Presentation on theme: "The Shell Model of the Nucleus 2. The primitive model"— Presentation transcript:
1 The Shell Model of the Nucleus 2. The primitive model [Sec. 5.3 and 5.4 Dunlap]
2 Reason for Nuclear Shells ATOMNUCLEUSType of particles Fermions FermionsIndentity of particles electrons neutrons + protonsCharges all charged some chargedOccupancy considerations PEP PEPInteractions EM Strong + EMShape Spherical Approximately sphericalThe atom and nucleus have some differences – but in some essential features (those underlined) they are similar and we would expect similar quantum phenomenonATOM – SPECIAL NUMBERS: , 10, 18, 36, 54, 86NUCLEUS – SPECIAL NUMBERS: 2, 8, 20, 28, 50, 82, 126where there is extra strong binding.
3 Atomic Shell Modeln=1n=2n=3Principle Quantum No =
4 Atomic Shell ModelThe 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
5 Nuclear Atomic Shell Model Starting with the Solution of the Schrodinger Equation for the HYDROGEN ATOMThe natural coordinate system to use is spherical coordinates (r, , ) – in which the Laplacian operator isand the central potential being “felt” by the electron is the Coulomb potentialnlWe 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).
6 Nuclear Shell ModelFor 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 theare the spherical harmonicswhere the are Associated Legendre Polynomials made up from cos and sin terms.THE RADIAL EQUATION is most important because it gives the energy eigenvalues.
7 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:
8 Looking at the Centrifugal Barrier s p dThe 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
10 Solutions to the Infinite Square Well The solutions to this equation are the Spherical Bessel Functionsl
11 Solutions to the Infinite Square Well The zero crossings of the Spherical Bessel Functions occur at the following arguments for knl rSo that the wavenumber knl is given by:And the energy of the state as: