Microscopically-constrained Fock energy density functionals from chiral effective field theory. I. Two-nucleon interactions

The density matrix expansion (DME) of Negele and Vautherin is a convenient tool to map finite-range physics associated with vacuum two- and three-nucleon interactions into the form of a Skyrme-like energy density functional (EDF) with density-dependent couplings. In this work, we apply the improved formulation of the DME proposed recently in arXiv:0910.4979 by Gebremariam et al. to the non-local Fock energy obtained from chiral effective field theory (EFT) two-nucleon (NN) interactions at next-to-next-to-leading-order (N²LO). The structure of the chiral interactions is such that each coupling in the DME Fock functional can be decomposed into a coupling constant arising from zero-range contact interactions and a coupling function of the density arising from the universal long-range pion exchanges. This motivates a new microscopically-guided Skyrme phenomenology where the density-dependent couplings associated with the underlying pion-exchange interactions are added to standard empirical Skyrme functionals, and the density-independent Skyrme parameters subsequently refit to data. A link to a downloadable Mathematica notebook containing the novel density-dependent couplings is provided.     

On the Coulomb displacement energy

The Coulomb displacement energies of the $\text{T =}\text{1}\text{2}$ mirror nuclei (A = 15, 17, 27, 29, 31, 33, 39 and 41) are re-examined with the best available HF wave functions (the DME and the Skyrme II interaction), with the inclusion of all electromagnetic corrections. The results are compared with the experimental s.p. charge dependent energies extracted from the experimental data taking into account admixtures of core-excitation corrections with the help of present shell-model and co-existence model calculations. Although the so-called Nolen-Schiffer anomaly is not removed by these improvements, it is found that the remaining observed anomalies in the ground states of s.p. and s.h. systems can be resolved with the introduction of a simple, phenomenological charge symmetry breaking nucleon-nucleon force. This force can also account for the observed anomalies in the higher excited s.p. states, while those of the deeper s.h. states need further explanation.     

The quadrupole vibrational inertial function in the adiabatic time-dependent Hartree-Fock-Bogolyubov approximation

The adiabatic time-dependent Hartree-Fock-Bogolyubov (ATDHFB) approximation for the determination of the collective inertial functions is discussed and analysed in detail in the example of the quadrupole vibrational inertial function B_ββ. The collective path is constructed as a sequence of paired states of particles occupying the levels of the deformed Woods-Saxon potential. The Skyrme force is used in the particle-hole channel and the pairing force in the particle-particle interaction channel. The numerical calculations are performed for the axial shapes of samarium and barium nuclei. For the Skyrme force SIV the calculated B_ββ inertial function values and the cranking ones differ the most, the former values being about 30% larger than the latter.     

Time-dependent Hartree-Fock study of 136Xe + 209Bi collisions

We have performed axially symmetric Skyrme force time-dependent Hartree-Fock calculations for the ¹³⁶Xe + ²⁰⁹Bi system at E(lab) = 940 MeV and 1130 MeV. The evolution of the nuclear geometry is studied in detail. The calculated final fragment kinetic energies, scattering angles, and charges are in good agreement with experiment, but the fragment charge dispersion is greatly underestimated. Studies of head-on collisions for 900 MeV ? E(lab) ? 2000 MeV show no evidence for fusion and indicate a dramatic transparency for E(lab) ? 1500 MeV.     

The Skyrme-TQRPA calculations of electron capture on hot nuclei in pre-supernova environment

We combine the thermal QRPA approach with the Skyrme energy density functional theory (Skyrme–TQRPA) for modelling the process of electron capture on nuclei in supernova environment. For a sample nucleus, ⁵⁶Fe, the Skyrme–TQRPA approach is applied to analyze thermal effects on the strength function of GT₊ transitions which dominate electron capture at E _e ≤ 30 MeV. Several Skyrme interactions are used in order to verify the sensitivity of the obtained results to the Skyrme force parameters. Finite-temperature cross sections are calculated and the results are comparedwith those of the other model calculations.     

