The Angular Degrees of Freedom and Symmetries of Boundaries Between Grains of Layer Structures
John W. Cahn
The five angles which are needed to characterize interfaces between two crystals are reduced to three angles for interfaces between two layer structures with any of the five cylindrical point group symmetries. Two different sets of angles are suggested for describing any boundary; one leads to a periodic crystallographic space in which the angles serve as spheerical coordinates. Symmetry considerations identify the space and point groups of the two spaces. Each space has advantages for the description of how properties depend on the respective set of angles. The spaces have two different clearly defined l(superscript2) metrics for how close two boundaries are to each other, and two different gradients of properties such as interface energy.