43.3 Content
43.3.1 Radioactive Decay
Radioactive decay is the spontaneous emission of radiation from unstable nuclei.
| Type | Particle | Symbol | Charge | Mass | Penetration |
|---|---|---|---|---|---|
| Alpha | Helium nucleus | \(^4_2\alpha\) | +2 | 4 u | Paper stops it |
| Beta⁻ | Electron | \(^0_{-1}\beta\) | -1 | ~0 | Few mm aluminium |
| Beta⁺ | Positron | \(^0_{+1}\beta\) | +1 | ~0 | Few mm aluminium |
| Gamma | Photon | γ | 0 | 0 | Lead/concrete |
43.3.2 Interactive: Radioactive Decay Types
43.3.3 Decay Equations
Alpha decay: Nucleus loses 2 protons and 2 neutrons \[^A_Z X \rightarrow ^{A-4}_{Z-2} Y + ^4_2\alpha\]
Beta⁻ decay: Neutron → proton + electron + antineutrino \[^A_Z X \rightarrow ^{A}_{Z+1} Y + ^0_{-1}\beta + \bar{\nu}_e\]
Beta⁺ decay: Proton → neutron + positron + neutrino \[^A_Z X \rightarrow ^{A}_{Z-1} Y + ^0_{+1}\beta + \nu_e\]
Gamma emission: Nucleus loses energy without changing composition \[^A_Z X^* \rightarrow ^{A}_{Z} X + \gamma\]
In all nuclear reactions: - Mass number (A) is conserved - Atomic number (Z) is conserved - Charge is conserved - Energy is conserved (including mass-energy)
43.3.4 Exponential Decay
The number of undecayed nuclei decreases exponentially:
\[N_t = N_0 e^{-\lambda t}\]
where: - \(N_t\) = number of nuclei remaining at time t - \(N_0\) = initial number of nuclei - \(\lambda\) = decay constant (s⁻¹) - \(t\) = time (s)
43.3.5 Interactive: Exponential Decay
43.3.6 Half-Life
Half-life (\(t_{1/2}\)) is the time for half the nuclei to decay:
\[\lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.693}{t_{1/2}}\]
| After | Fraction Remaining |
|---|---|
| 0 half-lives | 1 (100%) |
| 1 half-life | 1/2 (50%) |
| 2 half-lives | 1/4 (25%) |
| 3 half-lives | 1/8 (12.5%) |
| n half-lives | 1/2ⁿ |
43.3.7 Activity
Activity (A) is the number of decays per second:
\[A = \lambda N = A_0 e^{-\lambda t}\]
Unit: Becquerel (Bq) = 1 decay per second
43.3.8 Interactive: Activity vs Time
43.3.9 Mass Defect
The mass defect is the difference between: - Mass of separated nucleons (protons + neutrons) - Actual mass of the nucleus
\[\Delta m = Zm_p + (A-Z)m_n - m_{nucleus}\]
This “missing mass” has been converted to binding energy.
43.3.10 Interactive: Mass Defect
43.3.11 Binding Energy
Binding energy is the energy required to separate all nucleons:
\[E_b = \Delta m \cdot c^2\]
Binding energy per nucleon indicates nuclear stability:
\[\frac{E_b}{A} = \frac{\Delta m \cdot c^2}{A}\]
43.3.12 Interactive: Binding Energy Curve
43.3.13 Key Features of the Binding Energy Curve
| Feature | Significance |
|---|---|
| Peak at Fe-56 | Most stable nucleus (~8.8 MeV/nucleon) |
| Light nuclei (left) | Fusion releases energy (moving toward peak) |
| Heavy nuclei (right) | Fission releases energy (moving toward peak) |
| H-2, H-3, He-3 | Low BE/A; unstable or fusion fuel |
Both fusion (of light nuclei) and fission (of heavy nuclei) release energy because the products are more tightly bound (higher BE/A) than the reactants.
43.3.14 Nuclear Fission
Fission is the splitting of a heavy nucleus into lighter nuclei:
\[^{235}_{92}\text{U} + ^1_0\text{n} \rightarrow ^{141}_{56}\text{Ba} + ^{92}_{36}\text{Kr} + 3^1_0\text{n} + \text{energy}\]
Key features: - Triggered by neutron capture - Releases ~200 MeV per fission - Produces 2-3 neutrons → chain reaction possible - Mass defect → energy release
43.3.15 Nuclear Fusion
Fusion is the combining of light nuclei into heavier nuclei:
\[^2_1\text{H} + ^3_1\text{H} \rightarrow ^4_2\text{He} + ^1_0\text{n} + 17.6\ \text{MeV}\]
Key features: - Requires extreme temperature (~10⁷ K) to overcome Coulomb repulsion - Powers the Sun and stars - Produces no long-lived radioactive waste - Mass defect → energy release
43.3.16 Interactive: Fission vs Fusion
43.3.17 Applications of Nuclear Physics
| Application | Type | Use |
|---|---|---|
| Nuclear power | Fission | Electricity generation |
| Nuclear weapons | Fission/Fusion | Military |
| Medical isotopes | Various | PET scans, cancer treatment |
| Carbon dating | β decay | Archaeology |
| Smoke detectors | α decay | Fire safety |