Revision of the Statistical Mechanics of Phonons to Include Phonon Linewidths

  • W. C. OvertonJr.
Conference paper
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 51)


Zubarev[1] in 1960 obtained the “smeared” Bose-Einstein (B-E) function in order to take into account the fact that the eigenenergy associated with a fixed phonon wave vector q and fixed polarization index j is not precisely defined but, instead, is smeared by phonon-phonon and phonon-electron interactions. The ratio Γ(qj)/ω(qj) is often quite small, i.e., of the order of 0.01 or less, where Γ is the phonon linewidth and ħ ω is the eigenenergy. However, in strongly anharmonic crystals Γ /ω may be as large as 0.3 at certain points of the Brillouin zone. In such dramatic cases, one would suspect that such phonon linewidths would have some observable effect on the thermodynamic properties. Zubarev represented the effect of “smearing” on the statistical properties by the infinite integral[1],
$$ \overline n = \int_{ - \infty }^\infty d \omega \;{\left[ {\exp (\frac{{h\omega }}{{kT}}) - 1} \right]^{ - 1}}\;L(\omega ;\overline \omega ,\Gamma )\;, $$
$$ L(\omega ;\overline \omega ,\Gamma )\; = \;(\Gamma /\pi ){\left[ {{{(\omega - \overline \omega )}^2} + {\Gamma ^2}} \right]^{ - 1}}, $$
in which we have deleted the indices (q,j) for convenience. The term in square brackets in (1) is the usual B-E function, while in L, the usual Lorentzian function, \( \overline \omega \) is the average or center frequency of the distribution.


Half Plane Lower Half Plane Summation Part Small Semicircle High Temperature Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. N. Zubarev: [translation; Soviet Phys.-Usp. 3, 320 (1960)].Google Scholar
  2. 2.
    P. M. Morse and H. Feshbach: Theoretical Physics, McGraw Hill, New York, (1953).MATHGoogle Scholar
  3. 3.
    A. A. Maradudin and E. A. Fein: Phys. Rev. 128, 2562 (1962).CrossRefGoogle Scholar
  4. 4.
    B. N. Brockhouse, T. Arase, G. Caglioti, M. Sikamoto, R. N. Sinclair, and A. D. Woods: Inelastic Scattering of Neutrons, IAEA, Vienna (1961).Google Scholar
  5. 5.
    W. C. Overton, Jr.: J. Phys. Chem. Solids 29, 711 (1968).CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • W. C. OvertonJr.
    • 1
  1. 1.Los Alamos National LaboratoryLos AlamosUSA

Personalised recommendations