Crystalline undulator radiation and sub-harmonic bifurcation of system

Looking for new light sources, especially short wavelength laser light sources has attracted widespread attention. This paper analytically describes the radiation of a crystalline undulator field by the sine-squared potential. In the classical mechanics and the dipole approximation, the motion equation of a particle is reduced to a generalized pendulum equation with a damping term and a forcing term. The bifurcation behavior of periodic orbits is analyzed by using the Melnikov method and the numerical method, and the stability of the system is discussed. The results show that, in principle, the stability of the system relates to its parameters, and only by adjusting these parameters appropriately can the occurrence of bifurcation be avoided or suppressed.     

等效磁场与弯晶沟道辐射

引入场指标和等效磁场概念,把粒子在弯晶沟道中的运动行为等效为回旋加速器中的粒子运动.结果表明,描写粒子的运动方程是一个非线性薛定谔方程.在小振幅近似下找到了系统的严格解和粒子振动周期,导出了瞬时辐射强度和最大辐射频率. The sine-squared potential and the equivalent magnetic field intensity have been introduced, the motion equation of the particle in bent crystal has been derived, the motion equation and the oscillation period of the particle have been solved exactly by Jacobian elliptic fuanction and the first kind of complet elliptic integrals, the intensity and the spectral properties of channelling radiation in bent crystal discussed.     

正弦平方势与环形加速器的弯晶束流引出

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固定力矩的摆方程。 利用Jacobian椭圆函数和椭圆积分分析了系统的相平面特征。 导出了弯晶的最大偏转能量、 退道系数和退道长度。 计算结果表明, 对于曲率半径为1 m, 能量为1.0 GeV的质子, Si(110)面沟道的引出效率为70%; 对于MeV范围的粒子, 退道长度大约是μm量级; 对于TeV范围的粒子, 退道长度可达1 m以上。 In the classical mechanics frame and the dipole approximation the particle motion equation in bent crystal is reduced to the pendulum equation with a constant momentum by using the sine squared potential.The phase planar properties are analysed by means of Jacobian elliptic function and the elliptic integral. The maximum deflected energy, the dechanneling factor and the dechanneled length are derived. The results show that the extrated efficiency is 70% for a proton beam with energy E=1.0 GeV moved in the channel on Si(110) if a radius of curvature R=1 m; and dechanneled length is of the order of μm for the particles with MeV energy; the dechanneled length is above 1 meter for the particle with TeV energy.     

形(应)变超晶格的退道效应与系统的全局分叉

把超晶格"折沟道"对粒子的作用等效为形状相似的弱周期调制.利用正弦平方势把粒子运动方程化为具有外周期弱调制的非线性微分方程,导出了退道系数与晶格畸变的关系.利用多尺度法研究了系统的主共振和子共振,并利用Melnikov方法分析了系统的全局分叉和出现Smale马蹄的临界条件. The effect of deflected channel on particle motion is equivalent to modulation with a weak potential having a periodic same as the deflected channel has. The motion equation of a particle has been reduced to the nonlinear differential equation with a weak periodic modulation by using sine squared potential. The dechanneling fraction has been derived for a strained superlattice. The main resonance and sub resonance have been inverstigated by using multi scale method, a global bifurcation and the critic...     

带电粒子的参数X射线辐射及弯晶系统的稳定性

寻找新光源, 特别是企图利用PXR作为激光光源引起了人们广泛关注。 要成功获得相干（衍射）的PXR光源, 粒子的运动必须是稳定的。 但是由于晶体弯曲,系统可能变得不稳定, 从而对PXR的反射、 衍射、 聚焦和谱分布均会产生直接影响。 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。 结果表明, 系统的稳定性与弯晶“曲率”Q有关, 当条件0≤Q<1满足时, 系统是稳定的。 An exploration of a new light source by PXR as Laser gived rise to extensive attention. The particle motion must be stable in order to obtain coherent PXR source. But, an instability of system is possible due to the effect of bent crystal, thus the reflection, the diffraction, the focus and the spectral distribution of PXR will be effected.In the classical mechanics frame and dipole approximation the particle motion equation in bent crystal is reduced to the pendulum equation with a constant momentum by the sine squared potential.The phase planar properties are analysed by means of Jacobian elliptic function and the elliptic integral. It shows that the stability of the system is related to the curvature Q of the bent crystal. When the condition 0≤Q<1 is satisfied, the system is stable。     

