Space Group from Absences
https://xraytools.com/absences
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.
What is missing?
Set the classes you have determined and this names every space group consistent with them. There are 230 groups and the eleven classes above separate them into 77 distinct answers, so the usual result is a short list rather than one group.
The space-group page puts the number of distinct absence patterns at 80, which is a larger number for a real reason: that one counts what the symmetry operations extinguish, and this one counts what the eleven printed conditions can tell apart. A condition table is the weaker reading — a handful of groups print the same table and still extinguish different reflections, which is what the single-reflection panel on this page is for.
One missing reflection is not a condition. A structure factor can fall to nothing because of where the atoms happen to sit, with no symmetry involved — which is why a condition is read off a whole class going quiet, and why this form asks for classes rather than for a list of reflections you could not find.
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.