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

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PULSED NUCLEAR MAGNETIC RESONANCE IN 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 of New Zealand, 1961. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of P h y s i c s . We accept t h i s t h e s i s as conforming to the 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 Bri t i sh 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 Bri t ish Columbia Vancouver 8, Canada Date Pes \Cjb*>  The University 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 LESLIE ALLAN McLACHLAN B.Sc, University of New Zealand M.Sc.(Hons), University of .New Zealand THURSDAY, DECEMBER 9th, 1965, AT 2:30 P.M. IN ROOM 100, HENNINGS BUILDING COMMITTEE IN CHARGE Chairman: I. McT, Cowan External Examiner: A. G. Redfield I.B.M„ Watson Laboratory, New York Research Supervisor: D. L l . Williams of 1960 1961. M.' Bloom L, W, Reeves C. F. Schwertfeger E; Teghtsoonian B. G. T u r r e l l D. L l . Williams PULSED NUCLEAR MAGNETIC RESONANCE IN A METAL SINGLE CRYSTAL-ABSTRACT A pulsed nuclear magnetic resonance spectrometer using phase s e n s i t i v e detection was constructed f o r use on metal s i n g l e c r y s t a l s . Its c a p a b i l i t i e s and l i m i t a -tions were established 3 both experimentally 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 function of temperature i n aluminium, niobium and vana-dium sin g l e crystals,, An unsuccessful attempt was made to measure the anisotropic s p i n - l a t t i c e r e l a x a t i o n time i n an i s o t o p i c a l l y pure t i n sin g l e c r y s t a l . The orien-t a t i o n dependence of the t i n spin-spin r e l a x a t i o n time was measured and analysed i n terms of the random f l u c -tuation model of Anderson and Weiss. Values for both the pseudo-dipolar and psuedo-exchange constants were obtained„ Spin echoes were observed i n the i s o t o p i c a l l y pure t i n and were used to measure the spin-spin r e l a x a t i o n time. This was shorter than that measured from the free induction decay. The reason f o r t h i s could not be determined, GRADUATE STUDIES F i e l d of Study; Nuclear Magnetic Resonance Quantum. Theory of Solids R, Barrie Advanced Topics i n So l i d State Physics D. L L Williams Advanced Magnetism M. Bloom Low Temperature Physics J 0 B„ Brown S t a t i s t i c a l . Mechanics R, Barrie Related Studies E l e c t r o n i c Instrumentation F„ K. Bowers PUBLICATIONS L. A. McLachlano "Thermoluminescent Emission Spectra of X-ray Irradiated A l k a l i Halides". J . Phys. Chem. Solids 23, 1344 (1962) . i i A B S T R A C T 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 metal s i n g l e c r y s t a l s w i t h a pulsed nuclear magr/etic reson-f ance apparatus at both room and l i q u i d nitrogen- temperatures. . I The values obtained f o r aluminium and vanadium *agreed w e l l w i t h the values given i n the l i t e r a t u r e f o r powdered samples. The niobium value was s l i g h t l y lower than the most r e l i a b l e powder value, 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 to see i f any anisotropy could be detected i n the s p i n - l a t t i c e r e l a x a t i o n time. No anisotropy could be detected, but the c r y s t a l o r i e n t a t i o n used was so unfavourable that an anisotropy of l e s s than about 50% could not be detected. The s p i n - s p i n r e l a x a t i o n time was measured i n the 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 o r i -e n t a t i o n s . These showed that exchange narrowing occurred. With a s u i t a b l e choice of operating c o n d i t i o n s , the apparatus measured the equivalent of the abso r p t i o n mode i n steady s t a t e nuclear magnetic resonance as a f u n c t i o n of magnetic f i e l d o r i e n t a t i o n . This was combined w i t h the s p i n - s p i n measurements to give the complete o r i e n t a t i o n dependence of the l a t t e r . These measurements gave a value of ( 2.lio . 3)Kc/s. f o r the pseudo-exchange constant i n t i n . The pseudo-dipolar second moment was found to be twice the d i p o l a r second moment. i i i S pin echoes were observed i n the i s o t o p i c a l l y pure t i n and were used to measure the s p i n - s p i n r e l a x a t i o n time. These gave values -which were much shorter than those measured by f r e e i n d u c t i o n decays. The reason f o r t h i s was not determined. TABLE OF CONTENTS i v PAGE .Afo S *t 3? cl C t $ e o o o » « * o f t « » o 0 o o » o o » » » » « o o XX TcibXs o f Cont©n.ts O D O O O Q O « O O O « O O O O O O O O O i v LiS*t Of TclfoX©S o o o o o o o o o o o o o o o o o o e « o « VX L i s t Of IXXU-Stj?3"bJL OnS o o o o o o o « o « o e e o o o o o o V X X Acknov/led gementSo » <> » o « o » » • e o o • * i x INTRODUCTION© 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 X CHAPTER 1. THE ELECTRONIC STRUCTURE OF METALS 5 1.1 The Wave Functions of Conduction E l e c t r o n s 6 1.2 The Magnetic S u s c e p t i b i l i t y of Conduction E l e c t r o n s 11 I I . NUCLEAR MAGNETIC RESONANCE IN METALS . 18 2.1 The Magnetic F i e l d at the Nucleus 18 2.2 The Spin Temperature 2k 2.3 S p i n - L a t t i c e R e l a x a t i o n 27 2. U- Spin-Spin 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 Line Width With a Quadrupolar I n t e r a c t i o n . . . 38 2.7 Pulsed NMR With a Quadrupole I n t e r a c t i o n . . . . . . 38 I I I . THE EXPERIMENTAL METHOD. . . . . . . 3^ 3.1 General D e s c r i p t i o n of the Apparatus. M+ 3.2 The Timing System ^6 3.3 The Gated Power A m p l i f i e r 5l 3 • *^ Th© PT* 63. nip X i f i© J? o o o o » « o o o o • o o « o « o o 3^ 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 Supplies and Noise Suppression. . . . . . . . 58 3.8 The Magnets and Magnetic F i e l d Measurements . . . . 60 3.9 The Low Temperature System. 61 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 . . . 62 3.11 Acou s t i c O s c i l l a t i o n s 7^  3.12 C a l c u l a t i o n of the S/N R a t i o . . . . . . . 79 V CHAPTER PAGE III . ( c o n t i n u e d ) 3.13 Measurement of S p i n - L a t t i c e R e l a x a t i o n Times . . . 83 3.11+ Measurements of Spin-Spin R e l a x a t i o n Times . . 87 3.15 Measurement of Absorption and D i s p e r s i o n Modes . . 88 3.16 P o s s i b l e Improvements to the Apparatus. . 89 IV. THE EXPERIMENTAL RESULTS . . . . . . . . 96 *+.l Aluminium S i n g l e C r y s t a l • 97 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 Metals With a Large Quadrupole I n t e r a c t i o n . . . . 103 !+.5 Copper Wire, and Other Spurious S i g n a l Sources. . . 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 Experimental S/N R a t i o s 135 C O N C X J T J S X O N O • • • • • • o o » o o « e o o » * 0 0 0 0 0 0 0 15*1 P O S T S C R I P T o o o o o o o o o o o o o o o o o o o o o o o o X^ j? B I B X i I O G H A P H Y o o o o o o o o o o o o o o o o o o o o o o o X 8 APPENDIX I . D i s t o r t i o n i n the Phase S e n s i t i v e D e t e c t i o n System 162 APPENDIX I I . D e t a i l s of the Samples Used I6h APPENDIX I I I . The S i g n a l Induced i n the Pickup C o i l . . . 168 APPENDIX IV. Measurement of Absorption and D i s p e r s i o n Modes w i t h Pulsed NMR Apparatus 173 APPENDIX V. C i r c u i t Diagrams 176 v i LIST OF TABLES TABLE PAGE ^.1 Spin-Spin R e l a x a t i o n Times by Free Induction D@C3.y • • o 0 • « e o o o o o © o o o o o e e e e « • 1X6 h-,2 Spin-Spin R e l a x a t i o n Times by Spin Echoes 129 !+.3 V a r i a t i o n of Spin Echo Spin-Spin R e l a x a t i o n Time w i t h Pulse Length 132 hoh Experimental and T h e o r e t i c a l . S/N R a t i o s 136 LIST OF ILLUSTRATIONS FIGURE PAGE 1.1 Energy Versus Wave Number Diagram f o r the d Band of a T r a n s i t i o n Metal 16 3.1 Block Diagram of the Apparatus 92 3.2 Block Diagram of the Timing System 93 3.3 Eq u i v a l e n t C i r c u i t of the 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 of the C o i l System. . . . . . . . 62 3.5 V a r i a t i o n of the Ac o u s t i c O s c i l l a t i o n Amplitude w i t h the Magnetic F i e l d . 9*+ 3.6 Diagram of a Two Pulse Sequence. 95 -^.1 T y p i c a l Sweeps w i t h a Boxcar Gate Through a Free Induction T & i l o o o » o * o o o « o « o « « o o e o 13^ 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. . . . 139 ^.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 Spin Echo i n I s o t o p i c a l l y Pure T i n 1^2 ^.6 V a r i a t i o n of Spin Echo Amplitude w i t h r f Pulse 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 • o o 1^"3 h,7 V a r i a t i o n of S p i n Echo Amplitude w i t h the Second J?f P i l l S © Wid "fctl o o o o o o o o o o o o o o o o o o o 1^  i ^ I *f.8 Free Induction Decay i n I s o t o p i c a l l y Pure T i n . . . . 1^5 Lor e n t z i a n Line 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 of the C r y s t a l With Respect to 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 « ^ . l l A n i s t r o p y of the Line Width i n I s o t o p i c a l l y Pure T i n V+Q W-.12 Logarithm of the Spin Echo Amplitude Versus Time . . 1^9 Ani s t r o p y of the Spin Echoes i n I s o t o p i c a l l y Pure T i n 150 1.1 Equiva l e n t C i r c u i t of the Phase S e n s i t i v e Detector . 162 IV. 1 The A m p l i f i e r Output 173 v i i i FIGURE PAGE IV.2 E f f e c t of the Deadtime . o o o o o o o o o o o 9 0 17^ " V . l The Gated Transmitter. . O O B O O O O O O O . . . 177 V.2 The P r e a m p l i f i e r o o o o o o o o o o . . . 178 v.3 The Boxcar I n t e g r a t o r . . Q O O O O O O O O O . . . 179 Mixer, Pulse A m p l i f i e r , and Quench P u l s e r . . . . . 180 v.5 Coincidence Timing U n i t . o o o o o o o o o o . . . 181 V.6 Slow Sawtooth Generator. o o o o o o o o o o 0 e 0 l82 v.7 Regulated Filament Power Supply . . . 183 ACKNOWLEDGEMENTS i x I t i s a pleasure to thank Dr. D. L I . W i l l i a m s f o r h i s constant and p a i n s t a k i n g help i n a l l aspects 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 things looked black. Dr. M. Bloom a l s o made important c o n t r i b u t i o n s to t h i s work, both d i r e c t l y through suggestions and an i c o n o c l a s t i c reading of t h i s t h e s i s and i n d i r e c t l y through a l l the nuclear magnetic resonance theory I have l e a r n t from him. For a l l t h i s I would l i k e to thank him. Of the other people who have co n t r i b u t e d to t h i s work, I would l i k e to acknowledge an in 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 the a c o u s t i c a l aspects of t h i s work and Mr. Riseborough f o r t a k i n g the X-rays. Dr. H.E. Schone i s to be thanked f o r the loan 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 support from the U n i v e r s i t y of B r i t i s h Columbia, the B r i t i s h Columbia Hydro A u t h o r i t y , and the N a t i o n a l Research C o u n c i l of Canada which made t h i s work pos-s i b l e , i s g r a t e f u l l y acknowledged. F i n a l l y , I must acknowledge my indebtedness to Mr. and Mrs. R. F. C a r s w e l l and to Mr. and Mrs. P. K. Diggle f o r t h e i r innumerable kindnesses and a l s o to my f e l l o w students of the 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 c h a o t i c years. INTRODUCTION Ever since the e a r l y days of nuclear magnetic resonance, experiments have been made on metals. These experiments were always made on f i n e l y ground powders suspended i n an i n s u l a -t i n g o i l . This i s because of the r f s k i n e f f e c t which prevents r f f i e l d s p e n e t r a t i n g more than a few microns i n a m e t a l l i c sample. Studies 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 the Knight s h i f t were made on s e v e r a l metals and were even extended i n t o the superconducting r e g i o n . These gave considerable i n f o r m a t i o n on the s p h e r i c a l l y symmetric part of the conduction 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 being due to an a n i s o t r o p i c Knight s h i f t . By a n a l y s i n g the asymmetric l i n e shape, the magnitude of the a n i s o t r o p i c Knight s h i f t could be obtained w i t h considerable accuracy. ' E x p e r i -mental work was a l s o extended to a l l o y s and to l i q u i d metals. Later on the a n a l y s i s of powder measurements was extended to cover the case of the presence of an a n i s o t r o p i c Knight s h i f t and a quadrupole i n t e r a c t i o n . 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 which d i r e c t l y give the pseudo-exchange strength. Despite the considerable success of the 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 to 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 metal s i n g l e c r y s t a l . This was f i r s t done i n t h i s l a b o r a t o r y s e v e r a l years ago, using a sample constructed from t h i n s i n g l e - c r y s t a l slabs 2 separated by i n s u l a t i n g l a y e r s . The apparatus used a conven-t i o n a l marginal 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 cooled to l i q u i d helium temperature to get an adequate S/N r a t i o . Under favourable circumstances t h i s was as high as 50. In the f i r s t measurements, both the a n i s o t r o p i c Knight s h i f t and l i n e widths were measured. Since then ex-periments 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 metals, both i n t h i s l a b o r a t o r y and elsewhere. Some of these experiments detected d e t a i l s which were obscured by the averaging o v e r a l l o r i e n t a t i o n s which occurs i n a powder measurement. E a r l y on i t was found that a c o i l f a i r l y t i g h t l y wound on a c y l i n d r i c a l sample was j u s t as good as the layered sample. The, aim of t h i s work was to b u i l d a pulsed NMR apparatus f o r s i n g l e c r y s t a l s which could be used i n con j u n c t i o n w i t h the steady s t a t e apparatus. Pulsed NMR measurements had never been made i n s i n g l e c r y s t a l s before, so th a t t h i s work was of an exp l o r a t o r y nature. In the e a r l y stages the apparatus worked a t 700Kc/s. This low frequency was chosen w i t h the i n t e n t i o n of u l t i m a t e l y doing experiments on super-conductors. However, i t soon became c l e a r that the S/N was going to be too small at t h i s frequency, so the apparatus was r e b u i l t to operate i n the frequency range 6Mc/s. to lOMc/s. The apparatus worked 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 . For various experimental reasons, measurements could not be made a t l i q u i d helium tem-peratures and t h i s r e s t r i c t e d the experiments which could be done. 3 The anisotropy i n %. had o f t e n been observed i n steady s t a t e experiments, Anisotropy i n T, had never been detected since steady s t a t e apparatus i s q u i t e u n s u i t a b l e f o r T, measure-ments. Detection of any T, anisotropy was thus the main e x p e r i -ment to be attempted. There i s no t h e o r e t i c a l estimate of the magnitude of such an e f f e c t . A l l the t h e o r i e s developed so f a r have only been appl i e d to a cubic l a t t i c e , or e l s e make assump-t i o n s about the conduction e l e c t r o n d i s t r i b u t i o n which may not 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 that 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 that a search f o r Ti anisotropy should be confined to metals w i t h a non-cubic l a t t i c e and an a n i s o t r o p i c Knight s h i f t . Because of the e x p l o r a t o r y nature of t h i s work, most of the emphasis has been placed on the experimental aspects of the t o p i c , r a t h e r than on the t h e o r e t i c a l s i d e . The fragmen-t a r y nature of the theory of NMR i n metals was another i n c e n t i v e f o r concentrating on the experimental nature of the problem. The rudimentary s t a t e of the t h e o r i e s of s p i n - 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 i n metals i s not s u r p r i s i n g when i t s complexity i s considered. The 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 metals have been known f o r about three decades. They are described 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 . Most of the mechanisms involved i n NMR i n metals can be q u a l i t a t i v e l y described i n terms of these models. 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 impossible since many of the NMR p r o p e r t i e s are very dependent on de-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 . A q u a n t i t a t i v e com -p a r i s o n of t h e o r e t i c a l and experimental r e s u l t s must thus await the compution of much more accurate e l e c t r o n wave func-t i o n s . Such a comparison would provide an extremely s e n s i t i v e t e s t of any computed wave f u n c t i o n . However, i t w i l l be many years before such a comparison i s p o s s i b l e f o r any but the simplest of metals. In the f i r s t two chapters t h i s b a s i c theory of NMR i n metals i s discussed and the p r o p e r t i e s of a l l the important mechanisms involved are l i s t e d . The t h i r d chapter e x h a u s t i v e l y covers the experimental d e t a i l s . Some of t h i s overflows i n t o the Appendices. The r e s u l t s are given i n the f o u r t h chapter and the t h e s i s concluded w i t h suggestions f o r f u t u r e work. F i n a l l y , a note on the u n i t s employed i n t h i s t h e s i s . For the c l a s s i c a l electromagnetic s e c t i o n s M.K.S. u n i t s are used, w h i l s t c.g.s. u n i t s are used i n a l l the quantum mech-a n i c a l expressions. In the s e c t i o n s i n v o l v i n g both e l e c t r o -magnetic and atomic c o n s i d e r a t i o n s , the choice of u n i t s depends upon the u l t i m a t e use of the expressions i n the s e c t i o n . CHAPTER I 5 THE ELECTRONIC STRUCTURE OF METALS 'Accid e n t a l and F o r t u i t o u s Concurrence of Atoms.' - Lord Palmerston. The purpose of t h i s chapter i s to summarise the proper-t i e s of the e l e c t r o n i c s t r u c t u r e of metals necessary f o r an understanding of t h e i r nuclear magnetic resonance. I t i s assumed that the reader i s f a m i l i a r -with the concept of B r i l l o u i n zones and Fermi su r f a c e s , as described i n the stan-dard t e x t s (99 365 37). The f i r s t s e c t i o n describes the standard 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 l i m i -t a t i o n s . The f i n a l s e c t i o n concerns the e f f e c t s of a s t a t i c magnetic f i e l d upon the conduction 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 can, i n p r i n c i p l e , be got by s o l v i n g the Schroedinger equation of a l l the e l e c t r o n s and n u c l e i i n the metal. This i s an impossible task so a much s i m p l i f i e d model i s adopted. The n u c l e i form a p e r i o d i c l a t t i c e and are surrounded by closed 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 to be to p a r t i a l l y s h i e l d the nuclear charge. There are a l s o l o o s e l y bound con-du c t i o n e l e c t r o n s shared to some extent by a l l the n u c l e i i n the l a t t i c e . The Schroedinger equation f o r a l l these e l e c t r o n s must then be solved. To do t h i s , the c r u c i a l assump-t i o n i s made that the e l e c t r o n s i n t e r a c t so weakly that they 6 can move independent ly of each o the r . The wave f u n c t i o n s and energy l e v e l s obtained are thus those f o r an independent e l e c t r o n . 1.1 The Wave F u n c t i o n of Conduct ion E l e c t r o n s The most important e f f e c t of the p e r i o d i c l a t t i c e p o t e n t i a l i s to r e s t r i c t the s o l u t i o n s of S c h r o e d i n g e r 1 s equat ion to the form = U(fe,£) e x p C i k . ^ ) . U ( k , r ) i s a f u n c t i o n , depending on the wave vec to r k, of the e l e c t r o n , which 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 expCik^rJ . There are other r e s t r i c t i o n s on the form tha t the con-duction e l e c t r o n wave f u n c t i o n can t ake . The coulomb a t t r a c t i o n i s very s t rong c l o s e to the nucleus and so the conduct ion e l e c t r o n must have a compensat ingly l a r g e k i n e t i c energy to avoid being cap tured . I t thus o s c i l l a t e s r a p i d l y , r a the r i n the manner of the wave f u n c t i o n of a valence e l e c t r o n i n a f ree atom. However, u n l i k e the f ree atom, there i s no e x p e r i -mental evidence i n most metals f o r the e l e c t r o n having any o r b i t a l angular momentum. Th i s " quenching " of the o r b i t a l angular momentum occurs because the e l e c t r o n moves i n a p o t e n t i a l f i e l d having the l a t t i c e symmetry, not s p h e r i c a l symmetry as i n a f ree atom (36). Th is means that 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 valence band from which i t o r i g i n a t e d . Between the ions the e l e c t r o n moved i n a more uniform p o t e n t i a l and behaves r a t h e r l i k e a f r e e e l e c t r o n . Even i f many-body e f f e c t s are ignored, the exact s o l u t i o n of Schroedinger's equation i s impossible. The f i r s t problem i s d e c i d i n g on the c o r r e c t form of the p e r i o d i c p o t e n t i a l . This i n i t s e l f i s a complex many-body problem and even i f i t were solved computational d i f f i c u l t i e s preclude an exact s o l u -t i o n of Schroedinger's equation. I t i s thus necessary to assume various 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 Schroedinger's equation. Three of the simplest types of approximate wave f u n c t i o n s w i l l now be described, ( i ) The Tight Binding Approximation.. This assumes that i n s i d e each i o n the wave f u n c t i o n i s s i m i l a r to the wave f u n c t i o n of 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 set of atomic wave f u n c t i o n s i s then chosen f o r each io n i n the l a t t i c e and then a l i n e a r combination of these taken to give a Bloch f u n c t i o n f o r the whole l a t t i c e of the form metal. For t h i s method to be s a t i s f a c t o r y , i t i s necessary that the atomic wave f u n c t i o n s on d i f f e r e n t atoms do not over-la p much, so that each e l e c t r o n i s predominantly i n the near neighbourhood of i t s parent atom. I t i s p a r t i c u l a r l y s u i t a b l e f u n c t i o n on the j th atom i n the 8 f o r 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 metals. There are more elaborate v e r s i o n s of t h i s p r i n c i p l e which give b e t t e r r e s u l t s , but they a l l have severe computa-t i o n a l d i f f i c u l t i e s . ( i i ) The Nearly Free E l e c t r o n Approximation E x a c t l y the opposite assumption to the t i g h t binding case i s made. I t i s th 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 which i s much l e s s than t h e i r k i n e t i c energy so that they can be described by plane waves and the p e r i o d i c p o t e n t i a l treated as a p e r t u r b a t i o n . This treatment leads d i r e c t l y to the concept of B r i l l o u i n zones. I t a l s o shows that e l e c t r o n s i n a metal can be treated as a simple 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 replaced by an e f f e c t i v e mass m* which 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. Using t h i s very simple model s u r p r i s i n g l y accurate Fermi surfaces 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 metals (22). Due to i t s s i m p l i c i t y and a b i l i t y to e a s i l y describe the e l e c t r o n i c s t r u c t u r e of a metal, 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 s u i t a b l e . ( i i i ) The Orthogonalised Plane Wave Method. In n e a r l y a l l metals 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 binding a p p r o x i -mation. Many ways have been devised f o r t r e a t i n g t h i s i n t e r -mediate s i t u a t i o n , but only the O.P.W. method w i l l be described 9 here. This i s because i t gives a reasonably accurate 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 the f r e e e l e c t r o n approximation. The O.P.W. method i s based on the requirement that the conduction e l e c t r o n wave f u n c t i o n be orthogonal to a l l the f i l l e d core wave f u n c t i o n s to s a t i s f y the 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 combination of core wave f u n c t i o n s i s sub-t r a c t e d from a plane wave i n such a way that the r e s u l t i n g conduction e l e c t r o n wave f u n c t i o n i s orthogonal to a l l the core wave f u n c t i o n s . Then a l i n e a r combination of these wave f u n c t i o n s i s found which best s a t i s f i e s Schroedinger 1s 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 the 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. This i s f a i r l y c lose to how a r e a l wave f u n c t i o n must look. This 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 having 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 to those of the co 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 corresponds to a plane 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 approxi-mation works as w e l l as i t does. The e l e c t r o n i s nearly f r e e between the i o n s , but c l o s e to an i o n gains enough k i n e t i c energy to approximately cancel the a t t r a c t i v e coulomb poten-t i a l . 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 and i s t y p i c a l l y l e s s than the k i n e t i c energy of 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 to those of the n e a r l y f r e e e l e c t r o n model. This i s why n e a r l y f r e e e l e c t r o n s are a r e a -sonably good approximation 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 are concerned. However, the O.P.W. method a l s o shows that the true wave f u n c t i o n u s u a l l y a l s o contains p,d, c o n t r i b u t i o n s which may be very important i n c a l c u l a t i n g some other proper-t i e s . Another reason why the n e a r l y f r e e e l e c t r o n model works i s that an e l e c t r o n f e e l s not only the coulomb p o t e n t i a l of a given i o n , but that of a l l the other e l e c t r o n s and ions i n the 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 that i t i s n e g l i g i b l e beyond about a l a t t i c e spacing (37). This considerably 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 assumption that the conduction e l e c t r o n s i n t e r a c t weakly. T h i s , at f i r s t , seems a poor assumption since 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, g i v i n g an average coulomb r e p u l s i o n energy of s e v e r a l ev. However, the screening e f f e c t of the e l e c t r o n s and ions converts the long range coulomb p o t e n t i a l i n t o a short range screened p o t e n t i a l . This reduces the c r o s s -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 to such an extent that 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 l i k e l y (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 -trons w i t h p a r a l l e l spins due to the e x c l u s i o n p r i n c i p l e , but i t does not d r a s t i c a l l y modify the screening e f f e c t . 11 These e f f e c t s are of t e n discussed i n terms of q u a s i -p a r t i c l e s (38). 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 . In the dense e l e c t r o n gas i n a metal, 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 w i t h 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 f r e e e l e c t r o n . 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 neglected. This i s the cou p l i n g of the e l e c t r o n s p i n and o r b i t a l angular momentum through r e l a t i v i s t i c magnetic e f f e c t s which increase 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. I t can have s e v e r a l e f f e c t s on the conduction e l e c t r o n s . O r d i n a r i l y the energy l e v e l s are doubly degenerate because of the e l e c t r o n s p i n . 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 Fermi surface s t r u c t u r 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 angular momentum quenched by the l a t t i c e p o t e n t i a l . 1.2 The Magnetic S u s c e p t i b i l i t y of Conduction 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 on the conduction band e l e c t r o n s w i l l ; now be considered. This 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 questions 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. These conceptual d i f f i c u l -t i e s w i l l be ignored i n the f o l l o w i n g d i s c u s s i o n which 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 Fermi gas i s . F = NE F-kT.Tr[gCH)] where gCH) =ln{l+exp[^(E r*H)]} and (3 =(kT)"' „ The magnetic s u s c e p t i b i l i t y X i s given by the thermodynamic r e l a t i o n Consider a s i n g l e e l e c t r o n i n a metal 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 derived from a vector po-t e n t i a l A,. The Hamiltonian i s assumed to 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 the 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 the symmetric gauge A=-^(Hxr) chosen, the Hamil-to n i a n can be div i d e d i n t o two pa r t s *H = U + % where U = - # V J + V ( r ) + - j | r A J % = yufi.CL + 2 § ) . L i s the e l e c t r o n o r b i t a l angular momentum operator 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 to be used i n e v a l u a t i n g >f are Bloch f u n c t i o n s . Since % *H0, and H, are p e r i o d i c i n the l a t t i c e p o t e n t i a l , these give a r e p r e s e n t a t i o n which i s diagonal i n the wave vector k,(9)« I f s p i n - o r b i t c o u p l i n g e f f e c t s are small the e l e c t r o n s p i n up and s p i n down stat e s can be considered separately. The wave f u n c t i o n c o r r e s -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 cor-respond to the same value of k,. If i t is assumed that < H » > S H I J Tr[gCH)] can be expanded in powers of H by means of McLaurin's theorem and standard perturbation theory to give TrfgCH)] =Tr[g(H0)] +]T-f(Eq )<qm,ld> HX^^.