mapp.dmd.atoms.ff_eam_fs

atoms.ff_eam_fs(fs_file, r_crd, C, elems=None)

Tabulated Finnis-Sinclair EAM

Assigns Finnis-Sinclair EAM force field to system. For explanation of the parameter see the Notes section.

Parameters:

fs_file : string

relative path to DYNAMO file with fs format

r_crd : double[nelems]

cutoff radius for mass exchange

C : double[nelems]

dimensionless factor for each element, see here

elems : string[nelems]

mapping elements

Returns:

None

Notes

This is tabulated form of Finnis-Sinclair Embedded Atom Method (EAM) potential

Consider the general form of EAM potential:

U=\frac{1}{2}\sum_{i}\sum_{j\neq i}\phi_{\gamma \delta}{(x_{ij}) }+\sum_i E_\gamma \left(\sum_{j\neq i} \rho_{\delta\gamma}(x_{ij}) \right),

From here on, greek superscipts/subscripts are used to refer to elements present the system; \gamma, and \delta denote type of atom i and atom j, repectively. E, \rho, and \phi are embedding, electron density, and pair functions, respectively. Also x_{ij} refers to distance between i and j. Now the multi component formulation of DMD free energy would be:

F=&\frac{1}{2}\sum_{i,\gamma,j\neq i,\delta}c_i^\gamma c_j^\delta \omega_{\gamma \delta}\left(x_{ij}\right)+\sum_{i,\gamma} c_i^\gamma E_\gamma \left(\sum_{j\neq i,\delta} c_j^{\delta}\psi_{\delta\gamma}\left(x_{ij}\right)\right)-3k_BT\sum_{i,\gamma} c_i^\gamma \log\left(\sqrt{\pi e}\alpha_i^\gamma/\Lambda_\gamma\right)\\
&+k_BT\sum_{i,\gamma} c_i^\gamma\log c_i^\gamma+k_BT\sum_{i}c_i^v\log (c_i^v),

where

\Lambda_\gamma=\frac{h}{\sqrt{2\pi m_\gamma k_BT}}, \quad c_i^v=1-\sum_{\gamma}c_i^\gamma,

\omega_{\gamma\delta}(x_{ij})= \frac{1}{\left(\alpha^{\gamma\delta}_{ij}\sqrt{\pi}\right)^{3}}\int d^3\mathbf{x} \exp{\biggl[-\left(\frac{\mathbf{x}_{ij} -\mathbf{x}}{\alpha^{\gamma\delta}_{ij}}\right)^2\biggr]}\phi_{\gamma\delta}(|\mathbf{x}|)

\psi_{\gamma\delta}(x_{ij})=\frac{1}{\left(\alpha^{\gamma\delta}_{ij}\sqrt{\pi}\right)^{3}}\int d^3\mathbf{x} \exp{\biggl[-\left(\frac{\mathbf{x}_{ij} -\mathbf{x}}{\alpha^{\gamma\delta}_{ij}}\right)^2\biggr]}\rho_{\gamma\delta}(|\mathbf{x}|)

\alpha^{\gamma\delta}_{ij}=\sqrt{{\alpha_i^{\gamma}}^2+{\alpha_j^{\delta}}^2},\quad \mathbf{x}_{ji}=\mathbf{x}_j-\mathbf{x}_i

The numerical evaluation of these integrals is discussed here.

Recalling that

\langle U \rangle &= \frac{\partial }{\partial \beta}\left(\beta F \right)

the average potential energy is

\langle U \rangle = \frac{1}{2}\sum_{i,\gamma,j\neq i,\delta}c_i^\gamma c_j^\delta \omega_{\gamma \delta}\left(x_{ij}\right)+\sum_{i,\gamma} c_i^\gamma E_\gamma \left(\sum_{j\neq i,\delta} c_j^{\delta}\psi_{\delta\gamma}\left(x_{ij}\right)\right) -\frac{3}{2} k_B T

Examples

Iron Hydrogrn mixture

>>> from mapp import dmd
>>> sim=dmd.cfg(5,"configs/FeH-DMD.cfg")
>>> sim.ff_eam_fs("potentials/FeH.eam.fs")