Classical Trajectory Monte Carlo Research Articles

Differential cross sections for state-selective electron capture in 25-100-keV proton-helium collisions.

Cross sections differential in the scattering angle of the projectile are presented for electron capture summed over all states and to the 2{ital s}, 2{ital p}, 3{ital s}, 3{ital p}, 4{ital s}, and 4{ital p} states of hydrogen in 25-, 50-, and 100-keV proton-helium collisions. The classical-trajectory Monte Carlo (CTMC) technique was employed for these calculations as well as to compute total cross sections as a function of impact energy. The latter are compared with experiment to display the behavior of the integral state-selective cross sections in this energy regime. Detailed comparison is also made between the calculated angular differential cross sections and the experimental measurements of Martin {ital et} {ital al}. (Phys. Rev. A 23, 285 (1981)) for capture summed over all states and of Seely {ital et} {ital al}. (Phys. Rev. A 45, R1287 (1992)) for capture to the 2{ital p} state. Very good overall agreement is found. Regarding the cross section for capture summed over all states, an improved agreement is demonstrated by using an alternate representation of the initial state in the CTMC method, which improves the electronic radial distribution but which cannot presently be applied to state-selective determinations.

Angular-differential cross sections for H(2p) formation in intermediate-energy proton-helium collisions.

Angular-differential cross sections for charge transfer with simultaneous emission of a photon in collisions of protons with helium atoms have been measured. The incident proton energies were 25, 50, and 100 keV and the center-of-mass scattering angles were between 0 and 2.0 mrad. In the experiment, hydrogen atoms that scattered through an angle \ensuremath{\theta} were detected in coincidence with photons emitted perpendicular to the scattering plane with a wavelength between 1140 and 1400 \AA{}. Differential cross sections for capture into the 2p state of the hydrogen atom were determined from the variation in the coincidence signal with \ensuremath{\theta}. The experimental results are compared with the results of a classical trajectory Monte Carlo (CTMC) simulation and with the results of a calculation for H(2p) capture using the Coulomb-Brinkman-Kramers (CBK) approximation. The agreement between the experimental results and the CTMC calculation is good at all three energies while the agreement between the shape of the data and the CBK calculation is good at 50 and 100 keV.

Coherence parameters of atomic excited states: comparison of results from the CTMC and the quantal calculations

The coherence parameters of excited states formed in collisions of atoms with electrons, positrons, protons and antiprotons are calculated using the Classical Trajectory Monte Carlo (CTMC) method and the results are compared to those obtained from the quantal distorted wave and close-coupling calculations. The orientation parameter and the components of the electric dipole moment of the electronic charge cloud after the collision are examined for excitation and for electron capture toH(n = 2) states. It is found that the CTMC results are in qualitative agreement with the quantal distorted wave and close-coupling results for some of the coherent parameters only.

Convoy electrons emitted at glancing angle incidence of MeV light ions

Abstract The energy distributions of convoy electrons produced at glancing angle incidence of fast ions are studied with Classical Trajectory Monte Carlo (CTMC) calculations. The image potentials both of the ion and of the convoy electron are taken into account. For light ions, the acceleration of the convoy electron by the ion's image potential is depressed by the image potential of the convoy electron, while for heavy ions this effect is negligible.

On the description of classical trajectory methods in a quantum mechanical language

Formally, quantum collision theory can be exactly formulated in terms of real phase-space distributions. This type of approach provides an intuitive picture of complex scattering problems which has proved very useful in different disciplines such as nuclear physics, chemical physics and quantum field theory. In this work, the classical trajectory Monte Carlo (CTMC) method, which is a well known technique in ion-atom collisions, is put into context with these more general phase-space approaches. The most important aspects of this approximation and its validity are briefly discussed.

