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Title:Field theory of electrons and phonons
Author(s):Simkin, David Alan
Director of Research:Bardeen, John
Doctoral Committee Chair(s):Bardeen, John
Department / Program:Physics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D. (doctoral)
Abstract:Electrons in an alkali metal are allowed to interact by multiple processes, including direct coulomb interactions and retarded phonon exchanges, by including in the Hamiltonian both coulomb interaction and electron-phonon interaction terms. Corrections to low temperature one-electron properties are discussed for actual metallic densities by obtaining expressions for the electron self-energy, Sigma(p), in terms of single-particle Green's functions. The phonon terms, including Umklapp processes, are handled by means of Bardeen's matrix elements for the electron-phonon interaction, using longitudinal coupling and subjecting all directionally dependent quantities to a spherical averaging procedure. A criterion is developed to determine when phonon terms are comparable in magnitude to coulomb terms, and when they are negligible. Migdal has shown that phonon processes often contribute small terms proportional to factors of sqrt(m/M) (where m is electron mass and M is ion mass). In fact, phonon terms are of the same size as coulomb terms only for transitions lying within a narrow band of energies about the Fermi surface, whose width is of the order of the longitudinal phonon frequency, omega q. This leads to the general result that physical quantities depending upon Sigma(p), such as correlation energy or paramagnetic susceptibility, should show no phonon contribution to order sqrt(m/M), while quantities such as specific heat, which depend on (dsigma / dp)p = pf (density of states at the Fermi surface), show phonon contributions comparable to the coulomb contributions. A calculation of the phonon contribution to the linear electronic specific heat for sodium is made using the Nozieres-Pines interpolation scheme. Adding this to Silverstein's calculation of the coulomb terms yields a net enhancement over the free-electron value of 21 percent. Actually, for small wave-number vectors q, q less than or equal to sqrt(m/M) pf, it is shown that the electron-phonon interaction becomes just large enough to exactly cancel the divergences in the coulomb interaction perturbation series. This result depends upon the use of "bare," or unrenormalized, quantities, and makes strong use of the longitudinal sum rule for phonon frequencies. A new "combined" form of perturbation series is proposed which consists of keeping coulomb and phonon terms together in any integration process, and which depends, for consistency, on a certain contour in the complex energy-transfer plane, chosen to avoid phonon poles. The series does not use any infinite sums (screening) and has the advantage of being consistently defined for all q (no interpolation) in the region of actual metallic densities. A first order specific heat calculation predicts an enhancement of 32 percent over the free-electron value for sodium.
Issue Date:1963
Publisher:Department of Physics. College of Engineering. Graduate College. University of Illinois at Urbana-Champaign.
Genre:Dissertation / Thesis
Rights Information:Copyright 1963 David Alan Simkin
Date Available in IDEALS:2014-10-30

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