Finn, J M

Manheimer, W M

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Particle Accelerators1983 Vol. 14 pp. 29-370031-2460/83/1401/0029$18.50/0© Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaSELF-CONSISTENT THEORY OF EQUILIBRIUMAND ACCELERATION OF A HIGH-CURRENTELECTRON RING IN A MODIFIED BETATRONWALLACE M. MANHEIMER and JOHN M. FINNtPlasma Theory Branch, Plasma Physics Division, Naval Research Laboratory,Washington, D.C., 20375(Received November 1,1982; in final form May 10,1983)A theoretical and computational scheme is developed to describe the self-consistent equilibrium andacceleration of a high-current electron ring in a modified-betatron configuration. The principal output of thecalculation is the set of nested drift surfaces of the electron beam.Recently, se-verallaboratories have been engaged in studies that are aimed at assessingthe feasibility of developing high-current accelerators. One particularly promisingscheme is the modified betatron, which is a betatron with a strong toroidal field. 1- 3 Aschematic of the device is shown in Fig. 1. The toroidal angle is 8, and the rz plane isdenoted the poloidal plane, as is the conventional notation in tokamaks and other hightoroidal-field plasma devices. A conventional betatron, with no toroidal field, fails toconfine particles as soon as the density gets so high that the outward self-forces exceedthe inward focusing forces. In the presence of a strong toroidal field, the particle orbitsdo again remain confined. But, at high current the self-fields become so strong that thedevice cannot be thought of as a collection of individual particles, but rather as a fluid.Previous theoretical studies of this accelerator have generally either assumed a densityand current profile and calculated the orbits of test particles in fixed fields,1,2,4,5 or elsehave used particle simulations to determine the self-consistent motion. 5,6 The formertechnique is limited because of the assumed charge and current profile, while the latteris limited by considerations of computing cost. For instance, it is not easy to simulatethe beam for one complete drift orbit. For the modified betatron planned at the NavalResearch Laboratory, a drift orbit takes several hundred nanoseconds, but the timescale for acceleration, flux diffusion, etc. is very much longer. This paper attempts toextend these available theoretical techniques by developing a computational andanalytical scheme for analyzing both the self-consistent equilibrium in the modifiedbetatron, and its adiabatic development as external parameters are varied slowlycompared with a drift period. The theory and numerical results are given only brieflyhere. More details will be published elsewhere. 7We utilize a cold-fluid theory to describe the self-consistent equilibrium and itsadiabatic evolution. Since the modified betatron has cylindrical symmetry, the totalmagnetic field can be written as. io ioB = B.l + Bolo = V'" x - + g-r rt Present Address-Plasma Fusion Center, University of Maryland, College Park, MD 2074229(1)30 W. M. MANHEIMER AND J. M. FINNzCONDUCTINGWALLELECTRONBEAM.........---------rFIGURE 1 A schematic of the modified betatron.where '!J is the poloidal flux, ('!J = rAo)a function of rand z, and 9 is r times the toroidalmagnetic field Bo. If the electron beam has density n and velocity v, then Amperes law isc ( * io io)- nev = - - A '!J- + V9 x -4n r r (2)where ~*'!J = r2V' r- 2 V'!J. The toroidal component of the momentum equation insteady state (a/at = 0) can be shown to reduce towherev'VPo = V.1·VPo = 0- e'!JPo = ymrvo - -,c(3)(4)the fluid toroidal canonical angular momentum. Taking the dot product of Eq. (2) withVPo, one can arrive at the result that(5)Equations (3) and (5) imply that Po labels the drift surfaces of the electrons in thepoloidal plane and that 9 is constant on these drift surfaces. Clearly, equilibrium canonly exist if these Po surfaces are closed surfaces in the poloidal plane. Thus, this Poplane must contain one 0 point around which the confined orbits drift. This 0 point willbe called the reference orbit. For instance, for a very low density beam, the Po sur-faces are defined only by the external fields. If the reference orbit has position (r = r0,Z = 0) and energy y, then the vertical field Bz at (r o, 0) is given by eBzro/ymc = vo,where Vo is the velocity of the electron in the edirection. Then if the confining verti-cal field has index 11, one can easily calculate that(6)THEORY OF A MODIFIED BETATRON 31where K is a constant which may be time dependent if other parameters (for instanceBz ) vary in time. As long as the field index is between 0 and 1, Po has a 0 point atr = ro, z = 0, and this 0 point corresponds to a maximum of Po. There are no othersingular points in the Po plane. As we will see shortly, when strong self-fields areincluded, the topology of the Po surfaces can change considerably.We now proceed with our fluid formulation. We assume that the toroidal magneticfield Bois much greater than the vertical field Bz and also that the density is sufficientlylow as to be well below Brillouin flow. 8 This means 2ro;e/yo.;e « 1, where rope is theelectron plasma frequency, ro;e = 4nne21m and o.ee is the nonrelativistic cyclotronfrequency in the toroidal magnetic field. In this case, poloidal inertia and centrifugalforce are small terms in the poloidal (i.e., r, z) momentum balance equation. Then, if thetotal energy isE = yme 2 - e4>, (7)where <p is the electrostatic potential, then after some manipulation one can show thatE is also a function of Po. We also assume, consistent with the neglect of poloidalinertial, that y = (1 - (vole)2)-1/2. Furthermore, one can show that(8)so that the left hand side of Eq. (8) is also a function of Po. For the particular case thatthe energy is independent of Po, then Eq. (8) gives a particular simple expression for thedensityg dgn = -- ----=-.4nrvo dPo(9)We now digress briefly on the implications of Eq. (9) for the drift surfaces (Po surfaces) inthe modified betatron. Clearly for positive density, dgldPo > o. Thus if the referenceorbit corresponds to a maximum in Po (as implied by Eq. (6»), then the toroidal fielddecreases away from the reference orbit. This means that the poloidal currents exert anoutward force on the beam. In plasma parlance, this means the beam is paramagnetic.Therefore, force balance implies that the inward focusing force (from the field index) isequal to the sum of the outward self-forces and outward force from J.L x Bo. Clearly,this implies a low-current beam. Self-consistent equilibria for low-current electronrings have been worked out using a Vlasov theory.9 As the beam density increases, theoutward self-forces ultimately exceed the inward focusing force. This means that forequilibrium the J.l x Bo force must be inwards, or the 0 point in the Po planecorresponds to a relative minimum. Again, in plasma parlance, this means that thebeam is diamagnetic. While these conclusions were derived from Q(Po) = 0, we findthat they still generally apply even if Q -=1= O.To reiterate, the reference orbit for a high-current beam corresponds to a relativeminimum of Po. But once the beam density falls to zero, the self-fields decrease rapidlywith distance from the reference orbit. Therefore, far from the beam, Po is as given inEq. (6), even for high-current beams. Hence, while the reference orbit corresponds to a~lative minimum, it is not an absolute minimum. This complicates the topology of thePo surfaces considerably. Somewhere near the 0 point is an x point, and a separatrixthen distinguishes the diamagnetic region of the beam from neighboring paramagnetic32 W. M. MANHEIMER AND J. M. FINNregions. Outside this separatrix there may be still other 0 points and x points. Possibletopologies of the Po surfaces are shown in Fig. 2. In the calculations presented here, weassume that the beam lies entirely within the separatrix.To continue with our fluid formulation, all that is needed is an equation for the selfelectric and magnetic fields <t> and "'. These are simply Poisson's and Ampere's lawv2 <t> = 41t neL\*'" = 41t nervo/c.(10)(11)On the conducting boundary, <t> must vanish. However, '" must be specified on theconducting wall. For times short compared with a flux-diffusion time, '" is specifiedexternally (the vertical field with the proper index) and the self-field contribution to '"vanishes on the wall. For times long compared with a flux-diffusion time, '" on the wallsis the sum of the imposed flux plus the flux of a current ring having specified current. Itmay also be that the diffused flux is partly compensated by external coils. 