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## Fermi energyThe ## Additional recommended knowledge
## Introduction## ContextIn quantum mechanics, a group of particles known as fermions (for example, electrons, protons and neutrons are fermions) obey the Pauli exclusion principle. This principle states that two identical fermions can not be in the same quantum state. The states are labeled by a set of quantum numbers. In a system containing many fermions (like electrons in a metal) each fermion will have a different set of quantum numbers. To determine the lowest energy a system of fermions can have, we first group the states in sets with equal energy and order these sets by increasing energy. Starting with an empty system, we then add particles one at a time, consecutively filling up the unoccupied quantum states with lowest-energy. When all the particles have been put in, the
## Advanced contextThe Fermi energy ( ## Illustration of the concept for a one dimensional square wellThe one dimensional infinite square well is a model for a one dimensional box. It is a standard model-system in quantum mechanics for which the solution for a single particle is well known. The levels are labeled by a single quantum number - .
Suppose now that instead of one particle in this box we have N particles in the box and that these particles are fermions with spin 1/2. Then only two particles can have the same energy i.e. two particles can have the energy of , or two particles can have energy - .
## The three-dimensional caseThe three-dimensional isotropic case is known as the Let us now consider a three-dimensional cubical box that has a side length - n
_{x}, n_{y}, n_{z}are positive integers.
There are multiple states with the same energy, for example If we introduce a vector then each quantum state corresponds to a point in 'n-space' with Energy The number of states with energy less than E
the factor of two is once again because there are two spin states, the factor of 1/8 is because only 1/8 of the sphere lies in the region where all n are positive. We find so the Fermi energy is given by Which results in a relationship between the fermi energy and the number of particles per volume (when we replace L The total energy of a fermi ball of ## Typical fermi energies## White dwarfsStars known as White dwarfs have mass comparable to our Sun, but have a radius about 100 times smaller. The high densities means that the electrons are no longer bound to single nuclei and instead form a degenerate electron gas. The number density of electrons in a White dwarf are on the order of 10 ## NucleusAnother typical example is that of the particles in a nucleus of an atom. The radius of the nucleus is roughly: - where
*A*is the number of nucleons.
The number density of nucleons in a nucleus is therefore: Now since the fermi energy only applies to fermions of the same type, one must divide this density in two. This is because the presence of neutrons does not affect the fermi energy of the protons in the nucleus, and vice versa. So the fermi energy of a nucleus is about: The radius of the nucleus admits deviations around the value mentioned above, so a typical value for the fermi energy usually given is 38 MeV. ## Fermi levelThe where The where This concept is usually applied in the case of dispersion relations between the energy and momentum that do not depend on the direction. In more general cases, one must consider the Fermi energy. The where Below the where ## Quantum mechanicsAccording to quantum mechanics, fermions -- particles with a half-integer spin, usually 1/2, such as electrons -- follow the Pauli exclusion principle, which states that multiple particles may not occupy the same quantum state. Consequently, fermions obey Fermi-Dirac statistics. The ground state of a non-interacting fermion system is constructed by starting with an empty system and adding particles one at a time, consecutively filling up the lowest-energy unoccupied quantum states. When the desired number of particles has been reached, the Fermi energy is the energy of the highest occupied molecular orbital (HOMO). Within conductive materials, this is equivalent to the lowest unoccupied molecular orbital (LUMO); however, within other materials there will be a significant gap between the HOMO and LUMO on the order of 2-3 eV. ## Free electron gasIn the free electron gas, the quantum mechanical version of an ideal gas of fermions, the quantum states can be labeled according to their momentum. Something similar can be done for periodic systems, such as electrons moving in the atomic lattice of a metal, using something called the "quasi-momentum" or "crystal momentum" (see Bloch wave). In either case, the Fermi energy states reside on a surface in momentum space known as the The Fermi energy of the free electron gas is related to the chemical potential by the equation where k. The characteristic temperature is on the order of 10^{5} K for a metal, hence at room temperature (300 K), the Fermi energy and chemical potential are essentially equivalent. This is significant since it is the chemical potential, not the Fermi energy, which appears in Fermi-Dirac statistics.
## See also- fermi gas
- semiconductors
- electrical engineering
- electronics
- thermodynamics
## References- Kroemer, Herbert; Kittel, Charles (1980).
*Thermal Physics (2nd ed.)*. W. H. Freeman Company.__ISBN 0-7167-1088-9__. - Table of fermi energies, velocities, and temperatures for various elements.
- a discussion of fermi gases and fermi temperatures.
Categories: Condensed matter physics | Statistical mechanics |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Fermi_energy". A list of authors is available in Wikipedia. |