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

 2m

 r  
u r 
 Y m  ,    X ms
r
V r   V0 /1  expr  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   Vs r     s    r   0

2

m


1 dV
Vs 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    Vs
2
V r  
 1
 Vs
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  Vs
 / 2  Vs
Woods-Saxon potential
The spin-orbit term
 1 2
Es 
 1 2
2   1 2
   Vs
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