Exotic nuclear excitations

The soft dipole isovector response was usually described as the so-called pygmy mode [1]: a soft oscillation of the neutron skin around a proton–neutron core. It was first thought to happen only in unstable neutron-rich nuclei. However, the understanding of this mode has considerably evolved over the last few years. First, several observations on these modes were reported in stable nuclei [5]. For instance, the analysis of the soft dipole response in ¹²⁴Sn showed that the low-energy branch of this mode was sensitive to isoscalar probes such as (α, α'), whereas the higher energy part (above 7 MeV) of the soft response was more sensitive to isovector probes such as (γ, γ') [6]. Therefore, some soft dipole modes have a non-negligible isoscalar component [7]. More generally, the analysis of the corresponding transition densities shows in some cases a non-trivial feature of the soft dipole mode, depending on the nucleus [7–9]. Figure 1 displays such transition densities in ⁷⁸Ni: the giant dipole resonance (GDR) exhibits a clear out-of-phase pattern between protons and neutrons, whereas the soft mode is of a mixed nature and is not purely neutronic on the surface, as would have been expected from a neutron skin oscillation. The non-trivial pattern of the soft dipole response is confirmed by recent calculations beyond the RPA approach, which include two-particle two-hole configurations in the excitation [10].

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The investigation of the isoscalar response is therefore relevant for the soft mode [7]. The isoscalar giant dipole resonance (ISGDR) is a second-order effect, built on 3ℏω or higher configurations [1]. It corresponds to a compression wave travelling back and forth through the nucleus along a definite direction. The isoscalar E1 strength distributions display a characteristic bimodal structure with two broad components: one in the low-energy region close to the isovector giant dipole resonance (IVGDR) (2ℏω) and the other at higher energy close to the electric octupole resonance (3ℏω). Theoretical analyses have shown that only the high-energy component represents compressional vibrations, whereas the broad structure in the low-energy region could correspond to a vortical nuclear flow associated with the toroidal dipole moment [1, 11]. However, a strong mixing between the compressional and vorticity vibrations in the isoscalar E1 states can be expected. The vortical, toroidal and compressional modes have also recently been studied in [12].

Another relevant question is related to the impact of the deformation on the soft mode. It is believed that for a strong neutron excess, the large diffuseness of the density induces a weakening of the spin–orbit potential and therefore the disappearance of the shell closure, allowing for deformed nuclei. This means that a model including both pairing and deformation is required in order to describe low-energy excitations in very neutron-rich nuclei. More generally, most nuclei in the nuclear chart are deformed, which also requires us to investigate the interplay between the low-lying E1 strength and the deformation when going towards the drip-line. The use of relativistic QRPA predictions on very neutron-rich Sn isotopes shows that deformation hinders the dipole strength in this region [13]. This effect is linked to the suppression of vibrations along an axis perpendicular to the symmetry axis (K^π = 1⁻ mode) for prolate deformed systems. It is explained by the reduction of the neutron skin in this direction, arising from the difference between the deformations of the neutron and proton densities. On the other hand, the low-lying E1 strength increases with the neutron number, and thus the interplay of these two effects determines the actual dipole response in the low-energy region of deformed nuclei. It should be noted that recent calculations using the Skyrme approach led to a different conclusion [14]: with increasing the deformation, the soft dipole mode is enhanced. However, this discrepancy with the relativistic approach could come either from the models (relativistic versus non-relativistic) or from the definition of the energy threshold below which the pygmy mode is considered.

2.2.1. The r-process.

The dipole response, being the most probable one, is expected to play a relevant role in astrophysical sites, compared with other excitations: photoabsorption or photodisintegration reactions shall involve the E1 strength. Various astrophysical sites are displayed in figure 2, summarizing different accelerators of the Universe, by showing their typical magnetic field B with respect to their curvature radius ρ. A constant B·ρ value provides the energy which can be reached by the accelerated particles, following the well-known relation B·ρ = p/Q, where p is the momentum of the particle and Q its charge.

