The Direct Monte Carlo Method for Calculating Alloy Phases

Abstract

A number of theories exist for predicting the structure of an alloy on the assumption that the energy of the alloy can be obtained from an Ising model Hamiltonian

$${2}\sum\nolimits_{ij} {[V_{ij}^{AA} p_i^A p_j^A \, + \,V_{ij}^{BB} p_i^B p_j^B \, + \,V_{ij}^{AB} (p_i^A p_j^B + p_i^B p_j^A )]}$$

((1))

where ${p_i^A }$ is one if the atom at site i is an A atom, and zero if it is not Theorists have made efforts over the years to write the exact expression for the total energy of an alloy in the form of this Hamiltonian, ^1,2,3,4 and to calculate the interatomic potentials, ${V_{ij}^{\alpha \beta } }$, from the electronic structure. The present work is based on the observation that it is not necessary to introduce interatomic potentials if the Monte Carlo (MC) method⁵ is used for the thermodynamics. The crux of the MC method for obtaining the equilibrium distributions of atoms in an alloy is a calculation of the energy required to replace an A atom on site i with a B atom when the configuration of the atoms on the neighboring sites, κ, is specified, $\delta H(A \to B)\, = \,E_B (\kappa )\, - \,E_A (\kappa )$. A random number Z between zero and one is generated and the atoms at site i are interchanged only if a condition is satisfied, such as ${\rm{Z < }}\frac{1}{2}\left[ {1 - \tanh \left( {{\rm{\delta H/2}}{{\rm{k}}_{\rm{B}}}{\rm{T}}} \right)} \right]$. In the MC calculations done to date, δH is obtained from the Ising model in Eq. (1), but we calculate it directly from the electronic structure of the alloy using the embedded cluster method (ECM) of alloy theory. The ECM has reached its highest stage of development in a paper by Gonis, Butler, and Stocks,⁶ and is in turn based on the coherent potential approximation (CPA) for the electronic states of random substitutional alloys.⁷