Continued Fraction Method Research Articles

AERODYNAMIC PARTICLE SIZE MEASUREMENT BY ELECTRODYNAMIC OSCILLATION TECHNIQUES

Four dynamical techniques for measuring the aerodynamic size of spherical and non-spherical aerosol particles are evaluated and compared. For spheres the classical method of angular light scattering is also included. One method based on particle dynamics, the so-called springpoint method, has been extensively used, but the other particle oscillation methods have been used rarely or not at all. All the dynamical methods involve imbalancing a particle in an electrodynamic balance (EDB) by changing the dc potential to produce particle oscillation. A linescan CCD camera and associated electronics were used to measure the amplitude of the oscillations, the offset of oscillation centers, and the phase lag relative to the ac drive. These measurements are compared with theoretical solutions of the equation of particle motion to establish the aerodynamic size of the particle. The stability characteristics of the particle are analyzed by solving the particle equation of motion using the method of continued fractions. The various techniques are compared for spheres, spheroids and crystalline or amorphous particles of irregular shape. All five methods are shown to be in good agreement for spheres (within 3.9%). For non-spherical germanium dioxide particles the three oscillation methods agree with the springpoint method within 3.4%.

Thermodynamic properties of the periodic nonuniform spin- [formula omitted] isotropic XY chains in a transverse field

Using the Jordan–Wigner transformation and the continued-fraction method we calculate exactly the density of states and thermodynamic quantities of the periodic nonuniform spin- 1 2 isotropic XY chain in a transverse field. We discuss in detail the changes in the behaviour of the thermodynamic quantities caused by regular nonuniformity. The exact consideration of thermodynamics is extended including a random Lorentzian transverse field. The presented results are used to study the Peierls instability in a quantum spin chain. In particular, we examine the influence of a nonrandom/random field on the spin-Peierls instability with respect to dimerization.

The stress state of a viscoelastic orthotropic half-plane loaded with a concentrated force

The operator continued-fraction method is used to solve the problem on the stress state of a viscoelastic orthotropic half-plane loaded at a momentt=0 with a force normal to the boundary of the half-plane. The stress field in the half-plane is determined based on the Volterra principle and the solution of the corresponding elastic problem. The influence of the rheological parameters on the stress state of the half-plane is shown by an example of a specific material.

Crack resistance of a viscoelastic orthotropic body under plane strain

A problem of the linear theory of viscoelasticity for an orthotropic body with a crack under plane strain is solved. The expression for the viscoelastic opening of the crack is obtained based on the Volterra principle and the method of continued fractions. The safe load is evaluated. The effect of the degree of material anisotropy on the safe-load domain is shown by examples of specific materials.

Delayed fracture of a transversally isotropic viscoelastic material with a crack under a cyclic load

The delayed fracture of a transversally isotropic viscoelastic material due to slow subcritical growth of a flat circular macrocrack of normal separation under a cyclic load is investigated. The analysis is based on the modified δC of fracture and the hypothesis of the constancy of the prefracture zone. The study is made within the framework of the Boltzmann-Volterra theory for bounded resolvent operators of difference type, which describe the transversal isotropy of the strain properties of the material. To determine the analytic form of the kernel for the irrational function of a linear combination of the above-mentioned integral operators, the method of continued fractions is used. Analytical and numerical calculations are carried out for the bounded resolvent operators of difference type with the kernel in the form of Rabotnov's fractional exponential function.

