Decay Constant — The First Meeting
Imagine you have a room full of 1000 identical, unstable nuclei. Each one is like a tiny time bomb, but with no clock — it could pop in the next second, or a million seconds from now. You cannot predict which one will go next, but you can measure how likely any single nucleus is to decay in a given interval of time.
That likelihood is the decay constant, denoted by the Greek letter λ (lambda).
The Intuition
If λ=0.1 s−1, it means: each nucleus has a 10% chance of decaying in the next second. Not that 10% of the nuclei will decay in exactly one second — but that the probability per unit time is 0.1.
This is a rate, not a count. It stays constant for a given isotope, no matter how many nuclei are left. Whether you have a billion nuclei or just ten, each one still "rolls the dice" with the same probability per second.
The decay constant does not depend on how many nuclei are present. It is an intrinsic property of the isotope — like its fingerprint.
The Precise Statement
For a sample containing N radioactive nuclei at time t, the rate at which they decay (the activity) is:
−dtdN=λN
The minus sign means N is decreasing. The equation says: the number of decays per second is proportional to the number of nuclei present, and λ is the constant of proportionality.
From this, we get the exponential decay law:
N(t)=N0e−λt
where N0 is the initial number of nuclei.
Connecting to Half-Life and Mean Life
The decay constant is the most fundamental of the three related quantities. The others are derived from it.
Half-life T1/2 is the time after which half the nuclei remain. Set N=N0/2:
2N0=N0e−λT1/2⇒21=e−λT1/2
Take natural logs:
ln(21)=−λT1/2⇒−ln2=−λT1/2
So:
λ=T1/2ln2=T1/20.693
Mean life τ is the average lifetime of a nucleus. It turns out to be:
τ=λ1
Three forms of the same idea:
λ=T1/20.693=τ1
If you know any one of λ, T1/2, or τ, you know all three.
A Concrete Example
Carbon-14 has a half-life of about 5730 years. Its decay constant is:
λ=5730 years0.693≈1.21×10−4 year−1
That is a very small number — each C-14 nucleus has only a 0.012% chance of decaying in a year. That is why carbon dating works: the decay is slow enough to measure over thousands of years, but fast enough to be detectable.
Common Misconception …