1. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 1 Shell model Notes: 1. The shell model is most useful when applied to closed-shell or near closed-shell nuclei. 2. Away from closed-shell nuclei collective models taking into account the rotation and vibration of the nucleus are more appropriate. 3. 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. 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.

2. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 2 Shell model Fermi Gas E F  n 2/3 Range ?

3. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 3 Shell model Transition probability? Nuclear reactions?

4. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 4 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

5. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 5 Shell model Near valley of  stability No spin- orbit coupling Harmonic oscillator Near drip line

6. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 6 Shell model

7. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 7 Shell model 17 p, 21 n. p in 1d 3/2 l  s   = + n in 1f 7/2 l  s   = - Rule 3  s p  s n   l p  l n  ½ + ½ + 3 – 2 = 2  total  = -

8. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 8 Shell model Excited states:

9. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 9 Extreme independent particle model!!! Does the core really remain inert? Shell model 1d 5/2 2s 1/2 1d 3/2 1p 1/2 ? l  pairing 

10. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 10 Shell model Extreme independent particle model  only 23 rd neutron. More complete shell model  all three “valence” nucleons. HW 21 Discuss the energy levels of nuclei with odd number of nucleons in the 1f 7/2 shell.

11. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 11 Shell model Dipole Magnetic Moment HW 22 HW 22 Show that and examine Eqs. 5.9 in Krane. In addition, work out problem 5.8 in Krane  Conclusion? Proton: g s (free) = 5.5856912 ?g l = 1 ? Neutron: g s (free) = -3.8260837 ?g l = 0 ? What about  + and  - ?

12. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 12 Shell model Electric Quadrupole Moment Refined QM  for a uniformly charged sphere Number of protons in a subshell Extremes Single particle: n = 1  - ive Q Single hole: n = 2j  +ive Q Examine Table 5.1 and Fig.5.10 in Krane

13. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 13 Shell model Validity A < 150 190 < A < 220 NuclideQ (b) 2 H (D)+0.00288 17 O-0.02578 59 Co+0.40 63 Cu-0.209 133 Cs-0.003 161 Dy+2.4 176 Lu+8.0 209 Bi-0.37

14. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 14 Collective model Large quadrupole moments  nucleus as a collective body (Liquid drop model). Interactions between outer nucleons and closed shells cause permanent deformation. Single-particle state calculated in a non-spherical potential  complicated. Spacing between energy levels depends on size of distortion. Doubly magic  1 st excited state away from GS. Near closure  single-particle states. Further away from closure  collective motion of the core  excited states.

15. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 15 Collective model A net nuclear potential due to filled core shells exists. Collective model combines both liquid drop model and shell model. Two major types of collective motion:  Vibrations: Surface oscillations.  Rotations: Rotation of a deformed shape.

16. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 16 Collective model Symmetry axis Symmetry  only even I GS (even-even)  0 +

17. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 17 Collective model HW 23 HW 23 compare measured energies of the states of the ground state rotational band to the calculations. Rigid body or liquid drop? Intermediate  Short range and saturation of nuclear force.

18. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 18 Collective model Spin parity E measured (keV) E/E(2 + )I(I + 1)/6 12 + 10 + 1518.0016.6118.33 8+8+ 6+6+ 7.00 4+4+ 299.443.283.33 2+2+ 91.41.0 0+0+ 0 164 Er Higher angular momentum  centrifugal stretching  higher moment of inertia  lower energy than expected. HW 23 (continued)

19. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 19 Collective model Odd-A