Fermi energy

In quantum mechanics, a group of particles known as fermions (for example, electrons, protons and neutrons) obey the Pauli exclusion principle. This states that two fermions cannot occupy the same quantum state. Since an idealized non-interacting Fermi gas can be analyzed in terms of single-particle stationary states, we can thus say that two fermions cannot occupy the same stationary state. These stationary states will typically be distinct in energy. To find the ground state of the whole system, we start with an empty system, and add particles one at a time, consecutively filling up the unoccupied stationary states with the lowest energy. When all the particles have been put in, the Fermi energy is the kinetic energy of the highest occupied state.

As a consequence, even if we have extracted all possible energy from a Fermi gas by cooling it to near absolute zero temperature, the fermions are still moving around at a high speed. The fastest ones are moving at a velocity corresponding to a kinetic energy equal to the Fermi energy. This speed is known as the Fermi velocity. Only when the temperature exceeds the related Fermi temperature, do the particles begin to move significantly faster than at absolute zero.

The Fermi energy is an important concept in the solid state physics of metals and superconductors. It is also a very important quantity in the physics of quantum liquids like low temperature helium (both normal and superfluid ³He), and it is quite important to nuclear physics and to understanding the stability of white dwarf stars against gravitational collapse.

The Fermi energy for a non-interacting ensemble of identical spin-1⁄2 fermions in a three-dimensional (non-relativistic) system is given by[1]

$${\displaystyle E_{\text{F}}={\frac {\hbar ^{2}}{2m_{0}}}\left({\frac {3\pi ^{2}N}{V}}\right)^{2/3},}$$

where N is the number of particles, m₀ the rest mass of each fermion, V the volume of the system, and ${\displaystyle \hbar }$ the reduced Planck constant.

Using this definition of above for the Fermi energy, various related quantities can be useful.

The Fermi temperature is defined as

$${\displaystyle T_{\text{F}}={\frac {E_{\text{F}}}{k_{\text{B}}}},}$$

where ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant, and ${\displaystyle E_{\text{F}}}$ the Fermi energy. The Fermi temperature can be thought of as the temperature at which thermal effects are comparable to quantum effects associated with Fermi statistics.[3] The Fermi temperature for a metal is a couple of orders of magnitude above room temperature.

Other quantities defined in this context are Fermi momentum

$${\displaystyle p_{\text{F}}={\sqrt {2m_{0}E_{\text{F}}}}}$$

and Fermi velocity

$${\displaystyle v_{\text{F}}={\frac {p_{\text{F}}}{m_{0}}}.}$$

These quantities are respectively the momentum and group velocity of a fermion at the Fermi surface.

The Fermi momentum can also be described as

$${\displaystyle p_{\text{F}}=\hbar k_{\text{F}},}$$

where ${\displaystyle k_{\text{F}}}$, called the Fermi wavevector, is the radius of the Fermi sphere.[4]

These quantities may not be well-defined in cases where the Fermi surface is non-spherical.

• Fermi–Dirac statistics: the distribution of electrons over stationary states for non-interacting fermions at non-zero temperature.
• Fermi level
• Quasi Fermi level

• Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. ISBN 978-0-7167-1088-2.