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Analysis of the 16O(p,gamma)17F capture reaction using the shell model embedded in the continuum

We apply the realistic shell model which includes the coupling between many-particle (quasi-)bound states and the continuum of one-particle scattering states, called the Shell Model Embedded in the Continuum, to the spectroscopy of mirror nuclei: 17F and 17O, as well as to the description of low energy cross section (the astrophysical S-factor) for E1, E2 and M1 components in the capture reaction: 16O(p,gamma)17F. With the same microscopic input we calculate the phase shifts and differential cross-section for the elastic scattering of protons on 16O.


I. INTRODUCTION
Microscopic description of weakly bound exotic nuclei close to the drip-lines such as, e.g., 8 B or 11 Li in their ground state (g.s.), or nuclei close to the β-stability line in the excited configurations like, e.g., 17 F in the first excited state J π = 1/2 + , is an exciting theoretical challenge due to the proximity of particle continuum. Closeness of the scattering continuum implies that virtual excitations to continuum states cannot be neglected as they modify the effective interactions and cause the large spatial dimension of nuclear density distribution and, in particular, the existence of halo structures. Even basic concepts of the nuclear collective model hardly apply for those weakly bound configurations as the particle motion in the weakly bound orbit is presumably decoupled from the rest of the system [1]. The microscopic description of such exotic configurations should treat properly the residual coupling of discrete configurations and the scattering continuum. Recently, it was proposed to approach this difficult problem in the quantum open system formalism which does not separate the subspaces of (quasi-) bound (the Q subspace) and scattering (the P subspace) states [2,3]. Such a formalism can be provided by the Continuum Shell Model (CSM) [4] which in the restricted space of configurations generated using the finite-depth potential has been studied for the giant resonances and for the radiative capture reactions probing the microscopic structure of these resonances [4][5][6].
In this context, capture reaction 16 O(p, γ) 17 F is interesting for several reasons. First of all, precise experimental data in the energy range from 200 keV to 3750 keV are now available [7] and the decays to the g.s. (J π = 5/2 + ) and to the first excited state (J π = 1/2 + ) of 17 F have been accurately resolved [7]. The strikingly different behavior of the astrophysical S-factor for the proton capture into the 5/2 + state and into the 1/2 + weakly bound state (Q = 105 keV) has been explained by the existence of proton halo in the 1/2 + state of 17 F [7]. Different theoretical approaches, including the potential model [7,8], the model based on the Generator Coordinate Method (GCM) [9], or the K-and R-matrix analysis [10], have been tried to describe this reaction. Moreover, an expected simplicity of the low energy wave functions in 16 O, 17 O and 17 F, allows to test certain salient features of models such as the effective interactions between Q and P subspaces and the possible quenching of matrix elements of the residual coupling which depends on the model space used in the calculations.
Secondly, the exact knowledge of rate for the reaction 16 O(p, γ) 17 F is necessary for modelling of nucleosynthesis process in the hydrogen-burning stars. Explosive hydrogen burning is believed to occur at various sites in the Universe, including novas, X-ray bursts, or the supermassive stars [11,12]. Hydrogen burning of second-generation stars proceeds mainly through proton -proton chain and CNO cycle. The changeover from the pp chain to the CNO cycle happens near T ≃ 2 · 10 7 K. The reaction 16 O(p, γ) 17 F is of particular interest in this context as it provides a link to the higher branches of the CNO cycle. In particular, it starts the sidebranch : 16 O(p, γ) 17 F(p, γ) 18 Ne(β + ν) 18 F(p, α) 15 O. The contribution of CNO cycles to the total amount of produced energy in the Sun is small and CNO neutrinos account only for about 0.02 of the total neutrino flux [13]. Moreover, most of them are coming from the decay of 13 N and 15 O in the main CNO cycle CNO-I. Flux of ' 17 F-neutrinos', which is again two orders of magnitude smaller than the flux of neutrinos from 13 N and 15 O in CNO-I reactions, is controlled by the reaction 16 O(p, γ) 17 F in the CNO-II cycle. Hence, the measurement of CNO neutrinos coming from different sources provides an accurate handle on the thermonuclear reaction process in stars like the Sun and allows in principle to distinct between different branches of the CNO cycle.
The Shell Model Embedded in the Continuum (SMEC) model, in which realistic Shell Model (SM) solutions for (quasi-)bound states are coupled to the one-particle scattering continuum, is a recent development of the Continuum Shell Model (CSM) [4][5][6] for the description of complicated low energy excitations of weakly bound nuclei. The SMEC approach is based on the realistic SM which is used to generate the N-particle wave functions. This deliberate choice implies that the coupling between SM states and the one-particle scattering continuum has to be given by the residual nucleon -nucleon interaction. In SMEC, like in the CSM, the bound (interior) states together with its environment of asymptotic scattering channels form a quantum closed system. Using the projection operator technique, one separates 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.) wave functions and by the s.p. resonance wave functions. P subspace is assumed to contain (N − 1)-particle states with nucleons on bound s.p. orbits and one nucleon in the scattering state. Also the s.p. resonance wave functions outside of the cutoff radius R cut are included in the P subspace. The resonance wave functions for r < R cut are included in the Q subspace. The wave functions in Q and P are then properly renormalized in order to ensure the orthogonality of wave functions in both subspaces. The application of the SMEC model for the description of structure for mirror nuclei: 8 B, 8 Li, and capture cross sections for mirror reactions: 7 Be(p, γ) 8 B, 7 Li(n, γ) 8 Li has been published recently [2,3].
