Structure and astrophysical role of the neutron-rich $55 \le Z \le 92$ isotopes: status and perspectives

Abstract

Heavy neutron-rich nuclei are of great interest. Phenomena like shell evolution ($N\sim 126$, $Z\sim 82$), prolate–triaxial–oblate–spherical shape evolution ($Z=$ 70–80), possible deformed shell closures or structure change in the rare-earth region are under intense scrutiny. This latter is closely linked to the rare-earth r-process peak, while the $N \sim 126$ nuclei are connected to the third r-process peak at $A \sim 195$. Recent technical developments (e.g. increasing beam intensities at fragmentation facilities, new detection systems) provided huge amount of new experimental data, published in the last decade, allowing to probe structure and astrophysical models. Experimental methods and recent results are reviewed and future opportunities discussed.

1 Introduction

The nuclide chart showing all discovered isotopes consisted of 3338 isotopes at the end of 2022 [1]. There are about 3000 further isotopes expected to exist, the majority of these on the neutron-rich side. Unsurprisingly, the investigation of the properties of neutron-rich nuclei remains in the focus of modern nuclear physics research. This manuscript reviews the current status of our knowledge on heavy neutron-rich nuclei, defined here as the region between elements with $Z=55$ (cesium) and $Z=92$ (uranium). This review is timely, due to (i) the huge amount of experimental information obtained in the last 10–20 years, and (ii) the several new developments expected to address the region in the near future.

In the heavy neutron-rich region of the nuclide chart phenomena like shell evolution ($N \sim 126$, $Z \sim 82$), prolate–triaxial–oblate–spherical shape evolution ($Z=$ 70–80), possible subshell closures or structure change in the rare-earth region ($A\sim 165$), isomerism, etc. are under intense scrutiny. What connects this extended region is its importance for nuclear astrophysics. In order to get a deeper understanding of the rapid neutron-capture (r) process nucleosynthesis [2], the process which synthesizes about half of the isotopes heavier than iron in the solar system, it is paramount to understand the properties of the heavy neutron-rich nuclei. The solar system r-process abundance yield shows several peaks [2], which can be linked to nuclear stucture phenomena, the most prominent ones being at masses $A \sim 80$, 135 and 195, with a smaller one at $A \sim 165$. The $A \sim 195$ peak is the consequence of the magicity of the $N=126$ neutron number, very far from stability. The origin of the rare-earth abundance peak at $A \sim 165$ is debated, and possibly it is related to nuclear structure effects. These aspects makes the understanding of the region important beyond the field of nuclear physics. In 2017, gravitational waves were detected following a neutron-star merger. The light curve of its kilonova remnant was firmly linked to the r-process, providing further impetus to the study of the heavy neutron-rich region [3]. The progress achieved in the study of the region is illustrated in Fig. 1, showing the limit of our experimental knowledge, highlighting the more than 150 isotopes newly identified since 2010 [1, 4,5,6,7,8,9,10] (and references therein) at GSI Helmholtz Centre for Heavy Ion Research (GSI), RIKEN Nishina Center for Accelerator-Based Science (RIKEN), National Superconducting Cyclotron Laboratory (NSCL), Facility for Rare Isotope Beams (FRIB), Argonne National Laboratory (ANL), Istituto Nazionale di Fisica Nucleare-Laboratori Nazionali di Legnaro (INFN-LNL), Isotope Separator On Line Device (ISOLDE) and JAERI-ISOL (Japan Atomic Energy Research Institute-Isotope Separator On-line).

Fig. 1

Part of the Segrè chart, highlighting the isotopes first identified since 2010. These are color-coded according to the laboratory where they were first synthesized [1, 4,5,6,7,8,9,10] (and references therein). Black and grey boxes indicate the stable and unstable isotopes known before 2010, respectively

The extended neutron-rich $Z=$ 55–92 region is connected by technology. These nuclei can be populated by fragmentation, fission, multi-nucleon transfer (deep-inelastic, neutron-transfer etc) reactions. As a general statement, the most exotic isotopes were synthesized at fragmentation facilities, via in-flight fission of $^{238}$U (up to Z$\sim$68) and fragmentation of $^{208}$Pb, $^{238}$U and occasionally $^{198}$Pt. GSI was the leading facility addressing the region for a long time, relying on its high energy primary beams, needed for a clean identification of the reaction products. More recently, RIKEN with it higher primary beam intensity of the $^{238}$U beam, took over in the case of the fission products, and started to address also the region around $Z \sim 82$. Very recently the freshly commissioned FRIB facility produced new Tm–Yb–Lu isotopes using a primary $^{198}$Pt beam [10]. Such type of experiments allowed the study of the structure of the most exotic species via decay spectroscopy. Ground-state lifetimes were measured following $\beta$ decay, while short-lived isomeric states (<1 ms) provided information on excited states via internal decay.

Isotope Separation On-Line (ISOL) type facilities contributed with lots of physics in the case of moderately neutron-rich nuclei. For example, the molten lead target at ISOLDE, allowed the use of mercury isotopes in both decay experiments and in reactions after re-acceleration with HIE(High Intensity and Energy Upgrade)-ISOLDE. Similarly, decay experiments on the heavy end of the region ($Z>80$) provided lots of information. However, the extraction of nuclei from thick targets is chemistry dependent. Since refractory elements have extremely low volatility, their extraction is particularly difficult.

Multi-nucleon transfer reactions (deep-inelastic) offer an alternative production mechanism to fragmentation/spallation. The large cross sections predicted theoretically [11] and later (indirectly) determined experimentally [12] led to a revival of this technique. The best example so far is the KEK Isotope Separation System (KISS) facility at RIKEN [13, 14]. Following deep inelastic reactions using $^{136}$Xe beam on W and Pt targets, so far isotopes of Hf, Ta, Re, Os, Ir, Pt were successfully laser ionized and mass separated. The isomeric and ground-state decays of the extracted species were studied. In-gas-cell-laser spectroscopy were performed for Os, Ir and Pt isotopes [15]. The method works well for long lifetimes, allowing also the identification of long lived isomeric states and their spectroscopy. Deep-inelastic reactions using thick targets were also used. The most successful of these were performed at ANL, using the Gammasphere array, where a range of experiments were performed using different targets (W, Re, Os, Pt, Pb) (see e.g. [16]). These are high statistics experiments providing information on high-spin states, following prompt and short-lived isomeric decays. Notable is that there were several cases when the first information obtained from isomer spectroscopy following fragmentation were later used as a tool of ion selection in deep-inelastic experiments, thus extending the level schemes considerably (e.g. $^{190}$W, $^{191}$Re, $^{209}$Tl). Other techniques were also employed. For example using beams of $^7$Li and $^{18}$O allowed the transfer of two neutrons to the most neutron-rich stable isotopes, enabling the measurement of lifetimes of 2$^+$ states (see e.g. [17]) in Horia Hulubei National Institute for R & D in Physics and Nuclear Engineering (Bucharest, Romania) and Institute of Nuclear Physics of Orsay (Orsay, France).

We divide the region addressed in the present review article in four parts. Each of these are discussed in an individual section, followed by conclusions and outlook.

2 Nuclear physics in the $55 \le Z \le 70$ region

Considerable scientific attention has been focused on the neutron-rich rare earth region ($55 \le Z \le 70$) in the last decade. It is important to highlight from the astrophysical point of view that spectroscopic observations of old, metal-poor stars located at the halo of the Milky Way have revealed that the heavy element ($Z \ge 56$) abundances—including the rare-earth isotopes—are consistent with the scaled solar system abundances [2]. Additionally, atomic lines corresponding to rare-earth elements such as barium, lanthanum, and cerium were recently shown to be present in the near-infrared spectrum of GW170817/AT2017 [3]. Triggered by these observations, series of experiments were performed recently to provide nuclear physics data for studying to synthesis of these elements; a summary on the experiments are detailed in the following sections.

The rare-earth mass region has received special attention also due to nuclear physics phenomena. Namely, theoretical calculations (e.g. [18,19,20]) predict a deformation maximum close to $N \approx 104$ and $Z \approx 66$, however, there are experimental indications [21] showing that this lies at significantly lower proton and neutron numbers. Since deformation in this region can give rise to K isomerism, the neutron-rich midshell domain was examined in the last decade searching for metastable states and studying nuclear structure. These measurements provided also information on the deformed single-particle structure around $N = 100.$

2.1 The rare-earth r-process abundance peak

The abundance distribution of the astrophysical r-process is characterized by two main peaks located at $A \approx 130$ and $A \approx 195$. This feature was already in 1957 attributed to closed neutron shells occurring at magic neutron numbers in the pioneering works of Burbidge et al. [22] and Cameron [23]. However, there is a smaller, but distinct peak at $A \approx 165$, known as the rare-earth r-process abundance peak (REP). The REP is extremely sensitive on the nuclear physics input and thus offers an unique tool to probe r-process astrophysical models. However, at first the nuclear physical processes that determine the formation of the peak must be identified. Namely, since the abundance maximum is located further away from closed neutron shells, another mechanism must play a decisive role in the synthesis of these nuclei than in the case of the main peaks located at $A \approx 130$ and $A \approx 195$. Several peak formation mechanism were discussed in the literature, here a brief summary is given in order to show what nuclear physics inputs are needed for astrophysics calculations.

It was hypothesized by Surman [24] that the REP takes shape in later phases of the r-process when the density of available neutrons and the temperature significantly drop and thus the material starts to decay toward the valley of stability. Depending on the astrophysical scenario, different nuclear physics inputs are important in the peak formation. For example, if the temperature is high enough to maintain significant photo-disintegration flow [24] the combination of nuclear deformation and $\beta$-decay results in a mechanism which “funnels” the material into the peak region. Alternatively, in a lower temperature scenario the REP may form by a trapping mechanism involving slower neutron capture rates in the region compared to that above or below it [25]. The influence of the theoretical and individual neutron capture rates on rare-earth r-process abundance pattern was studied in details [26]. Furthermore, it was shown that the REP formation is sensitive to changes occurring at late times, e.g., to nonequilibrium effects, when the density of the free neutrons become very low ($Y_n/Y_{seed}\approx$ 10$^{-5}$) [27]. According to this complex picture, the understanding of the peak formation may allow to probe the freeze-out conditions [28]. To check all these models, experimental data is clearly needed. The mass number range to be investigated has been identified [25, 29] and numerous measurements aimed at studying nuclear masses and $\beta$-decay properties were carried out. These efforts and their conclusions will be detailed in the following sections and an insight into the ongoing experiments will be provided.

It should be mentioned that the REP may form also by fission cycling. This, perhaps less favored [30] scenario was addressed in several works [31,32,33]. Even today, large, model-dependent uncertainties are characterizing the calculated fission probabilities and fragment distributions, and thus further development of these models are clearly needed.

2.2 Nuclear data for astrophysical applications

We had limited knowledge on the $\beta$-decay parameters of the very exotic neutron-rich rare-earth isotopes before the 2010’s. The increase in the amount of available nuclear physics data is well indicated by the fact that in the last one and a half decade more than 80 new rare-earth isotopes were discovered at the world leading accelerator centers. For example, in 2012 a large scale survey was carried out at GSI and about 50 new neutron-rich isotopes with $60\le Z \le 78$ were identified for the first time and their production cross sections measured [5].

Given that REP is formed in the freeze out phase of the r-process, determining the half-life of nuclei in the nucleosynthesis pathway is of prime importance. For this purpose, since 2014 two large-scale $\beta$-decay surveys were carried out at RIKEN in the $55 \le Z \le 70$ mass region and as a result, the size of the $\beta$-decay database doubled. As an example Fig. 2 shows the available half-life data for neutron-rich ($A\ge 153$) promethium isotopes compared with the most recent theoretical predictions [37,38,39]. In the first experimental campaign, using the EUROBALL RIKEN Cluster Array (EURICA) [40], the structure of the rare earth isotopes were studied and the half-lives of more than hundred isotopes were derived [34, 41].

Fig. 2

Experimental half-lives of neutron-rich Promethium ($Z=61$) isotopes taken from the literature [34,35,36]. The data points shown with blue and red markers were obtained in the last decade). Lines show the theoretical values from three models [37,38,39]

Beta-delayed neutron emission also plays an important role in the formation of the REP [27]. However, before the experiments performed at RIKEN the heaviest isotope in this mass range with known $\beta$-delayed neutron emission probability (P$_{1n}$ value) was $^{150}$La, measured in the 90s [42]. The delayed neutron emission phenomenon was systematically studied in the rare-earth mass region in 2018 at RIKEN. The neutron-rich radioactive ions were implanted in the AIDA (Advanced Implantation Detector Array) system [43]. The AIDA implantation detector was surrounded by BRIKEN, the largest and most efficient neutron counter array ever built, to study the beta-delayed neutron emission phenomenon [44]. Details on conceptual design of the BRIKEN detector and further information on the analysis techniques can be found in [44,45,46]. The half-lives and P$_{1n}$ values were derived using an iterative approach, fitting the time distribution of the implant-$\beta$ particle and implant-$\beta$ particle-neutron (i-$\beta$-n)correlations. As an example, Fig. 3 shows the time distribution of the measured i-$\beta$-n correlations after implanting $^{162}$Pm isotopes into AIDA. The solid lines correspond to the total fit function (black) including an exponential component (green) and a fixed background (red) extracted from a linear fit to the backward time distribution of i-$\beta$-n correlations. As a result of this experiment the P$_{1n}$ value of more than 50 isotopes heavier than $^{150}$La were derived for the first time [35, 47].

Furthermore, recently the $\beta$-decay of neutron-rich Cs isotopes were studied also at the Isolde Decay Station [48, 49], while the P$_{1n}$ values of $^{144,145}$Cs were measured at the Californium Rare Isotope Breeder Upgrade (CARIBU) facility using a beta-decay Paul trap [50]. The significance of this last measurement is that it was done with a new experimental procedure and thus the data acts as consistency check and help reducing systematical uncertainties of the results derived at the fragmentation facilities.

Fig. 3

Fit to implant-$\beta$-1n time correlation histograms for the decay of $^{162}$Pm. The green line represents the total fit function, the red line show the background, respectively, and the black line indicates the parent decay

Figure 4 shows the available $\beta$-decay-related database available at present [34, 35, 42, 47,48,49]. The measured $\beta$-decay properties were used to probe the formation mechanisms for the high-mass wing of the rare-earth peak through astrophysical reaction network calculations. Using the variance-based sensitivity analysis method, variables—such as the half-lives of $^{168}$Sm and several gadolinium isotopes—determining the uncertainties of the calculated abundances under different astrophysical scenarios were identified [35]. Furthermore, the comparison between the measured $\beta$-decay data with theory shows the importance of systematic measurements along isotopic chains toward as neutron-rich isotopes as possible because at present theoretical models cannot consistently reproduce the trends of the experimental data (see e.g. Fig. 2) [34, 35, 47]. Most probably the reason for the discrepancy between the measured data and the theoretical predictions is the inaccurate knowledge of the Gamow–Teller transition strengths. Accordingly, it is essential to refine our knowledge on the strength distributions to increase the predictive power of the theoretical models and to provide solid ground for the astrophysical calculations. Measurements for this purpose are underway at RIKEN. It worth mentioning that the measured strength distributions can be used also to study the shape of the parent isotopes in special cases [51, 52], thus information on the deformation will be gained also from the new experimental data.

Fig. 4

The $\beta$-decay data available for modeling the formation of the REP. Stable isotopes are shown by black squares. The $t_{1/2}$ available at the beginning of the 2010s is indicated by gray boxes. The new $t_{1/2}$ data, measured in the last decade, is framed by the red line [34, 35, 41, 47,48,49]. The isotopes whose P$_{1n}$ value (or at least its upper limit) was known even before the 2010s are indicated by black circles [42] and the isotopes whose delayed neutron emission probabilities have been recently measured the first time are marked with green triangles [35, 47]. The astrophysically relevant mass region as presented in [25] is framed by orange lines

Neutron separation energies are one of the most influential input quantities for r-process calculations and thus the primary motivation of mass measurements. Indeed precise knowledge on the neutron separation energies, on the structure and presumable on the deformation of the exotic rare earth isotopes is crucial for modeling the peak formation. In the last decade the masses of about twenty rare-earth isotopes have been measured precisely at the JYFLTRAP double Penning trap (University of Jyväskylä) and at the Canadian Penning Trap mass spectrometer at the CARIBU facility [53,54,55,56,57]. Figure 5 shows two-neutron separation energies for the neutron-rich rare-earth isotopes. The data suggest that the neutron pairing in the very neutron-rich isotopes might be weaker than predicted by the theoretical models [53, 56]. In order to gain an insight into the nuclear structure features influencing the r-process, new data on more exotic isotopes are clearly needed [56, 57]. New mass measurements in this region using the ZD-MRTOF (Multiple reflection time of flight mass spectrometers installed at the Zero-Degree spectrometer) device are in progress at RIKEN.

Most probably during the r-process freeze-out the competition between $\beta$-decays and neutron captures shapes the REP while the material decays back toward the valley of stability. Therefore, to model the peak formation, it is necessary to have information on the cross sections of the neutron capture reactions. The effect of neutron capture rates (even on individual isotopes) on the peak formation has been extensively studied in the past (e.g. [24, 26] and references therein), however experimental data in the $A\sim 165$ mass region is still missing. In recent years innovative experimental techniques have been developed and used successfully in different parts of the nuclide chart, which in the future will enable the investigation of the (n,$\gamma$) reactions in this heavy mass region [58,59,60]. Traditionally, the theoretical (n,$\gamma$) estimates, used in the r-process simulations, are based on the statistical Hauser-Feshbach model that uses average quantities: nuclear level densities, photon strength functions and transmission coefficients. Using the $\beta$-OSLO technique, developed at NSCL [59, 60], from experiments performed with a total absorption gamma spectrometer the shape of the photon strength function is directly derived. An alternative approach, aiming the experimental determination of the $\Gamma _{\gamma }/\Gamma _{neutron}$ width ratio by measuring the $\gamma$-emission from states located above the neutron separation energy, was developed at the Valencia Group [58].

Fig. 5

Two-neutron separation energies for neutron-rich rare-earth isotopes. Data taken from [53,54,55,56,57]

2.3 Examination of the excited states

The synthesis of the rare-earth abundance peak can probably be linked to some kind of nuclear structure effect present in the mass region. However, even nowadays, only a fraction of the isotopes playing a key role in the peak formation are accessible experimentally. Thus understanding the structure of the extreme neutron-rich isotopes is crucial to develop nuclear models which predict the parameters of the relevant isotopes located outside the experimental reach.

The rare-earth mass region is at the forefront of interest also for strictly nuclear structure reasons. Namely, the doubly-midshell nucleus $^{170}$Dy lies exactly halfway between the closed neutron shells at $N = 82$, 126 and proton shells at $Z = 50$, 82. Accordingly, it has the largest N$_{\pi }$N$_{\nu }$ value (N$_{\pi }$ and N$_{\nu }$ are the number of proton and neutron pairs outside the closest magic number. This product is considered to be a good indicator of deformation [128]) of any isotope below $A = 208$ and thus it is naively expected to have the largest ground-state deformation. It can be assumed that the nuclear quadrupole collectivity increases with increasing values of N$_{\pi }$N$_{\upsilon }$ and the smoothly reducing E(2$^+_1$) energies and increasing E2 transition strengths (B(E2; 2$^+_1 \rightarrow 0^+_1$)) would indicate the onset of deformation [61]. Not surprisingly the properties of these isotopes were studied extensively using different theoretical frameworks. Indeed, mean field [62], Hartree–Fock [63] calculations were performed and also the region was studied using the finite range liquid drop model [18]. These calculations consistently predict the deformation maximum below the midshell at $N \le 104,$ however there are differences regarding the location. For example the the pseudo-SU(3) model expect a maximum collectivity at $Z=66$ and $N=100$ ($^{166}$Dy) or at $Z=66$ $N=102$ ($^{168}$Dy) [64] but the finite range liquid drop model [18] anticipate a peak in deformation at $Z=62$ or $Z=64$ and at $N=102$ ($^{164}$Sm or $^{168}$Gd)

Recently, using the EURICA germanium detector array the structure and isomeric states in neutron-rich neodymium, promethium, samarium, gadolinium, terbium and dysprosium isotopes have been studied (see e.g. [65,66,67,68,69,70,71,72,73]). Combined mass- and decay property measurement of $^{160,162}$Eu were carried out at the CARIBU facility at the ANL [55]. Furthermore, multi-nucleon transfer have been also used to access these nuclei [21, 74]. The resulted E(2$^+$) values for neutron-rich even–even $60 \le Z \le 70$ isotopes are shown on Fig. 6. The hypothesized onset of deformation approaching $N = 104$ is not observable but instead, an irregular behavior characterizes the measured data. A reduction in the E(2$^+$) values are clearly seen at $N=98,$ resulting in a local minimum for Dy and Gd isotopes. Even today, it is not clear whether the maximum collectivity is found in midshell isotopes or in nuclei with lighter mass numbers due to e.g. a subshell closure. Thus, to understand the macroscopic properties of the rare-earth isotopes, significant extension of the knowledge on their structure is clearly needed and experiments with this aim are in progress e.g. at GSI [75].

Fig. 6

Energies of the first excited 2$^+$ states in even-even isotopes of neodymium ($Z=60$), samarium ($Z=62$), gadolinium ($Z=64$), dysprosium ($Z=66$), erbium ($Z=68$) and ytterbium ($Z=70$) [36]

This region also offers an opportunity to study the interplay of collective and single-particle degrees of freedom at the neutron-rich side far from stability. So-called K isomers, which are non collective states, are predicted to appear in deformed neutron-rich nuclei with A $\ge$ 150. K is the projection of the total angular momentum on the symmetry axis. K isomerism arises from axially symmetric deformation, enabling the nucleus to be “trapped” in an aligned spin orientation relative to its symmetry axis [76]. The decay is K forbidden if the multipole order ($\lambda$) of the transition to a lower energy state with a different K is smaller than the change in the K value, i.e. $\Delta K \ge \lambda$. K isomers are characterized by the reduced hindrance factor of this transition: f$_{\nu }$ = ($T^{\gamma }_{1/2}$/$T^{W}_{1/2}$)$^{1/\nu }$, where $T^{\gamma }_{1/2}$ is the experimentally determined partial half-life of the isomeric state, and $T^{W}_{1/2}$ is the single-particle Weisskopf estimate. The hindrance factor is strongly correlated with forbiddenness of the symmetry breaking process, $\nu = {\Delta }K-\lambda$ and thus provides an insight into $\gamma$ softness (degree of rigidity towards triaxiality of a deformed nucleus) and deformation. In order to study these phenomena, the region was scanned, looking for isomeric states [65, 69, 70, 77, 78]. This systematical study revealed that the hindrance factors of K$^{\pi }$ = 4$^-$ isomers are very similar among $N = 100$ isotones. This suggests that there is no significant change in the single particle structure of neutrons for the $Z =$ 62–70 isotones at $N=100$ and the systematic of the excitation energies of these isomers can be explained without the predicted $N=100$ shell gap [70].

3 Shape transition in the $Z=70$–78 region

The overarching physics in the neutron-rich $Z=70$–80 region is shape evolution, see for example the global calculations of [79,80,81]. There is some astrophysical motivation as well, such as the possible effect of long-lived isomeric states during the second stage of the r-process, after the freeze out. Special case is that of the only isotope found naturally in its isomeric state on Earth, that of $^{180m}$Ta at 77 keV, the synthesis of which is under intense scrutiny [82].

This part of the nuclide chart is characterized by the presence of nuclei with competing prolate, oblate, triaxial and spherical shapes. The lighter isotopes are prolate deformed, and by adding more and more neutrons the shape becomes oblate, finally reaching spherical shape around the $N=126$ shell closure. Naturally, in the shape transition region shape coexistence is widespread. Shape transitional nuclei are difficult to treat theoretically, with the exact place and nature of the transition being dependent on the details of the specific model. Consequently, this region is considered to be an ideal testing ground for nuclear theories, and several calculations involving different models exist. While the detail varies, there is an overall picture emerging: the shape transition is smoother at higher Z, while the role of triaxiality also increases with Z. This is illustrated with the example of the shape calculations shown in Fig. 7 [83]. For lower Z nuclei, such as $_{70}$Yb and $_{72}$Hf, a sudden shape change is expected from well deformed prolate to oblate shape as more and more neutrons are added. In these cases triaxiality plays little role. In tungsten ($Z=74$) isotopes a transitional region with high $\gamma$ deformation or softness (depending of the theoretical calculation) is expected. Osmium ($Z=76$) and especially platinum ($Z=78$) isotopes are more $\gamma$ soft when compared to tungsten. While calculations vary with their predictions (compare for example the global calculations of [83] and [84]), what is clear is that variations are smaller with respect to Z and larger in N. In other words, the nature of the shape transition depends on the element, while the neutron number where it is predicted to happen it is highly theory dependent. Therefore, here we discuss each element individually, highlighting what is known experimentally up to date, and prediction for the heavier isotopes.

Nuclear shape cannot be measured directly, information on it is inferred from different observables. These are energies of low-lying states including those of the so called $\gamma$ bands (rotational band built on the $K=2$ vibrational state, typically the second 2$^+$ state in even–even deformed nuclei), transition strengths, observation of isomeric states, mean square charge radii etc.

Fig. 7

Example of shape calculation for even–even $Z=70$–78, $N=110$–122 nuclei, The data is from [83]. The $Q_0$-$\gamma$ planes were computed with the Gogny D1S interaction. $Q_0$ is the quadrupole moment roughly proportional to the $\beta _2$ deformation parameter. $\gamma$ indicates the degree of triaxiality, with $\gamma$ = 0$^{\circ }$ and $\gamma$ = 60$^{\circ }$ corresponding to prolate an oblate shapes, respectively. The contour lines are 0.25 MeV apart

In the case of $Z\le 72$ the shape transition region was not reached yet; all experimentally studied neutron-rich isotopes have well deformed prolate shape. The even–even hafnium ($Z=72$) isotopes are characterized by a roughly constant $E(2^+)$ energy in the region of 90–110 keV (see Fig. 8). They also exhibit 8$^-$ isomeric states interpreted to have two quasi-proton $\pi$7/2$^+$[404]9/2$^-$[514] configuration. The most neutron-rich isotope with such information available is $^{184}$Hf$_{112}$, although there are hints also for $^{186}$Hf from a NSCL experiment [8]. In addition, long lived (> 1 s), high-energy (> 2 MeV) isomeric states were identified in the $^{183,184,186}$Hf isotopes at the Experimental Storage Ring (ESR) at GSI [85, 86]. In $^{183}$Hf and $^{184,186}$Hf these were interpreted as prolate three- and four-quasiparticle isomers, respectively, based on multi-quasiparticle and potential-energy-surface (PES) calculations [85, 86]. Very recently, an experiment to search for the decay of the $^{183,184}$Hf isomers was performed at the KISS facility [13, 14] at RIKEN. The nuclei of interest were populated in the $^{136}$Xe+$^{nat}$W reaction. The reaction products were stopped in an argon gas cell, and the Hf atoms laser ionized and then mass selected [87]. The most neutron-rich hafnium isotope with experimental information is $^{187}$Hf. Within the RISING (Rare Isotope Spectroscopic INvestigation at GSI) campaign two $\gamma$-ray transitions were identified, but no interpretation was offered [88,89,90]. Moving into the so far uncharted region, the prolate 8$^-$ isomers based on the two quasi-proton $\pi$7/2$^+$[404]9/2$^-$[514] configuration are predicted to persist to higher masses [91]. Intriguingly, in $^{190,192}$Hf they are expected to decay into the oblate ground-state band and possible also $\beta$ decay [91]. In addition, a very long-lived 18$^+$ isomer in $^{188}$Hf is predicted. Based on its expected level of yrastness, its lifetime might be comparable with the 31 y $K=16$ isomer in $^{178}$Hf [92, 93]. Its four-particle structure contains the 8$^-$ two quasi-proton $\pi$7/2$^+$[404]9/2$^-$[514], which as discussed, is isomeric in the Hf chain and the 10$^-$ two quasi-neutron $\nu$9/2$^-$[505]11/2$^+$[615], which is isomeric in the heavier $N=116$ isotones $^{190}$W [94] and $^{192}$Os [36].

Tantalum isotopes up to $A=185$ are considered to have prolate shape. Testimony to this are the observed K isomers and the regularity of their rotational bands (e.g. for $^{185}$Ta studied in deep-inelastic reaction using a $^{186}$W target see [95]). $^{187}$Ta$_{114}$ is the most neutron-rich Ta isotope with extensive spectroscopic information. In the aforementioned ESR experiment two long-lived isomeric states were observed in it [85, 86] at 1789(13) and 2935(14) keV, respectively. In addition to their energies, lifetimes (for fully stripped ions) and decay modes were established. More recently, $^{187}$Ta was studied at the KISS facility at RIKEN [129], where it was populated in the $^{136}$Xe+$^{nat}$W reaction. The decay of an (25/2$^-$) isomer at 1778 keV with a half-life of 7.3(9) s was observed [96]. This is the same isomer as the lower energy one observed at ESR, where it was observed to both $\beta$ and $\gamma$ decay with a half-life of 22(9) s [85, 86]. The lifetime difference between the two experiments is explained by the fact that in the ESR measurement the ions are bare (without any electrons), therefore internal decay via conversion electron emission cannot take place. Internally the isomer decays into the rotational band based on the $\nu$9/2$^-$[505] configuration. The regularity of a rotational band is linked to the role of triaxility; regular rotational bands are characteristic of axially deformed rotors, while irregularities are the effect of departure from axial symmetry. The irregularities manifest themself as signature splitting [97], and the staggering can be quantified with the term $\Delta E$/2I, where $\Delta E$ is the energy of the $I\rightarrow I-1$ transition, is shown as function of spin on Fig. 9 for several $\nu$9/2$^-$[505] based rotational bands in the region. The $N=112$ isotones $^{185}$Ta and $^{187}$Re exhibit smooth behaviour, characteristic for rigid quadrupole rotors. In contrast, triaxiality causes staggering, with triaxiality increasing from $^{187}$Ta$_{114}$ to $^{189}$Re$_{114}$ to $^{191}$Re$_{116}$. Triaxial particle-plus-rotor model calculations performed for the $\nu$9/2$^-$[514] band quantify the triaxiality parameter $\gamma$, with values of 5, 18, 25 degrees for $^{187,189,191}$Re, respectively [98]. Total Routhian surface (TRS) calculations performed for $^{187}$Ta$_{114}$ and $^{189}$Re$_{114}$ [96] are in line with the above conclusions, but indicate $\gamma$ softness, instead of rigid $\gamma$ deformation. The increased role of triaxiality with increasing neutron number is evident also from the reduced hindrance determined for the E2 transition decaying from the isomeric state at $f_{\nu }=27$ in $^{187}$Ta compared to 71 in $^{185}$Ta. Note, that the ground-state of $^{187}$Ta was assigned spin-parity 7/2$^{+}$ based on $\pi$7/2$^+$[404] Nilsson orbital [99], determined from the $\beta$-decay work performed at KISS. Similarly to the case of $^{187}$Ta, in $^{186}$Ta a low energy long-lived isomeric state was observed at the ESR at GSI [85, 86], and its internal decay subsequently elucidated [100] at the KISS facility. The most neutron-rich Ta isotope with experimental information is $^{189}$Ta$_{116}$, in which isomeric decays were observed in two different RISING experiments [88, 101], and in one more recent one within the DESPEC (DEcay SPECtroscopy) campaign [102] at GSI. While the interpretation is presently still missing, most probably there are multiple isomers, suggested by the different extracted lifetimes.

Fig. 8

$E(4^+)/E(2^+)$ energy ratio and $E(2^+)$ energy systematic for even–even $_{72}$Hf, $_{74}$W, $_{76}$Os and $_{78}$Pt nuclei [36]

Fig. 9

Signature splitting as function of spin for rotational bands built on the $\nu$9/2$^-$[514] configuration. Data is from [95, 96, 98]

Tungsten ($Z=74$) isotopes have prolate shapes up to $^{188}$W$_{114}$. Coulomb excitation result of stable $^{184,186}$W are consistent with axially symmetric deformation [103]. $^{188}$W is also prolate deformed, based on its $E(4^+)/E(2^+)$ = 3.1 energy ratio, K-isomeric states and rotational bands built on them [94]. Its $B(E2;2^+ \rightarrow 0^+)$ transition strengths was determined from a lifetime measurement using the fast-timing technique. The ROSPHERE (ROmanian array for SPectroscopy in HEavy ion REactions) array consisting of high-resolution HPGe and fast LaB$_3$(Ce) scintillators [104] was used, while the nucleus was populated in the two-neutron adding $^{186}$W($^7$Li,$\alpha p$) reaction [17]. The decreasing transition strength, and increasing $E(2^+)$ energy, with increasing neutron number indicate that the transition region is being approached. $^{190}$W$_{116}$ is the most neutron-rich tungsten isotope with extensive experimental information, and it seems to be in the middle of the shape transition region. The energies of the low-lying yrast states were established following the decay of an isomeric state at GSI [105]. When compared to the region, the $E(4^+)/E(2^+)$ = 2.7 ratio is anomalous, being lower than expected (see Fig. 8). It seems that the origin of this low ratio is the unexpectedly large energy of the 2$^+$ state compared to the ground-state, as indicated by the systematic of the E(2$^+$) energies, also shown of Fig. 8 (note that one possibility is that actually the ground-state energy is anomalously low, however the uncertainties in the measured masses are too large to know whether this is the case [106]). A following $\beta$ decay work confirmed the energies of the $E(4^+$) and E(2$^+$) states, and tentatively identified a second 2$^+$ state at 426 keV [88]. This is the lowest value in the tungsten isotopic chain, and also suggests maximization of the $\gamma$ softness. The special status of $^{190}$W in the shape evolution region attracted a lot of interest. Experimentally the level scheme was extended, and the isomeric structures elucidated following its high-statistic population via deep-inelastic experiments with $^{136}$Xe beam on $^{186}$W and $^{192}$Os targets at Argonne using the Gammasphere array [94]. An 8$^+$ two-proton and a longer lived 10$^-$ two-neutron isomeric states were firmly identified. The decreasing K hindrance with increasing neutron number is consistent with a trend toward increasing triaxial softness. Recently, the lifetime of the 2$^+$ state in $^{190}$W was also measured following fragmentation of a $^{208}$Pb beam at GSI within the DESPEC collaboration [107]. Furthermore, masses of tungsten isotopes were measured up to $A=191$ [106]. There is a clear discontinuity at $^{189}$W, with the two neutron separation energy $S_{2n}$ value being lower than expected (see Fig. 5 of [106]), corresponding to higher mass than the trend would suggest. This discontinuity is a further attestation of a change in the ground-state structure of the tungsten isotopes. In conclusion, there are several experimental indications suggesting that $^{190}$W is a key nucleus for the understanding of the shape transition region.

$^{190}$W was the target of several theoretical works employing different approaches. The evolution of the ground-state deformation using self-consistent axially-deformed Hartree–Fock calculations, with a separable monopole interaction, of nuclei in the vicinity of $^{190}$W was examined [108]. It was concluded that the neutron number $N=116$ causes a coexistence of oblate and prolate shapes, with a weak dependence on proton number, thereby hindering the development of these isotones as well-deformed rotors. A further study based on energy-surface calculations, and allowing triaxial deformation, predicts that collective rotation of a deformed oblate shape is an energetically favored mode, especially for higher angular momenta at $I> 12$ [109]. It suggested that the 10$^-$ isomer may be an oblate shape isomer, decaying into prolate states. Similar conclusion was arrived in a work employing the projected shell model, which goes beyond the usual mean-field approximation. A change from prolate to oblate was suggested at spin $I=10$ driven by the rotation alignment of neutrons in the high-j $i_{13/2}$ orbital [110]. The tungsten and osmium region was studied also in the framework of the Interacting Boson Model (IBM). The IBM parameters were deduced by mapping the potential energy surface (PES) of the Gogny-EDF with quadrupole degrees of freedom onto the corresponding PES of the IBM system [111]. Maximum role of $\gamma$ deformation (softness), manifesting in minimum 2${_2}^+$ energies, were predicted for the $N=116$ isotones $^{190}$W and $^{192}$Os. One should note the difference between the Total Routhian Surface (TRS) and mean-field calculations regarding the role of triaxiality. While mean-field calculations predict a triaxial shape [112] with minimum at $\gamma \sim$ 30 degrees for $^{190}$W, the TRS calculations predict two minima, one prolate and one oblate [105, 109].

Several theoretical works puts the core of the shape transition region at $^{190}$W [105, 109, 112]. However, some studies put it at higher neutron number, to $^{192}$W$_{118}$ or even $^{194}$W$_{120}$ [84]. For $^{192}$W the only experimental information is the 2$^+$ energy obtained from the $\beta$ decay of $^{192}$Re [88] (see Fig. 8). In addition, an isomeric state in the odd-mass $^{191}$W was observed in two RISING experiments at GSI [88,89,90], but no interpretation was offered.

The odd-Z $_{75}$Re isotopes were partially discussed together with Ta. As indicated in Fig. 9, the rotational bands of the $^{187,189,191}$Re isotopes show increasing staggering, therefore increasing $\gamma$ deformation. All these data came from deep-inelastic experiments. Isomeric decays following fragmentation of a $^{208}$Pb beam were identified in $^{191,192,193,194,196}$Re [88, 101, 113]. The expanded level scheme of $^{191}$Re [98] obtained from a deep-inelastic experiments (using thick $^{198}$Os, $^{187}$Re, $^{186}$W targets at ANL with Gammasphere) is based on the knowledge of $\gamma$-ray energies from the fragmentation work. In $^{193}$Re the isomer is suggested to have $\pi$ 9/2$^-$[514] character decaying by a 145 keV transition into the $\pi$ 5/2$^+$[402] state [101]. However, in the case of the heavy odd-odd $^{192,194,196}$Re isotopes no level schemes were suggested. $\beta$ decay experiments provided further evidence of the role of triaxiality. In a KISS experiment, when for the first time rhenium was successfully extracted from an argon stopping cell, the ground-state spin-parity of odd–odd $^{192}$Re was tentatively determined as 0$^-$ [114]. Based on comparison with theoretical calculations a $\gamma$ triaxiality of over 30 degrees was inferred. Considering that the $\gamma$ value is increasing with neutron number (as discussed earlier), this suggests that the deformation will become oblate in heavier Re isotopes (note that $\gamma$ = 0$^{\circ }$ and $\gamma$ = 60$^{\circ }$ correspond to prolate and oblate shapes, respectively). The decay of the more neutron-rich $^{194}$Re was studied at GSI and three $\beta$-decaying states were identified [89, 90]. Nilsson multi-quasiparticle calculations predict both prolate (high-spin) and oblate (low-spin) configurations with spin-parities consistent with the observed decays, thus establishing shape coexistence in this nucleus.

In the osmium $(Z=76)$ isotopes the $\gamma$ softness is further increased. The 2$^+$ energy increases smoothly from $N=110$ to $N=118,$ followed by drastic increases for $^{196}$Os$_{120}$ and again for $^{198}$Os$_{122}$, see Fig. 8. The $B(E2; 2^+ \rightarrow 0^+)$ transition strength is decreasing with N. The most neutron-rich, $^{194}$Os, was measured using the ROSPHERE HPGe-LaBr3(Ce) array [104], with the nucleus populated in the $^{192}$Os($^{18}$O,$^{16}$O) reaction [115]. $^{198}$Os was interpreted as oblate, based on the observation of a 7$^-$ and 12$^+$ isomeric states with $\pi h_{11/2}d_{3/2}$ and $\nu i_{13/2}^2$ configurations, respectively [116]. Its level scheme is very similar to that of the $^{200}$Pt isotone. The claim of oblate shape was supported by TRS calculations. For $^{196}$Os the exact energies of the yrast states were measured relatively recently at the Legnaro National Laboratory [117]. The $^{82}$Se+$^{198}$Pt reaction with a thin target was used, allowing the identification of the beam-like particle with the PRISMA spectrometer. The $\gamma$ rays were detected with the AGATA (Advanced GAmma Tracking Array) demonstrator [118]. The interpretation of the shape in even-mass $^{188-198}$Os isotopic chain was based on yrast energies and potential energy surfaces calculated with the Gogny D1S interaction: $^{198}$Os was found indeed oblate, with $^{192-196}$Os rather $\gamma$ soft. Regarding the odd-mass osmium isotopes, in $^{193}$Os an isomeric state populated in the fragmentation of $^{208}$Pb was reported [101]. Subsequently, from a $^{192}$Os($^7$Li, $^6$Li)$^{193}$Os work, the isomer was identified as based on the $\pi$ 9/2-[505] Nilsson configuration, and a rotational band on the top of it was observed [119]. This means that $^{193}$Os is prolate. Conversely, a recent work suggests oblate shape supported by Interacting Boson-Fermion Model calculations based on nuclear density functional theory [120]. $^{195}$Os is also considered to be prolate, based on its three-qausiparticle K isomer [101, 113, 121]. The energy of its one-quasiparticle long-lived isomer was determined at the ESR [85, 86]. Within the KISS project, using the $^{136}$Xe+$^{198}$Pt reaction, the decay of its K = 13/2 $\nu$13/2$^+$[606] isomer into the $K=3/2$ $\nu$ 3/2$^-$[501] ground-state was established, and its half-life measured as 47(3) s [122]. The ground-state spin-parity was confirmed, from the observation of its $\beta$ decay [123], also at KISS. In contrast, $^{197}$Os is considered to be oblate [101, 121], with a level scheme very similar to that of odd-mass Pt isotopes, and also $^{198}$Os, but built on the 13/2$^+$ $\nu i_{13/2}$ state. Similarly, $^{199}$Os$_{123}$, which is the most-neutron-rich osmium isotope with spectroscopic information [101], must have an oblate shape. The observed isomeric transitions were suggested to originate from the existence of a high-spin $\nu i_{13/2} \pi h_{11/2}^2$ isomer, but no decay scheme was proposed [101]. Therefore, the shape-transition happens between the prolate $^{195}$Os and oblate $^{197}$Os [121], with $^{196}$Os being in the middle of it. In-gas-cell laser ionisation spectroscopy measurements at KISS were performed on $^{194,196}$Os [124]. The obtained mean-square charge radii were used to determine the $<\beta ^2>^{1/2}$ quadrupole deformation. The 0.147(10) value for $^{194}$Os is in agreement with the 0.14(1) determined from the 2$^+$ lifetime measurement [115]. Interestingly a larger value was extracted for $^{196}$Os, at 0.157(15), which is unexpected since the $E(2^+)$ energy is higher in this nucleus. Also at KISS, the $\beta$ decays of $^{197,198}$Os were observed, resulting in the first measurement of the lifetime of the $^{198}$Os ground-state [125].

As indicated, nuclei crucial to the understanding of the shape transition region are $^{190}$W and $^{196}$Os. Both are rather close to stability, just two protons away from stable $^{192}$Os and $^{198}$Pt, respectively. Therefore, they can be reached by Multi Nucleon Transfer (MNT) reactions. The inverse plunger method developed at the Legnaro National Laboratory might be able to provide the lifetime of their 2$^+$ state. The method relies on MNT reactions using a lighter beam, $^{136}$Xe or $^{82}$Se, on a heavy target. The target like nuclei recoil backwards, explaining the name of the technique.

