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Nuclear magnetic resonance in single crystals of tin and aluminum. 1962

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NUCLEAR MAGNETIC RESONANCE IN SINGLE CRYSTALS OF TIN AND ALUMINUM by EDWARD PETER JONES B.A.Sc., 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 , 1958 M . S c , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1959 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department o f PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA November, 1962 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission 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 publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia, Vancouver 8, Canada. Date i 1 . vnCS - i i - ABSTRACT N u c l e a r magnetic r e s o n a n c e s t u d i e s i n s i n g l e c r y s t a l s of aluminum and t i n have been done a t l i q u i d h e l i u m t e m p e r a t u r e s * The K n i g h t s h i f t i n t i n has been s t u d i e d as 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 i n a c o n s t a n t magnetic f i e l d f o r d i f f e r e n t v a l u e s of f i e l d and t e m p e r a t u r e . The a n 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 was o b s e r v e d d i r e c t l y f o r t h e f i r s t t i m e . The l i n e w i d t h o f t h e t i n r e s o n a n c e was a l s o s t u d i e d and f o u n d t o depend on t h e c r y s t a l o r i e n t a t i o n i n t h e magnetic f i e l d . The second moment of t h e l i n e has been c a l c u l a t e d i n terms o f d i p o l e - d i p o l e i n t e r a c t i o n s and i n d i r e c t exchange i n t e r a c t i o n s between n u c l e i o f d i f f e r e n t m a gnetic moments and compared w i t h t h e e x p e r i m e n t a l r e s u l t s . The K n i g h t s h i f t was s t u d i e d as a f u n c t i o n o f e x t e r n a l f i e l d f o r b o t h t i n and aluminum i n a s e a r c h f o r de Haas-van Alphen t y p e o s c i l l a t i o n s . No i n d i c a t i o n o f t h e s e was f o u n d . An upper l i m i t f o r t h i s e f f e c t was d e t e r m i n e d f o r each sample. - v i - ACKNOWLEDGMENT 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 Dr. Myer Bloom f o r t h e v a l u a b l e i n s t r u c t i o n and i n s p i r a t i o n he p r o v i d e d t h r o u g h o u t my c a r e e r as a g r a d u a t e s t u d e n t , and f o r h i s s u p p o r t i n o b t a i n i n g f i n a n c i a l a s s i s t a n c e f o r my s t u d i e s . I w i s h a l s o t o e x p r e s s my g r a t i t u d e t o Dr. D. L I . W i l l i a m s f o r h i s h e l p i n t h e measurements r e p o r t e d i n t h i s work and f o r many u s e f u l s u g g e s t i o n s p a r t i c u l a r l y as r e g a r d s work i n s i n g l e c r y s t a l s . My th a n k s a r e due t o Dr. Roger Howard who i n t r o - duced me t o t h e IBM 1620 computer and who c a r r i e d out t h e co m p u t a t i o n d e s c r i b e d i n Chapter 4. I w i s h t o acknowledge t h e f i n a n c i a l s u p p o r t p r o v i d e d by a f e l l o w s h i p g i v e n by t h e I n t e r n a t i o n a l N i c k e l Company of Canada and r e s e a r c h f u n d s made a v a i l a b l e by 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 . - i i i - TABLE OF CONTENTS CHAPTER Page 1 INTRODUCTION 1 2 THEORY OF THE KNIGHT SHIFT 4 3 EXPERIMENTAL APPARATUS AND TECHNIQUES 17 4 THE KNIGHT SHIFT AND LINE WIDTH OF THE TIN SINGLE CRYSTAL 27 5 THE POSSIBILITIES OF DE HAAS-VAN ALPHEN TYPE OSCILLATIONS IN THE KNIGHT SHIFT 45 APPENDIX A THE SECOND MOMENT OF THE RESONANCE LINE DUE TO MISALIGNMENT OF THE CRYSTAL SLICES 52 B THE ANISOTROPY OF THE NUCLEAR MAGNETIC RESONANCE IN WHITE TIN: E. P. JONES AND D. LI. WILLIAMS 55 C CIRCUIT DIAGRAMS OF PARTS OF THE SPECTROMETER 57 REFERENCES 62 - i v - L IST OF ILLUSTRATIONS FIGURE Page 1 B l o c k Diagram o f t h e S p e c t r o m e t e r 18 2 Schematic Diagram of t h e Low Temperature System 21 3 D e r i v a t i v e o f t h e Sn Resonance 26 4 D e r i v a t i v e of t h e A l Resonance 26 5 The K n i g h t S h i f t i n T i n as a F u n c t i o n o f t h e C r y s t a l O r i e n t a t i o n i n t h e M a g n e t i c F i e l d 28 6 P l o t s o f £2LI1 and d ( X ' + Z X 1 " ) f o r d d L o r e n t z i a n and G a u s s i a n L i n e Shapes 31 7 The L i n e Width of t h e T i n Resonance as a F u n c t i o n of t h e C r y s t a l O r i e n t a t i o n i n t h e M a g n e t i c F i e l d 33 A 2 8 P l o t o f as a F u n c t i o n o f kt? 41 A2 * 9 The K n i g h t S h i f t i n T i n as a F u n c t i o n of Temperature 43 10 The K n i g h t S h i f t i n T i n as a F u n c t i o n o f M a g n e t i c F i e l d 43 11 The K n i g h t S h i f t i n Aluminum as a F u n c t i o n o f M agnetic F i e l d 50 - V - 12 The K n i g h t S h i f t i n T i n as a F u n c t i o n of M a g n e t i c F i e l d 51 C l The Pound-Knight-Watkins O s c i l l a t o r 58 C2 M o d i f i e d T e k t r o n i x 162 Waveform G e n e r a t o r ' 59 C3 Phase S h i f t e r and H o r i z o n t a l A m p l i f i e r f o r the T e k t r o n i x 360 O s c i l l o s c o p e 60 C4 Phase S e n s i t i v e D e t e c t o r 61 CHAPTER I INTRODUCTION The change i n t h e n u c l e a r m agnetic r e s o n a n c e f r e - quency o f a n u c l e u s i n a m e t a l f r o m t h a t of t h e same n u c l e u s i n a n o n - m e t a l l i c s t a t e was f i r s t e x p l a i n e d i n terms o f an e l e c t r o n - n u c l e u s i n t e r a c t i o n by Townes, H e r r i n g and K n i g h t 1 . T h i s change i n r e s o n a n c e f r e q u e n c y , t h e K n i g h t s h i f t , has been s t u d i e d e x p e r i m e n t a l l y and t h e o r e t i c a l l y q u i t e e x t e n - s i v e l y f o r many m e t a l s and a l l o y s . The r e s u l t s o f much of t h i s work a r e summarized i n a r e v i e w a r t i c l e by K n i g h t ^ and a r i g o r o u s t h e o r e t i c a l t r e a t m e n t o f t h e K n i g h t s h i f t i s g i v e n by Abragam 3. Because o f t h e s m a l l s k i n d e p t h o f a l m o s t a l l m e t a l s , n u c l e a r m agnetic r e s o n a n c e e x p e r i m e n t s have been almost e n t i r e l y c o n f i n e d t o s t u d i e s i n m e t a l powders whose i n d i v i d u a l p a r t i c l e s i z e s a r e l e s s t h a n t h e s k i n d e p t h o f t h e m e t a l . I n any m e t a l w i t h c u b i c symmetry, t h e K n i g h t s h i f t i s independent o f t h e o r i e n t a t i o n of t h e m e t a l c r y s t a l 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 agnetic f i e l d . However, i f t h e symmetry o f t h e c r y s t a l i s l e s s t h a n c u b i c , t h i s i s not t r u e . The n u c l e a r m agnetic r e s o n a n c e s i g n a l w i l l be -2- broadened when observed in a powder whose particles are ran- domly oriented in the magnetic f i e l d or shifted when observed in a single crystal. The purpose of this work has been to study the nuclear magnetic resonance in metal single crystals, in particular, 31 white tin and aluminum. The advantages of using a single crystal to study the nuclear magnetic resonance of t i n ares (a) the anisotropy of the Knight shift due to the tetragonal symmetry of the t i n crystal can be studied directlyj (b) the line shape of the resonance can be studied without the large broadening caused by the anisotropic Knight shifty and (c) the measurements of the resonance frequency can be made more accurately because the resonance line i s narrower. 117 11Q The Sn and Sn isotopes were studied in a white t i n single crystal as a function of crystal orientation in the magnetic f i e l d , as a function of the magnetic f i e l d i t s e l f , and as a function of temperature in the liquid helium range. The two parameters which describe the Knight shift in a metal with tetragonal symmetry were determined. A slight dependence of the Knight shift on temperature and external magnetic f i e l d was observed. The line width of the t i n resonance was studied and found to have contributions from both the dipole-dipole interactions of nearest neighbours and from indirect exchange interactions which couple two nuclei of different magnetic moments by way of the conduction electrons. -3- A s e a r c h f o r a v a r i a t i o n o f t h e K n i g h t s h i f t caused by o s c i l l a t i o n s i n t h e d i a m a g n e t i c s u s c e p t i b i l i t y as t h e e x t e r n a l f i e l d i s v a r i e d (de Haas-van Alphen e f f e c t ) was made i n s i n g l e c r y s t a l s o f b o t h t i n and aluminum. An upper l i m i t t o t h e e f f e c t has been s e t f o r b o t h m e t a l s . CHAPTER 2 THEORY OF THE KNIGHT SHIFT The K n i g h t s h i f t i n m e t a l s w i t h t e t r a g o n a l symmetry can be e x p l a i n e d i n terms 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 w h i c h i n v o l v e s e s s e n t i a l l y t h r e e terms i n t h e s p i n H a m i l t o n i a n and w h i c h c o u p l e s t h e e l e c t r o n s p i n s t o t h e n u c l e a r s p i n s . The c o u p l i n g energy i s s m a l l compared t o t h e ato m i c energy s p l i t - t i n g s so t h a t p e r t u r b a t i o n t h e o r y can be used. F u r t h e r , t h e e l e c t r o n s a r e assumed t o be n o n - i n t e r a c t i n g , i . e . , t h e f r e e e l e c t r o n model i s used. The H a m i l t o n i a n f o r t h e magnetic i n t e r a c t i o n of t h e 4 e l e c t r o n s w i t h t h e n u c l e u s can be w r i t t e n as X i s the nuclear spin # i s the electron o r b i t a l quantum number J3 i s the electron spin X i s the radius vector from the nuclear spin to the electron spin (1) *U - 2 & h Y I $L S & ( r ) where £ i s t h e Bohr magneton y i s t h e n u c l e a r g y r o m a g n e t i c r a t i o - 5 - The f i r s t term of t h i s H a m i l t o n i a n w i l l g i v e almost no c o n t r i - b u t i o n t o t h e e l e c t r o n - n u c l e u s i n t e r a c t i o n i n most m e t a l s because t h e o r b i t a l a n g u l a r momentum i s quenched or n e a r l y so ( b i s m u t h i s a n o t a b l e e x c e p t i o n ) . I n o t h e r words, t h e diamag- n e t i c c o n t r i b u t i o n i s u s u a l l y s m a l l compared t o t h e p a r a m a g n e t i c c o n t r i b u t i o n . The d i a m a g n e t i c c o n t r i b u t i o n might be d e t e c t e d , however, by t h e o b s e r v a t i o n o f o s c i l l a t i o n s i n t h e K n i g h t s h i f t a t low t e m p e r a t u r e s as t h e m agnetic f i e l d i s v a r i e d . T h i s p o i n t w i l l be d i s c u s s e d i n C hapter 5 . F o r c r y s t a l s w i t h no lower t h a n c u b i c symmetry, t h e second and t h i r d t e rms, t h e d i p o l a r t erms, of t h e H a m i l t o n i a n g i v e no c o n t r i b u t i o n t o t h e K n i g h t s h i f t 5 . F o r c r y s t a l s of t e t r a g o n a l symmetry, t h e y account f o r t h e a n i s o t r o p i c p a r t o f t h e K n i g h t s h i f t . The l a s t term i n t h e H a m i l t o n i a n c o u p l e s o n l y s - e l e c t r o n s t o t h e n u c l e u s and g i v e s r i s e t o t h e K n i g h t s h i f t i n c r y s t a l s o f c u b i c symmetry o r t o the i s o t r o p i c p a r t of t h e K n i g h t s h i f t i n c r y s t a l s w i t h t e t r a g o n a l symmetry. The s t a n d a r d d e r i v a t i o n of t h e K n i g h t s h i f t i s now g i v e n i n d e t a i l f o l l o w i n g m a i n l y t h e t r e a t m e n t g i v e n by S l i c h t e r 5 . T h i s i s b e i n g done f o r c o m p l e t e n e s s . S i n c e t h i s work r e p r e s e n t s t h e f i r s t d i r e c t o b s e r v a t i o n o f t h e a n i s o - t r o p i c c o n t r i b u t i o n s t o the K n i g h t s h i f t , by g i v i n g t h i s d e r i v a t i o n t h e t y p e of i n f o r m a t i o n o b t a i n a b l e i n s u c h an e x p e r i m e n t s h o u l d be more c o m p r e h e n s i b l e . F o r s i m p l i c i t y , f i r s t c o n s i d e r t h e i n t e r a c t i o n s -s- i n v o l v i n g o n l y t h e l a s t term, o r t h e c o n t a c t term, i n t h e H a m i l t o n i a n (2) CU1 - i f E e y h I • S cf Cr) Because t h e e l e c t r o n s and n u c l e i a r e o n l y w e a k l y i n t e r a c t i n g , t h e complete wave f u n c t i o n V can be a p p r o x i m a t e d by t h e p r o d u c t o f t h e many p a r t i c l e e l e c t r o n i c and n u c l e a r wave f u n c t i o n s , V e and  [Vn The a n t i s y m m e t r i z e d e l e c t r o n i c wave f u n c t i o n i s c o n s i d e r e d t o b e 6 (3) r e 1 f N ( - 1 ) P ^ k S ( l ) U ' « ' < 2 > •••• V fcD^ ( N ) k's k * s r where ^ s - u f e e*~* r , t h e B l o c h f u n c t i o n i n c l u d i n g t h e s p i n term, and p i s an i n d e x t o i n d i c a t e an odd o r even number o f i n t e r c h a n g e s i n t h e p e r m u t a t i o n . The c o n t r i b u t i o n o f t h e e l e c t r o n - n u c l e a r i n t e r - a c t i o n s f r o m a l l e l e c t r o n s t o one n u c l e u s i s (4) -7- S i n c e t h e o p e r a t o r S cf ( r ) i n v o l v e s o n l y one e l e c t r o n , t h e r e a r e no c o n t r i b u t i o n s f r o m t e r m s i n w h i c h e l e c t r o n s a r e e x c h a n g e d . Thus u s i n g e q u a t i o n (3), we c a n r e w r i t e (4) ( 5 ) H r ^ ^ i j - f j [ V " J (V * [ V k s (1) ... ] d T 1 d T 2 ... I f t h e e l e c t r o n s a r e q u a n t i z e d a l o n g t h e z - d i r e c t i o n b y t h e e x t e r n a l s t a t e f i e l d H Q , t h e o n l y c o n t r i b u t i o n t o e q u a t i o n ( 5 ) comes f r o m S z ^ . The i n t e g r a l t h e n becomes (6) % j = ^ ^ ^3 I M ° ) ! 2 » B F & > S > where t h e sum i s o v e r a l l and s , and f ( k , s ) i s t h e F e r m i d i s t r i b u t i o n f u n c t i o n g i v i n g t h e p r o b a b i l i t y t h a t a g i v e n s t a t e d e s c r i b e d b y t h e wave v e c t o r k_ and s p i n c o o r d i n a t e s i s o c c u p i e d . The f a c t o r m s i s j u s t + £ o r - \ and u k ( o ) i s t h e s p a t i a l p a r t o f t h e e l e c t r o n i c wave f u n c t i o n e v a l u a t e d a t t h e n u c l e u s . F o r a g i v e n k , e q u a t i o n (6) c a n be w r i t t e n ,k (7) X~~ - f 1 ? * I z J 2 & < i ) f ( f e , i ) + 2 $ < - J ) x f ( k , - i ) J | u k ( o ) | -8- The factor i n square brackets i s the average contribution of the state to the z component of magnetization of the sample. If t h i s factor i s c a l l e d u , z k , the t o t a l z magnetization of the electrons i n a sample of a unit volume i s fiz ( 8 ) " k" ^ ~ k The t o t a l spin s u s c e p t i b i l i t y for a unit volume can be defined as O) i£ " * s H o and the spin s u s c e p t i b i l i t y for one value of k ( 1 0 ) x k  H° which means that s <io») X s - \x A k The t o t a l e f f e c t i v e i n t e r a c t i o n for the j t h nuclear spin i s then -9- I n o r d e r t h a t t h i s sum may be e v a l u a t e d , t h e f o l l o w - i n g q u a n t i t y must be c o n s i d e r e d . > A ) dE^ dA i s d e f i n e d as t h e number of a l l o w e d J k - v a l u e s l y i n g w i t h i n a s m a l l c y l i n d r i c a l volume of Js: space h a v i n g a c r o s s - s e c t i o n a r e a dA anci l y i n g between t h e energy s u r f a c e s E^ and E k + dE^ . The c o o r d i n a t e s of t h e s u r f a c e a r e denoted by A. The t o t a l number o f s t a t e s dN between E^ and E^ + d E k i s g i v e n by Slimming t h e c o n t r i b u t i o n s over t h e whole c o n s t a n t energy s u r f a c e . (12) dN - dEj^ g ( E k , A) dA E, - c o n s t k S P \ ) d \ P ( E ) i s a d e n s i t y of s t a t e s p e r u n i t energy i n t e r v a l a t t h e energy s u r f a c e E^ f o r a u n i t volume. F o r c o n v e n i e n c e l a t e r , N(Ep) can be s i m i l a r l y d e f i n e d as t h e d e n s i t y o f s t a t e s p e r u n i t energy i n t e r v a l a t t h e F e r m i s u r f a c e f o r one a t o m i c volume V Q . The sum i n e q u a t i o n (11) can now be e v a l u a t e d i n terms of t h e s e q u a n t i t i e s i f t h e sum i s r e p l a c e d by an i n t e g r a l (13) £L | u k ( o ) | Z X S - K (o)| 2 X S g < E k > A ) d % d A • s j -10- depends on t h e F e r m i f u n c t i o n s f ( k , £) and f ( k ,-£) and on t h e d i f f e r e n c e i n energy between two s p i n s i n t h e k s t a t e , one w i t h s p i n p a r a l l e l t o t h e magnetic f i e l d and t h e o t h e r w i t h s p i n a n t i p a r a l l e l . C l e a r l y , ^ C S w i l l be t h e same f o r any s t a t e s Jk h a v i n g the same v a l u e o f . I t i s ther e - s f o r e assumed t h a t DC can be w r i t t e n as a f u n c t i o n o n l y of t h e energy Ê . d 4 ) xt - X s < \ > k E q u a t i o n (13) can now be w r i t t e n (15) 2. | u k ( o ) l 2 X S - f l u , . ( o ) l 2 X S <E k ) x g ( E f c ,A) dA d E k \ i 2 The average v a l u e o f |u, ( o ) \ over a s u r f a c e of c o n s t a n t energy i s d e f i n e d as (16) <(lu k (o) l 2 ) E k I 2 u k (o)l g ( E k ,A) dA g(E, ,A) dA \ I 2 W k ( o ) l g ( E k ,A) dA ? < E k > -11- so t h a t <i7) sujoii !x;- k)v°)12K X* (V Because t h e e l e c t r o n s p i n s a r e e i t h e r p a i r e d o f f or t h e e l e c t r o n s p i n energy s t a t e s a r e u n o c c u p i e d , f o r a l l Ek not near (not w i t h i n a r e g i o n o f t h e o r d e r o f kT) t h e F e r m i s u r - f a c e Ep, t h e c o n t r i b u t i o n s of 3XLS(Ek ) t o t h e t o t a l s p i n s u s c e p t i b i l i t y w i l l come o n l y f r o m energy s t a t e s near t h e F e r m i s u r f a c e . I f i t i s assumed t h a t <^luk ( ° ) \ 2 ^ E v a r i e s s l o w l y w i t h E^ , t h i s i n t u r n can be t a k e n o u t s i d e t h e i n t e g r a l a s ) s Uk ( o ) i 2 X ^ - < K w i \ [ x B < \ ) ? ^ ) ^ U s i n g e q u a t i o n ( 1 0 ' ) , we can w r i t e Xs - £ Xs k k j X g ( E k ,A) dEk dA k — Xs (Ek > e ( E k >A> ^ d A or -12- (19) s Combining e q u a t i o n s ( 1 1 ) , ( 1 8 ) , and ( 1 9 ) , we g e t t h e i n t e r - a c t i o n w i t h t h e j t h n u c l e a r s p i n t o be (20, x • i„ [ f 0* <°>i2 > % xsK T h i s i n t e r a c t i o n r e p r e s e n t s an e x t r a m agnetic f i e l d added t o t h e a p p l i e d magnetic f i e l d HQ w h i c h g i v e s r i s e t o t h e K n i g h t s h i f t . The K n i g h t s h i f t can t h e n be w r i t t e n (21) A H H_ I f t, i s d e f i n e d as t h e r a t i o l«A<»>\ : \ I 2 where \ u A ( o ) \ i s t h e p r o b a b i l i t y d e n s i t y of e l e c t r o n s i n a f r e e atom, t h e n t h e e x p r e s s i o n f o r t h e K n i g h t s h i f t can be w r i t t e n i n terms o f t h e f r e e atom h y p e r f i n e c o u p l i n g c o n s t a n t . (22) a ( s ) - J f L -yn £\u A(o)\ E q u a t i o n (21) t h e n becomes -13- (23) £ S = a ( s ) ! ^ P M I n o r d e r t o conform w i t h more u s u a l n o t a t i o n 2 , X . s n a s been r e w r i t t e n as X pM, where " X p i s t h e P a u l i s u s c e p t i b i l i t y per u n i t mass and M i s t h e a t o m i c mass. E q u a t i o n (23) i s t h e e x p r e s s i o n f o r t h e K n i g h t s h i f t i n a c r y s t a l w i t h a t l e a s t c u b i c symmetry. I f t h e c r y s t a l has t e t r a g o n a l symmetry as does w h i t e t i n , t h e n e q u a t i o n (23) r e p r e s e n t s t h e i s o t r o p i c p a r t o f t h e K n i g h t s h i f t . As was mentioned e a r l i e r , t h e d i p o l e - d i p o l e i n t e r a c t i o n s i n t h e H a m i l t o n i a n o f e q u a t i o n (1) w i l l c o n t r i b u t e an a n i s o t r o p i c 7 term t o t h e e x p r e s s i o n f o r t h e K n i g h t s h i f t . C o n s i d e r t h e H a m i l t o n i a n < 2 4> °V4anis - - J . 1 J 3r (S • j;) L r 3 r 5 I f t h e a n g l e between H Q and t h e r a d i u s v e c t o r £ i s ©c , e q u a t i o n (24) becomes (25) ^ a n i s " " ^ m I ( 1 " 3 c o s 2 ^ ) r " 3 where t h e + or - s i g n i s d e t e r m i n e d by t h e e l e c t r o n s p i n b e i n g p a r a l l e l o r a n t i p a r a l l e l t o t h e magnetic f i e l d H Q. -14- To d e t e r m i n e t h e energy d i f f e r e n c e due t o t h i s H a m i l t o n i a n f o r a n u c l e u s o f s p i n J ( A i j => 1 ) , we must i n t e g r a t e e q u a t i o n (25) over t h e e l e c t r o n i c wave f u n c t i o n s i n a way s i m i l a r t o t h a t done f o r t h e i s o t r o p i c c a s e . F o r a u n i t volume, o n l y t h e 2 Q H Q N ( E F ) e l e c t r o n s w i t h u n b a l a n c e d s p i n near t h e F e r m i s u r f a c e w i l l c o n t r i b u t e t o t h e i n t e r a c t i o n a r i s i n g f r o m ^ a n i S * I f v o i s t n e a t o m i c volume, t h e energy d i f f e r e n c e due t o ^ H a n i g i s (26) A W a n i s " 7* * $ ' 2 $ H 0 V 0 N ( E F ) < I f * ek x (3 c o s 2 * k - 1) |r 1-3 dv) * / E F where V e k i s t h e e l e c t r o n i c wave f u n c t i o n o f an e l e c t r o n i n t h e t h s t a t e . I f we i n t r o d u c e an average wave f u n c t i o n Y so t h a t r e p r e s e n t s t h e average e l e c t r o n d e n s i t y i n space o f t h e c o n d u c t i o n e l e c t r o n s near t h e F e r m i s u r f a c e , t h e n e q u a t i o n (26) g i v e s f o r t h e a n i s o t r o p i c p a r t o f t h e K n i g h t s h i f t (27) A H a n i s - 2 6 2 V N ( E F ) 1 V * ( 3 c o s 2 * . - 1) | r \ " 3 ^ d x d y d z H o ° J To e v a l u a t e t h i s i n t e g r a l , we l e t t h e f i e l d have t h e p o l a r a n g l e s © and (p and t h e r a d i u s v e c t o r _r have p o l a r -15- a n g l e s and $ w i t h r e s p e c t t o t h e x, y, z c o o r d i n a t e 2 system. The term ( 1 - 3 c o s ex. ) can t h e n be e x p r e s s e d i n terms of t h e s e a n g l e s by a w e l l known a d d i t i o n theorem i n terms o f t e s s e r a l harmonics i n t h e a n g l e s © and a? and © and i> r e s p e c t i v e l y 2 ( 2 8 ) ^ ( 3 cos 2 =c - 1) - £ ( - l ) m P 2 m ( c o s 6 ) P 2 " " m ( c o s 0) m=-2 x e i m ( $ - £> ) We assume t h a t V can be r e p r e s e n t e d by a m i x t u r e o f p-wave f u n c t i o n s o f t h e form 4^ * (W. + W ) and IT2 -~— ( ( sV + - V ). These wave f u n c t i o n s a r e r e a l and t h e r e f o r e r e p r e s e n t quenched p o r b i t a l s . F o r a x i a l symmetry, t h e e l e c t r o n d e n s i t y can be w r i t t e n as ( 2 9 ) Y V * - g ( r ) [ A ( X 2 + y 2 ) + G z 2 ] = r 2 g ( r ) + ( C - A ) x. c o s 2 © j where g ( r ) i s a r a d i a l d i s t r i b u t i o n f u n c t i o n whose form i s u n i m p o r t a n t f o r t h i s c a l c u l a t i o n . S u b s t i t u t i n g e q u a t i o n s ( 2 9 ) and ( 2 8 ) i n t o e q u a t i o n ( 2 7 ) , we get (30) A H a n l S - £ 2 V Q N(Ej.) q(3 c o s 2 0 - 1 ) H o -16- where © i s t h e a n g l e between t h e z a x i s and t h e a p p l i e d magnetic f i e l d , and q i s g i v e n by ^ ( 3 cos 2© - 1) l r \ " 3 V dV bo (C - A) f r g ( r ) d r q i s t h e e l e c t r i c f i e l d g r a d i e n t a t t h e n u c l e u s caused by e l e c t r o n s near t h e F e r m i c s u r f a c e . I n o r d e r t o e x p l o r e t h e o r i e n t a t i o n dependence of th e K n i g h t s h i f t w i t h r e s p e c t t o t h e magnetic f i e l d , we can r e w r i t e t h i s e x p r e s s i o n as A H J „ (31) §=== - c o n s t a n t • (3 c o s z © - 1) H o The t o t a l K n i g h t s h i f t i n c l u d i n g t h e a n i s o t r o p i c p a r t i s t h e n g i v e n by e q u a t i o n s (23) and (31) and can.be w r i t t e n 8 (32) - ~ - K + $ KJ (3 c o s 2 © - 1) H o -17- CHAPTER 3 EXPERIMENTAL APPARATUS AND TECHNIQUE The n u c l e a r m agnetic r e s o n a n c e s p e c t r o m e t e r used i n t h i s work i s e s s e n t i a l l y of s t a n d a r d d e s i g n . F i g u r e 1 i s a s c h e m a t i c diagram o f t h e s p e c t r o m e t e r . The c i r c u i t diagrams a r e a l l g i v e n i n Appendix C. S i n c e t h e o b j e c t o f t h e e x p e r i m e n t s has been t o measure r e s o n a n c e f r e q u e n c i e s and l i n e shapes, a Pound-Knight- W a t k i n s 9 m a r g i n a l o s c i l l a t o r was chosen f o r t h i s work: The o s c i l l a t o r was v e r y s l i g h t l y m o d i f i e d f r o m t h e o r i g i n a l d e s i g n . I n s t e a d o f a 6J6 t u b e , a 396A was used as t h e o s c i l l a t o r t u b e . A l s o t h e r f a m p l i f i e r was f e d f r o m t h e ca t h o d e s o f t h e 396A i n s t e a d o f f r o m t h e g r i d t o w h i c h t h e sample c o i l was a t t a c h e d . The c a p a c i t a n c e f o r t h e r e s o n a n t c i r c u i t was p r o - v i d e d by a V a r i c a p ( v a r i a b l e c a p a c i t y d i o d e ) . The f r e q u e n c y of t h e o s c i l l a t o r was t h e n swept by v a r y i n g t h e v o l t a g e a c r o s s t h e d i o d e . The v o l t a g e s o u r c e f o r t h e d i o d e was p r o v i d e d by t h e s a w t o o t h waveform f r o m a m o d i f i e d T e k t r o n i x waveform g e n e r a t o r 1 0 ' 1 1 . The m o d i f i e d waveform g e n e r a t o r can p r o v i d e a n e g a t i v e g o i n g s a w t o o t h v o l t a g e f rom 100 t o 0 v o l t s w i t h p e r i o d s w h i c h v a r y f r o m s e v e r a l seconds t o s e v e r a l h o u r s . Power A mptif tar Au<Aio Osci HaW Ho ri70f\+al Arripli-Ficr PUce Sk.fW CRO P. K. W Occi I l a t e r 142 P. K . w . Co jrvffcr ar\d" Prir+'er £e.r\e.i"Vwe D i r e c t o r ^tco rder I M 0 0 I Figure 1. Block Diagram of the Spectrometer -19- Any v o l t a g e between 10 and 100 v o l t s can be used as t h e i n i t i a l v o l t a g e o f t h e saw t o o t h and t h e s a w t o o t h rundown can be s t o p p e d a t any t i m e w i t h t h e v o l t a g e r e t u r n i n g t o i t s i n i t i a l s e t v a l u e . U s i n g a V a r i c a p i n s t e a d of a v a r i a b l e a i r condenser e l i m i n a t e s n o i s e f r o m a d r i v i n g motor and p r o v i d e s a more u n i f o r m f r e q u e n c y sweep, e s p e c i a l l y a t t h e s l o w e r r a t e s . F o l l o w i n g t h e m a r g i n a l o s c i l l a t o r i s a model 216 White t w i n t e e narrow band audio; a m p l i f i e r . Two bandwidths a t 15 cps were used i n t h i s work: a 23% ba n d w i d t h (about 4 c p s ) network and a 1.3% (about .2 c p s ) network. The l a t t e r band- w i d t h c o r r e s p o n d s t o a t i m e c o n s t a n t of about 5 seconds and t h u s was used o n l y f o r c o m p a r a t i v e l y s l o w f r e q u e n c y sweeps. The phase s e n s i t i v e d e t e c t o r used i s i n p r i n c i p l e 12 t h e same as S i n i s t e r ' s , a l t h o u g h d i f f e r e n t t u b e s were employed. The s i g n a l r e c o r d e r was a V a r i a n r e c o r d e r , model G11A. The f r e q u e n c y o f t h e o s c i l l a t o r as i t swept t h r o u g h t h e s i g n a l was m o n i t o r e d by a 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 , model 524C, w i t h t h e a p p r o p r i a t e p l u g - i n u n i t . The f r e q u e n c y measured was i n a l l c a s e s t h e average f r e q u e n c y over .1 second. The f r e q u e n c y r e a d i n g was r e c o r d e d by a 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 , model 516B. The d i g i t a l r e c o r d e r i n t u r n a c t i v a t e d an i n d i c a t o r pen on t h e s i g n a l r e c o r d e r each time t h e e l e c t r o n i c c o u n t e r measured t h e f r e q u e n c y . T h i s method o f measuring and r e c o r d i n g f r e q u e n c y gave f a s t , a c c u r a t e , i and f r e q u e n t m o n i t o r i n g of t h e o s c i l l a t o r f r e q u e n c y as i t -20- p a s s e d t h r o u g h t h e n u c l e a r magnetic r e s o n a n c e s i g n a l . The magnet used t h r o u g h o u t t h i s work was a V a r i a n r o t a t i n g magnet w i t h t w e l v e i n c h p o l e f a c e s and a 2 1/4 i n c h gap and was c a p a b l e of g i v i n g a magnetic f i e l d o f 11.4 k i l o - gauss. M o d u l a t i o n o f t h e magnetic f i e l d was a c c o m p l i s h e d u s i n g two m o d u l a t i o n c o i l s each wound w i t h 60 t u r n s of No. 18 copper w i r e on b a k e l i t e forms w h i c h were mounted around t h e p o l e c a p s . The m o d u l a t i o n c o i l s were s u p p l i e d by an a u d i o o s c i l l a t o r t h r o u g h a 20 w a t t power a m p l i f i e r . A s c h e m a t i c dia g r a m of t h e low t e m p e r a t u r e system i s g i v e n i n F i g u r e 2. The h e l i u m c r y o s t a t i s a s t a n d a r d d o u b l e g l a s s dewar system. Temperatures down t o 1.15°K were o b t a i n e d by pumping on t h e l i q u i d h e l i u m w i t h a 3 i n c h K i n n e y pump. The t e m p e r a t u r e was c o n t r o l l e d by c o n t r o l l i n g t h e pump- i n g speed w i t h a n e e d l e v a l v e i n p a r a l l e l w i t h a l£ i n c h vacuum v a l v e , and was measured by o b s e r v i n g t h e vapour p r e s - s u r e o f t h e h e l i u m w i t h a mercury manometer i n p a r a l l e l w i t h one c o n t a i n i n g d i - b u t y l p t h a l a t e . The dewars and dewar cap were mounted s o t h a t t h e y c o u l d s l i d e i n and out from between the magnet p o l e f a c e s . The sample was mounted on t h e end of a 3/8 i n c h t h i n w a l l s t a i n l e s s s t e