UCSD NANO106 - 05 - Group Symmetry and the 32 Point Groups
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About This Presentation
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
Slide Content
Group symmetry and the 32 Point Groups
Shyue Ping Ong
Department of NanoEngineering
University of California, San Diego
Shyue Ping Ong
Department of NanoEngineering
University of California, San Diego
An excursion into
group theory
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
2
group theory
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
2
Definition
In mathematics, a group is a set of elements
together with an operation that combines any two
of its elements to form a third element satisfying
four conditions called the group axioms, namely
closure, associativity, identity and invertibility.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
3
In mathematics, a group is a set of elements
together with an operation that combines any two
of its elements to form a third element satisfying
four conditions called the group axioms, namely
closure, associativity, identity and invertibility.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
3
Group axioms
¡Closure
¡For all a, b in G, the result of the operation, a • b, is also in G.
¡Associativity
¡For all a, b and c in G, (a • b) • c = a • (b • c).
¡Identity
¡There exists an element e in G, such that for every element a in G, the
equation e • a = a • e = a holds.
¡Invertibility
¡For each a in G, there exists an element b in G such that a • b = b • a =
e, where e is the identity element.
¡(optional) Commutativity.
¡a • b = b • a . Groups satisfying this property are known as Abelianor
commutative groups.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
4
¡Closure
¡For all a, b in G, the result of the operation, a • b, is also in G.
¡Associativity
¡For all a, b and c in G, (a • b) • c = a • (b • c).
¡Identity
¡There exists an element e in G, such that for every element a in G, the
equation e • a = a • e = a holds.
¡Invertibility
¡For each a in G, there exists an element b in G such that a • b = b • a =
e, where e is the identity element.
¡(optional) Commutativity.
¡a • b = b • a . Groups satisfying this property are known as Abelianor
commutative groups.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
4
A very simple symmetry example
¡Let’s consider a simple 4-fold rotation axis.
We can construct a full multiplication table
(Cayleytable) for this set of symmetry
operations as follows:
¡How do you identify the inverse of each
member?
¡Is this group Abelian?
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
5
a
b
e 4 4
2
4
3
e e 4 4
2
4
3
4 4 4
2
4
3
e
4
2
4
2
4
3
e 4
4
3
4
3
e 4 4
2
¡Let’s consider a simple 4-fold rotation axis.
We can construct a full multiplication table
(Cayleytable) for this set of symmetry
operations as follows:
¡How do you identify the inverse of each
member?
¡Is this group Abelian?
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
5
a
b
e 4 4
2
4
3
e e 4 4
2
4
3
4 4 4
2
4
3
e
4
2
4
2
4
3
e 4
4
3
4
3
e 4 4
2
More complicated example
¡Quartz has configuration 223, i.e., it has a 2-fold
rotation axis and a 3-fold rotation axis that are
mutually perpendicular. As a consequence of
Euler’s theorem, the 2
uand 2
yrotations are
automatically determined by the combination of
2
xand 3.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
6
32 (D
3) e 3 3
2
2
x 2
y 2
u
e e 3 3
2
2
x 2
y 2
u
3 3 3
2
e 2
u 2
x 2
y
3
2
3
2
e 3 2
y 2
u 2
x
2
x 2
x 2
y 2
u e 3
2
3
2
y 2
y 2
u 2x 3 e 3
2
2
u 2
u 2
x 2
y 3
2
3 e
¡Quartz has configuration 223, i.e., it has a 2-fold
rotation axis and a 3-fold rotation axis that are
mutually perpendicular. As a consequence of
Euler’s theorem, the 2
uand 2
yrotations are
automatically determined by the combination of
2
xand 3.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
6
32 (D
3) e 3 3
2
2
x 2
y 2
u
e e 3 3
2
2
x 2
y 2
u
3 3 3
2
e 2
u 2
x 2
y
3
2
3
2
e 3 2
y 2
u 2
x
2
x 2
x 2
y 2
u e 3
2
3
2
y 2
y 2
u 2x 3 e 3
2
2
u 2
u 2
x 2
y 3
2
3 e
Properties of a group
¡Order: # of elements in group
¡Isomorphism: 1-1 mapping between two groups
¡Homomorphous groups: Two groups are homomorphous if there
exists a unidirectional correspondence between them.
¡Cyclic groups: A group is cyclic if there is an element Osuch that
successive powers of Ogenerates all the elements in the group.Ois
then called the generating element.
