NUCLEAR MAGNETIC RESONANCE STUDY OF SINGLE CRYSTALS OF GALLIUM METAL by • MARKO IVAN VALIC "Diploma of Physics",University of Ljubljana,1966 . Yugoslavia A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR. THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics We accept t h i s t hesis as confirming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February,1970 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t he U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment 7 The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date ABSTRACT The nuclear magnetic resonance spectrum of gallium metal has been investigated i n si n g l e c r y s t a l specimens i n both low and high magnetic f i e l d s from 4.2°K to the melting point ( T ^ = 300°K) and then extended to the l i q u i d phase.Precise determinations f o r the two non-equivalent nuclear s i t e s have been made of (a) the e l e c t r i c f i e l d gradient (EFG) tensor and (b) the Knight s h i f t (K) tensor. The r e l a t i o n s h i p of these r e s u l t s to the c r y s t a l structure of gallium metal i s discussed i n d e t a i l . s o l The i s o t o p i c Knight s h i f t i n the s o l i d , K^ s o> increases l i n e a r l y with temperature from (0.132±0.004)% at 4.2°K to (0.155±0.004)% j u s t be-low T > m . In the l i q u i d , j u s t above T i m , K"!;iq = (0.453±0.003)% and decrea-MP . . ' MP iso ses very slowly with increasing temperature. These r e s u l t s are discussed i n terms of the Korringa r e l a t i o n s i n the s o l i d and l i q u i d phases,res-p e c t i v e l y . Implications of these r e s u l t s are developed with regard to changes i n the e l e c t r o n i c structure of gallium upon melting. The angular dependence of K agrees with the predicted angular depen SLXX dence f o r the Knight s h i f t anisotropy i n an orthorhombic enviroment and i s described with two anisotropy parameters defined as K^ and K^. No K-anisotropy i s found along the Y (B c r y s t a l ) axis. The p r i n c i p a l axes of the K tensor have d i f f e r e n t signs from the quadrupolar p r i n c i p l e axes. K (X) i s found to be very large, i . e . K (X)/K. = -18% at 77°K, an an iso and temperature dependent.Increasing the temperature (T) from 4.2°K, K (X) increases to a maximum at 77°K and then slowly decreases as T an J approaches T ^ where i t s t i l l r e t a i n s a large value. This paradoxical behavoir i s assumed to be a r e s u l t of p a r t i c u l a r and unusual d e t a i l s of the gallium Fermi surface. - i i i -TABLE OF CONTENTS PAGE ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF ILLUSTRATIONS v i i ACKNOWXEDGF__-NTS ' x i CHAPTER I. Introduction -' ]_ II. Review of the Theory of the Quadrupole Interaction 4 A. The Quadrupole Hamiltonian 4 B. The Pure Quadrupole Spectrum 6 C. The Zeeman Effect 7 1. Low Magnetic Field (^-cSc^^) 9 2. Arbitrary Field III. Review of the Theory of NMR Frequencies in Pure Metals ^4 A. The Theory of the Knight Shift ^4 B. Alkali Metals 16 C. Core Polarisation ^.6 D. Orbital Paramagnetism E. The Anisotropy of the Shift -^ g F. Liquid Metals 20 G. Spin-Lattice Relaxation 21 IV . Some Previous Investigations of the Properties of Gallium 25 A. Crystal Structure 25 B. Physical Properties 25 C. The Nuclear Quadrupole Resonance 27 - i v -V. Experimental Aspects 32 A. Apparatus 32 B. Sample and Crystal Holder 36 VI. Experimental Results 43 A. EFG Tensor Results 43 B. Knight Shift Results 57 C. Lineshape Considerations 77 D. Knight Shift in Liquid Gallium 83 VII. Discussion 87 A. EFG Tensor 87 B. Knight Shift 94 SUGGESTIONS FOR FURTHER EXPERIMENTS 124 BIBLIOGRAPHY 129 APPENDIX 1 3 5 A. The P o s s i b i l i t i e s of de Hass-van Alphen Type Oscillations of the Knight Shift in a Monocrystal of Gallium 135 B. Second Moment , ... 140 C. The Knight Shift in Gallium Metal: M.I. Valic, S.N. Sharma, and D.LI. Williams. 145 D. A Nuclear Magnetic Resonance Study of Gallium Single 147 Crystals-I. Low Field Spectra: M.I. Valic and D. JJL, Wiili«ms. E. Modulation Effects on the NMR Lineshapes in Metals 159 1. Introduction 159 2. Mathematical Treatment 160 3. Li s t of Illustrations 164 F. Collective Electron Effects on Nuclear Spin-Lattice Relaxation V 'Rate and Knight Shift. ,. .... 1 9 1 - V -LIST OF TABLES TABLE I. I I . I I I . IV. V. VI. VII. VIII. IX. XI. XII. XIII. XIV. XV. The relevant p h y s i c a l properties of gallium. The d i r e c t i o n s and the weighting f a c t o r s of the bonds. The EFG tensor r e s u l t s . The influence of d i f f e r e n t parameters on the p o s i t i o n of the central l i n e for some (selected) f i e l d d i r e c t i o n s . The Knight s h i f t along the , C^, A^ and A2 d i r e c t i o n . The uncorrected Knight s h i f t i n three d i r e c t i o n s measured at d i f f e r e n t magnetic f i e l d s . The corrected experimental Knight s h i f t values f o r several o r i e n t a t i o n s and temperatures. K^ s o > the anisotropy parameters K^, K2 and the p r i n c i p a l values of the K -tensor for several temperatures, an The experimental r e s u l t s of the temperature dependence cf K iso" The i o n i c c o n t r i b u t i o n to the EFG tensor as a function of the number of atoms considered. The l a t t i c e parameters of gallium at three temperatures. The comparison of the i o n i c c a l c u l a t i o n with the experimental r e s u l t f o r the EFG tensor at 4.2°K. The temperature v a r i a t i o n of the i o n i c c o n t r i b u t i o n to the EFG tensor. The temperature v a r i a t i o n of the q ^ f f tensor. The temperature v a r i a t i o n of the s p i n - l a t t i c e r e l a x a t i o n PAGE 27 30 52 61 61 71 71 73 85 88 88 89 91 92 94 time T, ^ , K. and the quantity 3T, ^TK. appropriate f o r IQ i s o IQ i s o - v i -the magnetic hyperfine r e l a x a t i o n of the pure quadrupole resonance for 1 = 3 / 2 . XVI. The paramagnetic s u s c e p t i b i l i t i e s obtained from several 99 methods. XVII. The temperature c o e f f i c i e n t (^ K/I>T) f o r several metals. I l l P XVIII. A comparison of selected physical properties and the 113 Knight s h i f t data for copper and gallium. 1 - v i i -LIST OF ILLUSTRATIONS FIGURE PAGE 1. Quadrupole energy levels for I = 3/2. 8 2. a) The angular dependence of the quadrupole energy levels io with H q in the XZ principal plane, b) The predicted frequency pattern with H q rotated in XZ plane. A method for obtaining VQ and T| is indicated. 3. Gallium crystal structure: 26 a) The orthorhombic unit ceil. The double lines connect the nearest neighbours. b) The projection of the unit cell onto the AG (010) plane. Heavy lines correspond to the atoms lying in one plane and light lines indicate the atoms lying in a plane a distance 1/2 B away. The nearest neighbours are indi-cated by lettered vectors. 4. Apparatus-schematically. 33 5. The low temper a tare system. 37 6. a) Schematic diagram of the copper holder for the crystal. 39 b) Schematic diagram of the teflon mold for growing the crystal. 7. The X-ray back reflection Laue photographs with the beam: 49 (a) along A, (b) close to A, (c). along C, (d) close to B (the crystal was rotated for 0.3 degrees counter-clockwise about an axis perpendicular to b), (e) along B, (f) close to B (the rotation opposite to (d)). 69 8. Ga spectrum as a function of orientation of the magnetic 44 field "H in the AC crystal plane. Each point corresponds to •.-' , o an absorption resonance signal. - v i l i -High field angular dependence of the central line for both isotopes close to the Z principal axis. The field strength H is quoted in kG. o n Angular dependence of the central line for both isotopes as a function of the square of the misalignment 8 Q between the magnetic field and the Z principal axis. Schematic presentation of the procedure for accurate deter-mination of the crystal orientation within the magnet using high field properties of the gallium NMR spectra. Orientation of the two sets of principal axes with respect to the crystal structure of gallium. The possible orientations of the two Z axes with respect to "diatomic axis'. 69 The splitting A of Ga along the Z and X principal axes in the field of 300 G. The effect of the magnetic field on the determination of T|. 71 _> The angular variation of the central line for Ga with H o rotated in the AC crystal plane. The curve was calculated from the matrix of Equation (26) using parameters as noted (K = 0). The two curves correspond to the two non-equivalent sites. The angular variation of the central line for Ga^"*" with in the XY principal plane calculated from the matrix of Equation (26). The frequency of this transition is constant i f T| = 0. The curve for the second site is not shown. The method for extracting the Knight shift at the 'turning points'. Theoretically, the position of the resonance is - i x -below i t s experimental value (Fig. 17a). The r e l a t i v e s h i f t i n the f i e l d at the nucleus (Knight s h i f t ) produces the ob-served resonance frequency (Fig. 17b). 18. The experimental angular dependence of the Knight s h i f t i n 65 the XZ p r i n c i p a l plane at three d i f f e r e n t temperatures. The s o l i d l i n e i s the t h e o r e t i c a l l y predicted angular v a r i a -t i o n f o r the Knight s h i f t anisotropy for an orthorhombic symmetry. 19. F i g . 18 with corrected Knight s h i f t values. Included i s 66 also the angular dependence at 195°K. 20.. The angular dependence of the Knight s h i f t i n the AC 67 c r y s t a l plane. The two curves i n d i c a t e that the two Knight s h i f t tensors are symmetric with respect to the EFG p r i n c i p a l system. The f u l l curve i s the t h e o r e t i c a l p r e d i c t i o n for K-anisotropy i n an orthorhombic c r y s t a l due to the d i p o l a r term. 21. F i g . 20 with corrected Knight s h i f t values. 68 22. The temperature dependence of K^, K Y, K- and K^ g o. 74 23. The temperature dependence of K„_(X), K (Y) and K (Z). 76 * —n cii\ an 24. G a ^ resonance signals ( f i r s t d e r i v a t i v e s ) at (1) 285°K, 79 (2) 77°K and (3) 4 .2°K obtained from low f i e l d experiments • f o r Ij"3/2M>-**j+l/2M> t r a n s i t i o n s . The numbers speci f y the frequency scale i n kHz. 25. The experimental d e r i v a t i v e (crosses) taken at 77°K f i t t e d SQ to an equal mixture of normalized Gaussian absorption and d i s p e r s i o n modes. -X-2 6 . Ga^ resonance signals ( 1 ) first derivative ( 2 ) second 8 1 derivative, at 4.2°K obtained from high field experiments for the central line. 2 7 . The derivative signal of liquid gallium. The numbers 8 4 specify the frequency scale in kHz. 28 . The temperature dependence of the Ga^ NQR frequency. 9 0 The solid line represents the results of Pomerantz [ 1 7 ] . The points present the experimental result of the present work. 2 9 . T^ vs-temperature (full circles) in Ga 7 1. The solid line 1 0 3 is a prediction of the unenhanced Korringa relation. The broken-line curve is the prediction of the Korringa relation with an empirical f i t for the enhancement. Notice a change in the enhancement at T^p. - x i -ACKNOWLEDGEMENTS I wish to express my sincere gratitude to Dr.D.LI.Williams for the valuable i n s t r u c t i o n and i n s p i r a t i o n he provided throughout my career as a graduate student,and for h i s support i n obtaining f i n a n c i a l assistance for my studies. I wish to express my gratitude to Dr.M.Bloom who enabled me to study i n Canada.I am indepted to him for h i s valuable help throughout the whole of my studies. I acknowledge my dept to Dr.E.P.Jones who suggested the study of gallium to Dr.D.LI.Williams i n 1964. I am g r a t e f u l to Dr.S.N.Sharma for h i s assistance with the spectrometer. I am indepted to the U n i v e r s i t y of B r i t i s h Columbia for the Un i v e r s i t y Studentship and to the National Research Council of Canada f o r the subsequent award of an NRC Studentship. CHAPTER I INTRODUCTION The change in the nuclear magnetic resonance (NMR) frequency of a nucleus in a metal from that of the same nucleus in a non-metallic state was first explained in terms of an electron-nucleus interaction by Townes, Herring and Knight [ l ] . This change in resonance frequency, the Knight shift, has been studied experimentally and theoretically quite extensively for many metals and alloys. The results of much of this work are summarized in a review article by Drain [2] and a rigorous treatment of the Knight shift is given by Abragam [3]. Because of the small skin depth of almost all metals, NMR ex-periments have been almost entirely confined to studies in metal powders whose individual particle sizes are less than the skin depth of the metal. In any metal with cubic symmetry, the Knight shift is independent of the orientation of the metal crystal with respect to the external magnetic fields. However, i f the symmetry of the crystal is less than cubic, this is not true. The NMR signal will be broadened when observed in a powder whose particles are randomly oriented in the magnetic field. It was demonstrated by Sagalyn and Hofmann [4] that with proper attention to the relevant considerations, NMR signals can be seen in single crystals of metals. Since their original work on Cu and Al, work has been reported on the non-cubic metals Sn [5], Cd [6], Tl [7], Mg [8], Hg [9], Pb [10] and Ga [11]. The advantages of using a single crystal for NMR studies are: (a) the anisotropy of the Knight shift due to the crystal symmetry can be studied directly, (b) the line shape of the resonance can be studied without the large broadening caused by the anisotropic Knight shift, (c) the measurements of the resonance frequency can be 'This statement is valid only if spin-orbit coupling is negligible. -2-made more accurately because the resonance l i n e i s narrower and (d) i n some cases (gallium I) the use of a si n g l e c r y s t a l i s unavoidable. The purpose of t h i s work has been to study the NMR properties of the gallium metal. Gallium i s one of the most ani s o t r o p i c of metals and a d e t a i l e d NMR study could reveal useful informations concerning i t s e l e c t r o n i c properties. The measureable parameters are those of the elec-t r i c f i e l d gradient (EFG) tensor at the nuclear s i t e and the three d e s c r i -bing the o r i e n t a t i o n dependence of the Knight s h i f t i n an orthorhombic c r y s t a l . The work was i n i t i a t e d with the main aim of determining the Knight s h i f t i n the s o l i d with a view to understanding the temperature dependence of the s p i n - l a t t i c e r e l a x a t i o n time observed by Hammond, Wilkner and K e l l y [12], I t was hoped that a comparison with the measure-ments i n the l i q u i d state [13] would c l a r i f y the i n t e r p r e t a t i o n of the l i q u i d . The determination of these parameters however required an accu-rate knowledge of the EFG tensor. This work i s presented i n the following way: Chapter I I contains a review of the theory of the quadrupole i n t e r a c t i o n with a s p e c i a l emphasis to the case of I = 3/2 spin. The method, which was used i n this work to obtain the EFG tensor parameters i s discussed. In Chapter I I I a b r i e f t h e o r e t i c a l review of the NMR frequen-c i e s i n pure metals i s given. The d i f f e r e n t mechanisms for the Knight s h i f t are described. It follows, i n Chapter IV, a b r i e f d i s c u s s i o n of some previous i n v e s t i g a t i o n s of the relevant properties of gallium. Included i s a . d e s c r i p t i o n of the c r y s t a l - s t r u c t u r e and a review of the l i t e r a t u r e -3-about previous NMR studies of t h i s metal. Chapter V outlines the experimental aspects. A d e s c r i p t i o n of the apparatus used i n t h i s i n v e s t i g a t i o n together with the problems of growing and mounting the samples i s given. The experimental r e s u l t s . a r e summarized i n Chapter VI. The f i r s t Section of t h i s Chapter o u t l i n e s the EFG tensor r e s u l t s . They present the f i r s t d e t a i l e d study of the quadrupole coupling i n the me-t a l . At the end of t h i s Section a b r i e f d i s c u s s i o n of the method f or the exact alignment of the c r y s t a l w i t h i n the magnet i s discussed. The second Section summarizes the Knight s h i f t r e s u l t s . The Knight s h i f t i n gallium has not been previously studied since the complexity of the NMR spectrum i n a powder sample i n h i b i t e d any attempt. The t h i r d Section includes the l i n e shape considerations and the l a s t one i s concerned with the Knight s h i f t i n the l i q u i d gallium. It f o l l o w s , i n Chapter VII, the d i s c u s s i o n on the experimental r e s u l t s . An attempt to decompose the EFG tensor by subtracting the calcu-l a t e d i o n i c c o n t r i b u t i o n has been made. The i s o t r o p i c Knight s h i f t has been separated from the experimental s h i f t and compared with the Korringa r e l a t i o n . Some speculations about the changes i n the e l e c t r o n i c struc-ture of gallium upon melting are given. The p o s s i b i l i t y of the f i e l d de-pendence of the Knight s h i f t i n gallium i s deferred to Appendix A. The t h e s i s i s concluded with suggestions f o r future work. In standard theses i t has been a common procedure to write introductory chapters on the background theory. However, I found t h i s s c h o l a r l y review as superfluous, since the background material can be found i n standard texts [14]. -4-CHAPTER II REVIEW OF THE THEORY ON THE QUADRUPOLE INTERACTION A. The Quadrupole Hamiltonian The interaction energy of a quadrupole moment in an external electric field is (1) where Q is the quadrupole moment tensor representing the asymmetric charge distribution of the nucleus and VE the electric field gradient (EFG) tensor of the charges external to the nucleus. The tensor VE written in rectangu-_ —» lar coordinates is symmetric (since vxE=0) and is traceless (since V_=0). Therefore the EFG tensor has only five independent components. Assume an arbitrary set of axes xyz which is fixed with respect to the crystal. (This set may be externally easily identifiable; for example, one involving some of the crystallographic axes). The tensor will then have the general form q q XX ^xy xy q ^yz (2) zx V q z z with q 0 defined as CYB 3 X1 - 6 -- ~ r ~ P ( r - : ) • (3) In Equation (3) the p(r^) represents the charge density (of electrons and nuclei) at the point r\. For a continuous charge distribution the summa-tion is to be replaced by an integration. The prime over the summation implies that the charge on the nucleus at r\=0 is excluded from the sum-mation. -5-A set of axes can be found which put this tensor in the diagonal form • %(1-TD q z z 0 0 0 -%(1+TD q z z 0 0 0 qzz (4) Here the axes have been chosen such that q z z has the largest and q ^ the smallest absolute value. This restricts the value of the asymmetry para-meter T|= (^YY -^xx^I QZZ T O 0^ T]<1. values q z z, T| and any three quanti-ties (e.g. the three Eulerian angles) specifying the orientation of the principal X Y Z system with respect to the xyz frame constitute five inde-pendent quantities defining the tensor. It is desirable to study in detail the effect of a nuclear quadrupole interaction on the energy levels of a nucleus. In general, the calculation of the energy levels is performed either by a perturbation theory calculation or by the solution of the secular equation which results from the matrix elements of the Hamiltonian when the quadrupole contribu-tion of Equation (1) is included. In either case, the matrix elements of the quadrupole terms must be calculated. The representation commonly used is based on the angular momentum operators of the nucleus. In this way the matrix elements can be easily calculated from the standard formulae. The correct procedure is rather tedious and can be found elsewhere [15]. Equation (1) in the angular momentum representation then takes the form H Q = H° + H^ = A[3I Z - 1(1+1) + \ T|(I^ +1*)] , (5) where ' ' ^ 2 * '*ZZ A = e 2 q 7 7 Q/4l(2i-l) . : ' ^ '-"v-.. • • - ' • r r . ^ - ^ - d ^ y (6) The symbol Q is the nuclear quadrupole moment and e the electronic charge. B. The Pure Quadrupole Spectrum The non-vanishing matrix elements of between the 21+1 dege-nerate levels of the nucleus for the case of half integral spin are, by Equation (5), ^m|HQ|m>= A[3m2 - 1(1+1)] (7) <Tm + 2 | H Q l m > = 3/2 T| A f ^ t m) f _ ( l t m) , (8) where f_(m) = [l-m)(I+m+l)]% . (9) Thus a finite value of T| leads to the mixing of states \m> diffe ring by Am = 2. The eigenstates of Hp may be written as linear combina-tions of the degenerate states |m>; i.e., |mQ>= Z |m+2|i><jn+2l4mQ> . (10) The notation |mQ"> denotes an eigenstate of H^ which reduces to the state \my for T] = 0. Because of the mixing of states the Hamiltonian matrix may be broken up into two submatrices of degree 1+1/2. The secular equa-tion is then also of degree 1+1/2 which yields 1+1/2 doubly degenerate levels. It is possible to obtain explicit solutions for the secular equations only in the case of 1=3/2 for which the matrix table and the eigenstates are: -7-N. m m'\ +3/2 -1/2 1/2 -3/2 +3/2 3 V3T\ ; 0 0 -1/2 V3T\ -3 j 0 0 1/2 0 -3 >/3T\ -3/2 0 o j V3TI 3 (±3/2 Q>~ 3 A p E . = -3Ap, |+l/2 Q> H' ( I D where 2 h p = (1+1/311 ) (12) The transition between the energy levels excited by an r.f. field occurs at the quadrupole frequency VQ = 6Ap/h = v Q p , (13) where is the pure quadrupole frequency. The situation is schematically shown in Figure 1. It follows that in this case one cannot obtain both o V and T) from a measurement of the quadrupole frequency v . However, the Q Q orientation of the principal axes in a single crystal could, in principle, be obtained from a pure nuclear quadrupole resonance (NQR) study utilizing the fact that the intensity of the transition between the levels is a function of the orientation of the r.f. field in the principal system [16]. For I>3/2 there are in general two or more observable frequencies from which one can determine both parameters. C. Zeeman Effect The degeneracy of the pure quadrupole states |^"mQ^> is removed by applying a magnetic field H = H (sin 6 cos 0, sin 6 sin 0, cos 6) , o o (14) where 6 and 0 specify the orientation of the magnetic field in the prin-I I 3 / 2 Q > / I m > / ( 2 1 + I ) ^ \ It '/ 2 Q> ~K 7T | -3 / 2 M > I + \ M > I - ' / 2 M > r outer pair I + V2 M > H 0 " 0 inner pair H 0 * 0 Fig. 1: Quadrupole energy levels for I = 3/2 - 9 -c i p a l coordinate system. The-Hamiltonian f o r the system i s then H s = H Q + H Z , (15) where % - i0 +i0 H = -fvH = V T [ I cos 6 + % ( I e + I e ) s i n 6] (16) i s the Zeeman Hamiltonian, v, = V H /2TT i s the nuclear Larmour frequency i-» o and y x s t n e gyromagnetic r a t i o . The eigenstate problem i s d i f f i c u l t although i n p r i n c i p l e i t can be solved exactly. However, for the case of low magnetic f i e l d s one can obtain very useful information from a simple perturbation c a l c u l a t i o n . 1. Low Magnetic F i e l d ( v ^ < V ) L Q The energy l e v e l s lmM> can be c a l c u l a t e d to f i r s t order i n HD from the matrix elements of H- between the degenerate pure quadrupolar states |+ mQ^ > i n the standard way. The states |mM> denote the eigenstates of H which reduce to states jm> for T| = 0. The energy l e v e l s for any s value of T| and to f i r s t order i n H q are given by the expressions ] 2 2 2 2 E_ = % v n + -r- v T[(p+l -TT) s i n 6 cos 0 + (p+l+T0-l*V_M> 2p L J, 2 2 2 2 ^ + • s i n 6 s i n 0 + (2-p) cos 6] = % v - m i v L > ( 1 7 ) E_ . . v + i _ M [(p-l4 -TT; 2sin 25 cos 2 0 + ( p - l - T D 2 " 2D ^ s | i V t M> Q ' 2p• s i n 2 6 sin 2 0 + (2+p) 2cos 26] = -_ VQ t . (18) In place of a s i n g l e absorption l i n e there appear two p a i r s of l i n e s , as shown i n Figure 1. The low frequency t r a n s i t i o n s , shown as dotted l i n e s on t h i s f i g u r e , are neglected. Figure 2a presents the angular dependence -10-> >-o or - 6 0 - 3 0 0 30 DEGREES 60 150 8 F i g . 2(a) F i g . 2(b) The angular dependence of the quadrupole energy l e v e l s with H in the XZ p r i n c i p a l plane. The predicted frequency pattern with rotated i n XZ pl-inc, i s indicated, A luethod for obtaining VQ and -11-of the four energy levels as the magnetic f i e l d is rotated in the XZ principal plane. The four transitions correspond to the four frequen-cies v = v Q t j " m2) (19) The angular dependence of these frequencies for this rotation is repre-sented in Figure 2b. For an arbitrary direction four absorption lines are obtained. The angular variation of the 'inner pair 1 of lines (corresponding to the transitions |+3/2M>*-*> [+1/2M> and l-3/2M>- \- 1/2M>) yields a l l the parameters for the EFG tensor. In particular the following results should be noted: V|±3/2M>^|+i/2M> v n + v for H || Z (20) H L o VQ - <T°> \ f ° r =o I' X ( 2 1 ) v„ + vT for H || Y . (22) C» L o The mean sp l i t t i n g along the Z axis i s given by VQ = ^ (V|+3/2M>«-l+l/2M> + Vl-3/2M>**\-l/2M>)' ( 2 3 ) The s p l i t t i n g A = V|+3/2M>—l+l/2M>" V\-3/2M>«-~V-l/2M> ( 2 4 ) yields the asymmetry parameter T| from the relation Atf.H X _ 1 - Tl ^ (25) A f t 0 l l Z P The resulting T| is independent of the absolute value of the magnetic f i e l d for small f i e l d s . The positions of maximum and minimum splittings yield -12-the principal axes. The above results assume the orientation of the principal axes with respect to the crystal axes are known. In general, to find them, an orientational pattern similar to that in Figure 2b must be obtained from a rotation of the magnetic field in each of any three orthogonal planes. However, the considerations based on the crystal symmetry can simplify the interpretation considerably. All four transitions may not be observable because the intensi-ties are a function of the orientation of the r.f. field with respect to the principal axes as well as with respect to the orientation of the Zeeman field H . In fact, for certain orientations of HQ and of the r.f. field, the asymmetry parameter T| has a remarkable effect upon the spectrum. This, in turn, provides a different method for determination of T| and the orientation of the principal axes in a single crystal. How- . ever, the techniques which rely upon intensity measurements are not very accurate. The asymmetry parameter T| can also be extracted from a lineshape analysis obtained from a powdered sample in a small magnetic field. For further details the reader .is referred to reference [17]. 