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GOAT: a simulation code for high-intensity beams


A simulation code, GOAT, is developed to simulate single-bunch intensity-dependent effects and their interplay in the proton ring of the Electron-Ion Collider in China (EicC) project. GOAT is a scalable and portable macroparticle tracking code written in Python and coded by object-oriented programming technology. It allows for transverse and longitudinal tracking, including impedance, space charge effect, electron cloud effect, and beam-beam interaction. In this paper, physical models and numerical approaches for the four types of high-intensity effects, together with the benchmark results obtained through other simulation codes or theories, are presented and discussed. In addition, a numerical application of the cross-talk simulation between the beam-beam interaction and transverse impedance is shown, and a dipole instability is observed below the respective instability threshold. Different mitigation measures implemented in the code are used to suppress the instability. The flexibility, completeness, and advancement demonstrate that GOAT is a powerful tool for beam dynamics studies in the EicC project or other high-intensity accelerators.

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Data availability

The data that support the findings of this study are openly available in Science Data Bank at and


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The authors would like to thank Dr. W. W. Ge for a careful reading of the manuscript and helpful discussions and suggestions.

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Authors and Affiliations



All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Lei Wang, Jian-Cheng Yang, Ming-Xuan Chang, and Fu Ma. The first draft of the manuscript was written by Lei Wang and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Jian-Cheng Yang.

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Conflict of interest

Jian-Cheng Yang is an editorial board member for Nuclear Science and Techniques and was not involved in the editorial review, or the decision to publish this article.

Competing interests

All authors declare that there are no competing interests.

Additional information

This work was supported by the National Science Fund for Distinguished Young Scholars (No. 11825505) and the National Key R&D Program of China (No. 2019YFA0405400).

Appendix A: Estimation of the code performance

Appendix A: Estimation of the code performance

Table 3 The computation performance of each physical module under different numerical parameters

The code performance is a crucial aspect of simulation software. Due to the abundant physical modules implemented in GOAT, a substantial number of numerical parameters contribute to the computational speed. Therefore, three typical quantities, namely the number of macroparticles, slices, and mesh grids in the field solver, are selected as test quantities. Table 3 summarizes the performance of all modules mentioned in the paper. All tests are run on Intel(R) Core(TM) i9-9900K CPU @ 3.60GHz (RAM 16.0 GB) adopting single-core and single-thread. Some notes are attached to the test results. (1) The slicing is performed on the ring in the simulations of space charge effect and electron cloud build-up, while on the beam for the other three simulations. (2) The saturated electron cloud is usually established after the passage of multiple bunches within one revolution period, thus the time spent for each bunch passage is used to describe the elapsed time of the electron cloud simulation. Also, it is more intuitive and convenient to use the time step corresponding to each slice to describe the number of slices used in the electron cloud effect simulation. (3) As mentioned, in electron cloud simulations, the number of electron macroparticles and the amount of charge carried by each macroparticle are constantly changing due to secondary electron production, and there is no fixed number of macroparticles. Two typical preset values given by the user during the initialization stage are used. The first number indicates that: when the number of macroparticles exceeds this value, the coordinates and charges of all particles are mixed and regenerated, and the meaning of the second number is: the total number of macroparticles after regeneration.

For impedance induced collective effects, the main limitation comes from the number of macroparticles; for other effects, the increase in all three test quantities significantly increases the computation time.  

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Wang, L., Yang, JC., Chang, MX. et al. GOAT: a simulation code for high-intensity beams. NUCL SCI TECH 34, 78 (2023).

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