Shape isomers: status and perspectives across the nuclear chart

Abstract

The paper reviews the searches for shape isomers, which are “extreme” example of shape-coexistence phenomena in nuclei. They may appear when highly deformed structures, well localized in the nuclear Potential Energy Surface (PES) in the deformation space, undergo a significant change in shape to match the final state shape, thus leading to a substantially hindered $\gamma$ decay. This is in sharp contrast to the vast majority of shape-coexistence phenomena, where significant mixing in the initial and final state wave functions is found. So far, the most spectacular examples of shape isomerism are known in fissioning systems in the actinides, although only scarce experimental information is available from experiments conducted more than 20 years ago. Searches in lighter mass regions, guided by theory predictions based on different approaches and experimental investigations on superdeformed systems, are also presented. They point to $^{64,66}$Ni as additional examples of prolate deformed shape-isomer-like structures, although with much reduced hindrance with respect to the actinides. Their origin is strictly related to the action of the monopole component of the tensor force of the nucleon-nucleon interaction, which is expected to favor the appearance of deep secondary minima also in the PES of Sn nuclei with masses 112–118. Perspectives in current searches of shape isomerism with modern detection systems, in the actinides regions and in lighter masses, are discussed.

1 Introduction

Isomeric states may appear in a nucleus for different reasons, with half-lives covering a wide range, from less than a nanosecond up to years [1,2,3], or even stable in the rare case of $^{180m}$Ta [4]. In the case of the so-called spin traps, the nucleus has difficulties in changing its spin due to the large spin difference with respect to lower energy states—this implies the emission of high-multipolarity $\gamma$ rays. K isomers are instead associated to a significant change in spin orientation relative to a symmetry axis [5]. Shape isomers, the topic of this paper, form a special class of metastable states which may appear when the nucleus undergoes a significant change in shape, to match the final state shape. They arise when different shapes (i.e., spherical/normally deformed and strongly deformed) are well localized in minima of the Potential Energy Surface (PES), which may be separated by a high (up to few MeV) and thick barrier. As a consequence, they are characterized by a reduced overlap between initial and final state wave functions in the deformation space, what leads to a significantly hindered $\gamma$ decay.

According to Löbner [6], the Hindrance Factor (HF) for a $\gamma$ decay can be defined from the experimental B(E/M$\lambda$) reduced transition probabilities, relative to the Weisskopf estimates:

$$\begin{aligned} HF(E/M\lambda )= & {} \frac{T_{1/2\gamma }(exp)}{T_{1/2\gamma }(1 \ W.u. \ estimate)}\nonumber \\= & {} \frac{1}{B(E/M\lambda )(exp)_{W.u.}} \end{aligned}$$

(1)

where T$_{1/2\gamma }$(exp) is the experimental partial half-life of the $\gamma$ transition, T$_{1/2\gamma }$(1 W.u. estimate) is the Weisskopf estimate for the corresponding transition multipolarity, and B(E/M$\lambda$)(exp)$_{W.u. }$ is the experimental reduced transition probability in Weisskopf units (W.u.). From the systematics available in literature [6], typical HF values for E2 transitions (with $\varDelta K=0$) are of the order of 1, therefore HF>1 values correspond to retarded transitions, while HF<1 point to accelerated decays. This is the case, for example, of collective E2 transitions or decays between states with significant overlaps of wave functions.¹

In this paper, we focus on searching for shape isomers, which are “extreme” examples of shape-coexistence phenomena [7, 8], being associated to significantly hindered decay, i.e., HF>10 (this corresponds to B(E2;0$^+\rightarrow$2$^+$)<0.5 W.u. in the case of states with J$^{\pi }$=0$^+$, here considered to avoid effects of angular momentum). Such retarded decays are in sharp contrast with the modest hindrances (HF$\le$10) observed for the vast majority of shape-coexistence phenomena reported in literature, which are instead associated to a shape mixing [9, 10]. In this context, the very rare cases of fission shape isomers, in the actinides, are archetypal examples of shape isomerism, with hindrance of the $\gamma$ decay which can reach extremely large values (HF$\sim 10^7-10^9$ for E2 decays). Section 2 of this work is, therefore, devoted to a brief review of the study of fission shape isomers, including the current status of searches with modern detection setups and future perspectives.

At high spins, superdeformation is also an example of “extreme” shape coexistence. In these cases, significantly hindered decays (HF$\sim 10^2-10^4$) are observed to occur from superdeformed (SD) prolate states in the mass regions A>60 (with ps half-lives and located at several units of spins), towards spherical/normally-deformed lower-lying states. One may expect that the bandheads of these SD structures at spin 0, especially in the heavy A=190 mass region (where the barriers in the PES are predicted to be still sizable), can also result in remarkable examples of shape isomerisms, so far not observed. This subject is discussed in Sect. 3.1, taking as guidelines theory predictions based on different approaches.

In searching for additional examples of shape isomerism below the actinides region, one can start by investigating the decay properties of the 0$^+$ excitations between Z=18 (Ar) and Z=92 (U) isotopes [9, 10]. From such a survey, it is found that only a small number of 0$^+$ cases (about 30 out of more than 200 excitations) exhibit a firmly established and significantly hindered decay (i.e., HF>10). These cases are concentrated in classical regions of shape coexistence phenomena, namely around the Z=20 (Ca), 28 (Ni), 50 (Sn), 82 (Pb) proton shell closures and along the Z=36 (Kr), Z=38 (Sr) and Z=40 (Zr) isotopic chains. So far, for only two cases of 0$^+$ states, deexciting via hindered E2 decays, the retardation has been clearly attributed to a significant change of shape, involving configurations located in well-defined minima of the PES. They are the 0$^+_4$ states in $^{64}$Ni and $^{66}$Ni with half-lives >1 ps and $\sim$20 ps (corresponding to HF values >63 and 24), respectively [11, 12]. We refer to them as “shape-isomer-like” structures, due to their much-reduced hindrances with respect to the fission-shape isomers in the actinides. In these cases, the calculated potential barrier is still present, although less pronounced, and additional microscopic quantum mechanical effects may also contribute to vary the hindrance of the decay between states with different shapes, as discussed in Sect. 3.2.1. Finally, in Sect. 3.2.2, perspectives in the search for shape isomerisms in other regions of the nuclear chart, in particular around Zr and Sn proton shell closure, are discussed.

