Theory of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency excited by alternating electric field

Instability of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency (SCXM) in di-electric planar waveguide filled by plasma is studied in kinetic approximation. An external magnetic field is assumed parallel to the plasma surface. Doing that, two components of the SCXM wa...

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Бібліографічні деталі
Дата:	2010
Автори:	Girka, A.V., Girka, V.O., Pavlenko, I.V.
Формат:	Стаття
Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2010
Теми:	Нелинейные процессы в плазменных средах
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/17346
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Цитувати:	Theory of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency excited by alternating electric field / A.V. Girka, V.O. Girka, I.V. Pavlenko // Вопросы атомной науки и техники. — 2010. — № 4. — С. 274-276. — Бібліогр.: 9 назв. — англ.

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spelling	irk-123456789-173462011-02-26T12:04:23Z Theory of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency excited by alternating electric field Girka, A.V. Girka, V.O. Pavlenko, I.V. Нелинейные процессы в плазменных средах Instability of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency (SCXM) in di-electric planar waveguide filled by plasma is studied in kinetic approximation. An external magnetic field is assumed parallel to the plasma surface. Doing that, two components of the SCXM wave vector, which are perpendicular to external magnetic filed have been taken into the account in Vlasov-Boltzmann kinetic equation. Unlike the previous consideration an amplitude value of the alternating electric field is assumed to be either less or approximately equal to unit. Simple expressions for growth rates of the SCXM parametrical instability are calculated. The obtained results can be used for controlling of gas discharges sustained by surface waves under the regime of electron cyclotron resonance. Неустойчивость поверхностных циклотронных Х-мод на второй гармонике электронных циклотронных частот (ПЦХМ) в диэлектрических планарных волноводах с плазменным наполнением исследована в кинетическом приближении. Внешнее магнитное поле считалось параллельным поверхности плазмы. При этом две компоненты волнового вектора ПЦХМ, перпендикулярные внешнему магнитному полю, были учтены в кинетическом уравнении Власова-Больцмана. В отличие от предыдущего подхода к решению этой проблемы амплитуда внешнего переменного электрического поля предполагалась меньшей или приблизительно равной единице. Простые выражения для инкрементов параметрической неустойчивости ПЦХМ выведены. Полученные результаты могут быть использованы при управлении газовыми разрядами, которые поддерживаются поверхностными волнами в режиме электронного циклотронного резонанса. Нестійкість поверхневих циклотронних Х-мод на другій гармоніці електронних циклотронних частот (ПЦХМ) у діелектричних планарних хвилеводах із плазмовим наповненням досліджено у кінетичному наближенні. Зовнішнє магнітне поле вважалося паралельним межі плазми. При цьому дві компоненти хвильового вектора ПЦХМ, що є перпендикулярними до зовнішнього магнітного поля, були враховані у кінетичному рівнянні Власова-Больцмана. На відміну від попереднього підходу до розв’язання цієї проблеми амплітуда зовнішнього змінного електричного поля вважалася меншою або приблизно рівною одиниці. Прості вирази для інкрементів параметричної нестійкості ПЦХМ виведено. Здобуті результати можна використати при керуванні газовими розрядами, що підтримуються поверхневими хвилями у режимі електронного циклотронного резонансу. 2010 Article Theory of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency excited by alternating electric field / A.V. Girka, V.O. Girka, I.V. Pavlenko // Вопросы атомной науки и техники. — 2010. — № 4. — С. 274-276. — Бібліогр.: 9 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/17346 en Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах
spellingShingle	Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах Girka, A.V. Girka, V.O. Pavlenko, I.V. Theory of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency excited by alternating electric field
description	Instability of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency (SCXM) in di-electric planar waveguide filled by plasma is studied in kinetic approximation. An external magnetic field is assumed parallel to the plasma surface. Doing that, two components of the SCXM wave vector, which are perpendicular to external magnetic filed have been taken into the account in Vlasov-Boltzmann kinetic equation. Unlike the previous consideration an amplitude value of the alternating electric field is assumed to be either less or approximately equal to unit. Simple expressions for growth rates of the SCXM parametrical instability are calculated. The obtained results can be used for controlling of gas discharges sustained by surface waves under the regime of electron cyclotron resonance.
format	Article
author	Girka, A.V. Girka, V.O. Pavlenko, I.V.
author_facet	Girka, A.V. Girka, V.O. Pavlenko, I.V.
author_sort	Girka, A.V.
title	Theory of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency excited by alternating electric field
title_short	Theory of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency excited by alternating electric field
title_full	Theory of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency excited by alternating electric field
title_fullStr	Theory of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency excited by alternating electric field
title_full_unstemmed	Theory of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency excited by alternating electric field
title_sort	theory of surface cyclotron x-modes at the second harmonic of electron cyclotron frequency excited by alternating electric field
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2010
topic_facet	Нелинейные процессы в плазменных средах
url	http://dspace.nbuv.gov.ua/handle/123456789/17346
citation_txt	Theory of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency excited by alternating electric field / A.V. Girka, V.O. Girka, I.V. Pavlenko // Вопросы атомной науки и техники. — 2010. — № 4. — С. 274-276. — Бібліогр.: 9 назв. — англ.
work_keys_str_mv	AT girkaav theoryofsurfacecyclotronxmodesatthesecondharmonicofelectroncyclotronfrequencyexcitedbyalternatingelectricfield AT girkavo theoryofsurfacecyclotronxmodesatthesecondharmonicofelectroncyclotronfrequencyexcitedbyalternatingelectricfield AT pavlenkoiv theoryofsurfacecyclotronxmodesatthesecondharmonicofelectroncyclotronfrequencyexcitedbyalternatingelectricfield
first_indexed	2025-07-02T18:35:23Z
last_indexed	2025-07-02T18:35:23Z
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fulltext	THEORY OF SURFACE CYCLOTRON X-MODES AT THE SECOND HARMONIC OF ELECTRON CYCLOTRON FREQUENCY EXCITED BY ALTERNATING ELECTRIC FIELD A.V. Girka, V.O. Girka, I.V. Pavlenko Kharkiv National University, Kharkiv, Ukraine E-mail: v.girka@gmail.com Instability of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency (SCXM) in di- electric planar waveguide filled by plasma is studied in kinetic approximation. An external magnetic field is as- sumed parallel to the plasma surface. Doing that, two components of the SCXM wave vector, which are perpendicu- lar to external magnetic filed have been taken into the account in Vlasov-Boltzmann kinetic equation. Unlike the previous consideration an amplitude value of the alternating electric field is assumed to be either less or approxi- mately equal to unit. Simple expressions for growth rates of the SCXM parametrical instability are calculated. The obtained results can be used for controlling of gas discharges sustained by surface waves under the regime of elec- tron cyclotron resonance. PACS: 52.40.Fd; 52.77.-j; 52.25.Dg 1. INTRODUCTION The work develops theory of the surface electron cyclotron waves’ parametrical instability. Parametrical effect of the external alternating electric field on plas- mas is studied during a long time [1]. Thus parametrical excitation of the bulk cyclotron waves is examined ana- lytically rather well see e.g. [2,3] as compared with the case of surface waves’ (SW) excitation. The SW para- metrical instabilities have some peculiarities as com- pared with the case of bulk waves parametrical instabili- ties [4]. These instabilities are characterized by different dependences of their growth rates upon the alternating electric fields’ amplitude and plasmas’ parameters. Parametric excitation of the SW at the second har- monic of electron cyclotron frequency in planar dielec- tric waveguides with uniform plasma filling is examined here. The study is carried out in the framework of ki- netic description [5]. The SCXM dispersion properties were considered in monograph [6] using simple model of semi-bounded plasma. It was shown, that their skin- depth into plasma is approximately equal to their wave- length in the approach, if plasma density is supposed to be relatively large (Langmuir frequency is larger than electron cyclotron frequency: ee ω>Ω ) and plasma spatial dispersion is weak, i.e. Larmor radii of the plas- ma charged particles are essentially less than the SCXM wavelengths 1<<αρk r , here subscript α describes plasma species: electrons ( e=α ) and positively charged ions ( i=α ). But these results have been ob- tained by the aid of simplified assumption neglecting the wave vector component, which is oriented perpen- dicularly to the plasma surface. It was explained by ap- plication the approach of weak spatial dispersion of the plasma while solving the kinetic equation. Unlike the indicated simplified assumption of the [6], here we take into the account both components of the wave vector: (they are oriented along 21 kkk rrr += X r and coordinates, respectively), the wave vector is per- pendicular to an external magnetic field Y r 0B r , which is directed along the Z r axis. It allows us to make some correction of numerical coefficient in analytical expres- sion for the SCXM eigen frequency and to improve the- ory of parametrical influence of an external alternating electric field on these modes. The paper is organized as follows. In Section 2 we study dispersion properties of the SCXM propagating across an external steady magnetic field in plasma filled planar dielectric waveguides at the second electron cy- clotron harmonic. In Section 3 we examine the paramet- rical instability of the electron SCXM excited by exter- nal alternating electrical field. Simple analytical expres- sions for their growth rates’ values are derived. The paper is concluded by a short Summary. 2. DISPERSION OF THE SCXM Let’s consider uniform plasma that occupies the half-space x≤0 and that is bounded on the plane 0=x by a dielectric medium with dielectric coeffi- cient: dε . The model of uniform plasma is applicable if a scale of non-uniformity is much larger than wave's penetration depth into the plasma region. On the other hand, as far as we consider here electromagnetic pertur- bations just at the harmonics of electron cyclotron fre- quency then they are affected mainly just by an external magnetic field value but not by a plasma density non- uniformity. External steady magnetic field is parallel to the plasma-dielectric interface. The studied surface modes with components: , , 0B r xE yE zH propagate along the Y r direction across 0B r . Plasma particles behavior is governed by kinetic Vlasov-Boltzmann equation and the SCXM fields are described by Maxwell equations. De- pendence of the SCXM fields upon time and coordi- nates is chosen in the following form: yxE , , )exp()( 2 tiyikxfH z ω−∝ . We supposed that there is no dependence upon Z co- ordinate. One can't derive dispersion equation for the SCXM in general form (for implicit value of the elec- tron cyclotron harmonics’ number ). So let’s consider the case S 2=S because of its practical significance. Using method of Fourier transformation one can find out expression for Fourier coefficient of tangential _______________________________________________________________ ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2010. № 4. Серия: Плазменная электроника и новые методы ускорения (7), с.274-276. 274 component of SCXM electric field. By the aid of theory of residuals one can obtain expression for plasma im- pedance and then by the aid of linear boundary condi- tions for tangential component of the waves’ fields de- rive dispersion equation for SCXM propagating at the second harmonic of electron cyclotron frequency: 275 ])4(1)[( ))(( 1 12 1 22 2 22 2 2 0 01202 21 −Ω−+ + =++ hkkq iqqk ee ωερ ηε ηεε , (1) here 2,1ε are components of plasma permeability in approach of cold magneto-active plasma, 12ε is non- diagonal component of plasma permeability tensor that is obtained in a kinetic approach [5], is root of the equation 0q 0)( 0111 == qkε , which is located in upper semi-plane of the complex plane of wave vector com- ponent 1k r , ωω /21 eh −= is the shift of SCXM fre- quency from the second electron cyclotron harmonic. Appearance of multiplier in square brackets in the de- nominator of equation (1) makes it differ from that one, which was obtained in monograph [6]. Nevertheless its solution is not sufficiently differed from that one, which is indicated in [6]. It is explained by the following cir- cumstance: dispersion equation represented in [6] has been derived in supposition that: 12111 )( εε == kik , which is correct for the presented consideration. Be- cause of that relative error for the shift of SCXM fre- quency, which was obtained in [6] under the assumption about smallness of 0q , has been found small. In the present consideration, value of this root is 3/20 kiq ≈ for the approach of dense plasma ( ). The main distinguishing feature of the presented dispersion equation (1) is absence the wave, which is propagating along direction. So electron SCXM propagate along direction of Larmor precession for electrons lo- cated just nearby the plasma surface, i.e. their wave number k 22 ee ω>>Ω Y r + 2<0 and frequency value decreases with in- creasing ek ρ2 . Dispersion relation for the SCXM un- der considered conditions is as follows: )6/1/(2 22 2 ee k ρωω +≈ . (2) Using asymptotical expressions of the non-integrant kernel of the plasma conductivity tensor values [7] one can calculate the SCXM damping decrement caused by interaction between plasma particles and plasma bound- ary. Its value is approximately equal to 12/1 2 −chVk T , as it was indicated in [6]. Collisions between plasma parti- cles carry on their contribution into the process of the SCXM damping. The value of the collisional damping rate is proportional to the effective value of the collision frequency of electrons: )2/(2 hkcol ωνκ ≈ . Estimating the integral damping rate of these modes one can con- clude that they are weakly damped, the waves with long wavelengths damp more strongly than the modes with short wave lengths. 3. PARAMETRIC EXCITATION OF THE SCXM Let’s suppose in this section that considered wave- guide structure is affected by alternating electrical field: ZtE rr ⊥)cos( 00 ω . Solution of the kinetic equation allows one to drive two components of electrical conduc- tivity tensor of magneto-active plasma calculated in ki- netic approach, which are applied: ( ) ( ) ( ) ( ) 2 2 11 , , 4 ( ) y s m m n s m n n m i s e I y p J s y α α α α α α σ π ω ω −+∞ − =−∞ + Ω = × −∑ ∑ g J g ,(3) ( ) ∑ ∑ +∞ −∞= + − × − ΩΩ −= α α αα ωωππ σ α nms mn y s se p ,, 22 12 )(44 ( ) ( ) ( ) (gJgJyIyI nmmss − )−′× ][ αα , (4) here e kg m Eekg ρ αα α ωω ⊥ ⊥ = − = 022 0 0 )( , 1 2 22 <<= ⊥ α α ρk y , is Bessel function of the first type, summing up over indexes )(xJm nms ,, can be carried out independently from −∞ to +∞ , ( ) 0ωωω mpmp ++=+ , )(zIn and are modified Bessel function and its derivative over argument, accordingly. )(' yIm In order to derive set of equations, which describes excitation of electron SCXM one can apply the follow- ing boundary conditions: continuity of the tangential component of the SCXM electric field on the plasma- dielectric boundary and discontinuity of the magnetic field on the plane: 0=x , which is caused by nonlinear surface electric current flowing along the plasma sur- face. To formulate this non-linear boundary condition one can solve set of Maxwell equation using Fourier method. Doing that, reverse Fourier transformation can be carried out according to theory of residues. Thus dis- continuity of magnetic component of the SCMX field can be written in the following form: ∫ + − ≠ +−=≈= 0 0 0 )( 2 4),0(),0( l ln yy n z dxj c nxE kc inxH πω . (5) Application of the indicated boundary conditions al- lows one to derive infinite set of equations for satellite harmonics of tangential component of SCXM electric field, which include in themselves influence of external alternating electric field on plasma. Let’s supposed that the following resonant condition is realized: δωωω +Δ+= Te2 , (6) here TΔ is the frequency shift determined by the plas- ma particles thermal motion (see eq.(2)) and δω is cor- rection to the electron SCXM eigen frequency caused by affect of alternating electric field. Under the resonant condition (6), taking into the account only the nearest satellite harmonics of the tangential component of the electron SCXM field one can reduce the mentioned above infinite set of equations to the following one, which is formulated for determinant constructed of co- efficients herewith harmonics of the tangential compo- nent of the SCXM electric field: 0 )1()1,1()2,1( )1,0()0()1,0( )2,1()1,1()1( = +−+−+ +− +−+−− FFF FFF FFF , (7) here × − Ω = ∑ ∞+ ≠−∞= −+ 0,, 20 2 ),( )!(!2 )47.0( llm n lmm eeln lmm kg F ω ρ ] 2 3/293 [ 22 2 22 emn e emn mne k ωω ρ ωω ωω − + − + × ++ + , )1(/)1( 210221 )( +−+−+= εεεε qkiF n . Solution of the equation (7) for the limiting case of dense plasma ( ) has the following form: 22 ee ω>>Ω 276 TeT kgi Δ+Δ−≈ ρδω 203.03/2 . (8) Expression (8) coincides with the result obtained in the framework of the previous simplified model with accu- racy to within numerical multipliers. Thus general fea- tures of the SCXM growth rate remain the same and its value is larger than the SCXM damping rate for the range of relatively short wavelengths: 25.02 >ek ρ [8]. Analyzing expression (8) one can see, that paramet- rical influence of the external alternating electric field on the SCXM dispersion leads to decreasing of their frequency. The SCXM growth rates strongly decrease with increasing their wavelength and decreasing ampli- tude of the alternating electric field. It’s interesting to underline that: − the electron SCXM excitation by an alternating electric field can be realized only for the waves with (it is true both for the present model and for the previous model); − unlike the result obtained in [8], amplitude of an external alternating electric field can be relatively larger in the present approach. 02 <k 10 ≤g SUMMARY Motivated by problem of the plasma edge physics dispersion properties of the SCXM and their parametri- cal instability caused by affect of an external alternating electric field have been studied. We have derived equa- tions that consistently describe the SCXM dispersion and their parametrical excitation. Simple analytical ex- pressions for their eigen frequencies, their damping rates due to collisions between plasma particles and growth rates due to parametric instability are found out. The predictions of the presented theory can be interest- ing for the plasma edge physics and experiments de- voted to gas discharges sustained by surface waves [9]. REFERENCES 1. V.P. Silin. Parametrical Effect of the High Power Radiation on Plasmas. Moscow, 1973. 2. R.P. Sharma, A. Kumar, R. Kumar, V.K. Tripathi //Phys. Plasmas. 1994, v.1(3), p.522-527. 3. K.N. Stepanov. Nonlinear parametric phenomena in plasma during radio frequency heating in the ion cy- clotron frequency range // Plasma Phys. Control. Fusion. 1996, v.38, p.A13-A29. 4. N.A. Azarenkov, A.N. Kondratenko, K.N. Ostrikov. Parametric excitation of surface waves on the plas- ma-metal boundary // Zhurnal Tehnicheskoj Fiziki. 1990, v.69, №6, p.31-36 (in Russian). 5. N. A. Krall, A.W. Trivelpiece. Principles of Plasma Physics. McGraw-Hill Book Company, 1973. 6. A.N. Kondratenko. Surface and Bulk Waves in Bounded Plasmas. Moscow, 1985. 7. A.N. Kondratenko. Wave's Penetration into Plas- mas. Moscow, 1979. 8. V.O. Girka, O.E. Sporov. Parametrical instability of the surface waves on the second harmonic of elec- tron cyclotron frequency in plasma layer // Contribu- tions to Plasma Physics. 1997, v.37, №6, p.511-520. 9. Yu.M. Aliev, H. Schluter, A. Shivarova. Guided- Wave-Produced Plasmas. N-Y.: Springer, 2000. Статья поступила в редакцию 28.05.2010 г. ТЕОРИЯ ПОВЕРХНОСТНЫХ ЦИКЛОТРОННЫХ Х-МОД НА ВТОРОЙ ГАРМОНИКЕ ЭЛЕКТРОННОЙ ЦИКЛОТРОННОЙ ЧАСТОТЫ, ВОЗБУЖДАЕМЫХ ПЕРЕМЕННЫМ ЭЛЕКТРИЧЕСКИМ ПОЛЕМ А.В. Гирка, В.А. Гирка, И.В. Павленко Неустойчивость поверхностных циклотронных Х-мод на второй гармонике электронных циклотронных частот (ПЦХМ) в диэлектрических планарных волноводах с плазменным наполнением исследована в кине- тическом приближении. Внешнее магнитное поле считалось параллельным поверхности плазмы. При этом две компоненты волнового вектора ПЦХМ, перпендикулярные внешнему магнитному полю, были учтены в кинетическом уравнении Власова-Больцмана. В отличие от предыдущего подхода к решению этой пробле- мы амплитуда внешнего переменного электрического поля предполагалась меньшей или приблизительно равной единице. Простые выражения для инкрементов параметрической неустойчивости ПЦХМ выведены. Полученные результаты могут быть использованы при управлении газовыми разрядами, которые поддержи- ваются поверхностными волнами в режиме электронного циклотронного резонанса. ТЕОРІЯ ПОВЕРХНЕВИХ ЦИКЛОТРОННИХ Х-МОД НА ДРУГИЙ ГАРМОНІЦІ ЕЛЕКТРОННОЇ ЦИКЛОТРОННОЇ ЧАСТОТИ, ЩО ЗБУДЖУЮТЬСЯ ЗМІННИМ ЕЛЕКТРИЧНИМ ПОЛЕМ А.В. Гірка, В.О. Гірка, І.В. Павленко Нестійкість поверхневих циклотронних Х-мод на другій гармоніці електронних циклотронних частот (ПЦХМ) у діелектричних планарних хвилеводах із плазмовим наповненням досліджено у кінетичному на- ближенні. Зовнішнє магнітне поле вважалося паралельним межі плазми. При цьому дві компоненти хвильо- вого вектора ПЦХМ, що є перпендикулярними до зовнішнього магнітного поля, були враховані у кінетич- ному рівнянні Власова-Больцмана. На відміну від попереднього підходу до розв’язання цієї проблеми амп- літуда зовнішнього змінного електричного поля вважалася меншою або приблизно рівною одиниці. Прості вирази для інкрементів параметричної нестійкості ПЦХМ виведено. Здобуті результати можна використати при керуванні газовими розрядами, що підтримуються поверхневими хвилями у режимі електронного цик- лотронного резонансу.

Theory of surface cyclotron X-modes at the second harmonic of electron cyclotron frequency excited by alternating electric field

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