The effects of crystal symmetry
Matter tensors abide by a fundamental postulate of crystal physics known as Neumann's Principle. This principle states that:
'the symmetry elements of any physical property of a crystal must include the symmetry elements of the point group of the crystal' .
As we know, the physical properties of crystals are described by tensors and the point group of a crystal is the set of its macroscopic symmetry elements such as rotation axes, mirror planes, and centres of symmetry.
Taken with the 7 different crystal systems, the possible combinations of symmetry elements gives rise to the 32 crystal classes.
This postulate essentially puts conditions on the form of matter tensors depending on the crystal symmetry - the tensors describing the matter property must be invariant under its symmetry operations.
Effects of symmetry on first rank matter tensors
- The vectors describing the matter property must be invariant under the symmetry operations. Straight away we see that any crystal with a centre of inversion cannot hold a first rank property since for a general vector
p, ( p_{1}, p_{2}, p_{3} ) ≠ ( −p_{1}, −p_{2}, −p_{3} ).
- We can also see with a little thought that if there is a rotation axis the vector property must lie along the rotation axis. An immediate consequence of this is that if the crystal structure has more than one rotation axis, then once again the crystal cannot possess the vector property since it cannot lie along two different rotation axes.
- If the crystal includes a mirror plane , the vector must lie within the plane. If there is more than one mirror plane, the vector must lie in the intersection.
- Finally if the crystal system contains a mirror plane and a rotation axes, the vector is non-zero only if the rotation axis is contained within the crystal plane.
The crystal classes that can possess a first rank matter tensor property, along with the number of independent components and the form of the vector are shown below.
Crystal system |
Crystal class |
Number of independent components |
Form of the vector |
Triclinic |
1 |
3 |
(p1, p2, p3) |
Monoclinic (diad axis parallel to x2) |
2 |
1 |
(0, p, 0) |
Monoclinic |
m |
2 |
(p1, 0, p3) |
Orthorhombic |
mm2 |
1 |
(0, 0, p) |
Tetragonal |
4 |
1 |
(0, 0, p) |
Tetragonal |
4mm |
1 |
(0, 0, p) |
Triagonal |
3 |
1 |
(0, 0, p) |
Triagonal |
3m |
1 |
(0, 0, p) |
Hexagonal |
6 |
1 |
(0, 0, p) |
Hexagonal |
6mm |
1 |
(0, 0, p) |
Effects of symmetry on second rank tensors
Straight away we see that properties relating to a second rank tensor are centrosymmetric since by inverting the vectors in the equation p_{i} = T_{ij}q_{j}, the same T_{ij} satisfy the equation. So although the crystal may not have a centre of inversion, the tensor property does.
The best way to consider the conditions imposed by the crystal systems is to consider the representation surface and expressing its axes relative to the crystallographic axes. We shall consider rotations only as it can be demonstrated that mirror symmetries are covered by the rotation results.
The general representation surface has 3 mutually perpendicular diads, three planes of symmetry perpendicular to the diad axes and is centrosymmetric.
- Triclinic - Since there are no symmetry elements not possessed by the general representation surface, there are no restrictions on its components and so stays at 6 independent components. These components contain information on the magnitude of the principal values and the 3 angles required to define the orientation of the quadric axes to the crystallographic axes.
- Monoclinic - A diad of the representation surface must be aligned with the diad of the crystal system. Apart from this the surface is free to take any orientation, and so its independent components contain information about the three principal values and the one angle required to orientate the 2 free axes to the crystallographic axes.
- Orthorhombic - The crystal system contains 3 mutually perpendicular diads. On aligning the surface with the crystallographic axes we find only the principal value information is required. This also holds true for the mmm class.
- Uniaxial systems (tetragonal, triagonal and hexagonal) - The only way for the representation surface to possess 3-, 4- or 6-fold rotation symmetry is to align a diad along the crystallographic direction and revolve around it. This results in only 2 independent components since 2 of the principal values must be equal.
- Cubic - The four triad axes of the cubic system force the surface to become a sphere and so only a single component is required to define it.
Crystal system |
Number of independent components |
Form of the tensor |
Triclinic |
6 |
\[\left( {\begin{array}{*{20}{c}} {{T_{11}}}&{{T_{12}}}&{{T_{13}}}\\ {{T_{21}}}&{{T_{22}}}&{{T_{23}}}\\ {{T_{31}}}&{{T_{32}}}&{{T_{33}}} \end{array}} \right)\] |
Monoclinic (diad axis parallel to x2) |
4 |
\[\left( {\begin{array}{*{20}{c}} {{T_{11}}}&0&{{T_{13}}}\\ 0&{{T_2}}&0\\ {{T_{31}}}&{{T_0}}&{{T_{33}}} \end{array}} \right)\] |
Orthorhombic |
3 |
\[\left( {\begin{array}{*{20}{c}} {{T_1}}&0&0\\ 0&{{T_2}}&0\\ 0&0&{{T_3}} \end{array}} \right)\] |
Tetragonal, Triagonal, Hexagonal |
2 |
\[\left( {\begin{array}{*{20}{c}} {{T_1}}&0&0\\ 0&{{T_1}}&0\\ 0&0&{{T_3}} \end{array}} \right)\] |
Cubic |
1 |
\[\left( {\begin{array}{*{20}{c}} T&0&0\\ 0&T&0\\ 0&0&T \end{array}} \right)\] |