Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Pulsed nuclear magnetic resonance in metal single crystals Apps, Michael John 1971

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

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1971_A6_7 A66.pdf [ 6.17MB ]
Metadata
JSON: 831-1.0084871.json
JSON-LD: 831-1.0084871-ld.json
RDF/XML (Pretty): 831-1.0084871-rdf.xml
RDF/JSON: 831-1.0084871-rdf.json
Turtle: 831-1.0084871-turtle.txt
N-Triples: 831-1.0084871-rdf-ntriples.txt
Original Record: 831-1.0084871-source.json
Full Text
831-1.0084871-fulltext.txt
Citation
831-1.0084871.ris

Full Text

PULSED N U C L E A R MAGNETIC RESONANCE IN M E T A L SINGLE CRYSTALS by MICHAEL JOHN APPS B.Sc. , University of British Columbia, 1966 A THESIS SUBMITTED IN P A R T I A L F U L F I L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E O F M A S T E R OF SCIENCE in the Department of Physics We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA Apr i l , 1971 Iii presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Colunbia, I agree that the Library shall E a k e i t freely available for reference and study, I further agree that permission for extensive copying of this thesis for soholarly purposes nay be granted by the Head of ay Departaent or by his representative. It is understood that copying or publication of this thesis for financial gain shall not be allowed without ay written permission. MICHAEL JOHN APPS Department of Physics The University of British Columbia Vancouver 8, B.C. April 14, 1971 A b s t r a c t The study of pulsed n. m. r . i n single c r y s t a l m e t a l l i c samples, i n i t i a t e d by M c L a c h l a n , has been extended to l i q u i d 1 1 9 h e l i u m t e m p e r a t u r e s with s p e c i a l emphasis onSn . C o n t r a r y to M c L a c h l a n ' s b e l i e f i t was found that cooling to 4. 2°K and lower afforded si g n i f i c a n t i m p r o v e m e n t s to the s i g n a l to noise rat i o and i n many cases the n. m. r . signals (including spin 119 echoes i n Sn ) could r e a d i l y be seen on an o s c i l l o s c o p e without the use of a signal a v e r a g e r . The theory of magnetic resonance i n m e t a l l i c samples was studied i n some d e t a i l with p a r t i c u l a r emphasis on the e x p e r i m e n t a l situation where m a t t e r s are c o m p l i c a t e d by the high conductivity which modulates both the amplitude and the phase of the exciting r . f . magnetic f i e l d as i t penetrates into the sample. It i s shown t h e o r e t i c a l l y that s e v e r a l assumptions must be made to show that the conventional methods of pulsed n. m. r . used to m e a s u r e T (by either spin echo or f r e e induction decay) and T^ y i e l d true meaningful r e s u l t s . In p a r t i c u l a r i t i s found that the spin l a t t i c e r e l a x a t i o n time T^ i s obtained by the conventional two pulse sequence only when the magnetic f i e l d i s exactly on resonance; this was o b s e r v e d to be the case e x p e r i m e n t a l l y as w e l l . 1 1 9 In sharp d i s t i n c t i o n to M c L a c h l a n ' s findings f o r Sn , the s p i n - s p i n r e l a x a t i o n time T obtained by F I D methods (175 ± 18Lisec) was much s m a l l e r than that obtained by spin echo techniques (390 ± 48u,sec) in the present research 119 on Sn . The spin-lattice relaxation time was also measured at liquid helium temperatures and yielded a value of 56 ± 4 millisec deg for T^T, in excellent agreement with Dickson although twice as large as McLachlan's value. Acknowledgement To acknowledge all the help given to one on any research project can never be easy but yet must always be a pleasure. To all those at the University of B.C. and at the University of Bristol (where this thesis was written) who have in one way or another encouraged and assisted me during the work herein described I tender my most sincere thanks. Special honours must be paid to Dr. D.L1. Williams; never has there been a supervisor with such unfailing patience and sympathetic under standing. The financial support by the National Research Council of Canada in the form of graduate scholarships during the research is gratefully acknowledged. Page A B S T R A C T K A C K N O W L E D G E M E N T S i i : T A B L E O F C O N T E N T S i v L I S T O F I L L U S T R A T I O N S v C H A P T E R I : I N T R O D U C T I O N 1 C H A P T E R II : T H E O R Y O F N . M . R . IN M E T A L S 12 A , The Resonance C h a r a c t e r i s t i c s 12 (i) The effective field at the nucleus; Knightshift 15 (ii) Spin Lattice relaxation mechanisms 18 (iii) Spin-spin relaxation mechanisms 23 B . The E x p e r i m e n t a l Technique for Metals 29 (i) M o d e l of the metal and method of solution 31 (ii) The magnetization induced i n AV^ 32 (iii) The voltage induced i n coi l by AV^ 38 (iv) The total voltage induced in the coil 41 (v) Spin-spin relaxation time measurements 42 (vi) Spin-lattice relaxation time measurements 43 C H A P T E R HI : T H E A P P A R A T U S A N D T H E M E T H O D 48 A . The Apparatus 48 (i) The electronics 48 (ii) The cryogenic system 54 B . The Method 57 (i) Measurements of Absorption and D i s p e r s i o n 57 modes (ii) T measurements 58 (iii) T ^ measurements 60 C H A P T E R IV : E X P E R I M E N T A L R E S U L T S 80 A . Qualitative Results 81 (i) Apparatus 81 (ii) Magnetoacoustic oscillations 82 (iii) Other m a t e r i a l s , temperature dependence of S / N 83 , (iv) Descript ion of types of measurements 85 119 B . Quantitative Results : Isotopically pure S n 91 (i) Spin-lattice relaxation times 9It (ii) Spin-spin relaxation times 92 B I B L I O G R A P H Y 106 LIST O F ILLUSTRATIONS AND FIGURES Page Fig . III. 1 Block diagram of the spectrometer 64 Sch. 1 Schematic of repetition rate timer 65 F ig . III. 2 Block diagram of pulse sequence generator 66 Fig . III. 3 Ramp and trigger waveforms of timer unit 68 Sch. 2 Schematic of the oscillator and gate circuit 67 Sch. 3 Schematic of the r. f. power amplifier 69 Sch.4 Schematic of the preamplifier 70 Sch. 5 Schematic of the boxcar integrater 71/72 F ig . III. 4 Cryogenic system 73 Fig . III. 5 Details of the sample holder 74 119 Fig . III. 6 Typical T chart record (S n ) 75 F ig . III. 7 Typical T^ semilogarithmic plot 76 F ig . III. 8 Signal and reference waveforms 77 Fig . III. 9 Typical F . I . D. T chart record 78 F ig . HI. 10 Typical F . I . D . graph 79 63 65 Fig . IV. 1 Oscillograph of Cu and Cu resonances 96 F ig . IV. 2 Oscillograph of teflon resonance at room temperature and at 1.3°K 97 Fig . IV. 3 Niobium resonance at room temperature 98 Fig . IV. 4 Niobium resonance at liquid nitrogen temperature 98 F ig . IV. 5 Niobium resonance at 4. 2°K 99 F ig . IV. 6 Niobium resonance at 1 .3°K 99 Fig . IV. 7 S n 1 1 9 T chart record at 4. 2°K 100 119 o Fig . IV. 8 Oscillograph of Sn spin echo at 1.24 K 101 Fig . IV. 9 Semilogarithmic plot of echo T points 102 c* Fig . IV. 10 Chart record of spin echo shape 103 119 Fig . IV. 11 Field sweep (absorption mode) of Sn spin echo 104 119 Fig . IV. 12 T^ decay in Sn showing deviation from exponential dependence 105 Before looking at the particular aspects of nuclear magnetic resonance in metals, we shall review briefly some of the basic ideas involved in pulsed n. m. r. We first look at magnetic resonance qualitatively introducing the concepts of spin-lattice and spin-spin relaxation times and then proceed to solve the basic classical equations of motion for free spins. Drawing upon these results, we shall proceed in the following chapter to discuss the resonance characteristics peculiar to metallic single crystal specimens and the difficulties which arise when we attempt to measure the various resonance parameters. A l l nuclear magnetic resonance experiments involve the excitation of the nuclear spins from a given state to one of higher energy and the way in which the excited nuclei relax back to their original states. The 21+1 Zeeman energy levels of a spin I_ in a d. c. magnetic field being given by E = yti I H > I = -I, z o z —I "§*1, 4"I, it is possible to induce nuclear spin transitions by applying a time dependent magnetic perturbation at the resonant frequency W = AE/ft = -vH . In the absence of any o o such perturbation, the population of these Zeeman levels for a system of N spins in thermal equilibrium with their surroundings (called the lattice) at temperature T is given classically by the Maxwell-Boltzmann formulae. Thus the net magnetization for such a system in thermal equilibrium is given by: +1 yh I 7 H n / K T M = x H = N ~ - yfi £ I —o \) —o H I = -I 5 z o e z +1 yh I H / K T z o £ e I = -I z or f o r yh I H / K T « 1 z o to a good approximation by: M «.N y Z f i 2 l ( l + 1 ) H . 3 K T ~ ° A p e r t u r b a t i o n applied at the resonant frequency W =-yH fo r a time t = o to t = f, w i l l f o r c e t r a n s i t i o n s between the Z e e m a n l e v e l s d i s t u r b i n g the d i s t r i b u t i o n f r o m that d e s c r i b e d by the M a x w e l l - B o l t z m a n n f o r m u l a e . Thus at the time T , the net magnetization need no longer be aligned p a r a l l e l to H . A t the same time as the e x t e r n a l p e r t u r b a t i o n i s f o r c i n g t r a n s i t i o n s f r o m lower to higher energy l e v e l s , the coupling (that must e x i s t i n any r e a l m a t e r i a l ) of the spins with each other and with the la t t i c e i s giving r i s e to spontaneous downward t r a n s i t i o n s . In p a r t i c u l a r the coupling of the spins with the latt i c e at temperature T always tends to r e t u r n the spin population to i t s t h e r m a l e q u i l i b r i u m d i s t r i b u t i o n ; s u f f i c i e n t l y long after the e x t e r n a l p e r t u r b a t i o n has been turned off, the excited spin s y s t e m w i l l r e t u r n to t h e r m a l e q u i l i b r i u m ^ and exhibit the e q u i l i b r i u m magnetization given by I. 1 . The c h a r a c t e r i s t i c time for this decay, c a l l e d the sp i n - l a t t i c e r e l a x a t i o n time, i s thus a measure of the coupling that exists between the spin s y s t e m and the surrounding l a t t i c e . E x p e r i m e n t a l l y i n many cases, and t h e o r e t i c a l l y i n quite a few, 1 . F o r no n - f e r r o u s m a t e r i a l s , the assumption i s made that this ground state c o r r e s p o n d s to _M_ p a r a l l e l to H - independent of po s i t i o n within the sample. A n o t h e r way of stating the same thing i s to r e q u i r e that there be no h y s t e r e s i s i n the magnetization over time s c a l e s l a r g e r than T^. i t can be shown that the decay can be adequately d e s c r i b e d by a simple exponential with a unique time constant T ^ ^ . In addition to the s p i n - l a t t i c e i n t e r a c t i o n s , there ex i s t s i n m ost s o l i d s , and c e r t a i n l y most metals, v e r y strong i n t e r a c t i o n s between the i n d i v i d u a l n u c l e a r spins. T h i s tight coupling means that there can be a v e r y r a p i d t r a n s f e r of energy between the nucl e a r spins within the system. Thus when the ext e r n a l e x c i t a t i o n has been turned off, the spin s y s t e m can achieve an i n t e r n a l t h e r m a l e q u i l i b r i u m at a "t e m p e r a t u r e " T , the " s p i n temperature", i n a c h a r a c t e r i s t i c time T^ v e r y much sh o r t e r than the s p i n - l a t t i c e time constant T . When such a concept of (2) spin temperature i s v a l i d v i t i s p o s s i b l e to view this spin temperature T , est a b l i s h e d i n time T , as decaying to the S Ci l a t t i c e temperature T with an exponential time constant T^. The f i r s t step i n t r y i n g to d e s c r i b e the behaviour of the s y s t e m of spins i s to attempt to d e r i v e an a n a l y t i c a l e x p r e s s i o n for the net magnetization M i m m e d i a t e l y after a p p l i c a t i o n of the e x t e r n a l p e r t u r b a t i o n . We thus c o n s i d e r a sample containing N n u c l e a r spins i n a steady magnetic f i e l d H , turn on an r . f. magnetic f i e l d f o r a time t = o to t = ti and ask: what i s the resultant magnetization at time t = X.'-? It must be noted that the i m p l i c i t assumption i s u s u a l l y made that each spin i n the s y s t e m " s e e s " the same applied f i e l d s H and H^. If this were not the case, one method of approach would be to subdivide the total s y s t e m into subensembles of spins each having this p r o p e r t y although different subensembles may 1. Ref. 1. 2. See, f o r example, ref. 1 and ref. 40. see d i f f e r e n t applied f i e l d s ; the following a n a l y s i s may then be applied to each of the ensembles. T h i s method of approach w i l l be needed i n the next chapter when we analyse the e x p e r i m e n t a l techniques of n.m. r . i n single c r y s t a l s of m e t a l l i c s p e c i m e n s ^ . It i s now a ssumed that the e x t e r n a l p e r t u r b a t i o n - H^.M i s much stronger than the s p i n - l a t t i c e and s p i n - s p i n i n t e r a c t i o n s so that during the exc i t a t i o n (0 ^ t <, T ) the spin s y s t e m may be w e l l approximated as a c o l l e c t i o n of f r e e , n on-interacting spins. C l a s s i c a l l y the equation of motion i n the stationary l a b o r a t o r y N f r a m e of the magnetic moment _M = y h 1^= y fi S _I. i n a j = 1 3 magnetic f i e l d H i s found by equating the applied torque to the rate of change of angular momentum: dM — = yM x H . (I. 2) T r a n s f o r m i n g to a f r a m e rotating at an angular v e l o c i t y (ju : SM St = yM x { H + w/y} . (1.3) A We see that i f H i s a steady f i e l d H= H ."k". , by choosing — — o = (JD = -yH , the magnetization M becomes a constant vector; o o — BM/dt = 0. F o r this p a r t i c u l a r case, the magnetization would thus p r e c e s s at the L a r m o u r frequency yH about H i n the l a b o r a t o r y f r a m e . If now, at time t = 0, a s m a l l rotating f i e l d of frequency uu were applied i n the plane p e r p e n d i c u l a r to H , choosing 1. It i s i n t e r e s t i n g to note that this i s a l s o e s s e n t i a l l y the technique used by Hahn and by C a r r and P u r c e l l when f i r s t d i s c u s s i n g spin echoes (ref, 18 and ref. 10). T h e r e they subdivide into subensembles of spins each having the same d. c. f i e l d H . — z the x' axis of the rotating frame along at time t = 0, in that frame rotating at angular speed (ju the equation of motion 1.3 becomes: —— = Y M x VH + uu/v) K + at i o or 9M = V M x H at — - i e f f -(1.4) where —eff X H +U)/Y1 K +H£' \ o ' i -(I. 5) Thus the effective magnetic field seen by the magnetic moment in the rotating reference frame is time independent and so the magnetic moment M will pre cess about H ^ with an angular speed y| H ^ | . In the laboratory frame an observer would see this precession about H r r superimposed on the e f f precession about H at angular velocity rju. Application of a little standard geometry will show that the angle between eff H and H is given uniquely by: — O —off sin 9 H H 1 eff cos H + uu/y o H eff Moreover if the magnetic moment were initially aligned along H (M = M(o) K) then after a time t i t would have — o —o the vectorial components: M x,(t) M y I (t) M,,(t) = M(o) sin 0 cos 9 (1 ~ cos j ) s inj sin 9 2 2 cos J sin 9 + cos 9 (1.7) in the rotating reference frame where ^ = - H^^ t In p a r t i c u l a r the angle -1 ( " M ( T ) . H 1 a = cos J — v ' — o y 1 M ( f ) H ^o between and the magnetization M after the r . f . has been applied f o r a time T. i s given by: 2 2 cos a = c o s j s i n 0 + cos 8 ° r 2 2 cos a = 1-2 s i n 9 s i n [ i v H „ • x l . L 2 1 eff J T may be defined as that pulse length which gives r i s e a to a tipping of the magnetization through an angle a i n the rotating r e f e r e n c e f r a m e . F o r example i f the r . f . magnetic f i e l d H were applied for a time T • such that cos a = 0, 1 ir/ 2 then the magnetization at the end of the pulse would l i e e n t i r e l y i n the plane p e r p e n d i c u l a r to the d. c. f i e l d H . ( F o r obvious reasons such a pulse i s c a l l e d a it/2 pulse). F r o m equation I. 8 i t i s c l e a r that the pulse length T , •A-, a n e c e s s a r y to induce a tipping through an angle a depends through H . r and 8 on both the frequency d i f f e r e n c e cu — tu eff o and on the magnitude of the applied f i e l d s and H . That i s to say that T depends on the p a r a m e t e r ! ^ H,> H , cu and cu. a l o o While at pr e s e n t we have r e s t r i c t e d o u r s e l v e s to a spin s y s t e m i n which there i s no sp a t i a l v a r i a t i o n of these p a r a m e t e r s , i t must be noted that i n g e n e r a l a l l these p a r a m e t e r s , with the po s s i b l e exception of cu w i l l have s p a t i a l dependence so that the p i c t u r e of an angular pulse (so commonly used) i s not  1. In the m o d e l of free spins the d i s t i n c t i o n between cu and H i s of cour s e a r t i f i c i a l since cu = -yH . However when i n t e r a c t i o n s Eetween spins are allowed, there i s the p o s s i b i l i t y of s p a t i a l l y v a r y i n g f i e l d s due to these i n t e r a c t i o n s contributing to the magnetic f i e l d at the nu c l e a r site; H = H + AH. Thus there w i l l be a spread i n L a r m o u r fr e q u e n c i e s which i s not due to any spa t i a l v a r i a t i o n i n the applied f i e l d . such a clearly defined concept: a must be replaced by some sort of average over all the angles a (uu , H , m , H,) o o 1 1 resulting from a given pulse length T throughout the specimen. We shall be forced to return to these problems later when we consider in detail the experimental situation in metals; for the present we assume f is well defined. a Once the magnetic moment has been tipped to its new position and the perturbation turned off, if no mechanism existed for the nuclear spins to give up their increased energy, it would continue to pre cess about H at angle a and angular speed — o yH Q indefinitely. In a real piece of material, however, as has already been noted, mechanisms do exist for this transfer of energy - spin-spin and spin-lattice relaxation can take place. The tendency of the z-component of the net magnetization to relax to its thermal equilibrium value M= M £ given by I. 1 — z can often be described by a single exponential decay dM /dt = - (M - M )/T, z z o 1 Similarly the transverse components often decay according to dM /dt = - M /T ; dM /dt = - M /T . x x 2 y y 2 In the presence of both a direct field H and a small r.f. field — o Hj, the assumption is usually made that the overall motion can be expressed by the superposition of these relaxation equations on the free spin equation I. 2. giving the well-known Bloch equations: D M M l + M ? M - M f (i = ywl x H - x y - z o_ k VA dt T 2 TL R e l a x a t i o n due to f i e l d inhomogeneities: spin echoes. C o n s i d e r i n i t i a l l y a free non-inter acting spin s y s t e m which at t = 0 i s i n t h e r m a l e q u i l i b r i u m with the l a t t i c e . In the presence of a steady homogeneous magnetic f i e l d H the i n d i v i d u a l spins w i l l a l l p r e c e s s about H with the same L a r m o r frequency U) = v r l . The a p p l i c a t i o n of an intense r . f . f i e l d at this frequency w i l l f o r c e the p r e c e s s i o n a l motion of each spin into phase with each other and the applied f i e l d and excites i n d i v i d u a l spins to higher energy Z e e m a n l e v e l s i n the p r o c e s s . T h e r e i s thus induced a net m a c r o s c o p i c magnetization, the ensemble average of the i n d i v i d u a l spin magnetic moments. The t r a n s v e r s e component of this magnetization thus p r e c e s s e s about H with constant amplitude and at the p r e c e s s i o n a l frequency cu of i t s component spins. If on the other hand the magnetic f i e l d H i n c l u d e s a (1) s m a l l s p a t i a l inhomogeneity* ' over the spin system: HQ ( £ ) = S o ( r ) + J\.(r) A « H q (I. 10) then there w i l l be a c o r r e s p o n d i n g s p a t i a l d i s t r i b u t i o n of the 1. The sp a t i a l inhomogeneity can a r i s e f r o m s e v e r a l sources; i t can f o r example be due sol e l y to the applied magnetic f i e l d (poor magnet) or could a r i s e f r o m surface s t r a i n s on the sample. The important d i s t i n c t i o n between this inhomogeneity and the " l o c a l f i e l d s " a r i s i n g f r o m s p i n - s p i n coupling i s that the f o r m e r i s completely independent of the spin orientations so that i t i s incapable of effecting an energy t r a n s f e r between the spins (which would be a thermodynamic a l l y i r r e v e r s i b l e p r o c e s s ) . L a r m o r f r e q u e n c i e s of the i n d i v i d u a l spins i n the system: JQ ( £ ) = ^ + A(r) ( I (A) If an r . f . f i e l d i s applied at the mean resonant frequency uu , the ° (1) i n d i v i d u a l spins are again f o r c e d into coherence with the r . f . ' and there again r e s u l t s a net magnetization at the co n c l u s i o n of the r . f . pulse. The t r a n s v e r s e component, p r e c e s s i n g about H at the mean frequency cu now however begins to decay i n amplitude as the component spins, p r e c e s s i n g at di f f e r e n t L a r m o r f r e q u e n c i e s (2) s t a r t to get m o r e and m o r e out of phase with each o t h e r v . T h e r e i s thus, even i n the case of free spins, a r e l a x a t i o n of the t r a n s v e r s e magnetization when magnetic f i e l d inhomogeneities are allowed. It i s however important to note that, unlike the r e l a x a t i o n due to s p i n - s p i n coupling m echanisms, there i s no t r a n s f e r of energy between the i n d i v i d u a l spins of the system: each spin maintains i t s excited Z e e m a n state; the spin s y s t e m does not r e l a x to i n t e r n a l e q u i l i b r i u m at a common spin temperature due to magnetic f i e l d inhomogeneities. The i m p l i c a t i o n of this fact i s that the phenomenon i s a r e v e r s i b l e p r o c e s s . T h e r e a r e , i n fact, two ways of r e v e r s i n g the t r a n s v e r s e decay due to inhomogeneity of the d i r e c t f i e l d . If one were able to r e v e r s e the magnetic f i e l d H and TV. (£) at time t , then the motion of a l l the i n d i v i d u a l spins would a l s o be R r e v e r s e d and c l e a r l y at time 2t the spins would r e t u r n to R th e i r i n i t i a l d i s t r i b u t i o n - a l l i n phase. A second method, much (3) s i m p l e r e x p e r i m e n t a l l y , i s to apply a TT pulse at time t . T h i s pulse* ' R 1. In the rotating r e f e r e n c e f r a m e the r . f . f i e l d i s e f f e c t i v e l y the only f i e l d seen by the spins so that "intense r . f . f i e l d " means that i t be much l a r g e r than any inhomogeneities A. (r) i n the d i r e c t f i e l d . 2. A s the i n d i v i d u a l spins get out of phase with each other, so the net t r a n s v e r s e component, which may be r e g a r d e d as the vec t o r sum of the t r a n s v e r s e components of the i n d i v i d u a l spins, begins to decay u n t i l , when the phase d i s t r i b u t i o n of the spin s y s t e m i s completely random, i t d i s a p p e a r s completely. 3. Ref. 19 and Ref. 10. has the effect of i n v e r t i n g the phase shifts between the v a r i o u s spins. Thus i f spin A has got (tu . ~ tu „) t_ degrees ahead i n x oA oB R phase of;spin B i n i t s p r e c e s s i o n a l motion at to . i n time t , oA R then after the TT pulse, spin B w i l l be (to „ - cu ) t ahead of v oA oB' R spin A . Thus at time 2t , spin A w i l l have caught up to spin B R again and the spins w i l l again be i n phase giving r i s e to the same m a c r o s c o p i c t r a n s v e r s e magnetization at time 2t as R was p r e s e n t i m m e d i a t e l y after the i n i t i a l exciting pulse. T h i s reappearance of the t r a n s v e r s e magnetization i s c a l l e d a spin echo and f o r the case of f r e e spins can be made to reappear i n d e f i n i t e l y at ti m e s 2t , 4t , 6t ... by s u c c e s s i v e a p p l i c a t i o n R R R of TT pulses at times t , 3t , 5t ... R R R If the spin s y s t e m i s not f r e e but has, due to s p i n - s p i n coupling m e c h a n i s m s as p r e v i o u s l y d e s c r i b e d , an inherent line width (or d i s t r i b u t i o n of L a r m o r fr e q u e n c i e s i n field) and thus an inherent ( i r r e v e r s i b l e ) r e l a x a t i o n time T , the obse r v e d rate of decay of the t r a n s v e r s e magnetization following a single pulse would be the sum of the r e l a x a t i o n rates due to f i e l d inhomogeneities and the s p i n - s p i n coupling. If the r e l a x a t i o n rate due to the f i e l d inhomogeneity was made much l a r g e r ^ than the true s p i n - s p i n r e l a x a t i o n rate then the echo induced at time 2t by a TT pulse at t would have decayed f r o m the i n i t i a l R R (2) amplitude a c c o r d i n g to the s p i n - s p i n r e l a x a t i o n function . F o r 1. T h i s may be achieved by moving the sample into an inhomogeneous r e g i o n of the magnet, for example, 2. See chapter I I A ( i i i ) . the common case of an exponential r e t u r n to i n t e r n a l e q u i l i b r i u m within the spin system, the amplitude of the echo induced by a -2tr>/To pulse at time t would be reduced by the factor e K c f r o m R it s i n i t i a l value. Thus by m o n i t o r i n g the echo amplitude as a function of the two pulse s e p a r a t i o n t , an accurate R meas u r e m e n t of the true s p i n - s p i n r e l a x a t i o n time T^ may be made. C H A P T E R II ; T H E O R Y O F N.M.R. IN M E T A L S The p r e s e n c e of n o n l o c a l i z e d e l e c t r o n states, the conduction band, has v e r y pronounced effects both on the n, m. r . c h a r a c t e r i s t i c s of metals (the line shape, resonant frequency and the r e l a x a t i o n mechanisms) and on the e x p e r i m e n t a l techniques used to investigate these p r o p e r t i e s . In the f i r s t s e c tion of this chapter the effects of the magnetic coupling between the conduction e l e c t r o n s and the nuclear spins i s c o n s i d e r e d . T h i s i n t e r a c t i o n affects the c h a r a c t e r i s t i c s of the nuc l e a r magnetic resonance i n two ways: by effecting changes i n the n u c l e a r energy l e v e l s of the n u c l e a r spin s y s t e m and a l s o by p r o v i d i n g powerful r e l a x a t i o n m e c h a n i s m s for the coupling of the spins to each other and to the l a t t i c e . In the l a t e r section of this chapter some of the rather neglected e x p e r i m e n t a l p r o b l e m s inherent i n any measur e m e n t of n. m. r . p r o p e r t i e s i n meta l s w i l l be inve s t i g a t e d t h e o r e t i c a l l y . A. The Resonance C h a r a c t e r i s t i c s N e a r l y a l l the important d i f f e r e n c e s i n the resonance c h a r a c t e r i s t i c s of bulk metals and other solids a r i s e f r o m the magnetic hyperfine i n t e r a c t i o n between the n u c l e i and the conduction e l e c t r o n s . F o r a given nucleus of magnetic moment Li = v ft I i n a sample having N conduction e l e c t r o n s , this ^•N N —N i n t e r a c t i o n takes the fo r m : N I. N r. - 2(3s. 2ps. N -v H = u _ N . | Z 2(3 3 . + Z P l ^ g ^ - - ^ J - S - f ^ % ) ) _ j n . l ) i = l r. i = l r. r. i = l l i i where s. and r. a r e the o r b i t a l angular momentum, spin ~~1 i" 1 ~"1 th and coordinate v e c t o r ( measured f r o m the nu c l e a r site) of the i e l e c t r o n and p i s the Bohr magneton. It i s often convenient to r e g a r d the hyperfine i n t e r a c t i o n as a r i s i n g f r o m three d i s t i n c t parts: the f i r s t t e r m of II. 1 d e s c r i b e s the coupling of the n u c l e a r magnetic moment to the o r b i t a l magnetic moment of the conduction e l e c t r o n s ; the second t e r m i s the c l a s s i c a l d i p o l a r coupling between the n u c l e a r and e l e c t r o n i c spins and the f i n a l t e r m i s the i s o t r o p i c contact i n t e r a c t i o n between the n u c l e a r and e l e c t r o n i c spins. The same i n t e r a c t i o n II. 1 a l s o exists between the nucleus and the surrounding atomic or m o l e c u l a r ( i . e. core) e l e c t r o n s and i n non met a l s plays an important r o l e i n the resonance c h a r a c t e r i s t i c s . In met a l s this coupling w i l l be gr e a t l y over shadowed by that with the conduction e l e c t r o n s but i t i s s t i l l worth r e a l i s i n g what effect the core hyperfine i n t e r a c t i o n has on the n.m. r. p r o p e r t i e s . The core e l e c t r o n s , by de f i n i t i o n being tightly bound to the parent nucleus, have v e r y s m a l l o v e r l a p with neighbouring atoms and so the i n t e r a c t i o n would not be expected to affect g r e a t l y the r e l a x a t i o n mechanisms^ \ O n the other hand the actual f i e l d seen by a nucleus w i l l be mo d i f i e d by such i n t e r a c t i o n s . When a f i e l d H i s applied the L a r m o r p r e c e s s i o n — o of the e l e c t r o n i c charges, being equivalent to a c i r c u l a t i n g c u r r e n t , produces a s m a l l f i e l d H . at the n u c l e a r site which adds, and • — c i r c i s p r o p o r t i o n a l to, H . A t the same time H can p o l a r i z e the — o — o e l e c t r o n i c s h e l l s and thus produce a magnetic f i e l d H = v. H — p o l ^ms — o 1. The coupling II. 1 with non conduction e l e c t r o n s , however, admits of the p o s s i b i l i t y of s p i n - s p i n i n d i r e c t i n t e r a c t i o n s i n nonmetals. A n u c l e a r moment produces a f i e l d which d i s t o r t s i t s e l e c t r o n i c d i s t r i b u t i o n . T h i s d i s t o r t e d charge d i s t r i b u t i o n c r e a t e s a f i e l d H' p r o p o r t i o n a l to Uj at the site of a second nucleus u_2» Thus the coupling II. 1 i n non met a l s can give r i s e to a c o r r e l a t i o n of nuc l e a r spins and ji and so introduces a s p i n - s p i n r e l a x a t i o n m e c h a n i s m s . In met a l s this i n t e r a c t i o n would tend to be scr e e n e d out by the more mobile conduction e l e c t r o n s . at the nucleus. Thus the total magnetic f i e l d seen by the nucleus i s H = H + H , + H . and the resonant frequency — — o — p o l — c i r c 0) = yH i s thus d i f f e r e n t f o r the bare nucleus and the nucleus with i t s e l e c t r o n i c c o r e . T h i s shift i n resonant frequency, being dependent on the e l e c t r o n i c d i s t r i b u t i o n and hence the c h e m i c a l s t r u c t u r e of the mol e c u l e , i s a p p r o p r i a t e l y c a l l e d the c h e m i c a l shift. If the entire m e t a l were r e g a r d e d as a gigantic molecule then an exactly s i m i l a r a n a l y s i s could be applied to the i n t e r a c t i o n with the conduction e l e c t r o n s . T h e r e would thus be expected, i n addition to the above mentioned core shift, a shift i n the resonant frequency a r i s i n g f r o m the di a m a g n e t i s m of the conduction e l e c t r o n s . The diamagnetic s u s c e p t i b i l i t y of the conduction e l e c t r o n s having the w e l l known de Haas -van A l f r e n o s c i l l a t o r y dependence on the applied magnetic f i e l d , there should thus be an o s c i l l a t o r y dependence of the resonant frequency on the magnetic f i e l d ^ \ T h i s effect w i l l , however, be much s m a l l e r than the Knight shift to be d i s c u s s e d f u r t h e r below. The conduction e l e c t r o n s , being e x p r e s s e d as B l o c h functions so as to sat i s f y the t r a n s l a t i o n a l symmetry of the c r y s t a l l a t t i c e : \ k ( £ ) = e 1-- H^Cs) 5 H ^ ( £ + g n ) = H^(£) ( H . 2 ) w i l l have equal p r o b a b i l i t y of being found i n the v i c i n i t y of one (2) n u c l e a r site as at another* . C o n v e r s e l y , each n u c l e a r spin f e e l s the effect of a l l the conduction e l e c t r o n s simultaneously. 1. See ref. 16 and ref. 23. 2. In the case of a lat t i c e with b a s i s , the p r o b a b i l i t i e s are the same at equivalent n u c l e a r sites i n d i f f e r e n t c e l l s . Thus because of the non l o c a l i z e d c h a r a c t e r of the conduction e l e c t r o n s one would expect a l a r g e contribution f r o m the i n t e r a c t i o n II, 1 to the r e l a x a t i o n mechanisms: both s p i n - s p i n through i n d i r e c t coupling and sp i n - l a t t i c e through d i r e c t coupling are expected to be affected. M o r e o v e r , as hinted at above, since the conduction e l e c t r o n s w i l l exhibit P a u l i spin p a r a m a g n e t i s m i n an applied magnetic f i e l d , i t i s expected that each nucleus w i l l f e e l a f i e l d H , (in addition to the applied H ) due to the , ; — e l v — o' p o l a r i z a t i o n of the conduction e l e c t r o n s and thus give r i s e to a shift i n the resonant frequency. T h i s shift, c a l l e d the Knight shift after W. Knight who f i r s t o b s e r v e d i t , has both an i s o t r o p i c part coming f r o m the contact t e r m i n II. 1 and an a n i s o t r o p i c part coming f r o m the r e m a i n i n g t e r m s i n II. 1. (i) T he effective f i e l d at the nucleus; the Knight shift and  resonant frequency A s s u m i n g , as i s u s u a l l y the case i n metals, that the e l e c t r o n i c o r b i t a l angular momentum i s quenched, the i n t e r a c t i o n II. 1 may be r e c a s t i n the f o r m : H = - %' iff Cn 8 N N C H e £ £- - 2 P ^ 2 ^ 6^) - 2 P E { 3 £ I - - ] ( N 1=1 1=1 r. r. l l (not to be confused with the H ^ of Chapter I) i s to be i n t e r p r e t e d as an effective f i e l d o p erator. T o find the resultant energy l e v e l s and thence the resonant frequency shift, we must compute the expectation value of the energy of the above i n t e r a c t i o n ; we must average II. 4 over a l l the conduction e l e c t r o n s . The i s o t r o p i c contribution The f i r s t t e r m i n II. 4, having r o t a t i o n a l symmetry, i s c a l l e d the contact t e r m and can give r i s e only to an i s o t r o p i c f i e l d at the nucleus. Because of the D i r a c delta function f o r m , coupling w i l l take place only with those e l e c t r o n s having finite p r o b a b i l i t y of e x i s t i n g at the nucleus. Thus only the i s o t r o p i c s-type conduction e l e c t r o n s can contribute; e l e c t r o n s with p, d, f type symmetry, vanishing at the n u c l e a r si t e , cannot p a r t i c i p a t e . conduction e l e c t r o n s may be w e l l approximated as a degenerate F e r m i gas so that the summation over a l l conduction e l e c t r o n s becomes an ensemble average usi n g F e r m i D i r a c s t a t i s t i c s . Thus, because of the sharp cut off i n occupation number at the F e r m i l e v e l , only those s-type e l e c t r o n s within k T of the B F e r m i energy £ w i l l contribute. M o r e o v e r the degeneracy F between e l e c t r o n s sharing the same k state but opposite spin i s r e s o l v e d by the a p p l i c a t i o n of the magnetic f i e l d ; the magnetic f i e l d f o r c e s a r e d i s t r i b u t i o n of e l e c t r o n s between the two spin orientations and thus gives r i s e to a magnetic moment ( P a u l i spin paramagnetism): where G( £ ) i s the density of states at the F e r m i l e v e l . F Thus p o l a r i z a t i o n of the s-type conduction e l e c t r o n s i n t r o d u c e s an additional magnetic f i e l d at the nucleus: F o r the e l e c t r o n i c d e n s i t i e s that e x i s t i n m e t a l s , the M = 4(3 H G( E ) s v H — o F *p o "(II. 5) H. A I S O = T Xp ^ ( o ) | Z > ^ F {II. 6) and thus an i s o t r o p i c shift i n the resonant frequency All) u s u a l l y e x p r e s s e d as the f r a c t i o n a l change: K = H ° H o i££ = M = §2L X <| V / J 2 > c (n-7) co 3 P W  Z F where < | y ^ Q j | > £ p i S t h e averaged value of I V ^ 0 j l o v e r a l l e l e c t r o n s with F e r m i energy £ . F The a n i s o t r o p i c contribution The r e m a i n i n g t e r m s i n II. 4, co r r e s p o n d i n g to the c l a s s i c a l d i p o l a r i n t e r a c t i o n , need not have complete r o t a t i o n a l i n v a r i a n c e ; r a t h e r they w i l l , i n general, assume the symmetry of the e l e c t r o n i c charge d i s t r i b u t i o n . F o r the p a r t i c u l a r case of an a x i a l l y s y m m e t r i c d i s t r i b u t i o n , by c a r r y i n g out an ensemble avera g i n g p r o c e d u r e as i n the i s o t r o p i c case, i t i s f o u n d ^ that the effective f i e l d at the nucleus i s given by: H • = ^ X % - ( 3 C O S 2 6 - 1) H (II. 8) anis p " F ' o v 7 where 8 i s the angle between the symmetry axis and the applied f i e l d H . The quantity % i s a d i r e c t m e a s u r e of the anisotropy of the charge d i s t r i b u t i o n ^ ' and i s , i n fact, r e l a t e d to the e l e c t r i c q u a d r i p o l e moment of the e l e c t r o n s near the F e r m i s u r f a c e . Thus the shift i n resonant frequency depends both on the magnitude and the o r i e n t a t i o n of the d i r e c t f i e l d H . — o In a p o l y c r y s t a l l i n e m e t a l l i c sample, much of the anisotropy w i l l be averaged out by the random c o l l e c t i o n of c r y s t a l l i t e o r i e n t a t i o n s . The a n i s o t r o p i c contribution, however, can s t i l l 1. See ref. 5. 2. F o r a cubic l a t t i c e , % p = 0 s o that the di p o l a r i n t e r a c t i o n gives no contribution. give r i s e to a broadening of the resonance line and at the same time, can give r i s e to an a s y m m e t r y i n the line shape. ^ (ii) Spin L a t t i c e r e l a x a t i o n m e c h a n i s m s A t the beginning of this chapter i t was suggested that the non l o c a l nature of the conduction e l e c t r o n s can provide powerful r e l a x a t i o n m e c h a n i s m s f o r s p i n - l a t t i c e decay. A g a i n the p r i m a r y contribution i n met a l s comes f r o m the hyperfine i n t e r a c t i o n II. 1 but instead of the expectation value, which as has been shown gives r i s e to shifts i n the resonant frequency, i t i s the off-diagonal m a t r i x elements of the i n t e r a c t i o n , l i n k i n g d i f f e r e n t n u c l e a r spins, which d e s c r i b e s the r e l a x a t i o n phenomena. It i s e a s i e s t to r e g a r d the r e l a x a t i o n p r o c e s s v i a II. 1 as a sc a t t e r i n g p r o c e s s : the i n e l a s t i c c o l l i s i o n of an e l e c t r o n i n states | k, s> with a single nucleus causes the nuclear spin to f l i p f r o m | I = m> to | I = m'> and, to conserve energy and z z momentum, thus r e s u l t s i n a new e l e c t r o n i c state | k 1, s'>. Since the e l e c t r o n must thus be able to take up the s m a l l excess of energy due to the mutual spin f l i p s A£ ~ h(w -uu ) = f i H (v -y ), v e n o>'N e only a s m a l l f r a c t i o n of the e l e c t r o n s , those within k ^ T of the F e r m i s u r f a c e , w i l l be available to make the n e c e s s a r y t r a n s i t i o n | k, s >-* | k 1, s'> and thus contribute to the re l a x a t i o n . The p r o b a b i l i t y of such a simultaneous n u c l e a r and e l e c t r o n i c spin f l i p being induced by an i n t e r a c t i o n V i s given by time dependent p e r t u r b a t i o n theory as; 1. Abragam. When summed over a l l p o s s i b l e i n i t i a l and f i n a l e l e c t r o n i c states the total p r o b a b i l i t y per unit time of the n u c l e a r t r a n s i t i o n | m> -» | m'> i s obtained: k, s k*,s' '— — where f(k, s), the F e r m i D i r a c d i s t r i b u t i o n function, ensures that the i n i t i a l state | k, s> i s occupied and ^1 - f(k', s')^ ensures that the f i n a l state |k',s'> i s empty. It i s c l e a r that the s p i n - l a t t i c e r e l a x a t i o n time, which i s a m e a s u r e of the rate at which the d i s t u r b e d n u c l e a r spin s y s t e m re t u r n s to i t s t h e r m a l e q u i l i b r i u m d i s t r i b u t i o n , i s c l o s e l y r e l a t e d to the above t r a n s i t i o n p r o b a b i l i t y . In cases where the concept of a spin temperature i s meaningful i t can be shown that the s p i n -l a t t i c e r e l a x a t i o n time T i s r e l a t e d to P , b y ^ 1 m-»m' s P (£ -£ )2 1 _ _i m, n m, n m n  T = 2 2 1 S £ n n The concept of a spin temperature, as pointed out i n Chapter I, i s only u s e f u l when the p e r t u r b e d spin s y s t e m comes to i n t e r n a l e q u i l i b r i u m at a spin temperature T i n a time T S c* v e r y much l e s s than T^. When this , and c e r t a i n other g e n e r a l conditions are s a t i s f i e d , as they u s u a l l y are i n most metals, i t i s po s s i b l e to view the r e t u r n to e q u i l i b r i u m following an exci t a t i o n as a two stage p r o c e s s ; f i r s t the s p i n - s p i n coupling b r i n g s the spin s y s t e m to i n t e r n a l e q u i l i b r i u m at a "te m p e r a t u r e " T i n a s time <v> T , then the spin temperature decays to the l a t t i c e t emperature T, as the spin s y s t e m gives up i t s excess energy to the l a t t i c e , i n a time T^. In fact, i n the low f i e l d , high temperature i n 1. See r e f . 