Wigner–Seitz radius

The Wigner-Seitz radius, 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. In the more general case of metals having more valence electrons, 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. is typically calculated for bulk materials.

Formula

In a 3-D system with free valence electrons in a volume, the Wigner–Seitz radius is defined by
where is the particle density. Solving for we obtain
The radius can also be calculated as
where is molar mass, is the count of free valence electrons per particle, is the mass density, and 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
where n is the number of atoms.
Values of for the first group metals:

Element	_s in ₀
	3.25
	3.93
	4.86
	5.20
	5.62

Wigner–Seitz radius is related to the electronic density by the formula
where ρ can be regarded as the average electronic density in the outer portion of the Wigner-Seitz cell.