The U n i v e r s i t y 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 of SURENDRA NATH SHARMA M. Sc. Un i v e r s i t y of B r i t i s h Columbia, 1962 FRIDAY, APRIL 28th, 1.9.67.,.. at 3:30 P.M. IN ROOM 301, PHYSICS (HENNLNGS.) BUILDING COMMITTEE IN CHARGE Chairman.: S..D. Cavers M. Bloom E. Teghtsoonian L.W. Reeves B.G. T u r r e l l C.F. Schwerdtfeger D.L1-. Williams External Examiner: R.G. Barnes Department of Physics Iowa State Uni.ver.sity.. . Ames, U.S.A.' Research Supervisor: D..L1. Williams NUCLEAR MAGNETIC RESONANCE,IN SINGLE CRYSTALS OF TIN AND CADMIUM ABSTRACT A systematic study of the Knight s h i f t para-meters has been c a r r i e d out on single c r y s t a l specimens of t i n ..and .cadmium over a range from 1 K to just below the .melting-points. The l i n e widths observed i n cadmium are.,.approximately ha l f the widths quoted by other workers enabling more precise measure-ments to be made. These measurements show the aniso-tropy of the Knight s h i f t - i n cadmium..changes sign between 4 K- and 77. .K Pronounced ..changes i n the Knight s h i f t parameters are observed upon a l l o y i n g cadmium with mercury. The temperature...dependence of the iso.tropic Knight s h i f t i n t i n can be explained by the volume e f f e c t s and.the.phonon contributions when the.temp-erature exceeds the Debye ...temperature. Comparing the temperature dependence data with pressure dependence r e s u l t s of other workers.the,.explicit temperature-dependence i s extracted.. It appears that the v a r i a t i o n i n the anisotropy of the Knight s h i f t above the Debye temperature i s l a r g e l y caused by a change in l a t t i c e parameters, s p e c i f i c a l l y a change in,-c/a r a t i o . Measurements, of the l i n e width^a^ helium temperature i n i s o t o p i c a l l y .pure Sn and the. same isotope i n natural .tin as a function of,crys t a l o r i e n t a t i o n are reported. Second moments for d i f f e r e n t c r y s t a l orientations have .been .computed Analysis of the. i s o t o p i c a l l y .pure t i n data leads an evaluation of r e l a t i v e contributions•from the various s h e l l s , to the pseudo-dipolar i n t e r a c t i o n For the i s o t o p i c a l l y pure t i n , the measurements at the helium and room temperatures enable an eval-uation of the spin l a t t i c e r e l a x a t i o n time T^. T^T i s constant over the temperature range and c l o s e l y i s o t r o p i c . GRADUATE STUDIES i e l d of Study: Nuclear Magnetic Resonance Seminar in NMR -D.Ll. Williams Quantum Theory of Solids-R. Barrie S t a t i s t i c a l Mechanics -R; Barrie Advanced Magnetism -M. Bloom Low Temperature Physics-J.B. Brown E l e c t r o n i c Instrumentation -F.K.Bowers PUBLICATION "The Temperature Dependence of the Knight S h i f t i n T i n and Cadmium" S.N. Sharma and D . L l . -Williams, Proceedings of the International Conference on Magnetic Resonance and Relaxation (Ljubljana, September 1966.) (In press). NUCLEAR MAGNETIC RESONANCE I N S I N G L E C R Y S T A L S OF T I N AND CADMIUM b y SURENDRA NATH' SHARMA M . S c . , T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1962 A T H E S I S SUBMITTED I N P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF . DOCTOR OF PHILOSOPHY i n t h e D e p a r t m e n t Of > P H Y S I C S We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE U N I V E R S I T Y OF B R I T I S H COLUMBIA M a r c h , 1967 @ Surendra Nath Sharma 1967 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h s L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a nd s t u d y , I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t 'my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f ft(L^jn,6<^>f T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a - i i -A B STRACT A s y s t e m a t i c s t u d y o f t h e K n i g h t s h i f t p a r a m e t e r s h a s b e e n c a r r i e d o o u t o n s i n g l e c r y s t a l s p e c i m e n s o f t i n a n d c a dmium o v e r a r a n g e f r o m 1 K t o j u s t b e l o w t h e m e l t i n g p o i n t s . The l i n e w i d t h s o b s e r v e d i n c a dmium a r e a p p r o x i m a t e l y h a l f t h e w i d t h s q u o t e d b y o t h e r w o r k e r s , e n a b l i n g more p r e c i s e m e a s u r e m e n t s t o be made. T h e s e m e a s u r e m e n t s show t h a t t h e a n i s o -o o t r o p y o f t h e K n i g h t s h i f t i n c a dmium c h a n g e s s i g n b e t w e e n 4 K a n d 77 K. P r o n o u n c e d c h a n g e s i n t h e K n i g h t s h i f t p a r a m e t e r s a r e o b s e r v e d u p o n a l l o y -i n g c a d mium w i t h m e r c u r y . The t e m p e r a t u r e d e p e n d e n c e o f t h e i s o t r o p i c K n i g h t s h i f t i n t i n c a n be e x p l a i n e d b y t h e v o l u m e e f f e c t s a n d t h e p h o n o n c o n t r i b u t i o n s when t h e t e m p e r a t u r e e x c e e d s t h e D e b y e t e m p e r a t u r e . C o m p a r i n g t h e t e m p e r a t u r e d e p e n d e n c e d a t a w i t h p r e s s u r e d e p e n d e n c e r e s u l t s o f o t h e r w o r k e r s t h e e x p l i c i t t e m p e r a t u r e d e p e n d e n c e i s e x t r a c t e d . I t a p p e a r s t h a t t h e v a r i a t i o n i n t h e a n i s o t r o p y o f t h e K n i g h t s h i f t a b o v e t h e D e b y e t e m p e r a t u r e i s l a r g e l y c a u s e d b y a c h a n g e i n l a t t i c e p a r a m e t e r s , s p e c i f i c a l l y a c h a n g e i n c / a r a t i o . M e a s u r e m e n t s o f t h e l i n e w i d t h a t h e l i u m t e m p e r a t u r e i n i s o t o p i c -a l l y p u r e S n ^ ^ a n d t h e same i s o t o p e i n n a t u r a l t i n a s a f u n c t i o n o f c r y s t a l o r i e n t a t i o n a r e r e p o r t e d . S e c o n d moments f o r t h e d i f f e r e n t c r y s t a l o r i e n t a t i o n s h a v e b e e n c o m p u t e d . A n a l y s i s o f t h e i s o t o p i c a l l y p u r e t i n d a t a l e a d s t o a n e v a l u a t i o n o f r e l a t i v e c o n t r i b u t i o n s , f r o m t h e v a r i o u s s h e l l s , t o t h e p s e u d o - d i p o l a r i n t e r a c t i o n . F o r t h e i s o t o p i c a l l y p u r e t i n , t h e m e a s u r e m e n t s a t t h e h e l i u m a n d r o o m t e m p e r a t u r e s e n a b l e a n e v a l u a t i o n o f t h e s p i n - l a t t i c e r e l a x a t i o n t i m e , T]_. T^T i s c o n s t a n t o v e r t h e t e m p e r a t u r e r a n g e a n d c l o s e l y i s o t r o p i c . - i i i -TABLE OF CONTENTS CHAPTER Page 1 INTRODUCTION 1 2 EQUIPMENT 4 3 REVIEW OF THE THEORY OF THE KNIGHT SHIFT 3.1 The Knight S h i f t 11 3.2 O r b i t a l Paramagnetism 13 3.3 Core P o l a r i z a t i o n 15 3.4 Anisotropy 16 3.5 Temperature Dependence of the Knight s h i f t 18 4 EXPERIMENTAL RESULTS AND piSCUSSION 20 4.1 Tin , 20 4.2 Cadmium 24 4.3 Discussion 31 5 THEORY OF LINE-WIDTH AND LINE SHAPES 45 5.1 Dipolar Broadening 46 5.2 Indirect Exchange Interaction 48 5.3 Pseudo-dipolar Broadening 52 6 EXPERIMENTAL RESULTS AND DISCUSSION 56 SUGGESTIONS FOR FURTHER EXPERIMENTS 69 APPENDIX A Second Moment i n I s o t o p i c a l l y Pure T i n as a Function of Orientation 70 B Determination of the Correct Resonance Frequency i n Tin 72 REFERENCES 75 - I V -L I S T OF I L L U S T R A T I O N S F I G U R E Page 1 B l o c k D i a g r a m o f t h e S p e c t r o m e t e r 5 2 S c h e m a t i c o f Low T e m p e r a t u r e S y s t e m 8 3 The I s o t r o p i c K n i g h t S h i f t i n T i n a s a F u n c t i o n o f T e m p e r a t u r e 21 4 A n i s o t r o p y i n t h e K n i g h t S h i f t i n T i n a s a F u n c t i o n o f T e m p e r a t u r e 22 5 T h e K n i g h t S h i f t i n T i n , f o r V a r i o u s T e m p e r a t u r e s , a s a F u n c t i o n o f C r y s t a l O r i e n t a t i o n 23 6 T h e I s o t r o p i c K n i g h t S h i f t i n Cadmium a s a F u n c t i o n o f T e m p e r a t u r e 26 7 A n i s o t r o p y i n t h e K n i g h t S h i f t i n Cadmium a s a F u n c t i o n o f T e m p e r a t u r e 27 8 The K n i g h t S h i f t i n Cadmium, f o r V a r i o u s T e m p e r a t u r e s , a s a F u n c t i o n o f C r y s t a l O r i e n t a t i o n 28 9 The K n i g h t S h i f t i n Cadmium, a t L i q u i d N i t r o g e n a n d H e l i u m T e m p e r a t u r e s , a s a F u n c t i o n o f C r y s t a l O r i e n t a t i o n 29 10 D e r i v a t i v e s o f Cadmium R e s o n a n c e S i g n a l s 30 11 E x p l i c i t T e m p e r a t u r e D e p e n d e n c e o f t h e K n i g h t S h i f t i n T i n a s a F u n c t i o n o f T e m p e r a t u r e 34 12 E x p l i c i t T e m p e r a t u r e D e p e n d e n c e o f t h e K n i g h t S h i f t i n Cadmium a s a F u n c t i o n o f T e m p e r a t u r e 38 13 A n i s o t r o p y o f t h e K n i g h t S h i f t v s . c / a R a t i o f o r T i n 41 14 A n i s o t r o p y o f t h e K n i g h t S h i f t v s . c / a R a t i o f o r Cadmium . 43 15 The L i n e - W i d t h o f t h e I s o t o p i c a l l y P u r e T i n a s a F u n c t i o n 1 o f C r y s t a l O r i e n t a t i o n 5 7 16 The L i n e W i d t h o f t h e I s o t o p i c a l l y P u r e T i n i n t h e B a s a l P l a n e 58 119 17 T h e L i n e W i d t h o f Sn i n N a t u r a l T i n i n t h e B a s a l P l a n e 59 s2Q , \2r-6 ' W i j ; r i j 19 P l o t o f J~(1-3 c o s 2 9 . . ) 2 r T 6 v s . <$> ( i n - t h e B a s a l P l a n e ) 61 20 P l o t o f B ^ j a n d J ^ j v s . r ^ j 66 18 P l o t s o f 2~(l-3 c o s ^ Q , -)^r7^ v s . 