Octupole deformation in atomic nuclei is of interest for both nuclear structure and precision tests of fundamental symmetries, but identifying regions of octupole collectivity remains challenging. We analyze low-energy spectra of odd-mass nuclei and uncover a previously unrecognized empirical regularity that serves as a signature of octupole collectivity in neighboring even-even systems. The observed patterns, which can be understood within a core-coupling picture, are consistent with previous theoretical studies and lead to predictions for neutron-rich and proton-deficient nuclei. These findings provide a simple empirical guide for identifying promising candidates for future experiments and microscopic calculations.

Background: Understanding the experimental B(E2) values for Sn isotopes around ^110Sn has been a significant challenge in nuclear structure studies for over a decade. Both experimental data and many, though not all, calculations suggest a picture of the light Sn isotopes as being quadrupole-soft, that is, spherical, yet easy to deform. Purpose: To investigate the delicate interplay of quadrupole deformation and pairing correlations in these nuclides. In particular, by using slightly enhanced pairing, we ask: can we generate spherical mean-field solutions that describe the data? Method: First, we apply the standard spherical Skyrme HFBCS-QRPA calculation with default pairing parameters, allowing us to identify nuclides that are unstable against quadrupole deformation among Sn isotopes. Next, we moderately enhance the pairing strength to reproduce the experimental binding energy in the deformation-unstable isotopes. Result: Within our choice of Skyrme parameters and use of density-independent pairing, this moderate adjustment sufficiently stabilizes the HFBCS ground states against deformation, ensuring a successful QRPA calculation and, more importantly, leading to more realistic properties for the quadrupole 2^+ states. Conclusion: Careful attention to the sensitive interplay of pairing and shell effects in deformation-soft nuclides can be crucial to their correct descriptions. This sensitivity can be exploited to optimize the treatment of pairing in phenomenological approaches such as the present Skyrme-QRPA.

We give an updated view of the status and prospects of heavy-ion double charge exchange (HI-DCE) reaction studies performed at the Laboratori Nazionali del Sud of the Istituto Nazionale di Fisica Nucleare (INFN-LNS) in the context of the NUMEN project. The important role of HI-DCE for nuclear reaction, nuclear structure and double beta-decay investigations is outlined. A powerful way to scrutinize the nuclear response to HI-DCE is to consistently link it to the information extracted from the competing direct reactions pointing to a multi-channel description of the whole network of quasi-elastic processes. Indeed, these complementary studies are mandatory in order to minimize the systematic errors in the data analyses and build a many-facets and parameter-free representation of the systems under study.

We investigate the stability and softness of nuclei against quadrupole, octupole, and hexadecapole deformation. By applying the spherical Skyrme-force Hartree-Fock Bardeen-Cooper-Schrieffer quasiparticle random phase approximation, we diagnose ground-state deformation when imaginary solutions are obtained, i.e., the spherical ground state collapses. We also calculate the multipole polarizability in spherical nuclei with no collapse, as a measure of softness. This numerically light and theoretically sound method is found able to capture deformation patterns across the nuclide chart. The connection between the intrinsic shape of nuclei and the dynamics of their low-lying collective states is established and the role of shell structure is discussed.

Double charge exchange (DCE) reactions induced by heavy ions and other direct reactions characterized by same projectile and target are crucial tools to access information relevant for neutrinoless double beta decay nuclear matrix elements. In this context the NUMEN project aims to investigate, for each system of interest, not only the DCE channel but also the whole set of reactions promoted by the same projectile/target interaction in the same experimental conditions and within the same theoretical framework. An example of the application of such a multi-channel approach is presented here.

We review the status and prospects of heavy-ion double charge exchange (HI-DCE) reactions. Their important role for nuclear reaction, nuclear structure and double beta-decay investigations is outlined. From the experimental side the characteristically tiny cross sections for these processes and the high background generated by other more probable competing reactions is the main challenge, which has hindered HI-DCE spectroscopy until recent years. Modern magnetic spectrometers have proven to possess the right requisites to overcome past limitations, fostering the present and future development of the field. From the theory side, the description of the measured HI-DCE cross sections poses manifold challenges. Dealing with processes which involve composite nuclei, HI-DCE reactions can, in principle, proceed through several alternative paths. These, in turn, correspond to different reaction mechanisms probing competing aspects of nuclear structure, from mean field to various classes of nucleon–nucleon interactions and correlations. A powerful way to scrutinize the nuclear response to HI-DCE is to consistently link it to the information extracted from the competing direct reactions. Indeed, these complementary studies are mandatory in order to minimize the systematic errors in the data analyses and build a many-facets and parameter-free representation of the systems under study.

Double charge exchange (DCE) reactions induced by heavy ions are crucial tools to access information relevant for neutrinoless double beta decay nuclear matrix elements. In this context the NUMEN project aims to investigate, for each system of interest, the DCE reaction channel together with the whole set of reactions promoted by the same projectile/target interaction in the same experimental conditions and within the same theoretical framework.

