Space Group from Absences
https://xraytools.com/absences?h0l=l%3D2n&0k0=k%3D2n&00l=l%3D2n
The determination as it is actually done: you have a data set, some classes of reflection are systematically missing, and the question is which of the 230 space groups you are allowed to be in. Answer the classes you have determined and the page names every group that fits — and the measurement that would shorten the list.
- Before this
- A systematic absence is a reflection the symmetry forces to zero, as against one that happens to be weak. The reflection conditions are the forward direction of what this page runs backwards.
- You supply
- What you observe, class by class. Each menu offers only conditions some space group actually produces, so an answer can always be satisfied by something. Anything you have not determined is left alone and constrains nothing.
- Reading it
- A class left at not determined constrains nothing; answering it no condition asserts you looked and found nothing missing, which is what actually narrows the list. The strong answer is the one that can be wrong — and a wrong one removes the right group without saying so.
Worked examples: P21/c – the commonest determination there is · P21/c – the same data, empty classes too · Pbca – one pattern, two systems · P4cc – four groups, one reflection apart · Fd3m – diamond
Earlier on the path: Space Group Reflection Conditions Next on the path: Structure Factor Calculator On From planes to a powder pattern, step 6 of 7
Notation here: (hkl) — what each one means here
Terms here: centring · Laue class · setting · zone
See also: HKL Calculator · Powder Indexing · Space Group Reflection Conditions · Wilson Plot and E Statistics
What each input changes
- System
- How many groups are in the running before any absence is considered. Narrowing it does not make the answer more certain — it makes the question smaller.
Teaching with this page
- Objective
- After this page a learner can infer candidate space groups from which classes of reflection are missing, and say what the evidence cannot settle.
- Start from
- this worked example
- Ask first
- Every reflection with h + k odd is missing. Does that name a glide plane?
- Watch for
- “Yes — a whole class is gone, so an element removed it”
- Then
- Structure Factor Calculator
Check yourself: Every reflection with h + k odd is missing. Does that name a glide plane?
No — a condition on all reflections is a centring Yes — a whole class is gone, so an element removed it
Which reflections the condition applies to is what names the element. A condition on the general reflections comes from centring — a lattice point somebody could have chosen not to add. A condition on a zone (h0l) is a glide; one on an axis (00l) is a screw. Same shape of rule, three different objects.
Input
Answer only the classes you have actually determined. A class left at not determined constrains nothing, and that is the honest state for a zone you have not measured out far enough to be sure about — every answer you can give narrows the list, and a wrong one removes the right group without saying so.
11 space groups
These conditions are produced by 11 of the 230 space groups.
| No. | Symbol | System | Extinction symbol |
|---|---|---|---|
| 14 | P21/c | monoclinic | P 1 21/c 1 |
| 30 | Pnc2 | orthorhombic | P n c - |
| 38 | Amm2 | orthorhombic | A - - - |
| 39 | Aem2 | orthorhombic | A b - - |
| 56 | Pccn | orthorhombic | P c c n |
| 57 | Pbcm | orthorhombic | P b c - |
| 60 | Pbcn | orthorhombic | P b c n |
| 61 | Pbca | orthorhombic | P b c a |
| 130 | P4/ncc | tetragonal | P n c c |
| 138 | P42/ncm | tetragonal | P n c - |
| 205 | Pa3 | cubic | P b - - |
Systematic absences see the lattice centring, the glide planes and the screw axes and nothing else. They cannot see whether the structure has a centre of inversion, so two groups built from the same centring, glides and screws extinguish exactly the same reflections whether or not one of them has that centre — C2/c and Cc, Pnma and Pn21a, P1 and P1. Dropping the centre is not by itself enough to make two groups indistinguishable: P21/c is fixed by its own two conditions, and its subgroups P21 and Pc keep one each. Separating the genuinely ambiguous ones is a job for intensity statistics, for the structure solving, or for chemistry — never for more careful indexing.
