Nuclear magnetic resonance in liquid noble metal-tin alloys

‘OURNAL

OF MAGNETIC

RESONANCE

6, L 75-487 (1972)

Nuclear Magnetic Resonance iIn Liquid Noble Metal-Tin Alloy:s 1. P. HOSTAND

G. A. STYLES

School of Physics, University of Warwick, Coventrv, U.K. Presented at the Fourth International Symposium on Magnetic Resonance, Israel, August, 1971. Measurements have been made as a function of composition of the Knight shift K of the “%n resonance in the liquid alloy systems Sn-Cu, Sn--Ag and Sn-Au, and of the 63Cu resonance in Sn-Cu. K(“%n) increases monotonically with increasing noble metal concentration in all three alloys, the fractional change of shift increasing with the atomic number of the noble metal impurity. In Sn-Cu the fractional changes of K(Wu) and K(“%n) with composition are equal in magnilude and of opposite signs. The results are interpreted in terms of the variation with composition of the conduction electron probability density at the nucleus. This has been calculated in two ways. The first method was a partial wave analysis of the impurity scattering while in the second method the probability density is obtained at different compositions by explicitly orthogonalizing the conduction electron pseudo wavelunctions to the ionic core states. Both methods give qualitative agreement between thl:ory and experiment. 1. INTRODUCTION

Measured Knight shifts Kin those liquid binary alloys so 1Br investigated appear to fall into two categories. In the first and by far the larger categ;ory the solvent shift (and the solute shift where it can also be measured) shows a monotonic, usually linear, variation with alloy composition. These results have been interpreted as indicating that the liquid alloy is a random mixture of the two components and the variation of K is assumed to be due to the change of the conduction electron spin susceptibility with electron density in the alloy together with a change of the conduction electron wavefunctions at the nucleus accompanying the build up of screening charge around the solute ions. The attempts made to calculate the concentration dependence of K have been mainly concerned with the latter effect. The change of conduction electron probability density at the nucleus due to impurity scattering has been obtained by methods based either on the phase shift approach proposed by Blandin and Daniel (I) or on the pseudopotential formalism put forward by Faber (2). Neither of these methods has been particularly successful, the agreement between theory and experiment being particularly poor for alloys of polyvalent metals. In the second category the composition dependence of K is far from linear as, for example, in liquid In-Bi (3), where K(‘r51n) increases with increasing Bi concentration up to about 50 % Bi and then remains constant, while K( 209Eli)remains constant up to 50% Bi and then increases. This and similar behavior in o;her alloy systems can be explained by postulating that the atoms are not arranged randomly in the alloy. but ,Q 1972 by Academic

Press, Inc.

475

476

HOST

AND

STYLES

that rather some form of atomic clustering is occurring. The significant compositions for the clustering are sometimes those at which congruently melting intermetallic

compounds occur in the solid, and often anomalies appear at these compositions in other properties such as the electrical resistance and the X-ray form factors. There is much evidence (4) from measurements of various properties that the liquid noble metal-tin alloys are systems in which atomic clustering occurs. However, there is disagreement between different authors as to how this evidence is to be interpreted and the present NMR study was undertaken with the primary object of elucidating the atomic coordination in these alloys. The results of the investigation are presented in Section 3 and contain no indications at all of atomic ordering. The measurements fall into the first category mentioned above and we assume therefore that the alloys may be treated as random mixtures of the two constituents. In Section 4 we compare the results with two separate calculations of the change of K with alloy concentration. The first method involves a partial wave analysis of the impurity screening using phase shifts obtained from a model recently proposed by Asik et al. (5). In the second method the extension of the Faber theory outlined by Perdew and Wilkins (6) is used to obtain the probability density at the nucleus at different alloy compositions by explicitly orthogonalizing the conduction electron pseudo wavefunctions to the ionic core states. 2. EXPERIMENTAL

