Introduction

Exotic phenomena, including halos, have attracted considerable interest from physicists [1-4] and are one of the most popular frontier topics in radioactive nuclear beam physics. In isotope chains, with an increase in the neutron number, nuclei near the neutron-rich dripline may remarkably appear as neutron halos. As ¹¹Li was discovered to be an s-wave two-neutron (2n) halo nucleus [5], the formation of a halo is considered to be a large occupation of low-orbital-angular-momentum (l=0,1) levels close to the threshold of particle emission in weakly bound systems [6-8]. Research on halo structures, both theoretically and experimentally, is of great importance in understanding the structure of the nucleus.

On the experimental side, extensive research on halo nuclei with small mass numbers has led to a clear understanding of their halo structures; for example, ⁸B is a proton (1p) halo nucleus [9]; ⁶He [10, 11], ¹¹Li [5, 12], and ¹⁴Be [13, 14] are 2n halo nuclei; and ¹¹Be [15, 16] is a neutron (1n) halo nucleus. Subsequently, ²²C, ¹⁷B, and ¹⁹B were found to be s-wave 2n halo nuclei with large matter radii [17, 18], a small percentage of s-orbital spectroscopic factors [19], and an enhanced electric dipole strength [20], respectively. In the heavier mass region, ³⁷Mg [21] (the heaviest halo nucleus observed thus far) and ³¹Ne [22] and ²⁹F [23] (the heaviest Borromean nucleus to date) were found to be p-wave halos at or near the inversion island by single-neutron (1n) removal reactions and reaction-cross-section measurements, respectively.

On the theoretical side, the search for new 2n-halo systems [24] and new halo features, such as core–halo shape decoupling [25, 26] and the Efimov state [27, 28], have also attracted considerable interest. It can be seen that valence neutrons predominantly occupy p and s orbitals, so that correlations between f–p or s–d orbitals play an essential role in the formation of halo phenomena. These strong quadrupole correlations between f–p or s–d orbitals can drive the system to break spherical symmetry and undergo deformation, allowing the number of single-particle levels with low-l components to increase, resulting in abundant halos at the neutron dripline [29]. Specifically, Hove et al. [30] used the example of a medium-heavy nucleus ⁷²Ca and found that the presence of a slight, long-range s-wave tail is also sufficient to bring about halos in very weakly bound neutron-rich systems. Therefore, it is necessary to study the effects of quadrupole deformation and continuum effects on the halo structure of exotic nuclei.

In this study, we investigated the neutron halo structures of B isotopes. ¹⁵B and ¹⁷B neutron halo structures have been reported [31, 32]. For ¹⁵B, the 2n separation energies (S_2n) were 3.747 MeV [33], the quadruple deformation length δ was 1.17 [34], the quadruple deformation parameter β₂ was 0.61 [35], and the percentage of the s orbital was 63% [36].

For ¹⁷B, a strong mixing of s and d waves was observed based on the energy dependence observed by the interaction cross section [37]. Tanihata [24] considered large amplitudes of 2s_1/2 waves for all B isotopes with neutron number n≥8. For ¹⁷B, S_2n is 1.37 MeV [38], δ is 1.2 [34], β₂ is ∼0.6 [39] or 0.56 [35], the neutron skin thickness is 0.51 (0.11) fm [40], and the neutron has a large matter radius [41, 40]. The percentages of s orbital in ¹⁷B are 36(19)% [31], 69(20)% [32], 50(10)% [37], and 53(21)% [42], respectively. In 2021, Yang et al. [19] obtained spectral factors for the 1s_1/2 and 0d_5/2 orbitals, with the 1s_1/2 orbitals contributing only 9(2)% of the spectral factor, which is much smaller than the values given in Refs. [31, 42], revealing that ¹⁷B has a well-defined, but not dominant, neutron halo. The structure of this weak halo has been explained by the deformed relativistic Hartree–Bogoliubov theory in the continuum (DRHBc) [19], indicating the significance of deformation effects.

¹⁹B is the heaviest bound nucleus of the B isotopes, and, because ¹⁸B is unbound, ¹⁹B is the Borromean nucleus, suggesting a 2n halo structure. ¹⁹B has a particularly low two-neutron separation energy (S_2n) of ${0.089}_{-0.089}^{+0.560}$ MeV [38], enhanced interaction cross section [31], and large deformation [43-35]. Direct mass measurements revealed that the valence neutrons of ¹⁹B are dominated by d waves, which suppress halo formation [31, 42, 45]. However, Cook et al. [20] found a significant 2n halo in ¹⁹B using Coulomb dissociation. Recently, ¹⁹B has been suggested as a deformed 2n halo candidate with shape-decoupling effects [35]. The structures of ¹⁹B are also related to the newly discovered unbound isotopes ²⁰B and ²¹B.

