Thomas Peak Research Articles

Response to Thomas Peak’s Review of The Invention of International Order: Remaking Europe after Napoleon

An abstract is not available for this content. As you have access to this content, full HTML content is provided on this page. A PDF of this content is also available in through the ‘Save PDF’ action button.

Quasi-four-body treatment of charge transfer in the collision of protons with atomic helium: I. Thomas related mechanisms

The scattering of a completely bare ion by atoms larger than hydrogen is at least a four-body interaction, and the charge transfer channel involves a two-step process. Amongst the two-step interactions of the high-velocity single charge transfer in an anion-atom collision, there is one whose amplitude demonstrates a peak in the angular distribution of the cross sections. This peak, the so-called Thomas peak, was predicted by Thomas in a two-step interaction, classically, which could also be described through three-body quantum mechanical models. This work discusses a four-body quantum treatment of the charge transfer in ion-atom collisions, where two-step interactions illustrating a Thomas peak are emphasized. In addition, the Pauli exclusion principle is taken into account for the initial and final states as well as the operators. It will be demonstrated that there is a momentum condition for each two-step interaction to occur in a single charge transfer channel, where new classical interactions lead to the Thomas mechanism.

Single and double electron capture in p-He and α-He collisions

The differential and total cross sections for both single and double electron capture in collisions of and He2+ with ground state helium atom have been studied by means of the four-body model of target continuum distorted wave (TCDW-4B) approximation in the energy range from 30 to 1000 keV amu–1. In this model, distortion in the final channel related to the Coulomb continuum states of the active electron(s) in the field of residual target ion are included. The calculations are based on the independent electron model. The present computed results are compared with the available experimental and other theoretical results. Total cross sections are found to be in good agreement with the measurements. We have also analysed differential cross sections (DCS) for both single and double electron capture in the collision of proton and α-particles with helium atoms at different projectile energies. The present DCS data exhibits the typical steeply decreasing dependence on the projectile scattering angles, but neither oscillating structures characteristic of interference effects nor peaks reminiscent of the Thomas peak are observed at different projectile energies. The obtained results for the DCS into the ground state are compared with the experimental data and overall a satisfactory agreement has been found. Finally we have also studied the variation of double to single capture differential cross-section ratios with projectile scattering angles at different impact energies.

Angular differential cross sections for double and single capture in fast ion-atom collisions

SynopsisWe investigate angular differential cross sections for double-electron and single-electron capture in ion- atom collisions for the intermediate and high energy region. We see the diffraction pattern in the spectrum of differential cross sections in the high energy range, i.e. another peak at 0.75 mrad in addition to Thomas peak at 0.47 mrad have been observed.

Projectile angular distribution in single electron capture from helium by fast protons

In addition to the Thomas peak, occurs at 0.47 mrad, another peak has been observed at ≈ 0.75 mrad in a recent measurement for projectile angular differential cross sections of single electron capture in proton-helium collision at impact energies of MeV region [M. Gudmundsson et al., J. Phys. B 43, 185209 (2010)]. Stimulated by this report, the differential cross section is calculated based on the first Born approximation for single ionization of helium by proton impact. Calculations with and without inter-electron correlation give almost the same results, and the peak observed experimentally near 0.75 mrad is not accounted for in the treatment.

Two-centre interference effects on the Thomas two-step scattering mechanisms

The charge transfer process in the collision of fast protons with hydrogen molecules is theoretically investigated using the second-order Born approximation with correct boundary conditions. In addition to two first-order terms, the present calculations include the three second-order terms which correspond to the Thomas two-step scattering mechanisms. The interference effects, due to the scattering of the particles from two atomic centres, on the electron capture differential cross sections vary significantly with the orientation of the molecule and with the impact energy. After the averaging over all molecular orientations the interference patterns disappear but the Thomas peak becomes more pronounced. These patterns are also apparent in the differential cross sections as a function of the angle between the molecular axis and the incident beam direction. The integrated cross sections are calculated and the results are compared with available experimental data.

Closed-form expressions for state-to-state charge-transfer differential cross sections in a modified Faddeev three-body approach

The second-order Faddeev-Watson-Lovelace approximation in a modified form is applied to charge transfer from hydrogenlike target atoms by a fully stripped energetic projectile ion. The state-to-state, $nlm\ensuremath{\rightarrow}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}}{m}^{\ensuremath{'}}$, partial transition amplitudes are calculated analytically. The method is specifically applied to the collision of protons with hydrogen atoms, where differential cross sections of different transitions are calculated for incident energies of 2.8 and $5.0\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}$. It is shown that the Thomas peak is present in all transition cross sections. The partial cross sections are then summed and compared with the available forward-angle experimental data, showing good agreement.

