Following a previous paper [Y. Shi, Phys. Rev. C 98, 014329(2018)], we present an extension of the density-functional theory to allow for dynamic calculations based on the obtained static Hartree-Fock results. We perform extensive benchmark calculations, by comparing the calculated results with that of an existing code Sky3D. To perform linear-response calculations using the TDDFT method, comparisons have been made with the finite-amplitude quasiparticle random-phase approximation (FAM-QRPA) method. We plan to apply the TDDFT method to a systematic description of the IVD resonances in the Zr, Mo, and Ru isotopes.
The strengths of IVD resonances are calculated using two complementary methods: TDDFT and FAM-QRPA methods. For the TDDFT results, additional benchmark calculations have been performed using the well-tested code Sky3D. In these three models, the important ingredients which have major influence on the results, such as time-odd potentials, boundary conditions, smoothing procedures, spurious peaks etc., have been carefully examined.
The current TDDFT and the Sky3D codes yield almost identical response functions once both codes use the same time-odd mean fields and absorbing boundary conditions. The strengths of the IVD resonances calculated using the TDDFT and FAM-QRPA methods agree reasonably well with the same position of the giant dipole resonance. Upon seeing a reasonable accuracy offered by the implemented code, we perform systematic TDDFT calculations for spherical Zr and Mo isotopes near $N=50$, where experimental data exist. For neutron-rich Zr, Mo, and Ru isotopes where shape evolution exist we predict the photoabsorption cross sections based on oblate and triaxial minima.
The effective surface approximation is extended accounting for derivatives of the symmetry energy density per particle. Using the analytical isovector surface energy constants within the Fermi-liquid droplet model, one obtains energies and sum rules of the isovector dipole resonance structure in a reasonable agreement with the experimental data and other theoretical approaches.
A fully consistent relativistic random phase approximation is applied to study the systematic behavior of the isovector giant dipole resonance of nuclei along the $β$-stability line in order to test the effective Lagrangians recently developed. The centroid energies of response functions of the isovector giant dipole resonance for stable nuclei are compared with the corresponding experimental data and the good agreement is obtained. It is found that the effective Lagrangian with an appropriate nuclear symmetry energy, which can well describe the ground state properties of nuclei, could also re…
We present a superior proof that the results for summed strength isovector dipole, spin dipole, and orbital dipole excitations are independent of deformations at the Delta N = 0 level.
The nuclear isovector-dipole strength structure is analyzed in terms of the main and satellite (pygmy) peaks within the Fermi-liquid droplet model. Such a structure is sensitive to the value of the surface symmetry-energy constant obtained analytically for different Skyrme forces in the leptodermous effective surface approximation. Energies, sum rules and transition densities of the main and satellite peaks for specific Skyrme forces are qualitatively in agreement with the experimental data and other theoretical calculations.
A fully symmetry unrestricted Time-Dependent Density Functional Theory extended to include pairing correlations is used to calculate properties of the isovector giant dipole resonances of the deformed open-shell nuclei 172Yb (axially deformed), 188Os (triaxially deformed), and 238U (axially deformed), and to demonstrate good agreement with experimental data on nuclear photo-absorption cross-sections for two different Skyrme force parametrizations of the energy density functional: SkP and SLy4.
