Analysis of low-energy scattering and weakly-bound states through - PowerPoint PPT Presentation

Analysis of low-energy scattering and weakly-bound states through effective-range functions. Application to 12 C+ α . arez 1 Jean-Marc Sparenberg, Oscar Leonardo Ram´ ırez Su´ Nuclear Physics and Quantum Physics, Universit´ e libre de Bruxelles (ULB), Brussels, Belgium, EU June 2nd, 2015 Centre for Nuclear and Radiation Physics, University of Surrey, Guildford, England, UK ulbnorm 1 PhD student 2011-2014 Analysis of low-energy 12 C+ α scattering Jean-Marc Sparenberg (ULB) University of Surrey 2015 1 / 29

The 12 C ( α, γ ) 16 O reaction in nuclear astrophysics 1 (Weakly-)bound states and asymptotic normalization constant 2 (Low-energy) elastic scattering and phase shifts 3 Effective-range expansion(s) for scattering and bound states 4 Analysis of the 12 C + α experimental p -wave phase shifts 5 Analysis of the 12 C + α experimental d -wave phase shifts 6 Conclusions and perspectives 7 ulbnorm Analysis of low-energy 12 C+ α scattering Jean-Marc Sparenberg (ULB) University of Surrey 2015 2 / 29

The 12 C ( α, γ ) 16 O reaction in nuclear astrophysics The 12 C ( α, γ ) 16 O reaction in nuclear astrophysics 1 (Weakly-)bound states and asymptotic normalization constant 2 (Low-energy) elastic scattering and phase shifts 3 Effective-range expansion(s) for scattering and bound states 4 Analysis of the 12 C + α experimental p -wave phase shifts 5 Analysis of the 12 C + α experimental d -wave phase shifts 6 Conclusions and perspectives 7 ulbnorm Analysis of low-energy 12 C+ α scattering Jean-Marc Sparenberg (ULB) University of Surrey 2015 3 / 29

The 12 C ( α, γ ) 16 O reaction in nuclear astrophysics 12 C + α → γ + 16 O radiative capture in red-giant stars Importance: competes with 3 α → 12 C in red-giant helium burning ⇒ C/O in Universe Key quantity: cross section σ at “low” energy (Gamow peak) E c . m . = 300 keV Dominant transitions: E1 (resp. E2) ◮ from p (resp. d ) 12 C + α scattering state ◮ to s 16 O ground state (spin 0) Experiment: Coulomb repulsion Betelgeuse [Dupree, Gilliland, ⇒ very small cross section ⇒ data for E > 1 MeV only Hubble ST, NASA, ESA, 1996] Theoretical extrapolation to low energy: necessary but difficult because p and d subthreshold bound states enhance cross section but are not well known ulbnorm Analysis of low-energy 12 C+ α scattering Jean-Marc Sparenberg (ULB) University of Surrey 2015 4 / 29

The 12 C ( α, γ ) 16 O reaction in nuclear astrophysics 12 C + α → γ + 16 O direct measurements (recent) p -wave ( E 1 ) and d -wave ( E 2 ) capture to s -wave ground state Astrophysical S -factor S ( E ) = σ ( E ) E exp(2 πη ) η = Z 1 Z 2 e 2 / � v dimensionless Sommerfeld parameter � relative velocity v = 2 E c . m . / 2 µ ulbnorm [Plag et al. , PRC 2012] [Kunz et al. , PRL 2001; Hammer, Fey et al. , NPA 2005] Analysis of low-energy 12 C+ α scattering Jean-Marc Sparenberg (ULB) University of Surrey 2015 5 / 29

✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � � ✁ ✁ � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � ✁ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � The 12 C ( α, γ ) 16 O reaction in nuclear astrophysics Indirect information on subthreshold states Well-known energies (experiment) 2+ S(E) ◮ 1 − : E c . m . = − 45 keV 9.85 1- 9.59 ◮ 2 + : E c . m . = − 245 keV 2- 8.87 Structure rather-well understood (theory) 0.3 MeV 7.162 1- 7.117 12C+ α ◮ 1 − : 1 particule-1 hole shell-model state 2+ 6.917 3- 6.130 ◮ 2 + : 12 C+ α cluster structure, part of + 6.049 0 rotational band with 0 + 2 and 4 + E1,E2 π + 0 0 16O But asymptotic normalization constants [Buchmann, private com. 2004] (ANC ≈ width) not well known Radiative cascade transitions ⇒ (difficult) access to 1 − and 2 + states [Kettner et al. , ZPA 1982; Redder et al. , NPA 1987; Plag et al. , PRC 2012] 16 N β -delayed α decay ⇒ access to 1 − state [Azuma et al. , PRC 1994; Tang et al. , PRC 2010] α transfer reactions: 12 C( 6 Li, d) 16 O and 12 C( 7 Li, t) 16 O ⇒ access to 1 − and 2 + states [Brune et al. , PRL 1999] ulbnorm S E 1 (300 keV) = 101 ± 17 keV b and S E 2 (300 keV) = 42 +16 − 23 keV b Analysis of low-energy 12 C+ α scattering Jean-Marc Sparenberg (ULB) University of Surrey 2015 6 / 29

