Wigner–Seitz radius explained

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 (for first group metals).^[1] 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.^[2] 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

$$\backslash frac\; \backslash pi\; r\_^3\; =\; \backslash frac\; =\; \backslash frac\backslash ,$$

where is the particle density. Solving for

we obtain

$$r\_\; =\; \backslash sqrt[3].$$

The radius can also be calculated as$$r\_=\; \backslash sqrt[3]$$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$$R\_0\; =\; r\_s\; n^$$where n is the number of atoms.^{[3] [4]}

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$$r\_s\; =0.62035\; \backslash rho^$$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 . 1985PhRvB..31.6809P . 0163-1829. subscription .