Effects of pairing, continuum, and deformation on particles in the classically forbidden regions for Mg isotopes

TL;DR

This study investigates how pairing, continuum effects, and deformation influence particles in classically forbidden regions of Mg isotopes using a relativistic Hartree-Bogoliubov framework, revealing significant deviations in neutron halo nuclei.

Contribution

It introduces a detailed analysis of deformation and continuum effects on particles in forbidden regions for Mg isotopes within a relativistic mean-field approach.

Findings

Neutron and proton radii deviations are largest in $^{42}$Mg and $^{44}$Mg.

Deformation increases particles in forbidden regions below the continuum threshold.

Deeply bound states significantly contribute to particle increase due to deformation.

Abstract

Particles in the classically forbidden regions are studied based on the deformed relativistic Hartree-Bogoliubov theory in continuum with PC-PK1 for magnesium isotopes. By analyzing the neutron and proton radii, it is found that the largest deviations from the empirical values appear at the predicted neutron halo nuclei $^{42}$Mg and $^{44}$Mg. Consistently, notable increases at $^{42}$Mg and $^{44}$Mg are found in the total number of neutrons in the classically forbidden regions that includes the number of neutrons in continuum. It is shown that the deformation effect, in general, increases the number of particles in the classically forbidden regions below the continuum threshold. The most deeply bound single-particle states play the dominant roles in the increase caused by deformation.

Equations

$$f(\mathbf{r})=\sum _{\lambda }{f}_{\lambda }(r)P_{\lambda }(\cos \theta ),\lambda =0,2,4,\cdots ,$$

$$N_{\mathrm{CF}}^{\beta }={\int }_{{\mathbf{R}}_{\mathrm{CF}}^{\beta }}^{\infty }{\rho }_{\beta }(\mathbf{r})d^3\mathbf{r},$$

$$\begin{array}{rl}{N}_{\mathrm{CF}}^{\beta }& ={\int }_{{\mathbf{R}}_{\mathrm{CF}}^{\beta }}^{\infty }{\rho }_{\beta }(\mathbf{r})d^3\mathbf{r}\\ & ={\int }_0^{2\pi }d\phi {\int }_0^{\pi }\sin \theta d\theta {\int }_{R_{\mathrm{CF}}^{\beta }}^{\infty }{r}^2{\rho }_{\beta }(r)dr\\ & =4\pi {\int }_{R_{\mathrm{CF}}^{\beta }}^{\infty }{r}^2{\rho }_{\beta }(r)dr,\end{array}$$

$$V[R_{\mathrm{CF}}^{\beta }(\theta ),\theta ]=e_{\beta }.$$

$$\begin{array}{rl}{N}_{\mathrm{CF}}^{\beta }& ={\int }_{{\mathbf{R}}_{\mathrm{CF}}^{\beta }}^{\infty }{\rho }_{\beta }(\mathbf{r})d^3\mathbf{r}\\ & =\sum _{\lambda }{\int }_0^{2\pi }d\phi {\int }_0^{\pi }{P}_{\lambda }(\cos \theta )\sin \theta \left[{\int }_{R_{\mathrm{CF}}^{\beta }(\theta )}^{\infty }{r}^2{\rho }_{\lambda }^{\beta }(r)dr\right]d\theta \\ & =2\pi \sum _{\lambda }{\int }_0^{\pi }{P}_{\lambda }(\cos \theta )\sin \theta \left[{\int }_{R_{\mathrm{CF}}^{\beta }(\theta )}^{\infty }{r}^2{\rho }_{\lambda }^{\beta }(r)dr\right]d\theta .\end{array}$$

$$N_{\mathrm{CF}}=\sum _{\beta }{v}_{\beta }^2\cdot d_{\beta }\cdot N_{\mathrm{CF}}^{\beta },$$

Full text

I Introduction

The development of radioactive ion beam facilities worldwide Zhan et al. (2010); Gales (2010); Thoennessen (2010) stimulates greatly the study of so-called exotic nuclei far from the $\beta$ stability line in both experimental side Tanihata (1995); Sorlin and Porquet (2008); Alkhazov et al. (2011); Tanihata et al. (2013); Savran et al. (2013) and theoretical side Vretenar et al. (2005); Meng et al. (2006); Meng and Zhou (2015); Meng (2016); Chatterjee and Shyam (2018). Exciting discoveries in exotic nuclei including halo phenomena Tanihata et al. (1985), pygmy resonances Adrich et al. (2005), and changes of magic numbers Ozawa et al. (2000); Iwasaki et al. (2000); Kanungo et al. (2002); Liddick et al. (2004); Becheva et al. (2006); Bastin et al. (2007); Hoffman et al. (2008); Steppenbeck et al. (2013); Tanihata et al. (2019) have attracted a lot of attention.

