Wigner–Seitz radius

Formula

In a 3-D system with ${\displaystyle N}$ free valence electrons in a volume ${\displaystyle V}$, the Wigner–Seitz radius is defined by

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

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

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

The radius can also be calculated as

${\displaystyle r_{\mathrm{s}}={\left(\frac{3M}{4\pi \rho N_V{N}_{\mathrm{A}}}\right)}^{\frac{1}{3}},}$

where ${\displaystyle M}$ is molar mass, ${\displaystyle N_V}$ is count of free valence electrons per particle, ${\displaystyle \rho }$ is mass density and ${\displaystyle N_{\mathrm{A}}}$ 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

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

where n is the number of atoms.

Values of ${\displaystyle r_{\mathrm{s}}}$ for the first group metals:

Wigner–Seitz radius is related to the electronic density by the formula

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

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