Energy Density Functionals Research Articles

Monopole moments and the $ \beta$-vibration in deformed nuclei

I revisit the theory of the $ \beta$-vibration in deformed nuclei focusing on the transition matrix elements to the ground band. The relation between monopole and quadrupole matrix elements is derived from an incompressibility assumption and is validated by self-consistent mean-field theory with several energy density functionals. With estimates of the deformation-dependent mean-field potential energy function, it is found that the excited $ 0^+$ band in 156Gd comes fairly close to meeting the criteria to be classified as a $ \beta$-vibration.

Nuclear excitations within microscopic EDF approaches: Pairing and temperature effects on the dipole response

In the present work, the isovector dipole responses, both in the resonance region and in the low-energy sector, are investigated using the microscopic nuclear Energy Density Functionals (EDFs). The self-consistent QRPA model based on Skyrme Hartree Fock BCS approach is applied to study the evolution of the isovector dipole strength by increasing neutron number and temperature. First, the isovector dipole strength and excitation energies are investigated for the Ni isotopic chain at zero temperature. The evolution of the low-energy dipole strength is studied as a function of the neutron number. In the second part, the temperature dependence of the isovector dipole excitations is studied using the self-consistent finite temperature QRPA, below and above the critical temperatures. It is shown that new excited states become possible due to the thermally occupied states above the Fermi level, and opening of the new excitations channels. In addition, temperature leads to fragmentation of the low-energy strength around the neutron separation energies, and between 9 and 12 MeV. We find that the cumulative sum of the strength below E$\leq12$ MeV decreases in open-shell nuclei due to the vanishing of the pairing correlations as temperature increases up to T=1 MeV. The analysis of the transition densities in the low-energy region shows that the proton and neutron transition densities display a mixed pattern: both isoscalar and isovector motion of protons and neutrons are obtained inside nuclei, while the neutron transition density is dominant at the surface region.

Constraints on the inner edge of neutron star crusts from relativistic nuclear energy density functionals

The transition density nt and pressure Pt at the inner edge between the liquid core and the solid crust of a neutron star are analyzed using the thermodynami- cal method and the framework of relativistic nuclear energy density functionals. Starting from a functional that has been carefully adjusted to experimental binding energies of finite nuclei, and varying the density dependence of the cor- responding symmetry energy within the limits determined by isovector prop- erties of finite nuclei, we estimate the constraints on the core-crust transition density and pressure of neutron stars: 0.086 fm−3 ≤ nt < 0.090 fm−3 and 0.3 MeV fm−3 < Pt ≤ 0.76 MeV fm−3 [1].

Improvement of functionals in density functional theory by the inverse Kohn–Sham method and density functional perturbation theory

We propose a way to improve energy density functionals (EDFs) in density functional theory based on the combination of the inverse Kohn–Sham method and the density functional perturbation theory. Difference between the known EDF and the exact one is treated as the first-order perturbation. As benchmark calculations, we reproduce the theoretical exchange and correlation functionals in the local density approximation. Systems of noble-gas atoms are used for benchmark calculations, and the ground-state energies and densities, as well as the functionals, are reproduced with good accuracies.

Search for octupole-deformed actinium isotopes using resonance ionization spectroscopy

In-source resonance ionization spectroscopy of the neutron-rich actinium isotopes $^{225\ensuremath{-}229}\mathrm{Ac}$ has been performed at the ISAC facility in TRIUMF, probing a $^{2}D_{3/2}\ensuremath{\rightarrow}^{4}P_{5/2}^{\ensuremath{\circ}}$ atomic transition. New data on the magnetic dipole moments and changes in mean-square charge radii $\ensuremath{\delta}\ensuremath{\langle}{r}^{2}\ensuremath{\rangle}$ of $^{226,228,229}\mathrm{Ac}$ have been obtained. The comparison of the measured isotope shifts and magnetic dipole coupling constants $a(^{4}P_{5/2}^{\ensuremath{\circ}})$ of $^{225,227}\mathrm{Ac}$ with a high-resolution data set is used to identify systematic uncertainties on the deduced $\ensuremath{\delta}{\ensuremath{\langle}{r}^{2}\ensuremath{\rangle}}^{A,215}$ and magnetic dipole moment values. The charge radii odd-even staggering is evaluated for the odd-$N$ isotopes, showing that $^{226}\mathrm{Ac}$ has an inverted odd-even staggering that might be linked with a reflection-asymmetric shape. Comparison of the magnetic dipole moments of $^{225,227,229}\mathrm{Ac}$ with Nilsson-model estimates supports the assumption of octupole deformation in isotopes $^{225,227}\mathrm{Ac}$ and its gradual decrease toward isotope $^{229}\mathrm{Ac}$. The changes in mean-square charge radii are compared to self-consistent calculations employing multiple modern energy density functionals: SLy5s1, BSk31, and DD-MEB1. For SLy5s1 in particular, self-consistent time-reversal breaking calculations of odd-odd nuclei incorporating finite octupole deformation are reported for the first time. For these calculations, the overall best agreement is obtained when the octupole degree of freedom is taken into account.