As a microscopic quantum system, the atomic nucleus has many quantum characteristics and exhibits rich quantum phenomena. The quantum tunneling effect allows the penetration of nucleons in the classically forbidden (CF) regions and as a consequence has impacts on nuclear density distributions. For exotic nuclei that are very weakly bound systems, and particularly for halo nuclei that have very extended spatial density distributions, the tunneling effect is extremely important. Therefore, it is of particular interest to investigate particles in the CF regions for exotic nuclei.

In Refs. Im and Meng (2000a, b), particles in the CF regions for calcium isotopes were investigated with the spherical Skyrme Hartree-Fock theory. It was found that with increasing mass number $A$, the neutron number in the CF regions increases due to the increase of occupied neutrons in open shell while the proton number in the CF regions decreases due to proton orbits becoming more tightly bound. As the difference between the numbers of proton and neutron in the CF regions is very similar to the difference of the density distributions of proton and neutron, the particle number in the CF regions can give a signal for the halo or skin Im and Meng (2000b).

For exotic nuclei, pairing correlations and the coupling to continuum must be taken into account properly Dobaczewski et al. (1984); Meng and Ring (1996). Meanwhile, for open shell nuclei, one has to further deal with the deformation effect carefully. It is therefore necessary to investigate particles in the CF regions within a theoretical model that includes simultaneously the effects of pairing, continuum and deformation.

The covariant density functional theory (CDFT) describes nucleons as Dirac spinors which interact by exchanging effective mesons or point coupling in a microscopic and covariant way. The CDFT naturally includes the nucleonic spin degree of freedom and automatically results in a nuclear spin-orbit potential with empirical strength. It can give naturally the pseudospin symmetry in the nucleon spectrum Ginocchio (1997); Meng et al. (1998a, 1999); Chen et al. (2003); Ginocchio (2005); Liang et al. (2015) and the spin symmetry in the anti-nucleon spectrum Zhou et al. (2003a); Liang et al. (2015). Furthermore, the CDFT can include the nuclear magnetism Koepf and Ring (1989), which plays an important role in nuclear magnetic moments Yao et al. (2006); Arima (2011); Li et al. (2011a, b); Li and Meng (2018) and nuclear rotations Meng et al. (2013); König and Ring (1993); Afanasjev et al. (2000); Afanasjev and Ring (2000); Afanasjev and Abusara (2010); Zhao et al. (2011a, b, 2012a); Wang (2017, 2018). Due to its successful description of many nuclear phenomena, the CDFT has become one of the most important microscopic methods in theoretical nuclear physics and has attracted wide attention Ring (1996); Vretenar et al. (2005); Meng et al. (2006); Niksic et al. (2011); Meng et al. (2013); Meng and Zhou (2015); Meng (2016).

Based on the CDFT, taking advantage of the Bogoliubov transformation and solving the relativistic Hartree-Bogoliubov equations in the coordinate space, the relativistic continuum Hartree-Bogoliubov (RCHB) theory was constructed to consider pairing correlations and continuum in a unify and self-consistent way Meng and Ring (1996); Meng (1998). The RCHB theory has provided an interpretation of the halo in 11Li Meng and Ring (1996), predicted giant halos Meng and Ring (1998); Meng et al. (2002a); Zhang et al. (2002), reproduced the interaction cross section and the charge-changing cross sections in light exotic nuclei in combination with the Glauber theory Meng et al. (1998b, 2002b), better restored the pseudo-spin symmetry in exotic nuclei Meng et al. (1998a, 1999), and made predictions of exotic phenomena in hypernuclei Lu et al. (2003) and new magic numbers in superheavy nuclei Zhang et al. (2005).

