Acta Cryst. (2005). A61, 173–184 doi:10.1107/S0108767304025358 173 research papers Acta Crystallographica Section A Foundations of Crystallography ISSN 0108-7673 Received 8 July 2004 Accepted 7 October 2004 # 2005 International Union of Crystallography Printed in Great Britain – all rights reserved Clifford algebra approach to the coincidence problem for planar lattices M. A. Rodrı ´guez, a J. L. Arago ´n b * and L. Verde-Star c a Departamento de Matema ´ticas, Escuela Superior de Fı ´sica y Matema ´ticas, Instituto Polite ´cnico Nacional, Unidad Profesional Adolfo Lo ´ pez Mateos, Edificio 9, 07300, Me ´xico DF, Me ´xico, b Centro de Fı ´sica Aplicada y Tecnologı ´a Avanzada, Universidad Nacional Auto ´noma de Me ´xico, Apartado Postal 1-1010, 76000 Quere ´taro, Me ´xico, and c Departamento de Matema ´ticas, Universidad Auto ´ noma Metropolitana, Unidad Iztapalapa, Apartado Postal 55–534, CP 09340 Me ´xico DF, Me ´xico. Correspondence e-mail: aragon@fata.unam.mx The problem of coincidences of planar lattices is analyzed using Clifford algebra. It is shown that an arbitrary coincidence isometry can be decomposed as a product of coincidence reflections and this allows planar coincidence lattices to be characterized algebraically. The cases of square, rectangular and rhombic lattices are worked out in detail. One of the aims of this work is to show the potential usefulness of Clifford algebra in crystallography. The power of Clifford algebra for expressing geometric ideas is exploited here and the procedure presented can be generalized to higher dimensions. 1. Introduction Coincidence site lattice (CSL) theory has provided partial answers to the complex problem that arises in the description of grain boundaries and interfaces (see, for instance, Sutton & Baluffi, 1995). Most of the existing geometric models of grain boundaries idealize the two crystals that meet at a boundary as two interpenetrating lattices and it is assumed that grain boundaries with special properties arise when there is a high degree of good fit (or matching) between the lattices. The CSL model (Ranganathan, 1966) considers points common to both lattices (the intersection lattice) as points of good fit and assumes that special boundaries arise when the density of coincidence sites is high, because many atoms would occupy sites common to both grains. The experimental support for the CSL model is based on the special properties, such as changed migration rates, observed for boundaries with certain orien- tational relationships. Working with this model, we have to consider two identical copies of a lattice à (one of the 14 Bravais lattices) brought into coincidence. Next one lattice is rotated, relative to the other, by an angle  about an axis through a common lattice point. Then for different values of  two possibilities will arise: no lattice sites will coincide (except the site where the rotation axis passes through) or, owing to the periodicity of Ã, an infinite number will coincide forming a lattice. Such a lattice is a CSL and the ratio of the volumes of primitive cells for the CSL and for à is denoted by Æ . High densities of coincident sites correspond to low values of Æ. Since the advent of quasicrystals, it has been desirable to extend the mathematical theory of CSL to more dimensions. In general, the problem can be stated as follows. Let à be a lattice in R n and let R 2 OðnÞ be an orthogonal transforma- tion. R is called a coincidence isometry if à \ Rà is a sublattice of Ã. The problem is therefore to identify and characterize the coincidence isometries of a given lattice Ã. Several approaches have been used to tackle this problem. Fortes (1983) developed a matrix theory of CSL in arbitrary dimensions, including a method to calculate a basis for the coincidence lattice through a particular factorization of the matrix defining the relative orientation. The same matrix approach was implemented by Duneau et al. (1992), but they evaluated the parameters of the coincidence lattice using a method based on the Smith normal form for integer matrices. Baake (1997) used complex numbers and quaternions to solve the problem in dimensions up to 4. Finally, Arago ´n et al. (1997) proposed a weak coincidence criterion and used four-dimen- sional lattices to characterize coincidence lattices in the plane. Here, we analyze the problem using Clifford algebra with a twofold purpose. First, the power of this mathematical language for expressing geometric ideas is used to solve the coincidence problem, which is merely geometric. Second, we try to show that Clifford algebra, already used as a powerful language in several fields, can also be useful in geometrical crystallography. Although nothing new emerges, the results provide new insights in this and other problems in geometrical crystallography and the approach could be valuable for extension to arbitrary dimensions. In this approach, reflections are considered as primitive transformations and Clifford algebra emerges as a natural tool for this problem, without using matrices and only a minimum of group theory. It is found that any arbitrary coincidence isometry can be decomposed as a product of coincidence reflections by vectors of the lattice Ã, and the group of coincidence isometries is characterized by providing a way to generate it from vectors of Ã. The paper is organized as follows. In x2, we provide a brief introduction to Clifford algebra by considering the particular