Quasiparticle Random Phase Approximation Research Articles

Effects of Residual Tensor and Pairing Forces on the Gamow-Teller states in Magic Nuclei, 48Ca, 90Zr, 132Sn, and 208Pb

Abstract We investigate the effects of residual tensor force (TF) and pairing force on the Gamow-Teller (GT) transitions in four magic nuclei, 48Ca, 90Zr, 132Sn and 208Pb. The TF is taken into account by using the Brückner G-matrix theory with the charge-dependent (CD) Bonn potential as the residual interaction of charge-exchange quasiparticle random phase approximation (QRPA). We found that particle-particle (p − p) tensor interaction does not affect the GT transitions because of the closed shell nature in the nuclei, but repulsive particle-hole (p − h) residual interaction for the p − h configuration of spin-orbit partners dominates the high-lying giant GT states for all of the nuclei. It is also shown that appreciable GT strengths are shifted to lower energy region by the attractive p − h TF for the same jπ = jν configuration, and produce the low-lying GT peak about 2.5 MeV in 48Ca. Simultaneously, in 90Zr and 132Sn, the low-energy GT strength appears as a lower energy shoulder near the main GT peak. On the other hand, the shift of the low-lying GT state is not seen clearly for 208Pb because of the strong spin-orbit splitting of high j orbits, which dominates the GT strength.

Nuclear structure properties and weak interaction rates of even–even Fe isotopes

Nuclear structure properties and weak interaction rates of neutron-rich even–even iron (Fe) isotopes (A = 50 – 70) are investigated using the Interacting Boson Model-1 (IBM-1) and the proton–neutron Quasiparticle Random Phase Approximation (pn-QRPA) model. The IBM-1 is used for the calculation of energy levels and the B(E2) values of neutron-rich Fe isotopes. Later their geometry was predicted within the potential energy formalism of the IBM-1 model. Weak interaction rates on neutron-rich nuclei are needed for the modeling and simulation of presupernova evolution of massive stars. In the current study, we investigate the possible effect of nuclear deformation on stellar rates of even–even Fe isotopes. The pn-QRPA model is applied to calculate the weak interaction rates of selected Fe isotopes using three different values of deformation parameter. It is noted that, in general, bigger deformation values led to smaller total strength and larger centroid values of the resulting Gamow–Teller distributions. This later translated to smaller computed weak interaction rates. The current finding warrants further investigation before it may be generalized. The reported stellar rates are up to 4 orders of magnitude smaller than previous calculations and may bear astrophysical significance.

The TDHF code Sky3D version 1.2

The Sky3D code has been widely used to describe nuclear ground states, collective vibrational excitations, and heavy-ion collisions. The approach is based on Skyrme forces or related energy density functionals. The static and dynamic equations are solved on a three-dimensional grid, and pairing is been implemented in the BCS approximation. This updated version of the code aims to facilitate the calculation of nuclear strength functions in the regime of linear response theory, while retaining all existing functionality and use cases. The strength functions are benchmarked against available RPA codes, and the user has the freedom of choice when selecting the nature of external excitation (from monopole to hexadecapole and more). Some utility programs are also provided that calculate the strength function from the time-dependent output of the dynamic calculations of the Sky3D code. New version program summaryProgram Title: Sky3DCPC Library link to program files:https://doi.org/10.17632/vzbrzvyrn4.2Developer's repository link:https://github.com/manybody/sky3dLicensing provisions: GPLv3Programming language: Fortran, with one post-processing utility in Python.Journal reference of previous version:: Schuetrumpf, B., Reinhard, P.G., Stevenson, P.D., Umar, A.S., and Maruhn, J.A. (2018). The TDHF code Sky3D version 1.1. Comput. Phys. Commun. 229 (2018) 211–213.Does the new version supersede the previous version?: Yes.Reasons for the new version: The capability of reproducing the nuclear strength function for a variety of newly-coded external boosts has been added.Nature of problem: Calculating nuclear multipole strength functions is a crucial probe that can help model the nuclear system and its structure properties. A variety of models exist for this task, such as QRPA (Quasiparticle Random Phase Approximation) and its variants, but such approaches are often limited due to symmetry constraints. Time-dependent Hartree Fock (TDHF) has been used to simulate nuclear vibrations and collisions between nuclei for low energies without assuming any symmetry in the system. This code extends the TDHF to calculate the multi-pole strength functions of atomic nuclei. We showcase its reliability by comparing it with the established RPA codes for the calculation of such strength functions.Solution method: We extended previous versions of the Sky3D code [1,2] to include an external boost of multipole type where the user can provide custom input that decides the nature of the multipole (monopole, quadrupole, octupole, and so on) boost. The principal aim is to calculate the multipole strength function, which is the Fourier transform of the time-dependent expectation value of the multipole operator, which has the same form as the external boost. The proper unit conversion is done so that one can extract the exact unit of the thus calculated strength function, which is comparable to the available literature in the field. The boundary conditions are chosen such that a Woods-Saxon-like function cuts off the external field, driving it to zero at the boundary.Summary of revisions: The ability to give a more general external field for excitation has been implemented, which can be of a chosen multipolarity and isospin nature. Then, as a function of time, moments for both isospins and for multipolarities up to L=5 are followed. A new analysis program is included to calculate strength functions and energy weighted sum rules. The ability to specify a mixing parameter in the pairing force is included. This allows a mixed surface-volume pairing to be selected in a way more compatable with QRPA codes for comparison. An updated Makefile is included which makes compilation on Apple silicon computers easier.

