Description of longitudinal space charge effects in beams and plasma through dielectric permittivity
DDescription of longitudinal space charge effects in beams and plasma throughdielectric permittivity
Nikolai Yampolsky ∗ and Kip Bishofberger Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
We develop a universal framework which allows quickly solve a wide class of problems for lon-gitudinal space charge effects in beams and plasmas in cylindrical geometry. We introduce thelongitudinal dielectric permittivity for the beam of charged particles, which describes its collectivespace charge response. The analyis yields an effective plasma frequency, which depends on thetransverse geometry of the system. This dielectric permittivity mirrors the dielectric permittivityof plasma and matches the one dimensional (1D) expression once the transverse size of the beamis large. Several particle species can be included as additive terms describing susceptibility of eachspecie. The developed approach allows to study stability criteria for collective beam-beam andbeam-plasma instabilities for arbitrary transverse distributions in particle densities.
PACS numbers: 41.20.Cv, 41.75.-i, 52.25.Mq, 52.27.Cm, 52.27.Jt, 52.35.Fp, 52.35.Lv, 52.40.Mj, 52.59.Sa
I. INTRODUCTION
The problem of collective space charge effects is a ma-jor topic of research in plasma and accelerator physics.The research of longitudinal space charge effects coverssignificant portion of these studies. It ranges from lon-gitudinal space charge waves in plasma columns [1] andvacuum tubes [2], beam-cloud instability in ion trans-port [3], two-stream instability in vacuum electronics [4]and relativistic beams [5], longitudinal instability in in-tense beams [6], microbunching instability in high bright-ness linacs [7], free electron lasers in Raman regime [8],plasma wakefield accelerator in capillary [9] and manyothers. Despite evident similarity between these prob-lems and similar results obtained by different authors,there is no universal approach to addressing longitudinalspace charge effect in various beam/plasma systems incylindrical geometry. Essentially, every new problem issolved from the first principles repeating similar deriva-tions previously reported in literature.To date, there are two main approaches to address lon-gitudinal space charge in cylindrical geometry. The firstapproach is mainly used in accelerator physics and de-scribes electric fields as a product of impedance and beamcurrent [6, 10]. Another approach originated in plasmaphysics community [1, 3, 9, 11]. In that approach lon-gitudinal particle dynamics and Poisson’s equation aresolved simultaneously, which reduces to problem of find-ing eigenmodes of the second order differential equation,which represents transverse distribution of electric fieldin space. A drawback of these approaches is inability toinclude secondary species of particles such as backgroundplasma or additional beams. Moreover, simple analyticexpressions suitable for further studies can be obtainedonly for limited test transverse distributions, typicallyflattop [1, 3, 9, 11] or occasionally Gaussian [9, 12]. ∗ [email protected] Alternatively, collective effects in the medium can bedescribed in terms of polarization density and the re-sulting electric displacement field. In this approach eachspecie of particles results in the polarization density in-dependently from other species, reacting to the imposedelectric field. As a result, the displacement field of theoverall system can be found through additive contribu-tion of each specie. The overall system of multiple beamsand plasma species can be described with a single vari-able, i.e. dielectric permittivity, in the linear regimewhen the external electric field is small enough.