Deformation of nuclei close to the two-neutron drip line in Mg region

We present Hartree-Fock-Bogoliubov (HFB) calculations of the ground states of even Mg isotopes. A Skyrme force is used in the mean-field channel and a density-dependent zero-range force in the pairing channel. Our study shows that the ground states of^36,38,40Mg are strongly deformed with significantly different deformations for the neutrons and protons. Our study supports the disappearance of theN=28 shell gap in the Mg isotopes.     

Nuclear structure investigation of neutron-rich Mn isotopes

Structural evolution of odd-even and odd-odd Mn isotopes from the valley of stability up to neutron dripline is studied in the framework of the self-consistent mean-field theory of Hartree–Fock–Bogoliubov. Three Skyrme effective interactions, namely, SLy4, SLy5 and SLy5T, are employed to investigate the tensor force effect on the ground-state properties. It is shown that the calculated quantities with the SLy4 interaction correctly reproduce the available experimental data and agree well with finite range droplet model and relativistic mean-field predictions. An inconsistency has been observed between the curves of the separation energies and that of the charge radius around N = 40. It is explained by the flatness of the potential energy curves in this region. The SLy5 and SLy5T results point to the necessity to refit all the Skyrme parameters after including the tensor terms.     

Self-consistent calculation of the fission barrier of 240Pu

Completely self-consistent calculations using the Skyrme force have been carried out for the fission energy curve of ²⁴⁰Pu. We use a deformed oscillator basis including 13 major shells and convergence has been checked by extending the size of the basis to 15 shells. We obtain a double humped barrier with energies E_A = 9 MeV for the first barrier, E_B = 13 MeV for the second barrier and E_II = 4 MeV for the isomeric state. Corrections to our calculation, such as inclusion of non-axial and symmetric shapes and zero-point rotational motion, are likely to improve quantitative agreement with experimental data.     

Tensor force effect on single-proton energy levels of oxygen isotopes

In the framework of the Hartree-Fock approach the proton spin-orbital splittings of the 1p orbits and the shell gaps in the oxygen isotopes are investigated with the interactions SLy5+T,SLy5+Tw,SGII+Te1,SGII+Te2,SGII+Te3 and many sets of the TIJ interactions.All of the interactions are the Skyrme interactions and contain a tensor component(tensor force).It is shown that the evolution of the single-proton levels for the oxygen isotopes is sensitive to a parameterβTwhich is associated with the tensor force strength of the Skyrme interactions.To understand this phenomenon,we systematically analyze the dependence of the spin-orbit splittings and shell gaps on the parameterβTin terms of the spin-orbit potential and the corresponding wave function.We find that the Skyrme interactions can be classified into two groups:(a)T21,T32,T43,T54,SLy5+T,SLy5+Tw,SGII+Te1 and SGII+Te2,which can roughly reproduce the experimental shell gaps of the oxygen isotopes;(b)T1J and SGII+Te3,which can not reproduce the experimental shell gaps.     

Global dynamical correlation energies in covariant density functional theory: Cranking approximation

The global dynamical correlation energies for 575 even-even nuclei with proton numbers ranging from Z = 8 to Z = 108 calculated with the covariant density functional theory using the PC-PK1 parametrization are presented. The dynamical correlation energies include the rotational correction energies obtained with the cranking approximation and the quadrupole vibrational correction energies. The systematic behavior of the present correlation energies is in good agreement with that obtained from the projected generator coordinate method using the SLy4 Skyrme force although our values are systematically smaller. After including the dynamical correlation energies, the root-mean-square deviation predicted by the PC-PK1 for the 575 even-even nuclei masses is reduced from 2.58 MeV to 1.24 MeV.     

