half-life

A half-life is the time it takes for half of a sample of unstable atoms to decay. Wait one half-life and half remain. Wait another and a quarter remain. The clock never resets and it never runs out — it just keeps halving.

It is the most useful single number in radioactive decay, because it turns something completely unpredictable into something completely predictable.

One nucleus is random; a trillion are not

Nothing inside a nucleus counts down. A ²³⁸U atom that has sat in a rock for four billion years is exactly as likely to decay in the next second as one made yesterday. Nuclei do not age, and there is no way, even in principle, to pick out which one goes next.

What is fixed is the chance. Each nucleus has a constant probability per second of decaying, called the decay constant λ. That single assumption gives the whole exponential law:

N(t) = N₀ · e^(−λt)          activity A(t) = λN(t)
T½   = ln 2 / λ ≈ 0.693 / λ
τ    = 1 / λ = T½ / ln 2 ≈ 1.443 · T½

The mean life τ is longer than the half-life, and confusing the two is the classic error. The free Neutron is the standard trap: its half-life is 613.9 s (10.2 minutes), so its mean life is 1.443 times that — 885.6 s, or 14.8 minutes.

What survives, after n half-lives

half-lives elapsed	fraction left	left of 1 kg
1	50%	500 g
2	25%	250 g
3	12.5%	125 g
5	3.125%	31 g
7	0.781%	7.8 g
10	0.0977%	0.98 g
20	0.0001%	0.95 mg

The useful rule of thumb: ten half-lives leaves about a thousandth. That is where the familiar "wait ten half-lives" advice comes from — though a thousandth is only "gone" if what you started with was less than a thousand times a safe amount.

The range is absurd

Measured half-lives span roughly 48 orders of magnitude. Nothing else in physics is quoted over so wide a range with a straight face.

nuclide	half-life	in seconds	decay mode
⁸Be	8.2 × 10⁻¹⁷ s	8.19 × 10⁻¹⁷	α
²¹²Po	294.3 ns	2.943 × 10⁻⁷	α
²¹⁴Po	163.46 µs	1.635 × 10⁻⁴	α
free Neutron	613.9 s	613.9	β⁻
²²²Rn	3.8222 d	3.30 × 10⁵	α
¹³¹I	8.0252 d	6.93 × 10⁵	β⁻
⁶⁰Co	1925.28 d (5.27 y)	1.663 × 10⁸	β⁻
³H (tritium)	12.32 y	3.888 × 10⁸	β⁻
⁹⁰Sr	28.91 y	9.123 × 10⁸	β⁻
¹³⁷Cs	30.08 y	9.492 × 10⁸	β⁻
¹⁴C	5700 y	1.799 × 10¹¹	β⁻
²³⁹Pu	24 110 y	7.608 × 10¹¹	α
²³⁵U	7.04 × 10⁸ y	2.22 × 10¹⁶	α
⁴⁰K	1.248 × 10⁹ y	3.94 × 10¹⁶	β⁻ (89.28%)
²³⁸U	4.468 × 10⁹ y	1.41 × 10¹⁷	α
²³²Th	1.40 × 10¹⁰ y	4.42 × 10¹⁷	α
¹²⁸Te	7.7 × 10²⁴ y	2.43 × 10³² s	double β⁻

¹²⁸Te is the extreme case: a half-life more than a hundred trillion times the age of the universe. It is measurable only because a kilogram of tellurium contains so many atoms that a handful still decay each year.

The short end is just as strange. ⁸Be lives long enough for light to travel about 25 nanometres — a hundred atoms side by side. That flicker is the stepping stone stars use to build carbon: two helium nuclei make ⁸Be, and a third has to arrive before it falls apart.

Why the numbers are so spread out

For alpha decay, the nucleus has to tunnel out through an electric barrier it does not have the energy to climb. Tunnelling probability depends exponentially on how much energy is missing, so a small change in decay energy makes an enormous change in half-life. ²¹²Po and ²³²Th both decay by alpha emission. Their decay energies differ by barely a factor of two — 8954 keV against 4082 keV — and their half-lives differ by a factor of 1.5 × 10²⁴.

It does not care what you do to it

The half-life is a property of the nucleus, and the nucleus is shielded from the world by the electrons around it. Heating, freezing, crushing, dissolving, or chemically bonding the atom leaves the decay rate alone — this is why radioactive waste cannot be "neutralised" the way an acid can.

There are real exceptions, and all of them work by changing the electrons rather than the nucleus:

• Electron capture needs an electron to be present inside the nucleus. Change the electron cloud and you nudge the rate. A measured claim of a 0.9 ± 0.2% shift in ⁷Be between temperatures prompted a precision test on ⁹⁷Ru, which found no shift at all above 0.1% — so the effect, where real, is tiny and does not generalise.
• Strip the electrons entirely and the rules change outright. Neutral ²⁰⁵Tl is a stable Isotope that has sat in the Earth since it formed. Fully ionised ²⁰⁵Tl⁸¹⁺, circulating in a storage ring with all 81 electrons removed, decays by bound-state beta emission with a half-life of 291 (+33 / −27) days. Same nucleus. Different answer, because beta decay can now drop its electron into an empty orbital instead of having to eject it.

These are laboratory extremes. For anything in a rock, a reactor, or a body, the half-life is a constant.

What it is used for

• Dating. Compare how much of a parent nuclide is left against its daughter. Radiocarbon dating uses ¹⁴C for the last ~50 000 years; ²³⁸U and ⁴⁰K reach back billions.
• Medicine. ¹³¹I's 8-day half-life is long enough to ship and dose, short enough to clear the patient. Diagnostic tracers run shorter still — hours, not days.
• Waste and safety. ¹³⁷Cs and ⁹⁰Sr, both around 30 years, dominate the heat and hazard of spent fuel for the first few centuries. ²³⁹Pu, at 24 110 years, sets the long tail.
• Power. ²³⁸Pu's 87.7-year half-life is why it runs spacecraft generators: hot enough to be useful, steady enough to last a mission.

See also: Nuclear binding energy, gamma ray, alpha decay, beta decay, Atomic nucleus, Isotope.