¡Group generators: The minimal set of elements from which all group
elements can be constructed.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
7
¡Order: # of elements in group
¡Isomorphism: 1-1 mapping between two groups
¡Homomorphous groups: Two groups are homomorphous if there
exists a unidirectional correspondence between them.
¡Cyclic groups: A group is cyclic if there is an element Osuch that
successive powers of Ogenerates all the elements in the group.Ois
then called the generating element.
¡Group generators: The minimal set of elements from which all group
elements can be constructed.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
7
Subgroups and supergroups
¡If a subset of elements of a group form a group, this set
is called a subgroup of , and is denoted as
¡Note that the identity is always a subgroup of all groups.
¡If the subgroup is not the identity or itself, it is known
as a proper subgroup.
¡Can you identify all the subgroups in the quartz example?
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
8
Gk⊂GGkG
G
¡If a subset of elements of a group form a group, this set
is called a subgroup of , and is denoted as
¡Note that the identity is always a subgroup of all groups.
¡If the subgroup is not the identity or itself, it is known
as a proper subgroup.
¡Can you identify all the subgroups in the quartz example?
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
8
Gk⊂GGkG
G
Tips for Symmetry Table Construction
¡All symmetry operations must appear once, and
only once in each row and column (think of a
Sudoku table)
¡This means that once you get the table partially
filled, you can already work out the rest of the
table using the above constraints without doing
matrix multiplications.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
9
¡All symmetry operations must appear once, and
only once in each row and column (think of a
Sudoku table)
¡This means that once you get the table partially
filled, you can already work out the rest of the
table using the above constraints without doing
matrix multiplications.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
9
Derivation of the 32
3D-Crystallographic
Point Groups
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
10
3D-Crystallographic
Point Groups
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
10
Preliminaries
¡Crystallographic Point Groups:
¡Crystallographic –Only symmetries compatible with crystals, i.e., for
rotations, only 1, 2, 3, 4 and 6-fold.
¡Point: Symmetries intersect at a common origin, which is invariant
under all symmetry operations.
¡Group: Satisfy group axioms of closure, associativity, identity and
invertibility.
¡All point groups will be presented as:
¡We just saw our first point group!
¡C
1or identity point group.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
11
31 more to go….
¡Crystallographic Point Groups:
¡Crystallographic –Only symmetries compatible with crystals, i.e., for
rotations, only 1, 2, 3, 4 and 6-fold.
¡Point: Symmetries intersect at a common origin, which is invariant
under all symmetry operations.
¡Group: Satisfy group axioms of closure, associativity, identity and
invertibility.
¡All point groups will be presented as:
¡We just saw our first point group!
¡C
1or identity point group.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
11
31 more to go….
Notation and Principal Directions
¡The International or Hermann-Mauguinnotation for point groups
comprise of at most three symbols, which corresponds to the
symmetry observed in a particular principal direction. The principal
directions for each of the Bravaiscrystal systems are given below:
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
12
¡The International or Hermann-Mauguinnotation for point groups
comprise of at most three symbols, which corresponds to the
symmetry observed in a particular principal direction. The principal
directions for each of the Bravaiscrystal systems are given below:
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
12
Principal directions in a cube
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
13
Primary
Secondary
Tertiary
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
13
Primary
Secondary
Tertiary
Proper rotations
¡Notation:
¡All cyclic groups of order n (hence the “C”).
¡Principal directions given by directions of monoclinic,
trigonal, tetragonal and hexagonal systems respectively.
¡What is the generating element?
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
14
n Cn[]
¡Notation:
¡All cyclic groups of order n (hence the “C”).
¡Principal directions given by directions of monoclinic,
trigonal, tetragonal and hexagonal systems respectively.
¡What is the generating element?
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
14
n Cn[]
Dihedral groups
¡Notation:
¡Earlier, we derived the possible combinations of rotation axes. One
set of possible rotation combination contains a 2-fold rotation axis
perpendicular to another rotation axis.
¡How many unique 2-fold rotations are there in each group?
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
15n2(2) Dn[]
¡Notation:
¡Earlier, we derived the possible combinations of rotation axes. One
set of possible rotation combination contains a 2-fold rotation axis
perpendicular to another rotation axis.
¡How many unique 2-fold rotations are there in each group?
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
15n2(2) Dn[]
Rotations + Inversion
¡Notation:
¡We have already derived these in the previous lecture on
symmetry operations
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
16
=3/m
n Sn[]
¡Notation:
¡We have already derived these in the previous lecture on
symmetry operations
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
16
=3/m
n Sn[]
Rotation + Perpendicular Reflections
¡Notation:
¡mand 3/mare already derived in previous slide.
