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For coherent elastic diffraction, the intensity associated with a given scattering amplitude is proportional to its squared modulus. Integrated powder diffraction intensities additionally include the powder diffraction multiplicity, Lorentz-polarization factor, Debye–Waller factor, absorption and instrument corrections. A peak's height is not its integrated intensity unless its profile width and shape are also specified.

The high-pressure atomic basis of a crystal has the caesium chloride structure. Its chemically labelled Bravais lattice is simple cubic lattice: exchanging Na and Cl is not a translation symmetry. Its structure factor isBecause ionic bonding gives approximately Na${}^{+}$ and Cl${}^{-}$, their bound-electron counts are $10$ and $18$. In the stipulated electron-count approximation, odd-index-sum reflections have $|F|=8$, while even-index-sum reflections have $|F|=28$. In particular, the first ring $\{100\}$ is weak but present; it must not be discarded using a body-centering extinction rule.

The next ring is $\{110\}$, and Bragg's law givesThe powder diffraction multiplicities are six and twelve. Counting all symmetry-equivalent reflections, while neglecting other angular factors, therefore givesThe corresponding single-directed-reflection ratio would be $12.25$. Neutral-atom counts would give the wrong contrast because the material is ionic.

For measured integrated photon counts, the geometry must also be specified. More generally the ratio is $2|F_{110}/F_{100}{|}^2$ times the ratio of Lorentz-polarization factors, thermal-motion factors and detection efficiencies. For example, the common unpolarized symmetric-powder convention hasand gives $24.5LP(21.2506^{\circ })/LP(14.85^{\circ })\simeq 11.18$ if all other corrections are suppressed. Thus 24.5 is the multiplicity-and-structure-factor estimate, rather than a geometry-independent precision count ratio; actual angle-dependent ionic form factors are also not supplied.

Interpret the reported angles as total scattering angles $\psi =2\theta$. The photon energy givesCombining $2d\sin \theta =\lambda$ with the two index assignments yieldsThe slight difference is consistent with the rounded angular data, givingThe intensity estimate needs a stated convention. The geometric structure factors obey $|F_{111}{|}^2=32|f_{111}{|}^2$ and $|F_{220}{|}^2=64|f_{220}{|}^2$, so equal atomic factors give a single-directed-reflection ratio $1/2$. For a powder line, the powder diffraction multiplicities are eight for $\{111\}$ and twelve for $\{220\}$. Hence a multiplicity-and-structure-factor estimate isif other angle-dependent factors are neglected.

A measured integrated powder intensity also contains the Lorentz-polarization factor, atomic form factor, thermal-motion factor and instrument/sample corrections. For example, in the common unpolarized symmetric powder geometry,Including this factor but still taking equal atomic factors gives $I_{111}/I_{220}\simeq 0.9725$, rather than $1/3$. More generally,within that geometry, before other corrections. The actual form factors and measurement geometry are not supplied, so no unique precision experimental peak-height ratio follows. The values $1/2$, $1/3$ and $0.9725$ refer to distinct explicitly stated levels of approximation; a powder family must not be silently treated as one directed reflection.

The X-ray wavelength is $\lambda =hc/E=1.54980\times 10^{-10}\mathrm{m}$. Write the amplitude of one conventional unit cell as the face-centred cubic selection rule times the two-atom atomic basis amplitude:The first bracket is four when the indices have equal parity and zero otherwise. Among the allowed Bragg reflections, the zinc blende structure factor therefore givesHere the atomic scattering factors are taken as real. For the requested rough estimate set $f_{\mathrm{Ga}}\simeq 31$ and $f_{\mathrm{As}}\simeq 33$. The weak difference reflections remain allowed: unequal species do not have the extra exact extinction of the equal-species diamond limit.

