5. International Tables for X-Ray Crystallography, Volume I

This is an archive copy of the IUCr web site dating from 2008. For current content please visit https://www.iucr.org.

International Tables for X-Ray Crystallography contain a great deal of information concerning various aspects of crystallography, and the first volume deals with symmetry. Among other things, it introduces symmetry considerations in more detail and at a more advanced level than can be given here. It also lists information about all the space groups, and it is this that we will consider now. Fig. 5.1 reproduces a typical page from the space group listings, and we will use it to explain some of the important features.

**Fig. 5.1.** Page 151, Vol. I. reproduced from International Tables for X-Ray Crystallography, 1965 edition (by kind permission of the International Union of Crystallography).
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The space group symbol -- Pnma -- is given in bold type at the top of the page, together with the equivalent Schoenflies symbol. Replacing the translational elements n and a by the non-translational equivalent m gives crystal class mmm , system orthorhombic: this information is also given at the top of the page. On this same line are: (1) the full symbol P 2₁/n 2₁/m 2₁/a (which includes redundant twofold screw axes parallel to a, b and c created by the interaction of the mirror and glide planes) (2) the space group number, 62. The space groups are listed in a logical sequence in increasing symmetry of their crystal classes, beginning with No. 1, P 1, which has no symmetry at all, and ending with the highly symmetric cubic group Ia 3d , No. 230.

Immediately below this top line are two diagrams representing the space group in terms of asymmetric units (on the left) and as a collection of symmetry elements (on the right). They are projected onto the page down the z axis, with the y axis running horizontally from left to right across the page and the x axis downwards. The space group symbol tells us that there is an n -glide plane perpendicular to the x axis, a mirror plane perpendicular to the y axis and an a -glide plane perpendicular to the z axis (i.e. parallel to the plane of the paper); these appear on the right-hand diagram as dot-dash lines, heavy solid lines, and a right-angled line with an arrow, respectively. (The arrow shows the direction of the glide -- along a ; the small figure $\frac{1}{4}$ indicates that the glide plane is c /4 above the plane of the diagram.) The redundant twofold screw axes also appear, and a number of centres of symmetry are created.

The effect of these symmetry operations can be traced in the left hand diagram. The effect of the mirror plane is particularly easy to see (refer back to Fig. 1.4). After you have found this, look for the effect of the glide planes and screw axes; a key is given in Fig. 5.2. All the reflection planes occur in groups, separated by half a cell edge; here they are $\frac{1}{4}$ and $\frac{3}{4}$ of the way along because the origin has been chosen at a centre of symmetry (this has computational advantages).

**Fig. 5.2.** The equivalent groups generated by the operation of some of the symmetry elements in *Pnma]* . (a) An a -glide plane parallel to the plane of the paper. The group is moved down the diagram by a /2, and then reflected. (b) An n -glide perpendicular to a moves the same group b /2 to the right and c /2 vertically before reflection. (c), (d), (e) Screw axes move the group through a /2, b /2 and c /2, respectively, while rotating it through 180 $^{\circ}$ .
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The remaining feature of the page that concerns us here is the list of coordinates of equivalent positions. An asymmetric group placed at random in the cell (at x, y, z , for example, a position represented by the open circle at `+' in the top left hand corner of the left hand diagram) must be matched by seven other groups, a total of eight in all. Applied to a crystal structure, this means that an atom that occurs in such a general position must be one of a total of eight similar atoms in the unit cell; this can be very helpful in deciding chemical formulae. In the Tables, the coordinates corresponding to this general position are listed first: 8 is the number of equivalent positions, d is simply a letter assigned as a convenient means of referring to the set, and 1 shows that the site has no symmetry.

**Fig. 5.3.** Groups or atoms on special positions. (a) The effect of moving an asymmetric group onto a mirror plane; it no longer produces an equivalent group through the mirror, but must itself now possess a mirror plane of symmetry. (b) If an atom is placed on the special position c in *Pnma* there must be three more similar atoms in the cell, arranged as shown (not seven as in the general case).
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Suppose now that the group is moved from x, y, z to x , $\frac{1}{4}$ , z -- that is, onto the mirror plane (Fig. 5.3). This brings it into coincidence with its mirror image, and reduces the number of groups to four. This situation is summarised in the line beginning 4c ...; the symmetry of the site is now m , and there are only four equivalent positions instead of eight. The remaining two lines show the effect of selecting a position on one or other of the centres of symmetry. We make no use here of the rest of the information on the page.

This final paragraph gives a very brief illustration of one practical use of such symmetry information. Suppose that a metal sulfate, MSO₄, crystallizes in this space group with four formula units per cell. The four M and S atoms must lie on one of the positions, a, b or c ; the sulfur atom forms part of a sulfate group. SO₄ groups are tetrahedral, tetrahedra don't have centres of symmetry but do have mirror planes: therefore the sulphur atom must occupy position c , on the mirror planes (as also must two of the oxygen atoms attached to it). Combining this information with a knowledge of bond lengths should enable a trial structure to be worked out.