The EMT potential is included in the ASE package in order to have a
simple calculator that can be used for quick demonstrations and
tests.
Theory
In the following, the seven parameters \(E_{0i}\), \(s_{0i}\),
\(V_{0i}\), \(\eta_{2i}\), \(\kappa_{i}\), \(\lambda_{i}\),
and \(n_{0i}\) are specific for the species of atom \(i\).
Energy
In the effective-medium theory (EMT), the energy is given by
\[E = \sum_{i=1}^{N} (E_{\mathrm{c},i} + E_{\mathrm{AS},i})\]
The cohesive function \(E_{\mathrm{c},i}\) describes the energy in the
reference system, where we assume the face-centered cubic (fcc) structure and
given by
\[E_{\mathrm{c},i}
= E_{0i} f(\lambda_i (s_{i} - s_{0i}))
= E_{0i} f(\lambda_i \dot{s}_{i})\]
\[f(x) = (1 + x) \exp(-x)\]
where \(E_{0i}\) is the cohesive energy, \(s_{0i}\) is the Wigner–Seitz
radius in the equilibrium fcc state, and \(\lambda_i\) is related to the
curvature of the energy–volume curve and thus to the bulk modulus.
\(s_i\) is the neutral-sphere radius, and
\[\dot{s}_i
= s_{i} - s_{0i}
= - \frac{1}{\beta \eta_{2i}} \log \frac{\sigma_{1i}}{12 \gamma_{1i}}\]
where \(\beta\) is the constant related to the Wigner–Seitz radius and the
first nearest neighbor distance (cf. Tips).
\(\sigma_{1i}`\) is given by
\[\sigma_{1i}
= \sum_{j} \chi_{ij} w(r_{ij})
\exp(- \eta_{2j} (r_{ij} - \beta s_{0j}))
= \sum_{j} \dot{\sigma}_{1ij}^\mathrm{s}\]
The summation is over the neighbors of atom \(i\).
\(r_{ij}\) is the distance of atoms \(i\) and \(j\) and given
using their position vectors \(\mathbf{r}_i\) and \(\mathbf{r}_j\) by
\[r_{ij} = |\mathbf{r}_{ij}| = |\mathbf{r}_{j} - \mathbf{r}_i|\]
\(\chi_{ij}\) is given by
\[\chi_{ij} = \frac{n_{0j}}{n_{0i}}\]
The contribution from atom \(j\) is given by
\[\dot{\sigma}_{1ij}^\mathrm{s} = \chi_{ij} w(r_{ij})
\exp(- \eta_{2j} (r_{ij} - \beta s_{0j}))\]
For later convenience in Forces, the contribution from atom \(i\) to
atom \(j\) is also written as;
\[\dot{\sigma}_{1ij}^\mathrm{o} = \chi_{ji} w(r_{ij})
\exp(- \eta_{2i} (r_{ij} - \beta s_{0i}))\]
\(w(r)\) is the smooth cutoff function given by
\[w(r) = \frac{1}{1 + \exp(a (r - r_\mathrm{c}))}\]
\(\gamma_{1i}\) is a correction factor when considering beyond the first
nearest neighbor sites and given by (cf. Tips)
\[\gamma_{1i} = \frac{1}{12} (
n^\mathrm{1NN} w(d_0^\mathrm{1NN}) \exp(\eta_{2i} (d_0^\mathrm{1NN} - \beta s_{0i})) +
n^\mathrm{2NN} w(d_0^\mathrm{2NN}) \exp(\eta_{2i} (d_0^\mathrm{2NN} - \beta s_{0i})) +
n^\mathrm{3NN} w(d_0^\mathrm{3NN}) \exp(\eta_{2i} (d_0^\mathrm{3NN} - \beta s_{0i})) +
\cdots
)\]
which is \(1\) when considering only up to the first nearest neighbors of
the equilibrium fcc structure.
