The Nuclear Shell Model Toward the Drip
Lines
ALFREDO POVES
Departamento de Fı́sica Teórica and IFT, UAM-CSIC
Universidad Autónoma de Madrid (Spain)
Universidad Internacional del Mar
Aguilas, July 25-28, 2011
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
OUTLINE
Spherical mean field and Correlations
The physics of very neutron rich nuclei at N=20 and N=28
A new region of deformation south of 68 Ni
Interlude; Aligned T=0 pairs in 92 Pd
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
Monopole anomalies and Multipole universality
The different facets of the nuclear dynamics depend on the
balance of the two main components of the nuclear
hamiltonian; the Monopole which produces the effective
spherical mean field and the Multipole responsible for the
correlations
Large scale shell model calculations have unveiled the
monopole anomalies of the two-body realistic interactions,
i.e that they tend to produce effective single particle
energies which are not compatible with the experimental
data and which, if used without modifications, produce
spectroscopic catastrophes
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
Monopole anomalies and Multipole universality
Already in the late 70’s Pasquini and Zuker showed that
the Kuo Brown interaction could not produce neither a
magic 48 Ca nor a magic 56 Ni. In this last case it made a
nearly perfect rotor instead. A few monopole corrections
(mainly T=1) restored high quality spectroscopy
Holt, Otsuka and Schwenk have recently shown that the
monopole component of the three body force may explain
the monopole anomalies relevant for 28 O and 48 Ca .
The Multipole Hamiltonian of the realistic two body
interactions (dominated by L=0 pairings, quadrupole and
octupole) does not seem to require any substantial
modification and is ”universal” in the sense that all the
interactions produce equivalent multipole hamiltonoans
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
The fate of magic closures
Magic numbers are associated to energy gaps in the
spherical mean field. Therefore, to promote particles
above the Fermi level costs energy.
However, some intruder configurations can overwhelm
their loss of monopole energy with their huge gain in
correlation energy.
Several examples of this phenomenon exist in stable magic
nuclei in the form of coexisting spherical, deformed and
superdeformed states in a very narrow energy range,
Nuclear Allotropy? In the case of 40 Ca they have described
in tha spherical shell model framework
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
The Monopole Hamiltonian
X
1
aij ni (nj − δij )
(1 + δij )
Hm =
X
1
+ bij
2
X
3ni
δij
Ti · Tj −
+
Aijk ni nj nk
4
ǫi n i +
The coefficients a and b are defined in terms of the centroı̈ds:
P JT
J Vijij [J]
VijT = P
J [J]
as: aij = 14 (3Vij1 + Vij0 ), bij = Vij1 − Vij0 , the sums run over Pauli
allowed values.
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
The Monopole Hamiltonian
The evolution of effective spherical single particle energies with
the number of particles in the valence space is dictated by Hm .
Schematically:
ǫj ({ni }) = ǫj ({ni = 0}) +
X
i
Alfredo Poves
aij ni +
X
Aijk ni nk
i,k
The Nuclear Shell Model Toward the Drip Lines
Valence Spaces; sd-pf
The valence space of two major shells
1f5/2
2p1/2
2p3/2
1f7/2
pf -shell
1d3/2
2s1/2
1d5/2
sd-shell
can encompass the physics of nuclei between 18 O and 64 Ge
including the ”islands of inversion” which appear at N=20
(around 31 Na) and N=28 (around 42 Si) as well as the excited
superdeformed bands of N=Z magic nuclei like 40 Ca. And,
indeed, using essentially a single effective interaction, SDPF-U.
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
N=20 far from stability
The region around 31 Na provides a beautiful example of
intruder dominance in the ground states, known
experimentally since long (Detraz, Thibault, Guillemaud,
Klotz, Walter) .
Early shell model calculations (Poves and Retamosa (87),
Warburton, Becker and Brown (90)) pointed out the role of
deformed intruder configurations 2p-2h neutron excitations
from the sd to the pf -shell and started the study of the
boundaries of the so called “island of inversion” and the
properties of its inhabitants.
Similar mechanisms produce the other known “islands of
inversion” centered in 11 Li (N=8), 42 Si (N=28), and 64 Cr
(N=40)
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
The Drift of the Single Particle Energies: N=20
ESPE (MeV)
10
0
-10
-20
d5/2
s1/2
d3/2
f7/2
p3/2
p1/2
f5/2
8
14
16
20
Proton number
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
Quadrupole Collectivity vs. Magic Closures N=20
Four protons away from doubly magic 40 Ca, 34 Si is a new
doubly magic nucleus because the proton Z=14 and the
neutron N=20 gaps reinforce each other.
To go even more neutron rich, one needs to remove
protons from the 0d5/2 orbit.
This causes two effects; a reduction of the N=20 neutron
gap and the onset of proton collectivity.
Both conspire in the sudden appearance of an Island of
Inversion in which Deformed Intruder states become
ground states, as in 32 Mg, 31 Na and 30 Ne.
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
The drift of the single particle energies, N=28
ESPE (MeV)
10
0
-10
f7/2
p3/2
p1/2
f5/2
8
14
16
20
Proton number
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
Quadrupole Collectivity vs. Magic Closures N=28
As we remove protons from doubly magic 48 Ca, the N=28
neutron gap slowly shrinks. In 46 Ar the collectivity induced
by the action of the four valence protons in the almost
degenerate quasi-spin doublet 1s1/2 -0d3/2 , is not enough
to beat the N=28 closure. 46 Ar is non-collective.
