Shell model description of
16
O(p,γ)17F and
16
O(p,p)16 O reactions
arXiv:nucl-th/0004017v2 13 Jul 2000
K. Bennaceur1,2 , N. Michel1 , F. Nowack...

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16

O(p,γ)17F and

16

O(p,p)16 O reactions

arXiv:nucl-th/0004017v2 13 Jul 2000

K. Bennaceur1,2 , N. Michel1 , F. Nowacki3 , J. Okolowicz1,4 and M. Ploszajczak1 1. Grand Acc´el´erateur National d’Ions Lourds (GANIL), CEA/DSM – CNRS/IN2P3, BP 5027, F-14076 Caen Cedex 05, France 2. Centre d’Etudes de Bruy`eres-le-Chˆ atel, BP 12, F-91680 Bruy`eres-le-Chˆ atel, France 3. Laboratoire de Physique Th´eorique Strasbourg (EP 106), 3-5 rue de l’Universite, F-67084 Strasbourg Cedex, France 4. Institute of Nuclear Physics, Radzikowskiego 152, PL - 31342 Krakow, Poland

Abstract We present shell model calculations of both the structure of 17 F and the reactions 16 O(p,γ)17 F, 16 O(p,p)16 O. We use the ZBM interaction which provides a fair description of the properties of 16 O and neighbouring nuclei and, in particular it takes account for the complicated correlations in coexisting low-lying states of 16 O. 21.60.Cs, 24.10.Eq, 25.40.Lw, 27.20.+n

Typeset using REVTEX 1

A realistic account of the low-lying states properties in exotic nuclei requires taking into account the coupling between discrete and continuum states which is responsible for unusual spatial features of these nuclei. Within the newly developed Shell Model Embedded in the Continuum (SMEC) approach [1], one may obtain the uniﬁed description of the divergent characteristics of these states as well as the reactions involving one-nucleon in the continuum. This provides a stringent test of approximations involved in the SMEC calculations and permits to asses the mutual complementarity of the reaction and structure data for understanding of these nuclei. The quality of the SMEC description depends crucially on the realistic account of the conﬁguration mixing for coexisting low-lying structures and hence on the quality of the Shell Model (SM) eﬀective interactions and the SM space considered. In this work, we shall present the calculation for 17 F for which it is believed that conﬁgurations of up to four particles and four holes are necessary. Closed shell nuclei are never inert and multiple particle-hole excitations are always observed in their spectra. In 16 O and 17 O, Brown and Green described the low-lying spectra by mixing spherical and deformed states [2]. From the shell model point of view, Zuker-Buck-McGrory (ZBM) set an eﬀective interaction in the basis of 0p1/2 , 1s1/2 and 0d5/2 orbitals [3,4]. This valence space has the advantage to be practically non spurious and most of states at the p − sd interface around 16 O are nicely described through conﬁguration mixing of these three orbitals. In particular, the energy spectra, spectroscopic factors and correlations in the low-lying states of A = 16 and A = 17 nuclei are well reproduced [5]. The wavefunction components for the ﬁrst three 0+ states in 16 O are in a fair agreement with the recently developed interactions in the full p − sd shells [6]. The aim of this work is not to provide better new SM wavefunctions for 16 O but to build on them the continuum eﬀects and to investigate the consequences of this coupling both for the structure of 17 F and for the reactions 16 O(p,γ)17 F, 16 O(p,p)16 O. For that purpose, the ZBM interaction is satisfactory and we are going to use it in this study. In the SMEC formalism the subspaces of (quasi-) bound (the Q subspace) and scattering (the P subspace) states are not separated artiﬁcially [1]. (For the review of earlier works see also Ref. [7] ). Using the projection operator technique, we separate the P subspace of asymptotic channels from the Q subspace of many-body localized states which are build up by the bound single-particle (s.p.) wavefunctions and by the s.p. resonance wavefunctions. P subspace contains (N − 1)-particle localized states and one nucleon in the scattering state. The s.p. resonance wavefunctions outside of the cutoﬀ radius Rcut are included in the P subspace. The resonance wavefunctions for r < Rcut are included in the Q subspace. The wavefunctions in Q and P are then properly renormalized in order to ensure the orthogonality of wavefunctions in both subspaces. In the ﬁrst step, we calculate the (quasi-) bound many-body states in Q subspace. For that we solve the multiconﬁgurational SM problem : HQQ Φi = Ei Φi , using the code ANTOINE [8]. For HQQ ≡ QHQ we take the ZBM interaction which yields realistic internal mixing of many-body conﬁgurations in Q subspace. To generate the radial s.p. wavefunctions in Q subspace and the scattering wavefunctions in P subspace we use the average potential of Woods-Saxon (WS) type with the spin-orbit and Coulomb parts included: 1 df (r) + VC , r dr where λ ¯ 2π = 2 fm2 is the pion Compton wavelength and f (r) is the spherically symmetrical U(r) = V0 f (r) + VSO λ ¯ 2π (2l · s)

