Microscopic study of the $^{132,124}$Sn+$^{96}$Zr reactions: dynamic excitation energy, energy-dependent heavy-ion potential, and capture cross section
V. E. Oberacker, A. S. Umar, J. A. Maruhn, P. -G. Reinhard
We study reactions between neutron-rich $^{132}$Sn nucleus and $^{96}$Zr within a dynamic microscopic theory at energies in the vicinity of the ion-ion potential barrier peak, and we compare the properties to those of the stable system $^{124}$Sn+$^{96}$Zr. The calculations are carried out on a three-dimensional lattice using the density-constrained Time-Dependent Hartree-Fock method. In particular, we calculate the dynamic excitation energy $E^*(t)$ and the quadrupole moment of the dinuclear system, $Q_{20}(t)$, during the initial stages of the heavy-ion collision. Capture cross sections for the two reactions are analyzed in terms of dynamic effects and a comparison with recently measured data is given.
The microscopic effective reaction theory is applied to deuteron-induced reactions. A reaction model-space characterized by a $p+n+{\rm A}$ three-body model is adopted, where A is the target nucleus, and the nucleon-target potential is described by a microscopic folding model based on an effective nucleon-nucleon interaction in nuclear medium and a one-body nuclear density of A. The three-body scattering wave function in the model space is obtained with the continuum-discretized coupled-channels method (CDCC), and the eikonal reaction theory (ERT), an extension of CDCC, is applied to the calcu…
Radiative capture reactions for low energy protons have been theoretically studied for Ni and Cu isotopes using the microscopic optical model. The optical potential has been obtained in the folding model using different microscopic interactions with the nuclear densities from Relativistic Mean Field calculations. The calculated total cross sections as well as the cross sections for individually low lying levels have been compared with measurements involving stable nuclear targets. Rates for the rapid proton capture process have been evaluated for astrophysically important reactions.
Microscopic optical potentials obtained by folding the DDM3Y interaction with the densities from Relativistic Mean Field approach have been utilized to evaluate S-factors of low-energy $(p,γ)$ reactions in mass 60-80 region and to compare with experiments. The Lagrangian density FSU Gold has been employed. Astrophysical rates for important proton capture reactions have been calculated to study the behaviour of rapid proton nucleosynthesis for waiting point nuclei with mass less than A=80.
Low energy proton capture reactions have been studied in statistical model using the semi-microscopic optical potential in the mass range A=110-125. Nuclear density obtained from relativistic mean field calculation has been folded with nucleon-nucleon interaction to obtain the optical potential. Theoretical results have been compared with experimental measurements to normalize the potential. Results have also been compared with a standard calculation available in the literature.
The spectroscopic factor arises from short-range effects in the nuclear wave function. On the other hand, cross sections for external capture reactions, such as $^7$Be$(p,γ)^8$B at low energies, depend primarily on the asymptotic normalization of the bound-state wave function -- a long-range property. We investigate the relationship between potential models and the full many-body problem to illustrate how microscopic substructure effects arise naturally in the relevant transition matrix element and can be (in part) accounted for by a spectroscopic factor.
The understanding of the fusion probability is of particular importance to reveal the mechanism of producing superheavy elements. We present a microscopic study of the compound nucleus formation by combining time-dependent density functional theory, coupled-channels approach, and dynamical diffusion models. The fusion probability and compound nucleus formation cross sections for cold-fusion reactions $^{48}$Ca+$^{208}$Pb, $^{50}$Ti+$^{208}$Pb, and $^{54}$Cr+$^{208}$Pb are investigated and it is found that the deduced capture barriers, capture cross sections for these reactions are consistent w…
An isospin and spin dependent form of the equation of state for nuclear matter is presented. This form is used for the description of nucleon interaction in a new dynamic model. Preliminary calculations show that the new approach makes possible predicting the alpha structures appearing in the case of the ground state even-even nuclei.
We present a microscopic dynamical study of the reactions involving three $^{4}$He clusters. We show that the much discussed triple-$α$ linear chain configuration of $^{12}$C is formed with a certain lifetime and subsequently decays into a triangular configuration of $^{12}$C and then to a configuration near the ground state. Time-dependent Hartree-Fock (TDHF) theory coupled with a density constraint is used to study the properties of these configurations. We find a sequence of dynamical transitions analogous to the suggested astrophysical mechanism for the formation of $^{12}$C nuclei.
