Barletta, W A

Caporaso, G J

Neil, V K

CERN Document Server

Particle Accelerators1980, Vol. 11, pp. 71-790031-2460/80/1102-0071 $06.50/0© Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaTRANSVERSE RESISTIVE WALL INSTABILITY OF ARELATIVISTIC ELECTRON BEAM*G. J. CAPORASO, W. A. BARLETTA, and V. K. NEILLawrence Livermore National Laboratory, University of California, Livermore, CA 94550(Received June 19, 1980)The transverse stability of a relativistic electron beam propagating in an evacuated, smooth pipe offinite conductivityis examined. Exact and asymptotic solutions are obtained in the limit of continuous external focusing. An exactsolution for the transport through a single focusing period with discrete focusing is extended to provide an asymptoticsolution for a multiperiod transport system. It is found in both cases that focusing can only reduce the growth rate;it cannot suppress the instability. Applications to pulsed electric-power transmission are discussed.I. INTRODUCTIONThe displacement of the centroid of a relativisticelectron beam from the axis of an evacuated con-ducting pipe produces charges and currents in thepipe walls which will affect the subsequent mo-tion of the beam. The behavior of the beam inthe presence of these wall charges and currents,in addition to various focusing forces, is exam-ined.The surface charges which are induced on theinner pipe wall by the nonaxisymmetric compo-nent of the beam's electric field generate an elec-tric field which reacts on the beam to pull it fur-ther along in the direction of its initialdisplacement. The magnetic field of the displacedbeam generates surface currents in the pipewhich set up their own magnetic fields that actto push the beam in a direction opposite to thatof its original displacement. At the front, or headof the beam, these forces cancel to within a factorof 1/'Y2 resulting in a net destabilizing force. Here'Y is the particle's energy in units of the rest en-ergy.Gradually, the magnetic field of the beam dif-fuses through the conducting pipe wall. As thisoccurs, the induced current decreases, reducingthe magnetic restoring force on the beam. The* Work performed under the auspices of the u.s. Depart-ment of Energy by the Lawrence Livermore National Lab-oratory under contract number W-7405-ENG-48.This work is performed by LLNL for the Department ofDefense under DARPA (DOD) ARPA Order 3718, Amend-ment #12, monitored by NSWC under Contract No. N60921-80-PO-WOI88.71electric force, however, does not weaken. Thus,the net destabilizing force grows with distanceback from the head of the beam.In Section II we formulate the beam dynamicsproblem for continuous focusing. An exact so-lution is then presented in Section III. An asymp-totic solution is then given in Section IV and thegrowth of the beam displacement for parametersof interest for pulsed electric-power transmissionis discussed. In Section V the problem is refor-mulated for the case of discrete focusing ele-ments. An exact and an asymptotic solution areobtained. In Section VI the relationship betweenthe continuous focusing and discrete focusingsolutions is exhibited. Finally, the conclusionsare discussed in Section VII. The use of variousfocusing schemes will be shown to reduce thegrowth rate. However, focusing cannot suppressthe instability.II. FORMULATION OF THE PROBLEMFOR CONTINUOUS FOCUSINGConsider the geometry shown in Fig. 1. The beamis slightly displaced in the positive x-direction.The resulting beam current and charge densitiesmay be thought of as those arising from the un-displaced beam plus surface contributions of theform J = (I~/7ra2) 8(r - a) cos <t> ez and p =(I~/7ra2v) 8(r - a) cos <t>. Here a is the beam ra-dius, I the beam current and ~ is the displacementof the beam centroid from the pipe axis (z-axis).More precisely ~(z,t) = ~(z,t) ex, v is the z-com-ponent of the beam velocity and <t> is the azi-72 G. J. CAPORASO, WI A. BARLETTA, AND V. K. NEILThe equation of motion for the beam displace-ment now becomesOnly the case of positive focusing is of practicalinterest.