Re-examination of effects of pairing gaps on charge-changing transitions

We re-examine the effects of pairing gaps on charge-changing transitions and the associated β-decay half-lives. Calculations are performed using the proton-neutron quasiparticle random phase approximation (pn-QRPA) model for 250 nuclei possessing astrophysical significance. Pairing gaps between like nucleons are key model parameters in the pn-QRPA approach. Three different values of empirically computed pairing gaps were used as input parameters in our nuclear model. Changing the pairing gap values significantly altered the Gamow-Teller distributions including the total strength and centroid values. The β-decay half-lives also changed substantially. The half-lives computed via the three-term pairing formula, based on separation energies of nucleons, were in best agreement with the measured data.

Isoscalar giant quadrupole resonance of even-even 112-124Sn isotopes using BCS-QRPA

Using self-consistent Bardeen - Cooper - Schriffer + Hartree - Fock and quasiparticle random phase approximation, the isoscalar giant quadrupole resonance in the isotopes of 112,114,116,118,120,122,124Sn has been studied in this work. Five sets of Skyrme-type interactions of different values of the nuclear matter incompressibility coefficient KNM and effective mass m*/m are used in the calculations. Additionally, the impact of different types of pairing forces (i.e., volume, surface, and mixed) is examined. Comparisons are made between the computed strength distributions, centroid energies Ecen, scaled energies Es, and constrained energies Econ of the isoscalar giant quadrupole resonance and the available experimental data. Analysis is done on the relationships between KNM and m*/m, and the estimated properties.

Toward a new approach for the pygmy dipole resonance in even–even nuclei. Application to isotopes 144,148,150,152,154Sm

A many-body Hamiltonian consisting of a spherical shell model mean-field term, a pairing interaction for alike nucleons and a dipole–dipole interaction, with the dipole operator involving a cubic term in the radial coordinate, was studied within a quasiparticle random-phase approximation and applied numerically to five even–even isotopes of Sm. The resulting wavefunctions were further used to calculate the B(E1) values, which at their turn were employed to calculate the photoabsorption cross section, integrated moments of the cross section and energy weighted sum rule (EWSR). The calculated cross section and its integrated moments were compared with the available data, and a good agreement was observed. Two regions were distinguished: one corresponding to the pygmy dipole resonance (PDR) (1–10 MeV) and the other to the giant dipole resonance (GDR) (10–20 MeV), which were studied separately. The peaks belonging to each of the two ranges were analyzed in detail. The PDR states were located around the neutron separation energy and were mainly formed by the collective isoscalar and neutron collective states. The PDR states describe oscillations of the neutron excess against protons from the isospin-saturated core. The character of the states from the GDR region, isoscalar or isovector, is also pointed out. The PDR states carry only 0.8%–2.7% of the total EWSR and 0.4%–5.9% of the total E1 strength. The dependence of the dipole strength on nuclear deformation is evidenced. A comment on the cross section splitting into two branches for deformed isotopes is included. The r-cubic term and nuclear deformation have opposite effects on the dipole strength. In addition, it diminishes the effect of nonconservation of the center of mass momentum. The famous Thomas–Reiche–Kuhn sum rule formula is generalized to the case of the Schiff dipole momentum. The new sum rule is well satisfied. The projected spherical single-particle basis used in our formalism allows for a unified description of spherical transitional and deformed isotopes.

