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4. Space Groups

Just as the non-translational symmetry elements can be combined into point group symbols that describe the symmetry of finite groups, so the symmetry of infinite arrays can be summarized and symbolized. The addition of translation greatly increases the possibilities, so that instead of the 32 point groups 230 space groups are needed to describe the symmetries of infinite arrays. A complete list of these, with descriptions, is given in International Tables for X-Ray Crystallography (see Section 5). All that will be attempted here is to try to give some idea of what a space group symbol means and how to interpret it.

Typical space group symbols are: $P\bar{1}$, C2/m, Ibca, $R\bar{3}$, Fm3m, P2₁2₁2₁. You will notice that they all begin with a capital letter. This gives the lattice type , which tells you whether the unit cell is primitive or centred. P means primitive, A, B , or C means centred on the face perpendicular to the a, b or c axis, respectively, F means centred on all the faces, I means body centred -- centred in the middle of the cell (from the German, innenzentriert ) -- and R means rhombohedral, which is a special type of centring unique to the trigonal system. The group of symbols that follow give you the crystal class, and hence the system. Thus in $P\bar{1}$, the symbol $\bar{1}$ tells you that the system is triclinic (see Table 1). Likewise C2/m belongs to class 2/m, which you can see from Table 1 is monoclinic. $R\bar{3}$ and Fm3m are equally easy to assign to the trigonal and cubic systems, respectively. Ibca presents slightly more of a problem; the symbols b, c and a refer to glide planes, as explained in the previous section. To find the crystal class, simply replace any translational symmetry element by the equivalent non-translational element: this rule holds for both glide planes and screw axes. Thus Ibca belongs to class mmm , and is orthorhombic; the last example, P 2₁2₁2₁, belongs to class 222 and is also orthorhombic. The symbol also gives the positions of the various symmetry elements. Just as mmm implies mirror planes perpendicular to the three mutually perpendicular axes of the orthorhombic system, so Ibca tells us that in the body-centred array there are b glide planes perpendicular to the x axis, c glides perpendicular to y and a glides perpendicular to z .

The page of International Tables describing Pnma is reproduced in Fig. 5.1, from which it can be seen that the total collection of symmetry elements include many that are not listed in the space group symbol. Only the essential elements are given in the symbol; redundant ones are omitted, just as they are from point group symbols.

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