Two novel approaches to the Kramers rate problem in the spatial diffusion regime

At present, there are two general theoretical approaches to calculating the rate of thermally activated escape of a Brownian particle over a barrier out of a metastable well in the spatial diffusion regime. A direct approach involves techniques entirely based on the underlying Fokker–Planck equation, such as the Kramers flux over population method, the mean first passage time formalism, and the eigenmode expansion. An alternative consists of replacing the original one-dimensional stochastic dynamics by an infinite dimensional Hamiltonian system. The rate is then calculated using reactive flux methods. Both approaches are rather efficient when treating bistable potentials with high parabolic barriers. However, complications arise if the barrier is not parabolic. In such a case, large deviations of theoretical predictions from exact numerical rates are observed in the intermediate friction region. The latter holds true even though the barrier is infinitely high, to say nothing of low barriers for which the problem of finite barrier height corrections remains effectively unresolved. Based on the expansion of the Fokker–Planck equation in reciprocal powers of the friction coefficient, two novel methods for calculating analytically the rate of escape over an arbitrarily shaped barrier are presented. These are a continued fraction expansion method and a self-similar renormalization technique developed recently for summation of divergent field-theoretical series, respectively. In this way, two different rate expressions are constructed that agree in the limiting case of high friction with the rate following from the corresponding Smoluchowski equation and reduce to the transition state theory rate at zero damping. Comparison with a known rate expression for a purely parabolic barrier and from numerical simulations for bistable potentials with cusped and smooth barriers of different heights show excellent agreement between the present theories and exact numerical results. As long as the escape dynamics is dominated by spatial diffusion across the barrier top, the maximal relative errors attained with the continued fraction method and the self-similar renormalization technique are less than 3% and 7%, respectively. This is in drastic contrast to known rate formulas derived by other means, whose relative errors are larger by factors and even by orders of magnitude.

Transient nonlinear dielectric relaxation and dynamic Kerr effect from sudden changes of a strong dc electric field: Polar and polarizable molecules

The nonlinear transient response of polar and polarizable particles (macromolecules) diluted in a nonpolar solvent to a sudden change both in magnitude and in direction of a strong external dc field is considered. By averaging the underlying Langevin equation, the infinite hierarchy of differential-recurrence equations for ensemble averages of the spherical harmonics is derived for an assembly of polar and anisotropically polarizable molecules pertaining to the noninertial rotational Brownian motion. On solving this hierarchy, the relaxation functions and relaxation times appropriate to the transient dynamic Kerr effect and nonlinear dielectric relaxation are calculated. The calculations are accomplished using the matrix continued fraction method, which allows us to express exactly the solution of the infinite hierarchy of differential-recurrence relations for the first- and second-order transient responses of the ensemble averages of the spherical harmonics (relaxation functions). The results are then compared with available experimental data and solutions previously obtained for various particular cases.

Fokker-Planck dynamics in a bistable periodic potential

Dynamic properties of Brownian particles immersed in a periodic potential with two barriers V1 and V2 (symmetric bistable potential) are studied by using the Fokker-Planck equation which we solve numerically by the matrix continued fraction method. This study will therefore serve to demonstrate the influence of this form of potential, which is of great interest for superionic conductors and for many other solid systems, on the diffusion process. Thus, we have calculated the full width at half maximum (FWHM) \(\lambda (q)\)) of the quasi-elastic line of the dynamic structure factor, for a large range of values of the wave-vectors q. Our results show clearly that, by varying the ratio of the barriers \(\Delta = {V_2}/{V_1}\)strictly between and 1, the Fokker-Planck equation describes a diffusive process which has some characteristic of jump and liquid-like regimes. While in the limit cases, i.e. when Δ tends to or 1, the diffusion process can be described only by a simple jump motion. However, the jump-lengths corresponding to each limit case are not equal. In general the change of the ratio is found to have a significant effect on the character of the diffusive motion. We have also performed Fokker-Planck dynamics calculations of the diffusion coefficient in a bistable potential. We have found a good agreement between numerical calculations and analytical approximation results obtained in the high friction limit.

The imprint of the equation of state on the axial w-modes of oscillating neutron stars

We discuss the dependence of the pulsation frequencies of the axial quasi-normal modes of a nonrotating neutron star upon the equation of state describing the star interior. The continued fraction method has been used to compute the complex frequencies for a set of equations of state based on different physical assumptions and spanning a wide range of stiffness. The numerical results show that the detection of axial gravitational waves would allow to discriminate between the models underlying the different equation of states, thus providing relevant information on both the structure of neutron star matter and the nature of the hadronic interactions.

Electronic excitation of the a3 ∑ g+ and c3Πu states of H 2 by electron impact using the method of continued fractions

In the present work we use a multichannel extension of the method of continued fractions (MCF) to study the low-energy electron-impact excitation in linear molecules at the two-state close-coupling level. In particular we have calculated the excitation cross sections for the X 1 ∑ g + → a 3 ∑ g + and X 1 ∑ g + → c 3 Π u transitions in H 2 for incident energies from near threshold to 30 eV. In our work, in contrast with the early two-state studies of Chung and Lin, no orthogonality constraint between the bound and continuum orbitals is imposed and the one-electron exchange terms are explicitly considered. In general our calculated cross sections are in good agreement with the results of the two-state Schwinger multichannel method. Our results for the a 3 ∑ g + are also in quite good agreement with the available experimental data. However, the cross sections for excitation of the c 3 Π u state differ significantly from the measured values at 20 and 30 eV, where they are available. The disagreement is attributed to the multichannel effects, not accounted for in this study.

