Continuum Distorted-wave Research Articles

Post–prior discrepancies in the continuum distorted wave–eikonal initial stateapproximation for ion–helium ionization

We have explored post–prior discrepancies within continuum distortedwave–eikonal initial state theory for ion–atom ionization. Although there are nopost–prior discrepancies when electron–target initial and final states are exactsolutions of the respective Hamiltonians, discrepancies do arise for multielectronictargets, when a hydrogenic continuum with effective charge is used for the finalelectron–residual target wavefunction. We have found that the prior versioncalculations give better results than the post version, particularly for highlycharged projectiles. We have explored the reasons for this behaviour and foundthat the prior version shows less sensitivity to the choice of the final state.The fact that the perturbation potentials operate upon the initial statesuggests that the selection of the initial bound state is relatively moreimportant than the final continuum state for the prior version.

Ion–Atom/Argon—Calculation of ionization cross sections by fast ion impact for neutral target atoms ranging from hydrogen to argon

A FORTRAN 90 program is presented which calculates the total cross sections, and the electron energy spectra of the singly and doubly differential cross sections for the single target ionization of neutral atoms ranging from hydrogen up to and including argon. The code is applicable for the case of both high and low Z projectile impact in fast ion–atom collisions. The theoretical models provided for the program user are based on two quantum mechanical approximations which have proved to be very successful in the study of ionization in ion–atom collisions. These are the continuum-distorted-wave (CDW) and continuum-distorted-wave eikonal-initial-state (CDW-EIS) approximations. The codes presented here extend previously published codes for single ionization of target hydrogen [Crothers and McCartney, Comput. Phys. Commun. 72 (1992) 288], target helium [Nesbitt, O'Rourke and Crothers, Comput. Phys. Commun. 114 (1998) 385] and target atoms ranging from lithium to neon [O'Rourke, McSherry and Crothers, Comput. Phys. Commun. 131 (2000) 129]. Cross sections for all of these target atoms may be obtained as limiting cases from the present code. Program summary Title of program: ARGON Catalogue identifier: ADSE Program summary URL: http://cpc.cs.qub.ac.uk/cpc/summaries/ADSE Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: none Computer for which the program is designed and others on which it is operable: Computers: Four by 200 MHz Pro Pentium Linux server, DEC Alpha 21164; Four by 400 MHz Pentium 2 Xeon 450 Linux server, IBM SP2 and SUN Enterprise 3500 Installations: Queen's University, Belfast Operating systems under which the program has been tested: Red-hat Linux 5.2, Digital UNIX Version 4.0d, AIX, Solaris SunOS 5.7 Compilers: PGI workstations, DEC CAMPUS Programming language used: FORTRAN 90 with MPI directives No. of bits in a word: 64, except on Linux servers 32 Number of processors used: any number Has the code been vectorized or parallelized?: Parallelized using MPI No. of bytes in distributed program, including test data, etc.: 32 189 Distribution format: tar gzip file Keywords: Single ionization, cross sections, continuum-distorted-wave model, continuum-distorted-wave eikonal-initial-state model, target atoms, wave treatment Nature of physical problem: The code calculates total, and differential cross sections for the single ionization of target atoms ranging from hydrogen up to and including argon by both light and heavy ion impact. Method of solution: ARGON allows the user to calculate the cross sections using either the CDW or CDW-EIS [J. Phys. B 16 (1983) 3229] models within the wave treatment. Restrictions on the complexity of the program: Both the CDW and CDW-EIS models are two-state perturbative approximations. Typical running time: Times vary according to input data and number of processors. For one processor the test input data for double differential cross sections (40 points) took less than one second, whereas the test input for total cross sections (20 points) took 32 minutes. Unusual features of the program: none

Scaling behavior of the fully differential cross section for ionization of hydrogen atoms by the impact of fast elementary charged particles

Ionization of hydrogen atoms by the impact of fast charged particles can be accurately treated theoretically using simple two-center wave functions to describe the scattering system both initially and finally. For the final state, we use a continuum distorted wave (CDW) that contains the product of three Coulombic distortion factors (one for each two-body interaction), hence it is called the 3C wave function. This CDW (3C) wave function is ideal for studying fast collisions since it is asymptotically correct in all asymptotic domains of momentum space for all configurations of the three particles in coordinate space [S. Jones and D. H. Madison, Phys. Rev. A $62,$ 42 701 (2000)]. Coulomb distortion in the entrance channel is provided by an eikonal initial state (EIS), and this CDW-EIS approximation, introduced by Crothers and McCann [J. Phys. B $16,$ 3229 (1983)], has proven to be the most accurate perturbative method ever devised within a two-state approximation. The first fully differential cross sections for ion-atom ionization in the CDW-EIS approximation are presented. In addition, by considering projectiles of different mass and charge, it will be shown that although the charge of the projectile is important, the mass of the projectile plays virtually no role in the vast majority of fast ionizing collisions.

