Aperiodic crystals

Normally, solids are crystalline, i.e. they have a three-dimensional periodic order with three-dimensional translational symmetry. However, this is not always so. Aperiodic crystals do have a long-distance order, but no three-dimensional translational symmetry. In a formal (mathematical) way, they can be treated with lattices having translational symmetry in four- or five-dimensional space , the so-called superspace their symmetry corresponds to a four- or five-dimensional superspace group. The additional dimensions are not dimensions in real space, but have to be taken in a similar way to the fourth dimension in space-time. In space-time the position of an object is specified by its spatial coordinates x, y, z the coordinate of the fourth dimension is the time at which the object is located at the site x, y, z. 

Unlike crystals that are packed with identical unit cells in 3D space, aperiodic crystals lack such units. So far, aperiodic crystals include not only quasiperiodic crystals, but also crystals in which incommensurable modulations or intergrowth structures (or composites) occur [14], That is to say, quasiperiodicity is only one of the aperiodicities. So what is quasiperiodicity Simply speaking, a structure is classified to be quasiperiodic if it is aperiodic and exhibits self-similarity upon inflation and deflation by tau (x = 1.618, the golden mean). By this, one recognizes the fact that objects with perfect fivefold symmetry can exist in the 3D space however, no 3D space groups are available to build or to interpret such structures. 

Jassen T, Chapuis G, de Boissieu M eds (2007) Aperiodic crystals from modulated phases to quasicrystals. Oxford University Press, New York... 

Three types of interactions contribute to the lattice energies of the (small, aperiodic) crystals depicted in Fig. 10 kernel — electron-pair attractions (the only interactions that favor ion-aggregation) kernel-kernel repulsions and electron-pair — electron-pair repulsions. 

Aperiodic crystals from modulated phases to quasicrystals... 

The more detailed description of the non-conventional symmetry goes beyond the scope of this book as it has little use in powder diffraction, because even the three-dimensional diffraction from aperiodic crystals is quite complex. When the diffraction picture is projected along one dimension, its treatment becomes too complicated and the crystal structure of aperiodic crystals is rarely, if ever, completely studied by means of powder diffraction techniques beyond simple phase identification. Nevertheless, this section has been included here for completeness, and to give the reader a flavor of recent developments in crystallography. ... 

A more complete description of de Wolffs approach to treatment of various types of aperiodic crystals can be found in the International Tables for Crystallography, vol. B, Second edition, U. Shmueli, Ed., Published for the International Union of Crystallography by Kluwer Academic Publishers, Boston/Dordrecht/London (2001). 

The discovery of five-fold symmetry prompted the arf-interim Commission on Aperiodic Crystals of the International Union of Crystallography to change the definition of a crystal as a periodic three-dimensional arrangement of identical unit cells to the following ...by crystal we mean any solid having an essentially discrete diffraction diagram, and by aperiodic crystal we mean any crystal in which three-dimensional lattice periodicity can be considered to be absent . International Union of Crystallography. Report of the Executive Committee for 1991, Acta Cryst. A48,922 - 946 (1992). 

Wolynes, P. G., Aperiodic crystals Biology, chemistry and physics in a fugue with stretto, in Proceedings of the International Symposium on Frontiers in Science (S. S. Chan and P. G. Debrunner, Eds.), AIP Conf. Proc. 180,39 (1988). 

In the years after the discovery of aperiodic crystals (incommensurate modulated, intergrowth and quasicrystals) crystallography was for me a very rich and open field of research, but not mysterious. Even the surprising combination in snow crystals of sixfold circular rotations with hyperbolic rotations [1], leading to hexagrammal scaling symmetry, fitted into the whole because the atomic positions in ice are invariant with respect to both types of crystallographic rotations [2]. 

The concept of a pentagonal lattice (and in general that of a polygonal lattice) has been introduced by Yamamoto in the context of axially symmetric aperiodic crystals [18], As this term might be unfamiliar to many people, some explanations are required before showing that the form lattice of cyclophilin is isometric pentagonal. 

Z. Papadopolos, P. Kramer and W. Liebermeister, in Proc. of the Int. Conf on Aperiodic Crystals, Aperiodic 1997, edited by M. de Boissieu, J.-L. Verger-Gaugry and R. Currant (World Scientific, Singapore, 1998), p. 173. 

This book does not, however, pretend to present the state of the art in crystallographic research. Apart from a few rudimentary ideas, there is no discussion of the fascinating subject of quasi-crystals or aperiodic crystals because these are still quite rare materials. Although synchrotron radiation is the tool of choice for cutting-edge research, the classical sealed X-ray tube is the only source available in most universities and industrial laboratories and will certainly remain so. This book is not an introduction to structure determination, there being a number of modern texts already available in this area. 

Modulated crystal structures are by no means a rarity. Since the Calaverite mineral, many other examples of modulated structures and more generally aperiodic crystals were and continue to be discovered. They occur in all types of... 

At this point, one may wonder why the studies on modulated structures only recently acquired some momentum. One obvious answer lies in the availability of the tools required to perform these studies. The discovery of the 3D space groups that are fundamental for the symmetry description of crystalline material preceded by a quarter of a century the discovery of x-ray diffraction by crystals. Shortly after, a selected number of crystal structures were described for the first time by W.H. and W.L. Bragg. For aperiodic crystals and, in particular, modulated ones, all the tools and methodologies necessary to study chis new type of materials had to be developed before further progress could be made. The development of the mathematical theory of superspace was instrumental for further success. After more than three decades of development, the availability of good performing tools is just starting to appear and explains the relatively belated development of this speciality. 

Erwin Schrodinger (1887-1961), one of the founders of quantum mechanics, in his famous essay [37] on What is Life based on lectures delivered in Dublin in 1943, at the height of the war (by the way, in our opinion — a must reading for anyone interested in biological physics), coined the special term aperiodic crystal. This is intentionally an oxymoron. Aperiodic cr3 tal is a crystal in the sense that every atom or... 

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