Experimental measurement of the deformation through
the electromagnetic probe
Shape coexistence in exotic Kr isotopes.
E. Clément CNRS/GANIL
Kazimierz 2010
Shape coexistence in exotic Kr
Shape coexistence in the proper sense only if
(i) The energies of the states are similar, but separated by a barrier, so that
mixing between the different components of the wave functions is weak and
the states retain their character.
Single particule level scheme
(MeV)
(ii) The shapes involved are clearly distinguishable
74Kr
Shape coexistence in n-deficient Kr : an experimentalist view
What can we measure experimentally ?
Establish the shape isomer : 0+2
Collectivity in such nuclei : level scheme and B(E2) .
Shape (oblate - prolate ?) : Q0
Wave function mixing ? : r²(E0)
Shape isomer : systematic of 0+2 states
prolate
oblate
6+
6+
791
4+
2+
1233
611
0+
2+
710
0+
0+
918
4+
558
0+
456
858
824
4+
612
Transition strenght :
r²(E0).10-3
2+
2+
768
671
6+
6+
52
72Kr
74Kr
72(6)
84(18)
2+
508
0+
0+
346
770
424
76Kr
79(11)
664
562
2+
1017
455
0+
78Kr
47(13)
E. Bouchez et al.
Phys. Rev. Lett., 90 (2003)
•
0+
4+
Shape inversion
Maximum mixing of wave function in 74Kr
Collectivity measurement : the B(E2)
Recoil Distance Doppler Shift
Measure the B(E2) through the lifetime of the state ( ≈ ps ! )
Target and stopper at a distance d
During the desexcitation
of the nuclei, g are emitted :
 In flight  Shifted by the Doppler effect
 Stopped  E0
Collectivity measurement : the B(E2)
• The collectivity of the shape-coexisting
states are highly pertubated by the mixing
Weak mixing ≈ quantum rotor
74Kr
GSB
Strong mixing  perturbation of the collectivity
Collectivity measurement : safe coulomb excitation
1er order:
Reorientation effect
2nd order:
2+
2+
a(1)
a(1) a(2) a(2)
0+
0+
d

__
d
= __ Ruth Pif
dif d
a
(2)


a
I f M ( E 2) I j
(2)
 I f M ( E 2) I f
I j M ( E 2) I i
j
a
(1 )
 I f M ( E 2) I i
 B(E2)
 Q0
I f M ( E 2) I i
Static quadrupole moment sensitivity
minimisation du 2 :
74Kr


  0 . 70  0 .30


  1 . 02  0 .21
21 M ( E 2 ) 21
 0 . 33

I g ( 41 )
41 M ( E 2 ) 41

1
Ig (2 )
 0 . 59
Negative matrix element
(positive quadrupole moment Q0)
 prolate deformation
74Kr


2
Ig (2 )

I g ( 21 )

