Solutions of the matched KV envelope equations for a “smooth” asymmetric focusing channel

Abstract

In many particle accelerators the applied focusing forces may differ in two or three directions either by design in order to avoid resonances or for other reasons, such as design constraints, bunch compression/expansion, dispersion, etc. At high intensities, space charge effects and related collective forces may cause unwanted emittance growth via instabilities and equipartitioning (relaxation of temperature anisotropy). For the transverse two-dimensional case, such asymmetric (anisotropic) systems are described by the coupled, matched envelope equations of the Kapchinskij–Vladimirskij distribution with different focusing strengths and emittances in the x and y directions, which must be solved numerically for a periodic lattice. In this article, we present results for a “smooth” asymmetric focusing channel, in which case one obtains a set of two coupled algebraic equations for the envelopes X and Y. Though the algebraic equations can easily be solved numerically, the scaling with the physics parameters is usually obscured by the numerical procedures. We derived an approximate solution as well as a general, more accurate solution, both of which represent results that exhibit the scaling with the applied focusing, space-charge, and emittance terms. The accuracy of the approximate solution is in the range of a few percent for a channel with a small degree of asymmetry. The general solution is obtained by solving for the aspect ratio A=Y/X by an iteration method that yields results to any desired degree of accuracy. More importantly, to facilitate the comparison between systems with different particle species and/or operating parameters, the envelope equations in this general treatment are written in dimensionless form. This is accomplished by expressing the envelopes X and Y in terms of the “average radius” as, and by introducing dimensionless parameters, v and w, which measure the degrees of focusing and emittance asymmetries, and the ratios of the space charge to the external focusing forces, defined by the intensity parameter χ. The results are then used to obtain formulas for the frequencies, or wave numbers, of the betatron oscillations and the tune depressions due to space-charge forces in the x and y directions, which are of fundamental importance for understanding the beam physics. These dimensionless relations exhibit the desired beam physics scaling and represent mathematically convenient forms for calculating, tabulating or plotting both exact as well as approximate solutions for the various quantities of interest. Two examples, including figures, are presented to illustrate the practical use of the theoretical relations. Our results should be useful for machine designers, theorists and experimentalists.