ECEN 5005
Crystals, Nanocrystals and Device Applications
Class 10
Application of Group Theory to Crystals
• Crystal Symmetry Operators
• Crystallographic Point Groups
1
Crystal Symmetry Operators
• As defined in Class 1, a crystal is a periodic array of unit cells (that
may contain more than one atom) in such a way that it is invariant
under lattice translations by
T = n1a1 + n2a2 + n3a3
where n1 , n2 and n3 are integers and a1 , a2 and a3 are the primitive
unit vectors that define the unit cell of the crystal.
• As also discussed in Class 1, there are other symmetry operations that
leave the crystal invariant, such as rotation, reflection and inversion.
• The complete set of symmetry operations for a crystal is called the
space group.
- There are 230 possible space groups in total.
• If we set all translation elements in the space group equal to zero,
then we obtain the point group.
- The elements of point groups are those operations that have a point
(usually called the origin) fixed, which is why the group is called
the point group.
- There are only 32 point groups that are consistent with the
translational symmetry.
- In molecules, there is no translational symmetry and thus there are
infinite number of possible point groups. However, the most
frequently occurring point groups are among the 32
crystallographic point groups.
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Crystallographic Point Groups
• There are three types of fundamental symmetry operations we build
our point groups with.
- Rotation about an axis through the origin.
- Reflection in a plane that contains the origin.
- Inversion about the origin.
- Although inversion is a special case of a rotation followed by
reflection, it appears so frequently in molecules and crystals that
we consider it as a fundamental operation.
• The group multiplication for a point group is obviously the successive
application of two symmetry operations. It is worth noting that:
- The product of two rotations is also a rotation.
- The product of two reflections is a rotation by an angle 2ϕAB about
the line of intersection between the two reflection planes where
ϕAB is the angle between the planes.
- The product of a rotation and a reflection in a plane A that contains
the axis of rotation O is a reflection in another plane B that passes
through the axis of rotation. The angle between the planes is half
the angle of rotation.
- The product of two rotations by 180o about mutually intersecting
axes u and v is a rotation about an axis perpendicular to both u and
v with the angle of rotation being twice the angle between u and v.
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Successive Symmetry Operations
4
Relations Between Symmetry Elements
• Commuting operations (Recall the special meanings of commuting
operators in the quantum-mechanical application of group theory.)
- Two rotations about the same axis
- Two reflections in perpendicular planes
- Two rotations by 180o about perpendicular axes
- A rotation and a reflection in a plane perpendicular to the axis of
rotation – improper rotation or rotoflection.
- The inversion with any rotation or reflection.
• General rules between symmetry elements
- The intersection between two reflection planes must be a
symmetry axis. If the angle between the planes is 180o/n, then the
axis is n-fold.
- If a reflection plane contains an n-fold symmetry axis, there must
be (n-1) other reflection planes at angles of 180o/n.
- Two 2-fold axes separated by an angle 180o/n requires a presence
of a perpendicular n-fold axis.
- A 2-fold axis and an n-fold axis perpendicular to it require (n-1)
additional 2-fold axes separated by angles of 180o/n.
- An even-fold axis, a reflection plane perpendicular to it, and an
inversion center are interdependent. That is, the presence of any
two implies the existence of the third.
5
Notational Convention
• Symmetry operations
- E = Identity operation
- Cn = Rotation by 360o/n. As mentioned in Class 1, the only values
n can take in solids are 1, 2, 3, 4, and 6.
- σ = reflection in a plane
- σh = reflection in a horizontal plane, i.e. the plane perpendicular to
the highest rotational symmetry axis.
- σv = reflection in a vertical plane, i.e. the plane containing the
highest rotational symmetry axis.
- σd = reflection in a diagonal plane, i.e. the plane containing the
highest rotational symmetry axis and bisecting the angle between
the two two-fold axes perpendicular to the highest symmetry axis.
This is just a special kind of σv.
- Sn = improper rotation by 360o/n.
- i = inversion.
• Point Groups
- rotation group = Cn
- rotation group with symmetry planes = Cnh , Cnv
- dihedral group = Dn
- dihedral group with symmetry planes = Dnh , Dnd
- cubic group = T, Td, Th, O, Oh
- continuous group = C∞v, D∞h
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The Crystallographic Point Groups
• The 32 crystallographic point groups can be categorized into two
classes, the simple rotation groups and the groups with higher
symmetry.
- The simple rotation groups contain one axis that has higher
symmetry than any other.
- The groups with higher symmetry have no unique axis of highest
symmetry but more than one n-fold axis (n > 2).
- We may also define a third kind that contains C∞ . These groups
form the point groups of linear molecules but are not consistent
with the translational symmetry of crystalline solids.
• We begin our enumeration of 32 point groups with the rotation
groups, Cn.
- These groups contain only one axis of n-fold symmetry.
- These groups are cyclic and Abelian.
- The group C6 , for example, consists of , 6C ( )326 CC = ,
, ( )236 CC = ( )1
346
−= CC , ( )16
56
−= CC and ( )E=66C .
• Rotation groups are visualized by stereographic projections, which
show rotation axes and marks that are subject to various symmetry
operations.
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Stereographic Projections for Rotation Groups
• Projection of marks (+ or O) on a unit sphere onto xy plane. + is used
for positions above the xy plane and O for below the plane.
