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Register a new userThe Influence of Longitudinal Space Charge Fields on the Modulation Process of Coherent Electron Cooling
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The initial modulation in the scheme for Coherent electron Cooling (CeC) rests on the screening of the ion charge by electrons. However, in a CeC system with a bunched electron beam, inevitably, a long-range longitudinal space charge force is introduced. For a relatively dense electron beam, its force can be comparable to, or even greater than the attractive force from the ions. Hence, the influence of the space charge field on the modulation process could be important. If the 3-D Debye lengths are much smaller than the extension of the electron bunch, the modulation induced by the ion happens locally. Then, in that case, we can approximate the long-range longitudinal space charge field as a uniform electric field across the region. As detailed in this paper, we developed an analytical model to study the dynamics of ion shielding in the presence of a uniform electric field. We solved the coupled Vlasov-Poisson equation system for an infinite anisotropic electron plasma, and estimated the influences of the longitudinal space charge field to the modulation process for the experimental proof of the CeC principle at RHIC.
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We present analytic cooling and diffusion rates for a simplified model of coherent electron cooling (CEC), based on a proton energy kick at each turn. This model also allows to estimate analytically the rms value of electron beam density fluctuations in the kicker section. Having such analytic expressions should allow for better understanding of the CEC mechanism, and for a quicker analysis and optimization of main system parameters. Our analysis is applicable to any CEC amplification mechanism, as long as the wake (kick) function is available.
Longitudinal collective modes of a bunched beam with a repulsive inductive impedance (the space charge below transition or the chamber inductance above it) are analytically described by means of reduction of the linearized Vlasov equation to a parameter-less integral equation. For any multipolarity, the discrete part of the spectrum is found to consist of infinite number of modes with real tunes, which limit point is the incoherent zero-amplitude frequency. In other words, notwithstanding the RF bucket nonlinearity and potential well distortion, the Landau damping is lost. Hence, even a tiny coupled-bunch interaction makes the beam unstable; such growth rates for all the modes are analytically obtained for arbitrary multipolarity. In practice, however, the finite threshold of this loss of Landau damping is set either by the high-frequency impedance roll-off or intrabeam scattering. Above the threshold, growth of the leading collective mode should result in persistent nonlinear oscillations.
In preparation for a demonstration of optical stochastic cooling in the Cornell Electron Storage Ring (CESR) we have developed a particle tracking simulation to study the relevant beam dynamics. Optical radiation emitted in the pickup undulator gives a momentum kick to that same particle in the kicker undulator. The optics of the electron bypass from pickup to kicker couples betatron amplitude and momentum offset to path length so that the momentum kick reduces emittance and momentum spread. Nearby electrons contribute an incoherent noise. Layout of the bypass line is presented that accommodates optics with a range of transverse and longitudinal cooling parameters. The simulation is used to determine cooling rates and their dependence on bunch and lattice parameters for bypass optics with distinct emittance and momentum acceptance.
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