What is a crystal?

Abstract

Abstract. Almost 25 years have passed since Shechtman discovered quasicrystals, and 15 years since the Commission on Aperiodic Crystals of the International Union of Crystallography put forth a provisional definition of the term crystal to mean "any solid having an essentially discrete diffraction diagram." Have we learned enough about crystallinity in the last 25 years, or do we need more time to explore additional physical systems? There is much confusion and contradiction in the literature in using the term crystal. Are we ready now to propose a permanent definition for crystal to be used by all? I argue that time has come to put a sense of order in all the confusion.

Discussion of phasons in quasicrystals and their dynamics

“…, h D ) are the D Miller indices of the Bragg peak at wavevector G k . For a periodic crystal the indexing dimension D ¼ d.Incommensurate phases differ from periodic crystals in that additional lattice vectors are needed in order to index all the diffraction peaks, requiring D 4 d[7,8]. Ordinary incommensurate structures may have a single extra lattice vector such as…”

Ordres complexes et apériodicités

“…Très longtemps, les notions d'ordre à longue distance et de triple périodicité ont été systématiquement, implicitement et arbitrairement associées et le cristal a été défini par le triple empilement d'une maille élémentaire définie par 3 vecteurs de base linéairement indépendants a, b et c. Si l'on admet qu'un diagramme de diffraction, transformée de Fourier de la fonction d'autocorrélation de l'échantillon étudié, ne présentant que des signaux ponctuels, traduit intrinsèquement un ordre à grande distance, caractéristique de l'ordre cristallin [1,2], alors l'observation récente de figures de diffraction particulières ( Fig. 1) a clairement démontré que la triple périodicité en était curieusement absente (axes de symétrie incompatibles ( fig.…”

Localization properties at band edge of local density of states in two-dimensional Thue–Morse photonic systems

“…The lattice Fourier spectrum of an infinitely long ThMo structure is singular continuous in contrast to the pure-point Fourier spectra of periodic and quasiperiodic lattices [4]. This property of the Fourier lattice places the ThMo structures into a gray zone between the ordered and the random media [5]. In fact, the ThMo structures cannot be considered random because they are perfectly deterministic.…”

“…In fact, the ThMo structures cannot be considered random because they are perfectly deterministic. At the same time, they cannot be defined as crystals or quasicrystals because they have no pure-point spectrum parts and consequently no Bragg peaks into diffraction spectra [5]. The numerical investigation in electronic structures based on ThMo sequence suggests that the eigenstates are critical [6], i.e., neither are extended all over the lattice as are the Bloch states nor exponentially localized as are the strongly localized states in the Anderson model of localization.…”