The Feshbach theory of the optical potential naturally leads to a microscopic
description of scattering in terms of the many-body self-energy. We consider a
recent calculation of this quantity for 16O and study the possibility of
applying it at astrophysical energies. The results obtained for the phase
shifts and the 16O(p,\gamma) capture suggest that such studies are feasible but
the calculations require some improvement geared to this specific task.Comment: 4 pages, 3 figures; Proceedings of Nuclei In The Cosmos VIII, to
  appear in Nucl. Phys. 

B.K. Jennings

Barbieri

Baye

C. Barbieri

Dickhoff

Escher

Feshbach

Machleidt

Mahaux

Morlock

Müther

Salisbury

English

arXiv

Barbieri, C

Jennings, BK

Surrey Research Insight

Study of the O-16(p,gamma) reaction at astrophysical energies

Crossref

Study of the 16O(p,γ) Reaction at Astrophysical Energies

The Feshbach theory of the optical potential naturally leads to a microscopic description of scattering in terms of the many-body self-energy. We consider a recent calculation of this quantity for 16O and study the possibility of applying it at astrophysical energies. The results obtained for the phase shifts and the 16O(p,γ) capture suggest that such studies are feasible but the calculations require some improvement geared to this specific task

Barbieri C.

Jennings B. K.

AIR Universita degli studi di Milano

arXiv:nucl-th/0408017v1  5 Aug 20041Study of the 16O(p,γ) Reaction at Astrophysical EnergiesC. Barbieria∗ † B. K. JenningsaaTRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, Canada V6T 2A3The Feshbach theory of the optical potential naturally leads to a microscopic descriptionof scattering in terms of the many-body self-energy. We consider a recent calculation ofthis quantity for 16O and study the possibility of applying it at astrophysical energies. Theresults obtained for the phase shifts and the 16O(p,γ) capture suggest that such studiesare feasible but the calculations require some improvement geared to this specific task.1. ItroductionThe Feshbach theory of the optical potential provides a formal tool for developing adescription of the scattering of nucleons from nuclei in terms of the microscopic interaction.In its most standard application, this is done by reducing the Hilbert space to a subspacethat contains only the core nucleus and the additional particle in a scattering or boundsingle particle state. Since the scattered particle is not allowed to occupy the orbitals filledby the core nucleons, this subspace can only exhaust the part of the one-body Hilbertspace the lies above the Fermi energy). Several models of nucleon-nucleus scattering (suchas cluster model or folding potentials) implicitly use this approach and have been appliedwith significant success. However, the above limitation can be avoided by considering aFock space (without a definite number of particles) and applying the Feshbach formalismto a subspace that contains both the scattering of a particle on top of the nuclear core andthe possibility of propagating a hole excitation [ 2]. The resulting optical potential simplyreduces to the usual many-body self-energy. The properties of this particular choice ofthe optical model have been reviewed by Mahaux and Sartor [ 3].The above considerations suggest that the technique of using many-body Green’s func-tion can make reasonable predictions for nucleon-nucleus scattering. However, most ofits applications in nuclear physics have focused on the the study of nuclear correlations [4]. In this contribution, we consider the self-energy resulting from a recent applicationof the self-consistent Green’s function method the nucleus of 16O [ 5, 6] and explore itspredictions for proton-nucleus scattering.2. The modelIn Refs. [ 5, 6], the nuclear self-energy was computed within a model space P corre-sponding to the harmonic oscillator states of all orbitals up to the pf shell plus the 1g9/2∗Email: barbieri@triumf.ca†URL: http://www.triumf.ca/people/barbieri2*Σ (2p1h)R (2h1p)R= + +Figure 1. Feynman diagram representation of the self energy. The first diagram represent theHartree-Fock like contribution to the mean field. The remaining ones describe core polarizationeffects in the particle (R(2p1h)) and hole (R(2h1p)) part of the spectrum.orbital. A parameter b=1.76 fm was employed and a G-matrix interaction based on theBonn-C potential [ 7] was used in the calculation. When expressed in coordinate space,this self-energy takes the formΣ⋆(r, r′, ω) =∑α,β∈Pφα(r)ΣMFαβ (ω) +∑p(mp+α )∗mp+βω − εp+ + iη+∑hmh−α(mh−β)∗ω − εh− − iηφ∗β(r′) (1)where φα(r) are the harmonic oscillator wave functions, coupled to the nucleon spin withquantum numbers, α = {nα, lα, jα, mα} and the sum runs over all the orbits belongingto the model space 3. We note that this choice for the space P is adequate to studyproperties of the nuclear interior when considering low energy (long range) excitationsand accounting for the spectral fragmentation [ 6]. However the gaussian like functionsφα(r) are not optimal when one is concerned with properties sensitive to the nuclearsurface.In Eq. (1), the mean field part of the optical potential, ΣMF , corresponds to the Hartree-Fock diagram of Fig. 1. The remaining diagrams are usually referred as core polarizationcontributions [ 3]. They were computed in Ref. [ 6] by including the coupling of singleparticle motion (i.e. quasiparticles and quasiholes) to different types of collective motion,namely excitations of the A-particle core, two-particle and two-hole states. At low enoughenergies they can be expressed as a discrete sum of poles, see Eq. (1). There εi± determinethe excitation energies of the resonances that do not have a mean field character.Thus, Eq. (1) gives us a model for the optical potential that acts in the full singleparticle Hilbert space. However, before using it in practical applications one has tocorrect for the fact that the calculations of Ref. [ 6] did not include the electromagneticinteraction and were based on a two-body realistic interaction, which is not sufficient toaccount for spin-orbit splitting. Hence, in this work we augment the self-energy (1) withthe Coulomb potential for a uniformly charged sphere of radius Rc = 3.2 fm and add acorrection U(r, r′). The scattering equation takes a Schro¨dinger-like form [h¯=c=1 and µis the reduced mass]{−∇22µ+ VCoul.(r)}ψ(r) +∫dr′ {U(r, r′) + Σ⋆(r, r′, Ecm)} ψ(r′) = Ecm ψ(r) , (2)which, for Ecm < 0, also describes the bound states of17F. The potentialU(r, r′) =∑αδεα φα(r′)φ∗α(r′) (3)3The isospin degrees of freedom are not shown explicitly here.31 2 3 4 5 6Ecm [MeV]-202468δ N [rad]exp. d5/2exp. d3/2d3/2d5/2s1/2p - 16O scatteringFigure 2. Phase shifts for proton - 16Oscattering in the s1/2, d5/2 and d3/2 partialwaves, as a function of the center of massenergy. The experimental results are fromRef. [ 8].0 0.5 1 1.5 2Ecm [MeV]24681012S(E) [keV b]totald5/2s1/216O(p,γ)17FFigure 3. Astrophysical factor for the16O(p,γ) capture. The solid curves give thetheoretical results for capture to the d5/2 ands1/2 states and the total capture. The dashedcurves represent the same results after rescal-ing by a constant as described in the text.The experimental results are from Ref. [ 9].is chosen in analogy to the work of Ref. [ 6] and acts by simply shifting the energy ofthe principal mean field orbitals. In practice, this corresponds to modifying the depth ofthe optical potential independently for each different partial wave. In the present workwe found that one can fit the energy levels of the orbitals in the p and sd shells bychoosing δε1p3/2=-2.62 MeV, δε1p1/2=-2.25 MeV, δε2s1/2=+8.52 MeV, δε1d5/2=-3.75 MeVand δε1d3/2=+8.4 MeV.3. ResultsFigure 2 shows the phase shifts for the scattering of positive parity waves resultingfrom Eq. (2). The position of the bound states in 17F and of the resonances shown in theplot has been forced to agree with the experimental data employing the above choice ofU(r, r′). This is necessary for obtaining the correct asymptotic behaviour of the boundwave functions. The position of the first s1/2 resonance is determined by the first εp+ polein Eq. (1). This was originally predicted at 6.3 MeV, ∼0.5 MeV above the experimentalvalue. The background contribution to the phase shifts is instead a prediction of thetheory. As it can be seen, the mean field tends to be slightly too repulsive for thes1/2 partial wave and slightly too attractive in the d3/2 case. Considering the simpleexpansion of Eq. (1) in terms of a few harmonic oscillator wave functions, the model givesa reasonable description of the phase shifts. Eq. (2) was also solved for the valence orbitalsof the last proton bound states of 17F. The radial wave functions behave asymptoticallyasflj(r) −→r→∞ CljW−η,l+1/2(r)r, (4)4whereW−η,l+1/2 is a Whittaker function, η the Sommerfield parameter and the asymptoticnormalization constants (ANCs) are predicted to be Cs1/2= 98.2 fm−1/2 and Cd5/2= 1.59 fm−1/2.The astrophysical factor for 16O(p,γ) is computed from these wave functions and thesolutions of Eq. (2) for the relevant scattering waves. The results for the transitions tothe bound states of 17F are reported in Fig. 3 with solid lines. The curves overestimatethe experimental results, due to the large values obtained for the ANCs. In contrast, therelative spectroscopic factors are found to be 0.921 for the s1/2 state and 0.878 for the d5/2case. This agrees with what is expected from the halo nature of these orbitals. Moreover,we find that the shape of the theoretical curves agree well with the experimental data ifwe rescale the result for capture to the s1/2 state by 0.63 and the one for d5/2 by 0.49.This is depicted by the dashed lines in Fig. 3.At zero energy, the astrophysical factor is determined by ANCs. The analytical studyof the asymptotic wave functions done in Ref. [ 10] yields the following approximation,S(0) = 0.37C2s1/2 + 1.58× 10−3C2d5/2 . (5)After rescaling Cs1/2 and Cd5/2 as described above Eq. (5) gives S(0)=10.37 keV b, inagreement with their estimate [ 10].The discrepancies found between theory and experiment are not unexpected. Partof these are due to the fact that the expansion of Eq. (1), originally chosen to studyproperties sensitive to the interior of the nucleus, gives a poor description of the nuclearsurface to which halo states are particularly sensitive. A better choice of the basis wouldinvolve repeating completely the calculations of Ref. [ 6]. Nevertheless, it is plausible thatthis approach would lead to useful predictions of the scattering process.We would like to acknowledge useful discussions with J.-M. Sparenberg. This workwas supported by the Natural Sciences and Engineering Research Council of Canada(NSERC).REFERENCES1. Feshbach, Ann. Phys. (N.Y.) 5, 357 (1958); 19, 287 (1962).2. J. Escher and B. K. Jennings, Phys. Rev. C 66, 034313 (2002).3. C. Mahaux and R. Sartor, Adv. Nucl. Phys. 20, 1 (1991).4. W. H. Dickhoff and C. Barbieri, Prog. Part. Nucl. Phys. 52, 377 (2004).5. C. Barbieri and W. H. Dickhoff, Phys. Rev. C 63, 034313 (2001).6. C. Barbieri and W. H. Dickhoff, Phys. Rev. C 65, 064313 (2002).7. R. Machleidt, Adv. Nucl. Phys. 19, 191 (1989).8. S. R. Salisbury and H. T. Richards, Phys. Rev. 126, 2147 (1962).9. R. Morlock et al., Phys. Rev. Lett 79, 3837 (1997).10. D. Baye, P. Descouvemont and M. Hesse, Phys. Rev. C 58, 545 (1998).

Study of the 16O(p,gamma) Reaction at Astrophysical Energies

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