e l tube w h i c h formed t h e o u t e r conduc- t o r of t h e c o a x i a l l i n e c o n n e c t i n g t h e sample c o i l t o t h e m a r g i n a l o s c i l l a t o r . The c e n t r e c o n d u c t o r of t h e c o a x i a l l i n e was a No. 32 copper 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 . Kmney Pump He Cy ltr\der He Trorvs-fe «• JacUt (9) lo is| pWorv Fore. M Need le Valv/e 6 O i l Trap To Return Line X Oil Bottler- H 3 Oil Figure 2. Schematic. Diagram of the Low Temperature System -22- Three s e m i - c i r c u l a x r a d i a t i o n s h i e l d s made f r o m s h e e t m e t a l copper were mounted on the s t a i n l e s s s t e e l c o a x i a l l i n e , . , The t i n s i n g l e c r y s t a l was made from " E x t r a P u r e " t i n , VS-151, o b t a i n e d from t h e V u l c a n D e t i n n i n g Company. The p u r i t y quoted was 99.999+% t i n w i t h i m p u r i t i e s of .0008% l e a d and .000018% i r o n . The s i n g l e c r y s t a l was grown by p o u r i n g m o l t e n t i n i n t o a § i n c h by \ i n c h by 4 i n c h g r a p h i t e mold, th e n s l o w l y (one i n c h e v e r y 14 m i n u t e s ) w i t h d r a w i n g t h e mold from t h e f u r n a c e . In o r d e r t o grow a c r y s t a l w i t h t h e d e s i r e d o r i e n t a t i o n , t h a t i s w i t h t h e IpOl] d i r e c t i o n a p p r o x i m a t e l y p e r p e n d i c u l a r t o t h e c r y s t a l a x i s , a seed c r y s t a l w h i c h had the d e s i r e d o r i e n t a t i o n was used. The seed was j o i n e d t o the m o l t e n t i n j u s t o u t s i d e t h e f u r n a c e mouth by drawing out a s m a l l s t r e a m of t i n f r o m t h e main m e l t w i t h a g l a s s r o d . The i n t e r f a c e of t h e se e d and m o l t e n t i n was " p u d d l e d " or s t i r r e d u n t i l t h e seed began t o m e l t back f r o m t h e i n t e r f a c e . The mold was the n withdrawn from t h e f u r n a c e as d e s c r i b e d above. The c a s t t i n s l a b t h u s o b t a i n e d was e t c h e d e l e c t r o - l y t i c a l l y i n a f a i r l y d i l u t e HC1 s o l u t i o n , and i f no g r a i n b o u n d a r i e s were a p p a r e n t , an X-ray p h o t o g r a p h was ta k e n (Laue back r e f l e c t i o n photograph) t o d e t e r m i n e t h e c r y s t a l o r i e n t a - t i o n . T h i s method w i l l g i v e t h e o r i e n t a t i o n t o w i t h i n about one degree. The seeds were p r e p a r e d and a n a l y s e d i n a s i m i l a r manner. I n s t e a d o f d r a w i n g t h e m o l t e n t i n out t o j o i n a seed -23- as d e s c r i b e d a b o v e , i t was drawn o u t t o a f i n e p o i n t . The f i r s t t i p o f t h i s p o i n t t o s o l i d i f y does s o t o f o r m a s i n g l e c r y s t a l w h i c h i s t h e n p r o p a g a t e d a l o n g t h e w h o l e s p e c i m e n as i t i s w i t h d r a w n f r o m t h e f u r n a c e . B y u s i n g t h i s s e e d p l a c e d p r o p e r l y i n t h e m o l d , a new s e e d w i t h an o r i e n t a t i o n c l o s e r t o t h e d e s i r e d o r i e n t a t i o n c a n be g r o w n . T h i s p r o c e s s i s c o n t i n u e d u n t i l a s u i t a b l e s e e d c r y s t a l i s o b t a i n e d w h i c h i s t h e n u s e d f o r a l l s u b s e q u e n t c r y s t a l s . Once a c r y s t a l q u i t e n e a r t h e d e s i r e d o r i e n t a t i o n was o b t a i n e d , i t was s l i c e d a l o n g p l a n e s c o n t a i n i n g t h e [oOl] a x i s w i t h a j e w e l l e r ' s saw i n t o s l i c e s a b o u t 1 mm t h i c k . T h e s e s l i c e s were e t c h e d i n a m i x t u r e o f c o n c e n t r a t e d H C l and HNO3 f o r a few m i n u t e s u n t i l t h e y were a b o u t .3 mm t h i c k . They were t h e n g l u e d t o g e t h e r w i t h Q d o p e . The r e s u l t i n g l a m i n a t e d c r y s t a l was made up o f t h i r t y s l i c e s and was a b o u t ^ i n c h b y \ i n c h b y 3/8 i n c h . The c r y s t a l s l i c e s were a l i g n e d p r o b a b l y t o w i t h i n one d e g r e e and p o s s i b l y b e t t e r . The e r r o r o f a l i g n m e n t w o u l d seem t h e n t o be o f t h e same o r d e r o f m a g n i t u d e as t h e e r r o r i n d e t e r m i n i n g t h e c r y s t a l o r i e n t a t i o n f r o m t h e X - r a y p h o t o g r a p h . A l t h o u g h no s u b g r a i n b o u n d a r i e s were o b s e r v e d , t h e y may w e l l h a v e b e e n p r e s e n t and c o u l d g i v e a m i s a l i g n m e n t o f p e r h a p s o n e - h a l f d e g r e e . The e x p e r i m e n t s p e r f o r m e d w e r e m o s t l y n o t s e n s i t i v e t o m i s a l i g n m e n t s o f l e s s t h a n a few d e g r e e s , and no e v i d e n c e o f m i s a l i g n m e n t was o b s e r v e d . -24- The aluminum c r y s t a l was o b t a i n e d c o m m e r c i a l l y from M e t a l s R e s e a r c h L t d . , Cambridge. I t was a c y l i n d r i c a l s p e c i - men \ i n c h d i a m e t e r by 3/4 i n c h l o n g and was made f r o m 99.9999% p u r e aluminum. F o r e x p e r i m e n t s w i t h b o t h t h e aluminum and t i n c r y s t a l s , t h e sample c o i l was wound d i r e c t l y on the c r y s t a l r a t h e r t h a n h a v i n g t h e c o i l and c r y s t a l s e p a r a t e d by a l a y e r of some i n s u l a t o r l i k e e l e c t r i c a l t a p e o r m y l a r . I n o r d e r t o have t h e o s c i l l a t o r o s c i l l a t e a t t h e c o r r e c t f r e q u e n c y w i t h minimum c a p a c i t y , a f o u r s t r a n d 4^ t u r n c o i l wound f r o m No. 28 w i r e was used on t h e t i n c r y s t a l and a f i f t y t u r n c o i l o f No. 28 w i r e was wound around t h e aluminum c r y s t a l . Winding th e c o i l d i r e c t l y on t h e aluminum c r y s t a l improved t h e s i g n a l - t o - n o i s e r a t i o c o n s i d e r a b l y , presumably because t h e f i l l i n g f a c t o r o f t h e c o i l was i n c r e a s e d and because t h e i n d u c t a n c e of t h e c o i l was r e d u c e d , a l l o w i n g t h e c o i l t o have a g r e a t e r number o f t u r n s . U s i n g No. 32 w i r e f u r t h e r r e d u c e d t h e i n d u c - t a n c e , b u t w i t h more t h a n about 40 t u r n s i n t h e c o i l , t h e low Q of t h e i n d u c t a n c e p r e v e n t e d t h e o s c i l l a t o r f r o m o s c i l l a t i n g . Because t h e magnetic f i e l d d r i f t s somewhat d u r i n g an e x p e r i m e n t , t h e p r o c e d u r e f o l l o w e d was t o measure t h e magnetic f i e l d f r e q u e n t l y w i t h a probe c o n t a i n i n g D 20 mounted j u s t o u t s i d e t h e dewars. The d e u t e r o n r e s o n a n c e was o b s e r v e d d i r e c t l y on an o s c i l l o s c o p e t h r o u g h a wide band a u d i o a m p l i f i e r w h i l e a t t h e same t i m e t h e o s c i l l a t o r f r e q u e n c y was -25- measured w i t h 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 meter. For t h e measurements o f t h e a n i s o t r o p i c K n i g h t s h i f t , t h e magnetic f i e l d was measured b e f o r e and a f t e r each s e t o f r e a d i n g s t a k e n a t a p a r t i c u l a r c r y s t a l o r i e n t a t i o n . The second f i e l d measurement o f t e n agreed w i t h t h e f i r s t t o w i t h i n one p a r t i n 600,000 and never d i f f e r e d by more th a n f i v e p a r t s i n 10 f o r th e measurement of t h e t i n r e s o n a n c e t o be c o n s i d e r e d v a l i d . F o r t h e measurements d e s c r i b e d i n Chapter 5, somewhat more c a r e was t a k e n i n measuring t h e f i e l d . The f i e l d was measured b e f o r e and a f t e r e v e r y m e t a l resonance.and d i d not d i f f e r by more th a n two p a r t s i n 600,000 f o r t h e measurement t o be c o n s i d e r e d v a l i d . Almost a l l o f t h e r e s o n a n c e s were r e c o r d e d u s i n g 5 o r 10 second t i m e c o n s t a n t s i n t h e phase s e n s i t i v e d e t e c t o r . N e a r l y a l l o f t h e t i n r e s o n a n c e s were o b s e r v e d u s i n g a modu- l a t i o n o f l e s s t h a n o n e - t h i r d gauss. F o r t h e aluminum r e s o n a n c e s where t h e r e seemed t o be some problems of s a t u r a - t i o n and where o n l y t h e r e s o n a n c e f r e q u e n c y was wanted, t h e m o d u l a t i o n used was i n c r e a s e d f r o m l e s s t h a n two gauss t o about f o u r gauss f o r t h e l a s t s e t of measurements. F i g u r e s 3 and 4 show t y p i c a l t i n and aluminum s i g n a l s a t 1.15° K. F i g u r e 4. D e r i v a t i v e o f t h e A l Resonance F i g u r e 3. D e r i v a t i v e o f t h e Sn Resonance -27- CHAPTER 4 THE KNIGHT SHIFT AND LINE WIDTH OF ; THE TIN SINGLE CRYSTAL Measurements of t h e n u c l e a r magnetic r e s o n a n c e of 117 119 Sn and Sn i n t h e s i n g l e c r y s t a l o f w h i t e t i n d e s c r i b e d i n C h apter 3 were made as a f u n c t i o n o f t h e c r y s t a l o r i e n t a - t i o n i n t h e a p p l i e d m a g n e t i c f i e l d , as a f u n c t i o n o f t h e a p p l i e d m agnetic f i e l d , and as a f u n c t i o n of t e m p e r a t u r e i n t h e l i q u i d h e l i u m r a n g e . F i g u r e 5 summarizes some of t h e s e r e s u l t s . The d i f f e r e n t c u r v e s show t h e v a r i a t i o n o f t h e K n i g h t s h i f t as t h e e x t e r n a l f i e l d i s r o t a t e d t h r o u g h an a n g l e of 180° a t t e m p e r a t u r e s of 4.2°K and 1.15°K and a t f i e l d s o f 10.1 k i l o g a u s s and 6.13 k i l o g a u s s . The K n i g h t s h i f t was found t o be e q u a l w i t h i n e x p e r i m e n t a l e r r o r s f o r b o t h i s o t o p e s . Measurements o f t h e K n i g h t s h i f t as a f u n c t i o n of c r y s t a l o r i e n t a t i o n i n two d i f f e r e n t p l a n e s of 13 r o t a t i o n (appendix B) a s s u r e d t h a t t h e e l e c t r o n i c e n v i r o n - ment of t h e n u c l e u s showed t e t r a g o n a l symmetry. A l l o f t h e measurements r e c o r d e d i n t h i s c h a p t e r were made i n t h e c r y s t a l p l a n e c o n t a i n i n g t h e [00l] a x i s . T a b l e 1 g i v e s v a l u e s o f K and KJJ f o r d i f f e r e n t t e m p e r a t u r e s and a p p l i e d m a gnetic f i e l d s . -28- F i g . 