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What each group extinguishes

What each group extinguishes

What each group extinguishes. The extinction conditions of 8 space groups, each derived by summing the structure factor over that group's own operations and reading the surviving rule off the result: P1 — nothing; P2₁ — 0k0: k even; Pc — h0l: l even; P2₁/c — h0l: l even, 0k0: k even; C2 — hkl: h+k even; P2₁2₁2₁ — h00: h even, 0k0: k even, 00l: l even; Pna2₁ — h0l: h even, 0kl: k+l even; Fddd — hkl: h, k, l all even or all odd, 0kl: k+l divisible by four, h0l: h+l divisible by four, hk0: h+k divisible by four. 1 of the 8 extinguish nothing at all, and diffraction alone cannot distinguish those from each other.

The extinction conditions of 8 space groups, each derived by summing the structure factor over that group's own operations and reading the surviving rule off the result: P1 — nothing; P2₁ — 0k0: k even; Pc — h0l: l even; P2₁/c — h0l: l even, 0k0: k even; C2 — hkl: h+k even; P2₁2₁2₁ — h00: h even, 0k0: k even, 00l: l even; Pna2₁ — h0l: h even, 0kl: k+l even; Fddd — hkl: h, k, l all even or all odd, 0kl: k+l divisible by four, h0l: h+l divisible by four, hk0: h+k divisible by four. 1 of the 8 extinguish nothing at all, and diffraction alone cannot distinguish those from each other.

4 essays call absence-table. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

The h0l layer of P2₁/c. The h0l reflections of P2₁/c out to 4 in each index, with each spot decided by summing the structure factor over the group's operations: 44 survive and 36 vanish identically, whatever the atoms are. The pattern of holes is the condition h0l: l even, read back off the spots rather than imposed on them.

How it is known

The reflections that are not there

A screw axis and a glide plane leave no mark on the intensity of any reflection. What they do is delete some, exactly, for every possible arrangement of atoms — and the pattern of deletions is computed here from the sum a crystallographer writes down, rather than read from a table.

$What each group extinguishes. The extinction conditions of 9 space groups, each derived by summing the structure factor over that group's own operations and reading the surviving rule off the result: P1 — nothing; P1̅ — nothing; P2 — nothing; Pm — nothing; P2/m — nothing; P222 — nothing; Pmm2 — nothing; P4 — nothing; P23 — nothing. 9 of the 9 extinguish nothing at all, and diffraction alone cannot distinguish those from each other.$ How it is known

Where the experiment runs out

Absences narrow the space group down and often not to one. Two groups can extinguish exactly the same reflections and scatter with exactly the same symmetry, and telling them apart needs something the diffraction pattern does not contain.

$-2 0 -3 is forbidden and is reached 64 ways. A layer of the reciprocal lattice of Cc: pale spots are reflections the group extinguishes, solid ones are allowed. The path shows a detour — the beam diffracts once at -3 -3 -3 and again at 1 3 0, and the two together send it in exactly the direction a single reflection at -2 0 -3 would. That reflection is forbidden, so intensity arrives where the symmetry said none could. There are 64 such routes to this one spot inside this window alone.$ How it is known

The absence that fills itself in

A systematic absence is the strongest evidence this subject has: a whole zone of reflections cancelling exactly, for reasons of symmetry rather than of arithmetic accident. The exactness belongs to a model — that the beam scatters once. A beam that has already been diffracted can be diffracted again, and the two events together land where the group said nothing could.

342 unlabelled spots, cell volume 52. A bag of 342 reflection positions with no indices on them, collected out to a bound of 3 on each index. Their pairwise differences generate the reciprocal lattice; a basis of that is taken by integer elimination and then reduced, and the reduced basis is printed. Its determinant is 52, which is the volume of the cell the reflections were computed from — so the cell has been recovered from positions alone, with no intensity used anywhere.

Symmetry at work

A cell from a bag of spots

A single-crystal experiment returns a list of directions with no labels on them. Recovering the cell is recovering the lattice those directions generate, and the whole of it is take differences, reduce, read the answer. What no quantity of data settles is whether the lattice found is the true one or a sublattice of it.

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