Symmetries in Crystalline Solids

Classify the Bravais lattices and define the crystal lattice as a group of discrete translations.

Construct the reciprocal lattice and Brillouin zone from the translational symmetry group.

Derive Bloch's theorem from the translational symmetry of a periodic potential.

Define a space group and distinguish between symmorphic and non-symmorphic space groups.

Determine the irreducible representations of the point group at high-symmetry points in the Brillouin zone.

Apply time-reversal symmetry to derive constraints on electronic band structures, including Kramers' degeneracy.

Use group theory to determine selection rules for optical transitions between electronic bands.