The Collective Model

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, λis 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. / E 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

7. Photodisintegration spectrum of197Au • 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. deformed spherical Gamma induced emission of neutrons in neodymium • 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. Target Beam How to make a rotating nucleus • 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. z J R z’ y’ Coupling vibrational and rotational angular momentum • 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)