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 · C2/c – a glide inside a centred net · Pnma – every other reflection, in one zone

Input

Space group

Number or symbol

Any of 14, P2_1/c, P21/c or P 1 21/c 1. Subscripts may be written with an underscore or left out, and an inversion axis with a minus sign or without — Fd-3m and Fd3m both work.

Reflection (optional)

Position (optional)

Leave these empty for a general position. Put a point on a symmetry element — 0, 0, 0, or 1/4, 1/4, 1/4 — and the orbit collapses onto it.

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.

Pm3

Number	200
Full symbol	P 2/m 3
Schoenflies	T_h¹
Hall symbol	`-P 2 2 3`
Crystal system	cubic
Crystal class	m3, order 24 — projected
Laue class	m3, order 24
Patterson symmetry	Pm3 — the symmetry of the Patterson map, which is computed from the intensities and therefore needs no phases. Friedel's law makes it centrosymmetric whatever the crystal is, and a screw or a glide loses its translation, because the map is a function of interatomic vectors rather than of positions.
Chirality	some operation turns an object inside out — a mirror, an inversion or an inversion axis — so a structure built from one enantiomer cannot have this symmetry: the group would demand the other hand alongside it. Seen in the projection.
Lattice	P (primitive)
General position	24 equivalent points — the coordinates, drawn
Setting	standard

Twinning by merohedry

The lattice has 48 symmetry operations and the crystal has 24, so the lattice can be mapped onto itself in 2 ways for each way the crystal can. A twinned crystal can therefore take 2 orientations on one lattice, related by the 1 operation below.

The 1 twin law of this group
Operation	Matrix	Equivalent operations
2-fold rotation about [01-1]	-x,-z,-y	24

These are the twins by merohedry: every reflection of one orientation falls exactly on a reflection of the other, which is what makes them hard to spot and easy to refine wrongly. Two other kinds are not derived here and cannot be, from a space group alone. Reticular merohedry puts the two orientations on a common sublattice, so only some reflections overlap; finding those needs a search over superlattices and a decision about how much overlap counts. Pseudo-merohedry happens when a measured cell is accidentally close to a higher symmetry — that is a fact about one crystal and a tolerance, not about a space group.

Symmetry elements

Screw axes as

Look along

Point at an element to read what it is.

12 2-fold axes
7 3-fold axes
8 3 inversion axes
9 3₁ screw axes
9 3₂ screw axes
8 inversion centres
6 mirror planes

Point at any element for its type, its direction and where it sits. Axes are drawn thicker the higher their order; a screw axis or a glide plane is dashed, because its operation carries a translation the pure one does not. Switch a class off to see through a dense group.

The cell drawn here is a representative one: its shape obeys every constraint this group's symmetry imposes and nothing else, because a space group fixes the equalities among the cell constants and never their values. Read the right angles, the equal axes and the 120° where they appear; do not read the axial ratios.

The crystal class, projected

Every space group leaves a crystal class behind when its translations are taken away, and this is m3 — the 24 operations of Pm3 seen as directions rather than as places. On the left, one general direction and everywhere the class sends it; on the right, the same class written as its axes and mirrors.

Point at any pole or symbol to have it name itself. Click it to keep it on screen; click it again, click empty space, or press Escape to let go.

Both discs are the same sphere seen from the same place: every direction is joined to the far pole and marked where that line crosses the equator, so the centre of the disc is straight up c and the rim is ninety degrees away from it. The projection is conformal: two curves cross on the disc at the angle they cross at on the sphere. That is not the same as being able to read the angle between two poles off the disc — separation is not to scale, and measuring it needs a Wulff net or the direction vectors themselves. a points down the page and b across it.

A filled dot is a direction in the upper half of the sphere and an open circle one in the lower; where the group has a mirror across the page the two coincide, and the mark is a dot inside a circle. A comma means the operation that produced it turned the object inside out.

24 poles, drawn as 12 marks. m3 has a mirror across the page, so each pole in the upper half of the sphere lands exactly on top of one in the lower half. Every one of those pairs is drawn as the Tables draw it — a dot inside a circle — and pointing at it names both.

12 of the 24 poles carry a comma. Those images are of the opposite hand, produced by an operation that turns the object inside out — a mirror, an inversion, or an inversion axis. A structure built from a single enantiomer cannot have this symmetry, because the group would demand the other one alongside it.

The International Tables turn the page for monoclinic groups and draw the unique axis in the plane of the paper. This projects along c whatever the group is, so that all three diagrams on this page share one orientation — in a group with the unique axis b, the two-fold therefore lies across the disc instead of standing at its centre.

General position

The 24 symmetry operations of Pm3, written as coordinate triplets. Each one says where the group sends a point at x, y, z.

x,y,z
-x,-y,-z
-x,-y,z
x,y,-z
x,-y,-z
-x,y,z
-x,y,-z
x,-y,z
z,x,y
-z,-x,-y
z,-x,-y
-z,x,y
-z,x,-y
z,-x,y
-z,-x,y
z,x,-y
y,z,x
-y,-z,-x
-y,z,-x
y,-z,x
-y,-z,x
y,z,-x
y,-z,-x
-y,z,x

Wyckoff positions

A point of the cell is held in place by whichever operations send it to itself, and points held the same way form one Wyckoff position. Pm3 has 12, listed from the general position — the one no operation fixes — down to the most symmetric site it has.

