Magnetic giant resonances in the relativistic energy density functional theory

Magnetska pobuđenja predstavljaju fundamentalnu vrstu prijelaza u atomskim jezgrama bitnu i u modeliranju astrofizičkih procesa. U ovom radu istraženi su magnetski dipolni (M1) i kvadrupolni (M2) prijelazi u okviru relativističkog nuklearnog energijskog funkcionala gustoće (RNEDF) sa kontaktnim međudjelovanjem nukleona. Za opis M1 i M2 pobuđenja, uspostavljena je relativistička kvazičestična aproksimacija slučajnih faza (RQRPA) s uključenom izovektor-pseudovektor interakcijom i korelacijama sparivanja. Nakon provjere analitičkog pravila sume za M1 prijelaze, analizirana su M1 pobuđenja u jezgrama 48^Ca i 208^Pb. Istražena je evolucija snage M1 prijelaza duž izotopnih lanaca 36−64^Ca, 100−140^Sn i izotona Z=10-24, N=20, koja varira ovisno o M1 aktivnim nukleonima i spin-orbit cijepanju relevantnih stanja. Nedavno izmjerene snage M1 prijelaza izotopa 112−124^Sn kvalitativno se slažu sa teorijskim proračunom. Reproducirana je funkcija odziva eksperimentalnog spektra s dva vrha, a rezultati upućuju na potrebu manjeg gušenja žiromagnetskih faktora nukleona u jezgri u odnosu na prethodne teorijske rezultate. Razmatrana je uloga korelacija sparivanja u M1 prijelazima i pokazano je da su nužne za opis dva dominantna vrha iz eksperimentalnog spektra 50^Ti. Raspodjela prijelazne snage kvadrupolnih (M2) prijelaza pokazuje znatno fragmentiraniji spektar u odnosu na M1, dok analiza parcijalnih snaga prijelaza omogućava uvid u kolektivnost M2 pobuđenja.Electromagnetic excitations in finite nuclei represent one of the most important probes of relevance in nuclear structure and dynamics, as well as in nuclear astrophysics. In particular, various aspects of magnetic dipole (M1) mode have been considered both in experimental and theoretical studies [1–8] due to its relevance for diverse nuclear properties associated e.g., to unnatural-parity states, spin-orbit splittings and tensor force effects. Specifically, M1 spin-flip excitations are analog of Gamow-Teller (GT) transitions, meaning that, at the operator level, the dominant M1 isovector component is the synonym to the zeroth component of GT transitions, and can serve as probe for calculations of inelastic neutrino-nucleus cross section [9, 10]. This process is hard to measure but it is essential in supernova physics, as well as in the r-process nucleosynthesis calculations [7, 8, 11, 12]. The isovector spin-flip M1 response is also relevant for applications related to the design of nuclear reactors [13], for the understanding of single-particle properties, spinorbit interaction, and shell closures from stable nuclei toward limits of stability [14–18], as well as for the resolving the problem of quenching of the spin-isopin response in nuclei that is necessary for reliable description of double beta decay matrix elements [19]. In deformed nuclei, another type of M1 excitations has been extensively studied, known as scissors mode, where the orbital part of M1 operator plays a dominant role in a way that protons and neutrons oscillate with opposite phase around the core [3, 20–28]. In any nuclei undergoing experimental investigation, there are simultaneously present Eλ and Mλ multipole excitations, where the electric dipole (E1) and electric quadrupole (E2) responses [29–34] dominate over M1 response [35–43]. Thus, it is a rather challenging task to measure M1-related observables in a whole energy range. Even for the nuclides accessible by experiments, their full information on the M1 response has not been complete. The M1 transitions have been studied in various theoretical approaches. Various aspects of the M1 mode have been investigated in the shell model [9, 12, 14, 15, 22, 44, 45], including, e.g., scissors and unique-parity modes [22], tensor-force effect [14], low-energy enhancement of radiation [44], and the analogy with neutrino-nucleus scattering [9, 12]. The M1 energy-weighted sum rule has been discussed from a perspective of the spin-orbit energy [46]. The Landau-Migdal interaction has been one of the relevant topics in studies of M1 excitations [47, 48]. In order to reproduce a large fragmentation of the experimental M1 strength, the importance of including complex couplings going beyond the RPA level has also been addressed [26, 47–50]. Recently, the M1 excitation has been investigated in the framework based on the Skryme functionals [16–18], also extended to include tensor effects [51]. It has been shown that the results for the spin-flip resonance obtained by using different parameterizations do not appear as convincing interpretation of the experimental results. Additional effects have been explored in order to resolve this issue, e.g., the isovector-M1 response versus isospin-mixed responses, and the role of tensor and vector spin-orbit interactions [16, 17]. In recent analysis in Ref. [52], based on the Skyrme functionals, it has been shown that while modern Skyrme parameterizations successfully reproduce electric excitations, there are difficulties to describe magnetic transitions. In addition, some Skyrme sets result in the ambiguity that, by the same parameterization, the model cannot simultaneously describe one-peak and two-peak data for closed and open shell nuclei [16]. Thus, further developments of the Skyrme functional in the spin channel are called for [52]. Simultaneously, it is essential to explore the M1 response from different theoretical approaches to achieve a complete understanding of their properties, as well as to assess the respective systematic uncertainties. In this work we have introduced a novel approach to describe M1, 0+ ground state to 1+ excited state, transitions in even-even nuclei, based on the RHB + RQRPA framework with the relativistic point-coupling interaction, supplemented with the pairing correlations described by the pairing part of the Gogny force. In addition to the standard terms of