Reading a space group symbol

A space group name is a short code for the lattice plus the symmetry

Intuition

A space group symbol is like a licence plate. The first letter says what kind of box it is. The rest list the main moves the crystal is symmetric under, in a fixed order.

Lecture 5 recaps the idea: the unit cell, symmetry elements and atom positions are found by diffraction, and a space group is the point-group symmetry combined with translational symmetry. The lecture gives 32 point groups, 14 Bravais lattices and 230 space groups. Triclinic cells cannot have rotation axes or mirror planes. No 5-, 7- or 8-fold rotation is possible in a periodic crystal.

A space group is the set of symmetry elements and operations that fully describes the spatial arrangement of a 3D periodic pattern. In the symbol, the first letter is the Bravais lattice type: P (primitive), C (base-centred), I (body-centred) or F (face-centred). The remaining symbols are generators, read along the viewing directions.

The lecture’s example is low-temperature pyrite, FeS$_2$, space group $P\,c\,a\,2_1$ (orthorhombic, viewed along a, then b, then c). The $c$ is a glide plane perpendicular to a. The $a$ is a glide plane perpendicular to b. The $2_1$ is a screw axis parallel to c.

Exam detail

Rotation and screw axes lie parallel to the viewing direction. Mirror and glide planes lie perpendicular to it. Space group diagrams are 2D projections of the unit cell with symbols for the symmetry elements and the symmetry-equivalent positions.

Each crystal system has fixed viewing directions

Viewing axes are chosen to show new symmetry elements. For a monoclinic crystal, the direction is set by convention as b (c). For tetragonal, trigonal and hexagonal crystals, the highest-order rotation axis is parallel to c and is the first viewing direction. The cell edges satisfy $a = b$ for tetragonal, trigonal, hexagonal and cubic (and $a = b = c$ for cubic).

Crystal system	Primary	Secondary	Tertiary
Triclinic	N/A
Monoclinic	b (c)
Orthorhombic	a	b	c
Tetragonal	c	a	[110]
Trigonal	c	a	[210]
Hexagonal	c	a	[210]
Cubic	c	[111]	[110]

Optional aside

The lecture notes flag that the slide table layout was ambiguous in the raw text and that the cubic row follows the raw text order. Check the cubic row against the slide before relying on it.

Try it: worked example

Read the symbol P4/mmm for a tetragonal crystal.

Show answer and steps

AnswerPrimitive tetragonal cell. A 4-fold axis along c with a mirror plane across it (4/m), then a mirror plane viewed along a, then a mirror plane viewed along [110].

1. Tetragonal means $a = b \ne c$ and $\alpha = \beta = \gamma = 90^\circ$.
2. The first letter P says the cell is primitive.
3. Primary direction is c. There is a 4-fold rotation axis, so P4, and a mirror plane perpendicular to it, written /m.
4. Secondary direction is a. There is a mirror plane, so P4/m m.
5. Tertiary direction is [110]. There is a mirror plane, so P4/m m m.

The lecture’s example structures of this group are CuTi$_3$ and CuAu.

The lecture’s example space groups

• P1, triclinic primitive: only translation symmetry. Example: (R)-(−)-3-acetoxyquinuclidine methiodide.
• P2$_1$/a, monoclinic primitive: inversion centres, $2_1$ screw axes and glide planes. Example: caffeine hydrate.
• I4$_1$/amd, body-centred tetragonal: 32 symmetry elements.
• Fm$\bar{3}$m, face-centred cubic: 96 symmetry elements, and only 2½ carbon atoms define the structure.
• P3$_1$21 and P3$_2$21, primitive trigonal: only screw and rotation axes, no improper elements, so the crystals are chiral. Right-handed crystals are P3$_1$21 and left-handed P3$_2$21. Quartz is piezoelectric.

The slide on why symmetry matters (structures can be worked out and described, and properties depend on symmetry, with non-centrosymmetric groups highlighted) is marked beyond exam scope.

Pitfall

Look at which axes sit parallel and which planes sit perpendicular to the viewing direction before reading a symbol. Mixing the two up reads the symbol backwards.