Structure and reactions at the dripline with coupled-cluster theory. Gaute Hagen (ORNL) Collaborators: Thomas Papenbrock (UT/ORNL) Morten Hjorth-Jensen (UiO/CMA) Gustav Jansen (UiO/CMA) Ruprecht Machleidt (UI) Nicolas Michel (UT/ORNL) NN2012 San Antonio, May 31, 2012

Roadmap for Theory of Nuclei Main goal : To arrive at at comprehensive description of all nuclei and low-energy reactions from the basic interactions between the constituent nucleons Main goal : To arrive at at comprehensive description of all nuclei and low-energy reactions from the basic interactions between the constituent nucleons 2

3 Energy scales and relevant degrees of freedom Fig.: Bertsch, Dean, Nazarewicz, SciDAC review (2007) Chiral effective field theory Energy

Nuclear forces from chiral effective field theory [Weinberg; van Kolck; Epelbaum et al.; Entem & Machleidt; …] [Epelbaum, Hammer, Meissner RMP 81, 1773 (2009)] Low energy constants from fit of NN data, A=3,4 nuclei, or light nuclei. CECE CDCD

Effective three-nucleon forces By integrating over the third leg in infinite nuclear matter we can derive density dependent corrections to the nucleon-nucleon interaction. J. Holt. Phys.Rev.C81, , (2010) By integrating over the third leg in infinite nuclear matter we can derive density dependent corrections to the nucleon-nucleon interaction. J. Holt. Phys.Rev.C81, , (2010) CDCD CECE

Coupled-cluster method (in CCSD approximation) Ansatz: Correlations are exponentiated 1p-1h and 2p-2h excitations. Part of np-nh excitations included! Coupled cluster equations Scales gently (polynomial) with increasing problem size o 2 u 4. Truncation is the only approximation. Size extensive (error scales with A)  Most efficient for doubly magic nuclei Alternative view: CCSD generates similarity transformed Hamiltonian with no 1p-1h and no 2p-2h excitations.

Binding energy per nucleon Benchmarking different methods: Our CC results for 16 O agree with IT-NCSM (R. Roth et al PRL 107, (2011)) and UMOA (Fujii et al., Phys. Rev. Lett. 103, (2009) Benchmarking different methods: Our CC results for 16 O agree with IT-NCSM (R. Roth et al PRL 107, (2011)) and UMOA (Fujii et al., Phys. Rev. Lett. 103, (2009) NucleusCCSDΛ-CCSD(T)Experiment 4 He O Ca Ca Hagen, Papenbrock, Dean, Hjorth-Jensen, Phys. Rev. Lett. 101, (2008) Hagen, Papenbrock, Dean, Hjorth-Jensen, Phys. Rev. Lett. 101, (2008) Toward medium-mass nuclei Chiral N 3 LO (500 MeV) by Entem & Machleidt, NN only CCM(IT)-NCSMUMOA E/A 4 He-6.39(5) O-7.56(8)-7.48(4)-7.47

The Berggren completeness treats bound, resonant and scattering states on equal footing. Has been successfully applied in the shell model in the complex energy plane to light nuclei. For a review see N. Michel et al J. Phys. G 36, (2009). The Berggren completeness treats bound, resonant and scattering states on equal footing. Has been successfully applied in the shell model in the complex energy plane to light nuclei. For a review see N. Michel et al J. Phys. G 36, (2009). Open Quantum Systems

Physics of neutron rich nuclei Many-body Correlations: Coupled-Cluster theory Many-body Correlations: Coupled-Cluster theory Interactions: NN + 3NFs from Chiral EFT Interactions: NN + 3NFs from Chiral EFT Open channels: Gamow Shell Model Open channels: Gamow Shell Model Physics of neutron rich nuclei

Is 28 O a bound nucleus? Experimental situation “Last” stable oxygen isotope 24 O 25 O unstable (Hoffman et al 2008) 26,28 O not seen in experiments 31 F exists (adding on proton shifts drip line by 6 neutrons!? ) Shell model (sd shell) with monopole corrections based on three-nucleon force predicts 2 nd O as last stable isotope of oxygen. [Otsuka, Suzuki, Holt, Schwenk, Akaishi, PRL (2010), arXiv: ] 28 O?

