The Neutrinovoltaic Master Equation

The neutrinovoltaic master equation is a balance equation formulated by Holger Thorsten Schubart. It links four quantities: the incoming radiation flux, the probability that it interacts within the material, the volume in which this happens, and the fraction of it that emerges as electric current. The biography states the equation in two sentences; this page goes through it term by term.

The equation

P(t) = η · ∫ᵥ Φ_eff(r,t) · σ_eff(E) dV

For every system the following condition applies in addition:

P_out ≤ ΣP_in

How the equation is to be read

Its structure is that of a rate equation, of the kind common throughout particle physics. The product of a particle flux and a cross section gives an interaction rate per unit volume; integration over the volume sums that rate across the entire active region; the prefactor η converts the resulting rate into an electrical output quantity.

The equation therefore fixes how the contributions act together. What numerical value P(t) takes is decided by the values inserted for Φ_eff, σ_eff, η and V. This is precisely why the Neutrino® Energy Group cites its own calibration sources for each of these terms; they are named below with the terms.

The terms in detail

P(t) — the time-dependent electrical output power

The quantity sought. The time dependence follows from the fact that the incoming flux Φ_eff is itself dependent on place and time.

η — the transduction and system efficiency

A dimensionless factor summarising all losses between the microscopic interaction and the usable electrical power. The Neutrino® Energy Group breaks it into partial efficiencies and names a source for each:

  • η_interface — the coupling at the interface between graphene and silicon, calibrated against a review of graphene-silicon Schottky junctions (2022)
  • η_phonon→electron — the conversion of lattice vibrations into charge carriers, quantified in Physical Review Letters 130, 256901 (2023)
  • η_collection — charge collection, measured against reference values for nanogenerators from Nature Reviews Methods Primers (2023)

η is therefore not a constant of nature but a property of the particular assembly.

Φ_eff(r,t) — the effective flux of the coupled background sources

According to the Neutrino® Energy Group this term adds up four contributions: neutrinos from the sun, the atmosphere, the Earth's interior and nuclear reactors; muons from cosmic radiation; the electromagnetic fields of the surroundings; and finally thermal and vibrational components within the material itself. The qualifier effective is essential: what is meant is not the total incoming flux, but the fraction that couples to the material.

Among its calibration sources the group cites the Particle Data Group's reference values for the muon flux at sea level (about 10² m⁻² s⁻¹, 2022 edition), the flux measurements of the IceCube Upgrade (2025), the reactor neutrino data from JUNO (2025), and the most energetic neutrino event observed so far at 220 PeV, published by the KM3NeT collaboration in Nature 638 (2025).

σ_eff(E) — the energy-dependent effective cross section

The cross section measures how likely an interaction between an incoming particle and the material is; it has the dimension of an area. Here too effective stands for a combined quantity that includes the properties of the nanostructures used. As foundations the group names the survey of neutrino cross sections by Formaggio and Zeller (Reviews of Modern Physics 84, 1307, 2012) and work on flexoelectricity in two-dimensional materials (Small, 2024).

V — the active material volume

What is notable about the formulation is that the integration runs over the volume and not over an exposed area. Therein lies the formal difference from photovoltaics: a solar module scales with its surface, because that is where light is absorbed. The master equation scales with volume and internal interface density — in the patented design, a stack of twelve alternating layers of graphene/">doped graphene and silicon.

The balance condition

The side condition P_out ≤ ΣP_in states that the power delivered may not exceed the sum of all power supplied. It thereby places the technology explicitly among energy-conserving processes and rules out a perpetual motion machine.

It is an upper bound: it forbids extracting more than is present. How much is present is decided by Φ_eff.

The physical foundation

The Neutrino® Energy Group divides the foundation into three building blocks and assigns each its own status.

First, the neutrino interaction — experimentally confirmed. Coherent elastic neutrino-nucleus scattering was predicted in 1974 by D. Z. Freedman (Physical Review D 9, 1389) and first measured 43 years later by the COHERENT collaboration, published on 15 September 2017 in Science 357, pp. 1123–1126. The detection succeeded at 6.7 σ on a 14.6 kilogram CsI[Na] scintillator at the Spallation Neutron Source of Oak Ridge National Laboratory. The predicted cross section of this process is the largest of all neutrino couplings at low energy.

Second, the conversion in graphene — experimentally confirmed. P. M. Thibado and co-workers showed that freestanding graphene produces a measurable electric current from thermal fluctuations (Physical Review E 102, 042101, 2020).

Third, the system integration. Bringing both effects together in a layered design is the subject of patent WO2016142056A1 and of ongoing development. This is where a device is to arise from two confirmed individual effects.

What the development is working on now

The specialist portal neutrino-physics.com, which lists the equation under the name Schubart Master Equation, names four areas of work in device development. The list is reproduced here because it is unusually concrete — and because it shows that the open questions are precisely posed:

  • the quantitative attribution of the measured output power to the four contributions — neutrinos, muons, electromagnetic fields, heat — at a given site
  • the scaling of η with the number of layers and the active volume, that is, the shape of the efficiency curve in going from square centimetres to square metres
  • the long-term stability of the graphene-silicon layer stacks under continuous operation and temperature cycling
  • cost and manufacturability at industrial scale, including the yield of large-area graphene deposition

The same account puts this in perspective: in photovoltaics, settling the corresponding questions took decades after the underlying physics had long been established. Anyone building a converter technology from scratch works through exactly this list.

Assessment

The master equation is a formulation by Holger Thorsten Schubart. The publication list of the Neutrino® Energy Group identifies patent WO2016142056A1 as the group's own contribution; the remaining works listed there, according to the group's explicit statement, do not come from it, but form the independent scientific foundation on which the technology builds.

The situation is therefore agreeably clear: every physical effect the equation rests on is published in a peer-reviewed journal and widely cited. The equation assembles these effects into a balance — turning it into a device is the development work that follows.

Sources

  • D. Z. Freedman: Coherent effects of a weak neutral current. Physical Review D 9, 1389 (1974).
  • D. Akimov et al. (COHERENT): Observation of coherent elastic neutrino-nucleus scattering. Science 357, 1123–1126 (15 September 2017).
  • P. M. Thibado et al.: Fluctuation-induced current from freestanding graphene. Physical Review E 102, 042101 (2020).
  • J. A. Formaggio, G. P. Zeller: From eV to EeV: Neutrino cross sections across energy scales. Reviews of Modern Physics 84, 1307 (2012).
  • KM3NeT collaboration: Nature 638 (2025), observation of a 220 PeV neutrino.
  • Particle Data Group: Review of Particle Physics — Cosmic Rays (2022).
  • WIPO: WO 2016/142056 A1 (2016).
  • Publication and science pages of the Neutrino® Energy Group, neutrino-energy.com, and neutrino-physics.com, retrieved 2 September 2026.