5, 7Because the equations as written are highly implicit, an iterative scheme is necessaryfor their solution. The iteration scheme is as follows. First select a position for thereference orbit and the energy of the particle there. Further assume for purposes ofillustration Q = 0, so all particles have the same total energy. Then pick a first guess for(C) (0)FIGURE 2 Possible topologies of Po contours for a high-current modified betatron.THEORY OF A MODIFIED BETATRON 33the density and current profile. We generally pick v0 = constant and a density profilethat is parabolic, centered at the reference orbit and going to zero a certain distanceaway so that the total number of particles and maximum density are specified. Thenvalues of <I> and 0/ are calculated from Eqs. (10) and (11). From <1>, the particle's velocityVo and kinetic energy are calculated from Eq. (7). Then Po is calculated from Eq. (4) andthe Po surfaces are contoured. The 0 point will generally be different from the assignedvalue. The external vertical field is then adjusted so that the reference orbit is at theassumed position ro. This then gives a first guess for Po surfaces which have their 0 pointat the correct radius.To continue, we select the function g(Po) which according to Eq. (9) (or Eq. (8) ifQ i= 0) is equivalent to selecting the density. We used~ ={Q Po - POD for Po < POe9 d DOPOc - Poo£J (12)o otherwise,where Poo is the value of Po at the 0 point and Poc is the cutoff value of Po beyond whichthe density vanishes. We assume that Poc lies within the separatrix. The density given byEq. (12) will be parabolic near the 0 point if the Po surfaces are circular. Since we havethe first guess for the Po surfaces, the density and current as functions of Po can now becalculated as functions of rand z. In doing so, values of Qand Poc are selected so thatboth the maximum density and total number of particles have specified values. Thennew values of <I> and 0/ are calculated, and so are new Po surfaces, as well as values of Bz ,Q and Poc . The entire process is then iterated until a self-consistent equilibrium is found.In practice, the iteration scheme converges very quickly; about 10 iterations usuallygive extremely accurate results. Figure 3 shows final plots of Po surface for a 10-kAbeam with Yo = 7.9, beam radius 2 cm having Boo = 2kG. The boundary is arectangular conducting wall at z = ±7.5 em, r = 87.5 cm and 112.5 cm, and the fieldindex is 0.5. In Fig. 3, the innermost dotted curve is the Po surface at Po = Poco The nextdotted curve is the separatrix.We continue by investigating how this equilibrium evolves if external parameterschange on a time scale slow compared with a drift time. This could correspond toacceleration, flux diffusion, compression etc. We have shown that the equilibrium isspecified by the flux 'if on the walls and two other functions of Po, g(Po) and Q(Pt1). (Sofar we have considered only Q = 0.) Therefore the problem of adiabatic evolution is642O.-2-4-690 100 110FIGURE 3 The ~ surfaces for a self consistent 10 kA beam equilibrium having y = 7.9.34 W. M. MANHEIMER AND J. M. FINNsolved if at any time t, the functions g(Po, t) and Q(Po, t) can be specified from theirvalues at t = O.To accomplish this, we employ two invariants of the evolution, the number ofparticles within a Po drift surface and the toroidal flux within a drift surface. The first ofthese is trivially invariant because Po is a constant of motion for each particle, due to theesymmetry. Therefore the number of particles having each value of Po cannot change.The proof that the toroidal flux within a drift surface is an adiabatic invariant is moredifficult and is given elsewhere. 7 Our proof is based on calculating the fluid motion of adrift surface, and subtracting the change in poloidal flux from the change in total flux.Doing so, we arrive at the result that the difference of the two, the toroidal flux isconserved. From another viewpoint, this result is reasonable because the toroidal fluxthrough the drift orbit corresponds to the third adiabatic invariant of a particle'smotion. 10 In any case, we re-emphasize here that although Q(Po) may be zero at thestart of the acceleration, it does not remain zero as the beam is accelerated.Using the expression for n from Eq. (8), we find that the number of particles within aPo surface is given by(13)where I is the length along a Po contour in the poloidal plane. If the toroidal flux isdenoted <Ilt , the other adiabatic invariant is(14)(15)Thus, given N(Po) and <Ilt(Po), the procedure is to solve Eqs. (13) and (14) for g(Po) andQ(Po)·In our numerical computations, we have simplified this procedure somewhat byassuming Q is constant, and conserving toroidal flux only at the separatrix. A posteriorichecks on the validity of this approximation show that it is generally good to a fewpercent.To do the calculation, start with a reference equilibrium like that given in Fig. 3, butincrease the reference orbit energy to Y1 corresponding to acceleration of the beam.Then a guess for beam density and current give