= Solution
The sixth allowed family is $400$. For the two <lattice constants>, use $2\theta_{400}=2\sin^{-1}(2\lambda/a)$:
$$
2\theta_{400}(\mathrm{GaAs})=66.50250^\circ,\qquad
2\theta_{400}(\mathrm{AlAs})=66.40961^\circ.
$$
Consequently
$$
\boxed{\Delta(2\theta)\equiv2\theta(\mathrm{AlAs})-2\theta(\mathrm{GaAs})=-0.09290^\circ.}
$$
The 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,
$$
\frac{I_{200}}{I_{220}}=\frac6{12}\left(\frac{f_A-f_{\mathrm{As}}}{f_A+f_{\mathrm{As}}}\right)^2
=\begin{cases}0.0004883,&\mathrm{GaAs},\\0.09452,&\mathrm{AlAs}.
\end{cases}
$$
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.
\Image[/past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2017-zinc-blende-powder-contrast.png]
{title=Approximate powder-reflection contrast for GaAs and AlAs}
{description=Original stick patterns use the six derived angles, $f\simeq Z$, and multiplicity times squared structure amplitude. Each compound is normalized to its own $111$ reflection. A logarithmic intensity scale reveals the nearly cancelled $200$ and $222$ GaAs reflections. Geometry and form-factor corrections are omitted deliberately.}
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