-^^|<am,|q>f-where f (<H )=g'CH) = [ l+exp{§(E F - H )}] ' is the Fermi function. The summation is overall possible energy values for a l l pos-sible 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 ie ld. The second term requires a permanent magnetic moment to be present and so only occurs for ferromagnetic metals, a case which wil 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 En(k )^ which correspond to each value of k,. Substituting in the thermodynamic relation for X gives = 7iirTASI 57 r^j ,7 f ^L <hklL+2SI mkVmkl L+2S| nk> dk. X c o n s i s t s of three 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 exp re s s ion i n t o a more f a m i l i a r form, the assump-t i o n i s made tha t the Fermi surface i s s p h e r i c a l and that the e l e c t r o n s are n e a r l y f r e e . This term a r i s e s from a surp lus 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 , ( i i ) S p i n - o r b i t paramagnetism. f ? ; - " E f ^ [ < ^ \ L|mls><mk| as| nk> + / n k I 2S I mk> /mk L I nk> dk. The e f f e c t of s p i n - o r b i t c o u p l i n g i s to l i f t s l i g h t l y the quenching of the o r b i t a l angular momentum by mix ing i n other s t a t e s of appropr i a t e angular momentum and symmetry. The o r b i -t a l angular momentum i s approximate ly where X i s the s p i n -o r b i t c o u p l i n g cons tant and A i s the mean energy between the s t a te being considered and the s t a t e s being mixed i n (36). The arrangement of energy l e v e l s w i t h i n the conduc t ion band i s so complex tha t A i s almost imposs ib l e to c a l c u l a t e , but i s of the order of the bandwidth, a few ev. X i s about a t en th of an ev. or l e s s (22). I f the o r b i t a l m a t rix elements are now evaluated by using the t i g h t b i nding approximation, i t i s seen that the remainder of the expression c l o s e l y resembles that f o r . . J^io — ~g • Since -£« 1 i t i s not an important term, ( i i i ) Van Vleck paramagnetism. x""to$kf ^ f ' - i ^ ^ I M " * ) (m^\h\^) d£° This i s a second order e f f e c t a r i s i n g through the o r b i t a l angular momentum operator mixing unoccupied e x c i t e d s t a t e s i n t o an occupied ground s t a t e w i t h quenched o r b i t a l angular momentum. I t i s the same as the Van Vleck temperature independent induced o r b i t a l paramagnetism found i n some s o l i d s (36). I f a t i g h t b i nding approximation i s used JL only has matrix elements between s t a t e s w i t h the same value of k which d i f f e r i n the value of t h e i r magnetic quantum number m . Thus there are only c o n t r i b u t i o n s from matrix elements between l e v e l s i n the same p a r t i a l l y f i l l e d band. Most metals have mainly s e l e c t r o n s i n t h e i r conduction band and.so " X v i s n e g l i g i b l e . 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 give r i s e to a s i g n i f i c a n t value of " X v . Figure 1.1 shows the. t y p i c a l s t r u c t u r e of the d band and the types of t r a n s i t i o n s that give r i s e to "X v. 16 Energy (kltdron volts) W a v e n u m b a r k F i g . 1.1 Energy versus Wavenumber f o r the" d Band i n a T y p i c a l T r a n s i t i o n Metal where E i s the energy separation between the ground and ex-c i t e d s t a t e s , s u i t a b l e averaged over k space. There are a l s o two other c o n t r i b u t i o n s to the suscep-t i b i l i t y ; the core e l e c t r o n diamagnetism, and the Langevin paramagnetism of the o r b i t a l angular momentum caused by spin-o r b i t c o u p l i n g . These are always very small terms. The bulk s u s c e p t i b i l i t y i s due to a t l e a s t s i x terms. These u s u a l l y only have a secondary e f f e c t on the magnetic f i e l d a t the nucleus, which may be much d i f f e r e n t from that expected from the value of X. This i s since the bulk 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 throughout ,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 predominantly upon the cur r e n t s and magnetic moments i n i t s very near v i c i n i t y . 17 The magnetic f i e l d s h i f t s at the nucleus are u s u a l l y mainly due to 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 magnetic f i e l d p o l a r i s i n g the conduction e l e c t r o n s ( P a u l i s p i n para-magnetism). These then couple w i t h the nucleus through v a r i o u s mechanisms to produce a f i e l d s h i f t much l a r g e r than that d i r -e c t l y due to the P a u l i s p i n paramagnetism. The Van Vleck paramagnetism, and the Landau diamagnetism are the only d i r e c t terms which can sometimes be of importance i n determining the magnetic f i e l d a t the nucleus. 18 CHAPTER I I NUCLEAR MAGNETIC RESONANCE IN METALS 'My mind i s i n a st a t e of p h i l o s o p h i c a l doubt as to magnetism. 1 - C o l e r i d g e . I t i s the purpose of t h i s chapter to desc r i b e the pro-p e r t i e s of the NMR of metals which d i f f e r from those of other s o l i d s . I n accordance w i t h t h i s aim, the elementary aspects of pulsed and steady s t a t e NMR w i l l not be given, these being comprehensively discussed i n Chapter I I I of Abragam's book (1). The v a r i o u s reasons why the magnetic f i e l d a t a nucleus d i f f e r s from t h a t of the appli e d magnetic f i e l d are given. The s p i n temperature concept i s introduced and i t s importance i n the r e t u r n to e q u i l i b r i u m shown. There are two decays involved i n the r e t u r n to 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-in g energy t r a n s f e r to 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 , which 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 the spi n sys-tem. S p i n - l a t t i c e r e l a x a t i o n and the magnetic f i e l d s h i f t are often r e l a t e d . Both of these e f f e c t s are p r i n c i p a l l y due to the 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 r e l a x a t i o n . Many metals have a quadrupole i n t e r a c t i o n which a l t e r s the energy l e v e l s i n the system. The l a s t s e c t i o n shows how t h i s modifies the pulsed NMR i n s i n g l e c r y s t a l s . 2.1 The Magnetic F i e l d at The Nucleus The resonant frequency of a nucleus i n a metal d i f f e r s 19 from i t s value 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 nucleus which causes t h i s i s known as the Knight s h i f t . I t i s the sum of various c o n t r i b u t i o n s whose r e l a t i v e importance v a r i e s i n d i f f e r e n t metals. Two of these c o n t r i b u t i o n s a r i s e from the magnetic coupling of the nuclear d i p o l e moment^ to the e l e c t r o n d i p o l e moment ^ and to the current produced by i t s motion. This i n t e r a c t i o n can be treated n o n - r e l a t i v i s t i c a l l y , provided care i s taken to avoid divergences, to give as the Hamiltonian (1). where <H= + ^ ) r ' * ] ' 2 ^ h ^ • £ i s the separation between the nucleus and the e l e c t r o n . The operator H can be regarded as the magnetic f i e l d produced by the e l e c t r o n at the nucleus. The l a s t term can be ignored since the e l e c t r o n o r b i t a l angular momentum L, i s quenched. The other two terms each give r i s e to a c o n t r i b u t i o n to the Knight s h i f t , ( i ) The "Contact" term. This i s the dominant term i n most metals and i s exten-s i v e l y discussed i n the l i t e r a t u r e (1, 16, H-0). The Hamil-t o n i a n i s i The most important f e a t u r e s of t h i s i n t e r a c t i o n are i t s depend-ence on the s p i n o r i e n t a t i o n s and i t s Dirac b f u n c t i o n form. This means that there i s no i n t e r a c t i o n unless 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 value at the nucleus. 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 to t h i s i n t e r a c t i o n since they 20 a l l v a n ish at the nucleus. The coup l i n g i s only to the i s o t r o -p i c s e l e c t r o n s and i s given by 0)|* i s the s e l e c t r o n d e n s i t y at the nucleus, w h i l e the z a x i s i s along the applied magnetic f i e l d H 0. The conduction e l e c t r o n s are not l o c a l i s e d so the nucleus i s e q u a l l y i n f l u -enced by a l l of them. The summation thus becomes an ensemble average, using Fermi-Dirac s t a t i s t i c s , over a l l the conduction e l e c t r o n s . H 0 has p o l a r i s e d the conduction e l e c t r o n s ( P a u l i s p i n paramagnetism), so t h i s averaging gives a magnetic f i e l d AH 0 a t the nucleus. The r e s u l t i n g Knight s h i f t i s K C ^<i"t(o)r>x f. ( |^"\|/r(0)|1\ i s the averaged value of |^ J/'(0)|a' over a l l e l e c -trons w i t h the Fermi energy E F , wh i l e X P i s the n e a r l y f r e e e l e c t r o n P a u l i paramagnetism. A more accurate equation 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 are approximately taken i n t o account by using a many-body t h e o r e t i c a l , or an experimental, value of Xp. K c ranges from about 0.1$ to about.3$> i n c r e a s i n g s t e a d i l y w i t h i n c r e a s i n g atomic number. Although K c can be measured very a c c u r a t e l y , i t u s u a l l y does not give much i n -formation about j"ViyC0)j°* because i n most metals X p i s not a c c u r a t e l y known. This i s unfortunate since |\|/(0)| i s a parameter w i t h considerable t h e o r e t i c a l i n t e r e s t . K c i s temperature and magnetic f i e l d independent i n most metals, except f o r small e f f e c t s due to l a t t i c e expansion, ( i i ) The A n i s o t r o p i c Knight S h i f t , The Hamiltonian f o r t h i s i n t e r a c t i o n i s *H = ^ . Z P7* [j£ -3rt (^l.^)r: a] . This couples the nucleus to a l l the non-s e l e c t r o n s through t h e i r d i p o l e - d i p o l e i n t e r a c t i o n s . For an a x i a l l y symmetric c r y s t a l , a c a l c u l a t i o n s i m i l a r to that f o r the contact term gives (16) K«m = q' ( 3 e o ^ © - l ) X p , where q' = (J(|f [(3cosfe< - l ) r ' 3 ] ( f ) d x ) E . 0 i s the angle between H 0and the c r y s t a l a x i s of symmetry, <k the angle between H„ and j r , and (J) i s the t o t a l non-s e l e c t r o n wave f u n c t i o n , q' i s a measure of the s p a t i a l a n i -sotropy of the Fermi surface e l e c t r o n ' s charge d i s t r i b u t i o n . A p o s i t i v e q' means that the non-s e l e c t r o n d e n s i t y i s l a r g e s t along the symmetry a x i s , a negative q / t h a t i t i s l a r g e s t i n a plane perpendicular to the symmetry a x i s . A group theo-r e t i c a l treatment shows that q' i s zero f o r a cubic l a t t i c e , i r r e s p e c t i v e of the e l e c t r o n s t a t e s (*KL). K o n can e a s i l y be measured i n both powders and s i n g l e c r y s t a l s , but 'gives l i t t l e u s e f u l i n f o r m a t i o n , apart from the s i g n of q'. I t v a r i e s i n magnitude from zero to about 0.2$ and tends to increase w i t h i n c r e a s i n g atomic number, B i a 0 C J i s an anomalous case i n which K Q r » K c. 22 ( i i i ) Core P o l a r i s a t i o n Knight 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 ignored, apart from t h e i r e l e c t r o n i c screening 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 give the most important c o n t r i b u t i o n to the Knight s h i f t by means of the Heisenberg exchange i n t e r a c t i o n This i n t e r a c t i o n ' i s zero f o r conduction 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 spins and i s r e p u l s i v e i f they have p a r a l l e l spins (22). A conduction 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 inwards, i n c r e a s i n g 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 conduction e l e c t r o n s w i t h spins p a r a l l e l and a n t i p a r a l l e l to H Q . In a simple case t h i s causes an p o r t i o n a l to H 0 . This i s equ i v a l e n t to a small a d d i t i o n a l magnetic f i e l d a t the nucleus; the core p o l a r i s a t i o n Knight s h i f t . In a l l but the simplest metals the exchange i n t e r a c t i o n makes necessary a r e n o r m a l i s a t i o n of the wave f u n c t i o n s and t h i s allows the core p o l a r i s a t i o n to be e i t h e r p o s i t i v e or negative (^2). The importance of t h i s i n t e r a c t i o n has only r e c e n t l y been recognized and as yet there i s l i t t l e knowledge of i t s magni-tude, but i t i s probably present to a s i g n i f i c a n t extent i n a l l f o r t h i s s p i n o r i e n t a t i o n . from the e l e c t r o n s p a r a l l e l to H 0 pro-23 metals (*f2). In ?t^5 i t i s the dominant term, g i v i n g a Knight s h i f t of -3*5% (16). This i s one of the two known negative Knight s h i f t s and i s the l a r g e s t Knight s h i f t found so f a r . ( i v ) Van Vleck O r b i t a l Knight S h i f t . The o r b i t a l c urrents causing the Van Vleck 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 , there i s a long range demagnetising f a c t o r which i s small enough to be neglected. Secondly there i s the short range e f f e c t of the o r b i t a l c u r r ents which gives a Knight s h i f t 0*3) K v = 2n<r~3>Xv. fl i s the atomic volume and <r~3) i s the value of r ~ 3 averaged over a l l the occupied conduction 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 very important. I t cannot be measured a c c u r a t e l y , nor can <^ r"3) or Xv be separately determined, so l i t t l e i n f o r m a t i o n i s gained from t h i s 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 the o r b i t a l quenching. The e f f e c t s of t h i s are u n c e r t a i n since only very simple c a l c u l a t i o n s can be done and there are no experimental measurements i n metals of s p i n - o r b i t e f f e c t s alone. I t i s known that i t enhances the a n i s o t r o p i c Knight s h i f t (^ -1) and gives a c o n t r i b u t i o n to the i s o t r o p i c Knight 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 to become a n i s o t r o p i c , as i n c r y s t a l f i e l d theory ( M + ) . I f X i s the s p i n - o r b i t coupling 2h constant and A i s the average separation between e l e c t r o n s t a t e s , then X / A i s approximately the f r a c t i o n of unquenched o r b i t a l angular momentum ( M + ) . I t i s g e n e r a l l y assumed that the e f f e c t s of s p i n - o r b i t coupling are n e g l i g i b l e i f X i s very much l e s s than the conduction band width. This seems to be the case f o r most metals. There are va r i o u s other diamagnetic and paramagnetic terms, but these are u s u a l l y n e g l i g i b l e . Paramagnetic, or ferromagnetic, i m p u r i t i e s are the only other important cause of l i n e s h i f t s . With metals of f i v e nines p u r i t y , or b e t t e r , there should be no e f f e c t s from t h i s source. The Knight s h i f t thus seems to be predominantly due to fou r i n t e r a c t i o n s . Of these the contact term i s important i n ne 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 to some extent i n a l l metals, but i s dominant i n only a few t r a n s i t i o n metals. The Van Vleck term i s probably only im-portant i n some t r a n s i t i o n metals, w h i l e the a n i s o t r o p i c Knight s h i f t only occurs f o r non-cubic l a t t i c e s . 2.2 The Spin Temperature The Hamiltonian f o r a system of 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; -3$k (yjdk.iik)*?]. Only the case where the Zeeman term " H z > H i w i l l be considered here. The a p p l i e d magnetic f i e l d H 0 s p l i t s the ground stat e energy l e v e l of a nucleus wi t h s p i n I i n t o 21+1 e q u a l l y spaced energy l e v e l s separated by pE0 . I f the sp i n system i s i n thermal e q u i l i b r i u m w i t h the l a t t i c e then the populations p„,, pm_, of the m t h and (m-l)th l e v e l s are given by the Boltzmann d i s t r i b u t i o n where T i s equal to the l a t t i c e temperature TL . Under c e r t a i n circumstances the s p i n system can s t i l l be described i n t h i s way by a. s p i n temperature Ts , which can be d i f f e r e n t from the l a t t i c e temperature (^0). I f such a system i s perturbed, the Ts r e l a x e s towards TL a t a r a t e given by (1) dt~ ( T s H } = " TT ( i " " " T ) o Since Mz= C H ^ / T (Curie's l a w ) , t h i s becomes d M = !_ CM -M ) .'. M zoC l - e x p ( - \ ) > T, i s the 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 the ra t e at which energy i s t r a n s f e r r e d from the s p i n system to the l a t t i c e energy r e s e r v o i r . The Hamiltonian >id f o r the magnetic d i p o l e - d i p o l e i n -teraction between the n u c l e i can be w r i t t e n as H = Tj \ r; k 3 5~(A+B+C+D+E+F), where B = - ± ( l f IjT +1" ifXl-Scos^G ) . 26 The other terms are unimportant i n the f o l l o w i n g d i s c u s s i o n which i s based on that of S l i c h t e r (^ -0). The term B plays a d e c i s i v e r o l e i n the establishment of a s p i n temperature. I t couples two neighbouring 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 other one down. I f Ifj =%, the mutual sp i n f l i p conserves the Zeeman energy of the system, yet a l t e r s the population d i s t r i b u t i o n of the l e v e l s . Because of these p r o p e r t i e s , i t can be shown that 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 r e s t o r e the s p i n system to a Boltzmann d i s t r i b u t i o n . This decay of a perturbed 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 manifests i t s e l f e x t e r n a l l y as a decay of the com-„ ponents M x and M a , of M=^/A\ which are perpendicular to H 0 , towards zero. This decay of M x and i s c h a r a c t e r i s e d 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 appl i e d to the s p i n system, i t undergoes i n t e r n a l r e -o r g a n i z a t i o n f o r a time of about T a, u n t i l a q u a s i - e q u i l i b r i u m s t a t e d e s c r i b a b l e by a s p i n temperature has been reached. The sp 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 con-stant T, , towards the l a t t i c e temperature. This s i t u a t i o n r e q u i r e s the spins to be much more t i g h t l y coupled to each other than they are to the l a t t i c e , i . e . T|> Tx. Most metals s a t i s f y t h i s c o n d i t i o n . I f the energy l e v e l s are not e q u a l l y spaced, the mutual s p i n f l i p s no longer 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 disturbance no i n t e r n a l r e l a x a t i o n towards a spi n temperature can occur, but instead each l e v e l independently t r a n s f e r s i t s energy to the l a t t i c e . The s p i n - l a t t i c e r e l a x a t i o n i s then no longer des-cr i b e d by a s i n g l e e x p o n e n t i a l . This s i t u a t i o n w i l l be returned to l a t e r . 2.3 S p i n - L a t t i c e R e l a x a t i o n , In a metal the strongest c o u p l i n g of the n u c l e i to 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 e l e c -t r o n can be regarded as being 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 nucleus. In the process i t changes the nuclear s p i n from I m to I m_, and undergoes compensating changes i n i t s own angular momentum and k i n e t i c energy, as required by the conservation laws. This i s only an approximation since an e l e c t r o n wave packet extends over s e v e r a l n u c l e i , so i t can be scattered from an i n i t i a l to a f i n a l s t a t e by simultaneous c o l l i s i o n w i t h more than one nucleus (^0). This i s a r a r e occurrance i f H > H * (^ 5). A subsequent 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 to the l a t t i c e . Since the e l e c t r o n must be able to change i t s k i n e t i c energy by small amounts, only the e l e c t r o n s near the Fermi surface can take part i n the r e l a x a t i o n , ( i ) Contact R e l a x a t i o n . R e l a x a t i o n i s caused by the nucleus s c a t t e r i n g s e l e c t r o n s . 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 contact 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 gives T, a t a temperature T as (^ +0) (T.T)"' = - ^ r f k r / ^ l y C O ) ! * ) * Zj ( E F ) o 7Te , Tn are the e l e c t r o n i c and nuclear gyromagnetic r a t i o s r e s p e c t i v e l y , and Z s ( E F ) i s the average s e l e c t r o n d e n s i t y of s t a t e s at the Fermi surface. The most important f e a t u r e of t h i s expression i s that T, T i s a constant.' This has been experimentally 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 small de-v i a t i o n s observed being due to the e f f e c t s of thermal expansion of the l a t t i c e . 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 expression s i m p l i f i e s to the "Korringa r e l a t i o n " A more accurate expression which allows f o r i n t e r a c t i o n s be-tween the conduction e l e c t r o n s i s XF° i s the f r e e e l e c t r o n value, while X P can be e i t h e r an experimental value, or a c a l c u l a t e d value using q u a s i -p a r t i c l e theory. This expression d i f f e r s from the c o r r e s -T i T i s r e l a t e d to the Knight s h i f t due to the contact term by T, TK, •c - 4 t T k ^ T « ' ' 29 ponding one i n Abragam (1) or S l i c h t e r (^O), s ince recent t h e o r e t i c a l work shows tha t i t i s the f ree e l e c t r o n d e n s i t y of s t a t e s tha t i s i nvo lved in . s p i n - l a t t i c e r e l a x a t i o n , not the 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 (^6) . The K o r r i n g a r e l a t i o n , or i t s more accura te v e r s i o n , i s commonly used to es t imate unknown va lues of T, T from the known Knigh t s h i f t , or e l s e the va lue of T| de r ived from the Kor r i nga r e l a t i o n can be compared w i t h the exper imenta l v a l u e . A l a r g e d i sc repancy between the two va lues i n d i c a t e s tha t other i n t e r -a c t i o n s bes ides the contac t i n t e r a c t i o n are important i n the m e t a l . ( i i ) D i p o l a r R e l a x a t i o n . The i n t e r a c t i o n between the nuc lea r and e l e c t r o n i c d i p o l e moments p rov ides the s c a t t e r i n g mechanism. Th i s g ives M) (T| T)~' = 4 - r r ( T e Tn ) % 3 Z* (E , ) <I"3>*C. C i s a term whose va lue depends on the l a t t i c e and e l e c t r o n i c s t r u c t u r e . I t has a va lue of about u n i t y and no an i so t ropy f o r a cub ic l a t t i c e . < r^~v> i s averaged over a l l the conduct ion band e l e c t r o n s . 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 e l e c t r o n s . U n l i k e the a n i s o t r o p i c Knigh t s h i f t , T, can be non-zero i n a cub ic l a t t i c e . There i s no K o r r i n g a - l i k e r e l a t i o n between T, and K c f o r t h i s i n t e r a c t i o n . ( i i i ) O r b i t a l R e l a x a t i o n . The s c a t t e r i n g i s caused by the 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 the conduction e l e c t r o n and the nuclear magnetic d i p o l e . U n l i k e the previous two s c a t t e r i n g processes, there i s no r e o r i e n t a t i o n of the e l e c t r o n s p i n ; the nuclear s p i n change being compensated f o r by an equal change i n o r b i t a l angular momentum. T, T i s given by the same expression as the d i p o l a r r e l a x a t i o n , except th a t C has d i f f e r e n t values (V7). Dip o l a r and o r b i t a l r e l a x a t i o n have only been c a l c u -l a t e d f o r a cubic l a t t i c e , f o r which no anisotropy i s pr e d i c t e d . Theory has not yet shown whether there should be a s i g n i f i c a n t a n i s o t r o p i c T, i n non-cubic l a t t i c e s . ( i v ) Core P o l a r i s a t i o n R e l a x a t i o n . A non-s e l e c t r o n p o l a r i s e s the core e l e c t r o n s when i t i s s cattered by them. The t r a n s i e n t p o l a r i s a t i o n acts on the nucleus through the contact term to give nuclear r e l a x a -t i o n . In 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 to conserve angular 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 only been done f o r t r a n s i t i o n metals w i t h a cubic l a t t i c e (27). For t h i s case, T, T has a form s i m i l a r to that of the contact r e l a x a t i o n . I t a l s o s a t i s f i e s the K o r r i n g a - l i k e r e l a t i o n F i s a f a c t o r of about u n i t y which depends on the degeneracy of the conduction band. These r e l a x a t i o n mechanisms v i a the conduction e l e c -trons are the most important ones. Contact r e l a x a t i o n i s the dominant mechanism i n most metals, but core 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 metals. D i p o l a r r e l a x a t i o n i s never important. These r e l a x a t i o n mechanisms are stronger than those i n most other types of s o l i d s ; T, T t y p i c a l l y being about one sec-deg. There are probably 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 various 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 separated (27). In p r a c t i c e t h i s i s . very d i f f i c u l t and r e q u i r e s an extensive s e r i e s of measurements of T, , the Knight 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 the use of s e v e r a l not very 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 . Various r e l a x a t i o n mechanisms which d i r e c t l y couple the nucleus to the l a t t i c e v i b r a t i o n s through magnetic d i p o l e , or e l e c t r i c quadrupole, i n t e r a c t i o n s a l s o occur (1). These are a l l unimportant i n metals. I m p u r i t i e s can some-times cause n o t i c e a b l e r e l a x a t i o n . Paramagnetic, or ferromagnetic, ions have a l a r g e magnetic f i e l d near them which s t r o n g l y couples neighbouring n u c l e i to the l a t t i c e v i b r a t i o n s , thus forming q u i t e an e f f i c i e n t r e l a x a t i o n mechanism (1). In very impure samples a t low temperatures, t h i s paramagnetic r e l a x a t i o n could be important. 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 la r g e long range e l e c t r i c f i e l d gradient near the impurity (22). I f the nucleus has an e l e c t r i c quadrupole moment i t can couple to t h i s and hence to the l a t t i c e through the v i b r a t i n g i m p urity atom. This mechanism i s probably not very strong since d e l i b e r a t e i n t r o d u c t i o n of i m p u r i t i e s i n t o aluminium l e f t T, unaffected by an impurity c o n c e n t r a t i o n of 0.2% (^8). 2.If Spin-Spin R e l a x a t i o n I t i s w e l l known that a f t e r the a p p l i c a t i o n of a r f pulse the transverse magnetisation has a decreasing ampli-tude which i s the F o u r i e r transform of the l i n e shape (1). The s p i n - s p i n r e l a x a t i o n time T a governs the decay of the transverse magnetisation. Ta. can thus be r e l a t e d to the 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 to the second moment. The second moment i s one of the few p r o p e r t i e s of the l i n e shape which can of t e n be c a l c u l a t e d e x a c t l y . For 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 l i n e widths and second moments, r a t h e r than Tj, d i r e c t l y . In metals there are three main c o n t r i b u t i o n s to the l i n e width, ( i ) The D i p o l a r Line Width. This i s due to the nuclear magnetic d i p o l e - d i p o l e i n t e r a c t i o n . For t h i s i n t e r a c t i o n the second, and higher, moments can be c a l c u l a t e d f o r a given l a t t i c e s t r u c t u r e (1). From these the l i n e shape can, i n p r i n c i p l e , be got. The 33 l i n e shape should be approximately gaussian w i t h a width of a few gauss. This corresponds to T a ^ l O C y u s . ( i i ) Pseudo-Exchange (Ruderman-Kittel) Coupling. In t h i s i n t e r a c t i o n two adjacent n u c l e i couple by means of the conduction e l e c t r o n s . A conduction e l e c t r o n i s s c a t -tered from a nucleus by the contact term £\ <>,§6Cr) to an ex c i t e d s t a t e . The sp 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 that of the nucleus. I f the e l e c t r o n i s then scattered o f f a second nuclear s p i n , the nucleus f e e l s the s p i n of the ex 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 of the f i r s t nucleus. An elaborate second order p e r t u r b a t i o n theory c a l -c u l a t i o n , using the 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 , gives the pseudo-exchange Hamiltonian as (>+) *Hex = Jij .I; . I j • The constant J,-j i s given by Jij (Sy ) = ^ V T f ^ f i i * < l Y ( 0 ) r > E , rj; 4[2k,r u cos(2kFn;) - s i n ( 2 k p r y ) ] . The important f e a t u r e s i n t h i s expression are i t s o s c i l l a t o r y nature and i t s dependence on ri*<(|\|/(0)|4'^E . The l a t t e r a l s o occur i n the expression f o r the i s o t r o p i c Knight s h i f t i n a d i f f e r e n t combination, ^|"\|/"(0)|a^ can thus be obtained by comparing measurements of 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 of the value obtained. The e f f e c t of the o s c i l l a t o r y 3^  p a r t has not been worked out f o r ordinary metals and i t i s u s u a l l y ignored; J being assumed p r o p o r t i o n a l to the asymptotic form r ~ 3 . The 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 s t r u c -ture of the metal. 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 . However, i f there are s e v e r a l i s o t o p e s , or a quadrupolar i n t e r a c t i o n , so that there are two or more l i n e s , the i n t e r a c t i o n broadens the l i n e s . ( i i i ) Pseudo-Dipolar Broadening This 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 that e x c i t a t i o n of the e l e c t r o n i s through the i n t e r a c t i o n of the nuclear and e l e c t r o n i c magnetic moments. The d e - e x c i t a t i o n occurs through the con-t a c t i n t e r a c t i o n . A s i m i l a r c a l c u l a t i o n to that of the pseudo-exchange c o n t r i b u t i o n gives the pseudo-dipolar Hamiltonian as (50) By i s a complicated expression which i n v o l v e s an i n t e g r a t i o n over the non-s e l e c t r o n s at the Fermi surface and so i s r e l a t e d to the a n i s o t r o p i c Knight s h i f t . I t a l s o contains the same o s c i l l a t o r y term that J does. I f a s p h e r i c a l Fermi surface i s assumed, group theory shows that *HPd must have the d i p o l a r form (50). I t can a l s o be shown t h a t higher order c o n t r i b u t i o n s to the i n t e r a c t i o n 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 assumption i s not made there i s no group t h e o r e t i c a l proof tha t Vidimust be d i p o l a r ( 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 . Both of these i n t e r a c t i o n s inc rease r a p i d l y i n s t r eng th w i t h i n c r e a s i n g atomic we igh t . Below an atomic weight of about 80 the i n t e r a c t i o n s are unde tec tab le , but f o r very h igh atomic numbers they can inc rease the second moment by over an order of magnitude. Th i s corresponds to decreas ing T a to below 10/us. U s u a l l y both i n t e r a c t i o n s are s imul taneous ly p resen t ; t h e i r r e l a t i v e magnitudes depending upon the f r a c -t i o n s of s and non-s e l e c t r o n s p resen t . The only other l i n e broadening mechanisms of importance are T, broadening and quadrupolar e f f e c t s i n a s t r a i n e d or impure c r y s t a l . 