1 INTRODUCTION

The discovery of exotic states—states with halo and cluster states—has led to a revision of many established ideas in nuclear physics. The search and study of such exotic states were carried out in isobar-analog and mirror nuclei. The most suitable for studying such states are direct reactions (elastic and inelastic scattering, charge exchange and nucleon transfer). There are important classes of short-lived unstable nuclear states whose sizes cannot be measured in traditional ways. Two independent methods were developed.

At energies of 10–50 MeV/nucleon, the angular distributions of the reaction products have a characteristic pattern. At small angles, there is a pronounced (‘‘main’’) diffraction maximum followed by a minimum and, often, a second maximum. The research is based on measuring the radii of states in which the formation of exotic states is possible. We use two different methods to determine the radii: the Modified Diffraction Model (MDM) [1, 2] and the Asymptotic Normalization Coefficients (ANC) method [3, 4]. The first approach uses the diffraction part of the differential cross sections for inelastic and elastic scattering and makes it possible to determine the radii of many excited states, including states with a neutron halo. The MDM method is applied to inelastic scattering and charge exchange reactions. Its first application made it possible to determine the proton halo in the unbound state of \({}^{13}\)N [5]. The second approach also uses the first few oscillations of differential cross sections for transfer reactions, but is limited to consideration of peripheral reactions. It allows one to determine the radius of the valence nucleon, as well as the probability of its being outside the radius of the potential [6].

Theoretical analysis of such reactions in the framework of the distorted wave method in the Born approximation (DWBA) or the coupled-channel method (CC) can provide information about the wave function of this state by testing various nuclear models through the reaction form factor, which is the integral of the overlap of the wave functions of the initial target nucleus and the final product nuclei in the state under study. In addition, in cases where it is impossible to separate the contribution of close levels or the contributions of several components with different transferred angular momenta in the experimental data, such a theoretical analysis makes it possible to do this.

The development of methods for measuring the radii of nuclei in their short-lived excited states led to the discovery of new classes of states with abnormal radii, which we have named as the size isomers. Such states are weakly bound and have an exotic structure (cluster states, halo/‘‘skin’’). It should be noted that even 50 years ago exotic nuclear states with abnormally large radii located close to the thresholds of emission of nucleons or clusters were predicted [7].

Examples of cluster states, for example, are \({}^{6}\)Li\((\alpha+d)\), \(S_{\alpha}=1.47\) MeV; \({}^{7}\)Li\((\alpha+t)\), 2.47 MeV; \({}^{9}\)Be\((\alpha+n)\), 1.66 MeV, while Halo state examples are \({}^{11}\)Be(\({}^{10}\textrm{Be}+n)\), \(S_{\alpha}=0.50\) MeV; \({}^{6}\)He\((\alpha+2n)\), \(S_{\alpha}=0.98\) MeV.

Next, we consider cluster states and states with a halo.

2 RESULTS AND DISCUSSION

2.1 Cluster States in Light Nuclei

A key object for testing modern cluster theories is the 0\({}^{+}_{2}\), 7.65 MeV ‘‘Hoyle’’ state in \({}^{12}\)C. Hoyle state is responsible for existence in Universe of elements heavier than Helium. Exotic cluster structure was predicted for this state. A hypothesis about possible existence of \(\alpha\)-particle Bose–Einstein condensation (APC) [8] in finite nuclei which predicted appearance of nuclear states with unusual dilute \(\alpha\)-cluster structure resembling a gas of almost non-interacting \(\alpha\)-particles with considerably enhanced radii, has attracted plenty of attention in the two last decades. This idea gave impetus to further development of the theoretical approaches concerning \(\alpha\)-particle clustering initiated more than 50 years ago, and the experimental search for the diluted nuclear excited states. The direct measurement of the radius of this short-lived nuclear state is impossible, because of its very short half-life [\(\tau_{1/2}(0^{+}\), 7.65 MeV\()\sim 2\times 10^{-16}\) s]. Estimates of the APC model (\(R_{\mathrm{rms}}(0^{+}_{2})=4.31\) fm) [9] are nearly twice as large as the radius of the ground state (2.3 fm). Nevertheless, the recent calculations in the framework of the ab initio Lattice Effective Field Theory (L-EFT) [10] predicted the rms radius (2.4 fm) almost equal to the rms radius of \({}^{12}\)C in the g.s. (\(R_{\mathrm{rms}}=2.3\) fm). Application of our method, the Modified Diffraction Model [2], to the inelastic scattering of \({}^{2}\)H, \({}^{3}\)He, \({}^{4}\)He, \({}^{6}\)Li, and \({}^{12}\)C on \({}^{12}\)C in a wide range of energies allowed us to determine the consistent values of the rms radii of \({}^{12}\)C in the excited states up to \(E_{x}=11\) MeV. In particular, we showed that in line with the expectations, the Hoyle state has an abnormally large radius, (\(2.89\pm 0.04\)) fm, approximately 1.25 times larger than that for the ground state. The best agreement with this experimental result showed the AMD calculations [11] \(R_{\mathrm{rms}}(0^{+}_{2})=2.90\) fm, while the APC model result overestimates the radius of the Hoyle state.

Moreover, in the framework of the coupled-channel method, the transfer of a heavy \({}^{8}\)Be cluster with different values of \(L\) in the region of large angles for the 0\({}^{+}_{1}\) ground state and the 7.65 MeV, 0\({}^{+}_{2}\) state was considered on the basis of experimental data on the scattering of alpha particles at an energy of 110 MeV by \({}^{12}\)C [12]. It was shown that the main configuration of the 7.65 MeV, 0\({}^{+}_{2}\) state is \(L=0\) (60\({\%}\)) [13].

Prediction of the APC that all three alpha particles in \({}^{12}\)C should predominantly occupy the lowest s-orbit also was confirmed by experiment giving for the occupation probability \(W_{s}(\alpha)=0.6\) (to be compared with the theoretical value 0.7–0.8 [14]).

Calculations of the single-\(\alpha\) orbital behavior in 0\({}^{+}\) states as a function of the nuclear radius \(R_{N}\) demonstrated a correlation between the values of the radius and \(W(\alpha)\) shown in Fig. 1. \(W(\alpha)=70{\%}\) corresponds to a radius as large as 4.3 fm, whereas \(W(\alpha)=60{\%}\) suits to \(R_{N}=3\) fm, the value obtained experimentally. These findings resolve apparent inconsistence between both the most importance characteristics (the rms radius and \(W(\alpha))\). Taking into account all these results, we can conclude that the 0\({}^{+}_{2}\) Hoyle state possesses only some rudimentary features of \(\alpha\)-condensation if any [15].

Fig. 1
figure 1

Dependence of the occupation probabilities of the single-\(\alpha\) orbits in the Hoyle state of \({}^{12}\)C on its radius are presented. Contribution of the relative angular momenta \(L=0\), 2, and 4 are shown by the solid, dotted, and dashed lines. The experimental data corresponding to the Hoyle state radius \(R_{\textrm{rms}}\approx 2.9\) fm are noted by points. Vertical rectangle shows a prediction of the condensate model.

Full size image

Naturally, the question arose about analogs of the Hoyle state. First of all, they were searched for in the neighboring nuclei \({}^{11}\)B (one alpha-particle is replaced with a triton) and \({}^{13}\)C (one neutron is added), shown in Fig. 2.

Fig. 2
figure 2

The analogs of the Hoyle state in the neighboring \({}^{11}\)B and \({}^{13}\)C nuclei. Alpha-cluster states near \(\alpha\)-thresholds are dilute. Hoyle state is less dilute than condensate model predicts. Large enhancement of the 1/2\({}^{+}\), 3.09 MeV state radius was observed. No evidence provided for the existence of the dilute state 12.5 MeV, 1/2\({}^{-}_{4}\).

Full size image

The study of the reaction \({}^{11}\)B (\(\alpha,\alpha^{\prime}\)) \({}^{11}\)B, \(E_{\alpha}=65\) MeV [16] made it possible to reveal the 8.56 MeV, 3/2\({}^{-}_{2}\) state and measure its radius \(R(8.56)=(2.87\pm 0.12)\textrm{fm}\approx R\)(Hoyle). Moreover, the 1/2\({}^{+}\) state at 12.56 MeV with an anomalously large radius was predicted as Hoyle’s analogue. Our data showed that the giant state at 12.56 MeV is not observed.

Analogs of the Hoyle state are expected in high-lying states \({}^{13}\)C near \(\alpha\)-emission thresholds. Indeed, our experiments on the elastic and inelastic scattering of alpha particles at energies of 65 and 90 MeV by\({}^{13}\)C [17] and \({}^{13}\)C [18] made it possible to detect the 8.86 MeV, 1/2\({}^{-}_{2}\) state in \({}^{13}\)C, which was analogous to the Hoyle state in \({}^{12}\)C 7.65 MeV, 0\({}^{+}\). In this case, the measured radius for the state 8.86 MeV, 1/2\({}^{-}_{2}\) (\(R_{\mathrm{rms}}=(2.7\pm 0.1)\) fm) was very close to the Hoyle radius (\(R_{\mathrm{rms}}=(2.9\pm 0.1)\) fm). It should be noted that similar data were obtained by Kazakh colleagues at INP Almaty [19]. Moreover, another analog of Hoyle was found in the \({}^{13}\)C nucleus at 11.08 MeV, 1/2\({}^{-}_{3}\) [20].

