Energy Density Functional Approach Research Articles

Isospin Symmetry Breaking in Atomic Nuclei

In this paper, the importance of isospin symmetry and its breaking in elucidating the properties of atomic nuclei is reviewed. The quark mass splitting and the electromagnetic origin of the isospin symmetry breaking (ISB) for the nuclear many-body problem is discussed. The experimental data on isobaric analogue states cannot be described only with the Coulomb interaction, and ISB terms in the nucleon–nucleon interaction are needed to discern the observed properties. In the present work, the ISB terms are explicitly considered in nuclear energy density functional and spherical shell model approaches, and a detailed investigation of the analogue states and other properties of nuclei is performed. It is observed that isospin mixing is largest for the N = Z system in the density functional approach

Nuclear clustering over the nuclear chart from a covariant EDF approach

Nuclear clustering has been studied with covariant Energy Density Functional approaches for more than a decade. It recently allowed to bridge microscopically the description of cluster states in light nuclei with the one of cluster and alpha decays in heavy ones. The proper use of theoretical tools, such as the density and the nucleonic localization function, allows to shed light on the mechanisms of formation and identification of clusters in nuclei. Finally, a global analysis of nuclear clustering is discussed, in order to identify control parameters for a nuclear cluster phase.

Self-Consistent Calculation of Nuclear Charge Radii in K Isotopes

Fully self-consistent study of the charge radii in Ca region is exemplified by the calculation in the long chains of K isotopes. The neutron-deficient and neutron-rich nuclei with pairing in both neutron and proton sectors, as well as the (semi-) magic nuclei around the closed neutron shells at N = 20, 28, 32 are treated within the Energy Density Functional (EDF) approach with the Fayans functional DF3-a. The performance of the DF3-a is analysed in describing the odd-even staggering effects found both in previous and in more recent CERN-ISOLDE experiments for 36–52K isotopes.

Addressing energy density functionals in the language of path-integrals I: comparative study of diagrammatic techniques applied to the (0 + 0)-D O(N)-symmetric $$\varphi ^{4}$$-theory

The energy density functional (EDF) method is currently the only microscopic theoretical approach able to tackle the entire nuclear chart. Nevertheless, it suffers from limitations resulting from its empirical character and deteriorating its reliability. This paper is part of a larger program that aims at formulating the EDF approach as an effective field theory (EFT) in order to overcome these limitations. A relevant framework to achieve this is the path-integral formulation of quantum field theory. The latter indeed provides a wide variety of treatments of the many-body problem well suited to deal with non-perturbative interactions and to exploit a Lagrangian resulting from an EFT as a starting point. While developing the formalism in a general setting, we present below a comparative study of such techniques applied to a toy model, i.e. the (0 + 0)-D O(N)-symmetric $$\varphi ^{4}$$ -theory. More specifically, our focus will be on the following diagrammatic techniques: loop expansion, optimized perturbation theory and self-consistent perturbation theory. With these methods, we notably address the spontaneous breakdown of the O(N) symmetry with care especially since spontaneous symmetry breakings play a paramount role in current implementations of the EDF approach.

Isospin symmetry breaking in the charge radius difference of mirror nuclei

Isospin symmetry breaking (ISB) effects in the charge radius difference $\mathrm{\ensuremath{\Delta}}{R}_{\mathrm{ch}}$ of mirror nuclei are studied using the test example of $^{48}\mathrm{Ca}$ and $^{48}\mathrm{Ni}$. This choice allows for a transparent study of ISB contributions since pairing and deformation effects, commonly required for the study of mirror nuclei, can be neglected in this specific pair. The connection of $\mathrm{\ensuremath{\Delta}}{R}_{\mathrm{ch}}$ with the nuclear equation of state and the effect of ISB on such a relation are discussed according to an energy density functional approach. We find that nuclear ISB effects may shift the estimated value for the symmetry energy slope parameter $L$ by about 6 to 14 MeV while Coulomb corrections can be neglected. ISB effects on the ground-state energy and charge radii in mirror nuclei have been recently predicted by ab initio calculations to be relatively small, pointing to a negligible effect for the extraction of information on the nuclear EoS. These contrasting results call for a dedicated theoretical effort to solve this overarching problem that impacts not only the neutron-skin thickness or the difference in mass and charge radii of mirror nuclei but also other observables such as the isobaric analog state energy.

