Post on 27-Apr-2021
12/21/2020
1
Group Theory
Symmetry
1
2
12/21/2020
2
3
4
12/21/2020
3
Improper Axis of Rotation
5
6
12/21/2020
4
Improper Axis of Rotation
Elements of Point Symmetry
7
8
12/21/2020
5
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.)
9
10
12/21/2020
6
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
11
12
12/21/2020
7
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
13
14
12/21/2020
8
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
15
16
12/21/2020
9
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
17
18
12/21/2020
10
Example of 1-D
Example of 1-D
19
20
12/21/2020
11
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.
21
22
12/21/2020
12
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
23
24
12/21/2020
13
2D-LatticesOblique Lattice
γArbitrary
2D-Lattices
Primitive
Rectangle
a≠b
γ = 90°
a
b
25
26
12/21/2020
14
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.
27
28
12/21/2020
15
2D-Lattices
a
bHexagonal
a = b
γ = 60° or 120°
Symmetry of 2-D Lattices
29
30
12/21/2020
16
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
31
32
12/21/2020
17
Other Centered Lattices in 2-D
●
●
●
●Add a center
Produce a smaller denser
Primitive Lattice
Other Centered Lattices in 2-D
Add a center
33
34
12/21/2020
18
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
35
36
12/21/2020
19
Other Centered Lattices in 2-D
●
●
● ●Add a center
Destroys the
symmetry of the
Hexagonal cell and
lowers symmetry.● ●
● ●
● ●
2-D Space Group
37
38
12/21/2020
20
2-D Space Group
2-D Space group
39
40
12/21/2020
21
2-D Space Group
2-D Space Group
41
42
12/21/2020
22
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
43
44
12/21/2020
23
2-D Symmetry Diagrams
2-D Symmetry Diagrams
4-fold
2-fold
generated
45
46
12/21/2020
24
2-D Symmetry Diagrams
2-D Space Group
Determination Walk Through
47
48
12/21/2020
25
49
50
12/21/2020
26
2-D Example
2-D Example
4-Fold axisUnit Cell
Mirror
Mirror
51
52
12/21/2020
27
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
53
54
12/21/2020
28
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
55
56
12/21/2020
29
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
57
58
12/21/2020
30
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
59
60
12/21/2020
31
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
61
62
12/21/2020
32
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
63
64
12/21/2020
33
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
65
66
12/21/2020
34
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
67
68
12/21/2020
35
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)
69
70
12/21/2020
36
14 Bravais Lattice
32 Point Groups
71
72
12/21/2020
37
Glide Planes
Glide Planes
73
74
12/21/2020
38
Screw Axis
Screw Axis
75
76
12/21/2020
39
3-D Symmetry Diagrams
77
78
12/21/2020
40
3-D Symmetry Diagrams
General point
Comma represents object
is inverted
79
80
12/21/2020
41
Space Group
Diagram
81
82
12/21/2020
42
Grid
Grid Divisions
83
84
12/21/2020
43
2-D Miller Indices
Define a set of planes that divide the lattice.
2-D Miller Indices
1
1-1 plane
1
85
86
12/21/2020
44
2-D Miller Indices
1
3
3
3-1 Plane
1
1-1 plane
1
2-D Miller Indices
1
3
13
3-1 Plane
1 -2
1-(-2) plane1-1 plane
1
1
87
88
12/21/2020
45
3-D Miller Indices
Seven Crystal
Systems
89
90
12/21/2020
46
3-D Centering
Triclinic and Monoclinic
91
92
12/21/2020
47
Orthorhombic
93
94
12/21/2020
48
31 Screw Axis
Mirror Plane
95
96
12/21/2020
49
Inversion Screw Axis
41 Screw Axis
97
98
12/21/2020
50
Symmetry Space
Groups Relationships
3-D Diagram for Space
group equivalent positions
99
100
12/21/2020
51
Matrix Symmetry Operations
Symmetry Operations and Matrices
101
102
12/21/2020
52
P 21/c
Symmetry Operations Must Close
103
104
12/21/2020
53
Seven Crystal Systems
3-D Space Group Symbols
105
106
12/21/2020
54
Monoclinic Example
Orthorhombic Examples
107
108
12/21/2020
55
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
109
110
12/21/2020
56
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
111
112
12/21/2020
57
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
113
114
12/21/2020
58
Bragg’s Law
d-spacing and Bragg’s law
115
116
12/21/2020
59
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
117
118
12/21/2020
60
Reciprocal Space
Bragg’s Law
119
120
12/21/2020
61
Page 2 International
Systematic Absences
121
122
12/21/2020
62
Missing Diffraction Lines
123
124
12/21/2020
63
Lattice Centering
125
126
12/21/2020
64
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
127
128
12/21/2020
65
Generate hkl using space group Lattice symmetry
• Monoclinic
• Triclinic
• Orthorhombic
129