Lawrence Livermore National Laboratory Ab initio reactions of nucleons on light nuclei 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 (LLNL) Collaborators: Sofia Quaglioni (LLNL), Robert Roth (TU Darmstadt) UNEDF SCIDAC meeting, Pack Forest, WA, 6/22/2009

2 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

3 LLNL-PRES Lawrence Livermore National Laboratory The ab initio 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 standard potentials Unitary transformation in n-body cluster approximation (n=2,3)  Importance-truncated N max h  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 N max for bound states. No coupling to continuum.

4 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 Fully implemented for nucleon-nucleus basis. Work on deuteron-nucleus basis under way.

5 LLNL-PRES 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)

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

7 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

8 LLNL-PRES Lawrence Livermore National Laboratory NCSM/RGM ab initio calculation of n- 3 H and p - 3 He phase shifts  NCSM/RGM calculations with n+ 3 H(g.s.) and p+ 3 He(g.s.), respectively.  Benchmark with Alt, Grassberger and Sandhas (AGS) results [ PRC75, (2007) ] What is missing? - n+ 3 H(ex), 2 n+d, p- 3 He(ex), 2 p+d configurations The omission of three-nucleon partial waves with 1/2 < J ≤ 5/2 leads to effects of comparable magnitude on the AGS results. Need to include target excited states!   EFT N 3 LO NN potential: convergence reached with two-body effective interaction 3 He p 3H3H n

9 LLNL-PRES Lawrence Livermore National Laboratory n- 4 He & p- 4 He phase shifts from accurate NN interactions  4 He states: g.s.,     Reasonable agreement with experiment for 2 S 1/2, 2 P 1/2, 2 D 3/2 channels  Coulomb under control n- 4 He and p- 4 He phase shifts Insufficient spin-orbit strength: 2 P 3/2 underestimated  NNN needed CD-Bonn the best description of 2 S 1/2 phase shifts Insufficient spin-orbit strength: 2 P 3/2 underestimated  NNN needed CD-Bonn the best description of 2 S 1/2 phase shifts

10 LLNL-PRES Lawrence Livermore National Laboratory First ever ab initio calculation of A y in for a A=5 system. Strict test of inter-nucleon interactions. n + 4 He differential cross section and analyzing power  Neutron energy of 17 MeV beyond low-lying resonances  Polarized neutron experiment at Karlsruhe  NCSM/RGM calculations n+ 4 He(g.s,0 + 0) SRG-evolved N 3 LO NN potential  Good agreement for angular distribution  Differences for analyzing power A y puzzle for A=5? 4 He n

11 LLNL-PRES Lawrence Livermore National Laboratory NCSM/RGM ab initio calculation of n + 7 Li scattering  7 Li  Full NCSM up to N max =10 (12 possible)  IT-NCSM up to N max =18  Convergence of both ground and excited states  8 Li NCSM predicts unobserved low-lying 0 + and 2 + states NCSM/RGM with 7 Li 3/2 - and 1/2 - bound states included Up to N max =14 so far ( hΩ=20 MeV used)  Moderate changes from N max =6 to N max =14 Bound states  2 + state bound by 1.07 MeV (expt 2.03 MeV)  1 + state bound by 0.18 MeV (expt 1.05 MeV) 7 Li n PRC 73, (2006)

12 LLNL-PRES Lawrence Livermore National Laboratory NCSM/RGM ab initio calculation of n + 7 Li scattering  7 Li 3/2 - and 1/2 - bound states included  SRG-N 3 LO NN interaction with Λ=2.02 fm -1 Up to N max =14 so far (hΩ=20 MeV used) Moderate changes from N max =6 to N max = and 1 + states bound Other states unbound S-wave scattering length Expt: a 01 =0.87(7) fm a 02 =-3.63(5) fm Calc: a 01 =1.31 fm a 02 =-0.18 fm 7 Li n Qualitative agreement with experiment. Calculated broad 1 + resonance, predicted narrow 0 + resonance. The 3 + resonance not seen when 7 Li 7/2 - state not included.

13 LLNL-PRES Lawrence Livermore National Laboratory NCSM/RGM ab initio calculation of n + 7 Li scattering  7 Li 3/2 -, 1/2 - and 7/2 - states included  Result for N max =8 shown  2 + and 1 + states bound (slightly more)  0 + and 1 + resonances not affected  3 + and 2 + resonances appear  Improvement of S-wave scattering length S-wave scattering length Expt: a 01 =0.87(7) fm a 02 =-3.63(5) fm Calc: a 01 =0.73 fm a 02 =-1.42 fm 7 Li n Good match of bound states and narrow resonances with the 8 Li NCSM result. Predicted narrow 0 + and 2 + resonance. Seen at recent p+ 7 Be FSU experiment.