9 f o r V a r i o u s V a l u e s o f 60 . n s2 - - v -ACKNOWLEDGEMENT I w i s h t o e x p r e s s my s i n c e r e g r a t i t u d e t o D r . D. L I . W i l l i a m s f o r h i s c o n s t a n t h e l p a n d e n c o u r a g e m e n t t h r o u g h o u t t h i s w o r k . I am i n d e b t e d t o P r o f . M y e r B l o o m f o r h i s v a l u a b l e h e l p i n i n t e r -p r e t i n g some o f my r e s u l t s . I am g r a t e f u l t o D r . B. G. T u r r e l l f o r h e l p f u l d i s c u s s i o n s , t o D r . R. H o w a r d a n d K r i s h n a M. A g r a w a l f o r h e l p i n g i n t h e c o m p u t a t i o n o f some o f my d a t a o n t h e c o m p u t e r a n d t o D r . H. E . S c h o n e f o r t h e l o a n o f t h e i s o t o p i c a l l y p u r e t i n s i n g l e c r y s t a l . T h e c h e e r f u l a s s i s t a n c e o f t h e P h y s i c s W o r k s h o p s s t a f f i n t e c h n i -c a l m a t t e r s i s h i g h l y a p p r e c i a t e d T h e f i n a n c i a l a s s i s t a n c e o f t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a i n t h e f o r m o f summer r e s e a r c h g r a n t s i s g r a t e f u l l y a c k n o w l e d g e d CHAPTER I INTRODUCTION When a n a s s e m b l y o f i d e n t i c a l n u c l e i , e a c h p o s s e s s i n g a m a g n e t i c moment a n d a n g u l a r momentum I , i s s u b j e c t e d t o a d . c . m a g n e t i c f i e l d , t h e n u c l e i p r e c e s s a r o u n d t h e f i e l d w i t h t h e L a r m o r f r e q u e n c y — I f now a n r . f . f i e l d i s a p p l i e d p e r p e n d i c u l a r t o t h i s d . c . f i e l d s u c h t h a t t h e r a d i o f r e q u e n c y c o r r e s p o n d s t o t h e L a r m o r f r e q u e n c y , t h e s y s t e m a b s o r b s e n e r g y . T h i s i s t h e m a g n e t i c r e s o n a n c e p henomenon. I n p r a c t i c e , h o w e v e r , t h e f i e l d s e e n b y a g i v e n n u c l e u s i s d i f f e r e n t f r o m t h e a p p l i e d f i e l d b e c a u s e i t a l s o f e e l s a ' l o c a l ' f i e l d f r o m s u r r o u n d i n g e l e c t r o n s a n d n e i g h b o u r i n g n u c l e i . The L a r m o r f r e q u e n c y t h e n i s d e f i n e d i n t e r m s o f t h e v e c t o r sum o f t h e a p p l i e d a n d l o c a l f i e l d s , a n d t h e n u c l e u s c a n be u s e d a s a p r o b e t o m e a s u r e t h i s l o c a l m a g n e t i c f i e l d . The e x p e r i m e n t c o n s i s t s o f o b s e r v i n g t h e r a t e o f e n e r g y a b s o r p t i o n w i t h v a r y i n g r a d i o f r e q u e n c y a n d a r o u n d t h e r e s o -n a n c e f r e q u e n c y a s p e c t r a l ' l i n e ' i s o b s e r v e d . The s h a p e o f t h i s l i n e a n d t h e r e s o n a n c e f r e q u e n c y a r e p a r a m e t e r s o f p h y s i c a l i n t e r e s t . 20 W h i l e i n v e s t i g a t i n g NMR i n m e t a l s W.D. K n i g h t f o u n d t h a t t h e r e s o n a -n c e f r e q u e n c y o f a n y p a r t i c u l a r i s o t o p e i n a m e t a l was s h i f t e d f r o m t h e r e s o n a n c e f r e q u e n c y m e a s u r e d i n a n o n - m e t a l l i c c ompound i n t h e same e x t e r n a l m a g n e t i c f i e l d . T h i s s h i f t i n f r e q u e n c y i s known a s t h e K n i g h t s h i f t a n d i s u s u a l l y p o s i t i v e , I t c a n be e x p l a i n e d b y t a k i n g i n t o a c c o u n t t h e h y p e r f i n e i n t e r a c t i o n b e t w e e n t h e n u c l e u s a n d t h e c o n d u c t i o n e l e c t r o n s . T h e a p p l i e d m a g n e t i c f i e l d i n d u c e s a p o l a r i z a t i o n o f t h e c o n d u c t i o n e l e c t r o n s p i n s w h i c h i s t h e n f e l t b y t h e n u c l e u s . T h u s t h e K n i g h t s h i f t c a n p r o v i d e v a l u -a b l e i n f o r m a t i o n a b o u t t h e c o n d u c t i o n e l e c t r o n s n e a r t h e F e r m i s u r f a c e i n t h e m e t a l s . i I n t h e p a s t t h e e x p e r i m e n t s o n t h e K n i g h t s h i f t m e a s u r e m e n t s w e r e l i m i t e d t o p o w d e r e d s a m p l e s w i t h p a r t i c l e s i z e l e s s t h a n t h e e l e c t r o -m a g n e t i c s k i n d e p t h , i n o r d e r t o a c h i e v e s u b s t a n t i a l p e n e t r a t i o n o f t h e r . f . f i e l d i n t o t h e s a m p l e . F o r a c u b i c c r y s t a l t h e K n i g h t s h i f t i s i n d e p e n d e n t o f c r y s t a l o r i e n -t a t i o n w i t h r e s p e c t t o t h e e x t e r n a l m a g n e t i c , f i e l d . T h i s i s n o t t r u e f o r m e t a l s w i t h c r y s t a l s y m m e t r y l o w e r t h a n c u b i c . E x p e