o n . I f t h e l i n e shape changes w i t h t e m p e r a t u r e , t h e method becomes v a l u e l e s s . The e x a c t v a l u e o f E and, p a r t i c u l a r l y , o f ^ , w h i c h can be o b t a i n e d b y e x t r a p o l a t i o n , may n o t c o r r e s p o n d e x a c t l y t o " t r u e " a c t i v a t i o n energy and l i m i t i n g c o r r e l a t i o n f r e q u e n c y . E v e n s o , f r u i t f u l c o m p a r i s o n s may be made f r o m r e l a t i v e v a l u e s o f a c t i v a t i o n e n e r g i e s o f a s e r i e s o f r e l a t e d compounds» , I n o r d e r t o s t u d y b e t t e r t h e dynamics i n v o l v e d i n m o l e c u l a r r e o r i e n t a t i o n s , t e m p e r a t u r e s t u d i e s on T^, t h e " s p i n - l a t t i c e " o r " l o n g i t u d i n a l " r e l a x a t i o n t i m e , may be c a r r i e d o u t . T^ may be d e f i n e d as th e c h a r a c t e r i s t i c t i m e - c o n s t a n t w i t h w h i c h t h e n u c l e i r e t u r n to t h e i r n o rmal B o l t z m a n n d i s t r i b u t i o n o f s p i n - s t a t e p o p u l a t i o n s a f t e r t h i s d i s t r i b u t i o n has been t e m p o r a r i l y u p s e t . F o r a p a i r o f l i k e n u c l e i w i t h s p i n \ w h i c h undergo m u t u a l m a g n e t i c d i p o l e - d i p o l e c o u p l i n g , T^ may be 102 106 107 w r i t t e n as a l i n e a r c o m b i n a t i o n o f .spesitraU F o u r i e r covapsmsmts ' ' as f o l l o w s : 9 4,2 .(26) When t h e m o t i o n s o f t h e m a g n e t i c n u c l e i can be c h a r a c t e r i z e d b y a s i n g l e c o r r e l a t i o n time j , T^ may be c a l c u l a t e d f r o m t h e e x p r e s s i o n * ^ ' 44 ^ K A 2 ; , - * («) The c o n s t a n t K depends on t h e geometry o f t h e m o t i o n a l p r o c e s s ( e . g . , r e o r i e n t a t i o n about one a x i s o r random t u m b l i n g ; K = 0.225 f o r t h r e e f o l d m e t h y l r e o r i e n t a t i o n * * ^ ) and T C = T where, as b e f o r e , , E i s t h e a c t i v a t i o n e n e r g y of t h e r e l a x a t i o n p r o c e s s and T q i s a l i m i t i n g c o r r e l a t i o n t i m e . I f T^ i s p l o t t e d v s . l / ^ , , t h e c u r v e p a s s e s t h r o u g h a minimum when GO T = 0.6158. Under o r d i n a r y e x p e r i m e n t a l c o n d i t i o n s , o c 8 —1 T^ minima o c c u r when r e o r i e n t a t i o n s t a k e p l a c e a t f r e q u e n c i e s ~10 sec . From a d i s c u s s i o n of t h e r e l a t i v e m agnitudes o f T^ e f f e c t s i n p r o t o n s o f t h e same m e t h y l group and t h o s e of d i f f e r e n t m e t h y l g r o u p s , Y u k i t o s h i e t a l . , " ' ' * * c o n c l u d e t h a t r e l a x a t i o n i n t h e f o r m e r i s f a r and away t h e most 112 dominant one. More r e c e n t l y , " i n t r a " - m o l e c u l a r r e l a x a t i o n c a u s e d by i n t e r n a l m o t i o n s w i t h f l u c t u a t i n g r^.. v a l u e s has been s t u d i e d . The l i n e w i d t h depends upon the e s s e n t i a l l y s t a t i c component of t h e l o c a l m a g n e t i c f i e l d s p e c t r u m , J Q ( y j ) • I t has been n o t e d e a r l i e r t h a t BPP's c u t - o f f f r e q u e n c y f o r t h i s component i s a/m » where a i s a c o n s t a n t A 2 " o f o r d e r u n i t y " . Under t h e s e c o n d i t i o n s , one may r e c a s t e q u a t i o n ( 2 l ) i n t o t h e f o r m 1 3 . 4 , 2 T , T :,x f ^ T , = 3_ Y V l ( l + l ) A 2 J0(u))d«, w h i c h i n t e g r a t e s t o ^ ' • ' " =(-0 ^ t a n " 1 ^ . , 1 9 T • • • J TT T d i 2 r i g i d A 2 s i n c e f l : ) f o r . t h e ' r i g i d l a t t i c e o c c u r s i n the l i m i t T -*°> . S ^ r i g i d -'.:; , c' 45 S u b s t i t u t i n g t h e v a l u e cv = ( 8 1 n 2 ) ~ * and r e a r r a n g i n g , one o b t a i n s 1 1 x -1 T c t a n T 2 T 2 — 21n2 2 r i g i d I f T « ( T 0 ) . . , , e.g. case o f d i f f u s i o n , c v 2 ' r i g i d 1 = 1 — ( 2 8 ) T 2 2 21n2 2 r i g i d 3.8 S a t u r a t i o n , c h o i c e o f H-^  f i e l d s , and f a s t passage As t h e f i e l d i s i n c r e a s e d , t h e r a t e o f n u c l e a r t r a n s i t i o n s t o t h e upper s p i n s t a t e and t h e degr e e o f phase co h e r e n c e b o t h i n c r e a s e , t e n d i n g t o i n c r e a s e the s i g n a l a m p l i t u d e . S i m u l t a n e o u s l y , however, t h e d i f f e r e n c e between t h e s p i n - s t a t e p o p u l a t i o n d e c r e a s e s , t e n d i n g t o r e d u c e t h e s i g n a l a m p l i t u d e . A t l o w v a l u e s o f H-^  t h e a m p l i t u d e - i n c r e a s i n g f a c t o r s p r e d o m i n a t e and a t h i g h v a l u e s t h e p o p u l a t i o n change p r e d o m i n a t e s . F i n a l l y , a t v e r y h i g h , t h e p o p u l a t i o n o f t h e energy " s t a t e s " w i l l be e s s e n t i a l l y e q u a l and no f u r t h e r a b s o r p t i o n o f power c a n o c c u r . A t t h i s p o i n t ; t h e a b s o r p t i o n s i g n a l d i s a p p e a r s e n t i r e l y . T h us, t h e a b s o r p t i o n s i g n a l a m p l i -tude r i s e s t o a maximum and t h e n f a l l s t o z e r o as i n c r e a s e s c o n t i n u o u s -l y . T h i s l o s s i n s i g n a l i n t e n s i t y i s c a l l e d " s a t u r a t i o n " , and, a t h i g h H j v a l u e s t h e a b s o r p t i o n s i g n a l depends s i g n i f i c a n t l y on a s a t u r a t i o n p a r a m e t e r S = y ^ j T j T g i Under t h e 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 o f " s l o w p a s s a g e " , where one s c a n s t h r o u g h r e s o n a n c e s l o w l y such t h a t t h e r e s o n a n t n u c l e i a r e a l w a y s " r e l a x e d " and a t e q u i l i b r i u m w i t h t h e i r s u r r o u n d i n g s , v e r y low f i e l d s a r e m a i n t a i n e d . A s p e c i a l case r e s u l t s , however, f r o m o b s e r v a t i o n s o f the " d i s p e r s i o n " mode s i g n a l , v i z . , t h e s i g n a l t h a t i s r ; i n - p h a s e w i t h t h e a p p l i e d r f - f i e l d ( a n d 90° o u t - o f - p h a s e w i t h t h e a b s o r p t i o n s i g n a l ) , u n d e r dH c o n d i t i o n s o f h i g h H-^  a m p l i t u d e and s c a n - r a t e s , __£ , so r a p i d t h a t ...'» < n e g l i g i b l e r e l a x a t i o n t a k e s p l a c e d u r i n g passage t h r o u g h r e s o n a n c e . T h i s t e c h n i q u e , c a l l e d " a d i a b a t i c f a s t p a s s a g e " , may be employed f o r t h e s t u d y o f T,, and s h a l l be d i s c u s s e d f u r t h e r i n C h a p t e r V. 47 CHAPTER TV APPARATUS 4.1 The S t a i n l e s s S t e e l P r e s s u r e C e l l F o r t h e q u a n t i t a t i v e f o r m a t i o n o f a u n i f o r m specimen o f a gas h y d r a t e , advantage may be t a k e n o f t h e f a c t t h a t t h e h y d r a t e can be formed u n d e r a h i g h p r e s s u r e o f t h e gas o v e r w a t e r even a t t e m p e r a t u r e s s l i g h t l y h i g h e r t h a n t h e f r e e z i n g p o i n t o f w a t e r . E x p e r i m e n t a l l y , t h i s \ c o n d i t i o n was r e a l i s e d i n a s p e c i a l l y d e s i g n e d c e l l , o f a p p r o x i m a t e c a p a c i t y 20ml., machined out o f s t a i n l e s s s t e e l . I t c o n t a i n e d a M a t h i e s o n 316 h i g h -p r e s s u r e n e e d l e - v a l v e r e g u l a t o r and a l e n g t h of l / 8 " s t a i n l e s s s t e e l g a s -i n l e t t u b e . The c e l l c o u l d be c o n n e c t e d e i t h e r t o a c y l i n d e r o f t h e h y d r a t e f o r m e r by monel-metal c o u p l i n g s o r t o a g l a s s vacuum m a n i f o l d b y means o f "Swagelok" m e t a l - t o - g l a s s c o m p r e s s i o n f i t t i n g s . I n t h e n e x t c h a p t e r , a b r i e f ; t t e n t i o n w i l l be made of t h e use o f t h i s c e l l f o r f o r m i n g 48 and e s t i m a t i n g some o f t h e c l a t h r a t e s s t u d i e d i n t h i s work« 4.2. The NMR s p e c t r o m e t e r E a r l i e r work on t h e d e u t e r a r e d h y d r a t e s o f propane and t e t r a - i -amyl ammonium f l u o r i d e was done on a V a r i a n 4200/4300 d u a l purpose s p e c t r o -m e t e r e q u i p p e d w i t h a 56.4 M c/sec. c r y s t a l r f . o s c i l l a t o r . L a t e r on, an e x c l u s i v e "wide l i n e " s p e c t r o m e t e r o p e r a t i n g a t 30Mc/sec. was made a v a i l -a b l e , and t h e m a j o r b u l k of t h e e x p e r i m e n t s h e r e i n r e p o r t e d was c a r r i e d o u t w i t h t h i s s p e c t r o m e t e r . F o r measurements of T^, the n u c l e a r 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 , however, the 30 M c / s e c . r f . o s c i l l a t o r was f o u n d i n a d e q u a t e s i n c e o n l y v e r y moderate a m p l i t u d e s o f t h e f i e l d were a v a i l -a b l e , and c o n s e q u e n t l y a 16 M c / s e c . r f . t r a n s m i t t e r , w h i c h had a l a r g e r power o u t p u t and a w i d e - r a n g e p r e c i s i o n a t t e n u a t o r , was p r e f e r r e d . A b l o c k d i a g r a m o f the s p e c t r o m e t e r i s shown i n F i g . 11. The V-4331 probe c o n t a i n e d m u t u a l l y o r t h o g o n a l t r a n s m i t t e r and r e c e i v e r c o i l s 6 T and p i c k e d up r e s o n a n c e s i g n a l s by t h e method o f n u c l e a r i n d u c t i o n . The probe a l s o c o n t a i n e d " p a d d l e s " . These were m e c h a n i c a l d e v i c e s t o improve t h e o r t h o g o n a l i t y , o r " b a l a n c e " , o f t h e t r a n s m i t t e r and r e c e i v e r c o i l s and t o i n t r o d u c e a " l e a k a g e " v o l t a g e i n t h e r e c e i v e r c o i l i n phase w i t h t h e d e s i r e d mode — a b s o r p t i o n o r d i s p e r s i o n — of t h e o u t p u t s i g n a l . A 10 M c / s e c . c r y s t a l , t r i p l e d i n f r e q u e n c y by a t w o - s t a g e o s c i l l a t o r i n t h e V-4310C u n i t , f e d t h e t r a n s m i t t e r c o i l o f t h e p r o b e . A h i g h l y homogeneous m a g n e t i c f i e l d , H q, was p r o v i d e d b y a V a r i a n V-4007 s i x - i n c h w a t e r - c o o l e d e l e c t r o m a g n e t w i t h a T j " p o l e gap. Temperature r e g u l a t i o n o f t h e c o o l a n t + The i n s t a l l a t i o n and a s s e m b l y o f H h i s s p e c t r o m e t e r was due t o D. Ware (113) . V-4310 C RECEIVER R F amplifier I F amplifier detector gain local oscillator 30 Mc/sec TRANSMITTER fine attenu-ator RF amplifier coarse attenu-ator RF oscillator crystal —m— MODULATION INPUT PREAMPLIFIER z V-4007 6-INCH V-4331 WIDE-LINE PROBE ELECTRO-MAGNET V -4295 SELECTOR PANEL K 3—*E scope presentation selector filters gain V - 4 2 7 0 B OUTPUT CONTROL UNIT amplifier audio _^synchro-verter fir twin-T filter frequency selector D.C. amplifier low-pass filter synchroverter phasing V-4250 B SWEEP UNIT coarse attenuator PEN-AND-CHART RECORDER synchroverter drive amplifier scope phasing fine attenuator sweep amplifier U phase inverter frequency selector audio amplifier V-2200A POWER SUPPLY f > gate tubes chopper amplifier ~ ~ T ~ reference battery reference resistors rectifier V -4280A FIELD SCANNING UNIT helipot scan resistor battery motor scan selector FIGURE II. BLOCK DIAGRAM OF 30 Mc/sec WIDE LINE NMR SPECTROMETER w a t e r , d e v i a t i n g l e s s t h a n 1 C., was p r o v i d e d b y a t h e r m i s t o r - a c t u a t e d e l e c t r o n i c c o n t r o l l e r . I n r e c o r d i n g w i d e - l i n e d e r i v a t i v e s i g n a l s o f p r o t o n and f l u o r i n e a b s o r p t i o n , t h e r f . t r a n s m i t t e r was h e l d a t a c o n s t a n t f r e -quency w h i l e t h e H f i e l d was v a r i e d b y a f i e l d - s c a n u n i t ( V - 4 2 8 0 A ) . T h i s u n i t c o n s i s t e d o f a s t a n d a r d 1.34 v o l t c e l l , a s e t of r e s i s t o r s , and a m o t o r - d r i v e n h e l i p o t . A s e t o f g e a r s g o v e r n e d the speed o f t h e motor. A v o l t a g e - d i v i d e r had been w i r e d i n p a r a l l e l w i t h t h e h e l i p o t i n such a way t h a t t he o u t p u t v o l t a g e was z e r o a t t h e m i d - p o i n t o f t h e h e l i p o t and r a n to p o s i t i v e and n e g a t i v e v a l u e s on e i t h e r s i d e . When t h e h e l i p o t was s w i t c h e d on t o r e v o l v e , t h e s e v o l t a g e changes were s e n s e d by a h i g h - g a i n DC chopper a m p l i f i e r s i t u a t e d i n t h e V-2200A power s u p p l y u n i t , w h i c h t h e n v a r i e d t h e c u r r e n t t o t h e e l e c t r o m a g n e t and t h e r e f o r e t he v a l u e of H q. I n t h i s way one was a b l e t o scan H o , a t v a r i o u s r a t e s , o v e r a range o f about 15 gauss on e i t h e r s i d e o f the c e n t r e o f r e s o n a n c e . I n o r d e r t h a t t h e re s o n a n c e s i g n a l p i c k e d up by t h e r e c e i v e r c o i l be an A. C , s i g n a l , a c o n d i t i o n o f t e n p r e f e r r e d f o r p u r p o s e s o f a m p l i f i -c a t i o n , t h e main magnet f i e l d was "swept" b y a s m a l l a u d i o m o d u l a t i o n f i e l d Hffl = h ^ S i n u ^ t . Sweep m o l d u l a t i o n was a c h i e v e d b y a p a i r of H e l m h o l t z c o i l s , mounted c o n c e n t r i c t o t h e p o l e s o f t h e e l e c t r o m a g n e t and d r i v e n b y an a u d i o f r e q u e n c y sweep o s c i l l a t o r V-4250B and a on e - s t a g e p u s h - p u l l sweep a m p l i f i e r V-4240B ( t h i s u n i t i s n o t numbered i n F i g . l l ) . The r e c e i v e r n e t w o r k , V-4310C, was p h a s e - s e n s i t i v e w i t h r e s p e c t t o t h e m o d u l a t i o n f r e q u e n c y , y j ^ , and was e q u i p p e d w i t h an o u t p u t a u d i o a m p l i f i e r and a c o h e r e n t d e t e c t o r . The c o h e r e n t , o r " p h a s e - s e n s i t i v e " , d e t e c t o r i s one o f t h e b e s t l i n e a r s i g n a l d e t e c t o r s . I d e a l l y , t h i s t y p e o f d e t e c t o r m u l t i p l i e s t h e i n c o m i n g s i g n a l v o l t a g e by a s i n u s o i d a l r e f e r e n c e f u n c t i o n o f a s i n g l e f r e q u e n c y and t h e r e q u i r e d p h a s e , and t h e n 51 f i l t e r s t o a c h i e v e t h e t i m e - a v e r a g e o v e r many c y c l e s of r e f e r e n c e f r e q u e n c y . The r e f e r e n c e f r e q u e n c y i s o r d i n a r i l y t h e same as t h e f u n d a m e n t a l o f t h e s i g n a l . I d e a l l y , a g a i n , t h i s d e t e c t o r w i l l p a s s o n l y t h o s e f r e q u e n c y components w i t h i n the band p a s s o f t h e o u t p u t f i l t e r w h i c h a r e c e n t r e d around u^. The low-pass o u t p u t f i l t e r may be o f t h e L-R o r t h e R-C t y p e , and t h e l a t t e r was u s e d ( r e s p o n s e t i m e constant=RC) i n t h e 30 Mc/sec. s p e c t r o m e t e r . Response c o u l d be v a r i e d b y means o f a s e t o f r e s i s t o r s f o u n d i n t h e R C - c i r c u i t . These components were i n c o r p o r a t e d i n t h e V-4270B o u t p u t c o n t r o l u n i t . The a m p l i f i e d a b s o r p t i o n s i g n a l f r o m the o u t p u t o f t h e r e c e i v e r c o u l d e i t h e r be d i s p l a y e d on a Du Mont '304' o s c i l l o s c o p e , o r f e d t h r o u g h t h e o u t p u t c o n t r o l u n i t t o a low t i m e - c o n s t a n t m i l l i v o l t p e n - a n d - c h a r t r e c o r d e r t o o b t a i n f i r s t - d e r i v a t i v e t r a c e s . F r e q u e n c y m o d u l a t i o n o f t h e t r a n s m i t t e r f o r c a l i b r a t i o n p u r p o s e s was p o s s i b l e by c o n n e c t i n g a H e w l e t t - P a c k a r d a u d i o - o s c i l l a t o r t o t h e " m o d u l a t i o n i n p u t " o f t h e V-4310C u n i t . The a u d i o - o s c i l l a t o r c o u l d i n t u r n be c a l i b r a t e d a c c u r a t e l y by an e l e c t r o n i c f r e q u e n c y c o u n t e r . F o r t h e a d i a b a t i c f a s t passage measurements of T^ c a r r i e d o u t i n t h i s work, a H e w l e t t - P a c k a r d (NO.202A) square-wave g e n e r a t o r was u t i l i z e d . 4.3 Low Temperature Equipment A l l s p e c t r a a t 77°K. were r e c o r d e d by i m m e r s i n g s e a l e d sample t u b e s i n l i q u i d n i t r o g e n i n s i d e a s i m p l y d e s i g n e d dewar v e s s e l w i t h a wicle t o p - p o r t i o n and a t u b u l a r b o t t o m w h i c h f i t t e d i n s i d e t h e r e c e i v e r c o i l o f t h e probe . Thermal e q u i l i b r i u m was a t t a i n e d i n about h a l f an h o u r , d u r i n g w h i c h p e r i o d t h e dewar c o o l e d down u n i f o r m l y t o t h e t e m p e r a t u r e o f l i q u i d n i t r o g e n . I t was n e c e s s a r y t o r e f i l l t h e dewar w i t h the c o o l a n t o n l y once 52 4 \ >. p e r s p e c t r u m ( a p p r o x i m a t e l y 1 h o u r ) . The d^war v e s s e l was s i l v e r e d o n l y where t h e r e w o u l d be no i n t e r f e r e n c e w i t h t h e r f . f i e l d - i n t h e probe. F o r t e m p e r a t u r e s above 77°K., a c o l d - g a s f l o w v a r i a b l e t e m p e r a t u r e Dewar a s s e m b l y p i c t u r e d i n F i g . 12 was u s e d . The a p p a r a t u s had been d e v i s e d t o c a r r y a r a p i d c u r r e n t o f c o l d gas p a s t t h e sample and ex h a u s t i t t o t h e s u r r o u n d i n g s o u t s i d e the magnet gap. A s t r e a m of d r y n i t r o g e n , w i t h d r a w n f r o m a c y l i n d e r a t r e g u l a t e d r a t e s by means o f a n e e d l e v a l v e , was c h i l l e d b y passage t h r o u g h a co p p e r s p i r a l immersed i n a c o o l a n t ( l i q u i d n i t r o g e n f o r l o w t e m p e r a t u r e s ; d r y i c e - a c e t o n e s l u s h f o r t e m p e r a t u r e s above 200°K) and b l o w n on the sample by means of s i l v e r e d dewar l e a d s . F l o w ; o f t h e c o o l a n t gas was m o n i t o r e d by means of a c a l i b r a t e d f l o w - m e t e r ; by r e g u l a t i n g t h e n e e d l e v a l v e t o a c h i e v e a c o n s t a n t f l o w - m e t e r l e v e l , i t was p o s s i b l e t o r e p r o d u c e t e m p e r a t u r e s b e l o w 200°K. a c c u r a t e t o w i t h i n ±2 d e g r e e s . T h r o u g h -o u t a s e r i e s o f r u n s , sample t e m p e r a t u r e s n e a r t h e t o p and b o t t o m ends o f t h e sample tube i n s i d e t h e dewar a p p a r a t u s were m o n i t o r e d by means of c o p -p e r - c o n s t a n t a n t h e r m o c o u p l e s c o n n e c t e d t o a Leeds and N o r t h r u p 10 mv. r e c o r d e r . S e v e r a l t i m e s d u r i n g each s p e c t r u m , t h e v o l t a g e o f each thermo-c o u p l e was r e c o r d e d w i t h a " R u b i c o n " p o t e n t i o m e t e r , and t h e c o r r e s p o n d i n g t e m p e r a t u r e r e a d o f f a c a l i b r a t i o n c h a r t . THERMOCOUPLE LEAD F i g . 12 V a r i a b l e Temperature Dewar Assembly co CHAPTER V MATERIALS AND METHODS 5.1 M a t e r i a l s Compressed c a r b o n t e t r a f l u o r i d e o f 98.0+ mole$ p u r i t y , p u r c h a s e d f r o m C o l u m b i a O r g a n i c C h e m i c a l s Co., I n c . , and r e s e a r c h grade propane o f 99.9+ mole % p u r i t y , p u r c h a s e d f r o m P h i l l i p s P e t r o l e u m Company, were u s e d s t r a i g h t f r o m t h e i r c o n t a i n e r s w i t h o u t f u r t h e r p u r i f i c a t i o n . A c l y i n d e r o f r e g u l a r grade s u l p h u r h e x a f l u o r i d e , o b t a i n e d f r o m M a t h i e s o n o f Canada, L t d . , was r e v e a l e d by mass s p e c t r o m e t r i c a n a l y s i s t o be o n l y 96$ p u r e , and was 31 f u r t h e r p u r i f i e d b y s u b l i m i n g i t i n t o a c o n t a i n e r immersed i n a d r y i c e -a c e t o n e b a t h and pumping on t h e s u b l i m a t e f o r a few h o u r s . E t h y l e n e o x i d e , a l s o o b t a i n e d f o r m M a t h i e s o n o f Canada, L t d . , was of u n s p e c i f i e d p u r i t y , and was t r e a t e d b e f o r e use by f r e e z i n g the m a n u f a c t u r e r ' s specimen i n t o a c o l d t r a p and pumping on t h e s o l i d t o remove n o n - c o n d e n s a b l e i m p u r i t i e s . The 55 t h r e e monohalomethanes and d i c h l o r o m e t h a n e were Eastman K o d a t " S p e c t r o G r a d e " c h e m i c a l s o f p u r i t y i n e x c e s s of 99.5$ and were u s e d as s u c h . F r e s h l y -d i s t i l l e d specimens of Eastman Kodak 'w h i t e l a b e l ' t r i n - b u t y l a m i n e , n - b u t y l i o d i d e , t r i i - a m y l a m i n e , and i - a m y l i o d i d e were u s e d f o r t h e p r e p a r a t i o n o f t h e q u a t e r n a r y a l k y l ammonium i o d i d e s . To c o n v e r t t h e s e t o t h e c o r r e s p o n d i n g f l u o r i d e s , a sample o f s o l i d s i l v e r f l u o r i d e , o b t a i n e d f r o m A.D. McKay, I n c . , N.Y., U.S.A., and e s t i m a t e d 95$ pur e by g r a v i m e t r i c a n a l y s i s , was u s e d . DgO of 99.8$ p u r i t y , s u p p l i e d by G e n e r a l Dynamic C o r p o r a t i o n ( L i q u i d C a r b o n i c s D i v i s i o n ) and d o u b l e d i s t i l l e d w a t e r were u s e d f o r t h e f o r m a t i o n of t h e v a r i o u s c l a t h r a t e s . 