An ab-initio calculation of the inelastic neutral reaction of neutrino on 4He is presented, including realistic nuclear force and full final state interaction among the four nucleons. The calculation uses the powerful combintation of the Lorentz integral transform (LIT) method and the hyperspherical-harmonic effective interaction approach (EIHH). The neutrino - nucleus interaction is taken in the impulse approximation. With respect to previous calculations, the current work predicts an increased reaction cross-section by 10%-30% for neutrino temperature up to 15 MeV.
On the base of potential theory of light nuclei cluster photodisintegration the characteristics of Lithium nuclei cluster photodisintegration reactions are considered in the range of low and intermediate energies. For the inverse reaction the interference has an opposite character: in forward semisphere it is deconstructive, and in backward semisphere it is constructive. In the energy range above several MeV the E2-multipole becomes dominating. The effect of appearance of the node structure of the wave function of relative motion has been considered in the characteristics of Li cluster photodi…
Collective inertial mass coefficients with respect to translational, relative, and rotational motions are microscopically calculated, along the collective reaction path self-consistently determined, based on the adiabatic self-consistent collective coordinate (ASCC) method. The impact of the time-odd component of the mean-field potential on the inertial masses are investigated. The results are compared with those calculated with the cranking formulae. The inertial masses based on the ASCC method reproduce the exact total nuclear mass for the translational motion as well as the exact reduced ma…
The breakup cross sections in the reaction $^6$He+$^{12}$C are calculated at about 40 MeV/nucleon using the high-energy approximation (HEA) and with the help of microscopic optical potentials (OP) of interaction with the target nucleus $^{12}$C of the projectile nucleus fragments $^4$He and 2n. Considering the di-neutron $h$=2n as a single particle the relative motion $hα$ wave function is estimated so that to explain both the separation energy of $h$ in $^6$He and the rms radius of the latter. The stripping and absorbtion total cross sections are calculated and their sum is compared with the …
In this work we want to study quasielastic (p, n) exchange reactions using a semi-microscopic optical model derived in a previous work [1] based on a nuclear matter approach where the real and imaginary parts are given by the first and second order terms, respectively, of the mass operator obtained by a Brueckner-Hartree-Fock calculation using a G-matrix built up with an effective Gogny interaction. The study of these quasielastic reactions is performed within a Distorted Wave Born Approximation (DWBA) to evaluate the wave functions in the entrance and exit channels, which in turn are used to …
A microscopic model for three-cluster configurations in light nuclei is presented. It uses an expansion in terms of Faddeev components for which the dynamic eqations are derived. The model is designed to investigate binary channel processes in a compound system. Gaussian and oscillator bases are used to expand the wave function and to represent appropriate boundary conditions. We study the effect of cluster polarization on ground and resonance states of 7Be, and on the astrophysical S-factor of the reaction 6Li(p,3He)4He.
We analyse pion-nucleus reactions in a microscopic transport model of the BUU type, which propagates nucleons, pions, deltas and N(1440)-resonances explicitly in space and time. In particular we examine pion absorption and inelastic scattering cross sections for pion kinetic energies T(pi) =85-315MeV and various target masses. In general, the mass-dependence of the experimental data is well described for energies up to the delta-resonance (\approx 160 MeV) while the absorption cross sections are somewhat overestimated for the higher energies. In addition we study the possible dynamical effects…
We derive in-medium PROTON-PROTON cross sections in a microscopic model based upon the Bonn nucleon-nucleon potential and the Dirac-Brueckner approach for nuclear matter. We demonstrate the difference between proton-proton and neutron-proton cross sections and point out the need to distinguish carefully between the two cases. We also find substantial differences between our in-medium cross sections and phenomenological parametrizations that are commonly used in heavy-ion reactions.
