Rearrangement Term Research Articles

Comprehensive examination of the elastic scattering angular distributions of 10C+4He, 27Al, 58Ni and 208Pb using various potentials.

The recently measured elastic scattering angular distributions for 10C + 4He, 10C + 27Al, 10C + 58Ni and 10C + 208Pb nuclear systems are investigated in the current study using various potentials based on phenomenological, semi microscopic as well as microscopic approaches. The implemented potentials are: optical potential, double folding potentials based on both Sao Paulo and CDM3Y6 interactions with and without taking into account the effect of the rearrangement term, as well as the cluster folding potential. The cluster nature of 10C as a core of 9B with a valence proton orbiting this core is applied to generate the cluster folding potentials for the different considered systems. The concerned experimental data is fairly reproduced with all the aforementioned potentials.

Elastic scattering of 7Li+58Ni: a phenomenological and microscopic analysis

Motivating by examining the break-up effect of 7Li projectile into t + α cluster structure in the field of 58Ni nucleus, the available experimental angular distributions for 7Li + 58Ni elastic scattering system at energies ranging from 13 and up to 42 MeV are studied utilizing different phenomenological as well as microscopic potentials. Data analysis utilizing the Sao Paulo potential revealed that in order to reproduce the data, the strength of the real folded potential had to be reduced by ~ 36 %. While, data analysis utilizing the double folding CDM3Y6 potential with and without the rearrangement term revealed that the potential strength needed to be reduced by ~ 63 and 62 %, respectively. Cluster folding model based on the t + α cluster structure for 7Li is applied to reproduce the considered data. Similar results were obtained showing the necessity to reduce real cluster folding potential strength by about 49 %. The reported reduction in potential strength from the different implemented potentials supports the strong 7Li break-up impact. Finally, the full microscopic continuum discretized coupled channels approach is applied with a great success in reproducing the considered data. 

Analysis of 4,6,8He+208Pb elastic scattering at E=22 MeV using various potentials

Using phenomenological and microscopic potentials, the experimental angular distributions for the 4,6,8He nuclei elastically scattered from a 208Pb target at are investigated. Both the modified version of CDM3Y6 interaction based on the inclusion of the rearrangement term (RT) and those obtained from the Sao Paulo Potentials are used for the microscopic potentials. The cluster folding potential for 6He+208Pb is calculated using the triton + triton cluster structure for 6He. This analysis revealed that the real cluster folding potential strength must be reduced by about 90%. Using the extracted potentials, the total reaction cross sections were successfully reproduced.

Analysis of d + 6,7,9,11Li elastic scattering using different densities and calculation procedures

The elastic scattering differential cross-sections of d + 6,7,9,11Li systems have been reanalyzed using three different calculation methods and four different density distributions for the deuteron nucleus. The phenomenological optical model and the microscopic optical model incorporating the CDM3Y6 interaction with and without the inclusion of the rearrangement term have been used as the calculation models. A good agreement between the theoretical results and the experimental data has been achieved. The performed analyses emphasize the weak binding nature of the 11Li isotope.

Further investigation of 10,11B + 58Ni elastic scattering

The recently measured 10,11B + 58Ni elastic scattering angular distributions are subjected to thorough analysis using different potentials such as optical and double folding, based on the CDM3Y6, with and without taking into account the effect of the rearrangement term, as well as the cluster folding potential. The cluster structure for 10B (11B) as composed of 6Li (7Li) core with an alpha particle orbiting this core is hence studied. The experimental data for the 10,11B + 58Ni nuclear systems are fairly reproduced with all the aforementioned potentials. The renormalization factors for the real double folded potential (N r ) indicate the need to reduce the strength of the produced potentials by ∼ 50% and ∼ 36% to reproduce the considered data for the 10B + 58Ni and the 11B + 58Ni nuclear systems, respectively. As a result of applying a rearrangement term to the double folding potential, a small change can be observed in the N r values. Cluster folding analysis for the 10B + 58Ni elastic scattering cross sections reveals a similar trend; the strength of the renormalization factor for the real part of the potential constructed based on the cluster folding model ( N RCF ) needs to be reduced by ∼ 48%. The observed reduction in the strength of the constructed double and cluster folding potentials reflects the weak binding nature of the 10B and 11B nuclei. It was somewhat surprising that the extracted N RCF = 1.24 ± 0.064 for the 11B + 58Ni system.

