Impact Parameter Approximation Research Articles

Ion-molecule charge exchange in Io's extended atmosphere: Velocity dependence

The Io plasma torus is mostly composed of electrons and ions of O and S, singly and multiply-charged. These ions interact with the neutral species of Io's atmosphere (S, O, SO2 and SO) through symmetrical (e.g., O+ + O ⇒ O + O+) and asymmetrical (e.g., S+ + O ⇒ S + O+) charge exchange reactions. Moreover, molecular ions (SO2+, SO+) produced in Io's atmosphere by ionization or charge exchange are accelerated as they are picked up by the local plasma flow and eventually also charge exchange with the atmospheric neutrals (e.g., SO2+ + SO2 ⇒ SO2 + SO2+).Cross sections for many charge exchange reactions were estimated in Johnson and Strobel (1982) and McGrath and Johnson (1989) for ion speeds of 60 km/s appropriate for torus/neutral cloud interactions where the plasma is approximately at corotation in Io's reference frame. But close to Io, the flow velocity and ion temperature are drastically reduced (Vflow < 10 km/s and Tion < 50 eV) and greatly increased on its flanks (∼80 km/s and 400 eV). Thus, velocity-dependent charge exchange cross sections are critical for numerical simulations of the Io plasma/atmosphere interaction.We provide estimates of charge exchange cross sections of ions on molecular neutrals and of molecular ions on neutrals for the range of velocity [5–120] km/s representative of ion velocities at Io. We combine several methods valid at the low and higher velocities in this range: Two-state impact parameter approximation, quasi-thermal rate constant extrapolation, Langevin orbiting cross sections and scaling from experimental data. We describe these methods in details.Our estimates at 60 km/s are 2–3 times smaller or larger than previously published values, depending on the reaction. Most are generally large (∼60–90 Å2) at small velocities. We discuss the most important reactions relevant to Io's interaction and where improvements and experimental values are needed. We anticipate the effect of these velocity-dependent cross sections in numerical simulations of Io's interaction and show that some of these reactions are of critical importance regarding the plasma composition close to Io, the plasma supply to the torus and the neutral supply to Io's corona and to the giant neutral clouds that extend along Io's orbit.

Energy straggling cross section for antiproton-atom collisions

SynopsisWe present benchmark simulations of the energy straggling cross section for antiprotons colliding with hydrogen and helium. We solve the one- and two-particle Schrödinger equation numerically within the framework of the straight-line impact-parameter approximation. Our results provide first evidence for the convergence of the ab-initio simulation toward the classical Bohr straggling cross section proposed a century ago.

Wannier threshold theory for the description of the two-electron cusp in the ion-induced double ionization of atoms

The double electron capture into the continuum states of the projectile (double ECC) is investigated theoretically in collisions of 100keV He2+ ions with He atoms. The process is described within the framework of the impact parameter and frozen-correlation approximations where the single-electron events are treated by the continuum distorted wave method. On the other hand, the Wannier theory is employed for describing the angular correlation between both ejected electrons. This treatment substantially improved the agreement between the theory and experiment as compared to the previous version of the theory (Gulyás et al., 2010) in which the correlation between the ejected electrons was taken into account by the Coulomb density of states approximation.

Capture of K-shell electrons by α-particles in α-decay of superheavy nuclei: I. Charge transfer in collisions of α-particles with He+(1s) ions

We consider a process in which α-particles after α-decay ionize K-shell of daughter atoms and capture electrons. We first have performed similar calculations for collisions of α-particles with He+(1s) ions. This charge transfer process is studied for the collision energy region 0.04–400 keV. The transition amplitudes are calculated in the impact parameter approximation with the use of the close coupling equations method for a basis consisting of eight molecular states. Total and partial charge-transfer cross sections to the 1s and, for the first time, to the 2s and 2p 0, 2p ±1 states of the He+ ion are obtained. Angular differential cross sections for the elastic scattering and charge-transfer processes are calculated in the eikonal approximation for a wide range of scattering angles for the collision energy of 1 keV. Results of calculations are in an excellent agreement with experimental data.

