The first excited single-proton resonance in 15F by complex-scaled Green's function method

  • The complex-scaled Green's function (CGF) method is employed to explore the single-proton resonance in 15F. Special attention is paid to the first excited resonant state 5/2+, which has been widely studied in both theory and experiments. However, past studies generally overestimated the width of the 5/2+ state. The predicted energy and width of the first excited resonant state 5/2+ by the CGF method are both in good agreement with the experimental value and close to Fortune's new estimation. Furthermore, the influence of the potential parameters and quadruple deformation effects on the resonant states are investigated in detail, which is helpful to the study of the shell structure evolution.
  • The nuclear tensor force is one of the most important components of the bare nucleon-nucleon interaction [1, 2]. In recent decades, the effects of the tensor force in nuclear mediums have also been intensively investigated, after being neglected for a long time. It has been shown that the tensor force plays a crucial role in shell-structure evolution [3-25], spin-isospin excitations [26-31], and giant resonances [32-34]. In spite of such achievements, questions remain about the properties of the in-medium tensor force, i.e., the effective tensor force. Among the most intricate challenges is the problem surrounding the constraint of the strength of the tensor force [2]. To achieve this goal, one needs to determine the observables that are sensitive to the effective tensor force [2, 8, 35-37]. By fitting these observables based on certain many-body theories, one can expect to pin down the sign and strength of the tensor force.

    The nuclear density functional theory (DFT) [38-43] is currently the only candidate that can be applied to almost the whole nuclear chart, except for very light nuclei. Within both nonrelativistic and relativistic DFT, the parameters of the effective interactions are usually determined by fitting to the bulk nuclear properties, such as the masses and radii of finite nuclei, as well as the empirical knowledge of the infinite nuclear matter. However, these bulk properties are found to be, in general, not sensitive to the tensor components in effective interactions. In particular, by adding the Fock term of the pion exchange, which is one of the most important carriers of the tensor force in the relativistic nuclear forces, on top of the conventional relativistic mean-field (RMF) theory, Lalazissis et al. [44] found that the bulk properties of spherical finite nuclei and infinite nuclear matter disfavor the tensor force, i.e., the optimal fit is achieved for the vanishing pion field.

    Moreover, the tensor force has the characteristic property of spin dependence. It can significantly affect the shell structure of nuclei, especially those located far away from the stability line [4]. One of the most famous benchmarks is the evolution of the energy difference between the proton states 1h11/2 and 1g7/2 in the Sn (Z=50) isotopes and that between the neutron states 1i13/2 and 1h9/2 in the N=82 isotones [45]. Based on the Skyrme Hartree-Fock (SHF) [46], Gogny Hartree-Fock (GHF) [6], and the relativistic Hartree-Fock (RHF) [9] theories, it was found that the tensor force plays a crucial role in reproducing the empirical trend of the shell structure mentioned above. Accordingly, by reproducing the shell-structure evolution, one can expect to calibrate the strength of the effective tensor force.

    Nevertheless, the single-particle states observed experimentally are usually fragmented [47-50] due to, for example, the coupling with low-lying vibrations, which is related to the quenching of the spectroscopic factors. The distraction arising from the beyond-mean-field correlations makes it ambiguous to directly compare the single-particle energies calculated by DFT with the corresponding experimental data. A possible solution to eliminate this kind of distraction is to take into account the particle-vibration coupling (PVC) in the theoretical calculations [51-54]. By doing so, the descriptions of the energies, as well as the wave functions, can be improved, although the fragmentation of the single-particle states may remain a problem.

    Another option to avoid the distraction of the beyond-mean-field correlations is to seek for the ab initio calculations which can serve the meta-data, instead of the experimental data. In the last decade, the ab initio calculations have progressed greatly [55-58]. In particular, Shen et al. have established the self-consistent relativistic Brueckner-Hartree-Fock (RBHF) theory for the finite nuclei and achieved a much better agreement with the experimental data employing only the two-body interaction [59-61], in contrast to the previous nonrelativistic Brueckner-Hartree-Fock calculations. Like other ab initio calculations, the RBHF calculation is computationally consuming for heavy and even medium-mass nuclei.

    In a neutron drop, a collection of neutrons confined by an external field, for example, a harmonic trap, only the neutron-neutron interaction exists, and the equations are easier to be solved compared with real finite nuclei. It thus draws great attention [62-68] and provides an ideal platform to link the ab initio and DFT calculations [69-75]. More importantly, both the single-particle energies calculated by the RBHF theory and those calculated by the DFT are quantities on the pure mean-field level, which ensures that one can make a fair comparison between the two.

    Great successes have been achieved in nuclear physics with the nuclear covariant density functional theory (CDFT) [39-42]. As a branch of CDFT, the RHF theory shares the common advantages of it [76-82]. In addition, the RHF theory can take into account the tensor force via the Fock term without extra free parameters [9, 13, 78, 83-88]. In particular, the quantitative analysis of tensor-force effects in the RHF theory was recently performed [89]. According to the famous mechanism revealed by Otsuka et al. [4], spin-orbit (SO) splittings are sensitive to the tensor force. Taking the SO splittings calculated by the RBHF theory as meta-data, the strength of the tensor force was explored in the RHF theory [68, 74, 90, 91]. Nevertheless, the contributions of the tensor force were not quantitatively extracted in these works. In the present work, we will first quantitatively verify the tensor-force effects on the SO splittings in neutron drops within the RHF theory. In addition, the strength of the tensor force will be further explored. Motivated by the idea of renormalization persistency of the tensor force [92, 93], particular attention will be paid to the density dependence of the tensor force in the nuclear medium.

    This paper is organized as follows. In Section II, the RHF theory and the method to evaluate the tensor force contributions are briefly introduced. In Section III, we clarify the tensor-force effects on the evolution of the SO splittings in neutron drops and then further explore the strength of the tensor force. A summary is provided in Section IV.

    In the CDFT, the nucleons are considered to interact with each other by exchanging various mesons and photons [39-42, 94-98]. Starting from the ansatz of a standard Lagrangian density, which contains the degrees of freedom associated with the nucleon field, the meson fields, and the photon field, one can derive the corresponding Hamiltonian as

    H=d3xˉψ(x)[iγ+M]ψ(x)+12ϕd3xd4yˉψ(x)ˉψ(y)Γϕ(x,y)Dϕ(x,y)×ψ(y)ψ(x),

    (1)

    where ψ is the nucleon-field operator, and ϕ denotes the meson-nucleon couplings, including the Lorentz σ-scalar (σ-S), ω-vector (ω-V), ρ-vector (ρ-V), ρ-tensor (ρ-T), ρ-vector-tensor (ρ-VT), and π-pseudovector (π-PV) couplings, as well as the photon-vector (A-V) coupling. Here, Γϕ(x,y) and Dϕ(x,y) are the interaction vertex and the propagator of a given meson-nucleon coupling ϕ, respectively; their explicit expressions can be found in Refs. [78, 83, 84, 99-103]. Evidently, the photon field is not considered in the case of neutron drops.

