group theory - michigan state university...elementary solid state physics, by m. ali omar (addison...
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Group Theory
Symmetry
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Improper Axis of Rotation
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Improper Axis of Rotation
Elements of Point Symmetry
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Diagram to generate Point
Group
One-Dimensional Symmetry
The seven Classes of one-dimensional symmetry ( adapted from I. Hargittai and G. Lengyel, J. Chem., Educ., 1984, 61, 1033.)
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One-Dimensional Symmetry
The seven Classes of one-dimensional symmetry ( adapted from I. Hargittai and G. Lengyel, J. Chem., Educ., 1984, 61, 1033.)
Translation
One-Dimensional Symmetry
The seven Classes of one-dimensional symmetry ( adapted from I. Hargittai and G. Lengyel, J. Chem., Educ., 1984, 61, 1033.)
Translation
Mirror Plane along
Translation
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One-Dimensional Symmetry
The seven Classes of one-dimensional symmetry ( adapted from I. Hargittai and G. Lengyel, J. Chem., Educ., 1984, 61, 1033.)
Translation
Mirror Plane along
Translation
2-Fold on the Line of
Translation
Generates 2-fold
via Translation
One-Dimensional Symmetry
The seven Classes of one-dimensional symmetry ( adapted from I. Hargittai and G. Lengyel, J. Chem., Educ., 1984, 61, 1033.)
Translation
Mirror Plane along
Translation
2-Fold on the Line of
Translation
Generates 2-fold
via Translation
Transverse Mirror
Line
Second Mirror
generated by
Translation
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The seven Classes of one-dimensional symmetry ( adapted from I. Hargittai and G. Lengyel, J. Chem., Educ., 1984, 61, 1033.)
One-Dimensional Symmetry
Transverse
Mirror and
translational
Mirror
Generates
2-fold at Mirror
Intersection
Does not matter order in which you place the symmetry elements.
The seven Classes of one-dimensional symmetry ( adapted from I. Hargittai and G. Lengyel, J. Chem., Educ., 1984, 61, 1033.)
One-Dimensional Symmetry
Transverse
Mirror and
translational
Mirror
Generates
2-fold at Mirror
Intersection
Does not matter order in which you place the symmetry elements.
Glide Reflection
Reflection followed by ½ unit translation
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The seven Classes of one-dimensional symmetry ( adapted from I. Hargittai and G. Lengyel, J. Chem., Educ., 1984, 61, 1033.)
One-Dimensional Symmetry
Transverse
Mirror and
translational
Mirror
Generates
2-fold at Mirror
Intersection
Does not matter order in which you place the symmetry elements.
Glide Reflection
1)Unit Translation 2) Transverse Mirror 3) 2-Fold 4) Glide Reflection
Remember Generated Symmetry
Chart Style Determination of 1-D Symmetry Groups
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Example of 1-D
Example of 1-D
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Concept of a Lattice in 2-Dimensional2 directions to build an array
Built by translations of a
certain Unit in a certain
direction!
Concept of a Lattice in 2-Dimensional2 directions to build an array
LatticeLet a Dot represent each
Position where an Object
is Found.
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Concept of a Lattice in 2-Dimensional2 directions to build an array
Lattice
Infinite number of ways to generate
a lattice
1) Two shortest Vectors
2) Angle γ
Lattice is not a physical thing, it is simply an abstraction, a collection of points
where on real objects may be placed.
-Infinite array
of identical
points
2D-Lattices
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2D-LatticesOblique Lattice
γArbitrary
2D-Lattices
Primitive
Rectangle
a≠b
γ = 90°
a
b
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2D-Lattices
a
bSquare
a = b
γ= 90°
2D-Latticesa
b a = b
γ = arbitrary
Redefine
Rectangle
a ≠ b
γ = 90°
Centered Lattice
a
b
We Prefer 90° because Sine and Cosine are simply 1 and 0.
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2D-Lattices
a
bHexagonal
a = b
γ = 60° or 120°
Symmetry of 2-D Lattices
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Rotation axis limitations
ℓ must be an integer value of a
Therefore must be 0,1, or 1/2
Other Centered Lattices in 2-D
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Other Centered Lattices in 2-D
●
●
●
●Add a center
Produce a smaller denser
Primitive Lattice
Other Centered Lattices in 2-D
Add a center
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Other Centered Lattices in 2-D
●
●
● ●
Add a center
Produce a smaller
denser Lattice with no
change in symmetry
● ●
● ●
●
Other Centered Lattices in 2-D
Add a center
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Other Centered Lattices in 2-D
●
●
● ●Add a center
Destroys the
symmetry of the
Hexagonal cell and
lowers symmetry.● ●
● ●
● ●
2-D Space Group
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2-D Space Group
2-D Space group
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2-D Space Group
2-D Space Group
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2-D Symmetry Diagrams
2-Fold Axis
2-D Symmetry Diagrams
Mirror
plane
Glide plane
Caused by
Lattice
Centering
Glide
plane
2-Fold Axis formed by 2-mirror
planes intersection
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2-D Symmetry Diagrams
2-D Symmetry Diagrams
4-fold
2-fold
generated
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2-D Symmetry Diagrams
2-D Space Group
Determination Walk Through
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2-D Example
2-D Example
4-Fold axisUnit Cell
Mirror
Mirror
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2-D Example
Glide plane
Glide plane
4-Fold axisUnit Cell
Mirror
Mirror
2-Fold axis
2-D Example
Glide plane
Glide plane
4-Fold axisUnit Cell
Mirror
Mirror
2-Fold axis
P4gm
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M.C. Escher
• http://escher.epfl.ch/escher/
• Escher Sketch was originally created for the purpose of designing periodic decorations.
• Use as a Teaching tool, the Web version was created.
3-Dimensional SymmetryTransitional Effects and Angle Between them
a≠b≠c
α ≠ β ≠ γ
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3-Dimensional SymmetryTransitional Effects and Angle Between them
a≠b≠c
α ≠ β ≠ γ
a≠b≠c
α ≠ γ ≠ 90°
β = 90°
3-D Lattices Orthorhombic
a≠b≠c
α = γ = β = 90°
І centered,
object in center of
cell
F centered,
object in center of
all Faces of cell
Centered,
object in center of
a face of cell.
A , B , C
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3-D Lattices Orthorhombic
a≠b≠c
α = γ = β = 90°
І centered,
object in center of
cell
F centered,
object in center of
all Faces of cell
Centered,
object in center of
a face of cell.
A , B , C
3-D Lattices Orthorhombic
a≠b≠c
α = γ = β = 90°
І centered,
object in center of
cell
F centered,
object in center of
all Faces of cell
Centered,
object in center of
a face of cell.
A , B , C
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3-D Lattices Orthorhombic
a≠b≠c
α = γ = β = 90°
І centered,
object in center of
cell
F centered,
object in center of
all Faces of cell
Centered,
object in center of
a face of cell.
A , B , C
3-Dimensional SymmetryTransitional Effects and Angle Between them
a=b≠c
α = γ = β = 90°
a=b=c
α = γ = β = 90°
F centered,
object in center of
all Faces of cell
І centered,
object in center of
cell
І centered,
object in center of
cell
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3-Dimensional SymmetryTransitional Effects and Angle Between them
a=b≠c
α = γ = β = 90°
a=b=c
α = γ = β = 90°
F centered,
object in center of
all Faces of cell
І centered,
object in center of
cell
І centered,
object in center of
cell
3-Dimensional SymmetryTransitional Effects and Angle Between them
a=b≠c
α = γ = β = 90°
a=b=c
α = γ = β = 90°
F centered,
object in center of
all Faces of cell
І centered,
object in center of
cell
І centered,
object in center of
cell
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3-Dimensional SymmetryTransitional Effects and Angle Between them
a=b≠c
α = γ = β = 90°
a=b=c
α = γ = β = 90°
F centered,
object in center of
all Faces of cell
І centered,
object in center of
cell
І centered,
object in center of
cell
3-Dimensional SymmetryTransitional Effects and Angle Between them
a=b≠c
α = γ = β = 90°
a=b=c
α = γ = β = 90°
F centered,
object in center of
all Faces of cell
І centered,
object in center of
cell
І centered,
object in center of
cell
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3-Dimensional SymmetryTransitional Effects and Angle Between them
І centered,
object in center of
cell
a=b≠c
α = β = 90°
γ = 120°
Trigonal-Hexagonal
3-Dimensional SymmetryTransitional Effects and Angle Between them
І centered,
object in center of
cell
a=b≠c
α = β = 90°
γ = 120°
Cubic
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Properties of 3D Lattices
The Bravais Lattices Song• Walter F. Smith 1-22-2002
The Bravais Lattices Song
Walter F. Smith 1-22-02
If you have to fill a volume with a structure that’s repetitive,
Just keep your wits about you, you don’t need to take a sedative!
Don’t freeze with indecision, there’s no need for you to bust a seam!
Although the options may seem endless, really there are just fourteen!
There’s cubic, orthorhombic, monoclinic, and tetragonal,
There’s trigonal, triclinic, and then finally hexagonal!
There’s only seven families, but kindly set your mind at ease—
‘Cause four have sub-varieties, so there’s no improprieties!
(Chorus:
‘Cause four have sub-varieties, so there’s no improprieties.
‘Cause four have sub-varieties, so there’s no improprieties.
‘Cause four have sub-varieties, so there’s no impropri-e, prieties!)
These seven crystal systems form the fourteen Bravais lattices.
They’ve hardly anything to do with artichokes or radishes –
They’re great for metals, minerals, conductors of the semi-kind –
The Bravais lattices describe all objects that are crystalline!