6.0 MeV u−1 carbon ion (C6+ and C4+)-induced secondary electron emission from water vapor

We have precisely measured absolute double differential cross sections (DDCSs) for secondary electron emission produced in collisions of 6.0 MeV u−1 C6+ and C4+ ions with water vapor. Theoretical calculations of the DDCSs were made for C6+ ions using the continuum distorted wave-eikonal initial state model (CDW-EIS) in its straight-line version of the impact parameter approximation, showing general good agreement with experimental data, except in the intermediate- and high-energy region (> 50 eV), particularly at the backward angles (> 110°). On the other hand, the single differential cross section (SDCS), which was obtained by integrating the measured DDCSs over the solid angle, showed fairly good agreement with the CDW-EIS in the low-energy region (< 200 eV), while a significant discrepancy between the observed SDCS and the Rudd-model scaling (× 36) can be seen, suggesting that a simple Z2-scaling law (i.e. first Born approximation) is not applicable for high-Z bare projectiles such as C6+ ions. The SDCS of C4+ ions was observed to be smaller than that of C6+ ions by ∼ 50% in the low-energy region (< 200 eV) due to the screening effect of its bound electrons in C4+ ions, which could be explained quantitatively by taking account of an effective projectile charge.

ADAS tools for collisional–radiative modelling of molecules

New theoretical and computational tools for molecular collisional–radiative models are presented. An application to the hydrogen molecule system has been made. At the same time, a structured database has been created where fundamental cross sections and rates for individual processes as well as derived data (effective coefficients) are stored. Relative populations for the vibrational states of the ground electronic state of H2 are presented and this vibronic resolution model is compared electronic resolution where vibronic transitions are summed over vibrational sub-states. Some new reaction rates are calculated by means of the impact parameter approximation. Computational tools have been developed to automate process and simplify the data assembly. Effective (collisional–radiative) rate coefficients versus temperature and density are presented.

Two-electron cusp in the double ionization of helium

We analyze the double ionization of He under the impact of 100 keV He{sup 2+} projectiles. The process is described within the framework of the impact parameter and frozen-correlation approximations where the one-electron events are treated by the continuum distorted wave method. Correlation between the emitted electrons, which plays an important role in forming the shape of the differential distribution of the electron emission, is described by the Coulomb density of states approximation (CDS). Special attention is paid to the region of the two-electron cusp that has been observed in a recent experiment for 100 keV He{sup 0}+He collisions [L. Sarkadi and A. Orban, Phys. Rev. Lett. 100, 133201 (2008)]. In the cusp region the correlated motion of the two electrons is influenced dominantly by the outgoing projectile, that is, the correlation function of the CDS treatment is expected to depend on the electron momenta measured relative to the projectile rather than to the target nucleus. A qualitative agreement with the experiment is achieved with a CDS model based on the use of such a projectile-centered correlation function that applies effective charges as given in the dynamically screened three-Coulomb wave function.

Semiclassical fully differential cross section calculations for the ionization of small molecules

Recently we have developed a fully differential ionization cross section calculation method based on the semiclassical, impact parameter approximation. The method was successfully used to reproduce experimental data of kinematically complete experiments for single ionization of helium by the impact of fast C6+ ions. In this work the semiclassical method is used to calculate fully differential cross sections for the ionization of small molecules like H2, N2 and CH4 using simple wavefunctions. The obtained results will be compared to other theoretical results and experimental data available.

Ionization of helium by Au24+ and Au53+ projectiles

A coupled pseudostate impact parameter approximation has been used to calculate triple differential cross sections for single ionization of He by Au24+ and Au53+ projectiles.

Antiproton impact ionization of atomic hydrogen and helium

We shall present results for antiproton ionization of H and He ranging from fully differential cross sections to total ionization. The calculations have been made in a coupled pseudostate impact parameter approximation. It will be shown that the interaction between the antiproton and the target nucleus is very important at low energies.