Inner crust of a neutron star at the point of crystallization in a multicomponent approach (Corrigendum)

The possible presence of amorphous and heterogeneous phases in the inner crust of a neutron star is expected to reduce the electrical conductivity of the crust, with potentially important consequences on the magneto-thermal evolution of the star. In cooling simulations, the disorder is quantified by an impurity parameter which is often taken as a free parameter. We aim to give a quantitative prediction of the impurity parameter as a function of the density in the crust,performing microscopic calculations including up-to-date microphysics of the crust. A multi-component approach is developed at finite temperature using a compressible liquid drop description of the ions with an improved energy functional based on recent microscopic nuclear models and optimized on extended Thomas-Fermi calculations. Thermodynamic consistency is ensured by adding a rearrangement term and deviations from the linear mixing rule are included in the liquid phase. The impurity parameter is consistently calculated at the crystallization temperature as determined in the one-component plasma approximation for the different functionals. Our calculations show that at the crystallization temperature the composition of the inner crust is dominated by nuclei with charge number around $Z \approx 40$, while the range of the $Z$ distribution varies from about 20 near the neutron drip to about 40 closer to the crust-core transition. This reflects on the behavior of the impurity parameter that monotonically increases with density up to around 40 in the deeper regions of the inner crust. Our study shows that the contribution of impurities is non-negligible, thus potentially having an impact on the transport properties in the neutron-star crust. The obtained values of the impurity parameter represent a lower limit; larger values are expected in the presence of non-spherical geometries and/or fast cooling dynamics.

Analysis of strong refractive effect within 11Li projectile structure

In the context of the double folding optical model, the strong refractive effect for elastic scattering of 11Li + 12C and 11Li + 28Si systems at incident energies of 29, 50, and 60 MeV/n is studied. Real folded potentials are generated based on a variety of nucleon-nucleon interactions with the suggested density distributions for the halo structure of 11Li nuclei. The rearrangement term (RT) of the extended realistic density dependent CDM3Y6 effective interaction is considered. The imaginary potential was taken in the traditional standard Woods-Saxon form. Satisfactory results for the calculated potentials are obtained, with a slight effect of the RT in CDM3Y6 potential. Successful reproduction with a normalization factor close to one for the observed angular distributions of the elastic scattering differential cross section has been achieved using the derived potentials. The obtained reaction cross-section is studied as a guide by extrapolating our calculations and previous results.

Inner crust of a neutron star at the point of crystallization in a multicomponent approach

Context. The possible presence of amorphous and heterogeneous phases in the inner crust of a neutron star is expected to reduce the electrical conductivity of the crust, potentially with significant consequences on the magneto-thermal evolution of the star. In cooling simulations, the disorder is quantified by an impurity parameter, which is often taken as a free parameter. Aims. We aim to give a quantitative prediction of the impurity parameter as a function of the density in the crust, performing microscopic calculations including up-to-date microphysics of the crust. Methods. A multicomponent approach was developed at a finite temperature using a compressible liquid-drop description of the ions with an improved energy functional based on recent microscopic nuclear models and optimized on extended Thomas-Fermi calculations. Thermodynamic consistency was ensured by adding a rearrangement term, and deviations from the linear mixing rule were included in the liquid phase. Results. The impurity parameter is consistently calculated at the crystallization temperature as determined in the one-component plasma approximation for the different functionals. Our calculations show that at the crystallization temperature, the composition of the inner crust is dominated by nuclei with charge number around Z ≈ 40, while the range of the Z distribution varies from about 20 near the neutron drip to about 40 closer to the crust-core transition. This reflects on the behavior of the impurity parameter that monotonically increases with density reaching up to around 40 in the deeper regions of the inner crust. Conclusions. Our study shows that the contribution of impurities is non-negligible, thus potentially having an impact on the transport properties in the neutron-star crust. The obtained values of the impurity parameter represent a lower limit; larger values are expected in the presence of nonspherical geometries and/or fast cooling dynamics.