II. COLLECTIVE LONGITUDINAL SPACECHARGE EFFECTS IN BEAMS AND PLASMAS
We consider a long beam of charged particles travelingalong z direction inside a beam pipe of circular cross-section as outlined in Fig. 1. The beam is considered tobe axially symmetric and being matched into the focus-ing channel. The external longitudinal electric field willcause collective response of particles in the beam, whichis proportional to the external field in linear regime. Thebeam density n ( r ) does not need to be transversely uni-form and it is localized close to the pipe axis.This problem clearly needs to be solved in two dimen-sions (2D). It is possible to carry out analysis using theformalism developed in Refs. [3, 13]. In that analysis thebeam can be viewed as a layered dielectric with dielectricpermittivity matching that of local plasma parameters (cid:15) ( r ). Then the Laplace equation ∇ ( (cid:15) ( r ) ∇ φ ) = 0 for theelectrostatic potential φ can be reduced to a problem offinding an eigen-mode. However, this approach can onlybe done numerically for complicated enough transversebeam profiles. Moreover, inclusion of additional particlespecies ( e.g. electron cloud, background plasma, secondbeam of particles with different energy, etc. ) requiressolving different eigen-mode problem.The external electric field causes density modulationalong the beam. That modulation, in turn, results in a r X i v : . [ phy s i c s . p l a s m - ph ] F e b FIG. 1: Schematics of the axisymmetric beam propagatinginside a vacuum pipe. the induced electric field, which can be viewed as po-larization density of the beam. The induced longitudi-nal electric field has traverse dependence of its ampli-tude, which reflects the 3D nature of the problem. How-ever, the transverse scale of the field mirrors that of thelongitudinal scale of the beam modulation in the beamframe [14]. Therefore, long wavelength modulations inthe beam, kσ r /γ (cid:28) γ is therelativistic mass factor, k is the longitudinal wavenumberof modulation, and σ r is the rms beam size. Under thatcondition, the uniform external electric field results inuniform polarization density. As a result, one can treatthe beam as an effective dielectric medium and describeits properties with longitudinal dielectric permittivity (cid:15) || D z = (cid:15) || E z . (1)The longitudinal dielectric permittivity describes col-lective space charge effects in the beam. A. Longitudinal dynamics of particles
The longitudinal particle dynamics inside the beamcan be found using conventional fluid cold plasma equa-tions ∂ t n + ∂ z ( βcn ) = 0 , (2)( ∂ t + βc∂ z ) (cid:32) β (cid:112) − β mc (cid:33) = eE z , (3)where n is the density of charged particles in the beam, β ≡ v z /c is their normalized longitudinal fluid velocity, m and e are the mass and charge of particles, respec-tively, ∂ t and ∂ z are the partial derivatives over time t and longitudinal coordinate z , c is the speed of light.Equations (2) — (3) are derived under the assumptionthat the longitudinal and transverse particle motion arenot relativistic in the beam frame, which allows decou-pling transverse and longitudinal dynamics. We solveEqs. (2) — (3) in a linear limit of density and velocitymodulations n = n + δn, β = β + δβ, (cid:12)(cid:12)(cid:12)(cid:12) δnn (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) δββ (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) γ . (4) We solve linearized fluid equations using Fourier trans-form δn, δβ, E ∝ e iωt − ikz . (5)Then the linearized Eqs. (2) —(3) result in the solutionfor the perturbation δn = n − ikeγ m ( ω − kβ c ) E z , (6) δβ = δnn ω − kβ ckc = − ieγ mc ( ω − kβ c ) E z . (7)The linearized fluid equations (6) — (7) are correctfor arbitrary transverse profiles of the beam density n ( r ) and the applied electric field E z ( r ). We considerthe case of long wavelength modulations, kσ r /γ (cid:28) δβ is also trans-versely uniform (see Eq. (7)) and the transverse profileof density modulation matches that of the beam profile, δn ( r ) ∝ n ( r ) (see Eq. (6)). This kind of solution en-sures that the modulation will not be washed out bytransverse particle motion, which mixes particles acrossthe beam. The opposite case of short wavelength mod-ulations, kσ r /γ (cid:29)
1, requires taking transverse motioninto account since each individual particle will observevarying electric field during betatron motion.
