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 descriptionof all nuclei and low-energyreactions from the basicinteractions between the constituent nucleons

2

3

Energy scales and relevant degrees of freedom

Fig.: Bertsch, Dean, Nazarewicz, SciDAC review (2007)

Chiral effective field theory

Ener

gy

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.

CECD

Effective three-nucleon forces

By integrating over the third leg in infinite nuclear matter we can derive density dependentcorrections to the nucleon-nucleon interaction.J. Holt . Phys.Rev.C81, 024002, (2010)

CD CE

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 o2u4 .

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 16O agree with IT-NCSM (R. Roth et al PRL 107, 072501 (2011)) and UMOA (Fujii et al., Phys. Rev. Lett. 103, 182501 (2009)

Nucleus CCSD Λ-CCSD(T) Experiment4He 5.99 6.39 7.0716O 6.72 7.56 7.9740Ca 7.72 8.63 8.5648Ca 7.40 8.28 8.67

Hagen, Papenbrock, Dean, Hjorth-Jensen,

Phys. Rev. Lett. 101, 092502 (2008)

Toward medium-mass nucleiChiral N3LO (500 MeV) by Entem & Machleidt, NN

only

CCM (IT)-NCSM UMOAE/A E/A E/A

4He -6.39(5) -6.3516O -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, 013101 (2009).

Open Quantum Systems

Physics of neutron rich nuclei

Many-body Correlations:

Coupled-Cluster theory

Interactions:NN + 3NFs from

Chiral EFT

Open channels:Gamow Shell Model

Physics of neutron rich

nuclei

Is 28O a bound nucleus?

Experimental situation• “Last” stable oxygen isotope 24O• 25O unstable (Hoffman et al 2008)• 26,28O not seen in experiments• 31F exists (adding on proton shifts drip line by 6 neutrons!?)

Shell model (sd shell) with monopole corrections based on three-nucleon force predicts 2nd O as last stable isotope of oxygen. [Otsuka, Suzuki, Holt, Schwenk, Akaishi, PRL (2010), arXiv:0908.2607]

28O?

Oxygen isotopes from chiral interactions

Chiral three-nucleon force at order N2LO. kf=1.05fm-1, CD = 0.2, CE = 0.71 (fitted to to the binding energyof 16O and 22O).

• NN only underbinds and wrongly places dripline beyond 28O

• Inclusion of effective 3NF places dripline at 25O.

• Overall the odd-even staggering in the neutron rich oxygen is well reproduced.

• We find 26O to unbound with respect to 24O by ~100keV, this agrees with recent experiments on 26O by E. Lunderberg et al., Phys. Rev. Lett. 108 (2012) 142503

• We find 28O to be unbound with a resonance width of ~2MeV

CD CE

G. Hagen, M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, T. Papenbrock, accepted for Publication in PRL, arXiv:1202.2839 (2012)

Oxygen isotopes from chiral interactions

Excited states in 24O computed with EOM-CCSD and compared to experiment

Matter and charge radii for 21-24O Computed from intrinsic densities andCompared 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+

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?

Calcium isotopes from chiral interactions

Main Features:

1. Total binding energies agree well with experimental masses.

2. Masses for 40-52Ca are converged in 19 major shells.

3. 60Ca weakly bound/unbound

4. 60Ca is bound with respect to 1n emission.

5. 61-62Ca are located right at threshold.

G. Hagen, G. Jansen, M. Hjorth-Jensen, R. Machleid,T. Papenbrock, arXiv:1204.3612 (2012)

kF=0.95fm-1, cD=-0.2, cE=0.735Nmax = 18, hw = 26MeV

2+ systematics in Calcium isotopes

48Ca 52Ca 54Ca

2+ 4+ 4+/2+ 2+ 4+ 4+/2+ 2+ 4+ 4+/2+

3.58 4.20 1.17 2.19 3.95 1.80 1.89 4.46 2.36

3.83 4.50 1.17 2.56 ? ? ? ? ?CCExp

Main Features: 1. Very nice agreement

between theory and experiment.

2. Our calculations for 2+

and 4+ in 54Ca do not point to a new magic shell closure.

G. Hagen, G. Jansen, M. Hjorth-Jensen, R. Machleid, T. Papenbrock, arXiv:1204.3612 (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 53,55,61Ca

2. We find the ground state of 61Ca 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 51,52Ca A. T. Gallant et al arXiv:1204.1987v1 (2012)

Microscopic definition of Spectroscopic Factors SF is the norm of the overlap function and quantifies the degree of correlationsSFs are not observables and depend on the resolution scale

Asymptotic properties of the one-nucleon overlap functionsThe 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 functionsElastic 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 24O show strong “quenching” pointing to large deviations from a mean-field like picture.G. Hagen et al Phys. Rev. Lett. 107, 032501 (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 ~0.6-0.7 in 22,24,28O

Threshold effects and spectroscopic factors

Near the scattering threshold for one-neutrondecay the spectroscopic factors are Significantly influenced by the presence ofThe continuum. The standard shell model

approximation to spectroscopic factors completely fails in this region.

N. Michel et al Phys. Rev. C 75, 031301 (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 40Ca.

Good agreement between theory and Experiment for low-energy scattering.

Elastic proton/neutron scattering on 40Ca

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 densityB. G. Giraud, Phys. Rev. C 77, 014311 (2008)

Center of mass part Intrinsic densityLab. 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 basisGH, T. Papenbrock and D. Dean et al, Phys. Rev. Lett. 103, 062503 (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 21-24O depend weakly on the resolution scale

2. We clearly see shell structure appearing in the matter densities for 21-24O

3. Matter and charge radii depend on the resolution scale, however relative difference which is relevant for isotope shift measurements does not

23O interaction cross section (scattering off 12C target @ GSI)

Experimental radii extracted from matter distribution within Glauber model.Main result of new measurement: 23O follows systematics; interaction cross section consistent with separation energies. R. Kanungo et al Phys. Rev. C 84, 061304 (2011)

Experiment

Ozawa et al. (2001)

Kanungo et al (2011)E ≈ 900A MeV

Resolving the anomaly in the cross section of 23O

The anamoly of 23O

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

Summary 1. Interactions from Chiral EFT probed in nuclei2. 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 24O.

4. Predict weak sub-shell closure in 54Ca. 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 23O

Ab initio description of proton-halo state in 17F

• 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 with16O core [K. Bennaceur, N. Michel, F. Nowacki, J. Okolowicz, M. Ploszajczak, Phys. Lett. B 488, 75 (2000)]

Bound states and resonances in 17F

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 s1/2 d5/2 and d3/2 single-particle states

• Harmonic oscillator states for other partial waves

[G. Hagen, TP, M. Hjorth-Jensen, Phys. Rev. Lett. 104, 182501 (2010)]

• Gamow basis weakly dependent on oscillator frequency• d5/2 not bound; spin-orbit splitting too small• s1/2 proton halo state close to experiment

Variation of cutoff probes omitted short-range forces

• Proton-halo state (s1/2) only weakly sensitive to variation of cutoff• Spin-orbit splitting increases with decreasing cutoff

17F

[G. Hagen, TP, M. Hjorth-Jensen, Phys. Rev. Lett. 104, 182501 (2010)]