TABLE II. Coefficients a and b + r of log(ac,) vs log[2/( p 1/p0 +Pel ACKNOWLEDGMENTS p,)J. We thank Jeanette Nelson for a careful reading of the

log(ac1 ) =a+ b log(2/( pc/p1 + p 1/p0 ) l when a = 10 manuscript and G. Pollarolo for useful discussions about the NAG FORTRAN LIBRARY. The numerical computations were 0.1

1 where r0 is the Lorentz factor and spans between 1 and 10. A. E. Gill, Phys. Fluids 8, 1428 (1965). The presence of a shear in cylindrical symmetry has not 2 A. Ferrari, B. Trusooni, and L. Zaninetti, Mon. Not. R. Astron. Soc. 196, yet been considered and will presumably produce a cutoff in !OSI (1981). 3H. Cohn, Astrophys. J. 269, 500 (1983). the instabilities where a> 21'Id, with d (in jet radius units) 4 D. G. Payne and H. Cohn, Astrophys. J. 287, 29S (1984). the length characterizing the thickness of the shear. ' A. M. Anile, J.C. Miller, and S. Motta, Phys. Fluids 26, 14SO (1983).

Effect of quaslconflned particles and I= 2 stellarator fields on the negative mass lnstablllty in a modified betatron G. Roberts and N. Rostoker Department ofPhy sics, University ofCalifomia, Irvine. California 92717 (Received 5 February 1985; accepted 10 October 1985) A sufficient stability condition for the negative mass instability is derived. This condition is used to show that quasiconfined particles in a modified betatron create fields that can stabilize the negative mass instability. Additionally, it is shown that stellarator fields can inhibit this instability.

Recently there have been efforts to develop high current To derive a sufficient stability condition, consider the accelerators. Experiments with plasma betatrons have ob­ motion of a beam particle in a toroidal configuration as seen tained currents below their design limits. Some attribute this in Fig. 1. If the beam develops a region of higher density 1 2 to the negative mass instability. • Another proposed accel­ (clump), then the negative mass instability will exist if the 3 4 erator is the modified betatron. • The modified betatron is resulting particle motion is toward the clump. To see this basically a betatron with a magnetic field along the beam consider Fig. 2. In Fig. 2 it can be seen that a clump in the axis. Calculations show that negative mass may also limit beam will create electric fields that cause a beam particle to current in this device. •.s-7 These calculations, though, do not have a perturbed toroidal velocity away from the clump. consider quasiconfined particles in banana-like orbits, nor However, a perturbed toroidal velocity away from the clump do they consider stellarator fields. In recent experiments will not necessarily result in a perturbed position away from with the University of California at Irvine (UCI) Modified the clump. Specifically, 6v? = R 0 60 - B6x and a suitable 8 9 10 Betatron • and at Maxwell Laboratories, inductive charg­ choice of 6x will cause a 60 that will move the particle to-­ ing' 1· 12 was used. Quasiconfined particles appear and are far ward the clump (in 0). The criteria for this not to happen is more abundant than the accelerated beam particles. Addi­ clearly SiJ /Sv8 > 0, which is just tionally, I = 2 stellarator fields have been proposed for beam focusing6 and have been added to the (UCI) Modified Beta­ SiJ = (iJ Sx + t)-1->0, (1) tron. Sv8 6v8 R 0

A sufficient stability condition for negative mass is de­ where the previous expression for 6v8 has been used. Self­ rived. This condition can be used to examine the effect of consistent Vlasov and fluid treatments find that when the quasiconfined particles and stellarator fields. It is found that above equation is evaluated for a modified betatron it de­ even the quasiconfined particle densities already obtained at scribes a sufficient condition, but not a necessary condition, 7 2 UCI and Maxwell Laboratories will stabilize a beam to an for stability.S- • Beam temperature is shown to relax this energy of about 50 MeV. Also it is found that a stellarator criteria. The above has the advantage that as long as the field can stabilize negative mass. fields that a beam particle will experience are known, then