Solution of the time-dependent Schrödinger equation employing a basis of explicit discrete-coordinate eigenfunctions: spherical and azimuthal symmetry, adiabaticity, and multiphoton excitation of a rotating Morse oscillator

We have recently presented a method for solving the time-dependent Schrödinger equation by collocation on various members of a family of spatial grids and have shown that this method is interpretable in terms of the expansion of the wave function in a basis of discrete coordinate eigenfunctions. The method is applicable to molecular systems with one or multiple spatial dimensions, and also to bound or unbound molecular systems. We now place the method in the more general setting of spatial grids belonging to the family of scaled Gauss-Markov quadrature points. Further, we now treat molecular systems described in terms of the more general case of a curvilinear coordinate system. This allows us to treat, for example, a system with spherical or azimuthal symmetry and to employ the information regarding the symmetry, i.e. transform from Cartesian coordinates to spherical polar coordinates and obtain the Schrödinger equation for a wave function for a configuration space with a lower number of dimensions. In addition, we describe a modified adiabatic approach for decreasing the number of spatial dimensions in the Hamiltonian appearing in the time-dependent Schrödinger equation for the wave function. We then obtain a coupled set of differential equations describing the time evolution of the molecular system, including the time-dependent Schrödinger equation in terms of the Hamiltonian with the lower number of spatial dimensions, and we apply our method for solving the time-evolution problem for the wave function by employing the Hamiltonian with that smaller number of spatial dimensions. Finally, we demonstrate the application of this extension of our collocation method to the nonlinear problem of multiple-photon excitation of a rotating anharmonic diatomic molecule by the explicitly time-dependent term describing irradiation of the molecule by an intense classical electromagnetic field. We compare our results for the quantal approach with our results for two approximate approaches: a modification of the adiabatic quantal approach and a variation of the Monte Carlo classical trajectory approach.

State-selective capture in collisions between ions and ground- and excited-state alkali-metal atoms.

Total cross sections for state-selective electron capture in collisions between ions and alkali-metal atoms have been calculated by means of a three-body classical-trajectory Monte Carlo (CTMC) method using model potentials to describe the electron\char21{}ionic-core interactions. Calculations have been performed for ${\mathrm{Na}}^{+}$-Na(28d) collisions and for ${\mathrm{N}}^{5+}$ and ${\mathrm{Ar}}^{8+}$-Cs(6s) collisions. The collision velocity range corresponds to 0.5\ensuremath{\lesssim}${\mathit{v}}_{\mathit{p}}$/${\mathit{v}}_{\mathit{e}}$\ensuremath{\lesssim}2, where ${\mathit{v}}_{\mathit{p}}$ is the projectile velocity in the laboratory frame and ${\mathit{v}}_{\mathit{e}}$ is the initial orbital velocity of the electron bound to the alkali-metal core. In the case of ${\mathrm{Na}}^{+}$+Na(28d) collisions, calculations of the final n,l,m distributions show the importance of the electron-capture cross sections into states with mg1. For the case of multiply charged ion\char21{}Cs(6s) collisions, a predominance of electron capture to nearly circular states (large l values) is predicted for cross sections near the maximum of the n distribution. When the ${\mathit{e}}^{\mathrm{\ensuremath{-}}}$-${\mathrm{Cs}}^{+}$ interaction is described by a realistic model potential, the CTMC calculations are found to be in good agreement with recent measurements of the final n values that are predominantly populated after single-electron capture.

Semiclassical phase space description of ionisation and capture for ions colliding with hydrogen-like targets

Semiclassical calculations were performed for 10 keV amu-1-1 MeV amu-1 collisions of protons with H, He+ and Li2+ as well as for He2+, Li3+, C6+ and Ne10+ colliding with hydrogen. The calculated integrated cross sections for global ionisation and capture agree well with classical trajectory Monte Carlo (CTMC) calculations, other theories for capture and with experiment where available. The semiclassical method for the description of the electron processes encounters difficulties in the description of low-energy capture from highly charged hydrogen-like ions.

