Self-consistent time dependent two dimensional and three dimensional space charge distributions with linear force

Abstract

Finding self-consistent distributions of beam particles interacting with each other via the space charge force is one of the challenges of accelerator physics. Exactly solvable models are used for simulation benchmarks, instability threshold calculations, etc. Since such distributions have been found only in one and two dimensions (Kapchinsky-Vladimirsky distribution), it is not possible to apply them to a general three dimensional motion. This paper shows how to construct new sets of self-consistent distributions, extending even to the three dimensional case.

Dynamics of a three-dimensional charged particle dense bunch

“…One should note that this simple distribution is a particular case of Kapchinsky-Vladimirsky (KV) distribution [2]. In [5] another distribution was found leading to uniform density inside the beam too and in our notation it can be written as…”

Injection of a self-consistent beam with linear space charge force into a ring

“…With all this in mind, an excellent candidate distribution for painting into a ring was identified in Ref. [9]. Longitudinally, the distribution is uniform, so that the distribution's selfconsistency comes from its transverse properties.…”

“…In the nomenclature of Ref. [9], it was classified as a f2; 2g distribution, meaning a 2D distribution (4D in transverse phase space) with two linear relations between phase space variables expressed as delta functions, one relating x to y 0 and the other relating y to x 0 . In this distribution, there are two constants of motion, which reduce to the energy and angular momentum in the time-independent case.…”

“…A successful simulation of the injection of a 2D selfconsistent rotating distribution as a coasting beam, including space charge, into a linearized SNS lattice was conducted using the ORBIT code [12] and presented in Ref. [9]. The simulation demonstrated that all three of the conditions for self-consistent painting had to be satisfied to obtain a selfconsistent distribution.…”

“…Round or spherical beams yield the first examples of more general self-consistent time-dependent space charge distributions with linear forces, where the dependence is now on two invariants, one of which corresponds to the angular momentum. Reference [9] shows how to construct such distributions, including cases without special symmetries. This paper was primarily a mathematical exercise.…”