Modulation parameters in incommensurate modulated structures with inflation symmetry

Inflation symmetry is one of the peculiar features of the diffraction patte rn of a quasicrystal. However, ii is not an exclusive property of quasicrys talline structures and it may be present in incommensurately modulated stru ctures, as shown recently in the AI-Mg system (Donnadieu P st al 1996 J. Ph ysique I 6 1153-64). The conditions that a single modulation parameter of a n incommensurate structure must fulfil in order to have inflation symmetry are determined. Although the number of possible distinct inflation-symmetri c quasilattices is infinite, from physical/experimental arguments it can be concluded that, in practice, only a few of them can be experimentally obse rved, 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 t han one single modulation parameter is also discussed. Finally, the inflati on parameters of diffraction patterns with rotational point groups of finit e order, C-N, are compared with the parameters of the one-dimensional case.