Distorted-wave Approximation Research Articles

R-matrix calculations for opacities: I. Methodology and computations

An extended version of the R-matrix methodology is presented for calculation of radiative parameters for improved plasma opacities. Contrast and comparisons with existing methods primarily relying on the distorted wave approximation are discussed to verify accuracy and resolve outstanding issues, particularly with reference to the opacity project (OP). Among the improvements incorporated are: (i) large-scale Breit–Pauli R-matrix calculations for complex atomic systems including fine structure, (ii) convergent close coupling wave function expansions for the (e + ion) system to compute oscillator strengths and photoionization cross sections, (iii) open and closed shell iron ions of interest in astrophysics and experiments, (iv) a treatment for plasma broadening of autoionizing resonances as function of energy-temperature-density dependent cross sections, (v) a ‘top-up’ procedure to compare convergence with R-matrix calculations for highly excited levels, and (vi) spectroscopic identification of resonances and bound (e + ion) levels. The present R-matrix monochromatic opacity spectra are fundamentally different from OP and lead to enhanced Rosseland and Planck mean opacities. An outline of the work reported in other papers in this series and those in progress is presented. Based on the present re-examination of the OP work, opacities of heavy elements might require revisions in high temperature-density plasma sources.

Fine structure resolved excitation cross sections of singly ionized Ga for the modeling and diagnostics of Ga plasmas

Calculation of electron impact excitation cross sections for singly charged Ga ions plays a crucial role in plasma modeling, facilitating the comprehension of plasma behavior, characteristics, and dynamics in diverse domains, such as astrophysics, fusion research, the semiconductor industry, etc. In the available literature, there is a notable scarcity of, or even a complete absence of, these cross sections. Hence, in the present work, electron impact excitation cross sections are calculated for the transitions from the fine structure resolved energy levels of the configurations 4s2 and 4s4p to the fine structure resolved energy levels of the configurations 4s4p , 4s5s , 4p2 and 4s4d of the singly charged Ga ion (Ga+) using the relativistic distorted wave approximation theory with the target states represented by multi configurational Dirac Fock wavefunctions. The cross sections are calculated for projectile electron energy varying from threshold to 500 eV. Furthermore, the electron impact excitation rate coefficients for all the transitions under investigation are also calculated for electron temperatures ranging from 0.5 to 5 eV. In addition, analytic fitting of the rate coefficients is also performed, providing a practical resource for directly utilizing in plasma modeling applications.

Fully relativistic distorted wave calculations of electron impact excitation of gallium atom: Cross sections relevant for plasma kinetic modeling

Electron impact excitation cross sections of neutral Ga are calculated using the relativistic distorted wave approximation with the aid of multiconfigurational Dirac-Fock wavefunctions. The cross sections are calculated from the fine structure resolved levels of the 4s24p manifold, i.e., from the ground 2P1/2 and the first excited state 2P3/2, to the fine structure resolved levels of the manifolds 4s24pdf, 4s4p2(4P1/2,4P3/2, and 4P5/2), 4s25spdf, 4s26spdf, 4s27spdf, and 4s28spdf for incident electron energies ranging from threshold to 500 eV. Where available, a comparison of the calculated oscillator strengths, transition probabilities, and cross sections with the previously reported results has been performed. The cross sections for the excitation from 4s24p2P1/2 to 4s24f, 4s4p2(4P1/2,4P3/2, and 4P5/2), 4s25f, 4s26f, 4s27df, and 4s28spdf and all the excitation cross sections from 4s24p2P3/2 are calculated and reported for the first time for a wide range of electron energies. Furthermore, the rate coefficients are also calculated using the present cross sections for an electron temperature varying from 0.5 to 5 eV. An analytic fitting of the calculated rate coefficients is provided to facilitate their inclusion in various plasma kinetic models.

