Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Pulsed nuclear magnetic resonance in metal single crystals McLachlan, Leslie Allan 1965

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1966_A1 M3.pdf [ 8.6MB ]
Metadata
JSON: 831-1.0085527.json
JSON-LD: 831-1.0085527-ld.json
RDF/XML (Pretty): 831-1.0085527-rdf.xml
RDF/JSON: 831-1.0085527-rdf.json
Turtle: 831-1.0085527-turtle.txt
N-Triples: 831-1.0085527-rdf-ntriples.txt
Original Record: 831-1.0085527-source.json
Full Text
831-1.0085527-fulltext.txt
Citation
831-1.0085527.ris

Full Text

PULSED NUCLEAR MAGNETIC RESONANCE I N METAL SINGLE CRYSTALS  by L e s l i e A l l a n McL^chlan M.Sc.(Hons), U n i v e r s i t y o f New Z e a l a n d ,  1961.  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department of Physics.  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard.  THE UNIVERSITY OF BRITISH COLUMBIA November,  1965  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference and study.  I further agree that per-  mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives„  It is understood that copying or publi-  cation of this thesis for financial gain shall not be allowed without my written permission.  Department of The University of British Columbia Vancouver 8, Canada Date  Pes  \Cjb*>  The U n i v e r s i t y  of B r i t i s h  Columbia  FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR. THE  DEGREE OF  DOCTOR OF PHILOSOPHY  of  LESLIE ALLAN McLACHLAN  B.Sc,  U n i v e r s i t y of New Zealand 1960 M.Sc.(Hons), U n i v e r s i t y of .New Zealand 1961. THURSDAY, DECEMBER 9th, 1965, AT 2:30 P.M. IN ROOM 100, HENNINGS BUILDING  COMMITTEE IN CHARGE Chairman: I . McT, Cowan M.' Bloom L, W, Reeves C. F. Schwertfeger External  Examiner:  E;  Teghtsoonian B. G. T u r r e l l D. L l . W i l l i a m s A. G. R e d f i e l d  I.B.M„ Watson L a b o r a t o r y , New York Research S u p e r v i s o r :  D. L l . W i l l i a m s  PULSED NUCLEAR MAGNETIC  RESONANCE IN A  METAL SINGLE CRYSTALABSTRACT A p u l s e d n u c l e a r magnetic  resonance  spectrometer  u s i n g phase s e n s i t i v e d e t e c t i o n was c o n s t r u c t e d f o r use on metal s i n g l e c r y s t a l s . t i o n s were e s t a b l i s h e d  3  I t s c a p a b i l i t i e s and l i m i t a -  b o t h e x p e r i m e n t a l l y and t h e o r e t i  cal.ly The s p i n - l a t t i c e r e l a x a t i o n time was measured as a f u n c t i o n of temperature dium s i n g l e c r y s t a l s , ,  i n aluminium,  niobium and vana-  An u n s u c c e s s f u l attempt was made  to measure the a n i s o t r o p i c s p i n - l a t t i c e r e l a x a t i o n i n an i s o t o p i c a l l y pure t i n s i n g l e c r y s t a l .  The o r i e n -  t a t i o n dependence of t h e t i n s p i n - s p i n r e l a x a t i o n was measured and a n a l y s e d i n terms t u a t i o n model of Anderson  time  time  of t h e random f l u c -  and Weiss.  V a l u e s f o r both  the p s e u d o - d i p o l a r and psuedo-exchange c o n s t a n t s were obtained„ Spin echoes were observed i n t h e i s o t o p i c a l l y t i n and were used t o measure t h e s p i n - s p i n time.  pure  relaxation  T h i s was s h o r t e r than that measured from t h e  f r e e i n d u c t i o n decay. determined,  The reason f o r t h i s c o u l d not be  GRADUATE STUDIES Field  of Study;  N u c l e a r Magnetic  Resonance  Quantum. Theory of S o l i d s Advanced T o p i c s  i n Solid  R, B a r r i e State Physics  D. L L  Advanced Magnetism  M. Bloom  Low Temperature P h y s i c s  J  0  S t a t i s t i c a l . Mechanics  Related  Williams  B„ Brown R, B a r r i e  Studies  Electronic  Instrumentation  F„ K. Bowers  PUBLICATIONS  L. A. McLachlano  "Thermoluminescent  Emission  of X-ray I r r a d i a t e d A l k a l i H a l i d e s " . Solids  23,  1344 (1962) .  Spectra  J . Phys. Chem.  ii ABSTRACT  S p i n - l a t t i c e r e l a x a t i o n times have been measured i n m e t a l s i n g l e c r y s t a l s w i t h a p u l s e d n u c l e a r magr/etic  f  ance apparatus  reson-  a t both room and l i q u i d n i t r o g e n - temperatures.  .  I  The v a l u e s o b t a i n e d f o r aluminium and vanadium *agreed  well  w i t h the v a l u e s g i v e n i n t h e l i t e r a t u r e f o r powdered samples. The n i o b i u m v a l u e was s l i g h t l y lower than t h e most r e l i a b l e powder v a l u e , p o s s i b l y because of i m p u r i t i e s .  Measurements  were made on i s o t o p i c a l l y pure t i n t o see i f any a n i s o t r o p y c o u l d be d e t e c t e d i n t h e s p i n - l a t t i c e r e l a x a t i o n time. a n i s o t r o p y c o u l d be d e t e c t e d , b u t t h e c r y s t a l used was so u n f a v o u r a b l e  No  orientation  t h a t an a n i s o t r o p y o f l e s s  than  about 50% c o u l d n o t be d e t e c t e d . The  s p i n - s p i n r e l a x a t i o n time was measured i n t h e  i s o t o p i c a l l y pure t i n f o r f i v e d i f f e r e n t magnetic f i e l d entations.  These showed t h a t exchange n a r r o w i n g  ori-  occurred.  W i t h a s u i t a b l e c h o i c e o f o p e r a t i n g c o n d i t i o n s , t h e apparatus measured t h e e q u i v a l e n t o f t h e a b s o r p t i o n mode i n steady s t a t e n u c l e a r magnetic resonance as a f u n c t i o n o f magnetic field  orientation.  T h i s was combined w i t h t h e s p i n - s p i n  measurements t o g i v e t h e complete o r i e n t a t i o n dependence of the l a t t e r . for  These measurements gave a v a l u e of ( 2 . l i o . 3 ) K c / s .  t h e pseudo-exchange c o n s t a n t i n t i n .  second moment was found moment.  The p s e u d o - d i p o l a r  t o be t w i c e t h e d i p o l a r second  iii S p i n echoes were observed i n t h e i s o t o p i c a l l y pure t i n and were used t o measure the s p i n - s p i n r e l a x a t i o n t i m e .  These  gave v a l u e s -which were much s h o r t e r t h a n t h o s e measured by f r e e i n d u c t i o n decays.  The r e a s o n f o r t h i s was n o t d e t e r m i n e d .  iv  TABLE OF CONTENTS  PAGE .Afo  S *t 3? cl C t  TcibXs  $  e  o  o  o  »  o f Cont©n.ts  LiS*t  Of  List  Of  TclfoX©S  o  «  *  o  D  O  O  O  o  o  o  o  o  o  Acknov/led gementSo  »  <> »  INTRODUCTION©  0  0  0  t  O  IXXU-Stj?3"bJL OnS  0  f  0  o  0  Q  o  o  o  «  »  o  0  o  O  «  O  O  o  o  o  o  « 0  o  o  o  o  o  »  O  «  o  «  o  o  o  «  o  »  0  0  0  0  0  iv  o  o  e  «  o  «  VX  e  o  o  o  o  o  o  VXX  0  0  0  0  o  e  0  0  O  XX  o  •  O  o  o  »  O  o  o  e  O  «  O  o  O  »  O  » 0  »  O  o 0  »  o 9  O  • 0  *  i X  0  CHAPTER 1.  5  THE ELECTRONIC STRUCTURE OF METALS  1.1  The Wave F u n c t i o n s o f C o n d u c t i o n E l e c t r o n s  1.2  The M a g n e t i c S u s c e p t i b i l i t y o f C o n d u c t i o n E l e c t r o n s  II.  6  NUCLEAR MAGNETIC RESONANCE IN METALS .  11 18  2.1  The Magnetic F i e l d a t t h e Nucleus  18  2.2  The S p i n Temperature  2k  2.3  Spin-Lattice Relaxation  27  2. U- S p i n - S p i n R e l a x a t i o n  32  2.5  The Quadrupolar I n t e r a c t i o n  35  2.6  The L i n e W i d t h W i t h a Quadrupolar I n t e r a c t i o n . . .  38  2.7  P u l s e d NMR W i t h a Quadrupole I n t e r a c t i o n . . . . . .  38  III.  THE EXPERIMENTAL METHOD. . . . . .  .  ^3  3.1  G e n e r a l D e s c r i p t i o n of t h e A p p a r a t u s .  M+  3.2  The T i m i n g System  ^6  3.3  The Gated Power A m p l i f i e r  3 • ^*  T h © PT* 63. nip X i f i© J? o  3.5  The Main A m p l i f i e r . .  9+  3.6  The Boxcar I n t e g r a t o r .  56  3.7  Power S u p p l i e s and N o i s e S u p p r e s s i o n . . . . . . . .  58  3.8  The Magnets and Magnetic F i e l d Measurements . . . .  60  3.9  The Low Temperature System.  61  o  o  o  »  5l «  o  o  o  o  •  o  o  «  o  «  o  o  ^3  3.10 The C o i l System f o r a M e t a l l i c S i n g l e C r y s t a l . . . 3.11 A c o u s t i c O s c i l l a t i o n s  62 7^  3.12 C a l c u l a t i o n o f t h e S/N R a t i o . . . . . . .  79  x  V  CHAPTER  PAGE  III.(continued) 3.13  Measurement o f S p i n - L a t t i c e R e l a x a t i o n Times . . .  83  3.1 +  Measurements of S p i n - S p i n R e l a x a t i o n Times  . .  87  3.15  Measurement of A b s o r p t i o n and D i s p e r s i o n Modes . .  88  1  3.16 P o s s i b l e Improvements t o the A p p a r a t u s . . IV.  89 96  THE EXPERIMENTAL RESULTS . . . . . . . .  97  *+.l  Aluminium S i n g l e C r y s t a l  h.2  Vanadium S i n g l e C r y s t a l  98  *+.3  Niobium S i n g l e C r y s t a l  102  k.h  M e t a l s W i t h a L a r g e Quadrupole I n t e r a c t i o n . . . .  103  !+.5  Copper Wire, and Other S p u r i o u s S i g n a l S o u r c e s . . . 109  k-,6  I s o t o p i c a l l y Pure S i n g l e T i n  110  ^-.7  The E x p e r i m e n t a l S/N R a t i o s  135  CONCXJTJSXONO POSTSCRIPT B I B X i I O G H  APPENDIX  •  •  •  •  •  o  o  o  o  o  APHY  o  o  o  o  •  o  o  o  o  o  o  o  o  »  o  o  o  o  o  •  o  o  «  o  o  e  o  o  o  o  o  o  o  o  »  o  o  *  o  o  0  o  o  o  o  o  0  0  o  o  o  0  0  o  o  0  o  o  X^j?  o  o  o  I I . D e t a i l s of the Samples Used  APPENDIX I I I . APPENDIX APPENDIX  X  8  I . D i s t o r t i o n i n t h e Phase S e n s i t i v e D e t e c t i o n System  APPENDIX  15*1  0  The S i g n a l Induced i n the P i c k u p C o i l . . .  IV. Measurement o f A b s o r p t i o n and D i s p e r s i o n Modes w i t h P u l s e d NMR A p p a r a t u s V. C i r c u i t Diagrams  162 I6h 168 173 176  vi LIST OF TABLES TABLE ^.1  PAGE S p i n - S p i n R e l a x a t i o n Times by F r e e D@C3.y • •  Induction  o 0 • « e o o o o o © o o o o o e e e e « •  h-,2  S p i n - S p i n R e l a x a t i o n Times by S p i n Echoes  !+.3  V a r i a t i o n of S p i n Echo S p i n - S p i n  hoh  Time w i t h P u l s e L e n g t h E x p e r i m e n t a l and T h e o r e t i c a l . S/N R a t i o s  1X6  129  Relaxation 132 136  LIST OF ILLUSTRATIONS FIGURE 1.1  PAGE Energy V e r s u s Wave Number Diagram f o r t h e d Band of a T r a n s i t i o n M e t a l  16 92  3.1  B l o c k Diagram of the A p p a r a t u s  3.2  B l o c k Diagram of t h e T i m i n g System  93  3.3  E q u i v a l e n t C i r c u i t o f t h e Boxcar I n t e g r a t o r  57  3.^  E q u i v a l e n t C i r c u i t o f the C o i l System.  62  3.5  V a r i a t i o n o f the A c o u s t i c O s c i l l a t i o n A m p l i t u d e  . . . . . . .  w i t h t h e Magnetic F i e l d .  9*+  3.6  Diagram o f a Two P u l s e Sequence.  95  ^-.1  T y p i c a l Sweeps w i t h a Boxcar Gate Through a F r e e Induction T & i l  o  o  o  »  o  *  o  o  o  «  o  «  o  «  «  o  o  o  13^  . . .  139  e  h,2  Vanadium S p i n - L a t t i c e R e l a x a t i o n Measurements.  ^.3  A h i s t r o p y i n T i n S p i n - L a t t i c e R e l a x a t i o n Time. . . . lU-0 Induction T a i l Height versus r f Pulse Length . . . .  lM-l  *+.5  A S p i n Echo i n I s o t o p i c a l l y Pure T i n  ^.6  V a r i a t i o n of S p i n Echo A m p l i t u d e w i t h r f P u l s e W i(31 t l S O O O O O 0 9 O O « O O 9 o o e 0 o o o • V a r i a t i o n o f S p i n Echo A m p l i t u d e w i t h t h e Second  o o  1^"3  J?f P i l l S © Wid  o  1^ i ^ I  h,7  *f.8  "fctl  o  o  o  o  o  o  o  o  o  o  o  1^2  o  o  o  o  o  o  o  F r e e I n d u c t i o n Decay i n I s o t o p i c a l l y Pure T i n . . . . 1^5 L o r e n t z i a n L i n e Shape i n I s o t o p i c a l l y Pure T i n . . . lh6  h,10  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 the Met  gn© "tic F i s l d  o  t  t  o  o  «  o  o  o  o  o  o  o  o  o  o  o  o  *  o  «  V+Q  ^.ll  A n i s t r o p y o f t h e L i n e W i d t h i n I s o t o p i c a l l y Pure T i n  W-.12  L o g a r i t h m o f t h e S p i n Echo A m p l i t u d e V e r s u s Time . . 1^9 A n i s t r o p y o f t h e S p i n Echoes i n I s o t o p i c a l l y P u r e T i n 150  1.1 IV. 1  E q u i v a l e n t C i r c u i t o f t h e Phase S e n s i t i v e D e t e c t o r The A m p l i f i e r Output  . 162 173  viii PAGE  FIGURE IV.2  E f f e c t o f t h e Deadtime .  o  o  o  o  o  o  o  o  o  o  V.l  The Gated T r a n s m i t t e r .  O  O  B  O  O  O  O  O  O  O  V.2  The P r e a m p l i f i e r  o  o  o  o  o  o  o  o  o  v.3  The Boxcar I n t e g r a t o r . .  Q  O  O  O  O  O  O  O  O  Mixer, Pulse  .  A m p l i f i e r , and  o  9  0  17^"  . . .  177  o  . . .  178  O  . . .  179  Quench P u l s e r . . . . .  180 181  v.5  C o i n c i d e n c e Timing U n i t .  o  o  o  o  o  o  o  o  o  o  . . .  V.6  Slow Sawtooth G e n e r a t o r .  o  o  o  o  o  o  o  o  o  o  0  v.7  R e g u l a t e d F i l a m e n t Power S u p p l y  e  . . .  0  l82 183  ix ACKNOWLEDGEMENTS  I t i s a p l e a s u r e t o thank Dr. D. L I . W i l l i a m s f o r h i s c o n s t a n t and p a i n s t a k i n g h e l p i n a l l a s p e c t s of t h i s -work and e s p e c i a l l y f o r h i s encouragement when t h i n g s l o o k e d b l a c k . Dr. M. Bloom a l s o made i m p o r t a n t c o n t r i b u t i o n s t o t h i s work, b o t h d i r e c t l y through s u g g e s t i o n s and an i c o n o c l a s t i c r e a d i n g of t h i s t h e s i s and i n d i r e c t l y t h r o u g h a l l t h e n u c l e a r magnetic resonance t h e o r y I have l e a r n t f r o m him.  For a l l  t h i s I would l i k e t o thank him. Of t h e o t h e r people who have c o n t r i b u t e d t o t h i s work, I would l i k e t o acknowledge  an i n f o r m a t i v e d i s c u s s i o n w i t h  Dr. J.B.Brown on t h e a c o u s t i c a l a s p e c t s of t h i s work and Mr. Riseborough f o r t a k i n g the X-rays.  D r . H.E. Schone i s t o be  thanked f o r t h e l o a n of the i s o t o p i c a l l y pure s i n g l e c r y s t a l . The f i n a n c i a l s u p p o r t f r o m t h e U n i v e r s i t y of B r i t i s h Columbia, t h e B r i t i s h Columbia Hydro A u t h o r i t y , and t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada w h i c h made t h i s work possible, i s gratefully  acknowledged.  F i n a l l y , I must acknowledge  my i n d e b t e d n e s s t o Mr. and  Mrs. R. F. C a r s w e l l and t o Mr. and M r s . P. K. D i g g l e f o r t h e i r innumerable k i n d n e s s e s and a l s o t o my f e l l o w s t u d e n t s of t h e Lower M a l l f o r two i n t e l l e c t u a l l y s t i m u l a t i n g and s o c i a l l y chaotic years.  INTRODUCTION Ever s i n c e t h e e a r l y days of n u c l e a r e x p e r i m e n t s have been made on m e t a l s .  magnetic r e s o n a n c e ,  These e x p e r i m e n t s were  always made on f i n e l y ground powders suspended i n an i n s u l a ting o i l .  T h i s i s because of t h e r f s k i n e f f e c t w h i c h p r e v e n t s  r f f i e l d s penetrating sample.  Studies  more than a few m i c r o n s i n a m e t a l l i c  of s p i n - l a t t i c e r e l a x a t i o n , s p i n - s p i n r e l a x -  a t i o n , and t h e K n i g h t s h i f t were made on s e v e r a l m e t a l s and were even extended i n t o t h e s u p e r c o n d u c t i n g r e g i o n . considerable  information  These gave  on t h e s p h e r i c a l l y symmetric p a r t of  the c o n d u c t i o n e l e c t r o n d i s t r i b u t i o n .  I t was soon found  that  some l i n e s were asymmetric and t h i s was c o r r e c t l y i n t e r p r e t e d as b e i n g due t o an a n i s o t r o p i c K n i g h t s h i f t .  By a n a l y s i n g t h e  asymmetric l i n e shape, t h e magnitude of the a n i s o t r o p i c K n i g h t s h i f t c o u l d be obtained  with considerable  accuracy. ' Experi-  m e n t a l work was a l s o extended t o a l l o y s and t o l i q u i d  metals.  L a t e r on t h e a n a l y s i s of powder measurements was extended t o cover t h e case o f the presence of an a n i s o t r o p i c K n i g h t and  a quadrupole i n t e r a c t i o n .  shift  More r e c e n t l y s p i n echo e x p e r i -  ments have been made on powders w h i c h d i r e c t l y g i v e t h e pseudo-exchange  strength.  Despite the considerable  s u c c e s s of t h e powder method i n  measuring a n i s o t r o p i c p r o p e r t i e s , i t seemed obvious t o t r y and d i r e c t l y measure a n i s o t r o p i c p r o p e r t i e s i n a m e t a l s i n g l e crystal. ago,  T h i s was f i r s t done i n t h i s l a b o r a t o r y  u s i n g a sample c o n s t r u c t e d  several years  from t h i n s i n g l e - c r y s t a l s l a b s  2 s e p a r a t e d by i n s u l a t i n g l a y e r s .  The a p p a r a t u s used a conven-  t i o n a l m a r g i n a l o s c i l l a t o r and phase s e n s i t i v e d e t e c t i o n w i t h the sample c o o l e d t o l i q u i d h e l i u m temperature adequate S/N r a t i o . h i g h as 50.  t o g e t an  Under f a v o u r a b l e c i r c u m s t a n c e s t h i s was as  I n t h e f i r s t measurements, b o t h 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 l i n e w i d t h s were measured.  S i n c e then ex-  p e r i m e n t s of t h i s type have been made on s e v e r a l d i f f e r e n t m e t a l s , both i n t h i s l a b o r a t o r y and elsewhere.  Some of these  experiments d e t e c t e d d e t a i l s which were obscured by the a v e r a g i n g o v e r a l l o r i e n t a t i o n s w h i c h o c c u r s i n a powder measurement.  E a r l y on i t was found t h a t a c o i l f a i r l y  tightly  wound on a c y l i n d r i c a l sample was j u s t as good as t h e l a y e r e d sample. The, aim of t h i s work was t o b u i l d a p u l s e d NMR for  apparatus  s i n g l e c r y s t a l s which c o u l d be used i n c o n j u n c t i o n w i t h the  steady s t a t e a p p a r a t u s .  P u l s e d NMR measurements had never been  made i n s i n g l e c r y s t a l s b e f o r e , so t h a t t h i s work was of an exploratory nature. at  700Kc/s.  I n t h e e a r l y s t a g e s t h e a p p a r a t u s worked  T h i s low f r e q u e n c y was chosen w i t h t h e i n t e n t i o n  of u l t i m a t e l y d o i n g experiments  on s u p e r - c o n d u c t o r s .  However,  i t soon became c l e a r t h a t t h e S/N was going t o be t o o s m a l l at  t h i s f r e q u e n c y , so t h e a p p a r a t u s was r e b u i l t t o operate i n  the f r e q u e n c y range 6Mc/s. t o lOMc/s. s a t i s f a c t o r i l y a t these f r e q u e n c i e s .  The a p p a r a t u s worked For various experimental  r e a s o n s , measurements c o u l d n o t be made a t l i q u i d h e l i u m temp e r a t u r e s and t h i s r e s t r i c t e d done.  t h e experiments w h i c h c o u l d be  3 The  anisotropy  state experiments,  i n %. had  Anisotropy  o f t e n been observed i n steady i n T,  had  never been  detected  s i n c e steady s t a t e a p p a r a t u s i s q u i t e u n s u i t a b l e f o r T, ments.  Detection  of any  ment t o be a t t e m p t e d .  T,  anisotropy  There i s no  magnitude of such an e f f e c t . have only been a p p l i e d  was  measure-  thus the main e x p e r i -  t h e o r e t i c a l e s t i m a t e of  the  A l l the t h e o r i e s developed so f a r  t o a c u b i c l a t t i c e , or e l s e make assump-  t i o n s about the c o n d u c t i o n e l e c t r o n d i s t r i b u t i o n w h i c h may be v a l i d  i n a non-cubic l a t t i c e .  I t seems u n l i k e l y t h a t  not an  a n i s o t r o p i c T, would occur f o r a s p h e r i c a l e l e c t r o n d i s t r i b u t i o n so t h a t a s e a r c h f o r Ti  anisotropy  w i t h a n o n - c u b i c l a t t i c e and  should be c o n f i n e d  an a n i s o t r o p i c K n i g h t  to m e t a l s  shift.  Because of the e x p l o r a t o r y n a t u r e of t h i s work, most of the emphasis has been p l a c e d  on the e x p e r i m e n t a l a s p e c t s of  the t o p i c , r a t h e r t h a n on the t h e o r e t i c a l s i d e . t a r y n a t u r e of the t h e o r y of NMR f o r concentrating The and  i n m e t a l s was  The  another i n c e n t i v e  on the e x p e r i m e n t a l n a t u r e of the problem.  r u d i m e n t a r y s t a t e of the t h e o r i e s of  s p i n - s p i n r e l a x a t i o n i n m e t a l s i s not  complexity i s considered.  The  spin-lattice  s u r p r i s i n g when i t s  main f e a t u r e  of the e l e c t r o n i c  s t r u c t u r e of m e t a l s have been known f o r about t h r e e They are d e s c r i b e d  Most of the mechanisms i n v o l v e d  i n m e t a l s can be q u a l i t a t i v e l y d e s c r i b e d  these models.  decades.  i n terms of e i t h e r f r e e e l e c t r o n s , or e l s e  by t i g h t l y bound e l e c t r o n s . i n NMR  fragmen-  i n terms of  However, a q u a n t i t a t i v e d e s c r i p t i o n i s  s i n c e many of the NMR  p r o p e r t i e s are v e r y dependent on  t a i l s of the e l e c t r o n wave f u n c t i o n s . p a r i s o n of t h e o r e t i c a l and  impossible de-  A q u a n t i t a t i v e com  e x p e r i m e n t a l r e s u l t s must thus  -  a w a i t t h e compution o f much more a c c u r a t e e l e c t r o n wave f u n c tions.  Such a comparison would p r o v i d e an e x t r e m e l y  t e s t o f any computed wave f u n c t i o n . y e a r s b e f o r e such a comparison  However, i t w i l l  sensitive be many  i s p o s s i b l e f o r any b u t t h e  s i m p l e s t of m e t a l s . In  the f i r s t two c h a p t e r s t h i s b a s i c t h e o r y o f NMR i n  m e t a l s i s d i s c u s s e d and the p r o p e r t i e s of a l l t h e i m p o r t a n t mechanisms i n v o l v e d a r e l i s t e d . c o v e r s the e x p e r i m e n t a l d e t a i l s . the Appendices.  The t h i r d c h a p t e r e x h a u s t i v e l y Some of t h i s o v e r f l o w s i n t o  The r e s u l t s a r e g i v e n i n t h e f o u r t h c h a p t e r  and the t h e s i s concluded w i t h s u g g e s t i o n s f o r f u t u r e work. F i n a l l y , a note on t h e u n i t s employed i n t h i s  thesis.  For the c l a s s i c a l e l e c t r o m a g n e t i c s e c t i o n s M.K.S. u n i t s a r e used, w h i l s t c.g.s. u n i t s a r e used i n a l l t h e quantum mecha n i c a l expressions.  I n the s e c t i o n s i n v o l v i n g both  electro-  magnetic and atomic c o n s i d e r a t i o n s , t h e c h o i c e of u n i t s depends upon the u l t i m a t e use o f the e x p r e s s i o n s i n the section.  5 CHAPTER I THE ELECTRONIC STRUCTURE OF METALS  ' A c c i d e n t a l and F o r t u i t o u s Concurrence of Atoms.' - Lord  Palmerston.  The purpose o f t h i s c h a p t e r i s t o summarise t h e p r o p e r t i e s of t h e e l e c t r o n i c s t r u c t u r e of m e t a l s n e c e s s a r y f o r an u n d e r s t a n d i n g of t h e i r n u c l e a r magnetic resonance.  It is  assumed t h a t the r e a d e r i s f a m i l i a r -with t h e concept of B r i l l o u i n zones and F e r m i s u r f a c e s , as d e s c r i b e d dard t e x t s (9 standard tations.  9  365  37).  i n the stan-  The f i r s t s e c t i o n d e s c r i b e s t h e  forms of e l e c t r o n i c wave f u n c t i o n s and t h e i r  limi-  The f i n a l s e c t i o n concerns t h e e f f e c t s of a s t a t i c  magnetic f i e l d upon the c o n d u c t i o n e l e c t r o n s . The e l e c t r o n energy l e v e l s c a n , i n p r i n c i p l e , be got by s o l v i n g t h e S c h r o e d i n g e r e q u a t i o n and  n u c l e i i n the metal.  of a l l the e l e c t r o n s  T h i s i s an i m p o s s i b l e  much s i m p l i f i e d model i s adopted.  t a s k so a  The n u c l e i f o r m a p e r i o d i c  l a t t i c e and a r e surrounded by c l o s e d s h e l l s of t i g h t l y bound e l e c t r o n s whose only e f f e c t i s assumed t o be t o p a r t i a l l y s h i e l d the n u c l e a r  charge.  There a r e a l s o l o o s e l y bound con-  d u c t i o n e l e c t r o n s shared t o some e x t e n t by a l l t h e n u c l e i i n the l a t t i c e .  The S c h r o e d i n g e r e q u a t i o n  e l e c t r o n s must t h e n be s o l v e d .  f o r a l l these  To do t h i s , t h e c r u c i a l assump-  t i o n i s made t h a t t h e e l e c t r o n s i n t e r a c t so weakly t h a t they  6 can move i n d e p e n d e n t l y of each o t h e r .  The wave f u n c t i o n s and  energy l e v e l s o b t a i n e d are thus those f o r an  independent  electron. 1.1  The Wave F u n c t i o n o f C o n d u c t i o n E l e c t r o n s The most i m p o r t a n t e f f e c t  p o t e n t i a l i s to r e s t r i c t  of the p e r i o d i c  lattice  the s o l u t i o n s of S c h r o e d i n g e r s 1  e q u a t i o n to the form = U(k,r)  U(fe,£)  expCik.^).  i s a f u n c t i o n , depending on the wave v e c t o r k, of the  e l e c t r o n , w h i c h has the p e r i o d i c i t y of the l a t t i c e .  These  s o l u t i o n s are known as B l o c h f u n c t i o n s and are s i m i l a r i n form to the plane wave e x p C i k ^ r J . There are o t h e r r e s t r i c t i o n s  on the form t h a t the c o n -  duction e l e c t r o n wave f u n c t i o n can t a k e .  The coulomb a t t r a c t i o n  i s v e r y s t r o n g c l o s e to the n u c l e u s and so the c o n d u c t i o n e l e c t r o n must have a c o m p e n s a t i n g l y l a r g e k i n e t i c energy avoid being captured.  to  I t thus o s c i l l a t e s r a p i d l y , r a t h e r i n  the manner of the wave f u n c t i o n of a v a l e n c e e l e c t r o n i n a f r e e atom.  However, u n l i k e the f r e e  atom, t h e r e i s no e x p e r i -  mental e v i d e n c e i n most m e t a l s f o r the e l e c t r o n h a v i n g any o r b i t a l a n g u l a r momentum.  T h i s " quenching " of the  orbital  a n g u l a r momentum o c c u r s because the e l e c t r o n moves i n a potential field  h a v i n g the l a t t i c e symmetry, n o t  symmetry as i n a f r e e atom (36).  spherical  T h i s means t h a t the wave  f u n c t i o n depends on the l a t t i c e symmetry, as w e l l as on the  7 v a l e n c e band from which i t o r i g i n a t e d .  Between the i o n s the  e l e c t r o n moved i n a more u n i f o r m p o t e n t i a l and behaves r a t h e r l i k e a free  electron.  Even i f many-body e f f e c t s a r e i g n o r e d , t h e e x a c t s o l u t i o n of S c h r o e d i n g e r ' s e q u a t i o n i s i m p o s s i b l e .  The f i r s t  i s d e c i d i n g on the c o r r e c t form o f t h e p e r i o d i c  problem  potential.  T h i s i n i t s e l f i s a complex many-body problem and even i f i t were s o l v e d c o m p u t a t i o n a l d i f f i c u l t i e s p r e c l u d e an e x a c t s o l u t i o n of S c h r o e d i n g e r ' s e q u a t i o n .  I t i s thus n e c e s s a r y to  assume v a r i o u s approximate forms f o r the wave f u n c t i o n and then see how w e l l they s a t i s f y S c h r o e d i n g e r ' s e q u a t i o n . Three of the s i m p l e s t types of approximate wave f u n c t i o n s w i l l now be d e s c r i b e d , ( i ) The T i g h t B i n d i n g A p p r o x i m a t i o n . . T h i s assumes t h a t i n s i d e each i o n t h e wave f u n c t i o n i s s i m i l a r t o t h e wave f u n c t i o n o f an e l e c t r o n i n a f r e e atom. A s u i t a b l e s e t of atomic wave f u n c t i o n s i s then chosen f o r each ion  i n the l a t t i c e and then a l i n e a r c o m b i n a t i o n of these  t a k e n t o g i v e a B l o c h f u n c t i o n f o r t h e whole l a t t i c e  of the  form  f u n c t i o n on t h e j t h atom i n the metal.  F o r t h i s method t o be s a t i s f a c t o r y , i t i s n e c e s s a r y  t h a t t h e atomic wave f u n c t i o n s on d i f f e r e n t atoms do n o t overlap  much, so t h a t each e l e c t r o n i s p r e d o m i n a n t l y i n the near  neighbourhood  of i t s p a r e n t atom.  I t i s particularly  suitable  8 for  d e s c r i b i n g the narrow d band i n the t r a n s i t i o n There a r e more e l a b o r a t e v e r s i o n s of t h i s  principle  which g i v e b e t t e r r e s u l t s , but they a l l have severe tional  metals.  computa-  difficulties.  ( i i ) The N e a r l y F r e e E l e c t r o n  Approximation  E x a c t l y the o p p o s i t e assumption t o the t i g h t b i n d i n g case i s made.  I t i s t h a t the e l e c t r o n s move i n a p e r i o d i c  p o t e n t i a l w h i c h i s much l e s s t h a n t h e i r k i n e t i c energy so t h a t they can be d e s c r i b e d by plane waves and the p e r i o d i c p o t e n t i a l t r e a t e d as a p e r t u r b a t i o n .  T h i s treatment  the c o n c e p t of B r i l l o u i n zones.  leads d i r e c t l y to  I t a l s o shows t h a t e l e c t r o n s  i n a m e t a l can be t r e a t e d as a s i m p l e e l e c t r o n gas moving i n a constant  p o t e n t i a l , provided  the e l e c t r o n mass m i s r e p l a c e d by  an e f f e c t i v e mass m* w h i c h depends on the way the e l e c t r o n energy v a r i e s w i t h wave number.  U s i n g t h i s v e r y simple model  s u r p r i s i n g l y a c c u r a t e F e r m i s u r f a c e s and energy bands can be c a l c u l a t e d f o r q u i t e a few m e t a l s  (22).  Due t o i t s s i m p l i c i t y and a b i l i t y  t o e a s i l y d e s c r i b e the  e l e c t r o n i c s t r u c t u r e of a m e t a l , the n e a r l y f r e e e l e c t r o n model i s the one most commonly used, even i n cases where i t i s c l e a r l y not very  suitable.  ( i i i ) The O r t h o g o n a l i s e d  P l a n e Wave Method.  I n n e a r l y a l l m e t a l s the s i t u a t i o n l i e s somewhere between the n e a r l y f r e e e l e c t r o n case and the t i g h t b i n d i n g a p p r o x i mation.  Many ways have been d e v i s e d f o r t r e a t i n g t h i s  inter-  mediate s i t u a t i o n , but o n l y the O.P.W. method w i l l be d e s c r i b e d  9 here.  T h i s i s because i t g i v e s a r e a s o n a b l y a c c u r a t e wave  f u n c t i o n and a l s o shows the l i m i t a t i o n s of t h e f r e e  electron  approximation. The  O.P.W. method i s based on t h e r e q u i r e m e n t t h a t t h e  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 be o r t h o g o n a l t o a l l the filled  core wave f u n c t i o n s t o s a t i s f y t h e e x c l u s i o n p r i n c i p l e .  To do t h i s a l i n e a r c o m b i n a t i o n of c o r e wave f u n c t i o n s  i s sub-  t r a c t e d from a p l a n e wave i n such a way t h a t t h e r e s u l t i n g 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 i s o r t h o g o n a l t o a l l the c o r e wave f u n c t i o n s . functions  Then a l i n e a r c o m b i n a t i o n o f these wave  i s found w h i c h b e s t s a t i s f i e s S c h r o e d i n g e r s 1  equation. The  e l e c t r o n wave f u n c t i o n thus appears as a plane  wave between t h e i o n s , but o s c i l l a t e s r a p i d l y near an i o n core.  T h i s i s f a i r l y c l o s e t o how a r e a l wave f u n c t i o n must  look.  T h i s wave f u n c t i o n i s c o n v e n t i o n a l l y  described  by sep-  a r a t i n g i t i n t o a sum of s , p , d , — - c o n t r i b u t i o n s , these h a v i n g s p a t i a l symmetry p r o p e r t i e s  s i m i l a r t o those of the c o r r e s -  ponding atomic wave f u n c t i o n s .  The s e l e c t r o n wave f u n c t i o n  c o r r e s p o n d s t o a p l a n e wave. I t i s now c l e a r why the n e a r l y f r e e e l e c t r o n m a t i o n works as w e l l as i t does.  approxi-  The e l e c t r o n i s n e a r l y  free  between the i o n s , but c l o s e t o an i o n g a i n s enough k i n e t i c energy t o a p p r o x i m a t e l y c a n c e l tial. and  the a t t r a c t i v e coulomb poten-  Thus the e f f e c t i v e p o t e n t i a l of an i o n i s q u i t e  small  i s t y p i c a l l y l e s s than t h e k i n e t i c energy o f the e l e c t r o n  10 so t h a t c o n d i t i o n s are s i m i l a r t o those of the n e a r l y e l e c t r o n model.  T h i s i s why  free  n e a r l y f r e e e l e c t r o n s are a r e a -  s o n a b l y good a p p r o x i m a t i o n as f a r as band s t r u c t u r e c a l c u l a t i o n s a r e concerned.  However, the O.P.W. method a l s o shows t h a t  t r u e wave f u n c t i o n u s u a l l y a l s o c o n t a i n s p,d, which may  the  contributions  be very i m p o r t a n t i n c a l c u l a t i n g some o t h e r  proper-  ties. Another r e a s o n why  the n e a r l y f r e e e l e c t r o n model works  i s t h a t an e l e c t r o n f e e l s not o n l y the coulomb p o t e n t i a l of a g i v e n i o n , but t h a t of a l l the o t h e r e l e c t r o n s and l a t t i c e as w e l l .  The  l a t t e r screen  the i o n p o t e n t i a l so  i t i s n e g l i g i b l e beyond about a l a t t i c e s p a c i n g considerably  i o n s i n the  (37).  that  This  reduces the e f f e c t of the i o n p o t e n t i a l on a f a s t  moving e l e c t r o n . A l l the c a l c u l a t i o n s r e l y on the a s s u m p t i o n t h a t conduction e l e c t r o n s i n t e r a c t weakly.  This, at f i r s t ,  the seems a  poor assumption s i n c e the average e l e c t r o n s e p a r a t i o n i s about a l a t t i c e spacing, of s e v e r a l ev. and  g i v i n g an average coulomb r e p u l s i o n energy  However, the s c r e e n i n g  ions converts  e f f e c t of the  electrons  the l o n g range coulomb p o t e n t i a l i n t o a  s h o r t range screened p o t e n t i a l .  T h i s reduces the  cross-  s e c t i o n f o r e l e c t r o n - e l e c t r o n c o l l i s i o n s t o such an  extent  t h a t e l e c t r o n - l a t t i c e i m p e r f e c t i o n s c a t t e r i n g i s f a r more likely  (37).  There i s a l s o a r e p u l s i v e f o r c e between e l e c -  t r o n s w i t h p a r a l l e l s p i n s due  to the e x c l u s i o n p r i n c i p l e ,  i t does not d r a s t i c a l l y modify the s c r e e n i n g e f f e c t .  but  11 These e f f e c t s a r e o f t e n d i s c u s s e d p a r t i c l e s (38).  i n terms of q u a s i -  For a d i l u t e e l e c t r o n gas the q u a s i - p a r t i c l e s  are i d e n t i c a l w i t h the r e a l p a r t i c l e s .  I n the dense  electron  gas i n a m e t a l , the q u a s i - p a r t i c l e s behave l i k e e l e c t r o n s  with  an e f f e c t i v e mass m*which i s l a r g e r than the " bare *' mass of a free electron. So f a r the s p i n - o r b i t i n t e r a c t i o n has been T h i s i s the c o u p l i n g  neglected.  of the e l e c t r o n s p i n and o r b i t a l a n g u l a r  momentum t h r o u g h r e l a t i v i s t i c magnetic e f f e c t s w h i c h r a p i d l y i n s t r e n g t h w i t h i n c r e a s i n g atomic number. s e v e r a l e f f e c t s on the c o n d u c t i o n e l e c t r o n s .  increase I t can have  O r d i n a r i l y the  energy l e v e l s a r e d o u b l y degenerate because of the e l e c t r o n spin.  The s p i n - o r b i t i n t e r a c t i o n can l i f t  some of t h i s de-  generacy and t h i s s l i g h t l y a l t e r s the F e r m i s u r f a c e (22).  I t can a l s o r e i n s t a t e some of the o r b i t a l a n g u l a r  momentum quenched 1.2  structure  by the l a t t i c e p o t e n t i a l .  The Magnetic S u s c e p t i b i l i t y of C o n d u c t i o n Band E l e c t r o n s . The e f f e c t of a magnetic f i e l d  e l e c t r o n s w i l l ; now  be c o n s i d e r e d .  on the c o n d u c t i o n band  T h i s i s a c t u a l l y a problem  i n v o l v i n g d i f f i c u l t q u e s t i o n s of gauge i n v a r i a n c e and the v a l i d i t y of p e r t u r b a t i o n methods. t i e s w i l l be ignored  These c o n c e p t u a l d i f f i c u l -  i n the f o l l o w i n g d i s c u s s i o n w h i c h i s  based on the work of Kubo and Obata  (39)»  The f r e e energy F of a h i g h l y degenerate F e r m i gas i s . F = NE -kT.Tr[gCH)] F  where gCH) =ln{l+exp[^(E *H)]} and (3 =(kT)"' „ r  The magnetic s u s c e p t i b i l i t y X i s g i v e n by the thermodynamic relation  C o n s i d e r a s i n g l e e l e c t r o n i n a m e t a l i n f l u e n c e d by a s t a t i c e x t e r n a l magnetic f i e l d JH d e r i v e d f r o m a v e c t o r pot e n t i a l A,.  The H a m i l t o n i a n i s assumed t o be  V(i;) i s the p e r i o d i c e l e c t r o s t a t i c l a t t i c e p o t e n t i a l , S i s the e l e c t r o n s p i n , -^-jg.g,. i s the Zeeman energy, and A i s the s p i n - o r b i t c o u p l i n g  operator.  I f , f o r t h e moment, the s p i n - o r b i t i n t e r a c t i o n i s ignored  and t h e symmetric gauge A=-^(Hxr) chosen, the H a m i l -  t o n i a n can be d i v i d e d i n t o two p a r t s *H = U +  %  U = - # V + V(r) + - j | r A  where  J  J  % = yufi.CL + 2 § ) . L i s t h e e l e c t r o n o r b i t a l a n g u l a r momentum o p e r a t o r and has the l a t t i c e p e r i o d i c a l l y , w h i l e ^ i s the Bohr magneton. The wave f u n c t i o n s t o be used i n e v a l u a t i n g >f a r e Bloch functions.  Since  %  *H , and H, a r e p e r i o d i c i n t h e 0  l a t t i c e p o t e n t i a l , these g i v e a r e p r e s e n t a t i o n w h i c h i s diagonal  i n the wave v e c t o r k,(9)«  I f spin-orbit coupling  e f f e c t s a r e s m a l l the e l e c t r o n s p i n up and s p i n down s t a t e s can be c o n s i d e r e d  separately.  The wave f u n c t i o n  corres-  13 ponding to the energy E*(kJ is thus |.n,k) .  It is important  to note that because of Brillouin zones, excited states, and the applied magnetic field several different energies can correspond to the same value of k,. If i t is assumed that  <  H»> H J S  I  Tr[gCH)] can be expanded  in powers of H by means of McLaurin's theorem and standard perturbation theory to give TrfgCH)] =Tr[g(H )] +]T-f(E )<qm,ld> 0  q  HX^^.