The crystallographic point groups

A solid can belong to one of an infinite number of general three-dimensional point groups. However, if the rotation axes are restricted to those that are compatible with the translation properties of a lattice, a smaller number, the crystallographic point groups, are found. The operators allowed within the crystallographic 

There is no significant symmetry present in the triclinic system, and the symmetry operator 1 or 

1 is placed parallel to any axis, which is then designated as the a-axis. 

In the monoclinic system, the point group symbols refer to the unique axis, conventionally taken as the b-axis. The point group symbol 

2 means a diad axis operates parallel to the b-axis. The improper rotation axis 2, taken to run parallel to the unique b-axis, is the same as a mirror, m, perpendicular to this axis. The symbol 2/m, indicates that a diad axis runs parallel to the unique b-axis and a mirror lies perpendicular to the axis. Because there are mirrors present in two of these point groups, the possibility arises that crystals with this symmetry combination will be enantiomorphous, and show optical activity, (see Sections 4.7, 4.8, below). Note that in some cases the unique monoclinic axis is specified as the c-axis. In this case the primary position refers to symmetry operators running normal to the direction [001], as noted in Table 4.3. 

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Table 15 The crystallographic point groups (crystal classes).

The environment of an ion in a solid or complex ion corresponds to symmetry transformations under which the environment is unchanged. These symmetry transformations constitute a group. In a crystalline lattice these symmetry transformations are the crystallographic point groups. In three-dimensional space there are 32 point groups. 

Fig. 10.7. The crystallographic point groups arranged according to their order, ms, shown on the left, and linked to show sub- and supergroup relations (adapted from International Tables for Crystallography Vol. A, (1996) Table 10.3.2).

For crystals, the point group must be compatible with translational symmetry, and this requirement limits n to 2,3,4, or 6. (This restriction applies to both proper and improper axes.) Thus the crystallographic point groups are restricted to ten proper point groups and a total of... 

We will show how a finite group "works" by using the crystallographic point group D3h (6m2) as a working example and considering the operations... 

Figure 2.25 Stereographic projections of the regular orbits of the Crystallographic point groups onto an inscribed sphere, showing vertices of the orbits (open-circles in the Northern hemisphere and crosses in the Southern) and the locations of symmetry elements as described in the text. [From Symmetry in Physics, J.P. Elliot and PG. Dawber, Macmillan, London, 1979.]...

Table B.3. Characters of the representations spanned by polar and axial vectors for the different symmetry operations of the crystallographic point groups...

To determine the crystallographic point group of a planar shape it is only necessary to write down a list of all of the symmetry elements present, order them following the rules set out above, and then compare them to the list of ten groups given. 

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. 

In order to discuss the rest of the crystallographic point groups, one further has to consider the dihedral rotation groups D4 and D, and the cubic rotation group O. Their character tables, standard basis functions, and a useful choice of group generators are displayed in Tables 6,7, and 8. In this way the material required for symmetry considerations is directly available. 

In this table all the crystallographic point groups have been collected together. They can be briefly characterized as all the sub-groups of O plus the sub-groups of Dgh of class 6 of this table. 

I 1.9. Determine the crystallographic point group for each of the following crystals, where the rotational axes and mirror planes are indicated. Use both the Schoenflies and Hermann-Mauguin notations. 

In the molecular-cluster approach a crystal with a surface is modeled by a finite system consisting of the atoms on the surface and of some atomic planes nearest to it. The diperiodicity of the surface is not taken into account. The symmetry of such a model is described by one of the crystallographic point groups. 

CONTEXT X-ray diffraction is the most common method for determining molecular structures within a crystal, but other methods are capable of faster, less detailed information about the crystal. For example, electron backscatter diffraction (also called backscatter Kikuchi diffraction), from a scanning electron microscope, measures the diffraction patterns of electrons that scatter off more than one plane in the crystal. From the patterns, the crystallographic point group, the orientation of the crystal, and the exposed Miller indices of the surface can be determined. Copper crystals, which have the advantage of simple structure, have been used to test the strengths and limitations of this method. 

To determine the crystallographic point group, we isolate the smallest part of the crystal that shares all the symmetry properties of the crystal. For all lattice systems except the hexagonal, this part is the unit cell. Once we ve identified and isolated this subunit of the crystal, we analyze it according to the flow chart in Fig. 6.7. 

PROBLEM By examining the symmetry elements of the lattice, determine the crystallographic point group for the fee lattice of a pure metal, such as copper, having a basis of one atom. 

Find the crystallographic point group of a primitive tetragonal lattice, assuming the basis can he treated as one atom per lattice point. 

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