Coincidence-site lattice (CSL) theory provides a fundamental framework for classifying commensurate crystal orientations and plays a central role in describing grain boundaries. In particular, Ranganathan's formula determines all CSL orientation relationships. However, it is not possible to avoid duplication and also to determine the equivalence of the duplicates, and number theoretical analysis can be required to address these issues. In this study, we provide a simple necessary condition to find all CSL orientation relationships in the cubic system by only using linear algebra. The necessary condition is given by calculating whether the (hkl) plane is commensurate to the (001) plane. We also found that the number of orientation relationships without symmetric tilt grain boundaries (GBs) increases as the Σ value increases by comparing the number of solutions of the equation for constructing symmetric tilt GBs with that of the necessary condition.
Kawahara et al. (Tue,) studied this question.