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Nuclear spin-lattice relaxation in solid methane at low temperatures. De Wit, Gerald Aloysius 1966

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NUCLEAR SPIN-LATTICE RELAXATION IN SOLID METHANE AT LOW TEMPERATURES by Gerhardus A. de Wit B. Sc., The University of B r i t i s h Columbia, 1961 M. Sc., The University of B r i t i s h Columbia, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE .OF DOCTOR OF PHILOSOPHY in the Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1966 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Bri t ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that per-mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives,, It is understood that copying or publ i -cation of this thesis for financial gain shall not be allowed without my written permission. Department of P h y s i c s  The University of Bri t ish Columbia Vancouver 8, Canada Date March 7 , 1966 The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY B.Sc, The University of B r i t i s h Columbia, 1961 M.Sc, The University of B r i t i s h Columbia, 1963 WEDNESDAY, MARCH 2, 1966, AT 3:30 P.M. IN ROOM 301, HENNINGS BUILDING COMMITTEE IN CHARGE Chairman: I. McT. Cowan External Examiner: K. Tomita Kyoto University, Kyoto, Japan of GERHARDUS A. de WIT M. Bloom L. G. Harrison K. B. Harvey P. W. Matthews B. G. T u r r e l l D. L l . Williams Research Supervisor: M. Bloom NUCLEAR SPIN-LATTICE RELAXATION IN SOLID METHANE AT LOW TEMPERATURES ABSTRACT The s p i n - l a t t i c e r e l a x a t i o n time T^ has been measured i n the temperature range 1.2 to 55°K at 28.5 mcs. for the proton resonance, and at 4°4 mcs for the deuteron resonance using N.M.R. pulse techniques. The proton T^ has been measured for CH^, CH^D, CD-jH, 50%CH,-50%Kr, 90%CH,-10%Kr, 67%CD,-33%CH,, 10%CD -4 4 4 4 4 90%CH., and also for CH. at 4.4 mcs. The deuteron T. 4 4 1 has been measured for CD,, CD„H, and 67%CD.-33%CH. 4 3 4 4. It i s found that a d r a s t i c change i n the tempera-ture dependence of T^ occurs i n the temperature region below the phase t r a n s i t i o n s and that at most of the phase t r a n s i t i o n temperatures there i s either a d i s -continuous change- i n T^ or a change i n the slope of T^ versus T. A minimum i n i s found at low tempera-tures for a l l the systems studied. An analysis of the data based on conventional N.M.R. theory shows i n most cases that the c o r r e l a t i o n time ^ o c T ^ i n the neigh-bourhood of 20°K, and that \ i s almost independent of temperature near 1.2°K. It i s postulated that phonon-molecular i n t e r a c t i o n s , involving d i r e c t and Raman processes, can account for the temperature depen-dence of T. . The values of T.. at the minimum are c 1 completely determined by conventional theory. In most cases, however, the predicted values are of the order of 20 times too short. An unexplained minimum i n T^ was observed i n CH, , CH'-CD., and CH,-Kr mixtures above 4 4 4 4 the upper phase t r a n s i t i o n s . To investigate the o r i g i n of some of the inadequa-cies of the conventional theory, the two energy l e v e l scheme proposed by Colwell, G i l l , and Morrison (1965) i s used, where each of the two l e v e l s may be degenerate. Simple rate equations are used to c a l c u l a t e the condi-t i o n a l p r o b a b i l i t i e s and the c o r r e l a t i o n functions for the two l e v e l model, Tt i s found that the e f f e c t i v e i n t e r a c t i o n strength i s temperature dependent, that the c o r r e l a t i o n function can be described by a simple exponential under c e r t a i n conditions, and that the i n t e r a c t i o n strength has no simple r e l a t i o n s h i p with the c l a s s i c a l value. GRADUATE STUDIES F i e l d of Study: Nuclear Magnetic Resonance Quantum Theory of Solids R, Barrie Advanced Magnetism M. Bloom Low Temperature Physics Electromagnetic Theory G. M. Volkoff J . B. Brown S t a t i s t i c a l Mechanics R. Barrie Related Studies: Quantum Chemistry J . A. R. Coope PUBLICATIONS Gerald A. de Wit and Meyer Bloom "Nuclear Spin Relaxation i n L i q u i d and S o l i d Methane: Isotope E f f e c t s " , Can. J . Phys. 43, 986 (1965). A B S T R A C T i 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 T ^ h a s b e e n m e a s u r e d i n t h e t e m p e r a t u r e r a n g e 1.2 t o 5 5 ° K a t 2 8 . 5 m c s . f o r t h e p r o t o n r e s o n a n c e , a n d a t 4 .4 m c s f o r t h e d e u t e r o n r e s o n a n c e u s i n g N . M . R . p u l s e t e c h n i q u e s . T h e p r o t o n T ^ h a s b e e n m e a s u r e d f o r C H ^ , C R " 3 D , C D 3 H , 5 0 ^ C H ^ - 5 O ^ K r , 9 0 ^ C H ^ - 1 0 ^ K r , 6 7 ^ 0 ^ - 3 3 $ ^ , 1 0 $ C D ^ - 9 0 $ C H ^ , a n d a l s o f o r C H ^ a t 4 .4 m c s . T h e d e u t e r o n h a s b e e n m e a s u r e d f o r C D ^ , C D ^ H , a n d 6 7 $ C D ^ - 3 3 $ C H ^ . I t , i s f o u n d t h a t a d r a s t i c c h a n g e i n t h e t e m p e r a t u r e d e p e n d e n c e o f T - ^ o c c u r s i n t h e t e m p e r a t u r e r e g i o n b e l o w t h e p h a s e t r a n s i t i o n s a n d t h a t a t m o s t o f t h e p h a s e t r a n s i t i o n t e m p e r a t u r e s t h e r e i s e i t h e r a d i s c o n t i n u o u s c h a n g e i n T ^ o r a c h a n g e i n t h e s l o p e o f T - ^ v e r s u s T . A m i n i m u m i n T - j ^ i s f o u n d a t l o w t e m p e r a t u r e s f o r a l l t h e s y s t e m s s t u d i e d . A n a n a l y s i s o f t h e d a t a b a s e d o n c o n v e n t i o n a l N . M . R . t h e o r y s h o w s i n m o s t c a s e s t h a t t h e c o r r e l a t i o n t i m e f c o c T i n t h e n e i g h b o u r h o o d o f 2 0 ° K , a n d t h a t i s a l m o s t i n d e p e n d e n t o f t e m p e r a t u r e n e a r 1 . 2 ° K . I t i s p o s t u l a t e d t h a t p h o n o n -m o l e c u l a r i n t e r a c t i o n s , i n v o l v i n g d i r e c t a n d R a m a n p r o c e s s e s , c a n a c c o u n t f o r t h e t e m p e r a t u r e d e p e n d e n c e o f t c . T h e v a l u e s o f T ^ a t t h e m i n i m u m a r e c o m p l e t e l y d e t e r m i n e d b y c o n v e n t i o n a l t h e o r y . I n m o s t c a s e s , h o w e v e r , t h e p r e d i c t e d v a l u e s a r e o f t h e o r d e r o f 20 t i m e s t o o s h o r t . A n u n e x p l a i n e d m i n i m u m i n T 1 w a s o b s e r v e d i n C H ^ , C H ^ - C D ^ , a n d C H ^ - K r m i x t u r e s a b o v e t h e u p p e r p h a s e t r a n s i t i o n s . i i To i n v e s t i g a t e the o r i g i n of some of the inadequacies of the conventional theory, the two energy l e v e l scheme proposed by C o l w e l l , G i l l , and Morrison {1965) i s used, where each of the two l e v e l s may be degenerate. Simple r a t e equations are used to c a l c u l a t e the c o n d i t i o n a l p r o b a b i l i t i e s and the c o r r e l a t i o n f u n c t i o n s f o r the two l e v e l model. I t i s found that the e f f e c t i v e i n t e r a c t i o n strength i s temperature dependent, that the c o r r e l a t i o n f u n c t i o n can be described by a simple exponential under c e r t a i n c o n d i t i o n s , and that the i n t e r a c t i o n strength has no simple r e l a t i o n s h i p w i t h the c l a s s i c a l value. TABLE OP CONTENTS Page Abstract i i L i s t of Tables v i i L i s t of I l l u s t r a t i o n s v i i i Acknowledgements • x CHAPTER 1. INTRODUCTION 1 1:1 C l a s s i c a l Treatment 5 2. PROPERTIES OP THE METHANES 11 2:1 Properties of the Methane Molecules 11 2:2 Nuclear Magnetic Resonance Studies 17 2:3 Sp e c i f i c Heat Studies 20 2:4 Miscellaneous Measurements . . 23 3. EXPERIMENTAL METHOD AND APPARATUS 26 3:1 Relaxation Time Measurements 26 3:2 The Spectrometer 27 3:2:1 Timing Units 28 3:2:2 Transmitter 28 3:2:3 Sample C i r c u i t 30 3:2:4 Receivers ' 34 3:3 Cryostat 38 3:3:1 The Vacuum Can and the Dewar Head 38 3:3:2 Sample Holder 4l 3:4 Temperature Measurement and Control 42 3:4:1 Temperature Measurement 42 3:4:2 Temperature Control 45 3:5 Sample Preparation 46 i v V CHAPTER Page 4. THE THEORY OP RELAXATION 52 4:1 Conventional Theory 52 4:2 C l a s s i c a l Theory of Molecular Reorientations . 60 4:3 Line Shape 65 5. THE EXPERIMENTAL RESULTS . 69 5:1 Coupled Spin Systems 70 5:2 CH3D 72 5:3 The Analysis of the Data 5:4 CD^ 80 5:5 CD3H 83 5:6 CH 4 87 5:7 CH^-Kr Mixtures 96 5:8 CD^-CH^ Mixtures 97 5:9 A D i s t r i b u t i o n of Correlation Times 100 5:10 Line Shape 104 5:11 Summary 105 6. FURTHER DISCUSSION OF THE EXPERIMENTAL - RESULTS . . 108 6:1 Review of the C l a s s i c a l Calculation of the Correlation Functions 108 6:2 The Low Lying States of Methane 110 6:3 Calculation of the Correlation Functions for Molecules having Discrete Energy Levels . . 112 6:4 Two Energy Level Case . . . . 114 6:5 The Temperature Dependence of the Transition P r o b a b i l i t i e s 118 6:6 Temperature Dependence of ^ for Some Special Cases 120 v i CHAPTER Page 6:7 Spin Temperature 130 6:8 Some Comments about Matrix Elements of the Intramolecular Interactions 135 7. SUMMARY 139 ! BIBLIOGRAPHY 142 LIST OF TABLES Table Page 1. Properties of Methane 16 v i i LIST OP ILLUSTRATIONS Figure Page 1. Energy Levels of CH^ and CD^ for Different C r y s t a l l i n e F i e l d s 18 2. Block Diagram of Spectrometer 29 3. Pulse Sequences 31 4. Transmitter C i r c u i t Diagram 33 5. Sample C i r c u i t 35 6. Receiver C i r c u i t Diagram 37 7. Diagram of Dewar Head 39 8. Vacuum Can and Sample Holder 40 9. The Temperature Regulator 47 10. C i r c u i t Diagram of Temperature Control Unit . . . 48 11. The Sample Geometries 50 12. Relaxation: a two-step Process 6 l 13. Induction T a i l with Lowe Beats 67 14. T1 Versus T for CH-^ D 74 15. y and l/T.^ Versus x 78 16. l / c o 0 x c Versus T for CH^D 8 l 17. x Versus l/T for CE^D 82 18. T x Vs.T for CD^ " 84 19. 1/x Versus T f o r CD^ 85 20. Proton 1^ Versus T for CD^H 88 21. Deuteron T 1 Versus T for CD^H ., 89 22. Proton 1^ Versus T for CH^ at 28.5 Mcs 91 v i i i Figure Page 23. Proton T1 Versus T for CH^ at 4.% Mcs , . 92 24. Plot of 1/x Versus T for CH^ at 2 8 . 5 Mcs . . . . 94 25. Typical Plot Showing Non-exponential Relaxation . 95 26. Proton T x Versus T f o r 10$ Kr-90$ CH^ 98 27. Proton T x Versus T for 50$ Kr-50$ CH^ 99 28. Proton T1 Versus T f o r 67$ CD^-33$ CH^ 101 29. Deuteron T]_ Versus T for 67$ CD^-33$ CH^ . . . . 102 30. Proton T x Versus T f o r 10$ CD^-90$ CH^ 103 31. Energy Level Diagram for CH^D and CD^H 113 32. The Temperature Dependence of the Interaction Strength 125 33. T,/(T,) . Versus T/T . f o r Several Values of ' ' 1 34. T1 Versus T Including both Raman and Direct processes 129 35. l / T 1 Versus T ^ T for the Case co0%»/ 132 36. Coupling of Two Spin Systems to the L a t t i c e . . . . 133 ix ACKNOWLEDGEMENTS I would l i k e to express my deepest appreciation to Dr. Myer Bloom, my supervisor, f o r his constant encouragement and guidance. Without the many s a c r i f i c e s and the everready assistance of my wife, t h i s thesis would have never been written. I would l i k e to thank Dr. K. W. Gray and Dr. L. McLachlin for many stimulating discussions. I am indebted to Mr. John Lees for his indispensable services, as our glassblower. It i s my pleasure to have had the assistance of Mr. Peter Haas in the construction of the equipment and the preparation of the drawings. My fellow-students and the faculty members in the Nuclear Magnetic Resonance group are hereby thanked for the many ways in which they have contributed to t h i s thesis. The National Research Council i s g r a t e f u l l y ack-nowledged fo r providing f i n a n c i a l assistance throughout my post-graduate education. x CHAPTER 1 . INTRODUCTION T h i s t h e s i s i s a s t u d y of the low temp e r a t u r e magnetic r e l a x a t i o n p r o p e r t i e s of s o l i d methane, i t s d e u t e r a t e d modi-f i c a t i o n s , and some m i x t u r e s of methane (CH^) w i t h CD^ or w i t h K r . As i s w e l l known, the methanes, CH^ nD n> i n t h e s o l i d a r e a l l c h a r a c t e r i z e d by two A s i n g u l a r i t i e s i n t h e s p e c i f i c h e a t . The u l t i m a t e aim of t h i s work i s t o o b t a i n more c o n c l u s i v e knowledge about the n a t u r e of t h e s e phase t r a n s i t i o n s . N u c l e a r magnetic resonance i s c a p a b l e of p r o v i d i n g much i n f o r m a t i o n about t h e dynamics of any systems w i t h w h i c h the s p i n s i n t e r a c t and exchange energy; t h e s e systems a r e c o l l e c t i v e l y c a l l e d t h e " l a t t i c e " . The i n t e r a c t i o n s between the l a t t i c e and the s p i n system a r e weak and d i s t u r b the " l a t t i c e " n e g l i g i b l y . The " l a t t i c e " c o n s i s t s of the m o l e c u l a r r o t a t i o n a l and v i b r a t i o n a l degrees of freedom. In t h e next few pages, a b r i e f d e s c r i p t i o n w i l l be p r e s e n t e d of how c e r t a i n n o n e q u i l i b r i u m s t a t e s of the s p i n system can be p r e p a r e d , of how the approach t o e q u i l i b r i u m can be m o n i t o r e d , and of how s t u d i e s of t h i s r e c o v e r y can be r e l a t e d t o m i c r o -s c o p i c p r o p e r t i e s of t h e ' s p i n system and t h e l a t t i c e . F i r s t , some of t h e p r o p e r t i e s of a n u c l e a r s p i n w i l l be d e s c r i b e d . Many at o m i c n u c l e i p o s s e s s a non-zero s p i n a n g u l a r momentum fiT and a magnetic moment y t i T c o l i n e a r w i t h i t . The a p p l i c a t i o n of a magnetic f i e l d H Q produces 2 a Z e e m a n i n t e r a c t i o n e n e r g y of t h e n u c l e a r s p i n H = - JU.K ( 1 - 1 ) = ~ Y * Hol2r > i f H i s t a k e n t o b e i n t h e z - d i r e c t i o n . T h e a l l o w e d o e n e r g i e s o f a s i n g l e s p i n a r e , E m = - y f c H 0 m m = I , I - 1 , . . . , - I . N e x t , c o n s i d e r a n e n s e m b l e o f w e a k l y i n t e r a c t i n g s p i n s i n t h e r m a l e q u i l i b r i u m w i t h t h e i r e n v i r o n m e n t , t h e " l a t t i c e " . T h e t e m p e r a t u r e o f t h e s p i n s y s t e m w i l l b e t h e same a s t h a t of t h e " l a t t i c e " . > T h e p o p u l a t i o n s o f t h e e n e r g y l e v e l s E m a r e g i v e n b y a B o l t z m a n d i s t r i b u t i o n , Rrv, = &1 eon./ 1ST T h e m a g n e t i z a t i o n o f a n e n s e m b l e o f N s p i n s i s i n t h e z -d i r e c t i o n a n d i s g i v e n b y r M = M 0 = N IE y ( 1 - 2 ) = N y z ^ E ( r t i ) Ho 3 k T w h e r e t h e h i g h t e m p e r a t u r e a p p r o x i m a t i o n h a s b e e n m a d e , T h i s i s t h e w e l l k n o w n C u r i e l a w . T o d e t e c t t h e e x i s t e n c e o f t h i s s e t o f e n e r g y l e v e l s , a n i n t e r a c t i o n , w h i c h c a n c a u s e t r a n s i t i o n s b e t w e e n t h e s e l e v e l s , i s r e q u i r e d . A n a l t e r n a t i n g m a g n e t i c f i e l d 2 H ^ c o s c u t p e r p e n d i c u l a r t o H Q p r o v i d e s s u c h a n i n t e r a c t i o n , s i n c e i t h a s t h e f o l l o w i n g p r o p e r t i e s : ( l ) i t h a s n o n - z e r o m a t r i x e l e m e n t s f o r m - m ' - - 1 ( 2 ) i t c o n s e r v e s e n e r g y i f i t s f r e q u e n c y OJ= y H . N u c l e a r m a g n e t i c r e s o n a n c e i s c o n c e r n e d w i t h t h i s r e s o n a n t e x c h a n g e o f e n e r g y b e t w e e n a r a d i o f r e q u e n c y f i e l d a n d a s y s t e m o f n u c l e a r s p i n s i n a m a g n e t i c f i e l d H . C o n s i d e r a s y s t e m o f n o n - i n t e r a c t i n g n u c l e i p o s s e s s i n g s p i n \ , a n d s p e c i f y t h e n u m b e r o f n u c l e i i n t h e t w o m s t a t e s •+4 a n d - \ b y N + a n d N " r e s p e c t i v e l y . Now t h e a p p l i c a t i o n o f a n a l t e r n a t i n g f i e l d g i v e s r i s e t o a t r a n s i t i o n p r o b a b i l i t y W + _ ^ _ b e t w e e n s t a t e s m = +•§• a n d m = - \ a n d W _ ^ + f o r t h e i n v e r s e t r a n s i t i o n . T h e e q u a t i o n s g o v e r n i n g t h e p o p u l a -+ -t i o n s N a n d N c a n b e w r i t t e n a s A n T = N~ w _ _> . — N"*" \ A / + _ ( 1 - 3 ) d - t P r o m t i m e d e p e n d e n t p e r t u r b a t i o n t h e o r y , i t i s k n o w n t h a t i . e . d™- _ _ 2 W ( T V w h e r e /rv^N^-N" ( l - 4 ) oL - t T h i s e q u a t i o n e m p h a s i z e s t h e f a c t t h a t f o r a n e t a b s o r p t i o n o f e n e r g y ' n ' m u s t b e n o n - z e r o , a n d t h a t t h e a b s o r p t i o n o f e n e r g y f r o m t h e a l t e r n a t i n g f i e l d w o u l d e v e n t u a l l y v a n i s h a s n—*• 0. T h e c o n d i t i o n n = 0 i m p l i e s t h a t t h e s p i n s y s t e m i s c h a r a c t e r i z e d b y a n i n f i n i t e t e m p e r a t u r e . I n a c t u a l f a c t , e n e r g y i s a l w a y s a b s o r b e d . T h e r e m u s t t h e r e f o r e e x i s t a n o t h e r m e c h a n i s m f o r i n d u c i n g t r a n s i t i o n s b e t w e e n m = + | a n d m = - | . T h i s m e c h a n i s m a r i s e s t h r o u g h t h e c o u p l i n g o f t h e s p i n s w i t h 4 a n o t h e r s y s t e m , t h e l a t t i c e . L e t W s ^ s ' d e n o t e t h e p r o b a b i l i t y t h a t t h i s c o u p l i n g w i t h t h e l a t t i c e i n d u c e s a t r a n s i t i o n b e t w e e n t w o w e l l d e f i n e d s t a t e s s a n d s ' o f t h e s p i n s y s t e m a n d t h e r e v e r s e p r o c e s s b y W s » _ j , s • T h e t r a n s i t i o n p r o b a b i l i t i e s W s i ^ s a n d ^ ^ - * & > d e p e n d i n t i m a t e l y u p o n t h e i n i t i a l a n d f i n a l s t a t e o f t h e l a t t i c e . T h e i n i t i a l a n d f i n a l s t a t e s o f t h e s p i n s y s t e m a n d t h e l a t t i c e a r e r e s t r i c t e d t o t h o s e c o n s e r v i n g t h e e n e r g y o f t h e c o m b i n e d s y s t e m . T h e t r a n s i t i o n p r o b a b i -l i t y w i l l d e p e n d o n t h e m a t r i x e l e m e n t s c o n n e c t i n g t h e i n i t i a l a n d f i n a l s t a t e s o f t h e c o m b i n e d s y s t e m , a n d a l s o o n t h e p r o b a b i l i t y t h a t t h e l a t t i c e w i l l b e i n a s t a t e t h a t p e r m i t s t h e t r a n s i t i o n . D e n o t e b y P f t h e p r o b a b i l i t y o f f i n d i n g t h e l a t t i c e i n s t a t e I f ) , a n d W ^ s ^ t s i t h e p r o b a b i l i t y o f g o i n g f r o m a s t a t e | f ) f s ) o f t h e e n t i r e s p i n - l a t t i c e s y s t e m t o a n o t h e r | f ) I s 1 } , s u c h t h a t Ef + ES = F +. +- E s . . A s t h e s p i n -l a t t i c e s y s t e m i s a c l o s e d s y s t e m , a c c o r d i n g t o t h e g e n e r a l p r i n c i p l e s o f q u a n t u m m e c h a n i c s W £. •& f * ' = W I n a d d i t i o n , i t i s u s u a l l y a s s u m e d t h a t t h e l a t t i c e i s a l w a y s i n t h e r m a l e q u i l i b r i u m w i t h i t s e l f , a n d h a s a n i n f i n i t e s p e c i f i c h e a t , s u c h t h a t P f = e x p - (E.C / ^ T) I t i s e a s i l y s h o w n t h a t W s - » » ' = ST* P f W j s ^ f S ' _ e x p ( & s _ E 4.)/l,T ( 1 - 5 ) w s . _ ^ s ^ p r wt.'-si-%.fs T h i s - c o n d i t i o n w i l l l e a d t o a B o l t z m a n d i s t r i b u t i o n f o r t h e p o p u l a t i o n s o f t h e s p i n s y s t e m c h a r a c t e r i z e d b y t h e s a m e 5 t e m p e r a t u r e a s t h a t o f t h e l a t t i c e . T h i s a p p r o a c h t o e q u i -l i b r i u m h a s a s s o c i a t e d w i t h i t a t i m e c o n s t a n t T ^ , 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 ; i t w i l l i n t i m a t e l y d e p e n d o n t h e t r a n s i t i o n p r o b a b i l i t i e s W j ^ s , . . T h e a p p r o a c h t o e q u i -l i b r i u m i n m a n y c a s e s i s e x p o n e n t i a l , b u t t h i s i s n o t a l w a y s t h e c a s e . T h e n a t u r e o f t h e i n t e r a c t i o n b e t w e e n t h e s p i n s y s t e m a n d i t s s u r r o u n d i n g s , u n i q u e l y d e t e r m i n e s t h e m a t r i x e l e m e n t s i n v o l v e d i n t h e e x p r e s s i o n f o r W^-^si . T h e s e i n t e r a c t i o n s c a n b e m a g n e t i c , d i p o l e - d i p o l e i n t e r a c t i o n s b e t w e e n t h e n u c l e i a n d i n t e r a c t i o n s b e t w e e n t h e n u c l e i a n d o t h e r l o c a l m a g n e t i c f i e l d s ; o r , i n t h e c a s e o f a n u c l e u s s u c h a s t h e d e u t e r o n n u c l e u s , w h i c h h a s s p i n g r e a t e r t h a n \ , b e t w e e n t h e e l e c t r i c f i e l d g r a d i e n t s a n d t h e n u c l e a r q u a d r u p o l e m o m e n t s . I n a d d i t i o n , t h e t r a n s i t i o n p r o b a b i l i t i e s d e p e n d o n d e t a i l s o f t h e d e n s i t y o f s t a t e s f o r t h e n o r m a l m o d e s o f t h e l a t t i c e . T h u s , m e a s u r e m e n t s o f i m p l i c i t l y p r o v i d e i n f o r m a t i o n a b o u t t h e n a t u r e o f t h e s p i n - l a t t i c e i n t e r a c t i o n s a n d t h e l a t t i c e m o d e s . C l a s s i c a l T r e a t m e n t 1 : 1 M a n y m a g n e t i c r e s o n a n c e p h e n o m e n a a n d t e c h n i q u e s c a n b e u n d e r s t o o d c l a s s i c a l l y . A c c o r d i n g t o c l a s s i c a l e l e c t r o m a g n e t i c t h e o r y , t h e e q u a t i o n o f m o t i o n o f M i s , • 6 A 3 _ V M A H d t r (1-6) The Quantum M e c h a n i c a l e x p e c t a t i o n v a l u e s c a n be shown t o obey t h i s e q u a t i o n o f m o t i o n ; a l l t h e r e s u l t s d e r i v e d b e l o w c a n a l s o be d e r i v e d Quantum M e c h a n i c a l l y f o r an ensemble o f w e a k l y i n t e r a c t i n g n u c l e a r s p i n s . T e m p o r a r i l y , a l l r e l a x -a t i o n e f f e c t s w i l l be n e g l e c t e d . E q u a t i o n ( l - 6 ) can be s o l v e d when b o t h t h e main m a g n e t i c f i e l d H o and t h e r o t a t i n g r . f . f i e l d p e r p e n d i c u l a r t o H a r e p r e s e n t . I t i s c o n v e n i e n t t o t r a n s f o r m t o a o r o t a t i n g c o - o r d i n a t e frame w i t h a n g u l a r v e l o c i t y co a b o u t t h e z - a x i s . Now t h e e q u a t i o n o f m o t i o n becomes [H + " / y ] (1-7) 6 t T h i s has t h e same f o r m as e q u a t i o n (1-6) p r o v i d e d t h a t t h e m a g n e t i c f i e l d H i s r e p l a c e d by an e f f e c t i v e f i e l d H = H C o n s i d e r t h e c a s e t h a t H c o n s i s t s o n l y o f t h e s t a t i c f i e l d H Q = H Qk. I n t h i s f r a m e |JEL = 0, when to = - Y H 0 . I t can t h e r e f o r e be s a i d t h a t M p r e c e s s e s w i t h r e s p e c t t o l a b o r a t o r y f r a m e a t a f r e q u e n c e co = - y H , t h e L a r m o r f r e q u e n c y . Now l e t us add t o H , a f i e l d r o t a t i n g p e r p e n d i c u l a r t o H Q w i t h f r e q u e n c y co . Choose t h e x - a x i s o f t h e r o t a t i n g r e f e r e n c e f r a m e t o be i n t h e d i r e c t i o n o f H^.le.H^ = i . The e q u a t i o n o f m o t i o n f o r M i s &W _ y fyj A H&ff-wher e Hef|= (H 0 + -yY) + H, L (1 - 8 ) 7 T h i s i m p l i e s t h a t i n the r o t a t i n g frame M p r e c e s s e s about H ^ a t a n g u l a r f r e q u e n c y - y, . U s u a l l y i s much s m a l l e r than H Q , c o n s e q u e n t l y o n l y when [ H Q + 6 w i l l the r . f . f i e l d have a pronounced e f f e c t . I t w i l l change the d i r e c t i o n of H g f f f r o m the z - a x i s to the x - a x i s as | H Q + ^/y\ —** 0. At r e s o n a n c e , when t o = - y H Q , the m a g n e t i z a t i o n M p r e c e s s e s about the x - a x i s w i t h a n g u l a r f r e q u e n c y y H ^ . I f the f i e l d i s a p p l i e d as a s i n g l e p u l s e of w i d t h t w , then at t = 0 the m a g n e t i z a t i o n M w i l l s t a r t f r o m M = M k and w i l l ° o have p r e c e s s e d about the x - a x i s t h r o u g h an a n g l e Q = y H-^tw a t the end of the p u l s e . Thus , i n the s p e c i a l case t h a t 0 = 7 f /2 the m a g n e t i z a t i o n , as seen f r o m the l a b o r a t o r y f rame, w i l l be p r e c e s s i n g i n the x - y p l a n e a t an a n g u l a r f r e q u e n c y to = - y H Q about the z - a x i s . In N . M . R . measurements r . f . p u l s e s , such t h a t 9 = .7t/2 or Tt > are f r e q u e n t l y used and are r e f e r r e d t o as " T T / 2 or TT p u l s e s " , r e s p e c t i v e l y . The e q u a t i o n of motion has to be m o d i f i e d t o i n c l u d e the r e l a x a t i o n e f f e c t s , w h i c h were n e g l e c t e d a b o v e ; these were a l l o w e d f o r i n a p h e n o m e n o l o g i c a l f a s h i o n by B l o c h (1946) . E a r l i e r , i t was d e s c r i b e d how M z r e l a x e s t o M Q i n the absence of the r . f . f i e l d ; t h i s p r o c e s s was c h a r a c -t e r i z e d by the t ime c o n s t a n t T ^ . F u r t h e r m o r e , as was d e s c r i b e d i n the l a s t p a r a g r a p h , the a p p l i c a t i o n o f an r . f . p u l s e can g i v e r i s e t o a m a g n e t i z a t i o n s o l e l y i n the x - y p l a n e . When i n t h e r m a l e q u i l i b r i u m , the s p i n system p o s s e s s e s o n l y m a g n e t i z a t i o n i n the z - d i r e c t i o n ( t h i s was shown at the 8 b e g i n n i n g of the chapter)„ C o n s e q u e n t l y , t h e r e must e x i s t a n o t h e r r e l a x a t i o n mechanism, s p i n - s p i n r e l a x a t i o n , c h a r