The embedded atom method (EAM) potential

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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 [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.
  • The density [math]F (\rho_i)[/math] is contributed by the neighbors of [math]i[/math]:

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

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

The version by [Zhou 2004]

  • 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 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 \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

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 [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..
  • 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

References