Quels degrés de liberté pour quels phénom`enes? Part II. La

Quels degrés de liberté pour quels phénomènes?
Part II.
La coexistence de formes par les méthodes
au delà du champ moyen
Michael Bender
Institut de Physique Nucléaire de Lyon, CNRS/IN2P3, Université de Lyon, Université Lyon 1
69622 Villeurbanne, France
Journées SFP–BTPN sur
les grandes questions en physique nucléaire fondamentale
Campus Michel-Ange Paris-16
21-22 Juin 2016
M. Bender, IPN Lyon
La coexistence de formes
Which are the irreducible ingredients of a (minimal) predictive model of
shape coexistence and its experimental signatures? ?
I
What is there to be modeled?
I
I
I
I
I
I
Distinguish
I
I
I
I
sequence of levels and their excitation energies
E 0 transition matrix elements
E 2 transition matrix elements
(charge) radii (and isotopic shifts)
masses (and mass differences)
deformation softness (states spread over a wide
range of deformations)
shape coexistence (distinguishable states that might
be directly mixed)
shape entanglement (distinguishable states that can
only be mixed via third states) Poves, JPG 43 (2016) 020410.
role of np-n hole excitations involving intruder /
extruder states?
M. Bender, IPN Lyon
La coexistence de formes
Which are the irreducible ingredients of a (minimal) predictive model of
shape coexistence and its experimental signatures? ?
Heyde et al, PRC44 (1991) 2216
I
What is there to be modeled?
I
I
I
I
I
I
Heyde & Woods, RMP 83 (2011) 1467
sequence of levels and their excitation energies
E 0 transition matrix elements
E 2 transition matrix elements
(charge) radii (and isotopic shifts)
masses (and mass differences)
Distinguish
I
I
I
deformation softness (states spread over a wide
range of deformations)
shape coexistence (distinguishable states that might
be directly mixed)
shape entanglement (distinguishable states that can
only be mixed via third states) Poves, JPG 43 (2016) 020410.
I
role of np-n hole excitations involving intruder /
extruder states?
I
Early ad-hoc model of shape coexistence: estimate
excitation energy of 0+ states from the difference in
(spherical) single-particle energies, the change in
pairing energy, a monopole correction and the
quadrupole correlation energy.
M. Bender, IPN Lyon
La coexistence de formes
State-of-the-art modeling of shape coexistence
I
Shell model:
−
+
+
−
I
Poves, JPG 43 (2016) 020410.
shape remains implicit
good quantum numbers
band mixing
intruder states require two major shells
Interacting boson model:
Nomura et al JPG 43 (2016) 020408.
− shape remains implicit
+ good quantum numbers
+ band mixing
I
(Self-consistent) mean field:
+
−
−
−
I
energy surfaces with multiple minima
no quantum numbers, nor slection rules
non-orthogonal states
no mixing of bands
”beyond mean field” by projected GCM:
+ projection → quantum numbers & selection rules
+ Generator Coordinate Method → band mixing
− computationally intensive
I
”beyond mean field” with collective Hamiltonians
+ quantum numbers & selection rules
+ band mixing
M. Bender, IPN Lyon
La coexistence de formes
State-of-the-art modeling of shape coexistence
J. Phys. G: Nucl. Part. Phys. 43 (2016) 024010
Poves, JPG 43 (2016) 020410.
I
Shell model:
−
+
+
−
I
Poves, JPG 43 (2016) 020410.
shape remains implicit
good quantum numbers
band mixing
intruder states require two major shells
Interacting boson model:
Nomura et al JPG 43 (2016) 020408.
− shape remains implicit
+ good quantum numbers
+ band mixing
I
(Self-consistent) mean field:
+
−
−
−
I
”beyond mean field” with collective Hamiltonians
+
+
Figure 3. Energies (in MeV)
of the lowest states
at fixed np–nh
Shell-model
analysis
of in0+Calevels
black dots, energy of the most bound Slater determinant in each np–n
40 energy of the lowest 0 in the full diagonalization in each n
squares,
in
Ca
Energies of the three lowest 0 states in the full diagonalization in the
40
+
+
(blue lozenges) (from [14]).
black: lowest Slater determinant in
given np-nh subspace
the monopole hamiltonian play a very important role. In other words, the e
particle gap is configuration dependent. Next we diagonalize the full hamilton
the np–nh subspaces to get the red squares in figure 3. And now, the power of th
appears in full glory. Obviously the 0p–0h configuration, a single Slater determ
projection → quantum numbers & selection rules
gain any correlation energy in this step. The coherent 2p–2h bandhead gains m
Generator Coordinate Method → band mixingthe 4p–4h, 6p6h and over all the 8p–8h. The definition of the correlation energy
with our choice the 4p–4h bandhead gains about 10 MeV and the 8p–8h close t
computationally intensive
this moment of the process, the 0p–0h. 4p–4h and 8p–8h bandheads are near
The intrinsic quadrupole moments of their yrast bands are consistent with a nor
band (4p–4h) and superdeformed bands (6p–6h and 8p–8h). Finally, the fu
quantum numbers & selection rules
mixes all the configurations with the results gathered in figure 3 as blue lozen
the physical states to be compared with the experiment, and indeed the comp
band mixing
standing. It is seen that the closed shell gains about 5 MeV by pairing mixin
states. The 0p–0h content of the ground state is about 70%, hence we can spea
magic 40Ca,deorformes
perhaps of a doubly magic ground state in 40Ca? The second 0+
M. Bender, IPN Lyon
La coexistence
”beyond mean field” by projected GCM:
+
+
−
I
energy surfaces with multiple minima
no quantum numbers, nor slection rules
non-orthogonal states
no mixing of bands
red: lowest mixed state in given
np-nh subspace
blue: full shell model calculation
Which model ingredients are really relevant?
HAPE COEXISTENCE IN NEUTRON-DEFICIENT Kr M.
...
M. B.et al, PRC74 (2006) 024312.
Girod et al. / Physics Letters
B 676 REVIEW
(2009) 39–43
PHYSICAL
C 74, 024312 (2006)
Girod et al, PLB676 (2009) 39.
Clément et al, PRC75 (2007) 054313.
41
.
.
.
.
.
...
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FIG. 16. Comparison between the theoretical and experimental level schemes for the oblate and prolate bands in
.
76
Kr (top) an
FIG. 1. Nilsson plot of neutron single-particle energies with
ositive (solid lines) and negative (dotted lines) parity as a function (bottom). The excitation energies of the states are drawn to scale and the widths and labels of the arrows represent the calculated and me
74 1. Potential energy surfaces for 72 Kr, 74 Kr, and 76 Kr.
Kr.
the intrinsic mass deformation
B(E2) values, respectively.
I β2(i) , as obtained forFig.
.
.
.
Which are the irreducible ingredients of a (minimal) predictive microscopic
model
shape
coexistence
and
experimental
hese modifications have a strong
effect on theof
deformed
gaps,
calculation.
The very large
matrixits
element
between the prolate signatures?
states, the negative sign of the diagonal matrix elem
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
hich may correspond to quite different deformations and +
76
2 and the excited 0+
Kr corresponds to oblate shape, which would expla
ary in size. The spin-orbit splittings
2 state is well reproduced by the Gogny
Ifor the f and p levels in 1
e three potentials are very similar to ours. The differences calculation, while it is strongly underestimated by the Skyrme
predominant decay to the 0+
2 state. The positive sign fou
74
etween the single-particle spectraImust thus be related to calculation. It should be noted that the B(E2) values of Fig. 16
the 2+
Kr corresponds to prolate shape und
2 state in
e relative position of states with different orbital angular
are always given in the direction of the transition and may
assumption of K = 2. This difference is not reproduc
omentum within a given shell.
the Gogny calculation, where the 2+
Figure 2 shows the mean-field and J = 0 projected en- therefore differ by a factor of 2Ii + 1 in some cases where the
2 states can be interp
gy curves for 72–78 Kr obtainedI
with the SLy6 Skyrme ordering is reversed.
as gamma bands in the usual sense built on the g
+
arametrization. Throughout this paper, the projected energy
A
grouping
of
the
non-yrast
states
above
the
0
state
into
state.
2
I
plotted as a function of the intrinsic
β2(i) value of the
FIG. 2. Mean-field
(dotted)
and J = 0 projected
is not
straightforward.
Thedeformation
excitation energies
A detailed comparison of the spectroscopic quadr
ean-field states from which they are obtained. In our opinion, band structures
–78 Kr (see text).
