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A nuclear magnetic resonance study of the quadrupolar interaction in dilute aluminum alloys Campbell, Graham Roderick 1972

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c A NUCLEAR MAGNETIC RESONANCE STUDY OP THE QUADRUPOLAR INTERACTION IN DILUTE ALUMINUM ALLOYS by GRAHAM RODERICK CAMPBELL B.Sc., University of Waterloo, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1972 In present ing th is thes is in p a r t i a l f u l f i l m e n t o f the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y sha l l make i t f r e e l y a v a i l a b l e for reference and study. I fu r ther agree that permission for extensive copying o f th is t h e s i s for s c h o l a r l y purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or p u b l i c a t i o n o f th is thes is f o r f i n a n c i a l gain sha l l not be allowed without my wr i t ten permiss ion . Department of P h y s l ftg  The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date July ?$. 197?. - i i -ABSTRACT S a t e l l i t e l i n e s o r i g i n a t i n g from the f i r s t nearest neighbours have been observed on the N.M.R. spectrum i n d i l u t e single c r y s t a l samples of Al-Mn and Al-Cr. The f i e l d and orient a t i o n dependence of the s a t e l l i t e p o s i t i o n has revealed the quadrupole coupling constant and asymmetry parameter of \ = 60 + 1kHz and 7 = -0.01 t 0.005 f o r Al-Cr, and VQ = 19 t 1kHz, if = -0.08 t 0.02 f o r Al-Mn. An anisotropic magnetic perturbation of A K = -0.05 + 0.01% and A K = -(0.07 - 0.1)% i n Al-Mn i s at t r i b u t e d to a spin fooil density perturbation at the impurity s i t e s . The Al-Cr r e s u l t s agree with those of Janossy and Gruner [18] and Minier and Berthier [22] . The s a t e l l i t e l i n e observed i n the Al-Mn system was also seen by Launols and A l l o u l [z] , with appro-ximately the same values of >^  and A K. - i i i -TABLE OP CONTENTS PAGE ABSTRACT i i TABLE OP CONTENTS i i i LIST OP TABLES i v LIST OP ILLUSTRATIONS v ACKNOWLEDGEMENTS v i i i CHAPTER I. Introduction 1 II. Review of Theory k A. System Hamiltonian k 1. Impurity Po t e n t i a l Parameters 8 2. Eigenvalues of the System Hamiltonian 10 B. Radial Dependence of the E l e c t r i c F i e l d Gradient q 19 C. Knight S h i f t 26 1. Isotropic Knight S h i f t 26 III. Previous Investigations 32 IV. Experimental Aspects 3^ A. Experimental Apparatus 3^ B. Sample Preparation 46 V. Experimental Results and Discussion 51 VI. Conclusions 6? BIBLIOGRAPHY 68 LIST OP TABLES TABLE PAGE 1. Summary of the s a t e l l i t e s p l i t t i n g s i n the three p r i n c i p a l orientations, using equation (kZ). 16 - V -LIST OP ILLUSTRATIONS FIGURE PAGE 1. Pace-centered cubic structure of Aluminum. The impurity i s shown as • . The s h e l l s of nearest neighbours to the impurity are indicated by 1,2,3,... . 14 2. S a t e l l i t e s p l i t t i n g s due to a f i r s t order quadru-polar perturbation ("^  =0) as a function of the orientation of the magnetic field.© i s the angle between the [ooi] axis and H0* i n the (110) plane. 17 3. Relationship between the r a t i o of the s p l i t t i n g s i n the [00l| and \ l l l j d i r e c t i o n s and the value ofy . 18 4. Scattering of a Bloch wave by a solute atom. I IT I = 21 5 . Co-ordinate transformation. 21 6. Asymptotic value of q vs k p r . The positions of the neighbouring s h e l l s are indicated by arrows. 36 7. Experimental apparatus. 44 8. Copper sample holder. 49 9. (a) Typical trace i n Al-Cr showing high and low side s a t e l l i t e s , as indicated by the arrows. Recorded at 1.2°K and 8.1kG, with a 30 second time constant. The sweep rate was increased by a f a c t o r of two f o r t h i s i l l u s t r a t i o n . 52 (b) Typical trace i n Al-Cr showing one well resolved - v i -s a t e l l i t e below the main l i n e . Recorded at 1.2°K and 15.5kG, with a 30 second time constant. 53 10. S a t e l l i t e s p l i t t i n g versus the magnitude of the magnetic f i e l d , HQ, i n Al-Cr, with HQ i n the [ l l i ] d i r e c t i o n . 5^ 11. S a t e l l i t e s p l i t t i n g s as a function of the orien-t a t i o n of the magnetic f i e l d , In Al-Cr at 15«5kG. © i s the angle between the [00lj axis and HQ i n the (110) plane. 55 12. (a) T y p i c a l trace i n Al-Mn showing high and low side s a t e l l i t e s as indicated by arrows. Recorded at 1.2°K and 5.1kG, with a 30 second time constant. The sweep was increased by a fac t o r of two f o r th i s i l l u s t r a t i o n . 58 (b) Typical trace i n Al-Mn, showing a we l l resolved s a t e l l i t e above the main l i n e . Recorded at 1.2°K and 15.5kG with a 30 second time constant. 60 13. S a t e l l i t e s p l i t t i n g versus the magnitude of the magnetic f i e l d HQ, i n Al-Mn, with "HJ m the [ i l l ] d i r e c t i o n . The arrows Indicate data points that have been transfered from below the main l i n e to above the main l i n e . © indicates data taken with HQ i n the [ooij d i r e c t i o n . \ indicates data taken with HQ* i n the [ i l l ] d i r e c t i o n . 6 l 14. S a t e l l i t e s p l i t t i n g s as a function of the orien-t a t i o n of the magnetic f i e l d i n Al-Mn at 9.0kG. - v i l -© i s the angle between the (boij axis and HQ i n the (110) plane. © indicates data points correc-ted f o r the d i f f e r e n t i a l Knight s h i f t . 62 15. From Launols and A l l o u l [2]} . S p l i t t i n g of the high f i e l d peak of t h e i r s a t e l l i t e l i n e . The s o l i d l i n e s indicate d i f f e r e n t i a l Knight s h i f t s of A K = -0.076% and A K = -0.09#, and a zero f i e l d s p l i t t i n g of 19 1 1 gauss. A = Zero f i e l d s p l i t t i n g of the s a t e l l i t e observed i n t h i s i n v e s t i g a t i o n . B = A plus a k gauss d i p o l a r broadening correction. 66 - v i i i -ACKNOWLEDGEMENTS I g r a t e f u l l y acknowledge the guidance of Dr. D. LI. Williams throughout the course of t h i s work. His valuable i n s t r u c t i o n , dedication, and enthusiasm were a constant source of encouragement. I would also l i k e to thank Dr. M.I. V a l i c f o r his assistance with the experimental apparatus. Dr. Aktar of the Department of Metallurgy and Dr. H.J. Trohdahl provided help-f u l suggestions during the development of the sample prepara-t i o n technique. CHAPTER I INTRODUCTION If a foreign atom i s placed in a pure metal, there is usually a redistribution of electronic charge in the vi c i n i t y of the impurity of asymptotic form f i r ) ^ A cos(2k,r -v ( 1 ) The constants A and S are determined from the phase shifts of p a r t i a l waves scattered from the impurity potential. The oscillating charge density results in an electric f i e l d gradient of the impurity potential with a z-component given by V z 2= ecp c o S ( 2 H r r + S) ( g ) and x- and y-components related to the asymmetry parameter i? . The nuclear magnetic resonance method provides a technique for examining these features. If the quadrupolar interaction between the electric f i e l d gradient and the quadrupole moment of the host nuclei is a small perturbation on the Zeeman Hamiltonian, distinct s a t e l l i t e lines appear superimposed on the host spectrum, q and ip can be determined from the dependence of the satellites* position on the magnitude and orientation of the d.c. magnetic f i e l d . These effects were f i r s t investigated by Rowland [ i j by measuring the decreasing intensity of the main line as a function of increasing impurity concentration in powder -2-samples. The r e s u l t s were expressed i n terms of a "wlpe-out number" representing the number of host nuclei per Impurity atom whose resonance i s s h i f t e d outside the mam l i n e . This number gave an estimate of the range of the f i e l d gradient, but no absolute value f o r q.. S a t e l l i t e l i n e s were observed i n several a l l o y systems using powder or f o i l sam-ples (for example: Launols and A l l o u l [2] ), but i t was not possible to f i n d 1? or to determine the s h e l l of host nuclei that was producing the s a t e l l i t e . Recently, the use of single c r y s t a l samples by Jorgensen, Nevald and Wllliams [3l has resulted i n a thorough study of the i n t e r a c t i o n and an ac-curate assignment of the contributing s h e l l i n some a l u -minum a l l o y s . If the solute atom i s a non-transition element, then the phase s h i f t £ i s determined l a r g e l y by the phase s h i f t s S9 , £, of the & - 0, J2 = 1 p a r t i a l waves. However, f o r tran-s i t i o n metal (3d) solutes, the ^ = 2 phase s h i f t S£ domi-nates since the = 2 p a r t i a l wave undergoes resonance scat-t e r i n g with a v i r t u a l state on the impurity atom. In general, £ z > o e , S, , and thus, one would expect a stronger e f f e c t f o r the 3d solutes. The purpose of t h i s work was to determine the e l e c t r i c f i e l d gradient parameters i n the a l l o y s Al-Mn and Al-Cr by using single c r y s t a l samples, and to assign t h e i r observed s a t e l l i t e s to the appropriate nuclear s i t e . The r e s u l t s f o r these systems could then be compared with r e s u l t s