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The four FCC translations give a lattice structure factor that is nonzero only when $h,k,l$ are all even or all odd. The identical two-atom motif then multiplies it byFor $(200)$ the FCC part is allowed, but the motif factor is $1+e^{i\pi }=0$. HenceIn the projection, equal numbers of atoms occur at $x/a=0,1/4,1/2,3/4$. Their $(200)$ phases are respectively $+1,-1,+1,-1$. The paired layers cancel exactly for equal silicon scattering amplitudes. This diamond-basis diffraction extinction is additional to the FCC centring rule; it is not evidence that no atoms lie on the nominal reflecting planes.

Measure ring centers and radii against the printed scale, rather than comparing raw squared radii. A representative reading of the four rings is $R\simeq 6.0,7.0,10.6,13.2\mathrm{mm}$, with broad photographic lines. For a flat plate at distance $L$,With $L=16\mathrm{mm}$ the ${\sin }^2\theta$ ratios are approximately $1:1.33:2.67:3.67$, givingFor identical atoms at the four conventional FCC positions,The structure factor is $4f$ for indices all even or all odd, and zero for mixed parity. Thus, for example, 100, 110 and 210 are systematic absences. The allowed squared-index sums begin $3,4,8,11,12,16,\dots$. Only the first four rings fit within the displayed field; 222 and later allowed reflections are not forbidden merely because they are outside the image. The central shadow and its holder are not extra diffraction peaks.

For a cubic crystal, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$ and first-order bragg diffraction requires $2d_{hkl}\sin \theta =\lambda$. The face-centred lattice sum isIt is four if $h,k,l$ are all even or all odd, and zero for mixed parity. With the NaCl basis displaced by $(\frac{1}{2},0,0)$, the allowed amplitudes are $F_{hkl}=4[f_{\mathrm{Cl}}+(-1{)}^{h+k+l}{f}_{\mathrm{Na}}]$, up to the arbitrary choice of origin/species. Both allowed parity classes survive because the two ionic scattering factors are not identical. Intensities depend on $|F{|}^2$, geometry, absorption and scattering vector.

There are three separate tests: nonzero structure factor, accessible Bragg angle, and accessible plane orientation. In a symmetric reflection scan the scattering vector is normal to the specimen face, selecting the surface-parallel family and its allowed orders. Other families need a different mounting, independent orientation control or transmission/another facet. The original diagrams distinguish these geometric requirements; angles in the reciprocal-space panel are between plane normals, not Bragg angles.

The 020 reflection is extinguished by the diamond basis, even though it satisfies the face-centered lattice parity rule. Write the diamond structure as an FCC lattice with identical atoms at basis positions $(0,0,0)$ and $(1/4,1/4,1/4)$.

The structure factor separates into lattice and basis terms:At $(h,k,l)=(0,2,0)$, all indices are even, so the first bracket is $4$. The second bracket is $1+e^{i\pi }=0$. The two identical sublattices scatter with opposite phases and cancel:For all-even reflections of diamond, the additional allowed condition is $h+k+l$ divisible by four. This is a systematic absence, not an absence of the corresponding family of geometrical planes.

The solid trace is wurtzite; the dashed trace is zinc blende. The isolated strong dashed peak is the cubic $111$ reflection. The solid trace has hexagonal $100$, $002$ and $101$ reflections in this angular interval; its central $002$ peak is the one at $28.65^{\circ }$, while cubic $111$ occurs at $28.53^{\circ }$. These assignments are consistent with the absence of lower-angle reflections.

In the ideal close-packed model, the cubic $111$ and hexagonal $002$ reflections probe the same local alternating layer geometry. Their scattering strength per suitably normalized amount of material and their angular factors are therefore very similar, but powder diffraction multiplicity differs: the cubic family has eight directed normals, while hexagonal $002$ has two. This explains the approximately fourfold greater intensity of the dashed peak. Comparing unnormalized conventional-cell structure factors alone would be misleading because the cells contain different numbers of formula units. Peak heights additionally depend on line widths; the multiplicity argument applies most directly to integrated intensities.

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.