5.2 P r e p a r a t i o n and E s t i m a t i o n ( a ) The P r e s s u r e C e l l Method: S i m p l e and d e u t e r a t e d h y d r a t e s of CF^, S F g , CgHg, and CgH^O — t h e 62 ' t r u e 'gas' h y d r a t e s — were formed i n t h e s t a i n l e s s s t e e l p r e s s u r e c e l l ( d e s c r i b e d i n s e c t i o n 4.1 o f t h i s t h e s i s ) by b u b b l i n g t h e gaseous g u e s t component i n t o t h o r o u g h l y degassed H^O o r D^O u n t i l a gas p r e s s u r e of 5 t o 6 atmospheres a c c u m u l a t e d i n s i d e t h e c e l l . The h y d r a t e formed as a snowy mass of c r y s t a l s upon p r o l o n g e d c o o l i n g o f t h e c e l l a t 0°C. The u n c l a t h r a t e d g u e s t was t h e n s i m p l y a l l o w e d t o escape f r o m t h e c e l l . Maximum r a t i o s o f g u e s t - t p - h o s t were o b t a i n e d i n t h e s e p r e p a r a t i o n s . The mass of c r y s t a l s was c r u s h e d w i t h an i c e - p i c k , p a c k e d i n t o la. 10mm. O.D., n.m.r. sample, t u b e and s e a l e d . The s e a l e d sample was a n n e a l e d a t -5°C i n a r e f r i g e r a t o r f o r a few weeks. Exchange o f DgO w i t h a t m o s p h e r i c m o i s t u r e was p r e v e n t e d by h a n d l i n g t h e samples i n a n i t r o g e n - s a t u r a t e d d r y - b o x . Q u a n t i t a t i v e e s t i m a t e s o f the c o m p o s i t i o n o f t h e s e h y d r a t e s were made by a s t a t i c method d e s c r i b e d b e l o w . The o v e r a l l l i m i t s o f e x p e r i m e n t a l e r r o r i n v o l v e d i n the v a r i o u s measurements was ±10$ j and r e p r o d u c i b i l i t y f o r 56 i a p a r t i c u l a r h y d r a t e specimen was a l w a y s w e l l w i t h i n t h e s e l i m i t s . ' A sample of t h e gas h y d r a t e t o be e s t i m a t e d was a l l o w e d t o de.com-pose i n t h e h i g h p r e s s u r e c e l l by l e t t i n g i t warm up t o room t e m p e r a t u r e ; The p r e s s u r e o f t h e gaseous component e v o l v e d was t r a n s m i t t e d t h r o u g h a s m a l l - b o r e g l a s s t u b i n g t o a m e r c u r y manometer. The volume of t h e manometer s e c t i o n of t h e m a n i f o l d was known by a p r e v i o u s c a l i b r a t i o n . The p r e s s u r e of t h e gas was measured by r e a d i n g t h e h e i g h t o f t h e m e r c u r y column t o 10.05cm. Gas p r e s s u r e s were t y p i c a l l y of t h e o r d e r of 0,2 t o 0.4 atmos^. p h e r e s , and were c o r r e c t e d f o r t h e v a p o u r p r e s s u r e o f w a t e r a t t h e tempera-t u r e of t h e measurements. The c o r r e c t e d p r e s s u r e s were t h e n c o n v e r t e d t o volumes a t STP, assuming t h e g a s e s to behave i d e a l l y . A f t e r c o m p lete d e c o m p o s i t i o n of t h e h y d r a t e i n t h e c e l l , e n s u r e d by no f u r t h e r change i n manometric l e v e l , t h e r e s i d u a l w a t e r i n t h e p r e s s u r e c e l l was d r a i n e d o f f q u a n t i t a t i v e l y i n t o a t a r e d b o t t l e and w eighed. The volume of gaseous g u e s t and t h e w e i g h t of w a t e r were c o n v e r t e d t o mokes, and t h e m o l e c u l a r r a t i o s a r r i v e d a t f o r t h e h y d r a t e s . T a b l e V I I Some T y p i c a l E x p e r i m e n t a l V a l u e s o f C l a t h r a t e C o m p o s i t i o n G u e s t M o l e c u l e Volume o f Gaseous G u e s t , m l . a t STP M o l e s o f G u e s t W e i g h t o f W a t e r , gms. M o l e s of W a t e r C l a t h r a t e C o m p o s i t i o n C F 4 131.0 0.0056 0.707 0.0393 ( C F 4 ) ^ 7 H 2 0 S F 6 112.0 0.0050 1.544 0.0857 ( S F 6 ) ~ 1 7 H 2 0 C 3 H 8 141.3 0.0063 2.030 0.1127 ( C g R g M S I ^ O W 80.0 0.0035 0.502 0.0274 (C2H 40)~7H 2Q The d a t a shown i n T a b l e V I I a r e r e p r e s e n t a t i v e of t h e method u s e d . Heavy w a t e r h y d r a t e s were e s t i m a t e d by an i d e n t i c a l method, and t h e i r c o m p o s i t i o n s a g r e e d t o w i t h i n 10$ w i t h t h o s e t a b u l a t e d f o r t h e c o r r e s p o n d i n g o r d i n a r y h y d r a t e s . (b) D i r e c t F o r m a t i o n i n Sample Tubes; Where c l a t h r a t e f o r m i n g g u e s t s were a v a i l a b l e as l i q u i d s above 0°C., e.g., t h e t h r e e monohalomethanes and d i c h l o r o m e t h a n e , c l a t h r a t e specimens w i t h c o m p o s i t i o n s c o r r e s p o n d i n g t o o v e r 90$ o f f i l l e d c a v i t i e s were o b t a i n e d b y s l o w l y f r e e z i n g s t o i c h i o m e t r i c m i x t u r e s o f t h e gu e s t and t h o r o u g h l y d e g a s s e d DgO a t 0 ° C , i n s e a l e d sample t u b e s . C r y s t a l l i t e s o f u n i f o r m c o m p o s i t i o n c o u l d be formed i n t h e sample t u b e s when warming up and c o o l i n g down o p e r a t i o n s were done s e v e r a l t i m e s w i t h t h e samples n e a r l y a t t h e i r m e l t i n g p o i n t s . These were t h e n a n n e a l e d a t - 5 ° C , f o r a few weeks b e f o r e t h e i r s p e c t r a were r e c o r d e d . These c l a t h r a t e s were e s t i m a t e d i n an a n a l y s i s t r a i n a c c o r d i n g t o 34 a method o f due t o B a r r e r and E u z i c k a , u s i n g a c u r r e n t o f d r y n i t r o g e n as t h e c a r r i e r gas t o s e p a r a t e t h e o r g a n i c l i q u i d f r o m w a t e r . T h i s method d i f f e r e d f r o m t h e o r i g i n a l one, however, i n t h a t a c o a r s e - g r a i n e d sample of s i l i c a g e l was u s e d i n s t e a d o f z e o l i t e as a m o l e c u l a r s i e v e f o r w a t e r * The a b s o l u t e amounts of w a t e r and o r g a n i c l i q u i d i n a h y d r a t e c o u l d be d e t e r m i n e d t o ±5$ by t h i s method. C o m p o s i t i o n s o f ( C H g C l ) ~ 7 . 9 DgO, ( C H g B r ) ~ 8 . 1 5 DgO, ( C H g l ) ~ 1 8 . 0 D 2 0 , and ( C H 2 C l 2 ) - 7 . 7 DgO were a r r i v e d a t f o r t h e f o u r c l a t h -r a t e specimens a n a l y s e d by t h i s method. ( c ) T e t r a i - A r a y l Ammonium F l u o r i d e H y d r a t e : The p r e p a r a t i o n o f t h i s c l a t h r a t e c o n s i s t e d o f two s t e p s . F i r s t , t e t r a i - a m y l ammonium i o d i d e was made fr o m f r e s h l y - d i s t i l l e d samples o f 114 t r i i - amylamine and i - a m y l i o d i d e a c c o r d i n g t o K r a u s and F u o s s . The p a l e y e l l o w q u a t e r n a r y s a l t was r e p e a t e d l y r e c r y s t a l l i z e d f r o m a 1:8 a c e t o n e -w a t e r m i x t u r e and d r i e d i n vacuo . ' : a t 60 C. i n a d r y i n g p i s t o l . I n t h e second s t e p , t h e i o d i d e was suspended i n an e x c e s s o f 99.8$ DgO and shaken w i t h a s t o i c h i o m e t r i c w e i g h t o f AgF ( t h i s method was a 16 m o d i f i c a t i o n o f t h e one r e p o r t e d b y M c M u l l a n and J e f f r e y , i n w h i c h a m i x t u r e o f BaFg and Ag^SO^ was u s e d t o produce AgF i n s i t u ) . The p r e c i p i -t a t e d A g l was f i l t e r e d o f f , and t h e f i l t r a t e c o n c e n t r a t e d t o a s y r u p y c o n s i s t e n c y i n a d r y box by means o f an i n f r a - r e d r a d i a t o r . L a r g e , c o l o u r -l e s s c r y s t a l s o f t h e d e u t e r a t e d h y d r a t e o f t e t r a i - amyl ammonium f l u o r i d e s e p a r a t e d upon c o o l i n g t h e s y r u p t o 5°C. They were f i l t e r e d o f f and r e -c r y s t a l l i z e d f r o m a s a t u r a t e d h e a v y w a t e r s o l u t i o n . The d r i e d c r y s t a l s , m.p.~30°C, were t h e n sampled i n a b s o l u t e l y ^ d r y g l a s s t u b e s and s e a l e d . The g l a s s w a l l s o f t h e sample t u b e s had been p r e v i o u s l y t h i n n e d o u t t o enhance t h e p r o t o n p a c k i n g f a c t o r o f t h e sample. The c o m p o s i t i o n o f t h e c l a t h r a t e was d e t e r m i n e d ( b y d e h y d r a t i o n t o c o n s t a n t w e i g h t ) t o be ( i - C g H ^ ^ N F . S S D g O . I t was p o s s i b l e t o fo r m c r y s t a l s o f t h e o r d i n a r y h y d r a t e , c o r r e s p o n d i n g r o u g h l y t o t h e same c o m p o s i t i o n , by u s i n g H^O i n l i e u of D^O i n t h e above p r e p a r a t i o n . ( d ) T e t r a n - B u t y l ammonium F l u o r i d e H y d r a t e : The method of p r e p a r a t i o n was i d e n t i c a l t o t h e one d e l i n e a t e d above f o r t h e i - a m y l a n a l o g u e . D i s t i l l e d t r i n - b u t y l a m i n e and n - b u t y l i o d i d e were u s e d as t h e s t a r t i n g p r o d u c t s t o o b t a i n the q u a t e r n a r y i o d i d e . T h i s was t h e n t r e a t e d w i t h s t o i c h i o m e t r i c q u a n t i t i e s i n two s e p a r a t e l o t s — one i n HgO s o l u t i o n and t h e o t h e r i n DgO s o l u t i o n — t o o b t a i n o r d i n a r y and deu-t e r a t e d m o d i f i c a t i o n s of t h e c l a t h r a t e . I n b o t h c a s e s , c r y s t a l s formed w i t h a v e r y r a p i d r a t e o f c r y s t a l l i z a t i o n . The o r d i n a r y h y d r a t e , m.p., 25°C, was f o u n d t o c o n t a i n 31.5 H p 0 , w h i l e t h e d e u t e r a t e d h y d r a t e was a n a l y s e d t o c o n t a i n 31.8 DgO. The l a t t e r m e l t e d a t —24°C. 5.3 S p e c t r o m e t e r C a l i b r a t i o n s : ( a ) C a l i b r a t i o n of t h e s c a n n i n g f i e l d : The s c a n - r a t e o f t h e main m a g n e t i c f i e l d was c a l i b r a t e d as a f u n c t i o n of d i s t a n c e a l o n g t h e b a s e l i n e of t h e r e c o r d e r c h a r t by t u n i n g the magnet t o t h e r e s o n a n c e p o s i t i o n o f a sample of l i q u i d w a t e r and t h e n f r e q u e n c y - m o d u l a t i n g t h e t r a n s m i t t e r w i t h a H e w l e t t - P a c k a r d a u d i o o s c i l l a t o r . The r e c o r d e d s i g n a l c o n s i s t e d of a s t r o n g c e n t r a l peak f l a n k e d by a s e r i e s o f e q u i - s p a c e d s i d e b a n d s . By m e a s u r i n g t h e s p a c i n g between a p a r t i c u l a r s e t o f s i d e b a n d s i n c e n t i m e t e r s and by d e t e r m i n i n g the e x a c t f r e q u e n c y o f t h e a u d i o o s c i l l a t o r w i t h an e l e c t r o n i c f r e q u e n c y c o u n t e r , t h e s c a n n i n g f i e l d was c a l i b r a t e d i n gauss p e r cm. o f c h a r t p a p e r . Use was made of the e q u a t i o n 2rrv = u) = Y^o i n c o n v e r t i n g f r e q u e n c y u n i t s t o g a u s s . I n a few i n s t a n c e s where h i g h r e s o l u t i o n s p e c t r a were r e c o r d e d , t h e n a r r o w s i g n a l of t h e e x p e r i m e n t a l sample i t s e l f g e n e r a t e d t h e s i d e b a n d s n e c e s s a r y f o r c a l i b r a t i o n o f t h e s c a n n i n g f i e l d , (b) C a l i b r a t i o n o f H m , t h e sweep a m p l i t u d e s : T h i s was done f o r each s p e c t r o m e t e r s e t t i n g o f t h e sweep f i e l d by s l o w l y s c a n n i n g t h r o u g h t h e s i g n a l f r o m a sample of w a t e r — t h e o r e t i c a l l y a v e r y narrow l i n e — and m e a s u r i n g t h e l i n e w i d t h . T h i s w i d t h was t a k e n t o be H m , t h e sweep a m p l i t u d e , i n g a u s s . The sweep f i e l d f r e q u e n c y was k e p t c o n s t a n t a t 80 c p s . f o r a l l w i d e - l i n e e x p e r i m e n t s ( ( ^ = 2nX80 sec 5.4 Measurements ( a ) L i n e w i d t h s : The l i n e w i d t h , 6H, i s d e f i n e d as the w i d t h between s l o p e extrema o f a r e s o n a n c e a b s o r p t i o n c u r v e , and c o r r e s p o n d s t o the h o r i z o n t a l d i s t a n c e between p o s i t i v e and n e g a t i v e peaks o f a d e r i v a t i v e c u r v e . T h i s d i s t a n c e was measured f o r each w i d e - l i n e s p e c t r u m and r e c o r d e d i n u n i t s o f g a u s s . Where a b s o r p t i o n mode s i g n a l s were r e c o r d e d under c o n d i t i o n s o f h i g h r e s o l u t i o n , t h e l i n e shape ap p e a r e d L o r e n t z i a n i n t h e c e n t r a l r e g i o n . F o r s u c h c a s e s , a d e f i n i t i o n o f 5H w h i c h w o u l d be c o n s i s t e n t . w i t h t h e above 115 has been r e c e n t l y g i v e n b y S m i t h . F o l l o w i n g t h i s d e f i n i t i o n , t h e s i g n a l w i d t h a t h a l f t h e s i g n a l a m p l i t u d e was measured a c c u r a t e l y and m u l t i p l i e d 115 by a f a c t o r o f 0.6 b e f o r e b e i n g e x p r e s s e d as &H. F o l l o w i n g S m i t h a g a i n , m o d u l a t i o n a m p l i t u d e s , H , were chosen t o be <0.36H f o r t h e w i d e - l i n e * m s p e c t r a i n o r d e r t h a t t h e e r r o r i n gH be l i m i t e d t o 5$ o r l e s s . ( b) Second Moments: F o r a d e r i v a t i v e t r a c i n g o f t h e a b s o r p t i o n wide l i n e , t h e second moment e q u a t i o n g i v e n b y (13) may be r e - w r i t t e n as " +" f ' ( H ) ( H r H j 3 d H 3 ° f ( H ) ( H - H Q ) d H —00 where f '(H) i s t h e a b s o r p t i o n i n t e n s i t y i n a r b i t r a r y u n i t s . A c o n v e n i e n t n u m e r i c a l e x p r e s s i o n t h a t c o r r e s p o n d s t o the above i s 3 ( M 2 ) - ( s c a n ) 2 y n ( 3 0 ) VP ''EXP 3 £ny where y & i s t h e v a l u e o f f ' ( H ) , n u n i t s a l o n g f r o m t h e c e n t r e o f t h e l i n e , and " s c a n " i s t h e c a l i b r a t e d w i d t h o f a u n i t i n g a u s s . Any s m a l l b r o a d e n i n g o f the e x p e r i m e n t a l s p e c t r a due t o " o v e r -m o d u l a t i o n " by t h e H ^ f i e l d was c o r r e c t e d a c c o r d i n g t o t h e w e l l k n o w n 61 e x p r e s s i o n o f Andrew*"^: ( ^ T R T j E ^ Z P ' W - i H 2 ( 3 1 ) . A " F o r t r a n " programme, w r i t t e n by Mr. W.R. J a n z e n f o r the IBM 7040 computer a t t h i s U n i v e r s i t y , employed t h e summation e x p r e s s i o n (30) and t h e m o d u l a t i o n c o r r e c t i o n (31) t o y i e l d a c c u r a t e second moments fr o m t h e d e r i v a t i v e t r a c i n g s . ( c ) A c t i v a t i o n E n e r g i e s f r o m L i n e w i d t h D a t a : To o b t a i n a c c u r a t e e s t i m a t e s o f a c t i v a t i o n e n e r g i e s f r o m changes i n l i n e w i d t h s , e q u a t i o n ( 2 4 ) , s e c t i o n 3.7, was u s e d . A F o r t r a n I I computer 86 programme t o s o l v e t h i s e q u a t i o n was o b t a i n e d t h r o u g h the c o u r t e s y of D r . G.W. S m i t h of t h e G e n e r a l M o t o r s C o r p o r a t i o n , M i c h i g a n , U.S.A., and r e - w r i t t e n i n F o r t r a n IV l a n g u a g e , w i t h a p p r o p r i a t e m o d i f i c a t i o n s f o r t h e IBM 7040 computer u s e d t h r o u g h o u t t h i s work. The programme s o l v e d e q u a t i o n ( 2 4 ) f o r a number o f l i n e w i d t h s , &H^, o b t a i n e d a t v a r i o u s t e m p e r a t u r e s , T^, i n t h e l i n e - n a r r o w i n g r e g i o n , made a l e a s t s q u a r e s f i t of t h e s e d a t a p o i n t s -E/ t o v. = v e 'RT, e v a l u a t e d v and E f r o m t h e s t r a i g h t l i n e f i t , and p r i n t e d -C 00 co out a t a b l e of 6H v s . T f o r t h e b e s t f i t t o d a t a . The m o d i f i e d programme a p p e a r s i n i t s e n t i r e t y i n A p p e n d i x I . ( d ) T^ measurements by S i g n a l s a t u r a t i o n : These measurements i n v o l v e d a m o d i f i c a t i o n o f t h e p r o g r e s s i v e 117 s a t u r a t i o n t e c h n i q u e employed by McE.all and D o u g l a s s . The method i n -v o l v e d p l o t t i n g t h e a m p l i t u d e of one o f t h e d e r i v a t i v e s i g n a l extrema 68 118 a g a i n s t i n t e n s i t y o f f i e l d p r o d u c i n g a b s o r p t i o n . E x i s t i n g t h e o r i e s ' have shown t h a t the s i g n a l a m p l i t u d e i s m a x i m i z e d when t h e s a t u r a t i o n f a c t o r y'Tl^TjTg = 1 and t h a t f u r t h e r i n c r e a s e i n p r o d u c e s s u p p r e s s i o n o f t h e s i g n a l . F o r t h e V a r i a n 4210A ( l 6 M c / s e c . , ) r f . t r a n s m i t t e r u s e d i n t h i s e x p e r i m e n t , t h e r f , f i e l d m e ter r e a d i n g u, ( i n micro-amperes) was c a l i b r a t e d 119 and f o u n d d i r e c t l y p r o p o r t i o n a l t o by J a n z e n who u s e d a method d e s -120 c r i b e d b y A n d e r s o n . A n e s t a b l i s h e d a s s u m p t i o n w h i c h h o l d s n o t o n l y f o r p u r e l y ' L o r e n t z i a n ' and ' G a u s s i a n ' l i n e s h a p e s , b u t a l s o f o r t h e n o t i n -f r e q u e n t c a s e s o f l i n e shapes " i n t h e m i d d l e ground", i s t h a t Tg 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 a c o n s t a n t t i m e s t h e l i n e w i d t h as l o n g as t h e l i n e shape i t s e l f does n o t change d r a s t i c a l l y o v e r t h e t e m p e r a t u r e r e g i o n b e i n g i n v e s t i g a t e d . T h e r e f o r e , one may j u s t i f i a b l y w r i t e t h e s a t u r a t i o n f a c t o r as 2 T 1 h m = c o n s t a n t ( 3 2 ) on-U s i n g , t h e n , a sample whose l i n e w i d t h and T^ a r e a c c u r a t e l y known a t a p a r t i c u l a r t e m p e r a t u r e , one s h o u l d be a b l e t o e s t i m a t e t h e c o n s t a n t i n (32) and t h e n p r o c e e d t o d e t e r m i n e p, f o r an e x p e r i m e n t a l sample a t any t e m p e r a -t u r e as t h e r f , c u r r e n t c o r r e s p o n d i n g t o maximum e x c u r s i o n o f t h e o u t p u t meter l e v e l . A t each t e m p e r a t u r e o f t h e T^ measurement, t h e a c c u r a t e l i n e -w i d t h o f the sample was known, so t h a t T p t h e o n l y unknown i n ( 3 2 ) , was e a s i l y c a l c u l a t e d . F i g . 21, c h a p t e r V I I , shows t h e a c t u a l r e s u l t s f r o m a T^ measurement. V a l u e s o f T^ r e p o r t e d a r e r e l a t i v e t o t h a t o f a v e r y p u r e sample o f s t e a r i c a c i d — a r e l i a b l e " s t a n d a r d " u s e d i n t h i s l a b o r a t o r y — w h i c h had a c o n v e n i e n t and r e p r o d u c i b l e T^ of 7.6 s e c , a t 77°K, as d e t e r m i n e d by a d i r e c t method o f " s i g n a l r e c o v e r y a f t e r s a t u r a t i o n " . F u r t h e r , t h e c h o i c e o f t h i s sample as t h e T^ s t a n d a r d was a d v e n t i t i o u s i n t h a t i t had a l i n e shape v e r y s i m i l a r t o t h e l i n e shapes o f t h e e x p e r i m e n t a l samples. —3 The above method was l i m i t e d t o T^ > 10 s e c , by t h e maximum a v a i l a b l e r a d i o - f r e q u e n c y power o f t h e t r a n s m i t t e r u s e d . A s m a l l m o d u l a t i o n a m p l i t u d e , H , o f 0.1 g a u s s , and a m o d u l a t i o n f r e q u e n c y o f 80 c p s . were 121 u s e d unchanged i n o b s e r v i n g s i g n a l s a t u r a t i o n . B e d f i e l d , and, more 122 r e c e n t l y , Goldman , have emphasized t h e i m p o r t a n c e of o p t i m i z i n g modula-t i o n f r e q u e n c i e s i n o b s e r v i n g c o r r e c t l y - p h a s e d o u t p u t s i g n a l s i n s a t u r a t i o n measurements t y p i f i e d by t h e above method. ( e) measurements by A d i a b a t i c P a s t P a s s a g e : The f a c t t h a t the n e t m a g n e t i z a t i o n , 3, o f a sample o f n u c l e i f o l l o w s t h e d i r e c t i o n o f t h e m a g n e t i c f i e l d when t h e l a t t e r changes d i r e c -t i o n s u f f i c i e n t l y s l o w l y has f r e q u e n t l y been r e f e r r e d t o by t h e t e r m " A d i a b a t i c " . To a c h i e v e t h i s e x p e r i m e n t a l l y , one i m p o r t a n t c o n d i t i o n t o he s a t i s f i e d i s * 2 2 U « Y H l ( 3 3 ) at where 2H^ i s t h e r a d i o f r e q u e n c y f i e l d a m p l i t u d e a n d ( ^ ) i s t h e r a t e o f change o f t h e main m a g n e t i c f i e l d , i . e . , t h e "sweep r a t e " . The term " f a s t p a s s a g e " , however, r e c o g n i z e s t h e e x i s t e n c e o f 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 t h a t e n t e r the e x p r e s s i o n f o r th e fl v e c t o r . Thus, f o r example, i f ~ - exceeds y ^ p t h e r e l a x a t i o n p r o c e s s c o r r e s p o n d i n g t o T^ w i l l . c a u s e fl t o d e c a y even b e f o r e fl c o m p l e t e s one p r e c e s s i o n a l c y c l e a r o u n d t h e e f f e c t i v e m a g n e t i c f i e l d A g ^ f * * T h e r e -f o r e , t h e p a s s a g e , o r ^.sweep" t h r o u g h r e s o n a n c e , must be " f a s t " enough s u c h t h a t SH. (Time o f P a s s a g e Through Resonance) ~ « T p Tg .......(34) *fi - (3 - — ) i n a c o o r d i n a t e s y s t e m w h i c h r o t a t e s a t the r a d i o -e f f • o y' f r e q u e n c y , <o» Upon combining (33) and (34), one gets ^ « f « • - * 2 . , (35) which nearly completely sums up the conditions of a t y p i c a l adiabatic f a s t passage experiment where ~ T 2 v -JiBased on these conditions, very u s e f u l experiments were developed by D r a i n * 2 3 and C h i a r o t t i et a l . , * 2 ^ f o r measuring r e l a x a t i o n times i n l i q u i d s . In some cases, however, where / Tg, an a d d i t i o n a l condition may have to be imposed on the above, namely, that 1 • 2 which would appear to require extremely high f i e l d s . However, as d i s -118 cussed by Abragam , t h i s l a s t i s f a r too stringent a t h e o r e t i c a l require-ment which experiments r e a l l y do not demand, and adiabatic f a s t passage responses have i n f a c t been successfuly demonstrated for s o l i d s even when 121 125 weaker H^ f i e l d s were employed ' 124 As C h i a r o t t i et a l . , have shown, the dispersion (in-phase) component of magnetization, under the above rapid passage conditions, has the form M = M o 1 - expCVV V 1 + expr*/"^ ) y where M q i s the equilibrium magnetization i n the absence of an r f . f i e l d , and t i s the time between passages through resonance. If the r e p e t i t i v e main f i e l d sweep frequency i s " f " , then t = ( 2 f ) ~ * . Since M O i n the experiment was unknown, r e l a t i v e values of M were obtained as signal amplitudes f o r two settings of sweep period t . T^, the only unknown, was obtained from 65 1 - expCVTi) .... (37) M x _ 1 + exptrV^) M 2 1 - e x p ( " t 2 / ) _l_ 1 + expC'^/T^ M l I n p r a c t i c e , t h e r a t i o - j - was o b t a i n e d f o r t g = 2 t ^ and f o r t g = 4 t ^ . Measurements were c a r r i e d out w i t h t h e 16 M c / s e c . t r a n s m i t t e r s e t a t maximum o u t p u t ( H ^ = 0.2 g a u s s ) . The c r o s s e d - c o i l probe was s e t t o r e c e i v e t h e " d i s p e r s i o n " mode. A H e w l e t t - P a c k a r d (No.202A) s q u a r e wave g e n e r a t o r was u s e d t o modulate t h e H q f i e l d . S q uare wave m o d u l a t i o n was p r e f e r r e d so t h a t passage t h r o u g h r e s o n a n c e w o u l d be r a p i d and i n d e p e n d e n t o f t h e w a i t i n g t i m e a t each end o f the p a s s a g e , and t h e e x t r e m a i n H q w o u l d be s u f f i c i e n t l y c l o s e t o r e s o n a n c e so t h a t t h e m a g n e t i z a t i o n ( p r o p o r t i o n a l t o H ) w o u l d r e l a x t o e s s e n t i a l l y t h e same v a l u e of M . E l s e w h e r e , o' J o 126 P o w l e s has u s e d a t r a p e z o i d a l wave-form f o r t h e same r e a s o n . The range of sweep a m p l i t u d e s u s e d l a y between 120 and 200 g a u s s , v a l u e s s m a l l compared to H q ( 2 . 6 2 k i l o g a u s s ) . The minimum p e r i o d of a square wave p u l s e t o w h i c h t h e V-2100 power s u p p l y r e s p o n d e d s a t i s f a c t o r i l y was 11 m i l l i -s e c o n d s , and t h e r e f o r e v a l u e s o f T^ t h a t c o u l d be measured b y t h i s t e c h -n i q u e were l i m i t e d t o a minimum above t h i s v a l u e . The i n - p h a s e component of the m a g n e t i z a t i o n was r e c t i f i e d and d i s p l a y e d on a Du Mont '304' o s c i l l o s c o p e w i t h e x t e r n a l c a p a c i t o r s (~ 20..u-f, e l e c t r o l y t i c " ) t o p r o l o n g th e t i m e — b a s e t o about 30 s e c o n d s , so t h a t v i s u a l r e a d — o u t s o f s i g n a l a m p l i t u d e s were p o s s i b l e . The s i g n a l d i s p l a y was s y n c h r o n z i e d e x t e r n a l l y w i t h t h e square wave g e n e r a t o r w h i c h m o d u l a t e d t h e main m a g n e t i c f i e l d . 66 CHAPTER V I THE HYDRATES OF CARBON TETRAFLUORIDE, SULPHUR HEXAFLUORTDE, ETHYLENE OXIDE, AND PROPANE 6.1 O u t l i n e . I n t h i s c h a p t e r , we d i s c u s s t h e r e s u l t s o f o u r e x a m i n a t i o n o f f o u r gaseous g u e s t m o l e c u l e s i n t h e i r h y d r a t e h o s t s t r u c t u r e . F o r t h e f i r s t two c l a t h r a t e s , i . e . , t h o s e o f c a r b o n t e t r a f l u o r i d e and s u l p h u r h e x a f l u o r i d e , 19 F NMR l i n e shapes and t h e i r t e m p e r a t u r e dependence have been d e s c r i b e d f r o m 77°K u p t o v e r y n e a r t h e r e s p e c t i v e m e l t i n g p o i n t s . F o r t h e o t h e r two h y d r a t e s , namely, t h o s e o f e t h y l e n e o x i d e and p r o p a n e , H* r e s o n a n c e has been i n v e s t i g a t e d . The b e h a v i o u r o f e t h y l e n e o x i d e i n t h e Type I h y d r a t e l a t t i c e was examined o n l y a t t h r e e t e m p e r a t u r e s , and t h e r e s u l t s compared w i t h e x i s t i n g d a t a on t h e c l a t h r a t e a t t h e s e t e m p e r a t u r e s . F o r t h e propane h y d r a t e , second moment and l i n e w i d t h measurements were f o l l o w e d by T, i n v e s t i g a t i o n s using progressive signal saturation and adiabatic f a s t passage techniques over a l i m i t e d temperature range. 