Multiplicativity in the theory of coincidence site lattices

TL;DR

This paper explores the property of multiplicativity in the multiplicity functions of coincidence site lattices (CSLs), linking it to lattice decompositions and providing criteria for when multiplicativity holds or fails.

Contribution

It establishes a connection between multiplicativity of CSL multiplicity functions and lattice decompositions, offering criteria for multiplicativity and analyzing its limitations.

Findings

Multiplicativity is linked to specific lattice decompositions.

Criteria for when CSL multiplicity functions are multiplicative.

Supermultiplicativity often holds even when multiplicativity fails.

Abstract

Coincidence Site Lattices (CSLs) are a well established tool in the theory of grain boundaries. For several lattices up to dimension $d=4$, the CSLs are known explicitly as well as their indices and multiplicity functions. Many of them share a particular property: their multiplicity functions are multiplicative. We show how multiplicativity is connected to certain decompositions of CSLs and the corresponding coincidence rotations and present some criteria for multiplicativity. In general, however, multiplicativity is violated, while supermultiplicativity still holds.