Wigner–Seitz radius

The Wigner–Seitz radius $r_{s}$, 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 $N$ particles in a volume $V$, the Wigner–Seitz radius is defined by^[1]

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

Solving for $r_{s}$ we obtain

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

where $n$ 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 $M$ is molar mass, $\rho$ is mass density, and $N_{A}$ is the Avogadro number.

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

Values of $r_{s}$ 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

See also

• Wigner–Seitz cell

References

1. Girifalco, Louis A. (2003). Statistical mechanics of solids. Oxford: Oxford University Press. p. 125. ISBN 978-0-19-516717-7.
2. • Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics. Holt, Rinehart and Winston. ISBN 0-03-083993-9.