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UBC Theses and Dissertations

N.M.R. study on single crystals of lead and thallium Schratter, Jacob Jack 1973

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AN N.M.R. STUDY ON SINGLE CRYSTALS OF LEAD AND THALLIUM by JACK SCHRATTER Diplomat U n i v e r s i t a r , U n i v e r s i t y of Bucharest, 1957 M.Sc,, U n i v e r s i t y of B r i t i s h Columbia, 1968 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department o f P h y s i c s We accept t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF J u l y , BRITISH COLUMBIA 1973 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n 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 e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d 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 b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y 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 b e 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 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 ABSTRACT A study o f the Knight s h i f t and i n d i r e c t c o u p l i n g c o n s t a n t s has been c a r r i e d out on s i n g l e c r y s t a l s of l e a d and t h a l l i u m . In l e a d , the exchange-narrowing model of Anderson and Weiss has been shown t o be i n a p p l i c a b l e . The angular v a r i a t i o n o f the second moment has e s t a b l i s h e d t h a t the c o n t r i b u t i o n from the second-nearest neighbours i s ve r y s m a l l , and v a l u e s are o b t a i n e d f o r the pseudoexchange and p s e u d o d i p o l a r c o e f f i c i e n t s f o r the f i r s t - n e a r e s t neighbours, which are i n good agreement w i t h the t h e o r e t i c a l v a l u e s o f T t e r l i k k i s , Mahanti, and Das. A s m a l l a n i s o t r o p y o f the Knight s h i f t which has the ex-pected a n g u l a r dependence has been observed f o r the f i r s t time i n a c u b i c m e t a l , In t h a l l i u m , the i s o t r o p i c Knight s h i f t decreases w i t h i n -c r e a s i n g temperature, showing a n o n l i n e a r v a r i a t i o n which i s most pronounced between 50° and 150°K. T h i s behaviour i s c o r r e l a t e d w i t h the anomalous p r e s s u r e dependence of the superconducting t r a n s i t i o n temperature a t low p r e s s u r e s . From the superconducting d a t a on pure T l and Tl-Hg a l l o y s we e x t r a c t the volume dependence of the e l e c t r o n i c d e n s i t y of s t a t e s , and use i t to d e r i v e the temperature dependence of the Knight s h i f t . The r e s u l t s agree w e l l w i t h the observed d a t a . We have measured the angular dependence of the l i n e w i d t h i n t h a l l i u m and r e l a t e d i t t o the second moment to e s t i m a t e the p s e u d o d i p o l a r and pseudoexchange c o n t r i b u t i o n s f o r the f i r s t two s h e l l s o f neighbours. These v a l u e s are ve r y d i f f e r e n t , d e s p i t e the f a c t t h a t both s h e l l s have n e a r l y the same r a d i u s , and t h i s I l l e s t a b l i s h e s e x p e r i m e n t a l l y f o r the f i r s t time an o r i e n t a t i o n dependence o f the i n d i r e c t c o u p l i n g c o n s t a n t s . IV TABLE OF CONTENTS PAGE CHAPTER I: INTRODUCTION 1 CHAPTER I I : THEORETICAL BACKGROUND 2.1 The I s o t r o p i c Knight S h i f t 5 2.1.1 F e r m i - c o n t a c t - K n i g h t - s h i f t 5 2.1.2 Other c o n t r i b u t i o n s t o the i s o t r o p i c K n i g h t s h i f t 9 2.2 The A n i s o t r o p i c Knight S h i f t 12 2.2.1 The d i p o l a r c o n t r i b u t i o n 12 2.2.2 The c o n t a c t c o n t r i b u t i o n 16 2.2.3 Other c o n t r i b u t i o n s 18 2.3 I n d i r e c t N u c l e a r I n t e r a c t i o n s 18 2.4 The Lineshape 26 CHAPTER I I I : THE EQUIPMENT 28 CHAPTER IV: EXPERIMENTAL CONSIDERATIONS 32 CHAPTER V: LINESHAPE ANALYSIS 39 CHAPTER VI : EXCHANGE INTERACTIONS IN LEAD 6.1 I n t r o d u c t i o n 45 6.2 Ex p e r i m e n t a l D e t a i l s and R e s u l t s 46 6.3 D i s c u s s i o n 52 6. 3a C r i t i c a l a n a l y s i s o f o t h e r work 61 6.4 - C o n c l u s i o n 64 CHAPTER V I I : KNIGHT SHIFT ANISOTROPY IN SINGLE--CRYSTAL-CUBIC-LEAD 7.1 I n t r o d u c t i o n 65 7.2 Exp e r i m e n t a l D e t a i l s and R e s u l t s 66 V 7.2.1 H i g h - f i e l d i n v e s t i g a t i o n 67 7.3 D i s c u s s i o n 70 7.3a C r i t i c a l a n a l y s i s of o t h e r work 75 CHAPTER V I I I : CORRELATION BETWEEN THE ANOMALOUS PRESSURE DEPENDENCE OF THE SUPER-CONDUCTING TRANSITION TEMPERATURE IN THALLIUM AND ITS TEMPERATURE DEPENDENCE OF THE KNIGHT SHIFT 8.1 I n t r o d u c t i o n - 81 8.2 Ex p e r i m e n t a l D e t a i l and R e s u l t s 84 8.3 D i s c u s s i o n 88 8.3.1 The i n t r i n s i c temperature dependence of K 94 8.3.2 The volume dependence of the Knight s h i f t 95 8.3.3 The t o t a l T-dependence of K 108 8.4 C o n c l u s i o n s 114 CHAPTER IX: INDIRECT COUPLING CONSTANTS IN A THALLIUM SINGLE CRYSTAL 9.1 I n t r o d u c t i o n 115 9.2 Experimental R e s u l t s 117 9.3 D i s c u s s i o n 120 9.4 C o n c l u s i o n 133 BIBLIOGRAPHY 134 APPENDIX I ' 139 APPENDIX I I 14 2 APPENDIX I I I CARROUSEL . 145 VI LIST OF ILLUSTRATIONS FIGURE TITLE PAGE 3.1 C i r c u i t diagram of the P.K.W. spectrometer 29 3.2 Block diagram of the spectrometer • 30 5.1 F i t t e d e x p e r i m e n t a l l i n e ( P b 2 0 7 ) 44 6.1 Angular dependence of the l i n e w i d t h W 49 6.2 An g u l a r dependence of the second moment 53 6.3 T h e o r e t i c a l angular dependence of S1 and S 2 as d e f i n e d i n the t e x t 59 6.4 Repr o d u c t i o n from Luders and H e c h t f i s c h e r 63 6.5 Repr o d u c t i o n from Luders and H e c h t f i s c h e r 63 7.1 Angular dependence of the resonance frequency 68 8.1 The temperature-dependence of the i s o t r o p i c K n i g h t s h i f t i n T l 2 0 5 86 8.2 Angular dependence of the Knight s h i f t 87 8.3 Repr o d u c t i o n from Makarov and Baryakhtar (1965) 103 8.4 The temperature-dependence of dlnN/dlnV, as e x t r a c t e d from the s u p e r c o d u c t i n g data 105 8.5 D e r i v e d temperature-dependence of (dlnK/dT) xlO 1* 109 P 8.6 D e r i v e d temperature-dependence of the Knight s h i f t 110 8.7 The f u n c t i o n g ( x ) , a s ' d e f i n e d i n the t e x t 112 9.1 S e p a r a t i o n of modes f o r a t y p i c a l t h a l l i u m l i n e 118 VII 9.2 The o r i e n t a t i o n - d e p e n d e n c e of the l i n e w i d t h i n T l 2 0 5 119 9.3 C o n t r i b u t i o n s to the second moment 130 9.4 The o r i e n t a t i o n - d e p e n d e n c e o f the second moment 132 ACKNOWLEDGEMENT I wish to express my s i n c e r e thanks to Dr. D. L l . W i l l i a m s f o r h i s guidance throughout t h i s work. H i s phy-s i c a l i n t u i t i o n has helped me a v o i d b l i n d a l l e y s , w h i le a t the same time s p u r r i n g on the work along many a d i f -f i c u l t path t h a t proved u s e f u l i n the end. I wish to thank Dr. M. Bloom, Dr. A.V. Gold, and Dr. B.G. T u r r e l l f o r t h e i r work while s e r v i n g on my com-m i t t e e . I am t h a n k f u l to Dr. M. Bloom f o r the c o n s t a n t en-couragement and a m i c a b i l i t y which I have had the p r i v i l e g e t o enjoy d u r i n g my s t a y a t U.B.C. I am indebted to Dr. S.N. Sharma f o r h i s guidance and h e l p i n b u i l d i n g the spectrometer. My w i f e M a r g i t has shown courage and a b i l i t y i n t y p i n g t h i s t h e s i s , and I wish to thank her f o r her d e v o t i o n . The f i n a n c i a l a s s i s t a n c e of the U n i v e r s i t y of B r i t i s h Columbia and the Mc. M i l l a n S c h o l a r s h i p Fund i s g r a t e f u l l y acknowledged. CHAPTER I INTRODUCTION N u c l e a r magnetic resonance s t u d i e s i n metals, i n p a r t i c u l a r i n v e s t i g a t i o n s o f the Knight s h i f t , the i n d i r e c t c o u p l i n g c o n s t a n t s , and the r e l a x a t i o n mechanisms, y i e l d i n f o r m a t i o n about the e l e c t r o n i c w a v e f u n c t i o n s , which o f t e n cannot be o b t a i n e d from ot h e r s o u r c e s . T h i s i n f o r m a t i o n complements the b a n d - s t r u c t u r e data o b t a i n e d from e.g. de Haas van Alphen e f f e c t , o r c y c l o t r o n resonance s t u d i e s , thus p r o v i d i n g a more complete p i c t u r e o f the e l e c t r o n i c s t r u c t u r e i n the metal. Most o f the NMR-work i n metals was performed on p o l y c r y s t a l -l i n e powder specimens. In r e c e n t years s i n g l e - c r y s t a l s t u d i e s were made on a few m e t a l l i c systems (e.g. Jones and W i l l i a m s (1962), Sharma e t . a l . (1969) on white t i n ; Sagalyn and Hoffman (1962) on aluminum; Sharma and W i l l i a m s (1967) on cadmium; S c h r a t t e r and W i l l i a m s (1967) on t h a l l i u m ; Dougan e t . a l . (1969) on magnesium; V a l i c and W i l l i a m s (1969) on g a l l i u m ; Schone (1969) on niobium; Luders and H e c h t f i s c h e r (1972), S c h r a t t e r and W i l l i a m s (1972) ( c h a p t e r 6 and 7) , de C a s t r o and Schumacher (1973) on lead). T h i s t h e s i s r e p o r t s an i n v e s t i g a t i o n o f Knight s h i f t s and i n d i r e c t c o u p l i n g c o n s t a n t s i n t h a l l i u m and l e a d s i n g l e - c r y s t a l s . These two heavy metals occupy a d j o i n i n g s i t e s i n the p e r i o d i c t a b l e , a r e both o f s p i n H, and r e l a t i v i s t i c e f f e c t s are expected t o p l a y an important r o l e i n t h e i r e l e c t r o n i c wavefunctions and b a n d - s t r u c t u r e . Lead has an e x t r a p-valence e l e c t r o n w i t h r e s p e c t t o t h a l l i u m , but both have a p p r e c i a b l e p ^ c h a r a c t e r a t the Fermi 1 s u r f a c e , and can be c o n s i d e r e d f r e e - e l e c t r o n - l i k e metals t o a f i r s t a pproximation. On the oth e r hand, t h a l l i u m has a hexagonal c l o s e d packed s t r u c t u r e , w h ile l e a d i s face c e n t e r e d c u b i c . The former has two magnetic i s o t o p e s (70% T l 2 0 5 , 30% T l 2 0 3 ) , whereas the l a t t e r i s m a g n e t i c a l l y "pure" (21% P b 2 0 7 ) From the N.M.R-point of view, l e a d , being c u b i c , i s not ex-pected to e x h i b i t an a p p r e c i a b l e a n i s o t r o p i c Knight s h i f t , and s i n c e i t c o n t a i n s o n l y one magnetic i s o t o p e , t h e r e i s no exchange broadening, and i t s l i n e w i d t h i s narrow. T h a l l i u m , on the o t h e r hand, e x h i b i t s both a wide l i n e as w e l l as a l a r g e a n i s o t r o p i c Knight s h i f t . S i n c e both metals are of s p i n % th e r e are no d i f f i -c u l t i e s a r i s i n g from quadrupolar i n t e r a c t i o n s , As can be seen, the two metals show enough s i m i l a r i t i e s and d i f f e r e n c e s to make a comparative study worthwhile. They a l s o show s u f f i c i e n t l y p r o -nounced a n i s o t r o p i c e f f e c t s i n t h e i r NMR p r o p e r t i e s t o warrant a s i n g l e - c r y s t a l i n v e s t i g a t i o n . Two important t o o l s were developed f o r t h i s p r o j e c t : a s e n s i t i v e spectrometer, d e s c r i b e d i n chapter 3, and an a c c u r a t e l i n e s h a p e a n a l y s i s f o r bulk m e t a l l i c specimens, d e s c r i b e d i n v chapter 5. As a r e s u l t we o b t a i n e d r e l i a b l e data on the l i n e s h a p e and resonance f r e q u e n c i e s which i n t u r n l e a d t o f o u r developments (1) The angular dependence of the l i n e w i d t h i n l e a d was not the same as t h a t o f the second moment. To o b t a i n c o u p l i n g c o n s t a n t s , second moments had to be measured. T h i s i n v e s t i g a t i o n i s r e p o r t e d i n chapter-- 6 (2) Our l i n e s h a p e a n a l y s i s f u r n i s h e d an unexpected angular dependence of the resonance frequency i n l e a d , i n d i c a t i n g an an-i s o t r o p i c Knight s h i f t . Such an e f f e c t can come about through 3 , s p i n - o r b i t c o u p l i n g which c o u l d be l a r g e enough i n t h i s heavy metal. We t h e r e f o r e c a l c u l a t e d the expected angular dependence of t h i s c o n t r i b u t i o n and obtained f o r the f i r s t time an a n i s o -t r o p i c K n i g h t s h i f t i n a c u b i c metal. T h i s work i s r e p o r t e d i n chapter 7. (3) The temperature dependence of the i s o t r o p i c Knight s h i f t (K ,: ) i n t h a l l i u m showed an unusual behaviour. K. de-i s o i s o c r e a s e d w i t h i n c r e a s i n g temperature showing a s i g n i f i c a n t l y non-l i n e a r behaviour which i s most pronounced a t l i q u i d n i t r o g e n temperatures. T h i s behaviour turned out to c o r r e l a t e w i t h an ab-normal p r e s s u r e dependence of the superconducting t r a n s i t i o n temperature , which had been e x p l a i n e d as due to the f o r m a t i o n of s m a l l pockets i n the Fermi s u r f a c e of t h a l l i u m . T h i s i n v e s t i g a t i o n i s d e s c r i b e d i n chapter 8. (4) The l i n e w i d t h i n t h a l l i u m showed the same angular de-pendence as t h a t expected f o r the second moment. However, the s i z e of the a n i s o t r o p y i n d i c a t e s t h a t the f i r s t two s h e l l s , even though of equal r a d i u s , have d i f f e r e n t c o u p l i n g c o n s t a n t s as-s o c i a t e d w i t h them. T h i s i s a r e s u l t of t h e i r d i f f e r e n t symmetry p r o p e r t i e s i n the l a t t i c e . I t i s f o r the f i r s t time t h a t such a d i s t i n c t i o n has been i n v e s t i g a t e d . The r e s u l t s of t h i s study are r e p o r t e d i n chapter 9. ^ S i n c e the aforementioned f o u r c h a p t e r s e s s e n t i a l l y d e s c r i b e d i f f e r e n t i n v e s t i g a t i o n s they have been t r e a t e d i n dependently i n t h i s t h e s i s . Each, of them i s p r e s e n t e d i n a p u b l i c a t i o n - l i k e manner, c o n t a i n i n g f o u r s e c t i o n s : (1) I n t r o d u c t i o n ; ( 2 ) E x p e r i -mental d e t a i l s and r e s u l t s ; (3) D i s c u s s i o n ; (4) C o n c l u s i o n . Only i n s e c t i o n (2) was t h e r e danger of unnecessary r e p e t i t i o n o f some o f the e x p e r i m e n t a l c o n s i d e r a t i o n s . T h i s problem was s o l v e d by-w r i t i n g a s e p a r a t e chapter (4) on experimental d e t a i l s p e r t a i n i n g t o the whole work, and r e p o r t i n g i n s e c t i o n s (2) o n l y those a s p e c t s which were c l o s e l y r e l e v a n t to the r e s p e c t i v e c h a p t e r s . In t h i s way the o v e r l a p of i n f o r m a t i o n was reduced to a minimum. F i n a l l y , i n c h a p t e r 2, we d e s c r i b e the r e l e v a n t t h e o r e t i c a l background r e l a t e d t o t h i s work , The emphasis i s p l a c e d here on a p h y s i c a l u n d e r s t a n d i n g o f the l e s s f a m i l i a r i n t e r a c t i o n s , and textbook d e r i v a t i o n s have been avoided (with the e x c e p t i o n of an o u t l i n e on the i s o t r o p i c Knight s h i f t and the Rudermann-Kittel i n t e r a c t i o n ) . T h i s c h a p t e r a l s o p l a y s the r o l e of s e p a r a t i n g the t h e o r e t i c a l background i n f o r m a t i o n from our own i n t e r p r e t a t i o n s . CHAPTER I I THEORETICAL BACKGROUND. 2.1 The I s o t r o p i c Knight S h i f t The K n i g h t s h i f t i s due to the magnetic f i e l d produced •. mainly by the c o n d u c t i o n e l e c t r o n s a t the s i t e o f the nucleus through t h e h y p e r f i n e i n t e r a c t i o n . The i s o t r o p i c s h i f t can be w r i t t e n as the sum o f f o u r c o n t r i b u t i o n s : K. = K(Pauli)+K(c.p.)+K(orb)+K(dia) K ( P a u l i ) i s due t o the Fermi c o n t a c t term and i s p r o p o r t i o n a l to the s - c h a r a c t e r o f the co n d u c t i o n e l e c t r o n wavefunctions a t the Fermi s u r f a c e . K(c.p.) stands f o r the' c o r e - p o l a r i z a t i o n - K n i g h t s h i f t and i s caused by the p o l a r i z a t i o n of core s - s t a t e s by con-d u c t i o n e l e c t r o n s w i t h u n p a i r e d s p i n , v i a exchange i n t e r a c t i o n s , K(orb) i s the o r b i t a l paramagnetic c o n t r i b u t i o n , and K(dia) i s the d i a m a g n e t i c s h i e l d i n g of n u c l e i v i a the Landau diamagnetic s u s c e p t i b i l i t y . 2.1.1 F e r m i - c o n t a c t - K n i g h t - s h i f t I n most metals t h i s i s the major c o n t r i b u t i o n to the i s o -t r o p i c K n i g h t s h i f t . The e x c e p t i o n s are t r a n s i t i o n metals where co r e p o l a r i z a t i o n and o r b i t a l e f f e c t s are of a p p r e c i a b l e im-p o r t a n c e . The o r i g i n a l c a l c u l a t i o n o f t h i s term was made by Townes and K n i g h t (1950). S i n c e t h i s i s the term used i n chapter 8 t o determine the temperature dependence of the Knight s h i f t i n t h a l l i u m , we g i v e a s u c c i n t d e r i v a t i o n o f i t , p o i n t i n g out the 5 6 e s s e n t i a l approximations. The s-state hyperfine contact coupling i s given by the well known expression 8 u . — (fr e.fr n)6(r"-R) (2.1) where u e i s the magnetic moment associated with the electron spin, y n i s the moment associated with the nuclear spin, r i s the po-s i t i o n vector of the electron, and that of the nucleus. Since ue=-Ye#S and u n=y n#I, the coupling over a l l n u c l e i and electrons i s of the form *en= ^ Wn^W'V-V (2'2) 3 3 SL In the presence of an external magnetic f i e l d H the ele c -trons are p a r t l y polarized, giving r i s e to an average nonvanishing coupling at the s i t e of each nucleus. Because of the r e l a t i v e weakness of the hyperfine coupling the problem can be treated by perturbation theory. In the unperturbed system one neglects i n t e r -actions between electrons and n u c l e i . This implies that the t o t a l unperturbed Hamiltonian can be written as a sum, H e + H n , and the unperturbed wavef unctions can be expressed as products V ^ e ^ n * F i r s t order perturbation theory y i e l d s H'en = ^ e H e n ^ e ' <2.3) where d x ^ i n d i c a t e s i n t e g r a t i o n over s p a t i a l and s p i n e l e c t r o n c o o r d i n a t e s . I f the e l e c t r o n s are t r e a t e d as weakly i n t e r a c t i n g t h e i r w a v e f u n c t i o n can be taken as a product o f s i n g l e - e l e c t r o n B l o c h and s p i n f u n c t i o n s . F u r t h e r one r e p l a c e s s t a t e o c c u p a t i o n numbers by the Fermi f u n c t i o n and assumes the e l e c t r o n s q u a n t i z e d by the magnetic f i e l d H along the o z - d i r e c t i o n . T h i s y i e l d s f o r the j ^ t h n u c l e u s i n t e r a c t i n g w i t h an e l e c t r o n i n s t a t e k (2.4) , t 8 TT + ^ { H e n } j = " J " Y n # I j z U l / 2 ) t f Y e f (k,l/2) - (1/2) J*y ef (k ,-1/2) } . | ^  (0) | 2 # where f ( k , s ) i s the Fermi f u n c t i o n f o r an e l e c t r o n i n B l o c h s t a t e ic, and s p i n s, and i|/^(0) i s i t s Bloch f u n c t i o n a t the s i t e o f the nucleus j , (Rj=0). S i n c e the quantity- i n b r a c k e t s i s minus the average c o n t r i b u t i o n of s t a t e R" to the z-component of the e l e c t r o n m a g n e t i z a t i o n u"2£, and the l a t t e r i s r e l a t e d t o the s p i n s u s c e p t i -b i l i t y x s by y z £ = X s H f t n e e f f e c t i v e e l e c t r o n i n t e r a c t i o n w i t h the j - t h n u c l e a r s p i n becomes { H ' e n } j = ~ Y „ K l j z H Z j * £ ( 0 ) | 2 X 3 < 2- 5) k I f i t i s assumed t h a t the f u n c t i o n i n the sum v a r i e s s l o w l y enough i n Ic-space, one can use the e l e c t r o n d e n s i t y f u n c t i o n and r e p l a c e the summation by an i n t e g r a l . To separate the wavefunction term from the s u s c e p t i b i l i t y one must make the f u r t h e r assumption t h a t the d i f f e r e n c e i n energy between s p i n up and s p i n down i s the same f o r two s t a t e s k and k' i f t h e i r t r a n s l a t i o n a l energy ( k i n e t i c t p o t e n t i a l ) i s the same (E£ = E+ (), In f a c t t h i s im-p l i e s t h a t Xg depends on E^ o n l y which, i n t u r n i m p l i e s an i s o -t r o p i c g f a c t o r on the Fermi s u r f a c e . With these approximations Eqn. (2.5) becomes 8TT <K K = yjil. H / x s ( E ^ ) < | ^ ( 0 ) |2> N(Ej)dE£ (2.6) e n J 2 n J k K k £ K K where < | i/>k (0) | 2 >E-»- i s the average value o f the e l e c t r o n d e n s i t y over the s u r f a c e E->=const and N(E^) i s the e l e c t r o n d e n s i t y o f s t a t e s a t E£. S i n c e f o r E^<<E'F (the Fermi l e v e l ) both s p i n s t a t e s are 100% p o p u l a t e d and f o r E£>>E n e i t h e r s p i n s t a t e i s occup i e d , X s (Ej^) i s zero f o r a l l v a l u e s o f E£ which are not near E p . As-suming t h a t < 11|>£ (0) | 2 >E_>. v a r i e s s l o w l y i n t h i s r e g i o n one o b t a i n s rC {H; n >. = - ^ - Y n ^ J 2 H < k s ( 0 ) | 2 > F X s (2.7) where <|tp s(0) | 2 > F i s the s - s t a t e e l e c t r o n d e n s i t y averaged over the Fermi s u r f a c e , and x s i s t n e s p i n s u s c e p t i b i l i t y g i v e n by X s = / x s ( E k ) N ( E i c ) d E k = N ( E p ) / X s ( E £ ) d E £ = 2u 2N(E p) (2.8) where ^ i s the e l e c t r o n magnetic moment, Eqn, (2.7) has the form of an i n t e r a c t i o n o f the j - t h n u c l e a r s p i n w i t h a magnetic f i e l d 8 TT A H = _ < | ^ s ( 0 ) | 2 > p X sH , (2.9) 9 so t h a t the Knight s h i f t i s g i v e n by AH 8 TT K ( P a u l i ) = H - < | ^ ( 0 ) |2 > p X s (2.10) I f exchange i n t e r a c t i o n s between cond u c t i o n e l e c t r o n s are taken i n t o account the e l e c t r o n s p i n s u s c e p t i b i l i t y takes the form where f ( a ) i s an enhancement f a c t o r (see e.g. S i l v e r s t e i n (1963)) 2.1.2 Other c o n t r i b u t i o n s to the i s o t r o p i c Knight s h i f t . S i n c e i n the case of t h a l l i u m and l e a d these c o n t r i b u t i o n s are o f minor importance we w i l l not t r e a t them i n d e t a i l . Heine (1957) has shown t h a t i n the case of t r a n s i t i o n metals t h e r e e x i s t s an important c o n t r i b u t i o n to the h y p e r f i n e s t r u c t u r e coming from core s - e l e c t r o n s due to exchange i n t e r a c t i o n s w i t h p o l a r i z e d d - e l e c t r o n s . C l o g s t o n and J a c c a r i n o (1961) used the same mechanism to e x p l a i n temperature dependent and n e g a t i v e Knight s h i f t s an V - i n t e r m e t a l l i c compounds. Depending on the d e t a i l s of the exchange i n t e r a c t i o n , core s - e l e c t r o n s whose s p i n i s , say, p a r a l - r l e i t o the p o l a r i z e d o u t e r s p i n s , may be e i t h e r p u l l e d away from the n u c l e u s or pushed towards i t . Thus c o r e - p o l a r i z a t i o n may ex-h i b i t n e g a t i v e or p o s i t i v e c o n t r i b u t i o n s to the Knight s h i f t . Because of degenerate, temperature dependent, narrow d-bands, t h i s e f f e c t proved to be important i n most t r a n s i t i o n metals. Very few i n v e s t i g a t i o n s of t h i s c o n t r i b u t i o n f o r n o n - t r a n s i t i o n metals e x i s t i n the l i t e r a t u r e ; and the e f f e c t i s c u s t o m a r i l y n e g l e c t e d X s = 2 u 2 N ( E F ) f (a) (2.11) (a) The " c o r e - p o l a r i z a t i o n " Knight s h i f t 10 f o r these m e t a l s . T t e r l i k k i s e t . a l , ( 1 9 7 0 ) attempt a r e l a t i v i s t i c c a l c u l a t i o n o f the i s o t r o p i c Knight s h i f t i n l e a d . They f i n d good agreement w i t h experiment when i g n o r i n g c o r e - p o l a r i z a t i o n . In our work t h i s e f f e c t c o u l d be of importance o n l y i f the exchange mechanism i t s e l f were temperature dependent i n t h a l l i u m . S ince t h i s i s v e r y u n l i k e l y we w i l l i g n o r e i t i n our a n a l y s i s . (b) O r b i t a l Knight s h i f t Kubo and Obata ( 1 9 5 6) have shown t h a t the Van-Vleck induced o r b i t a l paramagnetism can be an important c o n t r i b u t i o n t o the magnetic s u s c e p t i b i l i t y i n meta l s . In g e n e r a l the o r b i t a l mag-n e t i c moments u Q L are quenched by the c r y s t a l f i e l d ; a s m a l l o r b i t a l c o n t r i b u t i o n to the s u s c e p t i b i l i t y i s due to a second ->--»-o r d e r mechanism i n which the p e r t u r b a t i o n -y f lLH lowers the energy ->-of the " e l e c t r o n s . P h y s i c a l l y , the magnetic f i e l d H p a r t l y un-quenches the a n g u l a r momentum. Si n c e the e f f e c t i s due to second o r d e r p e r t u r b a t i o n , the r e s u l t i s p r o p o r t i o n a l t o ( A E ) - 1 where A E i s the mean s e p a r a t i o n o f the energy l e v e l s connected by the o r b i t a l a n g u l a r momentum. One can see t h a t the e f f e c t w i l l be ag a i n l a r g e s t i n t r a n s i t i o n metals w i t h non-s degenerate bands. For n o n - r t r a n s i t i o n metals the e f f e c t i s s m a l l . Moreover, s i n c e t h i s c o n t r i b u t i o n i s temperature-independent we are w e l l j u s t i f i e d i n n e g l e c t i n g i t i n the temperature dependence of the Knight s h i f t . (c) Diamagnetic Knight s h i f t s Das and Sondheimer ( 1 9 6 0 ) 'made a c a l c u l a t i o n o f the d i a -magnetic c o n t r i b u t i o n o f c o n d u c t i o n e l e c t r o n s to the Knight s h i f t . The o r b i t a l motion of c o n d u c t i o n e l e c t r o n s i n a magnetic f i e l d i s w e l l known and g i v e s r i s e t o the Landau diamagnetic s u s c e p t i b i l i t y 11 X^. For a f r e e e l e c t r o n gas Xd =~ d/3) X s an<3 the r a t i o between the two K n i g h t s h i f t s t u r n s out to be K(dia ) X d = 2 (2.12) K(Pauli) X S < I * S ( ° ) I ^ F V a where V i s the atomic volume. Since f o r a f r e e e l e c t r o n gas a < I (0) I2 > = V - 1 one o b t a i n s -1/3 f o r t h i s r a t i o . In r e a l metals s t a an e s t i m a t e by the same authors y i e l d s K ( d i a ) 1 (m/m e f f) 2 = , (2.13) K ( P a u l i ) 3 <U (0) | 2 > FV t> c* They p o i n t out t h a t even though i n r e a l metals the denominator i s much l a r g e r than u n i t y , one c o u l d o b t a i n a p p r e c i a b l e e f f e c t s i f the e f f e c t i v e mass o f the c o n t r i b u t i n g e l e c t r o n s i s very s m a l l . In the case o f l e a d K ( P a u l i ) ~ l % and x s ~ l ° ~ 6 C<3S u n i t s ( T t e r l i k k i s e t . a l . (1970)). T h i s i m p l i e s < | T|>s (0) | 2 > pV a-10 3. On the o t h e r hand m eff~m (Anderson and Gold (1965)). Thus K ( d i a ) / K ( P a u l i ) ~ 1 0 ~ 3 , so t h a t K(dia) can be s a f e l y n e g l e c t e d . A s i m i l a r e s t i m a t e l e a d s t o the same c o n c l u s i o n r e g a r d i n g t h a l l i u m . We conclude t h a t the P a u l i term i s the main c o n t r i b u t i o n to the i s o t r o p i c K n i g h t s h i f t i n both t h a l l i u m and l e a d . I t s temper-at u r e dependence i s t r e a t e d i n d e t a i l i n chapter 8, and i t w i l l t h e r e f o r e not be c o n s i d e r e d here. 2.2 The A n i s o t r o p i c Knight S h i f t The a n i s o t r o p i c Knight s h i f t K a n i s the component of the t o t a l s h i f t which depends on the o r i e n t a t i o n o f the d.c. magnetic f i e l d H w i t h r e s p e c t t o the c r y s t a l l o g r a p h i c axes. U s u a l l y K a n i s s m a l l e r than K^ s o. For i n s t a n c e , we f i n d i n the case of t h a l l i u m K = ^/K • „. ~-6*10~ 2 , The main term i n the 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 H a m i l t o n i a n r e s p o n s i b l e f o r K a n i s the d i p o l a r i n t e r a c t i o n between e l e c t r o n and n u c l e a r s p i n s (Bloembergen and Rowland (1953)): -> s 3r (s. r) X an (2.14) The n o t a t i o n i s the same as i n s e c t i o n (2.1.1). In s p e c i a l c a s e s , which w i l l be d i s c u s s e d below, one can o b t a i n an a p p r e c i a -b l e c o n t r i b u t i o n t o K__ from the c o n t a c t term (2.10). Since chapter 7 i s concerned w i t h the a n i s o t r o p i c Knight s h i f t i n c u b i c l e a d , and s i n c e a p h y s i c a l u n derstanding o f the same i s hard to come by,, we w i l l t r y to pursue i t here i n a somewhat d e t a i l e d d i s c u s s i o n . 