However, the deformation of B isotopes close to the neutron dripline has not been experimentally determined. The shape of the nucleus is determined by the shell structure of protons and neutrons. Therefore, it is of interest to study quadrupole deformation and continuum effects on halo formation in B isotopes.

The continuum and particularly the resonant states in the continuum play a crucial role in the formation of exotic phenomena. Therefore, it is essential to reasonably handle the resonant states in the continuum, especially those near the threshold. Given the significance of resonant states in studying exotic phenomena, physicists have developed many theoretical resonance models, such as the Green’s function method [46], the complex scaling method (CSM) [47], and the complex momentum representation (CMR) method [48].

To study the halo phenomenon, we used the CMR method based on the coupled-channel approach to solve the equations of motion of the deformed nucleus. It is worth noting that the quadrupole–quadrupole force in the spherical mean field is included in the deformation mean field and would make the single-particle figure work much better in the deformation nucleus than in the spherical nucleus. To determine the appearance of the halo phenomenon more accurately, it is essential to use models in which the radial wavefunctions of the weakly bound particles are treated appropriately in this study.

The CMR method can be used to obtain the bound states, resonant states, and continuum on an equal basis [48, 49]. This method is not only very effective for narrow resonances but also can be reliably applied to broad resonances. In particular, the results of the calculations have been proven to be independent of any nonphysical parameters, which ensures that high-precision energies and widths can be obtained. The CMR method is widely used to study the properties of weakly bound nuclei [50, 51]. We have recently studied in detail the structures of exotic spherical nuclei using the CMR method combined with the relativistic meson exchange model [48, 52] and relativistic point coupling model [53]. Furthermore, pairing correlations are treated with the BCS approximation, coupling bound states to resonant states and describing exotic phenomena such as halos and giant halos very well [54, 55].

In addition, we extended the CMR method to a deformed system. We have systematically investigated experimentally discovered and undiscovered halo structures, establishing that ³¹Ne [50], ³⁷Mg [49], ⁵³Ar [56], ³⁴Na [57], and ^{29, 31}F [58, 59] (or s waves or a mixture of s and p waves) are p-wave halo nuclei and that ¹⁹C [60], ⁷⁵Cr [56], and ⁷⁷Fe [56] are s-wave halo nuclei. Recently, Wang et al. investigated neutron-rich ⁷⁵Cr using the CMR method in a relativistic mean-field framework, which applies to deformed nuclei. They found that ⁷⁵Cr is a d-wave deformed halo nucleus [61]. The ground-state properties of Ti isotopes were also investigated using the relativistic mean-field framework CMR method [62]. The predictions of halo phenomena in the neutron-rich region close to the neutron dripline will help to experimentally explore the deformation halos of more nuclei in medium-mass regions. The present research demonstrates the advantages of the CMR method, which will help us explore further deformed halo nuclei using the CMR method.

In this study, we systematically investigated the exotic phenomena of ^15,17,19B using the CMR method. The energies and widths of the single-particle bound and resonant states, component ratios of the orbitals occupied by valence neutrons, and radial density distribution were obtained. The results revealed that ^17,19B may be prolate halo nuclei, which can be well understood because of the neutron halo phenomenon. In other words, observing the halo phenomena in ^17,19B can be used as critical evidence for the prolate deformation of ^17,19B ground states. The theoretical formalism is described in Sect. II. Numerical details and results are presented in Sect. III. Finally, a summary is provided in Sect. IV.