first guesses for Po surfaces as before.Generally these Po surfaces (a) will not have the 0 point at ro, (b) will not have the cor-rect value of Po = Poo at the 0 point, and (c) will not conserve flux within theseparatrix. These can be corrected as follows (a) the position of the 0 point can bechanged by changing the external vertical field, (b) the value of Po at the 0 point can bechanged by adding an arbitrary constant to '11. This corresponds to the change in ,fluxthrough the orbit. The relation between flux change and vertical field change then givesthe betatron conditions corrected for self-fields, (c) finally, the toroidal flux through theseparatrix can be changed by varying Q. The details of the iterative scheme are as givenin Ref. 7. Once the Po surfaces are modified so that conditions a-c are accounted for,g(Po) is determined from Eq. (13)._d g2 _ 2_dN [i _d_l l_J-ldPo - dPo J IVPol vo- rQand Eq. (15) for g2 is integrated from POe, where g is known as the external applied field,THEORY OF A MODIFIED BETATRON 35inward in Po to Poo . From g, one has the density from Eq. (8) and the entire process isiterated until it converges. As was the case in calculating the reference equilibrium, theiteration scheme converges relatively quickly.Figures 4 and 5 show two cases of beam acceleration. Figure 4 shows the Po surfacesfor a 10 kA beam with Bo = 2kG and having a field index of 11 = 0.5, and Fig. 5 shows110110110100(a)100(b)100(c)6~~0-2-4-690642o ~-2-4-6906420:-2-4-690FIGURE 4 The Po surfaces for a 10kA beam having field index 11 = 0.5 as it accelerates from y = 3.85 to6.96 to 11.05. Notice that as the inner part of the separatrix approaches the beam edge, the outer part of theseparatrix is well confined within the walls.36 W. M. MANHEIMER AND 1. M. FINN642o-2-4-690 100(a)110110110100(b)100(c)20-2-4-690:tI2rOr-2-4-690FIGURE 5 Same as Fig. 4 but for a field index 11 = 0.4. Notice that the outer part of the separatrix is notconfined as the inner part scrapes the beam. This implies large beam losses as the beam accelerates further.the same thing but for a field index of 11 = 0.4. In Fig. 4 the y's are 3.85, 6.96 and 11.05,corresponding to an increase in vertical field from 114.5 G to 238.0 G. In Fig. 4 the stepsin yare 3.91, 7.02 and 10.07 corresponding to an increase in vertical field from 114.6 Gto 220.6 G. In each case, the separatrix moves in as the beam accelerates. At somecritical y, the separatrix touches the outer edge of the beam, and these outer particlesTHEORY OF A MODIFIED BETATRON 37scrape off and begin to populate the paramagnetic region on the other side of theseparatrix. Thus, in order not to lose the beam, the outer part of the separatrix must bewell confined within the walls when the inner part scrapes the beam. This is so in Fig. 4,but not in Fig. 5. Thus for the shape walls we consider, the beam will be able to make thediamagnetic to paramagnetic transition (i.e., go through the region of orbit insta-bility)1,2,4 for a field index of 0.5, but not 0.4.To conclude, we have shown that self-consistent equilibria and adiabatic evolutionof these equilibria can be calculated in a high-current modified betatron. The simula-tion code developed can also provide important design information, such as how tomaximize the chances of passing through the region of orbital instability withoutdisruption.ACKNOWLEDGMENTWe have benefited from discussions with Dr. C. A. Kapetanakos and Dr. D. P.Chernin. This work was supported by the Office of Naval Research.REFERENCES1. P. Sprangle and C. A. Kapetanakos, J. Appl. Phys., 49, 1 (1978).2. N. Rostoker, Comm. Plasma Phys., 6,91 (1980).3. C. A. Kapetanakos, P. Sprangle and S.1. Marsh, Phys. Rev. Lett., 49, 741 (1982).4. D. P. Chernin and P. Sprangle, Particle Accelerators, 12, 85 (1982).5. C. A. Kapetanakos, P. Sprangle, D. Chernin, S.1. Marsh and I. Haber, Particle Accelerators, to bepublished. (Also see NRL Memo Report 4905.)6. P. Sprangle, C. A. Kapetanakos and S. 1. Marsh, Proc. of Intern. Topical ConL on High-Power Electronand Ion Beam Research and Technology, Palaiseau, France, June 29-July 3, 1981, p. 803.7. J. M. Finn and W. M. Manheimer, Phys. Fluids, to be published.8. R. C. Davidson, Theory of Non Neutral Plasmas (W. A. Benjamin, Inc., 1974), p. 7.9. Ibid, Sec. 3.5, and references therein.10. T. G. Northrup, The Adiabatic Motion of Charged Particles (Wiley Interscience, New York, 1963).

Self-consistent theory of equilibrium and acceleration of a high- current electron ring in a modified betatron

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