We aim in SMEC at a microscopic description of low lying, complicated many-body bound and resonance states. For that reason, the description of particle continuum is restricted to the subset of one-nucleon decay channels. This should be a reasonable approach for describing the structure of mirror nuclei 17 F and 17 O around 16 O, and the capture reactions: 16 O(p, γ) 17 F and 16 O(n, γ) 17 O. One expects that at higher excitation energies, e.g., above α emission threshold, the one-particle continuum approximation is too restrictive and the residual correlations generated in bound state wave functions by the coupling to those channels cannot be described. Effects of such correlations have been seen in the structure of J π = 3 + resonances in 8 B and 8 Li [3] which strongly couples to the three-particle decay channels. In this case, one should try to employ methods based on the cluster expansion of the wave function and the three-body continuum models [14][15][16]. More complicated twoparticle channels like, e.g., the α -decay channel, can be treated in SMEC, following the approach of Balashov et al. [17].
The paper is organized as follows. In Sect. II, we remind certain elements of the SMEC formalism and, in particular, those features of the S-matrix in SMEC which are involved in the calculation of the elastic cross-sections, phase shifts and the capture cross-sections. Sect. III is devoted to the discussion of specific properties of 17 F and 17 O. In particular, properties of the matrix elements of the effective operator which couple Q and P subspaces in 17 F and 17 O will be discussed in Sect. III.B. Features of the self-consistent average potentials in Q subspace for different many-body states of 17 F will be presented in Sect. III.C. Sect. IV is devoted to the discussion of 16 O(p, γ) 17 F capture cross sections for different multipolarities and different final states of 17 F. The differential elastic cross-section and the elastic phaseshifts in p + 16 O scattering will be compared to the experimental data in Sect. V. Summary and outlook will be given in Sect. VI.

II. THE FORMALISM OF SHELL MODEL EMBEDDED IN THE CONTINUUM
The full solution of SMEC approach is constructed in three steps. In the first step, one calculates the (quasi-) bound many-body states in Q subspace. For that one solves the multiconfigurational SM problem : where H QQ ≡ QHQ is the SM effective Hamiltonian which is appropriate for the SM configuration space used. For solving (1), we use the code ANTOINE [18] which employs the Lanczos algorithm and allows for the diagonalization in large model spaces.
For the coupling between bound and scattering states around 16 O, we use either a combination of Wigner and Bartlett forces (WB force) [3] : or the density dependent interaction (DDSM1) : (2) is the spin exchange operator and ρ(r) in (3) is : with r 0 = 2.64 fm and d = 0.58 fm. The DDSM1 force (3) has four parameters determined as:v in 00 = 36.45,v ex 00 = −297.9,v in 01 = 109.4 andv ex 01 = 115.5, all in units MeV·fm 3 . Similar force has been used by Schwesinger and Wambach [19] for the description of giant resonances in (1p -1h) -(2p -2h) space. However, as compared to their interaction which is close to the Landau -Migdal type of interactions, the DDSM1 parametersv in 00 ,v ex 00 ,v in 01 andv ex 01 , are reduced by a constant factor 0.68. Also the parameter r 0 has been somewhat reduced to better fit the experimental matter radius in oxygen. For the WB residual interaction (2), the overall strength V (0) 12 is adjusted to reproduce as good as possible the energy spectra and the decay widths of states in nuclei around 16 O. A reasonable compromise is provided by V (0) 12 = 300 MeV·fm 3 . The relative contribution of spin exchange term has been discussed for the p-shell nuclei 8 B and 8 Li [3]. It was argued that the spin exchange part is very small ((1 − α) ≃ 0) for most p-shell nuclei but is expected to increase for heavier nuclei. In present studies, we use (1 − α) = 0.27, which is a standard value in the 1s0d mass region [4]. It should be stressed also that matrix elements of H QQ are not modified by the residual coupling between P and Q because they are fitted to experimental discrete levels and narrow resonances in the continuum and, therefore, they are believed to contain already those coupling effects.
The SM wave function has an incorrect asymptotic radial behavior for unbound states. Therefore, to generate both the radial s.p. wave functions in the Q subspace and the scattering wave functions in P subspace we use the average potential of SW type with the spin-orbit part included: whereλ 2 π = 2 fm 2 is the pion Compton wavelength and f (r) is the spherically symmetrical SW formfactor : The Coulomb potential V C in (5) is calculated for the uniformly charged sphere with radius R 0 . For the continuum part, one solves the coupled channel equations: where index c denotes different channels and H P 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. The channel -channel coupling potential in (7) is : where T is the kinetic-energy operator and υ J cc ′ is the channel-channel coupling generated by the residual interaction. The potential for channel c in (8) consists of 'initial guess', U(r), and diagonal part of coupling potential υ J cc which depends on both the s.p. orbit φ l,j and the considered many-body state J π . This modification of the initial potential U(r) change the generated s.p. wave functions φ l,j defining Q subspace which in turn modify the diagonal part of the residual force, etc. This means that the solution of coupled channel equations (7) is accompanied by the self-consistent iterative procedure which yields for each channel independently the new self-consistent potential : and consistent with it the new renormalized matrix elements of the coupling force. The parameters of the initial potential 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 inhomogeneous coupled channel equations: with the source term w i which is primarily given by the structure of N -particle SM wave function Φ i . The explicit form of this source is given in [3]. These equations define functions ω (+) i , which describe the decay of quasi-bound state Φ i in the continuum. The source w i couples the wave function of N-nucleon localized states with (N − 1)-nucleon localized states plus one nucleon in the continuum. Formfactor of the source term is given by the self-consistently determined s.p. wave functions.