The structure of the isovector dipole resonance in neutron-rich calcium isotope, $^{60}Ca$, has been investigated by implementing a careful treatment of the differences of neutron and proton radii in the continuum random phase approximation ($RPA$). The calculations have taken into account the current estimates of the neutron skin. The estimates of the escape widths for direct neutron decay from the pygmy dipole resonance ($PDR$) were shown rather wide, implicating a strong coupling to the continuum. The width of the giant dipole resonance ($GDR$) was evaluated, bringing on a detailed discussi…
The self-consistent random phase approximation (RPA), based on the framework of relativistic energy density functionals, is employed in the study of isovector and isoscalar dipole response in $^{68}$Ni, $^{132}$Sn, and $^{208}$Pb. The evolution of pygmy dipole states (PDS) in the region of low excitation energies is analyzed as a function of the density-dependence of the symmetry energy for a set of relativistic effective interactions. The occurrence of PDS is predicted in the response to both the isovector and isoscalar dipole operators, and its strength is enhanced with the increase of the s…
The remaining uncertainties of isovector nuclear interactions call for reliable experimental measurements of isovector probes in finite nuclei. Based on the Bayesian analysis, although the neutron-skin thickness data or the isovector giant dipole resonance data in $^{208}$Pb can constrain only one isovector interaction parameter, correlations between other parameters are built. Using combined data of both the neutron-skin thickness and the isovector giant dipole resonance helps to constrain significantly all isovector interaction parameters, thus serves as a useful way in the future analysis.
We argue that a substantial increase in the cross section for Coulomb excitation in the region of the Double Giant Dipole Resonance should be expected from Coulomb excitation of excited states involved in the spreading of the one-phonon resonance, in a manifestation of the Brink-Axel phenomenon. This generates an additional fluctuating amplitude and a corresponding new term to be added incoherently to the usual cross-section. The appropriate extension of an applicable reaction calculation is considered in order to estimate this effect.
Isovector and isoscalar dipole excitations in $^9$Be and $^{10}$Be are investigated in the framework of antisymmetrized molecular dynamics, in which angular-momentum and parity projections are performed. In the present method, 1p-1h excitations on the ground state and large amplitude $α$-cluster mode are incorporated. The isovector giant dipole resonance (GDR) in $E>20$ MeV shows the two peak structure which is understood by the dipole excitation in the 2$α$ core part with the prolate deformation. Because of valence neutron modes against the $2α$ core, low-energy E1 resonances appear in $E<20$…
The E1 strength is systematically analyzed in very neutron-rich Sn nuclei, beyond $^{132}$Sn until $^{166}$Sn, within the Relativistic Quasiparticle Random Phase Approximation. The great neutron excess favors the appearance of a deformed ground state for $^{142-162}$Sn. The evolution of the low-lying strength in deformed nuclei is determined by the interplay of two factors, isospin asymmetry and deformation: while greater neutron excess increases the total low-lying strength, deformation hinders and spreads it. Very neutron rich deformed nuclei may not be as good candidates as stable spherical…
We study the information content carried by the electric dipole strength with respect to isovector and isoscalar indicators characterizing bulk nuclear matter and finite nuclei. To separate isoscalar and isovector modes, and low-energy strength and giant resonances, we analyze the E1 strength as a function of excitation energy $E$ and momentum transfer $q$. We use the self-consistent nuclear density functional theory with Skyrme energy density functionals, augmented by the random phase, to compute the E1 strength, and covariance analysis to assess correlations between observables. Calculations…
We investigate the isoscalar and isovector E1 response of neutron-rich nuclei, within a semi-classical transport model employing effective interactions for the nuclear mean-field. In particular, we adopt the recently introduced SAMi-J Skyrme interactions, whose parameters are specifically tuned to improve the description of spin-isospin properties of nuclei. Our analysis evidences a relevant degree of isoscalar/isovector mixing of the collective excitations developing in neutron-rich systems. Focusing on the low-lying strength emerging in the isovector response, we show that this energy region…
We perform Time Dependent Hartree-Fock (TDHF) calculations to investigate the small amplitude dipole response of selected neutron-rich nuclei and Sn isotopes. A detailed comparison with the dipole strength predicted by Random-Phase Approximation (RPA) calculations is presented for the first time. TDHF results are also confronted to Vlasov calculations, to explore up to which extent a semi-classical picture can explain the properties of the nuclear response. The focus is on the low-energy response, below the Giant Dipole Resonance region, where different modes of non negligible strength are ide…
The dipole response function of nuclear matter at zero and finite temperatures is investigated by employing the linearized version of the extended TDHF theory with a non-Markovian binary collision term. Calculations are carried out for nuclear dipole vibrations by employing the Steinwedel-Jensen model and compared with experimental results for $^{120}Sn$ and $^{208}Pb$.
