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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