Deformation of nuclei close to the two-neutron drip line in the Mg region

We present Hartree-Fock-Bogoliubov (HFB) calculations of the ground states of even Mg isotopes. A Skyrme force is used in the mean-field channel and a density-dependent zero-range force in the pairing channel. ⁴⁰Mg and ²⁰Mg are predicted to be at the two-neutron and two-proton drip lines respectively. A detailed study of the quadrupole deformation properties of all the isotopes shows that the ground states of ^36,38,40Mg are strongly deformed with significantly different deformations for the neutrons and protons. Our study supports the disappearance of the N = 28 shell gap in the Mg and Si isotopes.     

Collective gyromagnetic ratio and moment of inertia from density-dependent Hartree-Fock calculations

The collective gyromagnetic ratio and moment of inertia of deformed even-even axially symmetric nuclei are calculated in the cranking approximation using wave functions obtained with the Skyrme force S-III. Good agreement is found for g_R, while the moment of inertia is about 20 % too small. The cranking formula leads to better agreement than the projection method.     

Abnormal odd-even staggering behavior around 132Sn studied by density functional theory

In this work, we have performed Skyrme density functional theory (DFT) calculations of nuclei around ¹³²Sn to study whether the abnormal odd-even staggering (OES) behavior of binding energies around N = 82 can be reproduced. With the Skyrme forces SLy4 and SkM*, we tested the volume- and surface-type pairing forces and also the intermediate between these two pairing forces, in the Hartree-Fock-Bogoliubov (HFB) approximation with or without the Lipkin-Nogami (LN) approximation or particle number projection after the convergence of HFBLN (PLN). The Universal Nuclear Energy Density Function (UNEDF) parameter sets are also used. The trend of the neutron OES against the neutron number or proton number does not change significantly by tuning the density dependence of the pairing force. Moreover, for the pairing force that is favored more at the nuclear surface, a larger mass OES is obtained, and vice versa. It appears that the combination of volume and surface pairing can give better agreement with the data. In the studies of the OES, a larger ratio of surface to volume pairing might be favored. Additionally, in most cases, the OES given by the HFBLN approximation agrees more closely with the experimental data. We found that both the Skyrme and pairing forces can influence the OES behavior. The mass OES calculated by the UNEDF DFT is explicitly smaller than the experimental one. The UNEDF1 and UNEDF2 forces can reproduce the experimental trend of the abnormal OES around ¹³²Sn. The neutron OES of the tin isotopes given by the SkM* force agrees more closely with the experimental one than that given by the SLy4 force in most cases. Both SLy4 and SkM* DFT have difficulties in reproducing the abnormal OES around ¹³²Sn. Using the PLN method, the systematics of OES are improved for several combinations of Skyrme and pairing forces.     

Systematic study of the heavy-ion fusion barrier in the frozen approximation

The height and position of the fusion barrier for the real part of the interaction potential between two colliding nuclei are studied as a function of the Coulomb strength. The calculations are performed in the framework of the spherical constrained HF + BCS formalism using the frozen approximation. The effective Skyrme type force SKa is used. The model contains no free parameters. For the height of the fusion barrier good agreement is found with the values obtained using the empirical Bass potential that is known to reproduce well the data for many nuclear systems. In contrast, the recently observed apparent rise of the fusion barrier (‘extra-extra-push’) for very massive nuclear systems with a product of nuclear chargesZ ₁ xZ ₂ above 1600 cannot be reproduced. We also give estimations for the overlap of the mass densities of both colliding nuclei at the fusion barrier distance.     

Developing a force field for simulation of poly(ethylene oxide) based upon ab initio calculations of 1,2-dimethoxyethane

The relative conformational energies in the 1,2-dimethoxyethane (DME) molecule have been extensively studied using B3LYP and MP2 ab initio methods, employing a range of commonly used basis sets. These conformational energies have been used to fit new O–C–C–O and C–O–C–C torsional interaction parameters for the OPLS-AA force field. The resulting force field (DMEFF) shows some improvement in conformational populations, calculated from molecular dynamics simulation of bulk DME, compared to two other commonly used force fields. Extensive reverse-engineering of the OPLS-AA energy function has also allowed the development of additional sets of torsion parameters for these two dihedral types, resulting in a force field that reproduces the conformational behaviour of DME in the liquid phase extremely well.     

