Radioactive Decay

Decay types

Type	Equation	What changes	Examples
Alpha (α)	(A,Z) → (A-4,Z-2) + ⁴He	A−4, Z−2	U-238, Ra-226, Th-232
Beta-minus (β⁻)	n → p + e⁻ + ν̄	A same, Z+1	C-14, Sr-90, Cs-137
Beta-plus (β⁺)	p → n + e⁺ + ν	A same, Z−1	C-11, F-18, Na-22
Electron capture (EC)	p + e⁻ → n + ν	A same, Z−1	K-40, Be-7
Gamma (γ)	Nucleus* → Nucleus + γ	A, Z same	Ba-137m → Ba-137 + γ
Spontaneous fission	Heavy nucleus → fragments	Various	Cf-252, U-238 (rare)

Exponential decay law

N(t) = N₀ · e^(-λt)

where λ is the decay constant (per second). Half-life τ:

τ = ln(2) / λ ≈ 0.693 / λ

Activity (decays per second):

A(t) = λ · N(t) = A₀ · e^(-λt)

Time	Fraction remaining
0	100%
1 half-life	50%
2 half-lives	25%
5 half-lives	3.1%
10 half-lives	0.098%
20 half-lives	~10⁻⁶

JavaScript — radioactive decay

// Decay constant from half-life
function decayConstant(half_life) {
  return Math.log(2) / half_life;
}

// Number remaining after time t
function remaining(N_0, half_life, t) {
  return N_0 * Math.exp(-Math.log(2) * t / half_life);
}

// Activity at time t (decays/s) for sample with N atoms
function activity(N, half_life) {
  return N * decayConstant(half_life);
}

// Carbon-14 dating
function carbon14Age(remaining_fraction) {
  // remaining = exp(-λt) → t = -ln(remaining) / λ
  const half_life = 5730;  // years
  return -Math.log(remaining_fraction) / decayConstant(half_life);
}

console.log(`50% C-14: ${carbon14Age(0.5).toFixed(0)} years`);  // 5730
console.log(`25% C-14: ${carbon14Age(0.25).toFixed(0)} years`); // 11460
console.log(`12.5% C-14: ${carbon14Age(0.125).toFixed(0)} years`); // 17190

// Tc-99m for medical imaging (6 hr half-life)
function medicalDoseRemaining(initial_activity_MBq, hours) {
  const half_life = 6;  // hours
  return initial_activity_MBq * Math.exp(-Math.log(2) * hours / half_life);
}

console.log(`Tc-99m after 12 hr: ${medicalDoseRemaining(740, 12).toFixed(0)} MBq`);
console.log(`After 24 hr: ${medicalDoseRemaining(740, 24).toFixed(0)} MBq`);  // ~46
console.log(`After 48 hr: ${medicalDoseRemaining(740, 48).toFixed(2)} MBq`);  // ~3

// Ra-226 (1600 yr half-life): activity from 1 g
function radiumActivity(mass_g = 1) {
  // Ra-226 atomic mass = 226 g/mol; 6.022e23 atoms/mol
  const N = mass_g / 226 * 6.022e23;
  const half_life_s = 1600 * 365.25 * 24 * 3600;
  return activity(N, half_life_s);
}

console.log(`1 g Ra-226: ${(radiumActivity() / 1e9).toFixed(1)} GBq`);  // 36.6 (= 1 Ci)

// Half-lives of key isotopes
const isotopes = {
  'C-14': { halflife_yr: 5730, decay: 'beta-minus' },
  'I-131': { halflife_yr: 0.022, decay: 'beta-minus + gamma' },
  'Tc-99m': { halflife_yr: 0.000685, decay: 'gamma' },  // 6 hr
  'Sr-90': { halflife_yr: 28.8, decay: 'beta-minus' },
  'Cs-137': { halflife_yr: 30.2, decay: 'beta-minus + gamma' },
  'Pu-239': { halflife_yr: 24100, decay: 'alpha' },
  'U-235': { halflife_yr: 7.04e8, decay: 'alpha' },
  'U-238': { halflife_yr: 4.5e9, decay: 'alpha' },
  'Po-212': { halflife_yr: 9.5e-15, decay: 'alpha' }
};
console.log(isotopes);

Where decay matters

• Radiometric dating. C-14 (recent organics), U-Pb (rocks, billions of years), Ar-Ar (volcanic), tritium (water).
• Medical imaging. SPECT (Tc-99m), PET (F-18), nuclear medicine.
• Cancer therapy. Brachytherapy seeds; external beam (Co-60, accelerator-based).
• Nuclear power. Reactor fuel cycle, fuel longevity, waste handling.
• Industrial. Sterilization, gauging (thickness, density), tracers in pipelines.
• Smoke detectors. Most use small Am-241 source (alpha emitter).
• Geochemistry. Dating of sediments, magmatic events, climate proxies.

Common mistakes

• Treating all radiation as equally dangerous. Alpha — heavy, short range; only dangerous if ingested/inhaled. Beta — moderate range, can be stopped by aluminum. Gamma — long range, needs lead/concrete shielding.
• Assuming half-life can be controlled. Half-life is constant; not affected by chemistry, temperature, pressure (within physical limits).
• Linear extrapolation of N vs t. Decay is exponential, not linear. After 10 half-lives, ~0.1% remains, not 0%.
• Confusing activity with dose. Activity (Bq, Ci) — decays per second. Dose (Sv, rem) — biological effect. Same activity gives different dose depending on radiation type and exposure conditions.
• Forgetting parent-daughter chains. U-238 decays through 14 daughter isotopes before reaching stable Pb-206. All members contribute to total radioactivity.
• Treating radiation phobia as proportional to risk. Background and medical exposures small fraction of cancer risk. Major events (Chernobyl, Fukushima) had quantified, geographically-limited impacts.