Symmetries in the representation of grain boundary-plane distributions

Symmetries play a crucial role in the theoretical analysis and visualization of the five macroscopic grain boundary parameters, including the misorientation (three parameters) and the orientation of the boundary-plane (two parameters). The symmetry aspects of the misorientation spaces are very well documented and in this article all possible boundary-plane symmetries are enumerated for the 32 crystallographic point groups. It is observed that the boundary-plane spaces exhibit a wide variety of point group symmetries, which depend both on the crystallographic point group and on the corresponding misorientation (i.e. location in the fundamental zone). The list of symmetries presented here should serve as a guide for graphical representations of not only the distributions of boundary-plane orientations but also for the representation of boundary-plane related properties such as energy, mobility etc.