67 6 . 2 Results and Conclusions: (a) Carbon T e t r a f l u o r i d e i n the "Type I" host l a t t i c e : Two samples of t h i s clathrate were prepared, one with ordinary water and the other using heavy water. These were estimated to correspond within 1 0 $ to the i d e a l c l a thrate formula, 6 G . 4 6 H 2 O . D e r i v a t i v e l i n e shapes of F?!?< resonance were recorded between 77°K and 272°K for the two samples. Line shapes were roughly Gaussian and s t r u c t u r e l e s s . A gradual narrowing of the resonance linewidth with increasing temperature, without any perceptible change i n the l i n e shape, was a feature of a l l the spectra recorded f o r the two modifications of the c l a t h r a t e . Moderate amplitudes of s i g n a l modulation, a slow field-scanning rate, and r f . power l e v e l s w e l l below saturation were used while recording the spectra. Experimental r e s u l t s were obtained for warming as;well as cooling temperature cycles. A sample was e q u i l i b r a t e d f o r an hour at each temperature, and at l e a s t four spectra were recorded each time. At 77°K, the ordinary hydrate had an F"^ resonance linewidth of 4 , 5 ± 0 . 2 G . This narrowed gradually to ~ 2 . 8 G at 270°K. For the deuter-ated hydrate, where the F*^ vs. H* in t e r a c t i o n s were absent, the F*^ l i n e -width at 77°K was 3 . 6 - 0 . 1 G , and at 270°K t h i s reached a very small l i m i t i n g value of ~ 0 . 1 G which, presumably, was c h a r a c t e r i s t i c of the magnetic f i e l d inhomogeneity over the entire sample. The errors quoted i n the above r e s u l t s are standard d e v i a t i o n s . In Table VIII i s found a comparative presentation of the mean experimental second moments (cor r e c t e d f or modulation broadening) of CF. i n t h e HgO and DgO l a t t i c e s o v e r t h e whole t e m p e r a t u r e range s t u d i e d . The a c t u a l e x p e r i m e n t a l d e v i a t i o n s f r o m t h e mean second moment f o r upward and downward s c a n o f t h e m a g n e t i c f i e l d were l e s s t h a n 5$ a t any t e m p e r a t u r e . The l a s t column of T a b l e V I I I i n d i c a t e s a f a i r l y c o n s t a n t v a l u e o f 2 19 1 ~ 2.5G f o r t h e F v s . H i n t e r a c t i o n between " g u e s t " and " c a g e " . T h i s w i l l be d i s c u s s e d l a t e r i n t h i s c h a p t e r . T a b l e V I I I E x p e r i m e n t a l Second Moments f o r CF. c l a t h r a t e H y d r a t e s T m p e r a t u r e , ( C F 4 ) ~ 6 H 2 0 , ( C F 4 ) ~ 6 D 2 0 , 19 * 1 D i f f e r e n c e = F vs H °K G 2 G 2 "Guest-Cage" I n t e r a c t i o n , 77 6.10 3.60 2.50 91 — 3.20 — 100 5.50 3.05 2.45 117 5.15 2.71 2.44 138 4.53 2.15 2.38 150 4.50 — -156 4.49 — -170 4.49 2.10 2.39 190 4.42 1.99 2.43 230 4.04 1.64 2.40 251 2.59 0.21 2.38 270 2.57 ) 2.37 271 - 0.20) 272 2.55 0.20 2.35 To r a t i o n a l i z e the t r e n d i n t h e second moment r e s u l t s , p r e c i c a l c u l a t i o n s o f t h e o r e t i c a l s econd moments f o r v a r i o u s d y n a m i c a l models o f th e g u e s t m o l e c u l e s , c o m p r i s i n g " i n t r a - g u e s t " c o n t r i b u t i o n s f r o m a s i n g l e h o s t l a t t i c e c a v i t y as w e l l as " i n t e r - g u e s t " c o n t r i b u t i o n s f r o m n e i g h -b o u r i n g c a v i t i e s , a r e n e c e s s a r y . A s s u m i n g C-F=1.323A and t e t r a h e d r a l FCF 127 a n g l e s f o r c a r b o n t e t r a f l u o r i d e , anvintramolecular second moment o f 2 78 9.35G i s computed f r o m t h e powder f o r m u l a of Van V l e c k . The t e t r a -h e d r a l C F 4 m o l e c u l e p o s s e s s e s Cg- and C g r axes o f symmetry, and r e o r i e n t a -t i o n s a b out t h e s e axes must t h e r e f o r e be h i g h l y f a v o u r e d . Reduced second 69 moments c o r r e s p o n d i n g t o r e o r i e n t a t i o n s o f t h e m o l e c u l e a l o n g t h e s e axes have t h e r e f o r e been computed (C-F = 1.323A) a c c o r d i n g t o t h e method o f K r o o n 81 . I n T a b l e I X , t h e s e v a l u e s a r e seen t o compare w e l l w i t h the c o r r e s -128 p o n d i n g v a l u e s r e p o r t e d f o r t h e s e m o t i o n s by A s t o n e t a l . , " T a b l e I X 49 T h e o r e t i c a l F Se c o n d Moments f o r CF. i n t h e D e u t e r a t e d 4 H y d r a t e L a t t i c e Type of M o t i o n " I n t r a - G u e s t " second Moment, G 2 From A s t o n e t a l ( l 2 8 ) ^ C a l c u l a t e d f o r C-F=1.323A • I n t e r - G u e s t " Second Moment, G 2 l . 9 ( H b ; i.io( c> 1.52< c) T o t a l Second Moment, G 2 R i g i d M o l e c u l e -no m o t i o n R e o r i e n t i n g about Cg A x i s R e o r i e n t i n g a b out Cg A x i s R e o r i e n t i n g about Random Axes I s o t r o p i c R o t a t i o n 9.52 1.19 2.38 1.90 9.35 1.11 ( a ) 2.34 ( a ) 1 . 5 6 ( a ) 0.27 0>) 1.0<d> 11.35+0.10 2.25±0.04 3^88±0.02 2.02+0.15 ~ 1.0 v a l u e s c a l c u l a t e d a c c o r d i n g , t o the methods o f [a) K r o o n ( 8 l ) >b) S m i t h (86) c) M i l l e r and Gutowsky (130) 'd) D m i t r i e v a and M o s k a l e v ( 9 l ) The " i n t e r - g u e s t " i n t e r a c t i o n s c o u l d be c a l c u l a t e d s i n c e the e x a c t l o c a t i o n s o f t h e c a v i t i e s i n the Type I h y d r a t e c r y s t a l s a r e known f r o m X - r a y c r y s t a l s t r u c t u r a l a n a l y s i s ( s e e 'fa b l e I I I ) . Methods a t p r e s e n t a v a i l a b l e f o r c a l c u l a t i n g " i n t e r - m o l e c u l a r " second moments o r i g i n a t e f r o m an e l e g a n t c o n c e p t , f i r s t p r o p o s e d as a rough a p p r o x i m a t i o n by Andrew and 84 Eades . These a u t h o r s c o n s i d e r e d two i n t e r a c t i n g m o l e c u l e s as two s p h e r e s w i t h s p i n s c o n c e n t r a t e d a t t h e i r r e s p e c t i v e c e n t r e s . T h a t t h i s c o n c e p t i s i n d e e d c o r r e c t f o r s p h e r i c a l and c r u a s i - s p h e r i c a l m o l e c u l e s has been borne 81 117 o u t by l a t e r work by K r o o n , M c C a l l and D o u g l a s s , and D m i t r i e v a and 91 M o s k a l e v . A l l t h e s e a u t h o r s have assumed t h a t t h e s p i n s o f each i n t e r -a c t i n g m o l e c u l e a r e homogeneously smeared o u t o v e r t h e s u r f a c e o f a sphere s i t u a t e d a t i t s l a t t i c e s i t e . However, a method o f more g e n e r a l v a l i d i t y , a p p l i c a b l e t o two i n t e r a c t i n g m o l e c u l e s w h i c h a r e e i t h e r a t r e s t o r r e -85 o r i e n t i n g about p r e f e r r e d a x e s , has been p r o p o s e d . T h i s c o n s i d e r s a c u b i c a r r a y o f s p h e r e s whose s u r f a c e s a r e c o v e r e d w i t h a c o n t i n u o u s , , u n i f o r m d i s -2 2 / 6 t r i b u t i o n o f t h e m a g n e t i c moment. The term, (3Cos f - l ) / r , a p p e a r i n g i n 129 th e Van V l e c k e x p r e s s i o n , i s t h e n a v e r a g e d o v e r t h e s e s p h e r e s . F o r a p o l y c r y s t a l l i n e s pecimen, where t h e a n g u l a r f a c t o r above may be a v e r a g e d o u t , t h e r e s u l t o f t h i s method may be p u t i n t h e f o r m X N INTER" INTER where * S ^ e (*ou*>"''e a v e r a £ e > p v e r b b o t h s p h e x i c a L l s u r f a c e s , o f the v e c t o r f r o m s p i n j o f an o r i g i n m o l e c u l e t o s p i n k o f i t s i t h n e a r e s t n e i g h b o u r . R ^ ^ j ^ . i s the r a d i u s o f t h e o r i g i n m o l e c u l e s , and- R p ^ ^ i s t h e - 86 ' c e n t r e - t o - c e n t r e d i s t a n c e between t h e two s p h e r e s . S m i t h has r e c e n t l y -T^INTBA 1 shown t h a t , f o r cas.es where R ^ > 2R T X , t h e f u n c t i o n f\ss y has R 2 ™ i m B A 4 INTER t h e f o r m ' " T m T^" " " """""" " c  Q- - R l N T M ^ + - (^INTBA ^ INTRA" = INTER INTER (39) ENTER ( l - ^ W ) 3 R I N T E R The above e x p r e s s i o n may be a p p l i e d t o t h e t y p e I and t y p e I I gas h y d r a t e s . ^JJ^TRA. m a ^ ^ e ^ o u n o - ^ r o m t h e m o l e c u l a r d i a m e t e r s o f g u e s t m o l e c u l e s l i s t e d i n T a b l e V and RT.TimnT,, t h e c e n t r e - t o - c e n t r e d i s t a n c e between two a d j a c e n t o c c u p i e d c a v i t i e s , may be computed f r o m t h e s t r u c t u r a l d a t a q u o t e d i n T a b l e I I I . R e s u l t s o f " i n t e r - g u e s t " second moments o b t a i n e d t h r o u g h the methods o u t l i n e d above a r e p r e s e n t e d i n t h e t h i r d column o f T a b l e I X . A 2 v a l u e of 1.90 G i s o b t a i n e d f o r the i n t e r a c t i o n between a " r i g i d " CF^ and f i v e " r i g i d " n e i g h b o u r s . S i n c e CF^ i s q u a s i - s p h e r i c a l , t h e t e a r l i e r -84 91 me n t i o n e d methods ' have been a p p l i e d f o r t h e case o f i s o t r o p i c r o t a t i o n . F o r t h e o t h e r m o d e l s , where r e o r i e n t a t i o n s o f each g u e s t m o l e c u l e about chosen axes a r e c o n s i d e r e d , " i n t e r - g u e s t " second moments have been 130 c a l c u l a t e d a c c o r d i n g t o a method by M i l l e r and Gutowsky . Comparing the t o t a l t h e o r e t i c a l s econd moments g i v e n i n T a b l e I X w i t h the e x p e r i m e n t a l v a l u e s p r e s e n t e d i n T a b l e V I I I , i t i s a t once a p p a r e n t t h a t t h e CF^ m o l e c u l e s a r e d e f i n i t e l y n o t r i g i d i n t h e i r c a v i t i e s even a t 77°K. The p r i n c i p a l l i n e - n a r r o w i n g m o t i o n o c c u r r i n g a t t h i s t e m p e r a t u r e i s v e r y l i k e l y r e o r i e n t a t i o n o f CF^ about a Cg a x i s ( t h e o r -2 2 e t i c a l second moment: 3.88 0.02G ; e x p e r i m e n t a l v a l u e : 3.60G ) . A s t h e t e m p e r a t u r e i s i n c r e a s e d , g r e a t e r m o t i o n o f t h e e n c l a t h r a t e d m o l e c u l e i s i n d i c a t e d b y a g r a d u a l d e c r e a s e i n second moment, and i n t h e v i c i n i t y o f 140°K, random r e o r i e n t a t i o n s appear t o have s e t i n . I t i s q u i t e l o g i c a l t o c o n s i d e r t h i s m o t i o n as a p r e c u r s o r t o f r e e , i s o t r o p i c r o t a t i o n , w h i c h p r e s u m a b l y s e t s i n around 200°K a t f r e q u e n c i e s a p p r e c i a b l e enough f u r t h e r t o n a rrow the l i n e . F o r t h i s u n i f o r m m o t i o n , t h e " i n t r a - g u e s t " second moment v a n i s h e s owing t o t h e a v e r a g i n g out o f the f a c t o r c o n t a i n i n g t h e E u l e r i a n a n g l e s , and t h e o b s e r v e d second moment i s e n t i r e l y " i n t e r - g u e s t " i n o r i g i n . Above 250°K, a sh a r p d r o p i n t h e e x p e r i m e n t a l second moment of (CF^J-J^O f r o m 1.60G&' to 0.21G2 o c c u r s , w i t h a c o n c o m i t a n t change o f a s i m i l a r magnitude f o r t h e o r d i n a r y h y d r a t e . These f a c t s i n d i c a t e e i t h e r d i f f u s i o n o f C F ^ t h r o u g h t h e l a t t i c e o r a sudden e x p a n s i o n o f t h e l a t t i c e w i t h the s i m u l t a n e o u s " o p e n i n g up" of t h e hydrogen-bonded ca g e s . The l a t t e r p o s s i b i l i t y seems i n d i c a t e d by t h e sudden appearance o f s i g n i f i c a n t decrements i n t h e " g u e s t - c a g e " i n t e r a c t i o n l i s t e d i n t h e l a s t column o f T a b l e V I I I as t h e d i f f e r e n c e between t h e second moments of o r d i n a r y and d e u t e r a t e d h y d r a t e s . I n d e e d , a s m a l l , g r a d u a l d e c r e a s e between s u c c e s s i v e v a l u e s i n t h i s column may be n o t i c e d over, t h e e n t i r e t e m p e r a t u r e r a n g e , and t h i s p e r h a p s i n d i c a t e s a s t e a d y , i s o t r o p i c e x p a n s i o n o f t h e l a t t i c e w i t h t e m p e r a t u r e . E x p e r i m e n t a l v a l u e s above 250°K a r e somewhat v i t i a t e d b y s m a l l H o f i e l d i n h o m o g e n e i t i e s , and one i s n o t w a r r a n t e d i n d i s c u s s i n g t h e r e s u l t s beyond t h i s t e m p e r a t u r e any t o o s e r i o u s l y . The c l a t h r a t e decomposes a t ~ 275°K. I t i s i n t e r e s t i n g t o compare t h e r e s u l t s o f t h i s s t u d y on CF^ 128 t r a p p e d i n a w a t e r - l a t t i c e w i t h t h o s e o b t a i n e d by A s t o n e t a l . , f o r 19 c r y s t a l l i n e CF^. These w o r k e r s r e p o r t a b r o a d F re s o n a n c e l i n e g r o w i n g a t t h e expense o f a narrow l i n e on c o o l i n g CF^. The l a t t i c e i s r i g i d below 55°K. The na r r o w l i n e a p p e a r i n g above 70°K has been a t t r i b u t e d b y t h e s e a u t h o r s t o m o l e c u l a r r e o r i e n t a t i o n about a Cg symmetry a x i s . T h e i r 130 r e s u l t s have been r e - i n t e r p r e t e d , however, by M i l l e r and Gutowsky , who s u g g e s t t h a t t h e CF^ m o l e c u l e s r e o r i e n t r a n d o m l y i n t h e i r c r y s t a l l a t t i c e a t 70°K, and t h a t t h e s e soon s t a r t d i f f u s i n g above a s e c o n d - o r d e r t r a n s i t i o n (•J-point) o c c u r r i n g a t 76°K. I n s h a r p c o n t r a d i s t i n c t i o n t o the above r e s u l t s , t h e p r e s e n t work c l e a r l y i n d i c a t e s t h a t CF^ m o l e c u l e s t r a p p e d s i n g l y i n hydrogen-bonded w a t e r c a v i t i e s a r e r e o r i e n t i n g about f i x e d c e n t r e s o f mass a t 77°K. These r e o r i e n t a t i o n s a r e p r e s u m a b l y around a Cg - a x i s o f symmetry. I s o t r o p i c r o t a t i o n does n o t become p r o m i n e n t u n t i l t h e t e m p e r a t u r e r e a c h e s above 230 K. D i f f u s i o n s e t s i n , i f a t a l l , o n l y a t t e m p e r a t u r e s c l o s e t o t h e d e c o m p o s i t i o n p o i n t o f t h e c l a t h r a t e s . I n o t h e r words, once t r a p p e d i n i t s c a v i t y , the m o l e c u l e e x h i b i t s l i t t l e t r a n s l a t i o n a l f r e e d o m . T h i s o b s e r v a -t i o n i s i n l i n e w i t h some o f t h e h e a t - c a p a c i t y r e s u l t s of S t a v e l e y e t a l . , 40 41 ' o b t a i n e d f o r B - q u i n o l c l a t h r a t e s . The above c o n c l u s i o n p r o v i d e s u n e q u i v o c a l e v i d e n c e f o r t h e e x i s t e n c e o f a v e r y s y m m e t r i c a l f o r c e - f i e l d w i t h i n the c a v i t y o f t h e CF^-c l a t h r a t e h y d r a t e . A t l o w t e m p e r a t u r e s , a t any r a t e , t h e r e e x i s t s a v e r y e f f e c t i v e c a g i n g a c t i o n by the hydrogen-bonded l a t t i c e , and a g u e s t m o l e c u l e spends most o f i t s time r e o r i e n t i n g i n t h e c e n t r e o f t h e c a v i t y w i t h l i t t l e o r no t r a n s l a t i o n a l d i f f u s i o n a t n.m.r. f r e q u e n c i e s . A c e n t r a l f o r c e - f i e l d a p p r o x i m a t i o n , of t h e Lennard-Jones(12-6) or K i h a r a (12-6) t y p e , s h o u l d t h e r e f o r e a d e q u a t e l y d e s c r i b e t h e thermodyna-m i c s o f suc h a c l a t h r a t e , as has i n f a c t been v e r i f i e d by McKoy and 42 S i n a n o g l u . These a u t h o r s have f u r t h e r shown t h a t , f o r a c l a t h r a t e s u c h a s CF^— h y d r a t e , a b a r r i e r t o r o t a t i o n of t h e g u e s t m o l e c u l e i n i t s c a v i t y i s u n l i k e l y . The r e s u l t s o f t h i s NMR s t u d y p r o v i d e i n d e p e n d e n t s u p p o r t o f t h e i r c o n c l u s i o n . ( b ) S u l p h u r H e x a f l u o r i d e i n the "Type I I " h o s t l a t t i c e : 19 H e r e , as i n t h e p r e v i o u s c a s e , the F r e s o n a n c e a b s o r p t i o n was s t u d i e d f o r b o t h t h e o r d i n a r y and d e u t e r a t e d m o d i f i c a t i o n s o f the Type I I h y d r a t e o f SFg. These samples were p r e p a r e d u n d e r i d e n t i c a l e x p e r i m e n t a l c o n d i t i o n s and had c o m p o s i t i o n s v e r y c l o s e t o t h o s e r e p o r t e d b y o t h e r 74 7, 31 wo r k e r s ' . S p e c t r a were r e c o r d e d i n two s e p a r a t e e x p e r i m e n t a l r u n s , a g a i n f o r warming and c o o l i n g c y c l e s . Thermal h y s t e r e s i s e f f e c t s were a b s e n t f o r b o t h t h e samples, and no f i n e - s t r u c t u r e was a p p a r e n t . L i n e scan" and " l o w r f . " c o n d i t i o n s a l r e a d y s t a t e d i n t h e p r e v i o u s s e c t i o n , t h e m o d u l a t i o n a m p l i t u d e was a d j u s t e d t o a bare minimum and a h i g h t i m e -c o n s t a n t was i n c l u d e d i n t h e R-C c i r c u i t o f t h e s p e c t r o m e t e r o u t p u t u n i t . The h i g h t i m e - c o n s t a n t l e d t o n e g l i g i b l y l ow n o i s e components i n t h e spect r u m , so t h a t t he " s k i r t s " o f the narrow d e r i v a t i v e s i g n a l s c o u l d be r e c o r d e d above n o i s e . A t y p i c a l s p e c t r u m a t 180°K, r e c o r d e d f o r ( S F g ) ~ D^O under the above c o n d i t i o n s , i s shown i n F i g . 13. shapes, w h i c h were v e r y n a rrow and G a u s s i a n a t 77°K, became a l m o s t L o r e n t z i a n above ~ 180°K. T h e r e f o r e , i n a d d i t i o n t o c h o o s i n g t h e " s l o w — t e m p e r a t u r e I8( s e c o n d m o m e n t F i g . 13 A t y p i c a l F l i n e d e r i v a t i v e f o r ( S F g ) ~ DgO a t 180°K. A t l e a s t f o u r s p e c t r a were r e c o r d e d a t each t e m p e r a t u r e , and second moments computed t h e r e f r o m were f o u n d t o have a s m a l l s c a t t e r o f £ 3$ around t h e mean v a l u e . A t 77°K, t h e second moment of ( S F g ) ~ H g 0 , 2 c o r r e c t e d f o r any f i e l d b r o a d e n i n g e f f e c t s , was 1.96 +- 0.05G . The 2 c o r r e s p o n d i n g v a l u e f o r t h e d e u t e r a t e d c l a t h r a t e was 1.60 ± 0.05G . These v a l u e s d i d n o t appear t o change a p p r e c i a b l y beyond t h e e r r o r l i m i t s q u o t e d u n t i l t h e t e m p e r a t u r e r e a c h e d 150°K. A t t h i s t e m p e r a t u r e , l i n e w i d t h s n a r r o w e d r a t h e r n o t i c e a b l y f o r b o t h m o d i f i c a t i o n s o f t h e c l a t h r a t e ; t h e 2 o r d i n a r y h y d r a t e had a c o r r e c t e d second moment o f 1.22 ~ 0.05G , and t h e 2 d e u t e r a t e d h y d r a t e a c o r r e c t e d second moment of 1.0 +- 0.05G . Once a g a i n , t h e s e v a l u e s Btayed f a i r l y c o n s t a n t u p t o t e m p e r a t u r e s i n t h e v i c i n i t y o f 240°K. Above t h i s t e m p e r a t u r e , t h e r e s o n a n c e l i n e f o r t h e d e u t e r a t e d specimen narrowed c o m p l e t e l y t o s h a r p " L o r e n t z i a n " s p i k e s and had an a p p a r e n t w i d t h o f ~ 0.1G. A t 255°K, t h e o r d i n a r y h y d r a t e s t i l l had a d e f i n i t e l i n e w i d t h g r e a t e r t h a n t h e e x p e r i m e n t a l e r r o r s , and t h e se c o n d 2 moment c o r r e s p o n d e d t o ~ 0.35G a f t e r a l l o w i n g f o r t h e e f f e c t s of the m o d u l a t i n g f i e l d . The second moment v e r s u s t e m p e r a t u r e c u r v e f o r t h e d e u t e r a t e d h y d r a t e t h u s has two b r o a d p l a t e a u x , and two d i s t i n c t m o t i o n a l t r a n s i t i o n s of the g u e s t s p e c i e s a r e i n d i c a t e d w i t h i n the t e m p e r a t u r e range of t h i s s t u d y . The dependence of t h e second;moments on t e m p e r a t u r e f o r t h e two specimens i s shown i n F i g . 14. The s i m p l e s t m o t i o n s w h i c h w o u l d narrow t h e f l u o r i n e r e s o n a n c e a p p r e c i a b l y a r e r e o r i e n t a t i o n s o f t h e SFg m o l e c u l e s . SFg has 2-, 3-, and 4- f o l d symmetry axes about w h i c h r e o r i e n t a t i o n s m i g h t o c c u r , and, c o r r e s p o n d i n g l y , t h e a n g u l a r dependences of t h e " i n t r a - g u e s t " v a l u e w o u l d be r e d u c e d by t h e f a c t o r T_ 2 2 yr ( 3C 0 S @"~l) . » • « « . a o . e » . e . a . . . . ( 40) CM CO CO < r -z Ixl o 5 o o UJ CO 2 . 0 0 I .75 I .50 1.25 1.00 0 . 7 5 0 . 5 0 0 . 2 5 0 i—r i—i—i—I—i—r i i r •=(SF 6 )~H 20 o=(SF 6)-D 20 i i i i i ;i I I ii i i i i ' ' I I I I I I I L l I I I I I I f I iM.R 100 150 200 250 TEMPERATURE, °K F i g . 14 ( SFJ HYDRATE : SECOND MOMENT VERSUS TEMPERATURE 6 -a OS 77 where g, as d e f i n e d i n F i g . 10, i s t h e r a n g l e between t h e i n t e r n u c l e a r v e c t o r and the r e o r i e n t a t i o n a x i s . T h i s f a c t o r h o l d s f o r s t e p w i s e r e o r i e n t a t i o n about an n - f o l d a x i s w i t h n ^, 3 and f o r c l a s s i c a l " r o t a t i o n " about any 106 a x i s . An " i n t r a - m o l e c u l a r " second moment was c a l c u l a t e d f o r r i g i d SFg by a ssuming an S-F d i s t a n c e of 1.58| ' • The r e d u c t i o n f a c t o r (40) 1 81 was u s e d a f t e r t h e manner of K r o o n f o r r e o r i e n t a t i o n of t h e SFg m o l e c u l e about Cg- , Cg-, and C^- a x e s . V a l u e s c a l c u l a t e d by t h i s p r o c e d u r e a r e p r e s e n t e d i n T a b l e X. T a b l e X T h e o r e t i c a l F Second Moments f o r SF f i i n t h e D e u t e r a t e d H y d r a t e L a t t i c e . T ype-of " I n t r a - G u e s t " ^ M o t i o n Second Moment, G , f o r S-F=1.58A " I n t e r - G u e s t " S econd Moment, G 2 T o t a l s e c o n d Moment, G R i g i d M o l e c u l e -no m o t i o n 12.60 1.41<b> 14.01 R e o r i e n t i n g a b out / \ C 2 - A x i s 2 . 1 0 ^ a ) 0 . 3 5 ^ 2.45 R e o r i e n t i n g / \ about Cg A x i s 2.63^ a ' 0 . 1 6 ^ * 2.79 R e o r i e n t i n g , \ about A x i s 1.5P ' 0 . 0 9 ^ 1.60 R e o r i e n t i n g / \ about Random Axe s ~ 1.00^ ' 0 . 