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Encyclopedia > Fermi energy

The Fermi energy is a concept in quantum mechanics referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature. This article requires a basic knowledge of quantum mechanics. For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Probability densities for the electron at different quantum numbers (l) In quantum mechanics, the quantum state of a system is a set of numbers that fully describe a quantum system. ... In particle physics, fermions are particles with half-integer spin, such as protons and electrons. ... Absolute zero is the lowest possible temperature where nothing could be colder, and no heat energy remains in a substance. ... For other uses, see Temperature (disambiguation). ...

Context

The Fermi energy (EF) of a system of non-interacting fermions is the increase in the ground state energy when exactly one particle is added to the system. It can also be interpreted as the maximum energy of an individual fermion in this ground state. The chemical potential at zero temperature is equal to the Fermi energy. In particle physics, fermions are particles with half-integer spin, such as protons and electrons. ... In physics, the ground state of a quantum mechanical system is its lowest-energy state. ... In thermodynamics and chemistry, chemical potential, symbolized by Î¼, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a...

Illustration of the concept for a one dimensional square well

The 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 n and the energies are given by In physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a very simple problem consisting of a single particle bouncing around inside of an immovable box, from which it cannot escape, and which loses no energy when it collides with...

$E_n = frac{hbar^2 pi^2}{2 m L^2} n^2 ,$.

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 particle can have energy $E_1=frac{hbar^2 pi^2}{2 m L^2}$, two particles can have energy E2 = 4E1 and so forth. The reason that two particles can have the same energy is that a spin-1/2 particle can have a spin of 1/2 (spin up) or a spin of -1/2 (spin down), leading to two states for each energy level. When we look at the total energy of this system, the configuration for which the total energy is lowest (the ground state), is the configuration where all the energy levels up to n=N/2 are occupied and all the higher levels are empty. The Fermi energy is therefore In quantum mechanics, spin is an intrinsic property of all elementary particles. ...

$E_f=E_{N/2}=frac{hbar^2 pi^2}{2 m L^2} (N/2)^2 ,$.

The three dimensional case

The three dimensional isotropic case is known as the fermi ball. Isotropic means independent of direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. ...

Lets now consider a three dimensional cubical box that has a side length L (see infinite square well). This turns out to be a very good approximation for describing electrons in a metal. The states are now labeled by three quantum numbers nx, ny, and nz. The single particle energies are In physics, the particle in a box (or the square well) is a simple idealized system that can be completely solved within quantum mechanics. ...

$E_{n_x,n_y,n_z} = frac{hbar^2 pi^2}{2m L^2} left( n_x^2 + n_y^2 + n_z^2right) ,$
nx, ny, nz are positive integers.

There are multiple states with the same energy, for example E100 = E010 = E001. Now let's put N non-interacting fermions of spin 1/2 into this box. To calculate the Fermi energy, we look at the case for N is large.

If we introduce a vector $vec{n}={n_x,n_y,n_z}$ then each quantum state corresponds to a point in 'n-space' with Energy

$E_{vec{n}} = frac{hbar^2 pi^2}{2m L^2} |vec{n}|^2 ,$

The number of states with energy less then Ef is equal to the number of states that lie within a sphere of radius $|vec{n}_f|$ in the region of n-space where nx, ny, nz are positive. In the ground state this number equals the number of fermions in the system.

$N =2 frac{1}{8} frac{4}{3} pi n_f^3 ,$
The free fermions that occupy the lowest energy states form a sphere in momentum space. The surface of this sphere is the Fermi surface.

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 Image File history File links Fermi_energy_momentum. ... Image File history File links Fermi_energy_momentum. ... A sphere is a symmetrical geometrical object. ... This article is about momentum in physics. ... In condensed matter physics, the Fermi surface is an abstract boundary useful for predicting the thermal, electrical, magnetic, and optical properties of metals, semimetals, and doped semiconductors. ...

$n_f=left(frac{3 N}{pi}right)^{1/3}$

so the Fermi energy is given by

$E_f =frac{hbar^2 pi^2}{2m L^2} n_f^2$
$= frac{hbar^2 pi^2}{2m L^2} left( frac{3 N}{pi} right)^{2/3}$

Which results in a relationship between the fermi energy and the number of particles per volume (when we replace L2 with V2/3):

 $E_f = frac{hbar^2}{2m} left( frac{3 pi^2 N}{V} right)^{2/3} ,$

The total energy of a fermi ball of N0 fermions is given by

$E = {int_0}^{N_0} E_f(N) dN = {3over 5} N_0 E_f$

Typical fermi energies

White dwarfs

Stars 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 1036 electrons/m3. This means their fermi energy is: A white dwarf is an astronomical object which is produced when a low to medium mass star dies. ... Sol redirects here. ... Degenerate matter is matter which has sufficiently high density that the dominant contribution to its pressure arises from the Pauli exclusion principle. ... In physics, the free electron model is a possible model for the behaviour of electrons in a crystal structure. ...

$E_f = frac{hbar^2}{2m_e} left( frac{3 pi^2 (10^{36})}{1 mathrm{m}^3} right)^{2/3} approx 3 times 10^5 mathrm{eV} ,$

Nucleus

Another typical example is that of the particles in a nucleus of an atom. The radius of the nucleus is roughly: The size of an atomic nucleus is of the order of metres. ...

