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**Coulomb-excitation of 112, 114, 116Sn**

Pieter Doornenbal

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**The three faces of the shell model**

Pairing interaction: large spin-orbit splitting implies a jj coupling scheme.

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**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:

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**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:

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**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

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**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:

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**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

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**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

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**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

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**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

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**Reduced transition probability in a single J-shell**

≈Nparticles*Nholes (2j+1) ≡ nucleons/orbital

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**Reduzierte Übergangswahrscheinlichkeit in einer einzelnen J-Schale**

≈NTeilchen*NLöcher (2j+1) ≡ Anzahl der Nukleonen in der j-ten Schale

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**Reduced transition probability in a complex shell**

≈Nparticles*Nholes number of nucleons between shell closures.

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**Reduzierte Übergangswahrscheinlichkeit in einer komplexen Schale**

≈NTeilchen*NLöcher Anzahl der Nukleonen zwischen den Schalenabschlüssen

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**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

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**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

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**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

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**Choosing the right target**

2+ 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+

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**Important to know: Transition Ratio 90º-140º 2+→0+ 1 4+→2+ 0.017 116Sn**

θγ = 25º 116Sn→58Ni Secure energy:

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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.

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