Lie algebraic analysis for the nonlinear transport of intense bunched beam in electrostatic quadrupoles

In this paper, the nonlinear transport of intense bunched beams in electrostatic quadrupoles is analyzed using the Lie algebraic method, and the results are briefly presented of the linear matrix approximation and the second order correction of particle trajectory in the state space. Beam having K-V distribution and Gaussian distribution approximation are respectively considered. A brief discussion is also given of the total effects of the quadrupole and the space charge forces on the evolution of the beam envelope.     

Numerical simulation for space charge dominated beam transport

To simulate the intense bunched beam transport, a computer program LEADS-3D has been developed. The particle trajectories are analysed with the Lie algebraic method. The third order approximation of the trajectory solutions is made with space charge forces off, and the second order approximation is made with space charge forces on. The particle distribution in the 3D ellipsoid is uniform or Gaussian. Most of the conventional beam optical elements are incorporated in the code. The optimization procedures are provided to fit the beam lines to satisfy the given optical conditions.     

离散速度方向模型在微尺度气体流动中的应用

离散速度方向模型是一种简化Boltzmann方程的新方法。该方法通过减少Boltzmann方程的维数来降低数值求解的计算量。在DVD模型中,分子速度的方向是离散的,而分子的速率仍然是连续的,这样就可以用一组三维的速率分布函数来代替Boltzmann方程中六维的速度分布函数。由于减少了三个动量维,同Boltzmann方程相比,DVD模型的数值计算量可以降低几个数量级。本文用数值的方法对DVD模型进行了研究。数值结果显示,在广泛的Knudsen数下,DVD方法可给出精确的计算结果。同线性化Boltzmann方程的计算结果相比,最大的误差不超过6%,在连续介质领域中,误差甚至不超过1%。     

Non-equilibrium approach to transport phenomena in quantum crystals

A pure dielectric quantum crystal subjected to an external mechanical force is described by non-equilibrium Green’s functions. In equilibrium the leading approximation leads to the definition of elementary excitations, the phonons in the renormalized harmonic approximation. Their temperature dependent energies are to be determined as solutions of an integral equation. For hydrodynamic disturbances a generalized transport equation for a phonon number density is derived. A similar approximation for the spectral function yields an integral equation for space and time dependent quasiparticle energies which are expressed as functionals of the displacement field and the phonon distribution. The Boltzmann equation for the latter includes the quasi-particle interaction.     

Classical projected phase space density of billiards and its relation to the quantum neumann spectrum

A comparison of classical and quantum evolution usually involves a quasiprobability distribution as a quantum analogue of the classical phase space distribution. In an alternate approach that we adopt here, the classical density is projected on to the configuration space. We show that for billiards, the eigenfunctions of the coarse-grained projected classical evolution operator are identical to a first approximation to the quantum Neumann eigenfunctions. However, even though there exists a correspondence between the respective eigenvalues, their time evolutions differ. This is demonstrated numerically for the stadium and lemon-shaped billiards.     

Propagation properties of vectorial elliptical Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh diffraction integral formula and without using the far-field approximation, a solution of the wave equation beyond the paraxial approximation is found, which represents vectorial non-paraxial elliptical Gaussian beams in free space. The far-field expressions for non-paraxial Gaussian beams and elliptical Gaussian beams can be regarded as special cases treated in this paper. Some basic propagation properties of vectorial non-paraxial elliptical Gaussian beams, including the irradiance distribution, phase term, beam widths and divergence angles are studied. Numerical results are given and illustrated.     

An identity for the non-equilibrium phase space distribution function

A new identity satisfied by the phase space distribution function is derived and the Zubarev statistical operator is obtained as an approximation to this identity.     

The Δ- Method for the Boltzmann Equation

A new numerical method is proposed to solve the Boltzmann equation. A frame is set up by using a discrete velocity approximation in the infinite velocity space, but by considering only those distribution function points which are not too small. The distribution function points may occur anywhere in the infinite discrete velocity space and are not constrained to a pre-specified region. A fourth-order finite difference is used for the convection terms. A Monte Carlo-like method is applied to the discrete velocity model of the collision integral. The effort of the method is proportional to the number of discrete points. Numerical examples are given for the full Boltzmann equation and results for some benchmark problems are compared with analytical or prior solutions.     

电推力器流动模拟中的电子处理方法

 分析了电子的准中性假设、玻耳兹曼分布假设、粒子模型在电推力器流动模拟中的适用性和优劣性,提出了一种新的电子处理方法——电子漂移扩散近似,采用该方法模拟了离子发动机栅极光学系统等离子体运动过程。结果表明:该方法得出的电势分布、离子相空间分布及电子数密度分布与经典的电子玻耳兹曼分布假设处理方法计算结果一致,验证了该方法可以很好地应用于电推力器栅极光学系统模拟。     

Partially coherent non-paraxial vectorial Hermite-Gaussian beams and their far-field properities

The far-field theory of partially coherent vectorial non-paraxial beams is expanded and their analytical propagation expressions of the Wigner distribution function matrices and the cross-spectral-density matrices in free space are derived using the far-field approximation. Some interesting cases, in particular the vectorial non-paraxial partially coherent Hermite-Gaussian (HG) beams, are discussed and treated as special cases of our general expressions. It is shown that the f_σ and f parameters play an important role in determining vector and non-paraxiality of partially coherent HG beams. When two parameters are small enough, scalar and paraxial vectorial approximation is allowed; otherwise, non-paraxial vectorial approximation is applied. But the decided parameters additionally affect their far-field divergence angles.     

