The definition of quasicrystals

It is argued that the prevailing definition of quasicrystals, requiring them to contain an axis of symmetry that is forbidden in periodic crystals, is inadequate. This definition is too restrictive in that it excludes an important and interesting collection of structures that exhibit all the well-known properties of quasicrystals without possessing any forbidden symmetries.

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 l...

The experimental discovery of quasicrystals by D Shechtman, D Gratias, I Blech, and J W Cahn in 1984 provided the paradigm for a new type of long-range order of solid matter in nature. This discovery stimulated an explosion of new experimental and theoretical research. In years prior to the discovery, there was a very active development of various gateways to quasicrystals in theoretical and mathematical physics. Without this conceptual basis, it would have been impossible to...

A new kind of aperiodic tiling is introduced. It is shown to underlie a structure obtained as a superposition of waves with incommensurate periods. Its connections to other other tilings and quasicrystals are discussed.

Crystals are paradigms of ordered structures. While order was once seen as synonymous with lattice periodic arrangements, the discoveries of incommensurate crystals and quasicrystals led to a more general perception of crystalline order, encompassing both periodic and aperiodic crystals. The current definition of crystals rests on their essentially point-like diffraction. Considering a number of recently investigated toy systems, with particular emphasis on non-crystalline or...

Mathematicians have been interested in non-periodic tilings of space for decades; however, it was the unexpected discovery of non-periodically ordered structures in intermetallic alloys which brought this subject into the limelight. These fascinating materials, now called quasicrystals, are characterised by the coexistence of long-range atomic order and 'forbidden' symmetries which are incompatible with periodic arrangements in three-dimensional space. In the first part of th...

With the rapid development of topological states in crystals, the study of topological states has been extended to quasicrystals in recent years. In this review, we summarize the recent progress of topological states in quasicrystals, particularly focusing on one-dimensional (1D) and 2D systems. We first give a brief introduction to quasicrystalline structures. Then, we discuss topological phases in 1D quasicrystals where the topological nature is attributed to the synthetic ...

A new class of self-similar ordered structures with non-crystallographic point symmetries is presented. Each of these structures, named superquasicrystals, is given as a section of a higher-dimensional "crystal" with recursive superlattice structures. Such structures turn out to be limit-quasiperiodic, distinguishing themselves from quasicrystals which are quasiperiodic. There exist a few real materials that seem to be promising candidates for superquasicrystals.

Quasicrystals have intrigued and stimulated research in a large number of disciplines. Mathematicians, physicists, chemists, metallurgists and materials scientists have found in them a fertile ground for new insights and discoveries. In the quarter century that has ensued since the publication of the experimental observation of a quasiperiodic Al-Mn alloy \cite{shecht}, many different kinds of quasiperiodic alloys have been manufactured and studied. The physical properties of...

The theory of magnetic symmetry in quasicrystals is used to characterize the nature of magnetic peaks, expected in elastic neutron diffraction experiments. It is established that there is no symmetry-based argument which forbids the existence of quasiperiodic long-range magnetic order. Suggestions are offered as to where one should look for the simplest kinds of antiferromagnetic quasicrystals.