Oxygen isotopes from chiral interactions Chiral three-nucleon force at order N2LO. k f =1.05fm -1, C D = 0.2, C E = 0.71 (fitted to to the binding energy of 16 O and 22 O). Chiral three-nucleon force at order N2LO. k f =1.05fm -1, C D = 0.2, C E = 0.71 (fitted to to the binding energy of 16 O and 22 O). NN only underbinds and wrongly places dripline beyond 28 O Inclusion of effective 3NF places dripline at 25 O. Overall the odd-even staggering in the neutron rich oxygen is well reproduced. We find 26 O to unbound with respect to 24 O by ~100keV, this agrees with recent experiments on 26 O by E. Lunderberg et al., Phys. Rev. Lett. 108 (2012) We find 28 O to be unbound with a resonance width of ~2MeV NN only underbinds and wrongly places dripline beyond 28 O Inclusion of effective 3NF places dripline at 25 O. Overall the odd-even staggering in the neutron rich oxygen is well reproduced. We find 26 O to unbound with respect to 24 O by ~100keV, this agrees with recent experiments on 26 O by E. Lunderberg et al., Phys. Rev. Lett. 108 (2012) We find 28 O to be unbound with a resonance width of ~2MeV CDCD CECE G. HagenG. Hagen, M. Hjorth-Jensen, G. R. Jansen,M. Hjorth-JensenG. R. Jansen R. Machleidt, T. Papenbrock, accepted forR. MachleidtT. Papenbrock Publication in PRL, arXiv: (2012)arXiv: G. HagenG. Hagen, M. Hjorth-Jensen, G. R. Jansen,M. Hjorth-JensenG. R. Jansen R. Machleidt, T. Papenbrock, accepted forR. MachleidtT. Papenbrock Publication in PRL, arXiv: (2012)arXiv:

Oxygen isotopes from chiral interactions Excited states in 24 O computed with EOM-CCSD and compared to experiment Matter and charge radii for O Computed from intrinsic densities and Compared to experiment. The effects of three-nucleon forces decompress the spectra and brings it in good agreement with experiment. We predict the newly observed resonance at ~7.5MeV in 24O to be a super position of several states with spin and parity 4 +,3 +,2 + The effects of three-nucleon forces decompress the spectra and brings it in good agreement with experiment. We predict the newly observed resonance at ~7.5MeV in 24O to be a super position of several states with spin and parity 4 +,3 +,2 +

Evolution of shell structure in neutron rich Calcium isotopes How do shell closures and magic numbers evolve towards the dripline? Is the naïve shell model picture valid at the neutron dripline? How do shell closures and magic numbers evolve towards the dripline? Is the naïve shell model picture valid at the neutron dripline?

Calcium isotopes from chiral interactions Main Features: 1.Total binding energies agree well with experimental masses. 2.Masses for Ca are converged in 19 major shells Ca weakly bound/unbound Ca is bound with respect to 1n emission Ca are located right at threshold. Main Features: 1.Total binding energies agree well with experimental masses. 2.Masses for Ca are converged in 19 major shells Ca weakly bound/unbound Ca is bound with respect to 1n emission Ca are located right at threshold. G. Hagen, G. Jansen, M. Hjorth-Jensen, R. Machleid, T. Papenbrock, arXiv: (2012) G. Hagen, G. Jansen, M. Hjorth-Jensen, R. Machleid, T. Papenbrock, arXiv: (2012) k F =0.95fm -1, c D =-0.2, c E =0.735 N max = 18, hw = 26MeV

2 + systematics in Calcium isotopes 48 Ca 52 Ca 54 Ca / ????? CC Exp Main Features: 1.Very nice agreement between theory and experiment. 2.Our calculations for 2 + and 4 + in 54 Ca do not point to a new magic shell closure. Main Features: 1.Very nice agreement between theory and experiment. 2.Our calculations for 2 + and 4 + in 54 Ca do not point to a new magic shell closure. G. Hagen, G. Jansen, M. Hjorth-Jensen, R. Machleid, T. Papenbrock, arXiv: (2012) G. Hagen, G. Jansen, M. Hjorth-Jensen, R. Machleid, T. Papenbrock, arXiv: (2012)

