collective properties of even-even nuclei – miscellaneous topics vibrators and rotors

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