The space groups of axial crystals and quasicrystals

Abstract

We compute all the three-dimensional quasicrystallographic space groups with $n$-fold axial point groups and standard lattices by a method that treats crystals and quasicrystals on an equal footing. We do not rely on projecting higher-dimensional crystallographic space groups, our results are valid for arbitrary $n$, and our analysis is elementary. We regard space groups as a scheme for classifying diffraction patterns to be carried out in three-dimensional reciprocal space. The familiar three-dimensional crystallographic space groups with axial point groups emerge simply and directly as special cases of the general $n$-fold three-dimensional quasicrystallographic treatment with $n=3,4,\mathrm{and} 6$. We give a general discussion of extinctions in quasicrystals and give a simple (three-dimensional) geometrical specification of the extinctions for each axial space group. The paper is intended both for people trying to systematize quasicrystal diffraction patterns and for people interested in a simple alternative approach to the computation of crystallographic or quasicrystallographic space groups.