2 Shape isomers in the actinide region

The most extreme form of nuclear shape co-existence in the nuclear chart can be found in the actinide region, where a highly deformed second minimum in the nuclear potential energy can co-exist alongside a ground state with much lower deformation. The phenomenon was first discovered accidentally by Polikanov and collaborators in the 1960s [13], where it was observed that these isomeric states decayed via spontaneous fission with half livers typically more than 25 orders of magnitude shorter than the nuclear ground state. More puzzling still was the severe hindrance of the internal decay of these states via $\gamma$ emission from the second minimum to the first minimum. Later it became clear that a strong selection rule was in operation due to the highly deformed shape of the isomeric state sitting in the middle of a double-humped fission barrier potential [14].

An island of around 30 of such “fission isomers” is known today [15, 16] to extend from Uranium (Z = 92) to Berkelium (Z = 98) for neutron numbers between 141 to 151, with spontaneous fission half-lives typically ranging from several ns to several ms (see Fig. 1). However, since the preferred decay mode is spontaneous fission, it is difficult to gain information on the nature of these states beyond the half-life. Very little is known about the nuclear structure of these isomers, the excited states that exist in the deformed minimum and the $\gamma$-decay pathways which de-excite to the first minimum. Information on the inner and outer barrier heights and the potential energy pathway to fission are needed for benchmarking models of actinide nuclei which are important for energy applications.

Fig. 1

Portion of the nuclear chart showing the actinide region and the known and predicted shape isomers. On the right, schematic drawings of the Potential Energy Surface (as a function of the axial deformation parameter $\beta$) displaying the main weakly deformed minimum, the strongly elongated secondary minimum where the shape isomer resides, and the fission barrier. The latter increases in size moving towards lighter systems, resulting in a progressive favorable competition for the $\gamma$-decay mode against fission

For shape isomers in actinides of the highest Z’s, the outer fission barrier is lower than the inner barrier due to Coulomb effects and the higher fissility parameter (Z$^2$/A) of these nuclei. Fission completely dominates as the decay mode and the $\gamma$/fission decay branching ratio here is expected to be extremely small. However, for the shape isomers with the lowest Z’s the situation is reversed and the $\gamma$ branch begins to dominate at Z$\le$ 93 (Np). It is in this region that significant knowledge could potentially be gained through high-resolution prompt and delayed $\gamma$-ray spectroscopy. Below Uranium (Z<92), many fission shape isomers are predicted [17,18,19], but nothing has yet been discovered. In this region the internal $\gamma$ back-decay will completely dominate and be the only detectable decay mode for these shape isomers. Hence $\gamma$-ray spectroscopy is the only tool available to study them.

Until present, the $\gamma$-ray spectroscopy of fission isomer back-decays were mostly performed during the late 1980s using the Darmstadt-Heidelberg crystal ball [16]. This is a 4$\pi$ NaI array in experiments especially designed to populate and select these rare decays (10$^{-6}$) from the huge background of events not passing through the shape isomer and from other unrelated decays.

2.1 Experimental techniques

The excitation energy of the second 0$^+$ shape isomeric state is typically around 2.5$-$3.5 MeV above the ground state. Hence, transfer reactions, especially (d,p) or (d,pn) around the Coulomb barrier (12–15 MeV) are the most well-suited since they leave the compound system with roughly the correct amount of energy to have a chance of populating the isomer. There is a narrow window in excitation energy needed to achieve this; not too much to provoke prompt fission, but sufficiently much to overcome the inner barrier of the second potential well at (4–5 MeV). Inelastic neutron scattering is also an excellent method, since there is no Coulomb barrier to overcome and the incident neutron energy can be tuned to the correct window [20].

Population of the shape isomeric state occurs very rarely and is typically 10$^{-5}$ to 10$^{-6}$ of the total reaction cross section for transfer reactions. The experimental signature of fission shape isomer decays is the detection of delayed radiation with respect to the nuclear reaction employed to populate them. Delayed fission fragments typically have 80 MeV of kinetic energy and hence deposit a large signal in a detector which is very easy to discriminate from other particles. Furthermore, since there are two fragments emitted per fission, detection efficiencies of close to 100$\%$ can easily be achieved with an appropriate fission-fragment detector geometry. Detecting delayed fission events from fission shape isomeric decays can either be achieved by shadowing prompt fission from the target with an appropriate collimator while catching the recoiling nuclei, or using accelerator beam pulsation to look for delayed fission events between beam bursts. However, detection of delayed $\gamma$ rays of 1–2 MeV from fission shape isomer decay is much more difficult due to typical photo-peak detection efficiencies of a few percent, even for large detector arrays. Furthermore, the large background from unwanted processes such as $\beta$ decay and isomeric decay of fission fragments makes detection very hard.

