Applications: power reactors, half-life, dating, nuclear weapons, X-rays, and microwaves
1. Half-life and decay law
Radioactive decay is random and spontaneous: we cannot predict when a single nucleus will decay, but for a large number \( N \) of identical nuclei, the rate of decay (activity) is proportional to \( N \): \( dN/dt = -\lambda N \), where \( \lambda \) is the decay constant (probability per unit time).
\[ \frac{dN}{dt} = -\lambda N \quad \Rightarrow \quad N(t) = N_0\, e^{-\lambda t} \]
The half-life \( t_{1/2} \) is the time for half of the nuclei to decay: \( N(t_{1/2}) = N_0/2 \). So \( e^{-\lambda t_{1/2}} = 1/2 \), hence \( \lambda t_{1/2} = \ln 2 \).
\[ N(t) = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}} = N_0\, e^{-\lambda t}, \quad \lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.693}{t_{1/2}} \]
Derivation of \( N(t) = N_0 e^{-\lambda t} \): From \( dN/dt = -\lambda N \) we get \( dN/N = -\lambda\, dt \). Integrate: \( \ln N = -\lambda t + C \). At \( t = 0 \), \( N = N_0 \), so \( C = \ln N_0 \). Thus \( \ln(N/N_0) = -\lambda t \), i.e. \( N = N_0 e^{-\lambda t} \). The half-life is defined by \( N_0/2 = N_0 e^{-\lambda t_{1/2}} \), so \( e^{\lambda t_{1/2}} = 2 \), \( \lambda t_{1/2} = \ln 2 \), \( t_{1/2} = (\ln 2)/\lambda \).
After one half-life, \( N = N_0/2 \); after two, \( N = N_0/4 \); after \( n \) half-lives, \( N = N_0/2^n \). Activity (decays per second) is \( A = -\frac{dN}{dt} = \lambda N \), so \( A(t) = A_0 e^{-\lambda t} = A_0 (1/2)^{t/t_{1/2}} \); it also halves every \( t_{1/2} \). Unit: becquerel (Bq) = 1 decay/s; or curie (Ci) = \( 3.7 \times 10^{10}\,\text{Bq} \).
Example: A sample has \( 10^6 \) nuclei and \( t_{1/2} = 5 \) years. After 5 years, \( N \approx 5 \times 10^5 \). After 15 years (3 half-lives), \( N = 10^6/2^3 = 1.25 \times 10^5 \). Activity at \( t = 0 \): \( A_0 = \lambda N_0 = (\ln 2 / t_{1/2}) N_0 \). If \( t_{1/2} = 5 \) years \( \approx 1.58 \times 10^8\,\text{s} \), then \( \lambda \approx 4.4 \times 10^{-9}\,\text{s}^{-1} \), so \( A_0 \approx 4.4 \times 10^{-3}\,\text{Bq} \).
2. Dating (e.g. carbon-14)
Living organisms take in carbon from the environment (e.g. CO₂). A tiny fraction of carbon is \( {}^{14}\text{C} \) (created in the atmosphere by cosmic-ray neutrons hitting \( {}^{14}\text{N} \)). While the organism is alive, the ratio \( {}^{14}\text{C}/{}^{12}\text{C} \) in its body matches the atmosphere. After death, no new \( {}^{14}\text{C} \) is incorporated, and \( {}^{14}\text{C} \) decays by \( \beta^- \) with half-life \( t_{1/2} \approx 5730 \) years. By measuring the remaining ratio in the sample and comparing to the “living” ratio, we estimate the time since death.
\[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^{t/t_{1/2}} \quad \Rightarrow \quad t = t_{1/2}\,\frac{\ln(N_0/N)}{\ln 2} = t_{1/2}\,\frac{\ln(N_0/N)}{0.693} \]
Derivation of the age \( t \): We have \( N/N_0 = e^{-\lambda t} = (1/2)^{t/t_{1/2}} \). Taking natural log: \( \ln(N/N_0) = -\lambda t = -(t/t_{1/2})\ln 2 \). So \( t = t_{1/2}\, \ln(N_0/N) / \ln 2 \). Thus if we measure \( N \) (e.g. from the activity or from mass spectrometry) and know \( N_0 \) (the ratio when the organism died, assumed equal to the atmospheric ratio at that time), we get \( t \). Corrections are needed for changes in atmospheric \( {}^{14}\text{C} \) over time (calibration curves).
The same idea applies to other isotopes: e.g. uranium–lead dating (for rocks, using \( {}^{238}\text{U} \to {}^{206}\text{Pb} \) and half-lives of billions of years), potassium–argon, etc. The physics is always: exponential decay gives a unique relation between the remaining fraction (or ratio) and the elapsed time.
Example: A piece of wood has \( N/N_0 = 1/4 \) for \( {}^{14}\text{C} \). So \( (1/2)^{t/t_{1/2}} = 1/4 = (1/2)^2 \), hence \( t/t_{1/2} = 2 \), \( t = 2 \times 5730 \approx 11\,460 \) years. If \( N/N_0 = 0.1 \), then \( \ln(N_0/N) = \ln 10 \approx 2.30 \), so \( t = 5730 \times 2.30 / 0.693 \approx 19\,000 \) years.
