Longitudinal Space Charge Effects in Particle Accelerators

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Abstract
The modification of the single particle theory of particles subject to radio‐frequency acceleration caused by electrostatic repulsion between particles is calculated. With the aid of an appropriate Hamiltonian the Boltzmann equation is reduced to a one‐dimensional integral equation for the case of a uniform particle density in a limited region of phase space. For a range of parameters that includes most accelerators, this equation is approximated by an algebraic equation, and an analytic solution valid below the transition energy is obtained. Computations are given of the ``longitudinal space charge limit,'' as well as the effect of space charge on the azimuthal bunching of a beam by an accelerating cavity. Numerical examples of these two effects are given.

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