Solving the Schrödinger equation for a charged particle in a magnetic field using the finite difference time domain method

We extend our finite difference time domain method for numerical solution of the Schrödinger equation to cases where eigenfunctions are complex-valued. Illustrative numerical results for an electron in two dimensions, subject to a confining potential V(x,y), in a constant perpendicular magnetic field demonstrate the accuracy of the method.     

Voigt lineshape function as a solution of the parabolic partial differential equation

The goal of this paper is to show that the Voigt function may be found as a solution of a parabolic partial differential equation, like the heat conduction equation or other diffusion equations. A square of the Gaussian half-width of the Voigt function plays the role of ‘time’ and initial conditions are determined by a Lorentz function. Some questions concerning the practical application of the numerical grid methods for the calculation of the Voigt function are discussed. It is shown, that in some cases the offered calculation algorithm can be both faster and more accurate than other known algorithms.     

Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh

A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier–Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.     

Spectral dependence of the optical absorption temperature coefficient of CdS1−xSex nanocrystallites embedded in a silicate glass

The absorption temperature coefficient of CdS_1−xSe_x nanocrystallites embedded in a silicate glass has been studied in the temperature range above room temperature at different technological regimes and sizes of nanocrystals. To understand the optical properties of silicate glasses with semiconductor nanocrystallites, especially that at the initial stage of their formation, it is necessary to include the structural changes occurring in the nanocrystals during the heat treatment.     

Relativistic quantum particle in a homogeneous external field

The model of the relativistic quantum particle in a homogeneous external field is proposed. This model is realized in the one-dimensional relativistic configurational x-space and is described by the finite-difference equation. The momentum p-space in our case is the one-dimensional Lobachevsky space. We have found the wave functions and propagator for the model under study in both x- and p-representations.     

Two Kinds of Square-Conservative Integrators for Nonlinear Evolution Equations

Based on the Lie-group and Gauss-Legendre methods, two kinds of square-conservative integrators for square- conservative nonlinear evolution equations are presented. Lie-group based square-conservative integrators are linearly implicit, while Gauss-Legendre based square-conservative integrators are nonlinearly implicit and iterative schemes are needed to solve the corresponding integrators. These two kinds of integrators provide natural candidates for simulating square-conservative nonlinear evolution equations in the sense that these integrators not only preserve the square-conservative properties of the continuous equations but also are nonlinearly stable. Numerical experiments are performed to test the presented integrators.     

Silicon nanocrystals in silica—Novel active waveguides for nanophotonics

Nanophotonic structures combining electronic confinement in nanocrystals with photon confinement in photonic structures are potential building blocks of future Si-based photonic devices. Here, we present a detailed optical investigation of active planar waveguides fabricated by Si⁺-ion implantation (400 keV, fluences from 3 to 6×10¹⁷ cm⁻²) of fused silica and thermally oxidized Si wafers. Si nanocrystals formed after annealing emit red-IR photoluminescence (PL) (under UV-blue excitation) and define a layer of high refractive index that guides part of the PL emission. Light from external sources can also be coupled into the waveguides (directly to the polished edge facet or from the surface by applying a quartz prism coupler). In both cases the optical emission from the sample facet exhibits narrow polarization-resolved transverse electric and transverse magnetic modes instead of the usual broad spectra characteristic of Si nanocrystals. This effect is explained by a theoretical model which identifies the microcavity-like peaks as leaking modes propagating below the waveguide/substrate boundary. We present also permanent changes induced by intense femtosecond laser exposure, which can be applied to write structures like gratings into the Si-nanocrystalline waveguides. Finally, we discuss the potential for application of these unconventional and relatively simple all-silicon nanostructures in future photonic devices.     

Compact-2D FDTD for waveguides including materials with negative dielectric permittivity, magnetic permeability and refractive index

An efficient compact-2D finite-difference time-domain method is presented for the numerical analysis of guided modes in waveguides that may include negative dielectric permittivity, negative magnetic permeability and negative refractive index materials. Both complex variable and real variable methods are given. The method is demonstrated for the analysis of channel-plasmon-polariton guided modes in triangular groves on a metal surface. The presented method can be used for a range of waveguide problems that were previously unsolvable analytically, due to complex geometries, or numerically, due to computational requirements of conventional three-dimensional finite-difference time-domain methods. A three-dimensional finite-difference time-domain algorithm that also allows analysis in the presence of bound or free electric and equivalent magnetic charges is presented and an example negative refraction demonstrates the method.     

The two-level element free Galerkin method for MHD flow at high Hartmann numbers

In this Letter, a new element free Galerkin method, namely the two-level element free Galerkin method, is presented for solving the governing equations of steady magnetohydrodynamic duct flow. Because this element free Galerkin method makes use of the nodal point configurations which do not require a mesh, therefore it differs from FEM-like approaches by avoiding the need of meshing, a very demanding task for complicated geometry problems. Another distinguished feature of the proposed method is the resolving capability of high gradients near the layer regions without local or adaptive refinements. Numerical results indicate that no matter how large the Hartmann number is, this method has the ability to produce the satisfactory results for the velocity and the magnetic field simultaneously. That is to say, the presented method has some excellent properties, such as better stability and accuracy.     

Statistical properties of turbulence in a toroidal magnetized ECR plasma

The statistical analyses of fluctuation data measured by electrostatic-probe arrays clearly show that the self-organized criticality (SOC) avalanches are not the dominant behaviors in a toroidal ECR plasma in the SMT (Simple Magnetic Torus) mode of KT-5D device. The f⁻¹ index region in the auto-correlation spectra of the floating potential Vf and the ion saturation current Is, which is a fingerprint of a SOC system, ranges only in a narrow frequency band. By investigating the Hurst exponents at increasingly coarse grained time series, we find that at a time scale of τ>100 μs, there exists no or a very weak long-range correlation over two decades in τ. The difference between the PDFs of Is and Vf clearly shows a more global nature of the latter. The transport flux induced by the turbulence suggests that the natural intermittency of turbulent transport maybe independent of the avalanche induced by near criticality. The drift instability is dominant in a SMT plasma generated by means of ECR discharges.     

