Pygmy dipole resonances: open problems

The low-lying dipole states, known as Pygmy Dipole Resonances (PDR) have been studied at length for several nuclei and several isotopes. Many properties have been acquired and understood by the theoretical and experimental studies of the isovector and isoscalar response of the PDR. Nevertheless, some unresolved questions still remain and two of them are briefly discussed in this contribution: the long standing question about the collective nature of this new mode and the presence of the PDR in deformed nuclei.


Introduction
Nuclei with neutron excess shown several interesting properties which have been thoroughly investigated in the last few years.Among them the so-called Pygmy Dipole Resonances (PDR) was studied and identified as a group of dipole states whose excitation energy are well below the well known isovector Giant Dipole Resonance (IVGDR).They exhaust a small fraction of the Energy Weighted Sum Rule and have a small strength.They have been found in all the investigated nuclei with neutron excess and a lot of work -both experimental and theoreticalhas been dedicated to their study [1,2,3,4,5].Its excitation has been related to the symmetry energy parameter of the Equation of State (EOF) [7,8] and a better knowledge of the PDR mode contributes to a precise theoretical description of the dipole Photon Strength Function which is very important to estimate the neutron capture rates in the astrophysical r-process [9,10,11].
The theoretical approaches employed to the study of the PDR can be roughly separated in two classes, macroscopic and microscopic.In ref. [12], a nucleus with neutron excess is considered as formed by incompressible three fluid components -say proton and neutron of the core, and neutron excess -which are supposed to move in according to the well known Steinwedel-Jensen model.A different version of the model is obtained when only two fluids, neutron excess and core, are considered moving as predicted by the Goldhaber-Teller prescription [13].
A more satisfactory description of the mode is achieved when microscopic many-body models are employed.Almost all the known approaches based on particle hole (p-h) excitation were implemented for the description of the PDR characteristics: the Hartree-Fock plus Random Phase Approximation (RPA) with Skyrme interactions; the Hartree-Fock-Bogoliubov (HFB) theory plus the Quasi-Particle RPA (QRPA) with Skyrme or Gogny effective interactions; the relativistic RPA (RRPA) and a relativistic quasi-particle RPA (RQRPA).A better description of the low-lying dipole excitation below the neutron emission threshold is obtained when the coupling up to three-phonon states are allowed as it is the case for the subtracted second RPA (SSRPA), the Quasi-particle Phonon Model (QPM) and the Relativistic Quasi-particle Time Blocking Approximation (RQTBA).More details can be found in the review paper of refs.[1,5,6].
Similar isoscalar and isovector response are found in almost all the microscopic approaches showing a strong isospin mixing of the PDR which can be considered the quiddity of this excitation mode.This is clearly seen in the shape of the transition densities where the proton and neutron transition densities have the same phase at the interior of the nucleus while at the surface only the neutrons give a contribution.This special characteristic of the PDR mode allows the studies of the low-lying dipole states with both isoscalar and isovector probes.Most of the experimental investigation has been devoted to the study of the PDR in stable nuclei with the (γ, γ ) reaction, or Nuclear Resonance Fluorescence (NRT) technique, at Darmstadt University.Isovector probes have been employed in the pioneering works on 132 Sn [14] and later on 68 Ni [15] at GSI with relativistic Coulomb excitation.The (p,p') reaction at zero degrees have been employed at the Research Center for Nuclear Physics, Osaka University, and also at iThemba LABS.Using the (α, α γ) at KVI laboratory, Groningen and ( 17 O, 17 O' γ) reaction at the INFN Legnaro laboratory, as isoscalar probes, a novel feature has been unveiled: the socalled isospin or PDR splitting.The energy range of the low-lying dipole states can be divided in two parts.The lower-energy part belong to states that have a strong isospin mixing while in the higher energy part the states are essentially excited by electromagnetic probes.This behaviour was found in all the investigated nuclei below the neutron emission threshold.An attempt to see whether the splitting can be observed also above the neutron emission threshold has been done at the INFN-LNS in Catania where the dipole excitation of the nucleus 68 Ni -impinging on a 12 C target at an incident energy of 28 MeV/A [16] -has been studied.The isospin splitting seems to be excluded at the energy above the neutron emission threshold; although more precise measurements are necessary to dconfirm this observation.More details on the experimental methods and results can be found in the review paper of refs.[2,3,4].

Open Problems
Even though the PDR has been studied at length for several nuclei and several isotopes, there are still some open problems that are worth investigation.Among them it is the long debated question whether this new mode can be considered collective or not.Another question is related to the presence of the PDR and in which form it manifest in deformed nuclei.Also the interplay between isoscalar and isovector excitation is still not completely clarified especially in connection with the experimental evidence of the isospin splitting.This last aspect should be further investigated with a particular attention to its possible presence at energies above the neutron emission threshold.In this paper, two of the above aspects will be briefly introduced.More detailed discussion about these points can be found in ref. [6].

About collectivity of the PDR
In macroscopic approaches for the PDR, the collectivity is implicitly assumed by the models: for instance, the two fluids (core and neutron excess) composing the nucleus are coherently moving one against each other.In a microscopic approach, like RPA, in order to decide about the collective property of a mode, one has to consider besides the number of p-h composing the considered state also the coherence of these configurations.One way to check the coherence property is to look at the reduced transition probability written as [17] where T λ ph are the 2 λ multipole transition amplitudes associated with the elementary p-h configurations of a state ν and the X and Y are the RPA amplitudes.The partial contribution b ph of the reduced transition probability are plotted in Fig. 1 as function of the order number n ph of the p-h configurations of an RPA calculation for the 68 Ni isotope.The b ph contributions are plotted as red bars and the contribution from proton and neutron p-h are separated by the vertical dashed line.The thin blue line indicates the cumulative sum of the partial contributions.Panel (a) and (b) are for the isovector response for two states belonging to the PDR and IVGDR region, respectively.For the IVGDR case it is clear that the contribution of the single p-h contributions are summing up coherently.For the low-lying dipole state (panel (a)) it is clear that, although there are several p-h contributions, they do not add coherently and therefore one can not infer that the state is collective.When analysing the isoscalar response for the same two states an evident collective behaviour is clearly found (see panels (c) and (d)).Similar analysis was performed for several nuclei with different Skyrme force obtaining similar results [18].Another way to investigate the collectivity is to use a correlation analysis based on a data analysis method as presented in ref. [19].The use of the energy distribution of the transition densities averaged over the energy allows to distinguish very well the energy regions where collective states are located as it is the case for the IVGDR and ISGDR.On the other hand, in the low energy region the energy distribution of the transition density show a complex multi-nodal behaviour in both isospin channels which is an indication of a weak collectivity.
Recently, this problem has been tackled also from the experimental point of view by looking at the single-particle character of low-lying dipole states.This has been done using a (d, p) transfer reaction for 208 Pb [20] and with a (d, p' γ) reaction for 120 Sn [21].They were done at different incident energies and with different experimental set ups.In both cases the experimental data were analysed with Energy Density Functional plus the QPM with the results that in the higher part of the PDR energy region most of the states have a strong contribution from complex configuration like two-or three-phonons.The dipole states can not be considered as single 1p-1h states even though two dominant single particle configurations were identified to contribute mainly to the states at lower energies.These experimental results, together with the theoretical analysis, strongly suggests that the isovector response can not be considered collective, although more work has to be done in this field to definitely confirm this picture.Another experiment that can contribute to clarify some aspect of this problem has been performed at INFN-LNS in Catania where the reaction 97 Mo(p,d) 96 Mo and its conjugate reaction 95 Mo(d,p) 96 Mo were used to populate the single-particle states of 96 Mo.Preliminary results for these experiments are reported in T. Khumalo contribution [22].

PDR in deformed nuclei
In a well deformed nucleus the dipole strength distribution splits in two peaks each one associated to the oscillation of neutrons against protons along the two symmetry axes.The separation between the two peaks depends on the nuclear deformation.The question is whether something similar may happen for the low-lying dipole peak.Theoretical studies of the PDR in deformed nuclei were based on Hartree-Fock-Bogoliubov (HFB) plus QRPA approach with Skyrme and Gogny effective interaction (for details see [6] and reference therein).In particular, two of such approaches reach different conclusions for the PDR win deformed nuclei.The relativistic Hartree-Bogoliubov (RHB) mean field plus a relativistic QRPA microscopic calculations have been performed for several Sn isotopes [23], taking into account -in a full self-consistent way -pairing correlations.The main quantities, like the transition densities, are calculated in the intrinsic frame and only after the projection to the dipole angular momentum one can obtained them in the laboratory frame.These transition densities -obtained for both quantum numbers K π = 0 + and K π = 1 − in the intrinsic frame -have the same structure of the ones for the spherical nuclei: proton and neutron transition densities are in phase inside the nucleus and at the surface only the neutrons give a strong contribution.Therefore, also for deformed nuclei it is possible to study the low-lying dipole states with both isovector and isoscalar probes.The summed of the electromagnetic reduced transition probability B(E1) in the PDR region plotted as function of the mass number for the Sn isotopes show a linear increase with the neutrons number.However, in correspondence to deformed nuclei the summed strength is reduced in comparison to the liner increase.This depletion in strength brought to the conclusion that nuclei that exhibit ground state deformation are not a good candidate for the study of the low-lying dipole states.
The approach based on an HFB plus QRPA with Skyrme interaction reaches different conclusions: the summed B(E1) is found much larger than in spherical nuclei.The studied nuclei are the Nd and Sm isotopes [24] and for the Mg isotopes a strong enhancement in the isoscalar B(E1) has also been found.The two approaches differ in many aspects.The pairing correlation are treated in a different way; the inclusion of the continuum states and the weakly bound states have two different treatments; and probably the most important difference is that while in ref. [23] both deformation and pairing were introduced in a full-consistent way, in ref. [24] this was not the case inducing a possible contamination of the spurious center-of-mass motion.These opposite results should be a strong incentive to further studies.
The Nd and Sm isotopes were also studied to investigate the evolution of the isoscalar response with deformation [25].The isoscalar E1 strength is plotted in Fig. 2 for several Sm isotopes and for the two projection K π = 0 − and K π = ±1 − of the total angular momentum onto the symmetry axis of the intrinsic (body-fixed) frame.Increasing the deformation, it is visible the separation between the K = 0 (black bars) and K = 1 (red bars) contributions.For the two spherical nuclei 144 Sm and 146 Sm the two contributions are equal.As for the IVGDR, the splitting is more pronounced as the deformation is increasing.This interesting aspect has been recently experimental investigated and some preliminary results can be found in H. Jivan The black and red bars correspond to the two K= 0 and K = 1 contributions, respectively.For the spherical nuclei, 144 Sm and 146 Sm, the two K-mode contributions are equal.Adapted from ref. [25].contribution [26] where the excitation of the PDR in the deformed 154 Sm has been studied via the (α, α γ) reaction.The role of the PDR in deformed nuclei is a very interesting problem and a challenge for both theoretical and experimental investigations.

Summary
The low-lying dipole states in nuclei with neutron excess, know as Pygmy Dipole Resonances (PDR), have been largely studied both experimentally and theoretically.Their strong isospin mixing allows the use of isovector and isoscalar probes and many properties have been acquired and understood in the last decades.However, there are still some unresolved questions that need to be addressed.One long standing question is about the collective nature of this new mode.To consider the PDR as collective, it should be show that many particles participate -in a coherent way -to build up the excitation.Many of the microscopic mean field approaches have found that while the isoscalar response can be considered collective, this is not the case for the isovector response.Indeed, in the latter the coherence property is lost.Some experiments, like (d, p) reactions, have been proposed to investigate the single-particle propriety of the low-lying dipole states.They show that only few single p-h configurations contribute to some of the low-lying dipole states.The majority of states, though, have many p-h configurations with some strong contribution of complex p-h configurations coming from two-and three-phonon states.
The other still unresolved problem is connected with the presence of the PDR in deformed nuclei.One of the question is whether the peak of the low-lying dipole states will split in two parts as for the IVGDR in very strongly deformed nuclei.There are very few experimental results with not conclusive answers, while several theoretical works devoted to this studies produced very different results.As for the spherical nuclei, the low-lying dipole states have a strong isospin mixing and the study with an isoscalar probes is possible.Theoretical calculation predicts a splitting when isoscalar probes are used even though not pronounced as in the IVGDR case.
The problems of collectivity and the role of deformation for the PDR, together with the isovector and isoscalar mixing not treated here, are a challenge for theoretical and experimental future investigations.

28thFigure 1 .
Figure 1.Partial contributions b ph of the reduced transition probability vs. the order number of the p-h configurations n ph used in the RPA calculations for the PDR (a and c) and IVGDR (b and d) dipole states of 68 Ni.The dashed vertical lines divide the protons from the neutron configurations.The order goes from the most to the less bound ones.The bars corresponds to the individual b ph contributions; the continuous thin line is the cumulative sum of the contributions.The panels (a) and (b) are for isovector dipole states responses whose excitation energy are written in the panels; in panels (c) and (d) the results for isoscalar responses.

28thFigure 2 .
Figure 2. The low-lying isoscalar E1 strength for several Sm isotopes.The black and red bars correspond to the two K= 0 and K = 1 contributions, respectively.For the spherical nuclei, 144 Sm and 146 Sm, the two K-mode contributions are equal.Adapted from ref.[25].