R-Matrix Methods and an Application to 12C(α,γ)16O Carl R. Brune Ohio University 5th European Summer School on Experimental Nuclear Astrophysics 24 September 2009

Outline/Goals of this Presentation Overview of R-Matrix and 12C(α,γ)16O (hopefully pedagogical!) Overview Recent Results Open Problems

R-Matrix Methods Developed by Wigner and collaborators in the 1940s Applicable to wide range of problems in nuclear and atomic physics A useful tool for both experimentalists and theorists Here we are concerned with the semi-empirical representation of nuclear physics data Many advances in R-Matrix techniques have been motivated by 12C(α,γ)16O

Why / When to use R-Matrix Methods ? Parameterization of data for reaction rate calculation Extrapolation (or interpolation) of data into regions without data When dealing with resonances - particularly when more than one - particularly when resonances are resolved with non-negligible widths * A < 25 for charged-particle reactions * neutron-induced reactions at low energies When at low energies (few partial waves) When incorporating information from multiple sources - cross section data - spectroscopic information (excitation energies, spins,…) - transfer reactions (ANCs / spectroscopic factors) - beta decay

Low-Energy Neutron Capture (S-Process) Fujii et al. (CERN / n-TOF)

Why R Matrix ? Red Giant T=(1-3)x108 K

1- and 2+ states of 16O

R-Matrix Method Exact implimentaton of 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 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 Major Approximation: TRUNCATION (levels / channels)

R-Matrix Parameters l – level label c – channel label (e.g. a-particle, g-ray) El – level energy glc – reduced width ampitude Bc – boundary condition constants (related to “level shift”) ac – channel radius

R-Matrix Calculations Level energies and reduced widths  R-matrix R-matrix (plus Bc and ac)  scattering matrix scattering matrix  total and differential cross sections

Breit-Wigner Formula General QM result (from atomic physics to Z0 bosons) G0c are observed partial widths ER and G0c are considered “Physical Parameters” Physical Parameters should be independent of boundary condition constants (Bc) and channel radius (a)

1-Level R-Matrix Formula Very similar to general Breit-Wigner formula Glc are formal partial widths

Make it look like the B-W Formula: 1-level R-matrix “standard” Breit-Wigner

What if we have more than one level? 1-level approximation very good when E is near El. Same procedure can be used. But Bc can only be set once [ recall Bc = Sc(El) ] Simple relation to physical parameters only for one level

How does one define a resonance energy? Peaks of excitation function ? Phase shift = p/2 ? (for elastic scattering) Complex poles of the S-matrix (or U-matrix) ? … Bottom line: There is no “right” answer, be careful about “apples and oranges”. I will describe how we can do this in the R-matrix formalism.

Can we change the Bc? Yes ! If Bc  Bc’ then if and El’ and glc’ are given by THEN the U matrix (i.e. cross sections, etc…) are not changed! F.C. Barker Aust. J. Phys 25, 341 (1972). In practice you could also re-fit the data with different Bc.

Definition of the A Matrix The A matrix determines the U matrix The U matrix determines observables (cross sections, etc…)

Extracting Physical Parameters is Iterative Another approach: Solve Eigenvalue Equation Non-linear! Eigenvalues are the resonance energies Eigenvectors yield the physical partial widths Note: G.M. Hale studies the complex eigenvalues of this equation:

Physical Parameters to R-Matrix Parameters Another eigenvalue equation The eigenvalues are El The eigenvectors can be arranged into a matrix b which diagonalizes M and N, and also yields gc

Working Directly with Physical Parameters Definition of the A matrix in terms of physical parameters Mathematically equivalent to Lane and Thomas (i.e. same U) See C.R. Brune, Phys. Rev. C 66, 044611 (2002).

1- and 2+ states of 16O We have two narrow 2+ states with well-known properties!

