Study of 17 O(p,α) 14 N reaction via the Trojan Horse Method for application to 17 O Nucleosynthesis Maria Letizia Sergi LNS-INFN, Catania 53-esimo Congresso della Societa' Astronomica Italiana (SAIt) “L'Universo quattro secoli dopo Galileo” Maggio 2009, PISA

1)It is one of the very few isotopes whose nucleosynthetic origin can be attributed to Novae, stellar explosion occurring in close binary system that contain White Dwarf (WD) as a compact object and a companion star. In novae, 17 O is produced in one of the two paths of CNO cycles leading to 18 F production which is of special interest for gamma ray astronomy. 2) The relative abundances of the oxygen isotopes Giant (RG) stars. Red Giant Mira have been observed at the surface of some Red Role of 17 O: astrophysical scenarios The change in the surface composition offers an opportunity to probe the “history” of the stellar interior. Nova Cygni 1992 γ-ray line fluxes measurement would shed light into the physical processes that occur in the early phases of the explosion.

In nova 17 O is produced starting with the 16 O isotope found at the surface of the WD progenitor. CNO2 cycle HCNO2 cycle 16 O nuclei can be processed in a two different competing cycles: 17 O production & destruction 16 O(p,  ) 17 F(  +) 17 O(p, α ) 14 N(p,  ) 15 O(  +) 15 N(p,  ) 16 O 16 O(p,  ) 17 F(  +) 17 O(p,  ) 18 F(p, α ) 15 O(  +) 15 N(p,  ) 16 O Production: 16 O(p,  ) 17 F : reaction rate well known in literature Destruction : 17 O(p,  ) 18 F: important for 18 F production in novae 17 O(p,  ) 14 N : dominant channel for 17 O destruction Stellar temperatures of primary importance for nucleosynthesis: T= GK for red giant, AGB, and massive stars; T= GK for classical nova explosion (peak temperatures of 0.35 GK can be easily achieved in explosion hosting very massive white dwarfs.) Stellar temperatures of primary importance for nucleosynthesis: T= GK for red giant, AGB, and massive stars; T= GK for classical nova explosion (peak temperatures of 0.35 GK can be easily achieved in explosion hosting very massive white dwarfs.) C. Iliadis, Nuclear physics of Stars, 2007 (NACRE)

(p,α) reaction is faster than the (p,γ) reaction over the entire temperature range and in particular by a factor ≈200 at temperature T=0.1 – 0.4 GK Stellar temperatures of primary importance for nucleosynthesis: T= GK for red giant, AGB, and massive stars; T= GK for classical nova explosion (peak temperatures of 0.35 GK can be easily achieved in explosion hosting very massive white dwarfs.) Stellar temperatures of primary importance for nucleosynthesis: T= GK for red giant, AGB, and massive stars; T= GK for classical nova explosion (peak temperatures of 0.35 GK can be easily achieved in explosion hosting very massive white dwarfs.) 17 O(p,  ) 18 F important for 18 F production in novae 17 O(p,  ) 14 N dominant channel for 17 O destruction 17 O destruction C. Iliadis, Nuclear physics of Stars, 2007

T= GK: 17 O(p,α) 14 N and 17 O(p,γ) 18 F reaction cross section have to be precisely known in the center-of-mass energy range E c.m. = MeV. In this energetic region, two resonant levels of 18 F are important for 17 O(p,α) 14 N reaction: E c.m. = 65.0 keV J π = 1 - E c.m. = keV J π = 2 - corresponding to E x = MeV and E x = MeV respectively.. Two sub-threshold levels at E X (J π )=5.605 MeV (1 − ) and E X (J π )=5.603 MeV (1 + ) could also play a significant role in the reaction rate through the high-energy tail of the levels. Possible interference effects between MeV level and MeV level Energetic Region of astrophysical interest for the 17 O(p,α) 14 N reaction