a c t e r i z e d by Tg. A l l t h e s e f a c t s can be i n c o r p o r a t e d i n the f o l l o w i n g s i m p l e e q u a t i o n s , the B l o c h e q u a t i o n s : AR = y M A H — l ^ x l H- M ^ J - (Mj, - M-) k ( 1 - 9 ) These e q u a t i o n s a r e p a r t i c u l a r l y w e l l s u i t e d t o the d e s c r i p t i o n of t r a n s i e n t e f f e c t s , w h i l e a t t h e same time p r o v i d i n g a d e s c r i p t i o n of s t e a d y s t a t e e f f e c t s b r o a d l y c o n s i s t e n t w i t h the p r o p e r quantum m e c h a n i c a l treatment,. F o r example, i f M = (M M # N M # \) a t t = o and no K v x ( o ) y ( o ) z ( o ) ' r . f . f i e l d i s p r e s e n t , t h e n t h e s o l u t i o n s o f the B l o c h e q u a t i o n s i n t h e l a b o r a t o r y frame a r e : M x ( t ) = M 1 ( 0 ) e " t / / T 2 cos( t + 0) M ( t ) = M 1 ( o ) e " t ' / T 2 s i n ( t + 0) ( 1 - 1 0 ) M„ ( t ) = M + (M / \ - M ) e " t / / T l zK ' o v z ( o ) o' where Mn = [M 2 ( o ) + M 2 ( o ) p • x x y Thus, a c c o r d i n g t o the B l o c h e q u a t i o n s , the r e t u r n t o e q u i l i b r i u m of the s p i n system i s e x p o n e n t i a l . T h i s b e h a v i o u r i s c e r t a i n l y not g e n e r a l l y t r u e . The B l o c h e q u a t i o n s p r o v i d e a u s e f u l a pproximate d e s c r i p t i o n o f t h e s p i n dynamics; however, t h e y s h o u l d not be r e g a r d e d as a s u b s t i t u t e f o r a more r i g o r o u s quantum m e c h a n i c a l t r e a t m e n t . Now, t h e concept of s p i n - s p i n r e l a x a t i o n w i l l be e x p l a i n e d i n more d e t a i l . As can be shown, u s i n g the B l o c h 9 e q u a t i o n , the d e s c r i p t i o n of the . Tt/2 p u l s e g i v e n above s t i l l h o l d s as l o n g as t w « I V , , ! ^ . Under those c o n d i t i o n s , at the end of the r . f . p u l s e , t h e r e e x i s t s coherence between the phases of the wave f u n c t i o n s d e s c r i b i n g the s p i n system b r i n g i n g about a t r a n s v e r s e m a g n e t i z a t i o n i n t h e x - y p l a n e . The random i n t e r a c t i o n s of the s p i n s w i t h the environment g r a d u a l l y d e s t r o y t h i s coherence and the t r a n s v e r s e m a g n e t i -z a t i o n ; t h i s p r o c e s s i s c a l l e d s p i n - s p i n r e l a x a t i o n . I t i s observed t h a t T^ i s a lways g r e a t e r than or e q u a l t o T g , s i n c e the decay of the t r a n s v e r s e m a g n e t i z a t i o n c o n s e r v e s energy i n the f i e l d H Q , u n l i k e the r e c o v e r y of the l o n g i t u d -i n a l m a g n e t i z a t i o n (M ) i n w h i c h energy has t o be t r a n s f e r r e d t o the l a t t i c e . How are T^ and Tg measured e x p e r i m e n t a l l y ? One method uses the p r o p e r t i e s of the TC/2 p u l s e . As was mentioned e a r l i e r t h i s h o l d s as l o n g as t « T - ^ T g . The 7t /2 p u l s e s e t s up a p r e c e s s i n g m a g n e t i z a t i o n i n the x - y p l a n e and the, M m a g n e t i z a t i o n i s then z e r o . I f a c o i l i s wound around z the sample w i t h i t s symmetry a x i s i n the x - y p l a n e , the p r e c e s s i n g m a g n e t i z a t i o n w i l l induce a v o l t a g e i n the c o i l d e c a y i n g w i t h a t ime c o n s t a n t T ^ ; t h i s i s c a l l e d the f r e e i n d u c t i o n t a i l . T h u s , the decay of the t r a n s v e r s e m a g n e t i -z a t i o n can be m o n i t o r e d . M o r e o v e r , the s i g n a l i n d u c e d i n the c o i l i s p r o p o r t i o n a l t o the M m a g n e t i z a t i o n at the z b e g i n n i n g of the p u l s e . T h i s f a c t i m m e d i a t e l y s u g g e s t s a method f o r measur ing T 1 : d i s t u r b the M z m a g n e t i z a t i o n f r o m i t s e q u i l i b r i u m v a l u e M Q by a p p l y i n g an r . f . f i e l d f o r a 1 0 s h o r t t i m e , a 7 t / 2 p u l s e , f o r example; m o n i t o r the r e c o v e r y of M at t ime t l a t e r hy a p p l y i n g a T T / 2 p u l s e ; r e p e a t t h i s sequence f o r many v a l u e s of 1 1 ' , a l l o w i n g the s p i n system t o a t t a i n e q u i l i b r i u m between s u c c e s s i v e sequences of p u l s e s . E x p e r i m e n t a l l y , the r . f . f i e l d i s produced by an r . f . c u r r e n t p a s s i n g t h r o u g h the c o i l s i t u a t e d around the sample . The a l t e r n a t i n g f i e l d , thus p r o d u c e d , can be decomposed i n t o two c i r c u l a r l y p o l a r i z e d components w i t h o p p o s i t e senses of r o t a t i o n . I f one component s a t i s f i e s the resonance c o n d i t i o n , OJ0 - — y H g , , then the e f f e c t o f the o t h e r component i s n e g l i g i b l e as i t w i l l be o f f - r e s o n a n c e by 2 a > 0 . In t h i s c h a p t e r , some of the c o n c e p t s i n v o l v e d i n N . M . R . have been i n t r o d u c e d , and i t has been shown how they are r e l a t e d t o the m i c r o s c o p i c p r o p e r t i e s of the s y s t e m . In the next c h a p t e r , e x i s t i n g knowledge about the methane system at low tempera tures w i l l be r e v i e w e d . ' C h a p t e r s 3 and 4 d e a l w i t h the e x p e r i m e n t a l t e c h n i q u e s and the t h e o r y , r e s p e c t i v e l y . CHAPTER 2. PROPERTIES OP THE METHANES I t i s w e l l known that s o l i d CH^ and a l l of i t s deuterated m o d i f i c a t i o n s e x h i b i t A anomalies i n the s p e c i f i c heat. This study of the low temperature nuclear magnetic r e l a x a t i o n p r o p e r t i e s was motived by ,(1) the l a c k of d e f i n i t i v e experiments regarding the nature of these J\ anomalies; (2) some i n t e r e s t i n g r e s u l t s obtained by us i n p r e l i m i n a r y experiments studying the r e l a x a t i o n p r o p e r t i e s of the deuteron spin system between 110 and 55°K. In t h i s chapter, some of the preceding experimental work w i l l be reviewed. Notwithstanding a wealth of e x p e r i -mental data, the nature of the phase t r a n s i t i o n s i s s t i l l very much a mystery. Many of the experiments appear to contra-d i c t each other; other experimental conclusions are too q u a l i t a t i v e t o be of much value. P r o p e r t i e s of Methane Molecules: 2:1 In the f r e e methane molecule, the deuterons and protons are s i t u a t e d at the corners of a tetrahedron w i t h the carbon atom at i t s center. I t f o l l o w s from the geometry, that CH^ and CD^ are s p h e r i c a l top molecules and possess a T^ symmetry group; CD^H and CH^D are symmetric top molecules w i t h symmetry group C y 11 12 The wave f u n c t i o n of p o l y a t o m i c m o l e c u l e s are c h a r a c t e r -i z e d by t h r e e quantum numbers K, and M. The square of the t o t a l a n g u l a r momentum i s q u a n t i z e d and i s e q u a l t o J ( J + l ) ti , where J i s the r o t a t i o n a l a n g u l a r momentum quantum number. The p r o j e c t i o n of the r o t a t i o n a l a n g u l a r momentum on the m o l e c u l a r a x i s i s q u a n t i z e d and e q u a l s Kti where K = J , . . . , - J i s an i n t e g e r . S i m i l a r l y , i t s p r o j e c t i o n on the p o l a r a x i s f i x e d i n space i s Mfc where M = J , . . . , - J . The energy l e v e l s o f f r e e s p h e r i c a l top m o l e c u l e s h a v i n g moment of i n e r t i a I depend o n l y on J and a r e (2J + if degenerate EJKM = \fro J ( J + i ) • ( 2 - D The symmetric top m o l e c u l a r energy l e v e l s a re s p e c i f i e d by J and K and are 2 ( 2 J + l ) degenerate f o r K / 0 E J K M = B J ( J + 1) + (C - B ) K 2 (2-2) where B and C depend on the v a r i o u s components of the moment of i n e r t i a t e n s o r of the m o l e c u l e . The wave f u n c t i o n of the methane m o l e c u l e i s u s u a l l y a p p r o x i m a t e l y e x p r e s s e d as a p r o d u c t of f u n c t i o n s of the v a r i o u s c o - o r d i n a t e s . z o v e v R S where the s u b s c r i p t s e , v , R , and S i n d i c a t e t h a t the c o r r e s p o n d -i n g f a c t o r i s a f u n c t i o n of t h e , e l e c t r o n i c ( i n c l u d i n g e l e c t r o n s p i n ) , v i b r a t i o n a l , r o t a t i o n a l and n u c l e a r s p i n c o - o r d i n a t e s . 13 T h i s w a v e f u n c t i o n i s n o t u n i q u e , s i n c e t h e p e r m u t a t i o n o f i d e n t i c a l n u c l e i g e n e r a t e s a n u m b e r o f o t h e r w a v e f u n c t i o n s . T h e c o r r e c t w a v e f u n c t i o n m u s t b e c o m p l e t e l y s y m m e t r i c a l o r a n t i s y m m e t r i c a l w i t h r e s p e c t t o i n t e r c h a n g e o f a n y p a i r o f i d e n t i c a l n u c l e i . S i n c e p r o t o n s o b e y t h e P a u l i e x c l u s i o n p r i n c i p l e , C H ^ h a s a n t i s y m m e t r i c a l w a v e f u n c t i o n s , w h e r e a s C D ^ h a s s y m m e t r i c a l w a v e f u n c t i o n s . T o o b t a i n w a v e f u n c t i o n s w i t h t h e p r o p e r s y m m e t r y c h a r a c t e r , i t i s n e c e s s a r y t o f o r m l i n e a r c o m b i n -a t i o n s o f t h e d e g e n e r a t e f u n c t i o n s d i s c u s s e d a b o v e . T h i s p r o b l e m i s s i m p l i f i e d s i n c e t h e e l e c t r o n i c w a v e f u n c t i o n j6e a n d t h e v i b r a t i o n a l w a v e f u n c t i o n $ i n t h e g r o u n d s t a t e a r e u s u a l l y c o m p l e t e l y s y m m e t r i c a l . T h u s , t h e s y m m e t r y p r o p e r -t i e s o f j ^ Q - j - d e p e n d o n l y o n t h e s y m m e t r y p r o p e r t i e s o f 0 R a n d f2fg. M o r e o v e r , a l l r o t a t i o n s o f t h e g r o u p T ^ c o r r e s p o n d t o " e v e n " p e r m u t a t i o n s o f t h e n u c l e i , h e n c e f o r C H ^ a n d C D ^ t h e t o t a l w a v e f u n c t i o n 0^Q^. b e l o n g s t o t h e t o t a l l y s y m m e t r i c i r r e d u c i b l e r e p r e s e n t a t i o n A . T h e s y m m e t r y o p e r a t i o n s w h i c h t r a n s f o r m t h e m o l e c u l e i n t o i t s e l f b e l o n g t o t h e t e t r a h e d r a l s y m m e t r y g r o u p T ^ . T h i s g r o u p c o n s i s t s o f t h r e e c l a s s e s o f o p e r a t i o n s g i v i n g r i s e t o t h r e e i r r e d u c i b l e r e p r e s e n t a t i o n s A , E , a n d T . E a c h r o t a t i o n a l w a v e f u n c t i o n 0R i s a s s i g n e d i n a c c o r d a n c e w i t h i t s s y m m e t r y p r o p e r t i e s t o o n e o f t h e s e i r r e d u c i b l e r e p r e s e n t a t i o n s a n d w i l l b e d e n o t e d b y 0R ( E ) , j # R ( A ) , a n d 0R[T). T h e 0R p r o v i d e t h e b a s i s f u n c t i o n s f o r t h e s e r e p r e s e n t a t i o n s . O n l y r o t a t i o n a l w a v e f u n c t i o n s w i t h c e r t a i n J q u a n t u m n u m b e r s b e l o n g t o a p a r t i c u l a r i r r e d u c i b l e r e p r e s e n t a t i o n . A 14 s i m i l a r a s s i g n m e n t o f t h e s p i n w a v e f u n c t i o n s c a n h e m a d e . I n g e n e r a l , t h e s p i n w a v e f u n c t i o n s f o r e a c h i r r e d u c i b l e r e p r e s e n t a t i o n h a v e d e f i n i t e v a l u e s o f t h e t o t a l s p i n o f t h e n u c l e i i n t h e m o l e c u l e . M o r e o v e r , t h i s r e l a t i o n i s n o t o n e -t o - o n e ; t h e £ f g o f e a c h r e p r e s e n t a t i o n c a n p o s s e s s m o r e t h a n o n e v a l u e o f t o t a l s p i n . I t c a n b e s h o w n , u s i n g g r o u p - t h e o r e t i c a l m e t h o d s ( M a u e , 1937)5 t h a t t h e o n l y c o m b i n a t i o n s o f 0 g ^ R p r o d u c i n g a t o t a l l y s y m m e t r i c w a v e f u n c t i o n a r e 0R ( A ) ^ s ( A ) 0R ( E ) 0S ( E ) (2-4) 0R ( T ) 0g ( T ) I n o t h e r w o r d s , e a c h r o t a t i o n a l s t a t e w i l l h a v e a s s o c i a t e d w i t h i t a r e s t r i c t e d s e t o f s p i n w a v e f u n c t i o n s . T h e s p i n w a v e f u n c t i o n s b e l o n g i n g t o a p a r t i c u l a r i r r e d u c i b l e r e p r e s e n t a t i o n T , A , o r E a r e s a i d t o b e l o n g t o t h e s p i n s p e c i e s o r t h o , m e t a , o r p a r a , r e s p e c t i v e l y . T h i s i s a n a -l o g o u s t o t h e f a m i l i a r c a s e o f h y d r o g e n ; i t p o s s e s s e s t w o s p i n s p e c i e s o r t h o ( i ^ = 1) a n d p a r a (1 .^ = 0 ) . I n a n a l o g y w i t h h y d r o g e n , c o n v e r s i o n b e t w e e n t h e s e t h r e e s p e c i e s i s v e r y s l o w , s i n c e t h e m a g n e t i c i n t e r a c t i o n s b e t w e e n m o l e c u l e s a r e n o r m a l l y t o o w e a k . U n l i k e h y d r o g e n , i n C D ^ f o r c e r t a i n r o t a t i o n a l s t a t e s t h e p e r m i s s i b l e s p i n w a v e f u n c t i o n s p o s s e s s m o r e t h a n o n e t o t a l s p i n q u a n t u m n u m b e r e . g . t h e ^ g ( A ) h a v e t o t a l s p i n q u a n t u m n u m b e r s 0 , 2 , 4 a n d a r e a s s o c i a t e d w i t h r o t a t i o n a l l e v e l s J = 0 , 3 * ^ e t c . T h e s p i n w a v e f u n c t i o n s f o r C H , . a r e c h a r a c t e r i z e d b y t o t a l s p i n q u a n t u m n u m b e r s 15 0,1, and 2 f o r E,T, and A spin species, r e s p e c t i v e l y . Similar-c o n s i d e r a t i o n s a l s o apply to the symmetric top molecules GD^H and CH^D, see Table I. So f a r , only the p r o p e r t i e s of the f r e e molecule have been considered. In a c t u a l f a c t , the coupling between the molecules w i l l modify the f r e e molecular wave f u n c t i o n s and energy l e v e l s . I t i s not known to what extent they are modi-f i e d ; not even the nature and the strength of these f o r c e s are known w i t h any c e r t a i n t y . One attempt t o describe the i n f l u e n c e of the i n t e r m o l e c u l a r i n t e r a c t i o n s has been made by Nagamiya (l95l)» He assumed that the i n t e r m o l e c u l a r f o r c e s could be described by a c r y s t a l l i n e e l e c t r i c f i e l d p o t e n t i a l possessing a d e f i n i t e symmetry. Using group t h e o r e t i c a l methods, he worked out t o what extent the va r i o u s degeneracies of the f r e e molecular energy l e v e l s would be removed by c r y s t a l l i n e f i e l d s of various symmetries. His r e s u l t s f o r CH^ and CD^ are shown i n Figure 1. These r e s u l t s have proven t o be extremely u s e f u l i n a n a l y z i n g the entropy of the s o l i d methanes at low temperatures. A d i f f e r e n t t h e o r e t i c a l approach t o the d e s c r i p t i o n of the i n t e r m o l e c u l a r i n t e r a c t i o n s was used by James and Keenan (1959). They assume that the i n t e r a c t i o n s are due to the octupole moments of the molecular charge d i s t r i b u t i o n s and have attempted t o provide a d e t a i l e d model of the low temp-erature phase t r a n s i t i o n s i n s o l i d methane i n terms of these i n t e r a c t i o n s . Since the theory of James and Keenan i s com-p l e t e l y c l a s s i c a l , they f e e l t h a t , of the i s o t o p i c m o d i f i c a t i o n s Modi-f i c a t i o n < (°K) \ (°K) Some allowed values of J and K Constants i n eqn .2 -2 Ortho (T) Met a (A) Para (E) B/k C/k (°K) CK 4 2 0.4 8 . 0 J = 1 = 2 = 3 i i J = 0 = 3 = 4 J = 2 = 4 = 5 7.68 7.68 2 7 . 0 2 1 . 9 3.84 3.84 CHgD 2 3 . 2 1 5 . 9 J = 1 K = 1 = 2 = 1 = 2 = 2 = 3 = 1 = 3 = 3 J = 0 K = 0 = 1 = 0 = 2 = 0 = 3 = 0 = 3 = 3 5 . 5 9 7.68 CD3H 2 6 . 1 1 9 . 9 4 . 7 3 3.8^ Table 1 . Some Properties of Free Methane Molecules 17 of methane, i t s p r e d i c t i o n s would a p p l y b e s t to C D ^ , which has the l a r g e s t moment of i n e r t i a . N u c l e a r M a g n e t i c Resonance S t u d i e s 2 :2 \ The e a r l i e s t low temperature N . M . R . s t u d i e s of methane were c a r r i e d out by Thomas et a l . (1950). In t h e i r e x p e r i m e n t a l measurements of the s p i n - l a t t i c e r e l a x a t i o n t i m e s , i n a d e q u a t e p r e c a u t i o n s were taken t o remove Og i m p u r i t i e s f r o m t h e i r sample . As a consequence , the r e l a x a t i o n p r o c e s s was dominated by the v e r y s t r o n g i n t e r a c t i o n s between the paramagnet ic Og i m p u r i t i e s and the n u c l e a r s p i n s on the methane m o l e c u l e s . Many d e t a i l s p e r t a i n i n g t o the n u c l e a r r e l a x a t i o n w h i c h are r e p o r t e d i n t h i s t h e s i s , were obscured by these i n t e r a c t i o n s i n the measurements of Thomas et a l . The f i r s t t h e o r e t i c a l t r e a t m e n t of N . M . R . i n s o l i d methane was p r o v i d e d by Tomita (1953). U n f o r t u n a t e l y , h i s t h e o r y was a p p l i e d t o the i n c o r r e c t e x p e r i m e n t a l r e s u l t s d i s -cussed above . However, h i s g e n e r a l a p p r o a c h , which b r i n g s out e x p l i c i t l y the r o l e of m o l e c u l a r symmetry w i t h r e s p e c t t o the e f f e c t s of the i n t r a m o l e c u l a r s p i n - s p i n i n t e r a c t i o n s on T^ and T 2 , w i l l be v e r y u s e f u l i n any attempt t o f o r m u l a t e a d e t a i l e d theory , of N . M . R . i n low temperature methane. In p a r t i c u l a r , he shows t h a t the d i p o l a r i n t e r a c t i o n s between the n u c l e a r s p i n s on m o l e c u l e s h a v i n g s p i n wave f u n c t i o n s 0^(A), g i v e no c o n t r i b u t i o n t o s p i n r e l a x a t i o n . Measurements of the l i n e - w i d t h i n C H ^ , c a r r i e d out by Wolf (1963) and Thomas et a l . (1950), i n d i c a t e d t h a t t h e r e was 18 1 \ \ \ \ * \ \ N \ \ \ \ \ 9 k \ 4 9 j = o L L : L I l FREE OCTA TETRA TRIGONAL ASYM-ROTOR HEDRAL HEDRAL METRIC F i g u r e 1 . T h e M o l e c u l a r E n e r g y L e v e l s i n C r y s t a l l i n e  F i e l d s w i t h D i f f e r e n t S y m m e t r i e s f o r C H ^ a n d C D ^ 19 no s i g n i f i c a n t change in the line-width at either of the two phase t r a n s i t i o n temperatures. The experimental values of the line-width were much smaller than were calculated assuming a r i g i d l a t t i c e . Actually, the observed values were in reasonable agreement with the calculated values, i f the molecules were assumed to be i s o t r o p i c a l l y reorienting. For s u f f i c i e n t l y rapid molecular reorientations, the contribution of the i n t r a -molecular dipolar interactions (the interactions between nuclei on the same molecule), to the line-width would be considerably reduced. The absence of a change in the line-width at the t r a n s i t i o n temperatures rules out the proposal by Pauling (1930) that the t r a n s i t i o n involved a change from o s c i l l a t i o n s of the molecules in the low temperature phase to free rotation i n the high temperature phase. Another set of pertinent r e s u l t s was reported by de Wit and Bloom (1965). Measurements of the deuteron s p i n - l a t t i c e relaxation time between 55-110°K indicated that the c o r r e l a t i o n time for molecular reorientations (to be defined in Chapter 4) was a very slowly decreasing function of temperature over the entire temperature range, with no change being observable at the melting point 90°K. Moreover, by measuring the spin-l a t t i c e relaxation times i n mixtures of CH^ and CD^, thus varying the intermolecular dipolar interactions, i t was con-c l u s i v e l y shown that below 65°K, the interactions causing relaxation are solely intramolecular. 20 S p e c i f i c Heat Studies 2:3 S p e c i f i c heat s t u d i e s of the methane system were f i r s t c a r r i e d out i n 1929. C l u s i u s , i n 1929, discovered the upper phase t r a n s i t i o n at 20.4°K i n s o l i d CH^. Subsequently, C l u s i u s and Popp (1937) , found that CD^ and CDH^ e x h i b i t e d two phase t r a n s i t i o n s . These occurred at 27.0°K and 21.9°K f o r CD^, and at 23.2°K and 15.9°K f o r CH3D. The other methanes CDgHg and CD^H have a l s o been found t o possess two phase t r a n s i t i o n s , see Table I (Sperandio, 1961). I t may be noted that the upper phase t r a n s i t i o n temperatures vary almost l i n e a r l y w i t h 'n' f o r CH^ n D n . This l i n e a r i t y holds f o r the lower phase t r a n s i t i o n s between CHgDg and CD^, but does not hold f o r CH^D and CH^ (as they do not l i e on the s t r a i g h t l i n e ) . In a d d i t i o n , the lower phase t r a n s i t i o n i s sharper than the upper phase t r a n s i t i o n , except f o r the case of CH^. Only r e c e n t l y has the lower phase t r a n s i t i o n i n CH^, which occurs at 8°K, been discovered by C o l w e l l , G i l l and Morrison (1962). This phase t r a n s i t i o n was not very pronounced and was ra t h e r d i f f i c u l t to observe. The discovery of the lower phase t r a n s i t i o n i n CH^ removed the p u z z l i n g f e a t u r e that CH^ was apparently the only methane m o d i f i c a t i o n which was not c h a r a c t e r i z e d by two s p e c i f i c heat anomalies. The obser-v a t i o n s of Morrison et a l . have been supported by a r e -i n v e s t i g a t i o n of the high pressure c a l o r i m e t r i c behaviour of CH^ by Rosenshein and Whitney (1964). They found, contrary to e a r l i e r experimental work (Stewart (1959) and Stevenson (1957))> that the lower phase t r a n s i t i o n at higher pressures 21 i n C H ^ when ext r a p o l a t e d to zero pressure, corresponded t o the phase t r a n s i t i o n temperature observed by Morrison et a l . As • a l l methanes possess two phase t r a n s i t i o n temperatures and can e x i s t i n three d i s t i n c t phases, the f o l l o w i n g conventions w i l l be adopted: T"^  and denote the upper and lower phase t r a n s i t i o n temperatures, r e s p e c t i v e l y ; and phase I w i l l be the phase above ; phase I I , the phase between and T-£ ; phase I I I , the phase below 1\ . Studies were a l s o c a r r i e d out on mixtures of C H ^ and C D ^ . I t was found that the upper t r a n s i t i o n temperatures v a r i e d l i n e a r l y as a f u n c t i o n of the mixing r a t i o between those of pure C H ^ and C D ^ . For a mixing r a t i o of C H ^ and C D ^ equal to the s t o i c h i o m e t r i c r a t i o of deuterons to protons i n a given i s o t o p i c m o d i f i c a t i o n of methane, i s roughly the same as that of the given i s o t o p i c modification.' For example, a 25$ 0^-75^01)^ mixture has T £ = 25.4°K, whi l e C H D ^ has T £ = 25.9°K. Moreover, i t was found that the lower phase t r a n s i t i o n tempera-ture v a r i e d l i n e a r l y w i t h the mixing r a t i o , i f the C D ^ content was l a r g e r than 60$; below 60$, the t r a n s i t i o n temperature dropped more r a p i d l y (Bartholome et al.- 1938). Methane and krypton can be mixed t o form s o l i d s o l u t i o n s (Stackelberg et a l . , 1936). S o l i d s of pure methane and pure krypton each have the same c r y s t a l s t r u c t u r e ( f . c . c . ) and t h e i r l a t t i c e constants d i f f e r by only 1 .5$. When Kr i s added, the s p e c i f i c heat maximum i s broadened, i t s peak value i s decreased, and the temperature at which the peak occurs i s s h i f t e d down-wards (Eucken and V e i t h , 1936). For Kr concentrations greater 22 than 16$, the maximum i n the s p e c i f i c heat versus temperature curve i s unobservable. A recent paper by C o l w e l l , G i l l and Morrison (1965) i s of s p e c i a l i n t e r e s t since the conclusions from s p e c i f i c heat measurements, and our conclusions from magnetic resonance agree i n p r i n c i p l e . They measured the heat c a p a c i t i e s of the par-t i a l l y deuterated methanes between 2 .5 and 27°K. In a d d i t i o n , to the two t r a n s i t i o n s f o r each methane, they observed l a r g e heat c a p a c i t y anomalies below 8°K f o r CH^D, CHgDg and CHD^. These anomalies are i n t e r p r e t e d as the hig h temperature t a i l s of Schottky anomalies (Rosenberg, 1964). In the a n a l y s i s of t h e i r data, they assume that no conversion takes place between the d i f f e r e n t species, and that i n s o l i d methane the coupling between the molecules i s weak enough to consider the energy s t a t e s t o be those of the f r e e molecule weakly perturbed .by c r y s t a l l i n e f i e l d s i n the s o l i d . In p a r t i c u l a r , they i n t e r p r e t these anomalies i n terms of removal of the r o t a t i o n a l degener-a c i e s by the c r y s t a l l i n e e l e c t r i c f i e l d . They are able t o draw some conclusions regarding the s i z e of the energy l e v e l s p l i t t i n g , the c r y s t a l f i e l d symmetry, and the degeneracies of the l e v e l s . N.M.R. measurements were c a r r i e d out on mixtures of CH^ w i t h CD^, and of CH^ w i t h Kr, i n a d d i t i o n t o measurements of CH^ and i t s i s o t o p i c m o d i f i c a t i o n s . These experiments were c a r r i e d out to c o r r e l a t e the magnetic r e l a x a t i o n behaviour w i t h the i n t e r e s t i n g r e s u l t s of the s p e c i f i c heat measurements discussed above. 