Circular attractors as heating mechanism in coherent electron cooling

We show that under certain conditions a longitudinal mismatch between an electron and a hadron beam in a coherent electron cooling (CeC) scheme creates a circular attractor in the longitudinal phase space of the cooled hadrons. Formation of an attractor completely stops the cooling and results in anticooling (“heating”) causing small synchrotron amplitudes to grow to the attractor’s radius, rather than being damped. We present a theory of this effect, compare the analytical results with simulations and derive tolerances to possible sources of the longitudinal mismatch. We further show that under certain conditions a “weak” attractor, affecting hadrons with large synchrotron amplitudes, can appear in the hadrons’ longitudinal phase space and explain that formation of such an attractor does not require the presence of any mismatches.

Figure 1

The force (blue solid line) experienced by an ion on a single pass through the cooling section in the presence of a systematic mismatch $z_1=1.6z_c$. The green dashed line represents a reflection (around the vertical axis) of the force for negative $z$. The green dot shows a “radius” ($z_A$) of the attractor’s projection on the z-space.

Figure 2

Initial distribution of the ion bunch in the phase space normalized by $R_{56}/z_0$.

Figure 3

Phase space of the ion bunch (normalized by $R_{56}/z_0$) after 200 synchrotron turns under the influence of the force with $z_1=1.5z_c$. Blue dots show the simulated i-bunch distribution. The solid red line represents the theoretically predicted attractor with radius $\sqrt{J_A}={\delta }_A$ given by Eq. (9).