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Knight shift and quadrupole interaction in single crystal magnesium Dougan, Patrick Daniel 1969

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KNIGHT SHIFT AND QUADRUPOLE INTERACTION IN SINGLE CRYSTAL MAGNESIUM PATRICK DANIEL DOUGAN B.Sc. (Hons.), University of British Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Apri l , 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I 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 and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It 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 thes, is f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . The U n i v e r s i t y o f B r r t i s h C o l u m b i a V a n c o u v e r 8, Canada D e p a r t m e n t 1 ABSTRACT The n u c l e a r m a g n e t i c r e s o n a n c e i n s i n g l e c r y s t a l magnesium has been s t u d i e d a t 4.2°K. The i s o t r o p i c K n i g h t s h i f t , t h e a n i s o -t r o p i c K n i g h t s h i f t and t h e q u a d r u p o l e i n t e r a c t i o n have been i n v e s t i -g a t e d . The v a l u e s f o r t h e p a r a m e t e r s measured a re K = (0.1127 + 0.0005)%, K' = (0.0004 + 0.0002)%, e 2qQ/h = 3 2 4 + 6 KHz. These r e s u l t s a r e compared w i t h t h o s e o f Rowland i n powdered magnesium t o r e v e a l t h e K n i g h t s h i f t i s i n d e p e n d e n t o f t e m p e r a t u r e . T h i s c o n t r a s t s d i r e c t l y w i t h t h e b e h a v i o r o f cadmium. The q u a d r u p o l e i n t e r a c t i o n i s shown t o i n c r e a s e by 4 0 % as t e m p e r a t u r e i s l o w e r e d t o 4.2°K f r o m room t e m p e r a -t u r e . TABLE OF CONTENTS Page ABSTRACT i LIST OF ILLUSTRATIONS i i i ACKNOWLEDGEMENTS i v CHAPTER I INTRODUCTION , 1 I I REVIEW OF THE THEORY A. K n i g h t S h i f t 3 1. The I s o t r o p i c S h i f t 4 2. A n i s o t r o p i c K n i g h t S h i f t 6 . B. Quadrupole I n t e r a c t i o n 7 1. The Quadrupole Moment 9 2. The E l e c t r i c F i e l d G r a d i e n t 11' 3. P e r t u r b a t i o n C a l c u l a t i o n s 12 . C. F i e l d G r a d i e n t o f t h e I o n i c M o d e l 19 I I I APPARATUS AND TECHNIQUE A. G e n e r a l D e s c r i p t i o n B. C r y s t a l v s . Powder C. E x p e r i m e n t a l T e c h n i q u e and C o n s i d e r a t i o n s •1. The Sample 2. M o d u l a t i o n and D e t e c t i o n 3. The M a g n e t i c F i e l d IV RESULTS AND DISCUSSION A. E x p e r i m e n t a l O b s e r v a t i o n s B. Comparison w i t h Powder D a t a C. R o l e o f t h e C o n d u c t i o n E l e c t r o n s V CONCLUSIONS A. Comparison w i t h Cadmium 44 B. F u r t h e r S t u d i e s 46 22 24 25 25 27 29 34 38 41 BIBLIOGRAPHY i i i L I S T OF ILLUSTRATIONS FIGURE Page 1 x, zy X and Z Axes 14 2 . Quadrupole S p l i t t i n g o f t h e M a g n e t i c Resonance o f a 16 N u c l e u s S p i n 3/2 3 A n g u l a r D e p e n d e n c i e s o f t h e S p l i t t i n g s o f t h e S a t e l l i t e s 18 o f t h e C e n t r a l L i n e 4 S c h e m a t i c D i a g r a m o f t h e S p e c t r o m e t e r 23 5 S p e c i a l Four-Sample Probe f o r F i e l d H o m o g e n i z a t i o n 31 6 O s c i l l o s c o p e T r a c e s f o r Probe i n Inhomogeneous and 31 Homogeneous F i e l d s 7 Resonance L i n e C o r r e s p o n d i n g t o t h e (\<^~\) T r a n s i t i o n 34 o f Mg25 i n a S i n g l e C r y s t a l 25 8 Measured F r e q u e n c y S h i f t o f C e n t r a l Component o f t h e Mg 37 Resonance 9 T h e o r e t i c a l - P o w d e r S i g n a l Shapes 40 i v ACKNOWLEDGEMENTS W i t h p l e a s u r e I t h a n k D r . D. L l . W i l l i a m s f o r h i s h e l p , i n s t r u c t i o n and p a t i e n c e w i t h t h i s work, as w e l l as f o r h i s s p o n s o r -s h i p and s u p p o r t i n o b t a i n i n g f i n a n c i a l a s s i s t a n c e . F o r h i s a s s i s t a n c e i n a l l p a r t s o f t h i s work, I t h a n k Dr. S. N. Sharma who a l s o c o n s t r u c t e d t h e s p e c t r o m e t e r used. I am g r a t e f u l t o D r . A. A k h t a r f o r t h e l o a n o f t h e c r y s t a l and f o r h i s h e l p w i t h i t s p r e p a r a t i o n . F o r f i n a n c i a l s u p p o r t , my t h a n k s go t o t h e N a t i o n a l R e s e a r c h C o u n c i l and t h e Dean's Committee on R e s e a r c h . CHAPTER I INTRODUCTION -Since n u c l e i have magnetic moments, they are s e n s i t i v e to magnetic f i e l d s o r i g i n a t i n g i n the spin and o r b i t a l currents of the electrons. Lacking an e l e c t r i c dipole moment,- n u c l e i are i n s e n s i t i v e to homogeneous e l e c t r i c f i e l d s but since some may possess e l e c t r i c • quadrupole moments, they can experience torques i n s u f f i c i e n t l y inhomogeneous e l e c t r i c f i e l d s produced i n part by the electron clouds Since these i n t e r a c t i o n s couple the nuclear spin system to the e l e c -tron system, the study of nuclear magnetic resonance can y i e l d v a l u -able information on the e l e c t r o n i c properties. I f a magnetic f i e l d UQ- i s applied i n the z d i r e c t i o n to a system of N n u c l e i and n conduction electrons, the r e s u l t i n g Hamiltonian, neglecting nuclear d i p o l a r and exchange i n t e r a c t i o n s , can be written^"^: H = H + /3H ( L ^ + 2 S ^ ) + H e 1=1; o v z z ' n + g U H 1^ ") + ?-t J-t H j = l & / o o z i = l j = l en where H i s the Hamiltonian f o r the electrons i n the f i e l d of the e ion cores; H n represents the Hamiltonian f o r the ion cores. The terms containing H q are the Zeeman energies of the electrons and n u c l e i . H , the Hamiltonian f o r the noncoulombic i n t e r a c t i o n en between the nuclear spin and the electron o r b i t a l and spin coordinate - 2 -can be a p p r o x i m a t e d by i n c l u d i n g o n l y t h e ma g n e t i c d i p o l e and e l e c t r i c q u a d r u p o l e p a r t s o f t h e i n t e r a c t i o n , H = 2aUl.\X -% + ^ l l + ^ s S ( r ) } + i %>. V..Q.. ( 2 ) en / I r r r 1 J 1 3 J The K n i g h t s h i f t and q u a d r u p o l e c o u p l i n g , t o be d i s c u s s e d i n c h a p t e r t w o , a r e t h e most marked m a n i f e s t a t i o n s o f t h e s e i n t e r -a c t i o n s . . ( 2 3 4 ) . S t u d i e s i n cadmiunv ' ' J have r e v e a l e d a l a r g e t e m p e r a t u r e v a r i a t i o n o f t h e K n i g h t s h i f t w h i c h has n o t y e t been s a t i s f a c t o r i l y e x p l a i n e d . The s p e c u l a t i o n t h a t t h i s b e h a v i o r might: be c h a r a c t e r i s t i c o f o t h e r d i v a l e n t h e x a g o n a l c l o s e - p a c k e d m e t a l s , n o t a b l y magnesium and z i n c , m o t i v a t e d t h i s s t u d y o f t h e n u c l e a r m a g n e t i c r e s o n a n c e o f magnesium. B e c a u s e o f t h e s e v e r a l a d v a n t a g e s o u t l i n e d i n c h a p t e r t h r e e , t h i s s t u d y was p e r f o r m e d on a s i n g l e c r y s t a l ; b e f o r e t h i s , t h e r e s o n a n c e had b e e n o b s e r v e d i n magnesium m e t a l o n l y i n a powdered s p e c i m e n . The i n v e s t i g a t i o n o f t h e K n i g h t s h i f t was c o m p l i c a t e d by t h e p r e s e n c e o f a q u a d r u p o l e moment o f t h e magnesium n u c l e u s . Con-s e q u e n t l y , a d e t e r m i n a t i o n o f t h e q u a d r u p o l a r i n t e r a c t i o n was a l s o a c h i e v e d . CHAPTER I I REVIEW OF THE THEORY A. K n i g h t S h i f t The m a g n e t i c f i e l d seen by a n u c l e a r s p i n i n a r e s o n a n c e e x p e r i m e n t on a m e t a l h a s , i n a d d i t i o n t o t h e a p p l i e d e x t e r n a l f i e l d , c o n t r i b u t i o n s f r o m t h e s u r r o u n d i n g n u c l e i and e l e c t r o n s . T h i s f i e l d i s a l m o s t a l w a y s h i g h e r t h a n t h a t w h i c h t h e same n u c l e u s would e x p e r i e n c e i n a n o n - m e t a l l i c s u b s t a n c e . The r e s u l t i n g s h i f t i n f i e l d i s c a l l e d t h e " K n i g h t s h i f t " , w h i c h i s q u a l i t a t i v e l y d e f i n e d as K = p (3) where H i s t h e - e x t e r n a l a p p l i e d f i e l d and H i s t h e a d d i t i o n a l f i e l d due t o t h e h y p e r f i n e c o u p l i n g . Of c o u r s e , t h i s change i n f i e l d c a u s e s a c o r r e s p o n d i n g s h i f t i n r e s o n a n c e f r e q u e n c y , so where a " c o n s t a n t f i e l d i s employed, K can be measured by d e t e r m i n i n g K = . (-0 where and 1^ a r e t h e r e s o n a n c e f r e q u e n c i e s i n t h e m e t a l and a n o n - m e t a l l i c r e f e r e n c e r e s p e c t i v e l y . S i m i l a r s h i f t s i n n o n - m e t a l l i c compounds, c a l l e d c h e m i c a l s h i f t s , a l t h o u g h much s m a l l e r , a r e p r e s e n t and n e c e s s i t a t e s p e c i f y i n g t h e r e f e r e n c e compound when q u o t i n g K n i g h t s h i f t v a l u e s . The o r i g i n o f t h e K n i g h t s h i f t l i e s i n t h e m a g n e t i c i n t e r -a c t i o n p a r t o f t h e H a m i l t o n i a n ( e q n . 