¡The /m notation indicates that the mirror is perpendicular
to the rotation axis.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
17
n/m Cnh[]
¡Notation:
¡mand 3/mare already derived in previous slide.
¡The /m notation indicates that the mirror is perpendicular
to the rotation axis.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
17
n/m Cnh[]
Rotations + Coinciding Reflection
¡Notation:
¡Note that the coincidence of a mirror with a n-fold rotation
implies the existence of another mirror that is at angle π/n
to the original mirror plane.
¡Generating elements are and
¡Which mirror planes are related by symmetry?
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
18nm Cnv[]
nm
¡Notation:
¡Note that the coincidence of a mirror with a n-fold rotation
implies the existence of another mirror that is at angle π/n
to the original mirror plane.
¡Generating elements are and
¡Which mirror planes are related by symmetry?
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
18nm Cnv[]
nm
Roto-inversions + Coinciding
Reflection
¡Notation:
¡Can you show that ?
¡Generators: Inversion rotation +
mirror plane
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
19nm Dnd[]1m ≡2/m2m ≡mm2
Reflection
¡Notation:
¡Can you show that ?
¡Generators: Inversion rotation +
mirror plane
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
19nm Dnd[]1m ≡2/m2m ≡mm2
Rotations with Coinciding and
Perpendicular Reflections
¡Notation:
¡Only even rotations result in new groups.
¡Exercise: What do the 1 and 3-fold rotations lead to
when we add coinciding and perpendicular mirror
planes?
¡Full vsshorthand symbol:
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
20n/m m (m) Dnh[]
2m2m2m≡mmm4m2m2m≡4mmm6m2m2m≡6mmm
n-fold rotation axes omitted if the
rotation axis can be
unambiguously obtained from the
combination of symmetry
elements presented in the
symbol.
Perpendicular Reflections
¡Notation:
¡Only even rotations result in new groups.
¡Exercise: What do the 1 and 3-fold rotations lead to
when we add coinciding and perpendicular mirror
planes?
¡Full vsshorthand symbol:
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
20n/m m (m) Dnh[]
2m2m2m≡mmm4m2m2m≡4mmm6m2m2m≡6mmm
n-fold rotation axes omitted if the
rotation axis can be
unambiguously obtained from the
combination of symmetry
elements presented in the
symbol.
Combination of Proper Rotations (not at right
angles)
¡Notation:
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
21n1n2 T[] or O[]
angles)
¡Notation:
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
21n1n2 T[] or O[]
Adding reflection to n
1n
2
¡Notation:
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
22n1n2 Td[], Th[]or Oh[]
Adding mirror
plane to 2-fold
rotation axes of
23.
Adding mirror
plane to 3-fold
rotation axes of
23.
Adding inversion
or mirror to 432.
1n
2
¡Notation:
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
22n1n2 Td[], Th[]or Oh[]
Adding mirror
plane to 2-fold
rotation axes of
23.
Adding mirror
plane to 3-fold
rotation axes of
23.
Adding inversion
or mirror to 432.
Laue Classes
¡Only 11 of the 32 point groups are
centrosymmetric, i.e., contains an
inversion center. All other non-
centrosymmetricpoint groups are
subgroups of these 11. Each row is
called a Laue class.
¡Polar point groups are groups that
have at least one direction that has
no symmetrically equivalent
directions. Can only happen in non-
centrosymmetricpoint groups in
which there is at most a single
rotation axis (1,2, 3, 4, 6, m, mm2,
3m, 4mm, 6mm) –basically all
single rotation axes + coinciding
mirror planes.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
23
¡Only 11 of the 32 point groups are
centrosymmetric, i.e., contains an
inversion center. All other non-
centrosymmetricpoint groups are
subgroups of these 11. Each row is
called a Laue class.
