The behavior of plasma with an arbitrary degree of degeneracy of electron gas in the conductive layer

A. V. Latyshev   ; N. M. Gordeeva

Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 3, pp. 506-522

Résumé

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.

Détail

Keywords: characteristic system, eigenfunction, Drude mode, decomposition of the solution with eigenfunctions.
Mots-clés : Debye mode, Van Kampen mode

Bibliographie

[1] A. A. Vlasov, “O vibratsionnykh svoistvakh elektronnogo gaza”, ZhETF, 8:3 (1938), 291–318 | Zbl

[2] L. D. Landau, “O kolebaniyakh elektronnoi plazmy”, Sobranie trudov, v. 2, Nauka, M., 1969, 7–25 ; ЖЭТФ, 16 (1946), 574–586 | MR | MR | MR | Zbl | Zbl

[3] A. V. Latyshev, A. A. Yushkanov, “Analiticheskoe reshenie granichnykh zadach dlya nestatsionarnykh modelnykh kineticheskikh uravnenii”, TMF, 92:1 (1992), 127–138 | DOI | MR | Zbl

[4] A. V. Latyshev, A. A. Yushkanov, “Plazma v vysokochastotnom elektricheskom pole s zerkalnym usloviem na granitse”, Izv. RAN. Ser. MZhG, 2006, no. 1, 165–177 | DOI

[5] A. V. Latyshev, A. A. Yushkanov, “Analiticheskoe reshenie zadachi o povedenii elektronnoi plazmy v poluprostranstve metalla v peremennom elektricheskom pole”, Poverkhnost. Fizika, khimiya, mekhanika, 1993, no. 2, 25–32

[6] A. V. Latyshev, A. A. Yushkanov, “Vyrozhdennaya plazma v poluprostranstve vo vneshnem elektricheskom pole”, TMF, 147:3 (2006), 487–502 | DOI | DOI | MR | Zbl

[7] A. V. Latyshev, A. A. Yushkanov, “Analiticheskoe reshenie zadachi o povedenii vyrozhdennoi elektronnoi plazmy”, Entsiklopediya nizkotemperaturnoi plazmy. Ser. B, v. VII-I, Matematicheskoe modelirovanie v nizkotemperaturnoi plazme, ed. V. E. Fortov, Yanus-K, M., 2008, 159–177

[8] J. M. Keller, R. Fuchs, K. L. Kliewer, “$p$-polarized optical properties of a metal with a diffusely scattering surface”, Phys. Rev. B, 12:6 (1975), 2012–2029 | DOI

[9] J. M. Kliewer, R. Fuchs, “$s$-polarized optical properties of metals”, Phys. Rev. B, 2:8 (1970), 2923–2936 | DOI

[10] V. M. Gokhfeld, M. A. Gulyanskii, M. I. Kaganov, A. G. Plyavenek, “Neeksponentsialnoe zatukhanie elektromagnitnogo polya v normalnykh metallakh”, ZhETF, 89:3(9) (1985), 985–1001

[11] V. M. Gokhfeld, M. A. Gulyanskii, M. I. Kaganov, “Anomalnoe proniknovenie prodolnogo peremennogo elektricheskogo polya v vyrozhdennuyu plazmu pri proizvolnom parametre zerkalnosti”, ZhETF, 92:2 (1987), 523–530

[12] A. A. Abrikosov, Vvedenie v teoriyu normalnykh metallov, Nauka, M., 1972

[13] B. B. Kadomtsev, Kollektivnye yavleniya v plazme, Nauka, M., 1976 | MR

[14] E. V. Chizhonkov, “K modelirovaniyu elektronnykh kolebanii v plazmennom sloe”, Zh. vychisl. matem. i matem. fiz., 51:3 (2011), 456–469 | DOI | Zbl

[15] E. M. Lifshits, L. P. Pitaevskii, Teoreticheskaya fizika, v. 10, Fizicheskaya kinetika, Nauka, M., 1979 | MR

[16] V. S. Vladimirov, V. V. Zharinov, Uravneniya matematicheskoi fiziki, Fizmatlit, M., 2000 | MR

[17] N. Ashkroft, N. Mermin, Fizika tverdogo tela, Mir, M., 1979 | Zbl

[18] A. V. Latyshev, A. A. Yushkanov, Granichnye zadachi dlya vyrozhdennoi elektronnoi plazmy, Izd-vo MGOU, M., 2006

[19] A. V. Latyshev, A. A. Yushkanov, “Nestatsionarnaya granichnaya zadacha dlya modelnykh kineticheskikh uravnenii pri kriticheskikh parametrakh”, TMF, 116:2 (1998), 305–320 | DOI | DOI | Zbl

[20] F. D. Gakhov, Kraevye zadachi, Nauka, M., 1977 | MR

[21] V. V. Vedenyapin, Kineticheskaya teoriya po Maksvellu, Boltsmanu i Vlasovu, Izd-vo MGOU, M., 2005