Molecular treatment of atomic collisions in the keV/amu energy range: Application to the Li3+-H system

The molecular treatment of ion-atom collisions is used for studying the electron capture from hydrogen by completely ionized lithium. The five molecular states considered (2pσ, 2pπ, 3dσ, 3dπ, and 4fσ) account for transitions to the n = 2 and n = 3 states of Li2+. The semiclassical impact parameter approximation is employed for obtaining the transition amplitudes. Total cross sections are compared with other theoretical and experimental results. Differential cross sections are shown. The possibility of extending the present analysis to other similar incident charged ions is discussed.

Theoretical Investigation on Excitation, Ionization and Capture in H(1s, 2s) + H(1s, 2s) Collisions

Cross sections of electron-loss in H(ls)+ H(ls) collisions and total collisional destruction of H(2s) in H(ls) + H(2s) collisions are calculated by four-body classical-trajectory Monte Carlo (CTMC) method and compared with previous theoretical and experimental data over the energy range of 4–100 keV. For the former a good agreement is obtained within different four-body CTMC calculations, and for the incident energy Ep > 10keV, comparison with the experimental data shows a better agreement than the results calculated by the impact parameter approximation. For the latter, our theory predicts the correct experimental behaviour, and the discrepancies between our results and experimental ones are less than 30%. Based on the successive comparison with experiments, the cross sections for excitation to H(2p), single- and double-ionization and H- formation in H(2s)+H(2s) collisions are calculated in the energy range of 4–100keV for the first time, and compared with those in H(ls)+H(ls) and H(ls)+H(2s) collisions.

On proton excitation of forbidden lines in positive ions

The semi-classical impact parameter approximations used by Bahcall and Wolf and by Bely and Faucher, for proton excitation of electric quadrupole transitions in positive ions, both fail at high energies, giving cross sections which do not fall off correctly as constant/E. This is in contrast with the pioneering example of Seaton for Fe+13 and of Reid and Schwarz for S+3, both of whom achieve the correct functional form, but do not ensure the correct constant of proportionality. By combining the Born and semi-classical approximations one can obtain cross sections which have the full correct behaviour as E → ∞, and hence, rate coefficients which have the correct high temperature behaviour (∼C/T1/2 with the correct value of C). We provide a computer program for calculating these. An error in Faucher's derivation of the Born formula is also discussed.

Electron emission characteristics inp−H(1s)collisions from numerical solutions of the time-dependent Schrödinger equation

The numerical integration of the three-dimensional time-dependent Schrodinger equation that appears in the impact-parameter approximation is carried out using a Fourier collocation method that is based on a splitting technique for the time evolution operator. The final wave function is analyzed in momentum space using a histogram approach. Singly and doubly differential cross sections for ionization as a function of polar emission angle and energy are calculated near the maximum in the total cross section, i.e., at 50-60-keV impact energy. They compare well with recent experimental results for intermediate and high electron emission energies, while some discrepancies remain at low energies due to the neglect of residual Coulomb interactions in the ionized wave packet. In order to achieve convergence and to reproduce experimental results at electron energies below 20 eV one needs to extend the calculations to final internuclear separations beyond 30 a.u.

On the semiclassical approach to cold atomic collisions

We extend the semiclassical description of two-state atomic collisions to low energies for which the impact parameter treatment fails. The problem reduces to solving a system of first-order differential equations with coefficients whose semiclassical asymptotes experience the Stokes phenomenon in the complex coordinate plane. Primitive semiclassical and uniform Airy approximations are discussed. The theory of collisions between two atomic systems goes back to the early days of quantum mechanics (Mott 1931; Teller 1937; Landau & Teller 1936) and the basic 1930s models of state interaction are detailed in the recent review of Nikitin (1999). Typically, the colliding atoms undergo electronic transitions and one needs to solve a number (in the simplest case, two) of coupled radial Schrodinger equations. It has been noted since the early thirties that the relative motion of the heavy nuclei can be described classically (Mott 1931; Rosen & Zener 1932). Semiclassically, Stueckelberg (1932) first suggested the analytical continuation of the JWKB (Jeffreys-Wentzel- Kramers-Brillouin) wave functions into the complex plane of the internuclear sepa- ration and a proper handling of the Stokes phenomenon. His solution of the avoided crossing time-independent problem (Stueckelberg 1932) shows why a description of the interference in terms of adiabatic quasiclassical phases fails if the phase difference accumulated during the adiabatic motion of the two atoms between the centre of the coupling region and the turning points is small. The vexatious problem of the choice of branch cuts and the determination of Stokes constants via the comparison equa- tion method, were largely resolved by Crothers (1971). Equally well, it was shown by Coveney et al . (1985) that the classically forbidden region is less amenable to generalized phase-integral analysis. The classical approximation of the nuclear motion leads to the impact parameter treatment (Gaunt 1927; Landau 1932; Zener 1932). The impact parameter approx- imation (IPA) offers a simple physical picture in which fast electronic transitions occur as the slow nuclei follow a classical trajectory. By its nature, the IPA neglects