1): _4_ L r r r J (5) A s s u m i n g f o r s i m p l i c i t y t h a t t h e e l e c t r o n o r b i t a l m o m e n t u m i s q u e n c h e d , t h e e x p e c t a t i o n v a l u e o f h y p e r f i n e c o u p l i n g f o r a g i v e n n u c l e a r s p i n c a n b e w r i t t e n rk rk * S i n c e o r b i t s w i t h t w o p a i r e d e l e c t r o n s c a n n o t c o n t r i b u t e t o t h e c o u p l i n g , t h e s u m m a t i o n i s o v e r t h e u n f i l l e d o r b i t s n e a r t h e t o p o f t h e F e r m i d i s t r i b u t i o n . H e r e a n d a r e t h e p o s i t i o n a n d s p i n o f t h e u n p a i r e d e l e c t r o n i n t h e o r b ' i t 1. T h e I s o t r o p i c S h i f t . I f t h e s y m m e t r y o f t h e e l e c t r o n e n v i r o n m e n t . o f t h e n u c l e a r s p i n i s c u b i c o r h i g h e r , o n l y t h e S - f u n c t i o n c o u p l i n g w i l l b e n o n -z e r o . S i n c e t h i s t e r m a r i s e s f r o m t h e c o n t a c t o f t h e s - s t a t e e l e c t r o n w a v e f u n c t i o n w i t h t h e n u c l e u s a t r = 0, w e n e e d o n l y c o n s i d e r T^^(0)-I n t h i s c a s e t h e c o u p l i n g (6) b e c o m e s - p C l . ' H ) ¥ < | f k ( » ) | 2 ) E F X / (.7) s w h e r e X g i s t h e t o t a l s p i n s u s c e p t i b i l i t y o f t h e e l e c t r o n s . T h i s i s e n t i r e l y e q u i v a l e n t t o t h e i n t e r a c t i o n o f t h e n u c l e a r s p i n w i t h a n e x t r a m a g n e t i c f i e l d -5-T h i s t e r m i s i n d e p e n d e n t o f t h e o r i e n t a t i o n o f t h e a p p l i e d f i e l d a n d i n i t s f r a c t i o n a l f o r m , AH/Hq, i s r e f e r r e d t o a s K ^ g o , t h e i s o t r o p i c K n i g h t s h i f t . T h e c o r r e s p o n d i n g r e s o n a n c e f r e q u e n c y s h i f t g i v e s O ^ K r E f f e c t s ' w h i c h h a v e b e e n i g n o r e d a b o v e b u t w h i c h a l s o c o n t r i -b u t e t o t h e s h i f t a r e c o r e p o l a r i z a t i o n a n d o r b i t a l p a r a m a g n e t i s m . C o r e p o l a r i z a t i o n , a n i m p o r t a n t e f f e c t i n t h e t r a n s i t i o n e l e m e n t s , a r i s e s f r o m a n u n p a i r e d o u t e r e l e c t r o n i n t e r a c t i n g w i t h t h e s - e l e c t r o n s i n t h e c l o s e d - s h e l l i o n c o r e o f t h e a t o m . T h i s i n t e r a c t i o n p r o d u c e s p e r t u r b a t i o n s o n t h e c o r e - e l e c t r o n w a v e f u n c t i o n s w h i c h d e p e n d o n w h e t h e r t h e s p i n o f t h e c o r e - e l e c t r o n i s p a r a l l e l o r a n t i - p a r a l l e l t o t h e s p i n o f t h e c o n d u c t i o n e l e c t r o n . B e c a u s e o f t h e c o n t a c t o f t h e s - e l e c t r o n a n d t h e n u c l e u s , a n e t m a g n e t i c f i e l d r e s u l t s f r o m t h e p e r t u r b e d e l e c t r o n s p i n d e n s i t y a t t h e n u c l e a r s i t e . T h i s i s a n i m p o r t a n t e f f e c t i n t h e t r a n s i t i o n m e t a l s . T h e p r e s e n c e o f a n a p p l i e d m a g n e t i c f i e l d m o d i f i e s t h e o c c u p i e d e n e r g y l e v e l s o f t h e s y s t e m , m i x i n g i n h i g h e r u n o c c u p i e d s t a t e s h a v i n g a n e t o r b i t a l m a g n e t i c m o m e n t i n t h e d i r e c t i o n o f t h e f i e l d . T h i s " u n q u e n c h i n g " o f t h e e l e c t r o n o r b i t a l p a r a m a g n e t i s m i s r e s p o n s i b l e f o r t h e c h e m i c a l s h i f t s w h i c h a r e a l s o p r e s e n t i n t h e d i a m a g n e t i c c o m p o u n d s u s e d f o r r e f e r e n c e . F o r s i m p l e m e t a l s , t h e s e e f f e c t s a r e m u c h s m a l l e r ' t h a n t h e K n i g h t s h i f t . -6-2. A n i s o t r o p i c K n i g h t S h i f t W h e r e t h e s y m m e t r y o f t h e e l e c t r o n e n v i r o n m e n t o f t h e n u c l e a r s p i n i s l o w e r t h a n c u b i c , t h e K n i g h t • s h i f t i s c o m p l i c a t e d b y t h e i n c l u s i o n o f a t e r m w h i c h i s d e p e n d e n t o n t h e o r i e n t a t i o n o f t h e c r y s t a l w i t h r e s p e c t t o t h e a p p l i e d f i e l d . O n e c o n t r i b u t i o n t o t h e a n i s t r o p y a r i s e s f r o m t h e m a g n e t i c d i p o l e t e r m o f t h e i n t e r a c t i o n H a m i l t o n i a n ( e q n . 2) H d 2/3 V^ I . r r J A s w a s d o n e f o r t h e i s o t r o p i c s h i f t , t h e e x p e c t a t i o n v a l u e o f t h i s t e r m i s summed o v e r t h e u n f i l l e d e l e c t r o n o r b i t a l s n e a r t h e t o p o f t h e F e r m i d i s t r i b u t i o n : 2 / 3^1-x r F A s i n t h e p r e v i o u s c a s e , t h i s e f f e c t c a n b e i n t e r p r e t e d a s a n a d d i t i o n a l f i e l d A H . e x p e r i e n c e d b y t h e n u c l e u s w h e r e V : ' t a l c i n g a n i s t h e z c o m p o n e n t , A H . = 2, a n i s ' ' |c r r • ' k/ H e r e a l s o we d e f i n e a f r a c t i o n a l s h i f t A H / H a n d l a b e l i t K . . a n i s C a r r y i n g o u t t h i s ' s u m m a t i o n f o r t h e c a s e i n w h i c h t h e c r y s t a l h a s a x i a l s y m m e t r y , B l o e m b e r g e n a n d R o w l a n d ^ ^ a r r i v e a t t h e e x p r e s s i o n i n s p h e r i c a l c o - o r d i n a t e s , -7-where B i s the angle between the c r y s t a l symmetry axis and the d i r e c t i o n of the applied f i e l d , V q i s the atomic volume and N ( E p ) i s the number of electron states near the Fermi surface. Here I F = ( f|3cos 2</>-l | f ) F (14) i s the quadrupole moment of the electrons near the Fermi surface where Y' i s the average wave function and the angle of the radius vector from the symmetry axis. The quadrupole moment can be determined i n part by the c r y s t a l structure and apparently accounts for the sign of K' which may be positive or negative. A further contribution may come from the anisotropy of the (7) magnetic s u s c e p t i b i l i t y which i n some cases, as i n scandium, ranges up to 30%. This effect has the same f i e l d and angular dependence as the dipolar e f f e c t making them phenomenologically indistinguishable. Since a l l the mechanisms of the Knight s h i f t anisotropy are not properly understood, t h e i r t o t a l effect i s generally grouped i n the single parameter A H . , 9 N Ha n i S = K . = i(K.. - K. >(3' cos -9 - 1) (15) H ams 3 II -J-• o K„ and K, are the s h i f t s corresponding to Q =0° and B - 90° respectively. With the quantity ^(K^ _ K j _ ) denoted by K'/2? the t o t a l Knight s h i f t can be conveniently expressed as ^ AH v K' /„ 2n ,s i - = -n- = K. + w- (3 cos H - 1) (16) H iso 2 v v J \ / O B. Quadrupole Interaction • Nuclei are unaffected by homogeneous e l e c t r i c f i e l d s because they lack a dipole moment. However, they do have quadrupole e l e c t r i c - 8 -moments w h i c h w i l l i n t e r a c t w i t h i n h o m o g e n e o u s e l e c t r i c f i e l d s and m a n i f e s t t h e m s e l v e s t h r o u g h c o u p l i n g w i t h t h e e l e c t r o n s y s t e m and t h e n u c l e a r s p i n s y s t e m . A d i r e c t method o f d e r i v i n g t h i s i n t e r a c t i o n i s t o c o n s i d e r t h e e l e c t r o s t a t i c i n t e r a c t i o n e n e r g y o f a n u c l e u s o f c h a r g e d i s t r i b u t i o n ^ o ( x ) w i t h a p o t e n t i a l V ( x ) due t o i t s s u r r o u n d i n g • e n v i r o n m e n t o f c h a r g e d p a r t i c l e s , E = j ^ ( x ) V ( x ) d 3 x (17) We c a n e x p a n d V ( x ) i n a T a y l o r s e r i e s a b o u t t h e o r i g i n . v « - ( 0 ) + f X i(-Ma) r = o + i J ( ^ : ( ) _ . . . . ( 1 8 ) N 1 / r=0 • * I j / r=0 t o g e t E = V ( o ) J / o ( x ) d 3 x + ^ V.Jx.^ )(5 ) d 3 x + i ^ V . ^ x . x ^ ^ ^ d 3 ^ + . . . (19) The f i r s t t e r m i s t h e e l e c t r o s t a t i c e n e r g y o f t h e n u c l e a r p o i n t c h a r g e t a k e n a t t h e c e n t e r o f mass o f t h e n u c l e u s . S i n c e n u c l e a r s t a t e s have d e f i n i t e p a r i t y , t h e s e c o n d t e r m , i n v o l v i n g t h e e l e c t r i c d i p o l e o f t h e /- 3_ (8) x^^o ' (x)d x , . - v a n i s h e s f o r symmetry r e a s o n s . The t h i r d t e r m i s t h e e l e c t r i c q u a d r u p o l e t e r m . We c a n a l w a y s f i n d p r i n c i p l e a x e s o f t h e p o t e n t i a l V ( x ) s u c h t h a t V . . = 0 i f i f- j . H e r e any e l e c t r o n i c c h a r g e w h i c h l i e s w i t h i n t h e n u c l e u s h a s b e e n n e g l e c t e d s i n c e , i f we ' c o n s i d e r a t o m s , o n l y s - e l e c t r o n s p e n e t r a t e t h e n u c l e u s and h a v i n g s p h e r i c a l s y m m e t r y , g i v e .' no q u a d r u p o l a r c o u p l i n g . - 9 -Terms h i g h e r t h a n t h e q u a d r u p o l e a r e i g n o r e d s i n c e e i t h e r t h e y v a n i s h f o r symmetry r e a s o n s o r t h e i r e f f e c t i s n e g l i g i b l y s m a l l . 