Photodisintegration of Light Nuclei
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Photodisintegration of light nuclei Nir Barnea The Hebrew University Jerusalem Giuseppina Orlandini Trento (Italy) Winfried Leidemann Trento (Italy) Sonia Bacca Darmstadt (Germany) Doron Gazit Jerusalem (Israel) Yael Ronen Jerusalem (Israel) Nuclear Structure and Astrophysics with Radioactive Beams WIS Rehovot, June 4-6, 2006 Photodisintegration of light nuclei Neutrino neutral reaction on 4He Low energy photodisintegration of 16O in an 4α-particle model (A comment) Some technical remarks on ab-initio calculation of inelastic reactions on light nuclei projectile photon target electron neutrino nucleus with A=N+Z nucleons or another nucleon N neutrons and Z protons The cross-section In general a reaction cross-section can be written as σ ()ω = Const × R(ω) where the response function is given by ˆ 2 R()ω = ∑ f O i δ (E f − Ei −ω) f Due to the summation over the final states That are usually continuum states, Calculation of reaction cross-sections Energy levels of 4He is much more difficult then calculating bound states. The Lorentz Integral transform (LIT) method In order to avoid continuum wave functions R(ω) we replace the response function with L()ω ,Γ = dω 0 ∫ 2 2 its transform, using a Lorentzian kernel, ()ω0 − ω − Γ 2 ||i Oˆ f = ∑ 2 2 f ()ω0 + Ei − E f − Γ Or in a different form, 1 1 L = ∑ i Oˆ f f Oˆ i f ()ω0 + Ei − E f +iΓ ()ω0 + Ei − E f − iΓ 1 1 = ∑ i Oˆ f f Oˆ i f ()ω0 + Ei − H +iΓ ()ω0 + Ei − H − iΓ See mom no f !!! 1 1 = i Oˆ Oˆ i = ψ~ ψ~ ()ω0 + Ei − H +iΓ ()ω0 + Ei − H − iΓ Therefore we have to solve the following Schroedinger like equation: ~ ˆ ~ (H − Ei − ω0 +iΓ )ψ = O ψ Few Remarks There is no solution to the homogeneous equation. Boundary conditions are the same as for bound state. The inversion is unique. Inversion of the transform is unstable and needs some regulation. Efros, Leidemann & Orlandini, PLB 408, 1 (1994). Solving the Schroedinger equation – bound states We use the effective interaction hyperspherical harmonics (EIHH) method Barnea, Leidemann and Orlandini, PRC 61, 054001 (2000) Replacing only 2-body forces with effective ones 1 ⎛ ∂ 2 3N −1 Kˆ 2 ⎞ H = − ⎜ + − ⎟ + V eff + V rr,rr ,rr +.... ⎜ 2 2 2 ⎟ ∑ 2 ∑ 3 ()i j k 2m ⎝ ∂ρ ρ ρ ⎠ i< j i< j<k The effective two-body potential is obtained through the Application of the Lee-Suzuki similarity transformation to the “2-body” equation 1 Kˆ 2 H ()ρ = +V ()rr 2 2m ρ 2 2 N −1,N Benchmark test calculation of a four nucleon bound state Kamada et. al. Phys. Rev. C 64, 044001 (2001) 4-body system with the MTV nucleon-nucleon potential BE EIHH bare Radius 4He total photoabsorption cross-section with the realistic forces AV18+UIX. Berman et al. (g,n) 1980 Feldman et al. (g,p) 1990 Wells et al. 1992 Nilsson et al. 2005 Shima et al. 2005 Gazit et. al., PRL 96, 112301 (2006) total photoabsorption cross-section For A=6 nuclei Soft dipole resonanace Giant dipole resonanace Bacca et. al., PRL 89 052502 (2002) 6He total photoabsorption cross-section MN - Minnesota potential MT - I-III Malfliet-Tjon Act only in the S=1,T=0 and S=0,T=1 channels. AV4' - Argonne AV4' Acts in all partial waves. Bacca