Fermi-Dirac integral

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In statistical physics , the Fermi-Dirac integral (after Enrico Fermi and Paul Dirac ), with index, is defined as

where is the gamma function . The lower limit of the integral is given as an argument of the function

then one speaks of the incomplete Fermi-Dirac integral .

Application for F 1/2

The function occurs, among other things, in solid-state physics in connection with the distribution of electrons in the crystal lattice. Often the integral has to be calculated there (see: density of states ). Substitute for the second equal sign as well so that :

Approximation for F 1/2

The integral can be solved approximately for different value ranges of :

The relative error of this approximate solution is a maximum of 3% (maximum deviation at and at ). For a large distance from the origin, two functions can be used to approximate:

  For  
  For  

Representation with polylogarithms

Using the polylogarithm , the Fermi-Dirac integral can be represented as

.

Because of

follows from this

.

Web links

literature

  • JS Blakemore: Approximations for Fermi-Dirac Integrals . Solid-State Electronics, 25 (11): 1067-1076, 1982.