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 , . I f T, ^  t h i s s i g n i -f i c a n t l y broadens the l i n e . Quadrupolar e f f e c t s s l i g h t l y smear the resonant frequency and t h i s appears as a broadening of the l i n e . 2.5 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. The quadrupole moment i n t e r a c t s w i t h the e l e c t r i c f i e l d g rad ien t s present i n a non-cubic l a t t i c e to g ive a s e r i e s of energy l e v e l s . Consider a metal w i t h an a x i a l l y symmetric l a t t i c e , so tha t i t produces an a x i a l e l e c -t r i c f i e l d g rad ien t V z l = ^ r . I f a magnetic f i e l d H 0 i s 36 a p p l i e d at an angle 6 to the c r y s t a l symmetry a x i s , the Hamiltonian f o r a s i n g l e nucleus becomes (^O) where H.= - Tft H„ Iz< , U* = •[31«fcos 10+ 3IV 31^0+1(1^1^+ 1^1/) s i n 2 9 - I 1 ] . When H z ^ H Q the energies and wave f u n c t i o n s of the Hamil-tonian have to be found by numerical computation (51). P e r t u r b a t i o n theory can be used outside t h i s r e g i o n . The low f i e l d case ( T V » H . ) w i l l not be considered 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 along the magnetic f i e l d so a p e r t u r b a t i o n theory c a l c u l a t i o n gives the resonant frequencies as (1) V^ - U.+ i^(m-i ) (3cos a 9 -1)+ higher terms, where the quadrupole frequency H = a i ^ i n " * VL i s the Larmor frequency, w h i l e V* i s the frequency of the t r a n s i t i o n from the m-1 to the mth energy l e v e l . There are s e v e r a l f e a t u r e s to note about the quadrupole i n t e r a c t i o n . The most important i s that unless I > 1 i t van-i s h e s . The other f e a t u r e i s that the 21+1 Zeeman energy l e v e l s are no longer e q u a l l y spaced. The r e s u l t of t h i s i s that the resonance l i n e i s s p l i t i n t o 21 separate l i n e s . I f only the f i r s t order term i s considered, the frequency of the c e n t r a l m-1 = -£r*m=-£- t r a n s i t i o n remains the same as VL» The 37 other 2 1 - 1 s a t e l l i t e l i n e s are symmetrically d i s p l a c e d from the c e n t r a l l i n e . Furthermore, i f the magnetic f i e l d i s rotated u n t i l 0=cos~' ( 3 ) , the 2 1 l i n e s coalesce i n t o a s i n g l e l i n e . I f higher order terms are considered, then the frequency of the c e n t r a l l i n e i s s h i f t e d and i t i s a l s o no longer 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 coalesce. The e l e c t r i c f i e l d gradient i s d e r i v e d from a l l the e l e c t r o n s and ions i n the metal. The closed e l e c t r o n s h e l l s are s p h e r i c a l l y symmetric and so do not d i r e c t l y c o n t r i b u t e to V 2 Z , even though they are the nearest charges to the nucleus. The p o t e n t i a l at a nuclear s i t e due to a l l the other ions 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 considerable accuracy. However, t h i s i s not the p o t e n t i a l gradient a c t u a l l y f e l t by the nucleus. The f i e l d from the other ions s l i g h t l y d i s t o r t s the closed s h e l l s . Because they are so c l o s e to the nucleus t h e i r d i s t o r t i o n magnifies the e f f e c t of the e x t e r n a l f i e l d by a f a c t o r 1+TW (k-0). %, i s the Sternheimer a n t i s h i e l d i n g f a c t o r and i s u s u a l l y at l e a s t ten. U n f o r t u n a t e l y , i t cannot be a c c u r a t e l y c a l c u l a t e d . 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)r3](|) dsx> . This i s r e l a t e d to the term q' i n the expression 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 since q' i s averaged over the Fermi surface e l e c t r o n s and i s averaged over a l l the conduction e l e c t r o n s . I f the con-ducti o n band has a complex s t r u c t u r e , i t i s not even necessary f o r them to have the same s i g n . The meagre evidence a v a i l a b l e suggests that most of the e l e c t r i c f i e l d gradient i s due to the conduction electrons (^ +9). 2.6 The Line Width With A Quadrupole I n t e r a c t i o n I t i s assumed that the quadrupole i n t e r a c t i o n i s l a r g e enough to c l e a r l y separate the 21 l i n e s . I f : 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 considered, there i s l i t t l e change i n the l i n e width (1, 52). The presence of a pseudo-exchange i n t e r a c t i o n causes a considerable change i n the l i n e width. When any of the l i n e s overlap t h i s i n t e r a c t i o n causes a narrowing of the l i n e s . However, when they do not overlap many of the mutual s p i n f l i p s no longer conserve Zeeman energy and so are sup-pressed. The e f f e c t of t h i s i s to allow the i n t e r a c t i o n to broaden the l i n e (56). The pseudo-dipolar i n t e r a c t i o n a l s o broadens the l i n e . The l i n e width i n metals w i t h an atomic weight above 80 and w i t h a quadrupole i n t e r a c t i o n i s thus always greater than the d i p o l a r width. 2.7 Pulsed.NMR With a Quadrupole I n t e r a c t i o n In the system to be discussed H x » eHa>v>'H(. There w i l l thus be 21 separate l i n e s i n the spectrum. Because of the i n e q u a l i t y i n energy l e v e l spacing, the only mutual s p i n f l i p s which are e n e r g e t i c a l l y allowed are those which do not change the energy l e v e l p o p u l a t i o n s . Thus i f the system i s ' perturbed i t i s no longer 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. This changes some of the NMR p r o p e r t i e s . 39 I n i t i a l l y assume that there i s no coup l i n g between separate energy l e v e l s . Thus a pulse a p p l i e d a t the resonant frequency of one l i n e does not a f f e c t the other l e v e l s so that they can be disregarded. This assumes that the pulse contains only one frequency, a s i t u a t i o n u n a t t a i n a b l e i n p r a c t i c e . The c o n d i t i o n f o r a 90° pulse f o r the m^ =i m+1 l i n e becomes (1) where T i s the r f pulse width. Since only two l e v e l s are i n -volved, the precessing magnetic moment i s much smaller than when a spi n temperature allows a l l 21+1 l e v e l s to be involved i n the t r a n s i t i o n s . The system can be assumed to have a f i c -t i t i o u s s p i n of £ ( D ? rat h e r than i t s true s p i n of I , so that the f r a c t i o n a l r e d u c t i o n i n magnetisation i s 3 A K I+D. A f t e r a p p l i c a t i o n of an r f pulse, the two l e v e l sub-system can be described by a spi n temperature which r e l a x e s towards the l a t t i c e temperature. The r e l a x a t i o n need not be a simple ex-p o n e n t i a l decay. Pulsed NMR i n a system of t h i s type thus r e q u i r e s shorter p u l s e s , but gives a weaker s i g n a l , than i n a normal Zeeman system.' I f there i s coupling between the l i n e s , the s i t u a t i o n can be very complicated. However, the bas i c f e a t u r e s of such a system can be understood from studying the simpler case of 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 temperature, coupled together. Energy conserving s p i n f l i p s are by f a r the strongest c o u p l i n g mechanism. However, some other terms i n the d i p o l a r Hamiltonian give a much weaker co u p l i n g . 1+0 Mutual s p i n f l i p s -which do not conserve Zeeman energy can a l s o occur i f phonons, or some other source, can supply the energy d i f f e r e n c e . This i s u s u a l l y a weak coupling mechanism because of the s c a r c i t y of phonons w i t h the required energy. Let both systems be perturbed and then set up the r a t e equations f o r the population changes of a l l the l e v e l s . When combined w i t h the p r i n c i p l e of d e t a i l e d balance, t h i s gives the r a t e equations f o r the spi n temperatures as (^ +0, 53) & (e; 1 ) = -T,;' (e; 1 - )-A e* (e; 1 - a " ) , G;1 ) = - T; 1 c e;1 - e;1 H A Q: C e;1 - e ; 1 ) . 9, and are the spi n temperatures, 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 . In both equations, the f i r s t term on the r i g h t i s the sp 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 B0i w h i l e the l a s t term i n v o l v e s an energy t r a n s f e r (cross r e l a x a t i o n ) to the other system at a r a t e depending on the temperature d i f f e r e n c e between them. The time constant governing t h i s cross r e l a x a t i o n can be found by p u t t i n g T„ = Txl = oo and combining the two equations to give The"cross r e l a x a t i o n time Tu i s thus given by T,a = A ( e* + el). 1+1 When T h , T^"^ T,a , the system f i r s t cross r e l a x e s to a common spi n temperature, which then r e l a x e s towards the l a t t i c e temperature. I t s behaviour can thus be described by two independent exponential decays. I f T i a» T(1 , T a a the systems decay almost independently toward the l a t t i c e tem-perature a t r a t e s described by Tn and Tal r e s p e c t i v e l y . For the intermediate case where T ( a ~ Tn , a simple d e s c r i p t i o n of the system i s no longer p o s s i b l e ; i t s decay depending on the three time constants and a l s o on the way i t i s perturbed. Although some d e t a i l s of the energy l e v e l s and couplings are 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 experimental i n v e s t i g a t i o n of such a system showed -that a cross r e l a x a t i o n theory based upon the idea of mutual s p i n f l i p s when the l i n e s overlapped was only moderately s u c c e s s f u l i n d e s c r i b i n g the system (5^). When the l i n e s overlapped e x t e n s i v e l y cross r e l a x a t i o n between the l e v e l s occurred i n a time l e s s than 60ms,, but when they were widely separated the cross r e l a x a t i o n time constant was J+7 seconds, not much l e s s than T, , I f the l i n e s only p a r t i a l l y overlapped the cross r e l a x a t i o n time was intermediate between these va l u e s , as expected t h e o r e t i c a l l y . However, there was often an extended period i n the middle of the cross r e l a x a t i o n when energy t r a n s f e r ceased. A more serious discrepancy i s that i f a s a t e l l i t e l i n e i s perturbed by a very short r f pulse 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 cross r e l a x a t i o n i s not a Boltzmann d i s t r i b u t i o n . P e r t u r b i n g 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 cross r e l a x a t i o n has occurred, i n agreement w i t h the theory. Recent t h e o r e t i c a l and experimental work (55) suggests th a t the f a i l u r e of the r a t e equation approach to cross r e l a x a -t i o n i s due to neglec t of the d i p o l e - d i p o l e system. The nuclear magnetic system a c t u a l l y c o n s i s t s of sub-systems described by the Zeeman terms and by the d i p o l e - d i p o l e terms of the Hamil-t o n i a n . Each of these sub-systems has i t s own energy and spi n temperature and can be weakly coupled to other sub-systems. Cross 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 from Zeeman to 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 bel i e v e d that a c a r e f u l consider-a t i o n of these.energy exchanges can e x p l a i n the 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! But Not Without a P l a n . 1 - Pope; The pulsed NMR spectrometer i s to be discussed here i s designed s p e c i f i c a l l y to measure the anisotropy of T| i n m e t a l l i c s i n g l e c r y s t a l s , but i t does have s u f f i c i e n t versa-t i l i t y to measure T\ and Ta 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 pulsed NMR system, the nuclear s p i n system, i n i t i a l l y a l i g n e d along H©, i s tipped by a huge uniform r f pulse a p p l i e d at r i g h t angles to H 0. At the end of the r f pulse a l l the n u c l e i are a l i g n e d at the same angle to H 0. The recovery of the s p i n system i s then studied by a m p l i f y i n g the short 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 the sample. Abragam (1) gives a general 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 , while C l a r k (2) l u c i d l y describes the compromises and experimental d e t a i l s involved i n the design and c o n s t r u c t i o n of a pulsed NMR apparatus. The w r i t e r ' s apparatus i s based on C l a r k ' s apparatus, both 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 -ferences i n design philosophy, and a l s o i n the c i r c u i t r y . In a no n - m e t a l l i c sample, or a very f i n e l y d i v i d e d m e t a l l i c powder, the r f f i e l d completely penetrates the sample, but i t can only penetrate a very short d i s t a n c e i n t o a m e t a l l i c sample. This i s because of tfre well-known s k i n e f f e c t (3). The s k i n e f f e c t has two main experimental 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 dist a n c e of about the s k i n depth & of the surface. These are u s u a l l y only about 1% of the t o t a l number of n u c l e i i n the sample, so that the s i g n a l s are much weaker than i n a normal NMR experiment. They are so weak that they are always ob-scured by noise when dis p l a y e d on an o s c i l l o s c o p e , so that a boxcar i n t e g r a t o r , a device f o r improving the 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 distance from the surface of the sample. Thus at the end of the r f pulse, the n u c l e i are not a l l a l i g n e d at the same angles to H 0 ? so that conventional pulse t r a i n s such as a ^ — T T pulse 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 of measuring T( had to be developed. 3.1 General D e s c r i p t i o n of the Apparatus The main f e a t u r e s of the apparatus ares ( i ) a b i l i t y to operate a t any frequency between about 5 and 10 Mc/s without extensive returning ( i i ) a r f magnetic f i e l d H, of 25 gauss ( i i i ) a recovery time of Ijjus ^5 ( i v ) phase s e n s i t i v e d e t e c t i o n (v) boxcar i n t e g r a t i o n ( v i ) a c o i l system designed s p e c i f i c a l l y f o r metal samples. Two s i g n a l s are taken from the master C o l p i t t s o s c i l -l a t o r . One i s used as a reference phase and i s passed through a phase s h i f t e r i n t o the a m p l i f i e r . The other s i g n a l passes i n t o the gated power a m p l i f i e r . This i s gated by p o s i t i v e pulses from a t i m i n g u n i t , and d e l i v e r s r f pulses of about 1.8 KV. peak to peak to the t r a n s m i t t e r c o i l . The induced s i g n a l from the c o a x i a l pickup c o i l i s a m p l i f i e d i n a tuned pre-a m p l i f i e r w i t h a bandwidth of about 0 . 5 Mc/s and then passed i n t o an Arenberg WA600D a m p l i f i e r . In 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 reference s i g n a l , are l i n e a r l y added to give a phase s e n s i t i v e system. This 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 that 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 which i s caused by the small 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 diodes. The s i g n a l plus reference 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 -grator improves the S/N r a t i o by a f a c t o r ranging from about 10 to 100. The output from the boxcar i s then recorded on a V a r i a n Model G-11A chart recorder ( F i g . 3.1). The t i m i n g u n i t can a l s o send synchronised pulses to the recorder event marker, so that timing pips at i n t e r v a l s of lOOyUs to one second can be recorded on the chart along v/ith the s i g n a l s . The timing u n i t can a l s o supply a quench pulse to the p r e a m p l i f i e r which helps reduce the recovery time. I t does t h i s by lowering the Q of the pickup c o i l w h i l s t the r f pulse i s on, and then r a i s i n g i t soon a f t e r the pulse has ceased. The t i m i n g u n i t a l s o provides a two pulse sequence of a r b i t r a r y widths, separation and r e p e t i t i o n r a t e f o r gating the power a m p l i f i e r . I t a l s o provides a boxcar gating pulse, and event marker pulses which are synchronised w i t h the basic r f pulse sequence. As w e l l as t h i s i t contains a sawtooth generator which can be used f o r l i n e a r l y sweeping the 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 the sepa r a t i o n between two p u l s e s , or l i n e a r l y v a r y i n g the separation between a r f pulse and the boxcar gating pulse. The dewer system was designed and b u i l t to operate at helium temperatures. However, no helium temperature measure-ments have been made and i t was subsequently necessary to modify the apparatus i n such a way that helium temperatures are now u n a t t a i n a b l e . L i q u i d n i t r o g e n temperature measure-ments are 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 of the pulse sequence i s determined by a f r e e running m u l t i v i b r a t o r which t r i g g e r s a Tekt r o n i x 162 sawtooth generator. This r e p e t i t i o n r a t e can be v a r i e d from (8ms.)' to (9 sec) - 1 . The pulse from the ^7 Gate Out te r m i n a l of the generator i s used to t r i g g e r the marker puls e , w h i l e the sawtooth output i s fed i n t o two Tekt r o n i x 163 pulse generators. One of the pulse generators i s set to t r i g g e r at the beginning of the c y c l e , w h i l e the second one i s set to t r i g g e r a t some l a t e r time i n the c y c l e . Thus the two pulses can be separated by any des i r e d i n t e r v a l up to nine seconds. The width of each of these pulses can be independently v a r i e d from ^ius up to many m i l l i s e c o n d s . The two pulses are then fed i n t o a pulse mixer which a l s o a m p l i f i e s them to the 90 v o l t s required to gate the 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 the ga t i n g pulses are l e s s than O.^us. Observation on an o s c i l l o s c o p e shows that 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 pulse 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 angle through which the n u c l e i are tipped 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 metal samples i s so poor that the e x t r a noise due to the pulse j i t t e r i s u s u a l l y unobservable. J i t t e r due to v a r i a t i o n s i n the t r i g g e r i n g time of the pulse genera-t o r s i s u s u a l l y a l s o unobservable. A pulse i s a l s o taken from the Pulse Out t e r m i n a l of the second T e k t r o n i x 163 pulse generator and used f o r gating the boxcar. This 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-tooth generator whose output i s fed i n t o a T e k t r o n i x 161 pulse generator. This can t r i g g e r on any p a r t of the saw-to o t h , so that i t s output pulses ( 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 ) , which gate the boxcar, can occur any time a f t e r the beginning of the second r f pulse ( F i g . 3.6). This completes the d e s c r i p t i o n of the bas i c p a r t of the timi n g u n i t . There are, however, s e v e r a l a u x i l i a r y p a r ts of the t i m i n g u n i t , some of which are not o f t e n used. In the Tektronix 161 and 163 pulse generators, a voltage comparator stage compares the instantaneous voltage of the ^nput sawtooth w i t h a voltage set by a potentiometer. When the decreasing sawtooth voltage equals the preset v o l t a g e , the pulse generator i s t r i g g e r e d . I f the preset voltage i s now var i e d slowly and l i n e a r l y , the time delay between the s t a r t of the sawtooth and the generator t r i g g e r i n g w i l l a l s o vary l i n e a r l y w i t h time. The preset voltage i s v a r i e d i n t h i s f a s h i o n by dis c o n n e c t i n g the comparator g r i d from the p o t e n t i o -meter and instead feeding a slowly v a r y i n g negative sawtooth voltage onto i t ( i f ) . A T e k t r o n i x 161 and a 163 pulse generator were modified i n t h i s way; a two-way switch being used so that e i t h e r the i n t e r n a l preset v o l t a g e or the e x t e r n a l sawtooth voltage can be fed onto the comparator g r i d . I f the e x t e r n a l sawtooth i s applied to the modified 163 pulse generator gating the power a m p l i f i e r , a two pulse se-quence i s obtained 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 pulses. This sequence enables the recovery of the mag-n e t i s a t i o n a f t e r a pulse to be d i r e c t l y recorded on a c h a r t , from which T, can be q u i c k l y obtained. **9 Applying the e x t e r n a l sawtooth to the modified 161 pulse generator gating the boxcar, sweeps the boxcar gate over the 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 pulse. From the re c o r d i n g of the boxcar output T a can be found. The comparison sawtooth i s got from a phantastron which s t a r t s a sweep when manually t r i g g e r e d . Normally the sawtooth decreases from 1*4-0 v o l t s to 20 v o l t s , but a bias voltage can be app l i e d so that the output voltage remains constant at any voltage between l*+0 v o l t s and 100 v o l t s , u n t i l the decreasing sawtooth reaches t h i s v o l t a g e . The output voltage then f o l l o w s the sawtooth v o l t a g e . This f e a t u r e allows f o r the f i n i t e width of the r f pul s e s , a necessary f e a t u r e when sweeping w i t h some pulse sequences. The sawtooth d u r a t i o n can be va r i e d from ten seconds to t h i r t y minutes i n seven steps. 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 the sweep, but then increases r a p i d l y to about 10% at the end of the sweep. A s e r i e s of measurements showed that the sawtooth l e n g t h was re p r o d u c i b l e to w i t h i n 2% of i t s l e n g t h . Neither 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 sweep i s not necessary, or e l s e c a l i b r a t e d timing pips are used. The sawtooth generator i s u s u a l l y used f o r sweeping the magnetic f i e l d through the resonant value. To do t h i s , an attenuated sawtooth voltage i s taken from the sawtooth generator and fed i n t o the magnet power supply. To measure T, or Ta i t i s necessary to measure time i n -t e r v a l s such as those between r f pul s e s , or between one r f 50 pulse and the boxcar gating p u l s e , with an accuracy of 2% or b e t t e r . Two a l t e r n a t i v e t i m i n g methods are a v a i l a b l e . I f the slow sawtooth i s being a p p l i e d to the 161, or the 163 generator, a pulse i s taken from the Gate Out t e r m i n a l of the r e l e v a n t 162 sawtooth generator t r i g g e r i n g i t and fed i n t o the Gate In te r m i n a l of the 162 sawtooth generator used as a marker pulse generator. This i s set to run at some convenient r e p e t i t i o n r a t e , such as l K c / s . When the pulse i s appl i e d to the Gate In t e r m i n a l , the generator gives out pulses a t , say, 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 the gating pulse ceases. This t r a i n of pu l s e s , which i s synchronised to the r f pulses, i s c then fed i n t o a coincidence u n i t . A pulse from the boxcar gate, or the second r f gate pulse, i s a l s o fed i n t o the co-incidence u n i t and when these two pulses c o i n c i d e , a current pulse actuates the event marker pen on the recorder. Thus the boxcar output and a s e r i e s of timing pips are recorded on the same chart. 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 can be used w i t h the t r a i n of marker pips displayed on one channel and the r f pulse sequence displayed on the other channel. The separ-a t i o n between r f pulses can then be manually adjusted to co i n c i d e w i t h the desired timing p i p . The pulse generator was c a l i b r a t e d w i t h a C.M.C. 707BN frequency counter. I t was found to be accurate to w i t h i n 1,5% 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 period of s e v e r a l hours. The coincidence u n i t r e q u i r e s the pulses 51 to overlap 0.6yus before i t t r i g g e r s . However, the minimum separation between marker pulses i s lOO^is, so that t h i s i s a n e g l i g i b l e systematic e r r o r . I f a double beam o s c i l l o s c o p e i s used the tim i n g e r r o r due to a l i g n i n g the two pulses v i s u a l l y i s s t i l l l e s s than 1%. This i s since the screen i s 10cm. wide and the f u l l width of the screen i s always used. As the l i n e s are l e s s than 1mm. wide, i t i s easy to make them c o i n c i d e to w i t h i n l e s s than 1% of the screen width. I t thus i s safe to assume an e r r o r l e s s than 2% i n a l l t i m i n g measurements. The 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 , so that the e r r o r i n the t i m i n g measurements i s s u f f i c i e n t l y s m a l l . The pulse mixer a l s o provides a two v o l t p o s i t i v e pulse which i s used f o r t r i g g e r i n g the o s c i l l o s c o p e and a minus twenty v o l t quenching pulse of v a r i a b l e width f o r the pre-a m p l i f i e r quenching c i r c u i t . 3.3 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 to 11 Mc/s. and has a long term frequency d r i f t of one part i n 5xl04 per hour. This i s adequate s t a b i l i t y f o r most measure-ments on broad metal l i n e s . The output from the o s c i l l a t o r passes through a cathode f o l l o w e r to a gating c i r c u i t designed by Blume (5) f o r n e g l i -g i b l e r f leakage when the gate i s o f f . 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 the gate i s switched on by a ni n e t y v o l t p o s i t i v e pulse from the pulse mixer, i t loads the cathode 52 f o l l o w e r s u f f i c i e n t l y to decrease i t s output by about 20$. In Blume's c i r c u i t , the phase reference 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 . In t h i s case, the drop i n output upsets the phase reference s i g n a l f o r about 50yws a f t e r the gate i s switched o f f . To cure t h i s t r o u b l e , a separate cathode f o l l o w e r was added f o r the phase reference s i g n a l 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 ampli-f i e r stages. These give high power a m p l i f i c a t i o n , good c a r r i e r suppression, and short r i s e and f a l l times w i t h reasonably high Q c i r c u i t s i n the f i r s t two stages. The f i n a l stage i s an 829B which i s operated w i t h 1500 v o l t s on the p l a t e and a screen voltage which can be v a r i e d from 5^0 to 600 v o l t s . Under these operating c o n d i t i o n s , the maximum power output i s about 2KW. As i n C l a r k ' s t r a n s m i t t e r (2), v a r i a b l e damping of the t r a n s -m i t t e r c o i l i s provided by biased diodes placed across i t . With l a r g e a p p l i e d r f voltages one of the diodes i s always open c i r c u i t e d so that n e g l i g i b l e damping occurs. However, when the r f pulse decays to about one v o l t , both diodes con-duct and shunt the c o i l w i t h an impedance of about 600A . Rise times and f a l l times are t y p i c a l l y about lyMs, while pulses up to 600/us long can be generated before sagging i n the r f output voltage becomes excessive. The p i e z o e l e c t r i c p r o p e r t i e s of ceramic condensers d i d not cause r i n g i n g i n the output c i r c u i t , provided they were used w e l l below t h e i r maxi-mum rated v o l t a g e . This i s contrary to the experience of C l a r k (2). 53 3.^ The P r e a m p l i f i e r This i s a tuned voltage a m p l i f i e r s l i g h t l y modified from one designed by C l a r k (2). I t has a gain of 16 and a band-width of about 0„5Mc/s. The main d i f f e r e n c e i s th a t crossed s i l i c o n diodes l i m i t the g r i d swing of the input stage to -1 v o l t , even f o r a p p l i e d voltages of s e v e r a l hundred v o l t s . They a l s o heavy damped the pickup 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 only the very small s i g n a l voltage i s present. Instead of crossed diodes, C l a r k used a quenching c i r c u i t i n which a quenching pulse derived from the pulse mixer and a m p l i f i e r c i r c u i t switched a low impedance loud across the pickup c o i l during the r f pulse, and f o r a v a r i a b l e time afterwards. However, i n the present apparatus the input r f pulses are so l a r g e that crossed s i l i c o n diodes were i n i t i a l l y added to pr o t e c t the quenching c i r c u i t . I t was then found that when the quenching c i r c u i t was switched o f f , a small t r a n s i e n t occurred which swamped the very weak induced s i g n a l . This t r a n s i e n t was due to the storage capacitance of the swi t c h i n g diode i n the quenching c i r c u i t and could not be e l i m i n a t e d . Thus the quenching c i r c u i t had to be disconnected f o r a l l measurements on s i n g l e c r y s t a l s and the crossed diodes alone used f o r damping the c o i l . I f powdered samples are 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 that 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 noise r e s i s t a n c e of the pre-a m p l i f i e r i s about 25011 so that f o r l i q u i d n i t r o g e n tempera-tures and above, the thermal n o i s e from the resonant pickup c o i l i s the dominant noise source. 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 that i t s noise output i s much l a r g e r than the thermal noise generated i n the input stage of the Arenberg a m p l i f i e r . 