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interactions:adp

The angular dependent potentials were developed for the Fe-Ni system. It is a generalization of the EAM model for the simulation of the covalent component of bonding.

The angular dependent potential (ADP) for [[http://imd.itap.physik.uni-stuttgart.de/|IMD]] was implemented by Franz Gähler. It was adapted for use in *potfit* by Daniel Schopf.

The angular dependent potenial model was suggested by Mishin et al.^{1)}

Mishin, Y., Mehl, M. J., and Papaconstantopoulos, *D. A. Acta Mater.*,** 53** (15), 4029-4041, (2005) )).
Since it is based on the EAM potential model, the first two terms in the expression for the energy are exactly the same as for EAM potentials:
$$E_{\text{total}}=\frac{1}{2}\sum_{i,j(j\neq i)}^N\Phi_{ij}(r_{ij})+\sum_iF_i(n_i)+\frac{1}{2}\sum_{i,\alpha}(\mu_i^\alpha)^2+\frac{1}{2}\sum_{i,\alpha,\beta}(\lambda_i^{\alpha\beta})^2-\frac{1}{6}\sum_i\nu_i^2$$
Here the indices $i$ and $j$ enumerate atoms and the superscripts $\alpha,\beta=1,2,3$ refer to the Cartesion directions. The first two terms are explained in detail on the [[EAM|EAM]] page.
The additional three terms introduce non-central components of bonding through the vectors
$$\mu_i^\alpha = \sum_{j\neq i} u_{ij}(r_{ij})r_{ij}^\alpha$$
and tensors
$$\lambda_i^{\alpha\beta} = \sum_{j\neq i}w_{ij}(r_{ij})r_{ij}^\alpha r_{ij}^\beta$$
The quantities $\nu_i$ are traces of the $\lambda$-tensor:
$$\nu_i = \sum_\alpha\lambda_i^{\alpha\alpha}$$
These additional terms can be thought of as measures of the dipole ($\mu$) and quadrupole ($\lambda$) distortions of the local environment of an atom.
===== Number of potential functions =====
To describe a system with *N* atom types you need *N*(3*N*+7)/2 potentials.

===== Order of potential functions =====
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:
$\Phi_{00}, \ldots, \Phi_{0N}, \Phi_{11}, \ldots, \Phi_{1N}, \ldots, \Phi_{NN}$

$\rho_0, \ldots, \rho_N$

$F_0, \ldots, F_N$

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

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

===== Special remarks ===== Tabulated ADP 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.

# atom types | $\Phi_{ij}$ | $\rho_j$ | $F_i$ | $u_{ij}$ | $w_{ij}$ | Total # potentials |
---|---|---|---|---|---|---|

$N$ | $N(N+1)/2$ | $N$ | $N$ | $N(N+1)/2$ | $N(N+1)/2$ | $N(3N+7)/2$ |

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

2 | 3 | 2 | 2 | 3 | 3 | 11 |

3 | 6 | 3 | 3 | 6 | 6 | 24 |

4 | 10 | 4 | 4 | 10 | 10 | 38 |

$\rho_0, \ldots, \rho_N$

$F_0, \ldots, F_N$

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

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

===== Special remarks ===== Tabulated ADP 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.

interactions/adp.txt · Last modified: 2018/01/22 17:48 by daniel

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