Wigner–Seitz radius

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


 * $$\frac{4}{3} \pi r_{\rm s}^3 = \frac{V}{N} = \frac{1}{n}\,,$$

where $$n$$ is the particle density. Solving for $$r_{\rm s}$$ we obtain


 * $$r_{\rm s} = \left(\frac{3}{4\pi n}\right)^{1/3}.$$

The radius can also be calculated as
 * $$r_{\rm s}= \left(\frac{3M}{4\pi \rho N_{V} N_{\rm A}}\right)^\frac{1}{3}\,,$$

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

$$R_0 = r_s n^{1/3}$$

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

$$r_s =0.62035 \rho^{1/3}$$

where, ρ can be regarded as the average electronic density in the outer portion of the Wigner-Seitz cell.