1. 1 Andreas Görgen Atelier ESNT 4.-6.2.2008 Nuclear Shapes, Shape Coexistence, and Electromagnetic Moments An Experimentalist’s Perspective on the Interaction between Experiment and Theory Andreas Görgen DSM / IRFU / SPhN CEA Saclay

2. 2 Andreas Görgen Atelier ESNT 4.-6.2.2008 Nuclear deformation quadrupoleoctupolehexadecapole 0°0° -60°-120° spherical oblate non-collective prolate collective oblate collective prolate non-collective   =60°  spectroscopic quadrupole moment Q s <0 Q s >0 collective non- collective reduced transition probability

3. 3 Andreas Görgen Atelier ESNT 4.-6.2.2008 Coulomb excitation Nuclear excitation by electromagnetic field acting between nuclei. b projectile target The excitation cross section is a direct measure of the E matrix elements. IfIf 1 st order: IiIi 2 nd order: IiIi IfIf ImIm reorientation effect: IfIf IiIi MfMf  lifetime measurement differential Coulomb excitation cross section  transition probability B(E2)  quadrupole moment Q s 44 Ar + 208 Pb

4. 4 Andreas Görgen Atelier ESNT 4.-6.2.2008 Shape coexistence oblateprolate 0+0+ 0+0+ 2+2+ 2+2+ 4+4+ 4+4+ 6+6+ 6+6+ 8+8+ Configuration mixing: 74 Kr   M. Girod M. Bender et al., PRC 74, 024312 (2006) electric monopole (E0) transition

5. 5 Andreas Görgen Atelier ESNT 4.-6.2.2008 Systematics of the light krypton isotopes  Inversion of ground state shape for 72 Kr  Coulomb excitation to determine the nuclear shapes directly 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 E. Bouchez et. al., Phys. Rev. Lett. 90, 082502 (2003)  energy of excited 0 +  E0 strengths  2 (E0)  configuration mixing Mixing of the ground state (two-level mixing extrapolated from distortion of rotational bands) 72 · 10 -3 85 · 10 -3 79 · 10 -3 47 · 10 -3  2 (E0)

6. 6 Andreas Görgen Atelier ESNT 4.-6.2.2008 Coulomb excitation of 74 Kr and 76 Kr SPIRAL beams 76 Kr 5  10 5 pps 74 Kr 10 4 pps 4.7 MeV/u EXOGAM Pb [24°, 55°][55°, 74°][67°, 97°][97°, 145°] 74 Kr Coulomb excitation at sub-barrier energies:  purely electromagnetic (‘safe’)  multi-step excitation possible  differential measurement: d  /d 

7. 7 Andreas Görgen Atelier ESNT 4.-6.2.2008 Matrix elements Fit matrix elements (transitional and diagonal) to reproduce experimental  -ray yields (as function of  )  14 B(E2) values  5 quadrupole moments E. Clément et al., PRC 75, 054313 (2007) first reorientation measurement with radioactive beam

8. 8 Andreas Görgen Atelier ESNT 4.-6.2.2008 prolate oblate Q s <0 prolate Q s >0 oblate experimental B(E2;  ) [e 2 fm 4 ] Comparison with ‘beyond-mean-field’ calculations K=2  vibration E. Clément et al., PRC 75, 054313 (2007) GCM (GOA) calculation q 0, q 2 : triaxial deformation Gogny D1S M. Girod et al. prolate oblate GCM calculation axial deformation Skyrme SLy6 M. Bender et al. PRC 74, 024312 (2006)

9. 9 Andreas Görgen Atelier ESNT 4.-6.2.2008 Shape transition in the light krypton isotopes 0+0+ 2+2+ 0+0+ 4+4+ 710 671 612 0+0+ 0+0+ 2+2+ 4+4+ 508 456 558 52 0+0+ 0+0+ 2+2+ 4+4+ 770 424 611 346 0+0+ 0+0+ 2+2+ 4+4+ 1017 455 664 562 72 Kr 74 Kr 76 Kr 78 Kr prolate oblate Experimental (direct and indirect evidence) 2+2+ 2 2+2+ 1 Theory Gogny D1S 5D GCM (GOA): M. Girod et al. 2+2+ 2 2+2+ 1 2+2+ 2 2+2+ 1 also mixing of K=0 and K=2 states Skyrme SLy6 GCM: M. Bender et al., PRC 74, 024312 (2006)

