Coulomb-projected Born Approximation Research Articles

Angular distribution in the double ionization of helium by fast electrons: Effect of target e-e correlations.

Some kinematical arrangements in He(e,3e) process have been explored to ascertain the sensitivity of continuum electron-electron angular correlation to the electron-electron (e-e) correlation in the target wave function. The calculation has been done in the Coulomb-projected Born approximation with suitably chosen effective charges. It is found that the variation in the cross section is quite sensitive to the target e-e correlation when (i) the two low-energy ejected electrons are detected at fixed angles ${\mathrm{\ensuremath{\theta}}}_{\mathit{b}}$ and ${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$ but their energies ${\mathit{E}}_{\mathit{b}}$ and ${\mathit{E}}_{\mathit{c}}$ are varied such that ${\mathit{E}}_{\mathit{b}}$+${\mathit{E}}_{\mathit{c}}$ is held fixed, or (ii) one of the electrons is detected at some large fixed angle (say, 90\ifmmode^\circ\else\textdegree\fi{}) and the other angle ${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$ is varied at fixed energies.

Effect of distortion in electron impact excitation in Coulomb-projected Born approximation

We have calculated total and differential cross-sections for 1s →ns (n = 2, 3, 4) electron impact excitation of hydrogen and hydrogenic ions at various energies in Coulomb-projected Born approximation. Distortion due to static interactions, target polarization and exchange effects has been incorporated in the initial channel. The present calculations have been compared with other theoretical and experimental results.

Electron impact excitation of 11 S–21 S transition in helium

The modified variable-charge Coulomb-projected Born approximation is applied to electron impact excitation of 11 S–21 S transition in helium. The results are compared with other theoretical and experimental results.

Electron impact excitation of hydrogen and helium in Coulomb-projected Born methods

Coulomb-projected Born methods for the theoretical study of electron impact excitation of hydrogen and helium are reviewed. The results obtained by using different forms of Coulomb-projected Born methods are compared with other theoretical and experimental results and analyzed. The inadequacy of the variable charge Coulomb-projected Born approximation (VCCPB)—the most recent form of the Coulomb-projected Born methods—in giving good results in processes where exchange is dominant is discussed in detail. The ‘modified’ VCCPB approximation obtained by modifying the VCCPB method to remove its shortcomings is also discussed and its application to electron impact excitation of 23 s state of helium is studied.

On the variable-charge Coulomb-projected Born approximation

Total cross-sections for electron impact excitation of 11S – 23S transition in helium have been calculated using variable-charge Coulomb-projected Born approximation and also using a distorted wave model in which the prior form of theT-matrix is used. The comparison of the two sets of results enables us to make certain observations about the suitability of the variable-charge Coulomb-projected Born approximation.

Threshold behaviour for ionisation by electrons and positrons

The threshold laws for atomic ionisation by electron and positron impact are derived in the Coulomb-projected Born approximation, and compared with other theories. There is a very large difference between the resulting threshold forms for electron and positron impact.

21S excitation of heliumlike ions

The variable charge, Coulomb-projected Born (VCCPB) approximation is applied to the calculation of cross sections for electron impact 11S–21S excitation in heliumlike ions (3 ≤ Z ≤ 8 and Si12+, Ca18+, and Fe24+). Differential and total cross sections have been reported for incident energies ranging from near threshold to five times threshold energy and are compared with other available theoretical estimates.

Elastic scattering of electrons by hydrogen atoms at intermediate energies

The elastic scattering of electrons with hydrogen atoms at incident electron energies of 200--680 eV is studied using a two-potential method and the variable-charge Coulomb-projected Born approximation. Effects of exchange and polarization are included. The results obtained are found to be in reasonable agreement with recent experimental measurements. Comparison is made with other available theoretical calculations.

2S1excitation of the helium atom

The variable-charge Coulomb-projected Born approximation proposed by Schaub-Shaver and Stauffer is used to analyze the electron-impact excitation of the 2 /sup 1/S state of helium. Differential and integrated cross sections for the transition are calculated and compared with the other available theoretical calculations and experimental measurements.

Charge-transfer cross sections forαparticles colliding with hydrogenic ions

The Coulomb-projected Born approximation has been used to calculate the total charge-transfer cross sections for ..cap alpha.. particles (He/sup 2 +/) colliding with hydrogenic systems at medium and high energies. The present results overestimate the recent experimental data of Hvelplund et al. for the system He/sup 2 +/+H in the energy range 0.5 to 5 MeV. Further, it is seen that the present scaled results (Z/sup 6//sub T/sigma) for capture into the n = 1 state follow the universal curve in the energy range E/Z/sup 2//sub T/> or =100 keV when plotted against E/Z/sup 2//sub T/ and are in agreement with the scaling formula of Nikolaev.

Precise test of the Coulomb-projected Born approximation for the 2 1S and 3 1S excitation of helium by electron impact

By use of the accurately known generalized oscillator strength (i.e., the result of the first Born approximation due to Kim and Inokuti), we have evaluated the differential cross sections for the 2 1S and 3 1S excitation of helium by electron impact, much more precisely than before, within the Coulomb-projected Born approximation. The results are in good agreement with measurements by Dillon and Lassettre reported in 1975. The method of the present calculation is based on the Fourier decomposition of the Coulomb potential for interactions between the incident electron and electrons in the target atom, and has been used in a few other contexts. We point out here that the same method can be readily extended to many other scattering approximations such as distorted-wave approximations, either with or without account for electron-exchange effects.

