Rotational excitations in near neutron-drip line nuclei: the birth and death of particle-bound rotational bands and the extension of nuclear landscape beyond spin zero neutron drip line

TL;DR

This paper uncovers mechanisms in rotating neutron-rich nuclei that enable the formation and disappearance of particle-bound rotational bands, extending the nuclear landscape beyond the neutron drip line.

Contribution

It introduces new rotational mechanisms involving Coriolis interactions that explain the birth and death of particle-bound bands near the neutron drip line.

Findings

Strong Coriolis interaction can bind unbound nucleonic states at high angular momentum.

Particle-bound rotational bands can form beyond the neutron drip line due to rotation.

Rotational effects can cause the disappearance of bound bands at high spins.

Abstract

Two new mechanisms active in rotating nuclei located in the vicinity of neutron drip line have been discovered. Strong Coriolis interaction acting on high-$j$ orbitals transforms particle-unbound (resonance) nucleonic configurations into particle-bound ones with increasing angular momentum. The point of the transition manifests the birth of particle-bound rotational bands. Alternative possibility of the transition from particle-bound to resonance rotational band (the death of particle-bound rotational bands) with increasing spin also exists but it is less frequent in the calculations. The birth of particle-bound rotational bands provides a mechanism for the extension of nuclear landscape to neutron numbers which are larger than those of the neutron drip line in non-rotating nuclei.

Full text

††journal: Physics Letters B

One of most fundamental questions in nuclear physics is what combinations of protons and neutrons form a nucleus as a bound system. For a given proton number, there is a maximum number of neutrons beyond which the formation of bound nuclear systems is impossible. This limit is known as a neutron drip line. Although significant experimental efforts have been dedicated to the investigation of very neutron rich nuclei [1, 2], the neutron drip line is definitely deliniated only up to $Z=10$. Future experimental facilities such as FRIB and FAIR will hopefully extend experimental neutron drip line to higher $Z$ up to mass number $A\sim 70$ (see Ref. [3] and Fig. 1 in Ref. [4]). These experimental efforts are accompanied by a significant amount of theoretical investigations of neutron-rich limit of the nuclear landscape (see Refs. [5, 6, 7] and references quoted therein). They provide the predictions for the position of neutron-drip line extracted from binding energies of non-rotating ground states. Such line is denoted here as the $I=0$ neutron drip line.

On the other hand, the questions of possible existence of particle-bound nuclear states in the nuclei located beyond the $I=0$ neutron drip line and physical mechanism leading to such states have not even been raised in the literature. For the first time we show that nucleonic configurations which are particle-unbound at zero or low spins can be transformed into particle-bound ones by collective rotation of nuclear systems. This leads to new features of rotational bands which are distinctly different from those seen before in known nuclei and which are discussed in the present manuscript for the first time. These features facilitate the extension of nuclear landscape to higher neutron numbers beyond those seen in the $I=0$ neutron drip line.

Static pairing correlations are present in atomic nuclei at spin zero. However, a reliable extrapolation of their properties towards neutron drip line still remains a challenge. As discussed in Ref. [4], even a careful fitting of the pairing force in known nuclei to experimental odd-even mass staggerings will not necessarily lead to a pairing force with a reliable predictive power towards the two-neutron drip line. Indeed, there is a substantial difference in the model predictions for pairing properties near two-neutron drip line which depends both on underlying functional and on the type of employed pairing interaction (see Ref. [4, 8, 9]). For example, relativistic Hartree-Fock calculations with separable pairing predict either similar or somewhat larger pairing in the nuclei near two-neutron drip line as compared with known nuclei (see Fig. 2 in Ref. [4]). On the contrary, the Skyrme DFT calculations for spherical nuclei show the reduction of neutron pairing towards the neutron drip line [8].

However, the role of pairing is expected to be substantially reduced with increasing spin because of Coriolis anti-pairing [10, 11] and blocking [10] effects. The angular momentum in the ground state bands of the light nuclei is very limited (see discussion in Ref. [12]); higher spin configurations are built by particle-hole excitations with many of them triggering the reduction of pairing by blocking effect. Moreover, the pairing drastically decreases after paired band crossings in the proton and neutron subsystems111The treatment of pairing correlations in rotating nuclei requires the methods of approximate (such as Lipkin-Nogami method [13, 14]) particle number projection [15, 16, 17, 18]. While their contribution is very important for the description of rotational properties at low and moderate spins (see Refs. [15, 16, 17, 18]), the role of pairing correlations is substantially reduced at high spin after paired band crossings in proton and neutron subsystems or for the configurations with blocked particles so that the results obtained in the calculations without pairing become a very good approximation to the ones calculated with the LN method [19, 12, 20]. Note that the configurations considered in the present manuscript belong to a such high spin range. [19]. As a consequence, it is expected that pairing correlations will have only a minor impact on rotational and deformation properties of the nuclei of interest at spins above $I\sim 10\hbar$. Weakening of pairing with angular momentum also leads to a significant reduction of the coupling with continuum. Note that neutron subsystem of Mg nuclei under study is very similar to the one of the $N\sim Z$ $A=40-50$ nuclei and in these nuclei pairing plays a negligible role above $I\sim 10\hbar$ [12].

Because of these reasons the pairing correlations are neglected in the present exploratory study which is focused on medium and high spin states. The calculations are performed in the framework of cranked relativistic mean field (CRMF) theory [21, 22, 23]222So far only few manuscripts have been dedicated to the study of rotating near neutron-drip line nuclei. Few ground state rotational bands of light nuclei have been studied in Skyrme DFT framework [24] and in qualitative model of Ref. [9]. Rotating cluster configurations of the C neutron-rich isotopes were investigated in the CRMF framework (Ref. [25]). Note that pairing correlations have been neglected in the studies of Refs. [24, 25].. It represents the realization of covariant density functional theory (CDFT) for rotating nuclei with no pairing correlations in one-dimensional cranking approximation [26]. It has been successfully tested in a systematic way on the properties of different types of rotational bands in the regime of weak pairing such as normal-deformed, superdeformed and smooth terminating bands as well as the bands at the extremes of angular momentum (see Refs.  [19, 26] and references therein). In the current study, we restrict ourselves to reflection symmetric shapes, which are dominant deformed shapes in nuclear chart.

The properties of neutron-rich nuclei are frequently studied in spherical shell motivated models (see, for example, Refs. [27, 28]). However, because of numerical reasons such models cannot handle the configurations built on intruder orbitals which are involved in the structure of all high spin states. This is seen for example in the $A\sim 60$ $N\sim Z$ region in which highly- and superdeformed configurations are based on intruder orbitals. Such configurations are successfully described in the cranked versions of relativistic and non-relativistic density functional theories [29, 30] but they are outside the applicability range of spherical shell motivated models (see, for example, Ref. [31]).

As discussed in detail below, many rotational bands in near neutron-drip line nuclei are built from resonance and particle-bound parts. In resonance part of the band, at least one single-particle state forming nucleonic configuration is embedded into continuum. On the contrary, all single-particle states have negative energy (are particle-bound) in particle-bound part of the rotational band. The CRMF theory uses basis set expansion for the solution of the underlying equations. Note that in the case of no pairing this is very good approximation for particle-bound parts of rotational bands. This is because all single-particle orbitals have negative energy and there is no coupling with continuum in the case of no pairing. Thus, particle-bound parts of rotational bands are properly described using basis set expansion method. This method would also be appropriate for the case when pairing is present but the Fermi level is located reasonably far away from continuum threshold333Note that all systematic state-of-the-art calculations of the position of the two neutron-drip line in the density functional and mean field theories have been performed in the computer codes employing basis set expansion in large basises (see Refs.  [5, 7, 32] and references therein).. This is indeed a case for a number of bands, the Fermi level of which is located at the energy $e_{F}\sim-2$ MeV for top parts of these bands (see discussion below).

The use of the basis set expansion is more questionable for the nucleonic configurations in which at least one neutron single-particle orbital is embedded in the continuum. This is because of possible couplings with continuum. As a result, the detailed properties of resonance parts of rotational bands are not subject of this study. Their investigation requires cranking codes formulated in coordinate space which to our knowledge are not available nowadays for density functional theories. It is however interesting to mention that in the present CRMF calculations the resonance and particle-bound parts of rotational band do form smooth rotational band without sharp changes on the transition from one part to another.

The CRMF calculations have been performed with the NL3* functional [33] which is state-of-the-art functional for nonlinear meson-nucleon coupling model [7]. It is globally tested for ground state observables in even-even nuclei [7, 34]. The CRMF and cranked relativistic Hartree-Bogoliubov calculations with this functional provide a very successful description of different types of rotational bands [33, 35, 36] both at low and high spins.

The CRMF equations are solved in the basis of an anisotropic three-dimensional harmonic oscillator in Cartesian coordinates characterized by the deformation parameters $\beta_{0}$ and $\gamma$ and oscillator frequency $\hbar\omega_{0}=41A^{-1/3}$ MeV for details see Refs. [23, 21]. The truncation of basis is performed in such a way that all states belonging to the major shells up to $N_{F}=18$ fermionic shells for the Dirac spinors and up to $N_{B}=20$ bosonic shells for the meson fields are taken into account. This truncation scheme provides very good numerical accuracy for the nuclei under study (see discussion of Fig. 4 below).

The rotational structures are studied in even-even 42,44,46,48,50Mg nuclei. Their nucleonic configurations are specified in the calculations by the number of proton/neutron orbitals of four combinations of parity $\pi$ and signature $r$ occupied from the bottom of the potential. Because the pairing correlations are neglected, the intrinsic structure of the configurations of interest can be described by means of dominant single-particle components of intruder/hyperintruder/megaintruder orbitals occupied. Thus, the calculated configurations are labeled by shorthand [$n_{1}$($n_{2}$)($n_{3}$),$p_{1}$] labels, where $n_{1}$, $n_{2}$ and $n_{3}$ are the number of neutrons in the $N=4$ intruder, $N=5$ hyperintruder and $N=6$ megaintruder orbitals and $p_{1}$ is the number of protons in the $N=3$ intruder orbitals. The $n_{3}$ and $n_{2}$ numbers are omitted from shorthand labels when respective orbitals are not occupied. Note that the configurations which are yrast or near-yrast in the $I\leq 30\hbar$ range are shown in Figs. 1 and 6 only up to the spin of first unpaired band crossing with the change of at least one label (either $n_{1}$ or $n_{2}$ or $n_{3}$). On the contrary, higher spin configurations are shown with such band crossings included with respective labelling of the before/after band crossing branches of the configuration.

According to relativistic Hartree-Bogoliubov calculations of Ref. [7], the 46Mg nucleus with 12 protons and 34 neutrons is last neutron bound nucleus in the NL3* functional444There are substantial uncertainties in the predictions of the position of two-neutron drip line in the Mg isotopes (see Refs. [5, 6, 7, 2]) which is typically located from 40Mg up to 46Mg. However, available experimental data does not allow to discriminate between these predictions (see Refs. [1, 2]).. The configurations of this nucleus, which appear as yrast or near-yrast at least in some spin range, are shown in Fig. 1. Its ground state band has the [0,0] configuration in the CRMF calculations. The rotational states of this band are neutron bound since all occupied neutron single-particle routhians have negative energy.

The angular momentum in the valence space configurations is limited (see Ref. [12]). Thus, higher spin states are build by means of particle-hole excitations to intruder and hyperintruder orbitals which increase both the maximum spin within the configurations and quadrupole deformation of the configurations (see discussion in Ref.  [12] and Fig. 1). Fig. 1 clearly shows that with increasing spin the yrast line of the 46Mg nucleus is gradually built by the configurations which have larger and larger intruder/hyperintruder content reaching [411,1] at the highest calculated spins.

This process leads to new physical mechanism which is not active in rotational bands of the nuclei in the vicinity of the $\beta$-stability line. This mechanism is best illustrated on the example of the [3,0] configuration the neutron routhian diagram of which is shown in Fig.  2(a). In this configuration, the $3/2[431](r=-i)$ orbital is the highest in energy occupied positive parity intruder orbital. Note that in the calculations without pairing, the energy of the highest occupied neutron orbital corresponds to the energy of neutron Fermi level 555The stability of the element with respect of one-neutron decay is defined by one-neutron separation energy $S_{n}$. Note that in the calculations with pairing, $S_{n}\approx-\lambda_{n}+\Delta_{n}$ in even $N$ nuclei with $\lambda_{n}$ and $\Delta_{n}$ being the neutron Fermi energy and pairing gap, respectively. In the case of no pairing, $S_{n}\approx-\lambda_{n}$ with the energy of last occupied neutron orbital equal to $\lambda_{n}$. Thus, the nucleonic configuration the energy of the last occupied neutron orbital of which is negative is expected to be particle bound. [10, 38, 40]. At rotational frequency $\Omega_{x}<1.03$ MeV, the $3/2[431](r=-i)$ orbital is particle unbound since its single-particle energy is positive. Above this frequency, its energy becomes negative and this orbital dives deeper into nucleonic potential with increasing rotational frequency. Above $\Omega_{x}\sim 2.0$ MeV, the structure of the [3,0] configuration changes because of the occupation of the lowest $N=5(r=+i)$ orbital (see Fig. 2(a)).

Fig.  1 shows the [3,0] configuration from low spin up to the spin at which the configuration change takes place. At low spin (up to $I\sim 16\hbar$), this rotational band can exist only as a band embedded in particle continuum (see Refs. [41, 42]) (further ’resonance band’) since last occupied orbital, namely, the $3/2[431](r=-i)$ orbital is particle unbound. The resonance width of the states in the resonance band is energy dependent and its existence depends on whether respective neutron or other decay channels are open or closed (see Refs.  [41, 42]). Because of the location in continuum, the detailed properties of resonance bands or the question of their existence are not the subjects of the present study. However, at spin $I>16\hbar$ (at $\Omega_{x}\geq 1.03$ MeV), the configuration [3,0] becomes particle bound. Thus, respective rotational band changes its character from particle-unbound resonance band (at $I<16\hbar$) to particle-bound rotational band (at $I>16\hbar$) with discrete rotational states of extremely narrow width.

So the transition from resonance part to particle-bound part of rotational band in the [3,0] configuration is defined solely by the evolution in energy of the $3/2[431](r=-i)$ orbital with respect of the continuum threshold as a function of rotational frequency (see Fig. 3a). This transition takes place when the single-particle state crosses the continuum threshold at frequency $\omega_{tran}$. The question is whether cranking description in rotating frame provides a proper reproduction of energy evolution of the single-particle state and its crossing of continuum threshold. It turns out that the situation discussed above and shown in Fig. 3a is extremely similar to the description of paired and unpaired band crossings in rotating nuclei. In cranking model description of rotating nuclei with pairing included, two-quasiparticle configuration which is located at some excitation energy with respect of quasiparticle vacuum at zero frequency gets lower in energy with increasing rotational frequency and becomes a vacuum configuration at some frequency $\omega_{pair}$ (see Fig. 3b). This change in vacuum configuration leads to a paired band crossing (Refs. [43, 39, 11, 37]). The experimental data related to the evolution of the energies of quasiparticle states with frequency and the properties of paired band crossings (the frequency of crossing and the gain in alignment at the crossing) in rotational bands are, in general, well described in cranking model with quasiparticle energies defined in rotating frame (see, for example, Refs. [43, 39, 15, 44, 18] for non-relativistic results and Refs.  [17, 19, 36, 45] for relativistic ones). Unpaired band crossings emerge from the crossing of two single-particle orbitals with the same quantum numbers which are separated by some energy at zero frequency/spin (see Fig. 3c and Refs. [38, 40]). Again the cranking models with single-particle orbitals, defined in rotating frame, well describe alignment properties of the single-particle orbitals (which define the change of single-particle energies with frequency/spin) extracted from experimental data (see Refs.  [46, 47, 19]) and experimental data on unpaired band crossings (see, for example, Refs.  [23, 40]).

These results clearly indicate that the cranking description in rotating frame well reproduces the change of the energies of the single-particle states with rotational frequency. Good agreement with experiment obtained in the cranking model proves that such a description is a good approximation to the description in the laboratory frame. This analysis also suggests that that the transition from resonance to particle-bound part of rotational band takes place at approximately the same frequency in rotating and laboratory frames. Note also that the description of this transition is simpler than the description of band crossings because the single-particle orbital does not interact with continuum threshold. On the contrary, there is an interaction of the single-particle states at and near band crossing resulting in the repulsion of the single-particle orbitals (see Fig. 3b,c). It is also important to mention the equivalence of the description of rotating nuclei in the intrinsic (rotating) and laboratory frames was studied in Ref. [48] on the example of 48Cr.

At highest spin displayed in Fig. 1, the configuration [411,1] is yrast. The routhian diagram of the combined [5,1]/[411,1] configuration is shown in Fig. 2b. At low rotational frequency, three occupied neutron orbitals from the $N=4$ shell are located in continuum; this corresponds to the [5,1] configuration. With increasing rotational frequency these orbitals become particle-bound at $\Omega_{x}\sim 1.6$ MeV. Thus, for the spins up to/beyond $I\sim 27.9\hbar$, the [5,1] configuration represents the resonance/particle-bound band. At even higher frequency of $\Omega_{x}\sim 1.8$ MeV ($I\sim 32.7\hbar$), the lowest $N=5$ and $N=6$ orbitals become occupied leading to the [411,1] configuration, which corresponds to particle-bound rotational band.

Fig. 4 illustrates the dependence of binding energies $E$ and neutron single-particle energies $e_{i}$ on the number of fermionic shells $N_{F}$ used in the calculations. One can see that with increasing $N_{F}$, the configuration [3,0] becomes more bound but each subsequent increase of $N_{F}$ leads to the decrease in the gain of the binding energy so that the $N_{F}=18$ and $N_{F}=20$ results differ by only $\approx 150$ keV; this difference represents 0.06% of total binding energy. Note that the slope and the curvature of $E-E_{RLD}$ curve defines the moment of inertia (see Ref. [40]); the similarities of these curves calculated with different values of $N_{F}$ indicate that the moments of inertia only weakly depend on $N_{F}$. Indeed, the difference of the moments of inertia calculated with $N_{F}=18$ and $N_{F}=20$ is of the order of 0.6%. The frequency $\omega_{tran}$ at which the occupied orbital (the $3/2[431](r=-i)$ orbital in the [3,0] configuration) crosses the continuum threshold is another important quantity. This is because it defines the frequency (spin) at which the transition from resonance to particle-bound part of rotational band takes place. Indeed, there is some dependence of $\omega_{tran}$ on $N_{F}$ at low values of $N_{F}$; $\omega_{tran}=1.27$, 1.17, 1.11 and 1.05 MeV at $N_{F}=10$, 12, 14 and 16, respectively. However, further increase of $N_{F}$ does not change $\omega_{tran}$. These features can be understood from Fig. 4b. The single-particle energies of bound states are the same at $N_{F}=18$ and $N_{F}=20$. However, the decrease of $N_{F}$ leads to some minor changes in the energies of the single-particle states. Since the accuracy of the description of the energies of the single-particle routhians is the same as for the states at no rotation, the crossing frequency $\omega_{tran}$ is well defined by the bound parts of occupied routhians in the $N_{F}=18$ calculations.

The situation is somewhat different for the single-particle states in the continuum. The energies of some of the states almost do not change with the increase of $N_{F}$ above 16. These are the 3/2[301], 3/2[312], 7/2[303], and 3/2[431] states (see Fig. 4b). On the contrary, although the energies of the 1/2[301] and 1/2[431] states show the trend to stabilization with increasing $N_{F}$, they do not completely stabilize at $N_{F}=20$ (see Fig. 4b). The extrapolation to higher $N_{F}$ suggests the modification of their energies in the range of few hundred keV as compared with the $N_{F}=20$ results. If these states are occupied in the configurations of interest, they may somewhat modify (as compared with the solution in larger size basises) the energies of resonance parts of the bands. However, these modifications are rather modest (above mentioned few hundred keV) since the rotational properties of the configurations are predominantly defined by bound states. Note also that among considered configurations only few of them involve the occupation of the 1/2[431] orbital in the continuum.

Fig. 5 compares proton and neutron density distributions obtained for the ground state [0,0] (at spin $I=0$) and excited [411,1] (at spin $I=41\hbar$) configurations. One can see substantial differences in the density distributions between these configurations and between proton and neutron density distributions within the same configuration. The later feature is due to the excess of neutrons over protons which also leads to the creation of neutron skin and larger and more spread out neutron density distribution as compared with proton one. The transition from the [0,0] to [411,1] configuration leads to a substantial elongation of density distributions and to a larger neutron skin in the axial direction. The density distributions of the configurations forming the yrast line in 46Mg are located in between of those shown in Fig.  5 and follow the simple rule: the elongation of the density distribution increases with the increase of intruder/hyperintruder content of the configuration.

Fig. 6 shows the results of the CRMF calculations for even-even 42-50Mg nuclei. The first three nuclei are bound at spin zero in the RHB calculations of Ref. [7], while the 48,50Mg nuclei are unbound. The upper panels show both resonance and particle-bound parts of rotational bands. Note, that many calculated resonance bands may not exist because of open decay channels. The bottom panels of Fig. 6 show only particle-bound parts of rotational bands. The ground state bands of the [0,0] structure in spin zero bound 42-46Mg nuclei are particle-bound in full calculated spin range. On the contrary, they consist mostly of resonance parts in spin zero particle unbound 48,50Mg nuclei; only two states at spin $I=4$ and 6 are particle-bound in the 48Mg nucleus and no such states exist in the [0,0] band of 50Mg. The rotational structures built on particle-hole excitations in the nuclei under study consist both of resonance and particle-bound parts. Although the balance of these two parts depends on nucleus and configuration, the general trend of decreasing spin range of particle-bound parts of rotational bands with increasing neutron number are clearly seen. Note that resonance parts of rotational bands exists also in bound nuclei located below the $I=0$ two-neutron drip line.

Fig. 6 reveals for the first time new mechanism of the extension of nuclear landscape towards higher neutron numbers beyond the $I=0$ two-neutron drip line. The 46Mg is last bound nucleus at spin zero. However, particle-bound rotational bands exist in more neutron rich 48,50Mg nuclei. These are the [0,0], [2,0], [3,0] [41,0] and [411,1] configurations in 48Mg (Fig. 6i) and the [511,1] configuration in 50Mg (Fig. 6j). Particle-hole excitations building these configurations increase their intruder/hyperintruder content and deformations. The combined effect of these two factors changes their character from resonance type at low spin to particle-bound at high spin. While this is quite frequent effect in 48Mg, this transition, which takes place at very high spin, requires significant deformation and high intruder/hyperintruder content in the 50Mg nucleus which is located four neutrons beyond the $I=0$ two-neutron drip line.

Note that this mechanism has large similarities with the one seen recently in proton-rich hyperheavy $(Z\geq 126$) nuclei (see discussion in Sect. IX of Ref.  [49]). Proton chemical potential $\lambda_{\pi}$ has a pronounced slope as a function of quadrupole deformation $\beta_{2}$: for a given nucleus the proton Fermi level is located at substantially lower energies at toroidal shapes as compared with ellipsoidal ones. This leads to the transition from proton-unbound ellipsoidal nuclear shapes to proton-bound toroidal ones. In a similar way in rotating nuclei the transition from unbound (resonance) to bound part of rotational band is triggered by an increase of other collective coordinate, namely, rotational frequency. Similar to rotating nuclei, above mentioned feature of $\lambda_{\pi}$ in hyperheavy nuclei is a source of unusual shift in the position of two proton-drip line toward more proton rich nuclei (as compared with general trend seen in the $(Z,N)$ plane for the $Z<120$ nuclei) (see Ref. [49]).

The current investigation neglects the pairing correlations. While this is crude approximation for the ground state bands, it becomes much more realistic with increasing of spin and the number of particle-holes excitations involved in the building of nucleonic configurations. Even if some minor pairing correlations will remain active in the high-spin configurations, the general features will not be strongly affected. This is because the neutron Fermi level will be located in the vicinity of last occupied neutron orbital (as defined in the calculations without pairing), and, thus it will also be diving into the region of negative energies with increasing rotational frequency.

The possibility to observe particle-bound parts of rotational bands by means of traditional gamma-spectroscopy depends on the competition of gamma-decay with neutron(s) emission. For such an observation, the half-lives of gamma-decays from the states within the rotational bands have to be shorter than the half-lives of neutron(s) emission. Gamma-decay half-lives are expected to be similar to the ones of experimentally known rotational bands; typically these half-lives are extremely short being in nanosecond ($10^{-9}$ s) range. On the contrary, neutron(s) emission from rotational states of particle-bound parts of rotational bands has not been studied so far. In general, the suppression of the coupling of the bound states with continuum due to decreasing role of pairing with increasing spin is expected to lead to a substantial suppression of neutron emission at high spin. Note that weakly bound states along the proton drip line still form rotational bands which are seen in experiment. It is reasonable to expect that this will also be true for bound parts of rotational bands of near-neutron drip line nuclei. However, a quantitative answer on the question of the competition of gamma- and neutron(s) decays requires an additional investigation which will be performed in future. Note that the existence of particle-bound rotational bands will not be affected even in the case when neutron(s)-decay rates are shorter than gamma-decays; this will simply change the tool of their observation from gamma-spectroscopy to particle-spectroscopy. With nuclear timescale of $\tau\sim 10^{-23}$ s, there is huge range of half-lives in which the particle-bound rotational states can exist. For example, in nuclear world the analog of the Earth (extremely stable object inhabited by humans) would have existed only for time of $\sim 4.5\cdot 10^{-14}$ sec = $\tau$ times $4.5\cdot 10^{9}$ revolutions of Earth around Sun. This time is by orders of magnitude shorter than the typical gamma-decay half-lives of rotational states.

In conclusion, the investigation of rotational properties of the Mg isotopes in the vicinity of two-neutron drip line has been performed in the cranked relativistic mean field theory. It reveals two new physical mechanisms active in the vicinity of neutron drip line which have not been discussed before in the literature. Nucleonic configurations having occupied neutron single-particle orbitals in continuum lead to the formation of resonance bands in the calculations; many of them will not exist in nature because of open decay channels. However, occupied intruder/hyperinitruder orbitals of these configurations initially located in continuum dive (because of their high-$j$ content leading to strong Coriolis force) into the region of negative energies with increasing rotational frequency and become particle-bound. This mechanism leads to the birth of particle-bound rotational bands. Alternative possibility of the transition from particle-bound to resonance part of rotational bands (the death of particle-bound rotational bands) with increasing spin also exists but it is less frequent since it requires the occupation of the orbital, strongly up-sloping in rotational frequency, which raises from negative to positive energy with increasing rotational frequency. These features of the birth of particle-bound rotational bands provide a new mechanism of the extension of nuclear landscape at non-zero spin to neutron numbers which are larger than those for two-neutron drip line at spin zero.

Acknowledgements

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award No. DE-SC0013037 and KAKENHI Grant No. 17K05440.