 0 . 28
2 2 M ( E 2 ) 2 2   0 . 33  0 .23
Positive matrix element
(Negative quadrupole moment Q0)
 oblate Deformation
Radioactive beams experiment at GANIL
•
•
•
78Kr
The 74,76Kr RIB are produced by fragmentation of
a 78Kr beam on a thick carbon target.
Radioactive nuclei are extracted and ionized
Post-accelaration of the RIB
1 MeV/u
1 10 MeV/u
4.7 MeV/u
1.5104 pps
1
3
70 MeV/u
1012 pps
2
74Kr
6104 pps
Safe Coulomb excitation
g detection
Pb
Particle detection
E. Clément et al. PRC 75, 054313 (2007)
Very well known technique for stable nuclei but for radioactive one …
The differential Coulomb excitation cross section is sensitive to
transitionnal and diagonal E2 matrix elements
 GOSIA code
Safe Coulomb excitation results
74Kr
 13 E2 transitional matrix elements
76Kr
 16 E2 transitional matrix elements
In 74Kr and 76Kr, a prolate ground state coexists with an
: describe the coupling between states
oblateTransition
excited probability
configuration
 5 E2 diagonal matrix element
 5 E2 diagonal matrix element
Spectroscopic quadrupole moment : intrinsic properties of the nucleus
E. Bouchez PhD 2003
E. Clément PhD 2006
E. Clément et al. PRC 75, 054313 (2007)
Configurations mixing
Shape coexistence in a two-state mixing model
Pure states
Perturbed states
Extract mixing and shape parameters from set of experimental matrix elements.
Configurations mixing
Shape coexistence in a two-state mixing model
Pure states
Perturbed states
Extract mixing and shape parameters from set of experimental matrix elements.
• Energy perturbation of 0+2 states
76Kr
74Kr
72Kr
E. Bouchez et al. Phys. Rev. Lett 90 (2003)
cos2θ0
0.73(1)
0.48(1)
0.10(1)
0.69(4)
0.48(2)
*
• Full set of matrix elements :
E. Clément et al. Phys. Rev. C 75, 054313 (2007)
oExcited Vampir approach:
A. Petrovici et al., Nucl. Phys. A 665, 333 (00)
*
0.6
0.5
Model describes mixing of 0+ states well, but ambiguities remain for higher-lying states.
Two-band mixing of prolate and oblate configurations is too simple.
Vampir calculations
A. Petrovici et al., Nucl. Phys. A 665, 333 (00)
Beyond …
Several theoretical approaches, such as shell-model methods, selfconsistent triaxial mean-field models or beyond-mean-field models predict
shape coexistence at low excitation energy in the light krypton isotopes.
The transition from a prolate ground-state shape in 76Kr and 74Kr to oblate
in 72Kr has only been reproduced in the so-called excited VAMPIR approach,
This approach has only limited predictive power since the shell-model
interaction is locally derived for a given mass region.
On the other hand, no self-consistent mean-field (and beyond) calculation
has reproduced this feature of the light krypton isotopes so far.
Shape coexistence in mean-field models
• In-band reduced transition probability and spectroscopic quadrupole moments
GCM-HFB (Gogny-D1S)
E. Clément et al., PRC 75, 054313 (2007)
M. Girod et al. Physics Letters B 676 (2009) 39–43
GCM-HFB (SLy6) M. Bender,
P. Bonche et P.H. Heenen,
Phys. Rev. C 74, 024312 (2006)
Shape coexistence in mean-field models (2) Skyrme
HFB+GCM method
Skyrme SLy6 force
density dependent pairing
interaction
g
Restricted to axial
symmetry : no K=2
states
 Inversion of oblate and
prolate states
 Collectivity of the prolate
rotational band is correctly
reproduced
B(E2) values e2fm4
 Interband B(E2) are under estimated
E. Clément et al., PRC 75, 054313 (2007)
Shape coexistence in mean-field models (3) Gogny
HFB+GCM with Gaussian
overlap approximation
Gogny D1S force
Axial and triaxial degrees of freedom
E. Clément et al., PRC 75, 054313 (2007)
Shape coexistence in mean-field models (3) Gogny
The agreement is remarkable for excitation energy and matrix elements
 K=0 prolate rotational ground state band
 K=2 gamma vibrational band
 Strong mixing of K=0 and K=2 components for
2+3 and 2+2 states
 Grouping the non-yrast states above 0+2 state
in band structures is not straightforward
 2+3 oblate rotational state
E. Clément et al., PRC 75, 054313 (2007)
g
g
Shape coexistence in mean-field models (3) Gogny
M. Girod et al. Physics Letters B 676 (2009) 39–43
Potential energy surface using the Gogny GCM+GOA appraoch
Shape coexistence in mean-field models (3) Gogny
M. Girod et al. Physics Letters B 676 (2009) 39–43
Is the triaxiality the key ?
Difference #1: effective interaction
very similar single-particle energies
→ no big differences on the meanfield level
axial quadrupole deformation q0
(exact GCM formalism)
• Good agreement for in-band B(E2)
• Wrong ordering of states: oblate
shape from76Kr to72Kr
• K=2 outside model space
M. Bender and P. –H. Heenen
Phys. Rev. C 78, 024309 (2008)
↔
triaxial quadrupole deformation q0, q2
Euler angles Ω=(θ1,θ2,θ3)
→ 5-dimensional collective Hamiltonian
(Gaussian overlap approximation)
• Excellent agreement for Ex, B(E2), and Qs
• Inversion of ground state shape from prolate in
76Kr to oblate in 72Kr
• Assignment of prolate, oblate, and K=2 states
• When triaxiality is “off” same results than the
“old” Skyrme
Triaxiality seems to be the key to describe prolate-oblate shape coexistence in this region
Do the GCM (+GOA) approach
and the triaxiality key work
everywhere ?
In the n-rich side ?
2+1
 The n-rich Sr (Z=38), Zr (Z=40) isotopes
present one of the most impressive deformation
change in the nuclear chart
 Systematic of the 2+ energy
(Raman’s formula : b2~0.17  0.4)

