architecture

Variable Definitions

Precise definitions and units for every quantity in the P(t) master formula: flux density, cross-section, momentum transfer, phonon velocity, and efficiency.

Last updated: 2026-07-20

This chapter gives a precise, self-contained definition of every quantity that appears in the Schubart Master Formula, together with its physical units and its role in the integrand. It is intended as the authoritative reference for the meaning of each symbol; the compact glossary at the notation reference mirrors these definitions, and the formula itself is stated at the equation. For orientation, the master formula reads

P(t) = \iint \Phi(E,\theta,t)\,\sigma(E)\,\Delta p\,v_{\text{ph}}\,\eta_{\text{conv}}\;dE\,d\theta .

Each term below is defined so that the product of the integrand and the integration measure reduces to power, as discussed under dimensional consistency in mathematical architecture.

The output quantity

$P(t)$ — Power output. Units: watts [W]. This is the instantaneous usable electrical power delivered by the converter at time $t$ . It is the quantity the entire framework exists to compute. Because it carries a time argument, $P(t)$ is understood as a function that tracks the fluctuating ambient environment rather than a single steady number. It is always defined relative to a control surface; see system boundaries.

The integrand factors

$\Phi(E,\theta,t)$ — Radiation flux density. Units: incident radiation per unit area, per unit time, per unit solid angle, per unit energy (for example $\mathrm{W\,m^{-2}\,sr^{-1}\,J^{-1}}$ ). This is the ambient resource: the distribution of incident field energy over energy $E$ , incidence angle $\theta$ , and time $t$ . It is supplied by the environment and enters the framework as an externally imposed boundary condition, not as a property of the device. Its multi-argument, distributed character is the reason the framework is written as an integral; see multi-channel energy flux and the dedicated treatment in radiation flux density. Note that $\Phi$ may be expressed either as an energy flux density or, dividing by the quantum energy $E$ , as a number flux density; the two differ by a factor of $E$ and the choice must be kept consistent with the units carried by the other factors.

$\sigma(E)$ — Scattering cross-section. Units: square metres [m²]. This measures the coupling strength between the incident field and the converter material as a function of energy. A large $\sigma(E)$ at a given energy means the material interacts strongly with flux at that energy; a small value means the flux passes through largely uncoupled. Because it depends on $E$ , spectral matching between $\sigma(E)$ and $\Phi$ governs how much power can be harvested. The cross-section has a direct physical reading as an effective target area: the interaction rate per unit incident flux is $R = \Phi\,\sigma$ , so $\sigma$ is the area the material “presents” to the flux at that energy. Its full analytic behaviour and the dual meaning of the symbol $\sigma$ (cross-section here, standard deviation in statistics) are treated in scattering cross-section.

$\Delta p$ — Momentum transfer. Units: kilogram-metres per second [kg m/s]. This is the net directed momentum deposited into the converter per interaction: the symmetry-breaking term. It is defined after the isotropic part of the momentum transfer has been averaged away, so that $\Delta p$ represents only the directed push that can produce a net current. Formally it is the projection onto the collection axis $\hat{\mathbf{n}}$ of the mean deposited momentum, $\Delta p = \langle \mathbf{q}\cdot\hat{\mathbf{n}}\rangle$ , where the average runs over the (possibly asymmetric) distribution of outgoing channels. When the situation is fully symmetric or in equilibrium, this average vanishes, $\Delta p \to 0$ , and the output vanishes with it — the mechanism that keeps the framework thermodynamically honest; see non-equilibrium energy conversion and the kinematic details in governing relations.

$v_{\text{ph}}$ — Phonon velocity. Units: metres per second [m/s]. This characterises how quickly the crystal lattice transports the transferred momentum away from the interaction site. It sets the rate at which directed momentum can be collected before it is randomised, and so acts as a transport factor in the chain from microscopic interaction to macroscopic output. Physically it is of the order of the speed of sound in the material — for typical solids a few $\times 10^{3}\,\mathrm{m\,s^{-1}}$ (an illustrative order of magnitude, not a device value) — and it is bounded above by the requirement that the lattice response be causal. Its microscopic origin lies in the phonon dispersion relation, treated in the solid-state literature cited below.

$\eta_{\text{conv}}$ — Conversion efficiency. Units: dimensionless, constrained to the interval $0 \le \eta_{\text{conv}} \le 1$ . This is the fraction of collected, directed energy that is actually rectified into usable electrical power, as opposed to being lost to heat, back-scattering, or internal resistance. It is the single factor that absorbs all the aggregate losses of the real conversion process, and its bounded range enforces that the framework cannot report more output than the resource and the losses allow. Where a source–sink temperature difference can be assigned, $\eta_{\text{conv}}$ is further capped by a Carnot-style ceiling, as discussed in open thermodynamic systems and in conversion efficiency.

