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 valence 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. 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

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

where ${\displaystyle M}$ is molar mass, ${\displaystyle N_{V}}$ is count of free valence electrons per particle, ${\displaystyle \rho }$ 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.

Assuming that each atom in a simple metal cluster occupies the same volume as in a solid, the radius of the cluster is given by

${\displaystyle R_{0}=r_{s}n^{1/3}}$

where n is the number of atoms.^[3][4]

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

Wigner–Seitz radius is related to the electronic density by the formula

${\displaystyle r_{s}=0.62035\rho ^{1/3}}$

where, ρ can be regarded as the average electronic density in the outer portion of the Wigner-Seitz cell.^[5]

See also

• Wigner–Seitz cell
• Wigner crystal

References