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LLNL-PRES-XXXXXX This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA Lawrence Livermore National Security, LLC Ab Initio Description of Nucleon and Deuteron Scattering for Systems with up to A = 6 Nucleons «Understanding nuclear structure and reactions microscopically, including the continuum» FUSTIPEN topical workshop, GANIL Caen, March17 th Collaborators: S. Quaglioni (LLNL) P. Navrátil (TRIUMF) R. Roth (TU Darmstadt) J. Langhammer (TU Darmstadt) C. Romero-Redondo (TRIUMF) F. Raimondi (TRIUMF) J. Dohet-Eraly (TRIUMF) Guillaume Hupin Fusion based energy generation Nuclear astrophysics Quest for the nuclear interaction

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 2 Ab initio NCSM/RGM: formalism for binary clusters S. Quaglioni and P. Navrátil, PRL101 (2008); PRC79 (2009) Starts from: Schrödinger equation on channel basis: Channel basis RGM accounts for: 1) interaction (Hamiltonian kernel), 2) Pauli principle (Norm kernel) between clusters. NCSM accounts for: internal structure of clusters. Together with the same microscopic nuclear interaction. Relative wave function (unknown) Ex: n- 4 He scattering Exit channel Entrance channel Antisymmetrizer Cluster expansion technique

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 3 Going around the hard core problem E. Jurgenson, Navrátil, R. J. Furnstahl PRL103 (2009) In configuration interaction methods we need to soften interaction to address the hard core We use the Similarity-Renormalization- Group (SRG) method Bare potential Evolved potential Evolution with flow parameter Preserves the physics Decouples high and low momentum Induces many-body forces Unitary transformations k’(fm -1 ) k(fm -1 ) k’(fm -1 ) k(fm -1 ) Flow parameter

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 4 Demonstrated capability to describe binary-cluster reactions starting from NN interactions Nucleon-nucleus collisions n- 3 H, p- 3 He, N- 4 He, n- 10 Be scattering with N 3 LO NN (mod. Lee-Suzuki eff. Int.) Nucleon scattering on 3 H, 3,4 He, 7 Li, 7 Be, 12 C, 16 O with SRG- N 3 LO 7 Be(p,γ) 8 B radiative capture with SRG-N 3 LO Deuterium-nucleus collisions d- 4 He scattering and 6 Li structure with SRG-N 3 LO (d,N) transfer reactions 3 H(d,n) 4 He and 3 He(d,p) 4 He reactions with SRG-N 3 LO n- 4 He scattering Lee-Suzuki effective interaction Good reproduction, but still incomplete n-He 4 phase-shift NN only dependent

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 5 Including the NNN force into the NCSM/RGM approach nucleon-nucleus formalism Direct potential:Exchange potential: -

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 6 n - 4 He scattering: NN versus NNN interactions G. Hupin, J. Langhammer et al. PRC88 (2013) The NNN interactions influence mostly the P waves. The largest splitting between P waves is obtained with NN+NNN. The agreement of the P 3/2 phase-shifts between NN-only and NN+NNN forces is accidental. Three scenarii of nuclear Hamiltonians Comparison between NN+NNN -ind and NN+NNN at Nmax=13 with six 4 He states. More spin-orbit splitting NNN vs NN “bare” (NN+NNN-ind) n- 4 He scattering

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 7 n - 4 He scattering: study of the RGM convergence in the NNN case G. Hupin, J. Langhammer et al. PRC88 (2013) We have included the first 6 low-lying states of 4 He. Convergence is difficult to assess. Target polarization (g.s., low-lying states) 4 He low-lying states Convergence of the phase-shifts as a function of the 4 He excited states. Convergence with respect to the target polarization Exp.

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 8 4 He( d, d ) 4 He with NN+NNN interaction G. Hupin, S. Quaglioni and P. Navrátil, work in progress d- 4 He scattering Comparison of the d-α phase-shifts with different interactions d- 4 He(g.s.) scattering phase-shifts for NN, NN+NNN- induced and NN+NNN potential with 2.0 fm Li loses binding energy with NNN interactions. The largest splitting between 3 D 3 and 3 D 2 partial waves is obtained with NN+NNN.