The cubic is the most important one in my “exparience”,
It comes in simple and in face- and body-centered variants.
And next in line’s tetragonal, it’s not at all diagonal,
Just squished in one dimension, so it’s really quite rectagonal!
The orthorhombic system has one less degree of symmetry
Because an extra squish ensures that a not equals b or c.
If angle gamma isn’t square, the side lengths give the “sig-o-nal”
For monoclinic if they’re different, or, if equal, trigonal!
(Chorus (reprovingly):
Of course for trigonal, recall that alpha, beta, gamma all
Are angles that are equal but don’t equal ninety, tut, tut, tut!
Are angles that are equal but don’t equal ninety, tut, tut, tut, tut tut!)
If you squish the lattice up in every way that is conceivable,
You’ll get the least amount of symmetry that is achievable –
It’s called triclinic, then remains the one that really self explains –
Hexagonal gives us no pains, and so we now may rest our brains!
Element songFigure from
Elementary Solid State Physics,
by M. Ali Omar (Addison Wesley, 1993)
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14 Bravais Lattice
32 Point Groups
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Glide Planes
Glide Planes
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Screw Axis
Screw Axis
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3-D Symmetry Diagrams
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3-D Symmetry Diagrams
General point
Comma represents object
is inverted
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Space Group
Diagram
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Grid
Grid Divisions
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2-D Miller Indices
Define a set of planes that divide the lattice.
2-D Miller Indices
1
1-1 plane
1
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2-D Miller Indices
1
3
3
3-1 Plane
1
1-1 plane
1
2-D Miller Indices
1
3
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3-1 Plane
1 -2
1-(-2) plane1-1 plane
1
1
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3-D Miller Indices
Seven Crystal
Systems
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3-D Centering
Triclinic and Monoclinic
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Orthorhombic
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31 Screw Axis
Mirror Plane
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Inversion Screw Axis
41 Screw Axis
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Symmetry Space
Groups Relationships
3-D Diagram for Space
group equivalent positions
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Matrix Symmetry Operations
Symmetry Operations and Matrices
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P 21/c
Symmetry Operations Must Close
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Seven Crystal Systems
3-D Space Group Symbols
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Monoclinic Example
Orthorhombic Examples
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Name the Space Group
• Primitive
– 2-fold on a-axis
– 2(1) on b-axis and on c-axis
– b-glide on c-axis
– c-glide on b-axis
Name the Space Group• Primitive
– 2(1) on b-axis
– c-glide on b-axis
• C Centered
– 2 on b-axis
– c-glide on b-axis
• Primitive
– C-glide on a-axis
– c-glide on b-axis
– 2-fold on c-axis
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Examples of generation of equivalent positions.
• http://img.chem.ucl.ac.uk/sgp/large/sgp.htm
• http://www.uwgb.edu/dutchs/SYMMETRY/3dSpaceGrps/3dspgrp.htm
• http://homepage.univie.ac.at/nikos.pinotsis/spacegroup.html
• http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list
Diagrams
Matrix Method ī 0 0
0 ī 0
0 0 ī
ī 0 0
0 ι 0
0 0 ī
ī 0 0
0 ι 0
0 0 ι
ī 0 0
0 ī 0
0 0 ι
ι 0 0
0 ι 0
0 0 ī
ι 0 0
0 ī 0
0 0 ī
ι 0 0
0 ī 0
0 0 ι
-x
-y
-z
x
-y
-z
-x
y
-z
x
y
-z
x
-y
z
-x
y
z
-x
-y
z
x
y
z
2-foldMirror plane
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Matrix Method
ī 0 0
0 ι 0
0 0 ι
ι 0 0
0 ι 0
0 0 ī
ι 0 0
0 ī 0
0 0 ι
x
y
-z
x
-y
z
-x
y
z
glide plane
ī 0 0
0 ι 0
0 0 ī
ī 0 0
0 ī 0
0 0 ι
ι 0 0
0 ī 0
0 0 ī
x
-y
-z
-x
y
-z
x
-y
z
x
y
z
2(1) screw axis
0
0
½
0
0
½
0
0
½
0
0
½
Example use of Matrix Method
• P2(1)
• Pbca
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Bragg’s Law
d-spacing and Bragg’s law
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d-spacing and Bragg’s law
Remember, Volume of Cell is not simple if angles are not 90 degrees.
Example Calculate d-spacing
Wavelength of Mo = 0.7103
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Reciprocal Space
Bragg’s Law
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Page 2 International
Systematic Absences
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Missing Diffraction Lines
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Lattice Centering
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Space Group Determination examples
• Orthorhombic
0 0 l l=2n+1
• Orthorhombic
0 k l l=2n+1
h 0 l l=2n+1
h k 0 h+k=2n+1
h 0 0 h=2n+1
0 k 0 k=2n+1
0 0 l l=2n+1
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Generate hkl using space group Lattice symmetry
• Monoclinic
• Triclinic
• Orthorhombic
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