0 2 ^ ~ 1.02 I s o t r o p i c R o t a t i o n 0 ~ i.oo ( d> ~ .1.00 V a l u e s c a l c u l a t e d a c c o r d i n g t o t h e methods o f ( a ) K r o o n ( 8 l ) ( b ) S m i t h (86) (c) M i l l e r and Gutowsky (130) (d) D m i t r i e v a and M o s k a l e v ( 9 l ) I n c omputing t h e " i h t e r - g u e s t " c o n t r i b u t i o n s c o r r e s p o n d i n g t o t h e s e m o dels, t h e methods d e s c r i b e d i n s e c t i o n 6.2 ( a ) were u s e d . The e x p e r i m e n -t a l second moment o f 1.60 ± 0.05G a t 77°K f o r ( S F e ) ~ D o0 i s i n e x c e l l e n t ^ 6 7 2 agreement w i t h t h e t h e o r e t i c a l second moment e x p e c t e d f o r C ^ - r e o r i e n t a t i o n ' \ o f SF f i m o l e c u l e s i n t h e i r c a v i t i e s . T h i s i s p r e s u m a b l y t h e o n l y r e o r i e n t a -) \-t i o n c a u s i n g s p e c t r a l n a r r o w i n g u n t i 1 150°K, above w h i c h t h e r e o c c u r s a r a t h e r s h a r p m o t i o n a l t r a n s i t i o n , and the second moment i s r e d u c e d t o 2 1.0 - 0.05G . The m o t i o n a s s o c i a t e d w i t h t h i s new v a l u e may e i t h e r be a "jump" t y p e r e o r i e n t a t i o n of the g u e s t m o l e c u l e s about random axes a t 'n.m.r". f r e q u e n c i e s , o r i s o t r o p i c " r o t a t i o n " i n t h e c l a s s i c a l s e n s e . The e s t i m a t e s f o u n d i n T a b l e X f o r t h e s e two t y p e s of m o t i o n l i e w i t h i n t h e e r r p r l i m i t s o f t h e e x p e r i m e n t a l v a l u e , and t h e r e f o r e a c l e a r - c u t d i s t i n c -t i o n between t h e two m o t i o n a l t y p e s i s n o t p o s s i b l e . A t t e m p e r a t u r e s above 240°K, a f u r t h e r r e d u c t i o n i n t h e second moment t o a n e g l i g i b l e f i n a l v a l u e f o r t h e d e u t e r a t e d h y d r a t e s i g n i f i e s t h a t o n s e t o f t r a n s l a t i o n a l d i f f u s i o n . j n u s t be o c c u r r i n g r a p i d l y enough'^'nci n u l l i f y a l l " i n t r a - g u e s t " and " i n t e r - g u e s t " components of t h e se c o n d moment. How-e v e r , t h e i n t e r e s t i n g f a c t t h a t , f o r t h e o r d i n a r y h y d r a t e , a r e s i d u a l v a l u e as l a r g e as ~ 0.35G 2 was s t i l l o b s e r v e d a t 250°K s u g g e s t s t h a t t h e r e a r e s t i l l some una v e r a g e d d i p o l e - d i p o l e i n t e r a c t i o n s i n t h e c l a t h r a t e . T h i s a p p e a r s t o be i n c o n f l i c t w i t h t h e s u g g e s t i o n j u s t made r e g a r d i n g o v e r a l l d i f f u s i o n o f g u e s t m o l e c u l e s . On t h e o t h e r hand, one may s u s p e c t s a t u r a -133 o. t i o n b r o a d e n i n g t o have s e t i n f o r t h e ( S F g ) ~ H^O a t 250 A, p a r t i c u l a r l y 19 s i n c e t h e F a b s o r p t i o n mode s i g n a l u s u a l l y s a t u r a t e s e a s i l y as tem p e r a -t u r e goes up. However, the e x p e r i m e n t a l s i g n a l was r e c o r d e d a t v e r y l ow r f . power l e v e l s , and the peaks o f t h e d e r i v a t i v e s i g n a l d i d n o t appear t o be "rounded o f f " i n t h e manner c h a r a c t e r i s t i c o f s a t u r a t i o n . D i s r e g a r d i n g 2 t h e p o s s i b i l i t y of s a t u r a t i o n , t h e n , t h e v a l u e o f 0.35G measured f r o m t h i s s i g n a l may be a s c r i b e d t o t h e u n a v e r a g e d d i p o l e - d i p o l e i n t e r a c t i o n s a r i s i n g f r o m g u e s t ( F * ^ ) -cage (H*) i n t e r a c t i o n s . T r a n s l a t i o n a l " B a t t l i n g " o f SFg m o l e c u l e s i n t h e i r cages I n a c l a t h r a t e specimen w i t h as h i g h a degree of f i l l i n g as was 134 u s e d i n t h i s s t u d y , t h e r e e x i s t s l i t t l e p r o b a b i l i t y f o r a g u e s t m o l e c u l e l e a v i n g i t s own cage e n t i r e l y and d i f f u s i n g i n t o n e i g h b o u r i n g h y d r o g e n -bonded c a g e s . To e x p l a i n the e x p e r i m e n t a l v a l u e a t 250°K f o r ( S F g ) ^ H^O, 26 t h e n , one may c o n s i d e r t h e p o s s i b i l i t y o f " r a t t l i n g " , o r s e v e r e l y l i m i t e d random t r a n s l a t i o n a l movement, of a f r e e l y r o t a t i n g g u e s t w i t h i n i t s c a v i t y . F r e e r o t a t i o n of SFg by v i r t u e o f i t s h i g h symmetry n u l l i f i e s * i n t r a - g u e s t " i n t e r a c t i o n s , and " r a t t l i n g " m i n i m i z e s t h e " i n t e r - g u e s t " c o n t r i b u t i o n t o a : n e g l i g i b l e v a l u e i n v i e w of t h e c e n t r e - t o - c e n t r e d i s t a n c e (7.36A) between two h e x a k a i d e c a h e d r a l w a t e r c a v i t i e s . The r e s i d u a l i n t e r a c t i o n s (=0.35G ) , 19 1 w h i c h i s o f t h e F v s . H t y p e o n l y , may now be c o n s i d e r e d as a sum o f ( i ) G u e s t ( S F g ) - c a g e (HgO) i n t e r a c t i o n s i n a s i n g l e h e x a k a i d e c a h e d r a l 78 c a v i t y by summing t h e i n t e r a c t i n g v e c t o r s i n the u s u a l way u p t o a l i m i t of 6A, and ( i i ) a s m a l l i n t e g r a l t o c o v e r i n t e r a c t i o n s beyond 6A i n t h e u n i t c e l l . Of t h e s e two t e r m s , the second one may i n t u r n be e x p r e s s e d as a sum o f two s e p a r a t e i n t e g r a t e d a v e r a g e s o v e r t h e f l u o r i n e and p r o t o n d e n s i -t i e s i n t h e c e l l , t h e i n t e g r a l s b e i n g o b t a i n e d o v e r an assumed homogeneous d i s t r i b u t i o n o f n e i g h b o u r s . The number o f n e i g h b o u r s between r and r+dr i s 4TTr 2pdr, where p i s t h e number of n u c l e i p e r u n i t volume. The a p p r o -p r i a t e Van V l e c k . f o r m u l a ... . . . . 00 00 A v . = 1 [ 3 1 7 - 4 x 4 " P F j ' r ~ 4 d * ] + T 5 [ 3 5 8 ' ° X 4 T T PH . J r ^ d x j i 4 1 ) 6A 6A h o l d s f o r term ( i i ) , where p^ i s t h e average f l u o r i n e d e n s i t y ( i . e . , t h e number o f f l u o r i n e n u c l e i p e r AL.) and p t h e average p r o t o n d e n s i t y i n t h e u n i t c e l l o f 8 S F g . l 3 6 H 2 0 . The u n i t c e l l c o n s t a n t f o r 8 S F g . l 3 6 H 2 0 has been r e p o r t e d by S t a c k e l b e r g and J a h n s ^ t o be 17.21A a t 0°C. These a u t h o r s have n o t e d , however, t h a t a t -30°C, t h i s v a l u e c o n t r a c t s b y about 0.02A. T a k i n g t h i s c o n t r a c t i o n i n t o a c c o u n t , a c e l l volume o f 5088.54A i s o b t a i n e d , and t h i s y i e l d s t h e v a l u e s o f t h e p's r e q u i r e d i n e q u a t i o n ( 4 l ) . The v a l u e o f ^\pV^^ v above i s c a l c u l a t e d t o be 0.17G 2. S u b t r a c t i n g t h i s v a l u e f r o m the 2 2 0.35G o b t a i n e d e x p e r i m e n t a l l y , one o b t a i n s a r e s o n a b l e e s t i m a t e o f 0.18G f o r t h e " g u e s t - c a g e " i n t e r a c t i o n s i n S n i i n g l e cage. The SFg m o l e c u l e i s s p h e r i c a l , and i t s h o u l d be i n t e r e s t i n g to examine a model i n w h i c h SFg r o t a t e s i s o t r o p i c a l l y i n a h e x a k a i d e c a h e d r a l c a v i t y w h i c h i s a l s o s p h e r i c a l t o a f i r s t a p p r o x i m a t i o n . T h i s " s p h e r e — w i t h i n - a - s p h e r e " m o d e l , o f w h i c h F i g . 15 ( a ) r e p r e s e n t s an extreme case o f t r a n s l a t i o n o f t h e i n n e r sphere so t h a t t h e two c e n t r e s o f g r a v i t y a r e n o t c o i n c i d e n t , may now be t r e a t e d i n t h e f o l l o w i n g way. I n F i g . 1 5 ( a ) , R^ i s t h e r a d i u s of t h e SFg sphere w h i c h , f r o m i t s m o l e c u l a r d i a m e t e r r e p o r t e d i n T a b l e V I , i s 2.88A. Rg i s t h e r a d i u s o f a 25 26 h e x k a i d e c a h e d r a l c a v i t y , w h i c h i s known fr o m t h e l i t e r t u r e v a l u e ' o f 6.70A f o r t h e d i a m e t e r o f t h i s c a v i t y . Rg i s t h e d i s t a n c e f r o m t h e c e n t r e of SFg t o t h e p e r i p h e r y o f t h e c a v i t y , and ' c ' i s t h e c e n t r e - t o - c e n t r e d i s p l a c e m e n t between t h e two s p h e r e s . I n the l i m i t o f f r e e r o t a t i o n o f t h e i n n e r sphere o n l y , t h e F-H d i s t a n c e , .^r) , a v e r a g e d o v e r t h e e n t i r e i n n e r 84 s u r f a c e o f t h e l a r g e r s p h e r e , i s g i v e n by TT —3 1 P —3 2 / r >= 5 r . 2TjRn . S i n x d v From F i g . 15 ( b ) , r d r = RgR^Sin^dx ( a ) F i g . 15 " S p h e r e - w i t h i n - a - s p h e r e " Model of S F g - H y d r a t e ( F i g u r e n o t t o s c a l e ) . 82 so t h a t , f o r t h e l i m i t s x = 0 and ^ = rr> we have ~ ,+c-t~ '-2 y -3v 1 <r > = r d r 2(R 0+c)R, j 2 1 R g + c ^ ( R 2 + c ) [ ^ R 2 + c ) 2 - R 2 1 The t e r m i n t h e Van V l e c k e x p r e s s i o n (16) c o r r e s p o n d i n g t o F-H i n t e r a c t i o n s t h e n becomes E q u a t i n g e x p r e s s i o n (42) t o t h e 0.18G o b t a i n e d above, a v a l u e o f ~ 0.5A i s d e r i v e d f o r c. T h i s , t h e n , i s a measure o f t h e i s o t r o p i c " r a t t l -i n g " of an SFg m o l e c u l e i n i t s h y d r a t e cage. I t i s i n t e r e s t i n g to compare 135 t h i s v a l u e w i t h v a l u e s o f ~ 2A r e p o r t e d r e c e n t l y f o r t h e h i g h e r - t e m p e r a -t u r e " r a t t l i n g d i s t a n c e s " of some g u e s t m o l e c u l e s i n B - q u i n o l c l a t h r a t e s . Q u i t e o b v i o u s l y , r a t t l i n g of SFg i n i t s w a t e r - c a g e i s o c c u r r i n g f a s t enough t o l e a d to t h e h i g h l y smeared out e l e c t r o n d e n s i t y p eaks o b s e r v e d i n the 7 e a r l i e r x - r a y d i f f r a c t i o n s t u d y of S F g - k y d r a t e . ( c ) E t h y l e n e O x i d e i n t h e "Type I " H o s t L a t t i c e : A specimen o f t h i s gas h y d r a t e , p r e p a r e d u s i n g DgO i n p l a c e o f HgO, was a n a l y s e d t o be (CgH^O)^ TD^O, i n rou g h agreement w i t h t h e c o m p o s i -57 t i o n e s t a b l i s h e d by a r e c e n t x - r a y c r y s t a l s t r u c t u r a l s t u d y . P r o t o n r e s o n a n c e s p e c t r a were r e c o r d e d w i t h t h i s specimen h e l d a t t h r e e t e m p e r a -t u r e s ; ( i ) 77°K, t h e l o w e s t a t t a i n a b l e i n t h i s s t u d y , ( i i ) 158°K, t h e t e m p e r a t u r e a t w h i c h r e s u l t s o f d i e l e c t r i c s t u d i e s ^ a r e a v a i l a b l e , and ( i i i ) 250°K, a t w h i c h c r y s t a l s t r u c t u r a l d a t a a r e a v a i l a b l e i n c o m p l e t e d e t a i l f o r t h i s c l a t h r a t e . A v e r a g e v a l u e s f o r t h e l i n e w i d t h s and t h e second moments fr o m f o u r s p e c t r a r e c o r d e d a t each o f t h e above t e m p e r a t u r e s w i t h t h e i r l i m i t s o f d e v i a t i o n a r e q u o t e d i n T a b l e X I . T a b l e X I E x p e r i m e n t a l L i n e w i d t h s and Sec o n d Moments f o r E t h y l e n e O x i d e i n t h e Deu-t e r a t e d H y d r a t e L a t t i c e . T e m p e r a t u r e , °K L i n e w i d t h , G Second Moment, G 2 77 2.6±0.10 4.70 ± 0.20 158 1.0±0.10 0.90 i 0.25 250 0.91±0.05 - 0 . 3 2 The r e c o r d e d l i n e shapes were s t r u c t u r e l e s s and r o u g h l y G a u s s i a n a t a l l . t h e s e t e m p e r a t u r e s . C a l c u l a t e d second moments f o r C-C = 1.54A and C-H = 1.09A a r e l i s t e d i n T a b l e X X I f o r c e r t a i n m o t i o n a l modes o f e t h y l e n e o x i d e . T a b l e X I I l y l e n e 0} L a t t i c e . T h e o r e t i c a l H* Second Moments f o r E h y l e n e O x i d e i n the D e u t e r a t e d H y d r a t e Type of " I n t r a - G u e s t " 2 " I n t e r - G u e s t " 2 T o t a l second M o t i o n Second Moment, G Second Moment, G Moment, G R i g i d M o l e c u l e -no m o t i o n 13.70 - 2 . 0 0 - 1 5 . 7 0 R e o r i e n t a t i o n a r o u n d C-C bond 3.50 - 1.60 - 5.10 R e o r i e n t a t i on around C„ A x i s t h r o u g h 0 5.25 - 1.60 - 6.85 Random t u m b l i n g 0 ~ 1.00 — 1.00 Comparison o f t h e e x p e r i m e n t a l v a l u e s w i t h t h o s e found i n t h i s T a b l e r e v e a l s e x t e n s i v e m o t i o n o f t h e g u e s t i n i t s l a t t i c e s i t e , a r e s u l t 84 r e a d i l y t o be a n t i c i p a t e d f r o m t h e r e f e r e n c e s c i t e d above. A t 77°K, t h e absence o f f i n e s t r u c t u r e i n t h e l i n e shape and t h e c o r r e s p o n d i n g second 2 moment o f 4.70 ±- 0.20JEr s t r o n g l y s u g g e s t the o c c u r r e n c e o f r a p i d r e o r i e n t a -t i o n s o f e t h y l e n e o x i d e m o l e c u l e s around t h e i r C-C bond a x e s . A f u r t h e r 2 o d e c r e a s e .in second moment t o 0.90£0.25G a t 158 K e v i d e n t l y r e s u l t s f r o m r a p i d t u m b l i n g o f t h e s e m o l e c u l e s around random axes i n symm e t r i c f o r c e f i e l d s , a p p a r e n t l y w i t h l i t t l e o r no h i n d r a n c e f r o m the w a l l s o f the c a v i t y . T h i s m o t i o n , w h i c h w o u l d n e c e s s i t a t e a change i n t h e d i p o l e moment o f t h e m o l e c u l e , i s a l m o s t c e r t a i n l y t h e same m o t i o n w h i c h gave r i s e t o t h e -11 v e r y s h o r t d i e l e c t r i c r e l a x a t i o n o f about 10 s e c . , i n t h e s t u d i e s o f D a v i d s o n e t a l . , ^ . A t 250°K, t h e a b s o r p t i o n l i n e s t i l l p o s s e s s e d a v e r y d e f i n i t e w i d t h (0.91G ) d i s t i n g u i s h a b l e above f i e l d i n h o m o g e n e i t i e s , and t h e second moment 2 o f t h e s i g n a l was ~ 0.32G . By a n a l o g y w i t h SFg h y d r a t e , one w o u l d t h e r e f o r e e x p e c t t h a t a c e r t a i n degree of r a t t l i n g o f t h e g u e s t m o l e c u l e s has now s e t i n , w h i c h i s r a p i d enough t o be seen a t t y p i c a l n.m.r., f r e q u e n c i e s . The 57 x - r a y c r y s t a l s t r u c t u r e s t u d y a t t h i s t e m p e r a t u r e , however, has c l e a r l y e s t a b l i s h e d t h a t t h e g u e s t m o l e c u l e behaves as a h i n d e r e d a x i a l r o t o r between two d i s t i n g u i s h a b l e o r i e n t a t i o n s . These two r e s u l t s need h a r d l y be c o n s t r u e d t o c o n t r a d i c t each o t h e r because o f t h e v e r y d i f f e r e n t t i m e s c a l e s i n v o l v e d i n n.m.r., and x - r a y f r e q u e n c i e s . (d) P r o p a n e i n t h e "Type I I " H o s t L a t t i c e A sample of (CgHgj^DgO, w h i c h c o r r e s p o n d e d t o t h e maximum c o m p o s i -t i o n e x p e c t e d f o r a ty p e I I gas h y d r a t e , was p r e p a r e d . Q u i t e u n l i k e t he o t h e r gas h y d r a t e s r e p o r t e d i n t h i s c h a p t e r , ( C g H g ) ^ DgO seemed t o have a range o f g u e s t - t o - h o s t r a t i o s f o r w h i c h t h e s t r u c t u r e was s t a b l e . I t was p o s s i b l e , f o r example, t o pump s m a l l q u a n t i t i e s o f the g u e s t o f f t h e c l a t h -85 r a t e w i t h o u t d i s r u p t i n g the s t r u c t u r e o f t h e h o s t l a t t i c e . The c l a t h r a t e was m a i n t a i n e d f r o z e n i n a d r y - i c e b a t h , pumped on f o r a s h o r t p e r i o d , and t h e n a n n e a l e d a t d r y - i c e t e m p e r a t u r e f o r a few d a y s . R a t i o s o f g u e s t - t o -h o s t r a n g i n g between 65$ and 95$ c o u l d be a c h i e v e d by t h i s t e c h n i q u e . The c l a t h r a t e was n o t s t a b l e , however, when more t h a n 40$ o f g u e s t m o l e c u l e s had been pumped o f f t h e h o s t l a t t i c e . D e r i v a t i v e l i n e s h a p e s o f p r o t o n - r e s o n a n c e were r e c o r d e d f o r two samples o f t h e d e u t e r a t e d h y d r a t e , one o f w h i c h had i t s t h e o r e t i c a l complement o f t h e g u e s t s p e c i e s and t h e o t h e r was ^ 65$ w i t h r e s p e c t t o i t . N a r r o w and s t r u c t u r e l e s s G a u s s i a n l i n e s were o b t a i n e d f o r b o t h specimens, and s a t u r a t i o n e f f e c t s were a b s e n t . Second moments, o b t a i n e d by n u m e r i c a l i n t e g r a t i o n o f the s p e c t r a , 116 were c o r r e c t e d f o r t h e e f f e c t s o f t h e m o d u l a t i o n f i e l d and p l o t t e d a g a i n s t t e m p e r a t u r e as shown i n F i g . 16, w h i c h c o n t a i n s t h e e n t i r e s e t of d a t a p o i n t s c o l l e c t e d i n the e x p e r i m e n t e x c e p t i n g t h o s e a t 77°K. To h e l p r a t i o n a l i z e t h e b e h a v i o u r s o f ^ t h e e n c l a t h r a t e d s p e c i e s , c a l c u l a t i o n s were made of t h e o r e t i c a l " r i g i d " a n d " r e o r l f i r i j f c l n g " m o l e c u l a r second moments f r o m c a r t e s i a n hydrogen atom c o o r d i n a t e s g e n e r a t e d t r i g o n o m e t r i c a l l y w i t h t h e a i d o f a F o r t r a n I V computer programme ( s e e A p p e n d i x l l ) , assuming t h e 136 v a l u e s of C-H d i s t a n c e s and HCH a n g l e s r e p o r t e d by L i d e f r o m t h e m i c r o -wave s p e c t r u m o f propane. The hyd r o g e n atom c a r t e s i a n s t h u s computed a r e l i s t e d i n T a b i e X I I I , a n d t h e c a l c u l a t e d sums o f " i n t r a - g u e s t " and " i n t e r -g u e s t " second moments f o r v a r i o u s t y p e s o f m o t i o n i n T a b l e X I V . T a b l e X I I I C a r t e s i a n Hydrogen C o o r d i n a t e s G e n e r a t e d F o r t h e n-Propane C h a i n o H y d r o g e n Atom No. ~x y z~ 1 2 0.424 1.912 0,000 -0.526 0,000 -0.911 F i g . 16 00 OS 87 T a b l e X I I I C o n t d . Hydrogen Atom No. X y z 3 1.912 -0.526 0.911 4 1.681 1.978 -0.911 5 1.681 1.978 0.911 6 3.965 2.378 -0.911 7 3.965 2.378 0.911 8 3.965 3.956 0.000 T a b l e X I V T h e o r e t i c a l H* Second Moments f o r Propane i n t h e D e u t e r a t e d H y d r a t e L a t t i c e . Type of M o t i o n " I n t r a - G u e s t " g " I n t e r - G u e s t " 2 T o t a l Second Moment, G Second Moment, G Second 2 Moment,G R i g i d M o l e c u l e - no M o t i o n Cg R e o r i e n t a t i o n one M e t h y l group o f ) up ) Cg R e o r i e n t a t i o n of b o t h M e t h y l groups M e t h y l s r i g i d , b u t ) m o l e c u l e R e o r i e n t -i n g a b o u t C 2 v a x i s Cg M e t h y l R e o r i e n t a - ) t i o n s s u p e r p o s e d ) on M o l e c u l a r R o t a t i o n a bout C„ a x i s 2v Random T u m b l i n g 20.49 13.68 7.22 3.30 2.10 4.53 1.13 0.65 0.65 > 25.02 14.00 8.00 4.43 2.75 0.65 One n o t i c e s i n F i g . 16 t h a t , a t l o w e r t e m p e r a t u r e s , t h e r e i s l i t t l e d i f f e r e n c e i n t h e se c o n d moments of t h e " t h e o r e t i c a l l y - f i l l e d " and " 6 5 $ - f i l l e d " c l a t h r a t e s p e c i m e n s . A t 77°K, t h e second moment has an 88 2 aver a g e v a l u e o f 5.50G . I f t h i s i s compared w i t h t h e v a l u e s l i s t e d i n T a b l e X I V , i t becomes i m m e d i a t e l y a p p a r e n t t h a t propane i s n o t " r i g i d " i n o i t s h y d r a t e l a t t i c e p o s i t i o n s even a t 77 K. From t h e r a t h e r sharp s l o p e i n t h e second moment c u r v e c e n t r e d a r o u n d 8 0 ° ^ i t w o u l d appear t h a t t h e g u e s t m o l e c u l e s a r e , i n f a c t , i n the m i d s t of a m o t i o n a l t r a n s i t i o n a t 77°K. Q u i t e i n t e r e s t i n g l y , t h e t e m p e r a t u r e o f t h e m i d d l e r e g i o n o f t h i s t r a n s i t i o n c o r r e s p o n d s v e r y w e l l w i t h the known t r i p l e p o i n t t e m p e r a t u r e o f 85. 44°K 137 f o r propane . However, s i n c e s e c o n d moment v a l u e s a r e o n l y o f q u a l i t a t i v e s i g n i f i c a n c e i n t h e r e g i o n i n w h i c h m o t i o n a l n a r r o w i n g i s s t i l l o c c u r r i n g , a c l e a r - c u t c o m p a r i s o n between o b s e r v e d and c a l c u l a t e d v a l u e s o f t h e second moment i n t h i s r e g i o n i s n o t p o s s i b l e . I m m e d i a t e l y a f t e r t h e t r a n s i t i o n j u s t m e n t i o n e d , a f l a t p l a t e a u o c c u r s i n t h e second-moment v s . t e m p e r a t u r e c u r v e , c o r r e s p o n d i n g t o a v a l u e o f 2.70G ( l i n e w i d t h 1.10G), w h i c h r e m a i n s f a i r l y s t e a d y u n t i l t h e t e m p e r a t u r e r e a c h e s 150°K. From T a b l e X I V , i t i s seen t h a t t h i s m o t i o n may be v e r y w e l l a p p r o x i m a t e d by s u g g e s t i n g r e o r i e n t a t i o n s of t h e whole m o l e -c u l e s u p e r p o s e d on C g - r o t a t i o n of t h e two m e t h y l g r o u p s . T h i s c o o r d i n a t e d m o t i o n , t h e n , i s p r e s u m a b l y o c c u r r i n g a t t e m p e r a t u r e s between 100°K and 150°K f o r b o t h t h e specimens i n v e s t i g a t e d . I t i s perhaps of i n t e r e s t to 136 r e c a l l a t t h i s p o i n t t h e r e s u l t s o f L i d e ' s microwave s t u d i e s " on gaseous p r o p a n e . L i d e ' s s t u d i e s have e s t a b l i s h e d t h a t t h e e q u i l i b r i u m c o n f i g u r a t i o n o f propane has a r a t h e r t h a n a symmetry. I t has been i n f e r r e d f r o m t h i s h i g h e r symmetry t h a t t h e two o u t - o f - p l a n e hydrogen atoms o f one m e t h y l group a r e " e c l i p s e d " w i t h r e s p e c t t o t h e two c o r r e s p o n d i n g atoms on t h e o t h e r m e t h y l g r o u p . Hence t h e r e i s a b a r r i e r t o t h e r o t a t i o n of t h e m e t h y l g r o u p s . L i d e a r g u e s , however, t h a t any f a c t o r s w h i c h f a v o u r a s t r o n g r e p u l s i o n o f t h e m e t h y l hydrogen atoms w o u l d t e n d t o f o r c e m e t h y l group rotation and thus lead to a Cg molecular symmetry. The results of the present study, insofar as they provide evidence 'for rotation of both the end methyls of propane at radiofrequencies, should therefore lead one to the interesting conclusion that propane is forced to assume a Cg symmetry inside the hexakaidecahedral host cavity. Above 160°K, the slope of the second moment curve once again changes abruptly for both specimens. Clearly, the second moment value is < 1G at _ 210°K. Here, then, the guest molecules are very likely executing overall random tumbling motions, with the result that the "intra-2 guest" second moment vanishes, and only the "inter-guest" value of 0.65G , Table XIV, makes any measurable Contribution to the total second moment. A close inspection of the set of data points obtained for the two samples at temperatures above 210°K reveals that,at at least three temperatures, second moments for the sample with the lower guest content lie distinctly below the mean second moment curve. This is particularly strikingly illustrated by the cluster of points obtained beyond 240°K for the two specimens. A plausible explanation for these observations may be attempted in terms of guest molecules overcoming the caging effects of the hydrogen-bonded lattice and diffusing into neighbouring cavities. Assuming this to be the line-narrowing mechanism at 240°K for both specimens..one would indeed expect guest-diffusion to occur more rapidly in a "partially-empty" lattice than in a "filled" lattice. Furthermore, diffusion may be considered a very reasonable preamble to the eventual decomposition of the clathrates liberating propane at 276°K. To elucidate this point further, a few absorption spectra were obtained beyond 240°K for the two samples under "high resolution" conditions. These spectra were obtained at 56.4 Mc/sec, on the V-4200/4300 dual purpose spectrometer. Very low rf. levels were chosen, and the "slow sweep" unit of the instrument was used in conjunction with the flux stabilizer to sweep the magnetic field through resonance. The samples produced their own rather broad and weak side-bands when an audiomodulation of 2000 cps. was employed, and these were used to measure the linewidths of the Lorentzian signals. Since "spinning" the samples was not practicable owing to the use of the low-temperature wide-line probe the signals included some inhomegeneity broadening. Linewidths accurate to *10$,, could be measured. In Fig. 17 are shown the linewidths vs. temperature obtained for the two samples. From these results, it is possible to evaluate the activation energies for diffusion, ^ f t , for the two clathrate specimens. Since the absorption line shapes have been uniformly Lorentzian, equation (28) is applicable ' •'. , where the subscript "rigid" will now refer to the molecule "rigid" with respect only to diffusion. Since the linewidth 6H = const x f _1_ ^  for a given line shape, and since T = = 12 ' the correlation time at infinite temperature, the diffusional activation energy may be calculated by linearizing equation (28) to the form IngH = const + " (43) lei From the slopes of the experimental lines in Fig. 17, E^^^j values of 1.40 ± 0.02 kcal/mole and 0.75 * 0,05 kcal/mole were calculated for the "theoretically filled" and "65$ - filled" clathrate specimens, respectively, thus supporting the diffusion mechanism proposed above. 