23 Miscellaneous Measurements 2:4 A s e r i e s of i n f r a - r e d and Raman experiments have been reported drawing c o n t r a d i c t o r y conclusions from t h e i r e x p e r i -ments. Welsh et a l . (1952) have studied the Raman spectra of methane as a l i q u i d and as a s o l i d j u s t below i t s melting p o i n t . They concluded that r o t a t i o n was e s s e n t i a l l y f r e e i n the l i q u i d , and c e r t a i n aspects of the spectrum i n the s o l i d a l s o l e d them to conclude that the r o t a t i o n a l freedom was unchanged by s o l i d i f i c a t i o n . However, Sa v i t s k y and Hornig (1962) c o n t r a d i c t e d these r e s u l t s . They'examined the i n f r a -red spectra of the s o l i d phases I , I I , and I I I of CH^ and CDjj, and concluded from the absence of observable r o t a t i o n a l f i n e s t r u c t u r e that a b a r r i e r - ' i h excess of s e v e r a l hundred degrees K e l v i n must hinder molecular r o t a t i o n . I t should be mentioned that t h e i r measurements were l i m i t e d t o temperatures l e s s than 4o°K. Recently, Ewing (1964) has shown that the observed spectra of l i q u i d and s o l i d methane do not- c o n f l i c t w i t h the hypothesis that the molecules are undergoing hindered r o t a t i o n i n these phases. The above c o n t r a d i c t o r y r e s u l t s prompted a t h e o r e t i c a l .-study by Gordon (1965). He shows how the i n f r a - r e d and Raman l i n e shapes are the F o u r i e r transforms of the time c o r r e l a t i o n f u n c t i o n s of the f i r s t and second order s p h e r i c a l harmonics, r e s p e c t i v e l y . To compare the r e s u l t s of the Raman and i n f r a -red s t u d i e s w i t h each other, the c o r r e l a t i o n f u n c t i o n s should be compared. 24 To support h i s contention, he has c a l c u l a t e d the F o u r i e r transforms of the Raman spectrum obtained by Welch et a l . , and -1^ the i n f r a - r e d spectrum by Ewing. He f i n d s that f o r t <10 sec, the c o r r e l a t i o n f u n c t i o n s are i n d i s t i n g u i s h a b l e from those of the f r e e r o t a t i o n a l motion. Both the Raman and i n f r a r e d -1^ c o r r e l a t i o n f u n c t i o n s show the exponential decay f o r t > 3 x 10 sec that i s expected f o r r o t o r s hindered by random i n t e r m o l e c u l a r torques. However, the Raman, c o r r e l a t i o n f u n c t i o n decays f a s t e r than the i n f r a r e d f u n c t i o n ; i t has f a l l e n t o a much smaller value by the time the exponential d e s c r i p t i o n becomes v a l i d . Therefore, the Raman spectrum resembles the f r e e r o t a t i o n a l spectrum co n s i d e r a b l y more than the i n f r a r e d one does. X-ray s t u d i e s by Mooy (1931) and McLannan et a l . (1929), showed that s o l i d CH^ has f . c . c . l a t t i c e w i t h the same l a t t i c e constants above and.below the phase t r a n s i t i o n . On the other hand, Schallamach (1939) found that f r e q u e n t l y a d d i t i o n a l l i n e s not a s s o c i a t e d w i t h f . c . c . l a t t i c e would appear below the phase t r a n s i t i o n , which would i n v a r i a b l y be accompanied by a 1.6$ increase i n the l a t t i c e constants. Moreover, there are c o n t r a d i c t i n g r e p o r t s about the existence or absence of b i r e f r i n g e n c e below the phase t r a n s i t i o n ; b i r e f r i n g e n c e would not e x i s t i f the c r y s t a l i s f . c . c . Thus, i t i s not at a l l c l e a r whether or not there i s a small change i n the c r y s t a l s t r u c t u r e on passing through the phase t r a n s i t i o n . In the d i s c u s s i o n of the r e s u l t s t h i s point w i l l be r e f e r r e d t o again. Some int e r e s t i n g r e s u l t s have been obtained by Rosenshein et a l . (1964) from ultrasonic attenuation and sound v e l o c i t y measurements i n CH^. The v e l o c i t y of sound exhibits an appreciable change at the upper phase t r a n s i t i o n . Moreover, there appears to be a maximum in the attenuation a few degrees above the upper phase t r a n s i t i o n ; there i s an int e r e s t -ing correspondence between t h i s maximum and certa i n features in the spin l a t t i c e relaxation data of CH^, which w i l l be discussed i n Chapter 5. No explanation of t h e i r r e s u l t s i s provided by them. In e l a s t i c neutron scattering studies of CH^ have been reported by S t i l l e r and Hautecler (1962) and Dasannacharya et a l . (1964); they a r r i v e at the conclusion that hindered rotations are non-existent, and that the only motions of the molecules are librations, small angle o s c i l l a t i o n s about a certain axis. C H A P T E R 3 . E X P E R I M E N T A L M E T H O D A N D A P P A R A T U S R e l a x a t i o n T i m e M e a s u r e m e n t s 3-1 A l l r e l a x a t i o n t i m e m e a s u r e m e n t s r e p o r t e d i n t h i s t h e s i s w e r e c a r r i e d o u t a t 28 .5 a n d 4 .4 m c s u s i n g t h e p u l s e m e t h o d d e v e l o p e d b y H a h n (1950).In t h i s m e t h o d , a r a d i o f r e q u e n c y f i e l d i s a p p l i e d i n t h e f o r m o f p u l s e s ; w e a r e i n t e r e s t e d i n t h e t r a n s i e n t s i g n a l s a f t e r t h e p u l s e s . U n d e r c o n d i t i o n s p r e v i o u s l y d i s c u s s e d i n t h e I n t r o d u c t i o n , t h e p u l s e r o t a t e s t h e m a g n e t i -z a t i o n o f t h e s p i n s y s t e m i n t o t h e x - y p l a n e ; t h e e q u i l i b r i u m m a g n e t i z a t i o n b e f o r e t h e p u l s e i s a p p l i e d b e i n g a l o n g t h e z d i r e c t i o n . T h e p r e c e s s i o n o f t h e m a g n e t i z a t i o n a b o u t t h e m a i n m a g n e t i c f i e l d i n d u c e s a t r a n s i e n t s i g n a l , c a l l e d t h e i n d u c t i o n t a i l , i n t h e c o i l a r o u n d t h e s a m p l e o r i e n t e d i n t h e x - y p l a n e . A s d i s c u s s e d e a r l i e r , t h e r e t u r n o f M t o i t s e q u i l i b r i u m z v a l u e M 0 , i s u s u a l l y e x p o n e n t i a l a n d i s d e s c r i b e d b y t h e t i m e c o n s t a n t T-^, 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 . T o m e a s u r e T-^ o n e u s e s t h e f a c t t h a t t h e a m p l i t u d e o f t h e i n d u c t i o n t a i l , a f t e r a 90° p u l s e , i s p r o p o r t i o n a l t o t h e v a l u e o f M a t t h e b e g i n n i n g o f t h e p u l s e . O n e p r e p a r e s t h e s t a t e o f z t h e s p i n s y s t e m a t t = 0, s u c h t h a t M = 0, b y a p p l y i n g a 90° p u l s e . A f t e r a c e r t a i n t i m e % a n o t h e r 90° p u l s e i s a p p l i e d ; t h e a m p l i t u d e o f t h e i n d u c t i o n t a i l i s a m e a s u r e o f t h e e x t e n t o f t h e r e c o v e r y o f M t o w a r d s i t s e q u i l i b r i u m v a l u e M Q a f t e r z a t i m e t . T h i s s e q u e n c e i s r e p e a t e d m a n y t i m e s w i t h d i f f e r e n t 26 27 v a l u e s of T . B e f o r e r e p e a t i n g t h e sequence, the s p i n system has t o be a l l o w e d t o r e t u r n t o e q u i l i b r i u m ; t h i s i m p l i e s one has t o w a i t a t l e a s t 5 T-^  s, t o i n c u r an e r r o r o f l e s s than Vfo. The s p i n - l a t t i c e r e l a x a t i o n t ime i s o b t a i n e d by p l o t t i n g l n ( M 0 - MCP)) as a f u n c t i o n of X, t h e s l o p e of t h e s t r a i g h t l i n e i s - l / T - L . An a l t e r n a t i v e method, w h i c h saves time when T-^  i s l o n g , i n v o l v e s t h e a p p l i c a t i o n of a t r a i n o f 15 or more 90° p u l s e s w i t h a s e p a r a t i o n between them g r e a t e r t h a n T2', but much l e s s t h a n T n. T h i s p r e p a r e s t h e s p i n system i n a s t a t e w i t h M =0. As b e f o r e , t h e a p p l i c a t i o n of a 90° p u l s e w i l l measure t h e r e c o v e r y o f M towards e q u i l i b r i u m ; t h e p r o c e d u r e f o r o b t a i n i n g 2 T-j^  i s t h e same as above. U s i n g t h i s method i t i s not neces-s a r y t o w a i t between s u c c e s s i v e sequences. The r e p r o d u c i b i l i t y of measurements i s about 5$ i n t h e s e e x p e r i m e n t s . U n l i k e T-., t h e decay o f M i s not e x p o n e n t i a l , but J- x, y i n g e n e r a l e x h i b i t s a v e r y c o m p l i c a t e d time dependence. I n the s o l i d methanes, t h e l o c a l f i e l d s a r e alm o s t always much l a r g e r t h a n t h e magnetic f i e l d i nhomogeneity over t h e sample; t h e r e f o r e no c o r r e c t i o n had t o be a p p l i e d t o o b t a i n t h e shape of t h e i n d u c t i o n t a i l . The S p e c t r o m e t e r 3 : 2 The a p p a r a t u s used i n p u l s e d n u c l e a r magnetic resonance has been d e s c r i b e d i n many a r t i c l e s (Hahn 1950, C l a r k 1964), c o n s e q u e n t l y o n l y the' b a s i c f e a t u r e s w i l l be d e s c r i b e d i n t h i s s e c t i o n . Each of t h e major c o n s t i t u e n t s of t h e s p e c t r o m e t e r , as shown i n the b l o c k d i a g r a m ( F i g u r e 2 ) , w i l l now be d i s c u s s e d i n t u r n . T i m i n g U n i t s 3:2:1 As d e s c r i b e d i n the p r e v i o u s s e c t i o n , p u l s e d n u c l e a r magnetic resonance ( N . M . R . ) measurements of s p i n - l a t t i c e or s p i n - s p i n r e l a x a t i o n t imes r e q u i r e p u l s e sequences c o n s i s t i n g e i t h e r of two r a d i o f r e q u e n c y ( r . f . ) p u l s e s ( the t w o - p u l s e s e q u e n c e ) , or of a t r a i n of r . f . p u l s e s f o l l o w e d by a s i n g l e r . f . p u l s e ( the s a t u r a t i n g t r a i n s e q u e n c e ) . These two t y p e s of p u l s e sequences are shown i n F i g u r e (3). The s e p a r a t i o n and the number of p u l s e s i n the s a t u r a t i n g t r a i n can be c o n -t r o l l e d . F u r t h e r r e q u i r e m e n t s a re t h a t the w i d t h and the p o s i t i o n of the l a s t p u l s e can be v a r i e d i n d e p e n d e n t l y of the i n i t i a l p u l s e or p u l s e s . The e l e c t r o n i c c i r c u i t s used t o p r o -v i d e t i m i n g p u l s e s f o r the above p u l s e sequences and t o p r o v i d e g a t i n g p u l s e s f o r the r . f . o s c i l l a t o r a re c o m p l e t e l y s t a n d a r d (see F i g u r e (3)) arid are not shown h e r e . I t may be remarked, however, t h a t the output p u l s e s used t o gate the o s c i l l a t o r s tage have a m p l i t u d e s of about 80 v o l t s and r i s e and f a l l t i m e s of a p p r o x i m a t e l y O.lytt - s e c . T r a n s m i t t e r 3:2:2 ( F i g u r e (4)) The t r a n s m i t t e r has t o be c a p a b l e of d e l i v e r i n g v e r y i n t e n s e r . f . . p u l s e s , w i t h r i s e and f a l l t imes a p p r o x i m a t e l y 0.2yu ,sec, i n t o a low impedance l o a d . No r . f . l eakage can be t o l e r a t e d between p u l s e s . The s i m p l e s t method t o a c h i e v e t h i s 29 TIMING UNIT TRANSMITTER H.R FREQUENCY COUNTER MATCHING NETWORK SAMPLE COIL RECEIVER OSCILLOSCOPE F i g u r e 2 . A B l o c k Diagram of the Spect rometer 30 i s to gate the o s c i l l a t o r . The tank c i r c u i t of the o s c i l l a t o r i s situated in the cathode c i r c u i t of a gating tube. During the o off cycle, the gating tube i s biased on, A d.c. current flows through the c o i l and thus reduces the Q of the tank c i r c u i t so that i t cannot o s c i l l a t e . When a pulse, supplied by the timing unit, biases the gating tube off, the current through the tank c i r c u i t i s suddenly switched off and the tank c i r c u i t rings at i t s resonant frequency. If there i s s u f f i c i e n t feedback, i t w i l l o s c i l l a t e f o r the duration of the gating pulse. The o s c i l l a t o r i s an electron-coupled Hartley c i r c u i t to reduce the e f f e c t s of variat i o n s i n loading on the frequency. The o s c i l l a t o r stage i s followed by a class A amplifier and two class C amplifiers, a l l using 6 L 6 tubes. These stages were capacitance coupled with a tuned c i r c u i t i n the g r i d of the next stage. This resulted in better power transfer at 3 0 mcs, than when transformer coupling was used. The f i n a l output stage consisted of an 8 2 9-B with both sections in p a r a l l e l , which was capacitance coupled to an impedance matching network between the transmitter and the sample c i r c u i t . To achieve f a s t r i s e and f a l l times, the Q's of a l l tuned c i r c u i t s must be quite low. At 3 0 mcs, the peak to peak voltage across the sample c o i l was 1 5 0 0 v o l t s ; at 4 . 5 mcs, i t was 3 5 0 0 v o l t s . Sample C i r c u i t 3 : 2 : 3 As w i l l be discussed in the Theory chapter, accurate l i n e shapes are obtained'only i f the r . f . magnetic f i e l d i s considerably larger than the l o c a l magnetic f i e l d due to T E K 163 M U L T I V I B R A T O R T E K . 163 JI T E K . 163 JLTLTL P U L S E M I X E R JLJl J i n n SATURATING TRAIN SEQUENCE TWO PULSE SEQUENCE F i g u r e 3. The Two Pulse Sequences Used 32 i n t e r a c t i o n s of the s p i n w i t h n e i g h b o u r i n g s p i n s . In o r d e r t o a c c o m p l i s h t h i s , care had t o be t a k e n i n matching the sample c i r c u i t t o t h e t r a n s m i t t e r . The sample c i r c u i t i s shown i n F i g u r e ( 5 ) . The c r o s s e d d i o d e s on the t r a n s m i t t e r s i d e s e r v e d t h r e e p u r p o s e s : ( l ) they i n c r e a s e d the Q, of the sample c i r c u i t , (2) they p r e v e n t e d t r a n s m i t t e r n o i s e f r o m b e i n g f e d i n t o the r e c e i v e r a f t e r the r . f . p u l s e , by d e c o u p l i n g the t r a n s m i t t e r c i r c u i t r y ; (3) they improved the r e c o v e r y t ime of the r e c e i v e r by p r e v e n t i n g any r . f . f r o m e n t e r i n g " the sample c i r c u i t , a f t e r the r . f . l e v e l i n the p u l s e had dropped below \ v o l t . The i n n e r c o n d u c t o r of the t r a n s m i s s i o n l i n e c o n s i s t e d of s i l v e r p l a t e d s t a i n l e s s s t e e l t u b i n g , d i a m e t e r ( l / 8 " ) ; the o u t e r c o n d u c t o r , a s t a i n l e s s s t e e l t u b e , s e r v e d as the pumping l i n e as w e l l . These d i a m e t e r s were chosen such t h a t the impedance of the t r a n s m i s s i o n l i n e was as c l o s e as p o s s i b l e t o 90 ohms, the impedance o f the c o a x i a l c a b l e used o u t s i d e the dewar h e a d . At 28.5 mcs, a h a l f wave t r a n s m i s s i o n l i n e was u s e d between the sample c o i l and the impedance m a t c h i n g network; at 4.4 mcs, t h i s was e l i m i n a t e d , i n s t e a d the t r a n s m i t t e r and the r e c e i v e r were p l a c e d as c l o s e as p o s s i b l e t o the sample c o i l . S i n c e the r . f . v o l t a g e s i n the sample c i r c u i t were l a r g e , care had t o be e x e r c i s e d t o p r e v e n t e l e c t r i c a l breakdown; i n p a r t i c u l a r , t h e He exchange gas p r e s s u r e had t o be l e s s than 10 / 6 . A n o t h e r t roublesome e f f e c t , not n o t i c e d at 30 mcs, was l a r g e h i g h f r e q u e n c y s i n u s o i d a l t r a n s i e n t s a f t e r an r . f . -I50V +750 V U J U J F i g u r e 4. The T r a n s m i t t e r C i r c u i t D i a g r a m 34 p u l s e , w h i c h appeared o n l y i n the p r e s e n c e of the main magnetic f i e l d . T h e i r a m p l i t u d e and f r e q u e n c y c o u l d he a l t e r e d d r a s t i c a l l y by p o t t i n g or c o a t i n g the c o i l w i t h d i f f e r e n t s u b s t a n c e s , but they would u s u a l l y r e a p p e a r , i f the c o i l was r e p e a t e d l y c o o l e d t o 4.2°K or i f the c o a t i n g was c r a c k e d . We s u r m i s e d t h a t the i n t e r a c t i o n between the p u l s e d r . f . c u r r e n t , w h i c h may be 10-20 amps, and the main magnetic f i e l d e x c i t e d some s t r u c t u r a l resonance i n the c o i l a t u l t r a s o n i c f r e q u e n c i e s . These e f f e c t s were not seen at 30 mcs because the r . f . c u r r e n t s were s m a l l e r , the i n d u c t a n c e of the c o i l was l e s s , and the s t r u c t u r e was more r i g i d . The b e s t p o t t i n g agents were f o u n d t o be l i q u i d p o r c e l a i n " S a u e r e i s e n " and " S t y c a s t " epoxy. R e c e i v e r s 3:2:4 The s i n g l e c o i l sample c i r c u i t , d e s c r i b e d above, s u f f e r s f r o m the s e r i o u s d i s a d v a n t a g e of i n j e c t i n g a l a r g e r . f . p u l s e i n t o the i n p u t s tage of the r e c e i v e r . F o r f a s t r e c o v e r y , f r o m the l a r g e r . f . p u l s e , w h i c h s a t u r a t e s the r e c e i v e r , a l l tuned c i r c u i t s and RC c o u p l i n g networks must have s h o r t t ime c o n -s t a n t s . In p a r t i c u l a r , the l a s t s tage and the d e t e c t o r , w h i c h h a n d l e the maximum s i g n a l , must have v e r y f a s t r e c o v e r y . P a r a l y s i s , w h i c h i s a temporary c u t - o f f due t o g r i d c u r r e n t b e i n g drawn d u r i n g the p u l s e , can be a v o i d e d by making a l l g r i d and cathode c i r c u i t t ime c o n s t a n t s s h o r t ; t h i s s u f f e r s f r o m the s l i g h t d i s a d v a n t a g e of l o w e r i n g the g a i n p e r s t a g e . Supply v o l t a g e r e c o v e r y , i f not guarded a g a i n s t , can s e r i o u s l y d i s t o r t the output and l e n g t h e n the r e c o v e r y t i m e . r. f. Input o 10 pf HI- Output to Receiver Sample Coil F i g u r e 5 . The Sample C i r c u i t VJl 36 The g a i n c o n t r o l b i a s s u p p l y , i n p a r t i c u l a r , s h o u l d have an output impedance of l e s s than 2000 ohm's. The r e c e i v e r used at 2 8 , 5 mcs, was an A r e n b e r g Wide Band a m p l i f i e r (WA.600C) . The preamp had a cascode i n p u t , two s tages of narrow band a m p l i f i c a t i o n , and a cathode f o l l o w e r o u t p u t . The main a m p l i f i e r was wide band, 1 0 - 6 0 mcs. The o v e r a l l a m p l i f i e r had a 7 /U/S r e c o v e r y t i m e . The 4 . 4 mcs r e c e i v e r , .which we c o n s t r u c t e d , had a 1 mcs band w i d t h and a 6/fcs r e c o v e r y t i m e , ( s e e F i g u r e ( 6 ) ) . The preamp, b u i l t a c c o r d i n g t o C l a r k ' s d e s i g n ( 1 9 6 4 ) , uses a h i g h g m pentode 7722 i n the i n p u t w i t h a v e r y low e q u i v a l e n t n o i s e r e s i s t a n c e , (200 ohms); f o l l o w e d by a 6DJ8 a m p l i f i e r and cathode f o l l o w e r s t a g e . The main a m p l i f i e r c o n s i s t e d of f o u r s y n c h r o n o u s l y tuned a m p l i f y i n g s t a g e s , a d e t e c t o r , and a low impedance output s t a g e . The tuned r . f . s tages c o n s i s t e d of 6AK5 p e n t o d e s , w i t h the i n d i v i d u a l band w i d t h s of each stage b e i n g 1 .5 mcs. To improve the r e c o v e r y t ime of the r e c e i v e r , p a i r s of c r o s s e d - d i o d e s , FD 1 0 0 , were used i n the p l a t e c i r c u i t s of the f i r s t two a m p l i f y i n g s t a g e s . The l i n e a r i t y of b o t h a m p l i f i e r s was checked c a r e f u l l y . I t was f o u n d t h a t the l i n e a r i t y c o r r e c t i o n was n e g l i g i b l e . To keep the o p e r a t i n g c o n d i t i o n s the same at a l l t i m e s , d e s -p i t e a v a r i a t i o n of 80 i n the s i g n a l s t r e n g t h i n the range f r o m 1 . 2 - 8 0 ° K , we i n s e r t e d a w i d e - b a n d Hewlet t Packard a t t e n u a t o r between the p r e a m p l i f i e r and the main a m p l i f i e r . T h i s r e d u c e d the s i g n a l o u t p u t , but not the s i g n a l - t o - n o i s e r a t i o . — — - < ] F i g u r e 6. The k\k mcs. R e c e i v e r C i r c u i t Diagram 38 C r y o s t a t 3:3 The d e s i g n of the c r y o s t a t resembles t h a t of an a d i a b a t i c c a l o r i m e t e r , the main d e s i g n c r i t e r i o n b e i n g the m i n i m i z a t i o n of any heat l e a k s . The m o t i v a t i o n i s , however, d i f f e r e n t . In the case of the c a l o r i m e t e r , the purpose i s t o o b t a i n the b e s t p o s s i b l e a c c u r a c y ; i n our c a s e , the aim i s t o reduce the temperature g r a d i e n t s i n the sample and t o d e c r e a s e the h e l i u m b o i l - o f f r a t e . These c o n s i d e r a t i o n s are i m p o r t a n t as the sample may be a t any temperature between 4.2°K and 50°K. A f t e r t h e s e i n t r o d u c t o r y r e m a r k s , some of the d e t a i l s w i l l be d e s c r i b e d i n the f o l l o w i n g s e c t i o n s . The Vacuum Can and the Dewar Head 3:3:1 F i g u r e s (7) and (8) show the c o n s t r u c t i o n of the dewar head and the vacuum c a n . The pr ime concern i n the d e s i g n was t o keep the amount of m e t a l t o a minimum, so t h a t the c o n -sumption of l i q u i d h e l i u m was r e d u c e d . Only non-magnet ic m a t e r i a l s were used t o keep the magnetic f i e l d as homogeneous as p o s s i b l e near the sample ; t h i s e x c l u d e s c e r t a i n t y p e s of s t a i n l e s s ' s t e e l and b r a s s . The m a n i f o l d , shown as A i n F i g u r e (7), a l l o w e d a c c e s s t o the i n n e r c o n d u c t o r of the t r a n s m i s s i o n l i n e , w h i c h r a n down the c e n t e r of the pumping t u b e . The bot tom end of the t r a n s m i s s i o n l i n e was anchored at 4 . 2 ° K , t o e l i m i n a t e any heat l e a k down i t . The e l e c t r i c a l l e a d s f o r the thermometers and the h e a t e r were brought i n t o the vacuum can u s i n g two Manifold (A) Transfer siphon fitting pumping for He bath Connector for transmission line pumping line for Vacuum can Kovar seals for electrical leads O.OI S.S. tubing oo Nylon Thread ,S.S. Tubing 4-Wood's Metal Seal Glass Sample Tube Copper Sample Holder Teflon Spacer Vacuum Can tr -Thermometer Sample Coil F i g u r e 8 . A Diagram Showing the Vacuum Can and Sample H o l d e r 41 Housekeeper s e a l s f i t t e d w i t h a p i n c h c a r r y i n g f i v e t u n g s t e n w i r e s e a c h . The can was s o l d e r e d t o the f l a n g e u s i n g Woods m e t a l ; Indium and T e f l o n O - r i n g s e a l s were found t o be l e s s r e l i a b l e and i n v o l v e d the use of more m e t a l . W h i l e s o l d e r i n g t h e f l a n g e , the t a i l end of the can was kept i n l i q u i d n i t r o g e n . T h i s condensed the methane i n the sample t u b e , and thus r e d u c e d the danger of the sample e x p l o d i n g . Only copper was used i n the c o n s t r u c t i o n of the c a n , because i t has a b e t t e r t h e r m a l c o n d u c t i v i t y c o e f f i c i e n t than b r a s s , and i t does not s p o i l the homogeneity of the magnetic f i e l d . However, i n m a c h i n i n g the w a l l t h i c k n e s s t o 30 thousandths of an i n c h , i t became porous t o H e l l and had t o be c o a t e d w i t h a t h i n l a y e r of s o f t s o l d e r . The Sample H o l d e r 3^3:2 To r e d u c e the temperature g r a d i e n t s , a sample h o l d e r was machined f r o m a copper r o d . The sample h o l d e r c o m p l e t e l y e n c l o s e d the sample , except f o r the t a i l end, around w h i c h the sample c o i l was s i t u a t e d . Good t h e r m a l c o n t a c t between the sample h o l d e r and the sample was ensured by f i l l i n g the space between them w i t h A p i e z o n g r e a s e . To reduce heat l e a k s t o a minimum, the sample h o l d e r was suspended by two n y l o n t h r e a d s and h e l d i n p l a c e by a t e f l o n s p a c e r . In a d d i t i o n gauge 34 Magnanin w i r e was used f o r the thermometer l e a d s , because i t i s a poor t h e r m a l c o n d u c t o r . The r a d i a t i o n b a f f l e shown i n F i g u r e (8), s e r v e d t o i n t e r c e p t any room temperature r a d i a t i o n down the pumping t u b e , and at the same t ime anchored the bottom end of the t r a n s m i s s i o n l i n e a t 4 . 2 ° K . Two copper tubes s o l d e r e d i n t o the sample h o l d e r served as the thermometer w e l l s . The sample h o l d e r a l s o p r o v i d e d a mounting f o r a 700 ohm C o n s t a n t a n h e a t e r , w h i c h was b i f i l a r wound t o e l i m i n a t e any s t r a y magnetic f i e l d s a s s o c i a t e d w i t h the h e a t e r c u r r e n t . Thermal c o n t a c t between the h e a t e r and the sample h o l d e r was p r o v i d e d by two c o a t s of s h e l l a c . The h e a t e r was e v e n l y spaced over the whole sample h o l d e r t o a v o i d any hot s p o t s . To o b t a i n an e f f i c i e n t h e l i u m t r a n s f e r , a h e l i u m exchange gas p r e s s u r e of 3 cm was i n t r o d u c e d i n t o the vacuum c a n . I t was found t h a t a m o d e r a t e l y f a s t pumping system was needed t o i s o l a t e the sample f r o m the b a t h w i t h i n a r e a s o n a b l e l e n g t h of t i m e . U s i n g a Hyvac 6 and an o i l d i f f u s i o n pump c o n -n e c t e d by a 1-| i n c h pumping l i n e t o the dewar head , a u s e f u l vacuum of 0.1 fx or b e t t e r was o b t a i n e d a f t e r 30 minutes of pumping. Temperature Measurement and C o n t r o l 3:4 Temperature Measurement 3:4:1 Two thermometers were used t o cover the range f r o m 1.2-55°K, a 100 ohm Speer carbon thermometer Grade 1003* and a Hartman-Braun p l a t i n u m thermometer model W 4871. The thermometers were f i t t e d i n t o the copper thermometer w e l l s w i t h some A p i e z o n grease a s s u r i n g good t h e r m a l c o n t a c t . The p l a t i n u m thermometer , w h i c h had been c a l i b r a t e d by the N . B . S . , was used t o measure t e m p e r a t u r e s above 20°K; the carbon thermometer was used below 20°K, T h i s was n e c e s s a r y because the s e n s i t i v i t y of the p l a t i n u m thermometer d e c r e a s e s v e r y r a p i d l y below 20°K; and t h a t of the carbon thermometer becomes l e s s s e n s i t i v e above 25°K, C a l i b r a t i o n p o i n t s f o r the carbon thermometer were o b t a i n e d around 20°K, u s i n g the p l a t i n u m thermometer, and i o 4 around 4 .2 K u s i n g the vapour p r e s s u r e s c a l e of He . These c a l i b r a t i o n p o i n t s were then used t o e v a l u a t e the c o n s t a n t s A , B , and C i n the e q u a t i o n : -l o g R + ^ R = A + B / T T h i s i s a s e m i - e m p i r i c a l r e l a t i o n s h i p , w h i c h has been found t o d e s c r i b e the temperature v a r i a t i o n s of the carbon r e s i s -tance (Clement and Q u i n n e l l , 1952). Some of t h e s e c a l i b r a t i o n p o i n t s were checked f o r e v e r y He run and i t was f o u n d t h a t changes i n carbon r e s i s t a n c e i m p l i e d c o r r e c t i o n s of the o r d e r of 0 . 