Exotic calcium isotopes and three-nucleon forces
Javier Menéndez
Institut für Kernphysik (TU Darmstadt) and ExtreMe Matter Institute (EMMI)
15th Int. Symposium on Capture Gamma-Ray Spectroscopy and Related Topics
Dresden, 28 August 2014
Outline
Theoretical approach
Shell evolution in neutron-rich calcium
Detailed spectroscopy:
excitation spectra, electromagnetic moments and transitions
Summary
2 / 21
Outline
Theoretical approach
Shell evolution in neutron-rich calcium
Detailed spectroscopy:
excitation spectra, electromagnetic moments and transitions
Summary
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
3 / 21
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...
4 / 21
• 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
5 / 21
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
6 / 21
α
Normal-ordered 3N Forces
Treatment of 3N forces:
normal-ordered 2B: 2 valence, 1 core particle
⇒ Two-body Matrix Elements
N
N
N
N
N
N
N
N
N
N
N N
N
N
N N
π
π
π
N
N
normal-ordered 1B: 1 valence, 2 core particles
⇒ Single particle energies
N
N
N
N
N
N
N N
N
N
N N
π
π
π
N
O core
N
N
7 / 21
N
N
N
Outline
Theoretical approach
Shell evolution in neutron-rich calcium
Detailed spectroscopy:
excitation spectra, electromagnetic moments and transitions
Summary
Oxygen dripline anomaly and 3N forces
ty
ili e
ab lin
Z
st
Si 2007
Al 2007
Mg2007
Na 2002
Ne 2002
F 1999
8
Single-Particle Energy (MeV)
O isotopes: ’anomaly’ in the dripline at 24 O, doubly magic nucleus
Chiral NN+3N forces provided repulsion needed to predict dripline
4
-4
d 5/2
-8
B 1984
unstable fluorine isotopes
Be 1973
Li 1966
2
unstable oxygen isotopes
He 1961
neutron halo nuclei
H 1934
2
8
20
28
Based on many-body perturbation theory
N
Single-Particle Energy (MeV)
unstable isotopes
4
14 16
Neutron Number (N)
20
(c) G-matrix NN + 3N (∆) forces
0
d 3/2
-4
s 1/2
d 5/2
-8
NN + 3N (∆)
NN
Otsuka et al. PRL105 032501 (2010)
8
8 / 21
s 1/2
G-matrix
Vlow k
8
stable isotopes
C 1986
d 3/2
0
O 1970
N 1985
(a) Forces derived from NN theory
14 16
Neutron Number (N)
20
Neu
Me
-100
-125
-150
Ab-initio oxygen dripline
. -175
-50
æ
à
æ
à
-75
E [ MeV]
Oxygen
dripline
benchmarked
CSM
ground-state
energies
of the even
with few(a)ab-initio
approaches:
3N-induced
and NN+3N-full
Hamilfm . Solid lines indicate the energy
dotted
lines guide
the
No-coredata,
shell
model
(truncated)
ainties due to the importance truncation
In-medium
used
to represent SRG
the data. All energies
Coupled-cluster
(a) NN+3N-induced
ô
-100
-125
-150
. -175
ó
ô
à
æ
ó
MR-IM-SRG(2)
IT-NCSM
CCSD
Λ-CCSD(T)
à
ô
æ
ó
à
æ
à
æ
(b) NN+3N-full
the
cut is naturally compat-
9 / 21
max
12
14
18
à
ô
æ
ó
à
æ
20 22 24 26
A
Hergert
et al.ground-state
PRL110 242501
4. (Color online)
Oxygen
energies(2013)
for the NN+3Nour MR-IM-SRG(2) ground-state FIG.
induced (top) and NN+3N-full (bottom) Hamiltonian with 3N
em to results from the IT-NCSM,
MeVused,
. MR-IM-SRG(2),
CCSD, and -CCSD(T) results are
Sensitivity
the chiral
to be explored
CSM results
withintoquantified
un-interaction
obtained at optimal , using 15 major oscillator shells and max
et al.
rtance Simonis
truncation
[26,in preparation
33]. In the
. The IT-NCSM energies are extrapolated to infinite model space.
use the full 3N interaction without Experimental values are indicated by black bars [28, 29].
Self-consistent Green’s function
MeV observed in Refs. [23, 26,
16
à
ô
æ
ó
Ca isotopes: masses
Ca isotopes: explore nuclear shell evolution N = 20, 28, 32?, 34?
0
NN
NN+3N (emp)
NN+3N (MBPT)
-60
3N forces repulsive contribution,
chiral NN-only forces too attractive
5
-90
-120
Mass-differences
2+
1 energies
-150
40
44
48
52
56
60
Mass Number A
10 / 21
4
E2+ (MeV)
Probe shell
evolution:
64
68
Jones et al.
Nature 465 454 (2010)
a
Sn
Pb
3
2
1
0
b
20
S2n (MeV)
Energy (MeV)
-30
Ca measured from 40 Ca core
in pfg9/2 valence space
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
11 / 21
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
11 / 21
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
11 / 21
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)
12 / 21
Outline
Theoretical approach
Shell evolution in neutron-rich calcium
Detailed spectroscopy:
excitation spectra, electromagnetic moments and transitions
Summary
Excitation spectra
Spectra for neutron-rich calcium isotopes
Ca
Energy (MeV)
4
3/21/29/2-
5/2
-
3
-
(5/2 )
(7/2 )
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
+
+
0
0
KB3G
NN+3N
pfg9/2
0
Expt.
