Abstract
A computer program has been developed which simulates the two-dimensional transverse behaviour of a proton beam in a focusing channel. The model is represented by an assembly of a few thousand “superparticles” acted upon by their own self-consistent electric field and an external focusing force. The evolution of the system is computed stepwise in time by successively solving Poisson's equation and Newton's law of motion. Fast Fourier transform techniques are used for speed in the solution of Poisson's equation, while extensive area weighting is utilized for the accurate evaluation of electric field components. A computer experiment has been performed on the CERN CDC 6600 computer to study the nonlinear behaviour of an intense beam in phase space, showing under certain circumstances a filamentation due to space charge and an apparent emittance growth. The results obtained led to a generalization in a nonlinear region of the “beam envelope oscillations” so far developed only for uniform density distributions; many cross checks have been made between the new theory and numerical results, including total energy and other integral expressions. This is now leading to a progressive understanding of space-charge phenomena of stochastic type. Some results are displayed on a motion picture.
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