The low-lying isovector dipole strengths in neutron rich nuclei $^{26}$Ne and $^{28}$Ne are investigated in the quasiparticle relativistic random phase approximation. Nuclear ground state properties are calculated in an extended relativistic mean-field theory plus BCS method where the contribution of the resonant continuum to pairing correlations is properly treated. Numerical calculations are tested in the case of isovector dipole and isoscalar quadrupole modes in the neutron rich nucleus $^{22}$O. It is found that in present calculation low-lying isovector dipole strengths at $E_x < 10$ MeV …
Electroexcitations of the dominantly T=1 particle-hole states of C-12 are studied in the framework of the harmonic oscillator shell model. All possible T=1 single-particle-hole states of all allowed angular momenta are considered in a basis including single-particle states up to the 1f-2p shell. The Hamiltonian is diagnoalized in this space in the presence of the modified surface delta interaction. Correlation in the ground state wave functions by mixing more than one configuration is considered and shows a major contribution that leads to enhance the calculations of the form factors. A compar…
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.…
The determination of nuclear symmetry energy, and in particular, its density dependence, is a long-standing problem for nuclear physics community. Previous studies have found that the product of electric dipole polarizability $α_D$ and symmetry energy at saturation density $J$ has a strong linear correlation with $L$, the slope parameter of symmetry energy. However, current uncertainty of $J$ hinders the precise constraint on $L$. We investigate the correlations between electric dipole polarizability $α_D$ (or times symmetry energy at saturation density $J$) in Sn isotopes and the slope parame…
The effects of ground-state correlations on the damping of isovector giant dipole resonances in $LS$ closed shell nuclei $^{16}$O and $^{40}$Ca are studied using extended random-phase-approximation (RPA) approaches derived from the time-dependent density-matrix theory. It is pointed out that unconventional two-body amplitudes of one particle--three hole and three particle--one hole types which are neglected in most extended RPA theories play an important role in the fragmentation of isovector dipole strength.
The relativistic random phase approximation is applied in the analysis of the evolution of the isovector dipole response in nuclei with a large neutron excess. The self-consistent framework of relativistic mean-field theory, which has been very successfully applied in the description of ground-state properties of nuclei far from the valley of $β$-stability, is extended to study the possible onset of low-energy collective isovector dipole modes in nuclei with extreme isospin values.
The summed strength for transitions from the ground state of $^{12}C$ via the operators $\vec{s}t, \vec{\ell}t, rY't, r[Y's]^λt$ and $r[Y'\ell]^λt$ are calculated using the $ΔN = 0$ rotational model. If we choose the z component of the isospin operator $t_{z}$, the above operators are relevant to electromagnetic transitions; if we choose $t_{+}$ they are relevant to weak transitions such as neutrino capture. In going from the spherical limit to the asymptotic (oblate) limit the strength for the operator $\vec{s} t$ decreases steadily to zero; the strength for the operator $\vec{\ell}τ$ (scisso…
The electric dipole response of $^{140}$Ce is investigated using the fully consistent relativistic quasiparticle random phase approximation. By analyzing the isospin structure of the E1 response, it is shown that the low-energy (pygmy) strength separates into two segments with different isospin character. The more pronounced pygmy structure at lower energy is composed of predominantly isoscalar states with surface-peaked transition densities. At somewhat higher energy the calculated E1 strength is primarily of isovector character, as expected for the low-energy tail of the giant dipole resonan…
We study isovector collective excitations in nuclear matter by employing the linearized Landau-Vlasov equation with and without a non-Markovian binary collision term at finite temperature. We calculate the giant dipole resonance (GDR) strength function for finite nuclei using Steinwedel-Jensen model and also by Thomas-Fermi approximation, and we compare them for 120Sn and 208Pb with experimental results.