Total charge transfer cross-sections for protons colliding with hydrogen atom

Total charge transfer cross-sections for protons impinging on hydrogen has been calculated in Coulomb-projected Born approximation. It is seen that the present calculation compares well with the available experimental data.

Differential cross sections for excitation of the helium atom by proton impact

The differential cross sections for the excitation of helium atoms to the n=2 level are calculated in the high-energy region by applying the Coulomb-projected Born approximation as proposed by Geltman (1971). Good agreement in the angular distribution is obtained with recently published data.

Polarised Coulomb-projected Born approximation for the triple differential cross section for electron impact ionisation of atomic hydrogen

The Coulomb-projected Born (CPB) approach for the ionisation of atomic hydrogen is extended to include polarisation effects. The results (triple differential cross sections) are compared with the recent experimental data of Weigold et al. (1979) and with the distorted-wave results of Smith et al. (1979) in the energy range 100-250 eV. It is found that the present results show a substantial improvement over the usual CPB results for small scattering angles and are quite close to those of Smith et al. (1979).

The variable-charge Coulomb-projected Born approximation

The authors have modified the generalised Coulomb-projected Born approximation proposed by Stauffer and Morgan (1975) by introducing a screening parameter which varies with the coordinates of the incident particle. Total and differential cross sections have been calculated at various medium and high incident energies for 1s to ns (n=2,3,4) transitions in electron-hydrogen collisions. Elastic scattering results are also included for electron-hydrogen collisions.

Triple differential cross sections for the electron impact ionisation of helium at 35 eV collision energy

In a coincidence experiment with coplanar geometry, angular correlations of the two outgoing electrons have been studied for the ionisation of helium atoms by electrons with an impact energy 10.5 eV above the first ionisation threshold. The measured angular dependencies of the triple differential cross section are compared with calculated results of the Coulomb-projected Born approximation including exchange. Neither the measured data nor the calculated ones show the features that have been found at medium and high impact energies. The momentum transfer K0a loses its importance as a means of classification. For these low-energy collisions the amplitudes of direct scattering, exchange scattering and electron capture are of similar size and only the direct scattering amplitude has the momentum transfer vector K0a as a symmetry axis (in Born approximation). Therefore the angular dependence of the triple differential cross section are no longer symmetrical with respect to the momentum transfer. Rapid variations of the angular distributions for different scattering angles theta a of one of the two outgoing ionisation electrons indicate interference of the different scattering amplitudes. Moreover the measurements demonstrated that the angular correlations for fixed scattering angle theta a are independent of the ratio of the energies of the two emerging electrons.

Charge-transfer cross sections for protons colliding with hydrogenic ions. I

Theoretical angular distributions and total charge-transfer cross sections have been studied for protons impinging on highly ionised (hydrogenic) systems He1+, Li2+, Be3+, B4+, C5+, N6+, O7+, Fe25+ in the Coulomb projected Born approximation of Geltman (1971). The angular distribution curves exhibit sharp dips in the forward direction for all the ionised systems. The present calculations of Glembocki and Halpern (1977). ZT7 sigma treated as a function of E/ZT2 for all the cases studied are found to follow an almost identical curve which lies lower than the OBK universal curve. The total cross sections converge to hydrogen results at high equivalent incident proton energy.

Total cross sections for electron impact excitation of the 11S→23S transition in He-like ions

Total cross sections for collisional excitation of the 11S to 23S transition in helium-like ions with atomic number 2<or=Z<or=8 have been computed using a modified form of the Oppenheimer approximation which is equivalent to Geltman's Coulomb-projected Born approximation when exchange is taken into account. Although this approximation cannot be derived from a variational principle it nevertheless yields a result for helium which compares surprisingly well with most of the available experimental data for incident energies ranging from near threshold up to 300 eV. This suggests that the results obtained for the ionised members of the sequence may be quite reliable.

Angular distribution for electron capture by protons from hydrogen atoms in the energy range 15-200 keV

Several authors have, using a variety of approximations, predicted a sharp minimum in the angular distribution for the process $p+\mathrm{H}(1s)\ensuremath{\rightarrow}\mathrm{H}(1s)+p$, over a wide range of energies. I have used the results of recent impact-parameter coupled-state calculations, in conjunction with the eikonal approximation, to calculate differential cross sections for this process in the energy range 15-200 keV; no sharp minima were found, but at the lowest end of the energy range there is some indication of the emergence of a dip. The present results are qualitatively consistent with the results of Glembocki and Halpern who used a Coulomb-projected Born approximation.

Generalized Coulomb-projected Born-approximation cross sections for atomic excitation by charged-particle impact

An expression is given for the (${n}_{i}$, ${l}_{i}=0\ensuremath{\rightarrow}{n}_{f}$, ${l}_{f}=0$) excitation cross section in terms of a single one-dimensional integral with the aid of a transformation to prolate spheroidal coordinates. Results are given for the $1s\ensuremath{-}2s$ excitation of arbitrary-$Z$ hydrogenic targets. For our choice of wave functions the cross sections are found to scale as ${Z}^{\ensuremath{-}4}$ at fixed $\frac{{E}_{\mathrm{inc}}}{{Z}^{2}}$. The principal difference between the Coulomb-projected Born-approximation (CPBA) and Born results is in the large-angle portion of the differential cross section. This, however, leads to CPBA and Born momentum-transfer cross sections which differ by less than 20% at high energies for $1s\ensuremath{-}2s$ transitions.