Neutron Density Research Articles

Systematic researches on the ground state properties of the medium-mass neutron-rich nuclei

Abstract The recently developed relativistic-mean-field in complex momentum representation with the functional NL3* was used to explore the exotic properties of neutron-rich Pd, Cd, Te, and Xe isotopes. The results were compared with those obtained using the relativistic Hartree-Bogoliubov (RHB) calculations, and available experimental data. The single-particle levels were obtained for the bound and resonant states. The two neutron separation energies S2n and root mean square (rms) radii, agree with the experimental data. It is shown that a halo structure in the extremely neutron-rich 164-180Te and 164-182Xe, as well as a thick neutron skin in the extremely neutron-rich Pd and Cd isotopes. From the numbers of neutrons (Nλ) and (N0 ) occupying the levels above the Fermi surface and the zero-potential energy level, it was found that pairing correlations play an important role in the formation of halo phenomena. These findings are further supported by investigating S2n, rms radii, occupation probabilities, contributions of single-particle levels to the neutron rms radii, and density distributions. The neutron rms radii increased sharply, obviously deviating from the traditional rule r ∝A1/3, and the density distributions were very diffuse. Finally, the contributions of different single-particle levels to the total neutron density and wavefunction were discussed. It was found that the sudden increase in the neutron rms radii and diffuse density distributions mainly come from the resonant levels with a lower orbital angular momentum near the continuum threshold.

Asymptotic Normalization Coefficient Investigation of the 17O(d, p) Transfer for Astrophysical Application to the 17O(n, α)14C Reaction at Low Energies

Indirect methods have proven to be a complementary approach for extending our knowledge of nuclear structure and low-energy cross sections. Among these, the neutron-induced reaction cross sections appear to be of particular interest since their role both for unstable and stable beams. In view of this, we report here the combined study of the 17O(n, α)14C reaction accomplished by the Trojan Horse Method (THM) and the asymptotic normalization coefficient (ANC) method. The low-lying 8038, 8125, 8213, and 8282 keV resonances in 18O are studied, and their Γ n are derived. A comparison with recent direct data and recent THM experimental data is presented. The independent ANC investigation corroborates our previous THM results, confirms the consistence of the two indirect investigations, and shows new frontiers for neutron-induced reactions with radioactive ion beams. Moreover, we examined the impact of adopting the newly recommended 17O(n, α)14C reaction rate on asymptotic giant branch stars' nucleosynthesis. Our findings reveal significant variations (≳10%) in the production of the neutron-rich heavy isotopes sensitive to neutron density, underlining the neutron-poisoning effect of 17O on the s-process.

Responsibility of small defects for the low radiation tolerance of coated conductors

Abstract Rare-earth-barium-copper-oxide based coated conductors exhibit a relatively low radiation robustness compared to e.g. Nb3Sn due to the d-wave symmetry of the order parameter, rendering impurity scattering pair breaking. The type and size of the introduced defects influence the degrading effects on the superconducting properties; thus the disorder cannot be quantified by the number of displaced atoms alone. In order to develop degradation mitigation strategies for radiation intense environments, it is relevant to distinguish between detrimental and beneficial defect structures. Gadolinium-barium-copper-oxide based samples irradiated with the full TRIGA Mark II fission reactor spectrum accumulate a high density of point-like defects and small clusters due to n - γ capture reactions of gadolinium. This leads to a 14–15 times stronger degradation of the critical temperature compared to samples shielded from slow neutrons. At the same time both irradiation techniques lead to the same degradation behavior of the critical current density as function of the transition temperature J c ( T c ) . Furthermore, annealing the degraded samples displayed the same T c recovery rates, indicating the universality of the defects responsible for the degradation. Since the primary knock on atom of the n - γ reaction as well as the recoil energy is known, we used molecular dynamics simulations to calculate which defects are formed in the neutron capture process and density functional theory to assess their influence on the local density of states. The defects found in the simulation were mainly single defects as well as clusters consisting of Oxygen Frenkel pairs, however, more complex defects such as Gd C u antisites occurred as well.

