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 [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.