*Phys. Rev. Lett.*

**59**, 2666 (1987)

interactions:meam

The energy in potentials of the Modified Embedded Atom type behaves like the EAM type and consists of two parts, a pair potential term specified by the function $\Phi\left(r\right)$ representing the electrostatic core-core repulsion, and a cohesive term specified by the function $F\left(n\right)$ representing the energy the ion core gets when it is “embedded” in the “Electron Sea”. This Embedding Energy is a function of the local electron density, which in turn is constructed as a superposition of contributions from neighboring atoms. The “electron” density depends on the electron transfer, $\rho\left(r\right)$, and the 3-body terms $f\left(r\right)$ and $g\left(\cos\theta\right)$.

The embedding function $F_i\left(n\right)$ and the 3-body angular function $g_i\left(\cos\theta\right)$ both depend on the type of the embedded atom, the transfer function $\rho_j\left(r\right)$ depends on the type of the donating atom, and the pair potential $\Phi_{ij}\left(r\right)$ and the 3-body radial function $f_{ij}\left(r\right)$ both depend on the types $i$ and $j$ of both atoms involved. All potential functions are given in a single file, whose format is described here.

The Modified Embedded Atom Method (MEAM) was adapted for use in *potfit* by Jeremy Nicklas.
It is currently offered in the following Molecular Dynamics codes:
(untested and yet-to-be-released version) IMD
implemented by Jeremy Nicklas, ohmms implemented by
Jeongnim Kim, and a private version of LAMMPS implemented by Richard Hennig.

The Modified Embedded Atom Method was suggested by Baskes
^{1)}
^{2)}.
A simplified version of MEAM that employs cubic splines instead of predetermined analytic
functions was developed later by Lenosky
^{3)}.
The explicit 3-body term may improve upon the description of
materials with highly-directional bonds while only being about 3-5 times slower than EAM.

Total energy form for MEAM potential:

$$E_{\text{total}}=\frac{1}{2}\sum_{i,j}^{N}\Phi_{ij}\left(r_{ij}\right)+\sum_i^N F_i\left(n_i\right)$$ $$n_i=\sum_{j\neq i}^N\rho_j\left(r_{ij}\right)+ \frac{1}{2}\sum_{j,k\neq i}^N f_{ij}\left(r_{ij}\right)f_{ik}\left(r_{ik}\right)g_{i}\left(\cos\theta_{jik}\right)$$

The functions $\Phi_{ij}\left(r_{ij}\right)$, $\rho_j\left(r_{ij}\right)$, and $F_i\left(n_i\right)$ are taken from the EAM model. The explicit 3-body term is included in the local “electron” density through the dependence on $f_{ij}\left(r_{ij}\right)$ and $g_{i}\left(\cos\theta_{jik}\right)$, where $\theta_{jik}$ is the angle between atoms $j$, $i$, and $k$ centered on atom $i$.

The invariance properties of MEAM are:

$$\rho_j\left(r\right)\rightarrow \alpha\rho_j\left(r\right)$$ $$g_i\left(\cos\theta\right)\rightarrow \alpha g_i\left(\cos\theta\right)$$ $$F_i\left(n\right)\rightarrow F_i\left(\tfrac{n}{\alpha}\right)$$

and

$$f_{ij}\left(r\right)\rightarrow \beta f_{ij}\left(r\right)$$ $$g_i\left(\cos\theta\right)\rightarrow \tfrac{1}{\beta^2} g_i\left(\cos\theta\right)$$

To describe a system of $N$ atom types you need $N(N+4)$ potentials.

# atom types | $\Phi_{ij}$ | $\rho_j$ | $F_i$ | $f_{ij}$ | $g_i$ | Total # potentials |
---|---|---|---|---|---|---|

$N$ | $N(N+1)/2$ | $N$ | $N$ | $N(N+1)/2$ | $N$ | $N(N+4)$ |

1 | 1 | 1 | 1 | 1 | 1 | 5 |

2 | 3 | 2 | 2 | 3 | 2 | 12 |

3 | 6 | 3 | 3 | 6 | 3 | 21 |

4 | 10 | 4 | 4 | 10 | 4 | 32 |

The potential table is assumed to be symmetric, i.e. the potential for the atom types 1-0 is the same as the potential 0-1.

The order of the potentials in the potential file for $N$ atom types is:

$\Phi_{00}, \ldots, \Phi_{0N}, \Phi_{11}, \ldots, \Phi_{1N}, \ldots, \Phi_{NN}$

$\rho_0, \ldots, \rho_N$

$F_0, \ldots, F_N$

$f_{00}, \ldots, f_{0N}, f_{11}, \ldots, f_{1N}, \ldots, f_{NN}$

$g_0, \ldots, g_N$

Tabulated MEAM potentials require the embedding function $F_i$ to be defined at a density of $1.0$. This is necessary to fix the gauge degrees of freedom.

M. I. Baskes, *Phys. Rev. Lett.* **59**, 2666 (1987)

M. I. Baskes, J. S. Nelson, and A. F. Wright, *Phys. Rev. B* **40**, 6085 (1989)

T. J. Lenosky, B. Sadigh, E. Alonso, V. Bulatov, T. D. de la Rubia, J. Kim, A. F. Voter, and J. D. Kress, *Modeling Simul. Mater. Sci. Eng.* **8**, 825 (2000)

interactions/meam.txt · Last modified: 2018/09/27 08:00 by 170.79.176.67