Formalism

We briefly introduce the theoretical framework of the CMR method to explore the ground-state properties of neutron-rich nuclei. The details of the CMR method can be found in Refs. [48, 50]. The main formulas of the CMR method are provided for the convenience of later discussion. We introduce a theoretical framework using the Hamiltonian $H=T+V\mathrm{,}$ (1) where $T={\overrightarrow{p}}^2/2m$ denotes the kinetic operator with momentum $\overrightarrow{p}=\hslash \overrightarrow{k}$ (where $\overrightarrow{k}$ is the wavevector). Following Ref. [63], the interaction potential V consists of the central potential, deformation potential, and spin-orbit coupling potential: $\begin{array}{l}{V}_{\text{cent}}(r)=V_0(r)\mathrm{,}\\ V_{\text{def}}(\overrightarrow{r})=-{\beta }_2{V}_{0}r\frac{\text{d}f\left(r\right)}{\text{d}r}{Y}_{20}(\vartheta \mathrm{,}\phi )\mathrm{,}\\ V_{\text{sl}}(r)=-16V_0{\Lambda }^2\frac{1}{r}\frac{\text{d}f\left(r\right)}{\text{d}r}(\overrightarrow{s}\cdot \overrightarrow{l})\mathrm{,}\end{array}$ (2) where $\Lambda$ is the reduced Compton wavelength of the nucleon, $\hslash /m_{\text{r}}c$, and $f(r)=\frac{1}{1+e^{\frac{r-R}{a}}}\mathrm{.}$ The diffuseness a=0.67 fm, the radius R=r₀A^1/3 with r₀=1.27 fm, and the depth of potential $V_0=-51+33\frac{N-Z}{A}$ MeV depends on the mass number A, proton number Z, and neutron number N. To obtain the bound states, resonant states, and continuum on an equal footing, the Schrödinger equation is transformed from a coordinate representation to a momentum representation as ${\displaystyle \int }\text{d}\overrightarrow{k^{\prime }}\langle \overrightarrow{k}\left|H\right|\overrightarrow{k^{\prime }}\rangle \psi (\overrightarrow{k^{\prime }})=E\psi (\overrightarrow{k})\mathrm{,}$ (3) where $\psi (\overrightarrow{k})$ are the momentum wavefunctions. For axially deformed nuclei, the third component of the total angular momentum Ω and parity π are good quantum numbers. The wavefunctions ${\psi }_{\text{Ω}}(\overrightarrow{k})$ can be expanded as ${\psi }_{\text{Ω}}(\overrightarrow{k})={\displaystyle \sum _i{\Phi }_i}(\overrightarrow{k})={\displaystyle \sum _{lj}{f}^{lj}}(k){\varphi }_{lj\text{Ω}}({\text{Ω}}_k)\mathrm{,}$ (4) where flj(k) and ϕ_ljΩ(Ωk) are the radial and angular components, respectively. By substituting Eq. (4) into Eq. (3), the Schrödinger equation becomes $\begin{array}{c}\frac{{\hslash }^2{k}^2}{2M}{f}^{lj}(k)+{\displaystyle \int }{k}^{\text{'}}{}^2\text{d}{k}^{\text{'}}{V}_s(l\mathrm{,}j\mathrm{,}k\mathrm{,}{k}^{\text{'}})f^{lj}(k^{\text{'}})\\ -{\beta }_2{\displaystyle \sum _{l^{\text{'}}{j}^{\text{'}}}{\displaystyle \int }{k}^{\text{'}}{}^2}\text{d}{k}^{\text{'}}{V}_d(l\mathrm{,}j\mathrm{,}{l}^{\text{'}}\mathrm{,}{j}^{\text{'}}\mathrm{,}{m}_j\mathrm{,}k\mathrm{,}{k}^{\text{'}})f^{l^{\text{'}}{j}^{\text{'}}}(k^{\text{'}})\\ =E{f}^{lj}(k)\mathrm{,}\end{array}$ (5) with $V_s(l\mathrm{,}j\mathrm{,}k\mathrm{,}{k}^{\text{'}})=\frac{2}{\pi }{\displaystyle \int }{r}^2\text{d}r[V_{\text{cent}}(r)+C_{\text{sl}}{V}_{\text{sl}}(r)]j_l(kr)j_l(k^{\text{'}}r)$ and $\begin{array}{l}{V}_d(l\mathrm{,}j\mathrm{,}{l}^{\text{'}}\mathrm{,}{j}^{\text{'}}\mathrm{,}\text{Ω}\mathrm{,}k\mathrm{,}{k}^{\text{'}})\\ =i^{3l+l^{\text{'}}}\frac{2}{\pi }{\displaystyle \int }{r}^2\text{d}rk\left(r\right)j_l(kr)j_{l^{\text{'}}}(k^{\text{'}}r)\\ \cdot {\displaystyle \sum _{m_s}\langle lm\left|Y_{20}({\text{Ω}}_r)\right|l^{\text{'}}m\rangle }\langle lm\frac{1}{2}{m}_s|j\text{Ω}\rangle \langle l^{\text{'}}m\frac{1}{2}{m}_s|j^{\text{'}}\text{Ω}\rangle \mathrm{.}\end{array}$ The above Schrödinger equation is solved in complex momentum space by diagonalizing the Hamiltonian matrix. For the sake of computational simplicity, the integral of Eq. (5) is transformed into summation form, and Eq. (5) becomes the matrix equation. In this study, it was combined with the coupled channel method to describe the deformed nuclei. Details can be found in Refs. [60, 59].

For every Nilsson level, the wavefunction is extended with a set of spherical components. The wavefunctions in the coordinate representation can be obtained using the following transformation: ${\psi }_{\text{Ω}}\left(\overrightarrow{r}\right)=\frac{1}{{\left(2\pi \right)}^{3/2}}{\displaystyle \int }d\overrightarrow{k}{e}^{i\overrightarrow{k}\cdot \overrightarrow{r}}{\psi }_{\text{Ω}}(\overrightarrow{k})\mathrm{.}$ By assuming that ${\psi }_{\text{Ω}}(\overrightarrow{r})$ holds the form ${\psi }_{\text{Ω}}\left(\overrightarrow{r}\right)={\displaystyle \sum _{lj}{f}^{lj}}(r){\varphi }_{lj\text{Ω}}({\text{Ω}}_r)\mathrm{,}$ (6) where $f^{lj}(r)=i^l\sqrt{\frac{2}{\pi }}{\displaystyle \sum _{a=1}^N\sqrt{w_a}}{k}_a{j}_l(k_{a}r)f^{lj}(k_a)\mathrm{,}$ (7) the corresponding radial density distributions can be calculated as $\rho (r)={\displaystyle \sum _{lj}{f}^{lj\ast }}(r)f^{lj}(r)\mathrm{.}$ (8)

numerical details and results

The details of the numerical calculations are the same as those in Refs. [60, 59], where the number of coupling channels, Nc, momentum truncation maximum value kmax, and number of Gauss–Legendre grids, Nl are taken to be 8, 4.0 fm^-1, and 120, respectively. The CMR calculations achieved the required precision under these numerical conditions. Because quadrupole deformation and continuum effects play an essential role in the formation of the halo of exotic nuclei [35, 58, 64], the quadrupole deformation parameter β₂ was taken from -0.4 to 0.7 in steps of 0.02. Based on Refs. [48, 50], we found that the positions of the resonant states do not depend on the integration path. Therefore, we chose a sufficiently large contour (k₁=0.0 fm^-1, k₂=0.5-i0.5 fm^-1, k₃=1.0 fm^-1, and kmax=4.0 fm^-1) to expose all relevant resonances. Based on the numerical details presented, we first reveal the contributions of quadrupole deformations and continuum effects to the ground-state properties of ¹⁷B.

The nuclear radius is a significant physical quantity that describes the exotic properties of nuclei. It is known that the root mean square (RMS) radius of halo nuclei (nuclei that form a halo away from the periphery of the core) increases suddenly compared to that of the neighboring nucleus. It can be seen from the definition of halo nuclei that the RMS radii of the orbits occupied by the last few valence neutrons are of crucial importance in researching halo phenomena. Therefore, we calculated the RMS radii for all the single-particle levels in the considered deformation range. For convenience, as shown in Fig. 1, we show only the RMS radii of the levels occupied by the last two valence neutrons of ^13–19B for the quadrupole deformations β₂=-0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7.

Fig. 1Full-screen View

(Color online) RMS radius of single-particle levels occupied by the last valence neutron as a function of the mass number A for ^13–19B. The corresponding deformation β₂ and the quantum numbers of the level occupied by the last valence neutron are displayed in every subfigure.

The halo structure was experimentally confirmed for ^17,19B [19, 20]. The last two valence neutrons of ¹⁷B occupy the levels 5/2[202], 1d_5/2, 1/2[220], and 3/2[211] at β₂ =-0.1, β₂=0.0, β₂ = 0.1–0.2, and β₂ = 0.3–0.7, respectively (see Fig. 2). We can see that the RMS radius of level 3/2[211] occupied by the last two valence neutrons of ¹⁷B significantly increases compared to that of ¹⁵B at β₂=0.6 and β₂=0.7. However, we noticed that the RMS radius of ¹⁷B increases slightly at β₂=0.1–0.2, as a result of the valence neutrons occupying the resonant level 1/2[220]. This deformation region β₂=0.1–0.2 was excluded because ¹⁷B is a weakly bound nucleus. Similar phenomena were observed in ¹⁹B, where the RMS radius increased significantly compared to that of ¹⁷B at β₂=0.3–0.7. For ¹⁵B, there was no significant increase in the RMS radius over the range of deformations compared to the neighboring nuclei. Similar to the phenomena at β₂=0.1–0.2 in ¹⁷B, the increased RMS radius for ¹⁵B at β₂=-0.1–0.2 is caused by the last two valence neutrons occupying an extremely weakly bound level 1/2[200]. However, the deformation region β₂=-0.1–0.2 was excluded because the two-neutron separation energy of the weakly bound level 1/2[200] was much less than the experimental value of 3.747 MeV [33]. These phenomena suggest that ¹⁷B and ¹⁹B may be prolate halo nuclei with deformations β₂ = 0.6–0.7 and β₂ = 0.3–0.7, respectively. In comparison, ¹⁵B is only a prolate deformation nucleus, which is relatively close to the results obtained from DRHBc calculations [35].

