We have been demonstrated that a multiple excess of the accelerated bunch energy εmax over the energy of the exciting. REB is possible in a magnetoactive plasma at a certain relationship between the parameters of the “plasmabunch-magnetic field” system owing to a hybrid volume-surface character of REB-excited wake fields, even without using the REBs contoured in the longitudinal direction

Balakirev, V.A.

Karas, I.V.

Karas, V.I.

Sotnikov, G.V.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

                                 ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ 2000. №1.   Серия: Плазменная электроника и новые методы ускорения (2), с. 113-116.113UDK 533.9RELATIVISTIC ELECTRON BUNCH EXCITATION OF WAKE FIELDSIN A MAGNETOACTIVE PLASMAV.A. Balakirev, V.I. Karas’, I.V. Karas’, G.V. SotnikovNational Science Center “Kharkov Institute of Physics and Technology”, Kharkov 61108, Ukraine,karas@kipt.kharkov.uaWe have been demonstrated that a multiple excess of the accelerated bunch energy εmax over the energy of theexciting. REB is possible in a magnetoactive plasma at a certain relationship between the parameters of the “plasma-bunch-magnetic field” system owing to a hybrid volume-surface character of REB-excited wake fields, even withoutusing the REBs contoured in the longitudinal direction.     The ideas of using collective fields for accelerationin the plasma and noncom-pen-sated charged beamswere stated as early as in 1956 by V.I. Veksler, G.I.Budker and Ya.B. Fainberg [1-3].  The appearance anddevelopment of new powerful energy sources such aslasers, high-current relativistic electron beams,superhigh-power microwave generators, gave anotherimpetus to the development of the collective methods ofcharged particle acceleration. As a result, in 1979 [4]and 1985 [5], there appeared new modifications of themethod of charged particle acceleration in a plasma bycharge density waves (ref. [3]), where it was proposedthat the accelerating fields should be excited by laserpulses and relativistic electron bunches.     In our opinion, the excitation of accelerating fields ina plasma by an individual relativistic electron bunch(REB) appears most preferable, because it isnonresonant in character, and therefore, is little sensitiveto the longitudinal plasma density inhomogeneityobserved in experiment. Besides, to preclude theelectromagnetic filamentation slipping of instabilities,etc. [6-9], it is reasonable to use the stabilizing externallongitudinal magnetic field [10]. Aside fromstabilization, the magnetic field also gives rise to amultitude of new wave branches, and this, as will beshown below, essentially extends the potentialities of thewake-field method of charged particle acceleration.      The report presents the results from theoreticalstudies into the processes of REB excitation  of wakefields in a magnetoactive plasma, both the cases of anunbounded plasma and the waveguide with a partialplasma filling.     1. Let us define the wake-field by an axiallysymmetric REB moving in the magnetoactive plasmaalong the axis z. We neglect the thermal motion ofelectrons, and assume ions to be immobile. The REBcurrent density is written as( )002ext zr zj I T t eVψπ = − −  ! !,(1)where I0 is the peak value of the total current of bunch,ψ(r) is the function describing the current distribution inthe bunch cross section, the function T describes thelongitudinal profile of the bunch (maxT = 1), t is thetime, z and r are, respectively, the longitudinal andradial coordinates, V0 is the bunch velocity, ez is the unitvector in the direction of bunch motion and  an externalmagnetic field H0 = H0ez. Formula (1) describes theREB current density in the approximation of assignedmotion, i.e., the “rigid” bunch model is used. Thefunction ψ(r) satisfies the condition( )01brr rdrψ =∫ ,where rb is the maximum radius of REB.We express the electric field E, the magneticfield H and the current density of REB (1) in terms ofthe Fourier integral, then the set of Maxwell equationstakes the form0 rkE k Hωϕ ω= ,      ( )0 1 2rkH k E i Eωϕ ω ωϕε ε= + ,01zd rE ik Hr dr ωϕ ω= ,( )0 1 2r r rdH ikH ik E i Edrω ω ωϕ ωε ε− = − , (2)0zrdE ikE ik Hdrωω ωϕ− = ,( ) ( )00 3 21 z Id rH ik E r Tr dr cωϕ ωε ψ ω= − ,where 0k Vω= , 0k cω= ,( )( )21 2 21 peHeiiω ω νεω ω ν ω+= −  + − ,( )22 2 2pe HeHeiω ωεω ω ν ω=  + − ,( )23 1peiωεω ω ν= −+,ωpe,ωHe are, respectively, theLangmuir and Larmor frequencies of plasma electrons, vis the effective collision frequency, ϕ is the azimuthalcoordinate.     