    The nucleon-field operators, ψ(x) and ψ(x), can be expanded on a set of creation and annihilation operators defined by a complete set of Dirac spinors {φα(r)}, where r denotes the spatial coordinate of x. In this work, the spherical symmetry is assumed. Then, the energy density functional can be obtained through the expectation value of the Hamiltonian on the trial Hartree-Fock state under the no-sea approximation [94]. Variations of the energy density functional with respect to the single-particle wave functions give the Dirac equations,

    drˆh(r,r)φ(r)=εφ(r),

    (2)

    where ˆh(r,r) is the single-particle Hamiltonian. In the RHF theory, ˆh(r,r) contains the kinetic energy ˆhK, the direct local potential ˆhD, and the exchange nonlocal potential ˆhE; see Refs. [79, 83, 102, 104] for detailed expressions. Notice that the tensor force contributes only to the nonlocal potentials.

    For the RHF theory with density-dependent effective interactions, the meson-nucleon coupling strengths are taken as functions of the baryonic density ρb. For convenience, here we explicitly present the density dependence of the π-PV coupling, which reads

    fπ(ρb)=fπ(0)eaπξ,

    (3)

    where ξ=ρb/ρsat. with the saturation density of the nuclear matter ρsat., and fπ(0) corresponds to the coupling strength at zero density. The coefficient aπ determines how fast the coupling strength fπ decreases with the increasing density. The density dependence of the other meson-nucleon couplings can be found in Refs. [78, 84].

    The external field to keep neutron drops bound is chosen as a harmonic oscillator (HO) potential as

    Uh.o.(r)=12Mω2r2,

    (4)

    with ω=10MeV. It is worth noticing that the choice of the external field here is not completely arbitrary, but is optimal, as specifically discussed in Ref. [91].

    In Ref. [89], the tensor force in each meson-nucleon coupling was identified through the nonrelativistic reduction. They can be expressed uniformly as

    ˆVtϕ=1m2ϕ+q2FϕS12,

    (5)

    where mϕ is the meson mass, q is the momentum transfer, and S12 is the operator of the tensor force in the momentum space, which reads

    S12(σ1q)(σ2q)13(σ1σ2)q2.

    (6)

    The coefficient Fϕ associated with a given meson-nucleon coupling reflects the sign and the rough strength of the tensor force, as listed in Table 1 of Ref. [89].

    The method to quantitatively evaluate the contributions of the tensor force was also established in Ref. [89]. Using this method, one can first calculate the tensor-force contributions to the two-body interaction matrix elements; the explicit formulae are referred to Appendix C in Ref. [89]. Then, the contributions of the tensor force to the nonlocal potential can be obtained, and eventually, its contributions to the single-particle energies are quantitatively extracted.

    First, we calculate the binding energies and radii of neutron drops with the neutron number N from 4 to 50 using the RHF theory with the effective interaction PKO1 [78], as shown in Table 1. The effective interaction PKO1 is the first widely used RHF functional with density-dependent meson-nucleon coupling strengths. It has produced good descriptions for finite nuclei and nuclear matter, comparable with those of the RMF functionals. In particular, the tensor force is explicitly taken into account in PKO1, owing to the inclusion of the Fock terms of the relevant meson-nucleon couplings. The comparison between the energies and the radii of neutron drops given by the RBHF calculation with the interaction Bonn A and those by the RHF calculation with PKO1 has been presented in Ref. [68].

    Table 1

    Table 1.  Binding energies and radii of neutron drops calculated by the RHF theory with the effective interaction PKO1 [78], compared with the results without the tensor force (denoted by PKO1n.t.). See the text for more details.
    N E /MeV r /fm
    PKO1 PKO1n.t. PKO1 PKO1n.t.
    4 60.679 60.518 2.527 2.526
    6 93.123 92.629 2.589 2.593
    8 125.681 125.680 2.703 2.703
    10 175.632 175.518 2.836 2.836
    12 223.301 222.924 2.921 2.923
    14 268.870 268.180 2.981 2.987
    16 316.018 315.445 3.067 3.074
    18 365.329 365.206 3.150 3.151
    20 414.085 414.084 3.216 3.216
    22 477.173 477.112 3.281 3.281
    24 539.286 539.075 3.336 3.337
    26 600.468 600.065 3.383 3.386
    28 660.757 660.163 3.425 3.430
    30 725.978 725.415 3.485 3.491
    32 790.552 790.090 3.538 3.546
    34 858.068 857.635 3.594 3.598
    36 925.387 924.758 3.644 3.646
    38 992.335 992.686 3.692 3.692
    40 1059.237 1059.236 3.733 3.733
    42 1137.216 1137.188 3.770 3.770
    44 1214.829 1214.734 3.805 3.805
    46 1292.080 1291.902 3.837 3.838
    48 1368.977 1368.722 3.867 3.869
    50 1445.527 1445.220 3.895 3.899
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    Here, we also calculate the energies and radii of neutron drops with PKO1 without the tensor force, i.e., the contribution of the tensor force to the nonlocal mean field is excluded in each step of the iteration before convergence is reached, using the method proposed in Ref. [89]. The results are shown in Table 1 and compared with the results of the full calculation with PKO1. It is evident that the results with and without the tensor force are close together, which means that the tensor-force contributions to the energies and radii of neutron drops are negligible. In particular, the contributions of the tensor force almost vanish in the neutron drops with spin saturation, namely N=8, 20, and 40. This is in agreement with the common understanding of the properties of the tensor force [2, 4].

    The single-particle energies of neutron drops as a function of the neutron number N, calculated by the RBHF theory using the interaction Bonn A, are shown in Fig. 10 in Ref. [68]. In general, the single-particle energies calculated by the RHF theory (which are not explicitly shown here for simplicity) reproduce the results using the RBHF theory well for states near the Fermi energy. However, for those far away from the Fermi energy, the differences between the results of the two models are remarkable. It is known that the value of single-particle energy is determined by various components of the nuclear force, such as the central force and the SO force. In contrast, the evolution of SO splittings is mainly determined by the tensor force and largely free from the other components [2]. Thus, the evolution of SO splittings, rather than the single-particle energies themselves, should be adopted as a benchmark for the tensor force. As a typical representative, the relative change of SO splittings from the neutron-proton drop 4020 (Z=20, N=20) to 4820 one (Z=20, N=28), calculated by the RBHF theory using the Bonn A interaction, is employed to determine the tensor term in the Skyrme interactions [21].

