Shell evolution in neutron rich nuclei
Gustav R. Jansen1,2
[email protected]
1 University
2 Oak
of Tennessee, Knoxville
Ridge National Laboratory
March 18. 2013
Collaborators and acknowledgements
• Andreas Ekström (UiO, MSU)
• Christian Forrsen (Chalmers)
• Gaute Hagen (ORNL)
• Morten Hjorth-Jensen (UiO, MSU)
• Gustav R. Jansen (UTK, ORNL)
• Ruprecht Machleidt (UI)
• Hai Ah Nam (ORNL)
• Witold Nazarewicz (UTK, ORNL)
• Thomas Papenbrock (UTK, ORNL)
Outline
• Motivation.
• Shell evolution in oxygen isotopes.
• Dripline
• Magic nuclei.
• 26 O vs. 26 F.
• Shell evolution in calcium isotopes.
• The single particle picture.
• Magic numbers and shell closures.
• Consistent nuclear forces
Understanding matter
The nuclear shell model
• Maria Goeppert Mayer, Phys. Rev.
75 (1949).
• Otto Haxel, J. Hans D. Jensen,
and Hans E. Suess, Phys. Rev. 75
(1949).
• Nobel prize in physics in 1963:
Eugene Wigner, Maria Goeppert
Mayer, J. Hans D. Jensen.
• Defined by shell closures at magic
numbers, where a spin-orbit force
added to a harmonic oscillator
potential reproduce the magic
numbers.
The nuclear manybody problem
Need to solve the Schrödinger equation
Ĥ|Ψi = T̂ + V̂1 + V̂2 + V̂3 . . . |Ψi = E|Ψi
Two ingredients
1. The nuclear interaction.
2. A method to solve the many body problem.
Nuclear interactions
• Want an interaction between nucleons,
with the pion as force carrier.
• Not the fundamental degrees of freedom.
• Hadrons are color neutral, in much the
same way as atoms are charge neutral.
• The interaction between hadrons are
residual interactions, much like the van
der Waals interactions between molecules.
Complicated interaction with manybody components
Chiral effective field theory
D. R. Entem and R. Machleidt, Phys. Rev. C 68, 041001 (2003)
• Direct link to QCD.
• Perturbative
expansion in
momentum.
• Chiral symmetry is
spontaneously and
explicitly broken.
• The hierarchy of
nuclear forces unfolds
automatically.
Density dependent chiral threebody force
Finite basis expansion
• The wavefunction is
expanded in Slater
determinants
|Ψi =
D
X
ci |Φi i.
i
• The number of possible
n
Slater determinants is A ,
where n is the number of
single particle states and A
is the number of nucleons.
Curse of dimensionality
4
1052
10
Slater determinant basis
Be
16
O
40
Ca
48
1044
10
He
10
40
1036
1032
1028
1024
1020
1016
1012
108
104
100
0
50
100
Number of single particle states
150
200
The coupled-cluster method
Excited states using EOM-CC
Eigenvalues of H̄ = e −T̂ Ĥe T̂ − hΦ0 |Ĥ|Φ0 i
H̄R̂
c
= ω R̂
Properties of H̄.
• Non-symmetric (non-hermetian) operator.
• For CCSD and a twobody hamiltonian - six-body operator.
• The matrix representation is very sparse.
• Generally too large to store and diagonalize exactly.
Efficient implementation of H̄R̂
is key.
C
Open Quantum Systems
The oxygen dripline
Oxygen isotopes from chiral interaction
Evolution of single particle energies
S. M. Lenzi, F. Nowacki, A. Poves and K. Sieja, PRC 82 054301 (2010)
Main features
• Shell-model calculation in the sd-shell including 0f7/2 and 1p3/2 for neutrons.
• Inversion of the 0f7/2 and the 1p3/2 single particle states in 28 O.
Negative parity states in O25
Partial summary
•
28 O
is not a closed shell nucleus.
• Contiuum effects crucial for level ordering.
• New shell consisting of 0d3/2 , 1p3/2 and 1p1/2 ?
• Next doubly magic oxygen nuclei would be
34 O?
• Coupled cluster calculation in this region will be very difficult,
since we have no good closed shell reference.