Electroproduction of Hypernuclei

Abstract. The electroproduction of hypernuclei initiates a captivating domain of study, offering great insights into the characteristics of composite nucleon-hyperon systems. This inquiry assumes prime significance as it displays critical features of nuclear matter structure and tests the interactions between hyperons and nucleons. In this paper, we unveil the distorted-wave impulse approximation (DWIA) approach, aimed at explaining the cross sections in electroproduc-tion of Λ-hypernuclei. Our investigation examines how various model assumptions affect the results we get. We focus on two important factors: influence of the proton Fermi motion and description of the various models of nuclear and hypernuclear structure.

Electron and positron impact differential cross sections for the (e,2e) process on water molecule

We present a theoretical study for electron and positron-impact ionization of water molecule by using the three-body formalism of distorted-wave approximation. In this approximation, the final state contains one-Coulomb, two-Coulomb, and three-Coulomb distortion due to pairwise Coulombic interactions, respectively. In the entrance channel, the initial state is a product of two wavefunctions: the first one for the incident electron/positron described by a Coulomb wave and the second one for the ten bound electrons. The molecular wavefunction for these bound electrons of H2O are described by the linear combination of atomic orbitals. We investigate the angular distribution of the triple differential cross section (TDCS) for single ionization of the outer orbitals 1b1,3a1,1b2, and 2a1. The structure with two peaks around the binary and recoil region are clearly observed for 1b1, 3a1, and 1b2 orbitals. At the same time, a single peak in the binary region with no recoil peak is found for the 2a1 orbital. The present TDCS results are compared with available experimental and theoretical findings. Fair agreement regarding the structure of angular profile of the TDCS is observed for all orbitals except 2a1. Finally, an enhanced TDCS have been found for positron impact compared to the electron impact.

Pikoe: A computer program for distorted-wave impulse approximation calculation for proton induced nucleon knockout reactions

pikoe, Proton-Induced KnockOut reaction calculation for Exclusive processes, is a Fortran 90 program that calculates triple- and quadruple-differential cross sections, vector analyzing powers, and momentum distributions of reaction residues, for proton-induced nucleon knockout reactions in normal and inverse kinematics. The distorted-wave impulse approximation with the factorization approximation to the nucleon-nucleon transition matrix is adopted, and the distorted waves of the incoming proton and the outgoing two nucleons are calculated quantum mechanically. Kinematics of the reaction particles are treated in a relativistic manner, which gives the proper asymptotics of the three-body scattering wave in the plane-wave limit. Program summaryProgram Title: pikoeCPC Library link to program files:https://doi.org/10.17632/m594h58kck.1Licensing provisions: MITProgramming language: Fortran 90Supplementary material: See https://www.rcnp.osaka-u.ac.jp/~kazuyuki/pikoe/index.phpNature of problem: Calculation of triple- and quadruple-differential cross sections, vector analyzing powers, and momentum distributions of reaction residues, for nucleon knockout reactions with proton beams/targets in normal/inverse kinematics.Solution method: Numerov method for solving single-channel second-order differential equations; multi dimensional integration with the Gauss-Legendre quadrature rule.Additional comments including restrictions and unusual features: The global optical potential by Koning and Deraloche [1] for nucleons at energies from 1 keV to 200 MeV, for (near-)spherical nuclei with mass number between 24 and 209, is implemented in the program. Tables of nucleon-nucleon (NN) elastic differential cross sections and NN transition amplitudes calculated with the NN effective interaction by Franey and Love [2] are provided as external files for pikoe (included in the package). Users are encouraged to prepare tables for the optical potentials and the cross sections of the elementary process according to the reaction system of interest by themselves. With external files thus prepared, pikoe can be applied to various kinds of knockout processes such as α or deuteron knockout by proton, α-induced knockout reactions, etc. Calculation of spin observables, the vector analyzing power in fact, however, is restricted to nucleon-induced nucleon knockout processes with the coplanar kinematics.

Theoretical investigation of electron-impact ionization of W8+ ion

The electron-impact single ionization cross section for W8+ ion has been calculated using flexible atomic code, employing the level-to-level distorted-wave approximation. This calculations takes into account contributions form both direct ionization (DI) and excitation autoionization (EA). However, the theoretical predictions, based solely on the ground state, tends to underestimate the experimental values. This discrepancy can be mitigated by incorporation contributions from excited states. We extended the theoretical analysis, including the contributions from the long-lived metastable states with lifetimes exceeding 1.5 × 10−5 s. We employed two statistical models to predict the fraction of ground state ions in the parent ion beam. Assuming a 79% fraction of parent ions in ground configuration, the experiment measurements align with the predictions. Furthermore we derived the theoretical cross-section for the ground state as correlated plasma rate coefficients, and compared it with existing data. Despite the uncertainty in our calculations, our results are still acceptable.

Experimental study of (p, 2p) reactions at 197 MeV on 12C, 16O, 40,48Ca, and 90Zr nuclei leading to low-lying states of residual nuclei

Abstract We have measured the differential cross section and analyzing power for (p, 2p) reactions that lead to discrete states of residual nuclei at an incident energy of 197 MeV for five kinds of target nuclei, 12C, 16O, 40,48Ca, and 90Zr. The data are compared with distorted-wave impulse approximation calculations using two kinds of global optical potentials. The spectroscopic factors obtained from these calculations agree with those determined in (e, e′p) studies within 20%, except for one orbital of the 90Zr nucleus where the distortion effect is much greater than that for the other orbitals of 90Zr and for other target nuclei. We also observed a clear and visibly distinct j-dependence in the analyzing power.

Electron-impact single ionization for the Fe2+ ion

Electron-impact ionization was analyzed for all 34 levels of the ground configuration of the Fe2+ ion. Both direct and indirect ionization processes are included in the study. For the direct process, the potential of the ionized ion was used to explain experimental data. For the indirect process, electron-impact excitation with subsequent autoionization was considered. The convergence of the cross sections of the indirect process was determined by considering the excitations up to shells with the principal quantum number n ⩽ 20. The main contribution to the indirect process was determined by the 3p → 3d excitations. The scaled distorted-wave approximation was applied to explain measurements for the Fe2+ ion. Scaling of the cross sections lowers values by ~35% at the peak for the ground level and ~50% for the highest level of the ground configuration of the Fe2+ ion. The experimental value for the single ionization threshold was used to provide better agreement with the measurements for the cross sections. This led to the cross sections for the levels of the ground configuration diminishing in size by 15–25%. The direct process contributes ~70% and ~50% to the total ionization from the lowest and highest levels of the ion, respectively.

Relativistic calculation of electron impact excitation of tin: cross sections of importance in plasma modeling

The present work includes the calculation of electron impact excitation cross sections for the fine structure resolved transitions in Sn I and Sn II, using the relativistic distorted wave approximation with the aid of Dirac-Fock multiconfigurational wavefunctions. For Sn I, the cross sections are calculated for the excitations from the fine structure resolved energy levels of the manifolds, , , , and including the ground state, to the fine structure resolved levels of , , , , and manifolds for a wide range of projectile electron energy from 0 to 500 eV. Further, the cross sections are also calculated for the excitations of Sn II from the fine structure resolved energy levels of the manifolds, , , , and including the ground state, to the fine structure resolved levels of , , , , and manifolds. All the cross sections for Sn I except for the excitation from the ground state to the fine-structure resolved excited states of , and manifolds and all the cross sections for Sn II are reported for the first time. In order to ensure the consistency of the present calculations, a comparison of the oscillator strengths and transition probabilities is carried out with the available calculations and measurements. The present cross sections are also compared with previous calculations and observed to be in reasonable agreement. The calculated cross sections are of great interest in the modeling and diagnostic of various plasmas such as laser induced plasma, fusion plasma, etc. Further, to ease the use of calculated cross sections, the analytic fitting of the cross sections is also provided.

Flux-integrated semiexclusive cross sections for charged-current quasielastic and neutral-current elastic neutrino scattering off Ar40 and a sterile neutrino oscillation study

Flux-integrated semiexclusive differential cross sections for charged-current quasielastic and neutral-current elastic neutrino scattering on argon are analyzed. The cross sections are calculated using the relativistic distorted-wave impulse approximation with values of the nucleon axial mass ${M}_{A}=1\text{ }\text{ }\mathrm{GeV}$ and 1.2 GeV. The elastic scattering cross sections are also computed for different strange quark contributions to the neutral-current axial form factor. The flux-integrated differential cross sections as functions of reconstructed neutrino energy are evaluated for the far detector of the SBN experiment. The effects of the short baseline neutrino oscillations are taken into account in a $3+1$ framework. We found that cross sections depend on oscillation parameters and the ratio of the measured and predicted cross sections can be used in a sterile neutrino oscillation study.

Theoretical investigation of electron-impact double ionization of B-like C+ to Ne5+ ions

Electron-impact double ionization (DI) including direct and indirect processes of B-like ${\mathrm{C}}^{+}$ to ${\mathrm{Ne}}^{5+}$ ions is investigated using a two-step approach. The direct process is treated as a primary single ionization (SI) and subsequent knockout mechanism. The indirect process is considered as a primary SI of an inner-shell $1s$ electron followed by a single autoionization. Based on the two-step approach, the DI cross sections are calculated using flexible atomic code with distorted-wave approximation. The potential of the intermediate ion is approximately equivalent to the mean potential constructed by three ions involved for calculating the cross section of the direct process, while using the potential of the ionizing ion in each step of the present two-step approach overestimates results. The present cross sections agree with available theoretical and experimental data.

Extended p_{3/2} Neutron Orbital and the N=32 Shell Closure in ^{52}Ca.

The one-neutron knockout from ^{52}Ca in inverse kinematics onto a proton target was performed at ∼230 MeV/nucleon combined with prompt γ spectroscopy. Exclusive quasifree scattering cross sections to bound states in ^{51}Ca and the momentum distributions corresponding to the removal of 1f_{7/2} and 2p_{3/2} neutrons were measured. The cross sections, interpreted within the distorted-wave impulse approximation reaction framework, are consistent with a shell closure at the neutron number N=32, found as strong as at N=28 and N=34 in Ca isotopes from the same observables. The analysis of the momentum distributions leads to a difference of the root-mean-square radii of the neutron 1f_{7/2} and 2p_{3/2} orbitals of 0.61(23)fm, in agreement with the modified-shell-model prediction of 0.7fm suggesting that the large root-mean-square radius of the 2p_{3/2} orbital in neutron-rich Ca isotopes is responsible for the unexpected linear increase of the charge radius with the neutron number.

Fermi motion effects in electroproduction of hypernuclei

In a previous analysis of electroproduction of hypernuclei the cross sections were calculated in distorted-wave impulse approximation where the momentum of the initial proton in the nucleus was set to zero (the frozen-proton approximation). In this paper we go beyond this approximation assuming a non zero effective proton momentum due to proton Fermi motion inside of the target nucleus discussing also other kinematical effects. To this end we have derived a more general form of the two-component elementary electroproduction amplitude (Chew-Goldberger-Low-Nambu like) which allows its use in a general reference frame moving with respect to the nucleus-rest frame. The effects of Fermi motion were found to depend on kinematics and elementary amplitudes. The largest effects were observed in the contributions from the longitudinal and interference parts of the cross sections. The extension of the calculations beyond the frozen-proton approximation improved the agreement of predicted theoretical cross sections with experimental data and once we assumed the optimum on-shell approximation, we were able to remove an inconsistency which was previously present in the calculations.

The M1 transitions and visible spectra of W13+ ion

The theoretical visible transitions and spectra of W13+ ion from the Electron Beam Ion Trap are investigated by constructing and solving a collisional-radiative model (CRM) within the Quasi-Steady-State approximation. The necessary atomic data for constructing the CRM have been calculated using the relativistic configuration interaction (RCI) method and distorted wave approximation. The present results are in reasonable agreement with the available experimental observations and theoretical calculations. The relative intensities of the M1 transitions between the ground and low-lying excited fine-structure levels have been found to strongly depend on the electron density from ne=109 cm−3 to ne=1012 cm−3, which might be useful for the future diagnosis of the electron density of the plasma.

On the development of relativistic distorted wave approximation for the energies and collision dynamics of atoms or ions subjected to the outside plasma

In this manuscript, we present the development of a relativistic distorted wave method for determining the energies and collision dynamics of plasma-immersed atoms or ions. The methodology is based on the Dirac–Coulomb Hamiltonian, in which contributions from relativity and higher order effects, such as quantum electrodynamics and Breit interaction, are incorporated. The key element in this method is that a modified Debye–Hückel approximation is employed to represent the effect of plasma screening. In order to correctly describe the (bound and continuous state) wave functions, a self-consistent field calculation incorporating the shielding potential is performed within the fully relativistic framework. The particle interaction within the scattering matrix element of the excitation process is described by the shielded Coulomb interaction. The present technique is illustrated by calculations of energy, line shift, transition probability, electron-impact excitation/ionization cross section, and photoionization cross section of a few-electron system confined in plasma environments. The present model is tested and validated against a number of known cases (simulations are made for the He-like Al11+ ion) in the literatures. Numerical results demonstrate that the modifications to the Coulomb potential proposed in the spatial and temporal criteria of the Debye–Hückel approximation allow us to improve the theoretical description of the plasma shielding and thus the dynamical processes in dense plasmas. Comparisons of our computational predictions and the recent experimental measurements are performed. The current work not only has far-reaching implications for our understanding of the dense plasma screening, but also has potential applications in fusion, laboratory astrophysics, and related disciplines.

Electron impact excitation cross section calculations of the fine structure transitions of Mo and their applications in the diagnostics of the laser induced Mo plasma

The current work reports a detailed calculation of electron impact excitation cross sections for the fine structure transitions of Mo from the manifolds 4d 55s, 4d 45s 2, 4d 6 and 4d 55p to the manifolds 4d 55s, 4d 45s 2, 4d 6, 4d 55p, 4d 45s5p, 4d 55d, 4d 56s, 4d 45s6s, and 4d 57s, using the relativistic distorted wave approximation for the applications in plasma modelling. Multi-configurational Dirac–Fock wave functions are used in the calculations. The oscillator strength and cross section results are compared with the previous calculations and measurements. A comprehensive collisional radiative (CR) model is developed and used to characterize laser induced molybdenum plasma to ensure that the calculated cross sections can be used for various plasma modelling applications. The current CR model has taken into account the electron impact excitation and de-excitation processes using the calculated consistent cross sections. The electron induced processes are dominant kinetic processes in the laser induced plasma. Furthermore, the diagnostics of the laser induced Mo plasma is done by coupling the current CR model with the experimental laser induced breakdown spectroscopic measurements of Mal et al (2021 Appl. Phys. B 127 52). The plasma parameter, i.e., electron temperature has been calculated using nine measured intensities of the emission lines of Mo, with wavelengths 406.9, 423.3, 438.2, 453.7, 476.0, 550.7, 553.3, 557.0 and 592.9 nm. The results are also compared with the values reported in the Boltzmann plot at various delay times ranging from 0.5 to 5.0 μs.

Astrophysical N15(n,γ)N16 reaction rate from precision measurement of the N15(d,p)N16 angular distributions

