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

A representative half-turn-plus-translation operation about an axis parallel to $x$ isIt rotates about the line $y=1/4,z=0$ and translates by $a/2$ along $x$. It exchanges the two Fe sites, and maps $(1/2,0,\eta )$ onto $(0,1/2,-\eta )$ modulo a lattice translation. Likewise,is a $2_1$ screw about $x=1/4,z=0$. Including translated equivalents gives the principal-axis screw linesThus the cell contains lines at the quarter and three-quarter coordinates, in both the $z=0$ and $z=1/2$ levels.

For completeness, translated diagonal twofold operations also produce diagonal $2_1$ screw lines. The operations $(x,y,z)\mapsto (y+1,x,-z)$ and $(x,y,z)\mapsto (1-y,-x,-z)$ preserve the same basis. Their axial translations are $(1/2,1/2,0)$ and $(1/2,-1/2,0)$, half the primitive repeats along the corresponding diagonal directions. Their lines areDiagonal lines with integer $x\mp y$ instead support pure twofold rotations. Listing only rotational parts modulo primitive translations would hide these shifted screw lines. There is no fractional screw advance along $z$: every Fe site has integer $z$, so a symmetry taking the Fe layer into itself requires an integer vertical translation. Proper fourfold axes parallel to $z$ are therefore not $4_1$, $4_2$, or $4_3$ screws in this structure. All these statements can be checked directly on the four supplied coordinates without choosing an alternative tabulated origin.^[ref]