Asymmetric beam-beam effect study in a highly polarized electron-ion collider

The EicC (The highly polarized Electron-ion collider in China) project proposes a scheme with different numbers of bunches in the two rings, which leads to an asymmetric beam-beam effect, also known as the gear-change effect. In order to investigate the feasibility of asymmetric collision in EicC, a self-consistent simulation program named athenagpu, which uses GPU for high-performance computing, was developed. A linearized matrix model is used to account for the dipole instability that occurs in the simulation. The simulation results in different collision modes show that asymmetric collision can lead to instability, and the beam can be stabilized by adjusting the nominal tunes. In the process of scanning the beam intensity, instability may occur as the beam intensity decreases, and this instability can be suppressed by compensating the nominal tunes. Offset collision simulations show that the decoherence time of the displaced bunch is shorter in the asymmetric collision than in the symmetric collision. The study in this paper suggests that asymmetric collision is promising for EicC when only the beam-beam effect is considered.

Figure 1

The layout of the EicC.

Figure 2

Interpolation scheme of fields.

Figure 3

Performance of the athenagpu program. (a) Computation speed on different GPUs in the 1e vs 1p mode with 10 slices per bunch. (b) Computation speed on A100 in the 8e vs 9p mode. Seventy-two collisions in one superperiod.

Figure 4

The maximum value of the imaginary part of the eigentunes as a function of the horizontal ${\nu }_e$ and ${\xi }_e$ in the 3e vs 5p mode. ${\nu }_p$ and ${\xi }_p$ are 0.315 and 0.004, respectively. The longitudinal phase space is decomposed (bottom) or not (top). In the bottom figure, $N_r$ and $N_s$ are set to 5 and 33, respectively.

Figure 5

In the 4e vs 7p collision mode, $1\times {10}^5$ superperiods are simulated using athenagpu. FFT algorithm is used to get the spectrum from the bunch centroid. The values of ${\nu }^{\mathrm{eff}}$ and $1-{\nu }^{\mathrm{eff}}$ are indicated in each figure. (a) The horizontal spectrum of an electron bunch. (b) The horizontal spectrum of a proton bunch.

Figure 6

Evolution of the horizontal centroid (a) and size (b) of the proton and electron bunches in different collision modes.

Figure 7

Evolution of the luminosity in different collision modes.

Figure 8

The real part of the horizontal eigentunes in the 3e vs 5p mode. The matrix model is used to scan ${\xi }_e$ in steps of 0.002 without considering the synchrotron oscillation and the hourglass effect. ${\xi }_p$ is fixed at 0.004. (a) Rotating electron bunches in the matrix model. (b) Rotating proton bunches in the matrix model.

Figure 9

The maximum value of the imaginary part of the eigentunes as a function of the horizontal ${\nu }_e$ and ${\nu }_p$ in the 3e vs 5p mode. ${\xi }_e$ and ${\xi }_p$ are 0.088 and 0.004, respectively. The cross indicates ${\nu }_e=0.58$, ${\nu }_p=0.315$.

Figure 10

Projection of the maximum value of the imaginary part of the eigentunes corresponding to ${\xi }_e$ on the ${\nu }_e$ axis in the 3e vs 5p mode.

Figure 11

$D_L$ value as a function of electron bunch intensity (a) and proton bunch intensity (b) in 1e vs 1p and 4e vs 7p modes. $N_e$ and $N_p$ are the numbers of particles in each electron and proton bunch, respectively. $N_{e_0}$ and $N_{p_0}$ are the design parameters that have been listed in Table 1.

Figure 12

$D_L$ value as a function of the horizontal ${\nu }_e$ and ${\nu }_p$ in the 4e vs 7p mode. The cross indicates ${\nu }_e=0.58$, ${\nu }_p=0.315$.

Figure 13

$D_L$ value as a function of the horizontal ${\nu }_e$ and proton bunch intensity in the 4e vs 7p mode.

Figure 14

Schematic diagram of the resonance region and measures for compensating the nominal tune.

Figure 15

The maximum value of the imaginary part of the eigentunes as a function of the horizontal ${\nu }_e$ and ${\xi }_e$ in 1e vs 1p (top) and 4e vs 7p (bottom) modes. ${\nu }_p$ and ${\xi }_p$ are 0.315 and 0.004, $N_r$ and $N_s$ are 5 and 33.

Figure 16

Luminosity for various horizontal offsets $\mathrm{\Delta }x$ of the electron beam. In the 4e vs 7p collision mode, all four electron bunches have the same $\mathrm{\Delta }x$.

Figure 17

The orange color represents the 1e vs 1p collision mode. The blue, green, and red colors represent that 1, 2, and all 7 proton bunches are displaced in the 4e vs 7p collision mode, respectively. (a) Luminosity for various horizontal offsets $\mathrm{\Delta }x$ of the proton beam. (b) Evolution of the centroid of the displaced proton bunch for various horizontal offsets $\mathrm{\Delta }x$.