Critical energy in the cyclotron heating of ions in an ECR plasma source

The problem of plasma cyclotron heating in ECR plasma sources, to sustain the discharge, remains important at present. There are two methods for the analysis of this problem. The first one is the one particle stochastic mechanism and the second one is related with the non-linear interaction of waves...

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Опубліковано в: :	Вопросы атомной науки и техники
Дата:	2002
Автори:	Gutiérrez-Tapia, C., Hernández-Aguirre, O.
Формат:	Стаття
Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2002
Теми:	Low temperature plasma and plasma technologies
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/80325
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Critical energy in the cyclotron heating of ions in an ECR plasma source / C. Gutiérrez-Tapia, O. Hernández-Aguirre // Вопросы атомной науки и техники. — 2002. — № 4. — С. 165-167. — Бібліогр.: 6 назв. — англ.

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spelling	Gutiérrez-Tapia, C. Hernández-Aguirre, O. 2015-04-14T18:30:24Z 2015-04-14T18:30:24Z 2002 Critical energy in the cyclotron heating of ions in an ECR plasma source / C. Gutiérrez-Tapia, O. Hernández-Aguirre // Вопросы атомной науки и техники. — 2002. — № 4. — С. 165-167. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.50.-b https://nasplib.isofts.kiev.ua/handle/123456789/80325 The problem of plasma cyclotron heating in ECR plasma sources, to sustain the discharge, remains important at present. There are two methods for the analysis of this problem. The first one is the one particle stochastic mechanism and the second one is related with the non-linear interaction of waves where the collective behaviour of particles becomes the most important. In this work, in the Hamilton formalism, the stochastic mechanism of the cyclotron heating is analyzed. It is considered the case of an ECR plasma source where a TE11 electromagnetic wave is applied. The critical energy for the resonance mixing is calculated an by the “section-of-surface” method the inhomogeneity of the external magnetic field is analyzed. This work was partially supported by project CONACyT 33873-E en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Low temperature plasma and plasma technologies Critical energy in the cyclotron heating of ions in an ECR plasma source Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Critical energy in the cyclotron heating of ions in an ECR plasma source
spellingShingle	Critical energy in the cyclotron heating of ions in an ECR plasma source Gutiérrez-Tapia, C. Hernández-Aguirre, O. Low temperature plasma and plasma technologies
title_short	Critical energy in the cyclotron heating of ions in an ECR plasma source
title_full	Critical energy in the cyclotron heating of ions in an ECR plasma source
title_fullStr	Critical energy in the cyclotron heating of ions in an ECR plasma source
title_full_unstemmed	Critical energy in the cyclotron heating of ions in an ECR plasma source
title_sort	critical energy in the cyclotron heating of ions in an ecr plasma source
author	Gutiérrez-Tapia, C. Hernández-Aguirre, O.
author_facet	Gutiérrez-Tapia, C. Hernández-Aguirre, O.
topic	Low temperature plasma and plasma technologies
topic_facet	Low temperature plasma and plasma technologies
publishDate	2002
language	English
container_title	Вопросы атомной науки и техники
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
format	Article
description	The problem of plasma cyclotron heating in ECR plasma sources, to sustain the discharge, remains important at present. There are two methods for the analysis of this problem. The first one is the one particle stochastic mechanism and the second one is related with the non-linear interaction of waves where the collective behaviour of particles becomes the most important. In this work, in the Hamilton formalism, the stochastic mechanism of the cyclotron heating is analyzed. It is considered the case of an ECR plasma source where a TE11 electromagnetic wave is applied. The critical energy for the resonance mixing is calculated an by the “section-of-surface” method the inhomogeneity of the external magnetic field is analyzed.
issn	1562-6016
url	https://nasplib.isofts.kiev.ua/handle/123456789/80325
citation_txt	Critical energy in the cyclotron heating of ions in an ECR plasma source / C. Gutiérrez-Tapia, O. Hernández-Aguirre // Вопросы атомной науки и техники. — 2002. — № 4. — С. 165-167. — Бібліогр.: 6 назв. — англ.
work_keys_str_mv	AT gutierreztapiac criticalenergyinthecyclotronheatingofionsinanecrplasmasource AT hernandezaguirreo criticalenergyinthecyclotronheatingofionsinanecrplasmasource
first_indexed	2025-11-24T03:46:05Z
