Plane Group Identification
Is glide reflection present in an axis that is not a
reflection axis?
Are all centers of rotation on reflection axes?
Do reflections occur in axes that intersect at 45?
(Do reflections intersect at 45?)
Determining Plane Groups
I bought some Fe powder from Arcos
Organics for a reaction
Great the bottle says Iron, 99% powder!
Looks a little dark for Fe powder (which
is supposed to be grey).
An X-ray powder pattern
The disappointing result
More disappointment!
Fe 48%
Magnetite 38
Non Crystallographic Point
Groups
Cv
Linear object. "Infinite" rotor with vertical mirrors.
No other symmetry
Dh
Linear object. "Infinite" rotor. Horizontal mirror.
What other symmetries?
2, i
Icosahedral Symmetry
Flowchart for Determining Significant
Poi
Classification of Point Groups
and Space Groups
into Crystal Systems
A set of coordinate axes must be attached to the origin in
a point group so that points in objects can be described
mathematically.
In crystallography we use
a right-handed coordinate
sy
32 Crystallographic Point Groups
Point Groups
The 32 crystallographic point groups (point groups consistent
with translational symmetry) can be constructed in one of two
ways:
1. From 11 initial pure rotational point groups, inversion
centers can be added
Three Dimensional Symmetry
Symmetry Elements
Rotation Axis
Operation: Rotation about an axis
International Notation: n
Schoenflies Notation: Cn
4 (C4)
Mirror Plane
Operation: Reflection across a plane
International notation: m
Schoenflies notation:
m ()
Point Groups
Group Theory
Point Group
A collection of symmetry elements obeying the properties of
a mathematical group and having one point in common,
which remains fixed through all symmetry operations.
Point groups describe the allowable collection of s
Plane Lattices
Plane Lattices
We have seen the types of symmetries that a motif can
possess in two dimensions. This gives rise to the 10 plane
point groups.
When periodicity is added to the motif there is one important
criterion that must be met:
The latt
Lattices
Crystal
A periodic arrangement of atoms in three dimensions
Periodicity
Characteristic repetition distance (t) of a motif.
The collection of repetition distances defines a unit cell
Point Lattice
Replacement of repeated motif with an array of poi
Symmetry
Basic 2-D Symmetry
Weve already seen that periodicity (translational
symmetry) is very important in the study of arrays of
objects. There are two other basic operators that move
objects in the plane:
Mirrors
Rotations
Mirror Symmetry
2-fold Rot
Plane Point Groups
Point Groups
Symmetries created by symmetry operators passing
through a single point.
Actually, point group symmetries include such operators
as 5-fold symmetry, however, since our interest is in
objects that are periodic, only certain