Calculation of Order Parameters in a Binary Alloy by the Monte Carlo Method

A Monte Carlo sampling scheme similar to that used by Metropolis, Wood, and others in equations of state computations for gases has been used to investigate order-disorder phenomena in a face-centered cubic $A_{3}B$ alloy. The model of the alloy assumes that the structure of the lattice is fixed and that interactions exist between first neighbors and second neighbors only. In most of the calculations detailed consideration is given to an array consisting of five unit cells on an edge (five hundred sites) with periodic boundary conditions. The long-range order and short-range order for first and second neighbors has been computed above and below the critical temperature. Using the energy parameter, $v_n=\left[\frac{({V_{\mathrm{AA}}}^{\left(r_n\right)}+{V_{\mathrm{BB}}}^{\left(r_n\right)})}{2}\right]-{V_{\mathrm{AB}}}^{\left(r_n\right)}$, for $n\mathrm{th}$ neighbors it is found that $\frac{v_2}{v_1}=-0.25\mathrm{}$ and $v_1=816\mathrm{}$ cal/mole gives the best agreement with experiments on ${\mathrm{Cu}}_3$Au. The critical temperature appears to vary linearly with the ratio $\frac{v_2}{v_1}$.