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要成功获得相干(衍射)的参数X-射线辐射(PXR)光源,粒子的运动必须是稳定的。但是由于衰减、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PXR的反射、衍射、聚焦和谱分布产生直接影响。在经典力学框架内和偶极近似下,引入正弦平方势,把粒子在弯晶中的运动方程化为具有常数力矩的摆方程。利用Jacobian椭圆函数和椭圆积分分析了系统的相平面特征,并用能量法讨论了系统的稳定性,在相空间密度均匀分布的假设下,导出了系统的退道系数。以硅单晶为例,计算了能量为1.0 GeV的质子运动在曲率半径为1 m的(110)面沟道的退道系数η=0.3。
To successfully obtain a coherent (diffracted) parametric x-ray radiation (PXR) light source, the motion of the particles must be stable. However, due to the influence of attenuation, lattice thermal vibration, electron multiple scattering and the nonlinearity of interaction potential, the system will have unstable phenomena such as retreat, bifurcation, chaos and so on, so the reflection, diffraction, focusing and spectrum of PXR Distribution has a direct impact. In the framework of classical mechanics and the dipole approximation, a sine-squared potential is introduced to transform the equation of motion of particles in a bending crystal into a pendulum equation with a constant moment. The Jacobian elliptic function and elliptic integral were used to analyze the phase plane characteristics of the system. The stability of the system was discussed by the energy method. Under the assumption of the uniform distribution of phase space density, the system ’s exit coefficient was derived. Taking monocrystalline silicon as an example, the proton motion with energy of 1.0 GeV was calculated and the retreat coefficient η = 0.3 of the (110) plane channel with a radius of curvature of 1 m was calculated.