An unstable nucleus can transform spontaneously into other particles or nuclei. For a large ensemble of identical unstable nuclei, each undecayed nucleus can often be modeled as having the same constant probability per unit time of decaying.

This produces the shared first-order exponential-decay law. In nuclear physics the first-order rate constant is called the decay constant, $\lambda$:

$$N(t)=N_0e^{-\lambda t}.$$

The corresponding half-life is therefore

$$\boxed{t_{1/2}=\frac{\ln2}{\lambda}}.$$

The characteristic mean lifetime is

$$\tau=\frac1\lambda,$$

so

$$t_{1/2}=\tau\ln2.$$

Nuclear activity

The activity $A$ is the expected number of decays per unit time. Since each of the $N$ remaining nuclei has decay probability rate $\lambda$,

$$\boxed{A=\lambda N}.$$

Activity therefore decays exponentially with the same decay constant:

$$A(t)=A_0e^{-\lambda t}.$$

The SI unit of activity is the becquerel:

$$1,\mathrm{Bq}=1\text{ decay per second}.$$

Example

A sample initially contains $8000$ unstable nuclei and has a half-life of $5$ days. After $15$ days—three half-lives—the expected remaining number is

$$N=8000\left(\frac12\right)^3=1000.$$

The activity has also fallen to one eighth of its initial value.

Statistical meaning

The exponential law predicts ensemble behavior; it does not predict the exact decay time of one individual nucleus. Under the constant-$\lambda$ model, radioactive decay is memoryless: conditional on a nucleus having survived until now, its probability of surviving an additional interval does not depend on its age.

Ordinary nuclear half-lives are largely insensitive to macroscopic temperature and pressure because the interactions governing most decays occur on nuclear scales. Some decay modes involving atomic electrons can show environmental effects, but the simple constant-decay model is an excellent approximation for many radioactive processes.