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	~~Jay Arneson~~ is ~~selected his parents gave him but he doesn't~~ like ~~when people use his full name~~. ~~Years ago we relocated~~ to ~~Georgia nevertheless need~~ to ~~relocate~~ for ~~our kids~~. ~~Auditing~~ is ~~my profession~~. ~~Her friends say it isn~~'~~t good~~ for ~~her but what she loves doing~~ in order to ~~use play new bands~~ but ~~she hasn't developed~~ a ~~dime~~ by it. ~~If you want~~ to ~~find uot more check out his website: http~~://~~party-bus-philadelphia~~.~~com~~		{{TOCright}}
			'''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 charge]]s (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 [[Electric field screening\|screening]], [[plasmon]]s, [[Wigner crystal]]lization and [[Friedel oscillation]]s.

			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.<ref>{{cite journal\|doi=10.1162/posc.2006.14.4.457\|last=Hughes\|first=R. I. G. \|year=2006\|title=Theoretical Practice: the Bohm-Pines Quartet\|journal=Perspectives on Science\|volume=14\|issue=4\|pages=457–524\|url=http://muse.jhu.edu/journals/perspectives_on_science/v014/14.4hughes.pdf}}</ref>

			==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 [[Hamiltonian (quantum mechanics)\|Hamiltonian]] 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<ref>{{cite book\|last=Gross\|first=E. K. U.\|coauthors=Runge, E.; Heinonen, O.\|title=Many-Particle Theory\|publisher=Verlag Adam Hilger\|location=Bristol\|year=1991\|pages=79–80\|isbn=0-7503-0155-4}}</ref><ref>{{cite book\|last=Giuliani\|first=Gabriele\|coauthors=Vignale; Giovanni\|title=Quantum Theory of the Electron Liquid\|publisher=Cambridge University Press\|year=2005\|pages=13–16\|isbn=978-0-521-82112-4}}</ref>

			:<math> \hat{H}=\hat{H}_{\mathrm{el}}+\hat{H}_{\mathrm{back}}+\hat{H}_{\mathrm{el-back}},\,</math>

			where

			*''H''<sub>el</sub> is the electronic Hamiltonian consisting of the kinetic and electron-electron repulsion terms:
			::<math> \hat{H}_{\mathrm{el}}=\sum_{i=1}^N\frac{p_{i}^2}{2m}+\sum_{i<j}^N\frac{e^2}{\|\mathbf{r}_i-\mathbf{r}_j\|}</math>
			*''H''<sub>back</sub> is the Hamiltonian of the positive background charge interacting [[electrostatic]]ally with itself:
			::<math> \hat{H}_{\mathrm{back}}=\frac{e^2}{2}\int_{\Omega}\mathrm{d}\mathbf{R}\int_{\Omega}\mathrm{d}\mathbf{R}'\ \frac{n(\mathbf{R})n(\mathbf{R}')}{\|\mathbf{R}-\mathbf{R}'\|} = \frac{e^{2}}{2}\left(\frac{N}{\Omega}\right)^{2}\int_{\Omega}\mathrm{d}\mathbf{R}\int_{\Omega}\mathrm{d}\mathbf{R}'\ \frac{1}{\|\mathbf{R}-\mathbf{R}'\|}</math>
			*''H''<sub>el-back</sub> is the electron-background interaction Hamiltonian, again an electrostatic interaction:
			::<math> \hat{H}_{\mathrm{el-back}}=\int_{\Omega}\mathrm{d}\mathbf{r}\int_{\Omega}\mathrm{d}\mathbf{R}\ \frac{\rho(\mathbf{r})n(\mathbf{R})}{\|\mathbf{r}-\mathbf{R}\|} = -e^{2}\frac{N}{\Omega}\sum_{i=1}^{N}\int_{\Omega}\mathrm{d}\mathbf{R}\ \frac{1}{\|\mathbf{r}_{i}-\mathbf{R}\|}</math>

			''H''<sub>back</sub> is a constant and, in the limit of an infinite volume, divergent along with ''H''<sub>el-back</sub>. 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
			: <math> KE = \frac{3}{5} E_F = \frac{3}{5}\frac{\hbar ^2k_F^2}{2m_e} = \frac{2.21}{r_s^2} \textrm{Ryd} </math>
			where <math> E_F</math> is the Fermi energy, <math> k_F </math> is the Fermi wave vector, and the last expression shows the dependence on the [[Wigner-Seitz radius]] <math> r_s </math> where energy is measured in [[Rydberg constant\|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 <math> 1/r_{12} </math> (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 <math> r_s </math> (i.e. <math>1/r_s^2</math> being larger than <math>1/r_s</math>) and hence high electron density.. Unfortunately, real metals typically have <math>r_s</math> between 2-5 which means this picture needs serious revision.

