Coulomb Functions Research Articles

Partial expansion of the two-centre Coulomb Green's function for the eZZ system

The partial expansions of the two-centre Coulomb Green's function in terms of Coulomb spheroidal functions are obtained. Two types of expansions are built for regular and irregular radial Coulomb spheroidal functions in terms of the usual radial Coulomb functions and in terms of the solutions of the confluent hypergeometric equation. For the coefficients in these expansions, relatively simple three-term recurrence relations have been obtained; these relations are finite difference analogues of second-order differential equations. The calculation of the acceptable values νmℓ that single out the minimal solutions of the mentioned finite-difference equations and ensure convergence of the constructed expansions has been implemented by using the continued fractions method.

Laser-assisted Mott scattering in the Coulomb approximation

We investigate the influence of the Coulomb interaction on laser-assisted Mott scattering. This is done by using an approximation in which the initial state of the electron is described by a Coulomb function modulated by the laser in the same way as a relativistic Volkov state. Numerical calculations are carried out for medium field intensity, and compared with the corresponding results obtained within the first-order Born approximation (where Coulomb distortion effects are neglected). It is found that the differential cross sections of the present treatment agree well with the Born predictions at small angles. With increasing scattering angles, however, including the Coulomb interaction enhances the cross section, as compared to the first-Born results. Furthermore, we show that the Coulomb distortion of the electron motion leads to a qualitatively different dependence of the cross sections on the laser frequency at high frequencies. The dependences of the differential cross section on the other field parameters, impact energy and the charge state of the residual ion are also studied.

Stochastic computational modelling of highly heterogeneous poroelastic media with long‐range correlations

AbstractThe compaction of highly heterogeneous poroelastic reservoirs with the geology characterized by long‐range correlations displaying fractal character is investigated within the framework of the stochastic computational modelling. The influence of reservoir heterogeneity upon the magnitude of the stresses induced in the porous matrix during fluid withdrawal and rock consolidation is analysed by performing ensemble averages over realizations of a log‐normally distributed stationary random hydraulic conductivity field. Considering the statistical distribution of this parameter characterized by a coefficient of variation governing the magnitude of heterogeneity and a correlation function which decays with a power‐law scaling behaviour we show that the combination of these two effects result in an increase in the magnitude of effective stresses of the rock during reservoir depletion. Further, within the framework of a perturbation analysis we show that the randomness in the hydraulic conductivity gives rise to non‐linear corrections in the upscaled poroelastic equations. These corrections are illustrated by a self‐consistent recursive hierarchy of solutions of the stochastic poroelastic equations parametrized by a scale parameter representing the fluctuating log‐conductivity standard deviation. A classical example of land subsidence caused by fluid extraction of a weak reservoir is numerically simulated by performing Monte Carlo simulations in conjunction with finite elements discretizations of the poroelastic equations associated with an ensemble of geologies. Numerical results illustrate the effects of the spatial variability and fractal character of the permeability distribution upon the evolution of the Mohr–Coulomb function of the rock. Copyright © 2004 John Wiley & Sons, Ltd.

Charged-particle potential for boron nitrides, silicon nitrides, and borosilazane ceramics: Derivation of parameters and probing of capabilities

A classical pair potential, augmented by three-body interactions, for the modeling of borosilazane ceramics has been derived on the basis of both experimental and ab initio data. The primary goals were a good description of structural parameters and applicability in molecular dynamics. Furthermore, major challenges were to enable bond breaking and to avoid Coulomb collapse during simulations. This has been achieved by long-range, exponentially damped, analytical forms and the inclusion of short-range Coulomb tapering functions. We report on the fitting procedure, discuss the analytical forms employed, and demonstrate the abilities of the potential by comparing to ab initio calculations and experiments.

Resolution of phase ambiguities in electron-impact ionization amplitudes

The formal theory of atomic ionization by electron impact relates the breakup amplitude to an integral expression involving the exact wave function and an unperturbed final state that contains two Coulomb functions with effective, noninteger charges. This integral expression has an associated phase factor that diverges logarithmically on an infinite volume unless the effective charges are chosen to satisfy a kinematic relationship, the so-called ``Peterkop condition.'' We derive the explicit form of the Peterkop phase for two commonly used models of electron-hydrogen ionization, the Temkin-Poet model and the collinear model. We show that the formal theory can be used to identify and remove this physically insignificant, volume-dependent phase from amplitudes computed using numerically stable integral expressions that do not satisfy the Peterkop condition.

