EAM

from mdapy.calculators import EAM

potential = EAM.zhou(["Cu", "Ni"])  # or EAM("CuNi.eam.alloy")
potential.elements, potential.cutoff
potential.write("CuNi.eam.alloy")

The energy per atom of fcc copper and nickel against the lattice parameter, from the potential:

import numpy as np
from mdapy.build import build_crystal
from mdapy.visualization import plot

lattice = np.linspace(3.2, 4.4, 61)
with plot.style():
    fig, ax = plot.figure("single")
    for element in potential.elements:
        cells = [build_crystal("fcc", a, elements=element) for a in lattice]
        energy = [potential.compute(cell, ("energy",)).energy / 4 for cell in cells]
        ax.plot(lattice, energy, label=element)
    ax.set_xlabel("Lattice parameter (Å)")
    ax.set_ylabel("Energy (eV/atom)")
    ax.legend()
    plot.save(fig, "images/eam", formats=("png",))

The minima are the lattice parameters and cohesive energies of the potential, near the measured 3.615 Å and 3.52 Å; the curvature at the minimum gives the bulk modulus.

EAM(path) reads an embedded atom method potential in the eam/alloy (setfl) format of LAMMPS and computes every property.

Attribute	Type	Meaning
path	str or None	the potential file, None for a built potential
elements	tuple of str or None	the elements of the potential, in the order of the file; None for an average-atom potential
masses	(n,) float64	their masses in amu; the mass of the average atom for an average-atom potential
cutoff	float64	cutoff radius in Å

Method	Returns
compute(system, properties, *, neighbors=None, progress=None, cancel=None)	as every calculator, with the progress= and cancel= of Parallelism
write(path)	writes the potential as an eam/alloy file, which LAMMPS reads and which reads back to the same tables
EAM.zhou(elements, *, nr=2000, nrho=2000)	the potential of Zhou, Johnson and Wadley for the elements
average(concentrations)	the average-atom potential of an alloy of the elements

The atoms need an element column; elements the potential does not have raise InvalidParameter, and a malformed file raises FileFormatError with its line. A file may name species that are not elements, such as the A of an average atom; they are kept when the potential is written, but no atom of a system is one of them.

The energy of atom \(i\) is

\[ E_i = F_{s_i}\Big(\sum_j \rho_{s_j}(r_{ij})\Big) + \frac{1}{2}\sum_j \phi_{s_i s_j}(r_{ij}), \]

with the embedding energy \(F\), the electron density \(\rho\) and the pair potential \(\phi\) of the species \(s\) of the atoms, interpolated from the tables of the file as LAMMPS pair_style eam/alloy does. Each pair's virial is shared equally by its two atoms.

Please cite

Daw and Baskes (1984), and the paper of the potential.

Potentials of Zhou, Johnson and Wadley

potential = EAM.zhou(["Cu", "Ni", "Al"])
potential.write("CuNiAl.eam.alloy")

EAM.zhou builds the potential for any of Cu, Ag, Au, Ni, Pd, Pt, Al, Pb, Fe, Mo, Ta, W, Mg, Co, Ti, Zr and Cr, with the tables that the database tool of LAMMPS (tools/eam_database) writes for them: the parameters of Zhou et al. (2004), those of Lin et al. (2008) for Cr, and their rule for the pair potential between two elements,

\[ \phi_{ab}(r) = \frac{1}{2}\left(\frac{\rho_b(r)}{\rho_a(r)}\phi_{aa}(r) + \frac{\rho_a(r)}{\rho_b(r)}\phi_{bb}(r)\right). \]

Name	Type	Default	Unit	Meaning
elements	sequence of str			the elements, in the order of the file
nr	int32	2000		points of the tables in distance, up to the cutoff \(\sqrt{10}/2\) times the largest fcc lattice constant \(\sqrt 2\, r_e\)
nrho	int32	2000		points of the embedding tables in density

The parameters are fitted to the pure metals; the pair potentials between elements come from the rule above, not from a fit to alloys. An unknown or repeated element raises InvalidParameter.

Please cite

Zhou et al. (2004); with Cr, also Lin et al. (2008).

Average atom

average = potential.average({"Cu": 0.4, "Ni": 0.4, "Al": 0.2})
average.compute(system)  # every atom is the average atom
average.write("CuNiAl.average.eam.alloy")  # Cu, Ni, Al and the average atom A

average(concentrations) returns the average-atom potential of Varvenne et al. (2016) for a random alloy with the fractions \(c_a\) of the elements:

\[ F_A = \sum_a c_a F_a, \qquad \rho_A = \sum_a c_a \rho_a, \qquad \phi_{Ab} = \sum_a c_a \phi_{ab}, \qquad \phi_{AA} = \sum_{a,b} c_a c_b \phi_{ab}. \]

concentrations is a mapping from element to fraction, or the fractions in the order of elements; elements left out have none, and the fractions sum to 1. In MDAPY every atom of a system is the average atom, whatever its element. The written file holds the elements of the potential and the average atom, named A, last, so that LAMMPS can mix them, for example with pair_coeff * * CuNiAl.average.eam.alloy A Cu for a Cu solute in the average alloy. Invalid fractions raise InvalidParameter.

Please cite

Varvenne et al. (2016), and the paper of the potential averaged.