The extinction symbol is the name International Tables gives to a set of reflection conditions: the lattice letter, then what each symmetry direction’s conditions require of it — a glide letter, a screw axis, a dash where they require nothing, or a 1 where the group has no symmetry along that axis. It is derived from the absences and not from the symbol, which is why Pnma appears here as Pn-a: its two screw axes and its mirror are real symmetry that leaves no trace a diffraction pattern can see. The candidates above carry 10 different symbols, which means the conditions are not settled: two groups with different symbols differ in a class you have not answered, and answering it would drop one of them. Note the symbol is read within a crystal system — the Tables index by the Laue class and the symbol together, and this answer spans 4 systems, so the same string appears in more than one place. A screw axis is printed only where the conditions along it are stronger than the glide planes already force; the Tables print a redundant one in a few tetragonal and cubic entries, which names the same groups.
The same conditions on other axes — 37 non-standard settings
These are the same space groups with their axes labelled differently, so whether one of them is your answer depends on how you indexed rather than on the crystal. A symbol can appear twice here: two settings of one group can share a Hermann–Mauguin symbol and differ in the cell choice, which is why the number and the standard symbol are given beside it.
| No. | Space group | In this setting | System |
|---|---|---|---|
| 5 | C2 | A2 | monoclinic |
| 5 | C2 | A2 | monoclinic |
| 8 | Cm | Am | monoclinic |
| 8 | Cm | Am | monoclinic |
| 9 | Cc | Aa | monoclinic |
| 9 | Cc | An | monoclinic |
| 12 | C2/m | A2/m | monoclinic |
| 12 | C2/m | A2/m | monoclinic |
| 15 | C2/c | A2/a | monoclinic |
| 15 | C2/c | A2/n | monoclinic |
| 20 | C2221 | A2122 | orthorhombic |
| 21 | C222 | A222 | orthorhombic |
| 29 | Pca21 | Pbc21 | orthorhombic |
| 32 | Pba2 | P2cb | orthorhombic |
| 33 | Pna21 | P21cn | orthorhombic |
| 35 | Cmm2 | A2mm | orthorhombic |
| 36 | Cmc21 | A21ma | orthorhombic |
| 38 | Amm2 | Am2m | orthorhombic |
| 39 | Aem2 | Ae2m | orthorhombic |
| 40 | Ama2 | Am2a | orthorhombic |
| 41 | Aea2 | Ae2a | orthorhombic |
| 50 | Pban | Pncb | orthorhombic |
| 50 | Pban | Pncb | orthorhombic |
| 52 | Pnna | Pncn | orthorhombic |
| 53 | Pmna | Pncm | orthorhombic |
| 54 | Pcca | Pccb | orthorhombic |
| 54 | Pcca | Pbcb | orthorhombic |
| 55 | Pbam | Pmcb | orthorhombic |
| 60 | Pbcn | Pnca | orthorhombic |
| 62 | Pnma | Pmcn | orthorhombic |
| 63 | Cmcm | Amma | orthorhombic |
| 64 | Cmce | Aema | orthorhombic |
| 65 | Cmmm | Ammm | orthorhombic |
| 67 | Cmme | Aemm | orthorhombic |
| 67 | Cmme | Aemm | orthorhombic |
| 130 | P4/ncc | P4/ncc | tetragonal |
| 138 | P42/ncm | P42/ncm | tetragonal |
Measure this next
The 0kl class separates them best. Whichever of these 5 outcomes your data show, the list shortens to at most 4.
| If 0kl shows | Groups left |
|---|---|
| no condition | 1 |
| k + l = 2n | 2 |
| l = 2n; k = 2n | 1 |
| l = 2n | 3 |
| k = 2n | 4 |
Where this comes from
- International Tables for Crystallography
International Union of Crystallography · on the reading list under “The tables this site computes from”
The reflection conditions this page matches against are Volume A's, and the eleven classes it asks about are the ones the Tables print. What is computed here rather than copied is the conditions themselves — they are derived from each group's symmetry operations, which is how the page can answer for classes a group's own printed table leaves out.