2. I. Samples The samples were prepared from metals of 5Npurity supplied by Koch-Light Laboratories Ltd. The alloys were first cast in ingot form by melting the weighed constituents under an argon atmosphere and then quenching to room temperature in a copper mould. The ingots were subsequently broken up and remelted and then sprayed into a water bath using the technique described by Heighway et al. (7). The NMR specimens were made up from the resultant powder by mixing those particles which passed through a 75 pm sieve with an approximately equal volume of fused quartz powder. This ensures complete penetration of the metal by the rf field and prevents the droplets recoagulating during the experiment thus removing any rf skin effect problems. As explained in Ref. (7), this technique ensures that the specimen droplets are homogeneous and that a large coil filling factor is obtained. 2.2. Apparatus All the measurements up to 500°C were made using a Varian VF16B spectrometer and associated 12” electromagnet. To obtain these temperatures a gas-flow furnace based on that described by Schrieber (8) was employed. Above 500°C a Watkins-Pound spectrometer was used in conjunction with a specially constructed high-temperature furnace. The temperature was measured by means of a Pt/13 %Rh-Pt thermocouple placed at the bottom of the specimen ampoule. A subsidiary experiment to measure the temperature inhomogeneity over the sample volume indicated that this did not exceed 10°C at any of the temperatures used in the experiment. The positions of the resonances were determined by sweeping the magnetic field repetitively through the line at constant radiofrequency while recording the output.Ff

Nh .R IN LIQUID

TIN

ALLOYS

477

the spectrometer phase-sensitive detector in a 1000 channel signal averaging computer. Calibration of the field at the start and finish of the sweep was zccomplished by observation of the deuterium resonance in heavy water using a separate spectrometer system. In the calculation of the Knight shifts from the measured resonance positions, use was made of the nuclear gyromagnet c ratios given in the table published by Varian Associates. Problems of signal intensity and recoagulation of the droplets at high temperatures prevented results from being oljtained at high noble metal concentrations in Cu-Sn and Ag-Sn. 3. RESULTS

InlFig. 1 the experimental results for K(ii9Sn) in Cu-Sn, Ag-Sn and Au-Sn and of K(63Cu) in Cu-Sn are presented. For convenience in making comparisons with the

At. O/o Noble

Metal

FIG. 1. The fractional change of Knight shift 6K/K vs. noble metal concentration 63Cu resonances in liquid Cu-Sn, Ag--Sn and Au-Sn alloys.

for the ’ “%n and

theoretical predictions, the fractional changes of shift SK/Kcal :ulated using the formula. 6K K(alloy) - K(pure metal) - =: K(pure metal) K

are plotted as a function of alloy composition. The temperattues at which the shifts were measured are 470°C for Au-Sir, 507°C for Ag-Sn snd 527°C for Cu-Sn with the exceptions of the alloys Ag-35 ‘!@n and Cu-65 %Sn for which the temperatures were 560 and 55O”C, respectively. The temperature variation of the shift has been measured

478

HOST

AND

STYLES

for several of the Au-Sn alloys and found to be small. If it is assumed, as is likely. that similar temperature variations occur in the Cu-Sn and Ag-Sn alloys then the above changes of temperature can be shown to produce changes of shift which are negligible in comparison with the experimental errors indicated in the figure. Thus the results shown in Fig. 1 may be considered to be essentially at constant temperature. The measurements were made in magnetic fields of 10.8, 8.4, and 8.6 kG for Cu-Sn, Ag--Sn and Au-Sn, respectively. The three main features of the results are as follows : (a) K(‘19Sn) increases smoothly with increasing noble metal concentration for all three alloys. (b) The magnitude of the variation increases with the atomic number of the noble metal component. (c) The variations of K(‘i9Sn) and K(63C~) in Cu-Sn are of approximately equal magnitudes but of opposite signs. In the next section these three features of the results will be discussed in detail. 4. DISCUSSION

There are no irregularities in the variations of Kshown in Fig. 1 to suggest that atomic clustering is occurring in these alloys, either at those compositions, e.g., AuSn, AuSn,, etc., thought from other measurements to be associated with ordering effects, or indeed at any other composition. In the discussion which follows therefore it is assumed that the alloys are random mixtures of the two constituents. The Knight shift in a metal due to the contact interaction with the conduction electron spins is given by the well-known expression K = $rQP~ xp,

