1. Coulomb-excitation of 112, 114, 116Sn
Pieter Doornenbal

2. The three faces of the shell model
Pairing interaction:
large spin-orbit splitting implies a jj coupling scheme.

3. Symmetries of the shell model
Three bench-mark solutions:
No residual interaction  IP shell model.
Pairing (in jj coupling)  Racah’s SU(2).
Quadrupole (in LS coupling)  Elliott’s SU(3).
Symmetry triangle:

4. Racah’s SU(2) pairing model
Assume large spin-orbit splitting ls which implies a jj coupling scheme.
Assume pairing interaction in a single-j shell:
Spectrum of 210Pb:

5. Solution of pairing hamiltonian
Analytic solution of pairing hamiltonian for identical nucleons in a single-j shell:
Seniority  (number of nucleons not in pairs coupled to J=0) is a good quantum number.
Correlated ground-state solution (cfr. super-fluidity in solid-state physics).
G. Racah, Phys. Rev. 63 (1943) 367

6. Superfluidity in semi-magic nuclei
Even-even nuclei:
Ground state has =0.
First-excited state has =2.
Pairing produces constant energy gap:
Example of Sn nuclei:

7. Seniority scheme in Sn isotopes:2 6+
 = 0 2 4+
 = 0 2 2+
 = 2 0+
energy axis  jn J 6+
min   4+ j j j 2+ J j j j j J j J 0+
-residual interaction gives nice simple geometric rationale for Seniority Isomers from
E(j2J) ~ -V0Frtan(/2) for T=1, even J

8. Example of the 8+ ( p1h9/2)2 isomers in nuclei with Z > 82
Seniority Scheme
Seniority conserving Dn = 0
1-body even tensor
B(E2: I  I – 2, I  2)
Seniority changing Dn  0
B(E2: I  I – 2, I = 2) 2 8+ 2 6+ 2 4+ 2 2+
j = (9/2)n
n = 0 0+
B(E2)
Fractional Filling

9. e.g. Jp = (h9/2)2 coupled to 0+, 2+, 4+, 6+ and 8+. q q
d-interaction gives nice simple
geometric rationale for
Seniority Isomers from
DE ~ -VoFr tan (q/2)
for T=1, even J 8 6 4
DE(j2J) 2
e.g. Jp = (h9/2)2 coupled to
0+, 2+, 4+, 6+ and 8+.
180 90 q 2 4 6 8 q

10. d-interaction gives nice simple geometric rationale
for Seniority Isomers from DE ~ -VoFr tan (q/2)
for T=1, even J 2 4 6 8

11. Reduced transition probability in a single J-shell
≈Nparticles*Nholes
(2j+1) ≡ nucleons/orbital

12. Reduzierte Übergangswahrscheinlichkeit in einer einzelnen J-Schale
≈NTeilchen*NLöcher
(2j+1) ≡ Anzahl der Nukleonen in der j-ten Schale

13. Reduced transition probability in a complex shell
≈Nparticles*Nholes
number of nucleons between shell closures.

14. Reduzierte Übergangswahrscheinlichkeit in einer komplexen Schale
≈NTeilchen*NLöcher
Anzahl der Nukleonen zwischen den Schalenabschlüssen

15. Proton np-nh core excitations (t=n)
Theoretical interpretation
theory (neutron valence + proton core excitations and
90Zr as closed-shell core)
theory (neutron valence and 100Sn as closed-shell core)
Neutron/proton single-particle states
in a nuclear shell-model potential:
from A. Banu et al., cond. publ.
t=0
t=2
t=4
Neutron number
B(E2 ) e2 b2
This work
••••••••
Proton np-nh core excitations (t=n)
&
100Sn core is open

16. Previous measurements:
112Sn 2+
112Sn
(20) fs
≙ (13) e2b2
75Gr30
0.229 (5)
α, 16O
Coul Ex
81Ba05
0.256 (6)
16O
57Al43
0.180 (40) α
61An07
0.33 (6)
14N, 20Ne
70St20
Reference*
Measured Value
Projectile
Method 0+ 2+
114Sn
(60) fs
≙ 0.25 (5) e2b2
114Sn 0+
Coul Ex α
0.20 (7)
57Al43
14N, 20Ne
0.25 (6)
61An07
Coulex
16O
0.25 (5)
81Ba05
DSA
112Cd(α,2n)
0.238 (77)
91VIZW
RDDS
100Mo(18O, 4n)
0.189 (39)
01Ga52 2+
116Sn
(10) fs
≙ (5) e2b2 0+
*From NNDC

17. Coulomb excitation experiment
112,114,116Sn→58Ni at 3.6MeV/u
Ex=1257MeV, 1300MeV, 1294MeV
B(E2)↑=0.244(13), 0.25(5), 0.209(5)e2b2
Sn-excitation ~ 180 mb
Ni-excitation ~115 mb
γ-efficiency = 0.005
beam intensity = 1pnA
target thickness = 1mg/cm2
10 % duty factor
pγ-rate (Sn) = 1/s

18. Choosing the right target2+
1454
154 keV 2+
184W → 4.7 MeV/u
114Sn
1300 0+ 0+
58Ni
184W
1120
1250 2+
120Sn
1171
120Sn 2+ 0+
θγ = 25º 4+

19. Important to know: Transition Ratio 90º-140º 2+→0+ 1 4+→2+ 0.017 116Sn
θγ = 25º
116Sn→58Ni
Secure energy:

20. Conclusion:
Very easy to perform, yet leads to interesting physical results
All necessary equipment is already available at GSI.
Only feasible using Sn ion beams
We ask for a total of 3 times 7 shifts of beam time for the
isotopes 112, 114, 116Sn.