Il Nuovo Cimento D

, Volume 15, Issue 10, pp 1301–1314 | Cite as

Fokker-Planck equation and relevant Chapman-Cowling-Davydov expression modified for coulomb scattering

  • G. Cavalleri
  • G. Mauri
Article

Summary

TheP 1 approximation to the Boltzmann equation in the case of slightly ionized gases leads to the usual Fokker-Planck equation whose solution in steady-state condition is the Chapman-Cowling-Davydov (CCD) expression. We have extended the same procedure to include electron-ion interactions when the corresponding collision frequency is of the same order of, or larger than, the electron-neutral molecule collision frequency. We have considered a case where, after an initial ionization of a column of gas, the majority of the electrons have diffused to, and have been captured by, the walls of the chamber so that we may neglect the electron-electron interactions. We have found a modified CCD expression. However, numerical calculations show that the differences between the modified expressions and the standard ones in which the electron-ion collision frequency for momentum transfer is added to the electron-neutral collision frequency are beyond the present experimental accuracy.

PACS 52.25.Fi

Transport properties 

PACS 52.80.Dy

Low-field and Townsend discharges 

PACS 05.60

Transport processes: theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Chapman andT. G. Cowling:The Mathematical Theory of Non Uniform Gases, 3rd edition (Cambridge University Press, Cambridge, 1970), sect. 19.6, p. 386.Google Scholar
  2. [2]
    B. Davydov:Phys. Z. Sowjetunion,8, 59 (1935).MATHGoogle Scholar
  3. [3]
    For the use of d3 V′ instead of d3 V in the first integral, seeW. P. Allis: inHandbuch der Physik, edited byS. Flügge, Vol. 21 (Springer, Berlin, 1956), Chapt. III, sect. 28, p. 404. For the nomenclature seeG. Cavalleri andS. L. Paveri-Fontana:Phys. Rev. A,6, 327 (1972).Google Scholar
  4. [4]
    See, for example, the most recent treatises on electron-atom collisions, asA. Gilardini:Low Energy Electron Collisions in Gases (J. Wiley, New York, N.Y., 1972), eq. (2.3.27);L. B. Loeb:Recent Advances in Basic Processes of Gaseous Electronics (Berkeley University, Berkely, Cal., 1973);L. J. H. Huxley andR. W. Crompton:The Diffusion and Drift of Electrons in Gases (Wiley, New York, N.Y., 1974).Google Scholar
  5. [5]
    G. Cavalleri:Phys. Rev.,179, 186 (1969);D. K. Gibson, R. W. Crompton andG. Cavalleri:J. Phys. B,6, 1118 (1973);T. Rhymes, R. W. Crompton andG. Cavalleri:Phys. Rev. A.,12, 776 (1975).CrossRefADSGoogle Scholar
  6. [6]
    L. Landau:Phys. Z. Sowjetunion,10, 154 (1936);S. Chandrasekhar:Principles of Stellar Dynamics (University of Chicago Press, Chicago, Ill., 1939);Rev. Mod. Phys.,15, 1 (1943);R. S. Cohen, L. Spitzer jr. andP. Mc Routly:Phys. Rev.,80, 230 (1950);L. Spitzer jr. andR. Härm:Phys. Rev.,89, 977 (1953). However,L. Stenflo:Plasma Phys.,8, 665 (1966) uses the first approach.MATHGoogle Scholar
  7. [7]
    M. N. Rosenbluth, W. M. MacDonald andD. L. Judd:Phys. Rev.,107, 1 (1957).MATHCrossRefADSMathSciNetGoogle Scholar
  8. [8]
    G. Cavalleri andG. Mauri: inNonequilibrium Effects in Ion and Electron Transport, edited byJ. W. Gallagher, D. F. Hudson, E. E. Kunhardt andR. J. van Brunt (Plenum Press, New York, N.Y., 1990), p. 353.Google Scholar
  9. [9]
    The use of the marginal distribution functionf is convenient even for the calculation of diffusion coefficients: seeG. Cavalleri andG. Mauri:Phys. Rev. B,37, 6868 (1988-II).CrossRefADSGoogle Scholar
  10. [10]
    G. Cavalleri, E. Gatti andF. Gonzales-Gascon:Nuovo Cimento B,55, 291 (1980).CrossRefADSGoogle Scholar
  11. [11]
    L. C. Pitchford, S. V. Oneil andJ. R. Rumble:Phys. Rev. A,23, 294 (1981).CrossRefADSGoogle Scholar
  12. [12]
    G. Cavalleri:Aust. J. Phys.,34, 361 (1981), Appendix A1; alsoG. Cavalleri:Nuovo Cimento B,55, 360 (1980).ADSMathSciNetGoogle Scholar
  13. [13]
    H. Goldstein:Classical Mechanics (Addison-Wesley, London, 1959).Google Scholar
  14. [14]
    R. Balescu:Phys. Fluids,3, 52 (1960); see also:Statistical Mechanics of Charged Particles (Wiley-Interscience Publ., New York, N.Y., 1963), sect. 41.MATHCrossRefADSMathSciNetGoogle Scholar
  15. [15]
    E. M. Conwell andV. F. Weisskopf:Phys. Rev.,77, 388 (1950).MATHCrossRefADSGoogle Scholar
  16. [16]
    H. Brooks:Phys. Rev.,83, 879 (1951); inAdvances in Electronics and Electron Physics, edited byL. Marton, Vol.7 (Academic Press, New York, N.Y., 1955), p. 85;C. Herring andE. Vogt:Phys. Rev.,101, 944 (1956);105, 1933 (1957).Google Scholar
  17. [17]
    D. Chattopadhyay andH. J. Queisser:Rev. Mod. Phys.,53, 745 (1981), eqs. (26) and (34), translated into the Gauss system and puttingE=(1/2)m *v2.CrossRefADSGoogle Scholar
  18. [18]
    J. S. Blakemore:Semiconductor Statistics (Pergamon, Oxford, 1962).MATHGoogle Scholar
  19. [19]
    B. K. Ridley:J. Phys. C,10, 1589 (1977).CrossRefADSGoogle Scholar

Copyright information

© Società Italiana di Fisica 1993

Authors and Affiliations

  • G. Cavalleri
    • 1
  • G. Mauri
    • 2
  1. 1.Università Cattolica del Sacro CuoreBresciaItalia
  2. 2.Dipartimento di Matematica del Politecnico di MilanoMilanoItalia

Personalised recommendations