A numerical method to integrate the time-dependent Hartree-Fock Bogoliubov (TDHFB) equations with Gogny interaction is proposed. The feasibility of the TDHFB code is illustrated by the conservation of the energy, particle numbers, and center-of-mass in the small amplitude vibrations of oxygen 20. The TDHFB code is applied to the isoscalar quadrupole and/or isovector dipole vibrations in the linear (small amplitude) region in oxygen isotopes (masses A = 18,20,22 and 24), titanium isotopes (A = 44,50,52 and 54), neon isotope (A = 26), and magnesium isotopes (A = 24 and 34). The isoscalar quadrup…
The isoscalar and isovector particle densities and the surface tension coefficients at the average binding energy are used to derive analytical expressions of the neutron skin thickness and the isovector stiffness of sharp edged proton-neutron asymmetric nuclei. For most Skyrme forces these quantities are significantly larger than the well known ones. Using the analytical isovector surface energy constants in the framework of the hydrodynamical and the Fermi-liquid droplet models the mean energies and the sum rules of the isovector giant dipole resonances are in fair agreement with the experim…
We study the response of isovector orbital magnetic dipole (IOMD) transitions to the quadrupole-quadrupole ($Q \cdot Q$) interaction, to the isospin-conserving pairing interaction (ICP) and to combinations of both. We find qualitatively different behaviours for transitions in which the final isospin differs from the initial isospin versus cases where the two isospins are the same. For $N=Z$ even-even nuclei with $J^π=0^+, T=0$ ground states such as $^8Be$ and $^{20}Ne$, the summed $T=0 \to T=1$ IOMD from the ground state to all the $J=1, T=1$ states in the $0 \hbar ω$ space does not vanish whe…
The isovector dipole response in $^{208}$Pb is described in the framework of a fully self-consistent relativistic random phase approximation. The NL3 parameter set for the effective mean-field Lagrangian with nonlinear meson self-interaction terms, used in the present calculations, reproduces ground state properties as well as the excitation energies of giant resonances in nuclei. In addition to the isovector dipole resonance in $^{208}$Pb, the present analysis predicts the occurrence of low-lying E1 peaks in the energy region between 7 and 11 MeV. In particular, a collective state has been id…
Using an energy density functional (EDF) based on the thermodynamic Gibbs-Duhem relation, found equivalent to the standard Skyrme EDF for infinite nuclear matter, it is demonstrated that the parameters of this EDF are not uniquely determined from the fit of the empirical and theoretical data related to nuclear matter. This prevents an unambiguous determination of the nucleon effective mass ($\frac{m_0^*}{m}$) and its isovector splitting ($Δm_0^*$). Complementary information from dipole polarizability of atomic nuclei helps in removing this ambiguity and plausible values of $\frac{m_0^*}{m}$ an…
The excitation of the isovector giant dipole resonance (GDR) by alpha scattering is investigated as a method of probing the neutron excess in exotic nuclei. DWBA calculations are presented for 28O and 70Ca and the interplay of Coulomb and nuclear excitation is discussed. Since the magnitude of the Coulomb excitation amplitude is strongly influenced by the Q-value, the neutron excess plays an important role, as it tends to lower the energy of the GDR. The excitation of the compressional isoscalar dipole state in 70Ca by alpha scattering is also investigated. It is shown that the population of t…
The vibrational structure of the Pygmy Dipole Resonance (PDR) is investigated within a quantum many-body treatment with extended separable interactions able to encode the dependence of nuclear symmetry energy on density. A new picture of PDR is unveiled in terms of a combined dynamics of the neutron skin and of the core isovector polarization, which determines the isoscalar features of PDR while reproducing the isovector properties of Giant Dipole Resonance. The key role played by the variation with density of the symmetry energy on shaping the low-lying dipole response and its isoscalar-isove…
