On the Velocity Distribution Function of Light Ions in Heavy Gases in an Electric Field

The special properties presented by the Boltzmann collision operator in the case of an induced-dipole interaction (Maxwellian interaction) between ions and neutrals are exploited to obtain in such case a proper solution of the Boltzmann equation for light ions in heavy gases in an electric field. Meanwhile it is proved that consideration of more than two terms of the spherical harmonic expansion of the ion velocity distribution requires to improve the usual accuracy of the terms deriving from the collision integral; vice versa, the improvement of the accuracy of the collision terms requires to retain more than two terms of the spherical harmonic expansion. The consistency of our procedure and results in the approximation neglecting the square of the ion-neutral mass ratio with respect to unity is discussed. Finally, the most significant velocity averages are calculated on the basis of the obtained ion distribution. In the limit of the above approximation they are shown to agree with Wannier's results.     

Plasmon‐Enhanced Second‐ and Third‐Order Harmonic Generation in Spherical Free‐Electron Nanoclusters with Diffuse Surface

The nonlinear scattering of a laser pulse off spherical nanoclusters with free electrons and with a diffuse surface is examined in the collisionless hydrodynamics approximation in the framework of perturbation theory with respect to the laser pulse intensity, as well as of the steady‐state approximation. In a previous publication [S.V. Fomichev and W. Becker, Phys. Rev. A 81 , 063201 (2010)] we reported the full nonlinear hydrodynamic model of forced collective electron motion confined to a cluster with diffuse surface and introduced two different perturbation theories corresponding to different laser intensity regimes. In the current paper, in the framework of this hydrodynamic model we focus on the properties of plasmon resonance‐enhanced third‐harmonic generation in a spherical cluster and its dependence on the shape of its diffuse surface whose role increases for nonlinear processes. At the same time, the quadrupole second‐harmonic generation in a spherical cluster is also inspected as a necessary intermediate step. Both cold metal clusters in vacuum or in a dielectric surrounding and hot laser‐heated and laser‐ionized clusters are considered within the same approach for a wide range of the fundamental laser frequency. Nonlinear laser excitation of the dipole plasmon Mie resonance in spherical clusters, as well as of other respective multipole plasmon resonances is investigated analytically and numerically in detail (position, width, and strength) versus the cluster‐surface diffuseness, the outer ionization degree in charged clusters, the electron‐density diffuseness, and their interplay. Under certain conditions, depending on the various cluster parameters, different secondary nonlinear resonances are found. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spherical harmonics and cartesian tensor scalar product expansions of the distribution function

On the basis of the expansion of the distribution function in a sum of the spherical harmonics, the distribution functionf(v, r, t) is expanded in a series of scalar products of two Cartesian tensors term by term, i.e. The tensors and ^(l) (l=2, 3) are constructed in dependence on the spherical harmonic expansion coefficients (the tensors and ^(l) (l=0, 1) have been constructed by Jancel and Kahan [3]). On the basis of the knowledge of the analytic form off ₂ andf ₃ the equations forf ₁ f ₂ andf ₃ for the case of the Boltzmann's equation are determined.Technická 2, Praha 6, Czechoslovakia.     

采用分布式球形阵列的球谐波域声场分离方法

为从测得的混合声场信号中提取出需要的目标声场,提出一种基于分布式球形传声器阵列的声场分离方法。该方法依据声场的球谐波分解,利用阵列各传声器采集到的声场声压信号,获得目标声场与干扰声场的球谐波展开系数,进而估计目标声场。该方法利用声场以不同中心展开的球谐波系数之间的变换关系,直接建立传声器测量声压信号与整体坐标系下声场展开系数的方程,与传统的分布式球形阵列声场分离方法,即先求解局部坐标系下声场展开系数,再变换为整体展开系数的方法,进行比较。分别通过数值仿真和实验说明了提出方法的有效性。结果表明:该方法能够从混合声场中较准确地估计出目标声场,并且在干扰声场能量增大时,保持了较小的声场估计误差,相比于传统方法误差增加更少。     

Derivation of collective potential and mass from the generator coordinate method

The Hill-Wheeler equation of the generator coordinate method is approximated by a local collective Schrödinger equation. General expressions for the potential and the mass parameter are obtained by a symmetrized moment expansion. The validity of the approximation is tested for several examples where the exact solution is known. These include the Gaussian overlap with harmonic and anharmonic interaction, the Lipkin model, and monopole resonances of spherical light nuclei. In all cases, surprisingly close agreement with the exact solution is found. Other possible applications of the formalism are indicated.     