1. The Q u a d r u p o l e Moment By d e f i n i n g a q u a n t i t y Q^. = J^bc^x.. - S ^ r 2 j ^ j ( x ) d 3 x t o g e t Jx~x_. ^/0(x)d 3x = ^ [ Q j ^ - j + J ^ i j r /° ( ^ ) d 3 ^ 1, we can w r i t e f o r t h e q u a d r u p o l e i n t e r a c t i o n e nergy E = i £r V. . I x.x.d 3x 2 13 13 J i 3 = i £ fv. .Q. . + V. . $ . . / r 2 x ? ( x ) d 3 x l ( 2 1 ) 6 13 L 13 13 13 13 / / J S i n c e t h e p o t e n t i a l V(x) must s a t i s f y L a P l a c e ' s e q u a t i o n a t t h e o r i g i n , t h e second t e r m o f eqn. (8) v a n i s h e s , l e a v i n g E = i V. .Q. . • ( 2 2 ) 6 13 13^ 13 v J R e p l a c i n g t h e c l a s s i c a l d e n s i t y b y i t s o p e r a t o r , we g e t an H a m i l t o n i a n f o r t h e q u a d r u p o l e i n t e r a c t i o n H a = i X. V. .Q. . ( 2 3 ) nQ 6 13 i3 Hi3 v ' where Q.. . i s an o p e r a t o r o v e r t h e p r o t o n s i n t h e n u c l e u s , Q. . = e 13 XT- ( 3 X . X . . - h. .r, 2 ) ( 2 4 ) k i k 3 k i J k I n t h e r e s o n a n c e e x p e r i m e n t we are i n t e r e s t e d o n l y i n t h e ground s t a t e o f a n u c l e u s and s i n c e t h e s e p a r a t i o n i n energy between ground and e x c i t e d s t a t e s i s v e r y l a r g e compared t o t h e magnitude o f -10-HQ, f o r a p e r t u r b a t i o n c a l c u l a t i o n i n v o l v i n g we need knowledge o n l y o f m a t r i x e l e m e n t s o f t h e fo r m ^Im'| Q | Im^ . These e l e m e n t s a r e , e x c e p t f o r m, d i a g o n a l i n a l l quantum numbers such as t h e s p i n I , w h i c h c h a r a c t e r i z e t h e n u c l e a r ground s t a t e . Here m i s t h e m a g n e t i c quantum number c o r r e s p o n d i n g t o an o r i e n t a t i o n o f t h e n u c l e a r s p i n I . : By means o f t h e W i g n e r - E c k a r t Theorem, we can e x p r e s s t h e m a t r i x e l e m e n t s o f Q^ _. as e l e m e n t s o f an e q u i v a l e n t o p e r a t o r i n a n g u l a r momentum s p a c e . The theo r e m y i e l d s ( l m ' | Q i j j l m ) = c ( l m ' | | ( I . I . + I . I . ) - S . J2 \ L m ) (25) where C i s a c o n s t a n t i n d e p e n d e n t o f m o r m'. The c o n s t a n t C i s r e l a t e d t o t h e q u a n t i t y c a l l e d " t h e n u c l e a r e l e c t r i c q u a d r u p o l e moment" denoted by t h e l e t t e r Q and u s u a l l y measured i n u n i t s o f c h a r g e e. Q i s d e f i n e d by eQ = < ( l l | Q 3 3 | l l ) (26) as t h e e x p e c t a t i o n v a l u e o f Q^^ i n t h e s t a t e i n w h i c h t h e component o f I a l o n g t h e z a x i s i s a maximum (m = I ) . By w r i t i n g t h i s i n terms o f 8) eQ = J / ^ I ( ^ ) ( 3 z 2 " r 2 ) d 3 x (27) we see t h a t i t g i v e s us a measure o f t h e asymmetry o f t h e n u c l e a r c h a r g d i s t r i b u t i o n . Q has t h e u n i t s o f a r e a and i s commonly e x p r e s s e d i n ' /.n-24 2. b a r n s (10 cm ), - 1 1 -Th e c o n s t a n t C can now b e . e v a l u a t e d i n terms o f eQ 2Q = C ( l l \'SIz2 - i 2 I 11^ = C [ 3 I 2 - 1 ( 1 + 1)J ( 2 8 ) S O • .- 1(21=17 We t h e r e f o r e have a f i n a l e x p r e s s i o n f o r our q u a d r u p o l e H a m i l t o n i a n : H n = \ V.:Q. .= , T , ^ n s Lc V. . f j C l . I . + I . I . ) -% . .I21 ( Q 6 13 13^13 61(21-1) 13- 13 2 V 1 3 3 xJ 13 J v I n o r d e r t h a t t h e n u c l e u s have a q u a d r u p o l e moment, t h e s p i n must be I <t 1. T h i s -is a p p a r e n t s e m i c l a s s i c a l l y , f o r i f t h e s p i n were \ and t h e n u c l e a r charge d i s t r i b u t i o n were s p h e r o i d a l i n shape w i t h t h e s p i n a x i s b e i n g t h e s p h e r o i d ' s a x i s o f symmetry, t h e two p o s s i b l e s p i n o r i e n t a t i o n s would c o r r e s p o n d t o t h e same e f f e c t i v e e nergy. A n u c l e u s w i t h I = 0 i s , o f c o u r s e , s p h e r i c a l l y symmetric and has no p r e f e r r e d o r i e n t a t i o n a t a l l . 2. The E l e c t r i c F i e l d G r a d i e n t The magnitude o f t h e e l e c t r i c f i e l d g r a d i e n t a t t h e n u c l e u s can be d e s c r i b e d by t h e t e n s o r V. . i n t r o d u c e d e a r l i e r . T h i s t e n s o r i s 13 2 s y m m e t r i c ( c u r l E = 0) and t r a c e l e s s ( y V = 0 a t t h e o r i g i n ) so i t h a s , i n g e n e r a l , f i v e i n d e p e n d e n t components. These can be r e d u c e d t o t h r e e by t a k i n g t h e p r i n c i p l e a x i s ( X , Y, Z) as a b a s i s so V. . can be 2 e x p r e s s e d i n d i a g o n a l f o r m . The e q u a t i o n V V = V w + V w + V = 0 X X YY uLi must s t i l l be s a t i s f i e d so i t i s s u f f i c i e n t t o d e f i n e two symbols V| and q, c a l l e d t h e asymmetry p a r a m e t e r and t h e f i e l d g r a d i e n t , by -12-eq V, ZZ (3 0 ) ( 3 1 ) The. H a m i l t o n i a n ( 2 9 ) can now be w r i t t e n : (32) where 1+ I + i i x — y I f i n t h e magnesium sy s t e m we assume a x i a l symmetry and t a k e t h e a x i s t o be i n t h e z d i r e c t i o n so t h a t Y| = 0 ( s i n c e = V ^ ) , we a r r i v e a t our f i n a l f o r m , 3. P e r t u r b a t i o n C a l c u l a t i o n s I n some c a s e s , t h e n u c l e a r q u a d r u p o l e i n t e r a c t i o n i s so g r e a t i t i s r e s p o n s i b l e f o r t h e e n t i r e dependence o f t h e energy o f a n u c l e u s on i t s s p i n o r i e n t a t i o n . I n t h e s e c a s e s , r e s o n a n c e e x p e r i m e n t s i n z e r o o r v e r y l o w f i e l d s a r e p o s s i b l e . I n t h e s e " l o w - f i e l d " s t u d i e s t h e Zeeman s p l i t t i n g can o f t e n be t r e a t e d as a p e r t u r b a t i o n on t h e q u a d r u p o l e e n e r g y . We s h a l l be d e a l i n g h e r e w i t h t h e " h i g h - f i e l d " s t u d i e s where t h e q u a d r u p o l e i n t e r a c t i o n i s s m a l l enough t o be t r e a t e d as a p e r t u r b a t i o n on t h e m a g n e t i c e n e r g i e s . ( 3 3 ) The a p p l i c a t i o n o f a.magnetic f i e l d t o a s p i n system g i v e s r i s e t o t h e Zeeman e n e r g i e s = - g ^ H . I . W i t h z|j H, t h e a p p l i e d f i e l d , a c o o r d i n a t e system i s d e f i n e d i n w h i c h i s d i a g o n a l and has t h e e i g e n v a l u e s E ( o ) m -g/3Hm r - h ] | m , m.= - I , -1+1, 1 ( 3 4 ) There a r e t h u s 21+1" e q u a l l y spaced m a g n e t i c energy l e v e l s , s e p a r a t e d by h \JL , between w h i c h d i p o l e t r a n s i t i o n s can be i n d u c e d by t h e a p p l i c a t i o n o f a r a d i o - f r e q u e n c y f i e l d t r a n s v e r s e t o H. The s e l e c t i o n r u l e s A m = +1- g i v e r i s e t o a s i n g l e r e s o n a n c e l i n e a t t h e Larmor f r e q u e n c i e s ( f i g u r e 2 a ) . I n g e n e r a l t h e axes z and Z f o r t h e m a g n e t i c and q u a d r u p o l i n t e r a c t i o n d i f f e r . We have t h e n f o r our H a m i l t o n i a n d e s c r i b i n g t h e two e f f e c t s , H = «M. + H Q - - P V z + 4iry?iy [ 3 I z - - ^ " l . <35> I f t h e q u a d r u p o l e c o u p l i n g i s weak compared t o t h e m a g n e t i c i n t e r a c t i o n we c o n s i d e r t h e s p i n q u a n t i z e d a l o n g t h e z a x i s . D e f i n i n g t h e x a x i s t o l i e i n t h e p l a n e o f z and Z we have ( f i g . 1) I„ = I cos/) + I s i n # ( 3 6 ) Z z w x I n e q u a t i o n ( 3 5 ) we now have / -14-F i g . 1 x, z, X and Z axes By w r i t i n g 1^ = | ( l + + I~) and I = ( l / 2 i ) ( l + - l~) , eq. ( 3 7 ) gives*- 9-* i(3 cos 2e -1)(3I 2 - 1(1+1) £• 'Zi | s i n # c o s # [ l z ( l + + l " ) + ( l + + f ) l j ( 3 8 ) J 3 . 2 A ,_+2 , - 2 N T h i s i s w r i t t e n t o show up t h e d i a g o n a l (A m = 0) a n d o f f -d i a g o n a l (A m = +1, +2) m a t r i x e l e m e n t s o f H^ w h i c h w i l l come i n t o t h e p e r t u r b a t i o n t r e a t m e n t . The t o t a l H a m i l t o n i a n H = + has energy l e v e l s E = E (°> + E ^ f E ( 2) + ... m m m m ( p ) where E V J^' i s t h e c o n t r i b u t i o n t o t h e energy o f t h e p e r t u r b a t i o n o f o r d e r - 1 5 -The z e r o - t h o r d e r t e r m , t h a t of- t h e Zeeman c o u p l i n g a l o n e i s En/°) = - y t i l l m = - h l | m. The f i r s t o r d e r t e r m E n / 1 ^ = ^ m i J H ^ m ^ i s E m ( 1 ) = J h i ; Q ( 3 ^ 2 - l ) ( m - § a) (3 9 ) where f o r c o n v e n i e n c e t h e f o l l o w i n g t erms have been i n t r o d u c e d : Em ( 2 ) _ I f t h e p e r t u r b a t i o n i s c a r r i e d t o second o r d e r , we g e t Z I M s B l 2 n/m m | 2 ( l - > U 2 ) ( 8 m 2 - 4 a + l ) + | ( l - y U 2 ) (-2m 2+2a+l) (40) Because o f t h e s e changes i n t h e e n e r g y l e v e l s , t h e r e a r e now s e v e r a l r e s o n a n c e f r e q u e n c i e s : 1/ m h L m m I n f i r s t o r d e r , t h e m a g n e t i c r e s o n a n c e l i n e i s s p l i t i n t o 21 components w i t h r e l a t i v e i n t e n s i t y n e a r l y e q u a l t o l ( l + l ) - m(m-l) f o r t h e m -»m-l t r a n s i t i o n . F o r systems w i t h h a l f - i n t e g r a l s p i n s , t h e c e n t r a l (-^ — \ ) component i s u n a f f e c t e d and t h e s h i f t e d components, c a l l e d s a t e l l i t e s a r e a r r a n g e d s y m m e t r i c a l l y about t h e c e n t r a l t r a n s i t i o n i n p a i r s ( f i g u r e 2 b ) . The f r e q u e n c y o f t h e s p l i t t i n g i s g i v e n by V ( 1 ) E W - E W m-l m m w h i c h v a n i s h e s f o r m _ I 25 (41) The Mg . i s o t o p e , w i t h a n u c l e a r s p i n I = 5/2. would have i t s r e s o n a n c e l i n e s p l i t i n t o f i v e components. -16-m 2 a b F i g . 