Fundamentals of a Technique for Determining Electron Distribution Functions by Multi-Term Even-Order Expansion in Legendre Polynomials. 2. Comprehensive Investigation of a Model Plasma

Applying the new technique for finding the converged solution of the Boltzmann equation in a weakly ionized plasma, which was developed in the first part of this paper, a comprehensive study of the electron velocity distribution function for a model plasma with elastic and exciting collisions is performed by solving the Boltzmann equation with increasing order of approximation. The purpose of this investigation is that of calculating the isotropic distribution f₀, the first contribution f₁ to the anisotropy of the velocity distribution, the important macroscopic quantities and, more generally, that of studying the total anisotropy as well as the changes of all these quantities when the approximation degree is enlarged beyond the 2 terms of the conventional Lorentz approximation. By varying some parameters of the model plasma, that is the electric field strength, the magnitude of the excitation cross section and the excitation threshold, the main features of plasmas in inert as well as molecular gases are modelled and the impact of these parameters on the mentioned quantities is analysed. Some of the converged results are compared with results of corresponding Monte Carlo simulations. The approximation degree required to find the converged values of isotropic distribution, main macroscopic quantities and electron distribution in the velocity space (and thus its real anisotropy) is estimated by solving the Boltzmann equation over wide parameter ranges.     

Lie algebraic analysis for the nonlinear transportof intense pulsed beams in electrostatics lenses

The Lie algebraic method is applied to the analysis of the nonlinear transport of an intense pulsed beam in cylindrically symmetrical electrostatic lenses, and particle orbits in a six-dimensional phase space (x, p_x, y, p_y, τ, p_τ) are obtained in the second order approximation. They can also be acquired in the third or higher order approximation if needed. In the analysis, we divide the electrostatic lenses into several segments. Each segment is considered as a uniform accelerating field, and each dividing point is treated as a thin lens. The particle distribution in a three-dimensional ellipsoid is of Gaussian type.     

A Uniform Semiclassical Calculation of the Rings in Photodetachment Microscope

We describe a uniform semiclassical theory for the spatial distribution of the photodetached electron of H^- in the presence of a static electric field. In this theory we propagate the photodetached electron wavefunction using a mixed position and momentum coordinate method to large distance where a uniform approximation to the electron flux distribution is calculated. Details are given for the propagation of the mixed coordinate wavefunction with cylindric symmetry and the transformation between configuration space wavefunction and the mixed position and momentum wavefunction.     

Approximate solutions of the Liouville equation. IV. The two-body additive approximation

The two-body additive approximation on the time-dependent Liouville distribution, first introduced in part I of this series, is put into the conventional form of a self-contained kinetic equation for the doublet distribution. From this point of view the approximation consists in truncating the BBGKY chain by expressing the triplet distribution as a functional of lower distributions at the same value of the time variable. To accomplish this, it is necessary to study two associated purely spatial integral equations. The doublet kinetic equation can then be written in terms of solutions of these integral equations and comparison with conventional methods of truncating the BBGKY chain can then be made. For the purpose of comparison a method of truncating the chain based on the Kirkwood superposition approximation is introduced and discussed briefly. The momentum structure of the resulting doublet kinetic equation is similar, but the nonlocality in space of our truncation introduces distinct differences in the spatial structure. The inconsistency between conventional truncations and the exact initial conditions used for the calculation of time-dependent correlation functions is pointed out. This inconsistency is not shared by the two-body additive approximation.     

Space distribution and energy straggling of charged particles via Fokker-Planck equation

Summary The Fokker-Planck equation describing a beam of charged particles entering a homogeneous medium is solved here for a stationary case. Interactions are taken into account through Coulomb cross-section. Starting from the charged-particle distribution as a function of velocity and penetration depth, some important kinetic quantities are calculated, like mean velocity, range and the loss of energy per unit space. In such quantities the energy straggling is taken into account. This phenomenon is not considered in the continuous slowing-down approximation that is commonly used to obtain the range and the stopping power. Finally the well-known Bohr or Bethe formula is found as a first-order approximation of the Fokker-Planck equation. The authors of this paper have agreed to not receive the proofs for correction.     

From classical dynamics to continuous time random walks

The migration of a classical dynamical system between regions of configuration space can be treated as a continuous time random walk between these regions. Derivation of a classical analog of the quantum mechanical generalized master equation provides expressions for the waiting time distribution in terms of transition memory functions. A short memory approximation to these memory functions is equivalent to the well-known transition state method. An example is discussed for which this approximation seems reasonable but is entirely wrong.     

半导体深能级波函数在Bloch空间的分布特征

在Koster-Slater格林函数以及中心原胞缺陷势近似的基础上第一次给出并讨论了Si,GaAs,GaP深能级波函数在Bloch空间的分布特征。并指出,对于一定能带数目的近似晶体模型,计算深能级波函数的收敛性比计算深能级能量的收敛性要快。 关键词：     

弯晶的沟道效应

本文引入正弦平方势,将弯晶中的粒子运动方程化为常力矩作用下的摆方程,并用非线性力学方法,详细讨论了系统的相平面特征,分析了相平面上奇点分布和相对运动对系统稳定性的影响,从系统的临界状态出发导出了动力学允许的临界参数和可能偏转的最大能量;此外,还考察了小振幅近似下的粒子运动行为,发现该系统具有软弹簧特征,用Jacobian椭圆函数和第一类全椭圆积分严格求解了运动方程和振动周期,导出了低能带电粒子在弯晶中的空间分布和动量分布。 关键词：