-^^|<am,|q>f-  where f (<H )=g'CH) = [l+exp{§(E - H )}] ' is the Fermi function. F  The summation is overall possible energy values for a l l possible values of After considerable mathematical labour Tr gCHj]  can be  shown to give the diamagnetic susceptibility of the conduction band electrons (9)«  For nearly free electrons this is the  Landau diamagnetism (9),  (36).  This arises from the electron  moving in;,a circular orbit around the applied magnetic f i e l d . The second term requires a permanent magnetic moment to be present and so only occurs for ferromagnetic metals, a case which w i l l not be considered here. The last term gives the paramagnetic susceptibility. To evaluate this summation over a l l possible energy values is changed to an integration over k space and a summation over a l l the values of E (k^) which correspond to each value of k,. n  Substituting in the thermodynamic relation for X gives  =  7iirT SI A  57 r^j,7 ^L f  <hklL+2SI mkVmkl L+2S| nk> d k .  X c o n s i s t s of t h r e e c o n t r i b u t i o n s , ( i ) The P a u l i s p i n paramagnetism.  *P= T ^ Z ; r j - ^ E f ^  |2fi | <*> <-fc| 281 "is) «*.  To get t h i s e x p r e s s i o n i n t o a more f a m i l i a r f o r m , t i o n i s made t h a t the F e r m i s u r f a c e e l e c t r o n s are n e a r l y  the  assump-  i s s p h e r i c a l and t h a t  the  free.  T h i s term a r i s e s from a s u r p l u s of e l e c t r o n s w i t h t h e i r magnetic moments p a r a l l e l to H , (ii)  S p i n - o r b i t paramagnetism. f  ?;-" f ^ E  [ < ^ \ L|mls><mk| as| nk> / n k I 2S I mk> /mk  The e f f e c t  L I nk>  of s p i n - o r b i t c o u p l i n g i s to l i f t  +  dk.  slightly  the  quenching of the o r b i t a l a n g u l a r momentum by m i x i n g i n o t h e r s t a t e s of a p p r o p r i a t e a n g u l a r momentum and symmetry. t a l a n g u l a r momentum i s a p p r o x i m a t e l y  The o r b i -  where X i s the  o r b i t c o u p l i n g c o n s t a n t and A i s the mean energy between s t a t e b e i n g c o n s i d e r e d and the s t a t e s b e i n g mixed i n  spinthe  (36).  The arrangement  of energy l e v e l s w i t h i n the c o n d u c t i o n band i s  so complex t h a t  A i s a l m o s t i m p o s s i b l e to c a l c u l a t e , but  is  of the o r d e r of the b a n d w i d t h , a few e v . X i s about a t e n t h of  an ev. or l e s s  (22).  I f t h e o r b i t a l m a t r i x elements a r e now  e v a l u a t e d by u s i n g t h e t i g h t b i n d i n g a p p r o x i m a t i o n ,  i t i s seen  t h a t t h e remainder o f the e x p r e s s i o n c l o s e l y resembles t h a t f o r  ..  J^io —  ~g  •  S i n c e -£« 1 i t i s n o t an i m p o r t a n t  term,  ( i i i ) Van V l e c k paramagnetism. x  ""to$kf  ^f'-i^  ^ I M " * )  T h i s i s a second order e f f e c t  ( ^\h\^) m  a r i s i n g through t h e  o r b i t a l a n g u l a r momentum o p e r a t o r m i x i n g unoccupied s t a t e s i n t o an occupied a n g u l a r momentum.  excited  ground s t a t e w i t h quenched o r b i t a l  I t i s t h e same as t h e Van V l e c k temperature  independent induced (36).  £°  d  o r b i t a l paramagnetism found i n some s o l i d s  I f a t i g h t binding approximation  i s used JL o n l y has  m a t r i x elements between s t a t e s w i t h t h e same v a l u e of k w h i c h d i f f e r i n t h e v a l u e of t h e i r magnetic quantum number m .  Thus  t h e r e a r e o n l y c o n t r i b u t i o n s from m a t r i x elements between l e v e l s i n t h e same p a r t i a l l y f i l l e d  band.  Most metals  have  m a i n l y s e l e c t r o n s i n t h e i r c o n d u c t i o n band and.so " X v i s negligible.  However t r a n s i t i o n metals have a p a r t i a l l y f i l l e d  d band which can g i v e r i s e t o a s i g n i f i c a n t v a l u e of " X v . F i g u r e 1.1  shows the. t y p i c a l s t r u c t u r e of t h e d band and  the types of t r a n s i t i o n s  t h a t g i v e r i s e t o "X . v  16  Energy (kltdron volts) Wavenumbar  Fig.  1.1  k  Energy v e r s u s Wavenumber f o r the" d Band i n a Typical T r a n s i t i o n Metal  where E i s t h e energy s e p a r a t i o n between t h e ground and excited  s t a t e s , s u i t a b l e averaged over k space. There a r e a l s o two o t h e r c o n t r i b u t i o n s t o t h e suscep-  tibility;  t h e c o r e e l e c t r o n diamagnetism, and the L a n g e v i n  paramagnetism  of the o r b i t a l a n g u l a r momentum caused by s p i n -  orbit coupling.  These a r e always v e r y s m a l l terms.  The b u l k s u s c e p t i b i l i t y i s due t o a t l e a s t s i x terms. These u s u a l l y o n l y have a secondary e f f e c t on t h e magnetic field  a t t h e n u c l e u s , w h i c h may be much d i f f e r e n t from t h a t  expected f r o m the v a l u e of X .  T h i s i s s i n c e t h e b u l k sus-  c e p t i b i l i t y depends on the e l e c t r o n d i s t r i b u t i o n t h r o u g h o u t ,the  whole s o l i d , w h i l s t the magnetic f i e l d  a t the nucleus  depends p r e d o m i n a n t l y upon t h e c u r r e n t s and magnetic moments i n i t s v e r y near v i c i n i t y .  17 The magnetic f i e l d  s h i f t s a t the n u c l e u s a r e u s u a l l y  m a i n l y due t o i n d i r e c t e f f e c t s o c c u r r i n g through the field  magnetic  p o l a r i s i n g the c o n d u c t i o n e l e c t r o n s ( P a u l i s p i n p a r a -  magnetism).  These t h e n c o u p l e w i t h the n u c l e u s t h r o u g h v a r i o u s  mechanisms t o produce a f i e l d  s h i f t much l a r g e r than t h a t d i r -  e c t l y due t o the P a u l i s p i n paramagnetism. paramagnetism, and the Landau diamagnetism  The Van V l e c k a r e the o n l y d i r e c t  terms w h i c h can sometimes be of importance i n d e t e r m i n i n g the magnetic f i e l d  a t the n u c l e u s .  18 CHAPTER I I NUCLEAR MAGNETIC RESONANCE IN METALS 'My mind i s i n a s t a t e of p h i l o s o p h i c a l doubt as to magnetism. 1  - Coleridge. I t i s t h e purpose of t h i s c h a p t e r t o d e s c r i b e t h e p r o p e r t i e s of t h e NMR of m e t a l s w h i c h d i f f e r f r o m those of o t h e r solids. of  I n accordance w i t h t h i s a i m , t h e elementary a s p e c t s  p u l s e d and steady s t a t e NMR w i l l n o t be g i v e n , these b e i n g  c o m p r e h e n s i v e l y d i s c u s s e d i n Chapter I I I of Abragam's book The v a r i o u s r e a s o n s why t h e magnetic f i e l d from t h a t of the a p p l i e d magnetic f i e l d  (1).  a t a nucleus d i f f e r s  are given.  The s p i n  temperature concept i s i n t r o d u c e d and i t s importance i n t h e r e t u r n t o e q u i l i b r i u m shown.  There a r e two decays i n v o l v e d i n  the  r e t u r n t o e q u i l i b r i u m , the s p i n - l a t t i c e r e l a x a t i o n govern-  ing  energy t r a n s f e r t o the l a t t i c e and s p i n - s p i n r e l a x a t i o n ,  w h i c h i s concerned w i t h i n t e r n a l e q u i l i b r i u m of t h e s p i n s y s tem.  S p i n - l a t t i c e r e l a x a t i o n and t h e magnetic f i e l d  often related. the  B o t h of t h e s e e f f e c t s a r e p r i n c i p a l l y due t o  conduction e l e c t r o n s , which can a l s o a f f e c t the s p i n - s p i n  relaxation.  Many m e t a l s have a quadrupole i n t e r a c t i o n w h i c h  a l t e r s t h e energy l e v e l s i n t h e system.  The l a s t s e c t i o n shows  how t h i s m o d i f i e s t h e p u l s e d NMR i n s i n g l e 2.1  s h i f t are  crystals.  The Magnetic F i e l d a t The Nucleus The r e s o n a n t f r e q u e n c y o f a n u c l e u s i n a metal d i f f e r s  19 from i t s v a l u e i n an i n s u l a t o r .  The f r a c t i o n a l change i n the  magnetic f i e l d a t the n u c l e u s w h i c h causes t h i s i s known as the K n i g h t s h i f t . r e l a t i v e importance Two  I t i s the sum  of v a r i o u s c o n t r i b u t i o n s whose  varies i n different  metals.  of these c o n t r i b u t i o n s a r i s e from the magnetic  c o u p l i n g of the n u c l e a r d i p o l e m o m e n t ^ t o the e l e c t r o n d i p o l e moment ^  and  to the c u r r e n t produced by i t s motion.  This  i n t e r a c t i o n can be t r e a t e d n o n - r e l a t i v i s t i c a l l y , p r o v i d e d  care  i s taken t o a v o i d d i v e r g e n c e s , to g i v e as the H a m i l t o n i a n ( 1 ) .  where £  <H=  +  ^  )  r  '  *  ]  ' ^h^ 2  •  i s the s e p a r a t i o n between the n u c l e u s and the e l e c t r o n .  The  o p e r a t o r H can be regarded as the magnetic f i e l d produced by the e l e c t r o n a t the n u c l e u s .  The l a s t term can be i g n o r e d s i n c e  the e l e c t r o n o r b i t a l a n g u l a r momentum L, i s quenched.  The  two terms each g i v e r i s e to a c o n t r i b u t i o n t o the K n i g h t ( i ) The " C o n t a c t "  other shift,  term.  T h i s i s the dominant term i n most metals and s i v e l y d i s c u s s e d i n the l i t e r a t u r e ( 1 , 16, H-0).  The  i s extenHamil-  tonian i s i The most i m p o r t a n t f e a t u r e s of t h i s i n t e r a c t i o n a r e i t s dependence on the s p i n o r i e n t a t i o n s and i t s D i r a c b f u n c t i o n form. T h i s means t h a t t h e r e i s no i n t e r a c t i o n u n l e s s the e l e c t r o n wave f u n c t i o n has a f i n i t e v a l u e a t the n u c l e u s .  p,d,...  e l e c t r o n s make no c o n t r i b u t i o n t o t h i s i n t e r a c t i o n s i n c e they  20 a l l v a n i s h a t the n u c l e u s . pic  The c o u p l i n g i s o n l y t o the i s o t r o -  s e l e c t r o n s and i s g i v e n by  0)|*  i s the s e l e c t r o n d e n s i t y a t the n u c l e u s , w h i l e t h e  z a x i s i s a l o n g the a p p l i e d magnetic f i e l d H . The c o n d u c t i o n 0  e l e c t r o n s a r e n o t l o c a l i s e d so the n u c l e u s i s e q u a l l y i n f l u enced by a l l o f them.  The summation thus becomes an ensemble  average, u s i n g F e r m i - D i r a c s t a t i s t i c s , electrons.  H  0  over a l l the c o n d u c t i o n  has p o l a r i s e d the c o n d u c t i o n e l e c t r o n s ( P a u l i  s p i n paramagnetism), so t h i s a v e r a g i n g g i v e s a magnetic f i e l d AH  0  a t the n u c l e u s . K  (^|"\|/ (0)| \ r  1  C  The r e s u l t i n g K n i g h t s h i f t i s  ^<i"t(o)r>x . f  i s the averaged  v a l u e o f |^J/'(0)| ' a  over a l l e l e c -  t r o n s w i t h the Fermi energy E , w h i l e X i s the n e a r l y f r e e P  F  e l e c t r o n P a u l i paramagnetism.  A more a c c u r a t e e q u a t i o n r e -  s u l t s i f the e l e c t r o n - e l e c t r o n i n t e r a c t i o n s a r e  approximately  taken i n t o account by u s i n g a many-body t h e o r e t i c a l , o r an experimental, value of K  c  Xp.  ranges from about 0.1$ t o about.3$> i n c r e a s i n g  s t e a d i l y w i t h i n c r e a s i n g atomic number.  A l t h o u g h K can be c  measured v e r y a c c u r a t e l y , i t u s u a l l y does n o t g i v e much i n f o r m a t i o n about j"ViyC0)j°* a c c u r a t e l y known.  because i n most m e t a l s X p i s not  T h i s i s u n f o r t u n a t e s i n c e |\|/(0)|  parameter w i t h c o n s i d e r a b l e t h e o r e t i c a l i n t e r e s t .  K  isa c  i s  temperature and magnetic f i e l d  independent i n most m e t a l s ,  except f o r s m a l l e f f e c t s due t o l a t t i c e e x p a n s i o n , ( i i ) The A n i s o t r o p i c K n i g h t S h i f t , The H a m i l t o n i a n f o r t h i s i n t e r a c t i o n i s  *H = ^ . Z  P7* [j£  -3r (^l.^)r: ] . a  t  T h i s c o u p l e s t h e n u c l e u s t o a l l the non-s e l e c t r o n s through their dipole-dipole interactions.  F o r an a x i a l l y  symmetric  c r y s t a l , a c a l c u l a t i o n s i m i l a r t o t h a t f o r t h e c o n t a c t term gives  (16) K«m = q' ( 3 e o ^ © - l ) X p ,  where q' = (J(|f [(3cosfe< - l ) r ' ] ( f ) d x ) 3  0  E  .  i s the a n g l e between H and the c r y s t a l a x i s of symmetry, 0  <k the a n g l e between H„ and j r , and (J) i s the t o t a l non-s e l e c t r o n wave f u n c t i o n ,  q' i s a measure of the s p a t i a l a n i -  s o t r o p y of the Fermi s u r f a c e e l e c t r o n ' s charge  distribution.  A p o s i t i v e q' means t h a t t h e non-s e l e c t r o n d e n s i t y i s l a r g e s t a l o n g t h e symmetry a x i s , a n e g a t i v e q  /  that i t i s largest i n  a p l a n e p e r p e n d i c u l a r t o the symmetry a x i s .  A group  theo-  r e t i c a l t r e a t m e n t shows t h a t q' i s z e r o f o r a c u b i c l a t t i c e , i r r e s p e c t i v e of the e l e c t r o n s t a t e s K  on  (*KL).  can e a s i l y be measured i n b o t h powders and s i n g l e  c r y s t a l s , but 'gives l i t t l e u s e f u l i n f o r m a t i o n , a p a r t from t h e s i g n of q'.  0.2$  I t v a r i e s i n magnitude from z e r o t o about  and tends t o i n c r e a s e w i t h i n c r e a s i n g atomic number, i s an anomalous case i n which K  Q r  »  K. c  Bi  a 0 C J  22  ( i i i ) Core P o l a r i s a t i o n K n i g h t S h i f t . The f i l l e d  e l e c t r o n s h e l l s have so f a r been i g n o r e d ,  a p a r t from t h e i r e l e c t r o n i c s c r e e n i n g e f f e c t and t h e i r n e g l i g i b l e diamagnetism.  However, they can sometimes g i v e the  most i m p o r t a n t c o n t r i b u t i o n t o the K n i g h t s h i f t by means of the H e i s e n b e r g exchange i n t e r a c t i o n  T h i s i n t e r a c t i o n ' i s z e r o f o r c o n d u c t i o n and core e l e c t r o n s w i t h a n t i p a r a l l e l s p i n s and i s r e p u l s i v e i f they have p a r a l l e l s p i n s (22).  A c o n d u c t i o n e l e c t r o n thus pushes  an s core e l e c t r o n w i t h a p a r a l l e l s p i n i n w a r d s ,  increasing  for this spin orientation. I f a magnetic f i e l d H  0  i s a p p l i e d , there i s a population  d i f f e r e n c e between c o n d u c t i o n e l e c t r o n s w i t h s p i n s p a r a l l e l and a n t i p a r a l l e l t o H  Q  .  I n a s i m p l e case t h i s causes an  from the e l e c t r o n s p a r a l l e l t o H p o r t i o n a l to H magnetic f i e l d shift.  0  .  0  pro-  This i s equivalent to a small a d d i t i o n a l  a t t h e n u c l e u s ; t h e core p o l a r i s a t i o n K n i g h t  I n a l l but t h e s i m p l e s t metals the exchange i n t e r a c t i o n  makes n e c e s s a r y a r e n o r m a l i s a t i o n of t h e wave f u n c t i o n s and t h i s a l l o w s the c o r e p o l a r i s a t i o n t o be e i t h e r p o s i t i v e or negative The  (^2). importance  of t h i s i n t e r a c t i o n has o n l y r e c e n t l y been  r e c o g n i z e d and as y e t t h e r e i s l i t t l e knowledge of i t s magnitude, b u t i t i s p r o b a b l y p r e s e n t t o a s i g n i f i c a n t e x t e n t i n a l l  23 m e t a l s (*f2).  I n ?t^  i t i s the dominant term, g i v i n g a K n i g h t  5  s h i f t o f -3*5% ( 1 6 ) .  T h i s i s one o f t h e two known n e g a t i v e  K n i g h t s h i f t s and i s t h e l a r g e s t K n i g h t s h i f t found  so f a r .  ( i v ) Van V l e c k O r b i t a l K n i g h t S h i f t . The  o r b i t a l c u r r e n t s c a u s i n g t h e Van V l e c k paramagnetism  can be s p l i t i n t o two p a r t s .  F i r s t l y , t h e r e i s a l o n g range  d e m a g n e t i s i n g f a c t o r which i s s m a l l enough t o be n e g l e c t e d . Secondly t h e r e i s t h e s h o r t range e f f e c t of t h e o r b i t a l c u r r e n t s which g i v e s a K n i g h t s h i f t  K fl  0*3)  = 2n<r~ >X . 3  v  v  i s the atomic volume and <r~ ) i s t h e v a l u e o f r ~ 3  3  averaged  over a l l t h e occupied c o n d u c t i o n band wave f u n c t i o n s . For t r a n s i t i o n metals t h i s term can be v e r y i m p o r t a n t . I t cannot be measured a c c u r a t e l y , n o r can <^r") o r X v be 3  separately determined, this  so l i t t l e i n f o r m a t i o n i s gained  from  term. The s p i n - o r b i t i n t e r a c t i o n removes some of t h e o r b i t a l  quenching.  The e f f e c t s o f t h i s a r e u n c e r t a i n s i n c e o n l y v e r y  s i m p l e c a l c u l a t i o n s can be done and t h e r e a r e no e x p e r i m e n t a l measurements i n m e t a l s o f s p i n - o r b i t e f f e c t s a l o n e .  It i s  known t h a t i t enhances the a n i s o t r o p i c K n i g h t s h i f t (^-1) and g i v e s a c o n t r i b u t i o n t o the i s o t r o p i c K n i g h t s h i f t i n any l a t t i c e w i t h i n v e r s i o n symmetry (*f3)o  I t a l s o presumably  causes the e l e c t r o n g f a c t o r t o become a n i s o t r o p i c , a s i n crystal field  theory ( M + ) . I f X i s the s p i n - o r b i t coupling  2h c o n s t a n t and A i s t h e average s e p a r a t i o n between e l e c t r o n s t a t e s , then  X / A i s a p p r o x i m a t e l y t h e f r a c t i o n o f unquenched  o r b i t a l a n g u l a r momentum ( M + ) .  I t i s g e n e r a l l y assumed t h a t t h e  e f f e c t s of s p i n - o r b i t c o u p l i n g a r e n e g l i g i b l e i f X i s v e r y much l e s s than t h e c o n d u c t i o n band w i d t h .  T h i s seems t o be t h e  case f o r most m e t a l s . There a r e v a r i o u s o t h e r d i a m a g n e t i c and paramagnetic terms, b u t t h e s e a r e u s u a l l y n e g l i g i b l e .  Paramagnetic, or  f e r r o m a g n e t i c , i m p u r i t i e s a r e the o n l y o t h e r i m p o r t a n t cause of l i n e s h i f t s .  W i t h m e t a l s of f i v e n i n e s p u r i t y , o r b e t t e r ,  t h e r e should be no e f f e c t s from t h i s s o u r c e . The K n i g h t s h i f t thus seems t o be p r e d o m i n a n t l y due t o four interactions.  Of these t h e c o n t a c t term i s i m p o r t a n t i n  n e a r l y a l l metals w h i l e core p o l a r i s a t i o n probably occurs t o some e x t e n t i n a l l m e t a l s , b u t i s dominant i n o n l y a few t r a n s i t i o n metals.  The Van V l e c k term i s p r o b a b l y o n l y im-  p o r t a n t i n some t r a n s i t i o n m e t a l s , w h i l e 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 o n l y o c c u r s f o r non-cubic 2.2  lattices.  The S p i n Temperature The H a m i l t o n i a n f o r a system o f N i d e n t i c a l n u c l e i i s  where  K = - Hs-X/W' »  H =  *Md;  Only t h e case where t h e Zeeman term here.  -3$  k  (yjdk.iik)*?].  "Hz>Hi  w i l l be c o n s i d e r e d  The a p p l i e d magnetic f i e l d H  0  s p l i t s t h e ground  energy l e v e l o f a n u c l e u s w i t h s p i n I i n t o 21+1 energy l e v e l s s e p a r a t e d by pE  .  0  equally  state spaced  I f t h e s p i n system i s i n  t h e r m a l e q u i l i b r i u m w i t h t h e l a t t i c e then t h e p o p u l a t i o n s p„,, p _, m  o f t h e m t h and ( m - l ) t h l e v e l s a r e g i v e n by t h e Boltzmann  distribution  where T i s e q u a l t o t h e l a t t i c e temperature T . L  Under c e r t a i n c i r c u m s t a n c e s the s p i n system can s t i l l be d e s c r i b e d i n t h i s way by a. s p i n temperature T , which can s  be d i f f e r e n t from t h e l a t t i c e temperature (^0). system i s p e r t u r b e d , t h e T by  s  r e l a x e s towards T  L  I f such a a t a rate given  (1) dt~  ( T  s  H}  " TT i " " " T  =  (  ) o  S i n c e M = C H ^ / T ( C u r i e ' s l a w ) , t h i s becomes z  dM  =  !_ CM  -M  )  .'. M oC l - e x p ( - \ ) > z  T, i s t h e s p i n - l a t t i c e r e l a t i o n time and i s a measure of t h e r a t e a t which energy i s t r a n s f e r r e d f r o m t h e s p i n system t o the l a t t i c e energy r e s e r v o i r . The H a m i l t o n i a n >i f o r t h e magnetic d i p o l e - d i p o l e i n d  teraction between t h e n u c l e i can be w r i t t e n as H  = Tj \  r; 5~(A+B+C+D+E+F), 3  k  where B = - ± ( l f IjT +1" ifXl-Scos^G ) .  26 The o t h e r terms a r e u n i m p o r t a n t i n t h e f o l l o w i n g  discussion  w h i c h i s based on t h a t of S l i c h t e r (^-0). The term B p l a y s a d e c i s i v e r o l e i n t h e e s t a b l i s h m e n t of a s p i n temperature.  I t c o u p l e s two n e i g h b o u r i n g n u c l e i ,  f l i p p i n g one s p i n up i n energy and the o t h e r one down. Ifj =%,  the mutual  s p i n f l i p c o n s e r v e s the Zeeman energy of  the system, y e t a l t e r s t h e p o p u l a t i o n levels.  Because  If  d i s t r i b u t i o n o f the  of these p r o p e r t i e s , i t can be shown t h a t i f  the s p i n system i s d i s t u r b e d ,  the mutual s p i n f l i p s  restore  the s p i n system t o a Boltzmann d i s t r i b u t i o n . T h i s decay o f a p e r t u r b e d s p i n system towards i n t e r n a l e q u i l i b r i u m m a n i f e s t s i t s e l f e x t e r n a l l y as a decay of the com„  ponents M and M , of M=^/A\ x  towards z e r o .  which are perpendicular  a  T h i s decay of M  x  and  i s characterised  to H , 0  by the  s p i n - s p i n r e l a x a t i o n time Ta,. Thus a f t e r a p e r t u r b a t i o n has been a p p l i e d  t o the s p i n system, i t undergoes  internal re-  o r g a n i z a t i o n f o r a time of about T , u n t i l a q u a s i - e q u i l i b r i u m a  state describable  by a s p i n temperature has been reached.  The  s p i n temperature then e x p o n e n t i a l l y r e l a x e s , w i t h a time cons t a n t T, , towards the l a t t i c e t e m p e r a t u r e .  This s i t u a t i o n  r e q u i r e s the s p i n s t o be much more t i g h t l y coupled t o each o t h e r than they a r e t o the l a t t i c e , i . e . T|> T . x  satisfy this  Most metals  condition.  I f the energy l e v e l s a r e n o t e q u a l l y spaced, the mutual s p i n f l i p s no l o n g e r  conserve Zeeman energy and so have neg-  l i g i b l e p r o b a b i l i t y of o c c u r r i n g .  Thus a f t e r a d i s t u r b a n c e  no  i n t e r n a l r e l a x a t i o n towards a s p i n temperature can o c c u r , b u t instead each l e v e l i n d e p e n d e n t l y t r a n s f e r s i t s energy t o the lattice.  The s p i n - l a t t i c e r e l a x a t i o n i s t h e n no l o n g e r des-  c r i b e d by a s i n g l e e x p o n e n t i a l . T h i s s i t u a t i o n w i l l be returned to l a t e r . 2.3  Spin-Lattice Relaxation, I n a m e t a l the s t r o n g e s t c o u p l i n g of t h e n u c l e i t o the  l a t t i c e i s v i a the conduction e l e c t r o n s .  A conduction elec-  t r o n can be regarded as b e i n g i n e l a s t i c a l l y s c a t t e r e d by means of a c o l l i s i o n w i t h a s i n g l e n u c l e u s . i t changes t h e n u c l e a r s p i n from I compensating  m  I n the process  t o I _ , and undergoes m  changes i n i t s own a n g u l a r momentum and k i n e t i c  energy, as r e q u i r e d by the c o n s e r v a t i o n l a w s .  This i s only  an a p p r o x i m a t i o n s i n c e an e l e c t r o n wave p a c k e t extends  over  s e v e r a l n u c l e i , so i t can be s c a t t e r e d f r o m an i n i t i a l t o a f i n a l s t a t e by s i m u l t a n e o u s c o l l i s i o n w i t h more than one n u c l e u s (^0). subsequent  This i s a r a r e occurrance i f H  > H *  (^5).  A  i n e l a s t i c c o l l i s i o n between the e l e c t r o n and an  i o n t r a n s f e r s the excess energy t o t h e l a t t i c e .  Since the  e l e c t r o n must be a b l e t o change i t s k i n e t i c energy by s m a l l amounts, o n l y the e l e c t r o n s near the Fermi s u r f a c e can t a k e part i n the r e l a x a t i o n , ( i ) Contact Relaxation. R e l a x a t i o n i s caused by the n u c l e u s s c a t t e r i n g s electrons.  The s c a t t e r i n g i n t e r a c t i o n i s thus the c o n t a c t  28 term  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 then g i v e s T, (^+0)  a temperature T as (T.T)"'  at  = -^rfkr/^lyCO)!*)*  Zj ( E ) o F  7T , Tn a r e the e l e c t r o n i c and n u c l e a r gyromagnetic r a t i o s e  r e s p e c t i v e l y , and Z ( E ) i s the average s e l e c t r o n d e n s i t y of s  F  s t a t e s a t the F e r m i s u r f a c e . The most i m p o r t a n t f e a t u r e of t h i s e x p r e s s i o n i s t h a t T, T i s a c o n s t a n t . ' T h i s has been e x p e r i m e n t a l l y v e r i f i e d f o r s e v e r a l metals  over a wide temperature range, the s m a l l de-  v i a t i o n s observed of the  b e i n g due  to the e f f e c t s of t h e r m a l  expansion  lattice. T i T i s r e l a t e d to the K n i g h t s h i f t due  to the  contact  term by I f the n e a r l y f r e e e l e c t r o n model i s assumed, t h i s s i m p l i f i e s t o the " K o r r i n g a T, TK,•c  -  expression  relation"  4  tT  k  ^  T«  ' '  A more a c c u r a t e e x p r e s s i o n w h i c h a l l o w s f o r i n t e r a c t i o n s between the c o n d u c t i o n e l e c t r o n s i s  X ° i s the f r e e e l e c t r o n v a l u e , w h i l e X F  P  can be e i t h e r  an  e x p e r i m e n t a l v a l u e , or a c a l c u l a t e d v a l u e u s i n g q u a s i p a r t i c l e theory.  T h i s e x p r e s s i o n d i f f e r s from the c o r r e s -  29 ponding one i n Abragam (1) or S l i c h t e r ( ^ O ) , s i n c e r e c e n t t h e o r e t i c a l work shows t h a t i t i s the f r e e e l e c t r o n d e n s i t y of s t a t e s t h a t i s i n v o l v e d i n . s p i n - l a t t i c e r e l a x a t i o n , not q u a s i - p a r t i c l e d e n s i t y of s t a t e s  the  (^6).  The K o r r i n g a r e l a t i o n , or i t s more a c c u r a t e v e r s i o n ,  is  commonly used to e s t i m a t e unknown v a l u e s of T, T from the known Knight s h i f t ,  or e l s e the v a l u e o f T| d e r i v e d from the  Korringa  r e l a t i o n can be compared w i t h the e x p e r i m e n t a l v a l u e .  A large  d i s c r e p a n c y between the two v a l u e s i n d i c a t e s actions besides  t h a t other  the c o n t a c t i n t e r a c t i o n are i m p o r t a n t  inter-  i n the  metal. ( i i ) Dipolar Relaxation. The i n t e r a c t i o n between the n u c l e a r and d i p o l e moments p r o v i d e s the s c a t t e r i n g  electronic  mechanism.  This gives  M) (T| T)~'  =4-rr(T  T ) % Z* ( E , ) <I" >*C. 3  e  3  n  C i s a term whose v a l u e depends on the l a t t i c e and structure.  electronic  I t has a v a l u e of about u n i t y and no a n i s o t r o p y  for a cubic l a t t i c e . band e l e c t r o n s .  <^r~ > i s averaged v  over a l l the  conduction  Z i s here d e n s i t y of s t a t e s of a l l the non-s  electrons. U n l i k e the a n i s o t r o p i c K n i g h t s h i f t , zero i n a cubic l a t t i c e . between T, and K  c  for this  T, can be non-  There i s no K o r r i n g a - l i k e r e l a t i o n interaction.  ( i i i ) Orbital The  Relaxation.  s c a t t e r i n g i s caused by t h e i n t e r a c t i o n  between the d i p o l e magnetic f i e l d  generated by o r b i t a l motion  of t h e c o n d u c t i o n e l e c t r o n and t h e n u c l e a r magnetic d i p o l e . Unlike  t h e p r e v i o u s two s c a t t e r i n g p r o c e s s e s , t h e r e i s no  r e o r i e n t a t i o n of t h e e l e c t r o n s p i n ; t h e n u c l e a r s p i n change b e i n g compensated f o r by an e q u a l change i n o r b i t a l a n g u l a r momentum.  T, T i s g i v e n by the same e x p r e s s i o n as t h e d i p o l a r  r e l a x a t i o n , e x c e p t t h a t C has d i f f e r e n t v a l u e s ( V 7 ) . Dipolar  and o r b i t a l r e l a x a t i o n have o n l y been c a l c u -  lated f o r a cubic predicted.  l a t t i c e , f o r w h i c h no a n i s o t r o p y i s  Theory has n o t y e t shown whether t h e r e should be  a significant anisotropic ( i v ) Core P o l a r i s a t i o n  T, i n n o n - c u b i c l a t t i c e s .  Relaxation.  A non-s e l e c t r o n p o l a r i s e s the c o r e e l e c t r o n s when i t i s scattered  by them.  The t r a n s i e n t p o l a r i s a t i o n a c t s on  the n u c l e u s t h r o u g h t h e c o n t a c t term t o g i v e n u c l e a r tion.  relaxa-  I n the c o l l i s i o n the e l e c t r o n s u f f e r s a s p i n f l i p t o  c o n s e r v e a n g u l a r momentum.  The r e l a x a t i o n time c a l c u l a t i o n  i s d i f f i c u l t and has o n l y been done f o r t r a n s i t i o n metals with a cubic  lattice  (27).  F o r t h i s c a s e , T, T has a f o r m  s i m i l a r t o t h a t of t h e c o n t a c t r e l a x a t i o n . the K o r r i n g a - l i k e  I t also  satisfies  relation  F i s a f a c t o r of about u n i t y w h i c h depends on t h e degeneracy of t h e c o n d u c t i o n band.  These r e l a x a t i o n mechanisms v i a the c o n d u c t i o n e l e c t r o n s a r e the most i m p o r t a n t ones.  Contact r e l a x a t i o n i s  the dominant mechanism i n most m e t a l s , but c o r e or o r b i t a l r e l a x a t i o n can dominate i n some t r a n s i t i o n m e t a l s . r e l a x a t i o n i s never i m p o r t a n t .  Dipolar  These r e l a x a t i o n mechanisms  are s t r o n g e r than those i n most o t h e r t y p e s of s o l i d s ; T, T t y p i c a l l y b e i n g about one  sec-deg.  There are p r o b a b l y no s i g n i f i c a n t i n t e r f e r e n c e  terms  between the v a r i o u s r e l a x a t i o n mechanisms so they can, i n p r i n c i p l e , be unambiguously  s e p a r a t e d (27).  t h i s i s . v e r y d i f f i c u l t and r e q u i r e s  In practice  an e x t e n s i v e s e r i e s of  measurements of T, , the K n i g h t s h i f t , and the magnetic  sus-  c e p t i b i l i t y over a wide temperature range and t h e use of s e v e r a l n o t v e r y r e l i a b l e t h e o r e t i c a l parameters.  The  r e s u l t of t h i s type of a n a l y s i s shows which i s the dominant mechanism, but i s q u a n t i t a t i v e l y u n r e l i a b l e . V a r i o u s r e l a x a t i o n mechanisms w h i c h d i r e c t l y the n u c l e u s to the l a t t i c e v i b r a t i o n s t h r o u g h  couple  magnetic  d i p o l e , or e l e c t r i c q u a d r u p o l e , i n t e r a c t i o n s a l s o occur These a r e a l l u n i m p o r t a n t i n m e t a l s . times cause n o t i c e a b l e  relaxation.  Impurities Paramagnetic,  (1).  can someor  f e r r o m a g n e t i c , i o n s have a l a r g e magnetic f i e l d near them which s t r o n g l y c o u p l e s n e i g h b o u r i n g n u c l e i t o the l a t t i c e v i b r a t i o n s , thus f o r m i n g q u i t e an e f f i c i e n t mechanism (1).  relaxation  I n v e r y impure samples a t low  t h i s paramagnetic  r e l a x a t i o n could  temperatures,  be i m p o r t a n t .  Impuri-  t i e s a l s o d i s t u r b the l a t t i c e symmetry and t h i s r e s u l t s i n a l a r g e l o n g range e l e c t r i c f i e l d (22).  g r a d i e n t near t h e i m p u r i t y  I f t h e n u c l e u s has an e l e c t r i c quadrupole moment i t can  c o u p l e t o t h i s and hence t o t h e l a t t i c e through i m p u r i t y atom.  the v i b r a t i n g  T h i s mechanism i s p r o b a b l y n o t v e r y  strong  s i n c e d e l i b e r a t e i n t r o d u c t i o n o f i m p u r i t i e s i n t o aluminium l e f t T, u n a f f e c t e d by an i m p u r i t y c o n c e n t r a t i o n of 0.2%  (^8).  2.If S p i n - S p i n R e l a x a t i o n I t i s w e l l known t h a t a f t e r t h e a p p l i c a t i o n o f a r f p u l s e t h e t r a n s v e r s e m a g n e t i s a t i o n has a d e c r e a s i n g  ampli-  tude w h i c h i s t h e F o u r i e r t r a n s f o r m of t h e l i n e shape The  s p i n - s p i n r e l a x a t i o n time T  transverse magnetisation.  a  (1).  governs t h e decay o f t h e  Ta. can thus be r e l a t e d t o t h e  p r o p e r t i e s of the l i n e shape and i n p a r t i c u l a r t o t h e second moment.  The second moment i s one of t h e few p r o p e r t i e s of  the l i n e shape w h i c h can o f t e n be c a l c u l a t e d e x a c t l y . F o r t h i s reason, the f o l l o w i n g d i s c u s s i o n w i l l concern  line  w i d t h s and second moments, r a t h e r than Tj, d i r e c t l y . I n metals  t h e r e a r e t h r e e main c o n t r i b u t i o n s t o t h e  l i n e width, ( i ) The D i p o l a r L i n e W i d t h . T h i s i s due t o t h e n u c l e a r magnetic d i p o l e - d i p o l e interaction.  F o r t h i s i n t e r a c t i o n t h e second, and h i g h e r ,  moments c a n be c a l c u l a t e d f o r a g i v e n l a t t i c e s t r u c t u r e From these t h e l i n e shape c a n , i n p r i n c i p l e , be g o t . The  (1).  33 l i n e shape should be a p p r o x i m a t e l y g a u s s i a n w i t h a w i d t h o f a few gauss.  T h i s corresponds  to Ta^lOCyus.  ( i i ) Pseudo-Exchange ( R u d e r m a n - K i t t e l ) C o u p l i n g . I n t h i s i n t e r a c t i o n two a d j a c e n t n u c l e i c o u p l e by means of the c o n d u c t i o n e l e c t r o n s .  A conduction e l e c t r o n i s scat-  t e r e d from a n u c l e u s by the c o n t a c t term £\ <>,§6Cr) t o an e x c i t e d state.  The s p i n o r i e n t a t i o n of the e x c i t e d s t a t e depends on  t h a t o f the n u c l e u s .  I f the e l e c t r o n i s t h e n s c a t t e r e d o f f a  second n u c l e a r s p i n  , the n u c l e u s f e e l s the s p i n o f the  e x c i t e d s t a t e and hence senses the o r i e n t a t i o n o f the f i r s t nucleus.  An e l a b o r a t e second o r d e r p e r t u r b a t i o n t h e o r y c a l -  c u l a t i o n , u s i n g t h e n e a r l y f r e e e l e c t r o n model and a s p h e r i c a l Fermi s u r f a c e , g i v e s t h e pseudo-exchange H a m i l t o n i a n as (>+)  *Hex = The c o n s t a n t J,-j  Jij .I; . I j •  i s g i v e n by  Jij ( S y ) = ^ V T ^ f i i * < l Y ( 0 ) r > , f  rj; [2k,r 4  E  u  cos(2k n ) F  ;  -sin(2k r )] . p  y  The i m p o r t a n t f e a t u r e s i n t h i s e x p r e s s i o n a r e i t s o s c i l l a t o r y n a t u r e and i t s dependence on ri*<(|\|/(0)| '^ 4  E  . The l a t t e r a l s o  occur i n t h e e x p r e s s i o n f o r the i s o t r o p i c K n i g h t s h i f t i n a d i f f e r e n t c o m b i n a t i o n , ^|"\|/"(0)| ^ can thus be o b t a i n e d by a  comparing measurements o f these q u a n t i t i e s . The s i m p l i f i c a t i o n s i n the t h e o r i e s reduce the s i g n i f i c a n c e o f the v a l u e o b t a i n e d .  The e f f e c t o f t h e o s c i l l a t o r y  3^ p a r t has not usually  been worked out f o r o r d i n a r y m e t a l s and  ignored;  J  b e i n g assumed p r o p o r t i o n a l  i t is  to the  asymptotic  form r ~ . 3  The ture  e f f e c t of t h i s i n t e r a c t i o n depends upon the  of the m e t a l .  I f there i s only a s i n g l e isotope present,  the pseudo-exchange i n t e r a c t i o n narrows the l i n e . t h e r e are  several  t h a t t h e r e are  struc-  isotopes,  two  However, i f  or a q u a d r u p o l a r i n t e r a c t i o n ,  or more l i n e s , the  so  i n t e r a c t i o n broadens  the  lines. ( i i i ) Pseudo-Dipolar Broadening T h i s i s a s i m i l a r type of i n t e r a c t i o n to the  pseudo-  exchange i n t e r a c t i o n , except t h a t e x c i t a t i o n of the i s through the i n t e r a c t i o n of the n u c l e a r and de-excitation  electron  electronic  magnetic moments.  The  tact interaction.  A s i m i l a r c a l c u l a t i o n to t h a t of  pseudo-exchange c o n t r i b u t i o n H a m i l t o n i a n as  By  o c c u r s through the  g i v e s the  the  pseudo-dipolar  (50)  i s a complicated expression which involves  over the non-s e l e c t r o n s related  con-  an  a t the F e r m i s u r f a c e and  to the a n i s o t r o p i c  Knight s h i f t .  integration so i s  I t also contains  the  same o s c i l l a t o r y term t h a t J does. I f a s p h e r i c a l F e r m i s u r f a c e i s assumed, group t h e o r y shows t h a t *Hd must have the d i p o l a r f o r m (50). P  be  shown t h a t h i g h e r o r d e r c o n t r i b u t i o n s  t o the  I t can  also  interaction  do  35 not a l t e r t h i s form.  However, i f the s p h e r i c a l a s s u m p t i o n i s  n o t made t h e r e i s no group t h e o r e t i c a l p r o o f t h a t Vidimust be dipolar  (1). Because i t i s of the d i p o l a r form i t must always broaden  the l i n e .  B o t h of these i n t e r a c t i o n s i n c r e a s e r a p i d l y i n  s t r e n g t h w i t h i n c r e a s i n g atomic w e i g h t .  Below an atomic  w e i g h t of about 80 the i n t e r a c t i o n s are u n d e t e c t a b l e ,  but f o r  v e r y h i g h atomic numbers they can i n c r e a s e the second moment by over an o r d e r of magnitude. to below 10/us. present;  T h i s c o r r e s p o n d s to d e c r e a s i n g T  a  U s u a l l y b o t h i n t e r a c t i o n s are s i m u l t a n e o u s l y  t h e i r r e l a t i v e magnitudes depending upon the  t i o n s o f s and non-s e l e c t r o n s  frac-  present.  The o n l y o t h e r l i n e b r o a d e n i n g mechanisms of importance are T, b r o a d e n i n g and q u a d r u p o l a r e f f e c t s impure c r y s t a l .  i n a s t r a i n e d or  Because of the u n c e r t a i n t y p r i n c i p l e , T,  broadens the l i n e by an energy r v V T , . f i c a n t l y broadens the l i n e .  I f T, ^  this  Quadrupolar e f f e c t s  signi-  slightly  smear the r e s o n a n t frequency and t h i s appears as a b r o a d e n i n g of the 2.5  line.  