Intense nonneutral beam propagation through a periodic focusing quadrupole field II—Hamiltonian averaging techniques in the smooth-focusing approximation

Abstract

This paper considers a compact Paul trap configuration to model the transverse nonlinear dynamics of an intense charged particle beam propagating through a periodic focusing quadrupole magnetic field in the collisionless regime. 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 of the particles 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. Neglecting axial variations (∂/∂z=0), the Hamiltonian describing the transverse motion (assumed nonrelativistic) of a particle with charge q and mass m near the cylinder axis (rp≪rw) is given by H⊥(x,y,ẋ,ẏ,t)=(m/2)(ẋ2+ẏ2)+(m/2)κq(t)(x2−y2)+qφs(x,y,t), where φs(x,y,t) is the self-field electrostatic potential, and κq(t)≡8qV0(t)/πmrw2 is the (oscillatory) quadrupole focusing coefficient due to the applied field. Using a third-order Hamiltonian averaging technique [R. C. Davidson, H. Qin, and P. J. Channell, Physical Review Special Topics on Accelerators and Beams 2, 074401 (1999)], a canonical transformation is employed that utilizes an expanded generating function that transforms away the rapidly oscillating terms in the Hamiltonian H⊥(x,y,ẋ,ẏ,t). Formally, ε=|κ̂q|T2/(2π)2<1 is treated as a small dimensionless parameter, where κ̂q is the characteristic (maximum) amplitude of the applied quadrupole field, and the canonical transformation is carried out correct to order ε3. This leads to a Hamiltonian, H⊥(X̃,Ỹ,X̃̇,Ỹ̇,t)=(m/2)(X̃̇2+Ỹ̇2)+(m/2)ω̂q2(X̃2+Ỹ2)+φs(X̃,Ỹ,t), correct to order ε3 in the ‘slow’ transformed variables (X̃,Ỹ,X̃̇,Ỹ̇). Here, the transverse focusing coefficient in the transformed variables satisfies ω̂q2=const., leading to enormous simplification in the analysis of the nonlinear Vlasov-Poisson equations for F(X̃,Ỹ,XI1;,Ỹ̇,t) and φs(X̃,Ỹ,t).