Radioactive Decay Chains

Radioactive Decay Chains

Many radioactive nuclei do not decay directly to a stable state — they form a decay chain where each daughter nucleus is itself radioactive. Understanding the time evolution of each species requires the Bateman equations.

Two-Member Chain

Let $N_P$ be the number of parent nuclei and $N_D$ the number of daughter nuclei:

$\frac{dN_P}{dt} = -\lambda_P N_P$

$\frac{dN_D}{dt} = \lambda_P N_P - \lambda_D N_D$

The parent decays exponentially:

$N_P(t) = N_{P,0}\, e^{-\lambda_P t}$

The daughter population (Bateman solution, for $\lambda_D \neq \lambda_P$):

$N_D(t) = N_{P,0}\, \frac{\lambda_P}{\lambda_D - \lambda_P} \left(e^{-\lambda_P t} - e^{-\lambda_D t}\right)$

Secular Equilibrium

When the parent half-life is much longer than the daughter half-life ($t_{1/2,P} \gg t_{1/2,D}$), after sufficient time the activities become equal — secular equilibrium:

$\frac{N_D}{N_P} = \frac{\lambda_P}{\lambda_D} = \frac{t_{1/2,D}}{t_{1/2,P}}$

This ratio is always less than 1 (daughters are far less abundant than parents in secular equilibrium).

Your Task

All constants must be defined inside each function. Use import math for math.exp.

• parent_nuclei(N_P0, lambda_P, t) — parent population at time $t$
• daughter_nuclei(N_P0, lambda_P, lambda_D, t) — daughter population via Bateman solution
• secular_equilibrium_ratio(lambda_P, lambda_D) — equilibrium ratio $\lambda_P / \lambda_D$