Wigner–Seitz radius

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

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

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

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

The radius can also be calculated as

${\displaystyle r_{\rm {s}}=\left({\frac {3M}{4\pi \rho N_{V}N_{\rm {A}}}}\right)^{\frac {1}{3}}\,,}$

where ${\displaystyle M}$ is molar mass, ${\displaystyle N_{V}}$ is count of free valence electrons per particle, ${\displaystyle \rho }$ is mass density and ${\displaystyle N_{\rm {A}}}$ is the Avogadro constant.

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

Values of ${\displaystyle r_{\rm {s}}}$ for the first group metals:[2]

Element	${\displaystyle r_{\rm {s}}/a_{0}}$
Li	3.25
Na	3.93
K	4.86
Rb	5.20
Cs	5.62

• Wigner–Seitz cell
• Wigner crystal