10. 10 Andreas Görgen Atelier ESNT 4.-6.2.2008 Configuration mixing calculations protonsneutrons Skyrme SLy6  Gogny D1S Bender et al. Girod et al. very similar single-particle energies  no big differences on the mean-field level Difference #2: generator coordinates axial quadrupole deformation q 0  triaxial quadrupole deformation q 0, q 2 (exact GCM formalism) Euler angles  =(  1,  2,  3 )  5-dimensional collective Hamiltonian (Gaussian overlap approximation) Difference #1: effective interaction  excellent agreement for excitation energies, B(E2), and quadrupole moments  inversion of ground-state shape from prolate in 76 Kr to oblate in 72 Kr reproduced  assignment of prolate, oblate, and K=2 states  good agreement for in-band B(E2) and quadrupole moments  wrong ordering of states: oblate ground-state shape for 72 Kr  78 Kr  excited states dilated in energy  K=2 states outside model  triaxiality seems to be the key to describe prolate-oblate shape coexistence in this region

11. 11 Andreas Görgen Atelier ESNT 4.-6.2.2008 The light Se isotopes Coulex lifetimes  J. Ljungvall 70 Se 72 Se GCM - Gogny D1S (GOA) calculation  reproduces excitation energies  B(E2) values over-estimated  clarifies shape coexistence and shape transitions in Se J. Ljungvall et al., PRL in press B(E2;  ) [e 2 fm 4 ]

12. 12 Andreas Görgen Atelier ESNT 4.-6.2.2008 Deformation and shape coexistence in N=28 isotones Ca 40 96.94 Ca 42 0.65 Ca 44 2.08 Ca 46 0.003 Ca 48 0.19 Ar 38 0.07 Ca 50 13.9 s Ar 40 99.59 Ar 42 32.9 y Ar 44 11.9 m Ar 46 8.4 s Ar 48 0.48 s S 36 0.015 S 38 170 m S 40 8.8 s S 42 1.01 s S 44 123 ms S 46 50 ms Si 34 2.77 s Si 36 0.45 s Si 38 >1  s Si 40 33 ms Si 42 13 ms Si 44 10 ms Mg 32 86 ms Mg 34 20 ms Mg 36 3.9 ms Mg 38 >260 ns Mg 40 1 ms M. Girod Bruyères-le-Châtel 44 S, 42 Si, 40 Mg Coulomb excitation  dream experiments neutron-rich Ar isotopes from SPIRAL  Coulex of 44 Ar (2·10 5 pps)  EXOGAM + DSSD  on 208 Pb target at 3.68 MeV/u  on 109 Ag target at 2.68 MeV/u  differential measurement:  ( ,Z) 202224262830 20 18 16 14 12 R. Rodríguez-Guzmán et al, PRC 65, 024304 (2002) axial CGM with Gogny D1S

13. 13 Andreas Görgen Atelier ESNT 4.-6.2.2008 Coulomb excitation of 44 Ar at SPIRAL / GANIL Ag target, 35°  cm  70° Ag target, 70°  cm  130°Pb target, 30°  cm  130° M. Zielińska et al. 2 + 1.158 2 + 2.011 76(10) 4.6(8) 136(24) 0+0+ experiment B(E2;  ) in e 2 fm 4 (4 + ) 2.746 Q s =  8(3) e fm 2 0+0+ 102 156 2 + 1.729 2 + 3.597 theory HFB+GCM(GOA) 0 180 4.067 Q s =+7 e fm 2 Q s =  9 e fm 2 Q s =  14 e fm 2 4+4+  good agreement for B(E2) and Q  spectrum too spread out (collective masses from Inglis-Belyaev approximation)

14. 14 Andreas Görgen Atelier ESNT 4.-6.2.2008 Conclusions and Perspectives  Shape coexistence and transitions in light Kr and Se isotopes  Direct evidence through experimental quadrupole moments  Consistent theoretical description: Importance of triaxiality  Comparison with theory allows assigning the character of the states  Quadrupole moments and transition strengths  Sensitive probe for ‘beyond-mean-field’ calculations  Need to understand discrepancies for B(E2) in Se and E x in Ar  Experimental investigation of N=Z ( 68 Se, 72 Kr) and N=28 ( 46 Ar 28, 44 S 28, …)  Difficult but perhaps not unfeasible  Program to measure lifetimes after deep-inelastic collisions  Exogam + Vamos + Plunger: 208 Pb + 64 Ni in 2008  Agata Demonstrator + Prisma + Plunger: 70 Zn + 208 Pb LoI for 2009  Coulex of SPIRAL-2 (and HIE-ISOLDE) beams  Theory to guide experimentalists, e.g. for beam development

