TheWigner–Seitz radius${\displaystyle r_{\mathrm{s}}}$, named afterEugene Wigner andFrederick 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,${\displaystyle r_{\mathrm{s}}}$ is the radius of a sphere whose volume is equal to the volume per a free electron.^[2] This parameter is used frequently incondensed matter physics to describe the density of a system. Worth to mention,${\displaystyle r_{\mathrm{s}}}$ is calculated for bulk materials.

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

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

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

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

The radius can also be calculated as

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

where${\displaystyle M}$ ismolar mass,${\displaystyle N_V}$ is count of free valence electrons per particle,${\displaystyle \rho }$ ismass density and${\displaystyle N_{\mathrm{A}}}$ is theAvogadro constant.

This parameter is normally reported inatomic units, i.e., in units of theBohr 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

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

wheren is the number of atoms.^[3][4]

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

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

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

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

• Wigner–Seitz cell
• Wigner crystal

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