134 Recently, Goldberg has calculated the diameter of the rather oblate "escape orifice" of a Type II hydrate cavity to be ^ 3.OA. In1 Goldberg's calculation, the term "escape orifice" has been defined as the 91 10.0 r — i 1 1 1 1 — i 1 1 1 1 1 1 1 rq 5.0 h 3.0 2.0 1.0 co CO g 0.5 x \— 0.3 Q £ 0.2 ixJ z J 0.1 .05 .03 .02 • =Wide Line ^'Theoretically Filled" • •High Resolution)Specimen o«*Widel_ine ~\. ^ „ , ^65% Filled Specimen honj = High Resoluti  E d i f f ° 1 , 4 4 0 0 2 kcal /mole EM**"075±0.05 kcal/mole alTT .0 I L _ l I I J I I I I 1 1 1 1——I 1 2 0 0 210 220 230 240 2 5 0 260 270 T E M P E R A T U R E , °K TEMPERATURE DEPENDENCE OF LINE WIDTHS ( C 3 H 8 ) ~ D E U T E R A T E D HYDRATE F i g . 17 92 c i r c l e t h a t i s t a n g e n t i a l t o t h e oxygens i n t h e p l a n e o f an o b l o n g h e x a g o n a l f a c e , and, though i t i s n o t c l e a r how t h i s v a l u e has been a r r i v e d a t , t h e method has u n d o u b t e d l y t a k e n i n t o a c c o u n t the p h y s i c a l e x t e n t o f t h e e l e c t r o n wave f u n t i o n s a s s o c i a t e d w i t h t h e c a g i n g m o l e c u l e s . S i n c e ; i t i s h a r d t o e n v i s i o n t h e d i f f u s i o n t h r o u g h t h i s o r i f i c e o f a m o l e c u l e o f propane 0 135 whose d i a m e t e r i s as l a r g e as 6.28A ( T a b l e V i ) , t h e s u g g e s t i o n may be made t h a t t h e hydrogen-bonded w a l l s o f t h e h e x a k a i d e c a h e d r a l cage a r e pushed outward by the g u e s t m o l e c u l e , and s t r e t c h f a r enough t o w i d e n t h e 135 "escape o r i f i c e " . By a n a l o g y w i t h t h e r e c e n t l y r e p o r t e d c a s e o f the g -q u i n o l c l a t h r a t e of xenon, one wouldl^expect t o see t h e e f f e c t o f t h e h o s t l a t t i c e h y d r o g e n bonds i n t h e f a r - i n f r a r e d s p e c t r u m o f propane h y d r a t e . Meausrements o f T-^ , t h e p r o t o n 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 These measurements were made f o r t h e " t h e o r e t i c a l l y f i l l e d " c l a t h -r a t e f r o m 77°K upwards. B o t h a d i a b a t i c f a s t passage and p r o g r e s s i v e s i g n a l s a t u r a t i o n methods were u s e d , and f o u n d t o g i v e e s s e n t i a l l y t h e same t r e n d i n T^ v s . — • , as i l l u s t r a t e d i n F i g . 18. Owing t o t h e l i m i t a t i o n s imposed on t h e s e methods by t h e e x p e r i m e n t a l c o n d i t i o n s d e s c r i b e d i n c h a p t e r V, T^'s l e s s t h a n 50 m i l l i s e c o n d s c o u l d n o t be measured, and t h i s i n t u r n l i m i t e d measurements to 77°K - 110°K, i . e . , t h e r e g i o n o f r e o r i e n t a t i o n a l m o t i o n s o n l y . 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 s c o u l d n o t be o b t a i n e d f o r t h e d i f f u s i o n r e g i o n . S i n c e second moment d a t a have d e m o n s t r a t e d t h e e x i s t e n c e o f m o l e c u l a r r e o r i e n t a t i o n s about t h e Cg a x i s and s u p e r p o s e d on m e t h y l r e -o r i e n t a t i o n s i n t h e 77°K-110°K r e g i o n , i t may be s p e c u l a t e d t h a t s p i n -l a t t i c e r e l a x a t i o n o c c u r s by a c o m p o s i t e o f t h e s e two t y p e s o f r e o r i e n t a -t i o n s . I n d e e d , as i s w e l l k n o w n * * ^ , o v e r a l l m o l e c u l a r r e o r i e n t a t i o n p r o c e s s e s , when p r e s e n t , p r o v i d e a more p o w e r f u l means o f s p i n - l a t t i c e 1.0 0.8 0.6 0.4 0.2 -0 - 0 2 -0.4 O _ <5" O -1 -0.6 - Q 8 -1.0 -12 -14 T, MEASUREMENTS ON 8 (C H )~DEUTERATED HYDRATE • E A C T = 1.70 ± 0 . 0 8 K C A L / M O L E ADIABATIC FAST PASSAGE -O—PROGRESSIVE SATURATION 10 II 12 13 14 r 1 rKf ' x io 3 F i g . i s C D co r e l a x a t i o n t h a n s i m p l e m e t h y l r e o r i e n t a t i o n s . A c c o r d i n g l y , one w o u l d n o t e x p e c t t o see the i d e a l "V-shaped" v e r s u s ~ c u r v e c h a r a c t e r i s t i c " o f a s i n g l e c o r r e l a t i o n t i m e T c i n g o i n g f r o m ^T,. « 1 t o u ) o T c j ^ > l . T h a t a b r o a d and s h a l l o w minimum i s i n d i c a t e d i n t h e T j - c u r v e i n F i g . 18 a t t h e upper t e m p e r a t u r e l i m i t w o u l d seem t o l e n d s u p p o r t t o ;the argument, b u t t h i s p o i n t must be c o n s i d e r e d w i t h some r e s e r v a t i o n s s i n c e t h e s e t o f T^ measure-ments shown i n t h e n o n - l i n e a r r e g i o n o f F i g . 18 a r e s u b j e c t t o c o n s i d e r a b l e e x p e r i m e n t a l l i m i t a t i o n s . From the more r e l i a b l e l i n e a r r e g i o n ; i n F i g , 18, a t h e r m a l a c t i v a t i o n energy b a r r i e r o f 1.70 ± 0,08 k c a l / m o l e has been c a l -c u l a t e d f o r t h e c o m p o s i t e m o t i o n s d i s c u s s e d above. 95 CHAPTER V I I THE HALBHETHANE HYDRATES 7.1 O u t l i n e I n t h i s c h a p t e r , a c o m p a r a t i v e s t u d y has been made o f t h e p r o t o n m a g n e t i c r e s o n a n c e of t h r e e monohalomethanes (CHgX, X = C l , B r , l ) i n t h e i r h y d r a t e c a v i t i e s a t t e m p e r a t u r e s f r o m 77°K t o a l i t t l e b elow t h e i r m e l t i n g p o i n t s . I n a d d i t i o n , t h e b e h a v i o u r o f a d i h a l o m e t h a n e , C H ^ C l g , i n a "Type I " h y d r a t e h o s t l a t t i c e has been i n v e s t i g a t e d o v e r a l i m i t e d l o w - t e m p e r a t u r e r a n g e . 2.2 R e s u l t s and C o n c l u s i o n s ( a ) The Monohalomethane H y d r a t e s : Heavy w a t e r h y d r a t e s c o r r e s p o n d i n g t o t h e t h e o r e t i c a l c o m p o s i t i o n s were formed between DgO and pure g u e s t s p e c i e s . The a c t u a l c o m p o s i t i o n s and m e l t i n g p o i n t s o f t h e s e specimens a r e l i s t e d , a l o n g w i t h p r e v i o u s l y known d a t a , i n T a b l e XV. 96 T a b l e XV D a t a on t h e Monohalomethane H y d r a t e s Guest 1 (Type of H y d r a t e ) D i s s o c i a t i o n "EV.of H y d r a t e a t 273°K, mm. R e f . ( 1 2 ) D e c o m p o s i t i o n Temp, a t 1 atm., °K H y d r a t e X D e u t e r a t e d Ref.1 1 2A H y d r a t e , \ T h i s Work T h e o r e t i c a l C o m p o s i t i o n f r o m P r e v i o u s s t u d i e s Maximum C o m p o s i t i o n , t h i s work. C H g C l ( l ) 311 280.5 287t2 ( C H g C l ) , J . 7 H 2 0 ( C H g C l ) ^ 7 . 9 D 2 0 C H g B r ( l ) 187 284.1 285*2 (CHgBr)_7.7H 20 (CHgBr)_8.15D 20 C H g l ( H ) 74 277.3 278±2 ( C H g l ) _ 17B 20 ( C H g l ) _ 18D 20 A t low t e m p e r a t u r e s , v e r y b r o a d G a u s s i a n l i n e s were o b t a i n e d f o r c r u s h e d c r y s t a l l i n e specimens of a l l t h e t h r e e c l a t h r a t e s , and t h e s e d i s -p l a y e d c o n s i d e r a b l e n a r r o w i n g as the t e m p e r a t u r e was r a i s e d . A t 77°K, t h e l i n e shapes f o r the C H g C l - and C H g l - c l a t h r a t e s had no f i n e - s t r u c t u r e . V e r y i n t e r e s t i n g l y , t h e CHgBr- c l a t h r a t e a t 77°K y i e l d e d a d e f i n i t e t r i p l e t shape, w h i c h i n d i c a t e d t h r e e i n t e r a c t i n g p r o t o n s s p a t i a l l y a r r a n g e d a t t h e v e r t i c e s of an e q u i l a t e r a l t r i a n g l e . T h i s was by f a r t h e most r e m a r k a b l e f e a t u r e among t h e l i n e shapes o b s e r v e d a t low t e m p e r a t u r e . The t r i p l e t l i n e shape i s r e p r e s e n t e d i n F i g . 19 ( a ) . The s e t t i n g s o f t h e o u t p u t c i r c u i t o f t h e s p e c t r o m e t e r i n d i c a t e d i n t h i s d i a g r a m were t y p i c a l f o r a l l t h r e e c l a t h r a t e s w i t h i n t h e t e m p e r a t u r e range i n v e s t i g a t e d . F i g u r e s 19 (b) and 19 ( c ) i n d i c a t e n a r r o w e d l i n e s f o r the CHgBr- c l a t h r a t e a t two h i g h e r t e m p e r a t u r e s and t h e s e , a g a i n , were t y p i c a l , w i t h r e s p e c t t o b o t h s i g n a l - t o - n o i s e r a t i o and l i n e shape, o f thos e o b t a i n e d a t c o r r e s p o n d i n g t e m p e r a t u r e s f o r t h e o t h e r two c l a t h r a t e s . 7 7 ° K Output Time Constant: 3 sec i i i i i i i 5 Gauss (0 I 5 7 ° K Output Time Constant = 3 sec 10 L_i_l 0 I Gauss 276 ° K Output Time Constant = 10 sec F i g . 19 T y p i c a l H Resonance S p e c t r a o f t h e Bromomethane C l a t h r a t e . (Two-Headed A r r o w s R e p r e s e n t M o d u l a t i o n Used) The CHgBr- c l a t h r a t e a t 77°K had a second moment v a l u e o f / 2 2 21.64 ^ 0.40G , w h i c h i s q u i t e comparable t o t h e v a l u e o f 22.40G o b t a i n e d f o r an i s o l a t e d " r i g d " m e t h y l group ( T a b l e X V I ) f o r a normal C=H d i s t a n c e o o f 1.09A. T h i s f a c t , combined w i t h t h e t r i p l e t s t r u c t u r e o f the l i n e , w o u l d seem t o p r o v i d e a most p e r s u a s i v e argument f o r t h e e x i s t e n c e o f a " f i x e d " m e t h y l group i n t h e h y d r a t e c a v i t y a t 77°K. However, one ought t o c o n s i d e r " i n t e r - g u e s t " c o n t r i b u t i o n s t o t h e m e t h y l second moment fr o m m e t h y l n e i g h -b o u r s i n a d j a c e n t c a v i t i e s , and when t h i s i s done, an i n c r e a s e d t h e o r e t i c a l 2 second moment of 23.71G ( T a b l e X V l ) i s o b t a i n e d . C o n t r i b u t i o n s t o t h e second moment f r o m t h e d e u t e r i u m n u c l e i and t h e q u a d r u p o l e bromine n u c l e u s a r e h e r e assumed n e g l i g i b l e . The v a l u e o b s e r v e d f o r the CHgBr- c l a t h r a t e a t 77°K i s 2.07G 2 l e s s t h a n t h i s — r o u g h l y a 9$ r e d u c t i o n . T a b l e X V I T h e o r e t i c a l Second Moments f o r M e t h y l "Guest" i n D e u t e r a t e d H y d r a t e s . M e t h y l G r . , I n t r a m o l e c u l a r Second Moment, t h y l G r . , I n t e r m o l e c u l a r Second Moment, G 2. R i g i d C g - A x i s R o t a t i o n R i g i d N e i g h b o u r s R o t a t i n g N e i g h b o u r s Type I H y d r a t e Type I I H y d r a t e Type I H y d r a t e Type I I H y d r a t e 22.40 5.60 1.31 0.35 0.31 0.10 The v a l u e s o f the second moments o b t a i n e d a t 77°K. f o r t h e 2 2 C H g C l - and C H g l - c l a t h r a t e s were 20.80G and 19.98G , r e s p e c t i v e l y . S i g n i -f i c a n t l y , t r i p l e t s t r u c t u r e was a b s e n t i n t h e s e two c a s e s . The " r i g i d m e t h y l " t h e o r e t i c a l second moment f o r the C H g C l - c l a t h r a t e , i n c l u d i n g 99 2 2 "inter-guest" contributions, will again be 23.716 , but it will be^  22.75G (Table XVI) for the CHgl- clathrate, since neighbouring "guestfe" are a little farther apart in the Type II ice lattice. Clearly, the experimental values at 77°K are again less than those expected for fixed guest molecules in these structures, and some motion of these molecules is therefore indicated. It may be suspected that this motion may consist of oscillatory rather than reorientational movement of the methyl groups in the three clathrates, since reorientation of either a methyl group or a whole molecule would cause much smaller second moments than have been obtained at 77°K. Therefore, possible oscillatory motions of the methyl groups of the guests may now be examined. Zero-point intramolecular oscillations of the protons in the methyl groups will, of course, be the simplest kind of motion. Earlier investiga-tors*3^""*^ have shown about 10$ reduction in second moment due to such oscillations of simple groups of 3 and 4 resonant nuclei. A more refined 95 calculation of such effects is due to Ibers and Stevenson , who utilized a complete normal coordinate treatment for computing the effect of 2 ^ —6 vibrations on the (3Cos Yii-l) and r~ • factors in the second moment formula, equation (15). In their work, the changes in r. . arising from intra-molecular stretching anddeformational motions were expressed in terms of the normal coordinates, Q ,^ for the ground vibrational states. The were given by <<>i> -8 » f i v c / where I was the moment of inertia of the oscillating group. Reduction factors to the methyl seconds moment, calculated using the v values for the stretching (3000-2700 cm"*) and de format tonal (~1450 cm""*) modes of the 141 aliphatic methyl group , indicate that the ground state oscillatory 100 m o t i o n s h a r d l y a f f e c t t h e second moment; On t h e o t h e r hand, t h e m e t h y l " r o c k i n g " mode, w h i c h c o r r e s p o n d s t o movement o f t h e m e t h y l group as a whole u n i t w i t h o u t any i n t e r n a l d e f o r m a t i o n , might s t r o n g l y a f f e c t t h e " i n t e r -g u e s t " second moment. The f r e q u e n c y o f t h i s m o t i o n depends g r e a t l y on t h e r e s t o f t h e m o l e c u l e w h i c h i s m e c h a n i c a l l y i n v o l v e d i n t h e v i b r a t i o n . R e d u c t i o n s t o the o v e r a l l second moment computed f o r m e t h y l " r o c k i n g " 142 f r e q u e n c i e s i n the t h r e e halomethanes a c c o r d i n g t o t h e above method were fo u n d t o l i e between 1 and 2$ of t h e r e s p e c t i v e t o t a l v a l u e s . These r e d u c t i o n s a r e i n a d e q u a t e t o account f o r t h e o b s e r v e d d e c r e a s e i n second moment a t 77°K. R o t a t i o n a l o s c i l l a t i o n s of m e t h y l groups about C-Halogen bonds may be n e x t examined. F o l l o w i n g t h e u s u a l c o n v e n t i o n ^ ' t h e s e o s c i l l a t i o n s may be assumed t o be u n c o u p l e d f r o m the i n t r a — m o l e c u l a r s t r e t c h i n g and d e f o r m a t i o n a l v i b r a t i o n s . A r e d u c t i o n o f about 10$ i n t h e o 9 0 m e t h y l s e c o n d moment c o u l d o c c u r f o r r o t a t i o n a l a m p l i t u d e s o f ~50 , b u t su c h l a r g e a m p l i t u d e s a r e u n r e a l i s t i c s i n c e t h e y can s e r i o u s l y d i s t o r t t h e deuterium-bonded cage. A p o s s i b l e e x p l a n a t i o n , w h i c h a c c o u n t s f o r t h e se c o n d moment d e f i c i t as w e l l as t h e t r i p l e t s t r u c t u r e a t 77°K f o r t h e CHgBr- c l a t h r a t e , may be a t t e m p t e d as f o l l o w s i ' ^ i ) b y ass u m i n g s m a l l a m p l i t u d e " r o c k i n g " — o r l i b r a t i o n a l o s c i l l a t i o n s - o f t h e m e t h y l groups i n t h e i r c a v i t i e s , where-by t h e " i n t e r - g u e s t " s e c o n d moment i s g r e a t l y r e d u c e d , and ( i i ) by s u p p o s i n g , a d d i t i o n a l l y , t h a t each m e t h y l group i n i t s c a v i t y i s f l a t t e n e d o ut a l i t t l e a l o n g i t s C g - symmetry a x i s due t o a l o w - t e m p e r a t u r e c o n t r a c -t i o n o f t h e deuterium-bonded cage, w i t h a r e s u l t a n t i n c r e a s e o f t h e HCH a n g l e t o 110° 40' and t h e H-H d i s t a n c e t o 1.80A. T h i s new H-H v a l u e w o u l d r e d u c e t h e " i n t r a g u e s t " s e c o n d moment t o w i t h i n t he e r r o r l i m i t s o f t h e 101 e x p e r i m e n t a l v a l u e of 21.64i0.40G . I t may be p o s t u l a t e d a g a i n t h a t s i m i l a r l i b r a t i o n a l o s c i l l a t i o n s o f the m e t h y l groups may cause t h e o b s e r v e d d e c r e a s e f r o m t h e t h e o r e t i c a l second moment f o r t h e C H g C l - and C H g l - c l a t h r a t e s a t 77°K. I n a d d i t i o n , 46 f o r t h e s e two c a s e s one m i g h t t h i n k o f a " r a t t l i n g " m o t i o n o f e n t i r e en-caged m o l e c u l e s s u p e r i m p o s e d on t h e r o c k i n g m o t i o n s of t h e m e t h y l group. T h i s " r a t t l i n g " , o r v e r y l i m i t e d random t r a n s l a t i o n i n t h e ±oc, i y and i z d i r e c t i o n s , even t o the e x t e n t o f moving t h e c e n t r e of mass o f each m o l e c u l e o n l y a b o u t 0.2A f r o m the c e n t r e o f the cage, i s enough t o b r i n g t h e second moments i n l i n e w i t h t h e o b s e r v e d v a l u e s . A l so, t h e s m e a r i n g o u t of t h e t r i p l e t l i n e shape i n t h e s e two c a s e s i s e x p l a i n e d by t h e a s s u m p t i o n o f s u c h a c o m p o s i t e m o t i o n i n v o l v i n g " r o c k i n g " and " r a t t l i n g " . The l a r g e r r e d u c t i o n 2 i n second moment (2.90G ) f o r t h e C H g C l - compared t o the C H g l - c l a t h r a t e 2 (2.77G ) i s o n l y t o be e x p e c t e d i f the p i c t u r e of m e t h y l " r o c k i n g " s u g g e s t e d above i s c o r r e c t . The " r o c k i n g " V i b r a t i o n a l mode i s s t r o n g l y mass-dependent, and due t o t h e g r e a t e r mass o f t h e i o d i n e atom, r o c k i n g a m p l i -t u d e s w i l l be l e s s f o r CHgl and w i l l t h e r e f o r e cause a s m a l l e r r e d u c t i o n i n the s econd moment f o r t h e C H g l - c l a t h r a t e . Above 77°K, t h e r e d u c t i o n i n second moment was q u i t e g r a d u a l f o r a l l t n r e e c l a t h r a t e s . The t r i p l e t f o r t h e CHgBr- c l a t h r a t e was r e p l a c e d by the u s u a l b e l l - s h a p e d c u r v e a r o u n d 90°K. ( F i g . 19 b ) . P e r h a p s a t t h i s t e m p e r a t u r e t h e cage had expanded enough t o p e r m i t some m o l e c u l a r r a t t l i n g . A t 1 5 0 V t h e second moment f o r t h e bromomethane c l a t h r a t e f e l l o f f s h a r p l y and, a t ~155°K, had a t t a i n e d a v a l u e o f 4.70G . C e r t a i n l y , u n r e s t r i c t e d r e o r i e n t a t i o n s of t h e CHg- group about i t s Cg a x i s h a d now s e t i n . S i n c e , however, t h i s m o t i o n by i t s e l f w o u l d o n l y reduce t h e second moment t o 2 5.60G ( T a b l e X V l ) , a f u r t h e r r e d u c t i o n was v e r y l i k e l y c aused by some 102 random t u m b l i n g o f t h e m o l e c u l e as a w h o l e . M o l e c u l a r t u m b l i n g a p p e a r e d t o be t h e main m o t i o n i n t h i s p a r t i c u l a r c l a t h r a t e between 2 0 0 % and 240°K, 2 and a f i n i t e s e cond moment o f 0.80G , w h i c h p e r s i s t e d i n t h i s range a f t e r I c o r r e c t i n g the e x p e r i m e n t a l v a l u e s f o r any s p u r i o u s e f f e c t s due t o t h e 116 m o d u l a t i n g f i e l d , r u l e d o u t any f r e e t r a n s l a t i o n o r " d i f f u s i o n " o f g u e s t m o l e c u l e s . A n a l o g o u s l y , r e o r i e n t a t i o n s o f t h e m e t h y l groups about t h e C g -symmetry a x i s a r e i n d i c a t e d f o r t h e CH„C1- and CH„I- c l a t h r a t e s a t 132°K - - J J and 175°K, r e s p e c t i v e l y . These c l a t h r a t e s , t o o , h a d s m a l l b u t f i n i t e second moments (>0.30G ) a t t e m p e r a t u r e s i n t h e v i c i n i t y o f 260TC. A t the h i g h e s t t e m p e r a t u r e s s t u d i e d , w h i c h were j u s t b e l o w t h e m e l t i n g p o i n t s of t h e s e c l a t h r a t e s , t h e l i n e s became L o r e n t z i a n and had a f i n i t e w i d t h , and p a r t o f t h i s c o u l d have come f r o m a b r o a d e n i n g due t o t h e 100-milligaus3 modula-t i n g f i e l d u s e d . E f f e c t s o f d i f f u s i o n i n t h e s e two c l a t h r a t e s , i f p r e s e n t , were t h e r e f o r e m i s s e d and, a t t h i s s t a g e , t h e g u e s t m o l e c u l e s were a t l e a s t f r e e l y r a t t l i n g i n t h e i r c a g e s . The dependence o f t h e l i n e w i d t h s on t e m p e r a t u r e , and t h e m e l t i n g p o i n t s f o r t h e t h r e e c l a t h r a t e s , a r e shown i n F i g . 20. E a c h c u r v e d i s p l a y s two sharp d r o p s i n t h e l i n e w i d t h s ; t h e f i r s t o f e a c h o f t h e s e u n d o u b t e d l y c o r r e s p o n d i n g , on t h e s t r e n g t h o f t h e e v i d e n c e q u o t e d above, t o m e t h y l r e o r i e n t a t i o n . A c t i v a t i o n e n e r g i e s f o r t h i s m o t i o n were c a l c u l a t e d w i t h t h e a i d o f e q u a t i o n s (24) and ( 2 5 ) . The ' F o r t r a n IV' computer programme, a p p e a r i n g i n A p p e n d i x I , was u s e d . The c a l c u l a t i o n s were p e r f o r m e d on an IBM 7040 computer. V a l u e s o f t h e a c t i v a t i o n e n e r g i e s o b t a i n e d a r e 2.48±0.32, 9.30±0.25, and 6.80±0.50 k c a l s / m o l e f o r the CHgCl-, CHgBr-, and C H g l - c l a t h r a t e s , r e s p e c t i v e l y . I n c a l c u l a t i n g t h e above a c t i v a t i o n e nergy g a l u e s , t h e r e i s an 103 i n h e r e n t a s s u m p t i o n of a s i n g l e c o r r e l a t i o n t i m e f o r the r e o r i e n t a t i o n o f t h e m e t h y l groups i n t h e c a v i t i e s o f each c l a t h r a t e . A measurement o f t h e 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 v e r s u s t e m p e r a t u r e b y p r o g r e s s i v e s i g n a l s a t u r a t i o n was t h e r e f o r e u n d e r t a k e n f o r the C H g E l - c l a t h r a t e , and t h e T-^ -c u r v e , shown i n F i g . 21, was o b t a i n e d . The V-shape o f t h i s c u r v e , t y p i c a l o f a s i n g l e s h a r p c o r r e l a t i o n time f o r n u c l e a r r e o r i e n t a t i o n s , p r o v i d e s an adequate j u s t i f i c a t i o n of the a s s u m p t i o n m e n t i o n e d above. Though p r e c i s e measurements were n o t a t t e m p t e d i n t h e case of t h e CHgBr- and C H g l - c l a t h -r a t e s , a s i m i l a r t r e n d of T^ v s . 1 was seen r o u g h l y t o e x i s t , w h i c h s u p p o r t e d the above model. A n a c t i v a t i o n e nergy o f 2.35 k c a l / m o l e , c a l c u l a t e d f r o m t h e l o w - t e m p e r a t u r e s l o p e o f t h e T j - c u r v e f o r t h e C H g C l -c l a t h r a t e , i s i n r e m a r k a b l e agreement w i t h t h a t o b t a i n e d f r o m l i n e w i d t h d a t a . F o r t h e h i g h - t e m p e r a t u r e s i d e of the T^- c u r v e , a l o w e r energy b a r r i e r o f 1.50 k c a l / m o l e i s c a l c u l a t e d , i n d i c a t i n g t h a t m o t i o n s o t h e r t h a n m e t h y l r e o r i e n t a t i o n a r e now e f f e c t i v e i n r e l a x i n g t h e s p i n s y s t e m . I t may be seen f r o m F i g . 21 t h a t t h e e x t r a p o l a t e d T^ minimum f o r t h e C H g C l — c l a t h r a t e o c c u r s a t 125°K and has a v a l u e of 22.9 m i l l i s e c o n d s . T h i s v a l u e was u s e d i n the e x p r e s s i o n ( f ) - -0.321 ( Y ^ Z ^ min c a l c u l a t e d by A n d e r s o n and S l i c h t e r * * ^ f o r t h r e e f o l d r e o r i e n t a t i o n o f a m e t h y l group, where u) 0 i s the r a d i o f r e q u e n c y i n r a d i a n s and r . i s t h e J i n t e r p r o t o n d i s t a n c e i n t h e r e o r i e n t i n g m e t h y l group. S i n c e t h e above T^ measurements were made by s a t u r a t i o n o f t h e p o r t o n s i g n a l a t 1 6 M c / s e c , en =2TT X 16 x 10^ r a d i a n s , and i t t u r n s o u t t h a t r . = 2.19A. T h i s i s much l a r g e r t h a n t h e a c c e p t e d i n t e r p r o t o n d i s t a n c e of 1.79A f o r a m e t h y l groups 104 T 1 1 1 1 1 1 1 1 1 1 1 — i 1 1 1 1 1 i i i r Temperature, °K F i g . 20 P r o t o n L i n e w i d t h Changes f o r t h e Halomethane C l a t h r a t e s -21 • i i i i i ' I I 4 5 6 7 8 9 10 II 12 13 T - , ( ° K " ' ) x l 0 3 F i g . 