Plane-wave Born Approximation Calculations Research Articles

SBETHE: Stopping powers of materials for swift charged particles from the corrected Bethe formula

The Fortran program sbethe calculates the stopping power of materials for swift charged particles with small charges (electrons, muons, protons, their antiparticles, and alphas). The electronic stopping power is computed from the corrected Bethe formula, with the shell correction derived from numerical calculations with the plane-wave Born approximation (PWBA) for atoms, which were based on an independent-electron model with the Dirac–Hartree–Fock–Slater self-consistent potential for the ground-state configuration of the target atom. The density effect correction is evaluated from an empirical optical oscillator strength (OOS) model based on atomic subshell contributions obtained from PWBA calculations. For projectiles heavier than the electron, the Barkas correction is evaluated from the OOS model, and the Lindhard–Sørensen correction is estimated from an accurate parameterization of its numerical values. The calculated electronic stopping power is completely determined by a single empirical parameter, the mean excitation energy or I value of the material. The radiative stopping power for electrons, and positrons, is evaluated by means of Seltzer and Berger's cross section tables for bremsstrahlung emission. The program yields reliable stopping powers and particle ranges for arbitrary materials and projectiles with kinetic energy larger than a certain cutoff value Ecut, which is specific of each projectile kind. The program is accompanied by an extensive database that contains tables of relevant energy-dependent atomic quantities for all the elements from hydrogen to einsteinium. sbethe may be used to generate basic information for dosimetry calculations and Monte Carlo simulations of radiation transport, and as a pedagogical tool. Program summaryProgram title:sbetheCPC Library link to program files:https://doi.org/10.17632/7zw25f428t.1Licensing provisions: CC by NC 3.0Programming language: Fortran 90/95Nature of problem: The program calculates the stopping power of arbitrary materials for swift charged projectiles with small charges. The material is characterized by its chemical composition, mass density, and the empirical I value. The considered projectiles are electrons, positrons, negative muons, antimuons, protons, antiprotons, and alphas, which are described as point particles characterized by their mass and charge. If the actual I value of the material is known, the results from the program are expected to be reliable for projectiles with kinetic energy higher than a value Ecut, of the order of 1 keV for electrons and positrons, 150 keV for muons and antimuons, 0.75 MeV for protons and antiprotons, and 5 MeV for alpha particles.Solution method: The electronic stopping power is calculated by means of a corrected Bethe formula [1], which combines the conventional Bethe logarithm with the following corrections,1) the shell correction obtained from calculations based on the plane-wave Born approximation with the self-consistent Dirac–Hartree–Fock–Slater (DHFS) potential of neutral atoms in their ground-state configuration [2],2) the density effect correction, which accounts for the reduction of the stopping power caused by the dielectric polarization of the medium,3) a parameterization of the Lindhard–Sørensen correction, which generalizes the Bloch correction for relativistic projectiles, and4) the Barkas correction, which accounts for differences between the stopping powers of particles and their antiparticles.The density-effect and the Barkas corrections are calculated from a model of the optical oscillator strength (OOS) of the material, which combines the contributions of inner atomic subshells calculated with the DHFS potential, with a classical oscillator model for the contribution of valence electrons,A simple extrapolation formula is used to extend the calculated electronic stopping power to energies less than Ecut to allow the calculation of particle ranges.For electrons and positrons, the radiative stopping power is calculated from numerical tables prepared by Seltzer and Berger [3].Additional comments including restrictions and unusual features: The calculated stopping power is determined by a single parameter, the mean excitation energy or I value. The program assigns to each material a default I value, derived from the recommendations in the ICRU Report 37, which can be changed by the user. The distribution package includes text files with tables of atomic energy-dependent quantities (subshell optical oscillator strengths, shell corrections, scaled cross sections for bremsstrahlung emission) that are used in the calculations.

New measurements of Mαβ, Mγ and total M-shell X-ray production cross sections induced by carbon ions on Bi and Pt targets

In this work X-ray and elastic backscattered spectra from Pt and Bi thin targets were simultaneously measured using C4,3+ ions in the energy range 2.5 MeV to 5 MeV. Mαβ, Mγ and total M-shell X-ray production cross sections were extracted from collected X-ray and backscattered particle spectra using calibrated SDD X-ray and silicon solid state particle detectors. The obtained cross section values are compared to theoretical values employing commonly used datasets for fluorescence yields, emission rates, Coster-Kronig transitions and M-shell ionization cross sections based on the Semi-classical Approximation (United Atom and Separated Atom limits), PWBA (Plane Wave Born approximation), and ECPSSR models. The ISICS code was used for PWBA and ECPSSR calculations.

