Beam instability for longitudinally inhomogeneous systems

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Бібліографічні деталі
Дата:	2002
Автори:	Litovko, I., Goncharov, A.
Формат:	Стаття
Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2002
Назва видання:	Вопросы атомной науки и техники
Теми:	Plasma electronics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/77893
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Beam instability for longitudinally inhomogeneous systems / I. Litovko, A. Goncharov // Вопросы атомной науки и техники. — 2002. — № 5. — С. 95-97. — Бібліогр.: 4 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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record_format	dspace
spelling	irk-123456789-778932015-03-09T03:02:04Z Beam instability for longitudinally inhomogeneous systems Litovko, I. Goncharov, A. Plasma electronics 2002 Article Beam instability for longitudinally inhomogeneous systems / I. Litovko, A. Goncharov // Вопросы атомной науки и техники. — 2002. — № 5. — С. 95-97. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 52.35.-g; 52.59.-f http://dspace.nbuv.gov.ua/handle/123456789/77893 en Вопросы атомной науки и техники 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	Plasma electronics Plasma electronics
spellingShingle	Plasma electronics Plasma electronics Litovko, I. Goncharov, A. Beam instability for longitudinally inhomogeneous systems Вопросы атомной науки и техники
format	Article
author	Litovko, I. Goncharov, A.
author_facet	Litovko, I. Goncharov, A.
author_sort	Litovko, I.
title	Beam instability for longitudinally inhomogeneous systems
title_short	Beam instability for longitudinally inhomogeneous systems
title_full	Beam instability for longitudinally inhomogeneous systems
title_fullStr	Beam instability for longitudinally inhomogeneous systems
title_full_unstemmed	Beam instability for longitudinally inhomogeneous systems
title_sort	beam instability for longitudinally inhomogeneous systems
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2002
topic_facet	Plasma electronics
url	http://dspace.nbuv.gov.ua/handle/123456789/77893
citation_txt	Beam instability for longitudinally inhomogeneous systems / I. Litovko, A. Goncharov // Вопросы атомной науки и техники. — 2002. — № 5. — С. 95-97. — Бібліогр.: 4 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT litovkoi beaminstabilityforlongitudinallyinhomogeneoussystems AT goncharova beaminstabilityforlongitudinallyinhomogeneoussystems
first_indexed	2025-07-06T02:08:18Z
last_indexed	2025-07-06T02:08:18Z
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fulltext	BEAM INSTABILITY FOR LONGITUDINALLY INHOMOGENEOUS SYSTEMS I. Litovko1, A. Goncharov2 1SC “Institute for Nuclear Research” NASU, 39, pr. Nauki, Kiev, Ukraine; 2Institute of Physics NASU, 46, pr. Nauki, Kiev, Ukraine PACS: 52.35.-g; 52.59.-f INTRODUCTION One of the main problems of ion beam physics is transport of the compensated ion beam in a plasma. Often for these processes the plasma parameters are not constant along the ion beam line. For example, if beam is focused – beam density and compensate charge particles density increase. If beam from different reason is expanded than his density and so compensate charge density is decreasing under the ion beam line. For such processes is a very important to find evolution of initial perturbations of beam particles in beam-plasma systems with inhomogeneous dialectical permittivity along their propagation. In this article differential equation, describing spatial evolution of small perturbation for beam current density was obtained. It was made based on method presented firstly by Y. Fainberg and P. Blioch [1] for one- component beam. For some cases of beam-plasma systems analytical solutions this equation were founded. As an example computer simulations for transported ion beam and evolution of initial perturbation of plasma-beam instability on different parts of divided setup of isotopes electromagnetic separator were carried out. BASE EQUATIONS AND SOME ANALYTICAL SOLUTIONS We used linear two-liquid hydrodynamic model, describing space evolution