 E c.m. (keV) << E coul (MeV) Exponential decrease of cross section due to Coulomb barrier  ~ nano-picobarn Problem of direct measurement of low energy cross section Astrophysical energy S(E)  E  (E) exp(2  ) Extrapolations!! S(E) represents the intrinsically nuclear part of cross section and for non- resonant reaction is a smoothly varying function of energy. Introduction of Astrophysical factor tunnel effect Resonance’s tale subthreshold resonances ErEr E Extrapolations Direct Measurements 0 S(E) Non-resonant processes Uncertainties due to extrapolation: e.g., presence of a subthreshold resonance

 E c.m. (keV) << E coul (MeV) Exponential decrease of cross section due to Coulomb barrier  ~ nano-picobarn Problem of direct measurement of low energy cross section Astrophysical energy S(E)  E  (E) exp(2  ) Extrapolations!! S(E) represents the intrinsically nuclear part of cross section and for non- resonant reaction is a smoothly varying function of energy. Introduction of Astrophysical factor tunnel effect Resonance’s tale subthreshold resonances ErEr E Extrapolations Direct Measurements 0 S(E) Non-resonant processes NEW EQUIPMENT & NEW EXPERIMENTAL TECHNIQUES

Electron Screening: the needed of indirect techniques 3 He + d  4 He + p M. Aliotta et al., Nucl. Phys. A690, 790 (2001) Atomic electrons screen the nuclear charges thus determinig an enhancement of the cross section at the lowest energies, which is not related to nuclear physics. U e : Screening potential Even in those few cases where direct measurements have been possible (light nuclei) thanks to improved experimental techniques, the presence of the atomic electron has prevented to measure the bare-nucleus cross section, which constitutes the main nuclear input in astrophysisc. S(E) s = S(E) b exp(π  U e /E) AGAIN EXTRAPOLATION FROM HIGHER ENERGIES!!

Status of the Art I Even if several experimental determinations of 17 O(p,α) 14 N reaction cross section were made during the last decades, the literature adopted 17 O+p reaction rate shows still great uncertainties. REACTION RATE ADOPTED IN NACRE COMPILATION Because of the difficulty to measure at very low energy, until few years ago, only theoretical calculations of cross section (or astrophysical factor) were available at energies of astrophysical interest. V. Landre et al., Phys. Rev. C. 40, 1972, (1989) C. Fox et al., Phys. Rev. Lett. 93, , (2004)

The first direct measurement of the 17 O(p,α) 14 N at low energy LARGE UNCERTAINTIES !! In the last years several efforts to measure the cross section for the 17 O(p,α) 14 N at astrophysical energies were made in order to reduce the indetermination on reaction rate. Status of the Art J.C. Blackmon et al., Phys. Rev. Lett. 74, 2642, (1995) INDIRECT MEASUREMENT 65.0 keV keV To reduce the uncertainties A. Chafa et al., Phys. Rev. C 75, , (2007)

Status of the Art III Uncertainties reduced by orders of magnitude in the temperature range T= GK The most recent reaction rate calculation performed by Chafa’07 includes the six most important levels in 18 F and 12 additional levels at higher energy as done in NACRE compilation. INDIRECT MEASUREMENT To further reduce the uncertainties A. Chafa et al., Phys. Rev. C 75, , (2007) 65.0 keV keV Uncertainties reduced by orders of magnitude in the temperature range T= GK INDIRECT MEASUREMENT To further reduce the uncertainties 65.0 keV keV

Indirect Method in Nuclear Astrophysics Determination of astrophysically relevant cross section by selecting a precise reaction mechanism in a suitable chosen reaction and through the application of some theoretical consideration: Coulomb dissociation Asympotic Normalization Coefficient (ANC) Trojan Horse Method (THM) Determination of astrophysically relevant cross section by selecting a precise reaction mechanism in a suitable chosen reaction and through the application of some theoretical consideration: Coulomb dissociation Asympotic Normalization Coefficient (ANC) Trojan Horse Method (THM) …allows to deduce a charged- particle binary-reaction cross section inside the Gamow window by selecting the Quasi-Free (QF) contribuiton to an appropriate three-body reaction performed at energies well above the Coulomb barrier...

