Wigner–Seitz radius

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

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

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

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

The radius can also be calculated as

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

where ${\displaystyle M}$ is molar mass, ${\displaystyle Z}$ is amount of free electrons per atom, ${\displaystyle \rho }$ is mass density, and ${\displaystyle N_{A}}$ is the Avogadro number.

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

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

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

• Wigner–Seitz cell