FULLY RELATIVISTIC MULTI-SITE INTERACTIONS

P. Weinberger, 1. Udvardi* and R. Schneeweiss Technical University of Vienna, Austria

B. I. Bennett Los Alamos National Laboratory, L08 Alamos

INTRODUCTION

The concept of "re-interpretating" (mapping) the parameters of the Ising model in terms of (onto) effective pair-, triplet- and so on interactions, opens up new possibilities in computational solid state physics. In particular in connection with the Coherent Po­tential Approximation (CPA), which describes perfectly statistical chemical disorder, it appears that now a combination of methods is available that allows to describe with one and the same approach ground state properties such as transport properties, as well as uniquely determined parameters for statistical mechanics. One might as well add, that within the limitations of the local density approximation also an essentially parameterfree theoretical spectroscopy (e.g. XPS, XES or AES) for disordered or par­tially ordered systems is at hand.

METHOD OF CALCULATION

For a particular pair of sites i and j , occupied by atoms of species 0:, f3 = A, B the effective cluster interactions are given by 1,2,3

I'

V/jfi = 7r-11m J Tr In Qf!(E)dE, -00

where J1, is the chemical potential and

Qf! = 1 - X;"/(E)fC(E).

In (2) X;"/(E) is a site-diagonal supermatrix of the following form

OIfi() ( Xf(E) 0 ) Xi; E = 0 Xf (E) ,

·permanent adress: Technical University Budapest, Hungary

StaJics and Dynamics of Alloy Phase Transformations, Edited by P.E.A. Turchi and A. Oonis, Plenum Press, New York, 1994

(1)

(2)

(3)

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Xf(E) = {[ (ti(Er1 _ tnEr 1)] -1 _ T;i(E) } -1, (4)

where ti'{E)and tf(E)are the single site t-matrices for the coherent lattice and compo­nent C¥, respectively, and fC(E) is the nondiagonal part of the scattering path operator of the CPA medium

AC(E) _ ( 0 T;j(E) ) T - TJ;(E) 0 . (5)

The non-site-diagonal scattering path operators T;j(E) are obtained within the Embed­ded Cluster Method (ECM) by means of the following Brillouinzone integrals4

T;j(E) = TC(R; - Rj; E) = nB~ j[t't(Et1 - G(k,E)] exp{ik. (R; - Rj)}cPk, (6)

which can be reduced considerably for numerical purposes by using group theoretical methods4,5. In terms of (1) the well-known effective pair interactions are then given by (see, for example, Gonis et aP)

(7)

It should be noted that in a similar way also effective triplets or in general effective multi-site interactions can be obtained1 .

AN EXAMPLE: AU7oPd3o

Theoretically the AUcPd1- c system is quite demanding: the alloy partners are a "5d-" noble metal and a "4d-"metal with a nearly filled d-shell. This implies that by

25 I I I I I I

20 - + Au7()Pd30 -15 -

a = 7.6Qa.u. -

>: 10 - 0 -... .§,

+ '-' 5 > r 0 -

g + .6. .6. Q 0 u u u

-5 f- 0 --10 I I I I I I

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

distance

Figure 1. Effective pair interactions (crosses) up to the fifth nearest neighbors for AU7oPd3o, a = 7.60 a.u. The distance is units of the lattice constant. Au-Au pairs, Pd-Pd-pairs and Au-Pd pairs are shown as triangles, squares and open circles, respectively.

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adding Pd to Au, in a certain very narrow range of compositions the Fermi energy moves from a flat, almost free-electron like sp-band into the edge of the d-band. This kind of change, which occurs around 40% Pd, causes e. g. a rather sudden rise in the linear coefficient of the specific heat. All calculations refered to in this paper6 are fully relativistic selfconsistent and correspond to a consistent use of a maximal angular momentum quantum number of lmax = 2 .