Energy loss in the ECPSSR theory and its calculation with exact integration limits

ECPSSR cross sections for inner-shell ionization are compared with numerical calculations that use correct and wrong expressions for exact limits of momentum transfer. For very adiabatic collisions, the employment of the wrong expression, as done in several popularly utilized computer codes, overestimates the cross sections up to two orders of magnitude. By contrast, even in such collisions the original ECPSSR approach of Brandt and Lapicki that approximates the effect of projectile energy loss by a multiplicative correction function fS and the relativistic effects by a relativistically corrected electron mass yields cross sections that are in agreement within a factor of two with respect to the calculation with hydrogenic Dirac wavefunctions and exact limits of integration with no relativistic correction. For plane-wave Born approximation calculations, the relativistic corrected expressions of the exact limits of integration are presented. The cross section calculated with such limits differs from the ECPSSR calculation using the fS-function by less than 2% in the region of lowest collision velocities covered by experiments. A straightforward modification of the existing computer codes with the correct expression for exact momentum transfer is suggested.

Accurate cross sections for microanalysis

To calculate the intensity of x-ray emission in electron beam microanalysis requires a knowledge of the energy distribution of the electrons in the solid, the energy variation of the ionization cross section of the relevant subshell, the fraction of ionizations events producing x rays of interest and the absorption coefficient of the x rays on the path to the detector. The theoretical predictions and experimental data available for ionization cross sections are limited mainly to K shells of a few elements. Results of systematic plane wave Born approximation calculations with exchange for K, L, and M shell ionization cross sections over the range of electron energies used in microanalysis are presented. Comparisons are made with experimental measurement for selected K shells and it is shown that the plane wave theory is not appropriate for overvoltages less than 2.5 V.

Bound-free electron-positron pair production in relativistic heavy-ion collisions

We study the bound-free electron-positron pair production in relativistic heavy-ion collisions to different bound states in a full plane-wave Born approximation calculation. Exact Dirac wave functions are used for both the bound electron and the free positron in the final states. Results for the relativistic heavy-ion collider as well as the forthcoming large hadron collider are given. This process is one of the dominant beam loss processes and can become critical for the operation with heavy ions. A simple parametrization is given as well. We compare our results with calculations of other groups.

(e,2e) study of cadmium ionization in the4d95s25pautoionizing region

Cadmium $(e,2e)$ energy spectra have been measured for incident electron energy 150 eV and scattering angles 2\ifmmode^\circ\else\textdegree\fi{}--18\ifmmode^\circ\else\textdegree\fi{}, corresponding to momentum transfer 0.2--1 a.u. This choice of kinematics covers the transition from the dipole limit at 0.2 a.u. to the binary collisions regime at 1 a.u. The data are presented as the sum and difference of pairs of $(e,2e)$ spectra with ejected-electron directions 180\ifmmode^\circ\else\textdegree\fi{} apart. This analysis reveals trends in the data and also provides a sensitive test of theory. Data sets for three special ejected-electron directions are compared with plane-wave Born approximation calculations. The calculated spectral shapes are in good agreement with all experiments when a simple phase correction is incorporated. The calculated difference/sum relative intensities are in good agreement with one data set but are in very poor agreement with the other two.

Observation of Cd4d95s25pJ=3autoionizing levels in(e,2e)energy spectra

Cadmium $(e,2e)$ energy spectra have been measured for kinematics corresponding to a momentum transfer of 1 a.u. Two previously unknown cadmium autoinizing levels have been observed. Their energies are in excellent agreement with existing ab initio structure calculations of the ${4d}^{9}{5s}^{2}5p$ $J=3$ levels. One level is easily seen at an ejected-electron direction along the momentum-transfer axis, but is absent for a direction 39\ifmmode^\circ\else\textdegree\fi{} away from this axis. The opposite is true for the other level; it is absent in the former, but present in the latter case. This behavior is in agreement with a calculation that takes into account that the $J=3$ levels can autoionize into both singlet and triplet $5sEf$ continua. The intensity of the new levels, relative to the well-known ${4d}^{9}{5s}^{2}5p$ $J=1$ levels, agrees well with a plane-wave Born approximation calculation for the $J=3$ levels. The third ${4d}^{9}{5s}^{2}5p$ $J=3$ level is calculated to lie within the broad ${4d}^{9}{5s}^{2}5p{}^{1}{P}_{1}$ level and cannot be seen in the present experiments.

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.

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.

Investigation of complex ionization amplitudes in cadmium by (e,2e) spectroscopy.

