Special quasirandom structures¶

sqs_alloy places elements in given fractions on the sites of a perfect lattice so that the pairs of the nearest shells of neighbors are those of a random alloy, the best that a small cell can do for a random alloy.

from mdapy.build import build_crystal, sqs_alloy

sites = build_crystal("fcc", 3.6, repeat=(4, 4, 4))
cuni = sqs_alloy(sites, {"Cu": 1, "Ni": 1}, shells=3, seed=1)
cantor = sqs_alloy(sites, {"Co": 1, "Cr": 1, "Fe": 1, "Mn": 1, "Ni": 1}, seed=1)

import numpy as np
from mdapy.visualization import render, write_png

left = render(cuni, width=600, height=600, legend=True)
right = render(cantor, width=600, height=600, legend=True)
write_png("images/sqs.png", np.hstack([left, right]))

Special quasi-random structures

Left: equiatomic CuNi of 256 atoms whose pair correlations in the first three shells come as close as the cell allows to those of the random alloy. Right: the Cantor alloy on the same sites.

Parameters¶

sqs_alloy(system, fractions, *, shells=3, weights=None, steps=100000, temperature=1.0, seed=None, progress=None, cancel=None)

Name	Type	Default	Meaning
`system`	`System`		the sites, a perfect lattice such as from `build_crystal`
`fractions`	dict of str to float64		the fraction of each element; the counts are those of `random_alloy`
`shells`	int32	3	the number of shells of neighbors made random, 1 to 10
`weights`	sequence of float64	all 1	the weight of each shell in the deviation
`steps`	int64	100000	the number of swaps tried
`temperature`	float64	1	the starting temperature of the annealing, in units of the typical change of a swap; 0 keeps only improvements
`seed`	int64 or `numpy.random.Generator`	`None`	the starting alloy and the swaps
`progress`, `cancel`		`None`	as for every analysis

The shells are the distinct distances between sites, nearest first, so the lattice must be perfect.

Results¶

The system with a new element column, the best arrangement found.

Errors¶

sqs_alloy raises InvalidParameter for a number of shells out of range, weights that are not one per shell or negative, a negative number of steps or temperature, and the errors of random_alloy; Cancelled when cancelled.

Definition¶

With \(N^{(s)}_{ab}\) the ordered pairs of shell \(s\) from an atom of element \(a\) to one of \(b\), \(P_s\) all pairs of shell \(s\) and \(x_a\) the fractions, the deviation from a random alloy

\[ D = \sum_s w_s \sum_{a,b} \left( \frac{N^{(s)}_{ab}}{P_s} - x_a x_b \right)^2 \]

is zero when the Warren–Cowley parameters \(\alpha^{(s)}_{ab} = 1 - N^{(s)}_{ab} / (x_a x_b P_s)\) of every shell vanish. Starting from a random alloy, pairs of atoms of different elements swap places when the swap lowers \(D\), or with the probability \(e^{-\Delta D / T}\) otherwise, while \(T\) falls linearly to zero; the arrangement with the lowest \(D\) is returned.

Please cite

Zunger et al. (1990).