The effects of a single proton
• Adding a proton changes
the dripline by 6 neutrons.
• Not all fluorine isotopes
towards the dripline are
bound
• Interplay between
proton-neutron
interactions,
neutron-neutron
interactions, threebody
interactions and
continuum degrees of
freedom.
26
F – probing the proton-neutron interaction
• Simple structure –
24
O plus πd5/2 and νd3/2 .
• First approximation – J π = 1+ − 4+ .
• Weakly bound – Sn ≈ 0.8 MeV.
Threebody forces in
26
F
A. Lepailleur et al., Phys. Rev. Lett. 110, 082502 (2013)
Technical details
• Chiral interaction at
N3 LO.
• Identical threebody force
as established in the
oxygen chain.
• 17 major harmonic
oscillator shells with a
Gamow-Hartree-Fock
basis for νs1/2 and νd3/2
• CCSD with triples
corrections (Λ-CCSD(T))
for 24 O, with
2PA-EOMCC.
• 26 Ffree = B(25 O) +
B(25 F) − B(24 O)
Threebody forces are crucial for correct levelspacing.
Resonances in neutron-rich
24
O
Oxygen isotopes from chiral interaction
Evolution of single particle energies
Technical details
• J. Meng, H. Toki,
J. Y. Zeng, S. Q. Zhang
and S. -G. Zhou, PRC 65
041302(R) (2002).
• Relativistic mean-field
including continuum
effects.
Main features
• Bunching of
single-particle energies
outside the pf -shell.
• No shell-gap in 60 Ca 70
Ca.
• Large deformations and
no shell-closure.
• Continuum effects
responsible for bound
60
Ca - 72 Ca.
Evolution of single particle energies
S. M. Lenzi, F. Nowacki, A. Poves and K. Sieja, PRC 82 054301 (2010)
Main features
• Shell-model calculation in the pf -shell including 0g9/2 and 2d5/2 for neutrons.
• Inversion of the 0g9/2 and the 2d5/2 single particle states in 60 Ca.
• Bunching of levels including the 0f5/2 state indicates no shell-closure.
Binding energies in calcium isotopes
G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock,
PRL109 032502 (2012)
Technical details
-320
• Chiral interaction at
N3 LO.
-340
-360
NN + 3NFeff
Experiment
-380
E (MeV)
• Density dependent three
body force with
kF = 0.95fm−1 ,
cD = −0.2 and
cE = 0.735. Nmax = 18
and ~ω = 26 MeV.
-400
• Mass of 51 Ca and 52 Ca
from A. T. Gallant et al.,
PRL 109, 032506 (2012)
-420
-440
-460
Main features
-480
• Total binding energies
agree well with
experimental masses.
-500
-520 39 40 41 42
47 48 49 50 51 52 53 54 55 56
A
59 60 61 62
• 60 Ca is not magic.
• Three nucleon force is
repulsive.
Binding energies in calcium isotopes
G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock,
PRL109 032502 (2012)
Technical details
-320
• Chiral interaction at
N3 LO.
-340
-360
NN + 3NFeff
Experiment
NN only
-380
E (MeV)
• Density dependent three
body force with
kF = 0.95fm−1 ,
cD = −0.2 and
cE = 0.735. Nmax = 18
and ~ω = 26 MeV.
-400
• Mass of 51 Ca and 52 Ca
from A. T. Gallant et al.,
PRL 109, 032506 (2012)
-420
-440
-460
-480
Main features
NN only
• Total binding energies
agree well with
experimental masses.
-500
-520 39 40 41 42
47 48 49 50 51 52 53 54 55 56
A
59 60 61 62
• 60 Ca is not magic.
• Three nucleon force is
repulsive.
Shell evolution in neutron rich calcium isotopes.
Details
• J. D. Holt, T. Otsuka,
A. Schwenk and
T. Suzuki, J Phys G39
085111 (2012)..
• J π = 2+ systematics in
even calcium isotopes.
Main features
• Threebody forces needed
to make 48 Ca magic.
• Different models have
54
Ca magic, semi magic
and not magic at all.
J π = 2+ systematics in even calcium isotopes
G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock,
PRL109 032502 (2012)
5
NN+3NFeff
Exp
E2+ (MeV)
4
Technical details
• Chiral interaction at
N3 LO.
