Semiclassical Approximation Calculations Research Articles

Observation of internal structure of theL-shell x-ray hypersatellites for palladium atoms multiply ionized by fast oxygen ions

An observation of the internal structure of the L-shell hypersatellite x rays resulting from the one-photon decay of L{sup -2} double-vacancy states in palladium multiply ionized by oxygen ions is reported. The Pd L{sub 3}{yields}M{sub 4,5} x-ray spectrum was measured with a von Hamos high-resolution crystal spectrometer. The complex shape of the observed spectrum could be interpreted in detail using relativistic multiconfiguration Dirac-Fock calculations. The relative intensities of the measured x rays were found to be in good agreement with semiclassical approximation calculations using relativistic Dirac-Hartree-Fock wave functions.

High-resolution X-ray study of the multiple ionization of Pd atoms by fast oxygen ions

The multiple ionization of the L- and M-shells of Pd by fast oxygen ions has been studied by measuring with high-resolution the satellite structures of the Lα1,2 X-ray transitions. Relativistic multi-configuration Dirac-Fock (MCDF) calculations were used to interpret the complex X-ray spectrum, allowing to derive the number of L- and M-shell spectator vacancies at the moment of the X-ray emission. After correcting these numbers for the atomic vacancy rearrangement processes that take place prior to the X-ray emission, the ionization probabilities corresponding to the collision time were obtained. The latter were compared to predictions of the semiclassical approximation (SCA) and the geometrical model. The SCA calculations were performed using relativistic hydrogenic and self-consistent Dirac-Hartree-Fock (DHF) electronic wave functions. It was found that the use of the more realistic DHF wave functions in the SCA calculations leads to a much better description of the measured ionization probabilities for both the L- and M-shells.

X-ray study ofM-shell ionization of heavy atoms by 8.0–35.2-MeVOq+ions: The role of the multiple-ionization effects

The $M$-shell ionization in high-$Z$ atoms by ${\text{O}}^{q+}$ ions have been studied systematically in the energy range of 8.0--35.2 MeV in order to verify the available theoretical approaches describing the $M$-shell ionization by charged particles in asymmetric collisions. The measured $M$ x-ray spectra were analyzed taking into account the effects of x-ray line shifting and broadening caused by the multiple ionization in the $M$ and $N$ shells. The $M$-subshell ionization cross sections, derived by using the $M$-shell decay rates modified for the multiple ionization effects, have been compared with the theoretical predictions based on the plane-wave Born approximation (PWBA), the semiclassical approximation (SCA), and the binary-encounter approximation (BEA). In the PWBA approach two theoretical calculations were considered: the energy-loss Coulomb deflection perturbed stationary state relativistic (ECPSSR) theory and its recent modification called the energy-loss Coulomb deflection united and separated atoms relativistic (ECUSAR) theory, which corrects a description of the electron binding effect to account for the united and separated atoms (USA) electron binding energy limits. In the SCA calculations performed with relativistic hydrogenic wave functions the binding effect was included in the limiting cases of separated-atom (SA) and united-atom (UA) limits. The measured $M$-subshell ionization cross sections are the best reproduced by the SCA-UA calculations, with exception of the ${M}_{2,3}(3p)$-subshell cross sections which are strongly enhanced and cannot be reproduced by the discussed calculations.

M-shell ionization of heavy elements by0.1–1.0MeV∕amuH1,2andHe3,4ions

The $M$-shell ionization in high-$Z$ atoms by low-energy light $_{1}^{1}\mathrm{H}$, $_{1}^{2}\mathrm{H}$, $_{2}^{3}\mathrm{He}$, and $_{2}^{4}\mathrm{He}$ ions have been studied systematically in the energy range $0.1--1.0\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}∕\mathrm{amu}$ in order to verify the available theoretical approaches describing the $M$-shell ionization by charged particles in asymmetric collisions. The present low-energy data, combined with our earlier results reported for $M$-shell ionization by hydrogen and helium ions for higher energies, form a systematic experimental basis to test the theoretical predictions of $M$-shell ionization based on the plane-wave Born approximation (PWBA), the semiclassical approximation (SCA), and the binary-encounter approximation (BEA). In the PWBA based approaches the energy loss (E), Coulomb deflection (C), perturbed stationary state (PSS), and relativistic (R) effects were considered within the ECPSSR theory and its recent modification, called the ECUSAR theory, in which a description of the PSS effect was corrected to account for the united- and separated-atom (USA) electron binding energy limits. In the SCA calculations with relativistic wave functions the binding effect was included only in the limiting cases of separated-atom and united-atom limits. Possible contribution of the electron capture, multiple ionization, and recoil ionization to the $M$-shell vacancy production, which is dominated for light ions impact by direct single ionization process, are also discussed. The universal scaling of measured $M$-shell x-ray production and ionization cross sections was investigated in detail. Using the present data the isotopic effect has been studied by comparing the measured $M$-shell ionization cross-section ratios for equal-velocity hydrogen $_{1}^{1}\mathrm{H}$ and $_{1}^{2}\mathrm{H}$ as well as helium $_{2}^{3}\mathrm{He}$ and $_{2}^{4}\mathrm{He}$ isotopes. In addition, the ratios of measured ionization cross sections for $_{1}^{2}\mathrm{H}$ and $_{2}^{4}\mathrm{He}$ were used to investigate the role of the binding effect. The present results are of practical importance for the application of particle-induced x-ray emission technique in trace element studies.

