Group symmetry and the 32 Point Groups Shyue Ping Ong Department of NanoEngineering University of California, San Diego
Group symmetry and the 32 Point GroupsShyue Ping OngDepartment of NanoEngineeringUniversity of California, San Diego
An excursion into group theory
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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.
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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.
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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?
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a
b
e 4 42 43
e e 4 42 43
4 4 42 43 e42 42 43 e 443 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.
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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 32y 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.
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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?
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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.
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Derivation of the 32 3D-Crystallographic
Point Groups
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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.
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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:
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Principal directions in a cube
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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?
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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?
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n2(2) Dn[ ]
Rotations + Inversion¡ Notation:
¡ We have already derived these in the previous lecture on symmetry operations
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=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.
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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?
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nm Cnv[ ]
n m
Roto-inversions + Coinciding Reflection
¡Notation:
¡Can you show that ?
¡Generators: Inversion rotation + mirror plane
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nm 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 vs shorthand symbol:
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n / m m (m) Dnh[ ]
2m2m2m≡mmm
4m2m2m≡4mmm
6m2m2m≡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:
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n1n2 T[ ] or O[ ]
Adding reflection to n1n2
¡ Notation:
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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.
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Interpreting the full Hermann-Mauguin symbols¡O and Oh – Cubic System
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Primary
Secondary
Tertiary
432
4m3 2m
Interpreting the full Hermann-Mauguin symbols¡6mm and 6/mmm – Hexagonal System
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6mm
6mmm
Primary
SecondaryTertiary
SecondaryTertiary
Top view
Group-subgroup relations
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NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 5
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Linear?
Contains two or more unique C3
axes?
Contains an inversion center?
Containstwo or more
unique C5 axes?
C∞v
Yes
No
D∞h
No
Yes
Containsan inversion
center?Ih
Yes
I
Containstwo or more
unique C4 axes?
Contains one or more reflection
planes?
No
Containsan inversion
center?
Containsan inversion
center?
Yes
YesOh
T
No
No
Th
YesYes
Yes
No
No
O
Yes
Td
No
No
NANO106 Handout 4Flowchart for Point Group Determination
Continued onnext page
Red: Non-crystallographic point groupsGreen: Crystallographic point groups
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Yes
No
Containsa proper rotation
axis (Cn)?
Identify thehighest Cn. Are
there n ⟂C2 axes?
Containsa reflection
plane?
Containsan inversion
center?
C1
No
No
Contains ahorizontal
reflection plane ⟂to Cn axis (σh)?
Yes
Contains ndihedral (between
C2) reflection planes (σd)?
Dnh
Contains ahorizontal
reflection plane ⟂to Cn axis (σh)?
No
Yes
No
Dnd
Dn
No
Yes
Contains avertical reflection
plane (σv)?Cnv
No
Yes
Contains a2n-fold improper
rotation axis?
Cn
S2n
Yes
No No
Cnh
Yes
Cs
Ci
Yes
Yes
From previouspage
Molecular Point Group
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NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 5
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Methane (CH4)
SF5Cl
CO2BF3
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
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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.
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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.
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The 14 generator matrices
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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.
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Blackboard
Blackboard
Simple example: mmm¡ Using the generator matrices, we can now generate the 8
symmetry operations in this point group.
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1 0 00 1 00 0 −1
"
#
$$$
%
&
'''=m1
1 0 00 −1 00 0 1
"
#
$$$
%
&
'''=m2
−1 0 00 1 00 0 1
"
#
$$$
%
&
'''=m3
−1 0 00 1 00 0 −1
"
#
$$$
%
&
'''=m1 ⋅m3 = 2y
−1 0 00 −1 00 0 1
"
#
$$$
%
&
'''=m1 ⋅m2 = 2z
1 0 00 −1 00 0 −1
"
#
$$$
%
&
'''=m2 ⋅m3 = 2x
−1 0 00 −1 00 0 −1
"
#
$$$
%
&
'''=m1 ⋅m2 ⋅m3 = i
1 0 00 1 00 0 1
"
#
$$$
%
&
'''=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 im2 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%20Group.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
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