The atomic-sphere correction \(E_{\mathrm{AS},i}\) describes the derivation
from the reference fcc system and given by
\[E_{\mathrm{AS},i} = E_{\mathrm{AS},i}^{1} + E_{\mathrm{AS},i}^{2}\]
\(E_{\mathrm{AS},i}^{1}\) is the pair interactions of the real system,
and \(E_{\mathrm{AS},i}^{2}\) is the negative of the pair interactions of
the reference unary perfect fcc structure.
Both terms are described using the following pair interaction function;
\[V_{ij} (r) = -V_{0i} \cdot \frac{1}{\gamma_{2i}} \chi_{ij} w(r)
\exp(-\frac{\kappa_{j}}{\beta}(r - \beta s_{0j}))\]
\(\gamma_{2i}\) is a correction factor when considering beyond the first
nearest neighbor sites and given by (cf. Tips)
\[\gamma_{2i} = \frac{1}{12} (
n_\mathrm{1NN} w(d_\mathrm{1NN}) \exp(-\frac{\kappa_{2i}}{\beta} (r_\mathrm{1NN} - \beta s_{0i})) +
n_\mathrm{2NN} w(d_\mathrm{2NN}) \exp(-\frac{\kappa_{2i}}{\beta} (r_\mathrm{2NN} - \beta s_{0i})) +
n_\mathrm{3NN} w(d_\mathrm{3NN}) \exp(-\frac{\kappa_{2i}}{\beta} (r_\mathrm{3NN} - \beta s_{0i})) +
\cdots
)\]
Here, if we consider only up to the first nearest neighbors,
\[\gamma_{2i} \rightarrow 1\]
For \(E_{\mathrm{AS},i}^{2}\), only the interactions up to the first
nearest neighbors are considered, i.e., \(j = i\) and
\(r_{ij} = d^\mathrm{1NN} = \beta s_{i}\). Thus,
\[E_{\mathrm{AS},i}^{2} = \frac{1}{2} n^\mathrm{1NN} V_{ii}(d^\mathrm{1NN})
= -\frac{12}{2} V_{ii} (\beta s_{i})
= 6 V_{0i} \exp(-\kappa_{i} \dot{s}_i)\]
The first term \(E_{\mathrm{AS},i}^{1}\) is the pair interactions of the
real system. Here we consider the interactions up to a certain cutoff radius,
and we average the contribution from atom \(i\) to atom \(j\) and that
from atom \(j\) to atom \(i\). Thus,
\[E_{\mathrm{AS},i}^{1} = \frac{1}{2}
\sum_{j} \frac{1}{2} \left(V_{ij}(r_{ij}) + V_{ji}(r_{ij})\right)
= - \frac{V_{0i}}{2 \gamma_{2i}} \cdot \frac{1}{2} \sum_{j}
(\dot{\sigma}_{2ij}^\mathrm{s} + \dot{\sigma}_{2ij}^\mathrm{o})\]
where
\[\dot{\sigma}_{2ij}^\mathrm{s}
= \chi_{ij} w(r_{ij})
\exp(-\frac{\kappa_j}{\beta} (r_{ij} - \beta s_{0j}))\]
\[\dot{\sigma}_{2ij}^\mathrm{o}
= \chi_{ji} w(r_{ij})
\exp(-\frac{\kappa_i}{\beta} (r_{ij} - \beta s_{0i}))\]
and further for unary perfect fcc systems,
Forces
The forces on atom \(i\) can be computed as
\[\mathbf{F}_{i}
= -\nabla_i E
= \sum_j \frac{\partial E}{\partial r_{ij}} \frac{\mathbf{r}_{ij}}{r_{ij}}
= \sum_j \mathbf{f}_{ij}\]
where the force applied on atom \(i\) by atom \(j\) is given by
\[\mathbf{f}_{ij}
= \frac{\partial E}{\partial r_{ij}} \frac{\mathbf{r}_{ij}}{r_{ij}}\]
The derivative of \(E\) with respect to \(r_{ij}\) is further written
as
\[\frac{\partial E}{\partial r_{ij}} = \left(
\frac{\partial E_{\mathrm{c},i}}{\partial r_{ij}} +
\frac{\partial E_{\mathrm{c},j}}{\partial r_{ij}} +
\frac{\partial E_{\mathrm{AS},i}^2}{\partial r_{ij}} +
\frac{\partial E_{\mathrm{AS},j}^2}{\partial r_{ij}} \right) + \left(
\frac{\partial E_{\mathrm{AS},i}^1}{\partial r_{ij}} +
\frac{\partial E_{\mathrm{AS},j}^1}{\partial r_{ij}} \right)\]
Be careful that we also need to consider the contribution of the energy term
associated to atom \(j\).
The first terms depend on \(r_{ij}\) indirectly via \(s_{i}\) and \(s_{j}\).
\[\frac{\partial E_{\mathrm{c},i}}{\partial r_{ij}} =
\frac{\partial E_{\mathrm{c},i}}{\partial s_{i}}
\frac{\partial s_i}{\partial \sigma_{1i}}
\frac{\partial \sigma_{1i}}{\partial r_{ij}}\]
\[\frac{\partial E_{\mathrm{c},j}}{\partial r_{ij}} =
\frac{\partial E_{\mathrm{c},j}}{\partial s_{j}}
\frac{\partial s_j}{\partial \sigma_{1j}}
\frac{\partial \sigma_{1j}}{\partial r_{ij}}\]
\[\frac{\partial E_{\mathrm{AS},i}^2}{\partial r_{ij}} =
\frac{\partial E_{\mathrm{AS},i}^2}{\partial s_{i}}
\frac{\partial s_i}{\partial \sigma_{1i}}
\frac{\partial \sigma_{1i}}{\partial r_{ij}}\]
\[\frac{\partial E_{\mathrm{AS},j}^2}{\partial r_{ij}} =
\frac{\partial E_{\mathrm{AS},j}^2}{\partial s_{j}}
\frac{\partial s_j}{\partial \sigma_{1j}}
\frac{\partial \sigma_{1j}}{\partial r_{ij}}\]
They can be computed using
\[\frac{\partial E_{\mathrm{c},i}}{\partial s_i}
= - E_{0i} \lambda_i^2 \dot{s}_i \exp(-\lambda_i \dot{s}_i)\]
\[\frac{\partial E_\mathrm{AS}^2}{\partial s_i}
= -6 V_{0i} \kappa_i \exp(-\kappa_i \dot{s}_i)\]
\[\frac{\mathrm{d}s_i}{\mathrm{d}\sigma_{1i}}
= \frac{\mathrm{d}\dot{s}_i}{\mathrm{d}\sigma_{1i}}
= -\frac{1}{\beta\eta_{2i}} \frac{1}{\sigma_{1i}}\]
\[\frac{\partial \sigma_{1i}}{\partial r_{ij}}
= \chi_{ij}
\left(
\frac{\partial w}{\partial r_{ij}}
\exp(-\eta_{2j} (r_{ij} - \beta s_{0j})) -
w(r_{ij}) \eta_{2j}
\exp(-\eta_{2j} (r_{ij} - \beta s_{0j}))
\right)
= \left(
\frac{1}{w(r_{ij})} \frac{\partial w}{\partial r_{ij}} -
\eta_{2j}
\right) \dot{\sigma}_{1ij}^\mathrm{s}\]
\[\frac{\partial \sigma_{1j}}{\partial r_{ij}}
= \chi_{ji}
\left(
\frac{\partial w}{\partial r_{ij}}
\exp(-\eta_{2i} (r_{ij} - \beta s_{0i})) -
w(r_{ij}) \eta_{2i}
\exp(-\eta_{2i} (r_{ij} - \beta s_{0i}))
\right)
= \left(
\frac{1}{w(r_{ij})} \frac{\partial w}{\partial r_{ij}} -
\eta_{2i}
\right) \dot{\sigma}_{1ij}^\mathrm{o}\]
The second part directly depends on \(r_{ij}\) and given by
\[\frac{\partial E_{\mathrm{AS},i}^1}{\partial r_{ij}} +
\frac{\partial E_{\mathrm{AS},j}^1}{\partial r_{ij}}
= - \frac{1}{2} \left(