The isovector time-reversal- and parity-violating pion-nucleon coupling $\bar{g}_π^{(1)}$ is uniquely sensitive to dimension six interactions between right-handed light quarks and the Standard Model Higgs doublet that naturally arises in left-right symmetric models. Recent work has used the $\bar{g}_π^{(1)}$-induced one-loop contribution to the neutron electric dipole moment $d_n$, together with the present experimental $d_n$ bound, to constrain the CP-violating parameters of the left-right symmetric model. We show that this and related analyses are based on an earlier meson theory $d_n$ compu…
We consider the energy weighted sums (EWS) for isovector giant dipole resonances (IVGDR) in finite nuclei within Landau kinetic theory. The dependence of both IVGDR energy, $E_{IVGDR}$, and the EWS enhancement factor, $κ(A)$, on the mass number $A$ occurs because of the boundary condition on the moving nuclear surface. The values of $E_{IVGDR}A^{1/3}$ and $κ(A)$ increase with $A$. The obtained value of the enhancement factor is about 10% for light nuclei and reaches approximately 20% for heavy nuclei. A fit of the enhancement factor to the proper experimental data provides a value for the isov…
We consider the propagation and damping of isovector excitations in heated nuclear matter within the Landau Fermi-liquid theory. Results obtained for nuclear matter are applied to calculate the Giant Dipole Resonance (GDR) at finite temperature in heavy spherical nuclei within Steinwedel and Jensen model. The centroid energy of the GDR slightly decreases with increasing temperature and the width increases as $T^2$ for temperatures $T < 5$ MeV in agreement with recent experimental data for GDR in $^{208}$Pb and $^{120}$Sn. The validity of the method for other Fermi fluids is finally suggested.
By using the 1H(6Li,6Be)n charge-exchange reaction, continuum states in 6Be were populated up to E_t=16 MeV, E_t being the 6Be energy above its three-body decay threshold. In kinematically complete measurements performed by detecting alpha+p+p coincidences, an E_t spectrum of high statistics was obtained, containing approximately ~5x10^6 events. The spectrum provides detailed correlation information about the well-known 0^+ ground state of 6Be at E_t=1.37 MeV and its 2^+ state at E_t=3.05 MeV. Moreover, a broad structure extending from 4 to 16 MeV was observed. It contains negative parity stat…
Microscopic structure of the low-lying isovector dipole excitation mode in neutron-rich $^{26,28,30}$Ne is investigated by performing deformed quasiparticle-random-phase-approximation (QRPA) calculations. The particle-hole residual interaction is derived from a Skyrme force through a Landau-Migdal approximation. We have obtained the low-lying resonance in $^{26}$Ne at around 8.5 MeV. It is found that the isovector dipole strength at $E_{x}<10$ MeV exhausts about 6.0% of the classical Thomas-Reiche-Kuhn dipole sum rule. This excitation mode is composed of several QRPA eigenmodes, one is generat…
The isospin character, the collective or single-particle nature, and the sensitivity to the slope of the nuclear symmetry energy of the low-energy isovector dipole response (known as pygmy dipole resonance) are nowadays under debate. In the present work we study, within the fully self-consistent non-relativistic mean field (MF) approach based on Skyrme Hartree-Fock plus Random Phase Approximation (RPA), the measured even-even nuclei ${}^{68}$Ni, ${}^{132}$Sn and ${}^{208}$Pb. To analyze the model dependence in the predictions of the pygmy dipole strength, we employ three different Skyrme param…
Double-dipole excitations in ^{208}Pb are analyzed within a microscopic model explicitly treating 2p2h-excitations. Collective states built from such 2p2h-excitations are shown to appear at about twice the energy of the isovector giant dipole resonance, in agreement with the experimental findings. The calculated cross section for Coulomb excitation at relativistic energies cannot explain simultaneously the measured single-dipole and double-dipole cross sections, however.