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 9 Study of the influence of d* continuum G. Hupin, S. Quaglioni and P. Navrátil, work in progress d pseudo states 7d*+4d’*+7d`* 7d* d (g.s.) Pseudo-states in each deuteron channels d (g.s., 3 S D 1, 3 D 2, 3 D G 3 ) Comparison of the d-α phase-shifts with different interactions d- 4 He(g.s.) scattering phase-shifts with NN+NNN interaction and 2.0 fm -1 at Nmax=7. 3 S D 1 channel, 3 D 2 channel…

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 10 To overcome the difficulty: couple NCSM and NCSM/RGM (NCSMC) S. Baroni, P. Navrátil and S. Quaglioni PRL110 (2013) Channel basis Relative wave function (unknown) Antisymmetrizer Cluster expansion technique A-body harmonic oscillator states Mixing coefficients (unknown) Second quantization Methods develop in this presentation to solve the many body problem Can address bound and low-lying resonances (short range correlations) Design to account for the continuum of scattering state (long range correlations) The many body quantum problem best describe by superposition of both NCSMC

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 11 Including the NNN force into the NCSMC approach nucleon-nucleus formalism … A-body compound system Target and projectile in relative motion

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 12 n - 4 He scattering with NCSMC G. Hupin, S. Quaglioni and P. Navrátil, work in progress The convergence pattern looks good. The experimental phase-shifts are well reproduced. Study of the convergence with respect to the # of 4 He low-lying states n- 4 He scattering phase-shifts for NN+NNN potential with 2.0 fm -1 and 8 low-lying state of 5 He. Experimental low-lying states of the A=5 nucleon systems. n- 4 He scattering

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 13 How n - 4 He elastic cross-sections compare ? G. Hupin, S. Quaglioni and P. Navrátil, work in progress Differential cross-section at E neutron =0.84 MeV between NN+3N-ind and NN+3N. Comparison of the elastic cross-section between NN and NN+3N with 4 He (g.s.) n- 4 He elastic cross-section for NN+3N-induced, NN+3N potentials compared to expt. and ENDF evaluation. (NN+3N-induced) A better agreement with experiment is obtained with NN+NNN. The NNN force is essential to get the resonance right.

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 14 How n - 4 He elastic cross-sections compare ? G. Hupin, S. Quaglioni and P. Navrátil, work in progress A better agreement with experiment is obtained with NN+NNN. The NNN force is essential to get the resonance right. Differential cross-section at E neutron =1.79 MeV between NN+3N-ind and NN+3N. Comparison of the elastic cross-section between NN and NN+3N with 4 He (g.s.) n- 4 He elastic cross-section for NN+3N-induced, NN+3N potentials compared to expt. and ENDF evaluation. (NN+3N-induced)

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 15 p - 4 He scattering: NCSM/RGM and NCSMC G. Hupin, S. Quaglioni and P. Navrátil, work in progress Preliminary calculation, the NCSMC results are obtained with only the 4 He ground state. We can see an improvement in the reproduction of the experimental phase-shifts. p- 4 He scattering Comparison between NCSM/RGM and NCSMC at Nmax=13 and =2.0 with NN+NNN. NN+NNN NCSM/RGM and NCSMC

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 16 Analyzing power and differential cross section G. Hupin, S. Quaglioni and P. Navrátil, work in progress p- 4 He reaction observables compared to experiment

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 17 4 He(d,d) 4 He with NCSMC G. Hupin, S. Quaglioni and P. Navrátil, work in progress Preliminary results in a small model space (Nmax=9). The coupling to the compound nuclei addresses some missing correlation. d- 4 He scattering Effects of the short-range correlation of the compound nuclei on the d- differential cross-section d- 4 He(g.s.) differential cross-section for NN-only potential using NCSM/RGM andNCSMC. NN-only

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 18 4 He(d,d) 4 He with NN+NNN interaction G. Hupin, S. Quaglioni and P. Navrátil, work in progress The NCMSC weakens the dependence on the d* pseudo-states. Residual dependence could be attributed to the missing breakup channel. Comparison of the d- phase-shifts wrt the number of d* pseudo-states d- 4 He(g.s.) scattering phase-shifts for NN-only with different numbers of deuteron pseudo-states. before

