-
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(→s⋅→l).
(4) Here,
f(r)=11+exp(r−Ra) ,k(r)=rdf(r)dr . The parametermπ 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 byVcou={Zαr+3Zα5r(R0r)2β20Y20(ϑ,φ),r>R(ϑ,φ),3Zα5R0(rR0)2β20Y20(ϑ,φ)+Zα2R0(3−r2R02),r<R(ϑ,φ),
(5) here
α is the fine-structure constant. The Hamiltonian H and wave functionψ are transformed asHθ=U(θ)HU(θ)−1,
(6) ψθ=U(θ)ψ.
(7) Here,
U(θ) is a complex rotation operator defined by the transformation→r→→reiθ , andHθ(ψθ) is the complex scaled Hamiltonian (wave function) with the complex rotation angleθ in Refs. [34-36]. The corresponding complex scaled equation becomesHθψθ=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=1E−Hθ,
(9) in the coordinate representation
Gθ(E,→r,→r′)=⟨→r|1E−Hθ|→r′⟩.
(10) Then, with an extended completeness relation
Nb∑b|ψθb⟩⟨˜ψθb|+Nr∑r|ψθr⟩⟨˜ψθr|+∫dEθc|ψθc⟩⟨˜ψθc|=1,
(11) the density of states can be defined as
ρθ(E)=−1πIm∫d→r⟨→r|1E−Hθ|→r⟩=−1πIm∫d→r[Nb∑bψθb(→r)˜ψθ∗b(→r)E−Eb+Nr∑r ψθr(→r)˜ψθ∗r(→r)E−Eθr+∫dEθcψθc(→r)˜ψθ∗c(→r)E−Eθc].
(12) Employing the basis expansion method, the density of states can be approximately expressed as
ρNθ(E)=Nb∑bδ(E−Eb)+1πNr∑rΓr/2(E−Er)2+Γ2r/4+1πN−Nb−Nr∑cεIc(E−εRc)2+εI2c.
(13) In Eq. (12),
ψθb andψθr are the complex scaled wave functions for the bound and resonant states, respectively, whileψθc 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 , andEθc represent the energy eigenvalues ofHθ for the bound states, resonant states, and rotated continuum, respectively.Nb andNr 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 energiesEb(b=1,2,...,Nb) , the resonance complex energiesEθr=Er−iΓr/2(r=1,2,...,Nr) , and rotated continuum energiesεθc=εRc−iεIc(c=1,2,...,N−Nb−Nr) .As there are approximations in realistic calculations,
ρNθ(E) is slightly dependent onθ . The dependence can be weakened by subtracting the background ofHθ , which is defined as the density of continuum statesρ0Nθ(E) :ρ0Nθ(E)=1πN∑kε0Ik(E−ε0Rk)2+ε0I2k,
(14) where
ε0k(θ)=ε0Rk−iε0Ik are the eigenvalues of the asymptotic HamiltonianH0θ in the form ofHθ withr→∞ . After subtracting the background ofHθ , the continuum level density (CLD)Δρ(E) is as the difference between the density of statesρNθ(E) and the density of continuum statesρ0Nθ(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+Nr∑r=1{−cot−1(E−ErΓr/2)}+Nc∑c=1{−cot−1(E−εRcεIc)}−N∑k=1{−cot−1(E−ε0Rkε0Ik)}.
