Electron Capture Cross Sections Research Articles

Electron transfer from atomic hydrogen to multiply-charged nuclei at intermediate and high energies

We report on theoretical total cross sections for electron capture from the ground state of atomic hydrogen H(1s) by fast multiply-charged nuclei H+,He2+,Li3+,Be4+,B5+,C6+,N7+,O8+ and F9+. The prior form of the boundary-corrected continuum intermediate states (BCIS) method is used. For a single fixed initial ground-state (i=1s), comprehensive computations are carried out for a sequence of the transitions to the final state (n,l,m) with 1≤n≤nmax, including all the energy degenerate sub-levels (−l≤m≤l,0≤l≤n−1). The maximal quantum number nmax is respectively set to 4 for H+,He2+ and Li3+, to 5 for Be4+, to 6 for B5+, as well as to 7 for C6+,N7+,O8+ and F9+. The reported nine tables provide these state-selective cross sections (Qif) and their state-summed counterparts (QΣ). The cross sections QΣ for the sum over all the final states are evaluated using the exact results of Qif for n≤nmax and the scaled cross sections for n>nmax. The scaling is based on the Oppenheimer n−3 rule. In addition to these tables, at the same impact energies 20–3000 keV/amu, the cross sections from the BCIS method are presented graphically in eight figures for the nuclei Li3+,Be4+,B5+,C6+,N7+,O8+ and F9+. This complements our recently published cross sections for electron capture by protons (H+) and alpha particles (He2+) from H(1s). In the present figures, whenever available, the corresponding experimental data are compared with our results for QΣ. Also included in these comparisons are the corresponding results for QΣ from the continuum distorted wave (CDW) method. The cross sections tabulated here can be used in versatile applications ranging from plasma physics and astrophysics through fusion research to ion therapy.

Measurement of Absolute Single and Double Electron Capture Cross Sections for O6+ Ion Collisions with CO2, CH4, H2, and N2

Abstract The absolute electron capture cross sections for single and double charge exchanges (CEs) between the highly charged ion O6+ and CO2, CH4, H2, and N2, the dominant collision processes in the solar wind, have been measured in the energies from 7 keV · q (2.63 keV u−1) to 52 keV · q (19.5 keV u−1). These measurements were carried out in the new experimental instrument setup at Fudan University, and the errors of the cross sections for single and double CEs at the 1σ confidence level were about 11% and 16%, respectively. Limited agreement is achieved with single electron capture results calculated by the classical overbarrier model. These cross section data are useful for the simulation of ion–neutral processes in astrophysical environments and to improve the present theoretical model of fundamental atomic processes.

Lifetime limiting defects in 4H-SiC epitaxial layers: The influence of substrate originated defects

The minority carrier lifetime is a decisive factor to obtain 4H-SiC bipolar devices with a low forward voltage drop at high blocking voltages. The lifetime is directly correlated with the concentration of the so-called Z1/2 deep level and depends on the epitaxial growth process and post-epi processing like ion implantation, annealing, and thermal oxidation. The substrate has so far been attributed a subordinate role for the minority carrier lifetime. In this work, the influence of substrate quality on the minority carrier lifetime after epitaxial growth and post-epi processing is studied on substrates from different manufacturers. The investigation revealed a significant impact of the substrate contamination on the minority carrier lifetime of epitaxial layers and on the efficiency of lifetime enhancement by thermal oxidation. A deep level named SD2 was found in the samples which acts as an additional Shockley-Read-Hall recombination center. The deep level SD2 was traced back to the substrates themselves showing different levels of incorporation of recombination sites from the substrate into the epitaxial layer during the growth. A comparison of the energy levels and electron capture cross sections of the SD2 deep level with defects caused by a tungsten contamination shows a good agreement.

First-principles study on non-radiative carrier captures of point defects associated with proton generation in silica

The non-radiative carrier captures of point defects associated with proton generation in both crystalline (α-quartz) and amorphous silica (a-SiO2) have been investigated by using first-principles calculations based on the density functional theory. Major point defects related to proton generation are considered, including the dimer and puckered configuration of oxygen vacancy, doubly hydrogenated oxygen vacancy, and hydroxyl E′ center. The carrier capture cross sections of the defects are calculated in terms of a one-dimensional static coupling method. The neutral dimer and doubly hydrogenated oxygen vacancies show a large hole capture cross section, and the positively charged puckered oxygen vacancy and the hydroxyl E′ center show a large electron capture cross section. These results further support that dissociation of molecular hydrogen at a positively charged dimer oxygen vacancy and that of a positively charged doubly hydrogenated oxygen vacancy are the main reactions in silica to generate protons that play a key role in ionization damage of silicon devices.