High-resolution (e,2e) energy spectra are presented which enable the isolation of interference effects between J=0,1,2 multipoles in electron-impact ionization of cadmium. It is found that both resonant and nonresonant processes are important. Relative magnitudes and phases of ionization amplitudes are obtained which disagree with plane-wave Born approximation calculations.

Determination of complex ionization amplitudes by (e,2e) spectroscopy.

[ital (]e,2[ital e]) spectroscopy has been used to investigate interference between resonant dipole and nonresonant monopole electron-impact ionization of Cd. A technique is described which allows the experimental determination of the relative magnitude and phase of the amplitudes for these processes. The method makes use of the rapid variation with energy of the continuum phase in the region of autoionizing resonances. The experimental relative dipole/monopole amplitude for momentum transfer 0.18 a.u. is found to be in substantial disagreement with a plane-wave Born approximation calculation.

K-Shell ionisation of atoms by relativistic protons

The K-shell ionisation of atoms by relativistic proton has been calculated in plane wave Born approximation (PWBA) as well as by impact parameter method. In PWBA calculation the interaction between the incident proton and the atomic nucleus is considered in terms of virtual photon exchange and these virtual photons photoionise the atom. The ionisation cross section is expressed as a sum of the distant and close collisions. The virtual photon concept is used for evaluating the contribution due to the distant collisions and free electron proton scattering cross section for the close collisions. The calculations for photon energies 160 MeV, 3.672 GeV and 4.88 GeV for several atomic numbers (Z-25 to 95) are presented. The results are compared with the theoretical calculation of Davidovic et al and the available experimental data.

Binary-encounter electrons observed at 0 degrees in collisions of 1-2-MeV/amu H+, C6+, N7+, O8+, and F9+ ions with H2 and He targets.

The energy distribution of binary-encounter electrons (BEE) produced in collisions of 1--2 MeV/amu H{sup +} and bare C, N, O, and F ions with H{sub 2} and He gas targets is reported at 0{degree} with respect to the beam direction. These electrons result from ionization of the target due to hard collisions with the projectile and can thus be considered to be produced in a process analogous to elastic scattering of a free electron from a highly charged ion. An impulse-approximation (IA) model has been developed to describe this process in which quasifree'' target electrons undergo 180{degree} Rutherford scattering in the projectile frame. The measured BEE double-differential production cross sections for bare ions were well described by this model and were found to scale with {ital Z}{sub {ital p}}{sup 2} and {ital E}{sub {ital p}}{sup {minus}({similar to}2.6--2.7)} where {ital Z}{sub {ital p}} and {ital E}{sub {ital p}} are the charge and energy of the projectile, respectively. An energy shift of the BEE below 4{ital t}, where {ital t} is the cusp electron energy, is observed and is also predicted by the IA treatment. A plane-wave Born approximation (PWBA) calculation for BEE production is also found to be in overall agreementmore » with our data. However, the energy shift of the BEE peak could not be fully accounted for within this PWBA calculation.« less

Evidence for electron-electron interaction in projectile K-shell ionization.

Cross sections for projectile {ital K}-shell ionization were measured for 0.75--3.5-MeV/{ital N} C{sup 5+} and O{sup 7+} projectiles in collisions with H{sub 2} and He targets. The experimental results agree with plane-wave Born-approximation calculations which take into account the interaction between projectile and target electrons. We demonstrate that for energies where the target electrons have sufficient kinetic energy in the projectile frame to ionize the projectile electron, the electron-electron interaction can lead to a significant increase in the total ionization cross section.

Ionization of one-electron oxygen and fluorine projectiles by molecular hydrogen.

Cross sections for projectile ionization have been measured for hydrogenlike oxygen and fluorine ions incident on a molecular-hydrogen target over a projectile energy range of 0.5\char21{}2.5 MeV/amu. The experimental cross sections are compared to the plane-wave Born approximation (PWBA) and to the Glauber-approximation cross sections all of which were calculated for atomic hydrogen and multiplied by 2. The PWBA calculations have a projectile energy dependence similar to the measured cross sections but slightly underestimate them. The Glauber approximation also underestimates the measured projectile-ionization cross sections when the hydrogen target electrons are neglected, while it overestimates the measured cross sections when the effects of the hydrogen target electrons are included. The measured projectile-ionization cross sections for hydrogenlike ions incident on molecular hydrogen are approximately a factor of 2 smaller than previously reported projectile-ionization cross sections for hydrogenlike ions incident on helium. No cross sections are available for atomic hydrogen in this velocity and ion-charge regime.

Relativistic calculation of atomic N-shell ionization by protons.