15. 15 Andreas Görgen Atelier ESNT 4.-6.2.2008 Collaboration CEA Saclay – DSM / IRFU / SPhN:  E. Clément (Kr Coulex)  J. Ljungvall (Se Lifetimes)  M. Zielińska (Ar Coulex) W. Korten, A. Obertelli, Ch. Theisen A. Chatillon, E. Bouchez, A. Hürstel, Y. Le Coz CEA Bruyères-le-Châtel – DIF / DPTA / SPN:  M. Girod, J.-P. Delaroche (Theory) GANIL experiments: Warsaw, Liverpool, GSI, GANIL, Surrey, NBI Legnaro experiments: Köln, Padova, Legnaro, Oslo, Warsaw ISOLDE experiments: Liverpool, York, CERN, Leuven, Köln, München, Lund, Warsaw, Edinburgh, Legnaro, Oslo, Padova, Darmstadt, Heidelberg, Manchester Jeunots Jeunots ?

16. 16 Andreas Görgen Atelier ESNT 4.-6.2.2008 Electric monopole (E0) transitions  between states of the same spin and parity in particular 0 +  0 +  related to changes in the rms radius of the charge distribution  monopole matrix element r p radius vector of the protons R = 1.25 A 1/3 nuclear radius  non-radiative transitions  internal conversion or  internal pair creation (E >1.022 MeV)  example: two 0 + states of different shapes with mixing E0 transition strength E0 transitions proceed only in the presence of a sizeable deformation and mixing of components with different 0+0+ 0+0+ 2+2+ 2+2+ 4+4+ 4+4+ 6+6+ 6+6+ 8+8+

17. 17 Andreas Görgen Atelier ESNT 4.-6.2.2008 Quadrupole sum rules Model-independent method to determine shape of charge distribution from set of matrix elements 0101 + 2323 +2 + 2121 + needs complete set of matrix elements  only feasible for 0 + states

18. 18 Andreas Görgen Atelier ESNT 4.-6.2.2008 extrapolated states 218 515 772 996 1185 1355 1509 experimental states 710 612 792 996 1185 1355 1509 72 Kr 0+0+ 2+2+ 4+4+ 6+6+ 8+8+ 10 + 12 + 14 + 671 0+0+ Rotational band is distorted at low spin.  influence of mixing Regular rotational cascade at high spin. a |0 +  - b |0 +  o p a |0 +  + b |0 +  p o Mixed states 0 1 + 0 2 + 671 keV 0 p + |0 +  p  64 keV Pures states 0 o + 64 keV  |0 +  o V  Interaction V  mixing amplitudes a, b pure prolate Level mixing J (1) [ħ 2 MeV -1 ] (ħ  ) 2 [MeV 2 ] experimental states extrapolated 72 Kr

19. 19 Andreas Görgen Atelier ESNT 4.-6.2.2008 Coulomb excitation analysis : GOSIA* *D. Cline, C.Y. Wu, T. Czosnyka; Univ. of Rochester   -ray yields as function of scattering angle (differential excitation cross section)  experimental spectroscopic data (lifetimes, branching ratios)  least squares fit of ~ 30 matrix elements (transitional and diagonal)  Results inconsistent with transitions strengths from published lifetimes  New RDM lifetime measurement 74 Kr 2+2+ 4+4+ 6+6+

20. 20 Andreas Görgen Atelier ESNT 4.-6.2.2008 Lifetime measurement with GASP and the Köln Plunger 40 Ca( 40 Ca,a2p) 74 Kr 40 Ca( 40 Ca,4p) 76 Kr 124 MeV

21. 21 Andreas Görgen Atelier ESNT 4.-6.2.2008 Lifetime results 2+2+ 4+4+ Results consistent with Coulomb excitation. Lifetimes constrain GOSIA fit.  enhanced sensitivity for non-yrast transitions and diagonal matrix elements 74 Kr 2 + 4 + 76 Kr 2 + 4 + new33.8(6) 5.2(2) new41.5(8) 3.67(9)[ps] 28.8(57) 13.2(7) 35.3(10) 4.8(5) [ps] J. Roth et al., J.Phys.G, L25 (1984)B. Wörmann et al., NPA 431, 170 (1984) Eur. Phys. J. A 26, 153 (2005)  forward detectors (36˚)  gated from above 74 Kr