We study the couplings between collective vibrations such as the isovector giant dipole and isoscalar giant quadrupole resonances in tin isotopes in the framework of the time-dependent Hartree-Fock theory with a Skyrme energy density functional. These couplings are a source of anharmonicity in the multiphonon spectrum. In particular, the residual interaction is known to couple the isovector giant dipole resonance with the isoscalar giant quadrupole resonance built on top of it, inducing a nonlinear evolution of the quadrupole moment after a dipole boost. This coupling also affects the dipole m…
Proton inelastic scattering experiments at energy E_p = 200 MeV and a spectrometer scattering angle of 0 degree were performed on 144,146,148,150Nd and 152Sm exciting the IsoVector Giant Dipole Resonance (IVGDR). Comparison with results from photo-absorption experiments reveals a shift of resonance maxima towards higher energies for vibrational and transitional nuclei. The extracted photo-absorption cross sections in the most deformed nuclei, 150Nd and 152Sm, exhibit a pronounced asymmetry rather than a distinct double-hump structure expected as a signature of K-splitting. This behaviour can b…
We study the electric dipole polarizability $α_D$ in ${}^{208}$Pb based on the predictions of a large and representative set of relativistic and non-relativistic nuclear mean field models. We adopt the droplet model as a guide to better understand the correlations between $α_D$ and other isovector observables. Insights from the droplet model suggest that the product of $α_D$ and the nuclear symmetry energy at saturation density $J$ is much better correlated with the neutron skin thickness $Δr_{np}$ of ${}^{208}$Pb than the polarizability alone. Correlations of $α_D J$ with $Δr_{np}$ and with t…
We present recent investigations on dipole and quadrupole excitations in spherical skin nuclei, particular exploring their connection to the thickness of the neutron skin. Our theoretical method relies on density functional theory, which provides us with a proper link between nuclear many-body theory of the nuclear ground state and its phenomenological description. For the calculation of the nuclear excited states we apply QPM theory. A new quadrupole mode related to pygmy quadrupole resonance (PQR) in tin isotopes is suggested.
We have studied the uncharacteristic behavior of the measured values of the isoscalar and isovector centroid energies, ECEN, of giant resonances with multipolarity L=0-3 in 44Ca, 54Fe, 64,68Zn and 56,58,60,68Ni. For this purpose, we carried out calculations of ECEN within the spherical Hartree-Fock (HF)-based random phase approximation (RPA) theory with 33 different Skyrme-type effective nucleon-nucleon interactions. We have also determined the Pearson linear correlation coefficients between the calculated centroid energies and the various nuclear matter (NM) properties associated with each in…
We investigate the collective response function and the energy-weighted sums (EWS) $m_k$ for isovector mode in hot nuclei. The approach is based on the collisional kinetic theory and takes into consideration the temperature and the relaxation effects. We have evaluated the temperature dependence of the adiabatic, $E_1=\sqrt{m_1/m_{-1}}$, and scaling, $E_3=\sqrt{m_3/m_1}$, energy centroids of the isovector giant dipole resonances (IVGDR). The centroid energy $E_3$ is significantly influenced by the Fermi surface distortion effects and, in contrast to the isoscalar mode, shows much weaker variat…
The experimental and theoretical studies of Giant Resonances, or more generally of the nuclear collective vibrations, are a well established domain in which sophisticated techniques have been introduced and firm conclusions reached after an effort of several decades. From it, information on the nuclear equation of state can be extracted, albeit not far from usual nuclear densities. In this contribution, which complements other contributions appearing in the current volume, we survey some of the constraints that have been extracted recently concerning the parameters of the nuclear symmetry ener…
The fine structure of the IsoVector Giant Dipole Resonance (IVGDR) in the doubly-magic nuclei $^{40,48}$Ca observed in inelastic proton scattering experiments under $0^\circ$ is used to investigate the role of different mechanisms contributing to the IVGDR decay width. Characteristic energy scales are extracted from the fine structure by means of wavelet analysis. The experimental scales are compared to different theoretical approaches allowing for the inclusion of complex configurations beyond the mean-field level. Calculations are performed in the framework of RPA and beyond-RPA in a relativ…
Heavy Ion Collisions (HIC) represent a unique tool to probe the in-medium nuclear interaction in regions away from saturation. In this work we present a selection of reaction observables in dissipative collisions particularly sensitive to the isovector part of the interaction, i.e. to the symmetry term of the nuclear Equation of State (EoS). At low energies the behavior of the symmetry energy around saturation influences dissipation and fragment production mechanisms. We will first discuss the recently observed Dynamical Dipole Radiation, due to a collective neutron-proton oscillation during t…
Allowing for $2$$\hbar ω$ admixtures in $^4\mbox{He}$ we find that the summed magnetic dipole isovector orbital and spin strengths are equal. This indicates a symmetry which is associated with interchanging the labels of the spin with those of the orbit. Where higher admixtures are included, the orbital sum becomes larger than the spin sum, but the sums over the low energy region are still nearly the same.
