---
title: "Beta Energy Sharing Observatory Interactive Lab | Sandbox Physics"
description: "Why do electrons from beta decay come out with a whole range of energies? Because a third, unseen particle carries off part of the energy."
canonical_url: "https://sandboxphysics.com/labs/beta-energy-sharing/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/beta-energy-sharing/"
last_modified: "2026-10-27"
---

Neutrinos · D01 · The missing energy

# Beta Energy Sharing Observatory

**Why do electrons from beta decay come out with a whole range of energies?** Because a third, unseen particle carries off part of the energy. If the decay produced only the electron and the recoiling nucleus, energy and momentum conservation would give the electron the same energy every time, so recoil cannot explain the spread. Count the third particle in, and energy and momentum are conserved after all.

Replay decays inside an ideal spherical calorimeter. Build up the continuous electron spectrum, compare it with a two-body line, and reveal how an invisible particle carries away energy and momentum.

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

The decay is a fictional teaching example, not data from a real isotope or a historical experiment. The kinematics are exact relativistic three-body kinematics with a toy weight inspired by allowed decays; Coulomb, nuclear and material effects are not modeled. The calorimeter measures only the electron's energy, with an ideal Gaussian response. The recoil and neutrino shown on screen are generated truth. The large mass values illustrate the kinematics and are not realistic neutrino mass bounds. The shape deviances are diagnostics that depend on the assumed hypotheses; they give no discovery significance and no fitted mass limit.

From single decays to a full spectrum

## The missing share.

Measured counts Three-body template Two-body template

### One line or a continuum?

The spectrum has forty-eight equal energy bins, plus the two overflow bins, which are kept. The lines show the expected count in each bin under the current analysis assumptions.

### Read the same counts twice

- **Three-body deviance**: $0$
- **Two-body deviance**: $0$
- **Twice log-likelihood preference**: $0$
- **Recorded mean energy**: $0$
- **Assumed three-body endpoint**: $0$
- **Counts in both flows**: $0$

A positive preference favors the three-body shape. This conditional comparison gives no p-value, discovery significance or fitted mass bound. Where a Gaussian tail underflows numerically, a floor is applied only inside logarithms.

### Replay one recorded event

Event number 1 Show the generated particles and energy shares

Only electron energy was recorded. Gold and purple arrows, the energy ledger and momentum triangle below are generated truth. Paths and timing are schematic; recoil displacement is magnified.

### How the generated energy is shared

### Generated momentum triangle

Fixed horizontal-plane projection; vectors share one scale. Cyan electron, purple daughter, gold invisible particle close tip to tail. This is not detector reconstruction.

Recorded spectrum as a table, including both overflow bins

Energy edges in keV; recorded and expected counts

| Low | High | Counts | Three bodies | Two bodies |
| --- | --- | --- | --- | --- |

Conventions, approximations & primary sources

Fictional parent at rest. Exact relativistic phase space weighted by electron energy times neutrino energy in the parent frame; it approaches the leading allowed beta shape for a heavy daughter, with unit Coulomb factor. No nuclear spin, realistic angular correlations, Coulomb, molecular or radiative corrections. The two-body foil is a kinematic comparison, not a viable weak-decay theory.

Source at the origin; vertical axis upward. The ideal spherical collector has unit efficiency, arbitrary schematic radius and independent Gaussian energy sigma. Open shells expose the full-acceptance model. No material transport, background or detector angle measurement. Large neutrino and light daughter masses exaggerate the mechanism. A fixed event total yields multinomial counts; both normalized response templates retain Gaussian tails.

- [PDG · Kinematics §49.4](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-kinematics.pdf)
- [Geant4 · Beta decay physics](https://geant4.web.cern.ch/documentation/pipelines/master/prm_html/PhysicsReferenceManual/hadronic/Radioactive/radDecay.html)
- [Ellis & Wooster · 1927](https://doi.org/10.1098/rspa.1927.0168)

[Next: precision endpoint measurement →](https://sandboxphysics.com/labs/beta-endpoint-mass/) [Explore the worldwide history →](https://sandboxphysics.com/stories/neutrinos/)

Physics tutorial

## The invisible share

**Background** This Lab replays single decay events. It keeps two things apart: the full energy split known to the generator, and the energy that the one detector actually records.

**Why it matters** Beta-decay electrons come out with a continuous range of energies, which does not fit a simple two-body picture.

