Cavenago, M

Rostoker, N

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Particle Accelerators, 1987, Vol. 21, pp. 53-58Photocopying permitted by license only© 1987 Gordon and Breach Science Publishers, Inc.Printed in the United States of AmericaMODIFIED ELONGATED BETATRON ACCELERATORMARCO CAVENAGODepartment of Physics, University of California, Irvine, CA 92717andNORMAN ROSTOKERMaxwell Laboratories, Inc., San Diego, CA(Received January 3, 1986; in final form April 30, 1986)With a manifestly covariant treatment, self-consistent equilibria are obtained for the relativisticVlasov equation and the geometry of the MEBA. The equilibria are functions of the constant of themotion Po and the action integrals ~ Pz dz and ~ Pr dr, which are abiabatic invariants. A radialspace-charge limit is derived; comparisons are made with a longitudinal space charge limit and withrecent experiments. With a linear perturbation analysis, a classification is made of the waves resonantwith the beam accelerated in MEBA; the effects of particle motion along the symmetry axis areincluded. When the Doppler-shifted frequency Q = y(w ± lO) seen by a particle is much less than theshift due to longitudinal motion OJz = kz vz' a component of the perturbed current that is nearlyuniform along the symmetry axis is decoupled. This is the negative mass mode. With a spread of Poand I z in the equilibrium distribution, the growth rate is reduced, but the system is unstable for a finiterange of y. Only a spread of I r , the action variable of betatron oscillations, can produce stability forall values of y. The price for stability is a reduction in the space-charge limit and an increase inemittance.1. INTRODUCTIONThe Modified Elongated Betatron Accelerator (MEBA)l attempts to increase thebeam loading by storing more charge in more space. The elongated geometrygreatly simplifies injection and trapping, because electrons leave the injector,move to the opposite mirror, are reflected and travel the length of the acceleratoragain before they can hit the injector. There is sufficient time to change themagnetic field to prevent this. The MEBA is illustrated in Fig. 1. The vacuumspace that contains the electron beam has a hollow cylindrical shape of length L,inner radius r;, and outer radius rooThe magnetic fields consist of a poloidal field Bz in the direction of thesymmetry axis. It is produced by a long solenoid with additional turns at the endsto produce magnetic mirrors. There is an iron core or a separate flux coil toaccelerate the electrons while maintaining the betatron condition (Bz ) = 2Bzo ; Bzois the poloidal field at the electron location and (Bz ) is the mean field inside theelectron orbit. There is also a toroidal magnetic field Be produced by conductorsparallel to the symmetry axis. The magnetic fields increase together in time; thecoils may be connected in series. The MEBA is a direct descendant of the5354 M. CAVENAGO AND N. ROSTOKERMirror WindingElectron Beam Annulus InjectorToroidal FieldCoilFIGURE 1 Schematic diagram of the MEBA.Astron, which was developed at the Lawrence Livermore National Laboratoryunder the direction of N. Christofiles.2The usual geometric parameters l are L/ro = 20 and (r; - ro)/(r; + ro) = 1/5. Atinjection the spread in Pz and Pe are of order rl!1Pz / Pe =0.1, l!1Pe/Pe = 0.1, wherePe = (y2 - 1)1/2r, with y = 1.14. The poloidal and toroidal magnetic fields arealmost equal, i.e., Be/Bz =0.9-1.1.A small-scale experiment has been carried out that demonstrated injection,trapping, confinement, and acceleration. An experiment to demonstrate extrac-tion has been proposed.3 A theory to predict scaling to a large accelerator isdescribed in this paper. It has not previously been developed because of theunusual geometry that complicates the analysis.2. ANALYSISThe procedure is to solve self-consistently the relativistic Liouville equation,(1)and Maxwell's equations,(2)where (XJl, PJl) are the position and momentum four-vectors; f(XJl, PJl) is thedistribution function; h is the Hamiltonian, given by4h = -!gJlY(PJl + AJl)(Py + A y ) = -!, (3)gJlY is the metric tensor, and AJl is the vector potential.The only constant of the motion is Pe. For the radial and axial coordinates rMODIFIED BETATRON ACCELERATORand Z, we assume that all quantities are slowly varying, so thatIz =fPzdz,Ir=fPrdr55(4)are adiabatic invariants. Any function of In