The Quadrupolar I n t e r a c t i o n Many n u c l e i are n o n - s p h e r i c a l and so have an e l e c t r i c  quadrupole moment Q. electric field  The quadrupole moment i n t e r a c t s w i t h  the  gradients present i n a non-cubic l a t t i c e to  g i v e a s e r i e s of energy l e v e l s .  C o n s i d e r a m e t a l w i t h an  a x i a l l y symmetric l a t t i c e , so t h a t i t produces an a x i a l e l e c tric field  gradient V  z l  = ^ r .  I f a magnetic f i e l d  H  0  is  36 a p p l i e d a t an a n g l e 6 t o t h e c r y s t a l symmetry a x i s , t h e H a m i l t o n i a n f o r a s i n g l e n u c l e u s becomes (^O)  H.= - Tft H„ I < ,  where  z  •[31«fcos 0+ 3I  U* =  31^0+1(1^1^+ 1^1/)  1  V  sin29 - I ] . 1  When  Hz^HQ  t h e e n e r g i e s and wave f u n c t i o n s o f the H a m i l (51).  t o n i a n have t o be found by n u m e r i c a l computation  P e r t u r b a t i o n t h e o r y c a n be used o u t s i d e t h i s r e g i o n . field  case ( T V » H . ) w i l l n o t be c o n s i d e r e d  The low  here.  I f *Hi»Hi the a x i s of q u a n t i s a t i o n i s a l o n g t h e magnetic field  so a p e r t u r b a t i o n t h e o r y c a l c u l a t i o n g i v e s the r e s o n a n t  f r e q u e n c i e s as (1) V^ - U.+ i ^ ( m - i ) ( 3 c o s 9 -1)+ a  where t h e quadrupole V  L  frequency  H=  higher  terms,  ai^in" *  i s the Larmor f r e q u e n c y , w h i l e V* i s t h e f r e q u e n c y  t r a n s i t i o n from t h e m-1  of t h e  t o t h e mth energy l e v e l .  There a r e s e v e r a l f e a t u r e s t o note about t h e quadrupole interaction. ishes.  The most i m p o r t a n t i s t h a t u n l e s s I > 1 i t van-  The o t h e r f e a t u r e i s t h a t t h e 21+1 Zeeman energy  l e v e l s a r e no l o n g e r e q u a l l y spaced.  The r e s u l t of t h i s i s  t h a t the resonance l i n e i s s p l i t i n t o 21 s e p a r a t e l i n e s .  If  o n l y the f i r s t order term i s c o n s i d e r e d , the f r e q u e n c y o f t h e c e n t r a l m-1 = -£r*m=-£- t r a n s i t i o n remains t h e same as V » L  The  37 other 21-1  s a t e l l i t e l i n e s are s y m m e t r i c a l l y d i s p l a c e d from the  central line.  F u r t h e r m o r e , i f the magnetic f i e l d  u n t i l 0=cos~' ( 3 ) ,  i s rotated  the 2 1 l i n e s c o a l e s c e i n t o a s i n g l e l i n e .  h i g h e r order terms are c o n s i d e r e d , then the f r e q u e n c y c e n t r a l l i n e i s s h i f t e d and  of the  i t i s a l s o no l o n g e r p o s s i b l e f o r  the 2 1 l i n e s to e x a c t l y c o a l e s c e . The  electric field  e l e c t r o n s and  gradient  i o n s i n the m e t a l .  a r e s p h e r i c a l l y symmetric and V  2Z  i s d e r i v e d from a l l the The  closed e l e c t r o n s h e l l s  so do n o t d i r e c t l y c o n t r i b u t e t o  , even though they are the n e a r e s t charges to the  The p o t e n t i a l a t a n u c l e a r s i t e due  nucleus.  to a l l the o t h e r i o n s i n  the l a t t i c e can be c a l c u l a t e d w i t h c o n s i d e r a b l e  accuracy.  However, t h i s i s not the p o t e n t i a l g r a d i e n t a c t u a l l y f e l t the n u c l e u s .  The f i e l d from the o t h e r i o n s s l i g h t l y  the c l o s e d s h e l l s .  Because they are so c l o s e to the  t h e i r d i s t o r t i o n magnifies by a f a c t o r 1+T  W  f a c t o r and  (k-0).  distorts nucleus  the e f f e c t of the e x t e r n a l f i e l d %, i s the S t e r n h e i m e r a n t i s h i e l d i n g  i s usually at l e a s t ten.  be a c c u r a t e l y c a l c u l a t e d .  by  U n f o r t u n a t e l y , i t cannot  There i s a l s o a c o n t r i b u t i o n from  the n o n - s p h e r i c a l e l e c t r o n d i s t r i b u t i o n of V»  = e<j$*[(3cosV-l)r ](|) d x> . 3  s  T h i s i s r e l a t e d to the term q' i n the e x p r e s s i o n f o r the a n i s o t r o p i c Knight s h i f t .  They are not u s u a l l y i d e n t i c a l  s i n c e q' i s averaged over the F e r m i s u r f a c e e l e c t r o n s and i s averaged over a l l the c o n d u c t i o n  electrons.  I f the con-  d u c t i o n band has a complex s t r u c t u r e , i t i s not even  necessary  If  for  them t o have t h e same s i g n .  The meagre e v i d e n c e a v a i l a b l e  suggests t h a t most of the e l e c t r i c f i e l d the 2.6  g r a d i e n t i s due t o  c o n d u c t i o n e l e c t r o n s (^+9). The L i n e W i d t h W i t h A Quadrupole  Interaction  I t i s assumed t h a t t h e quadrupole i n t e r a c t i o n i s l a r g e enough t o c l e a r l y s e p a r a t e t h e 21 l i n e s .  If: j u s t the d i p o l a r  i n t e r a c t i o n i s c o n s i d e r e d , t h e r e i s l i t t l e change i n t h e l i n e w i d t h (1,  52).  The presence o f a pseudo-exchange  causes a c o n s i d e r a b l e change i n t h e l i n e w i d t h . the  interaction When any o f  l i n e s o v e r l a p t h i s i n t e r a c t i o n causes a n a r r o w i n g o f t h e  lines.  However, when they do n o t o v e r l a p many o f t h e mutual  s p i n f l i p s no l o n g e r conserve Zeeman energy and so a r e suppressed.  The e f f e c t o f t h i s i s t o a l l o w t h e i n t e r a c t i o n t o  broaden t h e l i n e (56). broadens t h e l i n e .  The p s e u d o - d i p o l a r i n t e r a c t i o n a l s o  The l i n e w i d t h i n m e t a l s w i t h an a t o m i c  w e i g h t above 80 and w i t h a quadrupole i n t e r a c t i o n i s thus always g r e a t e r t h a n t h e d i p o l a r w i d t h . 2.7  Pulsed.NMR W i t h a Quadrupole  Interaction  In t h e system t o be d i s c u s s e d  H x » Ha> >'H(. e  thus be 21 s e p a r a t e l i n e s i n t h e spectrum.  v  There w i l l  Because o f t h e  i n e q u a l i t y i n energy l e v e l s p a c i n g , the o n l y mutual s p i n f l i p s w h i c h a r e e n e r g e t i c a l l y a l l o w e d a r e those w h i c h do n o t change the  energy l e v e l p o p u l a t i o n s .  Thus i f t h e system i s ' p e r t u r b e d  i t i s no l o n g e r p o s s i b l e ,to e s t a b l i s h a s p i n temperature f o r the  whole system.  T h i s changes  some of the NMR  properties.  39 I n i t i a l l y assume t h a t t h e r e i s no c o u p l i n g between s e p a r a t e energy l e v e l s . frequency  Thus a p u l s e a p p l i e d a t the r e s o n a n t  of one l i n e does n o t a f f e c t t h e o t h e r l e v e l s so t h a t  they can be d i s r e g a r d e d . o n l y one f r e q u e n c y ,  T h i s assumes t h a t t h e p u l s e  contains  a situation unattainable i n practice.  c o n d i t i o n f o r a 90° p u l s e f o r the m^=i m+1  where T i s t h e r f p u l s e w i d t h .  The  (1)  l i n e becomes  S i n c e o n l y two l e v e l s a r e i n -  v o l v e d , t h e p r e c e s s i n g magnetic moment i s much s m a l l e r than when a s p i n temperature a l l o w s a l l 21+1 i n the t r a n s i t i o n s .  l e v e l s t o be i n v o l v e d  The system c a n be assumed t o have a f i c -  t i t i o u s s p i n o f £ ( D ? r a t h e r than i t s t r u e s p i n of I , so t h a t the f r a c t i o n a l r e d u c t i o n i n m a g n e t i s a t i o n  i s3AKI+D.  After  a p p l i c a t i o n of an r f p u l s e , t h e two l e v e l sub-system can be d e s c r i b e d by a s p i n temperature w h i c h r e l a x e s towards t h e l a t t i c e temperature. p o n e n t i a l decay.  The r e l a x a t i o n need n o t be a s i m p l e ex-  P u l s e d NMR i n a system o f t h i s type  thus  r e q u i r e s s h o r t e r p u l s e s , b u t g i v e s a weaker s i g n a l , than i n a normal Zeeman system.' I f t h e r e i s c o u p l i n g between t h e l i n e s , t h e s i t u a t i o n can be v e r y c o m p l i c a t e d .  However, t h e b a s i c f e a t u r e s of such  a system can be understood from s t u d y i n g t h e s i m p l e r case o f two d i f f e r e n t systems, each d e s c r i b a b l e by a s p i n t e m p e r a t u r e , coupled  together.  Energy c o n s e r v i n g  the s t r o n g e s t c o u p l i n g mechanism. i n the d i p o l a r Hamiltonian  s p i n f l i p s a r e by f a r  However, some o t h e r terms  g i v e a much weaker c o u p l i n g .  1+0 M u t u a l s p i n f l i p s -which do n o t conserve Zeeman energy can a l s o occur i f phonons, o r some o t h e r s o u r c e , can s u p p l y the energy difference.  T h i s i s u s u a l l y a weak c o u p l i n g mechanism because  of t h e s c a r c i t y of phonons w i t h the r e q u i r e d  energy.  L e t b o t h systems be p e r t u r b e d and then s e t up the r a t e e q u a t i o n s f o r t h e p o p u l a t i o n changes of a l l the l e v e l s .  When  combined w i t h t h e p r i n c i p l e of d e t a i l e d b a l a n c e , t h i s g i v e s the r a t e e q u a t i o n s f o r the s p i n temperatures as (^+0, &  (e; )  = -T,;'  (e; -  G; )  = -T;  c e; - e; H A  1  1  9, and  1  )-A e* (e; - a " ) ,  1  1  53)  1  1  Q:  C e;1 - e ; 1 ) .  a r e the s p i n t e m p e r a t u r e s , and T„  and T^  the  s p i n - l a t t i c e r e l a x a t i o n times i n systems 1 and 2 r e s p e c t i v e l y . I n b o t h e q u a t i o n s , the f i r s t term on the r i g h t i s t h e s p i n l a t t i c e r e l a x a t i o n towards the l a t t i c e temperature  B  0i  while  the l a s t term i n v o l v e s an energy t r a n s f e r ( c r o s s r e l a x a t i o n ) t o t h e o t h e r system a t a r a t e depending on t h e temperature d i f f e r e n c e between them.  The time c o n s t a n t g o v e r n i n g t h i s  c r o s s r e l a x a t i o n can be found by p u t t i n g T„ = T  xl  combining the two e q u a t i o n s t o g i v e  The"cross r e l a x a t i o n time T u  i s thus g i v e n by  T, = A ( e* + a  el).  = oo and  1+1 When T  h  , T^"^ T, , t h e system f i r s t c r o s s r e l a x e s t o a  a common s p i n temperature, l a t t i c e temperature.  which then r e l a x e s towards t h e  I t s behaviour  two independent e x p o n e n t i a l decays.  c a n thus be d e s c r i b e d by IfT » T ia  (1  , T  aa  the  systems decay a l m o s t i n d e p e n d e n t l y  toward the l a t t i c e tem-  p e r a t u r e a t r a t e s d e s c r i b e d by T  and T  n  the i n t e r m e d i a t e case where T ~ T (a  n  al  ,  respectively. For a simple d e s c r i p t i o n  of t h e system i s no l o n g e r p o s s i b l e ; i t s decay depending on the t h r e e time c o n s t a n t s and a l s o on t h e way i t i s p e r t u r b e d . Although  some d e t a i l s o f t h e energy l e v e l s and c o u p l i n g s  a r e d i f f e r e n t , a quadrupole s p l i t Zeeman system a l s o shows cross r e l a x a t i o n e f f e c t s .  An e x p e r i m e n t a l  i n v e s t i g a t i o n of  such a system showed -that a c r o s s r e l a x a t i o n t h e o r y based upon the i d e a o f mutual s p i n f l i p s when t h e l i n e s overlapped was o n l y m o d e r a t e l y s u c c e s s f u l i n d e s c r i b i n g t h e system When t h e l i n e s overlapped  (5^).  e x t e n s i v e l y c r o s s r e l a x a t i o n between  the l e v e l s o c c u r r e d i n a time l e s s than 60ms,, b u t when they were w i d e l y separated  t h e c r o s s r e l a x a t i o n time c o n s t a n t was  J+7 seconds, n o t much l e s s than T, , overlapped  I f the l i n e s only p a r t i a l l y  t h e c r o s s r e l a x a t i o n time was i n t e r m e d i a t e between  these v a l u e s , as expected  theoretically.  o f t e n an extended p e r i o d i n t h e middle when energy t r a n s f e r ceased.  However, t h e r e was  of the cross r e l a x a t i o n  A more s e r i o u s d i s c r e p a n c y i s  that i f a s a t e l l i t e l i n e i s perturbed  by a v e r y s h o r t r f p u l s e  the e q u i l i b r i u m d i s t r i b u t i o n a f t e r c r o s s r e l a x a t i o n i s n o t a Boltzmann d i s t r i b u t i o n .  Perturbing the c e n t r a l l i n e s gives a  \2 Boltzmann d i s t r i b u t i o n a f t e r c r o s s r e l a x a t i o n has o c c u r r e d , i n agreement w i t h t h e t h e o r y . Recent t h e o r e t i c a l and e x p e r i m e n t a l work (55)  suggests  t h a t the f a i l u r e of the r a t e e q u a t i o n approach t o c r o s s r e l a x a t i o n i s due t o n e g l e c t of t h e d i p o l e - d i p o l e system.  The n u c l e a r  magnetic system a c t u a l l y c o n s i s t s of sub-systems d e s c r i b e d by the Zeeman terms and by t h e d i p o l e - d i p o l e terms of the H a m i l tonian.  Each of these sub-systems has i t s own energy and s p i n  temperature and c a n be weakly coupled t o o t h e r sub-systems. C r o s s r e l a x a t i o n i n v o l v e s energy t r a n s f e r f r o m Zeeman t o d i p o l e - d i p o l e sub-systems, as w e l l as energy t r a n s f e r between Zeeman sub-systems.  I t i s believed that a c a r e f u l consider-  a t i o n o f t h e s e . e n e r g y exchanges can e x p l a i n t h e d i s c r e p a n c i e s i n the cross r e l a x a t i o n experiments.  CHAPTER I I I THE EXPERIMENTAL METHOD  'A Mighty Maze!  B u t Not W i t h o u t a P l a n .  1  - Pope; The p u l s e d NMR spectrometer designed  i s t o be d i s c u s s e d here i s  s p e c i f i c a l l y t o measure t h e a n i s o t r o p y o f T|  in  m e t a l l i c s i n g l e c r y s t a l s , b u t i t does have s u f f i c i e n t v e r s a t i l i t y t o measure T\  and T  a  i n m e t a l l i c powders, or non-  m e t a l l i c substances, w i t h only t r i v i a l m o d i f i c a t i o n s . In a standard p u l s e d NMR system, t h e n u c l e a r s p i n system, i n i t i a l l y a l i g n e d a l o n g H©, i s t i p p e d by a huge uniform r f pulse a p p l i e d a t r i g h t angles to H . 0  A t t h e end  of t h e r f p u l s e a l l the n u c l e i a r e a l i g n e d a t t h e same angle to H . 0  The r e c o v e r y o f t h e s p i n system i s t h e n s t u d i e d by  a m p l i f y i n g t h e s h o r t l i v e d f r e e i n d u c t i o n s i g n a l induced i n a c o i l wound round t h e sample.  Abragam (1)  gives a general (2)  d e s c r i p t i o n of the p r i n c i p l e s i n v o l v e d , w h i l e C l a r k l u c i d l y d e s c r i b e s t h e compromises and e x p e r i m e n t a l  details  i n v o l v e d i n t h e d e s i g n and c o n s t r u c t i o n of a p u l s e d NMR apparatus. The w r i t e r ' s a p p a r a t u s  i s based on C l a r k ' s a p p a r a t u s ,  b o t h i n p r i n c i p l e , and i n some e l e c t r o n i c c i r c u i t r y .  However,  the use of m e t a l l i c s i n g l e c r y s t a l s samples causes some d i f f e r e n c e s i n d e s i g n p h i l o s o p h y , and a l s o i n the c i r c u i t r y .  In a n o n - m e t a l l i c sample, o r a v e r y f i n e l y d i v i d e d m e t a l l i c powder, t h e r f f i e l d c o m p l e t e l y p e n e t r a t e s t h e sample, but i t can o n l y p e n e t r a t e a v e r y s h o r t d i s t a n c e i n t o a m e t a l l i c sample. The  T h i s i s because o f tfre well-known s k i n e f f e c t  s k i n e f f e c t has two main e x p e r i m e n t a l  (3).  consequences.  ( i ) A s i g n a l i s only obtained from n u c l e i w i t h i n a d i s t a n c e o f about t h e s k i n depth  & of the s u r f a c e .  These a r e  u s u a l l y o n l y about 1% o f t h e t o t a l number of n u c l e i i n t h e sample, so t h a t t h e s i g n a l s a r e much weaker than i n a normal NMR experiment.  They a r e so weak t h a t they a r e always ob-  scured by n o i s e when d i s p l a y e d on an o s c i l l o s c o p e , so t h a t a boxcar  i n t e g r a t o r , a d e v i c e f o r i m p r o v i n g t h e S/N r a t i o of  r e p e t i t i v e s i g n a l s , must always be used. ( i i ) The r f f i e l d 2H, v a r i e s r a p i d l y i n both s i z e and phase w i t h i n c r e a s i n g d i s t a n c e from t h e s u r f a c e of t h e sample. Thus a t t h e end o f t h e r f p u l s e , t h e n u c l e i a r e n o t a l l a l i g n e d at  t h e same a n g l e s t o H  0 ?  so t h a t c o n v e n t i o n a l p u l s e  trains  such as a ^ — T T p u l s e sequence cannot be used. Because of these f a c t s a r a t h e r l a b o r i o u s s p e c i a l method o f measuring T  (  3.1  had t o be developed.  G e n e r a l D e s c r i p t i o n of t h e A p p a r a t u s The main f e a t u r e s of t h e a p p a r a t u s (i) a b i l i t y  ares  t o operate a t any f r e q u e n c y  between about  5 and 10 Mc/s w i t h o u t e x t e n s i v e returning ( i i ) a r f magnetic f i e l d H, of 25 ( i i i ) a r e c o v e r y time o f Ijjus  gauss  ^5  ( i v ) phase s e n s i t i v e (v)  boxcar  detection  integration  ( v i ) a c o i l system d e s i g n e d s p e c i f i c a l l y f o r m e t a l samples. Two lator.  One  s i g n a l s are taken from the master C o l p i t t s  oscil-  i s used as a r e f e r e n c e phase and i s passed t h r o u g h  a phase s h i f t e r i n t o the a m p l i f i e r . i n t o the gated power a m p l i f i e r .  The o t h e r s i g n a l passes  T h i s i s gated by  positive  p u l s e s from a t i m i n g u n i t , and d e l i v e r s r f p u l s e s of about KV. peak t o peak t o the t r a n s m i t t e r c o i l . from the c o a x i a l p i c k u p c o i l i s a m p l i f i e d a m p l i f i e r w i t h a bandwidth of about 0 . 5  The induced s i g n a l i n a tuned p r e -  Mc/s  passed i n t o an Arenberg WA600D a m p l i f i e r .  1.8  and t h e n  I n the a m p l i f i e r  the s i g n a l and the much l a r g e r r e f e r e n c e s i g n a l , a r e l i n e a r l y added t o g i v e a phase s e n s i t i v e system.  T h i s improves the  S/N r a t i o s l i g h t l y , but i t s main advantage i s t h a t i t removes the n o n l i n e a r i t y  i n the Arenberg a m p l i f i e r w h i c h i s caused  by the s m a l l s i g n a l c h a r a c t e r i s t i c s of the r e c t i f i e r d i o d e s . The s i g n a l p l u s r e f e r e n c e i s then r e c t i f i e d and passed, a f t e r a m p l i f i c a t i o n , i n t o a boxcar i n t e g r a t o r .  The boxcar i n t e -  g r a t o r improves the S/N r a t i o by a f a c t o r r a n g i n g from about 10 t o 100.  The output from the boxcar i s t h e n r e c o r d e d on a  V a r i a n Model G-11A  c h a r t r e c o r d e r ( F i g . 3.1).  The t i m i n g u n i t can a l s o send s y n c h r o n i s e d p u l s e s t o the r e c o r d e r event marker, so t h a t t i m i n g p i p s a t i n t e r v a l s of lOOyUs t o one second can be r e c o r d e d on the c h a r t a l o n g  v/ith t h e s i g n a l s . The  t i m i n g u n i t can a l s o s u p p l y a quench p u l s e t o the  p r e a m p l i f i e r w h i c h h e l p s reduce t h e r e c o v e r y t i m e .  I t does  t h i s by l o w e r i n g t h e Q of t h e p i c k u p c o i l w h i l s t t h e r f p u l s e i s on, and then r a i s i n g i t soon a f t e r t h e p u l s e has ceased. The  t i m i n g u n i t a l s o p r o v i d e s a two p u l s e sequence of  a r b i t r a r y w i d t h s , s e p a r a t i o n and r e p e t i t i o n r a t e f o r g a t i n g the power a m p l i f i e r .  I t a l s o p r o v i d e s a boxcar  gating pulse,  and event marker p u l s e s w h i c h a r e s y n c h r o n i s e d w i t h t h e b a s i c r f p u l s e sequence.  As w e l l as t h i s i t c o n t a i n s a  sawtooth  g e n e r a t o r w h i c h can be used f o r l i n e a r l y sweeping t h e main magnetic f i e l d , f o r l i n e a r l y v a r y i n g t h e s e p a r a t i o n between two p u l s e s , o r l i n e a r l y v a r y i n g t h e s e p a r a t i o n between a r f p u l s e and t h e boxcar  gating pulse.  The dewer system was designed helium temperatures.  and b u i l t t o operate a t  However, no h e l i u m temperature  measure-  ments have been made and i t was s u b s e q u e n t l y n e c e s s a r y t o modify  t h e apparatus  i n such a way t h a t h e l i u m  a r e now u n a t t a i n a b l e .  temperatures  L i q u i d n i t r o g e n temperature  measure-  ments a r e s t i l l e a s i l y made. 3.2  The Timing System The b a s i c r e p e t i t i o n r a t e o f t h e p u l s e sequence i s  determined  by a f r e e r u n n i n g m u l t i v i b r a t o r which t r i g g e r s a  T e k t r o n i x 162 sawtooth be v a r i e d from (8ms.)'  generator. t o (9 s e c )  This r e p e t i t i o n r a t e can -1  .  The p u l s e f r o m t h e  ^7 Gate Out t e r m i n a l of t h e g e n e r a t o r  i s used t o t r i g g e r the  marker p u l s e , w h i l e t h e sawtooth output i s f e d i n t o two T e k t r o n i x 163 p u l s e g e n e r a t o r s .  One of t h e p u l s e  generators  i s s e t t o t r i g g e r a t t h e b e g i n n i n g of t h e c y c l e , w h i l e t h e second one i s s e t t o t r i g g e r a t some l a t e r time i n t h e c y c l e . Thus t h e two p u l s e s can be separated up t o n i n e seconds. independently  by any d e s i r e d  interval  The w i d t h of each of these p u l s e s can be  v a r i e d from ^ius up t o many m i l l i s e c o n d s .  The two  p u l s e s a r e then f e d i n t o a p u l s e mixer w h i c h a l s o a m p l i f i e s them t o the 90 v o l t s r e q u i r e d t o gate t h e power a m p l i f i e r ( F i g . 3.2). The r i s e t i m e and decay time of t h e g a t i n g p u l s e s a r e l e s s t h a n O.^us.  Observation  on an o s c i l l o s c o p e shows t h a t  t h i s causes h a l f a c y c l e j i t t e r i n the r f p u l s e l e n g t h .  This  j i t t e r causes s l i g h t v a r i a t i o n s i n the a n g l e through w h i c h t h e n u c l e i a r e t i p p e d and t h i s shows up as e x t r a n o i s e .  However,  the S/N r a t i o w i t h m e t a l samples i s so poor t h a t t h e e x t r a n o i s e due t o the p u l s e j i t t e r i s u s u a l l y u n o b s e r v a b l e . due  Jitter  t o v a r i a t i o n s i n t h e t r i g g e r i n g time o f t h e p u l s e genera-  tors i s usually also  unobservable.  A p u l s e i s a l s o taken from t h e P u l s e Out t e r m i n a l of the second T e k t r o n i x 163 p u l s e g e n e r a t o r and used f o r g a t i n g the b o x c a r .  T h i s i s done by t r i g g e r i n g a T e k t r o n i x 162 saw-  t o o t h g e n e r a t o r whose output i s f e d i n t o a T e k t r o n i x 161 pulse generator.  T h i s can t r i g g e r on any p a r t o f t h e saw-  t o o t h , so t h a t i t s output p u l s e s ( f i f t y v o l t s p o s i t i v e and  n e g a t i v e ) , w h i c h gate t h e b o x c a r ,  can occur any time a f t e r t h e  b e g i n n i n g of t h e second r f p u l s e ( F i g . 3 . 6 ) . T h i s completes t h e d e s c r i p t i o n of t h e b a s i c p a r t o f t h e timing unit.  There a r e , however, s e v e r a l a u x i l i a r y p a r t s o f  the t i m i n g u n i t , some o f which a r e n o t o f t e n used. I n t h e T e k t r o n i x 161 and 163 p u l s e g e n e r a t o r s , a v o l t a g e comparator stage compares t h e i n s t a n t a n e o u s v o l t a g e of t h e ^nput sawtooth w i t h a v o l t a g e s e t by a p o t e n t i o m e t e r . d e c r e a s i n g sawtooth v o l t a g e e q u a l s t h e p r e s e t pulse generator  i s triggered.  When t h e  v o l t a g e , the  I f the p r e s e t v o l t a g e i s now  v a r i e d s l o w l y and l i n e a r l y , t h e time d e l a y between t h e s t a r t of the sawtooth and t h e g e n e r a t o r t r i g g e r i n g w i l l a l s o v a r y l i n e a r l y w i t h time.  The p r e s e t v o l t a g e i s v a r i e d i n t h i s  f a s h i o n by d i s c o n n e c t i n g t h e comparator g r i d f r o m t h e p o t e n t i o meter and i n s t e a d f e e d i n g a s l o w l y v a r y i n g n e g a t i v e v o l t a g e onto i t ( i f ) .  sawtooth  A T e k t r o n i x 161 and a 163 p u l s e  generator  were m o d i f i e d i n t h i s way; a two-way s w i t c h b e i n g used so t h a t e i t h e r t h e i n t e r n a l p r e s e t v o l t a g e or t h e e x t e r n a l sawtooth v o l t a g e can be f e d onto t h e comparator g r i d . I f t h e e x t e r n a l sawtooth i s a p p l i e d t o t h e m o d i f i e d 163 pulse generator  g a t i n g t h e power a m p l i f i e r , a two p u l s e s e -  quence i s o b t a i n e d w i t h l i n e a r l y i n c r e a s i n g s e p a r a t i o n between the p u l s e s .  T h i s sequence enables  t h e r e c o v e r y o f t h e mag-  n e t i s a t i o n a f t e r a p u l s e t o be d i r e c t l y r e c o r d e d from which T, c a n be q u i c k l y  obtained.  on a c h a r t ,  **9 A p p l y i n g t h e e x t e r n a l sawtooth t o t h e m o d i f i e d 161 p u l s e g e n e r a t o r g a t i n g t h e b o x c a r , sweeps t h e boxcar gate over t h e whole i n d u c t i o n t a i l f o l l o w i n g a r f p u l s e . of t h e boxcar o u t p u t T  a  From t h e r e c o r d i n g  can be f o u n d .  The comparison sawtooth i s got from a p h a n t a s t r o n which s t a r t s a sweep when manually t r i g g e r e d .  N o r m a l l y t h e sawtooth  d e c r e a s e s from 1*4-0 v o l t s t o 20 v o l t s , b u t a b i a s v o l t a g e can be a p p l i e d so t h a t t h e output v o l t a g e remains c o n s t a n t a t any v o l t a g e between l*+0 v o l t s and 100 v o l t s , u n t i l t h e d e c r e a s i n g sawtooth r e a c h e s t h i s v o l t a g e . the  sawtooth v o l t a g e .  The output v o l t a g e t h e n f o l l o w s  T h i s f e a t u r e a l l o w s f o r the f i n i t e  w i d t h o f t h e r f p u l s e s , a n e c e s s a r y f e a t u r e when sweeping w i t h some p u l s e sequences. ten  The sawtooth d u r a t i o n can be v a r i e d from  seconds t o t h i r t y minutes i n seven s t e p s .  The sawtooth  l i n e a r i t y d e v i a t i o n i s l e s s than 1% over Q0% of t h e sweep, b u t t h e n i n c r e a s e s r a p i d l y t o about 10% a t t h e end o f t h e sweep. A s e r i e s of measurements showed t h a t t h e sawtooth l e n g t h was r e p r o d u c i b l e t o w i t h i n 2% of i t s l e n g t h .  N e i t h e r of these  imperfections a f f e c t s the r e s u l t s since e i t h e r a l i n e a r i s not necessary, or e l s e c a l i b r a t e d  sweep  t i m i n g p i p s a r e used.  The sawtooth g e n e r a t o r i s u s u a l l y used f o r sweeping t h e magnetic f i e l d  through t h e r e s o n a n t v a l u e .  To do t h i s , an  a t t e n u a t e d sawtooth v o l t a g e i s t a k e n from t h e sawtooth g e n e r a t o r and f e d i n t o the magnet power s u p p l y . To measure T, or T  a  i t i s n e c e s s a r y t o measure time i n -  t e r v a l s such as t h o s e between r f p u l s e s , or between one r f  50 p u l s e and t h e boxcar better.  g a t i n g p u l s e , w i t h an a c c u r a c y o f 2% o r  Two a l t e r n a t i v e t i m i n g methods a r e a v a i l a b l e . I f t h e slow sawtooth  163  i s b e i n g a p p l i e d t o t h e 161, or t h e  g e n e r a t o r , a p u l s e i s t a k e n from t h e Gate Out t e r m i n a l of  the r e l e v a n t 162 sawtooth  g e n e r a t o r t r i g g e r i n g i t and f e d i n t o  the Gate I n t e r m i n a l of t h e 162 sawtooth marker p u l s e g e n e r a t o r .  g e n e r a t o r used as a  T h i s i s s e t t o r u n a t some c o n v e n i e n t  r e p e t i t i o n r a t e , such as l K c / s .  When t h e p u l s e i s a p p l i e d t o  the Gate I n t e r m i n a l , t h e g e n e r a t o r g i v e s out p u l s e s a t , s a y , one m i l l i s e c o n d i n t e r v a l s u n t i l t h e g a t i n g p u l s e c e a s e s . t r a i n of p u l s e s , which i s synchronised c then f e d i n t o a c o i n c i d e n c e u n i t .  This  to the r f pulses, i s  A p u l s e f r o m t h e boxcar  g a t e , o r t h e second r f gate p u l s e , i s a l s o f e d i n t o t h e c o i n c i d e n c e u n i t and when these two p u l s e s c o i n c i d e , a c u r r e n t p u l s e a c t u a t e s t h e event marker pen on t h e r e c o r d e r .  Thus  the boxcar output and a s e r i e s of t i m i n g p i p s a r e recorded on the same c h a r t . A l t e r n a t i v e l y a double beam o s c i l l o s c o p e c a n be used w i t h t h e t r a i n o f marker p i p s d i s p l a y e d on one c h a n n e l and t h e r f p u l s e sequence d i s p l a y e d on t h e o t h e r c h a n n e l . a t i o n between r f p u l s e s can then be manually  The separ-  adjusted to  c o i n c i d e w i t h the desired timing p i p . The p u l s e g e n e r a t o r was c a l i b r a t e d w i t h a C.M.C. 707BN frequency 1,5%  counter.  I t was found  t o be a c c u r a t e t o w i t h i n  on a l l ranges and r e p e t i t i o n r a t e s a f t e r a warm-up p e r i o d  of s e v e r a l hours.  The c o i n c i d e n c e u n i t r e q u i r e s t h e p u l s e s  51 to  o v e r l a p 0.6yus b e f o r e i t t r i g g e r s .  However, t h e minimum  s e p a r a t i o n between marker p u l s e s i s lOO^is, so t h a t t h i s i s a n e g l i g i b l e systematic error.  I f a double beam o s c i l l o s c o p e i s  used t h e t i m i n g e r r o r due t o a l i g n i n g t h e two p u l s e s v i s u a l l y is  s t i l l l e s s than 1%.  T h i s i s s i n c e t h e s c r e e n i s 10cm.  and t h e f u l l w i d t h o f t h e s c r e e n i s always used. are  l e s s than 1mm.  wide  As t h e l i n e s  wide, i t i s easy t o make them c o i n c i d e t o  w i t h i n l e s s than 1% of t h e s c r e e n w i d t h .  I t thus i s s a f e t o  assume an e r r o r l e s s t h a n 2% i n a l l t i m i n g measurements. S/N r a t i o i s n e a r l y always l e s s than f i f t y , i n t h e t i m i n g measurements i s s u f f i c i e n t l y  The  so t h a t t h e e r r o r small.  The p u l s e mixer a l s o p r o v i d e s a two v o l t p o s i t i v e p u l s e w h i c h i s used f o r t r i g g e r i n g t h e o s c i l l o s c o p e and a minus twenty v o l t quenching p u l s e of v a r i a b l e w i d t h f o r the p r e a m p l i f i e r quenching 3.3  circuit.  The Gated Power A m p l i f i e r The C o l p i t t s o s c i l l a t o r can be tuned from about 5 t o  11 Mc/s. and has a l o n g term f r e q u e n c y d r i f t o f one p a r t i n 5xl0  4  p e r hour.  T h i s i s adequate s t a b i l i t y f o r most measure-  ments on broad m e t a l l i n e s . The output from t h e o s c i l l a t o r passes through a cathode f o l l o w e r t o a g a t i n g c i r c u i t designed by Blume (5) g i b l e r f l e a k a g e when t h e gate i s o f f .  for negli-  I t i s very s a t i s f a c t o r y  i n t h i s r e s p e c t , but when t h e gate i s s w i t c h e d on by a n i n e t y v o l t p o s i t i v e p u l s e from the p u l s e m i x e r , i t l o a d s t h e cathode  52 f o l l o w e r s u f f i c i e n t l y t o d e c r e a s e i t s output by about 20$.  In  Blume's c i r c u i t , t h e phase r e f e r e n c e s i g n a l i s a l s o taken from the same cathode f o l l o w e r .  I n t h i s c a s e , the drop i n output  u p s e t s t h e phase r e f e r e n c e s i g n a l f o r about 50yws a f t e r t h e gate is  switched  off.  To cure t h i s t r o u b l e , a s e p a r a t e  f o l l o w e r was added f o r the phase r e f e r e n c e s i g n a l  cathode channel.  A f t e r the g r a t i n g c i r c u i t there are three c l a s s C amplif i e r stages.  These g i v e h i g h power a m p l i f i c a t i o n , good c a r r i e r  s u p p r e s s i o n , and s h o r t r i s e and f a l l times w i t h r e a s o n a b l y Q c i r c u i t s i n t h e f i r s t two s t a g e s . w h i c h i s operated w i t h 1500  high  The f i n a l stage i s an 829B  v o l t s on t h e p l a t e and a s c r e e n  v o l t a g e w h i c h can be v a r i e d from ^50  t o 600 v o l t s .  o p e r a t i n g c o n d i t i o n s , t h e maximum power output As i n C l a r k ' s t r a n s m i t t e r (2),  Under these  i s about 2KW.  v a r i a b l e damping o f t h e t r a n s -  m i t t e r c o i l i s p r o v i d e d by b i a s e d d i o d e s p l a c e d a c r o s s i t . W i t h l a r g e a p p l i e d r f v o l t a g e s one of t h e d i o d e s i s always open c i r c u i t e d so t h a t n e g l i g i b l e damping o c c u r s .  However,  when t h e r f p u l s e decays t o about one v o l t , b o t h d i o d e s  con-  duct and shunt t h e c o i l w i t h an impedance o f about 600A . R i s e times and f a l l times a r e t y p i c a l l y about lyMs, w h i l e p u l s e s up t o 600/us l o n g can be generated the r f output v o l t a g e becomes e x c e s s i v e . p r o p e r t i e s of ceramic  b e f o r e sagging i n The p i e z o e l e c t r i c  condensers d i d n o t cause r i n g i n g i n t h e  output c i r c u i t , p r o v i d e d they were used w e l l below t h e i r maximum r a t e d v o l t a g e . Clark  (2).  T h i s i s c o n t r a r y t o t h e e x p e r i e n c e of  53 3.^  The P r e a m p l i f i e r T h i s i s a tuned v o l t a g e a m p l i f i e r s l i g h t l y m o d i f i e d f r o m  one designed  by C l a r k ( 2 ) .  w i d t h o f about 0„5Mc/s.  I t has a g a i n o f 16 and a band-  The main d i f f e r e n c e i s t h a t c r o s s e d  s i l i c o n d i o d e s l i m i t t h e g r i d swing o f t h e i n p u t stage t o -1 v o l t , even f o r a p p l i e d v o l t a g e s of s e v e r a l hundred v o l t s .  They  a l s o heavy damped t h e p i c k u p c o i l f o r l a r g e a p p l i e d r f v o l t a g e s , but have n e g l i g i b l e e f f e c t when o n l y t h e v e r y s m a l l s i g n a l voltage i s present.  I n s t e a d o f c r o s s e d d i o d e s , C l a r k used a  quenching c i r c u i t i n w h i c h a quenching p u l s e d e r i v e d from the p u l s e mixer and a m p l i f i e r c i r c u i t switched a low impedance l o u d a c r o s s t h e p i c k u p c o i l d u r i n g the r f p u l s e , and f o r a v a r i a b l e time a f t e r w a r d s . However, i n t h e p r e s e n t apparatus  the input r f pulses  a r e so l a r g e t h a t c r o s s e d s i l i c o n d i o d e s were i n i t i a l l y added t o p r o t e c t t h e quenching c i r c u i t .  I t was then found  t h a t when the  quenching c i r c u i t was switched o f f , a s m a l l t r a n s i e n t o c c u r r e d w h i c h swamped the v e r y weak induced  signal.  This t r a n s i e n t  was due t o t h e s t o r a g e c a p a c i t a n c e of t h e s w i t c h i n g diode i n the quenching c i r c u i t  and c o u l d n o t be e l i m i n a t e d .  Thus t h e  quenching c i r c u i t had t o be d i s c o n n e c t e d f o r a l l measurements on s i n g l e c r y s t a l s and the c r o s s e d d i o d e s a l o n e used f o r damping t h e c o i l .  I f powdered samples a r e used the much  l a r g e r s i g n a l a v a i l a b l e completely o b l i t e r a t e s the t r a n s i e n t so t h a t the quenching c i r c u i t can be used.  9* The e q u i v a l e n t s e r i e s n o i s e r e s i s t a n c e of t h e p r e a m p l i f i e r i s about 25011  so t h a t f o r l i q u i d n i t r o g e n tempera-  t u r e s and above, t h e t h e r m a l n o i s e f r o m the r e s o n a n t p i c k u p c o i l i s the dominant n o i s e s o u r c e .  The p r e a m p l i f i e r a m p l i f i -  c a t i o n i s such t h a t i t s n o i s e output i s much l a r g e r than t h e t h e r m a l n o i s e generated i n t h e i n p u t stage o f t h e Arenberg amplifier. 3.5  The Main A m p l i f i e r T h i s i s an Arenberg WA600D w h i c h has been m o d i f i e d t o  improve i t s r e c o v e r y t i m e .  O r i g i n a l l y the a m p l i f i e r had a  f r e q u e n c y response from 2 t o 65 Mc/s.  A u x i l i a r y tuning c o i l s  enabled t h e f r e q u e n c y response t o be a l t e r e d t o a passband about lOMc/s wide c e n t r e d \ on any f r e q u e n c y w i t h i n t h i s range. However, t h e r e was an annoying o v e r s h o o t p r e s e n t f o r about 20yus a f t e r t h e r f p u l s e .  To e l i m i n a t e t h i s , t h e low f r e q u e n c y  response was i n c r e a s e d t o 3.5Mc/s and t h e s c r e e n bypass densers reduced  i n value.  con-  L a t e r on t h e i n p u t s t a g e was r e b u i l t  a l o n g l i n e s suggested by t h e m a n u f a c t u r e r .  These improvements  reduced t h e o v e r s h o o t t o l e s s than ^ s i n d u r a t i o n . The low f r e q u e n c y response of t h e v i d e o s e c t i o n was decreased from 20 t o 2c/s t o a v o i d n o t i c e a b l e b a s e l i n e droop. T h i s s e c t i o n a l s o i n t r o d u c e d c o n s i d e r a b l e 60c/s p i c k u p from the f i l a m e n t s i n t o t h e o u t p u t , so t h a t t h e f i l a m e n t s were c o n v e r t e d t o r u n on r e g u l a t e d D.G. c u r r e n t . The r e c o v e r y time o f t h e whole system i s n e a r l y l O ^ s from t h e end o f t h e t r a n s m i t t e r g a t i n g p u l s e .  This i s  55 p r o b a b l y about t w i c e as l o n g as the minimum p r a c t i c a l  limit,  but s i g n i f i c a n t r e d u c t i o n i n the r e c o v e r y time would r e q u i r e a n - e x o r b i t a n t amount of time and  labour.  A l t h o u g h n e i t h e r the p r e a m p l i f i e r nor the main a m p l i f i e r has an automatic experienced  gain c o n t r o l , l i t t l e trouble i s  w i t h l o n g term d r i f t s i n g a i n .  l e s s than 1% per hour, p r o v i d e d  T h i s i s always  the a m p l i f i e r s have been  a l l o w e d to warm up f o r s e v e r a l hours.  However, t h e r e are  s h o r t term f l u c t u a t i o n s i n g a i n of about 10$, w i t h a p e r i o d of about t e n minutes' d u r a t i o n w h i c h cause some t r o u b l e . G.A.  DeWit has n o t i c e d s i m i l a r f l u c t u a t i o n s i n g a i n i n another  A r e n b e r g used i n t h i s l a b o r a t o r y ( p r i v a t e communication t o the w r i t e r ) . The l i n e a r i t y of the A r e n b e r g i s poor; the l i n e a r r e g i o n e x t e n d i n g from about 2 t o 12 v o l t s a t the output.  