collective model. nuclei z n character j q obs. q sp. qobs/qsp 17 o 8 9 doubly magic+1n 5/2 -2.6...
TRANSCRIPT
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
• 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
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