# The embedded atom method (EAM) potential

## Brief descriptions

• This is the EAM potential in some analytical forms as in [Zhou 2004] and [Dai 2006].
• It currently only applies to systems with a single atom type.

## 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 $\phi(r_{ij})$ is a pairwise potential and $F (\rho_i)$ is the embedding potential, which depends on the electron density $\rho_i$ at site $i$. The many-body part of the EAM potential comes from the embedding potential.
• The density $F (\rho_i)$ is contributed by the neighbors of $i$:

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

• Therefore, the form of an EAM potential is completely determined by the three functions: $\phi$, $f$, and $F$.

### The version by [Zhou 2004]

• The pair potential between two atoms of the same type $a$ 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 $a$ 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 $a$ and $b$ 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 2006]

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

## Parameters

See [Zhou 2004] and [Dai 2006].

## Potential file format

• The potential file for the version in [Zhou 2004] reads
   eam_zhou_2004 1
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 $cutoff$ 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..
• The potential file for the version in [Dai 2006] reads
   eam_dai_2006 1
A
d
c
c_0
c_1
c_2
c_3
c_4
B