Simulation of shot noise effects in the EIC strong hadron cooling accelerator using real number of electrons

In the electron ion collider design, in order to achieve the peak luminosity 1034/cm 2/s with a reasonable lifetime, an efficient coherent electron cooling scheme was proposed to reduce the hadron beam emittance growth. Such a cooling scheme requires a good electron beam quality with a small energy spread. However, the shot noise in the electron beam through the accelerator might be amplified due to the microbunching instability and degrades the electron beam quality in the modulator section of the strong hadron cooling channel and correspondingly cooling rate. In this study, we report on self-consistent simulations of these effects using a real number of electrons to capture the details of shot noise and analysis of the shot noise growth through the accelerator.


Introduction
The Electron-Ion-Collider (EIC) as the next generation nuclear physics collider demands high luminosity (10 34 /cm 2 /s) during the collision.For such a high luminosity, the intrabeam scattering effects inside the hadron beam and other collective effects can cause beam emittance growth and lifetime degradation.In order to maintain the high luminosity for sufficiently long collision time, the hadron beam has to be cooled.At present, the coherent electron cooling is the most promising method to cool the hadron beam in the EIC project due to the high cooling rate [1].An Energy Recovery Linac (ERL) has been actively studied to provide the intense electron beam needed for the coherent electron cooling [2].The shot noise inside the electron beam could be amplified in this accelerator through the process of microbunching instability.The amplification of the initial electron beam shot noise has been observed and studied in other accelerators such as LCLS [3,4].The amplified shot noise in the short wavelength regime will smear the hadron modulation signal and degrade the cooling rate.In this paper, we studied the shot noise evolution through the cooling accelerator using the self-consistent high precision macroparticle tracking with the real number of electrons and analyzed the shot noise growth through the accelerator.

Computational tool
The computational tool used in this study is a massive parallel beam dynamics simulation framework, IMPACT, code suite.It includes two parallel particle-in-cell tracking codes: IMPACT-T and IMPACT-Z [5,6,7].The IMPACT-T code is a three-dimensional macroparticle tracking code using time as the independent variable.It simulates the electron beam emission from the photocathode and the electron beam transport and acceleration through injector and accelerator including the self-consistent space-charge effects.Here, the space charge effects were computed by solving the three-dimensional Poisson equation in the moving beam frame using an integrated Greens function method [8].The fast Fourier transform (FFT) is used to compute the discrete convolution efficiently.The IMPACT-Z code is a parallel particle-incell code for modeling high intensity/high brightness beams in linear and circular accelerators using longitudinal position z as the independent variable.The computational model in the IMPACT-Z code includes exact transfer map through a drift, linear transfer map for the hard edge quadrupole with energy dependence, transfer map for dipole, linear transfer matrix through RF superconducting cavity, thin lens kick model for sextupole, self-consistent 3D space-charge effects, 1D steady state and transient coherent synchrotron radiation (CSR) effects, incoherent synchrotron radiation (ISR) effects through bending magnet, longitudinal structure and resistive wall wakefields, and uncorrelated energy increase from analytical intrabeam scattering (IBS) model.The IMPACT code has been successfully applied to the x-ray light source accelerator and benchmarked with both the other codes and experimental measurements.In this study, we have used real number of electrons in the high precision simulation to capture the initial shot-noise of the electron beam.

