Shell Model with residual interactions – mostly 2-particle systems

1. Shell Model with residual interactions – mostly 2-particle systems Simple forces, simple physical interpretation Lecture 2

2. Independent Particle Model • Some great successes (for nuclei that are “doubly magic plus or minus 1”). • Clearly inapplicable for nuclei with more than one particle outside a doubly magic “core”. In fact, in such nuclei, it is not even defined. Thus, as is, it is applicable to only a couple % of nuclei.

3. IPM cannot predict even these levels schemes of nuclei with only 2 particles outside a doubly magic core

5. IPM too crude. Need to add in extra interactions among valence nucleons outside closed shells. These dominate the evolution of Structure • Residual interactions – examples of simple forms • Pairing – coupling of two identical nucleons to angular momentum zero. No preferred direction in space, therefore drives nucleus towards spherical shapes • p-n interactions – generate configuration mixing, unequal magnetic state occupations, therefore drive towards collective structures and deformation • Monopole component of p-n interactions generates changes in single particle energies and shell structure

6. So, we will have a Hamiltonian H = H0 + Hresid.where H0 is that of the Ind. Part. ModelWe need to figure out what Hresid. does.Since we are dealing with more than one particle outside a doubly magic core we first need to consider what the total angular momenta are when the individual ang. Mon. of the particles are vector-coupled.

7. 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?) /

8. 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”.

10. Can we obtain such simple results by considering residual interactions?

11. Separate radial and angular coordinates

15. How can we understand the energy patterns that we have seen for two – particle spectra with residual interactions? Easy – involves a very beautiful application of the Pauli Principle.

17. x

18. This is the most important slide: understand this and all the key ideas about residual interactions will be clear !!!!!

20. R4/2< 2.0

28. Extending the Shell Model to 3-particle sysetms • Consider now an extension of, say, the Ca nuclei to 43Ca, with three particles in a j= 7/2 orbit outside a closed shell? • How do the 3 - particle j values couple to give final total J values? • If we use the m-scheme for 3 particles in a 7/2 orbit, the allowed J values are 15/2, 11/2, 9/2, 7/2, 5/2, 3/2. • For the case of J = 7/2, two of the particles must have their angular momenta coupled to J = 0, giving a total J = 7/2 for all three particles. • For the J = 15/2, 11/2, 9/2, 5/2, and 3/2, there are no pairs of particles coupled to J = 0. • What is the energy ordering of these 6 states? Think of the 2-particle system. The J = 0 lies lowest. Hence, in the 3-particle system, J = 7/2 will lie lowest.

29. Think of the three particles as 2 + 1. How do the 2 behave? We have now seen that they prefer to form a J = 0 state.

30. 43Ca Treat as 20 protons and 20 neutrons forming a doubly magic core with angular momentum J = 0. The lowest energy for the 3-particle configuration is therefore J = 7/2. Note that the key to this is the result for the 2-particle system !!

34. Multipole Decomposition of Residual Interactions We have seen that the relative energies of 2-particle systems affected by a residual interaction depend SOLELY on the angles between the two angular momentum vectors, not on the radial properties of the interaction (which just give the scale). We learn a lot by expanding the angular part of the residual interaction, Hresidual = V(q,f) in spherical harmonics or Legendre polynomials.

39. Probes and “probees”

45. Two mechanisms for changes in magic numbers and shell gaps • Changes in the single particle potential – occurs primarily far off stability where the binding of the last nucleons is very weak and their wave functions extend to large distances, thereby modifying the potential itself. • Changes in single particle energies induced by the residual interactions, especially the monopole component.