2: Q u a d r u p o l e S p l i t t i n g o f t h e M a g n e t i c Resonance o f a N u c l e u s S p i n 3/2 a: A p p l i e d F i e l d O n l y b; W i t h Q u a d r u p o l e I n t e r a c t i o n . ' The; e n e r g y l e v e l s shown a t t o p i n u n i t s o f h. The b o t t o m shows t h e l i n e shape for a p o l y c r y s t a l l i n e sample. -17-I f t h e p e r t u r b a t i o n i s c a r r i e d t o second o r d e r t h e c e n t r a l component i s no l o n g e r u n d e v i a t e d b u t i s s h i f t e d by p ( 2 ) F ( 2 ) ,2 r 1 -u T c -.. ie\ ^ ~ i ^ 1 " / ) < 9 / ~ 1 } . ( 4 2 ) The o t h e r components a r e o f c o u r s e s h i f t e d as w e l l , b u t w i l l n o t b e : c o n s i d e r e d h e r e . The a n g u l a r d e p e n d e n c i e s o f e q u a t i o n s ( 4 1 ) and ( 4 2 ) a r e shown i n f i g u r e 3. I n a powdered o r p o l y c r y s t a l l i n e sample t h e random d i s t r i -b u t i o n o f o r i e n t a t i o n o f t h e c r y s t a l a x i s w i t h r e s p e c t t o t h e m a g n e t i c f i e l d g i v e s r i s e t o a c o n t i n u o u s d i s t r i b u t i o n o f f r e q u e n c i e s . The e f f e c t t h e n o f t h e q u a d r u p o l e i n t e r a c t i o n i n s p l i t t i n g t h e r e s o n a n c e l i n e i n t o s a t e l l i t e s o r i n s h i f t i n g t h e c e n t r a l l i n e ( s e c o n d - o r d e r e f f e c t ) i s m a n i f e s t e d as a b r o a d e n i n g o f t h e l i n e . As a r e s u l t o f t h i s , t h e i n t e r a c t i o n must be weak t o be o b s e r v a b l e . Because o f t h e a d v a n t a g e s o u t l i n e d i n c h a p t e r f o u r , t h e e x p e r i m e n t may be p e r f o r m e d on a s i n g l e c r y s t a l w h e r e = cos Q i s w e l l d e f i n e d . I f t h e c r y s t a l i s r o t a t e d about an a x i s p e r p e n d i c u l a r t o t h e m a g n e t i c f i e l d , a p l o t o f t h e r e s o n a n c e f r e q u e n c y as a f u n c t i o n o f a n g l e o f r o t a t i o n w i l l g i v e t h e c u r v e s o f f i g u r e 3. The f i r s t ( f i g u r e 3 a) shows t h e f r e q u e n c y i n t e r v a l between t h e c e n t r a l l i n e and t h e s a t e l l i t e s f o r t h e f i r s t o r d e r , s h i f t s w h i l e t h e second ( f i g u r e 3b) d e m o n s t r a t e s t h e s e c o n d - o r d e r s h i f t o f t h e c e n t r a l component. The s a t e l l i t e l i n e s a r e v e r y s e n s i t i v e t o t h e e f f e c t s o f s t r a i n s and i m p e r f e c t i o n s ^ ^ i n t h e c r y s t a l and a r e t h u s c o n s i d e r a b l y w i d e r t h a n t h e c e n t r a l component and c o r r e s p o n d i n g l y more -18-I I 1 1 1 I 1 I- 1 . 0* 20° U0° 60° 80° 100° 120° 1MD° 160° 180° F i g . 3a The angular dependence of the Knight s h i f t anisotropy and/or the s a t e l l i t e l i n e s to f i r s t order 0° 20° i 0 ° 60° 80° 100° 120° •'1*0° 160° 180° F i g . 3b The angular dependence of the s h i f t of the central _L) t r a n s i t i o n due to the quadrupole interaction -19-d i f f i c u l t t o o b s e r v e o r measure a c c u r a t e l y . The p a r a m e t e r o f i n t e r e s t , 2 eq Q/h, commonly c a l l e d t h e q u a d r u p o l e c o u p l i n g c o n s t a n t , c a n t h e r e f o r e i n some c a s e s , be more a c c u r a t e l y d e t e r m i n e d f r o m measurement o f t h e s h i f t o f t h e c e n t r a l component. C. F i e l d G r a d i e n t o f t h e I o n i c M o d e l As d i s c u s s e d e a r l i e r , t h e p o t e n t i a l V p r o d u c i n g t h e f i e l d g r a d i e n t a r i s e s f r o m e x t e r n a l c h a r g e s o f o t h e r n u c l e i o r e l e c t r o n s . A c h a r g e e a t t h e p o i n t x, y, z o f t h e p r i n c i p l e axes p r o d u c e s a t t h e o r i g i n a f i e l d g r a d i e n t o f _ 3 2 V _ o ( 3 z 2 - r 2 ) - ,,„x e q - „ - e ^ g >- ( 43 ) C z r o r i n t e r m s o f s p h e r i c a l c o o r d i n a t e s q. V ( 3 COaf~ 1 > (44 ) r I f t h e i o n i c c r y s t a l e x i s t s as an a r r a y o f n o n - o v e r l a p p i n g , s p h e r i c a l l y s y m m e t r i c i o n s , t h e n t h e r e s u l t i n g l o c a l e l e c t r i c f i e l d g r a d i e n t , . V. w o u l d be t h a t a r i s i n g f r o m a l l t h e o t h e r i o n s . T h i s c o u l d be d e t e r m i n e d by summing t h e above q^ o v e r a l l l a t t i c e s i t e s a d i s t a n c e f . = ( x . , y., z.) f r o m t h e o r i g i n t o g e t x ^ x 1 x f 3 z . 2 - r . 2 i However, t h e i o n i s d i s t o r t e d b o t h by t h e q u a d r u p o l a r f i e l d o f t h e n u c l e u s and by t h e l o c a l e l e c t r i c f i e l d g r a d i e n t i t s e l f . Because o f - 2 0 -their 1/r dependence, the electrons in the closed shell of the ion make a substantial contribution to the total field gradient. This additional contribution is denoted by . where Yo> is termed the "antishielding factor". c This induced quadrupolar coupling is the sum of two terms: a) the nuclear quadrupole moment (N.Q.M.) polarizes the ionic shell and produces an electronic quadrupole moment which can then interact with the external gradient V. . , and 6 ' 13 b) the external gradient also polarizes the ionic shell which produces an additional gradient at the nucleus which wil l then inter-act with the N.Q.M. Formally the effects are treated as the interaction of a total field (1 + MV. )V. . with the nuclear moment Q. . . Physically, the antishielding is due to a rearrangement of electronic charge. If Q is positive, the potential energy is a minimum along the axis of Q and the electrons undergo an angular redistribution and concentrate there. This "angular excitation" is due to excitement of s-electrons into higher d states and of p-electrons into higher f states. Accompanying this may be a radial shift of charge inward along the axis and outward at right angles to i t . This "radiaL excita-tion", resulting from the excitation of p-electrons into higher p . states and d electrons into higher d states, tends to reinforce the effect of the N.Q.M. The angular redistribution wil l cause shielding "while the radial excitation is responsible for antishielding. The -21-a n t i s h i e l d i n g e f f e c t i s u s u a l l y r e l a t i v e l y l a r g e r where p r e s e n t , s i n c e t h e r a d i a l r e d i s t r i b u t i o n s h o u l d p r o p a g a t e t o t h e o u t e r r e g i o n s o f t h e i o n . The a n t i s h i e l d i n g f a c t o r f o r s m a l l i o n s can be p o s i t i v e o r n e g a t i v e d e p e n d i n g on t h e e l e c t r o n d i s t r i b u t i o n o f t h e c o r e . F o r h e a v i e r i o n s , ^ ^ i s a l w a y s l a r g e and p o s i t i v e . E f f e c t i v e l y t h e n , t h e i o n i c c o n t r i b u t i o n t o t h e e l e c t r i c f i e l d g r a d i e n t i s denoted l i o n = Z < 1 + ^ l a t t i c e ( 4 where Z i s t h e n o r m a l v a l e n c e o f t h e i o n . The e f f e c t s o f t h e v a l e n c e e l e c t r o n s has been e x c l u d e d h e r e . T hat t h e y can have a s u b s t a n t i a l e f f e c t w i l l be d i s c u s s e d i n c h a p t e r f o u r . CHAPTER I I I APPARATUS AND TECHNIQUE E f f o r t s h e r e were d i r e c t e d a t d e t e c t i n g t h e s t e a d y - s t a t e r e s o n a n c e s i g n a l t o d e t e r m i n e t h e f r e q u e n c y dependence as t h e m a g n e t i c f i e l d o r i e n t a t i o n was v a r i e d w i t h r e s p e c t t o a p a r t i c u l a r c r y s t a l a x i s . The e x p e r i m e n t a l a p p a r a t u s and t e c h n i q u e s used were t h o s e most commonly employed i n n u c l e a r m a g n e t i c r e s o n a n c e s p e c t r o s c o p y . ^ " ^ A. G e n e r a l D e s c r i p t i o n I n t h i s c a s e t h e s t e a d y - s t a t e r e s o n a n c e was d e t e c t e d by m o d u l a t i n g t h e m a g n e t i c f i e l d and sweeping t h e f r e q u e n c y o f t h e a p p l i e d r . f . f i e l d . The a n g u l a r o r i e n t a t i o n o f t h e sample w i t h r e s p e c t t o t h e m a g n e t i c f i e l d c o u l d be v a r i e d by r o t a t i n g t h e magnet about t h e sample w h i c h was r i g i d l y c o n n e c t e d t o a frame above t h e magnet. A c o i l wound about t h e c r y s t a l sample formed t h e i n d u c t i v e l o a d o f t h e t a n k c i r c u i t o f a m o d i f i e d P o u n d - K n i g h t - W a t k i n s o s c i l l a t i n g d e t e c t o r . The sample was h e l d w i t h i n ; " a c o p p e r bomb r i g i d l y f i x e d t o 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 l e a d i n g t o t h e o s c i l l a t o r . A l i q u i d h e l i u m c r y o s t a t e n c l o s e d t h e bomb and c o a x i a l l i n e . The o u t p u t o f t h e o s c i l l a t o r was f e d i n t o a l o c k - i n a m p l i f i e r and t h e s i g n a l r e c o r d e d on a s t r i p - c h a r t r e c o r d e r . An e l e c t r o n i c a l l y p r o duced s a w t o o t h waveform was a p p l i e d t o a v o l t a g e v a r i a b l e c a p a c i t a n c e d i o d e w h i c h was i n c o r p o r a t e d i n t o t h e t a n k c i r c u i t o f t h e o s c i l l a t o r t o p r o v i d e t h e f r e q u e n c y sweep. A s c h e m a t i c d i a g r a m o f t h e a p p a r a t u s i s shown i n f i g u r e 4 . Power Amplif ier Audio Oscillator PKW Oscillating Detector A SL PAR Lock-in Ampl i f ie r Hewlett Packard Electronic Counter Printer Varian Strip Chart Recorder i CO I F i g . 4 S c h e m a t i c D i a g r a m o f t h e S p e c t r o m e t e r -24-B. C r y s t a l v s . Powder The s t u d y b f n.m.r. i n m e t a l s i s more complex t h a n i n n o n - c o n d u c t o r s due t o t h e o r d i n a r y c o n d u c t i v e l o s s e s i n t h e sample. The sample can be p e n e t r a t e d by t h e r . f . f i e l d o n l y a d i s t a n c e g i v e n by t h e c l a s s i c a l s k i n d e p t h 5 - ^ ^ w h e r e i s t h e r e s i s t i v i t y , 1/ t h e f r e q u e n c y , and K t h e m a g n e t i c p e r m e a b i l i t y . T h i s " s k i n e f f e c t " c a u s e s n o t o n l y a t t e n u a t i o n o f t h e r . f . f i e l d as i t e n t e r s t h e m e t a l , b u t a l s o a phase s h i f t between t h e s u r f a c e and p o i n t s w i t h i n t h e sam p l e . To overcome t h i s p r o b l e m , t h e sample i s u s u a l l y p r e p a r e d i n t h e f o r m o f a powder, w i r e o r f o i l , h a v i n g a t l e a s t one d i m e n s i o n s m a l l compared w i t h t h e s k i n d e p t h . I f t h i s c o n d i t i o n i s n o t s a t i s f i e d , n o t o n l y does t h e a t t e n u a t i o n cause a r e d u c t i o n i n i n t e n s i t y , b u t t h e phase s h i f t c a u s e s a d i s t o r t i o n o f t h e o b s e r v e d r e s o n a n c e a b s o r p t i o n w h i c h can be d e s c r i b e d as a m i x i n g o f t h e a b s o r p -t i o n and d i s p e r s i o n modes o f t h e complex n u c l e a r s u s c