Crystallography and Crystal Defects. Anthony Kelly

groups that each point group and space group consists of a self‐consistent finite set of symmetry operations. Therefore, point groups and space groups can be described in the formalism of group theory [17–19]. A group G is a set of elements g₁, g₂, g₃ … g_n which conform to the following four conditions [19]:

      1 The product gigj of any two group elements must be another element within the group.

      2 Group multiplication is associative: (gigj)gk = gi(gjgk).

      3 There is a unique group element I called the identity element belonging to G so that its operation on a member of the group g is such that Igi = giI = gi.

      4 Each element within the group has a unique inverse; that is, for each gi there is a unique element gi−1 so that gigi−1 = gi−1 gi = I.

      Thus, for example, following the example of the point group mmm used by de Jong [5], p. 20, the eight group elements within this point group are: the identity, I; three 180° rotations about the x‐, y‐ and z‐axes; three reflections in the (100), (010) and (001) planes; and a centre of symmetry. A subgroup of these symmetry elements is found in 2mm: the identity, a 180° rotation about the x‐axis and two reflections in the (010) and (001) planes. Other subgroups are the point groups 222, 2/m, m, 2, , and 1. Conversely, the point group mmm is clearly a supergroup for 2mm and 222.

      Similar principles apply to subgroups and supergroups of space groups. Thus, for example, following Hans Wondratschek in chapter 8 in Volume A of the revised, fifth edition of the International Tables for Crystallography [20], the space group P2m is one member of the group whose minimal nonisomorphic supergroup is P6₃/mcm; it has as maximal nonisomorphous subgroups P, P321, P31m and Pm2m. (Groups are termed isomorphic if they have an equivalent group multiplication structure, but differ in the nature of the elements constituting them [19]. Thus, for example, the point groups 2/m, 222 and 2mm are isomorphic: all three have four elements within the group and the same multiplication structure within them.)

2.16 An Example of a Three‐Dimensional Space Group

To illustrate the principles we have outlined in Sections 2.12–2.14, an example of an entry from in Volume A of the revised, fifth edition of the International Tables for Crystallography [20] is shown in Figure 2.25. In the more recent sixth, revised edition of these Tables, the data tabulated in the revised, fifth edition on subgroups and supergroups of the two‐ and three‐dimensional space groups and shown in Figure 2.25 has been moved to Volume A1.

      

Figure 2.25 An example of a space group.

      Source: Taken from [20].

      It is apparent from the Hermann–Mauguin notation of this space group, Pnma, No. 62, that crystals with this space group symmetry belong to the orthorhombic crystal system, point group mmm. The Schoenflies notation indicates that the space group is dihedral (or two‐sided; from the ‘D’), that it has a twofold rotation axis (from the ‘2’), that there is in addition a mirror plane perpendicular to this twofold axis (from the ‘h’ subscript) and that it is the 16th of 28 such space groups … (space groups 47–74 inclusive), all of which are space groups derived from the mmm point group.

      The full Hermann–Mauguin notation for this space group is P2₁/n 2₁/m 2₁/a, showing that there are three mutually perpendicular sets of screw diads in addition to the mirror, m, and the diagonal, n, and axial, a, glide planes. The diagrams of the symmetry elements of the group follow the principles in Figure 2.24, with the use of standard graphical symbols for n (parallel to (100)) and a (parallel to (001)). The projections in the top left‐hand corner and the bottom right‐hand corner are both down [001], the one in the top right‐hand corner down [010] and the one in the bottom left‐hand corner down [100].

The asymmetric unit is ‘a (simply) connected smallest part of space from which, by application of all symmetry operations of the space group, the whole of space is filled exactly’ [20]. The listing under the heading ‘Symmetry Operations’ in Figure 2.25 is a summary of the various geometric descriptions of each of the eight symmetry operations in the space group, numbered (1)–(8). Thus, for example, here the second of these, 2, 0, z, defines the screw diad parallel to [001] intersecting the x–y plane at (, 0). The ‘Generators Selected' define the order in which the coordinates of the symmetrically equivalent positions are produced; for example, the coordinates related by symmetry to the general position x, y, z in the space group. (1) is the identity operation, showing that in this case integral lattice translations produce the equivalent point in adjacent unit cells. After the ‘Positions', which follow the protocol described in Section 2.12, and the ‘Symmetry of Special Projections', which are self‐explanatory, a summary is given of subgroups and supergroups related to the space group under consideration, following the methodology set out in chapter 2 of [20]. Finally, the reflection conditions listed under ‘Positions’ specify the systematic absences which occur under kinematical diffraction conditions, such as X‐ray diffraction.

2.17 Frequency of Space Groups in Inorganic Crystals and Minerals

Although there are 230 space groups, analyses of the crystal structures of organic compounds and inorganic crystals and minerals show that these structures clearly favour some space groups over others. Thus, for example, almost of inorganic crystal structures belong to one of 24 space groups [21]. For organic compounds just two space groups, P2₁/c and , account for over half the crystal structures solved by 1990 [22], while for monomeric proteins, the space group P2₁2₁2₁ accounts for almost one third of crystal structures solved by 1995 [23]. Packing and entropic considerations of atoms and molecules, discussed in Chapters 3 and 4, both help to rationalize these observations [21–23]. The consequence of these observations and considerations is that some space groups are markedly less important than others for crystal structure determination [21]. In this context, the choice of Pnma, No. 62, as the three‐dimensional