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Analytical investigation on the halo formation in space charge dominated beams

It is assumed that fermion particles, such as electron and proton, in a matched beam follow Fermi-Dirac statistics in the equilibrium state. Parametric resonances, chaotic motion, and halo formation in transverse direction of motion are investigated analytically. The analytical expressions for the parametric resonances and the maximum transverse position deviation, above which chaotic motion starts to occur due to transverse beam envelope oscillation, are derived. Analytical formula for the current loss rate is established.