Metal Potentials

DL_POLY_2 includes density dependent potentials suitable for calculating the properties of metals. The basic model is due to Finnis and Sinclair [28] as implemented by Sutton and Chen [7]. The form of the potential is: (stch)

$$U_{sc}=\epsilon\left[\sum_{i<j}\left(\frac{a}{r_{ij}}\right )^{n} -C\sum_{i}\rho_{i}^{1/2}\right ],$$ (2.87)

where the local density $\rho_{i}$ is given by

$$\rho_{i}=\sum_{j}\left(\frac{a}{r_{ij}}\right )^{m}$$ (2.88)

The Sutton-Chen potential has the advantage that it is decomposable into pair contributions and thus falls within the general tabulation scheme of DL_POLY_2 , where it is treated as a short ranged interaction. The same form of potential may be used in alloys, through the appropriate choice of parameters $\epsilon$ and $a$. The parameters $n$ and $m$ however must be the same for all component elements. DL_POLY_2 calculates this potential in two stages: the first calculates the local density $\rho_{i}$ for each atom; and the second calculates the potential energy and forces. Interpolation arrays are used in both these stages.

The total force $\mbox{$\underline{f}$}_{j}^{tot}$ on an atom $j$ derived from this potential is calculated in the standard way:

$$\mbox{$\underline{f}$}_{j}^{tot}=-\mbox{$\underline{\nabla}$}_{j} U_{sc},$$ (2.89)

which gives

$$\mbox{$\underline{f}$}_{j}^{tot}=-\epsilon\sum_{i\ne j}\left... ...\left(\frac{1}{r_{ij}^{2}}\right )\mbox{$\underline{r_{ij}}$},$$ (2.90)

which is recognisable as a sum of pair forces, for example the force on atom $j$ due to the atom $i$:

$$\mbox{$\underline{f}$}_{j}=-\epsilon\left [n\left (\frac{a}{... ...left (\frac{1}{r_{ij}^{2}}\right )\mbox{$\underline{r_{ij}}$},$$ (2.91)

where $\mbox{$\underline{r}$}_{ij}=\mbox{$\underline{r}$}_{j}-\mbox{$\underline{r}$}_{i}$. The force on atom $i$ is the negative of this.

With the pair forces thus defined the contribution to be added to the atomic virial from each atom pair is then

$${\cal W}=-\mbox{$\underline{r}$}_{ij}\cdot\mbox{$\underline{f}$}_{j}.$$ (2.92)

The contribution to be added to the atomic stress tensor is given by

$$\sigma^{\alpha \beta}=r_{ij}^{\alpha}f_{j}^{\beta},$$ (2.93)

where $\alpha$ and $\beta$ indicate the $x,y,z$ components. The atomic stress tensor is symmetric.

The long range correction for the system potential is in two parts. Firstly by analogy with the short ranged potentials the correction to the local density is obtained by

$$\rho_{i}=\rho_{i}^{o}+4\pi \bar{\rho}\int_{r_{cut}}^{\infty} \left(\frac{a}{r}\right)^{m}r^{2}dr,$$ (2.94)

where $\rho_{i}^{o}$ is the uncorrected local density and $\bar{\rho}$ is the mean particle density. Evaluating the integral yields

$$\delta \rho=4\pi\frac{\bar{\rho}a^{3}}{(m-3)} \left(\frac{a}{r_{cut}}\right )^{m-3}$$ (2.95)

which is the local density correction and is identical for all atoms. The correction is applied immediately after the local density is calculated. The density term of the Sutton Chen potential needs no further correction. The pair term correction is obtained by analogy with the short ranged potentials and is

$$U_{corr}=2\pi\frac{N\epsilon\bar{\rho}a^{3}}{(n-3)} \left(\frac{a}{r_{cut}}\right )^{n-3}.$$ (2.96)

The correction to the local density having already been applied.

To estimate the virial correction we assume the corrected local densities are constants (i.e. independent of distance - at least beyond the range $r_{cut}$). This allows the virial correction to be computed by the methods used in the short ranged potentials. The result is:

$${\cal W}_{corr}=-2\pi\bar{\rho}a^{3}\left\{\frac{nN\epsilon}... ...{a}{r_{cut}}\right )^{m-3} \sum_{i}^{N}\rho_{i}^{-1/2}\right\}$$ (2.97)

This correction may be used as it stands, or with the further approximation:

$$\sum_{i}^{N}\rho_{i}^{-1/2}=\frac{N}{<\rho_{i}^{1/2}>}$$ (2.98)

where $<\rho_{i}^{1/2}>$ is regarded as a constant of the system.

In DL_POLY_2 the metal forces are handled by the routine SUTTCHEN. The local density is calculated by routines SCDENS and DENLOC. The long range corrections are calculated by LRCMETAL.