Method

The Monte Carlo method is applied to study the effect of the interfaces and surface on the ordering of Cu₃Au. More specifically, we performed Monte-Carlo simulations using an N-body potential. No restrictions are applied to the ensemble except one, namely that the total number of the atoms is constant. This means that the atoms are free to mutate (this is controlled by the chemical potential) and every atom is allowed to move in any direction. The restriction of the number of atoms is not of major importance, since the temperature range of the simulations is far away from the melting point and vacancy formations are not expected to play a significant role at such temperatures. The specific potential used is the tight-binding in the second moment approximation. This potential consists of two terms, Eq. (7.1), one repulsive, which is a sum of pair-wise terms of the Born-Mayer type, and one binding term, produced from a simplified electronic structure model. The potential has contributions from different many-body terms. The specific form we use is:

 

$$\sum_\alpha \sum_{i_\alpha=1}^{N_\alpha} \Biggl( \sum_\beta \sum_... ...ha \beta} \left[ \frac {r^{\alpha \beta}}{d_{\alpha \beta}} -1\right ]\right \}$$ V =

$$\qquad \qquad - \sqrt{\sum_\beta \sum_{j_\beta=1, j_\beta \neq i_... ...eft [ \frac {r^{\alpha \beta}}{d_{\alpha \beta}} -1\right ]\right\} }\: \Biggr)$$ (7.1)

In the above equation $\alpha$ or $\beta$ denote either Cu or Au, $A, p, q, \xi$ are the potential parameter of the respective interaction terms, d is the nearest neighbor distance of the respective material, and $i_\alpha$ are the sites occupied by atoms of type $\alpha$. A detailed description of the method can be found in ref. (2).