16N(ba) Spectrum What fills in the interference minimum? 3- strength? What fills in the interference minimum? The reduced a width of the 6.1-MeV 3- state required is much larger than found theoretically (Descouvemont) or by transfer reactions…

Inclusion of Direct (External) Capture for Electromagnetic Reactions The direct (external) contribution to capture reactions, which depends of the reduced width of the final state, can be included in a consistent manner - essentially “direct capture” - F.C. Barker and T. Kajino, Aus. J. Phys. 44, 369 (1991) - R.J. Holt et al., Phys. Rev. C 18, 1962 (1978) Applicable to E2 transitions in 12C(α,γ)16O Reduced particle widths enter in two ways  more constraints on fitting

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

Data Relevant to 12C(a,g)16O 12C(a,g)16O cross section data (required!) * ground and excited states of 16O * wide range of energies 12C(a,a) elastic scattering data 16N b-delayed a spectrum Bound-state spectroscopy (Ex, Gx,…) Transfer reactions In some ways we are lucky: There are relatively few levels to be considered 12C and a are spin-0 nuclei

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

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!

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!

E1 Ground-State Cross Section Figure from Assuncao et al. (2006)

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

Cascade Cross Sections Probably provide 20-30% of the cross section at astrophysical energies. Help to constrain the ground-state cross section (same R-matrix parameters are involved) Limited data

New Total Cross Section Measurement ERNA/Bochum/Napoli (D. Schürmann et al. 2005), using a Recoil separator and inverse kinematics – all final states

New Direct Measurement Stuttgart (Assuncao et al. 2006), using Ge detectors

First Measurement of Cascade to 6.05-MeV State

First Measurement of Cascade to 6.05-MeV State TRIUMF/ISAC/DRAGON (C. Matei et al. 2006), BGO + recoil separator: S6.05=25(16) keV-b

DRAGON / ERNA Comparison

New Measurement of 16N(βα): ANL/ATLAS

New Measurement of 16N(βα): ANL/ATLAS W. Tang et al. 2007: SE1=74(21) keV-b

12C(a,g)16O(6.92 MeV) Cascades likely contribute significantly to the total cross section at astrophysical energies (this transition is estimated to be 5-10%). The direct capture contribution can (in principle) be analyzed to yield the reduced a width of the 6.92-MeV state (relevant for E2 ground-state cross section). Gg for the 7.12-6.92 MeV transition is unknown. We (primarily my former student, Catalin Matei) have attempted to measure the g-ray branching of the 7.12-MeV state.

Branching Ratio Measurement at Ohio University 19F(p,a)16O(7.12 MeV) Detect coincidence between 0.2- and 6.92-MeV g rays. Difficulty: small branching ratio, expected < 10-4.

Result for the 7.126.13-MeV transition Fit selected region to extract background and count events of interest. Calibrated sources and GEANT simulations used to estimate detectors efficiency Calculate 7.126.13-MeV branching ratio:

Result for the 7.126.92-MeV transition A limit for this transition can be set with a 2- confidence level:

Results for 12C(a,g)16O(6.92 MeV) Capture data are from Kettner et al. (triangles, 1982) and Redder et al. (circles, 1987) C. Matei et al. 2008: SE1=7.1(1.6) keV-b

New Measurement of Elastic Scattering

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

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

Summary of Determinations Result @ E=300 keV source SE1=79(21) keV-b 16N(ba), Buchmann et al. (1994) SE1=99(44) keV-b direct measurement, Roters et al. (1999) SE1=101(17) keV-b sub-Coulomb a transfer, Brune et al. (1999) SE1=74(21) keV-b 16N(ba), Tang et al. (2007) SE2=120(60) keV-b compilation, NACRE (1999) SE2=42+16-23 keV-b SE2=85(30) keV-b direct measurement, Kunz et al. (2001) SE2=53+13-18 keV-b 12C(a,a), Tischhauser et al. (2002) SC=16 keV-b theoretical, Barker and Kajino (91) SC=4(4) keV-b S6.05=25(16) keV-b Direct measurement, Matei et al. (2006) S6.9=7.1(1.6) keV-b Gamma branching, Matei et al. (2008) Stot ≅160 keV-b

Future Needs and Conclusions More accurate 12C(a,g)16O data at lower energies (but…) Measurements of ground-state capture above 3 MeV Additional measurements of cascade transitions Consistent assessment of uncertainties Further work on R-Matrix fitting (simultaneous fitting of cascade transitions in Barker-Kajino framework) The recoil separator + gamma detector approach needs to be fully exploited Indirect methods are still crucial Trust, but verify!