1. Experimental measurement of the deformation through the electromagnetic probe Shape coexistence in exotic Kr isotopes. Shape coexistence in exotic Kr isotopes. E. Clément CNRS/GANIL Kazimierz 2010

2. Single particule level scheme (MeV)‏ 74 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. (ii) The shapes involved are clearly distinguishable Shape coexistence in exotic Kr

3. What can we measure experimentally ? Establish the shape isomer : 0 + 2 Collectivity in such nuclei : level scheme and B(E2). Shape (oblate - prolate ?) : Q 0 Wave function mixing ? :  ²(E0) Shape coexistence in n-deficient Kr : an experimentalist view

4. 0+0+ 2+2+ 6+6+ 0+0+ 4+4+ 710 671 612 791 0+0+ 0+0+ 2+2+ 4+4+ 6+6+ 508 456 768 558 52 0+0+ 0+0+ 2+2+ 4+4+ 6+6+ 770 424 824 611 346 0+0+ 0+0+ 2+2+ 4+4+ 6+6+ 1017 455 858 664 562 72 Kr 74 Kr 76 Kr 78 Kr prolate oblate 72(6) 84(18) 79(11) 47(13) Transition strenght :  ²(E 0 ).10 -3 E. Bouchez et al. Phys. Rev. Lett., 90 (2003) 2+2+ 1233 2+2+ 918 Shape isomer : systematic of 0 + 2 states Shape inversion Maximum mixing of wave function in 74 Kr

5. During the desexcitation of the nuclei,  are emitted : Target and stopper at a distance d  In flight  Shifted by the Doppler effect  Stopped  E 0 Recoil Distance Doppler Shift Collectivity measurement : the B(E2) Measure the B(E2) through the lifetime of the state ( ≈ ps ! )

6. The collectivity of the shape-coexisting states are highly pertubated by the mixing Weak mixing ≈ quantum rotor Strong mixing  perturbation of the collectivity 74 Kr Collectivity measurement : the B(E2) GSB

7. 1 er order:  B(E2) 2+2+ 0+0+ a (1) ‏ 2+2+ 0+0+ a (2) ‏ 2 nd order: Reorientation effect  Q 0 dd d  Ruth PifPif = difdif dd __ Collectivity measurement : safe coulomb excitation

8. Static quadrupole moment sensitivity 74 Kr  prolate deformation minimisation du  2 : Negative matrix element (positive quadrupole moment Q 0 )  oblate Deformation 74 Kr Positive matrix element (Negative quadrupole moment Q 0 )

9. Radioactive beams experiment at GANIL 78 Kr 1 MeV/u 10 MeV/u 70 MeV/u 10 12 pps 74 Kr 6  10 4 pps 4.7 MeV/u 1.5  10 4 pps The 74,76 Kr RIB are produced by fragmentation of a 78 Kr beam on a thick carbon target. Radioactive nuclei are extracted and ionized Post-accelaration of the RIB 1 1 2 3

10.  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 excitation

11.  13 E2 transitional matrix elements  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 74 Kr 76 Kr In 74 Kr and 76 Kr, 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

12. Shape coexistence in a two-state mixing model Configurations mixing Perturbed states Pure states Extract mixing and shape parameters from set of experimental matrix elements.

13. Shape coexistence in a two-state mixing model Configurations mixing Perturbed states Pure 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 cos 2 θ 0 76 Kr 74 Kr 72 Kr 0.73(1) 0.48(1)0.10(1) E. Clément et al. Phys. Rev. C 75, 054313 (2007) o Excited Vampir approach: 0.6 0.5 A. Petrovici et al., Nucl. Phys. A 665, 333 (00)‏ * *

14. Vampir calculations

15. 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 76 Kr and 74 Kr to oblate in 72 Kr 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 …

16. 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

17. Restricted to axial symmetry : no K=2 states B(E2) values e 2 fm 4 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)

18. 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)

19. 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)

20. 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

21. Shape coexistence in mean-field models (3) Gogny M. Girod et al. Physics Letters B 676 (2009) 39–43

22. 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 76 Kr to oblate in 72 Kr Assignment of prolate, oblate, and K=2 states When triaxiality is “off” same results than the “old” Skyrme 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 Good agreement for in-band B(E2) Wrong ordering of states: oblate shape from 76 Kr to 72 Kr K=2 outside model space M. Bender and P. –H. Heenen Phys. Rev. C 78, 024309 (2008)

23. Do the GCM (+GOA) approach and the triaxiality key work everywhere ? Do the GCM (+GOA) approach and the triaxiality key work everywhere ?

24.  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 0+20+2 2+12+1 In the n-rich side ?

25. HFB Gogny D1S Shape coexistence between highly deformed and quasi-spherical shapes E [MeV] Both deformations should coexist at low energy 22 Shape transition at N=60

26. 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

27.  The Electric spectroscopic Q 0 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 e²fm4 Q s = -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

28. 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 94 Sr 96 Sr 98 Sr 100 Sr Gogny calculations

29. 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 ?

31. P. Möller et al Phys. Rev. Lett 103, 212501 (2009)

32. 40  2p 1/2 1g 9/2 1g 7/2 50 2p 1f 1g 28 2p 3/2 1f 5/2 2d 5/2 3s 1/2 0+20+2 0+10+1 40 1g 7/2 50 K. Sieja et al PRC 79, 064310 (2009) 2d 5/2  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

33. Shape coexistence in mean-field models (3) Gogny

34. 2+2+ 4+4+ 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) 74 Kr  5 lifetime known from the literature

35. Lifetime measurement 2+2+ 4+4+ A. Görgen, E. Clément et al., EPJA 26 (2005)

36. G. Rainovski et al., J.Phys.G 28, 2617 (2002) Similar j (1) in 68 Se & 70 Se : 70 Se oblate near ground state Prolate at higher spin Shape coexistence in Se isotopes

37. 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 70,72 Se 70,72 Se behaviors differ from neighboring Kr and Ge Isotopes 68 Se more “classical” compare to Kr and Ge

39. 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) 2d 5/2 g 7/2 h 11/2 g 9/2 from core  9/2+ isomer identified  g 9/2 [404]  extruder neutron orbital from 78 Ni core  Create the N=60 deformed gap   g   h 11/2 influence ?  Neutron excitation from d 5/2 to h 11/2  Octupole correlation ?