[II

where xp is the conduction electron spin susceptibility, Q is the atomic volume, and P, is the density at the nucleus of electrons at the Fermi surface. It is now known of course that there may be further contributions to the shift arising from the orbital magnetism of the conduction electrons and from the polarization of the cores electrons by the conduction electrons. We assume, however, that the contact term is dominant for the alloys under discussion and that the variation of K arises from changes of either or both the factors QP, and xP. The fractional change of shift may then be written

SK/K= NQP~)/fip,l+ [Sx,/xJ We begin by considering the effect of the change of QP, resulting from the formation of screening charge around the solute ions. 4.1.

PARTIAL-WAVE

ANALYSIS

OF

THE

IMPURITY

SCATTERING

It has been shown by Blandin and Daniel (I) using a partial wave analysis of the impurity scattering that the change in solvent Knight shift per unit atomic concentration cB (cs < 1) of solute ions is given by r=1~~=C(a,sin2?,+BIsin2?r), I3

NMI:

IN LlQUID

TIN

479

ALLOYS

where 7l is the phase shift of the I-th partial wave for electrons at the Fermi surface produced by the perturbing poten :ial of a single impurity ion (dissolved in the otherwise pure solvent) and the coefficients CQand /3* are given by 1 I mp(r)(n:(kF I) -j&r)> s 0

a, = (21-t

6’3r,

and Pl = 421 t 1) [p(r)

n:(k, r)j,(k, I) d3 r.

0

In these expressions, p(r)/po is t?e radial distribution function, and p. is the average number density of the ions. If dilatation around the solute ions is ignored, p(r) may be related to the structure factor S(cl) of the pure solvent by the usual transformation

co f(r)=PO +& os 45Tq2{S(q)

- I> %

dq.

In calculating the Q~‘Sand fll’s we have used for S(q) the model structure factors developed by Ashcroft and Lekn:r (9) for liquid metals at the melting point. The resultcoefficients for the alloys under investigation are given in Table 1. Also given in the table are the values for pure sodium metal and, as can be seen, good agreement exists with the values calculated by Thornton and Young (10). TABLE COEFFICIENTS

Solvent Sn CU Na Na”

‘%

a!

---0.057 0.625 0.69 0.68

-0.123 -1.38 -1.59 -1.52

’ Valuesfrom Thornton and

foung

1 CZ, AND p1

a2 --. -~~~-. 0.795 -1.78 -1.95 -1.86

PO

PI ..

82

-0.072 0.070 0.073 0.04

0.255 -0.697 --0.81 -0.70

-0.33 1.58 1.93 1.84

(IO).

The yl’s for use in Eq. [2] ar’e obtained using standard scattering theory [see, for example, Ref. (II)]. It is assumed that a conduction electron may be represented by a plane wave, written as a sum Df partial waves, which is scattered by a spherically symmetric ionic potential. This potential is taken to be zero beyond some radius a. Then the phase shifts produced in each partial wave by the scattering potential may be evaluated by solving the radial SchrBdinger equation inside the potential and matching this solution at the potential boundary to the solution outside which is given by the well-known expression R,(r)

= (2Zf

1) i’ ei@[cos 7jjj,(b)

- sinTr ,r,(kr)].

480

HOST

AND

STYLES

Since the Knight shift depends on electrons at the Fermi surface, we are only interested in solutions at the Fermi energy and k in the above equation may be taken as the Fermi wave vector k,. The oh’s evaluated are then those appropriate to the Fermi energy. We have used the model proposed by Asik er al. (5) to represent the ionic potential. In this model the potential inside a radius rT is identical to that of the free atom but shifted by a constant amount C so that it intersects the bottom of the conduction band at rT. Beyond rT the potential is flat and corresponds to the bottom of the conduction band. Using the free atom potentials tabulated by Herman and Skillman (12) together with a trial value of C the radial wave equation is solved numerically by integrating outwards from the origin using the Noumerov method. At rT the wavefunction and its derivative are matched to the above exterior solution and the phase shifts thus determined. The scattering must be such that the charge screen formed around the ion allows it to gain electrical neutrality. This requirement is described by the Friedel sum rule a-&21+ I

