A consistent microscopic theory of collective motion in the framework of an ATDHF approach

Based on merely two assumptions, namely the existence of a collective Hamiltonian and that the collective motion evolves along Slater determinants, we first derive a set of adiabatic time-dependent Hartree-Fock equations (ATDHF) which determine the collective path, the mass and the potential, second give a unique procedure for quantizing the resulting classical collective Hamiltonian, and third explain how to use the collective wavefunctions, which are eigenstates of the quantized Hamiltonian.

Nuclear Energy Density Optimization: UNEDF2

The parameters of the UNEDF2 nuclear energy density functional (EDF) model were obtained in an optimization to experimental data consisting of nuclear binding energies, proton radii, odd-even mass staggering data, fission-isomer excitation energies, and single particle energies. In addition to parameter optimization, sensitivity analysis was done to obtain parameter uncertainties and correlations. The resulting UNEDF2 is an all-around EDF. However, the sensitivity analysis also demonstrated that the limits of current Skyrme-like EDFs have been reached and that novel approaches are called for.

Universal trend of charge radii of even-even Ca-Zn nuclei

Radii of nuclear charge distributions carry information about the strong and electromagnetic forces acting inside the atomic nucleus. While the global behavior of nuclear charge radii is governed by the bulk properties of nuclear matter, their local trends are affected by quantum motion of proton and neutron nuclear constituents. The measured differential charge radii $\delta\langle r^2_c\rangle$ between neutron numbers $N=28$ and $N=40$ exhibit a universal pattern as a function of $n=N-28$ that is independent of the atomic number. Here we analyze this remarkable behavior in even-even nuclei from calcium to zinc using two state-of-the-art theories based on quantified nuclear interactions: t…

Shell structure and the fluctuation of the nuclear density distribution

We investigate the relation between the density-fluctuations in nuclei and their description by single-particle models, i.e. the shell model and the Hartree-Fock method; the question is whether every shell-structure necessarily leads to those fluctuations. We demonstrate the flexibility of the single-particle models by constructing a shell-model potential and an effective Hartree-Fock potential, respectively, which produce completely flat distributions without any density fluctuation; this means that “shell structure” is not sufficient an explanation for the fluctuations. Only the additional requirement that the dynamical features of nuclei are also met selects a subclass of “reasonable” po…

Description of giant resonances with Skyrme forces

Some representative cases of the description and analysis of giant resonances within the self-consistent separable random-phase-approximation model (SRPA) based on the Skyrme functional are discussed. It is shown that SRPA with SLy6 force well describes the dipole giant resonance in rare-earth and actinide regions. The sensitivity of the E1 strength near the particle thresholds to nuclear deformation is scrutinized. Finally, the open problems of description of spin-flip M1 resonance are discussed.

Measurement and microscopic description of odd-even staggering of charge radii of exotic copper isotopes

Isotopes with an odd number of neutrons are usually slightly smaller in size than their even-neutron neighbours. In charge radii of short-lived copper isotopes, a reduction of this effect is observed when the neutron number approaches fifty. The mesoscopic nature of the atomic nucleus gives rise to a wide array of macroscopic and microscopic phenomena. The size of the nucleus is a window into this duality: while the charge radii globally scale as $A^{1/3}$, their evolution across isotopic chains reveals unanticipated structural phenomena [1-3]. The most ubiquitous of these is perhaps the Odd-Even Staggering (OES) [4]: isotopes with an odd number of neutrons are usually smaller in size than …

Ground state correlations and the nuclear charge distribution

Combining a sum rule approach and the generator coordinate model we evaluate the influence of ground state correlations on the moments of the radial charge distribution. The isotopic and isotonic differences of the moments come out to be particularly sensitive to fluctuations of the ground state correlations due to changes in the low energy spectrum. A comparison with the experimental results for the isotopes of Ca, Fe, Ni and Zn shows a fair agreement and confirms the importance of the ground state correlations for isotopic and isotonic differences of radial moments.

Nuclear energy density optimization: Shell structure

Nuclear density functional theory is the only microscopical theory that can be applied throughout the entire nuclear landscape. Its key ingredient is the energy density functional. In this work, we propose a new parameterization UNEDF2 of the Skyrme energy density functional. The functional optimization is carried out using the POUNDerS optimization algorithm within the framework of the Skyrme Hartree-Fock-Bogoliubov theory. Compared to the previous parameterization UNEDF1, restrictions on the tensor term of the energy density have been lifted, yielding a very general form of the energy density functional up to second order in derivatives of the one-body density matrix. In order to impose c…

Skyrme-force parametrization: Least-squares fit to nuclear ground-state properties.

