1. Lawrence Livermore National Laboratory Scattering of light nuclei LLNL-PRES-XXXXXX 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 Sofia Quaglioni in collaboration with Petr Návratil 19 th International IUPAP Conference on Few-Body Problems in Physics Bonn, September 4, 2009

2. 2 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory 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

3. 3 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory 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!

4. 4 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory Microscopic  All nucleons are active  Exact Pauli principle Microscopic  All nucleons are active  Exact Pauli principle  Few-nucleon techniques using realistic NN (+ NNN) interactions Faddeev, AGS (Deltuva et al.), FY (Lazauskas et al.), HH (Viviani et al.), LIT (Bacca et al.), RRGM (Hoffman et al.), …  Many-body techniques using realistic NN (+ NNN) interactions GFMC (Nollett et al.), NCSM/RGM (Navrátil, SQ), FMD (Neff et al.), …  Cluster techniques using semi-realistic NN interactions RGM, GCM (Descouvemont et al.),... Reaction approaches Cluster few-body  N-nucleus interactions  (usually) inert core Cluster few-body  N-nucleus interactions  (usually) inert core  Techniques using local/non-local optical potentials Faddeev, AGS (Deltuva et al.), …  Techniques using local optical potentials CDCC (Moro et al.), XCDCC (Summers et al.), DWBA, adiabatic approaches (Baye et al.), …  Halo effective-field theories (Higa et al.), … PRC 79, 054007 (2009)

5. 5 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory 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  fm), 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

6. 6 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory 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,  EFT N 3 LO, … NNN interactions:  Tucson-Melbourne TM’,  EFT N 2 LO  Finite harmonic oscillator (HO) basis A-nucleon HO basis states  Jacobi relative or Cartesian single-particle coordinates complete N max ħ  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 N max

7. 7 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory Resonating-group method  Ansatz :  The many-body Schrodinger equation is mapped onto:  Input:,  Output (e.g., R-matrix method on Lagrange mesh):, scattering matrix Norm kernel Hamiltonian kernel  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. NCSM/RGM: NCSM microscopic wave functions for the clusters involved, and realistic (bare or derived NCSM effective) interactions among nucleons.

8. 8 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory Single-nucleon projectile: the norm kernel (A-1) (1)  (A-1)  (A-1) (1) (1,…,A-1) (A) (1,…,A-1) (A) “Direct term” treated exactly. “Exchange” term localized   expanded in HO radial w.f.

9. 9 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory Single-nucleon projectile basis: the Hamiltonian kernel  (A-1)   (A-1)(A-2)  “direct potential” “exchange potential” (A-1) (1) (1,…,A-1) (A) (1,…,A-1) (A) + terms containing NNN potential

10. 10 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory The RGM kernels in the single-nucleon projectile basis  (A-1)(A-2)   (A-1)  (A-1) (1)  + (A-1)  “direct potential” “exchange potential” In the A=5 system the 1/2 + ( 2 S 1/2 ) is a Pauli-forbidden state, therefore g.s. in P wave

11. 11 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory NCSM/RGM ab initio calculation of n- 4 He phase shifts  NCSM/RGM calculation with n + 4He(g.s.)  Low-momentum V lowk NN potential: convergence reached with bare interaction   EFT N 3 LO NN potential: convergence reached with two-body effective interaction 4 He n Is everything else under control? … need verification against independent ab initio approach! No fit. No free parameters. Convergence in N max under control. No fit. No free parameters. Convergence in N max under control.

12. 12 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory The A=4 system as a test ground for the NCSM/RGM approach within the single-nucleon-projectile basis  NCSM/RGM calculation with n + 3 H(g.s.) and p + 3 He(g.s.), respectively   EFT N 3 LO 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 ≤ 5/2 leads to effects of comparable magnitude on the AGS results. Need to include target excited (here breakup) states! The omission of A = 3 partial waves with 1/2 < J ≤ 5/2 leads to effects of comparable magnitude on the AGS results. Need to include target excited (here breakup) states! 3H3H n 3 He p

13. 13 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory n- 4 He phase shifts with  EFT N 3 LO NN interaction  Very mild effects of J  T = 0 + 0 on 2 S 1/2  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 2 P 1/2 2 - 0 and 2 - 1 affect 2 P 3/2  NCSM/RGM calculation with n + 4 He(g.s., ex.)   EFT N 3 LO NN potential: convergence with 2-body effective interaction 4 He n The resonances are sensitive to the inclusion of the first six excited states of 4 He