Spatial variation in the crustal anisotropy and its temporal variation associated with a moderate-sized earthquake in the Tokai region, central Japan

Spatial and temporal variations in seismic anisotropy in the crust are investigated using earthquakes in the crust and the subducting Philippine Sea Plate beneath the Tokai region, central Japan. We use a total of 351 high-quality waveform data recorded from 1986 December until 1999 August by the microearthquake observation network of the Research Center for Seismology and Volcanology, Nagoya University. The leading shear waves are polarized in an approximately E–W direction, independent of the focal depth, showing consistency with the direction of regional horizontal maximum compressive stress in the Tokai region. The time delay increases for greater focal depth down to 30 km. These results indicate that the regional compressive stress controls anisotropy not only in the upper crust but also in the lower crust. Assuming a uniform distribution of anisotropy, the degree of anisotropy is estimated to be 0.6 per cent in the lower crust. An increase in time delay is found after the Aichi-ken Tobu earthquake (M = 5.7) in 1997 at station STN. This variation is statistically significant and is not an apparent change due to the variation in hypocentre distribution or any kind of artificial effect. On the other hand, no temporal variation in time delay is found at station INU. The spatial pattern of the temporal variation in time delay is consistent with the change in the Coulomb failure function (ΔCFF) caused by both coseismic and post-seismic fault slips. We therefore conclude that the observed temporal change in time delays might be caused by increasing crack density as well as pore fluid pressure due to the static stress change accompanied with the Aichi-ken Tobu earthquake.

Kinetic correlation in the final-state wave function in photo-double-ionization of He

We evaluate the triply differential cross section (TDCS) for photo-double-ionization of helium. We use a final continuum wave function which correlates the motion of the three particles, through an expansion in products of two-body Coulomb functions. This function satisfies a set of appropriate physical conditions in the coalescence points, in addition to the correct asymptotic behavior condition. We analyze the effect of this correlation in the TDCS and compare our results with experimental data.

Thessaloniki–Gerakarou Fault Zone (TGFZ): the western extension of the 1978 Thessaloniki earthquake fault (Northern Greece) and seismic hazard assessment

Active faulting and seismic properties are re-investigated in the eastern precinct of the city of Thessaloniki (Northern Greece), which was seriously affected by two large earthquakes during the 20th century and severe damage was done by the 1759 event. It is suggested that the earthquake fault associated with the occurrence of the latest destructive 1978 Thessaloniki earthquake continues westwards to the 20-km-long Thessaloniki–Gerakarou Fault Zone (TGFZ), which extends from the Gerakarou village to the city of Thessaloniki. This fault zone exhibits a constant dip to the N and is characterised by a complicated geometry comprised of inherited 100°-trending faults that form multi-level branching (tree-like fault geometry) along with NNE- to NE-trending faults. The TGFZ is compatible with the contemporary regional N–S extensional stress field that tends to modify the pre-existing NW–SE tectonic fabric prevailing in the mountainous region of Thessaloniki. Both the 1978 earthquake fault and TGFZ belong to a ca. 65-km-long E–W-trending rupture fault system that runs through the southern part of the Mygdonia graben from the Strymonikos gulf to Thessaloniki. This fault system, here called Thessaloniki–Rentina Fault System (TRFS), consists of two 17–20-km-long left-stepping 100°-trending main fault strands that form underlapping steps bridged by 8–10-km-long ENE–WSW faults. The occurrence of large (M≧6.0) historical earthquakes (in 620, 677 and 700 A.D.) demonstrates repeated activation, and therefore the possible reactivation of the westernmost segment, the TGFZ, could be a major threat to the city of Thessaloniki. Changes in the Coulomb failure function (ΔCFF) due to the occurrence of the 1978 earthquake calculated out in this paper indicate that the TGFZ has been brought closer to failure, a convincing argument for future seismic hazard along the TGFZ.

