Setting up a cluster expansion on an augmented lattice

This page demonstrates a simple example of setting up a cluster expansion on an augmented lattice, with a focus on practical aspects. For detailed information on the underlying thermodynamic modeling using an augmented lattice and cluster expansion, please refer to the "Piecewise Polynomial Potential Partitioning (P4)" method introduced here.

Background
The main different of a standard lattice and an augmented lattice is that the latter contains "groups" of sites. The Euclidean distance between the sites within a group is artificially close. The purpose of creating these groups is to simulate the anharmonic vibrations by the hopping of an atom among the sites within each group. Consequently, the entities occupying these sites should always consist of vacancies. Therefore, when setting up a cluster expansion (CE) on an augmented lattice, it is important to inform the CE model about the existence of these "groups". Additionally, the CE model should enforce the condition that each group of sites is occupied by one and only one atom, preventing non-physical configurations where two atoms are in very close proximity. Considering these practical aspects is crucial when setting up a cluster expansion on an augmented lattice. In the following sections, we demonstrate how to utilize the cluster expansion utilities of the ATAT toolkit (link here) and the augmented lattice output by the P4 toolkit (link here ) to establish a cluster expansion on an augmented lattice.

Consideration 1: Artificially large effective cluster interactions (ECIs)
The effective cluster interactions for clusters formed within a group of sites should be artificially set to large values instead of being fitted to energy data. Unlike other cluster interactions, which are fitted to available data such as the DFT-calculated total enthalpy of different configurations on a lattice, the clusters within a group should have artificially large interactions to discourage the occupation of more than one atom within the group. This specific setup is only required for clusters formed within a group of sites. On the other hand, clusters formed among sites from different groups should be treated in the same way as clusters formed on a standard lattice, meaning they should be fitted to available data.
In summary, the clusters formed within a group necessitate artificially large interactions to enforce the single-atom occupation within the group, while clusters formed among sites from different groups should be fitted to available data.

Consideration 2: The energy of the configuration should remain intact with the addition of artificially large ECIs
In consideration 1, we mentioned that the ECIs associated with clusters formed within a group should be manually set to artificially large values. However, it's important to clarify that this setup should not alter the total energy of physically realistic configurations that occur on the augmented lattice. Instead, it should only lead to very large energy values when two atoms occupy sites within a group. In other words, the artificially large ECIs should not impact the energy of configurations where each group is occupied by only one atom. In the following explanation, we provide detailed steps to enforce this condition by explicitly enforcing a concentration and setting up penalty ECIs for clusters in a group.

Example 1 : Setting up the cluster expansion for the ordered B2 phase of PtTi
In this paper we elaborate the application of the P4 method for the ordered B2 phase of PtTi. In the following example, we demonstrate how one can setup large ECIs for the clusters formed within a group on the augmented lattice of cubic B2 PtTi.
The augmented lattice for the cubic B2 PtTi consist of groups of close by sites where 8 corner sites surrounds each standard B2 lattice site at its center. Below is the lat.in file

Example 2 : Setting up the cluster expansion for the chemically disordered B2 phase of NiTi
Unlike example 1, where the lattice sites for Pt and Ti were distinct, in a disordered phase each lattice site in the standard B2 phase can be occupied by both Ni and Ti. See more details in this paper . The augmented lattice associated with a disordered B2 NiTi is shown below: