*4.2. Nuclei with N > 50*

The Sn isotopes represent the longest chain of semi-magic nuclei, which makes them attractive for studies of shell-structure evolution as a function of the number of neutrons and how it can be related to collective as well to single-particle effects. The known, almost constant excitation energy of the first 2<sup>+</sup> state in the Sn isotopic chain has been a textbook example of the seniority scheme for a long time.

A sensitive probe for correlations of this kind is to measure transition probabilities for first excited selected states, which will manifest the configuration content of those states. With this approach the results of LSSM calculations based on microscopically-derived interactions can be tested through direct comparison with experiment. This approached was recognized with the availability of radioactive beams, and new measurements or new theory values have been seen frequently in recent years. A recent update on 6+ states in Sn isotopes is reported in [61] for the 6<sup>+</sup> state lifetime and effective charge analysis. The energy of the second 2+ state in 102Sn was recently claimed in Ref. [121].

However, the main focus in the studies of Sn isotopes is on the first 2<sup>+</sup> states since the first measurement of radioactive isotopes 16 years ago [122], and the last ones being the theoretical study of Togashi et al. [77] and experimental study of Siciliano et al. [29]. The study of Togashi et al., represents the first approach in which it is possible to reproduce remarkably well the whole Sn chain (shown in Figure 2 of [77]) in same calculation.

The method used for this unified description of the detailed nuclear structure is Monte Carlo Shell Model including isospin conserving interaction calculation of the *gds* HO shell as well as the lower part of the neighboring HO shells [77] (and reference therein). An alternative calculation with a small modification is shown by the authors as to give an Ansatz to the experimentalists for a more precise experimental answer to the values in the mid-shell.

Within the generalized seniority scheme this mid-shell valley would be interpreted as changing single-particle orbitals filled along the Sn chain (see also [123]). In Figure 3 of [77] the authors describe the complex wave functions including core excitation (and deformation) for the 2<sup>+</sup> as well as the 4+ states in Sn isotopes towards the mid shell modified by the quadrupole component of the proton-neutron interaction, which was first postulated in [124] by the LSSM analysis for the 2+ states. The quadrupole collectivity was also predicted in Ref. [12].

The results, presented in Figure 3 by Siciliano et al. [29], show an overview of experimental knowledge on *<sup>B</sup>*(*E2*:2+→0+) for the full Sn isotopic chain based on intense efforts of many laboratories and experimental groups. References [122,124–144] represent a complete up-to-date list. The usage of general Doppler methods, such as the Doppler shift attenuation (DSAM) and the recoil distance Doppler shift methods (RDDS) to measure the lifetimes and to extract the *B*(*E2*) values, were hampered until recently by the existence of higher-lying isomeric states.

Indeed, the authors of Ref. [29] managed to overcome the problem by using multinucleon transfer reactions and adjusting the excitation energy of the final product such that the 6<sup>+</sup> isomer feeding was minimized allowing for RDDS measurements. Moreover, the experimentalists harvested the first information on the lifetimes of 4<sup>+</sup> states, which opened up the systematics of *<sup>B</sup>*(*E2*:4+→2+) states below *N =* 60. This pioneering experimental work was accompanied by LSSM calculations, which could well reproduce experimental data using the new realistic effective interaction in the *gds* model space with a proper monopole treatment.

Another calculation indicating for the first time the double-hump shape associated to the quadrupole dominance, as shown in Ref. [77], refer to the importance of further investigations of the 4+ states, where pairing effects related to single particle energies dominate instead. The new findings, together with the recent theory calculation [145] request for further experimental and theoretical effort in this direction.

The systematics of the reduced transition probabilities, the *<sup>B</sup>*(*E2*:2+→0+) values, is expected to be completed soon with the inclusion of the 102Sn value [146]. The review dedicated to nuclear collectivity is in preparation [147], where updated figure will be presented.

Alternatively, the collective properties of 100Sn can be approached again by studying nuclei with slightly lower *Z*. Several dedicated attempts for such experiments were undertaken at ISOLDE and NSCL. In particular, light Cd isotopes were addressed already in earlier days and Coulomb excitation transition probabilities and quadrupole moments were extracted up to 102Cd [70]. The latest update on this can be found in [148], where *<sup>B</sup>*(*E2*:4+→2+) values were also measured, and the accompanying theory work, which attempted to explain particular conditions for collectivity in light Cd isotopes [70].

The beyond-mean-field calculations, presented in Ref. [148], reproduce the cadmium systematics, but also predict rotational structures for all of the *Z =* 48 isotopes, breaking the common view of the textbook example of vibrational nuclei. In addition, there (and in Ref. [85] therein), Z = 48 isotopes are predicted to be semi-magic deformed nuclei. The lightest Cd isotope for which *<sup>B</sup>*(*E2*:2+→0+) value measurement was attempted thus far is 100Cd [70].

## **5. Summary and Outlook**

Recent years have shown a great deal of interest from experimental and theoretical groups from all over the globe dedicated to investigations of the 100Sn region. This materialized in many publications (referred to here and with more to come) in the last decade, as well as active and waiting proposals and, presently, a large amount of as-yet unevaluated data. The primary reason for the particular excitement is the relevance of this region for understanding the nuclear force in general and various specific aspects that can be uniquely studied in this region of the heaviest doubly-magic *N=Z* nucleus.

To further encourage a steady level of development and the need for new data of key nuclei and particular states, two examples are mentioned here. The first one is the search for excited states in 100Sn, which could be determined from the decay of the predicted isomeric 6+ state [93] (and references therein). The second is excited states in 98Sn (predictions presented in Figure 5), a mirror nucleus of 98Cd [149]. Those two nuclei likely constitute the heaviest possibly bound mirror pairs of all nuclei.

**Figure 5.** Predictions of the 98Sn excited states according to the available interactions. All of them suggest an 8+ isomeric state as the one known in the mirror nucleus 98Cd [149]. The JUN45 spectrum is identical to that one of 98Cd because of the isospin symmetry of this interaction.

**Funding:** This research received no external funding.

**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.

**Acknowledgments:** Hubert Grawe is gratefully acknowledged post mortem for his mentoring until his very last days. Without him, this paper would not be possible. Andrey Blazhev, Piet Van Isacker, Frédéric Nowacki and Taka Otsuka are acknowledged for useful discussions and support. Kathrin Wimmer is acknowledged for introducing the author to the basics of Python libraries as well as for running calculations with JUN45 interaction for high spin states in *N=Z* nuclei on a larger computer than available for the author. Zsolt Podolyák and Helena May Albers are acknowledged for valuable comments and proofreading of this article.

**Conflicts of Interest:** The author declares no conflict of interest.
