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Gaussian-profile equilibrium electron beam in FEL

Based on the conservation of particle number, the analytical expressions of the Gaussian-profile density and the self-field of the equilibrium electron beam are deduced in terms of the macroscopic design parameters. Compared to the Gaussian-profile model, both the particle-distribution and the self-field effect of the equilibrium electrons are overestimated by the uniform-profile model. It is demonstrated by the experimental examination that a Gaussian-profile model is much better coincident with the actual fact of the equilibrium electron beam than a uniform-profile model.