Ground State Analysis on the FCC Lattice with Four Pair Interactions
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GROUND STATE ANALYSIS ON THE FCC LATTICE WIIT INTERACTIONS
FOUR ?Am
GERARDO D. GARBULSKY, PATRICK D. TEPESCH AND GERBRAND CEDER Massachusetts Institute of Technology, Department of Materials Science and Engineering, Cambridge, MA 02139 ABSTRACT We have partially solved the ground state problem of binary alloys on the fcc lattice with pair interactions up to the fourth nearest neighbor distance. Our results extend the study presented by Kanamori and Kakehashi [1], releasing the constraint they imposed on the nearest neighbor correlation. The solution we present increases the number of possible ground state structures by an order of magnitude with respect to previous results. We have applied both the polyhedron and the enumeration method. The latter proved more powerful when including interactions beyond the second nearest neighbor distance. 1 - INTRODUCTION First principles computations of alloy phase diagrams are becoming a predictive tool in materials science. This is due to the increase in both the computational power and the sophistication of the theoretical models describing the free energy of the alloy. Quantum and statistical mechanics are combined to get expressions for the free energy that do not rely on fitting parameters. With very few assumptions, the configurational dependence of the energy of a binary alloy can be reduced to the form of a generalized Ising Hamiltonian [2]
E({(o}) = 7 v,•o,((0))
(1)
where Va are the effective cluster interactions (ECI's), oa are the product of all the spin variables on cluster ot (a spin variable on a site of the lattice takes the values + 1 or -1 when the site is occupied by an A or B atom respectively), (a) indicates a configuration of A and B atoms on the lattice, and the sum is over all the clusters of sites on the lattice. This expansion is exact in the sense that the cluster functions form a complete set in the space of functions of configuration. The ECI's are not fitting parameters, but the coefficients of the expansion. Their values can be computed using first principles techniques (see for example [3]). Expression (1) for the energy of the system can be regarded as a cluster expansion containing pair and multiplet interactions of increasing range [4]. For real systems, this expansion is rapidly convergent and therefore a finite number of terms need to be considered. Taking the symmetry of the lattice into account and keeping the point and pairs interactions up to the fourth nearest neighbor distance, the expansion reads 4
E = Vl~l+j V2 , i=1
2 .i
(2)
where ýl is the lattice average of the spin (or point correlation), ý2,i is the lattice average of the spin product on the i-th nearest neighbor pair (or pair correlation), VI is the point interaction and V2,i are the pair interactions (see figure 1).
Mat. Res. Soc. Symp. Proc. Vol. 291. ,1993 Materials Research Society
260
2nd
Pairs
Tetrahedron
10 point tetrahedron Octahedron all the atoms in a of Figure 1. Cluster definitions. The 14 point cluster is composed conventional fcc unit cell and the 13
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