Week 3 - Nuclear Shell Model

Question 1

What are Magic Numbers in the the nuclear model?

Answer 1

Magic numbers 2 8 20 28 50 82 126 corresponds to very stable nuclei. If both Z and N are magic these are extra stable and are called doubly magic.

The concept is similar to atomic numbers which electrons complete the outer electron shells, such as the Noble Gas elements

Question 2

What observation provides evidence to the nature of magic numbers?

Answer 2

1. Enhanced Abundance: greater % of elements which contain magic N or Z numbers
2. Stable elements: (end of radioactive series) all have magic N / Z numbers (naturally occurring)
3. Neutron absorption cross-section: lower for isotopes containing N magic number than others not.
4. Increased BE: for final neutron with N magic number and much lower for surrounding isotopes
5. Electric quadruple moment: Near zero for magic number nuclei
6. Excitation energy: to 1st excited state much greater for closed shells

Question 3

What is the nuclear potential energy well and how does it differ between the nucleons?

Answer 3

An energy well describes how deeply bound a nucleon is within the nucleus, the lower down the nucleus the more stable it is.

The SNF affect both Z and N equally. The EMF affects Z’s only and is a repulsive force, hence the proton potential well is higher for protons (see diagram). This is also why the proton well has a raised lip, which needs to be overcome for protons to be added to the nucleus.

Particles have the tendency to fill the lowest energy levels first, with the difference between Z and N growing more pronounced for larger nuclei.

Well (below 0) = attractive

Outside well (above 0) = repulsive

Question 4

What is quantum tunnelling

Answer 4

A repulsive energy barrier is required to be overcome before protons can join the attractive energy well (via SNF). If the protons KE is high and/or the particle sufficiently travels up the barrier, it may ‘tunnel’ through it, thus not requiring the theoretical energy to pass through this barrier

Question 5

What is a Radioactive Decay Series and what is common element at the end?

Answer 5

Unstable heavy nuclei undergo a series of alpha and beta decay process until reaching a stable nucleus.

Neutron cross-section is high when nuclei single vacancy in closed shell (much lower for nuclei with magic number nucleons)

Isotopes of lead (Pb-82) is the stable product for Thorium (208Pb) & uranium (206Pb) —> due to doubly magic nuclei numbers.

Question 6

Briefly explain what atomic and nuclear spin-orbit interaction is and how it affects the formation of atoms / isotopes

Answer 6

They refer to the interaction between the intrinsic spin of a particle (electron and nucleons respectively) and it’s orbital motion within an atom or a nucleus. This leads to energy splitting and affects atomic / nuclear structure

Question 7

Explain in more detail how Nuclear spin-orbit interaction can explain the formation of magic numbers

Answer 7

Interaction between the spin magnetic moment of nucleons and their motion within the nucleus creates discrete energy levels.
• neutrons still exhibit magnetic moments due to their quarks having charges
• the splitting of these energy levels is 20x greater than the same in atomic theory

Spin-orbit interactions leads to the splitting of energy levels within the nucleus forming energy shells.

Nuclei with a closed shell configuration (corresponding to a magic number) exhibit greater stability due to a lower overall energy.

Parallel spins = lower energy level
Opposite spin = higher energy level
(Opposite for atomic case)