*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.

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* 2_{1}/*n* 2_{1}/*m* 2_{1}/*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* 3*d* , 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 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 and of the way along because the origin has been chosen at a centre of symmetry (this has computational advantages).

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.

Suppose now that the group is moved from *x, y, z* to *x* ,
, *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
4*c* ...; 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, MSO_{4},
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. SO_{4} 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.

**Copyright © 1984, 1997 International Union of
Crystallography**