Extended description for electron capture in ion-atom collisions: Application of model potentials within the framework of the continuum-distorted-wave theory

We perform an extension of the continuum-distorted-wave (CDW) approximation for single electron capture by introducing model potentials to describe the interaction of the active electron with the residual-target and projectile ions. The cross sections for electron transfer in collisions of bare and dressed projectile ions with H and He atoms are calculated and compared with experimental data and previous CDW calculations that make use of approximate analytical wave functions to represent the active electron initial and final states. For electron capture in proton-He collisions into definite final states the present calculations are in better agreement with experiments. We trace the differences in the results from both models to a different behavior at projectile scattering angles close to the Thomas peak. In the case of dressed projectiles with different charge states impinging on H and He the present version of CDW gives a better representation of the filling of the unoccupied orbitals of the dressed ion and, therefore, of the cross section as a function of the impinging ion charge state.

Two-center effect on low-energy electron emission in collisions of 1-MeV/u bare ions with atomic hydrogen, molecular hydrogen, and helium. I. Atomic hydrogen

We have investigated ionization mechanisms in fast ion-atom collisions by measuring the low-energy electron emission cross sections in a pure three-body collision involving bare carbon ions $(v=6.35\mathrm{a}.\mathrm{u}.)$ colliding with atomic hydrogen targets. The measurements have also been extended to molecular hydrogen and helium targets. In this paper we provide the energy and angular distributions of double differential cross sections of low-energy electron emission for atomic hydrogen targets. The Slevin rf source with a high degree of dissociation was used to produce the atomic H target. It is found that the two-center effect has a major influence on the observed large forward-backward angular asymmetry. A detailed comparison is presented with calculations based on the continuum distorted-wave (CDW) and CDW-EIS (eikonal initial-state) approximations. Both the continuum distorted-wave calculations provide a very good understanding of the data, whereas the first Born calculation predicts almost symmetric forward-backward distributions that do not agree with the data. The two-center effect is slightly better represented by the CDW calculations compared to the CDW-EIS calculation. The total cross sections are, however, in good agreement with the theories used. The results for molecular hydrogen and helium will be discussed in the following paper.

Two-center effect on low-energy electron emission in collisions of 1-MeV/u bare ions with atomic hydrogen, molecular hydrogen, and helium: II.H2and He

We have studied the energy and angular distributions of low-energy electron emission in collisions of bare carbon ions of 1-MeV/u energy with He and ${\mathrm{H}}_{2}$ targets. The double-differential cross sections (DDCS's) are measured for electrons with energies between 0.5 and 300 eV emitted within an angular range of 15\ifmmode^\circ\else\textdegree\fi{} to 160\ifmmode^\circ\else\textdegree\fi{}. The large forward-backward asymmetry observed in the angular distributions is explained in terms of the two-center effect. Single differential cross sections (SDCS's) and total cross sections are also derived by integrating the DDCS's over emission angles and energies. The data are compared with different theoretical calculations based on the first Born, CDW (continuum-distorted-wave), and CDW-EIS (eikonal-initial-state) approximations. The angular distributions of DDCS's and SDCS's are shown to deviate largely from the predictions of the B1 calculations, and are in much better agreement with both the continuum distorted-wave models. The CDW approximation provides a better agreement with the data compared to the CDW-EIS approximation, especially at higher electron energies. The total ionization cross sections for all three targets are shown to follow a scaling rule approximately.