2.2 Back-decay of the shape isomer in $^{236}$U

The key to isolating rare decays from the background is exploiting the energy balances of the reaction between prompt and delayed calorimetry, and deduction of the compound nuclear excitation energy from the energy of the outgoing charged particle in the transfer reaction. This allows a powerful event-by-event 2-Dimensional separation. The first parameter is the prompt missing energy (the difference between the Compound nucleus excitation energy and the total prompt $\gamma$-ray energy). The second parameter is the total delayed $\gamma$-ray energy. In a 2D histogram of these parameters, isomeric events show up on the diagonal at the energy of the E$_{0^+}$ shape isomer. After the selection of the peak along the diagonal, the prompt and delayed cascades can be separated into their constituent $\gamma$ rays. The energy spectrum of $\gamma$ rays in delayed cascades of multiplicity N=2 from the $^{236}$U shape-isomer population events can be seen in Fig. 2. It is from the coincidence relations of these $\gamma$ rays that the decay pathways can be deduced.

Fig. 2

Left panel: spectrum of N$_{\gamma }$=2 events which sum to the energy of the 0$^+$ state of the isomer at E$_{0^+}$=2814 keV in the $^{236}$U nucleus (with background events shown in the bottom part) [21, 22]. These rare events were previously selected in two dimensions using the prompt missing energy vs. delayed sum energy technique. Right panel: the corresponding deduced decay scheme from the 0$^+$ fission shape isomer (second minimum) to the first minimum in $^{236}$U

2.3 Back-decay of the shape isomer in $^{238}$U

The first report of a delayed $\gamma$ branch was from the known shape isomer in $^{238}$U by Russo et al., in 1975 [23]. They employed a single co-axial Ge(Li) detector of volume 25 cm$^3$ − the state-of-the-art at the time − to detect delayed decays after the $^{238}$U(d,pn) reaction. They reported a 0$^+$ to 2$^+$ transition of 2.514 MeV with the correct lifetime which appeared at a bombarding energy of 18 MeV, but was not present at 13 MeV. Two subsequent works also reported observation of a transition of such an energy, Kantele in 1984 using two 50 cm$^3$ Ge detectors and Steinmeyer in 1994 using a Ge of 180 cm$^3$ which detected $\gamma$ rays in coincidence with low-energy electrons [24, 25]. However, these previously reported decays were re-investigated with the Darmstadt-Heidelberg crystal ball. None of the previously reported transitions were observed. Again, in 2018 the reported decays were searched for with the nu-Ball1 high efficiency Ge spectrometer consisting of 106 Ge crystals (approx. 14,000 cm$^3$) [26] and exploiting the $^{238}$U(n,n’) reaction, known from fission studies to strongly populate the isomer at the millibarn level, relative to other reactions. None of the previously reported transitions were observed. Hence, doubts about the earlier results are justified.

2.4 $\gamma$-ray spectroscopy in the second minimum

Gamma-ray spectra of prompt decays within the shape isomer second minimum have only been measured for two nuclei, $^{236}$U and $^{240}$Pu (see Fig. 3). The first is from the $^{236}$U case using the Darmstadt-Heidelberg ball and has NaI resolution, so only a basic level-scheme could be proposed from the singles spectrum [16, 22]. The only high-resolution spectrum measured with Ge detectors was obtained by detecting prompt $\gamma$ rays in coincidence with delayed fission of the $^{240}$Pu shape isomer, which was populated with the $^{238}$U($\alpha$,2n) reaction [27]. A rich and complex decay scheme interpreted as configurations with different K-quantum numbers (K=0$^+$,1$^-$ and 2$^-$) was derived from the data. Rotational band transitions will be low in energy, strongly internally converted and hence unseen, but the groups of decays between different band heads and the 0$^+$ state appear to have energies in the 400–800 keV range and are hence easily observable.

Fig. 3

Left Panel: Spectrum of the prompt decay in the second minimum of the shape isomer in coincidence with delayed fission of $^{240}$Pu, taken from [27]. Right Panel: Spectrum of the prompt decay in the second minimum of the shape isomer in $^{236}$U in coincidence with identified isomeric $\gamma$ back-decay events in the crystal ball, taken from [16, 22]

2.5 K-isomerism

Evidence for the co-existence of both long and short components of delayed fission decay curves between beam pulses was presented for several U, Pu and Am isotopes [28]. This suggests that different metastable states exist in the second minimum which have a greater or a lesser stability against fission. This may be due to states with different K-quantum numbers. K, the projection of the single-particle angular momentum on the symmetry axis, is a good quantum number in this situation since the deformation is large and stable [29]. Hence, the existence of K-shape isomerism in the second minimum is certainly possible, and higher-K states may modify the barrier against fission due to the change in K-configuration that this requires. However, to gain information and to understand the nature of different possible K-states in the second minimum requires the use of high-resolution $\gamma$-ray spectroscopy. The signature of such decays in the lower-Z actinides would be extremely selective and proceed via “three step” (prompt-delayed-delayed) $\gamma$-ray cascades from K-isomer band heads (i.e., prompt, then delayed decay within the second minimum, followed by delayed decay from the second minimum to the first minimum). The study of K-isomerism in actinide shape isomers may also help understanding the stability of the super-heavy nuclei. Indeed, the survivability of high-K states in superheavy nuclei is currently a topic of great interest, where it has been observed that high-K states may have an extra stability than the ground state with a higher barrier against fission decay due to the added difficulty in changing configuration [30]. Hence, new spectroscopic information on K-isomeric states in the actinides could prove very useful since these states may be easier to populate and study than in the super-heavy nuclei.

