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Some NMR studies of NbSe₂ Abdolall, Khaled 1974

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SOME NMR STUDIES OF NbSe  2  by KHALED ABDOLALL B.Sc,  U n i v e r s i t y o f W a t e r l o o , 1972  A THESIS SUBMITTED  IN PARTIAL FULFILMENT OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e Department of Physics We accept t h i s t h e s i s as conforming t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1974  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 of the r e q u i r e m e n t s f o r  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the 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  I f u r t h e r agree 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 f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s .  study.  c o p y i n g of t h i s  be g r a n t e d by the Head of my  thesis  Department or  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 or p u b l i c a t i o n  of 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 not be a l l o w e d w i t h o u t written permission.  Department of The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8 , Canada  my  ABSTRACT  A s e n s i t i v e n u c l e a r magnetic resonance s p e c t r o m e t e r has been c o n s t r u c t e d u s i n g F i e l d E f f e c t T r a n s i s t o r s i n a Robinson c o n f i g u r a t i o n . The s p e c t r o m e t e r has been used t o s t u d y t h e anomalous n u c l e a r magnetic resonance s p e c t r a o f s i n g l e c r y s t a l s o f NbSe^.  An a n a l y s i s o f t h e f i e l d  dependence o f t h e l i n e w i d t h i n t h e low temperature phase has demonstrated t h a t t h i s r e s u l t s from a d i s t r i b u t i o n o f K n i g h t S h i f t s .  Such a d i s t r i b u t i o n  i s not c o n s i s t e n t w i t h a s t r u c t u r a l t r a n s f o r m a t i o n i n v o l v i n g o n l y two n o n e q u i v a l e n t s i t e s as p r o p o s e d by E h r e n f r e u n d e t a l . In a d d i t i o n a c c u r a t e measurements o f the K n i g h t S h i f t and e l e c t r i c f i e l d g r a d i e n t t e n s o r have been made i n t h e h i g h temperature phase a t 77K and 300K.  The K n i g h t S h i f t  has a v e r y l a r g e a n i s o t r o p i c component b u t an almost z e r o i s o t r o p i c  component  which i s i n d i c a t i v e o f n e g l i g i b l e s - e l e c t r o n c h a r a c t e r a t the: Fermi s u r f a c e .  TABLE OF CONTENTS  Abstract Table o f Contents L i s t o f Tables List of Illustrations'. Acknowledgements CHAPTER I :  Introduction  CHAPTER I I : T h e o r e t i c a l C o n s i d e r a t i o n s The Quadrupole H a m i l t o n i a n Energy L e v e l s i n s i n g l e c r y s t . a)  C r y s t a l s w i t h c y l i n d r i c a l symmetry: Magnetic f i e l d p a r a l l e l t o c r y s t a l  b)  axis.  C r y s t a l s w i t h l e s s than c y l i n d r i c a l symmetry: a r b i t r a r y o r i e n t a t i o n o f t h e magnetic  field  Theory o f t h e Knight S h i f t I s o t r o p i c Knight S h i f t A n i s o t r o p i c Knight S h i f t CHAPTER I I I :  Experimental A b r i e f i n t r o d u c t i o n t o s t e a d y s t a t e d e t e c t i o n o f NMR Apparatus 1.  The s p e c t r o m e t e r  I.  a)  The r . f . a m p l i f i e r  I.  b)  Noise f i g u r e o f the r . f . a m p l i f i e r  II.  The l i m i t e r  I I I . The d e t e c t o r IV.  The audio a m p l i f i e r  V.  Principle of operation  VI.  C o n s t r u c t i o n design o f the spectrometer  VII.  Performance  2.  Sample and c r y s t a l h o l d e r  3i  O r i e n t a t i o n of the c r y s t a l  4)  Temperature c o n t r o l  CHAPTER IV: R e s u l t s Temperature dependence o f t h e J9/2, l/2^> —i>J9/2, line .2 Determination of Discussion Conclusion References CHAPTER V:  Appendix  and t h e K n i g h t  Shifts  V  LIST OF TABLES  TABLE ( I ) :  Noise f i g u r e o f the r . f . a m p l i f i e r f o r 3 d i f f e r e n t  source  impedances.  TABLE ( I I ) :  The quadrupole i n t e r a c t i o n and t h e Knight S h i f t s a t room t e m p e r a t u r e and 77 K.  vi LIST OF ILLUSTRATIONS Page Fig.  (1):  93 Energy l e v e l diagrams f o r Nb  n u c l e u s o f s p i n 9/2  i n NbSe  2  s i n g l e c r y s t a l w i t h t h e magnetic f i e l d p a r a l l e l t o t h e h i g h symmetry a x i s o f t h e c r y s t a l Fig.  (.2):  O r i e n t a t i o n o f t h e magnetic f i e l d H w i t h r e s p e c t t o t h e principal  Fig.  C3):  5  (the c a x i s ) .  tensor.  6  absorption  18  axes 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  Schematic arrangement f o r s t e a d y s t a t e NMR experiments  Fig.  (4):  The r . f . a m p l i f i e r  19  Fig.  (5) :  A cascode a m p l i f i e r  20  Fig.  C6):  The  limiter  21  Fig.  C7):  The  detector  22  Fig.  (8):  Schematic diagram,of t h e d e t e c t i o n p r o c e s s u s i n g the n o n l i n e a r r e l a t i o n between t h e gate b i a s and d r a i n c u r r e n t o f a FET.  23  Fig.  C9):  The Audio a m p l i f i e r  Fig.  (.10):  Frequency r e s p o n s e o f the a u d i o a m p l i f i e r  25  Fig.  di):  The  26  Fig.  (12):  A b l o c k diagram o f t h e s p e c t r o m e t e r  Fig.  (13):  Resonance c u r v e o f a tuned LC C i r c u i t .  attenuator  27 I t shows how a n o i s e  v o l t a g e AV i s i n t r o d u c e d by a s m a l l change +Af where f i s the o s c i l l a t i o n f r e q u e n c y d i f f e r e n t from t h a t o f t h e resonance frequency f  of the tank.  Fig.  (.14):  C o n s t r u c t i o n layout of the spectrometer  29  Fig.  (15):  Low t e m p e r a t u r e c r y o s t a t and c r y s t a l h o l d e r  31  vii Page  Fig.  (16): A n g u l a r dependance o f t h e 1 9 / 2 , l/2> -> j 9/2, - l/2> resonance  Fig.  (17): Nb  i n NbSe  2  ]9/2, l/2>  32  |9/2, - l/2> l i n e shape a t 5.9 K  and 12.22 MHz w i t h S p a r a l l e l t o c.  35  Q 3  Fig.  (18): Nb  i n NbSe  2  |9/2, l/2> -> |9/2, - l/2> l i n e shape a t 21.3 K ->  ->  and 12.22 MHz w i t h H p a r a l l e l t o c.  36  Q 3  Fig.  (19): Nb  i n NbSe  2  19/2, l/2> -> 19/2, -l/2> l i n e shape a t 29 K ->  and 12.22 MHz w i t h H p a r a l l e l t o c. Fig.  (20): Nb  93  i n NbSe  I  2  j 9/2, l/2>  i  37  J 9/2, - l/2> l i n e shape a t 6.6 K  and 18.00 MHz w i t h H p a r a l l e l t o c. Fig.  (21): N b  9 3  i n NbSe  2  38  |9/2, l/2> •+ |9/2, - l/2> l i n e shape a t 12.6 K ->  and 18.00 MHz w i t h H p a r a l l e l t o c. Fig.  C22): Nb  93  i n NbSe  i  2  19/2, l/2>  39  i  19/2, - l/2> l i n e shape a t 20.6 K  and 18.00 MHz w i t h H p a r a l l e l t o c. Fig.  40  (23): Temperature and f i e l d dependence o f t h e l i n e w i d t h f o r t h e |9/2, l/2> -> |9/2, - l/2> l i n e w i t h H p a r a l l e l t o c.  Fig.  (24): A sample c a l c u l a t i o n o f K . " : x  f r e q u e n c y (E_-jy applied f i e l d .  - i/2^/^ * E  2  The c a l c u l a t e d s  41  resonance  p l o t t e d as a f u n c t i o n o f  The observed resonance f r e q u e n c y  v e x  p I  s  due t o an e f f e c t i v e f i e l d H  a t t h e s i t e o f t h e Nb eff n u c l e u s which i s lower t h a n t h e a p p l i e d f i e l d H . * exp  42  r  Fig.  (25): The v a r i a t i o n o f t h e |9/2, l/2> + |9/2, - l/2> f r e q u e n c y as a f u n c t i o n o f n.  Fig.  45  (26): F i e l d dependence o f t h e l i n e w i d t h o f t h e J9/2. l/2> |9/2, - l/2> l i n e w e l l below t h e t r a n s i t i o n .  46  ACKNOWLEDGEMENTS  I would l i k e t o e x p r e s s my s i n c e r e g r a t i t u d e t o Dr.DL'Ll. W i l l i a m s f o r his  v a l u a b l e i n s t r u c t i o n throughout t h e work. I would l i k e t o express my g r a t i t u d e t o Dr. M.I. V a l i c f o r h i s many  v a l u a b l e s u g g e s t i o n s and h i s h e l p i n t h e e x p e r i m e n t s . I am v e r y g r a t e f u l t o D r . W a l t e r N. Hardy f o r h i s g r e a t h e l p i n e x p l a i n i n g t h e c i r c u i t diagrams o f t h e s p e c t r o m e t e r , i t s t h e o r y o f o p e r a t i o n , and h i s v a l u a b l e s u g g e s t i o n s r e g a r d i n g i t s c o n s t r u c t i o n . I am a l s o i n d e b t e d t o him f o r r e v i e w i n g t h e s e c t i o n on t h e s p e c t r o m e t e r .  -1CHAPTER I INTRODUCTION Niobium d i s e l e n i d e i s a l a y e r e d t r a n s i t i o n metal d i c h a l c o g e n i d e which e x i s t s i n three stacking polytypes.  W i l s o n and Y o f f e  (1969) have p u b l i s h e d a t h o r o u g h  d e s c r i p t i o n o f t h e s e l a y e r e d s t r u c t u r e s and t h e i r p r o p e r t i e s .  A t room t e m p e r a t u r e  the 2H p o l y t y p e has a l a y e r e d hexagonal s t r u c t u r e t h a t has t h e symmetry o f t h e space group V6^/vmc. T h i s m a t e r i a l showed an anomalous s i g n r e v e r s a l o f t h e H a l l c o e f f i c i e n t a t 26K and a maximum i n t h e magnetic s u s c e p t i b i l i t y near 40K (Lee e t al  ( 1 9 6 9 ) ) , w h i c h were t a k e n t o be i n d i c a t i v e o f a phase t r a n s f o r m a t i o n . NMR s t u d i e s on powders o f 2H-NbSe2 were made by E h r e n f r e u n d e t a l (1971) .  They c o n c l u d e t h a t t h e anomalous b e h a v i o u r  o f NbSe2 a t low t e m p e r a t u r e s i s due  to a s t r u c t u r a l t r a n s f o r m a t i o n and an a s s o c i a t e d c o n d u c t i o n tribution.  electron redis-  I n t h e i r model, t h e y suggest t h a t a t and below 20K t h e r e a r e two  i n e q u i v a l e n t niobium s i t e s w i t h d i f f e r e n t e l e c t r i c f i e l d g r a d i e n t s .  Their analysis  i s based on t h e i n f o r m a t i o n e x t r a c t e d from t h e powder p a t t e r n l i n e shape, which i s assumed t o be t h e c o n v o l u t i o n o f a G a u s s i a n shape f u n c t i o n w i t h t h e f r e q u e n c y d i s t r i b u t i o n expected from a randomly o r i e n t e d m i c r o c r y s t a l i t e sample i n t h e p r e s e n c e o f quadrupole e f f e c t s and an i n i s o t r o p i c K n i g h t  shift.  The purpose o f t h i s work i s t o f u r t h e r study t h e p r o p e r t i e s o f NbSe2 by NMR methods u s i n g s i n g l e c r y s t a l s .  The advantages o f u s i n g s i n g l e c r y s t a l s  (as was demonstrated i n t h i s l a b o r a t o r y i n p r e v i o u s work on G a l l i u m and Aluminium) a r e : (a) a d i r e c t study o f t h e a n i s o t r o p y i n t h e Knight t o c r y s t a l symmetry, a n i s o t r o p i c Knight  s h i f t due  (b) e l i m i n a t i o n o f l i n e b r o a d e n i n g caused by t h e  s h i f t , and  (c) c o n s e q u e n t l y  more a c c u r a t e measurements o f  the Knight s h i f t s and t h e e l e c t r i c f i e l d g r a d i e n t p a r a m e t e r s .  - 2-  CHAPTER I I THEORETICAL CONSIDERATIONS The Quadrupolar H a m i l t o n i a n : The H a m i l t o n i a n o p e r a t o r f o r 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 q u a d r u p o l e moment w i t h t h e e l e c t r i c f i e l d g r a d i e n t e x i s t i n g a t t h e n u c l e u s i s g i v e n by [4]  where I  = I , +.iY ;  +  x  I ,, I  y l  x  Y  I  and I , z  a r e ' t h e components o f t h e n u c l e a r s p i n I ; eQC i s t h e s c a l a r q u a d r u p o l e moment and n i s an asymmetry parameter d e f i n e d a s : q  n =  X'X' " Y * Y ' q  where q  X ' X " Y'Y' q  a  n  d  q  Z*Z'  =  q  a r e d e f i n e d by 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 e n s o r ; 2  ( 1  -  n ) q  X'X< 1  0  J  0  O^qy'Y'  0  0  \  \ 0  0  "'^Z'Z  1  and X', Y', Z* a r e t h e p r i n c i p a l axes o f q. Energy  levels i n single  In t h e p r e s e n c e  crystals:  o f an a p p l i e d magnetic f i e l d S t h e H a m i l t o n i a n f o r a S p i n I  w i t h a quadrupole moment i s g i v e n by (Abragam P.232).  3 (a)  =  "  +  tf%  i - i ) < z< " 3I  1  C r y s t a l s w i t h c y l i n d r i c a l symmetry;  +  J + (I  +  :  -  )]  magnetic f i e l d p a r a l l e l t o c r y s t a l a x i s ,  - 3 -  For c r y s t a l s w i t h a x i a l symmetry, w h i c h i s t h e case o f NbSe^ a t temperatures above 24 K, n v a n i s h e s because  q  X'X'  =  q  Y'Y'  I f H i s chosen p a r a l l e l t o t h e symmetry a x i s o f t h e c r y s t a l  ( t h e Z' a x i s ) ,  t h e n t h e H a m i l t o n i a n t a k e s t h e form  £  -ThHI ,  =  4U2I-1) l h  +  "  31  z  1  so t h a t i n t h e | l , M > r e p r e s e n t a t i o n where I , i s d i a g o n a l , t h e e n e r g i e s of  t h e d i f f e r e n t l e v e l s a r e g i v e n by: 2  E or,  '  = - yhHm + 4 ^ , ^  m  [3m?- I  OD]  using the notations Y JJ 2TT '  L  V  _ 3e qQ h2I(2I-l) 2  V  Q  The c o r r e s p o n d i n g f r e q u e n c i e s o f t h e a l l o w e d t r a n s i t i o n s a r e v  , = m-l,m  =  E - E m-1 m n i  V  L  ^2  +  C  1  _  For I = ^ o f t h e N b  2  m  )  n u c l e u s t h e energy l e v e l diagram  9 3  Thus i n a n u c l e a r magnetic resonance  experiment  on N b  i s given i n F i g . ( l ) . 9 3  i n NbSe2 where t h e  magnetic f i e l d i s a l o n g t h e symmetry a x i s o f t h e c r y s t a l one would expect t o see n i n e l i n e s c o r r e s p o n d i n g t o t h e f r e q u e n c i e s :  v  r  =  V  L -  rv  Q'  r  =  °' ' ' ' 1  2  3  4  with a c e n t r a l l i n e corresponding t o the | j , y  I §"» ~ ^  t r a n s i t i o n and f o u r  s a t e l l i t e s on each s i d e e q u a l l y spaced by V Q t h e quadrupole  frequency o f t h e N b  >  nucleus. (b)  Crystals  w i t h l e s s t h a n c y l i n d r i c a l symmetry; a r b i t r a r y o r i e n t a t i o n s o f  the magnetic  field.  