1. More General IBA Calculations Spanning the triangle How to use the IBA in real life

2. Classifying Structure -- The Symmetry Triangle Most nuclei do not exhibit the idealized symmetries but rather lie in transitional regions. Mapping the triangle. Sph. Deformed

3. Mapping the Triangle with a minimum of data -- exploiting an Ising-type Model –The IBA Mapping the Triangle with a minimum of data -- exploiting an Ising-type Model –The IBA Competition between spherical-driving (pairing – like nucleon) and deformation-driving (esp. p-n) interactions H = aH sph + bH def Structure ~ a/b Sph. Def.

4. Relation of IBA Hamiltonian to Group Structure We will now see that this same Hamiltonian allows us to calculate the properties of a nucleus ANYWHERE in the triangle simply by choosing appropriate values of the parameters

5. V (γ) vs. χ H = -κ Q Q Only minimum is at γ = 0 o All γ excursions due to dynamical fluctuations in γ (γ-softness), not to rigid asymmetric shapes. This is confirmed experimentally !!! O(6) SU(3)U(5)  What is the physical meaning of  If you think about zero point motion in a potential like this, it is clear that depends on  For a flat potential the nucleus oscillates back and forth from 0 to 60 degrees so = 30 deg. For SU(3), will be small – nucleus is axially symmetric.

6. O(6) SU(3)U(5) 

7. Mapping the Entire Triangle with a minimum of data Mapping the Entire Triangle with a minimum of data 2 parameters 2-D surface  H = ε n d -  Q  Q Parameters: ,  (within Q) Awkward, though that varies from 0 to infinity  /ε Use of this form of the Hamiltonian, with T(E2) = aQ, is called the Consistent Q Formalism (or CQF). Roughly 94.68572382% of IBA calculations are done this way.

8. Spanning the Triangle H = c [ ζ ( 1 – ζ ) n d 4NB 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

9. CQF along the O(6) – SU(3) leg H = -κ Q Q Only a single parameter,  H = ε n d -  Q  Q Two parameters ε /  and 

12. Os isotopes from A = 186 to 192: Structure varies from a moderately gamma soft rotor to close to the O(6) gamma- independent limit. Describe simply with: H = -κ Q Q  : 0  small as A decreases

13. Universal O(6) – SU(3) Contour Plots in the CQF H = -κ Q Q χ = 0 O(6) χ = = - 1.32 SU(3) ( χ = - 2.958 ) O(6) SU(3)U(5)  SU(3) O(6)

17. Now, what about more general calculations throughout the triangle Spanning the triangle How do we fix the IBA parameters for any given collective nucleus?

18. 164 Er, a typical deformed nucleus

19. H has two parameters. A given observable can only specify one of them. What does this imply? An observable gives a contour of constant values within the triangle = 2.9 R4/2

20. At the basic level : 2 observables (to map any point in the symmetry triangle) Preferably with perpendicular trajectories in the triangle A simple way to pinpoint structure. What do we need? Simplest Observable: R 4/2 Only provides a locus of structure 3.3 3.1 2.9 2.7 2.5 2.2

21. Contour Plots in the Triangle 3.3 3.1 2.9 2.7 2.5 2.2 R 4/2 2.2 4 7 13 10 17 2.2 4 7 10 13 17 0.1 0.05 0.01 0.4

22. We have a problem What we have: Lots of What we need: Just one +2.9 +2.0 +1.4 +0.4 +0.1 -0.1 -0.4 -2.0-3.0 Fortunately:

23. VibratorRotor γ - soft Mapping Structure with Simple Observables – Technique of Orthogonal Crossing Contours Burcu Cakirli et al. Beta decay exp. + IBA calcs.

25. R 4/2 = 2.3 = 0.0 156 Er

26. Trajectories at a Glance R 4/2

27. Evolution of Structure Complementarity of macroscopic and microscopic approaches. Why do certain nuclei exhibit specific symmetries? Why these evolutionary trajectories? What will happen far from stability in regions of proton-neutron asymmetry and/or weak binding?