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For the four conventional face-centered cubic lattice sites $000$, $0\frac{1}{2}\frac{1}{2}$, $\frac{1}{2}0\frac{1}{2}$ and $\frac{1}{2}\frac{1}{2}0$, their centering factor isThis is the face-centred cubic selection rule. Enumerating the nonzero representatives with $h^2+k^2+l^2\le 16$ givesSymmetry-related permutations and signs belong to the same powder ring, not additional rings. For Al, the basis contains one atom, $S=F_{\mathrm{fcc}}{f}_{\mathrm{Al}}$. ThusFor the stated KCl basis,The all-even reflections add the two atomic factors; all-odd reflections subtract them. Ionic $K^{+}$ and $C{l}^{-}$ each have 18 electrons, giving similar X-ray amplitudes. Their difference is much smaller than their sum, so the 111 and 311 rings are weak. Consequently the four strong rings areThis is an accidental diffraction extinction from equal atomic scattering factors in the equal-factor limit, not an extra exact fcc selection rule. Isoelectronic ions can have different radial electron distributions, so real odd reflections need not vanish exactly. Even the cruder neutral-atom estimate $f_K\simeq 19$, $f_{\mathrm{Cl}}\simeq 17$ makes the odd/even basis-intensity ratio only $(2/36{)}^2\simeq 0.0031$, before other ring-intensity factors.

For Si's displaced identical basis atoms,For all-odd indices, the bracket is $1\pm i$, with squared magnitude 2. For all-even indices it vanishes when $h+k+l=2(\mathrm{mod}4)$, and has squared magnitude 4 when the sum is a multiple of four. Thus 200 and 222 vanish, whereas 111, 220, 311 and 400 survive:These are the basis extinctions of the ideal diamond cubic structure. Intensity strengths within each surviving set can still differ because of atomic factors, multiplicity and geometry.

The two body-centre/origin sites and the other six sites have different local environments. Their effective atomic scattering factors therefore need not be identical even in an elemental, chemically pure crystal. Write them as $f_A$ and $f_B$, respectively. The crystal basis structure factor becomesFor the low-angle families this givesThus $\{110\}$ can produce a weak extra powder diffraction ring below $\{200\}$. This is an accidental diffraction extinction from equal atomic scattering factors: its cancellation in part d is exact only in the identical-atom-scattering approximation. Small differences in the bonding electron density, or site-dependent Debye–Waller factors, can leave a small nonzero amplitude. For instance, taking $f_A=f+\delta f$ and $f_B=f$ gives $F_{110}=2\delta f$ while $F_{200}=4f+2\delta f$; the corresponding intensity suppression is quadratic in $|\delta f/f|$. No impurity is required. The $\{100\}$ and $\{111\}$ cancellations remain for arbitrary scalar $f_A,f_B$ in this two-site-class model.