Amatuni, A.Ts.

Elbakian, S.S.

Khachatryan, A.G.

Sekhpossian, E.V.

eScholarship - University of California

Lawrence Berkeley National LaboratoryRecent WorkTitleInvestigation of Focusing of Relativistic Electron and Positron Bunches Moving in Cold PlasmaPermalinkhttps://escholarship.org/uc/item/5d91m1jpAuthorsAmatuni, A.Ts.Elbakian, S.S.Khachatryan, A.G.et al.Publication Date1995-03-01eScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaLBL-36843 UC-414 IL~Wif®Iffi®® Iffi®rrilr®ll®y IL~lm@Ir~~@Iry UNIVERSITY OF CALIFORNIA Accelerator & Fusion Research Division Investigation of Focusing of Relativistic Electron and Positron Bunches Moving in Cold Plasma A.Ts. Amatuni, S.S. Elbakian, A.G. Khachatryan, and E. V. Sekhpossian March 1995 '; I Prepared for the U.S. Department of Energy under Contract Number DE-AC03-76SF00098 ::0 1"1"1 (") ., .... 0 1"1"1 -,o::o O(l)ITI s::cnz __, (") Ill ZITI t+O 11) t+ (") 0 aJ "0 __, -< 0.---CQ . U'l lSI .-.... C" (") ..., 0 Ill 'tJ ~ '< . ..... .-aJ .-I w en (XI -'=" w DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California. LBL-36843 UC-414 CBP Note - 138 "Investigation of Focusing of Relativistic Electron and Positron Bunches Moving in Cold Plasma" A. Ts. Amatuni, S. S. Elbakian, A. G. Khachatryan & E. V. Sekhpossian Center for Beam Physics University of California Lawrence Berkeley Laboratory Berkeley, California 94720 March,1995 *This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics, of the U.S. Department of Energy under Contract No. DE-AC 03-76SF00098. . 1 LBL /136843 FINAL REPORT CBP NOTE It 138 ON THE WORK "INVESTIGATION OF FOCUSING OF RELATIVISTIC ELECTRON AND POSITRON BUNCHES MOVING IN COLD PLASMA" Purchuase Order N 4596210, date 09.29.93 Lawrence Berkeley Laboratory Authors: A.Ts. Amatuni, s.s. Elbakian, A. G. Khachatryan, E.V. Sekhpossian. a) First stage - "FOCUSING IN DIFFERENT MODELS OF COLD OVERDENSE PLASMA-ELECTRON BUNCH SYSTEM". Focusing of relativistic electron (positron) bunches is considered in three different descriptions of cold overdense p_lasma-rigid electron bunch system. In all three models Coulomb component of field exists. For large values of the bunch Lorentz-factor it is negli-gible in comparison with the wake field component. Total charge and current densities in general are not compensated. For narrow bunches they are nearly proper-tional to each other. The resulting focusing force is a complex combination of magnetic and electric forces, which relative strength depends on bunch parameters. The obtained results in case of narrow bunches are practically independent from the considered models. The general formulae for focusing force are obtaine~ which can be used for estimates in planned experiments. Par~icular cases of narrow, short and long bunches dis-2 cussed and focusing gradient calculated for the experi-ments performed at ANL, Tokyo University-KEK and UCLA. Results of these stage were presented to LBL in Interim Report on March 1994, in talk on 6th Workshop on Advanced Accelerator Concepts, Abbey on Lake Geneva, Wiscononsin, June 1994, and also presented for publication in "Parti-cle Accelerators". b) Second stage - "INVESTIGATION OF THE FOCUSING AND WAKE FIELDS OF THE SEQUENCE OF RELATIVISTIC ELECTRON AND POSITRON BUNCHES MOVING IN COLD PLASMA". We considered the focusing of the bunch of electrons (positrons) moving in wake field generated by sequence of N-1 cylindrical bunches with uniform distribution of the charge with the densities f1.sk ( k = 1,~ 2,~ .. · ;tl- i ) , lengths dk , radius ~ and the distances in bet-ween 4 . We used the linear approximation when the condition is fulfiled. ( ft0 is the density of plasma electrons in equhibri~, plas-rna is neutral, ions are immobile). 1. THE GENERAL SOLUTION OF THE LINEAR EQUATIONS. The rigid bunch of electrons with the density fLg(~r),i=Z-Vof is moving in cold neutral plasma with the velocity Vo {0,0, Va' ). In linear approximation the components of the momenta of plasma elec-trans obeyed the system of eq-Introducing in {1) the Hankel transformations: oC 00 /_, ~ t-)= [ ~ (~r:Jr-2 (rJ)dr J ~ {~ ~) = !