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.

R

a
(

)
Y
   ( , );|  | 
Distortion of surface :


Y 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
:
(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 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)
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.
deformed
spherical
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 gammaray emission is
possible
Beam
Target
Coupling vibrational and rotational
angular momentum
z
J



Coupling vibrational angular
momentum K to the rotation R,
giving total angular momentum J.
The z projection of J, M  , is a
constant of motion.
Giving rotational angular
momentum,
R
M
y’
| R |2 | J |2 K 2 2   2[ J ( J  1)  K 2 ]
2
 And rotational energy, Erot ( J )  [ J ( J  1)  K ^ 2]
2I
 Where, I is the moment of inertia.
K
z’
Rotational band structure
2
Erot ( J )  [ J ( J  1)  K ^ 2]
2I



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+,...
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
I rig 
2
AmR 2
5
E. Grosse et al.,Phys rev. Lett. 31, 840 (1973)
Superdeformed bands


Super deformed rotational band of 152
66 Dy
Spins of up to 60 are observed
P.J. Twin et al.,Phys rev. Lett. 57, 811 (1986)