Fig. 2Full-screen View

(Color online) Neutron single-particle levels of ¹⁷B as a function of β₂. Every Nilsson orbital is labeled with the quantum numbers $\text{Ω}[N{n}_z\Lambda ]$. The bound and resonant levels are marked by solid and dashed lines, respectively.

The neutron halo structure can be identified by analyzing the valence neutron levels. In Fig. 2, the Nilsson diagram of ¹⁷B, including the resonant and bound levels, is shown to better study the effects of deformation and the continuum on halo structures. The bound and resonant states are marked with solid and dashed lines, respectively, and the corresponding Nilsson orbitals are marked with quantum numbers $\text{Ω}[N{n}_z\Lambda ]$. Levels 1p_1/2 and 2s_1/2, and 1d_5/2 and 1d_3/2 have large shell gaps between them when spherical, that is, in the spherical case, the old magic number N=8 is verified, and the new magic number N=16 is suggested.

For the spherical configuration (β₂ = 0), level 2s_1/2 is lower than 1d_5/2, which means s–d orbitals are inverted compared to the level order in the stable nuclei. This is similar to the case of ¹⁷B from DRHBc calculations [35]. In the spherical case, the resonant orbital 1d_5/2 is occupied by the last two valence neutrons of ¹⁷B, which have a large centrifugal barrier and negative two-neutron separation energy, suppressing the formation of halos. That is, the spherical shape does not support the formation of the halo phenomenon. Because the halo phenomena of ¹⁷B have been observed experimentally [19], this suggests that ¹⁷B is not a spherical nucleus but rather a deformed halo nucleus. Whether ¹⁷B is a prolate or oblate deformation halo nucleus is an problem to be solved.

Driven by the deformation, the spherical levels split and the degenerate single-particle levels become undegenerated. We can see that some of these levels split off, some fall with deformation, and some rise with deformation. The resonant state 1d_5/2 splits into Nilsson levels 1/2[220], 3/2[211], and 5/2[202], and the weakly bound level 2s_1/2 becomes 1/2[200].

Nilsson levels 1/2[200], 5/2[202], 1/2[220], and 3/2[211] are occupied by the last two valence neutrons of ¹⁷B at -0.4≤β₂<-0.14, -0.14≤β₂<0, 0<β₂<0.26, and 0.26≤β₂≤0.7, respectively. Given that ¹⁷B is a weakly bound nucleus, the last valence neutron may occupy the weakly bound level 1/2[200] with a deformation -0.4≤β₂<-0.14 or the weakly bound level 3/2[211] with a deformation 0.26≤β₂≤0.7.

On the oblate side, when pairing correlations are not considered, Nilsson level 1/2[200] is occupied by the last two valence neutrons of ¹⁷B at -0.4≤β₂&-0.14. It should be indicated that the pairing correlations play important role at halo formation [65, 66]. Given the impact of pairing correlations, the last two neutrons can partially occupy level 1/2[200]. The two-neutron separation energies of the weakly bound level 1/2[200] are <0.75 MeV, which is much lower than the experimentally obtained value of 1.37 MeV [38]. This phenomenon suggests that ¹⁷B is not an oblate nucleus.

In contrast, when pairing correlations are not considered, the Nilsson level 3/2[211] is occupied by the last two valence neutrons of ¹⁷B at 0.26≤β₂≤0.7. The S_2n value of level 3/2[211] reaches 1.3–1.6 MeV with a deformation 0.6≤β₂≤0.7, which is close to the experimentally obtained 1.37 MeV [38] and, when combined with the analysis in Fig. 1, suggests that favorable conditions for halo formation are met in this deformation region. Given the impact of pairing correlations, the last two neutrons can partially occupy the level 3/2[211]. This implies that ¹⁷B is a prolate deformation halo nucleus. This does not contradict the prediction of a prolate deformed halo nucleus for ¹⁷B.