    Displayed in Fig. 1 are the SO splittings of doublets 1p, 1d, 1f, and 2p in neutron drops, calculated using both the RMF and RHF theories. The results of the RBHF theory obtained using the Bonn A interaction [74] are also shown for comparison, serving as the meta-data. One can see that the meta-data present a nontrivial pattern: the SO splittings vary monotonously and even linearly between the neighboring (sub)shells N=8, 14, 20, 28, 40, and 50. Such a feature arises from the characteristic spin-dependent properties of the tensor force, as pointed out in Ref. [74].

    Figure 1

    Figure 1.  (color online) From top to bottom, the SO splittings of doublets 1p, 1d, 1f, and 2p in neutron drops. The results are calculated using the RHF theory with the effective interactions PKO1 [78], PKO2 [9], and PKA1 [84], as well as using the RMF theory with PKDD [105]. The results of RBHF [74] with the Bonn A interaction are shown as meta-data. The contributions of the HO potential in the RMF calculation with PKDD are also presented, denoted as “HO (PKDD).”

    For the calculation with PKDD [105], which is an RMF functional with density-dependent meson-nucleon coupling strengths, the results are evidently far away from the meta-data. This is mainly because of the absence of the explicit tensor force in the framework of RMF [9, 74], due to the lack of the Fock term. As a typical representative of the RHF effective interactions, PKO1 [78] reproduces the pattern of meta-data qualitatively, which is attributed to the tensor force in the π-PV coupling [74]. Interestingly, even though the tensor force is explicitly involved in PKO2 [9] and PKA1 [84], their results obviously deviate from the meta-data. In fact, the results of PKO2 and PKA1 are similar to that of PKDD rather than PKO1. To understand this phenomenon, one needs to examine the details of the tensor force arising from each meson-nucleon coupling, which are identified in Ref. [89].

    Among all the meson-nucleon couplings involved in the current RHF effective interactions, only the π-PV coupling produces a tensor force that matches the properties of spin dependence revealed in Ref. [4], i.e., it is repulsive (attractive) when the two interacting nucleons are parallel (antiparallel) in their spin states. The tensor forces in the other couplings are opposite to that in the π-PV coupling in sign, as shown in Table 2 of Ref. [89]. Meanwhile, it has been shown that the tensor force in the π-PV coupling dominates those in the other couplings. Thus, for PKO2, where the π-PV coupling is absent, the tensor force mainly comes from the ω-V coupling. This means that the sign of the net tensor-force contribution in PKO2 is opposite to that in PKO1. That is why PKO2 cannot reproduce the pattern given by RBHF, even qualitatively. PKA1 contains not only all the meson-nucleon couplings involved in the PKO series but also the ρ-T and ρ-VT couplings, which create tensor forces with considerable strengths. As mentioned above, the tensor-force contributions arising from these two couplings, partially cancel the corresponding contributions from the π-PV coupling. Therefore, PKA1 gives worse description of the meta-data than PKO1 does.

    It is noticeable that the RMF effective interaction PKDD also presents some kinks. To elucidate where these kinks arise from, we calculate the contributions of the external HO potential, denoted as “HO (PKDD)” in Fig. 1. By comparing the results of PKDD and “HO (PKDD),” one can find that the kinks given by PKDD are determined by the external HO potential. This also explains why the results given by different RMF effective interactions are so similar to each other, as shown in Fig. 2 of Ref. [74]. Definitely, the external potential can also affect the shell structure given by the RHF effective interactions.

    It has been shown that the meta-data can be better reproduced by PKO1 when the coupling strength of π-PV is enhanced properly [68, 74]. Actually, the RHF effective interaction PKO3 [9], which contains the same kinds of meson-nucleon couplings as PKO1 but slightly stronger π-PV coupling strength, can also provide a comparable description of the meta-data. According to the previous works and the discussion above, it can be shown that the π-PV coupling, essentially the embedded tensor force, plays a crucial role in determining the evolutionary trend of the SO splittings in neutron drops. Nevertheless, all these analyses regarding the tensor-force effects are intuitive to some extent, while the quantitative investigation is missing.

    Since the tensor forces arise from only the exchange terms of the relevant meson-nucleon couplings, we first calculate the contributions of the exchange term of each meson-nucleon coupling to the SO splittings of 1p and 1d doublets. The results are shown in Fig. 2. It can be seen that the evolution of the SO splittings calculated by PKO1 (black filled circles) is mainly determined by the contributions of the exchange terms (red triangles with cross). When the contribution of the exchange terms is excluded, i.e., simply subtracted from the results of the full calculation, the trend (black open circles) becomes almost the same as that given by PKDD shown in Fig. 1. Here, we remind that PKDD is an RMF effective interaction, which does not contain the exchange terms. Remarkably, the pattern given by the exchange term of the π-PV coupling (blue open stars) is almost the same as that by the total exchange term. The combined contributions of the exchange terms of the σ-S and ω-V couplings (green open triangles) are also considerable; however, in general, they are not considerably decisive for the kinks as those of the π-PV coupling. The contributions of the ρ-V coupling are negligible because of the small coupling strength.

    Figure 2

    Figure 2.  (color online) Contributions of the exchange terms to the SO splittings of 1p and 1d doublets in neutron drops, calculated by the RHF theory with the effective interaction PKO1. For comparison, the total SO splittings, the contributions of the tensor force from the π-PV coupling, and the results without the contribution of the Fock terms are also shown. See the text for more details.

    It is well known that the π-PV coupling contains not only the tensor force but also central components. Thus, it is of particular significance to quantitatively evaluate the contributions of the tensor force from the π-PV coupling. We calculate the tensor-force contributions using the method developed in Ref. [89]. The results are also shown in Fig. 2, denoted by the blue filled stars, which are almost hidden behind the blue open stars. One can find that the contributions of the π-PV coupling are almost totally determined by the tensor force, whereas the role of the central force is negligible. In other words, the π-PV coupling affects the evolutionary trend almost fully through the embedded tensor force. We thus certify quantitatively that it is reasonable to explore the tensor-force effects by varying the π-PV coupling in previous works.