e p t i b i l i t y . The (13") e f f e c t s on t h e l i n e shape o f t h i s m i x i n g i s u n d e r s t o o d v ' and can be c o r r e c t e d f o r i f n e c e s s a r y . A t l o w t e m p e r a t u r e s , a pure sample o f a good c o n d u c t o r may p a s s i n t o t h e "anomalous s k i n e f f e c t " r e g i o n where t h e mean, f r e e p a t h o f t h e e l e c t r o n s i s l a r g e compared t o t h e n o r m a l s k i n d e p t h . T h i s c o n d i t i o n f a c i l i t a t e s g r e a t e r p e n e t r a t i o n o f t h e r . f . f i e l d . E x p e r i m e n t s on s i n g l e c r y s t a l s a r e t h e n more d i f f i c u l t b e c a u s e o f t h e l o w s e n s i t i v i t y a f f o r d e d by t h e s k i n e f f e c t , so most work on m e t a l s i s p e r f o r m e d i n powdered samples. T h i s method has s e v e r a l a d v a n t a g e s . The l i n e i s much more i n t e n s e because e s s e n t i a l l y a l l t h e sample i s e x p o s e d t o t h e r . f . f i e l d . S e p a r a t i n g t h e modes i s no l o n g e r n e c e s s a r y s i n c e t h e o b s e r v e d s i g n a l w i l l be p u r e l y a b s o r p t i v e . -25-S i n c e t h e sample i s made up o f randomly o r i e n t e d c r y s t a l l i t e s , s e v e r a l d i f f i c u l t i e s a l s o a r i s e . The l i n e i s n e c e s s a r i l y broadened due t o t h e v a r i a t i o n o f r e s o n a n c e p o s i t i o n as a f u n c t i o n o f c r y s t a l o r i e n t a t i o n . When t h e e f f e c t u n d e r s t u d y i s o r i e n t a t i o n dependent, such as t h e K n i g h t s h i f t a n i s o t r o p y o r t h e q u a d r u p o l e e f f e c t s , t h e o b s e r v e d l i n e i s a s y m m e t r i c a l l y b r o a d e n e d , t h e r e s u l t o f a w e i g h t e d a v e r a g i n g o f (14) t h e i n d i v i d u a l p a r t i c l e o r i e n t a t i o n . These l i n e s can be i n t e r p r e t e d ' t o g i v e m e a n i n g f u l r e s u l t s b u t t h i s method becomes q u i t e complex when two o r more a n g u l a r l y dependent e f f e c t s a r e o p e r a t i n g t o g e t h e r ( C h a p t e r f o u r ) . W i t i i a s i n g l e c r y s t a l sample, t h e r e s o n a n c e f r e q u e n c y can be o b s e r v e d t o change as t h e a n g l e between t h e m a g n e t i c f i e l d and t h e c r y s t a l a x i s I s v a r i e d . I n t h i s way, v e r y s m a l l e f f e c t s can be o b s e r v e d w h i c b e s c a p e d e t e c t i o n i n t h e powdered t e c h n i q u e . U s i n g a s i n g l e c r y s t a l a l s o a l l o w s t h e measurement o f b u l k m a t e r i a l p r o p e r t i e s and n o t j u s t s u r f a c e e f f e c t s . C. E x p e r i m e n t a l T e c h n i q u e and C o n s i d e r a t i o n s 1... The Sample The sample i t s e l f was a c y l i n d r i c a l s i n g l e c r y s t a l 1.5 i n c h e s long, b y 0.5 i n c h e s i n d i a m e t e r . I t was s p a r k c u t f r o m a l a r g e r i n g o t : o f 99.999% p u r e magnesium so t h a t t h e c y l i n d r i c a l a x i s made a r i g h t a n g l e w i t h t h e ^ 0 0 0 1 ^ a x i s . S i n c e magnesium i s h e x a g o n a l , t h e e l e c t r i c f i e l d g r a d i e n t i s c o n s i d e r e d t o be a x i a l l y s y m m e t r i c ( 11 = 0) a b o u t "the h e x a g o n a l <0°°1/ a x i s o f t h e c r y s t a l . Thus when -26-t h e c r y s t a l was mounted w i t h i t s c y l i n d r i c a l a x i s v e r t i c a l , a h o r i -z o n t a l r o t a t i o n o f t h e magnet p r o d u c e d t h e r e q u i r e d a n g u l a r v a r i a t i o n f r o m 0° t o 360°. X - r a y a n a l y s i s d e t e r m i n e d t h e a c c u r a c y o f t h e c r y s t a l o r i e n t a t i o n t o w i t h i n one d e g r e e . A f t e r s p a r k - c u t t i n g and s l i g h t a n -n e a l i n g , t h e c r y s t a l was l i g h t l y e t c h e d i n a 1 0 % s o l u t i o n o f HC1 t o r e d u c e s u r f a c e s t r a i n s and i m p e r f e c t i o n s . The n a t u r e o f t h e c o i l was d e t e r m i n e d by t h e o p e r a t i n g c h a r a c t e r i s t i c s o f t h e o s c i l l a t o r . S i n c e t h e optimum s e n s i t i v i t y seemed t o r e q u i r e a s m a l l c a p a c i t a n c e i n t h e t a n k c i r c u i t w h i c h c o u l d be v a r i e d o v e r a v e r y l i m i t e d r a n g e , t h e f r e q u e n c y o f o s c i l l a t i o n was d e t e r m i n e d by t h e i n d u c t a n c e o f t h e sample c o i l . T r i a l and e r r o r p r o d u c e d a 2 cm l o n g c y l i n d r i c a l c o i l c o n s i s t i n g o f 450 t u r n s o f #44 c o p p e r w i r e wound on t h e c r y s t a l o v e r a s i n g l e l a y e r o f 0.0005" M y l a r . Too l o n g a c o i l w ould r e s u l t i n t h e sample o c c u p y i n g a l a r g e volume i n t h e m a g n e t i c f i e l d , p o s s i b l y l e a d i n g t o i n h o m o g e n e i t y ; t o o s h o r t a c o i l w ould d e c r e a s e t h e number o f n u c l e i a v a i l a b l e and r e d u c e t h e s i g n a l . . , • The s i n g l e l a y e r o f M y l a r was i n c o r p o r a t e d t o p r o t e c t t h e c r y s t a l s u r f a c e f r o m t h e c o i l w i n d i n g s and t o s l i g h t l y i n c r e a s e t h e i n d u c t a n c e so t h e c o i l l e n g t h c o u l d be k e p t down. However, t h e M y l a r i n c r e a s e d t h e q u a l i t y f a c t o r , Q, o f t h e c i r c u i t s l i g h t l y more t h a n was d e s i r a b l e , d e c r e a s i n g t h e f i l l i n g f a c t o r and t h u s t h e s e n s i t i v i t y . The f i l l i n g f a c t o r , t h e f r a c t i o n o f t h e t o t a l r a d i o f r e q u e n c y m a g n e t i c -27-e n ergy t h a t i s s t o r e d i n t h e sample, s h o u l d be k e p t as h i g h as p o s s i b l e f o r maximum s i g n a l . 2. M o d u l a t i o n and D e t e c t i o n The o s c i l l a t i n g d e t e c t o r was a P o u n d - K n i g h t - W a t k i n s t y p e , b u i l t and m o d i f i e d by S.N. Sharma. A f u l l d e s c r i p t i o n o f i t may be f o u n d i n h i s t h e s i s . An i n c r e a s e i n s e n s i t i v i t y was a t t a i n e d by use o f p h a s e -s e n s i t i v e d e t e c t i o n , a common and n e c e s s a r y t e c h n i q u e f o r weak s i g n a l d e t e c t i o n . I n t h i s , t h e f i e l d i s modulated so t h a t t h e s i g n a l a p p e ars a t t h e i n p u t o f t h e d e t e c t o r as a v a r i a t i o n i n a m p l i t u d e o f a p e r i o d i c f u n c t i o n . S ( t ) = G s i n ( w t + S ) ( 4 7 ) where i s t h e phase d i f f e r e n c e between t h e s i g n a l and r e f e r e n c e , R ( t ) , w i t h w h i c h t h e s i g n a l i s mixed. F o r s i m p l i c i t y assume t h a t t h e r e f e r e n c e i s s i n u s o i d a l , R ( t ) = A s i n wt. A f t e r m i x i n g , t h e r e s u l t i s i n t e g r a t e d by p a s s i n g t h r o u g h a l o w - p a s s f i l t e r o f i n t e g r a t i n g t i m e T t o g i v e an o u t p u t rttT E = ^ A  b o T J s i n wt s i n (wt + S ) d t °A c o s S ( 4 8 ) so t h a t maximum s i g n a l i s a t t a i n e d when r e f e r e n c e and s i g n a l a r e i n phase Or 180° o u t o f phase ( S = 0°, 180°). I n p r a c t i s e t h e d e v i c e behaves as a v e r y n a r r o w banded a m p l i f i e r . When t h e r e f e r e n c e s i g n a l i s a t t h e f r e q u e n c y o f t h e m o d u l a t i o n , t h e d e t e c t e d o u t p u t i s -28-p r o p o r t i o n a l t o t h e f i r s t d e r i v a t i v e o f t h e a b s o r p t i o n s i g n a l when t h e m o d u l a t i o n i s s u f f i c i e n t l y s m a l l . The o s c i l l a t o r f r e q u e n c y r a t h e r t h a n t h e m a g n e t i c f i e l d was swept t o d e t e r m i n e t h e r e s o n a n c e f r e q u e n c y . T h i s was n e c e s s i t a t e d by t h e h i g h m a g n e t o r e s i s t a n c e e n c o u n t e r e d i n t h e magnesium specimen. I t was f o u n d t h a t sweeping t h e f i e l d c aused v a r i a t i o n s i n r e s i s t a n c e o f t h e sample, t h u s a f f e c t i n g t h e Q o f t h e t u n e d c i r c u i t . T h i s caused l a r g e o s c i l l a t i n g d r i f t s o f t h e r . f . l e v e l w h i c h c o u l d n o t be c o n t a i n e d on t h e c h a r t r e c o r d e r w i t h o u t r e d u c i n g a m p l i f i c a t i o n and making s i g n a l d e t e c t i o n i m p o s s i b l e . The m o d u l a t i o n can a l s o be a p p l i e d , t o e i t h e r t h e r . f . f r e q u e n c y o r t h e m a g n e t i c f i e l d . F r e q u e n c y m o d u l a t i o n i s d i f f i c u l t b e c a u s e t h e r . f . l e v e l o f t h e t a n k c i r c u i t i s f r e q u e n c y dependent; a p e r i o d i c o s c i l l a t i o n o f t h e r . f . l e v e l a p p e ars w h i c h w o u l d , o f c o u r s e , n o t be f i l t e r e d by p h a s e - s e n s i t i v e d e t e c t i o n s i n c e i t i s o f t h e same f r e q u e n c y as t h e s i g n a l . A t t h e i n p u t o f t h e d e t e c t o r , i n s t e a d o f t h e f u n c t i o n o f e q u a t i o n ( 2 7 ) would be V ( t ) = C s i n (wt + 5 ) + D s i n (wt + & 2) ( 4 9 ) where C s i n (wt + i s t n e s i g n a l and D s i n (wt + ^ 2 ) i s t h e e f f e c t o f m o d u l a t i o n p i c k e d up on t h e r . f . l e v e l . Here <§ and § a r e phase d i f f e r e n c e s f r o m t h e r e f e r e n c e and a r e n o t i n g e n e r a l e q u a l . A f t e r m i x i n g and f i l t e r i n g , t h e d e t e c t e d r e s u l t i s -29-Thus t h e m o d u l a t i o n p i c k u p i s m a n i f e s t e d as a d.c. d r i f t w h i c h may be t o o l a r g e t o b a l a n c e i n t h