Wigner–Seitz radius

The Wigner–Seitz radius ${\displaystyle r_{\rm {s}}}$, 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).^[1] In the more general case of metals having more valence electrons, ${\displaystyle r_{\rm {s}}}$ 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, ${\displaystyle r_{\rm {s}}}$ is calculated for bulk materials.

Formula

In a 3-D system with ${\displaystyle N}$ free valence electrons in a volume ${\displaystyle V}$, the Wigner–Seitz radius is defined by

${\displaystyle {\frac {4}{3}}\pi r_{\rm {s}}^{3}={\frac {V}{N}}={\frac {1}{n}}\,,}$

where ${\displaystyle n}$ is the particle density. Solving for ${\displaystyle r_{\rm {s}}}$ we obtain

${\displaystyle r_{\rm {s}}=\left({\frac {3}{4\pi n}}\right)^{1/3}.}$

The radius can also be calculated as

${\displaystyle r_{\rm {s}}=\left({\frac {3M}{4\pi \rho N_{V}N_{\rm {A}}}}\right)^{\frac {1}{3}}\,,}$

where ${\displaystyle M}$ is molar mass, ${\displaystyle N_{V}}$ is count of free valence electrons per particle, ${\displaystyle \rho }$ is mass density and ${\displaystyle N_{\rm {A}}}$ 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

${\displaystyle R_{0}=r_{s}n^{1/3}}$

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

Values of ${\displaystyle r_{\rm {s}}}$ for the first group metals:^[2]

Element	${\displaystyle r_{\rm {s}}/a_{0}}$
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 formula

${\displaystyle r_{s}=0.62035\rho ^{1/3}}$

where, ρ can be regarded as the average electronic density in the outer portion of the Wigner-Seitz cell.^[5]

See also

• Wigner–Seitz cell
• Wigner crystal

References

1. Girifalco, Louis A. (2003). Statistical mechanics of solids. Oxford: Oxford University Press. p. 125. ISBN 978-0-19-516717-7.
2. • Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics. Holt, Rinehart and Winston. ISBN 0-03-083993-9.
3. Bréchignac, Catherine; Houdy, Philippe; Lahmani, Marcel, eds. (2007). Nanomaterials and nanochemistry. Berlin Heidelberg: Springer. ISBN 978-3-540-72992-1.
4. "Radius of Cluster using Wigner Seitz Radius Calculator | Calculate Radius of Cluster using Wigner Seitz Radius". www.calculatoratoz.com. Retrieved 2024-05-28.
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. 31 (10): 6809–6810. doi:10.1103/PhysRevB.31.6809. ISSN 0163-1829. PMID 9935571.