Global R-Matrix Analysis of the 12 C(α,γ) 16 O Reaction

1. Global R-Matrix Analysis of the 12C(α,γ)16O Reaction Carl Brune Ohio University, Athens Ohio NN2012 May 28, 2012

2. Origin of the Elements • Age of Universe: 13.7 Gyr • Age of Solar System: 4.5 Gyr Big Bang Star Formation Stellar Nucleosynthesis Novae Supernovae Image credit: NASA

3. Helium Burning Reactions • 3α 12C • -- α + α8Be (8Be ground state is just unboud) • -- 8Be + α12C + γ (via the 7.6-MeV “Hoyle” state) • rate known with ~10%, but still under intense scrutiny • ~ branching ratio measurements • ~ electron scattering measurement [TU Darmstadt, • Chernykh et al., PRL 98, 032501 (2007)] • 12C(α,γ)16O • -- rate only known within 30-50% • Rates of these reactions are roughly equal • Further α captures are hindered by the Coulomb barrier and • absence of resonances

4. Helium Burning Reactions Matter • They determine: • The 12C/16O ratio in massive stars • (and hence for the Universe) • Nucleosynthesis of heavier elements • The remnant mass after the supernova explosion • (black hole versus neutron star) • See recent studies: • Tur, Heger, and Austin, ApJ 671, 821 (2007) • Tur, Heger, and Austin, ApJ 702, 1068 (2009)

5. Model Calculations from Tur, Heger, and Austin ratio to solar  • Uses the Kepeler code (developed by Weaver, Woosley, and collaborators) • Tracks nucleosynthesis from zero-age main sequence through the explosion • Varies 12C(α,γ) rate, starting from Buchmann (1996): S(300 keV) = 146 keV-b • Preferred multiplier is 1.2, with an error of ≈ 25% • Other uncertainties: semiconvection, overshoot mixing, explosion mechanism,… • Tail wagging the dog?

6. The Problem: Red Giant The Lab T=(1-3)x108 K • cross section is abnormally small (E1 is isospin-forbidden) • subthreshold resonances

7. Thermonuclear Reaction Rate 12C(a,g)16O Charged Particles Coulomb barrier: S = “astrophysical S factor” EG is a constant - “Gamow Energy” Reaction Rate Formalism: T = temperature k = Boltzmann constant 2 x 108 K  E0=300 keV S(E0) essentially fixes the rate

8. 12C(,)16O Cross Section 12C(,) - extrapolation to helium burning energies E0≈300 keV 12C(,) cross section E1, E2 g.s. transitions thought to be largest cascade transitions Up to 30% contribution

9. Separating E1 and E2 Ground-State Components Dyer and Barnes (1974) A new parameter, the relative phase f, is introduced. Queens (1996)

10. The E1-E2 Phase and Elastic Scattering • The parameters d1 and d2 are scattering phases;h is the Coulomb parameter. • First derived by Barker assuming single-level R-matrix formulas; later derived for the fully general case (many levels and direct capture). • Also verified by L.D. Knutson in another context (1999) -- the formula is a consequence Watson’s Theorem (1954). • The only assumption is that the capture channels are weak – the same assumption that we make when using real phase shifts! There is no reason not to take the phase from elastic scattering!

11. Benefits of Fixing the Phase With Elastic Scattering • Smaller statistical errors, particularly when on of the capture components is small. • Less chance for systematic errors to drive the fit in the wrong direction. Speaking of systematic errors… • Kinematic effects on the g-ray distributions are often ignored (Assuncao et al. 2006?). • We have b~0.01 for normal kinematics. Seems small… • But ignoring it increases the extracted E2 cross section by 10-15% for E<2.5 MeV!