333 Phys. Fluids 29 (1). January 1986 0031 -9171/86/010333-03$01 .90 © 1986 American Institute of Physics 333 2 (zrs/R0 ) -O,,v8 + (Ca>;/2y) -a>;(a/b) /2)X0 ~>= . (a>;12y2) + Ca>;212) - + (sO,,v9/Ro) (5)

To evaluate the 6x/6v9 term in the sufficient condition for stability [ Eq. ( 1) ] the average position of the particles will be used. In the above, let (x) - (x) + 6x and

v8 - v9 + 6v8 • This yields the following expression for 6x/6v8 : 6x _. fn,, , <6> FIG. I. Local coordinate system. 2 2 6v8 [;(a/b) /2] + (; /2) - (1-s)o; where n,, = v /R = iJ has been assumed. Using the above 6x/6v can be found and this sufficient condition can be 8 0 8 in Eq. ( 1) , the following sufficient stability condition is ob­ evaluated. tained: In recent modified betatron experiments that use induc­ r202 tive charging to form a beam, quasiconfined particles have ,, + 1 >0. (7) been observed. To include the effect of these particles, con­ [a>; (a/b)2/2] + (a>;2/2) - (1 - s)n; sider the equation of motion for a beam particle in a back­ x The result for a conventional betatron can be recovered from ground of quasiconfined particles that have a constant den­ 2 the above if ;, a>; - 0, in which case the above becomes sity n': r2 < ( 1 - s), which is never satisfied since 0 < s < 1 is needed for particle orbit stability. 1 The beam self-fields and the qua· x= - zls (1 +2-) +fi,.v8 (t +~)-ne.Y Ro Ro Ro siconfined particles are stabilizing. In the UCI Modified Be­ tatron, the stabilizing term from the quasiconfined particles , .. 2 a>'2 (J)2 ( )2 ""P P P a is +-(x-x0 ) +-x+- - X 0, (2) quantitatively larger than the accelerated beam self-field 2y 2 2 b term. Specifically, the above stability condition is satisfied where a>;2 = 41rn'e2!(rm), a>;= 4mte21. n is the ac­ for a particle energy of celerated beam density (which has been taken to be a con­ (8) stant), v8 is the particle toroidal velocity, x0 is the center of the accelerated beam, the background beam position is taken whereO,,R0 = v8 a:cands =!have been assumed. In recent to be stationary and at x :::::0 (see Ref. 9), it has been assumed experiments in the UCI Modified Betatron, the quasicon­ that the chamber wall is slotted so that there is no magnetic fined particles have a line density of about 1012 cm - 1 (the 10 1 image field from the 8beam motion, a is the beam radius, bis accelerated beam line density was about 10 cm- ). Using theminorradiusofthetorus, O,. = eB,.l(rmc), 0 8 = eB8 / this the density of the quasiconfined particles (n') is about 10 3 (ymc), the betatron field is B,,(l +sx/R0 )y+B,,X(sy/ 2X 10 cm- ; sinceR0 = 40cm, Eq. (8) is then satisfied for R0 )x, sis the betatron field index, the toroidal magnetic field y$100, which corresponds to a particle energy of 50 MeV. 10 3 is B8 B, and lxl/R0 <1 has been assumed. Also, it has been In the experiment at Maxwell Laboratories n':::::: 10 cm - mra2e2 2 assumed that the Budker parameter /ymc is small so and R0 = 50 cm, so Eq. (8) is satisfied for y$120 or an that toroidal corrections are negligible. A prescription for energy of 60 MeV. It should also be noted that the injection including torodial effects is described in Ref. 13. method used in the UCI Modified Betatron is still under The above equation can be ensemble-averaged to find an 9 13 equation for the beam position. • This leads to the follow­ ing equation:

