Plane-wave Born Approximation Research Articles

Ion-atom inelastic scattering cross sections and energy loss in the plane-wave Born approximation

A procedure discussed by Bates for calculating ion-atom excitation and ionization cross sections and energy loss in the plane-wave Born approximation has been programmed. The procedure involves integration over the product of elastic scattering factors or generalized oscillator strengths for excitation or ionization from both projectile and target. The calculations are done in the center-of-mass system where the distinction between projectile and target is lost. Thus the cross sections and energy loss in the laboratory frames of both target and projectile emerge from a center-of-mass calculation symmetrical in nuclear and net charges. The traditional simple modeling of energy loss, using scaled proton stopping power (SP) and an effective projectile charge, is unsymmetrical, and therefore dubious as a guide for extrapolating to ion-ion energy loss. Because the energy loss includes processes that cannot be uniquely allocated to one ion or the other, e.g., mutual ionization, one does not expect, in general, a simple SP picture to suffice. However, it is found at intermediate and high center-of-mass energies that the energy loss is dominated by elastic scattering of one ion and ionization of the other. From this observation, and the requirement that hydrogenic ion results be recovered in the appropriate limit, a simple extension of the Bethe formula is obtained. Explicit subshell and total ionization cross sections are presented for Li ions interacting with neutral Zn and Au, and the calculated energy loss is compared with experiment. It is found that the simple extension of the Bethe formula is in surprisingly good agreement with the explicit calculations, suggesting that an improved extension of the Bethe formula may exist.

M-shell x-ray production cross section measurements in Pb and Bi due to the impact of protons and nickel ions

M-shell x-ray production cross sections in Pb have been measured using 58 - 82 MeV ions and those in Bi by 1 - 5 MeV protons and 58 - 82 MeV ions (q = 5, 6). The experimental results have been compared with the predictions of the plane-wave Born approximation (PWBA) and with the calculations of the perturbed stationary state theory (ECPSSR) accounting for the energy loss (E), Coulomb deflection (C) and relativistic (R) effects. The present results of proton impact on bismuth have also been compared with the experimental data available from other investigations. To the best of our knowledge, the M-shell x-ray production cross sections in Pb and Bi by Ni-ion impact are reported here for the first time. Good agreement of the present results with the predictions of the ECPSSR theory has been observed for the total M x-ray cross sections in Bi due to proton impact while discrepancies to a factor of 1.5 - 2.5 in the case of total M x-ray cross sections have been observed between the present experimental and theoretical (PWBA and ECPSSR, respectively) results for Pb and Bi in the case of Ni-ion impact. These discrepancies between the experimental and theoretical results could be attributed to the effect of multiple ionization on the x-ray emission probabilities. If the modified value of the average fluorescence yield accounting for the outer-shell multiple ionization is used, the theoretical results of total M-shell x-ray cross sections based on the PWBA theory show fair agreement with the experimental results (within experimental uncertainties) while the ECPSSR theory is found to underestimate the experimental results by about 40% for both Pb and Bi in the case of Ni-ion impact. Furthermore, the and x-ray cross sections have been evaluated experimentally for both Pb and Bi. The comparison of these results with the corresponding theoretical estimates have been done only for Pb as the emission rates for various M x-ray transitions of Bi are not available in the literature. The comparison of the cross sections show fair agreement with the PWBA-based theoretical values, while there are discrepancies of a factor of 10 and 20 in the cases of and x-ray transitions.

Electronic stopping power of 10B channeled into the Si 〈100〉 axial direction

In the present work we report measurements of 10B stopping powers in channeled Si 〈100〉 direction. The measurements were carried out with a SIMOX sample in the energy range of 500–9000 keV using a backscattering method. In the lower energy region (up to 800 keV) the results follow an dE dx ∝ E p regime with p = 0.55 ± 0.05 in good agreement with current theories. In addition, the obtained values show a maximum at around 5 MeV, which arises from the maximum contribution of the L shell to the total electronic stopping power. This behavior is in agreement with predictions by recent calculations using the Plane-Wave Born approximation.

Electron emission from binary collisions with ZProjectile ≤ ZTarget

We have measured doubly differential cross sections for electron emission in ion-atom collisions. In one set of experiments, 0.5 MeV/u He 2+ projectiles impinged on the target atoms He, Ne, Ar, Kr and Xe and molecules N 2 and O 2; in another set, H + (energy ranging from 2.0 to 6.0 MeV) and 0.75 MeV/u N q+ ( q = 2–7) collided with Ar targets. In both cases, emitted electrons were observed under 0°. Major interest was directed upon the position of the Binary Encounter Peak, which is commonly assumed to follow a simple kinematic model. Surprisingly the peak is significantly shifted towards lower energies. For comparison we calculated spectra with the Plane Wave Born Approximation and the Plane Wave Approximation. These do not show in general good agreement with our experimental data, presenting large discrepancies not only in absolute values but also in the behaviour with the target atomic number.

