Humphries, S

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Particle Accelerators1983 Vol. 13 pp. 249-2530031-2460/83/1303/0249$06.50/0© Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaEQUILIBRIA FOR FOIL-FOCUSEDRELATIVISTIC ELECTRON BEAMSSTANLEY HUMPHRIES, JR.Department of Chemical and Nuclear EngineeringUniversity o.f New MexicoAlbuquerque, New Mexico 87131 U.S.A.(Received February 23,1983)High-current relativistic electron beams can be focused effectively using a periodic array of transverseconducting foils. Image charges on the foils reduce the average radial electric fields acting on the beams sothey can be confined by self-magnetic fields. This type of transport has potential application in linearinduction accelerators. Calculations are presented which allow the determination of the foil geometrynecessary for radial-force balance in electron beams of any emittance.The space-charge electric fields of charged-particle beams propagating in vacuumcause them to expand. When the beams are relativistic, expansion is counteracted byself-magnetic-field forces. The magnetic force is a factor of - ~2 times the electricforce 1, where ~ = vz / c 1. The forces balance to within a factor of 1/y2. Various methodshave been used to balance the missing factor to transport constant-radius electronbeams. The most common is to apply an additional focusing force, usually by axialmagnetic fields or periodic fields from solenoidal magnetic lenses. Another approach isto reduce the radial electric field. This can be accomplished with a dilute background ofpositive ions with density> 1/y2 times that of the electron beam. 2An alternative method, suggested by Adler,3 is the use of a periodic array ofconducting foils transverse to the beam axis, as shown in Fig. 1.4 In this case, thepositive image charge on the foils reduces nearby radial electric fields. The averageelectric force on electrons passing through the array is reduced, and the electrons maybe confined magnetically. Two characteristics of relativistic electron beams make thisWall~ ~6r ~Foilrwrb5~Z~oFIGURE 1 Geometry for calculating foil-focusing of a relativistic electron beam. The system hascylindrical symmetry; one focusing cell is shown.249250 STANLEY HUMPHRIES, JRfocusing method feasible. First, at high y, focusing and defocusing forces are almostequal, so only a small electric-field reduction is needed. This can be provided withwidely spaced foils. Second, electrons have relatively small energy loss in materials.This means that substantial foils can be used' without significant degradation of thebeam energy or emittance. Furthermore, at low duty cycle, high-current pulsed beamscan propagate with little problem of foil heating. Foil focusing has the potential forlong-distance transport of multi-kiloampere electron beams above 10 MeV. It couldreplace magnetic focusing in high-current linear induction accelerators. The method ismore predictable than plasma focusing. It has the further advantage that shaped foilsmay be used to steer the beam and stabilize it on axis.The calculations of Adler3 treat the specific case of an emittance-dominated beamextracted from an immersed cathode. In this paper, general calculations of transportlimits as a function of foil geometry 'will be presented for beams of any emittance,including the laminar-flow case. The approach will be to seek a global balance bymatching forces on the beam envelope. Reference 3 shows that for the common case inwhich the distance between foils is large compared with the beam radius (D » rb), theaverage electric force is approximately linear for a uniform beam; the envelopeapproach therefore gives good estimates. The following assumptions will be made.Electron motion in the beam is paraxial and all electrons move with the same ~. Theaverage beam density does not vary axially; this is true if there is no accelerationbetween foils, or if the beams are highly relativistic. The beam has cylindrical symmetrywith radially uniform space charge and current density. Oscillations of the beamenvelope (see Fig. 1) are small; limits of validity of this assumption will be investigated.The foils are perpendicular to the system axis and radially uniform; the problem oftransverse stability will not be addressed.To begin, conditions for a laminar-flow equilibrium are derived. The assumptionsabove permit calculation of the average electric fields acting on the beam bydetermining the radial electric field of a uniformly charged cylinder of radius rb in aconducting pipe of radius rwand length D with conducting ends. The fields areaveraged over the length of the pipe at the envelope radius to estim.ate the averagedefocusing force acting on peripheral particles. The ratio of this force to the radialelectric force on the envelope of a beam of infinite length will be denoted by L\. Sincemagnetic fields are unaffected by the presence of foils, a laminar-flow force balance willoccur when (1 - L\) = 1/y2.The electrostatic potential can be calculated by expanding the potential and space-charge density in a Fourier-Bessel series of the formand<P = LL AmnJO(xmr/rw) cos (nrcz/D),m np(r, z) = LL {2enO(rb/rW):1(~( Si)n (T) Jo(Xrmwr) cos (nDTtz),}m n EoTt xmnJ1 X m(x m = 2.405, 5.520, ... ).