Integrated modeling of anisotropic neutron yields of classical and spherical tokamaks

Estimations of counting rates of neutron spectrometers in experiments on controlled fusion with magnetic confinement, as well as calculations of energy resolved flux densities of fusion neutrons from plasma to the walls of a reactor require spatial integration of the local, usually anisotropic function of the neutron source. The integrated modeling consists of three main stages. First, sources of fast particles in beam- or wave-heated plasma are calculated. The next stage deals with spatial, energetic, and angular velocity distributions of plasma ions. Finally, double differential rate coefficients of nuclear fusion reactions are computed. This article describes calculations of spatial distributions of nuclear fusion reaction rates in classical and spherical tokamaks and the anisotropy of the neutron yield and spectra. The results are based on analytical formulas for energetic and angular distributions of the local source of fusion products in plasma. Examples of energetic spectral densities of neutron fluxes on first walls are presented, as well as energy resolved counting rates of collimated neutron spectrometers for perpendicular and tangential lines of sight.

NEUTRON POINT KINETICS MODEL WITH A DISTRIBUTED-ORDER FRACTIONAL DERIVATIVE

In this paper, the solutions of an extended form of the Fractional-order Neutron Point Kinetics (FNPK) equation in terms of Caputo-time derivatives of the same order are investigated. Instead of using a Caputo derivative, a distributed-order fractional derivative in the Caputo sense was employed in the term of the FNPK equation which is multiplied by the reactivity. This term plays an important role in the description of neutron kinetics during the start-up, shutdown, and steady-state processes in nuclear reactors. The extended (DFNPK) model was solved using the beta, normal, bimodal and Dirac delta distributions to investigate their effect on the transient state solutions of the neutron density. Regardless of the distribution used, the most significant finding is that a destabilizing effect on the neutron density is induced when the mode (or the instant of application of the Dirac delta) of the distribution tends to one while maintaining the orders of the Caputo-time derivatives constant. What defines the destabilizing effect are large magnitude oscillations, a rapid decay, and an oscillation-free steady state with a monotonic increase that is parallel to but somewhat above the trend determined by the FNPK equation. The extended model is anticipated to be effective for modeling neutron density dispersion in a highly heterogeneous medium that may be described using distributed derivatives.

Calculation of Porosity and CO2 Content in CO2-Bearing Gas Formations Based on Density and Neutron Logging Forward and Inverse Methods.

When the gas reservoir contains CO2, the logging response tends to be complex. When calculating porosity with neutron and density curves, the accuracy of porosity calculation is reduced due to the uncertainty of fluid parameters, which in turn affects the evaluation of CO2 content. It increases the uncertainty of reserve evaluation. In this paper, a forward modeling model of density logging is established based on the equivalent volume model, and the method of neutron logging is formed by putting forward the nonlinear formula of the reciprocal of the comprehensive deceleration length. The key parameters and their variation rules of neutron logging and density logging forward modeling under different temperature and pressure conditions are obtained by experimental means and Monte Carlo method, respectively. The variation law of the influence of different CO2 content on neutron logging and density logging curves is simulated. The research shows that with the increase of CO2 content, the neutron logging value of gas reservoir decreases, while the density logging value increases, and the greater the porosity of gas reservoir, the more obvious this change trend is. Compared with CH4, CO2 has a more significant impact on the excavation effect of neutron logging. On this basis, combined with the conductivity model of the study area, the porosity and CO2 content of CO2 bearing gas reservoir are solved by the least-square method. The practice shows that the relative error of porosity calculation is within ±4% and the error of CO2 content calculation is within ±10%, which proves the reliability of this method.

The Neutron Skin-Thickness of 208Pb Determined by Electron and Proton Scattering

Abstract Electron as well as proton elastic scattering is not able to determine the point proton and point neutron densities, $\rho _\tau (r), (\tau =p,n)$, separately. If both scatterings are analyzed consistently, those densities would be determined uniquely, since the two densities are observed by different combinations from each other. Previous experiments have not provided $\rho _\tau (r)$ uniquely, but the values of the mean square radii of $\rho _p(r)$, $\langle \, r^2\, \rangle _p$, and of $\rho _n(r)$, $\langle \, r^2\, \rangle _n$, are shown to be determined consistently through the fourth moment of the observed charge density, $\langle \, r^4\, \rangle _c$, in 208Pb. The previous analyses of $(\gamma , \pi ^0)$ and $\bar{p}$-nucleus obtained a similar value of $\langle \, r^2\, \rangle _n$, but they do not yield the experimental value of $\langle \, r^4\, \rangle _c$ observed in electron scattering.