The existing and some preliminary experimental data on the total cross sections of the $^{4,6}$He, $^{6,7}$Li +$^{28}$Si reactions at energies E=5-50 A MeV are demonstrated. The data on $^{6}$Li,$^{6}$He+$^{28}$Si are analyzed in the framework of the microscopic optical potential with real and imaginary parts obtained with a help of the double-folding procedure and by using the current models of densities of the projectile nuclei. Besides, the microscopic double-folding Coulomb potential is calculated and its effect on cross sections is compared with that when one applies the traditional Coulo…
We present updated predictions of effective elastic nucleon-nucleon cross sections intended for use in nucleus-nucleus reactions. A novel characteristic of the present approach combines all microscopic medium effects included in the Dirac-Brueckner-Hartree-Fock G-matrix with a Pauli blocking mechanism which is more appropriate for applications in ion-ion reaction models as compared to a previous approach. The effective in-medium cross section is found to be quite sensitive to the description of Pauli blocking in the final configurations.
We investigate $^6$Li($n$, $n'$)$^6$Li$^*$ $\to$ $d$ + $α$ reactions by using the continuum-discretized coupled-channels method with the complex Jeukenne-Lejeune-Mahaux effective nucleon-nucleon interaction. In this study, the $^6$Li nucleus is described as a $d$ + $α$ cluster model. The calculated elastic cross sections for incident energies between 7.47 and 24.0 MeV are good agreement with experimental data. Furthermore, we show the neutron spectra to $^6$Li breakup states measured at selected angular points and incident energies can be also reproduced systematically.
We revisit the commonly accepted notion that the difference between interaction and reaction cross sections is negligible at relativistic energies, and show that, especially in small mass number region, it is large enough to help probe nuclear structure. For analyses of the difference, we construct "pseudo data" for the reaction cross sections using a phenomenological black-sphere model of nuclei since empirical data are very limited at high energies. The comparison with the empirical interaction cross sections suggests a significant difference between the reaction and interaction cross sectio…
The low-lying states of the $^{9}$Li nucleus are investigated with a unified framework of microscopic structure and reaction models. In the structure model, the wave function is fully antisymmetrized and the $^{9}$Li nucleus is described as an $α$ + $t$ + $n$ + $n$ four-body system, and low-lying 1/2$^{-}$, 3/2$^{-}$, 5/2$^{-}$, and 7/2$^{-}$ states are obtained by the stochastic multi-configuration mixing method. Using these wave functions, the quasi-elastic cross section at $E/A$ = 60 MeV and the elastic and inelastic cross sections at $E/A$ = 50 MeV on the $^{12}$C target are calculated in …
Using hadronic Lagrangians that include the interaction of pentaquark $Θ^+$ baryon with $K$ and $N$, we evaluate the cross sections for its production from meson-proton, proton-proton, and photon-proton reactions near threshold. With empirical coupling constants and form factors, the predicted cross sections are about 1.5 mb in kaon-proton reactions, 0.1 mb in rho-nucleon reactions, 0.05 mb in pion-nucleon reactions, 20 $μ$b in proton-proton reactions, and 40 nb in photon-proton reactions.
Within the framework of the improved quantum molecular dynamics model, the medium modifications on the free nucleon-nucleon elastic cross sections are investigated. By using various in-medium nucleon-nucleon elastic cross sections in the model, the nucleon-induced reactions on various targets are simulated, and the excitation functions of reaction cross sections in the energy range from 25 MeV to 1 GeV are calculated. By comparing the calculations with the experimental data, an isospin, density, and momentum-dependence medium correction factor on free nucleon-nucleon elastic cross sections is …
The relationship between the backward elastic scattering probabilities and reaction cross sections is derived. This is a a very simple and useful method to extract reaction cross sections for heavy ion systems. We compare the results of our method with those using the traditional full elastic scattering angular distributions, for several systems, at energies near and above the Coulomb barrier. From the calculated reaction and capture cross sections using the present method, we derive the cross sections of other mechanisms for nearly spherical systems.
The neutron capture cross sections and average radiative widths of neutron resonances for two double-magic nuclei 132Sn and 208Pb have been calculated using the microscopic photon strength functions, which were obtained within the microscopic self-consistent version of the extended theory of finite Fermi systems in the time blocking approximation. For the first time, the microscopic PSFs have been obtained within the fully self-consistent approach with exact accounting for the single particle continuum (for 208Pb). The approach includes phonon coupling effects in addition to the standard RPA a…
In the search for superheavy elements quasifission reactions represent one of the reaction pathways that curtail the formation of an evaporation residue. In addition to its importance in these searches quasifission is also an interesting dynamic process that could assist our understanding of many-body dynamical shell effects and energy dissipation thus forming a gateway between deep-inelastic reactions and fission. This manuscript gives a summary of recent progress in microscopic calculations of quasifission employing time-dependent Hartree-Fock (TDHF) theory and its extensions.