B. Space-charge impedance
The space charge caused by the density modulationdescribed by Eq. (6) results in the electric field. Analysisin this section resembles the one presented in Ref. [15]with one major difference. We look for the longitudi-nal electric field caused by the space charge wave ratherthan a moving particle. Essentially, the phase velocity ofthe wave serves as a velocity of the effective particles inthe beam. Presence of high density background plasmamay significantly reduce velocity of the space charge wavecompared to the velocity of charged particles in the beam.This may cause a major difference compared to conven-tional analysis in the accelerator physics.We start with Maxwell’s equations for electric andmagnetic fields ( E and B , respectively) in vacuum andinclude charge and current sources caused by beams ( ρ and j , respectively) ∇ × E = − ∂ t B /c, ∇ · E = 4 πρ ( r , t ) , (8) ∇ × B = (4 π j ( r , t ) + ∂ t E ) /c, ∇ · B = 0 , (9) j = e ( δnβ c + n δβc )ˆ z , ρ = eδn. (10)These equations can be combined to obtain the secondorder partial differential equation for the electric field ∇ E − c ∂ tt E = 4 π ∇ ρ + 4 πc ∂ t j . (11)We Fourier transform this equation assuming thatsources are monochromatic according to Eq. (5). We alsotake into account that the charge and current densitiesare related as follows from Eq. (2), j = ω/k · ρ (it is,essentially, the charge conservation law). Then the equa-tion for the longitudinal component of the electric fieldbecomes (cid:20) r ∂ r ( r∂ r ) − k + ω c (cid:21) E z = 4 πik (cid:18) k − ω c (cid:19) eδn. (12)We find the solution of Eq. (12) using Green’s functionapproach. We search for the electric field caused by a thinring of particles with radius r = r (cid:48) , i.e. δn ( r ) = δ ( r − r (cid:48) )(Fig. 2). The homogeneous solution of Eq. (12) can berepresented as linear combination of Bessel functions.These functions are either regular or modified Besselfunctions depending on the sign of k − ω /c . The ma-jority of problems including interaction of beams withplasmas result in a slow space charge wave having phasevelocity smaller than the speed of light, ω/k < c [16]. Inthis case the homogeneous solution of Eq. (12) can berepresented as a linear combination of modified Besselfunctions I and K . Then the Green’s function G ( r )can be found as G ( r ) = A I ( κ r ) + B K ( κ r ) , r < r (cid:48) , (13) G ( r ) = A I ( κ r ) + B K ( κ r ) , r > r (cid:48) , (14) κ = (cid:112) k − ω /c . (15) FIG. 2: Charge distribution corresponding to Green’s func-tion.
Equations (13) — (15) should be completed withboundary conditions | G ( r = 0) | < ∞ , (16) G | r (cid:48) + (cid:15) = G | r (cid:48) − (cid:15) , (17) ∂ r G | r (cid:48) + (cid:15) − ∂ r G | r (cid:48) − (cid:15) = 4 πie κ k , (18) G ( r = a ) = 0 , (19)where a is the radius of the vacuum pipe. After somestraightforward algebra, we find the amplitude of theelectric field on-axis G | r =0 = 4 πier (cid:48) κ k (cid:20) I ( κ r (cid:48) ) K ( κ a ) I ( κ a ) − K ( κ r (cid:48) ) (cid:21) . (20) The on-axis electric field caused by the space charge ofall particles in the beam E SC can be found through theconvolution of the particle distribution with the Green’sfunction E SC ( r = 0) = 4 πie κ k ×× a (cid:90) (cid:20) I ( κ r (cid:48) ) K ( κ a ) I ( κ a ) − K ( κ r (cid:48) ) (cid:21) δn ( r (cid:48) ) r (cid:48) dr (cid:48) . (21)Equation (21) is similar to the description of spacecharge in beams using impedance [6, 10, 15, 17] sincethe induced electric field is proportional to particle den-sity, i.e. current. The similarities and differences aredescribed in details in Sec. III A. C. Dielectric permittivity