e d e t e c t o r o r may o v e r l o a d t h e l o c k - i n a m p l i f i e r making i t n e c e s s a r y t o o p e r a t e a t a l o w e r a m p l i f i c a t i o n . U n f o r t u n a t e l y , t h e h i g h m a g n e t o r e s i s t a n c e caused t h e same e f f e c t on t h e r . f . l e v e l o f t h e t a n k c i r c u i t when m o d u l a t i o n was a p p l i e d t o t h e m a g n e t i c f i e l d . The d.c. d r i f t c a used by t h i s m o d u l a t i o n p i c k u p was overcome by d e t e c t i n g t h e second h a r m o n i c o f t h e r e s o n a n c e s i g n a l . The s i g n a l a p p e a r i n g a t t h e o u t p u t i s n o t p u r e l y s i n u s o i d a l and can be F o u r i e r decomposed i n t o - t e rms o f h i g h e r h a r m o n i c s . I t can be shown t h a t when m o d u l a t i o n o f s u f f i c i e n t l y s m a l l a m p l i t u d e i s a p p l i e d , t h e o u t p u t o f t h e phase s e n s i t i v e d e t e c t o r d e t e c t i n g a t t h e n*"^ h a r -monic o f t h e m o d u l a t i o n i s p r o p o r t i o n a l t o t h e n ^ d e r i v a t i v e o f t h e r e s o n a n c e l i n e shape S i n c e t h e m o d u l a t i o n p i c k u p has a l o w e r h a r m o n i c c o n t e n t t h a n t h e s i g n a l , t h e sys t e m can be us e d a t h i g h e r s e n s i t i v i t y and t h e p r o b l e m o f d.c. d r i f t s s i d e s t e p p e d . The m o d u l a t i o n a p p l i e d t o t h e f i e l d was one gauss p e a k - t o - p e a k a m p l i t u d e a t 20 Hz. The second d e r i v a t i v e was d e t e c t e d by u s i n g a r e f e r e n c e o f 40 Hz i n t h e l o c k - i n . Because t h e s i g n a l a m p l i t u d e i s dependent on t h e modula-t i o n a m p l i t u d e , l a r g e m o d u l a t i o n s may be a p p l i e d t o i n c r e a s e t h e s i g n a l -t o - n o i s e r a t i o . T h i s c a u s e s d i s t o r t i o n b r o a d e n i n g o f t h e l i n e w h i c h , however, can be c o r r e c t e d f o r ^ " * " ^ i f n e c e s s a r y . 3. The M a g n e t i c F i e l d •> The e x p e r i m e n t was p e r f o r m e d i n a m a g n e t i c f i e l d o f 20 -30-K i l o g a u s s . The c h o i c e o f f i e l d was d i c t a t e d by s e v e r a l f a c t o r s . The PKW o s c i l l a t o r p e r f o r m e d s e n s i t i v e l y o v e r a range o f about 8 t o 25 MHz, b u t w i t h some a l t e r a t i o n s and p r o p e r c h o i c e o f sample c o i l c o u l d be made t o p e r f o r m a d e q u a t e l y a t 5 MHz. I n a s t a t i c f i e l d o f 20 Kg t h e m a g n e t i c moment b f magnesium would produce, r e s o n a n c e a t a f r e q u e n c y o f 5.2 MHz. T h i s r e l a t i v e l y h i g h f i e l d was advantageous 2 s i n c e t h e s i g n a l i s p r o p o r t i o n a l t o H . A l t h o u g h t h e h i g h f i e l d magnet a v a i l a b l e would p r o v i d e up t o 24 Kg, i t s homogeneity d e c r e a s e d d r a s t i c -a l l y a t h i g h f i e l d s , p u t t i n g an upper l i m i t on t h e f i e l d a v a i l a b l e . The magnet, a r o t a t a b l e Magnion model, e q u i p p e d w i t h a f i e l d s e n s i n g d e v i c e and a c u r r e n t r e g u l a t o r , s u p p l i e d a v e r y s t a b l e f i e l d ( l e s s t h a n one gauss v a r i a t i o n ) o v e r a p e r i o d o f d a y s . I n t h e c e n t r e o f t h e p o l e f a c e s were a d j u s t a b l e c o r e s , c a l l e d J U — s h i m s , used t o o p t i m i z e t h e homogeneity o f t h e f i e l d . W i t h o u t a h i g h degree o f ho m o g e n e i t y , t h e n u c l e i i n v a r i o u s p a r t s o f t h e sample e x p e r i e n c e d i f f e r e n t m a g n e t i c f i e l d s and t h e o b s e r v e d r e s o n a n c e l i n e i s br o a d e n e d . The r e q u i r e d homogeneity, about one p a r t i n 1 0 ^ 3 was a c h i e v e d by u s i n g a s p e c i a l probe c o n s i s t i n g o f f o u r v i a l s o f l i t h i u m c h l o r i d e s o l u t i o n a r r a n g e d s y m m e t r i c a l l y about t h e c e n t r e o f t h e gap i n t h e p l a n e o f t h e p o l e f a c e s ( f i g . 5 ) . The p o l e f a c e s were a d j u s t e d t o produce t h e same f i e l d a t each o f t h e f o u r s a m p l e s ; t h i s was d e t e r m i n e d when t h e r e s o n a n t peaks f r o m each c o i n c i d e d ( f i g . 6 ) . The y £ / -shims were used as a f i n e a d j u s t m e n t t o y i e l d a homogeneity o f about one p a r t i n 10 ^ i n a r e g i o n 3 cm i n d i a m e t e r . -31-F i g . 6a O s c i l l o s c o p e t r a c e showing s i g n a l when probe i n inhomogeneous f i e l d F i g . 6b F i e l d homogenized -32-The f i e l d was measured b e f o r e and a f t e r t h e e x p e r i m e n t by t h e use o f a d e u t e r i u m o x i d e pr<obe t h e same s i z e as t h e sample and i n t h e e x a c t p o s i t i o n t h e sample was t o occupy. F u r t h e r enhancement o f t h e s i g n a l was a c h i e v e d by p e r -f o r m i n g t h e o b s e r v a t i o n s a t l i q u i d h e l i u m t e m p e r a t u r e , 4.2°K, s i n c e t h e a v a i l a b l e n u c l e a r m a g n e t i z a t i o n v a r i e s i n v e r s e l y w i t h t e m p e r a t u r e . The l o w t e m p e r a t u r e s y s t e m was o f a s t a n d a r d d e s i g n ; i t c o n s i s t e d o f an i n n e r l i q u i d h e l i u m dewar, c o n n e c t e d t o a vacuum system and c o n -t a i n e d w i t h i n an o u t e r l i q u i d n i t r o g e n dewar. CHAPTER IV RESULTS AND DISCUSSION A. E x p e r i m e n t a l O b s e r v a t i o n s F o r r e a s o n s d i s c u s s e d e a r l i e r , f r e q u e n c y measurements o f t h e c e n t r a l (-^ -*\) t r a n s i t i o n o f t h e magnesium s i n g l e c r y s t a l were t a k e n as t h e d i r e c t i o n o f t h e m a g n e t i c f i e l d was v a r i e d . The e f f e c t s o f o r i e n t a t i o n on t h i s f r e q u e n c y as t h e m a g n e t i c f i e l d i s r o t a t e d i n t h e p l a n e p e r p e n d i c u l a r t o t h e (oOOl^ a x i s i s g i v e n by c o m b i n i n g e q u a t i o n s ( 1 6 ) and ( 4 2 ) : TA = V + V IK. + ( 3 c o s 2 0 - i ) l * \ o o L i s o 2 v i .2 + I6y l ^ 1 " 1 ) - ! ( c o s 2 ^ - l ) ( 9 c o s 2 ^ - 1 ) ] ( 5 1 ) where T/L t h e r e f e r e n c e f r e q u e n c y and ~\) are t a k e n t o be e q u a l and d e n o t e d \ ) Q -Here we have added t o g e t h e r t h e e x p r e s s i o n s f o r t h e s e p a -r a t e i n t e r a c t i o n s o f t h e a n i s o t r o p i c K n i g h t s h i f t ( f i r s t o r d e r ) and t h e n u c l e a r q u a d r u p o l e i n t e r a c t i o n ( s e c o n d o r d e r ) . T h i s i s j u s t i f i e d s i n c e t h e h i g h e r o r d e r terms a r e n e g l i g i b l e and t h e c o n t r i b u t i o n r e s u l t i n g f r o m i n t e r f e r e n c e t e r m s o f t h e two i n t e r a c t i o n s v a n i s h e s ( 1 4 ) i d e n t i c a l l y . The m e a s u r e a b l e p a r a m e t e r s f r o m t h e o b s e r v a t i o n a r e and K', t h e K n i g h t s h i f t p a r a m e t e r s , and t h e magnitude o f t h e 2 q u a d r u p o l e c o u p l i n g c o n s t a n t , e qQ/h. The r e s o n a n c e l i n e o f f i g u r e 7 was o b s e r v e d a t 4.2°K. S i n c e a t t h i s t e m p e r a t u r e t h e l i n e was o b s e r v e d o n l y a t low r . f . f i e l d s oo in i CO I o cn P E A K - T O - P E A K MODULATION Fig. 7 Resonance Line Corresponding to the (\**-\) Transition 25 of Mg in a Single Crystal -35-and a t 1.2 K was b a d l y d i s t o r t e d by s a t u r a t i o n e f f e c t s , a l o n g s p i n -l a t t i c e r e l a x a t i o n t i m e i s p o s s i b l y i n d i c a t e d . The s i g n a l - t o - n o i s e r a t i o was t o o s m a l l a t l i q u i d n i t r o g e n t e m p e r a t u r e t o p r o v i d e a d i s t i n g u i s h a b l e l i n e . The s a t e l l i t e l i n e s f r o m t h e q u a d r u p o l e s p l i t t i n g c o u l d n o t be o b s e r v e d . T h i s i s n o t s u r p r i z i n g i n v i e w o f t h e s m a l l s i g n a l f r o m t h e more i n t e n s e c e n t r a l l i n e and t h e f a c t t h a t t h e w i d t h and i n t e n s i t y o f s a t e l l i t e l i n e s a r e v e r y s e n s i t i v e t o t h e e f f e c t s o f s t r a i n s and i m p e r f e c t i o n s i n t h e l a t t i c e . The l i n e i s d i f f i c u l t t o r e c o g n i z e as a second d e r i v a t i v e s i g n a l b ecause o f t h e b r o a d e n i n g and d i s t o r t i n g e f f e c t s mentioned i n t h e p r e v i o u s c h a p t e r . T h i s u n d e s i r a b l e d i s t o r t i o n d i d n o t a f f e c t t h e a c c u r a c y o f t h e measurements g r e a t l y b ecause o f t h e r e l a t i v e l y n a r r o w l i n e w i d t h o f about 230 Hz. The r e s o n a n c e f r e q u e n c y o f t h e l i n e was d e t e r m i n e d by co m p a r i n g t h e l i n e shape w i t h o v e r m o d u l a t e d l i n e shapes c a l c u l a t e d t h e o r e t i c a l l y and a p p l y i n g a c o r r e c t i o n t o a c c o u n t f o r t h e m i x t u r e o f a b s o r p t i o n and d i s p e r s i o n modes. T h i s c o r r e c t i o n c o u l d a f f e c t o n l y t h e a c c u r a c y o f t h e p a r a m e t e r > so an a c c o r d i n g l y l a r g e r u n c e r t a i n t y i n i t s v a l u e has been a t t a c h e d t o i t . S i n c e t h e l i n e w i d t h and shape d i d n o t v a r y s i g n i f i c a n t l y w i t h t h e a n g l e o f t h e f i e l d , vany c o n v e n i e n t r e f e r e n c e p o i n t on t h e l i n e would s u f f i c e t o d e t e r m i n e t h e a n i s o t r o p i c p a r t K' and t h e q u a d r u p o l e s p l i t t i n g w i t h