et. al., PRC 69 057001 (2004) 7Li total photoabsorption cross-section with AV4’ NN potential Bacca et. al., PLB 603, 159 (2004) Inelastic neutral neutrino reaction on 4He Neutrino-4He (ν−α) reactions are important in neutrino driven supernova, and for the nucleosyntesis during supernovae. Was first calculated by Haxton (1988), using shell model with a Sussex model. Using the LIT and EIHH methods we can now perform ab- initio calculation for this cross section. Doron Gazit and Nir Barnea, PRC, 70, 048801 (2004) Role in nucleosynthesis The nucleosynthesis of elements through neutrino induced interactions (The ν process): Epstein, Colgate, and Haxton (1988). Woosley and Haxton (1988) Woosley, Hartmann, Hoffman, Haxton (1990) … The production of 7Li in the 4He rich layer is through the reaction sequences: 37 4 He(νν , 'p) H( αγ ,) Li 377 4 He(νν , 'n) He( αγ ,) Be( n,p) Li Inelastic neutral neutrino reaction on 4He ν " 4He" μ μ k2 = (k2 ,k 2 ) Pf = (Ei ,Pi ) Lepton qμ = (ω,q) current Nuclear Z0 current μ μ k1 = (k1,k1 ) Pi = (Ei ,Pi ) ν 4He Inelastic neutral neutrino reaction on 4He The Impulse approximation (considering only 1-body currents) 90% of our cross-section is due to axial currents. Axial 2-body currents usually suppressed. Vector 2-body currents are protected in low energy by the Siegert Theorem. Leading transition operators The closed shell character of 4He, suppresses the usually leading operators: Gamow-Teller Fermi Instead, the leading operators are proportional to the momentum transfer. Cross-section Where the response functions, ˆˆ R ˆˆ()ωΨ=−+01O ΨΨffO 2Ψ 0δ EE f 0ω O,O12 ∑ f ( ) Are needed to be calculated using a model for the 4He nucleus – we used the AV8’ NN potential. Different multipole contributions to the cross-section. 0.20 5 E2 − 51% 5 M1 − 39% C1, L1, E1 − 6% 0.15 5 L2 − 2.5% 5 L0 − 1.5% cm/MeV] −42 [10 0.10 0.05 Multipole Strength 0.00 20 30 40 50 ω [MeV] Comparison with previous results equal 5% 15% 25% Woosley et. al, ApJ 356, 272 (1990) Haxton, PRL 60, 1999 (1988) Conclusions A first realistic calculation of the inelastic neutrino neutral reaction on 4He. Full Final state interaction and realistic inter-nucleon force are considered. The calculation is done in the impulse approximation. The numerical accuracy is about 1%. The new results facilitate stronger neutrino – matter interaction, i.e. enhanced cross-section. Stronger temperature dependence. Low energy photodisintegration of 16O in an 4α-particle model 4 He Low energy α 16O α Low energy α α α The model Even-even nuclei regarded as “Nα systems” The α’s are J=0 structureless particles Non-relativistic quantum mechanics Local interactions 1 2 H = ∑∑pi + Vc ()rij +∑ V2α ()rij +K 2mα iji< i< j Guideline Use force models that reproduce the 8Be, 12C ground states Available force models • Fedotov, Kartavtsev, Kochkin and Malykh, PRC. 70, 014006 (2004). • Fedorov and Jensen, Phys. Lett. B 389, 631 (1996). Introduced a force model of the form 2 Z r r r r V = ∑ +∑ V2α ()rij + ∑ V3α (ri , rj, rk ) i< j rij i< j i< j<k 2 2 2 2 (2r) −μ2rr (2a) −μ2a r V2α ()r = V0 e − V0 e 2 2 r r r (3) −μ3 ρ V3α (ri , rj, rk ) = V0 e ⎛ ⎞ 2 1 ⎜ r r 2 r r 2 r r 2 ⎟ ρ = ⎜ (r − r ) + (r − r ) + (r − r ) ⎟ ⎜ ⎟ 3 ⎝ i j j k k i ⎠ reproduce the 8Be Resonance, and also the BE and the first excited 0+ state of 12C Predictions for the 4,5-body systems What do we expect to find? Going from 3-body nuclei to 4-body one finds the “Tjon-line” i.e. potential models that reproduce 2,3-body data also reproduce the binding energy of the 4-body system !!! Same happens for He atoms ! What about α particles? Predictions for the 4,5-body systems Assuming that the 3-body force acts only on S-waves 12C 16O 20Ne FKKM1 -7.27 -31.15 -84.5 FKKM2 -7.27 -25.08 -57.1 FdJn96 -6.66 -30.32 -138.6 Exp -7.27 -14.44 -19.17 Conclusion: the current models leads to a collapse of the N-boson system A repulsive 4-body force must be added Conclusions The force models which include only 2 and 3 body terms tend to overbind the 16O. Adding a 4-body force, the ground state and 1st excited state of 16O can be reproduced fairly well. The 4-body force is not unique, there exist many possible sets of parameters. We managed to calculate the low energy 16O(γ,α)12C cross-section. However the results are very sensitive to the force model. Currently we try a new approach, fitting a non-local 2-body force. Thats all ... Conclusions Contrary to the common believe, reproducing the two body data and 3-body ground state properties are not enough to make a nuclear force model realistic. Photoabsorption reactions are a useful tool to differentiate between force models. The differences between models grow with A (for A=2,3,4). The theoretical tools and methods that we have developed allow an accurate description of nuclear structure and reactions (including FSI) for nuclei with A=2,3,...7 particles. What are the energy scales? E [MeV] 40 35 30 S_alpha 25 BE 20 BE/ N 15 E* 10 5 0 02468 N Excitation energy of the α particle: E* ~ 20 MeV In comparison For a system of N 4He atoms BE ≈ mk =10−7 eV <<1eV In nuclear physics the ratio between binding energy Per nucleon and nucleon excitation energy BE / A ≈ 8 MeV <<100MeV For N α particles the ratio between binding energy per α and α excitation energy BE / A ≈ 3 MeV < 20MeV The 16O with a four-body force The second excited level (Jπ=2+) experimental value This work The first excited level (Jπ=0+) experimental value This work Results for AV8’ −42 2 10−42 cm2 / MeV 10 cm 0.4 0.08 0.3 0.06 2 −40 0.2 0.04 / > 0.1 d< 0.02 T=10 MeV 0.0 0.00 20 30 40 50 60 ω [MeV] KE_0E_1 4 -7.1933 0.4536 12C - FKKM1 6 -7.2802 0.3147 Fedotov, Kartavtsev, Kochkin 8 -7.2723 0.3647 and Malykh – model 1 10 -7.2729 0.3697 12 -7.2739 0.3637 14 -7.2738 0.3670 16 -7.2738 0.3675 18 -7.2738 0.3667 20 -7.2738 0.3671 Exp -7.2747 0.3795 The convergence of the EIHH method is extremely fast For both, the GS and the excited state TheT full force model 2 Z r r r r r r r r V = ∑ +∑ V2α ()rij + ∑ V3α (ri , rj, rk )+ ∑ V4α (ri , rj, rk , rl ) i< j rij i< j i< j<k i< j<k<l The 2-body and 3- body Forces 2 2 2 2 V r = V(2r)e−μ2rr − V(2a)e−μ2a r 2α () 0 0 2 2 r = (ri − rj ) 2 2 r r r (3) −μ3 ρ3 V3α ()ri , rj, rk = L3 = 0 V0 e L3 = 0 1 ρ2 = ((vr − vr )2 + (vr − vr )2 + (vr − vr )2 ) 3 3 i j j k k i The 4-body force 2 2 r r r r (4) −μ4 ρ4 V4α ()ri , rj, rk , rl = V0 e 2 1 v v 2 ρ4 = ∑(ri − rj ) 4 i< j.