2. Arbitrary Magnetic Field For intermediate and high magnetic fields (VJJ^ -VQ) , the low field approximation used in the previous section is no longer valid. The problem must then be solved exactly when trying to locate the NMR lines in a given magnetic field. This is done by writing the Hamiltonian of Equation (5) in the representation that diagonalizes I „ . The following matrix is obtained -13-S ~2 3 cos 6 + y T/V3 y (3 sin 6 e"li8 T/V5 y -cos 6-y 3 sin 6 e + i 0 2 sin 6 e_i0 2 sin 6 e"1""""0 cos 6-y V3 sin 6 e" 1 0 T/ V3 y 0 sin 6 e+i0 y -3 cos 6+y (26) where y = v^/v^. The secular equation can be reduced to the form [(E 2-l)-3y 2-l/3 T | 2 ] 2 = 4y2[(E-l4-Tfj2sin26 cos20 +(E-1-T|)2sin26 sin20+(E+2)2cos26] . (27) In some cases, in particular for 6=0=0 (H II Z) and 6 0 = 0 (HQ \\ X) , Equation (27) can be solved exactly. But in all cases, i t may be diagona-lized with the computer keeping 6,0,T|,VQ° and H q as parameters. -14-•CHAPTER III THE REVIEW THEORY OF THE NMR FREQUENCIES IN PURE METALS A. The Theory of the Knight Shift In a given applied field, the NMR frequency for a nucleus in a metal is almost always higher than for the same nucleus in a non-metalTic substance. Quantitatively, one defines this 'Knight shift', K, as follows: A v - v k _ _ _ _ _ _ _ ^ V V r vr v and v are the resonance frequencies of the nuclei in the metal and a m r non-metallic reference, respectively. The mechanism of the shift is simply connected with the polarization of the conduction electrons by the magnetic field. Since these electrons have magnetic interactions with the nuclei, the latter see an extra field AHQ and so resonate at a different frequency. In general one can write K = — = -H Av o ^° bands <S*P + Vo> > ( 2 9 ) where the usual assumption has been made that the Pauli spin paramagnetism, X_,and the orbital paramagnetism, of the conduction electrons may be treated as independent terms (i.e. spin-orbit coupling is neglected). The coupling coefficient q and q depend on the wave functions of the electrons. P ° In general there are contributions arising from each partially f i l l e d band and a summation must be taken over all of these. Consider first only simple metals in which there is only one partially f i l l e d band and the orbital term is negligible. It is assumed -that only the dire'ct dipOle interactions are important (indirect interac-tions are important in transition metals). The magnetic field in the - 1 5 -direction of H q produced at the nucleus by the polarization of the electron cloud is fP °(cos 6) 2 AH Q = 2 [ H O X Q^lKr)! > p df1] . (30) r is the distance of the volume element dr at the point r from the nucleus, 6 is the angle r makes with H , ty(r) i s the electronic wave function norma-lized over an atomic volume Q,Oj? indicates an average taken over the Fermi surface and is the volume spin susceptibility of the' electrons. The expression enclosed in the square brackets is simply the magnetic mo-ment of the volume dr. Provided i|t(o) = 0, the integral may be evaluated directly but i f \jt(o) £ 0 the integral becomes indeterminate. It is usual therefore to exclude a sphere in the immediate neighbourhood of the nu-cleus from the integration. The contribution to K of the electron density inside the sphere may be shown by a variety of methods [14] to, be Kg = |2 Xp Q ^H(°)| 2> F • (3D This 'contact* contribution is of special importance, for con-duction electrons frequently have functions of atomic s-type that are strongly peaked at the nucleus. The 'dipole' term is obviously zero for atomic s-type electrons and vanishes in any case when the nucleus is in a position of cubic symmetry. The term becomes important for non-cubic symmetry where i t gives rise to the anisotropy of K. Kg tends to be very large (0.1%— 3%), increasing steadily with increasing atomic number and except for some transition metals is the dominant contribution to the Knight shift. -16-B. Alkali Metals Perhaps the best comparison of experimental results with the theory is provided by the alkali metals for their electronic structure is comparatively simple and well understood. The Knight shift here may be written as AH o _ 9 ^ = ir = r y^A ( o ) l >> ( 3 2 ) o 2 2 where £ = ^ I t C 0 ) ! ^p/^-^A^0^ 3 ^ a c t o r °^ o rder unity. It is pre-sented in this way since the wave functions in the alkali metal approximate those in the free atom. It is very unfortunate that Xp f° r m°st metals is not accurately known. This means the parameter <£\i|i(o)\ °f considerable theoretical interest, cannot be extracted from K^ . The only exceptions are lithium [18] and sodium [19]. The results for these f i t well with theory, values of § being 0.81 for Na and 0.46 for Li. The large differences in § are mainly due to the rather different Fermi surfaces of the two elements. Kg shows a slight temperature and pressure dependence [20]. The main effect for this comes from the variation in atomic volume which changes both x_ 2 and <^(ijr(o)| There is also an intrinsic temperature dependence resul-ting from an averaging of the wave functions by the atomic vibrations [20] which explains the isotopic dependence of K in metallic lithium [21], The only other example of the isotopic dependence is rubidium [22] where the effect is attributed to the finite size of the nucleus. C. Core Polarization The electrons of fil l e d shells in some cases give the most im-portant contribution to K by means of the Heisenberg exchange interaction of the unfilled shell electrons with s-electrons of the cores. Although -17-a n u n p a i r e d e l e c t r o n s p i n c a n n o t p r o d u c e a r e s u l t a n t s p i n moment on c o r e s , s i n c e t h e e l e c t r o n s t h e r e i n m u s t r e m a i n p a i r e d , p e r t u r b a t i o n s c o r e - e l e c t r o n wave f u n c t i o n s a r e p r o d u c e d . T h e s e p e r t u r b a t i o n s depei w h e t h e r t h e c o r e e l e c t r o n s p i n i s p a r a l l e l o r a n t i - p a r a l l e l t o t h e sr t h e e l e c t r o n o f t h e u n f i l l e d s h e l l s . The e l e c t r o n d e n s i t y a t t h e nu i s t h e n n o t t h e same f o r t h e two o r i e n t a t i o n s o f s - s p i n i n t h e c o r e , n e t m a g n e t i c f i e l d i s p r o d u c e d b y t h e n o n - c a n c e l l a t i o n o f t h e i r cont t e r a c t i o n s . T h i s a d d i t i o n a l c o n t r i b u t i o n i s , o f c o u r s e , p r o p o r t i o n , t h e number o f s p i n s w h i c h h a v e b e e n t u r n e d o v e r b y t h e m a g n e t i c f i e l t o X p , J u s t as i n E q u a t i o n ( 3 2 ) , t h e o r d i n a r y p a r t o f K . I t f o l l o w s t h i s i n d i r e c t c o n t a c t i n t e r a c t i o n , known as c o r e p o l a r i z a t i o n [ 2 3 ] . j u s t a s i f t h e r e w e r e a g e n u i n e c o n t a c t i n t e r a c t i o n f o r t h e e l e c t r o n u n f i l l e d s h e l l s . I t s c o n t r i b u t i o n t o K c o u l d be p o s i t i v e o r n e g a t i \ i s p r o b a b l y p r e s e n t i n a l l m e t a l s . I n some t r a n s i t i o n m e t a l s l i k e p d i u m and p l a t i n u m , [ 2 4 ] and [ 2 5 ] r e s p e c t i v e l y , t h i s i s t h e d o m i n a n t . I n p l a t i n u m , t h e t o t a l K amounts t o - 3 . 5 % , m o s t o f t h i s a c o n s e q u e n t t h e c o r e p o l a r i z a t i o n o f t h e i n n e r s - e l e c t r o n s by t h e d - b a n d e l e c t r o -I n l i t h i u m a n e t c o r r e c t i o n o f a b o u t - 5 . 3 % o f t h e c o n t a c t t e r m [ 2 6 ] i D. O r b i t a l P a r a m a g n e t i s m I n a n a l y z i n g t h e m a g n e t i c p r o p e r t i e s o f t r a n s i t i o n m e t a l s u s u a l t w o - b a n d m o d e l i s a d o p t e d . I t i s assumed t h a t t h e r e a r e two c l a p p i n g p a r t i a l l y f i l l e d b a n d s , a n a r r o w one u s u a l l y a s s o c i a t e d w i t h d - t y p e wave f u n c t i o n s , and a w i d e o n e , t h e s - b a n d c o n t a i n i n g m a i n l y a l s o p - w a v e f u n c t i o n s . Many d - b a n d m e t a l s h a v e c h a r a c t e r i s t i c a l l y 1 p a r a m a g n e t i c s u s c e p t i b i l i t i e s and e l e c t r o n i c s p e c i f i c h e a t s compared t h o s e o f s i m p l e m e t a l s . The d i f f e r e n c e i s t h a t t h e p a r t i a l l y f i l l e d -18-involved i s narrow, giving rise to a much greater density of states at the Fermi surface. A l l of the d-band paramagnetism i s not of spin origin, however. There i s an orbital contribution [27], that arises from a second-order per-turbation effect of the magnetic f i e l d which i s analogous in metals to the more familiar Van Vleck temperature-independent paramagnetism of free ions. The application of a f i e l d modifies the occupied energy levels of the sys-tem. This may be considered as a mixing i n of higher (unoccupied) states re-sulting in a net orbital magnetic moment for the f i n a l admixed state[l03[. Sine the amount of mixing depends inversely on the separation of the energy levels involved, the orbital paramagnetism i s particularly important when the energy separation of occupied and unoccupied levels i s small. This i s just the situation that occurs within the narrow bands of transition metals. The contribution to K i s K Q = 2 n^r-V d Xo , (33) where r i s the distance of an electron from the nucleus, the average being taken over the d-wave functions and i s t n e orbital susceptibility. It is of the same order as Pauli paramagnetism but differs i n not being pro-portional to the density of states at the Fermi surface. It tends to be largest for h a l f - f i l l e d bands, since there i s then the greatest probability of interaction between f i l l e d and unfilled levels. This i s the case that occurs in vanadium and rhodium, [28] and [29], respectively. E. Anisotropy of the Shift If the surroundings of the nucleus do not have cubic symmetry, there i s an additional complication due to the dependence of K on the -19-crystal orientation with respect to the magnetic field. There is no proper understanding of the mechanism of the K-anisotropy. One contri-bution certainly arises from the electron-spin susceptibility through the 'dipolar' term of Equation (30). This involves the non-s-type wave function but no anisotropy of the magnetic susceptibility. In the ab-sence of spin-orbit and electron spin-spin dipole coupling terms, the anisotropic K is zero in cubic materials as a general consequence of symmetry. Neglecting the same terms, one can derive formulae for the angular dependence of the shift on the orientation of the magnetic field relative to the crystallographic axes for the single crystals of the six non-cubic systems [30]. The result for the most frequent (in metals) case of the axial symmetry is where 6 is the angle between the crystal symmetry axis and the direction of the applied field. A is a constant. In the literature [31] A is often re-placed by 2/3 (Kn -K,.) where Ku and Kx are the shifts corresponding to 6=0° and 6 = 90°, respectively. In case of the orthorhombic symmetry the anisotropy is described with two parameters A and C^ . 6 and 0 denote the polar angles of the magnetic field direction. While in the case of the axial symmetry the choice of the axis is unique, in the orthorhombic case any of the three possibilities could be chosen. The anisotropy is most clearly shown in non-cubic metals having nuclei of spin 1/2, e.g. tin, cadmium, thallium, where there are no K = A P °(cos 6) , ah 2 % » (34) K = A P °(cos 6) + C„ P °(cos 6) cos 2 0 ar\ 2 x 2 2 (35) -20-additional complications due to the quadrupole interactions. It is known, however, that many of the metals investigated do exhibit appreciable magnetic anisotropy. In scandium, for example, the susceptibility along the C axis of the hexagonal structure is <~'30% greater than in the perpendicular directions [32]. This must lead to an additional contribution to the K-anisotropy. In fact, an anisotropic NMR relaxation rate, primarily due to the orbital interaction, in the single crystal of scandium has been reported [33]. The expression for the isotropic Knight shift contains the Lande g factor. If g is anisotropic, i t could also produce an anisotropy in the Knight shift as a result of an anisotropic Pauli spin paramagne-tism. F. Liquid Metals It is generally assumed that the electronic structure of liquid metals is comparatively simple owing to the absence of a periodic lattice structure. The free-electron approximation gives a reasonable account of a number of physical properties, whereas in solids this model generally has to be considerably modified to take account of the same structure im-posed by the crystal structure. It is thus rather surprising that, for almost a l l the metals, the change in K on melting is extremely small. Perhaps this can be understandable in alkali metals where the nearly free-electron (NFE) approximation is quite good in the solid state, but the changes in the Knight shift are also small for metals like tin and copper in which deviations from the free-electron model are quite appreciable. It appears as i f a solid-like structure is retained on a small scale within the liquid state [34]. For the metals cadmium, bismuth and gallium - 2 1 -(present work), the changes in K on melting are considerable. Recently the temperature variation of K in cadmium has been explained [35]. The effect of lattice vibrations upon the band structure decreases the effective strength of the lattice potential, thereby causing the energy bands to be more free-electron like, finally becoming completely free-electron in the liquid. G. Spin-Lattice relaxation The conduction electron-nucleus interaction leads also to a strong mechanism for the transfer of energy from the nuclear spins to the translational motion of the electrons and hence to the crystal-lat-tice vibrations. Assuming only contact interaction from one band, the relaxation time, T^ , is given by: 2 ~ = IT k T-n[M v e Y n < H ( o ) l 2 > F N(E p)] , (35a) where N(ET,) is the density of states at the Fermi level. The inverse pro-portionality of T^ to the temperature, T, is a very general characteristic of the spin-lattice relaxation in metals. There is an important difference between the above formula and the formula for the Knight shift of Equation (31). The Knight shift involves the spin susceptibility of the electrons, whereas the relaxation time involves the density of states at the Fermi surface, N(Ep), directly. In the band theory of metals with non-inter-acting electrons these quantities are related by the formula = % Y 2 ^ 2 N(Ep) . (35b) Using this relation and Equations (31) and (35a) , one can derive the well-known relation between the Knight shift and the spin-lattice relaxation -22-due to Korringa [36]: -2 -fi Ye 2 1 a 4nk v 'n Deviations from Korringa's relation may be due to the modification of the relation between x a n d N(E^ ) because of electron-electron exchange interaction [37], or to the presence of more than one contribution t o the Knight shift and the relaxation time. Another mechanism for the relaxation in metals is due to the. interaction of the charge on the electron with the nuclear electric quadrupole moments [38], The relative importance of this mechanism m a ; be tested when more than one isotope is available for the relaxation measurements. For purely magnetic interaction, T^ is inversely propor-tional to the square of the nuclear gyromagnetic ratio. This relation is valid to an accuracy of 2% for the ^ Ga and ^ Ga nuclei in gallium [12], For pure quadrupole interaction, T^ is inversely proportional t < . the square of the nuclear quadrupole moments. A nice example where b o ; contributions are important is the case of liquid gallium [13] or molyl denum metal [39]. It has been pointed out that T^ depends on the external magne. tic field [74]. In the theory of the field dependence of T^ the hyper-fine interaction is used in a perturbation calculation to transfer the energy between the nuclear spin system and the lattice via the conducts electrons. In a strong external field the zero-order nuclear spin Ham nian is taken as the Zeeman interaction, Hz, with the external field; conduction electrons are characterized by Bloch functions and a Fermi tribution at the lattice temperature. The result of the calculations Equation (35a). For an arbitrary external field the Zeeman interactic -23-H , plus the nuclear dipole-dipole interaction, Hn, has to be used as zero-order nuclear spin Hamiltonian, H. With the aid of spin temperature assumption a perturbation calculation yields [74] T (H ) = T (H2 + h 2)/(H 2 + 2h2) , (35d) ID o 1 o D o D where T is given by Equation (35a). H is the external magnetic field 1 o 2 and hp is of the order of the mean square local field at a nucleus due to the dipolar field of its neighbours. At zero field T 1 D(0) = T^2 (35e) Thus, T^p(O) is also inversely proportional to the temperature. Using the T^D(0) data instead of T^ , the Korringa relation given by Equation (35c) then reads 2 2T (0) TK = S (35f) ID s The calculations have also been carried out with quadrupole Hamiltonian, H , in place of H in the zero-order nuclear spin Hamilto-' Q D nian [74]. Expressions result which are similar to those of Equations (35a) and (35d). Labelling the relaxation time T to distinguish i t from T follows that ID T (H ) = T (H2 + h 2)/(H 2 + 3h2)-, (35g) IQ o 1 o Q o Q T (0) = Tx/3 (35h) 2 2 2 , 2 T is again given by Equation (35a) and h = H Tr(H )/Tr(H ) . Thus the 1 X • O x 2 form of the field dependence of T, „(H ) is similar to that of T ( H ). i<j o ID o 2 However, ^ (0)/^ = 1/3 and 1^(0)/^ = 1/2; also h Q could be much larg 2 than hp. Using the ^ ^(0) data the Korringa relation then reads er - 2 4 -3T (0) TK 2 = S - (35i) IQ s This r e l a t i o n w i l l be used l a t e r i n connection with the d i s c u s s i o n on the Knight s h i f t i n the s o l i d gallium. -25-~CHAPTER IV SOME PREVIOUS INVESTIGATIONS OF THE PROPERTIES OF GALLIUM A. Crystal Structure The crystal structure of solid gallium is orthorhombic with the unit cell dimensions A = 4.5156, B = 4.4904, C = 7. 6328 at 4.2°K and at atmospheric pressure [40], The eight atoms in the unit cell are located at the sites [4l]; (u,o, v) (u+1/2, 1/2, v) (u+1/2, 1/2, v) (u, o, v) (u, 1/2, v+1/2) (u+1/2, o, v+1/2) (u+1/2, o, v+1/2) (u, 1/2, v+1/2) where u = 0.0785 and v = 0.1525. The crystal structure together with its projection on to the AC (010) plane is shown in Figure 3. The structure may be visualized as a line of "diatomic molecules" lying with their axes in the AC plane at an angle 8 of 16° to the C axis. This line of mole-cules is displaced by half a lattice spacing along both B and C from the next line whose axes are rotated through 180° about the C axis. Each atom has one nearest neighbour at 2.429 A and six others varying in distance between 2.709 A to 2.791 A. The two next nearest neighbours are considerably further away at a distance of 3.54 A. The structure is isomorphic with the solid halogens. B. Physical Properties Gallium belongs to Group 1IIA of the periodic table and has an 2 outer electronic structure of 4s 4p. Its relevant properties are summa-rized in Table I. - 2 6 -3a. 3b. Lattice constants at 4 . 2 * K : • A • 4.5156 A B - 4 . 4 9 0 4 A C - 7 . 6 3 2 8 A Locations of the eight atoms in t h * unit c e l l : ( u . o . v ) ( u + '/ 2 , 'yfe , v) (u + '/fe , '/2 . V ) ( u . o . v ) ( u . ' / 2 , v + ' / 2 ) (u + !/2 , o . v + '/? ) ( u + % , o , v + l / j ) ( u , , / 2 f v + '/2 ) where u - 0 . 0 7 8 5 , v - 0 . 1 5 2 5 d , - 2 . 4 2 1 , d 2 - 2 . 7 0 4 , d 3 - 2 . 7 5 4 , d 4 - 2 . 7 7 4 Fig. 3: Gallium crystal structure: a) The orthorhombic unit cell. The double lines connect the nearest neighbours. b) The projection of r.he unit cell onto the AC(OIO) pl^ne. Heavy lines correspond to the atoms lying in one plane and light lines indicate the atoms lying in a plane a distance 1/2 B e\icy. The nearest neighbours are indicatec by loitered vet Lor*. — -27-TABLE I. The relevant physical properties of gallium Isotope Abundance (%) Nuclear Spin NMR Frequency in 10 KG (kHz) Pure NRQ Freq. (285.3°K) (kHz) Nucl. Quad. Moment (xl0- 2 4cm 2) Ga 6 9 60.8 3/2 10218 10823.8 0.19 Ga 7 1 39.2 3/2 12984 6820.7 0.12 The electric and magnetic effects on the nuclear resonance can be estimated separately in gallium, since the gyromagnetic ratios are related by y^/y^^ - 0.79 and the electric quadrupole moments by Q^/QJI = 1.58. Gallium displays highly anisotropic behaviour in its electrical resistivity [42], thermal resistivity [43], thermal expansion [42] and magnetic susceptibility [44], which, considered along with its crystal structure, suggests a complicated spatial distribution of the charges, indicative of covalency. ._ . C. The Nuclear Quadrupole Resonance The pure NQR in gallium was first observed in a polycrystalline sample by Knight, Hewitt and Pomerantz [45], who, at 0°C, found two re-sonance frequencies at 10.9 MHz for Ga^9 and 9.9 MHz for Ga^. An inves-tigation of the temperature dependence of the quadrupole frequency in a polycrystalline sample was subsequently undertaken by Pomerantz [17], He explained the origin of the electric field gradient on the basis of the ionic model but the agreement with this model was due to errors in his calculations. The asymmetry parameter was subsequently measured by Kiser and Knight [46] in a crystalline powder of gallium and i t was found to be (0.19 j" 0.03). The pure NQR measurements have been extended in the superconducting state by Hammond and Knight [47], who found only a very -28-small change in the quadrupole- frequency at the superconducting transi-tion. Some measurements on the pressure and temperature dependences of the quadrupole frequency in powder samples have been undertaken by Kushida and Benedek [48]. An investigation of the nuclear quadrupole coupling in a single metal crystal of gallium at 4.2°K was carried out by Kiser [ l l ] . Two physically inequivalent sites for the EFG tensor were found and the orientation of each set of principal axes was determined. The experimen-tal results obtained are considerably less accurate than ours particu-larly with regard to the orientation of the principal axes and the asym-metry parameter. Kiser has proposed a theory based on a covalent model in order to explain the above results. It is reasonable to explain the gallium results using features of oriented bonds characteristic of covalency be-cause in the solid gallium each atom appears to form a molecule with one rather close nearest neighbour. The theory of Kiser assumes that the EFG tensor arises from p-type bonding of a gallium atom with its first seven neighbours. His theory is a generalized theory of the EFG tensor in solid halogens [49]. A brief outline of the theory will now be given. Consider a free atom with a valence electron in a given atomic orbit tyn ^ m. This electron produces an EFG, 9zZ*atom' a t t n e nucleus whose magnitude depends on f , . s-states do not contribute to q 7 7 . because they are spherically symmetric. The next higher states to contri-bute are p-states, and of these, the lowest lying p-state makes the lar-gest contribution [49]. In a diatomic molecule, the EFG is due mainly to the valence electrons participating in the bond. The molecular wave function \Ji for -29-a valence electron may be expanded in terms of atomic orbitals i|rn ^ m . The dominant contribution to EFG comes from the electrons in the atomic orbitals surrounding the nucleus in question. If \jj is built from pure p-atomic states, then the EFG in the isolated atom, q„„ . , should be r ' ' ^ ZZ;atom' equal to q , the axially symmetric EFG in the molecule. This is ZZ;mol just the case in solid halogens [49]. In their work [49] the authors consider also the effects of the solid structure upon q^Z-mol' •^ie halogens crystallize in a gallium-like orthorhombic l a t t i c e except that the diatomic molecules are very well separated. In solid iodine, the main bond between a given atom and i t s molecular partner 2.7 A away is supplemented by two weak auxiliary bonds between the given atom and the two nearest neighbours at 3.54 A. The next-nearest neighbours exert negligible effects because of their larger distance of 4.35 A. Each of these bonds produces an axially EFG at the origin. Its magnitude is Qzz-atom roultiplied by some weighting factor. When choosing this weighting factor one has to follow the rule that the bonds which are nearly equally inclined to the main intermolecular axis have the same weighting factors. A l l these axially EFG are then transformed into a common (crystallographic) coordinate system, added together and diagonalized. Using the experimental values for 7] and the angle y between the Z principal axis and the C crystal axis one then obtains the theoreti-cal ratio 9 Z Z•solid^ qZZ-atom a n d t h e weighting factors. The theory agrees well with experiments. Although the difference between qZZ. atom a n c* ^ZZ-solid t' i e case of gallium is large compared to the halogens, the isomorphism of their crystal structures suggested to Kiser the possibility of interpre-ting q„„ .. . , in terms of q 7 7,,. . The gallium structure provided £&;solxci acorn - 3 0 -evidence that in the solid a given atom the strongest interaction is with its first seven nearest neighbours. The 8-th nearest neighbour is far enough away that i t would be expected to produce only weak effects on the quadrupole coupling. The number and distribution of the bonds suggest that the EFG tensor at a given site is considerably altered at the expense of the formation of the additional bonds. Briefly, Riser's calculation proceeds as follows: A coordinate system x'y'z1 is chosen with the origin at site 0 and with z 1 along the shortest intermolecular axis 0 - 1 and with the positive direction of the y1 axis emerging from the figure towards the viewer (see Figure 3b). A bond, let's say, i produces an axially symmetric EFG tensor E at the origin with q ^ = q y y = -_ q Z Z ; a t o m and q - - = q Z Z ; a t o m . These individual tensors in the X ^ Y ^ Z ^ systems are then transformed into x'y'z1 system by -1—' the transformation A. E A.. A. is the rotation matrix specified in terms i l l of the Eulerian angles cp, 6 and i|t. The bond directions and the assignment o_ the weighting factors of the bonds are given in Table IT. The convention used is that of Goldstein [ 5 0 ] , The resulting tensor,Z W. A. E A., ob-i 1 tained from the summation of all the individual terms multiplied by their corresponding weighting factors W\,is then diagonalized. Using the ex-perimental values for T| and the angle y between the Z and G axes he obtained K = 0 . 1 1 , X = 0 . 1 2 and q--. s o l i d/q z z. a t o m = 0 . 2 0 . This is to be compared with the experimental value of 0 . 1 9 . TABLE II. The directions and the weighting factors of the bonds. Bond Direction 6 9 Weighting of the Factor Bond 0 - 1 - 0° _ _ 1 - 3K - 3 0 - 2 140° 30' 90° K 0 - 3 106° 30' 90° X 0 - 4 100° 15' 33° 20' 0 - 4 100° 15' 146° 40' X 0 - 5 125° 54' 333° 37' K 0 - 5 125° 54' 206° 23' K -31-A good test for the model is the temperature dependence of the EFG tensor with the theoretical predictions. The theory, for example, predicts q n to be very sensitive to Ti: a 10% increase in Ti pro-r nzz;solid J " duces a 20% increase in q „ . 0 . I - J . However, the experimental results Z Z j S O I X Q presented in Chapter VII disagree; a 4% increase in T| produces only a 4% increase in q z z-solid' However the Riser's model is semi-empirical and gives no details of the EFG tensor. Deriving his theory the following important facts are ignored: (a) The effects of charges on neighbouring atoms are neglected. (b) The core electrons on the quadrupolar nucleus are of spherical symmetry. (c) A single orbital is assumed.The overlapping of orbitals centered on the same (or different) atom are completely neglected. -32-CHAPTER V EXPERIMENTAL ASPECTS A. Apparatus The NMR apparatus used for the experimental work is standard in design (Figure 4).A b r i e f description of each unit i s given i n t h i s section. All measurements were performed with a modified Pound-Knight-Watkins (PKW) variable frequency spectrometer [51]. The frequency of the oscillator was swept by applying a variable amplitude (0-100 V) linear sawtooth, available from a modified Tektronix waveform generator type 162. This could generate linear sweeps of from 1 msec to several hours duration. Alternatively the sweep could be provided by a 90 V dry cell connected to a fine potentiometer driven with a variable speed electric motor. By sweeping back and forth one could eliminate the distortion of signals due to the finite sweep rate. The audio output of the marginal oscillator was fed into a PAR model JB4 lock-in amplifier. The modulation was achieved using a pair of coils, each wound with 60 turns of #18 copper wire on bakelite forms, mounted around the magnet pole faces. The modulation current was supplied by a Williamson type power amplifier driven by the lock-in detector. Se-veral different modulation frequencies from 20 Hz to 150 Hz were used. Keeping all the variables of the PKW oscillator constant, i t was found that the optimum signal-to-noise (S/N) ratio was obtained at higher fre-quencies, though no drastic variations were observed over this frequency range. Due to large magnetoresistance pickup from the magnetic field modulation i t was necessary to detect the second derivative at 4.2°K. Another method of -overcoming this problem was to modulate the frequency of the marginal oscillator. This was accomplished with the use of a - 3 3 -POWER AMPLIFIER MAGNET ATTENUATOR P. K.W. OSCILLATOR SAWTOOTH GENERATOR BATTERY & HELIPOT FIELD MONITERING DETECTOR O O -Q PHASE SHIFTER PAR LOCK-IN AMPLIFIER CHART RECORDER COUNTER & PRINTER Fig. 4: Apparatus - schematically. -34-variable capacitor in the tank circuit. The signal recorder used was a Varian recorder, model G11A. The sample holder was mounted on a coaxial line. The outer conductor was a 3/8n O.D. thin walled stainless steel tube. The central conductor was #36 copper wire held in place with teflon spacers. Continuous cooling and warming up of the coaxial line stretched the central copper wire causing i t to loosen and thus become very sensi-tive to the smallest vibrations. The resulting microphonics caused a serious reduction on the (S/N) ratio. This problem was eliminated by re-placing the copper wire with 1/16" O.D. stainless steel tube. Such a configuration was very rigid and completely free of microphonics. S i l -verplating the stainless steel tubes might further improve the system. The external magnetic field was supplied by a rotatable Varian magnet with 12" diameter pole faces and a 2-1/4" gap. The magnet was mounted on a graduated table so that field orientation relative to the sample in the plane of rotation could be determined to less than 0.1 of 6 a degree. The field stability was of about 1 in 10 over a period of several hours. The maximum field which could be obtained with the magnet was 11 kG. The frequency measurements were made with Hewlett-Packard elec-tronic counters, models 524C and 5245L, which were connected to Hewlett-Packard digital recorders, models 516B and 5622B, respectively. These in turn activated an indicator pen on the signal recorder every time a fre-quency measurement was printed. The magnetic field measurements were made by displaying the proton signal on CRO II. The probe used consists of a glass vial, 5 mm in diameter, fill e d with glycerine, and with a copper coil wound around -35-i t . The probe was situated just outside the liquid nitrogen dewar. The relatively high homogeneity of the field made any special precautions in positioning the probe unnecessary. Nevertheless, the positions of the probe and the sample were kept constant throughout all the experiments. The circuit [52] used to monitor the field was assembled in a shock-mounted heavy brass box to avoid microphonics. The coaxial line used with the box was made of copper tubing about one foot in length and 1/2" outside diameter with #32 copper wire as the inner conductor. A BNC connector on one end of the coaxial line was attached directly to the box while the probe was permanently mounted on the other end. Several geometrically differing probes were necessary to cover the frequency range from 11 MHz to 60 MHz. An additional probe was made to measure fields as low as 250 G, the lowest field measurable by directly displaying the proton signal on CRO IT. The frequency of the field monitor was controlled by a small variable capacitor, 2 — 8 pf, in the tank circuit. An extremely fine frequency control was achieved by connecting in parallel with this capa-citor a series combination of a varicap and a 2 pf capacitor. The vari-able voltage for the varicap was obtained from a 90 V dry cell and a 100" K Helipot. The 2 pf capacitor was included to ensure that the change in the capacitance of the varicap produced a very small change in the total effective capacitance in the tank circuit. The proton resonance frequency was read from the H.P. frequency meter. The circuit had excellent frequency stability, 3 in 10^, and very high (S/N) ratio. In all cases, the magnetic field was monitored before and after each trace using very low modulation. Upon rotating the magnet the field was slightly changed from its previous value due to the steel frame around the magnet. This effect is quite large at high fields. - 3 6 -The low temperature system used is shown in Figure 5. An ordinary double dewar glass cryostat was used to achieve low tempera-tures. A water bath, whose temperature was regulated to better than 0.1°C was used to obtain temperatures close to the melting point of. gallium. The temperature of the crystal was monitored with a copper-constantan thermocouple. B. Samples and Crystal Holder All the crystals were grown from high purity 79 grade gallium supplied by Eagle Picher Industries Ltd. To obtain a seed crystal a single crystal of random orientation was first grown in a teflon mold. Its orientation was then determined with X-ray back reflection Laue photographs. A cut was made perpendicular to the Z crystal axis using a jeweler's saw while a flow of cold air was applied to prevent melting (+29°C). The surface damage caused by cutting was removed by etching in concentrated hydroflouric acid (HF). The reaction is slow and takes several hours to produce a damage-free surface, which is indicated when blurred X-ray spots become sharp. This seed crystal was oriented with the A crystal axis perpen-dicular to its length and adjusted so that its bottom surface was slightly separated from the neck of a teflon mold specially designed to provide a 1 star-shaped1 crystal. Liquid gallium was then injected into this mold. Once the temperature of the liquid decreased close to the melting point or lower (gallium can be easily supercooled), the seed was slowly lowered into contact. Solidification required about one hour. In such a way a 'star-like' seed crystal was obtained in which the arms of the star had a known orientation. TO KINNEY PUMP He CYLINDER < » (g) 0-HeTRANSFER DEWAR ^ | TO SIPHON JACKET (g) FORE PUMP (5?) k NEEDLE ^ I VALVE 0 0 TO RETURN ^ LINE 0 OIL TRAP MANOMETER OIL BUBBLER 0 0 Hg ® OIL DIBUTYL PTHALATE 1 oo I MANOMETERS Fig. 5: The low temperature system. -38-Using this seed, crystals of specified orientations could be easily grown in a teflon mold of the form shown in Figure 6b. These were all of cylindrical form, 3/8" in diameter and 3/4" in length. The long, narrow neck assured a monocrystal and provided a convenient way of handling , the crystal while X-raying and winding the coils. To avoid strains from the expansion of the metal at the transition from the liquid to solid, the mold was cut in two halves which were taped together. The crystals were examined with X-rays. The sharpness of the spots in photographs taken of different points on the crystal surface in-dicated a high degree of perfection of the crystal. There is some confu-sion in the literature regarding the distinguishability between the A and B axes. However, if a copper target is used, these different crystallo-graphic axes can be determined unambiguously [53], Several X-ray photo-graphs are shown in Figure 7. The orientation of the crystal can be de-termined with high accuracy merely by observing the intensities of the X-ray spots. An example of this is presented on the same figure. The centre picture in the lower row was taken with the beam parallel to the B axis. The intensities of all characteristic spots are equal. The crystal was then rotated 0.3 degrees clockwise about an axis perpendicu-lar to B. The X-ray photograph for this situation is represented with the last figure in the lower row. It is easily noticed that the two spots on the right side of the figure are less intense. The first.photograph in the lower row depicts a 0.3 degrees counterclockwise rotation but some of the difference in intensities has been lost in printing the picture. Si-milar effects occur with the beam along the two remaining crystal axes. In all experiments #40 copper wire was used for the sample coil. A layer of 0.001" thick mylar used as an insulator between the coil and 6 a . S b . F i g . 