Does stochastic resonance occur in periodic potentials?

The possibility of observing stochastic resonance (SR) in a periodic potential without a static bias, driven by noise and an oscillating field, is investigated in both limits of damping. It is shown from the matrix continued fraction method that, although an underdamped Brownian particle moving in a periodic potential displays a dynamical resonance facilitated by noise, there is no conventional SR irrespective of damping. The reason such SR cannot be observed in an overdamped regime is demonstrated by a hopping model. Due to the unbound motion in the periodic potential, transition probability decays algebraically and there is no persistent synchronized hopping. However, the noise-induced enhancement of the diffusion constant related to escape rate enhancement exhibits an SR-like behavior. A comparison with the dynamics on a circle as a different example of an unbound system is given.

Influence of hyperpolarizabilities on nonlinear Kerr effect relaxation responses

The nonlinear effects arising from the first and second hyperpolarizabilities on the Kerr effect relaxation response of an ensemble of polar and anisotropically polarizable molecules in a strong dc electric field are studied by solving the infinite hierarchy of differential-recurrence relations underlying the noninertial Fokker–Planck (Smoluchowski) equation. The step-on solution is obtained by using the matrix continued fraction methods. An expression for the Kerr effect relaxation time is derived as well following the procedure of Kalmykov, Déjardin, and Coffey [Phys. Rev. E 55, 2509 (1997)] which is in full agreement with the matrix continued fraction expression. For small values of the electric field, it is shown that the first hyperpolarizability contributes to increasing the Kerr effect relaxation time while the second hyperpolarizability shifts the maximum of the relaxation time toward the lowest electric fields on increasing the value of this maximum. The dual nature of essentially two relaxation processes is also emphasized by plotting the Kerr effect relaxation spectra for various values of the first and second hyperpolarizability parameters.

Rippled relaxation of ordered NiAl(110) surface

The electronic structure of ordered NiAl(110) surface has been calculated using the continued fraction method. The interaction energy between atoms is presented. On the basis of these results, the rippled relaxation, which previously has been observed in LEED experiments for the (110) surface of the NiAl alloy, is discussed.

An analytical approximation to the diffusion coefficient in overdamped multidimensional systems

An analytical approximation for the mobility of an overdamped particle in a periodic multi-dimensional system is presented. Attention is focused on two dimensions (quasi-2D approximation) in the most generic case of a 2D-coupled periodic potential in a rectangular lattice and of a position-dependent friction. The approximation is derived in the framework of the Linear Response Theory by fixing the value of one coordinate and solving the problem of diffusion along the other coordinate as strictly 1D. This is expected to be essentially correct if all the most relevant diffusion paths are straight lines. Two different specific applications have been considered: diffusion in a square egg-carton potential and diffusion in absence of potential in a 2D channel with unsurmountable periodic walls. Exact results are available in literature in the latter case and are obtained in the first case by solving the Smoluchowski equation (matrix continued fraction method). Comparisons with the quasi-2D approximation show that the agreement is excellent for the egg-carton potential but far less satisfying for migration in the 2D periodically shaped channel, characterized by important diffusion paths not being straight lines.

Analytical solutions for the dynamic Kerr effect: Linear response of polar and polarizable molecules to a weak ac electric field superimposed on a strong dc bias field

The infinite hierarchy of differential-recurrence relations for ensemble averages of the spherical harmonics pertaining to the noninertial rotational Brownian motion of an ensemble of polar and anisotropically polarizable molecules in a strong external dc electric field is derived by averaging the underlying Langevin equation. This procedure avoids recourse to the Fokker–Planck equation, the solution of which involves complicated mathematical manipulations. By calculating the Laplace transforms of the relaxation functions for the dynamic Kerr effect of symmetric top molecules, two equilibrium correlation functions are established, thus allowing one to express the corresponding birefringence ac responses by using linear response theory. Exact analytic solutions for the spectra of these correlation functions and relaxation times are first calculated for two limiting cases, namely, pure induced dipole moments and pure permanent moments, using the continued fraction method. The general case where both types of moments are taken into account, is then considered using matrix continued fractions. Furthermore, exact analytical expressions for the Kerr effect relaxation time are also derived in terms of integrals (which are evaluated exactly) and compared with the matrix continued fraction result. Plots of the relaxation time are presented for various values of the parameters ξ and σ characterizing the permanent and the induced dipole moments. Features of the relaxation behavior are emphasized in figures showing the real and imaginary parts of the spectra of the birefringence function. Moreover, Cole–Cole diagrams are presented for various values of ξ and σ in order to see how they deviate from the Debye-like spectra.