Good agreement with experiment when available,
comparable to standard phenomenological interactions
Holt, JM, Simonis, Schwenk PRC90 024312 (2014)
13 / 21
+
+
0
GXPF1A
0
KB3G
Excitation spectra
Spectra for neutron-rich calcium isotopes
-
7/25/27/23/25/27/2
Energy (MeV)
5
3/29/21/23/25/2-
4
7/2
3
-
-
3/2
5/2
-
(3/2 )
-
0
3/2
+
1+
2+
3
+
4
+
2
+
0
3
2
+
2
4
+
0+
2+
1
+
3
+
2
+
4
+
+
2
(5/2 )
+
-
1
+
2
+
3
+
1
+
4
+
4
-
-
Ca
+
4
1/23/23/29/27/2-
5/2
3/2
2
56
5
Ca
Energy (MeV)
53
6
5/2
-
4
+
2
1
+
2
5/2
-
-
1/2
0
NN+3N
pfg9/2
(1/2 )
Expt.
-
-
1/2
GXPF1A
1/2
+
+
0
0
KB3G
NN+3N
pfg9/2
(0 )
Expt.
0
GXPF1A
+
+
0
KB3G
Predictions in very neutron-rich nuclei, test in upcoming experiments
Holt, JM, Simonis, Schwenk PRC90 024312 (2014)
14 / 21
Carbon electromagnetic transitions
Electromagnetic transitions
can be calculated in p-shell
No need of effective charges
Still require consistent evolution
of transition operator
Second 2+
2 state decay sensitive to 3N forces
Decay not seen experimentally: 3N forces favored
→
+
2+
2 → 21
+
2+
2 → 01
Experiment
CDB2k
NN
NN + NNN
WBP
WBT
WBT*
>91.2%
<8.8%
32.4%
67.6%
21.6%
78.4%
97.6%
2.4%
92.2%
7.8%
93.2%
6.8%
97.6%
2.4%
Petri et al. PRC86 044329 (2012)
15 / 21
Oxygen electromagnetic transitions
B(E2)
20
O
20
O
O
22
0.04
O
USDB
NN
6
B(E2) (e2fm4)
B(M1)
22
0.03
NN+3N
4
0.02
2
0.01
0
0
21-->0 22-->21 22-->0
16 / 21
21-->0 22-->21 22-->0
Electromagnetic
transitions
(decay lifetimes)
22-->2122-->21
B(M1) (µN2)
8
B(E2) transitions
in 2+
2 states
20
O and 22 O
also sensitive
to 3N forces
Calcium B(E2) transition strengths
B(E2)s reasonable
agreement with experiment,
spread two over orders of
magnitude
Similar quality as
phenomenological
interactions
10
2
4
B(E2) [e fm ]
100
1
0.1
46
KB3G
GXPF1A
NN + 3N (MBPT)
Raman et al. (2001)
Montanari et al. (2012)
46
Ca +
+
2 →0
17 / 21
46
Ca +
+
4 →2
46
Ca +
+
6 →4
Ca: sd degrees of freedom?
Phenomenological
effective charges qn = 0.5e
47
Ca 3/2 →7/2
48
Ca +
+
2 →0
49
Ca 7/2 →3/2
50
Ca +
+
2 →0
Holt, JM, Simonis, Schwenk
PRC90 024312 (2014)
Calcium quadrupole moments
Electric quadrupole moments in ground states of calcium isotopes
very recently measured by COLLAPS at ISOLDE
Good agreement to experiment, up to neutron-rich systems
0.1
Q(eb)
0.05
Consistent description of
ground-state masses and
spectroscopy
NN+3N
GXPF1A
KB3G
Literature
ISOLDE
Comparable to
phenomenological
interactions
0
Phenomenological effective
charges qn = 0.5e
-0.05
-0.1
21
23
25
27
N
18 / 21
29
31
Garcia Ruiz et al.,
to be submitted
B(M1) transition in 48 Ca
B(M1) strength in 48 Ca compared to experiment
6
NN+3N calculation good
agreement with experiment
5
Experimental concentration
of the strength very sensitive
to effective interaction
2
B(M1) [µN ]
4
Experiment
NN + 3N (MBPT)
KB3G
GXPF1A
3
2
NN+3N interaction
associated to mildly
quenched g-factors
1
0
6
7
8
9
10
11
Excitation Energy (MeV)
Holt, JM, Simonis, Schwenk PRC90 024312 (2014)
19 / 21
12
Calcium magnetic moments
Predictions of magnetic moments in good agreement
with very recent measurements by COLLAPS at ISOLDE
Even improve agreement
of phenomenological
interactions
-1
µ(µN )
-1.2
Missing sd degrees of
freedom in lighter isotopes,
(sensitive to unpaired
particles)
-1.4
-1.6
ISOLDE
Literature
KB3G
GXPF1A
NN+3N
-1.8
-2
20 / 21
Bare g-factors!
Garcia Ruiz et al.,
21
23
25
N
27
29
31
to be submitted
Outline
Theoretical approach
Shell evolution in neutron-rich calcium
Detailed spectroscopy:
excitation spectra, electromagnetic moments and transitions
Summary
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 (ISOLDE)
• Shell structure: prominent closure established at N = 32
• Predicted 54 Ca 2+
1 in good agreement with measurement at RIKEN
• Excitation spectra reasonable agreement to experiment,
prediction of excited states in neutron-rich isotopes
• B(E2) and B(M1) transitions well described
• Quadrupole and Magnetic moments in neutron-rich isotopes
very good agreement to recent ISOLDE measurements
21 / 21
Collaborators
K. Hebeler, A. Schwenk, J. Simonis
J. D. Holt
TITAN Collaboration
ISOLTRAP Collaboration
COLLAPS Collaboration
21 / 21