$R = left(1.25 times 10^{-15} mathrm{m} right) times A^{1/3}$
where A is the number of nucleons.

The number density of nucleons in a nucleus is therefore: Nucleon is the common name used in nuclear chemistry to refer to a neutron or a proton, the components of an atoms nucleus. ...

$n = frac{A}{begin{matrix} frac{4}{3} end{matrix} pi R^3 } approx 1.2 times 10^{44} mathrm{m}^{-3}$

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. This article or section does not adequately cite its references or sources. ... For other uses, see Proton (disambiguation). ...

So the fermi energy of a nucleus is about:

$E_f = frac{hbar^2}{2m_p} left( frac{3 pi^2 (6 times 10^{43})}{1 mathrm{m}^3} right)^{2/3} approx 30 times 10^6 mathrm{eV} = 30 mathrm{MeV}$

Because the radius of the nucleus admits deviations around the value mentioned above, so would the fermi energy, and a typical value usually given is 38 MeV. The size of an atomic nucleus is of the order of metres. ... An electronvolt (symbol: eV) is the amount of energy gained by a single unbound electron when it falls through an electrostatic potential difference of one volt. ...

Fermi level

The Fermi level is the band level at absolute zero into which the valence electrons are placed. Since fermions cannot exist in identical energy states (see the exclusion principle), at absolute zero, electrons pack into the lowest available energy states and build up a "Fermi sea" of electron energy states. [1] In this state (at 0 K), the average energy of an electron is given by: The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ... For other uses, see Kelvin (disambiguation). ...

$E_{av} = frac{3}{5} E_f$

where Ef is the Fermi energy.

The Fermi momentum is the momentum of fermions at the Fermi surface. The Fermi momentum is given by: This article is about momentum in physics. ... In particle physics, fermions are particles with half-integer spin, such as protons and electrons. ... In condensed matter physics, the Fermi surface is an abstract boundary useful for predicting the thermal, electrical, magnetic, and optical properties of metals, semimetals, and doped semiconductors. ...

$p_F = sqrt{2 m_e E_f}$

where me is the mass of the electron.

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 relation between the energy of a system and its corresponding momentum is known as its dispersion relation. ... This article is about momentum in physics. ...

The Fermi velocity is the average velocity of an electron in an atom at absolute zero. This average velocity corresponds to the average energy given above. The Fermi velocity is defined by:

$V_f = sqrt{frac{2 E_f}{m_e}}$

where me is the mass of the electron.

Below the Fermi temperature, a substance gradually expresses more and more quantum effects of cooling. The Fermi temperature is defined by:

$T_f = frac{E_f}{k}$

where k is the Boltzmann constant. The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...

Quantum mechanics

According 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. In mathematics, a half-integer is a number of the form , where is an integer. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... For other uses, see Electron (disambiguation). ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ... Probability densities for the electron at different quantum numbers (l) In quantum mechanics, the quantum state of a system is a set of numbers that fully describe a quantum system. ... Fermi-Dirac distribution as a function of Îµ/Î¼ plotted for 4 different temperatures. ... The electronvolt (symbol eV) is a unit of energy. ...

Free electron gas

The Fermi energy of the free electron gas is related to the chemical potential by the equation In thermodynamics and chemistry, chemical potential, symbolized by Î¼, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a...

$mu = E_F left[ 1- frac{pi ^2}{12} left(frac{kT}{E_F}right) ^2 - frac{pi^4}{80} left(frac{kT}{E_F}right)^4 + cdots right]$

where EF is the Fermi energy, k is the Boltzmann constant and T is temperature. Hence, the chemical potential is approximately equal to the Fermi energy at temperatures of much less than the characteristic Fermi temperature EF/k. The characteristic temperature is on the order of 105 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. The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... For other uses, see Temperature (disambiguation). ... For other uses, see Kelvin (disambiguation). ... Fermi-Dirac distribution as a function of Îµ/Î¼ plotted for 4 different temperatures. ...

A Fermi gas is a collection of non-interacting fermions. ... A semiconductor is a solid whose electrical conductivity is in between that of a conductor and that of an insulator, and can be controlled over a wide range, either permanently or dynamically. ... Electrical Engineers design power systemsâ€¦ â€¦ and complex electronic circuits. ... This article is about the engineering discipline. ... Thermodynamics (from the Greek Î¸ÎµÏÎ¼Î·, therme, meaning heat and Î´Ï…Î½Î±Î¼Î¹Ï‚, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ...

References

Results from FactBites:

 Fermi energy - Wikipedia, the free encyclopedia (936 words) The Fermi energy is one of the central concepts of condensed matter physics. This is because the presence of neutrons does not affect the fermi energy of the protons in the nucleus. The Fermi momentum is the momentum of fermions at the Fermi surface.
 Fermi level and Fermi function (934 words) The concept of the Fermi energy is a crucially important concept for the understanding of the electrical and thermal properties of solids. The Fermi energy also plays an important role in understanding the mystery of why electrons do not contribute significantly to the specific heat of solids at ordinary temperatures, while they are dominant contributors to thermal conductivity and electrical conductivity. The Fermi function gives the probability of occupying an available energy state, but this must be factored by the number of available energy states to determine how many electrons would reach the conduction band.This density of states is the electron density of states, but there are differences in its implications for conductors and semiconductors.
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