Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 1 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 

Core Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 2 Shell model Extreme independent particle model  only 23 rd neutron. More complete shell model  all three “valence” nucleons. 20

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 3 Shell model HW 26 and 43 Sc, 43 Ti. Discuss the energy levels of nuclei with odd number of nucleons in the 1f 7/2 shell.

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 4 Shell model Dipole Magnetic Moment HW 27 HW 27 Show that examine and examine Eqs. 5.9 in Krane. In addition, work out problem 5.8 in Krane  Conclusion? Proton: g s (free) = ?g l = 1 ? Neutron: g s (free) = ?g l = 0 ? What about  + and  - ?

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 5 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 In the xy-plane: Q  -  r 2 .

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 6

7 Shell model Validity A < < A < 220 NuclideQ (b) 2 H (D) O Co Cu Cs Dy Lu Bi-0.37

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 8 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.

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 9 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:  Rotations: Rotation of a deformed shape.  Vibrations: Surface oscillations.

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 10 Collective model Rotational motion observed for non-spherical nuclei. Deformed nuclei are mainly Ellipsoid of surface: Difference between semimajor and semiminor axes. Deformation parameter. HW 28 Problems 5.11 and 5.12 in Krane. Discuss effect on quadrupole moment.  > 0  < 0 R av

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 11 Collective model Symmetry axis GS (even-even)  0 + Symmetry  only even I

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 12 Collective model HW 29 HW 29 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.

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 13 Collective model Spin parity E measured (keV) E/E(2 + )I(I + 1)/ Er Higher angular momentum  centrifugal stretching  higher moment of inertia  lower energy than expected  additional evidence for lack of rigidity. HW 29 (continued)

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 14 Collective model Odd-A

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 15 Collective model Average shape Instantaneous shape

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 16 Collective model Instantaneous coordinate Symmetry Amplitude Spherical harmonics r 0 A 1/3 = 0 monopole = 1 dipole = 2 quadrupole = 3 octupole.

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 17 Collective model Both monopole and dipole excitations require high energy. R(t) = R avr +  00 Y 00 = 0 monopole = 1 dipole

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 18 Collective model = 2 quadrupole Quantization of quadrupole vibration is called a quadrupole phonon. A phonon carries two units of angular momentum and even parity (-1 2 ). This mode is dominant. For most even-even nuclei, a low lying state with J π =2 + exists. Octupole phonon.

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 19 Collective model l = 4  = +4, +3, +2, +1, 0, -1, -2, -3, -4 l = 2  = +2, +1, 0, -1, -2 l = 0  = 0 Triplet 0 +, 2 +, 4 +

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 20 Collective model Two-phonon triplet at twice the energy of the single phonon state. HW 30 Krane 5.10

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 21 Nuclear Reactions X(a,b)Y First in 1919 by Rutherford: 4 He + 14 N  17 O + 1 H 14 N( ,p) 17 O Incident particle may: change direction, lose energy, completely be absorbed by the target…… Target may: transmute, recoil…… b =   Capture reaction. If B.E. permits  fission (comparable masses). Different exit channels a + X  Y 1 + b 1  Y 2 + b 2  Y 3 + b 3 …….

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 22 Nuclear Reactions Recoil nucleus Y could be unstable   or  emission. One should think about:  Reaction dynamics and conservation laws i.e. conditions necessary for the reaction to be energetically possible.  Reaction mechanism and theories which explain the reaction.  Reaction cross section i.e. rate or probability.

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 23 Nuclear Reactions Conservation Laws Charge, Baryon number, total energy, linear momentum, angular momentum, parity, (isospin??) …….   a papa X pYpY pbpb Y b +ve Q-value  exoergic reaction. -ve Q-value  endoergic reaction. +ve Q-value  reaction possible if T a  0. -ve Q-value  reaction not possible if T a  0. (Is T a > |Q| sufficient?). Conservation of momentum ……

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 24 Nuclear Reactions Conservation of momentum. We usually do not detect Y. Show that: The threshold energy (for T a ): (the condition occurs for  = 0º). +ve Q-value  reaction possible if T a  0. Coulomb barriers…….!!! -ve Q-value  reaction possible if T a > T Th. HW 31

Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh). 25 Nuclear Reactions The double valued situation occurs between T Th and the upper limit T a \. Double-valued in a forward cone. HW 31 (continued)