6 : a) Schematic diagram of the copper holder for the crystal. b) Schematic diagram of the teflon mold for growing the crystal. -41-the specimen provided a high enough quality factor Q to make the oscillator marginal at a l l temperatures. Thus the crystal, once mounted into the magnet, could be kept in the same position. By adjusting the frequency of the coil to around 13 MHz at room temperature i t was possible to perform low and high field measurements on the same crystal without rewinding the coil. The fre-quency range required for this (from 11 MHz to 15 MHz) could be covered by connecting the PKW box to the coaxial line with cables of appropriate length. This had the effect of introducing various capacitances into the tank circuit. With a 0.5 m long cable the PKW box could s t i l l oscillate at the frequencies convenient for the NMR measurements but with intensities of the signals slightly reduced. The crystal was mounted into a vertical hole on the special copper holder as shown in Figure 6a. The top part of the crystal was accessible to the X-ray beam. A whole series of adjustments and of X-ray shots was re-quired to orient the crystal to within less than one degree with respect to the holder. This procedure required a combination of delicate handling and great patience. The holder was hooked tightly to the coaxial line with a BNC con-nector. Care was taken that the holder should not move (rotate) while the magnet was rotated. A long, thin glass tube, glued on the crystal holder parallel to the X-ray film (perpendicular to one crystal axis) was used to locate the position of the crystal axes within j l degree with respect to the magnet pole faces. Because of the rapid angular variation of the NMR lines of interest, the in i t i a l location of the crystal axes in the field plane must be known to within a few degrees. The usual crystal position was such that the magnetic field could be rotated in a plane formed by two crystal axes. The exact orientation within the magnet was determined by -42-using the properties of the high f i e l d NMR spectra of the two gallium isotopes. This procedure is discussed in Chapter VI. - 4 3 -CHAPTER VI EXPERIMENTAL RESULTS A. EFG Tensor Results The resonance frequencies of the important lines for a rotation of the magnetic f i e l d in a plane perpendicular to the B crystallographic axis are shown in Figure 8. The occurrence of two sets of resonances demonstrates that two physically non-equivalent nuclear sites exist in the crystal. Since the two sets of lines only d i f f e r in the relative orientation, i t follows that the EFG tensors dif f e r only in the orienta-tion of their principal axes and have the same coupling constants and asymmetry parameters. The form of the orientation dependence together with considerations of crystal symmetry indicates the positions of the Z and X axes of the two EFG tensors shown in Figure 8. The positions of the two Z axes were determined more precisely from the behaviour of the central line corresponding to the [+1/2M/'** \-l/2M^>transition for the case of the magnetic f i e l d close to the Z principal axis. A calculation _ i shows that the resonance frequency of this line for the case H II Z i s o given by the following expression which omits only a negligibly small term in T|^ . i 2 v = VL [1 - Ti /3 — i ] . (36) Q Here 2rrvT =YH (1 + K) where K is the Knight shift. A misalignment of the magnetic f i e l d by a small angle 6 from the Z axis causes a deviation 8Vin 2 the resonance frequency which is given to f i r s t order in 6 by V [kHa] 11400. If 3 0 0 II 200. B 1 H 0 ; <f> = * ( C , H 0 ) ; T=4.5°K; H 0 = 6 0 Gauss R 6 9 - 6 0 - 4 0 - 2 0 2 0 6 0 C DEGREES) 100 i i Fig 8- Ga 6 9 spectrum as a function of orientation of the magnetic field HQ in the-AC. crystal plane. Each point corresponds to an absorption resonance signal. - 4 5 -VL 2 6 Y= 3/2 - 6 , (37) Vr 2 where 6 is measured in radians. This expression was calculated using the perturbation theory. The same results may be obtained from the matrix of Equation (26). It is obvious from the expression that S'fis highly sensitive to misalignment when is close to VQ° but can be made com-paratively insensitive by a suitable choice of V J / V Q ° . I N principle, with the reasonable assumption that K is field independent, two sets of measurements at different magnetic fields would suffice to determine both K and the misalignment. However, this method would give accurate results only by employing very high magnetic fields. Fortunately.gallium has two isotopes with rather different quadrupole moments (Table I), resulting in considerably different quad-rupole frequencies. The maximum field available was 1 1 kG and the lowest frequency obtainable in our spectrometer was 9 MHz. With these parameters the ranges in which the ratio VjVVQ° could be experimentally varied were ( 0 . 8 — 1 . 1 ) for Ga 6 9 and ( 1 . 4 —2.2) for Ga 7 1. In the field of, let's say, 1 0 kG, y^/VQ° could be set to about 1 for Ga^ and to about 2 for Ga7'''. According to Equation (37) the angular variation of the Ga^9 central line for *H close to Z is very rapid and that of Ga7''' is rather slow. The o angular variations for both isotopes about' the Z axis are shown in Figure 9 2 and are again displayed in Figure 1 0 as a function of. 6 . From the slowly varying results on Ga7''' in Figure 9 , i t is possible to determine the Knight shift along Z using Equation (36). With this value and with the assump-tion that K is the same for both isotopes, one can then calculate the an-6 9 gular variation for Ga . Due to the very rapid angular variation for -46-[kc/sec] 11640. 71 B 1 H 0 ; 0= « (C,Ho); T =77 °K H 0 =8.932 lc& II62Q-11600.-i i II 180.-II 160-II 140.-II 120-II 100.-11080-6 9 11092,3 H 0 = 10.805 k& •26 - 2 5 -24 -23 0 23 6 ( D E G R E E S ) 24 25 26 F i g . 9: High field angular dependence cf the central line for both isotopes close to the Z principal a:;is. The field strength H is quoted in kG. o ^ Fig. 10: Angular dependence of the central line for both isotopes as a function of the square of the mis-alignment 8 Q between the magnetic field and the Z principal axis. -48-Ga , Equation (37) calculated in first order perturbation theory is inadequate. An exact calculation of £lf(8) using Equation (26) is neces-sary. From <Tv"(9) the misalignment of HQ with respect to the Z axis is determined and is then used to correct the value of the Ga7^ Knight shift. This procedure is repeated for self-consistent results. For the results shown in Figure 9, the procedure above yields a misalignment of 0.2 degrees. The agreement between theory and experiment for 6v -0.2 degrees is shown in Figure 10. For a l l the measurements the misalign-ment never exceeded 0.5 degree. The results show that both Z principal axes l i e in the AC plane of the crystal and they make possible a highly accurate determination of their relative orientation. It should be noted that the crystal has reflection symmetry in the AC plane which requi-res that the B axis of the crystal be one of the principal axes and that the other two axes l i e in the AC plane. As is seen from the experimental results, these two axes are the X and Z and the B axis is coincident with the Y principal axis for both crystal sites. This was checked experimen-tally by studying a crystal rotation in the AB plane. In this case the patterns from both sites are observed to coincide, and the 'inner pair' splitting along the B axis is equal to that along the Z axis. An important source of errors could arise from the misalignment of the crystal but fortunately there is a very elegant way for precise determination of the crystal orientation. In the following, the method presented schematically in Figure 11 will be briefly discussed. Most of the results were obtained from the NMR experiments per-formed in the AC crystal plane which appeared to be also the XZ principal plane. In an actual experiment the AC crystal plane (lightly shaded) was of course slightly inclined with respect to the magnetic field plane -49-(heavily shaded). The latter is also the (1)(2) plane of the laboratory frame. The mutual positions of the two planes shown in Figure 11 re-present just one possible situation and it can be absolutely separated from the others in the following way: The curve representing the angular variation of the central line for the Ga7''" nuclei, situated at the site (1) crosses the corresponding curve for the nuclei at the site (2) along the C and A crystal axes. Setting the magnetic field up to 10 kG, the points C" and A' could easily be established with an accuracy of less than 0.1 degrees. These two points appear at the closest distance to C and A in the crystal plane. The angle (o^ + a^) was usually 0.1 or 0.2 degrees larger than TT/2. Next, the points and Z^ were determined by 69 looking at the rapid variation of the Ga resonance as discussed pre-viously. The two points appear at the distance closest to Z^ and Z^ . With such a procedure the angles , a n c* deviations 6-^ and 6^ were determined. In addition, i t was necessary to know whether (Z^ ) was above or below Z^(Z^). This was achieved by winding a copper coil of 5" in length around the tail of the outerdewar. A d.e. current of vary-ing magnitude created a magnetic field of up to 200 G perpendicular to the main field. With this coil the main field could be tilted by a small angle and so directed exactly along Z^ and Z^ . With the data ob-tained from such an experiment the orientation of the crystal in the laboratory system was well defined. The values of the parameters determining the EFG are tabulated in Table III and the orientation of the two sets of principal axes is shown in Figure 12. -51--52-TABLE III. The EFG tensor results. Temperature C°K) 69 VQ (KHz) (&z) 2Y (degrees) 4.2 11312.2 + 0.4 7091.2 0.179 + 0.001 48. 7 + 0.2 78 11253.2 + 0.4 7053.6 0.179 + 0.001 48.6 +0.2 285.3 10877.2 + 0.5 6820.7 0.171 + 0.002 48.2 + 0.2 With regard to Figure 12 it should be pointed out that the assignment of a Z axis to a particular nuclear site is not unique. The only case considered in this work was the one which is schemati-cally presented in Fig. 12a (Case I). From the point of view of the symmetry, Case II is equally possible. However, there are physical arguments against i t . Perhaps the main one is that in the solid gallium each atom appears to form a molecule with one rather close nearest neigh-bour. A similar situation is characteristic for solid halogens as dis-cussed already. One expects then the Z axis to be close to the axis of the "diatomic molecule" (see Section C of Chapter IV). In order to get any reliable value for the Knight shift and its anisotropy, a very accurate value for 7] is required. The determination of T| using Equation (25) depends on the measurement of the splittings A and A which ai-t unaffected by the distortipns in the line shape due to z modulation or the effect of the skin depth in mixing the modes (see Sec-tion C of this chapter). It is merely necessary to measure between any two equivalent points on the resonance lines. The results of a typical experiment is represented in Figure 13. Even though the magnetic field used was small, its influence on the determination of T| from Equation (25) was not entirely negligible. This is illustrated by the results and theo--53-Figure 12a: The possible orientations of the two Z axes with respect to 'diatomic axis'. a;;es in the field of 300 G. -55-retical variation shown in Figure 14. The fact that this correction was considered at a l l indicates the high precision of the results. The determination of the quadrupole frequency does require a line shape analysis to determine the true resonance frequency. It should be noted that V q 7 * was calculated from measured using very 71 , 69 accurate values for the ratio Q /Q obtained from the atomic beam measurements of the quadrupole hyperfine interaction [54]. The values of K along the Z axis have been published in a pre-liminary note [11]. Fig. 14: The effect of the magnetic field on the deterra of. Tl. -57-B. Knight Shift Results The high field NMR has not been observed previously in a gallium crystal largely due to lack of knowledge of the EFG tensor. Once the EFG tensor data were obtained, i t was possible to observe the high field NMR lines and extract the Knight shift and its anisotropy. The methods will be now briefly discussed. The.main interest was in observing the central line corres-ponding to j 1 /2My>*->\-1 /2M^> transition because the effect of the quadru-pole interaction on this transition is the smallest. All further dis-cussions of the resonance line in high fields will refer to this transi-tion unless otherwise specified. Figure 15 represents the calculated angular variation for this transition with the magnetic field in the AC crystallographic plane (which is also the XZ principal plane) and Fi-gure 16 with the field rotating in the X Y principal plane of the EFG ten-sor of one nuclear site. The curve corresponding to the other site is of no interest for this special rotation and is not shown. Of course, the two curves coincide along the Y(HB) axis. In the AC plane the two curves for the angular dependence corresponding to the two non-equivalent sites are identical but shifted with respect to each other by the angle 2y (Figure 15). This fact is very important and could be used for de-termination of the symmetry of the Knight shift tensor. It is seen that the angular variation in this plane is very rapid and possesses a maximum along the Z and the X axis and a minimum at 38.5 degrees from the Z axis. The effect of the crystalline field on the position of the central line decreases with increasing ratio V J ^ Q ° a n f l for this reason i t is better to look for Ga^ resonance when measuring the Knight shift. The experimen-DEGREES Fig..'15: The angular variation of the central line for Ga with"H rotated in t h e AC crystal plane. The curve was calculated from the matrix of Equation (26) using" parameters as noted (K = 0). The two curves correspond to the two non-equivalent sites. tr 1 [~ kc/sec] H 0 = 10764.6 GAUSS 14730 -, 0 = 7053.6 kc/sec V = 0.179 14690 14650 14610 14570 14550 ! Cn I 30 30 60 90 120 DEGREES Ga with ~H in the XY principal plane calculated from the maLrix of Equation (26). The frequency of this tran-curvc for the second site is not shown. Fig. 16: The angular variation of the central line for f t sition is constant if T| = 0 The -60-tal errors of the EFG data (in particular T|) strongly influence the accuracy of the Knight shift (K). This influence could be decreased by applying higher magnetic fields which were limited to 11 kG in these experiments. In spite of very accurate EFG data, reliable measurements of K in this field (vT ~ 2vn°) could be obtained only along the turning L 4 points, i.e. along Z, X, Y and along the minimum at 38.5 degrees from the Z axis (Figure 15).. From now on the notation K^ , Ky, K z and K^ g is used for the Knight shift measured along X, Y, Z and 6 = 38.5; 0=0, respectively. The four (non-equivalent) experimental points were suffi-cient to determine the K-tensor. However, the application of a high magnetic field would partially swamp the otherwise large crystalline quadrupole coupling and so yield more experimental data. The position of the central line is a function of five inde-pendent variables; v = v (6, 0, V Q°, T|, H ). The influence of these variables on i t for some selected magnetic field directions is presented in Table IV. Table IV was calculated for two values of the magnetic field using o T| = 0.179 and V Q = 7053.6 kHz. The upper numbers refer to H^ = 10 kG and the lower to H =8 kG. Each row in Table IV represents a shift of o Vj^ due to a change of one variable at the time; i.e. Av^ = (6 + A6, 0, v °, Tl, H ) - vT(6, 0, v °, T|, H ) for the first row. Apart from the Q o L 0. o misalignment the largest errors are due to the inaccuracy of T|. For practical purposes Av is linear in Av °, AT] and AH . The position of L Q o the central line along the Z axis is very insensitive to the EFG tensor parameters. Due to the large Av along the C and the A axis the Knight L shift in these directions cannot be measured accurately. An example illustrating the method for obtaining the Knight shift is presented in Figure 17a. The curve with points represents the -61-TABLE IV. The influence of different parameters on the position of the central line, t+l/2> |-l/2^>, for some selected magnetic field directions. H 10000.0 (G) 8000.0 H0II Z H0II X H 0H Y A\>L *o 11 C l H II C o it II A o A 6 = 0.2 (degrees) -0.07 -0.18 -0.08 -0.11 -0.13 -0.13 0.30 1.45 -6.97 -7.71 10.83 13.01 A 6 = 0.5 (degrees) -0.55 -0.90 -0.55 -0.56 -0.66 -0.76 0.95 3.91 -17.30 -19.05 27.36 32.53 L\0= 1.0 (degrees) 0.00 0.00 -0.08 -0.08 0.01 0.02 -0.10 -0.15 -0.06 -0.06 -0.14 -0.18 AvQ° = 0.4 (kHz) 0.00 0.00 0.11 0.11 0.06 0.08 -0.10 -0.13 -0.11 -0.11 -0.01 -0.00 AT] = -0.001 -0.07 -0.16 -0.49 -0.58 0.43 0.53 -1.20 -1.50 -0.82 -1.02 -1.01 -1.24 AH =0.3 o (G) 0.39 0.39 0.37 0.36 0.43 0.37 0.45 0.43 0.46 0.45 0.38 0.38 TABLE V. The Knight shift along the C , C A and A directions. H (G) o K(%) H 11 A H il A H- 14 C 0 H II C o 2 o ' l o 2 o l 10965.62 0.124+.010 0.126+.010 0.124+.010 0.125+.010 -62-V (kc/sec) 14750 14700 14650, 8 4 <(> • 0 ; T • 77 °K ; B 0 • 1 0 7 6 4 8 Gouss Go 71 8 6 experiment theory ( K - 0 ) 14745 .0 kc /sec 8 8 9 0 D E G R E E S 17a. 0. <t>. yQ. -n ' c o n s t Fig. 17: The method for extracting the Knight shift at the "turning points'. The r e t i c a l l y -the position of the resonance is below its experimental value (Fig. 17a). The relative shift in the f i e l d at the nucleus (Knight sh if produces the observed resonance frequency (Fig. 17b). -63-experimental and the smooth curve the theoretical angular variation of the central line around the X axis and with the field rotating in the XZ plane. The experimental curve is shifted with respect to the theo- . retical because in the calculations of the latter, the Knight shift was not taken into account. In order to obtain K one has to evaluate the field which produces the shift in the resonance frequency v in the measured direction. This is done as follows: In the matrix given by Equation (26) one has to replace H = H 6 X p with H = H e f f = H 6 X p(l+K) n ' o o o o o exp keeping K as a variable. Using the known EFG tensor data and one th en calculates VL = "^(K) for a specified direction. The K value for which v (K) = v T e x p is then the true Knight shift. However, since the L t i p y n calculated v is linear in H for a given direction, one obtains the L o . same result in a manner sketched in Figure 17b. The results for the Knight shift in gallium are indeed unique. -Figure 18 presents the angular dependence of the Knight shift in the XZ plane at three different temperatures obtained from an experiment in the AC crystal plane. Each solid curve is a theoretical prediction for the angular variation of the anisotropy of K in case of an orthorhombic sym-metry given by Equation (35). The curves are the best f i t to the experi-mental results. The large error bars include the statistical error due to the signal strength with corresponding uncertainty in the resonance fre-quency, misalignment and especially asymmetry parameter T|. Each point in this diagram is an average of many absorption curves. It is seen from Table V that the Knight shift results depend strongly on the correct value of 7|. This particularly applies for K . JO However, since the allowed form for the angular variation of K is given by Equation (35) , one can improve the appearance and accuracy of the re--64-sults included in Figure 18. Perhaps the best way to present the results is the following: The values K^ , K and K^. lie on a reasonable straight 2 line when plotted versus cos 6. This is just the predicted angular depen-dence for the K anisotropy in the XZ plane (see end of this section). 2 Therefore, one chooses T| to give the best cos 6 variation. The correct value of T| is then a compromise between this chosen T| value and the va-lues of T| obtained from the low field experiments (Table III). The corrected T| values are 0.180, 0.177, 0.175, and 0.173 at 4. 2, 77,195 and 300°K, respectively. Note that they are s t i l l within the experimental range of values obtained from the EFG tensor measurements (see Figure 14). This procedure affects the magnitude of the measured Knight shift values very l i t t l e . However, the large inaccuracy, which is mainly due to T|, is reduced. These new K-values will be called corrected values. Figure 19 is then a replica of Figure 18 using the corrected Knight shift values. An interesting result, apparent from Figure 19 is a considerable increase in K at room temperature compared to the values obtained at 4.2°K. The relative increase is angular dependent. The difference (K^-K^) which can be regarded as a measure for the Knight shift anisotropy is 0.026%, 0.032% and 0.022% at 4.2, 77 and 300°K, respectively, and is large com-pared to the rest of the metals. The anisotropic Knight shift, K a n, is separated from the isotropic, K . a t the end of this section. The complete angular dependence of the Knight shift (uncorrected) in the AC crystal plane is presented in Figure 20. The two curves in this figure indicate that there are two K-tensors, one for each of the two physically non-equivalent nuclear sites. The strongest experimental evi-dence for this statement are the two points at X£ and A^(or X-^ and A^) in Figure 15, corresponding the two very close directions in the crystal. -65-. 2 0 0 o o o 2 9 4 °K @ © © "7 "7 (_ . 160 u . X CO I CD *1 J 2 0 . 0 8 0 - 2 0 O 2 0 4 0 6 0 8 0 100 DEGREES Fig. 18: The experimental angular dependence of the Knight shift in the.XZ principal plane at three different temperatures. The solid line is the theoretically predicted angular varia-tion for the Knight shift anisotropy for an orthorhombic symmetry. - 6 6 -• 2 0 0 u. X CO H I 2 z a: -160 0 0 0 2 9 4 °K A A A 1 9 5 o K © @& 7 7 A A A 4 - 2 ° K •120 >080L_ - 2 0 2 0 4 0 6 0 8 0 100 D E G R E E S Fig. 19: Fig. 18 with corrected Knight shift values. Included i also the angular dependence at 195 K. 4 - 4 0 0 4 0 8 0 120 D E G R E E S Fig. 20: The angular dependence of the Knight shift in the AC crystal plane. The two curves indicate that the two Knight shift tensors are symmetric with respect to the EFG principal system. The full curve is the theoretical prediction for K-anisotropy in an orthorhombic crystal due to the dipolar term. D E G R E E S Fig. 21: Fig. 20 with corrected Knight shift values. -69-The K values for the two sites along these directions obtained from a measurement in a constant magnetic field, are very different and the difference is easy to detect. Each of two curves for the angular dependence of K in the AC plane is symmetric with respect to the corresponding experimental EFG tensor principal axes. This rather surprising fact is strongly evident from the four experimental points for the Knight shift measured along the C^ , C^ , and A^ directions (Figure 20). The two experimental points measured along C^ and A^ are symmetric with respect to Z^ ; the points measured along and are symmetric with respect, to Z^ . The uncorrected Knight shift values for these four directions, obtained from a measurement in a constant magnetic field at 77°K, are given in Table V. It is seen that they are all equal. The large inaccuracy of these values is, as discussed previously, mainly due to the inaccuracy in T|. In a constant magnetic field the position of the central line for a given T) and assuming K = 0 is the same for all four directions. An in-accuracy AT] of T| causes an inaccuracy Av^ of \>L a n d hence an inaccuracy AK of K. This error AK, however, is the same for all four directions. Therefore, the large error bars for K^g in Figure 20 do not influence the argument about the K symmetry. Figure 21 is a reproduction of Figure 20 using the corrected Knight shift values. The result for the symmetry of K was unexpected and i t was thought that the crystal was affected by the magnetic field (changes in dimensions or its orientation^induced strains). The measurements, sepa-rately repeated for each direction at different magnetic fields (down to 1/3 of the maximum available field), gave the same K values, thus excluding the possibility of any magnetic field induced effects. Some of the results -70-are included in Table VI. Note the increasing errors of K in lower magnetic fields arising from the reduced signal intensities and larger influence of the EFG tensor parameter on (Table IV). At 4.2°K no experimental evidence for the magnetic field dependent Knightshift was found. The corrected experimental Knight shift values for different orien-tations and several temperatures in the range from 4.2°K to the melting point (TMP = 3 0 0 ° K ) a r e listed in Table VII. With regard to Table VII a few re-marks should be pointed out. The two EFG tensor parameters, T] and V Q ° , necessary to obtain the Knight shift, have not been measured at 195, 294 and 299°K. Since T| is changing very l i t t l e with temperature a linear interpolated value, T) = 0.175 at 195°K has been used. In the temperature interval close to the melting point T| does not change. This is evident from the measurement of the central line in the 6 = 38.5°and 0 = 0° direction for which AV L/AT| is extremely large and Av^/AVQ° = -0.26. The temperature shift of v L is small o ° and is accounted for by changes in only. The value VQ = 6941.9 kHz at 195°K has been obtained from Figure 21. The values close to the melting point have been obtained from the measured value at 288°K (Table III) and using the temperature coefficient Av^^/AT = -1.34 kHz/°K. When measuring the Knight shift in the Y direction (B crystal axis) the field was rotated in the BC crystal plane. Only KyCould be ob-tained from such a rotation. The BC rather than AC crystal plane was chosen because the angular dependence of v in. the former is more rapid. The correct form of the Knight shift anisotropy in gallium or-thorhombic symmetry is given by Equation (35) which can be written as 2 2 K = K. +K =K + K,(3cos 6-l)/2- K„(sin 6 cos 20 )/2. iso an iso 1 2 (37a) -71-TABLE VI. The uncorrected Knight shift in three directions measured at different magnetic fields. H (G) o K(70) H (G) o K(%) Ho(G) K(%) H 11 X 0 o 2 o 2 Ho II C l 10965.76 .163+. 