1 and ref. 40. l i m i t where the magnetization follows C u r i e ' s law, M 01 V T , z as given by I. 1, the B l o c h equation i n t r o d u c i n g (equation 1.9) becomes: -± (T- 1) = I*'1" T 0" 1) : tn. i i T i where T = T i s to be i n t e r p r e t e d as the temperature of the s excited spin s y s t e m and T i s the lattice t h e r m a l e q u i l i b r i u m temperature. In m e t a l s the concept of a spin temperature i s u s u a l l y very-u s e f u l and, because of the strong s p i n - s p i n c o u p l i n g s ^ , u s u a l l y quite a p p l i c a b l e . Once again i t i s convenient to look at the s e v e r a l contributions f r o m the hyperfine i n t e r a c t i o n separately. T h i s i s allowable because i t follows f r o m II. 11 that i f there are s e v e r a l independent r e l a x a t i o n p r o c e s s e s , each giving r i s e to a r e l a x a t i o n rate T ,\ , then the o v e r a l l r e l a x a t i o n rate i s iust the sum of the l(i) J i n d i v i d u a l r a t e s T " 1 = 7 T ~L i ? iw ( 2 ) J a c c a r i n o and Y a f et have shown that there a r e no i n t e r f e r e n c e t e r m s between the contact, o r b i t a l , dipolar and core p o l a r i z a t i o n r e l a x a t i o n m e c h a n i s m s so the tot a l r e l a x a t i o n rate i s just the sum of these i n d i v i d u a l contributions. M o r e o v e r i t i s found that there are substantial contributions f r o m a l l of these with the exception of the d i p o l a r i n t e r a c t i o n and no single one i s predominant for a l l m e t a l s . Contact t e r m If the contact t e r m , the l a s t t e r m i n II. 1, i s put into the 1. T y p i c a l l y T i s of the o r d e r of 100 m i c r o s e c o n d s while T i s of o r d e r of m i l l i s e c o n d s at low t emperatures. 2. See ref. 21. e x p r e s s i o n II. 9 f o r the t r a n s i t i o n p r o b a b i l i t y and the appropriate m a t r i x elements computed, i t i s found to give: Pm->n = T ^ ^ N ^ k ^ ^ F k B T G s V F > S l < m | l J ° > | 2 (II. 1 — a where G ( € „ ) i s the density of el e c t r o n s of s«type symmetry at s F the F e r m i s u r f a c e . F r o m II. 10, the r e s u l t i n g r e l a x a t i o n time i s r e a d i l y found: T 7 " T'VV>BT<I\M|2>CF < W ( n . » ) The quantity <|'Y,(o)| > a l s o o c c u r s i n the e x p r e s s i o n II. 6 fo r the i s o t r o p i c Knight shift. U sing that r e s u l t , II. 13 may be r e w r i t t e n as: In the case of a free e l e c t r o n m e t a l this reduces to the w e l l known K o r r i n g a r e l a t i o n ; 2 T T K 2 = —-J -^2 (H. 15) 1 c 4-rr k „ y ^ v ' ' It i s important to note that the contact i n t e r a c t i o n gives r i s e to a s p i n - l a t t i c e r e l a x a t i o n time which i s d i r e c t l y p r o p o r t i o n a l to the i n v e r s e of the temperature. T h i s i s actually the e x p e r i m e n t a l case for quite a few metals; the product T ^ T r e m a i n s constant over a l a r g e temperature range with the s m a l l deviations o b s e r v e d u s u a l l y explainable by t h e r m a l expansion effects. D i p o l a r and o r b i t a l t e r m s The r e l a x a t i o n m e c h a n i s m s r e p r e s e n t e d by the f i r s t two t e r m s of II. 1 a r e u s u a l l y v e r y m uch l e s s e f f i c i e n t i n metals than the contact t e r m just d i s c u s s e d ^ . It i s however to be noticed that although the contribution to the Knight shift by these i n t e r a c t i o n s w i l l v a n i s h f o r cubic m e t a l s , i t does not follow that the contribution to the r e l a x a t i o n time (which depends on the square of the off-diagonal m a t r i x elements) w i l l vanish. In fact (2) Obata* has shown that f o r cubic p-band metals, both the o r b i t a l and d i p o l a r i n t e r a c t i o n s give r e l a x a t i o n rates of the f o r m : ( T i T f 1 = a . V 3 ( y e y N ) 2 < r - 3 > d G d 2 ( e F ) (II. 16) where a, d i f f e r e n t f o r o r b i t a l and d i p o l a r m echanisms, i s a constant which depends on the e l e c t r o n i c d i s t r i b u t i o n at the -3 F e r m i surface and takes a value between 0 and 10: <r > i s -3 d the average value of r over a l l d-band states near the F e r m i energy; and G ( £ ) i s the density of d - b a n d states at the d F F e r m i s u r f a c e . Obata has shown fu r t h e r that a,, /a \£ V 3 dip o r b so that the d i p o l a r i n t e r a c t i o n i s l e s s than a t h i r d as effective as the o r b i t a l i n t e r a c t i o n . Since the o r b i t a l angular momentum i s u s u a l l y quenched i n most metals, i t follows that both these r e l a x a t i o n m e c h a n i s m s are u s u a l l y r e l a t i v e l y unimportant. 1. See r e f . 1. 2. See ref. 32. (iii) S p i n - s p i n r e l a x a t i o n m e c h a n i s m s It w i l l be r e c a l l e d that the s p i n - s p i n r e l a x a t i o n time i s a m e asure of the time i t takes the spin s y s t e m to come to i n t e r n a l e q u i l i b r i u m v i a a spin d i f f u s i o n p r o c e s s whereby neighbouring spins l e a r n of the states of t h e i r neighbours by a t r a n s f e r of energy and momentum. A l t e r n a t i v e l y , neglecting inhomogeneities i n the applied f i e l d (which as was pointed out give r i s e to potentially r e v e r s i b l e decay), the B l o c h equation 1.9 makes i t p o s s i b l e to r e g a r d T as the c h a r a c t e r i s t i c time of decay of the C* t r a n s v e r s e component of the magnetization. It can be shown that f o r m o s t spin coupling m echanisms, and c e r t a i n l y for the three dominant ones i n metals, the t r a n s v e r s e magnetization has a d e c r e a s i n g amplitude which i s r e l a t e d to the F o u r i e r t r a n s f o r m of the line shape. The s p i n - s p i n r e l a x a t i o n time can thus be r e l a t e d to the p r o p e r t i e s of the resonant line shape and so, following convention, we s h a l l d i s c u s s the f a c t o r s affecting the line shape, r a t h e r than T d i r e c t l y . CM The d i p o l a r line width p r i m a r y coupling between the spins i s v i a the d i r e c t d i p o l a r F o r a r i g i d l a t t i c e of i d e n t i c a l l i k e spins where the i n t e r a c t i o n : * 2 2 n y 3 r { - 3 ( V ^ j k ' 4 ' {II. 17) i t can be r e a d i l y shown that the t r a n s v e r s e magnetization M ^ ( t ) has a time dependence which i s p r o p o r t i o n a l to the r educed a u t o c o r r e l a t i o n function; {II. 18) where i s the x-component of the magnetization operator and ^ I'^j- 1 S a truncated f o r m of the i n t e r a c t i o n II. 17. ( T h i s e x p r e s s i o n w i l l r eappear i n the next section when the e x p e r i m e n t a l d i f f i c u l t i e s i n m e t a l l i c single c r y s t a l s a r e discussed.) A s stated above this e x p r e s s i o n can be reduced to the F o u r i e r t r a n s f o r m of the l i n e shape function f(ou) m e a s u r e d f r o m the c e n t r a l resonance •0) frequency:* + Q Q G(t) = \ % + u,') e i U J , t duu' (II. 19) 1 TTjfcr J O -00 where 5Jr i s a constant determined by the n o r m a l i z a t i o n of the shape function f(uu). F o r a resonance which can be w e l l d e s c r i b e d by the L o r e n t z function if v = - -3  1 ? (11.20) ( d i ) TT c , / \ o the t r a n s v e r s e magnetization decays a c c o r d i n g to II. 19 with time dependence; G (t) = ~~r * \ < 2_ 5 f 4 i r J +00 i.uj't d . . -00 5 + w +00 ioo't 6 duu' TTjfy TT J (i5 + uu')(-i6 + UU1) -00 P e r f o r m i n g a contour i n t e g r a t i o n (closing the path f o r positive t i n the lower half plane and i n the upper half plane fo r negative t), one sees that this r educes to: 1. See ref. 1. G ^ t ) = G ^ e " * 6 ^ G l ( o ) e - t / T 2 (n. 21) for a true L o r e n t z line the decay i s thus seen to be t r u l y exponential with a time constant T = 1/6. Resonances are never t r u l y L o r e n t z i a n but may often be w e l l approximated as such i n t h e i r c e n t r a l region. The quantity which i s u s u a l l y quoted as the line w i d t h ^ ' i s the width of the line Auu at the points of m a x i m u m slope. By finding the value of to which makes the slope df/dto a m a x i m u m one can e a s i l y show that {II. 22) fo r a L o r e n t z l i n e . It i s c l e a r that any i n t e r a c t i o n between the nuc l e a r spins which has the same f o r m as H.17 w i l l give r i s e to s i m i l a r time dependences of the t r a n s v e r s e components of the magnetization. We now investigate the other m a i n contributions to the s p i n - s p i n relaxation; p r i m a r i l y those which a r i s e f r o m i n d i r e c t coupling v i a the n o n - l o c a l i z e d conduction e l e c t r o n s . R u d e r m a n - K i t t e l (Pseudo-exchange) coupling B ecause of the spread out nature of the conduction e l e c t r o n s , the i n t e r a c t i o n s between them and the n u c l e a r spins would be expected to contribute quite a p p r e c i a b l y to the c o r r e l a t i o n 1. See ref. 33 ( T h i s i s the value quoted by S.N. Sharma (re f e r e n c e 37) f o r S n ^ ^ to which r e f e r e n c e w i l l l a t e r be made when the r e s u l t s are presented). of n u c l e a r spin states. F i n d i n g that, f o r elements of m a s s great e r than about 100, e x p e r i m e n t a l values exceed the line widths p r e d i c t e d by the f i r s t o r d e r d i r e c t coupling II. 17 by s i g n i f i c a n t l y l a r g e amounts, R u d e r m a n and K i t t e l ^ suggested the n e c e s s a r y i n c l u s i o n of this second o r d e r effect, the i n d i r e c t coupling of n u c l e a r spins v i a the conduction e l e c t r o n s . A n e l e c t r o n p a s s i n g through the l a t t i c e w i l l scatter f r o m one n u c l e a r spin (j.^ v i a the hyperfine i n t e r a c t i o n II. 1 and the r e s u l t i n g e l e c t r o n i c state w i l l thence depend on Hj's i n i t i a l state. The e l e c t r o n , subsequently s c a t t e r i n g off a second spin U^J thus ef f e c t i v e l y couples the two n u c l e a r spins together. A s might be expected, the dominant contribution comes f r o m the s c a l a r contact i n t e r a c t i o n . C a r r y i n g out the i n d i c a t e d second o r d e r p e r t u r b a t i o n c a l c u l a t i o n , R u d e r m a n and K i t t e l showed that this m e c h a n i s m i n the n e a r l y f r e e e l e c t r o n a p p roximation gives the pseudo-exchange i n t e r a c t i o n H a m i l t o n i a n of the f o r m : ^ P . E . = . A i j (11.23) where 2 ^ A = - y y. v .ft m*<W, . I >_ r . " -f 2k r. cos(2k r. . ) ~ s i n ( 2 k r .)]• i j 9rr e Y i Y j 1 T ( o ) ' £ F i j I F i j F 1 / v F i / J (n. 24) The o s c i l l a t o r y behaviour of the coupling constant A. . should be noted; the sign of the i n t e r a c t i o n depends both on the d i a m e t e r of the F e r m i surface k „ and the separation r. . of F i j the two n u c l e a r spins. It i s also important to note that the 1. See r e f . 34. p a r a m e t e r < | ' V ^ 0 j | > £ o c c u r s also i n the e x p r e s s i o n II. 7 F for the i s o t r o p i c Knight shift and i n the spin l a t t i c e r e l a x a t i o n t e r m II. 13 due to the contact i n t e r a c t i o n . (The second o r d e r nature of the above c a l c u l a t i o n i s r e f l e c t e d i n the fourth power of | \ ^ | . ) Thus i f the v a r i o u s contributions to the line width, or T^, can be distin g u i s h e d f r o m each other, c o m p a r i s o n with the Knight shift data should, i n p r i n c i p l e , give i n f o r m a t i o n as 2 to the values of < | 1 ^ ^ | >£• ^  . Being of the same f o r m as the H e i s e n b e r g exchange operator, and thus c a l l e d the "pseudo-exchange" contribution, the coupling II. 19 has the effect of sharpening the resonance line i n specimens of i s o t r o p i c p u r i t y and of broadening i t f o r i s o t opic m i x t u r e s ^ . M o r e o v e r , aside f r o m i t s having a dif f e r e n t c o e f f i c i e n t , the coupling has the same f o r m as the d i r e c t d i p o l a r coupling II. 17 f o r the s p e c i a l case of cubic s y mmetry so that the decay of the t r a n s v e r s e magnetization M^,(t) i s again e x p r e s s e d i n the f o r m II. 18. Pseudo d i p o l a r i n t e r a c t i o n The conduction e l e c t r o n s can al s o serve to couple the nucl e a r spins v i a the d i p o l a r p a r t of the hyperfine i n t e r a c t i o n II. 1. Although there i s no g e n e r a l proof to the often made a s s e r t i o n that this i n t e r a c t i o n may be reduced to the f o r m of a pseudodipolar coupling: D = = B J U - s H ^ j , (n.25) i > j 1 J l ^ 3 2 J R 1. See ref. 17 B l o e m b e r g e n and R o w l a n d ^ have shown that such a f o r m (2) holds i f the s p h e r i c a l approximation* ' holds. The co e f f i c i e n t s B. ., i n v o l v i n g i n t e g r a l s of non-s type wave functions over the i j band s t r u c t u r e , are rather m e s s y but B l o e m b e r g e n and -3 Rowland have shown that t h e i r r a d i a l dependence i s on R. . - as i n the case of pseudoexchange and as expected. M o r e o v e r they have managed to show that the r a t i o of B. ./A , where A. . i s the c o e f f i c i e n t of the pseudoexchange, H a m i l t o n i a n II, 19, has an o r d e r of magnitude equal to the hyperfine, spl i t t i n g r a t i o i n the c o r r e s p o n d i n g p- and s- type states times the r e l a t i v e amount of p with r e s p e c t to s c h a r a c t e r of the conduction wave function. F o r heavy metals of m a s s gr e a t e r than about 80, the pseudo-dipolar i n t e r a c t i o n II. 21 w i l l be a si z e a b l e f r a c t i o n of the exchange i n t e r a c t i o n . Thus, i n the case of i s o t r o p i c a l l y pure specimens, the p o s s i b l e n a r r o w i n g of the line by pseudoexchange i n t e r a c t i o n s w i l l tend to be masked by the pseudodipolar coupling so that the line widths w i l l be r a t h e r b r o a d e r , i n g e n e r a l , than the d i r e c t d i p o l a r coupling II. 17 would p r e d i c t alone. It should again be noted that, having f o r m a l l y the same f o r m as II. 17, the pseudodipolar i n t e r a c t i o n a l s o leads to a decay dependence of the t r a n s v e r s e magnetization M^,(t) of the same s o r t as II. 18. 1. See ref. 6. 2. See ref. 4. B. The E x p e r i m e n t a l Technique f o r M e t a l s A v e r y r e a l p r o b l e m a r i s e s because of the high conductivity when one attempts an e x p e r i m e n t a l measurement of any magnetic resonance p r o p e r t y i n a single c r y s t a l of m e t a l . In the f i r s t p lace, as i s w e l l known f r o m elementary e l e c t r o m a g n e t i c theory, when an r . f . magnetic f i e l d i s applied at the su r f a c e of the m e t a l , the magnetic f i e l d i s both attenuated and shifted i n phase as i t moves into the metal; the c h a r a c t e r i s t i c scale f o r this phenomenum i s the s k i n d e p t h ^ ^ Thus n u c l e a r spins at d i f f e r e n t depths i n the m e t a l see di f f e r e n t magnetic f i e l d s , d i f f e r e n t both i n amplitude and phase. The resultant net magnetization after the a p p l i c a t i o n of an r . f , pulse of fixed d u r a t i o n no longer has the simple f o r m 1,7 but ins t e a d w i l l be, as e a r l i e r pointed out, an average over a d i s t r i b u t i o n of angular tippings of the magnetization of groups of spins which have seen the same excitation. In addition to this p r o b l e m of non u n i f o r m excitation of the n u c l e a r sp i n system, there i s a l s o the p r o b l e m inherent i n the detection of the decay to e q u i l i b r i u m once the excit a t i o n has been turned off. T he existence of p r e c e s s i n g magnetic moments within the m e t a l induces c i r c u l a t i n g c u r r e n t s which i n turn set up counter magnetic f i e l d s H which, by Lenz's law, act so as to oppose the magnetic moments. T h e r e i s thus e f f e c t i v e l y a shielding of the p r e c e s s i n g magnetic moments (whose combined rate of decay we wi s h to measure) by the e l e c t r o n s 1, Throughout this a n a l y s i s the c l a s s i c a l skindepth w i l l be assumed. F o r low enough'temperatures and pure enough specimens i t would be n e c e s s a r y to use the anomalous skindepth. f r o m any e x t e r n a l detector, such as a c o i l wound around the specimen, and once again the c h a r a c t e r i s t i c distance f o r this shielding i s the skindepth. F o r the above reasons, i t i s not i m m e d i a t e l y obvious that the u s u a l techniques of pulsed n, m.r. - e x c i t a t i o n of the spin s y s t e m v i a a pulse of r . f . magnetic f i e l d and detection of the resonance through the voltage induced i n a c o i l wound at the surface - w i l l l e a d to the c o r r e c t values of the quantity under i n v e s t i g a t i o n . In p a r t i c u l a r i t would seem n e c e s s a r y to show that m e a surements of the s p i n - s p i n and s p i n - l a t t i c e r e l a x a t i o n t i m e s give the c o r r e c t values. F o r the f o r m e r , i t i s n e c e s s a r y to show that the total voltage induced i n the c o a x i a l detector c o i l following a pulse of r . f . decays i n time a c c o r d i n g to the a u t o c o r r e l a t i o n function II. 18 d e s c r i b i n g the time dependence of the t r a n s v e r s e magnetization; for the latt e r m e a s u r e m e n t the magnitude of the s i g n a l induced i n the c o i l following a second pulse by a fixed delay must be shown to i n c r e a s e exponentially with time constant T as the s e p a r a t i o n between the two r . f . (I) pulses i s i n c r e a s e d linear|>j v . F i n a l l y we seek to d i s c o v e r what effects such p a r a m e t e r s as pulse length, r e f e r e n c e s i g n a l phase and r . f . amplitudes have on the-signals and t h e i r i n t e r p r e t a t i o n . (It should be noted that the averaging p r o c e s s alluded to above i s not a t r i v i a l concept. It i n v o l v e s the m a c r o s c o p i c a v e r a g i n g of a m i c r o s c o p i c phenomenon; the r e l a x a t i o n m e c h a n i s m s are of m i c r o s c o p i c o r i g i n and we seek to m e a s u r e them with this m a c r o s c o p i c averaging technique. 