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.
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.
Cmmm
— 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.
The International Tables print every setting of a space group together, and #63 has 6 of them here. They are the
same symmetry described in different axes and cell choices, so they are not interchangeable:
the reflection conditions move with the axes, which is why this site answers for the setting
you asked about rather than sending you to the standard one.
This group’s point group already has the full symmetry of its lattice, so there is no operation left over to twin by: it cannot twin by merohedry. That is the ordinary case — a crystal twins by merohedry only when it is less symmetric than the lattice it sits on.
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
drag to rotate · scroll to zoom · click to pin
Look along
Point at an element to read what it is.
8 2-fold axes
16 21 screw axes
2b glides
2c glides
16 inversion centres
4 mirror planes
4n glides
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 mmm — the
8 operations of
Cmcm 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.
8 poles, drawn as 4 marks.mmm 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.
4 of the 8 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 16 symmetry
operations of
Cmcm, written as coordinate triplets. Each one
says where the group sends a point at x, y, z.
(0,0,0)+ (1/2,1/2,0)+
Add each of those to every triplet below. That is
8 ×
2 =
16 operations in all.
x,y,z
-x,-y,-z
-x,-y,z+1/2
x,y,-z+1/2
x,-y,-z
-x,y,z
-x,y,-z+1/2
x,-y,z+1/2
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.
Cmcm has
8, listed from the general position — the one no operation
fixes — down to the most symmetric site it has.
Multiplicity times the order of the site symmetry is
16, 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 Cmcm other than the identity leaves it where it is, so the group sends it to 16 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
16 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 16 points
#
x
y
z
From
1
0.13
0.21
0.31
x,y,z
2
0.63
0.71
0.31
x+1/2,y+1/2,z
3
0.87
0.79
0.69
-x,-y,-z
4
0.37
0.29
0.69
-x+1/2,-y+1/2,-z
5
0.87
0.79
0.81
-x,-y,z+1/2
6
0.37
0.29
0.81
-x+1/2,-y+1/2,z+1/2
7
0.13
0.21
0.19
x,y,-z+1/2
8
0.63
0.71
0.19
x+1/2,y+1/2,-z+1/2
9
0.13
0.79
0.69
x,-y,-z
10
0.63
0.29
0.69
x+1/2,-y+1/2,-z
11
0.87
0.21
0.31
-x,y,z
12
0.37
0.71
0.31
-x+1/2,y+1/2,z
13
0.87
0.21
0.19
-x,y,-z+1/2
14
0.37
0.71
0.19
-x+1/2,y+1/2,-z+1/2
15
0.13
0.79
0.81
x,-y,z+1/2
16
0.63
0.29
0.81
x+1/2,-y+1/2,z+1/2
Reflection conditions
Class
Condition
From
hkl
h + k = 2n
integral
0kl
k = 2n
zonal
h0l
l = 2n; h = 2n
zonal
hk0
h + k = 2n
zonal
h00
h = 2n
serial
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. Nineteen 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
Cmcm 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.
0kl. 6 of the 54 lattice nodes drawn are systematically absent. No complete line of the net is empty: they go one at a time. A further 66 of the drawn positions are not nodes of this centred lattice at all.1kl. No reflection here is systematically absent. But 55 of the drawn positions are not nodes of this centred lattice at all.
allowed systematically absent not a lattice point the origin
The upper layer is complete: every absence in this zone is in the zero layer. A condition that holds in one plane of the reciprocal lattice and not in the ones beside it is zonal, and a glide plane is what produces one.
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
3 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. A1981, 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.