Rotationally inelastic scattering of glyoxal by H2 at E=80 meV

Using the Monte Carlo classical trajectory (CT) method and the azimuthal close‐coupled, infinite‐order sudden (ACC‐IOS) method, we have calculated cross sections for rotational excitation of S1 trans‐glyoxal by H2 at E=80 meV. The cross sections σ(k=0, j→k’) calculated with the CT method are nearly independent of j. The classical values of σ(k=0, j=5→k’) are in good agreement with the quantum values of σ(k=0→k’) for 2≤k’≤12, although the quantum calculations show a slight preference for odd Δk transitions which is not found in the CT calculations. Both the CT results and the ACC‐IOS results are in good agreement with results obtained in a recent crossed beam experiment. Rotational excitation to high k’ (k’=11,12) occurs by collisions of H2 with one of the H atoms of glyoxal, and the initial value of the orbital angular momentum approximately equals the final value of k in such collisions. Since backward scattering is dominant in collisions leading to high k’, angular momentum constraints alone cannot explain the maximum observed in Δk experimentally (Δk=14).

Impact-parameter treatment of classical trajectory Monte Carlo calculations for ion-atom collisions.

A simplified classical trajectory Monte Carlo (CTMC) method is proposed for ion-atom collisions. With the introduction of the prescribed classical trajectories for the internuclear motion, the electronic motion in the time-dependent interaction is solved classically. It gives almost identical integrated cross sections with the ``exact'' CTMC calculations. Differential cross sections are also calculated for p+H\ensuremath{\rightarrow}H+p. The results are in excellent agreement with the experiments.

Formation and lifetime of metastable complexes in collisions of t r a n s-glyoxal with helium

Using the Monte Carlo classical trajectory method, we have calculated cross sections for complex formation and complex lifetimes for glyoxal–helium collisions at low relative translational energies (<40 cm−1). The cross section for complex formation depends exponentially on the initial translational energy, and is independent of the initial glyoxal rotational state over a wide range of rotational energies. The lifetime of the glyoxal–helium complex depends strongly on the total angular momentum of the complex. The dissociation of the complex (rotational predissociation) is not statistical. Some insight in the dynamics of complex formation is obtained.

N, l Distributions for Electron-Capture from H(1s) by C6+ and O8+

Total cross sections for electron-capture from the ground state of hydrogen by C6+ (40, 50, 60, 80, 100 and 120 keV/u) and O8+ (40, 50, 60, 80, 100, 120 and 140keV/u) have been calculated using the classical-trajectory Monte Carlo (CTMC) technique. We tabulate these cross sections as a function of projectile energy for (i) capture to all product states, (ii) capture to product n-levels, and (iii) capture to product n, l-levels. The results of these calculations agree with and extend previous CTMC n, l distribution results.

Single-electron-removal processes in collisions of positrons and protons with helium at intermediate velocities.

Total cross sections for single ionization and charge transfer have been calculated using the classical-trajectory Monte Carlo (CTMC) technique for collisions of both positrons and protons with helium. Analysis of the classical trajectories has helped to explain the differences in the collision mechanisms responsible for the observed relative magnitudes of the positron and proton electron-removal cross sections. In the intermediate collision velocity range (1.5 a.u.<v<4.5 a.u.) it is found that because the positron is much smaller in mass than the proton, two dynamical effects occur leading to differences in their efficiency in electron removal. First, positrons are less likely to singly ionize helium than are protons since they possess less kinetic energy above the ionization threshold and accordingly there is a lower probability for ionization. Second, positrons are more likely to remove an electron from helium by charge transfer than are protons since they may be deflected by the target to large positive or negative scattering angles and be accelerated or decelerated to more readily momentum-vector match with an orbital electron. In the large-velocity regime (v>4.5 a.u.) positrons and protons are found to be equally likely to singly ionize helium, but positrons remain at least half an order of magnitude more likely to remove an electron by charge transfer.