In order to describe deformed nuclei properly, the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) was developed by solving the deformed relativistic Hartree-Bogoliubov equation in a Dirac Woods-Saxon basis Zhou et al. (2010); Li et al. (2012a, b). The DRHBc theory was applied to study the chain of magnesium isotopes and an interesting shape decoupling between the core and the halo was predicted in 44Mg and 42Mg Zhou et al. (2010); Li et al. (2012a). Later, the DRHBc theory was extended to incorporate the blocking effect which is required for the description of odd-nucleon systems Li et al. (2012b), and the density-dependent meson-nucleon couplings Chen et al. (2012). Recently, using the DRHBc theory, the puzzles concerning the radius and configuration of valence neutrons in 22C were resolved and 22C was predicted to be a new candidate for deformed halo nuclei with shape decoupling effects Sun et al. (2018).

In this work, the DRHBc theory is applied to investigate particles in the CF regions for magnesium isotopes. The results are compared with those from the RCHB theory and the RCHB calculations without pairing. The effects of pairing, continuum and deformation on particles in the CF regions are explored. Particles in the CF regions for a given single-particle state are also investigated.

II Theoretical framework

The starting point of the CDFT is a Lagrangian density where nucleons are described as Dirac spinors which interact via exchange of effective mesons or point coupling. With the mean-field and the no-sea approximations, one can obtain the energy density functional for the nuclear system. According to the variational principle, one obtains the equation of motion for nucleons by minimizing the energy density functional with respect to the densities. Without pairing correlations, the relativistic Hartree equation can be solved either in the basis space or in the coordinate space Ring (1996); Zhou et al. (2003b); Ren et al. (2017, 2019).

In the RCHB theory, the pairing is taken into account by the Bogoliubov transformation and the relativistic Hartree-Bogoliubov equations are solved in the coordinate space with spherical symmetry Meng and Ring (1996); Meng (1998). The details of the RCHB theory can be found in Ref. Meng (1998).

In the DRHBc theory, the potentials and densities are expanded in terms of the Legendre polynomials,

[TABLE]

to include the deformation degree of freedom. The deformed relativistic Hartree-Bogoliubov equations have been solved in a Dirac Woods-Saxon basis, in which the radial wave functions have a proper asymptotic behavior in large $r$ Zhou et al. (2010); Li et al. (2012a, b). The details of the DRHBc theory can be found in Ref. Li et al. (2012a).

Following Ref. Im and Meng (2000b), the number of particles in the CF regions for a single-particle state $\beta$, $N_{\mathrm{CF}}^{\beta}$, is defined by

[TABLE]

in which $\bm{R}_{\mathrm{CF}}^{\beta}$ is the position where the mean field potential is equal to the single-particle energy $e_{\beta}$, i.e., $V(\bm{R}_{\mathrm{CF}}^{\beta})=e_{\beta}$, and $\rho_{\beta}$ is the density of the single-particle state $\beta$.

In the spherical RCHB theory, $N_{\mathrm{CF}}^{\beta}$ can be calculated by

[TABLE]

in which $R_{\mathrm{CF}}^{\beta}$ is the radial coordinate where the mean field potential is equal to the single-particle energy, i.e., $V(R_{\mathrm{CF}}^{\beta})=e_{\beta}$. The boundary of the CF regions is a spherical surface with radius $R_{\mathrm{CF}}^{\beta}$ correspondingly.

In the DRHBc model, the boundary of the CF regions is no longer a sphere but a surface determined numerically. For every fixed $\theta$, there exists a radial position $R_{\mathrm{CF}}^{\beta}(\theta)$, and they satisfy

[TABLE]

As a result, $N_{\mathrm{CF}}^{\beta}$ is calculated by

[TABLE]

It is noted that the states in continuum have energies higher than the threshold, which means they cannot be occupied from the classical perspective, therefore the number of particles in continuum, $N_{\mathrm{continuum}}$, should contribute also to the number of particles in the CF regions. Finally, the number of particles in the CF regions is calculated by

[TABLE]

where $v_{\beta}^{2},d_{\beta}$ are the occupation probability and the degeneracy of single-particle state $\beta$, respectively. The summation runs over all of the occupied single-particle states. For the bound states, $N_{\mathrm{CF}}^{\beta}$ is calculated by Eq. (3) or Eq. (5), and for the states in continuum, $N_{\mathrm{CF}}^{\beta}=1$.

In the following, we use $N_{\mathrm{CF}}^{n}$ ($N_{\mathrm{CF}}^{p}$) to represent the neutron (proton) number in the CF regions that includes the neutron (proton) number in continuum, and $N_{\mathrm{CF}}^{n}-N_{\mathrm{continuum}}^{n}$ to represent the number of neutrons in the CF regions below the continuum threshold. The sum of $N_{\mathrm{CF}}^{n}$ and $N_{\mathrm{CF}}^{p}$ gives the number of total nucleons in the CF regions for a nucleus.

III Numerical details

To explore the effects of pairing, continuum and deformation, the even-even Mg nuclei from proton drip line to neutron drip line are investigated using the DRHBc theory and the RCHB theory. For the particle-hole channel, the relativistic density functional PC-PK1 Zhao et al. (2010), which has turned out to be very successful in providing good descriptions of the isospin dependence of the binding energy along either the isotopic or the isotonic chain Zhao et al. (2012b); Zhang et al. (2014), is adopted. For the particle-particle channel, the pairing is taken into account by using a density-dependent zero-range pairing force Meng (1998); Li et al. (2012a). In the present DRHBc calculations, we fix the box size $R_{\mathrm{box}}=20$ fm, the mesh size $\Delta r=0.1$ fm, and the angular momentum cutoff $J_{\mathrm{max}}=23/2~{}\hbar$. The parameters in pairing force follow Ref. Li et al. (2012a), i.e., the saturation density $\rho_{\mathrm{sat}}=0.152~{}\mathrm{fm}^{-3}$, and the pairing strength $V_{0}=-380~{}\mathrm{MeV}~{}\mathrm{fm}^{3}$ together with the cutoff energy of $60$ MeV for the pairing window. It was shown that these numerical conditions give results accurately enough in energies and radii for light nuclei Zhou et al. (2003b); Li et al. (2012a). The maximal order $\lambda_{\mathrm{max}}=4$ is adopted in the Legendre expansion Eq. (1) of the deformed potentials and densities. For determination of the Woods-Saxon basis an energy cutoff $E^{+}_{\mathrm{cut}}=300$ MeV for positive-energy states is used, and the number of negative-energy states in the Dirac sea is the same as that of positive-energy states in the Fermi Sea Zhou et al. (2003b). The numerical details in the RCHB calculations are the same as those in Ref. Xia et al. (2018).

IV Results and Discussion

In Fig. 1, the binding energies per nucleon for magnesium isotopes from the DRHBc calculations are shown as functions of the the neutron number. It can be seen that the binding energies per nucleon decrease as the neutron number moves away from $12$, i.e. 24Mg, which is the most stable nucleus in Mg isotopes. The deviations between the calculated neutron (proton) root-mean-square (rms) radii $R_{n}(R_{p})$ and the empirical values $r_{0}^{n}N^{1/3}(r_{0}^{p}Z^{1/3})$ denoted by the size of each point are shown. $r_{0}^{n}$ and $r_{0}^{p}$ are the lowest ratios $R_{n}/N^{1/3}$ and $R_{p}/Z^{1/3}$ determined from 26Mg (star). In general, as the nucleus moves away from 26Mg, the deviation from the empirical value becomes larger for both neutron and proton. For neutron, the largest deviations from the empirical values appear at 42Mg and 44Mg; this is consistent with the suggested halo phenomena in Refs. Zhou et al. (2010); Li et al. (2012a). For protons, the empirical value $r_{0}^{p}Z^{1/3}$ is a constant, and after 26Mg the proton radius increases gradually except for 46Mg. Instead of the increase in neutron radius from 44Mg to 46Mg, the proton radius slightly decreases, as 44Mg has a prolate shape whereas 46Mg is spherical in the DRHBc calculations.