The polarization density is proportional to the electricfield, E SC ∝ E z , as follows from Eq. (6) and Eq. (21).That allows to introduce the effective dielectric permit-tivity for the medium (cid:15) || = E z + E SC E z = 1 − ˜ ω p ( ω − kβ c ) , (22)˜ ω p = κ a (cid:90) (cid:20) K ( κ r ) − I ( κ r ) K ( κ a ) I ( κ a ) (cid:21) ω p ( r ) rdr, (23)where ω p ( r ) = (cid:112) πe n ( r ) / ( γ m ) is the relativisticplasma frequency of the beam.Equation (22) shows that the dielectric permittivity ofthe beam has the same functional dependence as of auniform plasma. At the same time, the effective plasmafrequency ˜ ω p depends on the geometry of the problem:beam pipe radius, beam density distribution and the ef-fective transverse wavelength of modulation κ . Note thatpresence of a pipe always results in the reduced valuefor the effective plasma frequency since the second termin brackets in Eq. (23) (the term which depends on thepipe radius a ) is always negative. The effective plasmafrequency can be significantly smaller than the charac-teristic on-axis plasma frequency ω p (0) in case of a longwavelength modulation. That reflects the fact that thespace charge field is mostly transverse in this case andlongitudinal particle interaction is strongly reduced com-pared to 1D case. The effective plasma frequency canbe complex if the resulting system is unstable ( e.g. self-modulation or two-stream instability). D. Applicability limits
As discussed above, the results for the description ofcollective space charge effects using dielectric permittiv-ity (22) are strictly valid only when the the longitudi-nal electric field is transversely uniform across the beam.That is achieved in the limit κ r b (cid:28)
1. In general case,extending findings beyond that limit is not appropriate.For example, consider a short wavelength longitudinalspace charge wave excited in such a beam. To first or-der, the excited space charge wave follows the local dis-persion relation (cid:15) || ( r ) = 0. The frequency of the lon-gitudinal wave parametrically depends on radius. Thatcauses dephasing between waves at different transverselocations. Transverse mixing of particles causes destruc-tive interference of the waves resulting in Landau damp-ing [6]. Moreover, longitudinal and transverse dynamicsare parametrically coupled, which can result in the para-metric instability [18].However, the application of longitudinal permittivityof the beam can be cautiously extended into the regime,where κ r b (cid:38)
1. Equation (22) can be used as an estimatefor the permittivity on axis. This formalism can be usedif transverse mixing of particles in the beam is limited.For example, it can be achieved in a laminar flow or whenthe beta function of the beam is the largest scale forthe beam dynamics ( e.g. it is larger than the growthrate of the resulting instability). In addition, plasma orbeams can be strongly magnetized so that the gyroradiusof particles is much smaller than their transverse sizes.Not that solution for the Maxwell equations described with Eq. (21) is obtained without any assumptions forelectrostatic limits for the space charge fields. It is validat any frequencies and wavenumbers, even for superlumi-nal waves having phase velocity larger than the speed oflight. In this case the transverse wavevector κ is imag-inary and the modified Bessel functions I and K ofimaginary arguments can be rewritten as regular Besselfunctions J and Y of real arguments. III. LIMITING CASES
In this section we present several important limitingcases which often serve as baseline models in variousstudies.The effective plasma frequency can be found explicitlyfor some beam profiles. The results are presented in Ta-ble I. In that table ω p ≡ ω p ( r = 0) is the on-axis plasmafrequency, I = (cid:82) en ( r ) β cd r is full current of the beam, I a = mc /e ≈ kA is Alfven current, Γ(0 , x ) is theupper incomplete gamma function, and γ E ≈ . TABLE I: Effective plasma frequency for beams with different density distributions
Density distribution ˜ ω /ω , ∀ κ a, κ r b ˜ ω / (cid:18) II a κ c β γ (cid:19) , /a (cid:28) κ (cid:28) /r b n ( r ) = n , r ≤ r b − κ r b (cid:18) K ( κ r b ) + I ( κ r b ) K ( κ a ) I ( κ a ) (cid:19) − κ r b /
2) + 1 − γ E n ( r ) = n (cid:18) − r r b (cid:19) − κ r b ) − + 2 K ( κ r b ) − I ( κ r b ) K ( κ a ) I ( κ a ) − κ r b /