Rearrangement term in the folding model of the nucleon optical potential

Based on the mean-field determination of the single-particle energy in nuclear matter that contains naturally a rearrangement term (RT) implied by the Hugenholtz–van Hove theorem, the folding model of the nucleon optical potential (OP) is extended to take into account the RT using the effective, density dependent CDM3Yn interaction. With the exchange part of the nucleon folded OP treated exactly in the Hartree–Fock manner, a compact nonlocal version of the folding model is suggested in the present work to determine explicitly the isospin-dependent, nonlocal central term of the nucleon OP. To solve the optical model (OM) equation with a complex nonlocal OP, the calculable R-matrix method is used to analyse the elastic neutron and proton scattering on 40,48Ca, 90Zr, and 208Pb targets at low energies. The inclusion of the RT into the folding model calculation of the nonlocal nucleon OP was shown to be essential for the overall good OM description of elastic nucleon scattering. To validate the nonlocal version of the folding model, the OM results given by the nonlocal folded nucleon OP are also compared with those given by the global parametrization of the nonlocal OP using the analytical nonlocal form factor suggested by Perey and Buck.

Consistent mean-field description of theC12 + C12optical potential at low energies and the astrophysicalSfactor

The nuclear mean-field potential built up by the $^{12}\mathrm{C}+^{12}\mathrm{C}$ interaction at energies relevant for the carbon burning process is calculated in the double-folding model (DFM) using the realistic ground-state density of $^{12}\mathrm{C}$ and the CDM3Y3 density dependent nucleon-nucleon (NN) interaction, with the rearrangement term properly included. To validate the use of a density dependent NN interaction in the DFM calculation in the low-energy regime, an adiabatic approximation is suggested for the nucleus-nucleus overlap density. The reliability of the nuclear mean-field potential predicted by this low-energy version of the DFM is tested in a detailed optical model analysis of the elastic $^{12}\mathrm{C}+^{12}\mathrm{C}$ scattering data at energies below 10 MeV/nucleon. The folded mean-field potential is then used to study the astrophysical $S$ factor of $^{12}\mathrm{C}+^{12}\mathrm{C}$ fusion in the barrier penetration model. Without any adjustment of the potential strength, our results reproduce very well the nonresonant behavior of the $S$ factor of $^{12}\mathrm{C}+^{12}\mathrm{C}$ fusion over a wide range of energies.

Mean-field description of heavy-ion scattering at low energies and fusion

The nuclear mean-field potential built up during the $${}^{12}\hbox {C}+{}^{12}\hbox {C}$$ and $${}^{16}\hbox {O}+{}^{16}\hbox {O}$$ collisions at low energies relevant for the carbon- and oxygen-burning processes is constructed within the double-folding model (DFM) using the realistic ground-state densities of $$^{12}\hbox {C}$$ and $$^{16}$$ O, and CDM3Yn density-dependent nucleon–nucleon (NN) interaction. The rearrangement term, indicated by the Hugenholtz–van Hove theorem for the single-particle energy in nuclear matter, is properly considered in the DFM calculation. To validate the use of the density-dependent NN interaction at low energies, an adiabatic approximation was suggested for the dinuclear overlap density. The reliability of the nucleus–nucleus potential predicted through this low-energy version of the DFM was tested in the optical model (OM) analysis of the elastic $${}^{12}\hbox {C}+{}^{12}\hbox {C}$$ and $${}^{16}\hbox {O}+{}^{16}\hbox {O}$$ scattering data at energies below 10 MeV/nucleon. These OM results provide a consistently good description of the elastic angular distributions and 90 $$^\circ$$ excitation function. The dinuclear mean-field potential predicted by the DFM is further used to determine the astrophysical S factor of the $${}^{12}\hbox {C}+{}^{12}\hbox {C}$$ and $${}^{16}\hbox {O}+{}^{16}\hbox {O}$$ fusions in the barrier penetration model. Without any adjustment of the potential strength, our results reproduce the non-resonant behavior of the S factor of the $${}^{12}\hbox {C}+{}^{12}\hbox {C}$$ and $${}^{16}\hbox {O}+{}^{16}\hbox {O}$$ fusions very well over a wide range of energies.