Three-body reaction: A + a  C + c + s    A + x  C + c a: x  s clusters Three-body reaction: A + a  C + c + s    A + x  C + c a: x  s clusters S c A a C Direct break-up x 2-body reaction Trojan Horse Method Quasi-free mechanism: A interacts only with x s is spectator  the spectator momentum distribution is strictly connected to the momentum distribution of the cluster x inside a E A > E Coul  NO Coulomb suppression, NO electron screening E cm = E Ax – B x-s ± intercluster motion B x-s binding energy of a=x  s E cm ~ 0

Basically… assuming that the QF mechanism is dominant, in simple PWIA one has: 3-body reactionVirtual decay Virtual reaction ( astrophysical process) A B S c d x = A x S x B c d  E Bx = E cd - Q 2b Measured at high energy d3σd3σ dΩ c dΩ d dE cm KF · |Φ (P s )| 2 Calculated e.g. by Montecarlos dσ N dΩ Deduced / =

Plain Wave Impulse Approximation (PWIA) formalism for THM Hypothesis: T The de-Broglie wave length associated to momentum of projectile A, is lower than the average distance between the clusters x and s (transparency hypothesis); The interaction between the projectile A with x is the same as x was free, that is the presence of s does not influence the interaction; The incident particle and the outgoing ones can be described by plane wave. KF: kinematical factor accounting for the final-state phase; |  (-k s )| 2 : Fourier transform of the radial wave function for the x-s relative motion, which in the case of deuteron break-up is well approximated by the Hulthén wave function. xas c A C

Experimental Set-up L.N.S - Catania 14 N n 17 O 2H2H α break-up diretto p 17 O+d 14 N+ α +n 17 O+p 14 N+ α Trojan Horse MethodDetectorsThickness[μm]θ[deg]r[mm]Δθ[deg] PSD ± PSD ± PSD ± PSD ± PSD ± PSD ± O CD E beam = 41 MeV Target Thickness ~ 150 μg/cm2 Two ionization chambers filled with 60 mbar of isobuthan gas as ΔE detector were in front of PSD1 and PSD4 detector

Selection of the 2 H( 17 O, α 14 N)n reaction channel N particles were selected with the standard ΔE-E technique in both telescopes 1 and 4 The loci events in E 1 vs E 5 and E 4 vs E 2 for the 2 H( 17 O, α 14 N)n reaction were deduced Good agreement with the theoretical value MeV Good detector calibration procedure!! Good reaction channel selection!!

Selection of the 2 H( 17 O, α 14 N)n reaction channel III Comparison between the experimental kinematical loci (red points) and the theoretical ones (black points) for different angular pairs chosen in the entire angular range covered by detectors. Very good agreement !!

For the selected events the Q-value spectra for the coincidence events a) b) were calculated. Selection of the 2 H( 17 O, α 14 N)n reaction channel IV Good agreement with the theoretical value MeV Good detector calibration procedure!! Good reaction channel selection!! Events inside the peak were selected for further analysis

The 14 N+α+n exit channel can be fed through different reaction mechanism. To evaluate the presence of SD and disentangle the QF mechanism from the SD ones: study of relative energy spectra study of the correlation between the three-body cross section and the spectator momentum Study of sequential mechanisms Quasi-Free mechanism (QF) Sequential Decay (SD)

Study of the presence of SD mechanism The clear horizontal loci in E 14N-α represent an evidence for the formation of the 18 F excited states. E 14N-α (MeV) E 14N-n (MeV) E α-n (MeV) The 14 N+α+n exit channel can be fed through different reaction mechanism: or Quasi-Free mechanism (QF). Sequential Decay (SD) Study of relative energy spectra:

Selection of the Quasi-Free mechanism: data as function of neutron momentum Coincidenza 1-5 a) |p s |<30 MeV/c the coincidence yield appears to be dominated by the decay of the several levels of 18 F between 0 and 0.7 MeV b) 30<|p s |<60 MeV/c c) 60<|p s |<90 MeV/c the coincidence yield decreases and such resonances become barely visible with respect to the background  necessary condition for the dominance of the quasi-free mechanism in the region approaching zero spectator momentum (s-wave relative motion) Study of coincidence yield divided by KF as function of E cm =E  -14N -Q two-body

