Review: Operations without translation.
Symmetry operations - rotation, reflection and inversion
Symmetry elements - rotation axis, mirror plane (m) and symmetry center ( i)
Rotational symmetry is designated by an integer (n) from 1 to infinity, (example is a beer can, i.e., bottom verses the top). Because of geometrical constraints, the internal order of a crystal is limited to 1-fold, 2-fold, 3-fold, 4-fold and 6-fold axes.
Rotational symmetry is always considered with respect to a full 360° rotation. (i.e., one full turn = 360°). The number of degrees in a turn must be exactly divisible into 360 degrees (must return to the starting point). This is why symmetry operations can be designated by integer values. (e.g., 360°/90° = 4).
Recall the following notation...
p = primitive cell --- 1/4 of motif is located at each corner
c = centered cell --- a two-step operation of translating motif 1/2 unit length a and 1/2 unit length b.
m = mirror --- imaginary plane that perpendicularly reflects motif (solid line ^{___________}).
g = glide plane --- a two-step operation of a reflection and translation (dashed line ----------------).
# = number of folds on perpendicular symmetry axis
2D single and combined operations that take place about a point
(i.e., not including translations) result in 10 possible point
groups (1, 2, m, 2mm, 4, 4mm, 3, 3m,
6, 6mm),
There are 17 plane groups
possible if translations and non-primitive unit cells are allowed.
Plane group symbols are defined by four parts: Unit cell type (p
or c); followed by rotational symmetry # (1,2,3,4,6);
followed by mirror or glide plane presence (m , g or
1 if they are absent in the x direction); followed by
either the symmetry in the y direction or symmetry at 45° in
square lattices or symmetry at 30° in hexagonal lattices.
Lattice Type |
Point Group (without translations) |
Plane Group (with translations) |
Click below for an example |
Oblique |
1 |
p1 |
Figure 1 |
2 |
p2 |
Figure 2 | |
Rectangular |
m |
pm |
Figure 3 |
pg |
Figure 4 | ||
cm |
Figure 5 | ||
2mm |
p2mm |
Figure 6 | |
p2mg |
Figure 7 | ||
p2gg |
Figure 8 | ||
c2mm |
Figure 9 | ||
Square |
4 |
p4 |
Figure 10 |
4mm |
p4mm |
Figure 11 | |
p4gm |
Figure 12 | ||
Hexagonal |
3 |
p3 |
Figure 13 |
3m |
p3m1 |
Figure 14 | |
p31m |
Figure 15 | ||
6 |
p6 |
Figure 16 | |
6mm |
p6mm |
Figure 17 |
Rotoinversion: Until this point, we have only discussed symmetry operation in two-dimensions. When combining operations of rotation and inversion through a center one must observe the elements in three space.
A 1-fold rotation with an inversion is designated as bar one (-1). In fact, it produces a center of symmetry : designated (i). therefore i = bar one.
(Figure 2.11, pg. 24)
Examples of projection of motifs on to equatorial planes.
bar-1 = center of symmetry (i)
bar-2= mirror plane (m) perpendicular to rotation axis
Rule: All symmetry operators must pass through a single point.
Example of 422 symmetry.
Combines both the rotation about an axis and the reflection across a plane perpendicular to the axis.
The symmetry operation is designated 4/m (four over m).
Conventions used in representing 3-dimensional symmetry elements in 2-dimensions.
1. solid points are motifs located above the mirror plane.
2. open points are motifs located below the mirror plane.
3. solid circles represent mirror planes on page.
4. solid lines represent mirror planes perpendicular to page.
Summary of Hermann-Mauguin symbols:
It is possible to classify the symmetry operation on the basis of increasing degree of rotational symmetry.
32 Point groups (degenerate operations) | Symmetry element(s) |
1, 2, 3, 4, 6 | rotation axis only |
-1, m, -3, -4, -6 | rotoinversion |
222, 32, 422, 622 | combination of rotation axes |
2/m , 4/m , 6/m (3/m = -6) | one rotation with perpendicular mirror |
2mm, 3m, 4mm, 6mm | one rotation with parallel mirror |
-32/m , -42/m, -6/m2 (-1/m=-1 2=2/m) (-22=-2m=2mm) | rotoinversion with rotation and mirror |
2/m2/m2/m, 4/m2/m2/m, 6/ m2/m2/m | three rotation axes and perpendicular mirrors |
combinations | elements in isometric patterns |
Crystal Systems | H-M notation |
Triclinic | 1 and -1 |
Monoclinic | 2, m and 2/m |
Orthorhombic | 222, 2mm, and 2/m2/m2/< i>m |
Tetragonal | 4, -4, 4/m, 422, 4mm, -42m an d 4/m2/m/2m |
Hexagonal |
3, -3, 3/m, 32, -32m, 6, -6, 6/m, 622, 6mm, -62m, and 6/m2/m2/m |
Isometric | 23, 2/m-3, 432, -43/m and 4/m- 32/m |