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Scattering of light nuclei Sofia Quaglioni in collaboration with PetrNávratil 19th International IUPAP Conference on Few-Body Problems in Physics Bonn, September 4, 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-XXXXXX
Nuclear reactions • Nuclear physics underlying many key astrophysical processes • Formation of the chemical elements • Solar neutrino problem • Stellar evolution • Tools for studying exotic nuclei • Structure inferred from breakup reactions • Most low-lying states are unbound • A formidable challenge to nuclear theory … • Main difficulty: scattering states
Disclaimer • As they deserve, nuclear reactions are attracting much attention • There are many interesting new developments … • … forgive me if I miss to mention some of them!
Reaction approaches Microscopic All nucleons are active Exact Pauli principle Cluster few-body N-nucleus interactions (usually) inert core Techniques using local/non-local optical potentials Faddeev, AGS (Deltuvaet al.), … Techniques using local optical potentials CDCC (Moro et al.), XCDCC (Summers et al.), DWBA, adiabatic approaches (Bayeet al.), … Halo effective-field theories (Higaet al.), … • Few-nucleon techniques using realistic NN (+ NNN) interactions • Faddeev, AGS (Deltuvaet al.), FY (Lazauskaset al.), HH (Vivianiet al.), LIT (Baccaet al.), RRGM (Hoffman et al.), … • Many-body techniques using realistic NN (+ NNN) interactions • GFMC (Nollettet al.), NCSM/RGM (Navrátil, SQ), FMD (Neff et al.), … • Cluster techniques using semi-realistic NN interactions • RGM, GCM (Descouvemontet al.), ... PRC 79, 054007 (2009)
Our goal:ab initio approach to low-energy reactions of light nuclei • Start with the ab initio description of the structure of light nuclei • The ab initio no-core shell model (NCSM) • A successful ab initio approach to nuclear structure • Capable of employing chiral effective field theory (EFT) NN + NNN potentials for A>4 • Covers nuclei beyond the s-shell • Incorrect description of wave-function asymptotic (r>5fm), no coupling to continuum • Add microscopic description of nucleus-nucleus scattering • The resonating-group method (RGM) • A successful microscopic cluster technique (also multi-cluster) • Preserves Pauli principle, includes Coulomb force • Describes reactions and clustering in light nuclei (also multichannel, transfer etc.) • Usually simplified NN interactions and internal description of the clusters • Combine: NCSM/RGM ab initio bound & scattering states in light nuclei • NCSM - single-particle degrees of freedom • RGM - clusters and their relative motion
The ab initio no-core shell model (NCSM) in brief The NCSM is a technique for the solution of the A-nucleon bound-state problem • Hamiltonian • “realistic” (= reproduce NN data with high precision) NN potentials: • coordinate space: Argonne … • momentum space: CD-Bonn, cEFT N3LO, … • NNN interactions: • Tucson-Melbourne TM’, cEFT N2LO • Finite harmonic oscillator (HO) basis • A-nucleon HO basis states • Jacobi relative or Cartesian single-particle coordinates • complete Nmaxħ model space • translational invariance preserved even with Slater-determinant (SD) basis • Constructs effective interaction tailored to model-space truncation • unitary transformation in a n-body cluster approximation (n=2,3) Convergence to exact solution with increasing Nmax
Norm kernel Hamiltonian kernel Resonating-group method • Ansatz: • The many-body Schrodinger equation is mapped onto: • Input: , • Output (e.g., R-matrix method on Lagrange mesh): , scattering matrix eigenstates of H(A-a), H(a) in the NCSM basis NCSM/RGM: NCSM microscopic wave functions for the clusters involved, and realistic (bare or derived NCSM effective) interactions among nucleons.
(A-1) (1) (1,…,A-1) (1,…,A-1) (A) (A) (A-1) (1) (A-1) Single-nucleon projectile: the norm kernel “Direct term” treated exactly. “Exchange” term localized d expanded in HO radial w.f.
(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
(A-1) (1) (A-1) “direct potential” +(A-1) “exchange potential” (A-1)(A-2) The RGM kernels in the single-nucleon projectile basis In the A=5 system the 1/2+ (2S1/2) is a Pauli-forbidden state, therefore g.s. in P wave
4He n NCSM/RGM ab initio calculation of n-4He phase shifts • NCSM/RGM calculation with n + 4He(g.s.) • Low-momentum VlowkNN potential: convergence reached with bare interaction • EFT N3LO NN potential: convergence reached with two-body effective interaction No fit. No free parameters. Convergence in Nmax under control. Is everything else under control? … need verification against independent ab initio approach!
p n 3H 3He The A=4 system as a test ground for the NCSM/RGM approach within the single-nucleon-projectile basis • NCSM/RGM calculation with n + 3H(g.s.) and p + 3He(g.s.), respectively • EFT N3LO NN potential: convergence with 2-body effective interaction • Benchmark: AGS results (+), Deltuva & Fonseca, PRC75, 014005 (2007) The omission of A = 3 partial waves with 1/2 < J ≤ 5/2leads to effects of comparable magnitude on the AGS results. Need to include target excited (here breakup) states!
4He n n-4He phase shifts with EFT N3LONN interaction • Very mild effects of JpT = 0+0 on 2S1/2 • The negative-parity states have larger effects on P phases (coupling to s-wave of relative motion) • 0-0, 1-0 and 1-1 affect 2P1/2 • 2-0 and 2-1 affect 2P3/2 • NCSM/RGM calculation with n + 4He(g.s., ex.) • EFT N3LO NN potential: convergence with 2-body effective interaction The resonances are sensitive to the inclusion of the first six excited states of 4He
Nucleon- phase-shifts with EFT N3LO NN interaction • NCSM/RGM calculation with N+4He(g.s., 0+00-01-01-12-02-1) • EFT N3LO NN potential: convergence with 2-body effective interaction • 2S1/2 in agreement with Expt. (dominated by N- repulsion - Pauli principle) • Insufficient spin-orbit splitting between 2P1/2 and 2P3/2 (sensitive to interaction) Fully ab initio, very promising results.The resonances are sensitive to NNN force.
4He n n+4He differential cross section and analyzing power • NCSM/RGM calculations with • N + 4He(g.s., 0+0) • SRG-N3LO NN potential with Λ=2.02 fm-1 • Differential cross section and analyzing power @17 MeV neutron energy • Polarized neutron experiment at Karlsruhe Good agreement for energies beyond low-lying resonances
7Li n NCSM/RGM ab initio calculation ofn+7Li scattering • Nmax = 8 NCSM/RGM calculation with n + 7Li(g.s.,1/2-, 7/2-) • SRG-N3LO NN potential with Λ = 2.02 fm-1 • Qualitative agreement with experiment: • Calculated broad 1+ resonance • 3+ resonance not seen when the 7/2- state of 7Li is not included 7Li Expt: a01=0.87(7) fm a02=-3.63(5) fm Calc: a01=0.73 fm a02=-1.42 fm Predicted narrow 0+ and 2+ resonances seen at recent p+7Be experiment at FSU
11Be 10Be n NCSM 1/2- 1/2+ 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 Parity-inverted g.s. of 11Be understood! E [MeV] Expt. NCSM/RGM 11Be bound states and n-10Be phase shifts • Exotic nuclei: vanishing of magic numbers, abnormal spin-parity of ground states, … • The g.s. of 11Be one of the best examples • Observed spin-parity :1/2+ • p-shell expected: 1/2- • Large-scale NCSM calculations, Forssenet al., PRC71, 044312 (2005) • Several realistic NN potentials • Calculated g.s. spin-parity: 1/2- • NCSM/RGM calculation with CD-Bonn • n+ 10Be(g.s.,21+,22+,11+) • Calculated g.s. spin-parity : 1/2+ What happens? Substantial drop of the relative kinetic energy due to the rescaling of the relative wave function when the Whittaker tail is recovered
(A-2) (2) (1,…,A-2) (1,…,A-2) (A-1,A) (A-1,A) The deuteron-projectile formalism: norm kernel
4He d NCSM/RGM ab initio calculation of d-4He scattering • Nmax = 8 NCSM/RGM calculation with d(g.s.) + 4He(g.s.) • SRG-N3LO potential with Λ = 2.02 fm-1 6Li preliminary • Calculated two resonances: 2+0, 3+0 • The 1+0 g.s. is still unbound: convergence moves towards bound state
r r r r r’ r’ n 3H Toward the first ab initio calculation of theDeuterium-Tritium fusion d 4He ✔ n n n n ✔ r’ r’ 3H 3H 3H 3H d d d d Work in progress on coupling between d + 3H and n + 4He bases
Conclusions and Outlook • With the NCSM/RGM approach we are extending the ab initio effort to describe low-energy reactions and weakly-bound systems • Recent results for nucleon-nucleus scattering with NN realistic potentials: • n-3H, n-4He, n-10Be and p-3,4He • S.Q. and P. Navrátil,PRL 101, 092501 (2008), PRC 79, 044606 (2009) • New results with SRG-N3LO: • N-4He, n-7Li, (also N-12C and N-16O, not presented here) • Initial results for d-4He scattering • First steps towards 3H(d,n)4He • To do: • Coupling of N+A and d+(A-1) • Inclusion of NNN force • Heavier projectiles: 3H, 3He, 4He • NCSM with continuum (NCSMC) • Three-cluster NCSM/RGM and treatment of three-body continuum
Thanks • PetrNavrátil, without whom much of this work would not have been possible • Our collaborators: • R. Roth, GSI, on the Importance-truncation NCSM • S. Bacca, TRIUMF, on the NCSMC Thank you for your attention!