The set of first-order differential equations (2) isconveniently reduced to coupled second-order114differential equations for longitudinal components ofelectric and magnetic fields( ) ( )2 2 30102112zH z zdHd r p H ikk Er dr drIk r Tcωω ωε εεε ψ ωε+ = − −−,( ) ( ) ( )2 2012 20 1 00 112zE z zdEd r p E ikk Hr dr drk k Ii r Tk cωω ωεεεψ ωε+ = +−+,(3)where 2 22 2 21 201Hp k kε εε−= − ,2 2 2 30 31Ep k kεεε= − .     The solution to the nonuniform set of equations hasthe form( )( )( ) ( )0 211 1 0 10 0 02 2 0 2,,zI kH i TcAG r rr dr rA G r rωεπ ωελ λψλ λ= ×− ×  −  ∫,( )( ) ( ) ( )00 31 1 0 1 2 2 0 2 0 0 01, ,zIE Tc kB G r r B G r r r dr rω π ω ελ λ λ λ ψ= − ×× −  ∫,(4)where 22 21 2iiAλλ λ= −,       ( )2 2 22 21 2i H iipBλ λλ λ−=−,λ1,2 are the transverse wave numbers of ordinary andinordinary waves, respectively, and are the roots of thebiquadratic equation( )24 2 2 2 2 2 20 312 2 0E HE Hp p k kp pελ λ εε − + − +  + =,(5)They can be written as:( )2 22 22 2 2 2 21,2 0 312 4E HE Hp pp p k k ελ εε−  += ± +    ,( )( ) ( ) ( )( ) ( ) ( )10 0 0 00 10 0 0 0,,,i ii ii iH r J r r rG r rH r J r r rλ λλ λλ λ >= <where J0(λir), H0(1) (λir) are the Bessel functions andHankel functions, respectively.     In the ultrarelativistic case, we can put V0 = c. Then,instead of eq.(5) we have( )22 01,2 3 1 2 311kλ ε ε ε εε = − ±   . (6)     In the frequency range ω2 < ω2pe the transverse wavenumbers are complex.     In the limiting case of a strong magnetic field, ω2He>> ω2pe, the expression for transverse wave numberstakes the form2 2 21 0k kλ = − ,      ( )2 2 22 0 3k kλ ε= − .     For simplicity sake we consider an infinitely thinannular bunch of radius rb ( ) 1T ω = . ( ) ( )bbr rrrδψ −= (7)     Then, for the electric field on the axis of the system r= 0 we obtain the following integral representation( ) ( ) ( )10 02 20 01 exp2z bQE i t H k r dVω ω ωγ∞⊥−∞=− −∫ , (8)where Q0 is the total charge of bunch, γ0 is therelativistic factor, k⊥2 = (k02 - k2)ε3. Note that in thefrequency range ω < ωpe the bunch emits anelectromagnetic field in the radial direction, becausek⊥2 > 0.     The argument of the Hankel function has two branchpoints ω = ±ωpe - iv/2 lying in the lower half-plane of thecomplex variable ω. Let us draw a cut along the line ofsegment connecting the branch points, and close theintegration contour in the lower half-plane over thesemi-circle of infinite radius. As a result, the initialintegral (8) reduces to the integral over the cut shores.Then, using the path-tracing rule for the Hankelfunction, we obtain the following expression for thelongitudinal component of the electric wake field( )( )2 102 2 20 0 1 0exp1pezisQE iV J s sdsτωγ µ−− ×= −× −∫ , (9)where0peztVτ ω = −   ,      0 0pe brVωµγ= .     After calculation of the integral we obtain from (9)115. 202 2 2 20 02 22 22 22sincospezQEVω τγ τ µτ µτ µτ µ= ×+ + − × +  − +  (10)     At large distances behind the bunch, the wake-wavefield decreases as 1/τ. This is due to the fact that theoscillations in the plasma placed in a strong magneticfield have the finite group velocity. The radiation ofplasma waves from the near-axis region causes the wakefield to decrease in the longitudinal direction.     2. Let us consider the plasma waveguide with apartial plasma filling in the external magnetic field, i.e.,the waveguide, where between the plasma boundary r =a and the conducting housing r = b there is a vacuumgap.     