1. The Collective Model
Aard Keimpema

2. Contents Vibrational modes of nuclei Deformed nuclei
Rotational modes of nuclei
Coupling between rotational and vibrational states

3. Nuclear vibrations
The absorbtion spectrum of nuclei can be understood in terms of vibrations and rotations of the nucleus.
Distortion of surface :
is spherical harmonic, λ is the multipolarity, aλ(μ) a constant
(λ=0) : monopole, (λ=1) : dipole, etc…
Oscilatations are quantized: vibrational quantum of frequency ωλ is called a phonon
Phonons of frequency ωλ has - energy :
- momentum :
- parity :

4. Isospin Nucleons can vibrate in two ways :
Protons and neutrons move in same direction, ΔI=0 (isoscalar)
Protons and neutrons move in opposing direction, ΔI=1 (isovector)

5. Vibrational modes λ=0 (monopole), radial vibrations
λ=1 (dipole), no isoscalar modes (no dipole moment in center of mass shift)
λ=2 (quadrupole), shape oscillations.

6. Microscopic interpretation of vibrational modes
Vibrations are identified with transitions between shell model states.
E.g. transition: 2p3/2(N=3) →2d5/2(N=4)
Transitions group around certain energies, Giant resonances / E

7. Photodisintegration spectrum of 197Au
Gold atoms are bombarded with high energy gamma rays. Prompting the gold to emit neutrons.
This is the first observed giant dipole resonance
S.C. Fulz, Phys. Rev. Lett. 127, 1273 (1963)

8. Deformed nuclei I
Nuclei around magic numbers are spherically symmetric.
Adding neutrons to a closed shell nucleus leads to suppression of vibrational states.
Nucleus becomes less compact, leads stable deformations.
In deformed nuclei, also rotational states are possible.
Not possible in spherical symmetric nuclei, because of indistinguishability of the angular parameters.

9. Deformed nuclei II
For low angular momentum nuclei can have either an Oblate (like the earth) or a Prolate (like a rugby ball) shape.
Rotations associated with valance nucleons.
For high angular momentum, deformations have a prolate shape.
Rotations associated with rotation of the core
Angular momenta can get very high.

10. Gamma induced emission of neutrons in neodymium
spherical
deformed
Cross-section for gamma induced emission of neutrons.
The neodymium progresses from spherically symmetric to deformed.
First peak in 150Nd, vibrations along symmetry axis.
Second peak in 150Nd, vibrations orthongonal to symmetry axis.

11. How to make a rotating nucleus
Beam
Target
A beam of ions is shot at a target
Peripheral collisions, may lead to fusion of two nuclei.
Initially the compound nucleus will emit light particles.
Finally, only gamma-ray emission is possible

12. Coupling vibrational and rotational angular momentumz J R y’ z’
Coupling vibrational angular momentum K to the rotation R, giving total angular momentum J.
The z projection of J, , is a constant of motion.
Giving rotational angular momentum,
And rotational energy,
Where, I is the moment of inertia.

13. Rotational band structure
For given J, the K for which [J(J+1)-K2] is a minimum defines the lowest energy.
Lowest energy states are called the yrast states
For a nucleus in the groundstate, the states are filled in opposing K’s, k and -k ( giving total K=0)
Angular momentum states : Jp=0+,2+,4+,...

14. Moment of inertia
When viewing the moment of inertia as function of energy, we find 3 zones.
Zone 1: As ω increases, the nucleus stretches and I increases
Zone 2: Coriolis force, work opposite on K and –K. Thus a preffered K direction is introduced. This will break the pairing. (backbending).
Zone 3: The moment of inertia assumes the rigid body value
E. Grosse et al.,Phys rev. Lett. 31, 840 (1973)

15. Superdeformed bands Super deformed rotational band of
Spins of up to are observed
P.J. Twin et al.,Phys rev. Lett. 57, 811 (1986)