A spectral method for the wave equation of divergence-free vectors and symmetric tensors inside a sphere

The wave equation for vectors and symmetric tensors in spherical coordinates is studied under the divergence-free constraint. We describe a numerical method, based on the spectral decomposition of vector/tensor components onto spherical harmonics, that allows for the evolution of only those scalar fields which correspond to the divergence-free degrees of freedom of the vector/tensor. The full vector/tensor field is recovered at each time-step from these two (in the vector case), or three (symmetric tensor case) scalar fields, through the solution of a first-order system of ordinary differential equations (ODE) for each spherical harmonic. The correspondence with the poloidal–toroidal decomposition is shown for the vector case. Numerical tests are presented using an explicit Chebyshev-tau method for the radial coordinate.     

Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements

We propose the P_N approximation based on a finite element framework for solving the radiative transport equation with optical tomography as the primary application area. The key idea is to employ a variable order spherical harmonic expansion for angular discretization based on the proximity to the source and the local scattering coefficient. The proposed scheme is shown to be computationally efficient compared to employing homogeneously high orders of expansion everywhere in the domain. In addition the numerical method is shown to accurately describe the void regions encountered in the forward modeling of real-life specimens such as infant brains. The accuracy of the method is demonstrated over three model problems where the P_N approximation is compared against Monte Carlo simulations and other state-of-the-art methods.     

Area-preserving diffeomorphisms and supermembrane Lorentz invariance

The Lie algebra of area-preserving diffeomorphisms on closed membranes of arbitrary topology is investigated. On the basis of a harmonic decomposition we define the structure constants as well as two other tensors which appear in the supermembrane Lorentz generators. We derive certain identities between these tensors and analyze their validity when the areapreserving diffeomorphisms are approximated bySU(N). One of the additional tensors can then be identified with the invariant symmetric three-index tensor ofSU(N), while the second has no obvious analog. We prove that the Lorentz generators are classically conserved in the light-cone gauge for arbitrary membrane topology, as a consequence of these tensor identities. This formulation allows a systematic study of the violations of Lorentz invariance in theSU(N) approximation.     

A novel approach to designing cylindrical-surface shimcoils for a superconducting magnet of magnetic resonance imaging

For a superconducting magnet of magnetic resonance imaging (MRI), the novel approach presented in this paper allows the design of cylindrical gradient and shim coils of finite length. The method is based on identification of the weighting of harmonic components in the current distribution that will generate a magnetic field whose z-component follows a chosen spherical harmonic function. Mathematical expressions which relate the harmonic terms in the cylindrical current distribution to spherical harmonic terms in the field expansion are established. Thus a simple matrix inversion approach can be used to design a shim coil of any order pure harmonic. The expressions providing a spherical harmonic decomposition of the field components produced by a particular cylindrical current distribution are novel. A stream function was applied to obtain the discrete wire distribution on the cylindrical-surface. This method does not require the setting of the target-field points. The discussion referring to matrix equations in terms of condition numbers proves that this novel approach has no ill-conditioned problems. The results also indicate that it can be used to design cylindrical-surface shim coils of finite length that will generate a field variation which follows a particular spherical harmonic over a reasonably large-sized volume.     

传感器电解质材料热膨胀性能和热稳定性变化规律研究

以ZrO2固体电解质材料为例,研究氧传感器电解质材料原子振动特点和热膨胀系数及其热稳定性随温度和时间的变化规律,探讨原子非简谐振动的影响。结果表明：原子振动的频率、阻尼系数,在简谐近似下为常数,在考虑到非简谐效应后随温度升高而增大;原子平均位移和热膨胀系数在简谐近似下为零,在考虑到非简谐效应后随温度升高而增大,随的时间的增长而减小;热膨胀性能稳定性温度系数随温度的升高而减小,随时间的增长而增大,即使用时间越长,材料的热膨胀性能稳定性越低;温度越高,热膨胀性能越稳定;非简谐情况下的原子振动的频率、阻尼系数和热膨胀系数与简谐近似下的差值随温度的升高而增大,即温度越高,非简谐效应越显著。     

