The complex momentum representation approach and its application to low-lying resonances in ¹⁷O and ^{29, 31}F

The complex momentum representation approach and its application to low-lying resonances in 17 O and 29, 31 F 1 first-author 1 corresp † zss76@buaa.edu.cn † zss76@buaa.edu.cn zss76@buaa.edu.cn https://orcid.org/0000-0003-3926-7151 1 2 corresp ‡ smithms@ornl.gov ‡ smithms@ornl.gov smithms@ornl.gov School of Physics, and International Research Center for Big-Bang Cosmology and Element Genesis, Beihang University, Beijing 100191, China School of Physics, and International Research Center for Big-Bang Cosmology and Element Genesis, Beihang University, Beijing 100191, China Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6354, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6354, USA Approaches for predicting low-lying resonances, uniformly treating bound, and resonant levels, have been a long-standing goal in nuclear theory. Accordingly, we explored the viability of the complex momentum representation (CMR) approach coupled with new potentials. We focus on predicting the energy of the low-lying 2p 3/2 resonance in 17 O, which is critical for s-process nucleosynthesis and missing in previous theoretical research. Using a Woods-Saxon potential based on the Koning-Delaroche optical model and constrained by the experimental one-neutron separation energy, we successfully predicted the resonant energy of this level for the first time. Our predictions of the bound levels and 1d 3/2 resonance agree well with the measurement results. Additionally, we utilize this approach to study the near-threshold resonances that play a role when forming a two-neutron halo in 29, 31 F. We found that the CMR-based predictions of the bound level energies and unbound 1f 7/2 level agree well with the results obtained using the scattering phase shift method. Subsequently, we successfully found a solution for the 2p 3/2 resonance with energy just above the threshold, which is decisive for halo formation. Neutron capture Low-lying resonance CMR Resonance energy Introduction 1 The weak s -process in massive stars [ 1 1 ], driven by neutron capture on heavy isotopes, is thought to be responsible for the synthesis of nuclides in the mass range of 60-90. However, 16 O is known to serve as a “neutron poison”, disrupting the weak s -process by capturing neutrons that would otherwise be consumed in neutron capture on heavier elements. Variations in the neutron capture rate on 16 O have been proven to affect the abundances predicted to be formed in the weak s -process [ 2 2 ]. Studies have demonstrated that contributions from low-lying ( i.e. , near-threshold) 17 O resonances make the main contribution to the total neutron capture on 16 O at energies beyond 70 keV [ 3 3 , 4 4 ]. The low-lying 1d 3/2 and 2p 3/2 levels in 17 O are especially crucial. Developing the theoretical tools to predict the properties of such low-lying, low angular momentum levels is critical, especially for nucleosynthesis processes relying on neutron capture cross-sections of unstable nuclei that have not been measured. The methods that can uniformly treat the bound and resonant levels are particularly important. One theoretical approach, the self-consistent microscopic RAB approach, provides structural information via the analytical continuation of the coupling constant (ACCC) method based on the relativistic mean field (RMF) theory with the Bardeen-Cooper-Schrieffer (BCS) pairing approximation. This method was utilized to predict the structure of 17 O [ 3 3 , 4 4 ] and successfully yielded solutions for bound levels as well as for the low-lying 1d 3/2 resonance. Specifically, for the 1d 3/2 level, the energy (width) of the level was predicted to be ~ MeV (~100 keV), which agrees well with the experimental value [ 5 5 ] of 0.944 MeV (88 keV). However, the RAB approach was unable to obtain a solution for the critical 2p 3/2 orbital, making it necessary to consider the experimental values for this level in the studies of 16 O( n,γ ) 17 O reaction in Refs. [ 3 3 , 4 4 ]. It is quite difficult to achieve a structure of low-lying excited states with negative parity, which was also proposed in the shell model [ 6 6 ] combined with the complex scaling method (CSM) [ 7 7 , 8 8 ]. The CSM is applicable to resonances with a small rotation angle θ , but it is difficult to detect those with larger rotation angles, such as the 2p 3/2 level in 31 Ne [ 8 8 ]. Another approach, based on a complex momentum representation (CMR) of the Schrödinger equation, sought to overcome the limitations of the CSM approach and describe both the bound and resonant levels in 17 O but failed to obtained a solution for the critical low-lying 2p 3/2 level in Ref. [ 9 9 ]. As described below, this study utilizes the CMR approach but is based on a new nuclear potential improving the result reported in Ref. [ 3 3 ] for 17 O to predict the structure of the important 2p 3/2 resonance. Near-threshold resonances with low angular momentum are not just important for astrophysical capture reactions. A number of studies have demonstrated that such resonances can be critical in the formation of halo structures in some exotic nuclei [ 10 10 - 12 12 10-12 ]. The study of halo nuclei and related exotic nuclear structure phenomena [ 13 13 - 17 17 13-17 ] are the main motivations for novel unstable beam facilities ( e.g. , FRIB [ 18 18 ], FAIR [ 19 19 ]). Moreover, understanding the exotic nuclear structure has been identified as a top priority for theoretical research communities ( e.g. , FRIB [ 20 20 ]). It is important to advance our understanding of the resonances that may drive the formation of nuclear halos. Numerous studies have focused on the possible halos in neutron-rich fluorine isotopes, especially for 29, 31 F. These studies have been driven by the importance of understanding the extent of the “island of inversion” as well as shell evolution across the fluorine isotopic chain [ 21 21 ]. Experimental studies have been conducted on halo structures [ 22 22 - 26 26 22-26 ] and theoretical studies [ 21 21 , 27 27 ]. Bagchi et al. [ 22 22 ] demonstrated the presence of a two-neutron Borromean halo in the ground state of 29 F, which was thought to be dominated by the p-orbital. Subsequently, a scattering phase-shift method was used in a theoretical study of the bound and resonant orbitals in 29 F [ 28 28 ]. This study utilized both spherical and deformed Woods-Saxon potentials and found solutions for three bound levels (1d 3/2 , 2s 1/2 , and 1d 5/2 ) and one resonant orbital (1f 7/2 ). However, there are hints in our calculations based on a deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) theory [ 29 29 ] that 29 F is spherical with deformation <math id="M1"><mrow><mi>β</mi><mo>≃</mo><mn>0</mn></mrow></math> . In the spherical calculation stated in Ref. [ 28 28 ], however, a solution could not be found for the single-particle 2p 3/2 resonant level. Because of the possible role this level could play in forming the 29 F or 31 F halo, it is critical to understand the properties of this resonant level. In this study, our CMR methodology is described in Sect. 2, whereas the results for the critical 2p 3/2 and 1d 3/2 resonances in 17 O are presented in Sect. 3. The latter section details the results of CMR predictions regarding the energies of the low-lying 2p 3/2 resonant orbitals in 29, 31 F, which affect the formation mechanism of a halo. Finally, a summary is given in Sect. 4. Methods 1 To describe the methodology of this study, theoretical details of the CMR framework, nuclear potential, integration approach, and convergence and accuracy of the integration are presented in this section. Throughout this section, we use the example of the low-lying resonant orbitals of 17 O to demonstrate the viability of this approach. Formalism 2 In this section, a unified description of the bound and unbound states described by the CMR method is given based on the spherical case, similar to the description presented in Ref. [ 9 9 ]. A more general formalism can be found in Ref. [ 30 30 ]. A Hamiltonian of the form <math display="block" id="M2"><mrow><mover accent="true"><mi>H</mi><mo>^</mo></mover><mo>=</mo><mover accent="true"><mi>T</mi><mo>^</mo></mover><mo>+</mo><mover accent="true"><mi>V</mi><mo>^</mo></mover><mn>,</mn></mrow></math> is assumed, where <math id="M3"><mover accent="true"><mi>T</mi><mo>^</mo></mover></math> is the kinetic energy and <math id="M4"><mover accent="true"><mi>V</mi><mo>^</mo></mover></math> is the nuclear potential with a Woods-Saxon shape. The foundation of the CMR approach [ 9 9 ] describes the Schrödinger equation with this Hamiltonian in momentum, rather than coordinate, is represented as: <math display="block" id="M5"><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>3</mn></msup><msup><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo>'</mo></msup><mrow><mo>〈</mo><mrow><mover accent="true"><mi>k</mi><mo>→</mo></mover><mrow><mo>|</mo><mover accent="true"><mi>H</mi><mo>^</mo></mover><mo>|</mo></mrow><msup><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo>'</mo></msup></mrow><mo>〉</mo></mrow><msub><mi>Φ</mi><mi>n</mi></msub><mo stretchy="false">(</mo><msup><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo>'</mo></msup><mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mi>n</mi></msub><msub><mi>Φ</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Here, <math id="M6"><mrow><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo>=</mo><mover accent="true"><mi>p</mi><mo>→</mo></mover><mo>/</mo><mi>ℏ</mi></mrow></math> is the wave vector, <math id="M7"><mrow><msub><mi>Φ</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo stretchy="false">)</mo></mrow></math> is the wave function, and E n is the energy of the n th state. If we assume spherical symmetry, the wave function can be stated as follows: <math display="block" id="M8"><mrow><msub><mi>Φ</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ϕ</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><msubsup><mi>Y</mi><mi>l</mi><mi>m</mi></msubsup><mo stretchy="false">(</mo><msub><mi>Ω</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></math> where ϕ n ( k ) is the radial portion of the momentum wavefunction and <math id="M9"><mrow><msubsup><mi>Y</mi><mi>l</mi><mi>m</mi></msubsup><mo stretchy="false">(</mo><msub><mi>Ω</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></math> is the angular portion with angular momentum quantum numbers l and m . The Schrodinger equation then becomes: <math display="block" id="M10"><mrow><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><msub><mi>ϕ</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>d</mi><msup><mi>k</mi><mo>'</mo></msup><msup><mi>k</mi><mrow><mo>'</mo><mn>2</mn></mrow></msup><msub><mi>V</mi><mi>l</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mn>,</mn><msup><mi>k</mi><mo>'</mo></msup><mo stretchy="false">)</mo><msub><mi>ϕ</mi><mi>n</mi></msub><mo stretchy="false">(</mo><msup><mi>k</mi><mo>'</mo></msup><mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mi>n</mi></msub><msub><mi>ϕ</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> with <math display="block" id="M11"><mrow><msub><mi>V</mi><mi>l</mi></msub><mo stretchy="false">(</mo><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><mi>d</mi><mi>r</mi><msup><mi>r</mi><mn>2</mn></msup><mi>V</mi><mo stretchy="false">(</mo><mi>r</mi><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><mn>,</mn></mrow></math> where j l ( kr ) denotes the spherical Bessel function of l th order. Equation (5) is the integral of the potential given in Sect. (2B) in a Woods–Saxon formulation over the coordinate space. Subsequently, this expression is used in Eq. (4) in an integral of the product of the potential and wavefunction over the momentum space in the CMR-based Schrödinger equation. The solution of these equations for eigenvalue level energies E n is discussed in Sect. (2c). Potential parameters 2 In Ref. [ 9 9 ], the CMR method was used to calculate the low-lying resonances in 17 O with a generic Woods-Saxon potential described in Ref. [ 31 31 ]. Because this potential does not reproduce the experimental one-neutron separation energy, the predicted energy and width of the resonant orbital 1d 3/2 are significantly overestimated. More importantly, a CMR-based solution for the energy level of the low-lying resonant 2p 3/2 orbital, whose contributions become progressively important and comparable to the direct capture process above 70 keV [ 3 3 , 4 4 ], cannot be found in that research. Therefore, we adopted the potential based on that presented in Ref. [ 3 3 ] to revisit the bound and resonant structures in 17 O. The interaction potential V includes the central part V C and spin-orbit part V SO , which takes the form of the Woods-Saxon type potential, <math display="block" id="M12"><mrow><msub><mi>V</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mn>0</mn></msub><msub><mi>f</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M13"><mrow><msub><mi>V</mi><mrow><mtext>SO</mtext></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msubsup><mi>V</mi><mrow><mtext>SO</mtext></mrow><mn>0</mn></msubsup><msup><mrow><mo stretchy="false">(</mo><mfrac><mi>ℏ</mi><mrow><msub><mi>m</mi><mi>π</mi></msub><mi>c</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">(</mo><mover accent="true"><mi>l</mi><mo>→</mo></mover><mo>⋅</mo><mover accent="true"><mi>s</mi><mo>→</mo></mover><mo stretchy="false">)</mo><mfrac><mn>1</mn><mi>r</mi></mfrac><mfrac><mrow><mtext>d</mtext><msub><mi>f</mi><mrow><mtext>SO</mtext></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mtext>d</mtext><mi>r</mi></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M14"><mrow><msub><mi>f</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>r</mi><mo>−</mo><msub><mi>R</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>/</mo><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn><mtext> </mtext><mo stretchy="false">(</mo><mi>i</mi><mtext> </mtext><mtext>refers</mtext><mtext> </mtext><mtext>to</mtext><mtext> </mtext><mtext>C</mtext><mtext> </mtext><mtext>or</mtext><mtext> </mtext><mtext>SO</mtext><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Here, the pion Compton wavelength <math id="M15"><mrow><mi>ℏ</mi><mo>/</mo><msub><mi>m</mi><mi>π</mi></msub><mi>c</mi><mo>=</mo><mn>1.414</mn></mrow></math> fm and potential depth V 0 for the central part and <math id="M16"><mrow><msubsup><mi>V</mi><mrow><mtext>SO</mtext></mrow><mn>0</mn></msubsup></mrow></math> for the spin-orbit part were adjusted to reproduce the one-neutron separation energy S n = 4.143 MeV for 17 O. The diffuseness and radius parameters are obtained from the Koning-Delaroche optical model potential [ 32 32 ]. The six parameters of the Woods-Saxon potential are listed in Table 1 Table 1 . 1 1 Parameter 1 1 Adopted value 1 1 V 0 (MeV) 1 1 -58.75 1 1 <math id="M17"><mrow><msubsup><mi>V</mi><mrow><mtext>SO</mtext></mrow><mn>0</mn></msubsup></mrow></math> (MeV) 1 1 -12.45 1 1 a 0 (fm) 1 1 0.675 1 1 a SO (fm) 1 1 0.590 1 1 r 0 (fm) 1 1 2.947 1 1 r SO (fm) 1 1 2.401 In the case of exotic neutron-rich fluorine isotopes 29, 31 F, a standard spherical potential [ 31 31 ] was adopted, which was also utilized in a recent study focusing on this nucleus using a scattering phase shift method [ 28 28 ]. By choosing this potential, a direct comparison can be made between the CMR method and scattering phase-shift approach. Integration method 2 The integral stated in Eq. (4) requires discretization to readily obtain the eigenvalue level energies E n . The integral over the momentum space can be approximated as a sum over a finite N q set of points k j with spacing dk and weights w j . The result is the following N q × N q matrix equation: <math display="block" id="M18"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mi>q</mi></msub></mrow></munderover></mstyle><msub><mi>H</mi><mrow><mi>i</mi><mn>,</mn><mi>j</mi></mrow></msub><msub><mi>ϕ</mi><mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mi>n</mi></msub><msub><mi>ϕ</mi><mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mn>,</mn><mtext> </mtext><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1,2,</mn><mo>⋯</mo><mn>,</mn><msub><mi>N</mi><mi>q</mi></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M19"><mrow><msub><mi>H</mi><mrow><mi>i</mi><mn>,</mn><mi>j</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup><msubsup><mi>k</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><mi>m</mi></mrow></mfrac><msub><mi>δ</mi><mrow><mi>i</mi><mn>,</mn><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>w</mi><mi>j</mi></msub><msubsup><mi>k</mi><mi>j</mi><mn>2</mn></msubsup><msubsup><mi>V</mi><mi>l</mi><mrow><mi>i</mi><mn>,</mn><mi>j</mi></mrow></msubsup></mrow></math> <math display="block" id="M20"><mrow><msubsup><mi>V</mi><mi>l</mi><mrow><mi>i</mi><mn>,</mn><mi>j</mi></mrow></msubsup><mo>=</mo><mfrac><mn>2</mn><mi>π</mi></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>d</mi><mi>r</mi><msup><mi>r</mi><mn>2</mn></msup><mi>V</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><msub><mi>j</mi><mi>l</mi></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mi>i</mi></msub><mi>r</mi><mo stretchy="false">)</mo><msub><mi>j</mi><mi>l</mi></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mi>j</mi></msub><mi>r</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Using the following transformations: <math display="block" id="M21"><mrow><msubsup><mi>ϕ</mi><mi>n</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><msub><mi>k</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><msub><mi>w</mi><mi>i</mi></msub></mrow></msqrt><msub><mi>k</mi><mi>i</mi></msub><msub><mi>ϕ</mi><mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mn>,</mn><mtext> </mtext><msubsup><mi>H</mi><mrow><mi>i</mi><mn>,</mn><mi>j</mi></mrow><mo>'</mo></msubsup><mo>=</mo><msqrt><mrow><mfrac><mrow><msub><mi>w</mi><mi>i</mi></msub></mrow><mrow><msub><mi>w</mi><mi>j</mi></msub></mrow></mfrac></mrow></msqrt><mfrac><mrow><msub><mi>k</mi><mi>i</mi></msub></mrow><mrow><msub><mi>k</mi><mi>j</mi></msub></mrow></mfrac><msub><mi>H</mi><mrow><mi>i</mi><mn>,</mn><mi>j</mi></mrow></msub><mn>,</mn></mrow></math> the system for the n th state is symmetrized as follows: <math display="block" id="M22"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mi>q</mi></msub></mrow></munderover></mstyle><msubsup><mi>H</mi><mrow><mi>i</mi><mn>,</mn><mi>j</mi></mrow><mo>'</mo></msubsup><msubsup><mi>ϕ</mi><mi>n</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><msub><mi>k</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mi>n</mi></msub><msubsup><mi>ϕ</mi><mi>n</mi><mo>'</mo></msubsup><mo stretchy="false">(</mo><msub><mi>k</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mn>,</mn><mtext> </mtext><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1,2,</mn><mo>⋯</mo><mn>,</mn><msub><mi>N</mi><mi>q</mi></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M23"><mrow><msubsup><mi>H</mi><mrow><mi>i</mi><mn>,</mn><mi>j</mi></mrow><mo>'</mo></msubsup><mo>=</mo><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup><msubsup><mi>k</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><mi>m</mi></mrow></mfrac><msub><mi>δ</mi><mrow><mi>i</mi><mn>,</mn><mi>j</mi></mrow></msub><mo>+</mo><msqrt><mrow><msub><mi>w</mi><mi>i</mi></msub><msub><mi>w</mi><mi>j</mi></msub></mrow></msqrt><msub><mi>k</mi><mi>i</mi></msub><msub><mi>k</mi><mi>j</mi></msub><msubsup><mi>V</mi><mi>l</mi><mrow><mi>i</mi><mn>,</mn><mi>j</mi></mrow></msubsup><mn>,</mn></mrow></math> <math display="block" id="M24"><mrow><msubsup><mi>V</mi><mi>l</mi><mrow><mi>i</mi><mn>,</mn><mi>j</mi></mrow></msubsup><mo>=</mo><mfrac><mn>2</mn><mi>π</mi></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>d</mi><mi>r</mi><msup><mi>r</mi><mn>2</mn></msup><mi>V</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><msub><mi>j</mi><mi>l</mi></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mi>i</mi></msub><mi>r</mi><mo stretchy="false">)</mo><msub><mi>j</mi><mi>l</mi></msub><mo stretchy="false">(</mo><msub><mi>k</mi><mi>j</mi></msub><mi>r</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> where k i,j denote the integral points along a contour in the momentum space with corresponding weights w i,j ; the potential is integrated over the coordinate space. It is known (see, e.g. , Refs. [ 33 33 - 35 35 33-35 ]) that bound states populate the imaginary axis in the momentum plane, whereas resonances are located within the fourth quadrant. By choosing the grid points k i, j along an appropriate path in the complex momentum plane that contains these levels, the eigenvalues - single particle energies - of this system for the bound, resonant, and continuum states can be determined. Because the integration stated in Eq. (4) ranges from zero to infinity, the sum stated in Eq. (13) must be carried out up to large momentum values. Therefore, we adopt a Gauss-Legendre quadrature approach to obtain solutions for the eigenenergies E n . Similarly, we apply the Gauss-Legendre quadrature to the coordinate-space integral stated in Eq. (15). In Ref. [ 9 9 ], a Gauss-Legendre quadrature method was used for the momentum integral, while a Gauss-Hermite quadrature method was used for the coordinate (potential) integral; the resulting accuracy for single-particle energies was 10 -4 MeV. The selection of the number of grid points and truncation in the momentum and coordinate space will influence the convergence of the summation (integration) for both the momentum and coordinate integrals, as well as the value and precision of the single-particle energy level solutions. The convergence and accuracy characteristics are provided in the next subsection. Convergence and accuracy 2 In coordinate space, we study the convergence and accuracy of the potential integral V i,j presented in Eq. (15) with variations in the number of grid points N p and truncation (maximum value) of the radius r cut in the integral. Integral accuracy is defined as the maximum accuracy for the real and imaginary parts of the integral. For a particular choice of momentum values k i = (0.3, -0.075) and k j = (0.4, 0), where the units of the real and imaginary components of the momentum grid points are given in fm -1 , Fig. 1 Fig. 1 demonstrates the smooth convergence of the coordinate-space potential integral. Moreover, an accuracy of 10 -10 MeV, which is well beyond the required precision of the calculation, can be reached when N p ≥90 and <math id="M25"><mrow><msub><mi>r</mi><mrow><mtext>cut</mtext></mrow></msub><mo>≥</mo><mn>30</mn></mrow></math> fm. To guarantee convergence with this accuracy, we utilize N p = 100 and r cut = 40 fm in our subsequent coordinate-space potential integral calculations. Similarly, in the momentum space, we study the convergence of the solutions to the system of equations presented in Eqs. (13) and (14) with variations in the number of grid points N q and truncation (maximum value) of the momentum k cut in the integral. The path for momentum integration is selected as a contour bounded by points (0, 0), (0.5, -0.1), (1, 0) and (10, 0) in the complex momentum plane. For each segment of the contour, the number of grid points for the