(1)we? < o.<002 = 0, andnegative focusing:positive focusing:critical focusing:d2~ 2t = OS/2 Jf a~ dt' (2)dt 2 + <00 ~ - x at'where <002 == kf32 V 2 - 2Je/(mvb2~3). The ~olutionsto Eq. (2) fall into three classes, depending uponthe value of <002 • We have:+ 4Iev Jf a~ dt'~1Tcb3(J'1/2m -x at lin which m and e are the electron rest mass andcharge, respectively, and k~1V2~ is a focusingterm similar to the restoring force in a simpleharmonic oscillator. This term represents the ef-fects of allowing the number of discrete ·~Ienses"to increase without limit in such a way that theproduct of the iht@t=lens ~istance ~nd th~ fo.callength remains constant. We consider this lIm-iting case itt Section VI. , _. .Notin~ that 4Iev/(~7rcb3(J'1/2m)has the dimen-sions of seC - 5/2 we define a frequency, 4evlOS/2 ==== """",-.-~,--.-=----=--~~- ~m1fcb3(J'1/2'The equation for ~ then can be written asJ2~ 2 2· _ 21e t-d·2 + k f3 V ~ - b2 3 . ~t v ~ m[21eF(z,t) = ex vb2-y2 ~(z,t)+ .4lev Jf a~ dt']'1Tcb3(J'1/2 - x at'The integral term in the expression for B rep-resents the weakening of the magnetic field withdistance back from the h€ad of the beam Gausedby the induced current diffusing into the pipewall. The resulting Lorentz force acting upon thebeam is2/ 4/B = -b2 ~(z,t) ey - b3 . 1/2 eyC 1T (J'Here band (1 are the pipe radius and conductiv-ity, respectively ,.and c is the speed of light. Gaus-sian units are used in this paper.2/E =: vb2 ~(z,t) ex.t It is assumed in the derivation that the fields which pre-cede the beam can be neglected either because the speed ofthe beam is so close to that of light or by noting that, in anycase, the fields leading the pulse cannot extend more than afew beam radii in front of the head. If aiL ~ I. where L isthe beam length, then these fields cannot have an appreciableeffect on the beam dynamics.x Jf a.~ ~dt'-x at'muthal angle measured from the positive x-axis.The quantity o(x) is the Dirac delta function.Only the fields due to the displacement of thebeam centroid from the pipe axis need be con-siderecL The fields produced by these beamcharge and current densities induce correspond-in~ ~harge and current densities in the walls. Thefields produced by these induced charge and cur-rent densities were computedt to be IFIG URE 1 Equivalent surface charg€ and current densitylayers of beam displaced frorft th~ axis of a conducting pipe.TRANSVERSE RESISTIVE INSTABILITY 73Some information on the growth rate of theinstability may be obtained from dimensional ar-guments. For simplicity consider the critical fo-cusing case:it nx - (t' - z/v)n12 - 1 (t - t') 112 dt, ,zlv 2where we have used the fact thata- u - I(t - z/v) = 5(t - z/v).atEquating coefficients of equal powers of T yieldsd 2go 2for n = 0: dz 2 (z) + ko go(z) = 0, (6)After rewriting Eq. (4) in terms of the retardedtime T = t - z/v, we can evaluate the integralon the right-hand side by use of the convolutiontheorem for Laplace transforms.The resulting differential equation for the func-tions gn (z) is(7)(5)for n > 0:(b ) 3/2X Ie; (meters).3 8 1/2 ( ) 1/4• ~ (TII( = Il!J.. 