\frac{V_{0i}}{2 \gamma_{2i}} \frac{\partial \dot{\sigma}_{2ij}^\mathrm{s}}{\partial r_{ij}} +
\frac{V_{0j}}{2 \gamma_{2j}} \frac{\partial \dot{\sigma}_{2ij}^\mathrm{o}}{\partial r_{ij}}
\right)\]
where
\[\frac{\partial \dot{\sigma}_{2ij}^\mathrm{s}}{\partial r_{ij}}
= \chi_{ij}
\left(
\frac{\partial w}{\partial r_{ij}}
\exp(-\frac{\kappa_j}{\beta} (r_{ij} - \beta s_{0j})) -
w(r_{ij}) \frac{\kappa_j}{\beta}
\exp(-\frac{\kappa_j}{\beta} (r_{ij} - \beta s_{0j}))
\right)
= \left(
\frac{1}{w(r_{ij})} \frac{\partial w}{\partial r_{ij}} -
\frac{\kappa_j}{\beta}
\right)
\dot{\sigma}_{2ij}^\mathrm{s}\]
\[\frac{\partial \dot{\sigma}_{2ij}^\mathrm{o}}{\partial r_{ij}}
= \chi_{ji}
\left(
\frac{\partial w}{\partial r_{ij}}
\exp(-\frac{\kappa_i}{\beta} (r_{ij} - \beta s_{0i})) -
w(r_{ij}) \frac{\kappa_i}{\beta}
\exp(-\frac{\kappa_i}{\beta} (r_{ij} - \beta s_{0i}))
\right)
= \left(
\frac{1}{w(r_{ij})} \frac{\partial w}{\partial r_{ij}} -
\frac{\kappa_i}{\beta}
\right)
\dot{\sigma}_{2ij}^\mathrm{o}\]
Note that
\[\frac{\mathrm{d}w}{\mathrm{d}r} = a w(r) (w(r) - 1)\]
Stress
The static part of the virial stress can be given as
\[\tau^{\alpha \beta}
= \frac{1}{\Omega} \frac{1}{2}
\sum_{i=1}^{N} \sum_{j \neq i} r_{ij}^{\alpha} f_{ij}^{\beta}
= \frac{1}{\Omega}
\sum_{i=1}^{N} \sum_{j > i} r_{ij}^{\alpha} f_{ij}^{\beta}\]
where \(\alpha\) and \(\beta\) are indices for Cartesian components.
When considering all the neighbors for each atom, we should not forget the
factor \(1/2\).
Tips
For the fcc structure, the numbers of neighbor sites and the
distances of first several shells are
\[ \begin{align}\begin{aligned}:wowrap:\\\begin{split}n^\mathrm{1NN} &= 12, & \quad d^\mathrm{1NN} &= \beta s_{i} \\
n^\mathrm{2NN} &= \phantom{0}6, & \quad d^\mathrm{2NN} &= \sqrt{2}\,d^\mathrm{1NN} \\
n^\mathrm{3NN} &= 24, & \quad d^\mathrm{3NN} &= \sqrt{3}\,d^\mathrm{1NN} \\
n^\mathrm{4NN} &= 12, & \quad d^\mathrm{4NN} &= \sqrt{4}\,d^\mathrm{1NN} \\
n^\mathrm{5NN} &= 24, & \quad d^\mathrm{5NN} &= \sqrt{5}\,d^\mathrm{1NN}\end{split}\end{aligned}\end{align} \]
where \(s_{i}\) is the Wigner–Seitz radius of the species of atom
\(i\) and \(\beta = 2^{-1/2} (16 \pi / 3)^{1/3} \approx 1.809\).