Lecture Notes: Q Equation and Neutron Scattering

Introduction

• The lecture will focus on the Q equation and its most general form.
• The Q equation will be used to explain ion/electron-nuclear interactions, neutron scattering, etc.
• Today, we will mathematically explore these interactions.
• Example: How much energy can a neutron lose when it hits something?

Q Equation Derivation

• Scenario Setup: Small nucleus 1 fires at large nucleus 2, resulting in small nucleus 3 and large nucleus 4.
• Angles: Particle 3 at angle theta (θ), particle 4 at angle phi (φ).
• Conservation of Mass and Energy:
  • Initial mass and kinetic energy equal final mass and kinetic energy.
  • Equation:
    • [ m_1c^2 + T_1 + m_2c^2 + T_2 = m_3c^2 + T_3 + m_4c^2 + T_4 ]
  • Simplification: Assume particle 2 (target) is at rest (T2 = 0).

Known and Unknown Quantities

• Likely known:
  • Mass of initial particles (m1, m2)
  • Initial kinetic energy (T1)
  • Mass of final products (m3, m4)
• Likely unknown: Kinetic energies (T3, T4), angles (θ, φ).

Momentum Conservation

• X and Y Momentum Equations:
  • [ m_1v_1 = \sqrt{2m_1T_1} ] for initial particles.
  • X Momentum:
    • [ \sqrt{2m_1T_1} = \sqrt{2m_3T_3}\cos(θ) + \sqrt{2m_4T_4}\cos(φ) ]
  • Y Momentum:
    • [ \sqrt{2m_3T_3}\sin(θ) = \sqrt{2m_4T_4}\sin(φ) ]
• Simplification by using trigonometric identities to eliminate angles.

Solving the Q Equation

• Quadratic Equation Form:
  • [ \text{Root } T_3 = s \pm \sqrt{s^2 + T} ]
  • Where S and T are derived from fractions related to kinetic energies and masses.
• Implications of Q Value:
  • Exothermic reaction: Q > 0, can occur without initial kinetic energy (T1 >= 0).
  • Endothermic reaction: Q < 0, requires initial kinetic energy to proceed.

Elastic Neutron Scattering

• Reduction of the Q equation for elastic scattering:
  • Neutron (m1 ≈ m3 = 1 AMU) hits a nucleus (m2 ≈ m4 = A).
  • Q = 0 for elastic scattering (no rest mass change).
• Simplified Q Equation:
  • [ T_{\text{in}}(1 - \frac{1}{A}) - 2\sqrt{T_{\text{in}}T_{\text{out}}}\cos(θ) + T_{\text{out}}(A + 1) = 0 ]

Additional Concepts

• Forward Scattering: Neutron continues on its path with no energy loss (cos θ = 0).
• Momentum Transfer: Depends on scattering angle and whether collision is elastic or inelastic.
• Practical Application: Understanding neutron moderation in nuclear reactors.

Conclusion

• The Q equation is a versatile tool in nuclear physics for understanding interactions.
• Future lectures will apply these concepts to specific nuclear phenomena like neutron slowing down and reactor designs.