Density of States - Symmetry and Density of States

Symmetry and Density of States

There are a large variety of systems and types of states for which DOS calculations can be done. An important property of a condensed matter system is the symmetry of the structure on its microscopic scale. Fluids, glasses or amorphous solids have dispersion relations with a rotational symmetry. In spherically symmetric systems the integrals of functions, for instance, are one-dimensional because all variables in the calculation depend only on the radial parameter of the dispersion relation.

Angular dependent calculations or measurements on systems consisting of a single crystal of a compound, for example, are anisotropic, meaning the density of states will be different in one crystallographic direction than in another. Anisotropic problems are more difficult to calculate, and the anisotropic density of states is more difficult to visualize, so methods such as calculating the DoS for particular points or directions only, or calculating the projected density of states (PDOS), are often used.

Measurements on powders or polycrystalline samples require evaluation and calculation functions and integrals over the whole domain, most often a Brillouin zone, of the dispersion relations the system of interest. Sometimes the symmetry of the system is high. The shape of the functions describing the dispersion relations of the system appears many times over the whole domain of the dispersion relation. In such cases the effort to calculate the DOS can be reduced by a great amount when the calculation is limited to a reduced zone or fundamental domain. The Brillouin zone of the FCC lattice in the figure on the right has the 48 fold symmetry of the point group O_h with full octahedral symmetry. This means that the integration over the whole domain of the Brillouin zone can be reduced to a 48-th part of the whole Brillouin zone. As a crystal structure periodic table shows, there are many elements with a FCC crystal structure, like Diamond, Silicon and Platinum and their Brillouin zones and dispersion relations have this 48 fold symmetry.

Two other familiar crystal structures are the BCC and HCP structures with cubic and hexagonal lattices. The BCC structure has the 24 fold pyritohedral symmetry of the point group T_h. The HCP structure has the 12 fold prismatic dihedral symmetry of the point group D_3h. A complete list of symmetry properties of a point group can be found in point group character tables.

In general it is easier to calculate a DOS when the symmetry of the system is higher and the number of topological dimensions of the dispersion relation is lower. The DOS of dispersion relations with rotational symmetry can often be calculated analytically. This is fortunate, since many materials of practical interest, such as steel and silicon, have high symmetry.