0 1 $ . A comparison of the phase t r a n s i t i o n t e m p e r a t u r e s o b t a i n e d f r o m the s p i n - l a t t i c e r e l a x a t i o n t ime data w i t h t h o s e o b t a i n e d f r o m s p e c i f i c heat measurements, p r o v i d e d , a v a l u a b l e check on the a c c u r a c y o f the thermometry . F u r t h e r -more, the f a c t t h a t the n u c l e a r s p i n m a g n e t i z a t i o n obeys the C u r i e Law, was a l s o used t o check r e l a t i v e t e m p e r a t u r e s of the sample between 4 .2 and 15°K w i t h an a c c u r a c y of about 3$. 44 Prom t h e s e c h e c k s , i t was c o n c l u d e d t h a t the thermometry was a c c u r a t e t o 0.1°K i n the range below 10°K, and a c c u r a t e t o 0 . 2 ° K f r o m 13°K t o 20°K. To a s s u r e o u r s e l v e s t h a t the magnet ic f i e l d dependence of the thermometers d i d not a f f e c t our temperature measurements, we measured the r e s i s t a n c e s of b o t h thermometers f o r a se t of t e m p e r a t u r e s w i t h the magnet ic f i e l d on and o f f . We found t h a t the c o r r e c t i o n was at the most 0.01°K. A R u b i c o n M u e l l e r b r i d g e was used t o measure the thermo-meter r e s i s t a n c e s , and a Brown n u l l i n d i c a t o r was used f o r the n u l l a m p l i f i e r . A M u e l l e r b r i d g e w i l l measure the r e s i s t a n c e of a f o u r l e a d thermometer i n the p r e s e n c e of s i z e a b l e l e a d r e s i s t a n c e s . 'However, t h i s b r i d g e c o u l d o n l y measure r e s i s -t a n c e s up t o 70 ohms; whereas the carbon r e s i s t a n c e assumes v a l u e s f r o m 200-500 ohms i n the temperature i n t e r v a l 20°K-1.2°K. At lower t e m p e r a t u r e s , the carbon r e s i s t a n c e changes w i t h temperature are f r a c t i o n a l l y v e r y l a r g e . C o n s e q u e n t l y , a s l i g h t l o s s i n a c c u r a c y and s e n s i t i v i t y c o u l d be t o l e r a t e d . We t h e r e f o r e d e c i d e d t o ex tend the range of the M u e l l e r b r i d g e a c c o r d i n g , t o the c i r c u i t shown i n F i g u r e ( 9 ) . A subsequent c h e c k of the s e n s i t i v i t y and the a c c u r a c y of the m o d i f i e d b r i d g e , u s i n g some a c c u r a t e l y measured s t a n d a r d r e s i s t a n c e s , showed t h a t t h e s e were q u i t e a d e q u a t e . In the measurement of the carbon r e s i s t a n c e below 4 . 2 ° K , care had t o be e x e r c i s e d t o p r e v e n t s e l f - h e a t i n g of the thermometer , w h i c h would l e a d t o e r r o n e o u s temperature r e a d i n g s (Berman, 1952). Below the 7\ p o i n t a measuring c u r r e n t t h r o u g h the r e s i s t o r of 1 ^tamp was u s e d ; a t 4 . 2 ° K and above a 10 //amp c u r r e n t was u s e d . A 1 ma. c u r r e n t was used f o r the p l a t i n u m thermometer . I t s h o u l d be noted t h a t a l l the r e s i s -t a n c e s were measured f o r b o t h d i r e c t i o n s of c u r r e n t t o e l i m i n a t e any t h e r m a l E . M . F . ' s . An at tempt was made t o measure the temperature g r a d i e n t a l o n g the sample h o l d e r at 20°K, u s i n g two c a l i b r a t e d carbon thermometers s i t u a t e d , 1 0 cms, a p a r t . The temperature d i f f e r e n c e was so s m a l l t h a t we c o u l d a s s i g n o n l y an upper l i m i t of 0 . 0 2 ° K . Temperature C o n t r o l 3 : 4 : 2 In these e x p e r i m e n t s , i t was i m p e r a t i v e t o c o n t r o l the temperature t o b e t t e r than 0 . 1 ° K over l o n g p e r i o d s of t ime at any d e s i r e d temperature between 4 .2 and 55°K. C o n s e q u e n t l y , we d e c i d e d t o use an e l e c t r o n i c f e e d - b a c k r e g u l a t o r . The temperature c o n t r o l u n i t i s shown i n F i g u r e ( 1 0 ) . The s e n s i n g element used was e i t h e r the carbon or the p l a t i n u m thermometer . The i n p u t t o the temperature c o n t r o l u n i t i s s u p p l i e d by the Brown n u l l i n d i c a t o r , w h i c h i s used here as a d . c . a m p l i f i e r . As the output v o l t a g e of the Brown n u l l i n d i c a t o r c o n t a i n e d much 60 c y c l e s / s e c , ' the f i r s t s tage of the c o n t r o l u n i t was a McFee d i f f e r e n c e a m p l i f i e r t o e l i m i n a t e the 60 c p s . component and t o a m p l i f y the s m a l l d . c . component. The next two s tages were cathode c o u p l e d d . c . a m p l i f i e r s d r i v i n g an output cathode f o l l o w e r , the 4 6 cathode l o a d of w h i c h i s the h e a t e r . The b i a s on the g r i d of the cathode f o l l o w e r d e t e r m i n e d the average c u r r e n t t h r o u g h the h e a t e r . Once the system s t a r t e d r e g u l a t i n g the t e m p e r a t u r e , we were a b l e t o a d j u s t the b i a s such t h a t the mean p o s i t i o n of the n u l l i n d i c a t o r was z e r o . At low t e m p e r a t u r e s , the g a i n of the system was t o o h i g h and the whole system would o s c i l l a t e . To remedy t h i s , we j u s t lowered the g a i n of the c o n t r o l u n i t u n t i l i t s topped o s c i l l a t i n g . To measure the t e m p e r a t u r e , the average c u r r e n t t h r o u g h the h e a t e r was m a i n t a i n e d , but i t was not r e g u l a t e d f o r the d u r a t i o n of the measurement. The d r i f t of the temperature d u r i n g the measurement was i n s i g n i f i c a n t . Because of the h i g h s e n s i t i v i t y , good e l e c t r i c a l i s o l a t i o n between the h e a t e r and thermometer w i r i n g was e s s e n t i a l , o t h e r w i s e the system would o s c i l l a t e . Sample P r e p a r a t i o n s 3-5 The p r o t o n and d e u t e r o n s p i n - l a t t i c e r e l a x a t i o n t imes i n the l i q u i d and i n the s o l i d methanes have been found t o be v e r y s e n s i t i v e t o s m a l l c o n c e n t r a t i o n s of Og i n the sample . S i n c e the Og m o l e c u l e i s h i g h l y p a r a m a g n e t i c , i t p r o v i d e s a p o w e r f u l mechanism f o r s p i n - l a t t i c e r e l a x a t i o n . As a consequence no i n f o r m a t i o n would have been o b t a i n e d about the n a t u r e o f the i n t e r a c t i o n s between the methane m o l e c u l e s , i f the 0o i m p u r i t i e s had not been removed. Decade Resistance Box (B To Battery T E R M I N A L S FOR P L A T I N U M a C A R B O N T H E R M O M E T E R S . Mueller Bridge v v Temperature Contro l Unit To Heater F i g u r e 9. Schematic of the Temperature R e g u l a t o r + 1 7 0 V current control F i g u r e 10. C i r c u i t Diagram of the Temperature C o n t r o l U n i t co The t e c h n i q u e found t o be most s u i t a b l e f o r the removal i n v o l v e d the use of a g e t t e r , M i s c h m e t a l , s u p p l i e d by L i n d s a y C h e m i c a l C o . The c o n c e n t r a t i o n i s p r o b a b l y r e d u c e d below 2.5x10"^$ u s i n g t h i s t e c h n i q u e (Sandhu et a l . i 9 6 0 ) . Two d i f f e r e n t g e o m e t r i e s , shown i n F i g u r e ( l l ) , were used d u r i n g the c o u r s e of t h e s e e x p e r i m e n t s . F o r b o t h geo-m e t r i e s , the r e l a x a t i o n t imes agreed t o w i t h i n e x p e r i m e n t a l e r r o r . In geometry ' a ' , the t u n g s t e n c o i l shown i n the diagram c o n t a i n s the M i s c h m e t a l . The sample tube was evacuated t o 10~^cm and baked s e v e r a l t i m e s w h i l e pumping. When the f l a s k had 'been baked o u t , i t was f i l l e d w i t h Argon t o a p r e s s u r e of 1 or 2 mmHg. The g e t t e r was t h e n f l a s h e d u n t i l a d i f f u s e l a y e r was d e p o s i t e d on the i n n e r s u r f a c e of the f l a s k . The argon was then pumped o u t , and the methane gas was i n t r o d u c e d ; the f l a s k was t h e n s e a l e d o f f f r o m the s y s t e m . The sample p r e p a r a t i o n was the same f o r geometry ' b ' , except t h a t the gas was l e f t i n c o n t a c t w i t h the g e t t e r f o r s e v e r a l d a y s , a f t e r w h i c h the methane was condensed i n t o the sample tube u s i n g l i q u i d n i t r o g e n and was s e a l e d o f f f r o m the g e t t e r i n g b u l b . At room t e m p e r a t u r e , the gas p r e s s u r e i n the sample tube was 25-30 a tmospheres , whereas the average b u r s t -i n g p r e s s u r e was measured t o be 70 a tmospheres . T h i s geometry was used f o r most of the low temperature measurements because i t meant t h a t t h e c r y o s t a t d e s i g n c o u l d be s i m p l i f i e d , and t h a t some of the heat l e a k and vacuum problems c o u l d be v i r t u a l l y " e l i m i n a t e d . T h i s p a r t i c u l a r geometry, however, i n t r o d u c e d the T U N G S T E N COIL ground glass f i t t ing 72 cm 0.5 cm 0.9cm 3 cm T sealed off here 0.9 cm F i g u r e 11. The Two Sample G e o m e t r i e s Used p o s s i b i l i t y of the sample e x p l o d i n g and of p o s s i b l e c o n t a m i n -a t i o n of the sample by 0 o i m p u r i t i e s due to the absence of the g e t t e r b u l b . The s p i n - l a t t i c e r e l a x a t i o n t imes were r e p r o d u c i b l e f r o m sample t o sample and over l o n g p e r i o d s of t ime f o r each sample . These measurements were c a r r i e d out over a p e r i o d of more than a y e a r and no change i n T^ w i t h t ime f o r any sample was ever o b s e r v e d . M i x t u r e s of CH^ w i t h CD^ or Kr were p r e p a r e d by m i x i n g a p p r o p r i a t e amounts of the gases a t room t e m p e r a t u r e . E s t i -mates of CH^ t o CD^ or t o Kr r a t i o s were made f r o m the p a r t i a l p r e s s u r e s , assuming the gases t o be i d e a l . CHAPTER 4. THE THEORY OP RELAXATION In Chapter 1, the s p i n - l a t t i c e r e l a x a t i o n was d e s c r i b e d as the exchange of energy between the s p i n system and the l a t t i c e , so t h a t t h e r m a l e q u i l i b r i u m w i l l be e s t a b l i s h e d between them, A d e t a i l e d i n t e r p r e t a t i o n of the r e l a x a t i o n b e h a v i o u r w i l l be a t tempted i n terms of the i n t e r a c t i o n s between the n u c l e i and the degrees of f reedom of the l a t t i c e . Many papers have appeared i n the l i t e r a t u r e on the t h e o r y of n u c l e a r magnetic r e l a x a t i o n ( R e d f i e l d 1957* B l o c h 1956, Abragam 1961) . Some of the g r o s s f e a t u r e s of the t h e o r y p e r t a i n i n g t o the systems under c o n s i d e r a t i o n w i l l be r e v i e w e d . C o n v e n t i o n a l : . t h e o r y 4:1 The p r o b l e m of c a l c u l a t i n g 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 , i n g e n e r a l , c o n s i s t s of e v a l u a t i n g the t r a n s i t i o n p r o b a b i l i t i e s between s p i n - s t a t e s due t o i n t e r a c t i o n s between the l a t t i c e and the s p i n sys tem. The H a m i l t o n i a n d e s c r i b i n g the s p i n - l a t t i c e i n t e r a c t i o n s f o r the systems under c o n s i d e r a t i o n i s : (4-1) 52 53 Tne f i r s t t h r e e terms d e s c r i b e the magnetic d i p o l e - d i p o l e i n t e r -a c t i o n s between l i k e and u n l i k e n u c l e a r s p i n s . The pr imed (unprimed) symbols denote those q u a n t i t i e s b e l o n g i n g to s p i n s I , f o r example p r o t o n s ( to s p i n s S, f o r example d e u t e r o n s ) . JO_ i s ' t h e d i r e c t i o n of the v e c t o r r . , between s p i n s i and k . i k i k ^ ^= *c« e = — "2-5" — + v 5" i r ^ i 2 T T where the Y 2 m (9,0) a re n o r m a l i z e d s p h e r i c a l h a r m o n i c s . The next two terms of the H a m i l t o n i a n d e s c r i b e the s p i n r o t a t i o n i n t e r a c t i o n , the i n t e r a c t i o n between the n u c l e a r s p i n and the c u r r e n t d i s t r i b u t i o n a s s o c i a t e d w i t h m o l e c u l a r r o t a t i o n . C and C are the s p i n - r o t a t i o n i n t e r a c t i o n t e n s o r s f o r the two t y p e s of n u c l e i . The l a s t term i s p r e s e n t when one of the t y p e s of n u c l e i has s p i n I > f-; i t d e s c r i b e s the i n t e r a c t i o n between the q u a d r u p o l e moment of the n u c l e u s and the e l e c t r i c f i e l d g r a d i e n t s at the s i t e of the n u c l e u s due t o the s u r r o u n d i n g e l e c t r o n i c and n u c l e a r c h a r g e s . The f u n c t i o n s Fm@U) are the same as above, except now sx. ^ s p e c i f i e s the o r i e n t a t i o n of the f i e l d g r a d i e n t a t the s i t e of n u c l e u s ' i ' w i t h r e s p e c t t o the l a b o r a t o r y f rame, and 5^ s i 2 = _ H _ x i . A l l of these c o n t r i b u t i o n s t o K i n v o l v e m o l e c u l a r p a r a -m e t e r s . As a consequence of those i n t e r a c t i o n s between the m o l e c u l e s w h i c h cause the m o l e c u l e s t o r e o r i e n t , these m o l e c u l a r parameters w i l l a l s o be random f u n c t i o n s of t i m e . In the g e n e r a l t h e o r y of n u c l e a r s p i n r e l a x a t i o n (Abragam p 3 0 0 ) , the r e l a x a t i o n p r o b a b i l i t y due t o random p e r t u r b a t i o n s i n v o l v e the F o u r i e r t r a n s f o r m of the " c o r r e l a t i o n f u n c t i o n " of these p a r a -m e t e r s . The " c o r r e l a t i o n f u n c t i o n " G ( ^ ) of s p h e r i c a l harmonics of o r d e r 2 , f o r example, i s denoted by where { y s i g n i f i e s an ensemble a v e r a g e . G ( X ) i s a f u n c t i o n of x o n l y i f , as i s g e n e r a l l y t r u e f o r an e q u i l i b r i u m ensemble , the c o r r e l a t i o n f u n c t i o n i s i n v a r i a n t under a change i n the o r i g i n of t i m e . The c o r r e l a t i o n f u n c t i o n G(t: ) —•* 0 as t*—*-°° . I f the c o r r e l a t i o n f u n c t i o n i s e x p o n e n t i a l , as i s o f t e n t h e case, : a c h a r a c t e r i s t i c t ime of the sys tem, the c o r r e l a t i o n t i m e , i s d e f i n e d by G ( T: ) = G(o) e~ / r t c . G e n e r a l p r o p e r t i e s of the F o u r i e r spectrum (or " s p e c t r a l d e n s i t y " ) j(oo); of G ( t ) a re t h a t the s h o r t e r the c o r r e l a t i o n t i m e , the b r o a d e r the f r e q u e n c y s p e c t r u m ; i n the l i m i t t h a t % — * • 0 i t approximates a " w h i t e s p e c t r u m " . More 55 s p e c i f i c a l l y , i f the c o r r e l a t i o n f u n c t i o n i s e x p o n e n t i a l , then the s p e c t r a l d e n s i t y i s and the F o u r i e r component J(co0) a t t a i n s i t s maximum a m p l i t u d e when U)Q KQ = 1. As we s h a l l see l a t e r , the f a c t t h a t j(co ) goes t h r o u g h such a maximum g i v e s r i s e t o a c h a r a c t e r i s t i c minimum i n the p l o t v e r s u s c o r r e l a t i o n t i m e . The t h e o r y of n u c l e a r s p i n r e l a x a t i o n o r i g i n a l l y d e v e l o p e d by Bloembergen, P u r c e l l and Pound ( B . P . P . , 1948), e x p r e s s e d the s p i n - l a t t i c e r e l a x a t i o n p r o c e s s i n terms of r a t e e q u a t i o n s i n v o l v i n g t r a n s i t i o n p r o b a b i l i t i e s between n u c l e a r s p i n s t a t e s i n d u c e d by the s p i n - l a t t i c e i n t e r a c t i o n s . T h i s t h e o r y was i n c a p a b l e of d e s c r i b i n g i n t e r a c t i o n s i n w h i c h the o f f - d i a g o n a l e lements of the d e n s i t y m a t r i x p l a y e d an i m p o r t a n t r o l e . A more g e n e r a l t h e o r y u s i n g the d e n s i t y m a t r i x was d e v e l o p e d by R e d f i e l d and by Wangness and B l o c h (Abragam p 2 7 6 ) . U s i n g t h i s f o r m a l i s m and n e g l e c t i n g c r o s s - c o r r e l a t i o n s a r i s i n g f r o m the f a c t t h a t as a m o l e c u l e moves as a r i g i d body t h e r e i s a c o r r e l a t i o n between the r e l a t i v e mot ions of v a r i o u s f o r i , j , / k , l , i t can be shown 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 due t o d i p o l a r i n t e r a c t i o n s can be d e s c r i b e d by a s i n g l e e x p o n e n t i a l and the e x p r e s s i o n f o r T-^ and T g due t o i n t e r a c t i o n s between l i k e s p i n s a r e : (4-5) p a i r s of s p i n s ) i . e . assume (4-6) 56 A s i m i l a r e x p r e s s i o n has been d e r i v e d f o r i n the case of d i p o l a r i n t e r a c t i o n s between u n l i k e s p i n s . F o r a n u c l e u s w i t h s p i n 1 = 1 and n o n - z e r o q u a d r u p o l e moment, the q u a d r u p o l e i n t e r a c t i o n w i t h an a x i a l l y symmetric f i e l d g r a d i e n t g i v e s r i s e t o a s p i n - l a t t i c e r e l a x a t i o n t ime \ 1 . T Bo 11 (4-7) In the t h e o r y l e a d i n g t o the a b o v e • r e s u l t s , i t has been e x p l i c i t l y assumed ( R e d f i e l d , 1957)> t h a t and t h a t ^H,)>=- O , where i s the p e r t u r b a t i o n c a u s i n g r e l a x a t i o n . T h i s a s s u m p t i o n i s e s s e n t i a l l y a s tatement of the f a c t t h a t the f r a c t i o n a l change i n the p o p u l a t i o n s has t o be s m a l l d u r i n g the t ime of i n t e r e s t (t ) o t h e r w i s e one does not get s i m p l e t i m e - i n d e p e n d e n t r a t e p r o c e s s e s . The above t h e o r y , w h i c h e s s e n t i a l l y t r e a t s the l a t t i c e c l a s s i c a l l y , s u f f e r s f r o m the d e f e c t t h a t the s p i n system r e l a x e s t o a s teady s t a t e d e s c r i b e d by an i n f i n i t e tempera-t u r e . I t can be shown t h a t a quantum m e c h a n i c a l d e s c r i p t i o n of the l a t t i c e w i l l l e a d t o a f i n i t e T f o r the s p i n system e q u a l t o t h a t o f the l a t t i c e . T h i s r e s u l t can r e a d i l y be u n d e r s t o o d , i f one remembers t h a t the l a t t i c e i s assumed t o have i n f i n i t e heat c a p a c i t y , i . e . a t a l l t i m e s , r e m a i n s i n t h e r m a l e q u i l i b r i u m , c o n s e q u e n t l y the t r a n s i t i o n p r o b a b i l i t y w i l l be w e i g h t e d by the B o l t z m a n f a c t o r s and w i l l t e n d t o b r i n g the s p i n system i n t o t h e r m a l e q u i l i b r i u m w i t h the l a t t i c e . The quantum m e c h a n i c a l d e s c r i p t i o n of the l a t t i c e becomes n e c e s s a r y at low t e m p e r a t u r e s when o n l y a s m a l l number of degrees of f reedom of the l a t t i c e a re e x c i t e d ; t h i s w i l l become apparent when we c o n s i d e r : a Quantum M e c h a n i c a l model i n Chapter 6. In the H a m i l t o n i a n d e s c r i b e d e a r l i e r , no d i s t i n c t i o n was made between i n t e r a c t i o n s o r i g i n a t i n g w i t h i n the m o l e c u l e ( i n t r a m o l e c u l a r i n t e r a c t i o n s ) , or o u t s i d e the m o l e c u l e ( i n t e r m o l e c u l a r i n t e r a c t i o n s ) . * - * i n t r a + " " i n t e r ^ F o r example, i n a m o l e c u l a r s o l i d the c o n t r i b u t i o n t o the e l e c t r i c f i e l d g r a d i e n t a t the s i t e of a n u c l e u s due t o n e i g h b o u r i n g m o l e c u l e s i s n e g l i g i b l e i n comparison w i t h t h a t due t o charges a s s o c i a t e d w i t h the m o l e c u l e on w h i c h the n u c l e u s i s s i t u a t e d . F o r the q u a d r u p o l a r i n t e r a c t i o n s , t h e r e f o r e , the o n l y m o l e c u l a r parameters e n t e r i n g i n t o the H a m i l t o n i a n of E q u a t i o n (4-1) w i l l be t h e a n g l e s of the e l e c t r i c f i e l d g r a d i e n t w i t h r e s p e c t t o the l a b o r a t o r y f r a m e . In a c t u a l f a c t , the d e u t e r o n s p i n - l a t t i c e r e l a x a t i o n i s dominated by the q u a d r u p o l a r i n t e r a c t i o n s , the d i p o l a r i n t e r a c t i o n s b e i n g much weaker (deWit and Bloom, 1965). The d e u t e r o n s p i n - l a t t i c e r e l a x a t i o n g i v e s i n f o r m a t i o n about the c o r r e l a t i o n f u n c t i o n of • The d i p o l a r i n t e r a c t i o n s are b o t h i n t r a - and i n t e r -m o l e c u l a r i n o r i g i n . The i n t r a - m o l e c u l a r d i p o l a r i n t e r a c t i o n s average t o z e r o under r o t a t i o n s of the m o l e c u l e . The c o r r e l a t i o n t imes f o r t h e s e r e o r i e n t a t i o n s are s u f f i c i e n t l y s h o r t a t a l l t h e e x p e r i m e n t a l t e m p e r a t u r e s so t h a t the assumptions (4-7a) a r e s a t i s f i e d and and Tg can be e v a l u a t e d u s i n g e q u a t i o n s (4-6). The i n t e r - m o l e c u l a r d i p o l a r i n t e r a c t i o n s a r e more com-p l i c a t e d because the i n t e r a c t i o n s t r e n g t h between the m o l e c u l e s i s det e r m i n e d by the r e l a t i v e s p a t i a l l o c a t i o n s of the m o l e c u l e s as w e l l as the o r i e n t a t i o n s . Because of t h i s f a c t , t h ey do not average t o z e r o under r o t a t i o n s a l o n e . Con s e q u e n t l y , the s p e c t r a l d e n s i t y i s i n g e n e r a l a f u n c t i o n o f two c o r r e l a t i o n t i m e s , one a s s o c i a t e d w i t h m o l e c u l a r r e o r i e n t a t i o n and t h e o t h e r w i t h t r a n s l a t i o n a l m otion. I f i t i s assumed t h a t t h e c o r r e l a t i o n time f o r r e o r i e n t a t i o n s i s much s h o r t e r than the c o r r e l a t i o n time f o r t r a n s l a t i o n a l m o t i ons, then H. , can be w r i t t e n as (Abragam, p453)> i n t e r — H K - r - U t where ( ^ r e p r e s e n t s an average over a l l o r i e n t a t i o n s si.x and si_^ of each o f the m o l e c u l e s . i s time dependent p r i m a r i l y t h r o u g h r e o r i e n t a t i o n s of t h e m o l e c u l e and depends o n l y on t h e r e l a t i v e p o s i t i o n s of t h e m o l e c u l e s so t h a t i t s time dependence i s due t o t r a n s l a t i o n a l motions o n l y . Now t h e c o r r e l a t i o n f u n c t i o n f o r ^ i s The l a s t two terms average t o z e r o , i f the t r a n s l a t i o n a l and the r o t a t i o n a l motions of the m o l e c u l e s a r e u n c o r r e l a t e d . 