On the validity of pre-equilibrium model assumptions

The Hartree-Fock calculation is performed for nuclear matter using the Skyrme interaction. It is shown that a stable density wave periodic in only one direction exists at densities below about one-third of the normal nuclear density. The critical densityρ _c below which the energy of the density wave is lower than that of the Fermi gas is determined for Skyrme interactions SKI to SKVI. It is further shown that even at densities slightly higher thanρ _c the density wave still exists as ametastable state in the sense that its energy is a local minimum in the variation parameter space. The density wave solution suddenly disappears at a higher density, since there the local minimum changes to an inflection point.     

On the use of Skyrme forces in self-consistent RPA calculations

Self-consistent random-phase (RPA) calculations including the continuum are presented using Skyrme forces. The density-dependent interpretation of the interaction is favoured as it does not violate the spin stability. A possible density dependence of the momentum-dependent S- and P-interaction is taken into account, which allows one to vary the incompressibility K and the effective mass $\text{m}{}^1\text{/m}$ independently. It is shown by analytic relations that these two quantities are the only degrees of freedom left in the parameterization of this Skyrme force,if the ground-state properties shall be reproduced, except for a still open degree of freedom in the spin exchange parameterization. The Landau parameters are discussed as a function of these degrees of freedom in order to find the best possible particle-hole interaction. Continuum calculations of the 1^?, 2 ⁺ and 3^? states in ¹⁶O are presented and compared with-discretized continuum calculations. It is found that the existing Skyrme forces do not show enough attraction and in addition cause relatively large isospin impurities, in ¹⁶O as well as in ²⁰⁸Pb. The influence of large configuration spaces is discussed. A systematic search for an interaction with a stronger particle-hole interaction is presented which seems to favour interactions with a high effective mass, but a low compression modulus.     

Stabilization of skyrmions via ϱ-mesons

It is shown that a skyrion can be stabilized by introducing ?-mesons into the chiral SU(2)_L × SU(2)_R lagrangian without higher-derivative terms like the Skyrme term. The ?-mesons are considered as dynamical gauge bosons associated with a hidden local symmetry of the non-linear sigma model. The lagrangian reduces to the Skyrme model in a limit of parameters. The Skyrmion mass M is found to be M = 1058 MeV when the parameters are fixed so as to satisfy the KSRF relation. It is also shown that a solition solution in a model with Ω-meson coupled with the baryonic current is a saddle-point solution.     

Skyrme force and the mass formula

Coefficients of the mass formula related to the property of the nuclear surface are calculated from the Skyrme forces SII, SIII and SV defined by the Orsay group. Comparing with Myers' mass formula parameters, it is found that the effective stiffness Q for separation of the neutron and proton surfaces is too large by a factor of about two for SIII. The discrepancy can also be stated by saying that the surface symmetry energy calculated from SIII is too small, and related to this is a too thin neutron skin for N > Z nuclei. The SV force gives a much too small value for Q. The SII force, with a repulsive term somewhere intermediate between SIII and SV gives a better value for Q. The strong density dependence of SIII is needed, however, to give good single-particle spectra (or the effective mass large enough). The SIII force has a density dependence between unlike nucleons only (between neutrons and protons). In this paper we modify the Skyrme force so as to allow for a density dependence also between like nucleons. A many-body theoretical justification for this modification is given in the paper. The mass formula parameter Q is reproduced with our new force maintaining a sufficiently strong density dependence. The neutron skin thickness is increased and the ⁴⁰Ca-⁴⁸Ca isotope shift of the charge radius is also well reproduced with this new force. A change of the density dependence from a linear to a one-third power dependence improves the agreement with the compressibility, and also affects the neutron-proton distributions in the surface.     

Effective Interactions in the Delta-Shells Potential

We determine two-body Skyrme force parameters from a nucleon–nucleon interaction as a function of the maximal momentum fitting NN scattering data. We find general agreement with V _{low k} interactions based on high quality potentials.