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

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

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

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

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 ! )

• 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

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

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)

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

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

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

Shape coexistence in a two-state mixing model

Configurations mixing

Perturbed statesPure states

Extract mixing and shape parameters from set of experimental matrix elements.

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) *

*

A. Petrovici et al., Nucl. Phys. A 665, 333 (00)

Vampir calculations

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 …

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

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)

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)

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)

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

Shape coexistence in mean-field models (3) Gogny

M. Girod et al. Physics Letters B 676 (2009) 39–43

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)

Do the GCM (+GOA) approach Do the GCM (+GOA) approach and the triaxiality key work and the triaxiality key work

everywhere ?everywhere ?

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 ?

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

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

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

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

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 ?

P. M

öller et al Phys. R

ev. Lett

103, 212501 (2009)

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

Shape coexistence in mean-field models (3) Gogny

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

Lifetime measurement

2+

4+

A. Görgen, E. Clément et al., EPJA 26 (2005)

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

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

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 ?