Multiplicity	Site symmetry	Order	Coordinates	Draw it
24	1	1	`x, y, z`	the 24 points
12	m	2	`x, y, 0`	the 12 points
12	m	2	`x, y, 1/2`	the 12 points
8	3	3	`x, x, x`	the 8 points
6	mm2	4	`0, 0, z`	the 6 points
6	mm2	4	`0, y, 1/2`	the 6 points
6	mm2	4	`x, 0, 1/2`	the 6 points
6	mm2	4	`x, 1/2, 1/2`	the 6 points
3	mmm	8	`0, 0, 1/2`	the 3 points
3	mmm	8	`0, 1/2, 1/2`	the 3 points
1	m3	24	`0, 0, 0`	the 1 point
1	m3	24	`1/2, 1/2, 1/2`	the 1 point

Multiplicity times the order of the site symmetry is 24, the number of operations, in every row — a point with more symmetry holding it has fewer copies, and the two numbers are computed separately here so that they have to agree.

The International Tables give each position a letter and print its site symmetry oriented against the directions of the space-group symbol — 4a, and m.mm rather than mmm. Neither is here. A letter is an assignment the Tables make, not a consequence of the symmetry, and the dots are a notation with no way of checking it short of copying Tables pages; the symmetry itself, the multiplicity and the coordinates all follow from the operations and are derived.

One member of each set is printed. Every position is several subspaces of the cell that the group carries onto one another, so the Tables may name a different one — the same position either way. Follow a row to put that point in the form and draw its orbit.

Where those points are

drag to rotate · scroll to zoom · click a point to keep it

Look along

Point at a position to read its coordinates, or click one to keep it.

0.13, 0.21, 0.31 is a general position. No operation of Pm3 other than the identity leaves it where it is, so the group sends it to 24 distinct points — one per operation, which is what makes the multiplicity equal to the order of the group.

That is the position x, y, z, of multiplicity 24 and site symmetry 1.

That is the general position the drawing opens on. Type a point into the form to move it — the picture and the multiplicity follow.

A filled circle is the point as you gave it; a hollow one is a mirror image of it, produced by an operation of the second kind — an inversion, a mirror, a glide or a rotoinversion. The International Tables mark the same distinction with a comma inside the circle. It matters for anything chiral: those copies are not superimposable on the one you typed.

The 24 points

#	x	y	z	From
1	0.13	0.21	0.31	`x,y,z`
2	0.87	0.79	0.69	`-x,-y,-z`
3	0.87	0.79	0.31	`-x,-y,z`
4	0.13	0.21	0.69	`x,y,-z`
5	0.13	0.79	0.69	`x,-y,-z`
6	0.87	0.21	0.31	`-x,y,z`
7	0.87	0.21	0.69	`-x,y,-z`
8	0.13	0.79	0.31	`x,-y,z`
9	0.31	0.13	0.21	`z,x,y`
10	0.69	0.87	0.79	`-z,-x,-y`
11	0.31	0.87	0.79	`z,-x,-y`
12	0.69	0.13	0.21	`-z,x,y`
13	0.69	0.13	0.79	`-z,x,-y`
14	0.31	0.87	0.21	`z,-x,y`
15	0.69	0.87	0.21	`-z,-x,y`
16	0.31	0.13	0.79	`z,x,-y`
17	0.21	0.31	0.13	`y,z,x`
18	0.79	0.69	0.87	`-y,-z,-x`
19	0.79	0.31	0.87	`-y,z,-x`
20	0.21	0.69	0.13	`y,-z,x`
21	0.79	0.69	0.13	`-y,-z,x`
22	0.21	0.31	0.87	`y,z,-x`
23	0.21	0.69	0.87	`y,-z,-x`
24	0.79	0.31	0.13	`-y,z,x`

Reflection conditions

None. This space group extinguishes no reflection systematically: it has a primitive lattice, no glide plane and no screw axis, so nothing in its symmetry forces a structure factor to zero. Thirty-seven of the 230 are like this, and for those, absences tell you nothing at all.

Is a reflection there?

Put three Miller indices into the form and this says whether Pm3 allows that reflection, and which symmetry operation removes it if it does not.

The conditions, drawn

Each picture is one plane of the reciprocal lattice, drawn as a net of the reflections that plane contains. The shape of the net is schematic — a space group is symmetry and not metric, so there are no cell constants here to draw — but the equalities among its lengths and angles are the group's own, which is why a monoclinic h0l net is oblique where its hk0 net is not.

Zonehk0h0l 0kl

hk0. No reflection here is systematically absent.

hk1. No reflection here is systematically absent.

allowed systematically absent the origin

Neither layer loses a reflection to the symmetry. Whatever conditions this group has, none of them reaches this zone.

Whatever the structure is, the pattern in the zero layer cannot break the symmetry the Laue group carries into it, which here is 2mm. The upper layer keeps the same 2mm. The reverse does not follow: a group with no conditions at all draws a perfectly symmetrical net, so a symmetrical picture is no evidence of a Laue class.

Drawn from the symmetry operations, not from the table above: the class labels there stand for whole symmetry orbits, so a picture built by applying them literally would show some genuinely absent reflections as present. The two are checked against each other rather than derived from each other.

What the absences do not tell you

These same conditions are produced by 36 other space groups. Systematic absences cannot separate them, and no amount of care with the data will change that: the choice between them is made on intensity statistics, on the structure solving, or on chemistry.

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 — which is why this table does not ask which one you are looking at, and the coordinates above it do.

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.