    It is notable that, in both the RHF and RBHF calculations, the filling approximation is adopted, i.e., the last occupied levels are partially occupied with equal probabilities for the degenerate states. Such an approximation may affect the levels which are not fully occupied but does not affect the neutron drops with closed (sub)shells. If we consider only the neutron drops with closed (sub)shells, i.e., those with N=8, 14, 20, 28, 40, and 50, we can avoid the distraction from the filling approximation. Here, we define the slope of the SO splittings in neutron drops with respect to the neutron numbers as

    LN1N2=ΔEs.o.(N2)ΔEs.o.(N1)N2N1,

    (7)

    where ΔEs.o.(N) is the SO splitting in the neutron drop with the neutron number N. For the neutron drops with closed (sub)shells, the combination of (N1-N2) has the following choices: (8-14), (14-20), (20-28), (28-40), and (40-50). Following the strategy proposed in Ref. [74], i.e., multiplying the fπ(0) in PKO1 by a factor λ (λ>1.0), we calculate once again the SO splittings in neutron drops. Then, we obtain the slopes defined above for different SO doublets and the root-mean-square (rms) deviations (denoted by Δ) with respect to the RBHF results as

    Δ=i(LRHFiLRBHFi)210,

    (8)

    where LRHFi (LRBHFi) is the slope calculated by the RHF (RBHF) theory, and i runs over all the possible combinations of (N1-N2) for different SO doublets. The results are presented in the upper half of Table 2. It can be seen that λ=1.42 gives the smallest Δ among the values of λ, which is 0.0362MeV. This, in general, agrees with the conclusion in Ref. [74].

    Table 2

    Table 2.  The rms deviations Δ (in the unit of MeV) of the slope of the SO splittings between the neighboring (sub)shells. The upper panel gives the results of PKO1 with fπ(0) multiplied by a factor λ (λ>1.0); the lower panel gives the results of similar calculations but with aπ multiplied by a factor η (0.0<η<1.0). The optimal values of λ and η as well as the corresponding Δ are shown in bold font. See the text for more details.
    fπλfπ λ 1.0 1.1 1.2 1.3 1.4 1.42 1.5 1.6
    Δ 0.0909 0.0758 0.0601 0.0456 0.0366 0.0362 0.0402 0.0561
    aπηaπ η 0.7 0.6 0.5 0.4 0.33 0.3 0.2 0.1
    Δ 0.0645 0.0543 0.0439 0.0356 0.0334 0.0341 0.0441 0.0643
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    To present a clearer comparison, we consider the results of λ=1.0, 1.2, 1.4, and 1.6 as examples and display them in Fig. 3. One can find that the results of λ=1.2 are slightly closer to the meta-data compared with the original PKO1. The meta-data can be reproduced better when λ=1.4, which is quite close to the optimal value of 1.42. If λ becomes significantly larger, the results get worse in general. Accordingly, it can be seen that the results of λ=1.6 evidently deviate from the meta-data.

    Figure 3

    Figure 3.  (color online) From top to bottom, the SO splittings of doublets 1p, 1d, 1f, and 2p in neutron drops. The calculations are performed by the RHF theory with the effective interaction PKO1, of which the π-PV coupling strength fπ(0) is multiplied by a factor λ (λ=1.0, 1.2, 1.4, and 1.6). The results of RBHF obtained using the Bonn A interaction are also shown for comparison.

    In addition to the strength of the tensor force, the properties of the tensor force in nuclear medium is also of great interests and still under discussion [2]. It has been argued that the bare tensor force does not undergo significant renormalization in the medium, which is denoted as the tensor renormalization persistency [92, 93]. In other words, the effective tensor force would be similar to the bare one. If this is true, one could also anticipate to verify such a property in the framework of DFT. Within the RHF theory with density-dependent effective interactions, the renormalization can be, to a large extent, reflected by the density dependence of the coupling strengths. Minor renormalization can naturally manifest as weak density dependence. For the π-PV coupling, which serves as the main carrier of the tensor force, tensor renormalization persistency requires that the coefficient of density dependence aπ should be small, as indicated by Eq. (3). In our previous work [102], we have shown that weakening the density dependence of the π-PV coupling, i.e., reducing aπ, can improve the description of the shell-structure evolution in the N=82 isotones and the Z=50 isotopes more efficiently, compared with enlarging fπ(0) with a factor. Inevitably, the beyond-mean-field correlations in the experimental single-particle energies make the comparison with the results on the mean-field level ambiguous. With the meta-data given by the RBHF theory, which are also on the pure mean-field level, we can further explore the tensor renormalization persistency in a more convincing way.

    Aiming at this goal, we recalculate the SO splittings in neutron drops with the RHF theory using PKO1, but the coefficient of density dependence aπ is multiplied by a factor η (0.0<η<1.0). We present in the lower half of Table 2 the rms deviations of the slope of the SO splittings between the neighboring (sub)shells. One can see that the smallest deviation is obtained when η=0.33, which is 0.0334MeV. We also find that the minimum value of Δ for modification with η is smaller than that with λ, which means that the modification with η is more adequate. Thus, one can conclude that weakening the density dependence is an available and efficient way to improve the description of the evolution of SO splittings. It should be stressed that this conclusion is reached by the comparison between the CDFT calculation and ab initio one, both of which belong to the pure mean-field level. Even though we did not perform a complete refitting procedure yet, we indeed provided a semiquantitative support for the renormalization persistency of the tensor force.

    To illustrate the discussion above more clearly, the results with η=0.0, 0.3, 0.6, and 0.9 are considered as examples and shown in Fig. 4, in comparison with the RBHF results. It can be seen that a smaller aπ, which means weaker density dependence, gives a better description of the evolution of SO splittings. When η=0.3, which is quite close to the optimal value of 0.33, the meta-data are reproduced quite well. If η is too small, the results become visibly worse. Eventually, when η=0.0, the deviation from the meta-data is much more significant.

    Figure 4

    Figure 4.  (color online) Similar to Fig. 3, but aπ is multiplied by a factor η (η=0.9, 0.6, 0.3, and 0.0).