Fission dynamics of Pu240 from saddle to scission and beyond

Calculations are presented for the time evolution of $^{240}\mathrm{Pu}$ from the proximity of the outer saddle point until the fission fragments are well separated, using the time-dependent density functional theory extended to superfluid systems. We have tested three families of nuclear energy density functionals and found that all functionals exhibit a similar dynamics: The collective motion is highly dissipative and with little trace of inertial dynamics, due to the one-body dissipation mechanism alone. This finding justifies the validity of using the overdamped collective motion approach and to some extent the main assumptions in statistical models of fission. This conclusion is robust with respect to the nuclear energy density functional used. The configurations and interactions left out of the present theory framework only increase the role of the dissipative couplings. An unexpected finding is varying the pairing strength within a quite large range has only minor effects on the dynamics. We find notable differences in the excitation energy sharing between the fission fragments in the cases of spontaneous and induced fission. With increasing initial excitation energy of the fissioning nucleus, more excitation energy is deposited in the heavy fragment, in agreement with experimental data on average neutron multiplicities.

Parameter optimization for the latest quark-meson coupling energy-density functional

The Quark--Meson--Coupling (QMC) model self-consistently relates the dynamics of the internal quark structure of a hadron to the relativistic mean fields arising in nuclear matter. It offers a natural explanation to some open questions in nuclear theory, including the origin of many-body nuclear forces and their saturation, the spin-orbit interaction and properties of hadronic matter at a wide range of densities. The QMC energy density functionals QMC-I and QMC$\pi$-I have been successfully applied to calculate ground state observables of finite nuclei in the Hartree-Fock + BCS approximation, as well as to predict properties of dense nuclear matter and cold non-rotating neutron stars. Here we report the latest development of the model, QMC$\pi$-II, extended to include higher order self-interaction of the $\sigma$ meson. A derivative-free optimization algorithm has been employed to determine a new set of the model parameters and their statistics, including errors and correlations. QMC$\pi$-II predictions for a wide range of properties of even-even nuclei across the nuclear chart, with fewer adjustable parameters, are comparable with other models. Predictions of ground state binding energies of even-even isotopes of superheavy elements with Z$>$96 are particularly encouraging.

Suddenly shortened half-lives beyond Ni78:N=50 magic number and high-energy nonunique first-forbidden transitions

$\beta$-decay rates play a decisive role in understanding the nucleosynthesis of heavy elements and are governed by microscopic nuclear-structure information. A sudden shortening of the half-lives of Ni isotopes beyond $N=50$ was observed at the RIKEN-RIBF. This is considered due to the persistence of the neutron magic number $N=50$ in the very neutron-rich Ni isotopes. By systematically studying the $\beta$-decay rates and strength distributions in the neutron-rich Ni isotopes around $N=50$, I try to understand the microscopic mechanism for the observed sudden shortening of the half-lives. The $\beta$-strength distributions in the neutron-rich nuclei are described in the framework of nuclear density-functional theory. I employ the Skyrme energy-density functionals (EDF) in the Hartree-Fock-Bogoliubov calculation for the ground states and in the proton-neutron Quasiparticle Random-Phase Approximation (pnQRPA) for the transitions. Not only the allowed but the first-forbidden (FF) transitions are considered. The experimentally observed sudden shortening of the half-lives beyond $N=50$ is reproduced well by the calculations employing the Skyrme SkM* and SLy4 functionals. The sudden shortening of the half-lives is due to the shell gap at $N=50$ and cooperatively with the high-energy transitions to the low-lying $0^-$ and $1^-$ states in the daughter nuclei. The onset of FF transitions pointed out around $N=82$ and 126 is preserved in the lower-mass nuclei around $N=50$. This study suggests that needed is a microscopic calculation where the shell structure in neutron-rich nuclei and its associated effects on the FF transitions are selfconsistenly taken into account for predicting $\beta$-decay rates of exotic nuclei in unknown region.

Energy-weighted sum rule for nuclear density functional theory

The expressions for the energy-weighted sum rule of the isoscalar and isovector coordinate operators are derived based on the second-order fluctuation of the local densities. Conventional derivation of the Thouless theorem for the energy-weighted sum rule is based on the double commutator of the Hamiltonian, while the present derivation does not assume a Hamiltonian operator and is applicable to nuclear energy density functionals. The expressions include the contribution of the local gauge symmetry breaking of the energy density functional. It is shown that the local gauge invariance of the kinetic and current densities and kinetic pair density is important, while all the other local densities do not contribute to the energy-weighted sum rule of the coordinate operators. The finite-amplitude method calculations are performed and the expressions for the energy-weighted sum rule are numerically examined for the isoscalar and isovector multipole operators up to $L=3$ for selected spherical and axially deformed nuclei.

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.

Fine structure of the pygmy quadrupole resonance in 112,114Sn

The electric quadrupole response in 112,114Sn isotopes is investigated by energy-density functional (EDF) and three-phonon quasiparticle-phonon model (QPM) theory with special emphasis on electric quadrupole excitations located above the first collective 2+ state and below 5 MeV. Additional quadrupole strength clustering as a sequence of states similar to the recently observed pygmy quadrupole resonance in 124Sn is found. The spectral distributions and transition densities of these 2+ states show special features being compatible with oscillations of a neutron skin against the isospin-symmetric nuclear core. Furthermore, two (p, p′γ) Doppler-shift attenuation (DSA) coincidence experiments were performed at the SONIC@HORUS setup. 2+ states with excitation energies up to 4.2 MeV were populated in 112,114Sn. Lifetimes and branching ratios were measured allowing for the determination of E2 transition strengths to the ground state. A stringent comparison of the new data to EDF+QPM theory in 112Sn and 114Sn isotopes hints at the occurrence of a low-energy quadrupole mode of unique character which could be interpreted as pygmy quadrupole resonance.

Analysis of nuclear structure in a converging power expansion scheme

In the framework of the KIDS generalized energy density functional (EDF), the nuclear equation of state (EoS) is expressed as an expansion in powers of the Fermi momentum or the cubic root of the density ($\rho^{1/3}$). Although an optimal number of converging terms was obtained in specific cases of fits to empirical data and pseudodata, the degree of convergence remains to be examined not only for homogeneous matter but also for finite nuclei. One goal of the present work is to validate the minimal and optimal number of EoS parameters required for the description of homogeneous nuclear matter over a wide range of densities relevant for astrophysical applications. The major goal is to examine the validity of the adopted expansion scheme for an accurate description of finite nuclei. To this end we vary the values of the high-order derivatives of the EoS, namely the skewness of the energy of symmetric nuclear matter and the kurtosis of the symmetry energy, at saturation and examine the relative importance of each term in $\rho^{1/3}$ expansion for homogeneous matter. For given sets of EoS parameters determined in this way, we define equivalent Skyrme-type functionals and examine the convergence in the description of finite nuclei focusing on the masses and charge radii of closed-shell nuclei. The EoS of symmetric nuclear matter is found to be efficiently parameterized with only 3 parameters and the symmetry energy (or the energy of pure neutron matter) with 4 parameters when the EoS is expanded in the power series of the Fermi momentum. Higher-order EoS parameters do not produce any improvement, in practice, in the description of nuclear ground-state energies and charge radii, which means that they cannot be constrained by bulk properties of nuclei.

Insights on Skyrme parameters from GW170817

The binary neutron-star merger event, GW170817, has cast a new light on nuclear physics research. Using a neutron-star model that includes a crust equation of state (EoS), we calculate the properties of a 1.4 solar-mass neutron star. The model incorporates more than 200 Skyrme energy density functionals, which describe nuclear matter properties, in the outer liquid core region of the neutron star. We find a power-law relation between the neutron-star tidal deformability, Λ, and the neutron-star radius, R. Without an explicit crust EoS, the model predicts a smaller R and the difference becomes significant for stars with large radii. To connect the neutron star properties with nuclear matter properties, we confront the predicted values for Λ against the Taylor expansion coefficients of the Skyrme interactions. There is no pronounced correlation between Skyrme parameters relevant to symmetric nuclear matter and neutron star properties. However, we find the strongest correlation between Λ and Ksym, the curvature of the density dependence of the symmetry energy at saturation density. At twice the saturation density, our calculations show a strong correlation between Λ and total pressure, providing guidance to laboratory nucleus-nucleus collision experiments.

