A Paul trap configuration to simulate intense non-neutral beam propagation over large distances through a periodic focusing quadrupole magnetic field

Abstract

This paper considers an intense non-neutral charged particle beam propagating in the z-direction through a periodic focusing quadrupole magnetic field with transverse focusing force, −κq(s)[xêx−yêy], on the beam particles. Here, s=βbct is the axial coordinate, (γb−1)mbc2 is the directed axial kinetic energy of the beam particles, qb and mb are the charge and rest mass, respectively, of a beam particle, and the oscillatory lattice coefficient satisfies κq(s+S)=κq(s), where S is the axial periodicity length of the focusing field. The particle motion in the beam frame is assumed to be nonrelativistic, and the Vlasov-Maxwell equations are employed to describe the nonlinear evolution of the distribution function fb(x,y,x′,y′,s) and the (normalized) self-field potential ψ(x,y,s)=qbφ(x,y,s)/γb3mbβb2c2 in the transverse laboratory-frame phase space (x,y,x′,y′), assuming a thin beam with characteristic radius rb≪S. It is shown that collective processes and the nonlinear transverse beam dynamics can be simulated in a compact Paul trap configuration in which a long non-neutral plasma column (L≫rp) is confined axially by applied dc voltages V̂=const on end cylinders at z=±L, and transverse confinement in the x−y plane is provided by segmented cylindrical electrodes (at radius rw) with applied oscillatory voltages ±V0(t) over 90° segments. Here, V0(t+T)=V0(t), where T=const is the oscillation period, and the oscillatory quadrupole focusing force on a particle with charge q and mass m near the cylinder axis is −mκq(t)[xêx−yêy], where κq(t)≡8qV0(t)/πmrw2.