In 44 S, the quadrupole collectivity sets in. The N=28
closure blows out and prolate and non collective states
coexist. The ground state and the first excited 2+ form the
germ of a prolate rotational band. The 0+ isomer,
predicted by the shell model calculations has been recently
found at Ganil.
In turn 42 Si is an oblate well deformed, rotor with a first 2+
state at 770 keV and 40 Mg is predicted to be a very
collective prolate rotor, with a 2+ at ∼680 keV. In addition it
could well develop a neutron halo because more than two
neutrons are, in average, in p wave.
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
The Magnesium isotopes from the proton to the
neutron dripline; SDPF-U interaction
+
2 excitation energy in MeV
2
EXP
TH
1.8
1.6
1.4
1.2
1
0.8
0.6
8
10
12
14
16
18
20
22
24
26
28
30
32
N
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
The Silicon isotopes from the proton to the
neutron dripline; SDPF-U interaction
EXP
TH
3
2
1
+
2 excitation energy in MeV
4
0
8
10
12
14
16
18
20
22
24
26
28
30
32
N
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
The Magnesium isotopes; B(E2)’s SDPF-U int.
150
2
e fm
4
100
50
0
B(E2) exp
B(E2) th
30
32
34
36
A
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
The Neon isotopes; SDPF-U int.
exp
th
2
1.5
1
+
2 excitation energy in MeV
2.5
0.5
8
10
12
14
16
18
20
22
24
26
28
N
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
The island of inversion south of 68 Ni
Figure credit, Carin Cain
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
The Valence Space for 68 Ni and its neighbors
2d5/2
1g9/2
1f5/2
2p1/2
2p3/2
1f5/2
2p1/2
2p3/2
1f7/2
Neutrons
48 Ca
Protons
acts as the inert core
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
The island of inversion south of 68 Ni
The situation at N=40 is similar to the one found at N=20
except that 68 Ni is not a “bona fide” magic nucleus.
Removing protons from the 0f7/2 orbit, activates the
quadrupole collectivity, which, in turn, favors the np-nh
neutron configurations across N=40, that take advantage
of the quasi-SU3 coherence of the doublet 0g9/2 - 1d5/2 .
Large scale SM calculations in the valence space of the full
pf -shell for the protons and the 0f5/2 1p3/2 1p1/2 0g9/2 and
1d5/2 orbits for the neutrons, predict a new region of
deformation centered at 64 Cr.
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
68
Ni
shell model
exp.
8+
6+
4399
4244
4+
5–
8+
6+
4208
3999
3184
4+
3147
2779
5–
2848
0+
2450
(0 + )
2511
2+
1990
0+
2+
0+
1770
1400
52
40
0+
2034
0
68
0+
0
Ni
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
The neutron ESPES at N=40 and N=20
10
5
N=40
N=20
0
ESPE (MeV)
ESPE (MeV)
5
0
-5
-10
-15
f7/2
p3/2
f5/2
p1/2
g9/2
d5/2
-5
-10
-15
-20
20
28
32
Z
d5/2
s1/2
d3/2
f7/2
p3/2
8
14
Z
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
16
0.5
0
20 22 24 26 28
Z
Alfredo Poves
4
1
2
N=40
1.5
500
400
+
2
(a)
+
E(2 ) (MeV)
SM
2.5 EXP
300
+
3
B(E2;2 -> 0 ) (e fm )
The N=40 isotones
200
100
(b)
SM
EXP
0
20 22 24 26 28
Z
The Nuclear Shell Model Toward the Drip Lines
+
0.8
2
E(2 ) (MeV)
(a)
4
1
B(E2; J+2 -> J) (e fm )
The Iron Isotopes
0.6
Fe
0.4
0.2
SM
EXP
0
36
38
40
42
N
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800
700
600
500
400
300
200
100
0
(c)
J=0
J=2
J=4
EXP, J=0
36
38
40
N
The Nuclear Shell Model Toward the Drip Lines
42
+
0.8
2
E(2 ) (MeV)
(a)
4
1
B(E2; J+2 -> J) (e fm )
The Chromium Isotopes
Cr
0.6
0.4
0.2
SM
EXP
0
36
38
40
42
N
Alfredo Poves
800
700
600
500
400
300
200
100
0
(c)
J=0
J=2
J=4
36
38
40
N
The Nuclear Shell Model Toward the Drip Lines
42
Ni
2
1.5
+
E(2 ) (MeV)
2.5
0.5
SM
EXP
0
36
38
40
42
N
4
(b)
200
150
+
1
250
2
(a)
+
3
B(E2;2 -> 0 ) (e fm )
The Nickel Isotopes
100
50
0
SM
EXP
36
38
40
N
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines
42
The yrast bands, Iron and Chromium
2.5
2
1.5
1
0.5
3
(b)
E(J+2)/E(J)
E(J+2)/E(J)
3
SM, J=2
SM, J=4
EXP, J=2
36
38
2.5
2
1.5
1
0.5
40
42
N
Alfredo Poves
(b)
SM, J=2
SM, J=4
EXP, J=2
36
38
40
N
The Nuclear Shell Model Toward the Drip Lines
42
Conclusions
State of the art Shell Model calculations encompassing two
major oscillator shells make it possible to describe
complete series of isotopes from the proton to the neutron
drip lines
They can also cross the ”islands of inversion” at N=8, 20,
and 28
And predict new ones as that south of 68 Ni, centered in
64 Cr
Alfredo Poves
The Nuclear Shell Model Toward the Drip Lines