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

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

2. + Binding energy 3fm 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 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. Collective motion: nuclear shapes shell structure: valence nucleons Cooper pairing: N s,d boson system The interacting boson approximation (IBA) 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: generates deformed shape generates spherical shape with with two structural parameters h and c and a scaling factor a.

5. 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) SU(3) limit U(6)  SU(3)  SO(3) SU(3) O(6) c = 0 SU(3) U(5) The simple hamiltonian has four dynamical symmetries The rich structure of this simple hamiltonian are illustrated by the Casten triangle h = 1 h = 0

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 b and g.

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

8. Experimental example: 196Pt the first O(6) nucleus 110Cd 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. First order phase transition with P = P0= const F(P0,T,xmin) xmin F(P0,T,x) T>Tc Tc T<Tc Tc T Tc x T Second order phase transition F(P0,T,x) T>Tc F(P0,T,xmin) xmin Tc T<Tc Tc Tc x T T

12. Landau theory of continuous phase transitions (1937) describes these shape phase transitions. Energy functional: P T L. Landau 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 Thermodynamic potential: Order parameter External parameters

13. In the case of our simple hamiltonian Landau theory gives the following Solution for bmin c h

14. first order transition second order transition prolate deformed b > 0 oblate deformed b < 0 SU(3) spherical b = 0 O(6) c = 0 SU(3) U(5) Triple point of nuclear deformation The new nuclear shape phase diagram J. Jolie, R.F. Casten, P. von Brentano, V. Werner, Phys.Rev.Lett.87 (2001)162501 h = 0 h = 1

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

16. Two-neutron separation energies have been used before to identify The phase transitions: S2(N) MeV Sm First order transition in U(5) to SU(3) N

17. The quadrupole moment corresponds to the control parameter b0: N=10 N=40 A sensitive signature is in particular the B(E2;22+-> 21+) N=40 N=10

18. 104 106 108 110 112 114 116 118 120 122 124 126 Pb Hg Pt Os W Hf Yb 200Hg 198Hg 194Pt 196Pt 190Os 188Os 192Os 184W 186W 182W R4/2 B(E2;2+2 ->2+1)[W.u] Q(2+1)[eb] 180Hf J.Jolie, A. Linnemann Phys. Rev. C 68 (2003) 031301. Experimental examples for the prolate-oblate phase transition

19. X(5) 152Sm R.F. Casten, V. Zamfir, Phys. Rev. Lett. 85 (2000)3584 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.

20. Recent plunger lifetime measurements seem to confirm the existence of X(5) in several nuclei: N=90 150Nd: R. Krücken et al. Phys. Rev. Lett. 88 (2002) 232501. 154Gd: D. Tonev et al. Phys. Rev. C69 (2004) 034334 156Dy: 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. • Dewald et al. AIP Conf. Ser. 831 (2006) 195 • Melon et al. to be publ.

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

24. A systematic study allows to place most nuclei in the two parameter extended Casten triangle. U(5) 156Dy 176,178Os 154Gd,150Nd X(5) X(5) 152Sm E(5) O(6) SU(3) SU(3) 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 h in the spherical-deformed transition for N = 30. From P. Cejnar and J. Jolie, Phys. Rev. E (2000) 6237

27. Can this be experimentally observed? 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). Eight nuclei in the rare earth region were systematically studied up to 3 MeV. (Yale/Köln/Bucarest/ Surrey/LMU-TU Munchen collaboration). Q3D Spectrometer Energy resolution: ~4 keV for 15-20 MeV tritons.

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

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

30. Level dynamics in the U(5) to O(6) phase transition v 9 0 6 3 0 6 3 0 3 0 0 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 Energy of 0+ states with U(6)  U(5)  O(5)  SO(3) U(6)  O(6) v L

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

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

33. Monodromy (Monodromia) Energy of 0+ states P. Cejnar, M. Macek, S. Heinze, J. Jolie and J. Dobes, Journ. of Phys. A 39, L515 (2006).

34. 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). D.D. Warner, Nature 420 614 (2002)

35. Thanks to: • 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) O(6) -U(5) N= N=80 N=40 N=20 N=10 P. Cejnar, S. Heinze, J. Jolie, Phys. Rev. C68 (2003) 034326.

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

38. with should be continuous everywhere. if discontinuous at x0 : first order phase transition. if discontinuous at x0: second order phase transition.

39. Solution: x0 B -A First order phase transitions at: Second order at: or

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

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

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

43. Landau&Lifschitz Statistical Physics §144

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