¡Polar point groups are groups that
have at least one direction that has
no symmetrically equivalent
directions. Can only happen in non-
centrosymmetricpoint groups in
which there is at most a single
rotation axis (1,2, 3, 4, 6, m, mm2,
3m, 4mm, 6mm) –basically all
single rotation axes + coinciding
mirror planes.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
23
Interpreting the full Hermann-
Mauguin symbols
¡O and O
h–Cubic System
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
24
Primary
Secondary
Tertiary
432
4m32m
Mauguin symbols
¡O and O
h–Cubic System
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
24
Primary
Secondary
Tertiary
432
4m32m
Interpreting the full Hermann-
Mauguin symbols
¡6mm and 6/mmm –Hexagonal System
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
25
6mm
6mmm
Primary
Secondary
Tertiary
Secondary
Tertiary
Top view
Mauguin symbols
¡6mm and 6/mmm –Hexagonal System
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
25
6mm
6mmm
Primary
Secondary
Tertiary
Secondary
Tertiary
Top view
Group-subgroup relations
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
26
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
26
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
27
Molecular
Structure
Is the molecule
linear?
Does the
molecule contain
two or more unique
C
3
axes?
No
Does the
molecule contain an
inversion center?
Yes
D
!h
C
!v
Yes
No
Does the
molecule contain
two or more unique
C
5
axes?
Does the
molecule contain
two or more unique
C
4
axes?
No
Does the
molecule contain an
inversion center?
Yes
I
hYes
I
No
Does the
molecule contain an
inversion center?
O
h
O
Yes Yes
No
Does the molecule
contain one or more
reflection planes?
Does the
molecule contain an
inversion center?
T
hYes Yes
T
dT
No
No No
Does the
molecule contain a
proper rotation axis
(C
n
)?
No
Does the
molecule contain a
reflection plane?
Does the
molecule contain an
inversion center?
Identify the
highest order C
n
.
Are there n perpen-
dicular C
2
axes?
Does the
molecule contain a
horizontal reflection
plane ("
h
)?
Yes Yes
Does the
molecule contain n
dihedral reflection
planes ("
d
)?
D
nd
D
n
D
nh
Does the
molecule contain a
horizontal reflection
plane ("
h
)?
C
s
C
i
C
1
Does the
molecule contain n
vertical reflection
planes ("
v
)?
Does the
molecule contain a
2n-fold improper
rotation axis?
C
nh
C
nv
S
2n
C
n
No
No
Yes Yes
Yes
No
No
Yes
Yes
No
No
Yes
Yes
No
No
No
Yes
27
Molecular
Structure
Is the molecule
linear?
Does the
molecule contain
two or more unique
C
3
axes?
No
Does the
molecule contain an
inversion center?
Yes
D
!h
C
!v
Yes
No
Does the
molecule contain
two or more unique
C
5
axes?
Does the
molecule contain
two or more unique
C
4
axes?
No
Does the
molecule contain an
inversion center?
Yes
I
hYes
I
No
Does the
molecule contain an
inversion center?
O
h
O
Yes Yes
No
Does the molecule
contain one or more
reflection planes?
Does the
molecule contain an
inversion center?
T
hYes Yes
T
dT
No
No No
Does the
molecule contain a
proper rotation axis
(C
n
)?
No
Does the
molecule contain a
reflection plane?
Does the
molecule contain an
inversion center?
Identify the
highest order C
n
.
Are there n perpen-
dicular C
2
axes?
Does the
molecule contain a
horizontal reflection
plane ("
h
)?
Yes Yes
Does the
molecule contain n
dihedral reflection
planes ("
d
)?
D
nd
D
n
D
nh
Does the
molecule contain a
horizontal reflection
plane ("
h
)?
C
s
C
i
C
1
Does the
molecule contain n
vertical reflection
planes ("
v
)?
Does the
molecule contain a
2n-fold improper
rotation axis?
C
nh
C
nv
S
2n
C
n
No
No
Yes Yes
Yes
No
No
Yes
Yes
No
No
Yes
Yes
No
No
No
Yes
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
28
Molecular
Structure
Is the molecule
linear?
Does the
molecule contain
two or more unique
C
3
axes?
No
Does the
molecule contain an
inversion center?
Yes
D
!h
C
!v
Yes
No
Does the
molecule contain
two or more unique
C
5
axes?
Does the
molecule contain
two or more unique
C
4
axes?
No
Does the
molecule contain an
inversion center?
Yes
I
hYes
I
No
Does the
molecule contain an
inversion center?
O
h
O
Yes Yes
No
Does the molecule
contain one or more
reflection planes?
Does the
molecule contain an
inversion center?
T
hYes Yes
T
dT
No
No No
Does the
molecule contain a
proper rotation axis
(C
n
)?
No
Does the
molecule contain a
reflection plane?
Does the
molecule contain an
inversion center?
Identify the
highest order C
n
.
Are there n perpen-
dicular C
2
axes?
