Past exam of the materials science course of the University of Cambridge / 2016 / Paper 1 / 1 / a / ii / Solution

There are $2_1$ screw axes parallel to $[100]$ through $(y/b,z/c)=(1/4,0),(3/4,0),(1/4,1/2),(3/4,1/2)$, with periodic repeats.

For the first axis the operation in fractional coordinates isIt rotates by $180^{\circ }$ about the axis and translates by $a/2$. For example, it sends the A center at $(0,0,0)$ to the A center at $(1/2,1/2,0)$, and sends B at $(1/2,0,1/2)$ to B at $(0,1/2,1/2)$ modulo integer translations. A half-turn preserves each unoriented diatomic bond. A second application gives one full $a$ translation.

There are also ordinary twofold axes through the molecular-center rows; those should not be confused with the screw axes halfway between the corresponding rows.