2.2.1 The d i p o l a r c o n t r i b u t i o n . We;first n e g l e c t s p i n - o r b i t c o u p l i n g and c o n s i d e r a. s p h e r i c c a l Fermi s u r f a c e . In t h i s case t h e r e e x i s t s no a p p r e c i a b l e mechanism by which the e l e c t r o n s p i n s can " f e e l " the l a t t i c e s t r u c t u r e . T h i s i m p l i e s t h a t t h e i r g - f a c t o r i s i s o t r o p i c , i . e . i t does not depend on the d i r e c t i o n o f the magnetic f i e l d H w i t h r e s p e c t t o the l a t t i c e . I f we c o n s i d e r the s p i n s p o l a r i z e d along H, then the i n t e r a c t i o n (2.14) i s of the form 13 H . d i p a n = Cr ~3 ( 3 c o s 2 a - l ) (2.15) where C i s independent o f s p a c i a l c o o r d i n a t e s , r i s the p o s i t i o n v e c t o r o f the e l e c t r o n w i t h r e s p e c t t o the nucleus and a i s the -»- -y angle made by r w i t h H. To o b t a i n the a n i s o t r o p i c Knight s h i f t t h i s i n t e r a c t i o n must be averaged over a l l e l e c t r o n s as i n the case o f t h e i s o t r o p i c s h i f t . S ince the average v a l u e over a l l space of ( 3 c o s 2 a - l ) i s zero one can see immediately t h a t under the assumptions made here, s - e l e c t r o n s do not c o n t r i b u t e to t h i s term. The c o n t r i b u t i o n from n o n - s - e l e c t r o n s w i l l i n g e n e r a l de-pend on t h e i r s p a c i a l d i s t r i b u t i o n i n the l a t t i c e , and thus on the d i r e c t i o n o f the magnetic f i e l d w i t h r e s p e c t t o the l a t t e r . For i n s t a n c e , l e t us c o n s i d e r a c r y s t a l w i t h a x i a l symmetry and p-e l e c t r o n s a t the Fermi s u r f a c e . Assume, e.g., p„^0 and p =p,r-0. When the f i e l d i s alo n g oz then a~0, 180° and 3 c o s 2 a - l ~ 2 . On the ot h e r hand, when the f i e l d i s i n the x-y plane, a~ T r/2 and 3 c o s 2 a - l ~ - l , In a c u b i c l a t t i c e , however, p x=p v=p z, and one ob-t a i n s a z e r o c o n t r i b u t i o n . To o b t a i n a fo r m a l e x p r e s s i o n f o r K one proceeds i n a s i m i l a r f a s h i o n as i n s e c t i o n (2,1,1). Again, o n l y e l e c t r o n s a t the Fermi s u r f a c e c o n t r i b u t e , and one o b t a i n s a f i e l d c o n t r i b u t i o n ->-a t the n u c l e u s due t o e l e c t r o n k g i v e n by The i n t e g r a l averaged over the Fermi s u r f a c e i s the analogue o f (2.16) <|^^(0)| 2> i n the i s o t r o p i c P a u l i t e r m . F o r a s p h e r i c a l Fermi K F su r f a c e i t i s meaningful t o d e f i n e an average wavefunction 4* such t h a t TpZi>„ r e p r e s e n t s the average conduction e l e c t r o n d e n s i t y a t the Fermi s u r f a c e . Then K 1 P = lyZfihWJ f i>* ( 3 c o s 2 a - l ) r ~ 3 i p dx (2.17) an e * p F where a i s now the angle made by the i n t e g r a t i n g v a r i a b l e r wit h H. Using the a d d i t i o n theorem f o r s p h e r i c a l harmonics, one changes from c o o r d i n a t e s c o n n e c t i n g the magnetic f i e l d w i t h e l e c t r o n po-s i t i o n v e c t o r s t o 2 separate s e t s of c o o r d i n a t e s : (1) C o o r d i n a t e s d e s c r i b i n g the e l e c t r o n p o s i t i o n v e c t o r i n the l a t t i c e (2) C o o r d i n a t e s d e s c r i b i n g the o r i e n t a t i o n o f the magnetic f i e l d i n the l a t t i c e . T h i s y i e l d s , f o r i n s t a n c e , i n the case o f a x i a l symmetry and p-wavefunctions (Bloembergen and Rowland (1953)) K d i p = 2u 2N(E ) q ( 3 c o s 2 6 - l ) (2.18) cin * where 0 i s the angle made by H w i t h the o z - a x i s o f the c r y s t a l and q = / i|>* ( 3 c o s 2 0 - l ) r ~ 3 ^ dx • (2.19) F p where © i s now an i n t e g r a t i n g v a r i a b l e , namely the p o l a r angle made by r w i t h the o z - a x i s . 15 Boon (1964) has made a g r o u p - t h e o r e t i c a l c a l c u l a t i o n o f K Q n c o n s i d e r i n g o n l y the d i p o l a r i n t e r a c t i o n . When he n e g l e c t s s p i n -o r b i t c o u p l i n g he o b t a i n s f o r c u b i c c r y s t a l s = 0 , i n d e -pendent o f the type o f e l e c t r o n wavefunctions. He a l s o f i n d s g e n e r a l e x p r e s s i o n s f o r the a n i s o t r o p i c s h i f t i n the case of the o t h e r 6 c r y s t a l systems. In the case of a hexagonal l a t t i c e K an i s o f the form K = A(3cos 20-1) ( 2 . 2 0 ) an where 6 i s again the angle made by S wit h the p r i n c i p a l a x i s . L e t us now s w i t c h on the s p i n - o r b i t c o u p l i n g . The e l e c t r o n s p i n s can now " f e e l " the l a t t i c e through t h e i r i n t e r a c t i o n w i t h the o r b i t a l a n g u l a r momenta. Due to t h i s i n t e r a c t i o n the g - f a c t o r becomes a n i s o t r o p i c , i . e . depending on the d i r e c t i o n o f H w i t h r e s p e c t t o the l a t t i c e . One f i n d s (Boon (1964)). <p|L |p'xp'|N.|p> + <p|N |p'xp'|L |p> g i i = g 6 i j + •£ - 1 •• - • '- J — • — 1 : — (2.21) where g i s the i s o t r o p i c e l e c t r o n g-f a c t o r , | p>=ij; (r) are the wave-A ^ f u n c t i o n s o f the unperturbed e l e c t r o n H a m i l t o n i a n c o n s i s t i n g o f the k i n e t i c and p o t e n t i a l energy, e p are the energy e i g e n v a l u e s , L i are the o r b i t a l a n g u l a r momenta, and N = ( J V 2 m 2 e 2 ) {gradV(r)xp} ' G . 2 2 ) where p i s the e l e c t r o n ' s l i n e a r momentum, 16 The a n i s o t r o p y of the g - f a c t o r , coupled w i t h the s p a c i a l dependence of the e l e c t r o n wavefunctions and w i t h the o r i e n t a t i o n dependence of the. i n t e r a c t i o n (2.15), y i e l d s an e x t r a a n i s o t r o p y i n the K n i g h t s h i f t which does not v a n i s h even i n the case of c u b i c symmetry. We do not g i v e here the formal e x p r e s s i o n f o r t h i s c o n t r i b u t i o n s i n c e we w i l l t r e a t i t i n d e t a i l i n chapter 7. S p i n - o r b i t c o u p l i n g makes one more c o n t r i b u t i o n to K ^ ^ by mixing d i f f e r e n t ^ n ( r ) s t a t e s i n t o the i n t e r a c t i o n i n t e g r a l . In c h a p t e r 7 we show by group t h e o r e t i c a l arguments t h a t t h i s c o n t r i -b u t i o n v a n i s h e s f o r c u b i c c r y s t a l s , and t h e r e f o r e we w i l l not c o n s i d e r i t here. R e c e n t l y de C a s t r o and Schumacher (1973) have made a some-what more g e n e r a l c a l c u l a t i o n of K ^ ^ which reduces i n the t i g h t ^ an -b i n d i n g l i m i t t o the r e s u l t o b t a i n e d by Boon (1964). They a l s o make an order-of-magnitude-estimate of t h i s term and f i n d i n d i p c u b i c metals K ~10~ 3 ><K^  c o . an • l s o 2.2.2 The c o n t a c t c o n t r i b u t i o n . The term (2.1) has s p h e r i c a l symmetry and i s non-zero o n l y i n the case of s - e l e c t r o n s . I t i s thus c l e a r t h a t f o r a s p h e r i c a l Fermi s u r f a c e and i n the absence o f s p i n - o r b i t c o u p l i n g , t h i s term cannot c o n t r i b u t e t o the a n i s o t r o p y of the Knight s h i f t , However, w i t h a p p r e c i a b l e s p i n o r b i t c o u p l i n g even the s p i n s of s - e l e c t r o n s w i l l " f e e l " the symmetry of the l a t t i c e . T h i s causes a g a i n an a n i s o t r o p i c g - f a c t o r , which, i n t u r n , c r e a t e s an a n i s o -t r o p y through the c o n t a c t term. 'Such an e f f e c t has been p r e d i c t e d as e a r l y as 1953 by Bloembergen and Rowland i n ' t h e i r o r i g i n a l paper on the a n i s o t r o p i c Knight s h i f t , but o n l y r e c e n t l y has a thorough t h e o r e t i c a l a n a l y s i s appeared i n the l i t e r a t u r e (de C a s t r o and Schumacher (1973)). Weinert and Schumacher (1968) p o i n t out i n an appendix to t h e i r paper t h a t i n the case o f heavy noncubic metals, where s p i n -o r b i t - c o u p l i n g i s a p p r e c i a b l e , one cannot i g n o r e the s - e l e c t r o n c o n t r i b u t i o n t o K a n > The c r u c i a l approximation l e a d i n g to 100% k i s o t r o p y i s the s t e p i n which the s p i n s u s c e p t i b i l i t y x s 1 S made dependent on the energy E-> alone (see s e c t i o n (2.1,1) f o l l o w i n g JC Eqn. ( 2 . 5 ) ) . They show t h a t the c o n t a c t c o n t r i b u t i o n d e r i v e d without the aformentioned assumption i s of the f o l l o w i n g form f o r an a x i a l c r y s t a l : 8 TT K ( P a u l i ) = < | i ^ (0) | 2 g (k ) > y 2 N (E ) (2.23) 3 k F F This, e x p r e s s i o n r e p l a c e s (2.10), I f the g - f a c t o r g(£p) i s a n i s o -t r o p i c a t the Fermi s u r f a c e , i , e , depends on the d i r e c t i o n of H w i t h r e s p e c t to the l a t t i c e , then c l e a r l y K ( P a u l i ) c o n t a i n s t h i s a n i s o t r o p y . The authors p o i n t out t h a t the a n i s o t r o p y of the K n i g h t s h i f t i s not the same as t h a t of e i t h e r x s o r 9/ because of the w e i g h t i n g f a c t o r | ^ ( 0 ) | 2 appearing i n the average. As mentioned b e f o r e , de C a s t r o and Schumacher (1973)make a d e t a i l e d t h e o r e t i c a l c a l c u l a t i o n o f the s p i n - o r b i t c o n t r i b u t i o n to the c o n t a c t term. They i n c l u d e the e f f e c t t o second order and show t h a t a n o n v a n i s h i n g but s m a l l c o n t r i b u t i o n e x i s t s even f o r c u b i c m e t a l s . An o r d e r of magnitude es t i m a t e shows t h a t f o r these metals K c o n t a c t ~ 1 0 ~ 1 * K. , i . e . an o r d d r of magnitude s m a l l e r than the. an i s o ' term c a l c u l a t e d by Boon. 18 2 . 2 . 3 Other c o n t r i b u t i o n s So f a r we have not taken i n t o account the dependence of the Fermi s u r f a c e on the f i e l d 3, T h i s i s a sm a l l e f f e c t , but i t can i n f l u e n c e any of the components measured a t the Fermi l e v e l ; i n p a r t i c u l a r , i t can produce o s c i l l a t i o n s i n the d e n s i t y o f s t a t e s ( i . e . i n x s ) (Das and Sondheimer ( I 9 6 0 ) ) , i n the wavefunction ( G l a s s e r ( 1 9 6 6 ) ) , or i n the diamagnetic s u s c e p t i b i l i t y c o n t r i -b u t i o n (Stephen ( 1 9 6 1 ) ) . Even though these e f f e c t s are c l a s s e d under o s c i l l a t i o n e f f e c t s of the Knight s h i f t , t h e y do depend on the o r i e n t a t i o n o f the magnetic f i e l d w ith r e s p e c t to the l a t t i c e , and i n t h i s sense they belong t o the a n i s o t r o p i c s h i f t . F i n a l l y , t h e r e w i l l a l s o e x i s t i n p r i n c i p l 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 from the core p o l a r i z a t i o n and o r b i t a l terms. In n o n - t r a n s i t i o n metals however these terms are expected t o be ve r y s m a l l . 2.3 I n d i r e c t ' N u c l e a r I n t e r a c t i o n s I n d i r e c t i n t e r a c t i o n s between n u c l e i were f i r s t t r e a t e d by Ruderman and K i t t e l ( 1 9 5 4 ) . L o c a l i z e d moments can be coupled v i a co n d u c t i o n e l e c t r o n s through a second o r d e r p e r t u r b a t i o n mechanism. A l o c a l i z e d moment y^ i n t e r a c t s w i t h the moment of a co n d u c t i o n ->-e l e c t r o n , say, i n ground s t a t e k, and s c a t t e r s i t i n t o an e x c i t e d -> s t a t e k'; the s c a t t e r e d e l e c t r o n , i n t u r n , i n t e r a c t s w i t h an oth e r l o c a l i z e d moment y_^, r e t u r n i n g t o the i n i t i a l s t a t e k, thus c o u p l i n g the two moments y^ and y . From the e l e c t r o n ' s p o i n t o f -y ->- -> view i t i s a mechanism o f double s c a t t e r i n g : k—>k'—>k. Ruderman and K i t t e l (R-K) t r e a t e d the s p e c i a l case where the o n l y c o u p l i n g was v i a the c o n t a c t term o f the h y p e r f i n e i n t e r -a c t i o n : A. ( 1 ) (I.S) = (8TT/3) y Y 6 (r ~R. ) (I ,S) ( 2 , 2 4 ) Z e n I i where the n o t a t i o n i s the same as the one used i n the p r e v i o u s two s e c t i o n s . They o b t a i n e d a n u c l e a r i n t e r a c t i o n H a m i l t o n i a n of the form (2.25) S i n c e the exchange i n t e r a c t i o n between e l e c t r o n s p i n s has the same form, w i t h J^j r e p l a c e d by exchange i n t e g r a l s , they c a l l e d t h i s mechanism the pseudo-exchange i n t e r a c t i o n , Bloembergen and Rowland (1955) t r e a t e d the same problem by c o n s i d e r i n g both the c o n t a c t term (2,24) as w e l l as the d i p o l a r i n t e r a c t i o n between the e l e c t r o n and n u c l e a r s p i n : T h i s term becomes important f o r n o n - s - e l e c t r o n s . The f i n a l r e s u l t y i e l d s b e s i d e the pseudo-exchange i n t e r a c t i o n , a l s o a pseudo-d i p o l a r i n t e r a c t i o n o f the form where R^^ i s the p o s i t i o n v e c t o r between nucleus i and nucleus j , As can be seen, t h i s term has the same angular dependence as the d i r e c t d i p o l a r i n t e r a c t i o n between n u c l e i , In o r d e r t o show the g e n e r a l f e a t u r e s appearing i n the c o u p l i n g c o n s t a n t s J^^ and b.. we g i v e a b r i e f o u t l i n e of the d e r i v a t i o n o f the R-K i n t e r a c t i o n . C o n s i d e r the c o n t a c t h y p e r f i n e c o u p l i n g f o r two n u c l e i i and j y y J# r " 3 l { S - 3 r - 2 ( S . r ) r } e n (2.26) (2,27) 20 i j ( i ) + * (j) + + H = Z A (1..S" ) + Z A n ( I . . S J = K! + H 2 (2.28) where A i s g i v e n by Eqn. (2,24), The second o r d e r change i n energy i s <0 |H*J |n><n|H *j|0> H = E — = (2.29) J n E -E„ o n where |0> and |n> are many-electron-wavefunctions. The e x p r e s s i o n (2.29) r e p r e s e n t s an e f f e c t i v e n u c l e a r i n t e r a c t i o n which must be added t o the t o t a l n u c l e a r H a m i l t o n i a n c o n s i s t i n g o f the n u c l e a r Zeeman terms, n u c l e a r d i p o l a r terms, e t c . When (2 . 2 8 ) i s s u b s t i -t u t e d i n t o (2.29) f o u r sums are o b t a i n e d , two of which are s e l f -e n e r g i e s which do not couple the n u c l e i . The o t h e r two sums can be w r i t t e n i n the form <0|Hi | n x n | H 2 1 0> H = Z : + c.c. (2.30) ! j n E -E o n The s t a t e s [0> and |n> are chosen t o be the u s u a l antisymmetric l i n e a r c o m b i n a t i o n o f products o f B l o c h wavefunctions and s p i n f u n c t i o n s . U s i n g the f a c t t h a t Hi and Hz a r e sums of o n e - e l e c t r o n o p e r a t o r s one can change from the many-electron s t a t e s to an ex-p r e s s i o n i n o n e - e l e c t r o n wavefunctions: ^ m l H j l ^ ^ n l H ^ V ... , «! ^ + C f C f H . . = Z — — 1 — 1 ~ + c,c, (2.31) 1 3 m-occ E -E n-«Hinocc m n 21 ( i ) -*• where H =A (I..S) are now o n e - e l e c t r o n o p e r a t o r s , and i|> = I k,s>=u^ .(r) e l k r I s> (2.32) m k are o n e - e l e c t r o n - e i g e n f u n c t i o n s . The r e s t r i c t i o n of occupancy i s removed by m u l t i p l y i n g the e x p r e s s i o n i n the sum by f i t , s ) { l - f ( k \ s')> (2.33) -»- -> where f ( K , S ) X S the Fermi f u n c t i o n of s t a t e k, s p i n s. Another ->- ->-s u b t l e s t e p a l l o w s the replacement of f ( k , s) and E £ ^ s by f ( k ) and Ej>, r e s p e c t i v e l y . F i n a l l y , u s i n g <k| 6 (r-R.) |k'>=u*(R.)u-^, (R.)exp{i(k-k')R.> (2,34) x k x k 1 1 and assuming a l l n u c l e a r s i t e s e q u i v a l e n t , i . e . u^. (R. ) =u->- (0) , one o b t a i n s + . cos{ (it-it 1) .R. .} + H . .=const(I. .1 .) Z |u+(0)|2|u-> (0)|2 i - L f (k) {1-f (k»)} 1 3 1 3 k k E r E i t . (2.35) T h i s e x p r e s s i o n i s s t i l l q u i t e r i g o r o u s , The o n l y important s i m p l i f i c a t i o n so f a r l i e s i n the c h o i c e of the wavefunctions. One c o u l d have chosen, f o r i n s t a n c e , Dirac-OPW-functions, e s -p e c i a l l y f o r heavy m e t a l s , F o r a g e n e r a l Fermi s u r f a c e the e x p r e s s i o n (2,35) i s q u i t e c o m p l i c a t e d . Ruderman and K i t t e l chose a s p h e r i c a l Fermi s u r f a c e , and E+ = (#2 /2in —)]<? ' (2.36) k e r r where m e £ f i s a k-independent e f f e c t i v e e l e c t r o n mass. They ob-t a i n a t zero temperature X i j = J i j ( 2 . 3 7 ) J - . = c o n s t { s i n (2kr,R. .)-2kT;,R cos(2k R )}R7\ (2.38) 3 i * i j F 1 3 1 3 T h i s c o u p l i n g i s c l e a r l y o s c i l l a t o r y i n R^ _. and v a r i e s l i k e cos (2k„R. .)/R3-; f o r l a r g e R. . . Note t h a t J . . o f (2.38) i s i s o -* 1 3 x 3 i j i j ->- -y t r o p i c i n R^ _. , i . e . does not depend on the d i r e c t i o n o f R^j • Com-p a r i n g (2,38) w i t h (2.35) we see t h a t the i s o t r o p y appeared through the c h o i c e o f a s p h e r i c a l Fermi s u r f a c e . For a r e a l metal J . s hould depend on the d i r e c t i o n o f R.., i . e . on the p o s i t i o n o f the two i n t e r a c t i n g n u c l e i w i t h i n the symmetry of the l a t t i c e . The c a l c u l a t i o n o f Bloembergen and Rowland (B-R) (1955) f o r the p s e u d o d i p o l a r i n t e r a c t i o n , e v e n though more co m p l i c a t e d , i s s i m i l a r i n p h y s i c a l c o n t e n t . When the d i p o l a r term (2,26) i s added t o the s c a l a r term (2.24), second o r d e r p e r t u r b a t i o n theory y i e l d s t h r e e types o f c o n t r i b u t i o n s : (1) A s c a l a r - s c a l a r c o n t r i b u t i o n (2.25) which i s the same as t h a t t r e a t e d by R-K. (2) A s c a l a r - t e n s o r c o n t r i b u t i o n o f the form (2.27) (3) A t e n s o r - t e n s o r c o n t r i b u t i o n , t r e a t e d o n l y q u a l i t a t i v e l y by B-R, and n e g l e c t e d i n a l l e x p e r i m e n t a l i n t e r p r e t a t i o n s t o date; i t c o n t a i n s both exchange-type as w e l l as p s e u d o d i p o l a r - t y p e terms. The c o u p l i n g c o n s t a n t s b^^ have s i m i l a r p r o p e r t i e s to those o f the pseudoexchange c o n s t a n t s . b^_. e x h i b i t s an o s c i l l a t o r y be-h a v i o u r w i t h R^j which i s dampened by R~?, and i f the Fermi s u r f a c e i s c o n s i d e r e d s p h e r i c a l , i t does not depend on the d i -r e c t i o n o f R^ j , The r a t i o b. ./J. . i s of the order of 13 xj { ( h f ) p / ( h f ) s > . ( P p / P S ) F (2.39) where ( h f ) p and ( h f ) s are p r o p o r t i o n a l to the h y p e r f i n e s p l i t t i n g f o r pure p and s s t a t e s r e s p e c t i v e l y , and ( P p / P S ) F stands f o r the r e l a t i v e amount of p and s - c h a r a c t e r of the wavefunctions a t the Fermi s u r f a c e . In metals e x h i b i t i n g a p p r e a c i a b l e p - c h a r a c t e r a t the Fermi s u r f a c e one can o b t a i n r a t i o s k > i j / J i j as l a r g e as 1/3. There have been no s e r i o u s attempts to c a l c u l a t e the i n d i -r e c t c o u p l i n g c o n s t a n t s f o r r e a l metals up to the l a t e s i x t i e s . S i n c e 1968, however, a few d e t a i l e d c a l c u l a t i o n s have appeared i n the l i t e r a t u r e , mainly due to T t e r l i k k i s , Mahanti, and Das, Mahanti and Das (1968) attempt a q u a n t i t a t i v e e v a l u a t i o n of J^j and b ^ j f o r Rb 8 5 and C s 1 3 3 , u s i n g o n e - o r t h o g o n a l i z e d - p l a n e -wave (OPW)-functions and c a l c u l a t e d band s t r u c t u r e s . T h e i r i n t e r -a c t i o n H a m i l t o n i a n c o n t a i n s the s c a l a r i n t e r a c t i o n (2.24), used by R-K, the s p i n - d i p o l a r i n t e r a c t i o n (2,26), used by B-R, p l u s an e x t r a term: 2 Y e y / ( I , t ) r " 3 ^ (2.40) which r e p r e s e n t s the n u c l e a r s p i n - e l e c t r o n o r b i t i n t e r a c t i o n . Using s t r a i g h t f o r w a r d second order p e r t u r b a t i o n t h e o r y (as i n (2,29)), they o b t a i n the f o l l o w i n g type of terras: J ^ j ( 1 ) r e p r e s e n t i n g the u s u a l R-K i n t e r a c t i o n due to the d o u b l e - c o n t a c t term (2,24) J j _ j ( 2 ) - a s c a l a r i n t e r a c t i o n due to the d o u b l e - o r b i t a l term (2.40) Jj_j (3) , due to the d o u b l e - d i p o l a r term (2,26) ( a l r e a d y d i s -cussed q u a l i t a t i v e l y by B-R) b ^ (1) r e p r e s e n t s the u s u a l B-R term (2.27) b ^ j (2) i s due to d o u b l e - d i p o l e i n t e r a c t i o n s ( a l s o d i s c u s s e d i n the o r i g i n a l B-R paper) The q u a n t i t a t i v e c a l c u l a t i o n s show t h a t , a t l e a s t f o r non-t r a n s i t i o n metals, the o n l y major terms are A^j(1) and b ^ j ( 1 ) ( b i j ( 2 ) may be as l a r g e as 10% of bj_j (1), but J ^ j ( 2 ) and J j ^ (3) are v e r y s m a l l ) . Moreover, when working out a l l the n o n - s - s t a t e c o n t r i b u t i o n s to b ^ j , i t i s found t h a t the p - e l e c t r o n s make by f a r the most important c o n t r i b u t i o n , as i n i t i a l l y c l a i m e d by B-R. In these r e s p e c t s then, not much has been gained. However, the d e t a i l e d computations of J ^ j (1) and b ^ j (1) r e p r e s e n t a g r e a t step forward. E^ i s not approximated by (]/i2/2me^^) k2 , but band s t r u c t u r e i n f o r m a t i o n i s used to c a l c u l a t e the energy denomi-. n a t o r s ( m e f f i s r e p l a c e d by the k-dependent "thermal mass"). S i m i -l a r l y , the k-dependence o f the o t h e r terms i n the sum (2.35) i s not r e s t r i c t e d t o Fermi s u r f a c e v a l u e s . The f i n a l r e s u l t depends now on the k-dependence o f the wavefunctions, the thermal mass, and o f the phase f a c t o r s over the whole Fermi sea. I t i s shown t h a t t h i s dependence i s q u i t e c r i t i c a l (with b^_. showing a l a r g e r k - v a r i a t i o n than J . . ) , and i t i s p o i n t e d out t h a t gross e r r o r s 25 are i n t r o d u c e d by approximations i n which the d i f f e r e n t com-ponents i n the sum are e v a l u a t e d a t the Fermi s u r f a c e . Exchange and Coulomb i n t e r a c t i o n s between e l e c t r o n s have a l s o been taken i n t o account, and i t was found t h a t they enhance the i n t e r a c t i o n by about 10%. There i s however an important p o i n t t o be made a t t h i s s t a g e . Even though q u i t e i n t r i c a t e , these c a l c u l a t i o n s s t i l l use s p h e r i c a l energy contours i n k space, and t h e r e f o r e t h i s ap-proach s t i l l suppresses the d i r e c t i o n a l a n i s o t r o p y o f the c o u p l i n g c o n s t a n t s , T t e r l i k k i s e t . a l . (1969) have made a r e l a t i v i s t i c c a l c u -l a t i o n o f the i n d i r e c t c o u p l i n g c o n s t a n t s (as w e l l as of the Knight s h i f t and the r e l a x a t i o n mechanism). T h i s study i s a c t u a l -l y a c o n t i n u a t i o n of the work d e s c r i b e d p r e v i o u s l y , but the i n t e r -a c t i o n H a m i l t o n i a n was r e p l a c e d by the r e l a t i v i s t i c o p e r a t o r , and D i r a c OPVJ's were used as wavef u n c t i o n s . The Fermi s u r f a c e was assumed a g a i n s p h e r i c a l . The c a l c u l a t i o n s become even more c o m p l i -c a t e d , b ut the q u a l i t a t i v e r e s u l t s are s i m i l a r t o those o b t a i n e d by n o n r e l a t i v i s t i c t h e o r y . For i n s t a n c e , the major c o n t r i b u t i o n t o J^j c a n be i d e n t i f i e d as due to the d o u b l e - c o n t a c t i n t e r a c t i o n , and the l a r g e s t c o n t r i b u t i o n t o b^.. i s due to the c o n t a c t - d i p o l e mechanism. Q u a n t i t a t i v e l y however, a p p r e c i a b l e d i f f e r e n c e s were found. I t was .concluded t h a t i n the case of heavy metals l i k e l e a d and t h a l l i u m o n l y a r e l a t i v i s t i c c a l c u l a t i o n can be expected t o g i v e q u a n t i t a t i v e agreement w i t h experiment. Such a c a l c u l a t i o n was made f o r l e a d ( T t e r l i k k i s e t . a l . (1968)) (see cha p t e r 6 ) . Sin c e i t i s a heavy metal the one-OPW's had to-be r e p l a c e d by many-OPW's. The pseudo-wavefunctions and the thermal mass were c a l c u l a t e d from the p s e u d o p o t e n t i a l parameters and energy bands ( r e s p e c t i v e l y ) o f Anderson and Gold (1965) F i n a l l y , Mahanti and Das (1971) have c a l c u l a t e d the "core-p o l a r i z a t i o n " c o n t r i b u t i o n to the c o u p l i n g c o n s t a n t s . J u s t as i n the case of the Knight s h i f t , c o n d u c t i o n e l e c t r o n s , which have been p o l a r i z e d through i n t e r a c t i o n s w i t h a ne i g h b o u r i n g n u c l e a r s p i n , can p o l a r i z e , v i a exchange i n t e r a c t i o n s , t h e i r core s - e l e c t r o n s , which i n t u r n i n t e r a c t w i t h t h e i r own nu c l e u s , v i a e i t h e r the c o n t a c t term o r the s p i n - d i p o l a r term. In the case o f rubidium and cesium they found s u r p r i s i n g l y l a r g e c o n t r i b u t i o n s (10-30% enhancements). 