Notes on Nuclear Shell Structure and Spin-Orbit Interaction

Introduction

• Discussed properties exhibited by nuclei suggesting a shell structure.
• Stability: Shell closures correspond to magic numbers in protons/neutrons.

Shell Structure Concepts

• Single particle potential concepts used similar to atomic physics.
• Energy levels filled according to the Pauli exclusion principle.
• Key question: What potential to choose for the single particle picture?

Potential Models Evaluated

1. Infinite Square Well Potential

   • Mathematically simple but unsuccessful for magic numbers beyond the first three.

2. Harmonic Oscillator Potential

   • Also failed to reproduce magic numbers beyond the first three.

3. Finite Square Well Potential

   • Pulls energy levels down slightly but does not change order of levels significantly.

4. Wood-Saxon Potential (most realistic)

   • Attractive potential characterized by a gradual tapering to zero, mimicking mass distribution within the nucleus.

   • Formula:

     V(r) = -V₀ / (1 + e^(r - R)/a)

   • Parameters:

     • R ≈ R₀ A^(1/3)
     • a ≈ 0.52 femtometers
     • V₀ ≈ 50 MeV

Spin-Orbit Interaction

• Introduced in 1949, inspired by hydrogen atom observations.
• Provides splitting in energy levels to account for fine structure, but much stronger in nuclei.
• Important for rearranging energy levels and providing correct magic numbers.

Hamiltonian Modifications

• Incorporates a term for spin-orbit interaction:

  H = H₀ + V(r) + (l·s) term

• Energy eigenstates defined in terms of j, where J = L + S.

• States retain definite l and s but get mixed m_l and m_s values due to non-commutativity with l_z and s_z operators.

Energy Level Calculations

• Energy contributions from l·s evaluated:

1. For j = l + 1/2:
   • Energy shift = (3/4)
2. For j = l - 1/2:
   • Energy shift = (-1/4)

Magic Numbers and Shell Closures

• Based on energy gaps between levels:
  • Magic numbers derived: 2, 8, 20, 28, 50, 82, 126.
• Shell closures occur at these numbers indicating stability in nuclei.

Energy Level Arrangement

• Harmonic oscillator levels organized showing splits due to spin-orbit interaction:
  • Example levels: 1s, 1p, 2s, 1d, etc.
  • Definitions of j values lead to a clearer understanding of energy separation.

Conclusions and Further Considerations

• The mechanism for magic numbers is largely due to l·s contributions.
• Energy levels depend critically on the potential chosen, particularly for heavier nuclei.
• Realistic modeling for atomic nuclei requires adjustments in mass and size parameters.

Future Directions

• More sophisticated models needed for accurate predictions in heavier nuclei.
• Importance of recognizing that potential parameters are not universal, but depend on nuclear size and composition.