Incommensurability in crystals

Organic molecular compounds with modulated crystal structures

“…They are called aperiodic crystals and are distinguished from the three-dimensional periodic ones -the normal crystals in the classical sense -by just this feature: their structures lack threedimensional translational symmetry but nevertheless posses three-dimensional long-range order. This is one of the main discoveries of the recent research of modern crystallography, that three-dimensional periodicity is not an essential condition for the existence of crystals [2]. Periodicity is not the only possible means to achieve long-range order.…”

“…The development of the concept, a historical overview and further explanation concerning aperiodic crystal structures and the higher-dimensional superspace approach might be found in the following review articles (and references therein): Janssen and Janner [2], Janssen [20], van Smaalen [21] and Yamamoto [22] discuss incommensurability, aperiodic and quasiperiodic crystals, incommensurate crystal structures and how to describe them in the higher-dimensional superspace. Bertaut [23] is treating commensurate and incommensurate crystals, while Chapuis [24] considers modulated structures in the context of phase transitions.…”

“…m ¼ AE2 in Eq. (2). In this example, the modulation wave vector runs parallel a * and can be written as q ¼ ðs 1 ; 0; 0Þ.…”

The Most Simple Icosahedral Quasi-Crystal with Continuous Phasons

“…Both can be described as cuts of higher-dimensional periodic structures [1]. They differ significantly in the behaviour under a phason mode.…”

“…The minimal dimension for periodic systems which possess Y as point group is six. So we commence with ( 1 ) No use is made of Y 1 = 53(2/m), which contains the inversion operation. As seems evident (and can be shown for our special model, but no general proof has been published yet) this symmetry does not permit structures with continuous phasons and hard cores.…”