Calculations for nuclear matter and finite nuclei within and beyond energy-density-functional theories through interactions guided by effective field theory

We propose a novel idea to construct an effective interaction under energy-density-functional (EDF) theories which is adaptive to the enlargement of the model space. Guided by effective field theory principles, iterations of interactions as well as enlargements of the model space through particle-hole excitations are carried out for infinite nuclear matter and selected closed-shell nuclei ($^{4}\mathrm{He}, ^{16}\mathrm{O}, ^{40}\mathrm{Ca}, ^{56}\mathrm{Ni}$, and $^{100}\mathrm{Sn}$) up to next-to-leading order. Our approach provides a new way for handling the nuclear matter and finite nuclei within the same scheme, with advantages from both EDF and ab initio approaches.

Axially-deformed solution of the Skyrme-Hartree-Fock-Bogoliubov equations using the transformed harmonic oscillator basis (IV) hfbtho (v4.0): A new version of the program

We describe the new version 4.0 of the code hfbtho that solves the nuclear Hartree-Fock-Bogoliubov problem by using the deformed harmonic oscillator basis in cylindrical coordinates. In the new version, we have implemented the restoration of rotational, particle number, and reflection symmetry for even-even nuclei. The restoration of rotational symmetry does not require using bases closed under rotation. Furthermore, we added the SeaLL1 functional and improved the calculation of the Coulomb potential. Finally, we refactored the code to facilitate maintenance and future developments. New version program summaryProgram title:hfbtho v4.0CPC Library link to program files:https://doi.org/10.17632/c5g2f92by3.2Code Ocean capsule:https://codeocean.com/capsule/5389629Licensing provisions: GPLv3Programming language: Fortran 2003Journal reference of previous version: R.N. Pérez, N. Schunck, R.-D. Lasseri, C. Zhang and J. Sarich, Comput. Phys. Commun. 220 (2017) 363Does the new version supersede the previous version: YesReasons for the new version: This version adds new capabilities to restore broken symmetries and determine corresponding quantum numbers of even-even nucleiSummary of revisions:1.Angular momentum projection for even-even nuclei in a deformed basis;2.Particle number projection for even-even nuclei in the quasiparticle basis;3.Implementation of the SeaLL1 functional;4.Expansion of the Coulomb potential onto Gaussians;5.MPI-parallelization of a single hfbtho execution;6.Code refactoring.Nature of problem:hfbtho is a physics computer code that is used to model the structure of the nucleus. It is an implementation of the energy density functional (EDF) approach to atomic nuclei, where the energy of the nucleus is obtained by integration over space of some phenomenological energy density, which is itself a functional of the neutron and proton intrinsic densities. In the present version of hfbtho, the energy density is derived either from the zero-range Skyrme or the finite-range Gogny effective two-body interaction between nucleons. Nuclear superfluidity is treated at the Hartree-Fock-Bogoliubov (HFB) approximation. Constraints on the nuclear shape allow probing the potential energy surface of the nucleus as needed, e.g., for the description of shape isomers or fission. A local scale transformation of the single-particle basis in which the HFB solutions are expanded provides a tool to properly compute the structure of weakly-bound nuclei. Restoration of the rotational, particle number, and reflection symmetry for even-even nuclei enables recovering the quantum numbers that are lost at the HFB approximation.Solution method: The program uses the axial harmonic oscillator (HO) or the transformed harmonic oscillator (THO) single-particle basis to expand quasiparticle wave functions. It iteratively diagonalizes the HFB Hamiltonian based on generalized Skyrme-like energy densities and zero-range pairing interactions or the finite-range Gogny force until a self-consistent solution is found. Lagrange parameters are used to impose constraints on HFB solutions, and their value is updated at each iteration from an approximation of the quasiparticle random phase approximation (QRPA) matrix. Symmetry restoration is implemented through standard projection techniques. Previous versions of the program were presented in [1-3].Additional comments including restrictions and unusual features: Axial and time-reversal symmetries are assumed in HFB calculations; y-simplex symmetry and even particle numbers are assumed in angular momentum projection.

