The Wigner–Seitz radius ${\displaystyle r_{s}}$, 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.[1] This parameter is used frequently in condensed matter physics to describe the density of a system.

## Formula

In a 3-D system with ${\displaystyle N}$ particles in a volume ${\displaystyle V}$, the Wigner–Seitz radius is defined by[1]

${\displaystyle {\frac {4}{3}}\pi r_{s}^{3}={\frac {V}{N}}.}$

Solving for ${\displaystyle r_{s}}$ we obtain

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

where ${\displaystyle n}$ is the particle density of the valence electrons.

For a non-interacting system, the average separation between two particles will be ${\displaystyle 2r_{s}}$. The radius can also be calculated as

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

where ${\displaystyle M}$ is molar mass, ${\displaystyle \rho }$ 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 single valence metals[2] are listed below:

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