参数X射线辐射与旋转周期轨道的局域分叉

The Parametric X-ray Radiation and Local Bifurcation in Rotated Periodic Orbit System

寻找新光源,特别是利用参数X射线辐射(PXR)、沟道辐射(CR)或晶体摆动场辐射(CUR)作为新的光源引起了人们极大关注。要成功获得这些新光源,粒子运动必须是稳定的。但是,由于辐射衰减、晶格热振动、电子多重散射和相互作用势的非线性等因素的影响,系统将产生退道、分叉、混沌等不稳定现象,从而对这些辐射的反射、衍射、聚焦和谱分布产生直接影响。讨论了旋转周期轨道的分叉对系统稳定性的影响。在经典力学框架内和偶极近似下,引入正弦平方势,把粒子运动方程化为了具有阻尼项、受迫项和常数力矩的广义摆方程。利用Jacobian椭圆函数和椭圆积分给出了系统的解和粒子运动周期,用Melnikov方法构造了旋转型周期轨道的Melnikov函数,并用这个方法分析了系统的局部分叉性质。结果表明,系统的临界条件与它的参数有关,只需适当调节这些参数,系统的分叉(或不稳定性)就可以原则上避免,从而进一步保证有效地输出PXR、CUR或CR辐射等。

An exploration by a parametric X-ray radiation, channeling radiation and crystalline Undulator radiation as a new light source has given rise to extensive attention. The particle motion must be stable in order to obtain these sources. But, because the radiation damp, the vibration of lattice, the electron multi-scattering and non-linearity of interaction potential of a particle with crystal, the instabilities by the dechanneling, birfucation and chaos exists in the system, thus the reflaction, the diffraction, the focus and the spectral distribution of these radiations will be effected. The effects of the stabilities by a bifurcationc of the rotated periodic orbit is regarded specifically in the paper. In the classical mechanics frame and the dipole approximation the particle motion equation is reduced to the generalized pendulum equation with a dampping, a force terms and the constant moment by the sine-squared potential. The solution of the equation and the period of the particle motion for a non-pertubated system are expressed exactly by means of Jacobian elliptic function and the 1-st kind elliptic integral. Melnikov function of the rotated periodic orbit is constructed, and also bifurcation of the system is analyzed by the method. It was shown that the critical condition of the system entered in a birfurcation is related to its parameters, provided regulating a parameters of the system, the birfurcation （an instability） can be avoided in principle. Thus, the effective export of the coherent, channeling radiation and crystalline undulator radiation can be assured furtherlly.