The R-matrix method and 12C(a,g)16OPierre DescouvemontUniversité Libre de Bruxelles, Brussels, Belgium • Introduction • The R-matrix formulation: elastic scattering and capture • Application to 12C(a,g)16O • Conclusions and outlook
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 12C(a,g)16O: nearly all recent papers • References: • A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30 (1958) 257 • F.C. Barker, many papers
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 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 • In practice limited to low energies (each Jp must be considered individually). well adapted to nuclear astrophysics
R-matrix parameters = poles Example: 12C+a 1- , ER=2.42 MeV, Ga=0.42 MeV 12C+a Reduced width g2 :Ga=2 g2 P(ER), with P = penetration factor 16O Physical parameters = “observed” parameters Resonances: R-matrix parameters = “formal” parameters Poles: Similar but not equal
Background pole High-energy states with the same Jp Simulated by a single pole = background Energies of interest Isolated resonances: Treated individually Non resonant calculations possible: only a background pole
Derivation of the R matrix (elastic scattering) • Hamiltonian: H Y=E Y • With, for r large: • Il, Ol= Coulomb functions • Ul = collision matrix (→ cross sections) • = exp(2idl) for single-channel calculations • b. Wave functions • Set of N basis functions ul(r) with • Total wave function
Bloch-Schrödinger equation: With L = Bloch operator (restore the hermiticity of H over the internal region) Replacing Yint(r) and Yext(r) by their definition:
Reduced width: proportional to the wave function in a ”measurement of clustering” Dimensionless reduced width “first guess”: q2=0.1 Total width: Solving the system, one has: R matrix P=penetration factorS=shift factor R-matrix parameters Depend on a =reduced width
Phase shift: • Two approaches: • Fit: The number of poles N is determined from the physics of the problem In general, N=1 but NOT in12C(a,g)16O : N=3 or 4 (or more) are fitted • Variational calculations (ex: microscopic calculations): • N= number of basis functions • are calculated (depend on a,but d should not)
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)
exp(-Kr) Capture cross sections in the R-matrix formalism • New parameters: Gg = gamma width of the polesel= interference sign between the poles • is equivalent to the Breit-Wigner approximation if N=1 • Relative phase between Mint and Mext : ±1 • Mint and Mext are NOT independent of each other: • a must be common • U in Mext should be derived from R in Mint • Sometimes in the literature:
Extension to 12C(a,g)16O: N>1 • Problem: many experimental constraints (energies, a and g widths)→ how to include them in the R-matrix fit? • Previous techniques: fit of the R-matrix parameters 11.52 2+ • 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!
Generalization of the Breit-Wigner formalism: link between observed and formal parameters when N>1C. 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 16O: energy a and g widths
Application to 12C(a,g)16O: E2 contribution Main goal: to reduce the number of free parameters Three 2+ states + background 11.52 2+ From phase shift From S factor 3 parameters + interference signs in capture 2 steps: 1) phase shifts: a widths 2) S factor: g width of the background the S-factor is fitted with a single free parameter
First step: fit of the 2+ phase shift 2 parameters:
Phase shift: 11.52 2+ Strong influence of the background!
Second step: fit of the E2 S-factor 1 remaining parameter: 4 poles→4 signs e1, e2, e3, e4, e1=+1 (global sign) e4=+1 (very poor fits with e4=-1) SE2(300 keV)=190-220 keV-b
Paper by Kunz et al., Astrophy. J. 567 (2002) 643 Similar analysis (with new data) SE2(300 keV)=85 ± 30 keV-b very different result
a-scattering does not provide without ambiguities! 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” known background Between 1~3 MeV, terms 1 and 4: have opposite signs are large and nearly constant Several equivalent possibilities Consistent with a recent work by J.M. Sparenberg
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 Original potential Transformed potential V V r r Supersymmetric transformation Both potentials have exactly the same phase shifts (different wave functions)
With this method: different potentials with • Same phase shifts • Different bound-state properties • Example: V(r)=V0 exp(-(r/r0)2)/r2, with V0=43.4 MeV, r0=5.09 fm No bound state V(r) Supersymmetric partners Identical phase shifts!
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!
The cross section to the 2+ state is proportional to 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 • Mint negligible
“Final” conclusions What do we know? • 12C(a,g)16O 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 (16N b-decay) • Elastic scattering is a useful constraint, but not a precise way to derive • Possible constraints from astrophysics? • New project 16O+g→a+12C (Triangle, North-Carolina)
Please avoid this! 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