    In this work, we have investigated the tensor-force effects on the evolution of SO splittings in neutron drops within the framework of the RHF theory. The corresponding results of the RBHF calculation with the Bonn A interaction were adopted as meta-data. Since the results of the RHF theory and the meta-data calculated by the RBHF theory are both within the pure mean-field level, fair comparisons can be made between them. Through a qualitative analysis of the results using the RHF effective interactions PKO1, PKO2, and PKA1, we confirmed that the tensor force in the effective interactions plays a crucial role in reproducing the meta-data. Meanwhile, for the RMF effective interactions, it was found that the evolution of SO splittings is mainly determined by the external HO potential. Moreover, we found that the contributions from the exchange terms almost fully determine the evolution of SO splittings. Among all the meson-nucleon couplings, the exchange term of π-PV coupling plays the dominant role. In particular, the tensor-force contribution was extracted, and it was found that the π-PV coupling affects the evolutionary trend through the embedded tensor force, while its central force has almost invisible effects. This conclusion verifies quantitatively that it is reasonable to explore the tensor-force effects by varying the π-PV coupling strength. In other words, the evolution of SO splittings belongs to the observables that can constrain the strength of the tensor force.

    The strength of the tensor force was explored by enlarging fπ in two different ways: (i) multiplying a factor λ (λ>1.0) as a whole and (ii) weakening the density dependence by multiplying a factor η (0.0<η<1.0) for the coefficient aπ. To avoid possible distractions from the filling approximation adopted in the RHF and RBHF calculations, we took into account only the neutron drops with (sub)shell closure and calculated the slopes of the SO splittings between the neighboring (sub)shells. Judging from the rms deviations of the selected slopes, with respect to those calculated from the meta-data, we found that when λ1.4 or η0.3, the meta-data are reproduced best. In particular, weakening the density dependence appears to be slightly better than enlarging fπ with a factor. In this manner, we provide a semiquantitative support for the renormalization persistency of the tensor force. Naturally, in practice, an overall refitting of the parameters in the RHF effective interaction is necessary. Work in this direction is in progress.

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Xin-Xing Shi, Quan Liu, Dong-Dong Ni, Jian-You Guo and Zhong-Zhou Ren. The first excited single-proton resonance in 15F by complex-scaled Green's function method[J]. Chinese Physics C. doi: 10.1088/1674-1137/44/5/054103
Xin-Xing Shi, Quan Liu, Dong-Dong Ni, Jian-You Guo and Zhong-Zhou Ren. The first excited single-proton resonance in 15F by complex-scaled Green's function method[J]. Chinese Physics C.  doi: 10.1088/1674-1137/44/5/054103 shu
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The first excited single-proton resonance in 15F by complex-scaled Green's function method

    Corresponding author: Jian-You Guo, jianyou@ahu.edu.cn
    Corresponding author: Zhong-Zhou Ren, zren@tongji.edu.cn
  • 1. School of Physics, Nanjing University, Nanjing 210093, China
  • 2. School of physics and materials science, Anhui University, Hefei 230601, China
  • 3. Space Science Institute, Macau University of Science and Technology, Macao, China
  • 4. School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
  • 5. Key Laboratory of Advanced Micro-Structure Materials, Ministry of Education, Shanghai 200092, China

Abstract: The complex-scaled Green's function (CGF) method is employed to explore the single-proton resonance in 15F. Special attention is paid to the first excited resonant state 5/2+, which has been widely studied in both theory and experiments. However, past studies generally overestimated the width of the 5/2+ state. The predicted energy and width of the first excited resonant state 5/2+ by the CGF method are both in good agreement with the experimental value and close to Fortune's new estimation. Furthermore, the influence of the potential parameters and quadruple deformation effects on the resonant states are investigated in detail, which is helpful to the study of the shell structure evolution.

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    1.   Introduction
    • The new generation radioactive ion beam facilities enabled the discovery of numerous exotic phenomena near the drip line, such as proton halo [1, 2], neutron halo [3-6], changes of nuclear magicity [7, 8], etc. Considering that the Fermi surface for exotic nuclei is very close to the continuum, the valence nucleons are easily scattered into the continuum. The contribution of the continuum, especially the resonance in the continuum, is particularly important. With the small Coulomb barriers, some of resonances are oriented to be very broad and the ground state (g.s.) of 15F is viewed as a (broad) s-wave resonance [9-11]. However, the first excited state (5/2+) of the unstable 15F is viewed as a narrow resonance with higher Coulomb barriers. Because the Coulomb barrier holds the proton for a long time, the 5/2+ state is very narrow [12]. Moreover, another important feature of 15F is a new resonant state 1/2, which was investigated by the Gamow Shell Model [13]. The very narrow width (36 keV) was pointed out for the first time with high precision. Further steps to explore exotic phenomena in 15F will be quite interesting.

      During the past decades, the proton-rich nuclei became accessible in the proton elastic resonance scattering reaction [14]. Various experiments on the resonances in 15F are in progress. The 20Ne (3He, 8Li) reaction had been used to investigate the unstable nucleus 15F in 1978 [15] and 0.24(3) MeV (the first excited state 5/2+) is close to the value indicated in the analysis (Γ0.2 MeV of 5/2+) of Ref. [16] and our predictions (Er=2.770 MeV and the width Γ=0.239 MeV) for the resonant state 1d5/2. Moreover, the radioactive beams of 14O were usually utilized to populate the resonant states (1/2+, 5/2+) in 15F [17, 18], where the peaks in the curve, that differential cross sections plotted as a function of the scattering angle, are used to present the resonant states.

      Considering the dominant role of the resonant states in thw proton drip line, numerous methods were developed for probing single particle resonances. The developed multi-channel algebraic scattering (MCAS) theory [12] is highly appropriate for narrow resonances. Based on the MCAS theory, three negative parity states (1/2, 5/2, 3/2) in 15F with the widths of only a few keV were found by Canton et al. [12]. Fortune and Sherr were committed to studying the ground and excited states of 15F for decades, by investigation with a potential model [19-21]. The R-matrix theory is usually used in elastic scattering experiments to analyze low-lying resonances [22], where the two lowest resonant states (1/2+ and 5/2+) in 15F were fitted by the R-matrix. The R-matrix theory can also be used to determine spectroscopic properties of the states [23]. The Gamow Shell Model (GCM) is likewise an efficient tool for exploring single-particle resonant states in the proton drip line [13], where the GCM provides relatively narrow resonances with high excitation energies of 15F. Meanwhile, several bound-state-like methods were developed for the single-particle resonant states. These include the S-matrix [24], analytic continuation in the coupling constant (ACCC) [25], complex scaling method (CSM) [26], the Green's function method [27], etc.

