Abstract
We obtain an analytic solution of the boundary problem for the behavior (fluctuations) of an electron plasma with an arbitrary degree of degeneracy of the electron gas in the conductive layer in an external electric field. We use the kinetic Vlasov–Boltzmann equation with the Bhatnagar–Gross–Krook collision integral and the Maxwell equation for the electric field. We use the mirror boundary conditions for the reflections of electrons from the layer boundary. The boundary problem reduces to a one-dimensional problem with a single velocity. For this, we use the method of consecutive approximations, linearization of the equations with respect to the absolute distribution of the Fermi–Dirac electrons, and the conservation law for the number of particles. Separation of variables then helps reduce the problem equations to a characteristic system of equations. In the space of generalized functions, we find the eigensolutions of the initial system, which correspond to the continuous spectrum (Van Kampen mode). Solving the dispersion equation, we then find the eigensolutions corresponding to the adjoint and discrete spectra (Drude and Debye modes). We then construct the general solution of the boundary problem by decomposing it into the eigensolutions. The coefficients of the decomposition are given by the boundary conditions. This allows obtaining the decompositions of the distribution function and the electric field in explicit form.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. A. Vlasov, “On vibration properties of electron gas,” J. Phys. USSR, 8, 291–318 (1938).
L. D. Landau, “On the vibrations of the electronic plasma [in Russian],” in: Collected Works, Vol. 2, Nauka, Moscow (1969), pp. 7–25; “On the vibrations of the electronic plasma [in Russian],” J. Phys. USSR, 10, 25–34 (1946).
A. V. Latyshev and A. A. Yushkanov, “Analytic solution of boundary-value problems for nonstationary model kinetic equations,” Theor. Math. Phys., 92, 782–790 (1992).
A. V. Latyshev and A. A. Yushkanov, “Plasma in a high-frequency electric field with a reflective condition on the boundary,” Fluid Dynamics, 41, 161–172 (2006).
A. V. Latyshev and A. A. Yushkanov, “Analytic solution of the problem of the behavior of an electronic plasma in the half-space of a metal in a variable electric field [in Russian],” Surface: Phys. Chem. Mech., No. 2, 25–32 (1993).
A. V. Latyshev and A. A. Yushkanov, “Degenerate plasma in a half-space under an external electric field,” Theor. Math. Phys., 147, 854–867 (2006).
A. V. Latyshev and A. A. Yushkanov, “Analytic solution of the problem of the behavior of a degenerate electronic plasma [in Russian],” in: Encyclopedia of Low-Temperture Plasma: Ser. B (V. E. Fortov, ed.), Vol. 7-1, Mathematical Modeling in a Low-Temperature Plasma, Yanus-K, Moscow (2008), pp. 159–177.
J. M. Keller, R. Fuchs, and K. L. Kliewer, “p-polarized optical properties of a metal with a diffusely scattering surface,” Phys. Rev. B, 12, 2012–2029 (1975).
J. M. Kliewer and R. Fuchs, “s-Polarized optical properties of metals,” Phys. Rev. B, 2, 2923–2936 (1970).
V. M. Gokhfel’d, M. A. Gulyanskii, M. I. Kaganov, and A. G. Plyavenek, “Nonexponential attenuation of electromagnetic fields in normal metals,” Soviet JETP, 62, 566–575 (1985).
V. M. Gokhfel’d, M. I. Kaganov, and G. Ya. Lyubarskii, “Anomalous penetration of longitudinal alternating electric field into a degenerate plasma with an arbitrary specularity parameter,” Soviet JETP, 65, 295–299.
A. A. Abrikosov, Introduction to the Theory of Normal Metals [in Russian], Nauka, Moscow (1972); English transl., Acad. Press, New York (1972).
B. B. Kadomtsev, Collective Phenomena in Plasmas [in Russian], Nauka, Moscow (1976).
E. V. Chizhonkov, “To the question of large-amplitude electron oscillations in a plasma slab,” Comput. Math. Math. Phys., 51, 423–434 (2011).
E. M. Lifshitz and L. P. Pitaevskii, Course of Theoretical Physics [in Russian], Vol. 10, Physical Kinetics, Nauka, Moscow (1979); English transl., Butterorth-Heinemann, Oxford (1981).
V. S. Vladimirov and V. V. Zharinov, Equations of Mathematical Physics [in Russian], Fizmatlit, Moscow (2000); English transl. prev. ed.: V. S. Vladimirov, M. Dekker, New York (1971).
N. W. Ashkroft and N. D. Mermin, Solid State Physics, HRW, Philadelphia (1976).
A. V. Latyshev and A. A. Yushkanov, Boundary Problems for a Degenerate Plasma [in Russian], Moscow Region State Univ. Press, Moscow (2006).
A. V. Latyshev and A. A. Yushkanov, “Nonstationary boundary problem for model kinetic equations at critical parameters,” Theor. Math. Phys., 116, 978–989 (1998).
F. D. Gakhov, Boundary Problems [in Russian], Nauka, Moscow (1977).
V. V. Vedenyapin, Kinetic Theory Acording to Maxwell, Boltzmann, and Vlasov [in Russian], Moscow Region State Univ. Press, Moscow (2005).
Author information
Authors and Affiliations
Corresponding author
Additional information
Deceased.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 192, No. 3, pp. 506–522, September, 2017.
Rights and permissions
About this article
Cite this article
Latyshev, A.V., Gordeeva, N.M. The behavior of plasma with an arbitrary degree of degeneracy of electron gas in the conductive layer. Theor Math Phys 192, 1380–1395 (2017). https://doi.org/10.1134/S0040577917090082
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577917090082