A novel approach for designing simple point charge models for liquid water with three interaction sites

A simultaneous improvement of the diffusion and dielectric properties of the simple point charge (SPC) model for liquid water appears to be very difficult with conventional reparametrization of the commonly used Lennard-Jones and Coulomb interaction functions and without including a self-energy correction in the effective pair-potential as is done in the SPC/E model. Here, a different approach to circumvent this problem is presented. A short-range interaction term, which corrects the oxygen-oxygen energy at small distances by small amounts of energy, was introduced in the nonbonded interaction function. This additional force-field term allows to derive new parameter sets for SPC-like water models that yield better agreement with experimental data on liquid water. Based on previous investigations of the force-field parameter dependence of the water properties of SPC-like models, the necessary parameter changes to obtain a lower diffusion coefficient and a larger dielectric permittivity were specified and accordingly six new models were developed. They all represent an improvement over SPC in terms of structural and diffusional properties, four of them show better dielectric properties also. One model, SPC/S, has been characterized in more detail, and represents most properties of liquid water better than SPC while avoiding the larger discrepancies with experimental values regarding density, thermal compressibility, energy, and free energy of the SPC/E model. We conclude that the use of a simple, short-ranged additional oxygen-oxygen interaction term makes a simultaneous improvement of the diffusion coefficient and the dielectric properties of water feasible.

A combined symbolic and numerical algorithm for the computation of zeros of orthogonal polynomials and special functions

A Maple algorithm for the computation of the zeros of orthogonal polynomials (OPs) and special functions (SFs) in a given interval [ x 1, x 2] is presented. The program combines symbolic and numerical calculations and it is based on fixed point iterations. The program uses as inputs the analytic expressions for the coefficients of the three-term recurrence relation and a difference-differential relation satisfied by the set of OPs or SFs. The performance of the method is illustrated with several examples: Hermite, Chebyshev, Legendre, Jacobi and Gegenbauer polynomials, Bessel, Coulomb and Conical functions.

Electrostatic Views of Stein-type Estimation of Location Vectors

Stein-type estimation of location vectors is discussed with the aid of the theory of electrostatics. We consider a class of estimating functions and assess the superiority of an estimating equation by its mean squared norm. The Coulomb potential function leads to a Pythagorean relationship with respect to this norm. By making full use of the Pythagorean relationship, we improve upon the likelihood estimating function. A further improvement is shown to be feasible under a certain condition which is described. We pursue possible strong relationships between the superiority over the likelihood estimating function and physical quantities appearing in the theory of electrostatics.

Computing the Zeros and Turning Points of Solutions of Second Order Homogeneous Linear ODEs

Algorithms to compute the zeros and turning points of the solutions of second order ODEs $y^{\prime\prime}+B(x)y^{\prime}+A (x)y=0$ are developed. Two fixed point methods are introduced. The first method consists of fixed point iterations that are built from the first order differential system relating the problem function with a contrast function w; the contrast function w has zeros interlaced with those of y, and it is a solution of a different second order ODE. This method, which generalizes previous findings [J. Segura, SIAM J. Numer. Anal., 40 (2002), pp. 114--133], requires the evaluation of the ratio of functions y/w. The second method is based on fixed point iterations stemming from the second order ODE; it requires the computation of the logarithmic derivative $y^{\prime}/y$. Both are quadratically convergent methods; error bounds are provided. The particular case of second order ODEs depending on one parameter, $y_n^{\prime\prime}+B_n (x) y_n^{\prime}+A_n (x) y_n (x)=0$, with applications to the computation of the zeros and turning points of special functions, is discussed in detail. The combination of both methods provides algorithms for the efficient computation of the zeros and turning points of a broad family of special functions, including hypergeometric and confluent hypergeometric functions of real parameters and variables (Jacobi, Laguerre, and Hermite polynomials are particular cases), Bessel, Airy, Coulomb, and conical functions, among others. We provide numerical examples showing the efficiency of the methods.

Non Resonant Two Photon Ionization of Atomic Hydrogen: Exact Expressions of Angular Distribution

Within the framework of second order time dependent perturbation theory and non relativistic dipole approximation, the exact expressions of the radial transition amplitude from 3d, 3p and 3s excited states, were obtained from the integral representation of the Green Coulomb function. Below threshold and above one photon ionization threshold, it can be expressed as an analytic combination of three Appell hypergeometric functions of first order, one of which is degenerated; while at threshold, it can be obtained in terms of an expansion of Gauss confluent hypergeometric functions. The angular coefficients are numerically evaluated and analysed for some available laser wavelengths.