      Although many experiments and theoretical studies [17, 20, 24] are devoted to the investigations for 15F, the width of the 5/2+ still has large uncertainty. In rare cases, the proton resonant state is extremely narrow at higher energies, even for the resonance with low angular momentum. 15F sets such an example in this study. Since Fortune's predictions in Ref. [16] and the single-neutron resonant states can be successfully located by the complex-scaled Green's function method (CGF) [28-32], here we use the CGF method to extend the investigation of single-proton resonant states with the solution of the Schrödinger equation, especially for the ground and first-excited states. The choice of 14O as the core of 15F is mainly attributed to two reasons. One is that the strong neutron shell closure effect also happens in 14O [33], the other is that Z = 8 (magic number) plus one proton could be better to investigate the single particle resonant states in neutron-deficient nuclei. The paper is constructed as follows: the general formalism of the CGF method is presented in Sec. 2; Sec. 3 presents the numerical details and results. Finally, the main conclusions are summarized in Sec. 4.

    2.   Formalism
    • To explore the single-proton resonances in 15F, we first sketch the theoretical formalism. The Hamiltonian of this system is written as

      H=p22M+Vcent+Vdef+Vcou+Vsl,

      (1)

      where the potential consists of the following four parts

      Vcent=V0f(r),

      (2)

      Vdef(r)=β2V0k(r)Y20(ϑ,φ),

      (3)

      and the spin-orbit coupling potential

      Vsl=Vsl02m2π1rdf(r)dr(sl).

      (4)

      Here, f(r)=11+exp(rRa), k(r)=rdf(r)dr. The parameter mπ is fixed at 135 MeV in Ref. [24] to investigate 15F. Because the Coulomb interaction potential is widely used in nuclear physics, the deformed Coulomb potential is given by

      Vcou={Zαr+3Zα5r(R0r)2β20Y20(ϑ,φ),r>R(ϑ,φ),3Zα5R0(rR0)2β20Y20(ϑ,φ)+Zα2R0(3r2R02),r<R(ϑ,φ),

      (5)

      here α is the fine-structure constant. The Hamiltonian H and wave function ψ are transformed as

      Hθ=U(θ)HU(θ)1,

      (6)

      ψθ=U(θ)ψ.

      (7)

      Here, U(θ) is a complex rotation operator defined by the transformation rreiθ, and Hθ(ψθ) is the complex scaled Hamiltonian (wave function) with the complex rotation angle θ in Refs. [34-36]. The corresponding complex scaled equation becomes

      Hθψθ=Eθψθ.

      (8)

      By solving the complex scaled Eq. (8), we can single out bound states and resonant states. Details are provided in the literature [29]. However, there are some disadvantages in CSM that are indicated in Ref. [31]. For example, to accurately determine resonance parameters, we need to repeat diagonalization of the Hamiltonian in complex scaling calculations, while a complicated loop integral is imposed on the Green's function method in search of resonant states. Therefore, the CSM is combined with the Green's function method by defining the complex-scaled Green's function (CGF) as

      Gθ(E)=U(θ)G(E)U(θ)1=1EHθ,

      (9)

      in the coordinate representation

      Gθ(E,r,r)=r|1EHθ|r.

      (10)

      Then, with an extended completeness relation

      Nbb|ψθb˜ψθb|+Nrr|ψθr˜ψθr|+dEθc|ψθc˜ψθc|=1,

      (11)

      the density of states can be defined as

      ρθ(E)=1πImdrr|1EHθ|r=1πImdr[Nbbψθb(r)˜ψθb(r)EEb+Nrr ψθr(r)˜ψθr(r)EEθr+dEθcψθc(r)˜ψθc(r)EEθc].

      (12)

      Employing the basis expansion method, the density of states can be approximately expressed as

      ρNθ(E)=Nbbδ(EEb)+1πNrrΓr/2(EEr)2+Γ2r/4+1πNNbNrcεIc(EεRc)2+εI2c.

      (13)

      In Eq. (12), ψθb and ψθr are the complex scaled wave functions for the bound and resonant states, respectively, while ψθc is the wave function of the rotated continuum. The bra states with tilde represent the bi-orthogonal counterparts of the ket states. Detailed explanations can be found in Ref. [37]. In Eq. (12), Eb,Eθr, and Eθc represent the energy eigenvalues of Hθ for the bound states, resonant states, and rotated continuum, respectively. Nb and Nr are the numbers of bound and resonant states, respectively. In Eq. (13), N represents the total number of states (the size of basis). Because of the normalization of the wave functions for bound and resonant states, the integration on r in Eq. (12) is unity. However, for the continuum, there appears singularity in the integration r, which can be eliminated using the basis expansion method in the discretization of the energy spectrum. Then, the density of states can be expressed by the bound state energies Eb(b=1,2,...,Nb), the resonance complex energies Eθr=EriΓr/2(r=1,2,...,Nr), and rotated continuum energies εθc=εRciεIc(c=1,2,...,NNbNr).

      As there are approximations in realistic calculations, ρNθ(E) is slightly dependent on θ. The dependence can be weakened by subtracting the background of Hθ, which is defined as the density of continuum states ρ0Nθ(E):

      ρ0Nθ(E)=1πNkε0Ik(Eε0Rk)2+ε0I2k,

      (14)

      where ε0k(θ)=ε0Rkiε0Ik are the eigenvalues of the asymptotic Hamiltonian H0θ in the form of Hθ with r. After subtracting the background of Hθ, the continuum level density (CLD) Δρ(E) is as the difference between the density of states ρNθ(E) and the density of continuum states ρ0Nθ(E):

      Δρ(E)=ρNθ(E)ρ0Nθ(E).

      (15)

      The CLD is also related to the scattering phase shift δ(E),

      Δρ(E)=1πdδ(E)dE.

      (16)

      By performing integration of every term, the phase shift δ(E) is obtained as:

      δ(E)=Nbπ+Nrr=1{cot1(EErΓr/2)}+Ncc=1{cot1(EεRcεIc)}Nk=1{cot1(Eε0Rkε0Ik)}.

      (17)

      With the definitions of δr, δc, and δk

      cotδr=EErΓr/2,cotδc=EεRcεIc,cotδk=Eε0Rkε0Ik,

      (18)

      the phase shift is then expressed as

      δ(E)=Nbπ+Nrr=1δr+Ncc=1δcNk=1δk.

      (19)

      Based on the relationship between the phase shift and the cross section, the partial cross section for spherical nuclei is expressed as

      σl(E)=4π(2l+1)k2sin2δl(E),

      (20)

      where k2=2Eμ/2 with the reduced mass μ.

    3.   Results and discussion
    • We explore the single-proton resonance in 15F using the formalism represented above. Based on the experiment and data analysis with the R-matrix, the model parameters are determined by reproducing the experimental single-proton separation energy Sp=1.51 MeV, suggested in Ref. [17]. The single-particle energy ε of the last valence proton is determined from the Sp energy, ε=Sp. Then, the 1/2+ (g.s.) resonance energy of 15F was determined to the one-proton decay energy 1.51 MeV, which correspond to the single-proton separation energy. For the broad resonance 1/2+ with Γ0.841.2 MeV [17, 38], the obtained width is 1.07 MeV using our method. These calculations confirm that the parameters are suitable for the following discussion. Under such circumstances, the potential geometry is fixed. For the central potential, V0=53.40 MeV, a=0.64 fm, and r0=1.17 fm for the radius R=r0A1/3. For the spin-orbit potential, the strength of the spin-orbit coupling Vsl0=7.46 MeV is applied. Moreover, the Coulomb potential with a radius Rc=R is assumed. The complex-scaled equation is solved by expansion in Laguerre polynomials. When the basis is truncated up to 100 shells, the size of all concerned subspaces is sufficiently large to study the single-proton resonant states in present calculations. With the fitted parameters, the pattern is similar to the neutron resonant states [39] in the following calculations. Fig. 1 shows the change of the eigenvalues of Hθ with θ from 3 to 7 by a step of 1. The resonant states are clearly isolated from the continuum with increase in the rotation angle. Possibly, the positions of the resonant state 5/2+ hardly change with increasing θ, however the real situation does not conform to this situation. More details are given in Fig. 2.

      Figure 1.  (color online) Variation of eigenvalues of Hθ with θ for the 1d5/2 state, where the complex-scaling parameter θ varies from 3 to 7 by steps of 1.

      Figure 2.  θ trajectory for resonant state 1d5/2.

      Although the resonant states are separated from the continuum, we still need to determine the most appropriate θ value for the investigation of the resonances in our calculations in Fig. 2, where the θ trajectory is plotted. Extending the ABC-theorem [34-36] in realistic calculations, the condition that the resonance parameters are independent on θ (dEθdθ=0) is not available. A common approach is considering that the obtained values for the resonance parameters depend slightly on θ by a finite basis expansion in Ref. [40]. Hence, the best estimate for the proton resonance parameters is |dEθdθ| at the minimal value. When θ is small (less than 6), the resonance position is very sensitive to the complex rotation angles. Once θ is larger than 6, the energies and widths of resonant states are almost independent of θ. As long as the complex rotation angle θ is sufficiently large, the obtained energies and widths are reliable. However, it is worth noting that θ is not infinite with a Woods–Saxon-type potential in the present work, and the effective range is 0<θ<tan1(πa/R). Hence, θ=6 as an optimal value is adopted in the following. Moreover, the corresponding width and energy are 0.240 MeV, 2.770 MeV, respectively, for the resonant state 1d5/2 with CSM, respectively.

      Comparing with the the resonance energies and widths by CSM, we apply the complex-scaled Green's function method to calculate the continuum level density to determine the optimal resonance locations. The results are exhibited in Fig. 3. Employing the same technique as in Ref. [32], the available energy is obtained at Er=2.770 MeV and the width Γ=0.239 MeV for the resonant state 1d5/2. These calculation results are close to the experimental value 0.24(3) MeV in Ref. [15], which is the first value close to Fortune's analysis (0.2 MeV) in Ref. [16]. By examining the spectroscopic factor, Fortune pointed out a serious problem that the width with 0.3 MeV obtained from Refs. [12,13,18] is too large compared with the one expected from spectroscopic factors for the lowest 5/2+ state in other A=15 nuclei. However, the current obtained width is 0.240 MeV with the CSM and 0.239 MeV with CGF. Hence, our calculations are closer to the width obtained by Fortune than previous results, and further details are provided in Table 1, which confirms that our method is reliable in determining the width of the resonance. At the same footing, we obtained the resonance energy 2.770 MeV, which is close to the average value (2.794 MeV) obtained by the previous method in Ref. [16]. It is further confirmed that the complex-scaled Green's function method hold the advantages for the study of single-particle resonant states near the drip line.

      Jπ Ep /MeV Γ /MeV Ref.
      1/2+(exp) 1.51±0.11 1.2 [17]
      1.56(13) 0.6(+0.8/0.4) [10]
      1.31(1) 0.853(146) [18]
      1.270(10)(10) 0.376(70) [13]
      (calc) 1.31 0.8 [12]
      1.39-1.51 0.8 [20]
      1.194 0.531 [24]
      1.51 1.07 CGF
      5/2+(exp) 2.78(1) 0.311(10) [18]
      2.853±0.045 0.34 [17]
      2.763(9)(10) 0.305(9)(10) [13]
      2.67(10) 0.5(2) [11]
      (calc) 2.785 0.2 [16]
      2.78 0.3 [12]
      2.780 0.293 [24]
      2.770 0.239 CGF

      Table 1.  Energies and widths calculated for 15F states with Jπ=1/2+ and 5/2+.

      Figure 3.  The continuum level density Δρ(E) for the 1d5/2 state with the complex-scaling parameter θ=6.

      To make the results reliable, the phase shift method (or δ=π/2 rule) will provide a check for studying the single-particle resonant states [24]. The phase shift δ of the first excited state 1d5/2 is shown in Fig. 4 with θ=6. The first excited state can be regarded as a 14O core plus a proton in the 1d5/2 orbit, which is viewed as a single-particle state in the system (15F = 14O+p). The resonant energy (one-proton decay) Er=2.789 MeV can be obtained from passing through δ=π2, which yields values that are very close to the CGF and other theories [16, 41]. In this study, we converted the obtained phase shifts into resonance cross sections using Eq. (20), which is plotted in Fig. 5. The resonance energy is determined by the cross section σ reaching its maximum value. Hence, the sharp peak value at E=2.787 MeV is considered to be the 1d5/2 resonance energy. The energy obtained is likewise in agreement with the results of CSM and CGF.

      Figure 4.  14O+p scattering phase shift for the 1d5/2 state of 15F generated by the Woods-Saxon potential. Remaining parameters are same as those in Fig. 3.

      Figure 5.  The cross section of l=2 partial wave in 15F. Remaining parameters are same as those in Fig. 3.

      Table 1 lists the results for the energies and widths of the low-lying states for 15F, which are then compared to each other. The ground state in 15F is a broad resonance, except for the value 0.376(70), and the narrower width is also supported by some theoretical predictions [42]. Our calculations 1.07 MeV (width) observed as a broad wave belong to Γ0.5311.2 MeV in Table 1, and 1.51 MeV is also in the energy range. For the 1d5/2 state, the width is as small as 0.239 MeV, which also belongs within the range (0.2–0.5 MeV) in Table 1. The suggested value 0.20 MeV by Fortune is slightly smaller than our results. This indicates the calculations by the CGF method are reliable.

      To further examine the applicability and validity of the current model, we explore the dependence of the resonant parameters on the shape of the potential. Using Woods-Saxon potential for the low-lying states for 15F, there are three parameters, namely depth of potential V0, surface diffuseness a, radius parameter r0. Keeping V0=53.40 MeV and the radius r0=1.17 fm fixed, we vary the parameter a to investigate how the energies and widths of the 2s1/2 and 1d5/2 states are sensitive to surface diffuseness a in Fig. 6. With the increase of a, the energies and widths of the resonant states decrease, which is expected because the potential becomes more dispersed. With the lower resonance energies and narrower resonance widths, the lifetime of single proton states would become longer with increasing a.

      Figure 6.  (color online) Parameters are same as Fig. 3, green circles represent resonant states with a=0.64 fm, r0=1.17 fm, V0=53.40 MeV.

      A similar trend is observed when we vary the central depths V0, keeping other parameters fixed. The results are displayed in the middle panel of Fig. 6. With the deeper potential, the energies and the widths of the 2s1/2 and 1d5/2 states both decrease. Compared with the 2s1/2 state, the energy of the 1d5/2 state drop faster, and there appears to be a crossover in the resonance energy with deeper potential. When the potential further deepens past 59 MeV, the resonant states show a disappearing trend, and the width would hence be too narrow. The lifetime of the resonant states is a reciprocal of the width, hence if one state has a narrower width, it would indicate that it is more stable. Because the depth of potential significantly influences the stability of resonant states, the relationship between the depth of potential and the resonant states can present the evolution of levels from unstable to stable nuclei.

      As r0 is particularly important for comparing resonant states of different isotopes, the influence of r0 on the resonance is shown in the right panel of Fig. 6. With increasing r0, the energies and widths of resonant states also decrease. The two states are degenerated around r=1.23 fm. The resonance energies become lower, and the resonance widths become narrower. This can be explained in terms of the increased r0. The potential becomes broader, which causes falling of the levels. Further increasing r0, the resonant states are likely to become the weakly bound states. The phenomena is in accordance with the effect of a and V0 on resonant states. This conclusion is useful to recognize the level structure beyond the drip line.

      From Fig. 6, we see that 2s1/2 level is lower than 1d5/2 level for 15F. Compared with the traditional shell structure in stable nuclei, these two states are inverted, which may occur in the exotic nucleus [43, 44]. It is worth to mention that 15C and 15F are mirror nuclei, and for 15C [4], the 2s1/2 orbit is likewise below the 1d5/2 orbital, which induces the one-neutron halo. Hence, the one-proton halo of 15F may occur. Among the parameters a,V0,r0, the energies and widths are less sensitive to the surface diffuseness a. Thus, it is inappropriate to obtain the energy of single-proton resonance by increasing a. With the increasing V0,r0, the energy of 1d5/2 state drops faster than that of the 2s1/2 state, which indicates if the resonant state 2s1/2 becomes a weakly bound state, the 1d5/2 orbit will be lower than 2s1/2 orbit, and the inversion of sd states is broken. Different from 15C, it is difficult to form one-proton halo for 15F in the spherical case. However, this may lead to new phenomena with the shift of levels.

      Notably, the energy difference between the 2s1/2 state and 1d5/2 state is only about 1.2 MeV, and the fact is that most nuclei around 15F are deformed, hence it is crucial to take the deformation effects into account. From this view, the single particle energies with the quadruple deformation β2 are exhibited in Fig. 7. There are two large gaps, i.e., the new magic number Z = 6 (1p3/2 and 1p1/2) and the conventional magic number Z = 8 (1p1/2 and 2s1/2) appear under spherical case. On the oblate side, the Z = 6 gap appears between 1/2[110] and 1/2[101]. On the prolate side, the gap is between 3/2[101] and 1/2[101]. With the deformation varying from β2=0.4 to 0.6, the gap (Z = 6) becomes smaller. A similar case occurs at the Z = 8 gap. From β2=0.4 to –0.23, the shell closure Z = 8 is related to the 5/2[202] and 1/2[101], while the gap is caused by 1/2[101] and 1/2[200] at β2>0.23. With the increasing deformation from β2=0.1 to β2=0.6, the energies of 1/2+ (1/2[200]) ground state spin of 15F decrease monotonously. The deformation effects destroy the Z = 8 shell closure. From Fig. 7 shows that the change of deformation from an oblate shape to a prolate shape, with the energy of the 5/2[202] orbit monotonically increasing, while the level 1/2[200] rapidly decreases from β2=0.23 to β2=0.6. It appears that the level 1/2[200] is lower than the other levels, which are split form 1d5/2 level from β2=0.23 to β2=0.6. Meanwhile, from β2=0.4 to β2=0.23, the 5/2[202] orbit is lower than 1/2[200] orbit. These results indicate that the deformation effect on the evolution of resonant levels structure cannot be ignored. Moreover, only one condition that the obtained Sp=0.215 to Sp=0.766 MeV from β2=0.3 to β2=0.35 agrees with the proton halo formation. Thus, it is not sufficiently evidenced to predict the formation of the proton halo in 15F. Therefore, the quadruple deformation effects provides us with more important information on the evolution of the shell structure.

      Figure 7.  (color online) Proton single-particle levels as a function of quadruple deformation β2.

    4.   Summary
    • The single-proton resonance in 15F is investigated by the CGF method, and the theoretical formalism is presented. We explored the single-proton resonant states, i.e., the ground state 2s1/2 and the first excited state 1d5/2 in 15F. The complex scaling parameter θ dependence is tested, which explained how the 1d5/2 state is isolated from the continuum. We compared the energy and width of the 1d5/2 state with those obtained by other methods. The present result, 0.239 MeV for the width, is close to the estimated decay width in Ref. [16] and also very close to the experimental value (0.24(3) MeV). Simultaneously, the energy of the 1d5/2 state is 2.770 MeV, which approaches the average calculation and experimental values. To confirm the reliability of results, we also employ the CSM, the scattering phase shift, and the cross section to investigate 1d5/2 with the same parameters. By the comparison of these methods, the differences in the calculation are found to be very small, and the correctness and universality of our method are confirmed, which provides an effective method for studying the resonances and nuclear structure. Further, we investigated the effect of the potential shape and quadruple deformation on the resonant states, which are helpful to recognize the shell structure of the exotic nuclei.

Reference (44)

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