S-wave log construction through semi-supervised regression clustering using machine learning: A case study of North Sea fields

Accurate prediction of S-wave velocity from well logs is essential for understanding subsurface geological formations and hydrocarbon reservoirs. Machine learning techniques, including clustering and regression, have emerged as effective methods for indirectly estimating S-wave logs and other rock properties. In this study, we employed clustering algorithms to identify similarities among well log datasets, encompassing depth, sonic, porosity, neutron, and apparent density, facilitating the discovery of correlations among various wells. These identified correlations served as a foundation for predicting S-wave values using a novel semi-supervised approach. Our approach combined clustering, specifically k-means clustering, with different types of regressors, including Least Squares Regression (LSR), Support Vector Regression (SVR), and Multi-Layer Perceptron (MLP). Our results demonstrate the superior performance of this integrated approach compared to traditional regression methods. We validated our methodology using various parametric and non-parametric regression techniques, showcasing its effectiveness not only on wells within the training region but also on wells outside the study area. We achieved a significant improvement in the R2 score metric, ranging from 2.22% to 6.51%, and a reduction in Mean Square Error (MSE) of at least 31% when compared to predictions made without clustering. This study underscores the potential of machine learning techniques for accurate prediction of S-wave velocity and other rock properties, thereby enhancing our comprehension of subsurface geology and hydrocarbon reservoirs.

New Wolf–Rayet wind yields and nucleosynthesis of Helium stars

ABSTRACT Strong metallicity-dependent winds dominate the evolution of core He-burning, classical Wolf–Rayet (cWR) stars, which eject both H and He-fusion products such as $^{14}$N, $^{12}$C, $^{16}$O, $^{19}$F, $^{22}$Ne, and $^{23}$Na during their evolution. The chemical enrichment from cWRs can be significant. cWR stars are also key sources for neutron production relevant for the weak s-process. We calculate stellar models of cWRs at solar metallicity for a range of initial Helium star masses (12–50 $\rm M_{\odot }$), adopting recent hydrodynamical wind rates. Stellar wind yields are provided for the entire post-main sequence evolution until core O-exhaustion. While literature has previously considered cWRs as a viable source of the radioisotope $^{26}$Al, we confirm that negligible $^{26}$Al is ejected by cWRs since it has decayed to $^{26}$Mg or proton-captured to $^{27}$Al. However, in Paper I, we showed that very massive stars eject substantial quantities of $^{26}$Al, among other elements including N, Ne, and Na, already from the zero-age-main-sequence. Here, we examine the production of $^{19}$F and find that even with lower mass-loss rates than previous studies, our cWR models still eject substantial amounts of $^{19}$F. We provide central neutron densities (N$_{n}$) of a 30 $\rm M_{\odot }$ cWR compared with a 32 $\rm M_{\odot }$ post-VMS WR and confirm that during core He-burning, cWRs produce a significant number of neutrons for the weak s-process via the $^{22}$Ne($\alpha$,n)$^{25}$Mg reaction. Finally, we compare our cWR models with observed [Ne/He], [C/He], and [O/He] ratios of Galactic WC and WO stars.

Neutron and fission decay width of superheavy compound nuclei

We have systematically studied the level density parameter, nuclear level density and decay width of neutron and fission of superheavy nuclei. This paper is validated by comparing with available experiments for [Formula: see text]U and [Formula: see text]Pu. The angular dependence of level density parameter, level densities and ratio of [Formula: see text] and [Formula: see text] the thermal excitation energy (U) have been studied in the superheavy region [Formula: see text]. The role of entrance channel parameter on decay width of neutron and fission has been studied both in cold and hot fusion reactions in the superheavy region [Formula: see text]. Systematic variation of [Formula: see text] and [Formula: see text] is observed in both the cases. Finally, predicted [Formula: see text] for the synthesis of superheavy nuclei [Formula: see text] and 120.

