Learning Objectives
- Understand atomic nucleus, its composition, and size
- Learn about nuclear binding energy and mass defect
- Study radioactivity and types of nuclear decay
- Understand nuclear fission and fusion
- Apply radioactive decay law and half-life calculations
Key Concepts
Nuclear Composition and Size
Nucleus: Contains protons (Z) and neutrons (N = A - Z). A = mass number, Z = atomic number.
Nuclear size: R = R₀A^(1/3), where R₀ = 1.2 × 10⁻¹⁵ m = 1.2 fm.
Nuclear density: ρ ≈ 2.3 × 10¹⁷ kg/m³ (constant for all nuclei).
Isotopes: Same Z, different A (e.g., ¹H, ²H, ³H). Isobars: Same A, different Z. Isotones: Same N, different Z.
Mass Defect and Binding Energy
Mass defect: Δm = [Zm_p + (A-Z)m_n] - M_nucleus
Binding Energy: BE = Δmc² (energy required to separate all nucleons).
Binding energy per nucleon: BE/A. Higher BE/A means more stable nucleus.
Maximum BE/A ≈ 8.75 MeV for Fe-56 (most stable nucleus).
1 u = 931.5 MeV/c².
Light nuclei (A < 56): fusion increases stability. Heavy nuclei (A > 56): fission increases stability.
Radioactivity
Spontaneous disintegration of unstable nuclei with emission of radiation.
Alpha decay (α): ⁴₂He emitted. A decreases by 4, Z by 2. Example: ²³⁸U → ²³⁴Th + ⁴He.
Beta-minus decay (β⁻): Neutron → proton + electron + antineutrino. Z increases by 1, A unchanged.
Beta-plus decay (β⁺): Proton → neutron + positron + neutrino. Z decreases by 1.
Gamma decay (γ): High-energy photon emitted. No change in A or Z (nucleus goes from excited to lower state).
Radioactive Decay Law
N(t) = N₀ e^(-λt), where λ is the decay constant.
Activity: A = dN/dt = λN = A₀ e^(-λt). SI unit: becquerel (1 Bq = 1 decay/s). 1 curie = 3.7 × 10¹⁰ Bq.
Half-life: t₁/₂ = 0.693/λ = ln2/λ. After n half-lives: N = N₀/2ⁿ.
Mean life: τ = 1/λ = t₁/₂/0.693.
Nuclear Fission
Heavy nucleus splits into two lighter nuclei with release of energy and neutrons.
Example: ²³⁵U + n → ¹⁴⁴Ba + ⁸⁹Kr + 3n + energy (~200 MeV)
Chain reaction: Neutrons from one fission cause more fissions. Controlled in nuclear reactors, uncontrolled in atomic bombs.
Critical mass: minimum mass needed to sustain a chain reaction.
Nuclear Fusion
Light nuclei combine to form heavier nuclei with release of energy.
Example: 4¹H → ⁴He + 2e⁺ + 2ν + 26.7 MeV (proton-proton chain in the Sun).
Requires extremely high temperatures (~10⁷ K) to overcome Coulomb repulsion (thermonuclear reactions).
Fusion is the source of energy in stars. Energy per nucleon released in fusion > fission.
Summary
Nuclei consist of protons and neutrons held by the strong nuclear force. Binding energy per nucleon peaks at Fe-56. Radioactive decay follows an exponential law with characteristic half-life. Alpha, beta, and gamma are the three types of radioactive emissions. Fission of heavy nuclei and fusion of light nuclei both release energy. Fission powers nuclear reactors; fusion powers the Sun.
Important Terms
- Mass Defect: Difference between total nucleon mass and actual nuclear mass
- Binding Energy: Energy equivalent of mass defect, BE = Δmc²
- Half-life: Time for half the radioactive nuclei to decay
- Decay Constant: Probability of decay per unit time, λ = 0.693/t₁/₂
- Fission: Splitting of heavy nucleus into lighter fragments
- Fusion: Combining of light nuclei to form heavier nucleus
Quick Revision
- R = R₀A^(1/3); nuclear density ≈ 2.3 × 10¹⁷ kg/m³
- BE = Δmc²; 1 u = 931.5 MeV/c²
- N = N₀e^(-λt); t₁/₂ = 0.693/λ; τ = 1/λ
- α: A-4, Z-2; β⁻: A same, Z+1; γ: no change
- Fission: ~200 MeV per event; Fusion: ~26.7 MeV (4H→He)
- Max BE/A at Fe-56: most stable nucleus