title: A projection-based reformulation of the coincident site lattice Σ for arbitrary bicrystals at finite temperature.
abstract: The coincident site lattice and, specifically, the `Σ value' of a grain boundary are a ubiquitous metric for experimental classification of grain boundaries. However, the mathematical nature of Σ - a pathological function taking values of either an integer or infinity - has been relatively unexplored. This work presents a framework for interpreting Σ as the inverse of a projection defined using the standard L2inner product over continuous fields that represent lattices. `Pre-mollifiers' are used to introduce thermal regularization in the context of the inner product, and a closed-form analytic result is derived. For all nonzero values of the regularization parameters, the formulation is mathematically smooth and differentiable, providing a tool for computationally determining experimental deviation from measured low-Σ boundaries at finite temperatures. It is verified that accurate Σ values are recovered for sufficiently low Σ boundaries, and that the numerical result either converges towards an integer value or diverges to infinity.
authors.affiliation: Department of Mechanical and Aerospace Engineering, University of Colorado Colorado Springs, CO, USA.
authors.forename: B
authors.initials: B
authors.lastname: Runnels
keywords: coincident site lattice; grain boundaries