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

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

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.

Atomic Shell Model 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 Starting with the Solution of the Schrodinger Equation for the HYDROGEN ATOM 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

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

Atomic Shell Model Radial solutions of the hydrogen atom wavefunction are complicated functions involving the associated Laguerre function n123n123 Principle Quantum No =

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

Atomic Shell Model

Atomic Shell Model Single electron separation energy

Atomic Shell Model Covalent Radius Atomic Radius

Evidence for Nuclear Shells Single neutron separation energy

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

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

Reason for Nuclear Shells Type of particles FermionsFermions Indentity of particles electronsneutrons + protons Charges all chargedsome charged Occupancy considerations PEP PEP Interactions EM Strong + EM Shape Spherical Approximately spherical ATOMNUCLEUS 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.