The odd-mass $^{193,195,197}_{77}$Ir nuclei have remarkably similar single quasiparticle level structures, with a 11/2$^-$ $\pi$11/2$^-$[505] isomer at $\sim$100 keV and a 3/2$^+$ $\pi$3/2$^+$[402] ground-state, with the 1/2$^+$ $\pi$1/2$^+$[400] state between them. In $^{195}$Ir, the observed three-quasiparticle isomer was interpreted as 27/2$^+$ with $\pi$11/2$^-$[505] $\nu$3/2$^-$[512]13/2$^+$[606] configuration [101]. Therefore, all these isotopes are thought to be prolate. In-gas-cell laser ionization of $^{196,197,198}$Ir was used to deduce magnetic moments [126], from where $^{197}$Ir was interpreted as being prolate. Isomeric states were reported in $^{198}$Ir [101, 113], as well as in $^{199,200,201}$Ir [101], but for none of these decay level schemes were suggested. Nevertheless, the several $\gamma$ rays observed in odd-mass $^{199,201}$Ir do not resemble rotational bands [101], suggesting that these are almost spherical nuclei. This is in line with theoretical calculations [79, 81], which suggest oblate deformation with $\beta <$ 0.1. Experimental information exists also on $^{202,203}$Ir [101]. These can be treated within the shell model, and they are discussed in the N $\le$ 126 section of this review. High-spin states, including isomeric ones were reported from deep-inelastic reactions in $^{191,193}$Ir [127]. While they were interpreted in terms of deformed Nilsson orbitals, the isomeric states did not show K hindrance. No information on high-spin states of heavier Ir isotopes were published, although the data must exist from the Gammasphere experiments performed on platinum targets.

Fig. 10

Ground-state quadrupole deformation in the $Z=$ 72–80 region using the FRDM (2012) model, taken from [81]. The FRDM(2012) model does not consider $\gamma$ deformation. Experimentally determined lifetime and mass limits, taken from [106, 131], are indicated with red and blue lines, respectively. The letters indicate the shape inferred from experimental data, as discussed in the text. pt indicates an elongated ellipsoid with some degree of triaxiality

Platinum $(Z=78)$ nuclei are characterized by much higher E(2$^+$) energies than their osmium isotones, increasing with N, and by a rather constant $E(4^+)/E(2^+)\sim 2.5$ ratio (see Fig. 8). They are $\gamma$ soft. $^{196}$Pt is the textbook example of the O(6) limit within the Interacting Boson Model [128] (the O(6) limit corresponds to a deformed $\gamma$-soft rotor). The structure of platinum isotopes changes little within A = 196–202. They all have a 7$^-$ isomeric state. The odd mass platinum isotopes up to mass $A=201$ have similar isomeric states, but built on the 13/2$^+$ $\nu i_{13/2}$ neutron level (see Fig. 13 of [101]). The 13/2$^+$ states are long-lived isomers, with their lifetime and internal decay measured up to $^{199}$Pt, but not in $^{201}$Pt and $^{203}$Pt [101]. These are all considered low deformation oblate nuclei. Laser spectroscopy measurements were performed within the KISS project for $^{199g,199m}$Pt [129] and $^{200,201}$Pt [130]. The extracted charge radii were interpreted to correspond to slightly deformed nuclei with $\beta _2 \sim$0.10. The higher mass $^{203,204}$Pt are expected to be spherical and can be discussed in the framework of the the shell model. This is also the case for the $Z>78$ elements, discussed in the next section.

Figure 10 summaries this shape transition section. The theoretical quadrupole deformation [81] predicted using the FRDM (2012) mass model is overlayed with shape characteristics inferred from experiments.

4 The $^{208}$Pb region

$^{208}$Pb is the heaviest doubly-magic nucleus. Although it is stable, it is difficult to populate neutron-rich nuclei in its vicinity. In addition to the interest in nuclear structure, the region is important also for nuclear astrophysics. The neutron $N=126$ magic number is linked to the r-process third abundance peak at mass $A \sim 195.$ This conveys a special status to the region, with facilities and projects devoted to its study. Experimental information during the last decades are from several laboratories. GSI with its $^{208}$Pb and $^{238}$U primary beams was “undisputed king” for the most neutron-rich nuclei (see Fig. 1). The first observation of large number of isotopes came from such fragmentation studies. Spectroscopic information were provided by the observation of isomeric states, abundant in this region. The technique is sensitive to isomeric lifetimes between roughly 50 ns and 1 ms. $\beta$ decays were also studied. Important contribution were obtained also from the use of MNT reactions, with the use of $^{208}$Pb usually as target, but occasionally as beam. Here the experiments performed at ANL using the Gammasphere array stand out. The ISOLDE facility made important contributions too, providing neutron-rich mercury beams from molten lead targets, both stopped and re-accelerated. UC$_x$ and ThC$_x$ targets were and will be advantageous for the study of other, mainly Z>82 nuclei (and possibly also gold) in the region. For the future, advances in identification at lower beam energies, combined with superior primary beam intensity will allow the fragmentation facilities RIKEN (first successful experiments already performed) and Facility for Rare Isotope Beams (FRIB) to study the most neutron-rich isotopes in the region. At longer timescales, the High Intensity Heavy-ion Accelerator Facility (HIAF) [132, 133] facility in China will also address this part of the nuclide chart.

The recent increased interest in this part of the nuclide chart resulted in the development of a new shell-model interaction [134]. The model space covers $50<Z \le 82$ for protons and $82< N \le 184$ for neutrons. A single one-particle-one-hole excitation is allowed across the $N=126$ gap. This reflects the fact that the core at $Z=82$ is robust, while in contrast, neutron core-excited states were observed at relativily low energies around 2 MeV in $^{209}$Pb and $^{208}$Tl. In $^{208}$Tl the lowest such state is at just 1807 keV with $\nu p_{1/2}^{-1} g_{9/2}^2 \pi s_{1/2}^{-1}$ configuration [135]. Masses, excited state energies, electromagnetic properties were examined with the new interaction, with the calculations giving a good reproduction of the known properties of nuclei in “below” lead. The new interaction was also used to address the competition of allowed and FF (First Forbidden) $\beta$ decay in the region [136].

4.1 Ground-state properties

Practically all works attempting to measure properties of neutron-rich $N \sim 126$ nuclei invoke astrophysical importance. Since the main motivation nowadays is the understanding of the r-process, we discuss the ground-state properties first. (We note that isomeric states can also play an important role in nuclear astrophysics, and such isomers are called astromers [137]).

The experimental information available on neutron rich nuclei around $^{208}$Pb is summarized in Fig. 11. The main nuclear physics input needed for r-process yield calculations are masses (which are used for (n,$\gamma$) and ($\gamma$,n) reaction cross-section calculations, $Q_{\beta }$ values etc.) and lifetimes. Along the $N=126$ isotonic chain, mass values are published “down” only to $^{206}$Hg [138], measured at the ESR at GSI, just two protons away from the doubly magic $^{208}$Pb. The mass of $^{205}$Au, both in its ground-state and long-lived isomer was measured in a recent MR-TOF measurement at GSI, with the data under analysis. Lifetimes along $N=126$ are known only for four nuclei “below” $^{208}$Pb, down to $^{204}$Pt [139]. In the last 15 years several measurements were performed at GSI, using both $^{238}$U and $^{208}$Pb beams, providing lifetimes of nuclei in the region [139,140,141,142,143]. The lifetimes and masses are the properties which are directly used to test global nuclear calculations. Other information on ground-state properties which are less used to constrain calculations are $\beta$-delayed neutron-emission probabilities and charge radii [144,145,146]. The $\beta$-delayed neutron emission probability was measured for only 7 nuclei (plus an upper limit) [142, 143], and with one exception there are all below 10%. The laser spectroscopy measurements for Bi, Pb, Hg [144, 145, 147] all show a steep increase in the charge radii after $N=126$, proving the magic nature of this neutron number (see Fig. 12). The rate of increase seems to be element independent.

Fig. 11

Status of experimental knowledge on the neutron-rich $N \sim 126$ nuclei. Determined lifetime and mass limits are indicated with red and blue lines, respectively [131]. The symbol $\times$ indicates that excited states were identified

Fig. 12

Charge radii normalized to the $N=126$ isotopes for Hg, Tl, Pb, Bi. The dashed lines shows the trend for the $N \le 126$ isotopes. The increase in the gradient after $N=126$ indicates that the $N=126$ is magic number. Data is from [144, 145, 147]

4.2 $\beta$ decay: allowed and first-forbidden transitions

The $\beta$ decay of the nuclei in the neutron-rich $N \sim 126$ region is different in two ways when compared to lighter nuclei. First-forbidden (FF) $\beta$ decays are expected to compete against allowed ones, and the effect of the $\Delta n=0$ selection rule for Gamow–Teller transitions becomes important (n is the number of nodes of the wave function). We discuss these two aspects one by one.

Global calculations show that on large part of the nuclide chart allowed $\beta$ decays (with selection rules $\Delta I=0$, ± 1 and $\Delta \pi$=No) dominate over first-forbidden ones ($\Delta I=0, \pm 1$, $\pm 2$ and $\Delta \pi$ = Yes). However, in the case of heavy nuclei, starting around $N=126$, FF transitions can become dominant, in stark contrast with lighter masses [39, 148]. The inclusion of the FF decays shortens the lifetimes of the nuclei on the r-process path, moving the calculated r-process abundances in the $A \sim 195$ peak towards higher masses [149]. Since the calculation of FF transitions is notoriously difficult, there are large discrepancies between lifetimes predicted by different theories.

The availability of molten lead targets at ISOLDE allows the production of clean mercury beams. In this way the $\beta$ decays of both $^{207}$Hg [150, 151] and $^{208}$Hg [135] were studied at the ISOLDE Decay Station. In a sense the decay of $^{208}$Hg into $^{208}$Tl is an ideal case for the study the competition between FF and allowed $\beta$ decays. $^{208}$Tl has a small number of both negative and positive parity levels with spin 0 or 1 below the $Q_{\beta }=3.48(3)$ MeV [203] energy, with simple and well-understood wave functions. Experimentally only three negative parity states, and no positive parity states were observed to be populated directly in the $\beta$ decay of $^{208}$Hg. Furthermore, the observation of the parity changing $0^+ \rightarrow 0^-$ $\beta$ decay where the daughter state is core excited is unique, and can provide information on mesonic corrections of effective operators [135]. Similarly to $^{208}$Hg, it was found that $^{207}$Hg also $\beta$ decays by FF transitions only [151].

Fig. 13

Shell model orbitals for $50< Z < 82$ and $126< N < 184$. The ordering and energy spacing of the orbitals are from the level schemes of $^{207}$Tl and $^{209}$Pb respectively [36]. Arrows link proton-neutron pairs which are forbidden in $\beta$ decay due to their $\Delta n=1$ condition [150]

A less well-known selection rule for the otherwise allowed $\beta$ decays is that the number of nodes (n) in the radial wave functions of the initial and final states has to be the same. This $\Delta n=0$ requirement plays a major role in the decay of heavy neutron-rich nuclei as there are several pairs of single particle $\Delta n=1$ orbitals for $N>126$ and $Z<82,$ see Fig. 13. The greatest impact of this rule is on nuclei where the Fermi level lies high above $N=126$ and/or much below $Z=82$, e.g. nuclei on the astrophysical r-process pathway, influencing the nucleosynthesis of heavy elements. On the other hand, this selection rule has little effect on isotopes which are proton-rich or close to the stability line. The most stringent test of the validity of the $\Delta n=0$ selection rule was provided by the $\beta$ decay of $^{207}$Hg, where the level of forbidenness of the $\Delta n=1$ $\nu 1g_{9/2} \rightarrow \pi 0g_{7/2}$ transition was studied. An upper population limit of $3.9 \times 10^{-5}$ was obtained, corresponding to log$ft>$8.8 (95% confidence limit) [150]. This selection rule is based on the property of the wave functions that their overlap integral vanishes, which is valid only if the proton and neutron feel the same potential. This is not strictly true as the Coulomb force is felt by protons only, and the different spin-orbit couplings (parallel and antiparallel) involved. While in $^{207}$Hg the selection rule was upheld, some global lifetime calculations suggest that in heavy nuclei about 20% of the decay is via these $\Delta n=0$ transitions [148]. In order to properly understand the relevance of this selection rule, experimental information from much more exotic nuclei is needed. The two available tests, those of the decays of $^{207}$Hg and $^{209}$Tl, are strongly reliant on small components of the wave functions obtained from shell model calculations. In the case of the $^{207}$Hg, the probability that there are two holes in the $\pi g_{7/2}$ orbital in its ground-state is needed to determine the logft limits, which is rather low at the level of $\sim$10$^{-4}$ [150].

