Shape coexistence in exotic nuclei studied
by low energy coulomb excitation
Emmanuel Clément
CERN-PH, Geneva
Shapes of exotic nuclei
Magnetic
and quadrupole
moments
are veryreflecting
sensitivethe
The
shapedipole
of a nucleus
is a fundamental
property
to all types
of correlations
spatial
distribution
of the nucleons
Important benchmarks for nuclear models / theory
B(E2) measurement
Static moment measurement (oblate-prolate)
Quadrupole deformation of the nuclear ground states
Prolate-oblatespherical shape
in a small
energy range
Shape coexistence
A=70-80, N=Z
n-rich Sr&Zr
M. Girod, CEA
n-rich Ar  See M. Zielińska’s talk
HFB+Gogny D1S M. Girod et al.,
To be published
Single particule level scheme (MeV)
Shapes coexistence in light Kr isotopes
74Kr
Deformation parameter 
Important constraints for modern nuclear structure theories :
Predicted values of 2
E (0+2), r²(E0), B(E2), Q0 …
Mixing of wave function  GCM
Once upon a time ….
prolate
2000 & 2001 Conversion electron
spectroscopy : E0 transition
oblate
6+
6+
2000 Coulomb excitation of 78Kr
2002 First Coulomb excitation of a
radioactive 76 Kr beam @ SPIRAL
+EXOGAM
791
2+
2+
4+
1233
611
558
0+
671
2004 Lifetime measurement
of 76Kr and 74Kr @ GASP
Transition strenght : r²(E0).10-3
0+
2+
710
0+
72Kr
72(6)
?
0+
456
52
End of 2006 Coherent analysis of all data from 76Kr and 74Kr
In the future Low-energy Coulomb excitation of 72Kr
beam development needed
2+
508
0+
74Kr
4+
0+
346
770
424
76Kr
0+
664
562
2+
1017
455
0+
78Kr
E. Bouchez
F. Becker et al.
E. Clément
84(18)
79(11)
47(13)
Thèse
Université
de
Nucl. Phys. A 770
Thèse Université de
Paris 11 (2006)
Complete measure of reduced transition
probability B(E2) and static quadrupole
moment
918
4+
4+
612
858
824
768
2+
2003 Coulomb excitation of a radioactive
74Kr beam @ SPIRAL+EXOGAM
6+
6+
Strasbourg 1 (2003)
(2006)
A. Görgen, E. Clément et al.
Eur. Phys. J. A 26, 153 (2005)
E. Bouchez et al. Phys. Rev. Lett., 90 (2003)
F. Becker et al., Eur. Phys. J. A 4 (1999)
A. Giannatiempo et al., Phys. Rev. C 52 (1995)
E. Clément, A. Görgen, W. Korten et al.
Submitted to Phys. Rev. C
Coulomb excitation analysis with GOSIA
74Kr
 14 E2 transitional matrix elements
In
74Kr
76Kr
 18 E2 transitional matrix elements
and 76Kr, a prolate ground state coexists with an oblate excited configuration
Transition probability : describe the coupling between states
First direct experimental proof of the shape
 5 E2 diagonal matrix coexistence
element
5 E2 diagonal matrix element
in light Kr 
isotopes
Spectroscopic quadrupole moment : intrinsic properties
of the
nucleus
E. Bouchez
Thèse
SPhN 2003
E. Clément Thèse SPhN 2006
E. Clément et al. Submitted to PRC
Configurations mixing (1)
For the shape-coexisting states,
prolate and oblate wave functions are highly mixed
Weak mixing ≈ quantum rotor
74Kr
Strong mixing  perturbation of the collectivity
Configurations mixing (2)
Shape coexistence in a two-state mixing model
Pure states
Perturbed states
Extract mixing and shape parameters from set of experimental matrix elements.
76Kr
74Kr
cos2θ0
0.73(1)
0.48(1)
cos2θ0
0.69(4)
0.48(2)
72Kr
Energy perturbation of 0+2 states
E. Bouchez et al. Phys. Rev. Lett., 90 (2003)
Full set of matrix elements :
E. Clément, A. Görgen, W. Korten el al.
Submitted to PRC
0.10(1)
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.
Shape coexistence in mean-field models (2) Skyrme
GCM-HFB (SLy6)
M. Bender, P. Bonche et P.H. Heenen,
Phys. Rev. C 74, 024312 (2006)
HFB+GCM method
Skyrme SLy6 force
density dependent pairing
interaction
Restricted to axial
symmetry : no K=2 states
Inversion of oblate and
prolate states
B(E2) values e2fm4
Collectivity of the prolate
rotational band is correctly
reproduced
Interband B(E2) are under
estimated
Same conclusion for 76Kr
Shape coexistence in mean-field models (3) Gogny
GCM-HFB (Gogny-D1S)
J-P. Delaroche et al.