2.6 Perspectives

Significant advances both in nuclear facilities and in multi-detector technology have taken place since shape-isomer studies were first performed 20–50 years ago. High resolution, high-efficiency tracking arrays such as AGATA [31,32,33] and GRETA [34, 35] will soon come online in their 4$\pi$ configurations providing unprecedented sensitivity. Hybrid spectrometers such as nu-Ball2 use fully digital electronics so data can be written in triggerless format at very high rates. This opens the possibility to study the rare low-multiplicity $\gamma$ decays associated with the population of fission shape isomers, while also being able to exploit prompt and delayed 4$\pi$ calorimetry for each event. Nuclei with predicted, but as yet undiscovered, shape isomers could potentially be studied in the low-Z actinides using transfer reactions on $^{232}$Th, $^{231}$Pa, $^{233}$U and $^{237}$Np targets and modern $\gamma$ spectrometer arrays. New facilities such as NFS (Neutrons For Science, at GANIL) could use high-energy neutron induced (n,xn) reactions to reach nuclei such as $^{231}$U or $^{230}$Th, which are otherwise inaccessible via transfer reactions with stable beam/target combinations. Alternatively new approaches of $^{238}$U fragmentation and implantation of actinide recoils at GSI and Jyvaskyla could open the door to the study of shape isomers in the sub-thorium region (Ra, Fr, Ac, etc.)[36].

3 Shape isomers below the actinide region

3.1 The Pb–Hg region

Apart from the previously discussed island of shape isomers in actinides, the presence of secondary minima in the Potential Energy Surface (PES) with significant deformation, particularly at low spin, was predicted by various theoretical models in other parts of the nuclear chart. The systematic and comprehensive theory investigations based on Hartree–Fock–Bogoliubov calculations, macroscopic-microscopic models, as well as Energy Density Functional approaches, which began already in the 1980s, have identified the Z = 82 region, particularly on the neutron-deficient side (A = 190–210), as one of the most intriguing areas where shape isomerism could potentially occur [37,38,39,40,41,42,43,44] They consistently predicted complex potential energy surfaces which exhibit distinctive principal and secondary minima characterized by significant axial deformation, for instance, with $\beta _2$ values around 0.5$-$0.6 in systems such as Os, Pt, Hg, and Pb. In several cases, these minima differ in energy from the main, spherical minimum by 2–5 MeV and are separated by barriers ranging from 2 to 4 MeV. Such barriers also exhibit considerable stiffness in the direction of triaxiality. In all considered cases, appearance of a second minimum was characterized by considerably smaller inner barrier relative to the outer barrier, implying a decay mode fully dominated by $\gamma$ emission. Examples of the above-mentioned calculations are illustrated in Fig. 4, where the one-dimensional potential energy surfaces for Pt, Hg, and Pb isotopes with neutron number $N=104-114$ are plotted as a function of the quadrupole deformation parameter $\beta _2$ (upper axis) based on macroscopic-microscopic calculations by Pomorski et al., [44]. These calculations utilize the Lublin-Strasbourg-Drop (LSD) macroscopic energy along with shell and pairing-energy corrections. Deep secondary minima are present (see Fig. 4), in particular in $^{192-194}$Hg and $^{194-196}$Pb.

Fig. 4

Potential energies of Pt, Hg, and Pb isotopes with 104$\le$N$\le$114, as a function of the elongation parameter $q_2$ or quadrupole deformation $\beta _2$ (upper axis). The predictions, obtained from the macroscopic-microscopic model of Pomorski et al., [44], refer to local minima corresponding to reflection symmetric shapes. Red dotes represent experimental energies of SD minima, extrapolated at spin 0$^+$ (see Ref. [45]). (Adapted from [44])

Existence of deep, secondary minima in PES at spin zero, of which shape isomers observed in actinide nuclei are a peculiar example, are closely associated with the phenomenon of superdeformation in rotating nuclei [45,46,47,48,49]. It turns out that also in a rotating nucleus, the shell corrections can create, at a certain rotational frequency, a secondary minimum in the potential energy surface, which competes with the main PES minimum when the spin of the nucleus increases − at high rotational frequencies this minimum becomes a main PES minimum (see Fig. 5). The nucleus, after the formation (for example in fusion reaction) and cooling, can be trapped in a superdeformed minimum and, being separated from the normal deformed states by the barrier, decays by a long series of highly-collective E2 transitions connecting members of superdeformed rotational bands. As spin decreases during the decay, the two minima interchange their energy, but the nucleus remains trapped in the SD minimum, even though its energy is above the normally-deformed yrast line by several MeV (Fig. 5). Eventually, being in such unfavored condition, the nucleus decays out of the SD minimum at the spin of around 10 $\hbar$ (in nuclei with A$\sim$190) with retarded transitions. Information on the decay along the SD band and on the decay-out allows for estimation of the 0$^+$ band head energy by using extrapolation, as shown in Fig. 5. In Fig. 4, red dots correspond to the experimental 0$^+$ band-head energies (4–6 MeV above the ground state), which were extracted from the data by using the method described above. Their position coincides extremely well with the energies of the calculated secondary minima. This supports the strong complementarity of theory searches for shape isomers at spin 0, located in deformed secondary potential minima and the study of superdeformation at high spins.

Fig. 5

Schematic representation of the $\gamma$-decay along the normal (ND) and superdeformed (SD) yrast lines, as a function of spin, with sketches of the ND and SD minima at different representative spin regions (i.e., at the entrance region at the highest spins, at the crossing between the SD and ND yrast, and at the decay out). Red dots indicate the extrapolation of the SD yrast to spin 0, which may correspond to a shape-isomer state laying between 4–8 MeV excitation energy, depending on the mass region

The observation of hypothetical 0$^+$ SD band heads, which would lie in the SD minimum and be candidates for shape isomers associated with very elongated (SD) shapes, might be at the limit or even beyond the present experimental capabilities due to the high density of states above 4 MeV excitation energy, where both theory and SD bands extrapolations place these states. However, as mentioned earlier, theoretical predictions for possible shape isomerism at lower excitation energy generally exists in the Z$\sim$82, A$\sim$200 region.

