A derivation of coincidence relations utilizing quaternion and matrix based on the hexagonal lattice for material engineers

Grain-oriented silicon steel is mainly used as the core material of transformers, and it is manufactured by applying secondary recrystallization. The driving force of this process is the grain boundary energy, based on the nature of the grain boundary, which is determined by coincidence site lattice (CSL) relations. CSL relations are determined by the arrangement of lattice points in three-dimensional space and have already been shown mathematically by using advanced mathematics. However, their derivation processes are abstract, making them difficult for material engineers to understand. Therefore, in this study, a derivation of CSL relations is attempted in order to enable material engineers to easily understand the derivation. This study contributes to industrial mathematics by helping material engineers understand the essence of the mathematical method in order to use it appropriately. Specifically, a derivation method for coincidence relations is proposed using the hexagonal lattice (in the case of an axial ratio of [Formula: see text]) as an example that avoids the need for advanced mathematics. This method involves applying the scale rotation of a quaternion, and it is thus named the quaternion-matrix method. The matrix specifying the [Formula: see text] coincidence relation of a certain lattice system is expressed by a similarity transformation using the matrix comprising its primitive translation vectors and is given as the following transformation matrix: [Formula: see text]. Based on the rational number property of the transformation matrix elements, the following formula is derived: [Formula: see text], [Formula: see text], [Formula: see text] value. Here, ([Formula: see text]) is specified by the integrality (lattice point) and irreducibility (unit cell) among the elements of [Formula: see text], and the quaternion for the CSL formation is thus derived. Finally, based on the polar form of this quaternion, the coincidence relation can be derived.