Abstract
In this paper, we discuss the electromagnetic interaction between a point charge travelling inside a waveguide of elliptical cross section, and the waveguide itself. By using a convenient expansion of the Mathieu functions, useful in particular for treating a variety of problems in applied mathematics and physics with elliptic geometry, we first obtain the longitudinal electromagnetic field of a point charge (Green's function) in free space in terms of elliptical coordinates. This expression allows, then, to calculate the scattered field due to the boundary conditions in our geometry. By summing the contribution of the direct or primary field and the indirect field scattered by the boundary, after a careful choice of some expansion expressions, we derive a novel formula of the longitudinal electric field, in any transverse position of the elliptical cross section, generated by the charge moving along the longitudinal axis of the waveguide. The obtained expression is represented in a closed form, it can be differentiated and integrated, it can be used to fully describe the radiation process of a particle beam travelling inside a waveguide of elliptical cross section, and it is valid for any elliptic geometry. The equations are used to evaluate the coupling impedance due to indirect space charge in case of elliptical geometry. In addition, they are useful as preliminary studies for the determination of the coupling impedance in different cases involving elliptic vacuum chambers, as, for example, the effect of the finite conductivity of the beam pipe wall or the geometrical variation of the vacuum chamber due to elliptic step transitions existing in some accelerators.
Highlights
A particle accelerator, from the electromagnetic point of view, can be thought as composed of several devices connected by a vacuum chamber
IV we derive the expression, for an arbitrary transverse position, of the longitudinal electric field produced by a point charge travelling on the axis of the elliptical beam pipe, and we show some numerical computation examples applied to two accelerators of the CERN complex, the Proton Synchrotron and the Super Proton Synchrotron
We have obtained a novel formula that describes the radiation process of a particle beam traveling along the longitudinal axis of a beam vacuum chamber of elliptical cross section
Summary
A particle accelerator, from the electromagnetic point of view, can be thought as composed of several devices (for example rf cavities, magnets, beam diagnostics) connected by a vacuum chamber. Unlike in the classical approach of propagation of electromagnetic fields in elliptical waveguides [11,12], in this paper we take, as field source, a charged particle beam moving with velocity v 1⁄4 βc, with c the speed of light, along the longitudinal axis (z-axis) of a perfectly conducting elliptical vacuum chamber Under this condition, if we consider a transverse magnetic (TM) mode propagating along z, the wave equation is given by [13]. In the limit F → 0, when the foci of the elliptic coordinates collapse into the origin, the angular and radial Mathieu equations become harmonic and Bessel equations, respectively It exists a countably infinite set of characteristic values aðqÞ which yield even periodic solutions of Eq (8). The product series is preferable for calculating the values of the functions Fek2nðμ; −qÞ
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