3. Nuclear power reactors
Reactors use controlled fission: heavy nuclei (e.g. \( {}^{235}\text{U} \)) absorb a neutron and split into two lighter nuclei (fission fragments) plus typically 2–3 fast neutrons. Those neutrons can cause further fissions in other \( {}^{235}\text{U} \) nuclei — a chain reaction. For a steady power output, on average one neutron from each fission triggers exactly one more fission (critical state).
Why fission releases energy: The binding energy per nucleon is largest for nuclei with \( A \approx 56 \) (iron region) and decreases for very heavy nuclei. So when a heavy nucleus (e.g. \( A \approx 235 \)) splits into two medium-mass fragments, the total binding energy of the products is greater than that of the parent. The mass of the parent is larger than the sum of the masses of the fragments (plus neutrons); that mass difference \( \Delta m \) appears as kinetic energy: \( E = \Delta m\, c^2 \). Typically \( \sim 200\,\text{MeV} \) per fission, mostly as kinetic energy of the fragments (which then heat the surrounding material).
Moderator: Fission neutrons are fast (MeV). \( {}^{235}\text{U} \) fissions more readily with thermal (slow) neutrons. A moderator (water, graphite) slows neutrons by elastic collisions. Slowed neutrons are more likely to cause fission and less likely to be captured without fission.
Control rods: Rods of material that absorb neutrons (e.g. boron, cadmium) are inserted or withdrawn to adjust the number of neutrons available for fission. Inserting them deeper reduces the reaction rate (subcritical); withdrawing increases it. The reactor is run at critical (one neutron per fission on average) for steady power.
Heat from the reactor core is removed by a coolant and used to produce steam, which drives turbines connected to generators. No combustion; the energy comes from nuclear binding energy.
Example: \( n + {}^{235}\text{U} \to {}^{236}\text{U}^* \to {}^{141}_{56}\text{Ba} + {}^{92}_{36}\text{Kr} + 3n \). Mass number: 1 + 235 = 236 = 141 + 92 + 3 ✓. Charge: 0 + 92 = 56 + 36 ✓. The three neutrons can each induce another fission if they hit \( {}^{235}\text{U} \) nuclei.
4. Nuclear bombs
A nuclear (fission) bomb uses an uncontrolled chain reaction: a supercritical mass of fissile material (e.g. \( {}^{235}\text{U} \) or \( {}^{239}\text{Pu} \)) is assembled quickly (e.g. by bringing two subcritical pieces together, or by compressing a sphere). Once supercritical, more than one neutron per fission on average causes another fission, so the number of fissions grows rapidly. A huge number of fissions (and thus a huge release of energy) occurs in a fraction of a second before the device blows apart and the reaction stops.
Criticality: If on average exactly one neutron from each fission causes another fission, the reaction is critical (steady). If the average is < 1, it is subcritical (reaction dies out). If > 1, it is supercritical (exponential growth). In a bomb, the goal is to make the assembly supercritical so that as many nuclei as possible fission before disassembly. Neutrons can escape from the surface; a larger mass has a smaller surface-to-volume ratio, so fewer neutrons escape — hence “critical mass” for a given geometry and enrichment.
Physics: Same fission process as in a reactor — \( E = \Delta m\, c^2 \) from the mass defect; each fission releases \( \sim 200\,\text{MeV} \). The difference is that no moderator or control rods are used to stabilise; the assembly is designed to be supercritical and to release as much energy as possible in a short time. Fusion weapons (hydrogen bombs) use a fission “primary” to create the high temperature and pressure needed to ignite fusion of light nuclei (e.g. deuterium, tritium); fusion releases additional energy (binding energy per nucleon is large for light nuclei fusing toward the iron region).
Scale: 1 kg of \( {}^{235}\text{U} \) fully fissioned releases \( \sim 8 \times 10^{13}\,\text{J} \) (tens of kilotons TNT equivalent). Only a fraction of the fuel actually fissions in a weapon before the device disassembles; the “efficiency” depends on design and how long the supercritical state is maintained.
5. X-ray machine
X-rays are high-energy photons (wavelength roughly 0.01–10 nm, energy roughly 0.1–100 keV). In an X-ray tube, electrons are emitted from a cathode, accelerated by a high voltage \( V \) (e.g. 20–150 kV), and strike a metal target (e.g. tungsten). Two processes produce X-rays:
- Bremsstrahlung (“braking radiation”): An electron is decelerated in the electric field of a target nucleus. Accelerated (or decelerated) charges radiate electromagnetic energy. The electron can lose any fraction of its kinetic energy in a single photon, so the spectrum is continuous from zero up to a maximum. The maximum photon energy equals the kinetic energy of the electron: \( E_\text{max} = eV \) (if the electron is stopped in one event). So \( \lambda_\text{min} = hc/(eV) \).
