# 7. Nuclear Shell Model Flashcards

Magic Numbers of Nucleons

- particularly stable nuclei are found with N or N = 2,8,20,28,50,82,126
- these are the magic numbers of nucleons
- if N and Z are both magic, the nucleus is ‘doubly magic’ and even more stable
- elements with a magic proton number tend to have many stable (or at least long-lived) isotopes

Determining the Shell Structure of the Nucleus

- analogous to finding the orbitals in an atom i.e we use the time-independent Schrodinger Equation
- but instead of the Coulomb potential we use a nuclear potential

Requirements of the Nuclear Potential

- the nuclear potential is created by the nucleon via the short range strong force
- this means that the potential should closely follow the nucleon distribution, it should be almost uniform for rRnucl over a short ‘surface thickness’ a

Woods-Saxon Potential

-used as the nuclear potential

-for a minimum potential Vo, the Woods-Saxon potential is given by:

V(r) = -Vo / [1+exp((r-Rnucl)/a)]

= -Vo / [1+exp((r-roA^(1/3))/a)]

-with a~0.5fm, ro~1.25fm and Vo~40-50MeV

How does the Woods-Saxon Potential explain the magic numbers?

- need the nuclear equivalent of hyperfine splitting: spin-orbit coupling
- in the nuclear case, the strong force is so strong that spin-orbital coupling gives large effects
- correctly matches observed magic numbers

Island of Stability

-as well as matching the observed magic numbers, the Woods-Saxon Potential also predicts a new region of stability above the valley of stability, a so called island of stability at larger values of N and Z

What is gamma decay?

- after some other decay, the nucleus can be left in an excited state
- de-excitation results in the emission of gamma radiation
- this is high-frequency electromagnetic radiation due to the transition of nucleons between energy levels, analogous to the emission spectrum lines given off by atoms when electron move energy levels
- N and Z are unchanged during decay

How many gamma photons are emitted during gamma decay?

- a gamma decaying nucleus can emit a single gamma photon or many gamma photons in a cascade as it makes multiple transitions down towards the ground state
- the difference in energy levels between the initial and final states is equal to the photon energy plus the recoil energy of the nucleus

Conversion Electrons

- in a similar fashion to the Auger effect, the energy released in the transition of the nucleus to a lower energy state can sometimes be absorbed by an electron
- these electrons are called conversion electrons and provide additional peaks in the beta decay spectrum

How is energy transferred to conversion electrons?

- the energy is not converted to a gamma ray that then ionises an electron, the nucleus transfers its energy directly to the electron, this is possible because the electron orbitals overlap with the nucleus
- since the lower energy states have the greatest overlap, it is the electrons from the inner shells that are most likely to be emitted

Nuclear Spin

Definition

- nuclear spin is the total angular momentum of an atom’s nucleus
- this may receive contributions from the spin angular momenta of the individual nucleons but also from their orbital angular momenta

How to Calculate Nuclear Spin

A even

- if both N and Z are even then the proton contributions will cancel to 0 and the neutron contributions will cancel to 0 so j=0
- if both N and Z are odd, the protons and the neutrons both give a non-zero contribution to the nuclear spin, these contributions can add in any combination and it Is almost impossible to calculate

How to Calculate Nuclear Spin

A Odd

- if A is odd then either N or Z is even
- for which ever one is even, its contribution to j will be 0
- for the odd one, fill up the energy levels, the last energy level with a single nucleon in will be the only one that doesn’t cancel and the j value can easily be read off

Metastable

Definition

- some gamma emitters are considerably longer lived than others
- those with a half life longer than a nanosecond are referred to as metastable an denoted with a subscript m

Why do metastable gamma emitters have longer lifetimes?

- the reason for the longer lifetime is the difference in angular momentum between the ground and excited states
- although the energy difference may be quite small, if the angular momentum difference is large, a single gamma photon cannot carry away the difference in angular momentum and the decay must occur through multiple emissions
- this suppresses its transmission amplitude