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About half of the nuclei with A > 60 observed in nature are formed by the so-called rapid neutron capture process (or r-process) of nucleosynthesis. Two main astrophysical sites may be candidates for hosting the r-process. The first one, which is well known, occurs in explosive stellar events, such as core collapse supernovae [15]. In this case, the r-process is believed to take place in environments characterized by high neutron densities (N_n ≃ 10²⁰ cm⁻³), so that successive neutron captures proceed into neutron-rich regions well off the β-stability valley forming exotic nuclei that cannot be produced and therefore studied in the laboratory. If the temperatures or the neutron-densities characterizing the r-process are low enough to break the (n,γ)–(γ,n) equilibrium, the so-called waiting point approximation is not valid anymore, and the r-abundance distribution directly depends on the radiative neutron capture rates [16], involving the E1 strength.

An alternative r-process scenario has generated renewed interest over the last few years. It is related to the decompression of cold neutron star matter, in particular its crust [15]. In this case, the production of heavy nuclei follows a totally different path, compared with the core-collapse supernovae scenario. The nuclear clusters immersed in the neutron gas are ejected from the neutron star crust, inducing a decrease of the matter density. The β equilibrium is broken, leading to the production of a variety of Z, typically ranging from 40 to 70. When the density reaches the drip density, drip-line nuclei are formed, which are immersed in a typical free neutron flux of N_n ≃ 10³⁵ cm⁻³. Competition between the neutron capture and the β decay is then ignited by these drip-line nuclei, leading to the production of heavy nuclei. This alternative scenario also shows that nuclear inputs are necessary to modelize the r-process, since this nucleosynthesis path is initialized from the drip-line, where no waiting point approximation is possible.

The nucleosynthesis during gamma ray bursts (GRB) has also recently been investigated [19]. During the burst, fusion reactions (involving light elements) can occur, and the neutron environment is crucial: if not neutron-rich, the GRB nucleosynthesis is similar to the stellar nucleosynthesis: ¹²C is produced through the triple α reaction, and the elements up to iron are synthesized. However, if the GRB happens in a neutron-rich environment, then a nucleosynthesis analogous to the r-process happens, with ¹²C produced through ⁹Be, bypassing the triple α reaction and leading to the production of heavy elements beyond iron. The neutron-rich environment is determined by the so-called Y_e factor, which plays a pivotal role in the description of core collapse of supernovae.

Predicting r-process abundances requires nuclear inputs from several thousands of nuclei [15]. Theoretical results therefore play a crucial role, and a universal description of this large variety of nuclei is necessary. Large-scale microscopic calculations were performed for astrophysical applications, predicting the masses [20] and dipole strength [17, 18] from a single Skyrme functional. The present era of large-scale microscopic applications opens the possibility to predict with a universal functional the various nuclear physics mechanisms implied in the r-process.

It should be noted that the soft dipole modes are expected to play an even more pivotal role in the r-process than the higher energy dipole strength. Low-energy dipole modes located close to S_n are directly fed by the one-neutron capture reaction. The relevance of the soft dipole modes has been shown in [16–18], where the presence of such states has been reported to have a strong impact on the r-process. The phenomenological prediction of a low-lying component of the E1 strength leads to an increase of the radiative neutron capture rate by a factor of 10–100 for nuclei with S_n between 2 and 4 MeV. The r-abundance distribution is affected: the soft dipole states accelerate the neutron capture and allow to produce heavy nuclei around A = 130.

The Maxwellian-averaged neutron capture rates obtained with the Hartree–Fock–Bogoliubov (HFB) + QRPA E1 strength are compared in figure 3 with those based on the hybrid phenomenological formula for all exotic neutron-rich nuclei with 8 ⩽ Z ⩽ 110. The QRPA-derived E1 strength gives an increase of the rate by a factor of up to 10 close to the neutron drip-line. The r-process nuclei characterized by S_n < 3 MeV have a neutron capture rate at least about twice faster than that predicted by the phenomenological Hybrid formula. This is partly due to the appearance of soft dipole modes at low energies. This effect tends to increase the E1 strength at energies below the GDR, i.e. in the energy window of relevance in the neutron capture process. For less exotic nuclei, the QRPA impact is relatively small, the differences being mainly due to the exact position of the GDR energy and the resulting low-energy tail. When compared with HFBCS + QRPA predictions [17], the HFB + QRPA model [18] gives very similar neutron capture rates even close to the neutron drip-line, as seen in figure 3 (lower panel). This shows the consistency of the results obtained from various microscopic approaches of the dipole strength.

