Space group symmetry and its mathematical representation

In X-ray crystallography atomic coordinates are usually given as fractions of the unit cell parameters in a reference system that coincides with the cell axes system. This choice has the disadvantage of accepting an oblique reference frame, at least in monoclinic and triclinic systems. The advantages will soon become clear. 

Given the position of an atom as x = (x,y,z), the translation of one period along the cell axis a can be written as 

For a screw axis operation, if the origin is placed on the screw axis running in the y direction, the corresponding coordinate transformation for a single and a double application of the operation are  

The double application of a screw axis operation is equivalent to one cell translation along the screw axis direction. 

Consider now the joint application of a center of symmetry and of a screw operation (see Fig. 5.8). The origin is kept in the center of symmetry, while the screw axis is at a distance s along the z axis. The two operations can be written as, respectively  

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