(455) HL Nuclear radius and closest approach [IB Physics HL]

Nuclear Radius

• Definition: Size of the nucleus, containing protons and neutrons, not the entire atom due to floating electrons.

• Important Variables:

  • Atomic Mass Number (A): Total number of nucleons (protons + neutrons).

  • Z (Atomic Number): Number of protons.

• Notation: Nucleids written as A/Z (e.g., for helium, A = 4, Z = 2).

Nuclear Radius Equation

• Formula: R = r₀A^(1/3)

  • r₀: Firm radius, approximately 1.2 × 10^-15 meters.

  • Applies to calculate the nuclear radius in meters based on the atomic mass number.

• Example Calculation:

  • For helium (A = 4), substitute into the formula.

Rutherford Scattering

• Concept: Positive alpha particles (helium-4) approaching a positively charged nucleus (e.g., gold-197) experience repulsion due to like charges.

• Distance of Closest Approach (R): The point where an alpha particle turns around due to repulsion from the nucleus.

Energy Considerations

• Kinetic Energy at Closest Approach: As the alpha particle approaches, its initial kinetic energy fully transitions to electric potential energy at the closest approach point.

• Energy Equations:

  • Kinetic Energy: KE = 1/2 mv²

  • Electric Potential Energy: PE = K * (Q * q) / R

    • Q: Charge of the nucleus (e.g., gold nucleus with 79 protons).

    • q: Charge of the alpha particle (2 × elementary charge).

Closest Approach Calculation

• Rearranged Equation: R = KQQ / EK

  • As initial kinetic energy (EK) increases, distance R decreases.

• Evidence for Strong Nuclear Force: At high energy levels, other forces come into play, leading to deviations from classical Rutherford scattering.

Example Scenario: Gold Nucleus Interaction

• Given Initial Kinetic Energy of Alpha Particle: 21 Mega electron volts (MeV).

• Calculate Nuclear Radius:

  • For gold (A = 197), apply formula: R = r₀ * 197^(1/3).

  • Result: Approximately 7.0 × 10^-15 meters.

Distance of Closest Approach Calculation

• Calculate using specific values:

  • K = 8.99 × 10^9 N·m²/C².

  • Q = 79 * 1.6 × 10^-19 C (charge of gold nucleus).

  • q = 2 * 1.6 × 10^-19 C (charge of alpha particle).

  • Convert kinetic energy from MeV to joules: 21 MeV to joules = 21 × 10^6 × 1.6 × 10^-19.

• Final Calculation: R ≈ 1.0 × 10^-4 meters, consistent with expected atomic scale.