3.5 The Main A m p l i f i e r This i s an Arenberg WA600D which has been modified to improve i t s recovery time. O r i g i n a l l y the a m p l i f i e r had a frequency response from 2 to 65 Mc/s. A u x i l i a r y tuning c o i l s enabled the frequency response to be a l t e r e d to a passband about lOMc/s wide centred\ on any frequency w i t h i n t h i s range. However, there was an annoying overshoot present f o r about 20yus a f t e r the r f pulse. To e l i m i n a t e t h i s , the low frequency response was increased to 3.5Mc/s and the screen bypass con-densers reduced i n value. Later on the input stage was r e b u i l t along l i n e s suggested by the manufacturer. These improvements reduced the overshoot to l e s s than ^ s i n d u r a t i o n . The low frequency response of the video s e c t i o n was de-creased from 20 to 2c/s to avoid n o t i c e a b l e b a s e l i n e droop. This s e c t i o n a l s o introduced considerable 60c/s pickup from the f i l a m e n t s i n t o the output, so that the f i l a m e n t s were converted to run on regulated D.G. cur r e n t . The recovery time of the whole system i s n e a r l y lO^s from the end of the t r a n s m i t t e r g a t ing pulse. This i s 55 probably about twice as long as the minimum p r a c t i c a l l i m i t , but s i g n i f i c a n t r e d u c t i o n i n the recovery time would r e q u i r e an-exorbitant amount of time and labour. Although n e i t h e r the p r e a m p l i f i e r nor the main ampli-f i e r has an automatic gain c o n t r o l , l i t t l e t r o u b l e i s experienced w i t h long term d r i f t s i n gain. This i s always l e s s than 1% per hour, provided the a m p l i f i e r s have been allowed to warm up f o r s e v e r a l hours. However, there are short term f l u c t u a t i o n s i n gain of about 10$, w i t h a period of about ten minutes' d u r a t i o n which 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 gain i n another Arenberg 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 to the w r i t e r ) . The l i n e a r i t y of the Arenberg i s poor; the l i n e a r r e g i o n extending from about 2 to 12 v o l t s at the output. To improve the l i n e a r i t y f o r small s i g n a l s the reference voltage from the o s c i l l a t o r i s added to the s i g n a l at the s i x t h stage of the a m p l i f i e r through a high pass f i l t e r and a r e s i s t i v e network. A phase s h i f t e r and a t t e n u t o r (2) are i n s e r t e d i n the reference channel between the o s c i l l a t o r and the a m p l i f i e r . The amplitude of the reference voltage i s much l a r g e r than the s i g n a l and i s adjusted so that i t i s near the centre of the l i n e a r r e g i o n . 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 small s i g n a l s and p r o v i d i n g phase s e n s i t i v e d e t e c t i o n . This simple 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 the s i g n a l (Appendix I ) . In these e x p e r i -ments, the reference s i g n a l i s at l e a s t ten times the s i g n a l , so that the 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 -able as i t i s l e s s than the e r r o r caused by no i s e . 3.6 The Boxcar I n t e g r a t o r This i s an e l e c t r o n i c device f o r improving the S/N r a t i o of a r e p e t i t i v e s i g n a l . 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 switch which i s switched on by a gating pulse oc-c u r r i n g at a set time a f t e r an ap p l i e d r f pulse, followed by a RC c i r c u i t which averages the voltages obtained during suc-c e s s i v e sampling i n t e r v a l s . The boxcar was b u i l t to the design of Blume (h). The only major m o d i f i c a t i o n i s the a d d i t i o n of a voltage a m p l i f i e r stage at the in 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 voltages up to ±20 v o l t s , but the voltage a m p l i f i e r at the input i s only l i n e a r f o r input voltages of about -8 v o l t s . This voltage swing i s adequate f o r the present experiments and can e a s i l y be increased i f necessary. The measured long term d r i f t i n the base l i n e c o r r e s -ponds to a d r i f t of 0.05 v o l t s per hour at the inpu t . This slow d r i f t i s hardly n o t i c e a b l e , even i n experiments t a k i n g many hours. In the i d e a l boxcar c i r c u i t ( F i g . 3.3) the switch S i s closed f o r a short time % at the r e p e t i t i o n r a t e T ' of the 57 0- 5 •AAAAAMA •0 F i g . 3.3 Equivalent C i r c u i t of the Boxcar I n t e g r a t o r r f pulses. The time constant E C » X S , so that the c i r c u i t acts as an i n t e g r a t o r . I f the r.m.s. input noise voltage i s v; and the output noise voltage i s E 0 then (6) where f c i s the noise c o r r e l a t i o n time. For a thermal noise source v,oCll> and <t toCEf' so that 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 . This equation has been derived under the assumption that 'ttj a c o n d i t i o n which i s e a s i l y s a t i s f i e d experimentally. I f vj i s the input s i g n a l , then the boxcar response i s the same as a low pass RC f i l t e r w i t h a time constant t'= RCT/T:t ( 6 ) . Unless care i s taken i n the choice of <t! considerable d i s t o r t i o n of the s i g n a l can occur. An experimental t e s t of these equations was made by using an a m p l i f i e r as a thermal noise source w i t h a frequency spectrum from about 2c/s to s e v e r a l megacycles. This was fed J . 58 i n t o the boxcar i n t e g r a t o r and the output measured on a chart recorder, 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, but v e r i f i e d that the noise output i s independent of the bandwidth and that the S/NoCRC, except f o r very l a r g e values of RC when the S/N became l e s s than p r e d i c t e d . The equations f o r the boxcar voltages ignore the e f f e c t of the recorder time constant, which i s about one second f o r the V a r i a n recorder. This time constant performs a f u r t h e r i n t e -g r a t i o n of the noise f o r small values of T, so that the S/N but decreases s t e a d i l y f o r T<~^70ms., being a f a c t o r of s i x smaller f o r T=lms. 3.7 Power Supplies and Noise Suppression The 1500 v o l t s f o r the power a m p l i f i e r i s supplied by a simple unregulated power supply, which a l s o gives ^50, 525 or 600 v o l t s f o r the screen regulated by voltage r e g u l a t i n g tubes. The l a r g e output impedance of these s u p p l i e s i s overcome by the use of l a r g e storage condensers. A 200ma. regulated supply gives the 350 v o l t s used i n an e a r l i e r stage of the power a m p l i f i e r . Apart from the Arenberg which has i t s own regulated supply, the r e s t of the apparatus i s powered by two Tektronix 160A regulated 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 current comes from a simple t r a n s i s t e r i s e d regulated power supply (7) fed by a Heathkit b a t t e r y e l i m i n a t o r . r a t i o i s increased. Experimentally 59 Various combinations 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 accumulator had been t r i e d e a r l i e r on but gave much poorer regu-l a t i o n than the t r a n s i s t e r i s e d supply. The 110 v o l t s A.C. f o r the whole apparatus i s s t a b i l i s e d by a General Radio Type 1570/A automatic l i n e v oltage r e g u l a t o r . Rf leakage from the o s c i l l a t o r to the a m p l i f i e r i s o f t e n a major source of tr o u b l e i n a coherent NMR system. However 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 of a l l power leads connected 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 completely 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 taken w i t h the Interstage f i l t e r i n g i n the am-p l i f i e r s to e l i m i n a t e any chance of regeneration o c c u r r i n g . In a l l of the r f f i l t e r i n g extensive use i s made of P h i l l i p s f e r r i t e beads along w i t h O.Olyuf ceramic bypass condensers. In a p e r f e c t apparatus the only noise source i s the thermal noise of the resonant pickup c o i l . This was some-times so w i t h the present apparatus, but o f t e n noise from e x t e r n a l sources was much l a r g e r than the thermal n o i s e . The most common e x t e r n a l noise 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 the l a b o r a t o r y , followed by noise from heavy e l e c t r i c a l machinery i n other parts of the b u i l d i n g . Both of these sources give r f pulses synchronised to the mains frequency. I t i s not c e r t a i n whether these pulses t r a v e l along the power mains, or are detected by the tuned pickup c o i l wrapped around the sample. 60 Because of the small magnet gap not enough space i s a v a i l a b l e to enclose the c o i l s i n an earthed metal s h i e l d , so the apparatus i s s u s c e p t i b l e to extraneous induced v o l t a g e s . An attempt was made to place an earthed s h i e l d around the outside 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 noise l e v e l , so the pickup c o i l has been l e f t unshielded. The whole e a r t h i n g system of the apparatus had to be arranged so that there were no ear t h i n g loops i n c r e a s i n g the noise l e v e l . This was mainly an e m p i r i c a l process that verged on black magic. 3.8 The Magnets and Magnetic F i e l d Measurements The apparatus 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 Var i a n l/hO07 6" electromagnet w i t h a two-inch gap. This gives a f i e l d of up to 7KG. which i s homogeneous to w i t h i n a gauss over the volume of the sample. P a r t way through the e x p e r i -ments, the apparatus was 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 w i t h a 2 .25-inch gap and a f i e l d s t rength of up to 11.2KG. This magnet has a homogeneity of about 0.1 gauss over the sample volume. Both magnets were r o t a t a b l e through over l 8 0 ° . The magnetic f i e l d s of both magnets were often swept through the resonance values by apply i n g a sawtooth voltage of 8 v o l t s to the E x t e r n a l Sweep terminals 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 . This v a r i e s the f i e l d by about 100 gauss as an approximately l i n e a r f u n c t i o n of time. A simple marginal o s c i l l a t o r was b u i l t f o r any magnetic f i e l d measurements needed. I t was used to c a l i b r a t e the 6" 61 electromagnet, but w i t h the 12" electromagnet a f i e l d c a l i b r a -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 accuracy f o r f i n d i n g resonances, so that the marginal 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 intended to make measurements at very low temperatures so a conventional glass dewar l i q u i d helium system was b u i l t . The dewars were b u i l t f o r the s i x magnet so the inner dewar has an i n t e r n a l diameter of only one i n c h , which severely r e s t r i c t s the c o i l dimensions. When the apparatus was s h i f t e d to the 12" electromagnets the dewar head already there was u n s u i t a b l e f o r pulsed NMR, so that the dewar system from the 6" magnet had to be used w i t h the 12" magnet. As the dewar head did not match the helium r e t u r n system already there, no l i q u i d helium temperature measurements were p o s s i b l e without extensive r e b u i l d i n g of the low temperature p a r t of the apparatus. This r e b u i l d i n g was not done f o r reasons to be given l a t e r . A l l the measurements have e i t h e r been made at room temperature, or at l i q u i d n i t r o g e n temperature. For l i q u i d n i t r o g e n measurements, the outer dewar i s f i l l e d w i t h l i q u i d 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 the inner dewar. I f the inner dewar alone i s f i l l e d w i t h l i q u i d n i t r o g e n , i t s bubbling causes v i b r a t i o n s which decrease the 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 S i n g l e C r y s t a l The most important part of the apparatus i s the c o i l system. I t i s a l s o the hardest part to design so that the c o i l s used, and the reasons.for using them, w i l l be discussed i n considerable 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 . The e q u i v a l e n t c i r c u i t of the c o i l system i s given by F i g . 3.^. 2 S i s the l a r g e s i g n a l output impedance of the f i n a l stage of the power a m p l i f i e r , w h i l e L* i s the inductance of the t r a n s m i t t e r c o i l . R , i s the eq 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 comparatively low when the r f pulse i s on, but i t i s much higher when i t i s o f f . R,' i s the r e s i s t i v e component of the impedance coupled back from the pickup c o i l by the mutual inductance M. L^ ., R i and R-L are the correspon-ding parameters f o r the pickup c o i l . R a i s l a r g e r during an r f pulse than when i t i s o f f . C l a r k (2) f u l l y discusses a l l the requirements f o r the c o i l s . B r i e f l y they are that the t r a n s m i t t e r c o i l must have the maximum number of ampere-turns 6 F i g . 3.*+ Equivalent C i r c u i t of the C o i l System 63 p o s s i b l e and a f a i r l y low Q, w h i l e the pickup c o i l should have the l a r g e s t number of turns and the highest Q p o s s i b l e . The mutual inductance should be as small as p o s s i b l e since i t causes the f o l l o w i n g undesirable e f f e c t s to occur. ( i ) The resonant c o i l pickup c o i l extracts a s i g n i f i c a n t f r a c t i o n of the power supplied to the t r a n s m i t t e r c o i l . This reduces H| by the same f r a c t i o n . I t i s assumed that both resonant c i r c u i t s are not very t i g h t l y coupled and that the t r a n s m i t t e r c o i l i s matched to the output impedance of the 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 accur-a t e l y gives the power d i s s i p a t e d i n the t r a n s m i t t e r c o i l as v I f the current passing through L( i s i , , then the power P absorbed by the resonant pickup c o i l i s t P' =^K,/ Thus the f r a c t i o n of power absorbed by the pickup c o i l i s P ~ fltf+MOCK'+fc,) by using Z^= Q,UJL, . This has a maximum value of 0 .17 when R,' = ^2 R . For the present apparatus L|=1.2/»h and Q i ^ l O so at 9Mc/s. R , - ^ r 7 A . L ^ 3 / i h and Q ^ 5 when the r f pulse i s on, so that R a ^r35n. R' = ^ where M=k-fL7L;, w i t h 0 < k ^ l , so i t 6tf i s imposs ib 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 absorp-t i o n . Represen ta t ive va lues of the f r a c t i o n of power absorbed are % f o r ksO. 1* and 2% f o r k=0.1. ( i i ) The t r a n s m i t t e r c o i l damps the p ickup c o i l . Q = tAJ Ls^  where R^ a . - ^ f L . . When the r f pulse i s o f f R ^ 6 n . and R , r £ h l 5 a . Thus Qa=^30 f o r k=0, Q a= 27 f o r k=0.1 and Qa = 20 f o r k'=0,^. ( i l l ) I f the c o l l s are very t i g h t l y coupled the two c i r c u i t s cannot be tuned independent ly of each o the r . This becomes n o t i c e a b l e f o r k ^ 0 . 5 . ( i v ) The r f pulse can induce a very l a r g e v o l t a g e which can damage the i npu t stage of the p r e a m p l i f i e r , or cause recovery time problems a f t e r the pu lse i s over . The induced vo l t age across the p ickup c o i l i s g iven approximate ly by Represen ta t ive va lues of are 0.8 f o r k=0.1 and 6 .25 f o r 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 tha 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. In the c o i l system f i n a l l y adopted, k can be as low as 0.03. The most unusual pa r t of the apparatus i s the sample c o i l w i t h the metal c y l i n d r i c a l sample i n s i d e i t . The f o l -65 lowing simple theory describes the behaviour of the sample c o i l w i t h s u f f i c i e n t accuracy f o r most experimental purposes. Assume that 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 metal 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 s k i n depth i s 8 at a frequency f . I t i s assumed that the metal c y l i n d e r i s longer than the c o i l . I f the metal core Is absent, 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 solenoid i s (3) H = n l where J i s the c u r r e n t i n the c o i l . This i s assumed to be uniform over the whole cross s e c t i o n a l area of the c o i l , so that the induced back emf i s v = -• bt = -mvW-where A =TTRa. I f I = I0exp(A.u/t) then Z = ~ j - =UJJ^A n* A l =iu>L7 , where L>= n^wAl*. This i s the i n f i n i t e solenoid formula f o r an a i r cored c o i l and i s accurate to w i t h i n ten percent provided 1 > 2R X. I f the metal core i s now i n s e r t e d , the s k i n e f f e c t pre-vents 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 metal so the e f f e c t i v e cross s e c t i o n a l area of the c o i l i s much l e s s . Thus i f i t i s assumed that H i s s t i l l uniform outside the metal core, becomes approximately 66 An^oiL aU(TT[R»-(R, - S f] . U s u a l l y R „ - R | ^ > £ > , so that the induc-tance w i t h a metal core becomes L ^ n ^ n ' l i (Rt -R*) The c o n d i t i o n s f o r the v a l i d i t y of t h i s equation are 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 c o n d i t i o n f o r an a i r cored c o i l . Experimental measurements show that t h i s formula i s accurate to w i t h i n 20$. The e r r o r i s probably due to 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 of the c o i l , consider what happens when a curren t j = j 0 e x p ( i m t ) i s passed through the c o i l . I t generates a magnetic f i e l d H=n»j p a r a l l e l to the surface of the metal c y l i n d e r . The magnetic f i e l d generates an eddy current which c i r c u l a t e s c l o s e to the surface i n the opposite d i r e c -t i o n to the appl i e d c u r r e n t . As f a r as power l o s s e s are con-cerned, the eddy cu r r e n t can be considered as a uniform c u r r e n t d e n s i t y equal to the a c t u a l c u r r e n t d e n s i t y at the surface of the metal and confined to a l a y e r 2~^b t h i c k at the surface of the metal ( 3 ) . Thus the eddy curren t c i r c u l a t e s i n a loop 2TTR, l o n g , 1 2 wide, and 2~*& t h i c k , so that the t o t a l r e s i s t a n c e of the eddy curren t path i s a/V5 rr Ri . The r.m.s. value of the eddy curren t i s Ir=a-*1aH Thus the power P disa p a t e d i n the sample i s 67 I f the r.m.s. value of j i s j r , then H ^ n * J r . . . r - grg Jr where R* = h * Rs 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 to the metal core so t h a t ' ( - f c - > ' - i - * The a c t u a l Q of the resonant c i r c u i t i s reduced by the r f r e s i s -tance Rvw of the wire i n the c o l l and the e x t e r n a l p a r t of the c i r c u i t to 0 = ^ L * Rs+Rw where Qs = t v L / R s and Q;w= w L / R „ . Thus i t i s easy to inc l u d e the wire r e s i s t a n c e i n the design 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 to a number of f a c t o r s which depend upon the r a t i o R , /Rx of the sample to c o i l r a d i u s . These f a c t o r s w i l l now be given and the optimum r a t i o of R i / R % c a l c u l a t e d from them. 68 The maximum f l u x change generated by the sample i s 9= -jufej^ jJ&'Qg' I f the c o i l i s not t i g h t l y wound on the sample magnetic f l u x leakage occurs, so that the f l u x 0 i n t e r -cepted by the pickup c o i l i s l e s s than ©. This f l u x leakage i s taken care 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 This depends only on the c o i l and sample geometry and can, 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 . In prac-t i c e such c a l c u l a t i o n s are unwieldy. For a short c o i l wound on a sample much longer than i t s diameter, a c a l c u l a t i o n gives ^ as *} = l - k [ ( \ f - l ] where k= ^^Vl?) and L i s the sample l e n g t h . Most of the c o i l c o n f i g u r a t i o n s approximated these c o n d i t i o n s . The induced voltage i s a l s o p r o p o r t i o n a l t o the number of turns n 2 i n the pickup c o i l . At a f i x e d frequency, the inductance L a = 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 perturbed magnetic moments. I f v i s the output v o l t a g e from the resonant pickup c o i l then VoGQ Tj nA 69 where x=Ra/R, and K= ^ ^ g - f e f i i . This has a maximum value when R ^ R , ( 1 + i b * . With t y p i c a l values of Q and K, R, /R x l i e s i n the r e g i o n of 0.7 to 0.8. In the c o i l s a c t u a l l y b u i l t R (/R % v a r i e d from about 0.6 to O.9o I t should be noted that s l i c i n g the sample i n t o t h i n slabs may not increase the Induced v o l t a g e . This i s because i n c r e a s i n g the surface area not only increases the number of n u c l e i e x c i t e d , but a l s o increases the path lengths of the c i r -c u l a t i n g c u r r e n t s and t h i s causes a compensating drop i n Q. Thus i f the Q of the resonant c i r c u i t i s determined by the 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 . 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 increases the s i g n a l 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 wire l o s s e s . The f u l l b e n e f i t s of s l i c i n g are not obtained unless the slabs are about 8 t h i c k , when the eddy current l o s s e s become considerably reduced. Manufacture and alignment of metal slabs 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 that no experiments along these l i n e s have been attempted. The most d i f f i c u l t part of b u i l d i n g the apparatus was f i n d i n g a s u i t a b l e c o i l c o n f i g u r a t i o n . C l a r k (2) l i s t s the r e l a t i v e merits of three d i f f e r e n t c o n f i g u r a t i o n s . Two of these were examined both t h e o r e t i c a l l y and e x p e r i m e n t a l l y , and one only t h e o r e t i c a l l y , before they were a l l r e j e c t e d . The e a s i e s t system to 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 simple c a l c u l a t i o n s showed there 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 con-f l i c t i n g requirements f o r the t r a n s m i t t e r and r e c e i v e r c o i l s . 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 and r e c e i v e r could not be used. A r f b r idge (8) was t r i e d and q u i c k l y r e j e c t e d . I t r e -duced the r f f i e l d 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 metal cored c o i l , s t i l l used a combined t r a n s m i t t e r -r e c e i v e r c o i l , and d i d not a l l ow the use of Q damping c i r c u i t s . Grossed c o i l s were s tudied more c a r e f u l l y , both e x p e r i -men ta l ly and t h e o r e t i c a l l y . They have the advantages tha t c o n f l i c t i n g t r a n s m i t t e r and r e c e i v e r c o i l requirements can s imul taneous ly 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 very 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 used. A c o u s t i c 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 ing crossed c o i l s . They have the major disadvantage tha t measurements of o r i e n -t a t i o n dependent NMR p r o p e r t i e s are very d i f f i c u l t . Th i s i s s ince only the component of H, pe rpend icu l a r to H 0 t i p s the nuc l ea 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 0 can make any angle between 0° and 90° w i t h H, as i t i s ro ta ted w i t h r e spec t to the sample. To e l i m i n a t e t h i s problem, the whole sample and c o i l assembly could be ro t a t ed w i t h respec t to Ho 5 a d i f f i c u l t mechanical and e l e c t r i c a l problem, es-p e c i a l l y a t l i q u i d he l ium temperatures . An e a s i e r s o l u t i o n , which was t r i e d , was to add another t r a n s m i t t e r c o i l which gave a r f f i e l d o r thogonal to both H 0 and the r f f i e l d from the f i r s t t r a n s m i t t e r c o i l . S w i t c h i n g the t r a n s m i t t e r output to the 71 approp r i a t e c o i l can always give a f a i r l y l a r g e r f f i e l d pe r -pendular to H 0 . The c y l i n d r i c a l sample shape and the. l i m i t e d space a v a i l a b l e meant the long 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 used. This system e a s i l y gave k<(0.5, even wi thout an e l e c t r o s t a t i c s h i e l d , but could not g ive H,") 10 gauss w i t h the power a v a i l a b l e from the t r a n s m i t t e r . Hi was a l s o non-uniform over the surface of the me ta l , dec reas ing to zero on some pa r t s of the su r f ace , so tha t not a l l the surface n u c l e i were pe r -turbed . For these reasons , the crossed c o i l system was aban-doned. Th 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 c o i l 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 ad ius R 3 , and l e n g t h l j as the only usab le c o n f i g u r a t i o n . The c o u p l i n g cons tant 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 d e r i v e d . L j =TTyU |^n£(Rt-R< ) so i f Vj i s the r f vo l t age a p p l i e d across the t r a n s m i t t e r c o i l then H = n^ A.j T T (K-K) • n,l> But f o r a r f t ransformer , = k -j* [RJ *;k = (Rl t x - R M 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 gives l c v 0 . 3 . Even though t h i s value was known to be too l a r g e , the c o n f i g u r a t i o n was t r i e d out and showed most of the defects p r e d i c t e d f o r i t . In order to reduce k, a bucking c o i l i s added i n s e r i e s w i t h the pickup c o i l i n such a way that the voltage induced i n i t opposes the voltage induced i n the pickup c o i l . The bucking c o i l has to be l o c ated where i t cannot p i c k up the l a r g e s i g n a l from the sample. The mutual c o u p l i n g between the bucking c o i l and the pickup c o i l should a l s o be as small as p o s s i b l e . Various p o s i t i o n s and types of bucking c o i l windings were t r i e d and 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 chosen. The d i r -e c t i o n s of the windings, and the p o s i t i o n s of a l l the wires were c a r e f u l l y arranged so that any c a p a c i t i v e c o u p l i n g opposed the i n d u c t i v e c o u p l i n g . I t was Impossible to get a p e r f e c t c a n c e l -l a t i o n , even a f t e r extensive experimentation, because of the c a p a c i t i v e c o u p l i n g and the i n d u c t i v e c o u p l i n g between the bucking and 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 capacit.ive~coupling. 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 metal sample. The eddy currents exclude f l u x from the i n t e r i o r of the sample and i n -crease the magnitude of H, a t the sample surface. This means that the bucking c o i l r e q u i r e s more turns than expected on the b a s i s of assuming a uniform f l u x i n t e n s i t y over 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 increase i n H, v a r i e d from sample to sample, but was t y p i c a l l y about 30$ and was never more than 50$. Since the inductance of the pickup c o i l plus that of the bucking c o i l should equal the maximum p o s s i b l e inductance f o r a pickup c o i l , the use of a bucking c o i l reduces the number of turns i n the pickup c o i l and hence reduces the s i z e of the induced s i g n a l . T h i s " r e d u c t i o n and the number of turns required i n the pickup c o i l , could i n p r i n c i p l e be c a l c u l a t e d from the above r e s t r i c t i o n on the inductances, plus the requirement that the t o t a l areas of the bucking and pickup c o i l s cut by the magnetic f l u x are equal to each other. However, the bucking c o i l has a l e n g t h much l e s s than i t s r a d i u s so that no simple formula f o r i t s inductance e x i s t s . The magnetic f l u x d i s t r i b u t i o n s i s a l s o uneven. I t i s thus i m p r a c t i c a l to do any c a l c u l a t i o n s i n v o l v i n g the bucking c o i l . The number of turns i n the bucking c o i l thus had to be found experimentally.. The c o i l c o n f i g u r a t i o n i s reasonably s a t i s f a c t o r y i n p r a c t i c e . I t t y p i c a l l y has k^0.05 s w h i l e reducing the s i g n a l by about 30$ of i t s maximum p o s s i b l e value. I f considerable t r o u b l e i s taken i n e m p i r i c a l l y f i n d i n g the optimum number of turns f o r the c o i l s , k can be reduced to about 0.02. This was done f o r the f i r s t two samples but then, f o r reasons of conven-ience and mechanical s t a b i l i t y , one bucking c o i l was used f o r a l l the subsequent samples. This caused l i t t l e d e t e r i o r a t i o n i n the performance of the system. •7h In the c o n f i g u r a t i o n f i n a l l y adopted there are two co-a x i a l c y l i n d r i c a l c o i l formers. These are made of Teflo n which has a high e l e c t r i c a l r e s i s t a n c e and the a b i l i t y to withstand repeated c y c l i n g to low temperatures without c r a c k i n g . The outer one has a diameter of 2 cm. and has the t r a n s m i t t e r and bucking 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 turns and an inductance of 1.2^uh and w i t h 1.8 KV peak to peak across i t , gives an H, of about 20 gauss. The bucking c o i l has h turns and an inductance of 0.9yUh. Both c o i l s were wound i n grooves cut i n the former and then imbedded i n epoxy r e s i n to give as much mechanical r i g i d i t y as p o s s i b l e . The diameter of the inner c y l i n d e r depends on the metal sample being used. The sample f i t s snugly i n s i d e the c y l i n d e r and the pickup c o i l i s wound on the o utside. None of the samples have the same dimensions, so that a d i f f e r e n t pickup c o i l has to 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 Polystyrene Q Dope No. 37-2 to prevent mechanical v i b r a t i o n . The c o i l system i s very e a s i l y assembled and mounted on the end of a s t a i n l e s s s t e e l tube, which a l s o acts as the outer conductor of 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 Ac o u s t i c O s c i l l a t i o n s Often i n pulsed NMR apparatus a troublesome damped os-c i l l a t i o n appears a f t e r the r f pulse. This o s c i l l a t i o n i s caused by the mechanical f o r c e generated by the i n t e r a c t i o n between H 0 and the la r g e c i r c u l a t i n g current i n the t r a n s -m i t t e r c o i l causing some part of the t r a n s m i t t e r c o i l , or sample, 75 to v i b r a t e at an u l t r a s o n i c frequency. The a c o u s t i c o s c i l -l a t i o n s p e r s i s t a f t e r the r f pulse has f i n i s h e d and somehow induce a voltage i n the pickup c o i l which o b l i t e r a t e s the induced nuclear s i g n a l (2). 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 lower part of the c o a x i a l cable and the^wire-leading 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 . Both the t r a n s m i t t e r lead and the inner wire of the c o a x i a l cable were o r i g i n a l l y very t h i n to reduce heat leakage during helium runs. However, they both v i b r a t e d badly and had to be replaced by heavy 2k- A.W.G. wi r e s . The bottom end of the c o a x i a l cable was a l s o f i l l e d w i t h poly-styrene glue. These measures eliminated a c o u s t i c o s c i l l a t i o n s from these w i r e s , but introduced such a l a r g e heat leakage that helium runs were rendered impossible. In a l l cases a c o u s t i c o s c i l l a t i o n s only occurred when the sample was present, so that they must come from the sample i t s e l f . Two experimental observations were made on rhenium which 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 that the amplitude 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) . This i s not very r e s t r i c t i v e since most p o s s i b l e mechanisms have t h i s Ho dependence. The second i s that the o s c i l l a t o r y frequency i s about 70Kc/s f o r a sample 3.15cm. long. Thus i f the sample i s assumed to be 76 long, the l o n g i t u d i n a l v e l o c i t y of sound i n rhenium i s about khOO metres/sec. 