Electronic Structure of Ordered and Disordered Ternary Intermetallics
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ELECTRONIC STRUCTURE INTERMETALLICS
OF ORDERED
C. WOLVERTONI AND D. DE FONTAINE
AND DISORDERED
TERNARY
2
1
Department of Physics, University of California at Berkeley (UCB), and Lawrence Berkeley Laboratory (LBL), Berkeley, CA. 2 Department of Materials Science and Mineral Engineering, UCB, and LBL, Berkeley, CA. ABSTRACT A cluster expansion for energetics is combined with a direct, real-space method of studying the electronic structure of ordered and disordered ternary intermetallics. The electronic structure calculations are based on an explicit averaging of local quantities over a small number of randomly chosen configurations. Quantities such as densities of states, one-electron energies, etc., are computed within the framework of the first-principles tight-binding linear muffin-tin orbital method (TB-LMTO). Effective pair interactions, which describe the ordering tendencies of the alloy, are computed for the full ternary alloy. With this technique, then, the effects on ordering trends of ternary additions to a binary alloy may be obtained. Results for Ag-Pd-Rh and Ni-Al-Cu are shown. The self-consistency of these calculations is checked against the fully self-consistent ordered LMTO calculations. INTRODUCTION-CLUSTER EXPANSIONS The study of intermetallic alloys is of practical and technological interest. Most of the theoretical effort to date has been concentrated on binary alloys. However, most alloys used in practice are multicomponent (ternary, quaternary, etc.), thus an extension of theoretical techniques to ternary systems is a crucial step towards making a predictive theory which is of use in alloy design. Consider a system of N lattice sites, each of which is occupied by an A-, B-, or C-atom. The description of such a ternary alloy is facilitated by an "Ising-like" model in which the atom at site p is designated by the "spin" variable 0 p = -1, 0, or +1, depending on whether the atom is A, B, or C, respectively. Any configuration of the system may be described by the N-dimensional vector, a = (al,"2..N). Sanchez et a1l showed that within this model, one may define a complete orthonormal basis of functions with respect to the inner product operation defined for arbitrary functions of configuration, f(s) and g(o): < f,g >=-3N f(o)g(o) 3N 10)
(1)
where the sum is over all of the 3N configurations of the lattice. The orthonormal basis functions are given by products of the first three Chebychev polynomials of the discrete variable rp: 0O(yp) = 1,
GI(up) =
O 0 2(0p)
- 2-2) 3
For a cluster of lattice points a = {p, p .... p") and a set of indices (s) = {n, n .... cluster function is defined as Oa=
On (Cp)O n'(Gp') ... On"(Up")
(2) n") the (3)
(The superscript s will be in parentheses to distinguish it from a power.) As shown in ref. 1, the cluster functions of Eq. (3) are not only orthonormal with respect to the operation of Eq. (1), but also satisfy a completeness relation, and therefore, any function of configuration may be exactly expanded in terms of these cluster functions. In particular,
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