It was suggested that the competition between allowed and FF $\beta$ decays can be studied also on the other, proton-rich, side of the line of stability [152]. The $N< 126,$ $Z > 82$ quadrant exhibits this competition (although for $\beta ^+$/EC instead of $\beta ^-$ decay), and has the advantage of easier population, therefore the nuclei can be studied in high statistics experiments. Consequently, it might be an excellent testing ground for $\beta$-decay theories.

As mentioned, lifetime of several neutron-rich nuclei in the vicinity of $^{208}$Pb were measured in the last decade [139, 141,142,143]. While lifetime measurements and allowed vs FF decays competition are motivated by astrophysics, the global calculations [39, 148, 153,154,155] used in the r-process abundance calculations are somewhat simpler than the most advanced ones (see Fig. 3 in [142] and Table 1 in [156]). They are not able to describe the decay properties of nuclei close to stability which depend strongly on individual nuclear states (same goes for $P_n$). Nevertheless, they might work very well for more exotic regions, which cannot be proven as no experimental information exist. Shell model might be able to reproduce much better the available information. Very recently the $\beta$ decay properties of the $N=126$ and $N=125$ nuclei were studied within the shell model, addressing the competition between allowed and first-forbidden transitions [157]. For $N > 126$ nuclei such calculations including FF transitions were performed recently for the first time. The presently available shell-model works cover the decays of $^{207}$Hg [158] and $^{208}$Hg, as well other nuclei in the region [136, 159, 160].

Finally, we note that in 2022 a $\beta$-decay strength measurement was performed at GSI. The decays of Hg–Pt isotopes with $N \sim 126$ were measured using the total absorption gamma-ray spectroscopy technique [161].

Fig. 14

Level schemes of $N=126$ nuclei “below” $^{208}$Pb [101, 163, 164, 166]. The 10$^+$ $\pi h_{11/2}^2$ states in the even mass $^{206}$Hg and $^{204}$Pt are drawn on the same level. Similarly, in the case of odd-mass $^{207}$Tl, $^{205}$Au, $^{203}$Ir the 11/2$^-$ $\pi h_{11/2}$ state was used as normalization. Isomeric states are shown by thick lines. Dominant configurations are also indicated

4.3 Neutron-rich $N \le 126$ nuclei

There is information on excited states of five neutron-rich N = 126 nuclei: $^{207}$Tl, $^{206}$Hg, $^{205}$Au, $^{204}$Pt and $^{203}$Ir. Here we discuss their structure (except core excited states, which are addressed later). The low-energy states of $^{207}$Tl provide the ordering and relative energies of the single-particle proton hole orbitals, which are used in shell-model calculations. These are $s_{1/2}$, $d_{3/2}$, $h_{11/2}$, $d_{5/2}$ and at much higher energy $g_{7/2}$, in this order, as shown in Fig. 14. The $s_{1/2}$, $d_{3/2}$ and $h_{11/2}$ proton-hole orbitals play the main roles in the yrast states of the lower Z isotones. The high-j $h_{11/2}$ orbital is responsible for the isomerism in the region, forming spin and seniority isomers. The observed E3 transitions are mediated by the $\Delta l=\Delta j$ = 3 $h_{11/2}-d_{5/2}$ proton-orbital pair, with the $d_{5/2}$ proton often providing only a small part of the wave-function of the yrast and close to yrast states [162]. Consequently these E3 transitions are not collective.

In its ground-state the $^{206}$Hg nucleus has a predominantly $s_{1/2}^2$ two-proton hole configuration. Its 10$^+$ isomeric state and its decay, shown on Fig. 14, was first observed following a deep-inelastic experiment at Gammasphere [163]. The 10$^+$ isomer (if core excitations are not considered) has clean $\pi h_{11/2}^2$ configuration. It decays by a low energy E2 transition into the 8$^+$ member of the same multiplet. In addition it decays also to the yrast 7$^-$ state by a $E_{\gamma }>$1 MeV E3 transition. The 7$^-$ state has a predominantly $h_{11/2}d_{3/2}$ configuration, with a $h_{11/2}d_{5/2}$ component, the latter one being responsible for the E3 decay. The 7$^-$ decays into the yrast 5$^-$ with $h_{11/2}s_{1/2}$. The 5$^-$ is also isomeric as the only identified excited state below it is the 2$^+$ yrast state with predominantly $s_{1/2}d_{3/2}$. The 5$^- \rightarrow$ 2$^+$ E3 decay is also mediated by the $h_{11/2}- d_{5/2}$ pair, relying on small $s_{1/2}d_{5/2}$ component of the 2$^+$ state. Note that the yrast 4$^+$ state predicted to have $\pi d_{5/2} d_{3/2}$ configuration is expected to lie above the 5$^-$ state. The first information on the three more exotic isotones, $^{205}$Au, $^{204}$Pt and $^{203}$Ir came from a single experiment performed in 2006 at GSI [101, 164, 165] using the fragmentation of a $^{208}$Pb beam and looking for the decay of isomeric states from a passive target. The long-lived 11/2$^-$ $\pi h_{11/2}$ isomer in $^{205}$Au was identified in a subsequent experiment, using an active DSSD stopper. The internal decay was identified by the measurement of the K and L conversion electron lines of the high-energy M4 transition [166]. The obtained level schemes are shown in Fig. 14. The structures of $^{206}$Hg and $^{204}$Pt are strikingly similar. They both have 10$^+$ and 5$^-$ isomeric states, with the same decay pattern. The difference is that the 2$^+$ state in $^{204}$Pt is of predominantly $d_{3/2}^2$ character, while in $^{206}$Hg it is mainly $s_{1/2}d_{3/2}$, and that the 7$^-$ in $^{204}$Pt is isomeric, with a relatively long lifetime. Its low-energy E2 decay transition into the 5$^-$ was not observed. In $^{205}$Au the 11/2$^-$ $h_{11/2}$ state decays both internally into the 3/2$^+$ $d_{3/2}$ ground-state by an M4 transition and by $\beta$ decay into $^{205}$Hg [166]. $^{205}$Au also has a 19/2$^+$ isomeric state, identified to have a $h_{11/2}^2 s_{1/2}$ configuration [164]. It decays by a low energy E2 and a $\sim$1 MeV stretched E3 transition. Note the similarity with the decay of the $h_{11/2}^2$ 10$^+$ isomers in $^{206}$Hg and $^{204}$Pt. The B(E2) and B(E3) transition strengths for the equivalent $^{206}$Hg, $^{205}$Au, $^{204}$Pt transitions are given in Table 1. The $B(E2; (h^2_{11/2}:10^+ \rightarrow h^2_{11/2}:8^+)$ transition strengths of $^{206}$Hg and $^{204}$Pt are very similar to the value of the analogous $B(E2; (h^2_{11/2}s_{1/2}:19/2^+ \rightarrow h^2_{11/2}s_{1/2}:15/2^+)$ in $^{205}$Au. The situation is similar for the case of the E3 transitions depopulating the same states. The B(E3) values are quite lower than 1 W.u. indicating that the $d_{5/2}$ orbital in the populated 7$^-$ and 13/2$^+$ states is not part of the main $h_{11/2}d_{3/2}$ and $h_{11/2}d_{3/2}s_{1/2}$ configurations, respectively. In $^{203}$Ir an isomeric state was also identified, by observing two transitions with energies $\sim$900 keV and, tentatively a much lower energy one. The level scheme is under revision following a higher statistics recent (2022) DESPEC experiment within the FAIR (Facility for Antiproton and Ion Research in Europe)-0 campaign [167].

Isomeric states are expected to abound also in the more neutron-rich $N=126$ isotones. 10$^+$ isomeric states are almost guaranteed in the even–even isotopes (due to their E3 decay, the lifetime is rather insensitive to the ordering of the 8$^+$ and 10$^+$ states). The existence of the 5$^-$ and 7$^-$ isomers depend on the ordering of the yrast 5$^-$, 7$^-$ and 4$^+$ states, which is difficult to predict. The odd-mass $^{203}$Ir must have a long-lived 11/2$^-$ isomer, similar to $^{205}$Au. Most likely it $\beta$ decays, but internal decay is also possible. In the more exotic odd-mass $N=126$ $^{201}$Re, $^{199}$Ta etc. (down to $Z=65$ $^{191}$Tb) the ground-state is expected to be 11/2$^-$ with $\pi h_{11/2}$ character. Here, low-spin 5/2$^+$ $\pi d_{5/2}$ isomers are expected. The ground-state properties of these nuclei are determined by the $\pi h_{11/2}$ proton, and the more exotic of them lie on the r-process path (exactly which ones depends on the astrophysical environment, see for example Fig. 1 of [149]). Therefore, the decay study of these nuclei will constitute the real test of the theoretical calculations of the nuclear properties, consequently measuring their lifetime and mass are very important. Large-scale shell model calculations predicting these quantities were published in [157]. The existing RIKEN and FRIB facilities and the future FAIR are well placed to provide such experimental information.

Table 1 Comparison of measured transition strengths in $N=126$ isotones $^{206}$Hg [163], $^{205}$Au [164] and $^{204}$Pt [165] mediated by the same single-proton proton orbitals. The values are given in Weisskopf units

Shell model is the preferred tool to address and predict the structure of nuclei close to doubly magic nuclei. The level scheme of $^{204}$Pt posed challenges to the standard shell-model parametrization using the so called Rydström interaction [168] (available in NuShellX and Oxbash). Therefore it was modified in three places in order to describe the known properties of $^{204}$Pt and $^{206}$Hg, namely to fix the ordering of the 5$^-$ and 4$^+$ states, to account for the very low $B(E2; 7^- \rightarrow 5^-)$ in $^{204}$Pt, and to fix the 7$^-$ to 5$^-$ energy difference in $^{206}$Hg, as described in [165]. Calculations using both the standard and modified parametrization are presented in [101], for both $N=126$ and $N< 126$ nuclei. Interestingly, the modified parametrization improved the description of other nuclei as well, e.g. the properties of odd-mass $^{205}$Au and $^{203}$Ir.