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The Rotation Groups, Cn
• Now we will show that the only possible values of n consistent with
crystal’s translational symmetry are 1, 2, 3, 4, and 6.
• For this, let us consider a translation operation T that is in
perpendicular direction to the rotation axis.
• Such an operation can always be found by applying a rotation
operation, Cn , to an arbitrary translation operation,
, which must yield another allowed translation
operation, . Then, the translation operation,
332211 aaaT nnn ++=′
T ′′ TT ′′−′ , which must
also be a symmetry element, is perpendicular to the rotation axis.
• Then we can choose the shortest of such translation operations and
call it, R. If we apply the symmetry rotation, Cn , to R, the resultant
R’ must be an allowed translational symmetry operation with the
same magnitude as R, just rotated by 360o/n.
• Now consider another translation operation, R - R’. The length of R -
R’ is given by 2Rsin(π/n). But by definition, R is the shortest
translation perpendicular to the rotation axis. Therefore, we must
have 2Rsin(π/n) < R, or sin(π/n) < ½. Thus, we have n ≤ 6.
• Furthermore, n = 5 is excluded because it would yield |R + R’| < | R |.
• Thus, we proved that the only allowed values of n for rotation axes
that are consistent with the translational symmetry of crystalline
solids are 1, 2, 3, 4 and 6.
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Rotation Groups, Cnv and Cnh
• Cnv : These groups contain a vertical reflection plane, σv , in addition
to the rotation axis, Cn.
- By the rules we established between symmetry operations, this
means that there always exist n reflection planes, separated by
angles 180o/n around the Cn axis.
- The vertical reflection planes are represented by solid radial lines
in the stereographic projections.
- Possible values of n are 2, 3, 4, 6.
- n = 1 is not possible for obvious reasons.
- The Cnv group contains 2n operations, that is n rotations, ,
belonging to the group C
mnC
n, plus n reflections, vσ .
• Cnh : These groups contain a horizontal reflection plane, σh , in
addition to the rotation axis, Cn.
- The existence of a horizontal reflection plane is indicated by solid
line circle in the stereographic projections.
- These reflection operations take + into O.
- Note that these groups always include the inversion operation
when n is even.
- Possible values of n are 1, 2, 3, 4, 6.
- The Cnh group contains 2n operations, that is n rotations, ,
belonging to the group C
mnC
hσn, plus n improper rotations, . mnC
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Rotation Groups, Sn and Dn
• Sn : These groups contain an n-fold axis for improper rotations.
- For odd n, these groups are identical to Cnh.
- If n is even (i.e. n = 2, 4 or 6), then they form distinct groups.
- S2 is equivalent to simple inversion symmetry, consisting only
of E and i.
- S4 consists of 4 symmetry operations, E, S4, and . )( 242 SC = 3
4S
- S6 consists of 6 symmetry operations, E, S6, , ,
and .
)( 263 SC = )( 3
6Si =
)( 46
23 SC = )( 1
656
23
−== SSC hσ
- Note that S6 is the direct-product group of C3 and S2.
• Dn (dihedral group): These groups contain a vertical Cn axis plus n
horizontal C2 axes intersecting at angles of π/n.
- The groups Dn have no symmetry planes.
- D1 is equivalent to C2.
- D2 consists of three mutually orthogonal C2 axes, plus E. This
group is often referred to as Vieregruppe.
- For odd n, all horizontal axes are equivalent.
- For even n, however, not all horizontal axes are equivalent but they
form two distinct classes of equivalent rotations.
11
Finite Group of Order 4
• Finite group of order 1: Obviously, there can be only one kind, that is
a group consisting solely of the identity element, E.
• Finite group of order 2: Again, there can be only one kind, that is a
group consisting the identity element, E, and another element A which
satisfies A2 = E.
- Physical example is the group S2 which consists of E and i (i2 = E).
- Another example is C2, consisting of E and C2 ( ). EC =22
• Finite group of order 3: Start with E and A and add another element
B. In order to form a group, the only possibility is the cyclic group
consisting of A, A2 (=B), and A3 (=E).
- The rotation group C3 is an example.
• Finite group of order 4: We now start having two distinct kinds.
- A natural extension of previous arguments gives a cyclic group,
consisting of A, A2, A3 and A4 (=E).
- Examples for this kind of group are C4 and S4.
- However, we also have another kind of group defined by the
multiplication table below.
E A B C E E A B C A A E C B B B C E A C C B A E
- This unique kind of finite group of order 4 is called Vieregruppe.
- D2 is a physical example of Vieregruppe.
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Rotation Groups, Dnd and Dnh
• Dnd : These groups contain all the elements of Dn plus diagonal
reflection planes, σd , which bisect the angles between the two-fold
axes that are perpendicular to the main rotation axis, Cn.
- Only D2d and D3d form distinct groups.
- Dnd has twice as many elements as Dn, consisting of 2n elements
from Dn plus n diagonal reflection planes, σd, and n improper
rotations achieved by successive application of horizontal 2-fold
rotation and reflection in the diagonal plane, dC σ2′ .
• Dnh : These groups contain all elements of Dn plus a horizontal
reflection plane σh.
- Dnh also has twice as many elements as Dn.
- Dnh is a direct-product group of Dn and C1h, the group consisting of
the identity and the reflection in the horizontal plane.
- Dnh is also a direct-product group of Dn and S2, the inversion
group.
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