f o r non--3-t r a n s i t i o n solute a l l o y s of Al-Mg, Al-Ga, Al-Ge as reported i n [3] . This thesis i s presented i n the following way. Chapter II contains a review of the theory of the quadrupo-l a r i n t e r a c t i o n i n metals, a de r i v a t i o n of the asymptotic form of the r a d i a l dependence of the f i e l d gradient, and a discussion of the Knight s h i f t . The previous investigations of other experimenters are discussed i n Chapter I I I . A desc r i p t i o n of the experimental apparatus and sample prepara-t i o n i s given i n Chapter IV, The experimental r e s u l t s and a discussion are given i n Chapter V. CHAPTER II REVIEW OP THEORY The i n t e r a c t i o n of i n t e r e s t occurs between the e l e c t r i c quadrupole moment of the nucleus and the e l e c t r i c f i e l d gradient (E.P.G.) of the el e c t r o n i c charge d i s t r i b u t i o n at the nucleus. The e l e c t r i c quadrupole moment i s due to the non-spherical shape of the nuclear charge d i s t r i b u t i o n . The E.P.G. i s produced when an impurity atom i s placed i n a pure metal. The uniform fermi sea of electrons i s disturbed by the impurity and an el e c t r o n i c charge r e d i s t r i b u t i o n r e s u l t s . The e l e c t r o s t a t i c p o t e n t i a l produced by the charge perturba-t i o n i s expressed i n terms of i t s second derivatives as the e l e c t r i c f i e l d gradient, and a parameter characterizing the departure of the p o t e n t i a l from c y l i n d r i c a l symmetry. A. System Hamiltonian The Hamiltonian of the system consists of two componentse the Zeeman Hamiltonian and the quadrupolar Hamil-tonian. The Zeeman H a m i l t o n i a n ^ represents the i n t e r -a c t i o n between the t o t a l magnetic moment of the nucleus yu. and the applied magnetic f i e l d H 0. The in t e r a c t i o n energy, H> , Is expressed as a Hamiltonian, ^---^•TC= -*KI*-H: ( 3 ) where I Is the angular momentum operator and y i s the -5-gyromagnetic r a t i o . If Eq i s applied along the axis z, i n a frame oxyz, then The eigenvalues of ^ M are E m , the Zeeman l e v e l s , where Em=-V^H0m > m=X3I-l>...-X. (5) The unperturbed t r a n s i t i o n s between these energy l e v e l s correspond to the ce n t r a l l i n e i n the magnetic resonance spectrum. The quadrupolar Hamiltonian can be developed as follows. C l a s s i c a l l y , the int e r a c t i o n can be expressed very simply, following S l l c h t e r L7Q . If yo("r) represents the nuc-l e a r charge density, and V(r) the p o t e n t i a l due to external sources, then the i n t e r a c t i o n energy E i s e= v m d t ( 6) where dtt i s an element of volume. The integrand can be si m p l i f i e d by expanding V(r) = V(x,y,z) i n a Taylor's series at ~r = 0. That i s , V<-> = V £ ^ \ v ± i ^ . ^ - \ .... ( 7 ) where i = x,y,z , 3 = x,y,z. Set , . Now E has become = I + II + III. (8) Term I represents the e l e c t r o s t a t i c energy of the nucleus as a point charge. Assuming that the nucleus' centre of mass and centre of charge are coincident, term I I , representing - 6 -the electrio dipole moment of the nucleus, vanishes. Term I I I , the electric quadrupole term, i s of interest here. By defining a set of quantities Q J J , term I I I may be simplified further. Set This may be re-written as which can be substituted in term I I I to give Assuming that there i s no electronic charge at the nucleus, V(x,y,z) must obey Laplace*s equation ^ L ' c = ° ( 1 2 ) t Thus, the classical energy due to the electric quadrupolar Interaction i s . . u<* 4 l) ^ ( 1 3 ) The quantum mechanical H a m i l t o n i a n , , can be derived from E Q by replacing the nuclear charge density f> with the operator p1 , defined as / 3 ' l , ) , i ^ S C r * - ^ ) (14) where k = 1,..N over the N discrete nuclear particles of charge q^ = e for the protons, and q k = G otherwise. The quantities Qj^ must now be re-written as ] • ( 1 5 ) n r o t o n s ' - 7 -Pollowing the same arguments used i n the c l a s s i c a l d i s -cussion, the quadrupolar i n t e r a c t i o n Hamiltonian can be written When dealing with resonance phenomena, the nucleus may be described using the angular momentum representation. The nuclear eigenstates are characterized by the t o t a l an-gular momentum I, the 21+1 components of angular momentum with quantum number m, and other quantum numbers p. The form of can be extracted by considering matrix elements of Q J J i n t h i s representationt i . e . by evaluating <XmplQ!.\x'my > (17) In a magnetic resonance t r a n s i t i o n I = I'. Assume f o r the purposes of t h i s argument, that p = p*. The matrix elements can be re-written using the Wigner-Eckart theorem, as < X m p\ftV \ X m > > = C < X r o ? \ | . ( l c X J + I . T ^ - ^ T l | X m / p > ( 1 8 ) where C i s a constant. C can be evaluated exactly using the r e s t r i c t i o n s m=m* = 1 , i = J = z . Then, using equation (15) f o r Q ^ j , < X X p \ e | ( 3 ^ - v - k ^ \ x X p > - C < X i : p | 3 T ; - x M x X p > ( 1 9 ) p r * o s = c ( 3 i * - i ( i * n ) C X ( 2 X - 0 By convention, the quadrupole moment of the nucleus i s defined as e Q = < X X p | e ! ( 3 a £ - v N 0 | T X p > . (20) p r o " t o r > S -8-Solving equation (19) f o r C, the r e s u l t i s C - e Q " X ( 2 X - V ) (21) T h u s , ^ Q can be written i n the angular momentum representa-tions as The other non-zero term i n the classi c a l . e x p r e s -sion f o r the e l e c t r o s t a t i c i n t e r a c t i o n energy (equation (8)) i s simply a sca l a r quantity, which can be neglected. CV^ Q characterizes the i n t e r a c t i o n between the nucleus and the impurity p o t e n t i a l . Thus, the system Hamiltonian i s W ^ V t , (23) where j \ i s expressed with respect to the system Gxyz. If the Zeeman in t e r a c t i o n i s large with respect to the quadrupolar i n t e r a c t i o n , t h e n m a y be treated as a small perturbation on the system. This r e s t r i c t i o n i s s a t i s f i e d i n the experimental investigation i n t h i s thesis. The eigenvalues of the perturbed system can be found using f i r s t - and second-order perturbation theory. 1. Impurity Po t e n t i a l Parameters The cartesian tensor V j j m equation (11) i s sym-metric and t r a c e l e s s , since i t obeys Poisson's equation. If the axes are chosen to coincide with the p r i n c i p a l axes -9-XYZ of V 1 ; j , then Jem ^ X l i = 0 (2*) where A = XYZ and m = XYZ. If ^  i s re-written i n terms of OXYZ, then V 7 f f e [ ^ - ^ W - i , ) * ^ 3 ^ n ] ( 2 5 ) The three remaining terms of the tensor can be expressed i n two parameters. The p r i n c i p a l axes OXYZ can be chosen such that I I I Then, the e l e c t r i c f i e l d gradient parameter, eq, i s defined as e c \ = V 2 2 (27) and the asymmetry parameter as T| ( 2 8 ) irv- now becomes ^ r 3i*-K3>i) + ±?(i>r)] ( 2 9 ) where T + = X * + l 1 * • X - = T x " i r 3 ( 3 0 ) with OXYZ as the p r i n c i p a l axes of V ^ . The asymmetry parameter r e f l e c t s the departure of the tensor V^JO. from a x i a l symmetry. In keeping with the l a b e l l i n g of the p r i n c i p a l axes, o f 1} - 3-with -7 * 0 corresponding to the case of a x i a l symmetry. -10-2. Eigenvalues of the System Hamiltonian The system Hamiltonian i n equation (23) can be re-written i n terms of a set of parameters describing the tensor vlm» following Cohen and Reif [5], as K = - 3 * H 0 I • J2—_. f ( 3 I*-I')V* ( I I . I J )V , + d v i i w * i a v * r v 1 # $• +> • 4 1 J (3D where, V l K - ^ i i V , ^ (32) The sets of axes used may be denoted £ , with Eq along 0a i n t h i s set, and as j>p which are the p r i n c i p a l axes of the tensor V ^ . y(5 i s written i n terms of ^  . T o simplify f H the discussion to follow, the dimensionless quantities can be defined as where the superscript denotes the co-ordinate axes and " 4 r H Now, equation (31) may be re-written i n terms of j . The eigenvalues of the system can be found by c a l c u l a t i n g the matrix elements of ^ s using second-order perturbation theory. The r e s u l t i s where C m i s the unperturbed Zeeman energy, and u m and m are the f i r s t - and second-order corrections to Um 11-produced by the quadrupolar Interaction ^ Q . The r e s u l t i s (36) where • I f i ' l V - ^ + Z . - O ] (37) ! / , » 3 e ^ Q , a . m i l ) . z i ( 2 X - i ) l i (38) The t r a n s i t i o n frequencies are e a s i l y found from these eigenvalues as - I | r „ H | 2 - ( » a m ( m - 0 - - 4 * + 6 ) J (39) The angular dependence of the t r a n s i t i o n f r e -quencies can be found by considering the co-ordinate trans-r H . p formation properties of the quantities . Suppose that £ i s c a r r i e d i n t o ^ by the Euler angles Then 5^ . can f P be written i n terms of j by * J " . I D C * V < . f t * L ^ £f. (40) where the u terms are expressed i n terms of the spherical harmonics. In <. , V p = 0 t i -12-Then, by using equation (40), the r e s u l t i s L* J (^2) In second order, with = 0, and m = \S corresponding to the (i-+-£) t r a n s i t i o n , The angles .A* specify the orientation of the applied f i e l d H 0 with respect to the p r i n c i p a l axes of the e l e c -t r i c f i e l d