We investigate systematically the possibilities and the limits of the Skyrme force for reproducing nuclear ground-state properties in a spherical Hartree-Fock calculation. This investigation is performed by means of least-squares fits of the force parameters to the measured binding energy, diffraction radius, and surface width of eight selected nuclei. Particular emphasis is put on the density dependence of the interaction, which turns out to be determined mainly by the surface width. The least-squares fitting procedure yields the best-fit parameters together with uncertainties on them, and it also allows one to estimate the uncertainties of an extrapolation to other fields, e.g., nuclear m…

RPA in wavefunction representation

The RPA is formulated in subspaces of coordinate-like and momentum-like I ph operators. This allows to embed a large class of approximative schemes into a generalized RPA treatment. We give a detailed formulation in terms of wavefunctions in coordinate space which is ideally suited to practical programming. In particular, we work out the reduction to spherical tensors in the case of spherical symmetry which is most often the starting point in finite Fermion systems.

Effect of three-body cluster on the healing properties of the Jastrow Correlation function

A variational equation for the Jastrow Correlation function is derived from the energy functional expanded up to three-body cluster terms. The asymptotic behaviour of this nonlinear equation is studied. The solutions show a healing at least of the type cos(tαr)/r2. The influence of higher cluster contributions is studied. Finally, it is discussed, how one can reduce the many-body cluster contributions to healing conditions to be used in the two-body cluster treatment.

Quasi-free electron scattering in a relativistic model of the nucleus

Longitudinal and transverse response functions of quasi-free electron scattering have been calculated in mean field approximation for a relativisticσ-ω model, including non-linearities in theσ-field. As a consequence of the completely consistent and fully relativistic treatment of wave functions and current operators, gauge invariance is perfectly satisfied throughout the calculation. The results indicate that models with the same nuclear bulk properties lead to similar response functions within 10%. Our results agree with the experimental data for12C and238U, but cannot describe the longitudinal response in the Ca-Fe region. Predictions for208Pb are given for comparison with future experim…

A two-center-oscillator-basis as an alternative set for heavy ion processes

The two-center-oscillator-basis, which is constructed from harmonic oscillator wave functions developing about two different centers, suffers from numerical problems at small center separations due to the overcompleteness of the set. In order to overcome these problems we admix higher oscillator wave functions before the orthogonalization, or antisymmetrization resp. This yields a numerically stable basis set at each center separation. The results obtained for the potential energy surface are comparable with the results of more elaborate models.

Time-dependent ground-state correlations in heavy ion scattering

Using a time-dependent generator-coordinate method, we derive a theory for time-dependent collective ground-state correlations which account for some quantum fluctuations about a TDHF trajectory. This theory is particularly suited for evaluating spreading widths of collective one-body operators. As an application we study head-on collision of heavy ions in a one-dimensional model. As one of the prominent results we find a substantial enhancement of the spreading width of the internal excitation energy due to the correlations.

Laser Spectroscopy of Neutron-Rich Tin Isotopes: A Discontinuity in Charge Radii across the N=82 Shell Closure

Physical review letters 122(19), 192502 (2019). doi:10.1103/PhysRevLett.122.192502

From Calcium to Cadmium: Testing the Pairing Functional through Charge Radii Measurements of Cd100−130

Differences in mean-square nuclear charge radii of $^{100--130}\mathrm{Cd}$ are extracted from high-resolution collinear laser spectroscopy of the $5s\text{ }{^{2}S}_{1/2}\ensuremath{\rightarrow}5p\text{ }{^{2}P}_{3/2}$ transition of the ion and from the $5s5p\text{ }{^{3}P}_{2}\ensuremath{\rightarrow}5s6s\text{ }{^{3}S}_{1}$ transition in atomic Cd. The radii show a smooth parabolic behavior on top of a linear trend and a regular odd-even staggering across the almost complete $sdgh$ shell. They serve as a first test for a recently established new Fayans functional and show a remarkably good agreement in the trend as well as in the total nuclear charge radius.

SKYRME-RANDOM-PHASE-APPROXIMATION DESCRIPTION OF SPIN-FLIP AND ORBITAL M1 GIANT RESONANCES

The self-consistent separable random-phase approximation (SRPA) with Skyrme forces is extended to the case of magnetic excitations and applied to the description of spin-flip and orbital M1 giant resonances in the isotopic chain 142-152 Nd . The Skyrme forces SkT6, SkM*, SLy6 and SkI3 are used. The calculations show an onset of the scissors mode with increasing deformation. A specific three-peak structure of the spin-flip response is found and explained by particular neutron and proton spin-flip transitions. Although the employed forces provide an acceptable qualitative description, the Skyrme functional still needs further improvement to reproduce quantitatively the experiment for spin mo…

Charge radii of exotic potassium isotopes challenge nuclear theory and the magic character of N = 32

Nuclear charge radii are sensitive probes of different aspects of the nucleon-nucleon interaction and the bulk properties of nuclear matter; thus, they provide a stringent test and challenge for nuclear theory. The calcium region has been of particular interest, as experimental evidence has suggested a new magic number at $N = 32$ [1-3], while the unexpectedly large increases in the charge radii [4,5] open new questions about the evolution of nuclear size in neutron-rich systems. By combining the collinear resonance ionization spectroscopy method with $\beta$-decay detection, we were able to extend the charge radii measurement of potassium ($Z =19$) isotopes up to the exotic $^{52}$K ($t_{1…

Collective mass parameters and linear response techniques in three-dimensional grids

We discuss four prescriptions for evaluating a collective mass parameter suitable for translations, rotations and large amplitude collective motions. These are the adiabatic time dependent Hartree-Fock theory (ATDHF) and the generator coordinate method (GCM), both with and without curvature corrections. As practical example we consider the16O+16O collision using a recently developed density dependent interaction with direct Yukawa and Coulomb terms. We present a fast iteration scheme for solving the linear response equation in a three-dimensional coordinate or momentum space grid. As test cases we consider the rotational and translational inertia parameters for various distances between the…

The relativistic mean-field model at large densities

The extrapolation properties of the relativistic mean-field model are investigated in nuclear matter. The results of a recent relativistic Bethe-Brueckner-Gatdstone calculation are taken as ‘data”. Fits in a window of normal nuclear densities are extrapolated to large densities and compared with the ‘data’. We find large discrepancies which hint at an insufficiency of the model in its present parametrisation.

The zero-point energy for rotation

The Gaussian overlap approach (GOA) becomes inappropriate for describing the rotation of weakly deformed systems. A modification is proposed which allows to maintain the GOA for small deformations. The zero-point energy subtraction, derived from it, provides a simple and reliable approximation for angular momentum projection. It becomes obvious, however, that the projection complicates the equations which determine the motion along the deformation path. These effects are studied in some simple models and the results are condensed into a simple interpolation formula for the total zero-point energy.

A sum-rule approach to nuclear ground state correlations

By combining the sum-rule approximation to nuclear giant resonances with a generator-coordinate description of the collective ground-state we obtain a simple estimate of the collective ground-state correlations. We investigate the approach for a variety of nuclei and forces. The correlation effects are small but not negligible in view of the precision achieved in modern Skyrme-Hartree-Fock calculations.

General treatment of vortical, toroidal, and compression modes

The multipole vortical, toroidal, and compression modes are analyzed. Following the vorticity concept of Ravenhall and Wambach, the vortical operator is derived and related in a simple way to the toroidal and compression operators. The strength functions and velocity fields of the modes are analyzed in $^{208}$Pb within the random-phase-approximation using the Skyrme force SLy6. Both convection and magnetization nuclear currents are taken into account. It is shown that the isoscalar (isovector) vortical and toroidal modes are dominated by the convection (magnetization) nuclear current while the compression mode is fully convective. The relation between the above concept of the vorticity to …

Spin-flip M1 giant resonance as a challenge for Skyrme forces

Despite a great success of the Skyrme mean-field approach in exploration of nuclear dynamics, it seems to fail in description of the spin-flip M1 giant resonance. The results for different Skyrme parameterizations are contradictory and poorly agree with experiment. In particular, there is no parameterization which simultaneously describes the one-peak gross structure of M1 strength in doubly magic nuclei and two-peak structure in heavy deformed nuclei. The reason of this mismatch could lie in an unsatisfactory treatment of spin correlations and spin-orbit interaction. We discuss the present status of the problem and possible ways of its solution. In particular, we inspect i) the interplay o…

Density as a constraint and the separation of internal excitation energy in TDHF

We present a fast and efficient constrained Hartree-Fock iteration scheme which constraints the complete density distribution to remain constant. The scheme is particularly suited to a coordinate- or momentum-space representation. The technique is applied to separate the collective and the internal energy in a propagating TDHF state. We study the behavior of these two energies in an16O+16O collision.

Error Estimates of Theoretical Models: a Guide

This guide offers suggestions/insights on uncertainty quantification of nuclear structure models. We discuss a simple approach to statistical error estimates, strategies to assess systematic errors, and show how to uncover inter-dependencies by correlation analysis. The basic concepts are illustrated through simple examples. By providing theoretical error bars on predicted quantities and using statistical methods to study correlations between observables, theory can significantly enhance the feedback between experiment and nuclear modeling.

