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Nuclear Models

Nuclear Models. Nuclear force is not yet fully understood. No absolutely satisfying model, but models. Specific experimental data  specific model. Model  success in a certain range. Some are: Individual particle model. (Energy. states, static properties, …).

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Nuclear Models

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  1. Nuclear Models • Nuclear force is not yet fully understood. • No absolutely satisfying model, but models. • Specific experimental data  specific model. • Model  success in a certain range. • Some are: • Individual particle model.(Energy. states, static properties, …). • Liquid drop model.(Strong force, B.E., Fission, …). • Collective model. • -particle model. • Optical model. • Fermi Gas model. • Statistical model. • others ….. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  2. Shell model • Electron configuration…. • 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 …. • AtomicElectron magic numbers: 2, 10, 18, 36, 54,… • Common center of “external” attraction. • Well understood Coulomb force. • One kind of particles. • Clear meaning for electron orbits. • … • Nuclearmagic numbers: 2, 8, 20, 28, 50, 82,126, … • (for Z or N). Chemistry! Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  3. Shell model • Evidence: • End of radioactive series: • thorium series 208Pburanium series 206Pbactinium series 207Pbneptunium series 209Bi • At Z and N mn’s there are relatively large numbers of isotopes and isotones. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  4. Shell model Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  5. Shell model Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  6. Shell model • Natural abundances. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  7. Shell model • Neutron capture cross section. NEUTRON CAPTURE CROSS SECTION NEUTRON NUMBER Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  8. Shell model • Binding energy of the last neutron • (Separation Energy). • (The measured values are plotted relative to the calculations without ). Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  9. Shell model • Excited states. Pb (even-A) isotopes. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  10. Shell model All are even-even. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  11. Shell model • Nucleons are in definite states of energy and angular momentum. • Nucleon orbit ?? Continuous scattering expected ..!! • No vacancy for scattering at low energy levels. • Nuclear potential? • Infinite square well: • Harmonic oscillator: Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  12. Shell model ? ? 2(2l + 1) accounts correctly for the number of nucleons in each level. But what about magic numbers? ?    Infinite spherical well (R=8F) Harmonic oscillator Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  13. Shell model • More realistic! (Can it solve the problem?) • Finite square well potential: • Rounded well potential: • Correction for asymmetry and Coulomb repulsion. Adjusted by the separation energies. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  14. Shell model Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  15. Shell model Coulomb repulsion? Vc(r) = ?? Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  16. Shell model Nuclear reactions? Transition probability? Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  17. Shell model • Separation of variables: • For a given spherically symmetric potential V(r), the bound-state energy levels can be calculated from radial wave equation for a particular orbital angular momentum l. • Notice the important centrifugal potential. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  18. Shell model Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  19. ml ms • 2, 8, 20 ok. • What about other magic numbers? • Situation does not improve with other potentials. • Something very fundamental about the single-particle interaction picture is missing in the description…..!!!!! • Spin-orbit coupling. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  20. Shell model • So far, 2(2l + 1) accounts correctly for the number of nucleons in each level, since we already considered both orbital angular momentum, and spin, but still not for closed shells. Separate. Spherical Harmonics, Eigenfunctions of L2 and Lz. But this representation does not solve the problem. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  21. Shell model • Spin-Orbit Coupling • M. G. Mayer and independently Haxel, Jensen, and Suess. • Spin-Orbit term added to the Hamiltonian: Orientation No longer Spherically symmetric Central, attractive Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  22. Shell model L S LL antiparallel UL parallel J Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  23. Shell model 2j+1 2(2x3 + 1) = 14 l = 3 1f7/2 j First time Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  24. Shell model HW 7 Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  25. Shell model Notes: 1. The shell model is most useful when applied to closed-shell or near closed-shell nuclei. 2. Simple versions of the shell model do not take into account pairing forces, the effects of which are to make two like-nucleons combine to give zero orbital angular momentum. The pairing force increases with l. 3. Away from closed-shell nuclei collective models taking into account the rotation and vibration of the nucleus are more appropriate. 4. Shell model does not treat distortion effects (deformed nuclei) due to the attraction between one or more outer nucleons and the closed-shell core. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  26. Shell model Ground state:(near closed shells) 1. Angular momentum of odd-A nuclei is determined by the angular momentum of the last nucleon that is odd. 2. Even-even nuclei have zero ground-state spin, because the net angular momentum associated with even N and even Z is zero, and even parity. 3. In odd-odd nuclei the last neutron couples to the last proton with their intrinsic spins in parallel orientation. Provided that the ordering is known….!! A < 150 190 < A < 220 Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  27. Shell model Harmonic oscillator Near drip line No spin-orbit coupling Near valley of  stability Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  28. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  29. Shell model Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  30. Shell model • 17 p, 21 n. • p in 1d3/2l  s   = + • n in 1f7/2l  s   = - • Rule 3 sp  sn   lp  ln  • ½ + ½ + 3 – 2 = 2  total  = - Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  31. Shell model Excited states: Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  32. Shell model Extreme independent particle model!!! Does the core really remain inert? 1d3/2 1p3/2 ? ? 1p1/2? l  pairing  2s1/2 1d5/2 Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  33. Shell model 20 Core • Extreme independent particle model  only 23rd neutron. • More complete shell model  all three “valence” nucleons. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  34. Shell model Discuss the energy levels of nuclei with odd number of nucleons in the 1f7/2 shell. HW 8 and 43Sc, 43Ti. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  35. Shell model Dipole Magnetic Moment HW 9Show that and examineEqs. 5.9 in Krane. In addition, work out problem 5.8 in Krane  Conclusion? Proton: gs(free) = 5.5856912 ? gl = 1 ? Neutron: gs(free) = -3.8260837 ? gl = 0 ? What about + and -? Schmidt lines. Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  36. Shell model In the xy-plane: Q - r2. Electric Quadrupole Moment Refined QM  Extremes Single particle: n = 1  - ive Q Single hole: n = 2j +ive Q Number of protons in a subshell Examine Table 5.1 and Fig.5.10 in Krane <r2> for a uniformly charged sphere (0.03 – 0.3 b) Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  37. Why not zero? Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

  38. Shell model + vs. – Holes vs. particles. Validity A < 150 190 < A < 220 Nuclear Physics, JU, Second Semester, 2010-2011 (Saed Dababneh).

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