偶极场扰动与冷却效应对同步运动的影响

在同步加速器特别是冷却存储环中,偶极场扰动和冷却效应将会对粒子的Betatron振荡和同步振荡产生影响。 叙述了偶极场扰动对粒子同步运动的影响与高频相位调制等效, 重点研究了偶极场扰动与冷却效应同时存在的情况对粒子的同步运动产生的影响。 在经典力学框架内把粒子的同步运动方程化为广义的摆方程, 然后利用Melnikov方法对系统的稳定性进行了分析, 讨论了系统进入Smale马蹄混沌的物理意义,并导出了系统稳定的临界条件。根据稳定性判据, 给出了系统稳定性所需要的偶极场扰动的高频相位调制幅度和冷却系数的限制条件。 结果表明, 系统的稳定性与它的参数有关, 只需适当调节这些参数, 混沌便可原则上控制或避免。 The Betatron oscillation and the synchrotron oscillation for the particles are effected by the dipole field perturbation and the cooling effect. Both effects are considered in the paper and the synchrotron motion equation of the particle in the synchroton is reduced to the general pendulum equation in the classical mechanics frame. The stabilities of the system caused by dipole field perturbation and cooling are analyzed by using Melnikov method. The thresholds entered a Smale horseshoe chaos are derived in detail and the stability conditions for the system caused by dipole field perturbation and cooling are dicussed. The results show that the critical condition or the stability thresholds are related to the system parameters. The chaos or an instability can be avoided or controlled in principle by regulating some parameters of the system.     

Volume reflection of 1-GeV protons by a bent silicon crystal

The volume reflection of 1-GeV protons by a bent crystal has been observed. The crystal is made of single crystal silicon. The (111) atomic planes are bent owing to the elastic quasimosaicity effect, which makes it possible to reduce the crystal length for a beam to 30 μm. It is found that the probability of the reflection effect is higher than the probability of the channeling effect (0.71 vs. 0.63), and the deflection angle of the protons reflected inside the crystal is equal to 1.39 ± 0.04 in terms of the critical angle for channeling under the conditions of the experiment (170 μrad). The width of the reflected peak is equal to 1.76 ± 0.04 in the same units. The protons that are not involved in channeling at the angular position of maximum channeling undergo volume reflection and are deflected in the direction opposite to the channeled beam by the angle 1.01 ± 0.05 in terms of the critical angle for channeling. The width of the reflected peak is equal to 1.94 ± 0.08 in the same units. Original Russian Text ? Yu.M. Ivanov, N.F. Bondar’, Yu.A. Gavrikov, A.S. Denisov, A.V. Zhelamkov, V.G. Ivochkin, S.V. Kos’yanenko, L.P. Lapina, A.A. Petrunin, V.V. Skorobogatov, V.M. Suvorov, A.I. Shchetkovsky, A.M. Taratin, W. Scandale, 2006, published in Pis’ma v Zhurnal éksperimental’noĭ i Teoreticheskoĭ Fiziki, 2006, Vol. 84, No. 7, pp. 445–450.     

Nonlinearity and periodic solution of a standard-beam balance oscillation system

We present the motion equation of the standard-beam balance oscillation system,whose beam and suspensions,compared with the compound pendulum,are connected flexibly and vertically.The nonlinearity and the periodic solution of the equation are discussed by the phase-plane analysis.We find that this kind of oscillation can be equivalent to a standard-beam compound pendulum without suspensions;however,the equivalent mass centre of the standard beam is extended.The derived periodic solution shows that the oscillation period is tightly related to the initial pivot energy and several systemic parameters:beam length,masses of the beam,and suspensions,and the beam mass centre.A numerical example is calculated.     