Gauss-Legendre quadrature is N q , such that the total number of grid points for the entire path is 3 N q . The positions of the complex momentum for the resonant states do not change with the different paths of the contour. Figure 2 Figure 2 demonstrates an example of the convergence of the momentum-space integral for the 1d 3/2 resonant orbital in 17 O. An accuracy of 10 -8 MeV can be achieved when N q ≥40 and <math id="M26"><mrow><msub><mi>k</mi><mrow><mtext>cut</mtext></mrow></msub><mo>≥</mo><mn>7.5</mn><msup><mrow><mtext>fm</mtext></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> . To guarantee convergence with such accuracy, we utilize N q = 100 and <math id="M27"><mrow><msub><mi>k</mi><mrow><mtext>cut</mtext></mrow></msub><mo>=</mo><mn>10</mn><msup><mrow><mtext>fm</mtext></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> in our solution of the momentum-space integral. For this level, the converged values are 1.038 MeV and 0.149 MeV for the energy and width, respectively, which are consistent with the measured values of 0.944 ± 0.001 MeV and 0.088 ± 0.003 MeV [ 5 5 ]. Results and discussions 1 Energy levels for bound and resonant states in 17 O 2 The CMR approach described above was used to solve for the bound and resonant states in 17 O. The contour for the momentum integration, as shown in Fig. 3 Fig. 3 , is bounded by points (0, 0), (0.25, -0.5), (0.5, 0), and (10, 0) in the complex momentum plane. This contour was chosen to include the 17 O bound states (indicated with the blue open squares in Fig. 3 Fig. 3 ) on the positive imaginary k axis, resonant states (indicated with the red open triangles) confined and scattered inside the contour, and continuum states (indicated with the black open circles) distributed along the contour. The positions of the bound and resonant states remained unchanged with different contour shapes. The CMR-based solutions for the five bound states and two resonant states of 17 O are listed in Table 2 Table 2 . Experimental data from Refs. [ 5 5 , 36 36 - 38 38 36-38 ] are presented for comparison. Additionally, the Numerov numerical method (see, e.g. , Ref. [ 39 39 ]) can be used to solve the Schrödinger equation for negative-energy (bound) levels to an arbitrary level of precision, including a Woods-Saxon potential formulation (see, e.g. , Ref. [ 40 40 ]). The results of the Numerov approach for the five bound 17 O levels using the potential described in Sect. 2.2 are reported in Table 2 Table 2 , underlining that the CMR results agree with the Numerov approach within 0.001 MeV. The CMR results agree with the experimental value of the 2s 1/2 excited state energy within 0.1 MeV. a Reference [ 36 36 ] b Reference [ 37 37 ] c Reference [ 38 38 ] d Reference [ 5 5 ] 1 1   2 1 This work 1 1 Numerov 2 1 Experiment 1 1   1 1 E 1 1 Γ 1 1 E 1 1 E 1 1 Γ 1 1 1s 1/2 1 1 -33.985 1 1   1 1 -33.985 1 1 1p 3/2 1 1 -18.992 1 1   1 1 -18.992 1 1 1p 1/2 1 1 -13.872 1 1   1 1 -13.872 1 1 1d 5/2 1 1 -4.143 1 1   1 1 -4.143 1 1 -4.143 a 1 1 2s 1/2 1 1 -3.381 1 1   1 1 -3.381 1 1 -3.272 b 1 1 2p 3/2 1 1 0.498 1 1 4.417 1 1   1 1 0.411 c 1 1 0.040 c 1 1 1d 3/2 1 1 1.038 1 1 0.149 1 1   1 1 0.944 d 1 1 0.088 d For the 1d 3/2 resonant orbital, the CMR approach provides a solution at position (0.217, -0.008) in the complex momentum plane, corresponding to a resonance energy E R = 1.038 MeV and decay width Γ =0.149 MeV. The energy is within 10% of the measured value of <math id="M28"><mrow><msubsup><mi>E</mi><mtext>R</mtext><mrow><mtext>Exp</mtext></mrow></msubsup><mo>=</mo><mn>0.944</mn><mo>±</mo><mn>0.001</mn></mrow></math> MeV [ 5 5 ]; the width is within 0.06 MeV of the measured value of <math id="M29"><mrow><msup><mi>Γ</mi><mrow><mtext>Exp</mtext></mrow></msup><mo>=</mo><mn>0.088</mn><mo>±</mo><mn>0.003</mn></mrow></math> MeV [ 5 5 ]. For the critical 2p 3/2 low-lying resonant orbital, the solutions for which have not been obtained with previous theoretical approaches, the CMR approach using the potential stated in Sect. 2.2 yields a predicted energy of 0.498 MeV, which is within 22% of the measured value of <math id="M30"><mrow><msubsup><mi>E</mi><mtext>R</mtext><mrow><mtext>Exp</mtext></mrow></msubsup><mo>=</mo><mn>0.411</mn></mrow></math> MeV [ 38 38 ]. This result demonstrates the potential of the CMR method for predicting low-lying odd-parity levels. However, the CMR-based prediction of the width of this critical level is 4.417 MeV, which is much larger than the experimental value of 0.040 MeV. This overestimation of the low-lying resonance width has appeared in other CMR-based studies (see, e.g. , Refs. [ 9 9 , 41 41 ]). The measured 3/2 - orbital may originate from 16 O core excitation [ 6 6 , 42 42 ], which should be treated beyond the mean-field framework. In Ref. [ 42 42 ], the Shell Model Embedded in the Continuum (SMEC) was used to study the spectroscopy of mirror nuclei: 17 O and 17 F, where realistic SM solutions for (quasi-) bound states were coupled to the one-particle scattering continuum for both Wigner and Bartlett (WB) and density-dependent (DDSM1) residual interactions. All these SMEC predictions of the level energies with residual interactions for 3/2 - state in 17 O are approximately 0.5 MeV larger than the experimental data with widths of 40 keV. As mentioned in Ref. [ 6 6 ], the potential model does not include particle-hole excitation in the 16 O core in their calculations; thus, low-lying negative-parity states, such as 3/2 - , cannot be reproduced. Therefore, our evaluation of the width might be narrower when the core excitation is considered. Energy levels for bound and resonant states in 29 F and 31 F 2 The exotic nuclei 29 F and 31 F, which are candidates for a two-neutron Borromean halo structure [ 22 22 ], were studied using the same CMR approach as that used for 17 O. As the separation energy has not been measured in this nucleus, a standard set [ 31 31 ] of potential parameters is used for 29, 31 F, which is the same as that used in Ref. [ 28 28 ]. The integration contour in the complex momentum space is bounded by points (0, 0), (0.25, -0.5), (0.5, 0) and (10, 0). The results for the bound and resonant states in 29 F and 31 F, respectively, are shown in Fig. 4 Fig. 4 . Owing to the lack of experimental measurements regarding the excited states of this nucleus, a comparison is made in Fig. 4 Fig. 4 between our CMR-based predictions and those obtained using the scattering phase-shift method described in Ref. [ 28 28 ]. The energies for the bound orbitals and unbound 1f 7/2 orbital in 29 F agree well with those stated in Ref. [ 28 28 ] with a maximum difference of 0.15 MeV. Most importantly, our prediction of the critical low-lying negative-parity 2p 3/2 resonant orbital in 29 F is only 0.582 MeV above the neutron threshold; a solution for this level was not presented in Ref. [ 28 28 ]. In the spherical case of 29 F, the predicted 2p 3/2 lies above 1 d 3/2 , which is consistent with conditions B and C described in Ref. [ 21 21 ]. From the calculated spectrum in 31 F, we can determine that the last two neutrons occupy 2p 3/2 , which might cause the halo formation in 31 F. As was the case for predicting the width of the 2p 3/2 resonance orbital in 17 O. The prediction of the decay width of the 2p 3/2 orbital in 29 F ( 30 F) - 2.384 MeV (2.337 MeV) - is likely significantly overestimated by the CMR approach and warrants further study. If deformation is considered, the 2p 3/2 orbital might be lower than the 1d 3/2 orbital, which will be studied in our future research. Summary 1 Solving the Schrödinger equation in a complex momentum space, rather than coordinate space, enables the bound and resonant levels to be treated simultaneously in a uniform manner. The viability of this approach for predicting the energies of low-lying resonances above the particle threshold has been explored focusing on the critical levels in 17 O and 29, 31 F. Using a novel nuclear potential based on Koning-Delaroche optical model potential constrained by the experimental one-neutron separation energy. Subsequently, the energy of the low-lying resonant 2p 3/2 orbital in 17 O is determined as 0.498 MeV in agreement with the experimental results. The predicted energy and width of the 1d 3/2 level agree well with the measured data as well. These two levels are known to dominate neutron capture on 16 O which acts as a neutron poison in weak s-process nucleosynthesis. In addition, the CMR approach was utilized to examine the levels in 29 F and 31 F exotic nuclei with two possible neutron halo structures. The CMR approach predicts the energies of the bound levels and unbound 1f 7/2 level in 29 F, which agree well with the results of the phase-shift method. Most importantly, the CMR approach predicts that the 2p 3/2 orbital lies just above the threshold, which is critical for halo formation in 31 F. These results demonstrate that, when coupled with the proper nuclear potential, the CMR approach is a promising tool for determining resonance energies, especially for low-lying orbitals with negative parity that are crucial in capture reactions and halo formation. In the future, we will improve the overestimated decay widths with a dedicated formalism [ 43 43 ] wherein occupation probabilities can be estimated by BCS approximation with particle-hole residual interactions for core excitation. References N. Prantzos , M. Hashimoto , K. Nomoto , The s-process in massive stars-yields as a function of stellar mass and metallicity . Astron. Astrophys . 234 , 211 - 229 ( 1990 ). https://ui.adsabs.harvard.edu/abs/1990AA...234..211P https://ui.adsabs.harvard.edu/abs/1990AA...234..211P The s-process in massive stars-yields as a function of stellar mass and metallicity P. Mohr , C. Heinz , M. Pignatari et al. , Re-evaluation of the 16 O ( n, γ ) 17 O cross section at astrophysical energies and its role as a neutron poison in the s-process . Astrophys. J . 827 , 29 ( 2016 ). doi: 10.3847/0004-637x/827/1/29 http://doi.org/10.3847/0004-637x/827/1/29 Re-evaluation of the 16O (n, γ) 17O cross section at astrophysical energies and its role as a neutron poison in the s-process M. He , S.-S. Zhang , M. Kusakabe et al. , Nuclear structures of 17 O and time-dependent sensitivity of the weak s-process to the 16 O ( n, γ ) 17 O rate . Astrophys. J . 899 , 133 ( 2020 ). doi: 10.3847/1538-4357/aba7b4 http://doi.org/10.3847/1538-4357/aba7b4 Nuclear structures of 17O and time-dependent sensitivity of the weak s-process to the 16O (n, γ) 17O rate S. Zhang , S. Xu , M. He et al. , Neutron capture on 16 O within the framework of rmf+ accc+ bcs for astrophysical simulations . Eur. Phys. J. A 57 , 1 - 7 ( 2021 ). doi: 10.1140/epja/s10050-021-00434-7 http://doi.org/10.1140/epja/s10050-021-00434-7 Neutron capture on 16O within the framework of rmf+ accc+ bcs for astrophysical simulations T. Faestermann , P. Mohr , R. Hertenberger et al. , Broad levels in 17 O and their relevance for the astrophysical s process . Phys. Rev. C 92 , 052802 ( 2015 ). doi: 10.1103/PhysRevC.92.052802 http://doi.org/10.1103/PhysRevC.92.052802 Broad levels in 17O and their relevance for the astrophysical s process K. Yamamoto , H. Masui , K. Katō et al. , Radiative Capture Cross Section for 16 O( n,γ ) 17 O and 16 O( p,γ ) 17 F below Astrophysical Energies . Prog. of Theor. Phys . 121 , 375 - 390 ( 2009 ). doi: 10.1143/PTP.121.375 http://doi.org/10.1143/PTP.121.375 Radiative Capture Cross Section for 16O(n,γ)17O and 16O(p,γ)17F below Astrophysical Energies N. Michel , W. Nazarewicz , M. Płoszajczak et al. , Shell model in the complex energy plane . J. Phys. G 36 , 013101 ( 2008 ). doi: 10.1088/0954-3899/36/1/013101 http://doi.org/10.1088/0954-3899/36/1/013101 Shell model in the complex energy plane Q. Liu , J.-Y. Guo , Z.-M. Niu et al. , Resonant states of deformed nuclei in the complex scaling method . Phys. Rev. C 86 , 054312 ( 2012 ). doi: 10.1103/PhysRevC.86.054312 http://doi.org/10.1103/PhysRevC.86.054312 Resonant states of deformed nuclei in the complex scaling method Y.-J. Tian , T.-H. Heng , Z.-M. Niu et al. , Exploration of resonances by using complex momentum representation . Chin. Phys. C 41 , 044104 ( 2017 ). doi: 10.1088/1674-1137/41/4/044104 http://doi.org/10.1088/1674-1137/41/4/044104 Exploration of resonances by using complex momentum representation S.-G. Zhou , J. Meng , P. Ring et al. , Neutron halo in deformed nuclei . Phys. Rev. C 82 , 011301 ( 2010 ). doi: 10.1103/PhysRevC.82.011301 http://doi.org/10.1103/PhysRevC.82.011301 Neutron halo in deformed nuclei A. Moro , J. Lay , Interplay between valence and core excitation mechanisms in the breakup of halo nuclei . Phys. Rev. Lett . 109 , 232502 ( 2012 ). doi: 10.1103/PhysRevLett.109.232502 http://doi.org/10.1103/PhysRevLett.109.232502 Interplay between valence and core excitation mechanisms in the breakup of halo nuclei S.-S. Zhang , M. S. Smith , Z.-S. Kang et al. , Microscopic self-consistent study of neon halos with resonant contributions . Phys. Lett. B 730 , 30 - 35 ( 2014 ). https://www.sciencedirect.com/science/article/pii/S0370269314000355 https://www.sciencedirect.com/science/article/pii/S0370269314000355 Microscopic self-consistent study of neon halos with resonant contributions L. Zhou , S.M. Wang , D.Q. Fang , Y.G. Ma , Recent progress in two-proton radioactivity . Nucl. Sci. Tech . 33 , 105 ( 2022 ). doi: 10.1007/s41365-022-01091-1 http://doi.org/10.1007/s41365-022-01091-1 Recent progress in two-proton radioactivity B. Li , N. Tang , Y.H. Zhang , et al . Production of p-rich nuclei with Z=20-25 based on radioactive ion beams . Nucl. Sci. Tech . 33 , 55 ( 2022 ). doi: 10.1007/s41365-022-01048-4 http://doi.org/10.1007/s41365-022-01048-4 Production of p-rich nuclei with Z=20-25 based on radioactive ion beams D.Z. Chen , D.L. Fang , and C.L. Bai . Impact of finite-range tensor terms in the Gogny force on the β -decay of magic nuclei . Nucl. Sci. Tech . 32 , 74 ( 2021 ). doi: 10.1007/s41365-021-00908-9 http://doi.org/10.1007/s41365-021-00908-9 Impact of finite-range tensor terms in the Gogny force on the β-decay of magic nuclei W. Nan , B. Guo , C.J. Lin , et al . First proof-of-principle experiment with the post-accelerated isotope separator on-line beam at BRIF: measurement of the angular distribution of 23 Na + 40 Ca elastic scattering . Nucl. Sci. Tech . 32 , 53 ( 2021 ). doi: 10.1007/s41365-021-00889-9 http://doi.org/10.1007/s41365-021-00889-9 First proof-of-principle experiment with the post-accelerated isotope separator on-line beam at BRIF: measurement of the angular distribution of 23Na + 40Ca elastic scattering S.W. Bai , X.F. Yang , S.J. Wang , et al . Commissioning of a high-resolution collinear laser spectroscopy apparatus with a laser ablation ion source . Nucl. Sci. Tech . 33 , 9 ( 2022 ). doi: 10.1007/s41365-022-00992-5 http://doi.org/10.1007/s41365-022-00992-5 Commissioning of a high-resolution collinear laser spectroscopy apparatus with a laser ablation ion source B. M. Sherrill , Future opportunities at the facility for rare isotope beams , in: EPJ Web of Conferences , volume 178 , EDP Sciences , 2018 , p. 01001 . doi: 10.1051/epjconf/201817801001 http://doi.org/10.1051/epjconf/201817801001 Future opportunities at the facility for rare isotope beams T. Nilsson , Radioactive ion beams at fair-nustar, Eur . Phys. J-Spec. Top . 156 , 1 - 12 ( 2008 ). doi: 10.1140/epjst/e2008-00606-2 http://doi.org/10.1140/epjst/e2008-00606-2 Radioactive ion beams at fair-nustar, Eur A. Balantekin , J. Carlson , D. Dean et al. , Nuclear theory and science of the facility for rare isotope beams . Mod. Phys. Lett. A 29 , 1430010 ( 2014 ). doi: 10.1142/S0217732314300109 http://doi.org/10.1142/S0217732314300109 Nuclear theory and science of the facility for rare isotope beams J. Singh , J. Casal , W. Horiuchi et al. , Exploring two-neutron halo formation in the ground state of 29 F within a three-body model . Phys. Rev. C 101 , 024310 ( 2020 ). doi: 10.1103/PhysRevC.101.024310 http://doi.org/10.1103/PhysRevC.101.024310 Exploring two-neutron halo formation in the ground state of 29F within a three-body model S. Bagchi , R. Kanungo , Y. K. Tanaka et al. , Two-neutron halo is unveiled in 29 F . Phys. Rev. Lett . 124 , 222504 ( 2020 ). doi: 10.1103/PhysRevLett.124.222504 http://doi.org/10.1103/PhysRevLett.124.222504 Two-neutron halo is unveiled in 29F A. Revel , O. Sorlin , F. M. Marqués et al. , (SAMURAI21 collaboration), Extending the southern shore of the island of inversion to 28 F . Phys. Rev. Lett . 124 , 152502 ( 2020 ). doi: 10.1103/PhysRevLett.124.152502 http://doi.org/10.1103/PhysRevLett.124.152502 (SAMURAI21 collaboration), Extending the southern shore of the island of inversion to 28F D. S. Ahn , N. Fukuda , H. Geissel et al. , Location of the neutron dripline at fluorine and neon . Phys. Rev. Lett . 123 , 212501 ( 2019 ). doi: 10.1103/PhysRevLett.123.212501 http://doi.org/10.1103/PhysRevLett.123.212501 Location of the neutron dripline at fluorine and neon G. Christian , N. Frank , S. Ash et al. , Exploring the low- z shore of the island of inversion at n =19 . Phys. Rev. Lett . 108 , 032501 ( 2012 ). doi: 10.1103/PhysRevLett.108.032501 http://doi.org/10.1103/PhysRevLett.108.032501 Exploring the low-z shore of the island of inversion at n=19 L. Gaudefroy , W. Mittig , N. Orr et al. , Direct mass measurements of 19 B, 22 C, 29 F, 31 Ne, 34 Na and other light exotic nuclei . Phys. Rev. Lett . 109 , 202503 ( 2012 ). doi: 10.1103/PhysRevLett.109.202503 http://doi.org/10.1103/PhysRevLett.109.202503 Direct mass measurements of 19B, 22C, 29F, 31Ne, 34Na and other light exotic nuclei H. Masui , W. Horiuchi , M. Kimura , Two-neutron halo structure of f 31 and a novel pairing antihalo effect . Phys. Rev. C 101 , 041303 ( 2020 ). doi: 10.1103/PhysRevC.101.041303 http://doi.org/10.1103/PhysRevC.101.041303 Two-neutron halo structure of f 31 and a novel pairing antihalo effect I. Hamamoto , Deformed halo of 929f20 . Phys. Lett. B 814 , 136116 ( 2021 ). https://www.sciencedirect.com/science/article/pii/S0370269321000563 https://www.sciencedirect.com/science/article/pii/S0370269321000563 Deformed halo of 929f20 L. Li , J. Meng , P. Ring et al. , Deformed relativistic hartree-bogoliubov theory in continuum . Phys. Rev. C 85 , 024312 ( 2012 ). doi: 10.1103/PhysRevC.85.024312 http://doi.org/10.1103/PhysRevC.85.024312 Deformed relativistic hartree-bogoliubov theory in continuum G. Hagen , J. S. Vaagen , Study of resonant structures in a deformed mean field by the contour deformation method in momentum space . Phys. Rev. C 73 , 034321 ( 2006 ). doi: 10.1103/PhysRevC.73.034321 http://doi.org/10.1103/PhysRevC.73.034321 Study of resonant structures in a deformed mean field by the contour deformation method in momentum space A. Bohr , B. R. Mottelson , Nuclear Structure , volume I , World Scientific , 1998 . Nuclear Structure A. Koning , J. Delaroche , Local and global nucleon optical models from 1 keV to 200 MeV . Nucl. Phys. A 713 , 231 - 310 ( 2003 ). doi: 10.1016/S0375-9474(02)01321-0 http://doi.org/10.1016/S0375-9474(02)01321-0 Local and global nucleon optical models from 1 keV to 200 MeV C. V. Sukumar , Schrodinger equation in momentum space . J. Phys. A Mathematic. General 12 , 1715 - 1730 ( 1979 ). doi: 10.1088/0305-4470/12/10/018 http://doi.org/10.1088/0305-4470/12/10/018 Schrodinger equation in momentum space Y. R. Kwon , F. Tabakin , Hadronic atoms in momentum space . Phys. Rev. C 18 , 932 - 943 ( 1978 ). doi: 10.1103/PhysRevC.18.932 http://doi.org/10.1103/PhysRevC.18.932 Hadronic atoms in momentum space N. Li , M. Shi , J.-Y. Guo , et al. , Probing resonances of the dirac equation with complex momentum representation . Phys. Rev. Lett . 117 , 062502 ( 2016 ). doi: 10.1103/PhysRevLett.117.062502 http://doi.org/10.1103/PhysRevLett.117.062502 Probing resonances of the dirac equation with complex momentum representation M. Wang , W. Huang , F. Kondev et al. , The AME 2020 atomic mass evaluation (II) . tables, graphs and references, Chin. Phys. C 45 , 030003 ( 2021 ). doi: 10.1088/1674-1137/abddaf http://doi.org/10.1088/1674-1137/abddaf The AME 2020 atomic mass evaluation (II) R. B. Firestone , Z. Revay , Thermal neutron capture cross sections for 16,17,18 O and 2 H, Phys . Rev. C 93 , 044311 ( 2016 ). doi: 10.1103/PhysRevC.93.044311 http://doi.org/10.1103/PhysRevC.93.044311 Thermal neutron capture cross sections for 16,17,18O and 2H, Phys Y. Nagai , M. Kinoshita , M. Igashira et al. , Nonresonant p -wave direct capture and interference effect observed in the 16 O (n, γ ) 17 O reaction, Phys . Rev. C 102 , 044616 ( 2020 ). doi: 10.1103/PhysRevC.102.044616 http://doi.org/10.1103/PhysRevC.102.044616 Nonresonant p-wave direct capture and interference effect observed in the 16O (n, γ) 17O reaction, Phys M. Pillai , J. Goglio , T. G. Walker , Matrix numerov method for solving schrödinger equation, Am . J. Phys . 80 ( 2012 ) 1017 - 1019 . doi: 10.1119/1.4748813 http://doi.org/10.1119/1.4748813 Matrix numerov method for solving schrödinger equation, Am M. Bhat , A. P. Monteiro , Numerical solution of schroedinger equation using matrix numerov method with woods - saxon potential , Proceedings of the DAE-BRNS symposium on nuclear physics , Prasanthi Nilayam (India) ; 7-11 December 2015 Numerical solution of schroedinger equation using matrix numerov method with woods - saxon potential Y.-X. Luo , K. Fossez , Q. Liu et al. , Role of quadrupole deformation and continuum effects in the “island of inversion” nuclei 28,29,31 F, Phys . Rev. C 104 , 014307 ( 2021 ). doi: 10.1103/PhysRevC.104.014307 http://doi.org/10.1103/PhysRevC.104.014307 Role of quadrupole deformation and continuum effects in the “island of inversion” nuclei 28,29,31F, Phys K. Bennaceur , F. Nowacki , J. Okołowicz et al. , Analysis of the 16 O (p, γ ) 17 F capture reaction using the shell model embedded in the continuum, Nucl . Phys. A 671 , 203 - 232 ( 2000 ). doi: 10.1016/S0375-9474(99)00851-9 http://doi.org/10.1016/S0375-9474(99)00851-9 Analysis of the 16O (p, γ) 17F capture reaction using the shell model embedded in the continuum, Nucl G. Cattapan , E. Maglione , From bound states to resonances: Analytic continuation of the wave function . Phys. Rev. C 61 , 067301 ( 2000 ). doi: 10.1103/PhysRevC.61.067301 http://doi.org/10.1103/PhysRevC.61.067301 From bound states to resonances: Analytic continuation of the wave function All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Si-Zhe Xu, Shi-Sheng Zhang, Xiao-Qian Jiang and Michael Scott Smith. The first draft of the manuscript was written by Si-Zhe Xu, Shi-Sheng Zhang, Michael Scott Smith and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. The authors would like to express their gratitude for the valuable discussions with Jian-You Guo. 10.1007/s41365-022-01159-y This research was partially supported by the National Natural Science Foundation of China (Nos. 12175010, 11375022, 11775014, 11975237), Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDB34010000), and U.S. Department of Energy Office of Science, Office of Nuclear Physics, under Award Number DE-AC05-00OR22725. 2022-07-12 2022-11-15 2022-11-18 2023-01-09 2023-02-20 © 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/s41365-022-01159-y http://dx.doi.org/10.1007/s41365-022-01159-y . 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