last_indexed	2025-11-24T03:46:05Z
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fulltext	CRITICAL ENERGY IN THE CYCLOTRON HEATING OF IONS IN AN ECR PLASMA SOURCE.* César Gutiérrez-Tapia1 and Omar Hernández-Aguirre2 1Departamento de Física, Instituto Nacional de Investigaciones Nucleares, Apartado Postal 18-1027, México 11801, D. F. 2Facultad de Ciencias, UAEM, Instituto Literario 100, Toluca, México The problem of plasma cyclotron heating in ECR plasma sources, to sustain the discharge, remains important at present. There are two methods for the analysis of this problem. The first one is the one particle stochastic mechanism and the second one is related with the non-linear interaction of waves where the collective behaviour of particles becomes the most important. In this work, in the Hamilton formalism, the stochastic mechanism of the cyclotron heating is analyzed. It is considered the case of an ECR plasma source where a TE11 electromagnetic wave is applied. The critical energy for the resonance mixing is calculated an by the “section-of-surface” method the inhomogeneity of the external magnetic field is analyzed. PACS: 52.50.-b 1. AVERAGING METHOD The equation of motion for one particle can be written as ( ),BvEp ×+= c ee dt d (1) where vp m= is the momentum, e and m are the charge and the mass of the ions. The magnetic field B and the electric field E in eq. (1) are assumed as the sum of slowly varying in space fields plus rapidly varying fields in the form .~,~ 00 BBBEEE +=+= (2) The electromagnetic wave propagating along the Oz axis is represented as ( ) ( ){ } ( ) ( ){ },0,cos,sin~ ,0,sin,cos~ 0 0 kztkztE kztkztE −−−= −−= ωω ωω B E (3) where ω is the frequency of the electromagnetic wave, and ck /ω= is the wave vector. The stationary magnetic field 0B until the first order terms in relation with the deviation from the axis z is represented as .)(,)( 2 ,)( 20       −−= zB dz zdBy dz zdBxB (4) Now, introducing the new dimensionless variables ,,, ,,,, iYX c v d dxX mc kzZkyYkxXt x +==== ==== ρ τ ωτ pP (5) into (1), and using expressions (3)-(5), the momentum components along OX and OY can be obtained ( ) ( ) ( ) ( ) , 2 sin1 , 2 cos1 0 0 0 0      Ω+Ω−−−=      Ω+Ω−−−= dZ dXPPZgP d dP dZ dYPPZgP d dP zxz y zyz x τ τ τ τ (6) where ,/0 ωmceB=Ω and ωmceEg /0= is the small parameter. In order to use the Bogoliuvov’s averaging method with resonances [1] it becomes important to transform the system of equations (6) into the standard form. This procedure is obtained by the introduction of the complex variable ρ : ( ) .'' 2 exp 0 0     Ω= ∫ τ ττζρ di (7) In this case, the equation of motion of the perpendicular motion is ( ) ( )[ ],exp12 0 ZiZg −−−=+ θζωζ  (8) where ,2/00 Ω=ω and '.)'( 0 0 τττθ τ d∫ Ω−= The “surface-of-section” picture of equation (8) is shown in Fig. 1. From this figure, we can observe that as the amplitude of the wave increases the convergence to a point is more slowly. In the case of very high values of the amplitude we obtain high-order resonances (small circles). This picture illustrates that there exist critical values of the energy absorption in dependence of the power of the wave. ζ ζ Fig. 1. Poincare map obtained from equation (8), for different powers of the electromagnetic field. Near to the cycotron resonance, .1 00 Ω−≈ω (9) Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P. 165-167 165 Thus, an easy way to obtain the standard form of equations (6) consist of introducing the new variables ( ) ( ),cos,cos 222111 ψθζψθζ +=+= aa (10) where 2,1a and 2,1ψ are unknown functions to be obtained, and 1ζ is related with 2ζ by .21 ζζζ += Applying the averaging method [1] to equations (6) using variables (10), the system of equation for 2,1a and 2,1ψ are obtained in the limiting cases .11,1,1 0 0 < <−Ω< < Ω < < dZ dg (11) After a straightforward algebra we obtain the corresponding average energy [ ] '1)'(sin )()0( 1 sin )0( )( )0( 2)0( )( 2 0 00 00 02 0 0 2 0 0 0 22 0             −Ω++ ΩΩ −        − Ω Ω − Ω Ω =        + = ∫ ⊥ ⊥ ⊥ τ ττχ τ χ ττ dZZ gPPW PP WW yx  (12) where 0χ is the initial phase, )0(0Ω is the initial value of )(0 τΩ and 2 0 mcW = is the rest energy. To study the energy absorption at the cyclotron resonance, we will consider the parameters of an electron cyclotron resonance plasma source reported in [3]. The magnetic field )(zBz is calculated using the method described in [4]. The parameters used in calculations are summarized in Table 1. Table 1. Parameters for calculating the magnetic field and the perpendicular energy (12). J1=J2=50.0A r1=r2=8.0cm L1=L2=20 M1=M2=20 Z1=-60-0cm Z2=60.0cm 0χ =0.0 V(0)=2.65E8cm/s Using these parameters, in Fig. 2 are shown two cases of the energy (12) in dependence of tωτ = . The first one corresponds to the field density 5 0 1062.2 ×=E V/cm, and the second one corresponds to the case of 2 0 1062.2 ×=E V/cm. As can be seen from these charts, a maximum of the energy absorption is present. If the field density is increased, the energy absorption is stopped (the average is equal to zer). Here a big question arises, is there only one maximum in the energy absorption? Fig. 2. Energy axial distribution for two values of the wave amplitude 5 0 1062.2 ×=E (A) and 2 0 1062.2 ×=E (B). 