Re-Examination of the Effect of Pairing Gaps on Gamow–Teller Strength Distributions and β-Decay Rates

β-decay is one of the key factors for understanding the r-process and evolution of massive stars. The Gamow–Teller (GT) transitions drive the β-decay process. We employ the proton–neutron quasiparticle random phase approximation (pn-QRPA) model to calculate terrestrial and stellar β-decay rates for 50 top-ranked nuclei possessing astrophysical significance according to a recent survey. The model parameters of the pn-QRPA model affect the predicted results of β-decay. The current study investigates the effect of nucleon–nucleon pairing gaps on charge-changing transitions and the associated β decay rates. Three different values of pairing gaps, namely TF, 3TF, and 5TF, were used in our investigation. It was concluded that both GT strength distributions and half-lives are sensitive to pairing gap values. The 3TF pairing gap scheme, in our chosen nuclear model, resulted in the best prediction with around 80% of the calculated half-lives within a factor 10 of the measured ones. The 3TF pairing scheme also led to the calculation of the biggest β-decay rates in stellar matter.

Electric dipole transitions in the relativistic quasiparticle random-phase approximation at finite temperature

Electric dipole transitions in the relativistic quasiparticle random-phase approximation at finite temperature

Re-analysis of the Gamow-Teller distributions for [formula omitted] nuclei, 24Mg, 28Si, and 32S

The Gamow-Teller (GT) strength distributions of sd-shell N=Z nuclei (24Mg, 28Si, and 32S) are investigated within the framework of proton–neutron quasi-particle random phase approximation (pn-QRPA) using a deformed basis. The nuclear properties of these special nuclei were investigated by the Relativistic Mean Field (RMF) model. The RMF framework with density-dependent interactions (DD-PC1 and DD-ME2) was employed to compute ground-state deformation parameters (β2) using potential energy curves. The β2 values computed from the RMF and the finite range droplet model (FRDM) were employed as a free parameter in the pn-QRPA calculation to investigate the resulting GT strength distributions. The present model-based analyses compare well with the observed data.

An empirical formula of nuclear <i>β</i>-decay half-lives

Nuclear <i>β</i>-decay half-lives play an important role not only in nuclear physics, but also in astrophysics. The <i>β</i>-decay half-lives of many nuclei involved in the astrophysical rapid neutron-capture (r -process) still cannot be measured experimentally, so the theoretical predictions of nuclear <i>β</i>-decay half-lives are inevitable for r-process studies. Theoretical models for studying the nuclear <i>β</i>-decay half-lives include the empirical formula, the gross theory, the quasiparticle random phase approximation (QRPA), and the shell model. Compared with other theoretical models of <i>β</i>-decay half-lives, the empirical formula has high computational efficiency, and its prediction accuracy can be improved by introducing more and more physical information. In this work, an empirical formula without free parameters is proposed to calculate the nuclear <i>β</i>-decay half-lives based on the Fermi theory of <i>β</i> decay. By including the pairing effect, the shell effect, and the isospin dependence, the newly proposed empirical formula significantly improves the accuracy of predicting the nuclear <i>β</i>-decay half-life. For the nuclei with half-lives less than 1 second, the root-mean-square deviation of the common logarithms of the nuclear <i>β</i>-decay half-life predicted by the new empirical formula from the experimental data decreases to 0.220, which is improved by about 54% compared with that by the empirical formula without free parameters, even better than those by other existing empirical formulas and microscopic QRPA approaches. In the unknown region, the nuclear <i>β</i>-decay half-lives predicted by the new empirical formula are generally shorter than those predicted by the microscopic models in the light nuclear region, while those predicted by the new empirical formula in the heavy nuclear region are generally in agreement with those predicted by the microscopic models. The half-lives of neutron-rich nuclei on the nuclear chart are then predicted by the new empirical formula, providing nuclear <i>β</i>-decay half-life inputs for the r-process simulations.

Predicting nucleon-nucleus scattering observables using nuclear structure theory

Developing a predictive capability for inelastic scattering will find applications in multiple areas. Experimental data for neutron-nucleus inelastic scattering is limited and thus one needs a robust theoretical framework to complement it. Charged-particle inelastic scattering can be used as a surrogate for (n, γ) reactions to predict capture cross sections for unstable nuclei. Our work uses microscopic nuclear structure calculations for spherical nuclei to obtain nucleon-nucleus scattering potentials and calculate cross sections for these processes. We implement the Jeukenne, Lejeune, Mahaux (JLM) semi-microscopic folding approach, where the medium effects on nuclear interaction are parameterized in nuclear matter to obtain the nucleon-nucleon (NN) interaction in a medium at positive energies. We solve for the nuclear ground state using the Hartree-Fock-Bogliubov (HFB) many-body method, assuming the nucleons within the nucleus interact via the Gogny-D1M potential. The vibrational excited states of the target nucleus are calculated using the quasi-particle random phase approximation (QRPA). We demonstrate our approach for spherical nuclei in the medium-mass region, showing scattering results for the 90Zr nucleus.

