Self-consistent approaches in the microscopic theory of the nucleus: Static moments of odd-odd nuclei

Self-consistent mean-field theory and the method of the energy density functional, which are two modern self-consistent approaches in the microscopic theory of the nucleus that possess the highest predictive power for describing unstable nuclei, are briefly discussed. Themost recent results of calculations performed within these approaches are presented. The mean energies of E1 excitations in the range of 0–30 MeV are calculated for 15 stable and unstable tin isotopes (A = 100–176) on the basis of the self-consistent version of the generalized theory of finite Fermi systems by employing SLy4 Skyrme forces. A parameter-dependent expression that takes into account the existence of a pygmy dipole resonance is obtained for this quantity. The density-functional method is used within the self-consistent theory of finite Fermi systems on the basis of the Fayans-Tolokonnikov-Trykov-Zawischa functional in order to calculate the ground-state static quadrupole and magnetic moments of odd and odd-odd stable and unstable spherical near-magic nuclei. Good agreement with available experimental data is attained. The respective features are predicted for unstable nuclei.     

Pygmy dipole resonance in nuclei

A brief survey of the state of the modern microscopic theory of the so-called pygmy dipole resonance in nuclei is given—in particular, some unresolved problems are listed. It is emphasized that, in order to explain the pygmy dipole resonance, it is necessary but not sufficient to take into account the coupling of single-particle degrees of freedom to photon degrees of freedom. The results of the calculations performed for the first time for the isovector pygmy dipole resonance and the isovector electric giant dipole resonance in ¹²⁴Sn within a self-consistent approach involving, in addition to the standard quasiparticle random-phase approximation, a single-particle continuum and quasiparticle-phonon coupling of single-particle degrees of freedom to phonon degrees of freedom are presented. The results are found to be in satisfactory agreement with experimental data. The calculation of the isoscalar strength function in the energy region of the pygmy dipole resonance revealed that the nuclear-structure mechanism does not provide the isoscalar-strength suppression observed at energies in excess of 7 MeV in (α, α′γ) reactions; therefore, this suppression may stem from the reaction mechanism.     

Radiative strength function and the pygmy dipole resonance in <Superscript>208</Superscript>Pb and <Superscript>70</Superscript>Ni

The pygmy-resonance parameters and the E1 strength function are derived for ²⁰⁸Pb using a fully self-consistent microscopic formalism recently developed for magic nuclei, which takes into account quasiparticle phonon interactions (or coupling to phonons) in addition to the random phase approximation. For the radiative strength function of ²⁰⁸Pb at energies above 5 MeV, the experimental data of the Oslo group are adequately described by our predictions, whereby the important role of coupling to phonons is confirmed. By comparing the measurements based on the (³He, ³He′γ) and (γ, γ′) reactions, we discuss the physical properties of the radiative strength function measured for ²⁰⁸Pb. For the neutron-rich ⁷⁰Ni nucleus, predictions for the radiative strength function and the pygmy resonance are obtained using a partially self-consistent approach, which invokes the Skyrme forces in deriving the mean field, effective nucleon–nucleon interaction, and phonon characteristics.     

Impact of the phonon coupling on the radiative neutron capture

Inclusion of the coupling of quasiparticle degrees of freedom with phonon degrees is a natural extention of the standard QRPA approach. The paper presents the quantitative impact of this phonon coupling on the dipole strength and radiative neutron capture for the stable ¹²⁴Sn and very exotic ¹⁵⁰Sn isotopes, as an illustration, using the self-consistent version of the Extended Theory of Finite Fermi Systems. It was found that the phonon contribution to the pygmy-dipole resonance and radiative neutron capture cross section is increased with the (N − Z) difference growth. The results show that the self-consistent nuclear structure calculations are important for unstable nuclei, where phenomenological approaches do not work.     