2

WS formfactor : f (r) = [1 + exp((r − R0 )/a)]−1 . The Coulomb potential VC is calculated for the uniformly charged sphere with radius R0 . This ’ﬁrst guess’ potential U(r), is then modiﬁed by the residual interaction. We shall return to this problem below. For the continuum part, we solve the coupled channel equations : c(+)

(E (+) − HP P )ξE

≡

X c

′

c (+)

(E (+) − Hcc′ )ξE

=0,

′

where index c denotes diﬀerent channels and HP P ≡ P HP . The superscript (+) means that boundary conditions for incoming wave in the channel c and outgoing scattering waves in all channels are used. The channel states are deﬁned by coupling of one nucleon in the scattering continuum to the many-body SM state in (N − 1)-nucleus. For the coupling between bound and scattering states around 16 O, we use the density dependent interaction which is close to the Landau - Migdal type of interactions [9,10]. However, as compared to the original force of Schwesinger and Wambach [9], the radius parameter r0 of the WS density formfactor is somewhat reduced to better ﬁt the experimental matter radius in oxygen (r0 = 2.64 fm). This interaction provides external mixing of SM conﬁgurations via the virtual excitations of particles to the continuum states. The channel - channel coupling potential is : J ′ , Hcc′ = (T + U)δcc′ + υcc

(1)

J where T is the kinetic-energy operator and υcc ′ is the channel-channel coupling generated by the residual interaction. Reduced matrix elements of the channel - channel coupling, K which involve one-body operators of the kind : Oβδ = (a†β a ˜δ )K , depend sensibly on the amount of 2p − 2h and 4p − 4h correlations in the ground state of 16 O. The potential for channel c in (1) consists of initial WS guess, U(r), and of the diagonal part of coupling J potential υcc which depends on both the s.p. orbit φl,j and the considered many-body state π J . This modiﬁcation of the initial potential U(r) change the generated s.p. wavefunctions φl,j deﬁning Q subspace which in turn modify the diagonal part of the residual force, etc. In other words, the procedure of solving of the coupled channel equations is accompanied by the self-consistent iterative procedure which yields for each total J independently the new self-consistent potential : J(sc) U (sc) (r) = U(r) + υcc (r) ,

and consistent with it the new renormalized formfactor of the coupling force. U (sc) (r) diﬀers signiﬁcantly from the initial WS potential, especially in the interior of the potential [1,10]. Parameters of U(r) are chosen in such a way that U (sc) (r) reproduces energies of experimental s.p. states, whenever their identiﬁcation is possible. The third system of equations in SMEC consists of the inhomogeneous coupled channel equations: (+) (E (+) − HP P )ωi = HP Q Φi ≡ wi with the source term wi which depends on the structure of N - particle SM wavefunction Φi . Formfactor of the source term is given by the self-consistently determined s.p. wavefunctions. (+) (+) (+) The solutions : ωi ≡ GP HP Q Φi , where GP is the Green function for the motion of s.p. in the P subspace, describe the decay of quasi-bound state Φi in the continuum. Reduced matrix elements of the source term, which involve products of two annihilation operators 3

