The symmetry and interfacial structure of bicrystals

Abstract
It is shown that the point and space symmetry of bicrystals can be classified according to established schemes of symmetry groups. The symmetry of the pattern created by the lattices of two crystals comprising a bicrystal is considered first. This symmetry depends on the symmetry of the component lattices and their relative orientation and position. A space group can be assigned to such a pattern by using the schemes of crystallographic rods, crystallographic layers, or conventional space groups (Shubnikov & Koptsik 1974) respectively, according to whether one, two or three non-collinear translation symmetry axes are present in the pattern. Patterns are considered to be dichromatic by regarding one lattice arbitrarily as white and the other black; space groups can then be expressed by using colour symmetry formulation. The variation of the symmetry of dichromatic patterns as the component lattices are displaced relatively is discussed. For a pattern with fixed relative orientation of the component lattices the number of non-collinear translation axes is invariant to changes of relative position. However, the point symmetry of the pattern varies according to a conservation rule; the product n$_{j}$r$_{j}$ is invariant with relative displacement where n$_{j}$ is the numerical expression of the point symmetry for the pattern created by a given relative displacement away from a holosymmetric pattern and r$_{j}$ is the number of crystallographically equivalent patterns obtained by symmetry related displacements. The product is equal to n$_{\text{h}}$, the numerical expression of the point symmetry of the holosymmetric pattern, i.e. n$_{j}$r$_{j}$ = n$_{\text{h}}$. Bicrystals are supposed to be obtained from dichromatic patterns by choosing the orientation and location of an interface plane and locating white bases at white lattice points on one side of the interface and black bases at black sites on the other. Bicrystals are therefore regarded as three-dimensional objects containing a unique plane, the interface, and the adjacent crystals can have different compositions and structures. Depending on the symmetry of a dichromatic pattern and the choice of interface plane there can be two, one or no translation symmetry axes in the interface plane, and colour symmetry groups can be assigned to bicrystals according to the schemes of two sided layers, bands or rosettes (Shubnikov & Koptsik 1974) respectively. The variation of bicrystal point symmetry with relative displacement of the adjacent crystals follows a conservation rule analogous to that for the case of dichromatic patterns. The symmetry of the physical properties of bicrystals is considered by invoking Neumann's principle. Computer calculations indicate that the relative displacements of adjacent crystals in mechanically stable polymeric and metallic bicrystals are such that bicrystal symmetry is often lower than holosymmetric. The relative position of adjacent crystals in a bicrystal is an important additional degree of freedom compared to a single crystal. For bicrystals which have at least one-dimensional translation symmetry and point symmetry higher than 1, equivalent bicrystal structures can exist corresponding to crystallographically equivalent relative displacements away from a holosymmetric structure. The number of equivalent bicrystals in a set, r$_{j}$, depends on the symmetry of these bicrystals and the holosymmetric form, and is given by the rule r$_{j}$ = n$_{\text{h}}$/n$_{j}$. Such equivalent bicrystals have degenerate energy, and it is possible for domains of equivalent structures, separated by special interfacial dislocations, to exist in a physical bicrystal. The Burgers vectors of these dislocations can have very small magnitudes and the dissociation of perfect interfacial dislocations into special dislocations is discussed.