The cross-sections of proton-induced reactions on 184W at incident proton energy below 200MeV are calculated and analyzed including reaction cross-sections, elastic scattering angular distributions, energy spectra and double differential cross section. Nuclear theoretical models which integrate the optical model, distorted born wave approximation theory, the intra-nuclear cascade model, the exciton model, the Hauser-Feshbach theory and the evaporation model are used in the reactions. Theoretical results are compared with the existent experimental data.
Total reaction cross sections are predicted for proton scattering from various nuclei. A simple functional form has been used that reproduces the total reaction cross sections for the scattering of protons from (15) nuclei spanning the mass range from ${}^9$Be to ${}^{238}$U and for proton energies 10 to 300 MeV.
Spectroscopic amplitudes play an important role in nuclear capture reactions. These amplitudes are shown to include both single-particle and polarization effects: the former through their spatial dependence and the latter through their normalization (the spectroscopic factors). Coupled-channels equations are developed for the spectroscopic amplitudes. These equations serve as a convenient starting point for the derivation of several approximations: Hartree, Hartree-Fock and two different single-particle models. The single-particle models include antisymmetry in different ways, but both miss ma…
The $α$ inelastic scattering on $^{16}$O is investigated with the coupled-channel calculation using the $α$-nucleus coupled-channel potentials, which are microscopically derived by folding the the Melbourne $g$-matrix $NN$ interaction with the $^{16}$O and $α$ densities. The matter and transition densities of $^{16}$O are calculated by a microscopic structure model of the variation after the spin-parity projections combined with the generator coordinate method of $^{12}$C+$α$ in the framework of the antisymmetrized molecular dynamics. The calculation reproduces the observed elastic and inelast…
We study the scaling properties of proton-nucleus total reaction cross sections for stable nuclei and propose an approximate expression in proportion to Z^{2/3}σ_{pp}^{total} + N^{2/3} σ_{pn}^{total}. Based on this expression, we can derive a relation that enables us to predict a total reaction cross section for any stable nucleus within 10% uncertainty at most, using the empirical value of the total reaction cross section of a given nucleus.
The radiative thermal neutron capture cross sections over the range of thermal energies from 1 keV to 1 MeV are studied in statistical Hauser-Feshbach formalism. The optical model potential is constructed by folding the density dependent M3Y nucleon-nucleon interaction with radial matter densities of target nuclei obtained from relativistic-mean-field (RMF) theory. The standard nuclear reaction code TALYS1.8 is used for calculation of cross sections. The nuclei studied in the present work reside near the $Z=28$ proton shell closure and are of astrophysical interests taking part in p-, s-, and …
We present a microscopic formalism that extends the traditional formulation of Williams, Ericson and Bloch and permits to obtain the transition strengths (TS) of pre-equilibrium nuclear reactions directly from their quantum microscopic description. We calculate the TS without resorting to the Laplace transform approach and the use of the saddle point approximation. We also analyze some problems that may appear in connection with these mathematical tools and the Darwin-Fowler approach in this case. We show that, analogously to the nuclear densities, the strengths for transitions that change the…
Fusion reactions with a weakly bound projectile are studied using the double-folding model along with a repulsive interaction modifying term. Using this modified potential, including nuclear matter incompressibility effects, the fusion reaction cross sections and suppression parameters are calculated for 9Be +209Bi,208Pb,29Si and 27Al reactions. The results show that applying these effects at energies near the Coulomb barrier improves the agreement between the calculated and experimental cross sections, and modifies the mean values of the suppression parameter.
A simple functional form has been found that gives a good representation of the total reaction cross sections for the scattering from ${}^{208}$Pb of protons with energies in the range 30 to 300 MeV.