r i m e n t s o n t h e p o w d e r s o f s u c h m e t a l s r e s u l t i n a b r o a d l i n e due t o t h e r a n d o m o r i e n t a t i o n o f t h e c r y s t a l l i t e s . A n y o r i e n t a t i o n d e p e n d e n c e o f t h e K n i g h t s h i f t a n d l i n e s h a p e c a n c l e a r l y b e o b s e r v e d i f s i n g l e c r y s t a l s p e c i m e n s a r e u s e d . T h i s w o u l d r e s u l t i n c o n s i d e r a b l y n a r r o w e r l i n e s h e n c e i n c r e a s i n g t h e a c c u r a c y o f t h e m e a s u r e m e n t s . H o w e v e r , NMR e x p e r i m e n t s i n s i n g l e c r y s t a l s a r e n o t a s e a s y a s i t m i g h t a p p e a r f r o m r e a d i n g t h e l a s t p a r a g r a p h f o r t h e f o l l o w i n g r e a s o n s . ( i ) I n a s i n g l e c r y s t a l e x p e r i m e n t t h e f r a c t i o n o f t h e t o t a l number o f n u c l e i p a r t i c i p a t i n g i n t h e r e s o n a n c e p h e n o m e n o n i s v e r y s m a l l a s c o m p a r e d t o t h a t i n a p o w d e r s p e c i m e n . S i n c e o n l y n u c l e i w h i c h h a p p e n t o be s i t u a -t e d i n t h e s k i n d e p t h r e g i o n , w h i c h i s a f e w m i c r o n s o n l y , t a k e p a r t w h i l e t h e r e s t o f t h e n u c l e i i n t h e c r y s t a l a r e s i t t i n g i d l e . T h i s r e s u l t s i n a p o o r s i g n a l t o n o i s e r a t i o w h i c h n e c e s s i t a t e s a s e n s i t i v e s e t o f e q u i p -ment. ( i i ) T h e o b s e r v e d r e s o n a n c e f r e q u e n c y i s n o t t h e t r u e r e s o n a n c e f r e q u e n c y . T h i s i s a l s o c a u s e d b y s k i n d e p t h e f f e c t s a n d c o m p l i c a t e s t h e a n a l y s i s o f t h e d a t a . The f i r s t r e a s o n i s a m a j o r h u r d l e a n d o n l y b e c a u s e o f t h i s p r e v i o u s e x p e r i m e n t s h a v e b e e n l i m i t e d t o t h e p o w d e r s p e c i m e n s . A t t h e commencement o f t h i s w o r k two e x p e r i m e n t s ) h a d b e e n (33 ) r e c e n t l y p e r f o r m e d o n m e t a l s i n g l e c r y s t a l s b u t S c h o n e \ ' h a s s i n c e a l s o s t u d i e d cadmium. Th e o b j e c t o f t h e p r e s e n t w o r k was ( i ) t o s t u d y t h e K n i g h t s h i f t p a r a m e t e r s i n t i n a n d c a d m i u m f r o m h e l i u m t e m p e r a t u r e s t o j u s t b e l o w t h e -3-m e l t i n g p o i n t , a n d ( i i ) t o s t u d y t h e l i n e w i d t h s a n d s h a p e s a s a f u n c t i o n o f c r y s t a l o r i e n t a t i o n a t v a r i o u s t e m p e r a t u r e s . The t e m p e r a t u r e d e p e n -d e n c e o f t h e K n i g h t s h i f t a n d i t s a n i s o t r o p y , h a s b e e n s t u d i e d . F o l l o w i n g B e n e d e k a n d K u s h i d a , t h e v a r i a t i o n i n t h e K n i g h t s h i f t w i t h t e m p e r a t u r e h a s b e e n a t t r i b u t e d t o v o l u m e e f f e c t s a n d a n i n t r i n s i c t e m p e r a t u r e d e p e n d e n c e . The p r e s e n t r e s u l t s h a v e b e e n c o m b i n e d w i t h o t h e r w o r k e r s ' r e s u l t s , o n p r e -s s u r e d e p e n d e n c e o f t h e K n i g h t s h i f t , a n d t h e i n t r i n s i c t e m p e r a t u r e d e p e n d e n c e h a s b e e n d e r i v e d o v e r t h e e n t i r e t e m p e r a t u r e r a n g e . We. h a v e i n v e s t i g a t e d t h e l i n e - w i d t h a s a f u n c t i o n o f c r y s t a l o r i e n t a -119 t i o n i n i s o t o p i c a l l y p u r e Sn a n d l i n e s h a p e s o f t h e same i s o t o p e i n n a t u r a l t i n . A n a l y s i s o f t h e i s o t o p i c a l l y p u r e t i n d a t a h a s e n a b l e d t h e e v a -l u a t i o n o f t h e r e l a t i v e p s e u d o - d i p o l a r c o n t r i b u t i o n s f r o m v a r i o u s s h e l l s . A n a n a l y s i s o f t h e l i n e - w i d t h s a t l i q u i d h e l i u m a n d r o o m t e m p e r a t u r e s h a s e n a b l e d a n e v a l u a t i o n o f t h e s p i n - l a t t i c e r e l a x a t i o n t i m e , T^, a t r o o m t e m p e r a t u r e . C H A P T E R - I I EQUIPMENT T h e a p p a r a t u s u s e d f o r t h e e x p e r i m e n t a l w o r k i s c o n v e n t i o n a l a n d s t a n d a r d i n d e s i g n . F i g u r e 1 shows t h e s c h e m a t i c d i a g r a m . A b r i e f , d e s -c r i p t i o n o f e a c h u n i t u s e d i s g i v e n b e l o w . (3,9) A PKW t y p e o s c i l l a t i n g d e t e c t o r - . w i t h m i n o r c h a n g e s was c o n s t r u c t e d . A 6922 v a c u u m t u b e was u s e d i n p l a c e o f 6 J 6 i n t h e o s c i l l a t o r s e c t i o n . T h e c a t h o d e s o f t h e o