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

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, - κ_q(s) X[xê_x - yê_y], on the beam particles. Here, s = β_bct is the axial coordinate, (γ_bb - 1)m_bc² is the directed axial kinetic energy of the beam particles, q_b and m_b are the charge and rest mass, respectively, of a beam particle, and the oscillatory lattice coefficient satisfies κ_q(s + S) = κ_q(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 f_b(x,y,x′,y′,s) and the (normalized) self-field potential ψ(x, y, s) = q_bφ(x, y, s)/γ³_bm_bβ² _bc² in the transverse laboratory-frame phase space (x, y, x′, y′), assuming a thin beam with characteristic radius r_b≪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≫r_p) 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 r_w) 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 -mκ_q(t)[xê_x - yê_y], where κ_q(t)≡8q V₀(t)/πmr²_w.