Research on the influence of quadrupole deformation and continuum effects on the exotic properties of ^15,17,19B with the complex momentum representation method

Research on the influence of quadrupole deformation and continuum effects on the exotic properties of 15,17,19 B with the complex momentum representation method 1 corresp first-author † 120210067@aufe.edu.cn † 120210067@aufe.edu.cn 120210067@aufe.edu.cn https://orcid.org/0000-0002-3962-7391 1 1 2 School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China School of Mechanics and Optoelectronic Physics, Anhui University of Science and Technology, Huainan 232001, China School of Mechanics and Optoelectronic Physics, Anhui University of Science and Technology, Huainan 232001, China The properties of exotic nuclei are the focus of the present research. Two-neutron halo structures of neutron-rich 17,19 B were experimentally confirmed. We studied the formation mechanism of halo phenomena in 17,19 B using the complex momentum representation method applied to deformation and continuum coupling. By examining the evolution of the weakly bound and resonant levels near the Fermi surface, s – d orbital reversals and certain prolate deformations were observed. In addition, by analyzing the evolution of the occupation probabilities and density distributions occupied by valence neutrons, we found that the ground state of 15 B did not exhibit a halo and the ground states of 17 B and 19 B exhibited halos at 0.6≤ β 2 ≤0.7 and 0.3≤ β 2 ≤0.7, respectively. The low- l components in the valence levels that are weakly bound or embedded in the continuous spectrum lead to halo formation. Quadrupole deformation Resonant states Complex momentum representation Halo Introduction 1 Exotic phenomena, including halos, have attracted considerable interest from physicists [ 1 1 - 4 4 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 11 Li was discovered to be an s -wave two-neutron (2 n ) halo nucleus [ 5 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 6 - 8 8 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, 8 B is a proton (1 p ) halo nucleus [ 9 9 ]; 6 He [ 10 10 , 11 11 ], 11 Li [ 5 5 , 12 12 ], and 14 Be [ 13 13 , 14 14 ] are 2 n halo nuclei; and 11 Be [ 15 15 , 16 16 ] is a neutron (1 n ) halo nucleus. Subsequently, 22 C, 17 B, and 19 B were found to be s -wave 2 n halo nuclei with large matter radii [ 17 17 , 18 18 ], a small percentage of s -orbital spectroscopic factors [ 19 19 ], and an enhanced electric dipole strength [ 20 20 ], respectively. In the heavier mass region, 37 Mg [ 21 21 ] (the heaviest halo nucleus observed thus far) and 31 Ne [ 22 22 ] and 29 F [ 23 23 ] (the heaviest Borromean nucleus to date) were found to be p -wave halos at or near the inversion island by single-neutron (1 n ) removal reactions and reaction-cross-section measurements, respectively. On the theoretical side, the search for new 2 n -halo systems [ 24 24 ] and new halo features, such as core–halo shape decoupling [ 25 25 , 26 26 ] and the Efimov state [ 27 27 , 28 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 29 ]. Specifically, Hove et al. [ 30 30 ] used the example of a medium-heavy nucleus 72 Ca 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. 15 B and 17 B neutron halo structures have been reported [ 31 31 , 32 32 ]. For 15 B, the 2 n separation energies ( S 2n ) were 3.747 MeV [ 33 33 ], the quadruple deformation length δ was 1.17 [ 34 34 ], the quadruple deformation parameter β 2 was 0.61 [ 35 35 ], and the percentage of the s orbital was 63% [ 36 36 ]. For 17 B, a strong mixing of s and d waves was observed based on the energy dependence observed by the interaction cross section [ 37 37 ]. Tanihata [ 24 24 ] considered large amplitudes of 2 s 1/2 waves for all B isotopes with neutron number n ≥8. For 17 B, S 2 n is 1.37 MeV [ 38 38 ], δ is 1.2 [ 34 34 ], β 2 is ∼ 0.6 [ 39 39 ] or 0.56 [ 35 35 ], the neutron skin thickness is 