Stokes Harold T, Campbell Branton J, Hatch Dorian M
Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA.
Acta Crystallogr A. 2007 Jul;63(Pt 4):365-73. doi: 10.1107/S0108767307024890. Epub 2007 Jul 1.
Phase transitions that result in incommensurate structural modulations are widely observed in crystalline solids and are relevant to a broad range of physical phenomena in magnetic, electronic, optical and structural materials. While the (3+1)-dimensional superspace-group symmetries associated with one-dimensional modulations have been tabulated, the order parameters that produce these modulations have not been explored in detail. Here, using group-theoretical methods, we present a unique and exhaustive enumeration of the isotropy subgroups (and their corresponding order-parameter directions) belonging to irreducible representations of the (3+1)-dimensional superspace extensions of the 230 crystallographic space groups at all incommensurate k points. The vast majority of experimentally observed incommensurately modulated structures have order parameters belonging to one of these subgroups.