As an illustration one particular case is chosen, namely AU7oPd30 corresponding to a lattice spacing of 7.60 a.u. This lattice spacing is slightly bigger than the one at OK (7.565 a.u.) and refers to an experimental value at about room temperature7. In Fig.1 the effective pair parameters are shown as a function of distance (in units of the lattice spacing). One clearly can see from this figure that the effective pair interactions falloff quite rapidly, but also that this fall-off is by no means uniform.

Exactly the parameters shown in Fig.1 were used to perform a 10 x 10 x 10 Monte Carlo simulation with periodic boundary conditions. Fig. 2 clearly indicates that there is an instability at about 450 - 500 K due to the order-disorder phase transition. Since there seem to be no experimental data available concerning the order-disorder phase transition in Au-Pd, a theoretical critical temperature of about 500 K is what is to be expected from similar systems such as Cu-Pd and Cu-Au.

By mapping effective pair interactions onto the parameters of the Ising model, one has to keep in mind that the effective pair interactions do not contain corrections due to the double counting term and therefore do not include charge fluctuations. These corrections can be estimated from the effective charges "'lJa = Za - Qa within the Wigner-Seitz-radius, where Za and Qa is the atomic number and Wigner-Seitz charge for components 0, f3 = A, B, respectively. These corrections (atomic Rydberg units),

V.f = - ("'lJ a - "'lJ (J) 2 / lit - Rj I, decrease in magnitude in the present calculations for a given lattIce constant from about -2.5 mry for first neighbors to less than -1 mry for fifth neighbors. Going from small to large lattice constants, a change of 0.25 a.u. induces a decrease of about 0.5 mry.

Finally it should be noted that within the (single-site) KKR-CPA method all quantities related to the one-particle Greens function can be calculated as a function of

572 2.0

~ 570 Au70Pd30 a = 7.60a.u. Au70Pd30 a = 7.6Oa.u.

8 1.6 • ::. 568 • <U

• 0) 1.2 » •• ..s:: •

rf566 <.) •• !::l

c • ·13 0.8 • .., • 8-OJ 564 . .."", '" • • c ... • •• ~ 562 0.4 I . 5 •• , • 560 0.0

0 400 800 1200 0 400 800 1200 temperature [K] temperature [K]

Figure 2. Internal energy (left, [K]) and specific heat (right, [dimensionless units]) for AU70PdaO , a = 7.60a.u.

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concentration. Exactly the same KKR-CPA results on which the present calculations of the Vii'S are based, are used in a forthcoming papers to compute theoretical inten­sities for UPS, XPS, XES and AES or can be applied to obtain residual resistivities as discussed by Banhart et a19 •

ACKNOWLEDGEMENT

This paper was partially supported by grants from the Austrian Science Foundation (P7996) and the Austrian Ministery of Science (Zl. 49.731/2-24/91).

REFERENCES

1. A. Gonis, X.-G. Zhang, A. J. Freeman, P. E. A. Thrchi, G. M. Stocks and D. M. Nicholson, Phys. Rev. B 36: 4630 (1987).

2. 1. Szunyogh and P. Weinberger, Phys. Rev. B 43: 3768 (1991). 3. J. Klima, Z. Physik B, submitted for publication. 4. P. Weinberger, "Electron Scattering Theory of Ordered and Disordered Matter", Clarendon

Press, Oxford (1990). 5. S. L. Altmann, "Rotation, Quarternion and Double Groups", Clarendon Press, Oxford (1986). 6. P. Weinberger, C. Blaas, B. I. Bennett and A. M. Boring, Phys. Rev. B, submitted for

publication (1992). 7. M. Hansen, "Constitution of Binary Alloys", McGrawHill (1958). 8. P. Weinberger, L. Szunyogh and B. I. Bennett, Phys. Rev. B, submitted for publication (1992). 9. J. Banhart, R. Bernstein, P. Weinberger and J. Voitlander, Solid State Comm. 77: 107 (1991);

J. Banhart, P. Weinberger and J. Voitlander, J. Phys. C 1: 7013 (1989).

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