1. The Shell Model of the Nucleus1. Evidences [Sec. 5.1 and 5.2 Dunlap]

2. Alpha Particle Decay Q What is causing this bump around Z=82, N=126, A=208

3. The fission barrier on the SEMF To calculate the height of the fission barrier using the SEMF is fairly complex, but can be done as seen in this study – Fig12.3 Dunlap. The dotted lines show variations that are understood on the shell model. Note that the barrier is only small ~3MeV for A>250.

4. By the end of the 1920s, the laws of quantum mechanics had been worked out. They had been applied to the hydrogen atom. They had also been extended to the MULTI-ELECTRON ATOM. This gave the first full understanding of the PERIODIC TABLE OF THE ELEMENTS. Atomic Shell Model

5. Starting with the Solution of the Schrodinger Equation for the HYDROGEN ATOM Atomic Shell Model The natural coordinate system to use is spherical coordinates (r, , ) – in which the Laplacian operator is and the central potential being “felt” by the electron is the Coulomb potential

6. Angular solutions of the 3D Schrodinger Eqn. are the spherical harmonic functions Yl,m(,). l is the angular momentum quantum number, m is called the magnetic quantum number. l=0 l=1 l=2 l=3 Atomic Shell Model m +3 +2 +1 0 -1 -2 -3 Every l state has (2l+1) magnetic substates

7. Radial solutions of the hydrogen atom wavefunction are complicated functions involving the associated Laguerre function Atomic Shell Model n 1 2 3 0 1 0 2 1 0 Principle Quantum No =

8. The amazing thing about the 1/r potential is that certain DEGENERGIES (same energies) occur for different principal quantum no “n” and “l”. i.e. when n=2, l=0 and l=1 have the same energies When n=3, l=0,1 and 2 have the same energy. Atomic Shell Model

9. Occupancy However when we extend the model to MULTI-ELECTRON atoms the degeneracy is lost. The potential each electron moves in is now more complicated. Atomic Shell Model The potential seen by the electron changes from these two extremes as it moves about the nucleus.

10. 2 Atomic Shell Model 10 18 36 54 86

11. Single electron separation energy Atomic Shell Model

12. Atomic Shell Model Atomic Radius Covalent Radius Atomic Shell Model

13. Evidence for Nuclear Shells Single neutron separation energy

14. Evidence for Nuclear Shells The famous binding energy per nucleon (B/A) as predicted by the SEMF – does not get it quite right. There are ripples and bumps which occur at the nuclear MAGIC NUMBERS, 28, 50, 82 and 126

15. Evidence for Nuclear Shells Another evidence for EXTRA STRONG NUCLEAR BINDING at the special “MAGIC NUMBERS” is that the frequency of ISOTONES is greatest when N=20, 28, 50 and 82. THE NUCLEAR MAGIC NUMBERS are: 2, 8, 20, 28, 50, 82, 128, 184

16. Reason for Nuclear Shells ATOM NUCLEUS Type of particles Fermions Fermions Indentity of particles electrons neutrons + protons Charges all charged some charged Occupancy considerations PEP PEP Interactions EM Strong + EM Shape Spherical Approximately spherical The atom and nucleus have some differences – but in some essential features (those underlined) they are similar and we would expect similar quantum phenomenon - i.e. some kind of SHELL STRUCTURE.