The Four Parts of a Full H–M Symbol
A full Hermann–Mauguin (International) space-group symbol always has four positions: one letter for the lattice centring, then one entry for each of three symmetry directions.
Centring of the conventional unit cell: P, A/B/C, I, F or R.
2The symmetry seen along the primary direction.
3The symmetry seen along the secondary direction.
4The symmetry seen along the tertiary direction.
Take P12/c1, the full form of P2/c. The cell is Primitive, nothing is required along the primary direction (1), the secondary direction carries a 2-fold axis with a c-glide perpendicular to it (2/c), and nothing is required along the tertiary direction (1).
Which lattice directions count as primary, secondary and tertiary is not the same in every crystal system. ITC fixes them by convention, as set out in the next section.
Click a position in the symbol to see what it describes in the cell.
Full-Symbol Decoder
Type a full symbol with spaces between positions (write screw axes as 2_1, bars as -3), or pick an example.
Lattice Symmetry Directions in 3D
The meaning of “primary”, “secondary” and “tertiary” is set by the crystal system. The three lowest-symmetry systems show the pattern.
No direction needs to be named: the only point groups are 1 and 1, which have no axis or plane to orient. Symbols: P1, P1.
Only the direction carrying the 2 or 21 matters. By convention this ‘unique axis’ is b [010] or c [001]; the other positions are written as 1, e.g. P121 or P112.
All three directions carry symmetry. By convention primary = [100], secondary = [010], tertiary = [001].
Ref: Table 2.2.4.1, International Tables for Crystallography (2006), Volume A, Chapter 2.2. Braces enclose symmetry-equivalent directions. Click any row to see it in 3D below.
Symmetry-Direction Explorer
One Group, Six Settings, Six Symbols
An orthorhombic cell has three mutually perpendicular but unequal edges, and nothing forces which one we call a, b or c. There are \(3! = 3\times2\times1 = 6\) ways to assign them. Each choice is a setting, and each can give a differently written H–M symbol.
Fig. 2.2.6.5: diagrams for the ‘standard setting’ as used in the space-group tables (G = general-position diagram). Click a projection to turn the 3D cell below to that view.
Fig. 2.2.6.6: the three projections with the six setting symbols. For setting symbols printed vertically, the page has to be turned clockwise by 90° (or viewed from the side); the next section works through exactly this. In the actual space-group tables, the full H–M symbols appear in place of the setting symbols. Click a setting symbol.
Ref: 2.2 Contents and arrangement of the tables, Th. Hahn, A. Looijenga-Vos, ITC (2006), Vol. A, Chapter 2.2, pp. 17–41
Setting Transformer
Pick a setting and the box turns so that the new origin O is at the top left, [100] points down, [010] points right and [001] comes out of the screen. The arrows keep their standard labels a, b, c, so reading what lies along [100], [010] and [001] spells the setting. An arrow pointing against its direction means a bar.
Rule used: each position reads the symmetry along the new [100], [010] or [001]; glide letters a/b/c are renamed to follow their translation vector; A/B/C centring letters follow the axis normal to the centred face; m, n, d, e and rotation/screw axes are unchanged. A bar (e.g. c) reverses that axis to keep the new set right-handed.
Turning the Page: Deriving a New Setting
In the standard setting for orthorhombic space groups the projection has a fixed orientation on the page. The convention belongs to the page, not to the crystal, and that is what lets us read off a new setting.
Purple: the axis directions the convention assigns to the turned page.
After the turn, a points left along the top edge and b points down the right-hand edge; c still comes out of the page. Now apply the page convention again:
-
1
[100] is vertical, pointing down, i.e. along b.
First H–M position → b -
2
[010] is horizontal, pointing right, i.e. along −a.
Second H–M position → a -
3
[001] is still normal to the page, along c.
Third H–M position → c
Apply it to a symbol. In bac, the first position now reads the symmetry that was along b, the second what was along a, and the third what was along c. Any glide letter is renamed to its new axis.
Signs never change an H–M letter, so bac gives the same symbol as ITC’s vertically printed bac in Fig. 2.2.6.6. It is the same box seen from the other side.
Priority Rules for Full H–M Notation
One symmetry direction can carry several axes and planes at once, e.g. a glide, a mirror and a rotation axis. The symbol has room for only one axis and one plane per direction, so which do we write?
Planes, in descending order of priority:
Axes: rotations before screw axes, e.g. 2 > 21.
So if a 2 axis, a 21 axis, a b-glide and a mirror all lie along one direction, that position is written 2/m.
*A double glide plane occurs in centred cells only: one plane with two reflections, whose glide vectors are perpendicular. The 3D animation below shows why.
Ref: Chapter 4.1, Introduction to the synoptic tables, ITC (2006), Vol. A, pp. 56–60
Priority Resolver
Switch on the elements present along one symmetry direction.
Everything along the direction, and the winners
Axes are drawn parallel to the direction and planes perpendicular to it, using the ITC graphical symbols for the 2-fold axis (lens) and 21 screw axis (lens with tails). In a real structure these elements sit at different positions in the cell; the symbol keeps only the highest-ranked axis and plane.
