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Incommensurate systems show long‐range order, but lack translational periodicity. They cannot be described within the three‐dimensional lattice space groups. In these systems a local atomic property is modulated with a period that is incommensurate with the underlying lattice as the wavelength of the modulation is not an integral multiple of the unit‐cell edge. Here, we describe the dielectric and polarization properties as well as neutron, X‐ray scattering and NMR features of incommensurate systems. The elementary excitations of incommensurate systems, i.e. phasons and amplitudons, are discussed as well. In addition to plane wave also soliton structures appear in the ground state. One‐, two‐ and three‐dimensional modulated structures are treated. The basic theoretical understanding of incommensurate systems is discussed.

This chapter opens with a brief description of the very wide variety of magnetically ordered structures, both commensurate (with crystal structure) and incommensurate. The concepts of magnetic Bravais lattices and magnetic space groups are introduced. For an unpolarized incident neutron beam, magnetic and nuclear scattered intensities are additive — calculation of the latter involves a magnetic form factor, a magnetic interaction vector (depending on magnetic moment relative to scattering vector), and a magnetic structure factor. Example calculations are given for anti-ferromagnetic AuMn and the incommensurate heli-magnetic Au₂Mn. Methods for solving magnetic structures, i.e., establishing the nature of the magnetic ordering, then determining the magnitude and orientation of the magnetic moments, are discussed. The solution of magnetic structures from neutron powder data is illustrated with examples taken from the recent literature.

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.

Once the bond network is known, ideal bond lengths can be calculated from the theoretical bond valences described in Chapter 3, but these cannot always be embedded in three-dimensional space without being stretched or compressed. The degree of strain can be measured using the bond strain index and the global instability index, the latter being less than 0.20 valence units in stable structures. Various mechanisms allow at least a partial relaxation of this strain, and these can lead to phase transitions and incommensurate structures.

Extended (polymeric) inorganic structures present different problems to crystallographers than molecular structures. The samples are often tiny crystals, atomic scattering powers vary widely, and structures are frequently disordered or twinned, strongly absorbing, or prone to temperature-induced phase transitions. Pseudo-symmetry is also a frequent problem. Disorder includes atomic substitution on a single site and partial occupancy by atoms of variable valency; some forms of disorder lead to diffuse scattering. Phase transitions may cause incommensurate structures to form. Extended structures require different approaches to testing and validation; these include bond valence calculations. This chapter illustrates the issues with two detailed case histories of mixed oxide materials.

The non-stoichiometric compounds that we describe in this chapter are closely correlated with the classical non-stoichiometric compounds derived from point defects discussed in Chapter 1. For the past twenty years precise structural analyses on complex binary and ternary compounds have been carried out using X-ray and neutron diffraction techniques. Moreover, owing to the striking development of the resolving power of the electron microscope crystal structures can be seen directly as structure images. As a result, it has been shown that most complex structures can be derived by introducing extended defects regularly into a mother structure. A typical example is a ‘shear structure’, which is derived by introducing planar defects of anion rows into the mother lattice. A ‘block structure’ is derived by introducing two groups of planar defects. ‘Vernier structures’, ‘micro-twin structures’, ‘intergrowth structures’, and ‘adaptive structures’ are also described in detail in this chapter. At the beginning of 1950, Professor A. Magnéli’s group in Sweden started a systematic study of the crystal structures of the oxides of transition metal elements such as Ti, V, Mo, and W, mainly by X-ray diffraction techniques. As a result, they confirmed the existence of the homologous compounds expressed by VnO2n–1; TinO2n–1 etc. (n = 2, 3, 4, . . .) and also predicted that the crystal structure of these compounds could be derived from a mother structure, ‘rutile’. Figure 2.1 shows the X-ray powder diffraction patterns (CuKα) of compounds TiOx between Ti2O3 (x = 1.5) and TiO2 (x = 2.0).3 This clearly indicates the convergence of the diffraction patterns to that of TiO2 (rutile) with increasing x, which is why the Magnéli school predicted the mother structure to be rutile. This prediction was verified by the structure determinations of Ti5O95 and VnO2n–1.6 These compounds are called Magnéli phases after the main investigator, and similar compounds have been discovered.