α and one creation operator of the kind : Rjγδ(L)β = (a†β (˜ aγ a ˜δ )L )jα , are calculated between diﬀerent initial state wavefunctions in 17 F and a given ﬁnal state wavefunction in 16 O. It should be stressed that the matrix elements of the source term depend sensitively on the percentage of the shell closure in 16 O, i.e., on the amount of correlations both in the g.s. of 16 O and in the considered states of 17 F. Obviously, this kind of couplings are not accounted for by the spectroscopic amplitudes. The total wavefunction is expressed by three functions: Φi , ξEc and ωi [1,7] :

ΨcE = ξEc +

X

(Φi + ωi )

i,j

1 c ef f < Φj | HQP | ξE > E − HQQ

(+)

(2)

ef f where : HQQ (E) = HQQ + HQP GP HP Q , is the new energy dependent eﬀective SM Hamilef f (E), which is Hermitian tonian which contains the coupling to the continuum. Operator HQQ for energies below the particle emission threshold, becomes non-Hermitian for energies higher than the threshold. Consequently, the eigenvalues E˜i − 12 iΓ˜i are complex for decaying states and depend on the energy E of the particle in the continuum. The energy and the width of resonance states are determined by the condition: E˜i (E) = E. The eigenstates corresponding to these eigenvalues can be obtained by the orthogonal but in general non-unitary transformation [7,10]. Inserting them in (2), one obtains symmetrically the new continuum many-body wavefunctions modiﬁed by the discrete states, and the new discrete state wavefunctions modiﬁed by the coupling to the continuum states. The SMEC wavefunctions, can be used to calculate various spectroscopic and reaction quantities. These include for example the proton (neutron) capture data, Coulomb dissociation data, elastic (inelastic) proton (neutron) scattering data, energies and wavefunctions of discrete and resonance states, transition matrix elements between SMEC wavefunctions, static nuclear moments etc. [1,10,11]. The application of the SMEC model for the description of structure for mirror nuclei and capture cross sections for mirror reactions in p-shell has been published in Ref. [1]. The analysis of the structure of 17 F and the reactions 16 O(p, γ)17 F, 16 O(p,p)16 O in the (0p1s0d)-space, neglecting the 2p-2h, 4p-4h admixtures in16 O wavefunctions have been reported recently as well [10]. To correct for the missing correlations in the low-energy wavefunctions of 16 O and 17 F, the matrix elements of the Q − P coupling in this study [10] have been quenched by the factors related to the spectroscopic amplitudes for positive parity states in 17 F (17 O) and to the amount of 2p − 2h, 4p − 4h correlations in the g.s. of 16 O. This quenching correction of the eﬀective operator allowed to obtain a reasonable description of the spectrum of 17 F but failed in solving the problem of ’halo’ of discrete states for positive energies which was observed in the elastic cross section and in the phase shifts [10]. The whole problem results from the non-hermitean corrections to the eigenvalues for positive energies which generate the imaginary part and which are particularly large for pure single-particle (or single-hole) conﬁgurations. For this reason, the incorrect internal conﬁguration mixing in SM wavefunctions may lead to an unphysical enhancement of the resonant-like correction from bound states into the scattering data (e.g. the elastic phase-shifts). This aspect of the continuum coupling we want to investigate using the ZBM interaction which includes the most essential for the present studies conﬁguration mixing in low energy wavefunctions. In Fig. 1 we show SMEC energies and widths for positive parity (l.h.s. of the plot) and negative parity (r.h.s. of the plot) states of 17 F. The calculations were performed using either