New formulas for Maclaurin's series expansions of powers of arctangent and other related topics

The work proposes new formulas for Maclaurin's series expansions of natural and real powers of the arctangent and hyperbolic arctangent, a new explicit formula for the Meixner polynomial of the second kind, an explicit expression for the coefficients associated with the moment of neutron density of a point source in an infinitely scattering medium, an explicit formula for the coefficients that are related to the series expansion proposed by Rzadkowskii of the Riemann zeta function, as well as several other identities containing Bell polynomials of the second kind and Stirling numbers of the first kind.

A Method to Calculate the Maturity Time for Low Neutron Source Nuclear Reactor Startup Simulations

An approximate method for determining the maturity time is presented for applications in low neutron source nuclear reactor startup simulations. This new method relies only on the calculation of the mean neutron density and does not require the additional calculation of the variance in the neutron density as the traditional method does. The most accurate method for determining the safe neutron source strength, required to sufficiently mitigate the probability of a rogue transient during nuclear reactor startup, uses the Pál-Bell equations. However, as space and energy dependencies are included, the numerical computation become computationally demanding. Therefore, approximate methods that significantly reduce the computation time and improve the computational efficiency of the simulation while remaining very accurate are extremely useful. The approximate method for determining the maturity time presented in this study has shown excellent agreement with traditional methods while offering an order of magnitude reduction in computation time.

Neutron Skins: Weak Elastic Scattering and Neutron Stars

The recently completed PREX-2 campaign measured the density distribution of neutrons in the lead nucleus as a function of momentum transfer (the form factor), confirmed a relatively large extent of the neutrons beyond the protons in the nucleus (the neutron skin), and provided a precise determination of the density of protons and neutrons at the center of a heavy nucleus. In turn, the measured form factor can be related to various nuclear and neutron star properties. The NICER X-ray telescope has inferred the masses and radii of some X-ray pulsars (neutron stars), although complications arise when determining these quantities independently. Further improvements in NICER have enabled simultaneous mass–radius determinations that had not previously been possible. During the next decade, measurements in astrophysics, gravitational-wave astronomy, and nuclear physics are expected to provide a wealth of more precise data. In this review, we present an overview of the current state of neutron skin measurements and offer insights into prospects for the future.

Fractional neutron point kinetics model for reactivity transients of the NuScale and comparison with the classical kinetics approach

In this work, the behavior of the neutron and natural circulation parameters for Reactivity Transients in the PWR reactor called NuScale are analyzed through the novel system equations. It model includes non-integer and ordinary order differentials coupling the neutron point kinetic with the thermos-hydraulic core of the reactor. Transient processes of increase and decrease of the feed flow and the coolant inlet temperature are analyzed to identify the feedback of the thermohydraulic effects with the neutron phenomena in a non-integer order scheme. In the results, it is observed that for short times the power and reactivity have the same behavior with the classical model and the non-integer order model for all values of the anomalous diffusion coefficient simulated. However, for long times of simulation, there is more reactivity and less power, in contrast to the classical model, when reducing the value of the anomalous diffusion coefficient these counter-intuitive effects are increased. Additionally, a delay is observed between the change in reactivity and power. In conclusion, it is observed that sub-diffusive processes are relevant when there are slow changes in reactivity, which can be analyzed through the Fractional Order Neutron Density Equation.

