Wigner–Seitz radius

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 (for first group metals)[1]. In the more general case of metals having more valence electrons, ${\displaystyle r_{s}}$ is the radius of a sphere whose volume is equal to the volume per a free electron[2]. This parameter is used frequently in condensed matter physics to describe the density of a system. Worth to mention, ${\displaystyle r_{s}}$ is calculated for bulk materials.

Formula

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

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.