exploreYOUR WORLDFIELD NOTE 03 / CHANCE
A parent disappears. A daughter takes its place.
Solid: simulated sampleDashed: expected mean
Violet hollow circles: stable product.
Make a prediction. Change one thing. See what happens.
FOLLOW THE SCIENCE
A radioactive nucleus can transform into a daughter nucleus. If that daughter is also unstable, another transformation follows. A decay chain is a sequence of these changes, ending here in a stable product.
You cannot predict exactly when one nucleus will decay. For a large collection, the average behaviour follows precise equations. In this experiment, each marker tracks one original parent lineage through all its descendants.
A FEW QUESTIONS TO TRY
Remove radioactive daughters. Jump forward by one half-life and compare the sample with its expectation.
Use a fast parent and a slower daughter. When does the daughter population overtake the parent?
Add three daughters. Compare when each population peaks, then switch to activity.
POPULATION · N
The number of nuclei occupying one stage of the chain at this moment.
ACTIVITY · A = λN
The instantaneous expected decay rate for the nuclei currently present.
A daughter with a short half-life may have a small population but a large activity. A stable product can accumulate a large population while its activity remains zero. Switch the graph between the two quantities to see the difference.
After one parent half-life, the expected number of surviving parents is 288 out of 576. Individual runs fluctuate around that value. A parent that survives has the same decay probability per unit time as before.
The sample starts with 576 parents and no daughters. Each marker follows one original nucleus through a linear chain. Choose zero to three radioactive daughters, followed by a stable endpoint. Every transition creates the next stage; the new daughter then waits its own independently sampled exponential lifetime.
These coupled Bateman equations describe the expected populations. Daughters receive atoms from the preceding stage while losing atoms through their own decay. The stable endpoint has λ = 0. Total lineages remain 576, but their populations move between stages.
Population counts the atoms present. Activity is the instantaneous model decay rate λN in decays per simulation second. A shorter half-life gives each atom a larger decay probability per unit time, so a small daughter population can still have high activity. Stable atoms have zero activity. The thicker ivory trace adds the parent and every radioactive daughter to give total activity, for both the sample and the expected mean. The table reports the same total. The stable-product population is the violet trace with hollow-circle markers; it is omitted from the Activity plot because its activity is exactly zero.
The solid lines follow the stochastic sample. The dashed lines show ensemble means, calculated from the Bateman system with a matrix exponential. This calculation also works for equal or nearly equal half-lives, where formulas containing differences of decay constants need special care.
Daughter buildup: a parent half-life of 8 s and a daughter half-life of 24 s let daughters accumulate. Cascade: half-lives of 24, 8, 4, and 2 s show successive generations. Set equal half-lives to see delayed peaks without dividing by differences between decay constants.
This is a closed toy chain with 100% feeding of each next stage, no branching, no initial daughters, and no daughter loss. Markers track the descendant nucleus; emitted particles are symbolic and are not counted as additional lineages. Changing the chain or a half-life prepares a new sample. Jump advances the same sample by one parent half-life, including every intervening daughter decay. The simulation ends only when all lineages are stable.
The laboratory backdrop is imagined particle-chamber artwork, not measured tracks. Synthetic clicks include daughter events and are limited for comfortable playback; the activity display is not a measured detector count rate.