Non-crystallographic point groups

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

Table 2.1 presents the non-cubic crystallographic point groups with compared notation. 

Table 1.17. The 230 crystallographic space groups arranged according to seven crystal systems and 32 crystallographic point groups as they are listed in the International Tables for Crystallography, vol. A. The centrosymmetric groups are in bold, while the non-centrosymmetric groups that do not invert an object are in italic. The remaining are non-...

Thus physical properties can be used as a probe of symmetry, and can reveal the crystallographic point group of the phase. Note that Neumann s principle states that the symmetry elements of a physical property must include those present in the point group, and not that the symmetry elements are identical with those of the point group. This means that a physical property may show more symmetry elements than the point group, and so not all properties are equally useful for revealing tme point group symmetry. For example, the density of a crystal is controlled by the unit cell size and contents, but the symmetry of the material is irrelevant, (see Chapter 1). Properties similar to density, which do not reveal symmetry are called non-directional. Directional properties, on the other hand, may reveal symmetry. 

For a given molecule the combination of symmetry elements is known as its point group. There are 32 crystallographic point groups that are possible. The asymmetric unit volume can contain an object which may or may not have symmetry. It is quite common in protein crystallography that this object has symmetry of its own, known as non-crystallo-graphic symmetry (examples include two-fold, three-fold, four-fold, five-fold etc, i.e. any point symmetry is possible in the asymmetric unit because it need not obey the translational symmetry of the crystal). 

In a famous paper by Shechtman et al. (1984) electron diffraction patterns were shown of rapidly quenched and solidified aluminium-manganese alloys. Sharp diffraction peaks, suggesting long-range translational order, were observed with the presence however of five-fold symmetry (that is of a non-crystallographic symmetry see 3.6.1.1). By different orientation of the specimen five-fold axes (in 6 directions), three-fold axes (in 10 directions) and two-fold axes (in 15 directions) were identified with the subsequent observation of the existence also of an inver-sion centre the assignment of this phase to the icosahedral point group, m36, was defined. 

To describe the contents of a unit cell, it is sufficient to specify the coordinates of only one atom in each equivalent set of atoms, since the other atomic positions in the set are readily deduced from space group symmetry. The collection of symmetry-independent atoms in the unit cell is called the asymmetric unit of the crystal structure. In the International Tables, a portion of the unit cell (and hence its contents) is designated as the asymmetric unit. For instance, in space group P2 /c, a quarter of the unit cell within the boundaries 0asymmetric unit. Note that the asymmetric unit may be chosen in different ways in practice, it is preferable to choose independent atoms that are connected to form a complete molecule or a molecular fragment. It is also advisable, whenever possible, to take atoms whose fractional coordinates are positive and lie within or close to the octant 0 < x < 1/2,0 < y < 1/2, and 0 < z < 1 /2. Note also that if a molecule constitutes the asymmetric unit, its component atoms may be related by non-crystallographic symmetry. In other words, the symmetry of the site at which the molecule is located may be a subgroup of the idealized molecular point group. 

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