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Special Quasirandom Structures

Optional dependency

SqsGenerator requires the sqsgen extra.

uv add "materialsframework[sqsgen]"

pip install "materialsframework[sqsgen]"

Generates a finite supercell whose short-range order mimics that of a truly random substitutional alloy, so disordered-alloy properties can be modeled with a small, deterministic cell instead of a much larger explicitly-disordered one.

Overview

A randomly-decorated supercell of practical size never quite reproduces the pair (and higher-order) correlations of an infinite random alloy, since finite-size sampling noise leaves residual short-range order. A Special Quasirandom Structure (SQS) is a supercell whose atomic decoration is chosen, by optimization rather than by chance, to match the target random-alloy correlation functions as closely as the cell size allows. SqsGenerator wraps the sqsgenerator optimizer: given a target composition, crystal structure, and supercell size, it searches over atomic decorations for the one whose correlation functions best match the random-alloy target.

Theory

For a cluster \(\alpha\) on a given coordination shell (e.g. all first-nearest-neighbor pairs), the correlation function is the average, over all clusters of that type in the supercell, of the product of a site-occupation variable across the cluster's sites:

\[ \Pi_\alpha(\sigma) = \frac{1}{N_\alpha} \sum_{\alpha} \prod_{i \in \alpha} \sigma_i \]

For a binary A-B system with concentration \(x\) of A, the exact pair-correlation value for a truly random arrangement, on any shell, is \(\Pi^{\text{random}} = (2x-1)^2\) (multi-component systems generalize this via a discretized occupation encoding rather than a single closed-form expression). SqsGenerator searches for the atomic decoration that minimizes the weighted deviation from this target across the requested coordination shells:

\[ \text{objective} = \sum_{k} w_k \sum_{\alpha \in \text{shell } k} \left| \Pi_\alpha(\sigma) - \Pi_\alpha^{\text{random}} \right| \]

where \(w_k\) is the shell_weights entry for shell \(k\) (default \(\{1: 1.0, 2: 0.5\}\); the first-neighbor shell is weighted twice as strongly as the second). The search runs for iterations trial decorations, either by random swaps (mode="random") or by systematic enumeration (mode="systematic"); the returned objective is the lowest value found, with 0 meaning an exact match to the random-alloy target on every requested shell.

References