When the possibilities of introducing screw axes to replace pure rotational axes and glide planes to replace mirror planes are taken into consideration, the result is to produce the total of 230 different space groups [3]. Of particular note are the space groups in the trigonal crystal system. Of the 25 trigonal space groups, only seven have a rhombohedral unit cell (R3, R
In centred cells, the possibility of a ‘double’ glide plane arises, given the symbol e. This symmetry operation is recognized officially in the fifth and sixth editions of the International Tables for Crystallography and incorporated into the conventional space group symbols of five orthorhombic space groups, Aem2, Aea2, Cmce, Cmme, and Ccce. In this operation, two glide reflections occur through the plane under consideration, with glide vectors perpendicular to one another related by a centring translation [3]. Seventeen space groups have these ‘double’ glide planes, recognized in the footnotes to Tables 2.1.2.2 and 2.1.2.3 of [3], but in 12 of these cases the ‘double’ glide planes are parallel to mirror planes. Therefore, their existence is not immediately apparent from the space group symbol because the mirror planes are stronger symmetry elements than the ‘double’ glide planes.
The arrangements of all of the symmetry elements in the 230 space groups are listed in the various editions of the International Tables for Crystallography. The numbers allocated to the various space groups are such that the space groups are in the sequence introduced by Schoenflies in 1891 [13], starting with the two triclinic space groups and finishing with the 36 cubic space groups.
Of particular interest to most materials scientists are the coordinates of general and special positions within each unit cell and diagrams showing the symmetry elements. More recent editions of the International Tables for Crystallography have extensive descriptions of space groups, which also include listings of maximal subgroups and minimal supergroups; these concepts will be discussed in general terms in Sections 2.14 and 2.15.
2.12.4 The Relationship Between the Space Group Symbol and the Point Group Symmetry for a Crystal
The conventional space group symbol shows that the space groups have been built up by placing a point group at each of the lattice points of the appropriate Bravais lattice. Thus, for example, Fm
2.13 The 17 Two‐Dimensional Space Groups
The essential elements of how the assignment of a crystal structure to a particular space group imposes restrictions on the possible general and special positions of atoms or ions within a particular unit cell can be illustrated by considering the 17 two‐dimensional space groups (or plane groups). These space groups are shown in Figure 2.24 in a manner similar to how they are depicted in the International Tables for Crystallography [3]. If the plane groups are understood, the essential elements of the three‐dimensional space group tables in the International Tables can be understood without difficulty.
Figure 2.24 The 17 two‐dimensional space groups arranged following the International Tables for Crystallography [3]. The headings for each figure read, from left to right, Number, Short Symbol (see Table 2.4), Point Group, Net. Below each figure the columns give the number of equivalent positions, the point group symmetry at those positions and the coordinates of the equivalent positions. The coordinates of positions x and y are expressed in units equal to the cell edge length in these two directions
Space groups are obtained by the application of point group symmetry to finite lattices, the possibility of translation symmetry being taken into account. There are five lattices or nets in two dimensions, shown in Figure 1.14. Following the convention in the International Tables for Crystallography, we shall give them the symbol p for primitive and c for centred. The symbols for twofold, threefold, fourfold and sixfold axes and for the mirror plane are as in Section 2.1. In the most recent editions of the International Tables for Crystallography, the parallelogram net in Figure 1.14a is referred to as an oblique net, and the triequiangular net in Figure 1.14c is referred to as a hexagonal net.
The only additional symmetry element in two dimensions is the glide reflection line: symbol g and denoted by a dashed line in Figure 2.24. It involves reflection and a translation of one‐half of the repeat distance parallel to the line.
The two‐dimensional lattices and the two‐dimensional point groups are combined in Table 2.4 to show the space groups that can arise, which are shown in Figure 2.24. In all the diagrams, the x‐axis runs down the page and the y‐axis runs across the page, the positive y‐direction being towards the right. In each of the diagrams the left‐hand one shows the equivalent general positions of the space group; that is, the complete set of positions produced by the operation of the symmetry elements of the space group upon one initial position chosen at random. The total number of general positions is the number falling within the cell, but surrounding positions are also shown to illustrate the symmetry. The right‐hand diagram is that of the group of spatially distributed symmetry