3
• Density dependent three
body force with
kF = 0.95fm−1 ,
cD = −0.2 and
cE = 0.735. Nmax = 18
and ~ω = 26 MeV.
2
1
0
42
48
50
A
Ca
52
54
56
Main features
• Good agreement between
theory and experiment.
• Shell closure in 48 Ca.
• Sub-shell closure in 52 Ca.
• Predict weak sub-shell
closure in 54 Ca.
J π = 2+ systematics in even calcium isotopes
G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock,
PRL109 032502 (2012)
5
NN+3NFeff
Exp
NN only
E2+ (MeV)
4
Technical details
• Chiral interaction at
N3 LO.
3
• Density dependent three
body force with
kF = 0.95fm−1 ,
cD = −0.2 and
cE = 0.735. Nmax = 18
and ~ω = 26 MeV.
2
1
0
NN only
42
48
50
A
Ca
52
54
56
Main features
• Good agreement between
theory and experiment.
• Shell closure in 48 Ca.
• Sub-shell closure in 52 Ca.
• Predict weak sub-shell
closure in 54 Ca.
Spectra in calcium isotopes
G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock,
PRL109 032502 (2012)
4
7
4
+
Technical details
+
9/2
+
3
6
Energy (MeV)
+
52
Ca
+
4
5
5/2
54
+
-
4
3
1
2
2
55
+
7/2
1+
2+
3+
4
+
1+
2+
3+
4
Ca
53
• Chiral interaction at
N3 LO.
Ca
+
5/2
56
Ca
• Density dependent three
body force with
kF = 0.95fm−1 ,
cD = −0.2 and
cE = 0.735. Nmax = 18
and ~ω = 26 MeV.
Ca
+
2
+
+
-
3/2
+
3
+
2
-
5/2
• Continuum included for
selected weakly bound
and resonant states.
+
4
+
2
1
0
0
+
0
-
+
-
1/2
1/2
0
+
0
+
-
-
5/2
5/2
0
+
0
+
Exp
NN+3NFeff
Exp
NN+3NFeff
Exp
NN+3NFeff
Exp
NN+3NFeff
Exp
NN+3NFeff
Main features
• Inversion of g9/2 and
d5/2 .
• 1/2+ groundstate in
61
Ca.
• Continuum effects are
crucial.
Consistent nuclear forces
• Want to derive
consistent forces.
• All contributions at a
given order are
evaluated.
• Currently NNLO.
• Apply numerical
optimization
algorithms to find the
optimal parameters.
NNLO (POUNDerS)
POUNDerS
• Practical Optimization Using No Derivatives for sums of
Squares
• Part of TAO
(http://www.mcs.anl.gov/research/projects/tao/)
Finding a better solution
• POUNDerS is allowed to explore the NNLO parameter space
to find the best fit to phaseshifts.
• Compute χ2 /datum against available scattering data, to
evaluate the quality of the interaction.
Quality of the fit
Preliminary
Tlab (MeV)
0–35
35–125
pp χ2 /datum
1.11
1.56
np χ2 /datum
0.85
1.17
125–183
23.95
4.35a
1.87
183–290
aC
pp
N3 LO
-7.8188
NNLO-POUNDerS
-7.8174
C
rpp
2.795
2.755
aN
pp
N
rpp
aN
nn
N
rnn
anp
rnp
-17.083
2.876
-17.825
2.817
-18.900
2.838
-23.732
2.725
2.224575
1.975
0.275
4.51
-18.889
2.797
-23.749
2.684
2.224582
1.967
0.272
4.05
BD (MeV)
rD (fm)
QD (fm2 )
PD
29.26
6.09
0–290
17.10
14.03 a
2.95
Empirical
-7.8196(26)
-7.8149(29)
2.790(14)
2.769(14)
-18.95(40)
2.75(11)
-23.740(20)
2.77(5)
2.224575(9)
1.97535(85)
0.2859(3)
Triton binding energy
Preliminary
-6
POUNDerS-NNLO (NN+NNN)
POUNDerS-NNLO (NN)
Experiment
N3LO (NN)
ENCSM (MeV)
-6.5
-7
-7.5
-8
-8.5
-9
0
10
20
Nmax
30
40
4
He binding energy
Preliminary
-20
-22
ENCSM (MeV)
-24
-26
-28
-30
POUNDerS-NNLO(NN)
POUNDerS-NNLO(NN+NNN)
N3LO(NN)
Experiment
-32
-34
0
5
10
Nmax
15
20
Oxygen isotopes with NNLO (POUNDerS)
Preliminary
−100
0
-10
E (MeV)
−120
−130
−140
E (MeV)
−110
-20
-30
-40
-50
-60
-70
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
A
−150
−160
−170
Experiment
NNLO (POUNDerS)
3
N LO
15 16 17 18 19 20 21 22 23 24 25 26 27 28
A
Oxygen spectra with NNLO (POUNDerS)
Preliminary
+
18
7
Energy (MeV)
+
2+
3
8
O
4
0
3
+
+
22
3+
0+
2
+
+
3
+
2
+
0+
+
4
2+
3+
4
2
+
4
3
4
2
2
2
1
0
+
0
0
+
1
+
+
+
3/2
1
+
+
2
5/2
+
+
+
+
5/2
3/2
+
24
+
+
+
+
+
+
3
?