Stellar electron-capture rates based on finite-temperature relativistic quasiparticle random-phase approximation

The electron capture process plays an important role in the evolution of the core collapse of a massive star that precedes the supernova explosion. In this study, the electron capture on nuclei in stellar environment is described in the relativistic energy density functional framework, including both the finite temperature and nuclear pairing effects. Relevant nuclear transitions $J^\pi = 0^\pm, 1^\pm, 2^\pm$ are calculated using the finite temperature proton-neutron quasiparticle random phase approximation with the density-dependent meson-exchange effective interaction DD-ME2. The pairing and temperature effects are investigated in the Gamow-Teller transition strength as well as the electron capture cross sections and rates for ${}^{44}$Ti and ${}^{56}$Fe in stellar environment. It is found that the pairing correlations establish an additional unblocking mechanism similar to the finite temperature effects, that can allow otherwise blocked single-particle transitions. Inclusion of pairing correlations at finite temperature can significantly alter the electron capture cross sections, even up to a factor of two for ${}^{44}$Ti, while for the same nucleus electron capture rates can increase by more than one order of magnitude. We conclude that for the complete description of electron capture on nuclei both pairing and temperature effects must be taken into account.

Electron-capture cross sections from atoms to hydrogen-like projectiles

State-selective total and projectile angular-differential cross sections for single-electron transfer to the ground state from atoms in collisions with hydrogen-like projectiles are calculated by means of the four-body distorted wave approximation in the energy range from 25 to 2500 keV amu−1. In the initial channel, the effect of the dynamic electron–electron correlation is explicitly taken into account through the complete perturbation potentials. Distortion in the initial channel due to the Coulomb continuum states of the target and the active electron in the field of the residual projectile ion is included. The quantum-mechanical prior and post transition amplitudes for single-electron capture for symmetric and asymmetric collisions are then derived in terms of three-dimensional real integrals. The role of dynamic inter-electron correlation on the cross sections as a function of the impact energy is examined. Only for He+–He collisions, the projectile angular-differential cross sections for ground state capture are calculated at different impact energies. We investigate the effect of the total cross section as well as the projectile angular-differential cross sections to the choice of different ground state wavefunction of the projectile in the final state. It is observed that although the shape of the differential cross section does not change significantly for these functions, there is a significant change of the differential cross sections due to the choice of the target description. The differential cross section reveals pronounced minima and maxima in the projectile scattering angles as the projectile energy decreases. The observed structure demonstrates the analogy of atomic de Broglie’s matter-wave scattering and Fraunhofer-type diffraction. Finally the validity of our results is assessed by comparison with available experimental data.

Ionization and Electron Capture Cross Sections for Single-Electron Removal from Biological Molecules by Swift Ion

We have studied ionization and electron capture cross sections for single-electron removal from biological molecules (adenine, cytosine, guanine, thymine, and uracil) by proton impact at energies ranging from 30 keV/amu to 10 MeV/amu. In this investigation, the three-body distorted wave method is used. The calculations are based on the independent electron model. For electron capture process, distortion in the final channel related to the Coulomb continuum states of the active electron and the projectile in the field of residual target ion is included. In case of ionization, we take all the pair-wise Coulomb interactions, which treat all interactions on equal footing in the final channel. Moreover, the asymptotic Coulomb logarithmic phase for the relative motion of two colliding nuclei is included in the initial channel. In both processes, the molecular character of the biological target is assumed to be a linear combination of their atomic orbitals (LCAO) and, for all cases, the different atomic orbitals are described with the Roothaan-Hartree-Fock (RHF) approximation. We have also calculated the total capture cross sections using simple Bragg’s additivity rule. For electron capture, the contributions to the TCS from the core orbitals of the molecule have also been explicitly analyzed. The double differential cross sections (DDCS) for electron emission as well as the total cross sections for single ionization of only uracil molecule with fast proton impact are also calculated. The results for capture and ionization cross sections are compared with other theoretical calculations and existing experimental data. We find that our calculated results are in quite better agreement with available experimental data than the other theoretical results.

Capacitance Spectroscopy of Heteroepitaxial AlGaAs/GaAs p–i–n Structures

The temperature dependences of the capacitance-voltage (C–V) characteristics and deep-level spectra of a graded high-voltage AlxGa1 – xAs p0–i–n0 junction grown by liquid-phase epitaxy via autodoping with background impurities are studied. Changes in the C–V characteristics at varied measurement temperature and optical illumination demonstrate that the p0-, i-, and n0-type layers in AlxGa1 – xAs under study contain bistable DX centers. Spectra furnished by deep-level transient spectroscopy (DLTS), measured at various bias voltages Vr and filling-pulse voltages Vf, show a positive DLTS peak for the n0-type layer, with a thermal activation energy of Et = 280 meV and electron-capture cross section of σn = 3.17 × 10–14 cm2, which is unusual for a majority-charge-carrier trap. This peak is attributed to the negatively charged state of the Se/Te donor impurity, which is a bistable DX center with negative correlation energy U.

Radiative Cascade Repopulation of 1s2s2p 4P States Formed by Single Electron Capture in 2–18 MeV Collisions of C4+ (1s2s 3S) with He

This study focuses on the details of cascade repopulation of doubly excited triply open-shell C3+(1s2s2p)4P and 2P± states produced in 2–18 MeV collisions of C4+(1s2s3S) with He. Such cascade calculations are necessary for the correct determination of the ratio R of their cross sections, used as a measure of spin statistics [Madesis et al. PRL 124 (2020) 113401]. Here, we present the details of our cascade calculations within a new matrix formulation based on the well-known diagrammatic cascade approach [Curtis, Am. J. Phys. 36 (1968) 1123], extended to also include Auger depopulation. The initial populations of the 1s2snℓ4L and 1s2snℓ2L levels included in our analysis are obtained from the direct nℓ single electron capture (SEC) cross sections, calculated using the novel three-electron close-coupling (3eAOCC) approach. All relevant radiative branching ratios (RBR) for n≤4 were computed using the COWAN code. While doublet RBRs are found to be very small, quartet RBRs are found to be large, indicating cascade feeding to be important only for quartets, consistent with previous findings. Calculations including up to third order cascades, extended to n→∞ using an n−3 SEC model, showed a ∼60% increase of the 1s2s2p4P populations due to cascades, resulting, for the first time, in R values in good overall agreement with experiment.

Temperature dependent carrier lifetime, diffusion coefficient, and diffusion length in Ge0.95Sn0.05 epilayer

The development of new technology, which would be able to shift photosensitivity of Si devices to the mid-infrared range, preserving the benefits of cheap silicon readout circuits, is of high priority for short-wave infrared photo-detection in defense, medical, night vision, and material production applications. Group IV GeSn-based materials have recently shown promising optoelectronic characteristics, allowing extension of the detection range to the mid-infrared region. However, the electronic properties of the material are not well understood and need further research. In this work, we provide temperature dependent studies of carrier lifetime, diffusion coefficient, and diffusion length in Ge0.95Sn0.05 epilayer on silicon by applying contactless light induced transient grating technique. The observed temperature dependence of lifetime was explained by the recombination of carriers on vacancy-related defects. The electron and hole capture cross sections were calculated. The temperature dependence of the diffusion coefficient indicated hole mobility limited by phonon and defect scattering. Weakly temperature dependent diffusion length of ∼0.5 μm verified material suitability for efficient submicrometer-thick optoelectronic devices.

Understanding the metastable states in K-Na aluminosilicates using novel site-selective excitation-emission spectroscopy

