Experimental studies of circular attractors in the first rf-based electron cooler

In electron coolers, under certain conditions, a circular attractor (CA) can appear in either a longitudinal or a transverse phase space of cooled ion bunches. In the presence of an attractor, ions with small amplitudes of synchrotron or betatron oscillations are getting excited rather than being damped. This effect was observed at several nonrelativistic electron coolers with a direct current electron beam. Experimental studies of circular attractors at electron coolers utilizing rf-accelerated electron bunches, where this effect becomes particularly important, are lacking. We report CA studies at the low energy RHIC electron cooler (LEReC)—the first rf-based relativistic electron cooler. We derive formulas defining conditions for a CA appearance, perform dedicated experiments, and compare the measurements’ results to the theoretical predictions.

Figure 1

Dependence of the longitudinal friction force on an ion’s longitudinal velocity (relative to the electron bunch’s rms velocity spread) in the case of the nonzero longitudinal velocity offset (the blue solid line) and in the case of ${\mu }_z=0$ (the blue dashed line). It is assumed that the ion’s transverse velocity is equal to the rms transverse velocity spread of the ion bunch. The force is calculated for the LEReC and RHIC parameters used in the experiment listed in Table 1.

Figure 2

Evolution of the relative amplitude of synchrotron oscillations under the influence of the friction force shown in Fig. 1. The oscillations of the ion with initial amplitude $j_0=0.4$ are getting excited (the green line), and the oscillations of the ion with $j_0=2$ are getting damped (the blue line). Both ions reach a nonzero equilibrium oscillations amplitude in approximately $4\times {10}^5$ synchrotron turns.

Figure 3

Effective average friction force acting on the ions.

Figure 4

The ions’ distribution in the presence of the circular attractor. In the simulations, the ion bunch has a Gaussian initial distribution with the rms energy spread ${\sigma }_{\delta i0}$. The ions are tracked for $4\times {10}^5$ synchrotron turns under the influence of the offset friction force shown in Fig. 1. The solid red line shows the location of the attractor of radius $j_a=v_a/(\beta c{\sigma }_{\delta i0})$ with $v_a$ found from Eq. (9).

Figure 5

Schematics of LEReC layout (not to scale).

Figure 6

Longitudinal distribution of the ion bunch (dashed red line) overlapped with the electron macrobunch (solid blue line) containing 30 bunches.

Figure 7

Evolution of the longitudinal distribution of the ion bunch in the presence of the circular attractor.

Figure 8

Comparison of the experimentally measured final distribution of Fig. 7 (red trace) to the simulations (blue histogram) performed with the parameters used in the experiment. The intensity of the simulated distribution is an adjustable parameter.

Figure 9

Theoretical and experimentally measured radius of a circular attractor in the ions’ longitudinal phase space created by a constant offset of the electron beam energy.

Figure 10

Transverse and longitudinal components of the friction force in the EIC precooler (solid blue line). Respective velocity distributions of protons are shown for comparison (red dashed line). The values of critical velocity mismatches are marked by $v_c$.