Collective properties of even-even nuclei

1. Collective properties of even-even nuclei Vibrators and rotors With three Appendices

2. What happens with both valence neutrons and protons? Case of few valence nucleons: Lowering of energies, development of multiplets. R4/2  ~2 Vibrational modes, 1- and multi-phonon 2-particle spectra Intermediate

3. Lots of valence nucleons of both types R4/2 ~3.33

4. B(E2; 2+ 0+ )

5. Broad perspective on structural evolution: R4/2 Note the characteristic, repeated patterns

6. Development of collective behavior in nuclei • Results primarily from correlations among valence nucleons. • Instead of pure “shell model” configurations, the wave functions are mixed – linear combinations of many components. • Leads to a lowering of the collective states and to enhanced transition rates as characteristic signatures. • How does this happen? Consider mixing of states.

7. A illustrative special case of fundamental importance Lowering of one state. Note that the components of its wave function are all equal and in phase T Consequences of this: Lower energies for collective states, and enhanced transition rates. Lets look at the latter in a simple model.

8. W

24. Even-even Deformed Nuclei Rotations and vibrations

26. 8+ 6+ 4+ 2+ 0+ Rotational states built on(superposed on) vibrational modes Vibrational excitations Rotational states Ground or equilibirum state

27. Systematics and collectivity of the lowest vibrational modes in deformed nuclei

28. E2 transitions in deformed nuclei • Intraband --- STRONG, typ. ~ 200 W.u. in heavy nuclei • Interband --- Collective but much weaker, typ. 5-15 W.u. Which bands are connected? • Alaga Rules for Branching ratios

30. 0

31. Experimental B(E2) values in deformed nuclei

41. How to fix the model? Note: the Alaga rules assume that each band is pure – ground or gamma, in character. What about if they MIX ?? Bandmixing formalism

42. Mixing of gamma and ground state bands

46. Axially Asymmetric NucleiTwo types: “gamma” soft (or “unstable”), and rigid

47. First: Gamma soft E ~ L ( L + 3 ) ~ Jmax ( Jmax + 6 ) Note staggering in gamma band energies

48. Overview of yrast energies E ~ J ( J + 1 ) E ~ J ( J + 6 ) 8 E ~ J ~ J ( J + )

49. “Gamma” rigid or Davydov model Note opposite staggering in gamma band energies

50. Use staggering in gamma band energies as signature for the kind of axial asymmetry