A very important open question related to the pygmy dipole resonance is about its quite elusive collective nature. In this paper, within a harmonic oscillator shell model, generalizing an approach introduced by Brink, we first identify the dipole normal modes in neutron rich nuclei and derive the energy weighted sum rule exhausted by the pygmy dipole resonance. Then solving numerically the self-consistent Landau-Vlasov kinetic equations for neutrons and protons with specific initial conditions, we explore the structure of the different dipole vibrations in the $^{132}Sn$ system and investigate their dependence on the symmetry energy. We evidence the existence of a distinctive collective isoscalar-like mode with an energy well below the Giant Dipole Resonance (GDR), very weakly dependent on the isovector part of the nuclear effective interaction. At variance the corresponding strength is rather sensitive to the behavior of the symmetry energy below saturation, which rules the number of excess neutrons in the nuclear surface.
Inelastic $α$-scattering excitation cross sections are calculated for electric dipole excitations in $^{124}$Sn based on the transition densities obtained from the relativistic quasiparticle time-blocking approximation (RQTBA) in the framework of a semiclassical model. The calculation provides the missing link to directly compare the results from the microscopic RQTBA calculations to recent experimental data measured via the $(α,α'γ)$ reaction, which show a structural splitting of the low-lying E1 strength often denoted as pygmy dipole resonance (PDR). The experimentally observed splitting is …
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…
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…
Within schematic models based on the Tamm-Dancoff Approximation and the Random-Phase Approximation with separable interactions, we investigate the physical conditions which determine the emergence of the Pygmy Dipole Resonance in the E1 response of atomic nuclei. We find that if some particle-hole excitation manifests a different, weaker residual interaction, an additional mode will appear, with an energy centroid closer to the distance between two major shells and therefore well below the Giant Dipole Resonance. This state, together with Giant Dipole Resonance, exhausts all the transition str…
Motivated by a recent experiment, the distribution of electric dipole strength in the neutron-rich 68Ni isotope was computed using a relativistic random phase approximation with a set of effective interactions that - although well calibrated - predict significantly different values for the neutron-skin thickness in 208Pb. The emergence of low-energy "Pygmy" strength that exhausts about 5-8% of the energy weighted sum rule (EWSR) is clearly identified. In addition to the EWSR, special emphasis is placed on the dipole polarizability. In particular, our results suggest a strong correlation betwee…
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…
We study the correlation between the neutron skin development and the low-energy dipole response associated with the pygmy dipole resonance (PDR) in connection with the properties of symmetry energy. We perform our investigation within a microscopic transport model based on the Landau-Vlasov kinetic equation by employing three different equations of state in the isovector sector. Together with the giant dipole resonance (GDR) for all studied systems, we identify a PDR collective mode whose energy centroid is very well described by the parametrization E_{PDR}=41 A^{-1/3}. A linear correlation b…
The strength functions of giant dipole resonance (GDR) in oxygen $^{18 - 24}$O, calcium $^{50 - 60}$Ca, and tin $^{120 - 130}$Sn isotopes are calculated within the phonon damping model under three approximations: without superfluid pairing, including BCS pairing, and exact pairing gaps. The analysis of the numerical results shows that exact pairing decreases the two-neutron separation energy in light nuclei, but increases it in heavy nuclei as compared to that obtained within the BCS theory. In neutron-rich medium and heavy nuclei, exact pairing significantly enhances the strength located at t…
The pygmy dipole resonance has been studied in the proton-magic nucleus 124Sn with the (a,a'g) coincidence method at E=136 MeV. The comparison with results of photon-scattering experiments reveals a splitting into two components with different structure: one group of states which is excited in (a,a'g) as well as in (g,g') reactions and a group of states at higher energies which is only excited in (g,g') reactions. Calculations with the self-consistent relativistic quasiparticle time-blocking approximation and the quasiparticle phonon model are in qualitative agreement with the experimental res…