5. The Knight S h i f t i n Tin as a Function of the Cr y s t a l Orientation i n the Magnetic F i e l d (Orientation measured from the [001] axis) -29- TABLE 1 Temperature F i e l d K x 1 0 4 KJ x 1 0 4 1.15° K 10.1 kg 71.9 ± .1 5.4 ± .1 4.2° K 10.1 kg 71.6 + .1 5.4 ± .1 1.15° K 6.13 kg 70.7 ± .2 5.4 ± .2 As has been n o t e d by Bloembergen and Rowland , K° i s p o s i t i v e i n d i c a t i n g t h a t q o f e q u a t i o n (30) i s p o s i t i v e . F u r t h e r m o r e , K{( i s q u i t e l a r g e , a l m o s t t e n p e r c e n t o f K. Because t h e h y p e r f i n e i n t e r a c t i o n f o r p-wave f u n c t i o n s i s l e s s t h a n f o r s-wave f u n c t i o n s and because o n l y t h e a n i s o - t r o p i c p a r t o f t h e p-wave f u n c t i o n i n t e r a c t i o n c o n t r i b u t e s t o K', , t h e l a r g e v a l u e of K'u i n d i c a t e s a s u b s t a n t i a l p-wave f u n c t i o n component i n t h e e l e c t r o n i c wave f u n c t i o n . I n m e t a l s whose specimen s i z e i s l a r g e compared t o t h e i r s k i n d e p t h , t h e power absorbed by t h e sample i s p r o p o r - t i o n a l t o X' , t h e r e a l and i m a g i n a r y p a r t s of t h e n u c l e a r s p i n s u s c e p t i b i l i t y , r a t h e r t h a n t o X " a l o n e as would be t h e case f o r non-metals o r f o r m e t a l s whose p a r t i c l e s i z e i s s m a l l compared t o t h e i r s k i n d e p t h . Chapman, Rhodes, and Seymour* 4 have d e t e r m i n e d t h a t t h i s e f f e c t would d e c r e a s e t h e -30- measured z e r o o f t h e d e r i v a t i v e o f t h e o b s e r v e d a b s o r p t i o n s i g n a l by about .3 of t h e l i n e w i d t h as d e t e r m i n e d f r o m t h e maxima of t h e d e r i v a t i v e . T h i s would d e c r e a s e K by about -4 .2 x 10 o r l e s s t h a n o n e - h a l f p e r c e n t of t h e v a l u e o f K. I t would a l s o d e c r e a s e t h e v a l u e o f K[( by, about one p e r c e n t of i t s v a l u e . The c o r r e c t i o n s t o K and K(' may w e l l be even 15 s m a l l e r because, as n o t e d by Karimov and Shchegolev , f o r r e s o n a n t c i r c u i t s o f low Q as i s t h e c a s e when t h e c o i l o f t h e c i r c u i t i s wound on a m e t a l specimen, t h e c o n t r i b u t i o n of X ' t o t h e o b s e r v e d a b s o r p t i o n may be q u i t e s m a l l . That t h i s c o u l d be t r u e i n our c a s e i s s u b s t a n t i a t e d by t h e l a r g e degree of symmetry shown by t h e o b s e r v e d a b s o r p t i o n c u r v e . F i g u r e 6 shows t h e asymmetry e x p e c t e d i n t h e d e r i v a t i v e o f L o r e n t z i a n and G a u s s i a n l i n e shapes. The r a t i o o f t h e a m p l i t u d e s of t h e extrema o f t h e d e r i v a t i v e o f t h e L o r e n t z i a n c u r v e i s .39 and of t h e G a u s s i a n c u r v e i s .55, b u t t h e r a t i o of t h e o b s e r v e d d e r i v a t i v e was about .7. T h i s means e i t h e r t h e l i n e shape of the o b s e r v e d r e s o n a n c e i s " s q u a r e r " t h a n a G a u s s i a n l i n e shape o r t h a t t h e d i s p e r s i o n mode does not c o n t r i b u t e e q u a l l y w i t h t h e a b s o r p t i o n mode t o t h e o b s e r v e d r e s o n a n c e . S i n c e t h e c o r r e c t i o n s t o K and K'( computed on t h e b a s i s o f e q u a l con- t r i b u t i o n s f r o m b o t h t h e a b s o r p t i o n and d i s p e r s i o n modes a r e j u s t b a r e l y l a r g e r t h a n t h e e r r o r s i n t h e measured K and , when t h e a d d i t i o n a l p o i n t o f u n e q u a l c o n t r i b u t i o n s i s c o n s i d e r e d i t seems u n n e c e s s a r y t o make any c o r r e c t i o n s t o -31- Lorerctzian Lwe Skape. d» \ 1 1 G a u s s i a n Lir\e SKape dV /- — — dv \ i i ^ i i Figure 6. -32- t h e o b s e r v e d r e s u l t s . F i g u r e 7 shows t h e l i n e w i d t h of t h e t i n specimen as d e t e r m i n e d by t h e extrema i n t h e d e r i v a t i v e s of a b s o r p t i o n l i n e s . The l i n e w i d t h cannot be e x p l a i n e d i n terms of d i p o l e - d i p o l e c o u p l i n g between n e a r e s t n e i g h b o u r s a l o n e . The second moment, A o J 2 f f o r d i p o l e - d i p o l e c o u p l i n g f o r a s u b s t a n c e 16 w i t h two s p i n s i s g i v e n by Van V l e c k (33) where A G J j 2 " ( A 6 J I 2 ) i i + ( A 0 J I 2 ) I S 3 4 2 , , ̂  (1 - 3 c o s 2 O j k ) 2 ( ^ 6 0 T 2 ) T T - - j Y T 4 * K I + 1 ) S 6  A I I 4 I k °  r "Jk and ( 1 - 3 c o s 2 © j k ) 2 Jk a r e t h e c o n t r i b u t i o n s t o a n u c l e u s f r o m l i k e n u c l e i l a b e l l e d I and u n l i k e n u c l e i l a b e l l e d S. I n t h e case o f t i n , f o r ©..=»() o n l y t h e two n e a r e s t n e i g h b o u r s g i v e a s i g n i f i c a n t JK c o n t r i b u t i o n t o t h e second moment. When t h e n a t u r a l abund- ances of t h e two t i n i s o t o p e s a r e c o n s i d e r e d , t h i s f o r m u l a -33- Figure 7* The Line Width of the Tin Resonance as a Function df the C r y s t a l Orientation i n the Magnetic F i e l d (Orientation measured from the [OOl] axis) -34- g i v e s a second moment of .15 ( k c / s ) 2 a t fl = 0. T a k i n g t h e j k s econd moment of t h e o b s e r v e d s i g n a l t o be a p p r o x i m a t e l y e q u a l t o t h e s q u a r e of t h e s e p a r a t i o n o f t h e extrema of t h e d e r i v a - 2 t i v e of t h e re s o n a n c e s i g n a l , we get a r e s u l t of 2 ( k c / s ) , more than t e n t i m e s t h e t h e o r e t i c a l v a l u e . I f t h e l i n e shape i s i n f a c t G a u s s i a n , t h e e x p e r i m e n t a l second moment i s 1.6 ( k c / s ) 2 . I f t h e c a l c u l a t e d second moment due t o d i p o l e - d i p o l e i n t e r a c t i o n s i s s u b t r a c t e d f r o m t h e o b s e r v e d second moment, 2 t h e r e r e m a i n s an i s o t r o p i c second moment o f .6 ( k c / s ) . One p o s s i b l e o r i g i n of t h i s second moment i s b r o a d e n i n g due t o m i s a l i g n m e n t of t h e s l i c e s w h i c h make up t h e c r y s t a l specimen ( a l t h o u g h t h i s b r o a d e n i n g w i l l n o t be i s o t r o p i c ) . A s i m p l e c a l c u l a t i o n (see Appendix A) shows t h i s second moment t o be (34) ( V - U ) 2 - ( } ) 12 c o s 2 0 c o s 2 * s i n 2 0 s i n 2 * * + — s i n 4 0 s i n 4 o t - where A \ > [ =» K,{ 7 H Q, 0 i s t h e a n g l e between t h e [pOl] c r y s t a l a x i s and t h e magnetic f i e l d , and o C i s a parameter e x p r e s s i n g t h e degree of m i s a l i g n m e n t . The c o n t r i b u t i o n t o th e second moment w i l l be g r e a t e s t f o r 0 «• 45°. M i s a l i g n m e n t i s b e l i e v e d t o be t h e e x p l a n a t i o n f o r p r e v i o u s l y r e p o r t e d b r o a d l i n e s . I n o r d e r t o e x p l a i n t h e measured second -35- moment o f .6 ( k c / s ) , c * . must be s e t a t 18°, an i m p r o b a b l y h i g h f i g u r e . F u r t h e r , a s t u d y of t h e l i n e w i d t h as a f u n c t i o n o f m agnetic f i e l d f o r 0 =» 45° r e v e a l e d no change i n t h e l i n e w i d t h when t h e f i e l d went from 10.1 k i l o g a u s s t o 6.13 k i l o g a u s s . The u n c e r t a i n t y i n t h e measurement o f t h e l i n e w i d t h was about .1 k c / s . Assuming t h e change i n l i n e w i d t h over t h i s r e g i o n of magnetic f i e l d t o be t h e maximum v a l u e w i t h i n t h i s u n c e r t a i n t y , we f i n d t he c o r r e s p o n d i n g change i n t h e second moment would mean a m i s a l i g n m e n t of <=*. =• 4°, s t i l l a somewhat h i g h v a l u e . I f i t i s assumed t h a t m i s a l i g n m e n t o f t h e c r y s t a l s l i c e s (or m i s a l i g n m e n t due t o sub g r a i n b o u n d a r i e s ) does not c o n t r i b u t e t o the l i n e w i d t h , t h e n t h e r e must be some f a i r l y l a r g e i s o t r o p i c b r o a d e n i n g mechanism such as t h a t a r i s i n g f r om 17 18 i n d i r e c t exchange i n t e r a c t i o n s ' between t h e two t i n i s o t o p e s w h i c h c o n t r i b u t e s t o t h e l i n e w i d t h . Karimov and S h c h e g o l e v s u g g e s t e d t h a t t h i s mechanism i s p r e s e n t i n t i n and a t t r i b u t e d a second moment o f about .5 ( k c / s ) t o i t . I n d i r e c t exchange i n t e r a c t i o n s i n m e t a l s r e s u l t f r o m a c o u p l i n g between t h e magnetic moments of two n u c l e i by way o f t h e i r c o n d u c t i o n e l e c t r o n s and t h e h y p e r f i n e i n t e r a c t i o n . I f o n l y t h e c o n t a c t term o f t h e s p i n H a m i l t o n i a n of e q u a t i o n (1) i s c o n s i d e r e d , t h i s c o u p l i n g i s p u r e l y s c a l a r 1 9 . The exchange i n t e r a c t i o n energy between u n l i k e n u c l e a r s p i n s i n 1 7 a m e t a l i s o f t h e form -36- (35) °H - £ £ A ± i I ± . I . JL 6 Van V l e c k shows t h a t t h e second moment o f t h e a b s o r p t i o n l i n e of t h e n u c l e i w i t h one magnetic