1. Quantum Phase Transitions (QPT) in Finite Nuclei R. F. Casten June 21, 2010, CERN/ISOLDE

2. Themes and challenges of Modern Science Complexity out of simplicity -- Microscopic How the world, with all its apparent complexity and diversity can be constructed out of a few elementary building blocks and their interactions Simplicity out of complexity – Macroscopic How the world of complex systems can display such remarkable regularity and simplicity Degrees of freedom: nucleon coordinates Description: nucleon orbits, interactions Degrees of freedom: nuclear shape variables,  Description: shapes, symmetries, quantum numbers of the many-body system as a whole

3. Quantum (equilibrium) phase transitions in the shapes of strongly interacting finite nuclei as a function of neutron and proton number order parameter control parameter critical point

4. Broad perspective on structural evolution

7. Seeing structural evolution Different perspectives can yield different insights Onset of deformation as a phase transition mediated by a change in shell structure Mid-sh. magic “Crossing” and “Bubble” plots as indicators of phase transitional regions mediated by sub-shell changes

8. Microscopic mechanism of first order phase transition (Federman-Pittel, Heyde) Monopole shift of proton s.p.e. as function of neutron number Gap obliteration 2-space 1-space (N ~ 90 )

9. Vibrator RotorTransitional E β 1 2 3 4 Quantum phase transitions in equilibrium shapes of nuclei with N, Z For nuclear shape phase transitions the control parameter is nucleon number Potential as function of the ellipsoidal deformation of the nucleus

10. E β 1 2 3 4

11. Neutron Number S (2n) MeV E E β 1 2 3 4 

12. Nuclear Shape Evolution  - nuclear ellipsoidal deformation (  is spherical) Vibrational Region Transitional Region Rotational Region Critical Point Few valence nucleons Many valence Nucleons R 4/2 = 3.33R 4/2 = ~2.0

14. Nuclear Shape Evolution  - nuclear ellipsoidal deformation (  is spherical) Vibrational Region Transitional Region Rotational Region Critical Point Few valence nucleons Many valence Nucleons New analytical solutions, E(5) and X(5) R 4/2 = 3.33R 4/2 = ~2.0

15. Bessel equation Critical Point Symmetries First Order Phase Transition – Phase Coexistence E E β 1 2 3 4  Energy surface changes with valence nucleon number Iachello X(5)

17. Casten and Zamfir

19. Comparison of relative energies with X(5)

20. Based on idea of Mark Caprio

22. Li et al, 2009 Flat potentials in  validated by microscopic calculations

23. Minimum in energy of first excited 0 + state Li et al, 2009 E E β 1 2 3 4  Other signatures

24. Isotope shifts Li et al, 2009 Charlwood et al, 2009

25. Treating QPT with the IBA H = c [ ζ ( 1 – ζ ) n d 4N B Q χ ·Q χ - ] ζ χ U(5) 0+0+ 2+2+ 0+0+ 2+2+ 4+4+ 0 2.0 1 ζ = 0 O(6) 0+0+ 2+2+ 0+0+ 2+2+ 4+4+ 0 2.5 1 ζ = 1, χ = 0 SU(3) 2γ+2γ+ 0+0+ 2+2+ 4+4+ 3.33 1 0+0+ 0 ζ = 1, χ = -1.32

26. E(5) X(5) 1 st order 2 nd order Axially symmetric Axially asymmetric Sph. Def.

27. Order parameters Li et al., 2009 Bonatsos et al

28. Q-invariants: model independent shape determinations Good to better than 95% for k =3 ~ Cartoon interpretation of “crossing” of  values across a transitional region

29. Werner et al, 2009 Li et al, 2009

30. E0 transitions in the IBA and phase Transitional Regions For 35 years, it was thought that E0 transitions to the ground state should peak in shape/phase transition regions because of the radius change (E0s had long been associated with changes in radii). It was thought that they should be small elsewhere. The IBA predicts something completely different: E0 transitions should be small for spherical nuclei, should grow rapidly in the shape/phase transition region, and REMAIN large throughout deformed nuclei. This is a robust prediction of the IBA so it is crucial to test it with measurements of E0 transitions to the g.s. in deformed nuclei. Very difficult experimentally.

31. E0 transitions in IBA. Brentano et al, 2004 Li et al, 2009 Delaroche et al, 2009 Traditional model, for 35 years Crit. Pt. E0 0+i  0+00+i  0+0

32. E0 transitions: 0 + i  0 + 0 Data and IBA calculations

33. 0+0+ 0+0+ 2+2+ 2+2+ If inertial properties of ground and excited sequences are very similar, as is likely, it is very difficult to isolate the 0 +  0 + transition. Wimmer et al, Priv. Comm., 2009

34. Enhanced density of 0 + states at the critical point Meyer et al, 2006 E E β 1 2 3 4 

35. Where else? Look at other N=90 nulei

36.   NpNnNpNn p – n P N p + N n pairing What is the locus of candidates for X(5) p-n / pairing P ~ 5 Pairing int. ~ 1 MeV, p-n ~ 200 keV p-n interactions per pairing interaction Hence takes ~ 5 p-n int. to compete with one pairing int.

37. Comparison with the data

38. Summary QPT Geometrical and IBA treatments Microscopic calculations Principle collaborators: Victor Zamfir, E. A. McCutchan, Deseree Meyer, Jan Jolie, Peter von Brentano, Mark Caprio, Dennis Bonatsos, Volker Werner