gradient tensor, and correspond to the polar angles . q and ip can be found using equation (42) above by measuring the s a t e l l i t e s p l i t t i n g i n any two p r i n c i p a l orientations. In the d i l u t e a l l o y s (less than 1 At% solute) under consideration here, most of the host nuclei do not experience the impurity p o t e n t i a l V(x,y,z). Thus, )A9 vanishes and )% s- only. These nu c l e i contribute to the main l i n e of the N.M.R. spectrum. However, the host nuclei adjacent to a s u b s t i t u t i o n a l impurity atom are affected by V(x,y,z) at distances up to several l a t t i c e constants away. These nuclei have resonant frequencies given by equations (39), (42), (43), and as a r e s u l t , t h e i r contribution i s displaced away from the main l i n e . Thus, the main l i n e w i l l be broadened, or, i f the i n t e r a c t i o n i s strong enough, s a t e l l i t e l i n e s w i l l appear adjacent to the main line,. -13-For aluminum, " f ^ m - f ' (44) To f i r s t order, as In equation (42), i t can be seen that the frequency s h i f t f o r the (£-*-£) nuclear t r a n s i t i o n vanishes. Thus, only the remaining four t r a n s i t i o n s are affected. Equa-t i o n (43) reveals that the (|-*-i) Is s h i f t e d i n second order. t This s h i f t i s proportional to ( HQ ) " 1 , and can thus be iden-t i f i e d e a s i l y . The s a t e l l i t e spectrum that i s observed must be interpreted i n terms of the face-centred cubic c r y s t a l struc-ture of aluminum shown In Figure 1. There are twelve nearest neighbours situated i n <TlO^ -type d i r e c t i o n s . For these, c r y s t a l symmetry requires that the p r i n c i p a l axes of the E.F.G. tensor be [ l i d ] , [lT6] , and [OOl] f o r the (110) s i t e . S l m i l a r i l y , the next nearest neighbours are located In <100"> -type d i r e c t i o n s , and, the f i e l d gradient tensor i s a x l a l l y symmetric (four-fold symmetry) with the axis along [OOl] f o r the (001) s i t e . In f i r s t order, f o r the f i r s t nearest neighbours, and q =0, the s a t e l l i t e s p l i t t i n g > £ A as a function of the orie n t a t i o n of the magnetic f i e l d can be predicted as follows. Suppose that the magnetic f i e l d HQ i s rotated about the [lio] d i r e c t i o n i n a plane normal to i t , from the [OOl] to [llOJ . Now, 8 Is the angle between "HQ* and the radius vector f o r the neighbour atom i n question, and 4* i s the angle between HQ and the z-axis of the c r y s t a l l a t t i c e . With HQ along [OOl] , -14-Figure 1. Face-centered cubic structure of Aluminum. The impurity i s shown as • . The s h e l l s of nearest neighbours to the impurity are indicated by 1,2,3,... . -15-eight of the twelve nearest neighbours have O = 45° and four have © = 90°. The terms i n equation (42) can now be evaluated. For O = 45°, ( 3cos 0 - 0 = J (8 A l atoms). For © = 90°, { I cos e - - \ ( / + A 1 atoms). The t r a n s i t i o n s between the (tj t~) , (--J-*- ±-§ ) Zeeman le v e l s of the nucleus produce (m - it) values of tl9 tz. The r e s u l t s can be summarized i n the following Table 1. Thus, there are eight atoms with s p l i t t i n g i VQ ; there are twelve atoms with s p l i t t i n g \ i>a ; and there are four with s p l i t t i n g . Similar calculations can be made f o r the f u l l set of d i r e c t i o n s between (polj and [lio] , The r e s u l t s are summari-zed by p l o t t i n g the s p l i t t i n g s as a function of the d i r e c t i o n of the magnetic f i e l d i n the (110) plane, with© measured from the [00l] d i r e c t i o n . Figure 2 shows the positions of the s a t e l l i t e l i n e s , with respect to the main l i n e , as calculated using f i r s t - o r d e r perturbation theory, with -7 = 0. The s p l i t -tings are shown i n units of gauss, which can be e a s i l y conver-ted to a frequency s h i f t . For the (110) s i t e , the p r i n c i p a l axes X,Y,Z, are t>Ql| , [llo] and [lid] . Similar p l o t s can be produced tor any set of neigh-bours, f o r V ^ 0 a n d using the second order expressions. Figure 3 indicates how the [ooij and [ i l l ] s p l i t t i n g s can be used to determine *vj> . -16-"HQ Dir'n 0 0 Number of Atoms C W 9 - I ^ + ? sin*ecos20) Possible S p l i t t i n g s [ooij 90 0 4 -(1- ?) M»(l- , ) **<l-<? ) *5 90 8 ) * * < l - 9 ) i * t t - ? ) [ i l l ) 35.3 90 6 (1-0.34? ) M*U-0.34? ) *^Q(1-0.349 ) 90 5^ .6 6 -(1-0.65? ) ^(1-0.65? ) iMa(l-0.65ri ) [no] 0, 180 0 2 2 2 90 0 2 - U - ? ) VQ ( 1 - ? ) 1^(1-7 ) 60 45 8 i 4 Vo * * * 5 Table 1, Summary of the s a t e l l i t e s p l i t t i n g s l n the three p r i n c i p a l orientations, using equation ( 4 2 ) . S a t e l l i t e S p l i t t i n g (gauss) 2A A 0 -A -2A Figure 2; S a t e l l i t e s p l i t t i n g s due to a f i r s t order quadrupolar perturbation (17 as a function of the orientation of the magnetic f i e l d . 0 i s the angle between the [00lj axis and EQ i n the (110) plane. -18-Figure 3. Relationship between the r a t i o of the s p l i t t i n g s i n the [p°lj a n d C 1 1 ^ d i r e c t i o n s and the value of y . -19-B. Radial Dependence of E l e c t r i c F i e l d Gradient q The e l e c t r i c f i e l d gradient i n a metal i s produced hy a r e d i s t r i b u t i o n of the e l e c t r o n i c charge over a l o c a l region around a point defect. If the defect i s an impurity atom, i t w i l l produce a charge r e d i s t r i b u t i o n due to i t s d i f f e r e n t valency and the l o c a l l a t t i c e s t r a i n . The mechanism f o r the "valence e f f e c t " can be f o r -mulated i n terms of scattering theory. The impurity atom acts as a scattering centre f o r the conduction electrons of the metal. As a r e s u l t , the various outgoing waves i n the p a r t i a l -wave analysis are phase-shifted r e l a t i v e to the incoming waves. The phase s h i f t s r e s u l t i n a b u i l d up of e l e c t r o n i c charge around the impurity, and thereby screen out the excess nuclear charge, leaving the system e l e c t r i c a l l y neutral. The screening charge Z i s r e l a t e d to the various phase s h i f t s by the F r i e d e l sum rule [jS\, ^ J V J (45) where are the phase s h i f t s f o r the H - t h p a r t i a l wave and cr denotes the spin. The spin-up and spin-down phase s h i f t s are i d e n t i c a l i f the impurity i s non-magnetic. The asymptotic form of the f i e l d gradient can be r e a d i l y obtained following Kohn and Vosko [ y l . 1 Let the 1 The problem was simultaneously treated by Blandin and F r i e d e l [8] using plane waves. - 2 0 -solute produce a perturbing potential U'(r) acting on the Bloch conduction electrons. If U(r) i s the periodic poten-t i a l of the unperturbed l a t t i c e , the Schroedinger equation can be written as (46) The asymptotic form of called ^ , corresponding to an incident Bloch wave 4^  and an outgoing scattered wave is described by - i V f - ( j > J * ^ * ? 4\Cr) i m as formulated in the theory of plane wave scattering (Schiff [9]). f (k' 7?) depends on the nature of the scattering centre. As seen in Figure 4, using the assumption that the energy surfaces are spherical. Assume that f (k* ^k) depends only on |"k*1 and the angle 0 between k and k', and thus set Now, f j r ( © ) can be expanded in terms of the phase shift using sperical harmonics as 2 i k * o \ / J ( 5 0 ) For large r, the excess electron density contributed by is -21-Pigure 4. Scattering of a Bloch wave by a solute atom. Aluminum Nucleus Figure 5. Co-ordinate transformation. -22-where (52) (53) The t o t a l electron density at r due to a l l T K , such that k < k F , Is V k < k P ( 54) After introducing the Bloch form of the unperturbed conduc-t i o n electron wave function . — * t k • <r <fU - ix^)e. (55) VnC»-} turns out to be, (56) Note that ^ n ^ - * ^ , and that S " r \ ( r ) i s o s c i l l a t o r y i n r. This o s c i l l a t i n g e l ectron density produces e l e c t r i c f i e l d gradients which i n t e r a c t with the quadrupole moments of the host n u c l e i . For plane waves with - 1,5""^ ) m equation (51) becomes r-ree r 3 ( 5 7 ) Elections where A and are given i n equations (65) and (66) below. The e l e c t r o s t a t i c p o t e n t i a l at r due to ©"v-\(V) i s - 2 3 -glven by (58) where and r " i s the variable of integration. V(r*) can be expanded in spherical harmonics. The term of interest here i s V 2C?')