Simulation of shot noise through the cooling accelerator
A schematic plot of the nominal strong hadron cooling accelerator is shown in Fig. 1 [2].It consists of an injector, a merger, and a linac section.The injector consists of a DC gun, a We carried out the high precision simulation using about 6.25 billion macroparticles for 1 nC charge and 64 × 64 × 2048 grid points.Each macroparticle corresponds to a real electron.Using the real number of electrons in the simulation is to capture the details of shot noise inside the electron beam.In order to estimate the shot noise level of a group of macroparticles, we compute the relative current fluctuation level from the linear deposition of these macroparticles onto a one-dimensional grid.For a randomly sampled smooth distribution function, the relative RMS density fluctuation level on a grid point I can be obtained as: where N P I is the number of macroparticles in grid cell I.This density fluctuation level represents the shot noise level of a group of macroparticles from the random sampling of a smooth function without any amplification.Figure 2 shows the final current profile and the relative current  more clearly, Figure 3 shows the zoom-in current profile and relative current fluctuation for a small section of the electron beam.The relative RMS current fluctuation after removing the modulation is about 7.5 × 10 −4 , which is at the same level as the relative current fluctuation from direct random sampling of the smooth fitting function.This suggests that the initial high frequency shot noise level has not been amplified through the nominal strong hadron cooling accelerator.Figure 4 shows the power spectral density of the relative current fluctuation (after In the above high precision simulation, the longitudinal resolution is about 20 microns.This is larger than the longitudinal scale length interested in the strong hadron cooling (about a few microns).In order to further increase the longitudinal resolution, we selected a slice of 150 micron electron beam at 20 MeV, assuming a longitudinal periodic boundary condition with a transverse open boundary condition, and reran the simulation through the linac using the real number of electrons inside that slice of the beam and 64 × 64 × 513 numerical grid points.The resultant 0.3 µm longitudinal resolution is sufficient for the few micron cooling interested scale length.Figure 5 shows the final current profile and the relative current fluctuation from that simulation macroparticles and from the random sampling of the smooth fitting function.The relative current fluctuation level at the end of the accelerator is the same as the fluctuation level from the direct random sampling.Figure 6 shows the power spectral density of the simulated relative current fluctuation and the directly random sampled current fluctuation.The relative current fluctuation from the simulation macroparticles with improved longitudinal resolution shows the same level of power spectral density as the directly sampled fluctuation.This suggests that the high frequency shot noise down to the cooling length scale were not amplified through the linac.

Estimate of the microbunching instability gain
The shot noise inside the electron beam can be amplified through the accelerator due to the collective effects, especially, the space-charge effects.This is also called the microbunching instability.In the following, we use an analytical model based on the linearized Vlasov equation to estimate the gain of the microbunching instability.Assuming an electron beam with an initial current modulation factor b 0 at the entrance (s 1 ) to RF buncher cavity, the modulation factor at a location s of the accelerator can be obtained by solving the following integral equation [9,10]: where the kernel of the above integral equation is given as: where U (s, τ ) = C(s)R 56 (s)−C(τ )R 56 (τ ), k the modulation wavenumber and the Alfvén current I A 17.045 kA.The above integral equation can be solved iteratively.Assuming that the electron beam is longitudinally frozen inside the injector and the linac sections and neglecting collective effects inside the mergers, we can calculate the final modulation factor at the exit of L3 (s 7 ) [11]. Figure 7 shows the microbunching instability gain at the end of the accelerator as a function IOP Publishing doi:10.1088/1742-6596/2687/6/0620206 micron initial modulation wavelength.Given the factor of 3 compression through the accelerator, this suggests that the final modulation wavelength should be around 300 microns.This is in consistent with the modulation wavelength observed in the above high precision simulation.

Figure 1 .
Figure 1.A layout of the nominal strong hadron cooling injector and linac.

14th
International Particle Accelerator Conference Journal of Physics: Conference Series 2687 (2024) 062020IOP Publishing doi:10.1088/1742-6596/2687/6/0620203 fluctuation from the linear deposition of the macroparticles (real number of electrons) after transporting through the strong hadron cooling accelerator in the high precision simulation including both the space-charge effect and the CSR effect.Here, the relative current fluctuation is defined as (I s − I f it )/I f it .where I s is the current from simulation and I f it the current from fitting using a polynomial function.A relative current modulation with a wavelength of about 280 µm is seen from the relative current fluctuation.In order to view this modulation

Figure 2 .
Figure 2. Final current profile (top) and relative current fluctuation (bottom) from the linear deposition of macroparticles at the end of the accelerator.

Figure 3 .
Figure 3. Zoom-in final current profile (top) and relative current fluctuation (bottom) from the simulation macroparticles and from the direct random sampling of the smooth fitting function.

Figure 4 .
Figure 4. Power spectral density of the simulated relative current fluctuation and the random sampled relative current fluctuation.

Figure 5 .
Figure 5.The final current profile (top) and relative current fluctuation (bottom) from the improved longitudinal resolution simulation macroparticles and from the direct sampling.

Figure 6 .
Figure 6.Power spectral density of the simulated current fluctuation and the random sampled fluctuation.

Figure 7 .
Figure 7. Microbunching instability gain as a function of initial modulation wavelength.