 Radioactive decay constant

The radioactive decay constant $\lambda$ is the probability per unit time that one surviving atomic nucleus decays in the simple constant-rate model. It gives $d\overline{N}/dt=-\lambda \overline{N}$ and hence $\overline{N}(t)=N_0{e}^{-\lambda t}$ for the mean surviving population.

The radioactive activity is $\lambda \overline{N}$, the mean lifetime is $1/\lambda$, and the half-life is $\ln 2/\lambda$. These are different quantities: a large sample can have high radioactive activity even when each atomic nucleus has a small decay rate. Counting successive time intervals tests the exponential model, but background, detection efficiency and daughter radioactive activity affect the observed counts. A constant decay constant describes stochastic individual events, not a scheduled time at which each atomic nucleus must decay.