Collective Model. Nuclei Z N Character j Q obs. Q sp. Qobs/Qsp 17 O 8 9 doubly magic+1n 5/2 -2.6 -0.1 20 39 K 19 20 doubly magic -1p 3/2 +5.5 +5 1.1 175.

Collective Model

Collective Model

• Nuclei Z N Character j Qobs. Qsp. Qobs/Qsp

• 17O 8 9 doubly magic+1n 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 magic+1p 9/2 -35 -30 1.1

Shell model fails for electric quadrupole moments.

Many quadrupole moments are larger than predicted by the model.

Consider collective motion of all nucleons

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.

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

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.W=0 static deformation, due to Coulomb repulsion. A<150 it is negligible.

V V V

x x x

W=0V W>0,, <x>=0 <x>=0V W<0, <x>=0 deformation

Vibrations

Average shape is spherical but instantenous shape is not.

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 ,

φ),(θY (t)αRR(t) λμλ

λ

λ μλμavr

μ- λ,λμ

Due to reflection symmetry

Vibrations

=0, vibration:Monopole

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

Dipole Vibrations

• λ=1,Vibration:Dipole

cosθ2π

3

2

1 αR

)YY and αα ( 0μfor 0α YαR

YαYαYαR

φ)(θθYαRR(t)

3/2

10avr

111- 1,111- 1,1μ1010avr

1- 1,1- 1,10101111avr

1μ

1

1 μ1μavr

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 MeV=wg Eshell 40 A-1/3 MeV=w0

Quadrupole Vibrations

• λ=2,Vibration

1)-θ(3cosπ

5

4

1 αR

θ) offunction a is R shape, lellipsoida(for 0μfor 0α YαR

YαYαYαYαYαR

φ)(θθYαRR(t)

21/2

20avr

2μ2020avr

2- 2,2- 2,2121202021212222avr

1μ

1

1 μ1μavr

The shape of the surface can be described by Y2m m=±2, ±1, 0.In the case of an ellipsoid R=R(θ) hence m=0.

Quadrupole Vibrations

• Quantization of quadrupole vibration is called a quadrupole phonon, Jπ=2+. This mode is dominant. For most even-even nuclei, a low lying state with Jπ=2+ exists and near closed shells second harmonic states can be seen w/ Jπ=0+, 2+ , 4+ .

• A giant quadrupole resonance at

E2 63 A-1/3 MeV

Quadrupole Vibrations

• For a harmonic motion

1/2

N

2

μ2μ

2

μ2μ

222

B

Cω ω;)

2

5(NE

|α|C2

1|α

dt

d|B

2

1rmw

2

1mv

2

1H

of phonons E

1.132 --------------- 0

1.208 --------------- 2

1.283 --------------- 4 ω2 two-phonon triplet

0.558 --------------- 2 ω1 single-phonon state

0 --------------- 0 ω0 ground state

N=2

N=1

N=0

Quadrupole Vibrational Levels of 114Cd

3 vibrations :

Octupole modes with λ=3 w/ Jπ=3 can be observed in many nuclei.

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

2I

RH

2

rot

Nuclear Rotations

• Solutions

... 4, 3, 2, 1, 0,J 1)J(J2I

E

Y1)J(JYR

ΨEΨ2I

R

2

J

JM2

JM2

J

2

J1)(

... 4, 2, 0,J 1)J(J2I

E2

J

Parity But there is reflection symmetry so odd J is not acceptable. Allowed values of J are 0, 2, 4, etc.

Nuclear Rotations

energy excitedfirst of in terms . . . 4, 2, 0,J 1)EJ(J6

1E

E 6

1

2I energy excited 1

2I6 1)2(2

2IE

0E

2J

2

2st

22

2

0

0 --------------- 0

0.0447 --------------- 2

0.148 --------------- 4

0.309 --------------- 6

0.525 --------------- 8

Energy levels of 238U.

Nuclear Rotations

• 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 Rz=0. The valance nucleon produces an angular momentum j

Deformed nucleus with spin J

Nuclear Rotations

)jJjJ(I

1Hj

2I

1jJ2J

2I

1

HjJ2I

1H

2I

R

energy rotational

nucleon valance theofenergy

HHH

KjJ 0R

ΨKΨJ

Ψ1)J(JΨJ 0J ,J

CPR H

yyxx

H

nucleon2

H

zz2

nucleon

2

nucleon

2

nucleonrot

zzz

z

22

z2

Nuclear Rotations

E2K1)J(J2I

E

becomesenergy Total

KJ ; 2K1)J(J2I

EΨEΨH motion, rotational thedescribes jJ2J2I

1H

ΨEΨH nucleus, theof state lrotattiona theoft independen is Hj2I

1 H

1/2Kexcept neglected becan it term),coupling-(rotation termCoriolis : )jJjJ(I

1H

P2

2

KJ,

22

RRRzz2

R

PPnucleon2

P

yyxxC

K=0 is spinless. K≠0 spins of rotational bands are given

2J 1)EJ(J6

1E

0K ; . . . 2,K 1,K K,J

jKJ|j-K| ?J jRJ

Nuclear Rotations

• The ratio of excitation energies of the second to the first excited state is obtained by putting J=K+2 and J=K+1

1

12

)(E)1(E

)(E)2(E

total total

total total

KKJKJ

KJKJ