Effect of the potential-energy surface on the diatom dissociation rate constant for F2in Arat3000 K

Gas-phase dissociation of fluorine (1Ʃ+g) molecules in an agron bath at 3000 K was studied by using the 3D Monte Carlo classical trajectory (3DMCCT) method. To assess the importance of the potential energy surface (PES) in such calculations, three surfaces, with a fixed, experimentally determined F2dissociation energy, were constructed. These surfaces span the existing experimental uncertainties in the shape of the F2potential. The first potential was the widest and softest; in the second potential the anharmonicity was minimized. The intermediate potential was constructed to ‘localize’ anharmonicity in the energy range in which the collisions are most reactive. The remaining parameters for each PES were estimated from the best available data on interatomic potentials. By using the single uniform ensemble (SUE) method (Kutz, H. D. & Burns, G. J.chem. Phys. 72, 3652-3657 (1980)), large ensembles of trajectories (LET) were generated for the PES. Two such ensembles consisted of 30000 trajectories each and the third of 26200. It was found that the computed one-way-flux equilibrium rate coefficients (Widom, B.Science148, 1555-1560 (1965)) depend in a systematic way upon the anharmonicity of the potential, with the most anharmonic potential yielding the largest rate coefficient. Steady-state reaction-rate constants, which correspond to experimentally observable rate constants, were calculated by the SUE method. It was determined that this method yields (for a given trajectory ensemble, PES and translational temperature) a unique steady-state rate constant, independent of the initial, arbitrarily chosen, state (Tolman, R. C.The principles of statistical mechanics, p. 17. Oxford University Press (1938)) of the LET, and consequently independent of the corresponding initial value of the reaction rate coefficient. For each initial state of the LET, the development of the steady-state rate constant from the equilibrium rate coefficient was smooth, monotonic, and consistent with the detailed properties of the PES. It was found that, although the increased anharmonicity of the F2potential enhanced the equilibrium rate coefficients, it also enhanced the non-equilibrium effects. As a result, the steady-state rate constants were found to be insensitive to the variation of the PES. Thus, the differences among the steady-state rate constants for the three potentials were of the order of their standard errors, which was about 15% or less. On the other hand, the calculated rate constants exceeded the experimental rate constant by a factor of five to six. Because within the limitations of classical mechanics the calculations wereab initio, it was tentatively concluded that the discrepancy of five to six is due to the use of classical mechanics rather than details of the PES structure.

Formation of protonium in collisions of antiprotons with H and H-

Cross sections for formation of protonium (pp\ifmmode\bar\else\textasciimacron\fi{}) in low-energy collisions of antiprotons with hydrogen atoms (H) and negative hydrogen ions (${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$) are calculated using the classical-trajectory Monte Carlo (CTMC) method. Full four-body dynamics is performed for p\ifmmode\bar\else\textasciimacron\fi{}+${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$. Previously unpredicted differences between p\ifmmode\bar\else\textasciimacron\fi{}+H and p\ifmmode\bar\else\textasciimacron\fi{}+${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ collisions, stemming from the dynamics of the weakly bound electron in ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$, are exhibited. A maximum protonium formation cross section of nearly 2 A${\r{}}^{2}$ for p\ifmmode\bar\else\textasciimacron\fi{}+${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ is found, smaller than previous theoretical estimates, but the reaction window is found to extend to higher energies. The electron stripping cross section for p\ifmmode\bar\else\textasciimacron\fi{}+${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ collisions is also calculated and yields results in agreement with our previous three-body CTMC calculation and with a very recent experimental determination. The implications of these results for experiments with corotating beams of p\ifmmode\bar\else\textasciimacron\fi{} and ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ are discussed.

A Monte Carlo trajectory study of angular distributions in inelastic scattering of NO from graphite surfaces

A Monte Carlo classical trajectory study of NO/graphite surface scattering was carried out. NO was modeled as a rigid rotor interacting with a smooth vibrating surface through two Lennard-Jones 12-6 potential terms. The study simulated the experiments performed on the same system, first published by Frenckel et al. (1982). Angular distributions were calculated and found to generally reproduce the measured scattering lobes best at a simulated surface mass of 18 amu. Two potential well depths were used and at the larger of these all measured lobes could be quite well reproduced.

