Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 1 / c / d

One crystallographic inversion center isThe inversion maps $(x,y,z)$ to $(1/2-x,1/2-y,-z)$; it exchanges both the Fe pair and the Se pair modulo lattice translations.

A $\bar{4}$ axis passes along $z$ through the Fe site $(0,0,0)$, and likewise through $(1/2,1/2,0)$. A representative rotoinversion isIt leaves the first Fe site fixed, maps the other Fe onto a translationally equivalent site, and exchanges the Se height classes. The $\bar{4}$ operation includes a vertical reversal, so it must not be replaced by a pure $90$-degree rotation about an Fe site. Such a pure rotation would put a selenium atom at the wrong height.