Simultaneous transfer ionization of a negative hydrogen ion by positron impact

A model has been proposed to investigate theoretically, the simultaneous electron capture and the ejection of another electron by which a bare hydrogen ion and a positronium atom (Ps) are produced in a single collision between a positron and a negative hydrogen ion $({\mathrm{H}}^{\ensuremath{-}}).$ The angular distributions of the scattered Ps atom as well as of the ejected electron are studied in the intermediate- and high-energy regimes (40--200 eV) with respect to the threshold energy for this particular transfer ionization (TI) process. The electron-electron correlation effect that mainly governs such two-electron transition processes in this energy regime has been taken into account in both the initial and final channels. The long-range Coulomb attraction between the incident positron and the negative hydrogen ion in the initial channel has also been incorporated properly. Signature of the so-called Thomas peak (a double-peak structure) predicted for charge-transfer or transfer-ionization process for heavy-particle projectiles in the high-energy regime has also been noted in the present TI process (for light projectile ${e}^{+})$ and could be attributed to the correlated ${e}^{+}\ensuremath{-}e\ensuremath{-}e$ scattering mechanism. In addition, the fully differential (triple) cross sections for different kinematics reveal some structures unusual for a pure transfer or a pure ionization process which are to be verified by the future experiments.

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.

Time correlation in two-electron transitions produced in fast collisions of atoms with matter and light

Time connection between electrons in dynamic atomic systems is considered. We describe time correlation in terms of the Dyson time ordering operator T. In this paper we decompose T into an uncorrelated term Tunc , plus a correlated term Tcor5T2Tunc , which interconnects the time-dependent external interactions. We show that time correlation between electrons requires both Tcor and spatial electron-electron correlation. Two examples are analyzed. In transfer ionization the time correlation operator incoherently changes the shape of an electron-electron Thomas peak. In double excitation the influence of Tcor in amplitudes for coherently interfering pathways changes resonance intensities and profiles.

Electron-electron Thomas peak in fast transfer ionization

''Thomas process'' is a name used for a family of singular two-step processes that can lead to electron transfer. The Thomas process of the ''second kind,'' occurring in reactions with both transfer and ionization, utilizes the e-e scattering in the second step, so this Thomas process requires the dynamics of the electron-electron interaction. We calculate numerically the second order element of an S matrix and corresponding cross sections for the transfer ionization process. We find that the position and shape of the Thomas peak depend on both electron-electron and the electron-nucleus interaction. Also the direct and exchange amplitudes are equal at the peak position. We test the peaking approximation used for transfer ionization. Our results can be compared to experimental results for p{sup +}+He{yields}H+He{sup 2+}+e{sup -}. (c) 2000 The American Physical Society.

Anti-capture into the continuum in electron-atom ionization

Collisional ionization by electrons and positrons is a major unsolved problem in atomic collisions. Because the projectile can transfer significant momentum to the loosely bound atomic electrons and/or the target core, this cannot be treated as a semi-classical three-body collision between the projectile, the target core and a loosely bound electron, even at 100 times the ionization energy. To elucidate this problem, we are studying single and double binary effects in correlation experiments between outgoing projectile and ejected electrons as a function of projectile energy, and projectile and ejected electron exit angles. We have made, what we believe to be, the first observation of anti-capture to the continuum, the electron equivalent of the Thomas peak in ion-atom scattering. We also have observed a very sharp drop in the double differential cross section at 45°. This is probably due to the fact that two identical particles cannot have the same momentum and energy. In addition, our results show that the probability of scattering one electron to a particular angle or two is given only by the geometry in zero order. Thus, the double differential cross section is much larger than one might expect on the basis of a second Born calculation.

Projectile and electron spectra resulting from positron-argon collisions

A three-body classical trajectory Monte-Carlo calculation is used to evaluate the differential cross sections for the ejected electron, as well as the projectile scattering for 50, 100 and 150 eV e++Ar0 electron removal collisions. In the ionized electron spectra both the electron capture to the continuum (ECC) and binary peak structures are observed. The theoretical ECC peak is diffuse and shifted to lower energies than expected from simple arguments. The calculated double differential cross sections for electron ejection have been compared to experiment. When the experimental apparatus function of Moxom et al. (1992) is used to convolute the theoretical spectra, good agreement with their data is found that confirms a sharp ECC structure is not present in this system. Moreover, comparisons made at non-zero angles show substantial agreement between theory and experiment. The scattered positrons agree well with measurements made at 3 degrees by Kover et al. (1993), but differ at larger angles. In addition, the differential cross section for positronium formation is presented; no evidence of an observable Thomas peak is present.

Positronium formation in positron-atom collisions at intermediate energies

The formation of ground-state positronium in collisions of positrons with hydrogen and helium atoms is considered within the distorted-wave Born approximation. In particular the authors analyse the eikonal and continuum-distorted-wave models in both their post and prior forms. Differential and total cross sections are calculated over a broad range of intermediate energies extending from 40 to 500 eV. The results confirm that the total cross section decreases rapidly with increasing energy, as predicted by the simple first Born approximation, and do not agree with experimental evidence of a slower decay. The computed total cross sections for helium targets agree with those of the close-coupling model for energies above 100 eV. Below 100 eV the authors find a large post-prior discrepancy. The calculated differential cross sections show evidence of a Thomas peak at high energies for the CDW models but not for the eikonal models.