14 LLNL-PRES Lawrence Livermore National Laboratory Parity-inverted ground state of 11 Be  11 Be Disappearance of N=8 magic number with increasing N/Z ratio Ground state 1/2 + instead of the p-shell expected 1/2 -  Large-scale ab initio NCSM calculations with several accurate NN potentials do not explain the parity inversion PRC 71, (2005)  Problem: Extended n+ 10 Be configurations suppressed 11 Be

15 LLNL-PRES Lawrence Livermore National Laboratory n- 10 Be phase shifts with CD-Bonn NN interaction n- 10 Be phase shifts  Dramatic increase of 11 Be   binding energy  NCSM/RGM calculation: CD-Bonn 2000 NN potential two-body effective interaction N max h  = 13 MeV 10 Be states:  Inversion between   and   states reproduced 10 Be n

16 LLNL-PRES Lawrence Livermore National Laboratory n- 10 Be phase shifts with CD-Bonn NN interaction n- 10 Be phase shifts  n- 10 Be wave function extends to large distances  Relative kinetic energies decreases  Dramatic increase of 11 Be   binding energy  NCSM/RGM calculation: CD-Bonn 2000 NN potential two-body effective interaction N max h  = 13 MeV 10 Be states:  Inversion between   and   states reproduced 10 Be n The proper description of extended n- 10 Be configurations leads to parity-inverted 11 Be g.s E [MeV] 1/2 - 1/2 + Expt. NCSMNCSM/RGM

17 LLNL-PRES Lawrence Livermore National Laboratory n- 10 Be & parity-inverted ground state of 11 Be  What happens? n- 10 Be wave function extends to large distances Relative kinetic and potential energies decrease in absolute values The kinetic energy more dramatically  Net effect: Gain in binding energy 10 Be n NCSM /RGM T rel V rel E 10Be E tot Model- space Full The proper description of extended n- 10 Be configurations leads to parity-inverted 11 Be g.s.

18 LLNL-PRES Lawrence Livermore National Laboratory Nucleon- 12 C scattering with SRG-N 3 LO NN potential  12 C Full NCSM up to N max =8 IT NCSM up to N max =18(!)  Perfect agreement for both the 0 + ground- and 2 + excited state up to N max =8  Convergence of the IT-NCSM  h  =24 MeV used  13 N, 13 C within the NCSM 1/2 + state too high by ~ 6 MeV  13 N, 13 C within the NCSM/RGM up N max =16 with 12 C g.s. and 2 + included 13 C:  1/2 - bound by 5.34 MeV (expt 4.95 MeV)  3/2 - bound by 2.23 MeV (expt 1.27 MeV)  1/2 + bound by 0.03 MeV (expt 1.86 MeV)  Excitation energy 5.31 MeV (expt 3.09 MeV) Excitation energy of the 1/2 + state drops by 4 MeV when n- 12 C long-range correlations included

19 LLNL-PRES Lawrence Livermore National Laboratory p - 12 C scattering with SRG-N 3 LO NN potential  Experiments with a polarized proton target under way  NCSM/RGM up N max =16 12 C g.s. and 2 + included 1/2 - state bound by 2.9 MeV  13 N ground state Other states unbound 1/2 + resonance at ~1.2 MeV 5/2 + resonance Good stability: Moderate changes from N max =6 to N max =16 Minimal difference between N max =14 and N max =16 Qualitative agreement with experiment

20 LLNL-PRES Lawrence Livermore National Laboratory 16 O ground state, 17 O bound states  16 O ground state calculated within importance-truncated NCSM up to N max =18 (N max =22 possible!?), h  =24 MeV SRG-N 3 LO with  =2.66 fm -1  Less overbinding: E   -140 MeV Benchmarking with full NCSM  16 O binding energy up to N max =8  Perfect agreement  17 O within ab initio NCSM/RGM 1/2 + bound: E b =-0.88 MeV wrt 16 O 5/2 + bound: E b =-0.41 MeV wrt 16 O  N max =19, h  =24 MeV  Only 16 O ground-state included 16 O n

21 LLNL-PRES Lawrence Livermore National Laboratory n - 16 O scattering with SRG-N 3 LO NN potential  16 O ground state only  Phase-shift convergence very good  Essential to use large N max Target wave function Expansion of short- range parts of kernels IT NCSM for the target makes it possible 16 O n Combining the ab initio NCSM/RGM with the importance-truncated NCSM highly promising. Access to medium mass nuclei. N max =18 Done up to N max =18 converged ( )