Does the
molecule contain a
horizontal reflection
plane ("
h
)?
Yes Yes
Does the
molecule contain n
dihedral reflection
planes ("
d
)?
D
nd
D
n
D
nh
Does the
molecule contain a
horizontal reflection
plane ("
h
)?
C
s
C
i
C
1
Does the
molecule contain n
vertical reflection
planes ("
v
)?
Does the
molecule contain a
2n-fold improper
rotation axis?
C
nh
C
nv
S
2n
C
n
No
No
Yes Yes
Yes
No
No
Yes
Yes
No
No
Yes
Yes
No
No
No
Yes
28
Molecular
Structure
Is the molecule
linear?
Does the
molecule contain
two or more unique
C
3
axes?
No
Does the
molecule contain an
inversion center?
Yes
D
!h
C
!v
Yes
No
Does the
molecule contain
two or more unique
C
5
axes?
Does the
molecule contain
two or more unique
C
4
axes?
No
Does the
molecule contain an
inversion center?
Yes
I
hYes
I
No
Does the
molecule contain an
inversion center?
O
h
O
Yes Yes
No
Does the molecule
contain one or more
reflection planes?
Does the
molecule contain an
inversion center?
T
hYes Yes
T
dT
No
No No
Does the
molecule contain a
proper rotation axis
(C
n
)?
No
Does the
molecule contain a
reflection plane?
Does the
molecule contain an
inversion center?
Identify the
highest order C
n
.
Are there n perpen-
dicular C
2
axes?
Does the
molecule contain a
horizontal reflection
plane ("
h
)?
Yes Yes
Does the
molecule contain n
dihedral reflection
planes ("
d
)?
D
nd
D
n
D
nh
Does the
molecule contain a
horizontal reflection
plane ("
h
)?
C
s
C
i
C
1
Does the
molecule contain n
vertical reflection
planes ("
v
)?
Does the
molecule contain a
2n-fold improper
rotation axis?
C
nh
C
nv
S
2n
C
n
No
No
Yes Yes
Yes
No
No
Yes
Yes
No
No
Yes
Yes
No
No
No
Yes
Molecular Point
Group
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
29
Group
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
29
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
30
Ethylene (H
2CCH
2)
Methane (CH
4)
SF
5Cl
CO
2
BF
3
PF
6
30
Ethylene (H
2CCH
2)
Methane (CH
4)
SF
5Cl
CO
2
BF
3
PF
6
Practicing point group
determination
¡Set of molecule xyz files with different point groups are
provided at
https://github.com/materialsvirtuallab/nano106/tree/master
/lectures/molecules
¡Other online resources
¡https://www.staff.ncl.ac.uk/j.p.goss/symmetry/Molecules_pov.html
¡http://csi.chemie.tu-darmstadt.de/ak/immel/misc/oc-
scripts/symmetry.html
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
31
determination
¡Set of molecule xyz files with different point groups are
provided at
https://github.com/materialsvirtuallab/nano106/tree/master
/lectures/molecules
¡Other online resources
¡https://www.staff.ncl.ac.uk/j.p.goss/symmetry/Molecules_pov.html
¡http://csi.chemie.tu-darmstadt.de/ak/immel/misc/oc-
scripts/symmetry.html
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
31
Matrix representations of Point Groups
¡As we have seen earlier, all symmetry operations can be represented
as matrices.
¡As point symmetry operations do not have translation, we only need
3x3 matrices to represent these operations (homogenous coordinates
are needed only to include translation operations).
¡We have also seen how working in crystal reference frame simplifies
the symmetry operation matrices considerably, and can be obtained
simply by inspecting how the crystal basis vectors transform under the
symmetry operation.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
32
¡As we have seen earlier, all symmetry operations can be represented
as matrices.
¡As point symmetry operations do not have translation, we only need
3x3 matrices to represent these operations (homogenous coordinates
are needed only to include translation operations).
¡We have also seen how working in crystal reference frame simplifies
the symmetry operation matrices considerably, and can be obtained
simply by inspecting how the crystal basis vectors transform under the
symmetry operation.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
32
Generator matrices
¡From the group multiplication tables, we know that all
symmetry elements in a group can be obtained as the
product of other elements.
¡The minimum set of symmetry operators that are needed
to generate the complete set of symmetry operations in
the point group are known as the generators.
¡All point groups can be generated from a subset of the 14
fundamental generator matrices.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
33
¡From the group multiplication tables, we know that all
symmetry elements in a group can be obtained as the
product of other elements.
