Jellium
Encyclopedia
Jellium, also known as the uniform electron gas (UEG) or homogeneous electron gas (HEG), is a quantum mechanical model of interacting electron
s in a solid where the positive charges (i.e. atomic nuclei) are assumed to be uniformly distributed in space whence the electron density
is a uniform quantity as well in space. This model allows one to focus on the effects in solids that
occur due to the quantum nature of electrons and their mutual repulsive interactions (due to like charge)
without explicit introduction of the atomic lattice and structure making up a real material. Jellium is often used in solid-state physics
as a simple model of delocalized electrons in a metal, where it can qualitatively reproduce features of real metals such as screening
, plasmon
s, Wigner crystal
lization and Friedel oscillations.
At zero temperature, the properties of jellium depend solely upon the constant electronic density
. This lends it to a treatment within density functional theory
; the formalism itself provides the basis for the local-density approximation
to the exchange-correlation energy density functional.
The term jellium was coined by Conyers Herring
, alluding to the "positive jelly" background, and the typical metallic behavior it displays.
for N-electrons confined within a volume of space Ω, and with electronic density
ρ(r) and background charge density n(R) = N/Ω is
where
Hback is a constant and, in the limit of an infinite volume, divergent along with Hel-back. The divergence is canceled by a term from the electron-electron coupling: the background interactions cancel and the system is dominated by the kinetic energy and coupling of the electrons. Such analysis is done in Fourier space; the interaction terms of the Hamiltonian which remain correspond to the Fourier expansion of the electron coupling for which q ≠ 0.
where
is the Fermi energy, 
is the Fermi wave vector, and the last expression shows the dependence on the Wigner-Seitz radius
where energy is measured in Rydbergs
.
Without doing much work, one can guess that the electron-electron interactions will scale like the inverse of the average electron-electron separation and hence as
(since the Coulomb interaction goes like one over distance between charges) so that if we view the interactions as a small correction to the kinetic energy, we are describing the limit of small 
(i.e. 
being larger than 
) and hence high electron density.. Unfortunately, real metals typically have 
between 2-5 which means this picture needs serious revision.
The first correction to the free-electron model for jelium is from the Fock exchange contribution to electron-electron interactions. Adding this in, one has a total energy of
where the negative term is due to exchange: exchange interactions lower the total energy. Higher order
corrections to the total energy are due to electron correlation
and if one
decides to work in a series for small
, one finds
The series is quite accurate for small
but of dubious value for 
values found in actual metals.
s and the nuclei are modeled as the uniform positive background and the valence electron
s are treated with full rigor. Semi-infinite jellium slabs are used to investigate surface effects and adsorption
, in which case the electronic density varies at the edges and tends to a constant value in the bulk.
Within density functional theory
, jellium is used in the construction of the local-density approximation
, which in turn is a component of more sophisticated exchange-correlation energy functionals. From quantum Monte Carlo
calculations of jellium, accurate values of the correlation energy density have been obtained for several values of the electronic density, which have been used to construct semi-empirical correlation functionals.
The jellium model has been applied to superatoms, and used in nuclear physics
.
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s in a solid where the positive charges (i.e. atomic nuclei) are assumed to be uniformly distributed in space whence the electron density
is a uniform quantity as well in space. This model allows one to focus on the effects in solids that
occur due to the quantum nature of electrons and their mutual repulsive interactions (due to like charge)
without explicit introduction of the atomic lattice and structure making up a real material. Jellium is often used in solid-state physics
Solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from...
as a simple model of delocalized electrons in a metal, where it can qualitatively reproduce features of real metals such as screening
Electric field screening
Screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases and conduction electrons in semiconductors and metals....
, plasmon
Plasmon
In physics, a plasmon is a quantum of plasma oscillation. The plasmon is a quasiparticle resulting from the quantization of plasma oscillations just as photons and phonons are quantizations of light and mechanical vibrations, respectively...
s, Wigner crystal
Wigner crystal
A Wigner crystal is the solid phase of electrons first predicted by Eugene Wigner in 1934. A gas of electrons moving in 2D or 3D in a uniform, inert, neutralizing background will crystallize and form a lattice if the electron density is less than a critical value...
lization and Friedel oscillations.
At zero temperature, the properties of jellium depend solely upon the constant electronic density
Electronic density
In quantum mechanics, and in particular quantum chemistry, the electronic density is a measure of the probability of an electron occupying an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either...
. This lends it to a treatment within density functional theory
Density functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
; the formalism itself provides the basis for the local-density approximation
Local-density approximation
Local-density approximations are a class of approximations to the exchange-correlation energy functional in density functional theory that depend solely upon the value of the electronic density at each point in space . Many approaches can yield local approximations to the XC energy...
to the exchange-correlation energy density functional.
The term jellium was coined by Conyers Herring
Conyers Herring
Conyers Herring was an American physicist. He was Professor of Applied Physics at Stanford University and the Wolf Prize in Physics recipient in 1984/5.-Academic career:...
