Crystallography and Crystal Defects. Anthony Kelly

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rel="nofollow" href="#fb3_img_img_b37ae731-a372-5429-ad76-1922e7b873c5.png" alt="images"/>m2, , , Pm2, P2m, P312, P321, P3m1, P31m, Pm1 and P1m. Therefore, there are 73 such space groups, referred to as arithmetic crystal classes in Section 1.3.4 of the sixth edition of the International Tables for Crystallography [3].

      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, R32, R3m, R3c, Rm, Rc); the majority have a primitive hexagonal unit cell. Buerger [1] argues that ‘to refer one crystal of the trigonal class to rhombohedral axes, and another to hexagonal axes, because the lattice types are R and P respectively, leads to confusion’ (p. 106) and discourages the use of rhombohedral axes for this reason. It is certainly the case in the scientific literature that the use of rhombohedral axes for crystals where the space group is one of the seven rhombohedral space groups is markedly less prevalent than the use of hexagonal axes.

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, Fmm (in full, F 4/m 2/m) means the cubic face‐centred lattice with the point group mm associated with each lattice point; P6₃/mmc (in full, P 6₃/m 2/m 2/c) is a hexagonal primitive lattice derived from P6/mmm by replacing the sixfold rotation axis by 6₃ and one of the mirror planes by a c‐axis glide plane. P6₃/mmc and P6/mmm share the same point group symmetry. The point group symmetry of any crystal is derived immediately from the space group symbol by replacing screw axes by the appropriate rotational axes and glide planes by mirror planes in the space group symbol.

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

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