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.

Frozen scan counts $0$

Fitted mass squared $0\,\mathrm{eV}^2/c^4$

Fitted endpoint offset $0\,\mathrm{eV}$

Scan the end of the beta spectrum

Weigh the neutrino from the end of the spectrum

Injected effective mass

m_\beta

1.2\,\mathrm{eV}/c^2

Dwell-time multiplier

T

4\,\times

Acquired filter width

\Delta E

1\,\mathrm{eV}

Injected background rate

b

0.15\,\mathrm{s}^{-1}

Inject a calibration shift

Injected endpoint offset

\delta_{\mathrm{true}}

0\,\mathrm{eV}

Refit the same scan

Assumed filter width

\Delta E_{\mathrm{fit}}

1\,\mathrm{eV}

Calibration center

\delta_{\mathrm{cal}}

0\,\mathrm{eV}

Endpoint calibration uncertainty

\sigma_\delta

0.1\,\mathrm{eV}

Try the Calibration trap preset, then move the calibration center to 0.30 eV. The counts stay fixed.

Changes to the acquisition settings need a new scan. The analysis controls refit the existing counts.

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.

Sampled countsFitted meansRefitted massless referenceReveal 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.

Read the KATRIN story →Where does the missing energy go? →

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 misconceptionNeighboring voltage points split one set of events into separate energy slices.

Better mental modelEach 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 misconceptionThe shaded range has guaranteed confidence coverage.

Better mental modelThe 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

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.
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.
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.
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.

Beta Endpoint Mass Observatory

The last few electronvolts.

A threshold scan, not an energy histogram

Inspect the same counts near the endpoint

What the filter actually accepts

How much can the mass change?

Follow one threshold

Read the conditional fit

Weighing through the endpoint

Core mathematical model

Shape of the spectrum near the endpoint

Counting every electron above the threshold

Expected counts at each voltage

Profiling the mass

Common difficulties

A voltage point is not an energy bin

A contour is not a published mass limit

Run the experiment

Scan the endpoint

Look inside the apparatus

Change the analysis, not the data

Fix the calibration trap

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