Why the Double Glide e Needs a Centred Cell
A C-centred orthorhombic cell with a glide plane ⊥ c at \(z=\tfrac12\) (amber).
- 1Start from a motif at \((x,y,z)\).
- 2a-glide: reflect across the plane, then translate by \(\mathbf a/2\) → \((x+\tfrac12,\,y,\,1-z)\).
- 3b-glide: the same reflection, then \(\mathbf b/2\) → \((x,\,y+\tfrac12,\,1-z)\).
- 4The two images differ by \((\tfrac12,-\tfrac12,0)\equiv(\tfrac12,\tfrac12,0)\), exactly the C-centring translation. One plane, two perpendicular glide vectors: that is e.
In a primitive cell the two images would not be equivalent, so a single plane could not act as both an a- and a b-glide.
Analytical Representation of Space Groups in ITC
The diagrams show where the symmetry elements are. The continued page of every ITC entry says the same thing in numbers: which operations generate the group, every set of equivalent positions, and which X-ray reflections must vanish.
- Generators selected
- Positions: general and special positions, multiplicity, Wyckoff letter, site symmetry, coordinates
- XRD reflection conditions
- Worked examples: plane groups p2, pm, pg, cm, p2mm and space groups P2/c and Cmcm
What the Numbers Page Contains
Here is the continued page for Cmcm (No. 63), redrawn. Click the numbered markers to see what each block is for.
Wyckoff letter,
Site symmetry
a′ = a b′ = b
Origin at 0,0,z
a′ = ½b b′ = c
Origin at x,0,0
a′ = ½c b′ = ½a
Origin at 0,y,0
Redrawn from ITC (2006), Vol. A, p. 301 (Cmcm, No. 63).
Reflection-Condition Checker
Is reflection hkl allowed by the general-position conditions?
General and Special Positions
A set of symmetry-equivalent points, each left invariant only by the identity 1 (C1). A point P at (x, y, z) is a general position.
A set of points, each mapped onto itself by the identity and at least one further operation. A point Q at (a, y′, z′) lying on a mirror is a special position.
Special positions lie on symmetry elements, except glide planes and screw axes. The translation part always carries the point somewhere else, so there are no special positions on a glide plane or a screw axis.
Drag the slider to bring a point onto each element and watch what happens to its image.
Circles follow the ITC convention: a comma marks the mirror-image (opposite-handed) copy, and a split circle marks a point sitting on a mirror, where the object and its image coincide.
Building the General Position of Cmcm
ITC lists a short, fixed sequence of operations that generates the whole group: (1); t(1,0,0); t(0,1,0); t(0,0,1); t(½,½,0); (2); (3); (5). Apply them one at a time to a single point and every one of the 16 general positions appears.
+ / − give the height (½+ means ½ + z); a split circle holds two objects at the same x, y but different heights; a comma marks a mirror-image copy. In 3D, coral objects keep the original handedness and blue ones are mirror images.
Symmetry elements of Cmcm in one unit cell
Drawn in the highlighted cell; objects in the other three cells are faded. Switch each family on or off.
Why this order? ITC generates every group the same way: the identity first, then the lattice translations, then any centring translations, then the point-group generators in a fixed order for each crystal class.
Cmcm belongs to class mmm, whose generators are 2z, 2y, 1. These are exactly (2), (3) and (5) of Cmcm. (Here (2) is the 21 screw along c; a screw component doesn’t change which point-group generator it plays.)
For the full list, see Table 8.3.5.1, p. 737 of ITC (2006), Vol. A.
| Crystal class | Generators Gi (left to right) |
|---|
Excerpt of Table 8.3.5.1, ITC (2006), Vol. A. Subscripts give the direction of the operation: z = [001], y = [010].
Reading the Positions Block
The top row is the general position; every row below it is a special position. Pick a row to see its points in the cell, and click a column heading to see what it means.
Wyckoff Positions in p2, pm, pg, cm and p2mm
Drag the teal point anywhere in the cell. It snaps onto rotation points and mirror lines, and the page reports its Wyckoff position, site symmetry and multiplicity, just as the ITC table does.
Why is the site symmetry in pm written “.m.”? Site-symmetry symbols use the same positions as the H–M symbol, with a dot for a direction that carries nothing. For a rectangular lattice the primary position is the rotation point, the secondary direction is [10] and the tertiary is [01].
The mirror at x = 0 is perpendicular to [10], so the symbol has nothing in the primary position, m in the secondary and nothing in the tertiary: .m.
| 2D lattice | Primary | Secondary | Tertiary |
|---|---|---|---|
| Oblique | Rotation point in plane | ||
| Rectangular | [10] | [01] | |
| Square | {[10], [01]} | {[11], [11]} | |
| Hexagonal | {[10], [01], [11]} | {[11], [12], [21]} |
Lattice symmetry directions in 2D (ITC Table 2.1.3.1).
Did It Sink In? - 13
Test your mastery of symmetry-direction conventions, orthorhombic settings, priority rules, and the analytical description of space groups: generators, Wyckoff positions, site symmetry and reflection conditions.