1)77l=Z’,

where Z’ is the valency difference between solute and solvent ions, modified to take account of the solute-solvent size difference. As shown by Blatt the effective valence difference may then be written as

where Z, and Z, are the valencies and Q, and Q, the atomic volumes of the solvent and solute ions, respectively. The constant C (and consequently rT) is therefore adjusted by trial and error until the phase shifts satisfy the sum rule. In Table 2 we give the phase shifts determined in this way for the noble metal-tin alloys studied in the present investigation. Also given in the table are values of solvent TABLE

2

PHASE SHIFT

SolventA Solute B

RESULTS

Sn

Sn

Sn

cu

CU

&

Au

Sn

Z’

-0.703

-1.713

kF (a.u.)

0.864 2.07 -0.3909 -0.0001 -0.1418

0.864 2.04 -0.7270 -0.1830 -0.2873

rT (ad.) w 71 172

--1.653 0.864 2.46 -0.7984 -0.1757 -0.2569

1.651 0.720 2.90 0.5992 0.6332 0.0081

Fermi wave vector, the point rT at which the model potential is cut off and the effective solute-solvent valence difference. In calculating the valence difference we have assumed that the solute ions retain in solution the volume which they would have in the pure solute metal, namely, 86, 137,134 and 202 a.u. for copper, silver, gold, and tin, respect-

NM R IN LIQUID

TIN

481

ALLOYS

ively. Only the first three phase shifts are given since it was f >und that v3 is negligibly small in every case. Values for the initial slope I’ of the Knight shift versus s0hr.e concentration obtained when the above phase shifts are sobstituted into Eq. [I] are given in Table 3 together with the experimental values. It must Je remembered that the Blandin and Daniel formulation is only valid for small cono:ntrations of solute. The experimental value of r for TABLE INITIAL

SLOPES OF KNIGHT

Solvent A Solute B

3 SHIFT vs. CONCENTRATION

Sn CU

Sn Ag

Sq Au

cu Sn

1 dK k 7~ NW

0.133

0.150

0.281

0.137

1 dK (Screened atom K de potential)

0.150

0.198

0.168

--0.859

fg

(Square well)

-0.016

--0.089

-0.392

- 0.312

the 63Cu resonance in copper containing small amounts of tin must therefore be treated with some reservations since it assumes that the linear variaion observed for tin-rich solutions continues across to pi re copper. For comparison purposes we give also the values of robtained using a square well potential to represent the impurity ion as in the original work of Blandin and Daniel. lgnoring for the time being thz contribution of xa to changes in K (for reasons to be discussed later) we consider that the agreement between theory and experiment to be seen in rows 2 and 3 of Table 3 for the variation of the ‘19Sn Knight shift with noble metal solute concentration is reasonably satisfactory. Exact agreement between the numbers in the table is not to b: expected in view of the approximations made in the theory, particularly that used in calculating the effect of dilatation on the effective solute-solvent valence difference. As can be seen in Table 2 the volume correction is large in every case and the assurrption that the solute ions retain their pure solute metal volumes when in solution, which implies a linear variation of mean atomic volume across the alloy concentration range, is probably unsatisfactory. The correction could be improved by using ionic volumes deduced from alloy density measurements. Furthermore, no account has been taken of the possible variation of electron spin susceptibility. In contrast the results obtained using a square well potential are of opposite sign to the experimental results. As has been noted previously for other systems (13), this simple model is clearly not adequate to explain the experimental results in polyvalent alloys. The predicted variation of the .Y(‘j3Cu) due to the addition ol‘tin solutes is of the opposite sign to that observed experimentally. It is possible, of course, that the extrapolation of the experimental results used is not valid and that the 63C~ Knight shift does in fact increase again at compositions approaching pure copper. However, it seems more likely that the theory is inadequate in this case because of the pnlximity of the atomic 3d electrons to the Fermi surface in copper. It has been estimaled (14) that the resulting