Extrapolation of nuclear properties to the region near Z = 184

Mean field methods in large amplitude nuclear collective motion

The time dependent Hartree-Fock method (TDHF) is reviewed and its success and failure are discussed. It is demonstrated that TDHF is a semiclassical theory which is basically able to describe the time evolution of one-body operators, the energy loss in inclusive deep inelastic collisions, and fusion reactions above the Coulomb barrier. For genuine quantum mechanical processes as e.g. spontaneous fission, subbarrier fusion, phase shifts and the description of bound vibrations, the quantized adiabatic time dependent Hartree-Fock theory (quantized ATDHF) is suggested and reviewed. Realistic three-dimensional calculations for heavy ion systems of A1+A2<32 are presented. Applications to various …

The influence of a bag deformation on N-N-scattering

We present a model calculation to study the influence of an intrinsic nucleon deformation on the elastic scattering of nucleons. The nucleons are visualized as deformed sources for the exchange of bosons. Both static changes in the derived OBEP and the admixtures of rotational states give rise to only small effects even for large bag deformation, in addition, the two contributions tend to cancel each other.

A model study of Hartree-Fock and Linear Response in coordinate space

A fast procedure for spherical Hartree-Fock is obtained by coordinate space representation and a modification of gradient iteration. Along similar lines, the corresponding Linear Response equations are derived and solved, in order to achieve a fully consistent treatment. The Linear Response equations are applied to a change in particle numbers, i.e. to the description of isotopic differences. In a model study we look for their physical and numerical properties, i.e. linearity of the response, numerical stability and consistency requirements for the Hartree-Fock basis.

A time dependent RPA-theory for heavy ion reactions

The time dependent Hartree Fock theory (TDHF) is generalized by incorporating 2p-2h correlations into the TDHF Slater determinant in order to improve the description of two-body observables. To this end a time dependent RPA theory (TDRPA) is formulated using the quasi boson approximation. The approach turns out to be readily applicable requiring only minor changes in the present time TDHF codes. The theory is exemplified by considering the spreading width of the fragment particle number in a nucleus-nucleus collision. The TDRPA states are furthermore used to formulate a scattering theory for heavy ion collisions which incorporates the quantum corrections of orderh2 by means of a gaussian pa…

A comment on time-dependent variational-principles

Two time-dependent variational principles are compared; the one varies the action integral, the other minimises the deviation from the Schrodinger-equation. They are shown to be equivalent for a variation with complex parameters, but different for a restricted variation.

Nuclear ground-state properties in a relativistic Meson-Field theory

We investigate the ability of a relativistic Mean-Field theory to reproduce nuclear ground state properties by an exhaustive fit to experimental data. We find that the bulk properties of nuclei from16O to208Pb can be adjusted very well. There remain problems with level density and fluctuations in the charge density similar as in fits using the conventional Skyrme Hartree-Fock model.

Collective subspaces for large amplitude motion and the generator coordinate method

The collection path $|\ensuremath{\varphi}(q)〉$ to be used in a microscopic description of large amplitude collective motion is determined by means of the generator coordinate method. By varying the total energy with respect to $|\ensuremath{\varphi}(q)〉$ and performing an adiabatic expansion a hierarchy of equations is obtained which determines uniquely a hierarchy of collective paths with increasing complexity. To zeroth order the $|\ensuremath{\varphi}(q)〉$ are Slater determinants, to first order they include 2p-2h correlations. In both cases simple noninterative prescriptions for an explicit construction of the path are derived. For a correlated path their solutions agree at the Hartree…

Adiabatic Time-Dependent Hartree-Fock Calculations of the Optimal Path, the Potential, and the Mass Parameter for Large-Amplitude Collective Motion

The adiabatic time-dependent Hartree-Fock theory is reformulated in order to yield a simple differential equation for the collective path with accompanying simple expressions for the collective mass and the potential. With use of three-dimensional coordinate- and momentum-space techniques and density-dependent interactions, the new adiabatic time-dependent Hartree-Fock formalism is applied to $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ scattering and correspondingly to the fission mode of $^{8}\mathrm{Be}$. In the overlapping region the resulting collective mass deviates strongly from the reduced mass.

Time Dependent Hartree-Fock and Beyond

On the variational approach to Jastrow correlations in nuclei

The variational equation determining the Jastrow correlation function is investigated with particular emphasis on the healing problem for both nuclear matter and finite nuclei. The consequences of several healing conditions are discussed. Furthermore, influences from the choice of the single particle basis and from long range correlations are studied and are found to be small in the short range region.