The Master Equation.

The equation

Components

Each term packages a physical process that is independently established and measured in the contemporary neutrino and radiation literature.

P(t)

Electrical output power of the device as a function of time. In steady ambient conditions this is approximately constant; variations with solar activity, weather, and altitude follow from the corresponding variations in the flux components.

η

Dimensionless conversion efficiency of the material system — the fraction of recoil energy delivered to nuclei (or of EM-field energy delivered to electrons) that is ultimately collected as electrical output. Determined empirically for a given device configuration; depends on material choice, layer thickness, phonon and charge-carrier transport, and readout architecture.

Φ_eff(r, t)

Effective flux at position r and time t — a weighted sum of the physically relevant radiation components:

• Neutrino flux from solar, atmospheric, reactor, geoneutrino, and cosmological sources, interacting through coherent neutral-current scattering on target nuclei.
• Cosmic-ray muon flux, whose ionization signal is comparatively larger per event but less continuous in temporal coverage.
• Electromagnetic field fluctuations — ambient RF, thermal noise, and anthropogenic EM environment — coupled to charge carriers in the conducting layers.
• Thermal gradients across the material stack, contributing Seebeck-effect-like components to the output.

σ_eff(E)

Energy-dependent effective cross-section. For the neutrino channel this is the CEvNS cross-section (established experimentally by the COHERENT collaboration in 2017, showing the predicted N² dependence on neutron number). For the muon channel it reduces to ionization energy loss dE/dx. Other terms contribute the RF and thermal coupling strengths.

V

Active volume of the material system. Device geometry, layer count, and the fraction of deposited volume that contributes to readout all enter here.

Physical grounding

The formulation builds on three experimental foundations, each independently peer-reviewed and recognized.

2015 — neutrino oscillation and mass

The 2015 Nobel Prize in Physics to Takaaki Kajita and Arthur McDonald recognized the discovery of neutrino oscillations, establishing that at least two neutrino mass eigenstates are non-zero. Neutrinos therefore carry non-zero momentum and can transfer momentum in scattering — a necessary condition for any energy-conversion process that relies on neutrino interaction.

2017 — CEvNS observation

Coherent elastic neutrino-nucleus scattering was predicted by Freedman in 1974 and observed by the COHERENT collaboration at Oak Ridge in 2017. The measured cross-section showed the predicted N² scaling and is the largest neutrino cross-section at sub-50-MeV energies. This is the process parametrized by σ_eff(E) in the Master Equation for the neutrino contribution.

2025 — precision flux measurement

JUNO's 2025 first oscillation-spectrum data provide the most precise reference for reactor antineutrino fluxes at the 10-MeV scale, constraining the flux term Φ_eff with per-cent-level accuracy. Solar neutrino fluxes are separately constrained by SNO, Super-Kamiokande, and Borexino.

From equation to device

Translating the Master Equation into a physical device requires several engineering steps not captured in the equation itself:

1. Material choice: a target system that both couples efficiently to the relevant radiation components and permits charge collection. Graphene-silicon multilayer nanostructures are currently the active research direction, combining high carrier mobility with tuneable work functions.
2. Layer architecture: alternating conducting, insulating, and active layers at nanometre thickness, tailored to extract recoil-induced charge or phonon excitations before recombination.
3. Readout electronics: high-input-impedance, low-noise charge amplifiers, coupled to storage and conditioning circuitry for end-use output.
4. Environmental shielding: to manage unwanted components such as bulk thermal noise or intermittent high-intensity events (cosmic-ray air showers, lightning EMI).

Interactive exploration

The simulator below sweeps the scalar form of the Master Equation across the four parameters η, Φ_eff, σ_eff, and V, and shows the resulting output power with an adjustable ambient-fluctuation amplitude. Units are normalised to the orders of magnitude relevant for the low-energy neutrino, muon, RF and thermal contributions; the curve is illustrative rather than a detailed device simulation.

Established foundations

The physics entering the Master Equation rests on three Nobel-level results that are neither under dispute nor open to revision:

• The 2015 Nobel Prize established that neutrinos have mass and can therefore exchange momentum with matter — the necessary precondition for any conversion process on the neutrino channel.
• The 2017 COHERENT observation confirmed that coherent elastic neutrino-nucleus scattering follows the predicted N² scaling and is the dominant low-energy interaction channel — the σ_eff(E) term for neutrinos.
• The 2025 JUNO first oscillation spectrum provides per-cent-level reference fluxes at reactor energies — the Φ_eff term is now measured to the precision that device engineering requires.

The equation itself is a straightforward integral of these established quantities. Every variable it contains is a measured or measurable physical parameter of the radiation environment or the device material.

Device characterisation in progress

What remains the focus of ongoing work is the engineering side — the parameters that describe how a specific multilayer device couples to the four ambient channels and converts the deposited energy to usable output. This is normal for any new energy-conversion technology: the equivalent questions for photovoltaic cells took decades to map out empirically even after the underlying physics was settled.

• Quantitative per-channel attribution of the measured output among neutrino, muon, electromagnetic, and thermal contributions at a given deployment site
• Scaling of η with layer count and active volume — the shape of the conversion-efficiency curve as devices grow from cm² to m² scale
• Long-term stability of graphene-silicon multilayer stacks under continuous operation and thermal cycling
• Cost and manufacturability parameters for industrial-scale deployment, including yield metrics for large-area graphene deposition

Characterisation programmes addressing each of these questions are running at multiple university and industrial partner sites, with a continuous publication pipeline into the applied-physics and materials-science literature. The experimental picture will continue to sharpen in the same iterative way that photovoltaic efficiency records have been refined since the 1970s.