for small fluctuations of non- uniform plasma-beam media. In [1] it was presented method, describing one component beam with changeable along propagation coordinate х unperturbed velocity Vb (but under jb=const), the method was developed in [2, 3] for two-component media. Follow by this method we can show that differential equation describing spatial evolution of small fluctuations of changeable beam current density on frequency ω has next form (Vb =const): ′′ + ′ + =j x j x M x j x N x( ) ( ) ( ) ( ) ( ) 0 (1) where: M x i L V d n dx N x L V L V i L V d n dx b b ( ) ln ( ) ln = − = − − 2 0 0 2 2 0 2 2 2 0 2 0 ω ω ε ω ω (2) Here L – scale of inhomogeneity, ε ω ω = −1 2 2 p x( ) - dialectical permittivity of cold plasma. From (1) and (2) using substitution ∫−= )())( 2 1exp()( xIdxxMAxj we can derive: ′′ + =I x I x f x( ) ( ) ( ) 0 (3) where f x L V d n dx d n dx b b b( ) ln ln = −       + 2 2 0 2 2 2 2 1 4 1 2 ω ε (4) So, if we know spatial dependences of stationary distribution for beam nb(x) and plasma np(x) and determine boundary conditions for j(0) и j′(0), we can obtain distribution of oscillation amplitude current density along beam propagation on different frequencies. Functions nb(x) and np(x) can obtain different forms and equation (3) for common case must solve by numerically. But for some separate cases analytical solutions equation (3) is possible to get. One observe for example some dependency for nb(x) and ε(х)=1-α np(x). I. Let us observe case, when plasma distribution density stays constant but beam density change along perturbation direction. So, follow this ε(х)=const. Let us describe beam concentration by function n x a bxb ( ) ( ) = +1 2 , here а and b – constants that determining by special physical conditions. Note that such description of beam density is typical for some parts of ion beam in magnetic separator by isotope [4]. For this case from (3) and (4) one can derive: ′′ + + =I x I x a bx ( ) ( ) ( ) β ε 1 02 , where β ω = L V b 2 2 0 2 (5) (5) Take into account initial condition j(0)=j0 and j′(0)=0 expression for space evolution of initial perturbations amplitude of beam current density j(х) for frequency region where ε≤0 can be obtain:               +    −− ++    ++ + = − − γ γω γ ω γ γ ω γ bx bV i bx bV i bx ejxj V xi 1 2 11 1 2 11 1 )( 0 00 0 (6) here γ β ε = + 1 4 2 a b Problems of Atomic Science and Technology. 2002. № 5. Series: Plasma Physics (8). P. 95-97 95 Dependencies j(х) had shown on figures 1 and 2. One can see that for b>0 (beam density decrease along propagation) current density amplitude grows very quickly and than faster then \|ε\| smaller. For b<0, almost exponential grows for point х=1/b, than fall down for point х=2/b and future stable near initial value j0. II. Let us observe now case beam density and plasma particles density change both along propagation direction. This situation is typical for beam-plasma systems, in which plasma media is generating by beam. Setting n x a bxb ( ) ( ) = +1 2 and nb(x)∼np(x) one can derive equation (3) to hypergeometrical type: ′′ + + − =I x I x a bx a ( ) ( ) ( ) β α1 02 (7) here: β ω = L V b 2 2 0 2 , )(1)( xnx pαε −= . The solution can be obtain in form : 0)(,}; 2 3; 4 1 2 , 4 1 2 1 ; 2 1; 4 1 2 , 4 1 21 1{)( 2 11 0 <     +−+     +     −−− + = − xzF a C zF bx Cexj V xi εγγ α γγ ω 0)(,}1;1; 4 1 2 , 4 1 2 )1( 1;1; 4 1 2 , 4 1 2 )1({ 1 )( 1 2 1 12 0 >     −+−−−     ++      ++−     + + = −− x z F a bxC z F a bxC abx exj V xi εγγγ α γγγ αα γ γ ω where γ β = − 1 4 2 a b , a bxz α 2)1( += , F(a,b,c,z) – hypergeometrical function. С1, С2, С1′, С2′ are constants that seeking from initial conditions j(0)=j0, j′(0)=0 and condition j1(x)\|ε=0 = j2(x) \|ε=0 . We are not present here these values from huge expressions. You can see transaction from region with ε<0 to region with ε>0 for b>0 (extension beam) on figures 3. Current density amplitude grows alike exponential for region ε<0. Dependence j(x) on different b>0 in region ε>0 shown on Fig. 4. One can see that for small b (b<0.01) oscillations is fall dawn, but for b>0.01 oscillations are increasing significally. Fig 4. Behavior for