0.51 (0.11) fm [ 40 40 ], and the neutron has a large matter radius [ 41 41 , 40 40 ]. The percentages of s orbital in 17 B are 36(19)% [ 31 31 ], 69(20)% [ 32 32 ], 50(10)% [ 37 37 ], and 53(21)% [ 42 42 ], respectively. In 2021, Yang et al. [ 19 19 ] obtained spectral factors for the 1 s 1/2 and 0 d 5/2 orbitals, with the 1 s 1/2 orbitals contributing only 9(2)% of the spectral factor, which is much smaller than the values given in Refs. [ 31 31 , 42 42 ], revealing that 17 B 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 19 ], indicating the significance of deformation effects. 19 B is the heaviest bound nucleus of the B isotopes, and, because 18 B is unbound, 19 B is the Borromean nucleus, suggesting a 2 n halo structure. 19 B has a particularly low two-neutron separation energy ( S 2 n ) of <math id="M1"><mrow><msubsup><mrow><mn>0.089</mn></mrow><mrow><mo>−</mo><mn>0.089</mn></mrow><mrow><mo>+</mo><mn>0.560</mn></mrow></msubsup></mrow></math> MeV [ 38 38 ], enhanced interaction cross section [ 31 31 ], and large deformation [ 43 43 - 35 35 43-35 ]. Direct mass measurements revealed that the valence neutrons of 19 B are dominated by d waves, which suppress halo formation [ 31 31 , 42 42 , 45 45 ]. However, Cook et al. [ 20 20 ] found a significant 2 n halo in 19 B using Coulomb dissociation. Recently, 19 B has been suggested as a deformed 2 n halo candidate with shape-decoupling effects [ 35 35 ]. The structures of 19 B are also related to the newly discovered unbound isotopes 20 B and 21 B. 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 46 ], the complex scaling method (CSM) [ 47 47 ], and the complex momentum representation (CMR) method [ 48 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 48 , 49 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 50 , 51 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 48 , 52 52 ] and relativistic point coupling model [ 53 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 54 , 55 55 ]. In addition, we extended the CMR method to a deformed system. We have systematically investigated experimentally discovered and undiscovered halo structures, establishing that 31 Ne [ 50 50 ], 37 Mg [ 49 49 ], 53 Ar [ 56 56 ], 34 Na [ 57 57 ], and 29, 31 F [ 58 58 , 59 59 ] (or s waves or a mixture of s and p waves) are p -wave halo nuclei and that 19 C [ 60 60 ], 75 Cr [ 56 56 ], and 77 Fe [ 56 56 ] are s -wave halo nuclei. Recently, Wang et al. investigated neutron-rich 75 Cr using the CMR method in a relativistic mean-field framework, which applies to deformed nuclei. They found that 75 Cr is a d -wave deformed halo nucleus [ 61 61 ]. The ground-state properties of Ti isotopes were also investigated using the relativistic mean-field framework CMR method [ 62 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 15,17,19 B 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 17,19 B may be prolate halo nuclei, which can be well understood because of the neutron halo phenomenon. In other words, observing the halo phenomena in 17,19 B can be used as critical evidence for the prolate deformation of 17,19 B 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 1 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 48 , 50 50 ]. The main formulas of the CMR method are provided for the convenience of later discussion. We introduce a theoretical framework using the Hamiltonian <math display="block" id="M2"><mrow><mi>H</mi><mo>=</mo><mi>T</mi><mo>+</mo><mi>V</mi><mn>,</mn></mrow></math> where <math id="M3"><mrow><mi>T</mi><mo>=</mo><msup><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>2</mn></msup><mo>/</mo><mn>2</mn><mi>m</mi></mrow></math> denotes the kinetic operator with momentum <math id="M4"><mrow><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>=</mo><mi>ℏ</mi><mover accent="true"><mi>k</mi><mo>→</mo></mover></mrow></math> (where <math id="M5"><mover accent="true"><mi>k</mi><mo>→</mo></mover></math> is the