Transverse Space-Charge Effects in Circular Accelerators

Abstract

The particles in an accelerator interact with one another by electromagnetic forces and are held together by external focusing forces. Such a many-body system has a large number of transverse modes of oscillation (plasma oscillations) that can be excited at characteristic frequencies by errors in the external guide field. In Part I we examine one mode of oscillation in detail, namely the quadrupole mode that is excited in uniformly charged beams by gradient errors. We derive self-consistent equations of motion for the beam envelope and solve these equations for the case in which the space-charge force is much less than the external focusing force, i.e., for strong-focusing synchrotrons. We find that the resonance intensity is shifted from the value predicted by the usual transverse incoherent space-charge limit; moreover, because the space-charge force depends on the shape and size of the beam, the beam growth in always limited. For gradient errors of the magnitude normally present in strong-focusing synchrotrons, the increase in beam size is small provided the beam parameters are properly chosen; otherwise the growth may be large. Thus gradient errors need not impose a limit on the number of particles that can be accelerated. In Part II we examine the other modes of collective oscillation that are excited by machine imperfections. For simplicity we consider only one-dimensional beams that are confined by harmonic potentials, and only small-amplitude oscillations. The linearized Vlasov and Poisson equations are used to find the twofold infinity of normal modes and eigenfrequencies for the stationary distribution that has uniform charge density in real space. In practice, only the low-order modes (the dipole, quadrupole, sextupole, and one or two additional modes) will be serious, and the resonant conditions for these modes are located on a tune diagram. These results, which are valid for all beam intensities, are compared with the known eigenfrequencies for the stationary distribution that has uniform particle density in phase space, and are extrapolated to the Gaussian distribution observed in the Brookhaven AGS.