UCSD NANO106 - 05 - Group Symmetry and the 32 Point Groups

Group symmetry and the 32 Point Groups
Shyue Ping Ong
Department of NanoEngineering
University of California, San Diego

An excursion into
group theory
NANO 106 - Crystallography ofMaterials 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.
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 Abelian or
commutative groups.
4

A very simple symmetry example
¡ Let’s consider a simple 4-fold rotation axis.
We can construct a full multiplication table
(Cayley table) for this set of symmetry
operations as follows:
¡ How do you identify the inverse of each
member?
¡ Is this group Abelian?
5
a
b
e 4 42 43
e e 4 42 43
4 4 42 43 e
42 42 43 e 4
43 43 e 4 42

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 2u and 2y rotations are
automatically determined by the combination of
2x and 3.
6
32 (D3) e 3 32 2x 2y 2u
e e 3 32 2x 2y 2u
3 3 32 e 2u 2x 2y
32 32 e 3 2y 2u 2x
2x 2x 2y 2u e 32 3
2y 2y 2u 2x 3 e 32
2u 2u 2x 2y 32 3 e
Important note: For
Cayley tables, each cell is
given by row.column, e.g.,
the element in the red box
on the right implies that
D(3)D(2x) = D(2u)

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 O such that
successive powers of O generates all the elements in the group. O is
then called the generating element.
¡ Group generators: The minimal set of elements from which all group
elements can be constructed.
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?
8
Gk ⊂ G
Gk
G
G

Tips for Symmetry Table Construction
¡ You can basically quickly fill up the cyclic subgroups parts
of the table because those are simply powers of a rotation
matrix.
¡ The inversion operation can be treated as simply equal to
-1 multiplied by any matrix, because D(i) = -E.
¡ 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.
9

Derivation of the 32
3D-Crystallographic
Point Groups
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!
¡ C1or identity point group.
11
31 more to go….

Notation and Principal Directions
¡ The International or Hermann-Mauguin notation 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 Bravais crystal systems are given below:
12

Principal directions in a cube
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?
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?
15
n2(2) Dn[ ]

Rotations + Inversion
¡ Notation:
¡ We have already derived these in the previous lecture on
symmetry operations
16
=3/m
n Sn[ ]

Rotation + Perpendicular Reflections
¡ Notation:
¡ m and 3/m are already derived in previous slide.
¡ The /m notation indicates that the mirror is perpendicular
to the rotation axis.
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?
18
nm Cnv[ ]
n m

Roto-inversions + Coinciding
Reflection
¡Notation:
¡Can you show that ?
¡Generators: Inversion rotation +
mirror plane
19
nm Dnd[ ]
1m ≡ 2 / m
2m ≡ 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 vs shorthand symbol:
20
n / m m (m) Dnh[ ]
2
m
2
m
2
m
≡ mmm
4
m
2
m
2
m
≡
4
m
mm
6
m
2
m
2
m
≡
6
m
mm
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:
21
n1n2 T[ ] or O[ ]

Adding reflection to n1n2
¡ Notation:
22
n1n2 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-
centrosymmetric point 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-
centrosymmetric point 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.
23

Interpreting the full Hermann-
Mauguin symbols
¡O and Oh – Cubic System
24
Primary
Secondary
Tertiary
432
4
m
3
2
m

Interpreting the full Hermann-
Mauguin symbols
¡6mm and 6/mmm – Hexagonal System
25
6mm
6
m
mm
Primary
Secondary
Tertiary
Secondary
Tertiary
Top view

Group-subgroup relations
26

27Molecule
Linear?
Contains two or
more unique C3
axes?
Contains an
inversion center?
Contains
two or more
unique C5 axes?
C∞v
Yes
No
D
∞h
No
Yes
Contains
an inversion
center?
Ih
Yes
I
Contains
two or more
unique C4 axes?
Contains one or
more reﬂection
planes?
No
Contains
an inversion
center?
Contains
an inversion
center?
Yes
Yes
Oh
T
No
No
Th
YesYes
Yes
No
No
O
Yes
Td
No
No
NANO106 Handout 4
Flowchart for Point Group Determination
Continued on
next page
Red: Non-crystallographic point groups
Green: Crystallographic point groups

28
Yes
No
Contains
a proper rotation
axis (Cn)?
Identify the
highest Cn. Are
there n ⟂
C2 axes?
Contains
a reflection
plane?
Contains
an inversion
center?
C1
No
No
Contains a
horizontal
reflection plane ⟂
to Cn axis (σh)?
Yes
Contains n
dihedral (between
C2) reflection
planes (σd)?
Dnh
Contains a
horizontal
reflection plane ⟂
to Cn axis (σh)?
No
Yes
No
Dnd
Dn
No
Yes
Contains a
vertical reflection
plane (σv)?
Cnv
No
Yes
Contains a
2n-fold improper
rotation axis?
Cn
S2n
Yes
No No
Cnh
Yes
Cs
Ci
Yes
Yes
From previous
page

Molecular Point
Group
29

30Ethylene (H2CCH2)
Methane (CH4)
SF5Cl
CO2
BF3
PF6

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
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.
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.
33

The 14
generator
matrices
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.
35
Blackboard
Blackboard

Simple example: mmm
¡ Using the generator matrices, we can now generate the 8
symmetry operations in this point group.
36
1 0 0
0 1 0
0 0 −1
"
#
$
$
$
%
&
'
'
'
= m1
1 0 0
0 −1 0
0 0 1
"
#
$
$
$
%
&
'
'
'
= m2
−1 0 0
0 1 0
0 0 1
"
#
$
$
$
%
&
'
'
'
= m3
−1 0 0
0 1 0
0 0 −1
"
#
$
$
$
%
&
'
'
'
= m1 ⋅m3 = 2y
−1 0 0
0 −1 0
0 0 1
"
#
$
$
$
%
&
'
'
'
= m1 ⋅m2 = 2z
1 0 0
0 −1 0
0 0 −1
"
#
$
$
$
%
&
'
'
'
= m2 ⋅m3 = 2x
−1 0 0
0 −1 0
0 0 −1
"
#
$
$
$
%
&
'
'
'
= m1 ⋅m2 ⋅m3 = i
1 0 0
0 1 0
0 0 1
"
#
$
$
$
%
&
'
'
'
= m1 ⋅m2 ⋅m3 ⋅m1 ⋅m2 ⋅m3 = i⋅i = E
E i m1 m2 m3 2x 2y 2z
E E i m1 m2 m3 2x 2y 2z
i i E 2z 2y 2x m3 m2 m1
m1 m1 2z E 2x 2y m2 m3 i
m2 m2 2y 2x E 2z m1 i m2
m3 m3 2x 2y 2z E i m1 m3
2x 2x m3 m2 m1 i E 2z 2y
2y 2y m2 m3 i m1 2z E 2x
2z 2z m1 i m2 m3 2y 2x E
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
37

UCSD NANO106 - 05 - Group Symmetry and the 32 Point Groups

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