2.4 The Lineshape The n-th moment o f an o b s o r p t i o n l i n e f (ui) o f resonance frequency toQ i s d e f i n e d by oo 00 M n = / (w-co 0) nf (oj)dw/ / f (w)da) (2.41) —00 —00 Van V l e c k (1948) has made a r i g o r o u s quantum mechanical c a l c u l a t i o n o f the moments. Since t h i s work i s now w e l l d e s c r i b e d i n the textbooks we w i l l not d w e l l on i t here. Systems c o n t a i n i n g o n l y one s p i n s p e c i e s have t h e i r second moment determined by the t e n s o r i n t e r a c t i o n a l o n e . The s c a l a r i n t e r a c t i o n • ) com-mutes wi t h the x^component of the s p i n - angular momentum and does not c o n t r i b u t e t o M 2; i t does however c o n t r i b u t e t o the f o u r t h moment, thus i n t e n s i f y i n g the s i g n a l i n the wings and narrowing the l i n e . Lead has o n l y one magnetic i s o t o p e and we use Van V l e c k ' s e x p r e s s i o n f o r the second moment to analyze our r e s u l t s i n c h a p t e r 6. -The s c a l a r i n t e r a c t i o n between u n l i k e s p i n s c o n t r i b u t e s to both the second and f o u r t h moment, thus broadening the l i n e s h a p e . T h i s i s the case i n n a t u r a l t h a l l i u m which c o n t a i n s two magnetic i s o t o p e s and i s t r e a t e d i n chapter 9. Anderson and Weiss (1953, 1954) have developed a mathemati-c a l model f o r resonance l i n e s due to s t r o n g exchange i n t e r a c t i o n s . They assume t h a t the d i p o l a r i n t e r a c t i o n i s randomly v a r i e d by the exchange, i n a s i m i l a r manner as i n the case of mo t i o n a l narrowing. T h i s e f f e c t averages out the t e n s o r c o n t r i b u t i o n to the l i n e w i d t h and produces exchange-narrowed L o r e n t z i a n l i n e s , whose l i n e w i d t h A co i s r e l a t e d to the second moment by Aco'vM2/co , where toe i s the exchange frequency.. The Anderson-Weiss model can be found i n the textbooks on magnetic resonance. F i n a l l y , i n the l a s t few y e a r s , p u l s e methods have been used t o measure the i n d i r e c t c o u p l i n g c o n s t a n t s i n metal powder samples, by o b s e r v i n g the spin-echo envelope i n d i s o r d e r e d a l l o y s (see e.g. A l l o u l and Froidevaux (1967)). We use one of t h e i r r e -s u l t s i n the r e d u c t i o n o f the l e a d c r y s t a l data i n chapter 6, but othe r w i s e t h i s t h e s i s i s s t r i c t l y a " s t e a d y - s t a t e " e f f o r t . CHAPTER I I I THE EQUIPMENT The spectrometer b u i l t f o r t h i s work was a m o d i f i e d v e r s i o n of the apparatus used by Sharma (1967) . B a s i c a l l y i t i s a Pound-Knight-Watkins-type m a r g i n a l o s c i l l a t o r . The c i r c u i t diagram i s shown i n F i g . ( 3 . 1 ) . The c h a s s i s i s made of heavy bra s s p l a t e s , and so are the p a r t i t i o n s which d i v i d e the spectrometer i n t o i t s compartments. T h i s reduces microphonics and p r o v i d e s s h i e l d i n g . We used four: compartments: the power supply s e c t i o n , the o s c i l -l a t o r , the r . f . a m p l i f i e r , and the audio a m p l i f i e r . In a f u r t h e r attempt to reduce e l e c t r o n i c n o i s e feed-through c a p a c i t o r s were used.between s e c t i o n s , and metal r e s i s t o r s i n the more s e n s i t i v e p a r t s of the c i r c u i t (e.g. i n the o s c i l l a t o r s e c t i o n ) . The spectrometer proved t o be of f a i r l y h i g h s e n s i t i v i t y and s t a b i l i t y between 5 and 23 MHz. A b l o c k diagram of the whole assembly i s shown i n F i g . ( 3 , 2 ) . The sample w i t h a copper c o i l wound around i t was r i g i d l y f a s t e n e d i n a p l a s t i c h o l d e r , A copper bomb used i n p r e l i m i n a r y experi.-. '.. ments, proved u n s a t i s f a c t o r y f o r h i g h e r modulation amplitudes a t . l i q u i d h e l i u m temperatures. The copper s h i e l d i n g has a t low temperatures a s m a l l enough a u d i o - s k i n depth to reduce the modu-l a t i o n amplitude a p p r e c i a b l y w i t h i n the bomb. Si n c e the l a t t e r i s measured o u t s i d e the bomb, an unknown f a c t o r i s i n t r o d u c e d , which p r e v e n t s an e x a c t f i t t i n g a n a l y s i s of the resonance l i n e s . A 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 , 3/8" i n diameter l e a d to the margin-a l o s c i l l a t o r . Small-diameter s t a i n l e s s - s t e e l t u b i n g served as 28 F i g . 3.1 C i r c u i t diagram of the P.K.W. spectrometer Counter P.K.W. O s c i l l a t o r M odulation c o i l Magnet Z ample F i e l d Probe t A Magnet Power..Supply i—I Em-power A m p l i f i e r Motor L o c k - i n • A m p l i f i e r C R 0 I I A Robinson O s c i l l a t o r F i g . 3.2 Block diagram of the spectrometer • Audio O s c i l l a t o r C R 0 I Recorder Phase S h i f t e r Counter P r i n t e r 7 o the c e n t r a l c onductor i n the coax. T h i s proved to be g r e a t l y s u p e r i o r t o w i r e c e n t r a l leads i n r e d u c i n g m i c r o p h o n i c s . We used Hewlett-Packard e l e c t r o n i c c o u n t e r s , model 5245L. The l o c k - i n a m p l i f i e r s were P.A.R., model JB4 and HR-8. The modulation c o i l s were d r i v e n by a Bogen MO-30A power a m p l i f i e r . Peak-to-peak modu-l a t i o n s up t o 16 gauss were a v a i l a b l e . The V a r i a n magnet has 12" p o l e f a c e s and a 2.2 5" gap. The probe measuring the magnetic f i e l d c o n s i s t s o f a g l a s s v i a l , 5 mm i n diameter, f i l l e d w i t h g l y c e r o l , and having a copper c o i l wound around i t . A s h o r t r i g i d copper coax connects the probe t o a Robinson-type o s c i l l a t o r which i s mounted on the r o t a -t a b l e magnet. S e v e r a l such probes were b u i l t to cover the whole range o f a v a i l a b l e magnetic f i e l d s . A v a r i a b l e c a p a c i t o r , 2-8pF, i n p a r a l l e l w i t h a v a r i c a p was used i n the tank c i r c u i t o f the R o b i n s o n - o s c i l l a t o r . A 90 v o l t b a t t e r y over a 100k H e l i p o t p r o-v i d e d the v a r i a b l e v o l t a g e f o r the v a r i c a p . A v e r y f i n e c o n t r o l of the p r o t o n frequency c o u l d be achieved i n t h i s manner. The pro t o n s i g n a l was l a r g e enough t o be d i s p l a y e d c l e a r l y on the scope. The low temperature system was of standard d e s i g n . The inner" dewar, c o n t a i n i n g l i q u i d helium, was connected to a vacuum l i n e . \ The o u t e r dewar, f i l l e d w ith l i q u i d n i t r o g e n , was l e f t a t atmos-p h e r i c p r e s s u r e . The b l o c k diagram shown i n Fig.(3.2) served f o r our t h a l l i u m work. In the case o f l e a d , the magnetic f i e l d sweep was r e p l a c e d w i t h a fr e q u e n c y sweep. The d e t a i l s are d e s c r i b e d i n the next c h a p t e r . CHAPTER IV EXPERIMENTAL CONSIDERATIONS The t h a l l i u m samples were s i n g l e c r y s t a l s o f the n a t u r a l metal ( 7 0 . 5 % T l 2 ° 5 a n d 2 9 . 5%T1 2° 3), produced by Semi-Elements,Inc. T h i s metal has a hexagonal c l o s e d packed l a t t i c e below 5°3°K. I t i s s o f t ( s i m i l a r to l e a d ) , and must t h e r e f o r e be handled w i t h g r e a t c a r e . T h i s i s e s p e c i a l l y important i n the case of s i n g l e c r y s t a l work, where a l l the s i g n a l comes from a t h i n s u r f a c e l a y e r . I t was s t o r e d on foam rubber, and the s u r f a c e r e s e r v e d f o r the winding o f the c o i l was touched as l i t t l e as p o s s i b l e . I f s u p e r f i c i a l s c r a t c h i n g o c c u r s , i t can be removed by e t c h i n g . T h a l l i u m when exposed to a i r o x i d i z e s q u i t e r a p i d l y , l o s i n g i t s m e t a l l i c l u s t r e . T h i s can be pre-vented by s t o r i n g the sample i n b o i l e d water. Not only does the c r y s t a l m a i n t a i n p e r f e c t m e t a l l i c l u s t r e when kept t h e r e i n , but i t was found t h a t t h i n l a y e r s of oxide disappeared w i t h i n a few hours. Tape wound around the sample, a f t e r the a p p l i c a t i o n of the c o i l proved u s e f u l i n keeping the c o i l i n p l a c e , as w e l l as i n the p r e v e n t i o n o f o x i d a t i o n . When perf o r m i n g experiments the c r y s t a l was kept as much as p o s s i b l e i n vacuum or i n a helium atmosphere/ T h i s was e a s i l y accomplished, s i n c e a l l experiments were performed i n dewers t h a t were p a r t of a vacuum system. When x - r a y i n g the c r y s t a l , the s u r f a c e must be f r e e o f oxide i f a c l e a r p i c t u r e i s to be o b t a i n e d . I t was found t h a t even very t h i n l a y e r s of oxide b l u r r e d the f i l m . The e t c h i n g was done w i t h a s o l u t i o n of 80% a c e t i c a c i d and 32 33 20% hydrogen peroxide. Extreme care has to be taken in this process, since i t i s an exothermic reaction that becomes very violent i f the solution heats up, resulting in the rapid destruction of the crystal. Two thallium crystals were available for this work.One of them 11 It was cylindrical in shape, 3 / 4 in length and l/k in diameter. Only preliminary work was performed with this sample. The other crystal had a quoted purity of 9 9 . 9 9 % and formed a parallelipiped with dimensions 1 x 3 / 8 x 3 / 8 . Since our magnet is of high homogeneity, the large surface of this sample proved beneficial i n improving the signal -to-noise ratio. The lead crystal was supplied by Metals Research Ltd., was of 9 9 . 9 9 % purity, and cylindrical in shape (12mm diam x 25mm in length). Lead is handled in a similar way to thallium, but is less troublesome because i t does not form thick oxide layers as easily as thallium does. A l l specimens were x-rayed at different sites of the surface to establish their mo'nocrystalline nature as well as their orien-tation. The results were in agreement with the quoted character-i s t i c s , A molybdenum tube proved optimum for thallium x-raying. A variety of coils were tried out for the different experi-ments , When winding coils one is faced with the choice of the following parameters: (1) The gauge of the wire. (2) The f i l l i n g factor (determined by the number of mylar layers between c o i l and sample). (3) The total number of turns. 34 (A) The number of turns per unit length. When performing an experiment the d.c. magnetic f i e l d is usually chosen as high as the magnet w i l l permit. This determines the desired o s c i l l a t i n g frequency of the resonance c i r c u i t . Since the capacitance of the c i r c u i t located in the PKW detector varies over a relatively small range, and should be kept at a minimum to reduce the losses, i t is the inductance of the c o i l that has to be chosen such as to provide the desired frequency. Since a l l four of the above mentioned parameters determine this inductance, one i s faced with a multitude of choices. From the accumulated experi-ence the following optimizing procedure emerged. The f i l l i n g factor (determined by the number of layers of mylar) was chosen such as to render the c o i l with high enough quality factor Q to oscillate at the desired temperature, but not above i t (the Q of the c o i l decreases with increasing temperature). This brought the system very close to the ideal conditions of a marginal oscillator and seemed to yield the best performance of the detection apparatus. The wire was chosen of f a i r l y high gauge ( 3 8 - ^ 0 ) . The number of turns and their spread was chosen such as to yield the right frequency and have the c o i l cover a large surface area of the sample. This can always be accomplished, since the inductance in-creases with the number of turns and decreases with the spread between them. The orientation of the crystals with respect to the magnetic f i e l d was f i r s t performed visually. In this way one obtains an accuracy of about +5°. In the case of experiments with high signal-to-noise ratio this can be improved to an accuracy of 35 -about ± 2 ° , by using the results of the anisotropic Knight s h i f t . During liquid helium experiments, a very accurate method of determining the crystal orientation was found. It is known that at these temperatures the magnetoresistance pick-up becomes very strong. Since i t is in phase with the signal, i t cannot be f i l t e r e d out by the lock-in amplifier. It was found that the effect is strongly anisotropic having sharp maxima and minima at positions of high symmetry of the crystal. For instance in the case of t h a l l i -um there appears an extremum when the d.c. f i e l d is parallel with the principal axis. By applying high modulation amplitudes and using low time constants on the lock-in amplifier one can determine in this way very quickly the orientation of the crystal within ±1° Evidently these measurements must be performed outside the resonance l i n e . The detection method employed was somewhat different for the two metals. In the case of lead the frequency and f i e l d measurement had to be of very high accuracy. The reason for this l i e s in both the small linewidth of the signal as well as the nature of one of the lead investigations: the measurement of a very small aniso-tropic Knight s h i f t . One i s forced to sweep over small intervals of the spectrum (a few gauss) in time intervals of the order of an hour. Sweeping and monitoring the f i e l d under these circum-stances would be very inconvenient. We found that by keeping the magnet turned on for 2k- hours prior to the experimental run we could obtain fields that were stable within ~ 3 0 milligauss over an hour. During any of the runs the f i e l d was however continously monitored with a precision of 5mG, and d r i f t s of more than 15mG 36 were accounted for in the lineshape analysis. Lineshapes can be obtained by sweeping the frequency of the resonant c i r c u i t . This is accomplished by applying a linear sawtooth voltage ( 0 - 1 0 0 volts) from a modified Tektronix waveform generator to a voltage sensitive diode capacitor (varicap P C 1 1 6 ) , in parallel with the capacitance of the resonant c i r c u i t . Linear sweeps from 1 msec to several hours were available. The level of the marginal oscillator depends on the capacitance of the varicap, i.e. on the frequency of the c i r c u i t . A change in level modifies the lineshape of the signal. Fortunately, in the case of narrow lines one can sweep over the whole line with a very small change in the capacitance of the varicap and consequently no level drop. It is thus clear that NMR in lead should be performed by sweeping the frequency, providing one has a stable o s c i l l a t o r . The spectrometer had a dynamic sta-b i l i t y o f ^ 1 0 Hz and proved quite satisfactory for this type of detection. The frequencies were continously punched out by the crystal counter assembly, with corresponding "blips" on the chart recorder. In this fashion one obtains a point by point frequency and f i e l d value of the lineshape. In the case of thallium the situation is reversed. As was mentioned above, long frequency sweeps modify the lineshape of the signal. Since thallium has a very wide line i t is. much more con-venient to keep the frequency fixed and sweep the f i e l d . The moni-toring of the f i e l d was performed in the following manner. The proton signal given by the glycerol probe was displayed on the oscilloscope. The width of the horizontal sweep on the scope was a measure of the modulation amplitude, and the proton signal, when 37 at the centre, gave an accurate measure of the f i e l d . As the latter was swept by means of a motor, the proton signal appeared on the scope travelling from, say, l e f t to right. When passing through the centre the corresponding proton frequency was punched out. The ti p of the proton signal was not wider than 1mm and the sweep was slow enough to permit an estimation of the tip position within the same accuracy. Therefore, for, say, a modulation amplitude of 5 kHz proton, the f i e l d could be measured with an accuracy of ± 0 . 2 kHz proton. The proton signal, i f desired, could be brought back on the scope within seconds by means of a varicap control of the f i e l d monitoring osc i l l a t o r . This, however, turned out to be unnecessary, since the sweep of the f i e l d was linear enough to provide accurate results by interpolation, even when the proton frequencies were punched out every 5 - 1 0 minutes. The modulation amplitude was measured by recording the proton frequency at the two ends of the horizontal sweep on the scope. If the f i e l d modulation is not perfectly sinusoidal there can appear an error in the abso-lute measurement of the f i e l d . This can be corrected by taking the average of the two values obtained with opposite polarities on the modulation c o i l s . Finally, whenever necessary, a correction for the' i n s t a b i l i t y of the PKW spectrometer was carried out. The momentary frequency of the oscillator was recorded every 5 minutes on the recording chart. If i t happened to vary by appreciable amounts, the f i e l d values were normalized to the frequency recorded at the centre of the l i n e . The modulation frequency was chosen by t r i a l and error - the optimum frequency for our experimental set-up lay between 2 0 and 38 5 0 H z . The signal-to-noise ratio improves with modulation amplitude, however high modulation amplitudes distort the lineshape. Negligible distortion is obtained for amplitudes which are smaller than a tenth of the linewidth. Depending on the quality of the signals we used modulation amplitudes which varied from a tenth of the li n e -width up to values which were of the order of the linewidth. As w i l l be seen in the subsequent chapter, our analysis of the line-shape can account for this. An analysis of the influence of the experimental time constant on the shape and resonance frequency of the line was carried out by feeding the output of the PKW spectrometer simultaneously through a twin BNC connector into two separate lock-in amplifiers followed by recorders. In the case of experiments with single crystals, the time constant does not only affect the resonance frequency of the line, but also i t s linewidth and apparent mixing of the absorption and dispersion mode. This effect is partly due to the asymmetry of the l i n e . A fast sweeping rate coupled with a long time constant wi l l displace the shallow peak by a larger amount than the sharp peak, thus increasing or reducing the linewidth depending on the direction of the sweep. Again, this effect is accounted for.by our lineshape analysis. CHAPTER V LINESHAPE ANALYSIS The e x p e r i m e n t a l l i n e i n a bulk m e t t a l i c specimen i s a mixture o f a b s o r p t i o n and d i s p e r s i o n modes as a r e s u l t o f the v a r i a t i o n i n the phase o f the r a d i o frequency f i e l d w i t h depth o f p e n e t r a t i o n i n t o the metal (Chapman e t . a l , (1957)). The s m a l l s k i n depth reduces the e f f e c t i v e sample volume t o a t h i n o u t e r l a y e r o f the specimen, but w i t h a s e n s i t i v e spectrometer one can o b t a i n r e a s o n a b l e s i g n a l - t o - n o i s e r a t i o s . We have a n a l y z e d the l i n e s t o o b t a i n resonance f r e q u e n c i e s and l i n e w i d t h s by f i t t i n g a s e t o f modulated m i x t u r e s o f a b s o r p t i o n and d i s p e r s i o n modes of L o r e n t z i a n l i n e s u s i n g the formalism developed by Wahlquist (1961), The f i t s a re remarkably good f o r both t h a l l i u m and l e a d , s u g g e s t i n g t h a t the l i n e s are of L o r e n t z i a n shape over r e g i o n s of about 2 l i n e w i d t h s from the resonance frequency. I t i s not p o s s i -b l e t o f i t the l i n e s w i t h a Gaussian p r o f i l e . In the case o f t h a l l i u m t h i s has been known to us from p r e l i m i n a r y experiments (see F i g . i o f the paper by S c h r a t t e r and W i l l i a m s (1967)). The n o r m a l i z e d L o r e n t z i a n a b s o r p t i o n l i n e i s g i v e n by the f u n c t i o n TT - 1 W g (H) '= — r (5.1) a W 2+(H-H 0) 2 where H i s the i n s t a n t a n e o u s v a l u e o f the magnetic f i e l d , W i s the h a l f - w i d t h a t h a l f - i n t e n s i t y of the a b s o r p t i o n mode, and H Q i s the resonance f i e l d . In a s t e a d y - s t a t e NMR experiment the 39 40 ins t a n t a n e o u s f i e l d i s of the form H(t) = H (t) + h coswt (5.2) a m where H (t) i s the slow sweeping d.c. f i e l d , h i s the modulation, a m f i e l d amplitude, and w i s the modulation frequency. I f the sweep r a t e i s c o n s i d e r e d s m a l l enough t o assume t h a t H a ( t ) i s constant over time i n t e r v a l s of the order o f 2TT/O3 then one can w r i t e 00 g a ( t ) = (W / T r ) N Z a n(W, h m , H d)cosnajt (5.3) where H-, = H (t) - H. and the F o u r i e r c o e f f i c i e n t s a„ are gi v e n d a o n by the i n t e g r a l ir/o) cosnwt'dt' a (W, h , K ) = (OO/TT) / — (5.4) n ma _ . W 2 + ( H + h c o s u t . ) 2 d m I f one makes the change of v a r i a b l e wt 1=t and takes i n t o c o n s i -d e r a t i o n t h a t the s i g n a l a t the output of a l o c k - i n a m p l i f i e r i s p r o p o r t i o n a l t o a i one o b t a i n s f o r the a b s o r p t i o n mode the ex-p r e s s i o n TT Wcost dt A(W, h , H ) = Cj / - r — (5.5) M D —TT W2+(H +h c o s t ? d m where i s a co n s t a n t depending on the g a i n of the spectrometer, but not on w. 41 P r o c e d i n g i n a s i m i l a r manner f o r the d i s p e r s i o n mode g d(H) TT 1 (H-H0 ) W2+(H-HQ )' (5.6) one o b t a i n s the d i s p e r s i v e component of the s i g n a l TT (H +h cost) c o s t dt D(W, h , H,) = -C 2 / d m  M A —TT W2 + ( H d + h m c o s t ) 2 (5.7) Consequently, the mixed mode a t the output o f a l o c k - i n a m p l i f i e r o f i n f i n i t e l y s h o r t time c o n s t a n t i s of the form Y(W, h , H , C x , C 2 ) = / m a _^ TT {C 1Wcost-C 2 (H+h cost) cost} d t d m  W2 + (H-,+h c o s t ) 2 d m (5.8) where C i / C 2 determines the mixture o f the modes. To take i n t o account the d i s t o r t i o n of the l i n e due to f i n i t e time c o n s t a n t s , i t was assumed t h a t the time dependent l i n e s h a p e f u n c t i o n e n t e r s a simple R-C f i l t e r as shown below O -WWVv-R Y i n ( t ) 1 -O The e q u a t i o n governing t h i s c i r c u i t i s Y i n ( t ) = RC ( d Y Q u t / d t ) + YQUtCt) (5,9) 42 w i t h the s o l u t i o n t Y Q U t ( t ) = (RC)" 1 / Y i n ( t ' ) e x p { ( t ' - t ) / R C } d f (5,10) The i n f o r m a t i o n was f e d i n t o a computer program as f o l l o w s . The data c a r d c o n t a i n e d the l i n e w i d t h W, the modulation amplitude h , the m i x t u r e of the modes determined by C i and C 2 ( C i + C 2 = l ) , m the i n i t i a l p o i n t o f the l i n e ( i n f i e l d or frequency u n i t s ) HDI, the f i n a l p o i n t o f the l i n e HDF, the i n t e g r a t i n g increment D of the f i e l d v a r i a b l e H^, the time c o n s t a n t TC, the peak-to-peak i n -t e n s i t y o f the l i n e (measured i n c h a r t paper u n i t s ) , LA=5(TC)/D=an i n t e g r a l number o f increments equal to about 5time c o n s t a n t s , NN = an i n t e g r a l number which makes the computer w r i t e out each NN-th p o i n t o f the computed l i n e , and the resonance f i e l d HDR. Using t h i s i n f o r m a t i o n , the computer p r o v i d e s a p o i n t by p o i n t r e p r e s e n t a t i o n of the e x p e r i m e n t a l l i n e s h a p e which can be compared d i r e c t l y w i t h the l i n e on the c h a r t paper. The computer program i s shown i n Appendix 1. I t s v e r s a t i l i t y would a l l o w anyone (even w i t h no programming knowledge what so ever) to use i t immediately f o r L o r e n t z i a n l i n e s , and w i t h v e r y few m o d i f i c a t i o n s f o r any o t h e r p r o f i l e . U s i n g an e m p i r i c a l convergent method of f i t t i n g , one can o b t a i n resonance f r e q u e n c i e s , l i n e w i d t h s and mixture c o e f f i c i e n t s w i t h h i g h accuracy. The v a l u e s o b t a i n e d f o r these parameters were unique, i . e . no o t h e r combination of the parame-t e r s c o u l d f i t the l i n e . A mixture of about 61% a b s o r p t i o n and 39% d i s p e r s i o n proved s a t i s f a c t o r y f o r both t h a l l i u m and l e a d a t l i q u i d h e l i u m temperatures. A s m a l l a n i s o t r o p y (C =58% to 63%) of the m i x t u r e o f the modes was observed i n l e a d . These v a l u e s are c l o s e to those i m p l i e d by the anomalous s k i n e f f e c t ( A l l e n and Seymour (1963)). At h i g h e r temperatures C =C 2=0.5. An ex p e r i m e n t a l l i n e , f i t t e d by computer i s shown i n F i g . (5.1). I t i s worthwhile mentioning t h a t the q u a l i t y of the f i t t i n g shown i n t h i s f i g u r e i s c h a r a c t e r i s t i c o f most of the o t h e r l i n e s as w e l l . x CALCULATED — EXPERIMENTAL F i g . 5.1 F i t t e d e x p e r i m e n t a l l i n e ( P b 2 0 7 ) . CHAPTER VI EXCHANGE INTERACTIONS IN LEAD. 6 . 