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Experimental evidence for closed nuclear shells
2828 50
50
82
82126
NeutronProton
Deviations from Bethe-Weizsäcker mass formula:
mass number A
B/A
(M
eV p
er n
ucl
eon
)
242 He
8168O
204020Ca
284820Ca
12620882 Pb
very stable:
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)
Sn
Ba
SmHf
Pb
5
7
9
11
13
15
17
19
21
23
25
52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132
Neutron Number
S(2
n)
Me
V
N = 82
N = 84
N = 126
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
aRrVrV /exp1/ 00
02
22
rrV
m
smm XY
r
rur ,
A
jiji
A
i i
i rrVm
pH ,ˆ
2
ˆˆ1
2
A
ji
A
iiji
A
ii
i
i rVrrVrVm
pH
11
2
ˆ,ˆˆ2
ˆˆ
Woods-Saxon potential
Woods-Saxon gives proper magic numbers (2, 8, 20, 28, 50, 82, 126) Meyer und Jensen (1949): strong spin-orbit interaction
02
22
rsrVrV
m s
01
~ mitdr
dV
rrV s
dr
rdV
rV r
Spin-orbit term has its origin in the relativistic description of the single-particle motion in the nucleus.
Woods-Saxon potential (jj-coupling)
2
2222
1112
12
1
ssjj
sjssj
2/12
jforVrV s
The nuclear potential with the spin-orbit term is
spin-orbit interaction leads to a large splitting for large ℓ.
2/12
1
jforVrV s
2/1j
2/1j
2/1j
sV 2/1
sV 2/
Woods-Saxon potential
The spin-orbit term
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 nucleiss VE
2
2
1221
21Important 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:
Number of neutrons in each level: 122
2~ RE
½ Nobel price in physics 1963: The nuclear shell model
Experimental single-particle energies
208Pb → 209Bi Elab = 5 MeV/u
1 h9/2
2 f7/2
1 i13/2 1609 keV
896 keV
0 keV
γ-spectrumsingle-particle energies
12620983 Bi
Experimental single-particle energies
208Pb → 207Pb Elab = 5 MeV/u
γ-spectrum
single-hole energies
3 p1/2
2 f5/2
3 p3/2 898 keV
570 keV
0 keV
12520782 Pb
Experimental single-particle energies
209Pb209Bi
207Pb207Tl
)2()()( 2/9208209 gEPbBEPbBE
)3()()( 2/1208207 pEPbBEPbBE
energy of shell closure:
432.3
)(2)()()3(2 2082072092/12/9
PbBEPbBEPbBEpEgE
)1()()( 2/9208209 hEPbBEBiBE
)3()()( 2/1208207 sEPbBETlBE
MeV
PbBETlBEBiBEsEhE
211.4
)(2)()()3(1 2082072092/12/9
1 h9/2
2 f7/2
1 i13/21609 keV
896 keV
0 keV
12620882 Pb
particle states
hole states
proton
Level scheme of 210Pb
0.0 keV
779 keV
1423 keV
1558 keV
2202 keV
2846 keV
-1304 keV (pairing energy)
M. Rejmund Z.Phys. A359 (1997), 243
12720982 Pb
Level scheme of 206Hg
0.0 keV
997 keV
1348 keV
2345 keV
12/5
12/1
ds
12/5
12/3
dd
B. Fornal et al., Phys.Rev.Lett. 87 (2001) 212501
126207
81Tl
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.
n ℓ j → J (-)ℓ = π
Success of the extreme single-particle model
magnetic moments: The g-factor gj is given by:
with
Simple relation for the g-factor of single-particle states
jgsgg jsj
2222 2 ssjjsj
2222 2
jjjs
j
jj
jjgjjg sj
12
4/3114/311
2/1
12
jfor
gggg s
KernK
j
j
j
jsgg sj
ssj ggjj
ssggg
1
11
2
1
2
1
Success of the extreme single-particle model
magnetic moments:
g-faktor of nucleons:proton: gℓ = 1; gs = +5.585 neutron: gℓ = 0; gs = -3.82
proton:
neutron:
2/1
2
1
2
3
1
2/12
1
2
1
jfürgjgj
j
jfürgjg
Ks
Ks
z
2/1
1293.2
2/1293.2
jfürj
jj
jfürj
K
K
z
2/1
191.1
2/191.1
jfürj
jjfür
K
K
z
Magnetic moments: Schmidt lines
magnetic moments: neutron
magnetic moments: proton