Fermi-Dirac statistics

Fermi-Dirac statistics, class of statistics that applies to particles called fermions. Fermions have half-integral values of the quantum mechanical property called spin and are "antisocial" in the sense that two fermions cannot exist in the same state. Protons, neutrons, electrons, and many other elementary particles are fermions. See Bose-Einstein statistics; elementary particles; statistical mechanics.

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Fermi-Dirac statistics

In quantum mechanics, one of two possible ways (the other being Bose-Einstein statistics) in which a system of indistinguishable particles can be distributed among a set of energy states. Each available discrete state can be occupied by only one particle. This exclusiveness accounts for the structure of atoms, in which electrons remain in separate states rather than collapsing into a common state. It also accounts for some aspects of electrical conductivity. This theory of statistical behaviour was developed first by Enrico Fermi and then by P.A.M. Dirac (1926–27). The statistics apply only to particles such as electrons that have half-integer values of spin; the particles are called fermions.

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Fermi-Dirac statistics

The statistical description of particles or systems of particles that satisfy the Pauli exclusion principle. This description was first given by E. Fermi, who applied the Pauli exclusion principle to the translational energy levels of a system of electrons. It was later shown by P. A. M. Dirac that this form of statistics is also obtained when the total wave function of the system is antisymmetrical. See Exclusion principle

Such a system is described by a set of occupation numbers n_i which specify the number of particles in energy levels ε_i. It is important to keep in mind that ε_i represents a finite range of energies, which in general contains a number, say g_i, of nondegenerate quantum states. In the Fermi statistics, at most one particle is allowed in a nondegenerate state. (If spin is taken into account, two particles may be contained in such a state.) This is simply a restatement of the Pauli exclusion principle, and means that n_i ≤ g_i. The probability of having a set {n_i} distributed over the levels ε_i, which contain g_i nondegenerate levels, is described by Eq. (1), which gives just the number

(1) 

of ways that n_i can be picked out of g_i, which is intuitively what one expects for such a probability. The equilibrium state which actually exists is the set of n's that makes W a maximum, under the auxiliary conditions given in Eqs. (2a) and (2b).

(2{\em a}) 

(2{\em b}) 

These conditions express the fact that the total energy E and the total number of particles N are given. Equation (3)

(3) 

holds for this most probable distribution. Here A and β are parameters, to be determined from Eq. (3); in fact, β = 1/kT, where k is Boltzmann's constant and T is the absolute temperature. When the 1 in the denominator may be neglected, Eq. (3) goes over into the Boltzmann distribution.

Classical conditions pertain when the volume per particle is much larger than the volume associated with the de Broglie wavelength λ of a particle. For electrons in a metal at 300 K, the ratio of the volume per particle to λ³ has the value 10^-4, showing that classical statistics fail altogether. When the classical distribution fails, a degenerate Fermi distribution results. A somewhat lengthy calculation yields the result that in this case the contribution of the electrons to the specific heat is negligible. This resolves an old paradox, for, according to the classical equipartition law, the electronic specific heat C should be (3/2)Nk, whereas in reality it is very small. See Bose-Einstein statistics, Kinetic theory of matter, Quantum statistics, Statistical mechanics