# Fermi Level

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## Fermi level

[′fer·mē ‚lev·əl]## Fermi Level

the energy level below which all energy states of the particles of a degenerate gas that obey Fermi-Dirac statistics are occupied at a temperature of absolute zero (*see*STATISTICAL MECHANICS). Particles of a degenerate gas that obey Fermi-Dirac statistics are called fermions. The existence of the Fermi level is a consequence of the Pauli exclusion principle, according to which not more than (2*s* + 1) particles, where *s* is the spin of a particle, may occupy a state with a specified momentum **p**. The Fermi level coincides with the values of the chemical potential of a fermion gas at *T* = 0°K.

The Fermi level ℰ_{F} can be expressed in terms of the number *n* of gas particles per unit volume: ℰ_{F} = [(2πℏ)^{2}/2*m*] [3*n*/4π(2*s* + 1)]^{⅔}, where *m* is the particle mass. The quantity is called the Fermi momentum. At *T* = 0°K, the particles occupy all states with momenta *p* < *p*_{F}; all states with *p* > *p*_{F} are empty. In other words, at *T* = 0°K, fermions occupy states in momentum space that lie within a sphere *p ^{2}* = 2

*m*ℰ

_{F}of radius

*p*

_{F}; this sphere is called the Fermi sphere. Upon heating, some particles move from a state with

*p*<

*p*

_{F}to a state with

*p*>

*p*

_{F}. Vacant sites, called holes, occur within the Fermi sphere. The quantity is called the Fermi velocity and specifies the upper limit to the velocities of fermions at

*T*= 0°K.

A degenerate gas consisting of conduction electrons in a solid at *T* = 0°K occupies surfaces of more complex shape in momentum space (*see*).

### REFERENCE

Landau, L. D., and E. M. Lifshits.*Statisticheskaia fizika*, 2nd ed. Moscow, 1964. (

*Teoreticheskaia fizika*, vol. 5.)

M. I. KAGANOV