Abstract
For the purposes of this paper, we define self-consistent beams as those which give rise to linear internal electric fields and maintain this property under any linear transport. Their analytic tractability provides valuable insights into space-charge effects, and they would possess a number of ideal properties if realized in practice. Although the Kapchinsky and Vladimirsky distribution is the most famous example, a larger class of self-consistent beams exists. Here, we focus on a particular case which we call the Danilov distribution. The beam is characterized by an elliptical shape, uniform charge density, and linear relationships between the particle positions and momenta in the transverse plane. The dynamical beam behavior is more complicated than that of the Kapchinsky and Vladimirsky distribution due to space-charge-driven linear coupling between the two transverse dimensions. There is current interest in generating the Danilov distribution experimentally; however, the beam dynamics have not yet been studied in detail. In this paper, we present an iterative method to calculate the matched envelope of the Danilov distribution in both coupled and uncoupled lattices using an existing parametrization of coupled motion. We demonstrate the method by calculating matched envelopes and studying the resulting beam properties for a few simple lattices, thus laying the groundwork for future calculations to optimize the injection of a self-consistent beam in a more complicated focusing system.