005 10965.62 ;124+.010 10966.85 .125+.010 9164.89 .159+.006 9165.03 .118+.011 9165.81 .118+.011 7965.07 .166+.007 • 8175.29 .119+012 8175.29 .119+.012 6964.96 .168+.008 TABLE VII. The corrected experimental Knight shift values for several orientations and temperatures. T (°K) Kx * (%) KY K38 4.2 0.115+0.004 0.148+0.004 / 0.128+0.005 77 0.117+0.003 0.168+0.003 0.141+0.003 0.136+0.004 195 0.128+0.004 0.166+0.004 0.145+0.004 0.146+0.005 289.8 0.140+0.004 0.174+0.004 0.158+0.005 293.8 0.138+0.004 0.170+0.004 0.165+0.005 299 0.139+0.004 0.168+0.004 0.158+0.005 294.3 0.159+0.004 299.7 0.152+0.004 -72-6 and 0 denote the orientation of the magnetic f i e l d . While in the case of the axial symmetry the choice of the Z axis is unique, in orthorhombic symmetry any of the three p o s s i b i l i t i e s could be chosen. As pointed out earlier, the principal axes of the K-tensor and that of the EFG tensor f a l l on top of each other. Therefore, the EFG tensor principal system i s assumed throughout this work. & Q is the isotropic Knight shift re-lated to the measured values K^, K^ and K^ with the expression K. = (K +K -HC )/3 (37b) iso X y Z Equation (37b) is a consequence of the fact that the Knight shift i s a second rank tensor with zero trace. The two parameters K^ and K2 are re-lated to the constants A and C2 of Equation (35). They can be evaluated from the relations K_ = K -K. (37c) 1 Z iso K = K -K 2 Y X which immediately follow from Equation (37a). ^ - j [ s o > ^ a n d ^2 a t ^ ^ ^ e ~ rent temperatures were calculated using the data of Table VII. They are displayed in Table VIII together with K along the X, Y and Z principal 3X1 axes for which the notation K (X), K (Y) and K (Z), respectively, w i l l an^ '' an an ' ' be used. The temperature dependence of KY, Kv, K 7 and K is graphically A 1 L iso shown in Figure 22. Ky was not measured at 4.2°K. An extrapolated value of 0.133% was used for evaluation of K. . It is seen from Figure 22 that ISO K K at a l l temperatures. Thus, within the experimental error the iso Y anisotropy i n the-Y direction i s zero at a l l temperatures. K.„_ varies I S O linearly with temperature; the temperature coefficient 1/K(«^K/ <=>T) at P TABLE VIII. K , the anisotropy constants K. , K« and the p r i n c i p a l values of the K a - tensor f o r iso 1 ^ n several temperatures. T (°K) K i s o K 2 K l s Kan< Z> (%) K (Y) an K (X) an 4.2 0.132+0.004 -0.015+0.004 -0.017+0.004 +.001+.004 .016+.004 77 0.142+0.00? -0.027+0.003 -0.025+0.003 - . 0 0 1 + 0 0 3 .026+003 195 0.146+0.004 -0.02lt0.004 -0.018+0.004 -.001+.004 .020+.004 290. 0.157+0.004 -0.016±0.004 -0.017+0.004 +.001+.004 .017+.004 294 0.156+0.004 -0.011+0.004 -0.018+0.004 +.003+.004 .014+.004 299 0.153+0.004 -0.016+0.004 -0.014+0.004 -.001+.004 .015+.004 -74-Fig. 2 2 : The temperature dependence of Kx, Ky, K z and K i s Q. -75-300°K i s -K).45xl0" 6 ° K _ 1 . At the melting point K; = (0.155+0.004)% which iso i s by a f a c t o r of 2.8 smaller than the Knight s h i f t of 0.45% i n the l i q u i d . K (X), K (Y) and K (Z) as a function of temperature are plotted an an an i n Figure 23. From t h i s diagram follows that (a) K a n ( Y ) = 0 at a l l temperatures and (b) K a n ( X ) and K^CZ) are r a p i d l y varying with temperature with an extremum at around 77°K. -76-$620 cc O < LL I 0 0 .0 1 0 0 Kan(x> K a r P 2 0 0 3 0 0 ' TEMPERATURE (°K) F i g . 23: The temperature dependence of K (X),K (Y) and K (Z) • an' " a n w an v ' -77-C. Line Shape Considerations In the interpretation of the NMR absorption lines i t is usually assumed that the amplitude of the r.f. field is insufficient to cause saturation so that there is a linear relationship between the nuclear magnetization and the r.f. field, i.e. the real (x 1) and imaginary (x") part of the complex susceptibility are independent of the r.f. field. For gallium, this is a good approximation because the spin lattice rela-xation" time is sufficiently short that the saturation effects may be disregarded. This assumption was checked experimentally by looking at the line shapes for different amplitudes of the r.f. field. In metals, when at least one sample dimension is greater than the r.f. skin depth 6, the unsaturated absorption line may be affected in several ways [55], First of a l l , only a limited fraction of the sample is exposed to the r.f. field, so that the available signal in-tensity is relatively small. Secondly, the quality factor of the system is reduced by the r.f. conduction losses resulting in decreased sensiti-vity of the detection apparatus. In addition to a reduction of intensity, the resonances may be distorted, because the conduction losses have the effect of introducing a mixture of both x' a*id x" °f t n e complex susceptibility into the total absorption. The power absorbed in an NMR experiment is determined by the r.f. field and the out-of-phase component (normally x" when the skin effect is not important) of the complex susceptibility. In the penetra-tion of a conductor by an r.f. field the phase of the current is gradu-ally shifted becoming TT/2 at the skin depth 5, so that x' c a n contribute to the power absorbed. Several experimental methods for analyzing such distorted lines are given in the literature [56]. -78-For a good conductor, in the region of the normal skin effect, the measured absorption is proportional to equal mixtures of x1 a n ^ x"-Thus the absorption is asymmetric and its derivative has unequal maximum and minimum amplitudes. The indicated frequency, obtained from the cross-over point of the derivative with the apparent base line, is shif-ted down from the true resonance frequency. 69 Figure 24 shows the observed Ga resonance signals at three temperatures obtained from a cylindrical gallium crystal. They are slightly distorted by the modulation amplitude used. The resonance lines were found to be well represented by a Gaussian absorption line and the resonance frequency was determined by fitting the results to a modulation broadened superposition of dispersion and absorption modes. Figure 25 shows a comparison between the observed resonance at 77°K and the theoretical Gaussian line shape for an equal mixture of modulation broadened absorption and dispersion modes. The way this was done is briefly discussed in Appendix E. In the low field experiments, where one is most interested in the NMR lines corresponding to the |+3/2M>**»\jl/2M> transition, the line shapes at 285°K and 77°K were the same. The line shape at 4.2°K re-vealed a larger proportion of the dispersion mode in contradiction to the shape predicted in anomalous skin effect regime [57]. It is possible • that this prediction which applies for zero magnetic field is invalid for finite fields. A stronger support for this statement are the line shapes taken at high fields. They revealed a very high proportion cf the disper-sion mode which is clearly seen from Figure 26. The two lines represent the observed first and second derivative resonance signals corresponding to the /+l/2M>*-^ (-l/2M> transition of Ga 7 1 at 4.2°K. I I F i g . 24: G a 6 9 resonance signals ( f i r s t d e r i v a t i v e s ) at (1) 285°K, (2) 77°K and (3) 4.2°K obtained from low f i e l d experiments for |+3/2^^1+3/2M>transitions. The numbers S D C c i f v the freauencv scale in kllr.. 8 EXPERIMENTAL NMR LINESHAPE OF GALLIUM:. THEORETICAL DERIVATIVE OF NORMALIZED GAUSSIAN LINE i ABSORTION MODE "2 DISPERSION MODE 3 EQUAL MIXTURE OF MODES -9 =*TT = 0 =nr s. Fig. 25: The experimental derivative (crosses) taken at 77°K fitted to an equal mixture of normalized Gaussian absorption and dispersion modes. ( I ) (2) Fig. 26: Ga resonance signals (1) first derivative (2) second derivative, at 4.2 K obtained from high field experiments for the central line. -82-No evidence for f i e l d or orientation dependence of the signal shape or line width was found. The f u l l width 6 H . i . e . the dis-r meas tance between the minimum and maximum of the experimental derivative signal^is temperature independent. It should be noticed that a detailed analysis of the line shapes was out of the scope of this work. A calcu-lation of the second moment and a brief discussion of the possible me-chanisms for the line broadening are in the Appendix B. In Appendix B i t is shown that the observed linewidths are greater than those which would be predicted from a reasonable extension of Van Vleck's theory of the second moments.This descrepancy.could be due to the improper assumptions made in that calculation.(A rigorous theory of the se-cond moment appropriate to the case of gallium does not exist.)An obvious source of the excess observed broadening might be the inhomogeneous quadru-polar and magnetic frequency shifts associated with possible crystalline strains or imperfections.These crystalline imperfections would also affect the precision of the EFG and Knight shift tensor results depending on how 69 large they are.In case they were large i n our experiments,the Ga linewidth 71 69 71 would be broadend with respect to the Ga linewidth because Q >Q • .This was not observed at any temperature.A stronger evidence that the effects of . crystalline imperfections were negligible in our samples is the angular depen-dence of the frequency position of the central l i n e . I t has been shown that t h i s position can be either weakly or extremely strongly influenced by the EFG tensor parameters.The influence i s dependent on the orientation of HQ. Therefore,one would expect very different lineshapes for the two cases.However, none of these effects were observed even by using different samples.lt was concluded then that the quadrupoiar inhomogeneitis in the sample used were small and could not influence the precision of the results.For this reason the separate experiments for detecting such inhomogeneitis were not considered. -83-D. Knight Shift in Liquid Gallium It was found that the Ga isotropic Knight shift in the solid, K?0''", is by a factor of 2.9 smaller than K^^, the reported value for iso' J iso ^ i t in the liquid [75], In order to determine the ratio K 1<VKS°''' abso-iso iso l i q lutely, K. was measured in our magnet under the same geometrical con-J ' iso & b sol ditions as K iso The liquid sample, prepared from the material from which the single crystals were grown, was kept in a thin-walled glass tube. A typical signal is shown in Figure 27. The shape is a mixture of dis-persion and absorption modes of the Gaussian line brought about by the skin depth. The peak-to-peak separation, 6H, for Ga7''' is 0.38 G and 69 0.58 G for Ga compared to 0.3 G for both isotopes as obtained by 69 Cornell [13]. The di screpancy for the Ga case is ascribed to the magnetic field inhomogeneity. At about 8 kG the inhomogeneity is much 3 less than 0.1 G in a volume of 1 cm but i t rapidly increases in the region of the maximum obtainable fields (<~ 11 kG). However, the main interest was a precise measurement of for Ga7''". ISO The experimental results of the temperature dependence of K.^ ^ for Ga7''" are summarized in Table IX. They were obtained in a constant field monitored with a proton resonance with TO.02 G precision. The" temperature was controlled with a Corona Ultra thermostat. For each tem-perature two absorption curves were traced. The Ga 7 1 Knight shift is (0.453JTJ.002)7a compared to previously reported value [75] of (0.449T0.004)The shift decreases linearly with temperature. The temperature coefficient, (,)K/0T) , is -3x10"7 °K"1 compared, to -2.95xl0-7 °K~^ " as quoted by Cornell [13]. This coefficient is of the order of 10"^ to 10~7 for many metals. It is positive for all Fig. 27: The derivative signal of liquid gallium. The numbers specify the frequency scale i n kHz. -85-TABLE IX. The experimental results of the temperature dependence of K. Isotope T [°c] Field (+0.02 G) NMR frequency (+0.05 kHz) K a) 57.5 9475.29 12358.20 12358. 21 Ga 7 1 47.8 9475.29 12358.23 12358.22 0.453+0.002 39.1 9475.29 12358.25 12358.24 29.9 9475.29 12358. 25 Ga 6 9 29.9 10921.89 10921.70 12358.25 11211.7+.10 11211.6+.10 0.457+.003 -86-those metals in which i t has been measured except for liquid Ga and both phases of Cs. The reference compound for the Knight shift measurements was an acqueous solution of GaClg. The NMR frequency in the field of 7748.00 G was found to be 10059.97 kHz for Ga 7 1 (6H~0.1 G) which corresponds to v 7 1 = 12983.9 kHz in 10 kG. For Ga 6 9 i t is found that v£9 = 10218.6 kHz in 10 kG. • A few remarks about the reference compound GaCl^ should be mentioned. The substance is extremely soluble and one expects a strong NMR signal from a saturated sample. However, the efforts to detect the NMR,signal from this compound were in vain for some time. The reason for this is (a) a large quadrupole coupling in GaClg and (b) the satu-rated solution is very viscous. As a consequence of (b) the correlation time, T c , for the molecular motions is very long. According to the theory [3], this causes a broad line which is then difficult to observe. Dilu-tion with water (or heating) decreases T c and narrows the absorption line. The signal intensity is then a function of the dilution and a maximum signal intensity is found when one part of the saturated GaCl^ solution is diluted with four parts of water. Further dilution does not shift the resonance frequency. Such a concentration drift is known in many salts, e.g. various thallium salts. -87-CHAPTER VII DISCUSSION A. EFG Tensor The EFG tensor in a metal at a particular nuclear site has been considered to arise from three sources [58] . The f i r s t , denoted by ZQ\ ' J i l a t t arises from the nuclear and electronic charges external to an atomic sphere around the nucleus in question (where Z is the normal valence of gallium). The second, 9*^ j 1 S due to the conduction electrons within the atomic sphere. And finally the third source which results from the distortion of the closed shell electrons at the atomic site may be accoun-ted for in terms of Sternheimer antishielding factors y<*> and RQ [59] . The total EFG tensor, qtot, is given by: q t o t - ( 1 - V ) Z q l a t t + (1 - RQ) e f l o c . (38) The first term on the right hand side represents the field gradient based on the ionic model. The second term is due to the conduction electrons. q- a^tt_ for gallium has been evaluated in the approximation of a uniform distribution of conduction electrons by direct machine summation of the expression 8 + N , 3 X * ( ^ , ^ , 4 3 ) X J ( V A Z3). 6 rm 2( ^ , h , ^ ^ l a t P i i = 2_ 2 - — 5 - " — i t s (39) The summation over m includes the eight atoms in the unit cell designated by (j ^ . J^jA^). ^h e second summation includes all the atoms lying within a spherical volume of radius R which is a multiple of the shortest lattice parameter at a given temperature. The atom at the origin is excluded from the summation. Indices i and j stand for any of three symbols A, B and C. - 8 8 -(q^ ) is the (ij) component of *q in the crystallographic coordi-latt i j latt nate system. From the calculations i t follows that (q^att^AB = ^ l a t P c B = 0. Therefore the B crystallographic axis is one of the principal axes, which of course also follows from the symmetry bf the unit cell. Instead of tabulating the calculated components (.\att>AA, (^latt^BB' ^latt^CC and (q.. )A„, q.. is displayed in Table X in diagonalized form. X a 11 AO latt TABLE X. The ionic contribution to the EFG tensor as a function of the number of atoms considered. R N (q ) V H l a t t ' x X (5 ) (q ) ^latt \att 3 532 -0.02147 -0.00329 0.02476 31.7 -0.734 15 66105 -0.02087 -0.00378 0.02464 30.0 -0.694 25 306450 -0.02094 -0.00374 0.02468 30.0 -0.697 N is the total number of contributing atoms inside the sphere of radius -R in units of B. The lattice parameters used for this calculation are those from Table XI at 4.2°K. The principal components of q^ a t t a r e all 24 3 in the units of 10 cm . is the angle in degrees between the Z principal axis of q^ a t t and the C crystallographic axis. \ a t t I s *-ts asymmetry parameter. TABLE XI. The lattice parameters of gallium at three temperatures. T A B C (K) (A) (A) (A) 4.2 4.5156 4.4904 7.6328 78 4.516 - 4.493 7.636 285.3 4.5195 4.5242 7.6618 Since the sums in Equation (39) converge rapidly (in contrast to the case of indium, etc.), alternative methods for calculating "q^ a t t have not been considered. In order to see to what extent the ionic model describes the crystalline field gradient in gallium the experimentally determined parameters characterizing the EFG at 4.2°K are compared in Table XII with those calculated from the ionic model. The quadrupole frequency v is calculated from Q 2 % V Q = e 2 Q(l - V ) Z ( 9 l a t t ) z z (1 + 1/3 T| l a t f c ) /2h . (40 3+ An antishielding factor of -9.50 for free Ga ions [59] and Z = 3 was assumed. TABLE XII. The comparison of the ionic calculation with the experimental result obtained for the EFG tensor at 4. 2°K. EFG tensor vQ(MHz) B H Experiment 11.31 24.3 0.179 Ionic Model 2.82 30.0 -0.697 In particular one should note that the X axis is parallel to the B axis in the ionic model rather than to the Y axis as in the experimental case. This implies that the ionic model is inadequate to explain the quadrupolar coupling in gallium. The experimental results in Section A of Chapter VI are perhaps most striking for the very small temperature variation of both the asym-metry parameter and the orientation of the principal axes. The quadrupole frequency exhibits the temperature variation shown in Figure 28 and the [kc/sec] 11.200 II000 10800 0 100 200 T ( ° K ) 3 0 0 Fig. 28: The temperature dependence of the Ga^ 9 NQR frequency. The solid line represents the results of Pomerantz [17], The points present rnsn'l i: of the present work. -91-results may be combined with the asymmetry parameter to give the changes in all EFG components between 285.3°K and 4.2°K. The temperature depen-dence of q has been calculated using the temperature dependent values of the lattice parameters from Table XI but assuming parameters u and v (Figure 3) to be the same for all temperatures. Results for R = 22 are included in the Table XIII. All the parameters remain practically con-stant over the whole temperature range so that thermal expansion does not explain the small changes observed. TABLE XIII. The temperature variation of the ionic contribution to the EFG tensor. T ^latt^XX (q ) T-att/YY (q ) l a t t ZZ P l a t t \ a t t VQlatt (OK) (MHz) 4.2 -0.02090 -0.00373 0.02463 30.0 -0.697 2.77 78 -0.02086 -0.00377 0.02462 30.0 -0.694 2.77 285.3 -0.02063 -0.00453 0.02516 30.0 -0.640 2.80 Proceeding in the spirit of the assumed model (Equation 38), the diffe-rence tensor is defined as ^diff ~ <X - V ^loc = «exP " ( 1 " V ) Z q l a t t (41) q\.^ represents the conduction electron contribution to the EFG tensor, diff The results are given in Table XIV. It is very interesting to note that the angle 8, • is almost equal to the angle 8 between the nearest neigh-diff bour direction and the C axis. This angle denoted as 8p a^ r is 15.9 de-grees. The temperature variation of P a £ r is not available in the present literature and i t would be of interest to see i f its temperature variation is the same as P ^ f f . -92-. Since only the absolute value of q can be determined reversing the sign of q gives an alternative solution for q j . r r with an angle exp airr of 31.6° "and T) = 0.03 which is also sensibly temperature indepen-dent. However, in view of the close correlation between the angles 8p a£ r and B^iff mentioned above, i t is likely that the first value is the correct one. Case II of Figure 12a yields 8 d i f f = 31.2°, T\ = 0.11 and 3 d f f = 14.9°, T| = 0.30 by taking q with minus and plus sign, respectively. The angle is measured to the left from the C axis in Figure 12a. The case with plus sign gives an EFG tensor with its Z axis close to the axis of the other "diatomic molecule". Of course, this coincidence is accidental and has no meaning. Important is the angle between the Z axis of q ^ f f and the "diatomic axis". These angles are 31° and 48° for these two choices of 9exp. According to the discussions in Chapter II and Chapter IV this situation is unlikely to occur. TABLE XIV. The temperature variation of the Q . . , R tensor. r diff T (°K) (^difPxX ^difpYY ^ d i f P z z P dif f ^ i f f 4.2 1.210 2.123 -3.333 15.7 0.274 78 1.205 2.113 -3.318 15.6 0.274 285.3 1.115 2.059 -3.228 15.0 0.276 In conclusion, these results suggest that the dominant contri-bution to the gradient is q^ o c. Watson, Gossard and Yafet [58] estimate that the main contribution to q. results from the interaction between ^loc the ionic contribution and the electron states at the Fermi surface. -93-But in view of the small temperature variation of the f i e l d gradient parameters i n contrast to the large variation of the Knight shift and the spin l a t t i c e relaxation time, this seems unlikely. This conclusion is further supported by the fact that the quadrupole frequency changes very l i t t l e at the superconducting transition. The close correlation between B j . r r r : and B . is suggestive of a strong interaction in the Qir r pair direction of the nearest neighbour, possibly an indication of covalency. In this connection i t would be valuable to have more information on the temperature variation of 8 p a ^ r . Clearly a detailed theoretical treatment is required. -94-B. Knight Shift The values for the Ga^ isotropic Knight shift are tabulated together with the values of the Ga^ spin-lattice relaxation time T in IQ Table XV. The T^ data were obtained from a study of the relaxation of the pure quadrupole resonance [12], TABLE XV. The temperature variation of the spin-lattice relaxation time 2 T, K. and the quantity 3T „ TK. appropriate for the 1Q' iso H IQ iso ^ r magnetic hyperfine relaxation of the pure quadrupole resonance for I = 3/2. T T T n o r K i s o ^ I Q ^ i s o (°K) (sec °K) (%) 4.2 0.63 + 0.01 0.132 + 0.004 (33 + 2) x 10" 7 77 0.58 + 0.01 0.142 + 0.003 (35 + 1) x 10" 7 300 0.46 + 0.01 0.155 + 0.004 (33 + 2) x 10" 7 The quantity -^T^IK^^ given by Equation (35i) and appropriate for the magnetic hyperfine relaxation of the pure quadrupole resonance for 1=3/2 may be compared with the Korringa value S = 28 x 10 ^ (Equation (35c)). 2 Experimentally, i t is observed that the product T,TK. is larger than r . . J ' r 1 ISO 2 S = T TK . An obvious reason may be that K^so is not appropriate for use X s in the Korringa relation. The Korringa relation was improved by including the many-body effects between electrons [76] which destroy the simple relation between Xp and N(E-p) given by Equation (35b). By adding a subscript (°) to de-note that the relation applies to only non-interacting electrons, - 9 5 -o , /r, r 2 ^ 2 ,„o the Korringa relation takes a new form x_ = 1/2 N (Ep) ] , C44) TXTK^ = S{ (Xp/Xp) [N°(E F)]} 2 = SKCc)"1 . (45) 2 K (a) expresses the enhancement of Korringa product T^TK^; theoretically, K(a) 41.0. (The reader should be warned that the enhancement factor called K(ot) in this thesis has gone by several names (symbols). The symbol K(a) is used in recent literature [77] and [77a].However,Narath and Weaver [104] use the symbol a for a different but related quantity. See also Appendix F.) A calculation for Ga7^ ,based on Ref. 76 using m*/m = 0.96 [78] and r = 2.17 [76], yields K( a) = 0.75. r is the inter-s s electron spacing measured in units of the Bohr adius and related to the electron density [76], It should be noted that a value m*/m ^ 1 has l i t t l e effect upon K(cx). It was found that the value K(c<) = 0.75±0.10 is consis-tent with experiments for several nontransition liquid metals [77]..The new form of the Korringa relation predicts values of T^which are longer than observed, so now the inclusion of neglected interactions could im-prove the agreement between the theory and the experiments. The enhancement enters the Korringa relation through the Knight sh i f t , Kg, which witnesses the electron exchange enhancement of the electron spin susceptibility. In addition, i t was pointed out [37] that 1/T^, the relaxation rate, i s also enhanced. The enhancement of the Kor-2 2 ringa relation is in fact the ratio of unenhanced % (hence K ) and an 2 2 enhanced 1/T,. That T,TK shows a net enhancement means that X is more l i s p enhanced that 1/T^ -. The theory [37] predicted an enhancement of 1/T^ beyond the enhan-cement observed experimentally. This effect has been traced to the -96-assumpticn of a 6-function interaction between the electrons [104],The prediction is improved by applying an interaction potential of longer range.This theory [104] however, does not apply to gallium since i t assu-mes a spherical Fermi surface. A brief discussion of this theory is in Appendix F. Recently, i t was suggested [79] that the energy dependence of the enhancement might be observed in medium-to-heavy metals as a reduction of the enhancement factor at higher temperatures or at the melting point. (i.e. K ( a ) ~ c l ) . The reduction would occur when the correlation time o f disordering motions became comparable to the correlation time of the electron-electron interaction (as in liquid metal where the correlation - 1 3 - 1 4 time of the ionic motions is of the order of 10 to 10 sec). The reduction is in effect a "motional narrowing" phenomenom. In view of measurably large effects of the electron-electron inter-action on the Korringa product, and the large effect of the electron-phonon interaction on the specific heat, the question of possible electro phonon effects on the Korringa product becomes relevant. It was shown [ S O that the electron-phonon interactions can have no effect on y a n ( i P that T is also unaffected [81]. The electron-phonon interactions enhance the total quasi particle density at E p (and thus the "electronic" speci-f i c heat) but have l i t t l e influence on x^ . The density of states which appears in the theory of nuclear spin-lattice relaxation (Equation (45)), however, refers to the "bare" electronic density. Electron-phonon effects are therefore only of importance in the case when attempts are made to eva luate the enhancement of the spin susceptibility by comparing calculated susceptibilities with the values obtained from the specific heat. Besides the contact hyperfine interaction there are other interaction which can influence the relaxation rate: spin-dipolar and orbital terms - 9 7 -in the hyperfine Hamiltonian, core polarisation and for metals with spin >l/2, an electric quadrupole contribution. In the solid gallium the ratio of the relaxation times obeys the relationship 2 " T 1(Ga 6 9)/T 1(Ga 7 1) = (^/u^ 9) =1.61 . (47) at various temperatures to within 1" 2%. Clearly, the relaxation mechanism in the solid is purely magnetic in the origin. Quadrupole relaxations are excluded, for otherwise one should obtain 2 T 1(Ga 6 9)/T 1(Ga 7 1) = ( Q 7 1 / Q 6 9 ) = 0.338 (48) which obviously is not the case. However, the experimental ratio, Equa-tion (47), in the liquid, is 0.80 [13] meaning a significant electric quadrupole contribution to 1/T^ in the liquid state. Fortunately for elements with two magnetic isotopes, like gallium, the quadrupole contri-bution to 1/T^ can be separated from magnetic [13]. By subtracting the quadrupole contribution one is then sure, that quadrupolar effects do not affect the apparent enhancement of the Korringa relation. Obata [82] has considered relaxation by the spin-dipolar and orbital hyperfine interaction for p-conduction electrons in Bloch states in cubic metals. This work is not strictly applicable to gallium but i t can provide estimates of the importance of these interactions. The rela--98-xation rate due to these interactions is = ^ [ 1 . 3 ] [ Y vn-h2N(EF) ^ 1 / r V ] 2 kT ( 4 9 ) •"•1 spin dipolar n e n r & orbital In order to apply Equation ( 4 9 ) to gallium, values for the - 3 density of states N(Ep) and <Tr > are required. N(Ep) can be calculated from the paramagnetic susceptibility y^. The contribution of the electron spins in the conduction band to the total magnetization of gallium metal is not known. The bulk susceptibility y^oE single crystal and polycrys-talline gallium spheres were measured at 2 5 ° C [ 8 3 ] , The following ani-sotropic diamagnetic susceptibilities were found: A axis ( - 0 . 1 1 9 + 0 . 0 0 1 ) , B axis ( - 0 . 4 1 6 + 0 . 0 0 2 ) , and C axis ( - 0 . 2 2 9 + 0 . 0 0 1 ) . All the values for the - 6 susceptibility in this work are given in 1 0 cgs mass units. The sus-ceptibility of the polycrystalline spheres, assumed to be the average value for the bulk susceptibility in gallium, was ( - 0 . 2 5 7 + 0 . 0 0 3 ) at 2 5 ° C , and ( - 0 . 2 9 9 + 0 . 0 0 3 ) at - 1 9 6 ° C . This corresponds to an increase of 1 7 7 o in the diamagnetic susceptibility. The susceptibility of liquid gallium was ( + 0 . 0 0 3 + 0 . 