Spectra and shell evolution in Calcium isotopes 1.Due to continuum coupling, we find an inversion of the 9/2+ and 5/2+ resonant states in 53,55,61 Ca 2. We find the ground state of 61 Ca to be ½+ located right at threshold. 3.A harmonic oscillator basis gives the naïve shell model ordering of states. 4.Continuum coupling is crucial. 1.Due to continuum coupling, we find an inversion of the 9/2+ and 5/2+ resonant states in 53,55,61 Ca 2. We find the ground state of 61 Ca to be ½+ located right at threshold. 3.A harmonic oscillator basis gives the naïve shell model ordering of states. 4.Continuum coupling is crucial. New penning trap measurement of masses of 51,52 Ca A. T. Gallant et al arXiv: v1 (2012)arXiv: v1 New penning trap measurement of masses of 51,52 Ca A. T. Gallant et al arXiv: v1 (2012)arXiv: v1

Microscopic definition of Spectroscopic Factors SF is the norm of the overlap function and quantifies the degree of correlations SFs are not observables and depend on the resolution scale Asymptotic properties of the one-nucleon overlap functions The overlap functions satisfy a one-body Schrodinger like equation, and outside the range of the interaction the overlap function is proportional to a single-particle wave function Df d Bound states Scattering states Towards nuclear reactions with coupled-cluster theory One-nucleon overlap functions Elastic scattering, capture and transfer reactions of a nucleon on/to a target nucleus with mass A is determined by the one-nucleon overlap function

Quenching of spectroscopic factors for proton removal in neutron rich oxygen isotopes Spectroscopic factor is a useful tool to study correlations towards the dripline. SF for proton removal in neutron rich 24 O show strong “quenching” pointing to large deviations from a mean-field like picture. G. Hagen et al Phys. Rev. Lett. 107, (2011). Spectroscopic factor is a useful tool to study correlations towards the dripline. SF for proton removal in neutron rich 24 O show strong “quenching” pointing to large deviations from a mean-field like picture. G. Hagen et al Phys. Rev. Lett. 107, (2011). Strong asymmetry dependence on the SF for proton and neutron removal in neutron rich oxygen isotopes. SF~1 for neutron removal while protons are strongly correlated SF ~ in 22,24,28 O Strong asymmetry dependence on the SF for proton and neutron removal in neutron rich oxygen isotopes. SF~1 for neutron removal while protons are strongly correlated SF ~ in 22,24,28 O

Threshold effects and spectroscopic factors Near the scattering threshold for one-neutron decay the spectroscopic factors are Significantly influenced by the presence of The continuum. The standard shell model approximation to spectroscopic factors completely fails in this region. N. Michel et al Phys. Rev. C 75, (2007) N. Michel et al Nucl. Phys. A 794, 29 (2007) Near the scattering threshold for one-neutron decay the spectroscopic factors are Significantly influenced by the presence of The continuum. The standard shell model approximation to spectroscopic factors completely fails in this region. N. Michel et al Phys. Rev. C 75, (2007) N. Michel et al Nucl. Phys. A 794, 29 (2007) Top and middle: Bottom :

The one-nucleon overlap function: Beyond the range of the nuclear interaction the overlap functions take the form: Elastic proton/neutron scattering on 40Ca G. Hagen and N. Michel, in preparation (2012). Preliminary

Differential cross section for elastic proton scattering on 40 Ca. Good agreement between theory and Experiment for low-energy scattering. Differential cross section for elastic proton scattering on 40 Ca. Good agreement between theory and Experiment for low-energy scattering. Elastic proton/neutron scattering on 40Ca G. Hagen and N. Michel In Preparation (2012). G. Hagen and N. Michel In Preparation (2012). Preliminary

Densities and radii from coupled-cluster theory We can obtain the intrinsic density by a deconvolution of the laboratory density B. G. Giraud, Phys. Rev. C 77, (2008) Center of mass part Intrinsic density Lab. density The coupled-cluster wave function factorizes to a good approximation into an intrinsic and center of mass part, where the center of mass part is a Gaussian with a fixed oscillator frequency independent of single-particle basis GH, T. Papenbrock and D. Dean et al, Phys. Rev. Lett. 103, (2009) The coupled-cluster wave function factorizes to a good approximation into an intrinsic and center of mass part, where the center of mass part is a Gaussian with a fixed oscillator frequency independent of single-particle basis GH, T. Papenbrock and D. Dean et al, Phys. Rev. Lett. 103, (2009) We solve for the right and left ground state of the similarity transformed Hamiltonian The density matrix is computed within coupled-cluster method as:

