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 the
nuclear chart, A ∼ 2...300
Systematic ab initio
calculations only possible in
the lightest nuclei
Hard many-body problem:
approximate methods suited
for different regions
Shell Model:
Solve the problem choosing the relevant degrees of freedom
Use 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...
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• 4 He radius
Medium-mass nuclei: shell model
To keep the problem feasible, the
configuration 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
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Renormalization group (RG) and MBPT
Better
convergence of
chiral forces after
RG transformation
Two-Body Matrix Elements
Single Particle Energies
Many-body perturbation
theory to third order:
obtain effective
shell model interaction
in 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
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α
Ca isotopes: masses
Ca isotopes: explore nuclear shell evolution N = 20, 28, 32?, 34?
Ca measured from 40 Ca core
0
NN
NN+3N (emp)
NN+3N (MBPT)
3N forces repulsive contribution,
chiral NN-only forces too attractive
5
-60
Probe shell
evolution:
-90
-150
44
48
52
56
60
Mass Number A
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64
68
Jones et al.
Nature 465 454 (2010)
a
Sn
Pb
3
2
1
Mass-differences
2+
1 energies
-120
40
E2+ (MeV)
4
0
b
20
S2n (MeV)
Energy (MeV)
-30
10
0
–20
–10
0
N – Nmagic
Two-neutron separation energies
Compare S2n = −[B(N, Z ) − B(N − 2, Z )] with experiment
18
AME2003
TITAN
NN+3N (MBPT)
NN
16
Precision measurements
with TITAN changed AME 2003
∼ 1.74 MeV in 52 Ca
12
10
8
6
4
2
0
S2n(theo) − S2n(exp) (MeV)
S2n (MeV)
14
2
1
More flat behavior in 50 Ca–52 Ca
0
-1
-2
28
7 / 14
S2n in 52 Ca predicted in
disagreement with old
measurements
3N forces needed
30
29
31
32
33
34
30
31
32
33
34
Neutron Number N
35
36
Gallant et al.
PRL 109 032506 (2012)
54
Ca mass and N = 32 shell closure
Recent measurement of 53,54 Ca at ISOLDE
18
ISOLDE
NN+3N (MBPT)
16
S2n evolution:
52
Ca–54 Ca decrease
similar to 48 Ca–50 Ca
unambiguously establishes
N = 32 shell closure
12
10
6
4
2
0
S2n(theo) − S2n(exp) (MeV)
S2n (MeV)
14
8
2
1
0
-1
-2
28
7 / 14
Excellent agreement with
theoretical prediction
30
29
31
32
30
31
32
33
34
Neutron Number N
33
34
35
36
Two-neutron separation energies
Compare to other theoretical calculations
18
ISOLDE
NN+3N (MBPT)
CC (Hagen et al.)
KB3G
GXPF1A
16
12
10
8
6
4
2
0
S2n(theo) − S2n(exp) (MeV)
S2n (MeV)
14
2
0
-1
-2
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Coupled-Cluster calculations
good agreement
phenomenological 3N forces
Hagen et al. PRL109 032502 (2012)
1
28
Phenomenology
good agreement
masses/gaps as input
30
29
31
32
30
31
32
33
34
Neutron Number N
33
34
35
36
Shell closures and 2+
1 energies
Correct closure at N = 28
when 3N forces are included
+
energies characterize
shell closures
2 1 Energy (MeV)
7
2+
1
6
5
4
3
2
1
Holt et al. JPG39 085111(2012)
Holt, JM, Schwenk,
JPG40 075105 (2013)
0
NN
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 closure
E(2+
1 )=2.04 MeV Steppenbeck et al. Nature 502 207(2013)
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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 )
+
+
2
2+
1
2
-
1/2
1
1
-
-
3/2
0
NN+3N
pfg9/2
3/2
Expt.
-
-
3/2
GXPF1A
3/2
KB3G
+
+
0
0
NN+3N
pfg9/2
0
Expt.
+
+
0
GXPF1A
0
KB3G
Good agreement with experiment when available,
comparable to phenomenological interactions
Predictions in very neutron-rich nuclei, test in upcoming experiments
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Electromagnetic transitions
100
Similar quality as
phenomenological
interactions
2
1
0.1
Experiment
NN
NN + 3N (emp)
NN + 3N (MBPT)
GXPF1
4
3
7
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8
46
Ca +
+
4 →2
46
Ca +
+
6 →4
47
Ca 3/2 →7/2
48
Ca +
+
2 →0
49
Ca 7/2 →3/2
B(M1) strength in 48 Ca
NN+3N good agreement
experiment strength and energy
1
6
46
Ca +
+
2 →0
2
0
KB3G
GXPF1A
NN + 3N (MBPT)
Raman et al. (2001)
Montanari et al. (2012)
pfg9/2-shell
5
B(M1) [µN ]
10
2
4
B(E2) [e fm ]
B(E2)s in reasonable
agreement with experiment
span three orders of
magnitude
9
10
Excitation Energy (MeV)
11
12
50
Ca +
+
2 →0
Towards theoretical uncertainties
Estimate theoretical uncertainties
allows meaningful comparison to experiment
and 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 results
with respect to cutoff of RG evolution
of unevolved chiral Hamiltonian
Impose correct nuclear matter saturation
Consider different unevolved chiral
Hamiltonians (outlook)
• Theoretical uncertainties
associated to the many-body approach
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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 through
RG transformation
to improve the
convergence
in many-body
calculation
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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
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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
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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
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Collaborators
K. Hebeler, J. D. Holt,
A. Schwenk, J. Simonis
ISOLTRAP Collaboration
TITAN Collaboration
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