1. Quantum shape transitions in the shapes of atomic nuclei. J. Jolie, Universität zu Köln

2. How do complex systems emerge from simple ingredients Basic ingredients: two sets of indistinguishable fermions a complex short range force (Van der Waals typ) the possibility that one kind of fermions becomes the other kind + Binding energy 3fm The atomic nucleus forms a unique two-component mesoscopic system, which is hard to manipulate but genereous in the number of observables it emits.

3. The interacting boson approximation (IBA) shell structure: valence nucleons Cooper pairing: N s,d boson system Collective motion: nuclear shapes Once the atomic nucleus is formed effective (in-medium) forces generate simple collective motions.

4. Most nuclei are very well described by a very simple IBA hamiltonian: with two structural parameters  and  and a scaling factor a  with generates spherical shape generates deformed shape

5. U(5)  The rich structure of this simple hamiltonian are illustrated by the Casten triangle  SU(3) U(5) limit U(6)  U(5)  O(5)  SO(3) O(6) limit U(6)  O(6)  O(5)  SO(3) SU(3) limit U(6)  SU(3)  SO(3) The simple hamiltonian has four dynamical symmetries O(6)  SU(3) SU(3) limit U(6)  SU(3)  SO(3)

6. Nuclear shapes associated with the four dynamical symmetries The shapes can be studied using the coherent state formalism. using the intrinsic state (Bohr) variables: Then the energy functional: can be evaluated for each value of  and 

7. U(5) limit: irrelevant: spherical vibrator O(6) limit: flat:  -unstable rotor SU(3) limit: prolate rotor SU(3) limit: oblate rotor   SU(3)  E U(5) O(6) 0° 60°

8. Experimental example: 196 Pt the first O(6) nucleus 110 Cd a U(5) nucleus Discovered in 1978. Cizewski, Casten, Smith, Stelts, Kane, Börner, Davidson, Phys. Rev. Lett. 40 (1978) 167 Boerner, Jolie, Robinson, Casten, Cizewski Phys. Rev. C42 (1990) R2271 M. Bertschy, S. Drissi, P.E. Garrett, J. Jolie, J. Kern, S.J. Manannal, J.P. Vorlet, N. Warr, J. Suhonen, Phys. Rev C 51 (1995) 103

9. Besides the atomic nuclei representing a dynamical symmetry, the IBA is also able to describe transitional nuclei.

10. Shape phase transitions in the atomic nucleus. When studying the changes of the nuclear shape one might observe shape phase transitions of the groundstate configuration. They are analogue to phase transitions in crystals

11. T cc  c  c   min T cc  P 0,T,  min  T cc  P 0,T,  First order phase transition with P = P 0 = const cc  c  c   min cc  P 0,T,  min  T cc  P 0,T,  Second order phase transition

12. Thermodynamic potential: External parameters Order parameter Energy functional: L. Landau Landau theory of continuous phase transitions (1937) describes these shape phase transitions. J.Jolie, P. Cejnar, R.F. Casten, S. Heinze, A. Linnemann, V. Werner, Phys. Rev. Lett. 89 (2002) 182502. P. Cejnar, S. Heinze, J.Jolie, Phys. Rev. C 68 (2003) 034326 P T

13. In the case of our simple hamiltonian Landau theory gives the following Solution for  min  

14. spherical  = 0 prolate deformed  > 0 oblate deformed  < 0 The new nuclear shape phase diagram U(5)   SU(3) O(6)  SU(3) first order transition second order transition J. Jolie, R.F. Casten, P. von Brentano, V. Werner, Phys.Rev.Lett.87 (2001)162501 Triple point of nuclear deformation

15. The shape phase transitions can be seen by the groundstate energies.  E  SU(3) O(6) SU(3) U(5) (N=40)

16. Two-neutron separation energies have been used before to identify The phase transitions: Sm S 2 (N) MeV N First order transition in U(5) to SU(3)

17. The quadrupole moment corresponds to the control parameter  0 : N=10N=40 N=10 N=40 A sensitive signature is in particular the B(E2;2 2 + -> 2 1 + )

18. Experimental examples for the prolate-oblate phase transition J.Jolie, A. Linnemann Phys. Rev. C 68 (2003) 031301. R 4/2 B(E2;2 + 2 ->2 + 1 )[W.u] Q(2 + 1 )[eb] Pb Hg Pt Os W Hf Yb 104 106 108 110 112 114 116 118 120 122 124 126 200 Hg 198 Hg 196 Pt 194 Pt 192 Os 190 Os 188 Os 186 W 184 W 182 W 180 Hf

19. Following a collective model approach F. Iachello introduced new symmetries that describe certain nuclei at the phase transition: Critical point symmytries, i.e. X(5) and E(5) F. Iachello, Phys. Rev. Lett.85 (2000) 3580 and 87 (2001) 052502. 152 Sm R.F. Casten, V. Zamfir, Phys. Rev. Lett. 85 (2000)3584 X(5)

20. Recent plunger lifetime measurements seem to confirm the existence of X(5) in several nuclei: N=90 150 Nd: R. Krücken et al. Phys. Rev. Lett. 88 (2002) 232501. 154 Gd: D. Tonev et al. Phys. Rev. C69 (2004) 034334 156 Dy: O. Möller et al. Phys. Rev. C74 (2006) 024313

21. Where do we expect the shape transitions? E.A.McCutchan et al., Phys. Rev. C69 (2004) 024308

22. New examples were indeed found in the Osmium isotopes. A.Dewald et al. AIP Conf. Ser. 831 (2006) 195 B.Melon et al. to be publ.