Densities and radii from coupled-cluster theory 1.Relative energies in O depend weakly on the resolution scale 2.We clearly see shell structure appearing in the matter densities for O 3.Matter and charge radii depend on the resolution scale, however relative difference which is relevant for isotope shift measurements does not 1.Relative energies in O depend weakly on the resolution scale 2.We clearly see shell structure appearing in the matter densities for O 3.Matter and charge radii depend on the resolution scale, however relative difference which is relevant for isotope shift measurements does not

23 O interaction cross section (scattering off 12 C GSI) Experimental radii extracted from matter distribution within Glauber model. Main result of new measurement: 23 O follows systematics; interaction cross section consistent with separation energies. R. Kanungo et al Phys. Rev. C 84, (2011) Experimental radii extracted from matter distribution within Glauber model. Main result of new measurement: 23 O follows systematics; interaction cross section consistent with separation energies. R. Kanungo et al Phys. Rev. C 84, (2011) Experiment Ozawa et al. (2001) Kanungo et al (2011) E ≈ 900A MeV

Resolving the anomaly in the cross section of 23 O The anamoly of 23 O New measurements (R. Kanungo) of the 23 O cross section and coupled cluster calculations show that 23 O is not consistent with a one-neutron halo picture The anamoly of 23 O New measurements (R. Kanungo) of the 23 O cross section and coupled cluster calculations show that 23 O is not consistent with a one-neutron halo picture

Summary 1.Interactions from Chiral EFT probed in nuclei 2.CC calculations for oxygen and calcium with effects of 3NF and continuum give significant improvement in binding energy and spectra. 3.Predict spin and parity of newly observed resonance in 24 O. 4.Predict weak sub-shell closure in 54 Ca. 5.Level ordering in the gds shell in neutron calcium is reversed compared to naïve shell model. 6.Quenching of spectroscopic factors near neutron dripline show role of continuum induced correlations for protons 7.Densities and cross sections from coupled-cluster theory help resolve long-standing anomoly of 23 O 1.Interactions from Chiral EFT probed in nuclei 2.CC calculations for oxygen and calcium with effects of 3NF and continuum give significant improvement in binding energy and spectra. 3.Predict spin and parity of newly observed resonance in 24 O. 4.Predict weak sub-shell closure in 54 Ca. 5.Level ordering in the gds shell in neutron calcium is reversed compared to naïve shell model. 6.Quenching of spectroscopic factors near neutron dripline show role of continuum induced correlations for protons 7.Densities and cross sections from coupled-cluster theory help resolve long-standing anomoly of 23 O

Ab initio description of proton-halo state in 17 F Separation energy of halo state is only 105 keV Continuum has to be treated properly Focus is on single-particle states Previous study: shell model in the continuum with 16 O core [K. Bennaceur, N. Michel, F. Nowacki, J. Okolowicz, M. Ploszajczak, Phys. Lett. B 488, 75 (2000)]

Bound states and resonances in 17 F Computation of single-particle states via “Equation-of-motion CCSD” Excitation operator acting on closed-shell reference Here: superposition of one-particle and 2p-1h excitations Single-particle basis consists of bound, resonance and scattering states Gamow basis for s 1/2 d 5/2 and d 3/2 single-particle states Harmonic oscillator states for other partial waves [G. Hagen, TP, M. Hjorth-Jensen, Phys. Rev. Lett. 104, (2010)] Gamow basis weakly dependent on oscillator frequency d 5/2 not bound; spin-orbit splitting too small s 1/2 proton halo state close to experiment Gamow basis weakly dependent on oscillator frequency d 5/2 not bound; spin-orbit splitting too small s 1/2 proton halo state close to experiment

Variation of cutoff probes omitted short-range forces Proton-halo state (s 1/2 ) only weakly sensitive to variation of cutoff Spin-orbit splitting increases with decreasing cutoff 17 F [G. Hagen, TP, M. Hjorth-Jensen, Phys. Rev. Lett. 104, (2010)]