Shell evolution, spectroscopy, theoretical uncertainties
in neutron-rich calcium isotopes
studied with chiral three-body forces

Javier Menéndez
Institut für Kernphysik (TU Darmstadt) and ExtreMe Matter Institute (EMMI)
Advances is Radioactive Isotope Science, ARIS 2014
Tokyo, 6 June 2014

Nuclear landscape
Big variety of nuclei in thenuclear chart, A ∼ 2...300
Systematic ab initio
calculations only possible inthe lightest nuclei
Hard many-body problem:
approximate methods suitedfor different regions
Shell Model:
Solve the problem choosing the relevant degrees of freedomUse realistic nucleon-nucleon (NN) and three-nucleon (3N) interactions
2 / 14

Nuclear forces in chiral EFT
Chiral EFT: low energy approach to QCD for nuclear structure energies
Short-range couplings are fitted to experiment once
Systematic expansion of nuclear forces
pion exchanges
contact terms
NN fitted to:• NN scattering• π-N scattering
3N fitted to:• 3 H Binding Energy
Weinberg, van Kolck, Kaplan, Savage, Weise, Meißner, Epelbaum...
3 / 14• 4 He radius

Medium-mass nuclei: shell modelTo keep the problem feasible, theconfiguration space is separated into
• Outer orbits:
orbits that are always empty• Valence space: the space in which
we explicitly solve the problem• Inner core:
orbits that are always filled
Solve in valence space: H |Ψi = E |Ψi → Heff |Ψieff = E |Ψieff
Heff is obtained in many-body perturbation theory (MBPT)
includes the effect of inner core and outer orbits
4 / 14

Renormalization group (RG) and MBPT
Betterconvergence ofchiral forces after
RG transformation
Two-Body Matrix Elements

Single Particle Energies

Many-body perturbation
theory to third order:
obtain effective
shell model interactionin the valence space
Solve many-body problem with shell model code ANTOINE
Diagonalize up to 1010 Slater determinants Caurier et al. RMP 77 (2005)
X+ ++|φα i = ai1
ai2 ...aiA|0i|Ψieff =cα |φα i
5 / 14αCa isotopes: masses
Ca isotopes: explore nuclear shell evolution N = 20, 28, 32?, 34?
Ca measured from 40 Ca core
0NN
NN+3N (emp)
NN+3N (MBPT)

3N forces repulsive contribution,chiral NN-only forces too attractive
5-60

Probe shellevolution:-90-1504448525660

Mass Number A
6 / 146468

Jones et al.
Nature 465 454 (2010)

aSnPb

321

Mass-differences
2+1 energies-12040E2+ (MeV)

40b20
S2n (MeV)

Energy (MeV)

-30

100
–20–100
N – NmagicTwo-neutron separation energies
Compare S2n = −[B(N, Z ) − B(N − 2, Z )] with experiment
18AME2003
TITAN
NN+3N (MBPT)
NN

16Precision measurementswith TITAN changed AME 2003
∼ 1.74 MeV in 52 Ca

121086420S2n(theo) − S2n(exp) (MeV)

S2n (MeV)

1421More flat behavior in 50 Ca–52 Ca

0-1-228

7 / 14S2n in 52 Ca predicted indisagreement with oldmeasurements

3N forces needed
302931323334303132
3334
Neutron Number N

3536

Gallant et al.
PRL 109 032506 (2012)

54Ca mass and N = 32 shell closure

Recent measurement of 53,54 Ca at ISOLDE
18ISOLDE
NN+3N (MBPT)

16

S2n evolution:
52
Ca–54 Ca decreasesimilar to 48 Ca–50 Ca
unambiguously establishes
N = 32 shell closure

12106420S2n(theo) − S2n(exp) (MeV)

S2n (MeV)

148210-1-2

28

7 / 14

Excellent agreement withtheoretical prediction

3029313230

31323334
Neutron Number N

33343536

Two-neutron separation energies
Compare to other theoretical calculations

18ISOLDE
NN+3N (MBPT)CC (Hagen et al.)KB3G
GXPF1A

161210864
20

S2n(theo) − S2n(exp) (MeV)

S2n (MeV)

1420-1-2

7 / 14

Coupled-Cluster calculations
good agreement
phenomenological 3N forces
Hagen et al. PRL109 032502 (2012)128

Phenomenology
good agreement
masses/gaps as input302931323031323334
Neutron Number N

33343536

Shell closures and 2+1 energies

Correct closure at N = 28
when 3N forces are included+shell closures

2 1 Energy (MeV)

72+1 energies characterize

654
3
2
1

Holt et al. JPG39 085111(2012)
Holt, JM, Schwenk,
JPG40 075105 (2013)

0NN
NN+3N [emp]NN+3N [MBPT]

42 44 46 48 50 52 54 56 58 60 62 64 66 68

Mass Number A• 3N forces enhance closure at N = 32
• 3N forces reduce strong closure at N = 34
Expt: suggest N = 34 shell closureE(2+1 )=2.04 MeV Steppenbeck et al. Nature 502 207(2013)8 / 14

Excitation spectra
Spectra for neutron-rich calcium isotopes
Ca

Energy (MeV)

4-

3/21/29/27/2

5/2

-

3-

(5/2 )(7/2 )---(5/2 )-

2-

7/2(3/2 )

-

5/2
5/2

-

5/23/2

-

5/2
3/2

-

5/2

(1/2 )

-

1/2

Ca+-

3/2

-

3/29/27/21/23/2

52

6-

7/29/2-(9/2 )-

1/2

Energy (MeV)

51

5+4+2+4+
3+2

5

2+1

4

4+2+0+
1+3+

2++0+1+2+4+3+0+

1

3+2+(2 )

++22+
12-

1/2

11--

3/2

0NN+3N
pfg9/2

3/2

Expt.

--

3/2

GXPF1A

3/2

KB3G++00

NN+3N
pfg9/20Expt.

++0GXPF1A

0KB3GGood agreement with experiment when available,comparable to phenomenological interactions
Predictions in very neutron-rich nuclei, test in upcoming experiments
9 / 14

Electromagnetic transitions
100

Similar quality as
phenomenologicalinteractions

210.1

ExperimentNN
NN + 3N (emp)
NN + 3N (MBPT)GXPF143710 / 14846

Ca ++4 →246

Ca ++6 →447

Ca 3/2 →7/2

48

Ca ++2 →049

Ca 7/2 →3/2B(M1) strength in 48 Ca
NN+3N good agreement
experiment strength and energy1646

Ca ++2 →020KB3GGXPF1A
NN + 3N (MBPT)
Raman et al. (2001)
Montanari et al. (2012)

pfg9/2-shell

5B(M1) [µN ]1024B(E2) [e fm ]B(E2)s in reasonableagreement with experiment
span three orders ofmagnitude

910Excitation Energy (MeV)111250Ca ++2 →0

Towards theoretical uncertainties
Estimate theoretical uncertainties
allows meaningful comparison to experimentand better predictions of properties of non-accessible isotopes• Theoretical uncertainties
associated to nuclear force:−1

Λ = 1.8 fm−1
Λ = 2.0 fm−1
Λ = 2.2 fm−1
Λ = 2.8 fm

LO (500 MeV)

Explore sensitivity of resultswith respect to cutoff of RG evolutionof unevolved chiral Hamiltonian
Impose correct nuclear matter saturation
Consider different unevolved chiralHamiltonians (outlook)• Theoretical uncertainties
associated to the many-body approach
11 / 14

2.0 < Λ3NF < 2.5 fm−1

3rd order pp+hh
1.6 0.8

1.0

1.2

1.4

1.6

−1

kF [fm ]Hebeler et al. PRC 83 031301 (2011)Chiral EFT and RG
Original Hamiltonian
includes:Chiral NN force
up to N3 LO
Chiral 3N force
up to N2 LO

Evolve throughRG transformation
to improve theconvergencein many-body
calculation
12 / 14

Theoretical uncertainties in sd nuclei
Sensitivity to resolution-scale dependence of RG-evolved nuclear forces

Experimental trends very well reproduced in S2n ’s and S2p ’s
Uncertainties in S2n ’s ∼ 1 − 3 MeV, in spectra much smaller ∼ 500 keV
13 / 14

Theoretical uncertainties in sd nuclei
Sensitivity to resolution-scale dependence of RG-evolved nuclear forces

Experimental trends very well reproduced in S2n ’s and S2p ’s
Uncertainties in S2n ’s ∼ 1 − 3 MeV, in spectra much smaller ∼ 500 keV
13 / 14

Summary
Shell Model calculation based on chiral effective field theory
including NN+3N forces and many-body perturbation theory• Predicted neutron rich Ca S2n ’s with NN+3N forces agree with recent
measurements of 51,52 Ca (TRIUMF) and 53,54 Ca (ISOLTRAP)• Shell structure: prominent closure established at N = 32
• Predicted 54 Ca 2+
1 in good agreement with measurement at RIBF
• Shell structure: suggested shell closure at N = 34
to be complemented with mass measurements at 55 Ca and 56 Ca• Excitation spectra, B(E2) and B(M1) transitions• Towards theoretical uncertainty quantification

14 / 14

Collaborators
K. Hebeler, J. D. Holt,A. Schwenk, J. Simonis

ISOLTRAP Collaboration

TITAN Collaboration

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