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Jellium, also known as the uniform electron gas (UEG) or homogeneous electron gas (HEG), is a quantum mechanical model of interacting electrons within an infinite volume of space and neutralized with a uniformly distributed background positive charge. 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 plasmons 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.^[1]

[edit] 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 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^[2]^[3]

$\hat{H}=\hat{H}_{\mathrm{el}}+\hat{H}_{\mathrm{back}}+\hat{H}_{\mathrm{el-back}},\,$

where

H_el is the electronic Hamiltonian consisting of the kinetic and electron-electron repulsion terms:

$\hat{H}_{\mathrm{el}}=\sum_{i=1}^N\frac{p_{i}^2}{2m}+\sum_{i<j}^N\frac{e^2}{|\mathbf{r}_i-\mathbf{r}_j|}$

H_back is the Hamiltonian of the positive background charge interacting electrostatically with itself:

$\hat{H}_{\mathrm{back}}=\frac{1}{2}\int_{\Omega}\mathrm{d}\mathbf{R}\int_{\Omega}\mathrm{d}\mathbf{R}'\ \frac{n(\mathbf{R})n(\mathbf{R}')}{|\mathbf{R}-\mathbf{R}'|} = \frac{e^{2}}{2}\left(\frac{N}{\Omega}\right)^{2}\int_{\Omega}\mathrm{d}\mathbf{R}\int_{\Omega}\mathrm{d}\mathbf{R}'\ \frac{1}{|\mathbf{R}-\mathbf{R}'|}$

H_el-back is the electron-background interaction Hamiltonian, again an electrostatic interaction:

$\hat{H}_{\mathrm{el-back}}=\int_{\Omega}\mathrm{d}\mathbf{r}\int_{\Omega}\mathrm{d}\mathbf{R}\ \frac{\rho(\mathbf{r})n(\mathbf{R})}{|\mathbf{r}-\mathbf{R}|} = -e^{2}\frac{N}{\Omega}\sum_{i=1}^{N}\int_{\Omega}\mathrm{d}\mathbf{R}\ \frac{1}{|\mathbf{r}_{i}-\mathbf{R}|}$

H_back is a constant and, in the limit of an infinite volume, divergent along with H_el-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.

[edit] Applications

Jellium is the simplest model of interacting electrons. It is employed in the calculation of properties of metals, where the core electrons and the nuclei are modeled as the uniform positive background and the valence electrons 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.^[4]^[5]^[6]

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,^[7] which have been used to construct semi-empirical correlation functionals.^[8]

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

[edit] See also

[edit] References

^ Hughes, R. I. G. (2006). "Theoretical Practice: the Bohm-Pines Quartet". Perspectives on Science 14 (4): 457–524. http://muse.jhu.edu/journals/perspectives_on_science/v014/14.4hughes.pdf.
^ Gross, E. K. U.; Runge, E.; Heinonen, O. (1991). Many-Particle Theory. Bristol: Verlag Adam Hilger. pp. 79–80. ISBN 0-7503-0155-4.
^ Giuliani, Gabriele; Vignale; Giovanni (2005). Quantum Theory of the Electron Liquid. Cambridge University Press. pp. 13–16. ISBN 978-0-521-82112-4.
^ Lang, N. D. (1969). "Self-consistent properties of the electron distribution at a metal surface". Solid State Commun. 7 (15): 1047–1050. doi:10.1016/0038-1098(69)90467-0.
^ Lang, N. D.; Kohn, W. (1970). "Theory of Metal Surfaces: Work Function". Phys. Rev. B 3 (4): 1215–223. doi:10.1103/PhysRevB.3.1215.
^ Lang, N. D.; Kohn, W. (1973). "Surface-Dipole Barriers in Simple Metals". Phys. Rev. B 8 (12): 6010–6012. doi:10.1103/PhysRevB.8.6010.
^ D. M. Ceperley and B. J. Alder (1980). "Ground State of the Electron Gas by a Stochastic Method". Phys. Rev. Lett. 45: 566–569. doi:10.1103/PhysRevLett.45.566.
^ Perdew, J. P.; McMullen, E. R.; Zunger, Alex (1981). "Density-functional theory of the correlation energy in atoms and ions: A simple analytic model and a challenge". Phys. Rev. A 23 (6): 2785–2789. doi:10.1103/PhysRevA.23.2785.

[0] Hughes, R. I. G. (2006). "Theoretical Practice: the Bohm-Pines Quartet". Perspectives on Science 14 (4): 457–524. http://muse.jhu.edu/journals/perspectives_on_science/v014/14.4hughes.pdf.

[1] Gross, E. K. U.; Runge, E.; Heinonen, O. (1991). Many-Particle Theory. Bristol: Verlag Adam Hilger. pp. 79–80. ISBN 0-7503-0155-4.

[2] Giuliani, Gabriele; Vignale; Giovanni (2005). Quantum Theory of the Electron Liquid. Cambridge University Press. pp. 13–16. ISBN 978-0-521-82112-4.

[3] Lang, N. D. (1969). "Self-consistent properties of the electron distribution at a metal surface". Solid State Commun. 7 (15): 1047–1050. doi:10.1016/0038-1098(69)90467-0.

[4] Lang, N. D.; Kohn, W. (1970). "Theory of Metal Surfaces: Work Function". Phys. Rev. B 3 (4): 1215–223. doi:10.1103/PhysRevB.3.1215.

[5] Lang, N. D.; Kohn, W. (1973). "Surface-Dipole Barriers in Simple Metals". Phys. Rev. B 8 (12): 6010–6012. doi:10.1103/PhysRevB.8.6010.

[6] D. M. Ceperley and B. J. Alder (1980). "Ground State of the Electron Gas by a Stochastic Method". Phys. Rev. Lett. 45: 566–569. doi:10.1103/PhysRevLett.45.566.

[7] Perdew, J. P.; McMullen, E. R.; Zunger, Alex (1981). "Density-functional theory of the correlation energy in atoms and ions: A simple analytic model and a challenge". Phys. Rev. A 23 (6): 2785–2789. doi:10.1103/PhysRevA.23.2785.

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Contents

[edit] Hamiltonian

[edit] Applications

[edit] See also

[edit] References