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, ? is the
  multipolarity, a?(µ) a constant
• (?0) monopole, (?1) dipole, etc
• Oscilatations are quantized vibrational quantum
  of frequency ?? is called a phonon
• Phonons of frequency ?? has - energy
  
• 
  - momentum
• 
  - parity

4
Isospin

• Nucleons can vibrate in two ways
• Protons and neutrons move in same direction,
  ?I0 (isoscalar)
• Protons and neutrons move in opposing direction,
  ?I1 (isovector)

5
Vibrational modes

• ?0 (monopole), radial vibrations
• ?1 (dipole), no isoscalar modes (no dipole
  moment in center of mass shift)
• ?2 (quadrupole), shape oscillations.

6
Microscopic interpretation of vibrational modes

• Vibrations are identified with transitions
  between shell model states.
• E.g. transition 2p3/2(N3) ?2d5/2(N4)
• 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
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
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
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(J1)-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 Ks, k and -k ( giving total
  K0)
• Angular momentum states Jp0,2,4,...

14
Moment of inertia

• When viewing the moment of inertia as function of
  energy, we find 3 zones.
• Zone 1 As ? 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)
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Superdeformed bands

• Super deformed rotational band of
• Spins of up to are observed

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