4

the density dependent interaction of Ref. [9], called DDSM0, or the similar interaction with the overall strength reduced by a factor 0.67 , called DDSM1 [10]. (In both cases, radius of the density formfactor has been reduced as discussed above.) This latter interaction was used before in the SMEC calculations in 0p1s0d SM space [10]. However, since the external mixing of wavefunctions in the SM space of ZBM is smaller than in the 0p1s0d SM space, therefore one may consider bigger coupling strength of the residual interaction in SMEC-ZBM calculations. For that reason, we compare results obtained with both DDSM0 and DDSM1 interactions. For the comparison, the experimental data and the SM input in Q-space is shown in Fig. 1 separately for positive and negative parity states as well. The agreement of the SM-ZBM calculations with the experimental data is even better than obtained in the 0p1s0d SM space [10]. The agreement of SMEC calculations with experiment is encouraging for both density dependent interactions but the widths of states are better reproduced by SMEC-DDSM0. Only the coupling matrix elements between the J π = 0+ 1 g.s. wavefunction of 16 O and all considered states in 17 F are included. Zero of the energy scale for 17 F is ﬁxed by the experimental position of J π = 1/2+ 1 (E = −105 keV) with respect to the proton emission threshold. In Fig. 1 the theoretical prediction for 3/2+ state is compared with the experimental 3/2+ 2 state, because the s.p. orbit 0d3/2 is missing in SM space of ZBM. For those s.p. wavefunctions which in a given many body state are not modiﬁed by the selfconsistent renormalization, we calculate radial formfactors using the common reference s.p. potential U (ref ) of the WS type which is adjusted to reproduce + experimental binding energies of of 5/2+ 1 and 1/21 states for protons [10]. This potential has the following parameters : the radius R0 = 3.214 fm, the diﬀuseness a = 0.58 fm, the spin-orbit strength VSO = 3.683 MeV and the depth V0 = −52.46 MeV. The same potential without the Coulomb term is also used to calculate radial formfactors for neutrons. In the case when the self-consistent conditions in a J π state determine the radial function of a s.p. wavefunction φl,j , we readjust the depth of ’ﬁrst guess’ potential U(r) so that the energy of the s.p. state φl,j in the converged potential U (sc) (r) corresponds to the energy of this s.p. state in U (ref ) . The remaining parameters : R0 , a, VSO , of the initial potential are the same as in the reference potential U (ref ) . This readjustment of V0 in the initial potentials U(J π ) for a s.p. state (l, j) guarantees that the binding of self-consistently determined wavefunction φl,j is at the experimental value. Moreover, we have the same s.p. energies and, consequently, the same asymptotic behaviours of wavefunctions for large r for both residual density dependent interactions. This choice is essential for quantitative description of radiative capture cross-section. + B(E2) transition matrix element between 1/2+ 1 and 5/21 bound states is an interesting test of the wavefunction. In SMEC with ZBM interaction and DDSM0 residual coupling, the value for this transition : B(E2) = 79.61 e2 fm4 , which is obtained assuming the eﬀective charges : ep ≡ 1+δp , en ≡ δn , with the polarization charge δ = δp = δn = 0.2, agrees with the experimental value B(E2)exp = 64.92 e2 fm4 . For the range of δ in between 0.1 and 0.3, which is compatible with the theoretical estimates [12], the SM values with Harmonic Oscillator wavefunctions are much smaller. This diﬀerence reﬂects a more realistic radial dependence of the 1s1/2 s.p. orbit in J π = 1/2+ 1 many body state which is in part a consequence of the external mixing in SMEC wavefunctions due to the coupling to the scattering continuum and provides the halo structure of the 1s1/2 s.p. orbit. Similar eﬀect can be generated using the WS potential or P¨oschl-Teller-Ginocchio potential for the appropriately chosen 1s1/2 5