2) + 1 . − γ E n ( r ) = n exp (cid:18) − r σ r (cid:19) κ σ b (cid:18) κ σ b (cid:19) (cid:20) Γ (cid:18) , κ σ b (cid:19) − K ( κ a ) I ( κ a ) (cid:21) − κ σ r / − γ E A. Quasi-stationary waves in beam frame
Equation (21) essentially describes on-axisspace charge impedance of the beam, E z ( ω, k ) = Z ( ω, k ) I ( ω, k ). However, the result seem to be differentfrom what is reported in the literature [10, 15, 17]. Thedifference comes from source for the electric field beinga space charge wave rather than a moving particle.Moving particles can be viewed as a space charge wave,which is stationary in the beam frame. That wave hasthe dispersion relation in the lab frame ω = βck . As aresult, the transverse wavenumber defined in Eq. (15) isequal to κ = k/γ and conventional results for the spacecharge impedance in beams are recovered.This approximation can be used if relevant dynam-ics can be approximated as a quasi-static process in thebeam frame ( e.g. [4, 5, 7, 8, 20]). The exact condition forthis approximation is for the phase velocity of the wave in the beam frame to be much slower than the speed oflight. The phase velocity of the wave in the beam framecan be found through Lorentz transform of the wave 4-vector κ ≈ kγ , (cid:12)(cid:12)(cid:12)(cid:12) ω (cid:48) k (cid:48) c (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ω − βkckc − βω (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) , (24)where ω (cid:48) and k (cid:48) are the wave frequency and wavenumberin the beam frame, respectively. B. Beam in free space
In this limit the pipe is considered to be so wide thatits effect on the beam is not relevant. This limit canbe achieved for small enough wavelengths of the mod-ulation, κ a (cid:29)
1. This condition allows us to simplifyEq. (23) for the effective plasma frequency of the beamsince the effect of the vacuum pipe becomes negligible, K ( κ a ) /I ( κ a ) (cid:28) ω p = κ ∞ (cid:90) ω p ( r ) K ( κ r ) r dr, (25)The beam susceptibility in the long wavelength limit κ r b (cid:28) ω p ∝ κ .At the same time, the numerical pre-factor in the scalingdepends on the beams profile, and may vary for moderateratios of wavelength and beam size, ln( κ r b ) ∼ κ r b (cid:29) ω p ≈ ω p κ ∞ (cid:90) K ( κ r ) rdr = ω p . (26)The effective plasma frequency for different beam pro-files is presented in Fig. 3. The effective plasma frequencymatches the 1D solution at small wavelengths, ˜ ω p ≈ ω p at κ r rms (cid:29)
1. The effective plasma frequency reduces whilethe wavelength of modulation increases. It is a universalscaling that follows from the 2D geometry of the problem.The distributions presented in Fig. 3 have identical on-axis density and identical root mean square (rms) radii, r rms = (cid:82) r n ( r ) rdr/ (cid:82) n ( r ) rdr . Beams with differentprofiles have the same effective plasma frequency in the1D limit κ r rms (cid:29) κ r rms (cid:28) -2 -1 -4 -2 flattopparabolicGaussian FIG. 3: The effective plasma frequency for the flattop (blue),parabolic (red), and Gaussian (green) beam profiles with iden-tical on-axis density and rms radii.
C. Long wavelength limit
In this limit the effective transverse wavelength of themodulation is much larger compared to the pipe radius, κ a (cid:28)
1. The size of the beam is smaller that the piperadius, so κ r (cid:28) ω p ≈ r b (cid:90) [ K ( κ r ) − K ( κ a )] ω p rdr ≈≈ κ r b (cid:20) ln (cid:18) ar b (cid:19) + 12 (cid:21) ω p (0) . (27)The result matches precisely the one reported in the lit-erature [6]. IV. GENERALIZATION OF RESULTSA. Kinetic effects
The model can be easily expanded to include finitelongitudinal energy spread in the beam. This can bedone using the kinetic equation for particle distributionin phase space, f ( r, v z ) = n ( r ) f ( v z ), same way it is typ-ically done for homogeneous plasma. The fluid equationsof motion (2) — (3) but do not affect the expression forthe resulting electric field Eq. (21) which can be viewedas a modification to the Laplace equation to include 2Dgeometry of the beam. Then the longitudinal dielectricpermittivity of the beam including kinetic effects can bedescribed as (cid:15) || = 1 + ˜ ω p k (cid:90) L ∂ v z f ( v z ) ω − kv z dv z , (28)where f ( v ) is particle distribution in velocity, and L indicates integration over Landau contour. This ex-pression is valid for quasi-mono-energetic beams whichonly have non-relativistic particles in the beam frame, γ | β δ ∆ β | (cid:28) B. Multiple species of particles