the point coupling model with the DD-PC1 parameterization, the residual R(Q)RPA interaction has been extended by the isovector-pseudovector (IV-PV) contact type of interaction that contributes to unnatural parity transitions. This pseudovector type of interaction has been modeled as a scalar product of two pseudovectors. The strength parameter for this channel, αIV-PV, is considered as a parameter, which is constrained by the experimental data on M1 transitions of selected nuclei. We note that the IV-PV term does not contribute in the RHB calculation of the ground state, thus its strength parameter cannot be constrained together with other model parameters on the bulk properties of nuclear ground state. The pseudovector type of interaction would lead to the parity-violating mean-field at the Hartree level for the description of the 0+ nuclear ground state, and it contributes only to the RQRPA equations for unnatural parity transitions, i.e. 1+ excitation of M1 type. The coupling strength parameter αIV-PV is determined by minimizing the standard deviation σΔ (αIV-PV), where Δ is the gap between the theoretically calculated centroid energy and experimentally determined dominant peak position of measured M1 transition strength in 208Pb [38] and 48Ca [37] nuclei. It turns out that the optimal parameter value is αIV-PV = 0.53 MeVfm3 and in this case the energy gap Δ is less than 1 MeV both for 208Pb and 48Ca. In this way all the parameters employed in the present analysis are constrained and further employed in the analysis of the properties of M1 excitations. A recently developed non-pairing M1 sum rule in core-plus-two-nucleon systems [29] has been used as a consistency check of the present theory framework. In Ref. [29], the non-energy weighted sum (mk=0) of the M1 excitation was evaluated for some specific systems, which consist of the core with shell-closure and additional two valence neutrons or protons, e.g., 18O and 42Ca. If the pairing correlations between the valence nucleons are neglected, one advantage of that sum rule is that its non-energy weighted sum-rule value (SRV) is determined analytically for the corresponding system of interest. The sum of the M1-transition strength for 18O and 42Ca accurately reproduced the sum rule value (SRV), thus validating the introduced formalism and its numerical implementation for further exploration of M1 transitions. The present framework is firstly benchmarked on M1 transitions for two magic nuclei, 48Ca and 208Pb. The response functions RM1(E) have been explored in details, including their isoscalar and isovector components, that relate to the electromagnetic probe, as well as contributions of the spin and orbital components of the M1 transition operator. It is confirmed that, in nuclei without deformation, the spin component of the M1 transition strength dominates over the orbital one. Due to the differences in the gyromagnetic ratios, the isovector M1 transition strength is significantly larger than the isoscalar one, and they interfere destructively. It is shown that the major peaks of isovector spin-M1 transitions are dominated mainly by a single ph configuration composed of spin-orbit partner states. One of our interests was to investigate the role of the pairing correlations on the properties of M1 response functions in open shell nuclei, that has been addressed in the study of 42Ca and 50Ti. The RQRPA calculations show a significant impact of pairing correlations on the major peak by shifting it to the higher energies, and at the same time, by reducing the transition strength. In the 50Ti case, this effect is essential to reproduce the two-peak structure measured in the experiment [53]. The main effect of the pairing correlations is observed at the level of the ground state calculation, while it is rather small in the particle-particle channel of the residual RQRPA interaction. The M1 transition strengths from the present study appear larger than the values obtained from the experimental data. Therefore, it remains open question whether some additional effects should be included at the theory side, or some strength may be missing in the experimental data. In addition, the M1-excitation energies of light systems, e.g. 48Ca still have some deviation from the reference data. Recently, an experimental study based on the inelastic proton scattering provided novel data on E1 and M1 strength distributions along the even-even 112−124Sn isotope chain [54]. The resulting photoabsorption cross sections derived from the E1 and M1 strength distributions showed significant differences when compared to those from previous (γ, xn) experiments [55, 56]. This new experimental research enables us to explore the properties on M1 excitations in a broad range of even-even Sn nuclides, and examine the model calculations in a view of the mentioned experimental data from Ref. [54]. The quenching factors can be obtained by normalizing the calculations on M1 transitions to the experimental data. Accordingly, the free value of the g-factor (gfree) is often considerably quenched, leading to its effective value that was previously reported mainly as geff/gfree ≈ 0.6-0.75 [16, 17, 55, 57–59]. Therefore, in view of the quenching of the g-factors in finite nuclei it is interesting to explore how the novel inelastic proton scattering data [54] compare to the results on M1 transitions in the framework of the relativistic energy density functional. As previously discussed, one of the most important mechanisms responsible for the quenching is mixing with higher order configurations [48, 57, 59–62]. Additional effects have been suggested to arise from the core excitation [63], and the meson-exchange current effect [64, 65]. However, in our recent work [66] is shown that the M1 transition strength distribution is characterized by an interplay between single and double-peak structures, which can be understood from the evolution of