Wigner–Seitz radius

The Wigner–Seitz radius ${\displaystyle r_{\rm {s}}}$, 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, ${\displaystyle r_{\rm {s}}}$ 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, ${\displaystyle r_{\rm {s}}}$ 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

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

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

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

The radius can also be calculated as

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

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

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

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

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

See also

• Wigner–Seitz cell
• Wigner crystal

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.