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.
Highlights
Beams which have ellipsoidal shape and uniform charge density are often used in analytical and computational studies of beam dynamics
The transverse evolution of the Kapchinskij and Vladimirskij (KV) distribution is governed by two second-order coupled differential equations known as the KV envelope equations, which provide many insights into dynamical beam behavior and provide an analytic benchmark for computer simulations
The form and properties of the self-consistent Danilov distribution were presented, as well as the equations governing the evolution of the beam envelope
Summary
Beams which have ellipsoidal shape and uniform charge density are often used in analytical and computational studies of beam dynamics. The reduced nonlinear forces could lead to a reduced tune spread and maximum tune shift, as well as reduced halo formation in channels where the beam is matched This is only a hypothesis; it is known that external nonlinearities can have significant impact on these properties [5], that the KV distribution can be unstable in certain linear focusing channels [6], and that it is possible for particles in a self-consistent beam to be driven out of the core due to resonances between the particle and core oscillations [7,8]. Our interest is to calculate and describe the properties of the matched envelope of the Danilov distribution in simple focusing lattices as space charge is increased. The lattice is modified to explore the effects of unequal tunes and external coupling on the matched beam
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