ION-ATOM/NEON – Calculation of ionization cross sections by fast ion impact for neutral target atoms ranging from lithium to neon

The program presented calculates the total cross sections, and the electron energy spectra of the singly and doubly differential cross sections for the single target ionization of neutral atoms ranging from lithium up to and including neon. The code is applicable for the case of both high and low Z projectile impact in fast ion-atom collisions. The theoretical descriptions provided for the program user are based on two quantum mechanical models which have proved to be very successful in the study of ionization in ion-atom collisions. These are the continuum-distorted-wave (CDW) and continuum-distorted-wave eikonal-initial-state (CDW-EIS) approximations respectively.

Asymptotic expansions for three-body continuum and bound states relevant for ion–atom collision processes

The continuum and bound states of a target electron in the field of a projectile ion are derived within the continuum distorted wave (CDW) approach. Unlike the CDW approximation of Cheshire, in addition we take into account a three-body coupling between the electron–target and electron–projectile subsystems to the first order in a small parameter O(1/R) where R is internuclear separation. It is found that the derived wave functions satisfy the Schrödinger equation up to terms of the second order O(1/R2), and as a consequence, give a better account of the interaction zone. These analytical functions can be used to investigate numerically the first-order (and partially higher-order) three-body coupling effects in the processes of excitation, ionization and charge transfer of an atom by ion impact.

Critical validity assessment of theoretical models: charge-exchange at intermediate and high energies

Exact comprehensive computations are carried out by means of four leading second-order approximations yielding differential cross sections dQ/dΩ for the basic charge exchange process H++H(1s)→H(1s)+H+ at intermediate and high energies. The obtained extensive set of results is thoroughly tested against all the existing experimental data with the purpose of critically assessing the validity of the boundary corrected second-Born (CB2), continuum-distorted wave (CDW), impulse approximation (IA) and the reformulated impulse approximation (RIA). The conclusion which emerges from this comparative study clearly indicates that the RIA agrees most favorably with the measurements available over a large energy range 25 keV–5 MeV. Such a finding reaffirms the few-particle quantum scattering theory which imposes several strict conditions on adequate second-order methods. These requirements satisfied by the RIA are: (i) normalisations of all the scattering wave functions, (ii) correct boundary conditions in both entrance and exit channels, (iii) introduction of a mathematically justified two-center continuum state for the sum of an attractive and a repulsive Coulomb potential with the same interaction strength, (iv) inclusion of the multiple scattering effects neglected in the IA, (v) a proper description of the Thomas double scattering in good agreement with the experiments and without any unobserved peak splittings. Nevertheless, the performed comparative analysis of the above four approximations indicates that none of the methods is free from some basic shortcomings. Despite its success, the RIA remains essentially a high-energy model like the other three methods under study. More importantly, their perturbative character leaves virtually no room for further systematic improvements, since the neglected higher-order terms are prohibitively tedious for practical purposes and have never been computed exactly. To bridge this gap, we presently introduce the variational Padé approximant (VPA) as a novel non-perturbative theory which is valid at all energies. This is a variationally unified T-matrix, T(VPA)i,f=T′i,f+Si,f, comprised of a selected perturbative model T′i,f and a related stationary non-linear remainder Si,f conceived as an L2-basis set expansion of the total Green's function. The key input Si,f is a double series ∑n1,n2Cn1n2 with bound-free atomic form factors as rational coefficients Cn1n2. Convergence of this sum is significantly accelerated by the two-dimensional Padé approximant (2D-PA) implemented through the bi-variate Wynn's epsilon (ϵ) algorithm. The table ϵ(λ) for Si,f(λ) is evaluated at a sufficiently dense grid λ∈Λ≡[λmin,λmax] of any chosen variate λ, e.g. scattering angle, incident energy, coupling strength, etc. The roots λk of the inverse function E(λ)=1/ϵ(λ) directly lead to the poles of Si,f(λ) in the complex λ-plane. The availability of all λk's permits an easy obtaining of each of the magnitudes dϵk=1/E′(λk) of the poles of Si,f(λ) by the Cauchy residue calculus and a mere knowledge of the first derivative E′(λk) of E(λk) at λk∈Λ. This is a new signal processing method called the parametric epsilon spectral (PϵS) estimator which opens an affordable way to powerful numerical investigations of analytical properties of transition and scattering matrices for a general process including resonance phenomena as one of the the most interesting parts of scattering.