Charge-exchange pigmy resonances of tin isotopes

Charge-exchange states, the so-called “pigmy” resonances, which are below the giant Gamow–Teller resonance, have been studied in the self-consistent theory of finite Fermi systems. Microscopic numerical calculations and semiclassical calculations are presented for nine tin isotopes with the mass numbers A =112, 114, 116, 117, 118, 119, 120, 122, and 124, for which experimental data exist. These data have been obtained in the Sn(³He,t)Sb charge-exchange reaction at the energy E(³He) = 200 MeV. The comparison of calculations with experimental data on the energies of charge-exchange resonances gives the standard deviation δE < 0.40 MeV for microscopic numerical calculations and δE < 0.55 MeV for calculations by semiclassical formulas, which are comparable with experimental errors. The strength function for the ¹¹⁸Sn isotope has been calculated. It has been shown that the calculated resonance energies are close to the experimental values; the calculated and experimental relations between heights of pygmy resonance peaks are also close to each other.     

Photonuclear data and modern physics of giant resonances

The results of the development (“renaissance”) of giant-resonance physics are briefly discussed from the point of view of their application to creating a photonuclear database. It is indicated that part of the recommendations from corresponding libraries of data are not at the level of the present-day status of giant-resonance physics. A Lorentzian parametrization of the most reliable experimental data on isovector M1 resonances is constructed for seven spherical nuclei, and it is shown that the widths of M1 resonances are severalfold, sometimes an order of magnitude, smaller than the value of Γ₀ = 4 MeV, which was recommended for all nuclei. The need for microscopically taking into account configurations more complex than those that are included within the standard random-phase approximation or within the quasiparticle random-phase approximation is emphasized. To be more precise, it is necessary to take into account coupling to phonons, since this changes the temperature dependence of the resonance width in relation to that which was used earlier and since, without this, one cannot explain the properties of pygmy dipole resonances in the region of the nucleon binding energy. Our calculations of the average energies of the pygmy dipole resonances in the Ca and Sn isotopes within the microscopic extended theory of finite Fermi systems reveal that the inclusion of coupling to phonons reduces these energies considerably toward the improvement of agreement with experimental data. The idea of creating a library of photonuclear data for unstable nuclei, including fission fragments, on the basis of the extended theory of finite Fermi systems is discussed in connection with the fact that information necessary for fitting the parameters of phenomenological theories is absent or insufficient for such nuclei.     

On microscopic theory of radiative nuclear reaction characteristics

A survey of some results in the modern microscopic theory of properties of nuclear reactions with gamma rays is given. First of all, we discuss the impact of Phonon Coupling (PC) on the Photon Strength Function (PSF) because it represents the most natural physical source of additional strength found for Sn isotopes in recent experiments that could not be explained within the standard HFB + QRPA approach. The self-consistent version of the Extended Theory of Finite Fermi Systems in the Quasiparticle Time Blocking Approximation is applied. It uses the HFB mean field and includes both the QRPA and PC effects on the basis of the SLy4 Skyrme force. With our microscopic E1 PSFs, the following properties have been calculated for many stable and unstable even–even semi-magic Sn and Ni isotopes as well as for double-magic ¹³²Sn and ²⁰⁸Pb using the reaction codes EMPIRE and TALYS with several Nuclear Level Density (NLD) models: (1) the neutron capture cross sections; (2) the corresponding neutron capture gamma spectra; (3) the average radiative widths of neutron resonances. In all the properties considered, the PC contribution turned out to be significant, as compared with the standard QRPA one, and necessary to explain the available experimental data. The results with the phenomenological so-called generalized superfluid NLD model turned out to be worse, on the whole, than those obtained with the microscopic HFB + combinatorial NLD model. The very topical question about the M1 resonance contribution to PSFs is also discussed.     

A possible proton pygmy resonance in 17Ne

The low-lying electric dipole strengths in proton-rich nuclei ¹⁷F and ¹⁷Ne, which can be produced at HIRFL-CSR in Lanzhou, are investigated. In the framework of the covariant density functional theory the self-consistent relativistic Hartree Bogoliubov model and the relativistic quasiparticle random phase approximation with the NL3 parameter set and Gogny pairing interaction are adopted in the calculations. A pronounced dipole peak appears below 10 MeV in ¹⁷Ne, but does not occur in ¹⁷F. The properties of this low-lying E1 excitation in ¹⁷Ne are studied, which may correspond to a proton pygmy resonance with different characteristics from those of giant dipole resonance.     

