~~NOTOC~~
====== Embedded Atom Method (EAM) Potentials ======
----

The energy in potentials of the Embedded Atom type consists of two parts, a pair potential 
term specified by the function $\Phi(r)$ representing the electrostatic core-core repulsion, 
and a cohesive term specified by the function $F(n)$ representing the energy the ion core 
gets when it is "embedded" in the "Electron Sea". This Embedding Energy is a function of the 
local electron density, which in turn is constructed as a superposition of contributions from 
neighboring atoms. This electron transfer is specified by the function $\rho(r)$.

The embedding function $F_i(n)$ depends on the type of the embedded atom, the transfer 
function $\rho_j(r)$ depends on the type $j$ of the donating atom, whereas the pair potential 
$\Phi_{ij}(r)$ depends on the types $i$ and $j$ of both atoms involved. All potential 
functions are given in a single file, whose format is described [[potfiles:main|here]].

The Embedded Atom Method (EAM) for [[http://imd.itap.physik.uni-stuttgart.de/|IMD]] was 
implemented by Erik Bitzek. It was adapted for use in //potfit// by Peter Brommer.

=====  Basic Theory  =====

The Embedded Atom Method was suggested by Daw and Baskes
((M. S. Daw and M. I. Baskes, //Phys. Rev. B// **29**, 6443 (1984) ))
((S. M. Foiles, M. I. Baskes, and M. S. Daw, //Phys. Rev. B// **33**, 7983 (1986) ))
as a way to overcome the main problem with two-body potentials: the coordination 
independence of the bond strength, while still being acceptable fast (about 2 times slower 
than pair potentials).

Ideas from the Density Functional Theory or the Tight Binding formalism may lead to the 
following form for the total energy:

$$E_\text{total}=\frac{1}{2}\sum_{i<j}^N\Phi_{ij}(r_{ij})+\sum_iF_i(n_i)\qquad\text{where}\qquad n_i=\sum_{j\neq i}\rho_j(r_{ij})$$

While an identification of the pair potential term $\Phi_{ij}(r_{ij})$ with the electrostatic core-core repulsion, and of the cohesive term $F_i(n_i)$ with the energy the ion core gets when it is "embedded" in the local electron density $n_i$ may be tempting, it is nevertheless without physical justification. Due to invariance properties of the EAM potential, a embedding energy term linear in the "electron" density can be described by pair interactions, thus shifting contributions between embedding and pair energy. So an isolated consideration of either part is not possible - physical relevance only lies in the combination of both.

The local electron density is constructed as a superposition of contributions $\rho_j(r_{ij})$ from neighboring atoms.

Also belonging to this analytical form are models like the glue model, and the Finnis-Sinclair potentials.

=====  Number of potential functions  =====

To describe a system with $N$ atom types you need $N(N+5)/2$ potentials.

^ # atom types ^ $\Phi_{ij}$ ^ $\rho_j$ ^ $F_i$ ^ Total # potentials ^
^ $N$ ^ $N(N+1)/2$ ^ $N$ ^ $N$ ^ $N(N+5)/2$ ^
| 1 | 1 | 1 | 1 | 3 |
| 2 | 3 | 2 | 2 | 7 |
| 3 | 6 | 3 | 3 | 12 |
| 4 | 10 | 4 | 4 | 18 |

=====  Order of potential functions  =====

The potential table is assumed to be symmetric, i.e. the potential for the atom types 1-0 is the same as the potential 0-1.

The order of the EAM potentials in the potential file for //N// atom types is:

$\Phi_{00}, \ldots, \Phi_{0N}, \Phi_{11}, \ldots, \Phi_{1N}, \ldots, \Phi_{NN},$\\
$\rho_0, \ldots, \rho_N,$\\
$F_0, \ldots, F_N,$

=====  Special remarks  =====

Tabulated EAM potentials require the embedding function $F_i$ to be defined at a density of $1.0$.
This is necessary to fix the gauge degrees of freedom.