The 12 C ( α, γ ) 16 O reaction in nuclear astrophysics Indirect information from 12 C + α elastic scattering Access to 1 − and 2 + states? Motivated high-precision elastic scattering remeasurement [Tischhauser et al. , PRL 2002, PRC 2009] Analysis with R-matrix formalism but questions raised ◮ partial-wave decomposition: contradicts scattering inverse problem? ◮ background description: channel radius + subthrehsold pole + background pole [Sparenberg, PRC 2004] More reliable way? ulbnorm Analysis of low-energy 12 C+ α scattering Jean-Marc Sparenberg (ULB) University of Surrey 2015 7 / 29

(Weakly-)bound states and asymptotic normalization constant The 12 C ( α, γ ) 16 O reaction in nuclear astrophysics 1 (Weakly-)bound states and asymptotic normalization constant 2 (Low-energy) elastic scattering and phase shifts 3 Effective-range expansion(s) for scattering and bound states 4 Analysis of the 12 C + α experimental p -wave phase shifts 5 Analysis of the 12 C + α experimental d -wave phase shifts 6 Conclusions and perspectives 7 ulbnorm Analysis of low-energy 12 C+ α scattering Jean-Marc Sparenberg (ULB) University of Surrey 2015 8 / 29

(Weakly-)bound states and asymptotic normalization constant Two-body potential model Isolated two-nuclei system ⇒ center-of-mass-movement separation ◮ one-body Schr¨ odinger equation H = p 2 E = � 2 2 µ k 2 Hϕ ( r ) = Eϕ ( r ) , 2 µ + V ( r ) , ◮ relative coordinate r , reduced mass µ , unit choice � 2 / 2 µ = 1 ◮ (complex) wave number k Interacting through central potential (no spin), nuclear Bohr radius a B = � 2 Z 1 Z 2 e 2 1 4 πǫ 0 µ , η = a B k V Coulomb ( r ) = 2 ηk V ( r ) = V nuclear ( r ) + V Coulomb ( r ) ∼ r →∞ 0 , r { H, L 2 , L z } Rotational invariance ⇒ CSCO ϕ Elm ( r ) = u El ( r ) ◮ angular separation, partial waves Y m l ( θ, φ ) r ◮ one-dimensional radial Schr¨ odinger equation, effective potential V l ( r ) � � − d 2 dr 2 + l ( l +1) ulbnorm + V ( r ) u El ( r ) = Eu El ( r ) r 2 Analysis of low-energy 12 C+ α scattering Jean-Marc Sparenberg (ULB) University of Surrey 2015 9 / 29

(Weakly-)bound states and asymptotic normalization constant Bound states and asymptotic normalization constants Bound states: negative (discrete) energy E b = − κ 2 b Potential model: normalized wave functions u b ( r ) ◮ asymptotic behaviour e − κ b r C b e − κ b r for η = 0 � � u b ( r ) ≈ r →∞ C b (2 κ b r ) | η b | ◮ slowly decreasing for weakly bound states ( κ b ≪ 1 ) ◮ Asymptotic Normalization Constant (ANC) C b ◮ related to nuclear vertex constant More sophisticated models (coupled-channel, microscopic. . . ): ◮ unnormalized overlap functions � � � � ψ 12 C ⊗ ψ α ⊗ Y m δ ( r ) u b ( r ) ≡ A | ψ 16 O l r � ∞ dr | u b ( r ) | 2 � = 1 ◮ spectroscopic factor ulbnorm 0 ◮ same definition for ANC Analysis of low-energy 12 C+ α scattering Jean-Marc Sparenberg (ULB) University of Surrey 2015 10 / 29

(Low-energy) elastic scattering and phase shifts The 12 C ( α, γ ) 16 O reaction in nuclear astrophysics 1 (Weakly-)bound states and asymptotic normalization constant 2 (Low-energy) elastic scattering and phase shifts 3 Effective-range expansion(s) for scattering and bound states 4 Analysis of the 12 C + α experimental p -wave phase shifts 5 Analysis of the 12 C + α experimental d -wave phase shifts 6 Conclusions and perspectives 7 ulbnorm Analysis of low-energy 12 C+ α scattering Jean-Marc Sparenberg (ULB) University of Surrey 2015 11 / 29

(Low-energy) elastic scattering and phase shifts Stationary-scattering-state partial-wave expansion Stationary scattering states ϕ k 1 z ( r ) ◮ solutions of 3D Schr¨ odinger equation with positive (continuous) energy E = k 2 ◮ rotationally invariant around z ◮ asymptotic behaviour: scattering amplitude f k ( θ ) r →∞ e ikz + f k ( θ ) e ikr (2 π ) 3 / 2 ϕ k 1 z ( r ) → ( η = 0) r dσ ◮ elastic-scattering differential cross section d Ω ( E, θ ) = | f k ( θ ) | 2 ϕ klm ( r ) = u kl ( r ) Y m Partial waves l ( θ, ϕ ) r ◮ solutions of radial Schr¨ odinger equation u kl ( r ) ◮ asymptotic behaviour: scattering phase shifts δ l ( k ) � π e iδ l ( k ) sin 1 2 kr − l π � � u kl ( r ) → 2 + δ l ( k ) ( η = 0) k r →∞ (2 π ) 3 / 2 ϕ k 1 z ( r ) = � ∞ Expansion (few terms at low E) l =0 c l ϕ kl 0 ( r ) ulbnorm � ∞ 1 l =0 (2 l + 1)( e 2 iδ l − 1) P l (cos θ ) ⇒ f k ( θ ) = 2 ik Analysis of low-energy 12 C+ α scattering Jean-Marc Sparenberg (ULB) University of Surrey 2015 12 / 29