Formalism

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]. A more general formalism can be found in Ref. [30]. A Hamiltonian of the form $\widehat{H}=\widehat{T}+\widehat{V}\mathrm{,}$ (1) is assumed, where $\widehat{T}$ is the kinetic energy and $\widehat{V}$ is the nuclear potential with a Woods-Saxon shape. The foundation of the CMR approach [9] describes the Schrödinger equation with this Hamiltonian in momentum, rather than coordinate, is represented as: ${\displaystyle \int }{d}^3{\overrightarrow{k}}^{\text{'}}\langle \overrightarrow{k}\left|\widehat{H}\right|{\overrightarrow{k}}^{\text{'}}\rangle {\Phi }_n({\overrightarrow{k}}^{\text{'}})=E_n{\Phi }_n(\overrightarrow{k})\mathrm{.}$ (2) Here, $\overrightarrow{k}=\overrightarrow{p}/\hslash$ is the wave vector, ${\Phi }_n(\overrightarrow{k})$ is the wave function, and En is the energy of the nth state. If we assume spherical symmetry, the wave function can be stated as follows: ${\Phi }_n(\overrightarrow{k})={\varphi }_n(k)Y_l^m({\Omega }_k)$ (3) where ϕn(k) is the radial portion of the momentum wavefunction and $Y_l^m({\Omega }_k)$ is the angular portion with angular momentum quantum numbers l and m. The Schrodinger equation then becomes: $\frac{{\hslash }^2{k}^2}{2m}{\varphi }_n(k)+{\displaystyle \int }d{k}^{\text{'}}{k}^{\text{'}2}{V}_l(k\mathrm{,}{k}^{\text{'}}){\varphi }_n(k^{\text{'}})=E_n{\varphi }_n(k)\mathrm{,}$ (4) with $V_l(k\mathrm{,}{k}^{\text{'}})=\frac{2}{\pi }{\displaystyle \int }dr{r}^{2}V(r)j_l(kr)j_l(k^{\text{'}}r)\mathrm{,}$ (5) where jl(kr) denotes the spherical Bessel function of lth 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 En is discussed in Sect. (2c).

Potential parameters

In Ref. [9], the CMR method was used to calculate the low-lying resonances in ¹⁷O with a generic Woods-Saxon potential described in Ref. [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 [3,4], cannot be found in that research.

Therefore, we adopted the potential based on that presented in Ref. [3] to revisit the bound and resonant structures in ¹⁷O. 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, $V_{\text{C}}(r)=V_0{f}_{\text{C}}(r)\mathrm{,}$ (6) $V_{\text{SO}}(r)=-V_{\text{SO}}^0{(\frac{\hslash }{m_{\pi }c})}^2(\overrightarrow{l}\cdot \overrightarrow{s})\frac{1}{r}\frac{\text{d}{f}_{\text{SO}}(r)}{\text{d}r}\mathrm{,}$ (7) $f_i(r)=\frac{1}{1+\exp ((r-R_i)/a_i)}\mathrm{,}(i\text{refers}\text{to}\text{C}\text{or}\text{SO})\mathrm{.}$ (8) Here, the pion Compton wavelength $\hslash /m_{\pi }c=1.414$ fm and potential depth V₀ for the central part and $V_{\text{SO}}^0$ for the spin-orbit part were adjusted to reproduce the one-neutron separation energy Sn= 4.143 MeV for ¹⁷O. The diffuseness and radius parameters are obtained from the Koning-Delaroche optical model potential [32]. The six parameters of the Woods-Saxon potential are listed in Table 1.

In the case of exotic neutron-rich fluorine isotopes ^{29, 31}F, a standard spherical potential [31] was adopted, which was also utilized in a recent study focusing on this nucleus using a scattering phase shift method [28]. By choosing this potential, a direct comparison can be made between the CMR method and scattering phase-shift approach.

Integration method

The integral stated in Eq. (4) requires discretization to readily obtain the eigenvalue level energies En. The integral over the momentum space can be approximated as a sum over a finite Nq set of points kj with spacing dk and weights wj. The result is the following Nq× Nq matrix equation: ${\displaystyle \sum _{j=1}^{N_q}}{H}_{i\mathrm{,}j}{\varphi }_n(k_j)=E_n{\varphi }_n(k_i)\mathrm{,}(i=\mathrm{1,2,}\cdots \mathrm{,}{N}_q)\mathrm{,}$ (9) $H_{i\mathrm{,}j}=\frac{{\hslash }^2{k}_i^2}{2m}{\delta }_{i\mathrm{,}j}+w_j{k}_j^2{V}_l^{i\mathrm{,}j}$ (10) $V_l^{i\mathrm{,}j}=\frac{2}{\pi }{\displaystyle \int }dr{r}^{2}V(r)j_l(k_{i}r)j_l(k_{j}r)\mathrm{.}$ (11) Using the following transformations: ${\varphi }_n^{\text{'}}(k_i)=\sqrt{w_i}{k}_i{\varphi }_n(k_i)\mathrm{,}{H}_{i\mathrm{,}j}^{\text{'}}=\sqrt{\frac{w_i}{w_j}}\frac{k_i}{k_j}{H}_{i\mathrm{,}j}\mathrm{,}$ (12) the system for the nth state is symmetrized as follows: ${\displaystyle \sum _{j=1}^{N_q}}{H}_{i\mathrm{,}j}^{\text{'}}{\varphi }_n^{\text{'}}(k_j)=E_n{\varphi }_n^{\text{'}}(k_i)\mathrm{,}(i=\mathrm{1,2,}\cdots \mathrm{,}{N}_q)\mathrm{,}$ (13) $H_{i\mathrm{,}j}^{\text{'}}=\frac{{\hslash }^2{k}_i^2}{2m}{\delta }_{i\mathrm{,}j}+\sqrt{w_i{w}_j}{k}_i{k}_j{V}_l^{i\mathrm{,}j}\mathrm{,}$ (14) $V_l^{i\mathrm{,}j}=\frac{2}{\pi }{\displaystyle \int }dr{r}^{2}V(r)j_l(k_{i}r)j_l(k_{j}r)\mathrm{.}$ (15) where ki,j denote the integral points along a contour in the momentum space with corresponding weights wi,j; the potential is integrated over the coordinate space. It is known (see, e.g., Refs. [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 ki, 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 En. Similarly, we apply the Gauss-Legendre quadrature to the coordinate-space integral stated in Eq. (15). In Ref. [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

In coordinate space, we study the convergence and accuracy of the potential integral Vi,j presented in Eq. (15) with variations in the number of grid points Np 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 ki = (0.3, -0.075) and kj = (0.4, 0), where the units of the real and imaginary components of the momentum grid points are given in fm^-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 Np≥90 and $r_{\text{cut}}\ge 30$ fm. To guarantee convergence with this accuracy, we utilize Np = 100 and r_cut = 40 fm in our subsequent coordinate-space potential integral calculations.

Fig. 1Full-screen View

(Color online) The accuracy variation in the coordinate-space potential integrals Vi,j (in MeV) with the number of grid points Np and cut-off radius r_cut at ki = (0.3, -0.075), kj = (0.4, 0).

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 Nq 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 Nq, such that the total number of grid points for the entire path is 3Nq. The positions of the complex momentum for the resonant states do not change with the different paths of the contour.

Figure 2 demonstrates an example of the convergence of the momentum-space integral for the 1d_3/2 resonant orbital in ¹⁷O. An accuracy of 10^-8 MeV can be achieved when Nq≥40 and $k_{\text{cut}}\ge 7.5{\text{fm}}^{-1}$. To guarantee convergence with such accuracy, we utilize Nq = 100 and $k_{\text{cut}}=10{\text{fm}}^{-1}$ 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].

Fig. 2Full-screen View

(Color online) The accuracy variation in the energy and decay width (in MeV) for the 1d_3/2 single particle state in ¹⁷O with the momentum-space number of grid points Nq and cut-off of wave vector k_cut.

Energy levels for bound and resonant states in ¹⁷O

The CMR approach described above was used to solve for the bound and resonant states in ¹⁷O. The contour for the momentum integration, as shown in 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 ¹⁷O bound states (indicated with the blue open squares in 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.

Fig. 3Full-screen View

The single neutron spectra for ¹⁷O in complex momentum plane. The blue open squares, red open triangles, and black open circles represent the bound states, resonant states, and continuum states, respectively. The momentum integration contour is defined by the points (0, 0), (0.25, -0.5), (0.5, 0), and (10, 0).

The CMR-based solutions for the five bound states and two resonant states of ¹⁷O are listed in Table 2. Experimental data from Refs. [5, 36-38] are presented for comparison. Additionally, the Numerov numerical method (see, e.g., Ref. [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]). The results of the Numerov approach for the five bound ¹⁷O levels using the potential described in Sect. 2.2 are reported in 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.

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 $E_{\text{R}}^{\text{Exp}}=0.944\pm 0.001$ MeV [5]; the width is within 0.06 MeV of the measured value of ${\Gamma }^{\text{Exp}}=0.088\pm 0.003$ MeV [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 $E_{\text{R}}^{\text{Exp}}=0.411$ MeV [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. [9,41]). The measured 3/2^- orbital may originate from ¹⁶O core excitation [6,42], which should be treated beyond the mean-field framework. In Ref. [42], the Shell Model Embedded in the Continuum (SMEC) was used to study the spectroscopy of mirror nuclei: ¹⁷O and ¹⁷F, 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 ¹⁷O are approximately 0.5 MeV larger than the experimental data with widths of 40 keV. As mentioned in Ref. [6], the potential model does not include particle-hole excitation in the ¹⁶O 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 ²⁹F and ³¹F

The exotic nuclei ²⁹F and ³¹F, which are candidates for a two-neutron Borromean halo structure [22], were studied using the same CMR approach as that used for ¹⁷O. As the separation energy has not been measured in this nucleus, a standard set [31] of potential parameters is used for ^{29, 31}F, which is the same as that used in Ref. [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 ²⁹F and ³¹F, respectively, are shown in Fig. 4. Owing to the lack of experimental measurements regarding the excited states of this nucleus, a comparison is made in Fig. 4 between our CMR-based predictions and those obtained using the scattering phase-shift method described in Ref. [28]. The energies for the bound orbitals and unbound 1f_7/2 orbital in ²⁹F agree well with those stated in Ref. [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 ²⁹F is only 0.582 MeV above the neutron threshold; a solution for this level was not presented in Ref. [28]. In the spherical case of ²⁹F, the predicted 2p_3/2 lies above 1d_3/2, which is consistent with conditions B and C described in Ref. [21]. From the calculated spectrum in ³¹F, we can determine that the last two neutrons occupy 2p_3/2, which might cause the halo formation in ³¹F. As was the case for predicting the width of the 2p_3/2 resonance orbital in ¹⁷O. The prediction of the decay width of the 2p_3/2 orbital in ²⁹F (³⁰F) - 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.

Fig. 4Full-screen View

Levels in ²⁹F and ³¹F around the threshold with spherical assumption. Levels on the left for ²⁹F are taken from Ref. [28]. Our predictions are shown on the right, in which orbitals 2p_3/2 are plotted in red.