= J P 2 . ( C O * © ' ) ( 5 9 ) where @ is defined in Figure 5, and S n ( T £ * - f * 0 ( 3 c o S a e ' - | ^ v - ' . (60) The f i e l d gradient at r ^ consists of q* in equation (60) and another contribution due to the effect of core polariza-tion. Foley and Sternheimer [io] show that the nuclear electric quadrupole moment induces a large quadrupole moment in the electronic shells of the surrounding atoms. This effect adds a term with the same form as q*, multiplied by the anti-shielding factor, y(*-'), which i s the ratio of the induced quad-rupole moment to the nuclear quadrupole moment. Thus, the f i e l d gradient seen by the nucleus at r n i s v - ( s " ( 7 ; ^ , V i + y ( r o ] ( 3 c o 3 a e , n ) d i v / ) L (61) This equation can be simplified by replacing r by "r^ in equation (60) at large r, and by expressing Sr\ (*\+r ') l n terms -24-of the phase shift as, Using this expression for S<r\(r^-v~r') , equation (61) becomes where the enhancement fact or, *x , i s and i y a. YN (62) 3 >Tn 3 (63) (64) (65) (66) Kohn and Vosko \j\ give an example calculation of the phase shifts using the Priedel sum rule and the residual resistance parameters. Usually only the f i r s t two phase shifts S 0 , S , , are appreciable for non-transition metal solutes. An estimate of o< i s also shown. The contribution of the local lattice strains to the electric f i e l d gradient i s discussed by Beal-Monod and KohnpLl] . -25-When an impurity atom i s placed i n the l a t t i c e , the host atoms, denoted by i , are displaced a distance u^. The poten-t i a l at r due to the displaced atom i s Svj-t(T*°) - T t t . f t ^ F * - " ^ ( 6 7 ) where r denotes the host atom positions and f < ^ = .jEL g C ^ ) (68) g(r") has the symmetry of the l a t t i c e . Using a method s i m i l a r to Kohn and Vosko»s (V] , they obtain an o s c i l l a t o r y c o n t r i -bution proportional to ^ s t r a i n <* <*© (69) where e*^ — ^> with V*e as Poisson*s r a t i o , v Q the volume per atom and S o . / / a i s the f r a c t i o n a l change of the l a t t i c e constant per unit atomic concentration of impurity. Since i s small, the contribution of <l strain t o t h e f i e l d gradient i s neglected i n the asymptotic region. A c a l c u l a t i o n of t h i s e f f e c t i n Cu-Ge gives a n e g l i g i b l e contribution of one part i n 10^ to the e l e c t r i c f i e l d gradient. Thus the contribution from the s t r a i n e f f e c t i s usually neglected. -26-C. The Knight S h i f t As i s we l l known, the resonance frequency of a nucleus i n a metal i s somewhat d i f f e r e n t from that i n a diamagnetic substance, as f i r s t observed by Townes et a l [12J. This so-cal-led "Knight Shift* 1 i s c l a s s i f i e d into two categories depending on the electron band which interacts with the nuclei v i a the hyperfine coupling. The i s o t r o p i c Knight s h i f t i s related to the coupling between the nuclei and the s-electron band, exhi-b i t i n g the spherical symmetry of the s-type wave functions. The a n l s t r o p i c Knight s h i f t a r i s e s from the i n t e r a c t i o n of the nuc l e i with non-s-bands. 1. Isotropic Knight S h i f t The i s o t r o p i c Knight s h i f t i s assumed to aris e from the extra f i e l d at the nucleus due to the hyperfine coupling of the nuclear spins with the s-state conduction electrons. The electrons are p a r t i a l l y polarized by the applied f i e l d g i v i n g an electron-nucleus coupling of the form 3 e ^ (70) where are the magnetic moments associated with the electron spin and nuclear spin and R i s the l a t t i c e vector of the nucleus. Using the coupling summed over a l l nuclei and electrons i s -27-where , define the electron and nuclear positions. Due to the weakness of the hyperfine coupling, can be treated as a perturbation of the unperturbed Hamiltonian where n i s the nuclear Hamiltonian, including the Zeeman energy of the nuclei, and describes a gas of Interacting electrons. The unperturbed wave function i s taken as < It i s d i f f i c u l t to specify a rigorous electron gas wave function % due to the long range Coulomb interaction between electrons. It Is customary to make the approximation that the electrons are only weakly interacting, based on the theoretical j u s t i f i -cation by Bohm and Pines [l3J . Then, the individual electron wave functions are assumed to be of the Bloch form V~> • ° f " * 1 <?*> where U^C?)has the same periodicity as the l a t t i c e , and ^ % is a spin function. The interaction energy C can then be written from f i r s t order perturbation theory, where indicates integration over electron spatial and spin co-ordinates. Using equation (71) with the j-th nucleus i s N with ^j =° for convenience, the contribution from (Ee ~ ) j = " 3 ^ * ^ J e i » i ^ - * -* e e (74) Since the operator Sft^ti.) acts on one particle only, there are -28-no exchange terms, andH^ describing the electron gas can be expanded to give ^ „ [ V i ^ ' - I ^ t , ^ (75) If the f i e l d H 0 i s applied along the z - d i r e c t i o n , then the el e c t r o n spins w i l l be polarized along HQ a l s o . Then, 3^- ^ = ^ j e ^ (76) i s the only remaining term. At t h i s point, the summation jL^ i s changed to_£_ and an occupation function i s i n -eluded ( pO£>S^=. \ i f k.>s are occupied, and zero otherwise). Then, (E-^y becomes with ms = Afte r averaging t h i s expression over occupied states, pOtis) can be replaced by the Fermi function fOt»s), which i s also written as 1+ expDE-E F)/ k T] ( 7 8) where E i s the energy of an electron with wave vector k and spin s. A t y p i c a l term of (^ e n)- can then be written as (79) a f t e r summing over s. The quantity i n the square bracket can be written as - 2 9 -representlng the average contribution of state k to the ele c -tron magnetization. Then, the t o t a l z-magnetization of the electrons i s It i s convenient to define the t o t a l electron spin suscepti-b i l i t y by >* £ sT£ HO (82) and, f o r state k Then W o (83) (84) e "7* w On summing terms l i k e (79) over a l l k states, the t o t a l e f f e c -tiv e i n t e r a c t i o n f o r state j i s ( C v ^ ^ U ' v i ^ l " . ( 8 5 ) For non-free electrons, the constant energy surfaces i n k-space are not sp h e r i c a l , and take on other p e c u l i a r shapes depending on the magnitude of the l a t t i c e p o t e n t i a l . Thus, the iL* above can be performed by integrating over the sur-face E - £ » , with area A, and a l l the allowed k-vectors. A d i s -t r i b u t i o n function ± s defined such that g(- E£ ^*vZ* cLA (86) i s the number of allowed k-vectors i n a volume i n k-space be-tween corresponding surfaces and «^ »^. bounded by area dA. Also the density of state syo(E»\ on surface i s defined as 3 < V * > a l A (87) - 3 0 -Using these r e l a t i o n s = S * ^ V ^ A | V r t l ^ A (66) where %*">U^ has been assumed through i t s dependence on the Fermi terms f( t e>i) andf^'S-x) . Defining the average value of | 0 . 1 0 ^ over a surface E-»> = constant, by then (89) (90) The function % S ( ^ has non-zero values only near E L ^ C L E p i k-T since f o r both spin states are 100$ populated and f o r ^ j j * " * > E F neither spin state i s occupied. Assuming that < | « 4 - % t o * | J i s slowly varying over the region where ^ ' ( ^ i s non-zero, the integrand reduces to < l a u . l ^ J T t H ^ y o ^ ^ , - ' - e x ' e (91) Thus, the f i n a l r e s u l t f o r the i n t e r a c t i o n of the electrons with the j-th spin i s The form of t h i s Interaction i s equivalent to the i n t e r a c t i o n of the 3-th spin with a f i e l d A H given by -31-A H i s usually expressed as *\eo , which is called the Knight s h i f t , K, when related to frequency instead of f i e l d . It i s easy to see that the s-state electron den-sity at the nucleus can he calculated by measu-ring K. In particular, by measuring the di f f e r e n t i a l Knight shift ^ K of a sate l l i t e line, the s-state electron density at sites adjacent to an impurity can be found. This, of course, gives information about the nature of the charge redistribution in the v i c i n i t y of impurity. The treatment above considers only the hyperfine coupling due to the s-state electron spins at the nucleus. Another contribution fcd the shift comes from the core polari-zation effect. The polarized conduction electrons at the Fermi level tend to polarize the core electrons near the nucleus via the exchange interaction. A similar effect i s produced by any unpaired electron present in the atom, such as in the d-band of transition metals. -32-CHAPTER III PREVIOUS INVESTIGATIONS The quadrupolar interaction in metal alloys has been investigated in powder and f o i l samples by studying both the dependence of the main line Intensity on solute concentra-tion, and the sat e l l i t e structure of the host resonance spec-trum. The concentration dependence of the main line in-tensity was studied f i r s t . In a pure metal, a l l of the nuclei (within the skin depth) contribute to the main line intensity. However, when a small amount of alloying element is present the intensity i s reduced, since the host nuclei near the im-purity have their resonance shifted out of the main line by the quadrupolar Interaction. Bloembergen and Rowland [l4] analyzed this phenomenon in terms of a wwipe-out number", n, which equals the number of host nuclei per impurity atom which have their resonance completely displaced. If the f i e l d gradient, q, exceeded a value q.Q then the contribution from this site was displaced completely} i f q<q Q, i t contributed f u l l y , g i -ving rise to the so-called "all-or-nothing" model. If c i s the solute concentration, then (1-c) i s the host concentration. Thus, the intensity, I, from the alloy i s related to the inten-sity l 0 from the pure metal by i = ( l - c ) n . (94) -33-The expressions f o r the f i r s t - and second- order corrections to the resonant frequency are shown l n equations (42) and (43). In f i r s t order, a l l the nuclear t r a n s i t i o n s are s h i f t e d except the central (|-*-§) t r a n s i t i o n . In the second order, the ce n t r a l t r a n s i t i o n i s s h i f t e d also. As a r e s u l t , a f i r s t order wipe-out number, n s , i s defined which gives the number of host nuclei whose s a t e l l i t e t r a n s i t i o n s ( i . e . those other than ( i - * - i ) ) are displaced. For a stronger f i e l d gradient, the second order wipe-out number, n c , gives the number of host nuclei whose central t r a n s i t i o n s are sh i f t e d a l s o . The al l - o r - n o t h i n g model i s useful f o r determining the distance over which the quadrupolar i n t e r a c t i o n occurs, by means of the value of n. Although the model i s an appro-ximation, i t i s supported experimentally by the f a c t that a reduction i n i n t e n s i t y i s observed with the addition of a l l o y -ing elements without appreciable l i n e broadening. The s a t e l l i t e l i n e s have been observed by several investigators. In powder and f o i l samples, the e l e c t r i c f i e l d gradient can be calculated d i r e c t l y using equation (38), i f the s p l i t t i n g s are not inversely proportional to the mag-ne t i c f i e l d . The asymmetry parameter,-? , cannot be determined using non-single c r y s t a l l i n e samples, since an average i s taken over a l l angles. The e l e c t r o n i c structure of a l l o y s using t r a n s i t i o n - 3 4 -atom (3d) solutes l n simple metals has been the subject of considerable study l n recent years. The t r a n s i t i o n atom acts as an ordinary s c a t t e r i n g p o t e n t i a l , except that only the JL = 2 phase s h i f t s of the outgoing p a r t i a l waves are large. The reason f o r t h i s has been developed by Anderson [15] . A v i r t u a l bound d-state i s formed on the transition-atom impurity due to the mixing of the s-d o r b i t a l s . When a conduction electron i s Incident on the impurity with an energy (E p) near that of the v i r t u a l bound state, i t undergoes resonant scattering and the corresponding phase s h i f t f o r the resonant p a r t i a l wave i s large. Since the v i r t u a l states are d-states, the Si = 2 phase s h i f t , S 4 , dominates a l l others. A simple expression f o r S\ f o r the case where the t r a n s i t i o n impurity has no magnetic moment, i s i _ 1 ° (95) where N i s the number of electrons i n the u n f i l l e d d - s h e l l of the impurity. The work of B r e t t e l and Heeger Q.6] on the N.M.R. of A l 2 ? m d i l u t e a l l o y s of Al-Cu and Al-Mn i s of i n t e r e s t since copper has a f u l l d-band ( 3 d 1 0 ) and Mn has (3d^). A larger f i e l d gradient i s expected at solvent s i t e s i n Al-Mn than i n Al-Cu since S x i s much greater than and ^, . As ex-pected, they observe a strong decrease i n main l i n e i n t e n s i t y as the concentration Increases, due to the quadrupolar i n t e r -a ction. For small c, equation (94) can be approximated as -35-i = 1-nc. (96) The slope of t h e i r I / I Q versus c graph gives the t o t a l wipe-out number n, which i s n = 167 f o r Al-Cu impurities and n = 850 f o r Al-Mn. For the low concentration used, 0.049At.#Mn, 0.3At.#Cu, the l i n e width and shape at 1.6°K did not change as the f i e l d was varied, i n d i c a t i n g that there i s no l o c a l i z e d moment on the impurities. In the Al-Mn system, the v i r t u a l l e v e l overlaps the Fermi l e v e l , r e s u l t i n g i n a large value of 8* , and hence a large value of the f i e l d gradient q. The t h e o r e t i c a l value of q can be e a s i l y calculated using equation (63) and the va-lues of Sj. = 1.90,oc = 4.76, and 4> = 1.90, f o r Al-Mn. Figure 6 shows a p l o t i f q versus k p r . The o s c i l l a t i o n s r e t a i n an ampli-tude greater than the c r i t i c a l q Q to at lea s t the f o r t i e t h s h e l l of neighbours to the Impurity. The r e s u l t s can be checked by c a l c u l a t i n g a t h e o r e t i c a l value f o r the t o t a l wipe-out num-ber. The s a t e l l i t e wipe-out number i s found by summing the 21 -3 t o t a l number of host nuclei f o r which q >6.3x10 cm . Then a c r i t i c a l of q f o r the c e n t r a l t r a n s i t i o n , q^, i s calculated, and n c can be found. The t o t a l wipe-out number, n t , i s related to n s and n 0,by s c a l i n g each i n proportion to the i n t e n s i t y of the t r a n s i t i o n i n the main l i n e as n t = 26/35ns + 9/35nc. (97) B r e t t e l and Heeger obtain a calculated value of n t = 894, Figure 6. Asymptotic value of q vs kpr. The positions of the neighbouring s h e l l s are indicated by arrows. -37-which compares well with their experimental value of 850, Thus, for low concentrations ( < 0 . 1 At.%), the v i r t u a l hound state description provides a satisfactory picture for the electronic structure of Al-Mn. However, the absolute value of q cannot be found using this technique and no further i n -formation can be deduced about the quadrupolar interaction. Launois and Alloul [2] studied the Al-Mn system via the N.M.R. of both the A l 2 ? host nuclei and Mn55 impurity nuclei, m alloys of 0.2 At% to 0.7 At# Mn, the linewidth and position of the A l 2 ? resonance are unchanged with respect to pure Al. That i s , the Knight shift of most of the Al nuclei, 0 . l 6 l # , i s not affected in this range of impurity concentration. More importantly, they observe a high f i e l d s a t e l l i t e line which is completely resolved from the main line at high f i e l d s . A plot of the sa t e l l i t e line s p l i t t i n g versus H reveals that between 20kg and 30kg the s p l i t t i n g i s proportional to H. This i s a t t r i -buted to a d i f f e r e n t i a l Knight shift K of the A l 2 ? nuclei in the v i c i n i t y of the impurity of = -109% t 7%. KA1 One possible origin of this i s core polarization of the f i r s t nearest neighbours to the impurity by the d-electrons on the Mn. At fields below 20kg the s p l i t t i n g A H can be decomposed into A H = A H m + AHa + AHq -38-where A ] ^ i s the s p l i t t i n g due to -AK, ^ % represents the dipolar broadening and £Hq. i s related to the quadrupolar effects. A plot of AHq versus -p=- is a straight line of slope corresponding to 4^ - 2.4 MHz. n Thus, the observed sate l l i t e line i s caused by the second order s p l i t t i n g of the central transition. This reveals the general nature of the quadrupolar interactions in the sys-tem but the angular dependence of the lines cannot be studied, and hence, the asymmetry parameter cannot be calculated. The Mn55 results indicate a Knight shift of KMn = -2'6# - 0.1#> corresponding to a strong local d-susceptibility. Also, the linewidth of the Mn55 resonance increases rapidly with larger o, which i s presumably indicating that impurity-impurity inter-actions become important at C>0.1 At#. The Al-Mn and Al-Cr systems have also been investi-gated by Gruner et a l by measuring the intensity of the Al 2 ? mam line as a function of impurity concentration and magnetic f i e l d . For low impurity concentrations up to 300 ppm, the intensity ratio i s proportional to c, giving total wipe-out number of 1380 for Al-Mn and 1620 for Al-Cr. These can be analyzed using the "all-or-nothing" model, following Brettel and Heeger. In the concentration range 0.1 At% to 0.6 At#, - 3 9 -s a t e l l l t e components should be completely wiped out, and thus, the i n t e n s i t y should be described by the second order e f f e c t s (proportional to 1/H) and by a magnetic perturbation ( A K, proportional to H). They found that the i n t e n s i t y r a t i o was d i r e c t l y proportional to H i n the range 0 to 10kg. There are two contributions proportional to H. Since the phase s h i f t S t i s d i f f e r e n t f o r spin up and spin down electrons (due to the Impurity states) there i s a spin density disturbance around the impurity and a Knight s h i f t change of AVC _ \o -not c a t ( 2 ^ ^ T ~ ~ 4 - K - l a N 0 ^ r 3 ( 9 8 ) where i s the volume of the unit c e l l , N Q i s the density of s-states at the Fermi l e v e l , «< the enhancement f a c t o r , and A the width of the resonant state. This contribution i s dependent on the l o c a l i z e d Impurity s u s c e p t i b i l i t y . Also, the charge re-d i s t r i b u t i o n around the impurity r e s u l t s i n K = ( ZW?r)Z The slope of the graph of l / l 0 versus H i s one and a h a l f times as large f o r Al-Mn as f o r Al-Cr as are t h e i r respective s u s c e p t i b i l i t i e s . This reveals that the spin d i s -turbance mechanism i s responsible f o r A K i n these systems. Thus, t h i s work shows that the reduction i n i n t e n s i t y of the A l 2 ? main l i n e i s due to a f i r s t order quadrupolar i n t e r a c t i o n , which i s s h i f t e d by the spin perturbation at the impurity. The s a t e l l i t e l i n e structure of the A l 2 ? resonance -40-i n Al-Cr has recently been reported by Janossy and Gruner [l&Q. Using t h i n f o i l samples with 0.2 At% and 0.4 kt% Cr, they observe two s a t e l l i t e l i n e s whose positions do not change with H between 0 and 8kg. This r e s u l t corresponds to a pure f i r s t order quadrupole s p l i t t i n g of the (-"jr"~**-Jr ), ( - 2 ~**- ) nuclear t r a n s i t i o n s with no observable spin den-s i t y perturbation ( i . e . No d i f f e r e n t i a l Knight s h i f t ) at the observed s i t e s . Their assigned value f o r q i s found from 2 s 21(21-1 )h = 6 6 * l k H z » e q = 3 . 