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Theoretical Predictions of Thermonuclear Rates in Astrophysics - Overview and Applications of Various Models

Explore theoretical models for predicting thermonuclear reaction rates in astrophysics, including R-matrix method applications, potential problems, and conclusions. Learn about low and high-level densities, types of reactions, and the R-matrix method's goal. Discover which models are suitable for different compilations and the applications of the R-matrix in atomic and nuclear physics. Dive into the complexities of elastic scattering, capture reactions, and transfer processes in nuclear astrophysics research.

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Theoretical Predictions of Thermonuclear Rates in Astrophysics - Overview and Applications of Various Models

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  1. THEORETICAL PREDICTIONS OF THERMONUCLEAR RATESP. Descouvemont Reactions in astrophysics Overview of different models The R-matrix method Application to NACRE/SBBN compilations Typical problems/questions 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,g), (a,g): electromagnetic interaction • Non resonant • Isolated resonance(s) • Multi resonance • 2. Transfer: (p,a), (a,n), etc: nuclear process • Non resonant • Isolated resonance(s) • Multi resonance • Transfer cross sections always larger than capture cross sections

  4. 5/2- 5/2- 6Li+p 7/2- Non resonant a+3He 1/2- 3/2- 7Be 3+ Resonantlmin=0lR=1 1+ 7Be+p 2+ 8B 5/2+ 3/2- Resonantlmin=0lR=0 1/2+ 12C+p 1/2- 13N

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

  6. Theoretical models

  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. E<0 E>0 The R-matrix method Goal: treatment of long-range behaviour Internal region External region  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 • Complement a variational calculation with long-range wave functions • Fit data (nuclear astrophysics) • Main reference: Lane and Thomas, Rev. Mod. Phys. 30 (1958) 257.

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

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

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

  13. Capture reactions: more complicated Internal contribution: New parameter (g width) elastic Elastic: El, gl: pole energy and particle width Capture: + Ggl : pole gamma width 3 parameters for each pole (2 common with elastic)

  14. External contribution: • 3 steps: • Elastic scattering  R matrix, phase shift d • Introduction of C  , external contribution Mext • Introduction of gamma widths  Calculation of Mint If external capture [7Be(p,g)8B, 3He(a,g)7Be]:  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: 4th 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: 3He(a,g)7Be • 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? • 15O(a,g)19Ne • Very little is known exp. • 3/2+ resonance not described by a+15O models

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

  21. Error treatment • Assume n parameters pi, N experimental points • Define • Find optimal values pi(min) and c2(min)

  22. Define the range • Sample the parameter space (Monte-Carlo, regular grid) p2 p2(min) p1(min) p1 • Keep parameters inside the limit • Determine limits on the S factor

  23. Common problems: • c2(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 p2 p2 p1 p1 Dp1 No correlation: p1 given as p1(min)±Dp1 Strong correlation between p1 and p2  Need of the covariance matrix

  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

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