Solid Solution and Disordered Crystalline Systems:  I- Symmetry Analysis for Counting and Enumerating Configurations.

Sami Mustapha

Département de Mathématiques, Université Paris VI, 4 Place Jussieu, 75252, Paris Cedex, France

Modelization of the geometry and thermodynamical properties of solid solutions or disordered crystalline systems is a daunting task. In recent decades, the most promising approach describes these systems as a weighted average of ordered configurations. But the number of such configurations is so large that ab initio calculations were precluded to produce directly average properties. They have been used to calibrate either atomistic potentials or the so-called cluster expansion interactions [1]. These "simple descriptions allows to calculate properties from a large number of configurations.

Symmetry naturally partitions the set of configurations in equivalence classes or orbits. Thus to completely describe the configurational space, it is sufficient to know for each class a representative and its multiplicity or length.

Taking advantage of the crystal symmetry reduces computational costs and can allow the use of continuously increasing compute performances for full ab initio studies of disordered systems [2].

Finding the symmetry independent classes (SICs) forming the configurational space has attracted numerous efforts. Initial studies were devoted to "simple\ materials such as alloys. In these cases, all atomic positions are translationally equivalent. Recently, Hart and co-workers [3] used the so-called Hermite normal form of integer matrices coupled with the corresponding diagonal Smith form to determine independent configurations within super-cells of a given index. They needed to introduce the multilattice description to count and enumerate configurations in more complex structure.

However, the space groups are the natural symmetry tools for compounds characterized by large cell containing numerous atoms. Using space groups, the group action theory [4] is a well documented framework to count and enumerate SICs. The basics that supports the counting and enumerating options available in the CRYSTAL code will be presented.

The Polya's counting theorem will be explained using the action matrix that is a very important conceptual tool for further developments in the group action theory framework.

Polya's theorem gives the number of SICs all possible compositions or at fixed one. This approach works for any number of involved crystallographic positions and atom species. Its de Bruijn's generalization allows to count spin configurations. However, these theorems are tools neither to enumerate classes and representatives nor to estimate the SICs' multiplicity. Efficient enumeration is obtained combining orderly generation algorithm [5] and the so-called surjective resolution [6] if more than 2 atomic species are considered. The low-memory demanding implementation in CRYSTAL will be sketched.

The implemented method scales linearly on the total number of configurations in contrast

with the direct method that scales as N2. Single, multiple-site examples with two and more species will be used to illustrate the approach.

This work has been published : J. Phys. : Condens. matter 25 2013

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