Crystal classes crystallographic point

Thirty-two crystal classes (crystallographic point groups)... 

The thirty-two crystal classes (crystallographic point groups) described in Section 9.1.4 can also be classified into the same seven crystal systems, depending on the most convenient coordinate system used to indicate the location and orientation of their characteristic symmetry elements, as shown in Table 9.2.1. 

Crystal family Symbol Crystal system Crystallographic point groups (crystal classes) Number of space groups Conventional coordinate system Bravais lattices... 

Table 15 The crystallographic point groups (crystal classes).

All the possible combinations of these symmetry elements result in 32 crystallographic point-group symmetries or crystal classes their symbols are listed in Table 3.3. Notice that in putting together the symbols to denote the symmetries of any crystal classes the convention is to give the symmetry of the principal axis first for instance 4 or 4, for tetragonal classes. If there is a plane of symmetry perpendicular to the principal axis, the two symbols are associated as in 4 m or Aim (4 over m), then the symbols for the secondary axes, if any, follow, and then any other symmetry planes. In a symbol such as Almmm, the second and third m refer to planes parallel to the four-fold axis. 

For the monoclinic system it is essential to have one twofold axis, either 2(C2) or 2(m), and it is permitted, of course, to have both. When both are present the point group is that of the lattice, 2lm Cy). There are no intermediate symmetries. By proceeding in this way, we can arrive at the results shown in column 4 of Table 11.4, where each of the 32 crystallographic point groups (i.e., crystal classes) has been assigned to its appropriate crystal system. 

A single crystal, considered as a finite object, may possess a certain combination of point symmetry elements in different directions, and the symmetry operations derived from them constitute a group in the mathematical sense. The self-consistent set of symmetry elements possessed by a crystal is known as a crystal class (or crystallographic point group). Hessel showed in 1830 that there are thirty-two self-consistent combinations of symmetry elements n and n (n = 1,2,3,4, and 6), namely the thirty-two crystal classes, applicable to the description of the external forms of crystalline compounds. This important... 

Knowledge of the diffraction symmetry of a crystal is useful for its classification. If the Laue group is observed to be 4/mmm, the crystal system is tetragonal, the crystal class must be chosen from 422,4mm, 42m, and 4/mmm, and the space group is one of those associated with these four crystallographic point groups. 

TABLE 4.3. The 14 Bravais Lattices, 32 Crystallographic Point Groups (Crystal Classes) and Some Space Groups. 

Radiation and particles, i.e. x-rays, neutrons and electrons, interact with a crystal in a way that the resulting diffraction pattern is always centrosymmetric, regardless of whether an inversion center is present in the crystal or not. This leads to another classification of crystallographic point groups, called Laue classes. The Laue class defines the symmetry of the diffraction pattern produced by a single crystal, and can be easily inferred from a point group by adding the center of inversion (see Table 1.10). 

What relates a crystal class to a crystallographic point group ... 

As remarked in Chapter 1, the external shape of a crystal can be classified into one of 32 crystal classes by making use of the symmetry elements that are present. These crystal classes correspond to the 32 crystallographic point groups. The two terms are often used completely interchangeably, and for practical purposes can be regarded as synonyms. 

The listing of the non-crystallographic point groups is perfectly analogous to that of the crystal classes ... 

All crystalline materials may be categorized into 32 crystallographic point groups. Of the 21 classes that lack a centre of symmetry, 20 produce an electric dipole (i.e. polarization) when mechanically stressed. These materials are termed piezoelectric, the lack of centrosymmetry being a necessary condition to allow movement of the positive and negative ions in order to produce a dipole. Ten of these classes possess a permanent dipole and will respond to changes in temperature as well as stress. These are defined as pyroelectric. These classes can be subdivided further into ferroelectric crystals, in which the dipole moments of the individual crystalline units can be reversed by application of an electric field. 

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