9 x l 0 1 3 ( c g s esu), which has been corrected by 2kHz f o r dipole-dipole and i n -homogeneous quadrupole broadening. The peaks of the s a t e l -l i t e l i n e occur at t and away from the c e n t r a l reso-nance frequency, or are s p l i t from the main l i n e by +61 gauss and +32 gauss respectively. They estimate that the s a t e l l i t e s are o r i g i n a t i n g from the seventh or greater s h e l l of impuri-t i e s , and that the asymmetry parameter i s n e g l i g i b l e . The core enhancement f a c t o r , oC , f o r A l as a host has been calculated by Pukai and Watanabe [19) , using where i s a plane wave and H^ * i s the true wave function f o r the conduction electrons, \~\t \ = k p. They choose a pseudopotential wave function as -41-where <j>^  are core wave functions specified by t, Nj^ i s a normalization constant, W 0(<p is the pseudopotential of the host atom, and £ runs over a l l lattice sites R„ , the total JL X of which i s N. As a result, ot = 22.8 The angular dependence of the quadrupolar inter-action i s most effectively studied using single crystal samples. The major d i f f i c u l t y Involves obtaining a sufficiently high signal-to-noise ratio on the sate l l i t e lines. Usually, the d i -rection of the applied f i e l d i s rotated with respect to the aligned sample. Jorgensen, Nevald and Williams [ j f | made the f i r s t study of this nature on the systems Al-Mg, Al-Ga, and Al-Ge. Prom the orientation dependence of the splittings and intensi-ties of sate l l i t e lines, they were able to assign the resonances to host nuclei that are f i r s t nearest neighbours to the impu-r i t y . The experimental values of q were compared with the theoretical predictions of Pukai and Watanabe . Satisfac-tory agreement was obtained by using a scaling factor of twice the predicted value. The problem has also been approached using the f i e l d cycling method described by Redfield Q20~[ . In this technique, the nuclear spins are polarized in a f i e l d of 5kG for about 3 seconds at a temperature of approximately 1.3°K. The f i e l d i s then switched off for O.65 seconds while a search f i e l d (sf) -42-at frequency ws i s applied to the sample f o r 0 .5 seconds of t h i s period. Then the search f i e l d i s turned o f f , a lkG d.c. i s a d i a b a t l c a l l y applied to the sample f o r 0 .5 seconds and a sweep i s made through the magnetic resonance s i g n a l . The i n t e n s i t y of the observed s i g n a l indicates whether the sf f i e l d has had any e f f e c t on the spin system. In e f f e c t , pure quadrupole resonance (H 0 = 0) i s done on the system while the sf f i e l d i s on. If u>s equals the quadrupole resonance frequen-cy of a set of spins near the impurities, the energy absorbed by these spins w i l l be transmitted to the spin system, and the magnetic resonance s i g n a l i s reduced. If two search f r e -quencies are applied at U>S + A , «os - A , the detectable reso-nance range can be extended. It i s possible then to observe resonances from successive s h e l l s of host atoms as the f i e l d gradient varies from s h e l l to s h e l l , Mlnier (2lJ has applied t h i s technique to d i l u t e a l l o y s of aluminum. He has detected (22] pure quadrupole reso-nances at 5 8 , 7 0 , 115, 125, 3 9 5 , and 690 kHz i n Al-Mn and 7 0 , 135, and 520 kHz i n Al-Cr. However, i t i s very d i f f i c u l t to assign these resonances to the appropriate s h e l l of host nuc-l e i . S a t e l l i t e l i n e s corresponding to these resonances should be observable on the normal host N.M.R. s i g n a l , providing that t h e i r i n t e n s i t y i s high enough. -43-CHAPTER IV EXPERIMENTAL ASPECTS A. Experimental Apparatus The N.M.R. apparatus used f o r the experimental investigation i s shown i n Figure 7. A b r i e f description of each unit follows. The d.c. magnetic f i e l d was supplied by a Magnion electromagnet, with 12" diameter pole faces and a 2^" gap, capable of producing f i e l d s of 20kG. The magnet was rotatable so that the f i e l d could be oriented accurately with-i n 0.1° by using a vernier scale on the base. The f i e l d homo-geneity was f i e l d dependent and never worse than 1 gauss over the sample volume at the highest f i e l d s . The magnetic f i e l d was swept l i n e a r l y using the sweep feature on the magnet con-t r o l panel. The f i e l d was monitored by a s e n s i t i v e , tempera-ture-compensated probe placed near the sample i n the magnet gap. The probe output was proportional to the difference be-tween the f i e l d at a given constant and the pre-set f i e l d and was displayed on a panel meter i n gauss, accurate to l l gauss. In t h i s experiment only the value of the f i e l d f o r 1G0 gauss above and below the main l i n e was required, and t h i s f i e l d determination method proved s a t i s f a c t o r y . An accurate value of the d.c. magnetic f i e l d could be e a s i l y calculated using the resonance frequency from the spectrometer which was d i s --44-Power Amplifier Attenuator P.K.W. Os c i l l a t o r Magnet i F i e l d Monitor PAR Lock-in Amplifier Chart Recorder Counter /C.R.O, Figure 7. Experimental apparatus. - 4 5 -played on a Hewlett-Packard counter. A p a i r of modulation c o i l s were mounted d i r e c t l y on the poles of the electromagnet. The c o i l s consisted of 2 2 5 turns of 0 . 0 3 8 " enamelled copper wire, wound on p l e x i g l a s forms. The modulation s i g n a l used was provided hy the l o c k - i n a m p l i f i e r at 2 8 . 5 Hz and was amplified hy a Bogen audio power amp l i f i e r . The modulation amplitude was chosed by obtaining a compromise between the maximum s i g n a l amplitude and the o p t i -mum linewidth. The s a t e l l i t e l i n e s were observed within 3 0 gauss of the main l i n e , and thus, no modulation broadening could be tolerated. Usually, t h i s r e s t r i c t i o n required some s a c r i f i c e i n s i g n a l i n t e n s i t y . For the experiments performed here, the modulation amplitude was between 3 and 4 gauss. The heart of the system was a modified Pound-Knlght-Watkins [ 2 2 J spectrometer. This o s c i l l a t o r provided an r . f . s i g n a l which was fed to a c o i l wound d i r e c t l y on the sample producing the r . f . magnetic f i e l d . As the d.c. magnetic f i e l d was swept through resonance, the spectrometer produced an audio output proportional to the s i g n a l i n t e n s i t y that was sent to a PAR HR-8 l o c k - i n a m p l i f i e r . The output of the l o c k - i n a m p l i f i e r was proportional to the f i r s t derivative of the resonance lineshape. This out-put was sent d i r e c t l y to a s t r i p chart recorder, which provided the f i n a l output. A l l of the measurements were made at a temperature of 1 . 2°K. The sample was suspended i n a l i q u i d helium bath i n -46-the t a i l of a double dewar cryostat, consisting of an outer l i q u i d nitrogen dewar and an inner l i q u i d helium dewar. The temperature of the l i q u i d helium was reduced to 1.2°K by pum-ping on the surface to reduce the vapour pressure. The signal-to-noise r a t i o obtained on the main l i n e was always around 1200:1 or better. This was good enough to give a S/N on the s a t e l l i t e l i n e s from about 3tl on the high side s a t e l l i t e l i n e s to approximately 10:1 on some of the best low side s a t e l l i t e s . However, i t was d i f f i c u l t to measure the s a t e l l i t e positions exactly since they were superimposed on the sloping t a i l s of the host main l i n e . In the cases where both peaks of the s a t e l l i t e were c l e a r l y v i s i b l e , the base-l i n e was drawn i n and the p o s i t i o n was determined from the cen t r a l cross-over point of the s a t e l l i t e and the main l i n e . If only part of the s a t e l l i t e was resolved, the p o s i t i o n was measured from the equivalent peaks of the s a t e l l i t e and the main l i n e . B. Sample Preparation Single c r y s t a l samples of d i l u t e , homogeneous Al-Mn and Al-Cr a l l o y s were prepared from 69*s p u r i t y aluminum obtained from Comlnco. Due to the very low s o l i d solution s o l u b i l i t y of Mn and Cr i n aluminum, i t was necessary to add the solute d i r e c t l y to the melt and hold i t at temperature f o r three -47-hours. This allowed complete homogenizatlon of the alloy. The aluminum was held in a covered alumina container in an Induction furnace, where eddy currents assisted the mixing. The ingot was then made single crystalline by pas-sing i t through a zone leveller. This consisted of a horizon-t a l 30" x 3/4" dia. quartz tube mounted on a trolley that was pulled by a small 1 r.p.m. electric motor via a speed reducer. The sample was held in an alumina boat