482

HOST

AND

STYLES

S--U’hybridization produces 30 y0 d character in the wavefunctions of the Fermi surface electrons and this has not been taken into account in the model used. Furthermore. it has been shown (15) that there is probably an indirect s core polarization contribution to the 63Cu shift in pure copper amounting to some l&50% of the direct contribution. This is presumably also present in the alloy but it is difficult to see what variation of the shift would result. 4.2. The Pseudopotential Approach Faber (2) has shown that an alternative to the phase shift method for calculating the Knight shift in an alloy lies in the use of pseudopotential theory. The change in electron density at the nucleus for a Fermi surface electron is calculated in terms of the variation with composition of the pseudo wavefunction formed by a first order perturbation of the electron plane wave by the weak pseudopotentials of the solute and solvent ions. Perdew and Wilkins (6) have recently extended this theory by removing the approximation made by Faber that the conduction electron plane wave may be taken as constant over the ion core. The theory has the advantage that unlike the phase shift method it is not restricted to the calculation of Knight shifts at small solute concentrations. Following Perdew and Wilkins we have used the theory to calculate QP, for the ‘19Sn and 63Cu nuclei at different compositions in the noble metal-tin alloys and hence derive the variation of Knight shift. The method consists of first calculating the conduction electron pseudo wavefunction using first order perturbation theory with a weak local pseudopotential. The “true” electron wavefunction is then derived by orthogonalizing this pseudo wavefunction to the ionic core states and renormalizing. QP, may then be obtained by evaluating the square of the wavefunction at the nucleus and averaging over ionic arrangements. It is then found that the contact density for an A type nucleus in an alloy mixture of A and B type ions is given by

(QP,), = [d&W&d1

[I + ~&F) +-~A&F>cdl

The factor contained in the first pair of square brackets in this equation is simply the contact density for a single orthogonalized plane wave with y&)

= 1 - 2 ~39 (4s Ik,; > n

where u,“, is an ionic core state of s-symmetry and lkF> is a plane wave for a Fermi surface conduction electron. Nk, is the single OPW normalization factor. The second pair of square brackets contains the first order correction arising because the pseudo wavefunction rather than a plane wave is being orthogonalized to the ionic core states. The first term CA(kF) comes from the effect of a single ion A immersed in a uniform electron gas and is given by

while the second term 'AdkF)

=

"q'dq J' -gf

A Fo

$&j

[(I

-

cB) uA(q)

+

NM].

IN LIQUID

TIN

ALLOYS

483

represents the influence of the reinaining ions in the alloy. In these expressions a,(q) and uB(q) are the pseudopotentia s of the A and B ions, respectively, while es is the concentration of B ions. It has bten assumed in arriving at these expressions that the radial distribution functions for Pt.and B ions around an A type ion are the same apart from the concentration factors. 7 he structure factor S(q) is therefore that appropriate to a pure liquid having the same density as the alloy. It should be noted that we have not retained the small first-order correction to the normalizalion factor N,Jca) which was included in Perdew and Wilkins original formulation. As these authors have shown, the correction is very small and we feel justified in ignoring it since its inclusion introduces severe complications in the: computation methods we h,*ve used. The present calculations differ from those performed by Perdew and Wilkins for alkali alloys in that there are no suitable analytic core wavefv nctions available for the noble metal or tin ions so that thl: angular integrations cannoi be performed explicitly. Instead we have used Herman an1 Skillman tabulated wavefunctions for the core states and have performed the angular integrations numerically on an Elliot 4130 computer, together with the integrations over the magnitude of q. Ashcroft pseudopotentials (16) were used in the calculations together with the Ashcroft and Lekner (9) hard-sphere structure factors. As in the case cf the phase shift calculations, it has been assumed that the mean atomic volume varies linearly with concentration and k, has been calculated from this density using the free electron approximation. Values of QP, were calculated at 10% concentration intervals for the ‘19Sn nucleus in Sn-Cu, Sn-Ag and Sn-Au and for the 63Cu nucleus in Cu-Sn. The results for the pure metals and for the 50 % alloys are given in Table 4. Included in the table are the results obtained for pure sodiurr metal and a sodium-rubidium alloy which are to be directly compared with the values, taken from Perdew and \Vilkins, given in the last two columns of the table. It is er couraging that the two sets of results are very similar even though different ionic core state wavefunctions were used in the calculations. It is also interesting to note in passing that the values of Y:(kF)/Nk,(CB) for the pure metals are considerably different from those obtained recently by Heighway and Seymour (17) using an approximate method in their calculation of Knight shifts in pure liquid metals. This discrepancy will be the subject of a separate paper. The full variations of 6K/K calculated from the fi?PFvalues are plotted as a function of alloy composition in Fig. 2. Comparison of these curves with those of Fig. 1 shows that the magnitudes of the predicted variations of K(‘lgSn) are about three times too large. Otherwise all the essential features of the experimental results are reproduced. The shift increases with increasing noble metal concentration in all three alloys while the magnitude of the variation is largest for Au-Sn and smallest for Cu-Sn. Furthermore, the upward curvature of the Au-Sri experimental results is alsc to be seen in the theoretical curve. For the 63Cu shift the predicted variation, as in the results of the phase-shift analysis, is in the opposite direction to that observed experimentally. No extrapolation of the experimental results is involved in this case, however, and there is therefore no doubt about the disagreement. The final feature of the theorerical curves which requires same comment is the turnover at high noble metal concent:rations. Unfortunately the experimental results do not extend to this region, but some measurements made recently in our laboratory on noble metal-indium alloys do in fact turn over in this way though at slightly lower noble metal

4x4

HOST

AND

STYLES

concentrations. It would obviously be of interest to extend the present measurements to more dilute Sn alloys. There are several approximations made in the theory which could cause the predicted variations of Knight shift to be too large. Inspection of Table 4 shows that the major

At %

Noble

Metal

FIG. 2. Theoretical results for the fractional change of Knight shift 6K/K us. noble metal concentration for the alloys Cu-Sn, Ag-Sn and Au-Sn obtained using the pseudo-potential method. TABLE THE

VARIOUS

CONTRIBUTIONS

TO

QPF

4

CALCULATED

USING

THE PSEUDOPOTENTIAL

Present calculations Solvent A Solute B -__ CB

Sn

Y:(~FYN&i)

492

-

Sn Cu 0.5 515

Sn Ag 0.5

Sn Au

Cu Sn

Na ~

Na Rb

Perdew and Wilkins Na Na Rb

-

0.5

-

0.44

648

668

467

400

165

186

0.928

0.918

0.001

1.48 -0.14

-0.02

-0.31

-0.34

-0.29

614

626

555

121

188

102

1 -+ ZA,(kF)

0.781

0.834

~A&F,CB) QPF

0.061

0.002

-0.008

415

481

596

a Atomic units are used throughout.

0.5

Cu -

METHOD”

1.41

1.04

1.35

0.44 151 0.99

175 1.32 m-O.34 162

NMR

IN LIQUID

TIN

ALLOYS

485

contribution to the variation o * QP, comes from the OP'W factor r2(kF)/iVk,(cB). r2(kF) changes with composition through the variation of kF while NkF(cB)depends on the mean atomic volume. The frel: electron estimate of k, and the atomic volume calculated assuming a linear variatio 1 of density in the alloy may therefore not be good enough. A second possibility lies n the use of a local pseudopotential in the calculation whereas strictly a nonlocal pseudopotential should be employed. As pointed out by Perdew and Wilkins, however, this is not too serious for simple metals since the high q Fourier components of u*(q) and uB(q) are suppressed in the integrations over q in the above expressions. However, the presence of filled d-bands close to the conduction band in copper may require the use of a nonlocal pseudopotential. Harrison (18) has discussed this problem in detail and proposed a nonlocal pseudopotential for the transition and noble metals which is unfortunately, however, rather difficult to use in practice. Finally the assumption that the partial structure factors in the alloy are identical is certainly not correct It is not clear how serious an effect this has but calculations are at present under way which incorporate the Ashcroft-Langreth (19) partial structure factors for a fluid consisting of hard spheres of twc different diameters. 4.3. The Electron Spin Susceptibility The remaining factor in Eq. [I] which may produce a variation of the Knight shift on alloying is the conduction electron spin susceptibility xp. In the past, experimental information on this quantity has been considered unreliable (except where measured using conduction electron spin resonance) since the measurements are of the total susceptibility given by Xmas

= XP $- xd +

Xc

where xd and xc are the diamagnetic contributions of the conduction electrons and the ion cores, respectively. Recently, however, Dupree and Seymour (20) have shown that a consistent set of values of ;vpcan be deduced from the total susceptibility for most pure liquid metals by using suitable theoretical estimates cf xE and X& Using their procedure we have estimated the variation of xI, in the present alloys by linearly interpolating between the values of XI, for the pure metals. Unfortunately, there are considerable differences amongst the values of xmessfor the noble metals obtained by different authors and the estimated xa variations are probably not very accurate. However, taking the values of xmensconsidered most reliable by Dupree and Seymour the results shown in Table 5 are obtained. The theoretical xc values are taken from Angus (21) while Xd is TABLE MAGNETIC

SUSCEPTIBILITIES

cu

x (mea4

5 (x 1 O6 cgs VOLUME

Ag

Au

UNITS)

Sn

-1.2

-2.5

-4.0

-m0.23

XC

-m2.44

--3.60

-5.10

-1.39

Xd

~-0.38

--0.33

-0.33

xp (exptl)

1.6

1.4

I .4

1.6

xD (theor)

I .2X

1.14

1.14

1.51

__-

0.44

486

HOST

AND

STYLES

calculated from the free electron value xdor taking account of electron-electron tions by using the Kanazawa formula (22) .Xd _ l $ E!$(EJ xdo

interac-

.+.4]

and assuming that the effective mass ratio of the conduction electrons is unity. In this expression c( = (4/97r)‘13 and r, = (3/477r~,)“~ where n, is the number density of the electrons. The fractional changes of spin susceptibility per unit noble metal concentration deduced from the values in the table are 0 for Cu-Sn and -0.13 for Ag-Sn and Au-Sn. In view of the uncertainties in these numbers we include in the table the theoretical values of xp calculated by Rice (23) taking account of electron-electron interactions and again assuming m*/m = 1. The fractional changes of spin susceptibility given by these values are --0. I5 for Cu-Sn and -0.25 for Ag-Sn and Au-Sn. Bearing in mind the approximations involved, the agreement between the theoretical and experimental values must be considered satisfactory. The variations are thus of opposite sign to those observed experimentally for K(“9Sn) in all three alloys but of the same sign as the KP3Cu) variation in Cu-Sn. If the results are combined with the variation of QPF calculated above, the disagreement between theory and experiment is generally reduced using the pseudopotential results while it is increased using the results of the phase shift analysis. It is clear, however, that more reliable results for xP are required before a more detailed comparison can be made and susceptibility measurements on these systems are planned for the near future. 5. SUMMARY

The present investigation was begun with the aim of studying structural effects in the noble metal-tin alloys. It is concluded that there is no evidence in the NMR data to suggest that the alloys are in any way different from the majority of liquid alloys in which the two constituents are randomly mixed, and the interpretation of the observed Knight shift variations is based on this model of the liquid. The expected variations of the l19Sn Knight shift with alloy composition, resulting from changes of the probability density at the nucleus of the conduction electrons, calculated using (a) a phase shift analysis and (b) a pseudopotential method are both in semiquantitative agreement with experiment. On the whole, the latter method gives the better results. In view of the approximations for various parameters used in the calculations it is concluded that the measure of agreement obtained indicates that the principles of the methods of calculation are correct at least for K(‘19Sn). On the other hand, the theoretical predictions for the K(‘j)Cu) in Cu-Sn are opposite to that observed experimentally. This is considered to be due to s-d hybridization of the conduction electron wavefunctions which has not been taken into account in the calculations. There is evidence that this would affect K(63C~) more seriously than K(‘19Sn) from the observation that the absolute Knight shift of pure copper calculated from the value of GP, given by the pseudopotential method is badly out while good agreement between theory and experiment is achieved for pure tin. It would be interesting to extend the b3C~ measurements to higher Cu concentrations and to investigate whether the losAg and 19’Au Knight shifts behave in a similar fashion to that of copper.