The impact of overshoot on thei-process in AGB stars

Context.The production of neutron-rich elements at neutron densities intermediate to those of thes- andr-processes, the so-calledi-process, has been identified as possibly being responsible for the observed abundance pattern found in certain carbon-enhanced metal-poor (CEMP) stars. The production site may be low-metallicity stars on the asymptotic giant branch (AGB) where the physical processes during the thermal pulses are not well known.Aims.We investigate the impact of overshoot from various convective boundaries during the AGB phase on proton ingestion events (PIEs) and the neutron densities as a necessary precondition for thei-process as well as on the structure and continued evolution of the models.Methods.We therefore analyzed models of a 1.2 M⊙,Z = 5 × 10−5star. A fiducial model without overshoot on the AGB (overshoot was applied during the pre-AGB evolution) serves as a reference. The same model was then run with various overshoot values and the resulting models were compared to one another. Light element nucleosynthesis is also discussed. Additionally, we introduce a new timescale argument to predict PIE occurrence to discriminate between a physical and a numerical reason for a nonoccurrence. A comparison to observations as well as previous studies was conducted before finally presenting the most promising choice of overshoot parameters for the occurrence of thei-process in low-mass, low-metallicity models.Results.The fiducial model reveals high neutron densities and a persistent split of the pulse-driven convection zone (PDCZ). Overshoot from the PDCZ results in either temporary or permanent remerging of the split PDCZ, influencing the star’s structure and evolution. While both overshoot and non-overshoot models exhibit PIEs generating neutron densities suitable for thei-process, they lead to varied C/O and N/O ratios and notable Li enhancements. Comparison with previous studies and observations of CEMP-r/s stars suggests that while surface enhancements in our models may be exaggerated, abundance ratios align well. Though, for high values of overshoot from the PDCZ the agreement becomes worse.

Machine Learning for Predicting Neutron Effective Dose

The calculation of effective doses is crucial in many medical and radiation fields in order to ensure safety and compliance with regulatory limits. Traditionally, Monte Carlo codes using detailed human body computational phantoms have been used for such calculations. Monte Carlo dose calculations can be time-consuming and require expertise in different processes when building the computational phantom and dose calculations. This study employs various machine learning (ML) algorithms to predict the organ doses and effective dose conversion coefficients (DCCs) from different anthropomorphic phantoms. A comprehensive data set comprising neutron energy bins, organ labels, masses, and densities is compiled from Monte Carlo studies, and it is used to train and evaluate the supervised ML models. This study includes a broad range of phantoms, including those from the International Commission on Radiation Protection (ICRP-110, ICRP-116 phantom), the Visible-Human Project (VIP-man phantom), and the Medical Internal Radiation Dose Committee (MIRD-Phantom), with row data prepared using numerical data and organ categorical labeled data. Extreme gradient boosting (XGB), gradient boosting (GB), and the random forest-based Extra Trees regressor are employed to assess the performance of the ML models against published ICRP neutron DCC values using the mean square error, mean absolute error, and R2 metrics. The results demonstrate that the ML predictions significantly vary in lower energy ranges and vary less in higher neutron energy ranges while showing good agreement with ICRP values at mid-range energies. Moreover, the categorical data models align closely with the reference doses, suggesting the potential of ML in predicting effective doses for custom phantoms based on regional populations, such as the Saudi voxel-based model. This study paves the way for efficient dose prediction using ML, particularly in scenarios requiring rapid results without extensive computational resources or expertise. The findings also indicate potential improvements in data representation and the inclusion of larger data sets to refine model accuracy and prevent overfitting. Thus, ML methods can serve as valuable techniques for the continued development of personalized dosimetry.

Effect of gadolinium addition on microstructure, mechanical properties and corrosion behavior of titanium alloys for spent fuel storage and neutron shielding

Ti-Gd alloys with Gd contents of 2 wt%–8 wt% were prepared, and the influence of Gd content on the microstructure, mechanical properties, corrosion behavior, neutron absorption property and density of the alloy weas investigated. The microstructure changes from full lamellar α phase to fine equiaxed crystals, and the area fraction of Gd-rich phase decreases from 3.2% to 1.8% and then increases to 9.1%. Gd has three existing forms: pure Gd, compound oxide of Gd2TiO5 and/or Gd2O3 and solidifies in the Ti matrix. Ti-4Gd exhibits the best mechanical properties, its tensile strength and elongation is 102 MPa and 49%, respectively. The neutron transmittancy of Ti-8Gd alloy in water is the lowest, which is 3.75%. The corrosion rate of Ti-Gd alloy is 0.00097–0.00238 mm/a, which meets the corrosion standard of small-scale nuclear reactors and containers for spent fuel.