Classical calculation of high-energy electron capture in 5-MeV proton-hydrogen collisions.

The existence of the classical Thomas peak in the angular distribution of projectiles undergoing capture in collisions of 5-MeV protons with atomic hydrogen is explored using the three-body, three-dimensional classical-trajectory Monte Carlo technique. A method that selects only that portion of the initial phase space which yields capture at this energy was developed to make the calculation tractable due to the extremely small cross section. The spectrum obtained displays only a small shoulder near the angle predicted by Thomas on the basis of successive classical binary collisions and the total (integral) cross section is overestimated by a factor of 26 compared to recent experimental measurements. The overestimation originates from too large a contribution from velocity matching direct capture; the energy regime in which it is significant is discussed. In addition, the double-scattering events in this model which contribute significantly to the cross sections are found to differ substantially from the Thomas picture.

Second Born approximation differential cross sections for p+H and p+He charge-exchange collisions.

Charge-exchange differential cross sections are calculated by the second Born approximations without a recourse to any further approximation. The exact second Born approximation cross sections show satisfactory agreement with the experimental data for proton-helium collisions at all the energies of 2.82, 5.42, and 7.4 MeV while they are considerably smaller than the existing experimental cross sections in the Thomas peak region for proton-hydrogen collisions at 2.8 MeV. Discussions are given on how to choose the interactions for two-electron systems in order to satisfy the boundary condition of neutral-atom collisions.

Absence of the Thomas peak in the classical-trajectory Monte Carlo calculations for proton-hydrogen collisions in the MeV region.

Large-scale classical-trajectory Monte Carlo calculations are executed in three-dimensional space for proton-hydrogen collisions at 2.8 and 5 MeV. The Thomas peak that has been confirmed in the experimental data and in the quantal calculations is invisible in the classical calculations. This unexpected result is due to the peculiar character of classical bound states that have no minimum binding energy. The Oppenheimer-Brinkmann-Kramers-type capture by means of the velocity-matching mechanism dominates even in the MeV region owing to this classical peculiarity, and the Thomas double-scattering contribution is embedded in this background.

Second-order Born calculations for electron capture in relativistic U+U collisions.

Using an exact evaluation of the relativistic second-order Oppenheimer-Brinkman-Kramers approximation, total and differential cross sections for electron capture in ${\mathrm{U}}^{92+}$+${\mathrm{U}}^{91+}$ collisions are calculated for projectile energies between 0.5 and 100 GeV/u. It is shown that (a) at 100 GeV/u the asymptotic cross-section behavior is not yet reached, (b) spin-flip transitions and negative-energy intermediate states become important at high energies, and (c) the Thomas peak in the differential cross section is relativistically suppressed.

Closed-form expression for 1s-->1s electron-capture amplitude in a second-order Faddeev approximation.

Within a second-order approximation to the Faddeev treatment of the exact transition operator for 1s\ensuremath{\rightarrow}1s electron capture in ion-atom collisions, expressions are derived in terms of ${F}_{4}$ generalized hypergeometric functions for that part of the full amplitude involving electronic-nuclear potentials only. For totally symmetric collisions, the amplitude is evaluated in terms of $_{3}\mathrm{F}_{2}$ functions. When near-the-energy-shell anomalies of the Coulomb scattering are neglected, the amplitude is seen to reduce to a symmetric impulse-approximation form, and, when the explicit Coulomb nature of the problem is neglected, to a second-order Born approximation form. As a special case, the amplitude is evaluated at the momentum transfer value where the differential cross section exhibits a local maximum---the Thomas peak. The values of the various amplitude norms at the Thomas peak are compared.

Second-order singularities in transfer ionization.

In processes in which one electron is transferred from a target to a projectile and the other electron is ionized, called transfer ionization, two second-order terms arise in the transition amplitude at high collision velocities. There is a kinematic singularity and a weaker Thomas singularity. These second-order singular terms are formulated to first order in the projectile electron and first order in the electron-electron interactions. Using standard peaking approximations, the T matrix is reduced to an approximate closed-form expression. Our expression is qualitatively similar to the expression found earlier by J. S. Briggs and K. Taulbjerg [J. Phys. B 12, 2565 (1979)], who used a different formulation and evaluation and omitted an exchange contribution which we include. In this paper we evaluate cross sections for both the Thomas peak in the angle of scattering of the projectile and a kinematic ridge in the momentum distributions, ${\mathrm{k}}_{2}$, of the ejected electron, both of which in principle could be observed experimentally. The ridge in transfer ionization has not been previously discussed in the literature, nor has it been observed experimentally.