o u t a c o r r e c t i o n b e i n g n e c e s s a r y . -36-F i g u r e 8 shows t h e o r i e n t a t i o n a l dependence o f t h e r e s o n a n c e f r e q u e n c y o f t h e c e n t r e l i n e . The e x p e r i m e n t a l p o i n t s were f i t t e d t o t h e e x p r e s s i o n o f e q u a t i o n ( 5 1 ) by a l e a s t s q u a r e s f i t r o u t i n e o f f e r e d by t h e U.B.C. Computing .Centre. W i t h t h e e x p e r i m e n t a l p o i n t s i s shown t h e r e s u l t i n g f i t t e d c u r v e w h i c h y i e l d s t h e p a r a m e t e r s g i v e n i n t h e t a b l e b elow. The u n c e r t a i n t i e s a t t a c h e d t o t h e v a l u e s TABLE I K % K' % 2 e qQ/h ( e x p ) KHz 2 e q. Q/h ( c a l c ) u i o n • ' KHz c/a Rm. Temp. O . l l l 3 2 3 0 a 93 1.6235 4.2°K 0.1127 + '0.00.05 0.0004 +• 0.0002 324 + 6 100 1.6230 See r e f e r e n c e 17 a r e t h e s t a t i s t i c a l r o o t mean sq u a r e d e v i a t i o n s s u p p l i e d by t h e computer program. ( 1 7 ) The t a b l e compares t h e r e s u l t s w i t h t h o s e o f Rowland J f o r powdered magnesium a t room t e m p e r a t u r e . The c a l c u l a t e d v a l u e f o r t h e c o u p l i n g c o n s t a n t was a r r i v e d a t by use o f e q u a t i o n s ( 4 6 ) and (47) 2 2 \ L a t t i c e J 3 z . Z - r . • 1 c 1 ; q . = Z ( l +u ' e P ) q 1 l 5 ' ^ i o n v w e p y \ L a t t i ce The summation was c a r r i e d o u t d i r e c t l y o v e r 400,000 s i t e s and t h e same ( 1 8 ) r e s u l t s were o b t a i n e d w i t h an e x p r e s s i o n by Das and Pomerantz, F i g . 8 Measured F r e q u e n c y S h i f t o f C e n t r a l Component o f t h e Mg Resonance oo 5 217.60' 1 I I 1 1 1 1 - I J I J I J I > i 360° 0° 20° U0° 60° 80° 100° 120° ANGLE B E T W E E N H A N D SYMMETRY A X I S -38-q l a t t i c e ~ [ ° - 0 0 6 5 " 4 - 3 5 8 4 ( c / a - 1.633)^ / a 3 . These were e v a l u a t e d (19) a t b o t h t e m p e r a t u r e s u s i n g t h e c o e f f i c i e n t s o f e x p a n s i o n f o r magnesium. • F o r q. ' a n o r m a l v a l e n c e o f 2 was assumed and X-,, t h e a n t i s h i e l d i n g f a c t o r u s e d , was 3 . 2 . ^ 2 ^ The v a l u e o f t h e e l e c t r i c q u a d r u p o l e moment, -24 2 ( 2 1 ) Q = 0.22 x 10 cm , i s f r o m n u c l e a r beam d a t a . B. Comparison w i t h Powder D a t a B e f o r e c o m p a r i n g t h e q u a d r u p o l e c o u p l i n g c o n s t a n t t o Rowland's v a l u e some c h a r a c t e r i s t i c s o f powdered l i n e shapes w i l l be d i s c u s s e d . I n s i n g l e c r y s t a l magnesium, t h e f r e q u e n c y o f t h e c e n t r a l t r a n s i t i o n i s g i v e n by .2 (52) V  vo+ 2 ^ V - ^ y 2 -!)+ «*/2 - » as i n e q u a t i o n ( 5 1 ) b u t w i t h J J L = cos 0 , a = K'/2, and I = 5/2. To deduce t h e l i n e shape f r o m a p o l y c r y s t a l l i n e sample, c o n s i d e r P ( T ^ - V Q ) d ( l)~ V ^ ) / 1 1 ^ t h e p r o b a b i l i t y t h a t (i) - \)q) l i e s between ( V _ y o ) and (~\)-\) Q) + d(]/J->> o). S i n c e t h e p r o b a b i l i t y t h a t a p a r t i c u l a r c r y s t a l l i t e has i t s z a x i s a t an a n g l e t o t h e m a g n e t i c f i e l d i s p r o p o r t i o n a l t o s i n Q , we have P ( l ^ - i ; o ) d(l/-v> 0) = P ( 0 ) d ( 0 ) = ^ s i n £ d # = ^d// ( 5 3 ) so t h a t V(V-V Q) = ^ d V/d^f _ 1 f o r - l ^ i 1; t h e f a c t o r ^ o c c u r s b ecause + $ and -Q c o r r e s p o n d t o t h e same f r e q u e n c y . Now f r o m ( 5 2 ) d(>»- = ( 2 ^ Q 2 / y o ) ^ [ ( 9 y t t 2 - 5) + 3 i ; o 2 a / 2 / Q 2 ] ( 5 4 ) -39-T h i s shape f u n c t i o n has s i n g u l a r i t i e s o c c u r r i n g a t y [ A - 0 and^V ~ ' + ^5/9 - A VQ 2 / 3 VQ21 These JU ' a r e n o t d i s t i n g u i s h a b l e s i n c e 2 V - " V o depends o n l y on^J . B e s i d e s t h e s i n g u l a r i t i e s a d i s c o n t i n u i t y o c c u r s i n t h e shape f u n c t i o n 3X.JH - 1. The l i n e shape a r i s i n g f r o m t h i s t r e a t m e n t i s i n f i g u r e 9. Rowland a t t r i b u t e d t h e s p l i t t i n g o f t h e c e n t r a l t r a n s i t i o n t o t h e q u a d r u p o l e i n t e r a c t i o n a l o n e , n e g l e c t i n g any p o s s i b l e c o n t r i b u t i o n f r o m K n i g h t s h i f t a n i s o t r o p y ( i . e . , a = 0 i n t h e a b o v e ) . T h i s would produce t h e l i n e shape o f f i g u r e 9a u s i n g 2 o 2 h i s v a l u e , e qQ/h = 230 KHz. I f our v a l u e measured a t 4.2 K, e qQ/h = 324 KHz, i s u s e d w i t h no a n i s o t r o p y , t h e r e s u l t i n g l i n e would be as i n f i g u r e 9b. I f o u r v a l u e h o l d s a t room t e m p e r a t u r e , an a n i s o t r o p y o f 0.056% w o u l d be n e c e s s a r y i n a d d i t i o n t o t h e q u a d r u p o l e e f f e c t t o p r o d u c e t h e same s p l i t t i n g Rowland o b s e r v e d . S i n c e i t i s . t h e d e r i v a t e o f t h e l i n e w h i c h i s d e t e c t e d , l i n e 9a would n o t be d i s t i n g u i s h a b l e f r o m l i n e 9c i n a s y s t e m w i t h a poor' s i g n a l - t o - n o i s e r a t i o . T h i s i n c r e a s e f r o m 0.0004% a t 4.2°K t o 0.056% a t room t e m p e r a t u r e would be a s u r p r i z i n g r e s u l t . The a n a l y s i s o f l i n e 9c a l s o a f f e c t s t h e p o s i t i o n o f ~\)' as c a n be s e e n , and would r e q u i r e a 1 0 % i n c r e a s e . i n K. • I S O A l t e r n a t i v e l y , i f t h i s l a r g e change i n a n i s o t r o p y does n o t o c c u r , a h i g h l y t e m p e r a t u r e dependent q u a d r u p o l e i n t e r a c t i o n would p r o d u c e an agreement w i t h Rowland's r e s u l t s . Of c o u r s e , any s u i t a b l e c o m b i n a t i o n o f t h e t e m p e r a t u r e dependences o f t h e two e f f e c t s would ( 2 2 ) a l s o r e s o l v e t h e d i s c r e p a n c y . More r e c e n t l y , D r a i n has r e p o r t e d an a n i s o t r o p y t o o s m a l l , t o measure a t room t e m p e r a t u r e i n powdered -40-FIGURE 9 THEORETICAL POWDER SIGNAL SHAPES 9a 9b 9c F i g . 9 a Rowlands v a l u e , e q Q/h = 230 KHz w i t h K' = 0 F i g . 9b P r e s e n t v a l u e (4.2°K), eq 2Q/h = 324 KHz w i t h K' = 0 2 F i g . 9c eq Q/h = 324 KHz w i t h K' a d j u s t e d t o g i v e same s i g n a l w i d t h Rowland o b s e r v e d See r e f e r e n c e 17 -41-magnesium. A l t h o u g h no l i m i t s a r e g i v e n , we can assume t h a t t h e a n i s o t r o p y was n o t l a r g e enough t o e f f e c t h i s deduced c o u p l i n g c o n s t a n t w h i c h a g r e e s w i t h Rowland's. T h i s would i n d i c a t e t h a t i t i s t h e 2 c o u p l i n g c o n s t a n t , e qQ/h, w h i c h undergoes a l a r g e t e m p e r a t u r e v a r i a t i o n . As shown, a s i n g l e measurement i n powder can l e a d t o ambiguous r e s u l t s , e s p e c i a l l y where weak s i g n a l s a r e i n v e s t i g a t e d . However, s i n c e ' t h e q u a d r u p o l e s p l i t t i n g v a r i e s i n v e r s e l y w i t h "])1 and t h e s p l i t t i n g due t o t h e a n i s o t r o p y v a r i e s d i r e c t l y w i t h " ] / > i f i s p o s s i b l e t o s e p a r a t e t h e c o n t r i b u t i o n s o f each by m e a s u r i n g t h e (14) s p l i t t i n g as a f u n c t i o n o f t h e a p p l i e d f i e l d . ' C. R o l e o f t h e C o n d u c t i o n ' E l e c t r o n s 2 M e a s u r i n g t h e c o u p l i n g ' c o n s t a n t e qQ/h, g i v e s an e x p e r i -m e n t a l v a l u e f o r eq, t h e f i e l d g r a d i e n t p a r a m e t e r . We a l s o have, f r o m c h a p t e r 2,. a c a l c u l a t e d v a l u e , q i o n = Z ( l + Yc*)q l a t t t h e p a r a meter o b t a i n e d by c o n s i d e r i n g t h e i o n i c model o f t h e m e t a l . T e m perature, t h r o u g h t h e r m a l e x p a n s i o n a l t e r i n g t h e l a t t i c e p a r a m e t e r s , does e f f e c t t h e l a t t i c e c o n t r i b u t i o n t o t h e e l e c t r i c f i e l d g r a d i e n t , t hough an e v a l u a t i o n o f q . a t d i f f e r e n t t e m p e r a t u r e s shows f o r magnesium t h a t t h e e f f e c t i s o n l y 7%,from 300 t o 4.2°K. By c omparing t h e e x p e r i m e n t a l and c a l c u l a t e d v a l u e s ( s e e T a b l e 1) i t i s e v i d e n t t h a t q £ o n a c c o u n t s f o r o n l y p a r t o f t h e f i e l d g r a d i e n t a t e i t h e r -42-t e m p e r a t u r e . The r e m a i n d e r must be a t t r i b u t e d t o t h e e l e c t r i c f i e l d g r a d i e n t , c a l l e d < l c o n ( j 5 a r i s i n g f r o m a n o n - s p h e r i c a l d i s t r i b u t i o n o f t h e c o n d u c t i o n e l e c t r o n s . We see t h a t q , q. and seems t o cond i o n i n c r e a s e w i t h d e c r e a s i n g t e m p e r a t u r e f r o m about 2.5 q ^ o n a t room t e m p e r a t u r e t o about 3.2 < l ^ o n a t 4.2°K. 1 c o n c j a P P e a r s t o be p o s i t i v e b u t s i n c e t h i s e x p e r i m e n t does n o t d e t e r m i n e t h e s i g n o f q, we can make s t a t e m e n t s c o n c e r n i n g o n l y m a g n i t u d e s . An a c c u r a t e e v a l u a t i o n o f < l c o n ( j = ^ f | 3 c ° s 2 $ - ll"^/* , t h e q u a d r u p o l e moment o f t h e c o n d u c t i o n e l e c t r o n s g i v e s a measure o f t h e i r n o n - s p h e r i c a l n a t u r e and t h e i r c o n t r i b u t i o n t o t h e i r f i e l d g r a d i e n t . T h i s r e q u i r e s a knowledge o f t h e o r b i t a l c h a r a c t e r i s t i c s o f a l l t h e c o n d u c t i o n s t a t e s i n t h e m e t a l , whereas f o r t h e a n i s o t r o p i c K n i g h t s h i f t , qp = ^ p | 3 c o s 2 $ - l ] * ^ p ( e q n . 