moment due t o t h e exchange i n t e r a c t i o n w i t h t h e n u c l e i o f a n o t h e r magnetic moment i s (36) h 2 ( AV)2 - i 1(1 + 1) f 1 A,, 2 3 . xj where f i s t h e f r a c t i o n o f t h e l a t t i c e s i t e s o c c u p i e d by t h e n u c l e i w i t h t h e d i f f e r e n t m agnetic moment. He a l s o shows i n t h e same paper t h a t exchange i n t e r a c t i o n s between n u c l e i of l i k e m a gnetic moments do not c o n t r i b u t e t o t h e second moment of t h e re s o n a n c e l i n e . S e v e r a l assumptions a r e made t o c a l c u l a t e A ^ j . I t •*\ 2-1.2 i s assumed t h a t t h e energy E K i s g i v e n by E k = . , , where — 2m* m* i s t h e e f f e c t i v e mass of t h e e l e c t r o n s w i t h wave number k. The number of o r b i t a l s t a t e s Z ( k ) d 3 k i n t h e space o f wave Vd3k v e c t o r s i s assumed t o be _ , t h e number f o r f r e e ( 2 T T ) 3 e l e c t r o n s . A l s o t h e e x c i t e d s t a t e s E^, a r e assumed t o e x t e n d f r o m t h e F e r m i l e v e l , Ep = ^2^F } t o i n f i n i t y , t h a t 2m* i s , t h e r e a r e no energy gaps j u s t above t h e F e r m i s u r f a c e . F i n a l l y , i t i s assumed t h a t t h e main c o n t r i b u t i o n t o t h e exchange i n t e r a c t i o n comes f r o m v a l u e s o f such t h a t Ik 2 - k / l 2 i s v e r y s m a l l , and t h a t |jkj ~ |k»| jkpl, -37- b e i n g t h e wave v e c t o r o f a f i l l e d s t a t e and .k' of an u n f i l l e d s t a t e . Under t h e s e a s s u m p t i o n s , t h e s c a l a r c o u p l i n g c o e f f i c i e n t becomes ' (37) A..- : : ( — \ 2 ^ ^ , 0 0 8 ( 2 ^ % , ) 1 J 2 ^ ( 2 ^ + l ) ( 2 I j + 1) \ R i j j I  F 3 F 3 - s i n 2k FR .< where i s t h e ato m i c volume m* i s t h e e f f e c t i v e mass o f t h e e l e c t r o n i n v o l v e d i n t h e exchange i n t e r a c t i o n § i s t h e r a t i o of t h e h y p e r f i n e i n t e r a c t i o n i n c r y s t a l t o t h a t i n a f r e e atom 11 i s t h e o b s e r v e d a t o m i c h y p e r f i n e s t r u c t u r e s p l i t t i n g R t i i s t h e s e p a r a t i o n of t h e two n u c l e a r s p i n s 1^ and I j kp i s t h e wave v e c t o r of an e l e c t r o n a t t h e F e r m i s u r f a c e I t i s d i f f i c u l t t o a s c r i b e a v a l u e t o A i j f o r t i n g i v e n by t h i s f o r m u l a because t i n does not have even a n e a r l y s p h e r i c a l F e r m i s u r f a c e . However, assuming t h a t t h i s f o r m u l a f o r A ^ j i s not e n t i r e l y i n v a l i d , we can make an o r d e r of magnitude e s t i m a t i o n o f t h e second moment due t o exchange -38- b r o a d e n i n g . I t i s assumed t h a t m = m . T h i s i s not t r u e f o r a l l e l e c t r o n s as i s shown by t h e many p e r i o d s of t h e de Haas- OQ pp rn • van Alphen o s c i l l a t i o n s i n t i n ' w h i c h g i v e v a l u e s o f — v a r y i n g between .05 and 1.0. From e l e c t r o n i c s p e c i f i c heat m ̂  measurements. —— i s found t o be 1.2. L i k e w i s e t h e v a l u e of t h e m wave v e c t o r k F depends on w h i c h p a r t of t h e F e r m i s u r f a c e i s b e i n g c o n s i d e r e d , b u t some average v a l u e can be a p p r o x i m a t e d u s i n g t h e measured v a l u e s of t h e e l e c t r o n i c s p e c i f i c h e a t . W i t h i n t h e l i m i t s o f t h e s e a p p r o x i m a t i o n s , i t i s r e a s o n a b l e t o assume J = 1 , and t o assume some average v a l u e o f V a measured i n t h e d i f f e r e n t f i n e s t r u c t u r e l e v e l s of t h e ground s t a t e of 21 t h e f r e e atom of t i n . p e r f o r m t h e sum over t h e f i r s t n-1 s h e l l s of n e a r e s t n e i g h b o u r s , and r e p l a c e t h e sum by an i n t e g r a l f o r t h e r e s t . In o r d e r t o p e r f o r m t h e summation of e q u a t i o n ( 3 6 ) , we (38) s i n ' i ( 2 k F R i j ) - 4 k F R ± j c o s ( 2 k F R i j ) s i n ( 2 k F R i j ) dV 2- dV R i n -39- where A r e p r e s e n t s t h e c o n s t a n t terms i n e q u a t i o n ( 3 7 ) , dV = 47TRJLJ 2 d i * i j > a n d ^ *-s * h e n u m D e r o f atoms per u n i t volume. The t h r e e i n t e g r a l terms g i v e . . ^ . 2 , p. .c ,2 f c o s 2 a s i n 2 a (39) Z. A. . = 4T T ^ ( 2 k p ) 5 A ) — + 5 — — + s i n b j=n 1 J C 3 a 5 a x j _ _2 _ 16 1 5 b 2 5b 4" where . . . £ s i b = - \ dx and b = 2a = 4 k F R i n F o r l a r g e b ( i n our c a s e b 8 0 ) , s i b can be expanded i n a s e r i e s , and t h e sum can be w r i t t e n 2 5 \ c o s ^ a 4 cos b 4 s i n b (40) £ A 2 - 4 T r ? A 2 ( 2 k F ) 5 { j - n 1 J (. 3 a 3 3 b 3 b 4 + s i n f a + 16 cos b + 0(J^} 5 a 5 5 b 5 W 6 J -40- U s i n g an IBM 1620 computer, e q u a t i o n (38) was e v a l u a t e d f o r kp between .5 x 10® cm"-1 and 2.5 x 1 0 8 cm 1 a t i n t e r v a l s of .01 by p e r f o r m i n g t h e sum over f i f t e e n s h e l l s (84 n e a r e s t n e i g h b o u r s ) and r e p l a c i n g the sum by an i n t e g r a l f r o m t h e s i x t e e n t h s h e l l outwards. F i g u r e 8 shows t h e r e s u l t «r A, 2 o f t h i s c a l c u l a t i o n w i t h a p l o t o f 2- J-J vs kp. As can 3 A 2 be s e e n , t h e sum i s v e r y s e n s i t i v e t o kp. The v a l u e o f k F o b t a i n e d from t h e e l e c t r o n i c s p e c i f i c heat of t i n i s 8 —1 1.5 x 10 cm" , w h i c h happens t o be near a maximum p o i n t o f t h e c u r v e . Thus a s m a l l e r r o r i n kp, a l t h o u g h i t w i l l cause a c o n s i d e r a b l e e r r o r i n , w i l l g i v e l e s s e r r o r t h a n T A 2 a n o t h e r v a l u e might. F u r t h e r , t h i s v a l u e of kp w i l l g i v e t h e upper l i m i t t o t h e exchange b r o a d e n i n g c o e f f i c i e n t u n l e s s t h e c o r r e c t v a l u e o f kp i s c o n s i d e r a b l y l a r g e r t h a n t h a t e x p e c t e d f r o m e l e c t r o n i c s p e c i f i c heat measurements. F o r k P = 1.5 x 1 0 8 cm" 1, 2. A i j - 1,75 x 1 0 6 2 , The i n t e g r a l J IF p a r t o f e q u a t i o n (38) c o n t r i b u t e s o n l y about one p e r c e n t t o t h i s sum, and t h e terms n e g l e c t e d , t h o s e o f t h e o r d e r 4TT9 ( 2 k p ) 5 ^ ° ( ^ ) 6 ] a r e d o w n b y a f a c t o r o f 1 0 8 . I n f a c t , f o r t h i s v a l u e o f kp, t h e most s i g n i f i c a n t c o n t r i b u t i o n comes fro m o n l y t h e f i r s t two n e a r e s t n e i g h b o u r s i n t h e f i r s t s h e l l . 10 —54 Assuming V a t o be 10 c / s , we get A =» 2.66 x 10 and u s i n g e q u a t i o n ( 3 6 ) , we g e t ( A D ) 2 - .57 ( k c / s ) 2 . T h i s e x c e l l e n t agreement between t h e o r y and experiment can be no -41- • 4 6 8 1.0 1.2 1.4 1 6 1.8 2 . 0 2 . 2 2 . 4 k F x I O ' 8 Figure 8. •42 more th a n f o r t u i t o u s c o n s i d e r i n g t h e a p p r o x i m a t i o n s i n v o l v e d i n d e r i v i n g t h e f o r m u l a f o r A^^, l e t a l o n e c o n s i d e r i n g t h e a p p r o x i m a t i o n s i n t h e v a l u e of t h e c o n s t a n t s s u c h as V a and ̂  . However, t h e agreement i s e n c o u r a g i n g i n s o f a r as i t g i v e s some j u s t i f i c a t i o n f o r c h o o s i n g kj, i n n o n - c u b i c m e t a l s from measurements o f t h e e l e c t r o n i c s p e c i f i c heat and i n d i c a t e s t h a t t h e a p p r o x i m a t i o n o f a s p h e r i c a l F e r m i s u r f a c e i n t h e d e r i v a t i o n o f t h e e x p r e s s i o n f o r A-|j may not be p a r t i c u l a r l y r e s t r i c t i v e . Because t h i s r e s u l t can be c o n s i d e r e d o n l y an o r d e r o f magnitude c a l c u l a t i o n , no attempt t o s e p a r a t e t h e 2 q u a n t i t i e s " X p and J has been made . F i g u r e s 5 and 9 show t h e te m p e r a t u r e dependence o f t h e S n 1 1 ^ and S n 1 1 9 r e s o n a n c e . The measured change o f i n H g o i n g f r o m 4.2°K t o 1.15°K was o n l y - .002% or c o n s i d e r a b l y l e s s t h a n 1% o f t h e t o t a l K n i g h t s h i f t . I f a f r e e e l e c t r o n model h o l d s , i s e x p e c t e d t o v a r y as V Q , where V 0 i s 2 t h e a t o m i c volume . The a c c u r a c y o f t h e te m p e r a t u r e dependence measurements i s t o o poor over t h i s r a n g e of t e m p e r a t u r e t o e n a b l e any st a t e m e n t about t e m p e r a t u r e v a r i a t i o n of t h e K n i g h t s h i f t o t h e r t h a n i t i n c r e a s e s s l i g h t l y as t h e te m p e r a t u r e i s d e c r e a s e d . F i g u r e 5 shows t h a t o n l y K and not K*v i s tempera- t u r e dependent w i t h i n t h e a c c u r a c y of t h e ex p e r i m e n t . 