Appearance of a peak in the symmetry energy at N=126 for the Pb isotopic chain within the relativistic energy density functional approach

The newly derived relativistic energy density functional [\textcolor{blue}{ Phys. Rev. C \textbf{103}, 024305 (2021)}], which stems from the effective field theory motivated relativistic mean-field (E-RMF) is employed to establish the appearance of peak/kink in the symmetry energy over the isotopic chain of Pb-nuclei. The coherent density fluctuation model parametrization procedure for finite nuclei is adopted here to obtain the relativistic energy density functional at local density. The relativistic energy density functional from E-RMFT takes precedence over the Br\"uckner energy density functional as it accurately predicts the empirical saturation density and binding energy per nucleon $E/A$, so-called 'Coester Band Problem'. Interestingly, using the relativistic energy density functional, it is possible to predict the peak at $N=126$ for recently developed G3 and widely used NL3 parameter sets, which is not observed for Br\"uckener's functional in-spite of using the E-RMF density. From the present analysis, the newly fitted energy density functional is found to be minutely sensitive to the choice of the parameter sets employed.

Self-consistent energy density functional approaches to the crust of neutron stars

In order to study structure of the crust in neutron stars, we develop a finite-temperature Skyrme-Hartree-Fock method in the full three-dimensional coordinate space using the Fermion operator expansion method. It provides us with a possible order-N approach to non-uniform neutron-star matters.

Isovector giant monopole and quadrupole resonances in a Skyrme energy density functional approach with axial symmetry

[Background] Giant resonance (GR) is a typical collective mode of vibration. The deformation splitting of the isovector (IV) giant dipole resonance is well established. However, the splitting of GRs with other multipolarities is not well understood. [Purpose] I explore the IV monopole and quadrupole excitations and attempt to obtain the generic features of IV giant resonances in deformed nuclei by investigating the neutral and charge-exchange channels simultaneously. [Method] I employ a nuclear energy-density functional (EDF) method: the Skyrme-Kohn-Sham-Bogoliubov and the quasiparticle random-phase approximation are used to describe the ground state and the transition to excited states. [Results] I find the concentration of the monopole strengths in the energy region of the isobaric analog or Gamow-Teller resonance irrespective of nuclear deformation, and the appearance of a high-energy giant resonance composed of the particle-hole configurations of $2\hbar \omega_0$ excitation. Splitting of the distribution of the strength occurs in the giant monopole and quadrupole resonances due to deformation. The lower $K$ states of quadrupole resonances appear lower in energy and possess the enhanced strengths in the prolate configuration, and vice versa in the oblate configuration, while the energy ordering depending on $K$ is not clear for the $J=1$ and $J=2$ spin-quadrupole resonances. [Conclusions] The deformation splitting occurs generously in the giant monopole and quadrupole resonances. The $K$-dependence of the quadrupole transition strengths is largely understood by the anisotropy of density distribution.

Isovector spin susceptibility: Isotopic evolution of collectivity in spin response

Background: Response to spin-dependent operators has been investigated in the magnetic dipole and Gamow-Teller transitions, which provides magnetic properties of a nuclear system. Purpose: I investigate an isotopic dependence of the collectivity generated by the spin-dependent interactions in the Ca and Ni isotopes through the isovector (IV) spin-flip excitations. The responses in the neutral $(t_z)$ and charge-exchange $(t_{\pm})$ channels are considered in a unified way. Method: A nuclear energy-density functional approach is employed for calculating the response functions based on the Skyrme-Kohn-Sham-Bogoliubov method and the quasiparticle-random-phase approximation (QRPA). I adopt the like-particle QRPA and the proton-neutron QRPA for the neutral and charge-exchange channels, respectively. I consider the fluctuation of the proton-neutron pair fields. Results: The collective shift due to RPA correlations for the response in the neutral channel is explained by the occupation probability of neutrons in the $j_>=\ell+1/2$ orbital. Many particle-hole or two-quasiparticle excitations have a coherent contribution to form a giant resonance in neutron-rich nuclei for the charge-exchange channel. The IV spin susceptibility displays the isotopic evolution of the collectivity and the underlying shell structure. Conclusions: A repulsive character of the residual interaction in the spin-isospin channel diminishes the IV spin susceptibility due to the collectivity, while the dynamic ${}^{3}S$ pairing appearing in the charge-exchange channel opposes the reduction.

