Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 4 / 57

The useful feature is the selenium atom's wavelength-dependent anomalous X-ray scattering. Close to its absorption edge the atomic scattering factor is complex:Replacing a sulfur-containing side chain by its selenium analogue introduces relatively strong, spatially localized anomalous scatterers. This is the diffraction-physics reason for the label; no biological preparation procedure is needed for the calculation.

The Se substructure can be located from intensity differences or Patterson maps. Measurements at several wavelengths change its known complex contribution while leaving most other atoms' scattering nearly unchanged. Interference between that contribution and the unknown remainder constrains the otherwise missing structure factor phases. Both wavelength-dependent dispersive differences and differences between Friedel pairs contribute to multiwavelength anomalous diffraction. Several wavelength constraints resolve phase ambiguities that remain with one intensity difference. The resulting phased Fourier synthesis yields the electron-density map.

Patterson maps give interatomic displacement vectors rather than absolute positions. The three strong half-cell vectors are consistent with a face-centered four-point Se substructure. One permissible coordinate set isEach nonzero vector occurs four times among the twelve ordered Se-Se pairs, explaining equal weights. These positions form a valid four-site orbit of the specified space group in a suitable origin convention.

A common translation of all four sites is invisible to a Patterson map; absolute coordinates are therefore not uniquely fixed by these three peaks. Subsequent phases are stated in the origin convention above. The remaining atoms need not have face-centered translational symmetry merely because the dominant Se substructure does.

For reflection $222$, the positional phase factor is $e^{2\pi i(2x+2y+2z)}=1$ for each of the four sites just found. Treat the supplied anomalous increment as the imaginary term $i{f}^{\prime \prime }$, with $f^{\prime \prime }=4.8$ electrons per Se atom. ThenIts magnitude is $19.2$ electrons and its phase is $+90^{\circ }$ relative to the real axis in this structure-factor convention. Translating the origin rotates both this reference contribution and the ordinary structure factor consistently; the physically relevant phase difference is invariant.

Write the ordinary structure factor as $F_0=|F_0|e^{i\varphi }$ and use $F=F_0+19.2i$. The intensities implyConsequently $\sin \varphi =0.43893$, and the two possible ordinary phases areThe twofold ambiguity is expected from a single anomalous-intensity constraint. The calculation assumes that the specified imaginary increment is the only change requiring inclusion; an additional unspecified real dispersive change could not be reconstructed from these data alone.

Without significant anomalous scattering, the electron density is effectively real and Friedel's law gives $F_0(-\mathbf{h})=F_0(\mathbf{h}{)}^{\ast }$, hence $I(\overline{222})=13130$.

With anomalous scattering the atomic imaginary term retains its sign when the reflection index reverses. Here $\Delta F(-222)=19.2i$, soThus the second expected intensity is approximately $1.157\times 10^4{\mathrm{electrons}}^2$. Conjugating the complete anomalous structure factor would incorrectly impose Friedel equality on the second experiment.