Space Group Reflection Conditions

Ia3

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.

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

• 12 2-fold axes
• 12 2₁ screw axes
• 7 3-fold axes
• 16 3 inversion axes
• 9 3₁ screw axes
• 9 3₂ screw axes
• 4 a glides
• 4 b glides
• 4 c glides
• 16 inversion centres

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 Ia3 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.

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 48 symmetry operations of Ia3, written as coordinate triplets. Each one says where the group sends a point at x, y, z.

(0,0,0)+ (1/2,1/2,1/2)+

Add each of those to every triplet below. That is 24 × 2 = 48 operations in all.

1. x,y,z
2. -x,-y,-z
3. -x,-y+1/2,z
4. x,y+1/2,-z
5. x,-y,-z+1/2
6. -x,y,z+1/2
7. -x,y+1/2,-z+1/2
8. x,-y+1/2,z+1/2
9. z,x,y
10. -z,-x,-y
11. z,-x,-y+1/2
12. -z,x,y+1/2
13. -z+1/2,x,-y
14. z+1/2,-x,y
15. -z+1/2,-x,y+1/2
16. z+1/2,x,-y+1/2
17. y,z,x
18. -y,-z,-x
19. -y+1/2,z,-x
20. y+1/2,-z,x
21. -y,-z+1/2,x
22. y,z+1/2,-x
23. y+1/2,-z+1/2,-x
24. -y+1/2,z+1/2,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. Ia3 has 5, 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 48, 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

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 Ia3 other than the identity leaves it where it is, so the group sends it to 48 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 48 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 48 points

Reflection conditions

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.

Three things are easy to run together, and they are one sequence. A crystal class (point group) is the symmetry of the crystal’s external form and of its physical properties — there are 32. Add a centre of inversion to each, because diffraction cannot see whether one is there, and the 32 collapse to the 11 Laue classes, which is what decides which reflections are equivalent. Put a class on a lattice and add the operations that carry a translation — screw axes and glide planes — and you get the 230 space groups, which is what a structure is actually described in. Going up the ladder adds information; going down throws it away. A diffraction experiment hands you a Laue class outright, narrows the space group to a family through the absences, and never determines the crystal class on its own.

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 Ia3 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.

Zonehk0h0l0kl

allowed systematically absent not a lattice point the origin

Both layers lose reflections, so at least one condition here holds in every layer rather than only in the central one — which is what a lattice centring (integral) and a screw axis (serial) do.

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 one other space group. 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.

• #73 Ibca

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.