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 metal, but i t i s too high f o r a transverse v e l o c i t y (10). I t was a l s o n o t i c e d that the longer samples had a lower a c o u s t i c a l o s c i l l a t i o n frequency. The frequency a l s o increased on going to l i q u i d notrbgen temperature. Thus the o s c i l l a t i o n s are probably due to an a c o u s t i c a l l o n g i t u d i n a l standing wave being set up i n the sample. This suggests that the f o l l o w i n g sequence of events occurs. The r f pulse generates a c i r c u l a t i n g eddy current of about 50 amps, i n the sample which i n t e r a c t s w i t h the l a r g e s t a t i c magnetic f i e l d . This 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 to the c y l i n d r i c a l a x i s of the sample which set up an a c o u s t i c a l standing wave. There i s such a l a r g e a c o u s t i c mismatch at each end that 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 of the sound wave, so that the standing wave p e r s i s t s long a f t e r the r f pulse i s turned o f f . This standing wave causes a v a r i a t i o n Ap i n the den s i t y £ of the sample. I f N e i s the number of f r e e e l e c t r o n s / u n i t volume, then the standing wave produces a v a r i a t i o n £N e =Ne^-i n t h e i r number. The e l e c t r o n i c magnetic s u s c e p t i b i l i t y X<pCNeT*% , where Te i s the e l e c t r o n gyromagnetic r a t i o , so that AXe/DCe = A N e/N e = Ap/p . The pickup c o i l i s wound around the centre of the sample and has a voltage of v^dC^eAX e be= w e X e f induced i n i t . £ e i s the s k i n depth a t the frequency cje of the a c o u s t i c o s c i l l a t i o n s . 77 The nuclear s u s c e p t i b i l i t y XqOCNATQA , so that — U J No since Na=£rNe f o r a metal. The induced voltage from the nuclear spins i s VaaO0JaXob^ so that the r a t i o of the two voltages i s In a t y p i c a l case a;„ ^  100cj e so that ^ ^ x l O 5 ^ . A 6 With t h i s mechanism i t only r e q u i r e s T ? - . ^ 5X10" , a qu i t e 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 nuclear s i g n a l . With the c o a x i a l c o i l system i t i s impossible to stop the standing wave being generated i n the sample, so that the only method of e l i m i n a t i n g the o s c i l l a t i o n s i s to very q u i c k l y damp them. F r i c t i o n a l damping w i t h i n the metal i s very small so that the main a c o u s t i c a l energy l o s s i s by transmission through the ends of the sample, w i t h f r i c t i o n a l l o s s e s at the sides of the c y l i n d e r p l a y i n g some pa r t . Thus the only way to damp the 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 the a c o u s t i c l o s s e s through the ends and s i d e s . I f the 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 c o e f f i c i e n t 78 R a t the i n t e r f a c e i s (11) R _ Q,c, - ?xCa The media are not a c t u a l l y i n f i n i t e , but ins tead the s i t u a t i o n i s c l o s e r to that of 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. For t h i s case the a c o u s t i c impedance Z a i s of the form (11) Z ^ | p » cATT9(Ri /if f o r hrT R, 41, ^ •=0=c^ f o r rrR, > 1. These equat ions 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 us ing a sample w i t h a l a r g e r a t i o of 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 of sound much c l o s e r to those of a metal than a i r has. These ideas were expe r imen ta l l y tes ted by immersing a rhenium sample i n g l y c e r i n e so tha t the 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 to 0.9. The d u r a t i o n of the a c o u s t i c os-c i l l a t i o n s decreased by 3>0% and t h e i r i n i t i a l ampli tude decreased by 20%. The damping a l s o increased w i t h an inc rease 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 indium (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 both i n c r e a s i n g R, / l and dec reas ing r e f l e c -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 simple theory p r e d i c t s and suggests tha t surface f r i c t i o n a l l o s s e s p l ay a s i g n i f i c a n t pa 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 very low thermal conduc-79 t i v i t y and so the r f pulses can heat the sample to temperatures w e l l above that of the dewar system. For t h i s reason g l y c e r i n e was 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 . Instead the sample was embedded i n a p o r c e l a i n cement^ which was a good a c o u s t i c a l match. The cement i s water s o l u b l e , so that d i s -mantling the sample mounting i s easy. P a r t of the sample was always l e f t exposed so as to provide a good thermal contact. The 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 used w i t h . 3.12 C a l c u l a t i o n of the S i g n a l to Noise R a t i o Since no measurements were made below 78 K. i t i s assumed th a t the normal s k i n e f f e c t theory i s a p p l i c a b l e . The voltage induced i n the pickup c o i l by the precessing n u c l e i i s (Appendix I I I ) v = t r ^ nwM„R|( e x p ( - ^ ) s i n ( r t B , e v i f e)cos 3'(^|^)dz. Jo This i n t e g r a l has been evaluated and has a maximum value of about 0.7 S when TB.Tl—frr r a d i a n s . The maximum induced voltage i s thus v = O^n^no^MoR, 8 . The pickup c o i l i s resonated at the frequency o>, so that the voltage a t the input to the p r e a m p l i f i e r i s v =0.7n^yMn wM0QR, & . I f T n i s the noise temperature of the sharply tuned pickup c o i l , then the r.m.s. noise voltage \rn i s (12) Sauereisen Adhesive Cement No.l Paste, Saureisen Cement Co., P i t t s b u r g h 15, Perm., U.S.A. 80 v„ = ^ E , where C i s the capacitance resonating the pickup 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 are chosen so that thermal noise from the resonant c i r c u i t i s the dominant noise source. This 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 are greater than that of the tuned pickup c o i l , so that the S/N r a t i o at the a m p l i f i e r out-put i s This expression i n c l u d e s the improvement of -[2 i n the S/N r a t i o introduced by phase s e n s i t i v e d e t e c t i o n (12). The boxcar i n t e g r a t o r enhances the S/N by V t^^ " f o r r e p e t i t i o n r a t e s greater than about (50 ms.)"' , so that the f i n a l S/N r a t i o S i s The parameters i n t h i s equation which are l i s t e d below have a temperature dependence. ( i ) The nuclear magnetic moment/unit volume M0ccT~' ( 1 ) , where T i s the sample temperature. ( i i ) 8oCo-^ and o~ has a complicated temperature dependence. For s i m p l i c i t y the high temperature approximation a-oCT-' w i l l be used ( 9 ) ? even though the Debye temperature f o r most metals f a l l s w i t h i n the temperature range of i n t e r e s t . Thus bcCT"^. ( i i i ) Q="4r where R n , 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 temperature. I f the damping i s only due to 81 j. -J-eddy currents i n the metal then 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 cable forms a l a r g e p a r t of the damping r e s i s t a n c e . This v a r i e s i n temperature between room temperature a t one end and the temperature of the sample at the other end, so that the noise temperature Tn of the tuned c i r -c u i t i s u s u a l l y d i f f e r e n t from T, l y i n g between T and room temperature. The dependence of T h on Rn i s not known, but TnocRh _JL seems a reasonable assumption. Thus QoCT*1. ( i v ) fcoC(Aw)"1 and Q=i£so TtocQ cCTn^. (v) In a metal sample the Korringa r e l a t i o n T,T=constant holds ( 1 ) . The r e p e t i t i o n time f o r the boxcar i s T r= KT, where K i s a constant. . Tf-ccT . The e f f e c t i v e time constant "t = RbCb of the boxcar i s a tem-perature independent constant determined only by the sweep time. .". R^ Cb oC Tr oC T. Combining a l l these temperature dependences gives SoCT„"* . The important f e a t u r e of t h i s simple a n a l y s i s i s that there i s l i t t l e improvement i n S/N on going to low tempera-t u r e s ; the increase i n M0 being compensated f o r by a decrease i n 8 and the decreasing e f f e c t i v e n e s s of the boxcar i n t e g r a t o r . E xperimentally i t was found that S improved by about 50% on going from room to l i q u i d n i t r o g e n temperature. This gives T„oCT^ as the approximate temperature dependence i n t h i s 82 -3-r e g i o n , so SoCT D. I f t h i s temperature dependence i s e x t r a -polated to lower temperatures and allowance made f o r the onset of anomalous conduction, then c o o l i n g from n i t r o g e n temperature to lf.2°K would increase S t e n f o l d . A c t u a l l y the temperature dependence of T h on T i s even l e s s below n i t r o g e n temperature because even at t h i s temperature most of the noise i s coming from the parts of the tuned c i r c u i t near room temperature. Thus c o o l i n g the sample to a lower temperature does not reduce the noise temperature very much. These f a c t s , plus the ex-perimental 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 , are the reasons why no measurements a t l i q u i d helium temperatures have been attempted. The other parameter which can a f f e c t S Is the resonant frequency. I f i t i s assumed that the tuning capacitance C Is the same f o r a l l f r e q u e n c i e s , the frequency dependent para-meters vary as f o l l o w s . (I) The inductance L o c n a , but wwcL^, so noctu"1. ( i i ) 6 ©ecu"* . ( i i i ) M0=XHo and rH= U J , SO MaoCuj. ( i v ) Q i s u s u a l l y frequency dependent. However, bandwidth r e s -t r i c t i o n s r e q u i r e Q to be kept reasonably constant so that over a moderate frequency range Q i s frequency independent. This i s a good approximation since even i f Q does vary w i t h frequency i t i s almost cancelled by the opposing v a r i a t i o n of t c . Thus SoC uA 83 This i s i n reasonably good agreement w i t h the experiments. Changing frequencies from 6 to 9Mc/s. increased S by about 30$. In a d d i t i o n recovery from the r f pulses i s b e t t e r at the higher frequency since a higher low frequency c u t o f f can be used. Measurements were thus u s u a l l y made a t the highest convenient frequency. These conclusions are not v a l i d above about 20Mc/s since i n t h i s r % gion the gr i d noise of,the p r e a m p l i f i e r input stage increases and a l s o i t s decreasing input 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 to use a 180° pulse f o l l o w e d by a 90° pulse a t a v a r i a b l e time t l a t e r . The amplitude of the i n d u c t i o n t a i l f o l l o w i n g the second pulse v a r i e s as l-exp(--^=-), so that T, can e a s i l y be obtained. When a boxcar i n t e g r a t o r i s used the spacing between the pulses i s l i n e a r l y increased w i t h time so that the expo n e n t i a l increase of amplitude i s recorded d i r e c t l y on the 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 metal 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 that there are no 90° or 180° pulses of the conventional type because of s k i n e f f e c t s (Appendix I I I ) . There are pulse lengths which give maximum amplitudes and even pulse lengths which give no amplitude which could be used f o r the equivalent of a 180° -0 90 pulse t r a i n . However, the s i g n a l i s w e l l below the noise l e v e l . Therefore tuning the apparatus so that 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 pulse length;, i s an impossible job when a boxcar i n t e g r a t o r must be used. The other b i g d i f f i c u l t y i s that the s i g n a l i s t y p i c a l l y about one tenth of the noise l e v e l , so that 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 of the noise l e v e l f o r even a moderately accurate measurement of T, . This i s a f a r more s t r i n g e n t requirement than i s u s u a l l y required i n pulsed NMR apparatus and i s very d i f f i c u l t to a t t a i n . For these reasons, an a l t e r n a t i v e method of measurement was devised which e l i m i n a t e s these d i f f i c u l t i e s at the expense of being very l a b o r i o u s . Let a sp i n system have a la r g e s t a t i c magnetic f i e l d H 0 a p p l i e d along 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 to i t . In the r o t a t i n g reference frame there i s an e f f e c t i v e magnetic f i e l d H, = H$ + (Ho+¥)k making an angle © = tan'T ,, H' 1 w i t h the z a x i s . I f L He + .the nuclear magnetism M„ i s i n i t i a l l y a l i g n e d along H 0, then on a p p l i c a t i o n of the r f pulse the components of M0 perpendi-c u l a r to H e r e l a x towards i t w i t h a time constant somewhat longer than Ta(13)9 so that 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 gned along H e w i t h magnitude M 0 cos6. When the r f pulse i s switched o f f the components of the magnetism perpendicular to H 0 decay i n a time of the order of Ta . The magnetism along the z a x i s thus has an amplitude \. M ^ M ocosO|eos0 |. 85 I t has so f a r been assumed that no s p i n - l a t t i c e r e l a x -a t i o n occurso This r e q u i r e s the r f pulse l e n g t h to be much l e s s than T, while s t i l l being many times T i o I f a short second r f pulse i s applied at a time t l a t e r the height of i t s i n d u c t i o n t a i l i s p r o p o r t i o n a l to Mz= M0-0--exp(- -=j=~ )} + M0 cos0 |cos0 | e x p ( - ) . From t h i s T( i s e a s i l y obtained. This i s the b a s i s of the method used i n the metal s i n g l e c r y s t a l s . A r f pulse of 200/us, or longer, i s a p p l i e d to b r i n g the s p i n system to e q u i l i b r i u m i n the r o t a t i n g reference frame i n the manner j u s t described. Tx i s l e s s than 50^3 i n n e a r l y a l l the metals, while 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 at room temperature, so the i n e q u a l i t i e s concerning the pulse l e n g t h are e a s i l y s a t i s f i e d . The ra p i d decrease, of H, w i t h depth means that some n u c l e i w i l l not r e l a x i n the r o t a t i n g reference frame since Ht w i l l be much smaller than the l o c a l f i e l d s . However t h i s , along 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 the s t a t e the spi n system i s l e f t i n when the pulse i s switched o f f . Mz. i s measured w i t h a r f pulse about 15/WS long a p p l i e d at 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 described p r e v i o u s l y . The height 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 3 wide ( 6 ) . The magnetic f i e l d i s l i n e a r l y swept through the resonant value so 86 that M can be got from the recorder t r a c e . The reference phase i u s u a l l y adjusted so that the recorder trace bears some resemblanc to an absorption curve. T h i s - i s not necessary, but makes measure ments from the recorder chart 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 decreasing. The measurements f o r each value of t are then averaged. This approximately averages out any steady d r i f t i n gain of the system and a l s o decreases the s t a t i s t i c a l e r r o r of each p o i n t . T y p i c a l l y measurements are made f o r about 30 d i f f e r e n t values of t . There are s e v e r a l advantages of t h i s method. The f i r s t one i s that the apparatus does not have to be tuned f o r exact resonance, nor does i t have to stay e x a c t l y on resonance f o r the d u r a t i o n of the measurement. A phase s e n s i t i v e system i s very s e n s i t i v e to frequency, or magnetic f i e l d d r i f t s of one tenth of the 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 qu i t e a s t r i n g e n t one to f u l f i l . The second advantage i s that sweeping through the l i n e e l i m i n a t e s any e r r o r s from 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 to both the s i g n a l and the o f f resonance b a s e l i n e i t s amplitude i s measured from. The main disadvantage of the method i s that i t takes about three hours to measure T| , as compared to about h a l f an hour by more conventional methods. This means that the d r i f t i n gain of the apparatus must be s m a l l , or a t l e a s t a constant d r i f t i n the same d i r e c t i o n . This was u s u a l l y the case, but sometimes there would be sudden jumps i n gain causing some 87 e r r o r i n the f i n a l value of T , . The r e s u l t s were e i t h e r analysed by means of a conven-t i o n a l l o g p l o t , or by a l e a s t squares f i t using the U.B.C. Computing Centre I.B.M. 70>+0 computer. The e r r o r s quoted f o r each r e s u l t are standard d e v i a t i o n s estimated from the s c a t t e r , plus the 2% e r r o r i n ti m i n g . Much of the noise i n these experiments i s from non-random sources such as machinery switching on and hence the e r r o r s do not obey a normal d i s t r i -b u t i o n . The e r r o r s should thus only be regarded 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 Spin-Spin R e l a x a t i o n Times In aluminium powders T a was measured by applying a short r f pulse and l i n e a r l y sweeping i n time a narrow boxcar gate about ^us. wide through the i n d u c t i o n t a i l ( 6 ) . The measurements were made w i t h H 0 w e l l o f f resonance, so that the chart recording i s s i m i l a r to a damped sine wave. Doing t h i s avoids the d i f f i c u l t y of keeping t>he apparatus on exact r e s -onance and a l s o makes i t e a s i e r to reduce the e f f e c t s of ba s e l i n e d i s t o r t i o n . This method could not be used on s i n g l e c r y s t a l s because of t h e i r poor S/N r a t i o and a l s o because c l o s e to the r f pulse the b a s e l i n e was badly d i s t o r t e d . Instead, a s i m i l a r method to that used f o r measuring T, was used. A short r f pulse was applied and at 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 the magnetic f i e l d . The gate was then manually s h i f t e d to a d i f f e r e n t value of t and 88 the measurement repeated. This gives the s i g n a l s shown i n F i g . *+.l. I f t^> T4 , the peak to peak amplitude (AB or AC on F i g . l f . l b ) i s p r o p o r t i o n a l to 2M 0 to a high degree of accuracy. However i f t « T a , the peak to peak amplitude i s l e s s than 2M 0 because of the f i n i t e value of H, . This can be corrected f o r by making sweeps through resonance at s e v e r a l s l i g h t l y d i f f e r e n t times and superimposing them to get an accurate measurement of the envelope of the o s c i l l a t i o n s . This can then be used to c o r r e c t the peak to peak amplitude of the sweeps. 3«15 Measurement of Absorption and D i s p e r s i o n Modes A pulsed NMR apparatus 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 give recorder t r a c e s equivalent to the unsaturated absorption 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 apparatus. The b a s i s of the method, as developed by C l a r k ( 2 ) , w i l l be given here, while the mathematical d e s c r i p t i o n and instrumental d i s t o r t i o n s that occur are given i n Appendix IV. A short r f pulse i s applied to the sample. A very wide boxcar gate which completely covers 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 to be a l i n e a r combination of X' and X". By appropriate choice of the reference phase e i t h e r X' or X" can be obtained. I f the magnetic f i e l d i s now swept l i n e a r l y through the resonance value recordings are obtained which are equivalent to those obtained by steady s t a t e apparatus. This type of measurement i s e a s i l y done on the present apparatus, but 89 s u f f e r s from the disadvantage that Tft i n metals i s so short that there i s considerable instrumental d i s t o r t i o n (Appendix I V ) . I t does have the advantage that X'and X " can be unambiguously separated. This has not been p o s s i b l e i n any of the 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 mar-g i n a l o s c i l l a t o r s . To obta i n the 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 necessary to simultaneously a c c u r a t e l y measure the magnetic f i e l d . A simul-taneous measurement of the magnetic f i e l d using a simple mar-g i n a l o s c i l l a t o r was t r i e d , but f a i l e d because there was mutual pickup between the marginal o s c i l l a t o r and the pulsed NMR apparatus. P o s s i b l y completely s h i e l d i n g both c o i l systems would e l i m i n a t e t h i s problem. 3.16 P o s s i b l e Improvements to the Apparatus As i t stands at present the apparatus i s not as good as i t should be f o r measuring T, at 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 reasons. ( i ) The o r i g i n a l idea was to measure the anisotropy i n T, at very low magnetic f i e l d s near the superconducting t h r e s h o l d . Thus the apparatus was o r i g i n a l l y designed to work at 750 Kc/s. However, a f t e r q u i t e a few months t h i s idea was abandoned, at l e a s t t e m p o r a r i l y , as the experimental d i f f i c u l t i e s were too great. The apparatus was then converted to work i n the region of 5 to lOMc/s. There are 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 parts of the apparatus, notably the over l y elaborate gated power a m p l i f i e r c i r c u i t . 90 This does not d e t r a c t very much from the apparatus's performance but does decrease i t s r e l i a b i l i t y . ( i i ) The dewars and sample holders were b u i l t f o r the 6 M magnet so they are smaller than i s necessary f o r the 12 w magnet and a l s o have no electromagnetic s h i e l d i n g . ( i i i ) The samples a v a i l a b l e are of assorted s i z e s and a l s o many of the f a c t o r s involved had to be found out by experiment. Thus the c o i l s are u s u a l l y not the optimum design. From these c o n s i d e r a t i o n s , and a l s o some other p o i n t s , i t i s c l e a r that there are two major ways i n which the appar-atus can be improved. The most important improvement i s to 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 simpler and can give more power i n t o a lower impedance l o a d . This would enable a lower Q t r a n s -m i t t e r c o i l to 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,. At present a 90° pulse i s about l ^ n s long. A more powerful t r a n s m i t t e r could reduce t h i s to about 5/Ws and a lower Q; c o i l could reduce the recovery time by about 5jusa With the very short values of T^  o c c u r r i n g i n metals these improvements could e a s i l y increase the S/N r a t i o by 50$. The second improvement would be to b u i l d a metal dewar system and a sample holder s p e c i f i c a l l y f o r use w i t h the 12" magnet at l i q u i d n i t r o g e n , or room temperatures. This would enable l a r g e r diameter samples to be used. More m e c h a n i c a l - r i g i d i t y could be b u i l t i n t o the e l e c t r i c a l leads to the sample holder and a l s o to the sample holder 91 i t s e l f . 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 noise caused by bubbling of the l i q u i d n i t r o g e n . There would a l s o be room to completely sur-round the c o i l system by a metal s h i e l d to reduce r f pickup from e x t e r n a l sources without causing s i g n i f i c a n t d e t e r i o r a t i o n i n i t s e l e c t r i c a l performance. I t would a l s o be d e s i r a b l e to standardize the sample s i z e s , but u n f o r t u n a t e l y there i s o f t e n no choice i n the s i z e that samples are grown i n . There are no s i g n i f i c a n t improvements which 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 a l -c u l a t i o n s both show that i t i s already performing at the minimum ^possible noise l e v e l . Reference Signal. . ) — Oscillator and G-afed Power Amplifier. Grat'ini "Pulse. Timing Unit. ?hase Staffer and Attenuator. iskv. r.f Pulse > Induced r-f. Signal Tuned Prea rr\ pi \f\&n Arenberg W A 6 0 0 D Amplifier-Quenching Pulse. Bo/car 6ratinq Pulse-: ^ : External Trigger Pulse. Oscilloscope. D.C. Output Event Marker Pulse. ) — Figure 3.1. B l o c k Diagram of the Apparatus . Varian Chart Recorder. Free Running Multivibrator. Tektronix. 162 5aw+ooih Grenerator. Slow Sawtooth Generator-Tektronix l6x Sawtooth Gtenerqtor Tektronix 163 Pulse Grene rotor. Pulse i . Pulse a. Tektronix 1 6 3 P w l s e Grenerator. Eytemat Normal Tektronix l6a SqWtooth Grenerator. Pulse Mixer and Amplt-fi'er. Tektronix 161 Pulse Grenerator. External 7^ Normal Comcjdehce Tirwmcj Unit Figure 3 . 2 . Block Diagram of The "Timing Unit. To External Trigger of the Oscilloscope. To Grated ?o«/er ° Amplifier. -) 0T0 Preamplifier Quench. (To Boxcar Gratfe* To Maqnet S — 0 Field Sweep. To Recorder Event Marker. 10 3.0 !>0 Ho* (kiloejawss*) SO —1— no Figure as Variation of the Acoustic Oscillation Amplitude With Magnetic Field Strength. 95 M u l t i v i b r a t o r T r i g g e r i n g Pulse Sawtooth from Te k t r o n i x 162 Generator D i s c r i m i n a t o r l e v e l f o r Pulse rt-Pulse 1 (from T e k t r o n i x 163) Timing Saw-toot h f o r r f a m p l i f i e r gat-i n g pulses Pulse 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 Tektronix 162 Generator Boxcar gating pulses from T e k t r o n i x 161 pulse generator Complete Pulse Sequence as Seen on Monitor O s c i l l o s c o p e F i g . 3.6 Diagram of the Most Commonly Used Two Pulse Sequence 96 CHAPTER IV THE EXPERIMENTAL RESULTS 'To observations which ourselves we make, We grow more p a r t i a l f o r the observer's sake.' - Pope. Although the main aim of t h i s work was to search 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 times, a secondary aim was to determine the p o s s i b l e uses and l i m i t a t i o n s of pulsed NMR i n metal s i n g l e c r y s t a l s . This p a r t of the work v e r i f i e d the theory of the apparatus developed i n the preceeding chapter. Spin 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 of metals. Some of these were selected f o r d e f i n i t e reasons, but most were only t r i e d because they were a v a i l a b l e . This random approach to the s e l e c t i o n of samples was mainly because many metals cannot be grown i n conveniently sized c r y s t a l s , except at a p r o h i b i t i v e c o s t , so th a t one had to use whatever samples were r e a d i l y a v a i l a b l e . None of the metals w i t h l a r g e quadrupole i n t e r a c t i o n s had de t e c t a b l e s i g n a l s , but f o u r other metals gave good enough s i g n a l s f o r T, measurements •to be made. These were aluminium, vanadium, niobium and white t i n . An upper l i m i t was placed on the 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 these gave the strengths of the pseudo-dipolar 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 to use a scandium c r y s t a l f o r the main search f o r T, a n i s o t r o p y . This 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 to T, , and a small quadrupole i n t e r a c t i o n . These f e a t u r e s made i t an e x c e l l e n t candidate f o r t h i s search. U n f o r t u n a t e l y , the f i r m which agreed to supply the c r y s t a l were unable to grow bne a f t e r e i g h t attempts so that t h i s idea 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 at a frequency of 7Mc/s. a t both room and l i q u i d n i t r o g e n temperatures. The S/N r a t i o was u s u a l l y about 20 when a boxcar i n t e g r a t o r was used. This allowed f a i r l y accurate measurements of T, to be made. Ta was too short to measure. In the T» measurements the f i r s t pulse was 300^is long and the second was 20^is long. The 60/as wide boxcar gate s t a r t e d -^Oyuis a f t e r the beginning of the second pulse and used a time constant 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 temperature, and (90ms.) f o r the 7° K. measurements. On the trace of the sweep through resonance the amplitudes between the p o i n t s A,B and B,C ( F i g . -^.1) were measured and then averaged. Choosing these p o i n t s , r a t h e r than the s i g n a l amplitude from the b a s e l i n e EF, increases the S/N r a t i o and e l i m i n a t e s the need to sweep from a long way o f f resonance. T, was then got from a l o g p l o t of t h i s amplitude. 98 At 295°K., T, T=(1.8 ±0 .3 )sec.deg. 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 gives T,T= (1 .7*0 .1 )sec.cleg, over the temperature range 78°K. to 295°K. This agrees w e l l w i t h T,T=(l.8Q±0.05)sec.deg. obtained f o r a powder from l . 2°K. to 930°K. (15). The experimental value of T, T i s about 20$ longer than the value predicted from the experimental Knight s h i f t and the Korringa r e l a t i o n , but agrees w e l l w i t h the value c a l c u l a t e d using the Korringa r e l a t i o n modified to take e l e c t r o n cor-r e l a t i o n s i n t o account (1) . The l e n g t h of the f i r s t pulse was va r i e d from l50jus. to s. without any n o t i c e a b l e e f f e c t on the amplitude of the i n d u c t i o n decay a f t e r the second pulse. A T, measurement taken w i t h a second pulse l ^ t j s . long gave the same value as the e a r l i e r measurements w i t h a 20yHs. pulse. On the ba s i s of the theory given i n the l a s t chapter, t h i s l a c k of s e n s i t i v i t y of the r e s u l t s to the pulse lengths was expected. Aluminium has a cubic l a t t i c e , a n e a r l y s p h e r i c a l Fermi surfa c e , and a dominant contact i n t e r a c t i o n . Anisotropy i n T, T i s thus very u n l i k e l y and was not looked 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 5 1 a t both 295°K. and a t 78°K. These gave values of (0 . 7 9 ±0 . 0 3)sec.deg. a t 295°K. and (0 . 78-0.02)sec.deg. a t 78°K. which are i n e x c e l l e n t agree-ment wi t h the value of (0.788*0.007) sec.deg. obtained f o r powders over the temperature range 20°K. to 295°K. (20). 99 Because of the cubic l a t t i c e T, T was not expected to be a n i s o t r o p i c (21). However the measurements at 78°K. were taken 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 o r i e n t a t i o n s . No a n i s o -tropy was detected i n measurements made w i t h e r r o r s of ±3$« Ta. could not be measured, but seemed to be shorter than that of aluminium. I f the experimental value of T, T i s used i n the Korringa r e l a t i o n , i t gives a Knight s h i f t of 0.21$ instead of the experimental value of 0.56$. This discrepancy i s too l a r g e to be explained by many-body e f f e c t s . The reason 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 studied i n d e t a i l . 