From the phenomenological law to the exponential

Problem

Starting from the phenomenological law $\Delta N = -\lambda N\,\Delta t$, derive the decay law $N(t) = N_0\,e^{-\lambda t}$ and show the link between the constant $\lambda$ and the half-life $T_{1/2}$. Finally, discuss why decay is a statistical phenomenon rather than a deterministic one for the single nucleus.

Solution

From the phenomenological law to the differential equation. The relation $\Delta N = -\lambda N\,\Delta t$ says that the number of nuclei decaying in an interval is proportional to the number of nuclei present and to the length of the interval. Taking the limit $\Delta t \to 0$ gives a differential equation:

$\ddt{N} = -\lambda N$

Separation of variables. We move everything involving $N$ to one side and time to the other:

$\frac{\dd N}{N} = -\lambda\,\dd t$

Integration. Integrating both sides:

$\ln N = -\lambda t + \text{const.} \quad\Longrightarrow\quad N(t) = N_0\,e^{-\lambda t}$

where $N_0$ is the number of nuclei at time $t = 0$.

Link with the half-life. By definition, after a time $T_{1/2}$ the number of nuclei halves: $N(T_{1/2}) = N_0/2$. So:

$\frac{N_0}{2} = N_0\,e^{-\lambda T_{1/2}} \quad\Longrightarrow\quad e^{-\lambda T_{1/2}} = \frac{1}{2} \quad\Longrightarrow\quad \lambda T_{1/2} = \ln 2$

$\ev{N(t) = N_0\,e^{-\lambda t},\quad \lambda = \frac{\ln 2}{T_{1/2}}}$

Why it is a statistical phenomenon. The equation describes with precision the behaviour of a huge number of nuclei, but says nothing about a single one. Each nucleus decays at an instant governed by quantum laws, entirely random: there is no internal or external cause that “schedules” the moment of decay, and two identical nuclei can decay centuries apart. The regularity of the exponential emerges only for large numbers, as a consequence of the law of large numbers: the probability of decaying per unit time, $\lambda$, is the same for all of them, and averaging over billions of nuclei the individual fluctuations cancel out.

Connections

Topics: Nuclear physics Concepts: Radioactive decay Methods: Separation of variables