1 I n t r o d u c t i o n The NMR l i n e w i d t h i n heavy metals i s s u b s t a n t i a l l y l a r g e r than the w i d t h c a l c u l a t e d from the d i p o l a r i n t e r a c t i o n alone (Van Vleck ( 1 9 ^ 8 ) ) . The main c o n t r i b u t i o n s r e s p o n s i b l e f o r t h i s i n c r e a s e are the pseudoexchange (Ruderman and K i t t e l (195*0) and p s e u d o d i p o l a r (Bloembergen and Rowland ( 1 9 5 5 ) ) i n t e r a c t i o n s . The former i s of s c a l a r n a t u r e and produces the s o - c a l l e d exchange narrowing i n the case o f l i k e s p i n s and broadening f o r u n l i k e s p i n s . The l a t t e r has the same t e n s o r i a l form and symmetry as the d i p o l a r i n t e r a c t i o n and always broadens the l i n e . Van V l e c k ( 1 9 ^ 8 ) has shown t h a t the second moment of the ab-s o r p t i o n l i n e i n substances c o n t a i n i n g only one magnetic i s o t o p e i s not a f f e c t e d by the pseudoexchange i n t e r a c t i o n . T h i s makes i t p o s s i b l e t o s e p a r a t e out the 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 i n a second moment measurement. Lead i s a s u i t a b l e metal f o r t h i s type of i n -v e s t i g a t i o n s i n c e i t c o n t a i n s only one magnetic i s o t o p e (21% a-bundance), and, moreover, i s of s p i n I = l / 2 , thus e s c a p i n g the ad-d i t i o n a l d i f f i c u l t i e s which might a r i s e from quadrupolar i n t e r -a c t i o n s . P r e v i o u s e x perimental work on the l i n e w i d t h mechanism i n l e a d (Bloembergen and Rowland ( 1 9 5 5 ) ; Snodgrass and Bennett ( 1 9 6 3 ) ; A l l o u l and F r o i d e v a u x ( 1 9 6 ? ) ) was done on powdered samples and under the assumption t h a t the NMR l i n e obeys the Anderson-Weiss ( 1 9 5 3 ) model o f extreme exchange narrowing. The j u s t i f i c a t i o n 45 46 f o r t h i s assumption came from the f a c t t h a t the NMR l i n e i n l e a d had a c u t o f f L o r e n t z i a n shape, as p r e d i c t e d by t h i s t h e o r y . In t h i s case the l i n e w i d t h i s p r o p o r t i o n a l to the second moment d i -v i d e d by the r a t e o f exchange, (Anderson and W e i s s ( 1 9 5 3 ) . Anderson ( 1 9 5 * 0 ) . Thus one can o b t a i n i n f o r m a t i o n about the p e r t i n e n t i n t e r -a c t i o n s by measuring l i n e w i d t h s i n s t e a d of the e x p e r i m e n t a l l y more d i f f i c u l t second moments. I n t h i s work we s t u d i e d a l e a d s i n g l e c r y s t a l i n o r d e r to d i r e c t l y observe the angular dependence of the second moment and thus o b t a i n a d d i t i o n a l i n f o r m a t i o n not o b t a i n a b l e from powder r e -s u l t s . I t was i n i t i a l l y i n t e n d ed to t a c k l e t h i s problem f o l l o w i n g the method o f Sharma e t . a l . ( 1 9 6 9 ) i n t h e i r study of white t i n , where i t was found t h a t the angular dependence of the l i n e w i d t h had the form expected f o r t h a t of the second moment. T h i s was taken as e v i d e n c e f o r the p r o p o r t i o n a l i t y -between the second moment and the l i n e w i d t h p r e d i c t e d by the Anderson-Weiss model,, and the c o u p l i n g c o n s t a n t s were then l a r g e l y determined from the l i n e w i d t h . However, we f i n d t h a t the l i n e w i d t h i n the case of l e a d does not e x h i b i t the expected angular dependence c h a r a c t e r i s t i c of the second moment, a l t h o u g h the l i n e s are L o r e n t z i a n i n the c e n t r e , and we a r e thus f o r c e d to attempt d i r e c t d e t e r m i n a t i o n s of the second moments. D e f i n i t e c o n c l u s i o n s are presented r e g a r d i n g the r a d i a l dependence o f the exchange c o n s t a n t s and the v a l u e s are compared w i t h the previous experimental r e s u l t s and the t h e o r e t i c a l c a l c u l a t i o n r e c e n t l y made by T t e r l i k k i s e t . a l . ( 1 9 6 8 ) . 6.2 E x p e r i m e n t a l D e t a i l s and R e s u l t s The s i g n a l s were observed w i t h the PKW spectrometer whose 47 frequency was swept with the varicap PC116 mounted in the tank c i r c u i t . The magnetic f i e l d , kept constant at 9 . 5 kilogauss, was produced by a rotable 1 2-in. Varian magnet and was monitored con-tinuously by a glycerine probe situated near the sample. In a typical measurement, the f i e l d did not vary by more than 3°mG during the. time taken to record a resonance line .-• However cor-:, rections were made for variations of more than 15mG. A. set of Helmholtz coils mounted on the pole faces provided the modulation f i e l d needed for phase-sensitive detection. The modulation frequency was 38 Hz. The crystal specimen, supplied by Metals Research Ltd., was of 9 9 . 9 9 % purity, cylindrical in shape (12 mm d i a m x 2 5 mm in length), and had the cylinder axis along the ( 1 1 0 ) orientation. It was mounted with this axis perpendicular to the magnetic f i e l d so that, by rotating the magnet, one swept the d.c. f i e l d in the plane containing the ( 0 0 1 ) , ( 1 1 1 ) , and ( 1 1 0 ) axes. The crystal could be oriented with respect to the d.c. magnetic f i e l d , using the magneto-resistance pick-up as described in chapter 4 . A sharp ex-tremum in the pick-up is found when the f i e l d is along the ( i l l ) axis, The measurements were made at 1.2°K by pumping on liquid helium. At this temperature the T^ contribution to the linewidth is negligible. To obtain resonance frequencies and linewidths the lines were f i t t e d by computer, as described in chapter 5« The f i t s are satisfactory for modulation amplitudes up to about 30% of the linewidth (defined as the width at half-intensity of the absorption mode), and imply that the lines are of Lorentzian shape over 48 r e g i o n s up t o two l i n e w i d t h s from the resonance frequency. No l i n e s o f modulation amplitude l a r g e r than 25% of the l i n e w i d t h were used f o r the d a t a p r e s e n t e d throughout t h i s work. The c a l c u l a t e d l i n e shapes f i t t e d our experimental r e s u l t s v ery w e l l (see F i g . 5 - 1 ) • The angular dependence of the l i n e w i d t h i s shown i n F i g . ( 6 . 1 ) I t has been shown by O ' R e i l l y and Tsang ( 1 9 6 2 ) t h a t the most gener-a l a l l o w e d a n g u l a r dependence of the second moment M2 f o r a cubi c c r y s t a l c o n t a i n i n g only one magnetic i s o t o p e i s of the form: M 2=P(x 4+y i W j-Q) ( 6 . 1 ) where P and Q are two independent parameters and x,y,z are the d i -r e c t i o n c o s i n e s s p e c i f y i n g the o r i e n t a t i o n of the magnetic f i e l d w i t h r e s p e c t to the c r y s t a l l o g r a p h i c axes. T h i s angular dependence i s shown as the dashed l i n e i n F i g , 6 . 1 . I t w i l l be immediately noted t h a t the a n g u l a r v a r i a t i o n of the l i n e w i d t h i s not i n ac-cordance w i t h t h i s dependence, i n c o n t r a s t to the w h i t e - t i n r e -s u l t s . We are t h e r e f o r e f o r c e d to attempt a d i r e c t d e t e r m i n a t i o n of the o r i e n t a t i o n dependence of the second moment. T h i s i s d i f f i -c u l t s i n c e the resonance l i n e s are L o r e n t z i a n f o r the c e n t r a l r e g i o n . The second moment i s d e f i n e d as +00 / (v-v 0 ) 2 X"(v) dv —00 M = ^ (6,2) 2 +00 / X"(v) dv —00 where v i s the frequency, vo i s the resonance frequency, and X"(v) i s the a b s o r p t i o n mode of the s i g n a l . I f the observed l i n e W(kHz) 0 25 55 75 90 F i g . 6.1 Angular dependence of the l i n e w i d t h W. Tne e r r o r bars r e p r e s e n t the root-mean-sguare d e v i a t i o n s of the experimental r e s u l t s ; the dashed curve r e p r e s e n t s the p r e d i c t e d angular dependence of the second moment. 50 i s a mixture of an a b s o r p t i o n and d i s p e r s i o n mode, i . e . X = C1x"(\>)+ C 2x'(v) , the Eqn. (6.2) may be r e w r i t t e n r e p l a c i n g x " by X without a f f e c t i n g the r e s u l t , s i n c e x ' ( v ) i s an odd f u n c t i o n of (v-vo).Of course, e r r o r s i n the resonance frequency, determined from the p r e v i o u s curve f i t t i n g , w i l l i n t r o -duce a s y s t e m a t i c e r r o r +00 AM2 = 2C 2.6v 0 / (v-v 0) x' (v) dv —00 where 6v„is the e r r o r i n the resonance frequency. Taking a g a i n advantage of the f a c t t h a t x"(v)is symmetric w i t h r e s p e c t to ( v-v^ and i n t e g r a t i n g by p a r t s , we o b t a i n 00 00 dx AM = 26v 0 / (v-v 0)x(v) dv = -6v 0 / ( v -v 0) 2 dv ^ —00 —00 dV Computer e s t i m a t e s of t h i s e x p r e s s i o n on r e a l l i n e s g i v e v a l u e s of the o r d e r of AM 2=1% of M 2. I n t e g r a t i n g the numerator and denominator of Eqn. (6.2) by p a r t s one o b t a i n s +°° dx / ( v - v 0 ) 3 -^,:dV 1 -«> dv M = (6.3) 3 + ° ° dx / (v-v 0) — dv -00 dv T h i s r e l a t i o n i s more convenient s i n c e i t c o n t a i n s the d e r i v a -t i v e s of the l i n e s h a p e , i . e . the experimental signal a t the output o f the l o c k - i n a m p l i f i e r . To c a l c u l a t e second moments we used t h i s e x p r e s s i o n and f e d the computer the f o l l o w i n g parameters of the p r e v i o u s l y f i t t e d e x p e r i m e n t a l l i n e (as i l l u s t r a t e d i n F i g . 5.1). The l i n e w i d t h W, the modulation amplitude h m, and the mixture of the modes (C^, C2), as defined in chapter 5» the resonance frequency v0, the left-and right-hand-side points at which the line begins to deviate from ,. the Lorentzian f i t (Land R), and the two cut-off points L c and Rc determined by a linear extrapolation to the baseline. When the shape of the line showed nonlinear cutoffs, the t a i l contribution was calculated manually. The computer program is given in appendix 2; i t writes out the total second moment and also the different contributions to M2 from different parts of the l i n e . It turns out that the central part and l e f t hand side t a i l of the line represent the main contributions, (with the central contribution usually being twice as large as the l e f t t a i l contribution). On the other hand, the right-hand-side t a i l contribution is of minor importance. This comes about because on the l e f t side of the line the ab-.. sorption and dispersion modes add whereas they subtract on the right-hand-side. This result is indeed fortunate since Rc, the right hand cutoff, i s hard to determine. It is worthwhile pointing out here that this advantage was quite c r i t i c a l in determining cut-off points, and i t is directly due to the mixture of the modes caused by the use of bulk metallic samples. Consequently, one can see that L c i s the most important single parameter in the determi-nation of One may argue that W and v 0 have similar influences on M2s W- because i t determines the central (most important) contribution, and v Q - because i t i s the difference (L c-v 0) which enters the computation. However in practice the experimental error of v„ and V/ i s much smaller than that of L c, and in this sense L c i s the determining factor in the accuracy of IVL,. Finally, a l l second moments were ^ corrected for modulation by subtracting l/4 h^ 52 a c c o r d i n g to the c a l c u l a t i o n by Andrew (1953). The r e s u l t s are shown i n F i g . (6.2). As can be seen, there i s some s c a t t e r i n the e x p e r i m e n t a l p o i n t s , but a d e f i n i t e p a t t e r n emerges which al l o w s d e f i n i t e c o n c l u s i o n s to be drawn, as w i l l be shown i n the subse-quent d i s c u s s i o n . 6.3 D i s c u s s i o n The second moment of an NMR l i n e i n a s i n g l e - c r y s t a l , con-t a i n i n g o n l y one magnetic i s o t o p e , i s g i v e n by (Van V l e c k (19^8)) o v , „ (l-3cos 2e,') 2 MO=-2-YV K I + I ) f s ( I + B , , ) 2 ^ — (6 A) k k k R3 where i t i s assumed t h a t a l l c r y s t a l s i t e s are e q u i v a l e n t , and the sum i s c a l c u l a t e d w i t h r e s p e c t to one s i t e chosen as the o r i g i n . R~^  i s the p o s i t i o n v e c t o r of nucleus k from the o r i g i n , C£ i s the angle between Pt k and the d.c. magnetic f i e l d H",Y i s the n u c l e a r gyromagnetic r a t i o , f i s the abundance of the magnetic i s o t o p e , a n d B k = ( ^ k i^lcjA Y2-^) where b^ i s "the p s e u d o d i p o l a r c o u p l i n g constant d e f i n e d by (Bloembergen and Rowland (1955)). •nri -*• ->• 3 ( l A » R i - i M - ' - n , ^ i i ) Hx1=b-- < ( l i - I i ) 1 1 J 2 * J > (6.5) ^ I D 1 J R ; . -^ J t>d where Ht . i s the p s e u d o d i p o l a r i n t e r a c t i o n H a m i l t o n i a n between ->- ->- -> s p i n Ij_ and s p i n 1^ ( j o i n e d by R^j) v i a the c o n d u c t i o n e l e c t r o n s , A f a i r l y c o m p l i c a t e d t h e o r e t i c a l e x p r e s s i o n f o r b.. was g i v e n by Bloembergen and Rowland (1955) i n t h e i r o r i g i n a l paper, and more r e c e n t l y by Mahanti and Das (.1968) and T t e r l i k k i s e t . a l . (1969), M 2 (kHz) 2 6 F i g . 6.2 Angular dependence of the second moment. The e r r o r bars r e p r e s e n t the root-mean-square d e v i a t i o n s of the experimental r e s u l t s and the s o l i d l i n e i s the b e s t f i t i n keeping w i t h the allowed angular dependence. 54 who made a r e l a t i v i s t i c second-order p e r t u r b a t i o n c a l c u l a t i o n , u s i n g D i r a c orthogonalized-^plane-wave f u n c t i o n s . As f a r as we know the o n l y t h e o r e t i c a l c a l c u l a t i o n of b.. f o r the case of l e a d 1 3 has been attempted by the same authors ( T t e r l i k k i s e t . a l . ( 1 9 6 8 ) ) , u s i n g the f o r m a l i s m developed i n t h e i r aforementioned papers. Even though they have used some of the d e t a i l s of the l e a d Fermi s u r f a c e , which has been e x t e n s i v e l y i n v e s t i g a t e d w i t h de Haas-van Alphen o s c i l l a t i o n t echniques by Anderson and Gold (1965) , t h e i r c a l c u l a t i o n e s s e n t i a l l y p r o v i d e s an i s o t r o p i c r e s u l t f o r the c o u p l i n g c o n s t a n t b ^ j , as has been the case w i t h a l l t h e o r e t i c a l c a l c u l a t i o n s of b... i n metals to date. In r e a l metals, however, 1 3 . i t i s known (e.g. Bloembergen and Rowland (1955); Mahanti and Das (1968)) t h a t b „ does depend on the o r i e n t a t i o n of the p o s i t i o n v e c t o r j o i n i n g the two n u c l e i . Measurements on powders cannot es-t a b l i s h t h i s o r i e n t a t i o n dependence, We hope t h a t work on s i n g l e c r y s t a l s w i l l s t i m u l a t e c a l c u l a t i o n s t h a t take t h i s a n i s o t r o p y i n t o account. The o s c i l l a t o r y c h a r a c t e r o f the r a d i a l dependence of makes i t d i f f i c u l t t o choose the s h e l l s which have the major c o n t r i b u t i o n t o the second moment of the l i n e . I f , f o r example, b ^ j ( R ^ j ) happens to have a zero a t the nearest-neighbour d i s t a n c e , then i t i s the s h e l l formed by the next n e a r e s t - n e i g h b o u r s which g i v e s the major c o n t r i b u t i o n t o the l i n e w i d t h . S ince d i f f e r e n t s h e l l s belong, i n g e n e r a l , to d i f f e r e n t symmetries, i t i s a g a i n the work on s i n g l e c r y s t a l samples which can h e l p d e c i d e upon t h i s p o i n t . A l l o u l and Froidevaux (1967) and A l l o u l and Deltour(1969) have used the s p i n echo technique on powdered samples to determine 55 the c o u p l i n g c o n s t a n t s b^_. and J\_. i n l e a d . J\ i s the pseudoex-change c o u p l i n g c o n s t a n t (Ruderman and K i t t e l (1954)). The s p i n -echo measurement f u r n i s h e s f o r each s h e l l k a modulation frequency-g i v e n i n f i r s t approximation by 1 i y\ V= = {J.+—(1+B,)} (6.6) 2 T m 4TT R k where 2 t m i s the p e r i o d o f the modulation. In o r d e r to determine J k and B k i n d e p e n d e n t l y , the s p i n echo data must be combined wi t h c.w. data on second moments. Si n c e the l a t t e r were e x p e r i m e n t a l l y d i f f i c u l t t o -obtain, A l l o u l e t . a l . used the Anderson-Weiss model which p r e d i c t s ( A l l o u l and Froidevaux (1966)) 2.36/r" (y 2-il/R 3 + b n , ) 2  w = _ _ „ oj °i (6.7) 2TT J 1 where o n l y the f i r s t n e a r e s t - n e i g h b o u r i n t e r a c t i o n has been taken i n t o account. The e x p e r i m e n t a l echo envelopes showed o n l y one c l e a r l y d e f i n e d modulation p e r i o d ( A l l o u l and Froidevaux (1967)) 2 1^=3051151^360. I t was not c l e a r whether i t was produced by the f i r s t or second-nearest neighbours. The e x p e r i m e n t a l h a l f - l i n e -width i n the powder was W=l.22+0.04kHz. Using these data and the e q u a t i o n s (6,6) and (6.7), they o b t a i n e d two s e t s o f data f o r the c o u p l i n g c o n s t a n t s , c o r r e s p o n d i n g to the two p o s s i b l e c h o i c e s i n which \^ i s a t t r i b u t e d t o e i t h e r the f i r s t o r second s h e l l of n e i g h b o u r s . r e s p e c t i v e l y . O ' R e i l l y and Tsang (19 62) have shown t h a t two independent parameters are r e q u i r e d to s p e c i f y the most g e n e r a l allowed angu-l a r v a r i a t i o n o f the second moment i n c u b i c - c r y s t a l s t r u c t u r e s . In the case of l e a d , which has a f . c . c . s t r u c t u r e , t h i s dependenc i s o f the form g i v e n by Eqn.(6...1) . Eqn. ( 6 . 4 ) can be w r i t t e n i n the form ( O ' R e i l l y and Tsang ( 1 9 6 2 ) ) 2 5 k R 6 . - k 127T d + B k ) 2 — - Y " # 2 i ( i + l ) f £ -{xj (G k,<j) k)x; (6,4)) + 5 k + -|-Xj (9k,<f>k)xJ (Q,4>)} ( 6 . 8 ) where X^ are l a t t i c e harmonics of the c r y s t a l , (Q k/$ k) a r e the. s p h e r i c a l angles made by R k, and ( 0 , < { > ) are those made by H with the c r y s t a l l o g r a p h i c axes. The a p p r o p r i a t e e x p r e s s i o n s of the l a t t i c e harmonics i n c u b i c symmetry are Xj= J U X V = V — ( x ^ + yHzt—2-) ( 6 . 9 ) /4TT / 6 4 T T 5 where x, y, z are ag a i n the d i r e c t i o n c o s i n e s . Using these ex-p r e s s i o n s i n E q n . ( 6 . 8 ) , we f i n a l l y o b t a i n 3 M = Y ^ K I + D fZ (1+B ) 2 S K ( 6 . 1 0 ) 2 5 K 57 1 S, = — — {37.5(0, - 0.6)a-2.5a +17.5} (6.11) * k k k where a = x£+y£+z£ r e f e r s to the d i r e c t i o n c o s i n e s made by R k K K X K and a to those made by H wi t h the c r y s t a l l o g r a p h i c axes. The c r y s t a l l a t t i c e of l e a d was generated by computer and the e x p r e s s i o n s d 6Z.S. f o r the d i f f e r e n t s h e l l s were c a l c u l a t e d i i (d i s the l a t t i c e c o n s t a n t ) . The r e s u l t s f o r the f i r s t seven s h e l l s a re shown i n Table (6.1). TABLE (6.1) . R e s u l t s of computer c a l c u l a t i o n o f d 6Z^Sj L f o r the f i r s t seven s h e l l s . S h e l l d 6 f f i i 1 -90(a-1.67)" 2 22.5(a-0.33) 3 -6.67(a-1.67) 4 - 1.41(a-1.67) 5 3.17(0-0.115) 6 -0.74(0-1) 7 -1.05(0-1.67) We note t h a t d 6 E ^ S ^ f o r a l l s h e l l s i s of the form s=p(a-|q|) w i t h p p o s i t i v e f o r some s h e l l s and n e g a t i v e f o r o t h e r s . For i n s t a n c e , s h e l l Nos. 1,3,4, and 7 have the same symmetry c h a r a c -t e r i z e d by S!=— j p j | (0-1.67), whereas the second s h e l l i s c h a r a c -58 t e r i z e d by S 2=|p 2|(a-0,33). I t i s important to r e a l i z e t h a t the d i f f e r e n t symmetries cannot be changed by the f a c t o r s (1+B^f , which are always p o s i t i v e . In the (110) plane, 1 a = — s i n ^ + c o s 4 ^ 2 (where i s the angle between H and the (001) - a x i s ) , and Sj and S 2 have the form shown i n F i g . ( 6 . 3 ) . Comparing the dependence o f Sx and S 2 on with t h a t of M 2 g i v e n i n Fig." (6.2), we see immediately t h a t the second s h e l l cannot make a major c o n t r i b u t i o n to the second moment. F u r t h e r , as can be seen from F i g . ( 6 . 2 ) , the e x p e r i -mental a n i s o t r o p y i s ve r y pronounced. I f one were to f i t the ex-p e r i m e n t a l curve to the e x p r e s s i o n (6.1) without knowing the t h e o r e t i c a l e x p r e s s i o n s appearing i n Table (6.1), one would have t o p i c k 1<Q<1.67. However, c o n t r i b u t i o n s from s h e l l s Nos. 2 and 5 tend t o make Q>1.67. T h e r e f o r e , u n l e s s s h e l l 6 happens t o make a l a r g e c o n t r i b u t i o n t o M , which i s very u n l i k e l y , our experimental r e s u l t s f o r c e us to p i c k Q=l,67. T h i s r e s u l t i n d i c a t e s t h a t (1+B ) 2-~0, i . e . the atoms of s h e l l 2 c o n t r i b u t e a t most s l i g h t l y t o the p s e u d o d i p o l a r i n t e r a c t i o n . I f we now make the s i m p l i f y i n g assumption t h a t o n l y the f i r s t f o u r s h e l l s c o n t r i b u t e to M , and B 1=B 3=B l t=B ? then we o b t a i n from Eqn.(6.10) _ 6 10 M 1 kH z (2TT)2 The f i t t e d e x p e r i m e n t a l curve of Fig.(6.2) i s g i v e n by 1.4 F i g . 6 . 3 T h e o r e t i c a l a n g u l a r dependence of S i and S 2 as d e f i n e d i n the t e x t 60 M = -4.1(a-1.67)kHz 2 2 o Tak i n g y=5.59xl03rad s e c " 1 G ~ 1 , d=4.90A at 1°K, f=0.21, 1=1/2, we o b t a i n (1+B) 2=223, whence B=13.9 or -15.9. We p i c k the p o s i t i v e v a l u e i n keeping w i t h the t h e o r e t i c a l c a l c u l a t i o n of T t e r l i k k i s e t . a l . ( 1 9 6 8 ) . T h i s y i e l d s — — = B=1.75 kHz 2TT 2frd 3 i n f a i r agreement w i t h the t h e o r e t i c a l v a l u e of b 0^2^=1,48kHz obtained }by T t e r l i k k i s e t . a l . I t i s worthwhile mentioning t h a t the approximation i m p l i e d i n the r e l a t i o n 6 ^ 6 3 = 6 ^ i s q u i t e good due to the f a c t t h a t a l l t h r e e s h e l l s belong to the same symmetry and a l s o i n view o f the r a p i d drop of the c o n t r i b u t i o n s coming from the f u r t h e r s h e l l s , as can be seen from Table (6.1) (the combined c o n t r i b u t i o n from s h e l l s 3 and 4 i s o n l y 8% of the t o t a l ) . Combining t h i s r e s u l t , which i s now f r e e from the Anderson-Weiss-model assumption, w i t h the s p i n echo r e s u l t , we o b t a i n f o r the Ruderman-Kittel c o u p l i n g c o n s t a n t the v a l u e J 1 y2#/s — = - (1+B) = 4.64, kHz 2TT T 2 d 3 m The ^ t h e o r e t i c a l v a l u e o b t a i n e d by T t e r l i k k i s e t . a l . (1968) i s J 1/2TT=4.81 kHz. F i n a l l y , i f we take (1+B 2) 2- 0, we o b t a i n 61 b 0 2 Y 2#B 2 — — = ' = -0.05 kHz 2 TT 2TTd 3 C l e a r l y , t h i s number j u s t g i v e s an i n d i c a t i o n of the o r d er of magnitude o f b 0 2 , and cannot be regarded as a r e l i a b l e e x p e r i -mental v a l u e . 6.3a C r i t i c a l A n a l y s i s of o t h e r work. The work d e s c r i b e d i n t h i s chapter was r e c e i v e d i n J u l y 1971 by the P h y s i c a l Review and p u b l i s h e d i n t h e i r June 1972 i s s u e . Luders and H e c h t f i s c h e r (1972) p u b l i s h e d a paper on the same t o p i c i n the A p r i l .issue .of P h y s i c a S t a t u s S o l i d i ; t h e i r work was r e -c e i v e d i n December 1971. As f a r as we know, t h e i r i n v e s t i g a t i o n s were c o m p l e t e l y independent of o u r s . The r e s u l t s o b t a i n e d by Luders and H e c h t f i s c h e r d i f f e r s i g -n i f i c a n t l y from ours i n as much as they conclude t h a t the second-n e a r e s t - n e i g h b o u r s have a l a r g e c o n t r i b u t i o n to the second moment. A c r i t i c a l , comparative a n a l y s i s of t h e i r r e s u l t s and i n t e r p r e - . . t a t i o n f o l l o w s . They measure l i n e w i d t h s and second moments i n two d i r e c t i o n s o n l y : H[|(001) and HJ| (110). These two o r i e n t a t i o n s correspond to two p o i n t s (^=0° and Tp =90°) on our graphs of F i g . (6 .1) , (6 .2) , (6.3). T a b l e (6.II) c o n t a i n s our and t h e i r r e s u l t s f o r these o r i -e n t a t i o n s . We reproduce i n Fig,(6.4) a t y p i c a l l i n e a t the output o f t h e i r l o c k - i n a m p l i f i e r (time c o n s t a n t : 10 to 100 seconds). F i g . ( 6 . 5 ) reproduces t h e i r computer drawing of the a b s o r p t i v e p a r t of t h e s i g n a l , a f t e r computer a v e r a g i n g . The. l a t t e r y i e l d e d t h e i r l i n e w i d t h s and second moments. The d i s c r e p a n c y between our and t h e i r l i n e w i d t h v a l u e s c o u l d 62 be due t o a m u l t i t u d e o f reasons. Poor s i g n a l - t o - n o i s e r a t i o (see F i g . (6.4), inhomogeneity of the f i e l d , i n s t a b i l i t y of the s p e c t r o -meter are o n l y a few o f the p o s s i b l e e x p l a n a t i o n s . T h e i r second moments a r e c o r r e s p o n d i n g l y l a r g e r , e s p e c i a l l y i n the (001) d i -r e c t i o n . The poor s i g n a l i n the wings, apparent from F i g . ( 6 . 5 ) , makes c a l c u l a t i o n s o f second moments q u i t e d i f f i c u l t . T h e i r l i n e o f argument i s as f o l l o w s . I f o n l y f i r s t - n e a r e s t neighbours c o n t r i b u t e to the second moment the r a t i o M 2(^=90 0)/M^(^=0°)=1.75. I f f i r s t - and.second-nearest neighbours c o n t r i b u t e w i t h the "normal" r a d i a l dependence, t h i s r a t i o i s 1.45. S i n c e t h e i r e x p e r i m e n t a l r a t i o i s 1.15, they conclude t h a t the c o n t r i b u t i o n due to the second-nearest-neighbours i s much l a r g e r t han t h a t i m p l i e d by a normal r a d i a l dependence. In f a c t , t h i s c o n t r i b u t i o n would have to be ^3 times l a r g e r , i . e . the f a c t o r 22.5 i n the second-shell-row o'f Table (6.1) would have to be r e p l a c e d by 67.0. Our experimental v a l u e f o r the same r a t i o i s 1.73, i n k e e p i n g w i t h our assumption t h a t the second-nearest neighbours have a v e r y weak c o n t r i b u t i o n t o M 2. TABLE (6.II) L i n e w i d t h (kHz) ( t h i s work) L i n e w i d t h (kHz) (L. and H.) M 2(kHz 2) ( t h i s work) M (kHz 2) ( L 2 and H.) ^=0° 1.88 ± 0.04 3.3±0.3 2.6±0.5 4.8±0.5 '|=90° 2.7 ±0.04 4.4±0.4 4.5±0.5 5.5±0.5 63 644 K . LUDERS a n d J). IIECIITFISCIIER i I A - A A / J J | A j A. • w f—3 V y t J -f— \ \ / \ / f 1 \ X / V E g . 2- Single sweep recorder trace of an >'MIt signal in a lead single crystal (frequency increases towards the left) F i g . 6.4 Reproduction from Luders and H e c h t f i s c h e r F i g . 6.5 Reproduction from Luders and H e c h t f i s c h e r ;Let us now c a l c u l a t e the c o r r e s p o n d i n g r a t i o f o r the (111) a x i s , i . e . M 2 (ifj=550 )/M 2 (xp=00) . Our experimental r e s u l t f o r t h i s r a t i o i s 2.2, a g a i n i n keeping with our assumption r e g a r d i n g second-n e a r e s t - n e i g h b o u r s . However, u s i n g the r e s u l t o b t a i n e d by Luder and H e c h t f i s c h e r t h i s r a t i o would have t o be 1.19. In view of the t o t a l l i n e w i d t h and second moment o r i e n t a t i o n dependence shown i n F i g . (6..I) and (6.2), such an i n f e r e n c e would be q u i t e i m p o s s i b l e . We c o n c l u d e t h a t Luder and H e c h t f i s c h e r have s t r o n g l y over-e s t i m a t e d the second-nearest-neighbour c o n t r i b u t i o n . We f e e l t h a t the l a c k o f d a t a i n the (111) d i r e c t i o n and i t s neighbourhood, and, p o s s i b l y , the poor s i g n a l - t o - n o i s e r a t i o were e s s e n t i a l i n d e t e r m i n i n g t h e i r r e s u l t s . 