9 3  -  A  -  F o r t h e more g e n e r a l problem o f an a r b i t r a r y o r i e n t a t i o n o f t h e magnetic f i e l d H t h e H a m i l t o n i a n t a k e s t h e form , s i n '6 'cos <f> , s H = - yhH { = - ( I +1 ) TT  T  r  + ^fsQ  r  41(21-1)  3 I  _  2  I  f  I  +  n  T  2  +  T  Q  T  7  ri2 2-)i  + t  Z»  L  i s i n 6 s i r i <J> , . - i = ( I - I )+ cos 6 I }  T  +I  + - -l  1  J  where t h e a n g l e s 6 , <>j d e f i n e t h e o r i e n t a t i o n o f H w i t h r e s p e c t t o t h e p r i n c i p a l axes X', Y', Z' as shown i n F i g . ( 2 ) . Making t h e s u b s t i t u t i o n  v)  ,L  =  Y  -  —  2-n  ,t  H  V  \> Q  2 3e qQ = h2I(2I-l)»  V v  Y =6 L v  Q  gives H  =  Y {  S i n  ?  C  O  S  • (I  + 31*, - K I + l ) The appendix  +  I  )  -  1  S  I  N  I  S  i  n  * (I - I )  +  cos 6 I ,}  + j (I* + I J ]  i n c l u d e s an e x p l i c i t form o f t h e 10 x 10 m a t r i x o f H f o r a  9 spin 1 = 2 "  a n  d  a  computer programme f o r d i a g o n a l i z i n g  such a m a t r i x and f i n d i n g  t h e exact e i g e n v a l u e s and t h e c o r r e s p o n d i n g n o r m a l i z e d e i g e n v e c t o r s .  - 5 -  9/2  9/2  J-7/^  A  CD i— CL)  C LU  zero  |9/2 -'/2> 3  9/2,  >/£>  I9/2.,  9/2>5/2>  Fig.  C I ) : Energy l e v e l diagrams f o r N b n u c l e u s o f s p i n 9/2 i n NbSe s i n g l e c r y s t a l w i t h t h e magnetic f i e l d p a r a l l e l t o t h e h i g h symmetry a x i s o f t h e c r y s t a l (the c a x i s ) . 9 3  ?  Fig,  (2):  O r i e n t a 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 t h e p r i n c i p a l axes 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 t e n s o r .  THEORY OF THE KNIGHT SHIFT F o r a g i v e n f i x e d a p p l i e d f i e l d t h e n u c l e a r magnetic resonance for  frequency  a n u c l e u s i n a m e t a l l i c substance i s d i f f e r e n t o r ' s h i f t e d ' from t h a t o f  t h e same n u c l e u s field.  i n a non m e t a l l i c r e f e r e n c e compound f o r t h e same a p p l i e d  T h i s s h i f t , known as t h e Knight S h i f t , i s due t o t h e i n t e r a c t i o n o f t h e  nuclear spins with the conduction electrons i n the metal. The H a m i l t o n i a n o p e r a t o r f o r t h e i n t e r a c 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 [1];  w i t h a n u c l e a r s p i n I i s g i v e n by I J  o t,t V / ^ ^ 3 r ( S . r \ 8TT - ± , , - » - . . , 2$YhI'> { — r - —T + 3 + r r S Sfr)} D  H.  =  (lj  e where r and S a r e t h e p o s i t i o n and s p i n o f an e l e c t r o n e and I i s t h e o r b i t a l momentum.  The summation i s t a k e n over a l l t h e e l e c t r o n s .  I f t h e e l e c t r o n i c o r b i t a l momentum i s c o m p l e t e l y quenched t h e n i n t h e one e l e c t r o n d e s c r i p 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 of t h i s operator i s : + 3(r\ .% ) r . -yh I.I<k|26 V> I* k k  S.  _ |l? 6 0  k  = - yh ?•! -y  where T k  \'%  i s a tensor  +  k  (?)}| k >  k  (2)  s\ „  3?, r , = <k|2g {—^ - ~V k k r  the expectation value  r  i 3  + 1  «  |k>  k  w i t h components depending on t h e s t a t e s |k>.  The summation i s over u n f i l l e d s t a t e s |k> because e l e c t r o n s i n f i l l e d s t a t e s do n o t c o n t r i b u t e t o t h i s averaged  i n t e r a c t i o n because t h e i r t o t a l s p i n moments  a r e equal t o z e r o . If  has a p p r o x i m a t e l y t h e same v a l u e f o r a l l u n f i l l e d s t a t e s near t h e t o p  o f t h e Fermi d i s t r i b u t i o n t h e n  yh I-l t • S, = yh I'T-S  (3)  where  I  S =  s.k  -> - » - - » • I f t h e t e n s o r s x ^ a r e n o t i d e n t i c a l t h e n T i s an average o f x ^ o v e r a l l the s t a t e s  |k> near t h e Fermi  surface.  In t h e p r e s e n c e o f an a p p l i e d f i e l d H  3  =-L 2g  we have:  q  X X - H- H  A A  p o  where Xp i s t h e magnetic s u s c e p t i b i l i t y p e r u n i t volume and i t becomes a t e n s o r if  t h e o r b i t a l momentum i s n o t c o m p l e t e l y  quenched and V i s t h e volume.  S u b s t i t u t i n g f o r S i n (3) g i v e s  -h  YH  X  P  ( 4 )  That i s t h e n u c l e a r s p i n I sees an a d d i t i o n a l f i e l d Vx^ x*H /23superimposed on H o  causing  Q  a s h i f t i n t h e n u c l e a r resonance f r e q u e n c y .  In t h e one e l e c t r o n d e s c r i p t i o n t h e s p a t i a l c o r r e l a t i o n between t h e e l e c t r o n s i s not taken i n t o account. formalism  [1]  that r e l a t i o n  I n f a c t i t can be shown u s i n g t h e d e n s i t y  matrix  (4) i s independent o f t h e one e l e c t r o n d e s c r i p t i o n .  I s o t r o p i c Knight S h i f t : I f t h e symmetry o f t h e e l e c t r o n i c environment o f t h e n u c l e a r s p i n i s c u b i c or higher then only the s c a l a r part of t h e tensor  i s d i f f e r e n t from z e r o ,  i s , only S electrons w i l l contribute t o the s h i f t .  I n t h i s case (4) r e d u c e s t o  - Vyh CI-H)  X p  ' y- <  I  KO) | >  that  2  F  where ip(0) i s t h e v a l u e o f t h e wave f u n c t i o n o f an e l e c t r o n a t t h e n u c l e u s and the average < The  >p i s made o v e r a l l s t a t e s a t t h e t o p o f t h e Fermi d i s t r i b u t i o n .  t o t a l Hamiltonian  .yh f . 3  o  becomes  CI + V - y- < U Xp  CO) |  2  >) p  -  T h i s corresponds  t o a p o s i t i v e frequency  AH  iso  x  s h i f t K g i v e n by  8ir  .  H  A n i s o t r o p i c Knight If  9  Shift  t h e symmetry o f t h e  riuclear  environment i s l e s s than c u b i c t h e t e n s o r  i s not a s c a l a r and t h e K n i g h t s h i f t w i l l depend on 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 magnetic f i e l d w i t h r e s p e c t t o t h e c r y s t a l a x i s . to  T h e r e f o r e we must add  t h e i s o t r o p i c s h i f t an a n i s o t r o p i c s h i f t g i v e n by: yh i ' K ' t i yh  (I  Y  ,I  1  Y  I  K  , I ,) 7  o\  0  XX  0  K' YY  0  0  K'  H,  zz  H„  where I , K' and H a r e now d e f i n e d w i t h r e s p e c t t o t h e p r i n c i p a l axes X', Y', Z ' o f t h e t e n s o r  t.  Since the a n i s o t r o p i c i n t e r n a l f i e l d applied f i e l d H frequency H  t h e o n l y e f f e c t i v e component i n s h i f t i n g t h e resonance  i s that along H (K'  O  [9],  Q  cos  2  q  0 + K'  LtLt  i s u s u a l l y much s m a l l e r t h a n t h e  A  which can be shown t o be sin  6 cos  2  <j> + K'  2  A  sin  2  [1]  9 sin  2  a))  I I  where 6, <f> s p e c i f y t h e o r i e n t a t i o n o f H  q  w i t h r e s p e c t t o X' ,Y!  For a x i a l symmetry we have  K''  (K + K' ) XX YY  from t h e t r a c e l e s s p r o p e r t y o f K'  1  ZZ  t h e r e l a t - i v e - frequency,,, s h i f t r becomes  H  = K. +IK! i s o " 2 || = K. iso  C3 c o s  + k K . 2 aniso  2  l„  6 - 1)  (3 c o s  2  0-1)  ;  _^  ,11.  -  where K  . = K aniso ||  10  -  i s a measure o f t h e a n i s o t r o p y i n t h e charge d i s t r i b u t i o n  1  r }  6  and i s r e l a t e d t o the c o n d u c t i o n e l e c t r o n wave f u n c t i o n by [ 6 ] I , 1  ( 3 cos  2  2  6 - 1 ) ,  3  which i s a p o s i t i v e q u a n t i t y i f t h e charge d e n s i t y i s g r e a t e s t i n t h e d i r e c t i o n o f the Z a x i s . For t h e extreme cases we have K - = K. + K. . for 9 = II iso aniso K j_  = K. iso  - \ K . for 0 = \ 2 aniso 2  This gives K. = f CK„ iso 3 || V  K  aniso  -  0  +  CK„ 3 ^ ||  2  K, ) \_  - KJ JJ  - 11 -  CHAPTER I I I EXPERIMENTAL A b r i e f i n t r o d u c t i o n t o steady s t a t e d e t e c t i o n o f NMR: Steady s t a t e d e t e c t i o n o f n u c l e a r magnetic resonance a b s o r p t i o n c o n s i s t s o f o b s e r v i n g t h e r e s p o n s e o f t h e n u c l e a r s p i n system t o a c o n t i n u o u s l y a p p l i e d r a d i o frequency  field.  The specimen i s p l a c e d i n s i d e an r . f . c o i l w h i c h forms p a r t o f  a tuned c i r c u i t as shown i n F i g . ( 3 ) . A t resonance t h e n u c l e a r s p i n system absorbs energy from t h e r . f . f i e l d H^ i n s i d e t h e c o i l .  T h i s resonance  effect  i s d e t e c t e d by i t s r e a c t i o n on t h e c i r c u i t s u p p l y i n g t h e r . f . f i e l d . The n u c l e a r resonance a b s o r p t i o n by t h e s p i n s w i t h i n t h e c o i l changes t h e q u a l i t y f a c t o r , Q, o f t h e c o i l by [ 5 ] :  s CqD =  4  IT  -n  x  M  where n i s t h e f i l l i n g f a c t o r and x" i s t h e i m a g i n a r y p a r t o f t h e n u c l e a r susceptibility. I f i n t h e absence o f s i g n a l t h e r . f . v o l t a g e a c r o s s t h e c o i l i s  then  n e a r resonance a change by 6 (—) w i l l r e s u l t i n  6 V  1  = V  1  Q <5 ( i )  Thus d e t e c t i o n o f n u c l e a r magnetic resonance a b s o r p t i o n i s reduced t o d e t e c t i n g t h e change i n t h e r . f . l e v e l a c r o s s t h e c o i l c o n t a i n i n g t h e sample. A v a r i e t y o f c i r c u i t s f o r d e t e c t i n g n u c l e a r magnetic resonance a b s o r p t i o n have been used.  The two most s u c c e s s f u l o f t h e s i n g l e c o i l t y p e s a r e t h e one  by Robinson [ 5 ] , and t h e one by Pound, Knight and Watkins  [8].  APPARATUS 1:  The S p e c t r o m e t e r : For t h i s work a t r a n s i s t o r i z e d v e r s i o n o f t h e Robinson c i r c u i t u s i n g F i e l d  E f f e c t T r a n s i s t o r (FETS) was b u i l t .  The c i r c u i t was d e s i g n e d by P r o f e s s o r V. Frank  -  12  at t h e T e c h n i c a l U n i v e r s i t y o f Denmark.  -  The f o l l o w i n g i s a d e s c r i p t i o n o f  such a c i r c u i t , i t s p r i n c i p l e o f o p e r a t i o n , c o n s t r u c t i o n d e s i g n and an e v a l u a t i o n o f i t s performance. I. a) The  The r . f . A m p l i f i e r : r . f . a m p l i f i e r i s wide band and c o n s i s t s o f an i n p u t stage and f o u r  a m p l i f i e r s t a g e s t h a t a r e connected i n cascade w i t h t h e i n p u t s t a g e .  (4)  Fig.  i s t h e c i r c u i t diagram. The  FETS  , Q ,  and  form two cascodes i n p a r a l l e l .  A cascode  a m p l i f i e r c o n f i g u r a t i o n i s o b t a i n e d u s i n g a common source and a common gate FET  connected as shown i n F i g . (5). T h i s c o n f i g u r a t i o n o f f e r s good s t a b i l i t y , low n o i s e a m p l i f i c a t i o n , and  l a r g e power g a i n . The and  s t a b i l i t y i s due t o t h e reduced M i l l e r e f f e c t i n t h e common source  t h e s m a l l d r a i n source c a p a c i t a n c e  o f t h e common gate s t a g e .  stage  Both f a c t o r s  c o n t r i b u t e t o a very small reverse t r a n s f e r admittance f o r t h e combination. Its  low n o i s e a m p l i f i c a t i o n i s due t o t h e f a c t t h a t t h e v o l t a g e g a i n o f  the f i r s t stage i s u n i t y and t h e r e f o r e does n o t c o n t r i b u t e t o t h e n o i s e . the second stage a l l o w s a m p l i f i c a t i o n w i t h o u t a d d i t i o n i n n o i s e noise f i g u r e o f t h e combination  [3].  While  Thus t h e  corresponds t o the n o i s e f i g u r e o f the f i r s t  stage. The  l a r g e power g a i n o f t h e cascode i s a r e s u l t o f t h e d e c r e a s e d o u t p u t  admittance. The p a r a l l e l c o m b i n a t i o n  o f t h e two cascodes r e d u c e s t h e n o i s e v o l t a g e  o f t h e i n p u t stage by a f a c t o r o f two, n o i s e a m p l i f i c a t i o n when t h e source Q  5  and  g i v i n g f u r t h e r improvement i n low  impedance i s l e s s than t h e optimum v a l u e .  are i n the emitter follower configuration.  arrangement p r o v i d e s low output impedance.  The p a r a l l e l  - 13 -  12  The r e m a i n i n g f o u r a m p l i f i e r s t a g e s each c o n s i s t o f a common s o u r c e configuration  f o l l o w e d by an e m i t t e r f o l l o w e r c o n f i g u r a t i o n .  The e m i t t e r  f o l l o w e r s e r v e s as a power a m p l i f i e r as w e l l as an impedance matching d e v i c e . The t o t a l v o l t a g e g a i n o f t h e r . f . a m p l i f i e r can be v a r i e d 4.6  to  i n s t e p s from  1430.  I . b)  Noise f i g u r e of the r . f . a m p l i f i e r :  The n o i s e f i g u r e o f t h e r . f . a m p l i f i e r was measured f o r d i f f e r e n t v a l u e s o f s o u r c e impedance at a f r e q u e n c y o f 10 MHz.  The r e s u l t s a r e shown i n T a b l e  ( I ) below.  SOURCE IMPEDANCE 244  NOISE FIGURE fi  3.44  .984 K 7.3  T a b l e (.1) :  1.44  K  1,46  Noise f i g u r e of the r . f . a m p l i f i e r f o r 3 d i f f e r e n t source impedances.  II:  The  Limiter:  The c i r c u i t i s shown i n F i g . ( 6 ) . t h e h i g h dynamic r e s i s t a n c e c u r r e n t s u p p l y Q-^Q-  The s o u r c e f o l l o w e r Qg makes use o f  i n the s o u r c e c i r c u i t p r o v i d e d by the c o n s t a n t  T h i s improves the l i n e a r i t y and performance  o f the s o u r c e  follower. The i n p u t s i g n a l f o r t h e l i m i t e r i s the o u t p u t o f t h e r . f . a m p l i f i e r . The output o f t h e s o u r c e f o l l o w e r i s a p p l i e d t o a p a i r o f d i o d e s called  'clippers'.  The output waveform i s l i m i t e d t o the v o l t a g e s s e t by t h e  r e v e r s e b i a s e s o f D^ and D . 2  bias voltages.  The  and T)^  Diodes D^,  D^,  D^ and D^ p r o v i d e t h e n e c e s s a r y  l i m i t i n g a c t i o n can be a d j u s t e d by a l t e r i n g the r e v e r s e  - 14 -  bias voltages of  and  by means o f t h e s w i t c h i n g arrangement shown.  The arrangement o f inverter.  and  forms a s o u r c e c o u p l e d 0°/180° phase  S i n c e each a m p l i f i e r s t a g e changes t h e phase by 180°,  t h e phase  i n v e r t e r i s n e c e s s a r y t o ensure t h a t t h e t o t a l phase s h i f t i s z e r o when t h e number o f a m p l i f i e r s t a g e s i s odd.  