In preparation
HFB+GCM with Gaussian
overlap approximation
Gogny D1S force
The agreement is remarkable for
excitation energy and matrix elements
K=0 prolate rotational ground state band
K=2 gamma vibrational band
2+3 oblate rotational state
Strong mixing of K=0 and K=2 components
for 2+3 and 2+2 states
Axial and triaxial degrees of freedom
Grouping the non-yrast states above 0+2 state in
band structures is not straightforward
GCM is a good approach to treat shape coexistence
main differences between the two ‘beyond mean field’ calculations:
 Skyrme  Gogny
 axial
 triaxial
• It is important to include the triaxial degree of freedom
to describe shape coexistence in light krypton isotopes
Same conclusion for 76Kr
New area of investigation
All theoretical calculations predict a sudden onset of
quadrupole deformation at the neutron number N=60
Neutron rich Sr & Zr isotopes are accessible by fission of an UCx target
Coulomb excitation of such nuclei can be performed at REX-ISOLDE
Sr and Zr n-rich isotopes around N=60
E [MeV]
HFB Gogny D1S
M. Girod
CEA Bruyères-le-Châtel
96Sr
is a transitional
nucleus

Both deformations should coexist at low energy
Shape coexistence between highly deformed and quasi-spherical shapes
Electromagnetic matrix elements are stringent test for theory
Evidence for shape coexistence in Sr
N=58
N=60
The highly deformed band 0+32+34+2 becomes the ground state
band in 98Sr
Evidence for shape coexistence in Sr
Recent results :C. Y. Wu et al. PRC 70 (2004)
W. Urban et al Nucl. Phys. A 689 (2001)
Lifetime compatible with  = 0.25
N=58
N=60
The measure of transition strength and intrinsic quadrupole moments is essential to
understand the complex shape coexistence in Sr isotopes  Coulomb excitation
The onset of deformation around N=58 is maybe more gradual
Accepted experiment at REX-ISOLDE (IS451)
Conclusion
Coulomb excitation at low energy offers an unique opportunity to
understand the complex scenario of shape coexistence in exotic nuclei
Precise comparisons with HFB+GCM calculations are essential to understand
the shape coexistence
• GCM is a good approach to treat shape coexistence.
• It is important to include triaxial degree of freedom.
• Data from n-rich nuclei will provide more insight into shape coexistence.
Collaboration
E. Clément,1 A. Görgen,1 W. Korten,1 E. Bouchez,1 A. Chatillon,1 Y. Le Coz,1 Ch. Theisen,1 J.N.
Wilson,1 M. Zielinska,5,1 , J.-P. Delaroche8, M. Girod8, H. Goutte8, S. Péru8, C. Andreoiu,2 F.
Becker,3 J.M. Casandjian,4 W. Catford,9 T. Czosnyka,5 G. de France,4 J. Gerl,3 J. Iwanicki,5 P.
Napiorkowski,5 G. Sletten,6 C. Timis7
1DAPNIA/SPhN,
CEA Saclay
2Oliver Lodge Laboratory, University of Liverpool
3GSI Darmstadt
4GANIL
5Heavy
Ion Laboratory, Warsaw
Copenhagen
7University of Surrey
8CEA/DIF, DPTA/SPN, CEA Bruyère-le-Châtel
6NBI
E. Clément1, A. Görgen2 , J. Cederkäll1, P. Delahaye1, L. Fraile1, F. Wenander1, J. Van de Walle4,
D. Voulot1, C.Dossat2, W. Korten2, J. Ljungvall2, A. Obertelli2, Ch. Theisen2, M. Zielinska2, J.
Iwanicki3, J. Kownacki3, P. Napiorkowski3, K. Wrzosek3, P. Van Duppen4, T. Cocolios4, M. Huyse4,
O. Ivanov4, M. Sawicka 4, I.Stefanescu4, N. Bree4, S. Franchoo5, F. Dayras6, G. Georgiev6, A.
Ekström7, M. Guttormsen8, A.C. Larsen8, S. Siem8, N.U.H. Syed8, P.A. Butler9, A. Petts9, D.G.
Jenkins10, V. Bildstein11, R. Gernhäuser11, T. Kröll11, R. Krücken11, P. Reiter12, N. Warr12 ,
1CERN,
Geneva, Switzerland
2DAPNIA/SPhN,
CEA Saclay, France
7Department
of Physics, Lund University, Sweden
8Department
of Physics, University of Oslo, Norway
3HIL,
Warsaw, Poland
4IKS
Leuven, Belgium
10Department
5IPN
Orsay, France
11TU
6CSNSM
Orsay, France
9Oliver
Lodge Laboratory, University of Liverpool, UK,
of Physics, University of York, UK,
München, Germany
12IKP
Köln, Germany