22. 22 Andreas Görgen Atelier ESNT 4.-6.2.2008 Configuration mixing calculations  start with a basis of wave functions from self-consistent mean-field calculations protonsneutrons Skyrme SLy6  Gogny D1S Bender et al. Girod et al. The two interactions give very similar single-particle energies  no big differences on the mean-field level  symmetry restoration: projection on particle number and angular momentum collective coordinates q: axial quadrupole deformation q 0  triaxial quadrupole deformation q 0, q 2 rotation: Euler angles  =(  1,  2,  3 )  correlated states  determine weight coefficients of mixed states by solving Hill-Wheeler-Griffin equation Generator Coordinate Method (GCM) exact solution  Gaussian overlap approximation

23. 23 Andreas Görgen Atelier ESNT 4.-6.2.2008 70 Se beam at REX-ISOLDE  2·10 13 protons/s on ZrO 2 target at 1.4 GeV  70 Se diffuses to target surface and forms SeCO  70 Se 12 C 16 O ionized to 1+ and mass selected (A=98) at 30 keV  accumulated and cooled in Penning trap at 6  10 5 ions/s  every 58 ms released into EBIS charge breeder where molecule is broken up  further selection for A/q = 70/19 and accelerated as 70 Se 19+ at 10 4 ions/s

24. 24 Andreas Görgen Atelier ESNT 4.-6.2.2008 70 Se G. Rainovski et al., J. Phys. G 28, 2617 (2002) Rotational bands in 68 Se and 70 Se 68 Se  ( g 9/2 ) 2  ( g 9/2 ) 2 S. Fischer et al., PRC 67, 064318 (2003)  ( g 9/2 ) 2 ( g 9/2 ) 2  Very similar behavior in 68 Se and 70 Se:  shape transition from oblate to prolate  alignment of g 9/2 protons and neutrons

25. 25 Andreas Görgen Atelier ESNT 4.-6.2.2008 Systematics 74 Kr 72 Se  prolate GSB  oblate excited band  drop in B(E2) for 2 +  0 +  strong mixing for 0 + very similar in 72 Se:  low-lying 0 +  drop in B(E2) for 2 +  0 + 70,72 Se: J. Heese et al., Z. Phys. A 325, 45 (1986) opposite behavior in 70 Se  increase in B(E2) for 2 +  0 + why ? upper limit for 68 Se from GANIL intermediate-energy Coulex; E. Clément et al., submitted to NIM new measurement at MSU in 2008 unexplained staggering due to large B(E2) in 70 Se Reliable enough to base conclusion on this value?

26. 26 Andreas Görgen Atelier ESNT 4.-6.2.2008 Systematics

27. 27 Andreas Görgen Atelier ESNT 4.-6.2.2008 Coulomb excitation of 70 Se at CERN / ISOLDE  70 Se on 104 Pd at 2.94 MeV/u  integral measurement  excitation probability via normalization to known 104 Pd A.M. Hurst et al., PRL 98, 072501 (2007) P 2+ is function of  transitional matrix element B(E2)  diagonal matrix element Q 0 IfIf IiIi MfMf  one measurement, but two unknowns ! M. Bender

28. 28 Andreas Görgen Atelier ESNT 4.-6.2.2008 Matrix elements for 2 + in 70 Se A.M. Hurst et al., PRL 98, 072501 (2007) rotational model values (Q t = Q 0 ) lifetime measurement  = 1.5(3) ps  J. Heese et al., Z. Phys. A 325, 45 (1986) Coulomb excitation probability (1  )  only prolate shape consistent with both Coulex and lifetime measurement contrary to previous assumptions

29. 29 Andreas Görgen Atelier ESNT 4.-6.2.2008 Lifetimes in 70 Se and 72 Se revisited March 2007 GASP and Köln Plunger at Legnaro 40 Ca( 36 Ar,  2p) 70 Se 40 Ca( 36 Ar,4p) 72 Se 12 distances between 8 and 400  m [1] J. Heese et al., Z. Phys. A 325, 45 (1986) [2] J. Ljungvall et al., in preparation old [1]  (ps) new[2]  (ps) B(E2;  ) (e 2 fm 4 ) 70 Se 2+2+ 1.5(3)3.2(2)342(19) 4+4+ 1.4(3)1.4(1)370(24) 6+6+ 3.9(9)1.9(3)530(96) 72 Se 2+2+ 4.3(5)4.2(3)405(25) 4+4+ 2.7(4)3.3(2)882(50) 6+6+ 2.3(2)1.7(1)1220(76) 2 +  0 + in 70 Se

