Collective Model
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Transcript Collective Model
Experimental evidence for closed nuclear shells
Deviations from Bethe-Weizsäcker mass formula:
B/A (MeV per nucleon)
Neutron
Proton
28
28
50
50
82
very stable:
4
2
He2
16
8 8
40
20
20
48
20
28
208
82
126
O
126
82
Ca
Ca
Pb
mass number A
Shell structure from masses
•
Deviations from Weizsäcker mass formula:
Energy required to remove two neutrons from nuclei
(2-neutron binding energies = 2-neutron “separation” energies)
N = 82
25
23
21
N = 126
S(2n) MeV
19
17
15
13
Sm
11
Hf
9
Ba
N = 84
7
Pb
Sn
5
52
56
60
64
68
72
76
80
84
88
92
96
100
Neutron Number
104
108
112
116
120
124
128
132
Shell structure from Ex(21) and B(E2;2+→0+)
high energy of first 2+ states
low reduced transition probabilities B(E2)
The three faces of the shell model
Average nuclear potential well: Woods-Saxon
A
A
pˆ i2
ˆ
H
Vˆ ri , rj
i 1 2mi
i j
A
A
pˆ i2
A ˆ
ˆ
ˆ
H
V ri V ri , rj Vˆ ri
i 1 2mi
i 1
i j
2 2
V r r 0
2m
r
u r
Y m , X ms
r
V r V0 /1 expr R0 / a
Woods-Saxon potential
Woods-Saxon gives proper magic numbers
(2, 8, 20, 28, 50, 82, 126)
Meyer und Jensen (1949): strong spin-orbit interaction
2 2
V r Vs r s r 0
2
m
1 dV
Vs r ~
r dr
mit
0
dV r
dr
V r
Spin-orbit term has its origin in the relativistic description of the single-particle motion in the nucleus.
r
Woods-Saxon potential (jj-coupling)
1
s
j 2 2 s2 2
j s
2
1
j j 1 1 s s 1 2
2
The nuclear potential with the spin-orbit term is
V r Vs
2
V r
1
Vs
2
for
j 1/ 2
for
j 1/ 2
spin-orbit interaction leads to a large splitting for large ℓ.
j 1 / 2
j 1 / 2
j 1 / 2
1 / 2 Vs
/ 2 Vs
Woods-Saxon potential
The spin-orbit term
1 2
Es
1 2
2 1 2
Vs
2
reduces the energy of states with spin
oriented parallel to the orbital angular
momentum j = ℓ+1/2 (Intruder states)
reproduces the magic numbers
large energy gaps → very stable nuclei
Important consequences:
Reduced orbitals from higher lying N+1 shell
have different parities than orbitals from the N shell
Strong interaction preserves their parity. The reduced orbitals
with different parity are rather pure states and do not mix
within the shell.
Shell model – mass dependence of single-particle energies
Mass dependence of the neutron
energies: E ~ R 2
Number of neutrons in each level:
2 2 1
½ Nobel price in physics 1963: The nuclear shell model
Experimental single-particle energies
γ-spectrum
single-particle energies
1 i13/2
1609 keV
2 f7/2
896 keV
1 h9/2
0 keV
209
83
Bi126
208Pb
→ 209Bi
Elab = 5 MeV/u
Experimental single-particle energies
γ-spectrum
208Pb
single-hole energies
3 p3/2
898 keV
2 f5/2
570 keV
3 p1/2
0 keV
207
82
Pb125
→ 207Pb
Elab = 5 MeV/u
Experimental single-particle energies
particle states
209Bi
2 f7/2
1609 keV
896 keV
1 h9/2
0 keV
1 i13/2
208
82
209Pb
energy of shell closure:
Pb126
BE(209Bi) BE(208Pb) E(1h9 / 2 )
BE(207Tl ) BE(208Pb) E(3 s1/ 2 )
E 1h9 / 2 E (3 s1/ 2 ) BE( 209Bi) BE( 207Tl ) 2 BE( 208Pb)
4.211MeV
207Tl
207Pb
BE(209Pb) BE(208Pb) E(2 g9 / 2 )
hole states
proton
BE(207Pb) BE(208Pb) E(3 p1/ 2 )
E 2 g9 / 2 E (3 p1/ 2 ) BE( 209Pb) BE( 207Pb) 2 BE( 208Pb)
3.432
Level scheme of 210Pb
2846 keV
2202 keV
1558 keV
1423 keV
779 keV
0.0 keV
-1304 keV (pairing energy)
M. Rejmund Z.Phys. A359 (1997), 243
209
82
Pb127
Level scheme of 206Hg
2345 keV
d3/12 d5/12
s1/12 d5/12
207
81
Tl126
1348 keV
997 keV
0.0 keV
B. Fornal et al., Phys.Rev.Lett. 87 (2001) 212501
Success of the extreme single-particle model
Ground state spin and parity:
Every orbit has 2j+1 magnetic sub-states,
fully occupied orbitals have spin J=0,
they do not contribute to the nuclear spin.
For a nucleus with one nucleon outside a
completely occupied orbit the nuclear spin is
given by the single nucleon.
nℓj→J
(-)ℓ = π
Success of the extreme single-particle model
magnetic moments:
The g-factor gj is given by:
j g g s s g j j
2
with 2 j s j 2 2 j s s 2
j
j j
j g g s s
j j
2
s 2 j j 2 2 j 2
g j j 1 1 3 / 4 g s j j 1 1 3 / 4
j
2 j j 1
1
1 1 s s 1
g j g g s
g g s
2
2
j j 1
Simple relation for the g-factor
of single-particle states
g g
g Kern g s
K
2 1
for
j 1/ 2
Success of the extreme single-particle model
magnetic moments:
1 1
g j g s K
2 2
z
j g j 3 1 g
s
K
j 1
2 2
j 1/ 2
für j 1 / 2
für
g-faktor of nucleons:
proton:
gℓ = 1; gs = +5.585
neutron: gℓ = 0; gs = -3.82
proton:
für
j 2.293 K
j
z
j 2.293
K für
j
1
neutron:
für
1.91 K
j
z
1.91
K für
j 1
j 1 / 2
j 1/ 2
j 1 / 2
j 1/ 2
Magnetic moments: Schmidt lines
magnetic moments: proton
magnetic moments: neutron