0 0 1 ) at 3 0 ° C and 1 0 0 ° C . No change in this susceptibility was noted from room temperature up to the steam point. Therefore, solid gallium is highly diamagnetic and liquid is weakly paramagnetic. The ani-sotropy in the single crystals is explained by partial overlap of Bril-louin zone boundaries by the Fermi surface. The large change in the sus-ceptibility associated with the change in state is attributed to the ab-sence of effective mass influence in the liquid state. Although there are no direct measurements of Xp, i t can be eva-luated from several methods. The values calculated by several ways are given in Table XVI. -99-TABLE XVI. The paramagnetic susceptibilities obtained from several methods [34] ( (in Xp(APW) cgs mass units x 10"^) X P(Y) Xp M(calc.) 0.20 0.22 0.18 0.12 0.27 v (F) in Table XVI is derived from the complete free-electron model, x (F) ^ P includes the corrections due to the electron correlations, Xp(y) ^ s eva-luated from the experimentally determined electronic specific heat of gallium metal at low temperatures, Xp(APW) is calculated from theoretical density of states at the APW Fermi energy. The value in the last column is obtained by subtracting the appropriate values of the known contribu-tions to diamagnetism from the total measured susceptibility x a n d setting the remainder equal to X p ( c a l c . ) . Thus, Xp = X " X^ " X-p Xj_ i - s t n e contribution from ion cores, (x^ = -0.18 for Ga [90])and x^ i s the diamag-netic susceptibility of the conduction electrons. For perfectly free electrons xe = "X /3 and 3(x - X )/2 is then the desired paramagnetic * p i part, Xp( c a i c')- For solid gallium (x - x^) turns to be negative and i t makes this method completely useless. It is very likely that the assump-tion x = -X /3 for the solid is wrong. & P • The density of states derived from XpCf) x s •> N(Ep) = 17.5x10 -1 -1 erg atom . The standard formulae for the optical hyperfine splitting parameters of Ga'''" give <Tr >^ = 2.36x 10^ "* cm [84]. With those values Equation (49) yields (T, T) , , =18 sec °K. The experimental value 1 orbit + dip i s : (T T), . = 0.2 sec °K and (T, T) . = 0.5 sec °K. It is therefore con-1 h q 1 'sol eluded that the orbital contribution in Ga^ is negligible compared to the contact contribution. -100-It is possible that x (Y) in Table XVI is enhanced by the electron-phonon effects leading to an overestimation of N(E ) used in the evaluation r -3 of Equation (49). Also, the value of <r > used is taken from optical pro-perties of the atom and should be modified for possible changes"in the -3 metal. The 4p x^ ave functions are expanded in the metal, i.e. < r > metal -3 is less than <r . > . The amount of this expansion would be very d i f f i -atom r J cult to calculate. For-the purpose of the present discussion i t s actual -3 value is not necessary. (The value 3/4 <r > has been assumed for the atom 4d wave functions of V"^ [108].) The inclusion of the proper values for -3 N(E ) and < r >in the evaluation of Equation (49) would obviosly yield r ( T ^ T ) o r ^ i t + which is larger than previosly estimated. This is then confirming even stronger that the orbital contribution to the relaxation 71 rate in Ga is negligible as compared to the contact contribution. -101-Yafet and Jaccarino [85] considered the core polarization con-tribution to the Knight shift and relaxation rate. They obtained the following relation (neglecting spin-dipolar and orbital contribution).. 1/T. = (1/T ) + (1/T) , (50) 1 1 s 1 cp and the Korringa-like relation (TJ TK2 = 3S (51) v 1 cp cp ' for core polarization induced by p-character in the conduction electron wave function. Under the assumption that the same enhancement factor K(a) applies to this relation as to the Korringa relation for the con-tact hyperfine interaction (1/T 1) c p/(1/T 1) s = 1/3 (K c p/K s) 2 . (52) Calculations in some light metals give K I = 0.1 K [86], These shifts I cp ' s arising from the polarization of the core states by s, p, etc., electrons are calculated independently and there may be cancellation among these different terms. If IK I = K /10 in gallium then 1 cp 1 s ( 1 / Vc P - m <L/V. • <"> Consequently, within the accuracy of measurements, can be neglected compared to (1/T..) . These estimates make the following situation favourable i s in gallium: Solid Liquid (1/Tp = ( 1 / T 1 ) S (1/T^ = (1/T 1) g + (1/T^Q (54) K s o l = K s o l + K s o l K U q = K l i q + K l i q I S O s other iso s other - 1 0 2 -s and Q refer to the direct-contact hyperfine and quadrupole contribution, respectively. ^-0ther i n c l u d e s t n e orbital (K ) and core-pol arization (K ) contribution to K. „_,. cp l f a u In view of the Equations ( 5 4 ) , any temperature dependence of 2 the product T^TK.£SO which reveals the apparent enhancement, can be ascribed to either a temperature dependent fraction K^^^/K or to the temperature dependence of the true enhancement factor K(o'). Next, the possibilities of separating these two effects for Ga are considered. The T^ data for Ga^ are reproduced in Figure 29. The experimen-tal points (full circles) are measured with an accuracy of t 270 in the so-l i d . [ 1 2 ] and t 1 0 % in the liquid [ 1 3 ] . The error bars are omitted because they are too small. The T^ data in the liquid phase are those of the mag-netic hyperfine interaction, derived from the experimental T^ using Cor-nell's isotopic separation. It is possible to f i t the observed T-^ with a single value K(a) in the solid and another single value of K(a) in the liquid. The full curve in Figure 2 5 is the prediction of unenhanced Kor-ringa relation; thus K(o') = 1. The broken-line curve is the prediction of the interacting Korringa relation with an empirical f i t for the enhancement; K(a) = 0 . 8 0 in the solid and K(Q-) = 0 . 7 3 in the liquid. K(a) will be termed the "observed K(a)" to distinguish i t from the "true K(a)" which arises from the electron exchange only. Figure 2 5 shows quite clearly that the modified Korringa relation of Equation ( 4 5 ) adequately relates the observed K. and T, except for the change of K(a) at the melting iso i point. As pointed out earlier any variation of the observed K(o?) can have its origin either in a variation of the relative K 0.ther^ s o r t n e t r u e exchange enhancement. Therefore, the constancy of the observed K(a) in both the solid and liquid phases of Ga indicates that K , /K is tem-T E M P E R A T U R E (°K) Fig. 29: T.. vs-temperature (full circles) in Ga 7 1. The solid line is a prediction of the unenhanced Korringa relation. The brokcn-linc curve is the prediction of the Korringa relation with an empirical f i t for the enhancement. Notice a change in the enhancement at Tj^,. -104-perature independent in each phase or the true exchange enhancement is temperature independent in each phase. The observed enhancement K(a) of 0.73 in the liquid is in very good accord with the theoretical pre-diction of 0.75. Furthermore, both the prediction and the observation are in close agreement with the generalization that K(cv) = 0.75 is appropriate for liquid metals [77]. This generalization is based on the analysis of the NMR inNa 2 3, I n 1 1 5 , Sb 1 1 2' 1 1 3, B i 1 0 9 and also Ga 7 1' 6 9. In addition i t was concluded that among these liquid metals K - o t h e r / l £ s ^ C ) only for In^"* and B i ^ 9 for which values -10 and -25%, respectively, are appropriate. Consequently, it appears that the observed K(oD in the liquid Ga very nearly reflects the true exchange enhancement and that K , /K w 0 in the liquid state, other s In the solid state the observed enhancement K(o0 = 0.80. The change from K(a) = 0.80 in the solid to K(Q?) = 0.73 in the liquid is in the opposite direction predicted by motional narrowing hypothesis [79]. It could be attributed to a non-zero and negative K ^ r in the solid which becomes negligible in the liquid state. Since there is no strong evidence that significant narrowing of the true K(a) actually occurs, i t seems reasonable to assume that i t remains constant upon melting. With this assumption as well as the assumption that K t^ e r- = 0 in the liquid Ga, a constant value of K , that stands in the ratio of K , / K =-5% ' other other s in the solid is required to account for the difference between the observed K(cv) = 0.80 and the postulated true K(CY) = 0.73. Since K . = K + K , iso s otiriGir and K , / K =-5% in the solid, K = 1.05 x K . Thus, just below the other s ' s iso melting point K S = 0.163% rather than the observed 0.155%. Just above the melting point, K , 0 and K = K . = 0.453%. Therefore, the other s iso disappearance of K0(-her, i.e. the negative core polarization and orbital -105-paramagnetism contribution is far from being sufficient to explain the large increase of the Knight shift upon melting. Because the conclusions above are based upon an experimental evidence that K ' 0 in the liquid Ga, and i t was found that In'"'''"' other ' has K o tj i e r/K g = -10% in the liquid, an alternative hypothesis will be considered. It is presumed that Ga has K , /K = -5% in the liquid other s state. Then the observed K(a)in the liquid no longer represents the true K(a) arising from exchange phenomena only. The true and the ob-served K(a) are related with the expression K t r U e(a) = K°bs(ff)[l +K o t h e r/K s] . (55) A value of the true K(o') of 0.65 is then required to reconcile the assumed K /K =» -5% with the observed K(a). Continuing the assumption that other s the true K(a) is the same in both the solid and the liquid states implies, according to Equation (55), K , /K is -10% for solid Ga. The implied other s value, K(of) = 0.65 is significantly lower than the generalized value of 0.75, and on this ground this alternative hypothesis is thought unlikely, although not impossible. For K o t j i e r to remain constant in the solid requires the s-p mixture of the wave function to remain constant also. The relative in-crease of K. in temperature intervals from 4.2 to 300°K is 16%. This iso r increase is partly due to changes in y_ and partly to a change in s-p P mixture. Since K , is only -5% of K at 300°K i t is reasonable to other J s. assume that K remains constant. In fact, this assumption is not other ' really necessary. Even if K.other does change with temperature, i t would not affect the above conclusion since i t is so small. The mechanism necessary for K o t j j e r t o disappear upon melting would be a considerable -106-increase in s-character of the wave function at the expense of the p-cha-racter. This very likely occurs in Ga as the complicated band structure disappears upon melting and so considerably reduces the p-character in the wave function. These effects will be discussed in the next para-graphs. With the available relaxation data for solid and liquid Ga at Tj^ p and known Knight shift in the liquid one can predict the isotropic Knight shift in the solid from the relation 2 2 [3T._(0) TK. ] . «{(T,)TK. 1 . . . (56) ^ IQ iso i sol V N I s iso J l i q The experimental values are; T 1g(0)T M p = (0.46+0.01) sec ° K , ( T 1 ) s T^ = (0.63+0.06) x 10"3 sec °K and = (0.453+0.002)%. The predicted K ^ ° * value at T>rr> is 0.166% and agrees well with the observed value of 0.155%. MP sol The slight difference is due to the fact that K incorporates a small ISO K , . For this reason the predicted value of 0.166% should be compared other to 0.163% as derived previously for K ^ 0 1 at T^. Also, the ( T 1 i q ) s value is not very accurate. Since K I ^ is 2.9 times that in the solid, one is in a posi-xso tion to speculate about the changes in the electronic structure of Ga upon melting. The bulk magnetic susceptibility x changes from -0.257 in the diamagnetic solid to +0.003 in the paramagnetic liquid. The total change, Ax = V-\±q " X s o^ = +0.26, is a result of an increase in the paramagnetic susceptibility, Xp a n <^ decrease in the diamagnetic suscep-t i b i l i t y x^« There is no reason to assume that Xi is unchanged upon melting. According to Ziman [87] Ga becomes a free electron-like metal in the melt. In view of this hypothesis and the fact that x^( F)^Xp( c a i c.) X l i q = x^( F) i s assumed. This yields X^^ = -0.06. Assuming for the mo--107-sol ment that x-^ does not change upon melting one obtains a negative •y^ which obviously makes no sense. Therefore, a change of is also con-tributed to the total change of x upon melting. Since these two con-tributions, AXp and AXi> t o the total change, Ax> cannot be separated sol with the available experimental data, Xp = Xp(Y) i-s assumed, x ( Y ) was obtained at low temperatures and has to be corrected before comparing i t with A T • ' •MP* ' ^ N & am um Xp0''' i s temperature dependent. This is manifested in a 16% relative increase of K. in the temperature interval i s o 1 from 4.2 to 300°K and in a 17% relative increase of x i - n t n e interval from 77 to 300°K. None of these changes is solely due to the changes of X^°^-The.change of K is likely accompanied with a change in s-p mixture, iso This change is implicit in a 4% change of the quadrupole coupling constant and asymmetry parameter T|, and also in the changes of the anisotropic Knight shift. On the other hand, the change of x i - s accompanied by a s ol decrease of x^ . In order to account for this, Xp =1.1 Xp(Y) i-s assumed at T^Q,. In Cd, for instance, X^ 0''" A T T M P ^s a b o u t ^0% larger than that at low temperatures [35], The many-body exchange and correlation effects are taken into account by a constant, temperature independent enhancement factor of 1.1. The same factor was used for Cd [88] (According to Ref. [88] a larger factor should be used for Cd. In recent band calculations [35] a factor of sol 1.55 was used.). With these two corrections, Xp = 1«2 Xp(v) S. Xp(y) =0.^ l i q at T^ g,. £he comparison of this postulated value with Xp a t T^, shows an increase of Xp upon melting by a factor of 1.7. However, this factor is too small to account for the observed change of the Knight shift upon mel-ting. Using Fermi-Segre formula [89] the Equation (32) is conveniently written as -108-K = - T ~ — ta(s) X (57) where = u^/I and T = P^ /P^ . P is the average probability density at the nucleus for electrons at the Fermi surface, P^ is the probability density for the free atom and a(s) is the hyperfine coupling constant for the s-electron in the atom, "j^ is a correction which accounts for the degree of ionization in the pure metallic state [90]. The ratio , called Knight shift factor, is supposedly near unity or ought at least to be the same for similar metals. This would not be true when the wave function of the electrons in the conduction band contains appreciable admixture of p or higher states. The Knight shift was calculated from Equation (57) for several metals using the corrected hyperfine constants a*(s) ="X-a(s), Xp(v) a n d assuming J = 1 [90]. Although this procedure is of limited accuracy, i t provides a consistent means of comparison among several metals. The calcu-lated values are in fair agreement with those obtained from experiments; the differences are then ascribed to the J factor. For In and Al, the observed K value is consistent with Equation (57), i f one takes "j = 0.5 and J =0.6 for In and Al, respectively. The Knight shift of these two elements does not change at T^ p; thus the quantity (Xp"j ) which determines K retains its value upon melting. In case of gallium, however, the quan-tity (Xp^ ) * s changed by a factor of 2.9. Using a K(s) = 0.326 cm [90] and known K^SQ a t ^jp> o n e derives from Equation (57) ^ P l l q " 0 - 0 9 5 (57a) The problem is then to find out whether the above change is due solely to -109-or T or both. Xp or } Ziman [87] found that the quantity x which determines K, is best correlated to Xp(F)> thus the theoretical free-electron value of density of states rather than to XpCv) o r 'independently measured1 Xp (calc.). This comparison yields ^ = 1.2 for the alkali metals and -^0.5 for similar metals Hg, Tl, Sb and Sn. This is then suggestive that the density of states in many metals is quite close to the free-electron value in the solid as in the liquid phase. Indeed, the direct theore-tical computations of N(E ) in solid metals give numbers quite close to F the simple free-electron value. In gallium, for instance, with such a complicated Fermi surface, the band-structure effect would only reduce N(E-p) by about 10% as calculated with APW method [9l], The correction to N(Ep) due to the electron-electron interaction depends only on the electron density and would be nearly the same in both phases. This is experimentally confirmed for Li by direct ESR measurements of Xp which changes l i t t l e on melting but is somewhat larger than Xp(^) i n both pha-ses [92]. The question is how much would one expect J to change at Tj^ p.Pp differs from P^ for two quite different reasons. The first effect is the change in potentials and boundary conditions for the electron wave func-tion when the atom is incorporated in the metal. These changes are quite drastic, the outer valence-charge field is screened, but overlaps its neighbours; the exponential decay at large distances is replaced by a periodicity condition on the cell boundary. This type of effect however, ought to be the same in the liquid as in the solid metal. The second effect is the true band-structure effect arising from the diffraction of the NFE waves by the arrangement of ions. For example, -110-P is reduced to zero for a pure p-like Bloch state whose wave functions vanish automatically at each nucleus. Such effects have been calculated [933 and were found to be small for the alkali metals, although large enough to be measured as part of the temperature and volume dependence of K in the liquid and solid phase. No detailed computations for more complicated metals have been done. The conclusion is that the factors J as well as Xp that deter-mine the Knight shift need not change greatly when a metal is melted, even though the arrangement of ions is profoundly modified. N(Ep) and are, in the first instance, dependent mainly on the atomic volume, which often stays very constant at the melting. The large change in (Xp j^) f° r Ga i s obviously a band-structure effect. The only other two cases with such a large change of (Xpj) a r e Bi and Cd. The fractional change (K l i q - K s o l)/K S o 1 at T M r o is +0.33 for Cd, -2.1 for Bi (shift in iso iso iso M P solid at 4°K, the only measurement available; the sign of the shift in Bi is changed at the melting) and 1.8 for Ga obtained in the present work. A fair idea of what may happen in Ga upon melting is obtained by assuming Xp^ = X*(F) a n d X^ 0''" = Xp(v) a s discussed earlier. Since Xp(Y)/Xp(F) = 0.65 follows that the density of states is decreased by 35% in solid compared to liquid. In Cd, for instance, the theoretical calculations yield a 30% change [35]. Also, i t is known that solid Ga is a "semi-metal" with a quite low density of states. The corresponding ^ l i q ' e v a l u a t e d from Equation (57a), is 0.45 and is consistent with the value of 0.50 for similar metals In and Al. If one assumes the electron wave function i(f to be a mixture bf s- and p-type wave functions, thus of 2 2 the form ilt = ail( + a ill (|al +|a| =1) then ik in liquid Ga is a s Ys p T p x s p T 45% s- and 55% p-type mixture. In the solid § s o^= 0.22; thus the s-- I l l -character is reduced by a factor of 2 as compared to liquid. The cor-responding mixture is then 22% s- and 78% p-type. The fact that the p-character of the conduction electron of Ga in the Fermi surface is as large as 80% is not surprising when compared with the values: 80% for Cd [94], 65% for Na [95], and more than 50% for Sn [3l]. The conclusion of the above discussion is that the large change of gallium in K . . at the melting point is due to a large change I S O of direct contact shift K . This change is a result of an increase of s the density of states and a change of the s-p mixture upon melting. With the available data a fair estimation yields a 35% increase of the density of states and a 100% increase of the s-character of electron wave function. A decrease to virtually zero of K.other n a s a negligible effect on this change. K . in solid gallium decreases about linearly with increasing I S O temperature. The temperature coefficient for several metals ( OK / ^ T ) ^ , is collected in Table XVII. TABLE XVII. The temperature coefficient Q K / D T ) p for several metals. Metal 107 QKPT) ( V 1 ) 104 ( K ) G>K/>0 o -1 P ( K ) Na (s), (1) 1.7 1.5 Rb (s), (1) 11.0 1.7 Cs (s) -28.0 -1.9 Cs (1) -45.0 -3.0 Cu (s) 0.0+0.9 0.0+0.4 Pb (s) 13.0 0.87 Al (s) 1.0 0.61 Ga (s) 6.0 4.0 Ga (1) -3.0 -0.67 In (s) 14.0 1.8 Bi (1) 0.0+10.0 0.0+0.7 Cd (s) 48.0 12.0 Cd (1) 0 0 -112-The coefficients are of the order 10"^ -f 10"' °K"1 and are positive for all those metals in which i t has been measured except for Ga and Cs. In Cs, QHL/~t)T)a is negative in both solid and liquid phases. In Ga, apart r from a change in its magnitude, the sign is altered upon melting. A change in K with temperature, volume or pressure reflects changes in either or both of Xp a n a Pp AK/K = AXp/Xp + APF/PF (58) The free-electron approximation predicted a negative temperature coeffi-cient of the right order of magnitude for liquid Ga [75]. In this 2/3 -1 -1/3 approximation Xp ~ V and P^,—V , giving a V dependence for K, and the temperature dependence is assumed to be implicit in the thermal expansion. However, the experimental data may be described approximately - 2/3 according to a V law [13]. The close agreement between free-electron theory and experiment should not be expected in this case for two reasons: (1) the best agreement between the theory and experiment was obtained when a correction for electron correlation [96] was applied in the case of Na, which"should be more easily treated by free-electron model than gallium and (2) the behaviour of P^, with volume may be expected to be complex, because the thermal expansion is characterized by growth in the concentration of void spaces rather than a uniform dilatation of lattice constants [97]. So far the pressure experiments have been done only for the liquid Ga [13]. The results were examined in the light of the theory of Benedek & Kushida [20], whose approach has met with some success in the case of the alkali metals. The Knight shift at constant pressure may be separated into two terms: -113-1_ /^ Kv _ 1_ /OKv /^ V\ , 1_ /t)Kv (59} K V p K ^ F T ^ r p K ^ r v V ' The two terms are the partial dependences of K on V and T, respectively. The volume dependence arises from the change in density of states and renormalization of the electron wave functions with volume. The intrin-sic temperature dependence arises because the electronic wave functions follow the vibrations of the nuclei about equilibrium positions. Since the wave functions and Pj, may be functions of volume, and the mean relative volume fluctuations increase with temperature, the time average value of P F 2 2 will be increased by a value proportional to P-p/^V )„_„ ; V is the r V—VQ O equilibrium volume. It is assumed that Xp b-as n o intrinsic temperature dependence because the electron spin-lattice relaxation time is long compared to an atomic vibration period. The theory of Benedek & Kushida gives reasonable good agreement with the observations in Na and Rb, but not for Cs. Their results for solid Cu were reported and left undiscussed. A comparison of the numerical results for two closely related ions Cu and Ga are displayed in Table XVIII [13]. TABLE XVIII. A comparison of selected physical properties and Knight shift data for copper and gallium. Cu (solid) Ga(solid) Ga(liquid) 1. Volume compressibility 7.19 21 24 at 1 atm -lC^V 1 C>V/-)P)T(kg/cm2)"1 2. Thermal expansion co- 5.0 5.4 11.5 efficient at 25°C 103 V ^ C W T j p C C ) - 1 3. Density (g/cm3) at 25°C 8.96 5.90 * 6.10 4. K(V) at 24°C 0.00232(V/V ) ° - 7 + 3 0.00155(V/V ) 0.00449(V/V)"-1-5. 10 4K-}(>KPT) p( OK)- 1 0.0+0.4 4.0 -0.67+0.02 6. l o V^K/^TOV/^Op +0.4+0.2 ... -0.1 +0.1 7. 10V 1C3K / n>T) v(°K)- 1 -0.4+0.6 ... -0.57+0.12 * The volume dependence of K S°* is not known. iso - 1 1 4 -The crystal and the liquid of Ga are three times more com-pressible as Cu, possibly reflecting the weak bonds between layers of the gallium solid structure and the presence of void space in the liquid. In general, most liquid metals are about 5 0 % more compressible than so-lids, the difference being due to the presence of voids in the liquids. However, the alkali metals, which are relatively loose-packed solids, experience as l i t t l e as 1 0 % increase in compressibility upon melting. Row 4 gives the experimental shift as a function of volume at 24°C. Row 5 expresses the fractional change of shift with temperature at con-stant pressure, Row 6 the fractional change of K due to thermal expan-sion, and Row 7 the intrinsic temperature dependence of K. The Rows 4 and 5 show that the compressional effects in liquid Ga and solid Cu are approximately the same magnitude but opposite in sign. Another feature is the set of temperature coefficients in Rows 5 and 7. It is possible that the intrinsic temperature dependence in Cu is cancelled largely by the part due to the thermal expansion. By contrast, in liquid Ga almost all the observed variation appears to be intrinsic with temperature. The free-electron theory predicts for Row 4, K(V) = K(Vo)/V/Vo)~°-The comparison is not good with either liquid Ga or solid Cu. Taking the general expression PP(V) = P^ CV )-(V/V ) Benedek's formalism predicts *• a o o 1/K C D K / 9 T ) v = m(m-l) K ^ / 2 Q . ( 6 0 ) for values in Row 7. Krj. is thermal compressibility, kg is the Boltzman constant, and Q the atomic volume. The large negative intrinsic tempera-ture dependence in Ga is not easily accounted for. At one extreme, a free-electron model (m = - 1 ) yields 1/K O K / ) T ) V = + 1 . 6 X 1 0 " 4 ^ K ) - 1 in con--4 o - 1 trast to observed - 0 . 5 7 x 1 0 ( K) for Ga. At the other extreme (m = - 0 . 1 from the experiment) there should be no intrinsic temperature depen--115-dence. In fact neither extreme yields a negative temperature coefficient. The conclusion is that the available data are insufficient to elucidate the origin and details of the negative intrinsic dependence. Following Benedek & Kushida theory, Pp possesses a negative (^2Pp/^V2)^_y . In the literature there are some theories which allow for a negative tempers ture coefficient for the Knight shift in Ga, but unfortunately they are not completed. Recently, Micah, Stocks, and Young [98] calculated the nuclear contact densities for valence electrons in metals and applied i t to ex-plain pressure experiments on the Knight shift in monovalent metals [99], The theory was proved for Na and Li where Xp an^ ^±So w e r e measured se-parately. The agreement is good and better than that offered by OPW approximation [18]. It also gives a semi-quantitative explanation for the volume dependence of K. The approach in Ref. 98 represents a generalization of the OPW method, improving on the latter in that i t recognizes the dynamical re-sponse, via a Schroedinger equation, of valence electrons to the ion core potentials. The chief new feature is a renormalization of the valence electron wave functions due to the attractive core fields and this is qualitatively different from the OPW method which expels charge from the cores. Following the theory, the Knight shift could be written as 8TT X p ( F ) K = ~ 2 • (61) (Xp(F)/Xp(c) + 1/3 - a) (1 + 0") Xp(F) is the susceptibility for electrons interacting neither with each other nor the ions, Xp(c) ( c = correlated) is the susceptibility of a gas 2 of interacting electrons in a uniform positive charge background and -116-2 is the value of Pp for a single ion in a free electron gas ("5 /Q=P^ ). The denominator defined via a 4 l ( i ! + l ) ( k V k ) (62) 3TT 1 J F represents a renormalization resulting from the collective electron affinity around the ion. j ^ is the phase-shift of the large-r asymptotic form of the radial wave function, 1 is the orbital momentum. The band structure effects in K are reflected via (T. The correlation effects among valence electrons are also included. Under pressure a normal de-2 crease in K arises via Xp(F) y . The departures from normality enter through and primarily through its variation in the susceptibility term rather than in the renormalization factor (1 +a). This is because \ diverges as ^approaches (Xp(F)/Xp(c) + 1/3). The volume coefficient in liquid Ga is negative. Though the pressure experiments in the solid Ga have not been done i t is speculated from TABLE XVIII that i t changes sign upon melting. In the spirit of the above formalism this may be ascribed to a large change of a which is very sensitive to the changes of the electronic wave functions. And that's just what happens in Ga upon melting. The s- and p-bands of gallium have been calculated according to the APW method [91]. The calculation shows eight bands of which four are completely fil l e d and four are only partially occupied. The E(k) curves reveal a considerable number of accidental degeneracies which are a result of the rather low symmetry. In fact, these degeneracies occur throughout the Brillouin zone and not just at the zone boundaries. As a consequence, departure of E(k) from free-electron behaviour occurs at al l these places. It is then no surprise that the Fermi surface arising from APW picture bears l i t t l e apparent resemblance to the free-electron -117-surface. In addition, since the electronic wave functions have a con-siderable amount of p-character a large anisotropy of the Knight shift is expected. Is is seen from Figure 23 that there is no K-anisotropy along the Y (== B crystal) axis; i.e. K g n(Y) = 0 and K^ = ^ -^so a t a H tempera-tures. However, the anisotropy in the X and Z directions is very large, i.e. the ratio K (X)/K. = 18% at 77°K. Since the trace of K tensor an iso an is zero and K (Y) = 0 i t follows that K (X) = -K (Z) with K (X)X). anv an an anv K (X) is very temperature dependent (Figure 23). By increasing the an temperature from 4.2°KjKan(X) rapidly changes; i t reaches a maximum at about 77°K and then slowly decreases as the temperature approaches T^. At T,m i t s t i l l retains a large value. This rather paradoxical behaviour MP of K (X) is a result of particular and unusual details of the Ga Fermi anv ' v surface. This surface is extremely complicated and not yet completely known. A qualitative picture of what may happen to the gallium Knight shift when the crystal is being heated can be obtained by reviewing the theoretical and experimental results performed on cadmium. From the point of view of its NMR properties, Cd is behaving very much like Ga. The unusual properties of Cd are [35]: (1) K ISO increases strongly with increasing temperature; for an ordinary metal K^go is practically temperature independent, with changes of less than 10%. (2) Upon melting K suffers an abrupt increase; most metals show i s o • . l i t t l e change. (3) K a n has also an anomalous temperature dependence. Properties (1) and (3) seem paradoxical, since (1) indicates that the s-part of the wave functions on the Fermi surface increases with temperature and (3) seems to indicate that the non-s-part (and mostly the p-part) of the same wave functions also increases with temperature. Re--118-cently, the puzzling paradox and the temperature variation of K and iso of K have been explained [35], The essential part of the theoretical an r r calculation in Ref. 35 is the derivation of the temperature-dependent band structure by introducing the Debye-Waller factor. This accounts for the lattice vibrations which effectively decrease the strength of the lattice potential and make i t more isotropic. As the temperature in-creases the energy bands become thus more free-electron-like and the s-character of the wave functions on portions of the Fermi surface in-creases. At the same time the cancellation due to different p-contribu-tions in the various sheets of Fermi surface is destroyed, and the aniso-tropy also increases. Upon melting Cd becomes a free-electron-like metal and K. increases abruptly. I S O tr J ' These structural characteristics are also peculiar to Ga which makes i t a good candidate for a thorough investigation". The problems here are rather difficult theoretical conditions imposed by its Fermi surface. It was pointed out [100] that a calculation for the temperature dependence of the Ga band structure would take up too much computer time at'present.A detailed analysis of the temperature dependence of K must await the outcome of such a calculation. It is interesting to compare the K g n tensor with the EFG tensor. Both are second rank tensors with zero trace. The experiment has shown that the principal axes of the K a n tensor are parallel to those of the experimental EFG tensor, <T e X p. The principal values of K a n, arranged by their magnitude, follow the inequality K (X) >K (Y) > K (Z) . (63) anv ' an anv J v 1 Recall that XYZ in Equation (63) refer to XYZ of q . The EFG tensor * '- exp An anomalous behavoir of the Ga Knight shift is also found in k\xQ,n^\S)% This behavoir is explained in terms of a changing character in wave functions of the electrons at the Fermi surface as the temperature is increased. -119-convention for the choice of the XYZ principal axes is l^xxi <• KYI ^ l qzzl • ( 6 4 ) Here, the absolute values are taken because the sign of the quadrupole coupling constant cannot be determined experimentally. However, one can obtain the sign of the Knight shift anisotropy. Formally, both tensors are determined by the same expression [P°,(cos 8)/r 3]p(r)dr where p(r) is the charge density. In case of the EFG tensor p(r) = P c o n d ( r ) + P o ther^); P C O n d ^ i s t h e d e n s i t y o f con-duction electrons around the nucleus and p , (r) is the charge density 'other of a l l surrounding charges outside a small sphere around the nuclear site in question. In case of the K Q n tensor p(r) = P c o n d p(^) = ^ l l ' ^ ( ^ ) l / • £ • Thus K a n is a measure of the quadrupole moment of the Fermi surface con-duction electrons. Contrary, p ,(r) which matters in the EFG tensor cond evaluation is the charge density of all conduction electrons and not just those at the Fermi surface. Intuitively one expects the principal axes of the K tensor to be parallel to q, rather than q . Also, note the an r ^loc exp reversal of the inequalities given by Equations (13) and (64). A similar comparison of the K a n tensor can be made with calcu-lated ionic EFG contribution, ?Lia£f A n obvious difference is that the X. and Z, axes of q, do not coincide with the X and Z axes of KOT,. 