Equations for the envelope of a curvilinear charged-particle beam

Abstract

The necessity of developing self-consistent models for the motion of charged-particle flows in external electromagnetic fields is caused by practical problems of beam and electron-ring formation and transport. Employing the equations of an envelope is one of methods for taking into account the effect of the self-field on the beam transverse dynamics. These equations are known for rectilinear and circular beams of charged particles [1‐3]. In the present paper, a method for constructing equations of the envelope for a curvilinear charged-particle beam propagating in a nonuniform magnetic field is proposed. A specific example for the occurrence of such a configuration is provided by the electron-beam injection at a certain angle to the geomagnetic field. Such a statement of the problem is of practical significance in the context of employing the electron beam for studying the ionosphere [4]. An analytical solution to the problem of constructing a self-consistent model for a curvilinear chargedparticle beam can be found for a pencil beam, when the ratios of the beam transverse size to both the radius of curvature and the radius of torsion are small quantities. In this case, there exists an approximate solution to the Euler equation for charged-particle gas in an external magnetic field, which happens to be correct to firstorder terms with respect to the specified small parameter. Here, we imply self-similar solutions to gasdynamic equations related to the class of gas motions for charged-particles, velocities of which are proportional to the distance to the center of symmetry [5]. Particle motion at the beam axis depends solely on an external field. Therefore, the position of the beam axis is determined by the trajectory Y ( s ) of the axial particle, where s is the trajectory length measured from the beam injection point. Substituting the expression for the velocity, v = u t , into the equation of particle motion in an external magnetic field, we obtain the curvature of the trajectory Y ( s ) :