wavevector). Following Ref. [ 63 63 ], the interaction potential V consists of the central potential, deformation potential, and spin-orbit coupling potential: <math display="block" id="M6"><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mrow><mtext>cent</mtext></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr><mtr><mtd><msub><mi>V</mi><mrow><mtext>def</mtext></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msub><mi>β</mi><mn>2</mn></msub><msub><mi>V</mi><mn>0</mn></msub><mi>r</mi><mfrac><mrow><mtext>d</mtext><mi>f</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow><mrow><mtext>d</mtext><mi>r</mi></mrow></mfrac><msub><mi>Y</mi><mrow><mn>20</mn></mrow></msub><mo stretchy="false">(</mo><mi>ϑ</mi><mn>,</mn><mi>φ</mi><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr><mtr><mtd><msub><mi>V</mi><mrow><mtext>sl</mtext></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>16</mn><msub><mi>V</mi><mn>0</mn></msub><msup><mi>Λ</mi><mn>2</mn></msup><mfrac><mn>1</mn><mi>r</mi></mfrac><mfrac><mrow><mtext>d</mtext><mi>f</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow><mrow><mtext>d</mtext><mi>r</mi></mrow></mfrac><mo stretchy="false">(</mo><mover accent="true"><mi>s</mi><mo>→</mo></mover><mo>⋅</mo><mover accent="true"><mi>l</mi><mo>→</mo></mover><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> where <math id="M7"><mi>Λ</mi></math> is the reduced Compton wavelength of the nucleon, <math id="M8"><mrow><mi>ℏ</mi><mo>/</mo><msub><mi>m</mi><mtext>r</mtext></msub><mi>c</mi></mrow></math> , and <math display="block" id="M9"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow><mfrac><mrow><mi>r</mi><mo>−</mo><mi>R</mi></mrow><mi>a</mi></mfrac></mrow></msup></mrow></mfrac><mn>.</mn></mrow></math> The diffuseness a =0.67 fm, the radius R = r 0 A 1/3 with r 0 =1.27 fm, and the depth of potential <math id="M10"><mrow><msub><mi>V</mi><mn>0</mn></msub><mo>=</mo><mo>−</mo><mn>51</mn><mo>+</mo><mn>33</mn><mfrac><mrow><mi>N</mi><mo>−</mo><mi>Z</mi></mrow><mi>A</mi></mfrac></mrow></math> 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 <math display="block" id="M11"><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mtext>d</mtext><mover accent="true"><msup><mi>k</mi><mo>′</mo></msup><mo>→</mo></mover><mrow><mo>〈</mo><mrow><mover accent="true"><mi>k</mi><mo>→</mo></mover><mrow><mo>|</mo><mi>H</mi><mo>|</mo></mrow><mover accent="true"><msup><mi>k</mi><mo>′</mo></msup><mo>→</mo></mover></mrow><mo>〉</mo></mrow><mi>ψ</mi><mo stretchy="false">(</mo><mover accent="true"><msup><mi>k</mi><mo>′</mo></msup><mo>→</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mi>ψ</mi><mo stretchy="false">(</mo><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where <math id="M12"><mrow><mi>ψ</mi><mo stretchy="false">(</mo><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo stretchy="false">)</mo></mrow></math> 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 <math id="M13"><mrow><msub><mi>ψ</mi><mtext>Ω</mtext></msub><mo stretchy="false">(</mo><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo stretchy="false">)</mo></mrow></math> can be expanded as <math display="block" id="M14"><mrow><msub><mi>ψ</mi><mtext>Ω</mtext></msub><mo stretchy="false">(</mo><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder><mrow><msub><mi>Φ</mi><mi>i</mi></msub></mrow></mstyle><mo stretchy="false">(</mo><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>l</mi><mi>j</mi></mrow></munder><mrow><msup><mi>f</mi><mrow><mi>l</mi><mi>j</mi></mrow></msup></mrow></mstyle><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><msub><mi>ϕ</mi><mrow><mi>l</mi><mi>j</mi><mtext>Ω</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mtext>Ω</mtext><mi>k</mi></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where f lj ( k ) and ϕ lj Ω (Ω k ) are the radial and angular components, respectively. By substituting Eq. (4) into Eq. (3), the Schrödinger equation becomes <math