1. T h i s type of m e a s u r e m e n t w i l l be d e s c r i b e d i n m o re d e t a i l l a t e r i n this chapter. To do this p r o p e r l y , the f u l l t r a n s p o r t equation should be solved, a n o n - t r i v i a l p r o b l e m because of the (forced non u n i f o r m i t y of the system. We s h a l l , however, use only a c l a s s i c a l a n a l y s i s and hope that the m a i n features r e m a i n valid.) (i) M o d e l of the m e t a l and method of solution Although the situation i n m e t a l s i s , as the above suggests, somewhat messy, i t i s not i n t r a c t a b l e . We con s i d e r f o r s i m p l i c i t y an a x i a l l y s y m m e t r i c sample aligned i n the x d i r e c t i o n with a c o a x i a l pickup c o i l of N turns and f i l l i n g f a c t o r f\^\ By Faraday's law the e.m. f. induced i n this c o i l w i l l be p r o p o r t i o n a l to the time rate of change of flux i n the x d i r e c t i o n . The s i g n a l i s thus p r o p o r t i o n a l to the net x-component of the p r e c e s s i n g magnetization. T o find the net induced magnetization following an r . f. pulse (of known du r a t i o n and amplitude at the surface) and thence (2) the induced voltage we c o n s i d e r the following m o d e l v '. We r e g a r d the m e t a l as divided into s m a l l volume elements AV each of which have the p r o p e r t i e s that: a. T h e r e are sufficient n u c l e i i n each AV, that each may k be r e g a r d e d as a m a c r o s c o p i c system with the same n. m.r p r o p e r t i e s as the bulk system; b. the phase and amplitude of the exciting r . f . magnetic f i e l d are constant over a l l the n u c l e i i n AV, . k 1. The f i l l i n g f a c t o r , defined by volume of sample enclosed by the c o i l ^ volume of the c o i l takes into account the fini t e s e p a r a t i o n between the c o i l turns and the sample. 2. T h i s m o d e l of the m e t a l was used by G a r a who used a s i m i l a r s o r t of approach to that p r e s e n t e d h e r e . The r e s u l t i n g magnetization following an r . f . pulse i n each of these volume elements i s then found. With this known d i s t r i b u t i o n of magnetic moments, Maxwell's f i e l d equations may then be used to compute the contribution which each of these volume elements makes to the induced voltage. It i s at this point that AV must be m o r e c l o s e l y s p e c i f i e d : we take AV to be k k a thin sheath at a depth Z below the s u r f a c e . T h i s allows us to compute the phase shift and attenuated amplitude of the exciting r . f , and thence the resultant magnetization as a function of depth below the surface and f i n a l l y the contribution to the induced voltage which must be then summed over a l l such sheaths to obtain the total s i g n a l . (ii) The magnetization induced i n AV Ic The magnetic f i e l d applied just outside the sample i s g e n e r a l l y a l i n e a r i l y p o l a r i z e d f i e l d i n the a x i a l d i r e c t i o n : H l ( t ) = Z H ^ R f e ^ j |x| = 1. = 2H^ x cosuut. which, as i s w e l l known, may be decomposed into two counter rotating c i r c u l a r l y p o l a r i z e d f i e l d s each of magnitude H^. The assumption i s made that the r . f . magnetic f i e l d at the volume element AV can be w r i t t e n as: k fi (AV k) = . k £ R j * * *>] (II. 27) where 2B i s the attenuated amplitude and (J) i s the resu l t a n t phase shift at the volume element AV . k F o r the case where AV i s chosen to be a thin sheath k A ill. 28) at depth Z below the surface of an i d e a l metal, II. 27 k becomes the w e l l known e x p r e s s i o n : -Z / / 2 6 i(uut - Z / / 2 6 <i 5<V= 2 B i ( c) J e M e r where 8 i s the skindepth p a r a m e t e r . A g a i n the f i e l d at AV can be decomposed into the two k counter-rotating c i r c u l a r l y p o l a r i z e d f i e l d s ; B (AV ) = B f cos ± (out + CP )x + s i n ± (cut + {> )y V (II. 29) i k 1 , K V K I C J of which only one w i l l affect the nu c l e a r spins (the other w i l l be *v 2to off resonance and thus have negligible effect). We o assume B (AV, ) i s the one which s a t i s f i e s the resonant + k condition. If we now choose that r e f e r e n c e f r a m e which i s rotating at angular speed tu and with i t s x" axis p a r a l l e l to B (AV ) at t = 0 ; x 1 = x cos (tot + CP ) + y sin (tot + G{> ) k k K. vJ = -x s i n (tot + Q? k ) + y cos(tut +<? k ) (11.30) A * Zk= Z then i n this f r a m e the r . f . f i e l d i s the constant vector; V A V = Bi ,k x' <n-31> and the equation of motion f o r the spins i n A V ^ i s , i n the c l a s s i c a l case and assuming no re l a x a t i o n , i d e n t i c a l to 1.4 i n the ro ta t ing r e f e r e n c e f r a m e . W e can thus now obta in an e x p r e s s i o n for the m a g n e t i z a t i o n i n the v o l u m e e l e m e n t A V u s i n g the c l a s s i c a l e x p r e s s i o n s of chapter I . If the r . f . m a g n e t i c f i e l d was t u r n e d on at t = 0 and off at t = r[., a s s u m i n g the m a g n e t i z a t i o n was i n i t i a l l y a l i g n e d p a r a l l e l to the d i r e c t f i e l d H , the r e s u l t i n g t r a n s v e r s e m a g n e t i z a t i o n i n the ro ta t ing r e f e r e n c e f r a m e II. 30 would be (I. 7) M t ( T , A V K ) = M z ( o , A V k ) s i n ejxj^  cos 0 ^ 1 - c o s j ^ + y ^ s i n j ^ w h e r e 9 i s the angle between the effect ive f i e l d H = 1 H + u/y \ z + H . , x 1 i n the v o l u m e e l e m e n t A V , — eff o J l , k k and the a p p l i e d d i r e c t f i e l d H : — o H l k H + U > / Y s i n 8, = 1 ; cos 0 = H , i A V ) k H . . ( A V J eff* k eff* k T h e angle ? i s the angle through w h i c h the m a g n e t i z a t i o n k p r e c e s s e s about H ( A V ) d u r i n g the r . f . pu l se : e xx K. eff* k' T h u s i n the l a b o r a t o r y f r a m e , the t r a n s v e r s e component at any t i m e g r e a t e r than t = T , a s s u m i n g no r e l a x a t i o n , i s : M T ( t , A V k ) = M z ( 0 ; A V k ) s in 8^ ^ x [cos(u)t + 9K)COS 8 k ( 1 - c o s J k ) - s i n £ s i n (out )] + y [ s in (ujt +(^>k)cos 9^1 - cos J ) + s i n j cos (cut ) ] } . or using a standard identity: M T ( t , A V k ) = M z ( 0 , A V k ) s i n [ 2 c o s (u)t+({> k)cos 9 k s i n 2 | k / 2 - s i n ^ k s i n (out + 9 K) ] + y [2sin(cut+<D )cos 8 sin^ S k/2 + s i n £ k cos (cot+q) k )]}. In p a r t i c u l a r the resultant component i n the x d i r e c t i o n i s : M x ( t , A V k ) = M z ( 0 , A V k ) s i n 8 k | 2 cos (cut+9FC) s i n 2 J k/2 c o s 9 k - S i n J k s i n (cut + CJ>k) ^ . The magnetization i n volume element AV can a l s o be (1) obtained using a quantum m e c h a n i c a l approach* . T h i s method allows us to include the i n t e r a c t i o n between the n u c l e a r spins and thus given e x p l i c i t l y the r e l a x a t i o n of the n u c l e a r spins when the excitation i s turned off. The dominant coupling between the spins i s assumed to be of the f o r m of a d i p o l a r i n t e r a c t i o n (which as was shown i n section A i s the case f o r m o s t metals) and i t i s f u r t h e r a ssumed that the i n t e r a c t i o n i s v e r y much weaker than the i n t e r a c t i o n of the n u c l e a r magnetic moments (2) with the r , f , magnetic f i e l d v , Under these assumptions, the 1, See ref, 14, 2, B y this assumption i s meant the inequality |& , , T/ft|«|jft ' O. | iC J. j AC T h i s w i l l of course not be true f o r spins deep i n s i d e the m e t a l wher H i s v e r y s m a l l . However the motion of these spins has v e r y l i t t l e effect on the n, m. r, signal so such an assumption r e m a i n s meaningful. H a m i l t o n i a n f o r the n u c l e a r spin s y s t e m may be w r i t t e n i n the form: where d,k for - oe> ^  t <, 0 0 <; t <; T d,k T £ t. N, ~ vfi H . Z I. ~ ° i = 1 is the dominant Z e e m a n H a m i l t o n i a n for the nu c l e a r spins i n AV ; N M 2 k k k l - 3 c o s 0 -** d,: = G £ £ 1=1 m= 1 ,(1) (31 I - I . I ) iz mz —£ —m i s the truncated* 7 f o r m of the d i p o l a r i n t e r a c t i o n ; and * i.k • - v * H i , k s , { : x cosKl -1 k> - V s i n (V -f k>t ' <f. = 1 Jf. 7 J). i s the coupling of the nu c l e a r spins to the rotating f i e l d H l , k which for s i m p l i c i t y , i s applied at the resonant frequency UJ = -yH o o The equation of motion f o r the density operator: ifi oT = [ii . P T ] ; P X = P x ( * . A V k ) i s then solved i n each of the time domains of 11.37 and continuity at the boundaries o and T i s imposed. A s s u m i n g the s y s t e m was i n i t i a l l y (t = -°°) i n t h e r m a l e q u i l i b r i u m , the magnetization 1, The constant G i s an aggregate of the effects of both the true di p o l a r coupling and the i n d i r e c t couplings v i a the conduction e l e c t r o n s which as shown i n section II. A have the same f o r m as the d i r e c t d i p o l a r i n t e r a c t i o n . at a time t s T i s given by: where i s the magnetic moment operator and the t r a c e i s taken over the ensemble of spins i n AV . G a r a obtains the k e x p r e s s i o n : M x ( t , A V k ) = M Z ( 0 , A V k ) j ^ l ^ J s i n ( v H 1 ] k T ) c o s ( % t +« f -1, k where f i s the reduced auto c o r r e l a t i o n function (introduced i n II. 18) d e s c r i b i n g the t r a n s v e r s e r e l a x a t i o n i n the volume element AV . k It i s important to note that i f the sample i s a p e r f e c t single c r y s t a l , then: G 1 > k ( t - T ) G j f t - T ) as defined by 11,44, i s independent of the p a r t i c u l a r volume element k and, as G a r a points out, reduces to the u s u a l r e l a x a t i o n f u n c t i o n ^ f o r the r i g i d l a t t i c e ( n o r m a l i z e d to volume element AV^). Its value may, of course, depend on the o r i e n t a t i o n of the c r y s t a l with r e s p e c t to the d i r e c t f i e l d H . — o It w i l l be furt h e r noted that i f the resonant frequency co = to i s put into the c l a s s i c a l e x p r e s s i o n II. 36, the two o "(II. 44) 1. Ref. 1. r e s u l t s 11.36 and 11.44 agree at the end of the r . f . pulse (t = T ) but that unlike the c l a s s i c a l e x p r e s s i o n , 11.44 has indeed got the t r a n s v e r s e r e l a x a t i o n built into i t by the admittance of n u c l e a r spin c o r r e l a t i o n v i a II. 39. (iii) Voltage induced i n the c o i l by M at a depth Z below the surface In o r d e r to evaluate c o r r e c t l y the voltage which w i l l be induced i n the pickup c o i l (wound at the surface of the sample) due to the p r e c e s s i n g magnetic moments i n volume element AV - which we now take as a c y l i n d r i c a l sheath at a depth Z below the surface - we must f i r s t solve Maxwell's equations to determine the degree of shiel d i n g which w i l l o c c u r due to the high conductivity of the m e t a l . Maxwell's equations f o r a conductor give the wave equation V 2 H = n O 0 ™ ( M * H ) =(Y B (II where H i s the counter f i e l d e s t a b l i s h e d by the p r e c e s s i o n a l motion of the magnetic moment M whose sp a t i a l v a r i a t i o n i s assumed knownv**/. A s s u m i n g u n i f o r m i t y along the axis of the (2) sample, neglecting any edge effects and imposin g the time 1, It i s perhaps important to note that 11.45 makes the i m p l i c i t a ssumption that there i s neither time nor sp a t i a l dependence i n the f i e l d H . We are of cou r s e working here i n a time range where there i s no applied r . f . f i e l d . 2. In imposin g this time dependence we are invoking an adiabatic approximation; we are assuming that the r e l a x a t i o n of the n u c l e a r spins i s v e r y much slower than t h e i r p r e c e s s i o n a l motion. dependence e l a ) t , we can reduce 11.45 to: { - 2 * h )  HJy> z> = 9 ^ ( M x ( y ' z ) +  HJ y' z<} A s s u m i n g that the d i s t r i b u t i o n of magnetic moments depends only on the depth at which they l i e below the surface, t r a n s f o r m i n g to planepolar coordinates and assuming the skindepth 6= , 1 ( (II i s v e r y much l e s s than the sample radius R, we have .2 I 2 2 2 H x ( p ) c - i - [ M j p ) + H x ( p ) ] ; p =^y + z dp 6 The r a d i a l distance p i s .reinterpreted as the depth Z below the s u r f a c e and equation II. 48 i s solved by the standard technique of v a r i a t i o n of p a r a m e t e r s to give a solution containing two unknown constants. T h e s e constants of in t e g r a t i o n are evaluated f o r the contribution of the magnetic moments i n volume element AV, : k M x k { Z ) ~ M x ( Z k ' 5 ( Z " Z k }  by i m p o s i n g the two boundary conditions: a. the magnetic flux i n the x - d i r e c t i o n d s due to the motion of the magnetic moments II. 49 must r e m a i n finite f o r a l l values of Z, . k b. the induced emf; £, = -N 9^x/ot , k where N is the number of turns, induced in the coil by the sheath of magnetic moments at depth Z must go to zero as Z gets very large k K. (expressing the fact that the shielding must increase with depth). These boundary conditions then uniquely specify the solution to II. 48 and it is found that the induced shielding field is given by: "x< z> • - i Mr e M*<V <P so that the voltage induced in the coil by the sheath 11.49 is: £ = - N a ^ k k. . st = - Ne l a , t(>) U q \ { H x ( z ) + M x ( z ) } d 2 S CO ^ H x ( z ) + M x ( z ) | ds Substituting from II. 49 and II. 50 and evaluating the integral we obtain: -Zk//26 ' i(rj)t - Zk//26) = - Nicu u T) R TT e e M (Z. ) k no x k7 where r\ is the filling factor as defined earlier. Thus, as expected, the signal from volume element AV is attenuated and phase shifted - with c h a r a c t e r i s t i c scale 6 - just as the excitation of the spins i n AV was attenuated and phase k shifted with scale 6. (iv) The total voltage induced i n the c o i l T o obtain the total voltage induced i n the pickup c o i l , the contributions f r o m each c y l i n d r i c a l sheath such as II. 51 must be added together. The summation i s converted to an i n t e g r a l and, as the contributions f a l l off exponentially with a c h a r a c t e r i s t i c scale 6 « R, the l i m i t s of i n t e g r a t i o n are extended f r o m z e r o to i n f i n i t y thus giving: ^ T O T A L = j £ k ( Z k > d Z k 1+i or using II. 51: e y 6 M x ( z ) d z . (11.52) The r e s u l t s of section (ii) above suggest the magnetization 7, has the f o r m : k G l , k ( t - T ) j „. A i(wt + < K ) C . 2 I i n the volume element AV, k M j t , AV k) = Mz(0, A V k ) { ] s i n 6 K + * ><>{ 2 sin 2 * ± cos ^ - 2 s i n ? k e' " / 2 ] . (II. 53) If we now assume that the s y s t e m was i n i t i a l l y i n t h e r m a l e q u i l i b r i u m then we may write: M z ( 0 , AV k) = M Q A V k (n. 54) where M i s the e q u i l i b r i u m magnetization I. 1. F o r a single o at depth Z is (using the phase factor Cp and the attenuated 1 Z crystal where £ fc(t- T ) ^ k ( 0 ) | = [ G ^ t - T J/G^O)} is independent of the particular volume element, the magnetization i IC field amplitude from I I . 28): M x (z,t) = M J ^ 1 ^ ^ } s i n 0 ( z ) e i w t e 2 6 p s i n 2 - ^ - c o s 9 ( z ) - 2 s i n J ( z ) e W | A V ( t ) ( I I . 55) Hence the induced voltage I I , 52 becomes for times t ^ T : C T O T A l M * " N i ^ o R " « ^ M o ^ £ l ! ! f L ) } ^ ^ " l o ' f > 5 6> Gj(0) where , . l+i xz_ r=° ~ / 2 5 Z "/26 2 l(z1 ' F(UMM , H »Y ) = - \ e e 12sin —^-J-cos9(z) ' » - { ,/ . o irr/2 2sinf (z)eX"' " sin 0 ^ dz . ( I I . 57) is a function, independent of the time, which is essentially a measure of how much being "off resonance", (uu-tjj ) , and how o much the shielding, both during excitation and detection, affects the signal amplitude and phase. (v) Spin-spin relaxation time measurements A l l measurements of the spin-spin relaxation time T^, other than by spin echoes, were essentially based on the free induction decay ( F . I . D . ) method^. The spin system is exci by an r . f . pulse of fixed duration and the return to internal 1. Ref. 18. e q u i l i b r i u m - that i s , the decay of the t r a n s v e r s e m a g n e t i z a t i o n ^ 7 i s m o n i t o r e d v i a the voltage induced i n the pickup c o i l . The voltage induced i n the c o i l , upon r e c t i f i c a t i o n , v a r i e s d i r e c t l y as the r e l a x a t i o n function G^(t -T ) a c c o r d i n g to II. 56. Thus the m o n i t o r i n g of the F.I.D. t a i l gives d i r e c t l y a m e a s u r e of (2) T . In chapter III the actual methods used to monitor this c* voltage w i l l be d i s c u s s e d ; F i g u r e s IV1 - IV6 show o s c i l l o g r a p h s of t y p i c a l F.I.D. t a i l s after phase sensitive detection. (vi) S p i n - l a t t i c e r e l a x a t i o n time measurements In a non m e t a l l i c sample the most d i r e c t method of m e a s u r i n g the spin la t t i c e r e l a x a t i o n time T with a pulsed ( 3 ) 1 n. m. r. s p e c t r o m e t e r i s to apply a TT pulse followed by a -rr/2 pulse at a v a r i a b l e time t l a t e r . The amplitude of the F.I.D. - t l / T i t a i l following this second pulse w i l l then v a r y as ( l - e ) as the pulse s e p a r a t i o n t^ i s changed and thus T^ may be r e a d i l y m e a s u r e d . In single c r y s t a l m e t a l l i c samples, however, the def i n i t i o n of TT and TT/2 pul s e s i s m eaningless but a s i m i l a r method may s t i l l be used. C o n s i d e r the spins i n volume 1. The decay of the t r a n s v e r s e magnetization depends on the magnetic f i e l d inhomogeneity as we l l , of course. 2. A s i d e f r o m the d i f f i c u l t y of uniquely defining T^ when the re l a x a t i o n i s not exponential. 3. A s the subsequent theory shows, i t i s i n fact not n e c e s s a r y to have such p e r f e c t l y defined pulses; any angular pulse w i l l s u f f i c e . T hese TT, TT/2 pulse t r a i n s however give the m a x i m u m si g n a l amplitudes. element AV , again taken as the c y l i n d r i c a l sheath used i n pre v i o u s p a r a g r a p h s . The assumption i s made that the spins are i n i t i a l l y i n t h e r m a l e q u i l i b r i u m (so that M (0)= M (0) z) — k z, k which means that the l a s t e x c i t a t i o n of the s y s t e m o c c u r r e d many ti m e s T^ seconds p r e v i o u s l y , A pulse of dura t i o n and at frequency cu (not n e c e s s a r i l y the resonant frequency « y H ) applied at t = 0 gives a z-component of magnetization o'M z , k < T l > = M Z , k < ° ) c ° S a k whereas i n I. 8, 2 cos V = s i n e i , k c o s * i , k * c o s e i,k-T h i s z-component then decays under s p i n - l a t t i c e 0 ) r e l a x a t i o n p r o c e s s e s to • - ( t 1 - T 1 ) / T 1 M S f k ( t l J = M z 9 k ( 0 ) I 1 " b ( k ) e 3 > W where b(k) = 1 - cos : at a l a t e r time t , The t r a n s v e r s e components of the magnetization having died out i n a c h a r a c t e r i s t i c time T , the net magnetization * 2 at a time t » T w i l l c o n s i s t only of this z-component, 1, E q u a t i o n II, 59 i s the solution to the B l o c h equation 1,9 with the i n i t i a l condition M ( T,) = M (0) cos OL , ^ Z,& X Z ) & 1c — ^ j i J- -d M /dt = ( M -M )/T, =^=> M - M = Ge 1 z l ^ o z ' l z o - T / T M ( T.) = M (0) cos a, ~=> Ge 1 = M (0) [ c o s a - l ] . Z I Z y K. xC Z 9 iC AC and thus (t - T )/T T h u s a s econd r . f . pu l se of l ength T and the same * 2 f r e q u e n c y UJ a p p l i e d at t » T w i l l give an x - c o m p o n e n t of m a g n e t i z a t i o n i n the lab f r a m e w h i c h then d e c a y s a c c o r d i n g to II. 53 wi th M r e p l a c e d by M , ( t , ) ^ o z , k v 1' r G (t) i M x , kw • M z , k wlG77(r i ) l a ( k ) : t a S < n- 6 2 ) 1 j K 1 w h e r e a(k) = e s i n 9 k 1 2 s i n ^ 2 j c o s 9 k - 2 s i n Sfc 2 e J (11.63) the other v a r i a b l e s being as p r e v i o u s l y de f ined . U s i n g II. 60, we thus obtain: r -VTi K G i ") • *• MK, kw • MO A \ t1 - b<k>e H 5 ^ ) j »«• i * * *i — » • ) 1, k 1 w h e r e M = M ( o)/AV i s the e q u i l i b r i u m m a g n e t i z a t i o n p e r O Z f ic ic unit v o l u m e . T h e voltage i n d u c e d i n the c o i l fo l lowing this s econd pulse i s g i v e n by the i n t e g r a l i n II. 56.. A s s u m i n g aga in that the s a m p l e i s a s ingle c r y s t a l and c h o o s i n g the v o l u m e e l e m e n t s A V to be the u s u a l c y l i n d r i c a l sheaths at depth Z , we have Ic Ic I . In u s i n g II. 53 to d e s c r i b e the F . I . D . fo l lowing the second pu l se we a r e a s s u m i n g that just p r i o r to the s econd pu l se the sp in s y s t e m i s i n t h e r m a l e q u i l i b r i u m i n t e r n a l l y - i . e . wi th u n i f o r m sp in t e m p e r a t u r e T - but not n e c e s s a r i l y i n t h e r m a l e q u i l i b r i u m with the l a t t i c e . A l t n o u g h the net t r a n s v e r s e component of the m a g n e t i z a t i o n m a y be z e r o , because of an a p p l i e d f i e l d i n h o m o g e n e i t y ( T ^ ^ ^z)> S P L N s y s t e m m a y s t i l l have e x c e s s e n e r g y . T h u s G ^ ^ ( t j / G j j ^ i ) ' t ^ l e r e l a x a t i o n funct ion u s e d i n II. 62, i s not n e c e s s a r i l y the same funct ion as u s e d p r e v i o u s l y w h e r e t h e r m a l e q u i l i b r i u m i n t e r n a l l y and wi th the la t t i ce was a s s u m e d . fco i + i - t / T ] G , ( t ) 6(T)= C e m t \ e " / 2 6 M o | l - b ( Z ) e 1J } a(z)dz \ ) - G e « M 0 { ^ } \ { F - e ^ ' * ] (n.66) where F = F(o)-O) , H, , T ) i s as defined i n II. 57 and F(uu-U) . H , T , , T , ) s f e / 2 5 a(z)b(z)dz (U. 67) O 1 j o x 1 i s independent of the pulse s e p a r a t i o n t^, If we now make the assumption that G (t)/G (t ) depends only on t and not on t 1 , then the induced voltage depends on the V T i pulse s e p a r a t i o n t only through the exponential e Thus i f the voltage following the second pulse i s m o n i t o r e d always at the same value of t ( i . e, at a constant time r e l a t i v e to the second pulse), the induced voltage v a r i e s with pulse s e p a r a t i o n a c c o r d i n g to: r ' t i / T i C (t t) = A ( l - Be ) (H. 68) where A and B are constants which depend on the e x p e r i m e n t a l p a r a m e t e r s uo, T , T , t and H but not on 1 6 l o the pulse s e p a r a t i o n t ^ . Some evidence was found e x p e r i m e n t a l l y to suggest that the m e a s u r e d time constant of the decay depends on whether or not the resonant conditions are exactly met. Although the e x p e r i m e n t a l r e s u l t s i n this r e g a r d are not c o n c l u s i v e , i t would appear that the time constant of the decay drops quite noticeably as the d i r e c t magnetic f i e l d i s moved f r o m the resonant value. (See section I V d ( i i i ) . T h i s r a t h e r unexpected r e s u l t (which unfortunately was d i s c o v e r e d too late for f u r t h e r experiments to be c a r r i e d out) leads one to d i s t r u s t c e r t a i n of the assumptions made i n the above a n a l y s i s . In p a r t i c u l a r the assumption that G (t)/G (t ) ( d e s c r i b i n g the t r a n s v e r s e r e l a x a t i o n following the second pulse, t e r m i n a t e d at time t j , at time t) i s dependent on t only and not on t^, i s subject to s u s p i c i o n . A s was pointed out i n a footnote, the situation following the second pulse i s rather d i f f e r e n t f r o m that following the f i r s t i n that i n the l a t t e r case the spin system, although i n a state of i n t e r n a l e q u i l i b r i u m for t i m e s l a r g e r than , i s s t i l l i n an excited state - the " s p i n t e m p e r a t u r e " has not decayed to the lattice temperature. One should t h e r e f o r e investigate much m o r e c l o s e l y the t r a n s v e r s e r e l a x a t i o n that o c c u r s i n this excited s y s t e m following the second pulse. It i s f e l t that f u r t h e r theoretical a n a l y s i s must await m o r e comprehensive e x p e r i m e n t a l s t u d y ^ to help act as a guide to understanding the p r o c e s s ; any f u r t h e r a n a l y s i s at this point would de f i n i t e l y be c o n j e c t u r a l . 1. One such study would be to look at the F.I.D. following the second pulse as a function of the pulse separation; does the decay of the f r e e induction signal, have the same time constant T f o r a l l pulse separations t ? C H A P T E R III : T H E A P P A R A T U S A N D T H E M E T H O D (A) The apparatus The pulsed n. m. r . s p e c t r o m e t e r used i n the present r e s e a r c h was o r i g i n a l l y designed and built by L . A . M c L a c h l a n ^ s p e c i f i c a l l y f o r the study of T^ and T^ at r o o m and l i q u i d n i trogen t e m p e r a t u r e s i n m e t a l l i c single c r y s t a l s . Although no sweeping changes were made to the e l e c t r o n i c s of the s p e c t r o m e t e r , the c r y o g e n i c p a r t of the apparatus was e n t i r e l y r e d e s i g n e d and r e b u i l t . In this s e ction the b a s i c g e n e r a l d e s i g n of the s p e c t r o m e t e r i s presented, the e l e c t r o n i c c i r c u i t s (most of which have undergone slight m o d i f i c a t i o n s during the c o u r s e of the p r e s e n t r e s e a r c h ) are given and the d e s i g n of the c r y o g e n i c s y s t e m and sample mounting d e s c r i b e d . (i) The e l e c t r o n i c s (2) A b r a g a m gives a g e n e r a l d e s c r i p t i o n of the g e n e r a l r e q u i r e m e n t s of an n. m. r . s p e c t r o m e t e r and C l a r k gives an exce l l e n t a n a l y s i s of the c o m p r o m i s e s and s p e c i a l r e q u i r e m e n t s (3) of a pulsed n. m. r . apparatus* . M c L a c h l a n ' s apparatus follows the b a s i c design of C l a r k but, as M c L a c h l a n points out, the high conductivity of single c r y s t a l m e t a l l i c samples influences the design philosophy a p p r e c i a b l y . F i r s t l y only those n u c l e i within the skindepth r e g i o n (a v e r y s m a l l f r a c t i o n of the total number of nuclei) w i l l be both excited by the r . f . magnetic f i e l d and effective i n inducing a s i g n a l voltage i n the pickup c o i l following excitation. Hence, the s i g n a l f r o m these 1. Ref. 27, r e f . 28, ref. 29. 2. Ref. 1. 3. Ref. 11. conducting samples being v e r y much s m a l l e r than most other non conducting specimens, the s p e c t r o m e t e r must have v e r y high s e n s i t i v i t y . Thus the e l i m i n a t i o n of noise by c a r e f u l d e s i g n and c o n s t r u c t i o n and the enhancement of S/N by averaging techniques becomes not a l u x u r y but a n e c e s s i t y . Secondly because of the r a p i d changes i n amplitude and phase of the exciting r . f , magnetic f i e l d with depth i n s i d e the metal, i t i s i m p o s s i b l e to define definite " a n g u l a r " pulses (such as a TT/2 pulse) as one can f o r i n s u l a t o r s . Thus the conventional pulse t r a i n s , such as the -rr/2-Tr pulse sequence used by H a h n ^ , cannot be obtained and new methods must be d e v i s e d to m e a s u r e T and T , •L C» A block d i a g r a m of the s p e c t r o m e t e r i s shown i n F i g . II. 1. The r e p e t i t i o n rate t i m e r , a free running m u l t i v i b r a t o r with v a r i a b l e / p e r i o d (schematic 1), i n i t i a t e s the pulse sequence t r a i n r e p e t i t i v e l y at a chosen frequency between (8msec) * and (9sec) \ Thus this t i m e r repeats each excitation of the n u c l e a r spin s y s t e m and hence each m e a s u r e m e n t at a p r e s e t frequency thus allowing the resultant signals following each r e p e t i t i o n to be s t o r e d up i n the signal averaging device (the boxcar integrator) In g e n e r a l the r e p e t i t i o n rate i s set such that many T^ time constants have elapsed before r e p e t i t i o n of the pulse sequence o c c u r s . T h i s allows the n u c l e a r spin s y s t e m to r e t u r n to t h e r m a l e q u i l i b r i u m with the l a t t i c e before the next pulse sequence comes along. Thus at high temperatures, the 1. Ref. 18, ref. 19. r e p e t i t i o n rate can be r e l a t i v e l y high (for S n at r o o m temperature T^ - 0. 11 m i l l i s e c so the pulse t r a i n could not be repeated fast enough with the p r e s e n t t i m e r to get the m a x i m u m benefit f r o m the s i g n a l averager) whereas at low temperatures the r e p e t i t i o n rate must be c o n s i d e r a b l y s m a l l e r (^ , 100 m i l l i s e c , _ 119 . ,o„. for S n at 2 K ) . The pulse sequence generator, t r i g g e r e d by the re p e t i t i o n rate t i m e r , c o n s i s t s a l m o s t e n t i r e l y of T e k t r o n i x 160 s e r i e s g e n e r a t o r s and i s shown i n block f o r m i n F i g . IH, 2. The m a s t e r r amp generator produces a sawtooth r e f e r e n c e voltage (repeated at the frequency of the re p e t i t i o n rate t i m e r ) f r o m which the two pulse g e n e r a t o r s are t r i g g e r e d when the sawtooth voltage equals a p r e s e t i n t e r n a l comparator voltage i n the separate pulse generators as shown i n F i g . III. 3, Thus adjustable delay between the two pulses i s obtained and one of two modes i s p o s s i b l e . In mode 1 the i n t e r n a l c omparator voltage of the pulse 2 generator i s p r e s e t by changing an i n t e r n a l potentiometer. In this mode the delay between the two r . f . pulses i s kept constant. In mode 2 ^ i s e f u l f o r T^ measurements) an ex t r e m e l y slow sawtooth (of p e r i o d ranging f r o m seconds to hours) i s e x t e r n a l l y generated i n a m o d i f i e d T e k t r o n i x u n i t ^ and used as the i n t e r n a l c omparator voltage for the pulse 2 generator. Thus i n this mode the se p a r a t i o n between the two r . f . pulses i s swept v e r y slowly i n time - i n g e n e r a l many repetitions of the pulse t r a i n w i l l have o c c u r r e d before the pulse separation w i l l have changed a p p r e c i a b l y . 1. Ref, 7 and ref. 8. The gate output of either pulse 1 or pulse 2 generator i s used to t r i g g e r a second sawtooth generator which then p r o v i d e s a r e f e r e n c e ramp for a v a r i a b l e delay for the boxcar gater. T h i s boxcar gater p r o v i d e s the s y m m e t r i c positive and negative square pulses r e q u i r e d by the boxcar i n t e g r a t e r . When operated i n the n o r m a l mode of fixed delay r e l a t i v e to the r . f . pulse t r a i n , i t i s apparent that the s i g n a l a v e r a g e r (or o s c i l l o s c o p e when the n. m. r. s i g n a l i s l a r g e enough to be seen) w i l l be t r i g g e r e d i n s y n c h r o n i z a t i o n with the pulse sequence, at the rep e t i t i o n frequency, and hence i n syn c h r o n i z a t i o n with the n. m. r . s i g n a l . B y v a r y i n g the i n t e r n a l c o m p a r i s o n voltage of the boxcar gater manually i n mode 1, the s i g n a l a v e r a g e r may be used to look at any d e s i r e d r e g i o n of the n. m. r . s i g n a l . In mode 2 the boxcar gate i s swept l i n e a r l y away f r o m the r . f . pulse t r a i n . T h i s mode proves v e r y u s e f u l when m e a s u r i n g T v i a the F.I.D. t a i l o r when studying the shape of the spin echoes. In g e n e r a l the slopes of the sawtooth ramp generators are kept as steep as p o s s i b l e so as to m i n i m i z e the p o s s i b i l i t y of ji t t e r i n the delays of the v a r i o u s p u l s e s . The m a s t e r r . f . o s c i l l a t o r , of Co l p i t t s d e s i g n (schematic 2) and enclosed i n a spun copper can to reduce r . f . leakage, i s r e a d i l y tuneable over a f a i r l y l a r g e range (up to about 12 Mhz) but was u s u a l l y operated at about 6 Mhz. P a r t of the continuous output f r o m the o s c i l l a t o r passes v i a a buffer a m p l i f i e r into an attenuator p h a s e - s h i f t e r network to provide the r e f e r e n c e s i g n a l f o r the phase sensitive detection. The r e m a i n d e r of the o s c i l l a t o r output i s fed into the Gate circuit which consists of a grounded grid planar triode (as in schematic 2) as suggested by Blume^', The gate is turned on and off by the pulse train coming from the pulse sequence generators described above. From the gate circuit the pulses of r.f . , as shown in F ig , III. 3, pass into the r . f . power amplifier (shown in schematic 3) where they first pass through a cathode follower and a tuned prepower amplifier before passing into the power amplifier tube .829B. The pre power amplifier circuit differs considerably from McLachlan* s circuit. The output of the final (tuned) stage is such that during an r . f . pulse the input to the 829 B is on an adjustable positive pedestal chosen so as to give maximum linear gain in the 829 B. In the absence of the r .f . pulse the input to the 829 B is held well below the cut off voltage thus suppressing any possible r . f . leak through the 829 B between pulses. This biasing system worked quite satisfactorily for short pulses and r . f . bursts of up to 2 Kv peak to peak were obtained at the 829 B plate. However the shape of the bias pedestal begins to suffer for long pulses due to the RC time constants in its circuitry. As it was never found necessary to use pulses longer than 20-30 usee, this feature was not a point of annoyance. The transmitter coil, wrapped on a teflon coil former (see section III.A.ii), forms part of a tuned plate amplifier with the 829 B . The preamplifier used was that built by McLachlan 1. Ref. 7 and ref. 8. following C l a r k ' s design. (The quenching c i r c u i t d e s c r i b e d by C l a r k was never used and this p art of the c i r c u i t was disconnected). F r o m the p r e a m p l i f i e r (schematic 4), which contains two sets of c r o s s e d diodes to l i m i t the l a r g e induced voltage during an r . f . pulse (thus reducing the o v e r l o a d r e c o v e r y time), the si g n a l p a s s e s into the M a i n a m p l i f i e r . T h i s wideband a m p l i f i e r , an A h r e n b e r g WA600D m o d i f i e d as d e s c r i b e d by M c L a c h l a n , has five stages of cascaded tuned plate a m p l i f i e r s . A t the input to the sixth stage the s i g n a l i s added a l g e b r a i c a l l y to the r e f e r e n c e s i g n a l (coming f r o m the phase sh i f t e r and attenuator) and the subsequent s i g n a l plus r e f e r e n c e then undergoes another eight stages of r . f . gain before p a s s i n g into a f u l l wave detector and three stages of d. c. sig n a l a m p l i f i c a t i o n . Two m a j o r advantages are gained by this p r o c e s s : f i r s t l y the addition of a l a r g e r e f e r e n c e s i g n a l to the v e r y s m a l l n. m. r . s i g n a l moves the o v e r a l l voltage signa l into a l i n e a r response r e g i o n of the f i n a l r . f . stages and detector of the A h r e n b e r g a m p l i f i e r ; secondly, with the use of the r e f e r e n c e si g n a l , phase sensitive detection i s achieved having as a f i r s t benefit an immediate i m p r o v e m e n t of J~2 i n the S/N. T o i m p r o v e the S/N even f u r t h e r , the detected signa l i s u s u a l l y fed into a boxcar i n t e g r a t o r . The c i r c u i t (schematic 5) used for this device was e s s e n t i a l l y that of B l u m e ^ . The theory of i t s (2) o p e r a t i o n i s e x c e l l e n t l y presented by Hardy* '. The boxcar 1. Ref. 7 and r e f . 8. 2. Ref. 20. i n t e g r a t o r i s b a s i c a l l y an e l e c t r o n i c switch which i s turned on for a set time by the pulses f r o m the boxcar gater allowing the si g n a l to pass while the gate i s on, into a simple R C i n t e g r a t o r c i r c u i t which averages the voltages obtained d u r i n g s u c c e s s i v e sampling i n t e r v a l s . The voltage a c r o s s the capa c i t o r , which i s c a r e f u l l y guarded against p o s s i b l e low r e s i s t a n c e leaks to earth, i s m o n i t o r e d continuously by the v e r y high input impedance e l e c t r o m e t e r tube and, after a stage of d. c. gain and d. c. offset b i a s , d i s p l a y e d on a s t r i p c hart r e c o r d e r . (ii) The c r y o g e n i c s y s t e m A s shown i n F i g . III. 4., the sample h o l d e r - c o i l f o r m e r i s supported by a c e n t r a l v e r y thin w alled s t a i n l e s s s t e e l tube which s e r v e s a l s o as a ground r e t u r n line f o r the pickup and t r a n s m i t t e r c o i l s . Mounted on this tube a l s o i s a s e r i e s of copper baffle plates which act as heat exchangers with the h e l i u m gas. P o w e r i s c a r r i e d to the t r a n s m i t t e r c o i l f r o m the t r a n s m i t t e r v i a another thin w alled s t a i n l e s s s t e e l tube r i g i d l y h e ld f r o m the c e n t r a l tube by s e v e r a l p erspex s p a c e r s . The v e r y low l e v e l s i g n a l i s s i m i l a r l y c a r r i e d away f r o m the pickup c o i l by a t h i r d thin w alled tube supported i n s i d e the c e n t r a l grounded tube by perspex s p a c e r s , thus p r o v i d i n g some degree of shielding f r o m the t r a n s m i t t e r l e a d and other noise s o u r c e s . The three s t a i n l e s s s t e e l tubes were s i l v e r plated by a l o c a l e l e c t r o p l a t i n g f i r m so as to provide high e l e c t r i c a l conductivity i n the r . f . s k i n depth r e g i o n while maintaining the low t h e r m a l conductivity of the st a i n l e s s s t e e l . T hese tubes r e p l a c e d the 24 A W G copper w i r e used by M c L a c h l a n and provide for much higher m e c h a n i c a l s t a b i l i t y (and hence a reduction i n spurious noise such as that due to nitrogen bubbling as ob s e r v e d by M c L a c h l a n ) while giving l e s s l o s s y t r a n s m i s s i o n of r . f . power into and out of the c r y o s t a t without presenting a l a r g e t h e r m a l leak. F i g . I l l , 5 shows a detai l e d drawing of the sample holder and c o i l f o r m e r unit. The o v e r a l l dimensions of the unit are 0. 9" O, D. by 2, 0" height - a tight but not i m p o s s i b l e f i t into the n a r r o w t a i l p i e c e of the h e l i u m dewar. The entire unit i s e a s i l y detachable f r o m the s t a i n l e s s s t e e l support s y s t e m thus allowing ready a c c e s s f o r sample or pickup c o i l replacement. The t r a n s m i t t e r c o i l , c o n s i s t i n g of 7*/4 turns of 22 H F copper wi r e i s l a i d into machined grooves on the outside of the teflon f o r m e r so as to provide as much m e c h a n i c a l r i g i d i t y as p o s s i b l e . M a c h i n e d i n the same f o r m e r but w e l l below the sample a r e a i s a groove to accommodate a "bucking c o i l " . T h i s c o i l (4*/4 turns of 26 H F copper wire), wound i n s e r i e s with, but i n the opposite sense to, the pickup c o i l has the function of reducing the mutual inductance between 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 y s t e m s . Although this bucking c o i l has the disadvantage of reducing the total n. m. r , s i g n a l voltage p o s s i b l e at the p r e a m p l i f i e r , i t a l s o reduces the over l o a d i n g of the r e c e i v e r by the t r a n s m i t t e r and thus cuts down the r e c o v e r y t i m e . The sample i t s e l f i s positioned i n s i d e the t r a n s i t t e r c o i l f o r m e r by means of other t e f l o n h o l d e r s . T h e s e teflon h o l d e r s can be custom made to f i t the p a r t i c u l a r sample being studied. Shown i n F i g , III, 5 i s the simple s y s t e m used f o r the i i s o t o p i c a l l y pure t i n sample. F o r this sample the pickup c o i l was separated f r o m the sample by a thin l a y e r of m y l a r . The number of turns on the pickup c o i l was found e m p i r i c a l l y by r e q u i r i n g that the p r e a m p l i f i e r could be tuned at the d e s i r e d 1 1 9 frequency. ( F o r the 'i.sot o p i c a l l y pure S n sample, 1 0 0 turns of 4 0 SWG copper w i r e gave tuning of the p r e a m p l i f i e r at 6. 