Coulomb Wave Functions Research Articles

Double ionization of the hydrogen chloride molecule: Influence of the target orientation on multiple differential cross sections

We report in this work multiple differential cross sections for the double ionization of isolated hydrogen chloride molecules impacted by electrons. The calculations are performed within the first Born approximation by considering a target molecular state described by means of a single-center molecular wave function. In the initial state, the incident electron is described by a plane wave, while in the final state Coulomb wave functions are used for modeling the interaction between the two ejected electrons and the residual target ion, the scattered electron being described by a plane wave. Additionally, a Gamow factor is introduced to take into account the Coulomb repulsion between the two outgoing particles. The contributions to the various differential cross sections of the four outermost orbitals of the hydrogen chloride molecule, namely, 4σ, 5σ, 2πx, and 2πy, are then reported for specific target orientations and by considering only the case of two electrons ejected from the same molecular orbital. The obtained results point out the effect of the target orientation on both the (e, 3e) and the (e, 3−1e) collisions and clearly evidence the signature of the well-known double ionization mechanisms, namely, the shake-off and the two-step 1 processes for all the cases investigated.

Coulomb wave functions in momentum space

An algorithm to calculate non-relativistic partial-wave Coulomb functions in momentum space is presented. The arguments are the Sommerfeld parameter η, the angular momentum l, the asymptotic momentum q and the ‘running’ momentum p, where both momenta are real. Since the partial-wave Coulomb functions exhibit singular behavior when p→q, different representations of the Legendre functions of the 2nd kind need to be implemented in computing the functions for the values of p close to the singularity and far away from it. The code for the momentum-space Coulomb wave functions is applicable for values of |η| in the range of 10−1 to 10, and thus is particularly suited for momentum space calculations of nuclear reactions. Program SummaryProgram title: libcwfnCatalogue identifier: AEUQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEUQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 864503No. of bytes in distributed program, including test data, etc.: 7178021Distribution format: tar.gzProgramming language: Fortran 90, Fortran 77, Python, make (GNU Make dialect), GNU Bash shell interpreter (available as /bin/bash).Computer: Apple Powermac (Intel Xeon), ASUS K53U (AMD E-350 (Dual Core)), DELL Precision T3500 (Intel Xeon), NERSC Carver (Intel Nehalem Quad Core).Operating system: Linux, Windows (using Cygwin).RAM: less than 512 MbytesClassification: 17.8, 17.13, 17.16.Nature of problem:The calculation of partial wave Coulomb functions with integer l and all other arguments real.Solution method:Computing the value of the function using explicit formulae and algorithms.Running time:Less than 10−3 s.

Coulomb problem in momentum space without screening

Background: The repulsive Coulomb force poses severe challenges when solving the three-body problem for ($d,p$) reactions on intermediate mass and heavy nuclei. Recently, a new approach based on the Coulomb-distorted basis in momentum space was proposed.Purpose: In this work, we demonstrate the feasibility of using the Coulomb-distorted basis in momentum space for calculating matrix elements expected in a wide range of nuclear reactions.Method: We discuss the analytic forms of the Coulomb wave function in momentum space. We analyze the singularities in the Coulomb-distorted form factors and the required regularization techniques. Employing a separable interaction derived from a realistic nucleon-nucleus optical potential, we compute and study the Coulomb-distorted form factors for a wide range of cases, including charge, angular momentum, and energy dependence. We also investigate in detail the precision of our calculations.Results: The Coulomb-distorted form factors differ significantly from the nuclear form factors except for the very highest momenta. Typically, the structure of the form factor is shifted away from zero momentum due to the Coulomb interaction. Unlike the Yamaguchi forms typically used in three-body methods, our realistic form factors have a short high-momentum tail, which allows for a safe and efficient truncation of the momentum grid.Conclusions: Our results show that the Coulomb-distorted basis can be effectively implemented.

Orthogonal polynomials associated with Coulomb wave functions

A new class of orthogonal polynomials associated with Coulomb wave functions is introduced. These polynomials play a role analogous to that the Lommel polynomials have in the theory of Bessel functions. The orthogonality measure for this new class is described in detail. In addition, the orthogonality measure problem is discussed on a more general level. Apart from this, various identities derived for the new orthogonal polynomials may be viewed as generalizations of certain formulas known from the theory of Bessel functions. A key role in these derivations is played by a Jacobi (tridiagonal) matrix JL whose eigenvalues coincide with the reciprocal values of the zeros of the regular Coulomb wave function FL(η,ρ). The spectral zeta function corresponding to the regular Coulomb wave function or, more precisely, to the respective tridiagonal matrix is studied as well.

Hall viscosity of hierarchical quantum Hall states

Using methods based on conformal field theory, we construct model wave functions on a torus with arbitrary flat metric for all chiral states in the abelian quantum Hall hierarchy. These functions have no variational parameters, and they transform under the modular group in the same way as the multicomponent generalizations of the Laughlin wave functions. Assuming the absence of Berry phases upon adiabatic variations of the modular parameter τ, we calculate the quantum Hall viscosity and find it to be in agreement with the formula, given by Read, which relates the viscosity to the average orbital spin of the electrons. For the filling factor ν=2/5 Jain state, which is at the second level in the hierarchy, we compare our model wave function with the numerically obtained ground state of the Coulomb interaction Hamiltonian in the lowest Landau level, and find very good agreement in a large region of the complex τ plane. For the same example, we also numerically compute the Hall viscosity and find good agreement with the analytical result for both the model wave function and the numerically obtained Coulomb wave function. We argue that this supports the notion of a generalized plasma analogy that would ensure that wave functions obtained using the conformal field theory methods do not acquire Berry phases upon adiabatic evolution.5 MoreReceived 8 January 2014Revised 26 February 2014DOI:https://doi.org/10.1103/PhysRevB.89.125303©2014 American Physical Society

Ionization of water molecules by ion beams. On the relevance of dynamic screening and the influence of the description of the initial state

Single ionization from water molecules by impact of protons, alpha particles and C6 + ions is studied. The post- and prior-versions of the continuum distorted wave-eikonal initial state (CDW-EIS) model within an independent electron approximation are employed to compute double differential cross sections. To avoid the complexity of using numerical molecular continuum states in the cross-section calculations, effective Coulombic continuum wavefunctions are employed. However, this may lead to the appearance of post–prior discrepancies and this fact is examined in detail. Moreover, the influence of the dynamic screening on this behaviour is studied. In addition, the contribution of different molecular orbitals to the angular spectrum is analysed for several ejection electron energies. Finally, the sensitivity of CDW-EIS calculations to the representation of the initial bound molecular orbitals is investigated.

On the definite integral of two confluent hypergeometric functions related to the Kampé de Fériet double series

We study the Kampe de Feriet double series \( F_{1:1;1}^{1:1;1 } \) through the solution to the associated first-order nonhomogeneous differential equation. We show that the integral of tβ+lM(⋅; β; λt)M(⋅; β;−λt) over t ∈ [0, T], T ⩾ 0, \( l=0,1,\ldots,\Re \beta +l>-1 \), is a linear combination of functions \( F_{1:1;1}^{1:1;1 } \). The integral is a generalization of a class of the so-called Coulomb integrals involving regular Coulomb wave functions.

On Recent Analytical Results for Solution of the Scattering Problem for the Sharply Screened Coulomb Potential

Exact representations for the wave function and for the Green function of the Hamiltonian with the sharply screened Coulomb potential are given. The representations are obtained by summing up the partial wave series. The final form of the wave function and the Green function in the region of the coordinate space where the potential is not zero are given in terms of the Coulomb wave function and the Coulomb Green function, respectively. The exact representation has been obtained for the transition operator in the configuration space.

Electron-induced double ionization of isolated water molecules

Theoretical doubly differential and total cross sections for electron-induced double ionization of oriented water molecules are reported here. The calculations are performed within the first Born approximation by describing the initial molecular state by means of single-center wave functions. Furthermore, the incident (scattered) electron is described by a plane wave while a Coulomb wave function is used for modeling the two secondary ejected electrons. The contribution of each final state to the double-ionization process is analyzed, namely, in considering target electrons ejected either from similar or from different molecular subshells. Thus, secondary electron energetic distributions as well as total cross sections are reported for particular target configurations, pointing out orientation effects in the double-ionization process.

Nonlinear thermo-inertial stability of thin circular FGM plates

In this research, linear and non-linear stability behaviour of a thin circular FGM plate subjected to the uniform temperature rise and the constant angular velocity loadings is analyzed. Properties of the FGM media are distributed across the thickness based on a power law form. Each property of the metal or ceramic constituents is considered to be the function of temperature based on the Touloukian model. General equilibrium equations for such conditions are obtained based on the classical plate theory. At first, the non-linear governing equations are established in a complete asymmetrical form. After that, two different analytical methods are presented to study the bifurcation behaviour. Existence of bifurcation phenomenon is examined. Pre-buckling analysis is performed for a plate with the immovable clamped edge. Stability equations are obtained based on the adjacent equilibrium criterion. The resulted equations are solved via the two distinct methodologies, i.e. the exact solution in terms of Coulomb wave functions and the power series method. A non-linear solution is also presented to detect the equilibrium path of the heated rotating FGM plate. It is found that the angular speed may stabilize the homogeneous circular plate which buckles during uniform heating. Furthermore, snapping may occur for FGM plates under the simultaneous action of heating and uniform rotation.

Special functions and spectrum of Jacobi matrices

Several examples of Jacobi matrices with an explicitly solvable spectral problem are worked out in detail. In all discussed cases the spectrum is discrete and coincides with the set of zeros of a special function. Moreover, the components of corresponding eigenvectors are expressible in terms of special functions as well. Our approach is based on a recently developed formalism providing us with explicit expressions for the characteristic function and eigenvectors of Jacobi matrices. This is done under an assumption of a simple convergence condition on matrix entries. Among the treated special functions there are regular Coulomb wave functions, confluent hypergeometric functions, q-Bessel functions and q-confluent hypergeometric functions. In addition, in the case of q-Bessel functions, we derive several useful identities.

Modern methods for calculating photoionization and electron-impact ionization of two-electron atoms and molecules

A number of recently developed methods for calculating multifold differential cross sections for photoionization and electron impact ionization of atoms and molecules with two active electrons are reviewed. The methods are based on unique approaches to the calculation of three-body Coulomb wave functions. The exterior scaling method and the driven Schrodinger equation formalism are considered. The effectiveness of the time-dependent approaches to the scattering problem, such as the paraxial approximation and the time-dependent scaling, is demonstrated. A novel numerical method is formulated, which has been developed by the authors to solve the six-dimensional Schrodinger equation for an atom with two active electrons on the basis of the Chang-Fano transformation and the discrete variable representation. The threshold behavior manifested by the angular distributions of the two-electron photoionization of a negative hydrogen ion and a helium atom and the multifold differential cross sections for the electronimpact ionization of hydrogen and nitrogen molecules are analyzed on the basis of numerical simulations. The Wannier law for the angular distribution of double ionization is demonstrated to be incorrect even at very low energies.

Eigenvalues of the Breit equation

Eigenvalues of the Breit Equation {eqnarray*} [(\vec{\alpha}_{1} \vec{p} + \beta_{1}m)_{\alpha \alpha^{\prime}} \delta_{\beta \beta^{\prime}} + \delta_{\alpha \alpha^{\prime}} (-\vec{\alpha}_{2} \vec{p} + \beta_{2}M)_{\beta \beta^{\prime}} - \frac{e^{2}}{r} \delta_{\alpha \alpha^{\prime}} \delta_{\beta \beta^{\prime}}] \Psi_{\alpha^{\prime} \beta^{\prime}} = E \Psi_{\alpha \beta}, {eqnarray*} in which only the static Coulomb potential is considered, have been found. Here the detailed discussion on the simple caces, $^{1}S_{0},\ m=M$ and $m \neq M$ is given deriving the exact energy eigenvalues. The $\alpha^2$ expansion is used to find radial wave functions. The leading term is given by classical Coulomb wave function. The technique used here can be applied to other cases.

Theoretical study of the two-proton halo candidate 17Ne including contributions from resonant continuum and pairing correlations

With the relativistic Coulomb wave function boundary condition, the energies, widths and wave functions of the single proton resonant orbitals for $^{17}$Ne are studied by the analytical continuation of the coupling constant (ACCC) approach within the framework of the relativistic mean field (RMF) theory. Pairing correlations and contributions from the single-particle resonant orbitals in the continuum are taken into consideration by the resonant Bardeen-Cooper-Schrieffer (BCS) approach, in which constant pairing strength is used. It can be seen that the fully self-consistent calculations with NL3 and NLSH effective interactions mostly agree with the latest experimental measurements, such as binding energies, matter radii, charge radii and densities. The energy of $\pi$2s$_{1/2}$ orbital is slightly higher than that of $\pi1d_{5/2}$ orbital, and the occupation probability of the $(\pi$2s$_{1/2})^2$ orbital is about 20%, which are in accordance with the shell model calculation and three-body model estimation.

Production Mechanism of Pion and Kaon in S + Pb Reaction at 200 GeV/c

The correlation data for identical pions and kaons produced in S+Pb collisions at 200 GeV/c by NA44 Experiment are analysed by theoretical formulae with the long range correlation. It is found that interaction range (the root mean square (rms)) of K−K− is smaller than that of K+K+: Rrms (K−K−) ≈ (5.2 ~ 6 fm) < Rrms(K+K+) ≈ (7 ∼ 7.5 fm). This suggests that in nuclear collisions the anti u-quarks are produced in smaller space region than the u-quarks. Moreover, formulae of the Coulomb wave function correction are presented. The corrections to data of pions and kaons are also discussed, to confirm the above results.

High Z effects in accounting for radiative component of the electron magnetic moment in hydrogen-like atoms

The behavior of electron energy levels in hydrogen-like atoms is studied while taking into account the nonperturbative interaction between the radiative component of the magnetic moment of a free electron Δgfree and the Coulomb field of an atomic nucleus with charge Z, including those with Z > 137. It is shown that for Zα ≪ 1 the energy-level shift is rather effectively determined through the matrix elements of the corresponding Dirac-Pauli operator with relativistic Coulomb wave functions. At the same time, for superheavy nuclei with Z ∼ 170, this shift, generated by Δgfree, is genuinely nonperturbative, behaves like ∼Z5 near the threshold of negative continuum, exceeds all the estimates of radiative corrections coming from vacuum polarization and electron self-energy known so far, and turns out to be at least of the same order as the effects of nuclear charge screening by filled electron shells.

Confluent Heun equations: convergence of solutions in series of coulomb wavefunctions

The Leaver solutions in series of Coulomb wavefunctions for the confluent Heun equation are given by two-sided infinite series, that is, by series where the summation index n runs from minus to plus infinity (Leaver 1986 J. Math. Phys.27 1238). First we show that, in contrast to the D’Alembert test, under certain conditions the Raabe test ensures that the domains of convergence of these solutions include an additional singular point. We also consider solutions for a limit of the confluent Heun equation. For both equations, new solutions are generated by transformations of variables. Finally, we discuss the time dependence of the Klein–Gordon equation in two cosmological models and the spatial dependence of the Schrödinger equation to a family of quasi-exactly solvable potentials. For a subfamily of these potentials, we obtain infinite-series solutions which converge and are bounded for all values of the independent variable, in opposition to a common belief.

Coulombic Transformation in Momentum Space

We studied the Coulombic transformation of potential in momentum space. The Coulombic transformation is defined as a unitary transformation in momentum space, which is equivalent of the Coulomb–Fourier transformation in coordinate space. The analytic continuation scheme avoids the difficulty which is occurred from the singularity of the Coulomb wave function in momentum space. We adopted the point method to perform the analytic continuation. The validity of the new scheme is checked by comparing with the analytic solution for the Malfliet-Tjon potential. Numerical calculation of the integration was done by separating into four intervals. We demonstrate the high accuracy of our calculation.

Solving a Three-Body Continuum Coulomb Problem with Quasi-Sturmian Functions

The scattering problem of three particles interacting via Coulomb poten- tials is studied using generalized parabolic coordinates. The scattering solutions are obtained by solving a driven equation. The 'perturbation' operator appearing in the driven term is the non-orthogonal part of the kinetic energy operator. The approxi- mated solution appearing in the driven term is the product of two two-body Coulomb wave functions. As a test for our proposal, a simple two-dimensional model problem has been solved numerically by using so called parabolic quasi-Sturmian basis repre- sentation. Convergence of the solution has been obtained as the basis set is enlarged.

Binding Energy of D0 and D− Impurity Centers in CdTe/ZnTe Spherical Quantum Dot

Binding energy of neutral (D(0)) and negatively charged (D(-)) hydrogenic impurity located at the center of the CdTe/ZnTe spherical quantum dot has been investigated. Calculations are performed under the effective mass approximation on the basis of exact solution of the Schrödinger and Poisson equations. Eigenfunctions are expressed in terms of the Whittaker and Coulomb wave functions. Calculated results show that D(0) and D(-) impurity binding energies of ground 1s, 1S(1s2) and 2p, 3P(1s2p) excited states strongly depend on CdTe/ZnTe QD size if QD radius is less than one effective Bohr radius.