117 The magnetic f i e l d dependence of t h e Sn and S n 1 1 9 r e s o n a n c e s shown i n F i g u r e s 5 and 10 i s f a i r l y s m a l l over t h e range o f f i e l d s c o n s i d e r e d , but i s w e l l o u t s i d e -43- .770 H /o760 F i g u r e 9 . The K n i g h t S h i f t i n T i n as a f u n c t i o n of Temperature F i g u r e 10. The K n i g h t S h i f t i n T i n as a F u n c t i o n of Magnetic F i e l d ( M a g netic f i e l d measured i n terms o f t h e non-metal Sn r e s o n a n c e ) -44- experimental error. No simple explanation of the f i e l d dependence seems to exi s t since X n i s believed to be f i e l d X"' independent. The f i e l d dependence can be explained perhaps in terms of p o l a r i z a t i o n of the ion cores, or i n terms of some diamagnetic f i e l d dependence. -45- CHAPTER 5 THE POSSIBILITIES OF DE HAAS-VAN ALPHEN TYPE OSCILLATIONS IN THE KNIGHT SHIFT As mentioned i n C h a p t e r s 1 and 2, i f t h e o r b i t a l a n g u l a r momentum of t h e e l e c t r o n s i n a m e t a l i s not c o m p l e t e l y quenched, t h e r e e x i s t s t h e p o s s i b i l i t y of an o s c i l l a t o r y b e h a v i o u r of t h e K n i g h t s h i f t caused by o s c i l l a t i o n s i n t h e d i a m a g n e t i c s u s c e p t i b i l i t y as t h e e x t e r n a l f i e l d i s v a r i e d . T h i s e f f e c t would be s i m i l a r t o t h e de Haas-van Alphen 22 23 24 e f f e c t * ' w h i c h has y i e l d e d so much i n f o r m a t i o n about th e F e r m i s u r f a c e o f many m e t a l s . The i d e a f o r t h i s e x p e r i - ment f i r s t a r o s e i n d i s c u s s i o n s i n t h i s l a b o r a t o r y between Dr. R. B a r r i e and Dr. M. Bloom who mentioned i t t o T. P. Das. The problem o f c a l c u l a t i n g t h e d i a m a g n e t i c f i e l d a t t h e n u c l e u s i s r a t h e r f o r m i d a b l e , a l t h o u g h a t t e m p t s a t i t have 25 been made by Das and Sondheimer f o l l o w e d by o t h e r a u t h o r s 2 6 ' 2 7 , 2 8 , 2 9 , 3 0 and e s t i m a t e s of t h e change i n t h e K n i g h t s h i f t have been made. The de Haas-van Alphen e f f e c t i s a low t e m p e r a t u r e , h i g h magnetic f i e l d phenomenon. T y p i c a l l y , e x p e r i m e n t s a r e done a t 1.2°K i n magnetic f i e l d s between 15 and 100 k i l o g a u s s . -46- A l t h o u g h t h e t e m p e r a t u r e s r e q u i r e d a r e r e a d i l y a c c e s s i b l e , t h e s e magnetic f i e l d s a r e somewhat h i g h f o r t y p i c a l n u c l e a r m agnetic r e s o n a n c e s t u d i e s . To o b t a i n f i e l d s o f 100 k i l o g a u s s , 22 Shoenberg and o t h e r s have used p u l s e d magnets whose r a t e of g change of f i e l d i s something l i k e 5 x 10 g a u s s / s e c , q u i t e u n s u i t a b l e f o r n u c l e a r magnetic r e s o n a n c e work. The work d e s c r i b e d h e r e was performed i n a f i e l d o f about 10 k i l o g a u s s a t 1.15°K. The magnetic f i e l d a t t h e n u c l e u s due t o o r b i t a l m o t ion o f e l e c t r o n s a t t h e bottom of a non-degenerate band i s 26 g i v e n by Y a f e t t o be (41) A H d - (4TT - D) • Xd • H where ;X(j i s t h e d i a m a g n e t i c s u s c e p t i b i l i t y t e n s o r , D i s th e d e m a g n e t i z i n g c o e f f i c i e n t t e n s o r , and H i s t h e a p p l i e d m a g n e t i c f i e l d . The t e n s o r "Xd w i l l have a terra w h i c h i s independent o f H and a term w h i c h o s c i l l a t e s as H i s v a r i e d . These two terms w i l l be superimposed on t h e p a r a m a g n e t i c 27 term w h i c h g i v e s r i s e t o t h e K n i g h t s h i f t . Stephen has c a l c u l a t e d t h e o s c i l l a t o r y p a r t of t h e s h i e l d i n g f a c t o r , <Td •» 4TT , and has f o u n d i t t o be -47- ( 4 2 ) < T d ( o s c ) - - 1 2 N 7 r ^ o k T , £ ^ ^ ^ ^ ^ ^ J m ^ V ^ 0 H n=l ^ \v* H s i n h f™2™ » o H N where ^ i s t h e d e n s i t y o f e l e c t r o n s i n v o l v e d i n t h i s i n t e r a c t i o n m* i s t h e e f f e c t i v e mass o f t h e e l e c t r o n | i Q i s t h e " e f f e c t i v e " Bohr magneton (m* r e p l a c e s m) ^ Q i s t h e F e r m i energy I ( n ) i s an i n t e g r a l t o be e v a l u a t e d n u m e r i c a l l y k i s Boltzmann's c o n s t a n t T i s t e m p e r a t u r e As can be seen f r o m e q u a t i o n ( 4 2 ) , c5",j(osc) v a r i e s p e r i o d i c a l l y i n ̂ . F u r t h e r , s i n c e m* depends on t h e o r i e n t a - t i o n of t h e F e r m i s u r f a c e , t h a t i s t h e c r y s t a l o r i e n t a t i o n i n t h e magnetic f i e l d , t h e s i n / n i T T 0 ) f a c t o r means <5~d (osc) w i l l be a n i s o t r o p i c . F o r c o m p a r a t i v e l y low f i e l d s such as 10 k i l o g a u s s , where ^ ° 500, t h i s a n i s o t r o p y would ^o c o m p l e t e l y mask any e f f e c t i n a powder whose p a r t i c l e s a r e randomly o r i e n t e d i n t h e magnetic f i e l d . F o r f i e l d s of t h e o r d e r o f 10 k i l o g a u s s and f o r l i q u i d h e l i u m t e m p e r a t u r e s , t h e a m p l i t u d e o f t h e f i r s t term - 4 3 - of e q u a t i o n (42) i s .2 < 4 3> [ ^ d < o s c > ] m a x 12NTT ^ — - r . _ * where Stephen g i v e s 1(1) a v a l u e of about . 1. I f we s u b s t i t u t e i n s u i t a b l e v a l u e s f o r t h e p a r a m e t e r s , we f i n d t h a t — — — i s H of t h e o r d e r of IO*"6 f o r b o t h aluminum and t i n . T h i s i s l e s s t han one p e r c e n t of t h e K n i g h t s h i f t and would be an upper l i m i t t o t h e e f f e c t s i n c e D has been assumed t o be z e r o . The p e r i o d of t h e o s c i l l a t i o n s i s g i v e n by A ^L-^P-^ = 2w . U s i n g t h e e x p e r i m e n t a l v a l u e f o r t h e p e r i o d s of aluminum and t i n o o g i v e n by Shoenberg"" , we f i n d t h a t a t 10 k i l o g a u s s aluminum has a p e r i o d of about 30 gauss and t i n has one between 20 gauss and 60 gauss, depending on t h e c r y s t a l o r i e n t a t i o n i n t h e magnetic f i e l d . Because t h e t h e o r y f o r t h e s e d i a m a g n e t i c o s c i l l a - t i o n s i n t h e K n i g h t s h i f t i s a t b e s t o n l y an o r d e r o f magnitude t h e o r y , i t was f e l t w o r t h w h i l e t o s e a r c h f o r them i n s p i t e of t h e i r s m a l l p r e d i c t e d a m p l i t u d e . The measurements were a l l done a t 1 .5°K a t f i e l d s near 10 k i l o g a u s s . The method o f r e c o r d i n g t h e d a t a t o ensure g r e a t e s t p o s s i b l e a c c u r a c y has a l r e a d y been d e s c r i b e d i n Chapter 3. S i g n a l s were r e c o r d e d a t i n t e r v a l s of a few gauss over a range of magnetic f i e l d c o r r e s p o n d i n g t o a t l e a s t one p e r i o d . The -49- r e s u l t s of the experiments are shown i n Figures 11 and 12. Within the accuracy of the experiment, neither the aluminum nor t i n c r y s t a l gave any i n d i c a t i o n of these diamagnetic o s c i l l a t i o n s . An upper l i m i t to th e i r contribution to the Knight s h i f t was found to be about .002% for the aluminum c r y s t a l and .001% for the t i n c r y s t a l . Similar experiments were attempted on a bismuth singl e c r y s t a l which should have a much greater amplitude of cT^(osc) than either t i n or aluminum. However, no s i g n a l from bismuth was observed at l i q u i d helium temperatures, 26 probably because of the long relaxation time of bismuth . - 5 0 - AH0/ H *> .150 .149 148 . 1 4 6 . , 1 4 5 . . 1 4 4 . o o o o o o o o o o o 8 8 o ° o ° 8 ° •WL ° . 8 - 8 • o 8 S 8 o ° o 0 6.578 6.SM) 6.582 6J84 6.5S6 6558 6.595 6.592 6.594 6.596 6.598 (l^uW • - t^k Figure 11. The Knight S h i f t i n Aluminum as a Function of Magnetic F i e l d (Magnetic f i e l d measured i n terms of the deuteron resonance) -51- H k •772 .17 \ o o o 0 0 9 o 770h « « a e 769 - .768 .767 - .766 .765 - o o ° 0 ° o 0 0 0 J 1 1 1 1 1 1 1 L_V^ 15 "30 IS.9I 15.92 IS.93 15.94 15.95 15.96 15.97 IS.96 15.99 16.00 16.01 16.0P 16.03 F i g u r e 12. The K n i g h t S h i f t i n T i n as a F u n c t i o n of Magn e t i c F i e l d ( M a g netic f i e l d measured i n terms o f t h e non-metal Sn r e s o n a n c e ) -52- APPENDIX A THE SECOND MOMENT OF THE RESONANCE LINE DUE TO MISALIGNMENT OF THE CRYSTAL SLICES The re s o n a n c e f r e q u e n c y i n a s i n g l e c r y s t a l w i t h t e t r a g o n a l symmetry measured f r o m i t s average v a l u e ^ Q ( l + K ) i s V - I A (3 c o s 2 © - 1) where A ~d\ = Kf \)Q . T h i s e q u a t i o n f o l l o w s f r o m e q u a t i o n ( 3 2 ) . We assume t h a t t h e s l i c e s a r e m i s a l i g n e d s uch t h a t © v a r i e s between © 0 - oC and © Q + oC , where »C i s a measure o f t h e m i s a l i g n m e n t of t h e c r y s t a l s l i c e s . To c a l c u l a t e t h e second moment o f the a b s o r p t i o n s i g n a l due t o m i s a l i g n m e n t , we c o n s i d e r t h e f o l l o w i n g q u a n t i t i e s . The average v a l u e of V i s d e f i n e d by V - "V f ( V ) dV where f(.v) dy i s t h e p r o b a b i l i t y t h a t t h e reso n a n c e f r e q u e n c y i s between V and "J + dV . I f we assume an -53- i s o t r o p i c d i s t r i b u t i o n i n © between t h e two l i m i t s © Q - o c and © 0 + ©< , t h e n f (u)du - 7— - — 4 ^ civ - f ( v ) d v b-a b-a dv where u « cos ©, b = cos(© 0 - o c ) , and a » cos(© 0 + o c ) . Thus "v" can be w r i t t e n b ^ _ C ( 3 u 2 - 1) d u 2 J b - a a (b' 5 + ba + a'4 - 1) S i m i l a r l y ~y2~ can be w r i t t e n b V » . ( - ^ i i . ) 2 C O u 4 - 6 u 2 + !) d u a j | ( b 4 + b 3 a + a 2 b 2 + b a 3 + a 4 } _ 2 ( b 2 + ba + a 2 ) + l j The second moment due t o m i s a l i g n m e n t o f the c r y s t a l s l i c e s i s - 5 4 - ( V - V )' v 2 .PS i'2 ^ ( b 4 + a 4 ) 5 I ( b 3 a + b 2 a 2 + b a 3 ) 5 If b =» cos(© -ot) and a - cos(© + <*), then v 2 - ( U ) 2 2 1 1) ) 1 2 c o s 2 © c o s 2 * s in 2 © s i n 2 < * + i s i n 4 © s in 4oC Volume 1, number 3 P H Y S I C S L E T T E R S - 1 May 1962 T H E A N I S O T R O P Y O F T H E N U C L E A R M A G N E T I C R E S O N A N C E IN W H I T E T I N * E . P. JONES ** and D. L L E W E L Y N WILLIAMS *** Department of Physics, University of British Columbia, Vancouver 8, B.C., Canada Received 17 April 1962 The nuclear magnetic resonance signal in white tin has been studied in some detail in the pow- der 1 - 3 ) , and it has been established that the ob- served broad line is a consequence of the anisotro- py of the Knight shift in tin together with the effect of nuclear spin exchange between different iso- topes 4). Bloembergen and Rowland from ob- servations on thallium, have suggested that the ex- change interaction need not be