---
title: "Beta Endpoint Mass Observatory Interactive Lab | Sandbox Physics"
description: "How can electron energies near the end of the beta-decay spectrum reveal the neutrino’s mass?"
canonical_url: "https://sandboxphysics.com/labs/beta-endpoint-mass/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/beta-endpoint-mass/"
last_modified: "2026-10-27"
---

Neutrinos · D14 · Weigh the invisible

# Beta Endpoint Mass Observatory

**How can electron energies near the end of the beta-decay spectrum reveal the neutrino’s mass?** A neutrino mass slightly changes the shape of the electron spectrum just below its endpoint. A giant electrostatic filter passes only electrons above a set energy, so stepping that energy maps out the shape. A small calibration error at the endpoint can mimic a different mass. The result is only as good as the models of the spectrum, the detector response and the calibration.

Step an electrostatic threshold across the end of the beta spectrum, record the counts and fit a neutrino mass. See how measuring time, filter width and endpoint calibration change the result.

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

This teaching simulation is inspired by MAC-E spectroscopy. In place of the molecular tritium spectrum and a real apparatus, it uses one effective mass, a fixed electron prefactor and an ideal response for electrons inside the accepted cone. At each nonnegative mass squared, the fit adjusts the source strength, a positive background and the endpoint offset again. The endpoint calibration enters as a Gaussian penalty, and the response width stays at the analysis setting. The deviance contours have no calibrated coverage and are not experimental mass limits. The Lab does not reproduce KATRIN data or sensitivity.

One voltage scan gives a conditional result

## The last few electronvolts.

Sampled counts Fitted means Refitted massless reference Reveal generating reference

### A threshold scan, not an energy histogram

Each dot is an independent Poisson count divided by its dwell time. The vertical scale is logarithmic; bars use the square root of the count, not exact Poisson intervals. Click or use arrow keys to select a voltage.

### Inspect the same counts near the endpoint

Observed minus fitted count rate over the last ten electronvolts. The gray curve is the massless prediction, with the other parameters refitted, minus the best fit. The pink curve is the generating reference, shown separately.

### What the filter actually accepts

Normalized transmission against surplus energy, for the width used to acquire the data (blue) and the width assumed in the analysis (gold). Magnetic acceptance is folded into the source normalization. The curves show an ideal response, not reconstructed detector data.

### How much can the mass change?

At each nonnegative mass squared, the source strength, a positive background and the endpoint offset are fitted again. Shading marks the range within a deviance rise of one; values above twelve are cut off. The coverage is not calibrated, and the plot claims no mass limit.

### Follow one threshold

- **Retarding energy**: $0$
- **Distance below reference endpoint**: $0$
- **Observed count**: $0$
- **Dwell time**: $0$
- **Fitted total rate**: $0$

The analyzing plane and detector light up for the selected setpoint. The electron paths are sketches, not sampled events.

### Read the conditional fit

- **Deviance-rise-one mass-squared range**: $0$
- **Source multiplier**: $0$
- **Fitted background rate**: $0$
- **Fitted endpoint offset**: $0$
- **Minimum penalized deviance**: $0$

Accessible counts at every setpoint

Deficit in eV; dwell time in seconds; fitted mean counts

| Deficit | Seconds | Counts | Fitted mean |
| --- | --- | --- | --- |

Conventions, approximations & primary sources

The reference endpoint is 18,575 eV. A positive deficit lies below it, and a positive endpoint offset raises the true endpoint. Electrons travel from left to right: source, analyzing plane, detector. Spacing, coils, flux lines, helix radii and detector segments are schematic. The cutaway and camera views all show the same apparatus. The paths only illustrate electrons that pass or are reflected; no electromagnetic field is tracked and no optical hits are sampled.

Across this narrow window, the electron momentum, energy and Fermi factor are held fixed. The remaining phase space uses one effective mass. An eight-point angular quadrature averages the analytic phase-space integral over an isotropic accepted cone, with a source-to-pinch field ratio of 0.6. The width is fixed at its value at the reference energy. Molecular final states, scattering in the source, Doppler broadening, voltage jitter, radiative corrections, field inhomogeneity and three distinct masses are left out. Because of these omissions, the results cannot be read as a precision KATRIN analysis.

Twenty-four thresholds run from thirty eV below to four eV above the reference endpoint. Dwell times are longer near the endpoint, and each threshold is a separate acquisition, not part of a nested count. A flat background is both generated and fitted; it is not subtracted as a known value. The endpoint calibration penalty stands for an assumed external measurement. The mass-squared scan covers the physical range from zero to nine square electronvolts; it gives no negative estimates and builds no calibrated confidence intervals.

