The Collective Model

Aard Keimpema

Contents

Vibrational modes of nuclei Deformed nuclei Rotational modes of nuclei Coupling between rotational and

vibrational states

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 :

||);,()( YaR

Y

)1(

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)

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.

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

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)

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.

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.

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.

spherical

deformed

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

BeamTarget

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.

z

M

K

J

R

y’

z’

M

])1([|||| 222222 KJJKJR

]2^)1([2

)(2

KJJI

JErot

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+,...

]2^)1([2

)(2

KJJI

JErot

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

2

5

2AmRI rig

E. Grosse et al.,Phys rev. Lett. 31, 840 (1973)

Superdeformed bands

Super deformed rotational band of Spins of up to are observed

Dy15266

60

P.J. Twin et al.,Phys rev. Lett. 57, 811 (1986)