Ackland–Jones analysis¶

AcklandJonesAnalysis gives each atom the crystal structure of its neighborhood, fcc, hcp, bcc, icosahedral or other, from the distribution of the angles between the bonds to its neighbors.

import mdapy as mp
from mdapy.build import build_crystal
from mdapy.identify import AcklandJonesAnalysis

# bcc iron of 5 x 5 x 5 cubic cells
system = build_crystal("bcc", 2.866, elements="Fe", repeat=(5, 5, 5))

result = AcklandJonesAnalysis().compute(system)
result.counts()  # {'other': 0, 'fcc': 0, 'hcp': 0, 'bcc': 250, 'ico': 0}

In a relaxed polycrystal of iron:

from mdapy.build import build_crystal, build_polycrystal
from mdapy.calculators import EAM
from mdapy.simulation import minimize
from mdapy.visualization import render

grains = build_polycrystal(
    build_crystal("bcc", 2.866, elements="Fe"),
    mp.Box(50.0),
    5,
    seed=4,
    min_distance=1.6,
)
grains = minimize(grains, EAM.zhou(["Fe"]), max_force=0.05, max_steps=300).system
structures = AcklandJonesAnalysis().compute(grains)
render(
    grains,
    "images/ackland-jones.png",
    color=structures,
    width=800,
    height=600,
    legend=True,
)

Ackland-Jones analysis of an iron polycrystal

Five grains of bcc iron (blue) after a short relaxation. The atoms of the boundaries take any type, mostly other (white) but also fcc (green), hcp (red) and ico (yellow): the analysis looks at the angles between bonds and knows nothing of grains.

Parameters¶

AcklandJonesAnalysis() has no parameters. compute(system, *, neighbors=None, progress=None, cancel=None) takes the arguments of every analysis. A given neighbor list must hold the 14 nearest neighbors of every atom.

Results¶

A StructureResult with the types of common neighbor analysis: other, fcc, hcp, bcc and ico.

Errors¶

compute raises InvalidParameter for a neighbor list that does not hold the 14 nearest neighbors.

Definition¶

The analysis takes the 14 nearest neighbors of an atom. With \(r_0^2\) the mean squared distance of the 6 nearest, \(n_0\) and \(n_1\) count the neighbors closer than \(\sqrt{1.45}\, r_0\) and \(\sqrt{1.55}\, r_0\). The cosines of the angles between every two of the \(n_0\) bonds fall into eight bins with the bounds −1, −0.945, −0.915, −0.755, −0.195, 0.195, 0.245, 0.795 and 1, which count \(\chi_0, \dots, \chi_7\). The deviations from the ideal structures are

\[ \begin{aligned} \delta_\mathrm{bcc} &= \frac{0.35\, \chi_4}{\chi_5 + \chi_6 - \chi_4}, \\ \delta_\mathrm{cp} &= \Big|1 - \frac{\chi_6}{24}\Big|, \\ \delta_\mathrm{fcc} &= \frac{0.61}{6} \big(|\chi_0 + \chi_1 - 6| + \chi_2\big), \\ \delta_\mathrm{hcp} &= \frac{1}{12} \big(|\chi_0 - 3| + |\chi_0 + \chi_1 + \chi_2 + \chi_3 - 9|\big), \end{aligned} \]

with \(\delta_\mathrm{bcc} = 0\) when \(\chi_0 = 7\), \(\delta_\mathrm{fcc} = 0\) when \(\chi_0 = 6\), \(\delta_\mathrm{hcp} = 0\) when \(\chi_0 \le 3\), and no bcc when \(\chi_5 + \chi_6 = \chi_4\). The atom is then, checked from the top down:

Condition	Type
\(\chi_7 > 0\)	`other`
\(\chi_4 < 3\)	`ico` when \(11 \le n_1 \le 13\), else `other`
\(\delta_\mathrm{bcc} \le \delta_\mathrm{cp}\)	`bcc` when \(n_1 \ge 11\), else `other`
\(11 \le n_1 \le 12\)	`fcc` when \(\delta_\mathrm{fcc} < \delta_\mathrm{hcp}\), else `hcp`
otherwise	`other`

These are the rules of the original implementation, which LAMMPS runs with legacy yes. An atom with fewer than 6 neighbors, as in a system without periodic directions of fewer than 7 atoms, is other.