New experimental data on the neutron single-particle character of the Pygmy Dipole Resonance (PDR) in $^{208}$Pb are presented. They were obtained from $(d,p)$ and resonant proton scattering experiments performed at the Q3D spectrograph of the Maier-Leibnitz Laboratory in Garching, Germany. The new data are compared to the large suite of complementary, experimental data available for $^{208}$Pb and establish $(d,p)$ as an additional, valuable, experimental probe to study the PDR and its collectivity. Besides the single-particle character of the states, different features of the strength distri…
Pygmy and Giant Dipole Resonance (PDR and GDR) in Ni isotopes have been investigated by Coulomb excitation in the framework of the Isospin-dependent Quantum Molecular Dynamics model (IQMD). The spectra of $γ$ rays are calculated and the peak energy, the strength and Full Width at Half Maximum (FWHM) of GDR and PDR have been extracted. Their sensitivities to nuclear equation of state, especially to its symmetry energy term are also explored. By a comparison with the other mean-field calculations, we obtain the reasonable values for symmetry energy and its slope parameter at saturation, which gi…
The electromagnetic dipole strength below the neutron-separation energy has been studied for the xenon isotopes with mass numbers A = 124, 128, 132, and 134 in nuclear resonance fluorescence experiments using the ELBE bremsstrahlung facility at Helmholtz-Zentrum Dresden-Rossendorf and the HIgS facility at Triangle Universities Nuclear Laboratory Durham. The systematic study gained new information about the influence of the neutron excess as well as of nuclear deformation on the strength in the region of the pygmy dipole resonance. The results are compared with those obtained for the chain of m…
Scattering of protons of several hundred MeV is a promising new spectroscopic tool for the study of electric dipole strength in nuclei. A case study of 208Pb shows that at very forward angles J^pi = 1- states are strongly populated via Coulomb excitation. A separation from nuclear excitation of other modes is achieved by a multipole decomposition analysis of the experimental cross sections based on theoretical angular distributions calculated within the quasiparticle-phonon model. The B(E1) transition strength distribution is extracted for excitation energies up to 9 MeV, i.e., in the region o…
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.
The validity of the Brink-Axel hypothesis, which is especially important for numerous astrophysical calculations, is addressed for 116,120,124Sn below the neutron separation energy by means of three independent experimental methods. The $γ$-ray strength functions (GSFs) extracted from primary $γ$-decay spectra following charged-particle reactions with the Oslo method and with the Shape method demonstrate excellent agreement with those deduced from forward-angle inelastic proton scattering at relativistic beam energies. In addition, the GSFs are shown to be independent of excitation energies an…
The low-lying electric dipole (E1) strength in the semi-magic nucleus 136Xe has been measured which finalizes the systematic survey to investigate the so-called pygmy dipole resonance (PDR) in all stable even N=82 isotones with the method of nuclear resonance fluorescence using real photons in the entrance channel. In all cases, a fragmented resonance-like structure of E1 strength is observed in the energy region 5 MeV to 8 MeV. An analysis of the fragmentation of the strength reveals that the degree of fragmentation decreases towards the proton-deficient isotones while the total integrated st…
In order to explore the nature of collective modes in weakly bound nuclei, we have investigated deformation effects and surface flow patterns of isovector dipole modes in a shape-coexisting nucleus $^{40}$Mg. The calculations were done in a fully self-consistent continuum finite-amplitude Quasiparticle Random Phase Approximation (QRPA) in a large deformed spatial mesh. An unexpected result of pygmy and giant dipole modes having disproportionate deformation splittings in strength functions was obtained. Furthermore, the transition current densities demonstrate that the long-sought core-halo osc…
We carry out a systematic investigation on the low-energy electric dipole strength, which is often called pygmy dipole resonances (PDR), using the canonical-basis time-dependent Hartree-Fock-Bogoliubov (Cb-TDHFB) method. The Cb-TDHFB is a new method which is derived from TDHFB with an approximation analogous to the BCS theory that the pair potential is assumed to be diagonal in the time-dependent canonical basis. We apply the method to linear-response calculation for even-even nuclei. We report the neutron-number dependence of PDR in light (A < 70) and heavy isotopes (A > 100) around N = 82.