The full solution of SMEC equations is expressed by three functions: Φ i , ξ c E and ω i [3,4] : where is the effective SM Hamiltonian which includes the coupling to the continuum, and G (+) P is the Green function for the motion of s.p. in the P subspace. Matrix H ef f QQ is non-Hermitian (the complex, symmetric matrix) for energies above the particle emission threshold and Hermitian (real) for lower energies. The eigenvaluesẼ i − 1 2 iΓ i are complex for decaying states and depend on the energy E of particle in the continuum. The energy and width of resonance states are determined by the condition:Ẽ i (E) = E [4]. The eigenstates corresponding to these eigenvalues can be obtained by the orthogonal but in general non- where the coefficients β ij form a complex matrix of eigenvectors in the SM basis satisfying : Analogously, one can define :ω andΩ 4 He core are taken from generalized monopoles fit of energies of single-particle and singlehole states over the mass table [25]. Then the cross monopoles are adjusted to reproduce 15 N, 15 O and 17 O, 17 F spectra. The center of mass components are removed with the usual Gloeckner -Lawson prescription [26] adding a center of mass kinetic energy hamiltonian. For natural parity states, we consider only 0hω excitations and for non natural parity states, 1hω excitations. Of course, it is well known and has been studied in the past, that 16 O (and 17 O, 17 F) have significant amount of 2p − 2h (3p − 2h) and 4p − 4h (5p − 4h) admixtures in their low lying states [27,28]. Nevertheless, the shell model calculation of these admixtures is not straightforward: • Firstly, it has been already stressed in the past by different authors [29][30][31], that Nhω mixing converges slowly with N and, if stopped for example after N = 2, causes a strong lowering of 0p − 0h states due to 2hω pairing correlations which were already present in pure 2p − 2h states. One could overpass this artifact of the calculation by artificially lowering the Nhω unperturbed configurations through the monopoles of the interaction, but the control of convergence of the mixing would be impossible as higher order admixtures would destroy the picture.
• Secondly, the building blocks of our interaction are made of 0hω 0p and 1s0d phenomenological interactions, which in principle already contain, in particular through pairing renormalizations, the higher order correlations. For this reason, we have decided to fix the unperturbed (4p − 4h) 0 + state around its experimental energy (6.049 MeV).
Again, the derivation of precise correlated wave functions for 16 O and 17 O, 17 F is out of the scope of this paper and we will refer to the mixing derived by other studies [27,28].

B. The effective operator of the residual coupling
Expressions for the matrix elements of residual interaction which couples P and Q subspaces have been given in Refs. [4,3]. There are two kinds of coupling operators. The channel -channel coupling reduced matrix elements involve one-body operators : Reduced matrix elements of the source term w i in the inhomogeneous coupled channel equations (10) contain a product of two annihilation operators and one creation operator : We perform the SM calculations for 16 O ground state in which admixture of 2p − 2h and 4p − 4h configurations are neglected : i.e., the g.s. of 16 O is taken to be a pure (0p 12 ) 0p − 0h configuration and the second excited 0 + 2 state with predominantly 4p − 4h structure is not mixed with the ground state. How big are these np − nh admixture depends on the SM effective interaction and the effective s.p. space used [27,28]. In Table I we give results of different SM calculations for 0 + 1 in 16 O. Likewise in 17 O and 17 F, the g.s. J π = 5/2 + 1 and the excited states J π = 1/2 + 1 , 3/2 + 1 correspond in our approximation to the pure configuration of one particle in, respectively, 0d 5/2 , 1s 1/2 or 0d 3/2 shells outside of the (0p 12 )-core of 16 O, i.e.: The amount of these admixture calculated by Brown-Green [27] and Zuker-Buck-McGrory [28] can be read from the Table II. The experimental spectroscopic amplitudes for 5/2 + 1 and 1/2 + 1 states can be found in Table III. The negative parity states involve the excitations across the shell closure and involve one hole in the p shell and two particles in the 1s0d shells. In our model of 16 O where 2p − 2h and 4p − 4h configurations are neglected, the spectroscopic factors of negative parity SM states are equal zero. Hence the particle decay width for these states is coming from the functions ω i (10) which describe the continuation of Q space wave functions Φ i into P , i.e., from the modification of discrete states by the coupling to the continuum (18).
Neglecting higher order correlations in the g.s. of 16 O and in positive parity excited states of either 17 F or 17 O means that residual coupling between Q and P which involves those states should be properly rescaled, i.e., the matrix elements of O, R should be quenched : Amount of configuration mixing for negative parity states is large (two particles in 1s0d shell and 1 hole in 0p shell) and therefore matrix elements of R are not quenched. In our calculations, we take the values given by Brown -Green [27] for the amplitudes a 00 and a J i 10 (see Tables I and II).

