ECEN 5005

Crystals, Nanocrystals and Device Applications

Class 11

Group Theory For Crystals

• Point Groups with Higher Symmetry

• Crystal Systems

Point Groups with Higher Symmetries

• So far, we discussed the 27 groups that can be classified as simple

rotation group because in all of them we can find a main rotation axis.

We now complete our enumeration of the 32 crystallographic point

groups by discussing the remaining 5 groups that involve higher

symmetry.

• These groups, T, Td, Th, O and Oh, have no unique axis of highest

symmetry but have more than one axis which is of at least three-fold

symmetry.

- In order to make all those rotation axes consistent with the

translational symmetry, the crystal must belong to the cubic crystal

system.

- That is, the fundamental translation vectors of the crystal are

mutually perpendicular and of equal length.

- Naturally, these groups are collectively called the cubic groups and

it is the most convenient to associate all these 5 groups with a

cube.

- The cubic groups can be further categorized into the tetrahedral

groups (T, Td and Th) and the octahedral groups (O and Oh).

- The tetrahedral groups can be associated with a tetrahedron and the

octahedral groups with an octahedron.

The Tetrahedral Groups

• T : This is the smallest group among the cubic groups and consists of

12 rotation operations that take a regular tetrahedron into itself.

- These rotations are easily visualized by considering a tetrahedron

inscribed in a cube.

- The symmetry elements are

the identity operation, E,

3 two-fold rotations, , about the x, y, z axes

and 8 three-fold rotations ( and ) about the cube diagonals.

2C

3C 23C

- The 8 three-fold rotations form two distinct classes representing

clockwise and counter-clockwise rotations by 120o, respectively.

The Tetrahedral Groups

• Td (full tetrahedral group): The full tetrahedral group contains all

symmetry operations of a regular tetrahedron including reflections.

- This is the symmetry group for zinc blende crystals, for example.

- Td has a total of 24 elements.

- In addition to the 12 elements belonging to T, it has 6 diagonal

reflection planes normal to a cube face and passing through a

tetrahedral edge.

- It also contains 6 four-fold improper rotations, S4, about the x, y,

and z axes (positive and negative).

• Th : This group is formed by taking a direct-product of T and S2, the

inversion group consisting of E and i.

- This group is also of order 24.

- The nature of all elements are easily seen by the definition of

direct-product, that is, 12 elements from the group T and additional

12 by multiplying with the inversion operator, i.

- Note that a regular tetrahedron does not have Th symmetry because

it lacks the inversion symmetry.

The Octahedral Groups

• O : This is perhaps the most important point group.

- The octahedral group, O, consists of all proper rotations that bring

a regular octahedron into itself.

- It has 24 elements:

The identity element, E,

8 three-fold axes along the body diagonals of the cube,

6 four-fold axes along x, y, z axes,

3 two-fold axes along x, y, z axes,

and finally 6 two-fold axes through the origin and parallel to the

face diagonal.

Full Octahedral Group

• Oh : Consisting of 48 elements, the full octahedral group, Oh, is the

largest of all 32 crystallographic point groups.

- Oh can be formed by taking a direct product of O and S2, the

inversion group (E and i).

- This group represents the full symmetry of a cube or an octahedron

including improper rotations and reflections.

Continuous Groups

• We just completed our enumeration of all 32 crystallographic point

groups. But for completeness, we add two continuous groups,

and that represent the axial symmetry of linear molecules.

vC∞

hD∞

• : This is a group for a general linear molecule which has a

vertical symmetry axis.

vC∞

- It has full rotation symmetry about the molecular axis, thus this

group contains infinite number of rotation operations.

- It also has reflection symmetry in any vertical plane containing the

molecular axis, thus the group contains infinite number of vertical

reflection planes, as well.

• : This group contains a horizontal reflection plane passing

through the center of the molecule.

hD∞

- It also has two-fold rotation symmetry about any axes in the

horizontal plane.

- The above two automatically imply that this group contains

inversion symmetry.

- This group represents the symmetry of homonuclear diatomic

molecules or a symmetric linear molecule like CO2.

Crystal Systems

• Labeling convention for a unit cell

a

c

b

αβ

γ

a

c

b

αβ

γ

• Triclinic crystals

Crystal System Unit Cell Groups # of Symmetry Elements

Triclinic cba ≠≠ C1 1 γβα ≠≠ S2 (Ci) 2

Crystal Systems

• Monoclinic crystals

Crystal System Unit Cell Groups # of Symmetry Elements

Monoclinic cba ≠≠ C1h 2 βπγα ≠== 2/ C2 2 C2h 4

Crystal Systems

• Orthorhombic crystals

Crystal System Unit Cell Groups # of Symmetry Elements

Orthorhombic cba ≠≠ C2v 4 2/πγβα === D2 (V) 4 D2h (Vh) 8

\

Crystal Systems

• Tetragonal crystals

Crystal System Unit Cell Groups # of Symmetry Elements

Tetragonal cba ≠= C4 4 2/πγβα === S4 4 C4h 8 D2d (Vd) 8 C4v 8 D4 8 D4h 16

Crystal Systems

• Rhombohedral crystals

Crystal System Unit Cell Groups # of Symmetry Elements

Rhombohedral cba == C3 3 (Trigonal) 2/3/2 ππγβα ≠<== S6 (C3i) 6

C3v 6 D3 6 D3v 12 D4 8

• Hexagonal crystals

Crystal System Unit Cell Groups # of Symmetry Elements

Hexagonal cba ≠= C3h 6 3/2,2/ πγπβα === C6 6 C6h 12 D3h 12 C6v 12 D6 12 D6h 24

Crystal Systems

• Cubic crystals

Crystal System Unit Cell Groups # of Symmetry Elements

Cubic cba == T 12 2/πγβα === Th 24 Td 24 O 24 Oh 48