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The '''Wigner–Seitz radius''' <math>r_s</math>, 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.<ref name=Grifalco>{{cite book|last=Girifalco|first=Louis A.|title=Statistical mechanics of solids|year=2003|publisher=Oxford University Press|location=Oxford|isbn=978-0-19-516717-7|page=125}}</ref> This parameter is used frequently in [[condensed matter physics]] to describe the density of a system.

==Formula==
In a 3-D system with <math>N</math> particles in a volume <math>V</math>, the Wigner–Seitz radius is defined by<ref name=Grifalco/>

:<math>\frac{4}{3} \pi r_s^3 = \frac{V}{N}.</math>

Solving for <math>r_s</math> we obtain

:<math>r_s = \left(\frac{3}{4\pi n}\right)^{1/3}\,,</math>
where <math>n</math> is the [[particle density]] of the valence electrons.

For a non-interacting system, the average separation between two particles will be <math>2 r_s</math>. The radius can also be calculated as
:<math>r_s= \left(\frac{3M}{4\pi \rho N_A}\right)^\frac{1}{3}\,,</math>
where <math>M</math> is [[molar mass]], <math>\rho</math> is [[mass density]], and
<math>N_A</math> is the [[Avogadro number]].

This parameter is normally reported in [[atomic units]], i.e., in units of the [[Bohr radius]].

Values of <math>r_s</math> for single valence metals<ref name=Ashcroft>*{{cite book
| last = Ashcroft
| first = Neil W.
| last2 = Mermin
| first2 = N. David
| title = Solid State Physics
| publisher = [[Holt, Rinehart and Winston]]
| year = 1976
| isbn = 0-03-083993-9
| ref =Ashcroft, Mermin
}}</ref> are listed below:

{| class="wikitable"
|-
! Element !! <math>r_s/a_0</math>
|-
| Li || 3.25
|-
| Na || 3.93
|-
| K || 4.86
|-
| Rb || 5.20
|-
| Cs || 5.62
|}

==See also==
*[[Wigner–Seitz cell]]

==References==
{{Reflist}}

{{DEFAULTSORT:Wigner-Seitz radius}}
[[Category:Condensed matter physics]]

{{physics-stub}}

Functional-theoretic algebra - Revision history

en>Frietjes: cleanup (wikitables, html markup, layout, etc.)

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