Common neighbor parameter¶

CommonNeighborParameter measures how symmetrically the common neighbors of each atom and its neighbors are arranged: zero in perfect fcc, bcc and diamond, large at defects and surfaces, defined for any structure.

import numpy as np
import mdapy as mp
from mdapy.build import build_crystal
from mdapy.identify import CommonNeighborParameter

# fcc copper of 4 x 4 x 4 cubic cells, with one atom missing
crystal = build_crystal("fcc", 3.615, elements="Cu", repeat=(4, 4, 4))
system = crystal.filter(np.arange(len(crystal)) > 0)

result = CommonNeighborParameter(cutoff=3.0).compute(system)
result.cnp.max()  # Å^2, at the neighbors of the vacancy
system = system.with_columns(result)  # the column 'cnp'

In a polycrystal, cut in half to show the boundaries inside:

from mdapy.build import build_crystal, build_polycrystal
from mdapy.visualization import render

grains = build_polycrystal(
    build_crystal("fcc", 3.615, elements="Cu"), mp.Box(50.0), 5, seed=4
)
cnp = CommonNeighborParameter(cutoff=3.08).compute(grains).cnp
render(grains, "images/cnp.png", atoms=grains["positions"][:, 0] < 25, color=cnp,
       colormap="plasma", value_range=(0, 20), width=800, height=600, legend="CNP (Å²)")  # fmt: skip

Common neighbor parameter of a polycrystal

Zero inside the grains (dark blue), up to 20 Å² and more at the boundaries.

Parameters¶

Name	Type	Default	Unit	Meaning
`cutoff`	float64		Å	neighbors and common neighbors are closer than it

compute(system, *, neighbors=None, progress=None, cancel=None) takes the arguments of every analysis. A given neighbor list must reach the cutoff.

Results¶

A CommonNeighborParameterResult:

Attribute	Type	Unit	Meaning
`cnp`	(N,) float64	Å²	the parameter of each atom, NaN for an atom without neighbors; also its column `cnp`

Errors¶

The constructor raises InvalidParameter for a cutoff that is not positive; compute raises it for a neighbor list that does not fit.

Definition¶

For an atom \(i\) with \(N_i\) neighbors \(j\), the common neighbors \(k\) of \(i\) and \(j\) are the atoms closer than the cutoff to both. With \(\mathbf r_{ik} = \mathbf x_i - \mathbf x_k\),

\[ Q_i = \frac{1}{N_i} \sum_{j} \Bigl| \sum_{k} (\mathbf r_{ik} + \mathbf r_{jk}) \Bigr|^2 . \]

It is zero in perfect fcc, bcc and diamond, and \(2a^2/3\) in ideal hcp of lattice constant \(a\) with the cutoff between its first and second shells. Unlike the centro-symmetry parameter it needs a cutoff rather than a number of neighbors.

Please cite

Tsuzuki et al. (2007).