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 19 4 He(d,d) 4 He with NN+NNN interaction G. Hupin, S. Quaglioni and P. Navrátil, work in progress Preliminary results in a small model space (Nmax=9). The 3 D 3 resonance is not quite reproduced but the 3N force is helping to get the correct position. Comparison of the d- phase-shifts with different interactions d+ 4 He(g.s.) scattering phase-shifts for NN-only, NN+NNN-induced, NN+NNN potential with 2 fm

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 20 First step towards Ab initio many-body calculations of fusion P. Navrátil, S. Quaglioni, PRL (2012) Calculated S-factors converge with the inclusion of the virtual breakup of the deuterium, obtained by means of excited 3 S D 1 ( d* ) and 3 D 2 ( d’* ) pseudo-states. 3 H(d,n) 4 He astrophysical S-factor NCSM/RGM results for the 3 He( d, n ) 4 He astrophysical S- factor compared to beam-target measurements. 3 He(d,p) 4 He astrophysical S-factor e - lab screening Complete picture: includes break-up Incomplete nuclear interaction: requires NNN force (SRG-induced + “real”) Evidence of incomplete model (nuclear force) Pseudo excited states

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 21 First step towards Ab initio many-body calculations of fusion J. Dohet-Eraly, S. Quaglioni, P. Navrátil et al. work in progress p- 4 He bremsstrahlung (preliminary) p+ 4 He(g.s.,0 + ) bremsstrahlung for NN+NNN potential with 2 fm -1. NCSM/RGM exchange terms of a non-scalar operator most difficult part Transitions of electric dipole operator between scattering states Perspective to diagnose plasmas in fusion experiments from t(d,n ) 4 He radiative transfer reaction Provides another way to probe cluster wave functions

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 22 We are extending the ab initio NCSM/RGM approach to describe low-energy reactions with two- and three-nucleon interactions. We are able to describe: Nucleon-nucleus collisions with NN+NNN interaction Deuterium-nucleus collisions with NN+NNN interaction NCSMC for single- and two- nucleon projectile Work in progress Fusion reactions with our best complete ab initio approach The present NNN force is "incomplete”, need to go to N 3 LO Conclusions and Outlook Evolution of stars, birth, main sequence, death

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 23 4 He(d,d) 4 He with NCSMC G. Hupin, S. Quaglioni and P. Navrátil, work in progress Preliminary results in a small model space (Nmax=9). The coupling to the compound nuclei addresses some missing correlation. Some splitting between the 3 D 3 and 3 D 2 phase-shifts is missing. Experimental bound and low-lying states of the A=6 nucleon systems. d- 4 He scattering Effects of the short-range correlation of the compound nuclei on the d- phase-shifts d- 4 He(g.s.) scattering phase-shifts for NN+NNN potential using NCSM/RGM and NCSMC. NCSM/RGM NN-only

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 24 Study of the SRG flow parameter dependence G. Hupin, J. Langhammer et al. PRC88 (2013) The SRG evolution is to a good extend unitary. Comparison at Nmax=13 between NN-only, NN+NNN-ind and NN+NNN between =2.0 and 1.88 fm -1 Unitarity behavior of the calculated phase-shifts

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 25 n - 4 He scattering: NCSM/RGM and NCSMC G. Hupin, S. Quaglioni and P. Navrátil, work in progress NCSMC phase shift at Nmax=13 and =2.0 with NN+NNN. NN+NNN NCSMC

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 26 Exemple of an application to nuclear astrophysics P. Navrátil, R. Roth, and S. Quaglioni, PLB704 (2011) The 7 Be(p, ) 8 B is the final step in the nucleosynthetic chain leading to 8 B and one of the main inputs of the standard model of solar neutrinos ~10% error in latest S 17 (0) NCSM/RGM results with largest realistic model space (N max = 10) Parameter of SRG NN interaction chosen to reproduce separation energy: 136 keV (Expt. 137 keV) 7 Be(p, ) 8 B astrophysical S-factor Ab initio theory predicts simultaneously both normalization and shape of S 17 8 B 8 Be*+e + + e Footprints of pp-chain on earth solar neutrinos E n < 15 MeV Astrophysical S-factor: 7 Be+p 8 B+ Theory

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 27 1 st focus of the talk: Influence of the three-nucleon (3N) interaction A high precision nuclear Hamiltonian should include 3N force Without NNN force, the 10 B g.s. has a wrong spin. P. Navrátil et al. PRL99 (2007)G. Hagen et al. PRL108 (2012) The 3N force plays a significant role close to the particle continuum?

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 28 No Core Shell Model Nmax 2 st focus of the talk: Address together short- and long-range correlations … inbound and outbound waves cannot be described by finite number of basis states Within the R-matrix method, one can work the Schrodinger equation within an internal region See P. Descouvemont, D. Baye Rep. Prog. Phys.73 (2010) A good nuclear and reaction model Address efficiently short-range correlation Address efficiently long-range correlation

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 29 p- 4 He scattering: NCSM/RGM versus experiment G. Hupin, J. Langhammer et al. PRC88 (2013) The present NNN force is incomplete (N 2 LO only versus N 3 LO for NN). The missing d+ 3 He channel could account for the shift in the P 3/2 resonance? p- 4 He scattering Comparison between NN+NNN –ind, NN+NNN and experiment at Nmax=13 and =2.0. Initial NN and NN+NNN compared to experiment

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 30 n- 4 He scattering: NCSM/RGM versus experiment G. Hupin, J. Langhammer et al. PRC88 (2013) The present NNN force is incomplete (N 2 LO only versus N 3 LO for NN). The missing d+ 3 H channel could account for the shift in the P 3/2 resonance? Comparison between NN+NNN –ind, NN+NNN and experiment at Nmax=13 and =2.0. Initial NN and NN+NNN compared to experiment

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 31 Including the NNN force into the NCSM/RGM approach deuteron-nucleus formalism Direct Exchange four body density

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 32 4 He( d, d ) 4 He with NN-only P. Navrátil and S. Quaglioni PRC83 (2011) d- 4 He scattering N max = 12 d(g.s., 3 S D 1, 3 D 2, 3 D G 3 ) + 4 He(g.s.) SRG-N 3 LO NN potential with 1.5 fm -1, ħ =14 MeV. Comparison with experiment 4 He(d,d) 4 He phase shifts: convergence wrt to d continuum Pseudo-states in each channel

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 33 We are missing correlations in fact mostly short-range ( 6 Li is underbound) d- 4 He scattering Comparison Expt. R-matrix Poor convergence behavior with respect to Nmax d- 4 He(g.s.) scattering phase-shifts with NN+NNN interaction and 2.0 fm -1 for Nmax=7 to 11. Softer interaction ( 1.5 fm -1 ). A smaller ħ (14 MeV) better describes the d continuum.

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 34 Demonstrated capability to describe binary-cluster reactions starting from NN+NNN interactions ☑ Nucleon-nucleus collisions N- 4 He with SRG-N 3 LO and N 2 LO (3N) ☑ Deuterium-nucleus collisions d- 4 He scattering and 6 Li structure with SRG-N 3 LO (NN) and N 2 LO (3N) ☒ (d,N) transfer reactions ✕ 3 H(d,n) 4 He and 3 He(d,p) 4 He reactions 3N force effects on n- phase-shifts n- 4 He scattering

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 35 Translational invariance is preserved (exactly!) also with SD cluster basis Matrix inversion Matrix elements of translationally invariant operators What we calculate in the “SD” channel basis Observables calculated in the translationally invariant basis Advantage: can use powerful second quantization techniques

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 36 Ab initio NCSM/RGM: formalism for binary clusters Few details (Jacobi) channel basis Constrained by the asymptotic scattering solution We introduce Jacobi channel states in the HO space We use the closure properties of HO radial wave function This defines the RGM model space (Ok for localized parts of the kernels) The coordinate space channel states are given by

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Lawrence Livermore National Laboratory LLNL-PRES-XXXXXX 37 In practice, we made use of second quantization More insights 1. New basis: Slater Determinant channel basis The target (A>a nucleons) described by a Slater Determinant Vector proportional to the c.m. coordinate of the a nucleons Vector proportional to the c.m. coordinate of the A-a nucleons 2. With a basis change, we can recover a simple expression Then, the kernels can be calculated with the help of the second quantization

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