Longitudinal Dynamics of a Charged-Particle Bunch in a Traveling Wave Field

Abstract

and A. V. NesterovichIn this work, which is devoted to the longitudinal dynamics of charged particles in the fi eld of traveling wave, on which the theory of the resonance method of acceleration is based, a transition is made from the dynamics of individual charges to the dynamics of charge bunches as a whole in order to increase the effectiveness of mathematical modeling. A similar problem has already been solved for nonrelativistic bunches. Using the same method, the solution is extended to relativistic bunches. The method is based on the Lagrange equation for the longitudinal motion of a charged-particle bunch in the fi eld of traveling wave; the solutions are the longitudinal size of a bunch and the width of the energy spectrum. The self-fi eld of the bunch is taken into account in the equation and the limitations of linear approximations are eliminated.In the present work, the solution found previously for the problem of the longitudinal dynamics of nonrelativistic bunches of charges in the fi eld of traveling wave is extended to relativistic bunches [1]. The method used is based on the Lagrangian, whose derivation is based on three basic assumptions: the emittance of a bunch can be approximated by an ellipse; the charges in a bunch are distributed uniformly in the region of the emittance; and the bunch itself is approximated in the confi guration space by an ellipsoid of revolution.Lagrangians. We proceed directly to the derivation of the Lagrangian. Let N and M be, respectively, the vertical and horizontal semiaxis of the emittance and N b the number of particles in a bunch uniformly distributed over its emittance, so that the phase density ƒ is constant within a bunch and given by the expressionSpecifi cally, the number of particles in a bunch can be expressed in terms of the pulsed current I of the beam:where λ is the wavelength of the generator powering the accelerating system; Z e is the charge of the particles in the bunch; and c is the speed of light. Switching from phase to confi guration space, after normalization we obtain the particle density function ρ(x) within a bunch in the form: