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A vanishing crystal basis structure factor can depend on approximate equality of atomic scattering factors, rather than on an exact space group condition. For two inequivalent site classes, an amplitude $F=C(f_A-f_B)$ vanishes when $f_A=f_B$ but becomes weakly observable if bonding electron densities or thermal displacements differ. Chemical identity alone does not imply identical local atomic scattering factors. The diffraction intensity is quadratic in the small difference.

An angular factor in measured integrated diffraction intensity combining scattering geometry and the incident beam's polarization. For an unpolarized beam in a common symmetric powder-diffractometer geometry, it is proportional to $(1+{\cos }^{2}2\theta )/({\sin }^2\theta \cos \theta )$, with convention-dependent overall constants. Other geometries or incident polarizations require the appropriate factor rather than this particular expression.

The structure factor is the coherent scattering amplitude of the contents of a unit cell at a specified momentum transfer. Contributions from its atoms add with phases fixed by their positions. At a reciprocal lattice vector indexed by Miller indices, the crystal basis structure factor isChanging every phase sign conjugates the amplitude for real scattering factors and leaves the intensity unchanged. The positions here are fractions of the chosen cell edges. An atomic basis of a crystal should not be confused with a set of primitive translation vectors.

In kinematic elastic scattering the diffraction intensity of a Bragg reflection is proportional to $|S_{hkl}{|}^2$, after including geometrical, polarization, thermal-displacement and experimental factors. Powder integrated intensities also include multiplicity. The many-cell interference factor selects reciprocal-lattice vectors satisfying the diffraction geometry; the unit-cell factor determines their relative amplitudes and additional extinctions. Ring positions determine spacings through Bragg's law, while intensity patterns and systematic absences constrain lattice centering and atomic arrangements. Intensity loses the overall complex phase, so structure determination is not a direct measurement of every individual atomic phase.

X-rays scatter from electron density, giving an atomic factor that changes with momentum transfer and chemical charge distribution. Neutrons scatter coherently from nuclei with bound coherent scattering lengths, which need not track atomic number; magnetic neutron scattering is an additional channel in magnetic materials. Their different elemental/isotopic contrasts make the two probes complementary.

With the convention ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$, the reciprocal lattice of a face-centered cubic lattice is a body-centered cubic lattice. The reciprocal conventional cube is parallel to the direct conventional cube and has side $4\pi /a$. In direct-cubic coordinates its points areThe even triples supply the corners of that reciprocal cube and the odd triples its body centres. The atomic basis can change diffraction intensities through its structure factor; it does not change the reciprocal lattice determined by the translation lattice.

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.