s c i l l a t o r a n d t h e a m p l i f i e r , w h i c h f e e d s b a c k t o t h e o s c i -l l a t o r , w e r e c o u p l e d t h r o u g h a .1 p¥ l o w v o l t a g e m y l a r c a p a c i t o r i n o r d e r t o t i e t h e o s c i l l a t o r a t a s i n g l e o p e r a t i n g p o i n t a n d a t t h e same t i m e t h e g a i n t h e a m p l i f i e r c o u l d b e v a r i e d o v e r a w i d e r a n g e . A s m a l l v a r i a b l e c a p a c i t o r C ^ o f v a l u e 2 - 8 p f i n p a r a l l e l w i t h a v o l t a g e s e n s i t i v e d i o d e c a p a c i t o r C2 ( v a r i c a p ) PC116 was u s e d i n t h e r e s o n a n t c i r c u i t . C± was k e p t r o u g h l y m i n i -mum d u r i n g t h e e x p e r i m e n t s . S i n c e C2 was a l w a y s g r e a t e r t h a n t h e r e s o -n a n c e f r e q u e n c y was p r a c t i c a l l y d e t e r m i n e d b y C 2 . T h e o s c i l l a t o r was v e r y s t a b l e i n f r e q u e n c y (2 i n 10^ o v e r s e v e r a l h o u r s ) a n d w o r k e d v e r y w e l l a t ilOOmv p e a k - p e a k r f l e v e l w i t h q u i t e l o w n o i s e . E x p e r i m e n t s f r o m 1 . 1 5 ° K t o o 450 K w e r e p e r f o r m e d a n d t h u s v a r i o u s r f l e v e l s h a d t o be u s e d . I t was f o u n d t h a t t h e ab.ove m e n t i o n e d e q u i p m e n t was a d e q u a t e l y s e n s i t i v e e v e n f o r h i g h r f l e v e l s , up t o I V p - p , u s e d f o r t h e h i g h t e m p e r a t u r e e x p e r i m e n t s . T h e f r e q u e n c y o f t h e o s c i l l a t o r was s w e p t b y a p p l y i n g a v a r i a b l e amp-l i t u d e ( 0 - 1 0 0 v o l t s ) l i n e a r s a w t o o t h a v a i l a b l e f r o m a m o d i f i e d T e k t r o n i x (32 ) wave f o r m g e n e r a t o r t y p e 162 w h i c h c o u l d g e n e r a t e l i n e a r s w e e p s o f f r o m 1 m . s e c . t o s e v e r a l h o u r s d u r a t i o n . The a u d i o o u t p u t o f t h e m a r g i n a l o s c i l l a t o r was f e d i n t o a W h i t e t w i n -t e e n a r r o w b a n d a m p l i f i e r m o d e l 216A. To t e s t t h e p e r f o r m a n c e o f t h e d e t e c -t o r a n A l u m i n u m s i n g l e , c r y s t a l was u s e d a s t h e s p e c i m e n a t r o o m t e m p e r a t u r e . S e v e r a l 5% b a n d w i d t h W h i t e t w i n - t e e s w i t h c e n t r e f r e q u e n c i e s f r o m 15 t o 4 0 0 c Power A m p l i -f i e r A t t e n -u a t o r A u d i o O s c i l l a t o r PKW O s c i l l a t i n g D e t e c t o i M o d i f i e d T e k t r o n i x 162. M a g n e t H o r i z o n t a l Amp. a n d P h a s e S h i f t e r N a r r o w B a n d A m p l i f i e r C o u n t e r a n d P r i n t e r F i e l d M o n i t o r i n g O s c i l l a t o r — ( CRO)-fCRO P h a s e S e n s i -t i v e D e t e c t o r R e c o r d e r B a t t e r y a n d H e l i p o t -6-w e r e t r i e d a n d i t was f o u n d t h a t t h e s i g n a l t o n o i s e r a t i o (S/N) was m a x i -mum f o r a t w i n - t e e o f 2 0 c / s , k e e p i n g o t h e r v a r i a b l e s o f t h e PKW d e t e c t o r t h e same. T h i s r a t h e r p e c u l i a r b e h a v i o r i s n o t i n a a g r e e m e n t w i t h t h e W a t k i n ' s ( 1 5 ) f i n d i n g s a n d t h e N a v a l R e s e a r c h L a b . R e p o r t F o l l o w i n g t h e n a r r o w b a n d a m p l i f i e r i s t h e p h a s e s e n s i t i v e d e t e c t o r w h i c h (35 ) i n p r i n c i p l e i s t h e same a s t h a t o f S c h u s t e r ' s . The s i g n a l r e c o r d e r u s e d was a V a r i a n r e c o r d e r , m o d e l G11A. The f r e q u e n c y m e a s u r e m e n t s w e r e made w i t h H e w l e t t - P a c k a r d e l e c t r o n i c c o u n t e r s , m o d e l s 524C a n d 5 2 4 5 L . T h e f r e q u e n c y was r e c o r d e d b y H e w l e t t -P a c k a r d d i g i t a l r e c o r d e r s m o d e l s 516B a n d 5622A r e s p e c t i v e l y , w h i c h i n t u r n a c t i v a t e d a n i n d i c a t o r p e n o n t h e s i g n a l r e c o r d e r e v e r y t i m e i t made a p r i n t . T h e m e a s u r e d f r e q u e n c y was a n a v e r a g e o v e r a p e r i o d o f .1 s e c o n d . T h e e x t e r n a l m a g n e t i c f i e l d was s u p p l i e d b y a r o t a t a b l e V a r i a n m a gnet w i t h 1 2 " d i a m e t e r p o l e f a c e s a n d a 2 - l / 4 " g a p . T h e m a g n e t i s m o u n t e d o n a g r a d u a t e d t a b l e s o t h a t f i e l d o r i e n t a t i o n r e l a t i v e t o t h e s a m p l e i n t h e p l a n e o f r o t a t i o n c a n be d e t e r m i n e d . The magnet was s w i t c h e d o n a t l e a s t 24 h o u r s b e f o r e t h e a c t u a l e x p e r i m e n t s t a r t e d . T h e c o o l i n g w a t e r f o r t h e magnet was s u p p l i e d f r o m a r e s e r v o i r w h i c h i n t u r n was c o o l e d b y t h e r u n n i n g t a p w a t e r . 