6.4 C o n c l u s i o n Our s t u d y o f the n u c l e a r magnetic resonance i n s i n g l e -c r y s t a l l e a d has shown t h a t the exchange-narrowing model of Anderson and Weiss i s i n a p p l i c a b l e i n t h i s metal. We have e s -t a b l i s h e d t h a t the c o n t r i b u t i o n o f the second-nearest neighbours to the second moment i s ve r y s m a l l and we have o b t a i n e d v a l u e s f o r both pseudoexchange and p s e u d o d i p o l a r c o e f f i c i e n t s f o r the f i r s t - n e a r e s t n e i g h b o u r s . CHAPTER VII KNIGHT SHIFT ANISOTROPY IN SINGLE-CRYSTAL-CUBIC-LEAD 7.1. I n t r o d u c t i o n The f r a c t i o n a l d i f f e r e n c e i n resonance frequency between metal and r e f e r e n c e compound, c a l l e d the Knight s h i f t , depends i n g e n e r a l on the d i r e c t i o n of the a p p l i e d magnetic f i e l d Ii w i t h r e -s p e c t t o the c r y s t a l l o g r a p h i c axes. One commonly w r i t e s AH=KH, where K i s a second rank t e n s o r . The i s o t r o p i c p a r t of the s h i f t , g i v e n by K ^ S Q = ( l / 3 ) T r { K } , has been s t u d i e d e x t e n s i v e l y , mainly i n powder samples which can (under c e r t a i n circumstances) be ade-quate f o r t h i s type o f work. The a n i s o t r o p i c s h i f t has been i n -v e s t i g a t e d i n m e t als of symmetry lower than c u b i c , u s i n g both powder and s i n g l e - c r y s t a l samples. I f the a n i s o t r o p y i s l a r g e the powder samples may prove adequate enough; however, s i n g l e -c r y s t a l samples become of paramount importance when a c c u r a t e measurements are r e q u i r e d . U s i n g g r o u p - t h e o r e t i c a l arguments, Boon (1964) has shown t h a t i n the absence o f s p i n - o r b i t c o u p l i n g the a n i s o t r o p i c Knight s h i f t i n c u b i c metals i s i d e n t i c a l l y zero. A s m a l l a n i s o t r o p i c c o n t r i b u t i o n i s expected i n the case of s t r o n g s p i n - o r b i t c o u p l i n g . Boon's c a l c u l a t i o n i s somewhat r e s t r i c t e d s i n c e i t o n l y uses the d i p o l a r i n t e r a c t i o n H a m i l t o n i a n . R e c e n t l y , de C a s t r o and Schumacher (1973) made a more g e n e r a l i n v e s t i g a t i o n of 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 c u b i c symmetry i n which they i n c l u d e e f -f e c t s due to both the d i p o l a r as w e l l as the c o n t a c t term. An ac-count o f these c o n t r i b u t i o n s was g i v e n i n chapter 2. 65 No a n i s o t r o p y i n a c u b i c metal has been seen p r i o r to t h i s work. Lead i s again a c h o i c e c a n d i d a t e f o r a sea r c h f o r t h i s e f -f e c t s i n c e i t i s a heavy c u b i c metal w i t h a l a r g e i s o t r o p i c Knight s h i f t and a r e l a t i v e l y s m a l l l i n e w i d t h . We have c a r e f u l l y s t u d i e d the a n g u l a r v a r i a t i o n of the resonance frequency o f our resonance l i n e s and we f i n d a s m a l l a n i s o t r o p i c c o n t r i b u t i o n to the Knight s h i f t which i s c o n s i s t e n t w i t h the expected angular dependence. 7.2 Ex p e r i m e n t a l D e t a i l s and r e s u l t s The sample and d e t e c t i o n method were d e s c r i b e d i n the p r e v i -ous c h a p t e r . When measuring s m a l l a n i s o t r o p i c Knight s h i f t s the f i e l d measurement must be i n v a r i a n t w i t h r e s p e c t to r o t a t i o n s of the magnet. To check t h i s , we p l a c e d a deuteron probe a t the s i t e o f the l e a d sample and measured the r a t i o between i t s resonance frequency and t h a t o f the f i e l d monitor probe as a f u n c t i o n o f angle . T h i s r a t i o was found t o be co n s t a n t w i t h i n one p a r t i n 10 6,which corresponds t o lOmgauss and i s t h e r e f o r e p e r f e c t l y ade-quate f o r our measurements. The homogeneity over the specimen volume was 3 p a r t s i n 10 6 . The sample was chosen o f low p u r i t y (99.99%) to prevent p o s s i b l e de Haas-van Alphen effects,. A p r e l i m i n a r y experiment to check f o r these o s c i l l a t i o n s as w e l l as t o get an i d e a o f our ac-curacy was performed a t 1.2°K. We determined the p o s i t i o n o f the l a r g e r peak o f the d e r i v a t i v e o f the resonance l i n e f o r d i f f e r e n t f i e l d v a l u e s and s m a l l v a r i a t i o n s i n ip around 0° {i> i s the angle made by the d.c, magnetic f i e l d - w i t h the ( O O l ) - a x i s ) . The r o o t -mean-square d e v i a t i o n o f the peak p o s i t i o n f o r ' 9 l i n e s was 24 Hz i n 9 MHz. To o b t a i n resonance f r e q u e n c i e s the l i n e s were analyzed by 67 • computer as d e s c r i b e d i n chapter 5. They are f i t t e d v ery w e l l by L o r e n t z i a n p r o f i l e s over r e g i o n s of about two l i n e w i d t h s on e i t h e r s i d e of the resonance frequency (see F i g . ( 5 . 1 ) ) . The e x p e r i m e n t a l Knight s h i f t r e s u l t s are shown i n F i g . (7.1). The smooth curve r e p r e s e n t s the best f i t f o r the expected a n g u l a r dependence; i t y i e l d s f o r the resonance frequency v'0= 9004.41+0.285 ( ^ s i n ^ + c o s ^ - 0 . 6) kHz (7.1) -y where ij> i s the angle made by H w i t h the (001)-axis i n the (110) p l a n e . From F i g . (6.1) we see t h a t the l i n e w i d t h a t ^=25° i s ap-p r o x i m a t e l y the same as t h a t at ip=90° . The f a c t t h a t we o b t a i n r e a s o n a b l e d i f f e r e n c e s i n the resonance f r e q u e n c i e s a t these two o r i e n t a t i o n s i s an- i n d i c a t i o n t h a t the a n i s o t r o p y of the Knight s h i f t i s not due t o f i t t i n g e r r o r s i n t r o d u c e d by the a n i s o t r o p y o f the l i n e w i d t h . 2.2.1 H i g h - F i e l d I n v e s t i g a t i o n We r e p o r t here an o t h e r experiment i n which the same search was undertaken a t h i g h e r f i e l d s , u s i n g the same sample., but an o t h e r magnet - Magnion FFC-4, 12" p o l e f a c e s , 2%"-gap. T h i s magnet had a lower homogeneity at the s i t e of the sample, which d e t e r i o r a t e d q u i c k l y as one moved away from the c e n t r e of the gap. The d i f f i c u l t i e s a r i s i n g from t h i s e x p e r i m e n t a l drawback p o i n t e d out the v e r y important r o l e p l a y e d by the s t a b i l i t y and o v e r - a l l homogeneity of the magnet i n t h i s type of work. Because of the poor f i e l d homogeneity o u t s i d e the low temperature c r y o s t a t the p r o t o n s i g n a l d i d not e x h i b i t a l i n e s h a p e which was narrow enough v 0 (kHz) . 3 0 9 0 0 4 . 2o| ( — J J 0 25 55 90 F i g . 7.1 Angular dependence of the resonance frequency. The e r r o r bars r e p r e s e n t root-mean-square d e v i a t i o n s of the experimental r e s u l t s and the s o l i d l i n e i s the best f i t i n keeping with the allowed angular dependence. when d i s p l a y e d on the scope. Thus we were not able to monitor the f i e l d p r e c i s e l y and c o n t i n o u s l y while p a s s i n g through the l i n e , as was done when u s i n g the V a r i a n magnet. We f e e l t h a t such p r e -c i s e m o n i t o r i n g i s important, i f one t r i e s to d e t e c t very s m a l l , o r i e n t a t i o n - d e p e n d e n t , changes i n the resonance frequency (~100Kz inlOMHz). To minimize the e r r o r s i n t r o d u c e d by f i e l d o r i e n t a t i o n we r o t a t e d the sample i n s t e a d of the magnet. To reduce f i e l d - i n h o m o -g e n e i t y broadening, the c o i l was wound on a narrow r e g i o n of the sample. The f i e l d s t a b i l i t y was i n c r e a s e d by t u r n i n g on the magnet 4 8 hours b e f o r e the experimental runs. A deuteron sample was p l a c e d c l o s e to the dewer. The l e a d and deuteron resonance l i n e s were d e t e c t e d through two d i f f e r e n t spectrometers and l o c k -i n a m p l i f i e r s , but r e c o r d e d on the same two-pen-chart r e c o r d e r . The two resonance f r e q u e n c i e s were d i s p l a y e d on two counters and t h e i r r a t i o was kept c o n s t a n t . T h i s was accomplished manually w i t h a f i n e c o n t r o l on the v o l t a g e a c r o s s the v a r i c a p . The f i e l d was then swept through the resonance, d i s p l a y i n g the two s i g n a l s on the c h a r t r e c o r d e r . We chose o n l y two o r i e n t a t i o n s : ^=2 5° and ^=90°. At these v a l u e s of the l i n e w i d t h s and mixture of the modes happen to be the same, and s i n c e the l i n e s h a p e s are c l o s e l y L o r e n t z i a n i t i s a good approximation to c o n s i d e r them i d e n t i c a l . T h i s a l l o w s a study of the a n i s o t r o p y by c o n c e n t r a t i n g on one pronounced f e a t u r e of the l i n e s h a p e i n s t e a d of f i t t i n g the whole resonance l i n e . For obvious reasons we chose the sharp peak of the d e r i v a t i v e . As mentioned p r e v i o u s l y , we were a b l e to measure the l o c a t i o n of t h i s peak w i t h an accuracy of 24. Hz i n 9 MHz when u s i n g the V a r i a n magnet a t 9 kgauss. 70 We r e c o r d e d the resonance l i n e s i n one of the two o r i e n -t a t i o n s , then r o t a t e d the sample and recorded i t a g a i n , r o t a t e d back, and so f o r t h . In p r i n c i p l e t h i s procedure should d i s p l a y the a n i s o t r o p y p r o v i d i n g a l l sweep r a t e s are i d e n t i c a l . T h i s i s v e r y hard t o a c c o m p l i s h when sweeping at slow r a t e s because o f the magnet's l a c k of dynamic s t a b i l i t y . The r e s u l t s a t 16kgauss showed the same a n i s o t r o p y as t h a t o b t a i n e d a t 9 kgauss, i n d i c a t i n g t h a t the a n i s o t r o p y i s i n d e -pendent o f f i e l d . T h i s i n c r e a s e d our c o n f i d e n c e i n the low f i e l d e x p e r i m e n t a l r e s u l t s , however no formal use was made of the h i g h f i e l d measurements s i n c e the e r r o r was of the same magnitude as the a n i s o t r o p y i t s e l f . We conclude t h a t these type of measure-ments demand an a c c u r a t e , continuous f i e l d m o n i t o r i n g procedure, c o u p l e d w i t h frequency sweeps which can a l s o be monitored p o i n t by p o i n t . 2.3 D i s c u s s i o n Boon (1964) has shown t h a t when s p i n - o r b i t c o u p l i n g i s taken i n t o account, the predominant c o n t r i b u t i o n to the a n i s o -t r o p i c K n i g h t s h i f t , r e s u l t i n g from the n u c l e a r - e l e c t r o n s p i n d i -p o l a r i n t e r a c t i o n i s of the form (7.2) K = / d 3 r . r ~ 3 { F (r)P°(cose)+0 (r) P'(cos6) sin(J)-0 (r) P \ (cos6) cose})} an zz 2 x 2 T y 2 ^ P™(cos0) are the a s s o c i a t e d Legendre p o l y n o m i a l s . F^ (r) ( i , j = =x,y,z) i s a symmetric t e n s o r f i e l d i n v a r i a n t under the symmetry o p e r a t i o n s o f the c r y s t a l , and i s due to the e f f e c t on the e l e c t r o n s p i n alignment of the o r b i t a l motion of the e l e c t r o n s through the a n i s o t r o p i c g - f a c t o r . 0-(r) i s a pseudovector f i e l d , a l s o i n v a r i a n t under the symmetry.operations of the c r y s t a l , and i s due to c r o s s terms of the d i p o l e i n t e r a c t i o n and the s p i n -o r b i t c o u p l i n g . The v e c t o r r(r ,8,<|>) d e s c r i b e s the e l e c t r o n po-s i t i o n i n the l a b o r a t o r y frame whose z- a x i s i s p a r a l l e l t o the magnetic f i e l d H, The i n t e g r a l i s over a l l space. The symmetric t e n s o r i s g i v e n by F . . ( r ) = g y 2 Z a . (p) (df/de) U ( r ) | 2 (7.3) i j " P i j P P where g i s the f r e e - e l e c t r o n g - f a c t o r , y 0 i s the Bohr magneton, f ( e ) i s the Fermi d i s t r i b u t i o n f u n c t i o n , 1(1 (r) = |p> are the eigen-P f u n c t i o n s of the unperturbed H a m i l t o n i a n p 2/2m+V(r), and g^_. i s g i v e n by Eqn. (2.21). The pseudovector i s g i v e n by df <p.|iN-|p'> * 0. (r) = 35g 2y 2 Z ,( — ) — i> , (r) 1J1 (r) (7.4) 1 P'P'de P e -E , P P P^p' P P where e are the energy e i g e n v a l u e s of the s t a t e s \b (r) and P P N = ( M / 2 m 2 c 2 ) ( g r a d V ( r ) x p ) (7.5) where j$ i s the e l e c t r o n ' s l i n e a r momentum. The d e t a i l e d p e r t u r -b a t i o n theory l e a d i n g to t h i s r e s u l t can be found i n Boon (1964) , and a p h y s i c a l d i s c u s s i o n of these c o n t r i b u t i o n s was g i v e n i n c h a p t e r 2. Using the c h a r a c t e r t a b l e s (Heine (I960) ) f o r the c u b i c groups, we f i n d t h a t the pseudovectors 0^ form a t h r e e - d i m e n s i o n a l b a s i s f o r the i r r e d u c i b l e r e p r e s e n t a t i o n T l f whereas P 2(cos0) sin<J>~zy and P* (cos0) cos<{>~zx belong to the t h r e e - d i m e n s i o n a l b a s i s , f o r T 2 . We conclude t h a t the l a s t two terms i n the i n t e g r a l (7.2) are z e r o i n c u b i c symmetry. Le t us now w r i t e F z z = ( 1 / 3 ) ( 3 F z z - T r F ) + ( l / 3 ) T r F (7.6) S i n c e F" i s i n v a r i a n t under the symmetry o p e r a t i o n s of the c r y s t a l , we conclude t h a t the f i r s t term of (7.6) t r ansforms as 3 z 2 - r 2 ~ P 2 ( c o s G ) , whereas the second term forms a b a s i s f o r the i d e n t i t y r e p r e s e n t a t i o n of the group. Since the l a t t e r i s m u l t i -p l i e d by P°(cos0), which belongs to a b a s i s f o r the i r r e d u c i b l e r e p r e s e n t a t i o n E, i t s c o n t r i b u t i o n to the i n t e g r a l must v a n i s h , and we are l e f t w i t h K 3 r, = (1/3) / d 3 r . r ~ 3 { 3 F (r) -TrF}P° (cos9) (7.7) an z 2 2 L e t us d e f i n e A(r) by ( l / 3 ) { 3 F z z ( r ) - T r F } = A(r)P° (cos0) (7.8) A ( r ) w i l l then t r a n s f o r m a c c o r d i n g to the i d e n t i t y r e p r e s e n t a t i o n , and the a n i s o t r o p i c Knight s h i f t becomes K „ = (4IT/5) / d 3r.r~ 3A(r)'{Y° (6,(j>) } 2 (7.9) where Y m are the s p h e r i c a l harmonics. To determine the e x p l i c i t a n g u l a r dependence of t h i s e x p r e s s i o n we w i l l make use of the l a t t i c e harmonics i n c u b i c symmetry used by O ' R e i l l y and Tsang (1962) i n t h e i r second moment c a l c u l a t i o n s . Using the c o u p l i n g r u l e f o r s p h e r i c a l harmonics and the s p h e r i c a l harmonic a d d i t i o n theorem we can w r i t e Y ° ( 6 ) Y ° ( 6 ) = 5 ( 4 T T ) " ^ E ( 2 L + 1 ) ~ ^ { C ( 2 2 L , 0 0 ) } 2 Y 0 ( 6 ) = L = 0, 2* L 5 ( 470*5 Z E (2L+ir 1{C(22L,00)} 2 Y ^ * ( 6 ' , <j> ' ) Y M (0 , $) L = Q 2 * M = - L L (7.10) where 6',(j>' are the p o l a r angles of r , and 0, $ are those of H w i t h r e s p e c t t o the c r y s t a l l o g r a p h i c axes. C(22L,00) are Clebsch-Gordon c o e f f i c i e n t s . Expanding the s p h e r i c a l harmonics i n terms of l a t t i c e harmonics, one o b t a i n s K =/(4T T ) 3 Z ( 2 L + l f 1 {C(22L,00)} 2 / d 3 r ' (r 1 )~ 3 A' ( r 1 ) a n Lyai * uct i u ct i x X T M - (6 ,,(t) ,)X M (0,$) (7.11) L' L where u l a b e l s the r e p r e s e n t a t i o n , a l a b e l s d i f f e r e n t s e t s of f u n c t i o n s b e l o n g i n g t o the same r e p r e s e n t a t i o n , and i l a b e l s the -y d i f f e r e n t b a s i s f u n c t i o n s w i t h i n a r e p r e s e n t a t i o n . Since A'(rO belongs a g a i n t o the i d e n t i t y , o n l y those l a t t i c e harmonics which form a b a s i s f o r the i d e n t i t y r e p r e s e n t a t i o n of the c u b i c group w i l l s u r v i v e i n the sum. S i n c e t h i s b a s i s i s one-dimensional, the sum over i has one term o n l y , and we are l e f t w i t h ( c a l l y=l the i d e n t i t y r e p r e s e n t a t i o n ) 74 K =/(4TT) 3 Z {C(22L,00)} 2 X r l a(0,$) / d 3 r'(rf3A»(r' )X* a* (6 ' , <j>' ) a n La L L (7.12) The i d e n t i t y - r e p r e s e n t a t i o n l a t t i c e harmonics can be found i n the a r t i c l e by O ' R e i l l y and Tsang (1962). For L<4 t h e r e are o n l y two of them: X\ = ( 4 T r ) ~ J s X 1 = (525/64TT) 3 5 (x" +y"+2"-0.6) (7.13) where x, y, z are the d i r e c t i o n c o s i n e s . We denote C = (4T T ) - 1 / d 3 r ' (r ' ) " 3A' (r ' ) and D= / d 3 r ' (r ' J" 3 A' (r ') x\ ( 6 ' , c f > ' ) (7.14) T h i s y i e l d s K = (4TT/5)C+2Tr/(3/7)D(x'*+y' ,+z' t -3/5) (7.15) <nrx where x, y, z are now the d i r e c t i o n c o s i n e s made by the magnetic f i e l d H w i t h the p r i n c i p a l axes of the c r y s t a l . C xs i s o t r o p i c and t h e r e f o r e cannot be determined e x p e r i m e n t a l l y by an angular dependence study. We i n c o r p o r a t e i t i n K £ s o and w r i t e K = K i s o + 2 T r / ( 3 / 7 ) D ( x ' t +y"+z4 -3/5) (7.16) The average over a l l space of the second term i s zero, as i t should be. The a n g u l a r dependence i s completely determined by Eqn. (7.16) , I f we denote again by ip the angle made by H w i t h the (001)-axis i n the (110) plane, we have K = .K i s o+2ir/(3/7) D(%sin>+cos L |T|;-3/5) (.7.17) Our e x p e r i m e n t a l r e s u l t can be w r i t t e n i n the form K = K-I S O (7.18) where v r i s the resonance frequency of the r e f e r e n c e compound, measured i n MHz. We choose (see Appendix 3) v r=8875.4±2,6 KHz (Rocard e t . a l . (1959)) and o b t a i n S i n c e the e l e c t r o n e i g e n f u n c t i o n s as w e l l as the Fermi s u r f a c e o f l e a d are now a v a i l a b l e , D c o u l d be c a l c u l a t e d from the e x p r e s s i o n s (7.3) t o (7.14). 7.3.a C r i t i c a l A n a l y s i s of o t h e r Work The work d e s c r i b e d i n t h i s c h apter appeared i n the June. 1972 i s s u e o f the P h y s i c a l Review. De C a s t r o and Schumacher p u b l i s h e d a paper on the same t o p i c i n the January 1973 i s s u e of the same j o u r n a l . T h e i r work: was r e c e i v e d i n J u l y 1972. As mentioned b e f o r e , t h i s a r t i c l e makes a t h e o r e t i c a l c a l c u l a t i o n o f the a n i s o t r o p i c K n i g h t s h i f t c o n s i d e r i n g c o n t r i b u t i o n s from b o t h the c o n t a c t as w e l l as the d i p o l a r i n t e r a c t i o n , They e s t i m a t e however the c o n t a c t c o n t r i b u t i o n to be an o r d e r of magni tude s m a l l e r than the d i p o l a r term. One can t h e r e f o r e conclude =1.45% and D=7.8x10 - 6 I S O 76 t h a t a t t h i s stage o f the e x p e r i m e n t a l a r t the c o n t a c t term can be n e g l e c t e d . Moreover, s i n c e the l a t t e r t u r n s out to have the same symmetry p r o p e r t i e s as the d i p o l a r term, our o r i e n t a t i o n de-pendence study would not be a l t e r e d even i f i t made an a p p r e c i a -b l e c o n t r i b u t i o n t o the Knight s h i f t . The a uthors a l s o r e p o r t an experimental study i n the same a r t i c l e . We quote: ...."NMR measurements on s i n g l e c r y s t a l s of the c u b i c metals l e a d and p l a t i n u m have shown the a n i s o t r o p y i n our samples to be l e s s t h a n , r e s p e c t i v e l y 3.4 and 1.5xl0 - 1*of the i s o t r o p i c s h i f t s . The upper l i m i t on l e a d i s h a l f the a n i s o t r o p y i n l e a d r e p o r t e d by S c h r a t t e r and W i l l i a m s . " ,.."A crude e s t i m a t e i n the same s p i r i t as above of the s i z e o f t h e a n i s o t r o p y l e a d s to K a n = 1 0 - 3 x K ^ S Q f o r a metal, such as l e a d and p l a t i n u m , f o r which the c o n d u c t i o n e l e c t r o n s a t the Fermi s u r f a c e have s u b s t a n t i a l p - c h a r a c t e r . The r e p o r t e d r e s u l t o f S c h r a t t e r and W i l l i a m s f o r l e a d f a l l s w i t h i n the order o f - 3 magnitude o f these e s t i m a t e s , " (Our v a l u e i s K a n=2.2xl0 x K ^ S Q ) We show below the e x p e r i m e n t a l r e s u l t s o b t a i n e d by de C a s t r o and Schumacher ( F i g . 2, 3, and 5 of t h e i r p a p e r ) . The r e s u l t s shown i n F i g . 3 are o b t a i n e d as f o l l o w s . "The samples were c u t t o c y l i n d r i c a l shape, w i t h the c y l i n d r i c a l a x i s a l o n g the c r y s t a l l i n e (100) d i r e c t i o n . " "The magnetic f i e l d a t the sample was measured by m o n i t o r i n g the N.M.R. frequency o f A l 2 7 i n a r e f e r e n c e sample c o n s i s t i n g o f Epoxy mixed w i t h a l u m i -num powder," The l a t t e r was mounted c o a x i a l l y w i t h the l e a d sample. The d a t a were taken up t o 18 kgauss u s i n g f i e l d modulation and l o c k - i n d e t e c t i o n . The data i n d e r i v a t i v e form were i n t e g r a t e d u s i n g the F a b r i t e k i n t e r n a l , "hard-wired" i n t e g r a t i o n r o u t i n e . Kramers-Kronig transforms of the i n t e g r a t e d data were o b t a i n e d . by computer. F i g . 2 shows the r e s u l t s . The 3-rd p a i r of l i n e s are the symmetric and antisymmetric p a r t s of the i n t e g r a t e d l i n e 1. The l i n e w i d t h r e s u l t s are shown i n F i g . 5, They analyze F i g . 3 by t r y i n g to f i t a f u n c t i o n of the expected angular dependence to the e x p e r i m e n t a l p o i n t s and f i n d t h a t t h e i r data, "which to the eye might seem b a r e l y to r e v e a l an a n i s o t r o p y , do not i n f a c t c o n c e a l an a n i s o t r o p y w i t h the c o r r e c t angular dependence," They add: "The r.m.s. d e v i a t i o n of the 19 p o i n t s from t h e i r mean i s ( 4 x l 0 _ 1 * ) % which we quote as the r e s u l t f o r the upper bound on the a n i s o t r o p i c Knight s h i f t . " FIG. 2. (1) Integrated NMR signal for Pb. (2) Kramers-Kronig transform of (1). (3) Symmetric and antisymmetric parts of (1). Total sweep 25 G. 78 K % 1.2010 1.2000 [ 1.1990 [100] direction J I I I I I I L J L 1 L FIG. 3. Measured •, Knight shift in lead as a function of magnetic field orientation in the (100) plane. 20 40 60 80 100 120 140 160 180 0 S gouss 4.00 3.00 3.00 2.50 I T T I ^ ^ ^ k T T magnetic field orienta { FIG. 5. Full width at half-maximum for Pb as a function of tion in the (100) plane. i < 1 t t 1 1 1 I 1 I 1 1 1 1 1 1 1 • 20 40 60 80 100 120 140 160 !80 @ [lOO] direction 79 Our comments f o l l o w : (^ 1) O r i e n t i n g the c r y s t a l along a (110) a x i s i s important when i n v e s t i g a t i n g a s m a l l a n i s o t r o p y . The (100) plane does not e x h i b i t the whole e f f e c t . (2) I t i s not a d v i s a b l e to use powder aluminum as a r e f e r -ence when one t r i e s to e s t a b l i s h resonance f r e q u e n c i e s w i t h a p r e c i s i o n o f 50mgauss. The l i n e w i d t h of aluminum i s of the order of 8 gauss, (3) The s i g n a l averager l i m i t e d the accuracy to 5 0 mgauss per c h a n n e l . (4) We swept the frequency under continuous m o n i t o r i n g , De C a s t r o and Schumacher swept the f i e l d w i t h no m o n i t o r i n g of the sweep r a t e (see our comments i n s e c t i o n (2,2.1)), (5) T h e i r s i g n a l - t o - n o i s e r a t i o i s poor (see Fig.3) (6) The p a i r o f l i n e s i n F i g . 3 are supposed to be symmetric and a n t i s y m m e t r i c r e s p e c t i v e l y ; they are not. (7) The l i n e w i d t h s are by about one gauss wider than o u r s . S i n c e the K n i g h t s h i f t a n i s o t r o p y i s so s m a l l , we f e e l t h a t t h i s can be due t o e i t h e r a l a r g e f i e l d inhomogeneity or a gross e r r o r i n l i n e s h a p e a n a l y s i s . (8) The l i n e w i d t h a n i s o t r o p y o b t a i n e d by us i s 70% h i g h e r as t h a t o b t a i n e d by de C a s t r o and Schumacher. (9) We f e e l t h a t the r.m.s. d e v i a t i o n from the mean i n F i g . 3 i s too s m a l l an upper bound on the Knight s h i f t a n i s o t r o p y . F i n a l l y we g i v e below the comments of the authors them-s e l v e s r e g a r d i n g t h e i r e x p e r i m e n t a l r e s u l t s . A l l of the f o l l o w i n g comments a r e quoted from t h e i r paper: (a) "In b o t h metals the l i n e s were L o r e n t z i a n , a t l e a s t 80 near the c e n t e r . S i g n a l - t o - n o i s e l i m i t a t i o n s p r e c l u d e d e f f e c t i v e i n v e s t i g a t i o n of the t a i l s , " (b) " I t s h o u l d be noted t h a t S c h r a t t e r and W i l l i a m s have a l s o o b served a l i n e w i d t h a n i s o t r o p y of 0.17 i n l e a d (to be com-pared to our 0.103), a d i s c r e p a n c y which lends support to specu-l a t i o n t h a t our r e s u l t s f o r the K n i g h t - s h i f t a n i s o t r o p y might be sample dependent. However, we d i d not take the care to c o r r e c t f o r p o s s i b l e modulation broadening t h a t S c h r a t t e r and W i l l i a m s took; so t h e d i f f e r e n c e s i n the l i n e w i d t h and i t s a n i s o t r o p y may r e f l e c t s m a l l e r sample dependences than the data seem to show." (c) "Some anomalies i n the l i n e w i d t h s i n both metals (Pb and Pt) l e a d us t o f e a r t h a t e x i s t i n g a n i s o t r o p y c o u l d be obscured by c r y s t a l i m p e r f e c t i o n s . In l e a d , p a r t i c u l a r l y , the c r y s t a l s u r f a c e i s e a s i l y damaged, I f the l o c a l c u b i c symmetry f o r n u c l e i near the s u r f a c e were l i f t e d by s m a l l , randomly o r i e n t e d d i s t o r t i o n s , the normal f i r s t o r d e r d i p o l a r i n t e r a c t i o n c o u l d produce s m a l l l o c a l a n i s o t r o p y f i e l d s t h a t would not have to be l a r g e t o ob-scure an e f f e c t as s m a l l as 0.05% of the i s o t r o p i c Knight s h i f t , " (d) " I t i s c e r t a i n l y p o s s i b l e t h a t sample dependence i s r e s p o n s i b l e f o r the d i f f e r e n c e between our e x p e r i m e n t a l r e s u l t s on l e a d and those of S c h r a t t e r and W i l l i a m s . " CHAPTER V I I I CORRELATION BETWEEN THE ANOMALOUS PRESSURE DEPENDENCE OF THE SUPERCONDUCTING TRANSITION TEMPERATURE IN THALLIUM AND ITS TEMPERATURE DEPENDENCE OF THE KNIGHT SHIFT. 