Relativistic wave-function effect on theK-shell ionization of Sb, Gd, Yb, Au, and Bi by low- to intermediate-velocity F ions

We have measured absolute cross sections for the K-shell ionization of medium- and high-Z targets of Sb, Gd, Yb, Au, and Bi induced by low- to intermediate-velocity F ions having energies between 2.5 and 5.8 MeV/u. Our main interest is to see the effect of the relativistic nature of the K-shell electrons of these target atoms on the ionization cross sections. The information on the degree of relativistic effect has been obtained by comparing the measured data with different theoretical calculations with and without including the relativistic corrections. A comparative study of the two different models such as SCA (semiclassical approximation) and ECPSSR [perturbed stationary state (PSS) including the corrections for energy (E) loss, Coulomb (C) deflection, and relativistic (R) effects] is presented. It is shown that the SCA calculations with the relativistic wave function predict an ionization cross section that is at least an order of magnitude higher compared to that given by the nonrelativistic calculation for Bi target. This factor is reduced to about 2 in the case of Sb. The ECPSSR, however, predicts lower ratios for the relativistic to nonrelativistic calculations. The experimental results, in general, are in good agreement with the SCA calculations using relativistic wave functions as well as with the ECPSSR model. For high-Z targets the SCA gives slightly better agreement with the data compared to the ECPSSR. In addition, it is shown that in the ECPSSR formalism the ionization cross sections of high-Z (with large relativistic effect) as well as low-Z targets (with less relativistic effect) can be scaled approximately to follow a universal curve after including the relativistic correction.

K-shell ionization by low-energy 14N ions

K-shell ionization by slow, heavy projectiles has been studied for moderately asymmetric ( Z 1/ Z 2 ∼ 0.3) collisions by measuring K-X-ray production cross sections in Ca, Sc, Ti, Cr, Fe and Cu excited by 14N ions of energy 1.75–3.15 MeV. The influence of multiple ionization in L- and M-shells on measured data was estimated by analyzing K-X-ray energy shifts and K β/K α intensity ratios. The results are compared with the predictions of K-shell ionization theories, based on the plane-wave Born approximation (PWBA), semiclassical approximation (SCA), binary-encounter approximation (BEA), and adiabatic molecular orbital (MO) approach. Different corrections to these approaches, to account for the higher-order effects, are discussed. The data are not satisfactorily described in detail by discussed theories over the whole energy range, but the SCA calculations reproduce the present, low-energy data quite reasonably. The need for a more realistic description of the electron binding effect is discussed.

Projectile energy loss in multiply ionizing ion-atom collisions.

The projectile energy loss for 7.5--25-MeV ${\mathrm{C}}^{6+,5+}$ and ${\mathrm{F}}^{6+}$ ions was measured for single collisions with He, Ne, Ar, and Kr as a function of the recoil-ion charge state and the projectile scattering. This energy loss was measured for collisions in which the projectile captured an electron and for those involving just direct ionization. We investigated and found a large average energy transfer (100--250 eV/electron) to the continuum electrons. A strong increase of the scattering angle with recoil-ion charge state was observed for both capture and direct ionization. The results imply that, for smaller impact parameters, higher recoil-ion charge states are produced and that higher energy losses are obtained. We observed a weak target-Z dependence of the energy loss. The results are compared with n-body classical-trajectory Monte Carlo calculations by Olson, semiclassical-approximation calculations by Schuch et al. [Nucl. Instrum. Methods Phys. Res. Sect. B 42, 566 (1989)], and the energy-deposition model.

Impact Parameter Dependent Probability of Au L3-Shell Ionization by 40.7 MeV Ar Ions

The ionization probability of Au–L 3 shell by 1 MeV/u Ar ions is measured as a function of impact parameter by a scattered projectile-x ray coincidence method. The result is compared with semi-classical approximation calculations with separated-atom and united-atom electron wave functions for the initial state of the ionized electron. It is close to or larger than the separated-atom values at impact parameter larger than the radius of Au–L shell and it approaches to the united-atom value at smaller impact parameters. This behavior is qualitatively consistent with the binding effect.

A test of theoretical K-shell ionization cross sections for protons

Various sophisticated theories (relativistic plane wave Born approximation (RPWBA), semi-classical approximation (SCA), coupled-channels codes) are tested by comparison to our recent extended list of empirical K-shell ionization cross sections for protons. Differences between the theories are outlined. Surprisingly, the elaborate coupled-channels codes appear low compared to experiment, especially for light targets. Possible reasons for this are discussed. The RPWBA calculations describe the data better in a large projectile velocity range, in spite of the fact that binding and Coulomb effects are only treated as corrections. The SCA calculations by Hansteen et al. appear closer to the data than those by Trautmann; this is surprising since both codes should give identical results. Systematic comparisons are still severely hampered by the fact that most published cross sections describe different targets or different energies.

On the semiclassical description of Coulomb ionization of inner shells

The theory of ionization of inner shell electrons induced by point-like charged particles is presented in the framework of the semiclassical approximation (SCA) using hyperbolic orbits for the motion of the projectile in the Coulomb field of the target. The accuracy of the SCA results is investigated by comparison with quantum mechanical calculations. Different choices for the electron wave functions in the initial and final state and their influence on the corresponding radial form factor are discussed. Finally, SCA calculations are compared with recent experiments and possible additional corrections to this SCA model are stated.

Systematics of the averageL-shell ionization probability inK-shell ionizing collisions by light ions

The apparent average $L$-shell ionization probabilities in $K$-shell ionizing collisions by He, C, O, and Ne ions incident on Al, Si, and Cl targets at relative velocities ranging from 2 to 8.5 have been determined from high-resolution $K\ensuremath{\alpha}$ x-ray measurements. These data, in combination with data from previous work, enabled a detailed examination of the dependence of the single $K$- plus multiple $L$-shell ionization process on projectile and target atomic number, and on relative velocity. The He-ion-on-Al data are compared with the results of binary-encounter approximation and semiclassical approximation calculations, and a general semiempirical scaling law is deduced for light-ion collisions.