It was suggested that particles in the CF regions can also provide information on the appearance of halos or skins Im and Meng (2000b, a). Figure 2 shows the number of neutrons, protons and total nucleons in the CF regions as functions of the neutron number for magnesium isotopes. In general, with the neutron number, the evolution trends from different calculations are consistent. The number of protons in the CF regions ($N_{\mathrm{CF}}^{p}$) gradually decreases with the neutron number, whereas the number of neutrons in the CF regions ($N_{\mathrm{CF}}^{n}$) rapidly increases. As a result, the number of total nucleons in the CF regions also increases with the neutron number. Similar results have been found in a previous study with the Skyrme Hartree-Fock model Im and Meng (2000b). For the nuclei near the neutron drip line, the difference $N_{\mathrm{CF}}^{n}-N_{\mathrm{CF}}^{p}$ becomes as large as $2$. $N_{\mathrm{CF}}^{n}$ increases with the neutron number due to the increase in occupied neutrons in weakly bound single-particle levels whereas $N_{\mathrm{CF}}^{p}$ decreases due to proton orbits becoming more tightly bound, which is the reason for the large difference and has been analyzed carefully in Ref. Im and Meng (2000b). Although consistent in general, there are some differences in the results from the DRHBc theory vs the RCHB theory. Notable increases can be seen in $N_{\mathrm{CF}}^{n}$ from the DRHBc calculations at the predicted neutron halo nuclei 42Mg Li et al. (2012a) and 44Mg Zhou et al. (2010). Moreover, there is a prominent bump near 24Mg for both $N_{\mathrm{CF}}^{p}$ and $N_{\mathrm{CF}}^{n}$.

As for halo nuclei, occupation of the continuum becomes very important Dobaczewski et al. (1984); Meng (1998), to study the continuum effect, Fig. 3 shows the number of neutrons in the CF regions, the number of neutrons in the CF regions below the continuum threshold ($N_{\mathrm{CF}}^{n}-N_{\mathrm{continuum}}^{n}$), and the contributions from the continuum obtained from the RCHB and DRHBc calculations. As seen in Fig. 3(a) and (b), for nuclei with $8\leq N\leq 20$, the numbers of neutrons in continuum are very small or even [math], but for the neutron-rich side, the numbers of neutrons in continuum are remarkable. In panel (a), $N_{\mathrm{CF}}^{n}$ from the RCHB calculations without pairing are also shown for comparison. For isotopes with $8\leq N\leq 20$, $N_{\mathrm{CF}}^{n}$ values from the RCHB calculations without pairing are very close to $N_{\mathrm{CF}}^{n}$ and $N_{\mathrm{CF}}^{n}-N_{\mathrm{continuum}}^{n}$ from the RCHB calculations, but for neutron-rich nuclei, $N_{\mathrm{CF}}^{n}$ from the RCHB calculations without pairing are in between them. This is because for stable isotopes with $8\leq N\leq 20$, neutrons occupy the well bound single-particle levels which are slightly affected by pairing correlations, but on the neutron-rich side, pairing correlations can scatter valance neutrons into continuum, which significantly contributes to $N_{\mathrm{CF}}^{n}$. In panel (b), the notable increases in $N_{\mathrm{CF}}^{n}$ at 42Mg and 44Mg in the DRHBc calculations can be well understood now. It shows that the increases mainly come from the increases in neutron numbers in continuum. This phenomenon is consistent with the previous predictions of halo phenomena in the nuclei 42Mg Li et al. (2012a) and 44Mg Zhou et al. (2010).

In order to see the possible deformation effect, for Mg isotopes, the number of neutrons in the CF regions below the continuum threshold, $N_{\mathrm{CF}}^{n}-N_{\mathrm{continuum}}^{n}$, and the ground state deformations from the DRHBc calculations are shown in Fig. 4. As seen in Fig. 4(b), the ground states of 20Mg, 32Mg and 46Mg are spherical in the DRHBc calculations. As the neutron number increases from $8$ to $20$ and from $20$ to $34$, the shape of the nucleus goes from spherical to prolate, and then back to spherical. The ground state of 24Mg is well deformed with the largest quadrupole deformation $\beta_{2}\approx 0.60$, which is consistent with the experimental value $\beta_{2}=0.609(6)$ Pritychenko et al. (2016). Comparing (a) and (b), one finds that the appearance of the bump near 24Mg is caused by the strong deformation effect. Moreover, Mg isotopes with $20<N<34$ are also deformed with large deformations $\beta_{2}\approx 0.3\sim 0.5$, and $N_{\mathrm{CF}}^{n}-N_{\mathrm{continuum}}^{n}$ values from the DRHBc theory are larger than those from the RCHB theory in this region except for 44Mg. For 44Mg, as can be seen in Fig. 3(b), the coupling to continuum is extremely strong and more than $1.5$ neutrons are scattered into continuum in the DRHBc theory (only $\sim 0.5$ neutron is scattered into continuum in the RCHB theory).

