Defects, modulated structures and quasicrystals

In this chapter, the concepts associated with classical crystallography are gradually weakened. Initially the effects of introducing small defects into a crystal are examined. These require almost no modification of the ideas already presented. However, structures with enormous unit cells pose more severe problems, and incommensurate structures are known in which a diffraction pattern is best quantified by recourse to higher dimensional space. Finally, classical crystallographic ideas break down when quasicrystals are examined. These structures, related to the Penrose tilings described in Chapter 2, can no longer be described in terms of the Bravais lattices described earlier. 

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Since our surroundings are three-dimensional, we tend to assume that crystals are formed by periodic arrangements of atoms or molecules in three dimensions. However, many crystals are periodic only in two, or even in one dimension, and some do not have periodic structure at all, e.g. solids with incommensurately modulated structures, certain polymers, and quasicrystals. Materials may assume states that are intermediate between those of a crystalline solid and a liquid, and they are called liquid crystals. Hence, in real crystals, periodicity and/or order extends over a shorter or longer range, which is a function of the nature of the material and conditions under which it was crystallized. Structures of real crystals, e.g. imperfections, distortions, defects and impurities, are subjects of separate disciplines, and symmetry concepts considered below assume an ideal crystal with perfect periodicity. ... 

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