Optically stimulated luminescence (OSL) and thermoluminescence (TL) from the metastable states in solids are widely used in luminescent phosphors, dosimetry, geochronology, and thermo- and photo-chronometry. OSL and TL result from a combination of three different processes (charge detrapping, transport, and recombination) and are, therefore, not ideal for characterizing the charge trapping states. Therefore, despite many decades of research, the OSL and TL kinetics and the associated defect systems remain poorly understood in natural minerals. Recently, a radio-photoluminescence (RPL) signal has been discovered in feldspar (K-Na-Ca aluminosilicates occupying > 50% of Earth’s crust) which helps overcome this limitation. This site-selective signal termed infrared photoluminescence (IRPL) arises from radiative relaxation of the excited state of the main electron trapping center (principal trap).In this study, IRPL excitation and emission spectroscopy at cryogenic temperatures reveals two distinct electron-trapping centers (i.e. two principal traps) in feldspar, and helps to determine their trap depths and the excited-state energies. The two trapping centers show the same electron capture cross-sections and the excited-state relaxation lifetimes, but different ground- and excited-state energies. Based on this peculiar combination of trap characteristics, we conclude that that the principal traps consist of the same defect residing at two different crystal sites. The differences in the energy levels of the two principal traps explain their distinct optical and thermal bleaching behavior.

Solar wind charge exchange in cometary atmospheres

Solar wind charge-changing reactions are of paramount importance to the physico-chemistry of the atmosphere of a comet. The ESA/Rosetta mission to comet 67P/Churyumov-Gerasimenko (67P) provides a unique opportunity to study charge-changing processes in situ. To understand the role of these reactions in the evolution of the solar wind plasma, and interpret the complex in-situ measurements made by Rosetta, numerical or analytical models are necessary. We use an extended analytical formalism describing solar wind charge-changing processes at comets along solar wind streamlines. The model is driven by solar wind ion measurements from the Rosetta Plasma Consortium-Ion Composition Analyzer (RPC-ICA) and neutral density observations from the Rosetta Spectrometer for Ion and Neutral Analysis-Comet Pressure Sensor (ROSINA-COPS), as well as charge-changing cross sections of hydrogen and helium particles in a water gas. A mission-wide overview of charge-changing efficiencies at comet 67P is presented. Electron capture cross sections dominate and favor the production of He and H energetic neutral atoms, with fluxes expected to rival those of H+ and He2+ ions. Neutral outgassing rates are retrieved from local RPC-ICA flux measurements, and match ROSINA's estimates very well. From the model, we find that solar wind charge exchange is unable to fully explain the magnitude of the sharp drop of solar wind ion fluxes observed by Rosetta for heliocentric distances below 2.5 AU. This is likely because the model does not take into account the relative ion dynamics and, to a lesser extent, ignore the formation of bow shock-like structures upstream of the nucleus. This work also shows that the ionization by solar EUV radiation and energetic electrons dominates the source of cometary ions, although solar wind contributions may be significant during isolated events.

Three-body boundary-corrected continuum-intermediate-state method for single charge exchange with the general transition amplitude (1s→nlm) applied to the p–H(1s), α–H(1s), and p–He(1s2) collisions with n≤4

We derive a general quantum-mechanical transition amplitude in the prior version of the three-body boundary-corrected continuum-intermediate-state (BCIS-3B) method for single charge exchange in collisions of heavy nuclei with atomic hydrogenlike targets. The obtained expression is valid for all the transitions from the initial ground to any final hydrogenlike bound states $(1s\ensuremath{\rightarrow}nlm),$ where ${n,l,m}$ is the usual triple of the quantum numbers (principal, angular, magnetic). The outcome is semianalytical: a judicious combination of an analytical calculation (as far as feasible) and the subsequent numerical computations (quadrature for integrals that could not be reduced to closed formulas). The final results for the $T$-matrix elements are a compact double integration computed by numerical quadratures over real variables in finite intervals. The ensuing total cross section is a single numerical quadrature of the squared absolute value of this transition amplitude integrated over the magnitude of the transverse momentum transfer vector. This can provide detailed information on the cross sections for each individual subshell ${n,l,m}$ of the newly formed hydrogenlike atom or ion comprised of a projectile nucleus and the captured electron. Using these general expressions, we computed a large number of state-selective and state-summed cross sections (both differential and total) for electron capture in the $p\text{\ensuremath{-}}\mathrm{H}$, $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{H}$, and $p\text{\ensuremath{-}}\mathrm{He}$ collisions at intermediate and high impact energies. The energy dependence of the cross sections for the individual shells and their subshells with the fixed quantum numbers $n$, ${n,l}$, and ${n,l,m}$ are reported via detailed tables and figures. Also, differential cross sections (state selective and summed) are computed for the $p\text{\ensuremath{-}}\mathrm{H}$ collisions at 60, 125, and 5000 keV. All the present results from the BCIS-3B method are found to be in very good agreement with the existing experimental data on differential and total cross sections, both state selective (whenever available) and state summed. The provided cross-section database can find important applications in several investigative fields ranging from plasma physics, astrophysics, and heavy-ion transport physics through fusion research and technology to hadron therapy.

