Trajectory displacement due to statistical Coulomb interactions can play a major role in determining the performance of a charged particle beam system. Accurate estimation of the trajectory displacement is thus an important part of the design procedure of such an optical system. Traditionally, there are three approaches to determine the trajectory displacement: Monte Carlo simulation, the slice method, where trajectory displacement is integrated along the beam length and finally a full analytical formula describing transparently the dependence of the trajectory displacement on the parameters of the system. The latter two were developed thoroughly by Jansen and Jiang. We revise Jansen's slice method and the derivation of the integral formulae in Holtzmark and pencil-beam regimes. We show the integral formula fails to give accurate results in case a transition between the regimes occurs and we derive a new analytical expression unifying the Holtzmark and pencil-beam regime into a single formula. Furthermore, we generalize the slice method for arbitrary beam trajectory, hugely increasing its accuracy for non-ideal systems.

中文翻译：

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电子束轨迹位移解析公式及广义切片法

由于统计库仑相互作用而产生的轨迹位移在确定带电粒子束系统的性能方面可以发挥重要作用。因此，轨迹位移的准确估计是这种光学系统设计过程的重要组成部分。传统上，有三种方法来确定轨迹位移：蒙特卡罗模拟、切片法，其中轨迹位移沿光束长度积分，最后是一个完整的解析公式，透明地描述了轨迹位移对系统参数的依赖性。后两者是由詹森和蒋彻底开发的。我们修改了 Jansen 的切片方法以及 Holtzmark 和铅笔束状态下积分公式的推导。我们表明积分公式在状态之间发生转变的情况下无法给出准确的结果，并且我们推导出了一个新的解析表达式，将 Holtzmark 和铅笔束状态统一为一个公式。此外，我们将切片方法推广到任意光束轨迹，极大地提高了非理想系统的准确性。