display="block" id="M15"><mtable><mtr><mtd><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup><msup><mi>k</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>M</mi></mrow></mfrac><msup><mi>f</mi><mrow><mi>l</mi><mi>j</mi></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>k</mi><mo>'</mo></msup><msup><mrow/><mn>2</mn></msup><mtext>d</mtext><msup><mi>k</mi><mo>'</mo></msup><msub><mi>V</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mi>l</mi><mn>,</mn><mi>j</mi><mn>,</mn><mi>k</mi><mn>,</mn><msup><mi>k</mi><mo>'</mo></msup><mo stretchy="false">)</mo><msup><mi>f</mi><mrow><mi>l</mi><mi>j</mi></mrow></msup><mo stretchy="false">(</mo><msup><mi>k</mi><mo>'</mo></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>−</mo><msub><mi>β</mi><mn>2</mn></msub><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msup><mi>l</mi><mo>'</mo></msup><msup><mi>j</mi><mo>'</mo></msup></mrow></munder><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>k</mi><mo>'</mo></msup><msup><mrow/><mn>2</mn></msup></mrow></mstyle><mtext>d</mtext><msup><mi>k</mi><mo>'</mo></msup><msub><mi>V</mi><mi>d</mi></msub><mo stretchy="false">(</mo><mi>l</mi><mn>,</mn><mi>j</mi><mn>,</mn><msup><mi>l</mi><mo>'</mo></msup><mn>,</mn><msup><mi>j</mi><mo>'</mo></msup><mn>,</mn><msub><mi>m</mi><mi>j</mi></msub><mn>,</mn><mi>k</mi><mn>,</mn><msup><mi>k</mi><mo>'</mo></msup><mo stretchy="false">)</mo><msup><mi>f</mi><mrow><msup><mi>l</mi><mo>'</mo></msup><msup><mi>j</mi><mo>'</mo></msup></mrow></msup><mo stretchy="false">(</mo><msup><mi>k</mi><mo>'</mo></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><mi>E</mi><msup><mi>f</mi><mrow><mi>l</mi><mi>j</mi></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> with <math display="block" id="M16"><mrow><msub><mi>V</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mi>l</mi><mn>,</mn><mi>j</mi><mn>,</mn><mi>k</mi><mn>,</mn><msup><mi>k</mi><mo>'</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>2</mn><mi>π</mi></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>r</mi><mn>2</mn></msup><mtext>d</mtext><mi>r</mi><mo stretchy="false">[</mo><msub><mi>V</mi><mrow><mtext>cent</mtext></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>C</mi><mrow><mtext>sl</mtext></mrow></msub><msub><mi>V</mi><mrow><mtext>sl</mtext></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msub><mi>j</mi><mi>l</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mi>r</mi><mo stretchy="false">)</mo><msub><mi>j</mi><mi>l</mi></msub><mo stretchy="false">(</mo><msup><mi>k</mi><mo>'</mo></msup><mi>r</mi><mo stretchy="false">)</mo></mrow></math> and <math display="block" id="M17"><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mi>d</mi></msub><mo stretchy="false">(</mo><mi>l</mi><mn>,</mn><mi>j</mi><mn>,</mn><msup><mi>l</mi><mo>'</mo></msup><mn>,</mn><msup><mi>j</mi><mo>'</mo></msup><mn>,</mn><mtext>Ω</mtext><mn>,</mn><mi>k</mi><mn>,</mn><msup><mi>k</mi><mo>'</mo></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><msup><mi>i</mi><mrow><mn>3</mn><mi>l</mi><mo>+</mo><msup><mi>l</mi><mo>'</mo></msup></mrow></msup><mfrac><mn>2</mn><mi>π</mi></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>r</mi><mn>2</mn></msup><mtext>d</mtext><mi>r</mi><mi>k</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><msub><mi>j</mi><mi>l</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mi>r</mi><mo stretchy="false">)</mo><msub><mi>j</mi><mrow><msup><mi>l</mi><mo>'</mo></msup></mrow></msub><mo stretchy="false">(</mo><msup><mi>k</mi><mo>'</mo></msup><mi>r</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋅</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>m</mi><mi>s</mi></msub></mrow></munder><mrow><mrow><mo>〈</mo><mrow><mi>l</mi><mi>m</mi><mrow><mo>|</mo><mrow><msub><mi>Y</mi><mrow><mn>20</mn></mrow></msub><mo stretchy="false">(</mo><msub><mtext>Ω</mtext><mi>r</mi></msub><mo stretchy="false">)</mo></mrow><mo>|</mo></mrow><msup><mi>l</mi><mo>'</mo></msup><mi>m</mi></mrow><mo>〉</mo></mrow></mrow></mstyle><mrow><mo>〈</mo><mrow><mi>l</mi><mi>m</mi><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>m</mi><mi>s</mi></msub><mo>|</mo><mi>j</mi><mtext>Ω</mtext></mrow><mo>〉</mo></mrow><mrow><mo>〈</mo><mrow><msup><mi>l</mi><mo>'</mo></msup><mi>m</mi><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>m</mi><mi>s</mi></msub><mo>|</mo><msup><mi>j</mi><mo>'</mo></msup><mtext>Ω</mtext></mrow><mo>〉</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> 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 60 , 59 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: <math display="block" id="M18"><mrow><msub><mi>ψ</mi><mtext>Ω</mtext></msub><mrow><mo>(</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>π</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>d</mi><mover accent="true"><mi>k</mi><mo>→</mo></mover><msup><mi>e</mi><mrow><mi>i</mi><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo>⋅</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover></mrow></msup><msub><mi>ψ</mi><mtext>Ω</mtext></msub><mo stretchy="false">(</mo><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo stretchy="false">)</mo><mn>.</mn></mrow></math> By assuming that <math id="M19"><mrow><msub><mi>ψ</mi><mtext>Ω</mtext></msub><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo stretchy="false">)</mo></mrow></math> holds the form <math display="block" id="M20"><mrow><msub><mi>ψ</mi><mtext>Ω</mtext></msub><mrow><mo>(</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>l</mi><mi>j</mi></mrow></munder><mrow><msup><mi>f</mi><mrow><mi>l</mi><mi>j</mi></mrow></msup></mrow></mstyle><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><msub><mi>ϕ</mi><mrow><mi>l</mi><mi>j</mi><mtext>Ω</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mtext>Ω</mtext><mi>r</mi></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where <math display="block" id="M21"><mrow><msup><mi>f</mi><mrow><mi>l</mi><mi>j</mi></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>i</mi><mi>l</mi></msup><msqrt><mrow><mfrac><mn>2</mn><mi>π</mi></mfrac></mrow></msqrt><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mrow><msqrt><mrow><msub><mi>w</mi><mi>a</mi></msub></mrow></msqrt></mrow></mstyle><msub><mi>k</mi><mi>a</mi></msub><msub><mi>j</mi><mi>l</mi></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mi>a</mi></msub><mi>r</mi><mo stretchy="false">)</mo><msup><mstyle mathvariant="bold" mathsize="normal"><mi>f</mi></mstyle><mrow><mi>l</mi><mi>j</mi></mrow></msup><mo stretchy="false">(</mo><msub><mi>k</mi><mi>a</mi></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> the corresponding radial density distributions can be calculated as <math display="block" id="M22"><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>l</mi><mi>j</mi></mrow></munder><mrow><msup><mi>f</mi><mrow><mi>l</mi><mi>j</mi><mo>∗</mo></mrow></msup></mrow></mstyle><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><msup><mi>f</mi><mrow><mi>l</mi><mi>j</mi></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> numerical details and results 1 The details of the numerical calculations are the same as those in Refs. [ 60 60 , 59 59 ], where the number of coupling channels, N c , momentum truncation maximum value k max , and number of Gauss–Legendre grids, N l 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 35 , 58 58 , 64 64 ], the quadrupole deformation parameter β 2 was taken from -0.4 to 0.7 in steps of 0.02. Based on Refs. [ 48 48 , 50 50 ], we found that the positions of the resonant states do not depend on the integration path. Therefore, we chose a sufficiently large contour ( k 1 =0.0 fm -1 , k 2 =0.5- i 0.5 fm -1 , k 3 =1.0 fm -1 , and k max =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 17 B. 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 Fig. 1 , we show only the RMS radii of the levels occupied by the last two valence neutrons of 13–19 B for the quadrupole deformations β 2 =-0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7. The halo structure was experimentally confirmed for 17,19 B [ 19 19 , 20 20 ]. The last two valence neutrons of 17 B occupy the levels 5/2[202], 1 d 5/2 , 1/2[220], and 3/2[211] at β 2 =-0.1, β 2 =0.0, β 2 = 0.1–0.2, and β 2 = 0.3–0.7, respectively (see Fig. 2 Fig. 2 ). We can see that the RMS radius of level 3/2[211] occupied by the last two valence neutrons of 17 B significantly increases compared to that of 15 B at β 2 =0.6 and β 2 =0.7. However, we noticed that the RMS radius of 17 B increases slightly at β 2 =0.1–0.2, as a result of the valence neutrons occupying the resonant level 1/2[220]. This deformation region β 2 =0.1–0.2 was excluded because 17 B is a weakly bound nucleus. Similar phenomena were observed in 19 B, where the RMS radius increased significantly compared to that of 17 B at β 2 =0.3–0.7. For 15 B, there was no significant increase in the RMS radius over the range of deformations compared to the neighboring nuclei. Similar to the phenomena at β 2 =0.1–0.2 in 17 B, the increased RMS radius for 15 B at β 2 =-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 β 2 =-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 33 ]. These phenomena suggest that 17 B and 19 B may be prolate halo nuclei with deformations β 2 = 0.6–0.7 and β 2 = 0.3–0.7, respectively. In comparison, 15 B is only a prolate deformation nucleus, which is relatively close to the results obtained from DRHBc calculations [ 35 35 ]. The neutron halo structure can be identified by analyzing the valence neutron levels. In Fig. 2 Fig. 2 , the Nilsson diagram of 17 B, 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 <math id="M24"><mrow><mtext>Ω</mtext><mo stretchy="false">[</mo><mi>N</mi><msub><mi>n</mi><mi>z</mi></msub><mi>Λ</mi><mo stretchy="false">]</mo></mrow></math> . Levels 1 p 1/2 and 2 s 1/2 , and 1 d 5/2 and 1 d 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 ( β 2 = 0), level 2 s 1/2 is lower than 1 d 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 17 B from DRHBc calculations [ 35 35 ]. In the spherical case, the resonant orbital 1 d 5/2 is occupied by the last two valence neutrons of 17 B, 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 17 B have been observed experimentally [ 19 19 ], this suggests that 17 B is not a spherical nucleus but rather a deformed halo nucleus. Whether 17 B 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 1 d 5/2 splits into Nilsson levels 1/2[220], 3/2[211] , and 5/2[202], and the weakly bound level 2 s 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 17 B at -0.4≤ β 2 < -0.14, -0.14≤ β 2 < 0, 0 < β 2 < 0.26, and 0.26≤ β 2 ≤0.7, respectively. Given that 17 B is a weakly bound nucleus, the last valence neutron may occupy the weakly bound level 1/2[200] with a deformation -0.4≤ β 2 < -0.14 or the weakly bound level 3/2[211] with a deformation 0.26≤ β 2 ≤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 17 B at -0.4≤ β 2 & -0.14. It should be indicated that the pairing correlations play important role at halo formation [ 65 65 , 66 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 38 ]. This phenomenon suggests that 17 B 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 17 B at 0.26≤ β 2 ≤0.7. The S 2n value of level 3/2[211] reaches 1.3–1.6 MeV with a deformation 0.6≤ β 2 ≤0.7, which is close to the experimentally obtained 1.37 MeV [ 38 38 ] and, when combined with the analysis in Fig. 1 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 17 B is a prolate deformation halo nucleus. This does not contradict the prediction of a prolate deformed halo nucleus for 17 B. Figure 3 Figure 3 shows the widths of the single-particle resonant states in 17 B. For simplicity, the marks of the widths in Fig. 3 Fig. 3 are the same as those for the levels shown in Fig. 2 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 17 B. The width of the resonant state 3/2[211] is consistently small and