- Characteristic X-rays: A fast electron knocks an inner-shell electron (e.g. K-shell) out of a target atom. An electron from a higher shell (e.g. L or M) drops into the vacancy; the energy difference is emitted as a photon. These energies are characteristic of the target element (e.g. tungsten K\(_\alpha\) lines at \( \sim 59\,\text{keV} \)), so they appear as sharp lines on top of the continuous bremsstrahlung spectrum.
Why \( E_\text{max} = eV \): The electron gains kinetic energy \( K = eV \) from the accelerating voltage. The maximum X-ray photon energy cannot exceed \( K \), because the electron cannot give up more energy than it has. So \( E_\gamma^\text{max} = eV \). Hence \( \nu_\text{max} = eV/h \) and \( \lambda_\text{min} = hc/(eV) \). For \( V = 100\,\text{kV} \), \( E_\text{max} = 100\,\text{keV} \), \( \lambda_\text{min} \approx 0.012\,\text{nm} \).
Imaging: X-rays pass through soft tissue (low atomic number \( Z \), low absorption) but are absorbed or scattered more by dense material (bone: high \( Z \), more photoelectric absorption; metal even more). The shadow cast on a detector (film or digital) gives contrast: bones and metal appear lighter (less X-rays reached the detector); soft tissue appears darker. So the physics is differential absorption and scattering of EM waves by matter.
Example: A 80 kV tube produces bremsstrahlung with \( E_\gamma \) from 0 to 80 keV. The intensity is roughly flat up to \( E_\text{max} \) then drops. Tungsten target also gives characteristic K lines. In a chest X-ray, ribs and spine absorb more than lung tissue, so they appear whiter on the image.
6. Microwave (oven and waves)
Microwaves are electromagnetic waves with wavelength roughly 1 mm–1 m (frequency roughly 300 MHz–300 GHz), between radio and infrared. In a microwave oven:
- A magnetron produces microwaves (typically \( f \approx 2.45\,\text{GHz} \), wavelength \( \lambda = c/f \approx 12\,\text{cm} \)). This frequency is in an allowed band for industrial use and is absorbed reasonably well by water.
- Dielectric heating: Water molecules have a permanent electric dipole moment (positive H side, negative O side). The oscillating \( \vec{E} \) field of the wave exerts a torque on the dipoles, making them rotate to align with the field. As the field reverses every half cycle, the dipoles keep flipping. This rotation is resisted by collisions with neighbouring molecules — energy is transferred from the field to the random kinetic energy of the molecules (heat). So the food (especially water-rich parts) heats from within.
- Non-polar materials (e.g. many plastics, glass) absorb 2.45 GHz much less, so they heat less. Metal reflects microwaves (standing waves can form, and arcing can occur), so metal containers or foil are unsafe.
Why dipole rotation heats: The torque on a dipole in a field \( \vec{E} \) is \( \vec{\tau} = \vec{p} \times \vec{E} \). So \( \vec{E} \) does work on the dipole as it rotates. In a time-varying field, the dipole constantly tries to align with \( \vec{E} \); when \( \vec{E} \) flips, the dipole flips. The energy delivered to the material is \( \sim \omega \varepsilon'' E^2 \) per unit volume per unit time, where \( \varepsilon'' \) is the imaginary part of the permittivity (loss factor). Water has a high loss factor at 2.45 GHz, so it heats efficiently.
Physics summary: No nuclear processes — purely classical EM. The magnetron converts electrical energy into EM waves; the waves are absorbed by polar molecules (mainly water), converting EM energy into thermal energy. Microwaves do not “resonate” with water in a quantum sense; the heating is dielectric loss, not a single resonant transition.
Example: 2.45 GHz gives \( \lambda \approx 12\,\text{cm} \). In a typical oven cavity, standing waves can form, so some spots get higher \( E \) (hot spots) and others lower (cold spots). Rotating the food or using a turntable helps average the heating. Thick food heats mainly near the surface at first (penetration depth of microwaves in water is a few cm); conduction then spreads the heat inward.
7. Quick comparison
- Reactors: Controlled fission; \( E = \Delta m\, c^2 \); chain reaction kept critical by moderator and control rods; heat → steam → electricity.
- Half-life: Exponential decay \( N = N_0 e^{-\lambda t} \), \( \lambda = \ln 2 / t_{1/2} \); used for dating (C-14, U-Pb, etc.) and activity calculations.
- Nuclear bomb: Uncontrolled fission (supercritical assembly); same \( E = \Delta m\, c^2 \); fusion weapons add fusion of light nuclei for extra energy.
- X-ray: Accelerated electrons hit target → bremsstrahlung (continuous) + characteristic lines; imaging by differential absorption (bone vs tissue).
- Microwave: EM waves at \( \sim 2.45\,\text{GHz} \); dielectric heating of water (dipole rotation); no nuclear physics.
Units reminder: Activity in Bq (decays/s) or Ci. Half-life in seconds for \( \lambda \) in s\(^{-1}\). Dating: use consistent units (e.g. years for \( t \) and \( t_{1/2} \)). Fission energy \( \sim 200\,\text{MeV} \) per fission; 1 eV \( \approx 1.6 \times 10^{-19}\,\text{J} \).