1. THEORETICAL PREDICTIONS OF THERMONUCLEAR RATES P. Descouvemont 1.Reactions in astrophysics 2.Overview of different models 3.The R-matrix method 4.Application to NACRE/SBBN compilations 5.Typical problems/questions 6.Conclusions

2. Low level densities:Light nuclei (typically A < 20, pp chain, CNO) or close to the drip lines (hot burning) Then: In general data are available: problems for compilations: * extrapolation * coming up with “recommended” cross sections * computing the rate from the cross section Specific models can be used Each reaction is different: no systematics High level densities: Hauser-Feshbach * accuracy? * data with high energy resolution are required

3. Types of reactions: 1. Capture (p,  ), (  ): electromagnetic interaction Non resonant Isolated resonance(s) Multi resonance 2. Transfer: (p,  ), ( ,n), etc: nuclear process Non resonant Isolated resonance(s) Multi resonance Transfer cross sections always larger than capture cross sections

4. 3/2- 1/2- 7/2- 5/2-  + 3 He 6 Li+p 2+ 1+ 3+ 7 Be+p 1/2- 1/2+ 3/2- 5/2+ 12 C+p 13 N 8B8B 7 Be Non resonant Resonant l min =0 l R =1 Resonant l min =0 l R =0

5. 0+ 3- 2+ 1- 2- 1- 2+  + 12 C 16 O 0+0  + 22 Ne 26 Mg n+ 25 Mg 125 states 20 states Subthreshlod states 2 +, 1 - multiresonant  Many different situations

6. Theoretical models ModelApplicable toComments Light systems Low level densities Potential modelCaptureInternal structure neglected Antisymmetrization approximated R-matrixCapture Transfer No explicit wave functions Physics simulated by some parameters DWBATransferPerturbation method Wave functions in the entrance and exit channels Microscopic models Capture Transfer Based on a nucleon-nucleon interaction A-nucleon problems Predictive power Hauser-Feshbach Shell model Capture Transfer Capture Statistical model Only gamma widths Heavy systems

7. Question: which model is suitable for a compilation? Potential model: limited to non resonant reactions (or some specific resonances): –NO DWBA: limited to transfer reactions, too many parameters: –NO Microscopic: too complicated, not able to reproduce all resonances: –NO R-matrix: only realistic common procedure: –If enough data are available –If you have much (“unlimited”) time –MAYBE Conclusion: For a broad compilation (Caltech, NACRE): no common method! For a limited compilation (BBN): R-matrix possible Problems for a compilations: Data evaluation Providing accurate results (and uncertainties) Having a method as “common” as possible “Transparency” Using realistic durations and manpower This system has no solution  a compromise is necessary

8. The R-matrix method Goal: treatment of long-range behaviour Internal region External region E>0 E<0  The R matrix

9. ●Applications essentially in: ●atomic physics ●nuclear physics ●Broad field of applications ●Resonant AND non-resonant calculations ●Scattering states AND bound states ●2-body, 3-body calculations ●Elastic scattering, capture, transfer (Nuclear astrophysics) beta decay, spectroscopy, etc…. ●2 ways of using the R matrix 1.Complement a variational calculation with long-range wave functions 2.Fit data (nuclear astrophysics) ●Main reference: Lane and Thomas, Rev. Mod. Phys. 30 (1958) 257.

10. 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 Main idea of the R matrix: to divide the space into 2 regions (radius a) –Internal: r ≤ a: Nuclear + coulomb interactions –External: r > a: Coulomb only

11. Basic ideas (elastic scattering) Isolated resonances: Treated individually High-energy states with the same J  Simulated by a single pole = background Energies of interest Phenomenological R matrix: E,  are free parameters Non-resonant calculations are possible: only a background pole

12. Transfer reactions Poles E >0 or E <0 Elastic scattering Threshold 1 Threshold 2 Inelastic scattering, transfer Pole properties: energy reduced width in different channels (  more parameters) gamma width  capture reactions R matrix  collision matrix  transfer cross section

13. elastic New parameter (  width) Elastic: E,  : pole energy and particle width Capture:+    pole gamma width  3 parameters for each pole (2 common with elastic) Capture reactions: more complicated Internal contribution:

14.  3 steps: 1.Elastic scattering  R matrix, phase shift  2.Introduction of C , external contribution M ext 3.Introduction of gamma widths  Calculation of M int External contribution: If external capture [ 7 Be(p,  ) 8 B, 3 He(  ) 7 Be]:  A single parameter: ANC

15. R matrix fit: P.D. et al., At. Data Nucl. Data Tables 88 (2004) 203 Only the ANC is fitted S(0)=0.51 ± 0.04 keV-b Cyburt04: 4 th order polynomial: S(0)=0.386 keV-b  danger of polynomial extrapolations!

16. Comparison of 2 compilations: NACRE (87 reactions): C. Angulo et al., Nucl. Phys. A656 (1999) 3  previous: Fowler et al. (1967, 1975, 1985, 1988) SBBN (11 reactions): P.D. et al., At. Data Nucl. Data Tables 88 (2004) 203  previous:M. Smith et al., ApJ Supp. 85 (1993) 219 K.M. Nollett, S. Burles, PRD61 (2000) 123505. NACRE fits or calculations taken from literature Polynomial fits Multiresonance (if possible) Hauser-Feshbach rates Rough estimate of errors SBBN R matrix for all reactions Statistical treatment of errors

17. Example: 3 He(  ) 7 Be NACRE RGM calculation by Kajino Scaled by a constant factor S(0)=0.54±0.09 MeV-b SBBN Independent R-matrix fits of all experiment Determination of averaged S(0) S(0)=0.51± 0.04 MeV-b

18. Typical problems for compilations: Difficulty to have a “common” theory for all reactions Data inconsistent with each other  how to choose?

19. In resonant reactions, how important is the non-resonant term? Properties of important resonances? 15 O(  19 Ne Very little is known exp. 3/2 + resonance not described by  + 15 O models

20. Level density 19 F+p How to relate the peaks in the S-factor with the 20 Ne levels? How to evaluate (reasonably) the uncertainties?

21. Error treatment Assume parameters p i, N experimental points Define Find optimal values p i (min) and  2 (min)

22. p1p1 p2p2 p 1 (min) p 2 (min) Define the range Sample the parameter space (Monte-Carlo, regular grid) Keep parameters inside the limit Determine limits on the S factor

23. Common problems:  2 (min)>1 : then statistical methods cannot be applied different experiments may have very different data points (  overweight of some experiments) Giving the parameters with error bars p1p1 p2p2 p1p1 No correlation:  p 1 given as p 1 (min)±  p 1 p1p1 Strong correlation between p 1 and p 2  Need of the covariance matrix p2p2

24. Analytical fits  tables with rates “Traditional” in the Caltech compilations Useful to understand the physical origin of the rates Difficult to derive with a good precision (~5%) in the full temperature range Question for astrophysicists: Tables only? Fits only? Tables and fits?

25. Conclusion Compilations are important in astrophysics But Having a high standard is quite difficult (impossible?) –Large amount of data (sometimes inconsistent and/or not sufficient) –No systematics –No common model Ideally: should be regularly updated Then Long-term efforts Small groups: difficult to find time Big groups: difficult to find agreements Compromises are necessary