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Rotation matrices (**R**) and translation vectors (**t**) are very
powerful descriptions of the symmetry within the crystal and give aid in origin
specification, in determining phase restrictions, systematic absences,
systematic enhancement and seminvariants, in distinguishing centric and acentric
reflections, general and spatial reflections and are helpful in making correct
space group determinations.

Every space group has a number of equivalent positions. These may be from 1, as
in *P*1, to 192, as in *Fm*3*m*, *Fm*3*c*, *Fd*3*m*, *Fd*3*m* and *Fd*3*c*. Every atom at
a point (*x*, *y*, *z*) is also found as a result of the symmetry at position
(). The equivalent positions are listed for
all space groups in *International Tables for X-ray Crystallography* . The
equivalent positions are related to each other through **symmetry
operations**.
Every symmetry operation is a pair of **R** and **t**. One equivalent
position is derived from another through a rotation and a translation applied in
that order. The word rotation stands not only for 2-, 3-, 4- or 6-fold
rotation, but also for reflections in a point or in a plane. The translations
are along axes or diagonals of the unit cell. The relation between two
equivalent positions can be denoted as:

(1) |

*Example* :

The space group *P*1 has only one equivalent position (*x*, *y*, *z*). The only
symmetry operation in that space group is thus the unit matrix, **I**

*Example* :

The space group *P*3_{1} has 3 equivalent positions: (*x*, *y*, *z*), (-*y*, *x* - *y*,
+ *z*) and (*y* - *x*, -*x*, + *z*). The symmetry
operations are:

Note that

The different symmetry operations for each space group can always be derived from at the most 3 unique not further reducible symmetry operations (3 because space is 3-dimensional). All other symmetry operations can be derived from the unique ones through:

(2) |

**Copyright © 1981, 1998 International Union of
Crystallography**