Experimentally, in Pt, Hg, Pb and Po isotopes there are about 100 excited 0$^+$ states located at relatively low excitation energy, although information on the probability of their decay is available only in about 20 cases [9, 10]. For most of these decays, the hindrance factors are concentrated below 10, what is in line with a scenario of strong shape mixing. In four nuclei, however, there exist 0$^+$ excited states of which the E2 decay is hindered. They are the 0$^+_2$ excitations in $^{206}$Pb and $^{208,212,214}$Po. For these states a shell model interpretation exists, and their decays are rather not hindered by the barrier in a secondary potential minimum.

3.2 Other regions below mass A$\sim$200

The potential existence of deep secondary minima in the nuclear Potential Energy Surface (PES), which could give rise to the emergence of shape isomers in nuclei lighter than A$\sim$200, has been explored in the theoretical approaches mentioned earlier [37, 38, 40, 42], as well as in additional studies utilizing Hartree-Fock plus BCS calculations, where a comprehensive survey of numerous nuclei was conducted [50]. These investigations identified $^{66}$Ni and $^{68}$Ni isotopes (along with eight other Pt, Os, and Hg nuclei) where a deformed 0$^+$ state is separated from a spherical structure by a significantly high barrier. Other candidates were $^{74,76}$Kr, $^{78,80,98}$Sr, $^{80,82,100}$Zr, $^{86,100}$Mo, $^{88}$Ru and $^{116,118}$Sn. In recent years, Möller et al., [41] conducted a comprehensive study of potential energy surfaces across 7206 nuclei from A = 31 to A = 290, utilizing the well-benchmarked macroscopic-microscopic finite-range liquid-drop model of Ref. [40]. In this study, deep, secondary PES minima at spin 0 were identified in a few tens of nuclei, including $^{66}$Ni, $^{68}$Ni, as well as $^{114}$Cd and $^{116}$Cd. The most recent Hartree-Fock-Bogoliubov calculations based on the Gogny force, conducted by S. Hilaire and M. Girod [51, 52] also indicated the Z=28, 50 regions, in addition to Z $\sim$82 nuclei, as potential locations for deep secondary PES minima. Figure 6 illustrates potential energy surfaces for even-even isotopes $^{66}$Ni, $^{112}$Sn, and $^{190}$Hg, as predicted by these calculations.

As the $^{66}$Ni nucleus consistently emerges in all calculations as a system with a potentially significant secondary minimum, it was logical to examine the properties of this nucleus and its neighbouring isotopes in a more detailed manner. These investigations are discussed in Sect. 3.2.1, while Sect. 3.2.2 is devoted to ongoing searches in the Zr and Sn regions.

Fig. 6

Potential energy surfaces for even-even $^{66}$Ni, $^{112}$Sn and $^{190}$Hg, as predicted by most recent Hartree–Fock–Bogoliubov calculations based on the Gogny force, by Hilaire and Girod [51, 52]

3.2.1 Shape-isomer-like structures in the Ni region

As discussed in the previous section, since the 1980s, different mean-field models have predicted the appearance of shape isomerism in different regions of the nuclear chart, pointing to the neutron-rich $^{66}$Ni and $^{68}$Ni isotopes as the most promising systems for developing deep minima in the PES, associated with well-deformed prolate shapes [37, 38, 41]. In contrast, the remaining of the Ni isotopic chain was expected to be characterized by a main spherical minimum only. In recent years, taking advantage of the most powerful supercomputing systems, fully microscopic approaches have become available to make predictions for shape-coexistence phenomena in medium mass nuclei, in particular in the Ni region [53,54,55,56]. In the case of the Monte Carlo Shell Model (MCSM) calculations performed by the Tokyo group [54], the neutron-rich Ni isotopes $^{66,68}$Ni were again indicated as best candidates for the appearance of shape isomerism, together with the lighter $^{62,64}$Ni isotopes, although with reduced magnitude of the barrier separating the well-deformed shape from the main spherical minimum [54]. Indeed, recent experiments, performed at ISOLDE/CERN, MSU and RIKEN, confirmed the existence of coexisting spherical, oblate and prolate shapes along the Ni isotopic chain, although no evidence for shape isomers was reported in $^{68}$Ni and heavier systems [57,58,59,60]. In contrast, a series of measurements performed in Bucharest, ALTO IPN-Orsay, ILL and Argonne, guided by predictions from the MCSM, confirmed the existence of a complex scenario of excitations based on different spherical, oblate and prolate shapes also in A=62, 64, 66 Ni isotopes. Such results were obtained with the employment of different reaction mechanisms, e.g., sub-Coulomb barrier transfer reactions with heavy-ion stable beams, neutron-capture reactions with intense thermal-neutron beams, and Coulomb excitation [11, 12, 61, 62]. In the case of $^{66}$Ni, the two-neutron transfer reaction induced by an $^{18}$O beam on a $^{64}$Ni target, at the sub-Coulomb barrier energy of 39 MeV, was first used to populate all excited states up to $\sim$4.1 MeV excitation energy [11]. Furthermore, lifetime measurements of all three 0$^+$ excited states, located at 2443-, 2671- and 2974-keV excitation energies, performed with a plunger setup, yielded B(E2) transition rates of 4.3(5), 0.09(1), and 0.21(7) W.u., for the 0$^+_2$, 0$^+_3$, and 0$^+_4$ states, corresponding to HF factors of 1.2, 56 and 24, respectively. MCSM calculations give an interpretation of the retardation of the E2 decay from the 0$^+_3$ and 0$^+_4$ states which are predicted to display spherical and prolate shapes, respectively. In the first case, the retardation arises from cancellation effects in the E2 matrix elements, while for the 0$^+_4$ state the hindered E2 decay towards the first excited 2$^+_1$ state is a consequence of a sizable barrier in the potential energy surface, between the secondary prolate minimum and the spherical ground state minimum [11]. Experimentally, support to such a scenario also comes from the $\beta$-decay of the spherical 1$^+$ ground state of $^{66}$Co, which strongly feeds the 0$^+_1$ and 0$^+_3$ (supposedly spherical) states in $^{66}$Ni, while no direct feeding is observed for the 0$^+_2$ (oblate deformed) and 0$^+_4$ (prolate deformed) states [63]. This makes the 0$^+_4$ state in $^{66}$Ni a rare example of shape-isomer-like excitation, similar to the ones reported in the actinides region, although with much reduced hindrance.

