Early transverse decoherence of bunches with space charge

The transverse decoherence of injected bunches is an important phenomenon in synchrotrons and storage rings. The initial stage of this process determines the transverse emittance blowup, which should be taken into account for the design of feedback systems, for example. The interplay of different high-intensity effects can strongly affect the initial decoherence stage. We present a model that explains decoherence and emittance growth with chromaticity, space charge, and image charges within the first synchrotron period. We compare the model for different combinations of parameters with self-consistent particle tracking simulations and measurements in the SIS18 synchrotron at GSI Darmstadt. Generally, space charge slows down the decoherence process and can cause the loss of decoherence. Chromaticity and image charges can partly compensate this loss and restore the decoherence. We also analyze the single-particle excitation driven by space charge during the decoherence process. Particles gain large amplitudes from the coherent beam oscillation, which leads to halo buildup and losses.

Figure 1

Time evolution of the beam offset after the transverse kick from particle tracking simulations with the fixed-position monitoring for $Q_s=0.01$. Red lines are for $\delta Q_{\xi }=0.01$ and $\delta Q_{\text{eff}}=0.015$, blue lines are for $\delta Q_{\xi }=0.005$ and $\delta Q_{\text{eff}}=0.01$ ($\xi <0$ and $\eta <0$). The solid lines are the bunch offsets and the dashed lines are given by Eq. (1) with the effective tune spread from Eq. (5).

Figure 2

Transverse decoherence of the kicked beams with space charge from the 2D simulations. The solid lines are the beam offsets and the dashed lines are oscillation amplitudes.

Figure 3

Time evolution of the beam oscillation amplitude for 2D beams with different space charge, $\delta Q_{\text{eff}}=0.01$, and $A_0=0.1{\sigma }_0$. Colors and the solid lines (contour lines) are from simulations. Dashed lines are given by Eq. (12). The vertical white line represents $N_{\mathrm{dec}0}=22.5$ turns [Eq. (8)].

Figure 4

Stability diagram with and without space charge from Eq. (10).

Figure 5

Beam losses in simulations for the transverse decoherence in 2D beams with space charge. The particles with the amplitude above $3.5{\sigma }_0+A_0$ are counted as lost particles and are removed from further calculations. The losses saturate after 250 turn. The simulation scan is performed for $\delta Q_{\text{eff}}=0.01$.

Figure 6

Stroboscopic plot for 11 particles for two cases: the red points are for zero space charge, and the black points are for zero chromaticity. In the former case, the chromaticity tune shift $\mathrm{\Delta }{Q}_{\xi }$ is in the range of $[-0.03,0.03]$ and all particles have zero initial amplitude with respect to the beam center, $A_0=a$. The latter case is for $\mathrm{\Delta }{Q}_{\mathrm{sc}}=0.05$ and different initial offsets in the range $[0,a]$. The plot contains 100 betatron periods.

Figure 7

Stroboscopic plot for 21 particles for the combination of space charge and chromaticity: $\mathrm{\Delta }{Q}_{\xi }\in [-0.03,0.03]$, $\mathrm{\Delta }{Q}_{\mathrm{sc}}=0.05$, $Q_0=4.29$, and $A_0=a$. All particles have zero initial amplitude with respect to the beam center. The plot contains 500 betatron periods.

Figure 8

Time evolution of the oscillation amplitude from simulations for bunches with Gaussian transverse and longitudinal distributions. Simulation parameters: $q_{\text{eff}}=1$, $A_0={\sigma }_0$, $Q_s=0.01$.

Figure 9

Decoherence in bunches with space charge. Colors and solid lines are from simulations for bunches. Dashed contour lines are given by Eq. (17). The vertical white line is $N_{\mathrm{dec}0}=22.5$ turns [Eq. (8)]. Simulation parameters: $q_{\text{eff}}=1$, $A_0={\sigma }_0$, $Q_s=0.01$.

Figure 10

Loss rate for different space charge from the simulation scan shown in Fig. 9. The aperture radius is $3.5{\sigma }_0+A_0$.

Figure 11

Time evolution of the emittance growth for decoherence with space charge shown in Fig. 9. Colors and solid lines are from simulations of bunches. Dashed lines are calculated using Eq. (3) from the oscillation amplitude given by Eq. (17). The vertical white line is $N_{\mathrm{dec}0}=22.5$ turns [Eq. (8)].

Figure 12

Decoherence for different initial kick strength. The left-hand side plot is the time evolution of the beam oscillation amplitude. The dashed line is given by Eq. (17). The right-hand side plot is the time evolution of the emittance growth. The dashed line is calculated using Eq. (3) from the oscillation amplitude given by Eq. (17). The vertical black line is $N_{\mathrm{dec}0}=22.5$ turns [Eq. (8)]. Simulation parameters: $q_{\text{eff}}=1$, $Q_s=0.01$.

Figure 13

The decoherence time $N_{\mathrm{dec}}$ for difference space charge and the effective tune spread. Simulation parameters: $A_0={\sigma }_0$, $Q_s=0.01$.

Figure 14

Decoherence in bunches with space charge and image charges. Colors and solid lines are from simulations. Dashed lines are given by Eq. (21). Simulation parameters: $a/b=1/5$, $q_{\text{eff}}=1$, $A_0={\sigma }_0$, $Q_s=0.01$.

Figure 15

Time evolution of the bunch offset in the horizontal plane from BPM measurements in SIS18 after a transverse kick. One turn corresponds to $1.682\mu \mathrm{s}$, $A_0\approx 4\mathrm{mm}$.

Figure 16

Decoherence with space charge from the measurements: the solid red line is for $N_p=4\times 10^9$, the blue solid line is for $N_p=1.6\times 10^9$. The dashed lines are from simulations with the fitted $q_{\mathrm{sc}}$.

Figure 17

Decoherence with space charge from measurements in SIS18 for different kick strength.

Figure 18

Decoherence with space charge from measurements in SIS18: the solid blue line for the natural chromaticity (${\xi }_x=-1.3$, $q_{\text{eff}}=1.3$), the solid red line for the compensated chromaticity (${\xi }_x\approx 0$, $q_{\text{eff}}=0.5$). The dashed lines are from simulations with the fitted $q_{\mathrm{sc}}$.

Figure 19

Summary of the decoherence measurements in SIS18. The points correspond to the decoherence time obtained from the measured signals for different machine and beam parameters. Lines are the characteristic decoherence time from simulations.