Classical and quantal nonperturbative treatments of multiphoton and above-threshold ionization

L2 non-Hermitian Floquet studies of intensity- and frequency-dependent threshold shifts of atomic hydrogen in intense laser fields are presented. We also performed a Monte Carlo classical trajectory study of the motion of an electron under the influence of both the Coulomb and the laser fields. We found that classical results are consistent with quantum-mechanical predictions with regard to continuum threshold shift and above-threshold ionization (ATI). In addition, our classical treatment reveals detailed mechanisms responsible for ATI processes.

Inelastic scattering of Br2 from graphite surfaces: A Monte Carlo classical trajectory study

The experimental results from Holmlid et al. (1982) for Br2 inelastic scattering from graphite surfaces were used as a reference point for the Monte Carlo classical trajectory calculations presented here. A one-dimensional potential energy model of the scattering system was constructed. No vibrational or rotational degrees of freedom for Br2 were included. The trajectories were generally found to be ‘‘long-lived.’’ Good agreement with experiments was found for six simulated C atoms moving in phase, in the case of one simulated C layer, and for two to three C atoms in each layer, in the case of two simulated vibrating layers. In these cases one-fourth of the Br2 molecules made large jumps on the surface and all molecules were hit more than once by the vibrating C atoms. The results found are in agreement with the one-phonon transfer model by Holmlid et al. Both models demonstrate agreement with experiments by simulating long interaction times and a more or less ‘‘statistical’’ energy transfer.

First-order Born and first-order distorted-wave Born cross sections for micro++( micro-p)-->( micro+ micro-)+p.

As one of the possible ways to produce a (${\ensuremath{\mu}}^{+}$${\ensuremath{\mu}}^{\mathrm{\ensuremath{-}}}$) atom, we have theoretically investigated the cross section of the process ${\ensuremath{\mu}}^{+}$+(${\ensuremath{\mu}}^{\mathrm{\ensuremath{-}}}$p)\ensuremath{\rightarrow}(${\ensuremath{\mu}}^{+}$${\ensuremath{\mu}}^{\mathrm{\ensuremath{-}}}$)+p. The general behaviors of the cross sections obtained by the first-order Born approximation (FBA) and the first-order distorted-wave Born approximation (DWBA) are discussed. From the comparison of the present results with those of classical-trajectory Monte Carlo (CTMC) calculation and from the reliability of the present method which was discussed in the case of ${e}^{+}$+H\ensuremath{\rightarrow}(${e}^{+}$${e}^{\mathrm{\ensuremath{-}}}$)+p, we can conclude that the maximum cross section is of the order of ${10}^{\mathrm{\ensuremath{-}}20}$ ${\mathrm{cm}}^{2}$ at about 2.8 keV impact energy, which is of the order of the geometrical cross section. The cross sections by FBA are of the same order with those by DWBA, agreeing to within 10--20 %. The DWBA cross section is larger than that by CTMC by 30--50 % in the velocity range of 5--10 keV. Scaling relationships between the cross section and impact velocity are discussed in the cases of ${\ensuremath{\mu}}^{+}$+(${\ensuremath{\mu}}^{\mathrm{\ensuremath{-}}}$p) and ${e}^{+}$+H.

Classical-trajectory Monte Carlo calculation for collision processes of e++(e-p) and micro++( micro-p).

Classical-trajectory Monte Carlo (CTMC) method is applied to the pickup process, A+(BC)\ensuremath{\rightarrow}(AB)+C, of a bound light particle in a heavy particle by an incident light particle as well as the breakup process A+(BC)\ensuremath{\rightarrow}A+B+C. Calculation has been made for the cross sections of the processes in the case of A=${e}^{+}$,${\ensuremath{\mu}}^{+}$, B=${e}^{\mathrm{\ensuremath{-}}}$,${\ensuremath{\mu}}^{\mathrm{\ensuremath{-}}}$, and C=p. The cross sections for positronium formation agree with those of recent quantum-mechanical calculations except for low impact velocities. The cross sections for (${\ensuremath{\mu}}^{+}$${\ensuremath{\mu}}^{\mathrm{\ensuremath{-}}}$) formation are smaller in magnitude than those of the quantum-mechanical calculations using the first-order Born approximation and the first-order distorted-wave Born approximation.