An observable which turns out to be more sensitive to the reaction mechanism is the shape of the experimental momentum distribution Selection of the Quasi-Free mechanism: experimental momentum distribution |P n | < 30 MeV/c In a energy windows of 100 keV d  /d  const.  dividing the resulting three-body coincidence yield by the kinematic factor, the p-n momentum distribution in arbitrary units is obtained |  (k s )| 2 = N: normalization parameter a= fm -1 b=1.202 fm -1 The extracted experimental momentum distribution is compared with the theoretical one, given by the Hulthén wave function in momentum space: E c.m. =183±50 keV

17 O(p,α) 14 N cross section & angular distributions Extraction of nuclear part of the two body cross section by using the PWIA approach d3σd3σ KF · |Φ(Ps)| 2 dΩ α dΩ 14N dE cm dσNdσN dΩdΩ ∝ THM data Chafa 07 Legendre polinomyal fit of direct data reported in Chafa et al., 2007 W c.m. (θ c.m. )=a 1 +a 2 P 2 (cosθ c.m. ) Theoretical calculation based on Blatt (1952) theory

17 O(p,α) 14 N cross section & angular distributions Extraction of nuclear part of the two body cross section by using the PWIA approach d3σd3σ KF · |Φ(Ps)| 2 dΩ α dΩ 14N dE cm dσNdσN dΩdΩ ∝ THM data Chafa 07 Legendre polinomyal fit of direct data reported in Chafa et al., 2007 W c.m. (θ c.m. )=a 1 +a 2 P 2 (cosθ c.m. ) Theoretical calculation based on Blatt (1952) theory

Fit of the two-body cross section evaluated for different valus of θ c.m. The contribution of non-resonant part and of each resonance were taken into account in the fits function. In particular, the centroide of the Breit-Wigner functions were fixed, while their widths and the normalization constants were used as fitting parameters. Extraction of experimental angular distributions

For an isolated and narrow resonance: Г 1 and Г 2 represent the partial widths describing the formation and the decay of the compound nucleus; Г= Г 1 +Г 2 is the total width. Reaction rate STRENGTH OF THE RESONANCE: KEY PARAMETER FOR NUCLEAR RATE DETERMINATION!! where N A R is expressed in cm 3 mol -1 sec -1, E R and ωγ in MeV and S(O) in MeV b. Z 1 and Z 2 are the projectile and the target atomic number respectively. We focussed on the MeV energy region and in particular on both E c.m. =65 keV and E c.m. =183 keV, obtaining the strength of the resonance at E c.m. =65 KeV by using the available information in literature on the well measured E c.m. =183 keV resonance.

Trojan Horse cross section: horizontal error bar refers to the integration bin while the vertical one arise for the statistics (  25%) In order to separate the different contributions on this cross section, a fit of the nuclear cross section has been performed. Extraction of: Resonance energies: E R1 =65±5 keV and E R2 =183±5 keV. Peak value of the two resonances: N 1 =0.170±0.025 and N 2 =0.220±0.031, used to derive the resonance strengths ωγ (case of narrow resonances). Trojan Horse Cross section The extracted two-body differential cross section has been integrated in the whole angular range, assuming that in the region where no experimental angular distribution are available, their trend is given by the fit of the obtained experimental angular distribution. σ N THM (arb. un.) E c.m. (MeV)

By using the Breit-Wigner formula, the peak TH cross section taken at the resonance E Ri energy for the (p,α) reaction 17 O+p-> 14 N+ α is given by New approach to extract the ωγ parameter for a resonance In our case, we have two resonances: 1 2 where M i (E) is the direct transfer reaction amplitude for the binary reaction 17 O+d-> 18 F*+s populating the resonant state 18 F* with the resonance energy E Ri ; Γ αi (E) is the partial resonance width for the decay 18 F* -> 14 N+α; Γ i is the total resonance width of 18 F*. σ N THM (arb. un.) E c.m. (MeV)

1 2 If Г 1 <<Г ris and Г 2 <<Г ris Г 1 ~ 130 eV Г 2 ~ 7 eV Г ris ~ 20 keV Г 1 =Г ris Г 2 =Г ris But The strength of the resonance at 65 keV is given from the ratio between the peak value N 1 and N 2 through the relation: σ N THM (arb. un.) E c.m. (MeV)

New approach 1 2 where M i (E) is the direct transfer reaction amplitude for the binary reaction 17 O+d-> 18 F*+s populating the resonant state 18 F* with the resonance energy E Ri ; σ N THM (arb. un.) E c.m. (MeV) The strength of the resonance at 65 keV is given from the ratio between the peak value N 1 and N 2 through the relation: Reaction rate determination STRENGTH OF THE RESONANCE: KEY PARAMETER: We focussed on the MeV energy region and in particular on both E c.m. =65 keV and E c.m. =183 keV, obtaining the strength of the resonance at E c.m. =65 keV by using the available information in literature on the well measured E c.m. =183 keV resonance. La Cognata et al., PRL 101, , (2008)

This two values are in agreement each other; with the value ·10 -9 eV adopted in NACRE; with the (4.7±0.8)·10 -9 eV calculated by using the value of Γ α and Γ p reported in Chafa’07. ωγ RESULTS: NACRE: C. Angulo et al., Nucl. Phys. A 656, (1999) Moazen’07: B.H. Moazen et al., Phys. Rev. C 75, , (2007) Reaction rate determination II TOTAL REACTION RATE: Ratio of the THM reaction rate to the NACRE one (blu line). The THM reaction rate was calculated by considering the value of ωγ=(4.4±1.1)x10 -9 eV for the 65 keV resonance. Ratio between the reaction rate evaluated by Chafa’07 and NACRE. NACRE adopted reaction rate.

This two values are in agreement each other; with the value ·10 -9 eV adopted in NACRE; with the (4.7±0.8)·10 -9 eV calculated by using the value of Γ α and Γ p reported in Chafa’07. ωγ RESULTS: NACRE: C. Angulo et al., Nucl. Phys. A 656, (1999) Moazen’07: B.H. Moazen et al., Phys. Rev. C 75, , (2007) Reaction rate determination II TOTAL REACTION RATE: T= GK: the difference between the rate adopted in literature and the total rate calculated, if one considers the N A 65 THM extracted as explained before, are smaller than 10%. Agreement between the two sets of data

Reaction Rate calculation Total reaction rate: Ratio of the THM reaction rate to the NACRE one (blu line). The THM reaction rate was calculated by considering the value of ωγ=(4.4±1.1)x10 -9 eV for the 65 keV resonance. Ratio between the reaction rate evaluated by Chafa’07 and NACRE. NACRE adopted reaction rate. T= GK: the difference between the rate adopted in literature and the total rate calculated, if one considers the N A 65 THM extracted as explained before, are smaller than 10%. T>0.2 GK: no significant differences.

Conclusions 1. A clear evidence of both levels at E c.m. =65 and 183 keV is present in the excitation function. Main results: 2. Extraction of angular distributions for both levels at E c.m. =65 (for the first time!!) and 183 keV and comparison with theoretical calculation and direct measurement (only for E c.m. =183 keV). 3. The 17 O(p,α) 14 N reaction rate was extracted and compared with that one reported in Chafa’07, giving a difference of less than 10%. A deeper analysis of contribution of sub-threshold level is needed Our results are affected by a statistical error of  25%. A further experiment was performed at Physics Department of Notre Dame University (Indiana, USA) in November 2008 by using the same experimental apparatus adopted in the previous one. Data analysis in progress … in progress:

