Plots of the Fermi-Dirac Distribution

Fermi-Dirac statistics deals with identical and indistinguishable particles with half-integral spins. Electrons, protons, neutrons, and so on are particles (called fermions) that follow Fermi-Dirac statistics. Fermions obey the Pauli exclusion principle, which states that two fermions cannot occupy the same quantum state at the same time. The Fermi-Dirac distribution function gives the probability that a given energy level is occupied by a fermion for a system in thermal equilibrium.

The Fermi-Dirac distribution function of elements is given by f(ϵ)=1/(exp[(ϵ-(Subscript[ϵ, F]))/((Subscript[k, B]) T)]+1), where (Subscript[ϵ, F]) is the Fermi energy of the element, (Subscript[k, B]) is the Boltzmann constant, and f(ϵ) is the probability that a quantum state with energy ϵ is occupied by an electron. This Demonstration shows the variation of the Fermi-Dirac distribution function of representative metals with energy at different temperatures.

THINGS TO TRY

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DETAILS

The dashed orange lines are plots of the Fermi-Dirac distribution function as a function of energy. At 0.0 °K, the highest occupied energy level is (Subscript[ϵ, F]).The blue section shows unoccupied energy levels at elevated temperatures, in the neighborhood of (Subscript[ϵ, F]).The red section shows occupied energy levels with energies greater than (Subscript[ϵ, F]) at high temperatures. For comparison, the Fermi function at 0.001 °K is shown with bold purple lines.

The Fermi energy of elements is taken from N. W. Ashcroft and N. D. Mermin, Solid State Physics, New York: Holt, Rinehart and Winston, 1976.

The Fermi-Dirac distribution function details are taken from C. Kittel, Introduction to Solid State Physics, 5th ed., New York: Wiley, 1976.

Additional information on the free electron model can be found on Wikipedia.

Contributed by: Kallol Das (St. Aloysius College, Jabalpur, India)

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