Relativistic plane-wave Born-approximation calculations of cross sections for N-shell ionization of $_{83}\mathrm{Bi}$ and $_{92}\mathrm{U}$ by protons with incident energies from 0.02 to 5 MeV are reported. The calculations were carried out by using Dirac-Hartree-Slater wave functions. Binding-energy change and Coulomb deflection were taken into account. The relativistic cross sections are compared with values from nonrelativistic Hartree-Slater wave functions to study the effects of relativity. A test calculation with hydrogenic wave functions yields very different results. The only available measurements (for $_{74}\mathrm{W}$), revised with a corrected 4d fluorescence yield, agree with present theoretical predictions for ${E}_{1}$\ensuremath{\le}0.1 MeV but fall below theory by a factor of 6 above ${E}_{1}$=0.4 MeV. This glaring discrepancy invites further investigation.

"Stripping" reaction in heavy ion projectile dissociation: Extended Serber model.

The opaque Serber model is extended for ``stripping'' reaction in heavy ion projectile dissociation by incorporating a critical distance between constituent clusters in the projectile. This model describes the higher energy part of the fragment inclusive spectra. The results are discussed in comparison with the local-momentum plane-wave Born approximation calculations by McVoy and Nemes and with the Friedman model.

M-shell x-ray production cross sections in thick targets of Ir, Pt, and Pb by 0.4-2.2-MeV 4He+ ions.

M-shell x-ray production cross sections in thick targets of Ir, Pt, and Pb have been measured for $^{4}\mathrm{He}^{+}$ ions of energy from 0.4 to 2.2 MeV in steps of 0.1 MeV. The cross sections are compared with plane-wave Born approximation (PWBA) calculations and with perturbed-stationary-state calculations with energy-loss, Coulomb deflection, and relativistic corrections (ECPSSR). The present measurements, in general, agree well with the PWBA predictions at all energies, but agree with the ECPSSR theory only at projectile energies \ensuremath{\ge}1.5 MeV. The M-subshell x-ray cross sections are also calculated by a nonlinear (iterative) least-squares-fitting algorithm by extracting the relative line intensities of the experimentally observed spectra obtained with a Si(Li) detector. The subshell cross sections are compared with the scaled cross sections based on PWBA calculations. Good agreement is obtained for the case of ${M}_{1}$, ${M}_{2}$, and ${M}_{4}$, and a larger deviation is found for ${M}_{3}$ and ${M}_{5}$. The double nodal structure of the 3s-electron wave functions is confirmed, and the single inflection associated with the 3${p}_{3/2}$-electron wave functions is not clearly visible in our data. The M-subshell cross-section ratios for ${M}_{1}$/(${M}_{2}$+${M}_{3}$), ${M}_{1}$/(${M}_{4}$+${M}_{5}$), and (${M}_{2}$+${M}_{3}$)/(${M}_{4}$+${M}_{5}$) are also calcu- lated, compared with the PWBA and ECPSSR theories, and discussed.

C2contributions to back-angle inelastic electron scattering fromEr166andTa181

We present results of distorted-wave Born approximation calculations of longitudinal contributions to 180\ifmmode^\circ\else\textdegree\fi{} inelastic electron scattering, in the limit of zero electron mass and pure Coulomb distortion. A rough estimate of said contributions is given for the transitions ${0}^{+}$\ensuremath{\rightarrow}${2}^{+}$ in $^{166}\mathrm{Er}$, and ${\frac{7}{2}}^{+}\ensuremath{\rightarrow}{\frac{9}{2}}^{+}$ and ${\frac{11}{2}}^{+}$ in $^{181}\mathrm{Ta}$. Results are compared to plane-wave Born approximation calculations of transverse form factors.

Proton-stopping power of argon, krypton, and xenon

Parameters for the Bethe stopping-power theory for Ar, Kr, and Xe are obtained both via integration over optical oscillator strengths and from explicit plane-wave Born-approximation calculations. The former procedure produces a Bethe mean excitation energy ($I$) significantly lower than that measured, while the latter procedure produces $I$ values within 10 eV of the measured values for Ar and Kr. For Xe the experimental $I$ value lies midway between the two calculated $I$ values. Inner-shell corrections are obtained from the explicit calculations. The $K$-shell corrections (${C}_{K}$) are a factor of 2 larger than those obtained from hydrogenic calculations. The total inner-shell correction ($\ensuremath{\Sigma}{\mathrm{nl}}^{}{C}_{\mathrm{nl}}$) is in qualitative agreement for Ar, Kr, and Xe with results from a statistical model. However, the quantitative difference between the two sets of inner-shell corrections leads to significant increases in $I$ values inferred from experimental measurements below 1 MeV proton energy.