PHY.K02UF Molecular and Solid State Physics

Empty lattice approximation

The dispersion relation for (light/sound) in uniform medium is ω = c|k| where c is the speed of (light/sound) in this medium. The dispersion relationship forms a cone in k-space. When (light/sound) moves through a crystal, it diffracts at the Brillouin zone boundaries. If the speed of (light/sound) does not vary greatly throughout the crystal, the dispersion relation will be nearly a cone except near the Brillouin zone boundaries where it will bend to strike the boundaries at 90°. The dispersion relation in a reduced zone scheme can be approximated by placing the apex of a cone at every reciprocal lattice point, ω = c|k - G|.

Cross sections of this collections of cones are taken in the high symmetry directions of the Brillouin zone to produce the dispersion relation. The resulting (photonic/phononic) bandstructures for some crystals are ploted below in various high symmetry directions of k-space.


2-D square

2-D hexagonal

Simple cubic

Face centered cubic

Body centered cubic



Body Centered Tetragonal