and placed in the tube. The tube was evacuated and then flushed with hydrogen several times, and f i n a l l y sealed with an atmosphere (5 torr) of hydro-gen inside. The heater consisted of an Insulated c o i l of resis-tance wire wound on an 8W x 1" dia. quartz tube, with power supplied from a Variac. The turns were wound more tightly near the centre to give a molten zone about 1" in width. As the sample passed through the molten zone, directional s o l i d i f i c a -tion took place and a single crystal of known orientation was produced. Several t r i a l runs revealed that satisfactory crys-tals were obtained with a trolley speed of 2 cm. per minute. The formation of a solute-rich phase from the solid solution was avoided by using low solute concentrations, and by cooling the crystal quickly after i t emerged from the zone leveller. A cross section of the crystal was polished and examined in an electron microprobe. The concentration profile taken across the section revealed the presence of any undesir-able second phase. The solute concentration was determined -48-roughly on the mlcroprobe, and accurately by spectrophoto-m e t r y analysis (Mn), and by volume t i t r a t i o n and atomic ab-sorption (Cr). The c r y s t a l was tested f o r any substructure, such as grain structure or twinned regions, using the back r e f l e c -t i o n x-ray technique. It was then oriented exactly i n a holder on the QjLloJ axis and transfered d i r e c t l y to the spark cutter. A c y l i n d r i c a l sample (0.2" dla. x 0.5") was cut out of the sample and polished i n 100t50tl.5JHNO3tHCLsHP. It was x-rayed along the length to check again f o r any mlcrostructure. The r . f . c o i l was wound d i r e c t l y on the sample over one or two layers of 0.002" mylar. The desired o s c i l l a t i o n frequency could e a s i l y be adjusted by varying the number of turns on the c o l l , the turn spacing, and the number of mylar layers. For the experiment the sample was mounted i n plasticene and glue i n a copper bomb shown i n Figure 8. The glass rod attached to the bomb was used as a reference p o s i -t i o n f o r o r i e n t i n g the c r y s t a l i n the magnet. Using the back r e f l e c t i o n x-ray technique, the sample was oriented i n the bomb so that the | j l o | axis was exactly v e r t i c a l and the axis pointed nearly along the beam d i r e c t i o n . The angle between the [ i l l ] axis and the normal to the rod was obtained by measu-r i n g the angles between the glass rod and the f i l m plane, and the (111) spot and the beam d i r e c t i o n . -49-Co-axial cable Copper bomb Figure 8. Copper sample holder. -50-The bomb was then mounted at the end of the coax i n the magnet gap and aligned so that the glass rod was p a r a l l e l to the pole faces. The t113 d i r e c t i o n with respect to the graduated scale on the magnet was then known to with-i n tl°. The d.c. magnetic f i e l d could then be rotated i n the (110) plane by simply r o t a t i n g the magnet housing. -51-CHAPTER V EXPERIMENTAL RESULTS AND DISCUSSION A series of runs was completed on an Al-Cr sample containing 0 .2 At% chromium, Satellite lines were seen above and below the mam line. Figure 9(a) shows a typical recorder trace taken at 1.2K and a f i e l d of 8.1kG oriented along the [ i l l ] direction. The S/N ratio i s approximately 1 2 5 0 s l . Two features should be noted. There are two satel l i t e contributions superimposed on the low f i e l d side of the main lin e , and one sate l l i t e line on the high side. Also, the amplification has been reduced by a factor of forty while sweeping through the main line. The satel l i t e intensity on the low side i s approxi-mately 0.3% of the main line. Figure 9(b) shows one well resolved sat e l l i t e below the main line. The results are summarized in Figure 10, showing the satell i t e s p l i t t i n g versus H0, and Figure 11 , showing the orien-tation dependence of the satel l i t e lines. In Figure 10 , the lines are symmetrically s p l i t above and below the main line, and do not change position with H 0 within experimental error in this range. This result i s indicative of a pure first-order quadrupole s p l i t t i n g of the ( - - | — ) • (~t.'~*1z ) transitions of the host nuclei adjacent to the impurity. The Knight shift of the satel l i t e and main line are identical, to within 2%, in this range of f i e l d . Evidently there i s no change in the s-elec-Typical trace i n Al-Cr showing high and low side s a t e l l i t e s , as indicated by the arrows. Recorded at 1.2°K and 8.1kG, with a 30 second time constant. The sweep rate was increased by a f a c t o r of two f o r t h i s i l l u s t r a t i o n . Figure 9.(b), Typical trace in Al-Cr showing one well resolved satellite below the main line. Recorded at 1.2°K and 15.5kG, with a 30 second time constant. S a t e l l i t e S p l i t t i n g (gauss) 6 0 5 0 40 3 0 2 0 1 0 0 1 0 2 0 3 0 40 5 0 6 0 T 1 T JL 1 0 \til\ (kilogauss) T 1 T 1 2 0 Figure 1 0 . S a t e l l i t e s p l i t t i n g versus the magnitude of the magnetic f i e l d , H Q, l n Al-Cr, with H Q i n the [ i l l ] d i r e c t i o n . 60 T 1 T 1 S a t e l l i t e S p l i t t i n g ( g a u s s ) 40 20 0 20 40 10* 20' 30( 1 40° 50' 60( 70* 80° 90* e i I 60 L T 1 T 1 F i g u r e 11. S a t e l l i t e s p l i t t i n g s a s a f u n c t i o n o f t h e o r i e n t a t i o n o f t h e m a g n e t i c , „ f i e l d , i n A l - C r a t 15.5kG. Q i s t h e a n g l e b e t w e e n t h e [001J a x i s a n d H Q i n t h e (110) p l a n e . - 5 6 -tron density at the perturbed n u c l e i , or any spin density perturbation due to the impurity atoms. The data plotted i n Figure 1 1 reveals that the s a t e l l i t e positions are independent of the d i r e c t i o n of H G to within experimental error. The asymmetry parameter 7 must be nearly zero. By using the r a t i o of the s p l i t t i n g s In the [OOl] and the [ i l l ] d i r e c t i o n s and Figure 3 , YJ I S found to be r? = - o. OOS 1 00O5 The negative value of 7 Indicates that the x-component of the f i e l d gradient i s greater than the y-component. However, the e l e c t r i c f i e l d gradient i s very nearly a x i a l l y symmetrical! W 1 ~ \ V I • The s p l i t t i n g s of the f i r s t and second s a t e l l i t e s are 2 7 . 6 + 0 . 9 gauss and 5 5 + 3 gauss, or 3 0 + 1 kHz, 6 l t 3 kHz. The error values correspond to the standard deviation of equi-valent measured values. Due to the low i n t e n s i t y of the second s a t e l l i t e l i n e , there i s more uncertainty i n i t s p o s i t i o n . The s p l i t t i n g s correspond to frequency s h i f t s of a n d - Hs? ' Thus, can be solved f o r n e q M . The r e s u l t i s eq = 3 . 6 t 0 . 1 x 1 0 1 3 (egg esu). These r e s u l t s deviate s l i g h t l y from experimental values of Janossy and Gruner [ l 8 J , who obtained -57-before correction. However, our s p l i t t i n g s have an exact f a c t o r of two r a t i o as predicted by theory. Minier and Berthier £22] saw a pure quadrupole resonance l i n e at 70 kHz corres-ponding to the s a t e l l i t e observed here. It i s l i k e l y that the observed s a t e l l i t e l i n e s o r i g i -nate from host nuclei i n the f i r s t nearest neighbour s h e l l to the impurity. The p o s s i b i l i t y of second nearest neighbour i s ruled out since with H c i n the [ i l l ] d i r e c t i o n and the f i r s t order s p l i t t i n g vanishes. The p o s s i b i l i t y of more distant s h e l l s i s u n l i k e l y due to the large wipe-out number l n t h i s system (~900). Also, i f the s a t e l l i t e s originated at more distant s i t e s , then s a t e l l i t e l i n e s should also be seen from the nearer s h e l l s of neighbours to the impurity. No addition-a l s a t e l l i t e s were observed. The r e s u l t s from the Al-Mn a l l o y containing 0.7 At% manganese were more in t e r e s t i n g . One s a t e l l i t e l i n e was seen on e i t h e r side of the main l i n e , as shown on the recorder trace i n Figure 1 2 ( a ) , taken at 1 .2°K, with a f i e l d of 5 . 