Structure of neutron rich calcium isotopes from coupled cluster theory

Gaute Hagen (ORNL)

Collaborators: Andreas Ekström (MSU)Christian Forrsen (Chalmers)Morten Hjorth-Jensen (UiO/CMA)Gustav Jansen (UT/ORNL)Ruprecht Machleidt (UI)Witold Nazarewicz (UT/ORNL)Thomas Papenbrock (UT/ORNL)Jason Sarich (ANL)Stefan Wild (ANL)

CREX WorkshopJefferson Laboratory, March 18, 2013

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

Chiral interactions from Practical Optimization Using No Derivatives (for Squares)

NNLO(POUNDerS)NNLO(EGM 450/500)Nijmegen PWA

cD = -0.2, cE=-0.36

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.

Light nuclei from NN2LO-POUNDerS

• Rapid Convergence for ground states of oxygen isotopes with NNLO-POUNDerS.

• Already with N =12-14 major harmonic oscillator shells results are well converged.

• NNLO-POUNDerS is a “soft” potential • No dramatic overbinding is found for

light nuclei

Oxygen isotopes from chiral NN forces

16O 22O 24O

NNLO(POUNDerS) -130.28 -159.76 -168.45

NNLO(EGM 450/500) -156.76 -208.85 -225.65

Experiment -127.62 -162.06 -168.48

Shell model calculations of oxygen from chiral NN forces

• Shell model calculations done in

s-d model space• Effective interaction

from g-matrix and third order perturbation theory.

• Folded diagrams to infinite order

• hw = 14MeV, N = 12 for intermediate excitations.

1. NNLO(POUNDerS) gives remarkable agreement with experiment, and the dripline in oxygen is correctly placed at 24O.

2. Two-body forces alone get the structure to leading order right!

Evolution of shell structure in neutron rich Calcium

• How do shell closures and magic numbers evolve towards the dripline?

• Is the naïve shell model picture valid at the neutron dripline?

• 3NFs are responsible for shell closure in 48Ca

• Different models give conflicting result for shell closure in 54Ca.

J. D. Holt et al, J. Phys. G 39, 085111 (2012)

Evolution of shell structure in neutron rich CalciumInversion of shell order in 60Ca

S. M. Lenzi, F. Nowacki, A. Poves, and K. Sieja Phys. Rev. C 82, 054301 (2010)

• Inversion of d5/2 and g9/2 in 60Ca.

• Bunching of levels pointing to no shell-closure.

Evolution of shell structure in neutron rich Calcium

• Relativistic mean-field show no shell gap in 60-

70Ca• Bunching of single-

particle orbitals• large deformations

and no shell closure

J. Meng et al, Phys. Rev. C 65, 041302(R) (2002)

How many protons and neutrons can be bound in a nucleus?

Skyrme-DFT: 6,900±500systLiterature: 5,000-12,000

288~3,000

Erler et al., Nature 486, 509 (2012)

Description of observables and model-based extrapolation• Systematic errors (due to incorrect assumptions/poor modeling)• Statistical errors (optimization and numerical errors)

Including the effects of 3NFs (approximation!)[J.W. Holt, Kaiser, Weise, PRC 79, 054331 (2009); Hebeler & Schwenk, PRC 82, 014314 (2010)]

Parameters: For calcium we use kF = 0.95 fm-1, cE = 0.735, cD = -0.2 from binding energy of 40Ca and 48Ca (The parameters cD, cE differ from values proposed for light nuclei)

3NFs as in-medium effective two-nucleon forcesIntegration of Fermi sea of symmetric nuclear matter: kF

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 is not magic 4. 61-62Ca are located

right at threshold.

See also: Meng et al PRC 65, 041302 (2002), Lenzi et al PRC 82, 054301 (2010) and Erler et al, Nature 486, 509 (2012)

G. Hagen, M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, T. Papenbrock, Phys. Rev. Lett. 109, 032502 (2012).

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

A peninsula of weak stability?

Is 54Ca a magic nucleus? (Is N=34 a magic number?)

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. Good agreement between

theory and experiment. 2. Shell closure in 48Ca due to

effects of 3NFs 3. Predict weak (sub-)shell

closure in 54Ca.

G. Hagen, M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, T. Papenbrock, Phys. Rev. Lett. 109, 032502 (2012).

Spectra and shell evolution in Calcium isotopes

1. 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.

New penning trap measurement of masses of 51,52Ca A. T. Gallant et al Phys. Rev. Lett. 109, 032506 (2012)

Continuum coupling is crucial!

Calcium isotopes from NNLO-POUNDerS

1MeV per nucleon overall shift. Basic features such as separtation eneriges and spectra well reproduced at the NN level.

Calcium isotopes from NNLO-POUNDerS

Neutron matter from NNLO-POUNDerS

• Neutron matter from the NNLO-POUNDerS (red line) and the N3LO NN potentials (blue dashed line).

• Coupled-cluster calculations (particle-particle ladders and exact Pauli operator)

• The NNLO-POUNDerS results are agreement with results from I. Tews, T. Krüger, K. Hebeler, and A. Schwenk PRL110, 032504 (2013).

• Modelspace: N =14 major harmonic oscillator for the NN interaction while E3max= 12 for three-nucleon force.

• Binding energies of Calcium isotopes are not converged => need E3max~ 20.

Point radii from NNLO-POUNDerS16O 22O 24O 48Ca

NN-only 2.29 2.55 2.69 2.97

NN+3NF 2.57 2.85 3.01 3.49

Exp 2.54(2) 2.75(15) 3.19(13)

Neutron skin (NN+3NF) = 0.16fmNeutron skin (NN-only) = 0.17fmPreliminary

Summary/perspectives 1. Effects of 3NF and continuum give good agreement

with experiment for binding energy and spectra in oxygen and calcium

2. Predict weak sub-shell closure in 54Ca. 3. Level ordering in the gds shell in neutron calcium is

reversed compared to naïve shell model4. A proper optimized NNLO chiral interaction gets

leading order structure right from light nuclei to neutron matter

1. Compute covariance matrix of optimized LECs and study correlations between LECs

2. Quantify how uncertainties propagate from optimized interaction to observables in medium mass nuclei

3. Compute radii and neutron skin in calcium-48 with NN+3NFs and quantified uncertainties