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A projection‐based reformulation of the coincident site lattice Σ for arbitrary bicrystals at finite temperature
Author(s) -
Runnels B.
Publication year - 2017
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.742
H-Index - 83
ISSN - 2053-2733
DOI - 10.1107/s205327331700122x
Subject(s) - regularization (linguistics) , lattice (music) , mathematics , differentiable function , mathematical analysis , inverse , boundary value problem , finite set , inverse problem , grain boundary , physics , geometry , materials science , computer science , microstructure , artificial intelligence , acoustics , metallurgy
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 2 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.