Equilibrium morphology of incommensurately modulated crystals: a superspace description

The theory for the explanation of equilibrium morphologies of incommensurately modulated one-dimensional crystals, presented in a previous paper, is extended to the case of incommensurately modulated three-dimensional crystals. It is shown that, concerning the morphology, there exists a one-to-one correspondence between faces on the physical crystal and crystallographic hyperplanes of the embedded crystal in superspace. This holds for both main faces and satellite faces. The occurrence of the latter, however, is unique for incommensurately modulated crystals. It is shown that the stability of satellite faces, as well as main faces, can be attributed to a principle of selective cuts. The superspace approach that is developed leads to a calculation method for surface free energies that, in principle, can be applied to incommensurately modulated structures of arbitrary complexity. Equilibrium morphologies are constructed from the calculated surface free energies by means of a standard Wulff plot. The dependence of the equilibrium morphology on several structural parameters is studied for an incommensurately modulated simple cubic model crystal. This study allows for a basic understanding of the differences in morphology of AuTe₂ crystals and [(CH₃)₄N]₂ZnCl₄ crystals.