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2.2.2. The propagation of ultra-high energy cosmic rays.

2.2.3. The cooling of neutron stars.

The cooling of low-mass neutron stars is strongly influenced by the superfluid properties of the inner crust matter [28]. These superfluid properties and their influence on the specific heat were rather intensively analysed in past years by using various frameworks, e.g. semiclassical pairing models, Bogoliubov-type calculations based on a Woods–Saxon mean field, and the self-consistent HFB approach [1]. These calculations showed that the specific heat of baryonic inner crust matter can be reduced by orders of magnitude due to pairing correlations. However, in all the calculations mentioned above, the specific heat was evaluated by considering only non-interacting quasi-particle states. The specific heat may also be affected by the collective modes created by the residual interaction between the quasiparticles, especially if these modes appear at low excitation energies. This effect has been studied in [29]. It should be noted that the specific heat of the inner crust is also largely determined by the motion of electrons and by the lattice vibrations [28]. A recent approach using superfluid hydrodynamical predictions also showed that the neutron gas excitations induced by the periodic structure of the crust may generate an important contribution to the specific heat [30].

The inner crust of a neutron star is usually decomposed into non-interacting cells containing several protons with a majority of neutrons. The neutron response of the cell ¹⁸⁰⁰Sn is displayed in figure 4 for several multipolarities. The unperturbed HFB response, built up by non-interacting quasiparticle states, and the QRPA response are both shown in figure 4. When the residual interaction is introduced among the quasiparticles, the unperturbed spectrum, distributed over a large energy region, is gathered in a peak located at about 3 MeV. All multipolarities exhibit an intense low-energy mode (the so-called supergiant resonance (SGR)), which can be more or less damped. In the case of the quadrupole response, the SGR peak collects more than 99% of the total quadrupole strength. This mode is extremely collective since there are more than a hundred two-quasiparticle configurations contributing. However, as discussed above, there is a mixed contribution from the nuclear cluster states and neutron gas states. The excitation is mostly constructed with neutrons belonging to the cluster surface and the external free gas [31].

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Not only the dipole, but also other multipoles could exhibit pronounced low-energy strength in neutron-rich nuclei. The monopole response in very exotic nuclei such as ⁶⁰Ca was investigated using the self-consistent Hartree–Fock calculation plus the RPA with Skyrme interactions [32]. It was shown that, approaching the drip line, the monopole response could develop a very pronounced structure at low energy.

More realistic examples of nuclei should also be considered, in which the low-energy monopole strength could be measured in the near future. The low-lying E1 strength has recently been measured in ⁶⁸Ni [33]. A new technique has been developed that enables the measurement of the monopole strength in unstable nuclei [34]. An experiment was recently performed at GANIL to measure the monopole strength in ⁶⁸Ni [35], and the analysis is in progress. In figure 5, the isoscalar monopole strength in ⁶⁸Ni is calculated with the SLy4 (left) and SGII (right) functionals. The GMR is predicted at about 20 MeV, and both functionals predict a pronounced low-lying structure located between 13 and 16 MeV excitation energy. These states are not purely isoscalar: the transition densities exhibit neutron-dominated modes, and the proton and neutron densities are not in phase in the interior of the nucleus. Similar conclusions are reached using relativistic approaches [36]. The configuration analysis of these low-lying states shows that they correspond to almost pure single hole–particle excitations: a single configuration contributes more than 98% to the total strength. These very non-collective states could bring valuable information on the spin–orbit interaction, and detecting these mode in the near future is of interest.

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The GMR is a tool of choice to constrain the nuclear incompressibility. However, it still seems for now difficult to straightforwardly determine the nuclear matter incompressibility, which may indicate a defect of the method. The earliest microscopic analysis came to a value of K_∞ = 210 MeV [37], but with the advent of microscopic relativistic approaches, a value of K_∞ = 260 MeV was obtained [38]. It has been further shown that K_∞ is not accurately determined and that the density dependence of the energy density functional (EDF) as well as pairing effects (and therefore the shell structure) have an impact on the determination of K_∞ [39–41]. The method of determination of the nuclear incompressibility may have to be rethought.

