Dynamics of a Charged-Particle Bunch in a Travelling Wave Field with Focusing by a Uniform Magnetic Field

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This work is devoted to charged-particle dynamics in a traveling wave field that is central to the theory of the resonance method of acceleration. The equations of longitudinal and transverse motions of a bunch, focused by a longitudinal uniform magnetic field, are derived, for the first time, within the framework of the theory of envelopes. The Lagrangian of the longitudinal and transverse motions is central to the method used; the derivation of these functions is based on the following assumptions: the emittance of a bunch is approximated by an ellipse and the bunch itself in the configuration space is approximated by an ellipsoid of revolution. The solutions of test problems in which the results obtained by the method of individual particles are compared with the results obtained with the derived equations show satisfactory agreement. The equations of the envelopes could find application in the design of accelerating systems of linear resonance accelerators with focusing by a uniform magnetic field. This pertains to, first and foremost, linear electron accelerators in which focusing of this kind is often used.