To  improve the l i n e a r i t y f o r s m a l l s i g n a l s the r e f e r e n c e v o l t a g e f r o m the o s c i l l a t o r i s added to the s i g n a l a t the s i x t h of the a m p l i f i e r through a h i g h pass f i l t e r and network.  A phase s h i f t e r and  s i g n a l and  a resistive  a t t e n u t o r (2) a r e i n s e r t e d i n  the r e f e r e n c e channel between the o s c i l l a t o r and The a m p l i t u d e  stage  the a m p l i f i e r .  of the r e f e r e n c e v o l t a g e i s much l a r g e r than  the  i s a d j u s t e d so t h a t i t i s near the c e n t r e of the  linear region.  I t thus performs the double f u n c t i o n of making  the a m p l i f i e r l i n e a r f o r s m a l l s i g n a l s and p r o v i d i n g phase sensitive detection.  T h i s s i m p l e method of phase s e n s i t i v e  d e t e c t i o n improves the S/N  by a f a c t o r of {2  ( 1 ) , but  can  56 introduce considerable d i s t o r t i o n i f the reference s i g n a l i s not much l a r g e r than t h e s i g n a l (Appendix I ) .  I n these e x p e r i -  ments, the r e f e r e n c e s i g n a l i s a t l e a s t t e n times t h e s i g n a l , so t h a t t h e d i s t o r t i o n i s always l e s s than % .  This i s t o l e r -  a b l e as i t i s l e s s than t h e e r r o r caused by n o i s e . 3.6  The Boxcar I n t e g r a t o r T h i s i s an e l e c t r o n i c d e v i c e f o r i m p r o v i n g  r a t i o of a r e p e t i t i v e s i g n a l .  t h e S/N  I t b a s i c a l l y c o n s i s t s of an  e l e c t r o n i c s w i t c h w h i c h i s switched  on by a g a t i n g p u l s e oc-  c u r r i n g a t a s e t time a f t e r an a p p l i e d r f p u l s e , f o l l o w e d by a RC c i r c u i t w h i c h averages the v o l t a g e s obtained c e s s i v e sampling  during  suc-  intervals.  The boxcar was b u i l t t o t h e d e s i g n o f Blume (h).  The  o n l y major m o d i f i c a t i o n i s t h e a d d i t i o n of a v o l t a g e a m p l i f i e r stage a t t h e i n p u t . The  boxcar c i r c u i t i s l i n e a r w i t h i n 2% f o r v o l t a g e s up  to ±20 v o l t s , b u t t h e v o l t a g e a m p l i f i e r a t t h e i n p u t i s o n l y l i n e a r f o r i n p u t v o l t a g e s of about -8 v o l t s .  This  voltage  swing i s adequate f o r t h e p r e s e n t e x p e r i m e n t s and can e a s i l y be i n c r e a s e d i f n e c e s s a r y . The measured l o n g term d r i f t i n t h e base l i n e c o r r e s ponds t o a d r i f t of 0.05 v o l t s per hour a t t h e i n p u t .  T h i s slow  d r i f t i s h a r d l y n o t i c e a b l e , even i n e x p e r i m e n t s t a k i n g many hours. I n t h e i d e a l boxcar c i r c u i t  ( F i g . 3.3)  the switch S i s  c l o s e d f o r a s h o r t time % a t t h e r e p e t i t i o n r a t e T ' of t h e  57  0-  •AAAAAMA  5  F i g . 3.3  •0  E q u i v a l e n t C i r c u i t of the Boxcar Integrator  r f pulses. as an  The  time c o n s t a n t E C » X , so t h a t the c i r c u i t a c t s S  integrator. I f the r.m.s. i n p u t n o i s e v o l t a g e i s v; and  noise voltage i s E  then  0  the output  (6) J.  where f source  c  i s the n o i s e c o r r e l a t i o n t i m e .  v,oCll> and t oCEf' <  t  so t h a t  For a t h e r m a l  noise  i s independent of the  bandwidth of any a m p l i f i e r ahead of the boxcar i n t e g r a t o r . T h i s e q u a t i o n has been d e r i v e d under the assumption t h a t 'ttj a c o n d i t i o n w h i c h i s e a s i l y s a t i s f i e d  experimentally.  I f v j i s the i n p u t s i g n a l , t h e n the boxcar response i s the same as a low pass RC f i l t e r w i t h a time t'= RCT/T:  t  (6).  care i s taken i n the c h o i c e of <t!  Unless  c o n s i d e r a b l e d i s t o r t i o n of the s i g n a l can An e x p e r i m e n t a l  constant  occur.  t e s t of these e q u a t i o n s was  u s i n g an a m p l i f i e r as a t h e r m a l n o i s e source w i t h a spectrum from about 2c/s  to s e v e r a l megacycles.  made by frequency  T h i s was  fed  58 i n t o the boxcar i n t e g r a t o r and t h e o u t p u t measured on a c h a r t r e c o r d e r , w h i l e v a r i o u s parameters were s y s t e m a t i c a l l y v a r i e d . The measurements were crude, b u t v e r i f i e d  that the noise  output  i s independent of t h e bandwidth and t h a t t h e S/NoCRC, e x c e p t f o r v e r y l a r g e v a l u e s of RC when t h e S/N became l e s s than p r e d i c t e d . The e q u a t i o n s  f o r t h e boxcar v o l t a g e s i g n o r e the e f f e c t of the  r e c o r d e r time c o n s t a n t , which i s about one second f o r t h e Varian recorder.  T h i s time c o n s t a n t performs a f u r t h e r i n t e -  g r a t i o n o f t h e n o i s e f o r s m a l l v a l u e s of T, so t h a t t h e S/N r a t i o i s increased.  Experimentally  but d e c r e a s e s s t e a d i l y f o r T<~^70ms., b e i n g a f a c t o r of s i x s m a l l e r f o r T=lms. 3.7  Power S u p p l i e s and N o i s e The 1500  Suppression  v o l t s f o r the power a m p l i f i e r i s s u p p l i e d by a  simple unregulated  power s u p p l y , w h i c h a l s o g i v e s ^50,  525  or  600 v o l t s f o r the s c r e e n r e g u l a t e d by v o l t a g e r e g u l a t i n g t u b e s . The l a r g e output impedance of these use of l a r g e s t o r a g e condensers. the 350  s u p p l i e s i s overcome by the  A 200ma. r e g u l a t e d supply  gives  v o l t s used i n an e a r l i e r stage of t h e power a m p l i f i e r . A p a r t from t h e Arenberg w h i c h has i t s own r e g u l a t e d  s u p p l y , t h e r e s t of the a p p a r a t u s i s powered by two T e k t r o n i x 160A r e g u l a t e d power s u p p l i e s . The 6.3  v o l t D.C. f i l a m e n t c u r r e n t comes from a s i m p l e  t r a n s i s t e r i s e d r e g u l a t e d power s u p p l y battery eliminator.  (7)  f e d by a H e a t h k i t  59 V a r i o u s c o m b i n a t i o n s of b a t t e r y e l i m i n a t o r and/or 6 v o l t a c c u m u l a t o r had been t r i e d e a r l i e r on but gave much p o o r e r r e g u l a t i o n than t h e t r a n s i s t e r i s e d The 110  supply.  v o l t s A.C. f o r the whole a p p a r a t u s i s s t a b i l i s e d  by a G e n e r a l Radio Type 1570/A a u t o m a t i c l i n e v o l t a g e r e g u l a t o r . R f l e a k a g e from the o s c i l l a t o r t o t h e a m p l i f i e r i s o f t e n a major source o f t r o u b l e i n a c o h e r e n t NMR system. t h i s i s e a s i l y e l i m i n a t e d by a c a r e f u l f i l t e r i n g l e a d s connected  However  of a l l power  to e i t h e r the a m p l i f i e r s or the o s c i l l a t o r ,  and by c o m p l e t e l y e n c l o s i n g the o s c i l l a t o r i n a copper can. Care was a l s o t a k e n w i t h t h e I n t e r s t a g e f i l t e r i n g plifiers  i n t h e am-  t o e l i m i n a t e any chance o f r e g e n e r a t i o n o c c u r r i n g .  In a l l of t h e r f f i l t e r i n g e x t e n s i v e use i s made of P h i l l i p s f e r r i t e beads a l o n g w i t h O.Olyuf ceramic bypass  condensers.  I n a p e r f e c t a p p a r a t u s t h e o n l y n o i s e source i s t h e t h e r m a l n o i s e of t h e r e s o n a n t p i c k u p c o i l .  T h i s was some-  times so w i t h t h e p r e s e n t a p p a r a t u s , but o f t e n n o i s e from e x t e r n a l s o u r c e s was much l a r g e r than t h e t h e r m a l n o i s e .  The  most common e x t e r n a l n o i s e source i s f a u l t y f l u o r e s c e n t l i g h t s i n t h e l a b o r a t o r y , f o l l o w e d by n o i s e from heavy e l e c t r i c a l machinery  i n other p a r t s of the b u i l d i n g .  Both of these  s o u r c e s g i v e r f p u l s e s s y n c h r o n i s e d t o t h e mains f r e q u e n c y . I t i s n o t c e r t a i n whether these p u l s e s t r a v e l a l o n g t h e power mains, o r a r e d e t e c t e d by t h e tuned p i c k u p c o i l wrapped around  t h e sample.  60 Because of the s m a l l magnet gap not enough space i s a v a i l a b l e to e n c l o s e the c o i l s i n an earthed metal  s h i e l d , so  the a p p a r a t u s i s s u s c e p t i b l e to e x t r a n e o u s induced  v o l t a g e s . An  attempt was  made to p l a c e an earthed  s h i e l d around the o u t s i d e  of the n i t r o g e n dewar, but the r e s u l t i n g hum  loop increased  the  n o i s e l e v e l , so the p i c k u p c o i l has been l e f t u n s h i e l d e d . The whole e a r t h i n g system of the a p p a r a t u s had  to be arranged so  t h a t t h e r e were no e a r t h i n g l o o p s i n c r e a s i n g the n o i s e T h i s was 3.8  m a i n l y an e m p i r i c a l p r o c e s s  level.  t h a t verged on b l a c k magic.  The Magnets and Magnetic F i e l d Measurements The a p p a r a t u s was  o r i g i n a l l y b u i l t f o r use w i t h a  V a r i a n l/hO07 6" e l e c t r o m a g n e t a field  of up to 7KG.  w i t h a two-inch  This gives  which i s homogeneous to w i t h i n a gauss  over the volume of the sample. ments, the a p p a r a t u s was  P a r t way  s t r e n g t h of up to 11.2KG.  the e x p e r i -  w i t h a 2.25-inch gap and a f i e l d T h i s magnet has a homogeneity of  about 0.1 gauss over the sample volume. over  through  s h i f t e d and used w i t h a V a r i a n  V^012/313 12" electromagnet  r o t a t a b l e through  gap.  Both magnets were  l80°.  The magnetic f i e l d s of both magnets were o f t e n swept through  the resonance v a l u e s by a p p l y i n g a sawtooth v o l t a g e of  8 v o l t s to the E x t e r n a l Sweep t e r m i n a l s of t h e i r r e s p e c t i v e magnet power s u p p l i e s .  T h i s v a r i e s the f i e l d by about 100  gauss as an a p p r o x i m a t e l y  l i n e a r f u n c t i o n of t i m e .  A simple m a r g i n a l o s c i l l a t o r was f i e l d measurements needed.  I t was  b u i l t f o r any  magnetic  used t o c a l i b r a t e the 6"  61 e l e c t r o m a g n e t , b u t w i t h the 12" e l e c t r o m a g n e t a f i e l d t i o n p r e v i o u s l y done by S.N. Sharma had s u f f i c i e n t  calibra-  accuracy f o r  f i n d i n g r e s o n a n c e s , so t h a t the m a r g i n a l o s c i l l a t o r  was never  used w i t h i t . 3.9  The Low Temperature System O r i g i n a l l y i t was i n t e n d e d t o make measurements a t v e r y  low temperatures so a c o n v e n t i o n a l g l a s s dewar l i q u i d system was b u i l t . the  helium  The dewars were b u i l t f o r t h e s i x magnet so  i n n e r dewar has an i n t e r n a l d i a m e t e r of o n l y one i n c h , which  severely r e s t r i c t s  the c o i l dimensions.  When t h e a p p a r a t u s was s h i f t e d t o the 12" e l e c t r o m a g n e t s the  dewar head a l r e a d y t h e r e was u n s u i t a b l e f o r p u l s e d NMR, so  t h a t t h e dewar system from t h e 6" magnet had t o be used w i t h the  12" magnet.  As t h e dewar head d i d n o t match t h e h e l i u m  r e t u r n system a l r e a d y t h e r e , no l i q u i d  h e l i u m temperature  measurements were p o s s i b l e w i t h o u t e x t e n s i v e r e b u i l d i n g o f t h e low temperature p a r t o f the a p p a r a t u s .  T h i s r e b u i l d i n g was  n o t done f o r r e a s o n s t o be g i v e n l a t e r . All  t h e measurements have e i t h e r been made a t room  t e m p e r a t u r e , or a t l i q u i d n i t r o g e n temperature.  For liquid  n i t r o g e n measurements, t h e o u t e r dewar i s f i l l e d w i t h  liquid  n i t r o g e n , w h i l e a i r i s used as an exchange gas i n t h e i n n e r dewar.  I f t h e i n n e r dewar a l o n e i s f i l l e d w i t h  liquid  n i t r o g e n , i t s b u b b l i n g causes v i b r a t i o n s w h i c h d e c r e a s e t h e S/N by two.  62 3.10  The C o i l System f o r a M e t a l l i c  Single Crystal  The most i m p o r t a n t p a r t o f the apparatus system.  i s the c o i l  I t i s a l s o t h e h a r d e s t p a r t t o d e s i g n so t h a t t h e  c o i l s used, and t h e r e a s o n s . f o r u s i n g them, w i l l be d i s c u s s e d i n c o n s i d e r a b l e d e t a i l i n the f o l l o w i n g s e c t i o n .  6  F i g . 3.*+  The F i g . 3.^.  E q u i v a l e n t C i r c u i t o f t h e C o i l System  e q u i v a l e n t c i r c u i t o f t h e c o i l system i s g i v e n by 2  S  i s t h e l a r g e s i g n a l output impedance o f the f i n a l  stage of t h e power a m p l i f i e r , w h i l e L* i s t h e i n d u c t a n c e of the t r a n s m i t t e r c o i l .  R , i s t h e e q u i v a l e n t damping r e s i s t a n c e  of t h i s c o i l and i s c o m p a r a t i v e l y low when t h e r f p u l s e i s on, but i t i s much h i g h e r when i t i s o f f . component o f t h e impedance coupled by t h e mutual i n d u c t a n c e M.  back from t h e p i c k u p  coil  L^., R i and R-L a r e t h e c o r r e s p o n -  d i n g parameters f o r the p i c k u p c o i l . r f p u l s e than when i t i s o f f . the r e q u i r e m e n t s  R,' i s t h e r e s i s t i v e  f o r the c o i l s .  R  a  i s l a r g e r d u r i n g an  C l a r k (2) f u l l y d i s c u s s e s a l l B r i e f l y they a r e t h a t t h e  t r a n s m i t t e r c o i l must have the maximum number of ampere-turns  63 p o s s i b l e and a f a i r l y low Q, w h i l e t h e p i c k u p c o i l should have the l a r g e s t number o f t u r n s and t h e h i g h e s t Q p o s s i b l e . The mutual i n d u c t a n c e should be as s m a l l as p o s s i b l e s i n c e i t causes t h e f o l l o w i n g u n d e s i r a b l e e f f e c t s t o o c c u r . ( i ) The r e s o n a n t c o i l p i c k u p c o i l e x t r a c t s a s i g n i f i c a n t f r a c t i o n o f t h e power s u p p l i e d  to the t r a n s m i t t e r c o i l .  This  reduces H| by t h e same f r a c t i o n . I t i s assumed t h a t b o t h r e s o n a n t c i r c u i t s a r e n o t v e r y t i g h t l y coupled and t h a t t h e t r a n s m i t t e r c o i l i s matched t o the o u t p u t impedance o f t h e a m p l i f i e r i . e . Z^wL^Q,, where Q, =  •  S t r a i g h t f o r w a r d c i r c u i t a n a l y s i s then q u i t e a c c u r -  a t e l y g i v e s t h e power d i s s i p a t e d i n t h e t r a n s m i t t e r c o i l as  v  I f the c u r r e n t passing through L P absorbed  (  i s i , , then t h e power  by t h e r e s o n a n t p i c k u p c o i l i s  t  P'  =^K,  /  Thus t h e f r a c t i o n o f power absorbed  P by u s i n g  Z^=  R,' = ^2 R .  Q,UJL,  ~ fltf+MOCK'+fc,) T h i s has a maximum v a l u e o f 0 . 1 7  when  F o r t h e p r e s e n t a p p a r a t u s L|=1.2/»h and Q i ^ l O so  a t 9Mc/s. R , - ^ r 7 A . so t h a t R ^ r 3 5 n . a  .  by t h e p i c k u p c o i l i s  L ^ 3 / i h and Q ^ 5 R' = ^  when t h e r f p u l s e i s on,  where M=k-fL7L;, w i t h 0 < k ^ l , so i t  6tf  is  i m p o s s i b l e to s a t i s f y the c o n d i t i o n f o r maximum power a b s o r p -  tion.  R e p r e s e n t a t i v e v a l u e s of the f r a c t i o n of power absorbed  a r e % f o r ksO. * and 2% f o r k=0.1. 1  (ii)  The t r a n s m i t t e r Q  where R^ a . - ^ f L . . R,r£hl5a . for  c o i l damps the p i c k u p c o i l .  tAJ Ls^  =  When the r f p u l s e i s o f f R ^ 6 n .  and  Thus Qa=^30 f o r k=0, Q = 27 f o r k=0.1 and Q = 20 a  a  k'=0,^. (ill)  I f the c o l l s a r e v e r y t i g h t l y c o u p l e d the two  c i r c u i t s cannot be tuned i n d e p e n d e n t l y of each o t h e r . becomes n o t i c e a b l e f o r (iv)  This  k^0.5.  The r f p u l s e can i n d u c e a v e r y l a r g e  voltage  w h i c h can damage the i n p u t stage of the p r e a m p l i f i e r , cause r e c o v e r y time problems a f t e r  the p u l s e i s  or  over.  The induced v o l t a g e a c r o s s the p i c k u p c o i l  i s given  a p p r o x i m a t e l y by  Representative  v a l u e s of  a r e 0.8 f o r k=0.1 and 6 . 2 5  for  k=0.03.  From the above c o n s i d e r a t i o n s i t i s c l e a r t h a t k should be l e s s than 0.1 and p r e f e r a b l y below 0.05. c o i l system f i n a l l y  a d o p t e d , k can be as low as  I n the  0.03.  The most u n u s u a l p a r t of the a p p a r a t u s i s the c o i l w i t h the m e t a l c y l i n d r i c a l sample i n s i d e i t .  sample  The f o l -  65 l o w i n g s i m p l e t h e o r y d e s c r i b e s the b e h a v i o u r of the sample c o i l w i t h s u f f i c i e n t a c c u r a c y f o r most e x p e r i m e n t a l purposes. Assume t h a t a c o i l of r a d i u s R  a  and l e n g t h l i w i t h n  a  t u r n s / u n i t l e n g t h i s spaced from a m e t a l c y l i n d e r whose r a d i u s i s R> , e l e c t r i c a l c o n d u c t i v i t y i s cr, and frequency f .  s k i n depth i s 8 a t a  I t i s assumed t h a t the m e t a l c y l i n d e r i s l o n g e r  than the c o i l . I f the m e t a l core I s a b s e n t , then the magnetic f i e l d s t r e n g t h i n s i d e an i n f i n i t e  s o l e n o i d i s (3)  H = nl where J i s t h e c u r r e n t i n the c o i l .  T h i s i s assumed t o be  u n i f o r m over the whole c r o s s s e c t i o n a l a r e a of the c o i l , so t h a t the induced back emf i s v  = •  = where A =TTR . a  where L>=  bt  -mvW-  I f I = I exp(A.u/t) t h e n Z = ~ j -  n^wAl*.  0  T h i s i s the i n f i n i t e  =UJJ^A  n* A l =iu>L , 7  solenoid formula f o r  an a i r cored c o i l and i s a c c u r a t e t o w i t h i n t e n p e r c e n t p r o v i d e d 1 >  2R . X  I f the m e t a l core i s now  i n s e r t e d , the s k i n e f f e c t p r e -  v e n t s magnetic f l u x p e n e t r a t i n g more than about a d i s t a n c e 8 i n t o the m e t a l so the e f f e c t i v e c r o s s s e c t i o n a l a r e a of the c o i l i s much l e s s .  Thus i f i t i s assumed t h a t H i s s t i l l  u n i f o r m o u t s i d e the m e t a l c o r e ,  becomes a p p r o x i m a t e l y  66  f]  .  Usually  tance w i t h a m e t a l c o r e  becomes  An^oiL U(TT[R»-(R, a  -  S  R„-R|^>£>,  so t h a t t h e i n d u c -  L ^ n ^ n ' l i (Rt -R*) The c o n d i t i o n s f o r t h e v a l i d i t y o f t h i s e q u a t i o n a r e R * - R i » S  and l  ^ 2 ( R , . - R | ).  The l a t t e r c o n d i t i o n i s more  e a s i l y s a t i s f i e d than the corresponding cored  c o n d i t i o n f o r an a i r  coil. Experimental  measurements show t h a t t h i s f o r m u l a i s  a c c u r a t e t o w i t h i n 20$.  The e r r o r i s p r o b a b l y due t o the  non-uniform f l u x d i s t r i b u t i o n which a c t u a l l y  occurs.  To c a l c u l a t e the Q o f t h e c o i l , c o n s i d e r what happens when a c u r r e n t j = j e x p ( i m t )  i s passed through t h e c o i l .  0  g e n e r a t e s a magnetic f i e l d metal c y l i n d e r .  H=n»j  p a r a l l e l t o t h e s u r f a c e of t h e  The magnetic f i e l d  generates an eddy c u r r e n t  which c i r c u l a t e s c l o s e to the surface i n the opposite t i o n to the applied c u r r e n t . cerned,  It  direc-  As f a r as power l o s s e s a r e con-  t h e eddy c u r r e n t can be c o n s i d e r e d  as a u n i f o r m  current  d e n s i t y e q u a l t o t h e a c t u a l c u r r e n t d e n s i t y a t t h e s u r f a c e of the m e t a l and c o n f i n e d t o a l a y e r 2~^b t h i c k a t t h e s u r f a c e of the m e t a l ( 3 ) . Thus the eddy c u r r e n t c i r c u l a t e s i n a l o o p 2TTR, l o n g , 1  2  w i d e , and 2~*& t h i c k , so t h a t t h e t o t a l r e s i s t a n c e  of the eddy c u r r e n t path i s a/V5 rr Ri .  The r.m.s. v a l u e of the eddy c u r r e n t i s I =a-*1 H r  Thus the power P d i s a p a t e d  a  i n t h e sample i s  67  I f t h e r.m.s. v a l u e of j i s j , then H ^ n * J . r  .. r  s  so  grg  Jr  R* =  where R  -  r  h  *  i s the e f f e c t i v e r e s i s t a n c e of the c o i l due t o t h e metal c o r e that  '(-fc->'-i-* The a c t u a l Q of t h e r e s o n a n t c i r c u i t i s reduced by t h e r f r e s i s t a n c e Rvw of the w i r e i n t h e c o l l and t h e e x t e r n a l p a r t of t h e c i r c u i t to 0 = ^ * Rs+Rw L  where Q  s  = tvL/R  s  and Q;= w L / R „ .  Thus i t i s easy t o i n c l u d e t h e  w  w i r e r e s i s t a n c e i n t h e d e s i g n of an a c t u a l c o i l  system.  The induced s i g n a l i s p r o p o r t i o n a l t o a number of f a c t o r s which depend upon t h e r a t i o R , /R  x  of t h e sample t o c o i l  These f a c t o r s w i l l now be g i v e n and t h e optimum r a t i o of R i / R % c a l c u l a t e d from them.  radius.  68 The maximum f l u x change generated by t h e sample i s 9= -jufej^  jJ&'Qg'  I f  the c o i l i s n o t t i g h t l y wound on the  sample magnetic f l u x l e a k a g e o c c u r s , so t h a t t h e f l u x 0 i n t e r cepted by the p i c k u p c o i l i s l e s s than  ©.  This f l u x leakage  i s t a k e n c a r e of by d e f i n i n g an e f f i c i e n c y f a c t o r 7} as  it T h i s depends o n l y on t h e c o i l and sample geometry and c a n , i n p r i n c i p l e , be c a l c u l a t e d f o r any c o i l c o n f i g u r a t i o n . t i c e such c a l c u l a t i o n s a r e u n w i e l d y .  In prac-  F o r a s h o r t c o i l wound  on a sample much l o n g e r than i t s d i a m e t e r , a c a l c u l a t i o n g i v e s ^ as *} = l - k [ ( \ f  -l]  where k= ^ ^ V l ? ) and L i s t h e sample l e n g t h . c o n f i g u r a t i o n s approximated  Most of the c o i l  these c o n d i t i o n s .  The induced v o l t a g e i s a l s o p r o p o r t i o n a l t o t h e number of  turns n  2  inductance L  i n the pickup c o i l . a  At a f i x e d frequency, the  = m m j ^ l | E j - R f ) i s a constant. .'. n,.oC(R>-R* )~* •  I t i s a l s o p r o p o r t i o n a l to the cross s e c t i o n a l area A = 2TTR,8 of the p e r t u r b e d magnetic moments. I f v i s t h e output v o l t a g e f r o m t h e r e s o n a n t p i c k u p coil  then VoGQ Tj nA  69 where x=R /R, and a  K= ^ ^ g - f e f i i .  T h i s has a maximum v a l u e when R ^ R , v a l u e s of Q and  K, R, /R  x  (1+ib*  .  With t y p i c a l  l i e s i n the r e g i o n of 0.7  I n the c o i l s a c t u a l l y b u i l t R /R (  %  to  0.8.  v a r i e d from about 0.6  to  O.9o I t should be noted t h a t s l i c i n g the sample i n t o s l a b s may  n o t i n c r e a s e the Induced v o l t a g e .  thin  T h i s i s because  i n c r e a s i n g the s u r f a c e a r e a n o t o n l y i n c r e a s e s the number of n u c l e i e x c i t e d , but a l s o i n c r e a s e s the p a t h l e n g t h s of the c u l a t i n g c u r r e n t s and  cir-  t h i s causes a compensating drop i n Q.  Thus i f the Q of the r e s o n a n t  c i r c u i t i s determined by  sample, s l i c i n g i t w i l l have l i t t l e e f f e c t .  the  However, i f the  w i r e r e s i s t a n c e i s dominant, s l i c i n g i n c r e a s e s the  signal  u n t i l the stage i s reached where the sample l o s s e s become as l a r g e as the w i r e l o s s e s . obtained  The f u l l b e n e f i t s of s l i c i n g a r e  not  u n l e s s the s l a b s are about 8 t h i c k , when the eddy  c u r r e n t l o s s e s become c o n s i d e r a b l y reduced.  Manufacture and  a l i g n m e n t of m e t a l s l a b s l e s s than lO^cm, t h i c k i s such a formidable  t e c h n i c a l problem t h a t no e x p e r i m e n t s a l o n g  these  l i n e s have been a t t e m p t e d . The  most d i f f i c u l t p a r t of b u i l d i n g the a p p a r a t u s  finding a suitable c o i l configuration.  C l a r k (2) l i s t s  r e l a t i v e m e r i t s of t h r e e d i f f e r e n t c o n f i g u r a t i o n s . these were examined both t h e o r e t i c a l l y and one  Two  was the of  experimentally,  and  o n l y t h e o r e t i c a l l y , b e f o r e they were a l l r e j e c t e d . The  e a s i e s t system t o r e j e c t was  the combined t r a n s -  m i t t e r - r e c e i v e r c o i l , as s i m p l e c a l c u l a t i o n s showed t h e r e  was  70 no change of f i n d i n g a s a t i s f a c t o r y compromise between the c o n f l i c t i n g r e q u i r e m e n t s f o r the t r a n s m i t t e r  and r e c e i v e r  A l s o the Q damping c i r c u i t s on the t r a n s m i t t e r  coils.  and r e c e i v e r  c o u l d n o t be u s e d . A r f b r i d g e (8) was t r i e d and q u i c k l y r e j e c t e d . duced the r f f i e l d  re-  c o n s i d e r a b l y , d i d not balance very w e l l  w i t h a m e t a l cored c o i l , receiver c o i l ,  It  s t i l l used a combined  transmitter-  and d i d n o t a l l o w the use of Q damping c i r c u i t s .  Grossed c o i l s were s t u d i e d more c a r e f u l l y , both e x p e r i m e n t a l l y and t h e o r e t i c a l l y .  They have the advantages t h a t  conflicting transmitter  and r e c e i v e r c o i l r e q u i r e m e n t s  s i m u l t a n e o u s l y be met.  The c o u p l i n g c o e f f i c i e n t k i s v e r y  s m a l l , w h i l e Q damping c i r c u i t s can a l s o be u s e d .  can  Acoustic  o s c i l l a t i o n s are a l s o hard to e x c i t e when u s i n g c r o s s e d They have the major d i s a d v a n t a g e t a t i o n dependent NMR p r o p e r t i e s  coils.  t h a t measurements of o r i e n are v e r y d i f f i c u l t .  s i n c e o n l y the component of H, p e r p e n d i c u l a r to H  0  This i s tips  the  n u c l e a r magnetic moment,while w i t h t h i s c o n f i g u r a t i o n H make any a n g l e between 0° and 90° w i t h H, as i t i s with respect  to the sample.  rotated  To e l i m i n a t e t h i s p r o b l e m ,  whole sample and c o i l assembly c o u l d be r o t a t e d w i t h to Ho 5 a d i f f i c u l t  m e c h a n i c a l and e l e c t r i c a l p r o b l e m ,  p e c i a l l y at l i q u i d helium temperatures. w h i c h was t r i e d , was to add another gave a r f f i e l d first  transmitter  o r t h o g o n a l to b o t h H  transmitter  coil.  An e a s i e r  0  can  0  the  respect es-  solution,  c o i l which  and the r f f i e l d  S w i t c h i n g the t r a n s m i t t e r  from the  output t o  the  71 a p p r o p r i a t e c o i l can always g i v e a f a i r l y l a r g e r f f i e l d pendular to H .  per-  The c y l i n d r i c a l sample shape and the. l i m i t e d  0  space a v a i l a b l e meant the l o n g r e c t a n g u l a r t r a n s m i t t e r c o i l s had to be u s e d .  T h i s system e a s i l y gave k<(0.5, even w i t h o u t an  e l e c t r o s t a t i c s h i e l d , but c o u l d n o t g i v e H,") 10 gauss w i t h power a v a i l a b l e from the t r a n s m i t t e r .  the  Hi was a l s o n o n - u n i f o r m  over the s u r f a c e of the m e t a l , d e c r e a s i n g to z e r o on some p a r t s of the s u r f a c e , turbed.  so t h a t not a l l the s u r f a c e n u c l e i were p e r -  F o r these r e a s o n s ,  the c r o s s e d c o i l system was aban-  doned. T h i s l e f t a c o a x i a l c o i l system w i t h the sample  coil  i n s i d e a t r a n s m i t t e r c o i l of Inductance L ^ n ^ t u r n s , r a d i u s R , 3  and l e n g t h l j as the o n l y u s a b l e c o n f i g u r a t i o n . The c o u p l i n g c o n s t a n t of t h i s c o n f i g u r a t i o n i s e a s i l y derived. L j =TTyU |^n£(Rt-R< ) so i f Vj i s the r f v o l t a g e a p p l i e d a c r o s s the t r a n s m i t t e r  coil  then H  = n^A.j  TT But f o r a r f t r a n s f o r m e r ,  (K-K) • n,l> = k -j*  tx-RM [RJ  *;k  = (Rl  W i t h the c o i l c o n f i g u r a t i o n i n i t i a l l y used t h i s f o r -  72 mula g i v e s l c v 0 . 3 .  Even though t h i s v a l u e was  l a r g e , the c o n f i g u r a t i o n was  tried  out and  known t o be  too  showed most of  the  defects predicted f o r i t . In order  to reduce k, a b u c k i n g c o i l i s added i n s e r i e s  w i t h the p i c k u p c o i l i n such a way i t opposes the v o l t a g e  t h a t the v o l t a g e  induced i n the p i c k u p c o i l .  induced i n The  bucking  c o i l has t o be l o c a t e d where i t cannot p i c k up the l a r g e from the sample. and  The  mutual c o u p l i n g between the b u c k i n g  the p i c k u p c o i l should Various  t r i e d and  signal coil  a l s o be as s m a l l as p o s s i b l e .  p o s i t i o n s and  t y p e s of b u c k i n g c o i l w i n d i n g s were  e v e n t u a l l y a s i n g l e c o i l wound on the same c o i l former  as the t r a n s m i t t e r c o i l and  j u s t above i t was  e c t i o n s of the w i n d i n g s , and  I t was  b u c k i n g and  dir-  c a p a c i t i v e c o u p l i n g opposed  I m p o s s i b l e t o get a p e r f e c t  l a t i o n , even a f t e r e x t e n s i v e e x p e r i m e n t a t i o n , c a p a c i t i v e c o u p l i n g and  The  the p o s i t i o n s of a l l the w i r e s were  c a r e f u l l y arranged so t h a t any inductive coupling.  chosen.  the  cancel-  because of  the  the i n d u c t i v e c o u p l i n g between the  pickup c o i l s .  A Faraday s h i e l d s i g n i f i c a n t l y r e -  duced the c a p a c i t . i v e ~ c o u p l i n g . Another c o m p l i c a t i o n was  the d i s t o r t i o n of the magnetic  f i e l d caused by the presence of the m e t a l sample.  The  eddy  c u r r e n t s e x c l u d e f l u x f r o m the i n t e r i o r of the sample and i n c r e a s e the magnitude of H,  a t the sample s u r f a c e .  T h i s means  t h a t the b u c k i n g c o i l r e q u i r e s more t u r n s than expected on b a s i s of assuming a u n i f o r m f l u x i n t e n s i t y over the  the  cross  s e c t i o n of the t r a n s m i t t e r c o i l ; an assumption which gave good  73 r e s u l t s f o r a powdered sample.  The magnitude of the i n c r e a s e i n  H, v a r i e d from sample t o sample, but was was never more than  t y p i c a l l y about 30$  and  50$.  S i n c e the i n d u c t a n c e of the p i c k u p  c o i l p l u s t h a t of the  b u c k i n g c o i l should e q u a l the maximum p o s s i b l e i n d u c t a n c e f o r a p i c k u p c o i l , the use of a b u c k i n g c o i l reduces the number of t u r n s i n the p i c k u p c o i l and hence reduces the s i z e of the induced signal.  T h i s " r e d u c t i o n and the number of t u r n s r e q u i r e d i n the  p i c k u p c o i l , c o u l d i n p r i n c i p l e be c a l c u l a t e d from the above r e s t r i c t i o n on the i n d u c t a n c e s , p l u s the r e q u i r e m e n t t h a t the t o t a l a r e a s of the b u c k i n g and p i c k u p c o i l s c u t by the f l u x a r e e q u a l t o each o t h e r .  magnetic  However, the b u c k i n g c o i l has a  l e n g t h much l e s s than i t s r a d i u s so t h a t no s i m p l e f o r m u l a f o r i t s inductance e x i s t s . uneven.  The magnetic f l u x d i s t r i b u t i o n s i s a l s o  I t i s thus i m p r a c t i c a l t o do any c a l c u l a t i o n s  the b u c k i n g c o i l .  involving  The number of t u r n s i n the b u c k i n g c o i l  thus  had to be found e x p e r i m e n t a l l y . . The c o i l c o n f i g u r a t i o n i s r e a s o n a b l y s a t i s f a c t o r y i n practice. by about 30$  I t t y p i c a l l y has k ^ 0 . 0 5  s  w h i l e r e d u c i n g the s i g n a l  of i t s maximum p o s s i b l e v a l u e .  I f considerable  t r o u b l e i s t a k e n i n e m p i r i c a l l y f i n d i n g the optimum number of t u r n s f o r the c o i l s , k can be reduced  t o about 0.02.  This  was  done f o r the f i r s t two samples but t h e n , f o r reasons of conveni e n c e and m e c h a n i c a l s t a b i l i t y , one b u c k i n g c o i l was a l l the subsequent  samples.  i n the performance  of the  T h i s caused l i t t l e system.  used f o r  deterioration  •7h In the c o n f i g u r a t i o n f i n a l l y a x i a l c y l i n d r i c a l c o i l formers.  adopted t h e r e a r e two co-  These a r e made of T e f l o n w h i c h  has a h i g h e l e c t r i c a l r e s i s t a n c e and t h e a b i l i t y  to withstand  r e p e a t e d c y c l i n g t o low temperatures w i t h o u t c r a c k i n g .  The  o u t e r one has a d i a m e t e r of 2 cm. and has the t r a n s m i t t e r and b u c k i n g c o i l s wound on i t .  The t r a n s m i t t e r c o i l has 11 t u r n s  and an i n d u c t a n c e of 1.2^uh and w i t h 1.8 KV peak t o peak a c r o s s it,  g i v e s an H, of about 20 gauss.  and an i n d u c t a n c e o f 0.9yUh. cut  The b u c k i n g c o i l has h t u r n s  B o t h c o i l s were wound i n grooves  i n t h e f o r m e r and t h e n imbedded i n epoxy r e s i n t o g i v e as  much m e c h a n i c a l r i g i d i t y  as p o s s i b l e .  The d i a m e t e r o f the i n n e r  c y l i n d e r depends on t h e m e t a l sample b e i n g used.  The sample  f i t s s n u g l y i n s i d e t h e c y l i n d e r and the p i c k u p c o i l i s wound on the  outside.  None o f the samples have the same d i m e n s i o n s , so  t h a t a d i f f e r e n t p i c k u p c o i l has t o be wound f o r each  sample.  The c o i l i s coated w i t h G.C. E l e c t r o n i c s P o l y s t y r e n e Q Dope No. 37-2 t o p r e v e n t m e c h a n i c a l v i b r a t i o n . v e r y e a s i l y assembled  The c o i l system i s  and mounted on t h e end o f a s t a i n l e s s  s t e e l tube, which a l s o a c t s as the o u t e r c o n d u c t o r o f a c o a x i a l cable l e a d i n g to the p r e a m p l i f i e r . 3.11  Acoustic Oscillations O f t e n i n p u l s e d NMR a p p a r a t u s a troublesome damped os-  c i l l a t i o n appears a f t e r t h e r f p u l s e .  This o s c i l l a t i o n i s  caused by t h e m e c h a n i c a l f o r c e generated by the i n t e r a c t i o n between H  0  and t h e l a r g e c i r c u l a t i n g c u r r e n t i n t h e t r a n s -  m i t t e r c o i l c a u s i n g some p a r t of the t r a n s m i t t e r c o i l , o r sample,  75 to v i b r a t e a t an u l t r a s o n i c f r e q u e n c y .  The a c o u s t i c  l a t i o n s p e r s i s t a f t e r the r f p u l s e has f i n i s h e d and  oscilsomehow  induce a v o l t a g e i n the p i c k u p c o i l w h i c h o b l i t e r a t e s the induced  (2).  nuclear s i g n a l  A c o u s t i c o s c i l l a t i o n s were not o f t e n n o t i c e d when the 6"  magnet was  used.  However, w i t h the 12™  magnet a c o u s t i c  o s c i l l a t i o n s became a severe ^problem because of the  increased  magnetic f i e l d and a l s o because the l o w e r p a r t of the c o a x i a l c a b l e and  t h e ^ w i r e - l e a d i n g t o — t h e — t r a n s m i t t e r c o i l were not i n  the magnetic f i e l d .  B o t h the t r a n s m i t t e r l e a d and  the  inner  w i r e of the c o a x i a l c a b l e were o r i g i n a l l y v e r y t h i n t o reduce heat l e a k a g e d u r i n g h e l i u m r u n s . badly and  styrene  to be r e p l a c e d by heavy 2k- A.W.G. w i r e s .  had  bottom end  However, they b o t h v i b r a t e d  of the c o a x i a l c a b l e was  glue.  also f i l l e d with poly-  These measures e l i m i n a t e d a c o u s t i c  from these w i r e s , but i n t r o d u c e d t h a t h e l i u m runs were rendered  The  oscillations  such a l a r g e heat l e a k a g e impossible.  I n a l l cases a c o u s t i c o s c i l l a t i o n s o n l y occurred when the sample was itself.  Two  p r e s e n t , so t h a t they must come from the sample  experimental  o b s e r v a t i o n s were made on rhenium  w h i c h suggest the cause of these o s c i l l a t i o n s .  The f i r s t of  these i s t h a t the a m p l i t u d e of the a c o u s t i c o s c i l l a t i o n s i s p r o p o r t i o n a l to H*(Fig.3.5).  T h i s i s not v e r y  restrictive  s i n c e most p o s s i b l e mechanisms have t h i s Ho dependence.  The  second i s t h a t the o s c i l l a t o r y f r e q u e n c y i s about 70Kc/s f o r a sample 3.15cm. l o n g .  Thus i f the sample i s assumed t o be  76 l o n g , t h e l o n g i t u d i n a l v e l o c i t y o f sound i n rhenium i s about khOO m e t r e s / s e c .  This i s a t y p i c a l l o n g i t u d i n a l v e l o c i t y  of sound f o r a m e t a l , b u t i t i s t o o h i g h f o r a t r a n s v e r s e v e l o c i t y (10).  I t was a l s o n o t i c e d t h a t t h e l o n g e r  a lower a c o u s t i c a l o s c i l l a t i o n frequency. increased  on going  The f r e q u e n c y a l s o  t o l i q u i d notrbgen temperature.  o s c i l l a t i o n s are probably s t a n d i n g wave b e i n g  Thus the  due t o an a c o u s t i c a l l o n g i t u d i n a l  s e t up i n t h e sample.  the f o l l o w i n g sequence of e v e n t s The  samples had  T h i s suggests t h a t  occurs.  r f p u l s e g e n e r a t e s a c i r c u l a t i n g eddy c u r r e n t o f  about 50 amps, i n t h e sample w h i c h i n t e r a c t s w i t h t h e l a r g e s t a t i c magnetic f i e l d .  T h i s produces o s c i l l a t o r y d r i v i n g  f o r c e s p a r a l l e l t o t h e c y l i n d r i c a l a x i s of t h e sample w h i c h s e t up an a c o u s t i c a l s t a n d i n g wave.  There i s such a l a r g e  a c o u s t i c mismatch a t each end t h a t the,re i s n e a r l y p e r f e c t r e f l e c t i o n o f t h e sound wave, so t h a t t h e s t a n d i n g wave p e r s i s t s l o n g a f t e r t h e r f p u l s e i s turned  off.  T h i s s t a n d i n g wave causes a v a r i a t i o n Ap i n t h e d e n s i t y £ of t h e sample.  If N  e  i s t h e number o f f r e e e l e c t r o n s / u n i t  volume, t h e n t h e s t a n d i n g wave produces a v a r i a t i o n £ N i n t h e i r number.  