Pierre Descouvemont Université Libre de Bruxelles, Brussels, Belgium

1. Pierre DescouvemontUniversité Libre de Bruxelles, Brussels, Belgium The 12C(a,g)16O reaction: theoretical introduction dreams and nightmares

2. Stellar models Cross sections Masses b-lifetimes Fission barriers Etc…

3. Content of the talk • Cross sections, S-factors: general properties • Reaction rates, stellar energies • H and He burning • Specificities of the 12C(a,g)16O reaction • Theoretical models

4. Types of cross sections Cross sections • Transfer cross sections Examples: 3He(3He,a)2p6Li(p,a)3He Strong interaction22Ne(a,n)25Mg • Capture cross sections Examples: 3He(a,g)7Be7Be(p,g)8B Electromagnetic interaction12C(a,g)16O • Weak capture cross sections Examples: p(p,e+n)2H Weak interaction3He(p,e+n)4He • Others: fusion, spallation, etc..

6. Cross section – S factor potential Astrophysical energies Relative distance • Cross section below the Coulomb barrier: s(E)  exp(-2ph)h=Sommerfeld parameter (h=Z1Z2e2/ v) • Astrophysical S factor: S(E)=s(E)*E*exp(2ph) smooth variation with energy • Low angular momenta (centrifugal barrier)

7. E0

8. Reaction rate with: N(E,T)= Maxwell-Boltzmann distribution ~ exp(-E/kT)T = temperaturev = relative velocity Gamow-peak energy : E0 = 0.122 m1/3 (Z1Z2T9)2/3 MeV DE0 = 0.237 m1/6 (Z1Z2)1/3 T95/6 MeV

9. Examples:E0= Gamow peak energy Ecoul = Coulomb barrier • Essentially 2 problems in nuclear astrophysics: • Very low cross sections (in general not accessible in laboratories) • Need for radioactive beams

10. Cross sections: theory Starting point: Schrodinger equation: HYJMp = EYJMp c=channel • Scattering states: E>0: Ic,Oc=Coulomb functions F1c, F2c=internal wave functions of the colliding nuclei UJp=collision matrix (contains all information) • Bound states : E<0 W=Whittaker function (decreases exponentially)

11. Cross sections: • Transfer (nuclear interaction) small J values at low energies • Capture: (electromagnetic interaction):H=HN + Hg, with Hg=electromagnetic interaction • Hg is expanded in multipoles: electric (MEl) and magnetic (MMl)with •  one needs the matrix elements of the multipole operators (in general E1)

12. H and He burning 99,77% p + p  d+ e+ + e 0,23% p + e - + p d + e 84,7% d + p 3He + ~210-5 % 13,8% 3He + 4He 7Be +  13,78% 0,02% 7Be + e-7Li + e 7Be + p 8B +  3He+3He+2p 7Li + p ->+ 8B 8Be*+ e+ +e 2 3He+p+e++e pp I pp II pp III hep • pp chain (from G. Fiorentini)

13. CNO cycle The pp chain and the CNO cycle transform protons into 4He

14. 4He burning • 12C produced by the triple a process: 3a→8Be+a→12C • 8Be(a,g)12C • 12C production enhanced by the 0+2 resonance • 0+2 resonance predicted from observation of 12C abundance (Hoyle) • 16O produced by the 12C(a,g)16O reactionIn the CNO cycle 15N(p,g)16O  15N(p,a)12C • 12C(a,g)16O determines the 12C/16O ratio after He burning

15. Specificities of 12C(a,g)16O 16O spectrum • E1 (almost) forbidden • Two subthreshold states: 1-, 2+ • Interference effects

16. =0 if isospin T=0 • E1 almost forbidden: • In practice: E1 not negligible (dominant?) owing to • isospin impurities (small T=1 components)cross section : • higher-order terms in the E1 operator • E1 is enhanced by multipolarity 1 reduced by cancellation of first-order terms • Mixing of E1 and E2 • Angular distributions: W(q)=WE1(q) + WE2(q) +cos(d1-d2)(WE1(q)WE2(q))1/2

17. Two subthreshold states: • affect the S-factor at low energies • weak effect in measurements Ecm E0

18. Interference effects: E1 Ecm

19. Interference effects: E2 Ecm

20. Current situation: E1 at 300 keV NACRE (Azuma 94)

21. Current situation: E2 at 300 keV

22. “Astrophysical approaches” • Weaver and Woosley : Phys. Rep. 227 (1993) 65 Production factor a 14 isotopes (from O to Ca) in a supernova explosion

23. “Astrophysical approaches” • T. Metcalfe, Astrophys. J. 587 (2003) L43 • Influence of 12C(a,g)16O on the structure of white dwarfs (GD358 and CBS114)

24. Theoretical models • Always necessary! (to go down to 300 keV) • Require: very high precision use of experimentally known information • Two main “families”: • Based on wave functions: Potential model (“direct-capture” model) Microscopic models • Based on parameters to be fitted R matrix K matrix • “Hybrid” models

25. initial Ecm g final 1. The potential model • Structure of the colliding nuclei is neglected • Wave functions given by the radial equationV(r)=nucleus-nucleus potential (Gaussian, Woods-Saxon,etc.) • Cross section for a multipole l • Depth: Pauli principle → additional (unphysical) bound states • For 12C(a,g)16O no E1  limited to E2 only (no recent application)

26. 2. Microscopic cluster models • Internal structure of the nuclei is taken into account • Hamiltonian Ti=kinetic energy Vij=nucleon-nucleon force • Wave functions: (spins zero)A = antisymmetrization operatorF1, F2 = internal wave functions gl(r) = relative wave function (output) • Inputs of the model: nucleon-nucleon interaction internal wave functions F1, F2 f1 f2 r

27. Advantages: • Predictive power (little information is necessary) • Unified description of bound and scattering states (important for capture) → tests with spectroscopy • Applicable to capture and transfer reactions • Inelastic channels can be easily taken into account Problems: • Choice of the nucleon-nucleon interaction • Precise internal wave functions • Limited to low level densities → limited to A  25-30 • Computer times

28. Application to 12C(a,g)16O:P.D., Phys. Rev. C 47 (1993) 210 SE2 (300 keV) = 90 keV-b

29. 3. The R-matrix method • Main goal: to deal with continuum states • Main idea: to divide the space into 2 regions (radius a) • Internal: r ≤ a: Nuclear + coulomb interactions • External: r>a: Coulomb only • Example: 12C+a Exit channels 12C(2+)+a Entrance channel 12C+a Internal region 16O 12C+a 15N+p, 15O+n Nuclear+Coulomb:R-matrix parameters Coulomb Coulomb

30. The R-matrix method • Definition of the R-matrix • = pole i, j = channels • N = number of poles • El = pole energy (parameter) • = reduced width (parameter) • The R-matrix is defined for each partial wave • « Observed » vs « calculated » parameters R-matrix parameters physical parameters Similar but not equal

31. Subthreshold states • One pole: R-matrix equivalent to Breit-Wigner But: • Ga=total width: defined for resonances (ER>0) only • ga=reduced alpha width: defined for resonances (ER>0) AND bound states (ER<0) E E=0

32. Subthreshold states • Effect: enhancement of the S factor at low energies • Not due to the width of the state: Gg |ER| • Enhancement essentially given by: • ER= energy (“easy”): spectroscopy • Gg=radiative width (“easy”): spectroscopy • ga=reduced a width (difficult): indirect methods! • Transfer : 12C(7Li,3H)16O + DWBA analysis12C(6Li,d)16O • Phase shifts: derived from the a+12C elastic cross section

33. Application to 12C(a,g)16O: E1(Azuma et al, Phys. Rev. C50 (1994) 1194) simultaneous fit of • 12C(a,g)16O S factor • 12C+a phase shift • 16N b decay parameters of the 1-1 and 1-2 states (+background): • 12C+a: El, gl • 12C(a,g)16O : El, gl, Ggl (radiative width) • 16N b decay : El, gl,Al (b probabilities)  Constraints on common parameters El, gl

34. 16N b decay 1- phase shift Pole 2: E2,g2,Nb2 Pole 1: E1,g1 Pole 1: E1,g1,Nb1 Pole 2: E2,g2 S(0.3) = 79 ± 21 keV-b (Azuma et al., 1994) 12C(a,g)16O Pole 3: background Pole 1: E1,g1,Gg1 Pole 2: E2,g2,Gg2

35. Application to 12C(a,g)16O: E2 2+ phase shift 12C(a,g)16O : E2 Pole 2: E2,g2,Nb2 Pole 1: E1,g1,Nb1 Pole 1: E1,g1,Gg1 Pole 2: E2,g2,Gg2 Pole 3: background Can we determine g1 from elastic scattering? Probably NO!(J.-M. Sparenberg, Phys. Rev. C 69, 034601 (2004))

36. Cascade transitions Ground state:E1: 50-100 keV-b E2: 50-200 keV-b Cascade (Redder et al., 1987)0+ : 13 keV-b 3- : 0.29 keV-b 2+ : 7.0, 4.2 keV-b 1- : 1.3 keV-b→ small compared to the g.s. Cascade

37. For tomorrow R-matrix theory: General formulation Application to 12C(a,g)16O Discussion of the E2 component