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

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

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 IPM too crude. Need to add in extra interactions among valence nucleons outside closed shells. These dominate the evolution of Structure

So, we will have a Hamiltonian H = H 0 + H resid. where H 0 is that of the Ind. Part. Model We need to figure out what H resid. 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.

Coupling of two angular momenta 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?) /

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”.

Can we obtain such simple results by considering residual interactions?

Separate radial and angular coordinates

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.

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

R 4/2 < 2.0

Extending the Shell Model to 3-particle sysetms Consider now an extension of, say, the Ca nuclei to 43 Ca, 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.

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

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 !! 43 Ca

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, H residual = V(  ) in spherical harmonics or Legendre polynomials.

Probes and “probees”

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.