Nuclear mean field and double-folding model of the nucleus-nucleus optical potential

Realistic density dependent CDM3Yn versions of the M3Y interaction have been used in an extended Hartree-Fock (HF) calculation of nuclear matter (NM), with the nucleon single-particle potential determined from the total NM energy based on the Hugenholtz-van Hove theorem that gives rise naturally to a rearrangement term (RT). Using the RT of the single-nucleon potential obtained exactly at different NM densities, the density- and energy dependence of the CDM3Yn interactions was modified to account properly for both the RT and observed energy dependence of the nucleon optical potential. Based on a local density approximation, the double-folding model of the nucleus-nucleus optical potential has been extended to take into account consistently the rearrangement effect and energy dependence of the nuclear mean-field potential, using the modified CDM3Yn interactions. The extended double-folding model was applied to study the elastic $^{12}$C+$^{12}$C and $^{16}$O+$^{12}$C scattering at the refractive energies, where the Airy structure of the nuclear rainbow has been well established. The RT was found to affect significantly the real nucleus-nucleus optical potential at small internuclear distances, giving the potential strength close to that implied by the realistic optical model description of the Airy oscillation.

Extended Hartree-Fock study of the single-particle potential: The nuclear symmetry energy, nucleon effective mass, and folding model of the nucleon optical potential

The nucleon mean-field potential has been thoroughly investigated in an extended Hartree-Fock (HF) calculation of nuclear matter (NM) using the CDM3Y3 and CDM3Y6 density dependent versions of the M3Y interaction. The single-particle (SP) energies of nucleons in NM are determined according to the Hugenholtz--Van Hove theorem, which gives rise naturally to a rearrangement term (RT) of the SP potential at the Fermi momentum. Using the RT obtained exactly at the different NM densities and neutron-proton asymmetries, a consistent method is suggested to take into account effectively the momentum dependence of the RT of the SP potential within the standard HF scheme. To obtain a realistic momentum dependence of the nucleon optical potential (OP), the high-momentum part of the SP potential was accurately readjusted to reproduce the observed energy dependence of the nucleon OP over a wide range of energies. The impact of the RT and momentum dependence of the SP potential on the density dependence of the nuclear symmetry energy and nucleon effective mass has been studied in detail. The high-momentum tail of the SP potential was found to have a sizable effect on the slope of the symmetry energy and the neutron-proton effective mass splitting at supranuclear densities of the NM. Based on a local density approximation, the folding model of the nucleon OP of finite nuclei has been extended to take into account consistently the RT and momentum dependence of the nucleon OP in the same mean-field manner, and successfully applied to study the elastic neutron scattering on the lead target at the energies around the Fermi energy.

Rearrangement term with relativistic density-dependent hyperon potentials

An appropriate density dependence of hyperon potentials is important for the stiffening of the equation of state and massive neutron stars. To persist in covariance and thermodynamic consistency, the rearrangement term is indispensable. In this work, we derive the rearrangement term for hyperon potentials with arbitrary density-dependence. The importance of the rearrangement term is also exhibited in numerical instances.

Isospin and temperature dependence of the rearrangement term in Brueckner theory

In the framework of Brueckner theory, the mass operator is evaluated up to second order in the hole-line expansion and the single-particle potentials of neutrons and protons are determined in asymmetric nuclear matter for different temperatures and densities. We find an almost linear dependence of the single-particle potentials on the nuclear asymmetry and important effects of finite temperature, particularly at low asymmetries. It is therefore important to take into account the effect of the ground-state correlations when applying the single-particle potentials to intermediate- and high-energy heavy-ion collisions.

Single-Particle Properties of Isospin Asymmetric Nuclear Matter

The single-particle potentials in isospin asymmetric nuclear matter is investigated in the framework of the Brueckner theory by adopting the realistic Bonn B two-body interaction in combination with a consistent microscopic three-body force. The rearrangement contribution induced by the ground state correlations to single-nucleon potentials is calculated in terms of the hole-line expansion of the mass operator. With the modification of this rearrangement term the symmetry potential is discussed as a function of momentum for several isospin asymmetries.