			The first correction to the free-electron model for jelium is from the [[Hartree–Fock method\|Fock exchange]] contribution to electron-electron interactions. Adding this in, one has a total energy of
			: <math> E = \frac{2.21}{r_s^2} - \frac{0.916}{r_s}</math>
			where the negative term is due to exchange: exchange interactions lower the total energy. Higher order
			corrections to the total energy are due to [[Electronic correlation\|electron correlation]] and if one
			decides to work in a series for small <math>r_s</math>, one finds
			: <math> E = \frac{2.21}{r_s^2} - \frac{0.916}{r_s} + 0.0622 \ln (r_s) - 0.096 + O(r_s)</math>
			The series is quite accurate for small <math>r_s</math> but of dubious value for <math>r_s</math> 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 electron]]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 properties such as [[work function]] and surface effects such as [[adsorption]]; near surfaces the electronic density varies in an oscillatory manner, decaying to a constant value in the bulk.<ref>{{cite journal\|last=Lang\|first=N. D.\|year=1969\|title=Self-consistent properties of the electron distribution at a metal surface\|journal=Solid State Commun.\|volume=7\|issue=15\|pages=1047–1050\|doi=10.1016/0038-1098(69)90467-0\|bibcode = 1969SSCom...7.1047L }}</ref><ref>{{cite journal\|last=Lang\|first=N. D.\|coauthors=Kohn, W.\|year=1970\|title=Theory of Metal Surfaces: Work Function\|journal=Phys. Rev. B\|volume=3\|issue=4\|pages=1215–223\|doi=10.1103/PhysRevB.3.1215\|bibcode = 1971PhRvB...3.1215L }}</ref><ref>{{cite journal\|last=Lang\|first=N. D.\|coauthors=Kohn, W.\|year=1973\|title=Surface-Dipole Barriers in Simple Metals\|journal=Phys. Rev. B\|volume=8\|issue=12\|pages=6010–6012\|doi=10.1103/PhysRevB.8.6010\|bibcode = 1973PhRvB...8.6010L }}</ref>

			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,<ref>{{cite journal \| title = Ground State of the Electron Gas by a Stochastic Method \| author = D. M. Ceperley and B. J. Alder \| journal = Phys. Rev. Lett. \| volume = 45 \| pages = 566–569 \| year = 1980 \| doi = 10.1103/PhysRevLett.45.566 \| bibcode=1980PhRvL..45..566C \| issue = 7}}</ref> which have been used to construct semi-empirical correlation functionals.<ref>{{cite journal\|last=Perdew\|first=J. P.\|coauthors=McMullen, E. R.; Zunger, Alex\|year=1981\|title=Density-functional theory of the correlation energy in atoms and ions: A simple analytic model and a challenge\|journal=Phys. Rev. A\|volume=23\|issue=6\|pages=2785–2789\|doi=10.1103/PhysRevA.23.2785\|bibcode = 1981PhRvA..23.2785P }}</ref>

			The jellium model has been applied to [[superatoms]], and used in [[nuclear physics]].

			==See also==
			*[[Free electron model]] — a model electron gas where the electrons do not interact with anything.
			*[[Nearly free electron model]] — a model electron gas where the electrons do not interact with each other, but do feel a (weak) potential from the atomic lattice.

			==References==
			{{reflist\|2}}

			[[Category:Condensed matter physics]]
			[[Category:Density functional theory]]
			[[Category:Nuclear physics]]

en>777sms at 03:04, 14 April 2012

2012-04-14T03:04:00Z

New page

Jay Arneson is selected his parents gave him but he doesn't like when people use his full name. Years ago we relocated to Georgia nevertheless need to relocate for our kids. Auditing is my profession. Her friends say it isn't good for her but what she loves doing in order to use play new bands but she hasn't developed a dime by it. If you want to find uot more check out his website: http://party-bus-philadelphia.com

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en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q2103196

en>777sms at 03:04, 14 April 2012