It should be noted that the \({}^{13}\)C nucleus is a unique nucleus. It combines the coexistence of three-type structures: shell model, exotic clusters, neutron halo at 3.09 MeV, 1/2\({}^{-}\).

A question naturally arises: do analogs of the Hoyle state exist in more massive 4\(N\) nuclei. First possible candidate is the \({}^{16}\)O. Suggestions about the structure of the 15.1-MeV 0\({}^{+}_{6}\) state were proposed also in the framework of APC [21]. We did not confirm the existence of a dilute state with super-large radius associated with the 15.1-MeV 0\({}^{+}_{6}\) state (\(R_{\mathrm{rms}}=\)6.0 fm), which was predicted by the APC model. The rms radius of \({}^{16}\)O in this state was found similar to the radius of \({}^{16}\)O in the ground state. From this point of view, the 0\({}^{+}_{6}\) state cannot be considered as an analog of the Hoyle state in \({}^{12}\)C [22].

The next candidate is \({}^{20}\)Ne. It was widely studied both theoretically and experimentally in literature [23–26].

The literature differential cross sections of the inelastic \(\alpha+^{20}\)Ne scattering in the energy interval from a few tens MeV up to 400 MeV were analyzed. We determined directly the rms radii of \({}^{20}\)Ne in a number of states with excitation energies up to 7 MeV applying the MDM. No significant radius enhancement was observed for the members of \(K^{\pi}=\) 0\({}_{1}^{+}\) and \(K^{\pi}=2^{-}\) bands in comparison with the ground state. By AMD predictions, these bands have different structures: \({}^{16}\textrm{O}+\alpha\) for the \(K^{\pi}=0_{1}^{+}\) band and \({}^{12}\textrm{C}+2\alpha\) for the \(K^{\pi}=2^{-}\) band. At the same time 20\({\%}\) radius enhancement was obtained for the \(K^{\pi}=0_{1}^{-}\) band members. By AMD predictions, these band members have \({}^{16}\textrm{O}+\alpha\) structure.

Moreover, our estimates of the radius of the 0\({}_{2}^{+}\) state, the head of the \(K^{\pi}=0_{2}^{+}\) band, showed 25\({\%}\) radius increase. This state is located above \(\alpha\)-emission threshold. Obtained result can speak in favor of possible \(\alpha\)-condensate structure of the 0\({}_{2}^{+}\) state and can be considered as a possible analog of the famous 7.65-MeV 0\({}_{2}^{+}\) Hoyle state of \({}^{12}\)C [27].

2.2 States with a Halo in Light Nuclei

There is another exotic structure—halo.

Measurements of elastic and inelastic scattering of \(\alpha\)-particles on \({}^{9}\)Be with an excitation of 1.68-MeV 1/2\({}^{+}\) and 2.43-MeV 5/2\({}^{-}\)states at \(E_{\alpha}=30\), 40 and 90 MeV were carried out and MDM analysis was performed by us [15, 28]. It was shown the 1.68 MeV, 1/2\({}^{+}\) is halo state. It should be mentioned that this state is located above neutron emission threshold. So this state is example of halo in continuum. Moreover, radius enhancement was also observed for all members of the \(\sigma-\)band based on 1.68-MeV, 1/2\({}^{+}\) state.

We will discuss only new types of halos, namely those in continuum and with a neutron in a \(p\)—state. Such structures were observed in \({}^{11}\)Be and \({}^{9}\)Be [28 and references therein].

The \({}^{11}\)Be nucleus in the ground 1/2\({}^{+}\) state is the most striking example of a nucleus with one-neutron halo. For the two bound states of this nucleus, the 1/2\({}^{+}\) g.s. and the first 0.32-MeV 1/2\({}^{-}_{1}\) excited state, one can assume a single-particle structure of the \({}^{10}\)Be core in the 0\({}^{+}\) g.s. and the last neutron in the 2\(s_{1/2}\) and 1\(p_{1/2}\) sp orbitals, correspondingly. We showed that the halo radius of \({}^{11}\)Be in the 1/2\({}^{+}\) g.s. is very large, \(R_{h}=8.0\pm 0.2\) fm that agrees with the result obtained in recent measurements of the charge radii of the light halo nuclei. This value corresponds to the contribution of the asymptotic part of the wave function of the last neutron of about 98\({\%}\).

