Lawrence Livermore National Laboratory Ab initio many-body calculations of light-ion reactions LLNL-PRES Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 Petr Navratil Collaborators: Sofia Quaglioni (LLNL), R. Roth (TU Darmstadt), E. Jurgenson (LLNL) 6th ANL/MSU/JINA/INT FRIB Theory Workshop, Argonne National Laboratory, March , 2010

2 LLNL-PRES Lawrence Livermore National Laboratory Outline  Motivation  Ab initio no-core shell model (NCSM)  Extension of the no-core shell model by resonating group method (ab initio NCSM/RGM) Nucleon-alpha scattering n- 3 H, p- 3 He cross sections 11 Be parity-inverted ground state n- 7 Li, N- 12 C, n- 16 O scattering  Calculations with wave functions from importance-truncated NCSM d-T fusion  Outlook

3 LLNL-PRES Lawrence Livermore National Laboratory Our goal is to develop an ab initio theory to understand nuclear structure and reactions of light nuclei  Nuclei are quantum many-body systems with bound states, resonances, scattering states Bound-state techniques not sufficient  Our approach - combining the ab initio no-core shell model (NCSM) with the resonating group method (RGM)  ab initio NCSM/RGM NCSM - single-particle degrees of freedom RGM - clusters and their relative motion PRL 99, (2007) NCSM RGM The Hoyle state missing Preserves Pauli principle and translational invariance  Important as nucleons are fermions and nuclei self-bound Preserves Pauli principle and translational invariance  Important as nucleons are fermions and nuclei self-bound

4 LLNL-PRES 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  Realistic nuclear Hamiltonian High-precision nucleon-nucleon potentials Three-nucleon interactions  Finite harmonic oscillator (HO) basis A-nucleon HO basis states  Jacobi relative coordinates  Cartesian single-particle coordinates complete N max h  model space  Translational invariance preserved even with single-particle coordinate Slater-determinant (SD) basis  Effective interaction tailored to model-space truncation for NN(+NNN) potentials Lee-Suzuki-Okamoto unitary transformation in n-body cluster approximation (n=2,3)  Or a sequence of unitary transformations in momentum space: Similarity-Renormalization-Group evolved NN(+NNN) potential Soft: No further model-space dependent effective interaction needed  Variational calculation Convergence to exact solution with increasing N max for bound states. No coupling to continuum.

5 LLNL-PRES Lawrence Livermore National Laboratory NCSM Convergence: 4 He  Chiral N 3 LO NN plus N 2 LO NNN potential Bare interaction (black line)  Variational calculation  Strong short-range correlations  Large basis needed Similarity-renormalization group evolved effective interaction (red line)  Unitary transformation  Two- plus three-body components, four-body omitted  Softens the interaction  Variational calculation  Smaller basis sufficient

6 LLNL-PRES Lawrence Livermore National Laboratory The ab initio NCSM/RGM in a snapshot  Ansatz:  Non-local integro-differential coupled-channel equations: Hamiltonian kernelNorm kernel  Many-body Schrödinger equation:  eigenstates of H (A-a) and H (a) in the ab initio NCSM basis either bare interaction or NCSM effective interaction NCSM/RGM: NCSM microscopic wave functions for the clusters involved, and realistic (bare or derived NCSM effective) interactions among nucleons. Proper boundary conditions for scattering and/or bound states

7 LLNL-PRES Lawrence Livermore National Laboratory Localized parts of kernels expanded in the HO basis Single-nucleon projectile: the norm kernel (A-1) (1)  (A-1)  (A-1) (1) (1,…,A-1) (A) (1,…,A-1) (A)

8 LLNL-PRES 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

9 LLNL-PRES Lawrence Livermore National Laboratory NCSM/RGM ab initio calculation of n- 4 He phase shifts Fully ab initio. No fit. No free parameters. Good convergence with respect to N max 4 He n n- 4 He phase shifts: SRG-N 3 LO, =2.02 fm -1  Similarity-renormalization-group (SRG) evolved chiral N 3 LO NN interaction (R. Roth)  Low-momentum V lowk NN potential  convergence reached with bare interaction V lowk

10 LLNL-PRES 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 MeV neutron energy Polarized neutron experiment at Karlsruhe 4 He n NNN missing: Good agreement only for energies beyond low-lying 3/2 - resonance

11 LLNL-PRES Lawrence Livermore National Laboratory p + 4 He differential cross section and analyzing power

12 LLNL-PRES Lawrence Livermore National Laboratory Neutron-triton elastic scattering at 14 MeV  Important for the NIF physics deuteron-triton fusion generates 14 MeV neutrons  Experimental situation confusing  Good data for p+ 3 He elastic scattering Use NCSM/RGM calculation to relate the two reactions and predict n+ 3 H cross section

13 LLNL-PRES Lawrence Livermore National Laboratory B(E1;1/2 + ->1/2 - )=0.02 e 2 fm 2 11 Be bound states and n- 10 Be phase shifts 10 Be n NCSM/RGM NCSM 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, (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

14 LLNL-PRES Lawrence Livermore National Laboratory NCSM/RGM with Importance-Truncated (IT-NCSM)  IT-NCSM, Roth & Navratil, PRL99, (2007) makes possible: large N max fortarget g.s. + excited states good convergence for integration kernels  7 Li NCSM up to N max =10 (12 possible) IT-NCSM up to N max =18  12 C, 16 O NCSM up to N max = 8 IT-NCSM up to N max = 18(!)  Benchmark with NCSM in smaller model spaces: perfect agreement Combining the NCSM/RGM with the IT-NCSM highly promising. Access to medium mass nuclei.

15 LLNL-PRES Lawrence Livermore National Laboratory NCSM/RGM ab initio calculation of n + 7 Li scattering 7 Li n  N max = 12 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 8 Li bound states: 2 + and 1 + 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 = 1.24 fm a 02 = fm Expt: a 01 = 0.87(7) fm a 02 = -3.63(5) fm Calc: a 01 = 1.24 fm a 02 = fm

16 LLNL-PRES Lawrence Livermore National Laboratory 13 C bound states and n - 12 C scattering 12 C n  N max = 16 NCSM/RGM calculation with n + 12 C(g.s.,2 + 1 )  SRG-N 3 LO NN potential with Λ = 2.02 fm -1 Three 13 C bound states: 1/2-, 3/2-, 1/2+ ( 5/2+ still unbound ) 5/2+ narrow resonance Excitation energy of the 1/2 + state drops by 4 MeV when n- 12 C long-range correlations included

17 LLNL-PRES Lawrence Livermore National Laboratory 13 N ground state and p - 12 C scattering 12 C p  Experiments with a polarized proton target under way  N max = 16 NCSM/RGM calculation with n + 12 C(g.s.,2 + 1 )  SRG-N 3 LO NN potential with Λ = 2.02 fm N 1/2- ground state (bound by 2.9 MeV), other states unbound 1/2+ and 5/2+ narrow resonance

18 LLNL-PRES Lawrence Livermore National Laboratory ( ) N max =12 16 O n 17 O bound states and n - 16 O scattering  N max = 12 NCSM/RGM calc. with n+ 16 O( g.s., 3 -,1 -,2 - )  SRG-N 3 LO NN potential with Λ = 2.02 fm O bound states: 5/2+, 1/2+ ( 1/2-, 5/2- unbound ) Narrow resonances only when 16 O excited states included Impact of incomplete 16 O description 13 C+alpha not taken into account yet ( ) N max =18

19 LLNL-PRES Lawrence Livermore National Laboratory Deuterium-Tritium fusion: a future energy source  The d+ 3 H  n+ 4 He reaction The most promising for the production of fusion energy in the near future Will be used to achieve inertial-confinement (laser- induced) fusion at NIF, and magnetic-confinement fusion at ITER NIF ITER Resonance at E cm =48 keV (E d =105 keV) in the J=3/2 + channel Cross section at the peak: 4.88 b MeV energy released: 14.1 MeV neutron and 3.5 MeV alpha Resonance at E cm =48 keV (E d =105 keV) in the J=3/2 + channel Cross section at the peak: 4.88 b MeV energy released: 14.1 MeV neutron and 3.5 MeV alpha

20 LLNL-PRES Lawrence Livermore National Laboratory Toward the first ab initio calculation of the Deuterium-Tritium fusion 3H3H d 4He4He n r’  n r n r’  d r d  r’  n r  n r’  d r d  ✔ ✔ d+ 3 H  d+ 3 H norm kern Direct and exchange part S-wave channel: J=3/2 +,J=1/2 + d, 3 H spins parallel, anti-parallel

21 LLNL-PRES Lawrence Livermore National Laboratory Toward the first ab initio calculation of the Deuterium-Tritium fusion 3H3H d 4He4He n r’  n r n r’  d r d  r’  n r  n r’  d r d  ✔ ✔ d+ 3 H  n+ 4 He norm kernel S-wave channel: J=1/2 + d, 3 H spins anti-parallel d+ 3 H S-wave to n+ 4 He D-wave transition: J=3/2 + Important for fusion ✔ ✔ 2 x -3 x

22 LLNL-PRES Lawrence Livermore National Laboratory Toward the first ab initio calculation of the Deuterium-Tritium fusion: Phase shifts 3H3H d 4He4He n D-T fusion happens through the S-wave d+ 3 H to D-wave n+ 4 He transition Ab initio phase shift calculations of the d+ 3 H elastic scattering show resonance in the 4 S 3/2 channel No resonance in the 2 S 1/2 channel: Pauli principle Ab initio phase shift calculations of the d+ 3 H elastic scattering show resonance in the 4 S 3/2 channel No resonance in the 2 S 1/2 channel: Pauli principle Phase shift of the n+ 4 He elastic scattering show slight impact of the d+ 3 H channels on P-waves Effect of resonance in the 3/2 + D-wave just above the d- 3 H threshold Phase shift of the n+ 4 He elastic scattering show slight impact of the d+ 3 H channels on P-waves Effect of resonance in the 3/2 + D-wave just above the d- 3 H threshold

23 LLNL-PRES Lawrence Livermore National Laboratory Toward the first ab initio calculation of the Deuterium-Tritium fusion: Cross section 3H3H d 4He4He n First ab initio results of d-T and d- 3 He fusion: promising, correct physics, more work needs to be done… Correct features: Resonance just above threshold, lower for d-T S-factor of d+ 3 He flat as E  0: Experimental rise due to electron screening Correct features: Resonance just above threshold, lower for d-T S-factor of d+ 3 He flat as E  0: Experimental rise due to electron screening Incorrect features: Resonances higher than in experiment: 150 keV vs. 50 keV (d-T) 250 keV vs. 200 keV (d- 3 He) Cross sections way too low, it gets increased by including 4 He resonances (2 - 0 in particular) Incorrect features: Resonances higher than in experiment: 150 keV vs. 50 keV (d-T) 250 keV vs. 200 keV (d- 3 He) Cross sections way too low, it gets increased by including 4 He resonances (2 - 0 in particular) Still preliminary, incomplete: N max =13, SRG-N 3 LO NN (Λ=2.02 fm -1 ), no NNN, ground states of d, 3 H, 4 He only.

24 LLNL-PRES 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. Quaglioni and P. N., PRL 101, (2008), PRC 79, (2009)  New results with SRG-N 3 LO: N- 4 He, n- 7 Li, N- 12 C and n- 16 O  Breakthrough due to the importance- truncated NCSM approach First results for 3 H(d,n) 4 He Development for 3 H, 3 He projectiles  To do: Heavier projectiles: 4 He NCSM with continuum (NCSMC) Inclusion of NNN force Three-cluster NCSM/RGM and treatment of three-body continuum 7 Li n