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In the conventional cubic unit cell, the four face-centered cubic lattice positions contributeThe two-atom atomic basis of a crystal gives the zinc blende structure factorMixed-parity Miller indices have $S_{\mathrm{FCC}}=0$. With all indices even or all odd, $S_{\mathrm{FCC}}=4$. For real atomic scattering factors,Hence all same-parity reflections are generally permitted in GaAs, although the last class is weak when the two atomic scattering factors nearly coincide. These are the structural selection rules; a particular experiment must also satisfy the elastic Bragg scattering condition.

In the electron-count estimate $f_{\mathrm{Ga}}\simeq 31$, $f_{\mathrm{As}}\simeq 33$, the structure factors are $F_{111}=4(31-33i)$ and $F_{200}=4(31-33)$. Thus a single-directed-reflection estimate isA powder X-ray diffraction peak collects all equivalent reflections. The powder diffraction multiplicities are eight for $\{111\}$ and six for $\{200\}$, so the multiplicity-and-electron-count estimate isAt $(200)$ the two sublattice amplitudes are opposite in phase and nearly cancel; at $(111)$ their relative phase is a quarter cycle, so this cancellation does not occur. In silicon, the two atomic scattering factors are identical and the diamond cubic structure has an exactly forbidden $(200)$ reflection, while $(111)$ remains allowed in the kinematic approximation.

The numerical ratio neglects the angular dependence of the atomic scattering factors, the Lorentz-polarization factor and thermal-motion corrections. No wavelength or measurement geometry is supplied to refine those factors. This is therefore a stated structural estimate, rather than a unique precision ratio of measured peak areas.

The all-even indices are allowed, but $d_{640}=0.07766\mathrm{nm}$ is small. copper $K_{\alpha }$ givesThis is near backscattering and outside an ordinary portable instrument whose arm reaches only about $130^{\circ }$. The numerical angular limit should be checked on the actual model; the mathematical requirement is that its maximum $2\theta$ reach $165^{\circ }$. It also requires a normal inclined ${\tan }^{-1}(4/6)=33.69^{\circ }$ to $[100]$, so the coupled surface-normal scan is insufficient. atomic scattering factors are smaller at this larger scattering vector.

A suitable high-angle, independently orientable diffractometer could detect it. A shorter selected wavelength can instead reduce the angular requirement: with $\lambda =0.071\mathrm{nm}$, $2{\theta }_B\simeq 54.4^{\circ }$, while orientation still needs to be set appropriately. If $\lambda$ exceeded $2d=0.1553\mathrm{nm}$ there would be no first-order Bragg solution at all. The tube continuum and other characteristic lines must therefore be distinguished from the specifically calculated copper $K_{\alpha }$ peak.

Ordering reflections by $d_{hkl}^{-2}=(h^2+k^2)/a^2+l^2/c^2$ with $c\simeq 0.9a$, the first three nonzero families are $001$, $110$ and $111$. The $100$ family is extinct. Their structure factors areThe atomic scattering factors in each expression are evaluated at that reflection's scattering vector, so equal symbolic forms need not mean equal numerical amplitudes at different angles. In particular, the low-angle superlattice peaks need not be the strongest peaks.

Summing phase-weighted atomic scattering factors over the four positions givesIf $h,k$ have different parity, the reflection vanishes. If they have equal parity, this becomes $2[f_{\mathrm{Au}}+(-1{)}^{h+l}{f}_{\mathrm{Cu}}]$. The difference reflections reveal chemical order; they disappear when all sites instead have the same average scattering factor.

The zinc blende structure consists of a face-centered cubic lattice with two species displaced by a quarter of the body diagonal. Its conventional-cell structure factor vanishes for mixed-parity Miller indices. For equal parity, $S_{\mathrm{FCC}}=4$: odd indices give $|F{|}^2=16(f_A^2+f_B^2)$ for real atomic scattering factors, even index sums divisible by four give $16(f_A+f_B{)}^2$, and sums congruent to two modulo four give $16(f_A-f_B{)}^2$. The last reflections diagnose sublattice scattering contrast. They vanish in the equal-factor diamond limit but need not vanish for a binary compound.