The e l e c t r o n i c magnetic  =N ^-  e  e  susceptibility  X<pCN T* , where T i s t h e e l e c t r o n gyromagnetic r a t i o , so t h a t %  e  e  AXe/DCe = A N / N = Ap/p . e  e  The p i c k u p  c o i l i s wound around t h e  c e n t r e of t h e sample and has a v o l t a g e o f v ^ d C ^ e A X b = w X f e  induced i n i t . the a c o u s t i c  £  e  e  e  i s t h e s k i n d e p t h a t t h e f r e q u e n c y cj of  oscillations.  e  e  77 The  s u s c e p t i b i l i t y XqOCN TQ  nuclear  A  —UJ  s i n c e N =£rN a  The  e  A  , so t h a t  No  f o r a metal.  induced v o l t a g e from t h e n u c l e a r  s p i n s i s V aO0J Xob^ a  so  a  t h a t t h e r a t i o of t h e two v o l t a g e s i s  I n a t y p i c a l case a;„ ^ 100cj ^  so t h a t  e  ^ x l O  5  ^ . 6  A  W i t h t h i s mechanism i t o n l y r e q u i r e s T ? . ^ 5X10" -  , a quite rea-  sonable i n e q u a l i t y (11), f o r the a c o u s t i c o s c i l l a t i o n s to dominate the n u c l e a r  signal.  W i t h t h e c o a x i a l c o i l system i t i s i m p o s s i b l e the s t a n d i n g wave b e i n g  t o stop  generated i n t h e sample, so t h a t t h e  o n l y method of e l i m i n a t i n g t h e o s c i l l a t i o n s i s t o v e r y damp them.  F r i c t i o n a l damping w i t h i n t h e m e t a l i s v e r y  quickly small  so t h a t t h e main a c o u s t i c a l energy l o s s i s by t r a n s m i s s i o n through t h e ends o f t h e sample, w i t h f r i c t i o n a l l o s s e s a t t h e s i d e s of the c y l i n d e r p l a y i n g some p a r t .  Thus t h e o n l y way t o  damp t h e o s c i l l a t i o n s i s by i n c r e a s i n g t h e a c o u s t i c  losses  through t h e ends and s i d e s . I f t h e d e n s i t i e s and v e l o c i t i e s i n two i n f i n i t e media are  ^, , ^  and  c,  9  c  a  r e s p e c t i v e l y , the r e f l e c t i o n  coefficient  78 R a t the i n t e r f a c e  i s (11) _ Q,c, - ?xCa  R  The media a r e n o t a c t u a l l y i n f i n i t e , b u t i n s t e a d t h e s i t u a t i o n i s c l o s e r t o t h a t o f a p i s t o n r a d i a t i n g i n t o an i n f i n i t e medium.  F o r t h i s case t h e a c o u s t i c impedance Z  a  is  of t h e form (11) Z ^ | p »  c TT (Ri /if 9  A  •=0=c^  f o r hrT R, 41,  ^  f o r rrR, > 1.  These e q u a t i o n s have used the s u b s t i t u t i o n X =21. The e a s i e s t -way of damping the a c o u s t i c o s c i l l a t i o n s i s thus u s i n g a sample w i t h a l a r g e r a t i o o f R, / l immersed i n a medium w i t h a d e n s i t y and v e l o c i t y o f sound much c l o s e r t o those o f a m e t a l than a i r h a s . These i d e a s were e x p e r i m e n t a l l y t e s t e d by immersing a rhenium sample i n g l y c e r i n e so t h a t t h e r e f l e c t i o n c o e f f i c i e n t was reduced from 1 t o 0.9. c i l l a t i o n s decreased decreased  The d u r a t i o n o f the a c o u s t i c o s -  by 3>0% and t h e i r i n i t i a l  amplitude  by 20%. The damping a l s o i n c r e a s e d w i t h an i n c r e a s e  i n the r a t i o R , / l .  Rhenium (R, /1=0.05) and bismuth (R,/l=0.09)  had l a r g e a c o u s t i c o s c i l l a t i o n s , w h i l s t i n i n d i u m (R,/l=0.35) the o s c i l l a t i o n s were j u s t n o t i c e a b l e .  The r e d u c t i o n i n  o s c i l l a t i o n s w i t h b o t h i n c r e a s i n g R, / l and d e c r e a s i n g t i o n c o e f f i c i e n t i s much l a r g e r than the s i m p l e t h e o r y and suggests t h a t s u r f a c e f r i c t i o n a l  reflecpredicts  losses play a s i g n i f i c a n t  p a r t i n the damping. • U n f o r t u n a t e l y , g l y c e r i n e has a v e r y low t h e r m a l conduc-  79 t i v i t y and  so the r f p u l s e s can heat the sample to temperatures  w e l l above t h a t of the dewar system. was  For t h i s r e a s o n g l y c e r i n e  not o f t e n used to dampen a c o u s t i c o s c i l l a t i o n s .  the sample was  embedded i n a p o r c e l a i n cement^ w h i c h was  a c o u s t i c a l match. mantling  Instead  The  cement i s w a t e r s o l u b l e , so t h a t d i s -  the sample mounting i s easy.  P a r t of the sample  always l e f t exposed so as to p r o v i d e a good t h e r m a l The  was  contact.  cement reduced a c o u s t i c o s c i l l a t i o n s to a t o l e r a b l e l e v e l  i n n e a r l y a l l the samples i t was 3.12  a good  used w i t h .  C a l c u l a t i o n of the S i g n a l t o N o i s e R a t i o S i n c e no measurements were made below 78  K. i t i s  assumed t h a t the normal s k i n e f f e c t t h e o r y i s a p p l i c a b l e . The  v o l t a g e induced i n the p i c k u p  c o i l by the  precessing  n u c l e i i s (Appendix I I I ) v  =tr^  nwM„R|( e x p ( - ^ ) s i n ( r t B , e  Jo  T h i s i n t e g r a l has been e v a l u a t e d about 0.7  S when  vife  and  TB.Tl—frr r a d i a n s .  )cos '(^|^)dz. 3  has a maximum v a l u e The  maximum induced  of voltage  i s thus v = O^n^no^MoR, 8 . The  p i c k u p c o i l i s resonated  a t the f r e q u e n c y o>, so t h a t  the  v o l t a g e a t the i n p u t t o the p r e a m p l i f i e r i s v =0.7n^yMn wM QR, & . 0  If T  n  i s the n o i s e temperature of the s h a r p l y tuned  p i c k u p c o i l , t h e n the r.m.s. n o i s e v o l t a g e  \r  n  S a u e r e i s e n A d h e s i v e Cement N o . l P a s t e , Cement Co., P i t t s b u r g h 15, Perm., U.S.A.  is  (12)  Saureisen  80 v„ =  ^ E  ,  where C i s the c a p a c i t a n c e r e s o n a t i n g the p i c k u p c o i l . The parameters of the c o i l c i r c u i t and the p r e a m p l i f i e r a r e chosen so t h a t t h e r m a l n o i s e from the r e s o n a n t c i r c u i t i s the dominant n o i s e s o u r c e .  T h i s i s e a s i l y done.  The  bandwidths  of the p r e a m p l i f i e r and a m p l i f i e r a r e g r e a t e r t h a n t h a t of the tuned p i c k u p c o i l , so t h a t the S/N r a t i o a t the a m p l i f i e r o u t put i s  T h i s e x p r e s s i o n i n c l u d e s the improvement of -[2  i n the  S/N  r a t i o i n t r o d u c e d by phase s e n s i t i v e d e t e c t i o n ( 1 2 ) . The boxcar i n t e g r a t o r enhances the S/N by r e p e t i t i o n r a t e s g r e a t e r than about (50  V^t^"  f o r  ms.)"' , so t h a t the  f i n a l S/N r a t i o S i s  The parameters i n t h i s e q u a t i o n which a r e l i s t e d  below  have a temperature dependence. ( i ) The n u c l e a r magnetic moment/unit volume M ccT~' ( 1 ) , 0  where T i s the sample  temperature.  ( i i ) 8oCo-^ and o~ has a c o m p l i c a t e d temperature dependence. F o r s i m p l i c i t y the h i g h temperature a p p r o x i m a t i o n a-oCT ' w i l l -  be used  (9)  ?  even though the Debye temperature f o r most m e t a l s  f a l l s w i t h i n the temperature range of i n t e r e s t . (iii)  Q="4r where  R  n  Thus  bcCT"^.  , the t o t a l s e r i e s damping r e s i s -  tance v a r i e s w i t h t e m p e r a t u r e .  I f the damping i s o n l y due t o  81 j.  -J-  eddy c u r r e n t s i n the m e t a l t h e n QoCcrbcT \  However, i n p r a c t i c e  the r e s i s t a n c e of the c o a x i a l c a b l e forms a l a r g e p a r t of the damping r e s i s t a n c e .  T h i s v a r i e s i n temperature between room  temperature a t one end and the temperature of the sample a t the o t h e r end, so t h a t the n o i s e temperature T c u i t i s u s u a l l y d i f f e r e n t from T, temperature.  The dependence of T  of the tuned  n  cir-  l y i n g between T and room on R  h  n  i s n o t known, but TnocRh _JL  seems a r e a s o n a b l e assumption. ( i v ) fcoC(Aw)"  1  Thus QoCT* . 1  and Q=i£so T ocQ cCTn^. t  (v) I n a m e t a l sample the K o r r i n g a r e l a t i o n T,T=constant h o l d s ( 1 ) . The r e p e t i t i o n time f o r the boxcar i s T = KT, where r  K i s a constant. . Tf-ccT . The e f f e c t i v e time c o n s t a n t "t =  RC b  b  of the b o x c a r i s a tem-  p e r a t u r e independent c o n s t a n t determined o n l y by the sweep t i m e . .". R^Cb oC T  r  oC T.  Combining a l l these temperature dependences g i v e s SoCT„"* . The i m p o r t a n t f e a t u r e of t h i s s i m p l e a n a l y s i s i s t h a t t h e r e i s l i t t l e improvement i n S/N on g o i n g t o low temperat u r e s ; the i n c r e a s e i n M  0  b e i n g compensated f o r by a decrease  i n 8 and the d e c r e a s i n g e f f e c t i v e n e s s of t h e boxcar i n t e g r a t o r . E x p e r i m e n t a l l y i t was  found t h a t S improved by about 50% on  g o i n g from room t o 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 .  This gives  T„oCT^ as the approximate temperature dependence i n t h i s  82 -3r e g i o n , so SoCT . D  I f t h i s temperature dependence i s e x t r a -  p o l a t e d t o l o w e r temperatures and a l l o w a n c e made f o r t h e onset of anomalous c o n d u c t i o n , then c o o l i n g from n i t r o g e n temperature to f.2°K would i n c r e a s e S t e n f o l d . l  dependence of T  h  A c t u a l l y t h e temperature  on T i s even l e s s below n i t r o g e n temperature  because even a t t h i s temperature most o f the n o i s e i s coming from t h e p a r t s of the tuned c i r c u i t near room temperature. Thus c o o l i n g t h e sample t o a l o w e r temperature does n o t reduce the n o i s e temperature v e r y much.  These f a c t s , p l u s t h e ex-  p e r i m e n t a l d i f f i c u l t i e s w i t h a c o u s t i c o s c i l l a t i o n s , a r e the r e a s o n s why no measurements a t l i q u i d h e l i u m temperatures have been a t t e m p t e d . The o t h e r parameter w h i c h can a f f e c t S I s the r e s o n a n t frequency.  I f i t i s assumed t h a t t h e t u n i n g c a p a c i t a n c e C I s  the same f o r a l l f r e q u e n c i e s , t h e f r e q u e n c y dependent p a r a meters v a r y as f o l l o w s . ( I ) The i n d u c t a n c e L o c n , but wwcL^, so noctu" . a  1  (ii) 6 ©ecu"* . ( i i i ) M =XH 0  o  and rH= U J , S O M oCuj. a  ( i v ) Q i s u s u a l l y f r e q u e n c y dependent.  However, bandwidth  t r i c t i o n s r e q u i r e Q t o be k e p t r e a s o n a b l y c o n s t a n t so t h a t over a moderate f r e q u e n c y range Q i s f r e q u e n c y independent. T h i s i s a good a p p r o x i m a t i o n s i n c e even i f Q does v a r y w i t h f r e q u e n c y i t i s almost c a n c e l l e d by t h e opposing v a r i a t i o n of t . c  Thus  SoC uA  res-  83 This i s i n reasonably  good agreement w i t h t h e e x p e r i m e n t s .  Changing f r e q u e n c i e s from 6 t o 9Mc/s. i n c r e a s e d S by about 30$. I n a d d i t i o n r e c o v e r y from t h e r f p u l s e s i s b e t t e r a t t h e h i g h e r frequency  s i n c e a h i g h e r low f r e q u e n c y  c u t o f f can be used.  Measurements were thus u s u a l l y made a t t h e h i g h e s t  convenient  frequency. These c o n c l u s i o n s a r e n o t v a l i d above about 20Mc/s s i n c e i n t h i s r%gion the grid noise of,the p r e a m p l i f i e r input  stage  i n c r e a s e s and a l s o i t s d e c r e a s i n g i n p u t impedance becomes important. 3.13  The Measurement of S p i n - L a t t i c e R e l a x a t i o n Times The most common method of measuring T  (  i s t o use a 180°  p u l s e f o l l o w e d by a 90° p u l s e a t a v a r i a b l e time t l a t e r . amplitude  of t h e i n d u c t i o n t a i l f o l l o w i n g t h e second  The  pulse  v a r i e s as l-exp(--^=-), so t h a t T, can e a s i l y be o b t a i n e d .  When  a b o x c a r i n t e g r a t o r i s used t h e s p a c i n g between t h e p u l s e s i s l i n e a r l y i n c r e a s e d w i t h time so t h a t t h e e x p o n e n t i a l i n c r e a s e of a m p l i t u d e  i s recorded  d i r e c t l y on t h e c h a r t .  U n f o r t u n a t e l y t h i s method cannot be used f o r m e t a l s i n g l e c r y s t a l s f o r s e v e r a l reasons.  One of these i s t h a t t h e r e  are no 90° o r 180° p u l s e s of t h e c o n v e n t i o n a l type because of s k i n e f f e c t s (Appendix I I I ) . g i v e maximum a m p l i t u d e s  There a r e p u l s e l e n g t h s w h i c h  and even p u l s e l e n g t h s which g i v e no  a m p l i t u d e w h i c h c o u l d be used f o r t h e e q u i v a l e n t of a 180° 0  90  pulse t r a i n .  level.  Therefore  However, t h e s i g n a l i s w e l l below t h e n o i s e t u n i n g t h e apparatus  so t h a t i t i s f i r s t  e x a c t l y on resonance,  and then f i n d i n g the r i g h t p u l s e length;,  i s a n i m p o s s i b l e job when a boxcar i n t e g r a t o r must be used. The o t h e r b i g d i f f i c u l t y  i s t h a t the s i g n a l i s t y p i c a l l y about  one t e n t h o f the n o i s e l e v e l , so t h a t any b a s e l i n e d i s t o r t i o n must be l e s s than about one hundredth o f the n o i s e l e v e l f o r even a moderately  a c c u r a t e measurement o f T, .  more s t r i n g e n t requirement  This i s a f a r  than i s u s u a l l y r e q u i r e d i n p u l s e d  NMR apparatus and i s v e r y d i f f i c u l t t o a t t a i n . F o r these r e a s o n s , an a l t e r n a t i v e method o f measurement was d e v i s e d w h i c h e l i m i n a t e s these d i f f i c u l t i e s  a t the expense  of b e i n g v e r y l a b o r i o u s . L e t a s p i n system have a l a r g e s t a t i c magnetic f i e l d H  0  a p p l i e d a l o n g the z a x i s w i t h a l i n e a r magnetic f i e l d 2H, coswt normal t o i t .  I n the r o t a t i n g r e f e r e n c e frame t h e r e i s an  e f f e c t i v e magnetic f i e l d H, = H$  making an a n g l e © = tan'T L  ,, ' He + H  + (Ho+¥)k  1  w i t h the z a x i s . I f  .the n u c l e a r magnetism M„ i s i n i t i a l l y a l i g n e d a l o n g H , then 0  on a p p l i c a t i o n o f the r f p u l s e the components of M cular to H  e  0  perpendi-  r e l a x towards i t w i t h a time c o n s t a n t somewhat  l o n g e r than T (13) a  9  so t h a t e v e n t u a l l y the magnetism i s com-  p l e t e l y a l i g n e d a l o n g H w i t h magnitude M c o s 6 . e  0  When t h e  r f p u l s e i s switched o f f the components o f the magnetism perpendicular to H  0  decay i n a time o f the o r d e r o f T . The  magnetism a l o n g the z a x i s thus has an a m p l i t u d e \. M ^ M cosO|eos0 |. o  a  85 I t has so f a r been assumed t h a t no s p i n - l a t t i c e r e l a x a t i o n occurso  T h i s r e q u i r e s the r f p u l s e l e n g t h t o be much  l e s s than T, w h i l e s t i l l b e i n g many times T  i o  I f a short  second r f p u l s e i s a p p l i e d a t a time t l a t e r t h e h e i g h t of i t s induction t a i l i s proportional to M -0--exp(- -== j~ )} + M cos0 |cos0 | e x p ( - ) .  M=  0  z  From t h i s T  (  0  i s easily  obtained.  T h i s i s t h e b a s i s of the method used i n t h e m e t a l s i n g l e A r f p u l s e of 200/us, or l o n g e r , i s a p p l i e d t o b r i n g  crystals.  the s p i n system t o e q u i l i b r i u m i n the r o t a t i n g r e f e r e n c e frame i n t h e manner j u s t d e s c r i b e d .  T  x  i s l e s s than 50^3 i n n e a r l y  a l l the m e t a l s , w h i l e T, i s u s u a l l y s e v e r a l m i l l i s e c o n d s even a t room t e m p e r a t u r e , so the i n e q u a l i t i e s c o n c e r n i n g length are e a s i l y s a t i s f i e d .  the pulse  The r a p i d decrease, of H, w i t h  depth means t h a t some n u c l e i w i l l n o t r e l a x i n t h e r o t a t i n g r e f e r e n c e frame s i n c e H w i l l be much s m a l l e r than t h e l o c a l t  fields.  However t h i s , a l o n g w i t h phase e f f e c t s , has l i t t l e  p r a c t i c a l e f f e c t on t h e s t a t e the s p i n system i s l e f t i n when the p u l s e i s switched  off.  Mz. i s measured w i t h a r f p u l s e about 15/WS l o n g a p p l i e d a t a v a r i a b l e time l a t e r on.  The time l a g i s measured w i t h a  double beam o s c i l l o s c o p e , as d e s c r i b e d The  previously.  h e i g h t of the i n d u c t i o n t a i l i s measured by the  boxcar i n t e g r a t o r used w i t h a gate about T wide ( 6 ) . The 3  magnetic f i e l d  i s l i n e a r l y swept t h r o u g h t h e r e s o n a n t v a l u e so  86 that M  can be got from t h e r e c o r d e r t r a c e . The r e f e r e n c e phase i  u s u a l l y a d j u s t e d so t h a t t h e r e c o r d e r t r a c e bears some resemblanc to an a b s o r p t i o n c u r v e .  T h i s - i s not necessary,  b u t makes measure  ments from t h e r e c o r d e r c h a r t e a s i e r . A s e r i e s of measurements i s made, f i r s t l y w i t h t i n c r e a s i n g and then w i t h t d e c r e a s i n g . each v a l u e of t a r e then averaged.  The measurements f o r This approximately  out any steady d r i f t i n g a i n of t h e system and a l s o the s t a t i s t i c a l e r r o r of each p o i n t .  averages  decreases  T y p i c a l l y measurements  a r e made f o r about 30 d i f f e r e n t v a l u e s of t . There a r e s e v e r a l advantages of t h i s method. one  i s t h a t t h e apparatus  The f i r s t  does n o t have t o be tuned f o r e x a c t  r e s o n a n c e , n o r does i t have t o s t a y e x a c t l y on resonance f o r the d u r a t i o n o f t h e measurement. very s e n s i t i v e to frequency,  A phase s e n s i t i v e system i s  o r magnetic f i e l d d r i f t s of one  t e n t h of t h e l i n e w i d t h or more, so the l a t t e r c o n d i t i o n i s q u i t e a s t r i n g e n t one t o f u l f i l . sweeping through  The second advantage i s t h a t  t h e l i n e e l i m i n a t e s any e r r o r s f r o m b a s e l i n e  droop, or d i s t o r t i o n , as any d i s t o r t i o n i s common t o both the s i g n a l and t h e o f f resonance b a s e l i n e i t s a m p l i t u d e  i s measured  from. The main d i s a d v a n t a g e  of t h e method i s t h a t i t takes  about t h r e e hours t o measure T| , as compared t o about h a l f an hour by more c o n v e n t i o n a l methods. i n g a i n of t h e apparatus  T h i s means t h a t t h e d r i f t  must be s m a l l , or a t l e a s t a c o n s t a n t  d r i f t i n the same d i r e c t i o n .  T h i s was u s u a l l y the c a s e , b u t  sometimes t h e r e would be sudden jumps i n g a i n c a u s i n g some  87 e r r o r i n the f i n a l  v a l u e of T , .  The r e s u l t s were e i t h e r a n a l y s e d by means o f a convent i o n a l l o g p l o t , or by a l e a s t squares f i t u s i n g t h e U.B.C. Computing C e n t r e I.B.M. 70>+0 computer.  The e r r o r s quoted f o r  each r e s u l t a r e s t a n d a r d d e v i a t i o n s e s t i m a t e d from t h e s c a t t e r , p l u s the 2% e r r o r i n t i m i n g .  Much of the n o i s e i n these  e x p e r i m e n t s i s from non-random s o u r c e s such as machinery s w i t c h i n g on and hence t h e e r r o r s do n o t obey a normal bution.  distri-  The e r r o r s should thus o n l y be r e g a r d e d as an  i n d i c a t i o n of how r e l i a b l e each r e s u l t i s . 3• 1*+  Measurement of S p i n - S p i n R e l a x a t i o n Times In aluminium powders T  a  was measured by a p p l y i n g a  s h o r t r f p u l s e and l i n e a r l y sweeping i n time a narrow boxcar gate about ^us. wide t h r o u g h t h e i n d u c t i o n t a i l measurements were made w i t h H  0  ( 6 ) . The  w e l l o f f r e s o n a n c e , so t h a t t h e  c h a r t r e c o r d i n g i s s i m i l a r t o a damped s i n e wave. avoids the d i f f i c u l t y  Doing  this  o f k e e p i n g t>he a p p a r a t u s on e x a c t r e s -  onance and a l s o makes i t e a s i e r t o reduce t h e e f f e c t s of baseline d i s t o r t i o n . T h i s method c o u l d n o t be used on s i n g l e c r y s t a l s of the to  because  t h e i r poor S/N r a t i o and a l s o because c l o s e t o the r f p u l s e b a s e l i n e was b a d l y d i s t o r t e d .  I n s t e a d , a s i m i l a r method  t h a t used f o r measuring T, was used.  A s h o r t r f p u l s e was  a p p l i e d and a t a time t l a t e r , a narrow boxcar gate was swept through resonance by l i n e a r l y v a r y i n g t h e magnetic f i e l d .  The  gate was then manually s h i f t e d t o a d i f f e r e n t v a l u e of t and  88 the measurement r e p e a t e d . Fig.  *+.l.  T h i s g i v e s the s i g n a l s shown i n  I f t^> T , the peak t o peak a m p l i t u d e (AB or AC 4  F i g . f . l b ) i s p r o p o r t i o n a l t o 2M l  However i f t « T , a  0  t o a h i g h degree of a c c u r a c y .  the peak t o peak a m p l i t u d e i s l e s s than  because of the f i n i t e  on  v a l u e of H, .  by making sweeps through resonance  2M  0  T h i s can be c o r r e c t e d f o r at several s l i g h t l y  different  times and superimposing them t o get an a c c u r a t e measurement of the envelope  of the o s c i l l a t i o n s .  T h i s can then be used  to  c o r r e c t the peak t o peak a m p l i t u d e of the sweeps. 3«15  Measurement of A b s o r p t i o n and D i s p e r s i o n Modes A p u l s e d NMR  a p p a r a t u s w i t h phase s e n s i t i v e d e t e c t i o n  and a boxcar i n t e g r a t o r can g i v e r e c o r d e r t r a c e s e q u i v a l e n t t o the u n s a t u r a t e d a b s o r p t i o n and d i s p e r s i o n modes, X"(u>) and X'(w)  measured by steady s t a t e a p p a r a t u s .  method, as developed  The b a s i s of the  by C l a r k ( 2 ) , w i l l be g i v e n h e r e , w h i l e  the m a t h e m a t i c a l d e s c r i p t i o n and occur a r e g i v e n i n Appendix  instrumental distortions that  IV.  A s h o r t r f p u l s e i s a p p l i e d to the sample.  A v e r y wide  boxcar gate which c o m p l e t e l y c o v e r s the whole of the f r e e i n d u c t i o n decay i s used.  The output of the boxcar can be  shown t o be a l i n e a r c o m b i n a t i o n of X ' and X".  By a p p r o p r i a t e  c h o i c e of the r e f e r e n c e phase e i t h e r X' or X " can be o b t a i n e d . I f the magnetic f i e l d  i s now  swept l i n e a r l y through the  resonance v a l u e r e c o r d i n g s are o b t a i n e d which are e q u i v a l e n t to  those o b t a i n e d by steady s t a t e a p p a r a t u s .  T h i s type of  measurement i s e a s i l y done on the p r e s e n t a p p a r a t u s , but  89 s u f f e r s from t h e d i s a d v a n t a g e t h a t T  ft  i n m e t a l s i s so s h o r t  t h a t t h e r e i s c o n s i d e r a b l e i n s t r u m e n t a l d i s t o r t i o n (Appendix I V ) . I t does have t h e advantage t h a t X ' a n d X " can be separated.  unambiguously  T h i s has n o t been p o s s i b l e i n any of t h e steady  s t a t e measurements on s i n g l e c r y s t a l s which have a l l used marginal oscillators.  To o b t a i n t h e f u l l b e n e f i t s of t h i s  advantage over the steady s t a t e method i t i s a l s o n e c e s s a r y t o s i m u l t a n e o u s l y a c c u r a t e l y measure t h e magnetic f i e l d . taneous measurement of t h e magnetic  A simul-  f i e l d u s i n g a s i m p l e mar-  g i n a l o s c i l l a t o r was t r i e d , b u t f a i l e d because t h e r e was mutual p i c k u p between t h e m a r g i n a l o s c i l l a t o r and t h e p u l s e d NMR apparatus.  P o s s i b l y c o m p l e t e l y s h i e l d i n g b o t h c o i l systems  would e l i m i n a t e t h i s 3.16  problem.  P o s s i b l e Improvements t o the Apparatus As i t stands a t p r e s e n t t h e a p p a r a t u s i s n o t as good as  i t should be f o r measuring T, a t room and l i q u i d n i t r o g e n temperatures f o r the f o l l o w i n g r e a s o n s . (i)  The o r i g i n a l i d e a was t o measure t h e a n i s o t r o p y i n T, a t  v e r y low magnetic f i e l d s near t h e s u p e r c o n d u c t i n g t h r e s h o l d . Thus t h e a p p a r a t u s was o r i g i n a l l y designed t o work a t 750 K c / s . However, a f t e r q u i t e a few months t h i s i d e a was abandoned, a t l e a s t t e m p o r a r i l y , as t h e e x p e r i m e n t a l d i f f i c u l t i e s were too great. of  The a p p a r a t u s was then c o n v e r t e d t o work i n t h e r e g i o n  5 t o lOMc/s.  There a r e however s t i l l  some remnants of t h i s  i n i t i a l stage of development i n some p a r t s o f t h e a p p a r a t u s , n o t a b l y t h e o v e r l y e l a b o r a t e gated power a m p l i f i e r  circuit.  90 T h i s does n o t d e t r a c t v e r y much f r o m the a p p a r a t u s ' s but does decrease  its reliability.  ( i i ) The dewars and  sample h o l d e r s were b u i l t f o r the 6  so they are s m a l l e r than i s n e c e s s a r y f o r the 1 2 a l s o have no e l e c t r o m a g n e t i c ( i i i ) The  performance  w  M  magnet  magnet and  shielding.  samples a v a i l a b l e are of a s s o r t e d s i z e s and a l s o many  of the f a c t o r s i n v o l v e d had  to be found  out by  experiment.  Thus the c o i l s a r e u s u a l l y n o t the optimum d e s i g n . From these c o n s i d e r a t i o n s , and a l s o some o t h e r p o i n t s , i t i s c l e a r t h a t t h e r e a r e two major ways i n w h i c h the appara t u s can be improved. The most i m p o r t a n t improvement i s t o r e b u i l d  the  t r a n s m i t t e r so t h a t i t i s s i m p l e r and can g i v e more power i n t o a lower impedance l o a d .  T h i s would enable a l o w e r Q t r a n s -  m i t t e r c o i l t o be used, w h i l e a l s o g e t t i n g a l a r g e r H,. p r e s e n t a 90°  p u l s e i s about l ^ n s l o n g .  At  A more p o w e r f u l  t r a n s m i t t e r c o u l d reduce t h i s to about 5/Ws  and a lower Q; c o i l  c o u l d reduce the r e c o v e r y time by about 5jus  a  W i t h the v e r y  s h o r t v a l u e s of T^ o c c u r r i n g i n m e t a l s these improvements c o u l d e a s i l y i n c r e a s e the S/N The  r a t i o by  50$.  second improvement would be to b u i l d a m e t a l  dewar system and a sample h o l d e r s p e c i f i c a l l y f o r use w i t h the 12" magnet a t l i q u i d n i t r o g e n , or room T h i s would enable l a r g e r diameter  temperatures.  samples t o be used.  More  m e c h a n i c a l - r i g i d i t y c o u l d be b u i l t i n t o the e l e c t r i c a l l e a d s to the sample h o l d e r and a l s o to the sample h o l d e r  91 itself.  A l l these f e a t u r e s would reduce the e f f e c t of a c o u s t i c  o s c i l l a t i o n s , and  i n c i d e n t l y n o i s e caused by b u b b l i n g  of  liquid nitrogen.  There would a l s o be room t o c o m p l e t e l y  the sur-  round the c o i l system by a m e t a l s h i e l d to reduce r f p i c k u p from e x t e r n a l sources without i n i t s e l e c t r i c a l performance. standardize  causing  significant deterioration  I t would a l s o be d e s i r a b l e t o  the sample s i z e s , but u n f o r t u n a t e l y t h e r e i s o f t e n  no c h o i c e i n the s i z e t h a t samples are grown i n . There are no s i g n i f i c a n t improvements w h i c h can be made to the a m p l i f i e r and  r e c o r d i n g system.  Measurements and  c u l a t i o n s b o t h show t h a t i t i s a l r e a d y p e r f o r m i n g ^possible noise  level.  cal-  a t the minimum  Reference Signal. . ) —  Oscillator and G-afed Power  iskv. r.f Pulse  ?hase  Staffer and Attenuator.  Induced r-f. Signal  Tuned  >  Prea rr\ pi \f\&n  Amplifier.  Grat'ini  "Pulse.  Arenberg WA600D  Amplifier-  Quenching Pulse.  Timing Unit. Bo/car  6ratinq Pulse^ :  :  External Trigger Pulse.  Oscilloscope.  D.C. Output Event Marker Pulse.  )  Figure 3.1. B l o c k D i a g r a m of t h e  —  Apparatus.  Varian Chart Recorder.  Tektronix  Free Running  163 Pulse  Multivibrator.  T o External Trigger of  Grene rotor.  the Oscilloscope. Pulse Mixer  Pulse i .  To  and Pulse a.  Tektronix. 162  ° Amplifier.  Amplt-fi'er. Tektronix  5aw+ooih  163  Grenerator.  Grated ?o«/er  0T0 Preamplifier Quench.  -)  Pwlse  Grenerator.  Eytemat  Normal  Tektronix  Tektronix Slow Sawtooth  l6a  Generator-  Grenerator.  Grenerator. External  Tektronix l6x  Sawtooth  Gtenerqtor  (To Boxcar Gratfe*  161 Pulse  SqWtooth  7^  Comcjdehce Tirwmcj Unit  Figure 3 . 2 . Block Diagram o f The "Timing Unit.  Normal  To Maqnet S—  0  Field Sweep. To Recorder Event Marker.  10  !>0  3.0  Ho*  (kiloejawss*)  SO  —1—  no  Figure a s Variation of the Acoustic Oscillation Amplitude With Magnetic Field Strength.  95 Multivibrator Triggering Pulse Sawtooth from T e k t r o n i x 162 G e n e r a t o r D i s c r i m i n a t o r l e v e l f o r P u l s e rt-  P u l s e 1 (from T e k t r o n i x 163)  Timing Sawtooth f o r r f a m p l i f i e r gating pulses  P u l s e 2 (from T e k t r o n i x 163)  D i s c r i m i n a t o r leve]>f-Qr boxcar  gate  Timing Sawtooth from T e k t r o n i x 162 G e n e r a t o r  Boxcar g a t i n g p u l s e s f r o m T e k t r o n i x 161 p u l s e  generator  Complete P u l s e Sequence as Seen on M o n i t o r Oscilloscope  F i g . 3.6  Diagram o f t h e Most Commonly Used Two P u l s e Sequence  96 CHAPTER IV THE EXPERIMENTAL RESULTS 'To o b s e r v a t i o n s w h i c h o u r s e l v e s we make, We grow more p a r t i a l f o r the o b s e r v e r ' s sake.' - Pope. A l t h o u g h t h e main aim of t h i s work was t o s e a r c h f o r a n i s o t r o p i c 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 s , a secondary aim was to determine t h e p o s s i b l e uses and l i m i t a t i o n s of p u l s e d NMR i n metal s i n g l e c r y s t a l s .  T h i s p a r t of t h e work v e r i f i e d t h e  t h e o r y of the a p p a r a t u s developed i n t h e p r e c e e d i n g c h a p t e r . S p i n echoes were a l s o observed and t h e i r p r o p e r t i e s s t u d i e d . S p i n - l a t t i c e r e l a x a t i o n measurements were attempted i n a number o f m e t a l s .  Some of these were s e l e c t e d f o r d e f i n i t e  r e a s o n s , but most were o n l y t r i e d because they were a v a i l a b l e . T h i s random approach t o t h e s e l e c t i o n o f samples was m a i n l y because many m e t a l s cannot be grown i n c o n v e n i e n t l y s i z e d c r y s t a l s , e x c e p t a t a p r o h i b i t i v e c o s t , so t h a t one had t o use whatever samples were r e a d i l y a v a i l a b l e .  None of the m e t a l s  w i t h l a r g e quadrupole i n t e r a c t i o n s had d e t e c t a b l e s i g n a l s , b u t f o u r o t h e r m e t a l s gave good enough s i g n a l s f o r T, measurements •to be made. tin.  These were aluminium, vanadium, n i o b i u m and w h i t e  An upper l i m i t was p l a c e d on t h e T, a n l s o t r o p i e s i n  vanadium and t i n .  S p i n - s p i n r e l a x a t i o n measurements were a l s o  made i n t i n and t h e s e gave the s t r e n g t h s of the p s e u d o - d i p o l a r and pseudo-exchange i n t e r a c t i o n s .  97 Throughout t h i s work the i n t e n t i o n was t o use a c r y s t a l f o r the main s e a r c h f o r T, a n i s o t r o p y .  scandium  T h i s i s because  i t i s a t r a n s i t i o n metal w i t h a non-cubic l a t t i c e , h a s a l a r g e o r b i t a l c o n t r i b u t i o n t o T, , and a s m a l l quadrupole  interaction.  These f e a t u r e s made i t an e x c e l l e n t c a n d i d a t e f o r t h i s s e a r c h . U n f o r t u n a t e l y , the f i r m w h i c h agreed t o s u p p l y the c r y s t a l were unable t o grow bne a f t e r e i g h t a t t e m p t s so t h a t t h i s i d e a had to be abandoned. h.l  Aluminium S i n g l e C r y s t a l S i g n a l s were observed a t a f r e q u e n c y of 7Mc/s. a t b o t h  room and 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 s .  The S/N r a t i o  u s u a l l y about 20 when a boxcar i n t e g r a t o r was used.  was This  a l l o w e d f a i r l y a c c u r a t e measurements of T, t o be made.  T  a  was too s h o r t to measure. I n the T» measurements the f i r s t p u l s e was 300^is l o n g and the second was 20^is l o n g .  The 60/as wide boxcar gate  s t a r t e d ^-Oyuis a f t e r the b e g i n n i n g of the second p u l s e and used a time c o n s t a n t of 1ms.  The r e p e t i t i o n r a t e was (35ms.)"  1  f o r the room t e m p e r a t u r e , and (90ms.) measurements.  f o r the 7°  K.  On the t r a c e of the sweep through resonance  the a m p l i t u d e s between the p o i n t s A,B and B,C were measured and then averaged.  (Fig.  ^-.1)  Choosing these p o i n t s ,  r a t h e r than the s i g n a l a m p l i t u d e from the b a s e l i n e EF, i n c r e a s e s the S/N r a t i o and e l i m i n a t e s the need t o sweep from a l o n g way  o f f resonance.  p l o t of t h i s a m p l i t u d e .  T, was t h e n got from a l o g  98  A t 295°K., T, T=(1.8 ± 0 . 3 ) s e c . d e g . w h i l e a t 78°K. T, T=(l.7-0.1)  sec.deg.  Combining these r e s u l t s g i v e s T,T=  (1.7*0.1)sec.cleg, over t h e temperature  range 78°K. t o 2 9 5 ° K .  T h i s agrees w e l l w i t h T,T=(l.8Q±0.05)sec.deg. powder from l . 2 ° K . t o 930°K.  obtained f o r a  (15).  The e x p e r i m e n t a l v a l u e of T, T i s about 20$ l o n g e r  than  the v a l u e p r e d i c t e d from t h e e x p e r i m e n t a l K n i g h t s h i f t and t h e K o r r i n g a r e l a t i o n , b u t agrees w e l l w i t h t h e v a l u e  calculated  u s i n g the K o r r i n g a r e l a t i o n modified t o take e l e c t r o n corr e l a t i o n s i n t o account  (1).  The l e n g t h o f t h e f i r s t p u l s e was v a r i e d from l50jus. s. w i t h o u t any n o t i c e a b l e e f f e c t on t h e a m p l i t u d e i n d u c t i o n decay a f t e r t h e second p u l s e .  to  of the  A T, measurement  t a k e n w i t h a second p u l s e l ^ t j s . l o n g gave t h e same v a l u e as the e a r l i e r measurements w i t h a 20yHs. p u l s e .  On t h e b a s i s of  the t h e o r y g i v e n i n t h e l a s t c h a p t e r , t h i s l a c k of s e n s i t i v i t y of t h e r e s u l t s t o t h e p u l s e l e n g t h s was e x p e c t e d . Aluminium has a c u b i c l a t t i c e , a n e a r l y s p h e r i c a l s u r f a c e , and a dominant c o n t a c t i n t e r a c t i o n .  Fermi  Anisotropy i n  T, T i s thus v e r y u n l i k e l y and was n o t l o o k e d f o r . *+.2  Vanadium S i n g l e C r y s t a l A s e r i e s of measurements were made on V  and a t 78°K. and  5 1  a t b o t h 295°K.  These gave v a l u e s of ( 0 . 7 9 0 . 0 3 ) s e c . d e g . a t 295°K. ±  ( 0 . 7 8 - 0 . 0 2 ) s e c . d e g . a t 78°K. w h i c h a r e i n e x c e l l e n t agree-  ment w i t h t h e v a l u e o f (0.788*0.007) sec.deg. powders over t h e temperature  obtained f o r  range 20°K. t o 295°K. ( 2 0 ) .  99 Because of the c u b i c l a t t i c e T, T was a n i s o t r o p i c (21).  However the measurements a t 78°K. were  w i t h s e v e r a l d i f f e r e n t magnetic f i e l d t r o p y was  not expected t o be  detected  orientations.  No  taken  aniso-  i n measurements made w i t h e r r o r s of ±3$«  Ta. c o u l d not be measured, but seemed to be s h o r t e r than t h a t of aluminium. I f the e x p e r i m e n t a l  v a l u e of T, T i s used i n the  r e l a t i o n , i t g i v e s a K n i g h t s h i f t of 0.21$ experimental  v a l u e of 0.56$.  i n s t e a d of  This discrepancy  be e x p l a i n e d by many-body e f f e c t s .  Korringa the  i s too l a r g e to  The r e a s o n f o r i t becomes  c l e a r when the e l e c t r o n i c s t r u c t u r e of vanadium i s s t u d i e d i n detail. The f o l l o w i n g d e s c r i p t i o n of the e l e c t r o n i c s t r u c t u r e of t r a n s i t i o n m e t a l s i s based on an a r t i c l e by Mott The  conduction  (23).  band i s b e l i e v e d t o c o n s i s t of a narrow d band  w i t h a h i g h d e n s i t y of s t a t e s o v e r l a p p i n g an s band w i t h a low d e n s i t y of s t a t e s .  The F e r m i energy l i e s i n the r e g i o n where  the bands o v e r l a p .  The  s band i s u s u a l l y d e s c r i b e d i n terms  of n e a r l y f r e e e l e c t r o n B l o c h f u n c t i o n s , w h i l e the d band i s much more l o c a l i s e d and approximation. separate  so i s d e s c r i b e d by the t i g h t b i n d i n g  However, i t i s i m p o s s i b l e i n p r i n c i p l e to  the d e n s i t y of s t a t e s i n t o independent bands d e r i v e d  w h o l l y from s, p, or d f u n c t i o n s , even i n the t i g h t b i n d i n g approximation.  The m i x i n g of s t a t e s ( h y b r i d i z a t i o n ) w h i c h  o c c u r s can d r a s t i c a l l y a l t e r some p r o p e r t i e s of the metals.  The most i m p o r t a n t  transition  e f f e c t of h y b r i d i z a t i o n of the d  100 •wave f u n c t i o n s i s t o i n t r o d u c e a deep minimum i n the m i d d l e of the d e n s i t y of s t a t e s c u r v e f o r b.c.c. l a t t i c e s , f.c.c. lattices.  but n o t f o r  H y b r i d i e a t i o n o f the s and d bands does n o t  g r e a t l y a l t e r the d e n s i t y of s t a t e s c u r v e , b u t a f f e c t s o t h e r p r o p e r t i e s i n a manner w h i c h i s n o t c l e a r l y understood a t p r e sent.  I n most m e t a l s the s i t u a t i o n i s c o m p l i c a t e d  by the s  band c o n t a i n i n g a c e r t a i n amount of p, or h i g h e r , wave f u n c t i o n s as w e l l .  Due t o t h e i r coulomb r e p u l s i o n t h e r e a r e l a r g e  cor-  r e l a t i o n e f f e c t s between e l e c t r o n s i n the s and d bands whose r o l e i s unknown.  