The first six allowed values of $h^2+k^2+l^2$ are $3,4,8,11,12,16$. Bragg's law gives the measured scattering angle $2\theta =2{\sin }^{-1}(\lambda \sqrt{h^2+k^2+l^2}/2a)$. For powder diffraction, multiply the squared amplitude by the powder diffraction multiplicity $m_{hkl}$. The last two columns use $I_{111}=100$:The first intensity column is the geometry-independent multiplicity-and-structure estimate. The second additionally assumes an unpolarized beam and a common symmetric powder-diffractometer geometry, with Lorentz-polarization factorAn overall constant cancels in either normalization. A particular measured intensity set is not fixed by the atomic numbers alone: actual atomic scattering factors decrease with momentum transfer, anomalous scattering and absorption can matter, and the instrument geometry and polarization determine angular corrections. The two columns state the approximation rather than asserting identical peak heights for all instruments.^[ref]

Initially approximate every tungsten atom by the same atomic scattering factor $f$. Pairing the quarter-coordinate sites in the crystal basis structure factor givesFor cubic powder diffraction, increasing deflection corresponds to increasing $h^2+k^2+l^2$, since $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$. The first possibilities are

Hence the first two strong rings have Miller indicesThe powder diffraction multiplicity is six for $\{200\}$, from three choices of nonzero axis and two signs. For $\{210\}$ it is $3!2^2=24$, from six permutations and four sign choices. If the atomic scattering factors and angular corrections are taken as approximately equal for these nearby rings, their integrated diffraction intensity obeyThis is a multiplicity-based estimate. An exact measured ratio also depends on the momentum-transfer dependence of the atomic scattering factor, the Lorentz-polarization factor, the Debye–Waller factor, absorption and instrument geometry. In particular, the source does not provide the beta-phase lattice constant or instrument corrections needed to calculate those angular factors. Peak heights additionally depend on peak widths; the estimate refers to integrated ring intensities.

At each reflection, the argument of the atomic scattering factor isSubstitution into the given atomic scattering factor approximation givesThus the structure factors are $S_{110}=36.2620$ and $S_{111}=113.7340$. The powder diffraction multiplicity is $m_{110}=12$: choose which Miller index is zero in three ways and the signs of the other two in four ways. For $\{111\}$, the eight sign choices give $m_{111}=8$. With the specified instrumental and angular factors omitted, the integrated powder diffraction wave intensity is proportional to $m_{hkl}|S_{hkl}{|}^2$, soIt would be incorrect to use the same atomic scattering factors at the two different scattering vectors, or to omit the powder diffraction multiplicity.

The stoichiometry fixes the site probabilities to $1/4$ gold and $3/4$ copper. Every site in the average crystal structure therefore has the same atomic scattering factorThe average structure factor isAt $(110)$ the bracket vanishes, while at $(111)$ it is four. HenceThe $(111)$ powder diffraction multiplicity is unchanged, so its coherent Bragg diffraction wave intensity equals the ordered value under the stated equal-lattice constant approximation.

The zero refers to the coherent Bragg diffraction peak of the average crystal structure. A random alloy can still give occupational diffuse scattering at the same scattering vector: the average of the squared amplitude includes fluctuations as well as the square of the average amplitude. Such background is not a surviving $(110)$ Bragg diffraction peak.

Use the same atomic basis phases as in the ordered structure factor, replacing each atomic scattering factor by its site average. For the reflection with equal integer parity,Conservation of the total composition removes the alloy sublattice order parameter from this amplitude. For the reflection with mixed integer parity,The coherent powder diffraction wave intensities consequently satisfyHere the lattice constant, atomic scattering factors, powder diffraction multiplicities, and omitted experimental factors are held fixed. The ratio of a superlattice reflection to a fundamental reflection can therefore measure the magnitude of the alloy sublattice order parameter.

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 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.

Diffraction from a specimen containing many crystallite orientations. Equivalent reciprocal-lattice spacings produce powder rings or corresponding peaks in an angular scan. Powder diffraction multiplicity counts the equivalent directed normals contributing to a reflection family.

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.