¡The minimum set of symmetry operators that are needed
to generate the complete set of symmetry operations in
the point group are known as the generators.
¡All point groups can be generated from a subset of the 14
fundamental generator matrices.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
33
The 14
generator
matrices
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
34
generator
matrices
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
34
Simple example: mmm
¡Consider the mmm point group with order 8. Let’s choose the
three mirror planes as the generators (note that these are not the
same as the ones from the 14 generator matrices! I am choosing
these to illustrate how you can derive these from first principles).
What are the generator matrices?
¡Using the generator matrices, we can now generate the 8
symmetry operations in this point group.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
35
Blackboard
Blackboard
¡Consider the mmm point group with order 8. Let’s choose the
three mirror planes as the generators (note that these are not the
same as the ones from the 14 generator matrices! I am choosing
these to illustrate how you can derive these from first principles).
What are the generator matrices?
¡Using the generator matrices, we can now generate the 8
symmetry operations in this point group.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
35
Blackboard
Blackboard
Simple example: mmm
¡Using the generator matrices, we can now generate the 8
symmetry operations in this point group.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
3610001000−1"#$$$ %&'''=m11000−10001"#$$$ %&'''=m2−100010001"#$$$ %&'''=m3−10001000−1"#$$$ %&'''=m1⋅m3=2y−1000−10001"#$$$ %&'''=m1⋅m2=2z1000−1000−1"#$$$ %&'''=m2⋅m3=2x−1000−1000−1"#$$$ %&'''=m1⋅m2⋅m3=i100010001"#$$$ %&'''=m1⋅m2⋅m3⋅m1⋅m2⋅m3=i⋅i=E
E im
1m
2m
32
x2
y2
z
E E im
1m
2m
32
x2
y2
z
i i E 2
z2
y2
xm
3m
2m
1
m
1m
12
zE 2
x2
ym
2m
3 i
m
2m
22
y2
xE 2
zm
1 im
2
m
3m
32
x2
y2
zE im
1m
3
2
x2
xm
3m
2m
1 i E 2
z2
y
2
y2
ym
2m
3 im
12
zE 2
x
2
z2
zm
1 im
2m
32
y2
xE
http://nbviewer.ipython.org/github/materialsvirtuallab/nano106/blob/master/lectures/lecture_4_point_
group_symmetry/Symmetry%20Computations%20on%20mmm%20%28D_2h%29%20Point% 20Gro
up.ipynb
¡Using the generator matrices, we can now generate the 8
symmetry operations in this point group.
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
3610001000−1"#$$$ %&'''=m11000−10001"#$$$ %&'''=m2−100010001"#$$$ %&'''=m3−10001000−1"#$$$ %&'''=m1⋅m3=2y−1000−10001"#$$$ %&'''=m1⋅m2=2z1000−1000−1"#$$$ %&'''=m2⋅m3=2x−1000−1000−1"#$$$ %&'''=m1⋅m2⋅m3=i100010001"#$$$ %&'''=m1⋅m2⋅m3⋅m1⋅m2⋅m3=i⋅i=E
E im
1m
2m
32
x2
y2
z
E E im
1m
2m
32
x2
y2
z
i i E 2
z2
y2
xm
3m
2m
1
m
1m
12
zE 2
x2
ym
2m
3 i
m
2m
22
y2
xE 2
zm
1 im
2
m
3m
32
x2
y2
zE im
1m
3
2
x2
xm
3m
2m
1 i E 2
z2
y
2
y2
ym
2m
3 im
12
zE 2
x
2
z2
zm
1 im
2m
32
y2
xE
http://nbviewer.ipython.org/github/materialsvirtuallab/nano106/blob/master/lectures/lecture_4_point_
group_symmetry/Symmetry%20Computations%20on%20mmm%20%28D_2h%29%20Point% 20Gro
up.ipynb
Procedure for constructing a symmetry
multiplication table
¡Identify point group
¡Identify compatible crystal system (if not
provided)
¡Align symmetry elements with crystal axes
¡Derive a set of minimal symmetry matrices
¡Iteratively multiply to get all the symmetry
matrices
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
37
multiplication table
¡Identify point group
¡Identify compatible crystal system (if not
provided)
¡Align symmetry elements with crystal axes
¡Derive a set of minimal symmetry matrices
¡Iteratively multiply to get all the symmetry
matrices
NANO 106 -Crystallography of Materials by Shyue Ping Ong -Lecture 5
37
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