Towards systematic large scale Quasiparticle Random-Phase Approximation calculations with covariant and chiral interactions

Towards systematic large scale Quasiparticle Random-Phase Approximation calculations with covariant and chiral interactions

Impact of the Brink-Axel hypothesis on unique first-forbidden β-transitions for r-process nuclei* *J.-U. Nabi is supported by the Higher Education Commission Pakistan through Project (0-15394/NRPU/R&D/HEC/2021)

Key nuclear inputs for the astrophysical r-process simulations are the weak interaction rates. Consequently, the accuracy of these inputs directly affects the reliability of nucleosynthesis modeling. The majority of the stellar rates, used in simulation studies are calculated by invoking the Brink-Axel (BA) hypothesis. The BA hypothesis assumes that the strength functions of all parent excited states are the same as for the ground state, only shifted in energies. However, the BA hypothesis has to be tested against microscopically calculated state-by-state rates. In this project, we study the impact of the BA hypothesis on calculated stellar -decay and electron capture rates. Our investigation include both unique first forbidden (U1F) and allowed transitions for 106 neutron-rich trans-iron nuclei ([27, 77] ≤ [Z, A] ≤ [82, 208]). The calculations were performed using the deformed proton-neutron quasiparticle random-phase approximation (pn-QRPA) model with a simple plus quadrupole separable and schematic interaction. Waiting-point and several key r-process nuclei lie within the considered mass region of the nuclear chart. We computed electron capture and -decay rates using two different prescriptions for strength functions. One was based on invoking the BA hypothesis and the other was the state-by-state calculation of strength functions, under stellar density and temperature conditions ([10, 1] ≤ [ ( ), T( )] ≤ [1011, 30]). Our results show that the BA hypothesis invoked U1F rates are overestimated by 4–5 orders of magnitude as compared to microscopic rates. For capture rates, more than two orders of magnitude differences were noted when applying the BA hypothesis. It was concluded that the BA hypothesis is not a reliable approximation, especially for -decay forbidden transitions.

Isoscalar monopole response in the neutron-rich molybdenum isotopes using self-consistent QRPA

The isoscalar giant monopole resonance (ISGMR) of even molybdenum isotopes 92,94,96,98,100Mo has been studied within the Skyrme self-consistent Hartree - Fock - Bardeen, Cooper, and Schrieffer and quasi-particle random phase approximation. Ten sets of Skyrme-type interactions of different values of the nuclear matter incompressibility coefficient KNM are used in the calculations. The calculated strength distributions, centroid energies Ecen, scaled energies Es and constrained energies Econ of ISGMR are compared with available experimental data. Due to the appropriate value of the nuclear matter incompressibility KNM, several types of Skyrme interactions were successful in describing the ISGMR strength distribution in the 92,94,96,98,100Mo isotopes. As a result, high correlations between Ecen and KNM were found.

What can we learn from recent 2ν β β experiments?*

Abstract Recent measurements of the two neutrino double beta decay high precision electron spectra, combined with charge exchange or β-decay experimental data, have revealed severe constraints across current nuclear many body calculations. Our calculations show that the quasi-particle random phase approximation (QRPA) approach can adequately reproduce the measured spectra for the two open shell nuclei, Se and Mo. For the closed shell nucleus Xe, QRPA can also reproduce the spectra with proper treatments. Considering the high-lying state reduction, we also find that the nuclear shell model can also adequately reproduce the spectra and Gamow-Teller transition strength under a unique quenched axial-vector coupling constant . For Xe, we find that flipping the sign for the decay strength causes the spectra to go beyond the so-called high-lying state dominance hypothesis. These results call for future high precision measurements of charge-exchange reactions.

Neutrinos from pre-supernova in the framework of TQRPA method

Abstract We propose a new method for calculating spectra and luminosities for (anti)neutrinos produced in the pre-supernova environment by weak processes with hot nuclei. It is based on the thermal quasiparticle random phase approximation (TQRPA), that allows microscopic thermodynamically consistent calculations of the weak-interaction response of nuclei at finite temperatures. For realistic representative pre-supernova conditions from the stellar evolution code MESA, we compute (anti)neutrino luminosities and spectra arising from neutral- and charged-current weak reactions with hot 56Fe and compare them with the contribution of thermal processes. We find that the TQRPA approach produces not only a higher total luminosity of electron neutrinos (mainly born in the electron capture reaction), compared to the standard technique based on the large-scale shell model (LSSM) weak-interaction rates, but also a harder neutrino spectrum. Besides, applying the TQRPA and LSSM, we find that in the context of electron antineutrino generation, the neutral-current nuclear de-excitation (ND) process via neutrino-antineutrino pair emission is at least as important as the electron-positron pair annihilation process. We also show that flavor oscillations enhance the high-energy contribution of the ND process to the electron antineutrino flux. This could potentially be important for pre-supernova antineutrino registration by the Earth’s detectors.

