High Temperature

, Volume 56, Issue 1, pp 10–19 | Cite as

Analytical Theory of Energy Relaxation Upon Propagation of a High-Energy Electron Beam in Gas

Plasma Investigations


This study is devoted to the development of an analytical theory for calculating the spatial distribution of the energy release upon propagation of a high-energy (1–100 keV) electron beam in a gas (based on the example of air). Based on the analysis of data on the cross sections of elastic and inelastic interactions between electrons and molecules of gases that are in the air composition, it was suggested that inelastic interaction causes energy relaxation, whereas elastic interaction leads to momentum relaxation. The model cross section of inelastic collisions of electrons with molecules is used for solving the Boltzmann kinetic equation for electrons; this cross section provides adequate description of the experimentally found energy dependence of the mass stopping power of electrons. The results for the dependence of the mean electron energy on the number of inelastic collisions are in good agreement with the results of calculation based on expansion of the distribution function in the number of collisions and solution by the Monte Carlo method. The calculations we performed show that the consideration of elastic collisions increases the spatial density of the energy release due to narrowing of the region where the main part of the energy of fast electrons is released, in comparison with calculations where only inelastic deceleration is taken into account.


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  1. 1.
    Fraishtadt, V.L., Kuranov, A.L., and Sheikin, E.G., Tech. Phys., 1998, vol. 43, no. 11, p. 1309.CrossRefGoogle Scholar
  2. 2.
    Sheikin, E.G. and Kuranov, A.L., MHD control in hypersonic aircraft, AIAA Pap. 2005-1335, 2005.CrossRefGoogle Scholar
  3. 3.
    Bityurin, V. and Bocharov, A.N., High Temp., 2010, vol. 48, no. 6, p. 874.CrossRefGoogle Scholar
  4. 4.
    Park, C., Bogdanoff, D.W., and Mehta, U.B., J. Propul. Power, 2003, vol. 9, no. 4, p. 529.CrossRefGoogle Scholar
  5. 5.
    Palm, P., Plonjes, E., Adamovich, I.V., and Rich, J.W., E-beam sustained low power budget air plasma, AIAA Pap. 2002-0637, 2002.CrossRefGoogle Scholar
  6. 6.
    Sheikin, E.G., Nucl. Instrum. Methods Phys. Res., Sect. B, 2014, vol. 319, p. 1.ADSCrossRefGoogle Scholar
  7. 7.
    Sheikin, E.G., Phys. Scr., 2010, 81045702.Google Scholar
  8. 8.
    Sheikin, E.G. and Sukhomlinov, V.S., Calculation of space distribution of energy deposited by E-beam for flow control applications, AIAA Pap. 2006-1369, 2006.CrossRefGoogle Scholar
  9. 9.
    Marchuk, G.I., Metody rascheta yadernykh reaktorov (Methods for Calculating Nuclear Reactors), Moscow: Gosatomizdat, 1961.Google Scholar
  10. 10.
    Kol’chuzhkin, A.M. and Uchaikin, V.V., Vvedenie v teoriyu prokhozhdeniya chastits cherez veshchestvo (Introduction to the Theory of the Passage of Particles through a Substance), Moscow: Atomizdat, 1978.Google Scholar
  11. 11.
    Stopping powers for electrons and positrons, ICRU Rep. no. 37, Bethesda, MD: Int. Commission on Radiation Units and Measurements, 1984.Google Scholar
  12. 12.
    Mayol, R.R. and Salvat, F., Atom. Data Nucl. Data Tables, 1997, vol. 65, no. 21, p. 55.ADSCrossRefGoogle Scholar
  13. 13.
    Akkerman, A.F., Modelirovanie traektorii zaryazhennykh chastits v veshchestve (Simulation of the Trajectory of Charged Particles in Matter), Moscow: Energoatomizdat, 1991.Google Scholar
  14. 14.
    Korn, G. and Korn, T., Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas, New York: McGraw-Hill, 1968.MATHGoogle Scholar
  15. 15.
    Macheret, S.O., Shnieder, M.N., and Miles, R.B., External supersonic flow and scramjet intel control by MHD with electron beam ionization, AIAA Pap. 2001-0492, 2001.Google Scholar
  16. 16.
    Dapor, M., Nucl. Instrum. Methods Phys. Res., Sect. B, 2011, vol. 269, p. 1668.ADSCrossRefGoogle Scholar
  17. 17.
    Bateman, H. and Erdélyi, A., Tables of Integral Transforms, New-York: McGraw-Hill, 1954, vol. 1.Google Scholar
  18. 18.
    Abramowits, M. and Stegun, I.A, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, 1964.Google Scholar
  19. 19.
    Cercignani, C., Theory and Application of the Boltzmann Equation, Edinburgh: Scottish Academic Press, 1975.MATHGoogle Scholar
  20. 20.
    Cercignani, C., Mathematical Methods in Kinetic Theory, New York, Plenum, 1969.CrossRefMATHGoogle Scholar
  21. 21.
    Von Grün, A.E., Z. Naturforsch., A: Phys. Sci., 1957, vol. 12, no. 2, p. 89.ADSCrossRefGoogle Scholar
  22. 22.
    Fröman, N. and Fröman, P.O., JWKB Approximation, Contributions to the Theory, Amsterdam: North-Holland, 1965.MATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of PhysicsSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.St. Petersburg Mining UniversitySt. PetersburgRussia

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