59 Thus, t h e s p e c t r a l d e n s i t y of H. , i s e a s i l y seen t o be the sum of two t e r m s — t h e s p e c t r a l d e n s i t y of H r i n v o l v i n g the r o t a t i o n a l c o r r e l a t i o n t i m e , and the s p e c t r a l d e n s i t y of rL_ i n v o l v i n g the c o r r e l a t i o n time f o r t r a n s l a t i o n a l d i f f u s i o n , 1 , e # J i n t e r ^ ) = J r ^ ) + J t ^  ) • I f the c o r r e l a t i o n t i m e s a r e s u f f i c i e n t l y s h o r t t o s a t i s f y the a ssumptions (4-7a). t h e n and Tg can be c a l c u l a t e d u s i n g e q u a t i o n s (4-6). When t h e c o r r e l a t i o n time f o r d i f f u s i o n i s l o n g , t he s p e c t r a l d e n s i t y J ( c0o ) i s s m a l l and the c o n t r i b u t i o n t o T^ i s n e g l i g i b l e . T h i s i s t h e case below 65°K, where s e l f -d i f f u s i o n i s v i r t u a l l y n o n - e x i s t e n t , i . e . a t r u l y r i g i d l a t t i c e . The s p i n - l a t t i c e t i me measurements i n m i x t u r e s of CH^ and CD^ (deWit and Bloom, 1965),, and e x p e r i m e n t a l r e s u l t s t o be d i s c u s s e d i n Chapter 5> showed c l e a r l y t h a t t h o s e i n t e r -a c t i o n s c a u s i n g r e l a x a t i o n below 65°K a r e i n t r a - m o l e c u l a r . In o t h e r words, t h e c o n t r i b u t i o n s of H and H, t o T, a r e r t 1 b o t h n e g l i g i b l e . T h i s r e s u l t i s i n agreement w i t h some t h e o r e t i c a l p r e d i c t i o n s by Hubbard (1963)5 r e g a r d i n g the i n f l u e n c e of i n t e r m o l e c u l a r i n t e r a c t i o n s on s p i n - l a t t i c e r e l a x a t i o n . What e f f e c t do t h e s e i n t e r a c t i o n s have on the s p i n - s p i n r e l a x a t i o n ? The c o n t r i b u t i o n t o T g due t o H^ i s s m a l l , s i n c e i t i s e q u a l t o the c o n t r i b u t i o n t o T-^  due t o H . Here, use i s made of the f a c t t h a t = T 2 when co0 -cc « / . The c o n t r i b u t i o n due t o H^ below 65°K i s much l a r g e r t h a n t h a t due t o H r. I t can not be c a l c u l a t e d u s i n g e q u a t i o n (4-6) because the assumptions (4-7a) a r e v i o l a t e d , t h e c o r r e l a t i o n GO-t ime f o r d i f f u s i o n X.Q b e i n g t o o l o n g , i . e . cot>vo»/ . Van V l e c k ' s moment a n a l y s e s has t o be used and i t w i l l be d i s c u s s e d i n s e c t i o n 4 : 3 . F o r u D 0 t 0 ^ > / , the c o n t r i b u t i o n of Hj. t o the l i n e shape i s temperature i n d e p e n d e n t . Thus , the e n t i r e dependence of the l i n e shape on t c i s due t o the s m a l l c o n t r i b u t i o n H . T h i s e x p l a i n s why our measure-ments of s p i n - l a t t i c e r e l a x a t i o n are more s e n s i t i v e t o changes i n t c than W o l f ' s (1963) measurements of s p i n - s p i n r e l a x a t i o n . In summary, o n l y the i n t r a - m o l e c u l a r i n t e r a c t i o n s cause a p p r e c i a b l e s p i n - l a t t i c e r e l a x a t i o n . One can thus t h i n k of 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 i n methane as a t w o - s t e p p r o c e s s , as shown i n F i g u r e ( 12) : f i r s t s t e p , the e l e c t r i c i n t e r a c t i o n s between a m o l e c u l e and i t s n e i g h b o u r s cause i t t o r e - o r i e n t ; second s t e p , the t ime dependence of i n t r a -m o l e c u l a r i n t e r a c t i o n s due t o m o l e c u l a r r e o r i e n t a t i o n s causes the s p i n s t o r e l a x by exchanging energy w i t h the m o l e c u l a r degrees of f r e e d o m . B o t h the i n t r a m o l e c u l a r d i p o l a r and the q u a d r u p o l a r i n t e r a c t i o n s i n v o l v e second order s p h e r i c a l h a r m o n i c s . C o n s e q u e n t l y , these i n t e r a c t i o n s are d e s c r i b e d by the same c o r r e l a t i o n f u n c t i o n s : these c o r r e l a t i o n f u n c t i o n s w i l l be d i s c u s s e d i n the next s e c t i o n . C l a s s i c a l Theory of M o l e c u l a r R e o r i e n t a t i o n s 4:2 In g e n e r a l , one can d e f i n e the r e d u c e d c o r r e l a t i o n f u n c t i o n g^ (% ) f o r the s p h e r i c a l harmonics of o r d e r Ji i n terms of the c o r r e l a t i o n f u n c t i o n G . ( T ) Spin System Rotational degrees of freedom Lattice Phonons F i g u r e 12. I l l u s t r a t i o n of the Idea t h a t R e l a x a t i o n i s a Two  Step Process o^  62 9& ( 4 - 9 ) Ivanov (1964) has shown t h a t f o r a g e n e r a l r o t a t i o n a l random walk w i t h no r e s t r i c t i o n s on the s i z e o f the i n d i v i d u a l a n g u l a r s t e p s , an e x p o n e n t i a l c o r r e l a t i o n f u n c t i o n i s o b t a i n e d . The r o t a t i o n a l d i f f u s i o n e q u a t i o n s of the Debye model (BPP, 1948) i s o b t a i n e d i n the l i m i t of s m a l l a n g u l a r s t e p s . The Debye model i s w i d e l y used by e x p e r i m e n t a l i s t s i n the f i e l d of NMR t o i n t e r p r e t t h e i r e x p e r i m e n t a l d a t a , a l t h o u g h t h e r e e x i s t s some doubt as t o the v a l i d i t y of the r o t a t i o n a l d i f f u s i o n e q u a t i o n when a p p l i e d t o m o l e c u l e s i n s i t u a t i o n s such as are e n c o u n t e r e d i n s o l i d methane. The e x i s t e n c e of an e x p o n e n t i a l c o r r e l a t i o n f u n c t i o n e n a b l e s one t o c a l c u l a t e some s o r t of c h a r a c t e r i s t i c t ime t c d i r e c t l y f r o m the e x p e r i -mental measurements of T ^ . Hubbard ( 1 9 6 1 ) , u s i n g the r o t a t i o n a l d i f f u s i o n equa-t i o n , c a l c u l a t e d the i n f l u e n c e of c r o s s - c o r r e l a t i o n s on the s p i n - l a t t i c e r e l a x a t i o n of C H ^ . He found t h a t the s p i n l a t t i c e r e l a x a t i o n was governed by t h r e e e x p o n e n t i a l s f o r a l l v a l u e s of t o t > b u t t h a t one e x p o n e n t i a l was dominant , o c and t h a t w i t h i n the p r e s e n t l y a v a i l a b l e e x p e r i m e n t a l t e c h -n i q u e s one s h o u l d observe o n l y one t ime c o n s t a n t e q u a l t o t h a t c a l c u l a t e d , n e g l e c t i n g c r o s s c o r r e l a t i o n s . The s p e c t r a l d e n s i t i e s f o r e q n . ( 4 - 6 ) a r e of the f o r m ( 4 - 1 0 ) J % ? ) : fU- = t i l t * 6 3 F o r the i n t r a m o l e c u l a r d i p o l a r i n t e r a c t i o n we o b t a i n u s i n g e q n . ( 4 - 6 ) ( 4 - 1 1 ) F i g u r e ( 1 5 ) i s a p l o t of T-^  v e r s u s a c c o r d i n g t o e q n . ( 4 - 1 1 ) . The minimum v a l u e of T-^ o c c u r s a t co0 Vc = 0 . 6 2 and the e x p r e s s i o n f o r ( T n ) . i s ^ v l ' m i n 1 min ^AE2^ = 8ms. f o r C H ^ f o r co0 = 2rt«z8.S + /oi where co0 i s Larmor f r e q u e n c y of the n u c l e a r s p i n s and ^ H * ) i s the mean square average of the magnetic f i e l d at the s i t e of a n u c l e a r s p i n due t o the d i p o l a r i n t e r a c t i o n s w i t h the o ther s p i n s on the same m o l e c u l e , i . e . the e f f e c t i v e s t r e n g t h of the i n t e r a c t i o n g i v i n g r i s e t o r e l a x a t i o n . S i m i l a r l y , the e x p r e s s i o n f o r T-^ due t o q u a d r u p o l a r r e l a x a t i o n a l s o i n v o l v e s the c o r r e l a t i o n f u n c t i o n s of the s p h e r i c a l harmonics Y 2 m L ( x i ) . F o r s p i n s 1 = 1 and a x i a l l y symmetric f i e l d g r a d i e n t s , such as i s the case f o r C D ^ , the e x p r e s s i o n f o r T1 i s : 6>o 4 - ( 4 - 1 2 ) An approximate v a l u e f o r (T-. ) i s o b t a i n e d u s i n g C . H . Anders 1 min on m o l e c u l a r beam measurement of the q u a d r u p o l e c o u p l i n g c o n s t a n t . and ( l i . l ) - (6.0) -L..\ — / ? O ~*~ v ' /w\.CIU s>ec. 64 The i m p o r t a n t f e a t u r e t o be noted i s t h a t i f the c o r r e l a t i o n f u n c t i o n s are e x p o n e n t i a l , then the v a l u e of a t the minimum d e t e r m i n e s the s t r e n g t h of the i n t e r a c t i o n c a u s i n g the r e l a x a t i o n of the s p i n sys tem; no a d j u s t a b l e parameters are i n v o l v e d . Y e t , the v a l u e c a l c u l a t e d f o r CH^ i s a f a c t o r of 20 s m a l l e r t h a n the e x p e r i m e n t a l v a l u e ; f o r C D ^ , the e x p e r i -mental v a l u e of 8.5ms. agrees r e a s o n a b l y w e l l w i t h the v a l u e c a l c u l a t e d above. T h i s d i s c r e p a n c y w i l l occupy our a t t e n t i o n f o r the remainder of t h i s t h e s i s ; we w i l l d i s c u s s the s h o r t c o m i n g s of the c o n v e n t i o n a l t h e o r y i n d e a l i n g w i t h a Quantum M e c h a n i -c a l system l i k e CH^ at low t e m p e r a t u r e s . I f one i n c l u d e d any o t h e r i n t e r a c t i o n s , l i k e the s p i n r o t a t i o n i n t e r a c t i o n s , the t h e o r e t i c a l T^ would become even s h o r t e r . The s p i n r o t a t i o n i n t e r a c t i o n i s u s u a l l y assumed to be n o n - e x i s t e n t i n the s o l i d and the dense l i q u i d s t a t e s . T h i s i s a consequence of the c r y s t a l l i n e e l e c t r i c f i e l d s p r e s e n t a t the m o l e c u l a r s i t e s . These are s a i d t o quench the m o l e c u l a r a n g u l a r momentum i . e . < ^ J " ^ > = 0 j u s t as i s e n c o u n t e r e d i n c r y s t a l f i e l d t h e o r y of paramagnet ic r e s o n a n c e . I t might be worth m e n t i o n i n g t h a t the s p i n - r o t a t i o n would be s t r i c t l y a d d i t i v e t o the d i p o l a r or q u a d r u p o l a r c o n t r i b u t i o n s , because any c r o s s terms would be z e r o , as they have d i f f e r e n t t r a n s f o r m a t i o n p r o p e r t i e s . As y e t , a t h e o r y p r e d i c t i n g the c o r r e l a t i o n t ime of m o l e c u l a r r e o r i e n t a t i o n s i n terms of the i n t e r m o l e c u l a r f o r c e s , the c o l l e c t i v e motions i n the s o l i d and the wave f u n c t i o n s of the m o l e c u l e i n the s o l i d , i s n o n - e x i s t a n t . Below the phase t r a n s i t i o n s p r o b a b l y , the o n l y c o r r e c t p r o -cedure would be t o d e r i v e the normal modes of the many body p r o b l e m ; as a A - s i n g u l a r i t y i n the s p e c i f i c heat u s u a l l y s i g n i f i e s a c o - o p e r a t i v e t r a n s i t i o n . The c r y s t a l l i n e e l e c t r i c f i e l d , w h i c h a r i s e s f rom a n i s o t r o p i c i n t e r m o l e c u l a r f o r c e s , i s sometimes used t o c a l c u l a t e s i n g l e m o l e c u l e wave f u n c t i o n s and energy l e v e l s i n s o l i d s . M o d u l a t i o n of the a n i s o t r o p i c i n t e r a c t i o n s by phonons can cause t r a n s i t i o n s between these s t a t e s . These i n t e r m o l e c u l a r f o r c e s a r e a l l due t o e l e c t r i c i n t e r a c t i o n s between the charge d i s t r i b u t i o n s of the v a r i o u s m o l e c u l e s . Examples are the a n i s o t r o p i c i n t e r a c t i o n s between the o c t u p o l e moments of the charge d i s t r i b u t i o n , and the Van der Waals d i s p e r s i o n f o r c e s , w h i c h c o n t r i b u t e b o t h t o the i s o t r o p i c and a n i s o t r o p i c i n t e r -m o l e c u l a r f o r c e s . L i n e Shape 4:3 As was mentioned at the end of s e c t i o n 4:1, the l i n e shape i s dominated by the t i m e - a v e r a g e d i n t e r m o l e c u l a r d i p o l a r i n t e r a c t i o n s , ^ H - j ^ t e r / * ' T h e P r o b l e m ° ^ c a l c u l a t i n g the l i n e shape due t o these i n t e r a c t i o n s i s the same as the u s u a l l i n e shape p r o b l e m i n w h i c h magnetic d i p o l e s a re s i t u a t e d i n the l a t t i c e s i t e s of a r i g i d l a t t i c e . Van V l e c k (Abragam pl08) has d e r i v e d g e n e r a l e x p r e s s i o n s f o r the moments of the l i n e shape f o r t h i s t y p e of p r o b l e m . I t i s g e n e r a l l y found 66 t h a t the r i g i d l a t t i c e a b s o r p t i o n l i n e shape f o r a n o n - m e t a l resembles a G a u s s i a n l i n e , but the v a l u e s of the f o u r t h and h i g h e r moments i n d i c a t e t h a t i t i s s q u a r e r than a G a u s s i a n . The e x p e r i m e n t a l l y observed i n d u c t i o n t a i l i s shown i n F i g u r e (13). A r i g o r o u s theorem can be d e r i v e d showing t h a t the t ime e v o l u t i o n of the f r e e i n d u c t i o n t a i l i s d e s c r i b e d by the F o u r i e r t r a n s f o r m of the a b s o r p t i o n l i n e shape . F o r t h i s theorem t o be v a l i d , the r . f . magnetic f i e l d d u r i n g the p u l s e has t o be much l a r g e r than the l o c a l f i e l d . T h i s c o n -d i t i o n i s s a t i s f i e d i f the w i d t h of the TT/2 p u l s e i s much l e s s than T 2 . I t f o l l o w s f r o m the above theorem t h a t the i n d u c t i o n t a i l can be e x p r e s s e d i n the f o l l o w i n g f o r m , c ( t ) = T. \ IT <4-13> where M n i s the n ^ h moment of the a b s o r p t i o n l i n e shape M n As the e x p r e s s i o n (4-13) converges r a t h e r s l o w l y , the s e m i -e m p i r i c a l a n a l y t i c e x p r e s s i o n P ( t ) = exp { Z ^ ~ ) T has been employed i n s e v e r a l i n s t a n c e s and found t o p r o v i d e a r e m a r k a b l e f i t of the d a t a . Once the parameters ' a ' and ' b ' i n F ( t ) have been o b t a i n e d by f i t t i n g i t t o the e x p e r i -mental i n d u c t i o n t a i l , the moments of the l i n e are g i v e n by 67 F i g u r e 13. Photographs Showing a T y p i c a l I n d u c t i o n T a i l f o r CH^ at 1.2"K (1) sweep 5 / ^sec / cm. (2) sweep 1 0 / * s e c / c m . (3) sweep 10 / ^sec /cm; i n c r e a s e d g a i n by f a c t o r of 15. (4) sweep 2 0 - A s e c / c m ; same g a i n as (3) (5) sweep 20 / ^sec /cm; i n c r e a s e d g a i n by f a c t o r of 6. 68 2 3 M i . = 3o? + za^L^ + 4 . h+ M / - = t5'*' v - fS-cfb2- + 3**- ti* +J. k6 6 7 One method f o r e v a l u a t i n g ' a ' and ' b ' f r o m the e x p e r i -mental data i s as f o l l o w s . As i s e a s i l y v e r i f i e d f r o m e q u a t i o n (4 - l4 ) , the i n d u c t i o n t a i l i s z e r o whenever b t = TTn where n i s an i n t e g e r . T h u s , by measur ing the t imes t , where the z e r o s occur i n the i n d u c t i o n t a i l , one can e v a l u a t e ' b ' . The maxima between the z e r o s occur at t imes t^ g i v e n by di -t — o U s i n g t h i s c o n d i t i o n , the f o l l o w i n g r e l a t i o n s h i p i s o b t a i n e d . ,2 a" — _b_ c o t L _ C a u t i o n s h o u l d be e x e r c i s e d t o check t h a t the e x p r e s s i o n (4-l4) u s i n g the e x p e r i m e n t a l l y e v a l u a t e d parameters ' a ' and ' b ' does i n d e e d f i t the complete e x p e r i m e n t a l i n d u c t i o n t a i l . In the next c h a p t e r , we w i l l r e v i e w the e x p e r i m e n t a l r e s u l t s and p o i n t out the d i s c r e p a n c i e s i n view of the c o n v e n -t i o n a l t h e o r y . In Chapter 6, we w i l l propose a model and d i s c u s s p o s s i b l e e x t e n s i o n s of the c o n v e n t i o n a l t h e o r y t o cope w i t h these d i s c r e p a n c i e s . CHAPTER 5. THE EXPERIMENTAL RESULTS The r e s u l t s of the c o n v e n t i o n a l t h e o r y quoted i n the p r e v i o u s c h a p t e r w i l l now be used t o a n a l y z e the e x p e r i m e n t a l r e s u l t s . There i t was observed t h a 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 ime depends on the s t r e n g t h of the s p i n -l a t t i c e i n t e r a c t i o n , and the c o r r e l a t i o n t ime d e s c r i b i n g the random c h a r a c t e r of the i n t e r a c t i o n s . Thus , under a p p r o p -r i a t e a s s u m p t i o n s , the temperature dependence of the c o r r e l a t i o n t ime can be e x t r a c t e d f rom the d a t a . The next and r a t h e r d i f f i c u l t t a s k i s the e x p l a n a t i o n of the tempera-t u r e dependence of the c o r r e l a t i o n t i m e . A model w h i c h p a r t i a l l y e x p l a i n s the temperature dependence and o t h e r f e a t u r e s of the data w i l l be p r e s e n t e d i n the next c h a p t e r . Some of the d i s c r e p a n c i e s between the e x p e r i m e n t a l r e s u l t s and the p r e d i c t i o n s f r o m c o n v e n t i o n a l t h e o r y , w h i c h w i l l become a p p a r e n t i n t h i s c h a p t e r , w i l l a l s o be d i s c u s s e d t h e r e . C e r t a i n s i m i l a r i t i e s between the v a r i o u s s e t s of data a re e v i d e n t , and w i l l now be summarized to a i d i n the d i s c u s s i o n of the i n d i v i d u a l s e t s of d a t a . The s p i n -l a t t i c e r e l a x a t i o n t ime v a r i e s s l o w l y w i t h temperature above 30°K. T h i s slow temperature dependence i s i n a g r e e -ment w i t h the e a r l i e r e x p e r i m e n t a l r e s u l t s between 5 5 - H O ° K (de Wit and Bloom, 1965). In c o n t r a s t , the temperature dependence below the upper phase t r a n s i t i o n temperature i s 69 70 v e r y s t r o n g . The s p i n - l a t t i c e r e l a x a t i o n t ime u s u a l l y d e c r e a s e s by two or t h r e e o r d e r s of magnitude i n t h i s r e g i o n . A l l but one of the c u r v e s of T^ v e r s u s T go t h r o u g h a minimum. The i n c r e a s e i n the s p i n - l a t t i c e r e l a x -a t i o n t ime on the low temperature s i d e of the minimum i s u s u a l l y l e s s than a f a c t o r of 1 0 ; except i n t h r e e cases where t h i s i n c r e a s e i s r a t h e r more s p e c t a c u l a r . At the lowest o b t a i n a b l e tempera tures ( l . 2 ° K ) , the temperature dependence of T^ g e n e r a l l y becomes much weaker . The tempera tures a t w h i c h the A - a n o m a l i e s i n the s p e c i f i c heat occur are e v i d e n t i n the s p i n - l a t t i c e r e l a x a t i o n t ime data i n many c a s e s , e i t h e r as changes i n the s l o p e or as d i s -c o n t i n u i t i e s i n the T^ v e r s u s T c u r v e . Now, the e x p e r i m e n t a l r e s u l t s w i l l be p r e s e n t e d . C o u p l e d S p i n Systems: 5=1 A d i s c u s s i o n of CD^H and CH^D s h o u l d be p r e f a c e d by a d i s c u s s i o n of the n a t u r e of the r e l a x a t i o n when a n u c l e u s , b e s i d e s i n t e r a c t i n g w i t h i d e n t i c a l n e i g h b o u r i n g n u c l e i h a v i n g s p i n s I , a l s o i n t e r a c t s w i t h n e i g h b o u r i n g u n l i k e n u c l e i h a v i n g s p i n s S. Because the s p l i t t i n g of the n u c l e a r Zeeman l e v e l s i s q u i t e d i f f e r e n t f o r the two t y p e s of n u c l e i , the e x p r e s s i o n s f o r the r e l a x a t i o n of one type of n u c l e u s as a r e s u l t of i n t e r a c t i o n s w i t h the o t h e r type of n u c l e u s are q u i t e d i f f e r e n t f r o m the e x p r e s s i o n s f o r i n t e r a c t i o n s between n u c l e i of the same t y p e . As a mat ter of f a c t , the t ime dependence of tne z-component of m a g n e t i z a t i o n of the s p i n system S, S , w i l l be a f u n c t i o n of I - I as w e l l a s of ^ ° ' z z o S -S , where I and are the e q u i l i b r i u m m a g n e t i z a t i o n s . The t ime dependence i s d e s c r i b e d by a se t of c o u p l e d d i f f e r e n t i a l e q u a t i o n s , as g i v e n by Abragam p295. dx - A x - A ' y dt dt - B ' x - B y (5-1) where x = 1 - 1 z o y = s z - s o where the A ' s and the B ' s a re r e l a x a t i o n r a t e s and w i l l be e x p l i c i t l y w r i t t e n i n the next p a r a g r a p h . The s o l u t i o n s of these e q u a t i o n s are x = y = _ c L & . 4 - f l ) ( ? u + a) r _ e + x ^ l (5-2) where \ , X 2 = -Cft + B) ±.[(ft-S^ +• * fl' 2 The s o l u t i o n s of e q u a t i o n s (5-1) were o b t a i n e d under the i n i t i a l c o n d i t i o n s t h a t a t t = 0, x = a , and y = 0. In o t h e r words , t h a t the s p i n system I has been d i s t u r b e d f r o m t h e r m a l e q u i l i b r i u m I / I , by an r . f . p u l s e f o r example; but the s p i n system S has not been d i s t u r b e d and t h u s the z-component of m a g n e t i z a t i o n i s s t i l l S 2 = S Q. T h i s se t of i n i t i a l c o n d i t i o n s a p p l i e s t o a l l the exper iments r e p o r t e d 7 2 h e r e ; r . f . p u l s e s are a p p l i e d o n l y a t the Larmor f r e q u e n c y of one of the s p i n sys tems . The above e x p r e s s i o n s w i l l be used t o determine t o what e x t e n t c r o s s r e l a x a t i o n e f f e c t s i n f l u e n c e the r e l a x a t i o n of a p a r t i c u l a r s p i n s y s t e m . Now the e x p r e s s i o n f o r the A ' s and B ' s w i l l be g i v e n . The e x p r e s s i o n f o r A and B i n v o l v e two c o n t r i b u t i o n s ; t h a t due t o r e l a x a t i o n by l i k e s p i n s w h i c h i s g i v e n by e q u a t i o n s ( 4 - 6 ) and ( 4 - 7 ) and t h a t due t o c o u p l i n g between u n l i k e s p i n s . The c o n t r i b u t i o n s due t o l i k e s p i n s w i l l be denoted by l/T^ and l / T 1 f o r the I and S s p i n systems, r e s p e c t i v e l y . A = \ , J A ' = ^ Y s ' ^ C t + O ^ - ^ J ^ ^ ) ^ ^ - ^ ) ! (5-3) B = ss where B 1 and l / T ^ are the c o r r e s p o n d i n g e x p r e s s i o n s g i v e n above w i t h the I and S i n t e r c h a n g e d . CH^D: 5:2 The p r o t o n r e l a x a t i o n data f o r CH^D a t 2 8 . 5 m c s . , shown i n F i g u r e ( l 4 ) , e x h i b i t the g e n e r a l f e a t u r e s o u t l i n e d above . The r e a s o n f o r d i s c u s s i n g CH^D f i r s t i s t h a t the b r e a k s i n T , , o c c u r r i n g at b o t h phase t r a n s i t i o n tempera tures i n CH^D 73 are the most pronounced of those f o r a l l the systems s t u d i e d . I t has been p r e v i o u s l y i n f e r r e d f r o m e n t r o p y measurements ( C o l l w e l l , G i l l and M o r r i s o n , 1965) t h a t these phase t r a n s i -t i o n s have t o do w i t h o r d e r i n g of the m o l e c u l a r o r i e n t a t i o n s . The r a p i d changes of near the phase t r a n s i t i o n s and the changes i n the temperature dependences of p r o v i d e , p e r -h a p s , the most d i r e c t c o n f i r m a t i o n of t h i s i n t e r p r e t a t i o n of the e n t r o p y d a t a . T h i s i s so because i t i s known t h a t below about 65°K, T 1 i s governed by i n t r a - m o l e c u l a r i n t e r a c t i o n s and i s t h e r e f o r e s e n s i t i v e o n l y t o the s t r e n g t h s o f these i n t e r a c t i o n s and the r a t e of m o l e c u l a r r e o r i e n t a t i o n . In t h i s c o n n e c t i o n note t h a t the h i g h temperature r e s u l t s between 25 and 55°K j o i n up w i t h the r e s u l t s o b t a i n e d by Bloom and Sandhu (1962) between 55 and 110 G K. The slow temperature dependence between 55 and 110°K, i m p l i e s t h a t the c o r r e l a t i o n t ime i s a l s o a s l o w l y v a r y i n g f u n c t i o n of temperature i n t h i s r a n g e . E a r l i e r i t was noted t h a t i f the r e l a x a t i o n i s domin-a t e d by one mechanism, and i f the c o r r e l a t i o n f u n c t i o n i s e x p o n e n t i a l , t h e n the e x p r e s s i o n f o r the minimum v a l u e i n the s p i n - l a t t i c e r e l a x a t i o n t ime i n v o l v e s no a d j u s t a b l e p a r a -m e t e r s . I t o n l y c o n t a i n s parameters s p e c i f y i n g the s t r e n g t h of the i n t e r a c t i o n and the Larmor f r e q u e n c y of the n u c l e i i n the a p p l i e d magnet ic f i e l d . F o r CH^D the dominant r e l a x a t i o n mechanism i s b e l i e v e d t o be the i n t r a m o l e c u l a r d i p o l a r i n t e r -a c t i o n s ; t h i s c o n j e c t u r e i s s u p p o r t e d by the p r e s e n t and e a r l i e r exper iments by us (see s e c t i o n 4 : 3 ) . 74 75 Now u s i n g the r e s u l t s of the p r e v i o u s s e c t i o n , the e f f e c t of the c o u p l i n g between the u n l i k e s p i n s can be t a k e n i n t o a c c o u n t . I t may be shown u s i n g e q u a t i o n s (4-6) and / x ^ l 1 , V * ! 