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Experimental measurement of the deformation through the Experimental measurement of the deformation through the electromagnetic probeelectromagnetic probe
Shape coexistence in exotic Kr isotopes.Shape coexistence in exotic Kr isotopes.
E. Clément CNRS/GANIL Kazimierz 2010
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Sin
gle
par
ticu
le l
evel
sch
eme
(MeV
)
74Kr
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.
(ii) The shapes involved are clearly distinguishable
Shape coexistence in Shape coexistence in exotic Krexotic Kr
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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 ? : ²(E0)
Shape coexistence in n-deficient Kr : an Shape coexistence in n-deficient Kr : an experimentalist viewexperimentalist view
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0+
2+
6+
0+
4+
710671
612
791
0+
0+
2+
4+
6+
508456
768
558
52
0+
0+
2+
4+
6+
770
424
824
611346
0+
0+
2+
4+
6+
1017
455
858
664562
72Kr 74Kr 76Kr 78Kr
prolate oblate
72(6) 84(18) 79(11) 47(13)
Transition strenght : ²(E0).10-3
E. Bouchez et al. Phys. Rev. Lett., 90 (2003)
2+
1233
2+
918
Shape isomer : systematic of 0Shape isomer : systematic of 0++22
statesstates
Shape inversion •
Maximum mixing of wave function in 74Kr
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During the desexcitation
of the nuclei, are emitted :
Target and stopper at a distance d
In flight Shifted by the Doppler effect
Stopped E0
Recoil Distance Doppler Shift
Collectivity measurement : Collectivity measurement : the B(E2)the B(E2)
Measure the B(E2) through the lifetime of the state ( ≈ ps ! )
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• The collectivity of the shape-coexisting states are highly pertubated by the
mixing
Weak mixing ≈ quantum rotor
Strong mixing perturbation of the collectivity74Kr
Collectivity measurement : Collectivity measurement : the B(E2)the B(E2)
GSB
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1er order:
B(E2)
2+
0+
a(1)
2+
0+
a(1) a(2) a(2)
2nd order: Reorientation effect
Q0
d dRuth Pif=dif d__ __
if IEIa )2()1( M
j
ijjf IEIIEIa )2()2()2( MMifff IEIIEIa )2()2()2( MM
Collectivity measurement : safe coulomb Collectivity measurement : safe coulomb excitationexcitation
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Static quadrupole moment Static quadrupole moment sensitivitysensitivity
)2(
)4(
1
1
I
I
74Kr
59.021.011
33.030.011
02.14)2(4
70.02)2(2
E
E
M
M
prolate deformation
minimisation du 2 :
Negative matrix element (positive quadrupole moment Q0)
28.023.022 33.02)2(2
EM
oblate Deformation
74Kr
)2(
)2(
1
2
I
I
Positive matrix element (Negative quadrupole moment Q0)
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Radioactive beams experiment at GANILRadioactive beams experiment at GANIL
78Kr1 MeV/u
10 MeV/u
70 MeV/u1012 pps
74Kr6104 pps
4.7 MeV/u1.5104 pps
• 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
1
2
3
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detection
Pb
The differential Coulomb excitation cross section is sensitive to transitionnal and diagonal E2 matrix elements
E. Clément et al. PRC 75, 054313 (2007)Particle detection
GOSIA code
Very well known technique for stable nuclei but for radioactive one …
Safe Coulomb excitationSafe Coulomb excitation
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13 E2 transitional matrix elements
5 E2 diagonal matrix element 5 E2 diagonal matrix element
16 E2 transitional matrix elements
Transition probability : describe the coupling between states
Spectroscopic quadrupole moment : intrinsic properties of the nucleus
74Kr 76Kr
In 74Kr and 76Kr, a prolate ground state coexists with an oblate excited configuration
E. Bouchez PhD 2003
E. Clément et al. PRC 75, 054313 (2007)E. Clément PhD 2006
Safe Coulomb excitation results Safe Coulomb excitation results
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Shape coexistence in a two-state mixing model
Configurations mixing
Perturbed statesPure states
Extract mixing and shape parameters from set of experimental matrix elements.
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Shape coexistence in a two-state mixing model
Configurations mixing
Perturbed statesPure states
Extract mixing and shape parameters from set of experimental matrix elements.
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.
• Full set of matrix elements :
0.69(4) 0.48(2)
E. Bouchez et al. Phys. Rev. Lett 90 (2003)
• Energy perturbation of 0+2 states
cos2θ0
76Kr 74Kr 72Kr
0.73(1) 0.48(1) 0.10(1)
E. Clément et al. Phys. Rev. C 75, 054313 (2007)
oExcited Vampir approach:0.6 0.5
A. Petrovici et al., Nucl. Phys. A 665, 333 (00) *
*
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A. Petrovici et al., Nucl. Phys. A 665, 333 (00)
Vampir calculations
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Several theoretical approaches, such as shell-model methods, self-consistent 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.