6 Mhz.) It was found quite e a r l y i n the low temperature e x p e r i m e n t a l p r o g r a m that the bothersome magnetoacoustic o s c i l l a t i o n s which plagued M c L i a c h l a n could be, to a l a r g e degree, damped out by coating the pickup c o i l and sample h o l d e r with g l y c e r i n e just p r i o r to pre cooling the s y s t e m f o r a h e l i u m t r a n s f e r . The g l y c e r i n e , which f r e e z e s above l i q u i d n i trogen t e m peratures, apparently has two functions; f i r s t l y , upon f r e e z i n g , i t p r o v i d e s f o r an additional m e a s u r e of m e c h a n i c a l r i g i d i t y (thus e l i m i n a t i n g p o s s i b l e motion of the pickup c o i l r e l a t i v e to the magnetic field) and secondly i t would appear to act as a good, l o s s y acoustic m a tch with the s p e c i m e n (thus p r o v i d i n g damping of any induced magnetoacoustic o s c i l l a t i o n s ) . The dewar s y s t e m i t s e l f c o n s i s t s of a r i g i d l y mounted standard n a r r o w t a i l e d glass h e l i u m dewar fitted i n s i d e a l a r g e r outer n i t r o g e n dewar. Pumping on the h e l i u m i s achieved with a 3" l i n e to a l a r g e remote pump while the temperature i s m e a s u r e d by mo n i t o r i n g the vapour p r e s s u r e with an o i l manometer. When the s y s t e m was not being pumped on, the c r y o s t a t i s opened to the h e l i u m r e t u r n line v i a an o i l bubbler which maintains a slight o v e r p r e s s u r e i n the r e t u r n l i n e . (B) The method In this s ection the salient features of the techniques adopted f o r the v a r i o u s measurements p e r f o r m e d are d i s c u s s e d . M u c h of the b a s i c theory u n d e r l y i n g the methods used has a l r e a d y been pr e s e n t e d i n chapter II; the d i s c u s s i o n here i s m a i n l y r e s t r i c t e d to the e x p e r i m e n t a l situation. A n a n a l y s i s of the s u c c e s s e s and f a i l u r e s of the techniques to be presented here w i l l be given i n the chapter on r e s u l t s , (i) M e a s u r e m e n t s of absorption and d i s p e r s i o n modes If the total F,I,D, t a i l following the excitation of the spin s y s t e m by a short r . f . pulse could be pa s s e d into the boxcar i n t e g r a t e r (by using a boxcar gate whose length was much l a r g e r than T ), the averaged output of the boxcar i n t e g r a t o r as the f i e l d (1) i s swept through the resonance, can be shown to be a l i n e a r combination of the unsaturated a b s o r p t i o n and d i s p e r s i o n modes X"(OJ) and x'C^)* suitably choosing the phase of the r e f e r e n c e r . f . s i g n a l either x'(txi) O R X"^) c a n ^ e m e a s u r e d d i r e c t l y -complete separation of the two modes i s p o s s i b l e , unlike the steady state e x p e r i m e n t a l case. Thus i f the magnetic f i e l d i s swept l i n e a r l y i n time, i t i s p o s s i b l e to plot either \'{{X)) o r x"(uu) d i r e c t l y . T h e s e measurements would be equivalent to those obtained by steady state techniques. Unfortunately the short T found i n meta l s means that much of the F.I.D. i s i r r e t r i e v a b l y l o s t i n the r e c o v e r y p e r i o d (2) of the r e c e i v e r following the r . f . pulse. A s M c L a c h l a n shows, 1. Ref. 20 and ref. 27. 2. Ref. 27, r e f . 28, r e f . 29. the loss of much of the initial part of the F . I . D . (the largest part of the signal) leads to considerable instrumental distortion -the signal begins to resemble the many-wiggled envelopes obtained when the field is swept with a narrow boxcar gate. As a result, no quantitative measurements were made of the line shapes (these had previously been carried out by M c L a c h l a n ^ (2) and by S .N . Sharma* ' on the same crystal in any case); it was decided to devote more time to the direct measurements of T 119 and T m S n , the previously obtained values of which were (3) m some controversy . measurements As has already been pointed out, the standard TT-TT/2 pulse sequence often employed in non conducting samples for measuring the spin-lattice relaxation time cannot be defined for metallic single crystal samples. McLachlan showed that when the delay between a pulse of sufficient length to saturate the nuclear spin system completely, (T > T » T ) and a second short pulse (T ) is swept linearly in time, the amplitude of any point on the F . I . D . tail following the second pulse increases with exponential time constant T^. As was shown in IIB(vi) however, the first pulse need not be of such long duration - it need not saturate the spin system. In fact, although there will be values of the pulse lengths which lead to maximum signal amplitudes, any pair of r . f . pulses would appear to give the exponential dependence on the pulse separation t^  indicated in II. 68 (provided the field is kept exactly on resonance). 1. Ref. 27, ref. 28, ref. 29. 2. Ref. 37. 3. See ref. 12, ref. 27 and ref. 35. Thus the b a s i c technique adopted for T^ m easurements was as f o l l o w s . A f t e r setting the magnetic f i e l d at the resonant value (as outlined i n the next section), a two pulse s e q u e n c e ^ was e s t a b l i s h e d with the length of the second pulse being chosen e m p i r i c a l l y so as to give the best F.I.D. s i g n a l amplitude i n the absence of the f i r s t pulse. The boxcar gate, t r i g g e r e d f r o m the pulse 2 generator was fixed m anually (mode 1) i n the F.I.D^s, of the second pulse so that i t was w e l l away f r o m r e c o v e r y time d i s t o r t i o n and yet giving a good clean s i g n a l . With the i n i t i a l s e paration (t = t ) of the two pulses being * held at some f i x e d value (much l e s s than T but l a r g e r than T ), 1 2 the length of the f i r s t pulse was adjusted so as to give m i n i m u m boxcar s i g n a l . The f i r s t pulse was then removed and the (2) r e s u l t a n t s i g n a l ( c orresponding to infi n i t e pulse separation ) r e c o r d e d . Thus the two endpoints t = t and t s <» 1 o 1 c o r r e s p o n d i n g to m i n i m u m and m a x i m u m boxcar s i g n a l were optimize d . The s e p a r a t i o n between the two r . f . pulses was then l i n e a r l y i n c r e a s e d i n time using the v e r y slow ramp generator with mode 2 f o r the comparator l e v e l of the pulse 2 generator as d e s c r i b e d i n section I I I . A . i . When the r e s u l t i n g exponential r i s e had e s s e n t i a l l y flattened out ( t ^ — 3 T ^ ) , the f i r s t pulse was again r e m o v e d i n o r d e r to check the t^ = 0 0 reading again - thus p r o v i d i n g a ready check on the baseline s t a b i l i t y of the whole system. The resultant c h a r t r e c o r d , a t y p i c a l t r a c e of which i s 1. See ref, 12, ref. 27 and ref, 35, 2, The r e p e t i t i o n p e r i o d being much longer than T the pulse se p a r a t i o n i s e f f e c t i v e l y i n f i n i t e (complete decay) when the f i r s t pulse i s removed. shown i n F i g . III. 6, was then plotted on s e m i l o g a r i t h m i c paper as shown i n F i g . III. 7 and the spin l a t t i c e r e l a x a t i o n time d e t e r m i n e d f r o m the slope and equation II. 68. (iii) T m e a s urements v ' 2  Two b a s i c methods were employed to m e a s u r e the s p i n -spin r e l a x a t i o n time. The f i r s t method was by studying the F.I.D. t a i l of a single pulse and the second was by m e a s u r i n g the decay of a spin echo with the pulse separation of a two pulse sequence. When an r . f . pulse i s applied to the n u c l e a r spin s y s t e m at a frequency (w + A) just off the resonant frequency UJ , the s i g n a l o o induced i n the pickup c o i l following the r . f . pulse i s g e n e r a l l y an exponentially damped sine wave at the L a r m o r frequency OJ of o the spin system. (See F i g . III. 8.) When this s i g n a l i s m i x e d with the r e f e r e n c e s i g n a l i n the m a i n a m p l i f i e r of the r e c e i v e r , a sequence of beats at the d i f f e r e n c e frequency A uu = A with the same exponential envelope, i s generated. Thus, when detected, the output s i g n a l c o n s i s t s of an exponentially damped sine wave at the beat frequency. The number of c y c l e s ( i . e . beats) appearing i n the decay i s c l e a r l y a reasonably sensitive m e a s u r e of how f a r the magnetic f i e l d i s off resonance. B y i n c r e a s i n g the f i e l d f r o m below resonance, the beat frequency starts to d e c r e a s e (when obs e r v e d on an o s c i l l o s c o p e , the o s c i l l a t i o n s seem to move away f r o m the pulse - the pattern appears to open out) and, as the resonant f i e l d i s n e a r e d the amplitude of the envelope i n c r e a s e s u n t i l , at resonance, the s i g n a l envelope i s at a m a x i m u m and no beat o s c i l l a t i o n s are p r e s e n t ^ . A s the f i e l d i s i n c r e a s e d above resonance, the beat  1. T h i s i s the case where the r e f e r e n c e phase i s set to give the a b s o r p t i o n mode. When operating i n the d i s p e r s i o n mode, the s i g n a l vanishes at the resonant f i e l d . frequency g r a d u a l l y i n c r e a s e s (the o s c i l l a t i o n s seem to co l l a p s e onto the pulse) and the envelope amplitude d e c r e a s e s , (in chapter I V B ( i i i ) , s e v e r a l o s c i l l o g r a p h s of this p r o c e s s are shown.) When working at l i q u i d h e l i u m temperatures i t was g e n e r a l l y p o s s i b l e to see the off resonance beat patterns quite e a s i l y on an o s c i l l o s c o p e plugged into the detected output of the m a i n a m p l i f i e r . The r a t h e r sensitive dependence of the beat frequency on magnetic f i e l d made for a rather convenient and quite accurate method of setting up the exact resonance conditions. (a) F.I.D. measurements The m o s t convenient method of m e a s u r i n g the F.I.D. T was to sweep a n a r r o w boxcar gate through the entire F.I.D. t a i l when the f i e l d was h e l d sl i g h t l y off resonance r e s u l t i n g i n a chart r e c o r d i n g s i m i l a r to that shown i n F i g . III. 9. A plot of the envelope function against sweep time on s e m i l o g a r i t h m i c paper then gave a straight lin e as i n F i g . III. 10, the slope of which gives the s p i n - s p i n r e l a x a t i o n time. When working at low temperatures, the S/N obtained was always s u f f i c i e n t l y high that i t was never n e c e s s a r y to use M c L a c h l a n ' s f a r m o r e tedious method of sweeping the magnetic f i e l d through resonance f o r many fi x e d values of boxcar delay and then plotting the envelope peak amplitude against the boxcar delay. M o r e o v e r , the present technique, allowing an e x p e r i m e n t a l d e t e r m i n a t i o n of T^ i n as l i t t l e as five minutes, m i n i m i z e d the e v e r present danger of slow d r i f t s i n s y s t e m gain, sample temperature and other e x p e r i m e n t a l p a r a m e t e r s , (b) Spin echo m e a surements With the spi n echo technique, one expects to be m e a s u r i n g only the true i r r e v e r s i b l e s p i n - s p i n r e l a x a t i o n time. In this r e g a r d the technique i s s u p e r i o r to the F.I.D. method but i t s u f f e r s , at l e a s t at present, f r o m the fact that i t i s somewhat more tedious to apply. By i n c r e a s i n g the applied f i e l d inhomogeneity by a f a i r l y l a r g e amount (by moving the sample into the f r i n g e f i e l d of the magnet, f o r example) one can f o r c e the F.I.D. s i g n a l to decay i n a s m a l l f r a c t i o n of the true time constant. The technique f o r o b s e r v i n g spin echoes i s then as follows: A f i r s t pulse i s applied for a time T which has p r e v i o u s l y been d e t e r m i n e d to give the best F.I.D. s i g n a l . A second pulse of width T 2 i s then applied a time t^ , l a t e r . A n echo should then appear at a time 2t^ f r o m the f i r s t pulse. The length of the second pulse i s then v a r i e d to give a m a x i m u m i n the echo S/N. (It was u s u a l l y found that the signals obtained were l a r g e enough to allow this o p t i m i z a t i o n to be c a r r i e d out on an o s c i l l o s c o p e . ) It was u s u a l l y found to be b e n e f i c i a l to adjust the phase of the r e f e r e n c e s i g n a l so that the echo shape looked l i k e two F.I.D. t a i l s back to back when at the resonant f i e l d . ( T h i s i s i l l u s t r a t e d i n F i g . IV. 10.) A g a i n as i n the F.I.D. case, as the f i e l d i s changed f r o m the resonant value, beat o s c i l l a t i o n s o c c ur - both to the right and l e f t of the center of the echo. The i d e a l method f o r m e a s u r i n g T^ using the spin echo decay would be to sweep the second pulse away f r o m the f i r s t and simultaneously, and at the same rate, to sweep the boxcar gate away f r o m the second so that the boxcar gate would r e m a i n always fi x e d on the same r e g i o n of the echo. T h e n a graph of the echo amplitude as m e a s u r e d by the boxcar against the pulse separation would i m m e d i a t e l y give T . However, there was i n s u f f i c i e n t time to rig such a system up so the changes in pulse separation and boxcar gate delay were carried out manually. The main difficulty which then arose was that of keeping the magnetic field constant throughout such a series of trials. Thus as an alternate method, at each pulse separation and boxcar gate delay, the magnetic field was swept through resonance and the peak amplitude of the resultant envelope plotted against pulse separation. FIGURE III 1 BLOCK DIAGRAM OF SPECTROMETER The figure shows a t y p i c a l two pulse sequence as might be used for either s p i n - l a t t i c e r e l a x a t i o n time or spin echo measurements. The waveforms, shown are exaggerated but indicate the appearance of the s i g n a l on an o s c i l l o s c o p e at the appropriate point following apulse f r o m the repetition rate t i m e r . A l l the waveforms shown are repeated at the frequency of the r e p e t i t i o n rate t i m e r . The lower turnings on fiiepickup c o i l r e p r e s e n t the bucking c o i l mentioned in the text. I Po\St •z. a, - =3=~ 5 W 1TCW h. ?<A<,<-winter A m p l i f i e r • Slou> C<.y\CraVor 4o fcoxcar iViVrcj cafe* S U K T C f t S . F I G U R E III 2 B L O C K D I A G R A M O F P U L S E S E Q U E N C E  G E N E R A T O R . The figure shows the n o r m a l arrangement of the pulse sequence generator as i t might be used for a f i e l d sweep with a fixed separation between the two r . f . pulses and a boxcar gate following afixed delay behind the second pulse. By switching A to mode 2, the second pulse may be swept away f r o m the f i r s t with the boxcar gate maintaining the same fixed delay after the secondpulse ; by switching B to mode 2, the boxcar gate may be swept away f r o m the second pulse. A l l the pulse and ramp generators are T e k t r o n i x units ; the type number in d i c a t e s the m odel . (^denotes vnits m o d i f i e d as d e s c r i b e d i n the text.) 6 ? 67. 6-3 v»t** A< FWArtlUITS 1 1 1 ol CM3 8 , -V70 00uT5. SCHEMATIC 2 R. F. OSCILLATOR AND GATE 1 IJ>E.ftLliE5> REPETITION I CarwPAKftVOt? 6JTPUT n ©uTPoT F I G U R E III 3 T I M I N G C I R C U I T W A V E F O R M S The fi g u r e shows i d e a l i z e d waveforms on a common time scale at v a r i o u s points i n the t i m i n g c i r c u i t as would appear when the second pulse and boxcar gate (with fixed delay ) are being swept away f r o m the f i r s t pulse. ( Switch A of F i g u r e I I I 2 in mode 2, switch B i n mode 1.) «•=•=>=<! 8 5 2.2-5" ooitf3 wwxerc HI' 6 6 « SCHEMATIC 3a R. F. A M P L I F I E R H.T. I. \ J 6 U T S . ( f — A A M 4 * K l - A M H >0^ 'W> "V<tM\siv«Hec S co\V. S SCHEMATIC 3b POWER A M P L I F I E R -€>.S ©• SCHEMATIC. 4 " PREAMPLIFIER TO M A * M fop? TO 2i« S C H E M A T I C 5a B O X C A R I N T E G R A T O R (Input and gate ) \12S OOV.T5 S C H E M A T I C 5b B O X C A R I N T E G R A T O R ( Integrator and output ) ]*" - — " -»— TO TRAt« M \ T T « F I G U R E III4 C R Y O G E N I C S Y S T E M The basic cryogenic and sample support sys tem i s shown. The t r ansmi t t e r tube ( mounted away f rom the others by perspex insu la tors ), together with the earth tube and(concentric ) pickup ceil tube, was made f rom s i l v e r plated s ta inless ste'eltube as desc r ibed i n the t ex t . (Not to sca le . ) tltKOT con- TOfte EM?.m -ruse J > U V T E TRftNSfAlTKJR. C O I L . 6 £ 5 „ »q SPrMPue TSPf>lSW\TTe«. C o n . SOCK reJOv c o t i . C^TMEMTEt) INTO 6f^W$ 3ttUP€T? INSULMDY? F I G U R E HI 5 S A M P L E H O L D E R The figure shows detai ls of the mounting used for the i so top ica l ly pure S n l l 9 sample . By unsolder ing the t ransmi t te r lead and the p i c k -up c o i l lead ( at the brass connector ) and removing the three mounting sc rews , the sample unit may eas i ly be removed f rom the cryosta t support sys tem, thereby a l lowing ready access to the sample . 0 *« 30 HO 5 o feo 70 80 <\o F I G U R E III6 T 1 C H A R T R E C O R D T L a figure shows a t r a c i n g of a t y p i c a l chart r e c o r d i n g of a T r u n on i s o t o p i c a l l y pure S n ^ ^ at 1.3 dK. The boxcar gate was 8t±sec. l o n g and delayed by 50 usee, f r o m the second pulse . The time scale (pulse s e p a r a t i o n i n c r e a s i n g f r o m x=2. 5 ) i s 1. 2 usee, per d i v i s i o n ; the v e r t i c a l scale ( si g n a l amplitude i n c r e a s i n g with y ) i s i n a r b i t r a r y units. The i n i t i a l flat p o r t i o n of the r e c o v e r y t a i l (x & 2. 5 ) was due to remenants of the F I D t a i l of the f i r s t pulse. The i n i t i a l (x< 2. 5 ) and f i n a l ( x> 100 ) segments of the chart give the 'infinite " pulse sep-a r a t i o n sign a l ( pulse 1 removed ) as d e s c r i b e d i n the text. F I G U R E III 7 P L O T O F D A T A T h e f i g u r e shows a t y p i c a l s e m i l o g a r i t h m i c p lo t of the s i g n a l a m p l i t u d e aga ins t^ l^e p u l s e s e p a r a t i o n fo r i s o t o p i c a l l y pu re Sn at 1. 3 K . ( T h e da ta for s u c h g r a p h s i s ex t ra ic t ed f r o m c h a r t r e c o r d s s u c h as s h o w n i n the p r e c e d i n g f i g u r e . ) T h e s lope of the g r a p h g i v e s the s p i n - l a t t i c e r e l a x a t i o n t i m e as d e s c r i b e d i n the t ex t . i dea l i zed p r eampl i f i e r | output; reference s ignal ; 3 - S 5 \ ^ C U J & * & H reference s ignal + nmr s ignal ( dropping the out of phase comp-onent cosu 0 t ); detected in phase s ignal ; FIGURE III 8 SIGNAL AND REFERENCE WAVEFORMS The figure shows the i dea l i z ed waveforns at var ious points i n the phase sensi t ive detector and i l l u s t r a t e s how the T FID waveforms are generated J ! ! | 3 ! i 1 ! i i ! i I ! I I I L x =s-F I G U R E I I I 9 T Y P I C A L T C H A R T R E C O R D T h e f i g u r e s h o w s a t r a c i n g t y p i c a l c h a r t r e c o r d o f a T^ FID t r i a l o n i s o t o p i c a l l y p u r e S n * a t a b o u t 1. 3 K „ T h e b o x c a r g a t e w a s 5 [.