'rl~~ee) 1017 (sec - I)III. EXACT SOLUTION FORCONTINUOUS FOCUSINGThe right-hand side of this expression representsthe effects of the diffusion of the B-field of thebeam into the pipe wall. If the pulse is 'T secondsin length then the maximum range of t' overwhich the integrand is nonzero is 'T. Thus, di-mensionally the equation becomes~/tR2 ~ a5/2~ 'T 1/2where tR is a characteristic growth time. HencetR ,..,..., a -514 'T- 1/4 and the corresponding growthlength IR is 19 ,..,..., vtR ,..,..., va ~514 'T- 1/4. Inserting nu-merical values, we obtainWe will compare this length to the growth lengthwith continuous focusing in Section IV and showthat focusing substantially increases the growthlength.In the equation of motion (2), we have d/dt= a/at + va/az. For the initial conditions a~/az(O,t) = 0; ~(O,t) = do, the solution is of the form~(z,t) = u - I(t - z/v) ~ (t - z/v)nI2gn(z), (3)n=Owhere ko == wo/v. The boundary conditions nowbecome go(O) = do, gn(O) = 0 for n > 0, andgn'(O) = 0 for all n, where a prime denotes dif-ferentiation with respect to z.With these boundary conditions, the solutionsof Eqs. (6) and (7) arewhere u - I(X) is the unit step function and thefunctions gn(Z) must be determined. SubstitutingEq. (3) into Eq. (2) yields11=0 11=0(4)74 G. J. CAPORASO, W. A. BARLETTA, AND V. K. NEIL(b)(c),2.50, "2.15, 1.80. 1.20 T (X 100)·0.60I l 1.00I,~/0.72, '0.48,"0.32 f(X 10°)0.12,0oLt)N(a)\II I':'\' IIi1[2.501.95i : V'1.50I ~" 0.90 T (X 10- 1)i [I ,'0.308-' gON NMaximum ~/do = 3.12 X 102Minimum ~/do = -1.35 X 102Maximum ~/do = 2.02 X 101 'Minimum ~/do = -3.40 X 101o 0 0 0 0Lt) 0 Lt) 0ci ~ ~ N~/dO = 0 zVdO~~/do = 0 z(8)kozjn - I (koz)x--------f(n +Of (~+ 1)'~(z,t) = dou - I (t - z/v):x; (Y1T(t - z/v) fl 5/2z)nX L 2n=O 4v koxwhich may be inverted to yieldx kozjn(koz)f(n + 2) f (~ + D'(V; OS/2 Z)n+ Ign+ t(Z) = do 4v 2 kowhere j n is the spherical Bessel function of ordern. Thus, the solution is(V; OS/2) n + Ign+ J (s) = do 2v 2where the tilde denotes the Laplace transformand s is the Laplace transform variable.This recursion relation may be solved to yieldgn + J(s) in terms of go(s). ThusUsing the convolution theorem for Laplace trans-forms, we obtainIf we define the transformed variables(8a)FIGURE 2 Normalized beam displacement for continuousfocusing solution Eq. (8c) as a function of i and i for (a) i= [0 to 25], i = [0 to 0.25], (b) Z = 10 to 25], i = [0 to 1]and (c) i = [0 to 25], i = [0 to 2.5].i == koz, (8b)TRANSVERSE RESISTIVE INSTABILITY 75the solution (8) can be written asXl (~i)n~(i;T) = dou-I(i) n~o -4-x zjn - I (Z) . (8c)f(n + l) r (~ + 1)by !£ - 1 and where J v is the cylindrical Besselfunction of order v. Use of the relation2(2 ) 1/2 00 tm- cosVz 2 -2zt= L ,Jm - I /2(z),7TZ m=O m.gives~(z,t) = dou - 1(I - zlv) !£ - 1The value of ~(i,T)ldo determined from Eq. (8c)is plotted vs i and T in Fig. 2. The displacementas a function of z and 1 may be found by applyingthe inverses of the transformations (8a) and (8b)to the i and T axes of the plots. In this mannerthe solution for many values of the parametersko and 0 5/2 may be determined from one seriesof plots.Using the Bromwich integral to invert the La-place transform gives1~(z,t) = dou - 1(t - zlv) -4.7Tlx [exp { {(koZ)2 - 2~J1/2}{ [2 2kozJ I/2} Jds+ exp - i (koz) - V; -;.If koz ~ A 2/3 ~ 1, the saddle points of the in-tegrand are found to be at s ~ (=+= i)2/3/(2A 2)2/3where the upper (lower) sign corresponds to thesaddle points of the first (second) term in Eq.