中文摘要：

为了提高粒子在四极磁场中输运轨迹计算的精度，基于李代数方法，分析和计算粒子柬在四极磁场中空间电荷的非线性效应，得到二级近似下粒子坐标的映射公式，编写了粒子束在四极磁场中传输的模拟程序QML。用模拟程序对四极磁场中的传输进行了模拟计算，并对模拟结果进行了分析。模拟结果表明，在空间电荷的非线性效应的影响下相图产生了明显的扭曲，对粒子束包络曲线横向发散作用明显。

英文摘要：

In order to improve the accuracy of trajectory of particles in quad - pole magnetic field, nonlinear effect of space charge was analyzed and calculated with Lie algebraic methods, the mapping formulas of particle coordinates through second order were obtained, and a computer code QML was written for the particle beam transmission in quad - pole magnetic field. QML was used to simulate particle beam transmission in quad - pole magnetic field, and the simulation results were analyzed. Simulation results indicated that, due to the influence of nonlinear effect of space charge, the beam phase diagram hadclear distortion, at the same time, the beam envelope had apparent lateral divergence.