1. Shell Model with residual interactions – mostly 2-particle systems Start with 2-particle system, that is a nucleus „doubly magic + 2“ Consider two identical valence nucleons with j 1 and j 2 Two questions: What total angular momenta j 1 + j 2 = J can be formed? What are the energies of states with these J values?

2. j 1 + j 2 all values from: j 1 – j 2 to j 1 + j 2 (j 1 = j 2 ) Example: j 1 = 3, j 2 = 5: J = 2, 3, 4, 5, 6, 7, 8 BUT: For j 1 = j 2 : J = 0, 2, 4, 6, … ( 2j – 1) (Why these?) Coupling of two angular momenta

3. 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”.

4. Coupling of two angular momenta

5. residual interaction - pairing  Spectrum 210 Pb:  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 0 2 4 6 8

6. pairing: δ-interaction wave function: interaction: with and A. de-Shalit & I. Talmi: Nuclear Shell Theory, p.200

7. pairing: δ-interaction wave function: interaction: with and A. de-Shalit & I. Talmi: Nuclear Shell Theory, p.200

8. δ-interaction (semiclassical concept)  forand θ = 0 0 belongs to large J, θ = 180 0 belongs to small J example h 11/2 2 : J=0 θ=180 0, J=2 θ~159 0, J=4 θ~137 0, J=6 θ~114 0, J=8 θ~87 0, J=10 θ~49 0

9. pairing: δ-interaction 0 2 4 6 8 δ-interaction yields a simple geometrical explanation for Seniority-Isomers:  E ~ -V o ·F r · tan (  / 2 ) for T=1, even J energy intervals between states 0 +, 2 +, 4 +,...(2j-1) + decrease with increasing spin.

10. 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 j n configuration, where n is the number of valence nucleons. energy spacing between ν=2 and ground state (ν=0, J=0): energy spacing within ν=2 states: independent of n

11. Generalized seniority scheme Seniority quantum number ν is equal to the number of unpaired particles in the j n configuration, where n is the number of valence nucleons. E2 transition rates: for large n Sn isotopes ≈ N particles *N holes

12. Generalized seniority scheme Seniority quantum number ν is equal to the number of unpaired particles in the j n configuration, where n is the number of valence nucleons. ≈ N particles *N holes number of nucleons between shell closures ≈ N particles *N holes

13. Signatures near closed shells Excitation energy Sn isotopesN=82 isotones N=50 isotones

14. Generalized seniority scheme Seniority quantum number ν is equal to the number of unpaired particles in the j n configuration, where n is the number of valence nucleons. E2 transition rates that do not change seniority (ν=2): Sn isotopes