isotropic. In the hope of clarifying these ideas, a direct study of the anisotropy of both the Knight shift and the line width has been carried out in a single crys- tal specimen of white tin. The specimen was con- structed in the form of a multiple layer sandwich of 0.1 mm thick oriented in tin layers separated by 0.05 mm layers of Mylar; the whole cemented to- gether with a silicone resin spray. The tin layers were formed by etching down 1 mm thick tin slices cut from a single crystal. The signals were observed with a Pound-Knight spectrometer, and in view of the comparatively weak tin signal, measurements were taken at the lowest available temperature (~ 1 .15°K) . A steady magnetic field of 10 kilogauss was produced by a Varian 12 inch rotatable magnet and was monitored by the deuteron resonance in D 2 O . Rotations were * Research supported by the National Research Coun- cil of Canada. ** Holder of International Nickel Company of Canada Research Fellowship. *** National Research Council of Canada Postdoctoral Fellow. Fig. 1. The (110) plane containing the [001] and the [111] directions, and the plane perpendicular to it. The magnetic field is rotated in either of these planes. performed in the mutually perpendicular planes shown in fig. 1. The observed anisotropy in the Knight shift and in the line width determined by the maxima in the derivative of the S n ^ " reso- nance are shown in figs. 2 and 3. The anisotropy was checked for 180° symmetry by two points at 2 2 5 ° and 3 1 5 ° . The Knight shift is closely expressible in terms 109 Volume 1, number 3 PHYSICS L E T T E R S 1 May 1962 76 74 ^1* 70 .68 o (jTo) plane * plane perpendicular to (110) K + K'„ (3 cos'& -/) 0° 45' 90' 135' 180° Fig. 2. The Sn 1 1^ Knight shift as a function of crystal orientation in the magnetic field. of the expected r e l a t i o n f o r t e t r a g o n a l s y m m e t r y ^ = K + \ K{ (3 c o s 2 6 - 1) , 6) w h e r e S i s the angle between the m a g n e t i c f i e l d and the (001) a x i s . O u r r e s u l t s a r e c o m p a r e d with other v a l u e s i n table 1. Table 1. Knight shift constants. K x 104 K\ x i o 4 Ref.1) Ref.2) Ref. 3) Present experiment 70.9 + 0.7 75.7 71.3 + 0.2 2.3 6.6 ± 0.6 5.4 + 0.2 60-\ SO _^4 0 ^3.0 % I J T h e a n i s o t r o p y of the l i n e width d i f f e r s i n both f o r m and m a g n i t u d e f r o m that expected f r o m d i p o - l a r b r o a d e n i n g alone. T h e m e a n v a l u e i s i n a g r e e - m e n t with that of K a r i m o v and Shchegolev f o r the s e c o n d m o m e n t (1.2 ( k c / s ) 2 ) a n d if the a d d i t i o n a l b r o a d e n i n g i s a t t r i b u t e d to the exchange i n t e r a c t i o n , t h i s i n t e r a c t i o n h a s a l a r g e a n i s o t r o p i c component. A d e t a i l e d e x p r e s s i o n f o r t h i s i s a v a i l a b l e 5) a n d c a l c u l a t i o n s on the i m p l i c a t i o n s of the o b s e r v e d a n i s o t r o p y a r e i n p r o g r e s s . In these e x p e r i m e n t s the s a m p l e s i z e i s m u c h g r e a t e r than the e l e c t r o m a g n e t i c s k i n - d e p t h . T h e o - r y 7> i n d i c a t e s that the l i n e - s h a p e s h o u l d then be an e q u a l m i x t u r e of a b s o r p t i o n and d i s p e r s i o n m o d e s . N o o b v i o u s c o n t r i b u t i o n f r o m the d i s p e r s i o n m o d e w a s o b s e r v e d , i n a g r e e m e n t with p r e v i o u s o b s e r - v a t i o n s 3 >8) that t h e r e s e e m s to be an effective s k i n - d e p t h f o r n u c l e a r m a g n e t i c r e s o n a n c e w h i ch i s g r e a t e r than the e l e c t r o m a g n e t i c s k i n - d e p t h . In o (l Io) plane * plane perpendicular to (llo) 'I [III] [00\] 0° 45' 90' 135' 180° Fig. 3. The Snll9 line width as a function of crystal orientation in the magnetic field. any c a s e , s u c h a m i x i n g w o u l d i n c r e a s e o u r v a l u e of A" b y o n l y one p e r c e n t and would have a n e g l i g i - ble effect on J£j[. One c a n r e a d i l y u n d e r s t a n d e r r o r s i n an a n a l y s i s of the powder l i n e shape s i n c e a weighting f a c t o r c o r r e s p o n d i n g to the a n i s o t r o p y of the l i n e width s h o u l d be u s e d . T h i s f a c t o r p r o b a b l y e x p l a i n s m o s t of the d i s c r e p a n c y between the r e s u l t s quoted in table 1. T h e o b s e r v e d l i n e - s h a p e showed s o m e a s y m m e - t r y w h i c h a p p e a r e d to be a function of o r i e n t a t i o n . W o r k i s continuing on a d e t a i l e d i n v e s t i g a t i o n of t h e s e p o i n t s together with a study of the S n 1 1 7 r e s - onance. W e would l i k e to thank D r . E . T e g h t s o o n i a n and M r . A . L . C a u s e y f o r p r o v i d i n g the t i n s i n g l e c r y s - t a l and D r . M y e r B l o o m f o r i l l u m i n a t i n g d i s c u s - s i o n s . References 1) B.R.McGarvey and H.S.Gutowsky, J. Chem. Phys. 21 (1953) 2114. 2) N. Bloembergen and T.J.Rowland, Acta Met. 1 (1953) 731. 3) Yu.S.Karimov and I. F.Shchegolev, J. Exptl. Theoret. Phys. (USSR) 40 (1961) 1289; translation: Soviet Phys. JETP 13 (1961) 899. 4) M.A.Ruderman and C.Kittel, Phys. Rev. 96 (1954) 99. 5) N.Bloembergen and T.J.Rowland, Phys. Rev. 97 (1955) 1679. 6) A.Abragam, The principles of nuclear magnetism (Oxford University Press, 1961), p. 205. 7) A.C.Chapman, P.Rhodes and E. F.W.Seymour, Proc. Phys. Soc. (London) B 70 (1957) 345. 8) A.G.Redfield, Phys. Rev. 101 (1956) 67. 110 -57- APPENDIX C T h i s appendix i n c l u d e s c i r c u i t diagrams o f t h e non- c o m m e r i c a l i t e m s o f t h e s p e c t r o m e t e r . One o r two comments on t h e a p p a r a t u s a r e a p p r o p r i a t e . The h e a t e r s o f t h e P ound-Knight-Watkins o s c i l l a t o r s and of t h e White a m p l i f i e r s a r e s u p p l i e d by a 6 v o l t s t o r a g e b a t t e r y i n p a r a l l e l w i t h a H e a t h k i t B a t t e r y E l i m i n a t o r . A dc s o u r c e f o r t h e h e a t e r s was f o u n d n e c e s s a r y e s p e c i a l l y f o r t h e White a m p l i f i e r s t o e l i m i n a t e 60 c/s i n t e r f e r e n c e . The Pound-Knight-Watkins o s c i l l a t o r used t o measure t h e d e u t e r o n r e s o n a n c e employed a V a r i c a p i n t h e same way as shown i n F i g u r e C l , e x c e p t t h a t a 90 v o l t b a t t e r y and a 100 K H e l i p o t p r o v i d e d t h e v o l t a g e sweep i n s t e a d o f t h e m o d i f i e d T e k t r o n i x 162 waveform g e n e r a t o r . The i n i t i a l v o l t a g e o f t h e sweep o f t h e m o d i f i e d T e k t r o n i x 162 waveform g e n e r a t o r i s s e t w i t h t h e c o n t r o l marked V e r n i e r , and the s a w t o o t h run-down i s begun or t e r m i n a t e d w i t h t h e s w i t c h marked Gate Out or P u l s e Out. F i g u r e C2 shows t h e m o d i f i c a t i o n s o f t h e waveform g e n e r a t o r i n d o t t e d s q u a r e s . The r e s t o f t h e c i r c u i t can be f o u n d i n t h e T e k t r o n i x 162 waveform g e n e r a t o r manual.  6 A6 2 Z.4 3.2 4- 5 6.5 8 10 /4 x I0&n Figure C2. Modified Tektronix 162 Waveform Generator .OO/ +/oo to Oy S<xv*toofA OuTpuT /O 225 V Phase S h i f t e r 2 5 0 v H o r i z o n t a l A m p l i f i e r f o r t h e T e k t r o n i x 360 O s c i l l o s c o p e F i g u r e C3. 2 2 5 V Figure C4. Phase Sensitive Detector - 6 2 - REFERENCES 1 C . H . Townes , C . H e r r i n g , W. D. K n i g h t : P h y s . R e v . 7 7 8 5 2 ( 1 9 5 0 ) 2 W. D. K n i g h t : S o l i d S t a t e P h y s i c s 2 9 3 ( 1 9 5 6 ) 3 A . Abragam: The P r i n c i p l e s o f N u c l e a r M a g n e t i s m , C h . I V 4 A . Abragam: The P r i n c i p l e s o f N u c l e a r M a g n e t i s m , p . 172 5 C. P . S l i c h t e r : N u c l e a r M a g n e t i c R e s o n a n c e , t o be p u b l i s h e d L . P a u l i n g , E . B . W i l s o n : I n t r o d u c t i o n t o Quantum M e c h a n i c s , p 2 3 2 7 N . B l o e m b e r g e n , T . J . R o w l a n d : A c t a M e t . 1. 7 3 1 ( 1 9 5 3 ) 8 A . Abragam: The P r i n c i p l e s o f N u c l e a r M a g n e t i s m , p 205 9 D. G . W a t k i n s : P h . D . T h e s i s , H a r v a r d U n i v e r s i t y ( 1 9 5 2 ) 1 0 R. B l u m e : R S I 32 743 ( 1 9 6 1 ) 1 1 E . S a w a t z k y : P h . D . T h e s i s , U n i v e r s i t y o f B . C . ( 1 9 6 1 ) 1 2 N . A . S h u s t e r : R S I 22 2 5 4 ( 1 9 5 1 ) 1 3 E . P . J o n e s , D. L I . W i l l i a m s : P h y s i c s L e t t e r s JL 109 ( 1 9 6 2 ) 1 4 A . C . Chapman, P . R h o d e s , E . F . W. Seymour: P r o c . P h y s . S o c . B L X X 345 ( 1 9 5 * ) 1 5 Y u . S . K a r i m o v , I . F . S h c h e g o l e v : J E T P (USSR) 4 0 1 2 8 9 ( 1 9 6 1 ) , t r a n s l : J E T P 13 8 9 9 ( 1 9 6 1 ) 1 6 J . H . Van V l e c k : P h y s . R e v . 74 1168 ( 1 9 4 8 ) 1 7 M . A . Ruderman, C . K i t t e l : P h y s . R e v . 9 6 99 ( 1 9 5 4 ) 1 8 N . B l o e m b e r g e n , T . J . R o w l a n d : P h y s . R e v . 9 7 1 6 7 9 ( 1 9 5 5 ) -63- Abragain: The Principles of Nuclear Magnetism, p 207 V, Gold, M. J, Priestly: Phil. Mag. 5 1089 (.1960) Tolansky, G. 0. Forester: Phil. Mag, 32 315 (1941) Shoenberg: Progress in Low Temperature Physics II, p 226 G. Chambers: Can. J. Phys. 34 1395 (1956) B. Pippard: Reports on Progress in Physics 22 176 (1960) P. Das, E. H. Sondheimer: Phil. Mag, 5 529 (1960) Yafet: J. Phys. Chem. Solids 21 99 (1961) J. Stephen: Phys. Rev, 123 126 (1961) I. Kaplan: J. Phys. Chem, Solids 23 826 (1962) J. Stephen: Proc. Phys. Soc. 79 987 (1962) E. Hebborn, M. J. Stephen: Proc, Phys. Soc. 80 991 (1962) L, Sagalyn, J. A. Hofmanni Bull. APS, Ser, II 7 226 (1962) Phys. Rev, 127 68 (1962)

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