Nuclear Decay and Half-Life — Alpha, Beta, Gamma

High School⏱ 10 min readModern Physics

Radioactive decay and half-life: alpha, beta, gamma decay, the exponential law N = N₀e^(−λt), half-life calculations, and carbon dating.

Quick answer

Radioactive decay is the spontaneous breakup of unstable nuclei, emitting alpha particles, beta electrons, or gamma rays. Half-life (t½) is the time for half a sample to decay — related to the decay constant by t½ = ln2/λ.

$N = N_0e^{-\lambda t}, \quad t_{1/2} = \frac{\ln 2}{\lambda}$

Unit: t½ in seconds/years; λ in s⁻¹

Example: Carbon-14's half-life is 5730 years. A wooden artifact with half the normal ¹⁴C content died ~5730 years ago; with a quarter remaining, ~11,460 years ago.

What you will learn

Define half-life and relate it to the decay constant
Apply N = N₀(1/2)^(t/t½) to dating and decay problems
Distinguish alpha, beta, and gamma radiation
Explain carbon-14 dating's principle and limits
State that decay rates are immune to external conditions

Three ways nuclei fall apart

Some atomic nuclei are restless. Too many neutrons, too few, or simply too much mass — and the nucleus rearranges itself, ejecting fragments and energy. This radioactive decay is spontaneous, random per nucleus, yet statistically clockwork for trillions of them.

Three decay modes dominate:

Alpha (α): ejects a helium nucleus (2 protons + 2 neutrons). Heavy and slow — stopped by paper, but devastating inside the body.
Beta (β): a neutron transforms into a proton, ejecting an electron (and an antineutrino). Stopped by a few millimeters of aluminum.
Gamma (γ): the nucleus sheds excess energy as high-energy photons. No particles change — pure electromagnetic energy, needing dense shielding.

The decay law

Individual decays are unpredictable, but a crowd obeys iron law. With λ the decay constant (probability per nucleus per second):

N = N_0e^{-\lambda t}

N₀ = initial nuclei, N = survivors after time t

Half-life ↔ decay constant

t½ = ln2/λ ≈ 0.693/λ. A large λ means frantic decay and a short half-life; a tiny λ means geological patience. Iodine-131: 8 days. Uranium-238: 4.5 billion years.

Half-lives by counting

Worked example. A sample contains 80 g of iodine-131 (t½ = 8 days). How much remains after 24 days?

24 days is exactly 3 half-lives:

$N = 80 \times \left(\tfrac{1}{2}\right)^3 = 80 \times \tfrac{1}{8} = 10\ \mathrm{g}$

Ten grams remain; 70 g became xenon-131 plus radiation. Note the pattern: you never need λ if you count half-lives — each one simply halves the remainder.

Reading ancient clocks

While alive, you exchange carbon with the air, keeping your carbon-14 fraction constant (cosmic rays manufacture it steadily). At death, exchange stops; the ¹⁴C clock starts ticking with its 5730-year half-life.

Measure the remaining ¹⁴C fraction $f$ and solve:

$t = -t_{1/2} \frac{\ln f}{\ln 2}$

A sample with $f = 0.25$ died about 11,460 years ago. Beyond ~50,000 years too little ¹⁴C survives to measure — older rocks need uranium-lead or potassium-argon clocks instead. Same exponential law, slower tick.

Decay put to work

Decay's randomness is tamed by statistics — and harnessed everywhere:

Medicine: technetium-99m's 6-hour half-life is long enough to image organs, short enough to vanish quickly.
Power: reactors manage chain reactions; waste repositories plan around half-lives up to 24,000 years (plutonium-239).
Smoke detectors: americium-241's alphas ionize air steadily for decades; smoke disrupts the current and sounds the alarm.

Unstable nuclei taught us that matter has an expiration date — and gave us tools from archaeology to oncology.

Variables and units

Symbol	Name	Unit	Meaning
$N$	Remaining nuclei	unitless	Number of undecayed nuclei remaining
$N₀$	Initial nuclei	unitless	Starting number of nuclei
$λ$	Decay constant	s⁻¹	Decay probability per nucleus per second
$t½$	Half-life	s, days, or years	Time for half the sample to decay

Common mistakes

✗ After two half-lives, everything has decayed.

Each half-life halves the remainder: 100% → 50% → 25% → 12.5%. It takes about 10 half-lives to drop below 0.1%. The decay is exponential, never truly finishing.

✗ Heating or pressure can speed up radioactive decay.

Decay rates are set by nuclear forces, utterly indifferent to temperature, pressure, or chemistry. You cannot hurry or slow a half-life — which is exactly why radiometric dating is trustworthy.

✗ Radioactive means dangerously radioactive.

All decay emits radiation, but danger depends on type, amount, and exposure. Potassium-40 in your own body decays constantly and harmlessly; concentrated spent fuel is another story.

Real-world applications

Carbon dating: archaeologists date ancient wood and bone from remaining carbon-14 (half-life 5730 years).
Nuclear medicine: technetium-99m (6-hour half-life) images organs; iodine-131 treats thyroid disease.
Nuclear power: reactor fuel and waste management revolve around half-lives from days to millennia.
Smoke detectors: americium-241's steady alpha decay provides the ionization current that smoke interrupts.

Frequently asked questions

What is half-life?

Half-life is the time for half the nuclei in a sample to decay. After one half-life 50% remains, after two 25%, after three 12.5% — the sample never quite reaches zero mathematically.

What is the radioactive decay law?

N = N₀e^(−λt), where λ is the decay constant. Equivalently, N = N₀(1/2)^(t/t½). Each half-life multiplies the remainder by one half — pure exponential decay.

What are alpha, beta, and gamma decay?

Alpha: a helium nucleus (2p+2n) ejected — heavy, slow, stopped by paper. Beta: a neutron becomes a proton, ejecting an electron — stopped by aluminum. Gamma: high-energy photons from an excited nucleus — needs lead or concrete.

How does carbon dating work?

Cosmic rays constantly make carbon-14 in the atmosphere; living things absorb it. After death, absorption stops and the ¹⁴C decays with a 5730-year half-life. Measuring the remaining fraction dates the sample — reliable back to ~50,000 years.