8.1 I n t r o d u c t i o n Thorough i n v e s t i g a t i o n s of the temperature dependence of the i s o t r o p i c Knight s h i f t K have been c a r r i e d out i n q u i t e a few metals. In the case of most n o n t r a n s i t i o n metals the Knight s h i f t i n c r e a s e s w i t h i n c r e a s i n g temperature (see e.g. Benedek and Kushida (1958) on a l k a l i , Kushida and Murphy (1971) on aluminum, Sharma and W i l l i a m s (1966) on t i n , Kasowski (1969) on cadmium, V a l i c and W i l l i a m s (1970) on g a l l i u m ) . P r i o r t o t h i s work th e r e has been no such i n v e s t i g a t i o n on t h a l l i u m . T h i s metal has a l a r g e a n i s o t r o p i c l i n e w i d t h as w e l l as a l a r g e a n i s o t r o p i c Knight s h i f t ( S c h r a t t e r and W i l l i a m s (1967)) which make i t very d i f f i c u l t - to say the l e a s t - t o undertake an a c c u r a t e K n i g h t - s h i f t a n a l y s i s i n powder samples. In t h i s chapter we r e p o r t Knight s h i f t measure-ments performed upon a s i n g l e c r y s t a l sample. We f i n d an unusual behaviour, i n as much as K decreases w i t h temperature and e x h i b i t s a n o n l i n e a r behaviour which i s most pronounced around l i q u i d n i t r o g e n temperatures. In our a n a l y s i s we c o r r e l a t e t h i s e f f e c t w i t h the unusual p r e s s u r e dependence of the superconducting t r a n -s i t i o n temperature i n t h a l l i u m . The l a t t e r has been i n t e r p r e t e d by Makarov and Baryakhtar (1965) as caused by a change i n the t o -pology of the Fermi s u r f a c e . They conclude t h a t the volume r e -d u c t i o n due to p r e s s u r e causes the i n c r e a s e o f sm a l l pockets of 81 82 the Fermi s u r f a c e which i n t u r n produces the n o n l i n e a r dependence of the t r a n s i t i o n temperature T . Experiments on the i m p u r i t y efi-f e c t on combined w i t h i t s p r e s s u r e dependence (e.g. Quinn and Budnick.(1961); Lazarev e t . a l . (1964); Ignateva e t . a l . (1968)) y i e l d r e s u l t s which can be i n t e r p r e t e d i n the same f a s h i o n . Holtham and P r i e s t l e y (1971) have made a thorough band s t r u c t u r e c a l c u l a t i o n of the Fermi s u r f a c e o f t h a l l i u m u s i n g a l o c a l p s e u d o p o t e n t i a l model which i n c l u d e s the e f f e c t of s p i n -o r b i t c o u p l i n g . They f i n d good agreement with de Haas-van Alphen data and c y c l o t r o n resonance experiments. Very r e c e n t l y Holtham (1973) has used the same model t o c a l c u l a t e the e f f e c t s o f p r e s s u r e on the Fermi s u r f a c e . His c a l c u l a t i o n s show t h a t the volume decrease caused by p r e s s u r e does indeed produce a c r i t i c a l i n c r e a s e o f some minute e l e c t r o n pockets s i t u a t e d i n the 4-th zone. These pockets have a l s o been observed independently by Ca p o c c i e t . a l . (1970) i n low f i e l d de Haas-van A l p h e n - e f f e c t ex-periments and by Shaw and E v e r e t t (1970) i n a c y c l o t r o n resonance study. Holtham f i n d s s e m i q u a n t i t a t i v e agreement between h i s c a l c u -l a t i o n s and the da t a o b t a i n e d from the p r e s s u r e and i m p u r i t y de-pendence o f T . These f i n d i n g s encouraged us to perform an a n a l y -s i s o f the su p e r c o n d u c t i n g data i n o r d e r t o e x t r a c t the s i n g u l a r component o f the d e n s i t y of s t a t e s a s s o c i a t e d w i t h the s m a l l pockets o f the Fermi s u r f a c e . Combined e f f e c t s o f i m p u r i t i e s and p r e s s u r e on.T c a l l o w us to c a l c u l a t e the volume dependence of t h i s component f o r n e g a t i v e p r e s s u r e s , i . e . volume expansion. The temperature i n c r e a s e i n our experiment e v i d e n t l y corresponds to a volume i n c r e a s e , which i n t u r n depresses the Fermi l e v e l . C o n s e -q u e n t l y , we are o b s e r v i n g the disappearance of the pockets. Our 83 ' a n a l y s i s of the superconducting data i n d i c a t e s t h a t the c r i t i c a l Fermi energy i s reached a t a volume i n c r e a s e of 0.4% (from the volume at T=1.2°K) which corresponds to a n e g a t i v e p r e s s u r e of ^-2 k i l o b a r s or a temperature of ^80°K, i n agreement wi t h our K n i g h t s h i f t r e s u l t s . 84 8.2 E x p e r i m e n t a l D e t a i l s and R e s u l t s . The measurements were made upon s i n g l e c r y s t a l t h a l l i u m specimens u s i n g a Pound-Knight spectrometer. The c r y s t a l was a p a r a l l e l i p i p e d w i t h the dimensions l"x3/8"x3/8"; i t was produced by Semi-Elements, Inc. and had a quoted p u r i t y of 99.99%. I t was mounted so t h a t the magnetic f i e l d which was s u p p l i e d by .a r o t a b l e V a r i a n 12 i n c h magnet c o u l d be r o t a t e d i n a plane c o n t a i n i n g the (0001) c r y s t a l l o g r a p h i c a x i s . The o r i e n t a t i o n of the c r y s t a l c o u l d be determined a c c u r a t e l y by r e c o r d i n g the magnetoresistance s i g n a l a t low temperatures, as d e s c r i b e d i n chapter 4. Since the r e s o -nance l i n e i s v e r y wide we p r e f e r r e d to work at f i x e d frequency, and sweep the magnetic f i e l d . The l a t t e r was c o n t i n u o u s l y moni-t o r e d by a p r o t o n magnetometer and was s u f f i c i e n t l y h i g h ('v-lO k i l o g a u s s ) t o s e p a r a t e the T l 2 0 5 from the T l 2 0 3 resonance. A l l our quoted r e s u l t s were o b t a i n e d from the s t r o n g e r T l 2 0 5 resonance, but i s o l a t e d measurements on the T l 2 0 3 resonance showed the same Knight s h i f t s . The homogeneity of the magnetic f i e l d over the specimen volume was of the o r d e r of 3 p a r t s i n 10 6, w i t h the ex-c e p t i o n o f the specimen used f o r room temperature measurements where the homogeneity was of the order of 1 p a r t i n 10 5. By room temperature our s i g n a l - t o - n o i s e r a t i o was too poor to use our s i n g l e c r y s t a l specimen, and we had to r e s o r t to a l a r g e r sample, which we sandwiched to improve the s i g n a l . T h i s sample was not a p e r f e c t s i n g l e c r y s t a l ; as can be seen from the e r r o r b a r s , we were not a b l e t o work wi t h h i g h accuracy a t these temperatures. Our r e f e r e n c e frequency was o b t a i n e d from a s a t u r a t e d aqueous so-l u t i o n o f t h a l l i u m a c e t a t e under the same c o n d i t i o n s as the sample. A s t r o n g narrow l i n e y i e l d e d the v a l u e v f = 24 .5720+ 85 +0.0001 MHz a t 10 k i l o g a u s s i n agreement w i t h the v a l u e of 24.57 MHz quoted by Rowland (1961). The c e n t r a l p a r t of the T l 2 0 5 resonance s i g n a l i s c l o s e l y L o r e n t z i a n ( S c h r a t t e r and W i l l i a m s (1967)) and an a n a l y s i s of the l i n e w i t h t h i s assumption a l l o w s the d e t e r m i n a t i o n of both the resonance frequency and l i n e w i d t h , as d e s c r i b e d i n c h a p t e r 5. Using a c c u r a t e parameters of the l i n e -shape o b t a i n e d a t low temperatures one can accomplish reasonable f i t t i n g a t the h i g h e r temperatures., where the s i g n a l - t o - n o i s e r a t i o i s lower. Our e x p e r i m e n t a l r e s u l t s f o r the Knight s h i f t are shown i n T a b l e (8.1). The temperature v a r i a t i o n of the i s o t r o p i c component i s shown i n F i g . (8.1) . I t i s important to r e a l i z e t h a t each of our e x p e r i m e n t a l p o i n t s a t a g i v e n temperature corresponds to a s e r i e s o f many measurements at d i f f e r e n t o r i e n t a t i o n s . T h i s y i e l d s an a n g u l a r dependence of the form shown i n F i g . (8.2). Knowing the f u n c t i o n a l form of t h i s dependence, i . e . (where 6 i s the angle between the f i e l d and the (0001) axis), one can determine the K n i g h t s h i f t s w i t h f a i r l y h i g h accuracy. In o r d e r to make a few p r e l i m i n a r y e s t i m a t e s of the data we draw a curve through the experimental p o i n t s as f o l l o w s . Up to about 4 0°K the temperatureO.dependence of the Knight s h i f t i s s t r o n g l y l i m i t e d by the low v a l u e of the thermal expansion c o e f -f i c i e n t . As wi11 become apparent from our a n a l y s i s , t h i s de-pendence i s w e l l known. T h i s a l l o w s us to determine the i n i t i a l v a r i a t i o n o f K w i t h temperature. From'then on we simply draw a H r e f AH (8.1) Fig, 8,1 The temperature-dependence of the isotropic Knight shift in T l 2 0 5 C O <7\ 300 360 420 480 540 600 88 smooth curve through the p o i n t s . We f i n d Knight s h i f t v a r i a t i o n s of (dlnK/dT) --4xl0~' t ( ° K ) - 1 around l i q u i d n i t r o g e n temperatures, t a p e r i n g o f f to about - l x l O - 1 * (°K) ~ 1 by room temperature. The ex-p e r i m e n t a l p o i n t a t 595°K i s taken from Moulson and Seymour. (1967) . They c l a i m to have observed no change i n Knight s h i f t upon m e l t i n g . One must however remember t h a t t h a l l i u m undergoes a phase t r a n s i t i o n from a h.c.p. to a b.c.c. l a t t i c e a t 503°K, TABLE ( 8 . 1 ) " T (°K) 1.2° . 77° 195° 295° 595° KxlO" 163.310.2 160.1±0.4 155.3±1.5 153.7±1.5 148* K x l O " -9.910.2 -8.510.4 -6.712.0 ax *Moulson and Seymour (1967) 8.3 D i s c u s s i o n The i s o t r o p i c Knight s h i f t of a.-=metal can be expressed as the sum of the f o l l o w i n g terms K = K ( P a u l i ) + K(dia) + K(c.p.) + K(orb) (8.2) K(orb) i s the o r b i t a l paramagnetic c o n t r i b u t i o n t o the Knight s h i f t caused by the o r b i t a l angular momentum (Clogston e t . a l . (1964)). Even though the l a t t e r i s quenched i n ' i t s ground s t a t e by the c r y s t a l f i e l d , i t can cause o r b i t a l paramagnetism through a second o r d e r mechanism f i r s t d e s c r i b e d by Kubo and Obata (1956). 89 In n o n t r a n s i t i o n metals K(orb) i s s m a l l . K(c.p.) i s due to the p o l a r i z a t i o n o f core s - s t a t e s by co n d u c t i o n e l e c t r o n s w i t h un-p a i r e d s p i n , v i a exchange i n t e r a c t i o n s (Clogston and J a c c a r i n o (1961)). The temperature dependence of K(c.p.) can be very p r o -nounced i n t r a n s i t i o n metals wi t h degenerate d-bands due to the s t r o n g temperature dependence of the d - s p i n paramagnetic sus-c e p t i b i l i t y . I t i s expected t o be sm a l l i n n o n t r a n s i t i o n metals, and, as f a r as we know, i t has been i g n o r e d i n a l l s t u d i e s to date. KCdia) i s the diamagnetic s h i e l d i n g of n u c l e i v i a the Landau 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 ( D a s and Sondheimer (I960)). In r e a l metals t h i s terra i s n e g l e c t e d . Thus we w i l l assume t h a t the predominant temperature de-pendence o f the Kn i g h t s h i f t i n t h a l l i u m i s due to the P a u l i term. The l a t t e r i s g i v e n by (Townes and Knight (1950)) 8 IT K ( P a u l i ) = P_X (8.3) 3 F s where P = <|^ (o) | 2>p .is the p r o b a b i l i t y d e n s i t y f o r s - e l e c t r o n s F s a t the n u c l e u s , averaged over the Fermi s u r f a c e , and X = 2 y 0 2 N ( E ) f ( a ) i s the exchange-enhanced paramagnetic s p i n S F s u s c e p t i b i l i t y f o r s - e l e c t r o n s . u 0 i s the e l e c t r o n magnetic moment, N ( E F ) i s the b a n d - s t r u c t u r e e l e c t r o n d e n s i t y o f s t a t e s per u n i t energy, e v a l u a t e d a t the Fermi s u r f a c e , and f ( a ) i s an en-hancement f a c t o r due to e l e c t r o n - e l e c t r o n exchange i n t e r a c t i o n s . A simple e x p r e s s i o n f o r f ( a ) i s 'given by the Stoner f a c t o r 1 f (a) = - (8.4) 1 - a 90 where a i s p r o p o r t i o n a l to N(E^,). A many body c a l c u l a t i o n made by S i l v e r s t e i n (1963) f o r n e a r l y - f r e e - e l e c t r o n metals y i e l d s fCa) = (8.5) n» Xo l+( 1) m e f f X fe where X 0 i s the enhanced s u s c e p t i b i l i t y i n the f r e e e l e c t r o n ap-p r o x i m a t i o n , m / m e £ £ = N ( E F ) f e / N ( E p ) , and f . e . stands f o r f r e e e-l e c t r o n . The temperature dependence of the Knight s h i f t can be due to volume expansion e f f e c t s as w e l l as to an i n t r i n s i c temper-a t u r e dependence (Benedek and Kushida (1958)): dlnK dLnK dlnV dlnK (; ) = ( r) ( ) +.(._: ) (8.6) dT P d l n V T dT P dT V In some me t a l s independent s t u d i e s o f the p r e s s u r e dependence of K have made i t p o s s i b l e t o separate the above terms (e.g. Kushida (1971)). In t h a l l i u m no such s t u d i e s have been made. The i n t r i n s i c temperature dependence of K i s predominantly due to l a t t i c e v i b r a t i o n s and has f i r s t been t r e a t e d by Benedek and Kushida (1958). S i n c e the e l e c t r o n s p i n s cannot f o l l o w the l a t t i c e v i b r a t i o n s , the i n t r i n s i c temperature v a r i a t i o n o f x c has been n e g l e c t e d and one w r i t e s dlnK d l n P w ( ) v = ( - ) v (8.7) dT dT 91 The e f f e c t of the v i b r a t i o n a l s t r a i n s on P p can be expressed f o r m a l l y by an averaged expansion around the e q u i l i b r i u m volume V a t temperature T: .{ ( V ( t ) - V 0 ) / V 0 } 2 +. . . (8.8) v=v o The term l i n e a r i n AV = V ( t ) - V 0 e v i d e n t l y does not e n t e r s i n c e AV=0. T a k i n g the d e r i v a t i v e w i t h r e s p e c t to temperature we c l e a r l y o b t a i n t o second o r d e r d . :(AV/V ) 2 (8.9) dT V=V0 " P p = P p ( t ) = P F ( V B ) + -d 2 P T d ( V / V n r dlnK 1 dP T dT P„ dT 1 2 I \ d 2P. d(V/V f l)' T h e o r e t i c a l c a l c u l a t i o n s o f P„ as a f u n c t i o n of volume are very d i f f i c u l t (see e.g. Kasowski (1969)) and no such c a l c u l a t i o n s have been performed f o r t h a l l i u m to date. In the 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 the o n l y volume e f f e c t on P F i s caused by the r e -n o r m a l i z a t i o n o f the wavefunction, i ,e. P„(V)°cV - 1, I f t h i s i s F so,then c l e a r l y (dlnP )/(dlnV) =-1 and 2P7 d 2P, d ( V / V 0 ) 2 = 1 v=v. (8.10) We w i l l w r i t e P p f V j ^ V * where x reme.ins to be determined e x p e r i -m e n t a l l y . The above term i s then x ( x - l ) / 2 . F o l l o w i n g Sharma and W i l l i a m s (1966) we w r i t e — — — - XU (AV/V 0) 2 = (8.11) V m where X i s the a d i a b a t i c c o m p r e s s i b i l i t y , U i s the i n t e r n a l ener-gy of the l o n g i t u d i n a l phonons per mole and V i s the molar volume. Assuming x temperature independent, we w r i t e dlnK x ( x - l ) v dU U dx XU dV ( ) = ( + SL.) (8.12) dT v 2 V dT V dT V 2 dT m m m dU To c a l c u l a t e we w i l l c o n s i d e r i t equal to the s p e c i f i c heat dT of the l o n g i t u d i n a l phonons. S i n c e we are not i n t e r e s t e d i n very low temperatures we w i l l use the Gruneisen r e l a t i o n throughout dU 3V = m (8.13) dT 3xrQ where 3 i s the temperature dependent thermal expansion c o e f f i c i e n t and TQ i s the Gruneisen c o n s t a n t f o r l o n g i t u d i n a l phonons. S i n c e the temperature dependence of the expansion c o e f -f i c i e n t and.the c o m p r e s s i b i l i t y i s q u i t e important f o r the whole of our subsequent a n a l y s i s we w i l l d i s c u s s them here. Swenson (1955) and Meyerhoff and Smith (1962) measured 'molar volumes of t h a l l i u m as a f u n c t i o n of temperature. The thermal expansion c o e f f i c i e n t s o b t a i n e d by Meyerhoff and Smith are by about 20% h i g h e r than those o b t a i n e d by Swenson. The d i s c r e p a n c y was i n -t e r p r e t e d as caused by the f a c t t h a t Swenson used p o l y c r y s t a l l i n e t h a l l i u m which may have e x h i b i t e d p r e f e r r e d o r i e n t a t i o n s i n the samples. We w i l l use the thermal expansion c o e f f i c i e n t s o b t a i n e d by Meyerhoff and Smith. Swenson (1955) has a l s o measured molar volumes a t zero and 10 kbar a t d i f f e r e n t temperatures r a n g i n g from 4.2°K to 300°K. From t h e i r data we f i n d the c o m p r e s s i b i l i t y r a n g i n g from 2.02x10" 1 2 to 2.51x10" 1 2 (dynes/cm 2)" 1. Vaidya (1970) measured T l - c o m p r e s s i b i l i t i e s a t room temperature as a f u n c t i o n of p r e s s u r e . H i s c o m p r e s s i b i l i t i e s range between 2.6x x l O " 1 2 (dynes/cm 2)" 1 a t low p r e s s u r e s (0-10 kbar) to 1 . 4 8 x l 0 ~ 1 2 (dynes/cm 2)" 1 a t 40-45 kbar. From the p o i n t of view of volume changes, thermal expansion i s e q u i v a l e n t t o n e g a t i v e compression. L e t us f i n d the n e g a t i v e p r e s s u r e s which would correspond to our thermal expansions; Using the data of Swenson (1955) and Meyerhoff and Smith (1962) we f i n d the f o l l o w i n g correspondences: 0-100° > 0.55% volume expansion > -2.6 kbar 0-200° > 1.35% " " > -6.2 " 0-273° > 1.98% " " > -8.7 " T h i s i m p l i e s t h a t our expansions correspond to Vaidya's low p r e s s u r e r e s u l t which, i n t u r n , agrees w e l l w i t h Swenson's r e s u l t a t room temperature. We w i l l t h e r e f o r e use Swenson's compressi-b i l i t i e s a t a l l temperatures. 94 8.3.1 The i n t r i n s i c temperature dependence of K . We f i r s t e s t i mate the term A = X U " 1 ) J L dJL (8.14) 2 V dT m Using the Gruneisen r e l a t i o n , x(x-l)B ( 8 > 1 5 ) A = 6 F G t h i s term c l e a r l y i n c r e a s e s with temperature. We w i l l t h e r e f o r e e s t i m a t e i t a t T=300°K. Subsequent a n a l y s i s w i l l show t h a t x c o u l d be as low as -6. At 300°K, 3=8.88xl0~ 5(°K)" 1; f o r l o n g i t u d i n a l phonons r G=2.1 ( G a l k i n e t . a l . (1971) ) . Thus, a t 300°K, A=3xl0~ l*. T h i s i s a l a r g e c o n t r i b u t i o n and we must take i t i n t o account. We w r i t e A=7.94xl0~ 2x(x-1) 3 . We now c a l c u l a t e the second term: x ( x - l ) U dX x ( x - l ) 3T dx (8.16) B E V m dT 6 X r G dT We e s t i m a t e i t a g a i n a t T=300°K. At t h i s temperature we o b t a i n from Swenson's data x= 2•51x10" 1 2(dynes/cm 2)~ 1 and dx/dT=l.5x10"15 (dynes ° K / c m 2 ) - 1 Thus B=0.53xl0 - 1 1. As can be seen, t h i s term i s j u s t l a r g e enough a t h i g h temperatures not to be n e g l e c t e d . We w r i t e B=4.7 xlO" 5x (x-l ) 3 T • (8.17) F i n a l l y , the 3-rd term, x ( x - l ) *u d V m x ( x - l ) 3 2 T (8.18) 2 V 2 dT 6T„ m G 95 At 300°K, C=-7.9xl0~ 6. T h i s terra i s two o r d e r s of magnitude s m a l l e r than our experimental v a l u e s and w i l l be n e g l e c t e d . From these e s t i m a t e s we conclude t h a t (dlnK/dT)^ c o u l d be as h i g h as 3. 5x10""* (°K) ~ 1. I t < should be noted t h a t t h i s q u a n t i t y i s p o s i t i v e i n a l l o t h e r metals as w e l l , (e.g. Mc Garvey and Gutowsky (1953); Kushida and Murphy (1969); Sharma and W i l l i a m s (1967)) . Consequently we must e x p l a i n our n e g a t i v e temperature d e r i v a t i v e as caused by volume expansion. 8.3.2 The volume dependence of the Knight S h i f t . Using Eqn. (8.3) f o r the Knight s h i f t we w r i t e (8.19) (dlnK/dlnV) = (dlnP /dlnV) +(dlnN (E p) / d l n V ) ( d l n f (a) /dlnV) „ T r *p *• i . -L For a f r e e e l e c t r o n gas, (8.20) 2 ( d l n P F / d l n V ) T = - l ; (dlnN (E p) /dlnV) T = — ; ( d l n f (a) /dlnV) T=0 , which y i e l d s (dlnK/dlnV) =-1/3. As one can see, our experimental r e s u l t s show a completely d i f f e r e n t behaviour, i n as much as ( d l n K / d l n V ) T i s s t r o n g l y nega-t i v e and n o n l i n e a r . In what f o l l o w s we w i l l t r y to c o r r e l a t e t h i s anomalous behaviour w i t h the unusual p r e s s u r e dependence of the supe r c o n d u c t i n g t r a n s i t i o n temperature (T ) i n t h a l l i u m . As a r u l e , i n n o n t r a n s i t i o n metals T decreases l i n e a r l y c J w i t h p r e s s u r e (see e.g. the review paper by Brandt and Ginzburg (1969)). T h a l l i u m i s an e x c e p t i o n i n t h i s r e g a r d , as T c f i r s t i n -c r e a s e s (up. to about 2 kbars) and then decreases w i t h p r e s s u r e . There e x i s t s a very l a r g e amount of experimental data on t h i s anomaly ( s t a r t i n g w i t h the work of Kan e t . a l . (1948)), and j u s t as many attempts to i n t e r p r e t i t (see e.g. Ignateva e t . a l . (1968)). Corresponding anomalies were found i n the volume change a s s o c i a t e d with the s u p e r c o n d u c t i n g t r a n s i t i o n (Olsen and Rohrer (1957); Cody (1958) ) , as w e l l as i n the p r e s s u r e dependence of the c r i t i -c a l f i e l d ( F i s k e (1957)). K a n . e t . a l . (1961) put forward the hy-p o t h e s i s t h a t the n o n l i n e a r p r e s s u r e dependence of T c may be due to a change i n the topology of the Fermi s u r f a c e under h y d r o s t a t i c p r e s s u r e . A g r e a t d e a l of e x p erimental and t h e o r e t i c a l work has appeared i n the l i t e r a t u r e which t r i e s to e l u c i d a t e the problem al o n g these l i n e s . As- mentioned i n the i n t r o d u c t i o n , i t seems t h a t the anomaly i s now f u l l y e x p l a i n e d . However th e r e e x i s t s no c a l c u l a t i o n of the e l e c t r o n d e n s i t y of s t a t e s N(E„) as m i r r o r e d F i n the n o n l i n e a r p r e s s u r e dependence of T . In the next s e c t i o n we w i l l e x t r a c t the volume dependence- of N(E F) from the combined i m p u r i t y and p r e s s u r e dependence of T . (a) The volume dependence of the d e n s i t y of s t a t e s The B.C.S. e x p r e s s i o n of the superconducting t r a n s i t i o n temperature i s g i v e n by Bardeen e t . a l . (1957)) 1.14<Jrfo» 1 T c = exp{ } (8.21) k B N ( E F ) u where k . i s the Boltzmann c o n s t a n t , Jrf <&:> i s an average phonon energy and u is.ari average m a t r i x element d e s c r i b i n g the e l e c t r o n -e l e c t r o n i n t e r a c t i o n v i a the phonons. T h a l l i u m would be c o n s i d e r e d an i n t e r m e d i a t e - c o u p l e d superconductor, f o r which the -B.C.S. ex^ p r e s s i o n i s o n l y an approximation (see e.g. M c M i l l a n (1968)). We w i l l take t h i s f a c t i n t o account i n our subsequent a n a l y s i s . Le t us take the l o g a r i t h m i c d e r i v a t i v e w i t h r e s p e c t to volume of the B.C.S. e x p r e s s i o n : dlnT dln<to> 1 dlnN(E ) dlnu , = + ( £_ + ) (8.22) or dlnV dlnV N(E„)u dlnV dlnV F dlnT 1.14<#OJ> d l n N ( E F ) dlnu £ = -r + l n ( ).( + ) (8.23) dlnV G k T dlnV dlnV B c L e t us w r i t e N(E p) = N 0 ( E p ) + N g (E p) where N 0 ( E ) i s the smooth c o n t r i b u t i o n to the d e n s i t y of s t a t e s which v a r i e s o n l y s l i g h t l y with volume and N (Ep) i s the s i n g u l a r c o n t r i b u t i o n which v a r i e s s t r o n g l y w i t h volume. Consequently we can w r i t e dlnN(Ep) V dN V dN dN V dN 0 dN = = — ( +— §) - ( + ) (8.24) dlnV N dV N dV dV N „ dV dV We see t h a t the volume dependence of T c s p l i t s i n t o two components: dlnT 1.14<Jrfw> d l n N 0 ( E ) dlnu ( - ) „ = - r + l n ( - ).( E - + ) (8.25) dlnV ^ k T dlnV dlnV B e 98 d l n T c 1.14<Jrfw> V dN ( ) = l n ( ).( -) (8.26) dlnV S k T N0dV B C ( a . l ) The volume dependence of the c o u p l i n g c o n s t a n t A To check the c o n s i s t e n c y of our approach and a l s o because such an e s t i m a t e i s i n t e r e s t i n g i n i t s own r i g h t , we c a l c u l a t e the q u a n t i t y ( d l n T c / d l n V ) 0 - c o r r e s p o n d i n g to the smooth p a r t of the d e n s i t y of s t a t e s and compare i t with the experimental v a l u e of the l i n e a r volume dependence of T . W<s use T G=2.27 (Gschneider (1964)) and T c=2.38°. Dynes (1972) has done thorough work on the m u l t i p l i c a t i v e f a c t o r i n the T c e q u a t i o n . We w i l l use h i s r e s u l t s : 1.14(^0)) 48° ln{ } = l n ( ) = 3.00 (8.27) k„T 2.38° U s u a l l y one uses Mc M i l l a n ' s (1968) e x p r e s s i o n f o r t h i s f a c t o r , i . e . r e p l a c e s 1.14<>lco>/kB by 0^/1.45 where 0 D Is the Debye temper-a t u r e . For low temperatures Van der Hoeven and Keesom (1964) measure 0^=78.6°. T h i s would y i e l d f o r the m u l t i p l i c a t i v e c o n s t a n t a v a l u e of 3,12. We w i l l now t r e a t the smooth component i n Mc M i l l a n ' s (1968) f a s h i o n , i . e . d l n N 0 dlnu d i n ( N u) , + = . (8.28) dlnV dlnV dlnV w i t h A-y*(l+0.62A) N 0u = = F (A) (8.29) 1.04(1+A) A i s the e l e c t r o n - phonon c o u p l i n g c o n s t a n t and y* accounts f o r the e l e c t r o n - e l e c t r o n r e p u l s i v e i n t e r a c t i o n and i s to a good ap-p r o x i m a t i o n volume independent. Thus dlnF(A) AF'(A) dlnA = (8.30) d l n V F(A) dlnV Using Dynes (1972), we choose A=0.78 and y*=0.127. Thus AF 1(A)/F(A)=0.773. In f i r s t approximation (e.g. Olsen e t . a l . (1968)) ( d l n A ) / ( d l n V ) = -2 ( d l n 0 D ) /dln.V) = 2T G = 4.54 A much more r e l i a b l e c a l c u l a t i o n of t h i s term was performed by C a r b o t t e and V a s h i s h t a (1971). They use the experimental phonon spectrum o f t h a l l i u m o b t a i n e d by t u n n e l l i n g experiments, s c a l e i t to volume changes and c a l c u l a t e the change i n A. They f i n d t h a t Z=l+A changes from 1.659 to 1.777 f o r a f r a c t i o n a l volume i n c r e a s e o f 5%. T h i s y i e l d s (dlnA)/(dlnV)=3.04, d i n ( N 0 u)/dlnV=2.35 and c o n s e q u e n t l y ( d l n T c / d l n V ) 0 = 4.78 ' (8.31) We see t h a t , j u s t as f o r o t h e r n o n t r a n s i t i o n metals, t h i s q u a n t i t y i s p o s i t i v e and predominantly determined by the e l e c t r o n phonon i n t e r a c t i o n , and not by the volume dependence of the e l e c t r o n d e n s i t y of s t a t e s . To compare t h i s r e s u l t w i t h experiment we must use the l i n e a r component of dT c/dp o b t a i n e d from h i g h 100 p r e s s u r e experiments, where the i n f l u e n c e due to the s i n g u l a r change i n the topology of the Fermi s u r f a c e i s absent. We.find (Brandt and Ginzburg (1965); Makarov and Baryakhtar (1965)) l i n e a r ( d T /dp) = -1.4 x 1 0 - 1 1 0K(dynes/cm 2)~ 1 (8.32) Usi n g Swenson's (1955) low temperature c o m p r e s s i b i l i t y x =2.02x x l 0 ~ l z(dynes/cm 2 ) ~~1 we o b t a i n { (dlnT /dlnV). } ^ = 2.9 (8.33) The f a c t t h a t our t h e o r e t i c a l r e s u l t i s by about 50% h i g h e r i s most pr o b a b l y caused by an o v e r e s t i m a t i o n of ( d l n X ) / ( d l n V ) . Indeed, i f we compare the t h e o r e t i c a l volume change of the super-c o n d u c t i n g gap o b t a i n e d by Ca r b o t t e and V a s h i s h t a (1971) wi t h t h a t o b t a i n e d by experiment ( G a l k i n e t . a l . (1971)) we f i n d t h a t the l a t t e r i s a p p r e c i a b l y s m a l l e r , i . e . the e l e c t r o n phonon i n t e r -a c t i o n i s " s t i f f e r " than t h a t i m p l i e d by the t h e o r e t i c a l s c a l i n g method. (a. 2) The volume dependence of N (E )_ We w i l l now t u r n to experimental data on the n o n l i n e a r p r e s s u r e and i m p u r i t y dependence of T c to e x t r a c t the v a r i a t i o n w i t h temperature of the volume dependence of N s ( E F ) c o r r e s p o n d i n g t o the disappearance of the s m a l l pockets i n the 4-th zone. The n o n l i n e a r p r e s s u r e dependence o f pure T l o b t a i n e d by Makarov and Baryak h t a r (1965) i s shown i n F i g . (8.3). S i n c e our Knight s h i f t measurements correspond to a volume i n c r e a s e r a t h e r than a p r e s s u r e i n c r e a s e we can, a t t h i s time, o n l y use the zero p r e s s u r e 101 v a l u e of dT„/dp f o r our c a l c u l a t i o n . We f i n d (dT /dp) =2.5x c' c c' ^ p=0 x l 0 _ n ° K ( d y n e s / c m 2 ) - 1 which corresponds to (dlnT /dlnV) =T5.19, and V dN § = (lnl.l4<Jrfo»/k_,T ) 1 . (dlnT /dlnV) = -1.73 (8.34) N 0 dV B c c s I f we take f o r the volume change of the smooth d e n s i t y of s t a t e s the f r e e e l e c t r o n v a l u e , i . e . (dlnN Q)/(dlnV)=2/3, then we have (dlnN ( E j / d l n V ) = -1.06 (8.35) F p=0 As can be seen, t h i s i s indeed a n e g a t i v e volume dependence of the d e n s i t y of s t a t e s of the r i g h t o r d e r of magnitude to e x p l a i n our Knight s h i f t r e s u l t s . I t however has the drawback of being a p o i n t v a l u e - c o r r e s p o n d i n g to the i n i t i a l volume expansion a t v e r y low temperature. We cannot e x t r a c t more i n f o r m a t i o n from pure T l low p r e s s u r e experiments s i n c e an i n c r e a s e i n p r e s s u r e corresponds to a volume decrease. F o r t u n a t e l y t h e r e e x i s t s a 0 g r e a t d e a l of e x p e r i m e n t a l data on T l c o n t a i n i n g i m p u r i t i e s , ( e , g . Ignateva e t . a l . (1968)). They .showed t h a t i m p u r i t i e s o f h i g h e r v a l e n c y i n c r e a s e d the Fermi energy by amounts which c o u l d be l a r g e enough to get away from the c r i t i c a l e n e r g i e s c o r r e s p o n d i n g to the s i n g u l a r d e n s i t y of s t a t e s . Subsequent a p p l i c a t i o n of p r e s s u r e swept the Fermi l e v e l to even h i g h e r v a l u e s and indeed t h e r e was no anomalous p r e s s u r e dependence of T c observed i n these samples. On the o t h e r hand, i m p u r i t i e s o f ' l o w e r v a l e n c y depressed the Fermi l e v e l below the c r i t i c a l v a l u e , such t h a t subsequent a p p l i -c a t i o n of p r e s s u r e e x h i b i t e d the whole n o n l i n e a r behaviour of T ^ c caused by the s i n g u l a r d e n s i t y of s t a t e s . 