1/2

orbit [13]. The rms radius for this orbit in SMEC is : < r 2 > = 5.24 fm, as compared to 1/2 : < r 2 > = 3.03 fm in SM. In Fig. 2, the calculated total astrophysical S-factor as a function of the c.m. energy, as 16 well as its values for the 16 O(p, γ)17 F(J π = 1/2+ O(p, γ)17 F(J π = 5/2+ 1 ) and 1 ) branches are compared with the experimental data [14]. Results shown in Fig. 2 have been calculated with DDSM1 interaction. The total S-factor depends weakly on the strength of the residual coupling and with DDSM0 interactions one obtains very similar results. The energy scale is adjusted to reproduce the experimental position of 1/2+ 1 state with respect to the proton emission threshold. Therefore, the energy scale for excitation energy is the same as c.m. energy in the p +16 O system. The photon energy is then given by the diﬀerence of c.m. energy of [16 O + p]Ji system and the experimental energy of the ﬁnal state Jf in 17 F. The + dominant contribution to the total capture cross-section for both 5/2+ 1 and 1/21 ﬁnal states, is provided by E1 transitions from the incoming p wave to the bound 0d5/2 and 1s1/2 states. We took into account all possible E1, E2, and M1 transitions from incoming s, p, d, f , and g waves but only E1 from incoming p - waves give important contributions. In the transition to the g.s., the E1 contribution from incoming f7/2 wave is by a factor ∼100 smaller than the contribution from p3/2 wave at ECM ∼ 0. This contribution however increases with the energy of incoming proton and becomes ∼ 0.3 at 3.5 MeV. The energy dependence of S-factor as ECM → 0 can be ﬁtted by a second order polynomial to calculated points obtained in the interval from 20 to 50 keV in steps of 1 keV. We have ′ S(0) = 9.32 × 10−3 MeV·b, and the logarithmic derivative is S (0)/S(0) = −4.86 MeV−1 . These results have been obtained for DDSM1 residual interaction but are practically identical for the DDSM0 force. The ratio of E2 and E1 contributions in the branch E2 E1 16 = 1.622 × 10−4 , 2.225 × 10−4 O(p, γ)17 F(J π = 1/2+ 1 ) for DDSM1 interaction is : σ /σ and 5.458 × 10−4 at 20, 100 and 500 keV, respectively. Also these ratios do not depend on the strength of the residual coupling. On the contrary, the ratio σ M 1 /σ E1 depends on the residual coupling and equals 3.90×10−4 and 1.087×10−3 at ECM → 0 for DDSM1 and DDSM0 interactions respectively. At 500 keV, this ratio equals 6.11×10−5 and 1.81×10−4 for both interactions, respectively. For the deexcitation to the g.s. 5/2+ 1 , the ﬁt of calculated S-factor for DDSM1 interaction as ECM → 0 yields: S(0) = 3×10−4 MeV·b and S ′ (0)/S(0) = 0.649 MeV−1 . Almost identical results are obtained with the DDSM0 interaction. The ratio of E2 and E1 contributions for DDSM1 interaction at 20, 100 and 500 keV is: σ E2 /σ E1 = 1.336 × 10−3 , 1.11 × 10−3 and 9.74 × 10−4 , respectively. These ratios show some sensitivity on the residual coupling. For DDSM0 one obtains : σ E2 /σ E1 = 2.25 × 10−3 , 1.458 × 10−3 and 1.043 × 10−3 at 20, 100 and 500 keV, respectively. Similarly as in the branch 16 O(p, γ)17 F(J π = 1/2+ 1 ), the ratio M1 E1 −3 −3 σ /σ depends on the residual coupling and equals 2×10 and 2.99×10 at ECM → 0 for DDSM1 and DDSM0 interactions respectively. In the total cross section, ratio of E2 and E1 contributions at 20, 100 and 500 keV equals : σ E2 /σ E1 = 2.03 × 10−4 , 2.63 × 10−4 and 5.85 × 10−4 for DDSM1 interaction , and σ E2 /σ E1 = 2.34 × 10−4, 2.8 × 10−4 and 5.92 × 10−4 for DDSM0 interaction. Elastic phase shifts and elastic cross-sections for diﬀerent proton bombarding energies are shown in Figs. 3 and 4. The elastic phase shifts data [15] are very well reproduced by the SMEC calculations for all considered partial waves including the 5/2+ for which a signiﬁcant discrepancy with the data has been reported in SMEC calculations using the 6