Ill:  The D e t e c t o r : D e t e c t i o n i s t h e p r o c e s s o f r e c o v e r i n g from a modulated  r . f . carrier a  s i g n a l t h a t v a r i e s i n accordance w i t h t h e m o d u l a t i o n p r e s e n t on t h e c a r r i e r . S e v e r a l methods o f d e t e c t i o n a r e a v a i l a b l e . s p e c t r o m e t e r makes u s e o f t h e n o n - l i n e a r r e l a t i o n  The method used i n t h i s (approximately quadratic)  between t h e g a t e b i a s and d r a i n c u r r e n t o f a FET. Fig.  (7) i s a c i r c u i t diagram o f t h e d e t e c t o r .  i s biased nearly to  p i n c h o f f and t h e r e f o r e i s o p e r a t i n g i n t h e n o n - l i n e a r r e g i o n .  When a modulated  r . f . c a r r i e r i s a p p l i e d t o t h e g a t e t h e averaged output s i g n a l w i l l v a r y a p p r o x i m a t e l y as t h e m o d u l a t i o n envelope..  This i s i l l u s t r a t e d i n F i g .  (8).  The D.C. component o f t h e r e c t i f i e d s i g n a l i s a m p l i f i e d by t h e D.C. a m p l i f i e r Q ^.  The r . f . l e v e l i s then m o n i t o r e d u s i n g a D.C. ammeter.  variable resistance R  IV:  q  The  s e r v e s as a D.C. o f f - s e t f o r t h e D.C. a m p l i f i e r .  The A u d i o A m p l i f i e r : The output o f t h e d e t e c t o r i s a m p l i f i e d by t h e low n o i s e a u d i o a m p l i f i e r  shown i n F i g . ( 9 ) .  The low f r e q u e n c y components o f t h e d e t e c t o r o u t p u t a r e  b l o c k e d by t h e h i g h pass f i l t e r a t t h e i n p u t o f t h e a m p l i f i e r . i n common source c o n f i g u r a t i o n and  Q^j. and  are  i s i n a source f o l l o w e r c o n f i g u r a t i o n .  The t h r e e s t a g e s a r e connected i n cascade and a g a i n o f about 3200 i s a c h i e v e d . The f r e q u e n c y response i s shown i n F i g .  (10).  - 15 -  The h i g h f r e q u e n c y r o l l o f f o f t h e a m p l i f i e r i s a t 1.1 kHz.  A higher  f r e q u e n c y r o l l o f f a t 12 kHz can be o b t a i n e d by means o f t h e 5 nF c a p a c i t o r shown i n the diagram. V:  P r i n c i p l e of Operation: A b l o c k diagram o f t h e s p e c t r o m e t e r i s shown i n F i g . ( 1 2 ) . O s c i l l a t i o n s i n t h e t a n k c i r c u i t a r e s u s t a i n e d by t h e p o s i t i v e  p r o v i d e d by the l i m i t e r t h r o u g h t h e a t t e n u a t o r .  feed-back  I f the phase s h i f t between  t h e i n p u t o f the r . f . a m p l i f i e r and t h e output o f t h e a t t e n u a t o r (as t h e r . f . s i g n a l goes t h r o u g h t h e r . f . zero,  o s c i l l a t i o n s w i l l o c c u r a t t h e c e n t r e o f t h e resonance c u r v e o f t h e  tuned c i r c u i t . to  a m p l i f i e r , l i m i t e r , and a t t e n u a t o r ) i s p r e c i s e l y  Under such c o n d i t i o n s t h e o s c i l l a t i o n a m p l i t u d e i s l e a s t  sensitive  s m a l l changes i n f r e q u e n c y . However, i f phase s h i f t s are i n t r o d u c e d by some elements o f t h e c i r c u i t  such as s t r a y c a p a c i t a n c e s , e t c . o s c i l l a t i o n s * w i l l s t i l l o c c u r , but i n o r d e r t o m a i n t a i n a z e r o phase s h i f t , the f r e q u e n c y o f o s c i l l a t i o n t h e c e n t r e o f t h e resonance c u r v e .  be o p e r a t i n g under optimum c o n d i t i o n s ; l e v e l across the c o i l  from  I n t h i s case a s m a l l change i n f r e q u e n c y  can cause a l a r g e change i n o s c i l l a t i o n  The r . f .  w i l l be s h i f t e d  a m p l i t u d e and t h e s p e c t r o m e t e r w i l l  not  This i s i l l u s t r a t e d i n F i g . (13).  i s determined by the amount o f feed-back.  Lower r . f . l e v e l s a r e o b t a i n e d by d e c r e a s i n g t h e amount o f feed-back and  adjusting  t h e g a i n o f the r . f . a m p l i f i e r such t h a t i t s o u t p u t i s s u f f i c i e n t t o d r i v e t h e limiter. N u c l e a r resonance a b s o r p t i o n changes t h e q u a l i t y f a c t o r o f t h e c o i l and t h u s changes t h e r . f .  l e v e l across i t .  I n t h e p r e s e n c e o f an audiomodulated  magnetic  f i e l d the s p i n system w i l l go i n and o u t . o f resonance p e r i o d i c a l l y w i t h a f r e q u e n c y equal t o t h a t o f t h e m o d u l a t i o n  frequency.  l e v e l a c r o s s the c o i l w i l l be a m p l i t u d e modulated. r.f.  Consequently the r . f .  T h i s i s a m p l i f i e d by t h e  a m p l i f i e r . a n d t h e n f e d i n t o t h e d e t e c t o r . The o u t p u t o f t h e d e t e c t o r i s  -  1,6 -  approximately p r o p o r t i o n a l t o the modulation by t h e a u d i o s t a g e .  envelope  and i s f u r t h e r a m p l i f i e d  The output i s then e i t h e r d i s p l a y e d on a scope o r , i n t h e  case o f weak s i g n a l s , i s f e d i n t o a phase s e n s i t i v e d e t e c t o r t o improve t h e signal t o noise r a t i o .  By t h e n a t u r e o f phase s e n s i t i v e d e t e c t i o n one o b t a i n s  t h e 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 r a t h e r than t h e s i g n a l VI:  itself.  C o n s t r u c t i o n D e s i g n o f t h e Spectrometer: Fig.  (14) i s a l a y o u t o f t h e s p e c t r o m e t e r .  Each s t a g e i s e n c l o s e d i n a  s e p a r a t e compartment t o s h i e l d i t from o t h e r s t a g e s and o u t s i d e i n t e r f e r e n c e . The r . f . a m p l i f i e r s t a g e s , l i m i t e r and a t t e n u a t o r a r e arranged  i n t h e way shown  t o e l i m i n a t e t h e u n d e s i r a b l e use o f a c o a x i a l l i n e i n t h e feedback  loop.  Feed t h r o u g h c a p a c i t o r s were used t o p r o v i d e t h e s u p p l y v o l t a g e as w e l l as r i g i d s u p p o r t s f o r mounting t h e FET's. to the c h a s s i s . each s t a g e . loops.  Ground l u g s were s o l d e r e d d i r e c t l y  Whenever i t was p r a c t i c a l o n l y one ground p o i n t was used f o r  T h i s has t h e e f f e c t o f r e d u c i n g s e l f o s c i l l a t i o n s and  ground  The l e a d s were kept as s h o r t as p o s s i b l e i n o r d e r t o reduce s t r a y p i c k  up and m i c r o p h o n i c s . Double p o l e double throw s w i t c h e s JMT 223 were used f o r c o n n e c t i n g r . f . a m p l i f i e r s t a g e s i n cascade.  I n one p o s i t i o n o f t h e s w i t c h t h e a m p l i f i e r  stage  i s c o n n e c t e d , i n t h e o t h e r i t i s bypassed. In t h e l i m i t e r a double p o l e f i v e p o s i t i o n r o t a r y s w i t c h A l c o s w i t c h MRA-25 i s used.  I t provides 5 l i m i t i n g options.  A l l t h e s i d e s o f the spectrometer  were made removable except t h e f r o n t  p a n e l where a l l t h e c o n t r o l knobs a r e mounted. t h e component s.  