Shell Model with residual interactions – mostly 2-particle systems
H  H 0  H residual
Start with 2-particle system, that is a nucleus „doubly magic + 2“
H residual  H12 r12 
Consider two identical valence nucleons with j1 and j2
Two questions:
What total angular momenta j1 + j2 = J can be formed?
What are the energies of states with these J values?
Coupling of two angular momenta
j1+ j2
all values from:
j1 – j2 to j1+ j2
(j1 = j2)
Example: j1 = 3, j2 = 5: J = 2, 3, 4, 5, 6, 7, 8
BUT: For j1 = j2:
J = 0, 2, 4, 6, … ( 2j – 1)
(Why these?)
How can we know which total J values are obtained for the coupling of
two identical nucleons in the same orbit with total angular momentum j?
Several methods: easiest is the “m-scheme”.
Coupling of two angular momenta
residual interaction - pairing
 Spectrum 210Pb:
g 92/ 2 ; J  2,4,6,8
 2
 Assume pairing interaction in a single-j shell
j JM J Vpairing r1 , r2  j JM J
2
2
 12 2 j  1  g ,   0, J  0

0,
  2, J  0

energy eigenvalue is none-zero for the ground state;
all nucleons paired (ν=0) and spin J=0.
 The δ-interaction yields a
simple geometrical expression
for the coupling of two particles
8
6
4
2
0
pairing: δ-interaction
E  j1 j2 J   j1 j2 JM V12 j1 j2 JM 
1
j1 j2 J V12 j1 j2 J
2J 1
wave function:  n  m   1 Rn (r )  Ym  ,  
r
interaction:
V12   
 V0
 r1  r2  cos1  cos 2  1  2 
r1r2
E  j1 j2 J   V0  FR n1 1n2 2  A j1 j2 J 
with FR n1 1n2 2   1
1 2
R r   Rn22 2 r  dr
2 n1 1

4 r
and
j2
J
 j

A j1 j2 J   2 j1  1  2 j2  1   1
1
/
2

1
/
2
0


A. de-Shalit & I. Talmi: Nuclear Shell Theory, p.200
2
δ-interaction (semiclassical concept)
J 2  j12  j22  2 j1 j2 cos


j1
J 2  j12  j22 J ( J  1)  j1 ( j1  1)  j2 ( j2  1)
cos 

2 j1 j2
2 j1 ( j1  1) j2 ( j2  1)

j2
J 2  2 j2
cos 
2 j2
for
j1  j2  j
and
j , J  1
θ = 00 belongs to large J, θ = 1800 belongs to small J

J
example h11/22: J=0 θ=1800, J=2 θ~1590, J=4 θ~1370,
J=6 θ~1140, J=8 θ~870, J=10 θ~490
J
sin   1  cos  
j
2
2

J2 
1  2 
 4j 
j
J 
 j
J2  1

  1  2 
1
/
2

1
/
2
0

  4 j 
1/ 2
sin

2
1
1/ 2

J2 
Jj 1  2 
 4j 
 1  cos  / 2
1/ 2
1/ 2

J2 
 1  2 
 4j 
sin 2  / 2 tan  / 2


2
  j sin 
  j2
pairing: δ-interaction
8
6
4
2
0
δ-interaction yields a simple geometrical explanation for
Seniority-Isomers:
E ~ -Vo·Fr· tan (/2)
for T=1, even J
energy intervals between states 0+, 2+, 4+, ...(2j-1)+ decrease
with increasing spin.
Generalized seniority scheme
Seniority quantum number ν is equal to the number of unpaired particles in
the jn configuration, where n is the number of valence nucleons.
energy spacing between ν=2 and ground state (ν=0, J=0):

 

n2
n
V0  V0
2
2
2
2
 j J V j J  V0
E j n ,  2, J  E j n ,  0, J  0  j 2 J V j 2 J 
independent of n
energy spacing within ν=2 states:
n2   2
n2 

E j n ,  2, J  E j n ,  2, J    j 2 J V j 2 J 
V0    j J  V j 2 J  
V0 
2
2

 


 

 j 2 J V j 2 J  j 2 J V j 2 J 
G. Racah et al., Phys. Rev. 61 (1942), 186 and Phys. Rev. 63 (1943), 367
independent of n
Generalized seniority scheme
Seniority quantum number ν is equal to the number of unpaired particles in
the jn configuration, where n is the number of valence nucleons.
E2 transition rates:
BE 2; J i  J f  
1
 J f Q Ji
2  Ji 1
2
2
2
 n  2  j  1  n 2
2
jnJ  2 Q jnJ  0  

j
J

2
Q
j
J

0

 2  2  j  1 
 2  j  12 
2
2

  f  1  f   j J  2 Q j J  0
 2  2  j  1
2
f 
B( E 2; 21  01 )  f  1  f 
≈ Nparticles*Nholes
Sn isotopes
n
 1 for large n
2  j 1
Generalized seniority scheme
Seniority quantum number ν is equal to the number of unpaired particles in
the jn configuration, where n is the number of valence nucleons.
B( E 2; 21  01 )  f  1  f 
B( E 2; 21  01 )  f  1  f 
≈ Nparticles*Nholes
f  n /  2 j  1
j
≈ Nparticles*Nholes
 2 j  1
f  1  f 
2 2 j  1
2
j
2
 2 j  1 
j
number of nucleons
between shell closures
Excitation energy
Signatures near closed shells
Sn isotopes
N=82 isotones
N=50 isotones
Generalized seniority scheme
Seniority quantum number ν is equal to the number of unpaired particles in
the jn configuration, where n is the number of valence nucleons.
E2 transition rates that do not change seniority (ν=2):
 2  j 1 2  n  2
jnJ Q jnJ  
 j J Q j2J 

 2 j  3 
2  j 1

 1  2  f  j 2 J Q j 2 J 
2 j 3
Sn isotopes