The Shell Model of the Nucleus 2. The primitive model [Sec. 5.3 and 5.4 Dunlap]

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 ATOM – SPECIAL NUMBERS: 2, 10, 18, 36, 54, 86 NUCLEUS – SPECIAL NUMBERS: 2, 8, 20, 28, 50, 82, 126 where there is extra strong binding.

Atomic Shell Model n=1 n=2 n=3 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”. Principle Quantum No = Radial node counter = nr

Nuclear 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 nl We must now , however, use the shape of the nuclear potential – in which nucleons move – this is the Woods-Saxon potential, which follows the shape of the nuclear density (i.e. number of bonds).

Nuclear Shell Model For a spherically symmetric potential – which we have if the nucleus is spherical (like the atom) – then the wavefunction of a nucleon is separable into angular and radial components. where as in the atom the are the spherical harmonics where the are Associated Legendre Polynomials made up from cos and sin terms. THE RADIAL EQUATION is most important because it gives the energy eigenvalues.

Nuclear Shell Model Solving the Radial Wave Equation [Eq. 5.7] Now make the substitution which is known as “linearization” The similarity with the 1D Schrodinger equation becomes obvious. The additional potential terms – is an effective potential term due to “centrifugal energy”. In the case of l=0, the above equation reduces to the famous 1D form. So what we really need to do is now to solve is:

Looking at the Centrifugal Barrier s p d The diagram shows the effect of the centrifugal barrier for a perfectly square well nucleus. The effect of angular momentum is to force the particle’s wave Unl(r) outwards. Centrifugal potential

Solutions to the Infinite Square Well

Solutions to the Infinite Square Well The solutions to this equation are the Spherical Bessel Functions l

Solutions to the Infinite Square Well The zero crossings of the Spherical Bessel Functions occur at the following arguments for knl r So that the wavenumber knl is given by: And the energy of the state as:

COMPARISON OF SCHRODINGER EQN SOLUTIONS Coulomb Infinite Square Well Harmonic Oscillator

58 Apart from 2,8 and 20 all the other numbers predicted by the primitive shell model are WRONG. 40 34 20 Note that the energy sequence is effective the same in all potential wells 8 2