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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 systemwith a bunched electron beam, inevitably, a long-range longitudinal spacecharge force is introduced. For a relatively dense electron beam, its force canbe comparable to, or even greater than the attractive force from the ions.Hence, the influence of the space charge field on the modulation process couldbe important. If the 3-D Debye lengths are much smaller than the extension ofthe electron bunch, the modulation induced by the ion happens locally. Then, inthat case, we can approximate the long-range longitudinal space charge field asa uniform electric field across the region. As detailed in this paper, wedeveloped an analytical model to study the dynamics of ion shielding in thepresence of a uniform electric field. We solved the coupled Vlasov-Poissonequation system for an infinite anisotropic electron plasma, and estimated theinfluences of the longitudinal space charge field to the modulation process forthe experimental proof of the CeC principle at RHIC.

The Influence of Longitudinal Space Charge Fields on the Modulation Process of Coherent Electron Cooling