Space Group Reflection Conditions

https://xraytools.com/spacegroups

What a space group extinguishes: the integral, zonal and serial reflection conditions of any of the 230 groups, in any of the settings the International Tables list. Ask about one reflection and the page names the symmetry operation that removes it.

Before this: A space group is the full symmetry of a crystal, written as a Hermann-Mauguin symbol, and one group can be written several ways depending on how the axes are chosen. That choice is the setting, and it changes which reflections are absent.
You supply: A space group number or symbol. Non-standard settings are included, so P2₁/n, Pbnm and I2/a answer for themselves rather than being redirected to a sibling with different conditions. A reflection is optional.
Reading it: Absences narrow the space group and rarely determine it: the 230 groups produce only 80 distinct patterns of absence, and 37 of them extinguish nothing at all. When your conditions fit more than one group, the page names the others.

Worked examples: P2₁/c – the commonest space group there is · P2₁/n – the same group, a different setting · P4₃2₁2 – screw axes and nothing else · R3c – rhombohedral, in hexagonal axes · Fd3m – diamond, and (2 0 0) missing · C2/c – a glide inside a centred net · Pnma – every other reflection, in one zone · Imma – body centring and an a-glide · Ibca – body centring and three glides

Earlier on the path: Reciprocal Cell Calculator Next on the path: Space Group from Absences On From planes to a powder pattern, step 5 of 7

Notation here: (hkl) · {hkl}, ⟨uvw⟩ — what each one means here

Terms here: centring · general position · Wyckoff position · point group · Laue class · multiplicity · setting · site symmetry · zone

What each input changes

Space group: Which group’s conditions are listed. Two settings of one group give different conditions on the same crystal, which is why the setting is part of the answer and not a formatting choice.
Zone: Which class of reflection is asked about. A condition that holds for a zone says nothing about the general reflections.

Input

Conditions belong to a setting, not to a space group number. Pnma, Pbnm and Pmcn are one space group with its axes labelled three ways, and the three tables differ: the glide that empties hk0 in one of them empties 0kl in another — a glide, because it is the fractional translation that makes a whole zone cancel, and a pure mirror carries none and empties nothing. Every setting in the International Tables is here with its own operations, so P2₁/n and P2₁/a answer for themselves.

Reflection conditions

Enter a space group above. There are 230 of them, in 531 settings, and every setting has its own table — the 530 the International Tables list, plus C1, which they do not: a triclinic lattice described on centred axes is a real thing to have and has no entry of its own.

Reading down the table: integral conditions come from the lattice centring and apply to every hkl; zonal conditions come from glide planes and empty one plane of reciprocal space; serial conditions come from screw axes and empty one row. The same three kinds are named under the stick pattern on the HKL page, which applies the integral ones only.

Where this comes from

Space-group notation with an explicit origin
S. R. Hall, Acta Cryst. A 1981, 37, 517–525 · doi:10.1107/S0567739481001228
The Hall symbol this page prints, and the reason it is there at all: it is the only name here that separates two settings differing in nothing but where the origin was put.