A hybrid fast-multipole technique for space-charge tracking with halos
The simulation of injection and accumulation of intense proton beams in synchrotrons and accumulator rings requires a flexible and robust treatment of space-charge effects. In particular, the simulation must be able to correctly incorporate the large-amplitude beam halo which is known to arise from a variety of mechanisms. This paper describes a technique for the efficient and accurate evaluation of transverse space-charge forces in a simulated beam that may include halo particles at arbitrarily large amplitudes. The technique uses the Fast Multipole Method (FMM), together with some elements of Particle-In-Cell (PIC) methods, to evaluate the space-charge field at the location of each macroparticle. Timing comparisons show that when a large-amplitude halo is present this hybrid technique is significantly faster than plain FMM and also offers performance advantages over PIC/FFT field calculations with equivalent spatial resolution. The hybrid FMM method has been implemented and tested in the multiparticle tracking and simulation code ACCSIM and some initial results are presented.
Betatron resonances with space charge
The point is made that betatron resonances do not occur at the incoherent value of the tune, but rather at the frequencies of the appropriate collective modes. This has important implications not only for the design of high intensity, low energy proton synchrotrons, but also for the interpretation of machine studies at existing synchrotrons of this type.
OSCILLATIONS IN A SYNCHROTRON UNDER SPACE CHARGE CONDITIONS
Space charge is a fundamental limitation in accelerators and is in practice the process which determines the intensity in present con­ ventional synchrotrons. For AG or strong fo­ cusing synchrotrons which are taking place of conventional ones for high energies, there must also be a similar space charge limitation, though it has not yet been experienced. For the design of new machines a formula has been computed which is based on the change in betatron frequency due to space charge forces and the condition that the v number should not cross any integral or half integral reso­ nance [1]. In fact, as shown by Morin [2], resonances induced by field errors do not occur on the modified v integers, since space charge forces are dependent of beam position and do not affect the middle of it. So that field error resonan­ ces always occur in the same place whether space charge is large or negligible. Gradient error resonances on the contrary are sensitive to space charge which modifies the external focusing forces. But one knows from the expe­ rience in CERN and Brookhaven that this type of resonance is not so dangerous as the other one and that it is possible to cross it. We intend to show here an effect which might even help in crossing these resonances and remove very far the true space charge limit in AG synchrotrons. Space charge forces depend on the transverse dimensions of the beam. If the beam injected into the synchrotron is matched to the focusing pattern, its transverse dimensions are nearly constant (neglecting the wriggling factor of the AG focusing) and space charge forces are also nearly constant. But if it is not matched or if, due to a resonance, larger oscillations are induced the beam enve­ lope scallopes in the synchrotron. In this case focusing forces vary along the orbit and can even alternate. Some kind of periodic focu­ sing is produced. The net result is an increase in overall focusing effect, which reduces the betatron wavelength and tends to restore v in its value v0 without space charge. Such a phenomenon is obviously to help removing space charge limit as computed with a per­ fectly matched beam.