The integration variables

$E$ — Energy variable. Units: energy [J] (or an equivalent spectral unit). This is the integration variable over the ambient spectrum. The energy integral accumulates contributions from every part of the spectrum where the flux and the cross-section overlap. Its lower and upper limits are set by the physical support of the integrand rather than chosen for convenience: formally one integrates from $0$ to $\infty$ , with the understanding that the integrand is negligible wherever either $\Phi(E,\theta,t)$ or $\sigma(E)$ vanishes. Care with the spectral unit is essential, since energy can be quoted in joules, electron-volts, or as a frequency or wavelength; converting between them rescales $\Phi$ correspondingly, and the units audit below must be repeated whenever the spectral variable is changed.

$\theta$ — Angular variable. Units: angle [rad] (or solid angle when the appropriate measure is included). This is the integration variable over incidence direction. The angular integral aggregates contributions arriving from different directions and is where any anisotropy of the ambient flux enters the balance. When $\theta$ is treated as a polar angle, a physically complete integration carries the usual $\sin\theta$ solid-angle weight and, for a flat collector, an additional $\cos\theta$ projection; the framework leaves these measures explicit so that the correct one for a given geometry can be supplied without altering the structure of the formula.

Established versus proposed parameter values

A crucial practical distinction runs through every one of these definitions. For an established converter — a photovoltaic cell, a thermoelectric junction — the numerical values of $\sigma$ , $\Delta p$ , $v_{\text{ph}}$ , and $\eta_{\text{conv}}$ are known from measurement, and the framework simply organises them. For a proposed converter whose coupling mechanism has not been experimentally confirmed — a neutrinovoltaic device being one such case — one or more of these parameters is a model input rather than a measured quantity. The framework does not distinguish the two cases in its structure, and that is deliberate: the same formula applies, but the epistemic status of the output differs entirely. When a factor is a placeholder, the resulting $P(t)$ is a conditional prediction (“if $\sigma$ takes this value, then the power is this”), not a demonstrated output. Keeping this distinction visible — never quietly promoting an assumed parameter to a measured one — is part of the discipline the framework is meant to enforce.

Interdependence of the factors

Although the factors are written as if independent — which is what makes the framework useful for isolating bottlenecks — several of them are physically linked, and a careful application keeps those links in view. The cross-section $\sigma(E)$ and the momentum transfer $\Delta p$ both derive from the same underlying interaction: the scattering physics that determines how strongly the field couples also determines how much and in what direction momentum is deposited. Similarly, the phonon velocity $v_{\text{ph}}$ and the conversion efficiency $\eta_{\text{conv}}$ are linked through the scattering time that governs both how far directed momentum travels and how much is lost to heat. Treating the five factors as independent is a first-order approximation, valid when the couplings between them are weak; where they are not, the factors must be co-determined from a common microscopic model rather than assigned separately. Stating the factors separately does not deny these links — it exposes them, so that an analyst knows which parameters cannot be varied in isolation. This is the difference between a framework that merely lumps everything into one coefficient and one that keeps the physics legible.

Reading the factors together

Taken together, the definitions describe a pipeline: a distributed resource ( $\Phi$ ) is coupled ( $\sigma$ ), converted into directed momentum ( $\Delta p$ ), transported ( $v_{\text{ph}}$ ), and rectified ( $\eta_{\text{conv}}$ ), with the total accumulated over energy and angle. Holding these definitions and units fixed across all chapters is what allows the framework to be applied consistently, whether to an established converter or to a proposed one — such as a neutrinovoltaic device — whose parameters are still uncertain.

Units audit at a glance

Collecting the units confirms that the integrand times the measure yields power. Writing $\Phi$ in $\mathrm{W\,m^{-2}\,sr^{-1}\,J^{-1}}$ , the product of the integrand factors and the measure $dE\,d\theta$ (in $\mathrm{J\,sr}$ ) is

\underbrace{\mathrm{W\,m^{-2}\,sr^{-1}\,J^{-1}}}_{\Phi}\cdot\underbrace{\mathrm{m^{2}}}_{\sigma}\cdot\underbrace{\mathrm{kg\,m\,s^{-1}}}_{\Delta p}\cdot\underbrace{\mathrm{m\,s^{-1}}}_{v_{\text{ph}}}\cdot\underbrace{1}_{\eta_{\text{conv}}}\cdot\underbrace{\mathrm{J\,sr}}_{dE\,d\theta} .

The $\mathrm{m^{-2}}$ and $\mathrm{m^{2}}$ cancel, as do $\mathrm{sr^{-1}}$ and $\mathrm{sr}$ and $\mathrm{J^{-1}}$ and $\mathrm{J}$ , leaving $\mathrm{W}$ multiplied by the residual $\mathrm{kg\,m^{2}\,s^{-2}}=\mathrm{J}$ from $\Delta p\,v_{\text{ph}}$ . A concrete realisation therefore supplies a proportionality constant (with units $\mathrm{J^{-1}}$ , i.e. an inverse-energy normalisation from the interaction kinematics) so that the full expression closes exactly to watts. The complete audit is carried out in dimensional analysis.