The Wigner–Seitz radius r s {\displaystyle r_{\rm {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 s {\displaystyle r_{\rm {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. r s {\displaystyle r_{\rm {s}}} is typically calculated for bulk materials.
Formula In a 3-D system with N {\displaystyle N} free valence electrons in a volume V {\displaystyle V} , the Wigner–Seitz radius is defined by
4 3 π r s 3 = V N = 1 n {\displaystyle {\frac {4}{3}}\pi r_{\rm {s}}^{3}={\frac {V}{N}}={\frac {1}{n}}\,}
where n is the particle density. Solving for r s {\displaystyle r_{\rm {s}}} we obtain
r s = 3 4 π n 3 . {\displaystyle r_{\rm {s}}={\sqrt[{3}]{\frac {3}{4\pi n}}}.}
The radius can also be calculated as
r s = 3 M 4 π ρ N V N A 3 {\displaystyle r_{\rm {s}}={\sqrt[{3}]{\frac {3M}{4\pi \rho N_{V}N_{\rm {A}}}}}}
where M is molar mass, NV is the count of free valence electrons per particle, ρ is the mass density, and NA is the Avogadro constant, 6.02214076×1023 mol−1. 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 {\displaystyle R_{0}=r_{s}n^{1/3}}
where n is the number of atoms. Values of r s {\displaystyle r_{\rm {s}}} for the first group metals:
Wigner–Seitz radius is related to the electronic density by the formula
r s = 0.62035 ρ 1 / 3 {\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.
See also Wigner–Seitz cell Wigner crystal
References