Kinetic theory of periodic holes in debunched particle beams

Observations as well as numerical simulations of coasting hadron beams in circular accelerators and storage rings have revealed the excitation of long-lived coherent structures superimposed on the beam. Such beams, which are interacting with the electromagnetic fields induced by the ring environment, can develop holes (or notches) in the longitudinal distribution function and in the associated line density at thermal velocities where linear wave theory would predict strong Landau damping. It is hence a nonlinear kinetic feature of the VLASOV-POISSON system which is responsible for this ubiquitous phenomenon in beam and plasma physics. In our contribution we report about recent progress [1-6] in the theoretical understanding of these holes, including those structures found recently at the CERN PSB [7]. We show how the VLASOV-POISSON system can be solved self-consistently in the small amplitude, steady state limit. The method, which was proposed by one of the authors earlier [8], consists first in solving the VLASOV equation in terms of the constants of motion (one of which is the single particle energy ), from which the line density as a functional of the electrostatic potential can be obtained by a velocity (momentum) integration of the distribution function. In the second step, POISSON’s equation, being a second order ordinary differential equation in the resistive or purely reactive case, can be solved easily for given boundary conditions. The analysis reveals new intrinsic modes which owe their existence to a deficiency of particles trapped in the self-sustained potential well, showing up as notches in the thermal range of the distribution function, see Fig. 1. Several conclusions can be drawn immediately: