Plane-wave Born Approximation Research Articles

Role of the Center of Mass Correction in Magnetic Form Factors for 19O and 21Na Exotic Nuclei with Different Interactions

The transverse magnetic electron scattering form factors of 19O and 21Na exotic nuclei were studied in the plane wave Born approximation (PWBA). The shell model calculations with three sd-shell interactions, USDA, USDB, and Wildenthal, were carried out for the ground and low-lying excited states. The exact center of mass correction in the Born approximation picture through the translation-invariant shell model (TISM) was included to modify the transverse form factors. The measured magnetic dipole moments were well reproduced with the core polarization (CP) effect introduced through the effective g-factors. The occupancies percentage with respect to the valence nucleons was also calculated.

Electromagnetic interaction models for Monte Carlo simulation of protons and alpha particles

Electromagnetic interactions of protons and alpha particles are modeled in a form that is suitable for Monte Carlo simulation of the transport of charged particles. The differential cross section (DCS) for elastic collisions with neutral atoms is expressed as the product of the DCS for collisions with the bare nucleus and a correction factor that accounts for the screening of the nuclear charge by the atomic electrons. The screening factor is obtained as the ratio of the DCS for scattering of the projectile by an atom with a point nucleus and the parameterized Dirac–Hartree–Fock–Slater (DHFS) electron density, calculated from the eikonal approximation, and the Rutherford DCS for collisions with the bare point nucleus. Inelastic collisions, which cause electronic excitations of the material, are described by means of the plane-wave Born approximation, with an empirical simple model of the generalized oscillator strength (GOS) that combines several extended oscillators with resonance energies and strengths determined from the atomic configurations and from the empirical mean excitation energy of the material. The contributions from inner subshells are renormalized to agree with realistic ionization cross sections calculated numerically from the DHFS self-consistent model of atoms by means of the plane-wave Born approximation. The resulting DCS allows analytical random sampling of individual hard inelastic interactions.

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.

Quadrupole deformation signatures in elastic electron scattering from oriented odd-A nuclei

Elastic electron scattering from oriented odd-A axially deformed nuclei is studied in the plane-wave Born approximation. The nuclear structure is described within a microscopic selfconsistent Skyrme deformed Hartree-Fock approximation with pairing correlations. The interference form factors between monopole and quadrupole Coulomb terms that characterize the nuclear response with aligned nuclear targets are shown to increase or decrease the unpolarized cross-section, depending on the sign of the quadrupole deformation. This feature provides valuable information on the nuclear deformation that can be used as a signature of the oblate or prolate character of the nuclear shape. Some selected cases of nuclei with different spins are presented that exemplify the scope of the method.

Collision-induced atomic alignment of silver after [formula omitted] ionization by 10–50 keV electron impact: Influence of Coster–Kronig transition effect

This work reports atomic alignment of silver after L3-subshell ionization in collisions with electrons in energy range 10–50 keV, i.e., (3∼15)EL3, where EL3(=3.351 keV) is the binding energy of electrons in the L3-subshell. The characteristic spectra are measured at different emission angles from -40∘ to 40∘ intervals of 10∘. The angular dependence of X-ray emission is studied for Lα(L3→M4,5) and Lβ2(L3→N5) lines respectively, using a thin target. The anisotropy parameter β is determined from two angular distribution measurements of differential X-ray intensity ratios dI(Lα,θ)/dI(Lβ1,θ) and dI(Lβ2,θ)/dI(Lβ1,θ). This served to reduce the experimental uncertainties. For the derivation of the alignment factor A20 we took into account the Coster–Kronig (CK) correction factor κ, which allows for, CK transition probabilities fij among L-subshells. We found that the sign of A20 is mostly negative, and the influence of CK transition is significant for large alignment parameters. In addition, the experimental results are compared to theoretical predictions within the framework of Plane Wave Born Approximation (PWBA) and reasonable agreement is found in the energy range studied in the present work. Simultaneously, good scaling property of atomic alignment is demonstrated when we consider also previous experimental results. Our measurement extends the scaled velocity V (=vp/vt) to a range of 1∼4 for medium-Zt targets, and the results of atomic alignment can provide a fundamental testing ground for ionization models.

Isomeric excitation of U235 by inelastic scattering of low-energy electrons

The nucleus of $^{235}\mathrm{U}$ has a low-lying isomeric state of energy about 76.7 eV. Excitation cross sections of the nucleus from the ground state to the isomeric state by inelastic electron scattering are calculated on the level of the Dirac distorted-wave Born approximation. The cross sections are about six orders of magnitude higher than known values based on the Dirac plane-wave Born approximation, emphasizing the vital importance of wave function distortion in the nuclear excitation. In addition, we find that even with low-energy electrons on the order of 100 eV, relativistic effects are significant without which the excitation cross sections drop by an order of magnitude.

Elastic Form Factors and Matter Density Distributions of Some Neutron-Rich Nuclei

The ground-state properties of exotic 18N and 20F nuclei, including the neutron, proton and matter densities and related radii are investigated using the two-body model of within Gaussian (GS) and Woods Saxon (WS) wave functions. The long tail is evident in the computed neutron and matter densities of these nuclei. The plane wave Born approximation (PWBA) is calculate the elastic form factors of these exotic nuclei. The variation in the proton density distributions due to the presence of the extra neutrons in 18N and 20F leads to a major difference between the elastic form factors of these exotic nuclei and their stable isotopes 14N and 19F. The reaction cross sections for these nuclei are investigated using the Kox and Glauber models. Furthermore, the Glauber model is employed to calculate the matter radii of these exotic nuclei. The calculated results for the selected exotic nuclei are in a good agreement with the experimental data.

Measurement of X-ray production cross sections for lower-energy (0.7-3 MeV) proton induced Cu_K, Ag_L and Au_M X-rays from thick samples

We conducted a series of fundamental research-based experimentations to measure the X-ray production cross sections of K-, L-, and M-shells from thick samples of Cu, Ag and Au, respectively, for proton beam in a lower energy range of 0.7 - 3.5 MeV utilizing particle induced X-ray emission (PIXE) analysis. The measurement theories based on the Merzbacher-Lewis (ML) relationship and procedure for thick samples are described. The detector efficiency and system calibration were first measured from Fe-55 and Am-241 X-ray sources and the standard sample, respectively. The X-ray yields as a function of the proton energy were measured and fitted for finding the derivatives which were used for the cross section measurement. The obtained results were compared with published data and predictions from the theoretical models of Plane Wave Born Approximation (PWBA) and the advanced ECPSSR. The measured results from Cu were in good agreement with the theoretical predictions, especially in the lower energy regime, and most of the reported data, while for the higher energy regime deviated by a few times but still in the same order of magnitude with the theoretical data. The Ag and Au results were in excellent agreement with published and theoretical values.

The ground state properties of some exotic nuclei studied by the two-body model

The ground state properties including the density distributions of the neutrons, protons and matter as well as the corresponding root mean square (rms) radii of proton-rich halo candidates 8B, [Formula: see text]N, [Formula: see text]Al and [Formula: see text]P have been studied by the single particle Bear–Hodgson (BH) wave functions with the two-body model of [Formula: see text]. It is found that the rms radii of these proton-rich nuclei are reproduced well by this model and the radial wave functions describe the long tail of the proton and matter density distributions. These results indicate that this model achieves a suitable description of the possible halo structure. The plane wave Born approximation (PWBA) has been used to compute the elastic charge form factors.

A model for Geant4-DNA to simulate ionization and excitation of liquid water by protons travelling above 100 MeV

Biological damage induced by ionizing radiation plays a major role in many application fields as radiotherapy and microdosimetry. Geant4-DNA Monte Carlo track-structure code has the capability to simulate the passage of radiation through liquid water, containing physical, physicochemical and chemical processes that lead the early DNA damage. For proton projectile, current models reach up to an incident energy of 100 MeV. In order to cover the entire energy regime involved in proton radiotherapy, this work presents a new model that extends proton ionization and excitation of liquid water up to 300 MeV. Calculation of cross section dataset is made for ionization of five ionization shells and five excitation levels of liquid water using the Relativistic Plane Wave Born Approximation (RPWBA). Implementation is validated through the spower and range examples of the official release, obtaining an agreement within 1% with respect to reference data published in ICRU90 report.

Inelastic collisions of fast charged particles with atoms: Bethe asymptotic formulas and shell corrections

The relativistic plane-wave Born approximation is applied to the study of inelastic collisions of charged particles with atoms, by considering atomic wave functions calculated from the independent-electron approximation with the self-consistent Dirac-Hartree-Fock-Slater potential. A database of longitudinal and transverse generalized oscillator strengths (GOSs) has been computed by using accurate numerical methods for all the subshells of the ground-state configurations of the elements with atomic numbers from 1 (hydrogen) to 99 (einsteinium). The calculated GOS do not satisfy the Bethe sum rule; departures from the sum rule are in accordance with previous theoretical estimates. Asymptotic high-energy formulas for the total cross section, the stopping cross section, and the energy-straggling cross section are derived with proper account of the relativistic departure from the Bethe sum rule. The shell correction is calculated as the energy-dependent term that, when added to the asymptotic formula, reproduces the value of the atomic cross section calculated by integrating the energy-loss differential cross section. Shell corrections to the stopping cross section obtained from the present approach are presented and compared with previous estimates.

Study of Matter Density Distributions, Elastic Electron Scattering Form factors and Root Mean Square Radii of 9C, 12N, 23Al, 11Be and 15C Exotic Nuclei

The ground state densities of neutron-rich (11Be,15C) and proton-rich (9C,12N,23Al) exotic nuclei are investigated using a two-body nucleon density distribution (2BNDD) with two frequency shells model (TFSM). The structure of the valence one-neutron of 11Be is in pure (1p1/2) and of 15C in pure (1d5/2) configuration, while the structure of valence one-proton configuration is in 9C,12N are to be in a pure (1p1/2) and 23Al in a pure (2s1/2) . For our studied nuclei, an efficient (2BNDD) operator for point nucleon system folded with two-body correlation operator's functions is used to investigate nuclear matter density distributions, elastic electron scattering form factors, and root-mean square (rms) radii. The effect of the strong tensor force (TC) in nucleon-nucleon forces is taken into account in the correlation. The wave functions of a single particle harmonic oscillator are used with two different oscillator size parameters, βc and βv, the former for core (inner) orbits and the latter for valence (halo) orbits. The measured matter density distributions of these nuclei clearly show the long tail results. The plane wave born approximation (PWBA) is used to investigate the elastic electron scattering form factors for these exotic nuclei.

W L-shell X-ray emission induced by C<sup>6+ </sup>ions with several hundred MeV/u

The L-shell X-ray emission of tungsten is investigated under the bombardment of C<sup>6+</sup> ions in a high energy range of 154—424 MeV/u. Compared with the atomic data, the energy of the X-ray is enlarged, and the relative intensity ratio of Lι, Lβ<sub>1,3,4</sub> and Lβ<sub>2,15</sub> to Lα<sub>1,2</sub> X-rays are enhanced. The L-subshell and the total X-ray production cross section are calculated from a well corrected thick target formula and compared with the theoretical estimation of binary encounter approximation (BEA), plane-wave Born approximation (PWBA) and ECPSSR (PWBA theory modified with Energy-loss, Coulomb-repulsion, Perturbed-Stationary-State and Relativistic corrections). On the whole, the experimental cross sections are all smaller than the prediction of PWBA and ECPSSR, but in rough agreement with that of BEA. It is indicated that the inner-shell ionization of W can be considered as a binary process between the high energy C<sup>6+</sup> ions acting as a point charge and the independent target electrons. With the L-shell ionization, the outer-shells are multiply ionized. The multi-ionization degree is approximately regard as a constant in the present work. This leads the X-ray energy to be blueshifted and the relative intensity ratios of Lι and Lβ to Lα X-ray to be enhanced. Using the atomic parameters corrected by multi-ionization, the X-ray production cross section can be estimated by the BEA model.

Collisions of Nucleons with Atoms: Calculated Cross Sections and Monte Carlo Simulation

After a summary description of the theory of elastic collisions of nucleons with atoms, we present the calculation of a generic database of differential and integrated cross sections for the simulation of multiple elastic collisions of protons and neutrons with kinetic energies larger than 100 keV. The relativistic plane-wave Born approximation, with binding and Coulomb-deflection corrections, has been used to calculate a database of proton-impact ionization of K-shell and L-, M-, and N-subshells of neutral atoms These databases cover the whole energy range of interest for all the elements in the periodic system, from hydrogen to einsteinium (Z = 1–99); they are provided as part of the penh distribution package. The Monte Carlo code system penh for the simulation of coupled electron-photon-proton transport is extended to account for the effect of the transport of neutrons (released in proton-induced nuclear reactions) in calculations of dose distributions from proton beams. A simplified description of neutron transport, in which neutron-induced nuclear reactions are described as a fractionally absorbing process, is shown to give simulated depth-dose distributions in good agreement with those generated by the Geant4 code. The proton-impact ionization database, combined with the description of atomic relaxation data and electron transport in penelope, allows the simulation of proton-induced x-ray emission spectra from targets with complex geometries.

Extraction of the second and fourth radial moments of nuclear charge density from the elastic electron-nucleus scattering

The expression for the radial moments of the nuclear charge density has been discussed under the plane wave Born approximation method recently, which is significant to investigate the nuclear surface thickness and neutron distribution radius. In this paper, we extend the studies of extracting second-order moment and fourth-order moment from the Coulomb form factors |F C(q)|2 by the distorted wave Born approximation (DWBA) at the small momentum transfer q region. Based on the relativistic mean-field calculations, the DWBA form factors are expanded into q 4, where the corresponding charge distributions are corrected by the contributions of neutron and spin–orbit densities. In the small q region, it is found that the experimental |F C(q)|2 can be well reproduced by considering the contributions of the at the small q region. Through further analyzing the second-order and fourth-order expansion coefficients of the , the relationship between the expansion coefficients and proton number Z is obtained. By the relationship, we extract the and from the limited experimental data of form factors at the small q region. Within the permissible range of error, the extracted are consistent with the experimental data in this paper.

A Brief Review on the Development of Theories of Deuteron Stripping Reactions

As one type of single nucleon transfer reactions, deuteron stripping reaction has been playing an important role in the study of the single particle structure of atomic nuclei. Also, as a typical three-body nuclear reaction, deuteron stripping reaction is also an important subject in the development of direct nuclear reaction theories. We briefly review the history of the study of deuteron stripping reactions, focusing on the evolution of its theoretical descriptions, and the differences and the relations of the Plane Wave Born Approximation(PWBA), the distorted wave Born approximation(DWBA), the Continuum Discretized Coupled Channel method(CDCC), the Adiabatic Wave Approximation(ADWA), and the approach based on the Faddeev equations. Some important problems being currently addressed in deuteron stripping reaction theories and those should be addressed in near future are also discussed, including deuteron strippings at intermediate and high energies, theory for transfer to continuum states of the residue nuclei, applications of dispersive optical model potentials and non-local potentials, etc. Some preliminary results of our theoretical calculations for deuteron stripping at intermediate and high energies are presented.

Matter Density Distributions, Root-mean Square Radii and Elastic Electron Scattering Form Factors of Some Exotic Nuclei (17B, 11Li, 8He)

The two-neutron halo-nuclei (17B, 11Li, 8He) was investigated using a two-body nucleon density distribution (2BNDD) with two frequency shell model (TFSM). The structure of valence two-neutron of 17B nucleus in a pure (1d5/2) state and in a pure (1p1/2) state for 11L and 8He nuclei. For our tested nucleus, an efficient (2BNDD's) operator for point nucleon system folded with two-body correlation operator's functions was used to investigate nuclear matter density distributions, root-mean square (rms) radii, and elastic electron scattering form factors. In the nucleon-nucleon forces the correlation took account of the effect for the strong tensor force (TC's). The wave functions of single particle harmonic oscillator are used with two different oscillator size parameters βc and βv, where the former is for the core (inner) orbits and the latter is for the valence (halo) orbits. The measured matter density distributions of these nuclei clearly showed long tail results. To investigate elastic electron scattering form factors the plane wave born approximation (PWBA) with two body nucleon density distribution (2BNDD's) was use.

Study of the proton halo structure of nuclei 23Al and 27P using the binary cluster model

The neutron, proton, and matter densities of the ground state of the proton-rich 23Al and 27P exotic nuclei were analyzed using the binary cluster model (BCM). Two density parameterizations were used in BCM calculations namely; Gaussian (GS) and harmonic oscillator (HO) parameterizations. According to the calculated results, it found that the BCM gives a good description of the nuclear structure for above proton-rich exotic nuclei. The elastic form factors of the unstable 23Al and 27P exotic nuclei and those of their stable isotopes 27Al and 31P are studied by the plane-wave Born approximation. The main difference between the elastic form factors of unstable nuclei and their stable isotopes is caused by the variation in the proton density distributions, especially the details of the outer part. Moreover, the Glauber model is used to calculate the matter rms radii and reaction cross-section of these exotic nuclei. The calculated results of the mentioned nuclei give a good accordance with the experimental data.

Simple electron-impact excitation cross-sections including plasma density effects

The modeling of non-local-thermodynamic-equilibrium plasmas is crucial for many aspects of high-energy-density physics. It often requires collisional–radiative models coupled with radiative-hydrodynamics simulations. Therefore, there is a strong need for fast and as accurate as possible calculations of the cross-sections and rates of the different collisional and radiative processes. We present an analytical approach for the computation of the electron-impact excitation (EIE) cross-sections in the Plane Wave Born (PWB) approximation. The formalism relies on the screened hydrogenic model. The EIE cross-section is expressed in terms of integrals, involving spherical Bessel functions, which can be calculated analytically. In order to remedy the fact that the PWB approximation is not correct at low energy (near threshold), we consider different correcting factors (Elwert–Sommerfeld, Cowan–Robb, Kilcrease–Brookes). We also investigate the role of plasma density effects such as Coulomb screening and quantum degeneracy on the EIE rate. This requires to integrate the collision strength multiplied by the Fermi–Dirac distribution and the Pauli blocking factor. We show that, using an analytical fit often used in collisional–radiative models, the EIE rate can be calculated accurately without any numerical integration, and compare our expression with a correction factor presented in a recent work.

Nuclear shape transitions and elastic magnetic electron scattering

Backward elastic electron scattering from odd-$A$ nuclear targets is characterized by magnetic form factors containing precise information on the nuclear structure. We study the sensitivity of the magnetic form factors to structural effects related to the evolution and shape transitions in both isotopic and isotonic chains. Calculations of magnetic form factors are performed in the plane-wave Born approximation. The nuclear structure is obtained from a deformed self-consistent mean-field calculation based on a Skyrme $\mathrm{HF}+\mathrm{BCS}$ formalism. Collective effects are included in the cranking approximation, whereas nucleon-nucleon correlations are taken into account in the coherent density fluctuation model. The evolution of the magnetic form factors is found to exhibit signatures of shape transitions that show up in selected isotopic and isotonic chains involving both stable and unstable nuclei. Several cases are identified as suitable candidates for showing such fingerprints of shape transitions. A new generation of electron scattering experiments involving electron-radioactive beam colliders will be available in the near future, leading to a renewed interest in this field.