A similar situation was found in $^{64}$Ni, with the three lowest 0$^+$ excitations reflecting the sequence of 0$^+$ excitations in $^{66}$Ni. In particular, the 0$^+_4$ state, located at 3463 keV, shows a hindered E2 decay to the 2$^+_1$ state of spherical nature, with HF>63, as follows from Coulomb excitations and lifetime measurements [12, 61]. Similarly to $^{66}$Ni, this 0$^+_4$ excitation in $^{64}$Ni is a candidate for a prolate deformed state lying in a secondary minimum in the PES. Again, experimental support for such a hypothesis comes from the $\beta$-decay of the spherical 1$^+$ ground state of $^{64}$Co, strongly feeding the supposedly spherical 0$^+_1$ ground state in $^{64}$Ni, to much lesser extent the 0$^+_2$ and 0$^+_3$ states, but not the 0$^+_4$ state [64]. Moreover, the MCSM calculations predict the 0$^+_4$ state as an excitation located in the deep prolate PES minimum, with a strongly hindered E2 decay. Looking closer to the 0$^+_4$ and 2$^+_1$ wavefunctions, one finds that not only the barrier separating the PES minima is responsible for this retarded E2 decay, but a significant contribution to the retardation arises also from cancellation effects among E2 matrix elements, similarly to the case of 0$^+_3$ in $^{66}$Ni. Partial level schemes of $^{64}$Ni and $^{66}$Ni, in comparison with MCSM theory predictions, are shown in Fig. 7.

Fig. 7

Partial level schemes of $^{64}$Ni (a) and $^{66}$Ni (c), with the decay of 0$^+_4$ states highlighted in red. Corresponding predictions from MCSM calculations (limited to the decay of 0$^+_{2,3,4}$ are also shown in panels (b) and (d). In all panels, $\gamma$-ray energies are in keV and B(E2) values in W.u. (in italic). Cartoons next to the calculated 0$^+$ excitations indicate the expected nuclear shapes (i.e., spherical, oblate and prolate), according to MCSM predictions [11, 12]

According to theory, the deep prolate secondary minimum in $^{64}$Ni and $^{66}$Ni arises from a sizable promotion of neutrons into the $g_{9/2}$ orbital, which causes a reduction of the proton spin-orbit splitting, favoring the promotion of protons across the Z = 28 shell gap. Such a phenomenon, called Type II shell evolution, is caused by the tensor and central forces acting coherently [54]. When moving toward lighter Ni isotopes, the effect is reduced as there are progressively fewer neutrons which can be excited across the N=40 subshell gap into the $g_{9/2}$ orbital. As a result, the deformed minimum rises in excitation energy. Figure 8 shows the systematics of the 2$^+_1$ and 0$^+$ (prolate) excitations in Ni isotopes, in comparison with MCSM predictions including or not the contribution of the tensor force. Instead of following a parabolic behavior typical for shape-coexisting states (with a minimum energy in the middle of the fp subshell, i.e., at N=34), the energy of the prolate 0$^+$ states raises with decreasing number of neutrons from N=42 in $^{70}$Ni to N=34 in $^{62}$Ni. The $^{62}$Ni nucleus also appears as the isotope marking the onset of shape coexistence in the neutron-rich Ni chain. Preliminary results based on transfer and neutron-capture experiments recently performed ad IFIN-HH and ILL seem to confirm such a scenario in $^{62}$Ni [62].

Fig. 8

Right: Excitation energies for even-even $^{64-70}$Ni. Symbols are experimental data: black circles for the first 2$^+_1$ states, red squares for 0$^+$ excitations associated to strongly prolate deformed states [9]. Lines are theoretical predictions for the prolate 0⁺ excitations: the blue (green) curve gives results from MCSM calculations including (or not) the contribution from the monopole tensor force. The black line indicates the simple modelling of deformed intruder states, with parameters fitted to the estimated values without the tensor-force effect. Left: Potential energy surfaces of $^{64}$Ni with (a) full, original interaction used in MCSM calculations [54], and (b) monopole-frozen interaction, showing the deepening of the prolate minimum in the latter case [12]

3.2.2 Searches for shape-isomer-like structures in the Zr and Sn regions

From the theory predictions and experimental investigations discussed above, which point to the Ni ($Z=28$) and Pb ($Z=82$) shell closures as favourable places for the appearance of shape isomerism below the actinides region, it is natural to expect that similar phenomena might also develop around the Zr ($Z=40$) and Sn ($Z=50$) proton closed shells, which are classical regions where shape-coexistence phenomena occur [7, 8, 10].

Indeed, the Zr region has been extensively investigated in recent years by a variety of different experimental techniques (see, e.g., [65,66,67,68,69]), including $\gamma$-spectroscopy studies of fission fragments [70]. Since quite some time, this region is well known for an abrupt shape transition from spherical ground states for isotopes with N<60, to deformed ground states in N$\ge$60 nuclei, because of the coexistence of well-isolated structures with different degree of deformation [71,72,73]. Out of more than 35 $0^+$ excitations identified along the Sr and Zr isotopic chain [9, 10], only five are found to exhibit a hindered E2 $\gamma$ decay, three of which are located near the $N=60$ isotone line. They are the 0$^+_4$ excitations in $^{96,98}$Zr, and the 0$^+_4$ state in $^{96}$Sr, with hindrance factors HF ranging between 17 and 180. While a theory interpretation is currently not available for such high-lying 0$^+$ excitations in $^{96,98}$Zr and $^{96}$Sr nuclei, microscopic predictions and detailed experimental investigation exist for the low-lying 0$^+_2$ and 2$^+_{1,2}$ states in $^{96}$Zr [68]. For example, MCSM calculations predict the 0$^+_1$ ground state and the 2$^+_1$ excitation to be spherical (in agreement with the low experimental value B(E2, 2$^+_1\rightarrow 0^+_1) = 2.3(3)$ W.u.), while a collective (triaxial) rotational structure is expected to be built on the 0$^+_2$ state, located experimentally at 1582 keV, below the 1750-keV 2$^+_1$ state. Indeed, high-resolution inelastic electron scattering provided a large transition probability for the 2$^+_2 \rightarrow$0$^+_2$ decay (B(E2)=36(11) W.u.), accompanied by a modestly retarded 2$^+_2 \rightarrow$0$^+_1$ decay $(B(E2)=0.26(8)$ W.u., corresponding to HF=3.8), what confirms the existence of two distinct structures with a certain degree of mixing in the wave functions (see Fig. 9).

Fig. 9

Left: partial level scheme of $^{96}$Zr showing the low-lying 0$^+$ and 2$^+$ states (with excitation energies in keV) and the corresponding B(EM$\lambda$) transition probabilities, as determined experimentally [9, 68]. Right: Potential Energy Surface from the Montecarlo Shell Model Hamiltonian, with circles showing the main components of the 0$^+_1$ and 2$^+_1$ states (spherical), and the 0$^+_2$ state and 2$^+_2$ states (triaxially deformed). The energy in the contour plot increases from blue (low) to red (high). Taken from Ref. [68]

For the semi-magic Sn nuclei with mass numbers in the range of 112–118, the appearance of relatively deep and well-separated secondary PES minima was predicted by a nonaxial Hartree-Fock-Bogoliubov approach already in 1988 [37, 50] by macro-microscopic mean-field calculations [74] and within the relativistic energy density functional approach [42]. According to some of these calculations, deep secondary PES minima are also expected to occur in the Cd and Pd region around N=66. This is the region where some 0$^+$ excitations have been interpreted in terms of phonons, although such interpretation has been questioned in light of recent experimental and theoretical investigations [75]. From a microscopic point of view, the appearance of relatively deep secondary minima in the $^{112-116}$Sn isotopes can be viewed in the context of Type II shell evolution mentioned above for the Ni isotopes (see Sect. 3.2.1). It reflects the action of the monopole tensor force between neutrons excited to the $h_{11/2}$ unique-parity orbital with protons promoted to the $g_{9/2}$ single-particle state, what induces the reduction of the proton $g_{7/2}$-$g_{9/2}$ spin-orbit splitting. In this way, proton excitations across the Z=50 shell gap are favoured that further stabilizes isolated, deformed local minima in the PES. This phenomenon has been demonstrated by Hartree-Fock-Bogoliubov (HFB) calculations of the PES with the interaction of Togashi et al., [76], which predicts a main spherical minimum, accompanied by the appearance of a prolate secondary one in $^{112-116}$Sn [10, 77]. This secondary minimum does not exist in Sn nuclei lighter than A=110 and essentially disappears in $^{118}$Sn and heavier isotopes.

The Sn isotopes exhibit numerous 0$^+$ excitations, as confirmed by a series of high-resolution transfer studies, predominantly (t,p) and (p,t), conducted over the past two decades using the Munich Q3D spectrograph [78,79,80,81,82,83,84]. However, associating some of these excitations with those residing in secondary deformed minima is difficult. Insight from the two-proton transfer reaction ($^3$He,n) on $^{106, 110, 112, 114}$Cd, and $^{116}$Cd targets offers potential help. This reaction has been observed to strongly and selectively populate 0$^+$ states in A = 108, 112, 114, 116, and 118 Sn isotopes [85], suggesting significant proton components in their wave functions. Further clarification about the nature of 0$^+$ excitations can be gained by comparing population cross sections of these states in ($^3$He,n) and two-neutron transfer processes [8]. Notably, differences in population between these processes, especially in the case of 0$^+$ states strongly populated in ($^3$He,n), may indicate their prolate nature. For instance, the 0$^+_2$ and 0$^+_3$ states in $^{114, 116}$Sn and $^{118}$Sn appear to be built on different configurations, with 0$^+_2$ predominantly dominated by protons (2p-2h) excitations.

Deformed rotational bands identified in even Sn nuclei with A=110–118 add to the issue of secondary minima existence. In $^{112,114,116,118}$Sn, these deformed bands start to decay out at spin-parity 6$^+$, nevertheless a small fraction of their intensity reaches the 0$^+$ band head. In the case of $^{118}$Sn, the 0$^+_2$ band head undergoes a decay to the spherical 2$^+_1$ state through a relatively fast E2 transition with a B(E2) value of 19 W.u. This observation suggests a substantial admixture of deformed and spherical components in the 0$^+_2$ wave function. In $^{116}$Sn the picture is not clear. Both 0$^+_2$ and 0$^+_3$, located at 1757 and 2027 keV, respectively, could be the band head of the deformed structure, due to the enhanced B(E2) values from the 2$^+_2$ state at 2112 keV to 0$^+_2$ and 0$^+_3$. An indication for considering 0$^+_2$ as band head comes from the population of this state in low-resolution, two-proton-transfer, ($^3$He,n) reaction [85]. However, recent $\beta$-decay and (n,$\gamma$) studies point to 0$^+_3$ as a more likely head of the rotational band [86, 87]. In addition, the (n,$\gamma$) investigation indicates a slightly delayed transition from 0$^+_3$ to 2$^+_1$ (B(E2) = 0.49(7) W.u.; HF = 10.2), while the 0$^+_2$ to 2$^+_1$ decay is faster (B(E2) = 18(3) W.u.; HF = 0.28), in agreement with predictions from the Interacting Boson Model [87]. In the case of $^{112}$Sn and $^{114}$Sn, comparable B(E2) information from the 0$^+_2$ band head is not currently available.

In conclusion of this section, to achieve a more profound understanding of the nature of 0$^+$ excitations in Sn isotopes around A=110–116, further studies of their properties, particularly their lifetimes, are needed. This keeps open the possibility of identifying shape-isomer-like structures among excited 0$^+$ states in Sn nuclei, characterized by a significant hindrance in their decay towards the spherical 2$^+_1$ state due to a substantial barrier in the Potential Energy Surface.

4 Conclusions

Shape isomers are fascinating examples of “extreme” shape-coexistence phenomena in nuclei, so far poorly investigated across the nuclear chart. One of their most characteristic and necessary fingerprints, regardless of the lifetime of the state, is a substantial hindrance in the $\gamma$ decay connecting the highly deformed structure, well localized in the nuclear Potential Energy Surface (PES) in the deformation space, to a lower-lying state of the spherical/weakly deformed main shape. Even though shape isomers were discovered already in the 1960s, following investigations of fissioning systems, information on their $\gamma$ decay is still very limited and restricted to candidates in the actinides, e.g., the 0$^+$ state at 2814 keV in $^{236}$U with exceptionally large hindrance factor (HF$\sim$10$^7$) for E2 decay, and in the much lighter Ni region, i.e., 0$^+_4$ in $^{64,66}$Ni with much-lower hindrances (HF$\sim$25). This is in spite of numerous examples of E2 $\gamma$ transitions between states characterized by different shapes, in nuclei in different regions of the nuclear chart, which display rather fast decays (with HF<10).

Significant advances in nuclear facilities and in multi-detector technology, which have taken place in recent years, open now the possibility for detailed studies of shape isomer decays in the historical actinides region, focusing on already known cases, as well as predicted but still undiscovered shape isomers in low-Z actinides systems. New instrumentation will allow also for extensive investigations throughout the entire nuclear chart, guided by state-of-the-art theory predictions. In the heavy mass Pb region, various candidates for shape isomerism have been proposed by mean-field-based models, although they will be hard to access even with modern detection systems, due to their high excitation energy (they are expected to be located in a region of very high density of states). In these cases, some help can be provided by the superdeformed rotational bands, well developed at higher spins, which, through the extrapolation to spin 0, could indicate the location of 0$^+$ band head, well deformed and possibly separated from normal states by a high potential barrier. In the past, transitions from excited states at low spin, populated using high-spin fusion-evaporation reactions, were discarded due to the necessity to impose high-multiplicity trigger conditions. This online event selection was needed to severely reduce the data volumes sent to the very limited disk (or tape) space at the time. However, very recently, fully digital, triggerless data acquisition systems have become available, where all events, including those of low-multiplicity, are written to disk. Examination of these previously-discarded events may reveal interesting information on shape isomeric states in nuclei.

Even more promising are searches for shape isomers in lighter nuclei, as for example around Sn. Here, in addition to mean-field models, microscopic shell models start to be applicable, like the Monte Carlo Shell Model of the Tokyo group. They provide access to the detailed composition of the wave functions of the initial and final states involved in the decay and to the origin of the decay hindrance. In this respect, an important role seems to be played by a specific term of the nuclear force, i.e., the monopole tensor part of the nucleon-nucleon interaction, which favors the existence of distinct and deep minima in the nuclear Potential Energy Surface, similarly to what was observed in the Ni region. Extensive searches for shape-isomer-like structures are currently undergoing in even-even Sn nuclei with $A= 112-118$, exploiting different reaction mechanisms (mainly based on transfer reaction with heavy ions).

Exploration of shape isomerism can be extended to exotic nuclei, presently beyond reach but accessible in the near future by radioactive beams. In such systems, deep secondary minima could be formed, possibly created by the action of tensor force. Predictions from mean-field-based approaches already indicate neutron-rich regions within the nuclear chart, particularly around Cd, Sn, Ba, Xe, and Yb (see, e.g., Ref. [88]), where extremely elongated shapes (with axis ratios significantly exceeding 2:1) are expected to manifest at high spins, as also supported by preliminary experimental investigations [89, 90]. Tracing these structures down to low spins or directly populating 0$^+$ states in these systems could help in identifying other examples of shape isomerism.