Start with the essentials

- **Focus question**: Can recoil alone explain the continuum?
- **One-sentence intuition**: In a two-body decay the electron always gets the same energy; with three bodies the energy is shared over a continuous range while every four-vector is still conserved.

### Core mathematical model

#### Energy conservation in the decay

$$
Q=K_e+T_D+E_\nu
$$

The neutrino term is its total energy, including rest energy; the electron and daughter terms are kinetic energies.

#### The three momenta close into a triangle

$$
\mathbf p_e+\mathbf p_D+\mathbf p_\nu=\mathbf 0
$$

The parent is at rest. The electron energy alone cannot tell you the other two momentum vectors, which are not measured. The ones shown here are generated truth.

#### Spectrum shape for a very heavy daughter

$$
\frac{d\Gamma}{dK}\propto p_eE_e w\sqrt{w^2-m_\nu^2},\quad w=Q-K
$$

This uses natural units and sets the Coulomb factor to one. It is the heavy-daughter limit of the toy weight used here, not its full density with recoil corrections and not a precision nuclear spectrum.

#### Two spectrum shapes compared on the same counts

$$
\Delta D=2\sum_i n_i\ln\frac{p_{3,i}}{p_{2,i}}
$$

A positive value favors the three-body template. Both normalized templates include the underflow and overflow counts produced by the Gaussian response. No distribution for judging significance is supplied.

### Common difficulties

#### Generated truth is not a reconstruction

**Typical misconception** The calorimeter measures the neutrino trajectory and recoil.

**Better mental model** It records only the electron's energy. The revealed vectors and the energy split come from the generator.

#### A preference is not a discovery

**Typical misconception** A large difference in deviance is a calibrated significance.

**Better mental model** The two fixed shape hypotheses come with no sampling distribution and no treatment of nuisance parameters, so the difference cannot be read as a significance.

### Run the experiment

1. 01

   #### Follow an electron

   Choose Three bodies, step through events with the slider, then reveal the generated particles.

   **What to observe:** The electron reaches the collector while the invisible particle passes straight through. The daughter carries momentum but little energy.

2. 02

   #### Remove the third body

   Choose Two-body foil and Light daughter, then compare the spectra and the energy split of each event.

   **What to observe:** Recoil shifts where the two-body line sits, but with a fixed parent mass it does not turn the line into a continuum.

3. 03

   #### Check the detector response

   Choose Resolution trap, then set the analysis sigma to 100 without acquiring again.

   **What to observe:** The same broad measured line fits a broad detector response. Blurring by the detector is not evidence for an extra particle.

4. 04

   #### Change the missing mass

   Choose Heavy invisible particle, then set the analysis mass to 150.

   **What to observe:** The endpoint and the spectrum move, and the template changes with the assumed mass. This is not a fitted mass bound.

Physical conclusion

The continuous spectrum points to an unseen particle that carries energy away, but conservation and the detector response have to be checked together.

## Related experiments and articles

Continue exploring

### Neutrino mass in beta-decay spectra

Beta decay shares its energy between an electron and a neutrino. How does the end of the electron spectrum limit the neutrino mass, and what would a peak in a double-beta decay spectrum mean?

1. [You are here **Beta Energy Sharing Observatory** Compare a continuous electron spectrum with a two-body line, and account for all of the generated energy.](https://sandboxphysics.com/labs/beta-energy-sharing/)
2. [Lab **Beta Endpoint Mass Observatory** Scan an endpoint and refit the same counts under calibration and response assumptions.](https://sandboxphysics.com/labs/beta-endpoint-mass/)
3. [Lab **Double Beta Decay Observatory** Freeze a rare-decay spectrum and separate a conditional bound from background-only sensitivity.](https://sandboxphysics.com/labs/neutrinoless-double-beta/)

Neutrinos · Reading path

### From missing energy to a new particle

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Where did the energy go? The puzzle that led to the neutrino**](https://sandboxphysics.com/stories/neutrinos/beta-decay-missing-energy/)
2. [Lab **Beta Endpoint Mass Observatory**](https://sandboxphysics.com/labs/beta-endpoint-mass/)
3. [Article **Fermi’s next question: how often should the unseen particle appear?**](https://sandboxphysics.com/stories/neutrinos/fermi-beta-decay-theory/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

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