(.seconds l o n g , t h e R C t i m e c o n s t a n t w a s 0. 1 m i l l i s e c o n d s a n d the s w e e p t o o k a b o u t 30 m i n u t e s . T h e t i m e s c a l e i s 20 ^seconds p e r d i v i s i o n ( the d e l a y a t x=2. 3 w a s 100 u s e c o n d s ) ; t h e s i g n a l 1 a m p l i t u d e i s i n a r b i t r a r y u n i t s . A t x=7. 5 a p h a s e c h a n g e i n t h e r e f e r e n c e s i g n a l i s a p p a r a n t ; s u c h c h a n g e s w e r e n o t u n c o m m o n bu t a s t h e y do n o t c h a n g e t h e e n v e l o p e h e i g h t , t h e y m a y be d i s r e g a r d e d . Sao F I G U R E III 10 BOXCMI; j>eusy_ T Y P I C A L F I D T G R A P H The f i g u r e shows a p lo t of the l o g a r i t h m of t h e e n v e l o p e a m p l i t u d e a g a i n s t the b o x c a i j d ^ l a y fo r a t y p i c a l F I D T_> t r i a l on i s o t o p i c a l l y pure . Sn at about 1,3 K , T h e s lope of the g r a p h ( the da ta p o i n t s fo r w h i c h a r e t a k e n f r o m c h a r t r e c o r d s l i k e that s h o w n i n the p r e v i o u s f i g u r e ) g i v e s the s p i n - s p i n r e l a x a t i o n t i m e as des c r i b e d i n the text. CHAPTER IV : RESULTS The principle objective of this research was to continue McLachlan's^ y measurements on single crystal metallic samples - particularly in tin where his results indicated a spin echo T which was shorter than the F.I.D. T and a value 2 2 for the spin-lattice relaxation time which supported Asayama (2) and Itoh's* 7 value but disagreed with that of Spokas and S l i c h t e r ^ and the more recent value obtained by Dickson^ 7. Although McLachlan believed that cooling to low temperatures would not improve the S/N, the results of Gara* 7 suggested that large improvements are possible if care is taken to reduce noise from magnetoacoustical effects. It was found in o the present study that cooling to 4.2 K, and lower, gave significant improvements in the signal resolution; so much so that most n. m. r, signals investigated (including spin echoes) could be seen directly on an oscilloscope after phase sensitive detection (and in some cases even before this stage). Much of the work carried out was devoted to a qualitative study of the behaviour of the n. m, r, signals obtained in single crystal metallic samples at low temperatures. This section thus begins with a discussion of the qualitative results (the successes and failures of the apparatus and techniques used, the amplitudes of the various signals observed and their temperature, field and pulse length dependences) and continues 1. See ref. 29. 2. See ref. 3. 3. See ref. 41. 4. See ref. 12. 5. See ref. 14. with a p r e s e n t a t i o n of the quantitative m easurements of 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 t i m e s on an °ivsot o p i c a l l y 119 pure tin single c r y s t a l ( S n ). A. Qu a l i t a t i v e R e s u l t s  (i) A p p a r a t u s The c r y o s t a t , as d e s c r i b e d i n section IIIA(ii), p e r f o r m e d i t s functions quite s a t i s f a c t o r i l y . A t y p i c a l t r a n s f e r of 3 l i t e r s of l i q u i d h e l i u m (of which 1 l i t e r would be l o s t during t r a n s f e r ) would l a s t under continuous input of pulsed r . f . power f o r about 5 to 6 hours i f l e f t at 4. 2 ° K and of the o r d e r of 3 to 4 hours when pumped below the lambda point. The s i l v e r plated c o a x i a l tubes p r e v i o u s l y d e s c r i b e d worked v e r y w e l l i n reducing the heat input while p r o v i d i n g low e l e c t r i c a l l o s s e s at r . f . f r e q u e n c i e s . When pumped on with the m a i n 3" pumping l i n e , the h e l i u m could be l o w e r e d to 1. 15°K, m e a s u r e d by vapour p r e s s u r e t h e r mometry. The only s e r i o u s f a i l u r e of the c r y o g e n i c s s y s t e m was that of temperature m e a s u r e m e n t and c o n t r o l . In m e a s u r i n g the s p i n - l a t t i c e r e l a x a t i o n time i n m e t a l s , one i s u s u a l l y c oncerned with product T^T, It i s thus n e c e s s a r y to keep T constant and a c c u r a t e l y known throughout the m easurement p e r i o d . The a n a l y s i s of the T ^ T data (T being m e a s u r e d by m o n i t o r i n g the vapour p r e s s u r e ) suggested that the temperature was not i n fact always constant but could d r i f t by *v 10% (at times) over a m e a s u r e m e n t of T^ (which u s u a l l y took about 30 minutes). The often l a r g e scatter i n s u c c e s s i v e m e a surements of T ^ T compared with the scatter of points within a given m e a s u r e m e n t of T ^ T leads one to d i s t r u s t the absolute m e a s u r e m e n t of T, A n independent m e asurement of the temperature, such as with a carbon r e s i s t o r , should have been c a r r i e d out; i t i s unfortunate that this f a i l i n g of the apparatus was not r e c o g n i z e d during the data taking and was only d i s c o v e r e d after detailed a n a l y s i s of the data had been c a r r i e d out. . A s p r e v i o u s l y stated, the t r a n s m i t t e r and r e c e i v e r worked quite s a t i s f a c t o r i l y p r o v i d e d c a r e was taken to tune the v a r i o u s stages p r o p e r l y . F u r t h e r comment at this point i s u n n e c e s s a r y . (ii) Magnetoacoustic O s c i l l a t i o n s T r u e magnetoacoustic v i b r a t i o n s were never as s e r i o u s a p r o b l e m as M c L a c h l a n ' s observations would suggest. It i s b e l i e v e d that much of the " m a g n e t o a c o u s t i c a l " noise experienced by M c L a c h l a n was i n fact "magnetomechanical" noise and was i m m e d i a t e l y e l i m i n a t e d i n the p r e s e n t apparatus by m o r e secure sample mounting and the much m o r e r i g i d mounting of the l e a d f r o m the pickup c o i l to the p r e a m p l i f i e r . In M c L a c h l a n ' s case this was a loose S.W.G. No. 34 gauge copper wire; i n the p r e s e n t case, i t was the s i l v e r plated thin w a l l c o a x i a l tube. E v e n with this improvement, however, a si z e a b l e p o r t i o n of the s i g n a l following an r . f . pulse r e m a i n e d i n a c c e s s i b l e due to the p r e s e n c e of o s c i l l a t i o n s a r i s i n g f r o m beats between a m a g n e t o a c o u s t i c a l s i g n a l and the r . f . r e f e r e n c e frequency. These o s c i l l a t i o n s were r e c o g n i z e d as a r i s i n g f r o m magnetoacoustical effects by t h e i r monotonic i n c r e a s e i n amplitude with i n c r e a s i n g magnetic f i e l d as w e l l as t h e i r definite temperature dependence. T h e r e being no s e r i o u s p r o b l e m presented by these remnant o s c i l l a t i o n s , a concentrated attempt to eliminate them was not c a r r i e d out. It was, however, r e a d i l y found that, i n the case of the isotropically pure tin sample, by bonding the copper substrate of the sample to the teflon holder with glycerine (which freezes solid at liquid nitrogen temperatures) a very large reduction in these oscillations could be achieved. This reduction is presumably due to a damping of the longitudinal magnetoacoustic waves by impedance mismatching at the ends of the sample as . _ (1) discussed by Gara* A check was made at one stage to discover whether any remnant of the magnetoacoustic oscillation might accompany the nuclear spin echo. At magnetic fields just above and just below the nuclear resonant field, no such effect was detectable, (iii) Other materials; temperature dependence of S/N 119 Although the bulk of the work was carried out on S n , in the early stages of the research, several other materials were looked at and for the sake of comparison of signal resolution with that obta briefly here. ained by McLachlan at 77°K, these are reported 119 In addition to the Sn resonances, n.m, r, signals were observed in niobium single crystals, powdered aluminium, 19 63 65 glycerine, teflon (F ) and in both isotopes (Cu and Cu ) in the copper windings of the coil. Of these the copper resonances were, of course, by far the weakest and could be observed on the oscilloscope only at the lowest temperatures. Fig. IV. 1 shows o the passage through resonance at about 1.3 K. On the other hand, the fluorine signal from the teflon coil former was very strong and could be readily observed even at room temperature. For the sake of comparison, the teflon signal at 1.3°K is shown in Fig. IV2a while in Fig. IV2b the signal at room temperature is shown. 1. Ref. 14. It is indicative of the general improvements made to the electronics that it was possible to observe the F.I.D. signal in the single crystal niobium specimen at room temperature on the oscilloscope. The first photo of Fig. IV3 shows the signal plus recovery tail when the magnetic field is well below resonance. The second photo shows the situation when just below the resonant field; it will be noticed that there is just the slightest hint of beat oscillations extending out into the recovery tail. In the third photo the beat oscillations have all disappeared and an approximately exponential decay is observed. At slightly higher fields, as in the fourth photo, the beat oscillations have reappeared and continue to move in towards the start of the recovery tail as the field is further increased. Finally, as the field is increased well beyond resonance, the beat pattern is lost entirely and only the receiver's "dead time" following the r.f. pulse is observed, as in the final photo of Fig. IV3. o In cooling the sample to 77 K a considerable improvement in S/N was obtained (so much so that the resonance could be observed on the oscilloscope even without phase sensitive detection). The series of photographs in Fig. IV4 show the passage through o (1) the resonant field for the niobium sample at 77 K . The improvement in cooling to liquid helium temperatures was even more striking. In Fig. IV5, a sweep through the resonant 1. It might be noticed that the resonance does not appear symmetrical in magnetic field - the signal appears larger above than below the resonant value. This was due to a misalignment of the sample during this trial; the sample was not in a homogeneous region of the magnet. field for niobium at 4, 2 K is shown. (It should be noted that signals better by a factor of 2 or 3 were later achieved by optimizing the pulse length and by careful tuning of the transmitter and preamplifier.) Upon cooling below 4. 2°K, even further improvement in the signal resolution could be obtained as o shown in Fig. IVo, the resonance at 1.3 K. (iv) Qualitative description of the types of measurement There were four basic types of measurement which could be made with the pulsed apparatus; measurements of absorption and dispersion modes, F.I.D. measurements of T , two pulse c* sequence measurement of T^, and measurements on the nuclear spin echoes. Each of these types of measurement have been discussed theoretically in an earlier section and in this section the successes and failures of these techniques are discussed. (a) Dispersion and Absorption modes As a method for determining T^ via the line width, the technique of recording the absorption line with a sweep of magnetic field and a long boxcar gate was abandoned very early in favour of the more direct F.I.D. method. The absorption-dispersion modes were only looked at in the very early stages with the niobium specimens and were of very little interest because of distortion due to the relatively long recovery time and lack of field homogeneity. Although consecutive sweeps of magnetic field gave recorder tracings which differed point by point by less than 1 % , it was never possible to obtain a good approximation to either a pure absorption or a true dispersion curve; there were always "wings" above and below the resonant field due to the finite receiver recovery time and often some a s y m m e t r y as w e l l which was a s c r i b e d to poor f i e l d ..homogeneity over the sample. S i m i l a r l y as a technique f o r m e a s u r i n g (as used by M c L a c h l a n ) by m o n i t o r i n g the a b s o r p t i o n peak m a x i m u m (following the second of a two pulse sequence) as a function of the pulse separation, the method f a l l s f a r short of the r e l i a b i l i t y and convenience of the m o r e d i r e c t method outlined e a r l i e r and d i s c u s s e d below i n ( i i i ) . (b) F.I.D. m e a s u r e m e n t of T P e r h a p s the e a s i e s t of a l l the m easurements to make, the F.I.D. m easurements of T (obtained by sweeping a n a r r o w boxcar gate through the F.I.D. tail) were also the m o s t s u c c e s s f u l . A s shown i n F i g , III. 9, a t y p i c a l r e c o r d e r t r a c e f o r . i s o t o p i c a l l y pure t i n at T = 1.3°K, the noise l e v e l i s v e r y low indeed compared with the s i g n a l . Sudden changes i n the phase of the beat pattern were o c c a s i o n a l l y o b s e r v e d and were i n t e r p r e t e d as a r i s i n g f r o m j i t t e r i n the t i m i n g c i r c u i t s of the t r a n s m i t t e r . It was r e a s s u r i n g to find that, to within the s t a t i s t i c a l scatter of s u c c e s s i v e t r i a l s , the m e a s u r e d value of T was independent of whether the magnetic f i e l d was exactly on resonance. S i m i l a r l y as f a r as could be determined, the length of the e x c i t i n g r . f . pulse d i d not influence the value obtained although, of course, the S/N f e l l off quite r a p i d l y as the pulse length was changed f r o m the optimum value. (c) Two pulse sequence m e a s u r e m e n t of T The two pulse sequence with a m o d e r a t e l y s i z e d boxcar gate situated i n the F.I.D. t a i l of the second pulse p r o v e d to be a simple, yet effective, method of measuring the spin-lattice relaxation time. Again the S/N was improved by going to the lower temperatures although, as with the other measurements, the longer accompanying the lower temperatures necessitated a smaller repetition rate. Fig. III. 6 shows a typical trace obtained on the "isotopically pure tin sample at 1.3°K while o Fig. IV. 7 shows a similar trace on the same sample, but at 4.2 K. To analyse the data meaningfully it was necessary to know the signal at infinite pulse separation. Experimentally this was found by simply removing the first pulse at the end of each sweep of pulse separation. As the repetition rate was several times larger than T^, the signal level with the first pulse removed provided an accurate reference level for this purpose. (To provide a check that the system gain had remained constant during a given trial, the first pulse was removed both before and after the sweep; any difference between the levels then indicated a change in the system parameters.) It was found experimentally that the time constant of the exponentially rising signal (which should of course be T^) depended noticeably on the magnetic field. If the magnetic field was either above or below resonance, the measured time constant was shorter than that when on resonance. It was found that for a beat oscillation period of 50(is in the F.I.D. (corresponding to a few gauss off the resonant field) the measured value of the exponential time constant was down by a factor of \ of that obtained when exactly on resonance 1^,  1. Unfortunately this field dependence of the measured time constant was not revealed until a complete analysis of the data had been carried out and further experimental work impossible to perform. For all the measurements of T^ made, with the exception of the few used to investigate this field dependence, the magnetic field was held as accurately as possible on resonance. It i s fortunate f o r this method that the F.I.D. s i g n a l following the second (sampling) pulse could be e a s i l y seen on the o s c i l l o s c o p e . T h i s feature was used to ensure that the magnetic f i e l d was on resonance at a l l t i m e s d u r i n g a T^ measurement. It i s s t i l l f e a r e d , however, that the i n a c c u r a c y of loc a t i n g and maintaining resonance was a contributing cause of the r e l a t i v e l y l a r g e scatter between s u c c e s s i v e t r i a l s . The effect upon the m e a s u r e d value of the exponential time constant by changing the length of the f i r s t d i s t u r b i n g pulse was a l s o checked and i t was concluded that within the s t a t i s t i c a l s catter of s u c c e s s i v e t r i a l s , the pulse length had no effect upon the value obtained. The data f r o m each T ^ T t r i a l was analysed n u m e r i c a l l y using a l e a s t squares f i t p r o gramme and the r e s u l t s f r o m many t r i a l s were averaged using the standard deviation of each i n d i v i d u a l t r i a l as a s t a t i s t i c a l weighting factor to give a weighted 119 m ean value of T ^ T f o r the 'isot;opically pure t i n Sn sample. (d) Spin E c h o e s P r o b a b l y the mos t exciting feature of operating with m e t a l l i c single c r y s t a l s at low tempe r a t u r e s was that i t enabled the spin echoes to be d i s p l a y e d v i s u a l l y on the o s c i l l o s c o p e . Although the echoes could be dis t i n g u i s h e d above the noise only at s m a l l pulse separations at 4. 2°K, they were much c l e a r e r even at extreme pulse separations (out to s e v e r a l hundred 119 m i c r o s e c o n d s i n S n ) when the temperature was low e r e d below 4. 