We review a recent progress in investigation of the isoscalar toroidal dipole resonance (TDR). A possible relation of the TDR and low-energy dipole strength (also called a pygmy resonance) is analyzed. It is shown that the dipole strength in the pygmy region can by understood as a local manifestation of the collective vortical toroidal motion at the nuclear surface. Application of the TDR as a measure of the nuclear dipole vorticity is discussed. Finally, an anomalous splitting of the TDR in deformed nuclei is scrutinized.
The electric dipole excitation of various nuclei is calculated with a Random Phase Approximation phenomenological approach. The evolution of the strength distribution in various groups of isotopes, oxygen, calcium, zirconium and tin, is studied. The neutron excess produces $E1$ strength in the low energy region. Indexes to measure the collectivity of the excitation are defined. We studied the behavior of proton and neutron transition densities to determine the isoscalar or isovector nature of the excitation. We observed that in medium-heavy nuclei the low-energy $E1$ excitation has characteris…
Relativistic quasiparticle time blocking approximation. II. Pygmy dipole resonance in neutron-rich nuclei
在相对论准粒子时间阻塞近似下，对偶偶球核中的低偶极强度进行了理论研究。作为自洽相对论准粒子无规相近似（RQRPA）的扩展，RQTBA可以在基于协变能量密度泛函理论的完全自洽计算框架内研究两准粒子激发对集体振动的耦合效应.用RQRPA和RQTBA方法计算的偶-偶$^{130}$ Sn - $^{140}$ Sn和$^{68}$ Ni - $^{78}$ Ni同位素的偶极谱显示出两种分离的集体结构：高位巨偶极共振（GDR）和低位侏儒偶极共振（PDR），这两种结构可以通过这些区域中态跃迁密度的不同行为来识别.
The occurrence of the low-lying charge-exchange non spin-flip dipole modes below the giant resonance in neutron-rich nuclei is predicted on the basis of nuclear density functional theory. The ground and excited states are described in the framework of the self-consistent Hartree-Fock-Bogoliubov and the proton-neutron quasiparticle-random-phase approximation employing a Skyrme-type energy density functional. The model calculations are performed for the spherical neutron-rich Ca, Ni, and Sn isotopes. It is found that the low-lying states appear sensitively to the shell structure associated with …
The low-lying spectra of $^{24,25,26}{\rm Ne}$ and the structure of the pygmy dipole resonance (PDR) in $^{26}{\rm Ne}$ have been theoretically studied by the antisymmetrized molecular dynamics (AMD) and its extended version called shifted-basis AMD. The calculated energy and strength of the PDR reasonably agree with the observation, and the analysis of the wave function shows that the PDR is dominated by neutron excitation coupled to the quadrupole excited core nucleus $^{25}{\rm Ne}$, which explains the observed unexpected decay of PDR to the excited states of $^{25}{\rm Ne}$. The large isos…
The pygmy dipole resonances (PDR) for even-even nuclei in 8=<Z=<40 are studied performing a systematic calculation of the random-phase approximation with the Skyrme functional of SkM*. The calculation is fully self-consistent and does not assume any symmetry in the nuclear shape of the ground state. In every isotopic chain, the PDR emerges by showing a peak of the E1 strength at energies less than 10 MeV. The E1 strength of the PDR strongly depends on the position of the Fermi level and shows a clear correlation with the occupation of the orbits with the orbital angular momenta less than 3\hba…
To complete earlier studies of the properties of the electric pygmy dipole resonance (PDR) obtained in various nuclear reactions, the excitation of the 1$^-$ states in $^{140}$Ce by $(e,e')$ scattering for momentum transfers $q=0.1-1.2$~fm$^{-1}$ is calculated within the plane-wave and distorted-wave Born approximations. The excited states of the nucleus are described within the Quasiparticle Random Phase Approximation (QRPA), but also within the Quasiparticle-Phonon Model (QPM) by accounting for the coupling to complex configurations. It is demonstrated that the excitation mechanism of the PD…
We study the effects of the phonon-phonon coupling on the low-energy electric dipole response within a microscopic model based on an effective Skyrme interaction. The finite rank separable approach for the quasiparticle random phase approximation is used. Choosing as an example the isotopic chain of Calcium, we show the ability of the method to describe the low-energy $E1$ strength distribution. With one and the same set of parameters we describe available experimental data for $^{48}$Ca and predict the electric dipole strength function for $^{50}$Ca.
