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Surrey Mini-School Lecture 2 R. F. Casten

Surrey Mini-School Lecture 2 R. F. Casten. Outline. Introduction, survey of data – what nuclei do Independent particle model and residual interactions Particles in orbits in the nucleus Residual interactions: results and simple physical interpretation Multipole decomposition

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Surrey Mini-School Lecture 2 R. F. Casten

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  1. Surrey Mini-SchoolLecture 2R. F. Casten

  2. Outline • Introduction, survey of data – what nuclei do • Independent particle model and residual interactions • Particles in orbits in the nucleus • Residual interactions: results and simple physical interpretation • Multipole decomposition • Seniority – the best thing since buffalo mozzarella • Collective models -- Geometrical • Vibrational models • Deformed rotors • Axially asymmetric rotors • Quantum phase transitions • Linking the microscopic and macroscopic – p-n interactions • The Interacting Boson Approximation (IBA) model

  3. Independent Particle Model – Uh –oh !!! Trouble shows up

  4. Shell Structure Mottelson (Nobel Prize for the Unified Model, 1975) – ANL, Sept. 2006 Shell gaps, magic numbers, and shell structure are not merely details but are fundamental to our understanding of one of the most basic features of nuclei – independent particle motion. If we don’t understand the basic quantum levels of nucleons in the nucleus, we don’t understand nuclei. Moreover, perhaps counter-intuitively, the emergence of nuclear collectivity itself depends on independent particle motion (and the Pauli Principle).

  5. Independent Particle Model • Some great successes (for nuclei that are “doubly magic plus 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. • Residual interactions and angular momentum coupling to the rescue.

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

  7. Residual Interactions • Need to consider a more complete Hamiltonian: H = H0 + Hresidual Hresidual reflects interactions not in the single particle potential. NOT a minor perturbation. In fact, these residual interactions determine almost everything we know about most nuclei. Start with 2- particle system, that is, a nucleus “doubly magic + 2”. Hresidual is H12(r12) Consider two identical valence nucleons with j1 andj2 . Two questions: What total angular momenta j1 + j2 = J can be formed? What are the energies of states with these J values?

  8. 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?) /

  9. 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

  12. Extending the IPM with residual interactions • 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 three particle angular momenta, j, couple to give final total J values? • If we use the m-scheme for three 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. • Since a J = 0 pair is the lowest configuration for two particles in the same orbit, that case, namely total J = 7/2, must lie lowest !!

  13. 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 results we have discussed for the 2-particle system !!

  14. 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.

  15. x

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

  17. R4/2< 2.0

  18. Backups

  19. Shell model too crude. Need to add in extra interactions among valence nucleons outside closed shells. These dominate the evolution of Structure • Residual interactions • 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

  20. 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.

  21. 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.

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