(1)(2)The potential of Eq. (1) satisfies the boundary conditions <P = Oatr = rw,z = -D/2and z = D/2. The coefficients in Eq. (2) apply to a uniform charge density. TheEQUILIBRIA FOR FOIL-FOCUSED RELATIVISTIC ELECTRON BEAMS 251coefficients A mn can be determined by substituting Eqs. (1) and (2) into Poisson'sequation. Thus(3)Taking - a<t>/ar at the en~elope, averaging over the length of the focusing cell anddividing by rbeno/2Eo (the radial electric field of an infinite-length beam) gives(4)This expression was evaluated numerically with 12 radial terms and 10 axial terms asa function of (D /rb) for various choices of (rw/rb)' Accuracy of the series expansiondeteriorates wi th large D/ r b' so it was used for field variations within 2r b of the foil, andthe uniform field approximation used in between. Results are plotted in Fig. 2, showing(1 -~) versus D/rb . The second vertical axis shows the value of y corresponding to alaminar-flow equilibrium. For instance, a 4-MeV electron beam of 2 cm radius in a 4-cm radius pipe can propagate with a foil spacing of 1.7 m.The results above are independent of the beam current. Nonetheless, the currentmust be included to determine if the conditions of the model are satisfied. It is requiredthat oscillations of the beam envelope between foils be small compared with the beamradius. The change in beam radius away from a beam neck (the midpoint in Fig. 1) to20.1 32 4 456780.01 l-----t-t--t-t-H-++t----+---f--f--...-.+--11-*+-'-9101 10 100FIGURE 2 Reduction of average radial electrical fields as a function of foil geometry. The parameter 1\ isthe ratio of the longitudinal average of radial fields at the beam envelope to the field on the periphery of aninfinite length beam. The vertical axis on the right shows the value of electron y for a laminar flowequilibrium.252the foil is approximatelySTANLEY HUMPHRIES, JR~r eI(D/2rb)2-;:;: = 4n£omoc 3 y· (5)Equation (5) is based on the assumption that ~ ~ 1 and ~r/rb « 1. For ~r/rb = 0.1,Eq. (5) becomesD f;3- < 82 -rb I '(6)where I is expressed in amperes. The limit is plotted in Fig. 3 as a function of d/rb forchoices of y. The model is valid for parameters in the region to the left of the lines.When the electric field is reduced below the laminar-flow conditions, beams withnon-zero emittance can be propagated. The emittance force on the envelope of a beamwith an elliptical distribution is given by 1(7)The emittance £ is defined as the area of the distribution ellipse divided by n. The netfocusing force is given by(8)for a foil-focused beam, where the quantity ~ can be found from Fig. 2. Force balance\ \ \ \ \ \ \\ \ \ \ \ \ \ \\ \ \ \ \ \ \~ \ \ \ \ \\ \ \ \\ \\ \ \\ \ \ \ \ \\ \ \ \ \\ \ \ \ \\ \ \ \ \\ \ \ \ \\ \ \ \\ \ \ \ \\ \ \ \\ \ \ \\ \ \ \\ \ \\ \ \\ \ \\ \ \\ \ \\\ \ \5 7 10 15 2010I(kA)0.116~ 2 310 100FIGURE 3 Limits of 'l.pplicability of the model as a function of foil geometry, beam current, and electronenergy. In the region to the left of the dashed lines, envelope oscillations are less than 10 per cent of the beamradius.EQUILIBRIA FOR FOIL-FOCUSED RELATIVISTIC ELECTRON BEAMS 253gives the conditionwhere fA is the Alfven current,€= (9)(10)The following example illustrates the determination of the propagation character-istics of a foil-focused beam. Assume a 10-kA beam with radius rb = 5 cm and energy9.5 MeV ("1 = 20). The distance between foils is 2.5 m. Under these conditions, thebeam can travel a distance of 42 m through 0.005 cm thick (2 mil) aluminum foils witha 5 per cent energy loss. An acclerating gradient of only 0.01 MV/m can balance theselosses. With a wall radius of 10 cm, Fig. 2 indicates that the average electric field of thebeam is reduced by a factor of 0.973. The magnetic force is 0.9975 times the electricforce of an infinite-length beam. Since ~2 > d, the beam is confined. The Alfven currentfor the beam is 168 kA. Thus, f / fA = 0.06. (This condition also guarantees that particlemotion is paraxial.) Substituting the quantities in Eq. (9), a beam with an emittance of270 mrad-cm will be matched. Finally, reference to Fig. 3 shows that the relative beamenvelope oscillations are less than 10 per cent.I would like to thank R. Adler for his suggestions on this material.REFERENCES1. Characteristics of relativistic electron beams are reviewed in J. D. Lawson, The Physics of ChargedParticle Beams (Clarendon Press, Oxford, 1977), and R. B. Miller, An Introduction to the Physics of IntenseCharged Particle Beams (Plenum Press, New York, 1982).2. R. J. Briggs, J. C. Clark, T. J. Fessenden, R. E. Hester, and E. J. Lauer, in Proc. 2nd Int'l. Topical Conf. HighPower Electron and Ion Beam Research and Technology, edited by 1. A. Nation and R. N. Sudan(Laboratory of Plasma Studies, Cornell University, 1977),319.3. R. J. Adler, Part. Accelerators, 12, 263 (1980).4. The type of focusing discussed in this paper differs from the process of "foil focusing" used in some low-current linear ion accelerators. In the low-current case, grids are placed across the downstream drift tubesat acceleration gaps to modify applied accelerating fields for increased electric-field focusing.

Equilibria For Foil Focused Relativistic Electron Beams

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