---
title: "Decay Chain Counter Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/decay-chain-counting/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/decay-chain-counting/"
last_modified: "2026-10-24"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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

> Interactive data plots; numerical values can be exported.

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.

Physics tutorial

## A daughter can grow inside a decaying sample

**Background** A radioactive parent can create a daughter that is itself radioactive. Their populations follow different time curves.

**Why it matters** A rising count does not necessarily mean material is being added from outside. The source can manufacture the atoms that later decay.

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 misconception** A lower count proves a slower nuclear decay.

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

#### Finite populations couple the bins

**Typical misconception** Every source bin is an independent Poisson experiment.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

Physical conclusion

Read a decay chain as production plus loss, then apply the detector response to those same decays.

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)