Relativistic wave and Green's functions for hydrogen-like ions

The Greens library is presented which provides a set of C++ procedures for the computation of the (radial) Coulomb wave and Green's functions. Both, the nonrelativistic as well as relativistic representations of these functions are supported by the library. However, while the wave functions are implemented for all, the bound and free-electron states, the Green's functions are provided only for bound-state energies ( E<0). Apart from the Coulomb functions, moreover, the implementation of several special functions, such as the Kummer and Whittaker functions of the first and second kind, as well as a few utility procedures may help the user with the set-up and evaluation of matrix elements.

Least squares B-spline solutions of the radial Dirac equation in the continuum

The B-spline least-squares approach for the determination of the continuum wavefunctions, which proves to be very accurate and numerically stable in the non-relativistic case has been generalized to the relativistic case. The least-squares scheme has been tested in the free-particle and Coulomb problems. Expression of relativistic Coulomb functions in terms of the corresponding non relativistic ones is spelled out in detail and employed for testing the spline solutions. The convergence properties of the algorithm with respect to the B-spline order and the radial grid are also analyzed.

Molecular three-continuum approximation for ionization ofH2by electron impact

A molecular three-continuum-type approximation is developed to study the $(e,2e)$ reaction for ${\mathrm{H}}_{2}$ targets. The molecular nature of the target is treated within the framework of a two-effective-center approximation. The correlate motion of the particles in the final channel of the reaction is taken into account by an adequate product of Coulomb functions. Triple differential cross sections are computed. A good agreement with the available experiments is obtained.

Analytical approach to electromagnetic processes in loosely bound nuclei: application to 8B

In this Letter we develop an analytical model in order to study electromagnetic processes involving loosely bound neutron-rich and proton-rich nuclei. We construct a model wave function, to describe loosely bound few-body systems, having the correct behaviour both at large and small distances. The continuum states are approximated by regular Coulomb functions. As a test case we consider the two-body Coulomb dissociation of 8B and, the inverse, radiative capture reaction. The difference between using a pure two-body model and the results obtained when incorporating many-body effects, is investigated. We conclude that the interpretation of experimental data is highly model dependent and stress the importance of measuring few-body channels.

Relation between the interaction potential, replacement collision sequences, and collision cascade expansion in iron

The binary collision approximation (BCA) grounded on molecular dynamics results is used to investigate the influence of the range and stiffness of interatomic potentials on the replacement collision sequence (RCS) length and frequency distributions as well as on the displacement cascade expansion and density. Different screened Coulomb potential functions are used in the Marlowe BCA program with suitably adjusted screening lengths. We show in this paper that for screened Coulomb potentials, the shorter the range, the lower the focusing threshold and the more important the RCS production. The cascade expansion and density is quite sensitive to the potential range at high interaction energies. The overall cascade expansion is found to be governed by the 10% highest-energy recoils. Their energy is above the RCS focusing energy threshold. The cascade density, i.e., the number of transient defects produced per unit volume, is suggested sufficient to interfere significantly with RCS propagation and thus with the spatial distribution of Frenkel pairs. Primary damage production thus involves the combined effect of high-energy collisions and RCS production. A careful choice of the short range potential has thus to be made when simulating displacement cascades.

Coulomb functions for attractive and repulsive potentials and for positive and negative energies

The paper gives a review of the mathematical properties of the Coulomb radial wave functions, for attractive and repulsive potentials and for positive and negative energies, together with recommendations on best methods for their computation. It includes discussions of analytic continuations (complex energies, radial co-ordinates and angular momenta) and of relativistic Coulomb functions.

Fgh, a code for the calculation of Coulomb radial wave functions from series expansions

The code fgh is an up-dated version of a code coulfg (see [Seaton, Comput. Phys. Comm. 25 (1982) 87]), used for the calculation of the Coulomb functions f and g, analytic in the energy, for attractive potentials. The new code works for attractive and repulsive potentials and also gives the functions h which have simple asymptotic forms. There is an option to use either the variables ( ϵ, r) customary in atomic physics, or (for positive energies) ( η, ρ) customary in nuclear physics. When ( η, ρ) are used, the code also gives the functions F ℓ( η, ρ) and G ℓ( η, ρ). Use of series solutions can lead to loss of accuracy due to cancellation effects. fgh provides an indication of the number of significant figures lost due to cancellations.