1 1 \Latt an From Table VIII i t follows; C q ^ ) ^ - 0 and ( 9 ^ ^ ) ^ = " C ^ a t t ^ with a negative sign for (?]_att)x^ X-|_* 0 N T N E O T N E R hand, K^^Y) = 0 and K (X) = -K (Z) with a positive K (X). an' an an It is not possible to draw some conclusions from a comparison of the anisotropy of the bulk susceptibility x with the EFG or K a n ten-sor. This is because the source of the x anisotropy is the anisotropic - I n -effective mass, m*. Landau diamagnetism, highly dependent upon m*, is the source of the diamagnetic (negative) part of the anisotropy, with anisotropic spin susceptibility (positive) of the conduction electrons constituting the remainder. The bulk susceptibility, x> is therefore a sum of several effects, each with a different dependence upon effective mass, m*. 2 1 The ground configuration of the gallium atom is (4s) (4p) . Therefore, i t is expected that the electrons at the Fermi level have a considerable p-type character. An 80% of p-type character was estimated earlier from the measured Knight shift and known hyperfine coupling con-stant. Following Masuda [94], the electron in the solid may adequately be described by one-electron wave function of the Bloch type, ^ = e 1 K' r U£(r) (65) o Uj^(r) has the periodicity of the lattice and, within the atomic polyhedron, it is assumed to be consisting of a mixture of s- and p-type functions of the form U ? ( r ) — > a i l f + a i l F = a \ l t + a . A + a„i|r + a 0ili , (66) k x ' s*s p y p — s Ys 1 P x *V Pz where hsi2 + h pi2 = 1 a n d l a i l 2 H a 2 l2 + N 2 = ! a P i2 • <67> In the case of orthorhombic symmetry one expects < r | a 1 l - 2 > ^ ^ | a 2 | 2 ^ | a 3 | 2 > , (68) where 4 >means the average over the Fermi surface. The Hamiltonian for the nuclear spin-electron interaction is given by - 1 2 1 -H = (16rr/3)^ Ng NIS6(r i)+2^g N i{Sr- 3-3r- 5(S?p (69) where r is the radius vector of the electron with the nucleus at the 1 - 4 - I origin. S and I are the electron and nuclear spins, respectively. The isotropic Knight shift is produced by the Fermi-type hyperfine interac-tion with the electron spin in s-state and is given by the first term of the Hamiltonian (69). It results in K i s o = (16TT/3)HgR|as| 2 <rUs(0)\ 2> QQN(EF) . (70) 2 R is the relativistic correction (1.10 for 4s Si state of Ga). It is -2 hard to obtain the exact wave functions ik and ill of metal Ga and as a vs yp crude approximation one replaces them by the atomic s- and p-type func-2 tions. Values of |tyg(0)| can be calculated using standard formulae [84] for non-hydrogenic wave functions. They can be also obtained from the hyperfine splitting of the atomic spectrum of Ga. Of course, care should be taken to account for the degree of ionization in metallic state which is different than that of a free ion [90]. The anisotropic Knight shift is produced by the nuclear spin-electron spin dipolar coupling when the charge distribution of the elec-trons is deviated from the' spherical symmetry about a nucleus and is deduced from the second term of Equation (69). It can be written in the form Kan = 2 ^ ( E F ) | f p - ( 3 cos 2cY-l)r" 3ik pd 3r . (71) cv is the angle between Ho(9,0) and r(e',0'). Using the addition formula for the second harmonics one derives K = AP°,(cos 9) + CJ?°(cos 9) cos 20 (72) - 1 2 2 -for the case of the orthorhombic symmetry for the electron charge dis-tribution. The two parameters A and C 2 are given by the following two expressions: 2 ^ A = 2^ BCN(EF)<:yp0:(cos e^r , (73) C 2 = 4u|o*(EF)^y P2(cos 9')cos 20' r~ 3|^> . (74) Using the wave function given by Equation (66) the charge density in an orthorhombic environment can be written as I 2 2 2 2 2 2 2 ijr*ilj = Ig (r)| r [ sin 61 ( I a I cos 0" + I a I sin 0') P P p 1 21 +|a 3l 2cos 29' J . (75) g (r) is the radial wave function. The parameters A and C9, are related P to a^ , a^ and a^ by the following expressions: A = (4Tr/15)|a ] 2/r" 3> [3 |b | 2 - l\ . { 2a2 N(E P)J , (76) p p J J j j r -C2= (48TT/15)|a p| V r " V (IbJ 2 - [ b2| 2 J. {2a2 N(Ep)J , (77) where |b.|2 = |a.| 2/|a p] 2 i = 1,2,3 . (78) Equations (76) and (77) were obtained by performing the averages in Equa-tions (73) and (74) using Equation (75). A and are also related to and of Equation (37a) whose values are given in Table IX. Using Equa-tions (76), (77), (70) and (37a) one derives the following two expressions: K./K. = (3/10).(|a | 2 / l a l 2).{R^r" 3> /[R^U (.0) | *>]]. J. l o U P S p o J . J 3 N 2 " l \ ' ( 7 9 ) -123-K./k. = (-36/5).(|a | 2 / | a j2). J R ^ r " 3 ? / { R ^ (0)|2>]S. 2 i s o 1 p 1 s p s J •U bii 2-i b 2i 2j- (80) 2 R1 i s a r e l a t i v i s t i c c o r r e c t i o n (for 4p, P^ st a t e ) . The quantities -3 -3 ,2 Z-3 R'<r > = 3.50 a and < | i b (0)| >= 8.2 a ; a i s the Bohr c l a s s i -P S ° , 2 ° , ,2 cal radius. I t was estimated e a r l i e r that la = 80% and a = 20%. 1 p 1 1 s' The r a t i o s K../K. and K./K. can be obtained from Table IX. Experi-1 i s o 2 i s o r 2 mentally, i t was shown that K-^ and are negative meaning [ b„j /1/3 2 2 2 2 and | b 1| >|1>2J . The calculated values are: |b3| = 26%, |b | = 37% and |b^| by about 1% larger than | b2f. In t h i s evaluation, the average values of K^/K^ s o and K 2/K^ g o, obtained from the l a s t three rows of Table IX were used. In terms of these r e s u l t s , Equation (66) i s then I W f ) -> 0.44 4 + 0.54 4 + 0.54 ill + 0.44 4 . (81) k s Px P y P z Expression (81) reads that the charge d i s t r i b u t i o n of the electrons i n the Fermi surface i s approximately tetragonal and contracted along the Z axis. Contrary, i n case of the EFG tensor the t o t a l charge density i s ortho-rhombic and i s elongated i n the Z d i r e c t i o n . This i s suggesting that there are l e s s free electrons i n the regions of the large e l e c t r i c f i e l d gradients. However, the large percentage of I|L i n Ur'(r) contradicts Py k t h i s naive p i c t u r e . -124-SUGGESTIONS FOR FURTHER EXPERIMENTS Recently some experiments on the f i e l d dependence of the Knight shift at 1.5°K have been done in some pure metals. The po s s i b i l i t i e s of de Haas-van Alphen type oscillations of the Knight shift in a monocrystal of gallium are discussed in Appendix A. An estimation of the amplitude of these oscillations for the Series A orbits [65], with the external f i e l d in the [010] direction, shows that they are far too small to be observed experimentally. This statement i s of course referred to a spe-ci a l Series of orbits for which the effective mass, m , is known. It would be worthwhile to perform experiments for different crystal orien-tations. Even a negative effort would give a result; i.e. an upper limit to the os c i l l a t i o n amplitudes. There are no experimental d i f f i c u l t i e s except for an enormous magnetoresistance pickup which could be avoided by using frequency modulation. A high, homogeneous magnetic f i e l d (pos-sibly >20 kG) is required. Since these measurements are relative, one can look for the oscillations in any crystal orientation regardless of an accurate knowledge of the EFG tensor. However, this i s no more true i f the absolute values of K are required. The measurements of the EFG and K tensors very close to T m MP could give some insights in the melting process and explain rather low melting points. It i s speculated in some literature [62] that most of the cohesive energy in gallium is due to 'tight binding 1 rather than the 'loose binding' which occurs in the a l k a l i metals. The bonds, presumably pure p-bonds, are rather weak as compared to, let's say, carbon atom. Also, recall that Ga has a layer li k e structure similar to that of solid iodine with two atoms appearing to form a 'Ga2 molecule 1. In accord with this picture, as the temperature approaches closer to T, — , the co--125-hesive forces among different layers start weakening first and then the layers start sliding relative to each other. Due to the increased lattice vibrations the layers gradually collapse into xGa^ molecule" in the liquid. Intuitively, i f the picture of this 'two step1 melting process is a correct one, then the quadrupole frequency, v^, and the asymmetry parameter, T|, should show some irregular temperature depen-dence. It could well be that the thermal vibrations are large enough to smear out these effects. It is believed that liquid gallium consists of Ga2 molecules [1013. However, this hypothesis is hard to justify in the light of re-cent results on the spin-lattice relaxation in the liquid Ga [133. It is found that the quadrupole coupling in the liquid is increasing with increasing temperature. This is/contradicting the existence of Ga^ mole-cules, since they should certainly dissociate with increasing temperature . Thus, the quadrupole coupling constant should decrease. It appears then that the (l/T^)^ data for the liquid Ga contradict the discussed picture for the melting process. . . • " In principle, information relating to the conduction-electron distribution could be extracted from a model which satisfactorily repre-sents the EFG at the nuclear site in a metal. However, i t is necessary that the criteria used to assess the merit of a model be detailed enough to be significant. A comparison between a theoretical value at a single temperature and pressure is insufficient. Therefore the effects of pres-sure on the EFG tensor parameters are of great interest. Owing to its complicated crystal structure, a complex set of independent measurements is required in order to separate the different contributions in gallium. .•.*"" "' 1 ' — ' ' ' ' . . . 1 — • - — ••• • - - — — — • "The observed temperature dependence could come from Ga2molecules i f heating produced molecular vibrational modes rather than dissociation.(Dr.Mahanti-private communications). -126-The EFG tensor at a nucleus in Ga metal does not have the axial symmetry. Therefore, the 'symmetry' of the EFG is not necessarily in-variant under the application of a uniform stress. Benedek & Kushida [48] measured the quadrupole frequency as a function of the hydrostatic pressure in the range from 1 to 8000 atm at 0,-29.8 and -75°C in a pow-der sample. It was found that VQ varies linearly with the pressure at all temperatures Qv^/h?)^ .is 16.1, 15.8 and 15.4xl0"3 kHz/atm at 0, -29.8 and T75°C, respectively. The expression relating the quadrupole frequency and hydrostatic pressure is V Q(P) = % [e 2Qq z z(P)/h]'[l + h i f o ) } • Thus, one should perform these experiments on a single crystal at diffe-rent temperatures in order to obtain the coefficients, (^ )q/c)p) , (DT|/^P) and also Qv/^P) ; y 1 S t n e angle between the Z axis and the C axis. The three parameters, q, T| and y, characterizing the EFG tensor are functions of a minimum four independent parameters. A possible choice of these parameters is; temperature, volume, the ratio 6 = - A / C and B = (u/v)6 with B being the angle between the diatomic axis and the C crystal axis. . Thus, 92z = qZz(T>v>&,B) , .'•11= T](T,V,6,B) , Y = Y(T,V,6,B) . The temperature and pressure dependences, in conjunction with the linear expansitivities and compressibilities provide only two of the four experi-ments necessary to separate the dependences of the EFG tensor upon these variables. The logical third experiment would be a measurement of the -127-effect of the uniaxial stress upon the EFG parameters. The fourth and perhaps most difficult experiment is the measurement of the angle 8 with X-rays under the pressure at different temperatures. The inference that the EFG tensor parameters are strongly correlated with changes of B is obvious. As outlined above the completion of such a program is in first instance a discouraging task for every experimentalist. However, the amount of data one would obtain from such a program is great. An imme-diate use of these data would be in the pressure experiments of the Knight shift on a single crystal of Ga. It was stressed that the K results are strongly dependent on the accurate values of the EFG tensor parameters. The K(P) measurements would give dependable information regarding the volume effects on K^go and its intrinsic temperature dependence, i f any. Also, re-call that («)K. c„/c)F)m changes its sign upon melting. This means that either I S O JL the volume dependence of the intrinsic temperature dependence or both change the sign upon melting. One could also determine the volume effects on the anisotropy over the entire temperature range. One should be able to tell whether the variation of K 0„ with temperature is purely due to changes in an 6 or 8 or to the explicit temperature dependence. In any case, all these phenomena could be sorted out. A third possible set of pressure experiments would be to study the structure, the EFG and K tensors in the three pressure induced phases of solid Ga [102]. Here, the problems are: (a) the requirement for high pressures up to 50 kb and (b) the difficulties in the observing broad absorption lines arising from the inhomogenous stresses over the sample. A further experiment recommended for the future is the tempera-ture and angular dependence of the linewidth and the second moment in -128-order to separate the different contributions causing the linew Most of the broadening in Ga is presumably due to the dipole-di teraction. An exact theoretical calculation of the second mome to this interaction and which could be applied to Ga has not be Also, Ga may show a field dependence of the linewidth at very 1 peratures [72]. Although the second moment problem was out of of this investigation a cursory glance on i t is made in Append! -129-BIBLIOGRAPHY 1. C. H. Townes, C. Herring, and W. D. Knight; Phys. Rev. 7_7, 852 (1950). 2. L. E. Drain; Metallurgical Reviews 119, 195 (1967). 3. A. Abragam; Principles of Nuclear Magnetism, Oxford U. Press ? 1961. 4. P. L. Sagalyn and J. A. Hofmann; Phys. Rev. 127, 68 (1962). 5. E. J. P. Jones and D. L l . Williams; Phys. Letters 1, 109 (1962); Can. Phys. 42, 1499 (1964). 6. H. E. Schone; Phys. Rev. Letters 13, 12 (1964). E. F. W. Seymour and G. A. Styles; Phys. Letters 10, 269 (1964). F. Borsa and R. G. Barnes; J. Chem. Phys. Solids 27_, 567 (1966). S. N. Sharma and D. Ll.Williams; Phys. Letters 25A, 738 (1967). 7. D. L l . Williams; Phys. Letters 13, 12 (1964). J. J. Schratter and D. L l . Williams; Phys. Letters 26A, 79 (1967). 8. P. D. Dougan, S. N. Sharma, and D. L l . Williams; Can. J. Phys. 47, 1047 (1969) . 9. R. W. Weinert and R. T. Schumacher; Phys. Rev. 172, 711 (1968). 10. D. L l . Williams (private communication). 11. S. R. Kiser; Ph.D. Thesis, University of California, Berkeley, (1965 -unpublished). M. I. Valic, S. N. Sharma, and D. L l . Williams; Phys. Letters 26A, 528 (1968). 12. R. H. Hammond, E. G. Wilkner, and G. M. Kelly; Phys. Rev. 143, 275 (1966) . 13. D. A. Cornell; Phys. Rev. 153, 208 (1967). 14. C P . Slichter; Principles of Magnetic Resonance, Harper and Row Publishing Co., N.Y. (1963). 15. T. P. Das and E. L. Hahn; Solid State Physics Supplement I, Chapter I, Academic Press (1958). C. Dean; Phys. Rev. 96, 1053 (1954). See Ref. 3, Chapter VII. -130-16. M. Toyama; J. Phys. Soc. Japan 14, 1727 (1959). 17. Y. Morino and M. Toyama; J. Phys. Soc. Japan 35, 1289 (1961). M. Pomerantz; Ph.D. Thesis, University of California, Berkeley, (1955 - unpublished). . 18. W. Kohn; Phys. Rev. 9_6, 590 (1954). 19. T. Kjeldass and W. Kohn; ib i d . 101, 66 (1956). 20. G. B. Benedek and T. Kushida; J. Phys. Chem. Solids 5_, 241 (1958). 21. T. Muto and S. Kobayasi; J. Phys. Soc. Japan 18, 1350 (1963). 22. W. E. Blumberg, J. Eisinger and M. P. Klein; Phys. Rev. 124, 206 (1961). 23. A. Abragam, J. Horowitz, and M. H. L. Pryce; Proc. Roy. Soc. 230, A169 (1955) . W. Marshall; Phys. Rev. 110, 1280 (1958). 24. J. A. Seitchik, A. C. Gossard, and V. Jaccarino; Phys. Rev. 136, A119 (1964) . 25. A. M. Clogston, V. Jaccarino and Y. Yafet; Phys. Rev. 134, A650 (1964). 26. M. H. Cohen, D. A. Goodir.gs and V. Heine; Proc. Phys. Soc. 7_3, 811 (1959). 27. R. Kubo and Y. Obata; J. Phys. Soc. Japan 11, 547 (1956). 28. L. E. Drain; Proc. Phys. Soc. 83, 755 (1964). 29. J. A. Seitchik, V. Jaccarino and J. H. Wernick; Phys. Rev. 138, A148 (1965) . 30. M. H. Boon; Physica 30, 1326 (1964). 31. N. Bloembergen and T. J. Rowland; Acta Metallurgica 1, 740 (1953). 32. V. I. Chechernikov, I. Pop and 0. P. Naumkin; Soviet Physics - JETP 71, 1228 (1963). 33. F. Y. Fradin, D. J. Lam, J. W. Ross and L. L. Isaacs; Bull. Am. Phys. Soc. 14, 332 (1969). F. Y. Fradin; Phys. Letters 28A, 441 (1968). 34. W. D. Knight, A. G. Berger, and V. Heine; Ann. Physics 8, 173 (1959). 35. R. V. Kasowski and L. M. Falicov; Phys. Rev. Letters 22, 1001 (1969). -131-36. J. Korringa; Physica 16, 601 (1950). ' 37. T. Moriya; J. Phys. Soc. Japan 18, 519 (1963). A. Narath; Phys. Rev. 162_, 320 (1967). 38. A. H. Mitchell; J. Chem. Phys. 26, 1714 (1957). 39. A. Narath and D. W. Aldermann; Phys. Rev. 143, 328 (1966). 40. S. C. Barrett; Advances in X-ray Analysis 5_, 33 (1961). 41. A. J. Bradley; Z. Krist. 91A, 302 (1935). 42. R. W. Powell; Proc. Roy. Soc (London) A209, 525 (1951). 43. H. M. Rosenberg; P h i l . Mag. 45, 767 (1954). 44. Pankey T., Jr; J. Appl. Phys. 31, 1802 (1960). 45. W. D. Knight, R. R. Hewitt and M. Pomerantz; Phys. Rev. 104, 271 (1956). 46. S. R. Kiser and W. D. Knight; Am. Phys. Soc. Bull. ]_, 613 (1962). 47. R. H. Hammond and W. D. Knight; Phys. Rev. 120, 762 (1960). 48. T. Kushida and G. B. Benedek; Am. Phys. Soc. Bull. 167 (1958 - May). 49. C. H. Townes and B. P. Dailey; J. Chem. Phys. 17, 782 (1949). 50. H. Goldstein; Classical Mechanics, Addison-Wesley Publishing Co., (1965) 51. D. G. Watkins; Ph.D. Thesis, Harvard University, Cambridge, Mass. (1952 - unpublished). S. N. Sharma; Ph.D. Thesis, University of British Columbia, Vancouver, (1967 -unpublished). 52. L. Buss and L. Bogart; Rev. Sci. Inst. 31, 204 (1960). 53. M. Yaqub and J. F. Cohran; Phys. Rev. 137_, A1182 (1965). 54. R. T. Daly, Jr., and J. H. Holloway; Phys. Rev. 96_, 539 (1954). 55. N. Bloembergen; J. Appl. Phys. 23, 1383 (1952). A. C. Chapman, P. Rhodes and E. F. W. Seymour; Proc. Phys. Soc. (London) 70, 345 (1957). 56. P. L. Sagalyn and J. A. Hoffmann; Phys. Rev. 127, 68 (1962). 57. P. S. Allen and E. F. W. Seymour; Proc. Phys. Soc. 82, 174 (1963). -132-58. R. A. Watson, A. C. Gossard, and Y. Yafet; Phys. Rev. 140, A375 (1965). 59. R. M. Sternheimer, Phys. Rev. 130, 1423 (1963). 60. B. R. McGarvey and H. S. Gutowsky; J. Chem. Phys. 21, 2114 (1953). 61. G. Seidel and D. H. Keesom; Phys. Rev. 112, 1083 (1958). ^ - o N. E. P h i l l i p s , Phys. Rev. 134, A385 (1964). 62. R. Bersohn; J. Chem. Phys. 36., 3445 (1962). 63. T. P. Das and E. H. Sondheimer; Ph i l . Mag. 5_, 529 (1960). 64. M. J. Stephen; Phys. Rev. 123, 126 (1961); see also S.Rodriquez, Phys. Letters 4_, 306 (1963) . 65. L. J. Neuringer and Y. Shapira; Phys. Rev. 16_5, 751 (1968). 66. A. Goldstein and S. Foner; Phys. Rev. 146, 442 (1966). 67. S. A. Khan, J. M. Reynolds and R. G. Goodrich; Phys. Rev. 163, 579 (1967). 68. M. L. Glasser; J. Math. Phys. 43, 158 (1964). 69. R. B. Dingle; Proc. Roy. Soc. (London) A211, 500 (1952); A211, 517 (1952); A212, 38 (1952); A212, 47 (1952). 70. J. H. Van Vleck; Phys. Rev. 74, 1168 (1948). 71. K. Kambe and J. F. Ollom; J. Phys. Soc. Japan 11, 50 (1956). 72. M. L. Glasser; Phys. Rev. 174, 424 (1968). 73. H. Wahlquist; J. Chem. Phys. 35, 1708 (1961). G. W. Smith; J. Appl. Phys. 35, 1217 (1964). J. E. Cousins, R. Dupree and R. H. Havill; B r i t . J. Appl. Phys. 16, 1687 (1965). 74. L. C. Hebel; Phys. Rev. 128, 21 (1962). 75. B. R. McGarvey and H. S. Gutowsky; J.' Chem. Phys. 21^ 2114 (1953). 76. D. Pines; Solid State Physics, (Academic Press, New York, 1955) Vol. I, p. 367. S. D. Silverstein; Phys. Rev. 128, 631 (1962); Ibid, 130, 912 (1963). 77. F. A. Rossini and W. D. Knight; Phys. Rev. 178, 641 (1969); E. M. Dickson; Phys. Rev. 184, 294 (1969). -133-77a. E. M. Dickson; Ph.D. Thesis, University of California, Berkeley, (1968). 78. D. Weaire; Proc. Phys. Soc. 92, 956 (1967). 79. W. W. Warren, Jr. , W. G. Clark, and P. Pincus; Solid State Commun. _6, 371 (1968). 80. J. J. Quinn; The Fermi Surface, ed. W. A. Harrison, M. B. Webb; (John Wiley and Sons, Inc., New York, 1960) p. 58. 81. A. E. Prange and L. P. Kadanoff; Phys. Rev. 134, A556 (1964). 82. Y. Obata; J. Phys. Soc. Japan 1_8, 1020 (1963). 83. Titus Dankey, Jr.; J. Appl. Phys. 31, 1802 (1960). 84. G. F. Koster; Phys. Rev. 86, 148 (1952). 85. Y. Yafet and V. Jaccarino; Phys. Rev. 133, A1630 (1964). 86. W. M. Shyu, T. P. Das, and G. D. Gaspari; Phys Rev. 152, 270 (1966); Ibid, 134, A852 (1964); ibid. 141, 603 (1966). 87. J. M. Ziman; Advan. Phys. 16, 421 (1967). 88. D. P. Tunstall and D. Brown; Phys. Letters 28A, 445 (1968). 90. W. D. Knight; Solid State Physics, (Academic Press, New York, 1956) Vol. 2, p. 93-136. 89. E. Fermi and E. Segre; Z. Physik J32, 729 (1933). 91. J. H. Wood; Phys. Rev. 146, 432 (1966). 92. J. E. Enderby, J. M. Titman, and G. D. Wignall; Phil. Mag. 10, 633 (1964) 93. F. Lackmann-Cyrot; Phys. Konst. Mat. 3, 75 (1964); M. Watabe, M. Tanaka, H. Endo, and B. K. Jones; Phil. Mag. 12, 347 (1965) 94. Y. Masuda; J. Phys. Soc. Japan 13, 597 (1958). 95. J. Jones and B. Schiff; Proc. Phys. Soc. A 441, 217 (1954) • 96. J. B. Campson and F. Seitz; Phys. Rev. 58, 633 (1940) • 97. P. 0. Simmons and R. W. Balluff; Phys. Rev. 119, 600 (1960) • 98. E. 2, T. Micah, G. M. Stocks, and W. H. Young; J. Phys. 1653 (1969). (Solid State Phys.) 99. E. 2t T. Micah, G. M. Stocks and W. H. Young; J. Phys. (Solid 1661 (1969). State Phys.) -134-100. W.A.Reed;(private communication). 101. D. R. Gustafson and A. R. MacKintosh; Phys. L e t t e r s _5, 23 (1963); A. T. Stewart, J. H. Kusmiss, and R. H. March; Phys. Rev. 132, 495 (1963); C. J. Pings; J. Chem. Phys. 42, 2435 (1965); T. E. Faber; S o l i d State Commun. 1_, 41 (1963). 102. J. E. I n g l e s f i e l d ; J. Phys. Soc. (Proc. Phys. Soc.) 1_, 1337 (1968). 103. C. K i t t e l ; Introduction to S o l i d State Physics, John Wiley & Sons,Inc., (1967). 104. A. Narath and H. T. Weaver; Phys.Rev. 175, 373 (1968). 105. T. "Izuyama, D. Kim, R. Kubo; J.Phys.Soc.Japan 18, lo25 (1963). 106. G. A l l a n , W. M. Lomer, R. D. Lowde, and C. G. Windsor; Phys.Rev.Letters 20, 933 (1968). 107. A. Narath; "Hyperfine Int e r a c t i o n s " , E d i t o r A. J . Freeman and R. B. Frankel, Academic Press (1967). 108. A. M. Clogston, A. C. Gossard, V. Jaccarino,and Y. Yafet; Phys.Rev.Letters 9, 262 (1962). 109. V. Jaccarino, MWeger, J . H. Wernick,and A. Menth; Phys.Rev.Letters 21, 1811 (1968). -135-APPENDIX A THE POSSIBILITIES OF DE HAAS-VAN ALPHEN TYPE OSCILLATIONS OF THE KNIGHT SHIFT IN A MONOCRYSTAL OF GALLIUM The possibility that the Knight shift might be field dependent in pure metals at low temperatures was first recognized in print by Das and Sondheimer [63], Since their suggestion, the effect has been calcu-lated on a free-electron model by several investigators, most completely by Stephen [64], It is interesting to employ his formulae for the field-dependent paramagnetic (o ) and diamagnetic (a,) shift to estimate the size of the effect in gallium. The calculations will be performed for the Series A oscillations [65] with the external field in the [010] di-rection. One is concentrating on this Series because the cyclotron mass is known just for this Series [66]. The results will be compared with those of B orbits in mercury [9] and 36^ orbits in white tin. The os-cillations have been recently observed in white tin [67], Stephen writes the following expressions for the paramagnetic and diamagnetic shifts for free electrons with effective mass m*: P (O •.kT (• 1 ) p osc d osc n=l (Al) (A2) sinh[(n rr H)kT] - 136 -with 2 1/3 2 5 0 ' < A 3 ) I(n) =1 d s [ s ( l - s ) ] s i n | n T T s | , (A4) where 5Q is the Fermi energy, k is the Boltzman constant, T is the abso-lute temperature, m* is the effective mass of carriers, \i * is the effec-tive Bohr magneton, N/V is the density of the electrons taking part in the interaction, and I(n) is an integral expression which has been eva-luated numerically by Glasser [68]. The ratio § /u, * is most usefully o o written as I l\i * = 2v ' (A5) o o where v is the DHVA frequency (in G). In all considerations the effec-tive scattering temperature (Dingle temperature) [69] will be ignored. This could be tolerated i f very pure metals are assumed. The shielding constant (ST ) has been related to the os-p osc d i l a t i n g part of the paramagnetic susceptibility by the equation (a ) = (x ) (A6) p osc J p osc This relationship is valid for a uniform electron density. In a real metal, a more detailed analysis shows that the factor V ^ in Equation .2 (Al) should be replaced by <^ ]\|i(o)\ >p = P^,, where i(r refers to the wave function of an electron at the Fermi surface and is normalized in V. Stephen recognizes that the use of V ^ in place of P substantially un-r derestimates the size of the paramagnetic term. Weinert and Schumacher [9] arbitrarily modified his (cfp) o s c by multiplying by P-pV. N/V is now the density of electrons participating in the orbit. - 137 -In order to apply formulae (Al) and (A2) t o a r e a l metal one defines a f i c t i t i o u s J * and (N/V)* using Equations (A3) and (A5) from DHVA data m* and V * . • , 2 The region of i n t e r e s t i s where rr kT/p, * H Z 1, and only the o f i r s t terms of the sum i n Equations (Al) and (A2) are important. This c o n d i t i o n i s met i n the liquid-helium-temperature range and i n f i e l d s of 10 G where Equations (Al) and (A2) are: < V°sc = °p cos[(2rr/H)v-l/4rr] , (A7) (a.) = a, sin[(2rr/H)v] , .' (A8) d osc d where " a - -Arr 2 ^ (P rV) (^) ( 2 v ) " 3 / 2 s i n ( T T f / m > , (A9) P V m s i n h ( n 2 k T / ^ H ) . a. - +3rr3 1(1) $ ^ cos(rr m*/m) . (A10) d V H v s i n h ( r r 2 k T / ^ - H) In Equation (A10), 1(1) = 0.3 [68]. a and a, are the amplitudes of (o ) n u j d p osc and (o\) . Their r a t i o i s d osc -P- = — - (1L) tan(n m*/m) P V . ( A l l ) a d 3rr 1(1) 2 V m F A l l the q u a n t i t i e s used i n the c a l c u l a t i o n s of (a ) and n P osc (o\) are dis p l a y e d i n Table I-A. The gallium DHVA [65] data are a osc taken from G o l d s t e i n and Foner [66] and Neuringer and Shapira [65], The data are r e f e r r e d to the low frequency Series A o s c i l l a t i o n (0.345 MG) •with the external magnetic f i e l d along the B c r y s t a l a x i s . The mercury data are r e f e r r e d to the 8 DHVA o s c i l l a t i o n s with the f i e l d p a r a l l e l to [211] d i r e c t i o n and are c o l l e c t e d i n the work of Ueinert and Schumacher [9], The white t i n data are r e f e r r e d to 36-^ o r b i t s with the f i e l d along -138-[001 ] direction and can be found in the Khan, Reynolds and Goodrich paper [67], The electron probability density at the nucleus, P_V, F 8TT is estimated from the isotropic Knight shift K = TpV. A reaso-nable estimate for Xp c a n be obtained from measurements of the elec-tronic part of the specific heat at low temperatures or else taking the free electron value. TABLE I-A. A collection of the quantities used in the calculation of ( O n - r and (a ) for Ga, Hg and Sn. P o s c d osc Quantity Units Ga Hg Sn I* ' o ergs -14 12.2 x 10 8.7 x IO" 1 4 32 x 10" 1 4 m*/m 0.051 0.23 0.095 V G 3.4 x 105 10.8 x 105 17.1 x 105 H G 16 x 103 16 x 103 16 x 103 T °K 1.2 1.2 1.2 K. ISO % 0.130 2.46 0.72 X cgs volume 0.95 x 10"6 1.5 x 10"6 2.5 x 10"6 P units P F V 160 2000 450 (N/Vf -3 cm 0.011 x 10 2 0 0.10 x 10 2 0 0.33 x 10 2 1 The comparison of the theoretical amplitudes a , a , (cr) = p o s c (cOnc- + f°r t n e oscillating Knight shift in Ga, Hg and Sn is p OoC Q osc given in Table II-A. It should be emphasized that i f the amplitudes of (o~p) o s c and (o" cj) o s c are comparable, the different phases of the trigono-metric factors they multiply-must be considered in assessing the over-all field dependence of the oscillatory shift. - 1 3 9 -TABLE II-A. A comparison of the theoretical amplitudes a , a, and (a) p' d osc in Ga, Hg and Sn. Metal a x 105 P a, x 105 d P a (a) osc (a ) exp osc Ga 0.02 0.13 0.14 0.14 Hg 0.8 .03 25 0.14 ^3.0 Sn 2.7 . 