The Wigner–Seitz radius r_{\rm 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 (for first group metals). In the more general case of metals having more valence electrons, r_{\rm s} is the radius of a sphere whose volume is equal to the volume per a free electron. This parameter is used frequently in condensed matter physics to describe the density of a system. Worth to mention, r_{\rm s} is calculated for bulk materials.

Formula

In a 3-D system with N free valence electrons in a volume V, the Wigner–Seitz radius is defined by where n is the particle density. Solving for r_{\rm s} we obtain The radius can also be calculated as where M is molar mass, N_{V} is count of free valence electrons per particle, \rho is mass density and N_{\rm A} is the Avogadro constant. This parameter is normally reported in atomic units, i.e., in units of the Bohr radius. Assuming that each atom in a simple metal cluster occupies the same volume as in a solid, the radius of the cluster is given by where n is the number of atoms. Values of r_{\rm s} for the first group metals: Wigner–Seitz radius is related to the electronic density by the formula where, ρ can be regarded as the average electronic density in the outer portion of the Wigner-Seitz cell.