Aperiodic crystals and their atomic structures in superspace: an introduction

periodic crystals are ordered matter without three-dimensional (3D) translational symmetry. They encompass quasicrystals, incommensurate composite crystals and incommensurately modulated structures. Their crystal structures are described by the so-called superspace theory, in which hidden periodicity is uncovered as translational symmetry in a space of dimensions higher than three. Here, I give an introduction to the structural features of aperiodic crystals and to the superspace theory for describing those structures. Applications, in particular t -plots, are demonstrated for the charge-density wave (CDW) materials Er _2 Ir _3 Si _5 and EuAl _4 . Graphical abstract