Intermolecular interaction effect on the line-shape of hyper-Rayleigh light scattering by molecular liquids

Scattering on molecules, correlated in regions of short-range quasi-ordering, is analysed with regard to its effect on the spectrum of hyper-Rayleigh light-scattering by molecular liquids. The general form of S ^2ω _c(?, Δω) derived using spherical tensors and expanding the orientational-positional-time pair correlation function in Wigner functions, is discussed on the assumption that Vineyard's [22] approximation can be applied to calculate G _c[τ₁(t), τ₂(0)]. Successive terms of the expansion are calculated assuming dipole-dipole interaction energy as predominant in the total molecular electric multipole interaction energy. In the present approximation the contribution from scattering on correlated molecules is shown to be given by the difference of two Lorentz lines, whose maxima depend on the magnitude of the interactions. The integral intensities, calculated from the formulae derived here, are in agreement with those of the literature.     

The Polarization Tensors of Z → 3γ and γγ → γγ via W-Loop in the Standard Model

The polarization tensors of Z → 3γ and γγ → γγ W-Boson loops are calculated in the standard model. The constrained equations for the tensors are deduced from the crossing symmetry and gauge invariance. These equations are numerically checked by the polarization tensor of γγ → γγ to the first three lowest orders in its low energy expansion.     

On the Solution of the Electron Boltzmann Equation in Higher Order Legendre Polynomial Expansion

The paper deals with the mathematical structure of the general solution and the selection of the physical relevant solution of the hierarchy resulting from the electron Boltzmann equation by the Legendre polynomial expansion for a weakly ionized plasma with elastic and exciting collisions and under the action of an electric field. At first the properties of the general solution for the two and four term approximation of the distribution function are analyzed and then the study is extended to arbitrarily even order of the hierarchy using the theory of weakly and strongly singular differential equation systems for the investigation of the solution behaviour of the truncated hierarchy at small and at large energies, respectively. In this way, especially the free parameters of the non-singular part of each general solution, which is obtained for small and large energies and is of interest for the construction of the desired solution, could be found, and the procedure for their final determination is explained. A first illustrative example is given of the application of these studies for the determination of the velocity distribution function of the electrons in four term approximation for a model plasma.     

Vibrational intensities in methane

The absorption intensities of the two infra-red active vibrations in methane have been obtained from a perturbation calculation on the equilibrium wave functions derived in the preceding paper. The perturbation field is the change in the potential field due to the nuclei which results from moving the nuclei in the vibrational coordinate concerned, and a simplified form of second order perturbation theory, developed by Pople and Schofield, is used for the calculation. The main approximation involved is the neglect of f and higher harmonics in the spherical harmonic expansion of the nuclear field. The resulting dipole moment derivatives are approximately three times larger than the experimental values, but they show qualitative features and sign relationships which are significant.

The experimental intensity measurements are interpreted and discussed in relation to these results.     

Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries

The inherent complexity of the radiative transfer equation makes the exact treatment of radiative heat transfer impossible even for idealized situations and simple boundary conditions. Therefore, a wide variety of efficient solution methods have been developed for the RTE. Among these solution methods the spherical harmonics method, the moment method, and the discrete ordinates method provide means to obtain higher-order approximate solutions to the equation of radiative transfer. Although the assembly of the governing equations for the spherical harmonics method requires tedious algebra, their final form promises great accuracy for any given order, since it is a spectral method (rather than finite difference/finite volume in the case of discrete ordinates). In this study, a new methodology outlined in a previous paper on the spherical harmonics method (P_N) is further developed. The new methodology employs successive elimination of spherical harmonic tensors, thus reducing the number of first-order partial differential equations needed to be solved simultaneously by previous P_N approximations (=(N+1)²). The result is a relatively small set (=N(N+1)/2) of second-order, elliptic partial differential equations, which can be solved with standard PDE solution packages. General boundary conditions and supplementary conditions using rotation of spherical harmonics in terms of local coordinates are formulated for the general P_N approximation for arbitrary three-dimensional geometries. Accuracy of the P_N approximation can be further improved by applying the “modified differential approximation” approach first developed for the P₁-approximation. Numerical computations are carried out with the P₃ approximation for several new two-dimensional problems with emitting, absorbing, and scattering media. Results are compared to Monte Carlo solutions and discrete ordinates simulations and a discussion of ray effects and false scattering is provided.     

