Wigner–Seitz radius explained

The Wigner - Seitz radius

r _{\rm} , 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).^[1] In the more general case of metals having more valence electrons, r _{\rm} is the radius of a sphere whose volume is equal to the volume per a free electron.^[2] This parameter is used frequently in condensed matter physics to describe the density of a system. Worth to mention, r _{\rm} is calculated for bulk materials.

Formula

In a 3-D system with

N free valence electrons in a volume V , the Wigner–Seitz radius is defined by(4)/(3) \pi r ^{3} _{\rm s} = (V)/(N) = (1)/(n) ,

where

n is the particle density. Solving for r _{\rm} we obtainr _{\rm} = \left( (3)/(4 \pi n) \right) ^{1/3} .

The radius can also be calculated as

r _{\rm} = \left( (3 M)/(4 \pi \rho N _{V} N _{\rm}) \right) ^{(1)/(3)} , where M is molar mass, N _{V} is count of free valence electrons per particle, \rho is mass density andN _{\rm} 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

R _{0} = r _{s} n ^{1/3}

where n is the number of atoms.^{[3] [4]}

Values of

r _{\rm} for the first group metals:

Element	r _{\rm} /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 formular _{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]

See also

• Wigner - Seitz cell
• Wigner crystal

Notes and References

1. Book: Girifalco, Louis A.. Statistical mechanics of solids. 2003. Oxford University Press. Oxford. 978-0-19-516717-7. 125.
2. • Book: Ashcroft. Neil W.. Mermin. N. David. Solid State Physics. Holt, Rinehart and Winston. 1976. 0-03-083993-9. Ashcroft, Mermin. registration.
3. Book: Nanomaterials and nanochemistry . 2007 . Springer . 978-3-540-72992-1 . Bréchignac . Catherine . Berlin Heidelberg . Houdy . Philippe . Lahmani . Marcel.
4. Web site: Radius of Cluster using Wigner Seitz Radius Calculator Calculate Radius of Cluster using Wigner Seitz Radius . 2024-05-28 . www.calculatoratoz.com . en.
5. Politzer . Peter . Parr . Robert G. . Murphy . Danny R. . 1985-05-15 . Approximate determination of Wigner-Seitz radii from free-atom wave functions . Physical Review B . en . 31 . 10 . 6809–6810 . 10.1103/PhysRevB.31.6809 . 9935571 . 0163-1829.