NhRINLIQUIDTIN

ALLOYS

487

The estimates of the effect of he variation of spin susceptibility on the Knight shift are considered to be unreliable b(:cause of the uncertainties in the measured susceptibilities available in the literature. E Lowever, when combined with the probability density variations they tend to improve t le agreement between theory- and experiment obtained using the results of the pseudopotential calculation. In order to improve the theoretical estimates of the Knight shift variations in these alloys, s-d hybridization must t’e incorporated and better values are required for the pseudopotentials, the atomic vclumes, the liquid structure -*actors and above all the electron spin susceptibility. .I\CKNOWLEDGMENTS This work was We wish to thank E. F. W. Seymour

carried out under an external research B. W. Mott for hi i support, J. Perdew for many valuabk discussions.

contract with AERE, Harewell, England. for his timeI5, private communications, and

REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. 10. II. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

A. BLANDIN AND E. DANIEL, J. Pt~ys. Chem. Solids 10,126 (1959). T. E. FABER, AduanPhys. 16,637 (1967). G. A. STYLES, Advan. Phys. 16,2"5 (1967). B. W. MOTT, UKAE report AERE-R5565 Her Majesty’s Stationery Office, London, (1967). J. R. ASIK, M. A. BALL, AND C. I’. SLICHTER, Phys. Rev. 181,645 (1969). M. A. BALL, J. R. ASIK, and C. P. SLICHTER, Phys. Rev. 181,662 (1969). J. P. PERDEW AND J. W. WILKINS, Solid State Commun. 8,204l (1970). J. HEIGHWAY, I. P. HOST, AND G. A. STYLES, J. Phys. E 3,391 (1970). D. S. SCHREIBER, Rev. Sci. Znsttwn. 35, 1582 (1964). N. W. ASHCROFT AND J. LECKNEII, Phys. Rev. 145,83 (1966). D. E. THORNTON AND W. H. YOIJNG, J. Phys. C 1,1097 (1968). L. 1. SCHIFF, “Quantum Mechanics,” 2nd ed., Chap 5, McGraw-Hill, New York, 1955. F. HERMAN AND S. SKILLMAN, “Atomic Structure Calculations,” Prentice-Hall, Englewood Cliffs, NJ, 1963. E. F. W. SEYMOUR AND G. A. STYLES, Proc. Phys. Sot. 87, 473 (1966). R. E. WATSON, H. EHRENREICH, AND L. HODGES, Phys. Rev. Lett. 24, 828 (1970). L. H. BENNETT, R. E. WATSON, PND G. C. CARTER, J Res. Nat. Bw. Stan USA A 74, 569 (1970). N. W. ASHCROFT, J. Phys. C 1,2: 2 (1968). J. HEIGHWAY AND E. F. W. SEYMOUR, Phys. Kond. Mate. 13, 1 (197’1). W. A. HARRISON, Phys. Rev. 181, 1036 (1969). N. W. ASHCROFT AND D. C. LANGRETH, Phys. Rev. 159,500 (1967). R. DUPREE AND E. F. W. SEYMOIIR, Phys. Kond. Mater. 12,97 (1970). W.R. ANGUS, Proc.Roy.Soc. A 136,569(1932). H. KANAZAWA AND N. MATSUDP.WA, Progr. Theor. Phys. 23,433 (I 960). T. M. RICE, Phys. Rev. 175, 858 I 1968).