Transport Calculations for Very High Current Beams
A computer program has been developed to calculate the effect of the transverse space charge forces on a particle beam. The method is to divide the beam into a large number, M, of "beamlets," and compute the force on each beamlet by summing over the forces due to all the others. M ¿ 1000 is needed to make spurious collisional effects negligible. To the space charge forces, we add the forces due to the physical components of the system such as quadrupoles, solenoids or accelerating gaps. Using this program we have verified the prediction that at high current, certain perturbations of the K-V distributions are unstable. In a FODO line tuned to 90° phase advance per cell at zero current, with the phase advance depressed to 30° by space charge, we found that the rms emittance of a perturbed K-V distribution grows by a factor of 2 in ~ 40 cells, and then remains stable, although no longer having the K-V form. For thin-lens focusing, this result is in good agreement with Haber's calculation. The use of finite quadrupoles rather than thin lenses has remarkably little effect.