A Model for Fermi-Dirac Integrals

​
k
/2
-1
1
3
5
Fermi–Dirac integrals arise in calculating pressure and density in degenerate matter, such as neutron stars; they also occur in the electronic density of semiconductors. A (semi-)closed form was not known until 1995, when Howard Lee noticed the application of the integral form of polylogarithms. We developed an alternative expression,
F
k/2
(η)=
1
Γ1+
k
2

∞
∫
0
k/2
x
1+
x-η
e
dx
(plotted here), using an exponential model that is accurate for
η<3
(arXiv:0909.3653v5 [math.GM]). Here we compare the model, in green, with the polylogarithm expression, in blue, for Fermi–Dirac integrals half-integer order
k
2
, with
k=-1,1,3,5
(orders commonly used in astrophysics and semiconductors).

Details

Briefly, the model is based on the idea of rewriting
k/2
x
1+
x-η
e
as a product of
k/2
x
1+
x
e
and a function that is a basic exponential model; for more details, see[1]. The dependence on the factor
η
e
explains the restriction to values
η<3
.
Written in terms of polylogarithms, the (normalized) Fermi–Dirac integral is[2]
F
k/2
(η)=-
Li
1+k/2
(-
η
e
)
.Meanwhile, the model is found to be
F
k/2
(η)=
η
e
(1-
-k/2
2
)ζ1+
k
2
-
η
e
-1
-η
e
+1
-k/2
2
ζ1+
k
2
,
c(k,η)+1
2
-ζ1+
k
2
,c(k,η)+1
,where
c(k,η)
is a function written in terms of the Lambert function (see[1] and references therein).

References

[1] M. H. Lee, "Polylogarithmic Analysis of Chemical Potential and Fluctuations in a d-Dimensional Free Fermi Gas at Low Temperatures," Journal of Mathematical Physics, 36, 1995 pp. 1217–1231.
[2] M. Morales, "Fermi-Dirac Integrals in Terms of Zeta Functions," arXiv:0909.3653v5 [math.GM].

External Links

Fermi–Dirac Distributions for Free Electrons in Metals
Plots of the Fermi–Dirac Distribution

Permanent Citation

Michael Morales
​
​"A Model for Fermi-Dirac Integrals"​
​http://demonstrations.wolfram.com/AModelForFermiDiracIntegrals/​
​Wolfram Demonstrations Project​
​Published: April 6, 2012