Wigner–Seitz radius
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
where n is the particle density. Solving for we obtain
![{\displaystyle r_{\rm {s}}={\sqrt[{3}]{\frac {3}{4\pi n}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f5520d52c284221562ae91e08a8a9620e01111f)
The radius can also be calculated as 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[3].
![{\displaystyle r_{\rm {s}}={\sqrt[{3}]{\frac {3M}{4\pi \rho N_{V}N_{\rm {A}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7455aeab372d9bcc194fb3660c99cf82ac4fc615)
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 where n is the number of atoms.[4][5]
Values of for the first group metals:[2]
| Element | rs in a0 |
|---|---|
| 7 Li |
3.25 |
| 23 Na |
3.93 |
| 39 K |
4.86 |
| 85 Rb |
5.20 |
| 133 Cs |
5.62 |
Wigner–Seitz radius is related to the electronic density by the formula where ρ can be regarded as the average electronic density in the outer portion of the Wigner-Seitz cell.[6]
See also
References
- ^ Girifalco, Louis A. (2003). Statistical mechanics of solids. Oxford: Oxford University Press. p. 125. ISBN 978-0-19-516717-7.
- ^ a b *Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics. Holt, Rinehart and Winston. ISBN 0-03-083993-9.
- ^ "2022 CODATA Value: Avogadro constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
- ^ Bréchignac, Catherine; Houdy, Philippe; Lahmani, Marcel, eds. (2007). Nanomaterials and nanochemistry. Berlin Heidelberg: Springer. ISBN 978-3-540-72992-1.
- ^ "Radius of Cluster using Wigner Seitz Radius Calculator | Calculate Radius of Cluster using Wigner Seitz Radius". www.calculatoratoz.com. Retrieved 2024-05-28.
- ^ 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. Bibcode:1985PhRvB..31.6809P. doi:10.1103/PhysRevB.31.6809. ISSN 0163-1829. PMID 9935571.