Crystallographic groups in space and time: III. Four-dimensional Euclidean crystal classes corresponding to generalized magnetic point groups

Abstract

To determine the non-isomorphic space groups one needs the arithmetic crystal classes. We limit ourselves to four-dimensional point groups which are 3 + 1-reducible in GL(4, R). These are called GM point groups. It is shown that there exists an one-to-one correspondence between n-dimensional GM point groups and ( n - 1)- dimensional magnetic point groups. So in four dimensions the GM point groups are known. For these groups corresponding arithmetic point groups are determined by constructing all lattices left invariant by such a group. The arithmetic crystal classes are found by eliminating equivalent arithmetic point groups. The arithmetic crystal classes and Bravais classes are explicitly given.