The shell model of the nucleus 2 the primitive model
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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 FermionsFermions Indentity of particles electronsneutrons + protons Charges all chargedsome charged

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The Shell Model of the Nucleus 2. The primitive model

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The Shell Model of the Nucleus2. The primitive model

[Sec. 5.3 and 5.4 Dunlap]

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

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.

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”.

Atomic Shell Model

Principle Quantum No =

Radial node counter = nr


and the central potential being “felt” by the electron is the Coulomb potential

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).


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

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

The solutions to this equation are the Spherical Bessel Functions


Solutions to the Infinite Square Well

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:



Infinite Square Well

Harmonic Oscillator


Apart from 2,8 and 20 all the other numbers predicted by the primitive shell model are WRONG.




Note that the energy sequence is effective the same in all potential wells



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