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 Workshop Jefferson Laboratory, March 18, 2013
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Structure of neutron rich calcium isotopes from coupled cluster theory
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Structure of neutron rich calcium isotopes from coupled cluster theory
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).