Coulomb Wave Functions Research Articles

High-energy double photoeffect and photoionization with excitation from 2 1S and 2 3S states of helium-like ions

Double ionization and ionization with excitation of helium-like ions with Z>>1 from 2 1S and 2 3S states on the absorption of a high-frequency photon have been considered. The analytical calculation is performed in the non-relativistic photon energy range in the lowest order of perturbation theory in the inter-electron interaction. Coulomb wavefunctions and the Coulomb Green function are used as a zeroth-order approximation. Differential and total cross sections of the processes are expressed via the corresponding values for the single photoionization. The photoelectron energy spectrum is obtained in the marginal energy range (i.e. for p1>>p2, p1 and p2 momenta of photoelectrons) for the double-ionization process. Simple relations between the cross sections of double ionization and ionization with excitation are derived.

Two Centre Coulomb Continua and the Multiply Differential Cross Section of the Double Ionisation of H2 by Electron Impact

The multiply differential cross section of the double ionisation of hydrogen molecule by electron impact, with coincidence detection of the two ejected and the scattered electron is determined by the application of a product of two two-centre Coulomb (TCC) wave functions describing the two ejected electrons in a first Born method. lt is shown that the double ionisation of H2 is more probable than that of a helium atom, in similar incidence and ejection conditions. The influence of the orientation of the internuclear axis as well as the role of the momentum transfer and the shared recoil momentum between the two nuclei are studied to find the optimal conditions for experimental detection.

Coulomb corrections to the three-body correlation function in high-energy heavy ion reactions

Starting from an asymptotically correct three-body Coulomb wave-function, we determine the effect of Coulomb final state interaction on the three-particle Bose-Einstein correlation function of similarly charged particles. We numerically estimate that the Riverside approximation is not precise enough to determine the three-body Coulomb correction factor in the correlation function, if the characteristic HBT radius parameter is 5–10 fm, which is the range of interest in high-energy heavy ion physics.

The multichannel Coulomb scattering theory and its applications to (e,2e) reactions

AbstracL The goal of this paper is to give a summary of methods of the rigorous multichannel Coulomb scattering theory and to formulate consistent approximations for the Coulomb wave functions and the Coulomb scattering amplitudes, in particular, for the (e,2e) reactions.

Role of electron-electron correlation in (e,2e) reactions

For the last several years, there has been significant attention paid to examining the role of final state correlation in (e,2e) collisions. Numerous calculations have been performed using wavefunctions that not only are asymptotically correct but that were also designed to incorporate other final state physical effects. While these elaborate wavefunctions gave improved agreement with experiment in some cases, none were consistently better than the original three coulomb wavefunction of Redmond, which was used by Brauner, Briggs and Klar in 1989. In al1 of these calculations, initial state correlation were ignored. In this paper, we will report on the importance of initial state correlation on the (e,2e) scattering process.

Ratio between π+π−-atom and free π+π− pair production rates with account of the strong interaction in final states

The ratio of π + π −-atom production rate to free π + π − pairs is obtained with account of the strong interaction in final states in the frame of the perturbation theory. It is shown that this ratio is expressed via a squared ratio of the well-known Coulomb wave functions and thus can be calculated with a high accuracy.

Analysis of correlation data ofπ+π−pair in thep+Tareaction using Coulomb wave function including momentum resolution and the strong interaction effect

We have proposed a new method for the Coulomb wave function correction which includes the momentum resolution for charged hadron pairs and applied it to the precise data on pi^+pi^- correlations obtained in p + Ta reaction at 70 GeV/c. It is found that interaction regions of this reaction (assuming Gaussian source function) are 9.8+-5.8 fm and 7.7+-4.8 fm for the thicknesses of the target 8 micro m and 1.4 micro m, respectively. We have also analyzed the data by the formula including strong interaction effect. The physical picture of the source size obtained in this way is discussed.