Proton-induced deuteron knockout reaction as a probe of an isoscalar proton-neutron pair in nuclei

The isoscalar $pn$ pair is expected to emerge in nuclei having the similar proton and neutron numbers but there is no clear experimental evidence for it. We aim to clarify the correspondence between the $pn$ pairing strength in many-body calculation and the triple differential cross section (TDX) of proton-induced deuteron knockout ($p,pd$) reaction on $^{16}$O. The radial wave function of the isoscalar $pn$ pair with respect to the center of $^{16}$O is calculated with the energy density functional (EDF) approach and is implemented in the distorted wave impulse approximation (DWIA) framework. The $pn$ pairing strength $V_0$ in the EDF calculation is varied and the corresponding change in the TDX is investigated. A clear $V_0$ dependence of the TDX is found for the $^{16}$O($p,pd$)$^{14}$N($1_2^+$) at $101.3$ MeV. The nuclear distortion is found to make the $V_0$ dependence stronger. Because of the clear $V_0$-TDX correspondence, the ($p,pd$) reaction will be a promising probe for the isoscalar $pn$ pair in nuclei. For quantitative discussion, further modification of the description of the reaction process will be necessary.

Nuclear magnetic transitions in the relativistic energy density functional approach

Recently a novel theory framework has been established for description of magnetic dipole (M1) transitions in finite nuclei, based on relativistic nuclear energy density functional with point coupling interactions. The properties of M1 transitions have been studied, including the sum rules, spin, orbital, isoscalar and isovector M1 transition strengths in magic and open shell nuclei. It is shown that pairing correlations and spinorbit interaction plays an important role in the description of M1 transition strength distributions. The analysis of the evolution of M1 transition properties in the isotope chain100-140Sn shows the interplay between single and double-peak structures, determined by the evolution of single-particle states, their occupations governed by the pairing correlations, and two-quasiparticle transitions involved. Comparison of the calculated B(M1) transition strength with recent data from inelastic proton scattering on112-124Sn, shows that quenching of thegfactorsgeff/gfree=0.80-0.93 is required to reproduce the experimental data. Further experimental investigations are needed to determine accurately the quenching factor.

Nucleonic localisation and alpha radioactivity

Relativistic energy density functional approaches are known to well describe nuclear states which involve alpha clusters. Here, alpha emitting nuclei are analyzed through the behavior of the spatial localisation of nucleonic states, calculated with an axially deformed relativistic Hartree–Bogoliubov approach over the nuclear chart. The systematic occurrence of more localised valence states, having an n = 1 radial quantum number, allows to pinpoint nuclei in agreement with experimentally known alpha-emitters. The cases of 212Po and 104Te are investigated, showing the concomitant contributions of the pseudospin symmetry and the presence of n = 1 states, on the alpha preformation probability. The impact of the localisation of valence states, on alpha preformation probability, is then analyzed. It allows to study shell effects on this probability, over isotopic and isotonic chains. Finally, a phenomenological law is also provided, relating this probability to the radial quantum number of the valence states.

Relativistic nuclear energy density functional approach to magnetic-dipole excitation

Magnetic-dipole (M1) excitations of 18O and 42Ca nuclei are investigated within a relativistic nuclear energy density functional framework. In our last work [1], these nuclei are found to have unique M1 excitation and its sum rule, because of their characteristic structure: the system consists of the shell-closure core plus two neutrons. For a more systematic investigation of the M1 mode, we have implemented a framework based on the relativistic nuclear energy density functional (RNEDF). For benchmark, we have performed the RNEDF calculations combined with the random-phase approximation (RPA). We evaluate the M1 excitation of 18O and 42Ca, whose sum-rule value (SRV) of the M1 transitions can be useful to test the computational implementation [1]. We also apply this RNEDF method to 208Pb, whose M1 property has been precisely measured [2, 3, 4, 5]. Up to the level of the M1 sum rule, our result is in agreement with the experiments, except the discrepancy related with the quenching factors for g coefficients.

Accessing the Single-Particle Structure of the Pygmy Dipole Resonance in ^{208}Pb.

New experimental data on the neutron single-particle character of the Pygmy Dipole Resonance (PDR) in ^{208}Pb are presented. They were obtained from (d,p) and resonant proton scattering experiments performed at the Q3D spectrograph of the Maier-Leibnitz Laboratory in Garching, Germany. The new data are compared to the large suite of complementary, experimental data available for ^{208}Pb and establish (d,p) as an additional, valuable, experimental probe to study the PDR and its collectivity. Besides the single-particle character of the states, different features of the strength distributions are discussed and compared to large-scale shell model (LSSM) and energy-density functional plus quasiparticle-phonon model theoretical approaches to elucidate the microscopic structure of the PDR in ^{208}Pb.

