The Wigner–Seitz radius
r
s
{\displaystyle r_{\rm {s}}}
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]
r
s
{\displaystyle r_{\rm {s}}}
[ 2] condensed matter physics to describe the density of a system. Worth to mention,
r
s
{\displaystyle r_{\rm {s}}}
In a 3-D system with
N
{\displaystyle N}
V
{\displaystyle V}
4
3
π π -->
r
s
3
=
V
N
=
1
n
,
{\displaystyle {\frac {4}{3}}\pi r_{\rm {s}}^{3}={\frac {V}{N}}={\frac {1}{n}}\,,}
where
n
{\displaystyle n}
particle density . Solving for
r
s
{\displaystyle r_{\rm {s}}}
r
s
=
(
3
4
π π -->
n
)
1
/
3
.
{\displaystyle r_{\rm {s}}=\left({\frac {3}{4\pi n}}\right)^{1/3}.}
The radius can also be calculated as
r
s
=
(
3
M
4
π π -->
ρ ρ -->
N
V
N
A
)
1
3
,
{\displaystyle r_{\rm {s}}=\left({\frac {3M}{4\pi \rho N_{V}N_{\rm {A}}}}\right)^{\frac {1}{3}}\,,}
where
M
{\displaystyle M}
molar mass ,
N
V
{\displaystyle N_{V}}
ρ ρ -->
{\displaystyle \rho }
mass density and
N
A
{\displaystyle N_{\rm {A}}}
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
R
0
=
r
s
n
1
/
3
{\displaystyle R_{0}=r_{s}n^{1/3}}
where n is the number of atoms.[ 3] [ 4]
Values of
r
s
{\displaystyle r_{\rm {s}}}
[ 2]
Element
r
s
/
a
0
{\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
r
s
=
0.62035
ρ ρ -->
1
/
3
{\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
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
^ 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. doi :10.1103/PhysRevB.31.6809 . ISSN 0163-1829 . PMID 9935571 .