For elastic scattering, $|{\mathbf{k}}_{\mathrm{in}}|=|{\mathbf{k}}_{\mathrm{out}}|=2\pi /\lambda$. Contributions from sites differing by any Bravais lattice translation $\mathbf{R}$ add in phase exactly whenfor every $\mathbf{R}$, namely when the momentum-transfer wavevector is a reciprocal lattice vector $\mathbf{G}$. For a family of planes of spacing $d$, the $n$th reciprocal vector normal to those planes has magnitude $2\pi n/d$. The scattering geometry gives $|\mathbf{G}|=2k\sin \theta$. Combining the two expressions provesConversely, reflection from that plane family satisfying Bragg's law supplies the corresponding normal reciprocal vector. The scalar angle condition alone needs this plane-normal geometry; it does not specify an arbitrary outgoing direction. A nontrivial atomic basis can extinguish an otherwise allowed reciprocal-lattice reflection through its structure factor.

The number of symmetry-equivalent directed reciprocal-lattice normals belonging to a powder-reflection family. For a cubic crystal, $\{111\}$ has multiplicity eight, corresponding to four unoriented plane orientations; hexagonal $002$ has multiplicity two. Multiplicity contributes to integrated powder intensity alongside the structure factor, angular factors and sample-dependent corrections.

The primary experimental-methods paper's section 4, equation 5, separates powder diffraction multiplicity, squared structure factor, polarization and geometric corrections. Sections 4.1–4.2 explain why powder diffraction intensity corrections depend on the measurement geometry rather than only the crystal structure.

For positional structure factors $A,B$, a real scattering factor $a$, and a complex factor $b+i{b}^{\prime \prime }$, the amplitudes are $F_{+}=aA+(b+i{b}^{\prime \prime })B$ and $F_{-}=a{A}^{\ast }+(b+i{b}^{\prime \prime })B^{\ast }$. Expanding their squared moduli gives the displayed difference. Resonance and lack of inversion allow a difference but do not force one: proportional positional sums with a real proportionality factor give zero.

Diffraction from an oriented single crystal. A reflection requires both an allowed nonzero structure factor and access to its reciprocal-lattice direction and Bragg angle.

Single isomorphous replacement estimates phases in X-ray crystallography by comparing a native crystal with one heavy-atom derivative. Under the ideal crystal isomorphism assumption, a reflection's derivative structure factor is $F_{PH}=F_P+F_H$, where the heavy-atom model supplies the complex reference $F_H$. The measured magnitudes constrain $F_P$ to two circles in the complex plane: one centered at zero with radius $|F_P|$, the other centered at $-F_H$ with radius $|F_{PH}|$. A general intersection gives two phase choices.^[ref]

An original exact example uses $|F_P|=5$, $F_H=3$ real and positive, and $|F_{PH}|=\sqrt{58}$. Writing $F_P=x+iy$ givesSubtracting the equations gives $6x+9=33$, hence $x=4$ and $y=\pm 3$. The alternatives areBoth alternatives predict exactly the same native and derivative intensities. Greater precision in these two magnitudes alone cannot select between them. The phase is the missing direction of a complex amplitude, not an unmeasured additive intensity.

Multiple isomorphous replacement can supply an additional circle from another derivative. The numerical example there resolves these particular candidates. In actual measurements, uncertainty, imperfect crystal isomorphism and an imperfect heavy-atom model broaden the constraints, so exact circle crossings are an explanatory idealization.

A split-site disorder model represents alternative atomic positions explicitly within an average crystal structure. Each alternative has its own coordinate and crystallographic site occupancy. For one atom distributed between exactly two alternatives, with no vacancy, the occupancies sum to1. Correlated alternative arrangements can be collected into disorder groups: atoms within one group occur together, while different groups in the same assembly represent alternatives.^[ref]

For an original scattering example, put the alternatives at fractional coordinates $(0,0,0)$ and $(1/2,0,0)$ with occupancies0.7 and0.3. Give both the same real atomic scattering amplitude $f$, and omit all other atoms and displacement attenuation. The coherent average contribution to a reflection $(h00)$ isFor an even integer $h$, this equals $f$; for an odd integer $h$, it equals $0.4f$. The corresponding squared amplitudes are $f^2$ and $0.16f^2$. Treating both alternatives as fully occupied would predict a different structure factor. This calculation concerns the periodic average contribution, not a complete description of diffuse scattering from local fluctuations.

A split-site disorder model is useful when an electron density map supports distinct alternative locations instead of one broad peak. An anisotropic atomic displacement parameter can describe directional broadening, but does not by itself encode two mutually exclusive chemical arrangements.

Occupancy constraints depend on the physical model: alternatives for a fully present atom sum to1, whereas a model admitting vacancies need not. An average crystal structure alone also cannot establish whether the alternatives exchange during the measurement, occur in different cells, or have particular local correlations.

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.