14. 14 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory Nucleon-  phase-shifts with  EFT N3LO NN interaction  NCSM/RGM calculation with N+ 4 He(g.s., 0 + 00 - 01 - 01 - 12 - 02 - 1)   EFT N 3 LO NN potential: convergence with 2-body effective interaction  2 S 1/2 in agreement with Expt. (dominated by N-  repulsion - Pauli principle)  Insufficient spin-orbit splitting between 2 P 1/2 and 2 P 3/2 (sensitive to interaction) Fully ab initio, very promising results. The resonances are sensitive to NNN force.

15. 15 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory n + 4 He differential cross section and analyzing power  NCSM/RGM calculations with N + 4 He(g.s., 0 + 0) SRG-N 3 LO NN potential with Λ=2.02 fm -1  Differential cross section and analyzing power @17 MeV neutron energy Polarized neutron experiment at Karlsruhe 4 He n Good agreement for energies beyond low-lying resonances

16. 16 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory NCSM/RGM ab initio calculation of n + 7 Li scattering 7 Li n  N max = 8 NCSM/RGM calculation with n + 7 Li(g.s.,1/2 -, 7/2 - )  SRG-N 3 LO 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 7 Li is not included 7 Li Predicted narrow 0 + and 2 + resonances seen at recent p+ 7 Be experiment at FSU Expt: a 01 =0.87(7) fm a 02 =-3.63(5) fm Calc: a 01 =0.73 fm a 02 =-1.42 fm Expt: a 01 =0.87(7) fm a 02 =-3.63(5) fm Calc: a 01 =0.73 fm a 02 =-1.42 fm

17. 17 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory 11 Be bound states and n- 10 Be phase shifts 10 Be n NCSM/RGM NCSM 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 E [MeV] Expt. 1/2 - 1/2 + Parity-inverted g.s. of 11 Be understood! 11 Be  Exotic nuclei: vanishing of magic numbers, abnormal spin-parity of ground states, …  The g.s. of 11 Be one of the best examples Observed spin-parity : 1/2+ p-shell expected: 1/2-  Large-scale NCSM calculations, Forssen et al., PRC71, 044312 (2005) Several realistic NN potentials Calculated g.s. spin-parity: 1/2-  NCSM/RGM calculation with CD-Bonn n + 10 Be(g.s.,2 1 +,2 2 +,1 1 + ) 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 What happens? Substantial drop of the relative kinetic energy due to the rescaling of the relative wave function when the Whittaker tail is recovered

18. 18 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory The deuteron-projectile formalism: norm kernel (A-2) (2) (1,…,A-2) (A-1,A) (1,…,A-2) (A-1,A)

19. 19 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory NCSM/RGM ab initio calculation of d - 4 He scattering  N max = 8 NCSM/RGM calculation with d(g.s.) + 4 He(g.s.)  SRG-N 3 LO potential with Λ = 2.02 fm -1 4 He d  Calculated two resonances: 2 + 0, 3 + 0  The 1 + 0 g.s. is still unbound: convergence moves towards bound state  Calculated two resonances: 2 + 0, 3 + 0  The 1 + 0 g.s. is still unbound: convergence moves towards bound state 6 Li

20. 20 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory Toward the first ab initio calculation of the Deuterium-Tritium fusion r’  n r n r’  d r d  r’  n r  n r’  d r d  3H3H d 4He4He n ✔ ✔ ✔ ✔ Work in progress on coupling between d + 3 H and n + 4 He bases

21. 21 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory 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- 3 H, n- 4 He, n- 10 Be and p- 3,4 He S.Q. and P. Navrátil, PRL 101, 092501 (2008), PRC 79, 044606 (2009)  New results with SRG-N 3 LO: N- 4 He, n- 7 Li, (also N- 12 C and N- 16 O, not presented here) Initial results for d- 4 He scattering First steps towards 3 H(d,n) 4 He  To do: Coupling of N+A and d+(A-1) Inclusion of NNN force Heavier projectiles: 3 H, 3 He, 4 He NCSM with continuum (NCSMC) Three-cluster NCSM/RGM and treatment of three-body continuum

22. 22 LLNL-PRES-XXXXXX Lawrence Livermore National Laboratory Thanks  Petr Navrátil, 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!