The s p i n - o r b i t i n t e r a c t i o n causes s m a l l  energy s h i f t s w h i c h a r e u s u a l l y Because of the d i f f i c u l t y  ignored. of t r e a t i n g h y b r i d i z a t i o n and  c o r r e l a t i o n e f f e c t s , they a r e u s u a l l y n e g l e c t e d  and the assump-  t i o n made t h a t t h e s and d bands can be t r e a t e d  independently.  T h i s i s c a l l e d the r i g i d band model. Vanadium l i e s i n the f i r s t l o n g t r a n s i t i o n p e r i o d and has f i v e e l e c t r o n s o u t s i d e the f i l l e d c u r v e has been e x p e r i m e n t a l l y  core.  determined and shows t h e b a s i c  f e a t u r e s of a ^s band c o n t a i n i n g about 0.5 and  I t s d e n s i t y of s t a t e s  a much narrower 3<3 band c o n t a i n i n g h,5  electrons/atom; electrons/atom  (25).'  T h i s h i g h d e n s i t y of d e l e c t r o n s i s the r e a s o n t h a t the K o r r i n g a r e l a t i o n does n o t h o l d . U s i n g the r i g i d band model, the K n i g h t s h i f t and 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 s i n vanadium powders have been t h o r oughly examined, both e x p e r i m e n t a l l y (27).  Of-necessity  (25)  and t h e o r e t i c a l l y  t h e r e a r e a number o f unknown f a c t o r s and  101 gross assumptions  i n the a n a l y s i s of the r e s u l t s so t h a t the  conclusions are only q u a l i t a t i v e .  Because of the low d e n s i t y  of s t a t e s i n the ks band, the c o n t a c t term p l a y s a minor r o l e . The dominant c o n t r i b u t i o n t o the K n i g h t s h i f t i s f r o m ' o r b i t a l paramagnetism, w i t h a secondary sation.  c o n t r i b u t i o n from core  The c o n t a c t term p r o v i d e s about 10$ of the  relaxation.  polari-  spin-lattice  The r e s t of the r e l a x a t i o n i s by means of o r b i t a l  and c o r e p o l a r i s a t i o n .  At the moment i t i s i m p o s s i b l e t o  d e c i d e which of these terms i s the l a r g e r , but i t i s p r o b a b l y the o r b i t a l term (25,27).  T h i s i s supported  by measurements on  s u p e r c o n d u c t i n g vanadium which show t h a t the K n i g h t s h i f t i s due t o a s p i n independent  term (28,25).  However some c a u t i o n  should be used i n the i n t e r p r e t a t i o n of t h i s type of  experiment  s i n c e the b e h a v i o r of the K n i g h t s h i f t i n some n o n - t r a n s i t i o n m e t a l s d i f f e r s from t h a t p r e d i c t e d on the b a s i s of the theory.  BCS  The most l i k e l y e x p l a n a t i o n f o r t h i s d e v i a t i o n i n -  v o l v e s s p i n - o r b i t c o u p l i n g and the sample s u r f a c e ( 6 1 ) .  s c a t t e r i n g of e l e c t r o n s from  I t i s n o t known t o what e x t e n t  these e f f e c t s occur i n s u p e r c o n d u c t i n g t r a n s i t i o n m e t a l s . The e x p e r i m e n t a l v a l u e of T T i s t w i c e the c a l c u l a t e d (  v a l u e (27).  B u t t e r w o r t h (29) showed t h a t t h i s d i f f e r e n c e  u n l i k e l y t o be caused parameters.  by e r r o r s i n c h o o s i n g the band  was  structure  The most p r o b a b l e r e a s o n i s t h a t the c a l c u l a t e d  r e l a x a t i o n time uses a d e n s i t y of s t a t e s d e r i v e d from e l e c t r o n i c s p e c i f i c heat (27,29).  the  This includes a c o n t r i b u t i o n  from e l e c t r o n - e l e c t r o n and e l e c t r o n - p h o n o n  interactions  which  102 do n o t c o n t r i b u t e t o r e l a x a t i o n C+6). r e l a x a t i o n time would be t o o s h o r t .  Thus t h e c a l c u l a t e d  Even i n c o m p a r a t i v e l y  s i m p l e m e t a l s t h e e l e c t r o n - p h o n o n i n t e r a c t i o n c a n double t h e e l e c t r o n i c s p e c i f i c heat (57) and so i s l a r g e enough t o e x p l a i n the d i f f e r e n c e between the e x p e r i m e n t a l and t h e t h e o r e t i c a l values,  s-d h y b r i d i z a t i o n e f f e c t s might a l s o c o n t r i b u t e t o  the d i f f e r e n c e . Niobium S i n g l e C r y s t a l Niobium I s a t r a n s i t i o n m e t a l w i t h a c u b i c l a t t i c e and e l e c t r o n i c and m e c h a n i c a l p r o p e r t i e s s i m i l a r t o those of vanadium.  I t a l s o has f i v e e l e c t r o n s o u t s i d e a f i l l e d  core,  but l i e s i n the second l o n g t r a n s i t i o n p e r i o d . A t room "jtemperature T, T was found t o be (0.3^-0.01)sec. deg. and a t 78°K. was (0.31-0.01) sec. deg.  These v a l u e s a r e  the average o f two measurements a t each t e m p e r a t u r e . The S/N  r a t i o was about 15 a t b o t h t e m p e r a t u r e s .  Acoustic o s c i l -  l a t i o n s caused some t r o u b l e 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 s . These v a l u e s d i s a g r e e w i t h the 0.19 sec.deg. measured by Asayama and I t o h i n the r e g i o n 2*K. t o 77 K. (58), b u t P  agree moderately w e l l w i t h the v a l u e o f (0.36*0.01)sec.deg. o b t a i n e d by B u t t e r w o r t h f o r t h e temperature range 20° K. t o 290°K. (29), effect.  He found t h a t i m p u r i t i e s d i d n o t have a s t r o n g  A powder sample contaminated by 1$ oxygen, 0.2$  hydrogen, and 0.08$ n i t r o g e n had a T T o n l y 10$ below t h a t t  of a v e r y pure f o i l sample.  The sample used by Aszyama  c o n t a i n e d 0.5$ of m e t a l l i c i m p u r i t i e s , as w e l l as t h e gaseous  103 i m p u r i t i e s , w h i c h were p r o b a b l y r e s p o n s i b l e f o r the l a r g e i n c r e a s e i n the r e l a x a t i o n r a t e .  Quadrupole e f f e c t s might a l s o  be i m p o r t a n t , a l t h o u g h B u t t e r w o r t h found  t h a t a n n e a l i n g a pow-  der sample made no d i f f e r e n c e t o e i t h e r the s i g n a l or to T i T .  intensity,  The d i f f e r e n c e between the p r e s e n t measurement and  t h a t of B u t t e r w o r t h i s h o t due t o a s y s t e m a t i c e r r o r s i n c e the v a l u e s measured f o r vanadium agreed error.  to w i t h i n experimental  I t i s a l s o u n l i k e l y to be o n l y a s t a t i s t i c a l  The n i o b i u m about 0.1$  variation*  sample used i n the p r e s e n t measurement c o n t a i n s of m e t a l l i c i m p u r i t i e s and n e g l i g i b l e gaseous  i m p u r i t i e s (Appendix c o u l d thus be due  II).  The d i f f e r e n c e between the r e s u l t s  to i m p u r i t i e s .  The  small  temperature  dependence of T, T s u p p o r t s t h i s , a l t h o u g h t h i s might be to h,k  statistical  due  fluctuations.  M e t a l s W i t h Large Quadrupole I n t e r a c t i o n s Measurements were a l s o attempted  w i t h l a r g e quadrupole a r e w e l l separated  interactions.  on a number of  metals  I n these m e t a l s the l i n e s  so t h a t c r o s s r e l a x a t i o n should be by  non-  s e c u l a r terms o n l y and hence of about the same magnitude as s p i n - l a t t i c e r e l a x a t i o n , or weaker. v a l u e i s about 2ms. than T , a  i n technetium  The  (60).  only  experimental  T h i s i s much l o n g e r  but i s c o n s i d e r a b l y s h o r t e r than T,.  However i t  seems s a f e t o assume t h a t the system has a f i c t i t i o u s of £ f o r a t l e a s t the i n i t i a l p a r t of the decay. t h a t the s i g n a l s should be q u i t e weak.  spin  T h i s means  I f e i t h e r p s e u d o - d i p o l a r , o r pseudo-exchange e f f e c t s o c c u r , T j w i l l be v e r y s h o r t .  T h i s decrease i n  becomes v e r y impor-  t a n t f o r metals w i t h atomic w e i g h t s o f about 100 o r more. of  Most  t h e metals s t u d i e d were i n t h i s r e g i o n . The l a r g e q u a d r u p o l a r i n t e r a c t i o n makes t h i s c l a s s o f  metals v e r y hard t o study u s i n g steady s t a t e NMR a p p a r a t u s and powdered samples so t h a t t h e r e have been v e r y few measurements made on t h i s c l a s s o f m e t a l s .  I t was thus c o n s i d e r e d w o r t h w h i l e  spending some time s e a r c h i n g f o r s i g n a l s i n them, even though the s h o r t T to (i)  a  and f i c t i t i o u s s p i n of £ would make them v e r y hard  find, Indium. A t a f r e q u e n c y o f 6Mc/s. a s e a r c h f o r a s i g n a l was made  from 3.0 KG. t o 6.5 KG. a t b o t h 295°K. and 78°K.  Searches were  made w i t h t h e magnetic f i e l d b o t h p a r a l l e l and p e r p e n d i c u l a r t o the c r y s t a l a x i s of symmetry.  A t 9Mc/s. sweeps were made from  *U0 KG. t o 11.2 KG. a t b o t h 78°K. and 295°K.  I n t h i s case t h e  magnetic f i e l d was p a r a l l e l t o t h e a x i s o f symmetry.  The r e p e -  t i t i o n r a t e was such t h a t s i g n a l s w i t h ^ T ^ I O sec.deg. should have been seen.  The room temperature  s i g n a l has been seen  w i t h steady s t a t e apparatus i n a powder (32),  so t h a t i t was  known t h a t the r i g h t r e g i o n was b e i n g searched.  Acoustic  o s c i l l a t i o n s gave o n l y minor t r o u b l e and v a n i s h e d when t h e sample was immersed i n g l y c e r i n e . S i g n a l s were n o t seen, even though c a l c u l a t i o n s showed t h a t t h e r e was a r e a s o n a b l y good chance o f s e e i n g them (Chp.^.?).  105 There a r e t h r e e p o s s i b l e e x p l a n a t i o n s f o r the f a i l u r e to see a s i g n a l .  The f i r s t of these i s t h a t T, T i s l o n g e r  than about lOsec.deg.  This i s not very l i k e l y since l i t h i u m  i s t h e o n l y metal known t o have T| T l o n g e r than 5sec.deg. and a l l n u c l e a r - c o n d u c t i o n e l e c t r o n i n t e r a c t i o n s become s t r o n g e r w i t h i n c r e a s i n g atomic number because  of the i n c r e a s e i n  e l e c t r o n d e n s i t y near the n u c l e u s . Indium i s v e r y s o f t and even l i g h t p r e s s u r e can cause the s u r f a c e t o become p o l y c r y s t a l l i n e .  A n i s o t r o p i c thermal  e x p a n s i o n , or some i n a d v e r t a n t l y rough h a n d l i n g , c o u l d thus cause the s u r f a c e t o become p o l y c r y s t a l l i n e .  T h i s would  render the s a t e l l i t e l i n e s u n o b s e r v a b l e and reduce the i n t e n s i t y of the c e n t r a l l i n e by over 50$, making i t u n o b s e r v a b l e .  thus p o s s i b l y  However X - r a y s taken b e f o r e and  d u r i n g the measurements showed no s i g n of a p o l y c r y s t a l l i n e surface l a y e r . Because of the a m p l i f i e r r e c o v e r y time of about  l^s.,  a weak s i g n a l w i t h T* l e s s than about 20yvs. i s u n o b s e r v a b l e . Indium has an atomic number of 115  and so p r o b a b l y has pseudo-  exchange and p s e u d o - d i p o l a r i n t e r a c t i o n s so t h a t T bably quite short.  a  i s pro-  T h i s i s t h e most l i k e l y r e a s o n t h a t no  s i g n a l was seen, (ii)  Rhenium. , A s e a r c h was made from 5.0  KG. t o 11.2  KG. a t a f r e -  quency of 9Mc/s. a t b o t h room and 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 s . No s i g n a l s were seen.  T h i s was n o t unexpected  puted S/N r a t i o was c o n s i d e r a b l y l e s s than one.  s i n c e the comThere were  106 a l s o v e r y l a r g e a c o u s t i c o s c i l l a t i o n s , even a f t e r immersion i n The atomic number i s 185  glycerine.  so t h a t T* should be much  l e s s than t h a t due t o d i p o l a r i n t e r a c t i o n s a l o n e , (ill)  Bismuth. Bismuth i s an u n u s u a l m e t a l w i t h some n o n - m e t a l l i c p r o -  perties.  These a r i s e because i t has a v e r y s m a l l number of  f r e e e l e c t r o n s , which gives i t a high e l e c t r i c a l r e s i s t a n c e and an e x t r e m e l y l a r g e m a g n e t o - r e s i s t a n c e (9>33). A room temperature s e a r c h was made f r o m 8.2KG. t o 11.2KG. a t a f r e q u e n c y of 7Mc/s. (0.2sec.)"'  The r e p e t i t i o n r a t e was  and the boxcar gate s t a r t e d 20yws. a f t e r the b e g i n -  n i n g of the r f p u l s e .  A s i m i l a r s e a r c h a t 78°K. used a r e p e -  t i t i o n r a t e of (O.SSsec.)"* . d i f f e r e n t magnetic f i e l d  A t both temperatures  o r i e n t a t i o n s were t r i e d .  several Acoustic  o s c i l l a t i o n s were seen a t 78°K., but were n o t l a r g e enough to cause t r o u b l e .  Powder measurements had been made a t  h.2°K. (3^)j  so t h a t the approximate p o s i t i o n of the l i n e s  was known.  Under these c o n d i t i o n s any s i g n a l of r e a s o n a b l e  intensity with T ^ a  been seen.  l ^ u s . and T, T ^50  sec.deg. should have  However, t h e r e was no s i g n of a s i g n a l .  A c a l c u l a t i o n of the s i g n a l a m p l i t u d e showed t h a t i t should have been seen. T h i s c a l c u l a t i o n of  the m a g n e t o - r e s i s t a n c e .  i g n o r e d the e f f e c t s  The change i n t u n i n g c a p a c i t a n c e  r e q u i r e d by a p p l i c a t i o n of a 10KG. magnetic f i e l d at  showed t h a t  78°K. the m a g n e t o - r e s i s t a n c e a p p r o x i m a t e l y doubled the  s k i n depth.  T h i s i s i n rough agreement w i t h t h e measured  m a g n e t o r e s i s t a n c e of bismuth (33).  I t was o r i g i n a l l y hoped  t h a t t h e S/N r a t i o might be improved by t h e i n c r e a s e i n s k i n d e p t h caused by t h e m a g n e t o r e s i s t a n c e .  This i s n o t neces-  s a r i l y so. The S / N r a t i o i s p r o p o r t i o n a l t o QS and i f Q depends o n l y on.the sample t h e n Qoc 8cr, so t h a t t h e S / N G C D V . However ^oecr-" , so t h a t the S/N r a t i o i s independent of <r 1  and  hence does n o t depend on any m a g n e t o r e s i s t i v e  This i s probably  effects.  t h e case i n bismuth s i n c e on t u n i n g t h e  a p p a r a t u s i t was n o t i c e d t h a t t h e Q was lower than f o r any other sample and t h a t t h e Q i n c r e a s e d T h i s t u n i n g was done w i t h o u t  on going t o 7° K.  t h e magnetic f i e l d  f i n a l t u n i n g b e i n g done w i t h t h e magnetic f i e l d  on, o n l y t h e applied.  I n a l l t h e o t h e r m e t a l s t h e Q depended on t h e c i r c u i t tance and was a p p r o x i m a t e l y  temperature independent.  resisIf  the Q had been l i m i t e d by t h e c i r c u i t r e s i s t a n c e i n the case of bismuth as w e l l , t h e n an i n c r e a s e i n 8 due t o magnetor e s i s t a n c e would have i n c r e a s e d From steady  the S / N r a t i o .  s t a t e measurements L.C. Hebel found t h a t  bismuth had a l i n e about 80 gauss wide w h i c h s a t u r a t e d (quoted i n r e f e r e n c e 35)•  From  t h i s one can deduce t h a t  T ~10/>is. and t h a t T,T> 25sec«deg. a  easily  These a r e both q u i t e  p l a u s i b l e v a l u e s ; t h e l o n g T, T r e s u l t i n g from t h e s m a l l number of f r e e e l e c t r o n s / a t o m  and t h e s h o r t T  exchange and p s e u d o - d i p o l a r  interactions.  ft  t h a t these u n s u i t a b l e v a l u e s of T, T and T t h a t no s i g n a l s were seen.  from pseudoI t i s most l i k e l y  a  are the reason  108  (iv)  Antimony. Antimony i s a metal w i t h s i m i l a r c h a r a c t e r i s t i c s t o those  of b i s m u t h , so t h a t i t was n o t expected b r i e f s e a r c h was made a t 78°K. 6.5KG. t o 11.2KG.  t o see a s i g n a l .  a t a frequency  A  of 9Mc/s. f r o m  No l i n e s were seen.  (v) G a l l i u m . T h i s metal has the v e r y low m e l t i n g p o i n t of 303°K. try at  To  t o a v o i d m e l t i n g t h e c r y s t a l a l l measurements were made 78°K.  The sample was immersed i n g l y c e r i n e t o dampen the  large acoustic o s c i l l a t i o n s  present.  A s e a r c h was made a t 78°K. w h i c h would d e t e c t s i g n a l s w i t h T > 15/<S. and T, T ^ I O sec.deg.  The f r e q u e n c y was 9Mc/s.  a  and  t h e f i e l d was swept f r o m 5.1KG. t o 10.3KG.  found a t about 6.7KG. w i t h a S/N of about 3. independent o f t h e magnetic f i e l d found t h a t eddy c u r r e n t s generated melted was  orientation.  One l i n e was T h i s l i n e was I t was l a t e r  by t h e r f p u l s e s had  t h e s u r f a c e of the c r y s t a l so t h a t t h e observed  probably  t h a t from m o l t e n Ga  71  .  line  The observed l i n e agreed  r e a s o n a b l y w e l l w i t h t h i s i d e n t i f i c a t i o n as f a r as both t h e p o s i t i o n and t h e c a l c u l a t e d S/N r a t i o were c o n c e r n e d .  I t was  too weak t o make any measurements on. T h i s s u r p r i s i n g m e l t i n g of t h e c r y s t a l p r o b a b l y  occurred  because g l y c e r i n e has a v e r y low t h e r m a l c o n d u c t i v i t y . a s o l i d a t 78°K. and even when l i q u i d so t h a t l i t t l e  It is  i t has a h i g h v i s c o s i t y ,  convective c o o l i n g can occur.  There i s thus  a poor t h e r m a l c o n t a c t between t h e sample and t h e l i q u i d  n i t r o g e n b a t h and so t h e h e a t i n g e f f e c t of t h e r f p u l s e s i s cumulative.  A c a l c u l a t i o n showed t h a t a f t e r a few hours t h e  sample temperature would r i s e f r o m 78°K. t o i t s m e l t i n g p o i n t . This i s approximately i s allowed  to r u n f o r , f o r s t a b i l i z i n g purposes, before a  measurement i s made. low  thermal  the l e n g t h of time t h a t t h e a p p a r a t u s  The h e a t i n g p r o c e s s i s a l s o a i d e d by the  c o n d u c t i v i t y of g a l l i u m w h i c h a l l o w s t h e s u r f a c e  temperature t o r i s e about 10° K. above t h a t of t h e i n t e r i o r for  many m i l l i s e c o n d s . A f t e r d i s c o v e r i n g t h i s sample h e a t i n g e f f e c t , t h e use  of g l y c e r i n e t o dampen a c o u s t i c o s c i l l a t i o n s was d i s c o n t i n u e d . 5  Copper W i r e and Other S p u r i o u s S i g n a l Sources Two resonances due t o t h e copper w i r e i n t h e p i c k u p  c o i l were o f t e n observed.  They were i d e n t i f i e d by t h e i r  p o s i t i o n s , r e l a t i v e i n t e n s i t i e s , and s p i n - l a t t i c e r e l a x a t i o n time. Cu  63  The s i g n a l s were q u i t e s t r o n g ; t h e a m p l i t u d e of t h e resonance b e i n g 70% of t h a t of t h e A l  resonance.  a 7  single crystal  T h i s i s because t h e s u r f a c e a r e a of t h e w i r e i n  the c o i l i s n e a r l y the same as t h e s u r f a c e a r e a of a s i n g l e c r y s t a l sample.  The e f f i c i e n c y f a c t o r i s n e a r l y u n i t y , w h i l e  copper has a l a r g e magnetic moment and so a s t r o n g s i g n a l i s got. V e r y s t r o n g s i g n a l s c o u l d a l s o be p i c k e d up f r o m f l u o r i n e and hydrogen n u c l e i i n t h e t e f l o n and i n s u l a t i o n near the p i c k u p c o i l .  These s i g n a l s l i m i t e d low magnetic  field  sweeps because they o b l i t e r a t e d s i g n a l s over a r e g i o n o f many  110 hundred of gauss i n t h e r e g i o n o f 2KG. k,6  I s o t o p i c a l l y Pure T i n S i n g l e C r y s t a l T h i s i s a t h i n c r y s t a l of i s o t o p i c a l l y pure S n  around a copper core (59).  u<?  wrapped  The S/N r a t i o was q u i t e good a t  78° K., w h i l e T^was about 200yws.  These f e a t u r e s made i t a good  sample t o use f o r s t u d y i n g some of the e x p e r i m e n t a l  details,  ( i ) V a r i a t i o n o f the I n d u c t i o n T a i l H e i g h t W i t h t h e r f P u l s e Lengths. The f i r s t experiment was t o study t h e v a l i d i t y o f t h e e x p r e s s i o n d e r i v e d f o r t h e induced  v o l t a g e o c c u r r i n g a f t e r ap-  p l i c a t i o n of a r f p u l s e (Appendix I I I ) .  To do t h i s t h e i n d u c -  t i o n t a i l h e i g h t was measured as c l o s e t o t h e r f p u l s e as p o s s i b l e w i t h a narrow boxcar gate.  The measurements were then  repeated f o r v a r i o u s r f p u l s e w i d t h s .  The i n d u c t i o n t a i l  h e i g h t was c o r r e c t e d f o r s p i n - s p i n r e l a x a t i o n . t i o n used Tj, measured a t t h e same magnetic f i e l d  This correcorientation.  S p i n - S p i n r e l a x a t i o n o c c u r r i n g d u r i n g t h e r f p u l s e was c o r r e c t e d f o r by t a k i n g t h e time o r i g i n as t h e c e n t r e o f t h e r f p u l s e (13).  T h i s c o r r e c t i o n i s e x a c t f o r a u n i f o r m H, much  l a r g e r than t h e l o c a l f i e l d , b u t i s o n l y an approximate c o r r e c t i o n i n the present case. always l e s s than 30$, The  However, t h e c o r r e c t i o n i s  so t h a t t h e e r r o r w i l l n o t be l a r g e .  c o r r e c t e d i n d u c t i o n t a i l h e i g h t s were then p l o t t e d  a g a i n s t t h e r f p u l s e l e n g t h ( F i g . U-.^-). p r e s s i o n was f i t t e d  The t h e o r e t i c a l ex-  t o t h e e x p e r i m e n t a l r e s u l t s on t h e  a s s u m p t i o n t h a t a -kit p u l s e a t t h e s u r f a c e of t h e metal was  Ill lOyus. l o n g .  T h i s v a l u e i s i n r e a s o n a b l e agreement w i t h  Hi =(13*2) gauss obtained from a f l u o r i n e resonance i n the  coil  mounting i f the i n c r e a s e of H, a t the s u r f a c e of the sample i s considered  (Chap. 3.10).  The p u l s e w i d t h s f i t the d a t a q u i t e w e l l f o r p u l s e l e n g t h s up t o about 50yws. l o n g .  The  small d e v i a t i o n f o r short pulse  l e n g t h s i s because the r f p u l s e s a r e n o t q u i t e r e c t a n g u l a r . I t i s not c l e a r whether the d i s c r e p a n c y beyond 50yUs. i s due  to  e x p e r i m e n t a l causes such as h e a t i n g of the sample, o r r e p r e s e n t s a breakdown of the t h e o r y .  The most l i k e l y cause of t h i s would  be e f f e c t s f r o m the r e g i o n i n w h i c h H , i s comparable to the local field.  I t was  found  t h a t i n the t h e o r e t i c a l e x p r e s s i o n  the r a t i o of the maximum p o s i t i v e going and n e g a t i v e  going  a m p l i t u d e s were v e r y s e n s i t i v e t o the form of the phase f a c t o r s . I f the phase term i s c o s * x the r a t i o i s 0.*+5, but i f i t i s coaxcos(^)  the r a t i o i s 1.2.  However a p a r t from the a m p l i -  t u d e s , the shape of the curve i s r e l a t i v e l y i n s e n s i t i v e t o the phases.  The experiment  thus v e r i f i e s the b a s i c f e a t u r e s  of the t h e o r y , but i n d i c a t e s t h a t i t might be too s i m p l e .  A  d i s c u s s i o n of the q u a n t i t a t i v e agreement of the e q u a t i o n w i t h the e x p e r i m e n t a l S/N  r a t i o s i s g i v e n a t the end  of t h i s  chapter. ( i i ) The S p i n - L a t t i c e R e l a x a t i o n Time. The  symmetry a x i s ( [001] a x i s ) makes an a n g l e of 28°  w i t h the c y l i n d r i c a l a x i s of the copper c o r e . f r o m the optimum a n g l e of 90°,  This i s f a r  so t h a t t i l t i n g the c r y s t a l  112 through 23° was t r i e d .  T h i s s u f f e r e d from the d i s a d v a n t a g e s  of r e d u c i n g t h e induced  v o l t a g e and of d r a s t i c a l l y i n c r e a s i n g  the a c o u s t i c o s c i l l a t i o n s .  The l a t t e r were t h e i m p o r t a n t  dif-  f i c u l t y as they s a t u r a t e d the a m p l i f i e r f o r about lOOyus. Mounting t h e sample i n p o r c e l a i n cement reduced t h e o s c i l l a t i o n s t o a more manageable s i z e . a t 78 K., b u t t h e e x p e r i m e n t a l o s c i l l a t i o n s was over 20%.  T, measurements were made  s c a t t e r caused by a c o u s t i c  There was thus l i t t l e chance of  measuring any a n i s o t r o p y i n T, .  The most a c c u r a t e measurement  gave T, T=( f5 l5)ms.deg., w h i l e t h e s c a t t e r of r e s u l t s showed l  ±  t h a t t h e r e was no a n i s o t r o p y g r e a t e r than 50% a t the magnetic f i e l d o r i e n t a t i o n s used. Measurements were then made a t 78°K w i t h t h e c r y s t a l mounted v e r t i c a l l y .  T h i s reduced the a c o u s t i c o s c i l l a t i o n s t o  o n l y s e v e r a l times t h e t h e r m a l n o i s e l e v e l . mately 500yMS. w h i l e T  was about 20CyUs.  a  to s p o i l t h e magnetic f i e l d u n t i l T* was  I t was thus  30yUs.  90yWs. l o n g and t h e second one 20ytis. l o n g .  approximately  a l o n g t h e [100]  T, was a p p r o x i necessary  The f i r s t  pulse  When H was e  a x i s T, T=(35 2)ms.deg., w h i l e ±  when H„ was i n t h e (010) plane and making an angle of 62° w i t h the  [00l] a x i s T, T=(33 3)ms.deg.  These r e s u l t s a r e g i v e n on  ±  F i g u r e *+.3 and show no a n i s o t r o p y w i t h i n t h e e x p e r i m e n t a l error. The average v a l u e of T, T=(3 +- 2)ms.deg. agrees w i t h t h e l  ±  v a l u e of T, T=(3 +-l)ms. deg. found by Asayama and I t o h 1  (58)  over t h e temperature range !+.2 K. t o 120°K. f o r powders. I t 0  n  d i s a g r e e s w i t h t h e measurement of Spokas and S l i c h t e r (15) 78°K. of T, T=(5 +± Oms.deg. i n powders. 1  3  at  I t i s impossible to  l  t e l l w h i c h of these measurements i s t h e more r e l i a b l e .  Spokas  and S l i c h t e r made o n l y one measurement which was i n c i d e n t a l t o t h e i r main experiment w h i l s t Asayama and I t o h ' s measurement of T, T i n n i o b i u m , w h i c h was made a t t h e same t i m e , i s unreliable. The form of the a n g u l a r dependence of the o r b i t a l and d i p o l a r r e l a x a t i o n i s n o t known f o r a t e t r a g o n a l l a t t i c e . (63),  C a l c u l a t i o n s u s i n g the f r e e e l e c t r o n ^ a p p r o x i m a t i o n the t i g h t b i n d i n g a p p r o x i m a t i o n ,  and  show t h a t f o r a c u b i c  lattice  i t i s a sum of terms of the form a'+b'cos(J>' , where $ i s t h e 4  a n g l e between H  0  and the [OOl] a x i s .  for a cubic l a t t i c e .  T h i s sum i s i s o t r o p i c  I t i s p l a u s i b l e that f o r a non-cubic  l a t t i c e the sum i s of the form (T, T)"* =a+bcos "(|) . 4  To t h i s  must be added the i s o t r o p i c c o n t r i b u t i o n c due t o t h e c o n t a c t and  c o r e p o l a r i s a t i o n terms.  I f the measured v a l u e s of T, T  are f i t t e d t o the above e x p r e s s i o n they  give  b =(1.5*3.0 )xlO" ms.* deg.' 3  However, b i s n o t expected about 70$  1  1  t o be n e g a t i v e , so t h a t i t has  p r o b a b i l i t y of b e i n g i n the range 0 t o  5x10 msl' deg," 3  From the p r e s e n t measurements i t has n o t been p o s s i b l e to d e t e c t any a n i s o t r o p y .  However, the r e q u i r e m e n t s  for a  u s e f u l measurement of any a n i s o t r o p y a r e now much c l e a r e r . any NMR experiment i t i s v e r y hard t o measure T, w i t h b e t t e r  In  ll*f than 2% a c c u r a c y and i n the p r e s e n t case 5$ a c c u r a c y b e s t t h a t can be o b t a i n e d .  i s the  The l e n g t h o f time r e q u i r e d f o r  such a c c u r a t e measurements make i t i m p r a c t i c a l t o measure T, a t more than a few a n g l e s .  The a n g l e s a t which T, must be  measured a r e f o r Ho a l o n g t h e [001] axis.  a x i s and a l o n g t h e [lOOj  S i n c e t h e assumed a n g u l a r v a r i a t i o n o f T, may be  wrong, measurements should a l s o be made w i t h H a l o n g t h e 0  [lio]  and t h e [lO]] axes as a check.  W i t h t h e assumed e r r o r  of 5$ i n T, an a n i s o t r o p y of W s . d e g . would be d e t e c t a b l e . An upper l i m i t c a n be p l a c e d on a by u s i n g t h e f r e e electron Korringa r e l a t i o n .  A r e c e n t c a l c u l a t i o n on t h e K n i g h t  t i n (61)  s h i f t i n superconducting  the i s o t r o p i c v a l u e of 0.713$ (6^) Van V l e c k paramagnetism.  i n d i c a t e s t h a t about 15$ of i s due t o s p i n - o r b i t and  Thus t h e r e l a x a t i o n due t o t h e con-  t r a c t term i s about 2xl0" ms.~'degT* a  T h i s g i v e s a ^SxlO'^msT degl* 1  I f t h e a n i s o t r o p i c r e l a x a t i o n i s due t o o n l y one mechanism, t h e r a t i o b/a depends o n l y on t h e symmetry of t h e e l e c t r o n wave f u n c t i o n s a t t h e F e r m i s u r f a c e . parameter o f c o n s i d e r a b l e t h e o r e t i c a l  I t i s thus a  importance,  ( i i i ) S p i n - S p i n R e l a x a t i o n Times., T  a  was measured from t h e f r e e i n d u c t i o n decay by t h e  method d e s c r i b e d i n Chapter 3.1*+»  Considerable  taken t h a t t h e resonance was n o t swept through  c a r e was so q u i c k l y t h a t  the o s c i l l a t i o n s were d i s t o r t e d by more than a few p e r c e n t . Timing was done by means of marker p i p s w i t h lOOyus. s e p a r a t i o n taken from t h e t i m i n g u n i t and d i s p l a y e d on t h e o s c i l l o s c o p e ,  115 a l o n g w i t h t h e r f p u l s e and boxcar gate.  The r f p u l s e was l ^ u s .  l o n g , w h i l s t t h e boxcar gate was lOyus. wide. Measurements were made a t 78°K. i n both t h e 12" and t h e 6" V a r i a n magnets. the [OOl] H  05  The sample was mounted v e r t i c a l l y so t h a t  a x i s made an angle of 62° t o t h e p l a n e of r o t a t i o n of  w h i l e the[010] a x i s l a y w i t h i n 15° o f t h i s p l a n e .  ments were made a t f i v e d i f f e r e n t o r i e n t a t i o n s of H r e s p e c t t o t h e (010)  0  Measurewith  plane.  The most i m p o r t a n t f e a t u r e of these measurements i s t h a t the f r e e i n d u c t i o n decay i s e x p o n e n t i a l f o r times up t o a t l e a s t 2.5T ( F i g . ^-.8). A  s t a t i s t i c s , but s t i l l to a t l e a s t 2T . (65)»  showed t h a t t h e decay was e x p o n e n t i a l out  The l i n e shape i n n a t u r a l t i n i s squarer  X  Gaussian  The o t h e r decays do n o t have such good  so t h a t these r e s u l t s a r e c l e a r evidence  extreme exchange n a r r o w i n g The  that  occurs i n i s o t o p i c a l l y pure t i n .  l i n e shape i s broadened by t h e f i n i t e time  s p i n - l a t t i c e r e l a x a t i o n allows a nucleus state.  than  that  t o remain i n a g i v e n  T h i s i s known as l i f e t i m e b r o a d e n i n g .  The c a l c u l a t i o n  of t h i s e f f e c t f o r t h e g e n e r a l case i s v e r y complex, b u t f o r a s p i n £ system w i t h a L o r e n t z i a n l i n e i t c a n r i g o r o u s l y by shown t h a t t h e i n d u c t i o n t a i l i s s t i l l e x p o n e n t i a l w i t h a decay c o n s t a n t T  a  g i v e n by  (1)  (Ta)"' = (Tn)"' + (2T, )"'. In a p p l y i n g t h i s c o r r e c t i o n t o t h e e x p e r i m e n t a l was  assumed t h a t T, was ^Oyus. a t 78°K.  neglected  results i t  The e r r o r i n T, i s  s i n c e i t c o n t r i b u t e s a t most a s y s t e m a t i c e r r o r of  116 5% t o T . x  I t should  be noted t h a t t h e l i f e t i m e  broadening  e f f e c t i s n o t l a r g e enough t o be t h e cause o f t h e e x p o n e n t i a l decay. T a b l e *+.!.  S p i n - S p i n R e l a x a t i o n Times by F r e e I n d u c t i o n Decay  A n g l e of Hofrom (010) p l a n e .  Ti  (yUS.)  -30°  200±7  260*10  -30°  190±7  2^-5*10  +10  i5o±io  i8o±i5  +10°  175*20  220*25  +30°  200*10  260±l5  +55°  170*10  210±15  +100°  150*7  180*10  6  I t t a k e s about a day t o make each measurement so t h a t a d e t a i l e d study o f t h e a n i s o t r o p y would take a l o n g I t was thus decided  t o use C l a r k ' s method o f measuring t h e  l i n e shape (Chapter 3.15) t o o b t a i n t h e r e l a t i v e and  t h e n use t h e f r e e i n d u c t i o n v a l u e s  solute anisotropy.  time.  of T  a  anisotropy  t o g e t t h e ab-  I t only took a day t o make a s t u d y o f t h e  l i n e shape a s a f u n c t i o n o f o r i e n t a t i o n so t h a t s e v e r a l weeks were saved. For t h e l i n e shape measurements a boxcar gate 1ms. wide was used.  T h i s s t a r t e d about 2C^us. from t h e t r u e time  o r i g i n and covered a l l o f t h e f r e e i n d u c t i o n decay.  Care was  t a k e n t h a t t h e l i n e was swept through s l o w l y enough t o a v o i d distortion.  Distortion s t i l l  o c c u r r e d due t o t h e s i g n a l n o t  b e i n g e x a c t l y i n phase w i t h t h e r e f e r e n c e s i g n a l .  The l i n e  shape i s v e r y s e n s i t i v e t o any e r r o r s i n t h i s phase s e t t i n g . A f t e r c o n s i d e r a b l e e x p e r i m e n t a t i o n , t h i s phase d i s t o r t i o n was reduced t o about the n o i s e l e v e l and t h e measurements then were made.  To a n a l y s e t h e l i n e shape t h e m i d p o i n t was chosen and  then t h e two a m p l i t u d e s w i t h t h e same f r e q u e n c y d e v i a t i o n from t h i s m i d p o i n t were averaged.  T h i s a p p r o x i m a t e l y averaged out  the r e s i d u a l phase d i s t o r t i o n and a l s o i n c r e a s e d t h e S/N r a t i o by «/2. The r e s u l t i n g c u r v e s a r e L o r e n t z i a n f o r a t l e a s t  several  l i n e w i d t h s from t h e c e n t r a l f r e q u e n c y ( F i g . >+.9)j a g a i n s t r o n g e v i d e n c e f o r exchange n a r r o w i n g .  The h a l f w i d t h s and maxi-  mum a m p l i t u d e s were b o t h measured.  As expected f o r exchange  n a r r o w i n g , when the h a l f w i d t h s i n c r e a s e d t h e a m p l i t u d e s dec r e a s e d by a p p r o x i m a t e l y t h e same f r a c t i o n . Two c o r r e c t i o n s t o t h e l i n e w i d t h s had t o be c o n s i d e r e d . The f i r s t of these was t h e d i s t o r t i o n i n t r o d u c e d by t h e deadtime  (Appendix I V ) .  T h i s reduced a l l t h e h a l f w i d t h s by  about 10$, b u t caused a much s m a l l e r e r r o r i n t h e i r  relative  values.  distor-  A p a r t from t h e n a r r o w i n g , i t caused l i t t l e  t i o n i n t h e l i n e shape.  F o r these r e a s o n s , i t was i g n o r e d .  A c o r r e c t i o n a l s o had t o be made f o r l i f e t i m e b r o a d e n i n g . The h a l f w i d t h i s t h e r e c i p r o c a l of T  x  so t h a t t h e l i f e t i m e  broadening c o r r e c t i o n s a l r e a d y obtained f o r T  a  were used t o  118 g i v e the h a l f w i d t h c o r r e c t i o n s . F u r t h e r a n a l y s i s of t h e l i n e w i d t h s r e q u i r e s a c l o s e r study of the exchange n a r r o w i n g p r o c e s s .  I n a r e f e r e n c e frame  r o t a t i n g a t the resonant frequency a s p i n only f e e l s the l o c a l f i e l d , which i t p r e c e s s e s about a t a f r e q u e n c y of t h e o r d e r of the l i n e w i d t h .  I f only a d i p o l a r i n t e r a c t i o n i s present,  the magnetic f i e l d f l u c t u a t i o n s occur a t about the i n s t a n t a n eous Larmor f r e q u e n c y so t h a t t h e s p i n s can f o l l o w them reasonably w e l l .  This gives the d i p o l a r l i n e w i d t h .  exchange i n t e r a c t i o n He* = } r  J,;I,-.I:  I f an  i s a l s o present the r a t e  of change of t h e d i p o l a r H a m i l t o n i a n H i s  If  Hx^H  the l o c a l f i e l d  f l u c t u a t e s a t a r a t e about e q u a l t o  the exchange i n t e r a c t i o n c o n s t a n t J . T h i s i s much f a s t e r  than  the Larmor f r e q u e n c y so t h a t t h e averaged f i e l d w h i c h t h e s p i n s f e e l i s much l e s s than t h e l o c a l f i e l d .  The l i n e i s thus much  narrower t h a n t h e d i p o l a r l i n e . The random f u n c t i o n model of Anderson and Weiss c a s t s t h i s p h y s i c a l p i c t u r e i n t o a q u a n t i t a t i v e form ( 1 , 6 6 ) .  In this  model i t i s assumed t h a t the random f l u c t u a t i o n s of t h e l o c a l field  A t o ( t ) from t h e r e s o n a n t f r e q u e n c y u„ a r e G u a s s i a n i n  a m p l i t u d e w i t h a mean square v a l u e OJ e q u a l t o t h e second 1  p  moment.  T h e i r time v a r i a t i o n s a r e d e s c r i b e d by t h e c o r r e l a -  tion function  119 A f o r m f o r g('t) must be chosen on t h e b a s i s o f p h y s i c a l p l a u s i b i l i t y and m a t h e m a t i c a l t r a c t i b i l i t y .  The p h y s i c a l r e s t r i c t i o n s  on g(T) a r e t h a t t h e second moment i s u n a f f e c t e d  by t h e exchange  i n t e r a c t i o n and t h a t t h e f o u r t h moment must remain f i n i t e . simplest expression  The  which s a t i s f i e s these r e q u i r e m e n t s i n t h e  Gaussian  g(T) = expC-irrwe'-ir ). 1  uJt^7 i s an average exchange f r e q u e n c y .  W i t h t h i s assumption 7  the f r e e i n d u c t i o n decay f o r t h e exchange narrowed  region  becomes G(-t)=exp(--glr This expression  h o l d s e x c e p t when t « w  tends towards a G a u s s i a n . G(t)  t ) .  i s a Lorentzian  _ l e  , where the decay  The l i n e shape c o r r e s p o n d i n g t o  out t o a f r e q u e n c y w h e r e  i t falls off  q u i t e r a p i d l y , k e e p i n g t h e second and f o u r t h moments f i n i t e . From G ( t ) , t h e f o u r t h moment i s <w*>=36Jp +-kTTtUeO^. 4  The  f o u r t h moment c a n a l s o be c a l c u l a t e d i n terms of the l a t -  t i c e s t r u c t u r e by means of Van V l e c k ' a method of t r a c e s . two  The  e x p r e s s i o n s have a s i m i l a r form so t h a t , by comparison,  io  can be w r i t t e n i n terms o f the l a t t i c e s t r u c t u r e and exchange and  dipolar interactions.  complicated that The  The r e s u l t i n g e x p r e s s i o n  u> i s c u s t o m a r i l y e  l i n e w i d t h i s u)?