1 k G along the [ i l l ] d i r e c t i o n . The S/N r a t i o i s about 1 0 0 0 : 1 . The a m p l i f i -cation was reduced by a f a c t o r of 5O while sweeping through the main l i n e . Only the low f i e l d peak of the low-side s a t e l l i t e l i n e i s resolved with an i n t e n s i t y of 1.3% of the main l i n e . Figure 12(a). Typical trace i n Al-Mn showing high and low side s a t e l l i t e s as indicated by arrows. Recorded at 1.2°K and 5.1kG, with a 30 second time constant. The sweep was increased by a fa c t o r of two f o r t h i s i l l u s t r a t i o n . -59-The second peak i s superimposed on the much larger t a i l of the main l i n e . In t h i s case, the p o s i t i o n of the s a t e l l i t e was determined by measuring from the f i r s t peak of the s a t e l -l i t e to the f i r s t peak of the main l i n e . The shape of the s a t e l l i t e should be roughly s i m i l a r to the main l i n e , with a strong p o s s i b i l i t y of broadening due to submicrostructure i n the c r y s t a l . An attempt was made to match the s a t e l l i t e shape to a broadened main l i n e . The base-line that resulted was a l -ways too discontinuous to be reasonable. The unusual shape could be due to a broadening e f f e c t associated with impurity-impurity inte r a c t i o n s . The high-side s a t e l l i t e i s c l e a r l y resolved even though i t i s superimposed on the sloping t a i l of the main l i n e . Measurements were taken from the cross-over point, indicated by the arrow, to the same point on the main l i n e . Figure 12(b) shows a well resolved s a t e l l i t e above the main l i n e . The r e s u l t s are summarized i n Figure 13, showing the s a t e l l i t e s p l i t t i n g versus H Q, and Figure 14, showing the orientation dependence of the s p l i t t i n g . C l e a r l y , Figure 13 shows that the s p l i t t i n g s above and below the main l i n e are not equal. However, i f the low side points are transfered to the high side the trend becomes more evident. The s a t e l l i t e s are being removed from the main l i n e by an e f f e c t of the form (a+bHQ) f o r the high side s a t e l l i t e s and (a-bH 0) f o r the low 12(b). Typical trace i n Al-Mn, showing a well resolved s a t e l l i t e above the main l i n e . Recorded at 1 .2°K and 15.5*G with a 30 second time constant. 30 r-20 10 S a t e l l i t e S p l i t t i n g (gauss) 10 20 30 5 —r 10 H 0| (kilogauss) —r~ 15 i O N I—1 I 9 I i L Figure 13. S a t e l l i t e s p l i t t i n g versus the magnitude of the magnetic f i e l d H, o* i n Al-Mn, with H Q i n the [ i l l ] d i r e c t i o n . The arrows indicate data points that have been transfered from below the main l i n e to above the main l i n e . © Indicates data taken with i n the tool] d i r e c t i o n , t indicates data taken with H Q i n the [ i l l ] d i r e c t i o n . 3 0 S a t e l l i t e S p l i t t i n g (gauss) 2 0 -1 0 -0 1 0 -© 10< ~1— 2 0 ° 3 0 ( - 1 — 4 0 ° e 50< 6 0 ° 7 0 * 8 0 * 9 0 ' 2 0 - 0 1 0 3 0 L Figure 1 4 . S a t e l l i t e s p l i t t i n g s as a function of the orientation of the magnetic f i e l d j . n Al-Mn at 9 . 0 k G . © i s the angle between the Cool] axis and HQ" i n the ( 1 1 0 ) plane. © indicates data points corrected f o r the d i f f e r e n t i a l Knight s h i f t . - 6 3 -side s a t e l l i t e s . Term "a" i s associated with a f i r s t order quadrupolar i n t e r a c t i o n and term "b" with a d i f f e r e n t i a l Knight s h i f t at host nuclei adjacent to the impurity. In addition, the l i n e s show that A K i s strongly anisotropic. The slope of the l i n e s f o r H G i n the [ill] d i r e c t i o n gives the va-lue of A K as ^ K [ l l l ] = - ( 0 » ° 5 - 0.01)% Expressed as a percentage of A l 2 ? i s o t r o p i c Knight s h i f t , i t i s - ( 3 1 + 6)% KA1 For H Q i n the | [ 00 l ] d i r e c t i o n , the value of AK i s A K [ 0 0 l ] = - ( ° » ° 7 - 0.01)% and ^ M 3 = - < 4 3 ± 6 ) * KA1 Both sets of values were obtained by using the Gaussian method of slope determination. The extrapolation of the pl o t s to H Q = 0 give a value of the quadrupolar coupling constant as y a = ££^ 3.. 2 \ t i k H a . This corresponds to a f i e l d gradient value of eq = 1 . 2 t 0.1 x 1 0 1 3 ( c g s e s u ) . There are two possible origins of AK. One i s that A K a r i s e s from core p o l a r i z a t i o n of host nuclei near the impurity by the d-electrons on the Mn. Or, the e f f e c t could be due to a spin -6k-density perturbation at the Impurity s i t e since spin up and spin down electrons have d i f f e r e n t phase s h i f t s . However, the expression f o r A.K/K f o r t h i s e f f e c t given by Gruner et a l [ 1 7 ] has r a d i a l symmetry and hence does not furnish an explanation f o r the observed anisotropy. C l e a r l y , a t h e o r e t i c a l treatment of the system i s required. It seems reasonable that the e f f e c t s should be strongest In the Al-Mn system as compared to Al-Cr, since Mn has f i v e 3 d - e l e c t r o n s versus four 3 d - e l e c t r o n s f o r Cr. Figure Ik shows the orientation dependence of the s p l i t t i n g s as H D i s rotated from [OOl] to [lio] . In the [OOl] and [ i l l ] d i r e c t i o n s the points have been corrected f o r the d i f f e r e n t i a l Knight s h i f t . The value of p f o r the Al-Mn system i s n = - 0 . 0 8 + - 0 . 0 2 as determined from Figure 3 . The observed s a t e l l i t e s a r i s e from host nuclei that are f i r s t nearest neighbours to the im-p u r i t y , since t h e i r orientation dependence i s very s i m i l a r to that shown i n Figure 2. The p o s s i b i l i t y of second nearest neighbours i s eliminated due to the non-vanishing s p l i t t i n g i n the [ i l l ) d i r e c t i o n . More distant s h e l l s are ruled out due to the large wipe-out number and the f a c t that no other reso-nances were observed. This value of 9 was used i n equation (42) to check the r e s u l t s f o r consistency. The predicted value f o r the s p l i t t i n g i n these two p r i n c i p a l orientations and the ex--65-perlmental data indicated i n Figure 14 agree within the error l i m i t s . Minler and Berthier [22] d i d not observe a pure quadrupole resonance below 58 kHz. However, an examination of t h e i r data i n [2l] and {22] reveals that the lowest frequency detectable using the f i e l d c y c l i n g technique i s approximately 40 kHz. The s a t e l l i t e observed here would not be seen using t h e i r technique. The other resonances that they observe must originate on more distant s h e l l s . Thus, the asymptotic form of the f i e l d gradient must have a d i s t i n c t l y d i f f e r e n t form than that shown In Figure 6. The r e s u l t s of Launols and A l l o u l \£\ are plotted i n Figure 15. Their high f i e l d data (8 kG to 30 kG) can be described by a f i r s t order i n t e r a c t i o n with a zero f i e l d s p l i t -t i n g of (19 t 1) gauss as observed here, increased by a 4 gauss di p o l a r broadening correction, and a d i f f e r e n t i a l Knight s h i f t A K as shown by the s o l i d l i n e s In Figure 15. The good agree-ment suggests that t h e i r high f i e l d data corresponds to the s a t e l l i t e observed i n t h i s i n v e s t i g a t i o n , a r i s i n g from f i r s t nearest neighbours. Their low f i e l d data indicates a second order s a t e l l i t e with 1^ = 360 kHz. Minier and Berthier [22] have observed a pure quadrupole resonance at 395 kHz that l i k e l y corresponds to t h i s second order l i n e . - 6 6 -H Q (kG) Figure 15. From Launois and A l l o u l [2] . S p l i t t i n g of the high f i e l d peak of t h e i r s a t e l l i t e l i n e . The s o l i d l i n e s indicate d i f f e r e n t i a l Knight s h i f t s of A K = -0.076% and A K = -0.09$, and a zero f i e l d s p l i t t i n g of 19 ± 1 gauss. A = Zero f i e l d s p l i t t i n g of the s a t e l l i t e obser-ved i n t h i s i nvestigation. B = A plus a 4 gauss di p o l a r broadening correction. - 6 ? -CHAPTER VI CONCLUSIONS The results presented in this thesis clearly illustrate the value of using single crystal samples to assign the observed satellite lines to their appropriate nuclear sites. For tran-sition metal solutes, the radial dependence of the asymptotic form of eq i s clearly different than that predicted for non-transition impurities. Also, the rather large di f f e r e n t i a l Knight shift in Al-Mn is in contrast with no dif f e r e n t i a l Knight shift in Al-Cr. Clearly a more thorough theoretical treatment of the simple metal-transition metal solute alloys i s required. - 6 8 -BIBL10GRAPHY 1. Rowland, T.J.j Phys. Rev. 1 1 £ , 900-912 ( i 9 6 0 ) 2. Launois, H., A l l o u l , H* j Sol. St. Comm. 2 » 525-528 ( I 9 6 9 ) 3 . Jorgensen, L., Nevald, R., Williams D. LI.j J. Phys. P: Metal Physics 1, 972-980 (1971) 4. S l i c h t e r , CP.; P r i n c i p l e s of Magnetic Resonance. Harper & Row Publishing Co., New York (1963) 5 . Cohen, M.H., Reif, P.; Sol i d State Phusics 5_, 321 (1957) 6. P r i e d e l , J.j P h i l Mag 4£, 153 (1952) 7. Kohn, W., Vosko, S.H.j Phys. Rev. 119_, 912-918 ( i 9 6 0 ) 8. Blandin, A., Pr i e d e l , J.j Le Journal de Physique et Le Radium 2 1 , 689-695 ( i 9 6 0 ) 9. S c h i f f , L.I.j Quantum Mechanics. McGraw-Hill Book Company, New York (1968) 10. Sternheimer, R. j Phys. Rev. 84, 244-253 ( 195D 11. Beal-Monod, M.T. , Kohn, W. : J. Phys. Chem. Solids 29_, 1877-1887 (1968) 12. Townes, C.H., Herring, C. , Knight, W.D.j Phys. Rev. 2 2 , 852 (1950) 13. Pines, D.j So l i d State Physics 1, 38 (1955) 14. Bloembergen, B., Rowland, T.J.j Acta. Met. 1, 731-746 (1953) 15. Anderson, P.W.j Phys. Rev. 124, 41 (1961) 16. B r e t t e l l , J.M. , Heeger, A.J, ; Phys. Rev. 15.2, 319-325 (1966) 17. Gruner, G., Kovacs-Csetenyi, E,, Tompa, K., Vassel, C.R.j Phys. Stat. Sol. (b) 45_, 663-667 (1971) 18. Janossy, A., Gruner, G. j Sol. St. Comm. 9_, 1503-I506 (1971) 19. Pukai, Y., Watanabe, K.j Phys. Rev. B 2 , 2353-2360 (1970) -69-20. Redfleld, A.G.t Phys. Rev. l^O, 589-595 (1963) 21. Mlnier, M.t Phys. Rev. 182, 437-4^5 (I969) 22. Mlnier, M., Berthier, C.; Private Communication 23. Watkins, D.G.j Ph.D. Thesis, Harvard University, Cambridge Mass. 

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