C. The self-consistent average potentials
Construction of the Q subspace in SMEC is achieved by the self-consistent, iterative procedure which for a given initial average s.p. potential (5) and for a given residual twobody interaction between Q and P yields the self-consistent s.p. potential U (sc) (r) which depends on the s.p. wave function φ l,j , the total spin J of the N-nucleon system as well as on the one-body matrix elements of (N − 1) -nucleon daughter system. As explained above, the positive parity states in 17 F and 17 O are described as a pure one particle configuration in 1s0d shells. Consequently, the spectroscopic factors for the states J π = 1/2 + , 5/2 + and 3/2 + are equal 1. Consistently with this approximation, we can identify position of proton (neutron) s.p. orbits 1s 1/2 , 0d 5/2 and 0d 3/2 with, respectively, J π = 1/2 + , 5/2 + and 3/2 + many body states of 17 F ( 17 O), i.e., we may ask that U (sc) (J π ) provides the energy of s.p. orbit at the energy of the corresponding many-body state with respect to the threshold. This choice is essential for quantitative description of radiative capture crosssection. Consequently, the initial SW potentials U(J π ) have all different depth parameters. Moreover, they depend on the choice of the residual interaction coupling P and Q. On the contrary, we keep a common radius R 0 = 3.214 fm, a common diffuseness a = 0.58 fm and a common spin-orbit strength V SO = 3.683 MeV for all of them. The depth parameters of initial SW potentials for [p 16 O] and [n 16 O] systems, are summarized in Table IV. For a given nucleus ( 17 F or 17 O), all those different central potentials correspond to the one and the same equivalent potential U (eq) (r) in the SW form which binds s.p. orbits at the same energy as obtained in the self-consistent potentials for neutrons and protons. For both WB and DDSM1 residual interactions, the depth of U (eq) is V 0 = −52.46 MeV in 17 F and V 0 = −52.49 MeV in 17 O. The potential radius, the diffuseness and the spin-orbit strength are the same as used for the initial potentials U(J π ). The spectrum of singleparticle (-hole) energies in this potential is given in Table V for 17 F and in Table VI for 17 O. These energies are determined from the data of 17 F, 17 O (for particles) and 15 N, 15 O (for holes). The difference of s.p. energies ǫ p (1s 1/2 ) − ǫ p (0d 5/2 ) = 0.495 MeV in 17 F and ǫ n (1s 1/2 ) − ǫ n (0d 5/2 ) = 0.871 MeV, which should be compared with the excitation energy of J π = 1/2 + 1 state in those nuclei, is a manifestation of the Thomas-Ehrman shift [38] and is included effectively in our equivalent and self-consistent s.p. potentials for all studied channels.
The self-consistent potential (the solid line) strongly deviates from the SW form. In the center U (sc) (r) has a strong maximum which is absent in the initial potential U(r) (the dashed line). The self-consistent potentials U (sc) (1/2 + ) and U (sc) (5/2 + ) are different (compare the solid curves on l.h.s. and r.h.s. of Fig. 1), in spite of the fact that up to spinorbit coupling the equivalent potential U (eq) (the dotted lines) in these states is identical. This clearly shows how strong is the state dependence of both the self-consistent average fields and the renormalized matrix elements of the coupling force. One should also notice the difference in surface region between U (sc) (1/2 + ), U (sc) (5/2 + ) and U (eq) . In U (sc) (5/2 + ) one may notice decrease of the potential radius in comparison with the radius of U (eq) . However, as compared to the initial average potentials U(5/2 + ) and U(1/2 + ), both the self-consistent potentials U (sc) (5/2 + ), U (sc) (1/2 + ), and the equivalent potential U (eq) are deeper and their effective radii are bigger.
In Fig. 2 we show examples of calculated potentials in 17 F for the proton s.p. orbital 0p 1/2 in the many-body states J π = 1/2 − . The calculations have been performed using the initial potentials U(1/2 − ) for the WB (l.h.s. of the plot) and DDSM1 (r.h.s. of the plot) residual interactions. For both interactions, the equivalent potential U (eq) (r) is the same. It is interesting to notice how different are the self-consistent U (sc) (r) potentials for the two considered residual interactions. As a rule, the renormalization of initial s.p. potential is weaker for the density-dependent DDSM1 interaction. Similarly as for the self-consistent potentials in J π = 5/2 + and J π = 1/2 + states (see Fig. 1 ), the effective radius of U (sc) (r) shrinks as compared to the radius of U (eq) (r) . Again this effect is stronger for the WB residual interaction.
In general, the surface region of average potential shows weak sensitivity to the selfconsistent correction. In our case, the radial dependence of self-consistent correction for all positive parity states in 17 F and 17 O is given by the 0p 1/2 and 0p 3/2 radial formfactors corresponding to the well bound single-hole state and not by the weakly bound single-particle states 1s 1/2 or 0d 5/2 . For that reason, even for the J π = 1/2 + 1 state in 17 F which is bound by 105 keV, induced renormalization of the surface in the self-consistent potential decreases the radius of potential in the surface region.

D. Discussion
In this section, we shall present the SMEC results for the spectrum of mirror nuclei: 17 F and 17 O. The results depend mainly on: (i) the nucleon -nucleon interaction in Q subspace, (ii) the residual coupling of Q and P subspaces, (iii) the self-consistent average s.p. potential which generates the radial formfactor for s.p. bound wave functions and s.p. resonances, and (iv) the cutoff radius for s.p. resonances. Below, we shall comment on their relative importance. The iterative procedure to correct U(r) and to include the diagonal part contribution of residual interaction has been described in the previous section. The self-consistently determined s.p. potential U (sc) (r) is then used to calculate radial formfactors of coupling matrix elements and s.p. wave functions. Different initial potentials U(J π ) (see Table IV) are used for the calculation of self-consistent average potentials for different many-body states in 17 F. These different self-consistent potentials correspond to a unique equivalent SW potential U (eq) (r). The renormalization of initial potential by the residual coupling of Q and P subspaces is the same for all states J π of the same spin and parity. For positive parity states, as a result of the restriction in SM calculations in Q subspace, the renormalization of average potential operates only for s.p. orbits which have the same spin j and parity π as those of the many body states J, π. E.g., for the many body state J π = 5/2 + of 17 F ( 17 O), only potential of the 0d 5/2 s.p. orbit is modified by the coupling to the continuum. Similarly for J π = 1/2 + many body state and the s.p. orbit 1s 1/2 . For those s.p. orbits which in a given many body state are not modified by the selfconsistent renormalization, we calculate radial formfactors using the 'universal' equivalent s.p. potential U (eq) (r) which does not depend neither on the many body state nor on the s.p. orbit. Of course, this special property is only due to a simple structure of 16 O and 17 F nuclei in our SM calculations. For neutrons in 17 F there is no renormalization of the average potential and we use the equivalent potential for neutrons to get the radial formfactors (see Table V). Supplementary informations concerning the results shown in Fig. 3 can be found in Table VII.
The spectrum of 17 F is insensitive to certain approximations in the SMEC. G.s. energy relative to the proton emission threshold is reasonably well reproduced by the SMEC. Coupling to the continuum induces strong relative shifts of 5/2 + 1 and 3/2 + 1 states with respect to the 1/2 + 1 state and the negative parity states. The position of g.s. with respect to the energy threshold for proton emission changes by about 500 keV due to the inclusion of coupling to the g.s. of 16 O (compare columns denoted 'SM' with those denoted 'SMEC (WB)' and 'SMEC (DDSM1)' in Fig. 3 and in Table VII). These shifts are generally smaller for the DDSM1 residual coupling. The coupling to excited 0 + 2 state at E * = 6.049 MeV in 16 O (0 + 2 has predominantly 4p − 4h structure with respect to the dominant 0p − 0h configuration in the g.s. 0 + 1 ) is expected to be unimportant in 17 F or 17 O. Calculated width of states depends on chosen residual couplings (compare 'SMEC (WB)' with 'SMEC (DDSM1)') and in general the agreement is better for the density dependent interaction. The effective quenching of the coupling matrix elements (35)(36)(37) for positive parity states does not solve this problem completely. To get a better agreement with the data, the energy separation of 5/2 + 1 and 1/2 + 1 SM states should be decreased while the splitting of 5/2 + 1 and 3/2 + 1 SM states should remain unchanged. This means that the separation of 1s subshell and the centroid od 0d subshells should be increased by few hundred keV. We did not however pursue the studies in this direction and we keep standard values of s.p. energies [25].
The effect of quenching of the coupling operator between Q and P can be seen in Fig. 4 where the comparison between the SMEC calculations with the quenching factors (entry 'SMEC (DDSM1)') and those without quenching factors (entry 'SMEC (DDSM1 * ') are shown. The strong downshift due to the continuum coupling is seen for the (5/2 + 1 -3/2 + 1 )pair with respect to both the negative parity states and the 1/2 + 1 state which moves downwards much less. The effect of quenching of the coupling operator on negative parity states is, as expected, very weak.
A useful measure of the radial wave function is the B(E2) transition matrix element between 1/2 + 1 and 5/2 + 1 bound states. In SMEC, we obtain 70.3 e 2 fm 4 for DDSM1 residual coupling. This value has been calculated using the effective charges : e p = 1.41 and e n = 0.47 and the BG quenching factors (Table II). The experimental value for this transition is B(E2) exp = 64.92 e 2 fm 4 . The SM prediction for this transition is B(E2) SM = 32.7 e 2 fm 4 assuming the oscillator length b = A 1/6 fm and the same effective charges as used in the SMEC calculations. The better agreement of calculated in SMEC B(E2) values with the data is mainly due to more realistic radial dependence of the 1s 1/2 s.p. orbit in J π = 1/2 + 1 many body state. The rms radius for this orbit in SMEC is < r 2 > 1/2 = 5.212 fm, as compared to the value of < r 2 > 1/2 = 3.629 fm for the 0d 5/2 s.p. orbit in J π 1 = 5/2 + state. Nuclear rms radius of 17 F is 2.664 fm in the g.s. and 2.814 fm in the first excited state 1/2 + .  Fig. 5, zero on the energy scale is fixed at the experimental position of J π = 5/2 − 1 state which is close to the neutron emission threshold.
The self-consistently determined s.p. potential U (sc) (r) is then used to calculate radial formfactors of coupling matrix elements and s.p. wave functions. Different initial potentials U(J π ) for [n 16 O] (see Table IV) are used for the calculation of self-consistent average potentials for different many-body states in 17 O. These different self-consistent potentials correspond to a unique equivalent SW potential U (eq) (r) (see Table VI). For protons there is no correction from the residual interaction modifying the average potential and we use the equivalent potential for protons to get the radial formfactors (see Table VI).
The energy intervals between the g.s. and the positive parity states 1/2 + 1 and 3/2 + 1 in SMEC(DDSM1) calculations perfectly reproduce the experimental energy sequence for these levels. The WB residual force leads to a too strong relative shift of 5/2 + 1 with respect to 1/2 + 1 and 3/2 + 1 states. The quenching factors have been introduced in the calculations for positive parity states similarly as discussed before for 17 F. One should remind that in 17 F, in spite of using the quenching factors, the 5/2 + 1 state was shifted downwards with respect to other positive parity states. Improved situation in 17 O is due to the larger separation of this state with respect to the particle continuum. Nevertheless, the coupling to the continuum is necessary here to bring closer the 3/2 + 1 and 1/2 + 1 SM states. The shift of the ground state energy in SMEC relative to the neutron emission threshold is solely due to the choice of zero on the energy scale at the experimental position of 5/2 − 1 weakly bound state. Calculated width of states depends on chosen residual couplings (compare 'SMEC (WB)' with 'SMEC (DDSM1)') and in general the agreement is better for the density dependent interaction.
As a measure of the radial wave function in 17 O we calculate the B(E2) transition matrix element between 1/2 + 1 and 5/2 + 1 bound states. In the SMEC, we obtain 5.1 e 2 fm 4 for DDSM1 residual coupling. Changing the residual coupling and using WB interactions leaves this value almost unchanged. The B(E2) value in SMEC has been calculated using the effective charges: e p = 1.41 and e n = 0.47, and the BG quenching factors (Table II). The experimental value for this transition is B(E2) exp = 6.2 e 2 fm 4 whereas the SM prediction is B(E2) SM = 3.62 e 2 fm 4 assuming the oscillator length b = A 1/6 fm and taking the same effective charges as in the SMEC calculations. The close agreement of calculated in SMEC B(E2) values with the data is due to a more realistic radial dependence of the 1s 1/2 s.p. orbit in J π = 1/2 + 1 many body state. The rms radius for this orbit is < r 2 > 1/2 = 4.184 fm, as compared to the value: < r 2 > 1/2 = 3.439 fm for the 0d 5/2 s.p. orbit in J π 1 = 5/2 + many body state. The dominant contribution to the total capture cross-section for both 5/2 + 1 and 1/2 + 1 final states, is provided by E1 transitions from the incoming p wave to the bound 0d 5/2 and 1s 1/2 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 f 7/2 wave gives is by a factor ∼100 smaller than the contribution from p 3/2 wave for energies of incoming proton up to 3.5 MeV. The E2 contribution is smaller by at least three orders of magnitude in both branches.
In the transition to the first excited state 1/2 + 1 , the dominant M1 contribution comes from the incoming s 1/2 wave and is by ∼2 orders of magnitude smaller than the E1 contribution. The M1 contribution is important, in particular, in the branch 16 O(p, γ) 17 F(J π = 5/2 + 1 ) for E CM > 3 MeV, where its contribution is of a similar order as the E1 contribution due to 3/2 + 1 resonance in the d 3/2 scattering wave. One should also notice that the energy dependence of M1 component in the capture cross section is totally different in 16 O(p, γ) 17 F(J π = 1/2 + 1 ) and 16 O(p, γ) 17 F(J π = 5/2 + 1 ) branches. The contribution of M1 for 16 O(p, γ) 17 F(J π = 1/2 + 1 ) is decreasing with increasing energy above E CM ∼ 1 MeV and approaches zero for E CM ∼ 3.5 MeV. The M1 contribution begins to grow again at higher energies. Since σ M 1 cross section for 1/2 + 1 final state originates only from the absorption of s-wave proton and, moreover, since by construction: < Ψ f |Ψ i > = 1 (the spectroscopic factor of 1/2 + 1 state equals 1), therefore the energy behavior of σ M 1 cross section is governed by the Breit-Wigner like factor coming from the 1/2 + 1 many-body wave function (see eq. (11) or (17)). So σ M 1 depends crucially on the energy dependence of this single eigenvalue. While the real part of E(J π = 1/2 + 1 ) eigenvalue behaves smoothly with energy, the imaginary part, on the contrary, after initial rise becomes almost equal to zero at E CM ≃ 3.5 MeV and starts to rise again for higher energies. Since we have only one J π = 1/2 + state in our SM space, the energy behavior of its eigenvalue determines directly the behavior of σ M 1 cross-section, because in the limit: Γ → 0, the Breit-Wigner factor behaves like the δ-function. In the more complete SM calculations which yield more J π = 1/2 + states, this unusual effect is expected to be reduced. Nevertheless, because its principal cause is the single-partial-wave characteristic of this transition, we believe that the trace of it remains.
In the branch 16 O(p, γ) 17 F(J π = 1/2 + 1 ), one may also notice small resonant part in E1 at the position of 1/2 − 1 resonance, but otherwise the cross-section has non-resonant behavior. The non-resonant E1 contribution is a good measure of spatial extension of 5/2 + 1 and 1/2 + 1 wave functions respectively. For the decay branch 16 O(p, γ) 17 F(J π = 1/2 + 1 ), due to the very small binding energy of J π = 1/2 + 1 state (Q = 105 keV) and its particularly simple structure, this cross section is a sensible measure of the extension of the proton 1s 1/2 orbit. It is essential for the calculated cross-section that the 1s 1/2 proton orbit in the self-consistent average potential for J π = 1/2 + is bound by 105 keV. Even small modification of this value by different choice of the depth parameter V 0 in U(r), introduce the modification of S E1 which is larger than any modification due to possible uncertainties in the potential radius R 0 or its surface diffuseness a. Fig. 7 shows the total S-factor as a function of the c.m. energy, as well as its values for the 16 O(p, γ) 17 F(J π = 1/2 + 1 ) and 16 O(p, γ) 17 F(J π = 5/2 + 1 ) branches. The overall agreement with the experimental data of Morlock et al [7] is very good. The resonance contributions of 1/2 − 1 in 16 O(p, γ) 17 F(J π = 1/2 + 1 ) is well reproduced even though their precise energy position is slightly shifted. This resonance contributes only through its P subspace continuation because the spectroscopic factor for this state is equal zero. The contribution of 5/2 − 1 resonance in the 16 O(p, γ) 17 F(J π = 5/2 + 1 ) branch is so narrow in energy that has been omitted in the figure. Energy of this resonance in SMEC (DDSM1) is E CM = 3.44 MeV.

V. ELASTIC CROSS-SECTION AND PHASE-SHIFTS
Another observable quantities that can be calculated using the solutions of the SMEC are the elastic phase shifts and elastic cross-sections for different proton bombarding energies. Results for the elastic phase shifts, shown in Figs. 9, and those for the elastic cross-section, shown in Fig. 10, have been obtained using the SMEC solutions for identical parameters of initial potentials and the DDSM1 residual coupling between Q and P subspaces as discussed before in Sect. III.D.1 for the spectra of 17 F and in Sect. IV for the capture reaction 16 O(p, γ) 17 F(J π ). The elastic phase shifts are well reproduced by the SMEC for all states except for the 5/2 + . In this case the calculated and experimental [39] values differ by about 5 • . This discrepancy disappears when the resonant contribution to this phase shift is removed from the SMEC solution (compare the solid and dashed lines in the plot for 5/2 + in Fig. 9). It may be surprising at first sight to see so large resonant contribution to the phase shift associated with the g.s. wave function. For energies below the proton emission threshold the coupling to the continuum introduces only the hermitean modifications of the Hamiltonian which shift the energy of 5/2 + 1 state in SMEC with respect to its initial position given by the SM but do not generate any width for this state. The coupling of Q and P subspaces is non-local and, hence, the effective Hamiltonian in SMEC is energy dependent. For excitation energies above the proton threshold, the coupling of Q and P subspaces generates the non-hermitean corrections to the effective Hamiltonian which yield the imaginary part of the eigenvalue and generates the resonant like behavior which is so well visible in the 5/2 + 1 elastic phase shift. The origin of this large 'halo' of 5/2 + 1 bound state in the continuum for positive energies and the large shift of the real part of the 5/2 + 1 eigenvalue for negative energies (see Fig. 3 and 4) is the same and should be traced back to the incorrect description of correlations in the SM wave function for this state. As discussed in the previous chapter, one expects that the 5/2 + 1 state in 17 F and 17 O is not a pure s.p. excitation outside of the 16 O core [27]. Therefore, it is also natural to quench the matrix elements of Q -P coupling as described in Sect. III.B. This procedure, which consists of using quenching factors for positive parity states to correct for missing higher order np − nh components in their wave function, is actually not sufficient as the example of elastic phase shift for 5/2 + 1 demonstrates clearly. The quenching procedure cures the problem of unphysically large energy shift for the 5/2 + 1 state due to the coupling to the continuum but does not correct the wave function for missing correlations which in turn lead to the disappearance of 'halo' of this state for positive energies. This deficiency of present SM calculations in 17 F and 17 O, which is seen consistently both in the energy spectrum as well as in the elastic phase shifts, demonstrates potentiality of SMEC approach for mass-regions far off the β -stability valley where the experimental information about exotic nuclei will be scarce and one will have to use both the spectroscopic information and the information from the scattering experiments to learn about the structure of those nuclei. This example shows also that in the SMEC approach which unifies description of discrete state properties and the scattering continuum, one may use different kinds of experimental data to fix those few parameters of the model such as the overall strength of the residual Q -P coupling or the radius and diffuseness of the initial average potential.
Elastic excitation functions at a laboratory angle of 166 • calculated in SMEC with DDSM1 residual interaction are compared with the experimental data [40] in Fig. 10. The calculated values at low energies are too low as compared to the data and this is again due to the too strong halo of 5/2 + 1 state for positive energies (compare the dashed and solid lines in Fig. 10 at low energies). Removal of the resonant contribution from this state to the elastic cross section brings the calculations close to the data. The agreement between calculated and experimental low energy cross-sections provides a supplementary check of the spatial extension of self-consistent potential and, hence, of the radial form factors of s.p. wave functions. One should however keep in mind that, in general, the information from the elastic cross-sections may be strongly perturbed by the non-locality effects in the SMEC effective Hamiltonian which depend strongly on the many-body correlations in the SM wave functions as the above example demonstrates.
On the average, the agreement between experimental and SMEC results for the elastic excitation functions is reasonable if one keeps in mind large sensitivity of this experimental measure to even small inaccuracies in the energy position and width of resonances. The model predicts correctly the interference pattern due to 1/2 − 1 resonance at E p ∼ 2.6 MeV and 3/2 − 2 resonance at E p ∼ 5.2 MeV. Also, the 5/2 − 2 resonance is correctly predicted by the calculations though its width is too narrow to be presented in the figure. For the same reason, this resonance was not plotted in the proton capture cross-section for the 16 O(p, γ) 17 F(J π = 5/2 + 1 ) branch in Fig. 7. The difference between interference patterns in the data and in the SMEC calculation for 3.8 MeV < E p < 5 MeV is mainly due to the reversed order of 3/2 − 1 and 3/2 + 1 resonances in SMEC calculations as compared to the data. Consequently, the hole in the experimental elastic excitation function at E p ∼ 4.2 MeV, due to 3/2 − 1 resonance interfering with the 3/2 + 1 resonance cuts sharply only the high energy tail of the 3/2 + 1 resonance.

VI. SUMMARY AND OUTLOOK
In this work we have applied the SMEC approach for the microscopic description of 17 F and 17 O spectra, the low-energy radiative capture cross sections in the reaction 16 O(p, γ) 17 F, and the elastic cross section for the reaction 16 O(p, p) 16 O. In the SMEC model, which is a development of CSM model [4,5] for the description of low energy properties of weakly bound nuclei, realistic SM solutions for (quasi-)bound states are coupled to the one-particle scattering continuum. For that reason, we use realistic SM effective interaction in the Q subspace and introduce residual force which couples Q and P subspaces. (For this residual coupling we take either a combination of Wigner and Bartlett forces or the density dependent DDSM1 interaction which is similar to the Landau -Migdal type of interactions.) This deliberate choice of interactions implies that the finite-depth potential generating P space and matrix elements of the residual Q − P coupling, have to be determined self-consistently. The self-consistent iterative procedure yields then new state-dependent average potentials and consistent with them new renormalized matrix elements of the coupling force. These renormalized couplings and average potentials are then consistently used both in Q and P subspaces for the calculations of spectra, capture cross-sections, elastic cross-sections, elastic phase-shifts, etc.