Background: Asymptotic giant branch stars are one of the possible sites of fluorine production as confirmed by astronomical observation. The $^{15}\mathrm{N}(n,\ensuremath{\gamma})^{16}\mathrm{N}$ reaction competes with $^{15}\mathrm{N}(\ensuremath{\alpha},\ensuremath{\gamma})^{19}\mathrm{F}$ on the reaction chain and thus influences the production of the only stable fluorine isotope $^{19}\mathrm{F}$. The $^{15}\mathrm{N}(n,\ensuremath{\gamma})^{16}\mathrm{N}$ reaction rate at low temperatures of astrophysical interest depends on the neutron spectroscopic factors of the four low-lying states in $^{16}\mathrm{N}$.Purpose: The $^{16}\mathrm{N}$ neutron spectroscopic factors from two previous measurements of $(d,p)$ reaction differ by a factor of $\ensuremath{\approx}2$. This work was intended to investigate these spectroscopic factors via a precision measurement of the $^{15}\mathrm{N}(d,p)^{16}\mathrm{N}$ angular distributions using a Q3D magnetic spectrograph.Methods: The high resolution of the Q3D magnetic spectrograph led to clear separation of these four closely spaced states. The neutron spectroscopic factors were extracted from the present angular distributions with the distorted wave Born approximation and adiabatic distorted wave approximation methods.Results: The neutron spectroscopic factors of the $^{16}\mathrm{N}$ ground, 0.120, 0.298, and 0.397 MeV states were extracted to be $1.32\ifmmode\pm\else\textpm\fi{}0.12, 1.33\ifmmode\pm\else\textpm\fi{}0.18, 1.10\ifmmode\pm\else\textpm\fi{}0.10$, and $1.30\ifmmode\pm\else\textpm\fi{}0.14$, respectively. The new results support the conclusion that these four states in $^{16}\mathrm{N}$ are all good single-particle levels. The cross section and reaction rate of $^{15}\mathrm{N}(n,\ensuremath{\gamma})^{16}\mathrm{N}$ were calculated based on the present spectroscopic factors.Conclusions: The present work provides a precision measurement of the spectroscopic factors of the $^{16}\mathrm{N}$ states and the resulting $^{15}\mathrm{N}(n,\ensuremath{\gamma})^{16}\mathrm{N}$ reaction rate. The present higher reaction rate of the $^{15}\mathrm{N}(n,\ensuremath{\gamma})^{16}\mathrm{N}$ reaction suggests higher flow through $^{15}\mathrm{N}(n,\ensuremath{\gamma})^{16}\mathrm{N}$ compared with previous expectation.

α knockout reaction as a new probe for α formation in α -decay nuclei

The $\alpha$-decay phenomennon has been studied for more than a century but the mechanism of the $\alpha$ particle formation and its tunneling process have not yet been fully understood. As an alternative to the $\alpha$-decay lifetime measurement, we propose the proton-induced $\alpha$ knockout reaction, ($p$,$p\alpha$), as a new probe for the surface $\alpha$ formation probability of $\alpha$-decay nuclei. The $^{210,212}$Po($p$,$p\alpha$)$^{206,208}$Pb reaction is described by the distorted-wave impulse approximation framework. It shows that the $^{212}$Po/$^{210}$Po ratio of the $\alpha$ knockout cross sections agrees with that of the surface $\alpha$ formation probabilities determined by lifetime measurements. The result implies that the ($p$,$p\alpha$) cross section is a direct probe for the surface $\alpha$ formation probability, which is an essential quantity to lead us to a complete understanding of the $\alpha$-decay phenomenon.

Probing for high-momentum protons in He4 via the He4(e,e′p)X reactions

Experimental cross sections for the He4(e,e′p)X reactions in the missing energy range from 0.017 to 0.022 GeV and up to a missing momentum of 0.632 GeV/c at xB=1.24 and Q2=2 (GeV/c)2 are reported. The data are compared to relativistic distorted-wave impulse approximation calculations for the He4(e,e′p)H3 channel. Significantly more events are observed for pm≥0.45 GeV/c than are predicted by the theoretical model, and striking fluctuations in the ratio of data to the theoretical model around pm=0.3GeV/c are possible signals of initial-state multinucleon correlations.5 MoreReceived 11 May 2019Revised 13 March 2022Accepted 14 April 2022DOI:https://doi.org/10.1103/PhysRevC.105.064003©2022 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBreakup reactionsFew-body systemsNucleon-nucleon interactionsPropertiesA ≤ 5Nuclear Physics