To better understand the deformation effect on the number of particles in the CF regions, taking 24Mg as an example, constraint DRHBc calculations with $\beta_{2}=0.1,0.2,0.3,0.4,0.5$ and $0.6$ are performed. Here pairing correlations are not taken into account to exclude the influence of the pairing effect. It is found that both the neutron number and the proton number in the CF regions increase with increasing $\beta_{2}$ and the increases in $N_{\mathrm{CF}}^{n}$ and $N_{\mathrm{CF}}^{p}$ are very close. From $\beta_{2}=0.1$ to $\beta_{2}=0.6$, the numbers of neutrons in the CF regions are $1.331,1.393,1.542,1.639,1.761$ and $1.847$, respectively. To be more explicit, the contributions of different single-neutron states are further investigated. Each state is labeled with $\Omega_{i}^{\pi}$ where $\Omega$ is the projection of the angular momentum on the symmetry axis, $\pi$ is the parity, and $i$ is used to order the level in the $\Omega^{\pi}$ block, as $\Omega$ and $\pi$ are good quantum numbers in the axially deformed system. It is found that the increase in $N_{\mathrm{CF}}^{n}$ mainly (more than $85\%$) comes from the contributions of the two most deeply bound single-neutron states $(1/2)_{1}^{+}$ and $(1/2)_{1}^{-}$, whereas for other occupied states, the neutron numbers in the CF regions increase or decrease very little.

Figure 5 shows the density distributions and the boundary of the classically forbidden regions for the single-neutron states $(1/2)_{1}^{+}$, $(1/2)_{1}^{-}$ and $(1/2)_{2}^{+}$ obtained from the constraint DRHBc calculations at $\beta_{2}=0.1,0.3$ and $0.5$. In each panel, the corresponding single-neutron energy, the neutron number in the classically forbidden regions and the two largest spherical components are also given. It can be seen that the three states become more bound with increasing deformation. For the states $(1/2)_{1}^{+}$ and $(1/2)_{1}^{-}$, the numbers of neutrons in CF regions increase by more than $0.2$, but for the state $(1/2)_{2}^{+}$, the number of neutrons in CF regions decreases only by about $0.03$. When the deformation increases, as seen in Fig. 5, on one hand, the intersection point between the boundary of the CF regions and the $z$ axis moves away from the center, whereas that between the boundary and the $x$ axis approaches the center, and as a result, the CF regions expand with the deformation. On the other hand, the density distributions of neutron states also evolve with the deformation as a result of the changes in the mean-field potential, the single-neutron energies and the component mixing. For the most deeply bound states $(1/2)_{1}^{+}$ and $(1/2)_{1}^{-}$, not only do the CF regions expand, but also the neutron densities at the boundary of the CF regions increase a bit, and then the numbers of neutrons in the CF regions increase a lot and dominate the increase in $N_{\mathrm{CF}}^{n}$. For other states like $(1/2)_{2}^{+}$, the boundaries of CF regions are farther away from the center and the density distributions are more diffuse, therefore the deformation effect on them is weak and their numbers of neutrons in the CF regions slightly change with deformation. Then it is concluded that, in general, the deformation will increase the particle number in the CF regions below the continuum threshold and the most deeply bound single-particle states play the dominant roles in this effect.

