The problem of self-consistent particle phase space distributions for periodic focusing channels

TL;DR

This paper investigates self-consistent phase space distributions for charged particle beams in periodic focusing channels, demonstrating that solenoid channels preserve beam quality while quadrupole channels cause emittance growth, supported by theoretical analysis and simulations.

Contribution

It introduces a method to determine Hamiltonians for self-consistent distributions and shows that solenoid channels can transmit beams without quality loss, unlike quadrupole channels.

Findings

Solenoid channels preserve beam quality during transmission.

Quadrupole channels cause a small, constant emittance growth.

Theoretical and simulation results confirm the preservation or growth of beam quality.

Abstract

Charged particle beams that remain stationary while passing through a transport channel are represented by ``self-consistent'' phase space distributions. As the starting point, we assume the external focusing forces to act continuously on the beam. If Liouville's theorem applies, an infinite variety of self-consistent particle phase space distributions exists then. The method is reviewed how to determine the Hamiltonian of the focusing system for a given phase space density function. Subsequently, this Hamiltonian is transformed canonically to yield the appropriate Hamiltonian that pertains to a beam passing through a non-continuous transport system. It is shown that the total transverse beam energy is a conserved quantity, if the beam stays rotationally symmetric along the channel. It can be concluded that charged particle beams can be transmitted through periodic solenoid channels without loss of quality. Our computer simulations, presented in the second part of the paper, confirm this result. In contrast, the simulation for a periodic quadrupole channel yields a small but constant growth rate of the rms-emittance.