102 Brandt e t . a l . (1966) i n v e s t i g a t e d the p r e s s u r e dependence of T c f o r pure T l , Tl-Hg(0.4 5 a t . % ) , and Tl-Hg(0.9 a t . % ) . We show t h e i r e x p e r i m e n t a l r e s u l t s i n F i g , (8,3), As can be seen from the e x p e r i m e n t a l r e s u l t s as w e l l as from t h e i r i n t e r p r e t a t i o n , the i m p u r i t y may be c o n s i d e r e d as having the s o l e e f f e c t o f d e p r e s s i n g the Fermi l e v e l below the c r i t i c a l energy. The subsequent a p p l i -c a t i o n of p r e s s u r e c r e a t e s then an image of the volume dependence of N(E F) as E F sweeps from lower to h i g h e r v a l u e s , p a s s i n g through the c r i t i c a l energy E c . I f t h i s i s so, then we can use the same curves (sweeping backwards) to i n f e r the volume de-pendence of N(E F) f o r i n c r e a s i n g volumes of pure Tl.We have done t h i s u s i n g a p o i n t by p o i n t a n a l y s i s of the Tl-0.9 at,%Hg d a t a . Our zero volume e v i d e n t l y corresponds to +8 kbars and we sweep towards lower p r e s s u r e s , u s i n g again Swenson's (1955) low temper-a t u r e c o m p r e s s i b i l i t y to change from a p r e s s u r e to a volume s c a l e and the temperature dependent expansion c o e f f i c i e n t s of Meyerhoff and Smith (1962) to change from the volume to the temperature s c a l e . The r e s u l t s are shown i n Table (8.II) and F i g . (8,4), As one can see the n e g a t i v e c o n t r i b u t i o n of the s i n g u l a r component N (E ) i s s t r o n g e s t around 80°K and g r a d u a l l y t a p e r s o f f as one approaches room temperature. T h i s behaviour does indeed c o r r e l a t e w i t h our Knight s h i f t data as can be seen from F i g . (8,1). However, to e s t a b l i s h a q u a n t i t a t i v e c o r r e l a t i o n we must e s t i m a t e the volume dependence of the o t h e r terms e n t e r i n g the Kni g h t s h i f t . 103 20 30 P, katm FIG. 3. Pressure dependence of the change in the super-conducting transition temperature of thallium and its alloys: I) pure T l ; 11) Tl—Hg (= 0.45 a t . % Hg); III) T l -Hg (= 0.9 a t . % Hg). Curves: a — linear components; b , , b 2 , b 3 — nonlinear com-ponents; dashed curves represent dT/dP for the nonlinear com-ponents. F i g . 8,3 Reproduction from Makarov and Baryakhtar (1965) . 104 TABLE (8.II) A V .10 3 V ( d T c / d p ) s x l O 1 l (dlnT / d l n V ) s dlnN/dlnV °K °K (dynes/cm 2) _ 1 0 .: 0. 09 0.34 0. 49 1. 08 1.40 2. 02 4.04 6.06 8. 08 10.10 12.12 14.14 16.16 22.16 1.5 12.5 22.5 27. 5 37.5 42 .5 49.0 78.0 105 131 156 181 205 229 300 2.5 2.8 3.1 2.5 1.4 0.8 0.4 0.1 0 0 -5.18 -1.06 -5.21 -1.07 -5.29 -1.09 -5.33 -1.11 -5.51 -1.17 -5.61 -1,20 -5.80 -1.26 -6.43 -1.47 -5.18 -1.06 -2.90 -0 .30 -1.66 0.12 -0.83 0.39 -0.21 0.60 0 0.67 0 0.67 dlnN/dlnV o 106 (b) The volume dependence of f(a) We w i l l assume 1 f ( a ) = f { N ( E F ) } = * (8,36) 1-CN(E F) where C i s a c o n s t a n t o f p r o p o r t i o n a l i t y . As can be seen from t h i s e x p r e s s i o n , a n o n l i n e a r decrease of N(Ep) w i t h volume w i l l produce a n o n l i n e a r decrease of f(a) thus enhancing the anomalous Knight s h i f t b e h aviour. The volume dependence of f ( a ) has been found of a p p r e c i a b l e magnitude i n other metals (e.g, El-Hanany and Zamir (1972); Weaver and Narath (1970)) and t h e r e f o r e we can not n e g l e c t i t . As f a r as we know, th e r e are no exp e r i m e n t a l or s p e c i f i c t h e o r e t i c a l data on t h a l l i u m . To o b t a i n an estimate we use one o f the r e c e n t t h e o r e t i c a l c a l c u l a t i o n s o f f ( a ) as a f u n c t i o n o f the d e n s i t y parameter r g -(Dupree and G e l d a r t (1971) ) . For T l , r g=2.48 which y i e l d s f(a)=1.39. We can estimate d l n f ( a ) / d i n V u s i n g a s i m i l a r approach as b e f o r e . We w r i t e d l n f Nf'(N) dlnN CN dlnN dlnN - r = = . = 0 .39 (8.37) dlnV f(N) dlnV 1-CN dlnV dlnV As one can see t h i s f a c t o r enhances dlnN/dlnV by 39% (c) The volume dependence of Pp There e x i s t no ex p e r i m e n t a l or t h e o r e t i c a l i n v e s t i g a t i o n s i n the l i t e r a t u r e which c o u l d e l u c i d a t e the volume dependence of P i n t h a l l i u m . We must t h e r e f o r e d e r i v e i t from our exp e r i m e n t a l F * d a t a . We c o l l e c t a l l terms. (8,38) (dlnK)/dT) =&{x(x-l) (4 .7 *10~ 5T+7 . 94 ><10~2) tx+1.39dlnN (Ej/dlnV) 107 where we used again the n o t a t i o n dlnP F/dlnV=x. To determine x we w i l l use a temperature f o r which our ex p e r i m e n t a l r e s u l t s are most r e l i a b l e . As can be seen from the data i n F i g . (8.1) a good c h o i c e i s T=80°K. For we can t r u s t K from above, s i n c e i t i s determined by our low temperature measure-ment and the thermal expansion c o e f f i c i e n t , and we can t r u s t i t below s i n c e we are c l o s e to our l i q u i d n i t r o g e n e x p e r i m e n t a l p o i n t . We f i n d i n t h i s r e g i o n (dlnK/dT) e x p=-3 , 75x10-" (°K) " 1 . At 80°K, 3 = 7 . 3 5 x l 0 ~ 5 ( ° K ) _ 1 and from our p r e v i o u s a n a l y s i s d l n N ( E F ) / /dlnV=-1.47. S u b s t i t u t i n g these v a l u e s i n the above e x p r e s s i o n we o b t a i n the f o l l o w i n g e q u a t i o n i n x: 0.61x 2 + 6.74x + 22.5 = 0 (8.39) I t s s o l u t i o n s are imaginary. The l i m i t i n g v alue which would y i e l d a r e a l s o l u t i o n i s (.dlnK/dT) =-3.40xl0 - 4, c e r t a i n l y not f a r away P from our ex p e r i m e n t a l p o i n t . I f we use t h i s v alue then xEdlnPp/ /dlnV=-5.6 which i s probably to low. Matzkanin and S c o t t (1966) es t i m a t e t h i s q u a n t i t y t o l i e between -1.9 and -6.1 i n l e a d and Kushida and Murphy (1971) f i n d -2.1 i n aluminum. The imaginary s o l u t i o n o f our equ a t i o n and the l a r g e n e g a t i v e v a l u e which we o b t a i n f o r dlnP„/dlnV w i l l be d i s c u s s e d i n the subsequent f i n a l a n a l y s i s , 108 8.3.3 The t o t a l T -dependence of K Our c a l c u l a t i o n s have j u s t l e a d to the f o l l o w i n g e x p r e s s i o n (dlnK/dT) = (1.39 dInN ( E p ) / d l n V + 1 . 74 x i o " 3 T - 2 . 6 6 )3 (8.40) P We p l o t t h i s r e s u l t as a f u n c t i o n of temperature i n F i g . (8.5). As can be seen from the graph, the e f f e c t of the s i n g u l a r component o f the d e n s i t y of s t a t e s produces the l a r g e d i p c e n t e r e d at 80°K. In F i g . (8.6) we show the temperature dependence of the Knight s h i f t as d e r i v e d from F i g . (8.5) by i n t e g r a t i o n . We use K(1.2 0)=163.3x10"* as the i n i t i a l i n t e g r a t i n g c o n s t a n t and no ot h e r a d j u s t a b l e parameters. Our experimental p o i n t s as w e l l as the v a l u e o b t a i n e d by Moulson and Seymour are shown on the same f i g u r e . As can be seen, agreement with experiment i s r a t h e r good, and the n o n l i n e a r behaviour due to Ng'(Ep) i s q u i t e apparent. I n t e r e s t i n g l y enough our theory seems i n q u i t e good agreement w i t h experiment even c l o s e to the m e l t i n g p o i n t . T h i s might be c o n s i d e r e d f o r t i t i o u s s i n c e t h a l l i u m undergoes a phase t r a n s i t i o n from hexagonal to body c e n t r e d c u b i c a t 503°K. On the o t h e r hand Moulson and Seymour (1967) r e p o r t no change on Knight s h i f t upon m e l t i n g . T h i s , as w e l l as the f a c t t h a t the Fermi energy at503°K i s f a r removed from the c r i t i c a l energy at which the pockets disappeared, p o i n t s t o a f r e e e l e c t r o n l i k e behaviour a t these temperatures which i n t u r n i m p l i e s l i t t l e change i n Knight s h i f t a t the phase t r a n s i t i o n . L e t us now d i s c u s s the r a t h e r l a r g e n e g a t i v e value which we used f o r the term xEdlnPp/dlnV. At t h i s p o i n t we must take a more c a r e f u l l o o k a t the temperature dependence of the d i f f e r e n t con-I l l t r i b u t i o n s to (dlnK/dT) . They are shown i n Table (8 .III) TABLE ( 8 . I l l ) (dlnK/dT) V {x(x-l) (4.7>=10~5T+7.94 X10~ 2}3 d l n P F / d l n V x. 3 dlnx /dlnV 1.39 ( d l n N ( E F ) / d l n V ) 3 3 m u l t i p l i e s a l l terms and i s t h e r e f o r e not c r i t i c a l . The term i n T (which i s due to the temperature dependence of the cmpr e s s i -b i l i t y ) i s s m a l l enough to be n e g l e c t e d i n a q u a l i t a t i v e d i s -cussion. Thus, the n e g a t i v e term x has two o p p o s i t e e f f e c t s on the Knight s h i f t v a r i a t i o n . I t enhances the p o s i t i v e c o n t r i b u t i o n o f (d l n K / d l n T ) y as x ( x - l ) and r e p r e s e n t s the n e g a t i v e c o n t r i b u t i o n d l n P F / d l n V =x. T h i s dependence i s g i v e n by In F i g . (8.7) we p l o t t h i s f u n c t i o n versus x. We can now under-stand our low d e r i v e d v a l u e o f x. The drop i n K d e r i v e d from the d e n s i t y of s t a t e s alone was not s u f f i c i e n t to d e s c r i b e the pro-nounced decrease w i t h temperature of our experimental Knight s h i f t s . I t was t h e r e f o r e the n e g a t i v e v a l u e o f the f u n c t i o n g(x) which had to account f o r the d i f f e r e n c e . U n f o r t u n a t e l y , as can be seen from F i g . (8,7), g(x) has a broad minimum a t x--6. T h i s ex-p l a i n s our i n i t i a l imaginary value o f x and a l s o i m p l i e s t h a t our g(x) E' 7.94 ; x 10 2 x ( x - l ) + x (8.41) 'g(x) -3 F i g . 8.7 The f u n c t i o n g ( x ) , as d e f i n e d i n the t e x t . d e r i v e d v a l u e of x i s u n r e l i a b l e . As a matter of f a c t any x l y i n g i n the i n t e r v a l ~(-9,-3) would f i t our experimental data. Most p r o b a b l y x~-3 or h i g h e r ; our data would then imply t h a t the anoma-l y i n the d e n s i t y of s t a t e s i s even s t r o n g e r than t h a t e x t r a c t e d from the p r e s s u r e dependence of the superconducting temperature. F i n a l l y , i t i s i n t e r e s t i n g t o compare our r e s u l t w i t h the t h e o r e t i c a l a n a l y s i s of Holtham (1973). The s i n g u l a r behaviour of the d e n s i t y of s t a t e s caused by the f o r m a t i o n or disappearance of a pocket i n the Fermi s u r f a c e has been t r e a t e d by L i f s c h i t z (1960). We have seen t h a t a volume i n c r e a s e i n t h a l l i u m produces a drop i n the Fermi l e v e l , which i n t u r n causes the disappearance of a s m a l l pocket i n the 4-th zone. L i f s c h i t z p r e d i c t s the f o l l o w -i n g behaviour f o r the s i n g u l a r p a r t of the d e n s i t y of s t a t e s i n t h i s case, 0 f o r E < E c /__ (8.4 2) a/E-E c f o r E > E c where a, i s a c o n s t a n t connected w i t h the band e f f e c t i v e masses and E c i s the c r i t i c a l energy a t which the pocket d i s a p p e a r s . C l e a r l y , then dN s dN dE a dE-= = — — - (8.43) dV dE dV 2/E-E dV c The mathematical s i n g u l a r i t y a t E=E i s of course smoothed out i n NS(E) = any p h y s i c a l experiment, but i t w i l l appear as a pronounced m i n i -mum i n any d e r i v a t i v e of the f u n c t i o n Ng.(E ) . From our data as w e l l as the superconducting data, we can see t h a t t h i s minimum i s found a t T~80°K which corresponds to a volume i n c r e a s e o f A V / V = =0.4% which i n t u r n corresponds to a n e g a t i v e p r e s s u r e o f -2 kbars. I f we compare t h i s r e s u l t w i t h the t h e o r e t i c a l p r e s s u r e dependence shown i n F i g . (4.b) of Holtham's paper we f i n d q u a l i -t a t i v e agreement, but a weaker dependence of the Fermi l e v e l w i t h volume. T h i s i s understandable, s i n c e i n the case o f the 4-th zone pockets he used a f r e e e l e c t r o n e s timate o f t h i s dependence. The p s e u d o - p o t e n t i a l - t y p e c a l c u l a t i o n s , c a r r i e d out i n the same paper g i v e a much s m a l l e r volume dependence as can, f o r example, be seen i n F i g . (4.a) (Holtham (1973)), which r e f e r s t o the "dumb-bell" r e g i o n o f the t h a l l i u m Fermi s u r f a c e . The a n i s o t r o p y of the Knight s h i f t presumably o r i g i n a t e s mainly i n the a n i s o t r o p y o f the charge d i s t r i b u t i o n of the p-con-d u c t i o n e l e c t r o n s . I t s n e g a t i v e s i g n i m p l i e s t h a t the e l e c t r o n wavefunctions have l e s s p„ than p or p c h a r a c t e r . A d e t a i l e d z x r y c a l c u l a t i o n a l o n g the l i n e s o f Kasowski (1969) may e l u c i d a t e the exp e r i m e n t a l temperature dependence, 8.4 C o n c l u s i o n s The i s o t r o p i c Knight s h i f t i n T l 2 0 5 was found t o decrease w i t h i n c r e a s i n g temperature. The e f f e c t i s most pronounced be-tween 50° and 150°K, and c o r r e l a t e s w i t h the p r e s s u r e dependence of the supe r c o n d u c t i n g t r a n s i t i o n temperature a t low p r e s s u r e s . From the su p e r c o n d u c t i n g data on pure T l and Tl-Hg a l l o y s we have e x t r a c t e d the volume dependence of the d e n s i t y o f s t a t e s , and used i t t o d e r i v e the temperature v a r i a t i o n of the Knight s h i f t . The r e s u l t s agree w e l l w i t h the observed d a t a . CHAPTER IX INDIRECT COUPLING CONSTANTS IN A THALLIUM SINGLE CRYSTAL 9.1 I n t r o d u c t i o n T h a l l i u m was the metal t h a t prompted Bloembergen and Rowland to p o s t u l a t e the p s e u d o d i p o l a r i n t e r a c t i o n i n 1955. N a t u r a l t h a l l i u m c o n t a i n s two magnetic i s o t o p e s (70.5% T l 2 0 5 , 29 . 5% T l 2 0 3 ), and e x h i b i t s s t r o n g pseudo-exchange i n t e r a c t i o n s . I t i s known (Van V l e c k (1948)) t h a t exchange i n t e r a c t i o n s between l i k e s p i n s do not c o n t r i b u t e t o the second moment, but do c o n t r i b u t e to the f o u r t h moment, thus narrowing the l i n e (exchange narrowing). On the o t h e r hand, exchange i n t e r a c t i o n s between u n l i k e s p i n s con-t r i b u t e t o both the second and f o u r t h moment, broadening the l i n e . Bloembergen and Rowland s t u d i e d the second moment i n t h a l l i u m as a f u n c t i o n of i s o t o p i c abundance and found t h a t even i n i s o -t o p i c a l l y pure samples the second moment exceeded the d i p o l a r v a l u e . The l a t t e r can be c a l c u l a t e d e x a c t l y (Van V l e c k (1948)). The broadening was e x p l a i n e d as due to the p s e u d o d i p o l a r i n t e r -a c t i o n and v a l u e s f o r both the pseudo-exchange and p s e u d o d i p o l a r c o u p l i n g c o n s t a n t s were determined. I t was a l s o p r e d i c t e d t h a t 2 0 5 ?03 a t low f i e l d s the two resonance l i n e s o f the T l and T l i s o -topes should c o a l e s c e i n t o one narrow l i n e . T h i s comes about be-cause of the s m a l l d i f f e r e n c e between the gyromagnetic r a t i o s of the two i s o t o p e s . A t low f i e l d s the d i f f e r e n c e i n the Zeeman i n t e r a c t i o n s as compared to the i n t e r a c t i o n s caused by l o c a l f i e l d s becomes s m a l l enough to e f f e c t i v e l y make a l l s p i n s behave a l i k e i n a resonance experiment. Karimov and Shchegolev (1962) 116 used t h i s e f f e c t t o determine again the c o u p l i n g c o n s t a n t s i n t h a l l i u m . They f i n d q u a l i t a t i v e agreement with Bloembergen and Rowland, i . e . s t r o n g s c a l a r i n t e r a c t i o n s versus weak t e n s o r i n t e r -a c t i o n n s , but t h e i r exchange c o u p l i n g constant i s twice as l a r g e as t h a t o b t a i n e d by Bloembergen and Rowland (19 55) . Both p r e v i o u s i n v e s t i g a t i o n s were performed on powder samples. T h a l l i u m i s a very i n t e r e s t i n g candidate f o r s i n g l e c r y s t a l work s i n c e i t s 12 n e a r e s t neighbours i n the almost i d e a l h.c.p. l a t t i c e , l i e i n two d i f f e r e n t s h e l l s of n e a r l y equal s i z e (1% d i f f e r e n c e i n the r a d i u s ) but of completely d i f f e r e n t symmetry. A l l o t h e r s h e l l s are s u f f i c i e n t l y f a r away to be c o n s i d e r e d as c o n t r i b u t i n g weakly to the c o u p l i n g c o n s t a n t s . T h i s o f f e r s a s p e c i a l o p p o r t u n i t y to check the a n i s o t r o p y of both the pseudo-exchange and p s e u d o d i p o l a r i n t e r a c t i o n s , The r a d i a l dependence of the c o u p l i n g c o n s t a n t s b e l o n g i n g to the n e a r e s t 12 neighbours cannot make any a p p r e c i a b l e d i f f e r e n c e i n t h e i r v a l u e s s i n c e the l e n g t h of t h e i r p o s i t i o n v e c t o r s i s n e a r l y the same. Thus i f t h e r e i s a d i f f e r e n c e between the two s h e l l s i t must be due to the f a c t t h a t they have d i f f e r e n t symmetries i n k-space. Of c o u r s e , the work i n p o l y c r y s t a l l i n e samples cannot d e t e c t such a d i f f e r e n c e and t h e r e f o r e both of the p r e v i o u s i n v e s t i g a t i o n s con-s i d e r e d the f i r s t s h e l l as c o n s i s t i n g of 12 n e a r e s t neighbours and n e g l e c t e d a l l o t h e r s h e l l s i n t h e i r a n a l y s i s . I t i s important to r e a l i z e t h a t when we speak of the f i r s t s h e l l i n t h i s work, we r e f e r o n l y to the f i r s t s i x neighbours, w h i l e s h e l l two c o n t a i n s the o t h e r s i x neighbours ( l y i n g i n the b a s a l p l a n e ) . 117 " 9.2 E x p e r i m e n t a l R e s u l t s The e x p e r i m e n t a l d e t a i l s are d e s c r i b e d i n the p r e v i o u s c h a p t e r s . To o b t a i n an a c c u r a t e angular dependence of the l i n e -width the l i n e s were an a l y z e d as d e s c r i b e d i n chapter 5. In F i g . (9.1) we show a resonance l i n e i n T l 2 0 5 a t 1.2°K, t o g e t h e r w i t h i t s L o r e n t z i a n t h e o r e t i c a l f i t . Most of the l i n e s used i n d e t e r m i -n i n g the d a t a were o f b e t t e r q u a l i t y , i . e . of h i g h e r s i g n a l - t o -n o i s e r a t i o . The L o r e n t z i a n f i t s were e x c e l l e n t i n a l l cases. The a n i s o t r o p y of the second moment i s s m a l l because of the l a r g e ex-change c o n t r i b u t i o n o f u n l i k e s p i n s which does not depend on the 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 to the c r y s t a l . F u r t h e r , second moments are hard t o measure and the t a i l of the T l 2 0 5 l i n e over-l a p s w i t h the t a i l o f the T l 2 0 3 l i n e . Because of these reasons i t would be v e r y d i f f i c u l t to o b t a i n an a c c u r a t e angular dependence of the second moment. On the o t h e r hand, as w i l l be shown i n the subsequent s e c t i o n , t h e r e are reasons to b e l i e v e t h a t the Anderson-Weiss model o f exchange narrowing a p p l i e s to t h a l l i u m and t h a t the a n g u l a r dependence of the second moment i s s i m i l a r to t h a t o f the l i n e w i d t h . The l a t t e r can be measured a c c u r a t e l y ( w i t h i n 3%) and c o n s e q u e n t l y i t s angular v a r i a t i o n i s w e l l d e t e r -mined. We show the r e s u l t s i n F i g , (9.2). Many of the p o i n t s are o b t a i n e d by r e f l e c t i o n , thus r e i n f o r c i n g the c o n f i d e n c e i n the e x p e r i m e n t a l d a t a . The e r r o r bars r e p r e s e n t root-mean-square d e v i a t i o n s . The second moments are of the order of 700±200 (kHz) 2. S i n c e the second moment i s p r o p o r t i o n a l i n f i r s t approximation to the square of the c o u p l i n g c o n s t a n t s , t h i s accuracy i s good enough f o r our r e d u c t i o n o f the data, as w i l l become c l e a r from the subsequent d i s c u s s i o n . F i g -.'.A-9 .1 ' Separation of Modes., for a typical Thallium Line. F i g . 9.2 The o r i e n t a t i o n - d e p e n d e n c e of the l i n e w i d t h i n T l 2 0 5 120 9.3 D i s c u s s i o n We must f i r s t d e r i v e the angular dependence f o r the second moment i n the case of a hexagonal c l o s e d packed l a t t i c e con-t a i n i n g two i s o t o p e s . There are two avenues a v a i l a b l e . One c o u l d use the approach taken by O ' R e i l l y and Tsang (1962) as we d i d i n the case of l e a d , or use "brute f o r c e " c r y s t a l symmetry arguments. Si n c e the h a r d e s t term i n the expansion has been analyzed i n my M.Sc, - t h e s i s we f e l t the second a l t e r n a t i v e to be the e a s i e r path; i t a l s o has the advantage of being more " p h y s i c a l " . The g e n e r a l e x p r e s s i o n f o r the second moment i n the case of two i s o t o p e s i s g i v e n by (see e.g. Bloembergen and Rowland (1955)). M 2 = (3/4) 2 I (1+1) (1-f) I (b . +Y2 h - 2 r r 3 ) 2 ( l - 3 c o s 2 0, ) 2 + k x k k + (1/3) ]/i~2 I (1+1) f £ {J. - (b. +y2 #2 r - 3 ) (3cos 2 0 -1) }2 k K * k k (9.1) where the nucleus undergoing resonance i s c o n s i d e r e d i n the c e n t r e of the c o o r d i n a t e system, I i s the n u c l e a r s p i n , y i s the gyro-magnetic r a t i o , r ^ i s the p o s i t i o n v e c t o r of nucleus k, 0^ i s the angle made by r k w i t h the magnetic f i e l d , i s the pseudoex-change c o u p l i n g c o n s t a n t , b^ i s the p s e u d o d i p o l a r c o u p l i n g c o n s t a n t , and f i s the r e l a t i v e abundance of the nucleus not undergoing resonance. I t i s assumed t h a t the f i e l d i s h i g h enough to s e p a r a t e the two resonance l i n e s , but the d i f f e r e n c e i n gyro-magnetic r a t i o s i s s m a l l enough not to i n f l u e n c e a p p r e c i a b l y the i n t e r a c t i o n s between the n u c l e i . Both of these assumptions are v e r y w e l l s a t i s f i e d i n our case. 121 We denote # V K I + 1 ) E C J kr3/y 2}* 2 E A k b k r k / y 2 ^ 2 = B k 3cos 2 0 k-l=2P 2 (cos0 k) = 2P 2 (k) ( V 1 ) 2 r k 6 ' 3k ( B k + D A k r k 6 = a k <9'2> and o b t a i n M = {3-(5/3)f}C Z B {P2 (k)} 2 - ( 4 / 3 ) f C Z a,P 2 (k) + k k k K + (1/3)fC Z A 2 r ~ 6 (9.3) k K K . To change from the c o o r d i n a t e system c o n n e c t i n g n u c l e a r p o s i t i o n v e c t o r s t o the magnetic f i e l d , t o a c o o r d i n a t e system o f the l a t t i c e we use the a d d i t i o n theorem of s p h e r i c a l harmonics. T h i s i s a standar d procedure i n group t h e o r e t i c a l a n a l y s e s ; the de-t a i l s , as a p p l i e d t o second moments, can be found i n my M . S c -t h e s i s . 2 c P, (cose, ) = (4U/5) Z Y m*(8,(|))y m(0^,(i) ) (9.4) k . m=-2 K K where (Q,^) are now the c o o r d i n a t e s o f the magnetic f i e l d w i t h c c r e s p e c t t o the l a t t i c e , ( ^ J ^ ) a r e the c o o r d i n a t e s of r k i n the l a t t i c e , and OZ i s p a r a l l e l to the p r i n c i p a l a x i s of the hexago-n a l c r y s t a l . We choose OX such as to have r e f l e x i o n symmetry 122 w i t h r e s p e c t t o the plane Z OX ; t h i s i s always p o s s i b l e f o r a h.c.p. l a t t i c e . L e t us a n a l y z e the second term i n e x p r e s s i o n (9.3): 2 E a k P 2 (k) = E a, P 2 (cos6 k) = (4TT/5) E E a, Y™* (6 , <j>) Y m ( ef , ^ k K k k X k m=-2 k 2 k k 2 (2-[m|)! c E E a k exp(im<j>£-im<j>)P2 (cos6)P, ( c o s G J (9. 