Wigner–Seitz radius

The Wigner–Seitz radius r_s, named after Eugene Wigner and Frederick Seitz, is the radius of a sphere whose volume is equal to the mean volume per atom in a solid. This parameter is used frequently in condensed matter physics to describe the density of a system.

Formula

In a 3-D system with N particles in a volume V, the Wigner-Seitz radius is defined by

$\frac{4}{3} \pi r_s^3 = \frac{V}{N}.$

Solving for r_s we obtain

$r_s = \left(\frac{3}{4\pi n}\right)^{1/3}\,,$

where n is the particle density of the valence electrons.

For a non-interacting system, the average separation between two particles will be 2r_s. The radius can also be calculated as

$r_s= \left(\frac{3M}{4\pi \rho N_A}\right)^\frac{1}{3}\,,$

where M is molar mass, ρ is mass density, and N_A is the Avogadro number.

This parameter is normally reported in atomic units, i.e., in units of the Bohr radius.

See also

• Wigner–Seitz cell