To derive a dispersion equation of eigenwaves of thisplasma wave guide, it is necessary to find theelectromagnetic fields in the vacuum gap b<r<a andjoin them with the plasma fields through the use ofboundary conditions on the plasma surface. Theboundary conditions are standard, i.e., the continuity ofelectromagnetic-field tangential components. As aresult, we obtain the dispersion equation, which can beconveniently written as a determinantDet 0A = , (11)where the matrix A has the following components11 1A = , 12 1A = , 13 1A = − , 14 0A = ,( )( )1 121 11 0 1J aAc J aλωλ λ= Γ ,( )( )1 222 22 0 2J aAc J aλωλ λ= Γ ,23 0A = , 24A cwω= ,31 1A = Γ , 32 2A = Γ , 33 0A = , ( )34A Q wa= ,( )( )1 141 31 0 1J aAc J aλωελ λ= ,( )( )1 242 32 0 2J aAc J aλωελ λ= ,( )43 1A F wacwω= − , 44 0A = ,where( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )0 1 0 11 1 1 1I wa K wb K wa I wbQ waI wa K wb I wb K wa+=−,( )( )1 3221 2201 2221,2 1 3 1 21 022222 3 20124kkkkkkε εε εε ε εεεε ε− ×    − − ±           Γ =   − − −   × +        ± −         +       ( )( )21 3 1 201 2220 1 3 1 21,2 01 2 22 2222 3 2024kkkkkkkε ε εε ε ελεε εε ε  + − −           − − −  =  +      − ±−       +    0 010 0( ) ( , );( ) ( , )p bpI w r wr wbFI w a wa wb∆=∆12 2 20 3( ) ;pw k k ε= −2 2 20 ;w k k= −0 0 0 0 0( ) ( ) ( ) ( );I wr K wb I wb K wr∆ = −Character of field distribution in the cross section for thewaveguide is determined by the transverse wavenumbers. If λ1,22 > 0, then the wave is three-dimensional(volumetric). However, if λ1,22 < 0, then the wave ispertai pertaining to surface. And finally, if λ1,22 are thecomplex variables, then the wave is hybrid. Theboundaries of the region, where λ1,22 become complex,are defined by the inequalities 1 2ω ω ω> > ,where( )1,21 22 2 4 2 2 2 22 2 224pe He pe He pe HeHekck ck cωω ω ω ω ω ωω= ×+ ± + −×+ (12)     To excite the REB of the hybrid wave, the relativisticfactor must satisfy the following condition0 2Hepeωγω> . (13)      The field pattern and the frequency of the hybridwave being in synchronism with the bunch were foundby the numerical methods.116     We attained the characteristic radial distribution ofthe longitudinal electric-field component at thefollowing plasma and waveguideparameters: 6.3Hepeωω= , 23.3peacω= , 2.4ba= ,0 4.6γ = . The wake hybrid wave frequency is hereequal to 0.35 ωpe. It is shown that for the radius ra= 0.8the magnitude of the longitudinal electric-fieldcomponent has a deep maximum corresponding to theenergy transformation coefficient( )max037zEz rERE== = .     Note that a great value of the transformationcoefficient corresponds to a significant ( ER  times)excess of the maximum energy obtained by theaccelerated bunch as compared to the energy of thebunch exciting the wake field, because the energytransformation coefficient RE is equal to the ratio of theelectric field amplitude accelerating the guided bunch tothe electric field amplitude decelerating the bunch thatexcites the accelerating wake field.So, it has been demonstrated that a multiple excess ofthe accelerated bunch energy εmax over the energy of theexciting. REB is possible in a magnetoactive plasma at acertain relationship between the parameters of the “plasma-bunch-magnetic field” system owing to ahybrid volume-surface character of REB-excited wakefields, even without using the REBs contoured in thelongitudinal direction, namely:( )2max 0 1Emc Rε γ= − . (14)References1. V.I.Veksler //Proc. Symp.CERN, 1956, vol. 1.p. 80.2. G.I.Budker //Proc. Symp.CERN, 1956, vol. 1, p. 68.3. Ya.B.Fainberg//Proc. Symp .CERN, 1956, vol. 1, p. 84.4. T Tajima, J.M.  Dawson // Phys. Rev. Lett., 1979,vol. 43,p. 267.5. Chen P., J.M. Dawson J.M., T Katsouleas., et al. //Phys. Rev. Lett., 1985, vol. 54, p. 6936. T.Katsouleas // Phys. Rev. A, 1986, vol. 33. p. 2066.7. R.,Keinigs, M.E.Jones // Phys. Fluids, 1987, vol. 30,# 1, p.252.8. Ya.B. Fainberg Ya.B. // Plasma Physics Reports,1997, vol. 23, # 4, p. 251.9. Ya.B. Fainberg // Plasma Physics Reports, 2000, vol.26, # 4, p.324.10. A.Ts.Amatuni., E.V.Sekhposyan., A.G.Khachatryan,and S.S.Elbakyan // Plasma Physics Reports, 1995,vol. 21, # 11, p. 945.

Relativistic electron bunch excitation of wake fields in magnеtoactive plasma

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Relativistic electron bunch excitation of wake fields in magnеtoactive plasma

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)