Space Group Notation
From examination of a space group in "The International Tables
for Crystallography" Vol. A, one can ascertain the following information:
- Herman-Mauguin (HM) Symbol (Long, Short)
- Point Group (HM, Schoenflies)
- Locate and identify symmetry elements
- Wyckoff site multiplicity and symmetry
- Distinguish general and special positions
- Extinction conditions
- Identify possible subgroups and supergroups
The HM space group symbol can be derived from the
symmetry elements present using the following logic.
The first letter identifies the centering of the lattice :
- P – Primitive
- I – Body centered
- F – Face centered
- C – C-centered
- B – B-centered
- A – A-centered
The next three
symbols denote symmetry elements present in certain directions,
those directions are as follows:
| Crystal
System |
Symmetry Direction |
|
Primary |
Secondary |
Tertiary |
| Triclinic |
None |
|
|
| Monoclinic |
[010] |
|
|
| Orthorhombic |
[100] |
[010] |
[001] |
| Tetragonal |
[001] |
[100]/[010] |
[110] |
Hexagonal/
Trigonal |
[001] |
[100]/[010] |
[120]/[1(-1)0] |
| Cubic |
[100]/[010]/
[001] |
[111] |
[110] |
[100] – Axis parallel or plane perpendicular to the
x-axis.
[010] – Axis parallel or plane perpendicular to
the y-axis.
[001] – Axis parallel or plane
perpendicular to the z-axis.
[110] – Axis parallel or
plane perpendicular to the line running at 45° to the x
and y axes.
[110] – Axis parallel or plane
perpendicular to the long face diagonal of the ab face of a hexagonal cell.
[111] – Axis parallel or
plane perpendicular to the body diagonal.
The long Herman-Mauguin symbol will show both rotation/screw
axes which run parallel to the specified direction, as
well as any mirror/glide plane which is perpendicular to the same direction. The two symbols will be separated by a
forward slash (/).
For example :
- C 1 2/m 1
- P 21 21 2
- P 2/m 2/n 21/a
- I 41/a 2/m 2/d
- P 63/m 2/m 2/c
- F 4/m 3 2/m
The short
Herman-Mauguin symbol will typically contain only the mirror/glide plane component corresponding to primary, secondary and tertiary
directions, or in the absence of a mirror/glide plane
the rotation/screw axis present will be shown. In certain cases (monoclinic, tetragonal and hexagonal) however, both the
rotation/screw axis and the
mirror/glide plane will be
retained in the short Herman-Mauguin symbol (for the primary direction only). To illustrate this consider the short H-M symbols
for the previous collection of space groups:
- C2/m
- P21212
- Pmna
- I41/amd
- P63/mmc
- Fm3m
With no
knowledge of the symmetry diagram we can identify the
crystal system from the space group symbol.
- Cubic – The secondary symmetry symbol will always be
either 3 or 3 (i.e. Ia3, Pm3m, Fd3m)
- Tetragonal – The primary symmetry symbol will always
be either 4, 4, 41, 42 or 43 (i.e.
P41212, I4/m, P4/mcc)
- Hexagonal – The primary symmetry symbol will always
be a 6, 6, 61, 62, 63, 64 or
65 (i.e. P6mm, P63/mcm)
- Trigonal – The primary symmetry symbol will always be
a 3, 3, 31 or 32 (i.e P31m, R3, R3c, P312)
- Orthorhombic – All three symbols following the
lattice descriptor will be either mirror planes, glide planes, 2-fold rotation
or screw axes (i.e. Pnma, Cmc21, Pnc2)
- Monoclinic – The lattice descriptor will be followed
by either a single mirror plane, glide plane, 2-fold rotation or screw axis or
an axis/plane symbol (i.e. Cc, P2, P21/n)
- Triclinic – The lattice descriptor will be followed
by either a 1 or a (1).