Wigner–Seitz radius

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The Wigner–Seitz radius r_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.[1] This parameter is used frequently in condensed matter physics to describe the density of a system.


In a 3-D system with N particles in a volume V, the Wigner–Seitz radius is defined by[1]

\frac{4}{3} \pi r_s^3 = \frac{V}{N}.

Solving for r_s we obtain

r_s = \left(\frac{3}{4\pi n}\right)^{1/3}\,,

where n is the particle density of the valence electrons.

For a non-interacting system, the average separation between two particles will be 2 r_s. The radius can also be calculated as

r_s= \left(\frac{3M}{4\pi \rho N_A}\right)^\frac{1}{3}\,,

where M is molar mass, \rho is mass density, and N_A is the Avogadro number.

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

Values of r_s for single valence metals[2] are listed below:

Element r_s/a_0
Li 3.25
Na 3.93
K 4.86
Rb 5.20
Cs 5.62

See also[edit]


  1. ^ a b 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.