超晶格量子阱的沟道辐射及其谱分布

在经典物理框架内和偶极近似下, 导出了超晶格量子阱沟道辐射频率和辐射谱分布。指出了对于自发辐射谱分布, 存在一个普适的线型因子, 而粒子的最大辐射能量与相对论因子γ 有关, 且与γ3/2成正比。以正弦平方势为例进行了具体讨论。结果表明, 由于势阱深度和噪音的影响, 谐波数l只取少数几个值。超晶格量子阱沟道辐射只存在不多的几条谱线, 为进一步应用提供了可能。最后, 还给出了一种可能的实验方案, 讨论了如何利用弯晶把超晶格量子阱的沟道辐射改造为相干辐射。 In the frame of classical physics and the dipole approximation the radiation frequency and the spectral distribution are derived for the channeling radiation of a charged particle in a superlattice quantum well. It indicated that there is a line type factor f(ξ) suited to various cases in the spontaneous radiations spectrum. Results also show that the maximum radiation energy is proportional to γ3/2 , but the relativistic effects have double effects in the spontaneous radiation of a charged particle. The case for the sine squared potential is discussed specifically. The harmonic number can be defined as a few variable values by the effects of the potential well depth and noise. In general there is a few spectral lines in the channeling radiation spectrum for the superlattice quantum well, and possibilities are provided for further application. Finally, a possible experimental scheme is proposed, and it is discussed that how to transform the channeling radiation in the quantum well into the cohenent radiation by the bent crystal.     

The density wave in a new anisotropic continuum model

In this paper the new continuum traffic flow model proposed by Jiang et al is developed based on an improved car-following model, in which the speed gradient term replaces the density gradient term in the equation of motion. It overcomes the wrong-way travel which exists in many high-order continuum models. Based on the continuum version of car-following model, the condition for stable traffic flow is derived. Nonlinear analysis shows that the density fluctuation in traffic flow induces a variety of density waves. Near the onset of instability, a small disturbance could lead to solitons determined by the Korteweg-de-Vries （KdV） equation, and the soliton solution is derived.     

Steering of Sub-GeV electrons by ultrashort Si and Ge bent crystals

We report the observation of the steering of 855 MeV electrons by bent silicon and germanium crystals at the MAinzer MIkrotron. Crystals with 15 \(\upmu \)m of length, bent along (111) planes, were exploited to investigate orientational coherent effects. By using a piezo-actuated mechanical holder, which allowed to remotely change the crystal curvature, it was possible to study the steering capability of planar channeling and volume reflection vs. the curvature radius and the atomic number, Z. For silicon, the channeling efficiency exceeds 35%, a record for negatively charged particles. This was possible due to the realization of a crystal with a thickness of the order of the dechanneling length. On the other hand, for germanium the efficiency is slightly below 10% due to the stronger contribution of multiple scattering for a higher-Z material. Nevertheless this is the first evidence of negative beam steering by planar channeling in a Ge crystal. Having determined for the first time the dechanneling length, one may design a Ge crystal based on such knowledge providing nearly the same channeling efficiency of silicon. The presented results are relevant for crystal-based beam manipulation as well as for the generation of e.m. radiation in bent and periodically bent crystals.     

弯晶沟道的混沌行为

本文预言了在周期场作用下,弯晶沟道的混沌行为。分析了系统的全局分叉性质和出现Smale马蹄的参数条件,指出系统通向混沌的过程就是粒子的退道过程,而相的沟道效应和沟道辐射都将表现出内在的随机性,为定量解释沟道效应和沟道辐射的某些特征提供了一种可能的理论分析。 关键词：     

Feasibility of an electron-based crystalline undulator

The feasibility to generate powerful monochromatic radiation of the undulator type in the gamma region of the spectrum by means of planar channeling of ultrarelativistic electrons in a periodically bent crystal is proven. It is shown that to overcome the restriction due to the smallness of the dechanneling length, an electron-based crystalline undulator must operate in the regime of higher beam energies than a positron-based one does. A numerical analysis is performed for a 50 GeV electron channeling in Si along the (111) crystallographic planes.     

Influence of quantum fluctuations on the stochastic dynamics of the channeling effect of relativistic electrons and positrons

A system of equations describing multiple scattering by crystal nuclei and electrons, and also quantum fluctuations of the coordinate and momentum operators of fast charged particles has been obtained. Quantum corrections to the classical equations of motion have been determined in quasiclassical approximations. A computer simulation of 855-MeV electron and positron motion in the (110) planar channel of a Si crystal has been carried out. The inclusion of quantum fluctuations in the equation of motion affects the dynamics of the electron channeling effect in planar crystal channels considerably; in particular, intense dechanneling (heating) occurs. Intense rechanneling (cooling) occurs in the case of positrons.     