6 T h i s g a v e a f i e l d s t a b i l i t y o f a b o u t 1 i n 10 o v e r a p e r i o d o f s e v e r a l h o u r s . F i e l d m o d u l a t i o n was a c h i e v e d b y a p a i r o f c o i l s , e a c h wound, w i t h 60 t u r n s o f No. 18 c o p p e r w i r e o n b a k e l i t e f o r m s , m o u n t e d a r o u n d t h e m a g n e t i c p o l e f a c e s . The m o d u l a t i o n c u r r e n t was s u p p l i e d b y a W i l l i a m s o n t y p e power a m p l i f i e r d r i v e n b y a H e w l e t t - P a c k a r d a u d i o o s c i l l a t o r , m o d e l 2 0 2 D . Th e l o w t e m p e r a t u r e s y s t e m u s e d i s shown i n f i g u r e 2. A n o r d i n a r y d o u b l e d e w a r g l a s s c r y o s t a t was u s e d t o a c h i e v e l o w t e m p e r a t u r e s . T h e l o w -e s t t e m p e r a t u r e a c h i e v e d was 1 . 1 5 ° K b y p u m p i n g o n t h e l i q u i d h e l i u m w i t h a t h r e e i n c h K i n n e y pump. The t e m p e r a t u r e was d e t e r m i n e d b y m e a s u r i n g t h e p r e s s u r e o v e r t h e l i q u i d h e l i u m b a t h u s i n g t h e o i l m a n o m e t e r . -7-T h e s a m p l e was m o u n t e d o n a c o a x i a l l i n e o f s t a i n l e s s s t e e l 3/8 i n c h i n d i a m e t e r . The c e n t r a l c o n d u c t i n g w i r e o f t h e c o a x i a l l i n e was a No. 36 c o p p e r w i r e h e l d i n p l a c e w i t h t e f l o n s p a c e r s . The m a g n e t i c f i e l d m e a s u r e m e n t s w e r e made b y d i s p l a y i n g t h e p r o t o n s i g n a l o n t h e o s c i l l o s c o p e d i r e c t l y . T he s a m p l e u s e d was g l y c e r i n e w h i c h was s i t u a -( 1 0 ) t e d j u s t o u t s i d e t h e l i q u i d n i t r o g e n d e w a r . T h e c i r c u i t u s e d t o m o n i t o r t h e f i e l d was a s s e m b l e d i n a s h o c k - m o u n t e d h e a v y b r a s s b o x , t o a v o i d m i c r o -p h o n i c s , a n d was m o u n t e d p e r m a n e n t l y o n t h e magnet u n t i l e a c h e x p e r i m e n t a l r u n was c o m p l e t e . The c o a x i a l l i n e u s e d w i t h t h e b o x was made o f c o p p e r t u b i n g a b o u t o n e f o o t i n > l e n g t h a n d h a l f i n c h o u t s i d e d i a m e t e r a n d h a d a No. 32 c o p p e r w i r e a s t h e i n n e r c o n d u c t o r . I t h a d a BNC c o n n e c t o r o n one e n d w h i c h was d i r e c t l y h o o k e d o n t o t h e b o x a n d o n t h e o t h e r e n d o f t h e c o a x i a l l i n e a g l y c e r i n e s a m p l e was m o u n t e d p e r m a n e n t l y i n a g l a s s t u b e 5 mm. o u t -s i d e d i a m e t e r . F o u r s u c h p r o b e s w e r e made w h i c h c o v e r e d t h e f r e q u e n c y r a n g e f r o m ( 1 1 - 6 0 ) M c / s . T h u s a s u i t a b l e p r o b e was a v a i l a b l e f o r e a c h f i e l d u s e d . A s m a l l c a p a c i t o r , 2 - 8 p f , i n p a r a l l e l w i t h a v a r i c a p No. 116 was u s e d i n t h e r e s o n a n c e c i r c u i t . T he v a r i a b l e v o l t a g e f o r t h e v a r i c a p was o b t a i n e d f r o m a 90 v o l t s d r y c e l l a n d a 100K H e l i p o t . A 2 p f c a p a c i t o r was u s e d i n s e r i e s w i t h t h e v a r i c a p . T h i s e n a b l e d e x t r e m e l y f i n e c o n t r o l o f t h e f r e q u e n c y a n d t h u s a v e r y a c c u r a t e m e a s u r e m e n t o f t h e f i e l d . T h e p r o t o n r e s o n a n c e f r e q u e -n c y was r e a d o n t h e H e w l e t t - P a c k a r d f r e q u e n c y m e t e r . T h e c i r c u i t h a d e x c e -6 l l e n t f r e q u e n c y s t a b i l i t y , 3 i n 10 , a n d v e r y h i g h s i g n a l t o n o i s e r a t i o . I n , a l l c a s e s t h e m a g n e t i c f i e l d was m e a s u r e d b e f o r e a n d a f t e r e a c h r u n f o r a p a r t i c u l a r f i e l d o r i e n t a t i o n w i t h r e s p e c t t o t h e c r y s t a l . A j l l t h e c r y s t a l s u s e d w e r e c y l i n d r i c a l i n s h a p e . The n a t u r a l t i n c r y s -t a l w i t h a x i s o f t e t r a g o n a l s y m m e t r y p a r a l l e l t o t h e s p e c i m e n a x i s , a x i s o f s y m m e t r y p e r p e n d i c u l a r t o t h e s p e c i m e n a x i s a n d t h e cadmium c r y s t a l w i t h h e x a g o n a l a x i s o f s y m m e t r y p e r p e n d i c u l a r t o t h e s p e c i m e n a x i s w e r e o b t a i n e d - 8 -Figure 2. Schematic of Low Temperature System. - 9 -c o m m e r c i a l l y f r o m M e t a l s R e s e a r c h