Predictions of identification and production of new superheavy nuclei with Z=119 and 120

Effective single-particle potentials obtained from the established nonrelativistic nuclear energy density functional approach are incorporated into the microscopic-macroscopic method to calculate the ground-state shell corrections, mass excesses, and ${Q}_{\ensuremath{\alpha}}$ values for heaviest nuclei. The low-lying one-quasiparticle states are studied in the isotonic chains with neutron numbers $N=175$, 176, and 177. The isomeric states are discussed in the odd isotopes. The possible $\ensuremath{\alpha}$-decay chains of nuclei $^{295}119$ and $^{295,297}120$ are analyzed and compared with the available experimental data. The termination of the $\ensuremath{\alpha}$-decay chains by spontaneous fission is analyzed. The production cross sections are calculated for superheavy nuclei with $112\ensuremath{\le}Z\ensuremath{\le}120$ in the actinide-based complete fusion reactions with projectiles $^{48}\mathrm{Ca}, ^{48,50}\mathrm{Ti}, ^{54}\mathrm{Cr}, ^{58}\mathrm{Fe}$, and $^{64}\mathrm{Ni}$.

Microscopically based energy density functionals for nuclei using the density matrix expansion

While ab initio many-body techniques have been able to successfully describe the properties of light and intermediate mass nuclei based on chiral effective field theory interactions, neutron-rich nuclei still remain out of reach for these methods. Conversely, energy density functional approaches can be used to calculate properties of heavy nuclei but rely mostly on phenomenological interactions. A usable form of the nuclear energy density functional that is rooted in the modern theory of nuclear forces was presented recently. The first component of this new set of functionals corresponds to the direct part (Hartree term) of the expectation value of local chiral potentials on a Slater determinant. The exchange term, which is a functional of the non-local density, is transformed into a local functional by applying the density matrix expansion. In order to reduce the computational cost due to the direct implementation of non-separable, local interactions in the Hartree term, we use an approximation to represent the regularized Yukawa functions in terms of a sum of (separable) Gaussian functions. These proceedings analyze the accuracy of such an approximation in terms of the number of Gaussian functions and look for an optimal value that gives an acceptable level of accuracy while maintaining the computational memory requirements in a many-body calculation as low as possible.

Моделирование нуклеации в бинарных сплавах на основе метода функционала плотности свободной энергии

In this study, the approach for nucleation rate calculation is developed on the basis of free energy density functional approach. The nucleation rate and characteristics of critical nucleus are calculated from results of composition field simulation, derived by the solution of deterministic or stochastic Cahn-Hilliard equation.

Microscopically based energy density functionals for nuclei using the density matrix expansion. II. Full optimization and validation

We seek to obtain a usable form of the nuclear energy density functional that is rooted in the modern theory of nuclear forces. We thus consider a functional obtained from the density matrix expansion of local nuclear potentials from chiral effective field theory. We propose a parametrization of this functional carefully calibrated and validated on selected ground-state properties that is suitable for large-scale calculations of nuclear properties. The first component of this functional is a non-local functional of the density and corresponds to the direct part (Hartree term) of the expectation value of local chiral potentials on a Slater determinant. A second component is a local functional of the density and is obtained by applying the density matrix expansion to the exchange part (Fock term) of the expectation value of the local chiral potential. We apply the UNEDF2 optimization protocol to determine the coupling constants of this energy functional. We obtain a set of microscopically-constrained functionals for local chiral potentials from leading-order up to next-to-next-to-leading order with and without three-body forces and contributions from $\Delta$ excitations. These functionals are validated on the calculation of nuclear and neutron matter, nuclear mass tables, single-particle shell structure in closed-shell nuclei and the fission barrier of $^{240}$Pu. Quantitatively, they perform noticeable better than the more phenomenological Skyrme functionals. The inclusion of higher-order terms in the chiral perturbation expansion seems to produce a systematic improvement in predicting nuclear binding energies. This result is especially promising since all the fits have been performed at the single reference level of the energy density functional approach, where important collective correlations such as center-of-mass correction have not been taken into account yet.