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]

$\frac{4}{3} \pi r_s^3 = \frac{V}{N}.$

Solving for $r_s$ we obtain

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

where $n$ is the particle density of the valence electrons.

For a non-interacting system, the average separation between two particles will be $2 r_s$. The radius can also be calculated as

$r_s= \left(\frac{3M}{4\pi \rho N_A}\right)^\frac{1}{3}\,,$

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	$r_s/a_0$
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.