L t d . , , C a m b r i d g e a n d h a d a p u r i t y o f 99.9997o. E a c h c r y s t a l was one i n c h i n l e n g t h a n d 3 / 8 " i n d i a m e t e r a n d was e t c h e d b e f o r e m a k i n g m e a s u r e m e n t s . T h e • i s o t o p i c a l l y p u r e t i n c r y s t a l was g r o w n i n t h e f o r m o f a h o l l o w o c y l i n d e r . T h e a x i s o f t e t r a g o n a l s y m m e t r y made a n a n g l e o f 30 w i t h t h e s p e c i m e n a x i s . Two s u i t a b l e ^ m o u n t s w e r e made t o make t h e m e a s u r e m e n t s i n t h e b a s a l p l a n e a n d w i t h t h e t e t r a g o n a l a x i s o f s y m m e t r y i n t h e p l a n e o f t h e f i e l d r o t a t i o n . T h e Cd-Hg a l l o y c r y s t a l , w i t h 12 a t o m i c p e r c e n t m e r c u r y was o b t a i n e d f r o m S e m i E l e m e n t s I n c . S a x o n b u r g , U.S.A. I t was .5 i n c h l o n g a n d . 4 mm. i n d i a m e t e r . The h e x a g o n a l a x i s o f s y m m e t r y was p e r p e n d i c u l a r t o t h e s p e c i m e n a x i s . I n a l l t h e e x p e r i m e n t s a c o p p e r w i r e No. 40 was u s e d f o r t h e c o i l a r o u n d t h e s p e c i m e n . One l a y e r o f .001 i n c h t h i c k m y l a r was u s e d a s i n s u l a t o r b e t -ween t h e c o i l a n d t h e s p e c i m e n f o r h e l i u m a n d l i q u i d n i t r o g e n e x p e r i m e n t s . F o r h i g h e r t e m p e r a t u r e e x p e r i m e n t s f i v e o r s i x l a y e r s o f m y l a r w e r e u s e d b e t w e e n t h e c o i l a n d t h e s p e c i m e n . T h e o s c i l l a t o r d i d n o t o s c i l l a t e a t r o o m t e m p e r a t u r e w i t h o n l y ,one l a y e r o f m y l a r a s i n s u l a t i o n b e t w e e n t h e c o i l a n d s p e c i m e n . I t was f o u n d n e c e s s a r y t o p u t t h e s p e c i m e n i n l i q u i d n i t r o g e n f o r t h e o s c i l l a t o r t o o p e r a t e a n d t o k n o w - t h e r e s o n a n c e f r e q u e n c y o f t h e c o i l . A f t e r two o r t h r e e t r i a l s a c o i l o f d e s i r e d r e s o n a n c e f r e q u e n c y c o u l d b e wound. F o r h i g h temp-e r a t u r e e x p e r i m e n t s , f i v e l a y e r s o f m y l a r made t h e Q o f t h e c i r c u i t h i g h e n o u g h f o r t h e o s c i l l a t o r t o w o r k a t h i g h t e m p e r a t u r e s . I t was e x p e r i m e n t a -l l y o b s e r v e d t h a t o p t i m u m s i g n a l t o n o i s e was o b t a i n e d w i t h t h e l o w e s t Q c o i l . T h e c o i l was wound a s c l o s e l y a s p o s s i b l e s o t h a t t h e o s c i l l a t o r w o u l d s t i l l o s c i l l a t e a t t h e t e m p e r a t u r e o f t h e e x p e r i m e n t . -10-The w i r e u s e d f o r t h e c o i l was n o t s t r o n g e n o u g h t o s u p p o r t t h e w e i -g h t o f t h e c r y s t a l a n d t o k e e p . i t i n t h e d e s i r e d p o s i t i o n . A c y l i n d r i c a l c o p p e r bomb was d e s i g n e d t o h o l d t h e c r y s t a l . T he m o u n t i n g o f t h e c r y s t a l i s v e r y i m p o r t a n t . I f t h e a x i s o f s y m m e t r y o f t h e c r y s t a l d o e s n o t l i e i n t h e p l a n e o f f i e l d r o t a t i o n , a n d i s o f f e v e n b y a c o u p l e o f d e g r e e s , i t w i l l g i v e w r o n g r e s u l t s o f t h e 1 K n i g h t s h i f t a n d a n i s o t r o p y . T h e • c o p p e r bomb n o t o n l y s e r v e d a s a s u p p o r t f o r t h e c r y s t a l b u t a l s o e n a b l e d t h e s a m p l e t o be r i g i d l y m o u n t e d i n - t h e c o r r e c t o r i e n t a t i o n . A s a l r e a d y d e s c r i b e d t h e PKW d e t e c t o r w o r k e d v e r y w e l l w i t h 20 c y c l e t w i n - t e e a n d s i g n a l t o n o i s e was no p r o b l e m f o r a n y o f t h e s p e c i m e n s u s e d . I n c a s e o f i s o t o p i c a l l y p u r e t i n t h e s i g n a l c o u l d e v e n b e d i s p l a y e d o n t h e o s c i l l o s c o p e . T i m e c o n s t a n t s o f .5 s e c , 2.5 s e c . a n d 5.0 s e c o n d s w e r e u s e d i n t h e p h a s e s e n s i t i v e d e t e c t o r f o r i s o t o p i c a l l y p u r e t i n , n a t u r a l t i n a n d c a d m i u m r e s p e c t i v e l y . I n a l l t h e e x p e r i m e n t s a t h e l i u m t e m p e r a t u r e s a m o d u l a t i o n l e s s t h a n o n e f i f t h o f t h e l i n e w i d t h was u s e d . H o w e v e r , f o r i s o -t o p i c a l l y p u r e t i n v e r y s m a l l m o d u l a t i o n was u s e d . F o r h i g h t e m p e r a t u r e e x p e r i m e n t s t i m e c o n s t a n t s up t o 30 s e c o n d s a n d h i g h e r m o d u l a t i o n a m p l i -t u d e s w e r e u s e d . CHAPTER I I I REVIEW OF THE THEORY•OF'THE KNIGHT S H I F T I n > t h e f o l l o w i n g s e c t i o n s a b r i e f r e v i e w o f t h e t h e o r y o f t h e K n i g h t (1) s h i f t i s g i v e n . A d e t a i l e d a n d c o m p r e h e n s i v e t r e a t m e n t i s g i v e n b y A b r a g a m a n d S l i c h t e r ^ ^ . I n t h e l a s t s e c t i o 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 h e K n i g h t s h i f t i s t r e a t e d . . • 3.1 • T h e K n i g h t S h i f t : T h e K n i g h t s h i f t h a s b e e n e x p l a i n e d i n t e r m s o f t h e h y p e r f i n e i n t e r a c t i o n b e t w e e n t h e n u c l e i a n d c o n d u c t i o n e l e c t r o n s . L e t u s c o n s i d e r a n a s s e m b l y o f n c o n d u c t i o n e l e c t r o n s a n d N i o n c o r e s . U n d e r t h e a d i a b a t i c a p p r o x i m a t i o n , t h e i o n c o r e s c a n be t r e a t e d a s s t a t i o n a r y l a t t i c e p o i n t s . T h e m o t i o n o f t h e e l e c t r o n s c a n be a s s u m e d t o be u n c o r r e l a t e d a n d t h e f r e e e l e c t r o n a p p r o -x i m a t i o n may be u s e d . The e l e c t r o n s f e e l t h e p o t e n t i a l due t o i o n c o r e s p l u s some o v e r a l l v a l u e o f e l e c t r o n . r e p u l s i o n . The H a m i l t o n i a n o f t h e s y s t e m i n , p r e s e n c e o f e x t e r n a l m a g n e t i c f i e l d H Q c a n be w r i t t e n a s , p ' n- n _,• • N n_ N . , ( i , j) M = ^ + * + H Q ( L Z + 2 S ^ ) + ^ Y ^ H b I z + Q ^ ^ i 3.1.1 w h e r e ^\ i s t h e H a m i l t o n i a n f o r t h e c o n d u c t i o n e l e c t r o n s i n t h e f i e l d o f i o n c o r e s , i s t h e H a m i l t o n i a n o f t h e i o n c o r e s . T h e t h i r d a n d f o u r t h t e r m s a r e t h e Zeeman e n e r g i e s o f t h e c o n d u c t i o n e l e c t r o n s a n d t h e n u c l e i a n d ( i , j ) ^) i s t h e i n t e r a c t i o n H a m i l t o n i a n b e t w e e n i t h e l e c t r o n a n d j t h n u c l e u s . I t r e p r e s e n t s t h e h y p e r f i n e . i n t e r a c t i o n e n e r g y a n d i s o f i n t e r e s t h e r e . The e n e r g y o f t h e j t h n u c l e u s c a n be w r i t t e n a s , 1 ( s ( V r \ i ) ^ | ( j ) = -Y* H 0 I z ( j ) + 2 S Y - ( r : 3 ) i q > i ; ( i ) Y * g > I ^ < - 2 _ - ' ) ' 1 L J i = T r^i r ^ i + 5 Z T ^ g ( l S ( i ) . T ( j ) R F i i ) 3.1.2 i = l 3 1 J The e l e c t r o n o p e r a t o r s a r e l a b e l l e d w i t h s u p e r s c r i p t i a n d n u c l e a r o p e r a t o r s w i t h j . T h e p r i m e o n t h e 3 r d t e r m means t h a t i t d o e s n o t c o n t a i n s i n g u l a r i t y . I t h a s a l r e a d y b e e n e v a l u a t e d a t t h e o r i g i n a n d t h e v a l u e i n c l u d e d i n t h e l a s t t e r m . T h e e x p e c t a t i o n v a l u e o f w i t h . r e s p e c t t o . t h e e l e c t r o n wave f u n c t i o n g i v e s t h e K n i g h t s h i f t . S i n c e t h e sum i s o v e r a l l t h e c o n d u c t i o n e l e c t -r o n s , ^ i s n o r m a l i z e d o v e r t h e c r y s t a l v o l u m e . 1 The e l e c t r o n wave f u n c t i o n s a r e s i m p l y B l o c h f u n c t i o n s l / / ( k " , f ) = U R ( r ) e i l c , r w h e r e U ^ C r ) i s a f u n c t i o n h a v i n g s a m e . p e r i o d i c i t y a s l a t t i c e . S i n c e t h e o r b i t a l a n g u l a r momentum i n m o s t m e t a l s i s q u e n c h e d , ( r a r e e a r t h s a r e e x c e p t i o n ) , t h e s e c o n d t e r m d o e s n o t c o n t r i b u t e . F o r m e t a l s w i t h c u b i c s y m m e t r y , t h e c o n t r i b u t i o n o f t h e (28) 1 p r i m e d t e r m i s z e r o b u t i t d o e s c o n t r i b u t e t o t h e K n i g h t s h i f t i n t h e m e t a l s w i t h s y m m e t r y l o w e r t h a n c u b i c a n d i s r e s p o n s i b l e f o r a n i s o t r o p y . T h i s w i l l be t r e a t e d i n one o f t h e f o l l o w i n g s e c t i o n s . T h e l a s t t e r m i n ( 3 . 1 . 2 ) known a s t h e c o n t a c t t e r m i s t h e o n l y t e r m , c o n t r i b u t i n g t o t h e i s o t r o p i c K n i g h t s h i f t . I f o n l y c o n t a c t t e r m i s c o n s i d e r e d , t h e e x p e c t a t i o n v a l u e o f may be w r i t t e n a s , ^ < j ) > - - W ^ ' + I f ? y * P I < J ) £ (0)f 3.1.3 w h e r e m ^ i s t h e e i g e n v a l u e o f S ^ a n d | 4 ^ [ i s t h e p r o b a b i l i t y o f t h e i t h e l e c t r o n b e i n g f o u n d a t t h e s i t e o f j t h n u c l e u s . The sum a p p e a r i n g i n . ( 3 . 1 . 3 ) g i v e s r i s e t o P a u l i s u s c e p t i b i l i t y . I t c a n be shown t h a t -. - ? ! j L X