Microscopic optical model potential results for reaction cross sections of proton elastic scattering are presented. The applications cover the 10-1000 MeV energy range and consider both stable and unstable nuclei. The study is based on in-medium g-matrix full-folding optical model approach with the appropriate relativistic kinematic corrections needed for the higher energy applications. The effective interactions are based on realistic NN potentials supplemented with a separable non-Hermitian term to allow optimum agreement with current NN phase-shift analyzes, particularly the inelasticities …
The main progresses in the multinucleon transfer reactions at energies close to the Coulomb barrier are reviewed. After a short presentation of the experimental progress and theoretical progress,the predicted production cross sections for unknown neutron-rich heavy nuclei and the trans-uranium nuclei are presented.
This paper presents reaction cross section predictions. These predictions are the result of a continuous pipeline which originates from a microscopic nuclear interaction. Density parameters and effective nucleon-nucleon cross sections (both involved in the reaction calculations) are by-products of the same equation of state. First, we perform tests of sensitivity to medium effects using reactions involving 208-Pb, a stable but weakly isospin-asymmetric nucleus. We also show predictions for collisions of some neutron-rich isotopes of Calcium and Argon. We observe significant sensitivity of the …
We calculate the differences between reaction and interaction cross sections in the collisions of relativistic light ions with $\text{A}<40$ in the framework of Glauber theory. Although, in the optical approximation of Glauber theory these differences are approximately 1% of the reaction cross sections or less, they increase up to 3-4% when all scattering diagrams of Glauber theory are included in calculation.
Reaction cross sections of carbon isotopes for proton scattering are calculated in large energy region. Density distributions of carbon isotopes are obtained from relativistic mean field results. Calculations are based on two procedures: the Glauber theory and relativistic impulse approximation, and results are compared with each other as well as with experimental data. A strong relationship between reaction cross section and root-mean-square radius is clearly shown for 12C using a model distribution.
A microscopic calculation of cross sections for fusion of oxygen isotopes $^{16}$O, $^{22}$O and $^{24}$O is presented. Fermionic Molecular Dynamics wave functions are used to describe the fully antisymmetrized and angular momentum projected nucleus-nucleus system. The same effective nucleon-nucleon interaction is used to determine the ground state properties of the nuclei as well as the nucleus-nucleus interaction. From the microscopic many-body wave function the corresponding wave function for the relative motion of two point-like nuclei is derived by a method proposed by Friedrich which lea…
A simple functional form has been found that gives a good representation of the total reaction cross sections for the scattering of protons from (15) nuclei spanning the mass range ${}^{9}$Be to ${}^{238}$U and for proton energies ranging from 20 to 300 MeV.
The heavy-ion fusion reactions induced by neutron-rich nuclei are investigated with the improved quantum molecular dynamics (ImQMD) model. With a subtle consideration of the neutron skin thickness of nuclei and the symmetry potential, the stability of nuclei and the fusion excitation functions of heavy-ion fusion reactions $^{16}$O+$^{76}$Ge, $^{16}$O+$^{154}$Sm, $^{40}$Ca+$^{96}$Zr and $^{132}$Sn+$^{40}$Ca are systematically studied. The fusion cross sections of these reactions at energies around the Coulomb barrier can be well reproduced by using the ImQMD model. The corresponding slope para…
Meson-meson nonresonant reactions governed by the quark-interchange mechanism are studied in a potential that is derived from QCD. S-wave elastic phase shifts for I=2 ππand I=3/2 K πscattering are obtained with wave functions determined by the central spin-independent term of the potential. The reactions include inelastic scatterings of two mesons in the ground-state pseudoscalar octet and the ground-state vector nonet. Cross sections for reactions involving pion, rho, K and K^* indicate that mesonic interactions in matter consisting of only K and K^* can be stronger than mesonic interactions …
We systematically analyze nuclear reaction data that are sensitive to nuclear size, namely, proton-nucleus total reaction cross sections and differential elastic cross sections, using a phenomenological black-sphere approximation of nuclei that we are developing. In this framework, the radius of the black sphere is found to be a useful length scale that simultaneously accounts for the observed proton-nucleus total reaction cross section and first diffraction peak in the proton elastic differential cross section. This framework is expected to be applicable to any kind of projectile that is stro…
$α$ elastic and inelastic scattering on $^{12}$C is investigated with the coupled-channel calculation using microscopic $α$-$^{12}$C potentials, which are derived by folding the Melbourne $g$-matrix $NN$ interaction with the matter and transition densities of $^{12}$C. These densities are obtained by a microscopic structure model of the antisymmetrized molecular dynamics combined with and without the $3α$ generator coordinate method. The calculation reproduces satisfactorily well the observed elastic and inelastic cross sections at incident energies of $E_α=130$~MeV, 172.5~MeV, 240~MeV, and 38…
The current status of transport theoretical models applicable to the physics of the Relativistic Heavy-Ion Collider is reviewed. The time evolution of microscopic reaction dynamics - from early, hard, partonic rescattering up to soft hadronic interactions close to freeze-out is analyzed and key observables linked to the different reaction stages are discussed.
