Coherent Electromagnetic Effects in High Current Particle Accelerators: I. Interaction of a Particle Beam with an Externally Driven Radio-Frequency Cavity

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Abstract
A calculation is made of the interaction of a beam of particles in an accelerator with the radio‐frequency cavity that provides the accelerating mechanism of the machine. A Hamiltonian for synchrotron motion is employed that makes possible the simultaneous solution of Maxwell's equations and the Vlasov equation, so that a self‐consistent distribution of particles in synchrotron phase space is determined. The effective voltage on the cavity due to the beam of charged particles is of the order of magnitude of the product of the total circulating current in the accelerator and the shunt impedance of the rf cavity. It has the net effect of producing a total voltage on the cavity which is both less than the applied voltage, and shifted in phase with respect to it. The increase in the stable phase angle required so the particles will remain in phase with the accelerating radio frequency is calculated. The decrease in total voltage and increase in stable phase angle result in a decrease in stable phase space available for acceleration, and convenient expressions are given for these quantities in terms of parameters of the accelerator. It is shown that the consequences of the induced voltage may be alleviated by increasing the voltage applied to the cavity. Non‐resonant interaction between the beam and cavity is not considered.

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