Coupled-channel calculations using continuum distorted-waves: capture to n = 3 in proton-hydrogen collisions

A 20-state coupled-channel symmetrized variational continuum distorted-wave (SVCDW) calculation for electron capture in proton-hydrogen collisions is presented. States with a principal quantum number , centred on both the target and the projectile, have been included in the calculation. Total cross sections for electron capture to the 3s, 3p and 3d levels are shown for the energy range 15-150 keV. The SVCDW results are compared with other large-scale coupled-channel calculations and with experiment.

Asymptotic expansions for three-body Coulomb scattering state

A wavefunction possessing the correct asymptotic behaviour in the region of configuration space where two particles are only slightly separated and a third particle is located far away is constructed within the quasiclassical eikonal, WKB and quantum-mechanical approach. This function cannot be represented as a product of independent two-particle distortion functions and in the limiting cases it passes into the continuum-distorted-wave (CDW) function and the asymptotic state of Kunikeev and Senashenko (1996 Sov. Phys.-JETP 82 839). The derived function gives a better account of the interaction zone and is used to investigate a fragmentation process. As an application, the ionization process of an atom by ion impact is considered. The amplitude and the double-differential cross section (DDCS) for the ionization process are developed. The problem to calculate the DDCS is reduced to a three-dimensional integration.

CDW theory of ionization by ion impact with a Hartree-Fock-Slater description of the target

We have carried out a generalization of the continuum-distorted-wave (CDW) model for single ionization of atoms by ion impact where the interaction of the active electron with the target is represented by a model potential. We apply this model to the ionization of He by proton and highly charged bare-ion impact. Doubly differential cross sections are calculated and compared with the available experimental data at high impact energy. Very good agreement is obtained for emission in the forward direction.

LMD — Calculation of longitudinal momentum distributions in the single ionization of helium by ion impact

A program is presented which calculates the final-state longitudinal momentum distributions of the ejected electron and recoil-ion for the single ionization of helium-like targets by both light and heavy-ion impact. The longitudinal momentum distributions are evaluated within a wave treatment. The continuum-distorted-wave (CDW) and continuum-distorted-wave eikonal-initial-state (CDW-EIS) approximations are used throughout. The program considered here extends a previously published program ION (D.S.F. Crothers, M. McCartney, Comput. Phys. Commun. 72 (1992) 288) which calculated the total, singly and doubly differential cross sections for the ionization of hydrogen-like targets by fully stripped projectile ions.

Dynamic electron correlations in single capture from helium by fast protons

We employ the four-body continuum distorted-wave (CDW-4B) method to compute the total cross sections for the single electron-capture process ${\mathrm{H}}^{+}+\mathrm{H}\mathrm{e}{(1s}^{2})\ensuremath{\rightarrow}\mathrm{H}(1s)+{\mathrm{He}}^{+}(1s)$ at incident energies ranging from 20 to 20 000 keV. The purpose of the present work is twofold: (i) to verify whether the CDW-4B approximation for one-electron transfer is as successful as for double charge exchange in fast proton-helium collisions and (ii) to search for evidence of dynamic electron correlation effects as a function of increased projectile energy. The validity of our findings is critically assessed in comparisons with the available experimental data in the quoted impact energy range.

Single ionization of He by ions

The final state longitudinal momentum distributions of the ejected electron, recoil ion and the projectile momentum transfer for are calculated within the continuum distorted-wave (CDW) model at an impact energy of . The results are discussed in the context of recent CDW eikonal-initial state calculations (CDW-EIS) and are compared with available experimental data. The calculated electron energy spectrum and total ionization cross sections within the CDW approximation are in good agreement with both the measurement and the theoretical CDW-EIS calculations.

Longitudinal momentum distributions in single ionization of He by Ni24+ ions.

The longitudinal momentum distributions for the recoil ion and ejected electron in single ionization of He by 3.6 MeV/amu Ni24+ ions are presented using the continuum distorted-wave (CDW-CDW) model. The results are discussed in the context of recent continuum distorted-wave eikonal-initial state (CDW-EIS) calculations [Rodriguez et al., J. Phys. B. 28, L471 (1995)] and compared with available experimental data [Moshammer et al., Phys. Rev. Lett. 73, 3371 (1994)].