£ {f1 t) J !fo (rf)df J {) 0 (2) 00 o<> T(J/t:)= [R(~r)r:{ (rf)dr J /2(r;t) = r nv) J J,(r;)dJ, 0 - 0 . where Yo (r J) and ~ (rf) are the Bessel function of the zero and first order, the system {1) become . 2 -J d~- _!_ dT +k 4 t/T o d r i2. -a ?.:~ t r > (.3) 0¢ 11 (I, r)= [ /l.'tf;-t:) r 2 {Jr)dr. 0 Excluding the function ~(S ~) from system {3) it is possible to / obtain the following eq. for l (f, i:} 2. 2 2. 2. where X =~ji + J . Solving the eq. {4) by Gteen function method for finite solution with the boundary condition 6 (t--+ + co)~ 0 from {2) we get 4 (5) where The components of electromagnetic fields and force, acting on elec-tron bunch are defined through (5) by formulae (6) 2. SELF-FOCUSING OF THE SINGLE BUNCH. For completness and comparison with the results obtained for the sequence of driving bunches, let us present here the expression for focusing force in case of the single uniform electron bunch, when 5 :e E- [-d o J r s a_ I I .I ' _.) 0 ) /1- n.,_ a ~ (a ) rr - Ylo f ../1 J ) and from (5-6) focusing force is where OJ ~>0, J(~) = 1.- C(}Skft:) -d -s;~~o-' - 2 sin kr ~ · Sin kf (t + -J-) ) -e <-d. / e-r)(. i: e- y,l( (~+d) 0 - ) ~> ) h (i~J )-= 2- e'x~ e-)'~(~ +d) d - ) - ~i~O.) e r)t(t+d) _ eJ'H.~ :e.<- el (7) ) . Consider the case when exp (-Y;B kf d /2 ) << 1 we have h 6) J )~ 2 - in the middle of the bunch and A (-l_, J)!::::::: i at the edges of the bunch (see fig.l). Using this approximation and performing the in- o tegration in (7) for r~a.. we obtain !;. = w; m R! tt {- j{-t) I. (kfr) k,(i<( ct) + (i~Z )[I, ("rr) k,(~ya. )--J,I, (k(Jr) 1(, (*ffa)]} _, (8) 6 where I 1 the factors and k. are the modified Bessel functions. In (8) (i>z) are taken for the middle of the bunch and for the edges consequent ly. For the case of narrow bunch when focusing force is linear on r, we have As it is evident from (9) for the region of the bunch, where (1.0) the force is defocusing) f.~ 0 t J...e (see fig. 2) due to presence of the Coulomb field, so self-focusing is absent for the short bunch-es . From (9) follows that coulomb component of y-Z the force, proportional to 0 , in ultrarelativistic limit is small, and can be neglegted, as it was done, for example, in the pri-mary work [ 4] . 3. FOCUSING IN THE WAKE FIELD OF THE TRAIN OF BUNCHES. For the chain of cylindrical bunches with the radius a IL '&J<. ) 7 where fltK -the electron density of k t-1, bunch, dl( -the length of k th bunch, ~ is the distance bet-ween k-fj and (k+!)-H bunches. Radial force, acting on electrons moving behind the chain of N-1 bunches with the velocity Vo from (5-6) is (i2) where ~1 ~~ c c~ J = f: fl-t" /f(t) ) H rs s) = {; 12-,J; ( (t, ; J) $_ (t)= - 2$-iYL k~ d" b'il1. Kp (u * + f /_,s ) ; S:: I K-1 t<.-1 hA=t}J)= f!J((:c+d,+l;ts) _ tfx(r+ if ls) • When bunch, which must be focused, is located on such a distance that the second term in {12) contained integrals is negligible and for the simple case, when n.,( = nl .) dl( = d J tJ:.:: e - follows from (12) (i3) From (13) it is evident that focusing force reached its maximum value when ( 1~) where n'f / hz = 1/ 2_, ... For the case of narrow bunches, ;c, C?. << 1 1 when condi-tions (14) are fulfilled, from (13) follows (see fig. 3): (1S) so the focusing force is enhanced by a factor N-1. It is necessary to add to (15) self focusing force of the single bunch, given by (9), which for large enough N can be neglected. By the same way it is possible to obtain the longitudinal force act-ing on the Nth bunch which follows the chain of N-1 bunches. Neglect-ing the Coulomb component from and (5) we obtain 9 (16) In case of narrow bunches which are suitable for transversal focus-ing (16) have the following form. where -is the Euler constant. The focusing bunch must be placed in the middle of the region with the length At/Z , located on the distance 3A.t/4+nA.1 (n=tz ~~ .. .) from the end of the bunches chain (see fig. 3). As it is seen from (17) and fig 3. at this case the longitudinal contraction of the Nth bunch is present, which however is weaker than than transwearsal one. It is also nessery to remember that all this results are valid only .Hi. · t' h' t' t t' t't' 1n 1near approx1ma 1on. T 1s assump 1on pu s cer a1n res r1c 1on on the number of driving bunches in the chain This condition is restrictive enough and in order to avoid it, we must consider the nonlinear effects. It is interesting to mention here also that when dl<::: 11.1< AI {!! .. ::: ~ 2, ... ) the wake fields are absent. 