/uj . &  i s so  assumed t o be i s o t r o p i c . The second moment f o r  w h i t e t i n has been c a l c u l a t e d by computer so t h a t t h e f o r m of  cjp- i s known. 4  I f to  e  i s i s o t r o p i c , the anisotropy  of t h e  t  120 measured l i n e w i d t h s should be p r o p o r t i o n a l t o  c*>p .  The f i t i s  r e a s o n a b l y good ( F i g . V . l l ) and, c o n s i d e r i n g t h e e x p e r i m e n t a l e r r o r s i n v o l v e d , shows t h a t c j i s i s o t r o p i c , or v e r y n e a r l y s o . e  The l a r g e s t e r r o r i s due t o s l i g h t misalignment The magnetic f i e l d  of the c r y s t a l .  o r i e n t a t i o n t r a c e s out a c o m p l i c a t e d  trajec-  t o r y ( F i g . if.10) and s l i g h t changes i n t h i s cause l a r g e changes i n t h e h i l l t o v a l l e y r a t i o of the second moment. of T  f o r f i v e d i f f e r e n t o r i e n t a t i o n s of H  a  The i n v e r s e s  are also plotted  0  on t h e same graph and agree w e l l w i t h t h e l i n e w i d t h v a r i a t i o n s . They a l s o g i v e t h e a b s o l u t e l i n e w i d t h s . tjp  1  i s t h e sum of t h e second moments due t o d i p o l a r and  pseudo-dipolar i n t e r a c t i o n s . dependence, so t h a t  These b o t h have t h e same a n g u l a r  6jp = <cj,|> (1+B), where  (00$  i s the d i p o l a r  second moment and B i s t h e f r a c t i o n of p s e u d o - d i p o l a r exchange present.  ,,. rp l  ~<Wcf>fHB)  a  U s i n g a l l of t h e measured T s , a l o n g w i t h t h e i r  corresponding  a  c a l c u l a t e d d i p o l a r second moments g i v e s =( 1+B) (7*1.5) 10  3  rad . a  (1)  —  The a b s o l u t e v a l u e of these parameters cannot be o b t a i n e d w i t h o u t making an independent t h i s case a measurement o f T  measurement i n v o l v i n g them. a  In  i n a n a t u r a l t i n c r y s t a l was made  o  a t 78 K.  The S/N r a t i o was poor, b e i n g about t h r e e , so t h a t  n o t h i n g c o u l d be s a i d about t h e f r e e i n d u c t i o n decay o t h e r than i t decayed f a s t e r than an e x p o n e n t i a l decay.  Steady  state  121 measurements show t h a t the l i n e i s n e a r l y G a u s s i a n  (65)  the l i n e was a n a l y s e d by assuming i t was G a u s s i a n .  so t h a t  This gives  T =(120-20^s., a f t e r a p p l y i n g a c o r r e c t i o n f o r T* b r o a d e n i n g . a  t o a second moment o f (1.3 O«2)Kc/s .  T h i s corresponds  This i s  i  i n e x c e l l e n t agreement w i t h t h e steady s t a t e v a l u e N a t u r a l t i n c o n t a i n s " two" i~sotopesy Sn" ;  a p p r o x i m a t e l y e q u a l abundances.  7  (68).  and Sn "' , of  The w i d t h o f t h e S n "  1  ?  line  thus c o n t a i n s a number o f c o n t r i b u t i o n s (1,52). (a) D i p o l a r broadening between l i k e and u n l i k e s p i n s . M< =i7 lfr(l+l)£'l£ 4  f  +  iT,*r * s  S(S+l)fi Jib*: , a  where t h e summations a r e o n l y taken over the l a t t i c e  b^ =*Vj" (3cos*c9,-j -1).  occupied by the a p p r o p r i a t e i s o t o p e , Tin  sites  3  has a p p r o x i m a t e l y e q u a l gyromagnetic  r a t i o s and abundances F  and so the e x p r e s s i o n s i m p l i f i e s t o  - ~~  *3  .  'j  where t h e summation i s now t a k e n over a l l the l a t t i c e (b) P s e u d o - d i p o l a r  sites.  broadening.  T h i s adds a term BM^ t o t h e second  moment.  (c) Pseudo-exchange broadening between u n l i k e n u c l e i . The exchange c o u p l i n g £Z.J?j (•'£* *§P  does n o t commute  w i t h Hf i f t h e s p i n s a r e u n l i k e and so c o n t r i b u t e s a term t o the second moment.  Fort i nthis i s -tfUJij (r),  i *  w i t h t h e summation over a l l l a t t i c e  sites.  122 (d) Exchange n a r r o w i n g between l i k e s p i n s . I n n a t u r a l t i n t h e exchange f r e q u e n c y i s F c j . e  about 0.08  Fis  and so t h e exchange f r e q u e n c y i s much l e s s than the  d i p o l a r f r e q u e n c y and can thus be n e g l e c t e d , ^b ^  has a l r e a d y been computed so t h a t t h e t o t a l  5  second moment i n n a t u r a l t i n c a n be c a l c u l a t e d . <Aeu > =7.5xlO (l+B)+2.5xlO a  t  In evaluating taken i n t o account. body-centred  J  H  It is  J . 1  t h e l a t t i c e s t r u c t u r e has been r o u g h l y  W h i t e t i n c o n s i s t s of two i n t e r p e n e t r a t i n g  t e t r a g o n a l l a t t i c e s w i t h a c/a r a t i o of 0.55  about 3«1A° away and 12  Each atom has s i x n e a r e s t n e i g h b o u r s next nearest neighbours  about hk° away.  than t h i s were n o t c o n s i d e r e d .  (67).  Atoms f u r t h e r away  J i s t h e v a l u e of the exchange  constant a t the nearest neighbours  and was assumed t o v a r y as  _3  r  i n summing over t h e o t h e r  sites.  Assuming a G a u s s i a n l i n e and u s i n g t h e measured v a l u e of T  ft  t o determine  <AU>*> g i v e s  (7 l)ylo =7.5xlo (l+B)+2.5ao" 7  ±  6  1  J \  —  ——(2)  The n e x t s t e p r e l a t e s J and u>& and i n v o l v e s t h e most d r a s t i c s t e p of t h e whole a n a l y s i s , j u s t i f i a b l e o n l y by expediency.  The r e l a t i o n between J and co should be o b t a i n e d &  by e q u a t i n g t h e f o u r t h moment 3&V  +%UpUj  % &  to the f o u r t h  moment c a l c u l a t e d from t h e l a t t i c e s t r u c t u r e by t h e method of the t r a c e s . «  The l a t t e r i n v o l v e s a t r i p l e summation over  123 a l l t h e l a t t i c e s i t e s and even f o r j u s t t h e 12 atoms out t o n e x t nearest neighbours  i s a p r o h i b i t i v e l y complicated  w h i l e summing over fewer atoms i s n o t p h y s i c a l l y The  calculation, justifiable.  relation 2=  , ,°: -  "  ,J J  i  = i.7/io"V o b t a i n e d f o r a s i m p l e c u b i c l a t t i c e (1)  thus had t o be used.  The l a t t i c e s p a c i n g d i s assumed t o be 3A° , t h e n e a r e s t spacing.  F o r <CJp>>  an average of the computed v a l u e s f o r t h e  w h i t e t i n l a t t i c e i s used. dipolar contribution.  neighbour  T h i s must a l s o i n c l u d e t h e pseudo-  T h i s was o b t a i n e d by i t e r a t i o n .  The  c a l c u l a t i o n was f i r s t c a r r i e d through w i t h o u t t h e p s e u d o - d i p o l a r c o n t r i b u t i o n t o o b t a i n an approximate v a l u e o f B and t h i s was then used i n o b t a i n i n g a more a c c u r a t e v a l u e o f ^uuf.y •  Because  of t h e l a r g e e x p e r i m e n t a l e r r o r s and t h e dominance o f t h e exchange term, i t was n o t n e c e s s a r y to r e p e a t t h e c y c l e .  The v a l u e  f i n a l l y adopted was <ujf> =(8.1*0.8) x l O rad . 7  a  This gives  W|  = (2.2±0.2)j\  (3)  T h i s n e g l e c t s t h e d i p o l a r c o n t r i b u t i o n t o t h e f o u r t h moment, a good a p p r o x i m a t i o n f o r extreme exchange n a r r o w i n g . S u b s t i t u t i n g t h i s expression i n the equation f o r the second moment i n n a t u r a l t i n and t h e n e l i m i n a t i n g 1+B g i v e s a  12*+ q u a d r a t i c e q u a t i o n i n GJ . The s o l u t i o n o f t h i s e q u a t i o n i s e  W  =(2.0±0.3)*10 radians. 4  e  T h i s c o r r e s p o n d s t o an exchange c o n s t a n t o f J=(2.2±0.3)Kc/s. U s i n g t h e above v a l u e of CJ g i v e s E  3=1.9*0.5. By comparing  t h e l i n e w i d t h they measured i n w h i t e t i n  powder by steady s t a t e methods w i t h t h e c a l c u l a t e d moment, Karimov and Schegolev  (68) o b t a i n e d  dipolar  J=(2.5 0.1)Kc/s. i  However, they n e g l e c t e d the p s e u d o - d i p o l a r c o n t r i b u t i o n .  If  t h e i r r e s u l t s are re-analysed w i t h the pseudo-dipolar c o n t r i b u t i o n from t h e p r e s e n t experiment i n c l u d e d , they g i v e J=(2.1±0.2)Kc/s.  There i s thus e x c e l l e n t agreement between t h e  two e x p e r i m e n t s .  Jones (65) a l s o attempted  are  t o measure J .  There  n u m e r i c a l e r r o r s i n h i s d e t e r m i n a t i o n o f t h e second moment  and an i n c o r r e c t l a t t i c e s t r u c t u r e was used i n h i s second moment c o m p u t a t i o n so t h a t h i s r e s u l t s a r e wrong.  A determin-  a t i o n o f B has n o t been made b e f o r e . The whole a n a l y s i s has r e s t e d on t h e t w i n assumptions t h a t t h e d i p o l a r f i e l d f l u c t u a t i o n s have a G a u s s i a n  correlation  f u n c t i o n and t h a t t h e d i p o l a r l i n e shape i s G a u s s i a n .  Little  can be s a i d about t h e r e l i a b i l i t y o f t h e c o r r e l a t i o n f u n c t i o n other than t h a t i t i s p l a u s i b l e . from t h e G a u s s i a n form i s unknown.  The e f f e c t o f d e v i a t i o n s More d e f i n i t e  statements  can be made about t h e assumption o f a G a u s s i a n l i n e The l i n e shape parameters  shape.  of i n t e r e s t a r e the second moment  125 of t h e d i p o l a r l i n e and t h e f o u r t h moment of t h e exchange narrowed l i n e .  The d i p o l a r second moment can be r i g o r o u s l y  c a l c u l a t e d i n terms o f t h e known l a t t i c e s t r u c t u r e .  The t o t a l  second moment has t o be measured i n n a t u r a l t i n b u t ,  provided  the S/N r a t i o i s good enough, t h i s can a c c u r a t e l y be done f o r non-Gaussian l i n e s by an a p p r o p r i a t e i n d u c t i o n decay.  a n a l y s i s of t h e f r e e  I n t h e case of extreme exchange n a r r o w i n g ,  the c o n t r i b u t i o n t o t h e f o u r t h moment from t h e exchange a c t i o n u s u a l l y dominates the d i p o l a r f o u r t h moment.  inter-  I f this  i s s o , t h e t o t a l f o u r t h moment i s I n s e n s i t i v e t o t h e d i p o l a r l i n e shape and so t h e v a l u e s of B and J o b t a i n e d unaffected shape.  should be  by moderate d e v i a t i o n s from a G a u s s i a n d i p o l a r i i n e  More e x p l i c i t l y , Anderson and Weiss found t h a t t h e  two most extreme d i p o l a r l i n e shapes l i k e l y , a square spectrum and  an e x p o n e n t i a l  one w i t h e q u a l second moments, had exchange  narrowed l i n e w i d t h s w h i c h d i f f e r e d  by about 50$.  I t thus  seems t h a t t h e measured v a l u e s of B and J do n o t c o n t a i n  large  e r r o r s due t o t h e assumption of a G a u s s i a n l i n e shape, but c o u l d p o s s i b l y c o n t a i n l a r g e e r r o r s due t o d e f e c t s  i n the ran-  dom f l u c t u a t i o n model. The  l a s t stage should be t o compare the e x p e r i m e n t a l  v a l u e s o f J and B w i t h v a l u e s c a l c u l a t e d from t h e t i n band structure. nor  However n e i t h e r t h e a c c u r a c y o f t h e measurements,  t h a t of the c a l c u l a t e d v a l u e s , w a r r a n t s such a step a t t h e  present time. B.  The main f e a t u r e t o n o t e i s t h e l a r g e v a l u e o f  T h i s i m p l i e s t h a t a l a r g e number o f t h e F e r m i  surface  126 e l e c t r o n s a r e o f p, o r h i g h e r c h a r a c t e r . a n i s o t r o p i c Knight s h i f t substantiates (iii)  The r e l a t i v e l y  large  this,  S p i n Echoes. At l i q u i d n i t r o g e n temperature t i n has T,/v/0.5ms. and  T --^OO^s. a  I fT  a  i s reduced by d e s t r o y i n g the homogeneity o f  the magnetic f i e l d , i t i s p o s s i b l e t o g e t s p i n echoes. I n the f i r s t measurement tv/o r f p u l s e s l ^ t / s . and 30^s. l o n g and s e p a r a t e d by 200yws were used. such t h a t  T^rsj^Ojuis.,  w h i l e the r e p e t i t i o n r a t e was (10ms.)  By v a r y i n g the magnetic f i e l d resonance a t K^ws. second p u l s e . the  The homogeneity was  .  the boxcar was swept through  i n t e r v a l s from 60yMs. t o 250yus. a f t e r t h e  A narrow boxcar gate o f lOyus. was used and a l l  times were measured between the c e n t r e s of the p u l s e s  and boxcar g a t e .  The magnetic moment measured by a sweep was  t h e n p l o t t e d a g a i n s t the time a f t e r the second p u l s e ( F i g . ^.3).  The graph shows a s p i n echo'which i s 180° out o f phase  w i t h the i n d u c t i o n t a i l and w i t h a s y m m e t r i c a l envelope whose time c o n s t a n t i s the same a s t h a t o f the i n d u c t i o n t a i l . I t s maximum o c c u r s a t 200,Hs. a f t e r the c e n t r e o f t h e second p u l s e . On removal o f the f i r s t p u l s e the s p i n echo v a n i s h e d . The measurements were r e p e a t e d w i t h s e v e r a l d i f f e r e n t  separations  between the r f p u l s e s .  increased  The a m p l i t u d e o f the echo  as the p u l s e s e p a r a t i o n d e c r e a s e d . The n e x t experiment was t o a p p l y two p u l s e s 200yWs. a p a r t w i t h a 30yUS. wide boxcar gate s e t on the echo maximum. The magnetic f i e l d was then swept t h r o u g h resonance w i t h  127 v a r y i n g v a l u e s o f t h e r e f e r e n c e phase.  As the phase was v a r i e d  by 90°, t h e p l o t of magnetic moment v e r s u s magnetic  field  changed from an a b s o r p t i v e t o a d i s p e r s i v e shape. So f a r t h e s p i n echo had behaved i n e x a c t l y t h e same f a s h i o n as a s p i n echo i n a n o n - m e t a l l i c substance w i t h a homogeneous H, •  I n t h e n e x t s e r i e s of measurements, the d i f f e r e n c e s  became a p p a r e n t .  T* was reduced  t o 20yMs.fifor these  experiments.  Two r f p u l s e s s e p a r a t e d by 150/as. were used a l o n g w i t h a boxcar gate hOj/Us. wide s e t on t h e echo maximum.  The w i d t h s  of the  two p u l s e s were then v a r i e d w h i l e k e e p i n g t h e r a t i o of t h e i r w i d t h s f i x e d a t 1:2.  The graph o f echo h e i g h t a g a i n s t t h e  f i r s t p u l s e w i d t h showed t h a t t h e maximum a m p l i t u d e  occurred  when t h e f i r s t p u l s e was about lO^us. wide ( F i g . ^ . ^ f ) .  The  f i r s t p u l s e w i d t h was then k e p t f i x e d a t lC^us. and t h e second p u l s e w i d t h v a r i e d f r o m ^ s . t o 90^\s. The maximum a m p l i t u d e o c c u r r e d when t h e second p u l s e was about 20^s. wide ( F i g . *+.5). These r e s u l t s c a n be e x p l a i n e d by a s i m p l e e x t e n s i o n of the c l a s s i c a l s p i n echo t h e o r y of Hahn (62) t o t h e case of a m e t a l i n t h e normal s k i n e f f e c t r e g i o n . the case of a phase c o h e r e n t  For s i m p l i c i t y , only  system s e t t o d e t e c t t h e a b s o r p t i o n  mode and o p e r a t i n g a t t h e r e s o n a n t f r e q u e n c y The  sample i s approximated by an i n f i n i t e plane so t h a t t h e r f  magnetic f i e l d cos(^g and  i s considered.  a t a depth z from t h e s u r f a c e i s H, =H  ) . R e p l a c i n g OJ by ou,(z)= t V , e x p ( - ^ ) X  )p  exp(-^)  i n Hahn's t h e o r y  t a k i n g a c c o u n t o f t h e phase coherence a l l o w s t h e magnetic  moment M  Ss  ( z , t ) p r o d u c i n g t h e s p i n echo a t a d e p t h z t o be  128 simply c a l c u l a t e d .  The s p i n echo induced  v o l t a g e i s then  (Appendix I I I ) (t) = J e x p ( - ^ ) c o s ( ^ - ) M  V  o  = M exp[>(^^)-ij  s e  (z,t)dz  e x p t - ^ s i n f o ; , (z)<l  0  Jo  sin [i(j,(z)'ta]cos ' ( ^ )dz. a  T  A  :1  i s t h e s p i n - s p i n r e l a x a t i o n time due t o d i p o l a r and exchange  e f f e c t s , w h i l e T* i s t h e t r a n s v e r s e r e l a x a t i o n time due t o t h e inhomogeneous magnetic f i e l d . the two r f p u l s e s of d u r a t i o n  T i s t h e time i n t e r v a l between and T.» seconds r e s p e c t i v e l y . By  w r i t i n g s i n ( c j , r , ) s i n ( i cj»f ) as s i n (  )+£ [ sin{oj,(T^-'t,)}  l  a  -sin{o;i (fa+'ti)} J , v  s e  becomes a sum o f i n t e g r a l s of t h e form  e x p ( - x ) c o s ( x ) s i n {co.-c.e" ) d * . a  ated made.  (Appendix I I I ) ,  These have a l r e a d y been e v a l u -  so t h a t a q u a n t i t i v e t e s t of v  can be  Je  The c a l c u l a t e d v a r i a t i o n o f s p i n echo a m p l i t u d e w i t h r f  p u l s e l e n g t h has been p l o t t e d i n F i g s . k,k the m e t a l l i c and n o n - m e t a l l i c c a s e s .  and *f.5 f o r both  I t was assumed t h a t a  £TT p u l s e a t t h e s u r f a c e of t h e sample was 9yus.  The t h e o r e -  t i c a l e x p r e s s i o n i s i n good q u a l i t a t i v e agreement w i t h the e x p e r i m e n t a l p o i n t s and would be i n q u a n t i t a t i v e agreement i f i t was assumed t h a t t h e &rr  p u l s e was about 8yus. l o n g .  This  i s p o s s i b l e s i n c e these measurements were n o t made under e x a c t l y the same e x p e r i m e n t a l c o n d i t i o n s as those of F i g . For the s p i n echo measurements of T , the f i r s t x  was lOyMS. l o n g and t h e second one was 20jus. l o n g . wide boxcar gate was used.  k.2.  pulse  A  I f the pulse s e p a r a t i o n (taken  between c e n t r e s )  was T, t h e gate c e n t r e was l o c a t e d a t a time  T a f t e r the c e n t r e  o f t h e second p u l s e .  A f t e r each sweep  through resonance T was a l t e r e d m a n u a l l y .  T i m i n g was by means  of t h e d o u b l e beam o s c i l l o s c o p e and marker p i p s from t h e t i m ing unit.  A l l the measurements were made a t 78 K. The s p i n  echo decays were a l l e x p o n e n t i a l  ( F i g . *+.13) w i t h a decay  constant %  f o r l i f e t i m e broadening i n  which was c o r r e c t e d  the same f a s h i o n as the f r e e i n d u c t i o n decays t o g i v e T . a  Table *+.2.  Spin-Spin Relaxation  Angle of H from (010) p l a n e . 0  %  Times by S p i n  (/AS.)  T (JJS.) a  +10°  120±10  1^0^12  +30°  170-15  210*20  +55  105±5  120*7  +100°  105*5  120*7  When t h e v a l u e s o f T obtained a  Echoes  by f r e e  induction  decay and by s p i n echoes were p l o t t e d a g a i n s t magnetic  field  o r i e n t a t i o n ( F i g . *+.13) i t was s u r p r i s i n g l y found t h a t T obtained  a  by s p i n echoes was always s h o r t e r than T measured a  by f r e e i n d u c t i o n decay.  The d i f f e r e n c e ranged from 20%  to 55% of t h e v a l u e o f the f r e e I n d u c t i o n  T .4  There a r e t h r e e p o s s i b l e reasons f o r t h i s l a r g e ference i n values. the s p i n g r a d i e n t For a s p i n  dif-  The f i r s t of these i s s p i n d i f f u s i o n i n caused by t h e l a r g e inhomogeneity i n H, .  system t h e s u r p l u s o f s p i n s i n one o r i e n t a t i o n  130 p(x,t)  obeys the d i f f u s i o n  equation ( 1 ) .  W i s the p r o b a b i l i t y / s e c o n d t h a t two n u c l e i undergo a mutual s p i n f l i p and a i s the s e p a r a t i o n between the n u c l e i . the exchange c o n s t a n t , sion alters T  so t h a t Wa*~> 10"'* c m t / s e c .  Spin  because the phase and a t t e n u a t i o n  a  W<~J, diffu-  of the induced  s i g n a l depend on the depth of the s p i n s from the s u r f a c e and so if  the s p i n c o n c e n t r a t i o n changes w i t h t i m e , T  For s p i n d i f f u s i o n  to reduce the f r e e  is  a  induction T  a  altered. by 10$  the  s p i n s must t r a v e l a d i s t a n c e of about 0.056 i n the time T . a  The d i s t a n c e t r a v e l l e d i n the time T  a  is  (2Wa T )\ 1  l  This i s  about one l a t t i c e s p a c i n g , so t h a t s p i n d i f f u s i o n e f f e c t s negligible.  are  Because o f t h i s v e r y slow v e l o c i t y s p i n d i f f u s i o n  should not have any e f f e c t on the s p i n echoes  either.  The second p o s s i b i l i t y i s t h a t two of the a p p r o x i m a t i o n s made i n d e r i v i n g the s p i n echo e q u a t i o n are n o t v a l i d and t h a t obscure phase e f f e c t s T . a  a s s o c i a t e d w i t h t h i s cause the e r r o r i n  I n the d e r i v a t i o n i t i s assumed t h a t the p u l s e  are n e g l i g i b l e compared to T  2  and T?  .  lengths  I f t h i s i s so,  then  r e l a x a t i o n i n the r o t a t i n g r e f e r e n c e frame o c c u r s d u r i n g the r f p u l s e and i t i s e a s i l y shown t h a t the o n l y e f f e c t  of  i s t o reduce the echo a m p l i t u d e w i t h o u t a f f e c t i n g the v a l u e of Ta..  However i n the p r e s e n t case the p u l s e  are about lOyus. w h i l e T  a  this  measured  lengths  i s about l50^s., so t h a t r e l a x a t i o n  i n the r o t a t i n g r e f e r e n c e frame o n l y o c c u r s w i t h i n about 6 of the s u r f a c e .  S p i n - s p i n r e l a x a t i o n f o r the case when  131 "2TH, <^ T*  (13).  has o n l y been c a l c u l a t e d f o r a two s p i n system  1  I n t h i s , n o n - s e c u l a r terms i n the r o t a t i n g r e f e r e n c e frame d r a s t i c a l l y modify r e l a x a t i o n d u r i n g the r f p u l s e , but have no e f f e c t on the f r e e i n d u c t i o n deca,y a f t e r the r f p u l s e i s o v e r , a p a r t from a l t e r i n g the i n i t i a l time o r i g i n and T h i s suggests t h a t t h e r e should be no e r r o r i n T  amplitude. a  obtained  from the f r e e i n d u c t i o n decay i n the case of an n s p i n A l t h o u g h they i n t u i t i v e l y seem independent  system.  of the p u l s e separ-  a t i o n , i t i s p o s s i b l e t h a t these n o n - s e c u l a r terms cause r e d u c t i o n i n T^  i n spin  echoes.  Hahn's t h e o r y a l s o c o n t a i n s the more d r a s t i c t h a t Ti;H,»  (Tj)" . 1  the  assumption  T h i s means t h a t the magnetic moments p r e c e s s  about the c o n s t a n t magnetic f i e l d H, , d i p o l a r f l u c t u a t i o n s h a v i n g n e g l i g i b l e e f f e c t on the p r e c e s s i o n .  I n these e x p e r i -  ments, t h i s requirement, does not even h o l d a t the s u r f a c e of the m e t a l .  I t i s q u i t e p o s s i b l e t h a t t h i s i s the r e a s o n f o r  the s m a l l e r T . a  The r e s t r i c t i o n on H, c o u l d be removed from the  t h e o r y i f i t were not f o r the e n s u i n g avalanche of a l g e b r a i c manipulations. The breakdown of these two a p p r o x i m a t i o n s suggests t h a t dividing v  s e  i n t o the p r o d u c t of a time dependent and a phase  dependent f a c t o r i s o n l y a p p r o x i m a t e l y c o r r e c t . e x p e r i m e n t a l checks of t h i s which can be made. would be to measure T £n  a  There a r e The f i r s t  two one  as a f u n c t i o n of H, u s i n g a p p r o x i m a t e l y  and rr p u l s e s a t the s u r f a c e of the m e t a l f o r each v a l u e  of H,•  U n f o r t u n a t e l y the a p p a r a t u s c o u l d not do t h i s .  However  132 the second check o f measuring T was attempted.  a  as a f u n c t i o n of pulse length  The f i r s t p u l s e l e n g t h was k e p t f i x e d a t  IC^MS.  w h i l e the second p u l s e l e n g t h was v a r i e d . Table *+.3  V a r i a t i o n of S p i n Echo Ta W i t h P u l s e Length  A n g l e of H from (010) p l a n e . 0  L e n g t h o f Second p u l s e (yus.).  S p i n echo. F r e e  +30°  10  l80±20  +30°  20  170*15  +55°  10  115*5  +55°  20  105*5  +55°  30  105±5  Although  Induction  200±10 - - - - -  170*10 —  t h e e x p e r i m e n t a l e r r o r s p r e v e n t a d e f i n i t e con-  c l u s i o n , i t seems t h a t t h e r e i s a s m a l l dependence o f T on 2  pulse length.  There c o u l d a l s o be a dependence on t h e f i r s t  p u l s e l e n g t h , b u t t h e poor S/N prevented a m i n a t i o n of t h i s .  any e x p e r i m e n t a l ex-  The experiment has f a i l e d  t o show the cause  of the b u l k of t h e descrepancy though. The f i n a l p o s s i b i l i t y  i s t h a t the d i f f e r e n c e i s due t o  the e f f e c t o f the exchange term. Hamiltonian  I n t h e l a b o r a t o r y frame t h e  of t h e system i s  The H a m i l t o n i a n i n a r e f e r e n c e frame r o t a t i n g a t a f r e q u e n c y to be comes U  =T-hZ(Ho-%)l  i z  + V  ,  133 where H"» T^±£r7jU-3eos»e,j ) (3IuIj»-I, . I j Before  the f i r s t p u l s e , t h e d e n s i t y m a t r i x d e s c r i b i n g the s p i n  system i s  The  Zeeman term i s much l a r g e r t h a n t h e d i p o l a r and exchange  frame r o t a t i n g a t t h e f r e q u e n c y u> o f t h e a p p l i e d r f p u l s e s . F o l l o w i n g Abragam ( p . M-98), t h e maximum a m p l i t u d e of the s p i n echo a t resonance becomes E(2T)  = Tr[exp(-I'H !r)exp(-i7H 'r, ) e x p ( - l t f l ! ) e x p ( - 1 7 H t ) 1  l  Ii  exp(lTH,t,)  l  exp(iVT)exp(i?H Tjexp(i7^)l. ] , l  e  where % and 1 are the f i r s t and second p u l s e l e n g t h s X  t i v e l y , and T i s t h e p u l s e s e p a r a t i o n . m a l l y s o l v e d ; the r e m a i n i n g exponential operators resulting traces.  l  respec-  The problem i s now f o r -  s t e p s c o n s i s t i n g of expanding t h e  i n a power s e r i e s and then e v a l u a t i n g t h e  The a t t e n u a t i o n and phase f a c t o r s due t o t h e  s p i n echo b e i n g i n a m e t a l c o u l d then be t a k e n c a r e of i n t h e same f a s h i o n as f o r t h e f r e e i n d u c t i o n decay. In p r a c t i c e , the mathematical complexity prevented E(2T) f r o m b e i n g e v a l u a t e d system.  i n v o l v e d has  f o r even a two s p i n  The r i g o r o u s quantum m e c h a n i c a l t h e o r y t h u s a t p r e s e n t  says n o t h i n g about t h e e f f e c t of the exchange i n t e r a c t i o n on s p i n echoes.  S t a t i s t i c a l t h e o r i e s , such as t h a t o f Anderson  13>+ and W e i s s , a r e n o t s u i t a b l e f o r d e s c r i b i n g t h e s p i n echoes s i n c e they c o n t a i n ad hoc assumptions and, by t h e i r n a t u r e ^ average out many o f the d e t a i l e d i n t e r a c t i o n s w h i c h might be expected t o modify t h e exchange e f f e c t s . Because of the d i p o l a r c o u p l i n g , equivalent  t h e n u c l e i a r e n o t an  s i t e s , so t h e r e i s no t h e o r e t i c a l p r o h i b i t i o n on  exchange e f f e c t s b e i n g observed by s p i n echoes  (1).  In the d e n s i t y matrix formalism the f r e e  induction  decay i s p r o p o r t i o n a l t o Tr [ e x p ( - i K t ) I e x p ( i ' H " t ) I ] . x  two  terms of t h i s have been e v a l u a t e d  exchange and d i p o l a r i n t e r a c t i o n s (1). moment) i s u n a f f e c t e d moment) i s i n c r e a s e d  i n t h e presence o f both The f i r s t  the  term (second  by exchange, but t h e second term ( f o u r t h by i t .  I n t h e case of exchange n a r r o w i n g  the s e r i e s must approximate an e x p o n e n t i a l obviously  The f i r s t  t  s e r i e s , so t h a t  t h e exchange must a l s o a f f e c t many h i g h e r  terms i n  expansion. The  t h e o r e t i c a l s i t u a t i o n i s t h a t t h e exchange i n t e r -  a c t i o n a f f e c t s t h e f r e e i n d u c t i o n decay i n a way c o n s i s t e n t w i t h t h e e x p e r i m e n t a l r e s u l t s i n both t h e d e n s i t y m a t r i x and s t a t i s t i c a l theories.  The d e n s i t y m a t r i x  about t h e s p i n echo case w h i l e g i v e s t h e same T  a  nothing  t h e Anderson-Weiss t h e o r y  f o r b o t h f r e e i n d u c t i o n and s p i n echo decays.  However t h i s i s n o t c o n c l u s i v e tions involved  t h e o r y says  i n t h e i r theory.  because o f t h e type o f assumpThese c o n s i d e r a t i o n s  suggest  t h a t t h e exchange i n t e r a c t i o n a f f e c t s t h e s p i n echoes, b u t i t i s impossible  t o say whether o r n o t t h e s p i n echo T  a  should be  135 the same as t h a t of the f r e e i n d u c t i o n decay. A l l of the obvious reasons f o r the d i f f e r e n c e between the s p i n echo and examined.  The  f r e e i n d u c t i o n r e l a x a t i o n times have now  only d e f i n i t e c o n c l u s i o n  e f f e c t s are n e g l i g i b l e .  been  i s that spin d i f f u s i o n  A l l of the o b v i o u s e x p e r i m e n t a l pos-  s i b i l i t i e s , such as a s y s t e m a t i c  e r r o r i n p o s i t i o n i n g the box-  car gate on the echo, were e l i m i n a t e d  by the  preliminary  e x p e r i m e n t a l i n v e s t i g a t i o n , or by the method used i n t a k i n g the d a t a .  P u l s e w i d t h e f f e c t s were e x p e r i m e n t a l l y  shown t o  be s m a l l .  There s t i l l remains the p o s s i b i l i t y of e x p e r i m e n t a l  e f f e c t s caused by H,  b e i n g too s m a l l .  the d i f f e r e n c e i n T  i s an i n t r i n s i c f e a t u r e of systems w i t h  a  a p p r o x i m a t e l y e q u a l d i p o l a r and The  I t i s also possible  pseudo-exchange i n t e r a c t i o n s .  o s c i l l a t o r y v a r i a t i o n of the pseudo-exchange i n t e r a c t i o n  w i t h d i s t a n c e might be of some importance i n c a u s i n g k-.Q  The  E x p e r i m e n t a l S/N  An e x p r e s s i o n (Chap. 3.12) S/N  that  Ratios  f o r the S/N  w h i c h can now  r a t i o has been d e r i v e d  be compared w i t h the  experimental  r a t i o s determined f o r s i x d i f f e r e n t m e t a l s under  conditions.  The  S/N  this.  varying  r a t i o a t the a m p l i f i e r output i s  S' =TT^ 6n utoM R, Q (-j^-)* . c  /  0  I n c a l c u l a t i n g t h i s Q i s a r b i t r a r i l y assumed to be 20 temperature independent, T p e r a t u r e and  n  i s assumed t o be the sample tem-  C i s t a k e n as 80pf.  on the i n d i v i d u a l sample and  and  The  c o i l and  o t h e r parameters depend are known, except f o r  _  F o r some m e t a l s  136  .  S c a n be a c c u r a t e l y c a l c u l a t e d a t t h e  temperatures o f i n t e r e s t , b u t f o r o t h e r m e t a l s t h e e l e c t r i c a l c o n d u c t i v i t y had t o be e s t i m a t e d .  The boxcar enhancement  f a c t o r i s r ^ ^ - j • f t was e s t i m a t e d from t h e bandwidth o f t h e tuned c i r c u i t t o be 2/as. ing  U s i n g these v a l u e s , a t a b l e compar-  t h e t h e o r e t i c a l and e x p e r i m e n t a l S/N r a t i o s can be con-  structed. Table h.h-  S/N a t A m p l i f i e r Output,S' .  Metal  Al  E x p e r i m e n t a l and T h e o r e t i c a l S/N R a t i o s F i n a l Experiment a l S/N S/N ratio  Boxcar Enhance ment f a c t o r  10 6  II  0.08 0.2  130 30  V'  O.U1  22 7  Nb*> n  1 3  30 30  30 90  Ga n n  0.006 O.OM-  k  a 7  s  II  8 7 ,  0.09  12  0.02 0.3 1  Sn"<  0.1  90  9  Natural tin  0.01  In'" it  11  7  Temperature "K.  15 10  295 78  30  ^5  295 78  80 50  295 78  3  295 78 303  15  78  3  78  150  1.5  o.o5 0.1  12 12  0.5  l  295 78  Bi*°< n  0.02 0.2  7 7  0.1 1.5  295 78  Sb n  0.02 0.06  7 7  0.15  O.Oh  --  295 78  Re  0.01 0.02  7 7  0.07 0.15  —  295 78  i  137 The agreement between t h e c a l c u l a t e d and e x p e r i m e n t a l v a l u e s i s v e r y good c o n s i d e r i n g the number o f i l l quantities involved.  defined  The e x p e r i m e n t a l S/N r a t i o was o n l y c r u d e l y  measured and was n o t c o r r e c t e d f o r s p i n - s p i n r e l a x a t i o n .  If  t h i s were done, i t would double most of t h e e x p e r i m e n t a l S/N ratios.  The r e d u c t i o n i n S/N caused  was n o t a l l o w e d f o r e i t h e r .  by a c o u s t i c o s c i l l a t i o n s  I n n i o b i u m t h i s i s the main n o i s e  source a t 78°K. I f s p i n - s p i n r e l a x a t i o n i s allowed f o r , the experimental S/N i s u s u a l l y about f o u r times the e x p e r i m e n t a l v a l u e . of t h i s d i f f e r e n c e i s undoubtedly  Part  due t o i n c o r r e c t v a l u e s of  some parameters and t o t h e s i m p l i f i c a t i o n s i n t h e t h e o r y . remainder  of t h e descrepancy  The  i s p r o b a b l y because the sample  s u r f a c e i s n o t p e r f e c t l y smooth, b u t c o n t a i n s i r r e g u l a r i t i e s w i t h dimensions  much l a r g e r than t h e s k i n d e p t h .  These i n -  c r e a s e the e f f e c t i v e s u r f a c e a r e a and hence i n c r e a s e t h e induced s i g n a l .  P r o v i d e d t h a t t h e Q i s l i m i t e d by t h e e x t e r -  n a l c i r c u i t r e s i s t a n c e , t h i s w i l l i n c r e a s e t h e S/N by some f a c t o r i n t h e r e g i o n o f two.  I n any c a s e , t h e d i f f e r e n c e  between t h e t h e o r e t i c a l and e x p e r i m e n t a l S/N r a t i o s i s now known, so t h a t t h e t h e o r y can c o n f i d e n t l y be used t o p r e d i c t the expected  S/N r a t i o of a sample to w i t h i n a f a c t o r of two  i n t h e temperature  range i n w h i c h t h e s k i n e f f e c t i s normal.  F i g u r e h.la  Sweep C l o s e t o t h e r f P u l s e B  Figure ^ . l b  Sweep a t a Time About T^. a f t e r the r f P u l s e  T y p i c a l C h a r t Records o f a Sweep w i t h a Narrow Boxcar Gate Through the F r e e I n d u c t i o n  Tail.  o  6-  «  °  H P in (010) Plane  *  Ho aloncj  CoioJ Axis  Induction ^" Tail Amplitudes After % e  Second  Vulse.  (Arbitrary  Unit*).  3  ,  o  —r  -  0-5  — I —  1-0  1-5  2-0  T i m e Between Pulses (Vnilliseeonals).  Figure 4.3 Spih-Lattice "Relaxationlime Anisotropy in Isotopically Pure I n .  s  -r  Time Between Boxcar fi-afeqnd Second Pbtee (microseconds).  Fiqure 4 5  A Spin Echo.  -r ro  T h e ratio of the. +wo pulse wtdtks is ua  \  \ 1  -i 5 W i d t h of First "Pulse  Figure 4.6  to  1  (microseconds).  Vqriqtion  \  '5 .  =•  1 a©  r-  •  .  •  as  o-f Spin Echo Amplitude With the rf Pulse Widths. £  SO-]  —'Metallic /""^ ^Theoretical Variation / \ —t^on-rnetqlhc J 40  30  Amplitude of 5 pin EcUoj (Arbitrqrt^  Units).  10  Width o-f the Second Pulse  (microseconds).  Figure 4.7 Variation of t h e Spin Echo Amplitude With Second Pulse Width.  54 Ln. of Induction Tail Height (Arbifrqry Units).  40-  3-0  Time  100  ^microseconds)  —i— a.oo  3O0  400  500  Figure 4.8 Spin-Spin "RelaxaTionTme by Free Induction Decay. i—•  -r  Amplitude CAi-m"h-airtj Units). The  C u r v e is L o r e r i t z J a n W i t h a H a l f W i d t h o f  S  Units.  Frequency Deviation (Arbitrary Units)  Figure 4.7 Lorenfzian Line S h a p e in IsoTopicallLj F u r e l i a  as  40-  0=4  "  "  r  ' '  At^cjle o-f Ho From (010) Plane-  45°  -l  r-  h•T 1  rHo Magnetic Field  Figure 4.io Relative Orientation of the Crystal and Magnetic Field.  _L  30  Experimental Line Widths.  Ume Width From Anderson-Weiss Theory  2S  ;  Line WVdth (Arbitrary Until).  ao-  i5-  IO  IF  -\— 0°  ~l  r  —i—  50°  1  r  too"  i  r  Angle of Ho From the Toio) "Plqhc  Figure 4.U Anisotropy of the Line Width In Isotopically Pure Snl'f--"  -r  00  Lr>. o f E c h o Amplitude  50  Time  (microseconds)  Figure 4,12.  T IOO  Spin-Spin Relaxation Time by 5pin Echoes. -r  \0  150  3T  o <U  £ .Q  -•5 3  •5  o o  0 O ' O  -o-  (Li  K  a?  u  j o  CO 0) C  cr  o o  -O-  Z5X PvO  o  I/)  it  c  C  CO  'co  0)  err.  O  I °  —i—  o  cr  «J1 cr  • * <u £  Lai J  ?  151 CONCLUSION • B e g i n a t the b e g i n n i n g , and go on t i l l ' you come to the ends t h e n s t o p . ' - A l i c e i n Wonderland.  I n t h i s t h e s i s the f e a s i b i l i t y  of measuring T, and T  i n m e t a l s i n g l e c r y s t a l s a t l i q u i d n i t r o g e n temperature been e s t a b l i s h e d .  I t has been shown t h a t r e a s o n a b l y  measurements can be made p r o v i d e d Ta^SO/Us. and T, & A r e a s o n a b l y complete t h e o r y of the a p p a r a t u s  x  has  accurate 2secdeg.  has a l s o been  developed and e x p e r i m e n t a l l y c o n f i r m e d . The b i g f a i l u r e i n t h i s work has been the i n a b i l i t y eliminate acoustic o s c i l l a t i o n s .  U n t i l a way i s found o f  d r a s t i c a l l y r e d u c i n g them, measurements l i q u i d helium temperatures.  to  cannot be made a t  The a p p a r a t u s  has r e s t r i c t e d  u s e f u l n e s s u n t i l such measurements can be made.  The g a i n i n  S/N would p r o b a b l y be moderate, but would c e r t a i n l y be enough t o a l l o w the s u c c e s s f u l c o m p l e t i o n o f s e v e r a l experiments w h i c h are a t p r e s e n t e i t h e r i n d e c i s i v e , or i m p r a c t i c a l . more i m p o r t a n t r e a s o n f o r g o i n g to h e l i u m temperature the p u l s e d a p p a r a t u s  i s that  c o u l d then be used i n c o n j u n c t i o n w i t h  steady s t a t e a p p a r a t u s  on the same sample, so t h a t more v a r i e d  and a c c u r a t e d a t a c o u l d be o b t a i n e d . pulsed apparatus  In p a r t i c u l a r ,  the  can g i v e a more a c c u r a t e i d e a of the  shape than the m a r g i n a l o s c i l l a t o r s used i n the steady work.  The  line state  L i f e t i m e b r o a d e n i n g i s a l s o u n i m p o r t a n t a t h e l i u m tem-  perature.  A l t e r n a t i v e experimental procedures  involving  152 d i g i t a l d e v i c e s f o r i m p r o v i n g the S/N  r a t i o , such as the  E n h a n c e t r o n , might be more s a t i s f a c t o r y t h a n the boxcar  inte-  g r a t o r a t low t e m p e r a t u r e s . There a r e s e v e r a l e x p e r i m e n t s which c o u l d be done on the o  p r e s e n t a p p a r a t u s - a t 78 K.  The most o b v i o u s ones a r e to  a t t e m p t t o measure the a n i s o t r o p y of T, i n scandium and i n i s o t o p i c a l l y pure t i n .  The l a t t e r experiment would  involve  r e g r o w i n g the c r y s t a l a t a more s u i t a b l e o r i e n t a t i o n .  The  s e r i e s of measurements done on i s o t o p i c a l l y p u r e , and n a t u r a l t i n t o o b t a i n the exchange c o n s t a n t s c o u l d r e a d i l y be done on cadmium.  