Systematics of subshell electron ionization cross sections

The excellent agreement between recent converged close-coupling calculations of electron ionization cross sections of atoms and ions and plane-wave Born approximation (PWBA) calculations leads to the hypothesis that PWBA calculations are accurate to 10% at high energy, and therefore can be used to renormalize experimental data to produce cross sections of 10% accuracy near threshold. Comparisons are made of various calculations of 2s and 3s electron ionization cross sections in low-Z atoms and ions, of PWBA calculations and experiments for atoms and ions up to , and of scaled peak cross sections to interrelate experiments and calculations on isoelectronic sequences.

Angular distribution of autoionization and Auger electrons ejected by electron impact from laser-excited and polarized atoms

A general formalism is developed for a description of the angular distribution of electrons ejected after the electron-impact excitation/ionization of the laser-excited and polarized atoms. Properties of the angular distributions are analysed in a general case as well as in some particular geometries of the experiment. Within the two-step approximation the formalism is applied to the resonant single and double ionization with ejection of autoionization and Auger electrons. As an example the angular distributions of autoionization electrons from laser-excited sodium atoms are calculated in the plane-wave and distorted-wave Born approximations. Angular correlations between autoionization and scattered electrons, which can be measured in a coincidence (e,2e) experiment with polarized targets, are also analysed.

On the single- and double-ionization of He by protons and anti-protons

In this paper we present the calculations of: (a) the single-ionization cross section of He by protons and anti-protons in the incident energy range of about 10 keV - 50 MeV, (b) the double-ionization cross sections by both projectiles in the energy rage of about 50 keV - 20 MeV and (c) the ratio of double- to single-ionization in the energy range of 50 keV - 20 MeV. The calculations have been performed in the two-coupled channel plane wave Born approximation, which is essentially the Born approximation that includes the final channel interaction between two electrons, in addition to the usual projectile - electron potential. The projectile has been represented by plane waves, and a correlated Hylleraas type of wavefunction has been used for the initial ground state. For the single-ionization case the final state electronic wavefunction is taken to be an anti-symmetrized product of a hydrogen-like (ls) and a continuum Coulomb wavefunction. For the double-ionization case, the final state wavefunctions of the two ejected electrons are taken to be Coulomb functions for calculating the matrix elements of the electron - projectile interaction, whereas plane waves are used for calculating the matrix elements of the electron - electron interaction. The calculated results are in reasonable agreement with the data in all three cases. The single-ionization cross section has also been calculated in the Lewis - Merzbacher approximation to determine the range of incident energy where it might represent a suitable approximation.

Photodisintegration of the triton with realistic potentials

The process $\ensuremath{\gamma}+t\ensuremath{\rightarrow}n+d$ is treated by means of three-body integral equations employing in their kernel the $W$-matrix representation of the subsystem amplitudes. As compared to the plane wave (Born) approximation the full solution of the integral equations, which takes into account the final state interaction, shows at low energies a 24% enhancement. The calculations are based on the semirealistic Malfliet-Tjon and the realistic Paris and Bonn $B$ potentials. For comparison with earlier calculations we also present results for the Yamaguchi potential. In the low-energy region a remarkable potential dependence is observed, which vanishes at higher energies.

L subshell and total M shell x-ray production cross sections of Hg for protons of energy 0.7 - 2.9 MeV

X-ray production cross sections of individual L lines, L and M total of Hg were measured for 0.7 to 2.9 MeV protons. The experimental results were compared with the theoretical predictions of the plane wave Born approximation (PWBA), ECPSSR and the semi-empirical formula of Orlic. Good agreements were obtained for L1 at lower energies and L2 and L3 at higher energies with Orlic and ECPSSR predictions, respectively. The influence of using different databases for x-ray fluorescence yields and Coster - Kronig transition probabilities is also discussed. The results of this work were not compared to any other reported data, since no experimental cross sections for mercury were found in the literature.

Measurement of M X-ray production cross sections in gold by proton, oxygen and nickel ion impact

Measurements of M-shell X-ray production cross sections in Au with proton, O 5+- and Ni 5+-ion impact have been performed. The energy of the incident protons range from 1 to 5 MeV, oxygen ions from 66 to 72 MeV and nickel ions from 58 to 87 MeV. The experimental results have been compared with the predictions of the plane wave Born approximation (PWBA) and the perturbed stationary state theory with energy loss, Coulomb deflection and relativistic corrections (ECPSSR). For proton impact the measured cross sections have been found to be in good agreement with both these theories. With O-ion impact the ECPSSR theory underpredicts the experimental data while the PWBA agrees fairly well. Both the theories, however, strongly underestimate the X-ray cross sections measured with Ni-ion impact. The discrepancy between the experimentally measured and theoretically calculated cross sections in case of heavy ion impact could be due to multiple ionization which changes the values of atomic parameters like sub-shell fluorescence yields, Coster-Kronig transition probabilities and emission rates.

K-, L-, and M-shell ionization of atoms by electron and positron impact

The plane-wave Born approximation with Coulomb, relativistic, and exchange corrections is employed to obtain the K-, L1-, L2-, L3-, and M-shell ionization cross sections of a number of atoms bombarded by electrons and positrons in the energy range varying from the threshold of ionization to 1 GeV. Transverse interaction of virtual photons with atoms is also included and it is found to be of great significance for impact energies greater than about 1 MeV. For K- and L-shell ionization, good agreement between the theoretical values and various experimental data for electron-impact cross sections is obtained. However, for the M shell, the theory overestimates the experimental cross sections. For positron impact the agreement between the present results and the limited experimental data is found to be quite satisfactory.

The K- and L-shell ionization of atoms by 1 GeV protons

Cross sections for K-vacancy and L-x-ray production by 1 GeV protons for elements ranging from Cu to U are measured and compared with calculations based on the plane-wave Born approximation. The analysis revealed the need to include the current - current interaction between the projectile and target electron and the spin-flip term.

Energy and angular distributions of electrons from ion impact on atomic and molecular hydrogen. III. 28-114-keV He++H2.

Absolute cross sections, differential in energy and angle of ejection of the secondary electrons, were determined for electron emission from ${\mathrm{He}}^{+}$+${\mathrm{H}}_{2}$ collisions at 28--114 keV by measuring electron energy spectra from 1.5 to 300 eV at several angles from 15\ifmmode^\circ\else\textdegree\fi{} to 160\ifmmode^\circ\else\textdegree\fi{}. A rotatable electrostatic analyzer was used with an energy resolution of 5% and an angular acceptance of 4.6\ifmmode^\circ\else\textdegree\fi{} full width at half maximum. The double-differential cross sections were integrated over angle, energy, or both to obtain single-differential and total ionization cross sections. The latter are in excellent agreement with previous experimental results. Agreement of the experimental double-differential cross sections with plane-wave Born approximation calculations is generally poor. The cross section for ejection of high-energy electrons by ${\mathrm{He}}^{+}$ is considerably larger than that for ${\mathrm{H}}^{+}$ at the same velocity. This is attributed mostly to electron emission from the projectile. The presence of prominent Doppler-shifted helium autoionization peaks in the spectra indicates an appreciable probability for electron capture by the helium ions with simultaneous double excitation. \textcopyright{} 1996 The American Physical Society.

Two-center electron emission in collisions of fast, highly charged ions with He: Experiment and theory.

Electron emission cross sections differential in energy and angle of the ejected electrons were measured in collisions of 5-MeV/{ital u} C{sup 6+}, O{sup 8+}, and Ne{sup 10+} ions on He. The experimental results deviate substantially from calculations based on the plane-wave Born approximation indicating that the ejected electrons are significantly affected by the two-center field of the target and projectile ion. The two-center effects were confirmed by comparison with results from the continuum-distorted-wave--eikonal-initial-state (CDW-EIS) approximation that has recently been improved by using Hartree-Fock-Slater wave functions for the ejected electrons. It is shown for nearly complete energy and angular ranges that the improved CDW-EIS method predicts cross sections in excellent agreement with experiment.

Excitation of K*(3p54s2 2P3/2, 2P1/2) by electron impact in the range from near threshold to 500 eV: alignment and cross section ratios

We have measured the alignment A20 of K*(3p54s2 2P3/2) and the ratio of cross sections R0 = sigma (3/2)/ sigma (1/2) of fine structure states K*(3p54s2 2P3/2, 2P1/2) for electron impact excitation in the range of incident energy E0 = 31.4-500 eV. We have also calculated these quantities in the distorted wave Born approximation (DWBA) and the plane wave Born approximation (PWBA). There is good agreement between experiment and the present DWBA and PWBA values for the alignment A20 whereas earlier DWBA values by Pangantiwar and Srivastava (1987) deviate systematically from experiment for E0 < 100 eV. For the ratio of cross sections R0 the present DWBA values agree only qualitatively with the experiment for E0 < 100 eV.

L-shell ionization in Au by O5+-and Ni5+-ion impact

TheL X-ray production cross sections in gold by 60 to 72 MeV O5+ ions and 58 to 87 MeV Ni5+-ions have been measured. TheL-subshell ionization cross sections derived from these experimental results have been compared with the direct ionization theories viz. plane wave Born approximation (PWBA) theory and modified perturbed-stationary-state theory with energy loss, Coulomb deflection and relativistic effects (MECPSSR). A new procedure has been described to account for the change in the yield ratioLβ1/Lβ2, 15 with energy, for Ni5+-ion impact on gold. TheL sub-shell ionization cross sections have been derived fromLβ1,Lβ4 andLβ2, 15 lines of theLβ group in addition to those calculated by the conventional Datz TRY3 technique usingLα,Lγ1 andLγ2, 3 X-ray lines. From the shift in the energies of variousL X-ray lines and changes in their intensities, 3 and 5 spectator vacancies in theM- andN-shells in gold have been estimated with O5+-ion impact while 7, 20 and 4 to 6 spectator vacancies have been inferred in theM-,N-andO-shells respectively with Ni5+-ion impact in the energy range of the projectiles undertaken in the present studies.

Inclusion of Final-State-Interaction Effects at Higher Energies in the Photodisintegration of the α-Particle

Final-state-interaction (FSI) effects are included in the treatment of the photodisintegration of the α-particle via the K-matrix approximation. These are applied to the exact AGS-type integral equation for this process at photon energies 50–100 MeV. Direct comparison with the plane-wave (Born) approximation indicates that these effects are small, which explains the good fit to the data we found previously. The FSI in the integro-differential equation treatment of the problem in configuration space are included by means of effective nucleon-trinucleon potentials constructed via an exact inverse scattering method and by using the phase shifts obtained from the K-matrix approach to the 3 + 1 → 3 + 1 scattering. Comparison between this approach and the integral-equation approach confirms the validity of this model previously employed at low energies using experimental phase shifts. Finally we show that the photonuclear cross sections are sensitive to the details of the boundstate wave functions of the 3He and 4He as calculated exactly via the AGS equations and in the integro-differential-equation approach, which for the 4He neglects higher-order correlations.

Tissue characterization of arteries with 4 MHz ultrasound

Reflections of 4 MHz ultrasound pulses from 60 sites on 15 human aorta specimens were analysed using the plane-wave Born approximation (PWBA) inverse scattering method. The recovered high-resolution acoustic impedance profile for each site was correlated with an analysis (by microscope) of sections taken from that site. The shape of the impedance profile was found to be sensitive to the pathology of the tissue at each site. The severity of the various conditions could be gauged from the recovered impedance profiles. Some of the insonifications were done on aorta specimens that were shielded from the ultrasound transducer by a human tissue specimen about 1.25 cm thick in order to study the effects of intervening tissue. We found that it was possible to compensate for the distortion due to the frequency dependent attenuation of the intervening tissue. We conclude that ultrasound imaging, with the reflections analysed by means of PWBA inverse scattering provides a promising non-invasive tissue characterization modality.

Systematic ion-atom interaction cross sections and stopping powers in the plane wave Born approximation

In Chapter 14 of Atomic and Molecular Processes, Bates (1962) outlines a procedure for calculating ion-atom cross sections in the plane-wave Born approximation (pwBa). The procedure involves integration over the product of elastic scattering factors or generalized oscillator strengths for excitation or ionization from both projectile and target. We have programmed this procedure to use our large database of excitation and ionization generalized oscillator strengths (GOS). The program calculates both cross sections (CS) and stopping power (SP) on a subshell basis. The calculations are done in the center of mass system where the distinction between projectile and target is lost. Thus, the SP in the laboratory frames of both target and projectile are symmetrical in nuclear and net charges. The traditional simple modeling of SP, using scaled proton SP and an effective projectile charge, is unsymmetrical, and therefore dubious as a guide for extrapolating to ion-ion SP. At high projectile energy, the SP curves, as a function of increasing projectile charge, approach the scaled protonic result from above, indicating that lowering the average charge raises the SP, in contradiction to the traditional picture that the projectile SP increases with increasing effective charge (assuming there is an underlying physical reality relating the effective and average charge). Comparison with experimental SP data (mostly from 30 years ago) shows generally poor agreement for Li ion projectiles in the 1–10 MeV range.

Effects of multiple ionization on the Kα spectrum of aluminum in intense lithium beam experiments

The effect of multiple ionization on Kα spectra is investigated for aluminum targets irradiated by intense lithium beams. Multiple ionization cross sections have been calculated using a formulation incorporating single-electron ionization probability in the binomial distribution. In contrast to conventional binary-encounter approximation (BEA) theory, the single-electron ionization probabilities for each atomic shell have been calculated using a combination of modified plane-wave Born approximation (MPWBA) and an empirical effective interaction radius which is dependent on both the target ion and the projectile. Our calculations show that the effect of multiple inner-shell ionization on aluminum Kα spectra observed in intense lithium beam experiments is important. Multiple ionization effects become less important as the target ionization state increases. Nevertheless, even for highly ionized species up through Be-like Al multiple ionization effects can be significant and must be considered in the analysis of spectra obtained in intense Li beam-plasma interaction experiments.