Why does the simulator renormalize N₁ + N₂ + N₃ to 1 in chain mode?

The continuous ODEs already conserve total atom count; small numerical drift from fixed-step RK4 can break that. Renormalization keeps the plotted curves interpretable as fractions of the original population and prevents slow leakage of probability mass.

What does the yellow dot on the daughter curve mean?

It marks the time at which the daughter fraction N₂(t) reaches its maximum on the current run. Comparing that time with the parent and daughter half-lives is a standard way to discuss transient equilibrium (similar half-lives) versus secular equilibrium (parent much longer-lived).

Are the activity values in becquerels?

No. They are proportional to λN in the same arbitrary time units as the graph, useful for comparing parent versus daughter activity shapes. Converting to SI activity requires a real N₀ in nuclei and consistent seconds-based λ.

Why restart the run when I change half-lives or switch modes?

The time axis and stored history describe one consistent parameter set. Mixing different λ values on the same curve would be misleading. Reset (or automatic restart on parameter change) rebuilds the history from t = 0.

Radioactive Decay & Chain

Radioactive decay follows an exponential law: the probability per unit time that a given nucleus decays is constant, which leads to N(t) = N₀ e^{−λt} with decay constant λ related to the half-life by λ = (ln 2)/T₁/₂. This simulator uses arbitrary but consistent time units; you choose T₁/₂ directly and see the corresponding λ in the sidebar. In single-nuclide mode it plots the surviving fraction of parent atoms versus time—identical to integrating dN/dt = −λN with N(0) = N₀. Chain mode adds the textbook serial decay parent → daughter → stable product, modeled as dN₁/dt = −λ₁N₁, dN₂/dt = λ₁N₁ − λ₂N₂, dN₃/dt = λ₂N₂ with independent half-lives for parent and daughter. After each numerical step the three populations are renormalized so N₁ + N₂ + N₃ = 1, preserving the interpretation as fractions of the original atom budget. A fourth-order Runge–Kutta integrator keeps the curves smooth. A yellow marker shows the time of maximum daughter fraction, a classic visual for transient equilibrium and secular equilibrium discussions when λ_parent and λ_daughter differ. Activities reported as proportional to λN use the same normalization (relative units, not becquerels). The model omits branching ratios, competing channels, chemistry in the sample, detector efficiency, and statistical fluctuations—appropriate for a first ODE picture in general chemistry or introductory modern physics.

Who it's for: High school and introductory undergraduate students learning exponential decay, half-life, activity, and serial radioactive chains before nuclear instrumentation or full Bateman formulas.

Key terms

Half-life (T₁/₂)
Decay constant (λ)
Exponential decay
Bateman equations
Parent and daughter nuclides
Activity
Secular equilibrium
Runge–Kutta integration

How it works

Exponential decay with half-life T₁/₂: λ = (ln 2)/T₁/₂ and N(t)/N₀ = e^{−λt} for a single species. In chain mode, parent decays to daughter and daughter to a stable product; the simulator integrates the normalized Bateman-style ODE system with RK4 and renormalizes to keep N_parent + N_daughter + N_stable = 1. Activities scale as λN for each radioactive step.

Key equations

Single: dN/dt = −λN, λ = ln 2 / T₁/₂

Frequently asked questions

Why does the simulator renormalize N₁ + N₂ + N₃ to 1 in chain mode?: The continuous ODEs already conserve total atom count; small numerical drift from fixed-step RK4 can break that. Renormalization keeps the plotted curves interpretable as fractions of the original population and prevents slow leakage of probability mass.
What does the yellow dot on the daughter curve mean?: It marks the time at which the daughter fraction N₂(t) reaches its maximum on the current run. Comparing that time with the parent and daughter half-lives is a standard way to discuss transient equilibrium (similar half-lives) versus secular equilibrium (parent much longer-lived).
Are the activity values in becquerels?: No. They are proportional to λN in the same arbitrary time units as the graph, useful for comparing parent versus daughter activity shapes. Converting to SI activity requires a real N₀ in nuclei and consistent seconds-based λ.
Why restart the run when I change half-lives or switch modes?: The time axis and stored history describe one consistent parameter set. Mixing different λ values on the same curve would be misleading. Reset (or automatic restart on parameter change) rebuilds the history from t = 0.

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Frequently asked questions