Exponential Decay and Half-Life

We cannot predict when a given nucleus will decay. But over a very large number of nuclei, a statistical law emerges: at every instant, the same proportion of the remaining nuclei decays. The result is an exponential decay of the number N of nuclei still present:

   N(t) = N₀ · e^(−λ·t)      (N₀ = initial number, λ = decay constant)

The key quantity for reading this curve is the half-life T: the time after which half of the nuclei have decayed. It is related to λ by:

   T = ln(2) / λ ≈ 0.693 / λ

With each half-life that passes, half as many nuclei remain:

   N
   N₀ |•
      |  •
   N₀/2|. . •
      |       •
   N₀/4|. . . . . •
   N₀/8|. . . . . . . . •._____
      +---|-----|-----|-----|----> t
          T    2T    3T    4T

After n half-lives, N₀/2ⁿ nuclei therefore remain. After 3 half-lives: N₀/8; after 10 half-lives: less than a thousandth. The half-life is a signature of the nucleus, and it varies dizzyingly: a few microseconds for some, 5,730 years for carbon-14, billions of years for uranium-238.

Remember the key point: decay is exponential, and the half-life is the halving time — a far more meaningful benchmark than λ for comparing two nuclei.