The first information on transition strengths involving the ground-state of any $N=126$ nucleus below $^{208}$Pb was provided by the Coulomb excitation of $^{206}$Hg [169]. The experiment was performed at a safe bombarding energy at the HIE-ISOLDE facility using the Miniball HPGe array. The 3$^-$ collective vibrational octupole phonon state was identified at 2705 keV, an energy similar to those of the analogous states in the Pb and other Hg even-mass isotopes. The deduced $B(E3)=30^{+10}_{-13}$ W.u value confirms that the octupole strength in Hg isotopes is reduced compared to the Pb ones, something observed for lighter isotopes. The lower value was attributed to a significant contribution of the $\pi s_{1/2}^{-1} - f_{7/2}$ excitation to the octupole phonon, which is missed in the mercury isotopes due to the lack of $s_{1/2}$ protons in their ground state. The $B(E2;2^+\rightarrow 0^+)$ reduced transition strength W.u. was also determined, the first such value for a nucleus “below” $^{208}$Pb. The experimental value is slightly lower than that predicted by shell-model calculations. The same situation is true for $^{204}$Hg$_{124}$, while the calculations are spot on for the two-neutron hole nucleus $^{206}$Pb$_{124}$. In addition, the measured $B(E3;5^- \rightarrow 2^+)$ transition strength in $^{206}$Hg (but also in $^{204}$Pt) is about an order of magnitude lower than the shell-model predicted value [101]. These all indicate a poor understanding of the wave function of the first excited state in the relatively simple two-proton hole nucleus $^{206}$Hg. It is not clear how this information can be used to improve the theoretical prescription of the more neutron-rich $N=126$ isotones on the r-process path.

Neutron-rich $N=125$ nuclei also exhibit isomerism, with isomeric states identified within the GSI campaign in $^{205}$Hg, $^{204}$Au, $^{203}$Pt and $^{203}$Ir. These were interpreted within the framework of the shell model [101]. In addition, $^{204}$Au was populated also following $\beta$-decay, at GSI, together with $^{203}$Au and $^{200-202}$Pt [170].

4.4 Neutron-rich Pb nuclei

The first orbital above the $N=126$ neutron shell-closure is the high-j $\nu g_{9/2}$. Consequently the even mass $^{210,212,214,216}$Pb nuclei exhibit an 8$^+$ seniority isomer with $\nu g^2_{9/2}$ configuration (seniority, denoted with v is the number of nucleons not paired to spin zero). Seniority is conserved for any interaction for single-j orbital with $j \le$7/2, while for $j\ge$9/2 orbitals seniority is conserved to a good approximation [171]. The isomers in $^{212,214,216}$Pb were all identified using the same technique, in decay spectroscopy following the fragmentation of $^{238}$U beams at GSI. $^{212}$Pb was observed in the pioneering work of Pfützner et al. [172], where the 6$^+ \rightarrow$4$^+$ and 4$^+\rightarrow$2$^+$ transitions were identified and the lifetime of the isomeric state measured with only 370 implanted ions. This work at GSI, together with that of Grzywacz et al. [173] at GANIL constitute the first isomeric decay spectroscopic studies following population in fragmentation reactions. The isomeric decays in $^{214,216}$Pb were identified much later [174], within the RISING campaign. The level schemes of all four lead isotopes are strikingly similar, as expected based on the seniority scheme. The reduced transition strength values $B(E2;8^+\rightarrow 6^+)$ as a function of the number of neutrons on the $\nu g_{9/2}$ orbital follow an inverse U shape, as shown in Fig. 15. This is the consequence of the properties of the matrix elements, as described in [128]. The transition strength is reduced in the middle of the U shape,with strength well below 1 W.u., in $^{212,214}$Pb. A similar trend can be observed for the odd-mass $^{211,213}$Pb nuclei, where the seniority $v=3$ 21/2$^+$ $\nu g_{9/2}^3$ state is isomeric [175, 176]. The analogue state in the more exotic $^{215}$Pb was not identified yet. Strikingly, the seniority scheme requires the seniority conserving 21/2$^+\rightarrow$17/2$^+$ transition (to the yrast 17/2$^+$ state at 1141 keV) vanish when the $g_{9/2}$ orbital is half filled, i.e. in $^{213}$Pb. Indeed, the measured $B(E2;21/2^+\rightarrow 17/2^+; \Delta v=0)=0.027(3)$ W.u. value is remarkably small [175]. The $v=3$ isomer decays predominantly into the non-yrast 17/2$^+$ state with seniority $v=5$ at 1260 keV. For this $\Delta v=2$ transition, which is expected to have maximum value for half-filled orbital nuclei, the strength is close to 1. W.u. The situation is somewhat similar for $^{95}$Rh where the $\pi h_{9/2}$ orbital is expected to be half filled, also shown on Fig. 15. One notes that the two 17/2$^+$ states do not mix in the single orbital approxition, even if they are close in energy, and that seniority is not expected to be conserved for a $j=9/2$ orbital.

Fig. 15

(Top) $B(E2; 8^+ \rightarrow 6^+)$ transition strength as function of the number of nucleons on the $j = 9/2$ $g_{9/2}$ orbitals for $Z=82$ [36, 174, 175, 177] and $N=50$ [36] nuclei. (bottom) $B(E2; 21/2^+ \rightarrow 17/2^+)$ strengths for $\Delta v = 0$ transitions in odd-mass nuclei. The $\Delta v =2$ seniority-changing transitions in $^{213}$Pb and $^{95}$Rh are also shown. In both figures the lead nuclei are shown with a thick blue line and their isomer half-lives are indicated. Data on $^{208,210}$Hg [178, 179] and $^{209}$Tl [180] are also shown, slightly shifted in X. The theory lines are also presented, based on the properties of matrix elements as given in [128]

4.5 The $Z<82$, $N>126$ region

It is difficult to populate $Z < 82,$ $N>126$ nuclei, consequently the experimental information is scarce. Fragmentation of $^{238}$U was used to populate nuclei in the region, and their subsequent $\beta$ and isomeric decays observed. In one such study, isomeric decays in $^{209}$Tl$_{128}$ and $^{208}$Hg$_{128}$ were identified [179], while a subsequent experiment provided information on $^{210}$Hg$_{130}$ [178], $^{211}$Tl$_{130}$ and $^{213}$Tl$_{132}$ [181]. In the even-even $^{208,210}$Hg isotopes the 8$^+$ $\nu g_{9/2}^2$ seniority isomers and their decays were observed. Their yrast structure upto 8$^+$ is dominated by the $g_{9/2}^2$ neutron excitations, therefore the level schemes are very similar to those in the even-mass Pb isotopes, although with slightly lower 2$^+$ energies. Similarly in $^{209}$Tl$_{128}$ the 17/2$^+$ isomer is interpreted as originating from the 8$^+$ $\nu g_{9/2}^2$ coupled to the $\pi s_{1/2}$ ground-state. The level scheme of $^{209}$Tl was later extended from a deep-inelastic experiment using the $^{136}$Xe+$^{208}$Pb reaction at Gammasphere [180]. Crucially, the low energy 47 keV 17/2$^+\rightarrow$13/2$^+$ transition was observed, together with the isomeric nature of the 13/2$^+$ state. The transition strengths determined for $^{208,210}$Hg (assuming that the unobserved 8$^+\rightarrow$6$^+$ energy is in the range of 20–80 keV) and $^{209}$Tl are also shown in Fig. 15. They show the same trend as the lead isotopes. Also, to note that in the case of the states with two particles on $\nu g_{9/2}$, by departing from the closed $Z=82$ shell closure the transition strength seem to increase, although the uncertainties are large. In the more neutron-rich $^{211}$Tl and $^{213}$Tl, only one and two delayed $\gamma$ rays were observed, respectively, indicating a deviation from the expected seniority-like scheme of $^{209}$Tl at a RISING experiment at GSI [181]. Very recently the $^{213,215}$Tl isotopes were populated at RIKEN and their isomeric decays studied. The reduction in energy of the 11/2$^-$ $\pi h_{11/2}$ state with neutron number provides the first experimental evidence of shell evolution in the $N>126$, $Z<82$ region [182]. We note that in $^{210}$Hg a second isomeric state was also observed [178]. It was speculated that this might be from a collective octupole 3$^-$ state, however a more recent experiment from RIKEN seems to contradict this [183].

As mentioned earlier, relatively clean neutron-rich mercury beams became available from molten lead targets ($^{207,208}$Hg $\beta$ decays [135, 150, 151], Hg radii [144, 145], $^{206}$Hg Coulex [169] and (d,p) [184]). Neutron excitations in $^{207}$Hg$_{127}$ were identified using the $^{206}$Hg(d,p) reaction in inverse kinematics at the ISOLDE Solenoid Spectrometer [184]. Spectroscopic factors were determined for a number of states involving the $g_{9/2}$, $d_{5/2}$, $s_{1/2}$ and $g_{7/2}$ neutron orbitals above $N=126,$ and their evolution deeper below the $Z=82$ proton core extrapolated.

The $\beta$ decay of $^{208}$Hg into $^{208}$Tl was discussed earlier[135]. The same work allowed building a level scheme for $^{208}$Tl. It is in agreement with the one obtained following the $\alpha$ decay of $^{212}$Bi [185], but contradicts the one from a previous $\beta$-decay study, where $^{208}$Hg was isolated from multi-nucleon reactions followed by chemical separation [186, 187]. The shell-model calculations describe well the level scheme deduced, validating the proton–neutron interactions used, with implications for the whole of the $N > 126,$ $Z < 82$ quadrant of neutron-rich nuclei [135].

4.6 Neutron-rich Bi and Po

8$^+$ isomeric states were observed in $^{212}$Po and $^{214}$Po. These are interpreted to have the same $\nu g_{9/2}^2$ structure then those in the lead and mercury isotopes. They follow the seniority rule, the $B(E2; 8^+ \rightarrow 6^+)$ value decreasing from 4.56(12) W.u. to 0.53(5) W.u [36, 188]. The 8$^+$ states are known in $^{216,218}$Po [189, 190] as well, from the $\beta$ decay of Bi isotopes performed at the ISOLDE facility, however they seem not to be isomeric (or possibly the lifetimes are below the limit of sensitivity, in the in order of ns-s). $\beta$ decays of odd-mass Bi isotopes $^{215}$Bi [191] and $^{217}$Bi [192] were also studied at ISOLDE, while the decay of more neutron-rich $^{219}$Bi at GSI within the RISING campaign [193]. Beta decays of neutron-rich Pb isotopes were studied both at GSI [193] and at ISOLDE. The most neutron-rich is that of $^{215}$Pb, studied at ISOLDE [194].

We note the existence of several $\alpha$ decaying high-spin isomeric states in the region, consequence of the high-j orbitals both above $Z=82$ and $N=126.$ In $^{211}$Po and $^{212}$Po these spin-trap isomers with spin-parity 25/2$^+$ and 18$^+$ [36] have $\pi h_{11/2}^2 \nu g_{9/2}$ and $\pi h_{9/2}^2 \nu g_{9/2} h_{11/2}$ configuration, respectively. Two isomeric states with microsecond lifetimes populating the 18$^+$ 45 s isomer in $^{212}$Po were observed [195]. In $^{212}$Bi [196] and $^{213}$Bi [197] long-lived isomeric states and there decays were observed at the ESR at GSI. Further from the beta stability line, a (25/2–29/2)$^{(-)}$ $^{215}$Bi [192] $\nu g_{9/2} i_{11/2} \pi h_{9/2}$ isomeric state and its $\beta$ decay were observed, populated in the spallation of a ThC$_2$ target at ISOLDE.

4.7 Core-excited states

Core excited states were identified in a number of neutron-rich nuclei in the vicinity of $^{208}$Pb. Generally, these are high-spin studies on nuclei populated in deep-inelastic reactions. The observed core-excited states involve, ussually, the breaking of the neutron core as the shell gap is smaller than that for protons. The Gammasphere array at ANL were used for the majority of these studies, providing results on $^{207}$Tl [198], $^{206}$Hg [163], $^{208}$Pb [199], $^{210}$Pb [177]. The information on core-excited states in $^{209}$Pb is much scarcer [36], although high-statistics data from Gammansphere experiments must exist. Core-excited states were observed also from $\beta$ decay in $^{207}$Tl [151] and $^{208}$Tl [135]. In $^{208}$Tl the 0$^-$ and 1$^-$ states with $\nu p_{1/2}^{-1}g_{9/2}^2 \pi s_{1/2}^{-1}$ configuration were observed. This 0$^-$ state is isomeric, a rarity for such a high-energy low-spin state. Noteworthy that the study of the timing properties of excited states in $^{210}$Pb provided evidence for a long-lived $\beta$ decaying isomer in $^{210}$Tl [177]. The half-life and properties of the isomer are not known, but shell-model calculations suggest spin-parity 11$^+$ with $\pi s_{1/2 }^{-1}\nu g_{9/2}^3$ configuration, or alternatively 14$^+$ with $\pi s_{1/2}^{-1} \nu i_{11/2}g_{9/2}^2$. To be noted that vibrational octupole collectivity plays an important role in the whole region, therefore nuclei exhibit large number of E3 transitions. For example in $^{208}$Pb [199] and $^{210}$Pb [177] there were 10 and 4 E3 transitions identified, respectively, with different level of collectivity.

The shell-model description of excited states requiring core excitation attracted some interest. In $^{207}$Tl both yrast states from deep-inelastic studies [198] and low-spin states from $\beta$ decay were identified [151]. Shell model calculations, using two different interaction involving different model spaces, were performed. The number of particles breaking the core, denoted with t, was limitedt to $t=1$. Mixing between valence ($t=0$) and core-excited ($t=1$) configurations was blocked. The shell-model results showed systematic spin-dependent deviations [198] when compared to the experimental level scheme. The three-particle states were predicted at lower energies, while octupole ones at higher ones (this latter depends strongly on the size of the model space [200]). While the shell-model calculations with one particle breaking the core predict correctly the ordering of the high-spin states, their energies are compressed at high spins. This compression is an intrinsic feature of shell-model calculations using two-body matrix elements developed for the description of two-body states. It was concluded that in order to replicate all $t=1$ state energies accurately the inclusion of mixing with $t=0,2,3$ excitations is necessary [151], however presently such calculations are not computationally feasible for this mass region. In addition, the isotopic chain of the Tl isotopes on both sides of the $N=126$ magic number, considering core excitations, was addressed in [201]. Furthermore, it is to be noted that shell model calculations using the KHM3Y interaction [202] in the model space covering four full shells ($50\le Z \le 126$, $82 \le N \le 184$) was successfully used to reproduce octupole–phonon states, although with the limitation explained above. The calculations of octupole states were computationally feasible not only in the doubly-magic $^{208}$Pb [202], but also for nuclei with one or two valence particles $^{207}$Tl [151, 198] and $^{206}$Hg [169].

5 The Z = 85–92 region

First one needs to establish what neutron rich means for elements between Pb and U, where all isotopes are unstable. We consider the isotopes which are unstable against $\beta$ decay, using their masses [203]; we define neutron-rich isotopes those which have more neutrons than the $\beta$-stable isotope(s) for that element. Therefore the first neutron-rich isotopes are: $^{210}_{83}$Bi, $^{217}_{84}$Po, $^{216}_{85}$At, $^{221}_{86}$Rn, $^{220}_{87}$Fr, $^{227}_{88}$Ra, $^{226}_{89}$Ac, $^{231}_{90}$Th, $^{232}_{92}$Pa and $^{239}_{92}$U.

The reflection asymmetric (“pear shaped”) nuclei attract a lot of interest; both due to their nuclear physics properties, and the connection to the search for permanent atomic electric-dipole moment (EDMs). In the $_{86}$Rn–$_{92}$U region the nuclei are quadrupole deformed, and they also exhibit octupole collectivity. The nature of this collectivity, static (rigid octupole shape) or dynamic (vibrational octupole) can be determined based on the level energies and transition strengths. Deep-inelastic experiments using thorium and uranium targets gave access to the heavier part of this region. In the work of Cocks et al. [204] using an $^{136}$Xe beam on a $^{232}$Th target, and detecting the $\gamma$-ray transitions with Gammasphere, the yrast structure of even–even isotopes upto $^{222}$Rn, $^{230}$Ra and $^{234}$Th were examined. The energies of states in interweaving bands of opposite parity provided information on the nature of the ocupole collectivity. The transition strengths can be obtained from Coulomb excitation experiments, in which the re-accelerated beams at ISOLDE play the leading role. The topic of pear shaped nuclei is not specific to the neutron-rich nuclei, as defined above, and we refer the reader to the recent review of Ref. [205]. We note that on the neutron-rich side all Coulomb excited even-even isotopes, $^{222,224,226}$Rn and $^{228}$Ra, are considered to be octupole vibrational nuclei. No doubt that these studies will be extended into the more neutron-rich region, including odd-mass nuclei and lower proton numbers.

Fragmentation of $^{238}$U was used to populate the most neutron-rich nuclei below thorium. The experiment which for the first time at the FRS-ESR (FRagment Separator-Experimental Storage Ring) facility at GSI combined the discovery of isotopes along with accurate mass measurement addressed this mass region. It discovered and measured the mass of $^{236}$Ac [197, 206], while also providing information on decay lifetimes. In a separate work at the FRS-ESR 40 new isotopes for elements between $_{78}$Pt and $_{87}$Fr were identified in a single experiment [207] (note that ESR is able to measure the mass of single ions, however the transmission between FRS and ESR is low, therefore it cannot measure the most exotic nuclei). The $_{84}$Po–$_{87}$Fr region was recently, in 2021, populated in a FAIR-0 experiment [208]. The work used the DESPEC setup [209], consisting of both HPGe detectors and the FATIMA (FAst TIMing Array) LaBr3(Ce) array [210] for fast-timing. The investigation of octupole deformation played an important role in the motivation of the experiment. The ground-state lifetime of several neutron-rich isotopes was measured for the first time [211].

The lack of isomeric states in the region (all isomeric states observed following fragmentation of $^{238}$U above $_{84}$Po are “proton-rich”, see Fig. 3 of [212]) somewaht hindered the structure studies. Information about excited states are therefore from prompt $\gamma$ and $\beta$-delayed $\gamma$ experiments. We already mentioned the deep-inelastic experiments using Th and U targets. At ISOLDE, ThC$_x$ and UC$_x$ ISOL targets were used to populate neutron-rich isotopes in $\beta$ decays. Concentrating on the most-neutron-rich ones, information on the level structure of $^{232}$Ra came from the decay of $^{232}$Fr (Fr is easily ionized and extracted) [213], while on $^{228}$Fr from the decay of $^{228}$Rn [214]. With the installation of the ISOLDE Decay Station as a permanent setup, as well as beam cleaning techniques such as the LIST mode [215], this method has the prospect of providing plenty of new information in the future.

The most neutron-rich nuclei in the Z = 90–92 region were populated in MNT reactions. $^{242}$U was identified and its ground-state band studied using a radioactive target in a $^{244}$Pu($^{18}$O,$^{20}$Ne) experiment [216]. Similarly $^{239}$Pa [217] and $^{238}$Th[218] where populated in the $^{18}$O+$^{nat}$U reaction, and following radiochemical separation, their $\beta$ decay studied. While both $^{239}$Pa and $^{238}$Th were identified in the last century, MNT reactions are experiencing a revival, based on the development of unambiguous identification of the reaction products. Recently the KISS collaboration started the study of this region using MNT reactions with $^{238}$U beams. Highlights are the discovery of the $^{241}$U isotope and the first measurement of the masses of $^{241,242}$U [9].

6 Conclusions and outlook

There was great progress on the study of neutron-rich nuclei in the recent decades, motivated by both nuclear structure and nuclear astrophysics. Future facilities as well improved experimental techniques are largely designed to address this region (to compare techniques see e.g. [219]). Fragmentation facilities have had the edge so far for the study of the most exotic species. The future FAIR facility is designed to increase the beam intensities compared to GSI, providing high energy beams at around $E/A=1$ GeV. RIKEN and the newly commissioned FRIB can provide higher beam intensities, but at lower energies. Lower beam energies complicate the identification of the reaction products. Nevertheless, RIKEN has recently shown that it can do it successfully for neutron-rich $Z \sim 82$ nuclei. Similarly, NSCL has proven to be able to do the same for primary beam energy as low as $E/A=85$ MeV, identifying new isotopes in the hafnium region [8]. Very recently further new isotopes were identified at FRIB, using the same $^{198}$Pt primary beam, but with a higher energy of $E/A=186$ MeV [10]. The newly commissioned FRIB facility, with its expected highest primary beam intensities, is expected to be a leader in this mass region. Different parts of the nuclide chart can be best addressed using different primary beams. The most promising ones for the heavy region are $^{238}$U, $^{208}$Pb, $^{198}$Pt and $^{186}$W. The first three were already used at NSCL, therefore available also for FRIB. RIKEN’s strength is in $^{238}$U beam, but the use of $^{208}$Pb beam is under consideration. GSI routinely employs $^{238}$U and $^{208}$Pb, and recently developed a $^{170}$Er beam. In slightly longer term future, the new HIAF [132, 133] facility in China will come online, as well as RAON (Rare isotope Accelerator complex for ON-line experiment) in South Korea.

There are several projects using multi-nucleon transfer to populate neutron-rich nuclei. While the production cross-sections are larger than in the case of fragmentation, the sensitivity of the method is lower. In the case of thick targets (when the product stays in the target) one requires gamma-gamma coincidences for identification. Here laboratories with heavy beams (Legnaro, GANIL, Argonne) and large HPGe arrays are best placed. In the case of thin targets the extraction efficiencies are low. The KISS facility at RIKEN will continue to provide data, by increasing the range of elements it can laser ionize and by improving its extraction efficiency. Works at IGISOL (Jyväskylä) and the FRS ion catcher [220, 221] at GSI will continue. At ANL, the $N=126$ factory [222] will start operation soon. Other techniques might also play a role. It was shown that chemical separation can be used with high efficiency to identify elements such as Os [223], while result of photon induced reaction experiments seems to indicate that several protons can be removed from the target, producing neutron-rich residues [224].

To give an example for the expected progress, we point out that the radioactive beam facilities will allow to measure directly the properties of some of the $N=126$ r-process path nuclei for the first time. This will help to determine how much material goes into the fission region, and get better understanding of the importance of fission recycling (for a given astrophysical scenario) [225]. This, together with experimental nuclear data on the $Z \sim 60$ region might clarify the origin of the rare-earth r-process peak, weather it is related to discontinuities of nuclear properties (mass, neutron capture rate) or is the result of fission. While this review focuses on experimental studies, theoretical predictions remain essential for the cases which cannot be reached experimentally.