4.3 0.64 3.0 1.5 From Table II-A one can immediately see the agreement with the experi-ment [67] in the case of white tin where the oscillations are predomi-nantly due to the diamagnetic shift. In case of mercury a »o~,. The P d theoretical oscillation amplitudes are less than Weinert and Schumacher [9] could have seen experimentally. From their experiment an upper limit on the amplitudes would be 1.5 x 10 One should be aware that 199 Hg resonance in a single crystal is very difficult to observe. Accor-ding to the theory the oscillations in gallium are solely due to the diamagnetic shift. ' However, the amplitudes are far too small to be ob-served experimentally. This statement is of course referred to a spe-cial Series of orbits for which m* is known. It would be worthwhile to perform experiments for different orientations of the magnetic field. Even a negative effort would give a result , i.e. an upper limit to the oscillation amplitudes. -140-APPENDIX B SECOND MOMENT ANALYSIS The first clue to the origin of the broadening comes from a comparison of the widths of the two isotopes for the same transition, temperature, magnetic field and orientation. For example, &H m e a s = 6.5 kHz for Ga 6 9 and 6H m e a s = 7.5 kHz for Ga71. The fact that Ga 6 9 has the narrower line and smaller magnetic moment (Table I) suggests that the width is associated with the magnetic moments and not the quadrupole moments of the nuclei. If, for example, there were large strains in the metal that produced inhomogeneous EFG over the sample, the line would be 69 broadened in proportion to the quadrupole moments and the Ga resonance would be widened with respect to the Ga7"*" line. This is not observed at any temperature. It is very fortunate that the isotope with larger quadrupole moment has the smaller magnetic moment so that the magnetic sources of broadening can be distinguished from electric, i.e. quadru-pole broadening. The question arises whether the observed width agrees with Van Vleck's [70] well known formula for the second moment produced by the dipole interaction. This formula is given by the expression: ^Av>Av = -h 2 Tr[H 1,S x] 2/Tr(S x) 2 , (Bl) where H is the Hamiltonian of the interaction and S the component of the x total spin vector operator along the direction of the high frequency field. Kambe and Ollom [71] calculated, using the above equation, the width of the NMR central line in the system of half-integral spins, where the electric quadrupole coupling is present. The coupling should be so -141-large that the central line is well resolved from the satellite lines. For instance, for 1=3/2 case the central line corresponds to the L% M^ >«-* l-%M^>transition and i t should be resolved from the satellite lines which result from the 1 3/2M^> <-* 1^ M^> or l-3/2y£> •** J.-%M> transi-tion. The interaction Hamiltonian H in Equation (Bl) takes different forms depending upon the type of interaction. Three types of interac-tions will be considered: (a) Interaction between resonant spins, (b) Interaction between unlike spins, (c) Interaction between semi-like spins. This terminology is commonly found in the literature [3], The.resonant spins are the same species of spins with the same quadru-pole coupling. Semi-like spins are the identical spins located at the different lattice sites, where the values of the quadrupole coupling constant are different. For semi-like spins the central line of the resonance coincides, since the central line is not affected by the crystalline field. Satellite lines, however, do not coincide. In the case of gallium the EFG tensor is asymmetric, i.e. T| £ 0, and the above theory could not be applied without speculations. The effect of T| on the NMR spectrum along the Z axis is negligible and an assumption is made that its effect on the second moment can be ignored since the interest is only in the order of magnitude of such a calcula-tion. The next question considers the choice of the correct type of interaction. To be more specific, second moment along the Z^ axis is calculated taking Ga^ with relative abundance F = 0.398 as the reso-nant nuclei. The different contributions to the second moment, using Abragam's notation [3], are: -142-(a) ZAvV = F . F L(I) T Z ( b ^ 2 . (B2) k m=l Equation (B2) represents the contribution due to the inter-71 action between like spins (Ga ). The summation, m, is over the four atoms in the unit cell belonging to the site (1). The other four belong to the site (2) and have to be treated differently. For the definition of the quantities F^(I) and b ^ the reader is referred to Ref. [3]. N 8 (b) <hv> = F . 1/3 I'(I'+1) v*. Y i ^ 2 ^ S ( b ™ ) 2 . (B3) k=l m=5 Equation (B3) represents the contribution due to the interac-tions with the spins of Ga7''" belonging to sites (2). These spins have the same quadrupole coupling constant and different orientations of the principal axes. Their spectrum is well resolved from the central line for the site (1), hence they should be treated as unlike and not as semi-like. In Equation (B3) I 1 = I and YT' = Yj« '^ie summation m, is over the four atoms belonging to the site (2). 2 1 2 2 2 N J - m 2 . (c) SL\V>= (1-F) I Yj, Y j * 2 S ^ (b ) (B4) k=l m=l ^ Equation (B4) represents the contribution from unlike spins 69 (Ga ). The total second moment is then the sum of all three contribu-tions. The result of a lengthy calculation performed on a computer is 2.21 G compared to 6.8 G from the experiment. The calculated value is by a factor of 3 smaller than the experimental one. It is possible that the assumption made in the calculation is wrong. However, other mechanisms for line broadening could be responsible for the difference. Another source of line width originating from the magnetic interaction among-the nuclei arises from the mechanism of indirect ex-change coupling [14], The process is of second order and has its origin -143-in the fact that the nuclear magnetic moments can be coupled to one an-other through the intermediary of the hyperfine interaction between the nucleus and a conduction electron. This process is more important in the heavier metals where the hyperfine interaction becomes larger. Our results on the EFG tensor and the Knight shift suggest a considerable decrease in the s-character of the electron wave functions in the solid compared to the liquid.- This fact favours a larger pseudo-dipolar in-teraction rather than an indirect exchange interaction. The former in-teraction comes about from the fact that nuclear magnetic moments can be coupled to one another through the intermediary of the dipole inter-action between the nucleus and a non s-conduction electron. Since the aim of the present work was the determination of the EFG and the Knight shift tensor parameters, the possible magnitude of both effects on the linewidth have not been considered in detail. Recently Glasser [72] has calculated the interaction between two spins interacting with a gas of independent electrons by a 6-function interaction in the presence of a uniform magnetic field in the effective-mass approximation. The interaction has both steady and oscillatory de-pendence on the magnetic field and is such as to contribute a field de-pendence to the NMR line shape in metals having low-mass carriers. As a promising substance to examine for field-dependent effects, gallium is the best candidate (HQ v 20kG). This could be an interesting experi-ment for future consideration. In many metals the linewidth is caused by the short relaxation time T-^. In solid gallium,however, the T-^ cannot be responsible for much line broadening. The relaxation times for the two isotopes at 300°K [12] and the "lifetime" broadenings are given in Table B-I. 144-TABLE B-I. The relaxation time T and the "lifetime" broadening for the two Ga isotopes. Isotope ^(sec) Av(kHz) Ga 6 9 Ga 7 1 0.00245. 0.41 0.00154 0.65 V o l u m e 26A. number 11 P H Y S I C S L E T T E R S 22 A p r i l 1968 T H E K N I G H T S H I F T I N G A L L I U M M E T A L * M . I . V A L I C , S . N . S H A R M A and D . L L E W E L Y N W I L L I A M S Department of Physics, University of British Columbia, Vancouver 8, B.C., Canada R e c e i v e d 28 M a r c h 1968 The f i r s t measurement of the Knight shift in s o l i d g a l l i u m is presented. The temperature var ia t ion i s s i m i l a r to that expected f rom re laxa t ion t ime measurements and a change by a factor of 3.3 i s ' obse rved at the me l t ing point . T h e Knight shift in ga l l ium metal has not been previously studied since the complexity of the nuc lear magnetic resonance spectrum in a pow-dered sample has inhibited any attempt. T h e present study of the ^ G a a n c j <lGa spectra has been made with a Pound-Knight spectrometer on single c r y s t a l specimens grown f r o m 79 purity Eagle P i c h e r ga l l ium, in both low and high m a g -netic fields at 4 . 2 ° K , 7 7 ° K and 2 8 5 ° K . The low f ield measurements have determined the p r i n c i -pal axis and asymmetry parameter of the e lec -t r i c f ie ld gradient tensor and wi l l be reported in a more complete paper [1]. T h e purpose of this letter is to report the Knight shift values ob-served at high fields for the specia l case where the magnetic f ield is para l l e l to the Z pr inc ipa l axis of the e lectr ic f ield gradient. The resonance frequency corresponding to the (5—• -i) transit ion with the magnetic f ield para l l e l to the Z axis of the e lectr ic f ield gradient is given by the following expression which is exact except for a negligibly s m a l l t e r m in 77^ expression " H 1 i„2 1 ~ 3 T7 1 (1) .1 " 4 ( ^ H / 1 ' Q ) 2 -where 2nv-^ = yH(l +K), H is the applied m a g -netic f ie ld, K i s the Knight shift, y i s the g y r o -magnetic ratio for the ga l l ium nucleus in ques-tion, TJ i s the a symmetry parameter , and I/Q i s the pure quadrupole resonance frequency. T h e main experimental difficulty lies in the accurate alignment of the magnetic f ie ld with respect to the Z pr inc ipa l ax is . A perturbation calculat ion shows that a misal ignment of the magnetic f ie ld by a s m a l l angle 9 causes a dev ia -tion A in the resonance frequency given by the * R e s e a r c h supported by the National R e s e a r c h C o u n c i l of Canada under Gran t N o . A - 1 8 7 3 . A = • ' H (2) where 0 is in radians . Fortunately the two i s o -topes and have markedly different nuclear quadrupole moments and the s p e c t r o m -eter frequency may be chosen so that the angular variat ion of the resonance frequency is very rap id for yet comparatively slow for 7 l G a . Some results are presented in fig. 1. Combining the results f r o m both isotopes indicates that the smal les t misal ignment was 0 . 3 ° which c o r r e s -ponds to a frequency correc t ion of 0.3 k H z for the "7lGa resonance frequency. T h e ""-Ga Knight shift values, measured with respect to the ga l l ium resonance frequency in an aqueous solution of G a C I 3 , are tabulated below together with the values of the 7 * G a spin- latt ice relaxation t ime T± obtained f r o m a study of the relaxation of the pure quadrupole resonance [2], T h e quantity ZT\T¥p- appropriate for the mag-netic hyperfine relaxation of the quadrupole r e s o -nance for / = § [3] may be compared with the K o r r i n g a value for 7 1 G a of 28 x 10" 7 s e c ° K . T h e resul ts a r e notable in two respects . F i r s t the constancy of the product T-^TK within experimental e r r o r over the temperature range studied may be taken as evidence that the source of the temperature variat ion of both T\T and K Tempera tu re (°K) ( s e c ° K ) 4.2 0 . 6 3 ± 0 . 0 1 0 . 1 1 5 ± 0 . 0 0 5 (25 ±2 .5 ) x 10" -7 77 0.58 ±0.01 0 . 1 1 6 ± 0 . 0 0 3 (23. 5 ± 1.5) x 10 -•7 285 0 . 5 0 ± 0 . 0 1 0.132 ± 0 . 0 0 8 (26 ± 3 ) X 1 0 " •7 528 Volume 26A, number 11 P H Y S I C S L E T T E R S 22 A p r i l 19C8 [sec] 10900 10850 10800 11200 11150 Go H = 10.500 Go H = 8.626 Z principal axis I 0 DEGREES -2 -3 F i g . 1. The angular va r i a t ion of the resonance frequen-cy c lose to the Z p r i n c i p a l ax i s for 6 3 G a and 7 1 G a . / / i s the magnet ic f i e ld quoted in k i l ogaus s . is the same. Secondly, the Knight shift in the solid extrapolated to the melting point is 0.136 which is smaller by a factor of 3.3 than the shift of 0.449% in the liquid [4]. This unusually large change however is also observed in the susceptibility (a factor of 3.5) and Cornell [5] has estimated an isotropic Knight shift of 0.17% from his relaxation time studies. Specific heat measurements [6] and a theoretical calculation [7] indicate that the density of states in the solid is between 60% and 90% of the free electron value so that it is likely that the effect is due to a considerable decrease in the s-character of the electron wavefunctions in the solid compared to the liquid. A large anisotropy in the Knight shift •would also be expected which would be consistent with both our results and Cornell's predicted value for the isotropic shift. It should be emphasized that the Knight shift has only been determined for one orientation of the magnetic field and a complete study as a func-tion of orientation which is presently in progress is necessary to separate the isotropic component which is relevant to the Korringa relation. How-ever, the isotropic contribution is normally the dominant one and its temperature variation is probably well represented by the present results. References (in p r e p a r a -M . K e l l y . P h y s . 1. M . I . Valic" and D . L l e w e l y n W i l l i a m s t ion) . 2 . R . H . H a m m o n d , E . G . W i l k n e r and G R e v . 143 (1966) 275. 3. A . H . M i t c h e l l . J . C h e m . P h y s . 26 (1957) 1714; L . C . H e b e l , P h y s . R e v . 128 (1962) 21 . 4. B . R . M c G a r v e y and H . S.Gutovvsky, J . C h e m . Phys 2] (1953) 2114. 5. D . A . C o r n e l l , P h y s . R e v . 153 (1967) 208. 6. G . S e i d e l and D . H . K e e s o m , P h y s . R e v . 112 (1958) 1083; N . E . P h i l l i p s , P h y s . R e v . 134 (1964) A 3 8 5 . 7. J . H . W o o d , P h y s . R e v . 146 (1966) 432. * * * * * 529 J. Phys. Chem. Solids Pcrgamon Press 1969. V o l . 30, pp. 2337-2348. Printed in d e a l Britain. A NUCLEAR MAGNETIC RESONANCE STUDY OF GALLIUM SINGLE CRYSTALS-I. LOW FIELD SPECTRA* M . I. V A L I C and D. L L E W E L Y N W I L L I A M S Physics Department, Universi ty of British Columbia , Vancouver 8. B . C . , Canada (Received 20 February 1969; in revised form 8 May 1969) Abstract —The low magnetic field splitting of the quadrupole resonance in gallium metal has been studied in single crystal specimens. Precise detcrminalions of the electric field gradient tensor have been made at 4-2°, 77° and 285°K. The results are only weakly temperature dependent and reveal two non-equivalent sites in the crystal which differ only in the relative orientation of their principal axes. A n attempt at a decomposition of the electric field gradient tensor by subtracting the calculated ionic contribution is highly suggestive of a strong contribution in the direction of the nearest neighbour and possibly indicative of some covalency. INTRODUCTION A CONSIDERABLE improvement in the re-solution of anisotropic effects in the nuclear magnetic resonance properties of metals has been achieved through their direct observation in single crystal specimens[l]. Gal l ium is one of the most anisotropic of metals and its nuclear magnetic properties can reveal use-ful information concerning its electronic properties. The measurable parameters are those of the electric field gradient ( E F G ) ten-sor at the nuclear site and the three describing the orientation dependence of the Knight shift in an orthorhombic crystal. The E F G tensor in gallium single crystals has been previously studied at 4-2°K by Kiser [2] but the results obtained are considerably less accurate than ours particularly with regard to the orientation of the principal axes. The quadrupole resonance has also been studied in the polycrystalline sample by Knight, Hewitt and Pomerantz [3], Pomerantz[4] and Kiser and Knight[5] over a wide temperature range and also in the superconducting state by Hammond and Knight [6]. Some measurements on the pressure and temperature dependence of the *Research supported by the National Research Counc i l of Canada through Grant A-1873 . quadrupole frequency in powder samples have been undertaken by Kushida and Benedek [7]. The present study was initiated with the main aim of determining the Knight shift parameters in the solid with a view to under-standing the temperature dependence of the spin-lattice relaxation time observed by .Hammond, Wilkner and Kelly[8] and also in the hope that a comparison with the measure-ments in the liquid state[9] would clarify the interpretation of the liquid. The determination of these parameters however requires an accurate knowledge of the E F G tensor and the present paper is confined to the results obtained for the E F G tensor over the temperature range from 4-2° to 285°K. The Knight shift results are deferred to a second paper [10]. The relevant properties of gallium are summarized in Table 1 and the crystal struc-ture is shown in Fig . 1. The crystal structure is orthorhombic with the unit cell dimensions A = 4-5156, B = 4-4904, C = 7-6328 at 4-2°K and atmospheric pressure[l I]. The eight atoms in the unit cell are located at the sites [12] (H.0, B) (it + hhv) (ii + i,i,B) (ii,0.v) (u,i,v + i) (n + kO,V + i) (ri + ^0,v+i) («, t,v + t) 2337 2338 M . I. V A L I C and D. L L E W E L Y N W I L L I A M S Table 1. Gallium magnetic resonance parameters Pure quadrupole N M R frequency N uclear quadrupole Nuclear Abundance frequency (285-3°K) in 10 kG moment Isotope spin (%) (kc/sec) (kc/sec) (x IO"-1 cm2) raGa 3/2 60-8 10823-8 10218 019 " G a 3/2 39-2 ' 6820-7 12984 012 Fig. 1. Gallium crystal structure. The double lines con-nect nearest neighbours. where // = 0-0785. v = 0-1525. The projection of the structure on to the AC plane is shown in Fig. 2. Tiie structure may be visualized as a line of 'diatomic molecules' lying with their axes in the AC plane at an angle /} of 16° to the C axis and displaced by half a lattice spacing along both B and C from the next line whose axes are rotated through 180° about the Caxes. Each atom has one nearest neighbour at 2-429 A and six others varying in distance between 2-709 A to 2-791 A . The two next nearest neighbours are considerably further away at a distance of 3-54 A . Fig. 2. Projection of the gallium unit cell onto the AC plane. Heavy lines correspond to atoms lying in one plane and light lines indicate atoms lying in planes a distance -J- B away. The various neighbours are indicated by the lettered vectors. T H E Q U A D R U P O L A R H A M I L T O N I A N The Hamiltonian describing the interaction between a nuclear quadrupole moment and an electric field, gradient together with the interaction between the magnetic dipole moment and a magnetic field H0[13,14]. = A { 3 / / - / ( / + 1) + irj (/+-' + IJ)} + lw„{L cos v + HK + / - e + i < t ) sin 9} (1) where A = g''Qg« • - f a " * ? " . 4 / ( 2 / - 1 ) ' 7 1 qa ' l<7xxl « \qm\ « k « | . ( 2 ) A N U C L E A R M A G N E T I C R E S O N A N C E S T U D Y 2339 / is the nuclear spin angular momentum, Q .is the scalar nuclear quadupole moment and 2-TTVI, = yHtt is the Larmor frequency. The coordinate system chosen is that of the principal axes system of the EFG tensor which then only has the diagonal components q2Z, q,,u and qxx. 17 is called the asymmetry parameter. 6 and d> are the polar angles specifying the orientation of the external magnetic field with respect to the principal system. The case of spin / = f in the absence of an external magnetic field is characterized by a single resonance line corresponding to a transition between two doubly degenerate energy levels with a frequency separation VQ^PVQ (3) where p = (1 + h ) 2 ) m and the pure quad-rupole frequency v°Q — 6Ajh, which cannot serve to determine either/} on? independent-ly. The application of a magnetic field however removes the degeneracy and the dependence of the spectra upon crystal orientation reveals the principal axes and the components of the EFG tensor uniquely, assuming that Q is also known. The frequencies for any value of 17 and correct to first order in the magnetic field are given by the expression [14] v= vQ±{m1±m2) v„ (4) where >"i = ip[(p +1 — i7)2sin20 cos2</> + (p+ 1 +17) 2 sin20sin20 + (2-p)2 cos20]1,2 m2 = |p[(p — 1 + 7))2sin20cos2(/>+ (p — 1 — 1 7 ) 2 sin20sin2(/)+(2 + p)2cos2(9]l/2. (5) In general four resonance lines will be ob-served and of these the angular variation of the 'inner pair' of lines (i.e. the lines which are least split from vQ by the magnetic field) con-tains sufficient information to determine accurately the required parameters. In par-ticular we may note the following for the 'inner pair'. // parallel to Z: v=vQ±v„ H parallel to X: v = vQ + vH{\--n)lp. (6) Thus the mean frequency for both cases is the quadrupole frequency and the frequency splittings AA- and Az between the 'inner pair' may be combined to determine the asymmetry parameter from the result A* = l~"n (7) Az p ' • It should be noted that this result does not require a knowledge of the magnetic field provided it is small. To obtain more accurate values of energy levels and frequencies we can solve the prob-lem exactly. The Hamiltonian (1) in the representation that diagonalized lz is given by the following matrix (8) where y = vHlv°0. In some cases, in par-ticular 0 = 0 = 0 (H0\\Z) and 0 = 90°, 0 = 0(H0\\X) the matrix is exactly soluble in closed form, but in all cases may be dia-gonalized by computor. E X P E R I M E N T A L D E T A I L S The experiments were performed with a modified Pound-Knight spectrometer in magnetic fields produced by a rotateable h cos 0 + y 77/V3}> V3 sin 0 e+i<" ^ = hvH j T)/V3y -cos0-y 2sin0e-w 0 I V3sin0e-'* \ 0 V3 sin 1 V3 sin 6 e+w> 2sin0e+'*cos0->' T)/V3y ** 7i/V3y -3cos(9 + y 2340 M . I. V A L I C and D . L L E W E L Y N W I L L I A M S Varian electromagnet. The specimens were single crystals of gallium grown from high purity 79 grade gallium supplied by Eagle Picher Industries Ltd. Cylindrical crystals of approximately f i n . dia. by f in . length were prepared by touching supercooled liquid gallium contained in a teflon mould with a seed crystal of the desired orientation. X : r a y back reflection Laue photographs enabled an unambigous determination of the crystal orientation [15] and the crystal was mounted in a coil of the spectrometer so that the magnetic field could be rotated in a plane containing two crystallographic axes. The coil was of No . 40 wire separated from the crystal surface by a layer of 0-0005 in. thick Mylar. This configuration was such that the oscillator could be made marginal at all temperatures. The applied magnetic field of 300 G was measured with a proton n.m.r. probe and a lower field of 60 G also used was determined by comparing the results with those obtained at the higher field. The experi-ments were carried out at the three tempera-tures 4-2°, 77° and 2 8 5 ° K . The former two temperatures were obtained with liquid helium and liquid nitrogen respectively whereas the latter was obtained with a water bath whose temperature was regulated to better than 0-1 °K in view of the rapid variation of the pure quadrupole resonance frequency with temperature close to the melting point of gallium. The magnetic field was modulated and the resonance signals observed with phase sensitive detection. R E S U L T S Figure -3 shows the observed ""Ga reson-ance signals at the three temperatures. The signal shapes reveal a mixture of absorption and dispersion modes brought about by the phase shifts involved in the penetration of the radio frequency field into the bulk metal and they are also slightly distorted by the modula-tion amplitude used. The resonance frequencies of the important lines for a rotation of the magnetic field in a plane perpendicular to the B crystallographic axis are shown in Fig. 4 and it is immediately seen from the occurrence of two sets of (3) 11368-2 ..I Fig . 3. m G a resonance signals at (1) 285°K (2) 77°K and (3) 4-2°K. The numbers specify the frequency scale in kc/s. A N U C L E A R M A G N E T I C R E S O N A N C E S T U D Y 2341 Bl H0 (C,H0); T-4-2°K; H0 =60 G 20 60 4, ( DEGREES) 100 140 Fig. 4. ""Ga spectrum as a function of orientation of the magnetic field H0 in the AC crystal plane. resonances that two physically non-equivalent sites exist in the crystal. Since the two sets of lines only differ in relative orientation it follows that the EFG tensors differ only in the orientation of their principal axes and have the same coupling constants and asymmetry parameters. The form of the orientation dependence together with considerations of crystal symmetry indicates the positions of the Z and X axes of the two EFG tensors shown in the figure. The positions of the two Z axes were determined more precisely in the following manner. A calculation shows that the resonance frequency corresponding to the (?—»—£) transition for the case of the magnetic field parallel to the Z principal axis is given by the following expression which omits only a negligibly small term inrj4. "L 3\l-4(vJv%r) (9) where 2TTV„ = y//»( 1 + K) and K is the Knight shift. A misalignment of the magnetic field by a small angle 0 from the Z axis causes a deviation J in the resonance frequency which is given by the following expression obtained by a perturbation calculation 5 = 1 2 1 - ( « „ / « £ ) o02 (10) where 0 is measured in radians. It is obvious from the expression that 8 is highly sensitive to misalignment when v„ is close to vQ, but can be made comparatively insensitive by a suitable choice of v„h°Q. ln principle, with the reasonable assumption that K is field inde-pendent, two sets of measurements at different magnetic fields would suffice to determine both A' and the misalignment. Fortunately gallium has two isotopes with rather different quadrupole moments so that with the assump-tion that K is not isotopically sensitive it is possible to avoid extensively retuning the spectrometer and yet observe the ,i:,Ga and 71Ga resonances under quite different condi-tions of equation (9). The angular variations are shown in Fig. 5 and are again displayed in Fig. 6 as a function of 6' 1. In practice the M . I. V A L I C and D . L L E W E L Y N W I L L I A M S 8 (DEGREES) Fig . 5. High field angular dependence of ( i - » — i ) transition frequency close to the Z principal axes. The field strength H0 is quoted in kg. ;.:hi shift is initially determined from the ly varying results on 7'Ga and is then j to predict the angular variation for (iMGa. _;tuse of the very rapid angular variation ;,e case of w,Ga, first order perturbation v , - y is inadequate and an exact calculation :cessary. The misalignment determined is i used to correct the 7lGa Knight shift and procedure repeated for self-consistent .sits. In the results shown, a misalignment .;•:'! deg is observed but in the whole set of tsurements the misalignment was never • erved to exceed 3-5 deg. The values of K . e been published in a preliminary note[19]. o results show that both Z principal axes i n the AC plane of the crystal and enable a hiy accurate determination of their relative entation. It should be noted that the stal has reflection symmetry in the AC ne which requires that the B axis of the stal to be one of the principal axes and that the other two axes lie in the AC plane. As is seen from the results the X and Z axes lie in the AC plane and the B axis is coincident with the Y principal axis for both crystal sites. This was checked experimentally by studying a crystal rotation in the AB plane. In this case the patterns from both sites are observed to coincide and the 'inner pair' splitting along the B axis is equal to that along the Z axes. Even though the magnetic field used was small, its influence on the determin-ation of T] from equation (7) was not entirely negligible and this is illustrated by the re-sults and theoretical variation shown in Fig. 7. The fact that this correction was considered at all indicates the high precision of the results. The accuracy in the determination of 17 using equation (7) is governed by the measurement of the splittings A,v and Az which is unaffected by the distortions in the line shape due to modulation or the effect of A N U C L E A R M A G N E T I C R E S O N A N C E S T U D Y , 2343 v [k'e/sec] 11630 11600 71 Ga H- • 8-932 II200 -2>" II150 perturbation theory exact theory .(80 = 0 and 0-2) experiment 11080. "O 0-5 10 B [degrees] F i g . 6. Angular dependence of ( i —> —A) transition frequency as a function of the square of the misalignment between the magnetic field and the Z principal axis. the skin depth in mixing the modes. It is merely necessary to measure between any two equivalent points on the resonance lines. However the determination of the pure quad-rupole frequency does require a line shape analysis to determine the true resonance frequency. The line was found to be well represented by a Gaussian absorption mode and the resonance frequency was determined by fitting the results to a modulation broadened superposition of dispersion and absorption modes. Figure 8 shows a compari-son between the observed resonance at 77°K and the theoretical Gaussian line shape for an equal mixture of modulation broadened absorption and dispersion modes. The line shape at 4-2°K revealed a higher proportion of dispersion mode in contradiction to the predictions of Allen and Seymour[18]. This however merely reflects the fact that the anomalous skin effect theory is only strictly valid in zero magnetic field and we may infer from our result that the imaginary part of the surface impedance is dominant at this field strength. It is also possible for the line to be 1-2300 l-2250r- - | 7 9 1-2150 — 1-2100 Y X |0 - 2 Fig . 