1 1 V T , 1 1 n ft-(5-3) t h a t ^ ^ 42, j « 5, and 1 « ° - 8 > l / T ^ 5 l / T ^ 1 t h e s e r e s u l t s were c a l c u l a t e d u s i n g the a p p r o x i m a t i o n t h a t f o r e x p o n e n t i a l c o r r e l a t i o n f u n c t i o n s the s p e c t r a l d e n s i t y J ( u ) j + u ) s ) » 5 j(to^) as 0 0 j = 28.5 mcs. and <-Og = 4.4mcs. T h u s , i f the t ime c o n s t a n t s are e v a l u a t e d and s u b s t i t u t e d i n the e q u a t i o n s (5-2), the c o n c l u s i o n i s t h a t the r e l a x a t i o n i s governed by one e x p o n e n t i a l w i t h t ime c o n s t a n t l / T ^ t o w i t h i n e x p e r i m e n t a l e r r o r . In o t h e r words , f o r a l l p r a c t i c a l p u r p o s e s the r e l a x a t i o n of p r o t o n system, and thus ( T i ) m j_ n > i s governed by the i n t e r a c t i o n s w i t h the o t h e r p r o t o n s i n the m o l e c u l e and i s g i v e n by e q u a t i o n (4-6) . Prom t h i s e q u a t i o n i t can be seen t h a t the o n l y d i f f e r e n c e between the c a l c u l a t i o n f o r ( T 1 ) m l n f o r CH^D and t h a t f o r CH^ i s t h a t the i n t e r a c t i o n s now i n v o l v e two n e i g h b o u r i n g n u c l e i r a t h e r than t h r e e as f o r C H ^ . Thus one f i n d s u s i n g e q u a t i o n (4-6), t h a t ( T - L ) m l n = 12 m i l l i s e c . f o r CH^D as compared w i t h 8ms f o r C H ^ . As i n the case of CH^ t o be d i s c u s s e d l a t e r , t h e r e i s a d i s c r e p a n c y o f a f a c t o r of 18 between the e x p e r i m e n t a l and t h e o r e t i c a l v a l u e s of C ^ ) l n . I f any other" c o n t r i b u t i o n s t o T ^ , due t o o t h e r r e l a x a t i o n mechanisms, a re i n c l u d e d , then the d i s c r e p a n c y would be even g r e a t e r . D e s p i t e the f a c t t h a t the p r e d i c t i o n s of ( T 1 ) m l n by the c o n v e n t i o n a l t h e o r y appear t o be wrong, the 76 r e l a t i o n s h i p between T n , TJ and (T n ) . g i v e n by the c o n -v e n t i o n a l t h e o r y w i l l be used i n the next s e c t i o n t o o b t a i n the c o r r e l a t i o n t i m e s . The A n a l y s i s of the D a t a : 5:3 In t h i s s e c t i o n a method of a n a l y s i s w i l l be d e s c r i b e d , w h i c h i s based on the d i s c u s s i o n i n Chapter 4. We r e c a l l the e x p r e s s i o n s (4-11) and (4-12) f o r the s p i n - l a t t i c e r e l a x a t i o n t ime due t o d i p o l a r and q u a d r u p o l a r i n t e r a c t i o n s , r e s p e c t i v e l y . These e q u a t i o n s can be r e w r i t t e n i n a f o r m i n v o l v i n g o n l y t r % , co„ and ( T . ^ min TT~ = - r r y (5-4) where y i s d e f i n e d as y = — ~ + — — (5-5) where x = oja xc The dependence of y on x i s shown i n F i g u r e (15). I t must be emphasized here t h a t owing to the p r e s e n c e of a maximum i n y v e r s u s x a t x = 0.62 and the r e s u l t i n g minimum i n T-^, d e f i n i t e v a l u e s can be a s s i g n e d t o % at any temperature u s i n g the v a l u e of X = 0.62/, 3 a t the temperature a t w h i c h the c ° minimum i n T^ o c c u r s . min 77 At a p a r t i c u l a r temperature *C can be o b t a i n e d f r o m the v a l u e of y . The v a l u e of y i s g i v e n by the r a t i o of T-^ a t the p a r t i -c u l a r temperature t o ( T - ^ ) m ^ n H a v i n g e v a l u a t e d y , two v a l u e s of x a re o b t a i n e d f r o m the graph of y v e r s u s x . One of these two v a l u e s i s e l i m i n a t e d u s i n g the c r i t e r i a t h a t x < 0.62 c o r r e s p o n d s t o the h i g h temp-e r a t u r e s i d e of the minimum,and x > 0.62 t o the low temperature s i d e . One i s f a c e d by the d i l e m n a t h a t the t h e o r e t i c a l and e x p e r i m e n t a l v a l u e s of ( T 1 ) m l n d i s a g r e e d r a s t i c a l l y . One p o s s i b l e e x p l a n a t i o n , w h i c h w i l l be j u s t i f i e d i n the next c h a p t e r , i s t h a t a t low tempera tures or f o r s m a l l J quantum numbers, the e f f e c t i v e i n t e r a c t i o n s t r e n g t h c a u s i n g the r e l a x a t i o n can not be c o r r e c t l y d e s c r i b e d c l a s s i c a l l y and has been r e d u c e d i n s t r e n g t h . I f the c o r r e l a t i o n f u n c t i o n i s s t i l l e x p o n e n t i a l and i f the s t r e n g t h of the i n t e r a c t i o n does  not change a p p r e c i a b l y i n the temperature range of i n t e r e s t , then the g e n e r a l f o r m of T , as a f u n c t i o n of X as g i v e n by e q u a t i o n (5-4) i s s t i l l c o r r e c t . I t i s t h i s g e n e r a l f o r m of the dependence of on X w h i c h i s of importance i n t h i s a n a l y s i s . The o n l y n e c e s s a r y m o d i f i c a t i o n t o e q u a t i o n (5-6) i s the r e p l a c e m e n t of the t h e o r e t i c a l v a l u e of ( T 1 ) m i n by the e x p e r i m e n t a l v a l u e , i n o r d e r t o c o r r e c t f o r the change i n the e f f e c t i v e s t r e n g t h of the i n t e r a c t i o n . The above two assump-t i o n s w i l l be examined i n d e t a i l i n the next c h a p t e r , where i t w i l l be seen t h a t they a r e not always v a l i d . There are two a d d i t i o n a l assumptions i m p l i c i t i n t h i s a n a l y s i s . One of the assumptions i s the n e g l e c t of e f f e c t s due t o c r o s s - c o r r e l a t i o n s between p a i r s of s p i n s on the m o l e c u l e . As was d i s c u s s e d e a r l i e r , Hubbard has shown t h i s assumption t o be c o r r e c t i n the range of v a l i d i t y of the c l a s s i c a l d i f f u s i o n e q u a t i o n , but i t s v a l i d i t y at low temperature i s d e b a t a b l e . The o t h e r assumption i s t h a t the r e l a x a t i o n of the s p i n system i s dominated by one mechanism. T h i s l a s t a s s u m p t i o n i s c o n s i s -t e n t w i t h the i n t e r p r e t a t i o n of the data between 55 and 110 ° K , and i s s u p p o r t e d by the p r e s e n t e x p e r i m e n t a l r e s u l t s ; i n p a r t i c u l a r the data on ^ v e r s u s T f o r m i x t u r e s of CH^ w i t h C D ^ , and the m i x t u r e s of CH^ w i t h Kr ( to be d i s c u s s e d l a t e r ) b e i n g e s p e c i a l l y r e l e v a n t . U s i n g the above o u t l i n e d method of a n a l y s i s , the v a l u e s of % versus T have been been o b t a i n e d f o r C H 0 D and are shown c 5 i n F i g u r e ( l 6 ) ( l o g l / o o / C e has been p l o t t e d v e r s u s l o g T ) . A s u r p r i s i n g f e a t u r e i s the slow temperature dependence a t the low temperature end . M o r e o v e r , i f i t i s assumed t h a t l / t oc T 7 then the r e s u l t i n g g r a p h i s i n d i c a t e d by the dashed l i n e . The b e h a v i o u r of the l o g l / x v e r s u s l o g T p l o t a t the h i g h e r t e m p e r a t u r e s sugges ts t h a t a T " 7 t emperature dependence i s not u n r e a s o n a b l e . T h i s l a t t e r f e a t u r e i s e n c o u n t e r e d i n most of the systems s t u d i e d and w i l l be d i s -c u s s e d i n terms of the c o u p l i n g between the phonons and the m o l e c u l e s i n the next c h a p t e r . The s o l i d c u r v e i s a t h e o r e t i -c a l f i t on the b a s i s of t h i s phonon c o u p l i n g m o d e l . 80 The f i r s t p o s s i b i l i t y t h a t s h o u l d be i n v e s t i g a t e d i s the p o s s i b i l i t y of d e s c r i b i n g the r e o r i e n t a t i o n a l c o r r e l a t i o n t ime by an a c t i v a t i o n energy as i t has o f t e n been found t o be the case (Andrew and E a d e s , 1 9 5 3 ) . The temperature dependence of the c o r r e l a t i o n time would then be of the f o r m where k T Q i s an a c t i v a t i o n energy 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 the h e i g h t of the r o t a t i o n a l b a r r i e r between d i f f e r e n t m o l e c u l a r p o s i t i o n s . To check whether the c o r r e l a t i o n time, can be d e s c r i b e d i n t h i s way i n F igure ( 1 7 . ) In toaXc has been p l o t t e d v e r s u s l / T . I t i s c l e a r l y demonstrated by t h i s f i g u r e t f iat f o r no a p p r e c i a b l e range of t e m p e r a t u r e s can the temperature dependence be d e s c r i b e d by an a c t i v a t i o n e n e r g y . E x a c t l y the same c o n c l u s i o n s can be drawn about t v e r s u s T i n each of the f o l l o w i n g s y s t e m s . C D ^ : 5 : 4 In F i g u r e ( 1 8 ) , the e x p e r i m e n t a l v a l u e s f o r the s p i n -l a t t i c e r e l a x a t i o n t ime as a f u n c t i o n of temperature of the d e u t e r o n s p i n system i n CD^ are shown. A d i s t i n c t i v e f e a t u r e of the data i s the r a p i d v a r i a t i o n of T^ on the low tempera-t u r e s i d e of the ( T n ) . . A n o t h e r d i f f e r e n c e between t h i s v l ' m i n se t of data and t h a t of CH^D i s the o c c u r r e n c e of o n l y a s i n g l e b r e a k i n v e r s u s T curve o c c u r r i n g at the upper phase t r a n s i -t i o n t e m p e r a t u r e . The absence of any d i s c o n t i n u i t y at the 81 ^ = . 3 5 7 coth ( - | | - ) + 5 . 8 5 X I O ~ 8 T 7 / I I 1 I I I i ' i I i i i i I 2 3 4 5 10 15 Temperature (°K) F i g u r e 16. P l o t of V a ^ T c V e r s u s T , The T h e o r e t i c a l P i t Obeys the, E q u a t i o n snown i n trie P i g u r e 1 82 10.0 1.0 3° 0.1 0.01 * J 0 T c = 2 .05 exp. ( 0 . 6 3 / T ) 10 w0 T c s 3 .95 x I 0 ~ 2 exp. ( 4 3 3 / T ) 2 0 3 0 4 0 I 0 0 / T ( ° K - ' ) 5 0 60 F i g u r e 17. P l o t of OJ0X^ V e r s u s 1 / T t o check whether M o l e c u l a r R e o r i e n t a t i o n i s Governed by an A c t i v a t i o n Energy 83 lower phase t r a n s i t i o n temperature may be due t o the e x p e r i -mental e r r o r s i n the T 1 measurements t o g e t h e r w i t h the v e r y s t e e p s l o p e of the 1^  v e r s u s T c u r v e a t t h a t p o i n t . The h i g h temperature v a l u e s of j o i n up w i t h the e a r l i e r r e p o r t e d measurement between 55 and 110°K. U s i n g c o n v e n t i o n a l t h e o r y and the q u a d r u p o l e c o u p l i n g c o n s t a n t f o r C D ^ , the minimum v a l u e of T^ i s c a l c u l a t e d t o be 8 m i l l i s e c " ' " ^ * ( s e e s e c " t i o n 4:3)> whereas the e x p e r i m e n t a l v a l u e i s 8.4 m i l l i s e c . Ramsey and Anderson i n a p r i v a t e communication have s t a t e d t h a t the u n c e r t a i n t y of t h e i r q u a d r u p o l e c o u p l i n g c o n s t a n t s h o u l d be even l a r g e r than t h a t quoted by Anderson (1961). A n d e r s o n ' s v a l u e s were used i n c a l c u l a t i n g the above l i m i t s on ( T ] _ ) m l n ' Thus the apparent agreement between the e x p e r i m e n t a l and t h e o r e t i c a l min m a ^ ^ e m i s l e a d i n g . The graph of l o g l / x v e r s u s l o g T o b t a i n e d u s i n g the above p r o c e d u r e i s shown i n F i g u r e .(19). T h i s curve i s not smooth as i s the one f o r C H ^ D . A c c o r d i n g t o some c o n j e c t u r e s the curve s h o u l d be smooth near the h i g h temperature e n d . The bumps i n l n l / x v e r s u s In T may be due t o the n a t u r e of the low l y i n g energy l e v e l s i n C D ^ , : C D o H : 5:5 F o r C D ^ H , b o t h the p r o t o n and d e u t e r o n s p i n - l a t t i c e r e l a x a t i o n t imes have been measured, see F i g u r e s (20) and 84 85 F i g u r e 19. P l o t of l / x V e r s u s T f o r CD^ ( 2 1 ) , The c u r v e s e x h i b i t changes i n s l o p e and b r e a k s i n T-^ a t the two phase t r a n s i t i o n s . The v a l u e of ( T , ) . f o r the d e u t e r o n resonance i s v l ' m i n 4 m i l l i s e c . as compared w i t h 8 m i l l i s e c . f o r C D ^ . I f one , r a t h e r n a i v e l y assumes t h a t the e l e c t r i c f i e l d g r a d i e n t a t the s i t e of d e u t e r o n i n CD^H has the same v a l u e as t h a t i n C D ^ , then the t h e o r e t i c a l v a l u e of ( T - L ) m l n f o r CD^H i s the same as t h a t f o r C D ^ , 8 ms. Now, the q u e s t i o n i s : how l a r g e i s the c o n t r i b u t i o n of the i n t r a m o l e c u l a r d i p o l a r i n t e r a c t i o n t o the r e l a x a t i o n of the d e u t e r o n , and can i t account f o r the a p p a r e n t l y low v a l u e of - ( T ^ ) m j _ n ^ov CD^H? I f a g a i n the a p p r o x i m a t i o n i s used t h a t j(od j + u) g) « J (OJ j ) , then the maximum c o n t r i b u t i o n t o the r e l a x a t i o n r a t e s due t o the c o u p l i n g between u n l i k e s p i n s SS IS c o r r e s p o n d t o t ime c o n s t a n t s T-^ and w h i c h s a t i s f y the r e l a t i o n s T ^ s s 800 m i l l i s e c and V ^ i ^ f & 0 .8 , and l / T S S l A l I S „ i-^Y- £Z 5. I t was noted above t h a t T n , the r e l a x a t i o n 1 / T 1 b l 1 t ime due t o q u a d r u p o l e i n t e r a c t i o n s , was a p p r o x i m a t e l y 8 ms. I t may be seen u s i n g these numbers and e q u a t i o n ( 5 - 2 ) t h a t the r e l a x a t i o n i s dominated by the q u a d r u p o l e i n t e r a c t i o n s . i . e . , T 1 = T 1 = 8 ms. S i m i l a r l y , the p r o t o n r e l a x a t i o n i s not a f f e c t e d by the d e u t e r o n s p i n system, because the c r o s s -IS r e l a x a t i o n t imes , e t c . , are so l o n g , and the d e u t e r o n r e l a x a t i o n t ime i s so s h o r t . In essence the z-component of the d e u t e r o n m a g n e t i z a t i o n S ( t ) remains always a p p r o x i m a t e l y z e q u a l t o S Q when the p r o t o n system i s d i s t u r b e d f r o m 8 7 e q u i l i b r i u m by an r . f . p u l s e i n the measurement of the p r o t o n T ^ . There i s another p r o b l e m i n c o n n e c t i o n w i t h CD^H, namely, i t has been found ( M o r r i s o n et a l . , 1 9 6 5 ) t h a t the CD^H s u p p l i e d by Merck, Sharp and Dohme c o n t a i n s s m a l l amounts of CD^ and CE^D^. T h i s f a c t d i d not come t o our a t t e n t i o n u n t i l i t was too l a t e t o have the sample a n a l y z e d , These i m p u r i t i e s i n the CD^H cause some doubt as t o the v a l i d i t y of our r e s u l t s f o r CD^H. In p a r t i c u l a r , the CHgDg i m p u r i t y i s l i k e l y t o have an a p p r e c i a b l e e f f e c t on the p r o t o n T.^ i n C D ^ H . T h i s has been d i s c u s s e d e lsewhere (de Wit and Bloom, 1 9 6 5 ) . C H . . : 5 = 6 The s p i n - l a t t i c e r e l a x a t i o n t ime data f o r the p r o t o n resonance at 2 8 . 5 mcs. i n CH^ i s shown i n F i g u r e ( 2 2 ) . A l t h o u g h a b r e a k i n the T^ v e r s u s T curve i s e v i d e n t a t the upper phase t r a n s i t i o n t e m p e r a t u r e , no change i s observed at the lower phase t r a n s i t i o n ; t h i s may be due t o the f a c t t h a t t h i s phase t r a n s i t i o n i s q u i t e broad and d i f f i c u l t t o observe ( M o r r i s o n et a l . 1 9 6 3 ) . The t h e o r e t i c a l v a l u e of ( T , ) . = 8 m i l l i s e c . was c a l c u l a t e d i n s e c t i o n 4 : 3 * and i s 2 0 t imes s h o r t e r than the e x p e r i m e n t a l v a l u e of 1 6 0 m i l l i s e c . The ( T , ) . i s r a t h e r s h a l l o w i n d i c a t i n g t h a t i n t h i s x 1 ' mm r e g i o n X i s a s l o w l y v a r y i n g f u n c t i o n of t e m p e r a t u r e . 0 That X c i s s l o w l y v a r y i n g around ( T 1 ) m l n l s I l l u s -t r a t e d by the s p i n - l a t t i c e r e l a x a t i o n t ime measurements c a r r i e d out a t 4.4 mcs, shown i n F i g u r e (23). A noteworthy f e a t u r e of t h i s s e t of d a t a i s the absence o f a minimum i n T^; a t t h e l o w e s t a t t a i n a b l e t e m p e r a t u r e s the s p i n - l a t t i c e r e l a x a t i o n t i m e i s s t i l l becoming s h o r t e r . I f one examines the c u r v e of the c o r r e l a t i o n t i m e s , l o g l / x v e r s u s l o g T, f o r the 28.5 mcs CH^ d a t a shown i n F i g u r e (24), t h e n one c o n c l u d e s t h a t t h e absence of a (T,) . a t 4.4 mcs. i s t o x l ' m i n be e x p e c t e d . U s i n g t h e v a l u e o f X g i v e n a t 1.2°K, one c f i n d s t h a t a t w 0= 2rc x 4,4 mcs. co0\= 0.18, whereas i t has been seen e a r l i e r t h a t ( T 2 ) m i n o c c u r s a t coe>Xc= 0.62. The h i g h t e m p e r a t u r e end o f t h e c o r r e l a t i o n t ime curve shows a T 7 b e h a v i o u r as i n d i c a t e d by t h e d o t t e d l i n e i n F i g u r e (24). In F i g u r e (23) t h e c o r r e l a t i o n t i m e s o b t a i n e d from the 28.5 mcs. d a t a have been used t o p r e d i c t t h e v a l u e s o f T-^  a t 4.4 mcs, so as t o check the c o n s i s t e n c y of t h e assumptions and method o f a n a l y s i s . The agreement between the p r e d i c t e d and e x p e r i m e n t a l r e s u l t s i s r e a s o n a b l y good. A p u z z l i n g f e a t u r e most c l e a r l y seen i n t h e 4.4 mcs. da t a i s t h e minimum i n w h i c h o c c u r s a t 23°K. T h i s second minimum i s a l s o p r e s e n t i n t h e d a t a f o r t h e CH^-Kr and CH^-CD^ m i x t u r e s . I t i s r a t h e r u n f o r t u n a t e t h a t t h e e x i s t e n c e of t h i s second minimum was not r e a l i z e d when the CH^ d a t a a t 28.5 mcs were t a k e n ; i n t h a t case i s d e c r e a s i n g , but the d a t a do not go t h r o u g h t h e second m i n i -mum. T h i s second minimum has not been observed f o r any o f the i s o t o p i c m o d i f i c a t i o n s o f CH^. Some remarks have t o be made about the d a t a o b t a i n e d i n the r e g i o n of t h i s upper minimum. The p l o t of l o g 92 10.0 o </> .02 Experimental Data • Predicted Values using 28.5 Mcs. Data phase transitions 10 50 60 20 30 40 Temperature (°K) F i g u r e 23. E x p e r i m e n t a l V a l u e s of 1^ V e r s u s T f o r CH^ a t 4.4Mcs. A l s o shown a r e t h e p r e d i c t e d v a l u e s u s i n g the 30 mcs. d a t a . Note the T]_ minimum above t h e upper phase t r a n s i t i o n . (A(CO ) -A^) v e r s u s t does not c o n s i s t of a s i n g l e e x p o n e n t i a l , i . e . , t h e r e i s no u n i q u e T-^, hut i t can be a n a l y z e d as the sum of the two e x p o n e n t i a l s . A t y p i c a l p l o t i s shown i n F i g u r e ( 2 5 ) . In view of the e x p e r i m e n t a l e r r o r s a s s o c i a t e d w i t h A(eo )-A^_ and the f a c t t h a t the two t ime c o n s t a n t s so o b t a i n e d u s u a l l y d i f f e r f r o m each o ther by l e s s than a f a c t o r of 4, i t i s i m p o s s i b l e t o t r y a m e a n i n g f u l s e p a r a t i o n of the two t ime c o n s t a n t s . The v a l u e s of T-^ shown on the graph are i n f a c t the s l o p e s of the i n i t i a l s t r a i g h t l i n e p o r t i o n of .the T-j^ p l o t s and the v a l u e of T-^ thus o b t a i n e d f a l l s i n between the v a l u e s of the two t ime c o n s t a n t s . One can say w i t h c e r t a i n t y t h a t t h e r e are two d i s t i n c t c o n t r i b u t i o n s t o the s p i n - l a t t i c e r e l a x a t i o n . Note t h a t the e f f e c t of the second minimum of s h o r t e n i n g T-^ i n the 4 .4 mcs. CH^ data p e r s i s t s even below the phase t r a n s i t i o n ; t h a t i s why the d e p a r t u r e of the e x p e r i m e n t a l v a l u e s of T^ f r o m t h e p r e -d i c t e d v a l u e s on the b a s i s of the 2 8 . 5 mcs. da ta i s g r e a t e r near the phase t r a n s i t i o n . The e f f e c t of t h i s second minimum on T^ can not be a c c o u n t e d f o r a t the p r e s e n t t i m e . U l t r a s o n i c a t t e n u a t i o n measurements by T h i e l e , Whitney and Chase (1964) were c a r r i e d out a t a f r e q u e n c y of 11.8 mcs. These measurements show a peak i n the a b s o r p t i o n at 2 3 . 5 ° K . T h i s peak i m p l i e s a c o r r e l a t i o n t ime of a p p r o x i -8 o mate ly 1.4 x 10" s e c . at 23 .5 K f o r whatever mechanism i s r e s p o n s i b l e f o r the u l t r a s o n i c a t t e n u a t i o n . The c o r r e l a t i o n t ime o b t a i n e d f r o m t h e i r d a t a i s t o be compared w i t h the Temperature (°K) F i g u r e 2k. l / x V e r s u s T f o r CHj, at 30 Mcs, T h e o r e t i c a l Fi , t Obeys E q u a t i o n Shown 0 4 8 12 16 20 24 28 32 t (seconds) F i g u r e 25. A T y p i c a l P l o t of A * - A ( t ) . V e r s u s t S h o w i n g ' t h e N o n - e x p o n e n t i a l R e l a x a t i o n Observed at the Upper (T, ) m l n i n CH^ c o r r e l a t i o n t ime of 2 , 2 x 10 s e c . a t 2 3 . 2 K i m p l i e d by the second T^ minimum d i s c u s s e d above . I t appears v e r y l i k e l y t h a t t h e r e i s a c o n n e c t i o n between these two o b s e r v a t i o n s , but a t the p r e s e n t t ime n e i t h e r the mechanism r e s p o n s i b l e f o r the anomalous u l t r a s o n i c a t t e n u a t i o n , nor t h a t a s s o c i a t e d w i t h the upper minimum i s known. CH^ - Kr M i x t u r e s : 5 : 7 The C H ^ - K r m i x t u r e s a g a i n e x h i b i t an upper minimum i n T 1 v e r y s i m i l a r t o t h a t observed i n pure C H ^ . The data are shown i n F i g u r e s (26) and (27) f o r the 10$ and 50$ Kr m i x t u r e s . However, no e v i d e n c e f o r n o n - e x p o n e n t i a l r e l a x a t i o n was o b s e r v e d . The e x p l a n a t i o n f o r t h i s d i f f e r e n c e may be c o n -t a i n e d i n the f a c t t h a t the measurements were t e r m i n a t e d when A ( t ) s s 0 . 1 5 A ( o o ) , a l t h o u g h i n the case of C H ^ , the non-e x p o n e n t i a l b e h a v i o u r was n o t i c e a b l e when the measurement was t e r m i n a t e d a t t h a t p o i n t . S u r p r i s i n g l y , the v a l u e s of the lower ( T ] _ ) m i n a r e v e r y s t r o n g l y dependent on the Kr c o n c e n -t r a t i o n , b e i n g 1 6 0 , 3 0 0 , and 5000 m i l l i s e c . f o r 1 0 0 $ , 9 0 $ , and 50$ C H ^ , r e s p e c t i v e l y . T h i s b e h a v i o u r of ( T 1 ) m l n i s s u r p r i s i n g i n view of the a s s u m p t i o n , w h i c h i s s u p p o r t e d by o t h e r e v i d e n c e , t h a t a l l s p i n - l a t t i c e r e l a x a t i o n i n t e r -a c t i o n s are i n t r a m o l e c u l a r , A p o s s i b l e e x p l a n a t i o n w i l l be p r o v i d e d i n the next c h a p t e r . M o r e o v e r , the temperature dependence below the (T]_)mj_n l s d r a s t i c a l l y changed by the a d d i t i o n of K r . The h i g h temperature v a l u e s above and below the upper ( T ]_) m l r i a r e the same f o r 10$ and 50$ Kr p r o v i d i n g a d d i t i o n a l e v i d e n c e t h a t a t l e a s t down t o 20°K the r e l a x a t i o n mechanisms a re i n t r a m o l e c u l a r . A n o t h e r p e c u l i a r f e a t u r e i s the l a r g e s h i f t t o h i g h e r tempera tures of t h e upper minimum as the Kr c o n c e n t r a t i o n i s i n c r e a s e d . CD^-CH^ M i x t u r e s : 5=8 The p r o t o n and d e u t e r o n s p i n - l a t t i c e r e l a x a t i o n t imes were measured f o r a 67$ CD^ - 33$ CH^ m i x t u r e , shown i n F i g u r e s (28) and (29). Only the 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 ime was measured f o r a 10$ CD^ - 90$ CH^ m i x t u r e shown i n F i g u r e (30). I f t h e r e l a x a t i o n due t o i n t e r m o l e c u l a r i n t e r a c t i o n s was a p p r e c i a b l e t h i s s h o u l d be r e f l e c t e d i n l o n g e r v a l u e s o f ( T , ) ." w i t h i n c r e a s i n g c o n c e n t r a t i o n of CDi.. In f a c t no v l ' r n m to 4 such e f f e c t was o b s e r v e d ; ( T 1 ) m l n l s 1-55* 195* and 175 m i l l i s e c f o r 100$ CH^, 90$ CH^ and 33$ CH^, r e s p e c t i v e l y . W i t h i n the e x p e r i m e n t a l e r r o r , the v a r i a t i o n i n T-^ minimum i s n e g l i g i b l e . The a d d i t i o n of CD^ t o CH^ seems t o have a marked e f f e c t on the temperature dependence of below the minimum. The upper minimum i n the T 1 data f o r the p r o t o n resonance i s a l s o e v i d e n t i n t h e m i x t u r e s as i t was i n pure CH^. As i n the case of CH^, the r e l a x a t i o n b e h a v i o u r f o r the p r o t o n resonance i n the r e g i o n of the upper minimum i s a l s o n o n -e x p o n e n t i a l . The temperature a t w h i c h the upper minimum 98 1000 99 100 o c c u r s i s a p p a r e n t l y independent of the CD^ c o n c e n t r a t i o n , whereas i t i s v e r y c o n c e n t r a t i o n dependent f o r the C H ^ - K r m i x t u r e s . A D i s t r i b u t i o n of C o r r e l a t i o n T i m e s : 5^9 I t has been observed i n s o l i d h i g h - c o n c e n t r a t i o n o r t h o Hg, t h a t the r e l a x a t i o n of M z towards i t s e q u i l i b r i u m v a l u e i s n o n - e x p o n e n t i a l . T h i s n o n - e x p o n e n t i a l b e h a v i o u r has been a t t r i b u t e d to a d i s t r i b u t i o n of c o r r e l a t i o n t imes e x i s t -i n g i n the s o l i d because i t i s inhomogeneous, i . e . , the c r y s t a l l i n e f i e l d s v a r y g r e a t l y f rom s i t e t o s i t e , t h e c r y s t a l l i n e f i e l d s b e i n g h i g h l y dependent on the l o c a l c o n -f i g u r a t i o n of the o r t h o and p a r a Hg m o l e c u l e s . C o n s e q u e n t l y , a " s p i n t e m p e r a t u r e " ( to be d i s c u s s e d i n the next c h a p t e r ) does not e x i s t , because the energy l e v e l s of the n e i g h b o u r i n g n u c l e a r s p i n s are not even a p p r o x i m a t e l y e q u i d i s t a n t . A d e t a i l e d e x a m i n a t i o n of the r e l a x a t i o n f u n c t i o n f o r M z shows (Sugawara e t a l . 1956) t h a t a d i s t r i b u t i o n of c o r r e l a t i o n t imes has the net e f f e c t of i n t r o d u c i n g a d i s -t r i b u t i o n of T - j / s . T h i s c o u l d have the e f f e c t of i n c r e a s i n g the mean v a l u e of at the minimum, and of b r o a d e n i n g the T^ minimum. N o n - e x p o n e n t i a l r e l a x a t i o n b e h a v i o u r was not observed i n any of our e x p e r i m e n t s i n the l o w e s t temperature phases i n methane. However, t h i s does not prove c o n c l u s i v e l y t h a t a d i s t r i b u t i o n of c o r r e l a t i o n t imes does not e x i s t , s i n c e , u n l i k e E0, i t may s t i l l be p o s s i b l e t o d e s c r i b e the s p i n system 101 102 0 10 20 30 40 T E M P E R A T U R E (°K) F i g u r e 30. Experimental Values of Proton T 1 Versus T f o r 10$ CD,,-90% CH,, Mixture 104 by a " s p i n t e m p e r a t u r e " . As f a r as our exper iments a re c o n -c e r n e d the l e n g t h e n i n g of c o u l d h e l p t o e x p l a i n t h e l o n g ^ T l ^ m i n f o r C H 4 * 0 n t h e o 1 ; h e r h a n d > w h y s h o u l d a d i s t r i b u t i o n of c o r r e l a t i o n t imes o n l y show, up f o r CH^ and not f o r CD^? N o n - e x p o n e n t i a l r e l a x a t i o n has o n l y been observed f o r C'CH^ and CH^ - CD^ m i x t u r e s , a l t h o u g h o n l y i n a s m a l l temperature range near the upper minimum. The L i n e Shape: 5:10 F o r a l l the systems s t u d i e d , o s c i l l o s c o p e photographs were t a k e n of the i n d u c t i o n t a i l a t a number of t e m p e r a t u r e s . As the shape of the i n d u c t i o n t a i l v a r i e d o n l y v e r y s l i g h t l y throughout the temperature range f o r a l l of the systems, the r e s u l t s w i l l not be p r e s e n t e d . The r e s u l t s agree i n p r i n c i p l e w i t h those r e p o r t e d f o r the p r o t o n s p i n system by Wolf (1964). The l i n e shape i n p a r t i c u l a r Tg of the d e u t e r o n resonance a l s o shows v e r y l i t t l e change w i t h t e m p e r a t u r e . A c u r s o r y e x a m i n a t i o n of the data does show t h a t t h e h i g h e r moments of the l i n e are s l i g h t l y temperature dependent . I t has a l s o been checked i n a c o u p l e of i n s t a n c e s , and shows t h a t the Lowe b e a t s can be f i t t e d u s i n g the e q u a t i o n ( 4 - l 4 ) . 105 Summary: 5:11 As has been shown i n t h i s c h a p t e r , the r e l a x a t i o n p r o p e r t i e s of the methane system at low t e m p e r a t u r e s "have many i n t e r e s t i n g f e a t u r e s . These r e s u l t s can be u s e d , i n p r i n c i p l e , t o a r r i v e a t a b e t t e r u n d e r s t a n d i n g of the na ture of the v a r i o u s phase t r a n s i t i o n s and of the p r o p e r t i e s t o f the t h r e e p h a s e s . The main q u a l i t a t i v e f e a t u r e s of the r e s u l t s a r e : 1. R a p i d changes of T-^ occur i n the immediate v i c i n i t y of many of the phase t r a n s i t i o n s . In cases where r a p i d changes of T 1 were not o b s e r v e d , i t was p o s s i b l e t o g i v e e x p e r i m e n t a l r e a s o n s f o r t h e i r n o n - a p p e a r a n c e . One can t e n t a t i v e l y c o n c l u d e t h a t a l l of the phase t r a n s i t i o n s have a s s o c i a t e d w i t h them l a r g e changes i n the o r i e n t a t i o n a l s t a t e s a n d / o r i n the r a t e s of m o l e c u l a r r e o r i e n t a t i o n . T h i s i s i n q u a l i t a t i v e agreement w i t h e n t r o p y measurements. 