21 T 1 D a t a f o r Chloromethane i n the D e u t e r a t e d H y d r a t e 105 P o s s i b l y , t h i s i s a measure of t h e e x t e n s i o n of t h e m e t h y l p r o t o n s f r o m t h e i r e q u i l i b r i u m p o s i t i o n s as the group r e o r i e n t s a t r a d i o f r e q u e n c i e s , b u t t h i s r a t h e r e m p i r i c a l s u g g e s t i o n w i l l be p u r s u e d no f u r t h e r h e r e s i n c e t h e T-^  measurements were n o t a b s o l u t e and t h e minimum was f o u n d o n l y by e x t r a -p o l a t i n g t h e two arms o f the c u r v e . I n d e e d , one s h o u l d a w a i t e x a c t measurements o f a b s o l u t e T-^  v a l u e s b y p u l s e techniques"'"'^ to c o n f i r m t h e m e t h y l d e f o r m a t i o n s u g g e s t e d above. The f o l l o w i n g o v e r a l l c o n c l u s i o n s may be made: o f the t h r e e " m e t h y l " g u e s t m o l e c u l e s s t u d i e d , , C H g B r i s t h e most l i m i t e d i n i t s c a v i t y by i t s own 12 m o l e c u l a r d i m e n s i o n (5.33A) and, a p a r t f r o m l i m i t e d a m p l i t u d e s o f m e t h y l " r o c k i n g " m o t i o n s , t h e m o l e c u l e i s w e l l o r i e n t e d i n i t s c a v i t y a t 77°K. A t th e same t e m p e r a t u r e , t h e s m a l l e r (5.06A)*2 CHgCl has more fre e d o m i n i t s c a v i t y t o p e r f o r m a l i m i t e d r a t t l i g g . Though CHg l has t h e l a r g e s t d i m e n s i o n (5.70A) , i t i s known f r o m x - r a y s t u d i e s t o r e s i d e i n a l a r g e r w a t e r c a v i t y and i s t h e r e f o r e l e s s r e s t r i c t e d t h a n t h e b r o m i d e . The r a t h e r low b a r r i e r o f 2.48 k c a l / m o l e a g a i n s t m e t h y l group r e o r i e n t a t i o n i n the 109 CH„C1- c l a t h r a t e s u g g e s t s t h a t m e t h y l b a r r i e r t u n n e l i n g i s p o s s i b l e . o CHgl i n i t s "Type I I " c a v i t y has a h i g h e r b a r r i e r , 6.80 k c a l / m o l e , a g a i n s t m e t h y l r e o r i e n t a t i o n . CHgBr i n i t s c a v i t y has t h e h i g h e s t b a r r i e r of 9.30 k c a l / m o l e a g a i n s t t h i s m o t i o n , and t h i s s i t u a t i o n i s r e m i n i s c e n t o f 52 t h e b e h a v i o u r o f CHgCN m o l e c u l e s e n c l a t h r a t e d i n a q u i n o l cage . A 42 p o t e n t i a l w e l l w i t h a b r o a d minimum and s t e e p s i d e s may be i m a g i n e d f o r the CHgBr i n i t s h y d r a t e c a v i t y a t 77°K, t h e m o t i o n of t h e m e t h y l group a p p r o x i m a t i n g t o s i m p l e harmonic " l i b r a t i o n " f o r t h e f i r s t few energy l e v e l s . ( b ) The D i c h l o r o m e t h a n e H y d r a t e : 106 A p o l y c r y s t a l l i n e specimen o f t h e d e u t e r a t e d h y d r a t e of s t o i c h i o -m e t r y (CHgClg) ~7.7D 20, asrhich m e l t e d s h a r p l y a t 3.0°C, was used f o r t h e p r o t o n resonance s t u d y . R a d i o f r e q u e n c y f i e l d s were weak enough n o t t o s a t u r a t e t h e s i g n a l s o b s e r v e d , and t h e m o d u l a t i n g f i e l d had a d e p t h of about 0.80 g a u s s . V e r y b r o a d G a u s s i a n l i n e s were o b t a i n e d a t low t e m p e r a t u r e s , and n a r r o w i n g of t h e l i n e shape was q u i t e pronounced as t h e t e m p e r a t u r e i n c r e a s e d . I n F i g . 22 a r e i l l u s t r a t e d t h r e e l i n e shapes o b t a i n e d d u r i n g a t e m p e r a t u r e r u n . S i n c e , f o r s p e c t r a o b t a i n e d a t 77°K, t h e r e was l i t t l e e v i d e n c e of "wings" a t t h e e x t r e m i t i e s , the second moment was f o u n d t o change v e r y l i t t l e between a n o i s y s i g n a l such as the one r e p r e s e n t e d i n F i g . 22 ( a ) and a l e s s n o i s y s i g n a l , r e c o r d e d a t t h e same t e m p e r a t u r e u s i n g a h i g h e r t i m e - c o n s t a n t i n t h e o u t p u t f i l t e r c i r c u i t . I n f a c t , t h e s p e c t r u m , F i g . 22 ( a ) , was f o u n d t o have t h e l e a s t d e v i a t i o n f r o m t h e second moment a v e r a g e d f r o m a t o t a l o f 4 s p e c t r a a t 77°K. F o r t h e h i g h e r t e m p e r a t u r e s p e c t r a , on the o t h e r hand, t h e " w i n g s " grew p r o g r e s s i v e l y more p r o m i n e n t , and a h i g h e r t i m e c o n s t a n t had of n e c e s s i t y t o be u s e d i n o r d e r n o t t o l o s e any of t h e a b s o r p t i o n i n t h e w i n g s . T h i s f a c t i s i l l u s t r a t e d i n F i g . 22 (b) and F i g . 22 ( c ) . One n o t i c e s t h a t t h e a b s o r p t i o n s i g n a l a t 77°K has a c l e a r l y r e s o l v e d d o u b l e t s t r u c t u r e c h a r a c t e r i s t i c of s t r o n g l y -c o u p l e d p r o t o n p a i r s . T h i s i s an i m p o r t a n t r e s u l t and w i l l be a n a l y s e d p r e s e n t l y . I n T a b l e XVTI a r e p r e s e n t e d t h e c o r r e c t e d e x p e r i m e n t a l second moment d a t a . I t i s seen t h a t the se c o n d moment r e a c h e s a c o n s t a n t u p p e r l i m i t v a l u e o f about 16.88G a t t e m p e r a t u r e s b e l o w 90°K. T h i s v a l u e may 2 be r o u g h l y d i v i d e d i n t o a n " i n t r a - g u e s t " c o n t r i b u t i o n of ~10G c a l c u l a t e d • 144 145 f o r an H-H d i s t a n c e o f 1.80A i n d i c h l o r o m e t h a n e ' , and an " i n t e r -107 I 1 I I I 0 2 4 6 8 Gauss I 1 I I I 0 2 4 6 8 Gauss F i g . 22 T y p i c a l H Resonance S p e c t r a o f t h e CHgClg- C l a t h r a t e (Two-headed A r r o w s R e p r e s e n t M o d u l a t i o n U s e d ) . 108 g u e s t " c o n t r i b u t i o n f r o m n e i g h b o u r i n g c a g e s . The f i r s t l i n e - h a r r o w i n g m o t i o n o f the g u e s t s i n t h e i r c a n i t i e s o c c u r s ibov.4 90°K, t h i s m o t i o n 2 c a u s i n g a r a t h e r s u b s t a n t i a l r e d u c t i o n o f n e a r l y 6.0G i n t h e second moment. A s i s se e n i n F i g . 22 ( b ) , t h e d o u b l e t d i s a p p e a r s when t h i s m o t i o n s e t s i n , i n d i c a t i n g t h a t the " p r o t o n - p a i r " c o n t r i b u t i o n t o t h e l i n e shape i s no l o n g e r t h e dominant one. The e x a c t n a t u r e o f t h i s m o t i o n c o u l d n o t , however, be i d e n t i f i e d . A t s t i l l h i g h e r t e m p e r a t u r e s , t h e r e i s a d i s c o n t i n u o u s d e c r e a s e i n t h e second moment, i n d i c a t i n g v a r y i n g d e g r e e s o f r e o r i e n t a t i o n a l and t u m b l i n g m o t i o n s . S i n c e t h e n a t u r e o f t h e s e m o t i o n a l p r o c e s s e s c o u l d n o t be r a t i o n a l i z e d i n terms o f s i m p l e models, t h e i n v e s t i g a t i o n was s t o p p e d a t 172°K. TABLE X V I I E x p e r i m e n t a l Second Moments f o r D i c h l o r o m e t h a n e i n t h e D e u t e r a t e d H y d r a t e L a t t i c e T e m p e r a t u r e , No. of S p e c t r a C o r r e c t e d mean e x p e r i m e n t a l 2 Second Moment, G 77 4 16.88±0.04 90 4 16.88±0.03 102 4 10.57*0.50 112 4 10.50*0.50 114 4 10.52*0.50 115 4 10.39±0.50 129 4 8.00±0.50 132 4 8.00*0.50 140 4 3.34±0.80 152 4 2.79±0.80 172 4 1.38±0.90 I t i s e v i d e n t t h a t no m o t i o n a l e f f e c t s a r e p r e s e n t b e l o w 90°K w h i c h c o u l d a f f e c t t h e a b s o r p t i o n l i n e shape. T h e r e f o r e , t h e d e r i v a t i v e o f the a b s o r p t i o n c u r v e r e c o r d e d a t 77°K r e p r e s e n t s t h e p r o t o n m a g n e t i c r e s o n a n c e a b s o r p t i o n o f g u e s t s h e l d r i g i d l y i n t h e i r h y d r a t e l a t t i c e p o s i t i o n s . The i n t e g r a t e d l i n e shape c o r r e s p o n d i n g t o t h i s c u r v e i s shown by the heavy l i n e i n F i g . 23, w h i c h i s t y p i c a l l y a p r o t o n d o u b l e t b r o a d e n e d CALCULATED FIT TO EXPERIMENTAL LINESHAPE (CH2CI2)~ deuterated hydrate Best Fit «=4.08g. *»4.28g. <*«4.00g. /*»l.25g. *»4.28g. /*«l.40g. <*«4.28g. /3>3X)4g. /*»l.70g. /**l.50g. EXPERIMENTAL 0 1 2 3 4 5 6 7 8 9 10 II DISTANCE FROM CENTRE OF RESONANCE, H (gauss) F i g . 23 110 by i n t e r a c t i o n s e x t e r n a l t o i t s e l f f r o m n e i g h b o u r i n g m o l e c u l e s . O n l y one h a l f o f t h e i n t e g r a t e d l i n e i s shown i n F i g . 23, t h e o t h e r h a l f b e i n g a m i r r o r image of t h e f i r s t about t h e y - a x i s . The a n a l y s i s o f such a l i n e shape has been f u l l y d i s c u s s e d i n s e c t i o n 3.2, where a f a c t o r a was d e f i n e d ( e q u a t i o n 10 ( a ) ) as a f u n c t i o n o f t h e d i s p l a c e m e n t , f ( ^ h ) , i n g a u s s , o f each of the d o u b l e t components o f a c r y s t a l l i n e specimen f r o m t h e m a g n e t i c f i e l d c e n t r e , and a n o t h e r f a c t o r 8 was d e f i n e d ( e q u a t i o n ( l 2 ) ) t o d e s c r i b e i n t e r m o l e c u l a r b r o a d e n i n g i n u n i t s o f g a u s s . A s s u m i n g 8 t o be c o n t a i n e d i n a G a u s s i a n t y p e o f b r o a d e n i n g f u n c t i o n , w h i c h i s a good a p p r o x i m a t i o n f o r d i p o l a r b r o a d e n i n g , e q u a t i o n (_12) may be r e - c a s t i n t o t h e f o r m r /• ,h-,h ^2-] 3 L 2 B 2 The shape t h a t a p l o t o f t h e above f u n c t i o n t a k e s depends on t h e r e l a t i v e v a l u e s o f a and 3. A " F o r t r a n IV" computer programme ( A p p e n d i x I I I ) was w r i t t e n t o o b t a i n t h e " b e s t f i t " t o the e x p e r i m e n t a l l i n e shape a c c o r d i n g to e q u a t i o n ( 4 4 ) , and an IBM 7040 computer was employed t o c a l c u l a t e l i n e shapes f o r 20 d i f f e r e n t v a l u e s of 3 r a n g i n g f r o m 1 gauss t o 3 g a u s s , f o r 5 s e t s of v a l u e s o f ot. Some o f t h e l i n e shapes t h u s d e r i v e d , a l o n g w i t h t h e " b e s t -f i t " l i n e shape, a r e shown i n F i g . 23, The " b e s t - f i t " c o r r e s p o n d e d t o a = 4.08G, 3 = 1.70G. F o r a p o l y c r y s t a l l i n e specimen, s u c h as t h e one u s e d 75 i n t h i s e x p e r i m e n t , t h e o b s e r v e d second moment may be e x p r e s s e d as 4 2 2 where the f a c t o r — r e p r e s e n t s t h e i s o t r o p i c a verage o f the (3Cos Y - l ) 2 2 term. The v a l u e of 16.21G f o r c a l c u l a t e d i n t h i s way i s i n good a g r e e -2 ment w i t h the v a l u e of 16.88 + 0.04G o b t a i n e d by n u m e r i c a l i n t e g r a t i o n of I l l t h e e x p e r i m e n t a l c u r v e a t 77°K. S i n c e , f r o m e q u a t i o n 10 ( a ) , a =— — , r a v a l u e o f r = 1.73A i s o b t a i n e d f o r t h e H-H d i s t a n c e of a " g u e s t " CHgClg m o l e c u l e . M i crowave s p e c t r o s c o p i c s t u d i e s on d i c h l o r o m e t h a n e i n t h e gas 144 145 phase ' have shown i t t o be a n e a r l y p r o l a t e symmetric t o p , p o s s e s s i n g a t e t r a h e d r a l s t r u c t u r e and a Cg-. a x i s o f symmetry. From t h e s e s t u d i e s , a mean H-H d i s t a n c e o f 1.814A may be d e r i v e d . Comparing t h i s v a l u e w i t h the o v a l u e o f 1.73A. o b t a i n e d above f o r a s i n g l e g u e s t ; m o l e c u l e t r a p p e d i n the h y d r a t e cage, i t i s r e a s o n a b l e t o c o n c l u d e t h a t t h e r e i s a s l i g h t s q u a s h i n g o r c o n t r a c t i o n o f t h e m o l e c u l e i n s i d e t h e t e t r a k a i d e c a h e d r a l c a g e , w i t h a c o n s e q u e n t s h o r t e n i n g of t h e gas-phase H-H d i s t a n c e t o t h e v a l u e n o t e d above. The shape of t h e t e t r a k a i d e c a h e d r a l cage i n a Type I h y d r a t e has 57 been shown t o have the n o n - s p h e r i c a l f o r m d e p i c t e d i n F i g . 24 ( a ) . I f i t i s assumed t h a t t h e d e u t e r a t e d h y d r a t e o f CHgClg has cages o f a s i m i l a r shape a t 77°K, i t i s p o s s i b l e t o s u g g e s t t h a t a m o l e c u l e of d i c h l o r o m e t h a n e i s a l i g n e d i n t h i s cage i n s u c h a way t h a t i t s H-H a x i s i s p a r a l l e l t o t h e s h o r t a x i s r a t h e r than the l o n g a x i s o f t h e p r o l a t e e l l i p s o i d a l cage. T h i s s u g g e s t e d a l i g n m e n t i s shown i n F i g . 24 ( b ) . 113 CHAPTER K i l l THE TETRA-ALKYLAMMONIUM "SALT" HYDRATE CLATHRATES 8.1 O u t l i n e The p r o t o n r e s o n a n c e o f two t e t r a ; , — a l k y l ammonium f l u o r i d e h y d r a t e s , ( i - £ 5 H 1 1 ) 4 N + F ~ ~ 38E>20 and ( n - C ^ 9 ) 4 N + F " ~ 31.8D 20, has been examined i n t h e t e m p e r a t u r e range 77°K-286°K. The r e s u l t s have been i n t e r p r e t e d i n terms of t h e known c r y s t a l s t r u c t u r a l a s p e c t s o f t h e s e two c l a t h r a t e s . 8.2 r R e s u l t s and C o n c l u s i o n s ; ( a ) ( i - C 5 H n ) 4 N + F ~ ~ 38D 20: P r o t o n r e s o n a n c e s p e c t r a f o r t h i s c l a t h r a t e were r e c o r d e d f r o m a t e m p e r a t u r e o f 77°K upwards, t a k i n g t h e u s u a l p r e c a u t i o n s t o a v o i d power s a t u r a t i o n and o v e r m o d u l a t i o n e f f e c t s . The o b s e r v e d t e m p e r a t u r e dependence o f second moments and l i n e w i d t h s f o r t h i s c l a t h r a t e i s shown i n F i g . 25. One n o t i c e s a s h a r p change i n t h e second moment a t 248°K, w i t h t h e c o r r e s -p o n d i n g l i n e w i d t h change c e n t r e d a t t h e same t e m p e r a t u r e . B e f o r e and a f t e r 114 t h i s t r a n s i t i o n , changes i n l i n e w i d t h and second moment a r e v e r y g r a d u a l and c o n t i n u o u s , t h i s t r e n d b e i n g r e p r o d u c i b l e f o r b o t h warming and c o o l i n g i l e x p e r i m e n t a l c y c l e s . These r e s u l t s may now be i n t e r p r e t e d i n terms of s i m p l e m o t i o n a l models o f the "guest" a l k y l c h a i n s . B e f o r e d o i n g t h i s , however, one ought t o r e c a l l c e r t a i n i n t e r e s t i n g a s p e c t s o f t h e x - r a y d i f f r a c t i o n s t u d y r e p o r t e d f o r t h e h y d r a t e a t a p p r o x i m a t e l y 25 C. T h i s c l a t h r a t e has an o r t h o r h o m b i c u n i t c e l l s t r u c t u r e a t room t e m p e r a t u r e , o f space group symmetry Pbmm. Two o f the f o u r i — a m y l c h a i n s o f a c a t i o n p r o j e c t i n t o t e t r a k a i d e c a h e d r a l c a v i t i e s , w h i l e t h e o t h e r two 17 p r o j e c t i n t o p e n t a k a i d e c a h e d r a l c a v i t i e s . F o l l o w i n g F e i l and J e f f r e y , the c a r b o n atoms o f t h e c h a i n s c o n t a i n e d i n p e n t a k a i - and t e t r a k a i d e c a h e d r a l c a v i t i e s s h a l l h e n c e f o r t h be d i s t i n g u i s h e d w i t h t h e s u p e r s c r i p t s p a n d ; t , r e s p e c t i v e l y . The two m i r r o r p l a n e s a r e m u t u a l l y p e r p e n d i c u l a r and cause the o r i e n t a t i o n a l d i s o r d e r of t h e c a t i o n s shown i n F i g s . 5 ( a ) and 5 ( b ) . The x - r a y d i f f r a c t i o n d a t a w a r r a n t e d o n l y a r e f i n e m e n t w i t h r e s p e c t t o t h e i s o t r o p i c t e m p e r a t u r e f a c t w r s , B, of t h e atom i c c o o r d i n a t e s . F o r t h e N + and F~ i o n s , w h i c h l i e a t the i n t e r s e c t i o n o f t h e two m i r r o r p l a n e s , B was .2 s i g n i f i c a n t l y l o w , a b out 1.4A« V a l u e s of B a s s o c i a t e d w i t h the c h a i n c a r b o n s i n t h e two t y p e s of c a v i t i e s a r e r e p r e s e n t e d i n F i g . 26. The f i r s t c a r b o n atoms, and , of the f o u r a l k y l c h a i n s have t e m p e r a t u r e f a c t o r s s i m i l a r t o t h o s e o f t h e N c a t i o n , and t h e r m a l m o t i o n i n c r e a s e s p r o g r e s s -° 2 i v e l y a l o n g t h e c h a i n to v a l u e s c l o s e t o B = 6A tfb.ro.the t e r m i n a l atoms, w i t h t h e e x c e p t i o n o f , w h i c h i s anomalous i n t h i s r e s p e c t . One a l s o n o t i c e s t h a t t h e r m a l m o t i o n i s s i g n i f i c a n t l y g r e a t e r i n s i d e a p e n t a k a i d e -c a h e d r o n , w h i c h i s t h e l a r g e r o f t h e two o c c u p i e d p o l y h e d r a . A s s u m i n g t h e above s t r u c t u r a l f e a t u r e s t o h o l d unchanged f o r the c l a t h r a t e made f r o m heavy w a t e r , one c a n u t i l i z e t he t e m p e r a t u r e - r e f i n e d 7r-e tn tn <9 UJ 1 SECOND MOMENT LINE WIDTH I 1 1 1 14 12 CM , u<0 < e 8! UJ o He, Az o o UJ tn 80 100 120 140 160 180 200 220 240" 260" 280 30' TEMPERATURE °K 8 LINE WIDTH AND SECOND MOMENT DATA, ( i - C 5 H M ) 4 N F ~ D20 CLATHRATE F i g . 25 GUEST CHAIN CARBON ATOMS AND THEIR ISOTROPIC TEMPERATURE FACTORS (KVy # N*F--381^0 PENTAKAIDECAHEDRON J L /*-TETRAKAIDECAHEDRON J L 1 2 3 4 CARBON ATOM NUMBER F i g . 26 a t o m i c c o o r d i n a t e s f o r C and t o compute t h e g u e s t c h a i n h y d r o g e n atom p o s i t i o n s f o r a " r i g i d " c l a t h r a t e , s t r u c t u r e . T h i s was done w i t h t h e a i d o f t h e computer programme shown i n A p p e n d i x I I f o r s t a n d a r d C-H d i s t a n c e s and t e t r a h e d r a l HCH a n g l e s , and t h e g e n e r a t e d hydrogen atom c o o r d i n a t e s a r e shown i n T a b l e X V I I I . When " i n t r a - c h a i n " i n t e r a c t i o n s u p t o 6A were c o n -2 2 s i d e r e d among t h e s e h y d r o g e n atoms, second moments o f 20.01G and 21.07G r e s u l t e d f o r t h e and c h a i n s , r e s p e c t i v e l y . A f u r t h e r c a l c u l a t i o n o f i n t e r p r o t o n d i s t a n c e s between c h a i n s i n s u c c e s s i v e c a v i t i e s i n t h e o r t h o -r h o m b i c u n i t c e l l r e v e a l e d t h a t o n l y t h e p r o t o n p a i r s on t h e f i r s t c a r b o n s o f t h e f o u r c h a i n s were c l o s e enough t o i n t e r a c t , t h e i r i n t e r a c t i o n g i v i n g 2 r i s e t o a second moment of 1.22G . O t h e r terms c o n t r i b u t i n g t o t h e t o t a l s e c ond moment a r e t h e f o l l o w i n g : Xi) a s m a l l i n t e g r a t e d c o n t r i b u t i o n f r o m i n t e r - p r o t o n v e c t o r s > 6 A » T h i s was c a l c u l a t e d f r o m an a p p r o p r i a t e f o r m o f the f o r m u l a (41) i n c h a p t e r V I , where p " i s R6w t h e a verage p r o t o n d e n s i t y due t o t h e 88 p r o t o n s i n the u n i t 2 c e l l . T h i s v a l u e o f t h e second moment e q u a l s 0.35G . ( i i ) c o n t r i b u t i o n by t h e d e u t e r o n s o f t h e l a t t i c e i n t e r a c t i n g w i t h the a l k y l p r o t o n s , c a l c u l a t e d a c c o r d i n g t o t h e e x p r e s s i o n &9,i = i ( l O - O J s O C o a ^ d - l ) 2 V where p, d r e f e r t o the p r o t o n s and d e u t e r o n s and d i s the a n g l e made by t h e i n t e r n u c l e a r v e c t o r w i t h t h e m a g n e t i c f i e l d a x i s . T h i s c o n t r i b u t i o n , when aveeaged f o r a p o l y c r y s t a l l i n e specimen, was n e g l i g i b l e . ( i i i ) c o n t r i b u t i o n f r o m t h e F"^9 n u c l e u s a t t h e l a t t i c e p o s i t i o n 6 . 1 9 A ( x ) , 5 . 4 0 A ( y ) , 6 . a U(z). T h i s e q u a l s 0.93G 2. 2 The t o t a l " r i g i d l a t t i c e " second moment i s t h e r e f o r e 23.04i0.50G . ( a ) c h a i n C^: T a b l e X V I I I COMPUTED HYDROGEN ATOM COORDINATES 118 (b) c h a i n C : X y (ANGSTROM UNITS) z 0.640 2.673 5.325 2.447 4.495 4.730 3.044 4.747 7.219 1.175 3.905 8.153 2.249 1.972 5.707 2.674 3.040 4.040) 4.768 3.034 5.434/ End M e t h y l 3.796 0.892 4,713) 2.157 1.623 8.953) 3.565 0.046 7.186) End M e t h y l 4.619 2.369 7.967) T h e o r e t i c a l Second Moment: 20.01G X y (ANGSTROM UNITS) z 12.074 6.669 4.021 12.376 5.391 6.288 10.977 3.544 4.504 9.364 4.538 5.862 9.802 6.646 3.127 9.609 8.028 4.227) 7.984 7.047 1.968/ E n d M e t h y l 7.316 6.192 4.500) 9.751 3.241 2.648) 7.647 3.852 3.525) End M e t h y l 8.193 4.551 1.454) T h e o r e t i c a l Second Moment: 21.07G ( r . j <6A) T h i s i s s e e n t o be w e l l above t h e e x p e r i m e n t a l v a l u e o f 12.50G o b t a i n e d a t 77°K. The " r i g i d l a t t i c e " v a l u e s h o u l d , o f c o u r s e , be c o r r e c t e d f o r t h e z e r o - p o i n t v i b r a t i o n o f t h e c h a i n atoms i n t h e i r l a t t i c e p o s i t i o n s , b u t a r e d u c t i o n o f t h i s t y p e w i l l n o t be l a r g e enough by i t s e l f t o a c c o u n t f o r t h e second moment o b t a i n e d a t 77°K. I t i s t h e r e f o r e i n f e r r e d t h a t t h e i - amyl c h a i n s a r e u n d e r g o i n g c o n s i d e r a b l e m o t i o n i n t h e i r c a v i t i e s a t the l o w e s t t e m p e r a t u r e o f t h i s i n v e s t i g a t i o n . P o s s i b l e m o t i o n s o f t h e i — a m y l c h a i n t h a t c a n a c c o u n t f o r t h e n a r r o w i n g o f t h e r e s o n a n c e l i n e , and the computed v a l u e s o f r e s i d u a l second moments f o r t h e s e m o t i o n s ^ a r e s e t f o r t h i n T a b l e X I X . 120 TABLE X I X P o s s i b l e i - A m y l C h a i n M o t i o n s and C o r r e s p o n d i n g R e s i d u a l Second Moments. No. M o t i o n a l Type R e s i d u a l Second Moment, G 2 1». " R i g i d R o t o r " M o d e l : Long c h a i n A x i s t u r n i n g o v e r a t 1 0 5 - 1 0 6 c p s : 5.20*0.10 2. C o - o p e r a t i v e R e o r i e n t a t i o n s o f End M e t h y l s added t o above M a i n C h a i n m o t i o n : )ne M e t h y l R e o r i e n t i n g 4.00*0.10 B o t h M e t h y l s R e o r i e n t i n g 2.82*0.10 a) Oi b) ( 3. T o r s i o n a l ( " t w i s t i n g " ) r e o r i e n t a t i o n s o f c h a i n segment about f i x e d M + -C, bond: a) B o t h M e t h y l s R i g i d 8.50±0.10 b) One M e t h y l R e o r i e n t i n g 7.35*0.10 c) B o t h M e t h y l s R e o r i e n t i n g 6.20*0.10 4. T o r s i o n a l ( " t w i s t i n g " ) R e o r i e n t a t i o n s of c h a i n segment about f i x e d N -Cg bond: a) B o t h M e t h y l s R i g i d 10.20t0.30 b) One M e t h y l R e o r i e n t i n g 9.10*0.30 c) B o t h M e t h y l s R e o r i e n t i n g 7.90*0.30 5. One E n d M e t h y l R e o r i e n t i n g i n a ' R i g i d 1 c h a i n . 12.00*0.30 6. B o t h End M e t h y l s R e o r i e n t i n g i n a ' R i g i d * C h a i n . 