Space Group Reflection Conditions
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.
- 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
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 mirror that empties hk0 in one of them empties 0kl in another. Every setting in the International Tables is here with its own operations, so P21/n and P21/a answer for themselves.
P21/c
| Number | 14 |
|---|---|
| Full symbol | P 1 21/c 1 |
| Schoenflies | C2h5 |
| Crystal system | monoclinic |
| Crystal class | 2/m, order 4 |
| Laue class | 2/m, order 4 |
| Lattice | P (primitive) |
| General position | 4 equivalent points |
| Setting | standard |
Other settings with this name
The name you typed fits more than one setting, and they do not have the same conditions.
Reflection conditions
| Class | Condition | From |
|---|---|---|
| h0l | l = 2n | zonal |
| 0k0 | k = 2n | serial |
| 00l | l = 2n | serial |
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.
A class in this table stands for its own zone and every zone the point group carries it onto. In P63/mmc the row for hhl also governs (−2, 1, l) and (1, −2, l); in a cubic group the row for hhl also governs hkh and hkk. Seventeen of the settings here extinguish reflections that no literal reading of a class label covers, which is why the check below asks the symmetry operations rather than this table.
Is a reflection there?
Put three Miller indices into the form and this says whether P21/c allows that reflection, and which symmetry operation removes it if it does not.
What the absences do not tell you
P21/c is the only space group with this pattern of absences. If your data show exactly these conditions and no others, the systematic absences alone identify the group — which is unusual, and is a large part of why the ones that manage it are so common in the literature.
Moving the origin multiplies every structure factor by a phase and changes no intensity, so the two origin choices the tables give for 24 space groups have identical conditions. Only one is listed here.