Injection into a circular machine with a KV distribution

摘要

In order to achieve a maximum space charge limit in the IPNS-II synchrotron it is desirable to inject a Kapchinskij-Vladimirskij (KV) distribution (1). We rederive the KV distribution, first starting from a smoothed Hamiltonian and then for the full alternating gradient case. The microcanonical distribution can be generalized slightly so as to allow one to alter the aspect ratio of the beam ellipse. The KV distribution requires that the injected particles all have the same total transverse oscillation energy, and also that they are distributed uniformly throughout the entire energy shell. This requires painting the injected beam uniformly in the three independent dimensions of the energy shell. We have devised two scenarios for doing this, one involving a suitable variation of the x and y injected amplitudes during the injection process, and the second involving introducing a small coupling between the x and y motions. We have written a program to simulate the injection process which includes the turn-to-turn forces between the (500) injected turns. If we omit the turn-to-turn forces then the resulting space charge density distributions are indeed very nearly uniform within a circular beam cross section for either KV injection scenario, but are neither uniform nor circular for other plausible scenarios. With turn-to-turn forces included, the interturn scattering can be fairly important and the resulting density distributions tend to develop lower density halos. If we add a gradient bump to simulate magnetic quadrupole errors in the lattice, then the effects of half-integral resonances can be clearly seen. When the space charge forces between turns depress the tune to a resonance, beam growth keeps the tunes constant at the edge of the stop band, unless the resonance is crossed quickly.