+ (solid)
+ +for 72+
+
energy curves
I
moments (see Tables VI, XI, and Fig. 15) is som
is quantity provides the most convenient and intuitive label of the 22 , 31 , 42 , 51 , and 62 states in both isotopes are
at can be defined for all states, irrespective
of the level of consistent with an interpretation of these sequences as gammahampered
by the rather large uncertainties of the experim
I
odeling. However, it should not be misinterpreted as an
oblate and large prolate deformations. For 76 Kr, the spherical
vibrational
bands. The Gogny-based calculation reproduces
values. The spectroscopic quadrupole moments for the 4
bservable. With our method, one calculates transition and
mean-field minimum is related to the large spherical N =
sequences
very
well the
andshallow
confirms
predominant
K=2
6+ states, in particular the very large experimental valu
pectroscopic multipole moments in the laboratory frame, these
40 subshell
closure,
whereas
oblatea and
prolate
I
+
76
hich can be directly compared to experimental data. How- character
structuresof
correspond
to two
N = 40the
gaps23in states
the
the states.
In deformed
this scenario
in both
Kr, should be taken with care since their non-yrast pa
2
Fig. 3.minimum
Electric
matrix
) for
the chain
light
Kraccount,
isotopesboth the Gogn
ver, the spectroscopic moments that characterize a state do isotopes
Nilsson diagram.
prolate
hasmonopole
moved to
smaller
can be The
interpreted
as rotational
states
with elements
oblate ρare(E0
not
known.
Takingofthis
into
74
ot provide useful coordinatesFor
to plot
potential
energy
curvesan effective
deformationfield
than intheory
Kr,
with
deformed
N rotating
= 38
gap being
recent
work
toward
of the
collectively
and/or
vibrating
deformedGCM
nuclei
see Papenbrock
etexperial,
+
in
comparison
with
results
from
the
axial
Skyrme
calculation
[17]inand
built on thethan
excited
excitation
the Skyrme calculations
are
reasonable
agreement wi
ecause they scale with angular momentum and are even character
at larger deformations
the N =0240states.
gap. TheThe
spherical
N = energies
mental
values
[8,9,32].
and
transition
strengths
found
in
the
Gogny
calculation
are
experiment.
entically zero for J = 0. ForNPA852
projected states
with
J
>
0
and
40
subshell
closure
is
strong
enough
to
stabilize
the
spherical
(2011) 36; Zhang et al, PRC87 (2013) 034323;
Coello-Pérez et al,PRC92 (2015) 014323
78
rge intrinsic deformation in nuclei with A larger than 100, in shape
up to the with
N = 42this
isotope
Kr.
agreement
interpretation,
as is the sign of the
It can be concluded that configuration-mixing calcul
(i)
(s)
As can be seen from the solid lines in Fig. 2, the energy +
2 is very close to the intrinsic deformation β2 determined experimental diagonal matrix elements for the 2
are a valid approach
to describe the shape-coexistence
3 states.
landscapes are qualitatively
modified
by the projection
om the laboratory-frame quadrupole moment Q(s) through
forstrong
all
isotopes
under on
study,
has a maximum
for 74 Kr. This supports
finds mean-field
mixing
of K a=J 0=
and K = 2
nomenon. The experimental set of matrix elements in 74 K
J =calculation
0. Since a spherical
state is already
e static rotor model, Eq. (3). Note that any coordinate might The
740
Bender,
IPN
Lyon
La
coexistence
de formes
76
calculated excitation spectrum of low-lying states
forthe72M.
Kr.for
interpretation
of
Kr showing
the
configuration
state,
energy
gain
by
on angular
momentum
components
the the
2+projecting
and
2+ itstates
in both
isotopes,
which
Kr strongest
represents a stringent
test of the mixtheoretical models.
.
.
quantum mechanics
shell structure and distinguishable configurations (that have different shape
or that can be associated with different shapes)
different mean fields (RPA-type methods fail for shape coexistence)
collectivity
configuration mixing (orthogonality, band mixing, . . . )
good quantum numbers (for selection rules of transitions).
.
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.
...
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.
Is there an effective field theory of shape coexistence?
Shell-model interpretation of beyond-mean-field states and vice versa
M. B., B. Bally, P.-H. Heenen, unpublished
M. Bender, IPN Lyon
La coexistence de formes
Shell-model interpretation of beyond-mean-field states and vice versa
M. B., B. Bally, P.-H. Heenen, unpublished
M. Bender, IPN Lyon
La coexistence de formes
Shell-model interpretation of beyond-mean-field states and vice versa
M. B., B. Bally, P.-H. Heenen, unpublished
M. Bender, IPN Lyon
La coexistence de formes
Shell-model interpretation of beyond-mean-field states and vice versa
collective wave function of the four lowest 0+ states
M. B., B. Bally, P.-H. Heenen, unpublished
M. Bender, IPN Lyon
La coexistence de formes
Shell-model interpretation of beyond-mean-field states and vice versa
collective wave function of the four lowest 0+ states
M. B., B. Bally, P.-H. Heenen, unpublished
M. Bender, IPN Lyon
La coexistence de formes
The European Physical Journal Special Topics
Neff, EPJST156 (2008) 69
I
6
4
C
6
4
-6
–6
-6 –4
–6
-4
I
6
z [fm]
0.001
0.5
z [fm]
0.1
0.0
clustering.
-4
–4
octupole?
-6
–6
I hexadecapole?
-2I 0tetrahedral
–2
2 4 6 or octahedral
-6 –4
–6
-4
-2 0
–2
shapes?
01 I
I
C
0
-2
–2
-4
–4
-6
–6
0.5
0.1
0.01
0.001
2 4 6
-6 -4
–6
–4 –2
-2 0 2
y [fm]
y [fm]
PHYSICAL REVIEW LETTERS
y [fm]
VOLUME 88, NUMBER 25
12
0.0101
0.0 0.1
2
1
0.
1
0.0
-4
–4
Are there coexistences 0driven by other
shape degrees of freedom?
-2
–2
1
0.0
-2
–2
12
2
I
0.001
0
0.1
2
All examples
shown so far concern the
12
6
C
coexistence
of
shapes with
different
0 00
0.0.1
0.01
1
quadrupole moment. 4
0.5
z [fm]
Exotic coexistences
Are they also driven by np-nh
12
12
6 else?
C something
C
excitations
the realistic
calculation
results of Fig. 1 where,
0closely
.001 by or
4
6
24 JUNE 2002
Dudek et al, PRL88 (2002) 252502
12
6
C
01
80
0.0 1
z [fm]
0. 5 . 1
0
Total Energy [MeV]
0.01
z [fm]
1.
0
0.01
0.5
0.001
0.5
z [fm]
1
00
0. 1
0.00.1
0.1
6
40 Zr 40
0 0.1
in addition, the extremes on the
4
4 horizontal axes have been
4
0.
108
0. in such a way that, by comparing the labels on the
40 Zr 68
chosen
01
160
2
2
2
left- and on the right-hand side of the figure, one can read
70 Yb 90
242
how quickly the parity mixing
4
0
0 sets in when the deforma0
100 Fm 142
tion increases (the curves are symmetric with respect to
-2
–2
-2j ! 0.15 is0so
–2
-2
–2
0.
0). The parity mixing at ja32
.1 strong that the
1
0.0
typical calculated parity expectation
values are "!60.5#.
1
0.00
-4
–4
-4
–4
-4
–4
2
1
0.0
Several observations deserve emphasizing. First of all,
01
-6
–6
-6 in the deformation depen–6
-6
–6
there is a qualitative difference
dence in the two studied cases as predicted by the consid-6 –4
–6
-4 –2
-2 0erations
2 4based
6 on the point-group
-6 symmetries
–6
-4 –2
–4
-2 0 presented
2 4 6
-6 -4
–6
–4 -2
–2 0 2 4 6
0
y [fm]
y [fm] can aly [fm]
above: At ja31 j . 0.15 the level distribution
0.0
0.1
0.2
0.3
0.4
ready be considered “nearly uniform” except for a relaElongation β2
12
56 describe
that decreases slowly
with row from the left: intrinsic state obtained
tively states
small gapused
at Z !to
nsities of intrinsic
C. First
increasing deformation. In contrast, the spectrum in func- +FIG. 2. Results of the multidimensional minimization
− of the
n (V), by variation
projection
onIPN
positive
(Vcoexistence
), nuclear
and
on negative
parity
(V
). Second
total
energies
projected on the
quadrupole
deformation
Bender,
Lyon
La
de formes
tion of aafter
increasing
gaps
at Z ! 32,
32 reveals strongly M.
subshells, and the mixing of the resulting proton and neutron
configurations are the essential ingredients to a unified view
of coexistence in nuclei. Figure 8 shows the regions of shape
coexistence that are discussed in this review and their location
with respect to magic numbers.
We present the experimental data that motivate this unified
view in a particular order. We first review mass regions for
which extensive data support the widespread and unequivocal
manifestation of shape coexistence, i.e., the regions centered
Coexistence in normal nuclei, exotic nuclei, and elsewhere
Heyde & Woods, RMP 83 (2011) 1467
FIG. 8 (color online). The main regions of nuclear shape coexistence discussed in Sec. III are shown in relationship to closed
shells. Regions A, F: see Sec. III.B.1; regions B, C, D, and E: see
Sec. III.B.2; region G: see Sec. III.A.8; region H: see Sec. III.A.5;
region I: see Sec. III.A.3; region J: see Sec. III.A.2; region K: see
Sec. III.A.4; and region L: see Sec. III.A.1.
M. Bender, IPN Lyon
La coexistence de formes
subshells, and the mixing of the resulting proton and neutron
configurations are the essential ingredients to a unified view
of coexistence in nuclei. Figure 8 shows the regions of shape
coexistence that are discussed in this review and their location
with respect to magic numbers.
We present the experimental data that motivate this unified
view in a particular order. We first review mass regions for
which extensive data support the widespread and unequivocal
manifestation of shape coexistence, i.e., the regions centered
Coexistence in normal nuclei, exotic nuclei, and elsewhere
Heenen, M. B., Bally & Ryssens, to be published
Heyde & Woods, RMP 83 (2011) 1467
FIG. 8 (color online). The main regions of nuclear shape coexistence discussed in Sec. III are shown in relationship to closed
shells. Regions A, F: see Sec. III.B.1; regions B, C, D, and E: see
Sec. III.B.2; region G: see Sec. III.A.8; region H: see Sec. III.A.5;
region I: see Sec. III.A.3; region J: see Sec. III.A.2; region K: see
Sec. III.A.4; and region L: see Sec. III.A.1.
M. Bender, IPN Lyon
La coexistence de formes
Summary
I
Profiting from high-performance computing, over the last few years the
range of applicability of the shell model and of beyond-mean-field methods
has been enlarged such that both methods begin to cover the physics
relevant for shape coexistence (intruder states, good quantum numbers,
configuration mixing, . . . ).
I
Shape coexistence emerges in both methods in similar situations: np-nh
excitations involving intruder states.
I
In the context of the shell these states are usually interpreted ”vertically”
in terms of occupations of spherical shells (”islands of inversion”).
I
In the context of self-consistent mean-field models ”and beyond” these
states are usually interpreted ”horizontally” in terms of gaps in the Nilsson
diagram.
M. Bender, IPN Lyon
La coexistence de formes
Summary
I
Profiting from high-performance computing, over the last few years the
range of applicability of the shell model and of beyond-mean-field methods
has been enlarged such that both methods begin to cover the physics
relevant for shape coexistence (intruder states, good quantum numbers,
configuration mixing, . . . ).
I
Shape coexistence emerges in both methods in similar situations: np-nh
excitations involving intruder states.
I
In the context of the shell these states are usually interpreted ”vertically”
in terms of occupations of spherical shells (”islands of inversion”).
I
In the context of self-consistent mean-field models ”and beyond” these
states are usually interpreted ”horizontally” in terms of gaps in the Nilsson
diagram.
The difference in interpretation appears to be more ”cultural” than physical.
M. Bender, IPN Lyon
La coexistence de formes
Bottom line
What is Shape Coexistence?
”Shape coexistence is a very peculiar nuclear phenomenon consisting in the
presence in the same nuclei, at low excitation energy, and within a very narrow
energy range, of two or more states (or bands of states) which: (a) have well
defined and distinct properties, and, (b) which can be interpreted in terms of
different intrinsic shapes.”
A. Poves, foreword to the 2015 special issue of JPG on ”Shape coexistence in nuclei”
I
Shape coexistence is a generic feature of atomic nuclei that in one way or
the other is exhibited by the majority of nuclei. It can come in many
flavours:
I
I
I
I
I
I
coexisting structures in regions of transitional nuclei (evolution with shapes
with filling of shells)
island(s) of inversion
rotational bands of ”spherical nuclei” including doubly-magic ones (16 O,
40 Ca, 56 Ni, . . . )
fission isomers / superdeformation / hyperdeformation
clustering
Shape coexistence imprints its presence on (the systematics of) virtually
all spectroscopic properties of nuclei at low excitation energy.
M. Bender, IPN Lyon
La coexistence de formes