A geometric algebra approach to coincidence site lattices

When a lattice in Rn is superimposed on a rotated copy with coincident origins, coincidence points may arise for specific rotations, forming a coincidence site lattice (CSL). In crystallography, particularly in two and three dimensions, CSL theory is fundamental for describing low-energy grain boundaries. Geometric (Clifford) algebra provides a natural framework for this problem, as orthogonal transformations can be expressed as products of reflections. This article presents a unified account of the geometric algebra approach to coincidence isometries, including a Cartan-type decomposition of coincidence rotations for hypercubic lattices Zn. We then summarize the analytical characterization of coincidence rotations, indices and bases for planar hexagonal lattices. The main new result is a rigorous extension of these results to honeycomb structures, such as graphene, which are periodic but not Bravais lattices. We prove that the coincidence structure of two rotated honeycombs is completely determined by the coincidence lattice of their associated hexagonal Bravais lattices. As an application, structures near the first magic angle in twisted bilayer graphene are analysed, yielding quantitative agreement with experimentally observed Moiré lattice parameters. This article is part of the theme issue 'Modern applications of geometric algebra'.