Fundamentals of a Technique for Determining Electron Distribution Functions by Multi-Term Even-Order Expansion in Legendre Polynomials 1. Theory

On the basis of our recent investigations concerning the mathematical structure of the hierarchy which results from the Legendre polynomial expansion of the electron velocity distribution function in Boltzmann's equation a new technique for solving this equation in multi-term even-order approximation is presented. This method is, even if more complex, the logical generalization of the well known technique for solving Boltzmann's equation by backward integration in the conventional two-term approximation. A weakly ionized, spatially homogeneous and stationary plasma with elastic and exciting electron-atom collisions is considered acted upon by a dc electric field. The technique, presented in detail, determines the distribution function in even order 2l of the expansion at the end by l-fold backward and 2l-fold forward integration of the hierarchy and by continuous connection of the resulting non-singular parts of the general solutions at low and high energies at an appropriate connection point. A first application of this method is made on a model gas for the even orders from 2 to 10 and under conditions with distinct anisotropy in the velocity space due to intensive exciting collisions. The converged macroscopic quantities and the corresponding first coefficients of the distribution expansion itself are compared with very accurate Monte Carlo simulations under the same conditions where a perfect agreement between the results obtained with both techniques was found confirming the high accuracy of the new technique to be presented.     

Lattice dynamics of quantum crystals

A theory of lattice dynamics for quantum crystals is developed. This is done by summing an infinite class of diagrams of the usual anharmonic expansion and by avoiding the harmonic approximation as starting point. The zero order of the expansion given in this paper corresponds to the harmonic approximation with an effective potential. Higher orders correspond to higher anharmonic corrections with the same potential. Since the new potential varies more slowly the expansion seems to converge more rapidly than the usual anharmonic expansion. Numerical calculations on bcc He³ show that the ground state energy is lowered by about 3–4 cal/mol by taking into account long range correlations due to phonons. The elastic constants and the Debye temperature are calculated in zero and second order. The lowering of the bulk modulus due to the second order is about 10%. Experiments agree quite well with the second order results.     

Temporal relaxation of plasma electrons acted upon by directcurrent electric and magnetic fields

On the basis of the time-dependent electron Boltzmann equation the temporal relaxation of the electrons in the presence of electric and magnetic fields in weakly ionized, collision dominated plasmas has been studied. The relaxation process is treated by using a strict time-dependent two-term approximation of the velocity distribution function expansion in spherical harmonics. A new technique for solving the time-dependent electron kinetic equation in this two-term approximation for arbitrary angles between the electric and magnetic fields has been developed and the main aspects of the efficient solution method are presented. Using this new approach and starting from steady-state plasmas under the action of time-independent electric fields only, the impact of superimposed DC magnetic fields on the electron relaxation is analyzed with regard to the control of a neon plasma. The investigations reveal an important effect of the magnetic field on the temporal relaxation process. In particular, it has been found that the relaxation time of the electron component with respect to the establishment of steady-state can be enlarged by some orders of magnitude when increasing the magnetic field strength     

Two Electron States in a Thin Spherical Nanolayer: Reduction to the Model of Two Electrons on a Sphere

Two electron states in a thin spherical nanolayer are discussed. Adiabatic approach is used to divide the system to fast (radial) and slow (angular) subsystems. This leads the Coulomb interaction to be dependent on angular variables, more precisely, on the relative angle between electrons. Approximated Coulomb interaction potential is discussed. Analytical solutions for angular part of Schrödinger equation as well as for energy spectrum for the case of harmonic approximation are obtained. Also the first order of correction energy is discussed by using perturbation theory. For the ground state an analytical expression for the first order correction energy dependent on effective radius of the nanolayer is obtained. Obtained results are compared with exact Coulomb interaction model presented by Loos and Gill.