On Matrix Equations of Twinning in Crystals

The algebraic equation of twinning involves a unimodular matrix, a general nonsingular matrix, a rotation, and two vectors. This paper generalizes the investigations of Ericksen, Pitteri, Zanzotto, and Gurtin by listing for each given unimodular matrix all the solutions of the twinning equation; that is, listing all the possibilities of the other variables. In an almost equivalent way, it assumes a rotation and a vector given and lists all the solutions. But in a different approach, it assumes that the nonsingular matrix is the identity and gives all the solutions. The last approach is linked to elementary number theory. One finding of the work here is that the twinning equation merely asks that a part of the orbit of the rotation matrix be synchronized with some part of the orbit of the unimodular matrix by the nonsingular matrix.