在经典力学框架内和偶极近似下,引入Lindhard势,把粒子运动方程转化为具有硬弹簧特性的达芬(Duffing)方程,利用多尺度法分析了系统的主共振和超共振;讨论了粒子在共振线附近运动的非线性行为;导出了系统的临界参数ac及满足系统稳定的条件。由于系统的临界条件与它的物理参数有关,只需这些参数适当,系统的不稳定性就可以原则上避免,从而保证系统能够稳定地输出晶体摆动场辐射,为寻找X激光或γ激光提供进一步理论分析。 In the framework of classical mechanics and the dipole approximation, the Lindhard potential is introduced to transform the particle motion equation into a Duffing equation with hard spring properties. The main resonance and super resonance of the system are analyzed by multi-scale method. The nonlinear behavior of moving around the resonance line; derived the critical parameter ac of the system and meet the conditions of system stability. As the critical condition of the system is related to its physical parameters, the system instability can be avoided as long as these parameters are appropriate, so as to ensure that the system can output the stable radiation of the crystal swing field and provide further search for X-laser or γ-laser theoretical analysis.