Objectives By the end of this section you should: be able to recognise rotational symmetry and...
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Transcript of Objectives By the end of this section you should: be able to recognise rotational symmetry and...
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)