p−sd interaction neglecting higher order p-h correlations in the SM wavefunction for positive parity states [10]. In Fig. 3 we show the calculations for the DDSM1 residual interaction. As we have already mentioned, 0d3/2 s.p. orbit is missing in the Q subspace. On the other hand, the 3/2+ partial wave contributes to the elastic cross-section. So we have decided to include this resonance in the P subspace, adjusting scattering potential for the d3/2 proton wave to place it at the experimental position. The encouraging agreement of SMEC results with the data for the spectrum of 17 F, the proton capture cross-section 16 O(p, γ)17 F(J π = 5/2+ 1 ) and the 5/2+ elastic phase shift is not accidental and shows importance of the np-nh excitations across the ’closed core’ N = Z = 8, which are taken into account in the present studies with the ZBM force. Calculated elastic excitation functions at a laboratory angle of 166◦ in SMEC with DDSM0 (the solid line) and DDSM1 (the dashed line) are compared with the experimental data [16] in Fig. 4. The agreement is very encouraging for both residual interactions in the almost entire energy range. where the interference pattern depending sensitively on the precise values of the energy and width of resonance states is absent. Earlier SMEC calculations [10], using the simpliﬁed wavefunctions for the g.s. of 16 O and 17 F, failed to reproduce the experimental elastic cross-section in the low-energy domain below the resonances. We have found that an unrealistic account for excitations from 0p to 1s0d shells leads to a large resonant contribution from the g.s. 5/2+ 1 wavefunction to the phase shift in the partial wave + 5/2 and implies a strong decrease of the elastic excitation function at low energies. This eﬀect is caused by the virtual coupling of discrete and continuum states. For energies below the proton emission threshold, coupling to the continuum introduces hermitean modiﬁcations of HQQ which shift the energy of 5/2+ 1 state with respect to its initial position given by the SM but do not generate any width for this state. For excitation energies above the proton threshold, the Q − P coupling generates non-hermitean corrections which yield the ef f and, hence, produce the resonant behavior. Internal imaginary part of the eigenvalue of HQQ mixing of conﬁgurations in the SM wavefunctions tend to reduce this resonant behavior for positive energies which, in general, is strongest for the states having little internal mixing, i.e. those having a s.p. nature. Actually, these strong resonant-like features associated with certain bound states for positive energies (above the particle-emission threshold) and the large shifts of the real part for certain eigenvalues at negative energies (below the particleemission threshold), have the same origin in the interplay between external (i.e. via the continuum coupling) and internal (i.e. within Q space) conﬁguration mixing in the SMEC wavefunction for this state. This example shows also that in the SMEC approach, one may use diﬀerent experimental observables to ﬁx those few parameters of the model such as the overall strength of the residual Q − P coupling or the radius and depth of the initial average potential. Moreover, the information about the amount of correlations in the low-lying coexisting states can be extracted not only from the spectroscopic data but also from the elastic excitation function. For nuclei far from the stability line where the amount of experimental data is strongly limited, this feature of the model is very attractive. We feel that the evidence presented in this work shows that the SM calculation extended to include the coupling to the continuum of the scattering states can go a long way towards providing a detailed explanation not only of the structure of 17 F, 16 O and neighboring nuclei, but also the reaction data involving one nucleon in the continuum. The corner-stone of this model is the eﬀective SM interaction providing a realistic internal mixing of conﬁgurations 7

in the Q subspace. Several problems remain, such as the missing conﬁgurations and/or more realistic asymptotic decay channels which show up in the decay width of resonances. In this latter case, the extension of the SMEC is being investigated.

Acknowledgments We thank E. Caurier for his help in the early stage of development of SMEC model. This work was partly supported by KBN Grant No. 2 P03B 097 16 and the Grant No. 76044 of the French - Polish Cooperation.