2. Above the upper phase t r a n s i t i o n (phase I ) , T-^ i s s l o w l y temperature dependent i n a l l c a s e s , except t h a t a pronounced minimum o c c u r s i n a l l measurements of T^ i n CH^ ( i n c l u d i n g e x p e r i m e n t s done on m i x t u r e s ) . The mechanism f o r these h i g h temperature minima i s not u n d e r s t o o d at a l l . The slow o v e r a l l temperature dependence a l s o remains t o be e x p l a i n e d . 3 . T 1 d e c r e a s e s v e r y r a p i d l y w i t h d e c r e a s i n g tempera-t u r e below the upper phase t r a n s i t i o n (phases I I and I I I ) i n a l l cases and goes t h r o u g h a minimum v a l u e i n phase I I I 106 In t h i s c h a p t e r , the measurements were d i s c u s s e d i n terms of the c o n v e n t i o n a l t h e o r y of r e l a x a t i o n by i n t r a -m o l e c u l a r i n t e r a c t i o n s due t o m o l e c u l a r r e o r i e n t a t i o n . In the c o n v e n t i o n a l t h e o r y the e f f e c t i v e s t r e n g t h of the i n t r a -m o l e c u l a r i n t e r a c t i o n s i s u s u a l l y t a k e n t o be independent of temperature so t h a t the v a r i a t i o n of v e r s u s T g i v e s the temperature dependence of the c o r r e l a t i o n t ime X Q f o r mole-c u l a r r e o r i e n t a t i o n . In the d i s c u s s i o n h e r e , i t was assumed t h a t the r e l e v a n t c o r r e l a t i o n f u n c t i o n s decay expon-e n t i a l l y i n t i m e . Some of the d i f f i c u l t i e s i n v o l v e d i n u s i n g t h i s type of i n t e r p r e t a t i o n t o o b t a i n q u a n t i t a t i v e i n f o r m a t i o n about the systems s t u d i e d are as f o l l o w s : 1. The minimum v a l u e s of T^ are known f r o m the c o n -v e n t i o n a l t h e o r y i f the i n t r a - m o l e c u l a r i n t e r a c t i o n s are known. F o r the p r o t o n s , the observed ( l ^ ) ^ a re a l l too l o n g , and i n many cases by f a c t o r s of 20 or even more. F u r t h e r m o r e , the v a l u e s of ( T - L ) m l n i n the C H ^ - K r m i x t u r e s a t low t e m p e r a t u r e s depend^ s t r o n g l y on the Kr c o n c e n t r a t i o n a l t h o u g h the temperature independent v a l u e s of T-^ a t h i g h t e m p e r a t u r e s are independent of Kr c o n c e n t r a t i o n . 2. The dependences of % v e r s u s T e x t r a c t e d f r o m c the T 1 data a re not governed by a s i n g l e a c t i v a t i o n energy as i s common f o r many c l a s s i c a l sys tems . 3. There i s no e x i s t i n g t h e o r y of m o l e c u l a r r e o r i e n -t a t i o n w h i c h i s c a p a b l e of d e s c r i b i n g the n a t u r e of the phase t r a n s i t i o n s i n methane. 107 4. The p r e s e n t i n a d e q u a t e knowledge of the c r y s t a l s t r u c t u r e of s o l i d methane above and below the phase t r a n s i t i o n (see s e c t i o n 2:4) does not e x c l u d e the p o s s i b i l i t y of a s m a l l change i n the c r y s t a l s t r u c t u r e on p a s s i n g t h r o u g h the phase t r a n s i t i o n . Such a change c o u l d account f o r the d i s c o n t i n u i t i e s i n T^ v e r s u s T t h r o u g h i t s e f f e c t on the m o l e c u l a r energy l e v e l s . Recent i n v e s t i g a t i o n s have r e v e a l e d such a s t r u c t u r e change i n s o l i d Hg a t i t s phase t r a n s i t i o n o c c u r r i n g at 1.5°K ( C l o u t e r e t a l . 1965, and M i l l s e t a l . 1965). The c o n v e n t i o n a l t h e o r y does g i v e a p p r o x i m a t e l y the c o r r e c t v a l u e s of (T^) f o r the d e u t e r o n s , but the u n c e r t a i n t y i n the q u a d r u p o l e c o u p l i n g c o n s t a n t i s g r e a t . In a d d i t i o n , the temperature dependence of T-^  i n CH^ a t 4.4 mcs can be p r e d i c t e d a p p r o x i m a t e l y f r o m the r e s u l t s a t 28.5 mcs. In the next c h a p t e r , the consequences of the d e t a i l e d model of the m o l e c u l a r energy l e v e l s i n phase I I I p r o p o s e d by M o r r i s o n e t a l . w i l l be examined. • When t h i s i s combined w i t h a s i m p l e p i c t u r e of the p h o n o n - m o l e c u l a r r o t a t i o n i n t e r a c t i o n s , t h e r e a r e i n d i c a t i o n s t h a t the r e s u l t s can be q u a n t i t a t i v e l y i n t e r p r e t e d i n terms of m o l e c u l a r p a r a m e t e r s . A l s o , some of the p o s s i b l e consequences of the quantum m e c h a n i c a l n a t u r e of the systems b e i n g c o n s i d e r e d w i l l be d i s c u s s e d . CHAPTER 6 . FURTHER DISCUSSION OF THE EXPERIMENTAL RESULTS Review of the c l a s s i c a l c a l c u l a t i o n of the c o r r e l a t i o n f u n c t i o n s : 571 As has been d i s c u s s e d i n C h a p t e r 4, i s r e l a t e d t o F o u r i e r t r a n s f o r m s of c o r r e l a t i o n f u n c t i o n s of f u n c t i o n s of the m o l e c u l a r o r i e n t a t i o n i n s o f a r as c o n t r i b u t i o n s of i n t r a -m o l e c u l a r i n t e r a c t i o n s to s p i n - l a t t i c e r e l a x a t i o n are c o n -c e r n e d . The c a l c u l a t i o n of such c o r r e l a t i o n f u n c t i o n s f o r a g e n e r a l c l a s s of systems i s d i f f i c u l t . However, u s e f u l r e s u l t s have been o b t a i n e d f o r s i m p l e models i n w h i c h f o r example, the m o l e c u l a r o r i e n t a t i o n s are t a k e n t o be random f u n c t i o n s of t i m e . As an example, suppose t h a t a g i v e n v e c t o r i n a mole-c u l e i s i n i t i a l l y o r i e n t e d i n the s o l i d a n g l e between X1. 0 and &-o+ drL0 w i t h p r o b a b i l i t y p ( j V o ) d r v o and t h a t r e o r i e n t a t i o n p r o c e e d s by d i s c r e t e random jumps w i t h e q u a l p r o b a b i l i t y t o any o t h e r element of s o l i d a n g l e between Si. and s i + d n . . Then the e x p r e s s i o n f o r the c o r r e l a t i o n f u n c t i o n of a f u n c t i o n of / ! ( * C ) , <^ .(n_ ( i r ) ) , i s g i v e n by <L&<LSL9 p ( £ y P ^ i A » ^ W (6-where P ( / v , J H - 0 , t )dn. i s the p r o b a b i l i t y t h a t the m o l e c u l e i s o r i e n t e d between n a n d n_+ d l l a t t ime T i f i t i s 108 109 i n i t i a l l y o r i e n t e d between £ \ . Q and f\.Q + d-&~Q. F o r the above model where 6 (-ft- - n Q ) i s the D i r a c 6 - f u n c t i o n and i s the average t ime spent by a m o l e c u l e i n a g i v e n o r i e n t a t i o n . U s u a l l y , i t i s a p p r o p r i a t e t o assume random a p r i o r i o r i e n t a t i o n s , i . e . , p<a.) = (6-3) i n w h i c h c a s e , s u b s t i t u t i o n of e q u a t i o n (6-2) i n t o e q u a t i o n (6-1) g i v e s = <| a f ) e ~ V T c (6-4) T h i s type of model has been used r e c e n t l y f o r the i n t e r -p r e t a t i o n of r e l a x a t i o n by q u a d r u p o l a r i n t e r a c t i o n s i n s o l i d s ( A l e x a n d e r , 1965). The o p p o s i t e l i m i t i n g case i s the r o t a t i o n a l d i f f u s i o n model i n w h i c h the i n d i v i d u a l jumps are assumed t o be v e r y s m a l l . As mentioned e a r l i e r , t h i s case has been examined i n g r e a t d e t a i l by Hubbard (1962) and o t h e r s ( S t e e l e , 1963). T h i s case would be most a p p l i c a b l e t o l i q u i d s composed of l a r g e . ( c l a s s i c a l ) m o l e c u l e s . The c o r r e l a t i o n f u n c t i o n of y(X) f o r t h e r o t a t i o n a l d i f f u s i o n model i s of the same form as e q u a t i o n (6-4) and % i n t h i s case i s e x p r e s s e d i n terms c of the r o t a t i o n a l d i f f u s i o n c o n s t a n t . The passage f r o m the s m a l l jump l i m i t t o the l a r g e jump l i m i t has been d i s c u s s e d by Ivanov (1964). 1 1 0 In the p r e c e d i n g c h a p t e r , the e x p e r i m e n t a l data have been a n a l y z e d under the assumption t h a t the c o r r e l a t i o n f u n c t i o n s are e x p o n e n t i a l as g i v e n i n e q u a t i o n ( 6 - 4 ) . How-e v e r , the u s u a l c l a s s i c a l i n t e r p r e t a t i o n of the parameters a p p e a r i n g i n e q u a t i o n ( 6 - 4 ) were found to be i n c o n s i s t e n t w i t h most of the d a t a . The y ( X l ) i n t r o d u c e d above i s i d e n t i f i e d w i t h the s p h e r i c a l harmonic of o r d e r 2 , namely Y 2 m ( j T l ) . The p r e d i c t e d minimum v a l u e of T^ f o r a g i v e n resonance f r e q u e n c y i s d e t e r m i n e d c o m p l e t e l y w i t h i n the above models by ^ | y | ^ , w h i c h i s f o r Y 2 m ( r i . ) i f a l l o r i e n t a -t i o n s are e q u a l l y p r o b a b l e . I t has been found t h a t the e x p e r i m e n t a l v a l u e s of ( T 1 ) m j _ n a r e c o n s i s t e n t l y l a r g e r than the t h e o r e t i c a l p r e d i c t i o n s on t h i s b a s i s . S e c o n d l y , the temperature dependences of Tj as g i v e n by the e x p e r i m e n t a l temperature dependence of T-^ assuming t h a t G ( T ) i s g i v e n by e q u a t i o n ( 6 - 4 ) cannot be d e s c r i b e d i n terms of an a c t i v a t i o n energy f o r most of the cases s t u d i e d , i . e . , the f o r m u l a i s not obeyed. F o r comparison w i t h o t h e r systems where e q u a t i o n ( 6 - 5 ) i s obeyed see A l e x a n d e r ( 1 9 6 5 ) . The Low L y i n g S t a t e s of Methane: 6 : 2 ( 6 - 5 ) The above mentioned d i s c r e p a n c i e s are not a t a l l s u r -p r i s i n g i n view of the f a c t t h a t the methane m o l e c u l e s have I l l v e r y s m a l l moments of i n e r t i a . F o r example, the r o t a t i o n a l c o n s t a n t of CH^ i n e q u a t i o n ( 2 - 2 ) i s B = ( 7 . 6 8 ° K ) k . T h e r e -f o r e , i t would not be s u r p r i s i n g i f i t were n e c e s s a r y t o take i n t o account the quantum m e c h a n i c a l p r o p e r t i e s of the r o t a t i o n a l energy l e v e l s i n o r d e r to d e s c r i b e a d e q u a t e l y the m o l e c u l a r r e o r i e n t a t i o n i n the temperature range i n w h i c h these exper iments were c a r r i e d o u t . There i s i n f a c t , some i n f o r m a t i o n a v a i l a b l e about the n a t u r e of the low l y i n g m o l e c u l a r s t a t e s on the b a s i s of a c c u r a t e heat c a p a c i t y measurements by M o r r i s o n and h i s c o l l a b o r a t o r s ( C o l w e l l , G i l l and M o r r i s o n , 1965). F i g u r e ( 3 1 ) i s a schemat ic r e p r e s e n t a t i o n of the lowest r o t a t i o n a l . energy l e v e l s of CH^D and CHD^ a c c o r d i n g to t h e i r measure-ments. Roughly s p e a k i n g , the 1 2 lowest r o t a t i o n a l s t a t e s are c o n s i d e r e d t o c o n s i s t of two groups of l e v e l s h a v i n g d e g e n e r a c i e s g Q and g^ r e s p e c t i v e l y , w i t h g ] / g 0 = 3 - The average s p l i t t i n g s between these groups are g i v e n as a p p r o x i -mately 1 . 7 7 ° K f o r CH"3D and i . 2 6 ° K f o r CRD^. F o r C H ^ , the r e s i d u a l e n t r o p y i n d i c a t e s t h a t the s o - c a l l e d T symmetry s p e c i e s ' 3 - f o l d s p a t i a l degeneracy i s p a r t i a l l y removed, the ground s t a t e c o r r e s p o n d i n g t o g Q = 2 and the e x c i t e d s t a t e t o g 1 = 1, The s p a t i a l d e g e n e r a c i e s of the A and E s p e c i e s are c o m p l e t e l y removed. F o r C D ^ , no removal of the 3 - f o l d s p a t i a l degeneracy of the T m o d i f i c a t i o n i s i m p l i e d by the e n t r o p y measurements. 1 1 2 C a l c u l a t i o n of the c o r r e l a t i o n f u n c t i o n s f o r  m o l e c u l e s h a v i n g d i s c r e t e energy l e v e l s : 6 : 3 Let the r o t a t i o n a l s t a t e s of a m o l e c u l e be denoted by the index m = 1 , 2 , - e t c . , and suppose t h a t the e x p e c t a t i o n v a l u e of the i n t r a - m o l e c u l a r i n t e r a c t i o n i n q u e s t i o n i n the s t a t e m i s y . The v a r i a b l e y i s c o n s i d e r e d t o be a random f u n c t i o n of t ime because of the f a c t t h a t t r a n s i t i o n s are i n d u c e d between m o l e c u l a r s t a t e s m , n , e t c . by i n t e r a c t i o n s which c o u p l e the m o l e c u l a r o r i e n t a t i o n s t o the l a t t i c e v i b -r a t i o n s and so f o r t h . I f y i s a " s t a t i o n a r y " , random f u n c t i o n of t i m e , the c o r r e l a t i o n f u n c t i o n G(t) of y may be w r i t t e n i n the f o r m (Abragam, 1 9 6 l , p 2 7 l ) G ( X ) - Z p.v. PJ& ^ ( 6 - 6 ) where p m i s the "a p r i o r i p r o b a b i l i t y t h a t a m o l e c u l e i s i n the s t a t e m, P m n ( t ) l s the p r o b a b i l i t y t h a t a m o l e c u l e i s i n the s t a t e n a t t ime X i f i t i s i n the s t a t e m at t ime X = 0, and i t has been assumed t h a t P_ (% ) = P (~X) . F o r t h i s mn mn c a s e , G ( T ) = G ( - T ) ( 6 - 7 ) In C h a p t e r 4 , the consequences of an e x p o n e n t i a l f o r m f o r the c o r r e l a t i o n f u n c t i o n G ( t ; ) have been examined. I t s h o u l d be noted t h a t an e x p o n e n t i a l form does not h o l d i n g e n e r a l f o r the G(X ) g i v e n by e q u a t i o n ( 6 - 6 ) even when P (x) a re governed by r a t e e q u a t i o n s , i . e . , 113 5 x E ^ J = 2 5 x E 5 x A x > -\ \ \ \ 5 x E ^^^~~--rs~~ J = 0 i x A T 5 c a , / M o l e UA+4E A + E F R E E ROTOR HINDERED ROTOR F i g u r e 31 . The Energy L e v e l Diagram Proposed by C o l w e l l ,  G i l l and M o r r i s o n (19&5) f o r C H ? D and C D 0 H 114 dP ^ — <r— _ -m= _ p — IE. w m^, + ^ - p w ( 6 - 8 ) where W , Is the p r o b a b i l i t y per u n i t t ime t h a t a m o l e c u l e undergoes a t r a n s i t i o n f r o m the s t a t e n t o the s t a t e n ' . The i n i t i a l c o n d i t i o n s a re P (0) = S ( 6 - 9 ) mnv ' ^ mn \ -> i Two Energy L e v e l C a s e : 6:4 In o r d e r t o examine the model of C o l w e l l , G i l l , M o r r i s o n i n more d e t a i l , assume a two energy l e v e l model i n w h i c h the lower energy l e v e l i s g Q f o l d degenerate and the upper l e v e l i s g-^  f o l d d e g e n e r a t e . The energy s e p a r a t i o n of the l e v e l s i s taken t o be k T Q L e t the i n d e x i denote the lower l e v e l s , 1 = l , 2 , . . . , g and the i n d e x j denote the upper l e v e l s , j=g Q +l , g + 2 , . . . , So + S i ' L e t y = y^ i n the l ^ * 1 s t a t e of the lower l e v e l , a n d . y = y . i n the s t a t e of the upper l e v e l . 3 T h e r e f o r e , p . = p independent of i 1 ^ e q u i l i b r i u m p r o b a b i l i t i e s P j = p.^ independent of j 115 p j / P o = e & S o p o + S l p l = 1 N o r m a l i z a t i o n (6-10) (6-11) To complete the model , the f o l l o w i n g s i m p l i f y i n g assumptions a r e made, W- = t r a n s i t i o n p r o b a b i l i t i e s between any p a i r of s t a t e s i n 0 , W-. = t r a n s i t i o n p r o b a b i l i t i e s between any p a i r of s t a t e s i n 1, W = t r a n s i t i o n p r o b a b i l i t i e s between any s t a t e i n 0 and any s t a t e i n 1, W = t r a n s i t i o n p r o b a b i l i t i e s between any s t a t e i n 1 and any s t a t e i n 0. T h e n , P Q W + = p^ jW , by d e t a i l e d b a l a n c e . Prom e q u a t i o n (6-6) and the above d e f i n i t i o n s , (6-12) (6-13) Now, by s u b s t i t u t i n g the t r a n s i t i o n p r o b a b i l i t i e s d e f i n e d above i n t o the r a t e e q u a t i o n s ( 6 - 8 ) , the c o n d i t i o n a l p r o b a b i -l i t i e s P k l i n the e x p r e s s i o n f o r G(TJ.) may be f o u n d . The c a l c u l a t i o n s a re v e r y l e n g t h y , but s t r a i g h t f o r w a r d and the r e s u l t s a r e : (6-14) 116 where use has been made of e q u a t i o n s (6-11) and (6-12) and 4 - = + 3 > w + T 0 a n d T , a re the average l i f e t i m e s of a m o l e c u l e i n an i n d i v i d u a l i and j l e v e l r e s p e c t i v e l y , w h i l e f 0 l i s the time c o n s t a n t f o r the e s t a b l i s h m e n t of a Bol tzman d i s t r i b u t i o n between the two groups of l e v e l s . S u b s t i t u t i n g e q u a t i o n s (6-11) , (6-12) and ( 6 - l 4 ) i n t o e q u a t i o n ( 6 - 1 3 ) , the c o r r e l a t i o n f u n c t i o n G ( t ) may be w r i t t e n i n the f o r m , + 3 . f t [ < ^ - < ^ ] ^ ' ( 6 " 1 6 ) + ^3.p•p.[<^> l-<^ j] ^ e ^' c ^ i , where and ^<^. are the average v a l u e s of y f o r a system of m o l e c u l e s r e s t r i c t e d to the i and j s t a t e s r e s p e c t i v e l y , e t c . <W-ir f *<• > etC' C6 - 1 7 ) U s i n g e q u a t i o n s (6-10) and (6-11)) (6-18). 1 1 7 I f a l l t h r e e terms i n e q u a t i o n ( 6 - l 6 ) a re n o n - z e r o , the a n a l y s i s of the e x p e r i m e n t a l data i n Chapter 5 i s i n v a l i d , s i n c e i t was assumed t h e r e t h a t the c o r r e l a t i o n f u n c t i o n c o u l d be e x p r e s s e d i n terms of a s i n g l e c o r r e l a t i o n t i m e . The e x p r e s s i o n s f o r the r e l a x a t i o n r a t e would be e x p r e s s e d i n terms of the F o u r i e r t r a n s f o r m of G ( t ) as i n e q u a t i o n s ( 4 - 6 ) and ( 4 - 7 ) , but the F o u r i e r t r a n s f o r m would then c o n s i s t of the sum of 3 terms of t h e f o r m of e q u a t i o n s ( 4 - 1 1 ) . As w i l l be d i s c u s s e d l a t e r , t h e r e are systems of i n t e r e s t i n w h i c h the f i r s t two terms i n e q u a t i o n ( 6 - l 6 ) a re z e r o or s m a l l . In t h i s c a s e , one o b t a i n s , where C , = JL r * f t * I ( I + 0 ( C H k , d i p o l a r i n t e r a c t i o n s ) see e q u a t i o n ( 4 - 1 1 ) ( 6 - 2 0 ) = - f - — —1 (CD,. , q u a d r u p o l a r i n t e r -ne <x>0 (_ M 4 a c t i o n s ) see e q u a t i o n ( 4 - 1 2 ) ( 6 - 2 1 ) i s the c l a s s i c a l c o e f f i c i e n t , x o l = u ^ t e , ( 6 - 2 2 ) The q u a n t i t y y i s i d e n t i f i e d w i t h the s p h e r i c a l harmonic . Whereas, i n the c l a s s i c a l case ^|Ya-^P^ = V ^ T * ( e q u a t i o n 4 - 9 ) ) 5 t h i s i s r e p l a c e d by <IY^= <iv-r> - ( Y - ) 1 ( 6 - 2 3 ) 1 1 8 f o r t h i s s p e c i a l c a s e . T h e r e f o r e , the e f f e c t i v e s t r e n g t h of the i n t e r a c t i o n i s temperature dependent i n g e n e r a l . The Temperature Dependence of the T r a n s i t i o n P r o b a b i l i t i e s : 575 In the p r e v i o u s s e c t i o n .the c o r r e l a t i o n t i m e s were e x p r e s s e d i n terms of the t r a n s i t i o n p r o b a b i l i t i e s W , W_, w\ and Wn. I f one b e l i e v e s t h a t the m o l e c u l a r r e o r i e n t -o 1 a t i o n s are caused by the c o u p l i n g between the m o l e c u l e s and the phonons, then the temperature dependences of the t r a n s i -t i o n p r o b a b i l i t i e s can be p r e d i c t e d u s i n g a f o r m a l i s m v e r y s i m i l a r t o t h a t used i n the t h e o r y f o r e l e c t r o n s p i n - l a t t i c e r e l a x a t i o n ( J e f f r i e s , 1953)• In e l e c t r o n s p i n - l a t t i c e r e l a x a t i o n the o n l y i m p o r t a n t r e l a x a t i o n p r o c e s s e s f o r a t w o - l e v e l system are the " d i r e c t " and the "Raman p r o c e s s e s . " H i g h e r - o r d e r r e l a x a t i o n p r o c e s s e s a r e a l s o p o s s i b l e , but the t r a n s i t i o n p r o b a b i l i t i e s due t o t h e s e p r o c e s s e s are n e g l i -g i b l e . In the d i r e c t p r o c e s s one imagines t h a t the m o l e c u l e s a re c o u p l e d t o the phonons by the c r y s t a l f i e l d i n t e r a c t i o n w i t h the m o l e c u l e . T h i s i n t e r a c t i o n a c t s as a t i m e -dependent r e l a x a t i o n p e r t u r b a t i o n i n d u c i n g a t r a n s i t i o n between the s t a t e s of the m o l e c u l e s i m u l t a n e o u s l y w i t h the c r e a t i o n or a n n i h i l a t i o n of a phonon of energy & =kT Q , t h u s , c o n s e r v i n g e n e r g y . U s i n g the p r o p e r t i e s of the m a t r i x e lements f o r the c r e a t i o n and a n n i h i l a t i o n of phonons, one 119 can d e r i v e the f o l l o w i n g : W_ = ft ( m 4-1) where /r»fe = ( e 1 ^ - / ) The c o n s t a n t A i s dependent on the n a t u r e of the r e l a x a t i o n p e r t u r b a t i o n . S u b s t i t u t i n g these i n e q u a t i o n ( 6 - 1 5 ) = fl 3r> +3. £ T ° / r (6-24) The u s u a l case c o n s i d e r e d i n E l e c t r o n S p i n Resonance i s g 0 = S1 = 1 , = A coth ( T O / 2 T ) 'Co, U n l i k e the d i r e c t p r o c e s s , the Raman p r o c e s s i n v o l v e s the s i m u l t a n e o u s a b s o r p t i o n of a phonon of energy fe( and the e m i s s i o n of a n o t h e r of energy S g - ^ ]_ + 6 a l o n g w i t h a t r a n s i t i o n of the m o l e c u l e f r o m s t a t e | j ) t o s t a t e j i ) and v i c e v e r s a . In c o n t r a s t t o the d i r e c t p r o c e s s where the phonon energy must e q u a l the energy l e v e l s p l i t t i n g , the o n l y r e q u i r e m e n t now i s t h a t d i f f e r e n c e of the two phonon e n e r g i e s be 8 f o r energy c o n s e r v a t i o n . S i n c e & « k ^ ( the Debeye t e m p e r a t u r e ) , the e n t i r e phonon spect rum i s a v a i l a b l e . 5 i & 2 d 6 . 120 Now I f $ « kT In the temperature range where Raman p r o c e s s Is dominant then W W + . M o r e o v e r , i f & « k T « . k9^ then (Abragam, 1961, p . 4o8) W + w_ oc T 7 (6-25) F o r the s p e c i a l case of n u c l e a r q u a d r u p o l a r r e l a x a t i o n i n a c u b i c c r y s t a l , i t . h a s been shown (van Kranendonk, 1954) 7 t h a t the T ' law o n l y h o l d s f o r T ^ 0 .02 9 D . U n t i l the f o r m of the p h o n o n - m o l e c u l e i n t e r a c t i o n i s known f o r s o l i d methane, 7 i t w i l l not be known over what range of tempera tures the T law i s v a l i d f o r the systems b e i n g c o n s i d e r e d . T r a n s i t i o n s between any p a i r s of l e v e l s w i t h i n the degenerate s e t s of l e v e l s , w h i c h are d e s c r i b e d by the t r a n s i -t i o n p r o b a b i l i t i e s W n and W can o c c u r as a resonant , p r o -1 o cess t h r o u g h the i n t e r m o l e c u l a r c o u p l i n g w h i c h e x i s t s between n e i g h b o u r i n g m o l e c u l e s i n these s t a t e s . S i n c e no energy i s exchanged w i t h the phonons i n such a p r o c e s s , i t i s l i k e l y t h a t W Q and W 1 would be temperature i n d e p e n d e n t . Temperature dependence of T , f o r some s p e c i a l c a s e s : 6:6 A . T w o - l e v e l s y s t e m : The s i m p l e s t case i s t h a t i n w h i c h the degeneracy of the m o l e c u l a r l e v e l s i s c o m p l e t e l y removed and i n w h i c h o n l y the lowest two l e v e l s a re a p p r e c i a b l y p o p u l a t e d . In t h i s c a s e , i t i s o b v i o u s t h a t 1 2 1 g 0 = gj = 1 ( 6 - 2 6 ) p. = ! P , = S - V r + e - V r At h i g h t e m p e r a t u r e s , T ^ T « 1 and hence |r, so t h a t e q u a t i o n s ( 6 - l 6 ) and ( 6 - 2 6 ) g i v e G( t ) d ^ e ' ^ ^ ' ( 6 - 2 7 ) where we have d e f i n e d y by 2 y = y Q - y 1 ( 6 - 2 8 ) . T h e r e f o r e , we c o n c l u d e t h a t a t h i g h t e m p e r a t u r e s , f o r a t w o - l e v e l system, the c o r r e l a t i o n time t c o b t a i n e d f r o m the a n a l y s i s o f C h a p t e r 5 i s t o be i d e n t i f i e d w i t h x o l . T h e r e f o r e , u s i n g e q u a t i o n s ( 6 - 1 5 ) , ( 6 - 2 4 ) and ( 6 - 2 5 ) J ^ - L - ~ +• W_ ( 6 - 2 9 ) « fl c o t l , ( r > ^ T ) - » - B T 1 I t has been shown i n F i g u r e s ( 1 6 ) and ( 2 4 ) t h a t e q u a t i o n ( 6 - 2 9 ) does f i t the p l o t of t t v e r s u s T d e r i v e d f r o m the e x p e r i m e n t a l d a t a . The c u r v e s shown i n F i g u r e s ( l 6 ) and ( 2 4 ) c o r r e s p o n d t o the v a l u e s of the parameters 122 A = 0 .88 s e c - 1 C H 4 0 .36 sec-"*" CH 3 D B = 1 .5 x i o - 7 s e c _ 1 ( 0 K ) - 7 C H 4 5 .9 x l O " 8 s . e c- 1(°K)- 7 CH^D 7.0 °K 8 . 8 °K C H 3 D . (6-30) T h i s a n a l y s i s i s now seen to be i n c o n s i s t e n t w i t h the assumption t h a t T Q << T i n the e x p e r i m e n t a l r e g i o n . I f one i n c l u d e s the temperature dependences of p Q and p ^ , e q u a t i o n (6-16) i s r e p l a c e d by G(X)= + <t* e~ T o / l" e-VI*'i (6-31) ( 1 4 . e - T o / ^ T h e r e f o r e , ^ | Ya^ l^ ^ i s n o w s t r o n g l y temperature dependent as i n d i c a t e d by e q u a t i o n s (6-23) and (6-31). I t i s t h e r e f o r e c o n c l u d e d t h a t i n s p i t e of the e x c e l l e n t agreement g i v e n by e q u a t i o n (6-29) and (6-27), the t w o - l e v e l model i s i n c o n -s i s t e n t w i t h the e x p e r i m e n t a l data f o r CH^ and CH" 3D. B . The two e n e r g y - l e v e l system w i t h d e g e n e r a c i e s There are many two e n e r g y - ] e v e l systems i n w h i c h the c o n d i t i o n s under w h i c h e q u a t i o n (6-19) i s v a l i d are s a t i s f i e d . 