7.10±0.40 I t w o u l d seem f r o m T a b l e X I X t h a t t o r s i o n a l o s c i l l a t i o n s o f t h e b r a n c h e d c a r b o n atom c a r r y i n g t h e end m e t h y l g r o u p s i s t h e t y p e o f m o t i o n t h a t comes c l o s e s t t o e x p l a i n i n g t h e second moment a t 77°K. However, such a m o t i o n , w h i c h c o n s i d e r s t o r s i o n o f a c h a i n segment i n p r e f e r e n c e t o t h e e n e r g e t i c a l l y more f a v o u r a b l e end m e t h y l r o t a t i o n , i s perhaps u n r e a l i s t i c . R a t h e r , one m i g h t t h i n k o f end m e t h y l r e o r i e n t a t i o n s i n v i e w of t h e l a r g e number of e x p e r i m e n t a l c a s e s * ^ ' ^ 4^' where s u c h m o t i o n has been e s t a b l i s h e d t o be t h e f i r s t l i n e - n a r r o w i n g m o t i o n a t low t e m p e r a t u r e s . A c l o s e r i n s p e c t i o n o f T a b l e X I X r e v e a l s t h a t f o r t h e r e o r i e n t a t i o n o f one end m e t h y l g r o u p o f t h e r i g i d i — a m y l c h a i n t h e r e s u l t a n t second moment w o u l d be 12.00±0.30G^. "When t h e " i n t e r - c h a i n " and o t h e r i n t e r a c t i o n s ( u n a f f e c t e d by t h i s m o t i o n ) a r e added as b e f o r e t o t h i s v a l u e , a second moment o f about 14.50G i s o b t a i n e d , w h i c h i s l a r g e r t h a n t h e e x p e r i m e n t a l v a l u e a t On t h e o t h e r hand, i f b o t h end m e t h y l groups were a l l o w e d t o r e o r i e n t , a second moment of about 9.60G r e s u l t s . The e x p e r i m e n t a l v a l u e a t 77 °K i s t h u s i n t e r m e d i a t e between the v a l u e s f o r o n l y one m e t h y l r e o r i e n t i n g and b o t h m e t h y l s r e o r i e n t i n g . I n v i e w of t h e f a c t t h a t t h e C p and C c h a i n s do n o t e n j o y t h e same freedom i n t h e i r r e s p e c t i v e c a v i t i e s , and remembering, p a r t i c u l a r l y , t h a t the Cg has a l r e a d y been known t o e x h i b i t an anomalous t h e r m a l m o t i o n ( F i g . 2 6 ) , t h e p l a u s i b l e s u g g e s t i o n may be made t h a t , f o r t h e C^- c h a i n , t h e l a r g e r v o i d p e r m i t s m o t i o n o f b o t h m e t h y l g r o u p s , w h i l e i n t h e C^- c h a i n t h e two m e t h y l g r o u p s a r e s l i g h t l y squeezed i n such t h a t one of them i s a b l e t o r e o r i e n t about i t s C g - a x i s w h i l e t h e o t h e r ; i s "gauche" o r " e c l i p s e d " w i t h r e s p e c t t o n e i g h b o u r i n g atoms. T h i s s u g g e s t i o n i s 17 s u p p o r t e d t o some e x t e n t by t h e f a c t t h a t F e i l and J e f f r e y r e p o r t a d e v i a t i o n of 3° i n t h e C^-Cg-Cg a n g l e when p a s s i n g f r o m t h e t e t r a k a i - t o the p e n t a k a i d e c a h e d r a l cage. Indeed, s u c h a s i t u a t i o n , where o n l y one o f 122 two m e t h y l groups on a c a r b o n r e o r i e n t s , h as been r e p o r t e d f o r t h e l o w -147 t e m p e r a t u r e m o t i o n a l model o f i - b u t y l bromide b y P o w l e s and K a i l . I t i s seen f r o m F i g . 25 t h a t the l i n e w i d t h s and second moments do n o t change a p p r e c i a b l y u n t i l t h e t e m p e r a t u r e r e a c h e s 240°K. P r e s u m a b l y , t h e e c l i p s e d m e t h y l group i n t h e s m a l l e r c a v i t y i s a l s o r e o r i e n t i n g a t t h i s 2 t e m p e r a t u r e , g i v i n g r i s e t o t h e o b s e r v e d second moment o f ~9.50G . The t r a n s i t i o n c e n t r e d around 248°K i n F i g . 25 may w e l l c o r r e s p o n d t o a t h e r m a l e x p a n s i o n of t h e c l a t h r a t e h o s t s t r u c t u r e r e s u l t i n g i n t h e i n c r e a s e i n the s i z e o f t h e deuteriumbonded v o i d s , whereby r e o r i e n t a t i o n s of whole a l k y l c h a i n s as w e l l as t o r s i o n a l o s c i l l a t i o n s about t h e N +-C^- a x i s a r e f a c i l i t a t e d . C o n s i s t e n t w i t h t h e t h e r m a l e x p a n s i o n o f the h o s t s t r u c t u r e 17 + — s u g g e s t e d above, F e i l and J e f f r e y r e p o r t a b n o r m a l l y l a r g e N -0 and F -0 d i s t a n c e s of ~ 4.20A and — 3.20A, r e s p e c t i v e l y . These i n c r e a s e d d i s t a n c e s l e a d t o n e g l i g i b l e " i n t e r - c h a i n " and o t h e r i n t e g r a t e d components of the second moment. From e n t r y 2 (b) of T a b l e X I X , i t i s seen t h a t c o m p l e t e r e o r i e n t a t i o n s o f t h e whole a l k y l c h a i n would r e s u l t i n a second moment of 2 2 2.82i0.10G . However, a much s m a l l e r s e c o n d moment of 1.85G was o b t a i n e d f o r t h e s p e c t r a between 2 5 5 \ and 286°K, and i t i s r e a s o n a b l e t o assume a d d i t i o n a l s m a l l - a m p l i t u d e o s c i l l a t i o n s o f the r e o r i e n t i n g i - amyl c h a i n s , w i t h the o n l y c o n s t r a i n t a t t h e n i t r o g e n - e n d o f t h e c h a i n due t o t h e t e t r a h e d r a l N + - c o n f i g u r a t i o n i n t h e l a t t i c e a r r a y . The a m p l i t u d e o f such a m o t i o n would, be e x p e c t e d t o i n c r e a s e p r o g r e s s i v e l y a l o n g t h e c h a i n u n t i l i t i s a maximum a t t h e b r a n c h e d c a r b o n . The p l o t o f t h e t e m p e r a t u r e f a c t o r s i n F i g . 26, showing t h a t t h e r m a l m o t i o n r e a c h e s a maximum a t C^, v e r i f i e s t h e above a s s u m p t i o n . (b) ( n - C 4 H 9 ) 4 N + F " ~ 31.8 D 20: The e x p e r i m e n t a l second moments o b t a i n e d d u r i n g a complete t e m p e r a -123 t u r e c y c l e a r e shown i n F i g . 27. A d e f i n i t e h y s t e r e s i s e f f e c t , r e p r e s e n t e d by t h e d o t t e d l i n e i n t h e f i g u r e , was n o t e d above 240°K. The s p e c t r a were s t r u c t u r e l e s s t h r o u g h o u t , and d i s p l a y e d pronounced n a r r o w i n g e f f e c t s i n t h r e e d i s t i n c t t e m p e r a t u r e r e g i o n s . C o n s i d e r i n g the e x t e n s i v e d i s o r d e r r e p o r t e d f o r t h e a l k y l c h a i n s i n the c r y s t a l s t r u c t u r e o f t h i s c l a t h r a t e , i t i s h a r d l y s u r p r i s i n g t h a t t h e o v e r a l l t r e n d o f t h e s e r e s u l t s i s m a r k e d l y d i f f e r e n t f r o m t h a t r e p o r t e d i n the p r e v i o u s s e c t i o n f o r t h e t e t r a i - amyl 21 ammonium a n a l o g u e . The e s s e n t i a l a s p e c t s o f t h e x - r a y s t r u c t u r a l a n a l y s i s may now be summarized. W i t h f i v e m o l e c u l e s o f t h e c a t i o n a r r a n g e d i n P4g/m symmetry i n a "pseudo b o d y - c e n t r e d " t e t r a g o n a l w a t e r l a t t i c e , t h i s s t r u c t u r e i s d e n s e r and l e s s s y m m e t r i c a l t h a n i t s t e t r a i — amyl ammonium a n a l o g u e . F o u r o f t h e f i v e c a t i o n s , c e n t r e d a t t h e f o u r f o l d l a t t i c e p o s i t i o n s ( x , y, 0 ) , are o n l y o r i e n t a t i o n a l l y d i s o r d e r e d , w h i l e t h e f i f t h i s d i s o r d e r e d o r i e n t a t i o n -a l l y as w e l l as p o s i t i o n a l l y o v e r t h e t w o f o l d l a t t i c e p o s i t i o n s , 0,,-J-). B e s i d e s t h i s d i s o r d e r e d s t r u c t u r e , t h e a l k y l c h a i n s of t h e g u e s t c a t i o n s and some o f t h e framework oxygens d i s p l a y e x t e n s i v e t h e r m a l m o t i o n , w h i c h r e s u l t e d i n c o n s i d e r a b l e d i s t o r t i o n s of t h e t e t r a g o n a l h o s t l a t t i c e . The d i s o r d e r e d c a t i o n s , numbered a c c o r d i n g t o the c r y s t a l d a t a o f M c M u l l a n , 21 e t a l . , , and t h e d i s t o r t i o n s o f t h e h o s t l a t t i c e i n t h e v i c i n i t y o f t h e N + - c a t i o n a r e shown i n F i g s . 6 ( a ) and 6 ( b ) . A s s u m i n g t h e t e m p e r a t u r e r e f i n e d a l k y l c h a i n c a r b o n c o o r d i n a t e s r e -p o r t e d i n t h e c r y s t a l s t r u c t u r e s t u d y , h y d r o g e n atom c o o r d i n a t e s f o r t h e a l k y l c h a i n s were g e n e r a t e d as i n t h e p r e v i o u s s e c t i o n , numbering t h e a l k y l c h a i n s as i n F i g . 6 ( a ) and 6 ( b ) . The hy d r o g e n s b e l o n g i n g t o t h e f o u r o r d e r e d c a t i o n s a r e l i s t e d as ( a ) , ( b ) , ( c ) , and ( d ) i n T a b l e XX, w h i l e t h o s e c o r r e s p o n d i n g t o t h e p o s i t i o n a l l y " o r d e r e d " c o n f i g u r a t i o n s o f t h e 16 15 14 13 12 II (M in v> 3 O 3 W 8 O Z 6 o O UJ tO 4 SECOND MOMENT VERSUS TEMPERATURE (n-Q. H Y N F ~ D O CLATHRATE \ 4 9 / 4 2 _L F i g . 27 _L 60 8 0 100 120 140 160 180 TEMPERATURE (°K) 200 220 240 260 280 3 0 0 f i f t h a l k y l c h a i n i n t h e u n i t c e l l o f t h e c l a t h r a t e a r e l i s t e d as ( e ) and ( f ) i n t h e same t a b l e . To o b t a i n t h e hydrogen c o o r d i n a t e s c o r r e s p o n d i n g t o t h e o t h e r , i . e . , p o s i t i o n a l l y " d i s o r d e r e d " , f o r m of t h i s c a t i o n , c a l c u l a -t i o n s were made on a s e t of c a r b o n c o o r d i n a t e s i n v e r t e d about a p l a n e p a s s i n g t h r o u g h the N + atom l o c a t e d a t the l a t t i c e p o s i t i o n (-g- 0 . A v e r a g e second moments, computed by u s i n g t h i s t a b l e f o r two t h e o r e t i c a l e xtremes o f c a t i o n i c m o t i o n , a r e s e t out i n T a b l e X X I . T a b l e XX Computed Hydrogen Atom C o o r d i n a t e s ( F o r k e y t o c h a i n number, see F i g . 6a and 6b) (a ) The N(30) - C ( 4 l ) t o C(44) C h a i n : y ( A n g s t r o m u n i t s ) -0.038 5.736 4.980 1.480 6.750 4.873 0.548 7.262 -1.059 -1.078 6.762 -0.839 -0.941 8.094 1.491 0.442 8.867 0.771 -1.641 10.120 0.220' -0.979 9.466 -1.314 -2.362 8.692 -0.594 T h e o r e t i c a l Second Moment: 1 8 . 8 3 G 2 ( r . .< 6A) ( b ) The N(30) - C(45) t o c(48) C h a i n : x y z ( A n g s t r o m u n i t s ) -0.279 4.610 -1.387 0.611 3.101 -0.797 -0.635 3.255 1.300 126 T a b l e XX ( C o n t d ) x y z ( A n g s t r o m u n i t s ) -1.475 4.673 0.741 -2.274 3.363 -1.309 -1.503 1.908 -0.622 -3.971 -1.947 -0.087) -2.995 2.066 1.388< E n d M e t h y l -3.767 3.251 0.701) ,2 T h e o r e t i c a l Second Moment: 19.95G ( r . .<6A) ( c ) The N(30) - C(49) t o C(52) C h a i n : x y ( A n g s t r o m u n i t s ) 1.249 4.036 1.971 2.322 3.196 0.737 3.891 5.062 0.942 2.879 5.778 2.197 3.051 3.638 3.524 4.112 2.970 2.283 5.449 3.911 4.175 ( 5.751 4.838 2.682< 4.690 5.505 3.922* T h e o r e t i c a l Second Moment: 20.22G 2 ( r . .<6A) (d) The N(30) - C(53) t o C(56) C h a i n : x y z ( A n g s t r o m u n i t s ) 1.631 5.772 -1.864 2.928 6.015 -0.583 3.643 3.746 -0.889 2.362 3.445 -2.063 127 T a b l e XX ( C o n t d ) X y z ( A n g s t r o m u n i t s ) 3.275 5.211 -3.561 4.513 5.612 -2.370 5.487 4.193 -4.211) 5.503 3.363 -2.643< 4.265 2.963 -3.834) T h e o r e t i c a l Second Moment: 20.22G ( r . . ^ A . ) ( e ) The N ( l 7 ) -C(57) t o C(60) C h a i n : x y (Angstrom u n i t s ) 10.582 -1.124 11.586 -2.103 13.579 -1.618 12.658 -0.607 11.039 -2.538 11.890 -3.548 12.644 -4.088 14.022 -3.416 13.171 -2.406 T h e o r e t i c a l S e c o n d Moment: 20.23G*(r. ^6A) ( f ) The N ( l 7 ) -€ ( 6 1 ) t o C(64) C h a i n : X y . z (Angs trom U n i t s ) 12.950 -1.710 3.255 13.358 -0.319 4.386 11.577 -0.961 5.792 11.119 -2.262 4.693 13.308 -3.303 4.969 T a b l e XX ( C o n t d ) ( A n g s t r o m u n i t s ) 13.831 -1.975 6.004 13.157 -4.017 7.462) 12.091 -2.669 7.735< End M e t h y l 11.568 -3.968 ).699) T h e o r e t i c a l Second Moment: 20.19G 2 ( r ; . < 6 A ) T a b l e X X I T h e o r e t i c a l H Second Moments: The D e u t e r a t e d H y d r a t e o f T e t r a n - rButy l Ammonium F l u o r i d e , Type o f S t r u c t u r e Second Moment f o r M o t i o n l e s s o C a t i o n s , G Second Moment f o r C a t i o n s o R e o r i e n t i n g i n C a v i t i e s , G " I n t r a -c h a i n " " I n t e r -c h a i n " ^ O t h e r I n t e r a c t i o n s T o t a l ' Intrar-c h a i n " " I n t e r -c h a i n " O t h e r I n t e r a c t i o n s * T o t a l C o m p l e t e l y O r d e r e d 50$ P o s i t i o n a l D i s o r d e r of one - N ( C 4 H g ) 4 u n i t . 19.50±0.70 19.50±0.70 2.0±.0.70 2.0±0.70 6.65±0.80 8,50 28.15±0.80 30,00±0.70 0 0 0 0 ~ 4.02 ~ .5.10 ~ 4.02 ~ 5.10 ^ V a l u e s i n t h i s column were o b t a i n e d b y a p p l y i n g a second moment " r e d u c t i o n f a c t o r " o f 0,63 a c c o r d i n g t o Andrew and E a d e s 8 4 , ' 130 C o n s i d e r i n g " i n t r a - c h a i n " i n t e r a c t i o n s , a mean second moment o f 2 19.50±0.70G was o b t a i n e d f o r t h e f o u r o r d e r e d c h a i n s . T h i s was i n c r e a s e d 2 to 21.50±0.70G when " i n t e r - c h a i n " i n t e r a c t i o n s among t h e s e c h a i n s were a l s o i n c l u d e d . A s s u m i n g a c o m p l e t e l y " o r d e r e d " c o n f i g u r a t i o n o f t h e f i f t h c a t i o n i n one of t h e two e q u i v a l e n t l a t t i c e p o s i t i o n s , a t o t a l second moment o f 2 28.15*0.80G was o b t a i n e d f o r f i v e i n t e r a c t i n g c a t i o n s i n t h e u n i t c e l l . On t h e o t h e r hand, when t h e " d i s o r d e r e d " c o n f i g u r a t i o n o f t h e f i f t h c a t i o n i n -v o l v i n g a 50-50 d i s t r i b u t i o n between i t s e q u i v a l e n t l a t t i c e p o s i t i o n s was c o n s i d e r e d , t h e number of " i n t e r - c h a i n " i n t e r a c t i o n s i n c r e a s e d , and t h e 2 o v e r a l l s econd moment was^j30 . 00G f o r " r i g i d " c h a i n s . S u c h i n c r e a s e s i n second moment c o r r e s p o n d i n g t o d i s o r d e r e d s t r u c t u r e s have been r e c e n t l y r e -148 p o r t e d i n a n o t h e r s t r u c t u r a l s t u d y . When t h e above e s t i m a t e s o f t h e o r e t i c a l second moments a r e compared to t h e v a l u e o f 15.60±0.15G 2 o b t a i n e d a t 77°K ( F i g . 2 7 ) , i t i s a t once a p p a r e n t t h a t t h e n - b u t y l c h a i n s a r e n o t r i g i d i n s i d e t h e c l a t h r a t i o n v o i d s a t t h i s t e m p e r a t u r e . »If t h e r e a s o n a b l e a s s u m p t i o n i s made t h a t t h e end m e t h y l 2 group o f each a l k y l c h a i n r e o r i e n t s , second moments o f 18.50±0.50G f o r an 2 " o r d e r e d " arrangement o f c a t i o n s and about 20.00G f o r the 50$ d i s o r d e r c o n s i d e r e d above a r e c a l c u l a t e d . A t 77°K, t h e e x p e r i m e n t a l v a l u e i s s m a l l e r t h a n e i t h e r of t h e s e ; i n d e e d , the l o w e s t t e m p e r a t u r e r e g i o n of t h e second moment c u r v e o f F i g . 27 d i s t i n c t l y r e s e m b l e s the c o r r e s p o n d i n g r e g i o n of F i g . 16 f o r the propane c l a t h r a t e , and, once a g a i n , t h e s u g g e s t i o n may be made t h a t a t 77°K the a l k y l g roups o f t h e t e t r a butylammonium c a t i o n a r e u n d e r g o i n g a m o t i o n a l t r a n s i t i o n w h i c h s t a r t e d a t some e a r l i e r t e m p e r a t u r e . A t 95°K, t h e n a r r o w i n g o f t h e a b s o r p t i o n l i n e due t o t h e above u n s p e c i f i e d m o t i o n a p p e a r s t o be c o m p l e t e , t h e narrowed l i n e now m a i n t a i n i n g a s t e a d y second moment o f 13.00±0.30G 2 upto 160°K. The s l o p e o f t h e second moment v e r s u s t e m p e r a t u r e c u r v e changed a b r u p t l y above 170°K, and l i n e - n a r r o w i n g was more r a p i d . There was now a n e a r l y l i n e a r d r o p i n t h e second moment, w h i c h r e a c h e d ,J5.00G a t 245 i £ . Beyond t h i s t e m p e r a t u r e , t h e r e was a f u r t h e r d i s c o n t i n u o u s b r e a k i n t h e c u r v e . I t i s supposed t h a t , i n t h e 170°K- 245°K r e g i o n , random t h e r m a l o s c i l l a t i o n s o f t h e n - b u t y l c h a i n o f p r o g r e s s i v e l y i n c r e a s i n g a m p l i t u d e s s e t i n , p o s s i b l y s uperimposed on t h e u n s p e c i f i e d segmental r e o r i e n t a t i o n s c a u s i n g the l i n e - n a r r o w i n g n o t e d e a r l i e r . Now, t h e t o t a l energy o f a harmonic o s c i l l a t o r i s p r o p o r t i o n a l t o t h e square o f t h e a m p l i t u d e o f m o t i o n , and t h e mean e n e r g y , p e r o s c i l l a t o r . . . , o f an a s s e m b l y o f such o s c i l l a t o r s i s p r o p o r t i o n a l t o t h e t e m p e r a t u r e . T h e r e f o r e , as has been a r g u e d e l s e w h e r e 149 f o r a s i m i l a r c h a i n m o t i o n , i f the ave r a g e m o t i o n o f t h e n - b u t y l c h a i n s i n t h e p o l y h e d r a l v o i d s i s r a t i o n a l i z e d as t h a t o f a system o f i n d e p e n d e n t o s c i l l a t o r s , w i t h t h e second moment d e c r e a s i n g l i n e a r l y w i t h i n c r e a s e i n th e s q u are o f o s c i l l a t i o n a l a m p l i t u d e , a l i n e a r d r op o f t h e t y p e s e e n i n the second moment c u r v e , F i g . 27, w o u l d i n d e e d be e x p e c t e d . The w i d t h o f the r e s o n a n c e a b s o r p t i o n l i n e changes from 5.70G t o 3.40G i n t h e 170 K-245 K r a n g e , and t h e d a t a p o i n t s i n t h i s t e m p e r a t u r e l i m i t , when t r e a t e d a c c o r d i n g t o A p p e n d i x I , y i e l d e d a v a l u e o f 5.31±0.22 k c a l / m o l e f o r t h e a p p a r e n t p o t e n t i a l energy b a r r i e r t o t h e above o s c i l l a t o r y m o t i o n . A v e r y i n t e r e s t i n g h y s t e r e s i s e f f e c t was n o t i c e d a t t e m p e r a t u r e s between 245°K and t h e m e l t i n g p o i n t o f t h e c l a t h r a t e . I n f i g u r e 27, t h e s o l i d l i n e i n d i c a t e s t h e t r e n d i n t h e second moment v a l u e s d u r i n g a True P o t e n t i a l E n e r g y B a r r i e r = A p p a r e n t P o t e n t i a l E n e r g y B a r r i e r + Ze r o P o i n t E n e r g y . "warming up" t e m p e r a t u r e r u n , and t h e c l a t h r a t e specimen m e l t e d t o a s y r u p y l i q u i d a t 295°K. The average r e s i d u a l second moment o f s i x s p e c t r a r e c o r d e d a t t h i s t e m p e r a t u r e , a f t e r c o r r e c t i o n f o r m o d u l a t i o n e f f e c t s , had a r a t h e r l a r g e s c a t t e r o f * 50$ . T h i s r e s u l t i s pr e s u m a b l y a measure of th e random, l i q u i d l i k e m o t i o n s o f t h e g u e s t c a t i o n s i n t h e w a t e r " s o l u t i o n " . D u r i n g the subsequent c o o l i n g c y c l e o f t h i s m o l t e n specimen, second moments c o r r e s p o n d i n g t o t h e d o t t e d l i n e shown i n F i g . 27 we rie o b t a i n e d . These marked d i f f e r e n c e s i n the second moment between warming and c o o l i n g r u n s may be i n t e r p r e t e d as b e i n g due t o a s u p e r - c o o l e d phase o f the t e t r a n - b u t y l ammonium f l u o r i d e s o l u t i o n i n D^O w h i c h r e v e r t s t o the c l a t h r a t e phase a t 248°K. I t i s an i n t e r e s t i n g c o i n c i d e n c e t h a t t h e x — r a y s t r u c t u r a l a n a l y s i s 21 O. of t h i s c l a t h r a t e has a l s o been c a r r i e d o ut a t 2 4 8 % and p o s i t i o n a l d i s -o r d e r d e m o n s t r a t e d f o r one o f t h e f i v e c a t i o n s o f the u n i t c e l l . The s h a r p upward t r a n s i t i o n i n t h e second moment c u r v e a t 248°K i n d i c a t e s some degree of " f r e e z i n g i n " o f t h e m o l e c u l a r d e g r e e s o f freedom o f a p o t e n t i a l g u e s t -f o r m i n g s t r u c t u r e i n w a t e r s o l u t i o n . The ave r a g e v a l u e of t h e second moment f o r t h e u p p e r l i m i t o f t h i s t r a n s i t i o n , o b t a i n e d f r o m f o u r s p e c t r a a t 248°K 2 d u r i n g t h e "warming up" e x p e r i m e n t a l r u n , i s 4.90*0.20G . From T a b l e X X I , i t i s seen t h a t t h i s v a l u e s u p p o r t s a d i s o r d e r e d r a t h e r t h a n an o r d e r e d s t r u c t u r e iior the f i f t h c a t i o n . The agreement between the e x p e r i m e n t a l and the c a l c u l a t e d v a l u e i s s u r p r i s i n g l y good, c o n s i d e r i n g t h a t o n l y a c r u d e m o t i o n a l model has been employed f o r t h e c a l c u l a t i o n s . The t r a n s i t i o n o f t h e s u p e r c o o l e d t o t h e d i s o r d e r e d c r y s t a l l i n e 150 phase i s i n l i n e w i t h a number o f r e c e n t t h e o r i e s on w a t e r - s t r u c t u r e . 151 I n p a r t i c u l a r , F r a n k has p o i n t e d out t h a t c l a t h r a t e f o r m a t i o n i n l i q u i d s l e a d i n g t o a s o l i d , i c e - l i k e s t r u c t u r e m i g h t w e l l c o n s i s t o f t r a n s i t o r y ( l O * " * * s e c , o r l o n g e r ) f r a g m e n t s , o r i c e - l i k e " c l u s t e r s " , o f p e n t a g o n a l 133 d o d e c a h e d r a . He has r e f e r r e d t o t h e v a r i e t y o f r e c e n t l y p r o p o s e d p o l y h e d r a l c l a t h r a t e s t r u c t u r e s i n v o l v i n g t h e t e t r a n - a l k y l ammonium c a t i o n , and has e x p l i c i t l y named t h e t e t r a n - b u t y l ammonium i o n as a s t r o n g s t r u c t u r e - m a k e r . 152 Diamond c o n s i d e r s a t e t r a alkylammonium i o n as a m o l e c u l e w i t h a c h a r g e 150 embedded i n i t , and i t has r e c e n t l y been a r g u e d t h a t such an i o n s h o u l d s t r o n g l y promote i c e - s t r u c t u r e . I t may be a r g u e d t h a t the s u g g e s t i o n o f extreme m o t i o n o f t h e g u e s t c a t i o n s w i t h i n t h e c l a t h r a t i o n v o i d s i s u n r e a l i s t i c i n v i e w o f t h e w e l l -r e s o l v e d F o u r i e r c o n t o u r s r e p o r t e d f o r t h e g u e s t b u t y l c h a i n s by M c M u l l a n 21 and c o - w o r k e r s . However, t h e t i m e s c a l e s c h a r a c t e r i s t i c o f the two t y p e s o f investigation.; a r e c r u c i a l t o such an argument. NMR a b s o r p t i o n i s p r o -f o u n d l y a f f e c t e d even f o r s u c h slow m o t i o n s as a r e c h a r a c t e r i z e d by a —8 c o r r e l a t i o n t i m e o f 10 s e c . From l i n e w i d t h d a t a f o r t h e d e u t e r a t e d h y d r a t e i n t h e t r a n s i t i o n t e m p e r a t u r e r e g i o n , a l i f e t i m e (T c) o f 10"** s e c , may be c a l c u l a t e d f o r a c e r t a i n c o n f i g u r a t i o n of each moving c h a i n a t 248°K. S i n c e t h i s p e r i o d i s s e v e r a l t i m e s t h a t o f a m o l e c u l a r v i b r a t i o n , i t i s l o n g enough t o c o n s t i t u t e a d i s t i n c t x - r a y d i f f r a c t i o n p a t t e r n * A g a i n , t h e q u e s t i o n might be a s k e d whether " v i o l e n t " m o t i o n s of t h e a l k y l c h a i n s o f the b u l k y q u a t e r n a r y i o n i n t h e c a v i t i e s w i l l n o t r e s u l t i n e x t e n s i v e r u p t u r e of s e v e r a l hydrogen-bonds i n t h e h o s t - s t r u c t u r e . The answer t o t h i s may be b a s e d on t h e f a c t t h a t hydrogen bonds a r e v e r y 153 f l e x i b l e . F o r example, i t has been p o i n t e d out b y P o p l e i n c o n n e c t i o n w i t h h i s s t u d i e s on a s t r u c t u r a l model o f w a t e r t h a t b e n d i n g o r d i s t o r t i o n o f h ydrogen bonds makes f o r an e x t e n s i v e a b s o r p t i o n o f energy and e n t r o p y . 153 P o p l e c o n s i d e r s s u c h " b e n d i n g " as a c o n t i n u o u s l y v a r i a b l e r o t a t i o n o f t h e hydrogen atom, o r t h e oxygen 2p l o n e - p a i r , o r b o t h , out o f t h e 0...0 l i n e o f c e n t r e s . I n d eed, once t h e p o s t u l a t i o n o f a s i m i l a r " b e n d i n g " o f t h e h y d r o g e n bonds i n t h e h y d r a t e s t r u c t u r e t o accommodate t h e r e o r i e n t i n g c a t i o n has been made, i t i s e a s y t o e x p l a i n t h e c o n s i d e r a b l e d i s t o r t i o n o f th e i d e a l i z e d t e t r a g o n a l s t r u c t u r e and of c e r t a i n h o s t - l a t t i c e oxygen atoms 21 o b s e r v e d i n t h e x - r a y c r y s t a l s t r u c t u r e . Xhe l a r g e a n i s o t r o p i c t h e r m a l 21 m o t i o n o f f i v e of t h e framework oxygens r e p o r t e d by M c M u l l a n e t a l . , — p a r t i c u l a r l y oxygen 5B i n t h e i r work — i s i n s u p p o r t o f s u c h f l e x e d h y d r o g e n bonds i n t h e c l a t h r a t e s t r u c t u r e . 135 CHAPTER I X SUMMARY AND FURTHER PROSPECTS The r e s u l t s o f o u r i n v e s t i g a t i o n s may now be summarized. The t e n c l a t h r a t e s s t u d i e d can be c o n v e n i e n t l y d i v i d e d i n t o t h e f o l l o w i n g c a t e g o r i e s . I n t h e f i r s t c a t e g o r y may be p l a c e d t h e h y d r a t e s o f CF^, S F g , and CgH^O ( e t h y l e n e o x i d e ) . The g u e s t m o l e c u l e s i n t h e s e c l a t h r a t e s e x p e r i e n c e b u t l i t t l e r e s t r i c t i o n f r o m t h e w a l l s of t h e h o s t c a v i t i e s , and f r e e l y r e -o r i e n t about a chosen a x i s o f symmetry a t l o w t e m p e r a t u r e s and undergo random m o t i o n a t h i g h t e m p e r a t u r e s . B e l o w 250°K, s p h e r i c a l l y - s y m m e t r i c f o r c e f i e l d s a r e i n d i c a t e d i n s i d e t h e c a v i t i e s o f t h e s e c r y s t a l l a t t i c e s . The g u e s t m o l e c u l e s spend a l l t h e i r t i m e r e o r i e n t i n g i n s i d e t h e s e c a v i t i e s b u t t r a n s l a t i o n a l m o t i o n i s q u i t e h i n d e r e d . F o r example, a t t e m p e r a t u r e s above 150°K, SFg undergoes a v e r y l i m i t e d i s o t r o p i c t r a n s l a t i o n a l ( " r a t t l i n g " ) m o t i o n u p t o a d i s t a n c e o f 0.50A f r o m t h e c e n t r e o f t h e c l a t h r a t e c a v i t y . F o r t h i s group o f c l a t h r a t e h y d r a t e s , t h e p a r t i a l m o l a r e n t h a l p y ( i . e . , 136 " b i n d i n g " energy) o f each g u e s t m o l e c u l e i n i t s l a t t i c e s i t e must d e r i v e 154 155 o n l y f r o m d i p o l e - i n d u c e d - d i p o l e i n t e r a c t i o n s ' . I n the l i m i t o f f u l l y q u a n t i z e d r e o r i e n t a t i o n s o f t h e g u e s t n u c l e a r s p i n s i n s i d e t h e i r c a v i t i e s , t h e dominant r e l a x a t i o n mechanism w o u l d be e x p e c t e d to be a s p i n -156 r o t a t i o n i n t e r a c t i o n . I n d eed, s u c h r e s u l t s have r e c e n t l y been i n d i c a t e d f r o m p r o t o n r e s o n a n c e s t u d i e s of HgS i n Type I and Type I I h y d r a t e l a t t i c e s between 0°C and -50°C. The h y d r a t e of propane would appear t o b e l o n g t o a c a t e g o r y of c l a t h a t a t e s i n w h i c h t h e t e m p e r a t u r e - i n d u c e d t r a n s l a t i o n a l f reedom of t h e g u e s t s i s a maximum. A m o t i o n a l t r a n s i t i o n o f propane f r o m r e o r i e n t a t i o n t o l a t t i c e - d i f f u s i o n has been o b s e r v e d f o r t h i s c l a t h r a t e a t t e m p e r a t u r e s c l o s e t o i t s d e c o m p o s i t i o n , and f r o m t h e specimen—dependent b e h a v i o u r o f t h e l i n e w i d t h v e r s u s t e m p e r a t u r e c u r v e s an u p p e r l i m i t d i f f u s i o n a l a c t i v a -t i o n energy o f 1.40*0.02 k c a l / mole has been c a l c u l a t e d . Among the c l a t h -r a t e s o f t h e q u i n o l s e r i e s , t h a t o f a r g o n has been shown t o e x h i b i t 15T 158 d i f f u s i o n o f the g u e s t . S i n c e a c e t o n e h y d r a t e i s known t o l o s e v a r i a b l e amounts o f a c e t o n e r a t h e r l i k e i t s propane a n a l o g u e , i t i s p o s s i b l e t h a t d i f f u s i o n o f a c e t o n e l i k e w i s e o c c u r s i n a c e t o n e - h y d r a t e , and one may p l a c e t h e s e two h y d r a t e s i n the same c a t e g o r y . A s i t u a t i o n i n s t r i k i n g c o n t r a s t t o the above i s p r o v i d e d by t h e l o w - t e m p e r a t u r e r e s u l t s ; o n t h e - h y d r a t e s o f CHgCl, CHgBr, C H g l , and C H g C l g . I n t h e s e h y d r a t e s ^ w h i c h make up a t h i r d c a t e g o r y , the g u e s t m o l e c u l e s have been f o u n d t o be i m m o b i l i z e d t o a degree a p p r o a c h i n g t h a t o f t h e s o l i d s t a t e a t low t e m p e r a t u r e s . E x p e r i m e n t a l d a t a o b t a i n e d a t 77°K f o r t h e m o n o c h l o r o - and monoiodomethane c l a t h r a t e s i n d i c a t e o n l y l o w - a m p l i t u d e o s c i l l a t o r y m o tions o f m e t h y l g r o u p s , w h i l e f o r t h e monobromomethane c l a t h r a t e a t 77°K a t r i p l e t l i n e shape, c h a r a c t e r i s t i c o f a r i g i d m e t h y l 137 group, has been r e c o r d e d . A t h i g h e r t e m p e r a t u r e s , t h e r e i s e v i d e n c e o f e x p a n s i o n o f t h e deuterium-bonded h y d r a t e l a t t i c e f o r t h e s e p a r t i c u l a r c l a t h r a t e s . Such r e s u l t s i n d i c a t e t h a t , a t l e a s t i n s p e c i f i c c a s e s , r e f i n e m e n t s t o t h e c u r r e n t l y e x i s t e n t a p p r o x i m a t i o n s o f " s p h e r i c a l l y — s y m m e t r i c " h o s t cages a r e n e c e s s a r y . I n f a c t , i t i s r a t h e r u n u s u a l t h a t t h e c o m p o s i t i o n o f th e C H g C l g - h y d r a t e u s e d i n t h i s work s h o u l d have c o r r e s p o n d e d t o t h a t o f a "Type I " h y d r a t e , s i n c e t h e g u e s t m o l e c u l e h e r e i s s l i g h t l y l a r g e r t h a n the upper l i m i t o f s i z e commensurate w i t h a t e t r a k a i d e c a h e d r a l c a v i t y o f t h e "Type I " l a t t i c e . F u r t h e r m o r e , the d o u b l e t p r o t o n l i n e s t r u c t u r e o f t h e g u e s t CHgClg a t 77°K c o r r e s p o n d s t o a s l i g h t l y "squeezed" m o l e c u l e w i t h an H-H d i s t a n c e o f 1.74A, so t h a t , a t low t e m p e r a t u r e s a t any r a t e , d i s t i n c t l y n o n - s p h e r i c a l h o s t c a v i t i e s , w i t h w e l l - o r d e r e d g u e s t m o l e c u l e s a l i g n e d a l o n g a s t e r i c a l l y f a v o u r a b l e a x i s i n s i d e them, a r e i n d i c a t e d . I n d e e d , i n s u c h extreme c a s e s o f " t i g h t - f i t t i n g " g u e s t m o l e c u l e s , i t i s even p o s s i b l e t o e n v i s i o n the h e r e t o f o r e n e g l e c t e d "exchange i n t e r a c t i o n s " a r i s i n g f r o m t h e o v e r l a p o f t h e e l e c t r o n i c wave f u n c t i o n s o f the g u e s t m o l e c u l e s w i t h p e r i -p h e r a l o r b i t a l s o f the h o s t - l a t t i c e oxygens. A l t e r n a t i v e l y , a l o w -t e m p e r a t u r e c o n t r a c t i o n of t h e h y d r a t e h o s t - l a t t i c e i n t h e d i r e c t i o n o f non-c u b i c symmetry i s a n o t h e r d i s t i n c t p o s s i b i l i t y , and i t s h o u l d be o f i n t e r e s t t o s t u d y t h e l o w - t e m p e r a t u r e x - r a y c r y s t a l s t r u c t u r e of t h e d i c h l o -romethane h y d r a t e . One m i g h t r e c a l l i n t h i s c o n t e x t t h a t e a r l i e r s t u d i e s See f o o t n o t e #2, T a b l e V 138 o f t h e q u i n o l c l a t h r a t e s o f s u f f i c i e n t l y l a r g e m o l e c u l e s ( e . g . SO^, CH^CN) 43 have l e d t o t h e p r o p o s a l o f d i s t o r t e d h o s t - l a t t i c e v o i d s . O t h e r i n t e r e s t i n g e x p e r i m e n t s t h a t c o u l d be p e r f o r m e d w i t h t h e above c a t e g o r y o f h y d r a t e s a r e t h e f o l l o w i n g : one c a n measure l i n e shapes, second moments, and 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 s o f t h e " g u e s t " f o r v a r i o u s o r i e n t a t i o n s o f a s i n g l e c r y s t a l o f monobromomethane o r d i c h l o r o -methane h y d r a t e . These s t u d i e s w o u l d perhaps l e a d t o a d e t e r m i n a t i o n o f t h e axes o f a l i g n m e n t o f t h e g u e s t s w i t h r e s p e c t t o t h e h y d r a t e c r y s t a l a x e s . A n o t h e r p r o m i s i n g avenue of i n v e s t i g a t i o n w o u l d be t h e s t u d y o f n u c l e a r q u a d r u p o l e r e s o n a n c e s p e c t r a , e t g . , t h e C l ^ ^ r e s o n a n c e i n h y d r a t e c r y s t a l s o f 35 CHgCl and CHgClg, and t h e t e m p e r a t u r e dependence o f t h e C I q u a d r u p o l e t e n s o r i n t h e s e c l a t h r a t e s . A s t u d y o f t h e Zeeman s p l i t t i n g s as a f u n c t i o n of c r y s t a l o r i e n t a t i o n s h o u l d h e l p l o c a t e t h e symmetry a x i s of t h e e l e c t r i c f i e l d g r a d i e n t t e n s o r , w h i c h i n t u r n w o u l d y i e l d t h e d i r e c t i o n o f t h e r e o r i e n t a t i o n a l a x es o f t h e g u e s t s . I n t h e f o u r t h and l a s t c a t e g o r y may be p l a c e d t h e t e t r a a l k y l a -mmonium f l u o r i d e h y d r a t e s , ( i - C g H^^NF.SSDgO and (n-C 4H 9) 4NF.31.8I> 20. I n t h e s e s t r u c t u r e s , d i s t o r t i o n s o f h o s t - l a t t i c e s f r o m i d e a l symmetries a r i s e due t o t h e p r e s e n c e o f p o l a r N + and F ~ i o n s , and t h e e n c l a t h r a t e d a l k y l c h a i n s e x h i b i t t e m p e r a t u r e - d e p e n d e n t o s c i l l a t o r y and r o t a t o r y m o t i o n s , 159 r a t h e r l i k e t h e a l k y l c h a i n s i n some r e c e n t l y s t u d i e d s t e a r i c a c i d soaps The p r o t o n r e s o n a n c e l i n e w i d t h and second moment r e s u l t s between 77°K and 286°K f o r i - amyl groups i n c l u d e d as " g u e s t s " i n the c l a t h r a t e h y d r a t e o f ( i - C g H ^ ) 4 N F may be c h a r a c t e r i z e d i n terms o f s i m p l e m o t i o n a l models o f t h e s e g u e s t m o i e t i e s . The r e s u l t s complement t h e r e p o r t e d x - r a y c r y s t a l s t r u c t u r e o f t h i s c l a t h r a t e . F o r the t e t r a n- butylammonium c l a t h r a t e , the l o w t e m p e r a t u r e second moments c o u l d n o t be i n t e r p r e t e d s a t i s f a c t o r i l y , a l t h o u g h m o t i o n of t h e a l k y l c h a i n s has been d e f i n i t e l y i n d i c a t e d . However, a p r i n c i p a l f i n d i n g f r o m t h e r e s u l t s o b t a i n e d w i t h t h i s c l a t h r a t e i s t h e phase t r a n s i t i o n o c c u r r i n g s h a r p l y a t 248°K. T h i s would seem t o encourage d i e l e c t r i c s t u d i e s i n t h i s t e m p e r a t u r e r e g i o n . A c c u r a t e thermodynamic measurements on t h e e n t r o p y of h y d r a t i o n and t h e m o l a r h e a t c a p a c i t y o f the t e t r a n-butylammonium c a t i o n i n aqueous s o l u t i o n s h o u l d p r o v i d e r e l e v a n t complementary d a t a t o t h o s e o b t a i n e d i n t h e p r e s e n t s t u d y . 140 BIBLIOGRAPHY 1. A. Clemm, Ann. Chem. 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Chem. P h y s . 34, 604 (1961) 159. B.A. D u n e l l and W.R. J a n z e n , W i s s . Z. d e r U n i v e r s i t S t J e n a M a t h. N a t u r w i s s . R e i h e _14, 191 (1965) 148 APPENDIX I Computer Programme "MODBPP" t o s o l v e e q u a t i o n 2T T V C = Q76H J t a n 'TT UH^ -BV"1-1  L 2 ( C 2 -B2) f o r a number of d a t a p o i n t s (6H\ , T\), make a l e a s t s q u a r e s f i t t o -E/RT v = v e c ~ g i v e and E f r o m t h e l e a s t s q u a r e s f i t and p r i n t out a t a b l e o f 6H v s . T f o r " b e s t - f i t " t o d a t a . The F o r t r a n IV programme s t a r t s on t h e f o l l o w i n g page. I n t h i s programme, P = C, Q = B, and AG _ .2Y _ i _ Y 2TT = "8TE2 = 7 6 7 , 5 f o r p r o t o n s , 149 FORTRAN SOURCE LI ST SOURCE STATEMENT • tIBFTC MOO BPP * C .__ ^MODIFIED B P P LINE NARROW I NG ANALYSES * C DIFFERS FROM ORIGINAL CASE BY INCLUSION OF TERM AG IN NUMERATOR OF * C BPP tQN PHYS REV L948 VOL73 P6f9.F0R PROTUNS AG=767.5 * C THIS PROGRAM USES THE MODIFIED BPP EON TO DERIVE THE CORRELATION * C FREQUENCY,MAKES A LEAST SQUARES FIT OF LN(COR.FRO) TO A ST.LINE * C WHEN PLOTTED V S . l / R T , DERIVES THE ACTIVATION ENERGY AND INF.TEMP. * C CORREL.FREQ.FROM THE FIT.PROGRAM THEN REVERSES THE PROCESS TO GIVE * C Thc-ORc 1 I CAL FIT TO THE LINEWIDTH VS. TEMP.DATA PLOT. t C DEFINITIONS-INPUT * C ALPHA = DAT A COMMENTS E.G. NAME OF COMPOUND,DATE. * C TEMP. I N UCG.KELVIN.,DELTAH=LINEWIDTH(GAUSS) IN NARROWING REGION * C (LESS THAN C, GREATER THAN B).C= A-VG.LINEWIDTH BELOW TRANSITION. *_C B 3 A V G . LINEWIOTH ABOVE TRANS ITI ON. AG=PARAMETER DEPENDENT ON NUCLR * C SPECIES,=767.6 FOR H. DEL M I N = M N LINEwIOTH IN REGION OF THEORt TIC t C FIT,GREATER THAN B.DELMAX=MAX LINEWIDTH IN REGION OF THEORETICFIT, * C LESS THAN C. DEL INC = LINEWIDTH INCREMENT FOR THEORY FIT.N = NO. OF * C DATA POINTS. * C CALCULATED QUANTITIES- OUTPUT * C C 0 R F R O = DERIVED CORREL. FREQ. R E C I P T = 1 / T EMP . X L N F R Q = L_N OF CORFRQ. * C RCPR'T" =1/( RVTEMP )". WHERE' R = '1.9869 CAL/DEG-MOLE """''" * C E AC T = ACTIVATION ENERGY(CAL/MOLE),EREACT= ERROR IN S AME » FRQMAX = * C INF.TEMP. CORREL.FREQ. IN THEOR.FIT. ERFRQ=ITS ERROR.THDEL= L I N E -* C WIDTH I N THEOR.FIT. THTEMP= TEMP.IN THEOR.FIT. THFRQ= CORREL.FREQ * C IN THEOR. F I T . * D_I M EJN1S_I_0 N TEMPI 500) , DELT AH ( 500 ), CORFRQ ( 500) , R E C I P T ( 5 0 0 ) , * lXLNFRQI500),XSIN!50 0),XCOS(500),XT AN(500)»RCPRT(500),ALPHA!12), * 2THFRQ(500),THTEMP(500),THDEL(500),THSNI500) t THCOI500),THTANI500) * PRINT 10 * 10 FORMAT( IX,49HNMR LINEWIDTH DATA TREATED ACCORDING TO G.W.SMITH) * 15 READ (5,20). (ALPHA! I ) ,1 = 1,12) * 20 FORMAT (12A6) * WR I T E' ( 6 , 30) (ALPHA ( I ) , I = 1, 12 ) * 30 FORMAT (IX,12A6) * 35 READ (5,40)N,C,B,AG * 40 FORMAT ( I 5 . 3 F 1 5 . 5 ) * READ < 5,50 ) (TEMPI I),DELTAH(I)» 1 = 1,N) * 50 FORMAT ( 6F12.5) * DO 60 I = 1» N * X S I N ( I ) = S I N ( 1. 5 707*(DELTAH(I)»»2-8»*2)/!C»»2-B»*2)> * XCOSI I )=COS( 1 . 5 7 0 7 * ( D E L T A H ( I ) » * 2 - B * * 2 ) / ( C » » 2 - B * « 2 ) ) * X T A N ( I ) = X S I N ( I ) /XC0 S1 I ) * CORFRQl I )=AG»DELTAH( I ) / X T A N ( I ) * XLNFRQ( I )=ALOG(CORFRQ( I ) ) • * R~ECIPT(I) = 1./TEMP! I) * RCPRT! I ) = !./[1.9869»TEMP( I ) ) * 60 CONTINUE * 65 WRITE(6,70)N,C,B,AG * 70 FORMAT(1H0,5X,2HN=,I 5,5X,25HRIGID LATTICE LINEWIDTH = ,FI 0.5,5X,20H * 1NARR0WED LINEWIDTH =,F10.5,5X,4HAG =,F15.5)  * PRINT 80 * 80 FORMAT (IHO,LX,16HTEMP(DEG KELVIN),4X,16HLINEWIDTH(GAUSS), 4X • * 114HC0R.FREG.(CPS),10X.6H1/TEMP,I OX,13HLN(COR.FREQ. ),1 1 X,4H1/RT) * W R I T E ( 6 , 9 0 ) ( T E M P ( I ) , D E L T A H ! I ) , C O R F R Q ( I ) , R E C I P T ( I ) , X L N F R Q ( I ) , * 1RCPRT! I ) , 1 = 1,N) F O R T R A N SOURCE L I S T MOO BP 1 5 0 SOURCE S T A T E M E N T * 9 0 F O R M A T ( I H , 2 X , F 1 2 » 5 , 8 X , F 1 2 . 5 , 7 X , E 1 4 . 6 , 9 X , F 1 0 . 7 , 9 X , F 1 2 . 7 , 9 X , F 1 0 . 7 ) * * SUMX = 0 . 0 SUMY = 0 . 0 * 0 0 2 0 0 J = 1 , N * SUMX * SUMX* R C P R T ( J ) * * SUMY = SUMY+ X L N F R Q ( J ) t * 2 0 0 C O N T I N U E * C A L L L S Q F I T ( N , . R C P R T , X L N F R Q , S U M X , S U M Y , Q , P , S T D E R Q , S T O E R P . X A V , * 1 Y A V . N 0 G 0 ) * FRQMAX = E X P ( Q ) * * E R F R Q = E X P ( Q ) » S T D E R Q * E A C T ( - l . ) » P * E R E A C T = ( - l . ) « S T D E R P * WRITE ( 6 , 2 1 0 ) F R Q M A X , E R F R Q , E A C T , E R E A C T * 2 1 0 FORMAT ( 1 H 0 . 2 5 H C 0 R . F R O . AT I N F . T E M P . = , E 1 2 . 5 , 2 X , 7 H E R R 0 R => ,E12.5 , * 1 5 X , 1 2 H A C T I V . E N . = , E 1 2 . 5 , 7 H C A L / M O L , 2 X , 7 H E R R 0 R = » E 1 2 . 5 t 7 H C A L / M 0 L ) R E A D ( 5 , 2 2 0 ) D E L M A X , D E L M I N , D E L INC * 2 2 0 F 0 R M A T J 3 F 1 0 . 5 ) * NOINC= ( O E L M A X - D E L M I N ) / D E L INC + 0 . 0 0 1 * NSW = N O l N C + l * DO 2 3 0 K= 1,NSW * AK = K * T H D E L ( K ) = D E L M I N + < A K - 1 . ) » D E L I N C * * CB = C » » 2 - B » » 2 t * T H S N ( K ) = S 1 N ( 1 . 5 7 0 7 » ( T H D E L ( K ) * » 2 - B » » 2 ) / C B ) * T H C O ( K ) = C O S ( 1 . 5 7 0 7 * ( T H D E L < K ) » « 2 - B « « 2 ) / C B ) * T H T A N ( K ) = T H S N ( K ) / T H C O ( K ) * T H F R Q ( K ) = ( A G * T H D E L ( K ) ) / T H T A N ( K ) t T H T E M P i ( K ) = E A C T / ( 1 . 9 8 6 9 » A L 0 G ( F R Q M A X / T H F R Q ( K ) ) ) * # 2 3 0 C O N T I N U E P R I N T 2 4 0 * * 2 * 0 F O R M A T ( I H O , 3 4 X » 4 7 H T H E 0 R E T I C A L M O D I F I E D BPP L E A S T S Q U A R E S DATA F I T ) * P R I N T 2 5 0 * # 2 5 0 F O R M A T ! 1 H 0 , 6 X , 2 8 H T E M P E R A T U R E ( D E G R E E S K E L V I N ) , 1 7 X , 1 8 H L I N E W I D T H ( G A * # 1USS ) , 1 7 X , 2 7 H C O R R E L A T I O N F R E Q U E N C Y ( C P S ) ) * WRITE I 6 , 2 6 0 ) ( T H T E M P ( K ) , T H D E L ( K ) , T H F R Q ( K ) , K = 1 , N S W ) * 2 6 0 F O R M A T ( I H r 1 4 X , F 1 1 . 5 , 3 0 X , F l 0 . 5 , 2 8 X , E 1 4 . 6 ) * GO TO 15 END 151 F O R T R A N SOURCE L I S T S O U R C E S T A T E M E N T r t I B F T C S U B R O U T I N E L S O F I T | N , X , Y , SUMX, SUMY, B f C S T O E R B . S T D E R C , X A V , YAV .NO '^O) * D I M t N S I ON X ( 5 C 0 ) , Y ( 5 0 0 ) I F I N - 2 ) 1 0 5 0 , 1 0 5 0 , 1 0 0 0 t 1 0 0 0 AN = N X A V = S U M X / A N Y A V = S U M Y / A N O I F X Y = 0 . D I F X S Q = 0 . DO 1 0 1 0 J = 1 , N D I F X Y = D I F X Y • ( X ( J ) - X A V ) « Y ( J ) D I F X S O = D I F X S Q + ( X ( J ) - X A V > » » 2 * 1 0 1 0 C O N T I N U E C = D I F X Y / 0 1 F X S G Q ^ Y A V - C « X A V DSQ = 0 . XSC =. 0 . DO 1 0 2 0 J= U N DSO = DSQ • I B + C » X ( J ) - Y ( J ) ) » » 2 XSQ = XSQ + X ( J ) * » 2 * 1 0 2 0 C O N T I N U E Q ^ S Q R T ( D S C / ( A N - 2 . ) ) DEE = A N » X S O - ( A N » X A V ) » » 2 I F ( D E E ) 1 0 5 0 , 1 0 5 0 , 1 0 3 0 * 1 0 3 0 QOVRTD= Q / S Q R T ( D E E ) S T D E R C - Q O V R T D » S Q R T ( A N ) S T D E R B = Q O V R T D ^ S Q R T < X S Q ) NOGO = 1 P R I N T 1 0 4 0 * 1 0 4 0 FORMAT ( IH , .52HN0G0 = 1, T H E R E F O R E L S Q F I T HAS MADE A S U C C E S S F U L F I T * 1) GO TO 1 1 0 0 * 1 0 5 0 NOGO = 2 P R I N T 1 0 6 0 * 1 0 6 0 F O R M A T ( I H , 8 4 H N 0 G 0 = 2 , L S Q F I T U N S U C C E S S F U L DUE TO DEE L E S S T H A N 0 4 1R = 0 , OR TO N L E S S THAN OR = 2 . ) * 1 1 0 0 R E T U R N END Step 2: Get and Hg as follows: RSinB AxB (AXB"| APPENDIX II 152 Generation of Hydrogen Atom Locations on a Carbon Chain Cp Cg, Cg are chain carbon atoms with (x,y,z) coordinates given, Step 1: Get A+B AxB |A+Bj ' JAxB| where HCH « 26, C-H m R. RCos9 H A+B 2 |A+B) P . «iRSin6AxB + RCoseA+B ** M j£J5| 153 F O R T R A N SOURCE L I S T SOURCE S T A T E M E N T * S I B F T C G E N H L O C * * C FORTRAN IV G E N E R A T I O N OF H ON CARBON C H A I N * * 43 C O N T I N U E * * C A L L S K I P TO ( 1 ) * 20 FORMAT ( l X f 4 8 H G E N E R A T I 0 N HATOM L O C A T I O N S P . R A G H U N A T H A N OCT 6 4 ) * 21 FORMAT ( 1 X . 4 1 H ) * P R I N T 20 * * READ 21 * * P R I N T 21 * * 23 C O N T I N U E * * D I M E N S I O N A ( 3 ) , B ( 3 ) , C ( 3 ) , P ( 3 ) , Q ( 3 ) , S ( 3 ) , D ( 3 ) , V E ( 3 ) * * D I M E N S I O N U ( 3 ) f V H U ( 3 ) , V H L ( 3 ) , V ( 3 ) , X ( 3 ) , E ( 3 ) , F ( 3 ) * * 22 FORMAT ( I 2 , 9 F 6 . 3 , F 8 . 5 , F 5 . 2 ) READ 2 2 , M , A m , A ( 2 ) , A ( 3 ) , B ( l ) , B ( 2 ) , B ( 3 ) , C ( L ) , C ( 2 ) , C ( 3 ) , A N G L t R * * IF ( M . G E . 1 ) GO TO 4 0 * 1 C O N T I N U E * * 0 0 2 1 = 1 , 3 * * P ( I ) = 2 . » B ( I ) - A ( I ) - C ( I ) * 0 ( I ) = B ( I ) - A ( I ) t * s m = e m - c m * * 2 C O N T I N U E * 6 PMAG = S Q R T ( P ( 1 ) » P ( 1 ) + P ( 2 ) » P ( 2 ) * P ( 3 ) « P ( 3 ) ) * * X ( l ) = Q ( 2 ) » S ( 3 ) - Q ( 3 ) » S ( 2 ) * * X ( 2 ) = Q ( 3 ) » S ( 1 ) - Q ( i ) » S ( 3 ) * X ( 3 ) = Q ( 1 ) « S ( 2 ) - Q ( 2 ) » S ( 1 ) * * XMAG = S O R T ! X I 1 ) » X ( 1 ) + X ( 2 ) * X ( 2 ) * X ( 3 ) » X ( 3 ) ) * * F A C I A = f t » C O S ( A N G L ) / P M A G * F A C T B = R » S I N ( A N G D / X M A G * * DO 3 1 = 1 , 3 * * D( I ) = P ( I ) » F A C T A * * U( I ) = X ( I ) « F A C T B V H U ( I ) = B ( I ) + D ( I ) + U < I ) * V H L ( I ) = B ( I ) + D ( I ) - U ( I ) * * 3 C O N T I N U E * * 24 FORMAT ( 1 X . 3 F 1 2 . 8 ) * * P R I N T 2 4 , V H L ( 1 ) , V H L ( 2 ) , V H L ( 3 ) * PUNCH 3 0 , V H L ( 1 ) , V H L ( 2 ) , V H L ( 3 ) * P R I N T 2 4 , V H U ( 1 ) , V H U ( 2 ) , V H U ( 3 ) * * PUNCH 3 0 , V H U ( 1 ) , V H U ( 2 ) , V H U ( 3 ) * GO TO 2 3 * * C M E T H Y L GROUP AT END , * 4 0 C O N T I N U E * * IF ( M - 2 ) 4 1 , 4 3 , 4 2 * * 41 C O N T I N U E * DO b 1 = 1 , 3 * P ( I ) = B l I ) - A ( I.) * * 5 C O N T I N U E * * PMAG = S Q R T ( P ( l ) » P ( l ) + P ( 2 ) » P ( 2 ) + P ( 3 ) « P ( 3 ) ) * F A C T C = R /PMAG * * F A C T D = 1 . 5 4 / P M A G * * DO 8 1 = 1 , 3 * V I I ) = C ( I ) + P ( I ) » F A C T C * V E ( I ) = P ( I ) » F A C T D * A ( I ) = B ( I ) * B ( I ) - c m • * SOURCE S T A T E M E N T F O R T R A N SOURCE L I S T G E N H L O 1 5 4 * C ( I ) = C ( I ) +VE( I ) + * 8 CON T I NUE * * 25 F O R M A T ( 1X « 3 F 1 2 • 8 , 2 0 X , I O H E N O M E T H Y L ) * * 3 0 FORMAT ( 1 X . 3 F 1 2 . 6 ) * * P R I N T 2 5 , V ( I ) , V ( 2 ) , V ( 3 ) * * PUNCH 3 0 , V ( 1 ) , V ( 2 ) , V ( 3 ) * * GO TO 1 * * 42 C O N T I N U E * * END * 155 APPENDIX I I I Computer Programme "DBLET" t o G e n e r a t e D o u b l e t L i n e Shapes, G i v e n a and B. +3a F ( ^ h ) = [ dp_ 1 exp [-(^J W f t ' ) "| d ^ ' ^ B / 2 T T 2 p 2 I n t h e F o r t r a n IV Programme, iJji = H = 0, 0.5, 1.0, 1.5, .. d£h = DX = 0.0125 F ( ^ h ) = D = P = FH (D) f o r - 3o<D<-cr d ^ i = FL ( D ) f o r - Q<D<Q' = FQ(D) f o r o<D<3of and |3/2n exp = SH(T) The programme s t a r t s on t h e f o l l o w i n g page 156 F O R T R A N SOURCE L I S T SOURCE S T A T E M E N T * S I B F T C D B L E T O I M E N S I O N C ( 1 4 ) * COMMON A L P H A , B E T A f X * * 21 R E A D I 5 . 2 2 ) ( C ( I ) , 1 = 1 , 1 4 ) * 22 F O R M A T ! I X , 1 3 A 6 , A 1 ) * W R I T E I 6 . 2 2 ) ( C ( I ) , I = 1 , 1 4 ) P R I N T 10 t t 10 F O R M A T ! I X , 3 4 H D 0 U 8 L E T L I N E S H A P E F I T ABRAGAM P 2 1 9 ) * R E A 0 ( 5 , 1 ) A L P H A , B E T A * DX= 0 . 0 1 2 5 X = 2 . * A L P H A * H = 0 . * 0 0 100 1 = 1 , 1 0 0 0 H= H + ( 4 0 . « D X ) * I F I H . E Q . ALPHA) . H = ( 4 0 . * D X ) • H * SUM = 0 . * 0 = - 3 . » A L P H A * 0 0 9 0 J= 1 , 4 0 0 0 D = D + DX * I F ! D . G T . ( 3 . " A L P H A ) . ) GO TO 91 * t T = H-D * r I F ! D . L T . ( — A L P H A ) ) P = F H ( D ) * t I F ( D . G T . A L P H A ) P = F O ( 0 ) * I F ! A D S ( D ) i L T . A L P H A ) P = F L ! D ) •  9 0 SUM = P » S H ( T ) » D X + S U M = 91 C O N T I N U E * W R I T E < 6 , 2 ) H , SUM * = 1 F 0 R M A T I 2 F 1 2 . 8 ) * = 2 FORMAT 1 1 X , 2 H H = . , F 1 2 . 8 , 1 0 X , 5HF (H ) = , F 1 3 . 4 ) I F I H . G T . ( 3 . * A U P H A ) ) GO TO 101 * = 1 0 0 C O N T I N U E * = 101 C O N T I N U E * GO TO 21 * ENO * 157 F O R T R A N SOURCE L I S T SOURGE S T A T E M E N T t M B F T C R HD * r F U N C T I O N F H ( D ) r COMMON ALPHA, .BETA< X r IF{ D . L T . X ) F H = . 0 . * t I F < D . G T . X ) F H = ( - D / A L P H A * 1 . > » » ( - 0 . 5 ) * t R E T U R N i END * F O R T R A N SOURCE L I S T 0 SOURCE S T A T E M E N T M B F T C SHT * F U N C T I O N S H I T ) * COMMON A L P H A . . B E T A , X * SH = E X P ( - T « T » 0 . 5 / ( B E T A * B E T A ) ) 7 ( 8 E T A » 2 . 5 0 5 ) * R E T U R N * END * F O R T R A N SOURCE L I S T 0 3 0 U R 0 E S T A T E M E N T * $ I B P T C F L D * F U N C T I O N F L I O ) * COMMON A L P H A , - B E T A i X * * A= D / A L P H A * * FL = I - A + l . ) » » ( - O J 5 ) + { A + l . ) . » » ( - 0 . 5 ) * 4 . R E T U R N * * END F O R T R A N SOURCE L I S T Oi SOURCE S T A T E M E N T * S I 8 F T C F L D * J F U N C T I O N F L I P ) *_ * COMMON A L P H A , B E T A , X * * A * D / A L P H A * * F L = ( - A + 1 . ) » * I - 0 . 5 - ) + ( A + l . ) » • ( - ( ) . 5 ) * * R E T U R N * * END * 

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