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Pierre Descouvemont Université 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<0 W=Whittaker function (decreases exponentially) Cross sections: theory
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2.Capture: (electromagnetic interaction): H=H N + H , with H =electromagnetic interaction H is expanded in multipoles: electric ( M E ) and magnetic ( M M ) with one needs the matrix elements of the multipole operators (in general E1) Cross sections : 1.Transfer (nuclear interaction) small J values at low energies
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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 <0) But: E=0 E
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Subthreshold states Effect: enhancement of the S factor at low energies Not due to the width of the state: |E R | Enhancement essentially given by: E R = energy (“easy”): spectroscopy =radiative width (“easy”): spectroscopy =reduced width (difficult): indirect methods! Transfer : 12 C( 7 Li, 3 H) 16 O + DWBA analysis 12 C( 6 Li,d) 16 O Phase shifts: derived from the + 12 C elastic cross section
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simultaneous fit of – 12 C( ) 16 O S factor – 12 C+ phase shift – 16 N decay parameters of the 1 - 1 and 1 - 2 states (+background): – 12 C+ : – 12 C( ) 16 O : (radiative width) – 16 N decay : ( probabilities) Constraints on common parameters Application to 12 C( ) 16 O: E1 (Azuma et al, Phys. Rev. C50 (1994) 1194)
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16 N decay 12 C( ) 16 O 1 - phase shift S(0.3) = 79 ± 21 keV-b (Azuma et al., 1994) Pole 1: E 1, 1, 1 Pole 1: E 1, 1, 1 Pole 1: E 1, 1 Pole 2: E 2, 2, 2 Pole 2: E 2, 2, 2 Pole 2: E 2, 2 Pole 3: background
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12 C( ) 16 O : E2 2 + phase shift Application to 12 C( ) 16 O: E2 Pole 1: E 1, 1, 1 Pole 1: E 1, 1, 1 Pole 2: E 2, 2, 2 Pole 2: E 2, 2, 2 Pole 3: background Can we determine 1 from elastic scattering? Probably NO! (J.-M. Sparenberg, Phys. Rev. C 69, 034601 (2004) )
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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
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For tomorrow R-matrix theory: General formulation Application to 12 C( ) 16 O Discussion of the E2 component
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