ALTRE DIAPOSITIVE

Roche Model Assumptions: 1.the third mass must be infinitesimal mass; 2.The two large masses must be in circular orbit. Solution of restricted three body problem L1,L2… L5 Lagrange points: points where there was not net force exerted on the third mass. Roche surface: equipotential surface where the sum of the rotational and gravitational potential energy is constant. Roche surface through L1: consists of two Roche lobes and form the inner critical potential. If one star completely fills its Roche lobe then it may loss matter to its companion star through L1. Roche surface through L2: it defines the outer critical potential. If a star has a potential greater than the outer critical potential mass may be transferred out of the system

Kopal Classification Comparison between the star potential and the inner critical potential. Detached system: neither star completely fill the Roche lobe. The stars evolve separately. Semi-detached system: only one of the two stars completely fills its Roche lobe.  Mass transfer (NOVA EXPLOSION). Contact system: both stars have the potentials greater than the inner critical potential but less than the outer critical potential. Both componetes of the binary fill their Roche lobe and a common envolope surrounds both stars.

Hydrogen-rich matter is tansferred via Roche lobe from a low-mass main sequence star to surface of WD. For effect of the high gravitational field created from WD, it draws on itself the matter that is in the envelope of the companion star. This transferred matter is accumulated in an accretion disc surrounding the WD with a accretion rates amount to ∼ −10 −9 M ⊙ per year. A fraction of this matter spirals inward and accumulates on the WD surface, where is heated and compressed by the strong surface gravity. At some point the bottom layers of the WD become electronic degenerate. Hydrogen starts to fuse to helium via the p-p chains during the accretion phase and the temperature increases gradually. The electron degeneracy prevents an expansion of the envelope and eventually a thermonuclear runaway occurs near the base of the accreted layers [Iliadis07]. At this stage the nuclear burning is dominated by explosive hydrogen burning via the CNO cycle. Both the compressional heating and the energy released from the nuclear burning heat the accreted material until an explosion occurs. A nova is a cataclysmic nuclear explosion caused by the accretion of hydrogen onto the surface of a white dwarf star.

L’altezza del picco della i-esima risonanza è legata la rapporto tra Γ αi e Γ i tot (E Ri ) della risonanza attraverso il quadrato dell’elemento di matrice che descrive il polo di break-up S i M i 2 (E Ri ): con S i fattore spettroscopico dell’i- esimo stato dell’ 18 F Strength della resonanza: Sostituendo: con “single particle width” ottenuta con calcoli di ANC M. La Cognata at al. PRL, in press arXiv: σ(E) THM (arb. un.) E c.m. (MeV)

Quindi: 1 2 Dividendo membro a membro: Rapporto dei parametri “model dependent” Calcolo “model independent” !! σ(E) THM (arb. un.) E c.m. (MeV)

Some details on used Blatt theoretical calculation Consider the reaction A+X->Y+b NOTATION Before collision: -channel index α (defines the type of incoming particles and the state of struck nucleus) -channel spin s (total spin angular momentum in the channel; it is the vector sum of intrinsic spin i of the incoming particle and the spin I of the struck nucleus) -orbital angular momentum l After collision: -channel index α’ (defines the type of outgoing particles and quantum state of the residual nucleus) -channel spin s’ (it is the vector sum of intrinsic spin i of the outgoing particle and the spin I of the residual nucleus) -outgoing orbital momentum l’

If the reaction A+X->Y+b procedes via a definite resonance level of the compound nucleus, with angular momentum J 0 and parity Π 0, the cross section for the α->α’ reaction is given by where

If the reaction A+X->Y+b procedes via a definite resonance level of the compound nucleus, with angular momentum J 0 and parity Π 0, the cross section for the α->α’ reaction is given by where P L are the Legendre polynomials

NOTATION where W is the Racah coefficients defined in Racah 1942

NOTATION, where Γ αsl is the partial widths of the resonant level.

