Proton Decay Rate Calculator

Enter detector and lifetime assumptions

Use scientific notation when needed, such as 1e34 for an assumed lifetime of 10³⁴ years.

Example data table

These hypothetical cases illustrate how exposure, efficiency, and lifetime assumptions influence expected signal counts.

Scenario	Mass (kg)	Molar Mass (g/mol)	Protons per Molecule	Lifetime (years)	Exposure (years)	Efficiency (%)	Expected Signal
Water target, long search	1000	18	10	1e34	10	85	2.84e-04
Oxygen-rich medium	5000	16	8	5e33	12	80	2.89e-03
Iron target study	15000	55.845	26	1e35	20	90	7.57e-04

Formula used

Targets: \(N_{targets} = \frac{m_{kg} \times 1000}{M} \times N_A \times f_{iso}\)

Target protons: \(N_p = N_{targets} \times Z_p\)

Decay constant: \(\lambda = 1 / \tau\)

Half-life: \(t_{1/2} = \tau \ln 2\)

Survival probability: \(P_{survive} = e^{-T/\tau}\)

Decay probability: \(P_{decay} = 1 - e^{-T/\tau}\)

Expected detected signal: \(S = N_p \times P_{decay} \times \varepsilon \times BR\)

Small-probability approximation: \(S \approx N_p \times T \times \varepsilon \times BR / \tau\)

Lifetime lower bound from null search: \(\tau_{limit} \approx N_p \times T \times \varepsilon \times BR / N_{limit}\)

How to use this calculator

Enter the detector sample mass and the target material molar mass.

Provide the number of protons available per atom or molecule.

Set active isotopic abundance if only part of the target is useful.

Enter the assumed proton mean lifetime for the decay model.

Add exposure time, detection efficiency, and channel branching ratio.

Choose a signal upper limit, such as 2.3 for a simple 90% null-search case.

Submit the form to view results above the calculator.

Use CSV or PDF buttons to export the displayed results.

FAQs

1. What does this calculator estimate?

It estimates target proton count, decay constant, half-life, survival probability, expected physical decays, expected detected signal, and a simple lower bound from null-search sensitivity assumptions.

2. Does this prove proton decay exists?

No. Proton decay remains unobserved. This calculator only models expected outcomes under hypothetical lifetimes and detector assumptions used in rare-event physics studies.

3. What should I enter for protons per molecule?

Use the total proton count available in one target unit. For a pure element, this is the atomic number. For compounds, sum the proton counts across the full molecule.

4. Why are the expected event counts so small?

Proton lifetime limits are extraordinarily large, often near 10³⁴ years. Even huge detectors and long exposures therefore predict tiny signals under most reasonable assumptions.

5. What does the signal upper limit mean?

It represents the allowed number of signal events when no clear discovery appears. A common simple choice is 2.3 for an idealized 90% one-sided Poisson limit.

6. Why include efficiency and branching ratio?

Not every decay is reconstructed, and experiments usually search a specific channel. Efficiency scales observable events, while branching ratio represents how often the chosen channel occurs.

7. Is the lower bound exact?

No. It is a compact sensitivity estimate. Real lifetime limits depend on full likelihood treatment, detector response, energy windows, channel acceptance, and measured background uncertainty.

8. Can I use scientific notation in the inputs?

Yes. The calculator accepts values like 1e34, 5e-6, or 6.022e23, which is useful when working with extreme particle-physics scales.