Iz, and Po is a solution of Eq. (1). Weselect a suitable function and calculate(5)Some examples of the distribution of I r are illustrated in Fig. 2, along with theresultant current distribution, which is proportional to rg(X) , where X = r - rAand rA is the orbit center. The result of the calculation is the average radialI I rI I rI-5-5-5oo,~ r93o5 xs x5 xFIGURE 2 The distribution function 11 and the corresponding current-density distribution rg( ill, x).56 M. CAVENAGO AND N. ROSTOKERoscillation frequencyro;A = (82h/8r2)r=rA={_ (8A4)2 + (8A z )2 + (Pe +Aef + [Po _~ (AO)]2}8r 8r r4 r2 8r r A__v_ (WrA)1I2,YArA Iwhere v = Ne2/mc2 is the Budker parameter;(6)(7)is the line density; n(r) is the electron density; rA is the beam average radius; and2s is the beam thickness. For an equilibrium to exist, W;A must be real, which setsa limit on v or on the circulating current. This is the space-charge limit.Stability is investigated by means of a linear perturbation expansion about theabove equilibria. The details· of the calculations are available elsewhere.53. RESULTS(8)y = 1 + [Bo(rA)]2 __1_ (/rA)2'Bz y2 -1 S2where BO(rA) is the toroidal magnetic field at the average radius, Bz is thepoloidal magnetic field, and Y= YA is the average relativistic energy factorrequired at injection to obtain the charge represented by v. I is the maximumvalue of Ir. Initially, we neglect 1=0. For y = 1, [rA/s(y2 - 1)]» 2 and theformula reduces to the usual scaling for a modified betatron. For y» 1, v scaleslinearly with y rather than with y3. The other new feature is that finite values of I rreduce the space-charge limit.The stability analysis shows that a cold beam would always have a range of yfor which the beam has a negative mass instability. A spread in Po and Iz in theequilibrium distribution reduces the growth rate, but the beam remains unstablefor a finite range of y. Complete stability can be obtained only with a finite valueof the radial action I. This, in turn, will modify the space-charge limit of Eq. (8).With the present theory, it is possible to determine the injection y and thequantity I corresponding to stable confinement of the charge v. From I theemittance can be determined.The space-charge limit is given byv[2 +s(y;~ 1)]MODIFIED BETATRON ACCELERATORTABLE IComparison of experimental results1 with predictions for a large MEBAExperiment Large MEBAAverage radius of beam (rA), cm 6 100Length of beam (L), cm 80 600Outside wall radius (ro), cm 9.5 102.5Inside wall radius (r;), cm 4.5 97.5Beam thickness (2s), cm 0.5 2Magnetic field ratio [Be (rA)/Bz ] 1 2Mirror ratio (M) 0.05 0.15Budker parameter (v) 0.041 14Total charge (Q)5 x 106Radial limit nC (Qr) 1850Longitudinal nC (Ql) 350 5 x 106Observed nC 100Total circulating current (I)Observed, A 100Required, kA 240InjectionRadial (Yr) 7.8Longitudinal (Yl) 5.4Observed Y 1.157(9)There is another space-charge limit due to action of the space-charge fields inthe axial direction. Confinement requires that e~<I> < Jl~Bz where Jl = Pi/2ymBzis the magnetic moment of the electron, ~Bz is the change in magnetic field overhalf the length, L12, of the beam. ~<I> is the corresponding change in electricpotential <1>. According to this criterion, the minimum injection y is4v In (rolrA)y = M [1 - (1/ y2)] ,where M = 1 + ~BzIBz is the mirror rate.In Table I, the predictions of Eqs. (8) and (9) are compared with experimentalresults. 1 Predictions of the y required for injection are made for a large MEBA.Some revision of these calculations will be required because of the radial action Inecessary for stability.ACKNOWLEDGMENTThis work was supported by the Naval Surface Weapons Center at MaxwellLaboratories, Inc., and the Office of Naval Research at the University ofCalifornia, Irvine.58REFERENCESM. CAVENAGO AND N. ROSTOKER1. A. E. Blaugrund, A. Fisher, R. Prohaska, and N. Rostoker, J. Appl. Phys. 57, 2474 (1985).2. R. Briggs et al., Astron Final Report, Lawrence Livermore National Laboratory report,UCRL-5184 (1972).3. A. Fisher, N. Rostoker, J. Pearlman, and K. Whitham, Proc. 1985 Particle AcceleratorConference, Vancouver, BC, Canada, May 13-16, 1985.4. H. Goldstein, Classical Mechanics, (Addison-Wesley, Cambridge, MA, 1950), p. 224.5. M. Cavenago and N. Rostoker, VCI Technical Report #85-36 (1985), available on request.

Modified elongated betatron accelerator

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