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

Masses Cross sections  lifetimes Fission barriers Etc… Stellar Stellar models

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

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

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)

E0E0

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

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 + p He + 3 He  + 12 C C + 12 C

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

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

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  %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 

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

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 resonance 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

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

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

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

Interference effects: E1 E cm

Interference effects: E2 E cm

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

Current situation: E2 at 300 keV

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

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

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

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

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 11 22 2. Microscopic cluster models

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  Computer times

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

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

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

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

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

simultaneous fit of – 12 C(  ) 16 O S factor – 12 C+  phase shift – 16 N  decay parameters of the and 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)

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

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, (2004) )

Cascade transitions Ground state: E1: keV-b E2: 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

For tomorrow R-matrix theory: General formulation Application to 12 C(  ) 16 O Discussion of the E2 component