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The Shell Model of the Nucleus 4. Applications – nuclear spin [Sec. 6.1 and 6.3 Dunlap] Switching to the nucleus ESO 3 50MeV 2/3 V0 1/3 lA 0.16 lA 10% 2/3 2 E 2(9 ) Mc 938MeV Even for large A and large l one is not going to get a big splitting. This was not able to change the magic numbers. Mayer, Jensen, Haxel and Suess knew this – they had done this calculation – it had no effect on the magic numbers. What they discovered was that this relativistic Spin Orbit energy was being swamped by another Spin-Orbit energy that was coming from a non-relativistic source. + The Full Energy Level diagram of the SHELL MODEL Remember the occupancy of each level nlj is (2j+1) – only depends on j The parity of each level nlj is (1) l only depends on l + + + + - 10 2 4 6 8 + + - + PARITY 2 4 6 - 2 4 + 2 Predicting nuclear spins UNDERSTANDING j-j COUPLING mj j HIGHER ENERGY STATE mj j nlj One nucleon in a j-state m j j 1 nlj LOW ENERGY STATE mj j nlj mj j A B Two nucleons in same j-state Both the states A and B (and any other mj substate) is ok by Pauli Principle. But state B is the lower energy state – because the strong interaction averaged over space is maximized. Predicting nuclear spins SPIN OF EVEN- EVEN NUCLEI mj j mj j nlj nlj mj j One nucleon in a j-state Two nucleons in same j-state CORROLARY: If a nucleus has an even number of neutrons – then these will couple to give spin zero If a nucleus has an even number of protons – then these will couple to give spin zero If a nucleus has N=EVEN, and Z=EVEN then J=0 Predicting nuclear spins SPIN OF ODD- EVEN NUCLEI mj j mj j mj j nlj nlj mj j One nucleon in a j-state 2 nucleons in same j-state nlj m j j 1 mj j 3 nucleons in same j-state CORROLARY: Since all EVEN-EVEN configurations are J=0 states – it follows that any EVEN-ODD nucleus must be getting its spin J from the single unpaired nucleon J junpaired Neutron and Proton levels 82 50 The potential as seen by a proton is different – especially in large nuclei – there is a Coulomb “tail” outside the nucleus and a “bump” in the center. This does not effect the lower energy levels up to magic (special) number 50. 50 28 20 20 8 8 2 2 Neutron and Proton levels For large nuclei (Z>50) there can be some subtle changes in the energy level sequencing due to the Coulomb potential. parity PREDICTING NUCLEAR SPINS Predicting nuclear spins Nuclide 40 20 Ca 41 20 Ca 42 20 Ca 43 20 Ca 44 20 Ca 45 20 Ca J 0+ 7 2 0+ 7 2 Ca 0+ 47 20 Ca 7 2 Ca 105y E.C. stable stable 46 20 48 20 stable 0+ 7 2 comment 0+ stable 165d stable 4.5d - stable PREDICTING NUCLEAR SPIN – Lead isotopes L=4 PREDICTING NUCLEAR SPIN – Lead isotopes Excited states of 41Ca 1 ? Excited states of 17O and 17F NOTE: As far as the strong force is concerned these two nuclei are THE SAME! The strong force does not distinguish n and p. But the EM force does.