10 c) IN ADDITION TO STAGES a) AND b) DESCRIBED ABOVE THE ATTEMPTS HAVE BEEN MADE TO ELABORATE SUITABLE TECHNIQUE FOR TREATMENT THE NONLINEAR EFFECTS IN PLASMA FOCUSING AND WAKE FIELD GENERATION. In particular, the analytical treatment of the case of underdense plasma was undertaken. It was also noticed that in all three models of cold plasma-rigid electron bunch system, considered in [3] in linear approximation the wake field component of magnetic field is zero. The physical rea-son for this effect is the absence of energy flow in wake fields, (see also results of work [5]). non If magnetic field is zero,i~linear approach too~electric field is potential. If plasma electrons are nonrelativistic (which also means that we neglect a significant part of nonlinearities, (caused by the relati-vism) and driving bunch is flat with the horizontal dimension much larger than vertical one, it is possible to show that potential func-tion obeys the following nonline-ar equation: ( i) (The Breizman-Tajima-Fisher-Chebotaev transformation of variables is used [6,2-3]). Exact solution of this eq. with separating arguments is the following }!(~; t:) = Y, ( ;;) Y1 (c)) (2) ~ (y):::: 11 where is the separation constant and must be found from equation: (3) First integral of eq. (3) is (4) The constant ~2 is connected with the boundary conditions: ~(-d)= jt'o 1 ~1 (-d)= y;; ,where -d is the posi-tion of the rare end. of the driving bunch, If X>O and G >o , the implicit solution of eq. (3) is (5) where i 0 I s:t'n.._ fX = 2 2 + fi ) .) and F is an elliptic integral of the first order. Solutions (2),{5) represent the nonlinear two dimensional wake field in the vi-cinity of the bunch symmetry plane y=O. ("Paraplanar" solution; it means that separation of variables is possible only in this region of space) . Further investigations of this particular solution as well as other solutions of eq. (1) will be the subject of the future work. The pos-sibility of taken into account the relativism of the plasma electrons also will be discussed. REFERENC ES A. Ts. Amatuni Ot .fJMtvfu,...; s. s. Elbakian S. {-(tCY~-~-?'7 A. G. Khachatryan 1/. K)tn)~ f-ry/IA.. E. V. Sekhpossian E. Se~~~ 1. W.B. Mori, T. Katzouleas Proc. EPAC-90, Nice-Paris, v.1, 603,1990. 2. A.Ts. Amatuni, s.s. Elbakian, E. v. Sekhpossian Preprint LBL-34836, UC-144,1993. 3. A.TS. Amatuni, s.s. Elbakian, A.G. Khachatryan, E. v. Sekhpossian 13 Proc. 6th Workshop on Adv. Ace. Concepts Abbey on Lake Geneva, Wisconsin, June 1994. 4. P. Chen Part. Ace. 20~71, (1986) 5. Kenings and Jones Physics of Fields 30,· 252, (1987). 6. B.N. Breizman, T, Tajima, P.L. Fisher, P.Z. Chebotaev INS preprint, University of Texas at Austin, (1993), 14 --- 2 -d -d/2 0 Fig. 1 Function h ( :c-, .f). 15 0 Fig. 2 Focusing force for single bunch. 16 ~ /(-1 ;V -1 ~-+--~-+~--~~---+--~--------~ ----+-------+~--~ L---------1 0 I L__ ~2A ---+---- 31l1 ; . ~ --~ Fig. 3 Forcies acting on Nth bunch, following the chain of N-1 bunches. 17 " LAWRENCE BERKELEY LABORATORY UNIVERSITY OF CALIFORNIA TECHNICAL AND ELECTRONIC INFORMATION. DEPARTMENT BERKELEY, CALIFORNIA 94720 ·-·~ ~ 

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Investigation of Focusing of Relativistic Electron and Positron Bunches Moving in Cold Plasma

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Investigation of Focusing of Relativistic Electron and Positron Bunches Moving in Cold Plasma

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