Compaction of Titanium Powders

Accurate modeling of powder densification has been an area of active research for more than 60 years. The earliest efforts were focused on linearization of the data because computers were not readily available to assist with curve-fitting methods. In this work, eight different titanium powders (three different sizes of sponge fines <150 μm, <75 μm, and < 45 μm; two different sizes of a hydride-dehydride [HDH] <75 μm and < 45 μm; an atomized powder; a commercially pure [CP] Ti powder from International Titanium Powder [ITP]; and a Ti 6 4 alloy powder) were cold pressed in a single-acting die instrumented to collect stress and deformation data during compaction. From these data, the density of each compact was calculated and then plotted as a function of pressure. The results show that densification of all the powders, regardless of particle size, shape, or chemistry, can be modeled accurately as the sum of an initial density plus the sum of a rearrangement term and a work-hardening term. These last two terms are found to be a function of applied pressure and take the form of an exponential rise.

Effect of repulsive and attractive three-body forces on nucleus-nucleus elastic scattering

The effect of the three-body force (TBF) is studied in nucleus-nucleus elastic scattering on the basis of Brueckner theory for nucleon-nucleon $(\mathit{NN})$ effective interaction (complex $G$ matrix) in the nuclear matter. A new $G$ matrix called CEG07 proposed recently by the present authors includes the TBF effect and reproduces a realistic saturation curve in the nuclear matter, and it is shown to well reproduce proton-nucleus elastic scattering. The microscopic optical potential for the nucleus-nucleus system is obtained by folding the $G$ matrix with nucleon density distributions in colliding nuclei. We first analyze in detail the $^{16}\mathrm{O}+^{16}\mathrm{O}$ elastic scattering at $E/A=70$ MeV. The observed cross sections are nicely reproduced up to the most backward scattering angles only when the TBF effect is included. The use of the frozen-density approximation (FDA) is essentially important to properly estimate the effect of the TBF in nucleus-nucleus scattering. Other prescriptions for defining the local density have also been tested, but only the FDA prescription gives a proper description of the experimental cross sections as well as the effect of the TBF. The effects of the three-body attraction and the $\ensuremath{\omega}$-rearrangement term are also analyzed. The CEG07 interaction is compared with CDM3Y6, which is a reliable and successful effective density-dependent $\mathit{NN}$ interaction used in the double-folding model. The CEG07 $G$ matrix is also tested in the elastic scattering of $^{16}\mathrm{O}$ by the $^{12}\mathrm{C}$, $^{28}\mathrm{Si}$, and $^{40}\mathrm{Ca}$ targets at $E/A=93.9$ MeV, and in the elastic scattering of $^{12}\mathrm{C}$ by the $^{12}\mathrm{C}$ target at $E/A=135$ MeV with great success. The decisive effect of the TBF is clearly seen also in those systems. Finally, we have tested CEG07a, CEG07b, and CEG07c for the $^{16}\mathrm{O}+^{16}\mathrm{O}$ system at various energies.

Effects of medium-inducedρ-ωmeson mixing on the equation of state in isospin-asymmetric nuclear matter

We reexamine effects of the $\rho-\omega$ meson mixing mediated by nucleon polarizations on the symmetry energy in isospin-asymmetric nuclear matter. Taking into account the rearrangement term neglected in previous studies by others, we evaluate the $\rho-\omega$ mixing angle in a novel way within the Relativistic Mean-Field Models with and without chiral limits. It is found that the symmetry energy is significantly softened at high densities contrary to the finding in earlier studies. As the first step of going beyond the lowest-order calculations, we also solve the RPA equation for the $\rho-\omega$ mixing. In this case, it is found that the symmetry energy is not only significantly softened by the $\rho-\omega$ mixing at supra-saturation densities, similar to the lowest-order $\rho-\omega$ mixing, but interestingly also softened at subsaturation densities. In addition, the softening of the symmetry energy at subsaturation densities can be partly suppressed by the nonlinear self-interaction of the $\sigma$ meson.

Density dependent parametrization models: Formalism and applications

In this work we derive a formalism to incorporate asymmetry and temperature effects in the Brown-Rho (BR) scaled Lagrangian model in a mean field theory. The Lagrangian density discussed in this work requires less parameters than the usual models with density dependent couplings. We also present the formalism with the inclusion of the eight lightest baryons, two lightest leptons, \ensuremath{\beta} equilibrium and charge neutrality in order to apply the BR scaled model to the study of neutron stars. The results are again compared with the ones obtained from another density dependent parametrization model. The role played by the rearrangement term at $T=0$ for nuclear or neutron star matter and at finite temperature is investigated. The BR scaled model is shown to be a good tool in studies involving density dependent effective masses and in astrophysics applications.