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Pierre Descouvemont Universit Libre de Bruxelles, Brussels, Belgium The 12 C( ) 16 O reaction: dreams and nightmares theoretical introduction

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Masses Cross sections lifetimes Fission barriers Etc Stellar Stellar models

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Content of the talk 1.Cross sections, S-factors: general properties 2.Reaction rates, stellar energies 3.H and He burning 4.Specificities of the 12 C( ) 16 O reaction 5.Theoretical models

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Transfer cross sections Examples: 3 He( 3 He, )2p 6 Li(p, ) 3 HeStrong interaction 22 Ne( ,n) 25 Mg Capture cross sections Examples: 3 He( ) 7 Be 7 Be(p, ) 8 BElectromagnetic interaction 12 C( ) 16 O Weak capture cross sections Examples:p(p,e + ) 2 HWeak interaction 3 He(p,e + ) 4 He Others: fusion, spallation, etc.. Cross sections Types of cross sections

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Cross section S factor potential Astrophysical energies Relative distance Cross section below the Coulomb barrier: (E) exp(-2 ) =Sommerfeld parameter ( =Z 1 Z 2 e 2 / v) Astrophysical S factor: S(E)= (E)*E*exp(2 ) smooth variation with energy Low angular momenta (centrifugal barrier)

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E0E0

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Reaction rate with: N(E,T)= Maxwell-Boltzmann distribution ~ exp(-E/kT) T = temperature v = relative velocity Gamow-peak energy :E 0 = 0.122 1/3 (Z 1 Z 2 T 9 ) 2/3 MeV E 0 = 0.237 1/6 (Z 1 Z 2 ) 1/3 T 9 5/6 MeV

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Examples: E 0 = Gamow peak energy E coul = Coulomb barrier Essentially 2 problems in nuclear astrophysics: oVery low cross sections (in general not accessible in laboratories) oNeed for radioactive beams ReactionT (10 9 K)E 0 (MeV)E coul (MeV) (E 0 )/ (E coul ) d + p0.0150.0060.310 -4 3 He + 3 He0.0150.0211.210 -13 + 12 C 0.20.3310 -11 12 C + 12 C12.4710 -10

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Starting point: Schrodinger equation: H JM = E JM c=channel 1.Scattering states: E>0: I c,O c =Coulomb functions 1c, 2c =internal wave functions of the colliding nuclei U J =collision matrix (contains all information) 2.Bound states : E

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

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

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

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

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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( )=W E1 ( ) + W E2 ( ) +cos( 1 - 2 )(W E1 ( )W E2 ( )) 1/2 E1 almost forbidden: =0 if isospin T=0

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Two subthreshold states:Two subthreshold states: affect the S-factor at low energies weak effect in measurements E cm E0E0

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Interference effects: E1 E cm

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Interference effects: E2 E cm

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

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Current situation: E2 at 300 keV

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

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

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Theoretical models Always necessary! (to go down to 300 keV) Require:very high precision use of experimentally known information Two main families: 1.Based on wave functions: Potential model (direct-capture model) Microscopic models 2.Based on parameters to be fitted R matrix K matrix 3.Hybrid models

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

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Internal structure of the nuclei is taken into account Hamiltonian T i =kinetic energy V ij =nucleon-nucleon force Wave functions: (spins zero) A = antisymmetrization operator 1, 2 = internal wave functions g l (r) = relative wave function (output) Inputs of the model:nucleon-nucleon interaction internal wave functions 1, 2 r 11 22 2. Microscopic cluster models

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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

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Application to 12 C( ) 16 O: P.D., Phys. Rev. C 47 (1993) 210 S E2 (300 keV) = 90 keV-b

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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: 12 C+ Internal region 16 O Entrance channel 12 C+ Exit channels 12 C(2 + )+ 15 N+p, 15 O+n 12 C+ Coulomb Nuclear+Coulomb: R-matrix parameters Coulomb

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

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Subthreshold states One pole: R-matrix equivalent to Breit-Wigner =total width: defined for resonances (E R >0) only =reduced alpha width: defined for resonances (E R >0) AND bound states (E R