To illustrate the effect of superfluidity on nuclear incompressibility [42], figure 6 displays the incompressibility in nuclei (K_A) as a function of the average pairing gap calculated with the HFB approach, from ¹¹²Sn to ¹³²Sn. A clear correlation is observed: the more superfluid the nucleus, the lower the incompressibility. Hence, it may be easier to compress superfluid nuclei. This may be the first evidence of the role of superfluidity in the compressibility of a fermionic system. A possible interpretation is that Cooper pairs can modify bulk properties, as is known from nuclear physics phenomenology [43].

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It should also be noted that the liquid drop expansion is not a perturbative one since the surface properties of the nucleus are almost as important as the bulk one. Therefore, it may be a misleading idea to consider the nucleus as mainly composed of nucleons at saturation density with a few at the surface: about one-third of the nucleons of the ²⁰⁸Pb nucleus lie in the saturation density area, whereas two-thirds are localized in the surface at a density lower than the saturation one [44]. Therefore, even in heavy nuclei, the contribution of the surface is larger with respect to the volume one, questioning the legitimacy of constraining EOS quantities at saturation density (such as K_∞) by measurements of nuclear observables. The measurement of an averaged observable in a nucleus is more properly related to a correlated EOS quantity defined around the mean density than at the saturation density.

Actually, the existence of a crossing density (ρ ≃ 0.11 fm⁻³) in EOS observables, close to the mean density, has been empirically noted in previous works on symmetry energy [45], pairing gap [46] or the neutron EOS [47, 48]: when designing EDF with nuclear observables, the corresponding EOS is constrained not at the saturation density but rather around the mean density (the crossing density). In the case of incompressibility, due to this crossing area, a larger K_∞ = K(ρ₀) value for a given EDF can be compensated for by lower values of K(ρ) at sub-crossing densities, so to predict a similar energy of the GMR in nuclei: the GMR centroid is related to the integral of K(ρ) over a large density range [42]. This allows us to understand how EDFs with different K_∞ can predict a similar energy of the GMR, as noted in [41].

When the GMR is measured and well reproduced by a given EDF, it shall therefore not be correlated with the incompressibility of EOS at saturation density but rather with its first derivative M around the crossing density:

$$\begin{equation} M=3\rho K'(\rho)|_{\rho=\rho_c}, \end{equation} \tag{ 1 }$$

where K'(ρ) is the derivative of the incompressibility density-dependent term [44]. The linear correlation observed in figure 7 is striking, since the interactions employed can have a symmetry energy spanning from 30 to 37 MeV. Another striking feature is that the Gogny and relativistic EDFs are found on the same linear correlation [44] as the Skyrme one, showing the universality of the E_GMR versus M correlation, unlike the E_GMR versus K_∞ one: it is well known that relativistic EDFs can predict a similar E_GMR in ²⁰⁸Pb but with a significantly larger value of K_∞ [38]. It is therefore argued that measurements in nuclei constrain the EOS around the crossing density (namely the derivative of the EOS quantity at the crossing density) and deducing the values at saturation density mainly remains a model-dependent extrapolation. In the case of nuclear incompressibility, it is proposed to change the usual E_GMR versus K_∞ correlation plot and replace it by a more reliable and universal E_GMR versus M plot (figure 7), where M is the derivative of the incompressibility at the crossing density.

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Measurement of the GMR in stable isotopic chains [50] and in unstable neutron-rich nuclei [34, 35] will certainly open up the possibility to study a part of the isospin dependence of incompressibility. Experimentally, the measurement method of GR is different between stable and unstable nuclei. In the case of stable nuclei, the probe is sent on the target nucleus (direct kinematics). The 0 degree recoiling alphas are measured in the focal plane of a spectrometer, whereas the beam is usually deviated in a Faraday cup in the focal plane. The position of the recoiling alphas in the focal plane detectors provides their energy and scattering angle, allowing us to deduce the GMR energy spectrum and its angular distribution. A 0 degree mode is relevant because the ISGMR cross section is peaked at this angle. In the case of unstable nuclei, the ISGMR is measured in inverse kinematics. The recoiling alphas have a very low energy (a few hundreds of keV) and therefore the use of an active target is useful [34, 35]. In such a device, the detection gas of a time projection chamber-like detector also plays the role of the target. The gas can be helium, to perform (α,α') scattering. This setup allows for a 4π solid angle coverage and also for a low alpha energy measurement: there is no such straggling as in a (solid target + external detector) setup.