Caution regarding ambiguities in similar expressions of orientation relationships

The orientation relationship (OR) between two phases is an essential parameter to characterize microstructure in multiphase crystalline materials. Among the numerous ways to define an OR, the expression in terms of the parallelism between lattice planes and between directions from different phases, referred to as the {hkl}/<uvw> expression, is the most commonly used. However, this expression scheme may be misleading because completely different ORs could be mistakenly regarded as equivalent ORs owing to the similarity of the expressions. In this article, the symmetries of the parallel elements were analyzed to clarify the conditions causing three cases. Namely, two similar expressions may describe two different ORs, two variants of a unique OR or exactly the same OR. This analysis helps not only to avoid ambiguities caused by the {hkl}/<uvw> expression but also to specify how variants are defined with respect to a different matrix phase. Based on the result of the analysis, the expressions of variants for the Kurdjumow-Sachs and Nishiyama-Wassermann ORs between face-centred cubic and body-centred cubic systems for different matrix phases are provided. Examples are given together with stereographic projections to demonstrate different cases.