8

REFERENCES [1] K. Bennaceur, F. Nowacki, J. Okolowicz, and M. Ploszajczak, J. Phys. G 24 (1998) 1631; Nucl. Phys. A 651 (1999) 289. [2] G.E. Brown and A.M. Green, Nucl. Phys. 75 (1966) 401. [3] A.P. Zuker, B. Buck and J.B. McGrory, Phys. Rev. Lett. 21 (1968) 39. [4] A.P. Zuker, Phys. Rev. Lett. 23 (1969) 983. [5] The problem encountered by the ZBM interaction in describing B(E1) transition strength is of no importance in the present analysis. [6] W.C. Haxton and C. Johnson, Phys. Rev. Lett. 65 (1990) 1325; E.K. Warburton, B.A. Brown and D.J. Millener, Phys. Lett. B 293 (1992) 7. [7] I. Rotter, Rep. Prog. Phys. 54 (1991) 635. [8] E. Caurier, code ANTOINE (unpublished). [9] B. Schwesinger and J. Wambach, Nucl. Phys. A 426 (1984) 253. [10] K. Bennaceur, F. Nowacki, J. Okolowicz, and M. Ploszajczak, Nucl. Phys. A 671 (2000) 203. [11] R. Shyam, K. Bennaceur, J. Okolowicz, and M. Ploszajczak, Nucl. Phys. A 669 (2000) 65. [12] M.W. Kirson, Ann. Phys. (NY) 82 (1974) 345. [13] J.N. Ginocchio, Annals of Physics (NY) 152 (1984) 203; 159 (1985) 467; K. Bennaceur, J. Dobaczewski and M. Ploszajczak, Phys. Rev. C 60 (1999) 034308. [14] R. Morlock, R. Kunz, A. Mayer, M. Jaeger, A. M¨ uller, J.W. Hammer, P. Mohr, H. Oberhummer, G. Staudt, and V. K¨olle, Phys. Rev. Lett. 79 (1997) 3837. [15] R.A. Blue and W. Haeberli, Phys. Rev. 137 (1965) B284. [16] S.R. Salisbury, G. Haradie, L. Oppliger, and R. Dangle, Phys. Rev. 126 (1962) 2143.

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FIGURES

17

π=+

F

π=–

6 3/2+

5

–

5/2–– 7/2– 1/2

1/2 7/2––, 5/2– 3/2– 9/2

energy [MeV]

4 3

3/2

–

5/2

–

1/2–

2 1 0

proton threshold

1/2+ 5/2

+

-1 SM

SMEC (DDSM1)

SMEC (DDSM0)

EXP

SM

SMEC (DDSM1)

SMEC (DDSM0)

EXP

FIG. 1. SM and SMEC energy spectra obtained with the ZBM effective SM interaction [3,4] are compared with the experimental states of 17 F nucleus. For the residual coupling between Q and P subspaces we use the density dependent DDSM0 [9] and DDSM1 [10] interactions. The proton threshold energy is adjusted to reproduce position of the 1/2+ 1 first excited state. The shaded regions represent the width of resonance states. For the details of the calculation see the description in the text.

10

0.01

5/2

0.008

+

0.006 0.004 0.002

Sp, γ [MeV·b]

0

1/2

0.008

+

0.006 0.004 0.002 0

total

0.008 0.006 0.004 0.002 0 0

1

2

3

ECM [MeV] FIG. 2. The astrophysical S-factor for the reactions 16 O(p, γ)17 F(J π = 5/2+ 1 ) and 16 O(p, γ)17 F(J π = 1/2+ ) is plotted as a function of the center of mass energy E . For the residual CM 1 coupling between Q and P subspaces we use DDSM1 density dependent interaction [10]. The ex16 O(p, γ)17 F(J π = 5/2+ ) perimental data are from [14]. The contribution of 5/2− 1 resonance in the 1 branch is very narrow in energy and has been omitted in the figure. The resonance is found at 5.29 MeV in the SMEC-DDSM1 calculations.

11

– 20

1/2

+

10

1/2

–

– 40 0 – 60 – 10

Phase shift δ [deg]

– 80 2

3

4

5

6

2 50

50

3/2

0

3

3/2

4

5

6

4

5

6

4

5

6

–

+

0

– 50

– 50 2

3

4

5

6

2

3

10

5/2

+

5

7/2

–

0 0 – 10 –5 – 20

2

3

4

5

6

2

3

Ep [MeV] FIG. 3. The phase-shifts for the p +16 O elastic scattering as a function of the proton energy Ep for different partial waves. The experimental data are from [15]. SMEC results have been obtained for the ZBM effective SM interaction [3,4] and the density dependent residual interaction DDSM1 [10] (the solid line).

12

500

θLAB = 166°

dσ/dΩ [mb/sr]

400 300 200 100 0

2

3

4

5

Ep [MeV] FIG. 4. The elastic cross-section at a laboratory angle θLAB = 166◦ for the p + 16 O scattering as a function of proton energy Ep . The SMEC calculations have been performed using either DDSM0 [9] (the dashed line) or DDSM1 [10] (the solid line) residual interaction. Experimental cross-sections are from Ref. [16].

13

6

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