Beyond …
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Shape coexistence in mean-field models
• In-band reduced transition probability and spectroscopic quadrupole moments
GCM-HFB (SLy6) M. Bender, P. Bonche et P.H. Heenen, Phys. Rev. C 74, 024312 (2006)
GCM-HFB (Gogny-D1S)E. Clément et al., PRC 75, 054313 (2007)M. Girod et al. Physics Letters B 676 (2009) 39–43
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Restricted to axial symmetry : no K=2 states
B(E2) values e2fm4
Shape coexistence in mean-field models (2) Skyrme
HFB+GCM method Skyrme SLy6 force density dependent pairing interaction
Inversion of oblate and prolate states
Collectivity of the prolate rotational band is correctly reproduced
Interband B(E2) are under estimated
E. Clément et al., PRC 75, 054313 (2007)
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Shape coexistence in mean-field models (3) Gogny
Axial and triaxial degrees of freedom HFB+GCM with Gaussian overlap approximation
Gogny D1S force
E. Clément et al., PRC 75, 054313 (2007)
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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
2+3 oblate rotational state
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
E. Clément et al., PRC 75, 054313 (2007)
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Shape coexistence in mean-field models (3) Gogny
Potential energy surface using the Gogny GCM+GOA appraoch
M. Girod et al. Physics Letters B 676 (2009) 39–43
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Shape coexistence in mean-field models (3) Gogny
M. Girod et al. Physics Letters B 676 (2009) 39–43
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axial quadrupole deformation q0 ↔ triaxial quadrupole deformation q0, q2(exact GCM formalism) Euler angles Ω=(θ1,θ2,θ3)
→ 5-dimensional collective Hamiltonian (Gaussian overlap approximation)
Difference #1: effective interaction
very similar single-particle energies
→ no big differences on the mean-field level
Is the triaxiality the key ?
• 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 When triaxiality is “off” same results than the “old” Skyrme“old” Skyrme
Triaxiality seems to be the key to describe prolate-oblate shape coexistence in this region
• 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)
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Do the GCM (+GOA) approach Do the GCM (+GOA) approach and the triaxiality key work and the triaxiality key work
everywhere ?everywhere ?
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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 : 2~0.17 0.4)
Low lying 0+ states were observed
E(0
+)
[keV
]
+ 2
+ 2
0+2
2+1
In the n-rich side ?
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HF
B G
ogny D1S
• Shape coexistence between highly deformed and quasi-spherical shapes
E [
MeV
]
Both deformations should coexist at low energy
2
Shape transition at N=60
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N=58 N=60
C. Y. Wu et al. PRC 70 (2004)W. Urban et al Nucl. Phys. A 689 (2001)
Shape transition at N=60
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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
462 (11) e²fm4
< 22 e²fm4Qs = -6 (9) efm²
399 ( -39 67) e²fm4
< 625 e²fm4
B(E2↓)
< 152 e²fm4
Shape transition at N=60 : Coulomb excitation
E. Clément et al., IS451 collaboration
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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
Gogny calculations
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Conclusion
We have studied the shape coexistence in the n-deficient 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 ?
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P. M
öller et al Phys. R
ev. Lett
103, 212501 (2009)
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40
2p1/2
1g9/2
1g7/2
50
2p
1f
1g
28
2p3/2
1f5/2
2d5/2
3s1/2
0+
2
0+
1
40
1g7/2
50
K. Sieja et al PRC 79, 064310 (2009)
2d5/2
Beyond N=60, the tensor force participates to the lowering 0+
2
state and to the high collectivity of 2+
1 state.
But in the current valence space, need higher effective charge to reproduce the known B(E2)
Shape transition at N=60
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Shape coexistence in mean-field models (3) Gogny
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2+
4+
Coulomb excitation analysis : GOSIA* Coulomb excitation analysis : GOSIA* *D. Cline, C.Y. Wu, T. Czosnyka; Univ. of Rochester
Lifetime incompatible with our coulex data
Lifetimes are the most important constraint because directly connected to the transitional matrix element B(E2)
74Kr 5 lifetime known from the literature
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Lifetime measurement
2+
4+
A. Görgen, E. Clément et al., EPJA 26 (2005)
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G. Rainovski et al.,J.Phys.G 28, 2617 (2002)
Similar j(1) in 68Se & 70Se :
• 70Se oblate near ground state• Prolate at higher spin
Shape coexistence in Se isotopes
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Qs from Gogny configuration mixing calculation
Shape coexistence in mean-field models (6) Gogny
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
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A. Jokinen WOG workshop Leuven 2009
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
W. Urban, Eur. Phys. J. A 22, 241-252 (2004)
2d5/2
g7/2
h11/2
g9/2 from core 9/2+ isomer identified g9/2[404] extruder neutron orbital from 78Ni core
Create the N=60 deformed gap
gh11/2 influence ?
Neutron excitation from d5/2 to h11/2 Octupole correlation ?