Semianalytical description of the modulator section of the coherent electron cooling

Abstract

In the coherent electron cooling, the modern hadron beam cooling technique, each hadron receives an individual kick from the electric field of the amplified electron density perturbation created in the modulator by this hadron in a copropagating electron beam. We developed a method for computing the dynamics of these density perturbations in an infinite electron plasma with any equilibrium velocity distribution-a possible model for the modulator. We derived analytical expressions for the dynamics of the density perturbations in the Fourier-Laplace domain for a variety of 1D, 2D, and 3D equilibrium distributions of the electron beam. To obtain the space-time dynamics, we employed the fast Fourier transform algorithm. We also found an analytical solution in the space-time domain for the 1D Cauchy equilibrium distribution, which serves as a benchmark for our general approach based on numerical evaluation of the integral transforms and as a fast alternative to the numerical computations. We tested the method for various distributions and initial conditions.

Dynamics of shielding of a moving charged particle in a confined electron plasma

“…Thereafter, we present our numerical results for shielding of a charged particle in the 1D, 2D, and 3D plasmas with the normal (Maxwell) spatial and velocity distributions. We compare our results to those obtained previously for the infinite plasma [6] and provide physical interpretation of the results. We then discuss application of these results to the CeC device now being constructed at Brookhaven National Laboratory.…”

“…The Vlasov-Poisson system (1)-(5) differs from the one for the infinite plasma [6] only by theα-term in H 0 . This term makes ∂H 0 ∂x nonzero and an extra term appears in the Vlasov equation, which makes it unsolvable by the methods that worked for the infinite plasma.…”

“…For the 1D infinite plasma with the Cauchy equilibrium velocity distribution, the exact solution can be obtained [6]. For the 3D infinite plasma with the Lorentz equilibrium velocity distribution, the solution can be expressed as a one-dimensional integral [5], and for other cases, the integral transforms must be inverted numerically [6].…”

“…For a particle moving in an infinite plasma, it is possible to solve the corresponding equations, i.e., the Vlasov-Poisson system, via the Laplace and Fourier transforms [5][6][7]. For the 1D infinite plasma with the Cauchy equilibrium velocity distribution, the exact solution can be obtained [6].…”

“…For a particle moving in an infinite plasma, it is possible to solve the corresponding equations, i.e., the Vlasov-Poisson system, via the Laplace and Fourier transforms [5][6][7]. For the 1D infinite plasma with the Cauchy equilibrium velocity distribution, the exact solution can be obtained [6]. For the 3D infinite plasma with the Lorentz equilibrium velocity distribution, the solution can be expressed as a one-dimensional integral [5], and for other cases, the integral transforms must be inverted numerically [6].…”