lies near zero potential energy. As β 2 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 17 B. The experimentally detected lifetime of 17 B will help us better understand the halo structure of 17 B because the widths are inversely proportional to the decay lifetime. The above discussions indicate that the last two valence neutrons of 17 B occupy the Nilsson level 3/2[211] and that 17 B is a prolate halo nucleus with a deformation 0.6≤ β 2 ≤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 Fig. 4 . It can be seen that the configuration d 5/2 exceeds 90%. Configuration d 3/2 increases slowly, reaching ∼ 6% at β 2 > 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 67 ]. Meanwhile, the occupation probabilities of level 1/2[200] were plotted (and, as these are similar to those in Fig. 7 Fig. 7 , they are not shown here). The occupations of configurations s 1/2 and d 5/2 remain almost constant at β 2 > 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 17 B. 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 Fig. 5 , we plotted the radial wavefunction for the 3/2[211] orbital at β 2 =0.64. As in Fig. 4 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 <math id="M25"><mrow><mi>r</mi><mo>≈</mo><mn>3</mn></mrow></math> 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 17 B occurs at 0.6≤ β 2 ≤0.7 requires further exploration. 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 17 B at 0.6≤ β 2 ≤0.7 is a reasonable approach. As shown in Fig. 6 Fig. 6 , we explored the density distribution of the single-particle orbital 3/2[211] at β 2 =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 Fig. 2 , we can see that the level 3/2[211] decreases slowly at 0.6≤ β 2 ≤0.7. These phenomena indicate that the last two valence neutrons of 17 B 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 17 B [ 19 19 ]. In summary, the observed halo phenomena suggest that 17 B 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 2 s 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. The reliability of the CMR method for studying exotic phenomena is well demonstrated by investigating the halo structure in 17 B. Hence, to obtain the ground-state properties of the other neutron-rich B isotopes, we further explored the exotic structures of neutron-rich 15 B and 19 B using the CMR method. Because 15 B has only two fewer neutrons than 17 B, 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 Fig. 2 , it can be seen that the last two valence neutrons of 15 B 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 β 2 ≥-0.4, and the S 2n value of level 1/2[200] is < 3.4 MeV and > 4 MeV at β 2 > 0.5 and β 2 < 0.56, respectively, which is not in agreement with the experimentally obtained 3.747 MeV [ 33 33 ]. These phenomena suggest that 15 B may be a deformed nucleus, with β 2 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 15 B. To confirm the results obtained above, the occupation probabilities of levels 1/2[200] are plotted in Fig. 7 Fig. 7 . The occupations of configurations s 1/2 and d 5/2 decrease and increase rapidly with an increase in β 2 , respectively. When β 2 > 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 Fig. 5 , we show the radial wavefunction at β 2 =0.52 for the single-particle orbital 1/2[200] of 15 B in Fig. 8 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 <math id="M26"><mrow><mi>r</mi><mo>≈</mo><mn>3</mn></mrow></math> fm. As r increases further, the wavefunctions of configurations s 1/2 and d 5/2 converge rapidly to zero at long distances. Meanwhile, in Fig. 9 Fig. 9 , we plotted the density distributions of level 1/2[200] in 15 B at β 2 =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, 15 B is a prolate nucleus, but no halo phenomenon exists. We performed a similar investigation on 19 B. Although 19 B has only two more neutrons than 17 B, 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 35 ](in Fig. 2 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 19 B. Given that 19 B is a weakly bound nucleus, the two-neutron separation energy is <math id="M27"><mrow><msubsup><mrow><mn>0.089</mn></mrow><mrow><mo>−</mo><mn>0.089</mn></mrow><mrow><mo>+</mo><mn>0.560</mn></mrow></msubsup></mrow></math> MeV [ 38 38 ]. Therefore, at 0.3≤ β ≤0.7, level 1/2[220] can be occupied by the last two valence neutrons of 19 B. 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 19 B. 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 19 B, we explore the density distribution of the single-particle orbital 1/2[220] at β 2 =0.36 in Fig. 10 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 19 B 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 20 ] and the predicted deformation [ 35 35 ]. In addition, we also provide the occupation probabilities of level 1/2[220] in Fig. 11 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 β 2 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 19 B. To better understand the effect of single-particle levels with low orbital angular momentum on the halo structure, in Fig. 12 Fig. 12 , we plotted the radial wavefunction for orbital 1/2[220] at β 2 =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 <math id="M28"><mrow><mi>r</mi><mo>≈</mo><mn>3</mn></mrow></math> 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 19 B suggest that 19 B 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. The present method allows us to identify 15 B as a possible prolate nucleus with β 2 between 0.50 and 0.56, and the halo phenomena in 17 B and 19 B may occur at quadrupole deformations 0.6≤ β 2 ≤0.7 and 0.3≤ β ≤0.7, respectively. The above results indicate that the presence of 17 B and 19 B 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 15 B are mainly the result of a certain degree of deformation. However, the ground-state structures of 17 B and 19 B are primarily the result of the interaction between s and d orbitals, large quadrupole deformation, and continuum effects. Summary 1 Inspired by the recently reported experimental work on the halo phenomena in 17,19 B [ 19 19 , 20 20 ], the role of quadrupole deformation and continuum effects on the halo structure of 17 B 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 β 2 =0, the neutron orbital 2 s 1/2 is lower than that of 1 d 5/2 , and the valence neutrons are mainly dominated by the d wave. Therefore, 17 B is not a spherical nucleus. Driven by the deformation, the ground state of 17 B is prolate (0.6≤ β 2 ≤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 17 B is a prolate halo nucleus, with 0.6≤ β 2 ≤0.7. As shown in Figs. 7 Figs. 7 , 8 8 , 9 9 , 10 10 , 11 11 and 12 12 , the ground state of 15 B 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 19 B appears as halos at 0.3≤ β 2 ≤0.7 and that the ground state consists mainly of the s configuration. 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Few-Body Syst. 62 , 32 ( 2021 ). doi: 10.1007/s00601-021-01612-5 http://doi.org/10.1007/s00601-021-01612-5 Structure of two-neutron halo in light exotic nuclei All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Xue-Neng Cao, Xian-Xian Zhou, Ming Fu and Xin-Xing Shi. The first draft of the manuscript was written by Xue-Neng Cao and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. The authors are indebted to Jian-You Guo and Tai-Hua Heng for their constructive guidance and valuable suggestions. 10.1007/s41365-023-01177-4 This work was partially supported by the National Natural Science Foundation of China (Nos. 12205001, 11935001, and 12204001) and the Scientific Research program of Anhui University of Finance and Economics (Nos. ACKYC22080 and ACKYC220801). 2022-09-22 2022-11-30 2022-12-02 2023-02-24 2023-03-20 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2023 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-023-01177-4 http://dx.doi.org/10.1007/s41365-023-01177-4 . 2023 Nuclear Science and Techniques 2 34