7. Effect of the magnetic field on the determination of 17. The experi-mental points correspond to the temperatures (I) 285°, (2) 77° and 4-2°K. 2344 M . I. V A L I C and D. L L E W E L Y N W I L L I A M S Fig. 8. Comparison between a theoretical Gaussian line shape and the experimen-tal line at 77°K. 1. Absorption mode. 2. Dispersion mode. 3. Equal mixture. The points are taken from the experimental line. influenced by saturation effects but the spin lattice relaxation time for gallium is probably sufficiently short that this may be disregarded. The values of the parameters determining the EFG are tabulated in Table 2 and the orientation of the two sets of principal axes is shown in Fig. 9. D I S C U S S I O N The EFG tensor in a metal at a particular nuclear site has been considered to arise from three sources[21], the first denoted by ZqhM, from the nuclear and electronic charges ex-ternal to an atomic sphere around the nucleus in question (where Z is the normal valence of gallium) and second q l o c . from the conduction electrons within the atomic sphere, and finally the third which results from the distortion of closed shell electrons at the atomic site may be accounted for in terms of Sternheimer antishielding factors yx and RQ so that the total EFG tensor q t o l is given by q t o t = (1 - y K ) Z q l o t t . + (l-R<p) q ioc. (11) We have evaluated q l a t t . for gallium in the Tabic 2. Electric field gradient tensor results Temperature 2y (°K) (kc/sec) V (deg) c/a 4-2 U312-2±0-4 0-179±0-00i 48-7 ±0-2 1-6903 78 11253-2 + 0-4 0-179 ±0-001 48-6 ±0-2 1-691 285-3 10877-2 ±0-5 0-171 ± 0 002 48-2 ±0-2 1 -6955 A N U C L E A R M A G N E T I C R E S O N A N C E S T U D Y 2345 F ig . 9. Orientation of principal axes with respect to the crystal structure of gallium. atom at the origin is excluded from the sum-mation. Indices / and j stand for any of three symbols A, B, C and (qi a t t.) ij is the (ij) com-ponent of the q i a t t . in the crystallographic coordinate system. From the calculations it follows that (qiatt.).« = (qiatt.)cB = 0 and therefore the B crystallographic axis is one of the principal axes, which of course also follows from the symmetry of the unit cell. Instead of tabulating the calculated com-ponents (qia„.Xu, (qiatt.W (qiatt.)cc and (qlatL)AC we present in Table 3 q l a t t. in diagonalized form already. N is the total number of contributing atoms inside the sphere of radius R in units of B. The lattice parameters used for this calcula-approximation of a uniform distribution of conduction electrons by direct machine summation of the expression 8 +N (q.att.)u=2 S 3jt M ' (/ ,/ a / 3 )x M J (/M) ~ ^ r j j l j ok ) X *•„*(/,/,/,) U 2 ) where the summation over m includes the eight atoms in the unit cell designated by C i . A>7 '3) and the second summation includes all the atoms lying within a spherical volume o f radius R which is a multiple of the shortest lattice parameter at a given temperature. The 2346 M . 1. V A L I C and D . L L E W E L Y N W I L L I A M S Table 3. Ionic contribution to the EFG tensor as a function of number of atoms considered R N (Qlat t.kv (qiatt .) l 'V (qiatt.)zz A a l t . TJlatt. 3 532 - 0 02147 - 0 00329 0 02476 31-7 -0-734 15 66105 - 0 02087 - 0 00378 002464 3 0 0 -0-694 25 306450 -0-02094 - 0 00374 0 02468 30 0 -0-697 tion are those from Table 4 at 4-2°K. The principal components of q l a t t . are all in the units of IO2'1 c m - 3 . /3 l a u . is the angle in degrees between the Z principal axis of q l a t t . and the C crystallographic axis. 7 j l a t t . is its asymmetry parameter. Table 4. Lattice parameters of gallium T A B C (°K) (A) ( A ) (A) 4-2 4-5156 4-4904 7-6328 78 4-516 4-493 7-636 285-3 4-5195 4-5242 7-6618 Since the convergence of the sums is quite fast (in contrast to the case of indium, etc.), alternative methods for calculating qlatt. have not been considered. In order to see to what extent our ionic model describes the crystal-line field gradient in gallium we compare in Table 5 the experimentally determined para-meters characterizing the E F G at 4-2°K with those calculated assuming an antishielding factor of —9-50 for free + : ! G a ions 17 and Z = 3, from the relation. vQ = e*0(\ - y « ) Z ( q l a t t . ) „ ( 1 - H a i t . 2 ) l , 2 / 2 A . (13) In particular one should note that the X axis is parallel to the B axis in the ionic model rather than the Y axis in the experimental case which we have indicated by assigning a negative sign to it. The most obvious feature of the experimen-tal results is the very small temperature varia-Table 5. Comparison of ionic cal-culation and experimental results for the EFG tensor E F G vQ (Mc/sec) Piat t . TJlatt. Experiment 11-31 24-3 -0-179 Ionic 2-82 3 0 0 -0-697 tion of both the asymmetry parameter and the orientation of the principal axes. The quad-rupole frequency exhibits the temperature variation shown in Fig. 10 and the results may be combined with the asy'mmetry parameter to give the changes in all E F G components between 285-3° and 4-2°K. We have cal-culated the temperature dependence of qi a l t . using the temperature dependent values of the lattice parameters from Table 4 and assuming the parameters // and v to be the same for all [ M %ec] O 100 200 300 T no Fig . 10. The temperature dependence of the TOGa N Q R frequency. The solid line represents the results of Pomerantz[4J; the experimental points are from the present work. A N U C L E A R M A G N E T I C R E S O N A N C E S T U D Y temperatures. Results for R = 22 are included in the Table 6. All the parameters remain practically constant over the whole tempera-ture range so that thermal expansion alone does not explain the small changes observed. However the explicit temperature dependence of the quadrupolar interaction which arises from the influence of the lattice vibrations is generally the dominant effect[20] and it is possible that this could account for the whole temperature variation. 2 3 4 7 qexp and obtain an alternative solution for qdiff. with an angle /3diff. of 31-6° and 17 = 0'03 and also sensibly temperature independent. However, in view of the close correlation between the angles mentioned above it is likely that our first value is the correct one. In conclusion, our results suggest that the dominant contribution to the gradient is qloc. Watson, Gossard and Yafet[21] estimate that the main contribution to qloc results from the interaction between the ionic and the Table 6. Temperature variation of the ionic contribution to the EFG tensor T ^ U l a t t . (°K) (q , a l , . ) . Y .V (qiatt.)zz (qiatl.)l 'V Aatt. r f l a t t . (Mc/sec) 4-2 - 0 - 0 2 0 9 0 - 0 - 0 0 3 7 3 0 0 2 4 6 3 30-0 - 0 - 6 9 7 2-77 78 - 0 - 0 2 0 8 6 - 0 - 0 0 3 7 7 0-02462 30 0 - 0 - 6 9 4 2-77 285-3 - 0 - 0 2 0 6 3 - 0 - 0 0 4 5 3 0-02516 30-0 - 0 - 6 4 0 2-80 We may now proceed in the spirit of the assumed model and determine the difference tensor qdiff. = (l —^)qioc. = qexP.— (i — y»)Zqiatt. (14) and the results are given in Table 7. It is very interesting to note that the angle /3diff. is al-most equal to the angle /3pair. between the nearest neighbour direction and the C axis which is 15-9 deg. The temperature variation of /3pair is not available in the present literature and it would be of interest to see if its tempera-ture variation is the same as /3diff. Since we can only determine the absolute value of qexp., we can reverse the sign of Table 7. Temperature variation of qdiff. T (°K) (qdlB.) .VA- (q<iirr.)i-j- (qdifrJzz ''Jdifr. 4-2 1-210 2-123 - 3 - 3 3 3 15-7 0-274 78 1-205 2-113 - 3 - 3 1 8 15-6 0-274 285-3 1 1 1 5 2 0 5 9 - 3 - 2 2 8 1 5 0 0-276 electron states at the Fermi Surface, but in view of the small temperature variation in contrast to the large variation of the Knight shift and the spin lattice relaxation time, and also the very small difference between normal and superconducting states, this seems un-likely. The close correlation between /3diff. and /3pair is suggestive of a strong interaction in the direction of the nearest neighbour, and /3pair is suggestive of a strong interaction in the direction of the nearest neighbour, possibly an indication of covalency. In this connection it would be valuable to have more information on the temperature variation of /3pair.. Clearly a detailed theoretical treatment is required together with a measurement of the Knight shift tensor to provide additional information on the Fermi Surface states. The large difference between the Knight shift in solid and liquid gallium [19] is certainly indi-cative of a large p-electron contribution. Measurements on the Knight shift are pre-sently in progress and preliminary measure-ment indicate a large anisotropy. We hope to further discuss these points in a subsequent paper when measurements are complete. 2348 M . I. V A L I C and D . L L E W E L Y N W I L L I A M S Acknowledgements —We wish to acknowledge our debt to D r . E . P. Jones who first suggested the study of gallium to one of us ( D . L I . W ) in 1964. A t that time Dr . Jones had already made a preliminary observation of the spectrum at 4-2°K in one crystal plane, but was unable to complete his work in the time available. We are also grateful to Dr . S. N . Sharma for his assistance with the spectrometer and one of us ( M . V . ) is indebted to the Universi ty of Brit ish Columbia for a Universi ty Studentship and to the Nat ional Research Counc i l of Canada for the subsequent award of an N R C Studentship. R E F E R E N C E S 1. J O N E S E . P. and L L E W E L Y N W I L L I A M S D . , Phys. Lett.X. 109(1962). 2. R I S E R S. R. , Thesis. Universi ty of California, Berkeley (1965-unpublished). 3. K N I G H T W . D . . H E W I T T R. R „ and P O M E R A N T Z M . . / ' / m . / i e i J . 104.271 (1956). 4. P O M E R A N T Z M . , Thesis. Universi ty of California. Berkeley (1955-unpublished). 5. K I S E R S. R. and K N I G H T W. D.,Am. Pliys. Soc. Bull. 7.613 (1962). 6. H A M M O N D R. H . and K N I G H T W . D . , Phvs. Rev. 120 ,762(1960). 7. K U S H I D A T . and B E N E D E K G . B . , Am. Phys. Soc. Bull. 167(1958). 8. H A M M O N D R. H . . W I L K N E R E . G . and K E L L Y , , G . M..Phvs. Rev, 143.275 (1966). 9. C O R N E L L D . A..Phvs.Rev, 153 .208(1967). 10. V A L I C M . I. and L L E W E L Y N W I L L I A M S D . , In preparation. 11. B A R R E T T C . S.. Advances in X-ray Analysis, Vol. 5. p. 33 (1961). 12. B R A D L E Y A . J . . Z . Kallogr. 91A, 302 (1935). 13. A B R A G A M A . . Principles of Nuclear Magnetism, Chap. V I I . Oxford Universi ty Press .Oxford(1961) . 14. D A S T . P. and H A H N E . L . . Solid State Physics, Suppl. I. Chap. I. Academic Press. N e w Y o r k (1969). 15. DEAN C. Phys. Rev. 96. 1053(1954). 16. Y A Q U B M . and C O C H R A N J . F . , Phys. Rev. 137, A l 182 (1965). 17. C H A P M A N A . C . R H O D E S P. and S E Y M O U R E . F . W . . Proc. phvs. Soc. 70. 345 (1957). 18. A L L E N P. S. and S E Y M O U R E . F . W..Proc.phys. Soc. 82. 174(1963). 19. V A L I C M . I.. S H A R M A S. N . and L L E W E L Y N W I L L I A M S D..Phvs. Lett. 26A, 528 (1968). 20. K U S H I D A T . . B E N E D E K G . B . and B L O E M B E R G E N N . . Pins. Rev. 104, 1364(1956). 21. W A T S O N R. A . . G O S S A R D A . C . and Y A F E T Y . , Phvs. Rev. 140. A375 (1965). 22. S T E R N H E I M E R R. M . , Phys. Rev. 130, 1423 (1963). -159-APPENDIX D MODULATION EFFECTS ON THE NMR LINE SHAPES IN METALS 1. Introduction The effects of magnetic field modulation on signal shape and amplitude have long been of interest in NMR where phase-sensitive de-tection methods are used. The output of the phase-sensitive detector is proportional to the coefficient of the first harmonic term in Fourier expansion of the resonance line shape. The degree to which the experi-mental derivative is a faithful reproduction of the true derivative de-pends on the amplitude (h ) of the magnetic field modulation. If the modulation is "small" compared to the resonance linewidth, the recorder tracing is an accurate representation of the derivative. If i t is large the signal is "overmodulated". There is an optimum modulation value for the maximum amplitude signal, depending on the line shape and modulation. NMR line shapes obtained from metal single crystals reveal a mixture of absorption and dispersion modes (Chapter VII) brought about by the phase shifts involved in the penetration of the radiofrequency field into the bulk metal [55], The problem in the NMR work with single metal crystals is then to find the correct resonance field. The theory and the experiments for the Lorentzian, Gaussian (pure modes) and Dysonian lines were already performed [73]. Some addi-tions which can be used in the work with metal single crystals are in-cluded in this Appendix. The theoretical calculations are presented in a form of a series of computer plots with the discussion limited just to the figure captions. Some experimental work has been performed in our -160-laboratory (Ga - present work, Mg [8], Tl [7], Sn [5], Cd [6]j; however, the results have not yet been put into a final form. 2. Mathematical Treatment The normalized unsaturated line shapes in dimensionless notation are given by: Absorption mode Gaussian line Lorentzian line 2 X " ( X ) E " X / T I (1) 1 + YT Dispersion mode X ' ( X ) ± e " / T T / eU'dtt (2) X/\Tff 2 -X2/TT ( U 2 X Vrf o 1 + X 2 The standard notation for the two absorption modes is given by: 2,2 r-4 Jin 2(H-H )' g(H) = L. (^1_2)2 exp % TT % 2 (Gaussian line) and (3) i V i ^ ^ - k , 2 ( 4 ) " {h\y + (H-H O) 2 2 (Lorentzian line) ( g(H) dH = 1 (5) H, is the true line width at the half maximum absorption signal inten-sity; H q is the magnetic field at the resonance center. If one modulates the magnetic field with h cos cut and scans i t linearly ° m and very slowly through the resonance with a field H (t), the total field 3 -161-is then H(t) = H a(t) + 1^ cos wt . (6) A signal will be generated proportional to g[H(t)] = I - ( i 2 ^ ) 2 ex P i r-4 in 2[H (t)-H +h cos cot]2 a o m (Gaussian line) (7) g[H(t)]=i * * H * 2- <8> (i> H i ) +[H (t)-H +h cos cut] (Lorentzian line) The sweep rate is assumed to be very small so that H (t) remains con-3 stant over time interval 2TT/CU (i.e. , H - H = Hr is constant over this a o ° interval. A Fourier analysis yields then g(t) = | - (-^-V T Q n C H ^ V H . ) c o s n • <9> Hi, TT n n -g HI o \ " n=0 (Gaussian line) H g(t) = ^o w ^ ' V c o s n • ( 1 0 ) (Lorentzian line) The Fourier amplitudes are given by +TT r-4 in 2[H +h cos v ] z ) Q (Hj,,h , H ) = - ( cos n 0) (exp 2 m n 2 m 6 TT _) I Hi )d0, (11) (Gaussian line) 0 (Hj,,h H ) = ~ cos n 5-I }dj . (12) ^ * m 6 " . i k% V +(H6+Hu> cos tf)2) (Lorentzian line) -162-where \) = cot. One can see that are independent of modulation fre-quency. The quantity of experimental interest is , since, in phase sensitive detection, the recorded signal is proportional to Q^ . In metals at low temperatures, one rather detects to reduce large mag-netoresistance pick-up. One wants to know thus: (a) The positions of the extreme of (or Q^ ) as a function of h /H, . That is, at what values of HR/R, is there a maximum (or minimum) of (or Q^ ) for a given value of hm/iy At these extrema, H& = | ( H F I ) P | ; |QI|= 1(0^ |. Since the measured linewidth between slope extrema, 6H m e a g, is equal to 21(Hg)^ J (the separation of the ma-ximum and minimum of Q-^) one obtains SlL^gg as a function of h from this result, m (b) The value of (Q ) , the maximum derivative signal as a 1 P function of h is also of interest. One wants to know m the optimum modulation amplitude for maximum derivative signal. (c) The shapes of the signals and the positions of the reso-nance field are of very interest for the work in metal single crystals where the signals are mixtures of absorp-tion and dispersion modes g(H) = [1-b] . X'(H) + b . X"(H) . (13) b is the mixing parameter. All the calculations and plots were done on IBM 7044 computer. A very general program involving numerical (single and double) integration for -163-the calculation of the quantity "r"TT = I I cos n^ [l-b].x'(X+Y cos v))+b.x"(X+Y cos dd (14) -TT and using normalized expression for x" a n ^ x" given by Equation (1) and Equation (2) was very easy to adjust to get different results. In all plots X means (unless otherwise specified) : X = 2 \|TT J2n 2 Ho/X = 2.952 H . / ^ for Gaussian line (15) o -g o -g X = H^/Hj^ for Lorentzian line (16) For comparison with the experiments one has to replace Hi by 6H using relations: Line Pure Model Mixed Modes(b=%) Gaussian 6H = .850 H2 6H = .847 E1 (17) "2 "2 Lorentzian 6H = .577 Ux 6H = -51T ^ -164-FIGURES 3. LIST OF ILLUSTRATIONS Gaussian line 1. Normalized (a) absorption (x"), (b) dispersion (x') mode, (h = 0), m 2. Fi r s t derivative (4~) of normalized (a) x"> (b) x'» dx (c) (0.5 x1 + 0.5x"). (h m - 0). 3. 4~ (x") for different modulations h /H ; (a) 0.1, (b) 0.2, (c) 0.3, dx m (d) 0.5, (e) 0.78,"(f) 1.0, (g) 1.5, (h) 2.0, (i) 2.5. '*. {—(x")> vs. h /H normalized to' { 3 - (x") >• * ^ +* dx • max m u dx A max for the optimum modulation. j. 4 - (x') f o r different modulations h /H. ; (a) 0.1, (b) 0.2, (c) 0.4, dx m >i (d) 0.5, (e) 0.7, (f) 1.0, (g) 1.2, (h) 1.5, (i) 2.0. 5. ~ (C l X" + C2x') with (a) C 1 = 1, C 2 = 0, (b) 3/4, 1/4, (c) 1/2, 1/2, (d) 1/4, 3/4, (e) 0, 1. (h = 0). m 1 ' 4~(0.5 x" + 0.5x') for different modulations h /H, ; (a) 1.5, (b) 1.0, dx m % (c) 0.5, (e) 0.2, (f) 0.1. The curve (d) presents normalized — (0.5X" + 0.5X') with h m = 0. 8' ^- (0.5x" + 0.5x') for different modulations h /Hj ; (a) 1.5, (b) 1.0, dx m >^ (c) 0.5, (d) 0.2. The curves are normalized with respect to the corresponding curve (e) with h = 0. m 9. (4-(0.5x" + 0.5x')} vs. h /H, normalized to {^ -(O.Sx" + 0.5X')} dx max m \ dx max for the optimum modulation. 10. (2.51 x 6H) vs. h /H, . 6H is the peak-to-peak distance of the m -1 ^ (0.5X" + 0.5X'). 11. (2.51 x 6H) vs. delta. Delta is the distance from the true resonance frequency of ^ (C^x" + ^ ^X1) to the cross-over point with the base-line . -165-d 2 Second derivative (—j) of normalized (a) x"> (b) X'J ( C ) dx (c) (0.5X" + 0.5X') . (h = 0). 2 m — 2 (C XM + C 2 x') with (a) C± = 1, C 2 = 0, (b) 3/4, 1/4, (c) 1/2, 1/2, dx (d) 1/4, 3/4, (e) 0, 1. (h =0). , 2 „ (0.5x" + 0.5x') for different modulations h /Hx ; (a) 2.0, dx (b) 1.5, (c).1.0, (d) 0.5, (e) 0.2. The curves are normalized with respect to the corresponding curve (f) with h =0. J ^-y (0.5x" + 0.5x') for different modulations h /l\ ; (a) 0.2, (c) dx 2 m 0.5, (d) 1.0, (e) 1.5, (f) 2.0, (g) 2.5. The curve (b) is normalized ,2 (0.5x" + 0.5X') with h = 0. , A m dx Lorentzian line 4~ (x") for different modulations h /HL ; (a) 1.0, (b) 0.8, (c) 0.6, dx m =s (d) 0.5, (c) 0.450, (f) 0.40, (g) 0.35, (h) 0.3, (i) 0.2, (j) 0.1. 4-(x') for different modulations h /H, ; (a) 0.1, (b) 0.2, (c) 0.3, dx m "5 (d) 0.4, (e) 0.5, (f) 0.6, (g) 0.8, (h) 1.0. 4-(x") for different modulations h /Hj ; (a) 2.5, (b) 2.0, (c) 1.5, dx m (d) 1.0, (e) 0.8, (f) 0.5, (g) 0.3, (h) 0.1. The curves are normalized with respect to the corresponding curve (i) with h^ = 0. 4~ (X') for different modulations h /Hx ; (a) 2.5, (b) 2.0, (c) 1.5, Q X TCI -"2 (d) 1.0, (e) 0.80, (f) 0.5, (g) 0.3, (h) 0.1. The curves are normalized with respect to the corresponding curve (i) with h^ = 0. 4-(0.5x" + 0.5xf) for different modulations h /H, ; (a) 2.5, (b) 2.0, dx m % (c) 1.50, (d). 1.0, (e) 0.80, (f) 0.60, (g) 0.50, (h) 0.45, (i) 0.4, (j) 0.35, (k) 0.2. -166-4-(0.5Y" + 0.5Y') for different modulations h /H. ; (a) 2.5, (b) 2.0, dx m ^ (c) 1.5, (d) 1.2, (e) 1.0, (f) 0.80, (g) 0.5, (h) 0.1. The curves are normalized with respect to the corresponding curve (i) with h = 0 •m (4- (0.5xM + 0.5x')} vs. h /H, normalized to {^-(0.5X" + 0.5x')} dx max m h dx A A max for the optimum modulation. 6H vs. h /H, for 4-(0.5x" + 0.5X'). m % dx 6H vs. delta. -167--168-fiBSORTION D E R I V A T I V E .0 ,166 .333 .5 .66S .033 1.0 ZT CD -691--170--171--172--173-E'~ 3 •H -174--175-in i n ZZ UJ J o ZD I CL. s: inCC rx ID in M 3 00 •H — r 9S9' 3 a n i n d i j y w £ - \ m d 991 0' -176--177--178-to Figure. 13. -180--181--182--183--184--185--186--187-PERK-PEAK AMPLITUDE .333 .5 J J-.665 .833 Ul Ul -88T--189--190--191-APPENDIX F COLLECTIVE ELECTRON EFFECTS ON NUCLEAR SPIN-LATTICE RELAXATION RATE AND KNIGHT SHIFT A correct treatment of the exchange enhancement effects on the Korringa relation between the Knight shift and spin-lattice relaxation time is discussed by Narath and Weaver f104]. The hyperfine interaction between a nuclear spin 1^ and the conduction-electron spins. . \t- ~ ^ C87T) y y h2 f. I t . «(?..) , (El) a ' e n l ^ i i i 3 gives rise to a shift in the field for nuclear resonances AH = (y b)" 1 I A.-^ . (E2) 'n L kkjacj k,0 . + • k,a m = - 1/2 is the'spin-proiection quantum number and f> is the a k,a Fermi distribution function for a state of wave vector l£ and spin orientation a - The generalized contact hyperfine coupling parameter A-*->, , , is defined in terms of the amplitudes y-*- (0) of the kkjoa' k,a conduction electron wave functions at the nucleus by * (ES^I The fluctuations associated with the transverse part of the hyperfine interaction give rise to a nuclear spin-lattice relaxation rate r+°° , T i " 1 = (2/h 2) dx cos(u) TT) < { X " ( T ) T ( 0 ) } > , ( E 4 ) J - c o L where CJ is the Larmor frequency and the bracketed term is the symmetrized, transverse autocorrelation function of the hyperfine interaction. -192-. In the i n d e p e n d e n t - p a r t i c l e approximation, Equations (E2) and (E4) reduce t o the standard form, provided only t h a t the hyperf.i i n t e r a c t i o n parameters are independent of a: 0 K 0 - AH/H = (2/| Y eh|) H h f g X ° , (E5) (Ti o ) " 1 = W „ 2 k k B T [ H h f s N°(E F)1 2 , (E6) where H, = - ( Y . h) 1 < A^Y > hfs .'n kk < A - ~ > " (E 7 ) and the average being taken over a l l s t a t e s at the Fermi l e v e l . Since the uniform c o n d u c t i o n - e l e c t r o n s p i n s u s c e p t i b i l i t y x° and the d e n s i t y of e l e c t r o n i c s t a t e s a t the Fermi l e v e l N°(E ) are r e l a t e d i n t h i s approximation by Equation (44) one obtains immediate' the K o r r i n g a r e l a t i o n given by Equation (35c). In the presence of e l e c t r o n - e l e c t r o n i n t e r a c t i o n s (E2) and (E4) can be s i m p l i f i e d e a s i l y only i f two important assumptions can be made: (1) The e f f e c t s due t o e l e c t r o n i c exchange and c o r r e l o can be represented by an e f f e c t i v e p o t e n t i a l V( q) whose magnitude-depends a t most on the momentum t r a n s f e r - h q between i n t e r a c t i n g e l e c t r o n s . (2)The wave-vector dependence of the h y p e r f i n e i n t e r -a c t i o n parameters can be approximated by A=> t = A-> . This assumption should be v a l i d only f o r metals i n which the Fermi surface i s n e a r l y s p h e r i c a l and i s contained w i t h i n the f i r s t B r i l l o zone. These severe r e s t r i c t i o n s can only be j u s t i f i e d i n the case o the a l k a l i metals and t o a l e s s e r extent the noble metals. With these two assumptions the Knight s h i f t K i s s t i l l g i ven by Equation (E3) provided t h a t the uniform s p i n s u s c e p t i b i l i t ; Y_ q i s r e p l a c e d by i t s many-body value x • Narath expresses the enhancement of ^ i n terms of the u s u a l Stoner f a c t o r (1 - a) \ -193-where a = 2V(0)/(. Y h ) xo • ' i s i n t e r a c t i o n P a r a D i e 1 : e r - Th e Knight shift may then be written as K = K o ( l - a ) - 1 . (E8) The spin-lattice relaxation time can be written in terms of susceptibility function ^ (^>tJ ) with the expression P L Tl~1 = h k B T (Y e ^)~ 2 b |A-| 2 0)^1 xJJCq,^) . (E9) The imaginary part of the susceptibility function Xp(q>wL), has the form of the reduced susceptibility [105] and in the l i m i t of small oj is given by 106 L X^(q,wL) = X^. (q,^) [1 -(2V(q)/Y e 2h 2) X p > (q,0) ] ~ 2 (E10) where ^ q and^'p Q a^e the imaginary and real part of the independent particle susceptibility. Making use of the Kramers-Kronig relation [ 3] and expression (E10) in Equation (E9) yields T l _ 1 = h " 2 k B T ™ L _ 1 I l A q ! 2 [ 1 -(2V(q)/Y e 2h 2) X 0F(?)]" 2 x q ( E H ) f=> and E^> refer to the independent-particle states and F(q) is the linear dielectric function [107J. Expression (Ell) can be simplified for two limiting cases. Case (a): The range of the electron-electron interaction i s great enough that the enhancement of x" is only significant for vanishly -194-small q values i.e., V(q)o. V(0),v5(cf). In this limit the electron-electron interactions have no effect on T^ . The Korringa relatic Lon therefore becomes K 2T XT = S(l - a ) ~ 2 . (E12) Case (b): The range of the interaction potential is of the § -function form i.e., V(q) - const. . In this limit the calculations yield T i " 1 = (TL o r 1 <[ 1 -xF(q)]" 2 > (E13) for a spherical Fermi surface. The pointed brackets denote an average over a l l q vectors spanning the Fermi surface. The Korringa relation appropriate for the present case is K?TT = K( a )~ 1 (E14) 2 2 K(a ) = <[ 1 - a F(q)]~ > ( l - a) • (E15) 2 Narath compares the experimental products K T^ T for the alkali and the noble metals with the predictions of the in-2 dependent particle model. In every case the ratio K T^ T/S exceeds unity by a substantial amount. Assuming that the observed deviations are caused entirely by electron-electron interaction effects, one 2 can use the measured K T^ T products to calculate the enhancement of the spin susceptibility for the two cases. From the comparison i t is concluded that the inclusion of an exchange-enhanced spin-lattice relaxation rate in the Korringa relation requires con-2 siderably larger values of- a to bring the theoretical K T^ T -195-products into agreement with experiment theory i n the case when only the Knight s h i f t i s assumed to be exchange-enhanced. The int e r a c t i o n parameters a can be estimated very accurately f o r L i andNa making use of the measured conduction-electron spin s u s c e p t i b i l i t i e s f o r these metals. The & values obtained from such a c a l c u l a t i o n f a l l i n the region predicted by the two cases. One i s therefore forced to conclude that the 5-function potential model for the conduction electrons leads to an overestimate of the associated s p i n - l a t t i c e relaxation rate. Thus, the electron-electron interactions, at least i n L i and Na, have a non-zero range and hence cannot be accurately represented by a wave-number-independent interaction constant.
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Nuclear magnetic resonance study of single crystals...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Nuclear magnetic resonance study of single crystals of gallium metal Valic, Marko Ivan 1970
pdf
Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.
Page Metadata
Item Metadata
Title | Nuclear magnetic resonance study of single crystals of gallium metal |
Creator |
Valic, Marko Ivan |
Publisher | University of British Columbia |
Date Issued | 1970 |
Description | The nuclear magnetic resonance spectrum of gallium metal has been investigated in single crystal specimens in both low and high magnetic fields from 4.2°K to the melting point (T[subscript MP] = 300°K) and then extended to the liquid phase. Precise determinations for the two non-equivalent nuclear sites have been made of (a) the electric field gradient (EFG) tensor and (b) the Knight shift (K) tensor. The relationship of these results to the crystal structure of gallium metal is discussed in detail. The isotopic Knight shift in the solid, K[superscript sol; subscript iso] increases linearly with temperature from (0.132±0.004)% at 4.2°K to (0.155±0.004)% just below T[subscript MP]. In the liquid, just above T[subscript MP], K[superscript liq; sucscipt iso] = (0.453±0.003)% and decreases very slowly with increasing temperature. These results are discussed in terms of the Korringa relations in the solid and liquid phases, respectively. Implications of these results are developed with regard to changes in the electronic structure of gallium upon melting. The angular dependence of K[subscript an] agrees with the predicted angular dependence for the Knight shift anisotropy in an orthorhombic environment and is described with two anisotropy parameters defined as K₁ and K₂. No K- anisotropy is found along the Y (B crystal) axis. The principal axes of the K tensor have different signs from the quadrupolar principle axes. K[subscript an] (X) is found to be very large, i.e. K[subscript an] (X)/K[subscript iso] = -18% at 77°K, and temperature dependent. Increasing the temperature (T) from 4.2°K, K[subscript an] (X) increases to a maximum at 77°K and then slowly decreases as T approaches T[subscript MP] where it still retains a large value. This paradoxical behaviour is assumed to be a result of particular and unusual details of the gallium Fermi surface. |
Subject |
Gallium |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-06-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085462 |
URI | http://hdl.handle.net/2429/35039 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
Download
- Media
- 831-UBC_1970_A1 V35.pdf [ 9.33MB ]
- Metadata
- JSON: 831-1.0085462.json
- JSON-LD: 831-1.0085462-ld.json
- RDF/XML (Pretty): 831-1.0085462-rdf.xml
- RDF/JSON: 831-1.0085462-rdf.json
- Turtle: 831-1.0085462-turtle.txt
- N-Triples: 831-1.0085462-rdf-ntriples.txt
- Original Record: 831-1.0085462-source.json
- Full Text
- 831-1.0085462-fulltext.txt
- Citation
- 831-1.0085462.ris
Full Text
Cite
Citation Scheme:
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>
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.831.1-0085462/manifest