Past exam of the condensed matter physics course of the University of Oxford / 2017 / 1 / e / Solution

The sixth allowed family is $400$. For the two lattice constants, use $2{\theta }_{400}=2{\sin }^{-1}(2\lambda /a)$:ConsequentlyThe larger lattice constant shifts the AlAs reflection to lower scattering angle. As a differential check, $\Delta (2\theta )\simeq -2\tan \theta \Delta a/a$ in radians.

If the angular displacement is unresolved, use the intensity contrast. With the same $f\simeq Z$ model, a difference reflection has basis amplitude $13-33=-20$ for AlAs, versus $31-33=-2$ for GaAs. Its squared amplitude is therefore one hundred times greater before overall sample normalization. More robustly, compare a weak and a strong reflection within each specimen. Neglecting their almost identical angular corrections,The contrast in this internal ratio is about a factor $194$. Element-specific X-ray fluorescence, for example testing for a Ga characteristic line with a suitably energetic incident beam, or an X-ray absorption spectroscopy scan through an element-specific absorption edge, also distinguishes composition. Such scans can change incident energy; one must not assume every core shell can be excited by the original $8\mathrm{keV}$ beam.