A projection-based reformulation of the coincident site lattice Σ for arbitrary bicrystals at finite temperature

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 L<sup>2</sup> inner 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.