Question

In a certain crystal the unit cell can be taken as six identical atoms lying at the corners of a regular octahedron. Convince yourself that these atoms can also be considered as lying at the centres of the faces of a cube and hence that the crystal has cubic symmetry. Use this result to prove that the conductivity tensor for the crystal, \(\sigma_{i j}\), must be isotropic.

Short answer

The six atoms form both an octahedron and lie at cube face centers, confirming cubic symmetry. Cubic symmetry implies that the conductivity tensor \( \sigma_{ij} \) must be isotropic.

Step-by-step solution

- Visualize the Octahedron

Imagine the six atoms positioned at the corners of a regular octahedron. An octahedron has 8 faces and 6 vertices.

- Identify Atoms at Cube Faces

Recognize that the vertices of a regular octahedron can be mapped onto the centers of the faces of a cube. Each face of a cube can be associated with one of the 6 vertices of the octahedron.

- Confirm Cubic Symmetry

To prove cubic symmetry, show that rotating or reflecting the cube leaves the octahedron's positions unchanged. The cube's symmetry operations map face centers to other face centers, preserving the arrangement of the atoms.

- Apply to Conductivity Tensor

Use the cubic symmetry to argue that the conductivity tensor must behave the same under these symmetry operations. For a tensor to be consistent with cubic symmetry, it must have the same value in all directions, meaning it is isotropic.

- Proof of Isotropy

Denote the conductivity tensor by \( \sigma_{ij} \). Any tensor invariant under all rotations and reflections of the cube must have the same value for all components. Hence, \ \sigma_{ij} = \sigma \delta_{ij} \, highlighting the isotropic nature.

Key concepts

Unit Cell

A unit cell is the smallest repeating structure of a crystal that, when repeated in all directions, forms the entire crystal lattice. Imagine it as the building block of the crystal structure.
In our crystal, the unit cell consists of six identical atoms positioned at the corners of a regular octahedron. This simple unit cell can recreate the entire crystal structure by repeating in three-dimensional space.

Regular Octahedron

A regular octahedron is a polyhedron with eight equilateral triangular faces, six vertices, and twelve edges. It's one of the five Platonic solids, known for its symmetry properties.
In the given problem, six atoms lie at the corners of this octahedron. Interestingly, these vertices of the octahedron can also be mapped to the centers of the faces of a cube. This dual representation showcases the versatile geometric nature of the octahedron.

Conductivity Tensor

A conductivity tensor \( \sigma_{ij} \ \) is a mathematical object describing how electrical current flows within a material in response to an electric field. It's a key concept in understanding how materials conduct electricity.
For our crystal, because of its cubic symmetry, the conductivity tensor simplifies. Instead of having different values in different directions, the symmetry dictates that the conductivity is the same in every direction. This means the tensor is isotropic.

Cubic Symmetry

Cubic symmetry refers to a crystal's symmetrical properties where the symmetry operations are those of a cube. These include rotations and reflections that leave the cube's appearance unchanged.
The six atoms of our crystal can also be thought of as being at the centers of the faces of a cube. This means the crystal possesses cubic symmetry. This symmetry is vital because it imposes restrictions on physical properties like conductivity, requiring them to be the same in all directions.

Isotropic Properties

Isotropic properties are those which are identical in all directions. When a property like conductivity is isotropic, it means that the material conducts electricity equally well in all directions.
Due to the cubic symmetry of our crystal, the conductivity tensor \( \sigma_{ij} \ \) must be isotropic. This isotropy ensures that no matter how you rotate or reflect the crystal, its electrical conductivity remains unchanged. Mathematically, this is captured by stating \( \sigma_{ij} = \sigma \delta_{ij} \ \), where \ \sigma \ is a constant and \ \delta_{ij} \ is the Kronecker delta.