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 face-

centred, 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 fromDr. 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

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 How symmetrical is a face?

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)°

We talk about the symmetry operation (rotation) about a symmetry element (rotation axis)

? Think of examples for n=2,3,4,5,6…

Rotational symmetry

n = 2 n = 5 n = 6

360/2 360/5 360/6

180o 72o 60o

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 axesTOTAL = 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 Cella = b c ; = = = 90

c < a, b c > a, b

elongated / squashed cube

Reduction in symmetry

Cubic TetragonalThree 4-axes 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

System Essential Symmetry Symmetry axes

Cubic 4 3-fold axes along the body diagonals

Tetragonal 1 4-fold axis parallel to c, in the centre of ab

Orthorhombic 3 mirrors or 3 2-fold axes perpendicular to each other

Hexagonal 1 6-fold axis down c

Trigonal (R) 1 3-fold axis down the long diagonal

Monoclinic 1 2-fold axis down the “unique” axis

Triclinic no symmetry

Essential symmetry is that which defines the crystal system (i.e. is unique to that shape).

Cubic Unit Cell

a=b=c, ===90

a

c

b

Many examples of cubic unit cells:

e.g. NaCl, CsCl, ZnS, CaF2, BaTiO3

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 LatticesCaesium 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

Atoms Shared Between: Each atom counts:corner 8 cells 1/8face centre 2 cells 1/2body centre 1 cell 1edge centre 4 cells 1/4

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

Question 4, handout

lattice type cell contentsP 1 [=8 x 1/8]IFC

e.g. NaClNa at corners: (8 1/8) = 1 Na at face centres (6 1/2) = 3Cl at edge centres (12 1/4) = 3 Cl at body centre = 1

Unit cell contents are 4(Na+Cl-)

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)]

SummarySummary

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)