Nuclear structure properties of even-even chromium isotopes and the effect of deformation on calculated electron capture cross sections

In this study, we investigate the role of the nuclear deformation on the calculated electron capture cross section (ECC) of even-even chromium (Cr) isotopes. We first determined the nuclear structure properties of these nuclei within the interacting boson model-1 (IBM-1). The energy spectra and E2 transition probabilities were calculated by fitting the parameters in the model formalism. The analysis of the potential energy surface were also performed to predict the geometric shape of the Cr nuclei by plotting their contour plot in the plane of (β, γ) deformation parameters. Later, we calculated the ECC within the proton-neutron quasiparticle random phase approximation (pn-QRPA) model. In particular we studied how the calculated ECC changed with different values of nuclear deformation parameter. The calculated Gamow-Teller (GT) strength distributions were widely spread among the daughter states. The total GT strength decreased with increasing value of the β parameter. The computed ECC values, however, increased with increase in the β value of the Cr isotopes.

Photoluminescence efficiency of Al-rich AlGaN heterostructures in a wide range of photoexcitation densities over temperatures up to 550 K

Time-resolved photoluminescence and light-induced transient grating techniques were applied for temperature and excitation dependent internal quantum efficiency and carrier diffusion investigation in silicon-doped epilayer and AlxGa1-xN MQWs (x > 0.6). Decrease of photoluminescence efficiency at high excitations correlated with increase of diffusion coefficient as well as nonradiative recombination rate, indicating excitation and temperature dependent localized exciton density reduction. Excitation-dependent lifetime decrease was less pronounced than the corresponding drop of the time-integrated photoluminescence efficiency dominated by thermal dissociation of excitons. At lowest excitations excitons were found captured to vacancy complexes. With increase of excitation exciton ionization appeared, leading to enhanced nonradiative recombination of holes on aluminum vacancies and simultaneous droop of PL efficiency due to saturated bimolecular free carrier plasma recombination. At initial recombination stages and high excitations diffusive recombination on dislocations was also revealed. Modelling provided free exciton binding energies of 104, 140 meV in the MQWs and in the layer; exciton localization energies in 12-35 meV range; localized exciton lifetimes in 2-4 ns range; free exciton and carrier radiative recombination rate coefficients of rex = (0.6 0.2)109(T/300)-1 1/s and Brad = (7 1)10-10(T/300)-3/2 cm3/s, respectively. Vacancy complex capture cross section for excitons was determined to be = (2 1)10-16(300/T)2 cm2. Electron and hole capture cross sections to aluminum vacancy were modeled (1.5 1)10-13 cm2 and (7 1)10-13 cm2, respectively. Dislocation Coulomb radius for free carrier recombination was found to be 12-15 nm.

Nonperturbative scaling behavior for net ionization of biologically relevant molecules by multiply charged heavy-ion impact

A recently developed model to describe proton collisions from molecules involving basic atoms such as hydrogen, carbon, nitrogen, oxygen and phosphorus (H, C, N, O, P) is extended to treat collisions with multiply charged ions. The ion-atom collisions are computed using the two-center basis generator method (TC-BGM), which has a proven track record of yielding accurate total cross sections for electron capture and ionization. The atomic net ionization cross sections are then used to assemble two models for ion-molecule collisions: an independent atom model (IAM) that follows the Bragg additivity rule (labeled IAM-AR), and also the so-called pixel-counting method (IAM-PCM). The latter yields reduced cross sections relative to IAM-AR near the maximum, since it takes into account the overlapping nature of effective cross sectional areas. The IAM-PCM for higher-charge projectiles leads to strong reductions of net ionization cross sections relative to the IAM-AR method, and is computed directly for projectile charges $Q=1, 2, 3$. The scaling behavior of the IAM-PCM is investigated over a wide range of energies $E$, and at high $E$ it converges towards the IAM-AR. An empirical scaling rule is established which allows to reproduce these results based on proton impact calculations. Detailed comparisons are provided for the uracil target ($\rm C_4 H_4 N_2 O_2$), for which other theoretical as well as experimental results are available. Based on the scaling model derived from the IAM-PCM data it is shown how the experimental data for uracil and water bombarded by multiply charged ions can be reduced to effective $Q=1$ cross sections respectively, and these are compared to proton impact data.