Orbital-free density functional theory correctly models quantum dots when asymptotics, nonlocality, and nonhomogeneity are accounted for

Million-atom quantum simulations are in principle feasible with orbital-free density functional theory (OF-DFT) because the algorithms only require simple functional minimizations with respect to the electron density function. In this context, OF-DFT has been useful for simulations of warm dense matter, plasma, cold metals, and alloys. Unfortunately, systems as important as quantum dots and clusters (having highly inhomogeneous electron densities) still fall outside OF-DFT's range of applicability. In this Rapid Communication, we make considerable progress in addressing this century old problem by devising and implementing an accurate, transferable, and universal family of nonlocal noninteracting kinetic energy density functionals that feature correct asymptotics and can handle highly inhomogeneous electron densities. We show that OF-DFT achieves close to chemical accuracy for the electronic energy and reproduces the electron density to about 5% of the benchmark for semiconductor quantum dots and metal clusters. Therefore, this work shows that OF-DFT can reliably simulate systems with highly inhomogeneous electron density, such as clusters and quantum dots, with applicability to the rational design of materials.

From homogeneous matter to finite nuclei: Role of the effective mass

Recent astronomical observations, nuclear-reaction experiments, and microscopic calculations have placed new constraints on the nuclear equation of state (EoS) and revealed that most nuclear structure models fail to satisfy those constraints upon extrapolation to infinite matter. A reverse procedure for imposing EoS constraints on nuclear structure has been elusive. Here we present for the first time a method to generate a microscopic energy density functional (EDF) for nuclei from a given immutable EoS. The method takes advantage of a natural Ansatz for homogeneous nuclear matter, the Kohn-Sham framework, and the Skyrme formalism. We apply it to the realistic nuclear EoS of Akmal-Pandharipande-Ravenhall and describe successfully closed-(sub)shell nuclei. In the process, we provide predictions for the neutron skin thickness of nuclei based directly on the given EoS. Crucially, bulk and static nuclear properties are found practically independent of the assumed effective mass value - a unique result in bridging EDF of finite and homogeneous systems in general.

Iterative approaches to the self-consistent nuclear energy density functional problem

Large-scale applications of energy density functional (EDF) methods depend on fast and reliable algorithms to solve the associated non-linear self-consistency problem. When dealing with large single-particle variational spaces, existing solvers can become very slow, and their performance dependent on manual fine-tuning of numerical parameters. In addition, convergence can sensitively depend on particularities of the EDF's parametrisation under consideration. Using the widely-used Skyrme EDF as an example, we investigate the impact of the parametrisation of the EDF, both in terms of the operator structures present and the size of coupling constants, on the convergence of numerical solvers. We focus on two aspects of the self-consistency cycle, which are the diagonalisation of a fixed single-particle Hamiltonian on one hand and the evolution of the mean-field densities and potentials on the other. Throughout the article we use a coordinate-space representation, for which the behaviour of algorithms can be straightforwardly analysed. We propose two algorithmic improvements that are easily implementable in existing solvers, heavy-ball dynamics and potential preconditioning. We demonstrate that these methods can be made virtually parameter-free, requiring no manual fine-tuning to achieve near-optimal performance except for isolated cases. The combination of both methods decreases substantially the CPU time required to obtain converged results. The improvements are illustrated for the MOCCa code that solves the self-consistent HFB problem in a 3d coordinate space representation for parametrisations of the standard Skyrme EDF at next-to-leading order in gradients and its extension to next-to-next-to-leading order.

Using Pauli energy to appraise the quality of approximate semilocal non-interacting kinetic energy density functionals.

