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 L 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.

Towards Generalized Noise-Level Dependent Crystallographic Symmetry Classifications of More or Less Periodic Crystal Patterns

“…The CSL index for a sufficiently good approximation may, however, be very high so that the actual grain size might set practical limits to the applicability of the bi-crystallographic symmetry theory to general grain boundaries with planar interfaces. When finite temperature effects are included, high CSL index grain boundaries reduce effectively to low CSL index grain boundaries [42] as the five macroscopic grain boundary parameters become less well defined.…”

“…Subjectivity in the experimental determination of the very basic Σ value (CSL index) has, for example, been recently discussed in ref. [42].…”

Approximating coincidence – turning a new page for bicrystallography

“…In fact, for all misorientations where tan =2 is irrational, a coincidence site lattice does not exist and AE ¼ 1. The atoms in this illustration are represented using the mollifier È ¼ exp½1=ð1 À x 2 Þ; x 2 ½À1; 1; 0 else È É (Runnels, 2017).…”

“…In this issue, Runnels presents a physically intuitive and mathematically elegant framework to overcome the challenges of the discrete nature of the CSL theory (Runnels, 2017). The central idea developed in this article is that AE can be computed by determining the extent of overlap between the atomic density fields of the two overlapping lattices.…”

“…Bump functions such as those shown in Fig. 1 of Runnels (2017) are example of such mollifiers. Truncated Gaussian functions typically suffice as mollifier approximants and are convenient for analysis.…”