5 ) k m=-2 (2+|m|)I where we have used the s p e c i a l p r o p e r t i e s of the s p h e r i c a l har-monics, and P™(x) are the a s s o c i a t e Legendre p o l y n o m i a l s . Denote c (2-|m|)i g. . (ef,9) = a . P' m i ( c o s e f ) p ' (cosG) (9.6) tml k k ( 2 + | m | ) ; . k 2 Then 2 c c E a, P 2 (k) = E E g m I (8 ,6) exp{im(<|>v-<|>)} = (9.7) k k m=-2 i m l k E 2g 2 (cos2<J>kcos2({)+sin2(|), s i n 2cj)) + 2g1(cos(j) coscf) + sin(f> sincj))+g } k •K k k -A l l terms i n v o l v i n g sin<J>^ are zero f o r each of the s h e l l s , c c because, f o r <(> =0/ir they v a n i s h , and f o r t h e r e e x i s t s i n the c c k expansion f o r each k-term an o t h e r term k' having ^ k ^ k ' a n d c c ' c <j)k=-(J) . A l l terms i n v o l v i n g coscj) a l s o v a n i s h due to the t h r e e -f o l d symmetry of the l a t t i c e : t h e r e e x i s t s i n the k-expansion, c c c f o r each k-term, o t h e r 2 terms k 1 and k", having ^ k i = ^ k n = ^ k a n d <J) (^=(|)^+1200 ; <J>^ „=(j)^ +24 0 0 . Thus, i n each s h e l l , t h e r e w i l l e x i s t 123 c c c sums of the form c o n s t . {coscj? +cos (<L +120° ) +cos (c}>v+ 2 4 0° ) } =0. The e x c e p t i o n s are those s h e l l s f o r which 0 =0,TT; but then the g j I m i c o n t r i b u t i o n i s zero because P 2 m ' ( 0 ) = P 2 m | ( - 1 ) = 0 . In the same way c c one can show t h a t the terms i n v o l v i n g cos2(J) and sin2<j> v a n i s h . Thus, one i s l e f t w i t h (9.8) E a v P 2 (k)= E g 0 (9?, 9) = (1/4) E a, (3cos 2 9 C-1) (3cos 2 9-1) k k k k k k The sum E B, { P 2 ( k ) } 2 i n v o l v e s much more t e d i o u s c a l c u l a t i o n s ; k * t h e r e are now 25 terms of products of f o u r s p h e r i c a l harmonics. However, the symmetry c o n s i d e r a t i o n s are s i m i l a r to those used above and t h e r e f o r e we j u s t quote the r e s u l t . The d e t a i l e d c a l c u -l a t i o n s can be found i n my M.Sc. - t h e s i s , and have been checked by computer i n a few a r b i t r a r y f i e l d d i r e c t i o n s . E e{p, (k) }2 = (1/4) E 3 { (315/8) z 2-45z +9) s i n " 9 + k k k k k k + (-45z 2+54z k-12)sin 2 0+9z 2-12z k+4} (9.9) where z k = s i n 2 0 k (9.10) (9.11) C o l l e c t i n g a l l terms we o b t a i n M 2 = C E {( (1/4) (3-5f/3) R ('3. 9375z 2-45z +9) ) s i n " 0 + + ( ( 1 / 4 ) ( 3 - 5 f / 3 ) 3 k ( - 4 5 z k + 5 4 z k - 1 2 ) + f a k ( 2 - 3 z k ) ) s i n 2 0 + + (1/4) (3-5f/3) 3 k ( 9 z k - 1 2 z k + 4 ) - ( 2 / 3 ) f a k ( 2 - 3 z k ) + ( l / 3 ) f A k r k 6 } 124 We have quoted t h i s t e d i o u s e x p r e s s i o n s i n c e i t i s completely g e n e r a l and a p p l i e s to any hexagonal l a t t i c e c o n t a i n i n g two (or one) i s o t o p e s . As can be seen, f o r each i n d i v i d u a l s h e l l , as w e l l as f o r the whole l a t t i c e , the second moment i s of the form M 2 = C i S i n 1 * G + C 2 s i n 2 9+C 3 (9.12) where 9 i s the angle made by the d.c. magnetic f i e l d w i t h the c-hexagonal a x i s . A g e n e r a l group t h e o r e t i c a l argument ( O ' R e i l l y and Tsang (1962)) p r e d i c t s t h a t the second moment i n a hexagonal l a t t i c e s h ould depend on th r e e c o n s t a n t s . As can be seen, our de-t a i l e d e x p r e s s i o n s a t i s f i e s t h i s c o n d i t i o n , and i m p l i e s t h a t a second moment measurement cannot determine more than t h r e e p h y s i -c a l parameters. At t h i s p o i n t i t i s u s e f u l to check i f our l i n e -width o r i e n t a t i o n can be expressed i n the form (9.12), i . e . h a l f width a t h a l f power g i v e n by W = a s i n " 9 + b s i n 2 9 + c (9.13) The smooth l i n e i n F i g . (9.2) r e p r e s e n t s W = J ^ ^ s i n 1 * 6- 1 6 , 5 s i n 2 9 +27.2 (9.14) As can be seen, the f i t i s q u i t e good. Anderson and Weiss (1953) have worked out the l i n e s h a p e i n the case of s t r o n g exchange i n t e r a c t i o n s . They f i n d L o r e n t z i a n l i n e s w i t h second moments p r o p o r t i o n a l t o the l i n e w i d t h . They w r i t e W=M2/we where we i s an a d j u s t a b l e parameter o f the order o f the exchange c o u p l i n g c o n s t a n t . T h i s model i s used e x t e n s i v e l y i n resonance work and we l i s t below the reasons which prompted us to 125 apply i t i n our case (1) The l i n e s are Lorentzian far out into the wings. (2) The exchange interactions are strong compared to the tensor i n t e r a c t i o n (see Bloembergen and Rowland (1955) and Karimov and Shchegolev (1962)) (3) The linewidth obeys the angular dependence of the second moment ('4) Bloembergen and Rowland have used the model successfully. As we have shown i n chapter 6, i n the case of lead, Lorentzian l i n e s are not a guarantee that the Anderson-Weiss model w i l l work. However lead has an extra p-electron and therefore, the pseudo-dipolar i n t e r a c t i o n (as compared to the exchange interaction) i s appreaciably l a r g e r than i n thallium. Further, the linewidth i n lead had the wrong angular dependence. The Anderson-Weiss model was derived under the assumption of l i k e spins, and as we w i l l show l a t e r t h i s may be the most serious objection to i t s use here. However, we w i l l accept i t for the time being and proceed under the assumption that the second moment i s proportional to the l i n e -width and has therefore the same angular dependence. This i s also the approach which was taken by Sharma e t . a l . (1969) i n the case of i s o t o p i c a l l y pure t i n . Using f=29.5% we obtain, from Eqn. (9.11), a f t e r some alge-bra, f o r each s h e l l i , 126 where n. i s the number of n u c l e i i n s h e l l i and we g i v e f o r com-1 p l e t e n e s s the l a t t i c e sums: C k ( l ) = (24.69z£-28.22z +5,64)r£ 6 C R(2) = (-28.22z£+33.86z k-7.52)r k 6 (9.16) C k ( 3 ) = (-0.885z k+0.59)r~ 6 C R ( 4 ) = (5.64z^-7.52z k+2.51)r~ 6 C k ( 5 ) = (0.59z -0.393)r~ 6 ^ k K C k ( 6 ) = 0.0983r k 6 The equal s i z e of the two major s h e l l s i m p l i e s equal v a l u e s of the r a d i a l p a r t of AY and A 2 . The same statement i s t r u e f o r B x and B 2 . We s t a r t w i t h the assumption t h a t t h i s , i n t u r n , im-p l i e s equal c o u p l i n g c o n s t a n t s f o r the two s h e l l s , i . e . A X=A 2 and 6 ^ 6 2 , T h i s i s the assumption t h a t has been made i n the p a s t f o r a l l metals i n both t h e o r e t i c a l and e x p e r i m e n t a l i n v e s t i g a t i o n s . F u r t h e r , s i n c e a l l o t h e r s h e l l s make o n l y a s m a l l c o n t r i b u t i o n , i t i s a good approximation to take A^=AX and B^=BX f o r a l l i . We have c a l c u l a t e d the l a t t i c e sums f o r the f i r s t 8 s h e l l s which comprise almost the whole c o n t r i b u t i o n to , We o b t a i n M 2 = 10'* 4C{25.15(B 1+1) 2 s i n " 0+(-3O .13 (Bi +1) 2 +0 . 64AX (Bj+DJsin 2 6+ + 49.36 (Bj+1) 2-0.39A! (Bx+1)+-8.33A2} (9.17) As can be seen, the c o n t r i b u t i o n due to the c r o s s terms i s v e r y s m a l l , as i t should be f o r a l a r g e number of s h e l l s , s i n c e 12 7 ( 3 c o s 2 6 - l ) averages out to zero. T h i s however does not imply t h a t the c r o s s terms f o r the i n d i v i d u a l s h e l l s are s m a l l , and t h i s i s important t o r e a l i z e . To check the angular dependence of the ex-p r e s s i o n (9.17) we w i l l s t a r t by u s i n g the s c a l a r - t o - t e n s o r r a t i o s found i n the two p r e v i o u s i n v e s t i g a t i o n s , Bloembergen and Rowland (BR) f i n d h - 1 ! ^ | =17 . 5 kHz and h" 1 | bj +y2 yiz r ~ 3 | =5 . 5 kHz. Karimov and Shchegolev (KS) f i n d h - 1|J 1|=37.5 kHz and h _ 1 | bx +y* tf2 r ^ 3 | = =2.7 kHz. Hence we o b t a i n two p o s s i b l e , but very d i f f e r e n t r a t i o s : R = 3.18 R = 13.19 (9.18) BR KS Using these r a t i o s we f i n d the f o l l o w i n g angular dependence of the second moment ( f o r comparison w i t h w, we normalize a t 6=0) M 2 ( B R ) a 5 . 1 7 s i n 4 6-5 . 7 7 s i n 2 6 + 27 .2 (9.19) M 2 (KS)<x0.4 6sin 1 ,6-0.4 0 s i n 2 6 + 2 7.2 T h i s dependence i s to be compared to t h a t o f the l i n e w i d t h (see (9.14)) W = 14.4sin'*0-16,5sin 28+27 .2 As can be seen immediately, the d i s c r e p a n c y i s l a r g e . The main d i f f e r e n c e l i e s i n a l a r g e r a n i s o t r o p y of W than t h a t o f M 2. T h i s comes about through the l a r g e exchange i n t e r a c t i o n and, as can be expected, produces a much l a r g e r d i s c r e p a n c y when we use R ^ than when we use R„ . BR As f a r as we can see, there e x i s t t h r e e p o s s i b l e e x p l a -n a t i o n s f o r t h i s d i f f e r e n c e . (1) The exchange i s s m a l l e r than t h a t p r e d i c t e d by the two 128 p r e v i o u s i n v e s t i g a t i o n s , (2) The Anderson-Weiss model i s i n a p p l i c a b l e , (3) The c o u p l i n g c o n s t a n t s of the f i r s t two s h e l l s are d i f f e r e n t . We w i l l f o l l o w each of these avenues i n t u r n . To o b t a i n even a h a l f r e a s o n a b l e f i t by r e d u c i n g the exchange c o u p l i n g c o n s t a n t one would have t o make i t of the order of the p s e u d o d i p o l a r c o n s t a n t . There c e r t a i n l y e x i s t s enough evidence f o r l a r g e ex-change i n t h a l l i u m and we t h e r e f o r e d i s c a r d t h i s p o s s i b i l i t y . L e t us now i n v e s t i g a t e the Anderson-Weiss model as a p p l i e d to our case. T h i s model was developed f o r an i s o t o p i c a l l y pure sample. The p h y s i c s i n i t c o n t a i n s the i d e a of exchange narrowing of the t e n s o r i n t e r a c t i o n . One o b t a i n s L o r e n t z i a n l i n e s h a p e s w i t h a l i n e w i d t h which i s p r o p o r t i o n a l to the second moment (which i s due t o the t e n s o r i n t e r a c t i o n alone) d i v i d e d by the exchange frequency, W=M2/toe, In our case, due to the u n l i k e s p i n s , t h e r e e x i s t s a s c a l a r c o n t r i b u t i o n to the second moment which i s not exchange narrowed. As i s w e l l known (Van Vl e c k (1948)), the ex-change due to u n l i k e s p i n s c o n t r i b u t e s t o both the second and the f o u r t h moment. T h e r e f o r e , t h i s i n t e r a c t i o n w i l l i n c r e a s e the l i n e -w idth. Hence, i n the case o f u n l i k e s p i n s , t h e r e w i l l e x i s t two c o n t r i b u t i o n s to the l i n e w i d t h : the exchange-narrowed t e n s o r con-t r i b u t i o n , and a s c a l a r c o n t r i b u t i o n from the u n l i k e s p i n s , which i s not exchange-narrowed. Since the l a t t e r does not c o n t r i b u t e to the a n i s o t r o p y (see Eqn. (9.15))-, we., expect the d e v i a t i o n from the Anderson-Weiss model, i f any, to l e a d to a s m a l l e r angular dependence. S i n c e our d i s c r e p a n c y goes the o t h e r way, we deci d e f o r the t h i r d a l t e r n a t i v e . 129 We show i n F i g , (9.3) the angular dependence of the i n d i -v i d u a l c o n t r i b u t i o n s of the f i r s t two s h e l l s , t o g e t h e r w i t h t h a t of the l i n e w i d t h (normalized a t 0=0°). The r ~ 6 - f a c t o r i s a p p r o x i -mately the same f o r s h e l l . 1 and 2; i t i s down by a f a c t o r of 8 when we r e a c h s h e l l 3. We p i c k R =3.18 and both A and (B.+l) BR 1 1 p o s i t i v e ; the same argument a p p l i e s f o r o t h e r c h o i c e s . M 2 ( s h e l l l ) o c ( - 3 1 . 2 s i n " 0+46 . 5 s i n 2 0 + 27 . 2) M 2 ( s h e l l 2 ) ^ ( 2 5 . S s x n h 0 - 3 4 , l s i n 2 0 + 2 7 . 2 (9.20) W = 14.4sin 40-16.5sin 20+27.2 One can immediately see t h a t each of the two s h e l l s e x h i b i t s a l a r g e a n i s o t r o p y - a c t u a l l y about twice as l a r g e as t h a t of the experiment. However, due to v e r y s t r o n g d e s t r u c t i v e i n t e r f e r e n c e between t h e two s h e l l s , the f i n a l r e s u l t shows a weak angular de-pendence. T h i s o f f e r s us then a simple e x p l a n a t i o n of the observed d a t a . I f one of the s h e l l s has a s m a l l e r p s e u d o d i p o l a r c o u p l i n g c o n s t a n t , the d e s t r u c t i v e i n t e r f e r e n c e i s reduced, and the a n i s o -t r o p y r e g a i n e d . From F i g . (9.3) i t i s a l s o c l e a r t h a t i t i s s h e l l 2 which must make the major c o n t r i b u t i o n t o the p s e u d o d i p o l a r i n t e r a c t i o n . There are f o u r c o n s t a n t s to be determined: A l f A 2, B x , B 2 . As has been mentioned b e f o r e , the angular dependence i n a hexago-n a l c r y s t a l cannot determine more than t h r e e parameters. Conse-q u e n t l y we can o n l y g i v e a p l a u s i b l e model of the i n t e r a c t i o n s . (1) C o n s i d e r a s t r o n g r e d u c t i o n o f the p s e u d o d i p o l a r c o n s t a n t i n s h e l l 1, namely (B x+1) =0,25 (B 2+1) ; a l l o t h e r B ^ B j . (2) Assume a s m a l l r e d u c t i o n of the pseudoexchange c o n s t a n t i n (kHz) F i g . 9.3 C o n t r i b u t i o n s to the second moment s h e l l 2, namely A 2 = 0.73Alf- a l l o t h e r A =AX . (3) A x = 3.75 (B 2 + l ) . (4) A l l B.+l and A- are p o s i t i v e . I t i s important to r e a l i z e t h a t c o n d i t i o n (3) does not imply t h a t the s c a l a r i n t e r a c t i o n i s M times l a r g e r than the t e n s o r i n t e r a c t i o n f o r a l l s h e l l s . Con-d i t i o n (2) shows t h a t i n the f i r s t s h e l l A x i s 15 times l a r g e r than (Bj+1). I f we use these f o u r c o n d i t i o n s and add the c o n t r i b u t i o n s of the f i r s t 8 s h e l l s we o b t a i n M 2 = lO"" 0(65.551^ 6-75.4sin 26 + 134 . 38) ( B 2 + l ) 2 M 2 « (13.25sin'*e-15.26sin 28 + 27. 2) (9. 21) We show t h i s a n g u l a r dependence i n F i g . (9.4) Our e x p e r i m e n t a l v a l u e f o r the second moment at 8=0° i s M2=700 ( k H z ) 2 . F o r t h a l l i u m , y=15.43xl03rad.sec-1.gauss"1, o o r x=3.4lA, r 2=3,45A , and, u s i n g the above 4 c o n d i t i o n s , we o b t a i n J1/h =24.9 kHz J 2 / h =18.2 kHz b j / h = 0.7 kHz b 2 / h = 5.42 kHz (9.22) The average i n t e r a c t i o n s f o r the two major s h e l l s are h " 1 ( b + y 2 # 2 r " 3 ) = 4 . 1 kHz and h _ 1J=21.5 kHz; they f a l l inbetween the v a l u e s o b t a i n e d by the p r e v i o u s two i n v e s t i g a t i o n s (see (9.18)). I f we use the Anderson-Weiss e x p r e s s i o n WroM2/a)e, we f i n d a3e/2ir= =25.7 kHz, which agrees w i t h our average v a l u e . F i n a l l y , i f the average exchange were as l a r g e as claimed by Karimov and Shchegolev, i . e . h - 1J=37.5 kHz, then, a c c o r d i n g to (9.17), t h i s Fig. 9.4 The orientation-dependence of the second moment term alone would c o n t r i b u t e twice our t o t a l e x p e r i m e n t a l value of M . S i n c e our observed M 2 agrees w e l l with the moments observed by both Bloembergen and Rowland (1955) and Karimov and Shchegolev (1962), we conclude t h a t the l a t t e r have o v e r e s t i m a t e d the aver-age exchange i n t e r a c t i o n . 9.4 C o n c l u s i o n Our study of the angular dependence of the l i n e w i d t h i n s i n g l e - c r y s t a l t h a l l i u m has shown t h a t the two major s h e l l s , even though of equal s i z e , make d i f f e r e n t c o n t r i b u t i o n s to the second moment. The p s e u d o d i p o l a r c o u p l i n g c o n s t a n t of the second s h e l l i s s i g n i f i c a n t l y l a r g e r than t h a t of the f i r s t s h e l l . We cannot overemphasize t h a t any theory attempting to ex-p l a i n these r e s u l t s must take i n t o account the a n i s o t r o p y of the Fermi s u r f a c e and p r o v i d e c o u p l i n g c o n s t a n t s which depend on the d i r e c t i o n of the p o s i t i o n v e c t o r S. between the i n t e r a c t i n g n u c l e i . The r a d i a l dependence alone cannot e x p l a i n the r e s u l t s i n t h a l l i -um, s i n c e from the r a d i a l p o i n t of view the two major s h e l l s are e q u i v a l e n t . We a l s o wish to p o i n t out t h a t our data w i l l p r o v i d e an u n u s u a l l y c r i t i c a l t e s t of the theory. 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Rev., 76, 350 (1949) APPENDIX I The program d e s c r i b e d i n t h i s appendix f i t s modulated mixed L o r e n t z i a n l i n e s h a p e s , c o r r e c t e d f o r time constant d i s t o r t i o n , as d e s c r i b e d i n chapter The n o t a t i o n i s the same as i n the t e x t , w i t h the e x c e p t i o n of h =HM, The output i s as f o l l o w s : HD2MA= the f i e l d v a lue of the shallow peak. HD2MI= the f i e l d v a lue of the sharp peak. HDMA and HDMI are the co r r e s p o n d i n g f i e l d v a l u e s of a l i n e of v e ry s h o r t time c o n s t a n t . Y2MA, Y2MI, YMA, and YMI are the co r r e s p o n d i n g i n t e n s i t i e s of the peaks, normalized to the peak-to-peak ' i n t e n s i t y of the experimental l i n e , i n c h a r t u n i t s . H D 3 ( I ) , Y l l ( I ) , Y22(I) are the ( p o i n t by p o i n t ) f i e l d v a l u e s , and the un-c o r r e c t e d and c o r r e c t e d i n t e n s i t i e s of the l i n e , r e s p e c t i v e l y . T h i s p a r t i c u l a r program f i t s the experimental s i t u a t i o n i n which the f i e l d was swept from low to high v a l u e s . The program h a n d l i n g frequency sweeps from h i g h to low f r e q u e n c i e s i s s i m i l a r , but w i t h the f o l l o w i n g minor changes: A f t e r e n t r y 22 change HDMA=HDR-HDMA HDMA=HDMA+HDR HDMI=HDR-HDMI HDMI=HDMI+HDR HD2MA=HDR-HD2MA to HD2MA=HD2MA+HDR HD2MI=HDR-HD2MI HD2MI=HD2MI+HDR 71 HD2(J)=HDR-HD1(J) 71 HD2(J)=HD1(J)+HDR The HD v a l u e s are now f r e q u e n c i e s . 139 140 X X X X X X X X X X XT X X X X VTX X X~X X XYX XXXYXXX X XX XX X XXXX X XX 'XX"JOOCX X XX X X X X X X X X X X X X X XX) C O M M O N W , K M , H D , C 1 , C 2 D I M E N S I O N H P 1 ( 1 0 0 0 ) , H 0 2 ( 1 0 0 0 ) , HD3( 1 0 0 0 ) ,"Y 1 ( 1 0 0 0 ) , Y2 ( 1 0 0 0 ) DTHE^ STON—Y r r r r o o o 2'2T r o 0 01 • : R E A D ( 5 , 6 9 ) w , H M , C 1 , C 2 , H D I , h D F , T C , D , P P 8 9 9 1 F O R M A T ( F 7 , 0 ) R t A D ( 5 r 9 1 ) L A , N N , H D R F O R M A T ( 2 1 3 / F 9 , 0 ) W R I T E ( 6 , 9 2 ) K , H M , C 1 f C 2 , H D I , H D F , T C , D , P P f 9 2 9 3 F O R M A T ( 9 F / e 3 ) W R I T E ( 6 , 9 3 ) I A , N N , H D R F O R M A T ( 2 I 3 , F i O t 3 ) P=3,14159 H D - H D I 1 = 1 / Y M A = O . 0 Y M I = O « 0 E X T E R N A L Y I 1 7 Y = C O S I M ( Y I ^ P , P , 0 , 0 5 , 2 ) H D J C I ) = H D Y 1 ( I ) = Y I P ( Y 1 ( I ) S G T , Y M A ) G U T O 30 I F ( Y I ( I ) , L T , Yhl )GO TO 3 1 GO T O 3 2 Y M A = Y 1 ( I ) H D M A = H D 1 ( I ) G O T O 3 2 • i i 3 2 Y M I = Y 1 ( 1 ) H D M I = H D 1 ( I ) HD=KD+ D 1 = 1 + 1 i I F ( H D . G T T H D F ) G O T O 1 9 GO T O 1 7 I V 1 1 = 1 - 1 L = l Y 2 M A = 0 , 0 « 3 Y 2 M I = 0 t 0 L L = L + U A . • . DO 3 9 K = L t L U 3 9 Y 2 ( K ) = Y1 ( K J L 1 = L + 1 D O m M s U l r L U 4 1 AM = M A L = L Y 2 ( U ) = Y 2 C L ) t Y 2 ( M ) * E X P ( ( - ( A . M ^ A L ) * D ) / T C ) Q=D/1€ Y 2 ( L ) = 0 * Y 2 ( L ) I F ( Y 2 ( L ) . G T , Y 2 M A ) G 0 T O 60 1 4 1 j F C Y 2 ( t ) 7CT7Y2 H rr G o-T 0-6-r GO TO 62 6 0 Y 2 M A = Y 2 ( L ) R 0 2 MT= H DXCC7 ~~~ GO TO 62 6 1 Y 2 M I = Y 2 ( L ) H D 2 M l = H D n n ' : 6 2 l = L + l I F ( L . G T , ( I I - L A ) ) G 0 TO 4 9 GO TO~try — <49 L F = l * l R 1 = P P / { Y 2 M A - Y 2 M I ) . YMA = R1 *YHA~ : — Y M l = « i * Y M I Y 2 M A = R 1 * Y 2 M A Y 2 M " I - T 2 " M T * H I :  D O 2 2 N s i ,IF Y l C N ) s R l * Y i ( N ) —• 2 2 — Y 2 ( N ) s R l " * Y 2 - ( N ) h D M A = H D R > H D M A H D M I s H O K - H D M I ' H"0"2"M-A-= H DR>" H D~2 M"A ; M D 2 M I a . H D R « H D 2 M I D O 7 i J s i i l l 7 1 H D 2 ( J ) = . H D R - H D 1 ( J ) . ' JJ=tF7N*l " " ' D O 3 5 H = 1 , J J f f = N T f *H : :  H D 3 (M).= H D 2 ( N ) Y U ( H ) = Y 1 ( N ) 35 Y 2 2 ( M ) = Y 2 ( N ) W R I T E ( 6 » 7 5 ) M 0 M A , H D M I f M D 2 M A # H D 2 M J « W K I T E ( 6 . i 7 5 ) Y M A , Y M I , Y 2 M A , Y 2 M I 75 FORMAT (iPai20771" DO 8 1 1 = 1 , J J W R I T E C t o , 7 6 ) H O i ( I ) , Y 1 1 ( I ) 1 Y 2 2 ( I ) ' " " 8 ' f CONTINUE ' ' 7 6 F O R M A T < 1 P 3 E 2 0 , 7 ) STOP  ENR3 : F U N C T I O N Y I ( T ) C O M M O N W # H M , H 0 , C 1 # C 2 Y J _ ( C J A W * C 0 S ( T ) « - C 2 * ( H D + H M * C O S ( T ) ")"*C'0S "CT") " ) 7 X w * W * " ( H D t H M * C 0 ' S ' ( T ) ) ' * * ' ? ^ ~ R E T U R N END APPENDIX I I T h i s program c a l c u l a t e s second moments of c u t - o f f , modu-l a t e d , mixed, L o r e n t z i a n l i n e s as used i n chapter 6. The n o t a t i o n i s as f o l l o w s : WS, HMS, e t c . stand f o r W, HM, e t c . of the l i n e -f i t t i n g program, as d e s c r i b e d i n appendix 1. F u r t h e r , AS=L C, HDAS=L, HDBS=R, ES=R C, as d e s c r i b e d i n chapter 6 and shown i n F i g , (5.1). The f o l l o w i n g output symbols h o l d : R R T = / ( v - v 0 ) 3 (dx/dv)dv B = / ( v - v 0 ) ( d X / d v ) d v L L These are the c o n t r i b u t i o n s to the numerator and denominator, r e -s p e c t i v e l y , o f the e x p r e s s i o n f o r M. r coming from the L o r e n t z i a n c e n t r a l p a r t o f the l i n e s h a p e . Sjand S 3 are the c o r r e s p o n d i n g c o n t r i b u t i o n s from the l e f t - h a n d - s i d e t a i l (from L t o L)and S, c 2 and S 4 are those of the r i g h t - h a n d - s i d e t a i l (from R to R ). F i n a l l y , T+S +S SEC = M 2 = (1/3) — B+S 3+S 4 The p a r t i c u l a r program of t h i s appendix c a l c u l a t e s the second moments of 13 exp e r i m e n t a l l i n e s 142 .. 1 4 3 IV G COMPILER MAIN 12-05-69 12 :44 :09 PAGE 0 COMMON w , H M T H D T C l , C 2 DIMENSION HD1(1000 ) ,Y1 (1000 ) ,Y2 (1000 ) DI MEMS I ON WS (80) t HMS( 80) f CIS ( 80) t C2S(80) , HP A'S (80) ,HDBS(80) DIMENSION P P S ( 8 0 ) , A S ( 8 0 ) , E S ( 8 0 ) M=l 64 READ (5 ,61 )WS (M ) ,HM S (M ) » CIS { M ) t C2-S ( M) ,HDAS (M) ,HOBS(M) , 1 P P S C H ) t A S ( M ) i E S ( M ) 61 FORMAT(9F7.0 ) WRITE(6 ,62)WS(M) ,HMS(M) ,C1S(M),C2S(M) ,HDAS(M) ,HDBS(M) , 1PPS(M)*AS(M) ,ES(M> 62 FORMAT(9F7.3 ) M=M+1 I F ( K . G T . 1 3 ) G 0 TO 63 GO TO 64 . 63 N = l 72 W=WS(N) HM=HMS(N) C1=C1S(N) C2=C2S(N) HDA=HDAS(N) HDB=HDBS(N) :pp = PPS(N) A=AS(N) E=ES(N) P = 3 „ i 4 1 5 9 HD=HDA 1 = 1 YMA =0.0 YM1=0.0 EXTERNAL YI 17 Y=COSIM(YIt -PfP»0.05 t2) HD1(I)=HD Y l { I ) =Y I F ( Y l ( I ) . G T . Y M A ) G O TO 30 I F ( Y l ( I J . L T . Y M I ) G 0 TO 31 GO TO 32 30 YMA=Y1(I) GO TO 32 31 YMI=Y1(I) 32 HD=HD+0.1 1=1+1 • IF (HD.GT .HDB)G0 TO 19 GO TO 17 19 HD1U)=HD r H=Y1( I -1 ) 1 4 4 R=PP/(YMA-YMI ) ~ ~ : H = R*H K = l 24 Y2.(K)=R*Y1 (K) ' " ~ K=K+1 I F (HD1 (K ),GT.HDB)GO TO 23 GO TO 2 4 ~ ~ ~ — 23 J = l T=0.0 20 T = T+0.0 6 2 5 *(HD 1 ( J +1)-H D 1 ( J ) ) * { Y 2 ( J ) + Y 2 ( J + 1 ) ) * ( H D 1 ( J + 1 ) + H D 1 ( J ) ) * * 3 J = J+1 I F ( H D 1 ( J + l I . G T . H D B I G O TO 21 GO TO 20 ' . 21 WRITE(6,22>T 22 F 0 R M A T ( 1 P E 2 0 . 7 ) L= l _ _ " B = 0.0 40 B=B+0.2 5 * ( K D K L + 1 ) - H D 1 ( L ) ) * (Y2 ( L ) +Y2 ( L+ 1) >*(HD1 ( L + D + H D l (L ) ) L = L + 1  I F ( H D I { L + l ) . G T . H D B ) G 0 TO 41 GO TO 40 41 WRITE ( 6 , 2 2 IB _ 1 , C = HDA G = Y 2 ( 1 ) D = HDB  S l = ( G / ( C - A ) )•#•(• (-C-**5—A**5 ) /5 . 0-A* ( C**4-A**4 ) /4 . 0 ) W R I T E ( 6 , 2 2 ) S 1 ' S 2 = ( H / ( E - D ) ) * { ( D * * 5 - E * * 5 ) / 5 . 0 - E * ( D * * 4 - E * * 4 ) / 4 . 0 ) _ W R I T E ( 6 , 2 2 ) S 2 S 3 = ( G / ( C - A ) I * ( ( C * * 3 - A * * 3 ) / 3 . 0 - A * ( C * * 2 - A * * 2 ) / 2 . 0 ) WRITE ( 6 , 2 2 ) S 3  S 4 = ( H / ( E - D ) ) * ( ( D * * 3 - E * * 3 ) / 3 . 0 - E * ( D * * 2 - E * * 2 ) / 2 . 0 ) W R I T E ( 6 , 2 2 ) S 4 S E C = ( T + S 1 + S 2 ) / { 3 . 0 * ( B + S 3 + S 4 ) ) ; WRITE(6,22>SEC N=N + 1 I F (N . GT »13 ) G 0 TO 71  GO TO 72 71 STOP . END _ . . _ ; ; ; FUNCTION Y I ( T ) COMMON W,HM tHD-,Cl,C2 YI=(C1*W*CQS(T)-C2*(H0+HM*C0S(T))»COS(T))/(W»W+(HD+HM*COS(T))»»2) RETURN END APPENDIX I I I CARROUSEL. Re c e n t l y T t e r l i k k i s e t . a l . (1970) have attempted a p r e c i s e t h e o r e t i c a l c a l c u l a t i o n of the i s o t r o p i c Knight s h i f t i n l e a d . They o b t a i n e d K=1.47% which seemed i n p e r f e c t agreement wi t h the e x p e r i m e n t a l v a l u e of Snodgrass and Bennett (1963).Since the l a t t e r o b t a i n e d f o r the resonance frequency of the metal v(metal)=9004.7±0.2 kHz a t 10 kgauss, which i s i n good agreement w i t h our metal measurement of v(metal)=9004.4±0.2, we s e t out to f i n d the a r t i c l e which p r o v i d e d the r e f e r e n c e compound frequency t h a t y i e l d e d K=1.47%. To our s u r p r i s e we found the f o l l o w i n g . P i e t t e and Weaver (1958) f i n d v(PbS0 4)=8848.2 kHz and v(metal)= =8978.4 kHz y i e l d i n g K=1.47%. Since t h e i r metal r e s u l t i s a p p r e c i -a b l y d i f f e r e n t from the more a c c u r a t e v a l u e s quoted above, one c o u l d d i s c a r d t h e i r measurements. An a l t e r n a t i v e would be to con-s i d e r the d i s c r e p a n c y between t h e i r and our metal r e s u l t due to a s y s t e m a t i c e r r o r i n t h e i r f i e l d measurement and s c a l e both of t h e i r r e s u l t s . T h i s would y i e l d v(PbSOj=8873.8 kHz. T r y i n g t o s o l v e t h i s dilemma we looked up the a u t h o r i t a t i v e t a b l e s on Knight s h i f t s , namely the V a r i a n t a b l e s , those g i v e n by D r a i n (1967), and those g i v e n by Rowland (1961), i n t h e i r r e - . s p e c t i v e review papers. A l l t h r e e quote v(PbS0 4)=8899.0 kHz and K=1.47% w i t h P i e t t e and Weaver's paper as r e f e r e n c e . T h i s seems odd indeed, s i n c e n e i t h e r P i e t t e and Weaver's measurement, nor the s c a l e d v a l u e of those measurements come c l o s e to t h i s number. The p u z z l e was f i n a l l y s o l v e d by the ever h e l p f u l Handbook 145 146 of Chemistry and P h y s i c s , 50-th e d i t i o n . On page E-77 we f i n d Vpk=8907.71 kHz w i t h Baker (1957) as l i t e r a t u r e r e f e r e n c e . Baker mentions t h a t h i s r e s u l t i s j u s t o u t s i d e the s t a t e d e r r o r s of the combined r e s u l t s of P r o c t o r (1950) and Zimmerman and W i l l i a m s (1949) who found v p b = 8 8 9 9 ± 4 kHz, T h i s must then be the number t h a t has c r e p t up i n a l l the t a b l e s and was quoted as P i e t t e and Weaver's r e s u l t . I f we were to use t h i s v a l u e , we would o b t a i n K=(1,19±0.05)%. I f we use the more a c c u r a t e v a l u e of Baker (1957) we o b t a i n K=1.09%. As can be seen, both of these v a l u e s are i n disagreement w i t h the t h e o r e t i c a l r e s u l t of T t e r l i k k i s e t . a l . L u c k i l y f o r the t h e o r e t i c i a n s , Rocard e t . a l . (1959) have made a more r e c e n t measurement of the same q u a n t i t i e s . They f i n d v (metal) =9012 . 7±2 . 6 and v (PbSO^ ) =8875 . 4±2 . 6 .. I f we use t h e i r v a l u e f o r the r e f e r e n c e compound we o b t a i n again K=1.4 5%. I f we s c a l e t h e i r v a l u e , we o b t a i n K=1.55%, 

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