cp's OEIS Frontend

We can subdivide the 230 crystallographically permissible 3D space groups into 16 subsets based on the number of general positions (i.e., coordinate triplets whose values describe points occupied by symmetry-equivalent atoms in 3D space) specified by the symmetry operators in each subset. These numbers range from 1 (corresponding to exclusively one primitive triclinic space group, P1) to 192 (corresponding to the four face-centered cubic space groups Fm-3m, Fm-3c, Fd-3m, and Fd-3c). Multiplicities 1 and 9 (corresponding to exclusively one rhombohedral space group, R3h) represent the smallest subsets, whereas the largest subset is formed by the 63 space groups with multiplicity 8.

Reconstruction of periodic density functions via Fourier synthesis is a cornerstone of crystallography. In a typical experiment, the angstrom-scale periodicity of a crystal lattice serves as a diffraction grating for a focused beam of incident radiation (e.g., X-rays, neutrons, or electrons whose wavelength is smaller than the unit-cell dimensions), leading to a complex interference pattern. Diffracted rays undergoing constructive interference coalesce into spots known as Bragg reflections, whose integrated intensities we can measure on a pixelated detector. Mutatis mutandis, these intensities correspond to Fourier amplitudes. However, since the frequency of the incident quanta is exceptionally high (on the order of 10^17 Hz), it is currently impossible to experimentally record the associated phase information, which is irreversibly lost. This is referred to as the "phase problem". We must instead deduce the missing phases through avenues such as direct methods before we can proceed to Fourier synthesis.

Depending on the specific symmetry operators in the space group of the 3D crystal under interrogation, certain pairs of Fourier amplitudes will have exactly one of two mathematically permissible phases. We typically refer to these as centric or phase-restricted reflections. For instance, if the space group in question contains inversion symmetry (e.g., one of the 92 centrosymmetric space groups), every single Fourier amplitude is trivially restricted to a phase of either 0 or 180 degrees. In non-centrosymmetric space groups, specific subregions of reciprocal space can show analogous restrictions. These powerful constraints can facilitate the process of phase retrieval.

This sequence contains every possible pair of restricted phase values permissible in 3D space groups obeying the crystallographic restriction theorem. Each number here is (i) an integer multiple of 15 degrees, (ii) related to its partner by an offset of 180 degrees, and (iii) separated from the next number in the sequence by an alternating pattern of 15, 15, 30, 30: (0, 180), (30, 210), (45, 225), (60, 240), (90, 270), (120, 300), (135, 315), (150, 330).