, alluding to the "positive jelly" background, and the typical metallic behavior it displays.
Hamiltonian
The jellium model treats the electron-electron coupling rigorously. The artificial and structureless background charge interacts electrostatically with itself and the electrons. The jellium HamiltonianHamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
for N-electrons confined within a volume of space Ω, and with electronic density
Electronic density
In quantum mechanics, and in particular quantum chemistry, the electronic density is a measure of the probability of an electron occupying an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either...
ρ(r) and background charge density n(R) = N/Ω is
where
- Hel is the electronic Hamiltonian consisting of the kinetic and electron-electron repulsion terms:
-
- Hback is the Hamiltonian of the positive background charge interacting electrostatically with itself:
- Hel-back is the electron-background interaction Hamiltonian, again an electrostatic interaction:
Hback is a constant and, in the limit of an infinite volume, divergent along with Hel-back. The divergence is canceled by a term from the electron-electron coupling: the background interactions cancel and the system is dominated by the kinetic energy and coupling of the electrons. Such analysis is done in Fourier space; the interaction terms of the Hamiltonian which remain correspond to the Fourier expansion of the electron coupling for which q ≠ 0.
Contributions to the total energy
The traditional way to study the electron gas is to start with non-interacting electrons which are governed only by the kinetic energy part of the Hamiltonian, which yields the free electron gas model. The kinetic energy per electron is given by
where
is the Fermi energy, is the Fermi wave vector, and the last expression shows the dependence on the Wigner-Seitz radiusWigner-Seitz radius
The Wigner–Seitz radius r_s, named after Eugene Wigner and Frederick Seitz, is a parameter used frequently in condensed matter physics to describe the density of a system...
where energy is measured in RydbergsRydberg constant
The Rydberg constant, symbol R∞, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to atomic spectra in the science of spectroscopy. Rydberg initially determined its value empirically from spectroscopy, but Niels Bohr later showed that its value could be calculated...
.
Without doing much work, one can guess that the electron-electron interactions will scale like the inverse of the average electron-electron separation and hence as
(since the Coulomb interaction goes like one over distance between charges) so that if we view the interactions as a small correction to the kinetic energy, we are describing the limit of small (i.e. being larger than ) and hence high electron density.. Unfortunately, real metals typically have between 2-5 which means this picture needs serious revision.The first correction to the free-electron model for jelium is from the Fock exchange contribution to electron-electron interactions. Adding this in, one has a total energy of
where the negative term is due to exchange: exchange interactions lower the total energy. Higher order
corrections to the total energy are due to electron correlation
Electronic correlation
Electronic correlation is the interaction between electrons in the electronic structure of a quantum system.- Atomic and molecular systems :...
and if one
decides to work in a series for small
, one finds
The series is quite accurate for small
but of dubious value for values found in actual metals.Applications
Jellium is the simplest model of interacting electrons. It is employed in the calculation of properties of metals, where the core electronCore electron
Core electrons are the electrons in an atom that are not valence electrons and therefore do not participate in bonding. An example: the carbon atom has a total of 6 electrons, 4 of them being valence electrons. So the remaining 2 electrons must be core electrons.They are so tightly bound to the...
s and the nuclei are modeled as the uniform positive background and the valence electron
Valence electron
In chemistry, valence electrons are the electrons of an atom that can participate in the formation of chemical bonds with other atoms. Valence electrons are the "own" electrons, present in the free neutral atom, that combine with valence electrons of other atoms to form chemical bonds. In a single...
s are treated with full rigor. Semi-infinite jellium slabs are used to investigate surface effects and adsorption
Adsorption
Adsorption is the adhesion of atoms, ions, biomolecules or molecules of gas, liquid, or dissolved solids to a surface. This process creates a film of the adsorbate on the surface of the adsorbent. It differs from absorption, in which a fluid permeates or is dissolved by a liquid or solid...
, in which case the electronic density varies at the edges and tends to a constant value in the bulk.
Within density functional theory
Density functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
, jellium is used in the construction of the local-density approximation
Local-density approximation
Local-density approximations are a class of approximations to the exchange-correlation energy functional in density functional theory that depend solely upon the value of the electronic density at each point in space . Many approaches can yield local approximations to the XC energy...
, which in turn is a component of more sophisticated exchange-correlation energy functionals. From quantum Monte Carlo
Quantum Monte Carlo
Quantum Monte Carlo is a large class of computer algorithms that simulate quantum systems with the idea of solving the quantum many-body problem. They use, in one way or another, the Monte Carlo method to handle the many-dimensional integrals that arise...
calculations of jellium, accurate values of the correlation energy density have been obtained for several values of the electronic density, which have been used to construct semi-empirical correlation functionals.
The jellium model has been applied to superatoms, and used in nuclear physics
Nuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...
.