Neutrinos · D14 · Weigh the invisible

Beta Endpoint Mass Observatory

Scan an electrostatic threshold, freeze the counts and fit a neutrino mass. Explore how exposure, transmission width and endpoint calibration change the inference.

Frozen scan counts $0$

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

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

SCAN · COUNT · INFER

Weigh the invisible.

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 FROZEN 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 Calibration trap, then move the calibration center to 0.30 eV. Keep the counts fixed.

Acquisition controls require a new scan. Analysis controls refit the existing counts.

Teaching simulation inspired by MAC-E spectroscopy. A single effective mass, a frozen electron prefactor and an ideal accepted-cone response replace the molecular tritium spectrum and real apparatus. Source strength, positive background and endpoint offset are refitted at each nonnegative mass squared. Calibration has a Gaussian penalty; response width is fixed to the analysis setting. Deviance contours have no calibrated coverage and are not experimental mass limits. No KATRIN data or sensitivity is reproduced.

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 in the last ten electronvolts. The gray curve is the refitted massless mean minus the best fit; pink is a separate generating reference.

What the filter actually accepts

Normalized transmission versus surplus energy for the acquired (blue) and assumed (gold) widths. Magnetic acceptance is absorbed into source normalization. This is an ideal response, not reconstructed detector data.

How much can the mass change?

Source strength, positive background and endpoint offset are refitted at each nonnegative mass squared. Shading marks a deviance rise of one; values above twelve are clipped. No coverage calibration or mass-limit claim.

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 highlight the selected setpoint. The electron paths are schematic illustrations, 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

Reference endpoint: 18,575 eV. Positive deficit is below it; positive endpoint offset raises the true endpoint. The electron travels left to right: source, analyzing plane, detector. Spacing, coils, flux lines, helical radii and detector segmentation are schematic. Cutaway and camera views expose the same apparatus. Paths only illustrate accepted and reflected electrons, without tracking an electromagnetic field or sampling optical hits.

The electron momentum, energy and Fermi factor are frozen across this narrow window. The remaining phase space uses one effective mass. An eight-point angular quadrature averages the analytic phase integral over an isotropic accepted cone with source-to-pinch field ratio 0.6. Width is frozen at the reference energy. Molecular final states, source scattering, Doppler broadening, voltage jitter, radiative corrections, field inhomogeneity and three distinct masses are omitted. These omissions preclude a precision KATRIN interpretation.

Twenty-four thresholds span thirty eV below to four eV above the reference endpoint. Dwell times grow near the endpoint; every threshold is a separate acquisition, not a nested count. A flat background is generated and fitted, not subtracted as known truth. The endpoint calibration penalty is an assumed external constraint. The physical mass-squared scan runs from zero to nine square electronvolts; negative estimates and calibrated confidence constructions are absent.

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

CONTINUE EXPLORING

From flavor to evidence

Propagation predicts a flavor probability. Which parts of the experiment turn an interaction into a usable record?

Start with the essentials

Focus question: Can a small endpoint calibration error resemble a different neutrino mass?
One-sentence intuition: A precise statistical contour remains conditional on the spectrum, response and calibration model.

Core mathematical model

The endpoint phase space

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

Natural units are used inside the model. The slowly varying electron prefactor is frozen. Molecular final states and three distinct masses are omitted.

First integrate, then average the response

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

For a sharp threshold, the phase-space primitive gives the integrated signal. The MAC-E response averages this primitive over residual transverse energies in an accepted isotropic cone.

Counts at different voltages

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

Each threshold is measured independently for a specified dwell time. The source multiplier, background and endpoint offset are nuisance parameters; the width is held at the analysis setting.

A conditional mass profile

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}

Refit all three nuisances at each nonnegative mass squared. A deviance rise of one defines the displayed contour. Coverage and physical-boundary behavior have not been calibrated.

Common difficulties

A voltage point is not an energy bin

Typical misconceptionAdjacent points divide one event sample into exclusive energies.

Better mental modelEach voltage has a separate dwell and independent Poisson count above threshold.

A contour is not a published mass limit

Typical misconceptionThe shaded range has guaranteed confidence coverage.

Better mental modelThe model uses a nonnegative mass scan and an assumed calibration penalty. It lacks calibrated confidence construction and real molecular or instrumental systematics.

Run the experiment

01
Scan the endpoint
Select Endpoint and inspect each threshold with arrow keys.
What to observe: Every voltage samples an integral spectrum for a different dwell time.
02
Expose the apparatus
Switch between cutaway, side, analyzing plane and detector.
What to observe: The analyzing plane rejects electrons with insufficient longitudinal energy. The downstream detector counts transmitted electrons.
03
Keep the record fixed
Change assumed width and endpoint uncertainty without acquiring.
What to observe: The counts stay fixed while the model and the profile change.
04
Repair the calibration trap
Select Calibration trap, then move calibration center to 0.30 eV.
What to observe: The known injected shift becomes compatible with the external calibration constraint; the mass inference changes on the same data.