Spreading Widths of Giant Monopole Resonance in the Lead Region: Random Matrix Approach

The microscopic calculation of the decay width of giant monopole resonance (GMR) anticipates the mixing of one-phonon states with configurations of increasing complexity. To this aim we develop the effective approach for description of monopole excited states that are obtained in the quasiparticle random phase approximation (QRPA), with regard of the coupling between one- and two-phonon states. Based on the QRPA one-phonon states, we generate the coupling and two-phonon states by means of the Gaussian orthogonal ensemble (GOE) distribution. Within our approach the spreading width of the GMRs in 204,206,208Pb are described by means of a random matrix approach on two energy scales. It is demonstrated that the main contribution into the decay of the GMR is determined by a small number of two-phonon states strongly coupled to low-energy surface vibrations. While a vast majority of the coupling matrix elements (that are small in value and following the GOE distribution) are responsible for the fine structure of the GMR spreading width. A remarkable agreement between the results of the full microscopic calculations (based on QRPA phonons coupled by means of the microscopic coupling matrix elements with calculated two-phonon states) with those of the developed approach confirms the vitality of the proposed ideas.

Dual nature of low-energy isoscalar monopole and dipole states in light nuclei

Isoscalar monopole (IS0) and dipole (IS1) states located below and around the alpha-particle threshold Sα in prolate 24Mg are investigated within fully self-consistent Skyrme Quasiparticle Random-Phase Approximation (QRPA) approach. It is shown that these states demonstrate strong mean-field (MF) effects: dipole vorticity, deformation-induced monopole-quadrupole (IS0/IS2) and dipole-octupole (IS1/IS3) coupling. At the same time, the density distributions show precursors of a cluster structure. So we get a cluster/MF duality with a strong dominance of MF impact. IS0 QRPA results are compared with iThemba Lab and RCNP (α, α′) data.

Validity of Brink-Axel Hypothesis for calculations of allowed stellar weak rates of heavy nuclei

The knowledge of beta-decay transitional probabilities and Gamow-Teller (GT) strength functions from highly excited states of nuclides is of particular importance for applications to astrophysical network calculations of nucleosynthesis in explosive stellar events. These quantities are challenging to achieve from measurements or computations using various nuclear models. Due to unavailability of feasible alternatives, many theoretical studies often rely on the Brink-Axel (BA) hypothesis, that is, the response of strength functions depends merely on the transition energy of the parent nuclear ground state and is independent of the underlying details of the parent state, for the calculation of stellar rates. BA hypothesis has been used in many applications from nuclear structure determination to nucleosynthesis yield in the astrophysical matter. We explore here the the validity of BA hypothesis in the calculation of stellar beta-decay (BD) and electron capture (EC) weak rates of fp- and fpg-shell nuclides for GT transitions. Strength functions have been computed employing the fully microscopic proton-neutron QRPA (quasi-particle random-phase approximation) within a broad density, ρY e = (10-1011) [g cm−3], and temperature, T = (1−30) [GK], grid relevant to the pre-collapse astrophysical environment. Our work provides evidence that the use of the approximation based on the BA hypothesis does not lead to reliable calculations of excited states strength functions under extreme temperature-density conditions characteristic of presupernova and supernova evolution of massive stars. Weak rates obtained by incorporating the BA hypothesis in the calculation of strength functions substantially deviate from the rates based on the state-by-state microscopically calculated strength functions. Deviation in the two calculations becomes significant as early as neon burning phases of massive stars. The deviation in the calculation of BD rates is even more pronounced, reaching up to three orders of magnitude.

ISOSCALAR DIPOLE RESPONSE IN 92Mo AND 100Mo ISOTOPES

In this work, the Isoscalar (IS) Giant Dipole Resonance (GDR) and Pygmy Dipole Resonance (PDR) of 92Mo and 100Mo Isotopes were calculated in the farmwork of the self-consistent quasi-particle random phase approximation (QRPA) based on the results of Hartree-Fock-Bardeen, Cooper and Schrieffer (HF-BCS) using 10 Skyrme-type interactions: KDE0v1, eMSL08, SKX, SGOI, v080, SKP, SIV, SIII, SKIII, and SGI. The strength distributions of isoscalar dipole response were compared with the available experimental data. Also, we discussed the statistical relation between the centroid energy and the nuclear matter incompressibility KNM.