Another interesting feature of \({}^{11}\)Be is the existence of the \(K^{\pi}=1/2^{+}\) rotational band based on its ground state. The quasielastic and inelastic \({}^{11}\textrm{Be}+^{12}\)C scattering leading to formation of the g.s. and the 5/2\({}^{+}\) and 3/2\({}^{+}\) unbound excited states belonging to this band in \({}^{11}\)Be (Fig. 3) has been measured and these scattering data were analyzed by the MDM. It was found that the diffraction radii for all three members of the \({}^{11}\)Be rotational band are quite similar and sufficiently enlarged. This means that the neutron halo exists in \({}^{11}\)Be not only in the bound states (g.s. and the first excited 1/2\({}^{-}_{1}\) state), but in the unbound excited states, members of the \({}^{11}\)Be rotational band, in spite of these states belonging to a continuum spectrum.

Fig. 3
figure 3

Plot of the energy levels of \({}^{11}\)Be and \({}^{9}\)Be belonging to the positive parity rotational bands. The moments of inertia are indicated in the bottom line. The diffraction radii from \({}^{11}\textrm{Be}+^{12}\)C scattering at \(E(^{11}\)Be\()=737\) MeV and \(\alpha+^{9}\)Be at \(E(\alpha)=30\) MeV are shown in the left and right columns.

Full size image

\({}^{9}\)Be is known to have two rotational bands, \(K^{\pi}=\) 3/2\({}^{-}\)based on the ground state and \(K^{\pi}=1/2^{+}\) based on the 1.68-MeV first excited state. Though the ground state is only 1.6-MeV below the neutron emission threshold, it does not have a halo contrary to the \({}^{11}\)Be case. The \(K^{\pi}=1/2^{+}\) band is very similar to the analogous band in \({}^{11}\)Be (Fig. 3): both bands have the inversed sequence of levels and similar moments of inertia, which are enhanced in comparison with the \({}^{9}\)Be g.s. band. However, the band head state (\(E_{x}=1.68\) MeV, 1/2\({}^{+})\) is situated in continuum, only 15 keV above the \({}^{8}\textrm{Be}+n\) threshold. Our MDM analysis of the inelastic \(\alpha\)-scattering data showed that all states belonging to the \(K=1/2^{+}\) band have the enlarged radii.

Significant progress has been made in recent decades in studying the phenomenon of the neutron and proton halo in light nuclei. Most of these states are near the particle emission threshold. One of the main characteristics of a nucleus in a halo state is the increased radius of the valence nucleon, and, consequently, the increased root-mean-square radius of the entire nucleus. Thus, the measurement and determination of the root-mean-square radius plays a decisive role in the problem of searching for nuclear halos. According to the type of valence nucleon, the halo can be proton or neutron. The proton halo is a rather rare phenomenon. We recently discovered a proton halo in the unbound \({}^{13}\)N state with energy of 2.37 MeV, 1/2\({}^{+}\), which is a mirror state with respect to the \({}^{13}\)C state with an energy of 3.09 MeV with a confirmed neutron halo. This was the third case of identification of the proton halo, previously proton halo was obtained only in \({}^{8}\)B and \({}^{17}\)F. The study of halo in isobar analog states (IAS) led to very interesting results. Such studies have hardly been carried out so far. The optimal reactions for searching for a halo in the IAS are nucleon transfer reactions and charge-exchange reactions. Nucleon transfer reactions are traditionally used to obtain information about single-particle (sp) states, spectroscopic factors (SF), asymptotic normalization coefficients and optical nucleus–nucleus potentials. In addition, these reactions are widely used to search for states with increased radii. Charge-exchange reactions, in particular, the reaction (\({}^{3}\)He, \(t\)), can also be used to obtain information about the radii of states. Their advantage lies in the possibility of studying unbound states in the continuous spectrum. It should be noted that the interpretation of the halo in excited states located above the particle emission threshold remains an open question. Since there is no possibility of quantitatively calculating the weight of the asymptotic part of the nucleon wave function, when describing the unbound state, difficulties arise when comparing its asymptotic and internal parts.