Calculations are made of the $K^{+}+^{12}$C, $^{40}$Ca differential elastic scattering cross sections at the beam momenta 0.635, 0.715, and 0.8 GeV/c. To this end the microscopic optical potential derived in the high-energy approximation was used where existing data on the kaon-nucleon amplitude and on the point-like density distributions of target-nuclei were utilized. The effect of different methods of relativization was studied and shown to play an important role. A good agreement with the experimental data on differential elastic cross sections was obtained.
A new framework for $A(d,p)B$ reactions is introduced by merging the microscopic approach to computing the properties of the nucleon-target systems and the three-body $n+p+A$ reaction formalism, thus providing a consistent link between the reaction cross sections and the underlying microscopic structure. In this first step toward a full microscopic description, we focus on the inclusion of the neutron-target microscopic properties. The properties of the neutron-target subsystem are encapsulated in the Green's function which is computed with the Coupled Cluster theory using a chiral nucleon-nuc…
Based on the statistical theory of light nucleus reactions (STLN), the description of the complicated emission processes of proton and light composite charged particles are further improved through considering the effects of Coulomb barriers both in incident and different outgoing reaction channels. And the analysis of the reaction channels including the sequential and simultaneous emission processes for $p + ^7$Li reaction is performed in detail. So the partial spectra of all of outgoing particles are also obtained for different reaction processes. The calculated double differential cross sec…
Isotope-dependence of measured reaction cross sections in scattering of $^{28-32}$Ne isotopes from $^{12}$C target at 240 MeV/nucleon is analyzed by the double-folding model with the Melbourne $g$-matrix. The density of projectile is calculated by the mean-field model with the deformed Wood-Saxon potential. The deformation is evaluated by the antisymmetrized molecular dynamics. The deformation of projectile enhances calculated reaction cross sections to the measured values.
Using the Glauber theory, we calculate reaction cross sections for the deformed halo nucleus $^{31}$Ne. To this end, we assume that the $^{31}$Ne nucleus takes the $^{30}$Ne + $n$ structure. In order to take into account the rotational excitation of the core nucleus $^{30}$Ne, we employ the particle-rotor model (PRM). We compare the results to those in the adiabatic limit of PRM, that is, the Nilsson model, and show that the Nilsson model works reasonably well for the reaction cross sections of $^{31}$Ne. We also investigate the dependence of the reaction cross sections on the ground state pro…
We studied the effect of cross section on mass asymmetric systems. The mass asymmetry of a reaction can be defined by the asymmetry parameter η = |(AT - AP)/(AT + AP)|; [1] where AT and AP are the masses of the target and projectile, respectively. The η = 0 corresponds to the symmetric reactions, whereas non-zero values of η define different asymmetries of a reaction. Reactions include 208Pb82 + 208Pb82 having η = 0, 40Ca20 + 208Pb82 having η = 0.6 and 12C6 + 197Au79 having η = 0.8 This work has been carried out within the frame work of isospin dependent quantum molecular dynamics (IQMD) model…
$^{50}$Ti-induced fusion reactions to synthesize superheavy elements are studied systematically with the two-step model developed recently, where fusion process is divided into approaching phase and formation phase. Furthermore, the residue cross sections for different neutron evaporation channels are evaluated with the statistical evaporation model. In general, the calculated cross sections are much smaller than that of $^{48}$Ca-induced fusion reactions, but the results are within the detection capability of experimental facilities nowadays. The maximum calculated residue cross section for p…