2. OBTANING OF THE HAMILTONIAN Following the procedure reported in [5] to describe the motion of a charged particle in magnetic mirror systems in presence of an electromagnetic field, where the perturbation method applied to the Hamiltonian of the particle seems to be highly suitable in using canonical transformations and thus finding the cyclic coordinates (momentums and frequencies) of motion of the particle. An appropriated form of the Hamiltonian is obtained by replacing the cylindrical coordinates by orthogonal curvilinear coordinates, defined by the lines of force of the magnetic field. By applying the perturbation method of multi-periodic systems to the Hamiltonian in these coordinates, the Hamiltonian of the guiding centers is obtained from which the integrals of motion and the frequencies of motion can be determined. The magnetic geometry for a mirror system is defined by two equations ,0,0 == BB divrot and by the boundary conditions. The motion of a particle in this field is described by the Hamiltonian in the field coordinates [5] ,0 HHH ∆+= (13) where 0H is the Hamiltonian which accounts for the external magnetic field, and H∆ is the Hamiltonian associated with the electromagnetic field. Introducing the magnetic field for a mirror system in the form [ ],)(1)(),(,0),( zkzBBB z αηξηξϕ +== (14) where ),(/1 zBk = and )(zB is the magnetic field along the Oz axis. Using the perturbative method described in [5] in relation with the small parameter k , and the initial and boundary conditions: if ,0=k ,ξ=r η=z ; ( ) ( ) 0,0,0,0 == ηη zr , as well as the components of the electric field for the case of a 11TE cylindrical wave- guide 166 ( ) [ ] ( ) [ ] ,0~ .,.(exp cos~ .,.(exp sin~ ' 0 20 = +− ×−= +− ×−= z lql lq H lql lq Hr E cctkzi lrJkZiCE cctkzi lrJ r lkZiCE ω ϕγ γ ω ϕγ γ ϕ (15) where Rj lqlq /'=γ , lqj' is the n-th root of equation 0)(' =xJ l , R is the internal wave-guide radius and HZ is the impedance. In the limit of linear terms respecting the electric field we obtain an expression for the Hamiltonian ( ) ( ) ( ),cossin ~~~ 2 02 0 2 2 102 2 1001 θωη ω +− ×+++++Ω= tklqE bkbkb mc ebkkbbpH r (16) where ib and ib~ (i=0, 1, 2) are functions of ( ηη pppqqq ,,,,, 2121 ). Here we have transformed canonically by the introduction of variables [5] ( )[ ] ( ) , , cos2sin2 sin2cos2 arctan ,cos2 ,sin22 12 2211 2211 21 2 21 22 212121 0 2 ppp qpqp qpqp qqppp qqpppp m −= + + = += +++ Ω = ϕ ξ ϕ ξ ξ (17) Where ),2,1( η=ipi are the actions and )2,1( =iqi are the angles having the meaning of the cyclotron and drift frequencies as is noticed in [5]. Expression (16) is obtained in the limit 1/ 21 < <pp . 3. STOCHASTIC ANALYSIS From the expression (20) for 0H we can determine η until zero-order terms with respect to k as follows ,0 01 ηη η + Ω = t p v (18) transforming the Hamiltonian into ( ) ( )( )[ ( )( )].qcos cos sin2 4 2 cos 002 002 201 0 01 2 21 2 01 θω θω ω η η η ++−Ω+ +−+−Ω+ ×Ω +         Ω ++Ω= tvk tvkq lqmp mc eE mp p qqpH r (19) Here it was assumed that ,01 tq Ω= and 01/ Ω= pkk . At exact resonances, when ( ) 00 =−±Ω ωηvk , from the last expression we obtain ( ) ( )[ ( ) ] .1sin 1sin2 8 2 cos 2 201 0 01 2 21 2 01res constEql qlmp mc eE mp p qqpH r ==− ++Ω +         Ω ++Ω≈ ω η (20) The last result shows that no other constant of motion in addition to the energy exist, then we can be sure that the system is stochastic. This result was obtained in [6], where an electrostatic wave was considered. In order to analyze the merging of the stochasticity it becomes important the construction of “surface-of-section” pictures of Hamiltonian (20) in dependence of the wave amplitude value. REFERENCES 1. N. N. Bogoliuvov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York (1961). 2. V. P. Milant’ev, Zh. Eksp. Theor. Fiz., 85, 132 (1983). 3. E. Camps, O. Olea, C. Gutiérrez-Tapia and M Villagrán, Rev.Sci.Instrum., 66, 3219 (1995) 4. C. Gutiérrez-Tapia, Proc. Int. Conf. & School on Plasma Phys. & Contr. Fusion, Alushta, Ukraine, 2002. 5. J. Lacina, Czech.J.Phys., B 13, 401 (1989). 6. G. R. Smith and A. N. Smith, Phys.Rev.Lett., 34, 1613 (1975). *This work was partially supported by project CONACyT 33873-E 167 REFERENCES

Critical energy in the cyclotron heating of ions in an ECR plasma source

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