Wigner–Seitz radius

The Wigner–Seitz radius $r_{\mathrm{s}}$, 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). In the more general case of metals having more valence electrons, $r_{\mathrm{s}}$ is the radius of a sphere whose volume is equal to the volume per a free electron. This parameter is used frequently in condensed matter physics to describe the density of a system. Worth to mention, $r_{\mathrm{s}}$ is calculated for bulk materials.

Formula

In a 3-D system with $N$ free valence electrons in a volume $V$, the Wigner–Seitz radius is defined by

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

where $n$ is the particle density. Solving for $r_{\mathrm{s}}$ we obtain

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

The radius can also be calculated as

$r_{\mathrm{s}}={\left(\frac{3M}{4\pi \rho N_V{N}_{\mathrm{A}}}\right)}^{\frac{1}{3}},$

where $M$ is molar mass, $N_V$ is count of free valence electrons per particle, $\rho$ is mass density and $N_{\mathrm{A}}$ 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

$R_0=r_s{n}^{1/3}$

where n is the number of atoms.

Values of $r_{\mathrm{s}}$ for the first group metals:

Element	$r_{\mathrm{s}}/a_0$
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

$r_s=0.62035{\rho }^{1/3}$

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

See also

• Wigner–Seitz cell
• Wigner crystal

References

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