Structure factors and extinction rules

For atomic positions R plus tau, the kinematic elastic amplitude separates into a translation sum and a basis factor. Define $S(\mathbf G)=\sum_s f_s(\mathbf G)e^{i\mathbf G\cdot\boldsymbol\tau_s}$, choosing the plus phase convention for the basis amplitude. Its complex conjugate convention gives the same intensity for the simple real form factors used here. The atomic factor f includes the probe's scattering response, while the phase includes the atom's position. Sum those complex amplitudes first; summing their individual intensities would erase interference.

The intensity contains the modulus squared of S multiplied by the lattice peak envelope and experimental factors. X-ray atomic factors generally depend on wavevector magnitude and electronic structure, while neutron scattering lengths have different species and isotope dependence. Thus a motif's geometric phases alone do not set every relative intensity. We will hold the atomic factors constant in the hand calculations to isolate interference. Exact extinction means an amplitude is forced to vanish under the stated symmetry and contrast assumptions, not merely that the measured peak is too weak to detect.

Represent an identical-atom bcc arrangement using a conventional simple-cubic grid and offsets zero and half the body diagonal. For $\mathbf G=(2\pi/a_c)(h,k,l)$, their phase difference is pi times the sum of the three indices. Even sums add and odd sums cancel. The conventional grid is deliberately nonprimitive for bcc: the extinction rule removes candidate simple-cubic reciprocal points that are not reciprocal vectors of the true bcc Bravais lattice. It does not mean the true primitive reciprocal lattice has arbitrary missing points.

The conventional fcc cell has offsets zero and the three face-centre representatives. Its structure factor is f times one plus three parity factors. All indices even or all odd give constructive addition; mixed parity cancels. A further motif can impose additional absences beyond this centring rule. For example, a diamond-type two-atom motif on an fcc Bravais lattice adds another phase factor. One should apply the Bravais selection and motif amplitude consistently rather than mixing a primitive reciprocal indexing scheme with a conventional cubic parity rule.

Now place different atoms at the corner and body-centre offsets. The amplitude becomes $f_A+(-1)^{h+k+l}f_B$. Odd sums give a difference instead of zero unless the two scattering factors happen to match. This CsCl-type decoration has a simple-cubic Bravais lattice with a two-atom basis, rather than an identical-atom bcc Bravais lattice. The apparent extinction has disappeared because the body-centred translation no longer preserves chemical identity. A probe with accidentally equal contrasts can still hide that distinction.

Observed violations of an ideal absence can also involve defects, partial occupation, disorder, multiple scattering, or experimental contamination. Conversely, an allowed peak can be weak because form factors or thermal motion suppress it. Diffraction inference therefore combines systematic peak locations, relative intensities, probe contrast, and assumptions about the scattering regime. A single missing spot rarely identifies a unique structure by itself. The quantitative structure-factor calculation is valuable because it states exactly which equality of phases and amplitudes is being tested and what changes would remove the cancellation.