To further study the relation between the halo phenomenon and particles in the CF regions, taking the neutron-rich nucleus 44Mg as an example, the numbers of particles in the CF regions for each bound single-particle state, $N_{\mathrm{CF}}^{\beta}$, from the RCHB calculations and the DRHBc calculations as functions of the single-particle energy are shown in Fig. 6. A similar trend of $N_{\mathrm{CF}}^{\beta}$ can be found for neutrons and protons, as they share the same mean field self-consistently. However, one cannot find a monotonic relationship between $N_{\mathrm{CF}}^{\beta}$ and the single-particle energy $e_{\beta}$ in either theoretical calculations. In Fig. 6(a), by comparing the $N_{\mathrm{CF}}^{\beta}$ with the corresponding orbital angular momentum $l$ of the single-particle state, one can find a smaller $N_{\mathrm{CF}}^{\beta}$ for the single-particle state with a larger $l$. This feature was also seen in Ref. Im and Meng (2000b) as a state with large $l$ has a high centrifugal barrier. A weakly bound single-particle state with a low orbital angular momentum $l$ is helpful in the formation of halos Meng and Ring (1996, 1998). It is shown that weakly bound $2p$ states have a larger $N_{\mathrm{CF}}^{\beta}$ than the others.

In Fig. 6(b), the evolutions of $N_{\mathrm{CF}}^{\beta}$ in the DRHBc calculations are similar to those in the RCHB calculations. It can be seen, the $(1/2)_{5}^{-}$ state and the $(3/2)_{3}^{-}$ state, that are near the threshold, have the largest $N_{\mathrm{CF}}^{\beta}$. These two weakly bound orbitals were suggested to give rise to the formation of the halo in 44Mg Zhou et al. (2010) and we find the $p$-wave components for the states $(1/2)_{5}^{-}$ and $(3/2)_{3}^{-}$ are $69.6\%$ and $81.5\%$, respectively. On the other hand, the weakly bound states $(5/2)_{1}^{-}$ and $(3/2)_{2}^{-}$ have much smaller $N_{\mathrm{CF}}^{\beta}$, as their dominant components are $f$ waves ($97.4\%$ and $80.1\%$, respectively) that corresponds to a high centrifugal barrier. Therefore, it is concluded that a weakly bound single-particle state with a low centrifugal barrier has a larger $N_{\mathrm{CF}}^{\beta}$.

V Summary

In summary, particles in the classically forbidden regions for magnesium isotopes are investigated within the relativistic continuum Hartree-Bogoliubov theory and the deformed relativistic Hartree-Bogoliubov theory in continuum with PC-PK1. It is found that, with an increasing neutron number, the number of neutrons in the classically forbidden regions rapidly increases, whereas the number of protons in the classically forbidden regions gradually decreases. Particles in the classically forbidden regions for a given single-particle state have been investigated and it is shown that a weakly bound single-particle state with a low centrifugal barrier has more particles in the classically forbidden regions. Similar conclusions have been obtained by the spherical Skyrme Hartree-Fock theory in Ref. Im and Meng (2000b).

In this paper, the effects of pairing, continuum and deformation on particles in the classically forbidden regions are studied for the first time. For the stable isotopes, as neutrons occupy the well bound single-particle levels, the neutron numbers in the classically forbidden regions are only slightly affected by pairing correlations, but on the neutron-rich side, pairing correlations can scatter valance neutrons into continuum, which significantly contributes to neutron numbers in the classically forbidden regions. Including neutrons in continuum, notable increases in the total number of neutrons in the classically forbidden regions at 42Mg and 44Mg have been found; this is consistent with the largest deviations from empirical values of radii at 42Mg and 44Mg shown in Fig. 1 and the predictions of halo phenomena in 42Mg Li et al. (2012a) and 44Mg Zhou et al. (2010). Comparing the results from the DRHBc theory and the RCHB theory, it is found that, in general, the deformation effect enhances the number of neutrons in the classically forbidden regions below the continuum threshold. The most deeply bound single-particle states play the dominant roles in the increase caused by deformation.

Acknowledgements.

The authors are indebted to J. Meng for constructive guidance and valuable suggestions. Helpful discussions with L. S. Geng, Z. X. Ren, S. B. Wang, Y. Zhang, B. Zhao and P. W. Zhao are greatly acknowledged. This work was partly supported by the National Science Foundation of China (NSFC) under Grants No. 11875075, No. 11621131001, No. 11935003 and No. 11975031, and the National Key R&D Program of China (Contract No. 2018YFA0404400, No. 2017YFE0116700).