Influence of quantum transitions in the continuum on ionization of atoms in strong fields

The tight-binding method is used to analyze the ionization of a hydrogenlike atom by an intense monochromatic laser field. The orthogonal and normalized basis in which the solution of the time-dependent Schrodinger equation is expanded contains unperturbed wave functions of the discrete spectrum and generalized Coulomb wave functions of the continuum. In the solution of the coupled equations we make use of the fact that the bound-free and free-free transitions are efficient in different regions of complex time. Simplified equations are constructed and investigated. Results of calculations for ionization of a hydrogen atom from its ground state and of the energy distribution of the electrons in strong and superstrong linearly polarized fields are presented. It is shown that in this case the ground state decays completely, and free-free transitions play a defining role in the dynamics of the process. It is established that the total probability of population of the upper Rydberg states abutting the continuum does not exceed 0.05. The range of applicability of the approach is discussed. A comparison with numerical results obtained by other authors is given.

Total Born cross sections for positronium and helium atom scattering

A thorough and systematic study has been performed for positronium (Ps) and helium (He) atom scattering using the first Born approximation. As a first step, the direct scattering has been studied. Here excitations of both the Ps atom and He atom, the ionization of the Ps atom with all the excitations of the He atom, the ionization of the He atom with all the excitations of the Ps atom and the ionizations of both the Ps and He atoms are taken into consideration. For the different states of the He atom, the wavefunctions suggested by Winter and Lin [Phys. Rev. A 12 (1975) 434] are used. For the ionization channels the continuum Coulomb wavefunctions have been taken for the ionized electrons.

Electron angular distributions in double photoionization: Use of effective Sommerfeld parameters

We present calculations of the fivefold differential cross-section (FDCS) for double photoionization of helium at excess energies of 6 and 20 eV above threshold. Our results are obtained using for the final double-continuum state a product of three Coulomb wave functions, with the Sommerfeld parameters modified to describe the strength of interaction of any two particles affected by the third particle. Our calculations are compared with recent absolute measurements by Dorner et al. (Phys. Rev. A 57, 1074 (1998)), both in coplanar and non-coplanar geometries. Very good agreement is obtained for the shape of the angular distributions, and differences in the absolute magnitude exist in comparison with the standard choice of Sommerfeld parameters.

Automatic Computation of Zeros of Bessel Functions and Other Special Functions

An efficient algorithm is developed for the automatic calculation of the zeros of Bessel functions of arbitrary order, Coulomb wave functions, and other special functions. Remarkably, this can be done without any actual functional evaluations. Limitations and generalizations are discussed.

Multipole moments for Coulomb and oscillator wavefunctions and the Heun equation

A new method for calculating multipole moments with eigenfunctions related to Sturm-Liouville problems is proposed. The method is based on an auxiliary third-order equation and its Laplace transform. The multipole moments for Coulomb and oscillator problems in quantum mechanics are calculated as an application of this method. The relation between the multipole moments for these problems and equations of Heun type is observed.

A Continuum Correlated Wave Model for Ion–Atom Collisions

We present analytical calculations of transition matrices in ion-atom ionizing collisions. The final correlated state is represented by a series expansion in terms of two-body Coulombic wave functions. We study the particular case when this series is the multivariable confluent hypergeometric function Φ2. Transition matrices are also represented by a series using undistorted and distorted wave approaches for the initial state. We show that this series is strongly convergent and analyze the contribution of their different terms to the double differential cross sections within the undistorted approach.

Semiclassical limit of the scattering Coulomb wave function

We present an alternative derivation of the semiclassical limit of the scattering Coulomb wave function. The result is expressed as a sum over classical paths. Each contribution is factorized in two terms, namely the square root of the classical particle density and a phase factor corresponding to the classical action.

K-shell photoionizations in strongly coupled plasmas

In strongly coupled plasmas, the photoionizations from the 1s ground state of hydrogenic ions are investigated using the ion-sphere model potential. The initial bound wave function and energy eigenvalue of the target ion are modified in the ion-sphere potential using the Ritz variation method. The final state of the ejected photoelectron is represented by a plane wave solution. The Coulomb correction is considered using the screened Coulomb wave function with the appropriate effective charge. The K-shell photoionization cross section is obtained for the acceleration form of the matrix element because of the convenience of the calculations. The plasma screening effect is obtained as a function of the ion-sphere radius and photon energy. The retardation correction is also considered in obtaining the total K-shell photoionization cross section in strongly coupled plasmas. The plasma screening effects on the total K-shell photoionization cross section for the ion-sphere radius, RZ⩾2aZ, are found to be less than about 25%. It is also found that the plasma screening effect is almost independent of the incident photon energy.

