Simulation study of the space charge limit in heavy-ion synchrotrons

The SIS100 synchrotron as a part of the new Facility for Antiproton and Ion Research (FAIR) accelerator facility at GSI should be operated at the “space charge limit” for light- and heavy-ion beams. Beam losses due to space-charge-induced resonance crossing should not exceed a few percent during a full cycle. The recent advances in the performance of particle tracking tools with self-consistent solvers for the 3D space charge forces now allow us to reliably identify low-loss areas in tune space, considering the full SIS100 accumulation plateau of one second (160 000 turns) duration. A realistic magnet error model, extracted from precise bench measurements of the SIS100 main dipole and quadrupole magnets, is included in the simulations. Previously, such beam dynamics simulations required non-self-consistent space charge models. By comparing to the self-consistent simulations results, we are now able to demonstrate that the predictions from such faster space charge models can be used to identify low-loss regions with sufficient accuracy. The findings are applied by identifying a low-loss working point region in SIS100 for the design FAIR beam parameters. The bunch intensity at the space charge limit is determined. Several countermeasures to space charge are proposed to enlarge the low-loss area and to further increase the space charge limit.

Figure 1

Incoherent tune footprint of a SIS100 ${\mathrm{U}}^{28+}$ bunch at injection with space charge (using the 2.5D PIC space charge model) and natural chromaticity as tune spread sources.

Figure 2

Simulated emittance exchange across the Montague resonance for different space charge models. Solid lines (top part) indicate the horizontal plane, dashed lines (bottom part) the vertical plane, and constant dash-dotted lines (center) slightly above 0.7 the transverse average.

Figure 3

Corrected warm quadrupole magnets. Beam-loss tune diagram with FFSC simulation results.

Figure 4

Tracking results without space charge for final rms beam sizes vs vertical tunes around the quadrupole resonance $Q_y=18.5$ at fixed $Q_x=18.75$. Dotted line: beam-loss curve for separate simulations including apertures.

Figure 5

Comparison between PIC and FFSC space charge models. Transverse emittance growth ratio vs vertical tunes around the quadrupole resonance at fixed $Q_x=18.75$. The resonance condition for the bunch center Eq. (8) is indicated by the dashed gray line; the gray area around it indicates the zero-intensity stop-band width.

Figure 6

Beam-loss tune diagram for vanishing space charge and all magnet field errors included at natural chromaticity.

Figure 7

FFSC results for full field error model as a function of the transverse tunes. (a) Beam losses. (b) Transverse emittance growth.

Figure 8

Loss-free tune areas for various included field error orders $n$ in both dipole and quadrupole magnets (FFSC simulations for 20 000 turns).

Figure 9

Self-consistent PIC results for beam loss after 20 000 turns using the full field error model.

Figure 10

Comparison of predicted loss contours for PIC and FFSC simulations.

Figure 11

High-resolution PIC results for the full field error model at working point $Q_x=18.97$, $Q_y=18.85$.

Figure 12

Space charge limit for nominal operation. The solid contour lines enclose the tune area with less than 2% beam loss after the injection plateau for various beam intensities, obtained from FFSC simulations with the full field error model. The dotted contour line shows the reference 5% beam-loss threshold as in Fig. 7.

Figure 13

Low-loss area contours for the cold symmetric lattice, the broken-symmetry lattice with the two warm quadrupole magnets, and the latter plus two corrector magnets. The full injection plateau is simulated with the magnet field error model and the FFSC model.

Figure 14

Tune area contours for a 5% beam-loss threshold during the injection plateau, plotted for various values of the gradient error $b_2$. The dark red star marks the asymptotic working point around $Q_x=18.97$, $Q_y=18.91$.

Figure 15

Size of low-loss area as a function of stochastically distributed $b_2$ in the quadrupole magnets (dipole magnets are set to $b_2=0$). The three curves show the $<1\%$, $<2\%$, and $<5\%$ beam-loss areas, respectively.

Figure 16

rms-equivalent bunch profiles for single-harmonic $h=10$ vs double-harmonic $h_1=10$, $h_2=20$ rf operation.

Figure 17

Double-harmonic rf operation. Beam-loss tune diagram after the injection plateau obtained from the FFSC simulations with the full field error model.

Figure 18

Space charge limit for double-harmonic rf operation. The solid contour lines enclose the tune area with less than 2% beam loss after the injection plateau for various beam intensities, obtained from the FFSC simulation model with the magnet field errors included. The dotted contour line shows the reference 5% beam-loss threshold as in Fig. 17.

Figure 19

Three pulsed electron lenses mitigate space charge. Beam-loss tune diagram after injection plateau, obtained from the FFSC simulation model with the magnet field errors included.

Figure 20

Vertical beam profiles for three working points above the $2Q_y=37$ resonance, comparing the PIC results (blue curves) to FFSC results (orange curves) after 20 000 turns.

Figure 21

Field error model based on cold bench measurements.

Figure 22

FFSC results for beam loss using the full field error model as a function of the transverse tunes above the integer resonances.

Figure 23

Simulations with 2.5D and 3D self-consistent PIC models for the full field error model. (a) Optimal working point $Q_x=18.97$, $Q_y=18.85$. (b) Resonance-affected working point $Q_x=18.84$, $Q_y=18.73$.