# The embedded atom method (EAM) potential

## Contents

## Potential form

### General form

- The site potential energy is

$$ U_i = \frac{1}{2} \sum_{j\neq i} \phi(r_{ij}) + F (\rho_i). $$

- Here, the part with [math]\phi(r_{ij})[/math] is a pairwise potential and [math]F (\rho_i)[/math] is the embedding potential, which depends on the electron density [math]\rho_i[/math] at site [math]i[/math]. The many-body part of the EAM potential comes from the embedding potential.

- This density is contributed by the neighbors of [math]i[/math]:

$$ \rho_i = \sum_{j\neq i} f(r_{ij}). $$

- Therefore, the form a an EAM potential is completed determined by the three functions: [math]\phi[/math], [math]f[/math], and [math]F[/math].

### The version by Zhou et al.

- The pair potential between two atoms of the same type [math]a[/math] is

$$ \phi^{aa}(r) = \frac{ A^a \exp[-\alpha(r/r_e^a-1)] } { 1+(r/r_e^a-\kappa^a)^{20} } - \frac{ B^a \exp[-\beta(r/r_e^a-1)] } { 1+(r/r_e^a-\lambda^a)^{20} }. $$

- The contribution of the electron density from an atom of type [math]a[/math] is

$$ f^a(r) = \frac{ f_e^a \exp[-\beta(r/r_e^a-1)] } { 1+(r/r_e^a-\lambda^a)^{20} }. $$

- The pair potential between two atoms of different types [math]a[/math] and [math]b[/math] is then constructed as

$$ \phi^{ab}(r) = \frac{1}{2} \left[ \frac{ f^b(r) } { f^a(r) } \phi^{aa}(r) + \frac{ f^a(r) } { f^b(r) } \phi^{bb}(r) \right]. $$

- The embedding energy function is piecewise:

$$ F(\rho) = \sum_{i=0}^3 F_{ni} \left( \frac{\rho}{\rho_n}-1\right)^i, \quad (\rho < 0.85\rho_e) $$ $$ F(\rho) = \sum_{i=0}^3 F_{i} \left( \frac{\rho}{\rho_e}-1\right)^i, \quad (0.85\rho_e \leq \rho < 1.15\rho_e) $$ $$ F(\rho) = F_{e} \left[ 1- \ln \left(\frac{\rho}{\rho_s}\right)^{\eta}\right] \left(\frac{\rho}{\rho_s}\right)^{\eta}, \quad (\rho \geq 1.15\rho_e) $$

### The version by Dai et al.

This is a very simple EAM-type potential which is an extension of the Finnis-Sinclair potential. The function for the pair potential is \begin{equation} \phi(r) = \begin{cases} (r-c)^2 \sum_{n=0}^4 c_n r^n & r \leq c \\ 0 & r > c \end{cases} \end{equation} The function for the density is \begin{equation} f(r) = \begin{cases} (r-d)^2 + B^2 (r-d)^4 & r \leq d \\ 0 & r > d \end{cases} \end{equation} The function for the embedding energy is \begin{equation} F(\rho) = - A \rho^{1/2}. \end{equation}

## Parameters

To be written.

## Potential file format

### Zhou et al.

- The potential file for this potential model reads

eam_zhou_2004 r_e f_e rho_e rho_s alpha beta A B kappa lambda F_n0 F_n1 F_n2 F_n3 F_0 F_1 F_2 F_3 eta F_e cutoff

- The last parameter [math]cutoff[/math] is the cutoff distance which is not intrinsic to the model. The order of the parameters is the same as in Table III of the paper by Zhou
*et al.*.

### Dai et al.

- The potential file for this potential model reads

eam_dai_2006 A d c c_0 c_1 c_2 c_3 c_4 B

## Tips

- We are still working on implementing a general spline-based version of the EAM potential.