1. The R-matrix method and 12 C(  ) 16 O Pierre Descouvemont Université Libre de Bruxelles, Brussels, Belgium 1.Introduction 2.The R-matrix formulation: elastic scattering and capture 3.Application to 12 C(  ) 16 O 4.Conclusions and outlook

2. Introduction Many applications of the R-matrix theory in various fields “Common denominator” to all models and analyses Can mix theoretical and experimental information Two types of applications:data fitting variational calculations Application to 12 C(  ) 16 O: nearly all recent papers References: A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30 (1958) 257 F.C. Barker, many papers

3. R-matrix formulation Main idea: to divide the space into 2 regions (radius a) Internal: r ≤ a: Nuclear + coulomb interactions External: r > a: Coulomb only 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 In practice limited to low energies (each J  must be considered individually).  well adapted to nuclear astrophysics

4. Example: 12 C+  Physical parameters = “observed” parameters Resonances: R-matrix parameters = “formal” parameters Poles: Similar but not equal 16 O 1 -, E R =2.42 MeV,   =0.42 MeV Reduced width  2 :   =2  2 P(E R ), with P = penetration factor 12 C+  R-matrix parameters = poles

5. Background pole Isolated resonances: Treated individually High-energy states with the same J  Simulated by a single pole = background Energies of interest Non resonant calculations possible: only a background pole

6. a.Hamiltonian: H  =E  With, for r large: I l, O l = Coulomb functions U l = collision matrix (→ cross sections) = exp(2i  l ) for single-channel calculations Total wave function b. Wave functions Set of N basis functions u (r) with Derivation of the R matrix (elastic scattering)

7. c.Bloch-Schrödinger equation: With L = Bloch operator (restore the hermiticity of H over the internal region) Replacing  int (r) and  ext (r) by their definition:

8. Solving the system, one has: R-matrix parameters R matrix =reduced width P=penetration factor S=shift factor Reduced width: proportional to the wave function in a  ”measurement of clustering” Dimensionless reduced width “first guess”:  2 =0.1 Total width: Depend on a

9. Penetration and shift factors P(E) and S(E)

10. Two approaches: 1.Fit: The number of poles N is determined from the physics of the problem In general, N=1 but NOT in 12 C(  ) 16 O : N=3 or 4 (or more) are fitted Phase shift: 2.Variational calculations (ex: microscopic calculations): N= number of basis functions are calculated (depend on a, but  should not)

11. Breit-Wigner approximation: peculiar case where N=1 One-pole approximation: N=1 Resonance energy: Thomas approximation: Then R-matrix parameters (calculated) Observed parameters (=data)

12. Capture cross sections in the R-matrix formalism  New parameters:   = gamma width of the poles   = interference sign between the poles  is equivalent to the Breit-Wigner approximation if N=1  Relative phase between M int and M ext : ±1  M int and M ext are NOT independent of each other:  a must be common  U in M ext should be derived from R in M int  Sometimes in the literature:  exp(-Kr)

13. Extension to 12 C(  ) 16 O: N>1 Problem: many experimental constraints (energies,  and  widths) → how to include them in the R-matrix fit? Previous techniques: fit of the R-matrix parameters 2+ 11.52 3 poles + background → 12 R-matrix parameters to be fitted + constraints (experimental energies, widths) New technique: start from experimental parameters (most are known) and derive R matrix parameters  strong reduction of the number of parameters!

14. Generalization of the Breit-Wigner formalism: link between observed and formal parameters when N>1 C. Angulo, P.D., Phys. Rev. C 61, 064611 (2000) C. Brune, Phys. Rev. C 66, 044611 (2002) idea: Information for E2: 2 + phase shift E2 S-factor spectroscopy of 2 + states in 16 O: energy  and  widths

15. 2+ 11.52 Three 2 + states + background Energy (MeV)  width (MeV)  width (eV) -0.24?0.097 2.683.68 x 10 -4 0.0057 4.361.39 x 10 -2 0.61 Backg.10??  3 parameters + interference signs in capture  2 steps:1) phase shifts:  widths 2) S factor:  width of the background  the S-factor is fitted with a single free parameter From phase shift From S factor Application to 12 C(  ) 16 O: E2 contribution Main goal: to reduce the number of free parameters

16. First step: fit of the 2 + phase shift 2 parameters:

17. Phase shift: Strong influence of the background! 2+ 11.52

18. Second step: fit of the E2 S-factor 1 remaining parameter: 4 poles→4 signs  1,  2,  3,  4,  1 =+1 (global sign)  4 =+1 (very poor fits with  4 =-1) S E2 (300 keV)=190-220 keV-b

19. Paper by Kunz et al., Astrophy. J. 567 (2002) 643 Similar analysis (with new data) S E2 (300 keV)=85 ± 30 keV-b  very different result

20. Origin: difference in the background treatment Here: background at 10 MeV Kunz et al.: background at 7.2 MeV R matrix: S factor at 300 keV“well” knownbackground Between 1~3 MeV, terms 1 and 4: have opposite signs are large and nearly constant  Several equivalent possibilities   -scattering does not provide without ambiguities!  Consistent with a recent work by J.M. Sparenberg

21. Recent work by J.-M. Sparenberg: Phys. Rev. C69 (2004) 034601 Based on supersymmetry (D. Baye, Phys. Rev. Lett. 58 (1987) 2738) acts on bound states of a given potential without changing the phase shifts VV Supersymmetric transformation Both potentials have exactly the same phase shifts (different wave functions) r r Original potential Transformed potential

22.  With this method: different potentials with  Same phase shifts  Different bound-state properties  Example: V(r)=V 0 exp(-(r/r 0 ) 2 )/r 2, with V 0 =43.4 MeV, r 0 =5.09 fm No bound state V(r) Supersymmetric partners Identical phase shifts!

23. Conclusion:  It is possible to define different potentials giving the same phase shifts but different  No direct link between the phase shifts and the bound-state properties  Consistent with the disagreement obtained for R-matrix analyses using different background properties (~ potential)   the background problem should be reconsidered!

24. One indirect method: cascade transitions to the 2 + state F.C. Barker and T. Kajino, Aust. J. Phys. 44 (1991) 369 L. Buchmann, Phys. Rev. C64 (2001) 022801 Weakly bound: -0.24 MeV Capture to 2 + is essentially external M int negligible The cross section to the 2 + state is proportional to

25.  12 C(  ) 16 O is probably the best example where the interplay between experimentalists, theoreticians and astrophysicists is the most important  Required precision level too high for theory alone  we essentially rely on experiment  E1 probably better known than E2 ( 16 N  -decay)  Elastic scattering is a useful constraint, but not a precise way to derive  Possible constraints from astrophysics?  New project 16 O+  →  12 C (Triangle, North-Carolina) “Final” conclusions What do we know?

26. What do we need? Theory: reconsider background effects Precise E1/E2 separation (improvement on E2) Capture to the 2+ state Data with lower error bars: precise data near 1.5 MeV are more useful than data near 1 MeV with a huge error Please avoid this!