+ 2n
 Low lying 0+ states were observed
E(0+) [keV]
0+2
+ 2n

Shape transition at N=60
E [MeV]
HFB Gogny D1S
b2
Both deformations should coexist at low energy
• Shape coexistence between highly deformed and quasi-spherical shapes
Shape transition at N=60
C. Y. Wu et al. PRC 70 (2004)
W. Urban et al Nucl. Phys. A 689 (2001)
N=58
N=60
Shape transition at N=60 : Coulomb excitation
B(E2↓)
< 625 e²fm4
< 152 e²fm4
399 (
-39
67)
e²fm4
< 22 e²fm4
Qs = -6 (9) efm²
462 (11) e²fm4
 The Electric spectroscopic Q0 is null as its B(E2) is
rather large  Quasi vibrator character ??.
No quadrupole ? but it doesn’t exclude octupole or
something else ??
 The large B(E2) might indicate a large contribution of
the protons
E. Clément et al., IS451 collaboration
Gogny calculations
Qualitatively good
agreement
The abrupt change not
reproduced
Very low energy of the 0+2
state is not reproduced
 overestimate the mixing ?
Highly dominated by K=2
configuration
94Sr
96Sr
98Sr
100Sr
Conclusion
We have studied the shape coexistence in the ndeficient Kr isotopes
Beyond the mean field calculations reproduce the
experimental results when the triaxiality degree of
freedom is available
Same calculations seem to not reproduce the shape
transition at N=60. What is missing ?
P. Möller et al Phys. Rev. Lett
103, 212501 (2009)
Shape transition at N=60
40
1g7/2
1g 2d5/2
1g9/2
2p
1f
2p1/2
1f5/2
2p3/2
1g7/2
3s1/2
50
40
ll ll ll
llll
28
2d5/2
0+
ll
ll
ll ll ll
50
 Beyond N=60, the tensor force
participates to the lowering 0+2
state and to the high collectivity
of 2+1 state.
2
0+
1
p
K. Sieja et al PRC 79, 064310 (2009)
n
 But in the current valence space,
need higher effective charge to
reproduce the known B(E2)
Shape coexistence in mean-field models (3) Gogny
Coulomb excitation analysis : GOSIA*
*D. Cline, C.Y. Wu, T. Czosnyka; Univ. of Rochester
Lifetimes are the most important constraint because directly connected to the
transitional matrix element  B(E2)
74Kr
 5 lifetime known from the literature
2+
Lifetime incompatible with
our coulex data
4+
Lifetime measurement
A. Görgen, E. Clément et al., EPJA 26 (2005)
2+
4+
Shape coexistence in Se isotopes
Similar j(1) in 68Se & 70Se :
• 70Se oblate near ground state
• Prolate at higher spin
G. Rainovski et al.,
J.Phys.G 28, 2617 (2002)
Shape coexistence in mean-field models (6) Gogny
Qs from Gogny configuration mixing calculation
Good agreement of B(E2)
Shape change in the GSB in 70,72Se
70,72Se
behaviors differ from neighboring Kr and Ge
Isotopes
68Se
more “classical” compare to Kr and Ge
 Clear evidence for neutron orbital playing an important role in the shape transition
 Established sign for extruder or intruder orbital
 Search for isomer in odd neutron Sr and Zr
A. Jokinen WOG workshop Leuven 2009
W. Urban, Eur. Phys. J. A 22, 241-252 (2004)
h11/2
g7/2
2d5/2
g9/2 from core
 9/2+ isomer identified  ng9/2[404]  extruder
neutron orbital from 78Ni core
 Create the N=60 deformed gap
 pg9/2 <> nh11/2 influence ?
 Neutron excitation from d5/2 to h11/2  Octupole
correlation ?