A high-intensity, low-energy heavy ion source for a neutron target proof-of-principle experiment at LANSCE

The capability to directly study neutron capture reactions on radionuclides with half-lives on the of order minutes would allow key cross section measurements in nuclear astrophysics and energy applications. However, such experiments with stationary targets are currently impossible because of signal detection and target sample fielding issues. To overcome these challenges, a neutron target facility is being developed to permit neutron capture experiments on unstable isotopes in inverse kinematics at the Los Alamos Neutron Science Center (LANSCE). This next-generation facility will consist of a heavily moderated, high-intensity spallation neutron target coupled with a radioactive ion beam storage ring. A proof-of-principle experiment is underway to demonstrate this neutron target concept at LANSCE in the near term with stable ions and without a storage ring. Here, mA-level beams of 10-50 keV heavy ions exhibiting large resonant neutron capture cross sections will be generated by a new ion source, transported through a large-volume neutron moderator surrounding an adjacent spallation target driven by the LANSCE accelerator, and collected downstream of the moderator for subsequent decay-counting measurements. The neutron density within the moderator will be obtained from these decay yields and compared with Monte Carlo N-Particle (MCNP) simulation results. The science application, operational requirements, and performance objectives for this heavy ion source will be presented.

A Precision Benchmark Suite for Nuclear Reactor Point Kinetics Equations via Converged Accelerated Taylor Series (CATS)

Extreme benchmarks of 10 or more places for the point kinetics equations for time-dependent nuclear reactor power transients are rare. Therefore, to establish an extreme benchmark, we employ a Taylor series (TS) with continuous analytical continuation to solve the ordinary differential equations of point kinetics including feedback. Nonlinear Wynn-epsilon convergence acceleration confirms the highly precise solutions for neutron and precursor densities. Through adaptive partitioning of time intervals, the proposed Converged Accelerated Taylor Series, or CATS algorithm in double precision, automatically performs successive mesh refinement to obtain high-precision initial conditions for each subinterval, with the intent to reduce propagation error. Confirmation of 10 to 12 places comes from comparison to the BEFD (Backward Euler Finite Difference) algorithm in quadruple precision also developed by the author. We report benchmark results for common cases found in the literature including step, ramp, zigzag, and sinusoidal prescribed reactivity insertions and insertions with nonlinear adiabatic Doppler feedback. We also establish a suite of new prescribed reactivity insertions and insertions with feedback, based on reactivities with Taylor series representations as suggested by the CATS algorithm.

Determination of Nuclear Matter Radii by Means of Microscopic Optical Potentials: The Case of 78\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{78}$$\\end{document}Kr

In this work we use microscopic Nucleon–Nucleus Optical Potentials (OP) to analyze elastic scattering data for the differential cross section of the 78\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{78}$$\\end{document}Kr (p,p) 78\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{78}$$\\end{document}Kr reaction, with the goal of extracting the matter radius and estimating the neutron skin, quantities that are both needed to determine the slope parameter L of the nuclear symmetry energy. Our analysis is performed with the factorized version of the microscopic OP obtained in a previous series of papers within the Watson multiple scattering theory at the first order of the spectator expansion, which is based on the underlying nucleon–nucleon dynamics and is free from phenomenological inputs. Differently from our previous applications, the proton and neutron densities are described with a two-parameter Fermi (2pF) distribution, which makes the extraction of the matter radius easier and allows us to make a meaningful comparison with the original analysis, that was performed with the Glauber model. With standard minimization techniques we performed data analysis and extracted the matter radius and the neutron skin. Our analysis produces a matter radius of Rm(rms)=4.12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_m^{\\mathrm{(rms)}} = 4.12$$\\end{document} fm, in good agreement with previous matter radii extracted from 76\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{76}$$\\end{document}Kr and 80\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{80}$$\\end{document}Kr, and a neutron skin of ΔRnp≃-0.1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta R_{np} \\simeq - 0.1$$\\end{document} fm, compatible with a previous analysis. Our factorized microscopic OP, supplied with 2pF densities, is a valuable tool to perform the analysis of the experimental differential cross section and extract information such as matter radius and neutron skin. Without any free parameters it provides a reasonably good description of the experimental differential cross section for scattering angles up to ≈\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\approx $$\\end{document} 40 degrees. Compared to the Glauber model our OP can be applied to a wider range of scattering angles and allows one to probe the nuclear systems in a more internal region.