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This chapter gives an introduction to the distinctive structural features of aperiodic crystals, including incommensurately modulated crystals, composite crystals, and quasicrystals. Atomic structures are discussed in relation to the structures of periodic crystals, while it is shown that translational symmetry is lost. Diffraction by incommensurately modulated structures is shown to give rise to Bragg reflections that can be indexed by four or more integers. The modulation wave vector is introduced as the vector defining the periodicity of the modulation functions, as well as being the reciprocal vector employed in the indexing of Bragg reflections.

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.

This chapter discusses the X-ray and neutron diffraction methods used to study the atomic structures of aperiodic crystals, addressing indexing diffraction patterns, superspace, ab initio methods, the structure factor of incommensurate structures; and diffuse scattering. The structure solution methods based on the dual space refinements are described, as they are very often applied for the resolution of aperiodic crystal structures. Modulation functions which are used for the refinement of modulated structures and composite structures are presented and illustrated with examples of structure models covering a large spectrum of structures from organic to inorganic compounds, including metals, alloys, and minerals. For a better understanding of the concept of quasicrystalline structures, one-dimensional structure examples are presented first. Further examples of quasicrystals, including decagonal quasicrystals and icosahedral quasicrystals, are analysed in terms of increasing shells of a selected number of polyhedra. The notion of the approximant is compared with classical forms of structures.

In this chapter, the two main classes of intermetallic QCs known so far are introduced: decagonal QCs and icosahedral QCs. The terminology “decagonal” and “icosahedral”, respectively, refers to the Laue symmetry (10/m, 10/mmm and m3¯5¯, respectively) of their diffraction patterns (intensity weighted reciprocal lattice) or, equivalently, to the symmetry of the interatomic vector map (auto-correlation function or Patterson map). It also refers to the “bond-orientational order” of a QC structure, what is nothing else but its vector map. The full spacegroup symmetry of a quasiperiodic structure can best be described in the framework of the nD approach. However, an equivalent description is also possible in 3D reciprocal space based on the symmetry relationships between the complex structure factors.

Symmetry arises not only in the invariance of an object to certain operations, but also in invariance of the equations governing motion of particles. Noether’s theorem connects continuous symmetries of equations of motion to conservation laws. The concept of broken symmetry arises in phase changes and topological defects, such as dislocations and disclinations. The principle of symmetry compensation reveals a deep sense in which symmetry is never destroyed – broken symmetries relate variants of an object displaying reduced symmetry. Symmetry plays a fundamental role in characterising the physical properties of crystals through Neumann’s principle. The concept of quasiperiodicity is introduced and it is shown how it is related to periodicity in a higher dimensional crystal.

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.