2 2 2 These c o n d i t i o n s a r e ^ = <^  y ) ^ = y Q , e t c . F o r example, a m o l e c u l e i n the J = 1 s t a t e s u b j e c t e d to an a x i a l l y sym-m e t r i c a l c r y s t a l l i n e f i e l d has i t s degeneracy p a r t i a l l y , removed such t h a t g ^ g - ^ = \ of 2. - S i m i l a r l y , a m o l e c u l e i n the J = 2 s t a t e s u b j e c t e d to a f i e l d h a v i n g c u b i c symmetry has two energy l e v e l s w i t h d e g e n e r a c i e s w h i c h s a t i s f y the e q u a t i o n gg/g-^ = 3 / 2 or 2 / 3 . F o r these and o ther s i m i l a r c a s e s , ,the e x p e c t a t i o n v a l u e s of Y 2 m a re i d e n t i c a l f o r a l l degenerate s t a t e s . T h e r e f o r e , the assumption t h a t the f i r s t two terms of e q u a t i o n ( 6 - l 6 ) v a n i s h i s v a l i d f o r such sys tems . T h i s i s not n e c e s s a r i l y t r u e of the s p i n - r o t a t i o n i n t e r a c t i o n as w i l l be d i s c u s s e d i n s e c t i o n 6 : 8 . As s t a t e d e a r l i e r , the model of C o l w e l l , G i l l and M o r r i s o n , 1 9 6 5 f o r CH^D and CHD^ c o r r e s p o n d s t o a two e n e r g y -l e v e l system w i t h u) = 3> where uj i s d e f i n e d as t o = g x / g 0 ( 6 - 3 2 ) The o t h e r parameters a re T Q = 1 . 7 7 ° K f o r CH^D T Q = 1 . 2 6 ° K f o r C H D 3 . ( 6 - 3 3 ) F o r t h e s e sys tems , the e x p r e s s i o n f o r T^ g i v e n by e q u a t i o n ( 6 - 1 9 ) becomes where D = 1 6 i t w y 2 c c l ( 6 - 3 5 ) f H * - 7 € ( 6 - 3 6 ) and y ( x o l ) h a s been g i v e n by e q u a t i o n s ( 5 - 5 ) and ( 6 - 2 2 ) . I t i s seen f r o m e q u a t i o n s ( 6 - 3 4 ) and ( 6 - 3 6 ) t h a t 1^ i s dependent on temperature because the e f f e c t i v e s t r e n g t h of the s p i n -l a t t i c e i n t e r a c t i o n i s temperature dependent as g i v e n by 124 f ( T Q / T ) and x ^ i s temperature dependent because of the temper-a t u r e dependence of the c o r r e l a t i o n t ime t e ( . The dependence of y on x has been p l o t t e d i n F i g u r e ,(15). The dependence of f on T g / T i s shown i n F i g u r e ( 3 2 ) . The l i m i t i n g v a l u e of f a t the h i g h and low temperature ends c o r r e s p o n d s t o L i m ^ ( V r ) =. L i m [ (JO/T) = exp(-T0/R) (6-37) Thus f ( T g / T ) d e c r e a s e s r a p i d l y w i t h d e c r e a s i n g temperature f o r T « T Q . At t e m p e r a t u r e s TQ<< T f ( T Q / T ) c l o s e l y a p p r o x i m a t e s i t s a s y m p t o t i c v a l u e w h i c h i s s e n s i t i v e t o co . F o r co<*\, f i s a m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n of T , but f o r co> l f goes t h r o u g h a maximum. As a consequence , the dependence of the e f f e c t i v e s t r e n g t h of the s p i n - l a t t i c e i n t e r a c t i o n on temperature i s a s e n s i t i v e f u n c t i o n of u> . From the d i s c u s s i o n i n s e c t i o n 6 : 5 , the e f f e c t of the i n t e r a c t i o n between the m o l e c u l e s and the phonons on Xc{ can be e x p r e s s e d i n terms of the " d i r e c t ' 1 ' and "Raman" p r o -c e s s e s as f o l l o w s J _ _ 0 ^ * ^ " T ° / l " + B T » (6-38) 1- i - <S.~ T°/T ' 0 | 7 where the a p p r o x i m a t i o n s made i n d e r i v i n g the T term (Raman term) are t h a t T Q « T « 9^. At low t e m p e r a t u r e s the d i r e c t p r o c e s s predominates and f o r T << T , i t f o l l o w s t h a t t o l , and hence x Q l , i s independent of t e m p e r a t u r e . T h e r e f o r e , f o r T « T , T^ 125 F i g u r e 32. The Temperature Dependence of the E f f e c t i v e  I n t e r a c t i o n S t r e n g t h f o r S e v e r a l V a l u e s of OJ exp (+ T ^ T ) . F o r T » T , on the o t h e r hand, f ( T ^ / T ) i s independent of temperature and l / T 1 oc y ( X q 1 ) . The tempera-t u r e T m l n a t w h i c h the minimum i n v e r s u s T o c c u r s depends on A, B , T Q , and the f r e q u e n c y oo , w h i c h can be v a r i e d e x p e r i m e n t a l l y . F i g u r e (33) shows how T - . / T , depends on 1 -'-min T / T m l n f o r d i f f e r e n t v a l u e s of T o / T m l n These p l o t s a re v a l i d f o r a s p e c i a l case c h a r a c t e r i z e d by to = 3, B = 0. T h i s assumes t h a t the temperature i s ' s u f f i c i e n t l y low t h a t the Raman term i s n e g l i g i b l e . F o r c o m p a r i s o n , the e x p e r i m e n t a l data f o r C H ^ , CH^D and CD^ are p l o t t e d on the same g r a p h . I t i s seen t h a t the e x p e r i m e n t a l v a l u e s of T-^ f o r CD^ and CH^D v a r y much more r a p i d l y w i t h temperature j u s t below T m l n than any of the t h e o r e t i c a l c u r v e s . The most p l a u s i b l e r e a s o n f o r t h i s i s t h a t the Raman term has been n e g l e c t e d s i n c e B has been chosen t o be z e r o . That the Raman term i s p r o b a b l y i m p o r t a n t f o r these ca.ses i s i n d i c a t e d by F i g u r e s (19) and ( l 6 ) i n C h a p t e r 5, where the term g i v i n g the h i g h temperature b e h a v i o u r t r ' 1 oc T ^ i s s t i l l i m p o r t a n t a t T ^ . The r e a d e r i s r e m i n d e d , however, t h a t f ( T Q / T ) has been taken t o be c o n s t a n t i n the a n a l y s e s of F i g u r e s (19) and ( l 6 ) . < F o r C H ^ , F i g u r e (24) i n d i c a t e s t h a t the T ^ term i s s m a l l near T m i n * i n d e e d , the e x p e r i m e n t a l v a l u e s of T^ f o r CH^ are i n r e a s o n a b l e agreement w i t h the t h e o r e t i c a l c u r v e s of F i g u r e (33) f o r T ^ T A between 0.05 and 0.175 f o r T / T m l n between 0.5 and 1.5. Because t h e r e a r e s e v e r a l parameters i n the t h e o r e t i c a l e x p r e s s i o n whose v a l u e s have been Only the D i r e c t P r o c e s s 128 chosen a r b i t r a r i l y , t h i s agreement has o n l y q u a l i t a t i v e s i g n i f i c a n c e . C o l w e l l , G i l l and M o r r i s o n (1965) g i v e the parameters CO= 3 and T Q = 1 . 7 7 °K f o r C H 3 D . F o r t h i s v a l u e of T Q , f f T ^ T ) v a r i e s by o n l y about 10$ f o r T > T m l r i ' I f t h i s i s so and i f the c o n t r i b u t i o n to I / T Q I d u e t o t h e Raman term r e a l l y does 7 7 v a r y as T , then u s i n g the e x t r a p o l a t i o n of the T curve i n F i g u r e ( 1 6 ) , the f r a c t i o n a l c o n t r i b u t i o n of the B T 7 term t o - t ^ - j - 1 i n e q u a t i o n ( 6 - 3 8 ) may be e s t i m a t e d at T m i n = 10.1°K. T h i s i s s u f f i c i e n t i n f o r m a t i o n t o determine Ag^ and B i n ° o e q u a t i o n ( 6 - 3 8 ) s i n c e i t i s known t h a t o0Q X o l = 0 . 6 2 a t T m i n * T h e n u s i n g the v a l u e of T Q = 1.77°K, we have c a l c u l a t e d ' ^ 0 1 ~ 1 v e r s u s T f o r d i f f e r e n t v a l u e s of to as g i v e n by e q u a t i o n ( 6 - 3 8 ) . U s i n g the c a l c u l a t e d v a l u e s of f ( T Q / T ) shown i n F i g u r e (32) and e q u a t i o n ( 6 - 3 4 ) , c a l c u l a t e d v a l u e s of T-j^  v e r s u s T have been p l o t t e d i n F i g u r e (34) f o r co= 1/3, 1> 3 , and 9 . The e x p e r i m e n t a l v a l u e s of T^ v e r s u s T f o r CH^D are shown on the same p l o t . A comparison between F i g u r e s (33) and (34) shows t h a t t h i s crude c o r r e c t i o n f o r the Raman term r e s u l t s i n b e t t e r agreement between the model and e x p e r i m e n t , but i t i s i m p o s s i b l e t o d i s t i n g u i s h between d i f f e r e n t v a l u e s of a ) on the b a s i s of the a v a i l a b l e d a t a . B e t t e r agreement w i t h exper iment c o u l d be o b t a i n e d between 5°K and 10°K f o r UJ = 3, i f B were chosen t o be a l i t t l e l a r g e r , but the v e r y slow dependence of on T below 5°K c o u l d not be r e p r o d u c e d f o r to = 3 and T Q = 1 . 7 7 ° K. I t would appear t h a t v e r y s i m i l a r remarks a p p l y t o C D ^ H . T (°K) F i g u r e 3 4 . T h e o r e t i c a l Curves of T^ V e r s u s T T a k i n g i n t o Account the Raman Term f o r C H 0 D . a s D e s c r i b e d i n the Text 130 I t would be v e r y d e s i r a b l e t o have an independent check u s i n g NMR of the parameters T Q and ui w h i c h appear i n the model of C o l w e l l , G i l l and M o r r i s o n . To do t h i s 0^ must be s t u d i e d i n the r e g i o n T « T Q where T-^ ocexp ( T ^ T ) . T h i s means g o i n g down t o t e m p e r a t u r e s of the o r d e r of 0.3°K u s i n g a l i q u i d He^ c r y o s t a t . In t h i s r e g i o n , the r e l a t i o n s h i p OJ0%,»1 s h o u l d be s a t i s f i e d so t h a t ^ i s g i v e n by 2 f ( T / T ) 2 D & T « T ^ (6-39) T l 0 T 0 | ^ min F i g u r e (35) shows a p l o t of l o g ^ l / T ^ ) v e r s u s T ^ T f o r d i f f e r e n t v a l u e s of co i n the r e g i o n T « , T m l n i f o n l y " d i r e c t " p r o c e s s e s are i m p o r t a n t . The s tudy of T.^ down t o 0.3°K i s a v e r y i m p o r t a n t experiment i n s o f a r as the e v a l u a t i o n of the parameters i n t h i s model i s c o n c e r n e d , , I f a r a p i d l y i n c r e a s i n g i s not observed at t h e s e t e m p e r a t u r e s , then the assumption t h a t (st^y^~ C})Z made e a r l i e r would have t o be r e - c o n s i d e r e d (see s e c t i o n 6:8). S p i n T e m p e r a t u r e : 6:7 In the d i s c u s s i o n of the e x p e r i m e n t a l r e s u l t s i n t h i s t h e s i s i t has been assumed t h a t the a p p r o a c h t o e q u i l i b r i u m of the n u c l e a r m a g n e t i z a t i o n can be d e s c r i b e d i n terms of a s i n g l e t ime c o n s t a n t T ^ . E x p e r i m e n t a l l y , t h i s has i n d e e d been f o u n d t o be t r u e , except f o r some cases i n t h e v i c i n i t y 131 of the u n e x p l a i n e d h i g h temperature " T ^ minimum" i n C H ^ . Y e t , t h e r e are a t l e a s t two r e a s o n s f o r e x p e c t i n g a p o s s i b l e d e v i a t i o n f r o m e x p o n e n t i a l b e h a v i o u r . Case ( l ) The s p l i t t i n g s of the m o l e c u l a r energy l e v e l s may v a r y f rom m o l e c u l e to m o l e c u l e because the c r y s t a l -l i n e e l e c t r i c f i e l d s may v a r y . Thus , t h e r e may not be a u n i q u e c o r r e l a t i o n t ime f o r t h i s sys tem. T h i s seems t o be the case i n s o l i d H 2 below 2°K, where a n o n - e x p o n e n t i a l r e l a x a t i o n i s observed (Sugawara et a l . , 1956 and H a r d y , 1965). Case (2) There may e x i s t d i f f e r e n t s p i n s p e c i e s each h a v i n g n o n - z e r o n u c l e a r s p i n s . I f s o , and i f these s p e c i e s have d i f f e r e n t o r i e n t a t i o n a l s t a t e s , t h e y may r e l a x a t d i f f e r e n t r a t e s . Indeed, one i m p l i c a t i o n of T o m i t a ' s (1953) a n a l y s i s of the low t e m p e r a t u r e , p r o p e r t i e s of CH^ i s t h a t the A s p e c i e s ( I = 2) cannot r e l a x . A c c o r d i n g t o h i s r e s u l t s , o n l y the T s p e c i e s (I = l ) can r e l a x . The two e f f e c t s d e s c r i b e d above need not m a n i f e s t them-s e l v e s i n a n o n - e x p o n e n t i a l r e l a x a t i o n b e h a v i o u r . The r e a s o n f o r t h i s i s i n d i c a t e d i n t h e f o l l o w i n g diagram ( F i g u r e 36) i n w h i c h we imagine two s p i n s p e c i e s a and b w h i c h have s p i n -a b l a t t i c e r e l a x a t i o n t imes T^ and T^ r e s p e c t i v e l y . The two s p i n systems can a l s o exchange energy w i t h each o t h e r a n d , i f not c o u p l e d to the l a t t i c e , t h e i r s p i n tempera tures ab a p p r o a c h a common s p i n temperature w i t h a t ime c o n s t a n t T 0 132 F i g u r e 35. T h e o r e t i c a l P l o t s of l / r . Versus T T~'in the Low Temperature L i m i t , <^ >p t r 0 > > l , I n c l u d i n g D i r e c t process uniy 133 SPIN SYSTEM a T2at SPIN SYSTEM b LATTICE F i g u r e 36. I l l u s t r a t e s the C o u p l i n g of Two S p i n Systems  ' a ' and ' b 1 ' t o the L a t t i c e and t o Each Other 134 I f T 2 << , , the s p i n tempera tures of the a and ID systems are a lmost e q u a l a t a l l t imes and i t may be shown t h a t the m a g n e t i z a t i o n of the composi te system approaches e q u i -l i b r i u m w i t h the l a t t i c e e x p o n e n t i a l l y w i t h a t ime c o n s t a n t 1 ° A 1 + — -TT (6-40) T l 6 f f " C a + C b T i a C a + C b T l b where 0 and C, a re the heat c a p a c i t i e s of the a and b systems a D r e s p e c t i v e l y . ab The t ime c o n s t a n t Tg s h o u l d be c l o s e l y r e l a t e d t o the " t r a n s v e r s e r e l a x a t i o n t i m e " Tg w h i c h may be measured f o r the composi te systems as the t ime c o n s t a n t f o r the decay of the x - y components of m a g n e t i z a t i o n . F o r CH^ at low t e m p e r a t u r e s Tg = 12 jlisecs, w h i l e f o r C D ^ , Tg = 5 0 0 /usees so t h a t the c o n -d i t i o n s r e q u i r e d f o r e q u a t i o n ( 6 - 4 0 ) t o be v a l i d would seem t o be w e l l s a t i s f i e d . The e x p e r i m e n t a l r e s u l t s g i v e a p o s t i o r i j u s t i f i c a t i o n of t h i s . I f the a and b systems a r e a s s o c i a t e d w i t h d i f f e r e n t l o c a l e n v i r o n m e n t s (Case ( i ) ) , the e f f e c t of a d i s t r i b u t i o n of c o r r e l a t i o n t imes would r e s u l t i n an i n c r e a s e i n the v a l u e of T , over t h a t p r e d i c t e d f o r a s i n g l e c o r r e l a t i o n t i m e . • xm±n F o r the s p e c i a l example of case (2) f o r CH^ g i v e n above, i f a i s a s s o c i a t e d w i t h A symmetry m o l e c u l e s and b w i t h T symmetry m o l e c u l e s , so that- l / T ^ A = o, then C m T 1 e f f C A H *°T T 1 T 135 w i t h C T N T I T ( I T + 1) = N A ZA ^ A + ^ where N ^ , are the number of m o l e c u l e s h a v i n g A and T symmetry r e s p e c t i v e l y and T_A T_T a re the t o t a l n u c l e a r angu-l a r momenta f o r m o l e c u l e s h a v i n g A and T symmetry r e s p e c t i v e l y . F o r C H ^ , i t i s known t h a t I^ = 2, 1^  = 1, w h i l e the h i g h temp-e r a t u r e r a t i o of N A / N T i s 5/9. I f t h i s r a t i o i s unchanged • a t low t e m p e r a t u r e s (no c o n v e r s i o n between d i f f e r e n t s p i n s p e c i e s ) 1 3 1 e f f " H T 1 1 T h i s would a l s o g i v e r i s e to an i n c r e a s e i n the p r e d i c t e d v a l u e of T . min Some Comments about M a t r i x E lements of the I n t r a m o l e c u l a r I n t e r a c t i o n s : 6 7 8 To c o n c l u d e t h i s d i s c u s s i o n of some of the m a n i f e s t a t i o n s of the quantum n a t u r e of the methane system, one more a s p e c t of t h i s p r o b l e m w i l l be d i s c u s s e d , namely, the e f f e c t t h a t r e s -t r i c t i n g the m o l e c u l e s t o a s m a l l se t of s t a t e s has on the c o r r e l a t i o n f u n c t i o n . In the c l a s s i c a l t rea tment of r o t a t i o n a l Brownian m o t i o n , the m o l e c u l e s a re assumed to have a d e f i n i t e o r i e n t a t i o n _n__( t ) , i . e . , the wave f u n c t i o n , i n v o l v e s a s u p e r p o s i t i o n of a l l J , rrij s t a t e s . The i n t r a m o l e c u l a r i n t e r a c t i o n s of the f o r m Y 2 m(-fl) a re c o m p l e t e l y s p e c i f i e d Toy jn - 0 ( t ) a t a p a r t i c u l a r t i m e . In the o p p o s i t e extreme, f o r an ensemble of non-i n t e r a c t i n g symmetric top m o l e c u l e s where J , K, M are good quantum numbers, the o r i e n t a t i o n of each m o l e c u l e i s i n d e t e r m i n -a t e . The e i g e n f u n c t i o n of the system of N m o l e c u l e s i s the p r o d u c t of the N f r e e r o t o r wave f u n c t i o n s \ J K M ^ . The' c o r r e l a t i o n f u n c t i o n of ^ m ^ " * " ^ i s "then where Y 2 m(.G - ( t ) ) i s a t i m e - d e p e n d e n t o p e r a t o r i n the H e i s e n b e r g r e p r e s e n t a t i o n and p i s the d e n s i t y m a t r i x d e s c r i b i n g the e q u i l i b r i u m p o p u l a t i o n s of the v a r i o u s l e v e l s . The c o r r e l a t i o n f u n c t i o n s can be w r i t t e n as f o l l o w s : co t r J" G ( * ) = 2 1 2 1 IT P r k M |<T'*'M'\ Y ^ I J ^ H ) ^ ^ - ^ , ^ O - J M - - J where E_ K = ^ J V = * and Pj- =• e x p -(E TKT/KT)  S I ^ + e x P - ( e ^ / K T ) F o r s p h e r i c a l top m o l e c u l e s , the e x p r e s s i o n may'be r e w r i t t e n Now the i n t e r a c t i o n s between the m o l e c u l e s can be i n t r o d u c e d i n an a r b i t r a r y manner by m u l t i p l y i n g G ( t ) by an e x p o n e n t i a l f u n c t i o n e~ where "ee i s then a measure of the t ime i n t e r v a l , d u r i n g w h i c h the m o l e c u l e s behave e s s e n t i a l l y as f r e e r o t o r s . Now, i f co0 and l / t ^ a r e a p p r e c i a b l y s m a l l e r than any f r e q u e n c i e s o c c u r r i n g i n the o s c i l l a t o r y terms , then these terms can be n e g l e c t e d and one sees t h a t G ( f ) has 1/5 of i t s c l a s s i c a l v a l u e . At s u f f i c i e n t l y low tempera tures the c o r r e l a t i o n t ime w i l l be l o n g enough f o r t h i s c o n d i t i o n t o be s a t i s f i e d . T h i s i s a n o t h e r p o s s i b l e c o n t r i b u t i n g f a c t o r to the l e n g t h e n i n g of (T-, ) ' > 1 min The above c o n s i d e r a t i o n s are i n t e n d e d o n l y t o i l l u s t r a t e how o f f - d i a g o n a l e lements of the i n t r a - m o l e c u l a r i n t e r a c t i o n s can be s u p p r e s s e d i n s o f a r as t h e i r c o n t r i b u t i o n t o s p i n -l a t t i c e r e l a x a t i o n i s c o n c e r n e d . They are not i n t e n d e d to i m p l y t h a t the wave f u n c t i o n s of methane m o l e c u l e s i n the s o l i d are w e l l approximated by f r e e r o t o r wave f u n c t i o n s . T h e model of C o l w e l l , G i l l and M o r r i s o n does i m p l i c i t l y assume t h a t the wave f u n c t i o n s of s o l i d methane can be w r i t t e n as a p r o d u c t of N s i n g l e m o l e c u l e wave f u n c t i o n s . No d e t a i l e d d e s c r i p t i o n of these s i n g l e m o l e c u l e wave f u n c t i o n s has y e t been g i v e n . W i t h i n the c o n t e x t of the above remarks t h i s i s an a p p r o p r i a t e t ime to-, make some comments about the s p i n - r o t a t i o n  i n t e r a c t i o n s . When the d e g e n e r a c i e s of the m o l e c u l a r r o t a t i o n a l l e v e l s a re c o m p l e t e l y l i f t e d by the c r y s t a l l i n e e l e c t r i c f i e l d , the r o t a t i o n a l a n g u l a r momentum J i s "quenched" (Abragam, 196l, p . 174). T h i s means t h a t the o n l y n o n - z e r o m a t r i x e lements of J a re between s t a t e s of d i f f e r e n t e n e r g y . F o r 138 t h i s r e a s o n , a c c o r d i n g t o the p r e c e d i n g argument, the c o n t r i -b u t i o n of the s p i n - r o t a t i o n i n t e r a c t i o n t o n u c l e a r s p i n r e l a x -a t i o n i s expec ted t o be s m a l l i n s o l i d s . In f a c t , the s p i n - r o t a t i o n i n t e r a c t i o n has been com-p l e t e l y n e g l e c t e d i n t h i s t h e s i s . F o r cases i n w h i c h degener-a c i e s do e x i s t , i t i s not c l e a r t h a t t h i s n e g l e c t i s j u s t i f i e d even i n s o l i d s i n w h i c h the m o l e c u l e i s s u b j e c t e d t o l a r g e c r y s t a l l i n e f i e l d s . In the d i s c u s s i o n of the model p r o p o s e d f o r s o l i d methane, i t has been assumed t h a t f o r s t a t e s i and j 2 2 w h i c h have d e g e n e r a c i e s g Q and g^ r e s p e c t i v e l y , <(y ^ = <^ y^ > ^ t h a t these e q u a l i t i e s a re s a t i s f i e d i n many c a s e s o f i n t e r e s t f o r y = Y 2 m , f o r example . I t i s q u i t e c l e a r t h a t f o r the examples g i v e n i n s e c t i o n 6:6B, these e q u a l i t i e s c o u l d not be s a t i s f i e d f o r m a t r i x e lements of the o p e r a t o r J i n the s p i n -r o t a t i o n i n t e r a c t i o n . Then , the assumption t h a t o n l y the l a s t of the t h r e e terms i n e q u a t i o n s (6-l6) i s n o n - z e r o , would not be v a l i d and the c o r r e l a t i o n t imes X0 and T., would e n t e r i n t o the e x p r e s s i o n f o r T , . and (see s e c t i o n 6 :6B). I t was p o i n t e d out CHAPTER 7. SUMMARY In t h i s t h e s i s , the r e s u l t s of a s tudy of the r e l a x a t i o n p r o p e r t i e s of the methane system a t low t e m p e r a t u r e s have been p r e s e n t e d . These were s t u d i e d w i t h the u l t i m a t e a im of g a i n i n g f u r t h e r i n s i g h t i n t o the n a t u r e of the phase t r a n s i -t i o n s . To a c h i e v e t h i s a i m , N . M . R . p u l s e t e c h n i q u e s were used w i t h the hope t h a t t h r o u g h the a p p l i c a t i o n of c o n v e n t i o n a l N . M . R . t h e o r y , i n f o r m a t i o n about the n a t u r e of the r e l a x a t i o n mechanisms and the r e o r i e n t a t i o n a l motions c o u l d be o b t a i n e d . I n s t e a d , i t was found t h a t the c o n v e n t i o n a l t h e o r y i s i n a d e -q u a t e . The e x p e r i m e n t a l r e s u l t s and the a n a l y s i s of the e x p e r i m e n t a l r e s u l t s u s i n g the c o n v e n t i o n a l t h e o r y have a l r e a d y been summarized a t the end of Chapter 5. The r e s u l t s d e f i n i t e l y i n d i c a t e t h a t the r e o r i e n t a t i o n a l motion of the methane mole-c u l e i s a t l e a s t p a r t i a l l y quenched at the phase t r a n s i t i o n s . As a consequence , an at tempt was made i n Chapter 6, t o modify the t h e o r y such as t o take account of some of the e f f e c t s due t o the quantum n a t u r e of the sys tem. The model f o r the low l y i n g energy l e v e l s of s o l i d methane p r o p o s e d by C o l w e l l , G i l l , and M o r r i s o n (1965) was used as the s t a r t i n g p o i n t f o r our c o n s i d e r a t i o n s i n C h a p t e r 6; the consequences of t h e i r model as f a r as the r e l a x a t i o n p r o p e r t i e s of methane are concerned were examined i n some d e t a i l . The f o l l o w i n g consequences were e s t a b l i s h e d t h e r e : 139 l4o (1) under c o n d i t i o n s s a t i s f i e d by many (but not a l l ) systems of i n t e r e s t , i t was found t h a t a s i n g l e e x p o n e n t i a l does i n d e e d d e s c r i b e the c o r r e l a t i o n f u n c t i o n . (2) The e f f e c t i v e s t r e n g t h of the i n t e r a c t i o n has been found t o be temperature dependent f o r a two l e v e l system, thus i n v a l i d a t i n g the a n a l y s i s of the e x p e r i m e n t a l r e s u l t s c a r r i e d out i n Chapter 5. (3) I t i s p o s t u l a t e d t h a t p h o n o n - m o l e c u l a r i n t e r a c t i o n s i n v o l v i n g d i r e c t (one phonon) and Raman (two phonon) p r o c e s s e s cause t r a n s i t i o n s between the energy l e v e l s . The observed temperature dependence of the c o r r e l a t i o n t ime i s c o n s i s t e n t w i t h t h i s p i c t u r e , a t l e a s t q u a l i t a t i v e l y . (4) Some c o n s i d e r a t i o n s l e a d i n g t o a r e d u c t i o n i n the s t r e n g t h of the r e l a x a t i o n p e r t u r b a t i o n and c o n s e q u e n t l y i n a l e n g t h e n i n g of were d i s -cussed i n Chapter 6. The f o l l o w i n g c o n c l u s i o n s can be drawn f r o m t h i s s tudy of the r e l a x a t i o n p r o p e r t i e s of the methane sys tem. The m o l e c u l a r r e o r i e n t a t i o n s are d e f i n i t e l y p a r t i a l l y quenched at the phase t r a n s i t i o n s , but not c o m p l e t e l y . I t seems l i k e l y t h a t the d i r e c t . a n d Raman p r o c e s s e s i n p a r t account f o r the observed temperature dependence of the observed c o r r e l a t i o n t ime at low t e m p e r a t u r e ; t h i s t o our knowledge i s the f i r s t t ime t h a t these t y p e s of p r o c e s s e s have been observed t o p l a y a r o l e i n the magnet ic r e l a x a t i o n i n d i a m a g n e t i c 141 s u b s t a n c e s . The c o n v e n t i o n a l t h e o r y appears i n c a p a b l e of e x p l a i n i n g the d e t a i l s of the r e l a x a t i o n phenomena i n s o l i d methane. In c o n c l u s i o n , some f u r t h e r exper iments w i l l be p r o -p o s e d . In Chapter 6, i t was p r o p o s e d t h a t c a r r y i n g the r e l a x a t i o n measurements down t o 0".3°K a l l o w s a d e t e r m i n a t i o n of T . A word of w a r n i n g i s a p p r o p r i a t e h e r e ; i f the s p i n -r o t a t i o n i n t e r a c t i o n i s an e f f e c t i v e r e l a x a t i o n p e r t u r b a t i o n , then the c o n s i d e r a t i o n s w h i c h l e d us t o propose t h a t e x p e r i -ment are not c o r r e c t . I t would be i n t r i g u i n g t o measure the r e l a x a t i o n p r o p e r t i e s of a C H ^ - K r m i x t u r e c o n t a i n i n g a v e r y s m a l l amount of CH^ i n v iew of the d r a s t i c changes e x h i b i t e d by the 50$Kr-50$CH^ d a t a . A more d e t a i l e d e x a m i n a t i o n of the upper minimum i n T-j_ f o r C H ^ , and CH^ m i x t u r e s w i t h CD^ of K r , i f p e r f o r m e d c a r e f u l l y , may a l l o w a s e p a r a t i o n of the two t ime c o n s t a n t s o b s e r v e d . 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