(10) . For each sign one of the saddle points lieson the branch cut of the integrand and is inac-cessible. The integration path is shown in Fig. 3.The result of the steepest-descent calculation is(10)(11 )[ 3 (A)2/3Jx exp 2 2" cos koz2do (2) 1/3~(z,t) ~ -- u - 1(t - zlv) -Th AThe solution for positive focusing (8) may bewritten as~(z,/) = dou - 1(t - zlv)i A nkozjn-1 (koz) (9)x n=O f(n + 1) r (~ + I)'whereIV. ASYMPTOTIC SOLUTION FORCONTINUOUS FOCUSINGOur aim is to derive an integral representation ofthis solution so that an asymptotic formula maybe obtained.Laplace transforming the sum in the variableA 2 gives~1TkoZ I~(z,/) = dou - 1(t - zlv) -2- :£-{ I Xl ( 1 )n 1 }x - L "r ,In-II2(koz) ,Sn=O vs n.where the inverse Laplace transform is denotedWe may now define a characteristic length forthe growth of the instability from the relation~=~ _ ~n5/2Zg - 2 - 8kov 2 Z76 G. J. CAPORASO, W. A. BARLETTA, AND V. K. NEILSaddle pointgate long high-current pulses over great dis-tances. The most sensitive parameter in thegrowth length expression is the pipe radius, whileone of the least is the pipe conductivity. A valueof 1017 sec - 1 corresponds to the (J of aluminum,which would be difficult to exceed in practicalsituations.Consider a pulsed electric-power transmissionexample. 3 For a 5 MeV, 2 kA beam with a pulselength of 100 nsec, the critical focusing case givesa growth length of ZI( ::::::: 15 m, while the positivefocusing expression gives ZI( ::::::: 63 (t 00 m/X-o) m,where (J and b have been taken as 1017 sec-I and10 cm, respectively. These growth lengths aredisturbingly short even though the latter growthlength can be extended by increased focusing(decreased X-o).Imaginarysvsaddle point/fSaddle pOint"\~X~-1'~-I" ~1'_~_~ +--_---=R-.:...::e:.=:.a:.....:1s:--V. FORMULATION OF THE PROBLEMFOR DISCRETE FOCUSINGFIGURE 3 Contour for Eq. (10).Consider the single transport cell shown in Fig.4 consisting of two half-lenses each 'of focallength 2f and a drift length L. The transport ma-trix for each half lens is(13 )in the sense that if a prime and a double primeo12f"Lens"° Lens"III8kov2ZI( = ~n5/2·(12)4 0( )1/2. "I (JZI(= 1/2 17 -II<kA) T< .....sec) 10 (sec )Inserting numerical values, we haveor( b<Cffi») 3 (1OO(m»)x -10 --- meters,~owhere ~o = 27T/ko, the net coherent betatronwavelength due to the external continuous fo-cusing. This expression should be compared withthat in Section II for the critical focusing case.The growth length with focusing grows more rap-idly with pipe radius, conductivity and "I, but lessrapidly with increasing beam current than thegrowth length in the absence of focusing. In ad-dition, the growth length for the focusing casevaries inversely with the coherent betatron wave-length. The inverse dependence of ZI( with beamcurrent and the square root of the pulse lengthillustrates the difficulty of attempting to propa----------HaIflensI II II II I~;.•..------- La ----------.~:Drift lengthxLzFIGURE 4 A single half-lens drift length half-lens combi-nation.TRANSVERSE RESISTIVE INSTABILITY 77denote quantities immediately before and afterpassing through the half-lens respectively, thenWe can thus write the unit-cell transport matrixM,MdM, as(14)sinh aLcosh o.L - 20.fcosh aL sinh aLa sinh aL - +f 40.f2In the drift-length region, the beam displace-ment varies according to the equation of motionwhere we have assumed that ~ ~ I. If we changevariables, from z, t to z, ,. = t - z/v, Eq. (15)can be written asm12 ]-mil ·d 2t Jf at-~ = OS/2 ~~dt'dt 2 -xat' 'a2~ OS/2 JT a~ "-=- -~dt.az2 V 2 - x a,.'