Uniqueness Theorem for the Non-Local Ionization Source in Glow Discharge and Hollow Cathode

Affiliation(s)

Kyiv National Taras Shevchenko University, Kiev, Ukraine,Moscow Institute of Physics and Technology, Moscow, Russia.

Kyiv National Taras Shevchenko University, Kiev, Ukraine,Moscow Institute of Physics and Technology, Moscow, Russia.

ABSTRACT

The paper is devoted to the proof of the uniqueness theorem for solution of the equation for the non-local ionization source in a glow discharge and a hollow cathode in general 3D geometry. The theorem is applied to wide class of electric field configurations, and to the walls of discharge volume, which have a property of incomplete absorption of the electrons. Cathode is regarded as interior singular source, which is placed arbitrarily close to the wall. The existence of solution is considered also. During the proof of the theorem many of useful structure formulae are obtained. Elements of the proof structure, which have arisen, are found to have physical sense. It makes clear physical construction of non-local electron avalanche, which builds a source of ionization in glow discharge at low pressures. Last has decisive significance to understand the hollow cathode discharge configuration and the hollow cathode effect.

The paper is devoted to the proof of the uniqueness theorem for solution of the equation for the non-local ionization source in a glow discharge and a hollow cathode in general 3D geometry. The theorem is applied to wide class of electric field configurations, and to the walls of discharge volume, which have a property of incomplete absorption of the electrons. Cathode is regarded as interior singular source, which is placed arbitrarily close to the wall. The existence of solution is considered also. During the proof of the theorem many of useful structure formulae are obtained. Elements of the proof structure, which have arisen, are found to have physical sense. It makes clear physical construction of non-local electron avalanche, which builds a source of ionization in glow discharge at low pressures. Last has decisive significance to understand the hollow cathode discharge configuration and the hollow cathode effect.

Cite this paper

V. Gorin, "Uniqueness Theorem for the Non-Local Ionization Source in Glow Discharge and Hollow Cathode,"*Journal of Modern Physics*, Vol. 3 No. 10, 2012, pp. 1647-1662. doi: 10.4236/jmp.2012.330202.

V. Gorin, "Uniqueness Theorem for the Non-Local Ionization Source in Glow Discharge and Hollow Cathode,"

References

[1] F. Paschen, “Bohrs Heliumlinien,” Annalen der Physik, Vol. 355, No. 16, 1916, pp. 901-940. doi:10.1002/andp.19163551603

[2] A. Engel and M. Shteenbeck, “Physics and Technology of Electric Discharge in Gases 2”, ONTI, Мoscow, 1936.

[3] Yu P. Rayzer, “Physics of Gas Discharge,” Nauka, Мoscow, 1987.

[4] B. I. Moskalev, “Discharge with Hollow Cathode,” Energia, Мoscow, 1969.

[5] K. Kutasi and Z. Donko, “Hybrid Model of a Plane-Parallel Hollow-Cathode Discharge,” Journal of Physics D: Applied Physics, Vol. 33, No. 9, 2000, pp. 1081-1089. doi:10.1088/0022-3727/33/9/307

[6] F. Sigeneger and R. Winkler, “Study of the Electron Kinetics in Cylindrical Hollow Cathodes by a Multi-Term Approach,” The European Physical Journal Applied Physics, Vol. 19, No. 3, 2002, pp. 211-223. doi:10.1051/epjap:2002068

[7] N. Baguer, A. Bogaerts and R. Gijbels, “Hollow Cathode Glow Discharge in He: Monte Carlo-Fluid Model Combined with a Transport Model for Metastable Atoms,” Journal of Applied Physics, Vol. 93, No. 1, 2003, pp. 47- 55. doi:10.1063/1.1518784

[8] F. Sigeneger, Z. Donko and D. Loffhagen, “Boltzmann Equation and Particle-Fluid Hybrid Modelling of a Hollow Cathode Discharge,” The European Physical Journal Applied Physics, Vol. 38, No. 2, 2007, pp. 161-167. doi:10.1051/epjap:2007067

[9] A. Derzsi, P. Hartmann, I. Korolov, J. Karacsony, G. Bano and Z. Donko, “On the Accuracy and Limitations of Fluid Models of the Cathode Region of DC Glow Discharges” Journal of Physics D: Applied Physics, Vol. 42 No. 22, 2009, Article ID: 225204. doi:10.1088/0022-3727/42/22/225204

[10] D. Hilbert, “Grundzüge Einer Allgemeinen Theorie der Linearen Integralgleichungen,” B. G. Teubner, Leipzig, 1912, Chelsea Publishing Company, New York, 1953.

[11] V. V. Gorin, “Non-Local Source Equation for Linear Stationary Kinetic Equation,” Proceedings of the Conference on Differential Equations and Topology, Moscow, 17- 22 June 2008, pp. 43-44.

[12] V. V. Gorin, “Non-Local Model of Hollow Cathode and Glow Discharge—Theory Calculations and Experiment Comparison,” European Physical Journal D, Vol. 59, No. 2, 2010, pp. 241-247. doi:10.1140/epjd/e2010-00165-9

[13] V. V. Gorin, “A Mathematical Model of Plane Glow Discharge and Hollow Cathode Effect,” Ukrainian Journal of Physics, Vol. 53, No. 4, 2008, pp. 366-372.