23. IBM 178 Os GCM η=0.766 χ=-√7/2 H= T + V(β, γ) Gneuss and Greiner D. Troltenier et al. Z. Phys. A338,261(1991 ) F. Iachello and A. Arima Cambridge University Press, 1987 B2=67.47 P3=0.0748 C2=174.9; C3=309.25;C4=3547.4; C5=0.0; C6=0.0; D6=3712.5 H = c[  n d – ( 1-  )/N Q·Q] ; Q(χ) H= T + V(β,γ) X(5) nuclei and the Interacting Boson Model

24. E(5) X(5) U(5) O(6) SU(3) 156 Dy 176,178 Os 152 Sm 154 Gd, 150 Nd A systematic study allows to place most nuclei in the two parameter extended Casten triangle. A. Dewald et al. AIP Conf. Ser. 831 (2006) 195

25. Level dynamics and phase transitions Up to now we concentrated only on the lowest states, what happens with higher excited states and the level density? U(5)-SU(3) first order shape phase transition Energies of 0 + states N=30 SU(3) U(5)

26. Energy of 0 + states up to 2.5 MeV as a function of  in the spherical-deformed transition for N = 30. From P. Cejnar and J. Jolie, Phys. Rev. E (2000) 6237

27. Q3D Spectrometer To excite the 0 + states the ideal and very complete way is using the (p,t) transfer reaction at the high resolution Q3D spectrometer (Garching). Can this be experimentally observed? Energy resolution: ~4 keV for 15-20 MeV tritons. (Yale/Köln/Bucarest/ Surrey/LMU-TU Munchen collaboration). Eight nuclei in the rare earth region were systematically studied up to 3 MeV.

28. Result: D.A. Meyer et al, Phys.Lett. B 638(2006) 44

29. # of 0+ states below 2.5 MeV 5 10 0 0 0.5 1.0  162 Dy 168 Er 158 Gd 176 Hf 154 Gd 152 Gd 180 W 184 W

30. Level dynamics in the U(5) to O(6) phase transition S. Heinze, P. Cejnar, J. Jolie, M. Macek, Phys. Rev. C73 (2006) 014306 M. Macek, P. Cejnar, S. Heinze, J. Jolie, Phys. Rev. C73 (2006) 014307 with 0 0 3 3 3 0 6 6 9 0 v 0 U(6)  U(5)  O(5)  SO(3) U(6)  O(6) v L Energy of 0 + states

31. After selection by v for 0+ states (N=80). v=0 v=18 Neighbour spacing v=0 v=18 Absolute energies

32.  0.0 1.0 Overlap with U(5) basis Here there are very interesting theoretical issues

33. Monodromy  Energy of 0 + states P. Cejnar, M. Macek, S. Heinze, J. Jolie and J. Dobes, Journ. of Phys. A 39, L515 (2006).

34. D.D. Warner, Nature 420 614 (2002) Conclusions -) The shapes of atomic nuclei undergo quantum phase transitions which are smoothed through the finite particle number. -)The IBM provides a realistic and very rich framework to study shapes and their relation to quantum phase transitions like Ising models do. -) New lifetime experiments in ground state bands allow to better identify X(5) nuclei and to confirm a strong correlation with the P-factor. -) The finite N and excited states phase transitions form an unknown field. -) Level dynamics exhibit bunching at E=0 reveals clues to the fixing of quantum numbers in the particular potential (monodromy).

35. Thanks to: A.Dewald, S. Heinze, A. Linnemann, (V. Werner), P. von Brentano, Universität zu Köln; R.F. Casten, (E. A. McCutchan), (V. Zamfir), Yale University; P. Cejnar, M. Macek Charles University Prague; General references: P. Cejnar and J. Jolie Prog. Part. Nucl. Phys.62 (2009) 210 P. Cejnar, J. Jolie, R.F. Casten, to be publ. in Rev. Mod. Phys. A. frank, J. Jolie, P. van Isacker, Springer Tracts in Modern Physics Vol 230 (2009)

36. The analogy with thermodynamics can be further investigated. Specific heat: SU(3) -U(5) P. Cejnar, S. Heinze, J. Jolie, Phys. Rev. C68 (2003) 034326. N= N=80 N=40 N=20 N=10 O(6) -U(5)

37. But also the U(5) wavefunction entropy can be used: with O(6) -U(5) SU(3) -U(5)

38. should be continuous everywhere. if discontinuous at  0 : first order phase transition. if discontinuous at  0 : second order phase transition. with

39. B -A 00 Solution: First order phase transitions at: Second order at: or

40. So we can absorb it by allowing negative  values ! and Energy functional in coherent state formalism

41. One obtains then: when we fix N: prolate-oblate The first order phase transitions should occur when spherical-deformed The isolated second order transition at:

42. spherical  = 0 prolate deformed  > 0 oblate deformed  < 0 Triple point of nuclear deformation : first order transition : isolated second order transition I III II P T J.Jolie, P. Cejnar, R.F. Casten, S. Heinze, A. Linnemann, V. Werner, Phys. Rev. Lett. 89 (2002)182502 Landau theory and nuclear shapes. Thermodynamic potential: External parameters Order parameter Energy functional:  (P,T;  ) E(N,  )

43. Landau&Lifschitz Statistical Physics §144

44. Analog system exists for the isotropic-nematic liquid crystal phases Isotropic phase Nematic phase Landau-de Gennes theory for unaxial phases. when D=E=0 B A 0 0 -0.2 -2 2