12. E1 Ground-State Cross Section Over 3 decades of work; figure from Assuncao et al. (2006)

13. E2 Ground-State Cross Section Measurements at higher energies will be helpful

14. Interference near the 2.68-MeV Resonance (E2) Narrow (0.6 keV) but important for 2 < E < 3.5 MeV !

15. One Approach to the Interference Question • Integrated (thick-target) yield of the resonance shows anisotropy • due to interference with underlying E1 cross section • Can be utilized to determine the interference sign • Measurement performed at Ohio University by Daniel Sayre

16. Results • Theoretical expectation: • - W(θ) = 1 + (5/7)P2 + (12/7)P4 + a (P1-P3) • - a = ±0.08, determined by Γα, Γγ, σE1, and phase shifts • - the sign of a is to be determined • Our result: a = +0.07(5) • Error is predominantly statistical; the systematic error is 0.01 • Our measurement favors the “positive a” interference scenario by three standard deviations

17. R-Matrix Method • Exact implementatonof quantum-mechanical symmetries and conservation laws (Unitarity) • Treats long-ranged Coulomb potential explicitly • Wavefunctions are expanded in terms of unknown basis functions • Energy eigenvalues and the matrix elements of basis functions are adjustable parameters, which are typically optimized via χ2 minimization • A wide range of physical observables can be fitted (e.g. cross sections, Ex, Gx,…) • The fit can then be used to determine unmeasured observables • Better than the alternatives (effective range, K-matrix,…) • Major Approximation: TRUNCATION (levels / channels)

18. R-Matrix Fits For L=2 • Take into account our findings regarding the interference of the 2.68-MeV resonance • Improvements over previous analyses: • - Allow normalizations of data sets to float where appropriate • - Discard data points which are clearly wrong (Chauvenet’s Criteria) • - Full simultaneous fitting • Start with E2 ground-state data: Dyer and Barnes (1974), Redder et a al. (1987), Ouellet et al. (1996), Roters et al. (1999), Kunz et al. (2001), Assuncao et al. (2006), Makiiet al. (2009), Tischhauser et al. (2009) • Use a 5-level fit, χ2 minimization • Error analysis: χ2 ≤ χ2min(1 + 9/ν)

19. Chauvenet’s Criteria Consider discarding points with χ2 ≥ [erfc-1(1/2N)]2

20. Best Fit

21. Results • SE2 = 64+10-11keV-b(at E=300 keV) • Similar to Tischhauser et al. analysis (SE2=53+13-18keV-b), except: • - new capture data included • - exclusion of a few data points • - interference structure above 2 MeV clarified

22. Global Fit to All data • Fit all multipolarities and transitions simultaneously • - capture data, elastic scattering, β-delayed α spectrum • Fit primary data (angular distributions) where possible • Investigate channel radius, level truncation sensitivity • Work in progress, particularly the error analysis

23. Example Ground-State Angular Distributions

24. Fit to Total Cross SectionSchürmann et al. (2005)

25. Summary of a Representative Fit

26. Conclusions and Outlook • Nature of the interference involving the 2.68-MeV resonance • has been determined • A global R-matrix analysis is underway • Presently, S(300 keV) ≈ 160 keV, with an error bar of ≈ 30% • Reasonable agreement with the astrophysical prediction • of ≈ 175 keV-b • More accurate 12C(a,g)16O data at lower energies (but…) • Measurements of ground-state capture above 3 MeV • Additional measurements of cascade transitions • More experimental attention to systematic errors • Indirect methods are still crucial • Trust, but verify! (Ronald Reagan)

27. Thanks to Collaborators • Former graduate students: CatalinMatei, Daniel Sayre • Others at OU: RemiAdekola, Don Carter, Chris Dodson, Steve Grimes, Zach Heinen, David Ingram, Devon Jacobs, Tom Massey, John O’Donnell, Moses Oginni, Alexander Voinov • Other experimental collaborators: Xiaodong Tang (ND), Ernst Rehm (ANL), LotherBuchmann (TRIUMF),… • R-matrix collaborators: Gerry Hale (LANL), Dick Azuma (U. of Toronto), E. Uberseder (ND),…