A Model for Fermi-Dirac Integrals

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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, (plotted here), using an exponential model that is accurate for (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 , with (orders commonly used in astrophysics and semiconductors).

Contributed by: Michael Morales (April 2012)
Open content licensed under CC BY-NC-SA



Written in terms of polylogarithms, the (normalized) Fermi–Dirac integral is [2] . Meanwhile, the model is found to be ,
where is a function written in terms of the Lambert function (see [1] and references therein).


[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].

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.