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This chapter discusses the physical properties of aperiodic crystals. Topics covered include tensorial properties, hydrodynamics, phonons, phasons, non-linear excitations, and electrons. It is shown that phonons and electrons in quasiperiodic crystals behave differently from those in periodic crystals. The most important differences include: (i) phonon eigenvectors and electron states occur as extended, as found in all periodic crystals, but they also occur as localized or as critical; (ii) quasiperiodic crystals show examples of absolute continuous, singular continuous, and point parts in their spectra; and (iii) electron and phonon dispersion curves have an infinite number of gaps, although these may be negligible for modulated phases and composites.

This chapter discusses the methods for determining the structure of crystals. Topics covered include diffraction techniques, determination of modulated phases and composites, structure determination of quasi-crystals, and diffraction by an imperfect crystal.

This chapter discusses aperiodic crystals. The structure and symmetry of quasiperiodic crystals can be described by embedding them into a higher-dimensional space as lattice periodic structures. Their intersection with the physical space gives the real structure, shifting the physical space parallel to the original one gives another possible realization of the crystal with the same energy. The Fourier transform of the quasiperiodic structure, and its diffraction pattern, are projections of the corresponding quantities in higher dimensions. The symmetry groups of quasiperiodic structures are superspace groups, higher-dimensional space groups for which the point group can be decomposed into a component in physical space and one in the additional, internal space. The structure determination reduces to the determination of number and positions of atomic surfaces in the higher-dimensional unit cell, and that of the shape of the atomic surfaces.

This introductory chapter discusses periodic and quasiperiodic crystals. Perfect quasiperiodic crystals and periodic crystals have sharp Bragg peaks, on the positions of a Fourier module. When the rank of the quasiperiodic structure is equal to the dimension of the space, the Fourier module is identical to the reciprocal lattice. A quasiperiodic crystal with a rank higher than the dimension of the physical space is called an aperiodic crystal.

The study of aperiodic crystals has been essentially based on diffraction methods. The characterization of diffracted intensities with a number of integer indices equal to its rank rather than its dimension led to the concept of superspace, which is now being systematically applied in the description of aperiodic crystals. When rank is higher than the dimension of the system, the crystal structure exhibits additional periodicities which are in general incommensurate with the others. Consequently, it should be possible to observe the additional periodicities on macroscopic crystal samples. This chapter discusses the morphology of aperiodic crystals, surfaces, and magnetic quasiperiodic systems.

This chapter presents the steps that are required to determine the superspace group of an aperiodic crystal from its diffraction pattern. This includes the analysis of the metric of the reciprocal lattice in superspace, the point symmetry of the diffraction pattern, and the reflection conditions.

This chapter begins with a discussion of the mechanisms for creating aperiodicity. It then discusses the Landau theory of phase transitions, semi-microscopic models, composites, electronic instabilities, and the numerical modeling of aperiodic crystals.

The law of rational indices to describe crystal faces was one of the most fundamental law of crystallography and is strongly linked to the three-dimensional periodicity of solids. This chapter describes how this fundamental law has to be revised and generalized in order to include the structures of aperiodic crystals. The generalization consists in using for each face a number of integers, with the number corresponding to the rank of the structure, that is, the number of integer indices necessary to characterize each of the diffracted intensities generated by the aperiodic system. A series of examples including incommensurate multiferroics, icosahedral crystals, and decagonal quaiscrystals illustrates this topic. Aperiodicity is also encountered in surfaces where the same generalization can be applied. The chapter discusses aperiodic crystal morphology, including icosahedral quasicrystal morphology, decagonal quasicrystal morphology, and aperiodic crystal surfaces; magnetic quasiperiodic systems; aperiodic photonic crystals; mesoscopic quasicrystals, and the mineral calaverite.

First a general description of the concept of crystalline structures is presented with some historical background information. The classical approach of periodic structures is presented along with the important topic of symmetry and its role characterizing physical properties. The limitations of the classical model are then introduced in view of the new experimental observations discovered since the 1970s. New forms of crystalline structures including incommensurately modulated and composite structures are presented along with quasicrystalline structures (quasicrystals). The necessity to extend the theory of space group symmetry is then discussed and the concept of superspace symmetry is introduced in order to describe these new forms of matters.