1 4 ) , r e q u i r e d o n l y t h e s t a t e s n e a r t h e F e r m i l e v e l . To c a l c u l a t e t h e i r e f f e c t on t h e f i e l d g r a d i e n t , i t i s sometimes u s e f u l t o c o n s i d e r t h e c o n d u c t i o n e l e c t r o n s w i t h a sphere c e n t e r e d about t h e n u c l e u s . I f t h e d e n s i t y w i t h i n t h i s s p h ere i s s p h e r i c a l l y s y m m e t r i c , t h e n t h e n u c l e u s a t the' c e n t r e w i l l f e e l no a d d i t i o n a l f i e l d g r a d i e n t b u t t h e o t h e r n u c l e i i n t h e l a t t i c e may t e x p e r i e n c e a d i f f e r e n t l a t t i c e ^ e c a u s e °^ ^ e e f f e c t i v e l y d i f f e r e n t c h a r g e a t t h e l a t t i c e s i t e . T h i s a d d i t i o n a l c h a r g e may be p o s i t i v e ( 2 3 ) o r , as i n t h e case o f B e r y l i u m , v J n e g a t i v e d e p e n d i n g on whether a n e t e l e c t r o n o r a n e t h o l e d e n s i t y i s c o n s i d e r e d . T h i s may a c c o u n t p a r t i a l l y f o r t h e f a c t t h a t q > q £ o n « - 4 3 -Watson, Gossard and Yafet^ ' suggest that q c o n ( j m a y be separated into 3 terms, Scond = q ° + + 1 " where q° is the local gradient obtained by constructing the Fermi surface on the assumption the Block functions are eigenfunctions of a spherical potential within the atomic sphere. The q' contribution arises from the external environment producing a potential within the atomic sphere which causes a first-order shift in the Block function energies, bringing about a redistribution of occupied states in the vicinity of the Fermi surface. The term q" is associated with the spatial distortion of the Block functions. The terms q° and q" are dependent on the behavior of the conduction states below the Fermi energy Ep, but i t is suggested that the dominant contribution, that of q' , is strongly dependent on N(Ep) the density of states near the Fermi surface only. The term q' is opposite in sign to qn . , therefore shielding i t , and in some cases (transition and some p-band metals) the ratio q 'Au m a y be as v ^ J • 1 ^lattice high as 100. Since i t is also shown that the Fermi-Dirac distribution introduces a large temperature dependence of q T , the total q may well be determined by q' and exhibit a significant temperature variation. Although the density of states has a significant variation with temperature only for metals with narrow conduction bands (d-band in transition metals) and does not vary significantly for s electrons, this mechanism may account in part for the temperature dependence observed in magnesium. CHAPTER V CONCLUSIONS A. . C o m p a r i s o n w i t h Cadmium B o t h t h e i s o t r o p i c and a n i s o t r o p i c s h i f t s i n cadmium v a r y (2 3 4) m a r k e d l y w i t h t e m p e r a t u r e . ' ' The same b e h a v i o r a p p a r e n t l y i s n o t r e p e a t e d i n magnesium; t h i s i s p r o b a b l y due t o t h e d i f f e r e n c e between t h e c/a r a t i o s i n t h e two m e t a l s and t o t h e r e l a t i v e l y s i m p l e r band s t r u c t u r e o f magnesium. The l a r g e t e m p e r a t u r e dependence o f K G N ^ S i n cadmium s u g g e s t s t h a t t h e q u a d r u p o l e moment o f t h e e l e c t r o n s may be r e l a t e d t o t h e change (25) i n t h e c/a r a t i o due t o t h e a n i s o t r o p i c t h e r m a l e x p a n s i o n . ' A d e c r e a s e i n t e m p e r a t u r e f r o m 300°K t o 4.2°K i s accompanied by a d e c r e a s e i n c/a from. 1.8857 t o 1.8640, a change o f 1.2%. W i t h i n t h i s same -4 t e m p e r a t u r e r a n g e , K A N ^ G d e c r e a s e s f r o m 5.0 x 10 t h r o u g h 0.0 a t about '60°K,, t o -1.0 x 1 0 - 4 a t 4.2°K/ 1 5^ The s p e c u l a t i o n i s t h a t t h e e l e c t r o n d i s t r i b u t i o n becomes s p h e r i c a l a t some p a r t i c u l a r c/a r a t i o . (2) To i n v e s t i g a t e t h i s p o s s i b l e dependence B o r s a and Bar n e s c o n s i d e r e d t h e e f f e c t o f a l l o y i n g magnesium w i t h cadmium. The p r e s e n c e o f 1.0 a t o m i c wt. % Mg i n Cd a c h i e v e d about t h e same change i n c/a as l o w e r i n g t h e t e m p e r a t u r e o f pure Cd. They o b s e r v e d no s i g n i f i c a n t change i n t h e p a r a m e t e r . I t i s n o t a p p a r e n t t h a t c o n c l u s i o n s drawn f r o m t h i s method a r e v a l i d s i n c e o t h e r e f f e c t s o f a l l o y i n g a re n o t s u f f i c i e n t l y u n d e r s t o o d . Cadmium does , however, e x h i b i t a l a r g e -45-temperature dependence in the anisotropy of the diamagnetic suscep-(25) t ib i l i t y . This also decreases greatly in the same temperature range but its relation to the Knight shift awaits explanation. The results from magnesium, though revealing a K ° ' ° anis which is perhaps too small to enable one to draw valid conclusions, tends to support the idea of a c/a dependence. From 300°K to 4.2°K, magnesium undergoes a change in c/a from 1.6235 to 1.6230, a decrease of only 0.03%, while displaying no significant change in K . (see r ° ° anis Table 1). The small ( i f any) variation is consistant with a model built up of perfectly spherical atoms. Such a lattice would have a c/a ratio of 1.6333, very close to magnesium's 1.6235, and would not be expected to display anisotropic features. The temperature dependences of the parameters in cadmium may be primarily due to the electron wave functions adjusting them-selves to accommodate the rapid changes of volume occupied by an atom as i t undergoes lattice vibrations. These variations in the wave-functions would manifest themselves more in cadmium than in magnesium because of the difference in complexity of their Fermi surfaces. The temperature variation in the quadrupole coupling constant would reflect a change in^fj3 cos 2$ - l j ^ ^ p , the electron quadrupole moment at the Fermi surface, i f the theory of Watson, Gosard and Yafet is applicable. Since this would suggest an accompanying variation in K . which was not observed, two conclusions may be anis drawn; either a substantial contribution to q arises from the electron -46-s t a t e s w e l l b e l o w t h e F e r m i l e v e l , o r t h e s t a t e s a t t h e s u r f a c e s a r e a n i s o t r o p i c b u t t h e i r c o n t r i b u t i o n i s n u l l i f i e d by a c o r e p o l a r i z a t i o n c o n t r i b u t i o n . B. F u r t h e r S t u d i e s No s t r o n g s t a t e m e n t s can be made about t h e t e m p e r a t u r e dependence on t h e b a s i s o f measurements a t o n l y two t e m p e r a t u r e s . I t would be d e s i r a b l e t o c o n t i n u e t h i s s t u d y i n magnesium f o r v a r i o u s t e m p e r a t u r e s . U n l e s s s i g n a l s f r o m t h e s i n g l e c r y s t a l can be i m p r o v e d , t h i s would n e c e s s a r i l y be done i n a powdered sample w i t h subsequent l o s s o f r e s o l u t i o n . The n e x t c a n d i d a t e f o r s t u d y i n t h i s s y s t e m o f b i v a l e n t 6 7 h e x a g o n a l c l o s e - p a c k e d m e t a l s i s z i n c . Zn , w i t h a low n a t u r a l abundance ( 4 . 2 % ) and l a r g e q u a d r u p o l e i n t e r a c t i o n ( e s t i m a t e d c o u p l i n g ( 2 7 ) c o n s t a n t as h i g h as 70ftHz ' ) , w o u l d , however, p r o v i d e a v e r y s m a l l , p e r h a p s u n d e t e c t a b l e , s i g n a l . These p r e s e n t measurements on magnesium s h o u l d p r o v i d e an i n t e r e s t i n g e x p e r i m e n t a l c o m p a r i s o n when a d e t a i l e d t h e o r e t i c a l t r e a t m e n t i s a v a i l a b l e . T h i s seems l i k e l y i n v i e w o f t h e p r e s e n t ( 2 8 ) d e t a i l e d knowledge o f i t s band s t r u c t u r e and i t s c o m p a r a t i v e l y l o w a t o m i c number. BIBLIOGRAPHY F . J . Milford, Am. J . Phys. 28, 521 (1960). F. Borsa and R.G. Barnes, J . Phys. Chem. of Solids 27, 567 (1966). E.F.W. Seymour and G.A. Styles, Physics Letters, 10, 269 (1964). H.E.. Schone, Phys. Rev. Letters 13, 12 (1964). C P . Slichter, "Principles.of Magnetic Resonance", Harper and Row, p. 97 (1963). N. Bloembergen and T . J . Rowland, Acta Met. 1, 731 (1953). V . I . Checkernikov, I. Pop, and O.P. Naumkin, Sov. Phys. - JETP 17, 1228; (1963). N.F. Ramsay, "Nuclear Moments", Wiley, p. 23 (1953). A. Abragam, "The Principles of Nuclear Magnetism", Oxford University Press, p. 233 (1961). H.E.. Petch,-N.G. Granna and G.M. Volkoff, Can. J . Phys. 31, 837 (1953). M.H. Cohen and P. Reif, Solid State Phys. 5, 321 (1957). G. E . Pake, Solid State Phys. 2, 1 (1956). A.C.. Chapman, P. Rhodes and E.F.W. Seymour, Proc. Phys. Soc. 70_, B 345 (1957). W.K.. Jones, J r . , T.P. Graham and R.G. Barnes, Phys. Rev. 132, 1898: (1963). S.N. Sharma, Ph.D. Thesis, University of British Columbia (1967). G.V.H. Wilson, J . App. Phys. 34, 3276 (1963). T. J.. Rowland, Nuclear Magnetic Resonance in Metals, Progress in Materials Science, vol. 9 (1961). T.P. Das and M. Pomerantz, Phys. Rev. 123, 2070 (1961). E. Goens and E . Schmid, Phys. Z. 37, 385 (1936); R.M. McCammon and G.K.. White, Phil . Mag. 11, 1125 (1965). G. Burns, J . Chem. Phys. 31, 1253 (1959). A. Lurio, Phys. Rev. 126, 1768 (1962). L . E . Drain, Metallurgical Reviews, Vol. 12, p. 195 (1967). M. Pomerantz and T.P. Das, Phys. Rev. 119, 70 (1960). R.E. Watson,''A.C. Gosard and Y. Yafet, Phys. Rev. 140, A375 (1965). E. Griieisen and E. Goens, Z. Physlk, 29, 141 (1924). J.A. Marcus, Phys. Rev. 76, 621 (1949). G. Seidel and P.H. Keesom, Phys. Rev. Lett. 2, 261 (1959). L.M. Falicov, Phil. Trans. Roy. A255, 55 (1962). 

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