Four-body model for transfer ionization in fast ion-atom collisions

Total cross sections for transfer ionization in fast collisions of a bare nucleus with helium are examined in the four-body distorted-wave formalism. A special emphasis is given to a proper inclusion of dynamic electron-electron correlation effects. For this purpose, the four-body continuum distorted-wave (CDW-4B) approximation with the correct boundary conditions is introduced. Along with the appropriate potential operators containing a single electron placed on one and/or both nuclear centers, accompanied by the corresponding Coulomb waves for continuum intermediate states, the dielectronic interaction ${\mathrm{V}}_{12}$=1/${\mathrm{r}}_{12}$\ensuremath{\equiv}1/|r-${\ensuremath{\rightarrow}}_{1}$-r-${\ensuremath{\rightarrow}}_{2}$| also explicitly appears in the perturbation potential of the transition probability amplitude. The inclusion of the potential ${\mathrm{V}}_{12}$ is essential for the description of the Thomas P-e-e scattering, through which one of the target electrons could be captured or ionized without ever experiencing any direct interaction with the projectile P. The total cross sections ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{if}}$ due to such a correlated CDW-4B theory are computed exactly and very efficiently by means of precise evaluation of certain seven-dimensional quadratures in momentum space. The proposed method is shown to be superior to the corresponding independent event model (CDW-IEM), which also proceeds through the same effort on multidimensional scattering integrals, but largely overestimates the measured values for ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{if}}$ . Comparisons between the present results ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{if}}$ and the available experimental data at E=30--600 keV/amu for transfer ionization in the ${\mathrm{He}}^{2+}$ -He collision yield satisfactory agreement at impact energies E\ensuremath{\geqslant}80 keV/amu. This is in full harmony with the well-known low-energy limit of the validity of the CDW-3B method assessed previously for the genuine three-body charge exchange in collisions between a fully stripped projectile and a hydrogenlike atomic target.

A theoretical description of electron capture by from ions using symmetrized variational continuum distorted waves

We present total cross sections in the laboratory energy range for electron capture by protons in collision with . We describe this process theoretically by performing a close-coupling (CC) calculation using the symmetrized variational (SV) continuum distorted-wave (CDW) collision ansatz. We discuss the SVCDW results in conjunction with another CC variational CDW calculation, with CDW perturbation theory, and also with close-coupling calculations which involve a representation of the continuum by a set of pseudo-states which may include pseudo-states of the united atom. Moreover, we compare the SVCDW results with the available experimental data. The SVCDW results are found to be in very good agreement with the experiments of Peart et al, Rinn et al and Watts et al at all energies, and consequently to be at variance with the experimental results of Angel et al. The SVCDW results are also found to be in good accord with a CC calculation involving Sturmian-type pseudo-states. We investigate this agreement further by comparing impact-parameter profiles at 50 and .

Rainbow and glory effects in the autoionization of atoms excited by the impact of heavy ions

When the autoionizing state formed by the impact of a slow positively charged ion decays, the line profile of the emitted electron is affected by the post-collision interaction with the Coulomb field of the outgoing projectile, displaying a strong enhancement in the forward direction. On the other hand, when the autoionization process is induced by a negatively charged projectile, we demonstrate that the electron spectrum displays a depletion in the forward direction and a sharp enhancement at a certain intermediate angle. We show that these structures can be interpreted as forward glory and rainbow effects, respectively. We discuss the similarities and discrepancies between a continuum-distorted wave (CDW) approximation and a classical description of these phenomena.

Third-order continuum distorted-wave double-scattering 1s - 1s transitions

A non-relativistic description of the differential cross section (DCS) for Thomas double-scattering electron capture at asymptotically high velocity is developed within the third-order continuum distorted-wave (CDW) perturbation theory for transitions in proton - hydrogen collisions. It is shown that at the critical proton scattering angles, namely the forward peak, Thomas maximum, small angles and the interference minimum, the CDW series has converged at second order. Moreover, it is proven that the third-order correction makes no contribution to the velocity-dependent and behaviour of the Thomas double-scattering total cross sections at the leading angles. The Oppenheimer - Brinkman - Kramers (OBK) travelling atomic orbital plane wave theory is compared to the CDW approximation. It is observed that the second-order OBK (OBK2) theory has not converged. It remains an open question as to whether fourth-order terms or higher in the OBK approximation contribute to the various differential cross sections. It is concluded that the CDW model gives a superior description of the Thomas double-scattering mechanism than the OBK model.