正弦平方势与小振幅近似下的弯晶沟道辐射

在理想情况下和经典力学框架内,引入正弦平方势,把粒子在弯晶中的运动方程化为具有外力矩的摆方程。并对系统的相平面特征进行了数值分析。在小振幅近似下,把粒子运动方程化为具有硬特性的弹簧-振子系统,用Jacobian椭圆函数和椭圆积分解析地给出系统的解和粒子运动周期。讨论了弯晶沟道辐射频率、无量纲偏转角和辐射谱的一般特征。指出利用沟道辐射作为激光的可能性。以正电子在碳单晶中沟道辐射为例进行了具体计算,得到了与其他工作基本一致的结果。     

晶体摆动场的辐射衰减与能量增益

由于晶体弯曲、电子多重散射和晶格热振动引起的粒子退道,辐射同原子相互作用引起的电离损失,以及系统非线性引起的全局分叉与混沌行为等,晶体摆动场辐射面临的问题比自由电子激光要复杂得多。首先,引入晶体摆动场自发辐射谱分布,并在简谐近似下讨论了运动方程、系统增益和能量衰减。结果表明:一般情况下辐射谱只有少数几条谱线,而在简谐近似下,谱线只有一条。其次,指出了辐射能量比较高时,衰减是主要的,选择的晶体长度必须小于衰减长度,且可选择正电子或质子作炮弹;当辐射能量比较低时,退道是主要的,选择的晶体长度必须小于退道长度,且可选择重离子作炮弹。     

Quantitative theory of channeling particle diffusion in transverse energy in the presence of nuclear scattering and direct evaluation of dechanneling length

A refined equation for channeling particle diffusion in transverse energy taking into consideration large-angle scattering by nuclei is suggested. This equation is reduced to the Sturm–Liouville problem, allowing one to reveal both the origin and the limitations of the dechanneling length notion. The values of the latter are evaluated for both positively and negatively charged particles of various energies. New features of the dechanneling dynamics of positively charged particles are also revealed. First, it is demonstrated that the dechanneling length notion is completely inapplicable for their nuclear dechanneling process. Second, the effective electron dechanneling length of positively charged particle varies more than twice converging to a constant asymptotic value only at the depth exceeding the latter.     

Dynamical chaos and stochastic mechanism of high-energy negatively charged particle deflection by bent crystals

Deflection of high-energy negatively charged particles in straight and bent crystals through multiple scattering by crystal atomic strings was considered for the case in which the initial angle between the particle momentum and one of the main crystallographic axes was approximately four critical angles of axial channeling. It was shown that in a bent crystal with a small crystal thickness, when the crystal bend was less than the beam incidence angle, the beam deflected in the direction opposite to the direction of the crystal bend. At larger crystal thicknesses, the large part of the beam starts to deflect in the direction of the crystal bend. In addition, there is a group of particles that follow the crystal axis bend in the angular region of approximately the critical angle of axial channeling with respect to the current direction of the crystal axis. It was shown that in all of these deflection processes, the periodicity of the location of atomic strings in the crystal does not influence the angular distributions of scattered particles. This fact is connected with the effect of dynamical chaos in particle motion in the periodical field of bent crystal atomic strings. It was also shown that observed in a recent CERN experiment effect of beam deflection, when the angle between the initial particle momentum and the crystal axis was approximately four critical angles of axial channeling, is due to peculiarities of the stochastic multiple scattering of particles by bent crystal atomic strings. These peculiarities are connected with the effect of dynamical chaos in particle motion in crystals.     

Critical parameters of channeling

Basic methods for calculating critical channeling parameters that play a decisive role in the problem of dechanneling of ions are reviewed. The analytic expressions derived for the threshold energy of channeling, the critical transverse energy, and the closest approach distance between particles and atomic chains or planes forming the channels, as well as critical angles of axial and planar channeling, are valid in a wide range of particle energies. The dependence of critical parameters of channeling on the temperature of crystals is analyzed.