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

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]

Reflection through $z=0$ interchanges the two Se height classes but, by itself, leaves their projected positions wrong. The required parallel translation isFor example, $(1/2,0,\eta )$ maps to $(1,1/2,-\eta )$, equivalent to the second Se site. The two Fe sites are also interchanged. This is a diagonal, or $n$, glide; adding an integer lattice vector gives the same fractional translation class. Its translation is expressed in the conventional primitive tetragonal lattice coordinates, not in a smaller Fe-only cell that would discard the selenium height pattern.

At $l=0$ a point and its glide partner have equal scattering amplitudes and relative phase $2\pi (h/2+k/2)$. Their joint contribution to the structure factor is proportional toIt vanishes for odd $h+k$. Hence the allowed-condition statement isThis is a systematic absence: it follows independently of the numerical Fe and Se atomic scattering factors. Explicitly, the supplied basis givesand both braces vanish when $h+k$ is odd. When the sum is even a reflection is symmetry-allowed, but an accidental cancellation for particular scattering factors or experimental conditions can still make its intensity small. The glide alone does not promise that every allowed reflection has a nonzero measured intensity.

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.