A new class of metrics for the macroscopic crystallographic space of grain boundaries

The macroscopic description of a defect-free, flat grain boundary in a pure material requires five degrees of freedom. There is a need to define the distance between boundaries in this five-dimensional space, because boundaries that are close together crystallographically should have similar properties. Morawiec has recently proposed such a metric, defined in terms of the misorientation of the two grains and their boundary normals. This approach has the disadvantage that there is no unique way of weighting the importance of the difference in disorientation compared to the difference in boundary normals, as was pointed out by Cahn and Taylor. In this work a metric is developed using a less familiar description of the crystallographic space which avoids this problem. Two technical results are proven, and a sample application to grain boundary properties is offered.