The resulting expression for the beam permittivity canbe easily modified to include other beams or the back-ground plasma. Multiple species of particles ( e.g. beamsof particles with different masses, charges, or average ve-locities) can be included as additive terms to the dielec-tric permittivity. Dielectric susceptibility for each speciedescribes the polarization of that beam in response tothe external electric field E z . The external electric fieldis identical for each specie, so its polarization does not de-pend on whether other types of particles are present. Atthe same time, the dielectric displacement D z includespolarization from each type of particles present in thesystem (cid:15) || = 1 − (cid:88) α ˜ ω pα ( ω − kβ α c ) , (29)where summation carries over all species α present in thesystem.Note that the effective plasma frequency for eachspecie ˜ ω pα should be calculated using the same trans-verse wavenumber κ = (cid:112) k − ω /c regardless of theenergy of each specie. As a result, different species ofparticles may fall into different limiting cases describedin Sec. III. For example, analysis of beam-cloud instabil-ity for the beam propagating through background plasmamay result in dielectric susceptibility for the beam beingwell approximated in the “free space long wavelength”limit ( κ r b (cid:28)
1) while dielectric susceptibility for thebackground plasma being well approximated in the “1D”limit ( κ a (cid:29) V. SAMPLE PROBLEM: TWO STREAMINSTABILITY
Several groups have proposed to use two-stream insta-bility for beam bunching in vacuum electronics [4, 21] orrelativistic electron beams [5, 22]. That scheme requirespresence of electrons with two or more distinct energiesco-propagating inside a focusing channel. The kineticinstability similar to the conventional two-stream insta-bility in plasma develops, which results in beam bunch-ing. The analysis of those schemes is typically done in1D geometry under assumption that the wavelength ofmodulation in the beam frame is much smaller than thetransverse beam size. However, this approximation isnot valid in most relativistic cases [5, 22]. The trans-verse beam profile also needs to be accounted for if morecomplicated geometry of beams is used [23].We consider two beams with close velocities β and β co-propagating along a focusing channel. Velocities ofthese beams are close to each other, so that¯ β = ( β + β ) / , (30)∆ = c ( β − β ) / E − E mc
12 ¯ βγ (cid:28) c ¯ β. (31)We consider two beams to have identical transverse den-sity profiles and currents for simplicity. The 1D analy-sis of the instability [4, 5] suggests rough scaling for thegrowth rate of the instability Im ( ω (cid:48) ) = γIm ( ω − kβc ) ∼ ω p and the wavenumber for the fastest growing mode k (cid:48) = γ ( k − βω/c ) ∼ ω p /γ ( E + E ) / ( E − E ). As a re-sult, the unstable wave can be viewed as quasi-stationary in the beam frame as discussed in Sec. III A. The effectiveplasma frequency depends only on wavelength of modu-lation since κ ≈ k/γ in this regime.The dielectric permittivity for the system of two beamscan be found as the additive contribution of two individ-ual beams (cid:15) || = 1 − ˜ ω p ( ω − k ¯ βc − k ∆) − ˜ ω p ( ω − k ¯ βc + k ∆) . (32)We search for the electrostatic plasma waves, which canbe supported by this dielectric medium. The electro-static modes satisfy the dispertion relation (cid:15) || = 0. Weintroduce the frequency Ω = ω − k ¯ βc , which describestime evolution in a frame co-moving with average beamvelocity. Then this frequency can be found to beΩ = ˜ ω p + k ∆ ± (cid:113) ˜ ω p + 4˜ ω p k ∆ . (33)In the dispersion relation described