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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.

A researcher who coauthored a quantitative powder diffraction method using a surface diffractometer and an area detector. Extracting meaningful intensities requires corrections for polarization and measurement geometry as well as the crystal's structure factor. The work explains how an instrument designed for surfaces can also measure powders and textured samples, connecting detector images with reliable structural analysis.^[ref]

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]

For the conventional body-centred cubic lattice, the crystal basis structure factor of identical atoms isIt vanishes for odd $h+k+l$ and equals $2f_{\mathrm{W}}$ for even $h+k+l$. Thus the first powder diffraction ring is the $\{110\}$ family, with interplanar spacing $d_{110}=a/\sqrt{2}$.

The measured deflection is the angle between incident and scattered rays, $2\theta =40.26^{\circ }$, so the Bragg angle is $\theta =20.13^{\circ }$. Using first-order Bragg diffraction, $2d\sin \theta =\lambda$, givesThe mass density follows from two atoms per conventional unit cell, with molar mass $M=0.1838\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$ and Avogadro constant $N_A$:Using $40.26^{\circ }$ as the Bragg angle would confuse $\theta$ with $2\theta$ and give an incorrect lattice constant and mass density.

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.

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.

For the cubic crystal system, the Miller indices satisfyThe inverse squared interplanar spacings have ratios approximately $1:2.004:3.012:4.010$. With the lattice constant close to $0.4\mathrm{nm}$, the appropriate squared-index sums are $1,2,3,4$. Thus the four powder diffraction rings areThe braces collect all symmetry-related lattice planes with the same interplanar spacing. In particular, $(100)$ is forbidden for both a body-centered cubic lattice and a face-centered cubic lattice, but occurs for a primitive cubic lattice. These observations identify the Bravais lattice as primitive cubic lattice, with a multi-atom atomic basis, consistent with a perovskite structure.

Each ring independently gives $a_j=d_j\sqrt{j}$:Their average is $0.42413\mathrm{nm}$. Allowing for the rounded measured angles, a suitable final value isA larger centred unit cell could relabel reflections, but would conflict with the stated approximate lattice constant.

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.

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.

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.