22 LLNL-PRES Lawrence Livermore National Laboratory n - 16 O scattering: Effect of 16 O excited states  Need to include 16 O excited states (1p-1h…)  IT NCSM for both the ground state & excited states  Done up to N max =12/13 g.s. in N max =12 3 -, 1 -, 2 - in N max =13 Significant increase of binding energies  1/2 + :   5/2 + :  Appearance of sharp resonances 16 O n Correct order Good stability ( )

23 LLNL-PRES Lawrence Livermore National Laboratory n - 16 O scattering: Open issues  16 O excited states with the SRG-N 3 LO 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?  Density too high? 12 C+alpha not included at present  n+ 16 O with the SRG-N 3 LO NN potential 5/2 +, 1/2 + underbound 1/2 -, 5/2 - not bound Resonances too high  Impact of incomplete 16 O description  13 C+alpha not taken into account

24 LLNL-PRES Lawrence Livermore National Laboratory The deuteron projectile: Norm kernel  2(A-2)  (A-2) (2) + (A-2)(A-3)/2  (A-2) (2) (1,…,A-2) (A-1,A) (1,…,A-2) (A-1,A)

25 LLNL-PRES Lawrence Livermore National Laboratory The deuteron projectile: Hamiltonian kernel (A-2) (2) (1,…,A-2) (A-1,A) (1,…,A-2) (A-1,A)

26 LLNL-PRES Lawrence Livermore National Laboratory d -  scattering: Progress so far  Norm kernel coded  Hamiltonian kernel partially coded Term with three-body density in progress  Convergence reached for d-  norm kernel (physics - Pauli principle) S-wave dominated D-wave dominated P-wave dominated

27 LLNL-PRES Lawrence Livermore National Laboratory d -  scattering: Progress so far  Norm kernel coded  Hamiltonian kernel partially coded Term with three-body density in progress  Convergence reached for d-  norm kernel (physics - Pauli principle) 1 + S-wave dominated 2 + D-wave dominated 1 - P-wave dominated

28 LLNL-PRES Lawrence Livermore National Laboratory Toward the first ab initio description of the Deuterium-Tritium fusion  Solve the many-body Schrödinger equation in the Hilbert space spanned by the RGM basis states:  Progress so far: Norm kernels calculated Method: To calculate the reaction D + T  n +  within the ab initio NCSM/RGM – Using the ab initio NCSM calculate eigenstates: – Solve the many-body Schrödinger equation in the Hilbert space spanned by the RGM basis states: – From the Scattering matrix calculate total and angular cross sections, polarization observablesIntroduction: – A fundamental theory of light-ion fusion reactions is the basis to enhance the predictive capabilities of astrophysics modeling, and provide precision diagnostics for prospective fusion-power plants – Presently there is no realistic theoretical model able to describe important processes such as the Deuterium-Tritium (D-T) fusion NCSM – Our approach combines the ab initio no-core shell model (NCSM) RGMab initio NCSM/RGM and the resonating-group method (RGM)  ab initio NCSM/RGM NCSM: single-particle degrees of freedom RGM: clusters and their relative motionr’ n  r n  r’ n  r n  r’ D  r D  r’ D  r D  Results: – Promising results using the NCSM/RGM nucleon-nucleus formalism The n- 4 He differential cross section (left) and analyzing power (right) for 17 MeV neutrons obtained within the ab initio NCSM/RGM (red solid line) compared to experimental data and ENDF/B-VI evaluations. More on many-body ab initio calculations of nucleon-nucleus scattering in S. Quaglioni and P. Navratil, PRL 101, (2008); PRC 79, (2009) – Approach applied also to n-T, n- 10 Be, n- 12 C, n- 16 O, p- 3,4 He   D r n  r D  r’ D  r D  D(g.s.)+T(g.s.) The D+T norm kernel: (a)diagrammatic representation of the “direct” and “exchange” components; exchange components for the spin-parity-isospin (b) 1/2 + 1/2 and (c) 3/2 + 1/2 channels (a) (b)(c) r D D(g.s.)+T(g.s.) The D+T norm kernel: (a)diagrammatic representation of the “direct” and “exchange” components; exchange components for the spin-parity-isospin (b) 1/2 + 1/2 and (c) 3/2 + 1/2 channels (a) (b)(c) r’ D  r D  D(g.s.)+T(g.s.) The D+T norm kernel: (a)diagrammatic representation of the “direct” and “exchange” components; exchange components for the spin-parity-isospin (b) 1/2 + 1/2 and (c) 3/2 + 1/2 channels (a) (b)(c) r’ D  r D  D(g.s.)+T(g.s.) The D+T norm kernel: (a)diagrammatic representation of the “direct” and “exchange” components; exchange components for the spin-parity-isospin (b) 1/2 + 1/2 and (c) 3/2 + 1/2 channels (a) (b)(c) r’ D  r D  D(g.s.)+T(g.s.) The D+T norm kernel: (a)diagrammatic representation of the “direct” and “exchange” components; exchange components for the spin-parity-isospin (b) 1/2 + 1/2 and (c) 3/2 + 1/2 channels (a) (b)(c) r’ n  r n  r’ n  r n  r’ D  r D  r’ D  r D 

29 LLNL-PRES Lawrence Livermore National Laboratory FY09 accomplishments  Development of ab initio many-body reaction theory by merging the NCSM and the RGM (P. Navratil and S. Quaglioni) Results with NN potentials used by UNEDF collaboration  n- 7 Li with SRG-N 3 LO using the importance-truncated NCSM  p- 12 C with SRG-N 3 LO using the importance-truncated NCSM  n- 16 O with SRG-N 3 LO using the importance-truncated NCSM  Collaboration with R. Roth (TU Darmstadt)  Deuteron-nucleus scattering under development  Development of the TRDENS transition density code  Distribution of two-body density structure over groups of processors  Similarity-renormalization-group evolution of NN+NNN interactions  Collaboration with D. Furnstahl (OSU) and E. Jurgenson (OSU)  A=14 nuclei with chiral EFT NN+NNN up to N max =8 Transformation of NNN to SD basis up to N max =8  Collaboration with J. Vary, P. Maris, H. Nam, E. Ormand and D. Dean

30 LLNL-PRES Lawrence Livermore National Laboratory Publications relevant to UNEDF in 2008/2009  E. D. Jurgenson, P. Navratil, and R. J. Furnstahl, Evolution of nuclear many-body forces with the similarity renormalization group, arXiv: [nucl-th], submitted to Phys. Rev. Lett.  I. Stetcu, S. Quaglioni, J.L. Friar, A.C. Hayes, and P. Navratil, Electric Dipole Polarizabilities of Hydrogen and Helium Isotopes, Phys. Rev. C 79, (2009)  P. Navratil, S. Quaglioni, I. Stetcu and B. R. Barrett, Recent developments in no-core shell- model calculations, J. of Phys. G 36 (2009) , invited topical review.  S. Quaglioni, P. Navratil, Ab initio many-body calculations of nucleon-nucleus scattering, Phys. Rev. C 79, (2009)  C. Forssen, E. Caurier, P. Navratil, Charge radii and electromagnetic moments of Li and Be isotopes from the ab initio no-core shell model, arXiv: [nucl-th], Phys. Rev. C 79, (R) (2009)  D. Gazit, S. Quaglioni, P. Navratil, Three-Nucleon Low-Energy Constants from the Consistency of Interactions and Currents in Chiral Effective Field Theory, arXiv: [nucl-th], submitted to Phys. Rev. Lett.  S. Quaglioni, P. Navratil, Ab initio many-body calculations of n- 3 H, n- 4 He, p-3He, 4 He, and n- 10 Be scattering, PHYSICAL REVIEW LETTERS 101, (2008)  R. Roth, P. Navratil, Comment on "Ab initio Study of Ca-40 with an Importance-Truncated No- Core Shell Model'' – Reply, PHYSICAL REVIEW LETTERS 101, (2008)

31 LLNL-PRES Lawrence Livermore National Laboratory Future plan  The rest of Year 3 Complete n- 7 Li calculations Begin n- 8 He investigation Continue work on deuteron-nucleus formalism (supported by LDRD)  Year 4 n- 8 He, n- 9 Li calculations Development of 3 H and 3 He – nucleus formalism (supported by LDRD) Development of the coupling of NCSM/RGM and NCSM  NCSMC Similarity-renormalization-group evolution of NN+NNN interactions  Application to p-shell nuclei (supported by DOE/SC/NP) Further development of importance-truncation NCSM scheme  Year 5 High profile science: Capture reactions - 3 He( ,  ) 7 Be  Computational challenges: n-body density (n>2) calculations  Distribution of structure allocation, parallelization

32 LLNL-PRES Lawrence Livermore National Laboratory Conclusions and outlook  Coming next: inclusion of NNN potential terms d, 3 H and 3 He, 4 He projectiles three-body continuum  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  We are extending the ab initio NCSM to treat low-energy light-ion reactions  Our recent achievements: n- 3 H, n- 4 He, n- 10 Be and p- 3,4 He scattering phase-shifts with realistic NN potentials ( PRL 101, (2008) )  n- 7 Li, N- 12 C, N- 16 O under way: Breakthrough due to the importance-truncated NCSM approach