We study the nature of the low-lying dipole strength in neutron-rich nuclei, often associated to the Pygmy Dipole Resonance. The states are described within the Hartree-Fock plus RPA formalism, using different parametrizations of the Skyrme interaction. We show how the information from combined reactions processes involving the Coulomb and different mixtures of isoscalar and isovector nuclear interactions can provide a clue to reveal the characteristic features of these states.
We put a stringent constraint on the isovector nuclear interactions in the Skyrme-Hartree-Fock model from the centroid energy $E_{-1}$ of the isovector giant dipole resonance in $^{208}$Pb as well as its electric polarizability $α_D$. Using the Bayesian analysis method, $E_{-1}$ and $α_D$ are found to be mostly determined by the nuclear symmetry energy $E_{sym}$ at about $ρ^\star=0.05$ fm$^{-3}$ and the isovector nucleon effective mass $m_v^\star$ at the saturation density. At $90\%$ confidence level, we obtain $E_{sym}(ρ^\star) = 16.4 ^{+1.0}_{-0.9}$ MeV and $m_v^\star/m = 0.79 ^{+0.06}_{-0.0…
We develop a new formulation of the continuum quasiparticle random phase approximation (QRPA) in which the residual interaction is derived directly from the Skyrme energy functional, keeping all the velocity dependent terms of the Skyrme effective interaction. Numerical analysis using the SkM$^*$ parameter set is performed for the isovector dipole and the isovector/isoscalar quadrupole responses in $^{20}$O and $^{54}$Ca. It is shown that the energy-weighted sum rule including the enhancement factors for the isovector responses is satisfied with good accuracy. We investigate also how the veloc…
Stable and unstable nuclei with neutron excess ($N>Z$) show - in the isovector dipole transition strength distribution - a small hump around the neutron emission threshold energy known as Pygmy Dipole Resonance (PDR). One of its main features is the isospin mixing allowing the experimental studies with both isovector and isoscalar probes. Different theoretical approaches and methodologies are used to deduce the characteristics of the PDR. In this Chapter, the various mean-field theories and their extensions, devoted to understand and reproduce the strength distribution of these low-lying dipol…
The isotopic dependence of the isovector Pygmy dipole response in tin is studied within the framework of the relativistic random phase approximation. Regarded as an oscillation of the neutron skin against the isospin-symmetric core, the pygmy dipole resonance may place important constraints on the neutron skin of heavy nuclei and, as a result, on the equation of state of neutron-rich matter. The present study centers around two questions. First, is there a strong correlation between the development of a neutron skin and the emergence of low-energy isovector dipole strength? Second, could one u…
Magnetic dipole (M1) excitations build not only a fundamental mode of nucleonic transitions, but they are also relevant for nuclear astrophysics applications. We have established a theory framework for description of M1 transitions based on the relativistic nuclear energy density functional. For this purpose the relativistic quasiparticle random phase approximation (RQRPA) is established using density dependent point coupling interaction DD-PC1, supplemented with the isovector-pseudovector interaction channel in order to study unnatural parity transitions. The introduced framework has been val…