Virtual excitation of the GDR mode in the subbarrier23Na(p, γ)24Mg reaction

Differential cross sections for nonresonant radiative capture of low energy protons (E _p = 1,348 keV and 1,370 keV) by²³Na nuclei exhibit features pointing to the virtual excitation of the giant dipole resonance (GDR) mode. Theoretical analysis carried out within the framework of the direct — semidirect capture model reveals an enhanced coupling of the GDR with the incident protonf-wave consistent with the microscopic structure of the GDR in thes-d shell nuclei.     

Fine Structure and Collectivity of the Levels of the Pygmy Dipole Resonance in <Superscript>208</Superscript>Pb in a Self-Consistent Model

A novel fully self-consistent microscopic approach based on the energy density functional method is employed to calculate the fine structure of the pygmy dipole resonance in ²⁰⁸Pb, i.e., the energies and reduced probabilities of E1 transitions for the states with energies below 10 MeV. The approach includes the random-phase approximation, quasiparticle–phonon interaction and the single-particle continuum. The theoretical results are compared to the available high-resolution data and found to agree with measured integral characteristics of the pygmy dipole resonance at energies above 5.7 MeV. Residual spin–spin forces are quantified, and their contribution is found to be significant at both low and high energies. A recently proposed criterion is employed to analyze the collectivity of the 1^–states in ²⁰⁸Pb.     

The pygmy dipole resonance in Ni and the neutron skin

A search of the pygmy resonance in ⁶⁸Ni was made using the virtual photon technique. The experiment was carried out using the radioactive beam ⁶⁸Ni at 600 A MeV, produced with fragmentation of ⁸⁶Kr at 900 A MeV on a ⁹Be target. The ⁶⁸Ni beam was separated by a fragment separator, and the γ-rays produced at the interaction with the Au target were detected with the RISING and FRS set-up at the GSI laboratory in Germany, also including the HECTOR array. The measured γ-ray spectra show a peak centered at approximately 11 MeV, whose intensity can be explained in term of an enhanced strength of the dipole response function (pygmy resonance). A pygmy structure of this type was also predicted by different models for this unstable neutron-rich nucleus. Correlations between the behavior of the nuclear symmetry energy, the neutron skins, and the percentage of energy-weighted sum rule (EWSR) exhausted by the pygmy dipole resonance (PDR) are investigated by using different random phase approximation (RPA) models.     

Effects of 2 particle–2 hole configurations on dipole states in neutron-rich <Emphasis Type="Italic">N</Emphasis> = 80–84 isotones

Starting from the Skyrme interaction SLy4 we study the effects of 2 particle–2 hole configurations on the low-energy electric dipole response in ^130–134Sn. It is shown that the pygmy dipole resonance properties are correlated with the neutron skin thickness. The two-phonon configurations give a considerable contribution to the low-lying E1 strength.     

Low-energy monopole and dipole response in nuclei at finite temperature

The multipole response of nuclei at temperatures T=0–2 MeV$T=0\text{–}2\text{MeV}$ is studied using a self-consistent finite-temperature RPA (random phase approximation) based on relativistic energy density functionals. Illustrative calculations are performed for the isoscalar monopole and isovector dipole modes and, in particular, the evolution of low-energy excitations with temperature is analyzed, including the modification of pygmy structures. Both for the monopole and dipole modes, in the temperature range T=1–2 MeV$T=1\text{–}2\text{MeV}$ additional transition strength appears at low energies due to thermal unblocking of single-particle orbitals close to the Fermi level. A concentration of dipole strength around 10 MeV excitation energy is predicted in ^60,62Ni. The principal effect of finite temperature on low-energy strength that is already present at zero temperature, e.g. in ⁶⁸Ni and ¹³²Sn, is the spreading of this structure to even lower energy and the appearance of states that correspond to thermally unblocked transitions.     

Unified theory of particle-phonon coupling: Application to Sb and In isotopes using a realistic interaction

A self-consistent, unified, microscopic theory ofparticle-phonon coupling is applied to the Sb and In isotopes, which are treated as proton-particle and proton-hole states respectively coupled to the ground and low-lying vibrational states of Sn. The particle-phonon coupling interaction is derived from the same realistic two-body interaction which gives rise to the vibrational excitations in Sn. Spectroscopic factors, level schemes and B(E2) values calculated with no adjustable parameters are shown to be in good agreement with experimental data.     

