Collective Model

1
Collective Model
2
Collective Model
Shell model fails for electric quadrupole moments.

• Nuclei Z N Character
  j Qobs. Qsp. Qobs/Qsp
• 17O 8 9 doubly magic1n
  5/2 -2.6 -0.1 20
• 39K 19 20 doubly magic -1p
  3/2 5.5 5 1.1
• 175Lu 71 104 between shells
  7/2 560 -25 -20
• 209Bi 83 126 doubly magic1p 9/2
  -35 -30 1.1

Many quadrupole moments are larger than predicted
by the model.
Consider collective motion of all nucleons
3
Collective Model

• Two types of collective effects nuclear
  deformation leading to collective modes of
  excitation, collective oscillations and
  rotations.
• Collective model combines both liquid drop model
  and shell model.
• A net nuclear potential due to filled core shells
  exists.
• Nucleons in the unfilled shells move
  independently under the influence of this core
  potential.
• Potential is not necessarily spherically
  symmetric but may deform.

4
Collective Model

• Interaction between outer (valence) and core
  nucleons lead to permanent deformation of the
  potential.
• Deformation represents collective motion of
  nucleons in the core and are related to liquid
  drop model.
• Two major types of collective motion
• Vibrations Surface oscillations
• Rotations Rotation of a deformed shape

5
Vibrations

• A nearly closed shell should have spherical
  surface which is deformable. Excited states
  oscillate about this spherical surface.
• Simplest collective motion is simple harmonic
  oscillation about equilibrium.W0 static
  deformation, due to Coulomb repulsion. Alt150 it
  is negligible.

6
Vibrations
Average shape is spherical but instantenous shape
is not.
7
Vibrations

• It is convenient to give the instantaneous
  coordinate R(t) of a point on the nuclear surface
  at (?, ?) in terms of the spherical harmonics ,

Due to reflection symmetry
8
Vibrations

• ?0, vibrationMonopole

R(t)Ravr ?00 Y00
Breathing mode of a compressible fluid.
The lowest excitation is in nuclei with A grater
than about 40 at an energy above the ground
state E0 ? 80 A-1/3 MeV
9
Dipole Vibrations

• ?1,VibrationDipole

10
Dipole Vibrations

• The dipole mode corresponds to an overall
  translation of the centre of the nuclear fluid.
  Proton and neutron fluid oscillate against each
  other out of phase. It occurs at very high
  energies, of the order 10-25 MeV depending on the
  nucleus. This is a collective isovector (I 1)
  mode. It has quantum numbers J?1- - in
  even-even nuclei, occurs at an energy
• E1 ? 77 A-1/3 MeV
• above the ground state, which is close to that of
  the monopole resonance
• Energy of the giant dipole resonance should be
  compared with shell model energy
• E1 ? 77 A-1/3 MeVwg Eshell ? 40 A-1/3
  MeVw0

11
Quadrupole Vibrations

• ?2,Vibration

The shape of the surface can be described by Y2m
m2, 1, 0.In the case of an ellipsoid RR(?)
hence m0.
12
Quadrupole Vibrations

• Quantization of quadrupole vibration is called a
  quadrupole phonon, Jp2. This mode is dominant.
  For most even-even nuclei, a low lying state with
  Jp2 exists and near closed shells second
  harmonic states can be seen w/ Jp0, 2 , 4 .
• A giant quadrupole resonance at

E2 ? 63 A-1/3 MeV
13
Quadrupole Vibrations

• For a harmonic motion

14
Quadrupole Vibrational Levels of 114Cd

• of phonons E
  

N2
two-phonon triplet
single-phonon state
N1
ground state
N0
15
? ? 3 vibrations
Octupole modes with ?3 w/ Jp3 can be observed
in many nuclei.
16
Nuclear Rotations
In the shell model, core is at rest and only
valance nucleon rotates. If nucleus is deformed
and core plus valance nucleon rotate
collectively.
The energy of rotation (rigid rotator) is given by
17
Nuclear Rotations

• Solutions

But there is reflection symmetry so odd J is not
acceptable. Allowed values of J are 0, 2, 4, etc.
Parity
18
Nuclear Rotations
Energy levels of 238U.
19
Nuclear Rotations
Deformed nucleus with spin J

• Let us now extend the arguments to a general
  case. Consider a nucleus with core plus one
  valance particle. The core give rise to a
  rotational angular momentum perpendicular to the
  symmetry axis-z so that Rz0. The valance nucleon
  produces an angular momentum j

20
Nuclear Rotations
21
Nuclear Rotations
K0 is spinless. K?0 spins of rotational bands
are given
22
Nuclear Rotations

• The ratio of excitation energies of the second to
  the first excited state is obtained by putting
  JK2 and JK1