The Wigner–Seitz radius formula_1, 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, formula_1 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, formula_1 is calculated for bulk materials.
Formula.
In a 3-D system with formula_4 free valence electrons in a volume formula_5, the Wigner–Seitz radius is defined by
formula_6
where formula_7 is the particle density. Solving for formula_1 we obtain
formula_9
The radius can also be calculated as
formula_10
where formula_11 is molar mass, formula_12 is count of free valence electrons per particle, formula_13 is mass density and
formula_14 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
formula_15
where "n" is the number of atoms.
Values of formula_1 for the first group metals: 
Wigner–Seitz radius is related to the electronic density by the formula
formula_17
where, "ρ" can be regarded as the average electronic density in the outer portion of the Wigner-Seitz cell.
References.
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