with Eq. (33), thebranch with the minus sign, corresponds to the unsta-ble mode. The instability occurs at large enough wave-lengths, k ∆ < ω p . However, the effective plasmafrequency scales with the wavenumber. In fact, the vac-uum pipe strongly suppresses large wavelength modes.We use the expression (27) for the effective plasma fre-quency to find the condition for the two stream instabilityto develop E − E mc < (cid:115) βγ II a (cid:20) ln (cid:18) ar b (cid:19) + 12 (cid:21) . (34)If the beams are not intense enough or their energies arenot sufficiently close to each other, then the two-streaminstability does not develop. The maximum growth ratefor the two-stream instability can be found ( ∂ k Ω = 0)if the beams are intense enough: (cid:34)(cid:115) k ∆ ˜ ω p − − k ∆ ˜ ω p (cid:35) ∂ k ˜ ω p ∆ = (35)= 2 − (cid:115) k ∆ ˜ ω p . (36)In the 1D limit the condition for the fastest grow-ing mode reduces to k ∆ = 3 / ω p since the effectiveplasma frequency does not depend on the wavelength asillustrated in Fig. 3. This result matches findings of otherstudies [5].The 2D effects need to be accounted for when the wave-length of modulation in the beam frame is comparableor larger than the beam radius, kr b /γ ∼
1. At the sametime, the effect of the pipe wall is small far enough fromthe threshold condition (34) and the beam can be approx-imated as propagating in free space. We approximate hebeams to have flattop distribution of density and use ex-pression for the plasma wave presented in Table I. Thenthe fastest growing mode and the growth rate of insta-bility Im (Ω) can be found as solutions of the followingtranscendental equation p K ( pκ ) (cid:34)(cid:115) κ − pκK ( pκ ) − − κ (cid:35) =2 − (cid:115) κ − pκK ( pκ ) , (37)Ω ω p = 1 − pκK ( pκ ) + κ −− (cid:112) (1 − pκK ( pκ )) + 4 κ (1 − pκK ( pκ ) , (38)where κ = k ∆ ω p , p = r b ω p γ ∆ = mc E − E (cid:114) βγ II a . (39) p p - i m a x / p0 FIG. 4: Wavenumber for the fastest growing mode (upperplot) and its growth rate (bottom plot) for different beamintensities.
The scaled wavenumber for the fastest growing mode κ depends on a single parameter p , which describes theintensity of the beam. Note that the scaled intensity ofthe beam cannot be significantly smaller than unity for realistic beam and pipe radii as described by inequality(34). The plots for the wavenumber of the fastest grow-ing mode and its growth rate are presented in Fig. 4. Theresults show a soft threshold for the instability at p ≈ . κ ≈ (cid:112) / p >
2. The growth rate of the instability also approaches1D result Ω = − ω p / pκ (cid:28) VI. SUMMARY
We have developed a general formalism describing lon-gitudinal space charge effects in beam-plasma systems.The self-consistent dynamics can be described in termsof effective longitudinal dielectric permittivity of themedium, which describes the response of the medium tothe external space charge wave. The permittivity hasa functional dependence matching the 1D plasma per-mittivity. The entire effect of the geometry (transversedensity profile and presence of the conducting cylindri-cal wall) results in the effective plasma frequency beingdifferent from the 1D plasma frequency. The effectiveplasma frequency is described with Eq. (23).The developed formalism provides a universal frame-work for studying longitudinal dynamics in various beam-plasma systems. Multiple species such as backgroundplasma, beams with different energies, and different typesof charged particles can be included in the analysis si-multaneously as additive terms to the dielectric permit-tivity of the medium. Each of these species may haveunique transverse profiles. Inclusion of kinetic effects isa straightforward generalization similar to the case of 1Dplasma. A particle beam propagating through plasma af-fects the background density distribution and generatesreturn current. These effects can also be included in thedeveloped framework through introduction of additionalparticle species, which describe the return current andmodified density profile.
