Collective properties of even-even nuclei

Vibrators and rotors

With three Appendices

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

Lots of valence nucleons of both types

R4/2 ~3.33

B(E2; 2+ 0+ )

Broad perspective on structural evolution: R4/2

Note the characteristic, repeated patterns

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

Even-even Deformed Nuclei

Rotations and vibrations

0+2+4+

6+

8+

Rotational states

Vibrational excitations

Rotational states built on(superposed on) vibrational modes

Ground or equilibirum

state

Systematics and collectivity of the lowest vibrational

modes in deformed nuclei

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

0

Experimental B(E2) values in deformed nuclei

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

Appendix A

on Intruder States

Another form of collective mode that sometimes appears in the low lying spectrum or can even

become the ground state equilibrium cofiguration

The basic idea behind Intruder States: a 2-

particle - 2-hole excitation that costs energy but gains it

back by added collectivity which

increases with increasing valence nucleon number.

Burcu Cakirli et al.Beta decay exp. + IBA calcs.

Appendix B

on development of collectivity

and lowering of collective

energies by configuration

mixing

Appendix C

on energies and transition

rates of 3-phonon states in terms of 2-phonon state

anharmonicities