Continued fraction method in inverse problem of photothermal diagnostics

The results of inverse problem solution for photothermal diagnostics of inhomogeneous medium are presented. The developed continued fraction method is shown to give fast and effective reconstruction of one-dimensional thermal parameter profiles. The only approximation of the algorithm is connected with discretization needed for numerical analysis. Because of ill-posed nature of the problem Tikhonov's regularization algorithm is used for account of surface temperature measurement errors.The numerical simulation results are also given.

Application of the method of continued fractions to low-energy electron scattering by the hydrogen molecule

The method of continued fractions (MCF) is applied to a systematic study of low-energy electron scattering by the hydrogen molecule. More specifically, we report differential, integral and momentum-transfer cross-sections for elastic e ̄ -H 2 collision in the 1.5–30 eV energy range. Our calculation is carried out using a static-exchange-polarization potential where the non-penetrating dipole-potential approximation is used to describe polarization effects. Calculated cross-sections are compared with the available experimental and theoretical data and show very good agreement at incident energies below the excitation threshold.

Nonlinear electro-optical response. I. Steady state Kerr effect relaxation arising from a weak ac electric field superimposed on a strong dc bias field

The dynamic Kerr effect (excluding inertial effects) of an assembly of both polar and anisotropically polarizable molecules acted on by a strong external dc electric field superimposed on a weak ac electric field is evaluated by starting from the Smoluchowski equation valid in the noninertial limit. The calculation proceeds by expanding the expectation value of the second Legendre polynomial which describes the Kerr effect relaxation, as a power series up to the second order in the small ac field strength, so that frequency components in ω and 2ω exist. This is accomplished using the matrix continued fraction method which allows one to express exactly the solution of the infinite hierarchy of differential-recurrence relations for the first and second order ac responses of ensemble averages of the Legendre polynomials (relaxation functions). In order to illustrate these results, diagrams showing the behavior of the real and imaginary parts of the complex birefringence functions are presented. In particular, the relaxation spectra clearly indicate the differences between the first order response (equivalent to that obtained using linear response theory) and the second order one. We present also the solutions for the particular cases where only either permanent or induced dipole moments are taken into account. The calculation then radically simplifies as the matrix continued fractions now reduce to scalar ones.

Nonlinear electro-optical response. II. Steady state dielectric relaxation in coupled fields

Nonlinear dielectric response of polar and polarizable molecules dissolved in a nonpolar solvent arising from a weak ac field superimposed on a strong dc bias field are established. The general case when both permanent and induced dipole moments are taken into account is considered. By expanding the expectation value of the first Legendre polynomial as a power series in the ac field strength up to the second order, we find two distinct components in ω and 2ω. Matrix continued fraction methods are used in order to express exactly the solution of the infinite hierarchy of the five-term differential recurrence relations for the ensemble averages of the Legendre polynomials (relaxation functions). The spectra of the real and imaginary parts of the first and second order complex dielectric response functions are presented in figures in order to show the essential differences between the linear and the nonlinear dielectric responses. The analytical expressions so obtained can be checked in so far as each component can be separated experimentally.

Detailed study of quasinormal frequencies of the Kerr black hole

We compute the quasinormal frequencies of the Kerr black hole using a continued fraction method. The continued fraction method first proposed by Leaver is still the only known method stable and accurate for the numerical determination of the Kerr quasinormal frequencies. We numerically obtain not only the slowly but also the rapidly damped quasinormal frequencies and analyze the peculiar behavior of these frequencies at the Kerr limit. We also calculate the algebraically special frequency first identified by Chandrasekhar and confirm that it coincide with the $n=8$ quasinormal frequency only at the Schwarzschild limit.