THE ULTRALAC : A COLLECTIVE WAVE ACCELERATOR FOR ULTRARELATIVISTIC PARTICLES J. A. Nation and A. P. Anselmo Laboratory of Plasma Studies and School of Electrical Engineering Cornell University, Ithaca, NY 148535401 Summary Collective wave accelerator schemes utilizing slow waves on an electron beam have an upper energy limit on the load particles set by the drive beam drift velocity. Typical intense electron beams operating near limiting current have drift veloc- ities between 0.7~ and O.Qc, setting an energy limit for protons of about 1 GeV. In order to collectively accelerate electrons or ions above this limit, the drive wave must have a phase velocity greater than electron beam drift velocity. The Ultralac concept utilizes the fast upper hybrid wave for particle acceleration. It has a phase velocity that is bounded only in the lower limit by the electron drift velocity. Calculations are presented showing the operating parameters of such a device including beam and wave parameters and achievable field gradients, load particle current limits and particle focusing. Introduction Up to now, attempts to build a collective particle accel- erator using intense electron beam technology has emphasized the use of slow waves. The first concept suggesting the use of beam slow waves for acceleration of particles was the autores- onant accelerator (ARA) described by Sloan and Drummond’ (1974). The ARA concept utilized the slow upper hybrid mode on an unneutralized relativistic electron beam for ion acceler- ation. The second collective wave concept forwarded was the slow space wave accelerator proposed by Sprangle, Manheimer and Drobot’ (1976). Since both techniques use slow waves, a velocity barrier will exist on the load particles, resulting in a final energy limit equal to the mass ratio times the kinetic en- ergy of the electrons. In actual practice, the ultimate velocity may be limited to a lower useful value by the available field gradient. Obviously slow waves on an electron beam are of no use for high energy applications. The Ultralac Concept Up to now, little or no work has been done on excitation of waves on electron beams in an effort to collectively accel- erate particles to ultra-relativistic energies. In a magnetized bounded electron beam, many different linear waves exist in the fluid model. For the lowest radial and azimuthally symmetric modes, eight different branches of the dispersion relationship can be plotted on a Brillouin diagram3. The electromagnetic waves have four branches, the modified TM01 and TEol waves, each with a positive and negative phase velocity branch. Four other eigenmodes of a electromechanical nature can exist in this system due to the collective aspects of the electron beam. These electrokinetic modes are the fast and slow space charge waves and the fast and slow upper hybrid waves. Of all these waves, only the fast upper hybrid mode satisfies all of the accel- erator criteria for ultra-relativistic particles. The upper hybrid modes have also been referred to as Trivelpiece-Gould4 modes, vortex waves’, and cyclotron modes3. For waves whose up- per hybrid frequency is less than the empty waveguide cutoff for the TMol electromagnetic wave, a quasi-static approxima- tion can be used where the electric field is derivable from the gradient of a scalar potential. Ideally, if the fast upper hybrid mode can be excited pre- cisely at the point which has phase velocity equal to c, this mode would then become extremely useful for the collective acceleration of ultra-relativistic particles. This is the Ultra- lac concept. Since the fast upper hybrid mode is a positive energy mode in a non-neutral particle beam, energy must be added in order to excite it. We propose using a three wave method employing a wiggler to couple energy from the neg- ative energy modes to the fast wave. The success of using wigglers to excite the positive energy electromagnetic modes in beam-guide systems is well understood both theoretically and experimentally’. In particular, the free electron laser has been successfully developed, using the parametric excitation of fast waves. In our case we want to excite a fast electrokinetic and not an electromagnetic mode. This condition can be en- sured if the wave is excited at a frequency below the lowest guide electromagnetic cutoff frequency. Using a linear, electrostatic fluid theory for a confined flow beam in a finite magnetic field, we can derive a relativistic dispersion relationship. If a relativistic rigid rotor equilibrium is used, we recover a dispersion relationship similar to that of Potz15 (1960). The use of this dispersion relation allows us to calculate the trapping electric field, the field structure and the power flow associated with the wave. Basic Theorv For cases when the hybrid wave frequency is lower than the TM01 cutoff frequency, the theoretical analysis can be greatly simplified by employing a quasi-static approximation. This requires V x E = 0. The analysis yields the electrokinetic modes that exist in the beam. For cases where the upper hybrid frequency is equal to or greater than the TM01 cutoff frequency, a fully electromagnetic model is necessary. This analysis, both linear and non-linear, is presently being undertaken by Seyle#. The quasi-static analysis employed here uses the continuity equation, the momentum conservation equation, and Poissons’ equation to yield a characteristic dispersion relation. The linear dispersion relation is derived from matching the jump conditions for beam-vacuum boundary and setting the wave potential at the wall equal to zero. For all values of azimuthal mode number, n, this characteristic equation is $(ka)K,(kb) - I,,(kb)K:,(ka) J:, (kl a) I,(ka)K,(kb) - I,(kb)K,(ka) = A(“‘k)~,Jkla) + B(w’k) where (1) A(w,k) = (w: + 72w; - w,‘)(wZ - w;, 1/Z (WV2 - w,“)w,” and B(w, k) = ny2w~w, ka(w,2 - wZ)w* In addition to these definitions, the perpendicular wavenum- ber, kl, is defined as kf = kt;;~2-+w~; (yw;);; ) P b To simplify the equations, the Doppler shifted eigenfrequency of the modes is defined as Wb = w - kvd - n&. Using the 145 CH2387-9/87KKKK-0145 $1.00 0 IEEE © 1987 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. PAC 1987