Mean field and beyond in nuclei far from stability lines

We calculate, for the first time, the state-dependent pairing gap of a finite nucleus (¹²⁰Sn) diagonalizing the bare nucleon-nucleon potential (Argonne v₁₄) in a Hartree-Fock basis. The resulting gap accounts for about half of the experimental gap. Going beyond the mean field in the particle-particle channel, the combined effect of the bare nucleon-nucleon potential and of the induced pairing interaction arising from the exchange of low-lying surface vibrations between nucleons moving in time reversal states close to the Fermi energy accounts for the experimental gap. Examples for light, halo nuclei are also reported. The more studied effects of the particle-vibration coupling in the particle-hole channel are discussed for the low-lying quadrupole vibration in ¹²⁰Sn and the giant dipole resonance in the unstable oxygen isotopes and ¹³²Sn.     

Isoscalar dipole coherence at low energies and forbidden E1 strength

In ¹⁶O and ⁴⁰Ca an isoscalar, low-energy dipole transition (IS-LED) exhausting approximately 4% of the isoscalar dipole (ISD) energy-weighted sum rule is experimentally known, but conspicuously absent from recent theoretical investigations of ISD strength. The IS-LED mode coincides with the so-called isospin-forbidden E1 transition. We report that for N = Z nuclei up to ¹⁰⁰Sn the fully self-consistent Random-Phase Approximation (RPA) with finite-range forces, phenomenological and realistic, yields a collective IS-LED mode, typically overestimating its excitation energy, but correctly describing its IS strength and electroexcitation form factor. The presence of E1 strength is solely due to the Coulomb interaction between the protons and the resulting isospin-symmetry breaking. The smallness of its value is related to the form of the transition density, due to translational invariance. The calculated values of E1 and ISD strength carried by the IS-LED depend on the effective interaction used. Attention is drawn to the possibility that in N 1 \neq Z nuclei this distinct mode of IS surface vibration can develop as such or mix strongly with skin modes and thus influence the pygmy dipole strength as well as the ISD strength function. In general, theoretical models currently in use may be unfit to predict its precise position and strength, if at all its existence.     

Description of cross sections for photonuclear reactions in the energy range between 7 and 140 MeV

A combination of the exciton and evaporation models is used to describe photonuclear reactions induced in light, medium-mass, and heavy nuclei by photons of energy in the range 7 ≤ E _γ ≤ 140 MeV. Two mechanisms of the photoexcitation of nuclei are considered. These are the formation of a giant dipole resonance at energies in the range E _γ ? 30 MeV and quasideuteron photoabsorption, which is dominant at energies in the region E _γ ? 40 MeV. The density of particle-hole states, which appears in the exciton model, is calculated on the basis of the Fermi gas model. The emission of two preequilibrium particles is taken into account in describing the quasideuteron reaction channel. The effect of isospin conservation on giant-dipole-resonance decay accompanied by photonucleon emission is examined. The model in question is used to describe cross sections for photon-induced reactions on ²⁶Mg, ⁵⁴Fe, ^{112,118,119,124}Sn, and ¹⁸¹Ta nuclei.     

Preequilibrium model of photonucleon reactions that is based on Fermi gas densities

A combination of the exciton and evaporation models is used to describe photonucleon reactions induced in heavy and medium-mass nuclei by photons of energy in the range 7 ≤ E _γ ≤ 140 MeV. The formation of a giant dipole resonance and quasideuteron absorption are considered as two mechanisms that are responsible for the photoexcitation of a nucleus in the energy regions E _γ ? 20 MeV and E _γ ? 40 MeV, respectively. As is well known, the densities of particle-hole states are employed in the exciton model, and these quantities are calculated on the basis of the Fermi gas model. This makes it possible to take into account the effect of the energy dependence of single-particle and single-hole densities of states on the rate of emission and intranuclear processes. The model in question is applied to describing partial photonucleon cross sections for ¹¹⁹Sn, ¹⁴⁰Ce, ¹⁸¹Ta, and ²⁰⁸Pb nuclei.