Physics:Wigner–Seitz radius

The Wigner–Seitz radius [math]\displaystyle{ r_{\rm s} }[/math], 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, [math]\displaystyle{ r_{\rm s} }[/math] 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, [math]\displaystyle{ r_{\rm s} }[/math] is calculated for bulk materials.

Formula

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

[math]\displaystyle{ \frac{4}{3} \pi r_{\rm s}^3 = \frac{V}{N} = \frac{1}{n}\,, }[/math]

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

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

The radius can also be calculated as

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

where [math]\displaystyle{ M }[/math] is molar mass, [math]\displaystyle{ Z }[/math] is amount of free electrons per atom, [math]\displaystyle{ \rho }[/math] is mass density, and [math]\displaystyle{ N_{\rm A} }[/math] is the Avogadro number.

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

Values of [math]\displaystyle{ r_{\rm s} }[/math] for the first group metals:

Element	[math]\displaystyle{ r_{\rm s}/a_0 }[/math]
Li	3.25
Na	3.93
K	4.86
Rb	5.20
Cs	5.62

See also

• Wigner–Seitz cell
• Wigner crystal