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Ab initio many-body calculations of nucleon scattering on light nuclei

Ab initio many-body calculations of nucleon scattering on light nuclei. Petr Navratil (LLNL ) Collaborators: Sofia Quaglioni (LLNL ), Robert Roth (TU Darmstadt) 2009 Mardi Gras Nuclear Physics Workshop, 2/18-20/2009.

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Ab initio many-body calculations of nucleon scattering on light nuclei

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  1. Ab initio many-body calculations of nucleon scattering on light nuclei Petr Navratil (LLNL) Collaborators: Sofia Quaglioni (LLNL), Robert Roth (TU Darmstadt) 2009 Mardi Gras Nuclear Physics Workshop, 2/18-20/2009 Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551 This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 LLNL-PRES-410642

  2. Ab initio nuclear reactions theory is one of the forefront topics in nuclear science • Nuclei are open quantum many-body systems • Bound states, resonances, scattering states • A realistic ab initio approach to light nuclei with predictive power must have the capability to describe all the above classes of states within a unified framework • Bound-state techniques not sufficient • Nuclear-reaction calculations provide a strict test for NN and NN+NNN interaction models • Understanding of weakly-bound exotic nuclei • Crucial for understanding nuclear reactions important for astrophysics that are hard or impossible to measure

  3. 12C(16O 2(12C ~ 10-16 s Ab initio nuclear reaction theory needed to help answer important questions: How are nuclei with A>8 made? • The two reactions, 2(12C, 12C(16O, strongly affect the production of all species made in C, Ne, O and Si burning stages and the size of iron cores in Type II supernovae • Impossible to measure at the energy relevant to astrophysics (300 keV) • Low cross section: 10-41 cm2 • Extrapolation from higher energies using theory required • Very complex system: “miraculous” resonances • Current uncertainty ~ 40% • No realistic microscopic theory for these clusters and reactions S. E. Woosley: Error bar ≤ 10% required!  Fundamental theory needed to enhance predictive capability of stellar modeling DOE milestone for 12C(16O

  4. Our goal is to develop an ab initio theory to understand nuclear structure and reactions in light nuclei The Hoyle state missing • … and calculate cross sections of reactions important for astrophysics (including those involving 12C and 16O) • How? - Combining the ab initio no-core shell model (NCSM) with the resonating group method (RGM) ab initioNCSM/RGM • NCSM - single-particle degrees of freedom • RGM - clusters and their relative motion • Ab initio theory of nuclear reactions for A>4 is new: • Lisbon: p+3He scattering publishedin 2007 (PRL 98, 162502 (2007)); A=4 is their limit • ANL: n+4He scattering published only recently (PRL 99, 022502 (2007)) • Our approach:readily extendable • p-3He, n-4He & p-4He scattering already calculated • promising results for p-shell nuclei: n-10Be, p-12C, n-16O • Inclusion of d, 3H, 3He and clusters under way NCSM PRL99, 042501(2007) RGM Preserves Pauli principle and translational invariance  Important as nucleons are fermions and nuclei self-bound

  5. The ab initio NCSM in brief • The NCSM is a technique for the solution of the A-nucleon bound-state problem • Hamiltonian • Standard high-precision nucleon-nucleon potentials: • Argonne, CD-Bonn, chiralN3LO … • Soft low-momentum interactions • Vlowk, SRG-N3LO, VUCOM, JISP … • Three-nucleon interactions: • Tucson-Melbourne TM’, chiral N2LO • Finite harmonic oscillator (HO) basis • A-nucleon HO basis states • Jacobi relative coordinates • Cartesian single-particle coordinates • complete Nmaxh model space • Translational invariance preserved even with single-particle coordinate Slater-determinant (SD) basis • Effective interaction tailored to model-space truncation for standard potentials • Unitary transformation in n-body cluster approximation (n=2,3) • Importance-truncated Nmaxh basis • Second-order many-body perturbation theory • Dimension reduction from billions to tens of millions • Access to nuclei beyond p-shell Convergence to exact solution with increasing Nmax for bound states. No coupling to continuum.

  6. Hamiltonian kernel Norm kernel The ab initio NCSM/RGM in a snapshot eigenstates of H(A-a) and H(a) in the ab initio NCSM basis • Ansatz: • Many-body Schrödinger equation:  either bare interaction or NCSM effective interaction • Non-local integro-differential coupled-channel equations: Fully implemented and tested for single-nucleon projectile (nucleon-nucleus) basis

  7. (A-1) (1) (1,…,A-1) (1,…,A-1) (A) (A)  (A-1)  (1) (A-1) Single-nucleon projectile: the norm kernel

  8. (A-1) (1) (1,…,A-1) (1,…,A-1) (A) (A) + terms containing NNN potential  (A-1) (A-1)(A-2) “exchange potential” “direct potential” Single-nucleon projectile basis: the Hamiltonian kernel

  9. 4He n NCSM/RGM ab initio calculation of n-4He phase shifts • Similarity-renormalization-group (SRG) evolvedchiral N3LO NN interaction (R. Roth) • Low-momentum VlowkNN potential • convergence reached with bareinteraction n-4He phase shifts: SRG-N3LO, =2.02 fm-1 Vlowk Fully ab initio. No fit. No free parameters. Good convergence with respect to Nmax Is everything else under control? … need verification against independent ab initio approach!

