E-X1 · Watch a daughter build up

Decay Chain Counter

Why can daughter counts rise while the parent source is decaying? The parent produces new daughter nuclei as it decays. At first that supply can exceed the daughter’s own losses, so daughter activity rises. Later the supply weakens and the daughter also fades. Detection probability and background change the counts you observe.

Watch a two-step decay chain change color and send pulses to a counter. Move the detector, save a new counting run and compare its fluctuations with the predicted time curve.

Recorded parent pulses $0$

Recorded daughter pulses $0$

Expected daughter peak $0$

Model assumptions

A hypothetical pure parent population decays through one radioactive daughter to a stable product, with independent exponential waiting times. The colors ideally identify the two decay channels. The circular detector has a three-centimeter radius; isotropic point-source geometric acceptance is multiplied by intrinsic efficiency. Source and detector face one another on their common axis. A homogeneous Poisson background is divided between channels. Finite source populations give correlated bin counts; the smooth reference integrates Bateman activities, including equal half-lives. The replay uses the saved individual decay times. Particle transport, pulse duration and nuclear size are schematic. No branching, dead time, pileup, energy spectrum or real isotope calibration is included.

Saved synthetic observations

Count the two stages in time

Default state

Parent counts

Daughter counts

Analytic source reference

Bars are saved detector pulses. Dashed curves integrate the two activities through the same detector response and background.

Start with the essentials

Focus question: Why can daughter counts rise while the parent source is decaying?
One-sentence intuition: The daughter grows when its supply from parent decays exceeds its own decay losses.

Core mathematical model

Parent loss and daughter supply

\frac{dN_A}{dt}=-\lambda_A N_A,\qquad\frac{dN_B}{dt}=\lambda_A N_A-\lambda_BN_B

The parent supplies one daughter per decay. The stable product retains the nuclei that finish the second step.

Half-life fixes the decay probability

\lambda=\frac{\ln2}{T_{1/2}}

Each nucleus has a constant decay probability per unit time; individual waiting times still vary.

Expected count in a time bin

\mu_{i,k}=p_{\rm det}\int_{t_k}^{t_{k+1}}\lambda_iN_i(t)\,dt+\frac{b\Delta t}{2}

The detection probability includes the disk solid angle and intrinsic efficiency. Half of the known background enters each ideal channel.

Common difficulties

Counts also depend on the apparatus

Typical misconceptionA lower count proves a slower nuclear decay.

Better mental modelThe same decay rate gives fewer detected pulses when the detector sees a smaller solid angle.

Finite populations couple the bins

Typical misconceptionEvery source bin is an independent Poisson experiment.

Better mental modelEach original nucleus can decay once in each chain step. Its event times connect the population and count histories.

Run the experiment

01
Watch the daughter build up
Use Daughter buildup. Follow the blue parent and amber daughter populations, then compare the colored count bins.
What to observe: The parent falls immediately. The daughter first gains more nuclei than it loses and reaches a later peak.
02
Move the detector
Drag the detector away from the sample, then record a new run.
What to observe: Its smaller solid angle lowers signal counts. The previously saved graph stays fixed until the new acquisition.
03
Match the lifetimes
Select Matched lifetimes and record a run.
What to observe: The daughter still rises and falls smoothly; equal half-lives are a regular physical limit.