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 metals i s based on an a r t i c l e by Mott (23). The conduction band i s believed to c o n s i s t of a narrow d band w i t h a high d e n s i t y of s t a t e s overlapping an s band w i t h a low d e n s i t y of s t a t e s . The Fermi energy l i e s i n the r e g i o n where the bands overlap. The s band i s u s u a l l y described i n terms of n e a r l y f r e e e l e c t r o n Bloch 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 so i s described by the t i g h t b inding approximation. However, i t i s impossible i n p r i n c i p l e to separate the d e n s i t y of s t a t e s i n t o independent bands derived wholly from s, p, or d f u n c t i o n s , even i n the t i g h t b inding approximation. The mixing of s t a t e s ( h y b r i d i z a t i o n ) which occurs 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 t r a n s i t i o n metals. The most important 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 to introduce a deep minimum i n the middle of the d e n s i t y of s t a t e s curve f o r b.c.c. l a t t i c e s , but not f o r f . c . c . l a t t i c e s . H y b r i d i e a t i o n of the s and d bands does not 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 curve, but a f f e c t s other p r o p e r t i e s i n a manner which i s not c l e a r l y understood at pre-sent. In most metals the s i t u a t i o n i s complicated 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 higher, wave f u n c t i o n s as w e l l . Due to t h e i r coulomb r e p u l s i o n there are 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 small energy s h i f t s which are u s u a l l y ignored. Because of the d i f f i c u l t y 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 are u s u a l l y neglected and the assump-t i o n made tha t the s and d bands can be treated independently. This 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 long t r a n s i t i o n period and has f i v e e l e c t r o n s outside the f i l l e d core. I t s d e n s i t y of s t a t e s curve has been experimentally determined and shows the 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 electrons/atom; and a much narrower 3<3 band c o n t a i n i n g h,5 electrons/atom (25).' This high d e n s i t y of d e l e c t r o n s i s the reason that the Korringa r e l a t i o n does not hold. Using the r i g i d band model, the Knight 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 times i n vanadium powders have been thor-oughly examined, both experimentally (25) and t h e o r e t i c a l l y (27). Of-necessity there are a number of 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 that 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 contact term plays a minor r o l e . The dominant c o n t r i b u t i o n to the Knight 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 c o n t r i b u t i o n from core p o l a r i -s a t i o n . The contact term provides about 10$ of the s p i n - l a t t i c e r e l a x a t i o n . 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 core p o l a r i s a t i o n . At the moment i t i s impossible to decide which of these terms i s the l a r g e r , but i t i s probably the o r b i t a l term (25,27). This i s supported by measurements on superconducting vanadium which show that the Knight s h i f t i s due to a s p i n independent term (28,25). However some caution 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 since the behavior of the Knight s h i f t i n some n o n - t r a n s i t i o n metals d i f f e r s from that predicted on the b a s i s of the BCS theory. 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 -volves s p i n - o r b i t coupling and s c a t t e r i n g of e l e c t r o n s from the sample surface (61). I t i s not known to what extent these e f f e c t s occur i n superconducting t r a n s i t i o n metals. The experimental value of T( T i s twice the c a l c u l a t e d value (27). Butterworth (29) showed that t h i s d i f f e r e n c e was u n l i k e l y to be caused by e r r o r s i n choosing the band s t r u c t u r e parameters. The most probable reason 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 dens i t y of s t a t e s derived from the e l e c t r o n i c s p e c i f i c heat (27,29). This i n c l u d e s 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 electron-phonon i n t e r a c t i o n s which 102 do not c o n t r i b u t e to r e l a x a t i o n C+6). Thus the c a l c u l a t e d r e l a x a t i o n time would be too short. Even i n comparatively simple metals the electron-phonon i n t e r a c t i o n can double the 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 to e x p l a i n the d i f f e r e n c e between the experimental and the t h e o r e t i c a l v alues, 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 to 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 Is a t r a n s i t i o n metal w i t h a cubic l a t t i c e and e l e c t r o n i c and mechanical p r o p e r t i e s s i m i l a r to those of vanadium. I t a l s o has f i v e e l e c t r o n s outside a f i l l e d core, but l i e s i n the second long t r a n s i t i o n p e r i o d . At room "jtemperature T, T was found to be (0.3^-0.01)sec. deg. and a t 78°K. was (0.31-0.01) sec. deg. These values are the average of two measurements at each temperature. The S/N r a t i o was about 15 at both temperatures. Acoustic o s c i l -l a t i o n s caused some t r o u b l e at l i q u i d n i t r o g e n temperatures. These values disagree w i t h the 0.19 sec.deg. measured by Asayama and It o h i n the r e g i o n 2*K. to 77PK. (58), but agree moderately w e l l w i t h the value of (0.36*0.01)sec.deg. obtained by Butterworth f o r the temperature range 20° K. to 290°K. (29), He found that i m p u r i t i e s did not have a strong e f f e c t . A powder sample contaminated by 1$ oxygen, 0.2$ hydrogen, and 0.08$ n i t r o g e n had a T tT only 10$ below that of a very pure f o i l sample. The sample used by Aszyama contained 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 the gaseous 103 i m p u r i t i e s , which were probably r e s p o n s i b l e f o r the l a r g e i n -crease 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 important, although Butterworth found that annealing a pow-der sample made no d i f f e r e n c e to e i t h e r the s i g n a l i n t e n s i t y , or to TiT. The d i f f e r e n c e between the present measurement and that of Butterworth i s hot due to a systematic e r r o r since the values measured f o r vanadium agreed to w i t h i n experimental e r r o r . I t i s a l s o u n l i k e l y to be only a s t a t i s t i c a l v a r i a t i o n * The niobium sample used i n the present measurement contains about 0.1$ 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 I I ) . The d i f f e r e n c e between the r e s u l t s could thus be due to i m p u r i t i e s . The small temperature dependence of T, T supports t h i s , although t h i s might be due to s t a t i s t i c a l f l u c t u a t i o n s . h,k Metals With Large Quadrupole I n t e r a c t i o n s Measurements were a l s o attempted on a number of metals w i t h l a r g e quadrupole i n t e r a c t i o n s . In these metals the l i n e s are w e l l separated so that cross r e l a x a t i o n should be by non-secular terms only 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. The only experimental value i s about 2ms. i n technetium (60). This i s much longer than T a, but i s considerably shorter than T,. However i t seems safe to assume th a t the system has a f i c t i t i o u s s p i n of £ f o r at l e a s t the i n i t i a l p art of the decay. This means that the s i g n a l s should be q u i t e weak. I f e i t h e r pseudo-dipolar, or pseudo-exchange e f f e c t s occur, Tj w i l l be very short. This decrease i n becomes very impor-ta n t f o r metals w i t h atomic weights of about 100 or more. Most of the metals studied were i n t h i s r e g i o n . The l a r g e quadrupolar i n t e r a c t i o n makes t h i s c l a s s of metals very hard to study using steady s t a t e NMR apparatus and powdered samples so that there have been very few measurements made on t h i s c l a s s of metals. I t was thus considered worthwhile spending some time searching f o r s i g n a l s i n them, even though the short T a and f i c t i t i o u s s p i n of £ would make them very hard to f i n d , ( i ) Indium. At a frequency of 6Mc/s. a search f o r a s i g n a l was made from 3.0 KG. to 6.5 KG. at both 295°K. and 78°K. Searches were made w i t h the magnetic f i e l d both p a r a l l e l and perpendicular to the c r y s t a l a x i s of symmetry. At 9Mc/s. sweeps were made from *U0 KG. to 11.2 KG. a t both 78°K. and 295°K. In t h i s case the magnetic f i e l d was p a r a l l e l to the a x i s of symmetry. The repe-t i t i o n r a t e was such that 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 that i t was known th a t the r i g h t r e g i o n was being searched. Acoustic o s c i l l a t i o n s gave only minor t r o u b l e and vanished when the sample was immersed i n g l y c e r i n e . S i g n a l s were not seen, even though c a l c u l a t i o n s showed that there was a reasonably good chance of seeing them (Chp.^.?). 105 There are three p o s s i b l e explanations 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 that T, T i s longer than about lOsec.deg. This i s not very l i k e l y s ince l i t h i u m i s the only metal known to have T| T longer than 5sec.deg. and a l l nuclear-conduction e l e c t r o n i n t e r a c t i o n s become stronger w i t h i n c r e a s i n g atomic number because of the increase i n e l e c t r o n d e n s i t y near the nucleus. Indium i s very s o f t and even l i g h t pressure can cause the surface to 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 expansion, or some i n a d v e r t a n t l y rough handling, could thus cause the surface to become p o l y c r y s t a l l i n e . This would render the s a t e l l i t e l i n e s unobservable 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$, thus p o s s i b l y making i t unobservable. However X-rays taken before and during 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 recovery 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 unobservable. Indium has an atomic number of 115 and so probably has pseudo-exchange and pseudo-dipolar i n t e r a c t i o n s so that Ta i s pro-bably q u i t e short. This i s the most l i k e l y reason that no s i g n a l was seen, ( i i ) Rhenium. , A search was made from 5.0 KG. to 11.2 KG. a t a f r e -quency of 9Mc/s. a t both room and l i q u i d n i t r o g e n temperatures. No s i g n a l s were seen. This was not unexpected since the com-puted S/N r a t i o was considerably l e s s than one. There were 106 a l s o very 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 g l y c e r i n e . The atomic number i s 185 so that T* should be much l e s s than that due to d i p o l a r i n t e r a c t i o n s alone, ( i l l ) Bismuth. Bismuth i s an unusual metal w i t h some no n - m e t a l l i c pro-p e r t i e s . These a r i s e because i t has a very small 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 extremely l a r g e magneto-resistance (9>33). A room temperature search was made from 8.2KG. to 11.2KG. a t a frequency of 7Mc/s. The r e p e t i t i o n r a t e was (0.2sec.)"' and the boxcar gate s t a r t e d 20yws. a f t e r the begin-ning of the r f pulse. A s i m i l a r search a t 78°K. used a repe-t i t i o n r a t e of (O.SSsec.)"* . At both temperatures s e v e r a l d i f f e r e n t magnetic f i e l d o r i e n t a t i o n s were t r i e d . A c o u s t i c o s c i l l a t i o n s were seen a t 78°K., but were not 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 that 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 reasonable i n t e n s i t y w i t h T a ^ l ^ u s . and T, T ^50 sec.deg. should have been seen. However, there 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 amplitude showed that i t should have been seen. This c a l c u l a t i o n ignored the e f f e c t s of the magneto-resistance. The change i n tuning capacitance required by a p p l i c a t i o n of a 10KG. magnetic f i e l d showed that a t 78°K. the magneto-resistance approximately doubled the s k i n depth. This i s i n rough agreement w i t h the measured magnetoresistance of bismuth (33). I t was o r i g i n a l l y hoped that the S/N r a t i o might be improved by the increase i n s k i n depth caused by the magnetoresistance. This i s not 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 to QS and i f Q depends only on.the sample then Qoc 8cr, so that the S / NG CD V . However ^oecr-" 1, so tha t the S/N r a t i o i s independent of <r and hence does not depend on any magnetoresistive e f f e c t s . This i s probably the case i n bismuth since on tuning the apparatus i t was noticed that the Q was lower than f o r any other sample and that the Q increased on going to 7° K. This tuning was done without the magnetic f i e l d on, only the f i n a l tuning being done w i t h the magnetic f i e l d a p p l i e d . In a l l the other metals the Q depended on the c i r c u i t r e s i s -tance and was approximately temperature independent. I f the Q had been l i m i t e d by the 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 , then an increase i n 8 due to magneto-r e s i s t a n c e would have increased the S/N r a t i o . From steady s t a t e measurements L.C. Hebel found that bismuth had a l i n e about 80 gauss wide which saturated e a s i l y (quoted i n reference 35)• From t h i s one can deduce that Ta~10/>is. and that T,T> 25sec«deg. These are both q u i t e p l a u s i b l e values; the long T, T r e s u l t i n g from the small num-ber of f r e e electrons/atom and the short Tft from pseudo-exchange and pseudo-dipolar i n t e r a c t i o n s . I t i s most l i k e l y t h at these u n s u i t a b l e values of T, T and T a are the reason that no s i g n a l s were seen. 108 ( i v ) 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 to those of bismuth, so that i t was not expected to see a s i g n a l . A b r i e f search was made at 78°K. a t a frequency of 9Mc/s. from 6.5KG. to 11.2KG. No l i n e s were seen. (v) G a l l i u m . This metal has the very low melting p o i n t of 303°K. To t r y to avoid melting the c r y s t a l a l l measurements were made at 78°K. The sample was immersed i n g l y c e r i n e to dampen the 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 present. A search was made at 78°K. which would detect s i g n a l s w i t h T a> 15/<S. and T, T^IO sec.deg. The frequency was 9Mc/s. and the f i e l d was swept from 5.1KG. to 10.3KG. One l i n e was found a t about 6.7KG. w i t h a S/N of about 3. This l i n e was independent of the magnetic f i e l d o r i e n t a t i o n . I t was l a t e r found that eddy currents generated by the r f pulses had melted the surface of the c r y s t a l so that the observed l i n e was probably that from molten Ga 7 1 . The observed l i n e agreed reasonably 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 the p o s i t i o n and the c a l c u l a t e d S/N r a t i o were concerned. I t was too weak to make any measurements on. This s u r p r i s i n g melting of the c r y s t a l probably occurred because g l y c e r i n e has a very low thermal c o n d u c t i v i t y . I t i s a s o l i d a t 78°K. and even when l i q u i d i t has a high v i s c o s i t y , so that l i t t l e convective c o o l i n g can occur. There i s thus a poor thermal contact between the sample and the l i q u i d n i t r o g e n bath and so the heating e f f e c t of the r f pulses i s cumulative. A c a l c u l a t i o n showed that a f t e r a few hours the sample temperature would r i s e from 78°K. to i t s melting p o i n t . This i s approximately the l e n g t h of time that the apparatus i s allowed to run f o r , f o r s t a b i l i z i n g purposes, before a measurement i s made. The heating process i s a l s o aided by the low thermal c o n d u c t i v i t y of g a l l i u m which allows the surface temperature to r i s e about 10° K. above that of the i n t e r i o r f o r 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 heating e f f e c t , the use of g l y c e r i n e to dampen a c o u s t i c o s c i l l a t i o n s was di s c o n t i n u e d . 5 Copper Wire and Other Spurious S i g n a l Sources Two resonances due to the copper wire i n the pickup c o i l were oft 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. The s i g n a l s were q u i t e strong; the amplitude of the Cu 6 3 resonance being 70% of that of the A l a 7 s i n g l e c r y s t a l resonance. This i s because the surface area of the wire i n the c o i l i s n e a r l y the same as the surface area 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 strong s i g n a l i s got. Very strong s i g n a l s could a l s o be picked up from f l u o -r i n e and hydrogen n u c l e i i n the t e f l o n and i n s u l a t i o n near the pickup c o i l . These s i g n a l s l i m i t e d low magnetic f i e l d sweeps because they o b l i t e r a t e d s i g n a l s over a re g i o n of many 110 hundred of gauss i n the reg i o n of 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 This 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 Sn u < ? wrapped around a copper core (59). The S/N r a t i o was q u i t e good at 78° K., whi l e T^was about 200yws. These f e a t u r e s made i t a good sample to use f o r studying some of the experimental d e t a i l s , ( i ) V a r i a t i o n of the Induction T a i l Height With the r f Pulse Lengths. The f i r s t experiment was to study the v a l i d i t y of the expression derived f o r the induced voltage 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 pulse (Appendix I I I ) . To do t h i s the induc-t i o n t a i l height was measured as clo s e to the r f pulse 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 var i o u s r f pulse widths. The i n d u c t i o n t a i l height was corrected f o r s p i n - s p i n r e l a x a t i o n . This c o r r e c -t i o n used Tj, measured at the same magnetic f i e l d o r i e n t a t i o n . Spin-Spin r e l a x a t i o n o c c u r r i n g during the r f pulse was cor-rected f o r by t a k i n g the time o r i g i n as the centre of the r f pulse (13). This c o r r e c t i o n i s exact f o r a uniform H, much l a r g e r than the l o c a l f i e l d , but i s only an approximate cor-r e c t i o n i n the present case. However, the c o r r e c t i o n i s always l e s s than 30$, so that the e r r o r w i l l not be l a r g e . The corrected i n d u c t i o n t a i l heights were then p l o t t e d a g a i n s t the r f pulse l e n g t h ( F i g . U-.^-). The t h e o r e t i c a l ex-pre s s i o n was f i t t e d to the experimental r e s u l t s on the assumption that a -kit pulse a t the surface of the metal was I l l lOyus. long. This value i s i n reasonable agreement w i t h Hi =(13*2) gauss obtained from a f l u o r i n e resonance i n the c o i l mounting i f the increase of H, at the surface of the sample i s considered (Chap. 3.10). The pulse widths f i t the data quite w e l l f o r pulse lengths up to about 50yws. long. The small d e v i a t i o n f o r short pulse lengths i s because the r f pulses are not 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 discrepancy beyond 50yUs. i s due to experimental causes such as heating of the sample, or represents a breakdown of the theory. The most l i k e l y cause of t h i s would be e f f e c t s from the r e g i o n i n which H , i s comparable to the l o c a l f i e l d . I t was found that i n the t h e o r e t i c a l expression the r a t i o of the maximum p o s i t i v e going and negative going amplitudes were very s e n s i t i v e to the form of the phase f a c t o r s . I f the phase term i s cos*x the r a t i o i s 0.*+5, but i f i t i s c o a x c o s ( ^ ) the r a t i o i s 1.2. However apart from the ampli-tudes, 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 to 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 theory, but i n d i c a t e s that i t might be too simple. A di 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 equation w i t h the experimental S/N r a t i o s i s given 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 angle of 28° wi t h the c y l i n d r i c a l a x i s of the copper core. This i s f a r from the optimum angle of 90°, so that t i l t i n g the c r y s t a l 112 through 23° was t r i e d . This suffered from the disadvantages of reducing the induced voltage 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 the important d i f -f i c u l t y as they saturated the a m p l i f i e r f o r about lOOyus. Mounting the sample i n p o r c e l a i n cement reduced the o s c i l -l a t i o n s to a more manageable s i z e . T, measurements were made a t 78 K., but the experimental s c a t t e r caused by ac o u s t i c o s c i l l a t i o n s was over 20%. There was thus l i t t l e chance of measuring any anisotropy i n T, . The most accurate measurement gave T, T=( lf5±l5)ms.deg., wh i l e the s c a t t e r of r e s u l t s showed that there was no anisotropy greater than 50% at the magnetic f i e l d o r i e n t a t i o n s used. Measurements were then made at 78°K w i t h the c r y s t a l mounted v e r t i c a l l y . This reduced the a c o u s t i c o s c i l l a t i o n s to only s e v e r a l times the thermal noise l e v e l . T, was approxi-mately 500yMS. w h i l e Ta was about 20CyUs. I t was thus necessary to s p o i l the magnetic f i e l d u n t i l T* 30yUs. The f i r s t pulse was 90yWs. long and the second one 20ytis. l o n g . When H e was approximately along the [100] a x i s T, T=(35±2)ms.deg., whil e when H„ was i n the (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 are given on Figure *+.3 and show no anisotropy w i t h i n the experimental e r r o r . The average value of T, T=(3l+-±2)ms.deg. agrees w i t h the value of T, T=(31+-l)ms. deg. found by Asayama and It o h (58) over the temperature range !+.20K. to 120°K. f o r powders. I t n 3 disagrees w i t h the measurement of Spokas and S l i c h t e r (15) at 78°K. of T, T=(51+±lOms.deg. i n powders. I t i s impossible to t e l l which of these measurements i s the more r e l i a b l e . Spokas and S l i c h t e r made only one measurement which was i n c i d e n t a l to t h e i r main experiment w h i l s t Asayama and Itoh's measurement of T, T i n niobium, which was made a t the same time, i s u n r e l i a b l e . The form of the angular 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 not known f o r a t e t r a g o n a l l a t t i c e . C a l c u l a t i o n s using the f r e e electron^approximation (63), and the t i g h t b i n d i n g approximation, show that f o r a cubic l a t t i c e i t i s a sum of terms of the form a'+b'cos4(J>' , where $ i s the angle between H 0 and the [OOl] a x i s . This sum i s i s o t r o p i c f o r a cubic l a t t i c e . 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+bcos4"(|) . 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 to the contact and core p o l a r i s a t i o n terms. I f the measured values of T, T are f i t t e d to the above expression they give b =(1.5*3.0 )xlO"3ms.*1 deg.'1 However, b i s not expected to be negative, so that i t has about 70$ p r o b a b i l i t y of being i n the range 0 to 5x10 3msl' deg," From the present measurements i t has not been p o s s i b l e to detect any anisotropy. However, the requirements f o r a u s e f u l measurement of any anisotropy are now much c l e a r e r . In any NMR experiment i t i s very hard to measure T, w i t h b e t t e r l l * f than 2% accuracy and i n the present case 5$ accuracy i s the best that can be obtained. The leng t h of time required f o r such accurate measurements make i t i m p r a c t i c a l to measure T, at more than a few angles. The angles a t which T, must be measured are f o r Ho along the [001] a x i s and along the [lOOj a x i s . Since the assumed angular v a r i a t i o n of T, may be wrong, measurements should a l s o be made w i t h H 0 along the [lio] and the [lO]] axes as a check. With the assumed e r r o r of 5$ i n T, an anisotropy of Ws.deg. would be d e t e c t a b l e . An upper l i m i t can be placed on a by usi n g the f r e e e l e c t r o n Korringa r e l a t i o n . A recent c a l c u l a t i o n on the Knight s h i f t i n superconducting t i n (61) i n d i c a t e s that about 15$ of the i s o t r o p i c value of 0.713$ (6 )^ i s due to s p i n - o r b i t and Van Vlec k paramagnetism. Thus the r e l a x a t i o n due to the con-t r a c t term i s about 2xl0"ams.~'degT* This gives a ^ SxlO'^msT1 degl* I f the a n i s o t r o p i c r e l a x a t i o n i s due to only one mech-anism, the r a t i o b/a depends only on the symmetry of the e l e c t r o n wave f u n c t i o n s at the Fermi surface. I t i s thus a parameter of considerable t h e o r e t i c a l importance, ( i i i ) Spin-Spin R e l a x a t i o n Times., Ta was measured from the f r e e i n d u c t i o n decay by the method described i n Chapter 3.1*+» Considerable care was taken t h a t the resonance was not swept through so q u i c k l y that 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 percent. Timing was done by means of marker pips w i t h lOOyus. separation taken from the timing u n i t and displayed on the o s c i l l o s c o p e , 115 along w i t h the r f pulse and boxcar gate. The r f pulse was l ^ u s . long, w h i l s t the boxcar gate was lOyus. wide. Measurements were made at 78°K. i n both the 12" and the 6" V a r i a n magnets. The sample was mounted v e r t i c a l l y so that the [OOl] a x i s made an angle of 62° to the plane of r o t a t i o n of H 0 5 w h i l e the[010] a x i s l a y w i t h i n 15° of t h i s plane. Measure-ments were made at 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 0 w i t h respect to the (010) plane. The most important f e a t u r e of these measurements i s that the f r e e i n d u c t i o n decay i s exponential f o r times up to a t l e a s t 2.5TA( F i g . ^-.8). The other decays do not have such good s t a t i s t i c s , but s t i l l showed that the decay was exponential out to at l e a s t 2TX. The l i n e shape i n n a t u r a l t i n i s squarer than Gaussian (65)» so that these r e s u l t s are c l e a r evidence that extreme exchange narrowing occurs i n i s o t o p i c a l l y pure t i n . The l i n e shape i s broadened by the f i n i t e time that s p i n - l a t t i c e r e l a x a t i o n allows a nucleus to remain i n a given s t a t e . This i s known as l i f e t i m e broadening. 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 the general case i s very complex, but f o r a sp i n £ system w i t h a L o r e n t z i a n l i n e i t can r i g o r o u s l y by shown th a t the i n d u c t i o n t a i l i s s t i l l e x p onential w i t h a decay constant T a given 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 to the experimental r e s u l t s i t was assumed th a t T, was ^Oyus. at 78°K. The e r r o r i n T, i s neglected since i t c o n t r i b u t e s at most a systematic e r r o r of 116 5% to T x . I t should be noted t h a t the l i f e t i m e broadening e f f e c t i s not l a r g e enough to be the cause of the exponential decay. Table *+.!. Spin-Spin R e l a x a t i o n Times by Free Induction Decay Angle of Hofrom (010) plane. Ti ( y U S . ) -30° 200±7 260*10 - 3 0 ° 190±7 2^ -5*10 +106 i5o±io i8o±i5 +10° 175*20 220*25 +30° 200*10 260±l5 + 5 5 ° 170*10 210±15 +100° 150*7 180*10 I t takes about a day to make each measurement so that a d e t a i l e d study of the anisotropy would take a long time. I t was thus decided to use C l a r k ' s method of measuring the l i n e shape (Chapter 3.15) to obtai n the r e l a t i v e anisotropy and then use the f r e e i n d u c t i o n values of T a to get the ab-so l u t e a n i s o t r o p y . I t only took a day to make a study of the l i n e shape as a f u n c t i o n of o r i e n t a t i o n so that s e v e r a l weeks were saved. For the l i n e shape measurements a boxcar gate 1ms. wide was used. This s t a r t e d about 2C^ us. from the true time o r i g i n and covered a l l of the f r e e i n d u c t i o n decay. Care was taken that the l i n e was swept through slowly enough to avoid d i s t o r t i o n . D i s t o r t i o n s t i l l occurred due to the s i g n a l not being e x a c t l y i n phase w i t h the reference s i g n a l . The l i n e shape i s very s e n s i t i v e to 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 considerable experimentation, t h i s phase d i s t o r t i o n was reduced to about the noise l e v e l and the measurements then were made. To analyse the l i n e shape the midpoint was chosen and then the two amplitudes w i t h the same frequency d e v i a t i o n from t h i s midpoint were averaged. This approximately 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 increased the S/N r a t i o by «/2. The r e s u l t i n g curves are L o r e n t z i a n f o r a t l e a s t s e v e r a l l i n e widths from the c e n t r a l frequency ( F i g . >+.9)j again strong evidence f o r exchange narrowing. The h a l f widths and maxi-mum amplitudes were both measured. As expected f o r exchange narrowing, when the h a l f widths increased the amplitudes de-creased by approximately the same f r a c t i o n . Two c o r r e c t i o n s to the l i n e widths had to be considered. The f i r s t of these was the d i s t o r t i o n introduced by the deadtime (Appendix I V ) . This reduced a l l the h a l f widths by about 10$, but caused a much smaller e r r o r i n t h e i r r e l a t i v e values. Apart from the narrowing, i t caused l i t t l e d i s t o r -t i o n i n the l i n e shape. For these reasons, i t was ignored. A c o r r e c t i o n a l s o had to be made f o r l i f e t i m e broadening. The h a l f width i s the r e c i p r o c a l of Tx so that the l i f e t i m e broadening c o r r e c t i o n s already obtained f o r T a were used to 118 give the h a l f width c o r r e c t i o n s . Further a n a l y s i s of the l i n e widths r e q u i r e s a c l o s e r study of the exchange narrowing process. In a reference frame r o t a t i n g at the resonant frequency a spi n only f e e l s the l o c a l f i e l d , which i t precesses about at a frequency of the order of the l i n e width. 