extension beam (b>0) in region with ε>0; 1- b=0.1, 2 – b=0.01, 3 – b=0.005 COMPUTER SIMULATION PROPAGATION OF ION BEAM WITH VARIABLE DENSITY Equation (3) was solved by numerically for different parts of electromagnetic separator of isotopes. Electrons drift stabilizes potentials of low frequency noise oscillations along propagation direction of beam and decreases effect 96 Fig 2. Dependence amplitude of current density under different b>0 and ε<0: 1,3 -b=1.5; 2,4 – b=0.5; \|ε 1 \|>\|ε 2 \| 1 2 3 4 j(x) x j(x) x 1 2 3 4 Fig 1. Dependence j(x) on different b<0, ε<0: 1) γ=1; 2) γ=0.5; 3) γ=0.1; 4) γ=0.05; ε1< ε4 Fig 3. Transition from region with ε<0 to region with ε>0 j(x) x j(x) 1 2 3 x of dynamical decompensation and essentially influents on formation and transportation intensive ion beams in specific condition of electromagnetic separator of isotopes [4]. Essential role plays ion-electron instability that has maximal increment of increasing oscillations propagating along beam and can lead as to additional decompensation, as to energetic spectrum extension of separating beam. For calculation was take into account as ion-ion as ion-electron interactions. According this dialectical permittivity in beginning part of separator was described by next function: )( 2 )()0( 1)( 2 2 xnxfx b b ω ω ε −= where 10 3 5 10)( + = x xnb ; 4)( x exf − ≈ . Thus equation for beginning part of separator can present in form: 0 )(4 1 2 2 2 0 22 =    + ++′′ bax a V L II b ε ω (8) here was set a=5/3, b=10. Fig. 5. Amplitude of current density for beginning part of separator for ω=0.5ωb (dotted line separate region ε (x)<0 from region with ε (x)>0) For middle part dialectical permittivity changes and can describe by next function for ion-ion interactions: )(103 )0( 1)( 3 2 2 xnx b b −⋅−= ω ω ε , here 21 10 3 5 10)( 2xx xnb −+ = . So equation for middle part has form: 0 )(4 4 2 2 2 0 22 =    −+ +++′′ cxaxb abc V L II b ε ω (10) Here a=5/3, b=10, c=1/21. For ion-electron interactions permittivity has form: )0,0( 3 510 10)( )03.05.01( )( 109.01)( 22 2 2 3 bb xb x x exx ωω ω ωε + = ++⋅−= − and equation follow: 0 )(4 )()( 2 2 2 0 22 =    + ++′′ bax a V LxIxI b ε ω Results of calculation for begin part presented on figure 5. We can see that after transaction from region ε <0 to region ε>0 oscillations can more increase. Results of calculation for middle with considering ion-ion and ion-electron interactions are on fig. 6. We can see that in middle part oscillations for some frequencies can significantly increase, especially for ion-electron interactions. Fig. 6. Dependence j(x) for separator of isotopes for middle part with considering a) ion-ion interactions; b) ion-electron interactions for ωb/ω=0.2 CONCLUSION Differential equation, describing spatial evolution of small perturbation of current density for beam-plasma system was obtained. It was shown that in some simple cases the equation could be solved analytically. For more common cases it can be solved using numerical methods. For expanded beam when it comes from region with negative dialectical permittivity to region with positive one essential rise of the beam current density oscillation take place. Experimentally these effects were observed for electromagnetic separator of isotope. REFERENCES 1. P.V. Blioch, Y. B. Fainberg // Zhurnal Technicheskoy Fiziki 1956, v. 26, pp. 530-534, (in Russian). 2. M. I. Gechtman, A.A. Goncharov // Zhurnal Technicheskoy Fiziki 1990, v. 60, pp. 72-76 (in Russian). 3. A.A. Goncharov, I.V. Litovko //Proceedings of Conference on plasma accelerators and ion 97 b) a) injectors. Kharkov, 1989, pp.155-156 (in Russian). 4. A.Goncharov, N.Datsko, I.Soloshenko // IEEE Trans. on Plasma Science, v. 23, #5, 1995. 98 Beam Instability For Longitudinally Inhomogeneous Systems I. Litovko1, A. Goncharov2 1SC “Institute for Nuclear Research” NASU, 39, pr. Nauki, Kiev, Ukraine; 2Institute of Physics NASU, 46, pr. Nauki, Kiev, Ukraine INTRODUCTION BASE EQUATIONS AND SOME ANALYTICAL SOLUTIONS CONCLUSION REFERENCES

Beam instability for longitudinally inhomogeneous systems

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