- [KATRIN · Measurement principle](https://www.katrin.kit.edu/79.php)
- [Kleesiek et al. · Spectrum, MAC-E response and statistical model](https://arxiv.org/abs/1806.00369)

[Read the KATRIN story →](https://sandboxphysics.com/stories/neutrinos/katrin-beta-endpoint-neutrino-mass/) [Where does the missing energy go? →](https://sandboxphysics.com/stories/neutrinos/beta-decay-missing-energy/)

Physics tutorial

## Weighing through the endpoint

**Background** The neutrino escapes unseen, but its rest energy changes the very end of the electron spectrum. A MAC-E filter counts only the electrons with enough energy to pass an adjustable electrostatic threshold.

**Why it matters** The Mainz, Troitsk and KATRIN experiments developed this direct way of measuring the neutrino mass from the endpoint. This Lab follows the same path from counts to mass, using an idealized response and simulated counts.

Start with the essentials

- **Focus question**: Can a small error in the endpoint calibration look like a different neutrino mass?
- **One-sentence intuition**: Even a tight statistical contour is only as good as the assumed spectrum, response and calibration.

### Core mathematical model

#### Shape of the spectrum near the endpoint

$$
\frac{d\Gamma}{dE}\propto w\sqrt{w^2-m_\beta^2}\,\Theta(w-m_\beta),\qquad w=E_0-E
$$

The model works in natural units. The electron prefactor varies slowly across this narrow window and is held fixed. Molecular final states are left out, and one effective mass replaces three distinct masses.

#### Counting every electron above the threshold

$$
S(d,m_\beta^2)=\frac{1}{3}(d^2-m_\beta^2)^{3/2}\Theta(d-m_\beta)
$$

For a perfectly sharp threshold, this integral of the phase space gives the signal above the threshold. The MAC-E response then averages it over the transverse energy that electrons keep, for electrons spread evenly in direction inside the accepted cone.

#### Expected counts at each voltage

$$
\mu_i=t_i[a\,R(d_i+\delta,m_\beta^2,\Delta E)+b]
$$

Each threshold is measured on its own for a set dwell time. The source multiplier, background and endpoint offset are nuisance parameters, fitted along with the mass; the width stays at the analysis setting.

#### Profiling the mass

$$
D=2\sum_i\left[\mu_i-n_i+n_i\ln(n_i/\mu_i)\right]+\frac{(\delta-\delta_{\mathrm{cal}})^2}{\sigma_\delta^2}
$$

At each nonnegative mass squared, all three nuisance parameters are fitted again. The displayed contour marks a rise of one in the deviance. Its coverage and its behavior near the physical boundary have not been calibrated.

### Common difficulties

#### A voltage point is not an energy bin

**Typical misconception** Neighboring voltage points split one set of events into separate energy slices.

**Better mental model** Each voltage is a separate measurement. It counts every electron above its threshold during its own dwell time, and that count is an independent Poisson number.

#### A contour is not a published mass limit

**Typical misconception** The shaded range has guaranteed confidence coverage.

**Better mental model** The model scans only nonnegative mass squared and uses an assumed calibration penalty. It builds no calibrated confidence interval and includes none of the real molecular or instrumental systematic effects.

### Run the experiment

1. 01

   #### Scan the endpoint

   Choose Endpoint and step through the thresholds with the arrow keys.

   **What to observe:** Each voltage counts every electron above its threshold, and each is measured for its own dwell time.

2. 02

   #### Look inside the apparatus

   Switch between the cutaway, side, analyzing plane and detector views.

   **What to observe:** The analyzing plane turns back electrons that lack enough energy along the field direction. The detector downstream counts the ones that get through.

3. 03

   #### Change the analysis, not the data

   Change Assumed filter width and Endpoint calibration uncertainty without acquiring again.

   **What to observe:** The counts stay the same; only the fitted model and the mass profile change.

4. 04

   #### Fix the calibration trap

   Choose Calibration trap, then move Calibration center to 0.30 eV.

   **What to observe:** The external calibration now agrees with the shift injected into the data, and the fitted mass changes on the same counts.

Physical conclusion

Measuring the absolute neutrino mass takes the shape of the spectrum at its endpoint and a well-controlled response; oscillations only probe differences between masses.

## 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. [Lab **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. [You are here **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 Energy Sharing Observatory**](https://sandboxphysics.com/labs/beta-energy-sharing/)

Neutrinos · Reading path

### Weigh the particle and test its nature

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

1. [Article **KATRIN: limiting neutrino mass at the end of the beta spectrum**](https://sandboxphysics.com/stories/neutrinos/katrin-beta-endpoint-neutrino-mass/)
2. [Lab **Neutrino Flavor Propagation**](https://sandboxphysics.com/labs/neutrino-flavor-propagation/)

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)