Suppression of beam-ion instability in electron rings with multibunch train beam fillings

The ion-caused beam instability in the future light sources and electron damping rings can be serious due to the high beam current and ultrasmall emittance of picometer level. One simple and effective mitigation of the instability is a multibunch train beam filling pattern which can significantly reduce the ion density near the beam, and therefore reduce the instability growth rate up to 2 orders of magnitude. The suppression is more effective for high intensity beams with low emittance. The distribution and the electric field of trapped ions are benchmarked to validate the model used in the paper. The wakefield of ion cloud and the beam-ion instability is investigated both analytically and numerically. We derived a simple formula for the buildup of ion cloud and instability growth rate with the multibunch train filling pattern. The simulation in NSLSII, PEPX, SuperKEKB, and the observation in SPEAR3 are used to compare with our analyses. The analyses agree well with simulations and observations.

Figure 1

Comparison of the ion distribution from analytical and numerical approaches. The electron beam profile is also shown in the plot for a better comparison. The rms beam size is 0.1 mm and $4.5\mu \mathrm{m}$ in horizontal and vertical planes, respectively.

Figure 2

Two-dimensional distribution of an ion cloud from simulation. $X$ and $Y$ are in the unit of rms beam size

Figure 3

Vertical electric field of a ion cloud shown in Fig. 2 (a) and a Gaussian distributed ion cloud with ${\sigma }_i={\sigma }_e/\sqrt{2}$ (b). The total charge is the same for the two distributions. Different lines in the figures are for different $x$. $Y$ is in the unit of rms beam size.

Figure 4

Simulated wake of ion cloud for various $\Delta y_{e0}$ ranging from $0.1\sigma$ to $1\sigma$ with a step of $0.1\sigma$. The beam parameters are ${\sigma }_x=1.19\mathrm{mm}$, ${\sigma }_y=7.56\mu \mathrm{m}$, $N_e=5\times {10}^9$, $S_b=0.5\mathrm{m}$, and $A=28$ (${\mathrm{CO}}^{+}$).

Figure 5

Wake of ion cloud for various bunch spacing with a constant beam line density and number of ions.

Figure 6

Comparison of the wake due to ion cloud given by numerical and analytical methods, $\Delta y_{e0}=0.13\sigma$ is used in the simulation and $Q=9$ is used in the analysis.

Figure 7

Dependence of the amplitude and frequency of the wake on the beam line density by simulation. Both the amplitude and frequency are normalized in order to clearly display both of them in one plot. The beam filling pattern is single bunch train.

Figure 8

Distributions of the speed of ion particles.

Figure 9

Buildup of ion cloud along a short bunch train and its decay during the train gap for various ions’ oscillation frequencies. The ion density is normalized so that the peak density at the end of the bunch train is 1. The calculated ion oscillation period $T_i$ and the simulated ion decay time ${\tau }_{\mathrm{diffusion}}$ during the bunch-train gap are also shown in the plot.

Figure 10

Variation of stable and unstable zones with the length of the bunch-train gap.

Figure 11

Buildup of ion cloud in PEPX. The pressure is 0.3 nTorr (${\mathrm{CO}}^{+}$). The beam consists of 183 bunch trains. The horizontal emittance is 81 pm and vertical emittance is 4.3 pm. The density near the beam is the ion density within $\sqrt{3}\sigma$.

Figure 12

Normalized ion-density near the beam for various numbers of bunch trains by simulation and analysis in NSLSII. The density is normalized by the density of one bunch-train filling pattern. The total number of bunches (1000), bunch current (0.5 mA), and bunch spacing (2 ns) are the same while the bunch-train gap is adjusted to keep the length of the ring constant. A diffusion time of 31.60 ns, which is equal to the ion oscillation period calculated using the average betatron function along the whole ring, is used in the calculation of $f_{\mathrm{gap}}$.

Figure 13

Growth of the beam’s vertical amplitude for various beam filling patterns in NSLSII. The amplitude is normalized by the vertical beam size. The total number of bunches is 1000 and the vacuum pressure is 1 nTorr of ${\mathrm{CO}}^{+}$.

Figure 14

Simulated vertical oscillation amplitude of different bunches (a), (b) and the unstable modes (c) in NSLSII. The amplitude is normalized by the beam size. The amplitude uses a linear scale in (a) to clearly show the oscillation amplitude of individual bunches and logarithmic scale in (b) to display the exponential growth. The exponential growth time is 1.0 ms. Different lines in (b) represent different bunches. There are 20 bunch trains. Each bunch train consists of 50 bunches followed by a bunch-train gap of 32 ns, which is about one ${\mathrm{CO}}^{+}$ oscillation period. The bunch spacing is 2 ns and bunch current is 0.5 mA. The pressure is 1 nTorr of ${\mathrm{CO}}^{+}$.

Figure 15

Growth of the beam’s vertical amplitude for various beam filling patterns in PEPX. The amplitude is normalized by vertical beam size. The vacuum pressure is 0.3 nTorr of ${\mathrm{CO}}^{+}$.

Figure 16

Instability growth rate in SuperKEKB (high charge option) as a function of the number of bunches per bunch train for different train gaps of 10, 15, and 20 buckets. The pressure is 1 nTorr of ${\mathrm{CO}}^{+}$. The number of bunch trains is also shown in the plot.

Figure 17

Calculated oscillation frequency of ${\mathrm{CO}}^{+}$ along the SPEAR3 ring. The beam current is 500 mA and bunch number is 280.

Figure 18

Measured beam’s vertical oscillation amplitude in SPEAR3 with single bunch-train filling pattern. The number of bunches is 280 and the total beam current is 200 mA.

Figure 19

Measured vertical lower sideband for different train gap length (a) 189 ns, (b) 38 ns, and (c) 17 ns. The beam filling pattern is a single bunch train with total beam current of 200 mA.

Figure 20

Measured beam’s vertical lower sideband for different beam filling patterns: (a) one bunch train; (b) four bunch train; (c) six bunch train. The total beam current is 500 mA with total bunch number of 280 in all cases.

Figure 21

Dependence of density of trapped ions on the beam filling pattern by simulation. The total beam current is 500 mA with total bunch number of 280 in all cases.

Figure 22

Simulated vertical beam instability in SPEAR3. The beam consists of a six bunch train with a total beam current of 500 mA. The total vacuum pressure is 0.5 nTorr with ion species shown in Table .

Figure 23

Measured beam’s vertical lower sideband for different vertical emittance with a six bunch-train beam filling pattern. The total beam current is 500 mA and total bunch number is 280. The vertical emittance is increased by a factor of 10 when all skew quadrupole magnets are off.

Figure 24

Measured beam’s vertical oscillation amplitude (a) and unstable mode (b) when the bunch-by-bunch feedback is on and off. The feedback is initially turned off and turned on around 20 ms. The beam consists of a six bunch train with total beam current of 448 mA.

Figure 25

Simulated vertical tune shift along the bunch train for SPEAR3 beam with a partial pressure of 1 nTorr ${\mathrm{CO}}^{+}$. The beam has a single bunch train with total bunch number of 280. A fast bunch-by-bunch feedback is applied to stabilize the beam. When feedback is turned off, the tune shift varies with time.

Figure 26

Dependence of vertical tune shift along the bunch train on the coupled motion in the linear regime. Here $i$ mode is one of the modes other than 0 and resonance modes.

Figure 27

One-dimensional distribution of an ion cloud and the electron bunch. The ion distribution with asymptotic form also is shown. The horizontal axis is in the unit of the rms beam size.