Crystallographic Point Groups |

The majority of the table is reference material.

**Crystallographic Point Groups**

The simplest crystallographic point groups are 1, 2, 3, 4, and 6 all of which
possess a single rotation axis only. Likewise the rotary-inversion axes are the
basis for the point groups -1, *m*, -3, -4, and -6. The remaining 22
crystallographic point groups result from the combination of the previous 10
point groups. With the exception of point groups belonging to the cubic crystal
system, only twofold rotation axes and/or mirror planes can be taken together
with other rotation or rotary-inversion axes: Twofold axes can be combined
perpendicular to other axes (e.g. point-group 422) while mirror planes can act
either perpendicular (e.g. 2/*m*) or parallel to another axis
(*mm*2). A slash ("/") character before the symbol *m* indicates a
mirror plane perpendicular to the main axis of rotation.

The crystallographic point groups may be classified according to the
crystal system with which they are associated. Thus the point
groups of the trigonal crystal system all possess a threefold axis, while
those of the tetragonal and hexagonal crystal systems possess a fourfold and
sixfold axis, respectively. The cubic point groups all have multiple threefold
axes (see below).
The orthorhombic point groups have twofold symmetry (either 2 or *m*
with respect to each of the X-, Y-, Z- directions of an orthogonal
axis system, while the monoclinic point groups are limited to
twofold symmetry with respect to a single axis direction.
Finally, the triclinic point groups can only have
an axis of order 1.

The cubic point groups are all characterised by the four threefold rotation axes which act along the body diagonals of a cube. This is indicated by the digit "3" in the cubic point-group symbols. In addition, the cubic point groups all contain at least three mutually perpendicular twofold rotation axes.

Crystal System | 32 Crystallographic Point Groups | ||||||
---|---|---|---|---|---|---|---|

Triclinic | 1 |
-1 | |||||

Monoclinic | 2 |
m |
2/m |
||||

Orthorhombic | 222 | mm2 |
mmm |
||||

Tetragonal | 4 |
-4 | 4/m |
422 | 4mm |
-42m |
4/mmm |

Trigonal | 3 |
-3 | 32 | 3m |
-3m |
||

Hexagonal | 6 |
-6 | 6/m |
622 | 6mm |
-62m |
6/mmm |

Cubic | 23 | m-3 |
432 | -43m |
m-3m |

Of the 32 crystallographic point groups,
those highlighted in magenta
possess a centre of inversion and are called centrosymmetric,
while those highlighted
in red possess only rotation axes
and are termed enantiomorphic.
A third type, highlighted in **bold type**, are referred to as polar.
The properties of these different types of point groups are explained
in more detail in the subsequent sections.

Note: this additional material is given for completeness.

The ordering of the symmetry elements that compose the point-group symbols
depends on the crystal system to which they are related. Thus, for the
orthorhombic crystal system, the order of the components of the point-group
symbol relates to the symmetry for the X,Y,Z directions, respectively.
For 222 and *mmm* the order is immaterial,
but for *mm*2 it is purely a convention
that the rotation axis is chosen along Z.
For the trigonal, tetragonal, and hexagonal point groups,
which possess rotation axes of order 3, 4, and 6, respectively, the convention
is to put the high symmetry rotation axis first in the symbol. The second
component of the
symbol refers to the symmetry with respect to the X axis (and also Y axis
assuming Y is chosen in the crystallographic sense for the trigonal/hexagonal
case), and again the highest-order rotation axis is chosen. Following these
simple rules, one obtains point-group symbols such as
-42*m* and -62*m*, but not -4*m*2 and -6*m*2.

However, the above rules only apply in the absence of a set of references axes.
With reference to a crystallographic unit cell, it may happen that the
twofold rotation axes, for example, in -42*m* refers to the diagonal
direction between the *a* and *b* axes,
but neither to *a* or *b*, while the normal for the
mirror plane may refer to the a and b axes. Under these conditions, it would
be correct to use the point-group symbol -4*m*2.
The use of -4*m*2 is illustrated in
a couple of point-group figures.
(Point group figures are discussed at length in the following page).
Note that there is no
difference in the properties of point groups -42*m* and -4*m*2
other than one of axes labelling, but there is a difference when it comes to
space groups (which form the subject of the next section on symmetry).

By contrast *m*-3 and -3*m* are quite different point groups.
The cubic crystal system is always characterised by four threefold rotation
axes. In addition,
the higher-symmetry cubic point groups possess three fourfold rotation
(or rotary-inversion) axes. If the highest-symmetry axis is put first, then
the logical order is, for example, 432 or -43*m*, with the symbol for the
threefold axis in second place. For consistency, all of the cubic point-group
symbols (and space-group symbols too) have the characteristic 3 (or -3) as the
second element of the point-group symbol.
In addition, this avoids confusion with the trigonal system where only
a single 3 (or -3) axis is present. Thus, for cubic systems the first component
of the point-group symbol
refers to symmetry with respect to the X, Y, and Z directions, the second
refers to symmetry with respect to the body-diagonal of a cube, and
the final component (if present) refers to symmetry with respect to the
face-diagonal directions.

© Copyright 1995-2006. Birkbeck College, University of London. |
Author(s):
Jeremy Karl Cockcroft |