[14] V. V. Gorin, “Integral Equation for Source of Ionization in Hollow Cathode,” Cornell University Library, in press. http://arxiv.org 0902.2655

[15] V. S. Vladimirov, “The Equations of Mathematical Physics,” Nauka, Мoscow, 1971.

[16] A. N. Kolmogorov and S. V. Fomin, “Elements of Function Theory and Functional Analysis,” Nauka, Мoscow, 1976.

[17] L. V. Kantorovich and G. P. Akilov, “Functional Analysis,” Nauka, Мoscow, 1984.

[18] J. P. Boeuf and E. Marode, “A Monte Carlo Analysis of an Electron Swarm in a Nonuniform Field: The Cathode Region of a Glow Discharge in Helium,” Journal of Physics D: Applied Physics, Vol. 15, No. 11, 1982, p. 2169. doi:10.1088/0022-3727/15/11/012

[1] F. Paschen, “Bohrs Heliumlinien,” Annalen der Physik, Vol. 355, No. 16, 1916, pp. 901-940. doi:10.1002/andp.19163551603

[2] A. Engel and M. Shteenbeck, “Physics and Technology of Electric Discharge in Gases 2”, ONTI, Мoscow, 1936.

[3] Yu P. Rayzer, “Physics of Gas Discharge,” Nauka, Мoscow, 1987.

[4] B. I. Moskalev, “Discharge with Hollow Cathode,” Energia, Мoscow, 1969.

[5] K. Kutasi and Z. Donko, “Hybrid Model of a Plane-Parallel Hollow-Cathode Discharge,” Journal of Physics D: Applied Physics, Vol. 33, No. 9, 2000, pp. 1081-1089. doi:10.1088/0022-3727/33/9/307

[6] F. Sigeneger and R. Winkler, “Study of the Electron Kinetics in Cylindrical Hollow Cathodes by a Multi-Term Approach,” The European Physical Journal Applied Physics, Vol. 19, No. 3, 2002, pp. 211-223. doi:10.1051/epjap:2002068

[7] N. Baguer, A. Bogaerts and R. Gijbels, “Hollow Cathode Glow Discharge in He: Monte Carlo-Fluid Model Combined with a Transport Model for Metastable Atoms,” Journal of Applied Physics, Vol. 93, No. 1, 2003, pp. 47- 55. doi:10.1063/1.1518784

[8] F. Sigeneger, Z. Donko and D. Loffhagen, “Boltzmann Equation and Particle-Fluid Hybrid Modelling of a Hollow Cathode Discharge,” The European Physical Journal Applied Physics, Vol. 38, No. 2, 2007, pp. 161-167. doi:10.1051/epjap:2007067

[9] A. Derzsi, P. Hartmann, I. Korolov, J. Karacsony, G. Bano and Z. Donko, “On the Accuracy and Limitations of Fluid Models of the Cathode Region of DC Glow Discharges” Journal of Physics D: Applied Physics, Vol. 42 No. 22, 2009, Article ID: 225204. doi:10.1088/0022-3727/42/22/225204

[10] D. Hilbert, “Grundzüge Einer Allgemeinen Theorie der Linearen Integralgleichungen,” B. G. Teubner, Leipzig, 1912, Chelsea Publishing Company, New York, 1953.

[11] V. V. Gorin, “Non-Local Source Equation for Linear Stationary Kinetic Equation,” Proceedings of the Conference on Differential Equations and Topology, Moscow, 17- 22 June 2008, pp. 43-44.

[12] V. V. Gorin, “Non-Local Model of Hollow Cathode and Glow Discharge—Theory Calculations and Experiment Comparison,” European Physical Journal D, Vol. 59, No. 2, 2010, pp. 241-247. doi:10.1140/epjd/e2010-00165-9

[13] V. V. Gorin, “A Mathematical Model of Plane Glow Discharge and Hollow Cathode Effect,” Ukrainian Journal of Physics, Vol. 53, No. 4, 2008, pp. 366-372.

[14] V. V. Gorin, “Integral Equation for Source of Ionization in Hollow Cathode,” Cornell University Library, in press. http://arxiv.org 0902.2655

[15] V. S. Vladimirov, “The Equations of Mathematical Physics,” Nauka, Мoscow, 1971.

[16] A. N. Kolmogorov and S. V. Fomin, “Elements of Function Theory and Functional Analysis,” Nauka, Мoscow, 1976.

[17] L. V. Kantorovich and G. P. Akilov, “Functional Analysis,” Nauka, Мoscow, 1984.

[18] J. P. Boeuf and E. Marode, “A Monte Carlo Analysis of an Electron Swarm in a Nonuniform Field: The Cathode Region of a Glow Discharge in Helium,” Journal of Physics D: Applied Physics, Vol. 15, No. 11, 1982, p. 2169. doi:10.1088/0022-3727/15/11/012