Energy Conversion from Environmental Radiation

We live inside a constant bath of radiation. Radio and mobile signals fill cities, every surface glows in the infrared, sunlight and artificial light reach most rooms, and particles from space, from cosmic-ray muons to solar neutrinos, cross every object without pause. It is tempting to see all of this as an untapped energy supply. The reality is more demanding. The intensities span many orders of magnitude, the energy per quantum ranges from micro-electronvolts for radio waves to millions of electronvolts for particles, and much of the radiation is in thermal equilibrium with its surroundings, which means it cannot be converted at all. Deciding which parts of the environmental radiation field can deliver usable power, and how much, requires both physics and careful bookkeeping.

The Established Approaches

Each part of the spectrum has its own established description.

Radio-frequency radiation. Received power follows the Friis equation, P_r = P_t · G_t · G_r · (λ / 4πd)², and rectennas convert it to direct current [1]. A city-wide survey of ambient RF power densities in London found usable levels mainly close to mobile base stations and broadcast transmitters, enough for low-power sensors near a source but falling off quickly with distance [2].

Infrared and thermal radiation. A surface at temperature T emits σ_SB · T⁴, about 460 W/m² at 300 K, where σ_SB is the Stefan-Boltzmann constant. Because the surroundings emit back almost the same amount, net exchange at uniform temperature is zero. Power can only be drawn from a difference: a cell that radiates to the cold night sky can generate a small output, as shown for thermoradiative “night-time” photovoltaics [3]. Optical rectennas aim to rectify infrared and visible radiation directly [4].

Visible light. Photovoltaics remain the most efficient route, bounded for a single junction by the Shockley-Queisser limit of about 33 per cent [5].

Particle fluxes. At sea level, roughly one cosmic-ray muon crosses each square centimetre per minute [6]. Solar neutrinos arrive at about 6 × 10¹⁰ per square centimetre per second [7], but interact very rarely; coherent elastic neutrino-nucleus scattering, observed in 2017, transfers a tiny push to whole nuclei [8]. The power these particles deposit in a device-sized volume is very small.

Each model works for its band. Together they show a hard boundary: radiation in equilibrium with the device cannot be converted, and only the non-equilibrium part is available.

The Open Question

The established models treat each band and each particle type on its own, with its own detector or converter. A material exposed to the real environment receives all of them at once: a little RF, a non-equilibrium share of infrared, some light, thermal fluctuations and a trickle of particles. What has not been described in one framework is how much of this mixed input a single material couples in, how the second law limits the convertible share, and how the contributions add up in one output, with the very small particle channels kept honestly in proportion.

A Generalized Approach

A generalized approach to this problem is represented by the Schubart Master Formula, which integrates all coupled inputs over the active volume:

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

The derivation, notation and open questions are published at pt-schubart.com. The model is introduced in Mathematical Models for Multi-Source Energy Harvesting.

Definitions

• P(t): instantaneous electrical output, the usable electrical power delivered at time t. Unit: watt (W). Energy over an interval is the time integral of P(t), in joules or watt-hours.
• η: total conversion efficiency, dimensionless, between 0 and 1. It combines coupling, resonance, rectification and electrical losses.
• Φ_eff(r, t): effective ambient flux density at position r and time t, the local strength of all ambient inputs acting on the material.
• σ_eff(E): energy-dependent effective interaction cross-section, how strongly the material couples to an input of energy E.
• ∫_V: volume integral over the active material.

Together, Φ_eff · σ_eff is the coupled power per unit volume (W/m³), so the integral over V yields watts. How that product divides between the two factors depends on the input channel, so no separate units are assigned to them here.

Assumptions and Boundaries

The Schubart Master Formula is a proposed generalized framework, not an established equation of physics. Its assumptions and limits:

• The material is an open, non-equilibrium system that exchanges energy with its surroundings.
• Output is bounded by coupled input power: P_out ≤ ΣP_in.
• Φ_eff counts only the non-equilibrium part of the radiation field. Radiation in equilibrium with the device contributes nothing.
• Channels with very low coupling, such as neutrino interactions, contribute correspondingly little; the framework keeps every channel in proportion rather than letting one dominate the narrative.
• Coupling values must be measured channel by channel, with each input isolated or shielded in turn.

Worked Example

All values below are illustrative values, chosen to show the method. They are not measured results.

A layered film of 1 m² with an active thickness of 10 µm has V = 1 × 10⁻⁵ m³. Assume these coupled power densities Φ_eff · σ_eff:

Input	Φ_eff · σ_eff	Coupled power
Thermal fluctuations (non-equilibrium share)	300 W/m³	3.0 mW
Infrared (net exchange)	200 W/m³	2.0 mW
Radio frequency	50 W/m³	0.5 mW
Particle interactions (muons, neutrinos and others)	1 W/m³	0.01 mW
Total ΣP_in		5.51 mW

With η = 0.1, the output is P ≈ 0.55 mW, within the boundary (0.55 mW ≤ 5.51 mW). In this example the particle channels contribute less than 0.2 per cent of the input. The table makes visible what a single-channel model hides: which inputs carry the result and which are present but small.

Where It Leads

Which of these inputs a material actually couples in depends on its structure; see Graphene in Energy Conversion. The smallest practical use, a sensor that changes batteries less often, is worked through in Powering Autonomous Sensors Without Batteries. The scaling to devices and buildings is described in From Sensor to Grid.

References

1. H. J. Visser and R. J. M. Vullers, “RF energy harvesting and transport for wireless sensor network applications: principles and requirements,” Proceedings of the IEEE 101, 1410–1423 (2013).
2. M. Piñuela, P. D. Mitcheson and S. Lucyszyn, “Ambient RF energy harvesting in urban and semi-urban environments,” IEEE Transactions on Microwave Theory and Techniques 61, 2715–2726 (2013).
3. T. Deppe and J. N. Munday, “Nighttime photovoltaic cells: electrical power generation by optically coupling with deep space,” ACS Photonics 7, 1–9 (2020).
4. G. Moddel and S. Grover (eds.), Rectenna Solar Cells, Springer (2013).
5. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” Journal of Applied Physics 32, 510–519 (1961).
6. Particle Data Group, “Cosmic rays,” in Review of Particle Physics, current edition.
7. Borexino Collaboration (M. Agostini et al.), “Comprehensive measurement of pp-chain solar neutrinos,” Nature 562, 505–510 (2018).
8. COHERENT Collaboration (D. Akimov et al.), “Observation of coherent elastic neutrino-nucleus scattering,” Science 357, 1123–1126 (2017).
9. Schubart Master Formula: derivation, notation and open questions, pt-schubart.com.

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