E x p e r i m e n t s of t h i s type on a s e r i e s of a l l o y s w i t h  s y s t e m a t i c a l l y v a r y i n g c o m p o s i t i o n s might g i v e q u i t e d e t a i l e d i n f o r m a t i o n on the pseudo-exchange i n t e r a c t i o n . S p i n echoes were a l s o observed and t h e i r b a s i c f e a t u r e s studied.  The f a c t t h a t they have a s h o r t e r T  a  t h a n the f r e e  i n d u c t i o n decay c l e a r l y r e q u i r e s a more i n t e n s i v e s t u d y . Measurements i n t i n o x i d e would  o b v i o u s l y be v e r y h e l p f u l .  Once the cause of t h i s has been f o u n d , s p i n echoes c o u l d u s e f u l f o r measuring T  a  have a s h o r t T* because f i e l d s i n the  be  i n those paramagnetic a l l o y s which of s p a t i a l l y v a r y i n g s t a t i c  magnetic  sample.  The a t t e n u a t i o n and phase s h i f t s of an r f f i e l d  pene-  t r a t i n g a m e t a l i n the normal s k i n e f f e c t r e g i o n a r e w e l l understood.  The t h e o r y f o r the r f p e n e t r a t i o n i n the case  where the e l e c t r o n mean f r e e p a t h approaches the s k i n depth i s n o t q u i t e so c l e a r c u t ( t h e anomalous s k i n e f f e c t ) .  Fairly  153 r i g o r o u s t h e o r i e s have been d e v e l o p e d , but they i n v o l v e assumpt i o n s c o n c e r n i n g the i n t e r n a l r e f l e c t i o n of e l e c t r o n s from the s u r f a c e of the m e t a l .  An e x p e r i m e n t a l i n v e s t i g a t i o n of the  anomalous s k i n e f f e c t r e g i o n would thus be d e s i r a b l e . NMR  Pulsed  i s c a p a b l e of d o i n g t h i s because the i n d u c t i o n t a i l h e i g h t  as a f u n c t i o n of r f p u l s e l e n g t h i s v e r y phase s h i f t s . t h e r e may  s e n s i t i v e t o the  I f a study of phase s h i f t s a l o n e was  contemplated,  be o t h e r ways of u s i n g the a p p a r a t u s w h i c h would  g i v e more, or b e t t e r , i n f o r m a t i o n on the phases.  Measurements  on p a r t i c l e s of v a r y i n g s i z e s would a l s o g i v e c o n s i d e r a b l e i n formation.  The r e s u l t s of steady s t a t e measurements a l s o  depend on the s k i n e f f e c t s , but they a r e l e s s s e n s i t i v e t o them and e x t r a c t i n g i n f o r m a t i o n on the phase s h i f t s i s even harder than i n the p u l s e d c a s e . I t i s l o g i c a l t o t r y t o extend experiments to the s u p e r c o n d u c t i n g r e g i o n . the s t a t i c magnetic f i e l d  of t h i s type  However, i n s u p e r c o n d u c t o r s  o n l y p e n e t r a t e s about  the r f f i e l d p e n e t r a t e s somewhat f u r t h e r .  500A , 0  while  Experiments u s i n g  s i n g l e c r y s t a l are thus n o t p o s s i b l e , except by u s i n g f i e l d c y c l i n g t e c h n i q u e s i n w h i c h they are a t a d i s a d v a n t a g e compared to powders.  However a study of the phase s h i f t s as the m e t a l  goes from the anomalous to the s u p e r c o n d u c t i n g r e g i o n i s q u i t e p o s s i b l e i n m e t a l s l i k e vanadium.  - C a l c u l a t i o n s show t h a t  t h e r e should be a s a t i s f a c t o r y S/N  r a t i o and temperatures  can  o  e a s i l y be h e l d t o w i t h i n 0.01 d e r s might g i v e a b e t t e r S/N  K. a t h e l i u m t e m p e r a t u r e s .  Pow-  r a t i o , but they do n o t have the  15*+ w e l l d e f i n e d and crystal.  NMR  e a s i l y s t u d i e d s u r f a c e s t r u c t u r e of a s i n g l e  measurements on p a r t i c l e s of v a r y i n g s i z e s ,  a l s o on f i l m s of v a r y i n g t h i c k n e s s e s , have r e v e a l e d a  and  Knight  s h i f t dependence on sample d i m e n s i o n s w h i c h i s i m p e r f e c t l y understood.  P u l s e d NMR  w i t h a c o h e r e n t system on t h i s type  sample might g i v e a d d i t i o n a l i n f o r m a t i o n . r e q u i r e a l o n g p e r i o d of c a r e f u l and to even see a  of  However, i t would  ingenious  experimentation  signal.  A f t e r the s p e c u l a t i v e n a t u r e  of the p r e c e e d i n g  para-  graphs, i t i s f i t t i n g t o c o n c l u d e w i t h some words of c a u t i o n . P u l s e d NMR  experiments are d i f f i c u l t ,  m i s e s , and  have a S/N  i n v o l v i n g many compro-  r a t i o w h i c h i s r a r e l y good enough t o g i v e  an answer of the r e q u i r e d a c c u r a c y .  A c r i t i c a l examination  any p r o s p e c t i v e experiment i s t h u s d e s i r a b l e .  In p a r t i c u l a r ,  most e x p e r i m e n t s can be done f a r more a c c u r a t e l y on powders. I t i s u s u a l l y o n l y where a n i s o t r o p i c p r o p e r t i e s are i n v o l v e d t h a t a s i n g l e c r y s t a l experiment may  be  of  worthwhile.  155  When the w r i t i n g of t h i s t h e s i s was  nearlng  completion  the a u t h o r became aware of the work of Gara (69). c e n t l y measured the s p i n - l a t t i c e and  He has r e -  spin-spin relaxation  times  i n metal s i n g l e c r y s t a l s by a p u l s e d method, but h i s e x p e r i mental procedure was  so d i f f e r e n t t h a t t h e r e has been l i t t l e  o v e r l a p w i t h t h i s work. He used an i n c o h e r e n t p u l s e system and a combined t r a n s mitter-receiver c o i l . system and  T h i s l a c k s the v e r s a t i l i t y of a  coherent  i s s u s c e p t i b l e t o a m p l i f i e r non-linearity„  s u f f e r s from the major d i s a d v a n t a g e the a m p l i t u d e  t h a t i n a T,  It  measurement  of the f r e e i n d u c t i o n decay f o l l o w i n g the second  p u l s e i s not a s i m p l e e x p o n e n t i a l f u n c t i o n of the p u l s e  separa-  t i o n , u n l e s s the two p u l s e l e n g t h s are i n a c e r t a i n r a t i o w h i c h i s determined  by a m i x t u r e of t h e o r y and experiment.  second p u l s e was  about -fit l o n g and  gave the maximum f r e e  i n d u c t i o n decay a m p l i t u d e w h i l e the f i r s t p u l s e was £mr  long,  necessary. boxcar  about  A c o m p l i c a t e d method of a n a l y s i n g the r e s u l t s The r e s t of the p u l s e c i r c u i t r y , a m p l i f i e r s ,  was and  i n t e g r a t o r were s i m i l a r t o those used i n t h i s work. Most of h i s measurements were made a t l i q u i d  temperatures.  helium  A c o u s t i c o s c i l l a t i o n s were a major problem, but  he a l m o s t e l i m i n a t e d them by s e v e r a l i n g e n i o u s One  The  of these was  techniques.  to c o a t the sample w i t h a l a y e r  r e s i n w i t h n y l o n f i l i n g s embedded i n i t .  of.epoxy  This quite e f f e c -  t i v e l y damped the a c o u s t i c o s c i l l a t i o n s , but had  the d e f e c t  156 t h a t t h e epoxy r e s i n was paramagnetic enough t o s i g n i f i c a n t l y s h o r t e n t h e f r e e i n d u c t i o n decay.  The o t h e r method was t o e t c h  narrow grooves about a m i l l i m e t r e deep i n the c r y s t a l . c o n s i d e r a b l y reduced the a c o u s t i c o s c i l l a t i o n s .  These  I t should be  noted t h a t i n s e v e r a l r e s p e c t s h i s a c o u s t i c o s c i l l a t i o n s behaved d i f f e r e n t l y t o those observed i n t h i s work.  This i s  p r o b a b l y because he mounts h i s sample by cementing one end of i t t o a h o l d e r , w h i l s t i n the p r e s e n t e x p e r i m e n t s the c e n t r e of the rod i s clamped and the ends a r e f r e e . He measured T, i n A l h.2°K.  a 1  and C u  6s  single crystals at  w i t h s e v e r a l d i f f e r e n t magnetic f i e l d  orientations.  W i t h i n h i s e x p e r i m e n t a l e r r o r of ~2% he d e t e c t e d no a n i s o t r o p y H i s v a l u e s were T, T = ( l 8 l - 0 « p ' 2 ) s e e . d e g . f o r a l u m i n i u m  i n T, .  a  and (1.275±O.Ol5)sec.deg. f o r copper.  B o t h o f these a r e i n ex-  c e l l e n t agreement w i t h t h e powder v a l u e s . when observed on the o s c i l l o s c o p e .  The S/N was about 20  F r e e i n d u c t i o n measurements  .were a l s o made on the c r y s t a l s and gave second moments w h i c h were i n good agreement w i t h v a l u e s measured by steady s t a t e methods.  E x p e r i m e n t s made w i t h v a r y i n g sample and c o i l  radii  showed t h a t the optimum S/N r a t i o o c c u r r e d when the r a t i o of the r a d i i was about 0 . 8 . t h e o r y developed i n t h i s  T h i s I s i n good agreement w i t h the thesis.  Gara's work has covered d i f f e r e n t e x p e r i m e n t a l a s p e c t s of the s u b j e c t t h a n those done i n t h i s work.  However, some  a s p e c t s of the t h e o r y of the a p p a r a t u s i n t h i s t h e s i s could be reviewed and extended i n the l i g h t o f h i s work.  In particular  157 some of the c o n c l u s i o n s and h e l i u m t e m p e r a t u r e s need  comments r e g a r d i n g w o r k i n g  modifying.  at l i q u i d  158 BIBLIOGRAPHY 'And out of o l d e bokes, i n good f e i t h , cometh a l l t h i s newe s c i e n c e t h a t men l e r e . 1  - Chaucer. (1)  Abragam, A., "The P r i n c i p l e s of N u c l e a r Magnetism," Oxford U n i v e r s i t y P r e s s , London (1961).  (2)  C l a r k , W.G., Rev. S c i . I n s t r . 3 £ , 316 (196*4-).  (3)  Sommerfeld, A., " E l e c t r o d y n a m i c s , " Academic P r e s s , New York  (196*0. OO  Blume, K . J . , Rev. S c i . I n s t r . 32, 1016 (1961).  (5)  Blume, R . J . , Rev. S c i . I n s t r . 32, 59+ (1961).  (6)  Hardy, W.N., Ph.D.  (7)  M i l i t a r y S t a n d a r d i z a t i o n Handbook S e l e c t e d Semiconductor C i r c u i t s , U.S. Department o f Defence, i960. Lowe, I . J . and B a r n a a l , D.E., Rev. S c i . I n s t r . 3}+, l*+3 (1963). M o t t , N.F. and J o n e s , H., "The Theory o f the P r o p e r t i e s o f M e t a l s and A l l o y s , " Dover P u b l i c a t i o n s I n c . , New Y o r k  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  U96\). (8) (9)  (1958).  (10)  "Handbook o f C h e m i s t r y and P h y s i c s , " ^ t h Ed., C h e m i c a l Rubber P u b l i s h i n g Co., C l e v e l a n d (1963).  (11)  H u n t e r , J . L , " A c o u s t i c s , " P r e n t i c e - H a l l I n c . , New York  (1957).  (12)  Van d e r Z i e l , A., " N o i s e , " P r e n t i c e - H a l l I n c . , New Y o r k (1956).  (13)  B a r n a a l , D. E. and Lowe, I . J . Phys. Rev. L e t t e r s 11, 258  (1963).  (ih)  Margenau, H. and Murphy, G.M., "The Mathematics of P h y s i c s and C h e m i s t r y , " 2nd Ed., D. Van Nostrand Co., I n c . , P r i n c e t o n (1955).  (15)  Spocas, J . J . and S l i c h t e r , C P . , Phys. Rev. 113, 1 +63(1959).  (16)  Rowland, T.J., " N u c l e a r Magnetic Resonance P e r g a m o n P r e s s , Oxford (1961).  1  i n Metals,"  159 (17)  Masuda, Y., J . Phys. Soc. Japan 12, 523 (1957).  (18)  Anderson, A.G.  and R e d f i e l d , A.G.,  (19)  O ' R e i l l y , D.E.  and Tsang, T., Phys. Rev. 128, 2639 (1962).  (20)  B u t t e r w o r t h , J . , Phys. Rev. L e t t e r s f , 305 ( I 9 6 0 ) .  (21)  Obata, Y., J . Phys. Soc. Japan 18, 1020 (1963).  (22)  K i t t e l , C , "Quantum Theory of S o l i d s , " John W i l e y and Sons I n c . , New Y o r k (1963).  (23)  M o t t , N.F.,  (25)  B u t t e r w o r t h , J . , P r o c . Phys. Soc. 83, 71 (196*+).  (26)  M a t t h e i s s , L.F., Phys. Rev. I2*±A, 971 (196*4-).  (27)  Y a f e t , Y. and J a c e a r l n o , V., Phys. Rev. 133A. 1630 (196*4-).  (28)  C l o g s t o n , A.M., G o s s a r d , A.C., J a c c a r i n o , V. and Y a g e t , Y., Phys. Rev. L e t t e r s £, 262 (1962).  (29)  B u t t e r w o r t h , J . , P r o c . Phys. Soc. 8 £ , 735 (1965).  (32)  Torgeson, D.R., 255 (1962).  (33)  F a w c e t t , E., Advances i n P h y s i c s ] J , 139 (196*4-).  (3*+)  H e w i t t , R.R. and W i l l i a m s , B.F., Phys. Rev. L e t t e r s 12, 216 (196*4-).  (35)  Y a f e t , Y., J . Phys. Chem. S o l i d s 21, 99 (1961).  (36)  K i t t e l , C , " I n t r o d u c t i o n to S o l i d S t a t e P h y s i c s , " 2nd Ed., John W i l e y and Sons, I n c . , New York (1956).  (37)  Ziman, J . M . , " E l e c t r o n s i n M e t a l s , " T a y l o r and F r a n c i s L t d . , London (1963).  (38)  Landau, L.D. and L i f s h i t z , E.M., " S t a t i s t i c a l Pergamon P r e s s , Oxford (1959).  (39)  Kubo, R. and Obata, Y., J . Phys. Soc. Japan 11, 5*4-7 (1956).  O+O)  S l i c h t e r , C P . , " P r i n c i p l e s of Magnetic Resonance," Harper and Row, New York (1963).  (*4-l)  Boon, M.H.,  Phys. Rev. 116, 583 (1959).  Advances i n P h y s i c s 12, 325 (196*0.  and Barnes, R.G.,  P h y s i c a 30, 1326  Phys. Rev. L e t t e r s  (196*0.  Physics,"  160 (4-2)  G a s p a r i , G.D., Shyu, W. and Das, T.P., Phys. Rev. 134-A, 852 (196*0.  0+3)  C l o g s t o n , A.M., J a c c a r i n o , V. and Y a f e t , Y., Phys. Rev. 13*fA. 650 (196*0.  (M+)  Pake, G.E., "Paramagnetic Resonance," W.A. Benjamin, I n c . , New York (1962).  (*f5)  W o l f f , P.A., Phys. Rev. 122, 8k (1963).  (4-6) K a d n o f f , L.P., Phys. Rev. 122, 2073 (1963). 0+7)  Obata, Y., J . Phys. Soc. Japan 18, 1020 (1963).  0+8)  H e b e l , L.C., Phys. Rev. 128, 21 (1962).  0*9)  B o r s a , F. and B a r n e s , R.G., Phys. Rev. L e t t e r s 12, 281 (1961+). ~~  (50)  Bloembergen, N. and Rowland, T . J . , Phys. Rev. £2> 1679 (1955).  (5D  T a b l e s o f E i g e n v a l u e s and E i g e n v e c t o r s o f t h e H a m i l t o n i a n D e s c r i b i n g t h e Combined S t a t i c Magnetic D i p o l e and E l e c t r i c Quadrupole I n t e r a c t i o n s o f a N u c l e a r L e v e l . , S t e f f a n , R.M., M a t t h i a s , E. and S c h n e i d e r , W., A.EG.-' (U.S.A.), D i v i s i o n o f T e c h n i c a l I n f o r m a t i o n TID-1574-9.  (52)  Kambe, K. and O l l o m , J . F . , J . Phys. Soc. Japan 11,50 (1956).  (53)  Schumacher, R.T., Phys. Rev. 112, 837 (1958).  (5*0  G o l d b e r g , W.I., Phys. Rev. ll£, 4-8 (1959).  (55)  J e e n e r , J . , E i s e n d r a t h , H. and Van S t e e n w i n k e l , R., Phys. Rev. 13 3A. 4-78 (196*4-).  (56)  Simmons, W.W. and S l i c h t e r , C P . , Phys. Rev. 121, 1580 (1961) .  (57)  A s h c r o f t , N.W., and W i l k i n s , J.W., Phys. L e t t e r s 14-, 285 (1965).  (58)  Asayama, K. and I t o h , J . , J . Phys. Soc. Japan 12, 1065 (1962) .  (59)  Schone, H.E, and O l s o n , P.W., Rev. S c i . I n s t r . ^6, 84-3 (1965).  (60)  Van Ostenburg, D.O., Spokas, J . J . and Lam, D.J., Phys. Rev. 132A, 713 (1965).  161 (61)  A p p e l , J . , Phys. Rev. 13 9A,  1536 (1965).  (62)  Hahn, E.L., Phys. Rev. 80, 580 (1950).  (63)  Masuda, Y., J . Phys. Soc. Japan ] J , 597 (1958).  (&)  J o n e s , E.P. and W i l l i a m s , D.LI., Phys. L e t t e r s 1, 109 (1962).  (65)  J o n e s , E.P., Ph.D.  T h e s i s , U n i v e r s i t y of B r i t i s h Columbia  (1962). (66)  Anderson, P.W. and W e i s s , P.R.,  (67) (68)  M i a s e k , M., Phys. Rev. 1^0, 11 (1963). Karimov, Y.S, and S c h l e g o l e v , J . F . , JETP JQ>  (69)  Gara, A.D.,  Ph.D.  Rev. Mod. Phys. 2 £ , 269 (1953). 908  (1961).  T h e s i s , Washington U n i v e r s i t y (1965).  162 APPENDIX I DISTORTION IN THE PHASE SENSITIVE DE&CTION SYSTEM A(tW<ut+<|))  «-  t  o-  $COSlU t 0  "Reference  t V*  i F i g u r e 1.1  V.ft)  i  i.  E q u i v a l e n t C i r c u i t of t h e Phase S e n s i t i v e D e t e c t o r  I n the phase s e n s i t i v e d e t e c t i o n system t o be a n a l y s e d a s m a l l time v a r y i n g s i g n a l of frequency u> and phase a n g l e l i n e a r l y added t o a much l a r g e r r e f e r e n c e voltage  (j) i s  s i g n a l to give a t o t a l  v(f) o f • v ( f l = A ( t ) e o s ( u » t + 4>)+Bcosu>ot = [A + B* +2ABcos (nt + $ )  cos [w»t  4  where  / I = OJ ~u  -tan"'{  0  (t)}] ,  0  P r o v i d e d u>o»ft, the time dependent phase a n g l e S ( t ) w i l l cause n e g l i g i b l e frequency t h a t the v o l t a g e a f t e r  or phase m o d u l a t i o n o f the s i g n a l , so  the s i g n a l has been r e c t i f i e d and the h i g h  f r e q u e n c y c a r r i e r f i l t e r e d out i s V(t)= = If  [ A ( t ) = o s ( n U $ )  i t i s assumed t h a t - ^ - ^ f ,  first  +  B ] [ l  +  ^ ± | ] .  t h e square r o o t can be expanded  order to give •V(t)  The  +B* +2ABcos(flt+ $ )]*  = [A(t)cosUU+(t>)+B]  [l+  , ?*ft"; £L» & l  t (  •  system i s A . C . c o u p l e d so the D . C . component i s removed,  to  g i v i n g the output s i g n a l as V (t) 0  =A(t)cosCat+ d?) + ^(a^)im^m^U3]  There i s thus an e r r o r e x c e p t when s i n ( i l t + 4>)=0. e r r o r can become s e r i o u s f o r s i n ( A t + (J) ) the e r r o r near i t s maximum v a l u e of  . This  ±1 s i n c e n o t o n l y i s but the  signal  Acos(/lt+ (J) ) i b O , so the f r a c t i o n a l e r r o r i s v e r y l a r g e .  I f the  a p p a r a t u s i s e x a c t l y on resonance s o i l = 0 the e r r o r i s reduced because t h e r e i s then no time dependence i n the term i n v o l v i n g B so the A.C. c o u p l i n g removes i t .  164APPENDIX I I DETAILS OF THE SAMPLES USED (i)  Aluminium S i n g l e C r y s t a l . T h i s was s u p p l i e d by Semi-elements, I n c . , P e n n s y l v a n i a  and i s . a c y l i n d e r 5 cm. l o n g by 0.7  cm. i n d i a m e t e r .  There were  a c t u a l l y two c r y s t a l s i n t h e sample, one b e i n g about one e i g h t h the s i z e of t h e o t h e r .  The s u r f a c e i s rough and unetched and t h e  p u r i t y i s unknown. Aluminium has a f a c e c e n t r e d p o l e moment. (ii)  c u b i c l a t t i c e and a quadru-  There i s one i s o t o p e w i t h s p i n  -|> .  Vanadium S i n g l e C r y s t a l . The zone r e f i n e d s i n g l e c r y s t a l was grown by t h e U.B.C.  M e t a l l u r g y Department.  The sample i s 5 cm. l o n g and has an  average d i a m e t e r o f 0.6  cm.  not etched.  I t had a smooth s u r f a c e and so was  A c h e m i c a l a n a l y s i s gave the i m p u r i t i e s i n p a r t s  per m i l l i o n as Oxygen Carbon Nitrogen Hydrogen V' s  160 136 318 7.2.  has an i s o t o p i c abundance  centred cubic l a t t i c e .  o f 99,7% and forms a body  I t i s a t r a n s i t i o n metal w i t h s p i n  and  a quadrupole moment. (iii)  Indium S i n g l e C r y s t a l . I t i s a 99.999$ pure c y l i n d r i c a l sample bought from  M e t a l s R e s e a r c h Co., Cambridge, E n g l a n d .  I t has t h e t e t r a g o n a l  a x i s p e r p e n d i c u l a r t o the sample a x i s t o w i t h i n 2 ° . l o n g c y l i n d e r was etched down t o 0.9  The 1.3  cm.  cm. d i a m e t e r by an e t c h of  16? one p a r t HC1 t o twenty p a r t s of e t h y l a l c o h o l .  X-raying  t h a t t h e e t c h i n g removed a s l i g h t l y p o l y c r y s t a l l i n e  showed  surface  structure. The main i s o t o p e i s In*' w h i c h has 96$ 5  and a quadrupole moment.  The l a t t i c e i s a f a c e c e n t r e d  gonal s t r u c t u r e w h i c h can a l s o be regarded l a t t i c e elongated (iv)  abundance, s p i n tetra-  as a f a c e c e n t r e d  cubic  by 7% a l o n g one a x i s .  Bismuth S i n g l e C r y s t a l . The  sample i s a c y l i n d e r 3 cm. l o n g by 0.65  cm. d i a m e t e r  purchased from M e t a l s R e s e a r c h Co* w h i c h was n o t e t c h e d .  The  t r i g o n a l a x i s i s w i t h i n 2° of t h e p e r p e n d i c u l a r t o the c y l i n d r i cal axis. I t is 100$ abundant w i t h  spin-4>  a  quadrupole moment and  c r y s t a l l i z e s i n a rhombohedral s t r u c t u r e . (v)  Rhenium S i n g l e C r y s t a l . T h i s was s u p p l i e d by Semi-elements and i s 2.8  0.3  cm. d i a m e t e r .  cm. l o n g by  The symmetry a x i s i s a l i g n e d t o w i t h i n 2 ° of  the p e r p e n d i c u l a r t o the c y l i n d r i c a l a x i s .  The c r y s t a l i s zone  r e f i n e d and was n o t e t c h e d . There a r e two i s o t o p e s , Re"*and Re * 1  i s o t o p i c abundances r e s p e c t i v e l y . a quadrupole moment. and (vi)  i sa transition  7  w i t h 37$  and 63$  They b o t h have s p i n \ and  I t has a hexagonal c l o s e packed s t r u c t u r e metal.  Antimony S i n g l e C r y s t a l . I t i s a c y l i n d e r 1.6  cm. l o n g by 0.9 cm. d i a m e t e r c u t  from a zone r e f i n e d s i n g l e c r y s t a l s u p p l i e d by Cominco L t d . ,  166 T r a i l , B.C.  The sample i s 99.999$ pure.  c r y s t a l i s unknown. one.part concentrated  The o r i e n t a t i o n of the  The c r y s t a l was etched w i t h a s o l u t i o n of n i t r i c acid to four parts hydrochloric a c i d .  T h i s e t c h i s v e r y f a s t i f the sample i s a l l o w e d (vii)  t o become heated.  Gallium Single Crystal. The c r y s t a l was grown i n t h i s l a b o r a t o r y by K. N i l s e n  from 99.9999$ pure g a l l i u m .  X - r a y i n g determined t h a t i t was a  s i n g l e c r y s t a l w i t h i t s l a t t i c e symmetry a x i s o r i e n t a t e d a t 30 t o the c y l i n d r i c a l a x i s .  I t i s 2 cm. l o n g by 0.9 cm.  B o t h of the i s o t o p e s Ga moment.  6<?  and Ga  71  i n diameter.  have a quadrupole  They a r e 60$ and 4-0$ abundant r e s p e c t i v e l y and c r y s -  t a l i z e i n an o r t h o r h o m b i c s t r u c t u r e . (viii)  Niobium S i n g l e C r y s t a l . The c r y s t a l i s a zone r e f i n e d c r y s t a l 5 cm. l o n g and  0.6  cm. i n d i a m e t e r grown by the U.B.C. M e t a l l u r g y  No e t c h i n g was done on i t .  Department.  The i m p u r i t i e s , i n p a r t s per m i l l i o n ,  are Oxygen Carbon Nitrogen Hydrogen Tantalum Zirconium Tungsten Nb  ,a  35 4-0 4-0 2 500 500 300  has 100$ l s o t o p i c abundance, a s p i n of  p o l e moment, and c r y s t a l l i z e s i n a body c e n t r e d c u b i c (ix)  , a quadrulattice.  Natural Tin Single Crystal. T h i s sample was  s u p p l i e d by M e t a l s R e s e a r c h L t d . i n the  form o f a c y l i n d e r 3 cm. l o n g and 0.9 cm. d i a m e t e r of 99-999$  167 purity.  The symmetry a x i s i s a l i g n e d  c y l i n d r i c a l a x i s o f t h e sample.  a t r i g h t angles t o the  An e t c h o f one p a r t HN0 , one 3  p a r t w a t e r , and two p a r t s acetone was used t o remove t h e s u r f a c e l a y e r p r i o r to the experiments. There a r e two main i s o t o p e s , each w i t h s p i n S n ' ^ w i t h 7,7% i s o t o p i c abundance and Sn"* w i t h 8*7%  These a r e abundance.  N e i t h e r o f the n u c l e i has a quadrupole moment, (x)  I s o t o p i c a l l y Pure T i n S i n g l e C r y s t a l . The c r y s t a l of i s o t o p i c a l l y pure t i n was grown by Schone  and O l s e n (59).  I t consists  of a l a y e r about 0.25  mm.  wrapped around a copper c o r e about 7 mm. i n d i a m e t e r . of t h e S n " Sn" . 7  9  thick The p u r i t y  i s unknown, b u t presumably the main i m p u r i t y would be  The symmetry a x i s i s a t an a n g l e of 28° t o t h e c y l i n d r i c a l  a x i s o f t h e copper c o r e .  The X - r a y s t a k e n t o d e t e r m i n e t h e  p o s i t i o n of t h i s a x i s showed t h a t t h e r e was n e g l i g i b l e  distortion  of the s i n g l e c r y s t a l and t h a t t h e r e was no p o l y c r y s t a l l i n e surface  layer.  168  APPENDIX I I I THE SIGNAL INDUCED IN THE PICKUP COIL (a) N o n - m e t a l l i c Sample. Assume t h a t t h e sample i s i s o t r o p i c and  non-ferromagnetic,  so t h a t t h e n u c l e a r magnetic moment a t any p o i n t i n t h e sample i s M, =X H 0  0  •where N i s t h e number of atoms/m , X i s t h e gyromagnetic r a t i o , 3  I t h e n u c l e a r s p i n , T t h e sample temperature,  H  0  the applied  s t a t i c magnetic f i e l d , and X i s the s t a t i c n u c l e a r magnetic sus-' 0  ceptibility. In e q u i l i b r i u m M  0  i s aligned along the z a x i s .  r f magnetic f i e l d B=2B, coswt, a t t h e r e s o n a n t p e r p e n d i c u l a r t o H„ causes M  s  A linear  frequencytu,\applied  t o precess i n the z y plane a t a  f r e q u e n c y TB, , so t h a t a f t e r a time t i t makes an a n g l e <*= TB.f w i t h the z a x i s ( 1 ) .  The p r o j e c t i o n o f M  p  i n the xy plane i s  M=M sine* 0  In t h e l a b o r a t o r y frame t h i s magnetic moment r o t a t e s a t a f r e quency w, i n d u c i n g a s i g n a l p r o p o r t i o n a l t o Mcoswt i n a p i c k u p coil. Now c o n s i d e r t h e sample i n s i d e a c o i l w i t h n t u r n s w h i c h i s perpendicular to H v o l t a g e v induced v  0  and has an e f f i c i e n c y f a c t o r 07. The  i n t h e c o i l i s g i v e n by F a r a d a y ' s e q u a t i o n as =r^E.dl  The s u r f a c e i n t e g r a l i s over t h e c r o s s s e c t i o n a l area of t h e c o i l .  169 The  line integral  i s taken over a path through  t h e sample p a r a l l e l  to the c o i l a x i s and then back v e r y c l o s e t o t h e o u t s i d e s u r f a c e of t h e c o i l .  Along  this £  path M s i n <A c o s ivt e  i n t h e sample and i s z e r o o u t s i d e i t , i f end e f f e c t s a r e n e g l e c t e d . .*. v^ori^nMoAsinricosu/t, where A i s t h e c r o s s s e c t i o n a l a r e a of t h e sample, (b) M e t a l l i c  Sample.  The problem i s t o c a l c u l a t e t h e d i s t r i b u t i o n and magnetic f i e l d s induced  of c u r r e n t s  i n t h e m e t a l by t h e o s c i l l a t i n g mag-  T h i s i n d u c e s a c i r c u l a t i n g c u r r e n t J=°% w h i c h  n e t i c moment M.  a magnetic f i e l d H o p p o s i n g M,.  generates  The  c i r c u l a t i n g c u r r e n t i s g i v e n by F a r a d a y ' s e q u a t i o n as crVxE^^CM+Hj,  w h i l e H i s g i v e n by  These e q u a t i o n s n e g l e c t t h e d i s p l a c e m e n t ' c u r r e n t moment X $ induced  by t h e c i r c u l a t i n g c u r r e n t .  and t h e magnetic  They a l s o assume  t h a t t h e metal i s n o t i n t h e anomalous c o n d u c t i o n r e g i o n .  Stan-  dard v e c t o r m a n i p u l a t i o n now g i v e s  Assume t h a t M^goCexpCiu/t). V * H = 1 S" (M+H), wkere  V = cv^cr. a  a  Now c o n s i d e r an i n f i n i t e p l a n e m e t a l  sample w i t h magnetic  moments M ( z ) a t a depth z below t h e s u r f a c e and w i t h M^=M = 0. x  The  t  equation s i m p l i f i e s to =A  a  [H,(z)+M (z)], s  170 where  A =  f' + {50  .  The s o l u t i o n of t h i s e q u a t i o n i s  (1*0  H (z) =exp (Az) [c* +£AJM( Z ) exp ( - A z ) d zj X  +exp(-Az)[C -£A|M(z)exp(Az)dz]. 3  C,. and C  3  are a r b i t r a r y i n t e g r a t i o n  t i o n s are now f i t t e d  f o r the  constants.  The boundary c o n d i -  s p e c i a l case of an i n f i n i t e l y  sheet of magnetic moments a t a d e p t h m below the i.e.  thin  surface.  M ( z ) = £>(m-z), x  where S(m-z) i s a D i r a c d e l t a H (z)  function.  =C exp(Az)+C exp(-Az)  x  a  f o r z < m,  5  = [C +£Aexp(-Am)] exp(Az)+[c>-£Aexp(Am)] e x p ( - A z ) _ f o r z > m, A  | H ( z ) d z =j 3C  [c exp(Az)+C e x p ( - A z ) ] dz+ ] f(C +-|-Aexp(-Am)) a  3  A  JO  J  0  •  *"  -  expCAzJ + CCj -iAexp(Am) ) e x p ( - A z ) j dz = •^Jo exp(Az)-C3exp(-Az)J + J^-^+£exp(-Am)j- exp(Az) a  -{•^--iexp(Am)} e x p ( - A z ) ] ^ . To keep the  total flux finite C  a  i t i s n e c e s s a r y to  put  =-|-Aexp(-Am). r  r*  H (z)dz =£exp(-Am)-l+-gx  The s i g n a l induced yxg  i n the p i c k u p c o i l i s o b t a i n e d  from.  - / ^ ( M + H ) .  The a s s u m p t i o n i s made t h a t S « R , where R i s the r a d i u s of c y l i n d r i c a l sample. round  the  i n a p i c k u p c o i l wound  sample i s  .'. If  The v o l t a g e induced  the  v *-i2rfRynjuu>  [M (Z)+HJC  vcG'lexp (-Am) +  .  x  (z)Jdz.  the magnetic moments are a t a l a r g e depth below the  s u r f a c e of  171 the m e t a l t h e r e should be no induced  v o l t a g e because the c i r c u -  l a t i n g c u r r e n t s w i l l , by Lenze's Law, c o m p l e t e l y  s h i e l d the  magnetic moments. i.e.  0+ C v  tX^ [-^- H e x p ( - A m ) ] . =0.  3  s-itp^n^cuRexpC-Am).  0  T h i s i s t h e v o l t a g e induced  by a d e l t a f u n c t i o n magnetic moment  a t a depth m below t h e s u r f a c e .  The v o l t a g e induced  by a d i s -  t r i b u t i o n M(m) of magnetic moments i n t h e m e t a l i s g i v e n by v =  I v (m)M(m)dm f* 0  = TTnyu«ntuRJM(z)exp[-(l+i)^-]dz.. I t i s now n e c e s s a r y field  t o c a l c u l a t e M(z) a f t e r a r f magnetic  . aBiexp(itut) has been a p p l i e d p a r a l l e l t o t h e s u r f a c e of F o r s i m p l i c i t y , assume t h a t to i s  the sample f o r a p e r i o d t . the n u c l e a r r e s o n a n t  frequency.  The magnetic f i e l d  a t a depth z  i n the m e t a l i s (3) B(z) = 2 B , e x p [ - ( l + i ) ^ ] e x p ( i c o t ) . In  t h e r o t a t i n g r e f e r e n c e frame i t i s a s t a t i c magnetic f i e l d  B , e x p ( - ^ ) l y i n g i n the plane p e r p e n d i c u l a r to H tan"  1  (vft-) t o t h e f i e l d B, (0)  0  a t t h e s u r f a c e o f the m e t a l .  magnetic moment M ( z , t ) a t r i g h t a n g l e s  t o B,(0) i s  M (z,t)=M,(t)sln[TB texp(-^)  ]cos(^).  s  a t an a n g l e  l  The  .*. v ( t ) =TTywtu ny RMo ( t ) £ e x p ( - ^ ) s i n ( Y T B , e x p ( - ^ ) ) c o s ( ^ ) d z . a  T h i s e x p r e s s i o n can be g e n e r a l i s e d t o the case when t h e a p p l i e d r f p u l s e i s o f f resonance and t h e phase r e f e r e n c e a x i s makes an a r b i t r a r y angle w i t h B , ( 0 ) .  An i m p o r t a n t f e a t u r e o f t h i s  equation  172 i s t h a t the induced v o l t a g e i s of the form M ( t ) F ( z ) , where M ( t ) x  x  i s the n u c l e a r magnetic moment a t the s u r f a c e seconds a f t e r  of the sample  a p p l i c a t i o n of an r f p u l s e and F ( z ) i s the  i n v o l v i n g time independent phase and a t t e n u a t i o n f a c t o r s . e v o l u t i o n depends o n l y on Ti and T  a  t  integral The time  so t h a t measurements of these  q u a n t i t i e s made on s i n g l e c r y s t a l s should g i v e the same v a l u e s  as  those made on powders.  is  I n p a r t i c u l a r the f r e e  n o t d i s t o r t e d by any phase  i n d u c t i o n decay  effects.  T h i s d e r i v a t i o n has i g n o r e d the d i p o l a r magnetic This i s v a l i d  i n the r e g i o n i n w h i c h the r f f i e l d  than the d i p o l a r f i e l d comes.  i s much g r e a t e r  and from w h i c h the m a j o r i t y of the s i g n a l  The o n l y measurements which might be a f f e c t e d  sence of the d i p o l a r f i e l d r o t a t i n g reference  field.  by the  pre-  are those i n v o l v i n g r e l a x a t i o n i n the  frame where the r f f i e l d  must be l a r g e .  The i n t e g r a l y= j e x p ( - x ) sin(T'£B e'" )cos 'xdx can be e v a ,t  a  (  l u a t e d by expanding sin(lPTB,e*) by t e r m .  i n a s e r i e s and i n t e g r a t i n g  term  This gives  This series  i s q u i t e s a t i s f a c t o r y f o r numerical computation u n t i l  TB,- t"-V 10 r a d i a n s when the number of terms, r  and the number of  s i g n i f i c a n t f i g u r e s i n each term i n c r e a s e s r a p i d l y .  In t h i s d e r i -  v a t i o n the sample has been t r e a t e d as a c y l i n d e r w i t h a p e r f e c t l y smooth s u r f a c e and a r a d i u s much g r e a t e r than i t s s k i n  depth.  T h i s i s a good a p p r o x i m a t i o n to the r e a l samples, p r o v i d e d t h a t they have no s u r f a c e i r r e g u l a r i t i e s whose d i m e n s i o n s are a p p r o x i mately e q u a l to the s k i n  depth.  173 APPENDIX IV MEASUREMENT OF ABSORPTION AND DISPERSION MODES WITH A PULSED NMR APPARATUS Let the free induction decay signal after a 9 0 r f pulse G(t)coscj t be fed into a phase sensitive detector using a ref0  erence frequency w and then r e c t i f i e d 8  0  The output signal i s  proportional to G(t) [ cos(flt)cos4>+sin(flt)sin(j) ]  s  where  is  the phase difference between the reference frequency and the induction s i g n a l , and f l = u>-U)  0o  The time o r i g i n for the signal  i s the midpoint of the r f pulse (13)?  not the end as stated by  <-THe rf Pulse  Clark ( 2 )  *rfrft)  T h e T r u e Time On£rn___  Figure IV.1  The Amplifier Output  The boxcar integrator integrates the applied signals for the duration of the gate width (6),. so that i f the gate covers the entire free induction decay the boxcar output i s V  = K  G(t) [cosAtcos()) + sinfit sinCJ)] dt,  Jo where K i s a constant.  "X"(fL)oC and  For most free induction decays  G(t)cosntdt  X (fl )oC\ G(t) sin(flt)dt, where X and X are the real and  imaginary parts of the r f s u s c e p t i b i l i t y . . A i s proportional to the absorption mode measured by steady state apparatus. V o C ' X ' t f D c o s ^ +X'Cn)  sin$.  By correctly setting (()either X ' o r X ' c a n be obtained.  Linearly  sweeping the magnetic f i e l d through resonance then gives "X' or  17h "X." on the c h a r t r e c o r d e r i d e a l case o u t l i n e d above.  C l a r k d e r i v e d the e q u a t i o n s f o r  the  However, t h e r e are a c o n s i d e r a b l e  number of i n s t r u m e n t a l d i s t o r t i o n s w h i c h l i m i t the u s e f u l n e s s  of  t h i s method,, (i) of  Because of the f i n i t e w i d t h of the r f p u l s e , and the dead time the a p p a r a t u s ,  the b o x c a r gate s t a r t s a t a t i m e . ' t ' s o t h a t  the  boxcar output i s a c t u a l l y  \/ a K | ° ° G ( t ) [ cos(nt)cos(J) +sin(fit)sin(}> ] d t . K' [X"(Q)CO8$  =  -rX/Cfl ) sin<J>] - K ' p a C t ) [ c o s ( n t )  eos$ +sin(Qt)sin(j) ]  dt.  U n l e s s G ( t ) i s known, the c o r r e c t i o n term cannot be e v a l u a t e d . However, i t i s easy to see the e f f e c t  of the i n s t r u m e n t a l d i s t o r -  t i o n when s i n (j)=0. .  v  A  y y-~M—Start  y  x  /  /  V  /  o-f iVie Boxccr Grate.  ' " -  /, Close to ftesoner^V--"  -  i£'  /  ^  Tirrvg  ^ _ ^ ^OfF Resonance  Figure IV.2  Effect  of the Deadtime  C l o s e to resonance the unmeasured area d e c r e a s e s q u i t e s l o w l y w h i l e the measured area d e c r e a s e s q u i t e r a p i d l y , s i n c e is  i n the t a i l of the decay which i s much more s e n s i t i v e  s l i g h t s h i f t s o f f resonance.  o f f resonance  However, a t a frequency  one p o s i t i v e q u a r t e r c y c l e i s unmeasured  t h a t the output goes n e g a t i v e . t o be n a r r o w e d .  to  Thus t h e r e w i l l be l i t t l e d i s t o r t i o n  of the boxear output c l o s e to r e s o n a n c e . (4T*)'  it  so  T h i s causes the a b s o r p t i o n s i g n a l  As the frequency goes f u r t h e r  and f u r t h e r  from  resonance i t i s e a s i l y seen t h a t the output w i l l c o n t i n u a l l y  175 oscillateo  T h i s i s the main form of i n s t r u m e n t a l d i s t o r t i o n , ,  The r e q u i r e m e n t f o r n e g l i g i b l e d i s t o r t i o n i s o b v i o u s l y t o have the f r e q u e n c i e s  f ± 0  f a l l i n g w e l l o u t s i d e the (ii)  I f X"(fl)  (t')' line.  i s t o be the u n d i s t o r t e d a b s o r p t i o n mode, of f l f o r a l l v a l u e s of f l w i t h i n  l i n e w i d t h s of the r e s o n a n c e . e  This requires  i n the r o t a t i n g r e f e r e n c e  t h a t TH, » T."o  frame.  G(0)  several  t h a t over t h i s r e g i o n ,  The c o n d i t i o n f o r t h i s  is  T h i s i s the normal c o n d i t i o n f o r p u l s e d NMR and i s  a c t u a l l y i m p l i e d by the p r e v i o u s c o n d i t i o n T » t ' >  must be r o t a t e d (iii)  goes n e g a t i v e  i . e . T* » T ' .  must be independent  H, =£= H  a t w h i c h the output  t h r o u g h 90° i n a time l e s s than  9  s i n c e the  spins  T'.  The method of phase s e n s i t i v e d e t e c t i o n used i n t h i s  ap-  p a r a t u s a l s o causes some d i s t o r t i o n . The i n p u t t o the boxcar integrator  i s (Appendix I) V ( t ) = G ( t ) c o s ( f l t + <J> ) + sm*ffltt$)C&ft)cosfllt+tfl-i-B3 ,« a[c.osf.nT.+$")-t-i^p  If f l -  o ( i . e . , a b s o r p t i o n mode;, e x a c t l y on resonance)  no d i s t o r t i o n , but as soon as the f r e q u e n c y the o u t p u t i s d i s t o r t e d ,  there i s  i s o f f resonance  g i v i n g a b a s e l i n e s h i f t . An e x a c t  (fl#o) cal-  c u l a t i o n of t h i s b a s e l i n e s h i f t i s messy s i n c e the b o x c a r gate p a r t i a l l y integrates is f a i r l y  it.  I t t u r n s out t h a t the b a s e l i n e  shift  s m a l l and o s c i l l a t e s q u i t e s l o w l y and so causes l i t t l e  trouble. If  desired,  the b a s e l i n e s h i f t can be e l i m i n a t e d by  measuring two a b s o r p t i o n s i g n a l s w i t h a phase d i f f e r e n c e between them $  ?  of  l80°  and then s u b t r a c t i n g one from the o t h e r . Changing  by 1 8 0 ° r e v e r s e s  the p o l a r i t y of the a b s o r p t i o n s i g n a l b u t  to a f i r s t approximations, does n o t a f f e c t w h i c h I s thus e l i m i n a t e d by the  the b a s e l i n e  subtraction.  shift,  9  APPENDIX  CIRCUIT  V  DIAGRAMS  0-O5 To  V  -I70V.  &crte Pube In  Figure 51.1 q. T h e Grated C o h e r e n t  Oscillator.  Tower  Amplifier  177b  178  I2AX7  ECC82  From the Boxcar Otate (O  W A V *--11/  n  M  1  4 H h (g  AAAAAA-  FDIOD  Frwnthe Timer Tulse. Grenerator  fl4* PPIOO  ?  7>  4, ^ To Event Marker  ure  The Coincidence Timinq Unit.  >/aa  183  3XOca9  Figure V.7  R e g u l a t e d F i l a m e n t Power S u p p l y  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085527/manifest

Comment

Related Items