The upgrade project POTLNS to produce high-intensity beams has already started at INFN- Laboratori Nazionali del Sud in Catania (Italy). The POTLNS project was triggered by the NUMEN physics case that aims to provide experimental information on the Nuclear Matrix Elements (NMEs) that enter in the expression of the neutrino-less double beta (0νββ) decay half-life. The tools proposed by NUMEN project are the cross-section measurements of nuclear Double Charge Exchange (DCE) reactions. The search for 0νββ decay is currently a key topic in physics, due to its possible wide implications for nuclear physics, particle physics and cosmology: the NUMEN project could provide a crucial contribution in this search.

Recent high-energy heavy-ion collision experiments have revealed that some atomic nuclei exhibit unusual softness and significant shape fluctuations. In this work, we use the fully self-consistent mean-field theory to identify all even-even nuclei that are unstable or soft against octupole deformation. All exceptional cases of enhanced octupole transition strengths in stable even-even nuclei throughout the nuclide chart are resolved and the origin is found in basic shell structure. The presence of atomic nuclei exhibiting significant softness to quadrupole-octupole deformation is suggested. These results represent a significant advance in our understanding of the underlying mechanisms of nuclear octupole deformation and have implications for further experimental and theoretical studies.

The excitation function of proton elastic scattering from $^{10}$Be at keV energy is calculated using the self-consistent Skyrme Hartree-Fock in the continuum method. The calculation successfully reproduces the narrow near-threshold proton-emitting resonance ($E_x = 11.4$ MeV, $\Gamma = 6$ keV, and quantum number $J^{\pi} = 1/2^+$) in $^{11}$B relevant to the $\beta$-delayed proton emission of $^{11}$Be. This supports the recent experimental result of Y. Ayyad \textit{et al.} at the ReA3 re-accelerator facility of the National Superconducting Cyclotron Laboratory (NSCL) at the Michigan State University. The resonance is interpreted as the $s_{1/2}$ single-proton resonance state in the Skyrme Hartree-Fock mean-field theory.

We study the influence of the Coulomb force on the Fermi beta-decays in nuclei. This work is composed of two main parts. In the first part, we calculate the Coulomb corrections to super-allowed beta decay. We use the notion of the isovector monopole state and the self-consistent charge-exchange Random Phase Approximation to compute the correction. In the second part of this work, we examine the influence of the anti-analog state on isospin mixing in the isobaric analog state and the correction to the beta-decay Fermi transition.

The use of double charge exchange reactions is discussed in view of their application to extract information that may be helpful to determinate the nuclear matrix elements entering in the expression of neutrinoless double beta decay half-life. The strategy adopted in the experimental campaigns performed at INFN - Laboratori Nazionali del Sud and in the analysis methods within the NUMEN project is briefly described, emphasizing the advantages of the multi-channel approach to nuclear reaction data analysis. An overview on the research and development activities on the MAGNEX magnetic spectrometer is also given, with a focus on the chosen technological solutions for the focal plane detector which will guarantee the performances at high-rate conditions.

The important step in understanding the nature of weak interactions is related to the idea of neutrino that restores conservation laws of energy and angular momentum. Weak interactions from the very beginning served as a natural laboratory for studies of symmetries. This chapter discusses the role of the weak interaction in nuclear structure and its use for the problems of fundamental symmetries when nuclei, atoms, and molecules can serve as natural laboratories where the complex many-body physics, in spite of its inevitable role as a complicated background, can enhance the manifestations of basic laws of nature. The non-conservation of parity in weak interactions was discovered in 1956 after analyzing the weak decay of K-mesons. Historically processes induced by weak interaction on nuclei such as beta-decay have been exploited to study the properties of the weak interaction and also investigate nuclear structure. One of the best examples were the experiments that determined the nonconservation of reflection symmetry in the weak interaction.

This white paper reports on the discussions of the 2018 Facility for Rare Isotope Beams Theory Alliance (FRIB-TA) topical program "From bound states to the continuum: Connecting bound state calculations with scattering and reaction theory". One of the biggest and most important frontiers in nuclear theory today is to construct better and stronger bridges between bound state calculations and calculations in the continuum, especially scattering and reaction theory, as well as teasing out the influence of the continuum on states near threshold. This is particularly challenging as many-body structure calculations typically use a bound state basis, while reaction calculations more commonly utilize few-body continuum approaches. The many-body bound state and few-body continuum methods use different language and emphasize different properties. To build better foundations for these bridges, we present an overview of several bound state and continuum methods and, where possible, point to current and possible future connections.

