Bicrystallography

In this paper a crystallographic framework and methodology are presented that enable the derivation of the symmetry of any bicrystal comprising a planar interface between two crystals of the same or different form. If the crystals are different the interface is an interphase boundary, and if they are the same the interface is a grain boundary. Special cases of the latter, corresponding to special relative orientations of the adjacent crystals, are domain and inversion boundaries and stacking faults and crystallographic shear faults. All possible symmetry groups for bicrystals are tabulated, and generic relations between the symmetry exhibited by the different types of interfaces are discussed. The crystallographic framework used is that devised by Pond & Bollmann (1979), in which the starting point in the treatment of a given bicrystal is the dichromatic pattern, i. e. the pattern created by the two crystal lattices with one regarded as white and the other black. An additional step has been introduced in order to be able to extend the treatment to bicrystals in which one or both of the crystals are non-symmorphic and/or non-holosymmetric. This additional step is the creation of a dichromatic complex, which is the pattern created by the lattice complexes of the two crystals. The analytical determination of the symmetry of dichromatic patterns and complexes is presented. The symmetry of a particular bicrystal can be obtained from the corresponding dichromatic complex by cross-sectioning. The methodology employed in this treatment of bicrystal symmetry is to use the theory of the symmetry of composites. A composite is regarded here as an entity comprising two components, which may be crystal lattices, as in a dichromatic pattern, or lattice complexes, as in a dichro­matic complex, or crystals, as in a bicrystal. The components of a com­posite may be equivalent or different, corresponding to the investigation of a bicrystal containing a grain or interphase boundary respectively. In the latter case, the symmetry of the composite is given by the intersection of the symmetry of the components, whereas in the former case additional symmetry may be present corresponding to symmetry operations relating the two equivalent components. The existence and disposition of such symmetrizing operations is revealed particularly clearly by using the dichromatic framework. Since one component is considered to be black and the other to be white, symmetrizing operations correspond to antisymmetry (or colour-reversing) operations. The methodology used in this work also elucidates the significance of crystallographically equivalent variants of a composite, and enables the interrelation of variants to be established. Crystallographically equivalent variants arise as a consequence of dissymmetrization, and, in this respect, the idea of regarding a relaxed bicrystal as having been created from a dichromatic pattern by a squence of imaginary steps is most helpful. Each step in the procedure causes dissymmetrization and therefore leads to the existence of variants. Four types of variants arise, and these have been designated orientation, complex, morphological and relaxational variants. Morphological v̇ariants, for example, arise as a result of cross-sectioning a dichromatic complex; the variants correspond to bicrystals having iden­tical symmetry (except possibly for orientation of their elements) and identical orientation of the adjacent crystals but different interfacial planes. The number and mutual disposition of such variants constitute the crystallographic aspect of the symmetry of grain boundary facetting and precipitate morphology for example.