T h i s p r o v i d e s easy a c c e s s t o a l l  - 17 -  VII:  Performance: The  c i r c u i t performed w i t h good s e n s i t i v i t y throughout t h e f r e q u e n c y  4 - 4 0 MHz.  range  T y p i c a l a b s o r p t i o n s p e c t r a a r e shown i n F i g s . ( 2 0 ) , ( 2 1 ) , and ( 2 2 ) .  No attempt was made t o compare t h e s e n s i t i v i t y o f t h i s c i r c u i t w i t h t h a t o f t h e marginal o s c i l l a t o r  (Pound Knight box) p r e s e n t l y used i n t h i s l a b .  However,  a c c o r d i n g t o B r o f e s s o r V. Frank t h e c i r c u i t i s c a p a b l e o f p e r f o r m i n g w i t h a s e n s i t i v i t y about a f a c t o r o f 2 b e t t e r t h a n t h a t o f t h e m a r g i n a l  oscillator.  A d d i t i o n a l advantages o f t h i s c i r c u i t o v e r t h e m a r g i n a l o s c i l l a t o r a r e [ 5 ] , ( a ) i t can be a d j u s t e d t o v e r y low r . f . l e v e l s ; c i r c u i t s o f v e r y low L/C r a t i o .  circuit  (b) i t can be used w i t h  I f t h e shunt impedance o f t h e tuned c i r c u i t i s  low, n o t o n l y i s t h e n o i s e f i g u r e o f a m a r g i n a l  o s c i l l a t o r impaired, but a l s o the  c i r c u i t may f a i l t o o s c i l l a t e a t a l l . In t h e case o f a metal sample t h e c o i l can be wound d i r e c t l y on t h e sample when u s i n g a Robinson c i r c u i t .  But f o r t h e m a r g i n a l o s c i l l a t o r i t i s o f t e n  n e c e s s a r y t o wrap a p i e c e o f m y l a r around t h e sample b e f o r e w i n d i n g t h e c o i l i n order f o r the c i r c u i t to o s c i l l a t e . affects the s e n s i t i v i t y .  T h i s r e d u c e s t h e f i l l i n g f a c t o r and t h u s  I n c o n c l u s i o n t h i s makes t h e Robinson c i r c u i t more  a t t r a c t i v e f o r NMR s t u d i e s o f m e t a l s i n g l e  crystals.  Circuit supp& I lying the r-f field  K  Fig.  (3):  Schematic arrangement f o r s t e a d y s t a t e NMR  Detector  absorption experiments.  + 30-35"  V  4-301/  100 —  I OIL 56  4.7 K  JL/w\ 10k  Ii-  T  /OOMF  /OO  ion  IZK  2X 8F/84  u,  2.X U(994  ST  T ^ 1  ALL FBSISi TORS ARE (/A/L£SS OT/tEXW/S£  Fig.  (4):  The r . f .  amplifier  1/2  WATT SPEC/F/ED  Fig,  C.5):  A cascode a m p l i f i e r  400 JJF —  1  ^  -  22.0SL _AAAA  I80IL  + 30V  5nF_ W W  R£S/STOKS 4Z£ V2 WATT UNLE5S Or//£ZW/SE SPEZI/ttD . Fig.  (6):  The l i m i t e r  -  2 2  -  .WVVvUI994  I  |0OK  UI9S4  180PF\ 13  OUTPUT TC ft.p. AMPLIFIER  V4 8-2 K  -301/ /JZ/! RESISTOAS AR£ UA/LESS 0T#£RW/S£  Fig-  C7).: The d e t e c t o r  V2  WATT SP£C/F/£1>  -  Fig.  (8). '<  23  -  Schematic diagrams o f t h e d e t e c t i o n p r o c e s s u s i n g the n o n l i n e a r r e l a t i o n between t h e gate b i a s and d r a i n c u r r e n t o f a FET.  - 24 -  4-301/  ALL RESISTORS f\R£ /2 WATT UNLESS OTHERWISE SPECIFIES l  Fig.  (9):  The Audio A m p l i f i e r  10  100 Fig.  (10):  Frequency (HZ )  Frequency r e s p o n s e o f t h e a u d i o a m p l i f i e r  1000  F"/?OM TANK  ,1 22 PF h  II  \ 2-2 PF 11  2.  2-2 P F 11  II  11  .4  < '3  11  .  5  2.2 PF 11 11  y\  \t /  -2.2PF  - - 2.2 PF  Fig.  (11):  -- 2.2 PF  The A t t e n u a t o r  z -2-2PF  --  3.3 Pf=  r-f amplifier  Fig.  • Detector  A-F amplifier  ( 1 2 ) : A b l o c k diagram o f t h e s p e c t r o m e t e r  to  out  -  Fig-  28  -  ( 1 3 ) : Resonance curve o f a tuned LC C i r c u i t . I t shows how a n o i s e v o l t a g e AV i s i n t r o d u c e d by a s m a l l change +Af where f i s t h e o s c i l l a t i o n f r e q u e n c y d i f f e r e n t from t h a t o f t h e resonance frequency f of the tank.  5  r  /  |  4  ^  Xy,pU+ k  s+acje [])  phase  I i mi ier OUT  Biasing  Fig.  (14):  netujo r K5  C o n s t r u c t i o n l a y o u t o f the s p e c t r o m e t e r  - 30 2.  Sample and C r y s t a l H o l d e r : A s i n g l e c r y s t a l o f NbSe2 was o b t a i n e d from Dr. R. F r i n d t o f Simon F r a s e r  U n i v e r s i t y , grown i n t h e form o f a t h i n sheet (about .1 mm t h i c k ) w i t h t h e c a x i s p e r p e n d i c u l a r to the plane of the sheet. To m i n i m i z e damage t o t h e c r y s t a l , the sample was p l a c e d between two sheets o f mylar before the r . f .  c o i l o f #40 copper w i r e was wound on i t .  thin The  whole sample assembly was g l u e d w i t h v a r n i s h onto a t h i n m i c r o s c o p e s l i d e which was t h e n g l u e d t o the copper b l o c k as shown i n F i g . ( 1 5 ) .  3 . . O r i e n t a t i o n o f the. C r y s t a l :•• ••/: il: I n i t i a l o r i e n t a t i o n o f t h e c r y s t a l c - a x i s was done v i s u a l l y .  The  final  o r i e n t a t i o n was then d e t e r m i n e d from the a n g u l a r dependence o f t h e | § - j > + 9  i  I y  - 2  the  >  resonance.  By r o t a t i n g t h e magnet and f i n d i n g t h e resonance  graph o f F i g . (16) was o b t a i n e d .  field  From t h i s graph t h e o r i e n t a t i o n o f t h e  c a x i s was d e t e r m i n e d t o w i t h i n h a l f a degree. 4 .  Temperatur.e.,Corit>r6T:; I . R e g u l a t i o n o f t e m p e r a t u r e between 4.2  K  and 50 K  i s a c h i e v e d by  electrical  h e a t i n g i n t h e low t e m p e r a t u r e c r y o s t a t , s h o w n i n F i g . ( 1 5 ) . D e s i r e d t e m p e r a t u r e s were o b t a i n e d by a manual c o n t r o l o f t h e h e a t i n g c u r r e n t and the amount o f exchange gas i n s i d e the c r y o s t a t . For t e m p e r a t u r e s above 20°K i t was n e c e s s a r y t o pump out a l l t h e exchange gas i n o r d e r t o reduce the  h e a t e r c u r r e n t and t h u s m i n i m i z e Helium b o i l - o f f .  As a r e s u l t t h e t i m e  r e q u i r e d f o r t h e Cu b l o c k and sample t o r e a c h t h e r m a l e q u i l i b i r u m was much l o n g e r (about 30 m i n u t e s ) . For  measuring t h e temperature a gold + .03% a t Fe Vs chromel  c o u p l e was used.  The r e f e r e n c e j u n c t i o n was kept a t 4.2  thermo-  K by e n s u r i n g t h a t i t  was i n c o n t a c t w i t h t h e brass b l o c k a t t h e bottom o f t h e c r y o s t a t which was i n t u r n i n d i r e c t c o n t a c t w i t h the h e l i u m b a t h . measured w i t h a K e i t h l e y 148 n a n o v o l t meter.  The thermocouple v o l t a g e  was  - 31 -  BNC connector  feed through  electrical connections  vto pump  L  thin wall stainless steel heater  sample thermocouple  brass F i g . (15):  copper block thin wall stainles£ steel tubing (heat leak) lead seal  Low temperature c r y o s t a t and c r y s t a l  holder  - 32 -  Fig.  (16):  A n g u l a r dependance o f t h e 19/2, resonance.  l/2> -> j 9/2,  - l/2>  - 33 CHAPTER IV RESULTS The p r e l i m i n a r y d a t a upon the sample was 19  1  T h i s d a t a showed t h a t t h e Ij,  19  taken by Dr. M.I.  Valic.  1  |y, T-—> l i n e o b s e r v e d w i t h t h e magnetic  f i e l d p a r a l l e l t o t h e c a x i s o f t h e c r y s t a l was unchanged as a f u n c t i o n of temperature f o r temperatures above 30K. broadened  At lower t e m p e r a t u r e s the  and a t r a n s i t i o n was o b s e r v e d n e a r 24K.  