Exponential law and mean life

Microscopically, a single decay is a probabilistic quantum phenomenon: it is not possible to predict when a given nucleus will decay, only its probability of decaying per unit time. It is like rolling a die, for every nucleus, at every instant. Individually unpredictable, the phenomenon nevertheless becomes perfectly regular when very many nuclei are observed together: the statistics of large numbers make an exponential law emerge.

Principle — Exponential decay

If $N(t)$ is the number of radioactive nuclei present at time $t$, the fraction decaying per unit time is constant: $\frac{\dd N}{\dd t} = -\lambda\,N \qquad\Longrightarrow\qquad \ev{N(t) = N_0\,e^{-\lambda t}}$ with $\lambda$ the decay constant (units $1/\text{s}$).

The equation $\dd N/\dd t = -\lambda N$ simply says that the more nuclei there are, the more of them decay per unit time, and this proportionality is what generates the exponential trend: separating the variables and integrating gives $N(t) = N_0 e^{-\lambda t}$.

From $\lambda$ two characteristic times are defined, proportional to each other:

• the mean life $\tau = 1/\lambda$: the average survival time of a single nucleus;
• the half-life $\tau_{1/2}$: the time in which $N$ drops to $N_0/2$. Setting $e^{-\lambda\tau_{1/2}} = 1/2$ and taking the logarithm gives $\tau_{1/2} = \frac{\ln 2}{\lambda} \approx 0{,}693\,\tau$

Key formula

$N(t) = N_0\,e^{-\lambda t}$ $\tau = \frac{1}{\lambda}, \qquad \tau_{1/2} = \frac{\ln 2}{\lambda}$ Activity: $\;\abs{\dfrac{\dd N}{\dd t}} = \lambda N$, measured in becquerel (1 Bq = 1 decay/s).

The activity $\lambda N$ is what a detector actually measures — the number of decays per second — and it itself decays exponentially in time, with the same constant $\lambda$ as the number of nuclei.

Example — Carbon-14 dating

All living organisms absorb carbon from the atmosphere, in which the ratio ${}^{14}\text{C}/{}^{12}\text{C}$ is kept constant by cosmic rays. At death, absorption ceases and the ${}^{14}$C in the body begins to decay via $\beta^-$ with $\tau_{1/2} = 5730$ years. If a sample is found to have an isotopic ratio equal to $1/8$ of the atmospheric one: $\frac{N}{N_0} = \frac{1}{8} = \left(\frac12\right)^3 \quad\Rightarrow\quad \Delta t = 3\,\tau_{1/2} \approx 17\,000\;\text{years}$ This method has been used to date the Similaun Man (Ötzi, 5300 years), the paintings of Lascaux (17,000 years), and the Shroud of Turin ($\sim 700$ years, a disputed result).

Collegamenti

Argomenti: Fisica nucleare Concetti: Decadimento radioattivo Competenze: Uso delle scale logaritmiche Metodi: Separazione delle variabili

Esercizi collegati: Esercizio svolto — decadimento del radon in cantina · Dalla legge fenomenologica all’esponenziale · Vita media di un campione misto