Figure 3 shows the widths of the single-particle resonant states in ¹⁷B. For simplicity, the marks of the widths in Fig. 3 are the same as those for the levels shown in Fig. 2. Similar to the levels, there is a shell structure in the widths. Over the deformation range under consideration, the gap at N = 16 is relatively large, not only in a spherical shape but also in a deformed shape, which means that a new magic number N = 16 is supported in ¹⁷B. The width of the resonant state 3/2[211] is consistently small and lies near zero potential energy. As β₂ increases, the width of resonant state 3/2[211] shrinks and the state eventually disappears, which is due to the resonance state 3/2[211] becoming a bound state. Therefore, it is relatively easy to form a halo at the level 3/2[211] in ¹⁷B. The experimentally detected lifetime of ¹⁷B will help us better understand the halo structure of ¹⁷B because the widths are inversely proportional to the decay lifetime.

Fig. 3Full-screen View

(Color online) Same as Fig. 2, but for the widths for all the resonances of interest.

The above discussions indicate that the last two valence neutrons of ¹⁷B occupy the Nilsson level 3/2[211] and that ¹⁷B is a prolate halo nucleus with a deformation 0.6≤β₂≤0.7. To reveal the spherical component of the Nilsson orbitals occupied by the last valence neutron, the occupation probabilities of the level 3/2[211] are shown in Fig. 4. It can be seen that the configuration d_5/2 exceeds 90%. Configuration d_3/2 increases slowly, reaching ∼6% at β₂>0.62. This suggests that the Nilsson level 3/2[211] is composed mainly of the d configuration, which is consistent with the results obtained in Ref. [67]. Meanwhile, the occupation probabilities of level 1/2[200] were plotted (and, as these are similar to those in Fig. 7, they are not shown here). The occupations of configurations s_1/2 and d_5/2 remain almost constant at β₂>0.3, both reaching ∼47%, indicating that level 1/2[200] is mainly composed of configurations d_5/2 and s_1/2. This suggests that the occupation of level 1/2[200] makes an important contribution to the distribution of diffuse density in ¹⁷B.

Fig. 4Full-screen View

(Color online) Occupation probabilities of major configurations as a function of β₂ for the single-particle level 3/2[211] in ¹⁷B.

Fig. 7Full-screen View

(Color online) Same as Fig. 4, but for ¹⁵B with the single-particle level 1/2[200].

To better understand the effects of quadrupole deformation and single-particle levels occupied by valence neutrons on the halo structure, as shown in Fig. 5, we plotted the radial wavefunction for the 3/2[211] orbital at β₂=0.64. As in Fig. 4, the radial wavefunctions are shown only for the real part of the configurations d_3/2, d_5/2, g_7/2, and g_9/2. The other spherical components and the imaginary parts can be ignored. The dependence of the wavefunctions of configurations d_3/2 and d_5/2 (l=2) on the radius r at long distances is not significantly different from that in the bound state. However, the wavefunctions of configurations d_3/2 and d_5/2 have large radial density distributions at $r\approx 3$ fm. It is well known that the configuration d-wave configuration does not support the formation of halos owing to its large centrifugal potential barrier. Therefore, whether the halo phenomenon in ¹⁷B occurs at 0.6≤β₂≤0.7 requires further exploration.

Fig. 5Full-screen View

(Color online) Wavefunctions of every component for the single neutron orbital 3/2[211] at β₂ = 0.64 for ¹⁷B.

The exotic characteristic of the halo is its density distribution with a long tail compared to its neighboring levels. To check whether the levels 3/2[211] are occupied by the last two valence neutrons of ¹⁷B at 0.6≤β₂≤0.7 is a reasonable approach. As shown in Fig. 6, we explored the density distribution of the single-particle orbital 3/2[211] at β₂=0.64. For comparison, the radial density distributions of levels 1/2[200] and 1/2[101] are also shown. The radial density distributions of the bound levels 1/2[200] and 1/2[101] did not diffuse at long distances. However, the weakly bound level 3/2[211] can be found trailing a longer tail at greater distances. According to Fig. 2, we can see that the level 3/2[211] decreases slowly at 0.6≤β₂≤0.7. These phenomena indicate that the last two valence neutrons of ¹⁷B occupy level 3/2[211] when 0.6≤β≤0.7 and weak halo phenomena are observed. These results are consistent with the halo observed for ¹⁷B [19]. In summary, the observed halo phenomena suggest that ¹⁷B is a deformation halo nucleus with a deformation parameter 0.6≤β≤0.7 and that its ground state consists mainly of the d configuration. However, we did not obtain the exact percentage of the 2s_1/2 orbital, which also shows that the prerequisite for the occurrence of neutron halos is not the dominance of components with smaller orbital angular momentum.

Fig. 6Full-screen View

(Color online) Radial density distributions multiplied by r² for the single-particle orbitals 3/2[211], 1/2[200], and 1/2[101] with β₂ = 0.64 in ¹⁷B.

The reliability of the CMR method for studying exotic phenomena is well demonstrated by investigating the halo structure in ¹⁷B. Hence, to obtain the ground-state properties of the other neutron-rich B isotopes, we further explored the exotic structures of neutron-rich ¹⁵B and ¹⁹B using the CMR method.

Because ¹⁵B has only two fewer neutrons than ¹⁷B, the order of the single-particle levels does not change, except for a slight overall upward shift. When pairing correlations are not considered, as shown in Fig. 2, it can be seen that the last two valence neutrons of ¹⁵B occupy levels 5/2[202] and 1/2[200], respectively. The S_2n value of the level 5/2[202] is <1.56 MeV at β₂≥-0.4, and the S_2n value of level 1/2[200] is <3.4 MeV and >4 MeV at β₂>0.5 and β₂<0.56, respectively, which is not in agreement with the experimentally obtained 3.747 MeV [33]. These phenomena suggest that ¹⁵B may be a deformed nucleus, with β₂ between 0.5 and 0.56. Given the impact of pairing correlations, the last two neutrons can partially occupy level 1/2[200]. This does not contradict the prediction of a prolate deformed nucleus for ¹⁵B.

To confirm the results obtained above, the occupation probabilities of levels 1/2[200] are plotted in Fig. 7. The occupations of configurations s_1/2 and d_5/2 decrease and increase rapidly with an increase in β₂, respectively. When β₂>0.3, their occupation remains almost constant, both reaching ∼50%, indicating that the level 1/2[200] is mainly composed of configurations s_1/2 and d_5/2.

As in Fig. 5, we show the radial wavefunction at β₂=0.52 for the single-particle orbital 1/2[200] of ¹⁵B in Fig. 8. The wavefunction of configuration s_1/2 decreases gradually as r increases from 0 to 3 fm. The wavefunction of configuration d_5/2 has a large radial density distribution at $r\approx 3$ fm. As r increases further, the wavefunctions of configurations s_1/2 and d_5/2 converge rapidly to zero at long distances.

Fig. 8Full-screen View

(Color online) Same as Fig. 5, but for ¹⁵B with the single-particle level 1/2[200].

Meanwhile, in Fig. 9, we plotted the density distributions of level 1/2[200] in ¹⁵B at β₂=0.52. For comparison, we also show the density distribution for the 3/2[211] and 1/2[101] levels. The density distribution of the bound level 1/2[200] converges rapidly at longer distances, exhibiting no halo tails, suggesting that halo formation is not supported when the valence neutrons occupy level 1/2[200]. That is, ¹⁵B is a prolate nucleus, but no halo phenomenon exists.

Fig. 9Full-screen View

(Color online) Same as Fig. 6, but for ¹⁵B at β₂=0.52.

We performed a similar investigation on ¹⁹B. Although ¹⁹B has only two more neutrons than ¹⁷B, the contribution of single-particle levels near the continuum threshold to the exotic phenomena becomes more significant, in addition to a slight downward shift at the single-particle level. This is consistent with the results obtained in Refs. [35](in Fig. 2). When pairing correlations are not considered, the Nilsson levels 3/2[211], 1d_5/2, 3/2[211], and 1/2[220] are occupied by the last two valence neutrons of ¹⁹B. Given that ¹⁹B is a weakly bound nucleus, the two-neutron separation energy is ${0.089}_{-0.089}^{+0.560}$ MeV [38]. Therefore, at 0.3≤β≤0.7, level 1/2[220] can be occupied by the last two valence neutrons of ¹⁹B. Given the impact of pairing correlations, the last two neutrons can partially occupy level 1/2[220]. This is not contradictory to the prediction of a prolate deformed halo nucleus for ¹⁹B. However, whether halo phenomena appear in this deformation region requires further exploration.

To verify whether quadrupole deformation with 0.3≤β≤0.7 is a reasonable conjecture in ¹⁹B, we explore the density distribution of the single-particle orbital 1/2[220] at β₂=0.36 in Fig. 10. For comparison, the radial density distributions of levels 5/2[202], 3/2[211], and 1/2[200] are also shown. The radial density distributions of the bound levels 3/2[211] and 1/2[200] were not very diffuse at large distances. It is noteworthy that the level 1/2[220] has a larger diffuse density distribution than the resonant level 5/2[202]. These phenomena indicate that the last two valence neutrons of ¹⁹B occupy level 1/2[220] when 0.3≤β≤0.7 and halo phenomena are observed. These results are consistent with the observed halo [20] and the predicted deformation [35].