An updated overview of recent results on Heavy-Ion induced reactions of interest for neutrinoless double beta decay is reported in the framework of the NUMEN project. The NUMEN idea is to study heavy-ion induced Double Charge Exchange (DCE) reactions with the aim to get information on the nuclear matrix elements for neutrinoless double beta (0v ss ss) decay. Moreover, to infer the neutrino average masses from the possible measurement of the half- life of 0v ss ss decay, the knowledge of the nuclear matrix elements is a crucial aspect.guarantees the above mentioned performance and its relevance in the research for heavy-ion physics [8-10], also taking advantage of its coupling to the EDEN neutron detector array [ 11,12]. The Ca-40(O-18,Ne-18)Ar-40 reaction was studied at 15 MeV/u, showing that high mass, angular and energy resolution energy spectra and accurate absolute cross sections are at our reach, even at very forward angles, see ref. [13]. In addition, a schematic analysis of the reaction cross sections demonstrated that relevant quantitative information on DCE matrix elements can be extracted from the data. This result demonstrated that the previous experimental limitations are almost overcome and that high resolution and statistically significant experimental data can be measured for DCE processes. In the present experimental conditions, due to the limitation arising from the tiny cross-sections of the processes of interest, only very few systems can be measured within the 5-years project. In order to systematically explore all the nuclei candidates for 0v ss ss, a beam intensity at least two orders of magnitude higher than the present must be achieved. The NUMEN project is conceived in a long-range time perspective, in view of a comprehensive study of many candidate systems for 0v ss ss decay. Moreover, the project promotes a renewal of the INFN-LNS research infrastructure [14] with a specific R&D activity on detectors, materials and instrumentation. As a consequence, major upgrades of the detector technologies (3D ion tracker, particle-identification wall, gamma-ray array,...) must be developed [15, 20]. Also the target technology must be upgraded, to avoid the damage of the thin films due to the high temperature involved [21, 23]. Moreover, a deep and complete investigation of the theoretical aspects connecting nuclear reaction mechanisms and nuclear matrix elements must be carried out [ 24, 26].

The idea of NUMEN project is to study heavy-ion induced Double Charge Exchange (DCE) reactions with the aim to get information on the nuclear matrix elements for neutrinoless double beta (0νββ) decay. To infer the neutrino average masses from the possible measurement of the half-life of 0νββ decay, the knowledge of the nuclear matrix elements is crucial. DCE reactions and 0νββ decay present some similarities, the initial and final-state wave functions are the same and the transition operators are similar. The experimental measurements of DCE reactions induced by heavy ions present a number of challenging aspects, since they are characterized by very low cross sections.

The NUMEN main goal is the extraction from measured cross-sections of "data-driven" information on Nuclear Matrix Elements for all the systems candidate for 0 nu beta beta. The idea is to use as experimental tool Heavy Ions-Double Charge Exchange (HI-DCE) reactions. Crucial for the experimental challanges is the INFN Laboratori Nazionali del Sud (LNS) facility, made by the Superconducting Cyclotron (CS) and the MAGNEX magnetic spectrometer. The experimental measurements of HI-DCE reactions present a number of challenging aspects, since they are characterized by very low cross sections. Here it is reported the new experimental campaign of NUMEN Project.

The NUMEN project aims at accessing experimentally driven information on Nuclear Matrix Elements (NME) involved in the half-life of the neutrinoless double beta decay (0v beta beta), by high-accuracy measurements of Heavy Ion (HI) induced Double Charge Exchange (DCE) reaction cross sections. In particular, the (O-18,Ne-18) and (Ne-20, O-20) reactions are used as tools for beta(+)beta(+) and beta(-)beta(-) decays, respectively. In the experiments, performed at INFN - Laboratory Nazionali del Sud (LNS) in Catania, the beams are accelerated by the Superconducting Cyclotron (CS) and the reaction ejectiles are detected the MAGNEX magnetic spectrometer. The measured cross sections are challengingly low (a few nb), being the total reaction cross section much larger (a few b), thus a high sensitivity and a large rejection capability are demanded to the experimental set-up. This limits the present exploration to few selected isotopes of interest in the context of typically low-yield experimental runs. A major upgrade of the LNS facility is foreseen in order to increase the experimental yield of at least two orders of magnitude, still keeping the high sensitivity of the present set-up, making it feasible a systematic study of all the cases of interest. Frontiers technologies are going to be developed, to this purpose, for the accelerator and the detection systems. In parallel, advanced theoretical models are being developed in order to extract the nuclear structure information from the measured cross sections.

In this work, the Coulomb mixing and the isospin mixing in the ground states of even-even nuclei are evaluated in perturbation theory. The calculation of the isospin mixing is performed by using the connection to isovector monopole resonance properties. The uncertainty in the results that depends on different choices of the Skyrme interactions is shown. While Coulomb mixing turns out to be large in the ground states of heavy nuclei, isospin mixing is very small.

The possibility to use a special class of heavy-ion induced direct reactions, such as double charge exchange reactions, is discussed in view of their application to extract information that may be helpful to determinate the nuclear matrix elements entering in the expression of neutrinoless double beta decay half-life. The methodology of the experimental campaign presently running at INFN - Laboratori Nazionali del Sud is reported andthe experimental challenges characterizing such activity are described.