At s t i l l  line  l o w e r temperatures  (10K and lower) an a d d i t i o n a l b r o a d base l i n e f e a t u r e appeared.  In c o n t r a s t 19 1  the  l i n e s a f f e c t e d by the quadrupole i n t e r a c t i o n , f o r example the |y, -j> ->-  19  3  12", 2"> l i n e , broadened as the t e m p e r a t u r e was d e c r e a s e d below 40K and were u n o b s e r v a b l y b r o a d a t 24K. To f u r t h e r s t u d y t h i s p r o b l e m , a new s e t o f d a t a was t a k e n a t a h i g h e r f i e l d and b o t h s e t s o f d a t a were a n a l y s e d .  F o r t h e purpose o f t h i s  thesis 19  1  we w i l l r e s t r i c t our a n a l y s i s t o the t e m p e r a t u r e dependence o f the \-^, y> -»•9  i  12", l i n e and t h e d e t e r m i n a t i o n o f t h e quadrupole i n t e r a c t i o n and t h e K n i g h t s h i f t s at room temperature and 77K. 19 1 19 1 Temperature dependence o f the ^ + |<f> line L i n e shapes were o b t a i n e d a t t e m p e r a t u r e s between 5.2K f i x e d f r e q u e n c i e s o f 101.443 MHZ, At temperatures below 5.OK the  l i n e s broadened  was  observed n e a r 24K.  12.217 MHZ,  and 18.000  t h e sample was a s u p e r c o n d u c t o r .  as t h e temperature was  MHZ.  At lower f i e l d s  lowered below 30K and a t r a n s i t i o n  The h i g h f i e l d d a t a showed a t r a n s i t i o n n e a r 18K  no broad base l i n e f e a t u r e . below the t r a n s i t i o n .  15.822 MHZ  and 50K f o r  F i g s . 17-22  show the l i n e shape above, n e a r , and  The v a r i a t i o n o f the l i n e w i d t h as a f u n c t i o n o f  t e m p e r a t u r e i s shown i n F i g . 23. D e t e r m i n a t i o n o f e q^- and the K n i g h t S h i f t s 2  The quadropole f r e q u e n c y  and  and t h e K n i g h t S h i f t s K^  and K^  c a l c u l a t e d from t h e s p e c t r a o b t a i n e d a t room temperature and 77K.  were The  - 34 i 9 1 s e p a r a t i o n between t h e \^-, j > -*  19  '1  •9  1  ,9  3  and \-^-, 2~ "~ 12' "2* •*-^ >  d i r e c t measurement o f v^. A sample c a l c u l a t i o n o f  K  nes  i s shownin  F i g . 24... The c a l c u l a t e d v a l u e s o f e q^- = 24v_., K. and K . h Q' i s o aniso 2  s  n  S  ..p  are l i s t  i n T a b l e ( I I ) below.  Room temperature 0.41  ISO  aniso e qQ h 2  Table ( I I ) :  + .04%  77 K 0.39  +_ .01%  -0.12  +_ .01%  -0.19  0.06  +_ .02%  0.00  +_ .02%  0.35  +_ .03%  0.39  + .02%  59853.6  + 6 0 . 0 kHz  61864.8  + .02%  + 32.0 kHz  The quadrupole i n t e r a c t i o n and t h e K n i g h t S h i f t s a t room temperature and 77 K.  a v e  10 gauss  Fig.  (17):  Nb  i n NbSe  0  j 9/2,  with H p a r a l l e l to  l/2> c  -> ]9/2,  - l/2>  l i n e shape at 5.9'K  and  12.22  MHz  - 37 -  1  (Fig.  23):  Temperature and f i e l d dependence o f t h e l i n e w i d t h f o r the |9/2, l/2> + ]9/2, - l/2> l i n e w i t h H p a r a l l e l t o c. The l i n e w i d t h s p e c i f i e d i s t h e peak-to-peak o f t h e d e r i v a t i v e spectrum.  - 42 -  (24):  A sample c a l c u l a t i o n o f K . The c a l c u l a t e d resonance f r e q u e n c y ( _ l / 2 - E ^ ) / h i s p l o t t e d as a f u n c t i o n o f a p p l i e d f i e l d . The x  E  1  2  resonance f r e q u e n c y v  i s due t o an e f f e c t i v e f i e l d H 6Xp  the Nb n u c l e u s which i s lower t h a n the a p p l i e d f i e l d  H  observed  at the s i t e of Git  - 43 -  Discussion I f t h e t r a n s i t i o n observed a t low temperatures i s a r e s u l t o f h a v i n g two i n e q u i v a l e n t Nb s i t e s w i t h d i f f e r e n t e l e c t r i c f i e l d g r a d i e n t s (EFG) as e x p l a i n e d by E h r e n f r e u n d and Gossard the  [ 2 ] , then i t s h o u l d be p o s s i b l e t o f i t  l i n e shape o b t a i n e d a t temperatures below t h e t r a n s i t i o n w i t h two l i n e s  c o r r e s p o n d i n g t o t h e two d i f f e r e n t s i t e s .  T h i s was t r i e d and c o u l d n o t be  done. Moreover, t h e observed t r a n s i t i o n c o u l d n o t be d u e i t o a low temperature s t r u c t u r a l d i s t o r t i o n alone.  19 frequency o f the parameter f).  19  1 j>  Fig.(24)shows t h e dependence o f t h e c a l c u l a t e d  ->  \j,  1 -j>  l i n e as a f u n c t i o n o f t h e asymmetry  I t i s c l e a r t h a t n must be much l a r g e r than i. 05, as quoted by  G o s s a r d , i n o r d e r t o have any e f f e c t c o m p a t i b l e w i t h o u r e x p e r i m e n t a l r e s u l t s . The c a l c u l a t e d  f r e q u e n c y s h i f t f o r n = .05 i s 0.5 KHZ and i s independent o f  f i e l d i n t h e range between 13 and 17.5 KG.  The e f f e c t o f t a k i n g n i n t o  account  i s n o t o n l y n e g l i g i b l e b u t a l s o y i e l d s t h e wrong f i e l d dependence. However i f t h e s p l i t t i n g below t h e t r a n s i t i o n i s p l o t t e d  as a f u n c t i o n  o f a p p l i e d f i e l d , as shown i n F i g . (26), i t can be seen t h a t t h e z e r o f i e l d s p l i t t i n g i s j u s t t h e l i n e w i d t h o b t a i n e d above t h e t r a n s i t i o n s .  This indicates  t h a t t h e l i n e shape i s not a s u p e r p o s i t i o n o f j u s t two l i n e s b u t i s a r e s u l t of a d i s t r i b u t i o n o f Knight S h i f t s .  I n o t h e r words t h e l i n e shape below t h e  t r a n s i t i o n i s a c o n v o l u t i o n o f some d i s t r i b u t i o n f u n c t i o n whose b r e a d t h i s l i n e a r i n f i e l d , and t h e l i n e shape f u n c t i o n above t h e t r a n s i t i o n . The change i n t h e t r a n s i t i o n temperature from 24K t o 18K a t h i g h f i e l d i s an i n d i c a t i o n t h a t t h e NMR p r o p e r t i e s o f t h e sample a r e dependent upon i t s m e c h a n i c a l and t h e r m a l h i s t o r y .  A p r e l i m i n a r y experiment on a specimen  w i t h 1% excess Nb showed no t r a n s i t i o n a t a l l . F i n a l l y we note t h a t t h e temperature 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  - 44 -  and e q^-between 77K and room temperature i s i n d i c a t i v e o f some change, 2  i n the e l e c t r o n wavefunctions w i t h temperature. e s t i n g t h a t K. b  It i s particularly  i s v e r y s m a l l and i n f a c t e s s e n t i a l l y zero a t 77K.  interThis  ISO  i n d i c a t e s that the S - e l e c t r o n c h a r a c t e r of the e l e c t r o n wavefunctions at t h e Fermi s u r f a c e i s v e r y s m a l l .  In c o n t r a s t the v e r y l a r g e a n i s o t r o p y o f  the K n i g h t S h i f t — as f a r as we know, t h e l a r g e s t y e t observed — i s s t r o n g evidence f o r a l a r g e P^. component i n t h e e l e c t r o n w a v e f u n c t i o n s is p a r a l l e l to c ) .  (where Z  T h i s s t r o n g d i r e c t i o n a l e f f e c t must be r e l a t e d t o t h e  s t r u c t u r e o f t h e Fermi s u r f a c e o f NbSe . 0  (25):  The v a r i a t i o n of the j9/2, l/2> + 19/2, - l/2> f r e q u e n c y as a f u n c t i o n o f n.  I  Fig.  I  I  I  1 1 I 5 10 15 Field ( expressed in MHz of Nb frequency) I  I  I  I  I  I  I  I  ( 2 6 ) : F i e l d dependence o f t h e l i n e w i d t h o f t h e 19/2, l/2> -> 19/2, transition.  I  I  I  I  1 20  - l/2> l i n e w e l l below t h e  - 47 -  CONCLUSION  In t h i s t h e s i s we have s u c c e s s f u l l y c o n s t r u c t e d an FET Robinson s p e c t r o m e t e r which we have used i n an i n i t i a l s t u d y o f NbSe2.  The  r e s u l t s o b t a i n e d demonstrate t h a t below a c e r t a i n t r a n s i t i o n t e m p e r a t u r e the  Nb n u c l e i e x p e r i e n c e a d i s t r i b u t i o n o f K n i g h t s h i f t s which i m p l i e s  t h a t some l o n g range f e a t u r e i s a s s o c i a t e d w i t h t h e low t e m p e r a t u r e phase which r e s u l t s i n many n o n - e q u i v a l e n t Nb s i t e s .  I t i s c l e a r that  to  r e s o l v e t h i s f e a t u r e r e q u i r e s a d e t a i l e d s t u d y o f NbSe^ as a f u n c t i o n  of  sample p u r i t y , magnetic f i e l d , and t e m p e r a t u r e .  In a d d i t i o n the study  of t h 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 and e l e c t r i c  field  g r a d i e n t parameters i n t h e hexagonal phase s h o u l d g i v e i n t e r e s t i n g i n f o r m a t i o n on t h e e l e c t r o n w a v e f u n c t i o n s . the  I n a l l o f t h e above,  use o f s i n g l e c r y s t a l specimens i s e s s e n t i a l .  clearly  - 48 -  REFERENCES  A. Abragam, The P r i n c i p l e s o f N u c l e a r Magnetism.  Oxford U n i v e r s i t y  P r e s s (1961) , p. 200-205. E. E h r e n f r e u n d , A.C.  G o s s a r d , F.R.  Gamble and T.H.  Gebale, J . of  A p l . Phys. 42, 4 (1971). W a l t e r N. Hardy, u n p u b l i s h e d l e c t u r e n o t e s , U n i v e r s i t y o f B r i t i s h Columbia  (.1972) .  C h a r l e s P. P o o l e , J r . , and H o r a c i o A. F a r a c h , The Theory o f M a g n e t i c Resonance . F. N.H.  John W i l e y and Sons (1972), p . 2 0 8  Robinson, J o u r n a l o f S c i e n t i f i c I n s t r u m e n t s , V o l ; 3_6_ (1959).  T . J . Rowland, P r o g r e s s i n M a t e r i a l s S c i e n c e , V o l . 9_, No. J.A. W i l s o n and A.D.  Y a f f e , Advances i n P h y s i c s 18^, 73  1 (1961), p . l . (1969).  G. D. W a t k i n s , T h e s i s , H a r v a r d U n i v e r s i t y . J . W i n t e r , M a g n e t i c Resonance i n M e t a l s  ;  Oxford U n i v e r s i t y Press  (1971)^  Unlisted references: M.I. V a l i c , T h e s i s , U n i v e r s i t y o f B r i t i s h  Columbia.  A. V a n d e r z e i l , N o i s e S o u r c e s , C h a r a c t e r i z a t i o n and Measurement.. Prentice Hall  (1970).  A. V a n d e r z e i l , N o i s e i n S o l i d S t a t e D e v i c e s and L a s e r s , I.E.E.E. P r o c , 58, 8 (August, R i c h a r d S.C. Sons (1970).  1970).  G o l b o l d , Theory and A p p l i c a t i o n o f F e t s .  John W i l e y  and  -49-  APPENDIX A  Explicit  form f o r t h e m a t r i x o f t h e H a m i l t o n i a n s  f o r a spin  9  a r b i t r a r y o r i e n t a t i o n o f the magnetic f i e l d .  F o r a r b i t a r y 6, <}> the t o t a l H a m i l t o n i a n (Zeeman p l u s i s g i v e n , i n the n o t a t i o n s o f Chapter  •H  =  [-  Y {  6  sinecosf 2  _ isii.esiiu (I 2  \(I  3I|, - 1(1 + 1) +  +  I I , by:  r  -  2  - I J  +  v  sinGcos* 2  =  + I )] 2  =  'Y  a  sin6sin<J> _ 2 cose  3  = 5.  Then hv H  6  Q  [- a ( I + I_) + i B ( I  Or i n d i m e n s i o n l e s s  +  +  - 1^  + J (I . + I ) ]  +  - I_) + j ( I + I ) ]  2  2  units,  j ^ - f ' H ) = [- a ( I + I_) + i B ( I +  The m a t r i x r e p r e s e n t a t i v e f o r  +  c o s  Z'  For b r e v i t y l e t v  quadrupolar)  2  t H ) i s g i v e n by  2  -  51-  APPENDIX B  C C C C C C C C C C C C  10 503  19  2  3  A COMPUTER PROGRAMME FOR DIAGONALIZING THE TOTAL HAMILTONIAN(ZEEM AN+QU ADROPOL AR) FOR A SPIN 9/2 FOR AN ARBITRARY ORIENTATION OF THE MAGNETIC FIELD. I T CALCULATES THE EXACT EIGENVALUES AND THE CORRESPONDING NORMALIZED EIGENVECTORS. THE INPUT PARAMETRS ARE: BZERO=APPLIED MAGNATIC F I E L D ZMDQ=THE QUDRU POLE FREQUENCY THETA,PHI SPECIFY THE ORIENTATION OF THE MAGNETIC FIELD WITH RESPECT TO THE PRINCPAL AXES OF THE ELECTRIC F I E L D GRADIENT TENSOR. ETA= ASYMMETRY PARAMETER.  I M P L I C I T SEAL*8(A-H,0-Z) REAL*8DSQRT DIMENSION AH(10,10) , A I ( 1 0 , 10) , ER (10) , E I ( 1 0 ) , VR (10,10) , VI (10,10) READ(5,2,END=999)BZERO,ZMUQ,THETA,PHI,IT A PRINT 5 0 3 , BZERO,ZMUQ,THETA, P H I , ETA FORMAT('1',5X,T15,'BZERO',T3 0,'ZMUQ*,T50,*THETA*, 1T69, ' P H I ' , ^ , »ETA»//6X, 2F15. 3, 3F20.6//) XMUL=1,0407D0*BZERO Y=6. DO *XMUL/ZMUQ WRITE (6,19) Y FORMAT (/, ' Y=',F15.5) ALFA=Y*DSIN (THETA) *DCOS (PHI) / 2 . DO BETA=Y*DSIN(THETA)*DSIN (PHI)/2.DO DELTA=Y* DCOS (THETA) ^ FORMAT(5F15.6) DO 3 J=1,10 DO 3 1=1,10 AR ( I , J) =0. DO AI(I,J)=0.D0 AR (1 , 1) =36. DO-4. 5D0*DELTA AR(2,2) =12.D0-3.5D0*DELTA AR (3, 3) =-6. DO-2. 5DO*DELTA AR(4,4) =-18.DO-1 .5DO*DELTA AR (5,5) = -24. D0-0.5D0*DELTA AR(6,6) =-24.D0+0.5D0*DELTA AR (7,7) =-18. DO + 1. 5D0*DELTA AR (8,8) =-6.D0+2. 5DO*DELTA AR (9 ,9) = 12. DO+3. 5DO*DELTA AR(10,10) =36.D0+4.5D0*DELTA AR (1 ,2)=-3. DO*ALFA A I { 1 , 2 ) =3.D0*BETA AR (2,3) =-4. DO*ALFA AI(2,3)=4.D0*BETA AR (3,4) =-DSQRT (2 1. DO) *ALFA A I (3,4) =DSQRT (21 .DO) *BETA AR (4,5)=-2. D0*DSQRT (6. DO) *ALFA  -6-2-  100  200  501  502 500 4 5 6 20  8 9 999 11  AI (4,5)=2.D0*DSQBT (6.D0)*BETA AR (5,6) =-5, DO*ALFA AI (5,6)=5.D0*BETA AR (6 ,7) =-2. DO*DSQRT (6. DO) * ALFA AI (6,7) =2.D0*DSQRT (6 » DO) *BETA AS (7,8) =-DSQRT (21. DO) * ALFA AI (7, 8) =DSQRT (21 . DO) *BET A AB (8,9)=-4.D0*ALFA AI (8,9) =4.D0*BETA AR (9 , 10) =-3. DO*ALFA AI (9,10)=3.D0*BETA DO 100 M=1,9 DO 100 N=2,10 AR (N , M) = A R (M, N) AI (N,M) =-1.D0*AI (M,N) AR (1,3) =6.D0*ETA AR (2,U) =2.D0*DSQBT (21 .DO) *ETA AR (3,5)=3.D0*DSQBT (14. DO) *ET A AR(U,6) =5.D0*DSQRT (6. DO)* ETA AR (5,7) =5.D0*DSQRT (6. DO) *ETA AR (6,8) =3.D0*DSQRT (14. DO) *ETA AB (7,9) =2. DO*DSQBT (21. DO) *ETA AB (8,10)=6.D0*ETA DO 200 MM=1,8 DO 200 NN=3,10 AR (NN,MM)=AB (MM, NN) CALL DCEIGN(AB , AI, 10 ,10,EB,EI,VB,VI,IEBBOB, 1,0) N1 = 10 IF (IEBBOB.GT.O) GO TO 20 DO 500 IB0W=1,10 SUM=0.D0 DO 501 IC0L=1,10 SUM=SUM + VB(IROW,ICOL) **2 + VI (IBOW,ICOL)**2 SUM=DSQBT (SUM) DO 502 ICOL=1,10 VR (IBOW,ICOL)=VR (IBOW,ICOL)/SUM VI (IROW,ICOL)=VI (IROW,ICOL)/SUM CONTINUE PBINT 4 FORMAT(«0',T5,«EIGENVALUES•,T50,'EIGENVECTORS'//) DO 5 1=1,10 PBINT 6 , EB (I) , (V B (I, J) , J= 1, 10) FOBHAT('0«,F11.5 lX 10F9.5) N1=IEBROB-1 DO 8 KK=1,10 DO 8 KKK=1,10 A R (KK , KKK) = 0 . D0 AI (KK,KKK)=0.DO DO 9 KJ=1,10 ER(KJ)=0.D0 GO TO 10 PBINT 11 FOBMAT ( * 1 ') STOP END #  f  

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