Xo=.. - -v; (i +-xo) + 0 ,.vs (i +-sxo) - 0 oYo. Ro Ro Ro a>;2 a>; ( 0 )i +Txo+T b X 0, (3) where it has been assumed that all particles have the same v8 • In Eq. ( 3) it can be seen that the solution for the beam posi­ tion will consist of oscillations of the form cos (wt) (see Ref. 9, 13) about an equilibrium position. The equilibrium beam position (X0 ) is the following:

Xo Czrs!R0 ) - O,,v8 2 2 (a>;(a/b) /2]+[a>; /2] - (v~/R~) + (s0,.v8 /R0 ) (4) FIG. 2. Negative mass instability. A particle will have a perturbed velocity Similarly, Eq. (2) shows that the average particle position away from the clump. Because ofthe geometry, this could cause a perturbed (x) is 88 toward the clump, which would create an instability.

334 Phys. Aulds, Vol. 29, No. 1, January 1986 Brief Communications 334 development and these particle densities should ultimately wherej>0, j

X. _ (~IR0 ) - fiyv9 1 2 2 2 R. Landau and V. Neil, Phys. Fluids 9, 2412 ( 1966). o- + j(B !By> + ]}' n -{n:1u 9 cn1ny> 2R. Landau, Phys. fluids 11, 205 ( 1968). (10) 3 N. Rostoker, Comments Plasma Phys. Controlled Fusion 6, 91 ( 1980). wherefl2 = [w!Ca/b)2/2] -(~/R~) + (sflyv /R )and •p. Spranglc and C. Kapctanakos, J. Appl. Phys. 49, 1 ( 1978). 9 0 sR. Davidson and H. Uhm, Phys. Fluids 25, 2089 ( 1982). it has been assumed that the appropriate stability conditions 6C. Roberson, A. Mondelli, and D. Chemin, IEEE Trans. Nucl. Sci. NS- 14 are satisfied. From the above, the ratio 8x/8v9 can be ob­ 30, 3162 (1983). tained and Eq. ( 1 ) can be used to find the following stability 'T. Hughes, M. Campbell, and B. Godfrey in the Proceedings of the 5th condition: International Conference on.High Power Electron and Ion Beam Research and Technology, San Francisco, California, 1983 (Lawrence Livermore r2n2 National Laboratory, Livermore, CA, 1983). p. 466. y + 1>0. 8H. Ishizuka, G. Lindley, B. Mandelbaum, A. Fisher, and N. Rostoker, 2 2 2 n - {n:1u + j(Be!By> + cn1ny> ]} Phys. Rev. Lett. S3, 266 ( 1984). (11) 9G. Roberts and N. Rostoker, Phys. Fluids 28, 1968 ( 198S}. Clark, P. Korn, A. Mondelli, and N. Rostoker, Phys. Rev. Lett. 37, 2 'oW. If n. = 0, then the condition w!Ca/b) >2(1-s)O;r2 is S92 (1976). sufficient for stability as it is in the modified betatron. How­ 11G. Janes, Phys. Rev. Lett. 15, 13S (1965). 2 12J. Daugherty, J. Eningcr, and G. Janes, Phys. fluids 12, 2677 ( 1969). ever, when yn; > (alb ) , then the stellarator fields can w; 13C. Kapctanakos, P. Sprangle, D. Chemin, S. Marsh, and I. Haber, Phys. still satisfy the stability criterion. Specifically, if the follow­ fluids 26, 1634 (1983 ). ing is satisfied then the above sufficient stability condition is 1•c. Roberson. A. Mondelli, and D. Chemin, Phys. Rev. Lett. 50, S07 satisfied: (1983). 15P. Spranglc and D. Cernin, Part. Acccl. 15, 35 ( 1984}. (12) 16B. B. Godfrey and T. P. Hughes, Phys. Fluids 28, 669 (198S}.

335 Phys. Fluids, Vol. 29, No. 1, January 1986 Brief Communications 335