A strong isoscalar dipole resonance is known to be excited in a variety of nuclei, including isospin symmetric ones, at approximately 6-7 MeV. A series of theoretical studies and accumulating experimental evidence support an interpretation of the above dipole resonance as an elementary surface vibration. Obviously, such a mode is potentially as interesting as any collective excitation for a variety of reasons. In addition, though, it is found to account for the observed isoscalar segment of pygmy dipole strength. As discussed here, this has important implications for pygmy-strength interpretat…
The pygmy dipole resonance (PDR) is studied in various medium-heavy nuclei by using a Gogny interaction in a self-consistent Hartree-Fock plus Random Phase Approximation method. We compare the details of the PDR structure with those of the giant dipole resonance (GDR). In the PDR protons and neutrons vibrate in phase, and the main contributions are given by particle-hole excitations involving the neutrons in excess. On the contrary, in the GDR protons and neutrons vibrate out of phase, and all the nucleons are involved in the excitation. The values of the parameters we used to define the colle…
A two-phonon version of the relativistic quasiparticle time blocking approximation (RQTBA-2) represents a new class of many-body models for nuclear structure calculations based on the covariant energy density functional. As a fully consistent extension of the relativistic quasiparticle random phase approximation (RQRPA), the two-phonon RQTBA implies a fragmentation of nuclear states over two-quasiparticle and two-phonon configurations. This leads, in particular, to a splitting-out of the lowest 1$^-$ state as a member of the two-phonon $[2^+\otimes3^-]$ quintuplet from the RQRPA pygmy dipole m…
A high-resolution study of the electromagnetic response of $^{206}$Pb below the neutron separation energy is performed using a ($\vecγ$,$γ'$) experiment at the HI$\vecγ$S facility. Nuclear resonance fluorescence with 100% linearly polarized photon beams is used to measure spins, parities, branching ratios, and decay widths of excited states in $^{206}$Pb from $4.9$ to $8.1$MeV. The extracted $Σ$$B$(E1)$\uparrow$ and $Σ$$B$(M1)$\uparrow$ values for the total electric and magnetic dipole strength below the neutron separation energy are 0.9$\pm$0.2e$^2$fm$^2$ and 8.3$\pm$2.0$μ_{N}^2$, respectivel…
The pygmy dipole resonance in neutron-rich nuclei is studied within the framework of the Vlasov equation which is solved numerically. The interaction used in the Thomas-Fermi ground state and in the Vlasov equation is derived from an energy functional which correctly describes the equation of state of nuclear matter and neutron matter. It is found that the pygmy resonance appears in the electric dipole response of all nuclei with strong neutron excess, the energies and transition probabilities being in reasonable agreement with experimental results. Since the Vlasov equation does not account f…
Starting from the quasiparticle random phase approximation based on the Skyrme interaction SLy5, we study the effects of phonon-phonon coupling~(PPC) on the low-energy electric dipole response in $^{40-58}$Ca. Using the same set of parameters we describe available experimental data for $^{40,44,48}$Ca and give prediction for $^{50-58}$Ca. The inclusion of the PPC results in the formation of low-energy $1^-$ states. There is an impact of the PPC effect on low-energy $E1$~strength of $^{40,44,48}$Ca. The PPC effect on the electric dipole polarizability is discussed. We predict a strong increase …
The properties of the two-quasiparticle-like soft E1-modes and PDR have been and are systematically studied with the help of inelastic and electromagnetic experiments which essentially probe the particle-hole components of these vibrations. It is shown that further insight in their characterisation can be achieved with the help of two-nucleon transferreactions, in particular concerning the particle-particle components of the modes, in terms of absolute differential cross sections which take properly into account successive and simultaneous transfer mechanisms corrected for non-orthogonality, a…
