Lecture notes for GEOL3010

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 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).


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 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


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


bar-3= 3-fold + i

bar-4= unique (no other combination can produce this symmetry).


bar-6 is equivalent to 3-fold + mirror plane perpendicular to rotation axis.


Combinations of rotations: Only one axis of rotation has been considered. It is possible to combine more than one axis of rotation into a symmetry operation and generate a pattern in 3-dimensions.

Rule: All symmetry operators must pass through a single point.

Example of 422 symmetry.

422

 


Rotation Axes and Mirrors

Combines both the rotation about an axis and the reflection across a plane perpendicular to the axis.

4overm

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