It is well-known that the kinetic energy density (KED) functional is the most difficult to approximate in density functional theory (DFT), yet to take full advantage of DFT with its density-based descriptive capability of molecular properties, an accurate account of KED is a must. To have a better idea of how an approximate KED formula behaves and where we should focus in the future development of better approximate KEDs, in this work we propose to employ the Pauli energy to assess their quality. We tested the performance of a total of 22 approximate semilocal noninteracting KED functionals from the literature for 18 neutral atoms and 20 small molecules. We found that generalized gradient approximation formulas of the KED functional can often reasonably accurately predict the total kinetic energy value for atoms and molecules but failed miserably to forecast the integrated values for Pauli energy related properties. The reason behind this is that presently available approximate KED functionals are unable to accurately account for the kinetic energy distribution in the medium range away from nuclei, where the Pauli energy plays a crucial role. Our results strongly suggest that the key information missing in approximate KED functionals comes from the medium regions, not nuclear cusps nor asymptotic areas, and the Pauli energy is a reliable measure of the quality of approximate KED functionals. Future efforts in developing better KED approximations should be invested in the regions of molecules where chemical bonds are formed in order to accurately account for the Pauli energy.

Impact of the surface energy coefficient on the deformation properties of atomic nuclei as predicted by Skyrme energy density functionals

In the framework of nuclear energy density functional (EDF) methods, many nuclear phenomena are related to the deformation of intrinsic states. Their accurate modeling relies on the correct description of the change of nuclear binding energy with deformation. The two most important contributions to the deformation energy have their origin in shell effects that are correlated to the spectrum of single-particle states, and the deformability of nuclear matter, that can be characterized by a model-dependent surface energy coefficient a_{surf}. With the goal of improving the global performance of nuclear EDFs through fine-tuning of their deformation properties, the purpose of this study is threefold. First, to analyze the impact of systematic variations of a_{surf} on properties of nuclei; second, to identify observables that can be safely used to narrow down the range of appropriate values of a_{surf} to be targeted in future parameter fits; third, to analyze the interdependence of a_{surf} with other properties of a nuclear EDF. Results for a large variety of relevant observables of deformed nuclei obtained from self-consistent mean-field calculations with a set of purpose-built SLy5sX parameterizations of the Skyrme EDF are correlated with the value of a_{surf}. The three main conclusions are that there is an evident preference for a comparatively low value of a_{surf}, as expected from the performance of existing parameterizations; that the isospin dependence of the surface energy also needs further finetuning in order to describe trends across the chart of nuclei; and that a satisfying simultaneous description of fission barriers and superdeformed states requires a better description of the single-particle spectra. [Note: The abstract has been abbreviated because of length restrictions imposed by the arXiv. See the paper for the full abstract.]

Nuclear structure and the nucleon effective mass: explorations with the versatile KIDS functional

The connection from the structure and dynamics of atomic nuclei (finite nuclear system) to the nuclear equation of state (thermodynamic limit) is primarily made through nuclear energy-density functional (EDF) theory. Failure to describe both entities simultaneously within existing EDF frameworks means that we have either seriously misjudged the scope of EDF or not fully taken advantage of it. Enter the versatile KIDS Ansatz, which is based on controlled, order-by-order extensions of the nuclear EDF with respect to the Fermi momentum and allows a direct mapping from a given, immutable equation of state to a convenient Skyrme pseudopotential for applications in finite nuclei. A recent proof-of-principle study of nuclear ground-states revealed the subversive role of the effective mass. Here we summarize the formalism and previous results and present further explorations related to giant resonances. As examples we consider the electric dipole polarizability of 68Ni and the giant monopole resonance (GMR) of heavy nuclei, particularly the fluffiness of 120Sn. We find that the choice of the effective mass parameters and that of the compression modulus affect the centroid energy of the GMR to comparable degrees.

KIDS Energy Density Functional and Mass–Radius Relation of Neutron Stars

Many efforts are made to determine the nuclear equation of state which governs the properties and evolution of neutron stars. Especially important is to constrain the parameters of the nuclear symmetry energy. In those efforts, nuclear energy density functional (EDF) theory has been a very useful tool, as it provides a unified framework for the description both of nuclei, which can be studied on Earth, and of infinite matter and its nuclear equation of state, which is a necessary input in the modelling of neutron stars. In the present study, a new nuclear EDF, the KIDS functional, is explored with a focus on the nuclear symmetry energy. The form of the functional allows us to vary at will the poorly constrained high-order derivatives of the symmetry energy and examine how the maximum possible mass of a neutron star is affected. Some tentative constraints on the skewness are presented, which will help guide further refinements. It is noteworthy that the pressure of neutron-rich matter is found strongly affected by skewness variations, both at low and high densities.