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 at about the i n s t a n t a n -eous Larmor frequency so that the spins 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 width. I f an exchange i n t e r a c t i o n rHe* = } J,;I,-.I: i s a l s o present the rat e of change of the d i p o l a r Hamiltonian H i s I f H x ^ H the l o c a l f i e l d f l u c t u a t e s at a r a t e about equal to the exchange i n t e r a c t i o n constant J . This i s much f a s t e r than the Larmor frequency so that the averaged f i e l d which the spins f e e l i s much l e s s than the l o c a l f i e l d . The l i n e i s thus much narrower than the 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 ca 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 (1,66). In t h i s model i t i s assumed that the random f l u c t u a t i o n s of the l o c a l f i e l d Ato(t) from the resonant frequency u„ are Guassian i n amplitude w i t h a mean square value OJ p 1 equal to the second moment. Their time v a r i a t i o n s are described by the c o r r e l a -t i o n f u n c t i o n 119 A form f o r g('t) must be chosen on the ba s i s of p h y s i c a l p l a u s i -b i l i t y and mathematical 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) are that the second moment i s unaffected by the exchange i n t e r a c t i o n and that the f o u r t h moment must remain f i n i t e . The simplest expression which s a t i s f i e s these requirements i n the Gaussian g(T) = expC-irrwe ' - ir 1 ). uJt^7 i s an average exchange frequency. With t h i s assumption 7 the f r e e i n d u c t i o n decay f o r the exchange narrowed r e g i o n becomes G(-t)=exp(--glr t ) . This expression holds except when t « w e _ l , where the decay tends towards a Gaussian. The l i n e shape corresponding to G(t) i s a L o r e n t z i a n out to a f r e q u e n c y w h e r e i t f a l l s o f f qu i t e r a p i d l y , keeping the second and f o u r t h moments f i n i t e . From G ( t ) , the f o u r t h moment i s <w*>=36Jp4 +-kTTtUeO^. The f o u r t h moment can 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 Vl e c k ' a method of t r a c e s . The two expressions have a s i m i l a r form so t h a t , by comparison, iot can be w r i t t e n i n terms of the l a t t i c e s t r u c t u r e and exchange and d i p o l a r i n t e r a c t i o n s . The r e s u l t i n g expression i s so complicated that u>e i s custo m a r i l y assumed to be i s o t r o p i c . The l i n e width i s u)?/uj& . The second moment f o r white t i n has been c a l c u l a t e d by computer so that the form of cjp4- i s known. I f toe i s i s o t r o p i c , the anisotropy of the 120 measured l i n e widths should be p r o p o r t i o n a l to c*>p . The f i t i s reasonably good ( F i g . V . l l ) and, c o n s i d e r i n g the experimental e r r o r s i n v o l v e d , shows that c j e i s i s o t r o p i c , or very n e a r l y so. The l a r g e s t e r r o r i s due to s l i g h t misalignment of the c r y s t a l . The magnetic f i e l d o r i e n t a t i o n t r a c e s out a complicated t r a j e c -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 the h i l l to v a l l e y r a t i o of the second moment. The inverses of T a 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 0 are a l s o p l o t t e d on the same graph and agree w e l l w i t h the l i n e width v a r i a t i o n s . They a l s o give the absolute l i n e widths. tjp 1 i s the sum of the second moments due to d i p o l a r and pseudo-dipolar i n t e r a c t i o n s . These both have the same angular dependence, so that 6jp = <cj,|> (1+B), where (00$ i s the d i p o l a r second moment and B i s the f r a c t i o n of pseudo-dipolar exchange present. ,,. rp l a ~<Wcf>fHB) Using a l l of the measured T a s , along w i t h t h e i r corresponding c a l c u l a t e d d i p o l a r second moments gives =( 1+B) (7*1.5) 103 rad a. — (1) The absolute value of these parameters cannot be obtained without making an independent measurement i n v o l v i n g them. In t h i s case a measurement of T a 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, being about three, so that nothing could be said about the f r e e i n d u c t i o n decay other than i t decayed f a s t e r than an exponential decay. Steady s t a t e 121 measurements show that the l i n e i s n e a r l y Gaussian (65) so that the l i n e was analysed by assuming i t was Gaussian. This gives Ta =(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* broadening. This corresponds to a second moment of (1.3 iO«2)Kc/s . This i s i n e x c e l l e n t agreement w i t h the steady sta t e value (68). N a t u r a l t i n contains" two" i~sotopesy ; Sn"7 and Sn1"' , of approximately equal abundances. The width of the Sn" ? l i n e thus contains a number of 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< =i74lfr(l+l)£'l£f + i T , * r s * S(S+l)fi a Jib*: , where the summations are only taken over the l a t t i c e s i t e s occupied by the appropriate i s o t o p e , b^ =*Vj"3 (3cos*c9,-j -1). T i n has approximately equal gyromagnetic r a t i o s and abundances F and so the expression s i m p l i f i e s to - ~ ~ * 3 . 'j where the summation i s now taken over a l l the l a t t i c e s i t e s . (b) Pseudo-dipolar broadening. This adds a term BM^ to the second moment. (c) Pseudo-exchange broadening between u n l i k e n u c l e i . The exchange co u p l i n g £Z.J?j (•'£* *§P does not commute w i t h Hf i f the spins are u n l i k e and so c o n t r i b u t e s a term to the second moment. For t i n t h i s i s - t f U J i j ( r ) , i * w i t h the summation over a l l l a t t i c e s i t e s . 122 (d) Exchange narrowing between l i k e s p i n s . In n a t u r a l t i n the exchange frequency i s Fcj e . F i s about 0.08 and so the exchange frequency i s much l e s s than the d i p o l a r frequency and can thus be neglected, ^ b 5 ^ has already been computed so that the t o t a l second moment i n n a t u r a l t i n can be c a l c u l a t e d . I t i s <Aeua> =7.5xlO t(l+B)+2 .5xlO H J 1 . In e v a l u a t i n g J the l a t t i c e s t r u c t u r e has been roughly taken i n t o account. White 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 body-centred 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 (67). Each atom has s i x nearest neighbours about 3«1A° away and 12 next nearest neighbours about hk° away. Atoms f u r t h e r away than t h i s were not considered. J i s the value of the exchange constant at the nearest neighbours and was assumed to vary as _3 r i n summing over the other s i t e s . Assuming a Gaussian l i n e and using the measured value of Tft to determine <AU>*> gives (7 ±l)ylo 7 =7.5xlo 6(l+B)+2 .5ao" 1 J \ — — — ( 2 ) The next step r e l a t e s J and u>& and i n v o l v e s the most d r a s t i c step of the whole a n a l y s i s , j u s t i f i a b l e only by ex-pediency. The r e l a t i o n between J and co& should be obtained by equating the 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 the l a t t i c e s t r u c t u r e by the 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 the l a t t i c e s i t e s and even f o r j u s t the 12 atoms out to next nearest neighbours i s a p r o h i b i t i v e l y complicated c a l c u l a t i o n , w h i l e summing over fewer atoms i s not p h y s i c a l l y j u s t i f i a b l e . The r e l a t i o n obtained f o r a simple cubic l a t t i c e (1) thus had to be used. The l a t t i c e spacing d i s assumed to be 3A° , the nearest neighbour spacing. For <CJp>> an average of the computed values f o r the white t i n l a t t i c e i s used. This must a l s o i n c l u d e the pseudo-d i p o l a r c o n t r i b u t i o n . This was obtained 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 without the pseudo-dipolar c o n t r i b u t i o n to ob t a i n an approximate value of B and t h i s was then used i n o b t a i n i n g a more accurate value of ^uuf.y • Because of the l a r g e experimental e r r o r s and the dominance of the ex-change term, i t was not necessary to repeat the c y c l e . The value f i n a l l y adopted was <ujf> =(8.1*0.8) xlO 7 rad a. This gives This n e g l e c t s the d i p o l a r c o n t r i b u t i o n to the f o u r t h moment, a good approximation f o r extreme exchange narrowing. 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 then e l i m i n a t i n g 1+B gives a 2= , ,°:,J-J " i = i.7 / io"V W| = ( 2 . 2 ± 0 . 2 ) j \ (3) 12*+ quadratic equation i n GJe. The s o l u t i o n of t h i s equation i s We =(2.0±0.3)*104radians. This corresponds to an exchange constant of J=(2.2±0.3)Kc/s. Using the above value of CJE gives 3=1.9*0.5. By comparing the l i n e width they measured i n white t i n powder by steady s t a t e methods w i t h the c a l c u l a t e d d i p o l a r moment, Karimov and Schegolev (68) obtained J=(2.5 i0.1)Kc/s. However, they neglected the pseudo-dipolar c o n t r i b u t i o n . I f 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 the present experiment i n c l u d e d , they give J=(2.1±0.2)Kc/s. There i s thus e x c e l l e n t agreement between the two experiments. Jones (65) a l s o attempted to measure J . There are numerical e r r o r s i n h i s determination of the 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 mom-ent computation so tha t h i s r e s u l t s are wrong. A determin-a t i o n of B has not been made before. The whole a n a l y s i s has rested on the twin assumptions that the 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 Gaussian c o r r e l a t i o n f u n c t i o n and that the d i p o l a r l i n e shape i s Gaussian. L i t t l e can be said about the r e l i a b i l i t y of the 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 . The e f f e c t of d e v i a t i o n s from the Gaussian form i s unknown. More d e f i n i t e statements can be made about the assumption of a Gaussian l i n e shape. The l i n e shape parameters of i n t e r e s t are the second moment 125 of the d i p o l a r l i n e and the f o u r t h moment of the 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 of the 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 to be measured i n n a t u r a l t i n but, 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 appropriate a n a l y s i s of the f r e e i n d u c t i o n decay. In the case of extreme exchange narrowing, the c o n t r i b u t i o n to the f o u r t h moment from the exchange i n t e r -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. I f t h i s i s so, the 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 to the d i p o l a r l i n e shape and so the values of B and J obtained should be unaffected by moderate d e v i a t i o n s from a Gaussian d i p o l a r i i n e shape. More e x p l i c i t l y , Anderson and Weiss found that the 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 exponential one w i t h equal second moments, had exchange narrowed l i n e widths which d i f f e r e d by about 50$. I t thus seems that the measured values of B and J do not con t a i n l a r g e e r r o r s due to the assumption of a Gaussian l i n e shape, but could 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 to defe 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 to compare the experimental values of J and B w i t h values c a l c u l a t e d from the t i n band s t r u c t u r e . However n e i t h e r the accuracy of the measurements, nor that of the c a l c u l a t e d values, warrants such a step a t the present time. The main f e a t u r e to note i s the l a r g e value of B. This i m p l i e s that a l a r g e number of the Fermi surface 126 e l e c t r o n s are of p, or higher character. The r e l a t i v e l y l a r g e a n i s o t r o p i c Knight s h i f t s u b s t a n t i a t e s t h i s , ( i i i ) Spin 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 Ta--^OO^s. I f Ta i s reduced by des t r o y i n g the homogeneity of the magnetic f i e l d , i t i s p o s s i b l e to get sp i n echoes. In the f i r s t measurement tv/o r f pulses l ^ t / s . and 30^s. long and separated by 200yws were used. The homogeneity was such that 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 va r y i n g the magnetic f i e l d the boxcar was swept through resonance at K^ws. i n t e r v a l s from 60yMs. to 250yus. a f t e r the second pulse. A narrow boxcar gate of lOyus. was used and a l l the times were measured between the centres of the pulses and boxcar gate. The magnetic moment measured by a sweep was then p l o t t e d against the time a f t e r the second pulse ( F i g . ^.3). The graph shows a spin echo'which i s 180° out of 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 symmetrical envelope whose time constant i s the same as that of the i n d u c t i o n t a i l . I t s maximum occurs at 200,Hs. a f t e r the centre of the second pulse. On removal of the f i r s t pulse the s p i n echo vanished. The measurements were repeated 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 pulses. The amplitude of the echo increased as the pulse separation decreased. The next experiment was to apply two pulses 200yWs. apart w i t h a 30yUS. wide boxcar gate set on the echo maximum. The magnetic f i e l d was then swept through resonance w i t h 127 v a r y i n g values of the reference phase. As the phase was v a r i e d by 90°, the p l o t of magnetic moment versus magnetic f i e l d changed from an absorptive to a d i s p e r s i v e shape. So f a r the s p i n echo had behaved i n e x a c t l y the same f a s h i o n as a s p i n echo i n a non- m e t a l l i c substance w i t h a homo-geneous H, • In the next s e r i e s of measurements, the d i f f e r e n c e s became apparent. T* was reduced to 20yMs.fifor these experiments. Two r f pulses separated by 150/as. were used along w i t h a boxcar gate hOj/Us. wide set on the echo maximum. The widths of the two pulses were then v a r i e d w h i l e keeping the r a t i o of t h e i r widths f i x e d at 1:2. The graph of echo height against the f i r s t pulse width showed that the maximum amplitude occurred when the f i r s t pulse was about lO^us. wide ( F i g . ^ . ^ f ) . The f i r s t pulse width was then kept f i x e d at lC^us. and the second pulse width v a r i e d from ^ s . to 90^\s. The maximum amplitude occurred when the second pulse was about 20^s. wide ( F i g . *+.5). These r e s u l t s can be explained by a simple extension of the c l a s s i c a l s p i n echo theory of Hahn (62) to the case of a metal i n the normal s k i n e f f e c t r e g i o n . For s i m p l i c i t y , only the case of a phase coherent system set to detect the absorption mode and operating at the resonant frequency i s considered. The sample i s approximated by an i n f i n i t e plane so that the r f magnetic f i e l d a t a depth z from the surface i s H, =H)p e x p ( - ^ ) c o s ( ^ g ). Replacing OJX by ou,(z)= tV,exp(-^) i n Hahn's theory and t a k i n g account of the phase coherence allows the magnetic moment M S s ( z , t ) producing the s p i n echo at a depth z to be 128 simply c a l c u l a t e d . The s p i n echo induced voltage i s then (Appendix I I I ) V ( t ) = J o e x p ( - ^ ) c o s ( ^ - ) M s e ( z , t ) d z = M 0 e x p [ > ( ^ ^ ) - i j e x p t - ^ s i n f o ; , (z)<l Jo s i n a [ i ( j , ( z ) ' t a ] c o s : 1 ' ( ^ )dz. TA i s the s p i n - s p i n r e l a x a t i o n time due to d i p o l a r and exchange e f f e c t s , w h i l e T* i s the transverse r e l a x a t i o n time due to the inhomogeneous magnetic f i e l d . T i s the time i n t e r v a l between the two r f pulses of d u r a t i o n 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 l ( i cj»fa) as s i n ( )+£ [ sin{oj,(T^-'t,)} -sin{o;i (fa+'ti)} J , v s e becomes a sum of i n t e g r a l s of the form exp(-x)cos a ( x ) s i n {co.-c.e" )d*. These have already been evalu-ated (Appendix I I I ) , so that a q u a n t i t i v e t e s t of v J e can be made. The c a l c u l a t e d v a r i a t i o n of s p i n echo amplitude w i t h r f pulse l e n g t h has been p l o t t e d i n F i g s . k,k and *f.5 f o r both the m e t a l l i c and non-metallic cases. I t was assumed that a £TT pulse at the surface of the sample was 9yus. The theore-t i c a l expression i s i n good q u a l i t a t i v e agreement w i t h the experimental 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 that the &rr pulse was about 8yus. long. This i s p o s s i b l e since these measurements were not made under e x a c t l y the same experimental c o n d i t i o n s as those of F i g . k.2. For the spi n echo measurements of Tx , the f i r s t pulse was lOyMS. long and the second one was 20jus. long. A wide boxcar gate was used. I f the pulse separation (taken between centres) was T, the gate centre was lo c a t e d at a time T a f t e r the centre of the second pulse. A f t e r each sweep through resonance T was a l t e r e d manually. Timing was by means of the double beam o s c i l l o s c o p e and marker pips from the tim-ing u n i t . A l l the measurements were made at 78 K. The spin echo decays were a l l exponential ( F i g . *+.13) w i t h a decay constant % which was corrected f o r l i f e t i m e broadening i n 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 to give T a . Table *+.2. Spin-Spin R e l a x a t i o n Times by Spin Echoes Angle of H 0 from (010) plane. % (/AS.) Ta (JJS.) +10° 120±10 1^ 0^ 12 +30° 170-15 210*20 +55 105±5 120*7 +100° 105*5 120*7 When the values of Ta obtained by f r e e i n d u c t i o n decay and by spin echoes were p l o t t e d a g a i n s t magnetic f i e l d 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 that Ta obtained by s p i n echoes was always shorter than Ta measured 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 the value of the f r e e Induction T 4. -There are three p o s s i b l e reasons f o r t h i s l a r g e d i f -ference i n values. 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 the s p i n gradient caused by the l a r g e inhomogeneity i n H, . For a spin system the surplus of spins 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 equa t ion ( 1 ) . W i s the p r o b a b i l i t y / s e c o n d tha 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 epa ra t i on between the n u c l e i . W<~J, the exchange cons tan t , so tha t Wa*~> 10"'* cmt / sec . S p i n d i f f u -s i o n a l t e r s T a because the phase and a t t e n u a t i o n of the induced s i g n a l depend on the depth of the sp ins from the surface and so i f the s p i n c o n c e n t r a t i o n changes w i t h t ime , T a i s a l t e r e d . For s p i n d i f f u s i o n to reduce the f r ee i n d u c t i o n T a by 10$ the sp ins 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 i s (2Wa 1T l)\ This i s about one l a t t i c e spac ing , so tha t s p i n d i f f u s i o n e f f e c t s are n e g l i g i b l e . Because of t h i s very 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 e i t h e r . The second p o s s i b i l i t y i s tha t two of the approximat ions made i n d e r i v i n g the s p i n echo equa t ion are not v a l i d and tha t obscure phase e f f e c t s a s soc ia t ed w i t h t h i s cause the e r r o r i n T a . I n the d e r i v a t i o n i t i s assumed that the pulse leng ths are n e g l i g i b l e compared to T 2 and T? . 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 reference frame occurs du r ing the r f pulse and i t i s e a s i l y shown tha t the only e f f e c t of t h i s i s to reduce the echo ampli tude wi thou t a f f e c t i n g the measured va lue of Ta.. However i n the present case the pu lse lengths are about lOyus. w h i l e T a i s about l50^s., so tha t r e l a x a t i o n i n the r o t a t i n g re ference frame only occurs w i t h i n about 6 of the su r f ace . 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*1 has only been c a l c u l a t e d f o r a two s p i n system (13). In t h i s , non-secular terms i n the r o t a t i n g reference frame d r a s t i c a l l y modify r e l a x a t i o n during 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 pulse i s over, apart 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 amplitude. This suggests that there should be no e r r o r i n T 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 system. Although they i n t u i t i v e l y seem independent of the pulse separ-a t i o n , i t i s p o s s i b l e that these non-secular terms cause the r e d u c t i o n i n T^ i n s p i n echoes. Hahn's theory a l s o contains the more d r a s t i c assumption that Ti;H,» ( T j ) " 1 . This means that the magnetic moments precess about the constant 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 having n e g l i g i b l e e f f e c t on the precession. In these e x p e r i -ments, t h i s requirement, does not even hold at the surface of the metal. I t i s quite p o s s i b l e that t h i s i s the reason f o r the smaller T a. The r e s t r i c t i o n on H, could be removed from the theory i f i t were not f o r the ensuing avalanche of a l g e b r a i c manipulations. The breakdown of these two approximations suggests that d i v i d i n g v s e i n t o the product of a time dependent and a phase dependent f a c t o r i s only approximately c o r r e c t . There are two experimental checks of t h i s which can be made. The f i r s t one would be to measure T a as a f u n c t i o n of H, using approximately £ n and rr pulses at the surface of the metal f o r each value of H,• Unfortunately the apparatus could not do t h i s . However 132 the second check of measuring T a as a f u n c t i o n of pulse l e n g t h was attempted. The f i r s t pulse l e n g t h was kept f i x e d at I C ^ M S . w h i l e the second pulse 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 Spin Echo Ta With Pulse Length Angle of H0 from (010) plane. Length of Second pulse (yus.). Spin echo. Free Induction +30° 10 l80±20 +30° 20 170*15 200±10 +55° 10 115*5 - - - - -+55° 20 105*5 170*10 +55° 30 105±5 — Although the experimental e r r o r s prevent a d e f i n i t e con-c l u s i o n , i t seems that there i s a small dependence of T 2 on pulse l e n g t h . There could a l s o be a dependence on the f i r s t pulse l e n g t h , but the poor S/N prevented any experimental ex-amination of t h i s . The experiment has f a i l e d to show the cause of the bulk of the descrepancy though. The f i n a l p o s s i b i l i t y i s that the d i f f e r e n c e i s due to the e f f e c t of the exchange term. In the l a b o r a t o r y frame the Hamiltonian of the system i s The Hamiltonian i n a reference frame r o t a t i n g at a frequency to be comes U = T - h Z ( H o - % ) l i z + V , 133 where H"» T^±£r7jU-3eos»e,j ) (3IuIj»-I, . I j Before the f i r s t p ulse, the de n s i t y matrix d e s c r i b i n g the spi n system i s The Zeeman term i s much l a r g e r than the d i p o l a r and exchange frame r o t a t i n g at the frequency u> of the ap p l i e d r f pulses. F o l l o w i n g Abragam (p. M-98), the maximum amplitude of the s p i n echo at resonance becomes E(2T) = Tr[exp(-I'H 1!r)exp(-i7H l 'r, )exp(-ltfl!)exp(-17H lt l) I i exp(lTH,t,) e x p ( i V T ) e x p ( i ? H l T j e x p ( i 7 ^ ) l . e ] , where % and 1X are the f i r s t and second pulse lengths respec-t i v e l y , and T i s the pulse s e p a r a t i o n . The problem i s now f o r -mally solved; the remaining steps c o n s i s t i n g of expanding the exponential operators i n a power s e r i e s and then e v a l u a t i n g the r e s u l t i n g t r a c e s . The a t t e n u a t i o n and phase f a c t o r s due to the sp i n echo being i n a metal could then be taken care of i n the same f a s h i o n as f o r the 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 involved has prevented E(2T) from being evaluated f o r even a two s p i n system. The rig o r o u s quantum mechanical theory thus at present says nothing about the e f f e c t of the exchange i n t e r a c t i o n on spin echoes. S t a t i s t i c a l t h e o r i e s , such as that of Anderson 13>+ and Weiss, are not s u i t a b l e f o r d e s c r i b i n g the s p i n echoes since they c o n t a i n ad hoc assumptions and, by t h e i r nature^ average out many of the d e t a i l e d i n t e r a c t i o n s which might be expected to modify the 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 , the n u c l e i are not an equiv a l e n t s i t e s , so there 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 being observed by s p i n echoes (1). In the d e n s i t y matrix formalism the f r e e i n d u c t i o n decay i s p r o p o r t i o n a l to Tr [exp(-iKt)I xexp(i'H"t)I t] . The f i r s t two terms of t h i s have been evaluated i n the presence of both exchange and d i p o l a r i n t e r a c t i o n s (1). The f i r s t term (second moment) i s unaffected by exchange, but the second term ( f o u r t h moment) i s increased by i t . In the case of exchange narrowing the s e r i e s must approximate an exponential s e r i e s , so that obviously the exchange must a l s o a f f e c t many higher terms i n the expansion. The t h e o r e t i c a l s i t u a t i o n i s that the exchange i n t e r -a c t i o n a f f e c t s the 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 the experimental r e s u l t s i n both the d e n s i t y matrix and s t a t i s t i c a l t h e o r i e s . The de n s i t y matrix theory says nothing about the s p i n echo case wh i l e the Anderson-Weiss theory gives the same Ta f o r both 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 not co n c l u s i v e because of the type of assump-t i o n s involved i n t h e i r theory. These c o n s i d e r a t i o n s suggest that the exchange i n t e r a c t i o n a f f e c t s the s p i n echoes, but i t i s impossible to say whether or not the spi n echo Ta should be 135 the same as that 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 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 been examined. The only d e f i n i t e c o n c l u s i o n i s t h a t s p i n d i f f u s i o n e f f e c t s are n e g l i g i b l e . A l l of the obvious experimental pos-s i b i l i t i e s , such as a systematic 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 el i m i n a t e d by the p r e l i m i n a r y experimental 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 data. Pulse width e f f e c t s were experim e n t a l l y shown to 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 experimental e f f e c t s caused by H, being too s m a l l . I t i s a l s o p o s s i b l e that the d i f f e r e n c e i n Ta i s an i n t r i n s i c f e a t u r e of systems wi t h approximately equal d i p o l a r and pseudo-exchange i n t e r a c t i o n s . The 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 causing t h i s . k-.Q The Experimental S/N R a t i o s An expression f o r the S/N r a t i o has been derived (Chap. 3.12) which can now be compared w i t h the experimental S/N r a t i o s determined f o r s i x d i f f e r e n t metals under var y i n g c o n d i t i o n s . The S/N r a t i o at the a m p l i f i e r output i s S' =TT^c6n/utoM0R, Q (-j^-)* . In 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 and temperature independent, T n i s assumed to be the sample tem-perature and C i s taken as 80pf. The other parameters depend on the i n d i v i d u a l sample and c o i l and are known, except f o r _ . 136 For some metals S can be a c c u r a t e l y c a l c u l a t e d at the temperatures of i n t e r e s t , but f o r other metals the e l e c t r i c a l c o n d u c t i v i t y had to be estimated. The boxcar enhancement f a c t o r i s r ^ ^ - j • f t was estimated from the bandwidth of the tuned c i r c u i t to be 2/as. Using these v a l u e s , a t a b l e compar-ing the t h e o r e t i c a l and experimental S/N r a t i o s can be con-s t r u c t e d . Table h.h- Experimental and T h e o r e t i c a l S/N R a t i o s Metal S/N a t Ampli-f i e r Output,S' . Boxcar Enhance ment f a c t o r F i n a l S/N Experimen-t a l S/N r a t i o Temp-erature "K. A l a 7 0.08 130 10 15 295 II 0.2 30 6 10 78 Vs' O.U- 22 8 30 295 II 1 7 7 , 5^ 78 Nb*> 1 30 30 80 295 n 3 30 90 50 78 Ga11 0.006 k 0.02 295 n O.OM- 7 0.3 78 n 0.09 12 1 3 303 Sn"< 0.1 90 9 15 78 N a t u r a l 0.01 150 1.5 3 78 t i n In'" o .o5 12 0.5 295 it 0.1 12 l 78 Bi*°< 0.02 7 0.1 295 n 0.2 7 1.5 78 Sb 0.02 7 0.15 295 n 0.06 7 O.Oh -- 78 Re 0.01 7 0.07 295 0.02 7 0.15 — 78 i 137 The agreement between the c a l c u l a t e d and experimental values i s very good c o n s i d e r i n g the number of i l l defined q u a n t i t i e s i n v o l v e d . The experimental S/N r a t i o was only crudely measured and was not corrected f o r s p i n - s p i n r e l a x a t i o n . I f t h i s were done, i t would double most of the experimental S/N r a t i o s . The re d u c t i o n i n S/N caused by a c o u s t i c o s c i l l a t i o n s was not allowed f o r e i t h e r . In niobium t h i s i s the main noise 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 four times the experimental value. P a r t of t h i s d i f f e r e n c e i s undoubtedly due to i n c o r r e c t values of some parameters and to the s i m p l i f i c a t i o n s i n the theory. The remainder of the descrepancy i s probably because the sample surface i s not p e r f e c t l y smooth, but contains 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 the s k i n depth. These i n -crease the e f f e c t i v e surface area and hence increase the induced s i g n a l . Provided that the Q i s l i m i t e d by the 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 increase the S/N by some f a c t o r i n the reg i o n of two. In any case, the d i f f e r e n c e between the t h e o r e t i c a l and experimental S/N r a t i o s i s now known, so that the theory can c o n f i d e n t l y be used to 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 the temperature range i n which the s k i n e f f e c t i s normal. Figure h.la Sweep Close to the r f Pulse B Figure ^ . l b Sweep a t a Time About T^. a f t e r the r f Pulse T y p i c a l Chart Records of a Sweep w i t h a Narrow Boxcar Gate Through the Free Induction T a i l . 6-o « Induction ^" Tail Amplitudes After % e Second Vulse. (Arbitrary Unit*). 3 , o ° H P in (010) Plane * Ho aloncj CoioJ Axis —r -0-5 — I — 1-0 Time Between Pulses (Vnilliseeonals). 1-5 2-0 Figure 4 . 3 Spih-Lattice "Relaxationlime Anisotropy in Isotopically Pure In. 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 \ \ \ -i 1 1 =• 1 r-5 to '5 . a© • . • as W i d t h of First "Pulse (microseconds). Figure 4.6 Vqr iqt ion o-f Spin Echo Amplitude With the rf Pulse Widths. £ SO-] 40 30 Amplitude of 5 pin EcUoj (Arbitrqrt^  Units). 10 —'Metallic / " " ^ ^Theoretical Variation / \ —t^on-rnetqlhc J Width o-f the Second Pulse (microseconds). Figure 4.7 Variation of the Spin Echo Amplitude With Second Pulse Width. 54 Ln. of Induction Tail Height (Arbifrqry Units). 