2°K. F i g . IV. 8 shows the appearance of a spin echo i n 119 the S n sample with pulse separations of 100, 150 and 200 useconds at 1.24°K. Needless to say the a b i l i t y to see the echoes so c l e a r l y pronounced at such l a r g e pulse separations (the echo i n the f i g u r e o c c u r s more than T seconds after the i n i t i a t i o n Ct of the s p i n - s p i n r e l a x a t i o n p r o c e s s ) made the ef f o r t of converting the s y s t e m to low temperature ca p a b i l i t y w e l l worthwhile. E a c h spin echo t r i a l took c o n s i d e r a b l y longer to p e r f o r m than did the F.I.D. t r i a l s because the pulse separation was i n c r e a s e d i n steps and at each step the boxcar gate had to be c a r e f u l l y adjusted m a n u a l l y so as to be centred a c c u r a t e l y on the echo. The u s u a l technique employed, and found quite s a t i s f a c t o r y though tedious, was to set up as a c c u r a t e l y as p o s s i b l e on the echo using the o s c i l l o s c o p e as an i n d i c a t o r and then to sweep the magnetic f i e l d through the resonance. The peak of the resultant envelope function could then be plotted on a l o g a r i t h m i c scale against the r . f . pulse separation to y i e l d the true T . A s a c* consistency check, at the end of each such s e r i e s of i n c r e a s i n g pulse separation, the pulse s e p a r a t i o n was r e t u r n e d to i t s s m a l l e s t value so that any changes i n s y s t e m gain or baseline d r i f t s , i f present, could be detected. (As the s e r i e s of t r i a l s n e c e s s a r y for a single m easurement of T could take up to an Ct hour, i t was f e l t n e c e s s a r y to p e r f o r m such a constancy check.) C l e a r l y a u s e f u l i m p r o v e m e n t to the apparatus would be to devise a t i m i n g c i r c u i t which would sweep both boxcar gate and the second pulse simultaneously so as to avoid this painstaking and tedious method. F i g . IV. 9 shows a t y p i c a l plot of the r e s u l t s obtained f r o m such a s e r i e s of f i e l d sweeps; the two data points at At = 420 us show a s m a l l change i n gain over the one hour this t r i a l took to p e r f o r m . The shape of the echo was b r i e f l y looked at by sweeping a n a r r o w boxcar gate through the echo. Although as f a r as could be d e t e r m i n e d f r o m the inhomogeneity reduced F.I.D. t a i l * following the f i r s t pulse (T 50 us) the f i e l d was at the resonance value and the r e f e r e n c e s i g n a l phase set to give the absorption mode, on either side of the echo peak the c h a r a c t e r i s t i c off resonance beat o s c i l l a t i o n could often be observed, A t y p i c a l t r a c e of such a t r i a l i s shown i n F i g . IV. 10 which shows the echo with a pulse s e p a r a t i o n of 205usec. It was often d i f f i c u l t to d i s t i n g u i s h the o s c i l l a t o r y pattern obtained with s l i g h t l y off resonant conditions f r o m that produced by allowing some of the d i s p e r s i o n mode si g n a l to be detected as w e l l . When the phase i s adjusted so that the r e c e i v e r detects only the d i s p e r s i o n mode, the echo vanishes at i t s center (when exactly on resonance) but has s y m m e t r i c a l o s c i l l a t i o n s on e i t h e r side. F o r a m i x t u r e of a b s o r p t i o n and d i s p e r s i o n modes on the other hand, the highest peak of the o s c i l l a t o r y pattern moves away f r o m the c e n t r a l p o s i t i o n and the resultant t r a c e may appear to be sl i g h t l y a s y m m e t r i c i n shape. When both f i e l d and phase are sl i g h t l y maladjusted i t i s v e r y d i f f i c u l t to decide which adjustment i s r e q u i r e d to r e t u r n to the optimum condition of pure ab s o r p t i o n mode (inherently the l a r g e s t echo s i g n a l for measurements) and exact resonant f i e l d . F o r the m e a surements p e r f o r m e d , i t was u s u a l l y found m o s t p r a c t i c a b l e to preadjust the magnetic f i e l d and r e f e r e n c e s i g n a l phase, using the single pulse decay as a v i s u a l m o nitor on an o s c i l l o s c o p e , to give the best F.I.D. t a i l . D u r i n g the subsequent s e r i e s of t r i a l s the r e f e r e n c e phase was le f t untouched although the f i e l d , i f i t were to be l e f t at the resonant value, was constantly checked before and after each t r i a l as there was a slight tendency fo r the magnet c u r r e n t supply to d r i f t . F o r completeness, the f i e l d dependent shape of the spin echo was a l s o looked at by situating a m o d e r a t e l y long boxcar gate over the echo center and sweeping the magnetic f i e l d through resonance. When the boxcar gate i s p r o p e r l y centered and the r e f e r e n c e phase c a r e f u l l y o p t imized, the expected * absorption mode s i g n a l (with the f i e l d s poiled T line width) 2 i s obtained as shown i n F i g . IV. 11. 119 B. Quantitative R e s u l t s ; i l s o t o p i c a l l y pure Sn The i L s o t o p i c a l l y pure t i n c r y s t a l , the same one as used by M c L a c h l a n , was a thin single c r y s t a l grown around a c y l i n d r i c a l copper c o r e . Unfortunately the symmetry axis £00l] makes angle of 28° with the c y l i n d r i c a l axis of the sample - f a r f r o m the optimum angle of 90°. It was decided to p r o c e e d with a v e r t i c a l l y mounted s p e c i m e n r a t h e r than attempt to mount a t i l t e d specimen, an or i e n t a t i o n which M c L a c h l a n found in t r o d u c e d l a r g e i n c r e a s e s i n a c o u s t i c a l noise while reducing at the same time the induced s i g n a l voltage. A l l m easurements were made at l i q u i d h e l i u m t e m p e r a t u r e s and, i n most cases, below the h e l i u m lambda point (2. 17 K ) . (i) S p i n - L a t t i c e R e l a x a t i o n T i m e The spin l a t t i c e r e l a x a t i o n time T was m e a s u r e d i n the manner d e s c r i b e d p r e v i o u s l y . A s the temperature was f a r lower than the 78°K used by M c L a c h l a n i t was not n e c e s s a r y to s p o i l the magnetic f i e l d to reduce T^ i n the manner he d e s c r i b e d ; indeed, while T was of the o r d e r of 50 m i l l i s e c o n d s , T was only s e v e r a l hundred m i c r o s e c o n d s . A weighted average over many t r i a l s at d i f f e r e n t t e m p e r a t u r e s ^ between 1.3°K and 4. 2°K gave the value T ^ T = 56 ± 4 m i l l i s e c deg. (2) i n good agreement with the recent m easurements of D i c k s o n o who obtained 52 ± 0.46 m i l l i s e c deg at 77 K. The present r e s u l t d i s a g r e e s quite m a r k e d l y with M c L a c h l a n 1 s value of 34 ± 2 m i l l i s e c deg (also at 77°K) and as the present r e s u l t was obtained f r o m the same c r y s t a l as M c L a c h l a n , i t i s strongly b e l i e v e d that Dickson's r e s u l t i s the c o r r e c t one. The r e l a t i v e l y l a r g e uncertainty i n the p r e s e n t r e s u l t , r e f l e c t i n g the s t a t i s t i c a l d e v iation between a l a r g e number of independent t r i a l s , i s b e l i e v e d to be p r i m a r i l y due to the p r o b l e m s p r e v i o u s l y mentioned i n m e a s u r i n g the temperature consistently and i n maintaining the exact resonant f i e l d . A n attempt to m e a s u r e the anisotropy i n T ^ T gave unconvincing r e s u l t s because of this l a r g e uncertainty. It i s however felt that with a few m i n o r improvements, the a c c u r a c y could be g r e a t l y i m p r o v e d and with the l a r g e S/N afforded by low temperatures, allowing a much more r a p i d m e a s u r e m e n t of T^T, an o r i e n t a t i o n study (given up by M c L a c h l a n as being p r o h i b i t i v e l y tedious with h i s m o re l a b o r i o u s method) should be quite fe a s i b l e and i n t e r e s t i n g . (ii) S p i n - s p i n R e l a x a t i o n T i m e The s p i n - s p i n r e l a x a t i o n time T^ was m e a s u r e d both by the F.I.D. method and by o b s e r v a t i o n of the spin echo decay. 1, It should be noted that, to within the s t a t i s t i c a l scatter of the t r i a l s , no temperature dependence of T ^ T over this range was observed. 2. Ref. 12. Although by f a r the e a s i e r m e a s u r e m e n t to p e r f o r m , the F.I.D. measurements showed cons i d e r a b l e scatter between s u c c e s s i v e t r i a l s although reasonable standard deviation of points within a given f r i a l . Weighted averages over many t r i a l s gave a value of T = 175 ± 18 us c* i n good agreement with M c L a c h l a n ' s value at the same orie n t a t i o n but at l i q u i d n itrogen t e m p e r a t u r e s . It should be noted that i n the present r e s u l t there was no n e c e s s i t y to c o r r e c t f o r l i f e t i m e broadening as was o r d e r s of magnitude l a r g e r than T at these low te m p e r a t u r e s . T h i s c o r r e c t i o n being c* f o r c e d upon M c L a c h l a n contributes to the uncertainty i n h i s r e s u l t p a r t i c u l a r l y as there i s strong evidence that h i s value of T^ was i n c o r r e c t . A s M c L a c h l a n had observed, i t was found that the deviation f r o m exponential decay m a n i f e s t s i t s e l f after a p e r i o d of the o r d e r of T i n the F.I.D. T h i s deviation, as shown i n F i g . III. 12 Ct o c c u r r e d quite c l e a r l y i n a l l of the F.I.D. measurements on the l i s o t o p i c a l l y pure t i n . A n o r i e n t a t i o n dependence of T having been c a r r i e d out by M c L a c h l a n , no attempt was made to a l t e r the sample mounting. M c L a c h l a n had found that the T m e a s u r e d by spin echo Ct decay was s h o r t e r than that obtained by the F.I.D. methods. One of the p r i n c i p a l objects of the p r e s e n t r e s e a r c h was to investigate f u r t h e r this unexpected state of a f f a i r s . F i g . IV. 9 shows a t y p i c a l 119 spin echo r e s u l t obtained f o r the ; i-sdt o p i c a l l y pure S n sample. It was r e a s s u r i n g , though perhaps a bit disappointing, to find that the value of T found f r o m the m o r e c a r e f u l l y p e r f o r m e d present spin echo m e a s u r e m e n t s was i n fact l a r g e r than the F.I.D. value. The mean of a s e r i e s of independent measurements, each weighted by th e i r s t a t i s t i c a l uncertainty, gave a value of T 2 = 390 ± 48 [is, Both M c L a c h l a n and S.N. Sharma, working with the same c r y s t a l as used i n the present r e s e a r c h , found that the 1 1 9 l i n e shape i n i s o t o p i c a l l y pure tin S n was L o r e n t z i a n f o r at l e a s t s e v e r a l line widths.Sharma has given the line width at m a x i m u m slope as a function of orientation r e l a t i v e to the magnetic f i e l d ; this quantity may be r e l a t e d to T = V§ by c* equation II. 22. Unfortunately i n the p r e s e n t work the or i e n t a t i o n was not known exactly although c l o s e to the b a s a l plane, so i n o r d e r to compare the p r e s e n t r e s u l t s with those of Sharma, the m a x i m u m and m i n i m u m values of T^ i n the b a s a l plane p r e d i c t e d by S harma 1 s values are calculated: T = 272 usee ± 11 usee 2 max ^ T . = 131 usee ± 6 usee 2 m m Although the F.I.D, value of T^ f a l l s n i c e l y into this range, the value m e a s u r e d by spin echoes l i e s w e l l above even the m a x i m u m value obtained by Sharma, One must thus conclude that the steady state and the F.I.D, determinations of line width (or T ) are i n e r r o r ; although one can envisage many defects i n the m e t a l l i c c r y s t a l which w i l l shorten the apparent r e l a x a t i o n time (without a c t u a l l y changing the i r r e v e r s i b l e s p i n - s p i n r e l a x a t i o n or true T ) i t i s not so easy to find one which w i l l d i n c r e a s e the r e l a x a t i o n time. A s the spin echo technique m e a s u r e s only the i r r e v e r s i b l e p a r t of the decay (which a r i s e s f r o m energy t r a n s f e r between the spins) i t i s felt that this p r e s e n t r e s u l t i s the m o r e r e l i a b l e m easurement of and that sample defects l e d to e r r o r s i n the previous determinations by M c L a c h l a n and Sharma. It i s not understood why M c L a c h l a n 1 s spin echo values f o r T were so low; one can only suppose that the low S/N with 2 which he had to work l e d to some unknown sys t e m a t i c e r r o r . H< H H< H 0 0 H > H o F I G U R E IV l a C u 6 5 R E S O N A N C E H ^ H Q The figure shows the s ignals o b s e r v e d at the input to the boxcar i n t e g r a t o r ^ var ious magnet ic f ie lds near the C u pj^pj r e s o n a n c e . T h e r . f . p u l s e , not shown 0 is at the right and t ime i n c r e a s e s to the left. ( It i s indicat ive of the sens i t iv i ty pj^pj of tbs spec trometer that these s ignals 0 could be seen on the ° s c i l l o s c o p e for the f i l l ing factor for the copper windings is e x t r e m e l y s m a l l . ) H > H 0 F I G U R E IV lb C u 6 3 R E S O N A N C E T H e f igure shows the s ignals observed H * H ^ a t the input to the boxcar integrator at var ious magnet ic f ie lds near the C u resonance . 0 H <H H«H 0 H< H, F I G U R E IV 2a T E F L O N A T The oscillograms-show the app. aranco of the teflon signal at various magnetic fields at a sample temperature of 1.3 K. The resonance is due to the Flourine in the sample holder. H£H 0 H i H 0 H«H. H < H 0 H«H H> H 0 0 H»H F I G U R E IV j b T E F L O N A T 293 K The oscil lograms show the Flourine resonance at various magnetic fields at room temperature. H«cH o H£H 0 H*H 0 IliH 0 F I G U R E IV 3 N b . A T R O O M T E M P E R A T U R E T h e f igure shows the de tec ted output of the m a i n a m p l i f i e r at v a r i o u s m a g n e t i c f ie lds for a N i o b i u m s a m p l e at r o o m t e m p e r a t u r e . T h e f i r s t and las t o s c i l l o g r a m s show the s i g n a l s below^and above the r e s o n a n c e f ie ld the Z and 4 show the s tar t of the beats just be low and above r e s o n a n c e and the 3 shows the F I D t a i l when a p p r o x i m a t e l y at r e s o n a n c e . S u p e r i m p o s e d on e a c h i s the r pulse and c l e a r l y v i s a b l e i s the r e c e i v e r dead t i m e of about 8 ^ s e c o n d s . r d H< H 0 F I G U R E IV 4 N b . A T 77*K 0 H£H0 T h i s f igure shows the de tec ted output of the m a i n a m p l i f i e r at v a r i o u s m a g n e t i c f ie lds for the N b . s a m p l e at l i q u i d N i t r o g e n t e m p -e r a t u r e s . A l t h o u g h the r e s o n a n c e i s m u c h m o r e c l e a r l y def ined( unfortunate ly a photo at the exact r e s o n a n c e was not taken ), the i n i t i a l par t of the s i g n a l i s o b l i t e r a t e d by l arge m a g n e t o a c o u s t i c o s c i l l a t i o n s . HfcH 0 nan 1*0 U<il I l s H . 0 I I » H I « H i I < H , H«H. ; I > H F I G U R E IV 5 N b . A I' T h i s f igure shows the N b . s i g n a l at v a r i o u s m a g n e t i c f i e lds but at 4. 2*K. T h e u p p e r m o s t photo was taken at z e r o m a g n e t i c f ie ld and shows the r e c e i v e r d e a d t i m e whiffh extends out about 8 usee . A l t h o u g h p o o r photographs , they indicate a m a r k e d i n c r e a s e in s i g n a l to no i se i f c o m p a r e d to F i g u r e IV 4. M o r e o v e r the m a g n e t o a c o u s t i c o s c i l l a t i o n s a r e seen not to be i m p o r t a n t ; they have been e f fec t ive ly d a m p e d by th$e g l y c e r i n e as d i s c u s s e d in the text. F I G U R E IV 6 N b . AT 1. 3 K T h i s f igure shows the s i g n a l at v a t i o u s m a g n e t i c f i e lds at a s a m p l e t e m p e r a t u r e of 1. 3°k. C o m p a r i s o n with F i g u r e IV 5 shows that s i g n i f i c a n t s i g n a l to no i se ra t io i m p r o v e m e n t s a r e p o s s i b l e by c o o l i n g the s a m p l e . J 1 1 ! I 1 ! ! 1 1 ! 1 I I I 1 I 1 i _ 5" io if ao F I G U R E IV 7 Sn T T R I A L A T 4. 2 K 119 • The figure shows a t y p i c a l chart r e c o r d i n g of a T^ t r i a l on Sn at 4. 2 K. The i n i t i a l 1.75 d i v i s i o n s give the " i n f i n i t e " s e p a r a t i o n s i g n a l ( pulse 1 r e m o v e d as d e s c r i b e d i n the text ). The time scale i s 3.43 m i l l i s e c o n d s per d i v i s i o n and the sweep of the pulse s e p a r a t i o n was initiated at x=2. 8. Pulse separat ion 100 usee . spin echo second pulse f irs t pulse * — 1 P u l s e separat ion ZOOusec. F I G U R E IV 8 S P I N E C H O E S IN Sn T h e o s c i l l o g r a m s show the detected output of the m a i n a m p l i f i e r at the boxcar in tegrator input. In ea^ch of the photographs a spin echo ( cons i s t ing of two T ^  F I D tai ls back to back ) o c c u r s at twice the pulse separat ion . T h e spiked o sc i l l a t i ons fol lowing each pulse are due to the magnetoacoust ic o sc i l l a t i ons s u p e r i m p o s e d upon the r e c e i v e r dead t i m e . ( Sn at IA»f*K. The time scale was 50 useconds per d i v i s i o n . ) F I G U R E IV 9 foo >usec. SPIN E C H O T_ IN Sn c 1 0 0 0 I 5 0 O 119 The figure shows a t y p i c a l graph of the spin echo amplitude ( obtained as d e s c r i b e d i n the text ) plotted against twice the r . f . pulse sep a r a t i o n . The two c i r c l e s at 420 ^seconds indicate the baseline d r i f t between the start ( y=9. 25 ) and the end ( y=9. 0 ) of the s e r i e s of t r i a l s . (Isotopically pure S n l l 9 at about 1. 3°K. ) ) i t i i i i t i I I ! I I 1 !_ ! 1 L 1 L 5 10 15~ . A«> -x. F I G U R E IV 10 SPIN E C H O S H A P E : S n 1 1 9 The figure shows a t r a c i n g of the chart r e c o r d i n g of the output f r o m the boxcar integr a t o r when a narr o w boxcar gate ( 5useconds ) i s swept through the spi n echo. The sample was i s o t o p i c a l l y pure S n ^ ^ at about 1.3°K. The pulse separation was ap p r o x i m a t e l y 20 5 useconds and the echo peak o c c u r s at x=10. 1 (415 ^ seconds after the i i r s t pulse ). The time s c a l e i s 16.4 useconds per d i v i s i o n . 119 F I G U R E IV 11 A B S O R P T I O N M O D E : SPIN E C H O IN Sn The f i g u r e shows the t r a c i n g of a chart recording o f the b o x c a r i n t e g r a t o r output when.the magnetic f i e l d was swept through resonance. The box c a r gate was much langer than the T * F I D length of the echo and the r e f e r e n c e phase was adjusted to give the abso r p t i o n mode as d e s c r i b e d i n the text. The pulse separation f o r this t r i a l was 250 useconds so that the spin echo o c c u r e d 500 useconds after the f i r s t pulse. ( S n ^ ^ at 1. 3*K ) F I G U R E IV 12 F I D T D E C A Y IN Sn 119 The figure shows the signal amplitude ( i n a r b i t r a r y units) of the FID t a i l against the boxcar cUelay i n the i s o t o p i c a l l y pure Sn 119 sample at about 1. 3 K. De v i a t i o n f r o m exponential decay becomes apparant after a p e r i o d of the o r d e r of T,. as mentioned i n the text. (q„8jusee Idvvi 1. A b r a g a m , A. " P r i n c i p l e s of N u c l e a r Magnetism", O x f o r d U. P r e s s (1961). 2. A n d e r s o n , A . G . and R e d f i e l d , P h y s . Rev. 116, 583 (1959). 3. A s a y a m a , K. and Itoh, J . , J.Phys. Soc. Japan, 1_7, 1065 (1962). 4. Bardeen, J . J . Chem. P h y s . 6_, 367 (1938). 5. Bloembergen, N. and Rowland, T. J. , P h y s . Rev. 97, 1679 (1955). 6. Bloembergen, N. and Rowland, T. J . , A c t a M e t a l , 731 (1953). 7. Blume, R. J. , Rev. S c i . In s t r . 32,, 1016 (1961). 8. Blume, R. J. , Rev. S c i . Instr. 32_, 554 (1961). 9. Butterworth, J . P h y s . Rev. L e t t e r s 5_, 305 (i960). 10. C a r r , H. Y. , and P u r c e l l , E.M. P h y s . Rev. 94. 630 (1954). 11. C l a r k , W.G. , Rev. S c i . In s t r . 3_5, 316 (1964). 12. D i c k s o n , E.M. , P h y s . Rev. 184, 294 (1969). 13. F r o i d e v a u x , C. , and A l l o u l , H. , P h y s . Rev. 163, 324 (1967). 14. G a r a , A.D. Ph.D. T h e s i s , Washington U n i v e r s i t y (1965). 15. G a s p a r i , G.D., Shyu, W. , and Das, T.P. , P h y s . Rev. 134A , 852 (1964). 16. G l a s s a r , M. L . , P h y s . Rev. 150, 234 (1966). 17. G o r t e r , C. J . , and Van V l e c k , J. H. , P h y s . Rev. TZ, 1128 (1947). 18. Hahn, E . L . , P h y s . Rev. TU 297 (1950). 19. Hahn, E . L . , P h y s . Rev. 80, 580 (1950). 20. Hardy, W.N. , Ph.D. T h e s i s , U n i v e r s i t y of B r i t i s h C o l u m b i a (1964). 21. J a c c a r i n o , V. and Yafet, Y . , P h y s . Rev. 133A, 1630 (1967). 22. Johnson, B.C., and G o l d s b e r g , W. J . , P h y s . Rev. 145, 380 (1966). 23. Kahn, S .A . , Reynolds, J . M . , and Goodrich, R . G . , Phys. Rev. 163, 579 (1967). 24. Kittel, C . "Quantum Theory of Solids" John Wiley and Sons Inc. , New York (1963). 25. Kubo, R. and Obata, Y . , J . Phys. Soc. Japan 11, 547 (1956). 26. Lowe, I. J . , and Norberg, R . E . , Phys. Rev. 107, 46 (1957). 27. McLachlan, L . A . , P h . D . Thesis, University of British Columbia (1966). 28. McLachlan, L . A . and Williams, D. L l . , Proc. of the XlVth Colloque Ampire, Ljubljana (1966). 29. McLachlan, L . A . , Can. J . Physics, 46_, 871 (1968). 30. Mansfield, P . , Phys. Rev. 151, 199 (1966). 31. Masuda, Y . , J . Phys. Soc. Japan 1_2, 523 (1957). 32. Obata, Y . , J . Phys. Soc. Japan 1_8, 1020 (1963). 33. Rowland, T . J . , Prog. Materials Science 9, 1 (1961). 34. Ruderman, M . A . , and Kittel, C . , Phys. Rev. 96, 99 (1954). 35. Schone, H . E . and Olsen, P . W . Rev. Sci. Instr. 36, 843 (1965). 36. Schumaker, R. T . Phys. Rev. 112, 837 (1958) 37. Sharma, S .N . , P h . D . Thesis, University of British Columbia (1967). 38. Shyu, W . M . , Gaspari, G . D . , and Das, T . P . , Phys. Rev. 141, 157 (1966). 39. Shyu, W . M . , Das, T . P . , and Gaspari, G . D . , Phys. Rev. 152, 270 (1966). 40. Slichter, C . P . , "Principles of Magnetic Resonance" Harper and Row Publishing Co. , New York (1963). 41. Spokas, J . J . , and Slichter, C . P . , Phys. Rev. 113, 1463 (1959). 42. Yafet, Y . , J . Phys. Chem. Solids 21, 99 (1961). 43. Ziman, J . M . "Electrons and Phonons" Oxford University Press (1963). 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0084871/manifest

Comment

Related Items