NOTATION, where Γ αsl is the partial widths of the resonant level. ξ l is the phase shifts for the potential scattering, in the hard sphere approximation, defined by equation where F l (R) and G l (R) are the regular and irregular Coulomb wave function (R is the channel radius and σl is the phase shift for Coulomb scattering from an impenetrable sphere of radius R)

First consideration PSD 1 -PSD 6 and PSD 4 -PSD 3 coincidences: PSD 3 and PSD 6 were placed in the scattering chamber to have an investigation of the whole kinematical locus reaction channel even if far away from the astrophysically relevant energy range. E c.m. >500 keV p s > 30 MeV/c E (MeV) Run 86 Run 79 Run 56 PSD1 The same for PSD4 Not possible to use coincidence 1-4 !! E1 and E4 obtained by kinematical calculation!

Reaction Rate Cross section is necessary input to know the stellar reaction rate: For an isolated and narrow resonance: where σ BW is the Breit-Wigner cross section Statistical factor depending by nuclear spin of compound nucleus J C*, target J X and projectile J a The product of the statistical factor ω and the width ratio γ=Γ 1 Γ 2 /Γ is referred as the strength of the resonance: Г 1 and Г 2 represent the partial widths describing the formation and the decay of the compound nucleus. Г= Г 1 +Г 2 is the total width

Reaction Rate Cross section is necessary input to know the stellar reaction rate: For an isolated and narrow resonance: where σ BW is the Breit-Wigner cross section Statistical factor depending by nuclear spin of compound nucleus J C*, target J X and projectile J a The product of the statistical factor ω and the width ratio γ=Γ 1 Γ 2 /Γ is referred as the strength of the resonance: KEY PARAMETER FOR NUCLEAR RATE DETERMINATION!!

Consideration on the extraction of the ωγ parameter for the 65 keV resonant level The first step of the reaction rate calculation is to evaluate the strengths of the resonances We focussed on the MeV energy region and in particular on both E c.m. =65 keV and E c.m. =183 keV, obtaining the strength of the resonance at E c.m. =65 KeV by using the available information in literature on the well measured E c.m. =183 keV resonance. To this aim, the extracted two-body differential cross section has been integrated in the whole angular range, assuming that in the region where no experimental angular distribution are available, their trend is given by the fit of the obtained experimental angular distribution.

Reaction Rate calculation I In the narrow resonance approximation, the reaction rate is deduced by relation where N A R is expressed in cm 3 mol -1 sec -1, E R and ωγ in MeV and S(O) in MeV b. Z 1 and Z 2 are the projectile and the target atomic number respectively. In the calculation of the 17 O(p,α) 14 N reaction rate, we followed the same procedure adopted in Chafa’07 by using for the resonance at E c.m. =65 keV the two value of ωγ extracted as explained before.

New approach to extract the ωγ parameter for a resonance I The THM cross section for the A+a(x+s)->c+C+s reaction proceeding through a resonance F i in the subsystem F=A+x=C+c is: where M i (E) is the direct transfer reaction amplitude for the binary reaction A+a->F i +s populating the resonant state F i with the resonance energy E Ri ; Γ cci (E) is the partial resonance width for the decay F i -> C+c; Γ i is the total resonance width of F i. The appearence of the transfer reaction amplitude M i (E) instead of the entry channel partial resonance width Γ (Ax)i (E) is the main difference between the THM cross section and the cross section for the resonant binary sub-reaction A+x->C+c A. M. Mukhamedzhanov et al., J. Phys. G: Nucl. Part. Phys. 35 (2008)

The peak TH cross section taken at the resonance E Ri energy for the (p,α) reaction A+x->C+c is given by New approach to extract the ωγ parameter for a resonance II Γ i /2 -Γ i /2 E Ri σ(E) NiNi In our case, we have two resonances: 1 2 Strength of resonance 2 (E c.m. =183 keV) well measured in two recent works !!

1 2 If Г 1 <<Г ris and Г 2 <<Г ris Г 1 ~ 130 eV Г 2 ~ 7 eV Г ris ~ 20 keV Г 1 =Г ris Г 2 =Г ris But The strength of the resonance at 65 keV is given from the ratio between the peak value N 1 and N 2 through the relation: