Affiliation:
1. Departamento de Matemáticas Instituto Politécnico Nacional Mexico City Mexico
2. Programa de Desarrollo Profesional en Automatización Universidad Autónoma Metropolitana Mexico City Mexico
3. Centro de Física Aplicada y Tecnología Avanzada Universidad Nacional Autónoma de México Santiago de Querétaro Mexico
Abstract
This work provides a comprehensive characterization of coincidence site lattices that arise from the compositions of the reflections of cubic lattices. For this purpose, we apply Cartan's theorem and use reflections as basic isometries, which allows the use of Clifford algebra as a powerful tool to handle reflections and products of reflections, leading to explicit expressions for coincidence indices and the basis of coincidence lattices. The three possible cases were studied: simple reflections, the product of two reflections (rotations), and the product of three reflections, where a factorization that produces improper rotations was provided. In the case of rotations, it is formally proven that Ranganathan's well‐known generating function of coincidence indices for cubic systems under rotations yields all possible coincidence lattices. Overall, this study provides valuable insights into the mathematical properties of coincidence site lattices and the use of Clifford algebras to describe orthogonal transformations.
Funder
Instituto Politécnico Nacional