research papers J. Appl. Cryst. (2007). 40, 559–569 doi:10.1107/S0021889807012770 559 Journal of Applied Crystallography ISSN 0021-8898 Received 9 January 2007 Accepted 18 March 2007 # 2007 International Union of Crystallography Printed in Singapore – all rights reserved Representation of orientation relationships in Rodrigues–Frank space for any two classes of lattice Youliang He a * and John J. Jonas b a Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, USA, and b Department of Materials Engineering, McGill University, Montreal, QC, Canada. Correspondence e-mail: yh254@cornell.edu The fundamental zones of Rodrigues–Frank (R-F) space applicable to misorientations between crystals of any two Laue groups are constructed by using a unified formulation in terms of quaternion algebra. Some of these regions are fully bounded by planes that are determined solely by the symmetries of the groups, while others have at least one unbounded direction. Each of the bounded fundamental zones falls into one of nine geometrically distinct configurations. The maximum symmetry-reduced angles and the corresponding Rodrigues–Frank vectors for these fundamental zones are evaluated. The use of Rodrigues–Frank space for the representation of orientation relationships between crystals of any two symmetry groups is also addressed. Examples concerning the transition of phases of the same symmetry group, i.e. from face-centered cubic to body-centered cubic, and of different groups, i.e. from body-centered cubic to hexagonal close-packed, are given to illustrate the usefulness of this space for representing orientation relationships during phase transformation or precipitation. 1. Introduction During phase transformation, a specific orientation relation- ship (OR) usually exists between the initial phase and the transformed product phase. In most cases, because of the symmetries of the crystals, a single initial orientation gives rise to a number of final orientations called crystallographic variants. For instance, it is well known that a single face- centered cubic crystal produces 24 product orientations during martensitic transformations if the Kurdjumov–Sachs (K-S) relationship (Kurdjumov & Sachs, 1930) is adopted. Experi- mentally, however, only a limited number of such variants are observed when a transmission electron microscopy (TEM) or X-ray diffraction method is used. This is because in a TEM analysis, usually only a small region containing a limited number of grains can be examined and no statistical data can be readily obtained. Using X-ray diffraction, normally only orientations of single crystals can be measured and it is diffi- cult to distinguish individual variants in a polycrystalline material. By contrast, if an electron backscatter diffraction (EBSD) technique is employed, almost all the variants can be detected relatively easily. The investigation of crystallographic orientation relationships by means of EBSD methods requires a tool to calculate all the variants of an orientation relation determined by TEM or X-ray diffraction and to compare the experimental data with those predicted according to the known orientation relation and a suitable variant selection model. This paper is aimed at developing such a tool for the representation of orientation relationships and of EBSD data in Rodrigues–Frank space. A typical way of presenting the data obtained by EBSD is to plot the orientations on stereographic pole figures and then to compare them with the predictions obtained from a particular orientation relationship. Alternatively, Rodrigues–Frank space can simply be used to present orientation and misor- ientation data. The advantages of this approach over other methods [such as orientation distribution functions (ODFs) in Euler space] have been addressed by several authors (e.g., Frank, 1987; Becker & Panchanadeeswaran, 1989; Neumann, 1991; Randle & Engler, 2000; Morawiec, 2004) and will not be repeated here. For the representation of orientation rela- tionships in Rodrigues–Frank space, it is necessary to establish the fundamental zones for specific crystal symmetries and to derive the crystallographic variants of the theoretical orien- tation relationships. For example, the classical orientation relationships for cubic crystals, i.e. the Bain (Bain, 1924), Kurdjumov–Sachs (Kurdjumov & Sachs, 1930), Nishiyama– Wassermann (Nishiyama, 1934; Wassermann, 1935), Pitsch (Pitsch, 1962) and Greninger–Troiano (Greninger & Troiano, 1949) relationships, have recently been identified and repre- sented in this space and compared with experimental data obtained by EBSD techniques (Godet et al., 2004; He et al. , 2005, 2006). For crystals of non-cubic symmetry, especially those with non-orthogonal crystal axes, misorientations and orientation relationships are rarely presented in Rodrigues–Frank space.