Wigner–Seitz radius

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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. Worth to mention, is calculated for bulk materials.

Formula

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In a 3-D system with   free valence electrons in a volume  , the Wigner–Seitz radius is defined by

 

where   is the particle density. Solving for   we obtain

 

The radius can also be calculated as

 

where   is molar mass,   is count of free valence electrons per particle,   is 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

 

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

Values of   for the first group metals:[2]

Element  
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

 

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

See also

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References

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  1. ^ Girifalco, Louis A. (2003). Statistical mechanics of solids. Oxford: Oxford University Press. p. 125. ISBN 978-0-19-516717-7.
  2. ^ a b *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.