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The identical two-atom motif adds a factor $1+\exp [i\pi (h+k+l)/2]$ to the FCC lattice structure factor. Among even-index FCC diffraction reflections, only those with $h+k+l$ divisible by4 survive. The allowed FCC reflection200 therefore vanishes for equal motif scattering amplitudes.

The useful feature is the selenium atom's wavelength-dependent anomalous X-ray scattering. Close to its absorption edge the atomic scattering factor is complex:Replacing a sulfur-containing side chain by its selenium analogue introduces relatively strong, spatially localized anomalous scatterers. This is the diffraction-physics reason for the label; no biological preparation procedure is needed for the calculation.

The Se substructure can be located from intensity differences or Patterson maps. Measurements at several wavelengths change its known complex contribution while leaving most other atoms' scattering nearly unchanged. Interference between that contribution and the unknown remainder constrains the otherwise missing structure factor phases. Both wavelength-dependent dispersive differences and differences between Friedel pairs contribute to multiwavelength anomalous diffraction. Several wavelength constraints resolve phase ambiguities that remain with one intensity difference. The resulting phased Fourier synthesis yields the electron-density map.

For reflection $222$, the positional phase factor is $e^{2\pi i(2x+2y+2z)}=1$ for each of the four sites just found. Treat the supplied anomalous increment as the imaginary term $i{f}^{\prime \prime }$, with $f^{\prime \prime }=4.8$ electrons per Se atom. ThenIts magnitude is $19.2$ electrons and its phase is $+90^{\circ }$ relative to the real axis in this structure-factor convention. Translating the origin rotates both this reference contribution and the ordinary structure factor consistently; the physically relevant phase difference is invariant.

Write the ordinary structure factor as $F_0=|F_0|e^{i\varphi }$ and use $F=F_0+19.2i$. The intensities implyConsequently $\sin \varphi =0.43893$, and the two possible ordinary phases areThe twofold ambiguity is expected from a single anomalous-intensity constraint. The calculation assumes that the specified imaginary increment is the only change requiring inclusion; an additional unspecified real dispersive change could not be reconstructed from these data alone.

Without significant anomalous scattering, the electron density is effectively real and Friedel's law gives $F_0(-\mathbf{h})=F_0(\mathbf{h}{)}^{\ast }$, hence $I(\overline{222})=13130$.

With anomalous scattering the atomic imaginary term retains its sign when the reflection index reverses. Here $\Delta F(-222)=19.2i$, soThus the second expected intensity is approximately $1.157\times 10^4{\mathrm{electrons}}^2$. Conjugating the complete anomalous structure factor would incorrectly impose Friedel equality on the second experiment.

With an origin at an inversion center, the structure factors are real, so their phases are either $0$ or $\pi$. Defining $S(h)=\mathrm{sgn}F(h)$ givesCentrosymmetry converts the phase problem into a sign problem; it does not imply that every structure factor is positive.

X-ray diffraction arises predominantly from electromagnetic scattering by the electronic charge distribution. For elastic scattering, contributions from different unit cells interfere constructively at a reciprocal lattice vector, while the unit-cell structure factor weights the different atoms and their positions. The atomic X-ray scattering factor generally decreases with increasing momentum transfer because an extended electron cloud scatters out of phase with itself. Heavy atoms often scatter more strongly, making a light atom difficult to locate beside a heavy one; near an absorption edge, anomalous dispersion changes the contrast and can provide additional chemical sensitivity.

In neutron diffraction, the strong interaction with atomic nuclei supplies a short-range nuclear scattering amplitude. Its coherent scattering length depends on isotope and does not vary monotonically with atomic number; isotope substitution can therefore give useful contrast, and some light elements can be prominent. A neutron's magnetic moment also couples to magnetic moments in a specimen, so magnetic diffraction measures ordered spin structures. Absorption and incoherent scattering depend strongly on isotope as well. The nuclear scattering length is approximately independent of momentum transfer in the usual thermal-neutron regime, whereas magnetic scattering has a form factor reflecting the electronic magnetization distribution.

Electron diffraction probes the electrostatic potential of both nuclei and electrons. The electron's charge makes scattering strong, allowing thin specimens or surfaces to be studied with intense signals. Strong interaction also makes multiple, dynamical scattering more important, so electron diffraction intensities often cannot be interpreted as a simple single-scattering structure factor squared. Transmission geometry needs electron-transparent material; reflected low-energy electrons provide a surface-sensitive geometry. All three methods use phase-sensitive interference to reveal periodic structure, but their scattering interactions, penetration, sample requirements and contrast differ.

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.

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.