Screw Axes

The combination of rotational symmetry and translational symmetry produces a new symmetry element, the screw axis.

A screw axis is constructed by applying a rotational symmetry element R to an object and then applying a fractional translation, t, parallel to the rotation axis.

The rotational element (and thus the transitional element) must be applied n times to rotate it a full 360 degrees and bring it back onto itself.  These n repetitions must also move the object by an integral number of unit cell periodicities.  That is nt = t_N

The translation, t, is along the rotation axis.  This makes Rt = t!  If the process is continued right to left the following result is obtained.

Comparing gives

General Symbol

Note: t = i/n