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ECEN 5005 Crystals, Nanocrystals and Device Applications Class 10 Application of Group Theory to Crystals Crystal Symmetry Operators Crystallographic Point Groups 1

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Page 1: Crystal Symmetry Operators Crystallographic Point …ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005...Crystal Symmetry Operators • As defined in Class 1, a crystal is a periodic

ECEN 5005

Crystals, Nanocrystals and Device Applications

Class 10

Application of Group Theory to Crystals

• Crystal Symmetry Operators

• Crystallographic Point Groups

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Page 2: Crystal Symmetry Operators Crystallographic Point …ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005...Crystal Symmetry Operators • As defined in Class 1, a crystal is a periodic

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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Page 3: Crystal Symmetry Operators Crystallographic Point …ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005...Crystal Symmetry Operators • As defined in Class 1, a crystal is a periodic

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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Page 4: Crystal Symmetry Operators Crystallographic Point …ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005...Crystal Symmetry Operators • As defined in Class 1, a crystal is a periodic

Successive Symmetry Operations

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Page 5: Crystal Symmetry Operators Crystallographic Point …ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005...Crystal Symmetry Operators • As defined in Class 1, a crystal is a periodic

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.

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Page 6: Crystal Symmetry Operators Crystallographic Point …ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005...Crystal Symmetry Operators • As defined in Class 1, a crystal is a periodic

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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Page 7: Crystal Symmetry Operators Crystallographic Point …ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005...Crystal Symmetry Operators • As defined in Class 1, a crystal is a periodic

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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Page 8: Crystal Symmetry Operators Crystallographic Point …ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005...Crystal Symmetry Operators • As defined in Class 1, a crystal is a periodic

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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Page 9: Crystal Symmetry Operators Crystallographic Point …ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005...Crystal Symmetry Operators • As defined in Class 1, a crystal is a periodic

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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Page 10: Crystal Symmetry Operators Crystallographic Point …ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005...Crystal Symmetry Operators • As defined in Class 1, a crystal is a periodic

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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Page 11: Crystal Symmetry Operators Crystallographic Point …ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005...Crystal Symmetry Operators • As defined in Class 1, a crystal is a periodic

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.

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Page 12: Crystal Symmetry Operators Crystallographic Point …ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005...Crystal Symmetry Operators • As defined in Class 1, a crystal is a periodic

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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Page 13: Crystal Symmetry Operators Crystallographic Point …ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005...Crystal Symmetry Operators • As defined in Class 1, a crystal is a periodic

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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