40-3-0 500 100 T i m e ^ m i c r o s e c o n d s ) —i— a.oo 3O0 4 0 0 Figure 4.8 Spin-Spin "RelaxaTionTme by Free Induction Decay. i—• -r Amplitude CAi-m"h-airtj Units) . T h e Curve is LoreritzJan W i t h a H a l f W i d t h o f S Units. Frequency Deviation (Arbitrary Units) as Figure 4.7 Lorenfzian Line Shape in IsoTopicallLj F u r e l i a 40-0=4 " r" ' ' At^cjle o-f Ho From (010) Plane- 45° - l r-h-1 • T rHo Magnetic Field Figure 4.io Relative Orientation of the Crystal and Magnetic Field. 3 0 2S; Line WVdth (Arbitrary Until). ao-i5-_L Experimental Line Widths. Ume Width From Anderson-Weiss Theory I O 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 Echo Ampl i tude T 50 IOO Time (microseconds) Figure 4,12. Spin-Spin Relaxation Time by 5pin Echoes. -r \ 0 150 3 T o <U £ .Q -•5 3 •5 (Li K o o - o --O-O I ° cr —i— o 0 O ' O 0) C cr o o a? u j -o CO Z5X Pv-O o I/) it c C CO 'co 0) err. «J1 cr • * <u £ L a i J ? 151 CONCLUSION •Begin a t the beg inn ing , and go on t i l l ' you come to the ends then s t o p . ' - A l i c e i n Wonderland. In 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 Tx i n metal s i n g l e c r y s t a l s at l i q u i d n i t r o g e n temperature has been e s t a b l i s h e d . I t has been shown that reasonably accura te measurements can be made provided Ta^SO/Us. and T, & 2 s e c d e g . A reasonably complete theory of the apparatus has a l s o been developed and expe r imen ta l l y conf i rmed. 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 to e l i m i n a t e a c o u s t i c o s c i l l a t i o n s . U n t i l a way i s found of d r a s t i c a l l y r educ ing them, measurements cannot be made a t l i q u i d he l ium temperatures . The apparatus has r e s t r i c t e d usefu lness u n t i l such measurements can be made. The ga in i n S/N would probably be moderate, but would c e r t a i n l y be enough to a l l o w the succes s fu l comple t ion of s e v e r a l experiments which are a t present 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 . The more important reason f o r going to hel ium temperature i s tha t the pulsed apparatus could then be used i n con junc t ion w i t h steady s ta te apparatus on the same sample, so tha t more v a r i e d and accura te data could be ob ta ined . In p a r t i c u l a r , the pulsed apparatus can g ive a more accura te idea of the l i n e shape than the marg ina l o s c i l l a t o r s used i n the steady s ta te work. L i f e t i m e broadening i s a l s o unimportant at hel ium tem-p e r a t u r e . A l t e r n a t i v e exper imenta l procedures i n v o l v i n g 152 d i g i t a l devices f o r improving the S/N r a t i o , such as the Enhancetron, might be more s a t i s f a c t o r y than the boxcar i n t e -g r a tor at low temperatures. There are s e v e r a l experiments which could be done on the o present apparatus-at 78 K. The most obvious ones are to attempt to measure the anisotropy 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 i n v o l v e regrowing the c r y s t a l at 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 pure, and n a t u r a l t i n to ob t a i n the exchange constants could r e a d i l y be done on cadmium. Experiments 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 compositions might give 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 . Spin echoes were a l s o observed and t h e i r basic f e a t u r e s s t u d i e d . The f a c t that they have a shorter T a than 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 study. Measurements i n t i n oxide would obviously be very h e l p f u l . Once the cause of t h i s has been found, s p i n echoes could be u s e f u l f o r measuring T a i n those paramagnetic a l l o y s which have a short T* because of s p a t i a l l y v a r y i n g s t a t i c magnetic f i e l d s i n the 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 metal i n the normal s k i n e f f e c t r e g i o n are w e l l understood. The theory 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 path approaches the s k i n depth i s not q u i t e so c l e a r c u t (the anomalous s k i n e f f e c t ) . F a i r l y 153 r i g o r o u s t h e o r i e s have been developed, but they i n v o l v e assump-t i o n s concerning 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 surface of the metal. An experimental 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 . Pulsed NMR i s capable of doing t h i s because the i n d u c t i o n t a i l height as a f u n c t i o n of r f pulse l e n g t h i s very s e n s i t i v e to the phase s h i f t s . I f a study of phase s h i f t s alone was contemplated, there may be other ways of using the apparatus which would give 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 give considerable i n -formation. The r e s u l t s of steady s t a t e measurements al s o depend on the s k i n e f f e c t s , but they are l e s s s e n s i t i v e to 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 pulsed case. I t i s l o g i c a l to t r y to extend experiments of t h i s type to the superconducting r e g i o n . However, i n superconductors the s t a t i c magnetic f i e l d only penetrates about 5 0 0 A 0 , w h i l e the r f f i e l d penetrates somewhat f u r t h e r . Experiments using s i n g l e c r y s t a l are thus not p o s s i b l e , except by using f i e l d c y c l i n g techniques i n which they are at a disadvantage compared to powders. However a study of the phase s h i f t s as the metal goes from the anomalous to the superconducting r e g i o n i s q u i t e p o s s i b l e i n metals l i k e vanadium. -Calculations show that there 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 held to w i t h i n 0.01 K. at helium temperatures. Pow-ders might give a b e t t e r S/N r a t i o , but they do not have the 15*+ w e l l defined and e a s i l y studied surface s t r u c t u r e of a s i n g l e c r y s t a l . NMR measurements on p a r t i c l e s of v a r y i n g s i z e s , and a l s o on f i l m s of v a r y i n g thicknesses, have revealed a Knight s h i f t dependence on sample dimensions which i s i m p e r f e c t l y understood. Pulsed NMR w i t h a coherent system on t h i s type of sample might give a d d i t i o n a l i n f o r m a t i o n . However, i t would r e q u i r e a long period of c a r e f u l and ingenious experimentation to even see a s i g n a l . A f t e r the s p e c u l a t i v e nature of the preceeding para-graphs, i t i s f i t t i n g to conclude w i t h some words of caution. Pulsed NMR experiments are d i f f i c u l t , i n v o l v i n g many compro-mises, and have a S/N r a t i o which i s r a r e l y good enough to give an answer of the required accuracy. A c r i t i c a l examination of any prospective experiment i s thus d e s i r a b l e . In p a r t i c u l a r , most experiments 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 only where a n i s o t r o p i c p r o p e r t i e s are involved that a s i n g l e c r y s t a l experiment may be 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 author became aware of the work of Gara (69). He has r e -c e n t l y measured the s p i n - 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 times i n metal s i n g l e c r y s t a l s by a pulsed method, but h i s e x p e r i -mental procedure was so d i f f e r e n t that there has been l i t t l e overlap w i t h t h i s work. He used an incoherent pulse system and a combined t r a n s -m i t t e r - r e c e i v e r c o i l . This l a c k s the v e r s a t i l i t y of a coherent system and i s s u s c e p t i b l e to a m p l i f i e r non-linearity„ I t s u f f e r s from the major disadvantage that i n a T, measurement the amplitude 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 pulse i s not a simple exponential f u n c t i o n of the pulse separa-t i o n , unless the two pulse lengths are i n a c e r t a i n r a t i o which i s determined by a mixture of theory and experiment. The second pulse was about -fit long and gave the maximum f r e e i n d u c t i o n decay amplitude w h i l e the f i r s t pulse was about £mr l o n g , A complicated method of a n a l y s i n g the r e s u l t s was necessary. The r e s t of the pulse c i r c u i t r y , a m p l i f i e r s , and boxcar i n t e g r a t o r were s i m i l a r to those used i n t h i s work. Most of h i s measurements were made at l i q u i d helium temperatures. 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 almost e l i m i n a t e d them by s e v e r a l ingenious techniques. One of these was to coat the sample w i t h a l a y e r of.epoxy r e s i n w i t h nylon f i l i n g s embedded i n i t . This q u i t e 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 defect 156 that the epoxy r e s i n was paramagnetic enough to s i g n i f i c a n t l y shorten the f r e e i n d u c t i o n decay. The other method was to etch 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 . These considerably reduced the a c o u s t i c o s c i l l a t i o n s . I t should be noted that i n s e v e r a l respects h i s a c o u s t i c o s c i l l a t i o n s be-haved d i f f e r e n t l y to those observed i n t h i s work. This i s probably because he mounts h i s sample by cementing one end of i t to a holder, w h i l s t i n the present experiments the centre of the rod i s clamped and the ends are f r e e . He measured T, i n A l a 1 and Cu 6 s s i n g l e c r y s t a l s at h.2°K. 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 o r i e n t a t i o n s . W i t h i n h i s experimental e r r o r of ~2% he detected no anisotropy i n T, . H i s values were T, T = ( l a 8l-0«p '2)see.deg. f o r aluminium and (1.275±O .Ol5)sec.deg. f o r copper. Both of these are i n ex-c e l l e n t agreement w i t h the powder value s . The S/N was about 20 when observed on the o s c i l l o s c o p e . Free 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 which were i n good agreement w i t h values measured by steady s t a t e methods. 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Rev. 133A. 1630 (196*4-). (28) Clogston, A.M., Gossard, A.C., J a c c a r i n o , V. and Yaget, Y., Phys. Rev. L e t t e r s £, 262 (1962). (29) Butterworth, J . , Proc. Phys. Soc. 8£, 735 (1965). (32) Torgeson, D.R., and Barnes, R.G., Phys. Rev. L e t t e r s 255 (1962). (33) Fawcett, E., Advances i n Physics ] J , 139 (196*4-). (3*+) Hewitt, 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 State P h y s i c s , " 2nd Ed., John Wiley and Sons, Inc., New York (1956). (37) Ziman, J.M.,"Electrons i n Metals," Taylor 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 P h y s i c s , " Pergamon Pre s s , Oxford (1959). (39) Kubo, R. and Obata, Y., J . Phys. Soc. Japan 11, 5*4-7 (1956). 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(5D Tables of Eigenvalues and Eigenvectors of the Hamiltonian D e s c r i b i n g the Combined S t a t i c Magnetic Dipole and E l e c t r i c Quadrupole I n t e r a c t i o n s of a Nuclear L e v e l . , S t e f f a n , R.M., Matthias, E. and Schneider, W., A.EG.-' (U.S.A.), D i v i s i o n of Tech n i c a l Information TID-1574-9. (52) Kambe, K. and Ollom, J.F., J . Phys. Soc. Japan 11,50 (1956). (53) Schumacher, R.T., Phys. Rev. 112, 837 (1958). (5*0 Goldberg, W.I., Phys. Rev. ll£, 4-8 (1959). (55) Jeener, J . , E i s e n d r a t h , H. and Van Steenwinkel, 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 Olson, 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) Appel, 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). ( & ) Jones, 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) Jones, E.P., Ph.D. Thesis, 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 Weiss, P.R., Rev. Mod. Phys. 2£, 269 (1953). (67) Miasek, M., Phys. Rev. 1^0, 11 (1963). (68) Karimov, Y.S, and Schlegolev, J.F., JETP JQ> 908 (1961). (69) Gara, A.D., Ph.D. Thesis, 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+<|)) «-$COSlU0t o-"Reference t i t V * V.ft) i i . F i g u r e 1.1 E q u i v a l e n t C i r c u i t of the Phase S e n s i t i v e Detec tor In the phase s e n s i t i v e d e t e c t i o n system to be analysed a sma l l time v a r y i n g s i g n a l of frequency u> and phase angle (j) i s l i n e a r l y added to a much l a r g e r re ference s i g n a l to g ive a t o t a l v o l t a g e v(f) of •v(fl=A(t )eos(u»t+ 4>)+Bcosu>ot = [ A 4 + B* +2ABcos (nt + $ ) cos [w»t -tan"'{ 0 ( t ) } ] , where / I = OJ ~u0 Provided u>o»ft, the time dependent phase angle S ( t ) w i l l cause n e g l i g i b l e frequency or phase modula t ion of the s i g n a l , so tha t the vo l t age a f t e r the s i g n a l has been r e c t i f i e d and the h igh frequency c a r r i e r f i l t e r e d out i s V(t)= +B* +2ABcos(flt+ $ )]* = [ A ( t ) = o s ( n U $ ) + B ] [ l + ^ ± | ] . I f i t i s assumed tha t - ^ - ^ f , the square roo t can be expanded to f i r s t order to g ive • V ( t ) = [A( t)cosUU+( t> )+B] [ l + , & l ? * f t " ; t ( £ L » • The system i s A . C . coupled so the D . C . component i s removed, g i v i n g the output s i g n a l as V 0 ( t ) =A(t)cosCat+ d?) + ^(a^)im^m^U3] . There i s thus an e r r o r except when s i n(ilt+ 4>)=0. This e r r o r can become se r ious f o r s i n(At+ (J) ) ±1 s i nce not on ly i s the e r r o r near i t s maximum va lue of but the s i g n a l Acos(/lt+ (J) )ibO, so the f r a c t i o n a l e r r o r i s very l a r g e . I f the apparatus 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 there 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 . 164-APPENDIX I I DETAILS OF THE SAMPLES USED ( i ) Aluminium S i n g l e C r y s t a l . This was supplied by Semi-elements, Inc., Pennsylvania and is. a c y l i n d e r 5 cm. long by 0.7 cm. i n diameter. There were a c t u a l l y two c r y s t a l s i n the sample, one being about one ei g h t h the s i z e of the other. The surface i s rough and unetched and the p u r i t y i s unknown. Aluminium has a face centred cubic l a t t i c e and a quadru-pole moment. There i s one isotope w i t h s p i n -|> . ( i i ) 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 the U.B.C. Metallurgy Department. The sample i s 5 cm. long and has an average diameter of 0.6 cm. I t had a smooth surface and so was not etched. A chemical a n a l y s i s gave the i m p u r i t i e s i n parts per m i l l i o n as Oxygen 160 Carbon 136 Nitrogen 318 Hydrogen 7.2. Vs' has an i s o t o p i c abundance of 99,7% and forms a body centred cubic l a t t i c e . 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. ( i i i ) 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 Metals Research Co., Cambridge, England. I t has the te t r a g o n a l a x i s perpendicular to the sample a x i s to w i t h i n 2 ° . The 1.3 cm. long c y l i n d e r was etched down to 0.9 cm. diameter by an etch of 16? one p a r t HC1 to twenty p a r t s of e t h y l a l c o h o l . X-raying showed that the etching 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 surface s t r u c t u r e . The main isotope i s In*' 5 which has 96$ abundance, s p i n and a quadrupole moment. The l a t t i c e i s a face centred t e t r a -gonal s t r u c t u r e which can a l s o be regarded as a face centred cubic l a t t i c e elongated by 7% along one a x i s . ( i v ) 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. long by 0.65 cm. diameter purchased from Metals Research Co* which was not etched. The t r i g o n a l a x i s i s w i t h i n 2° of the perpendicular to the c y l i n d r i -c a l a x i s . 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 . This was supplied by Semi-elements and i s 2.8 cm. long by 0.3 cm. diameter. The symmetry a x i s i s aligned to w i t h i n 2° of the perpendicular to 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 not etched. There are two i s o t o p e s , Re"*and Re1*7 w i t h 37$ and 63$ i s o t o p i c abundances r e s p e c t i v e l y . They both have sp i n \ and a quadrupole moment. I t has a hexagonal c l o s e packed s t r u c t u r e and i s a t r a n s i t i o n metal. ( v i ) 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. long by 0.9 cm. diameter cut from a zone r e f i n e d s i n g l e c r y s t a l supplied by Cominco L t d . , 166 T r a i l , B.C. The sample i s 99.999$ pure. The o r i e n t a t i o n of the c r y s t a l i s unknown. The c r y s t a l was etched w i t h a s o l u t i o n of one.part concentrated n i t r i c a c i d to f o u r parts h y d r o c h l o r i c a c i d . This etch i s very f a s t i f the sample i s allowed to become heated. ( v i i ) G a l l i u m S i n g l e C r y s t a l . 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-raying determined that i t was a si 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 to the c y l i n d r i c a l a x i s . I t i s 2 cm. long by 0.9 cm. i n diameter. Both of the isotopes Ga 6 < ? and Ga71 have a quadrupole moment. They are 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 orthorhombic s t r u c t u r e . ( v i i i ) 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. long and 0.6 cm. i n diameter grown by the U.B.C. Metallurgy Department. No e t c h i n g was done on i t . 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 35 Carbon 4-0 Nitrogen 4-0 Hydrogen 2 Tantalum 500 Zirconium 500 Tungsten 300 Nb , a has 100$ l s o t o p i c abundance, a sp i n of , a quadru-pole moment, and c r y s t a l l i z e s i n a body centred cubic l a t t i c e . ( i x ) N a t u r a l T i n S i n g l e C r y s t a l . This sample was supplied by Metals Research L t d . i n the form of a c y l i n d e r 3 cm. long and 0.9 cm. diameter of 99-999$ 167 p u r i t y . The symmetry a x i s i s al i g n e d a t r i g h t angles to the c y l i n d r i c a l a x i s of the sample. An etch of one part HN0 3, one part water, and two parts acetone was used to remove the surface l a y e r p r i o r to the experiments. There are two main i s o t o p e s , each w i t h s p i n These are Sn'^with 7,7% i s o t o p i c abundance and Sn"* w i t h 8*7% abundance. Neither of 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 Olsen (59). I t c o n s i s t s of a l a y e r about 0.25 mm. t h i c k wrapped around a copper core about 7 mm. i n diameter. The p u r i t y of the Sn" 9 i s unknown, but presumably the main impurity would be Sn" 7 . The symmetry a x i s i s at an angle of 28° to the c y l i n d r i c a l a x i s of the copper core. The X-rays taken to determine the p o s i t i o n of t h i s a x i s showed that there was n e g l i g i b l e d i s t o r t i o n of the s i n g l e c r y s t a l and that there was no p o l y c r y s t a l l i n e surface l a y e r . 168 APPENDIX I I I THE SIGNAL INDUCED IN THE PICKUP COIL (a) Non-metallic Sample. Assume that the sample i s i s o t r o p i c and non-ferromagnetic, so that the nuclear magnetic moment at any po i n t i n the sample i s M, =X 0H 0 •where N i s the number of atoms/m3 , X i s the gyromagnetic r a t i o , I the nuclear s p i n , T the sample temperature, H 0 the applied s t a t i c magnetic f i e l d , and X 0 i s the s t a t i c nuclear magnetic sus-' c e p t i b i l i t y . In e q u i l i b r i u m M 0 i s ali g n e d along the z a x i s . A l i n e a r r f magnetic f i e l d B=2B, coswt, at the resonant frequencytu,\applied perpendicular to H„ causes Ms to precess i n the zy plane at a frequency TB, , so that a f t e r a time t i t makes an angle <*= TB.f w i t h the z a x i s ( 1 ) . The p r o j e c t i o n of M p i n the xy plane i s M=M0 sine* In the l a b o r a t o r y frame t h i s magnetic moment r o t a t e s at a f r e -quency w, inducing a s i g n a l p r o p o r t i o n a l to Mcoswt i n a pickup c o i l . Now consider the sample i n s i d e a c o i l w i t h n turns which i s perpendicular to H 0 and has an e f f i c i e n c y f a c t o r 07. The voltage v induced i n the c o i l i s given by Faraday's equation as v =r^E.dl The surface i n t e g r a l i s over the cross s e c t i o n a l area of the c o i l . 169 The l i n e i n t e g r a l i s taken over a path through the sample p a r a l l e l to the c o i l a x i s and then back very close to the outside surface of the c o i l . Along t h i s path £ Me s i n <A c o s ivt i n the sample and i s zero outside i t , i f end e f f e c t s are neglected. .*. v^ori^nMoAsinricosu/t, where A i s the cross s e c t i o n a l area of the sample, (b) M e t a l l i c Sample. The problem i s to c a l c u l a t e the d i s t r i b u t i o n of currents and magnetic f i e l d s induced i n the metal by the o s c i l l a t i n g mag-n e t i c moment M. This induces a c i r c u l a t i n g c u r r e n t J=°% which generates a magnetic f i e l d H opposing M,. The c i r c u l a t i n g current i s given by Faraday's equation as c r V x E ^ ^ C M + H j , w h i l e H i s given by These equations n e g l e c t the displacement'current and the magnetic moment X $ induced by the c i r c u l a t i n g c u r r e n t . They a l s o assume that the metal i s not i n the anomalous conduction r e g i o n . Stan-dard vector manipulation now gives Assume that M^goCexpCiu/t). V * H = 1 S"a(M+H), wkere V a = cv^cr. Now consider an i n f i n i t e plane metal sample w i t h magnetic moments M x(z) at a depth z below the surface and w i t h M^=Mt= 0. The equation s i m p l i f i e s to =Aa [ H , ( z ) + M s ( z ) ] , 170 where A = f' + . The s o l u t i o n of t h i s equa t ion i s (1*0 {50 H X (z) =exp (Az) [c* +£AJM( Z ) exp (-Az) d zj + e x p ( - A z ) [ C 3 - £ A | M ( z ) e x p ( A z ) d z ] . 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 cons tan t s . The boundary c o n d i -t i o n s are now f i t t e d f o r the s p e c i a l case of an i n f i n i t e l y t h i n sheet of magnetic moments a t a depth m below the su r f ace . i . e . M x ( z ) = £>(m-z), where S(m-z) i s a D i r a c d e l t a f u n c t i o n . H x ( z ) =C a exp(Az)+C 5 exp(-Az) f o r z < m, = [CA +£Aexp(-Am)] exp(Az)+[c>-£Aexp(Am)] exp(-Az) _ f o r z > m, | H 3 C ( z ) d z =j [ca exp(Az)+C 3 exp(-Az)] dz+ ] f(CA+-|-Aexp(-Am)) J0 JO *" • -expCAzJ + CCj - iAexp(Am) )exp(-Az) j dz = •^Jo aexp(Az)-C3exp(-Az)J + J^-^+£exp(-Am)j- exp(Az) -{•^--iexp(Am)} exp(-Az) ]^ . To keep the t o t a l f l u x f i n i t e i t i s necessary to put C a =-|-Aexp(-Am). r* r H x ( z ) d z = £ e x p ( - A m ) - l + - g -The s i g n a l induced i n the p ickup c o i l i s obtained from. y x g - / ^ ( M + H ) . The assumption i s made tha t S « R , where R i s the r a d i u s of the c y l i n d r i c a l sample. The v o l t a g e induced i n a p ickup c o i l wound round the sample i s v *-i2rfRynjuu> [ M x ( Z ) + H J C ( z ) J d z . .'. vcG'lexp (-Am) + . I f the magnetic moments are a t a l a r g e depth below the surface of 171 the metal there should be no induced voltage because the c i r c u -l a t i n g currents w i l l , by Lenze's Law, completely s h i e l d the magnetic moments. i . e . 0+ tX^ [-^ - Hexp(-Am)] . C 3 =0. v 0 s-itp^n^cuRexpC-Am). This i s the voltage induced by a d e l t a f u n c t i o n magnetic moment at a depth m below the surface. The voltage induced by a d i s -t r i b u t i o n M(m) of magnetic moments i n the metal i s given by v = I v0(m)M(m)dm f * = TTnyu«ntuRJM(z)exp[-(l+i)^-]dz.. I t i s now necessary to c a l c u l a t e M(z) a f t e r a r f magnetic f i e l d . aBiexp(itut) has been a p p l i e d p a r a l l e l to the surface of the sample f o r a period t . For s i m p l i c i t y , assume that to i s the nuclear resonant frequency. The magnetic f i e l d a t a depth z i n the metal i s (3) B(z) = 2 B , e x p [ - ( l + i ) ^ ] e x p ( i c o t ) . In the r o t a t i n g reference 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 perpendicular to H 0 at an angle tan" 1 (vft-) to the f i e l d B, (0) at the surface of the metal. The magnetic moment M ( z , t ) at r i g h t angles to B,(0) i s M s ( z , t ) = M , ( t ) s l n [ T B l t e x p ( - ^ ) ] c o s ( ^ ) . .*. v ( t ) =TTywtu ny RMo ( t ) £exp(-^)sin(YTB, e x p ( - ^ ) ) c o s a ( ^ ) d z . This expression can be generalised to the case when the applied r f pulse i s o f f resonance and the phase reference a x i s makes an a r -b i t r a r y angle w i t h B , ( 0 ) . An important f e a t u r e of t h i s equation 172 i s tha t the induced vo l t age i s of the form M x ( t ) F ( z ) , where M x ( t ) i s the nuc l ea r magnetic moment a t the surface of the sample t seconds a f t e r a p p l i c a t i o n of an r f pu lse and F ( z ) i s the i n t e g r a l 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 . The time e v o l u t i o n depends only on Ti and T a so tha 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 ive the same va lues as those made on powders. In p a r t i c u l a r the f r ee i n d u c t i o n decay i s not d i s t o r t e d by any phase e f f e c t s . Th i s d e r i v a t i o n has ignored the d i p o l a r magnetic f i e l d . Th i s i s v a l i d i n the r e g i o n i n which the r f f i e l d i s much grea ter than the d i p o l a r f i e l d and from which the m a j o r i t y of the s i g n a l comes. The only measurements which might be a f fec ted by the p re -sence of the d i p o l a r f i e l d are those i n v o l v i n g r e l a x a t i o n i n the r o t a t i n g reference 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 exp(-x) sin(T'£B (e'" , t)cos a 'xdx can be eva-lua t ed by expanding sin(lPTB,e*) i n a s e r i e s and i n t e g r a t i n g term by term. T h i s g ives Th i s s e r i e s i s q u i t e s a t i s f a c t o r y f o r numer ica l computation u n t i l TB,-rt"-V 10 r ad ians when the number of terms, 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 inc reases 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 rea ted 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 surface and a r a d i u s much grea te r than i t s s k i n depth . Th i s i s a good approximat ion to the r e a l samples, provided that they have no surface i r r e g u l a r i t i e s whose dimensions are a p p r o x i -mately equal 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)coscj0t be fed into a phase sensitive detector using a ref-erence frequency w8 and then rec t i f i ed 0 The output signal is proportional to G(t) [ cos(flt)cos4>+sin(flt)sin(j) ] s where is the phase difference between the reference frequency and the induction signal, and f l = u>-U)0o The time origin for the signal is the midpoint of the rf pulse (13)? not the end as stated by <-THe rf Pulse *rfrft) Clark ( 2 ) 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 is V = K G(t) [cosAtcos()) + sinfit sinCJ)] dt, Jo where K is a constant. For most free induction decays "X"(fL)oC G(t)cosntdt and X (fl )oC\ G(t) sin(flt)dt, where X and X are the real and imaginary parts of the r f susceptibi l i ty . .A is proportional to the absorption mode measured by steady state apparatus. V o C ' X ' t f D c o s ^ + X ' C n ) s i n $ . By correctly setting (()either X ' o r X ' can be obtained. Linearly sweeping the magnetic f i e ld through resonance then gives "X' or 17h "X." on the cha r t r e c o r d e r C l a r k de r ived the equat ions f o r the i d e a l case o u t l i n e d above. However, there are a cons ide rab l e number of i n s t rumen ta l d i s t o r t i o n s which l i m i t the usefu lness of t h i s method,, ( i ) Because of the f i n i t e w id th of the r f p u l s e , and the dead time of the appara tus , the boxcar gate s t a r t s at a t i m e . ' t ' s o tha 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 ) C O 8 $ -rX/Cfl ) sin<J>] -K 'paCt) [ cos(nt ) eos$ +sin(Qt)sin(j) ] d t . Unless G( t ) i s known, the c o r r e c t i o n term cannot be eva lua t ed . However, i t i s easy to see the e f f e c t of the i n s t rumen ta l d i s t o r -t i o n when s i n (j)=0. V v . y y y-~M—Start o-f iVie Boxccr Grate. A x / / / ' " -/, Close to ftesoner^V--" - i£' ^ / Tirrvg ^ _ ^ ^OfF Resonance F i g u r e I V . 2 E f f e c t of the Deadtime C lose to resonance the unmeasured area decreases qu i t e s l o w l y w h i l e the measured area decreases qu i t e r a p i d l y , s ince i t i s i n the t a i l of the decay which i s much more s e n s i t i v e to s l i g h t s h i f t s o f f resonance. Thus there 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 lo se to resonance. However, a t a frequency (4T*) ' o f f resonance one p o s i t i v e quar te r c y c l e i s unmeasured so tha t the output goes n e g a t i v e . Th i s causes the a b s o r p t i o n s i g n a l to be narrowed. 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 tha t the output w i l l c o n t i n u a l l y 175 o s c i l l a t e o Th 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 requirement 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 to have the f requenc ies f 0 ± ( t ' ) ' a t which the output goes nega t ive f a l l i n g w e l l ou t s ide the l i n e . - i . e . T* »T ' . ( i i ) I f X"(fl) i s to be the u n d i s t o r t e d a b s o r p t i o n mode, G(0) must be independent of f l f o r a l l va lues of f l w i t h i n s e v e r a l l i n e w i d t h s of the resonance. Th i s r e q u i r e s tha t over t h i s r e g i o n , H, =£= H e i n the r o t a t i n g reference frame. The c o n d i t i o n f o r t h i s i s t ha t TH, » T."o This i s the normal c o n d i t i o n f o r pulsed NMR and i s a c t u a l l y i m p l i e d by the p rev ious c o n d i t i o n T > » t ' 9 s ince the sp ins must be ro ta ted through 90° i n a time l e s s than T ' . ( i i i ) 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-para tus a l s o causes some d i s t o r t i o n . The inpu t to the boxcar i n t e g r a t o r i s (Appendix I) V (t)=G(t ) cos ( f l t+ <J> ) + sm*ffltt$)C&ft)cosfllt+tfl-i-B3 ,« a[c.osf.nT.+$")-t-i^ p I f 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) there i s no d i s t o r t i o n , but as soon as the frequency i s o f f resonance (fl#o) the output i s d i s t o r t e d , g i v i n g a b a s e l i n e s h i f t . An exact c a l -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 ince the boxcar gate p a r t i a l l y i n t e g r a t e s i t . I t tu rns out tha t the b a s e l i n e s h i f t i s f a i r l y 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 t r o u b l e . I f d e s i r e d , 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 of l80° between them ? and then s u b t r a c t i n g one from the o the r . Changing $ by 180° reverses 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 9 to a f i r s t approximations, does not a f f e c t the b a s e l i n e s h i f t , which Is thus e l i m i n a t e d by the s u b t r a c t i o n . A P P E N D I X V C I R C U I T D I A G R A M S 0-O5 To Tower Amplifier &crte Pube In V-I70V. Figure 51.1 q. T h e Grated C o h e r e n t Oscillator. 177b 178 I2AX7 ECC82 From the Boxcar Otate (O W A V *--11/ n (g A A A A A A -M 1 4 H h Frwnthe Timer Tulse. Grenerator FDIOD fl-4* PPIOO ? 7> >/aa 4, ^ To Event Marker ure The Coincidence Timinq Unit. 183 3XOca9 F i g u r e V . 7 Regulated F i l amen t Power Supply 

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