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Shell Model with residual interactions – mostly 2-particle systems
residualHHH 0
Start with 2-particle system, that is a nucleus „doubly magic + 2“
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?
1212 rHH residual
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?)
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?)
Coupling of two angular momenta
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”.
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:
Assume pairing interaction in a single-j shell
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;22/9 Jg 2
0,2,0
0,0,12, 2
12
212
J
JgjJMjVJMj JJ
rrpairing
02
4
68
pairing: δ-interaction
JjjVJjjJ
JMjjVJMjjJjjE 2112212112212112
1
21212121
012 coscos
rr
rr
VV
,)(1
mn YrRr
mn
JjjAnnFVJjjE R 212211021
drrRrRr
nnF nnR22
22211 2211
1
4
1
2
212121 02/12/1
1212
Jjj
jjJjjA
wave function:
interaction:
with
and
A. de-Shalit & I. Talmi: Nuclear Shell Theory, p.200
δ-interaction (semiclassical concept)
2j
1j
J
cos2 2122
21
2 jjjjJ
)1()1(2
)1()1()1(
2cos
2211
2211
21
22
21
2
jjjj
jjjjJJ
jj
jjJ
2
22
2
2cos
j
jJ jjj 21
1, Jjfor and
θ = 00 belongs to large J, θ = 1800 belongs to small 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
2/1
2
22
41cos1sin
j
J
j
J 2/1
2
22/1
412/cos1
2sin
j
J
2/1
2
22
22
41
11
41
02/12/1
jJ
Jjj
JJjj
22
2 2/tan
sin
2/sin
jj
pairing: δ-interaction
02
4
68
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
G. Racah et al., Phys. Rev. 61 (1942), 186 and Phys. Rev. 63 (1943), 367
Seniority quantum number ν is equal to the number of unpaired particles in the jn configuration, where n is the number of valence nucleons.
0022
22
20,0,,2, V
nV
nJjVJjJjEJjE nn
022 VJjVJj
energy spacing between ν=2 and ground state (ν=0, J=0):
energy spacing within ν=2 states:
0
220
22
2
2
2
2,2,,2, V
nJjVJjV
nJjVJjJjEJjE nn
JjVJjJjVJj 2222independent of n
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.
222202
122
1202
JjQJjj
njnJjQJj nn
E2 transition rates:
222
2
021122
12
JjQJjff
j
j 112
j
nf
2
12
1;2 if
ifi JQJ
JJJEB
for large n
Sn isotopes
ffEB 1)02;2( 11
≈ Nparticles*Nholes
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.
ffEB 1)02;2( 11
≈ Nparticles*Nholes
number of nucleons between shell closures
ff 1
j
j 12
ffEB 1)02;2( 11
≈ Nparticles*Nholes
Signatures near closed shellsE
xcita
tion
ener
gy
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.
JjQJjj
njJjQJj nn
22
32
212
E2 transition rates that do not change seniority (ν=2):
JjQJjfj
j
222132
12
Sn isotopes
