Modulation parameters in incommensurate modulated structures with inflation symmetry

Abstract

Inflation symmetry is one of the peculiar features of the diffraction pattern of a quasicrystal. However, it is not an exclusive property of quasicrystalline structures and it may be present in incommensurately modulated structures, as shown recently in the Al-Mg system (Donnadieu P et al 1996 J. Physique I 6 1153-64). The conditions that a single modulation parameter of an incommensurate structure must fulfil in order to have inflation symmetry are determined. Although the number of possible distinct inflation-symmetric quasilattices is infinite, from physical/experimental arguments it can be concluded that, in practice, only a few of them can be experimentally observed, the reported phase of the Al-Mg system being one of these particular cases. A quantitative criterion to classify the modulation parameters that give rise to quasilattices with observable inflation symmetry is proposed. The generalization of the analysis of incommensurate structures with more than one single modulation parameter is also discussed. Finally, the inflation parameters of diffraction patterns with rotational point groups of finite order, CN, are compared with the parameters of the one-dimensional case.