30. 30 Andreas Görgen Atelier ESNT 4.-6.2.2008 Consequences old lifetime measurement Heese et al. new lifetime measurement J. Ljungvall et al. to be published  prolate shape can be excluded for 2 + in 70 Se  higher precision coulex measurement needed

31. 31 Andreas Görgen Atelier ESNT 4.-6.2.2008 Comparison with GCM calculations M. Girod, J.-P. Delaroche CEA Bruyères-le-Châtel  theoretical B(E2) values too large  oblate shape for 2 + in 70 Se confirmed by theory 1

32. 32 Andreas Görgen Atelier ESNT 4.-6.2.2008 Shape evolution in the light Selenium isotopes  oblate rotation prevails only in 68 Se  remains best example for shape coexistence in Se  matrix elements needed experiment theory J. Ljungvall et al., to be published

33. 33 Andreas Görgen Atelier ESNT 4.-6.2.2008 Radioactive beam production: SPIRAL 78 Kr 68.5 MeV/u 10 12 pps 74 Kr 4.7 MeV/u 10 4 pps SPIRAL: Système de Production d’Ions Radioactifs en Ligne ECRIS Target SPIRAL CIME CSS2 CSS1

34. 34 Andreas Görgen Atelier ESNT 4.-6.2.2008 Configuration mixing calculations protonsneutrons Skyrme SLy6  Gogny D1S Bender et al. Girod et al. very similar single-particle energies  no big differences on the mean-field level Difference #2: generator coordinates axial quadrupole deformation q 0  triaxial quadrupole deformation q 0, q 2 (exact solution) (Gaussian overlap approximation) Difference #1: effective interaction 0+0+ 2+2+ 0+0+ 4+4+ 0+0+ 0+0+ 2+2+ 4+4+ 0+0+ 0+0+ 2+2+ 4+4+ 0+0+ 0+0+ 2+2+ 4+4+ 72 Kr 74 Kr 76 Kr 78 Kr prolateoblate 0+0+ 2+2+ 0+0+ Gogny 0+0+ 2+2+ 0+0+ Skyrme 0+0+ 2+2+ 0+0+ 2+2+ 0+0+ 2+2+ 4+4+ 0+0+ Gogny 0+0+ 2+2+ 0+0+ Skyrme 2+2+ 0+0+ 2+2+ 4+4+ 0+0+ Gogny 0+0+ 0+0+ Skyrme 2+2+

35. 35 Andreas Görgen Atelier ESNT 4.-6.2.2008 Shape transition in the light Krypton isotopes 0+0+ 0+0+ 0+0+ 0+0+ 0+0+ 0+0+ 2+2+ 2+2+ 2+2+ 2+2+ 2+2+ 4+4+ 4+4+ 4+4+ 6+6+ 6+6+ 6+6+ there is also mixing between K=0 and K=2 states (and there are third 2 + states) experimental E x 72 Kr 74 Kr 76 Kr ? 2+2+ transition from prolate 78 Kr to oblate 72 Kr crossing of configurations in 74 Kr

36. 36 Andreas Görgen Atelier ESNT 4.-6.2.2008 The light Se isotopes 70 Se Coulex at Rex-ISOLDE 70,72 Se lifetimes from Legnaro  Joa Ljungvall 70 Se 72 Se

37. 37 Andreas Görgen Atelier ESNT 4.-6.2.2008 Shape evolution in the light Krypton isotopes 0+0+ 2+2+ 0+0+ 4+4+ 0+0+ 0+0+ 2+2+ 4+4+ 0+0+ 0+0+ 2+2+ 4+4+ 0+0+ 0+0+ 2+2+ 4+4+ 72 Kr 74 Kr 76 Kr 78 Kr prolateoblate 0+0+ 2+2+ 0+0+ Gogny 0+0+ 2+2+ 0+0+ Skyrme 0+0+ 2+2+ 0+0+ 2+2+ 0+0+ 2+2+ 4+4+ 0+0+ Gogny 0+0+ 2+2+ 0+0+ Skyrme 2+2+ 0+0+ 2+2+ 4+4+ 0+0+ Gogny 0+0+ 0+0+ Skyrme 2+2+ Skyrme: M. Bender, P. Bonche, P.-H. Heenen, PRC 74, 024312 (2006) 2+2+ 2 2+2+ 1 Gogny: M. Girod et al. 2+2+ 2 2+2+ 1 2+2+ 2 2+2+ 1 also mixing of K=0 and K=2 states