  10. 3He 3H p n • EFT N3LO NN potential: convergence reached with two-body effective interaction NCSM/RGM ab initio calculation of n-3H and p-3He phase shifts • NCSM/RGM calculations with n+3H(g.s.) and p+3He(g.s.), respectively. • Benchmark with Alt, Grassberger and Sandhas (AGS) results [PRC75, 014005(2007)] • What is missing? - n+3H(ex), 2n+d, p-3He(ex), 2p+d configurations The omission of three-nucleon partial waves with 1/2 < J ≤ 5/2leads to effects of comparable magnitude on the AGS results. Need to include target excited states!

  11. n-4He & p-4He phase shifts from accurate NN interactions • 4He states: g.s.,  • Reasonable agreement with experiment for 2S1/2 , 2P1/2, 2D3/2 channels • Coulomb under control n-4He and p-4He phase shifts Insufficient spin-orbit strength: 2P3/2 underestimated NNN needed CD-Bonn the best description of 2S1/2 phase shifts

  12. 4He n n+4He differential cross section and analyzing power • Neutron energy of 17 MeV • beyond low-lying resonances • Polarized neutron experiment at Karlsruhe • NCSM/RGM calculations • n+4He(g.s,0+0) • SRG-evolved N3LONN potential • Good agreement for angular distribution • Differences for analyzing power • Ay puzzle for A=5? First ever ab initio calculation of Ay in for a A=5 system. Strict test of inter-nucleon interactions.

  13. p-12C scattering with SRG-N3LO NN potential • 12C • Full NCSM up to Nmax=8 • IT NCSM up to Nmax=18(!) • 13N, 13C within the NCSM • 1/2+ state ~ 3 MeV too high • p+12C • Experiments with a polarized proton target under way • NCSM/RGM up Nmax=8 so far • 12C g.s. and 2+ included • 1/2- state bound by 2.5 MeV • 13N ground state • Other states unbound • 1/2+ resonance at ~1.5 MeV • 5/2+ resonance • Good stability: Moderate changes from Nmax=6 to Nmax=8 • Qualitative agreement with experiment

  14. 16O n 16O ground state, 17O bound states • 16O ground state calculated within importance-truncated NCSM • 4p-4h up to Nmax=18 (Nmax=22 possible!?), h=24 MeV • SRG-N3LO with =2.66 fm-1 • Less overbinding: E  -140 MeV • Benchmarking with full NCSM • 16O binding energy up to Nmax=8 • Perfect agreement • 17O within ab initio NCSM/RGM • 1/2+ bound: Eb=-0.88 MeVwrt16O • 5/2+ bound: Eb=-0.41 MeVwrt16O • Nmax=19, h=24 MeV • Only 16O ground-state included

  15. 16O n Nmax=16 n-16O scattering with SRG-N3LO NN potential • 16O ground state only • Phase-shift convergence very good • Essential to use largeNmax • Target wave function • Expansion of short-range parts of kernels • IT NCSM for the target makes it possible ( ) ( ) Done up to Nmax=18 converged Combining the ab initio NCSM/RGM with the importance-truncated NCSMhighly promising. Access to medium mass nuclei. ( )

  16. 16O n Correct order n-16O scattering: Effect of 16O excited states • Need to include 16O excited states (1p-1h…) • Test study: • g.s. in Nmax=8 • 3-, 1-, 2- in Nmax=9 • Significant increase of binding energies • 1/2+ : -0.66  -1.06 • 5/2+ : -0.46 -1.91 • Appearance of sharp resonances • IT NCSM for excited states in larger Nmax under way Done up to Nmax=12/13 ( ) ( )

  17. n-16O scattering: Open issues • 16O excited states with the SRG-N3LO NN potential too high • 3-, 1-, 2- calculated: 13.3, 15.9, 16.3 MeV • 3-, 1-, 2- experiment: 6.13, 7.12, 8.87 MeV • Importance of 3-body force? • 12C+alpha not included at present • n+16O with the SRG-N3LO NN potential • 5/2+, 1/2+underbound • 1/2-, 5/2- not bound • Resonances too high • Impact of incomplete 16O description • 13C+alpha not taken into account

  18. (A-2) (2) (1,…,A-2) (1,…,A-2) (A-1,A) (A-1,A) (2) (A-2) Including the deuteron projectile  2(A-2)  + (A-2)(A-3)/2 

  19. Conclusions and outlook • We are extending the ab initio NCSM to treat low-energy light-ion reactions • Our recent achievements: • n-3H, n-4He, n-10Be and p-3,4He scattering phase-shifts with realisticNN potentials (PRL 101, 092501 (2008)) • n-16O under way: • Breakthrough due to the importance-truncated NCSM approach • Coming next: • inclusion of NNN potential terms • d, 3H and 3He, 4Heprojectiles • Nuclei - complex open many-body systems • Bound states, resonances, continuum • A correct and efficient theoretical description must include all these features • Coupling of bound-state theory with cluster theory Ab Initio No-Core Shell Model with Continuum

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