+
2
3/2
+
2
2
0
+
+
+
4
1
2
+
2
2
3
+
+
O
O
3
4
23
+
4
+
2
6
5
2
+
3
O
+
0
+
5/2
+
0
+
0
+
+
1/2
+
+
1/2
0
1/2
+
0
+
0
+
Exp
3
NNLO
N LO
Exp
3
NNLO
N LO
Exp
NNLO
3
N LO
Exp
NNLO
3
N LO
J π = 2+ systematics in oxygen isotopes with NNLO
(POUNDerS)
Preliminary
5
3
N LO 3nfeff
Exp
3
N LO
NNLO (POUNDerS)
E2+ (MeV)
4
3
2
1
0
18
22
A
O
24
26
J π = 2+ systematics with NNLO (POUNDerS)
Preliminary
5
NN+3NFeff
Exp
NN only
NNLO (POUNDerS)
E2+ (MeV)
4
3
✹
2
1
N2LO (POUNDerS)
0
42
48
50
A
Ca
52
54
56
Calcium isotopes with NNLO (POUNDerS)
Preliminary
-320
-340
-360
NN + 3NFeff
Experiment
NN only
E (MeV)
-380
-400
-420
-440
-460
-480
NN only
-500
-520 39 40 41 42
47 48 49 50 51 52 53 54 55 56
A
59 60 61 62
Calcium isotopes with NNLO (POUNDerS)
Preliminary
-320
-340
-360
NNLO (POUNDerS)
NN + 3NFeff
Experiment
N3LO NN only
E (MeV)
-380
-400
-420
-440
-460
-480
NNLO (POUNDerS)
-500
-520 39 40 41 42
47 48 49 50 51 52 53 54 55 56
A
59 60 61 62
Calcium isotopes with NNLO (POUNDerS)
Preliminary
-320
-340
-360
NNLO (Shifted)
NN + 3NFeff
Experiment
N3LO NN only
E (MeV)
-380
-400
-420
-440
-460
-480
-500
-520 39 40 41 42
47 48 49 50 51 52 53 54 55 56
A
59 60 61 62
Calcium isotopes with NNLO (POUNDerS)
Preliminary
-320
-340
-360
NNLO (Shifted)
NN + 3NFeff
Experiment
N3LO NN only
E (MeV)
-380
-400
-420
-440
-460
-480
-500
-520 39 40 41 42
47 48 49 50 51 52 53 54 55 56
A
59 60 61 62
Questions?
Gustav R. Jansen
[email protected]
This work was partly supported by the Office of Nuclear Physics, U.S. Department of Energy (Oak Ridge National
Laboratory), under Contracts No. DE-FG02-96ER40963 (University of Tennessee) and No.DE-SC0008499
(NUCLEI SciDAC-3 Collaboration).
An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and
Experiment (INCITE) program. This research used resources of the Oak Ridge Leadership Computing Facility
located in the Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of
Energy under Contract DE-AC05-00OR22725 and used computational resources of the National Center for
Computational Sciences, the National Institute for Computational Sciences, and the Notur project in Norway.