VII. ACKNOWLEDGEMENTS
Authors are thankful to Petr Anisimov, Stanislav Ba-turin, Trevor Burris-Mog, Dima Mozyrsky, Derek Neben,and Vitaly Pavlenko for fruitful discussions. Work sup-ported by the US Department of Energy under contractnumber DE-AC52-06NA25396. [1] A. W. Trivelpiece and R. W. Gould, “Space ChargeWaves in Cylindrical Plasma Columns”, J. Appl. Phys. , 1784 (1959).[2] A. H. W. Beck, “XLIX. High Order Space Charge Wavesin Klystrons”, J. of Electronics and Control,
489 (1957).[3] Ronald C. Davidson, Igor Kaganovich, Hong Qin, Ed-ward A. Startsev, Dale R. Welch, David V. Rose, andHan S. Uhm,“Collective instabilities and beam-plasmainteractions in intense heavy ion beams ”, Phys. Rev. STAccel. Beams , 114801 (2004).[4] Bruce E. Carlsten, Kip A. Bishofberger, and Rickey J.Faehl, “Compact two-stream generator of millimeter-and submillimeter-wave radiation”, Phys. Plasmas ,073101 (2008).[5] A. Marinelli, E. Hemsing, and J. B. Rosenzweig, “Usingthe Relativistic Two-Stream Instability for the Genera-tion of Soft-X-Ray Attosecond Radiation Pulses”, Phys.Rev. Lett. , 064804 (2013).[6] A. Chao, “Physics of Collective Beam Instabilities inHigh Energy Accelerators”, Wiley, New York (1993).[7] Z. Huang, M. Borland, P. Emma, J. Wu, C. Limborg,G. Stupakov, and J. Welch, “Suppression of microbunch-ing instability in the linac coherent light source”. Phys.Rev. ST Accel. Beams , 074401 (2004).[8] N. M. Kroll and W. A. McMullin, “Stimulated emissionfrom relativistic electrons passing through a spatially pe-riodic transverse magnetic field,” Phys. Rev. A17 , 300(1978).[9] Y. Fang, J. Vieira, L. D. Amorim, W. Mori, and P. Mug-gli, “The effect of plasma radius and profile on thedevelopment of self-modulation instability of electronbunches”, Phys. Plasmas , 056703 (2014).[10] J. Rosenzweig, C. Pellegrini, L. Serafini, C. Ternienden,and G. Travish, DESY Report No. TESLA-FEL-96-15,1996.[11] C. B. Schroeder, C. Benedetti, E. Esarey, F. J. Gruner,and W. P. Leemans, “Growth and Phase Velocity of Self-Modulated Beam-Driven Plasma Waves”, Phys. Rev.Lett. , 145002 (2011).[12] F. J. Gr¨uner, C. B. Schroeder, A. R. Maier, S. Becker,and J. M. Mikhailova, “Space-charge effects in ultrahighcurrent electron bunches generated by laser-plasma accel-erators”, Phys. Rev. ST Accel. Beams , 020701 (2009). [13] Gianluca Geloni, Evgeni Saldin, Evgeni Schneidmiller,and Mikhail Yurkov, “Theory of space-charge waves ongradient-profile relativistic electron beam: An analysis inpropagating eigenmodes”, Nucl. Instrum. Meth. A ,20 (2005).[14] Petr M. Anisimov and Nikolai Yampolsky, “Spacecharge fields in azimuthally symmetric beams: integratedGreen’s function approach”, Internal LANL report, LA-UR-20-29774, arxiv:2102.01250 (2020).[15] Robert L. Gluckstern, “Analytic methods for calculat-ing coupling impedances’ CERN Yellow Report, CERN2000-011(2000).[16] Fast space charge waves having ω/k > c will result inexcitation of electromagnetic modes inside the vacuumpipe as they can be viewed as a source for Cherenkovradiation. Analysis of collective effects presented in thiswork will be incomplete in this case.[17] Marco Venturini, “Models of longitudinal space-chargeimpedance for microbunching instability” Phys. Rev. STAccel. Beams , 034401 (2008).[18] Robert L. Gluckstern, “Analytic Model for Halo Forma-tion in High Current Ion Linacs”, Phys. Rev. Lett. ,1247 (1994).[19] Dazhang Huang, Qiang Gu, and King-Yuen Ng, “PlasmaEffect in The Longitudinal Space Charge Induced Mi-crobunching Instability for Low Energy Electron Beams”,arXiv:1307.1190 (2018).[20] F. Mako and T. Tajima, “Collective ion acceleration by areflexing electron beam: Model and scaling”, Phys. Flu-ids , 1815 (1984).[21] S. Safari, B. Jazi and S. Jahanbakht, “Different roles ofelectron beam in two stream instability in an ellipticalwaveguide for generation and amplification of THz elec-tromagnetic waves”, Phys. Plasmas , 083110 (2016);[22] N. Yampolsky, G. L. Delzanno, C. Huang, andD. Shchegolkov, “Development of the two-stream insta-bility in a single bunch”, AIP Conference Proceedings1812