Classical impact parameter approximation: Application to atomic collisions with n independent electrons and to recoil analysis

We show that a classical Impact Parameter Method may be derived when taking fully into account the smallness of the ratio between the electron and nuclear masses. It allows to calculate, exactly as in the quantum version, projectile scattering and therefore recoil momenta required for the interpretation of recent measurements. We prove an additivity theorem which allows, in particular, to reduce the n-noninteracting electron problem to a set of n one-electron problems. Consequences for the interpretation of target recoil measurements are discussed.

Structure analysis of the WB 2(0001) surface

The structure of the WB 2(0001) surface has been investigated using impact–collision Li + ion-scattering spectroscopy combined with three-dimensinal three-atom computer simulation. We found that a graphitic boron layer is formed on the WB 2(0001) surface, where the strong W 5d–B 2p orbital hybridization and/or charge transfer from tungsten to boron may be responsible for stabilization of the graphitic boron layer. It is also found that the scattered Li + ions lose a considerable amount of their kinetic energy due to electron–hole pair excitation when passing through the negatively charged boron layer. The electronic stopping power is explained qualitatively by the impact-parameter approximation proposed by Firsov. But consideration of the electronic shell effect or the formation of molecular orbitals is suggested to improve the agreement between experiment and theory.

Structure Analysis of a Graphitic Boron Layer at theTaB2(0001)Surface

The structure of the ${\mathrm{TaB}}_{2}(0001)$ surface has been investigated by using impact-collision ${\mathrm{Li}}^{+}$ ion-scattering spectroscopy and three-dimensional three-atom computer simulation. We found for the first time that a graphitic boron layer is formed on the ${\mathrm{TaB}}_{2}(0001)$ surface and that the scattered ${\mathrm{Li}}^{+}$ ion loses significant energy due to electronic excitation when the ${\mathrm{Li}}^{+}$ ion passes through the graphitic boron layer. The electronic stopping power is qualitatively explicable by the impact-parameter approximation proposed by Firsov.

Ab initio calculation of the asymmetric electron transfer cross section in the low velocity collision

The one-electron molecule has been studied and the electronic energies of the states n = 1-5 have been determined as a function of the distance R between the two nuclei, in the Born-Oppenheimer approximation. The radial and rotational matrix elements for electron transfer in the collision have been computed and the system of 35 coupled equations describing the collision in the impact parameter approximation has been solved for an initial system composed of an incident ion and a target in the ground state. The signs of the different terms appearing in the radial matrix elements have been carefully re-examined and a clear resonance induced by the avoided crossing of the and states has been proven. The total electron transfer cross section is found to lie between those of the adjacent symmetric systems and and, in this respect, may be called `quasisymmetric transfer'.

Charge transfer in He++-He collisions at keV energies in an intense laser field: A theoretical close-coupled study using a diabatic molecular representation

We carry out a theoretical study of the effect of a strong laser field on charge transfer in collisions of doubly charged helium ions with helium at keV energies, using a close-coupled treatment with a diabatic molecular wave function basis in the semiclassical impact parameter approximation. Our results show significant effects of the laser intensity, wavelength and the orientation of the (linear) laser polarization on the charge transfer cross-sections. Our results also indicate the possibility of an experimental test for differentiating between the effects of the minimal coupling and the multipolar matter-radiation interactions on ion-atom collision processes.