Effect of deformation on gamow-teller strength and electron capture cross-section for chromium isotopes

In this work, we explore the role of deformation parameter (β) on the calculated Gamow-Teller (GT) strength distributions and electron capture cross-sections (ECC) for 46,48,50Cr isotopes within the formalism of the proton neutron-quasi-particle random phase approximation (pn-QRPA). Three different β parameters were used in the present study. Two of them were calculated by using the interacting boson model (IBM) and the macroscopic-microscopic (Mac-mic) models. The third one is the experimental β values obtained by employing its relation with the experimental B(E2)↑ values. The GT strength distributions were widely dispersed among all the daughter states of the given isotopes. They were found to have an inverse relation with the β parameter i.e decreasing with increasing the β value. The ECC was computed as a function of the β parameter and the results suggest that the calculated ECC decreased with decreasing value of the β for the selected cases.

Gamow-teller strength and electron capture cross-section calculation by pn-QRPA for selected fp-shell nuclei

The allowed Gamow-Teller (GT) strengths and associated weak interaction rates on fp-shell nuclides are the most familiar processes of spin-isospin (στ) type. These rates play a crucial role in several astrophysical processes, particularly in nuclear synthesis and supernova-explosions. As per simulation consequences, the electron capture cross-section on medium-heavy nuclei has a key impact on decreasing the ratio of electron-to-baryon of the stellar matter during the late stages of stars formation. Stellar model based on the theoretical approaches should be tested against the available measured data. In the current work we present calculated Gamow-Teller strength distributions by pn-QRPA model for selected fp-shell nuclei (42Ti, 46Cr, 50Fe and 54Ni) and compare our results with available measured data. The Gamow-Teller strength distributions are well fragmented over the energy range 0-12 MeV and have a good comparison with experimental data. We calculate the electron capture cross-section for selected nuclei at temperature 0.5 MeV, 1.0 MeV and 1.5 MeV that shows the temperature dependence of calculated electron capture cross-section for astrophysical applications.

Stripping of relativistic uranium-ion beams by foils

The dynamics of relativistic uranium-ion beams stripped by foils from Be to Au is investigated in the 100–500 MeV/u energy range. Calculations of the charge-state distributions of uranium fractions as a function of foil thickness are performed using the BREIT code, based on the analytical solution of the balance rate equations for the ion charge-state fractions. Relativistic electron-loss and electron-capture cross sections, required for BREIT as input data, are calculated for interactions of Uq+ ions, q = 65–92, colliding with Be, C, Al, Ti, Cu, Mo, W, and Au atoms. The beam energy losses are found to be rather small, less than 20%, and are not accounted for in calculations. Calculated charge-state distributions are compared with available experimental data, obtained at the GSI (Gesellschaft für Schwerionenforschung) and BEVALAC (Bevatron Linear Accelerator) facilities, and with calculations by the GLOBAL code created at GSI.The optimal conditions are predicted for production of bare, H-like, He-like and Li-like uranium ions with the largest probabilities. Dependencies of the equilibrium thicknesses for uranium charge-state fractions on the initial charge of incident ions are investigated.

One-center close-coupling approach to two-center rearrangement collisions

Calculations of ionization and electron-capture cross sections in ion-atom collisions usually require solving the Schrödinger equation governing the collision system by expanding the total scattering wave function in a basis of target- and projectile-centered pseudostates. This approach leads to the two-center close-coupling equations which in some cases may become ill-conditioned due to the non-orthogonality of the underlying combined basis. Here we develop a technique which allows to accurately extract the necessary collision information, including that for the rearrangement channels, from the computationally more convenient one-center close-coupling equations which are built from only target-centered pseudostates. The robustness of the method is demonstrated by considering the proton-hydrogen scattering problem across a wide incident energy range. The developed method is then applied to study proton scattering on multielectron target of lithium. The obtained results for the 2s → 2p excitation and the total electron-capture cross sections are in good agreement with corresponding experimental data. It is concluded that the presented technique could be a simpler alternative when integrated cross sections are required.