(15)(16)sinh o.Lasinh aLcosh 13L - 20.fThis unimodular matrix can be written as4M,MdM, = cos IL U~]+ sin fJ- [milm21In the problem under consideration we choosea~(O)/az = 0; therefore ~(N) = ~(O) cos N~ orand mil = O. After traversing N-cells,~ (N) ~ (0)= (M,MdM,)NHeresinh aL 1cos IL = cosh o.L - 20.f = 2Tr(M,MdM,)Now(M,MdM,)N = cos NIL r~ ~]+ sin NfJ- r 0m21m12 ]o .a~Bza~az- - 1 B~«»~(z) = ~(O) cosh az + - -- sinh az,a Bza~ - B~(O)- (z) = a~(O) sinh az + -- cosh az.az BzLaplace transforming in ,. givesa2~ \l;OS12_- - ~ = 0 (17)az 2 2V2~with solution ~ = D cosh az + E sinh az anda~/az = Da sinh az + Ea cosh az, where wehave defineda == 1T 1/40S/4/[V2I J(s) 1/4].Now at z = 0, ~ = ~«) = D and a~/az = Ea.ThereforeThen the drift-length transfer matrix is(h L sinh aL)M - cos ad - a,a sinh aL cosh aL- ~(O) { r~(N) = 2 exp iN COS-I( h L sinh aL)]x cos a - 2afsuch that after traversing one complete cell,~ ~a~ M,M"M, d~dZ dZ+ exp r-iN cos - I( sinh aL)]}x cosh o.L - . 20.f ·18 G. J. CAPORASO, W. A. BARLETTA, AND V. K. NEILNow i cos - I X = log [x ± w-=t], so thatdo !C+iX ds~(N) = -. - exp (TS)41Tt C -ix Sx sin21T 3V3----3 2x {exp[N log (x ± w-=t)]+ exp[ - N log(x ± w-=t)]},wheresinh aLx == cosh aL - 2af· ·( )2/3NUV;;05/2 1 - ~xFor large N the saddle points of the integrandoccur atwhere we have put ~(o) = do, the initial displace-ment of the beam.2/3VI. RELATION OF THE DISCRETEFOCUSING SOLUTION TO THECONTINUOUS FOCUSING SOLUTION\(0)a:)\azoo1----2ioNoI---2ioNxConsider the cell shown in Fig. 4. Let the numberof lenses increase such that the interlens distanceL = LoiN, while at the same time extending thefocal length so that i = foN. Then the displace-ment at the Nth cell is given by1/3where h = as 1/4 = 1T 1/40 5/4/(v2v) and the upper(lower) sign corresponds to the saddle points forthe first (second) term. For each term one of thesaddle points lies on a branch cut and cannot betraversed. The integration contour is basicallythe same as that for Eq. (10) and is shown in Fig.3. A steepest-descent evaluation of the integralyieldsL o- 2foN 2L o 14io2N 2 ioNLoNLo- 2foN 2x cos N tan -]L2iwhere we have chosen the beam to have an initialdisplacement but no initial slope; i.e., a~/ax.= 0at z = O. As in Section V, this may be writtenTRANSVERSE RESISTIVE INSTABILITY 79aswhere(N)In particular for z = L o, comparison of Eq. (19)with Eq. (20) gives1k~ = --. (21)YLofoIf we now put L = LoiN, f = foN in Eq. (18)and let N ~ 00, we obtain the continuous focusingasymptotic solution (11), as we should._ -I ( LO)J.1 - cos 1 - 2foN2 •ThusAs N ~ 00 this equation becomes~(Nl = ~(o) cos (N ~f~~2) = ~(O) cos ~.(19)Now in the continuous focusing limit we haveUsing the variables z, T = t - ziti this becomesa2~-2 + k~2~ = O.azFor the initial conditions ~«» =1= 0; a~(O)/az 0this gives~(z) = ~«» cos k~z. (20)VII. CONCLUSIONIn summary, an exact solution for the growth ofthe transverse displacement of a beam travelingin an evacuated pipe of finite conductivity withcontinuous focusing has been presented. In ad-dition, asymptotic growth rates for both contin-uous and discrete element focusing were ob-tained. The instability was shown to be significant-for the parameters of interest for pulsed electric-power transmission.For both continuous and discrete element fo-cusing, the instability grows. Focusing can onlyact to slow the growth: it cannot suppress theinstability.The authors wish to thank R. Melendez forperforming the numerical work and E. P. Lee forhelpful discussions.REFERENCES1. S. Bodner, V. K. Neil, and L. Smith, Particle Accelerators1, 327 (1970); K. W. Robinson, Stanford Linear Acceler-ator Center Storage Ring Summer Study, 1965 on Insta-bilities in Stored Particle Beams, A Summary Report.2. G. N. Watson, A Treatise on the Theory 0/ Bessel Func-tions, (Cambridge, 1962), 2nd ed., p. 140.3. A New Concept/or Electric Power Transmission Systems,Varian Associates (1973).4. E. D. Courant and H. S. Snyder, Ann. Phys. 3, 1 (1958).

Transverse Resistive Wall Instability of a Relativistic Electron Beam

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