We investigate the microscopic structure of the low-lying isovector-dipole excitation mode in neutron-rich $^{36,38,40}$Mg close to the drip line by means of the deformed quasiparticle-random-phase-approximation employing the Skyrme and the local pairing energy-density functionals. It is found that the low-lying bump structure above the neutron emission-threshold energy develops when the drip line is approached, and that the isovector dipole strength at $E_{x} < 10$ MeV exhausts about $6.0 %$ of the classical Thomas-Reiche-Kuhn dipole sum rule in $^{40}$Mg. We obtained the collective dipole mo…
We probe nuclear multipole resonances in the framework of the random-phase approximation by using the interaction obtained from the chiral effective field theory. The three-nucleon force is included in a form of the in-medium two-nucleon interaction which was derived from the chiral three-nucleon force. The isoscalar monopole, isoscalar dipole, isovector dipole and isoscalar quadrupole resonances of the closed-shell $^{56,68,78}$Ni have been investigated. The calculations reasonably reproduce the experimental multipole resonances of $^{56,68}$Ni, and well describe the pygmy dipole resonance an…
Microscopic description of the pygmy and giant electric dipole resonances in stable Ca isotopes
在\emph{Extended Theory of Finite Fermi Systems}（ETFFS）中计算了稳定的$^{40}Ca$、$^{44}Ca$和$^{48}Ca$同位素的侏儒共振（PDR）和巨偶极共振（GDR）性质。这种方法是基于无规相位近似（RPA），包括单粒子连续以及耦合到低的集体状态被认为是在一个一致的微观方式。对于$^{44}Ca$，我们还包括配对相关性。我们得到了很好的协议与实验数据的总属性的两个共振。它表明，最近测得的A-依赖的PDR低于10 MeV的强度是很好地理解在我们的模型：由于声子耦合的一些强度在$^{48}Ca$简单地移动超过10 MeV。可以在$(e,e')$和$(γ,γ')$实验中研究PDR的预测断裂。而PDR的等矢量偶极强度是小的，在所有的钙同位素，我们发现在这一地区令人惊讶的强大的isoscillate偶极状态，在协议与$(α,α'γ)$实验。我们的结论是，为了详细了解激发核的结构，如PDR和GDR的方法，像现在的一个是绝对必要的。
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) in 68Ni and 132Sn have been investigated by using different Random Phase Approximation (RPA) models for the dipole response, based on a representative set of Skyrme effective forces plus meson-exchange effective Lagrangians. A comparison with the experimental data has allowed us to constrain the value of the derivative of the symmetry energy at saturation. The neutron skin radius is deduced under this constraint…
The accurate characterization of the nuclear symmetry energy and its density dependence is one of the outstanding open problems in nuclear physics. A promising nuclear observable in order to constrain the density dependence of the symmetry energy at saturation is the neutron skin thickness of medium and heavy nuclei. Recently, a low-energy peak in the isovector dipole response of neutron-rich nuclei has been discovered that may be correlated with the neutron skin thickness. The existence of this correlation is currently under debate due to our limited experimental knowledge on the microscopic …
The information on pygmy resonances reveals new aspects on the isospin dynamics of the nucleus with important astrophysical consequences. In this connection, the precise knowledge of nuclear response functions plays a key role in the determination of photonuclear reactions cross sections which are of importance for the synthesis of heavy neutron-rich elements. For that purpose, a theoretical method based on density functional theory and multi-phonon approach is applied for investigations of nuclear excitations with different multipolarities and energies in stable and exotic nuclei. The possibl…
A possible relation of the low-lying E1 (pygmy resonance) and toroidal strengths is analyzed by using Skyrme-RPA results for the strength functions, transition densities and current fields in $^{208}$Pb. It is shown that the irrotational pygmy motion can appear as a local manifestation of the collective vortical toroidal dipole resonance (TDR) at the nuclear surface. The RPA results are compared to unperturbed (1ph) ones.
We study the Coulomb excitation of pygmy dipole resonances (PDR) in heavy ion reactions at 100 MeV/nucleon and above. The reactions 68Ni+197Au and 68Ni+208Pb are taken as a practical examples. Our goal is to address the question of the influence of giant resonances on the PDR as the dynamics of the collision evolves. We show that the coupling to the giant resonances affects considerably the excitation probabilities of the PDR, a result that indicates the need of an improved theoretical treatment of the reaction dynamics at these bombarding energies.