Multisymplectic Euler Box Scheme for the KdV Equation

We investigate the multisymplectic Euler box scheme for the Korteweg-de Vries （KdV） equation. A new completely explicit six-point scheme is derived. Numerical experiments of the new scheme with comparisons to the Zabusky-Kruskal scheme, the multisymplectic 12-point scheme, the narrow box scheme and the spectral method are made to show nice numerical stability and ability to preserve the integral invariant for long-time integration.     

An Explicit Scheme for the KdV Equation

A new explicit scheme for the Korteweg~：le Vries （KdV） equation is proposed. The scheme is more stable than the Zabusky Kruskal scheme and the multi-symplectic six-point scheme. When used to simulate the collisions of multi-soliton, it does not show the nonlinear instabilities and un-physical oscillations.     

Finite difference approximations for the fractional advection-diffusion equation

Fractional order diffusion equations are viewed as generalizations of classical diffusion equations, treating super-diffusive flow processes. In this Letter, in order to solve the two-sided fractional advection-diffusion equation, the fractional Crank-Nicholson method (FCN) is given, which is based on shifted Grünwald-Letnikov formula. It is shown that this method is unconditionally stable, consistent and convergent. The accuracy with respect to the time step is of order ²(Δt). A numerical example is presented to confirm the conclusions.     

Photoluminescence from heterogeneous SiGe/Si nanostructures prepared via a two-step approach strategy

A two-step approach of preparation for SiGe/Si heterogeneous nanostructures, which combined with ultra-high vacuum chemical deposition and electrochemical anodization techniques, is demonstrated. Uniformly distributed nanostructures with a quite uniform distribution of size and morphology are obtained. A strong room-temperature photoluminescence from the nanostructures was observed with a narrow full-width at half-maximum of around 110 meV. The possible origins of the two main peaks at around 1.6 and 1.8 eV have been discussed in detail. The two-step approach is proved to be a promising method to fabricate new Si-based optoelectronic materials.     

Two New Simple Multi-Symplectic Schemes for the Nonlinear Schrodinger Equation

We investigate the multi-symplectic Euler-box scheme for the nonlinear Schroedinger equation. Two new simple semi-explicit scheme are derived. A composition scheme based on the new derived schemes is also discussed. Some numerical results are reported to illustrate the efficiency of the new schemes.     

Thermal Diffusion Process Estimation for Fabrication of Graded Plastic Optical Fibre

Thermal diffusion of dopants is investigated in the process of generating the graded-index profile of plastic optical fibres. Because the diffusion coefficient in high polymers has been shown to depend strongly on dopant concentration, it is allowed in this work to vary with the radial coordinate of the multistep-core fibre. A novel multi-layer model is presented for solving the diffusion equation with the variable diffusion coefficient. It is solved by the finite difference method. The solution determines the dopant diffusion profile in the fibre. It is verified against a solution from the literature and two cases of fibres with diffused profiles. The application is demonstrated on two examples of graded-index plastic optical fibres, one originally with a two-step and the other with four-step core. The results indicate that closer to the core-cladding interface, the computed diffused profile with variable diffusion coefficient D is closer to target profile than the profile obtained with constant D for the same time of thermal process.     

Hybrid Lifting Wavelet-Like Transform for Solution of Electromagnetic Integral Equation

A hybrid lifting wavelet-like transform scheme is successfully applied to the solution of electric field integral equation using Rao-Wilton-Glisson basis functions. To speed up the matrix transform process, the lifting scheme is adopted. Numerical examples of different three-dimensional perfectly electric conducting objects are considered. Compared with the method of moments, the proposed matrix transform scheme can save considerable CPU time and memory.     

Substrate effects on ZnO nanostructure growth via nanoparticle-assisted pulsed-laser deposition

The effects of various substrate conditions on the morphology, crystal structure and photoluminescence of ZnO nanostructures synthesized by nanoparticle-assisted pulsed-laser ablation deposition were investigated. It is concluded that the sapphire substrate with a 1 h anneal at 1000 °C is the most favorable to the vertical growth of ZnO nanostructures. SEM analysis indicates that the well-aligned diameter-modulated ZnO nanonails with unique shape were successfully synthesized on the annealed sapphire substrate. The as-synthesized ZnO nanostructures exhibit an ultraviolet emission at around 390 nm and the absent green emission under room temperature, indicating that there is a very low concentration of deep-level defects inside ZnO lattices. The novel ZnO nanostructures could offer novel opportunities for both fundamental research and technological applications.     

Numerical tests of a new molecule-dependent momentum transport equation

It is shown that a new incompressible fluid equation is obtained by inclusion of a new dimensionless coupling parameter in the momentum transport equation derived in [L. Jirkovsky, L. Bo-ot, Momentum transport equation for the fluids using projection-perturbation formalism and onset of turbulence, Physica A 352 (2005) 241-251] from the Boltzmann kinetic equation where the Boltzmann collision integral includes inelastic interactions of quantum origin among the particles of the fluid. Numerical results from the equation for the plane and circular Poiseuille flows are consistent with the observations. The numerical tests also manifest a difference in the onset of turbulence between the flat plates and the circular pipe flow systems. Although all obtained velocity profiles are flattened at the center-a feature of turbulence-the results demonstrate greater stability of the velocity profiles in the circular pipe flow.