Our group developed MDM method for studying halos in isobar analog states, which gives only the radius of the considered state, while the ANC method allows one to obtain information about the structure of the state: the probability of a valence nucleon being outside the range of the potential (\(D_{1}\)), as well as the weight of the asymptotic part in the rms radius (\(D_{2}\)).

The first object of study was the IAS triplet 2\({}^{-}\) and 1\({}^{-}\) states with isospin \(T=1\) of the \({}^{12}\)B, \({}^{12}\)C, and \({}^{12}\)N nuclei. All these states are located near the nucleon emission thresholds. A series of experiments were carried out by us and data were obtained for the IAS states in the \(A=12\) triplet. IAS in \({}^{12}\)B was studied in the \({}^{11}\)B\((d,p)^{12}\)B reaction, IAS in \({}^{12}\)C in the \({}^{11}\)B\((^{3}\)He, \(d)^{12}\)C reaction, and IAS in\({}^{12}\)N in the\({}^{12}\)C\((^{3}\)He, \(t)^{12}\)N reaction at \(E\sim 10\) MeV/nucleon. The experiments were carried out at the K130 cyclotron at the University of Jyväskylä (Finland). It was found that the 1\({}^{-}\)IAS states in the \({}^{12}\)B, \({}^{12}\)C, and \({}^{12}\)N nuclei are a single-nucleon (neutron or proton) halo. Signs of a halo were also observed for the \(2^{-}\) IAS states of these nuclei. These states can be considered as possible halo candidates [29–31].

The multiplet \(A=14\) was studied on the basis of a review of the literature data (Fig. 4)

Fig. 4
figure 4

Halo in isobar analog states of multiplets \(A=12\) and \(A=14\). Isobar analog states in \({}^{12}\)B correspond to neutron halo, while all other correspond to proton halo or halo-like states.

Full size image

A neutron halo in the 1\({}^{-}\) IAS state in \({}^{14}\)C was confirmed. For the first time, a proton halo was found in the 1\({}^{-}\)IAS state in \({}^{14}\)N [32]. An increased root-mean-square radius was also obtained for the 1\({}^{-}\) IAS state in \({}^{14}\)O, which indicates a possible proton halo in this state.

Our results confirm that the halo phenomenon is universal and manifests itself not only in the ground states of exotic nuclei, but also in the excited states of ordinary light nuclei. This statement is based on the increased radii obtained in our work by different methods, as well as the obtained large values of the probability of finding a valence nucleon, neutron or proton outside the interaction potential. A great achievement was the development of the ANC method [29] for studying resonance states, which made it possible to identify new cases of a proton halo in isobar analogue states.

3 CONCLUSIONS

To conclude, our research has shown the nuclear size isomers contain two main classes:

\(\bullet\) States with halo (\({}^{9}\)Be, \({}^{11}\)Be, \({}^{12}\)B, \({}^{13}\)C, \({}^{13}\)N, \({}^{14}\)N, \({}^{14}\)O)

\(\bullet\) Alpha-cluster states with increased radii (\({}^{11}\)B, \({}^{12}\)C, \({}^{13}\)C, \({}^{20}\)Ne)

Despite APC predictions, we have not observed ‘‘giant’’ states in \({}^{11}\)B, \({}^{12}\)C, \({}^{16}\)O. Moreover, we haven’t found any size increase for 0\({}^{+}\) states in \({}^{16}\)O. We have shown that alpha-particle condensation exists in light nuclei only in rudimentary form.

The universal nature of the halo is demonstrated, which manifests itself not only in the ground states of exotic nuclei, but also in their excited states. Halos exist not only in particle-stable states, but in continuum as well.

Until now, the study of the halo in the IAS with the help of nuclear reactions has practically not been studied. We have shown that such a study makes it possible to investigate the manifestation of isotopic invariance in new objects and to relate the properties of the neutron and proton halo (or a halo with a larger number of nucleons). It is important to emphasize that more and more data are currently accumulating, indicating that the supposed halos in the continuum are no different from the halos located under the thresholds.

In present, many dozens of experimental and theoretical works are devoted to the halo problem, and these studies continue to develop intensively. It is now obvious that the neutron halo is one of the characteristic structures in the region of the stability boundary, but it can also exist outside it. In fact, the proton halo, which had previously been discovered only in three nuclei: \({}^{8}\)B, \({}^{17}\)F, and \({}^{13}\)N, the last one in our work, remained completely unexplored. This was the first case of its observation in a state located in the continuous spectrum. Within the framework of our research, new cases of a single-proton halo in \(A=12\) and \(A=14\) multiplets were discovered. At last, we should note that all our findings are done with stable beams.