Simulation studies of modulator for coherent electron cooling

Interaction of hadrons with electron beam in a modulator is an important part of coherent electron cooling (CeC), a novel cooling method for hadron beams. Being an untested technique, the CeC is undergoing a proof-of-principle test at Brookhaven National Laboratory (BNL). Simulation of this process for a realistic electron beam propagating through a realistic quadrupole beamline constitutes a very challenging problem. We successfully used the code space for these simulations and obtained accurate dependences of the modulation process on the position and velocity of ions. We obtained good numerical convergence of simulations and performed verification tests using theoretical predications available for a uniform infinite plasma with $\kappa -2$ velocity distribution. In this paper, we describe simulation methods and results, and report our findings for the CeC modulator in the BNL experiment.

Figure 1

Schematic of coherent electron cooling concept.

Figure 2

The CeC system at BNL. It includes the following equipment: 112 MHz SRF gun; 500 MHz RF buncher/pre-accelerator; 20 MeV 704 MHz SRF module; electron beam transport magnets (dipoles, quadrupoles, solenoids, trims); helical FEL wigglers; beam dump; and RHIC DX-, D0-, and triplet-magnets. The electron beam is merged with the ion beam using an achromatic dog-leg, cools the ions in the CeC, and then is discarded into a beam dump.

Figure 3

Comparison of theory and numerical simulations of density (a) and velocity (b) modulation by stationary ion with respect to uniform electron cloud.

Figure 4

Comparison of theory and numerical simulations of density (a) and velocity (b) modulation by ion moving with ${\sigma }_{v_z}$ velocity with respect to uniform electron cloud.

Figure 5

Longitudinal and transverse density and velocity modulation by reference energy ion in the center of the Gaussian electron beam in linear focusing field after 0.6 m (blue, solid line), 1.2 m (red, dash line), 1.8 m (green, dash-dot line), 2.4 m (yellow, dash-dot line), and 3 m (cyan, dash-dot line) of copropagation with electrons.

Figure 6

Density and velocity modulation by reference energy ion with various distances off the center of the Gaussian electron beam in linear focusing field after 3m of copropagation with electrons.

Figure 7

2D (x and y) plots of transverse density modulation of electrons at the exit of the modulator, measured in numbers of electrons per square meters, by resting ion at three transverse positions inside the Gaussian electron beam. The dashed yellow circle indicates the RMS size of Gaussian distribution of the electron beam, which is centered at the center of the beam.

Figure 8

2D (x and z) plot of density modulation, measured in numbers of electrons per square meters, by stationary ($v_{0,z}=0$) ion in the center of the Gaussian electron beam.

Figure 9

Longitudinal density and velocity modulation by stationary and moving ions, transversely located in the center of the Gaussian electron beam after 0.6 m (blue, solid line), 1.2 m (red, dash line), 1.8 m (green, dash-dot line), 2.4 m (yellow, dash-dot line), and 3 m (cyan, dash-dot line) length of copropagation.

Figure 10

CeC modulator section comprising four quadrupoles. The electron beam propagates from right to left.

Figure 11

(a) Evolution of $\beta$-functions in the modulator section obtained by mad-x code (red dots and black dots), space with the space charge turned off (blue and green dash-lines), and Impact-T with the space charge turned off (magenta and yellow dash-lines). (b) Same as in (a), but the space charge was turned on in both space and Impact-T codes. (c) Ratio between vertical and horizontal beam sizes in the CeC modulator.

Figure 12

Longitudinal density and velocity modulation by stationary ion on axis of the Gaussian electron beam in the CeC modulator: blue, solid line—after 0.6 m, red, dash line—after 1.2 m, green, dash-dot line—after 1.8 m, yellow, dash-dot line—after 2.4 m, and cyan, dash-dot line—after 3 m of copropagation through the modulator.

Figure 13

Transverse density and velocity modulation by stationary ion on axis of the Gaussian electron beam in the CeC modulator: blue, solid line—after 0.6 m, red, dash line—after 1.2 m, green, dash-dot line—after 1.8 m, yellow, dash-dot line—after 2.4 m, and cyan, dash-dot line—after 3m of copropagation through the modulator.

Figure 14

Product of diagonal elements of horizontal transport matrix as function of length along the length of the CeC modulator.

Figure 15

Longitudinal density modulation at the end of the modulator by stationary ion with various displacements from the axis in $x$ direction.

Figure 16

Longitudinal density modulation at $s=3\mathrm{m}$ in the CeC modulator induced by the moving ion with $v_{0,z}={\sigma }_{v_z}$ along the horizontally shifted off-axis trajectory.

Figure 17

Dependence of the bunching factor amplitude on the ion velocity at the end of the CeC modulator, electron velocity spread is fixed at 1e-3 and is used as a normalization constant.