Fig. 10Full-screen View

(Color online) Same as Fig. 6, but for ¹⁹B at β₂=0.36.

In addition, we also provide the occupation probabilities of level 1/2[220] in Fig. 11. This level consists mainly of the configurations s_1/2, d_3/2, and d_5/2, whereas the other configurations are negligible. Configuration s_1/2 is dominant compared to configurations d_3/2 and d_5/2. The occupation probability of configuration s_1/2 increases rapidly and then remains almost constant as β₂ increases, it still exceeds 65% on the prolate side, that is, the wavefunction of the weakly bound level 1/2[220] is composed mainly of configuration s_1/2, which supports the formation of the s-wave halo in ¹⁹B.

Fig. 11Full-screen View

(Color online) Same as Fig. 4, but for ¹⁹B with the single-particle level 1/2[220].

To better understand the effect of single-particle levels with low orbital angular momentum on the halo structure, in Fig. 12, we plotted the radial wavefunction for orbital 1/2[220] at β₂=0.36. The radial wavefunctions are shown here only for the real and imaginary parts of configurations s_1/2, d_3/2, and d_5/2. The wavefunctions of the real and imaginary parts of the configuration s_1/2 (l=0) extend to a considerable range in the coordinate space and exhibit significant halo phenomena. In contrast, the dependence of the wavefunctions of the real and imaginary parts of the configuration d_5/2 (l=2) on radius r is not significantly different from that in the bound state. Although the wavefunctions of the real and imaginary parts of configuration d_5/2 have a large radial density distribution at $r\approx 3$ fm, these local wavefunctions do not support halo formation because of the large centrifugal potential barrier and small occupation probabilities. The halo phenomena observed in ¹⁹B suggest that ¹⁹B is a deformation halo nucleus with a deformation parameter 0.3≤β≤0.7 and that it is the ground state consisting mainly of the s configuration.

Fig. 12Full-screen View

(Color online) Same as Fig. 5, but for ¹⁹B with the single-particle level 1/2[220].

The present method allows us to identify ¹⁵B as a possible prolate nucleus with β₂ between 0.50 and 0.56, and the halo phenomena in ¹⁷B and ¹⁹B may occur at quadrupole deformations 0.6≤β₂≤0.7 and 0.3≤β≤0.7, respectively. The above results indicate that the presence of ¹⁷B and ¹⁹B halo structures can be qualitatively explained in a valid model that considers quadrupole deformation and continuum spectral effects, despite its various limitations. The ground-state structures of ¹⁵B are mainly the result of a certain degree of deformation. However, the ground-state structures of ¹⁷B and ¹⁹B are primarily the result of the interaction between s and d orbitals, large quadrupole deformation, and continuum effects.

Summary

Inspired by the recently reported experimental work on the halo phenomena in ^17,19B [19, 20], the role of quadrupole deformation and continuum effects on the halo structure of ¹⁷B was studied using the CMR method.

Single-particle orbitals, including the bound and resonant states, were obtained, and an inversion of the s–d orbitals was found to be a critical factor in yielding the halo phenomena. At β₂=0, the neutron orbital 2s_1/2 is lower than that of 1d_5/2, and the valence neutrons are mainly dominated by the d wave. Therefore, ¹⁷B is not a spherical nucleus. Driven by the deformation, the ground state of ¹⁷B is prolate (0.6≤β₂≤0.7) and a halo appears. The calculated occupation probabilities, radial density distributions, and wavefunctions support the results obtained by using the Nilsson levels. This indicates that the ground state of ¹⁷B is a prolate halo nucleus, with 0.6≤β₂≤0.7.

As shown in Figs. 7, 8, 9, 10, 11 and 12, the ground state of ¹⁵B also exhibits a large deformation but no halo phenomenon. By observing the radial density distribution and the configurations of the orbitals occupied by the valence neutrons, we find that the ground state of ¹⁹B appears as halos at 0.3≤β₂≤0.7 and that the ground state consists mainly of the s configuration.

In this study, we used the CMR method to present some basic physics in a simple manner. By studying the quadrupole deformation and continuum effects in the ground-state properties of ¹⁵B, ¹⁷B, and ¹⁹B, we found that the ground-state structure in ¹⁵B may be significantly influenced by deformation effects and that the halo structures in ^17,19B may result from the interaction between s–d orbital reversals, large quadrupole deformations, and continuum effects. This work shows that further exploration of the relationship among halo features, deformation effects, and coupling of continuum spectra in more exotic nuclei is essential and interesting. Furthermore, the development of a CMR model is important, and future work will consider pairing correlations in this theoretical formalism.