Screw rotations and glide mirrors: crystallography in Fourier space.

Abstract

The traditional crystallographic symmetry elements of screw axes and glide planes are subdivided into those that are removable and those that are essential. A simple real-space criterion, depending only on Bravais class, determines which types can be present in any space group. This terminological refinement is useful in expressing the complementary relation between the real-space and Fourier-space formulations of crystal symmetry, particularly in the case of the two nonsymmorphic space groups that have no systematic extinctions (I212121 and I213). A simple analysis in Fourier space demonstrates the nonsymmorphicity of these two space groups, which finds its physical expression not in a characteristic absence of Bragg peaks, but in a characteristic presence of electronic level degeneracies.