Plasma screening effects on photoionizations in weakly coupled plasmas

Plasma screening effects on photoionizations from the 1s state of hydrogenic ions in dense weakly plasmas are investigated. The particle interaction potential in weakly coupled plasmas is obtained by the static Debye-Hückel model. The initial bound wave function and energy eigenvalue of the target ion are modified in the Debye-Hückel potential using the Ritz variation method. The final state of the ejected photoelectron is represented by a plane wave solution. The Coulomb correction is considered using the screened Coulomb wave function. The photoionization cross section is obtained the acceleration form of the matrix element in order to investigate the plasma screening effects on the interaction potential. The screening effect is obtained as a function of the Debye length and photon energy. The retardation correction effect is also considered in obtaining the total 1s photoionization cross section in plasmas. The plasma screening effects on the photoionization cross section for the interesting domain of the Debye length, Λ⩾10aZ, are found to be less than 10%. It is also found that the plasma screening effect is almost independent of the incident photon energy.

Inner-shell contribution to electron capture in positron-alkali collisions

Positronium formation through electron capture from ground state alkali targets is studied theoretically at intermediate and high impact energies by means of a continuum distorted-wave approximation. In this approximation, distortions in the final state are introduced by using two Coulomb wavefunctions associated with the positron- and electron-residual target interactions. Total cross sections for lithium and potassium targets are computed. An estimation of the capture from the inner shell of lithium and potassium is given. Comparison with other theoretical results and experiments is made. The contribution from the inner shells becomes important as the impact energy increases and seems to be necessary to describe the measured cross sections.

Influence of Coulomb boundary conditions for (e, 2e) processes on the K-shell of high- Z atoms

We study the relevance of physical boundary conditions for relativistic (e, 2e) processes on the K-shell of high-Z atoms. The relativistic distorted-wave Born approximation is extended to include the long-range Coulomb interaction in the final channel, allowing the two outgoing electrons to escape in the field of a singly charged positive ion. Besides the physical description we also construct the first fully relativistic Coulomb-Born approximation for (e, 2e) processes, which is the appropriate one to assess the validity of calculations based on semirelativistic Coulomb wavefunctions. The theoretical investigation is illustrated with calculated triple-differential cross sections for (e, 2e) processes on silver and gold, in both coplanar symmetric and asymmetric geometries. The comparison of the results obtained in the modified relativistic distorted-wave Born approximation with previous calculations indicates that the inclusion of asymptotic physical boundary conditions has only a limited effect. On the other hand, the fully relativistic Coulomb-Born approach is seen to yield results in very good agreement with the absolute experimental data, and demonstrates how the use of approximate semi-relativistic wavefunctions can be misleading.

A THEORETICAL STUDY ON ABSOLUTE TRIPLE DIFFERENTIAL CROSS SECTIONS FOR NEAR-THRESHOLD He+(e,2e) REACTIONS

A correlated three-body continuum wave function proposed by Brauner et al. has recently been modified further to allow for the three-body effects by introducing Sommerfeld parameters (Berakdar and Briggs),it is used to calculate the triple differential cross sections for electron impact ionization of He+ in coplanar equal-energy sharing and symmetric geometry. The initial channel wave function involves a Coulomb wave function due to the long range Coulomb attraction between the incident electron and the screened ionic nucleus. Since the present work deals with symmetric collisions, the electron exchang effect between the two continuum electrons has been taken into account properly. It is shown that the initial channel Coulomb field play a major role in determining the triple differential cross sections at near-threshold incident energy and small impact parameter values.

Two Coulomb waves theory for direct excitation

A theoretical model of Dewangan, in which the total scattering wave function is approximated by a distorted wave containing two Coulomb wave functions, is discussed and its relation with the Brauner-Briggs-Klar model for ionization is examined. An important feature of the theory is that it includes a second Born amplitude naturally and in addition, contains, albeit approximately, both real and imaginary parts of all higher order Born terms. The theory is applied to study the 1s→2s excitation of hydrogen by electrons in the energy range 54.4 to 400eV. The differential and integral cross sections predicted by the theory are compared with the results of other theories and experimental data at 54.4eV and a good agreement is found.