The embedded atom method (EAM) potential

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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.
• This density is contributed by the neighbors of $i$:

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

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

The version by Zhou et al.

• 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 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 $$\phi(r) = \begin{cases} (r-c)^2 \sum_{n=0}^4 c_n r^n & r \leq c \\ 0 & r > c \end{cases}$$ 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 $$F(\rho) = - A \rho^{1/2}.$$

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 $cutoff$ 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..

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.