We present a linearized method to study transverse instabilities due to electron clouds. It is based on an accurate and compact characterization of the cloud dipolar and quadrupolar forces, that can be easily obtained from quick single-pass numerical simulations. The long-term stability properties of the bunch are then predicted by solving the linearized Vlasov equation, taking into account the dipolar forces introduced by the e-cloud along the bunch as well as the betatron tune modulation with the longitudinal coordinate due to the e-cloud quadrupolar forces. The method is benchmarked against macroparticle simulations based on the same characterization of the e-cloud and the results are compared against conventional “brute-force” simulations based on the Particle-In-Cell method. The effect of transverse nonlinearities due to the e-cloud, which are neglected by the linearized method, is also analyzed.

中文翻译：

---

研究由电子云驱动的横向不稳定性的线性化方法

我们提出了一种线性方法来研究由于电子云引起的横向不稳定性。它基于对云的偶极和四极力的精确和紧凑的表征，可以通过快速的单遍数值模拟轻松获得。然后，通过解决线性化的Vlasov方程，并考虑到电子云沿束流引入的偶极力以及由于e-引起的纵坐标的自拍加速器调谐调制，可以预测束的长期稳定性。云四极力。该方法以基于电子云的相同特征的宏观模拟为基准，并将结果与​​基于“粒子内细胞”方法的常规“强力”模拟进行比较。电子云对横向非线性的影响，