Lecture notes for GEOL3010

Nesse p. 12-19
Klein p. 175-187, 213-228

Crystallography: Geometric operations

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 divisable into 360 degrees (must return to the starting point). This is why symmety operations can be designated by interger values. (e.g., 360°/90° = 4).

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Table of 2-D point groups and plane groups

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 lengtha and 1/2 unit lengthb.

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 perpedendicular 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 allowd.

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

Combined symmetry operations

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 perpe ndicular 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