single-particle states, their occupations governed by the pairing correlations, and two-quasiparticle transitions involved. Comparison with available experimental data shows that independent neutron and proton spin-flip spectra are correctly identified, single and double-peaked distribution of response function RM1(E) is reasonably well reproduced. The calculated peak positions Eth. peak show 1-2 MeV discrepancy, that could be further fine-tuned through additional adjustments of the strength parameter in the isovector-pseudovector channel, αIV-PV. In addition, it is shown that strength is considerably reduced than previously known, the quenching of the g-factors for the free nucleons needed to reproduce the experimental data on M1 transition strength amounts geff/gfree=0.80-0.93. Since some of the B(M1) strength above the neutron threshold may be missing in the inelastic proton scattering measurement, further experimental studies are required to confirm if only small modifications of the bare g-factors are actually needed when applied in finite nuclei. These data could allow us to establish an essential link between the M1 observables and theoretical models and improve our understanding of the role of M1 transitions in modeling radiative neutron capture cross sections of relevance for nucleosynthesis. A major interest in magnetic transitions is strongly focused on dipole (M1) excitations which was subject of our interest as well in our earlier work [66] and references therein. However, higher multipole, particulary 0+ ground state to 2− excited state transitions known as giant-quadrupole resonances (GQR), denoted as M2, are almost uncharted area of both theoretical and experimental research. Several available experimental references [67– 78] have elaborated highly fragmented M2 structure whose strength ∑/E BM2(E) is strongly suppressed compared to the few theoretical results [68, 73, 79–86] provided either in shell model or random-phase approximation (RPA) framework [87]. M2 quenching/suppression has been considered as a very important component in heavy stars modeling at later stage and before supernova collapse [88, 89] and nucleosynthesis processes [90]. Theoretical models which investigate nuclear dynamics with their basis in fluid-dynamics [91–94] predic activation of so called "twist mode" also experimentaly explored in Ref. [95] where 2− excitation in spherical nuclei is attributed to the orbital transitions caused by an effective rotation operator arround z axsis, rotation angle ”zα” is proportional to the z axis itself. In such picture one can imagine nucleon orbitals as different fluid layers which rotate in oposite dirrections, namely for z > 0 layers rotate counterclockwise while for z 2 [101] like M8 [102] or even higher spin states like M12 and M14 in Refs. [103, 104] have been part of experimental investigation. Exploration of M2 excitations showed consistent results when compared with earlier RPA and shell-model theoretical research, in particular for 16O, 48Ca, and 208Pb systems where strength values ∑/E BM2(E) are higher compared with the recent measurements. However, systematic partial strength analysis of complicated M2 fragmentation structure, in particular for 16O and 208Pb, showed very good agreement with the data. Pairing effects, modelled by the Gogny interaction with finite range, have shown tremendeous impact on spectra for open-shell nuclei by reducing the strength and shifting the centroid energy to higher values. This evidence convinced us that the model introduced in this work can be used for further thorough studies of M2 and higher multipole excitations. However, very fragmented structure of M2 transitions can not be explained without introducting a more sophisticated and technically demanding 2p2h excitations. In order to resolve still open questions, further theoretical and experimental developments, are needed, e.g., resolving the quenching effects in g factors, meson exchange effects, couplings with complex configurations, etc. Due to its relation to the spin-orbit interaction, M1 excitations could also provide a guidance toward more advanced RNEDFs. Recently, a new relativistic energy density functional has been constrained not only by the ground state properties of nuclei, but also by using the E1 excitation properties (i.e. dipole polarizability) and giant monopole resonance energy in 208Pb [105]. Similarly, M1 excitation properties in selected nuclei could also be exploited in the future studies to improve the RNEDFs. While in the previous studies the calculated ∑/E BM1(E) transition strengths considerably overestimated the experimental values, comparison of the RQRPA results for the total ∑/E BM1(E) strength with the new data on 112−124Sn isotopes from inelastic proton scattering [54] shows that differences are smaller than previously understood. Our analysis showed that discrepancy between model calculations and experiment are considerably reduced, i.e., the quenching of the g-factors for the free nucleons needed to reproduce the experimental data amount geff/gfree=0.80-0.93. Considering the fact that some of the ∑/E BM1(E) strength above the neutron threshold may be missing in the proton scattering experiment due to the reported limitations in accuracy [54], our analysis provides an indication that future experimental studies could confirm that actually very small modifications of the gfree factor are needed when applied in the nuclear medium in finite nuclei. Therefore, it is expected that this work can serve as guidance and motivation for the experimental community to systematically explore M1 resonant excitations, and in particular to reduce the uncertainties above the neutron threshold. Finally, complete understanding of M1 transitions within the framework used in this study, will also allow systematic large-scale calculations for the radiative neutron capture cross sections of relevance for the nucleosynthesis