Collective properties of even-even nuclei – Miscellaneous topics
Vibrators and rotors
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.
A illustrative special case of fundamental importance
T
Lowering of one state. Note that
the components of its wave function are all equal and
in phase
Consequences of this: Lower energies for collective states, and enhanced transition rates. Lets look at the latter in a
simple model.
W
The more configurations that mix, the stronger the
B(E2) value and the lower the energy of the
collective state. Fundamental property of
collective states.
Higher Phonon number states: n = 3
Even-even Deformed Nuclei
Rotations and vibrations
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
Note the very small B(E2)
values from the beta band to
the ground and gamma bands
0
g‘
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
Mixing of gamma and ground state bands
Axially Asymmetric Nuclei
Two types: “gamma” soft (or “unstable”), and rigid
First: Gamma soft
E ~ ( + 3 ) ~ Jmax ( Jmax + 6 )
Note staggering in gamma band
energies
E ~ J ( J + 6 )
E ~ J ~ J ( J + )
E ~ J ( J + 1 )
Overview of yrast energies
“Gamma” rigid or Davydov model
Note opposite staggering in gamma
band energies
Use staggering in gamma band energies as signature for the kind of axial asymmetry
Geometric Collective Model
Appendix
on energies and transition
rates of 3-phonon states in terms of 2-phonon state
anharmonicities