(17) With the definitions of
δr ,δc , andδk cotδr=E−ErΓr/2,cotδc=E−εRcεIc,cotδk=E−ε0Rkε0Ik,
(18) the phase shift is then expressed as
δ(E)=Nbπ+Nr∑r=1δr+Nc∑c=1δc−N∑k=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/ℏ2 with the reduced massμ . -
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 energyε of the last valence proton is determined from theSp energy,ε=−Sp . Then, the1/2+ (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 resonance1/2+ withΓ≈0.84−1.2 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, andr0=1.17 fm for the radiusR=r0A1/3 . For the spin-orbit potential, the strength of the spin-orbit couplingVsl0=7.46 MeV is applied. Moreover, the Coulomb potential with a radiusRc=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 ofHθ withθ from3∘ to7∘ by a step of1∘ . The resonant states are clearly isolated from the continuum with increase in the rotation angle. Possibly, the positions of the resonant state5/2+ hardly change with increasingθ , 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 withθ for the1d5/2 state, where the complex-scaling parameterθ varies from3∘ to7∘ by steps of1∘. 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 theθ trajectory is plotted. Extending the ABC-theorem [34-36] in realistic calculations, the condition that the resonance parameters are independent onθ (dEθdθ=0 ) is not available. A common approach is considering that the obtained values for the resonance parameters depend slightly onθ by a finite basis expansion in Ref. [40]. Hence, the best estimate for the proton resonance parameters is|dEθdθ| at the minimal value. Whenθ is small (less than6∘ ), the resonance position is very sensitive to the complex rotation angles. Onceθ is larger than6∘ , the energies and widths of resonant states are almost independent ofθ . As long as the complex rotation angleθ is sufficiently large, the obtained energies and widths are reliable. However, it is worth noting thatθ is not infinite with a Woods–Saxon-type potential in the present work, and the effective range is0<θ<tan−1 (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 state1d5/2 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 widthΓ=0.239 MeV for the resonant state1d5/2 . 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 lowest5/2+ 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 /MeVΓ /MeVRef. 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+ and5/2+. Figure 3. The continuum level density
Δρ(E) for the1d5/2 state with the complex-scaling parameterθ=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 shiftδ of the first excited state1d5/2 is shown in Fig. 4 withθ=6∘ . The first excited state can be regarded as a 14O core plus a proton in the1d5/2 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 throughδ=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 sectionσ reaching its maximum value. Hence, the sharp peak value atE=2.787 MeV is considered to be the1d5/2 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
Γ≈0.531−1.2 MeV in Table 1, and 1.51 MeV is also in the energy range. For the1d5/2 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
15 F, there are three parameters, namely depth of potentialV0 , surface diffuseness a, radius parameterr0 . KeepingV0=53.40 MeV and the radiusr0=1.17 fm fixed, we vary the parameter a to investigate how the energies and widths of the2s1/2 and1d5/2 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 the2s1/2 and1d5/2 states both decrease. Compared with the2s1/2 state, the energy of the1d5/2 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 ofr0 on the resonance is shown in the right panel of Fig. 6. With increasingr0 , the energies and widths of resonant states also decrease. The two states are degenerated aroundr=1.23 fm. The resonance energies become lower, and the resonance widths become narrower. This can be explained in terms of the increasedr0 . The potential becomes broader, which causes falling of the levels. Further increasingr0 , the resonant states are likely to become the weakly bound states. The phenomena is in accordance with the effect of a andV0 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 than1d5/2 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], the2s1/2 orbit is likewise below the1d5/2 orbital, which induces the one-neutron halo. Hence, the one-proton halo of 15F may occur. Among the parametersa,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 increasingV0,r0 , the energy of1d5/2 state drops faster than that of the2s1/2 state, which indicates if the resonant state2s1/2 becomes a weakly bound state, the1d5/2 orbit will be lower than2s1/2 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 and1d5/2 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 deformationβ2 are exhibited in Fig. 7. There are two large gaps, i.e., the new magic number Z = 6 (1p3/2 and1p1/2 ) and the conventional magic number Z = 8 (1p1/2 and2s1/2 ) 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 fromβ2=−0.4 to 0.6, the gap (Z = 6) becomes smaller. A similar case occurs at the Z = 8 gap. Fromβ2=−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] atβ2>−0.23 . With the increasing deformation fromβ2=0.1 toβ2=0.6 , the energies of1/2+ (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 fromβ2=−0.23 toβ2=0.6 . It appears that the level 1/2[200] is lower than the other levels, which are split form1d5/2 level fromβ2=−0.23 toβ2=0.6 . Meanwhile, fromβ2=−0.4 toβ2=−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 obtainedSp=0.215 toSp=0.766 MeV fromβ2=0.3 toβ2=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.
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 |