Simultaneous studies of spectroscopy in mirror systems as well as different reactions involving one nucleon in the continuum, allow for a better understanding of the role of different approximations and parameters in the model. The dependence on radius, diffuseness or spin-orbit coupling parameters of the initial potential U(r) is not very important and they can be taken from any reasonable systematics. On the contrary, the depth of U(r) has to be carefully adjusted so that the energies of s.p. orbits in U (sc) (r) for [n (N − 1)] and [p (N − 1)] systems, whenever their identification is possible, reflect the binding of manybody states near the particle emission threshold in the nucleus N. This is very important for the quantitative description of reaction cross-sections. In the case of 17 F ( 17 O), correct identification of 1s 1/2 and 0d 5/2 s.p. orbits and hence the determination of an appropriate depth parameter in U(1/2 + ) and U(5/2 + ) is unambiguous because the spectroscopic amplitudes in 1/2 + 1 and 5/2 + 1 are close to 1. Different binding of mirror nuclei 17 F/ 17 O leads for the same J π many-body states to different U (sc) (r) in these nuclei. This in turn causes breaking of mirror symmetry of SM spectra for these nuclei and is, e.g., an essential ingredient in understanding the difference between B(E2) values for the transition 5/2 + 1 → 1/2 + 1 in 17 F and 17 O, as discussed in Sect. III.
SMEC model in its present form includes the coupling to one-nucleon continuum. The wealth of experimental data can be described in a unified framework of SMEC in this approximation. These include: (i) the calculation of energy spectra, B(Πλ) transition matrix elements and various static nuclear moments such as the magnetic or mass/charge quadrupole moments etc., (ii) the calculation of various radiative capture processes: (p, γ), (n, γ), Coulomb breakup processes: (γ, p), (γ, n) and elastic or inelastic cross sections (p, p ′ ), (n, n ′ ); some of these observables have been discussed in this work. Problem of isospin symmetry breaking due to the coupling to the continuum can be addressed by comparing electromagnetic processes, e.g., B(Πλ) transition matrix elements for certain states in mirror nuclei, and weak interaction processes like the first-forbidden β-decay in mirror reactions. Finally, for nuclei close and beyond the proton (neutron) drip lines, the spontaneous proton (neutron) radioactivity can be studied in the microscopic framework of SMEC (SM). These unifying features of SMEC approach are extremely useful for understanding of the structure of exotic nuclei far from the β -stability for which the available experimental information will be scarce.
In this work we have studied nuclei close to the doubly magic 16 O in order to understand certain basic features of the SMEC and, in particular, of the Q -P coupling operator acting in the restricted SM configuration space. The resulting quenching of operators O and R ( Eqs. (31) and (32) respectively) could be 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. It was found also that this SM motivated correction of the effective operator does not solve the problem of 'halo' of discrete states for positive energies. This problem results from non-locality of the effective SMEC Hamiltonian and, more precisely, from the non-hermitean corrections to the eigenvalues for positive energies which generate the imaginary part. The imaginary part of eigenvalues and, hence, the size of this halo effect, is particularly large for pure single-particle (single-hole) configurations. For this reason, the simplification of structure of the many body states by neglecting the configuration mixing can in certain cases lead to an unphysical enhancement of resonant-like correction from bound states in, e.g., the elastic cross-section or the elastic phase-shifts. In this sense, 17 F and 17 O nuclei, with predominantly s.p. structure of positive parity states and extreme sensitivity to higher order correlations in the many-body wave functions are somewhat pathological. We believe that this problem will disappear in nuclei having more particles in the open shells, in which case SM will produce sufficient amount of mixing in the wave functions.
More complicated decay channels involving, e.g., α particle, 3 He or 3 H in the continuum, are beyond the scope of SMEC in its present form. The future extension of the SMEC for such cluster configurations is possible in a framework proposed by Balashov et al [17]. It is encouraging, however, that these possible shortcomings in the description of decay channels, are so unambiguously reflected in the calculated decay width for these states [3]. In general, the decay width is particularly sensitive to the details of the SM wave functions involved and to the values of matrix elements of residual coupling so they provide a sensible test of the quality of SMEC wave functions and/or approximations involved.
The present studies have shown that SMEC results depend sensitively on very small number of parameters. Some of them, like the parametrization of the residual interaction which couples states in Q and P subspaces, has been established in the present work for sd-shell nuclei. The others, related to the quenching of the effective coupling operator can be explained consistently with the SM analysis of spectroscopic amplitudes. Finally, the energies of s.p. states, which determine the radial wave function of many-body states, are bound by the SM spectroscopic factors and experimental binding energy in studied nuclei. This gives us a confidence that the SMEC can have large predictive power when applied to the nuclei in the less known regions of the mass table. The calculations can be performed on a similar level of sophistication as the SM which, with the recent progress in SM techniques and the effective interactions has been applied to the medium-heavy nuclei [41].  [31]. The entry WBP(4p4h) corresponds to the results of SM calculations using WBP interaction [32] and the '∆ 4p4h '-method [31]. Results of the Brown-Green (BG) [27], Zuker-Buck-McGrory (ZBM) [28] and Haxton-Johnson (HJ) [33] are also given.