Objectives
By the end of this section you should:
• be able to recognise rotational symmetry and
mirror planes
• know about centres of symmetry
• be able to identify the basic symmetry elements
in cubic, tetragonal and orthorhombic shapes
• understand centring and recognise facecentred, body-centred and primitive unit cells.
• Know some simple structures (Fe, Cu, NaCl,
CsCl)
Note for Symmetry experts!
• Crystallography uses a different notation
from spectroscopy!
In spectroscopy, this has ‘C4’ symmetry
In crystallography, it has ‘4’ symmetry
Symmetry everywhere
Pictures from
Dr. John Reid
Symmetry everywhere
Pictures from Dr. John Reid
Mirror Plane Symmetry
“Arises when one half of an object is the mirror image
of the other half”
Symbol
m
Mirror Plane Symmetry
How symmetrical is a face?
Left: Symmetrical face using the left half of the original face.
Middle: Original face. Right: Symmetrical face using the right
half of the original face.
http://www.uni-regensburg.de/Fakultaeten/phil_Fak_II/Psychologie/Psy_II/beautycheck/english/symmetrie/symmetrie.htm
Mirror Plane Symmetry
This molecule has two mirror
planes
One is horizontal, in the plane of
the paper - bisects the Cl-C-Cl
bonds
Other is vertical, perpendicular
to the plane of the paper and
bisects the H-C-H bonds
Symmetry
“Something possesses symmetry if it looks the same
from >1 orientation”
Rotational symmetry
Can rotate by 120° about the C-Cl
bond and the molecule looks
identical - the H atoms are
indistinguishable
This is called a rotation axis
- in particular, a three fold rotation axis, as rotate by
120° (= 360/3) to reach an identical configuration
All M.C. Escher works (c) Cordon Art-Baarn-the Netherlands.
All rights reserved.
In general:
n-fold rotation axis = rotation by (360/n)°
? Think of examples for n=2,3,4,5,6…
We talk about the symmetry
operation (rotation) about a
symmetry element (rotation
axis)
Rotational symmetry
n=2
n=5
360/2
360/5
360/6
180o
72o
60o
n=6
Centre of Symmetry
“present if you can draw a straight line from any point,
through the centre, to an equal distance the other side,
and arrive at an identical point” (phew!)
Centre of symmetry at S
No centre of symmetry
Combinations - the plane point groups
Carefully look at what symmetry
is present in the whole pattern
The blue pattern has rotational
symmetry, but the yellow dots
break this - therefore there are
two mirror planes perpendicular to
one another
= mm
Now try the examples on the sheet...
Combinations - the plane point groups
Symmetry in 3-d
In handout 1 we said that a crystal system is
defined in terms of symmetry and not by crystal
shape.
Thus we need to look at all the symmetry
arising from different shapes of unit cell.
From this we can deduce essential symmetry.
Unit cell symmetries - cubic
• 4 fold rotation axes
(passing through pairs of
opposite face centres,
parallel to cell axes)
TOTAL = 3
Unit cell symmetries - cubic
• 4 fold rotation axes
TOTAL = 3

3-fold rotation axes
(passing through cube
body diagonals)
TOTAL = 4
Unit cell symmetries - cubic
• 4 fold rotation axes
TOTAL = 3

3-fold rotation axes
TOTAL = 4
 2-fold rotation axes
(passing through
diagonal edge centres)
TOTAL = 6
Mirror planes - cubic
3 equivalent planes
in a cube
6 equivalent planes
in a cube
Tetragonal Unit Cell
a = b  c ;  =  =  = 90
elongated / squashed cube
c < a, b
c > a, b
Reduction in symmetry
Cubic
Three 4-axes
Tetragonal
One 4-axis
Two 2-axes
Four 3-axes
No 3-axes
Six 2-axes
Two 2-axes
Nine mirrors
Five mirrors
See Q3 in handout 2.
Essential Symmetry
Essential symmetry is that which defines the crystal system
(i.e. is unique to that shape).
System
Essential Symmetry
Symmetry axes
Cubic
4 3-fold axes
along the body diagonals
Tetragonal
Orthorhombic
Hexagonal
Trigonal (R)
Monoclinic
Triclinic
1 4-fold axis
3 mirrors or 3 2-fold axes
1 6-fold axis
1 3-fold axis
1 2-fold axis
no symmetry
parallel to c, in the centre of ab
perpendicular to each other
down c
down the long diagonal
down the “unique” axis
Cubic Unit Cell
a=b=c, ===90
c
Many examples of cubic
unit cells:
e.g. NaCl, CsCl, ZnS,
CaF2, BaTiO3


b

a
All have different arrangements of atoms within the cell.
So to describe a crystal structure we need to know:
 the unit cell shape and dimensions
 the atomic coordinates inside the cell (see later)
Primitive and Centred Lattices
Copper metal is
face-centred cubic
Identical atoms at
corners and at face
centres
Lattice type F
also Ag, Au, Al, Ni...
Primitive and Centred Lattices
-Iron is body-centred
cubic
Identical atoms at corners
and body centre (nothing at
face centres)
Lattice type I
from German, innenzentriert
Also Nb, Ta, Ba, Mo...
Primitive and Centred Lattices
Caesium Chloride
(CsCl) is primitive
cubic
Different atoms at
corners and body
centre. NOT body
centred, therefore.
Lattice type P
Also CuZn, CsBr, LiAg
Primitive and Centred Lattices
Sodium Chloride
(NaCl) - Na is much
smaller than Cs
Face Centred Cubic
Rocksalt structure
Lattice type F
Also NaF, KBr, MgO….
Another type of centring
Side centred unit cell
Notation:
A-centred if atom in bc plane
B-centred if atom in ac plane
C-centred if atom in ab plane
Unit cell contents
Counting the number of atoms within the unit cell
Many atoms are shared between unit cells
Unit cell contents
Counting the number of atoms within the unit cell
Thinking now in 3 dimensions, we can consider the
different positions of atoms as follows
Atoms
Shared Between: Each atom counts:
corner
8 cells
1/8
face centre
2 cells
1/2
body centre
1 cell
1
edge centre
4 cells
1/4
Question 4, handout
lattice type
P
I
F
C
cell contents
1
[=8 x 1/8]
2
[=(8 x 1/8) + (1 x 1)]
4
[=(8 x 1/8) + (6 x 1/2)]
2
[= 8 x 1/8) + (2 x 1/2)]
e.g. NaCl
Na at corners: (8  1/8) = 1
Na at face centres (6  1/2) = 3
Cl at edge centres (12  1/4) = 3
Cl at body centre = 1
Unit cell contents are 4(Na+Cl-)
Summary
 Crystals have symmetry
 Each unit cell shape has its own essential
symmetry
 In addition to the basic primitive lattice,
centred lattices also exist. Examples are
body centred (I) and face centred (F)