reference

Worked Example

From the formula to a number: an illustrative, order-of-magnitude evaluation of the Schubart Master Formula P(t) at a single representative slice, worked step by step.

Last updated: 2026-07-25

This chapter turns the Master Formula into a single number. The point is not to rate any device — no measured device values are used — but to show how the formula is evaluated and, more importantly, what the arithmetic reveals about the physics. Every value below is an illustrative order of magnitude in SI-like units, chosen only to make the mechanics of the calculation concrete. The exact statement remains the double integral derived in the full derivation; here we evaluate the integrand at one representative slice.

The representative-slice shortcut

The formula integrates a product of five factors over energy and angle,

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

For an order-of-magnitude estimate we replace the integral by a single representative slice: evaluate each factor at a typical energy and direction, and multiply by the effective widths $\Delta E$ and $\Delta\theta$ over which the integrand is appreciable. This is the same move one makes when estimating any integral by “peak value times width”:

P \;\approx\; \Phi\,\sigma\,\Delta p\,v_{\text{ph}}\,\eta_{\text{conv}}\,\Delta E\,\Delta\theta .

The estimate is only as good as the assumption that one slice represents the whole spectrum, which is why it is a teaching device, not a measurement.

Step 1 — representative values

Factor	Symbol	Illustrative value	Unit
Radiation flux density	$\Phi$	$10^{10}$	$\mathrm{m^{-2}\,s^{-1}\,sr^{-1}\,J^{-1}}$
Scattering cross-section	$\sigma$	$10^{-44}$	$\mathrm{m^{2}}$
Momentum transfer	$\Delta p$	$10^{-21}$	$\mathrm{kg\,m\,s^{-1}}$
Phonon velocity	$v_{\text{ph}}$	$10^{4}$	$\mathrm{m\,s^{-1}}$
Conversion efficiency	$\eta_{\text{conv}}$	$10^{-2}$	—
Energy width	$\Delta E$	$10^{-13}$	$\mathrm{J}$
Angular width	$\Delta\theta$	$1$	$\mathrm{sr}$

These numbers are deliberately round. The flux is enormous, the cross-section is minute, and the efficiency is a modest few percent — the qualitative profile that any continuous-ambient-conversion proposal must confront.

Step 2 — multiply

Because the estimate is a product, the cleanest way to evaluate it is to add the exponents. Collecting the powers of ten,

P \;\approx\; 10^{10}\cdot 10^{-44}\cdot 10^{-21}\cdot 10^{4}\cdot 10^{-2}\cdot 10^{-13}\cdot 10^{0}\ \text{W} .

Summing the exponents, $10 - 44 - 21 + 4 - 2 - 13 + 0 = -66$ , so

P \;\approx\; 10^{-66}\ \text{W}

for a single representative slice. The dimensional analysis confirms that the units of this product are indeed watts, so the number is at least dimensionally honest.

Step 3 — what the number means

Taken literally, $10^{-66}\ \text{W}$ from one slice is vanishingly small — and that is the point. The result is dominated by the tension between the two extreme factors: the flux $\Phi \sim 10^{10}$ pushes the estimate up, while the coupling $\sigma \sim 10^{-44}$ pulls it down by forty-four orders of magnitude. No amount of favourable engineering in the other three factors can rescue a slice whose coupling is that small.

This is exactly why the framework treats $\sigma$ and $\eta_{\text{conv}}$ as the decisive design levers, and why the scientific questions return repeatedly to the cross-section: the entire feasibility argument for any continuous-ambient proposal rests on whether the integrated coupling — summed over an enormous flux and a broad spectrum, possibly with coherent enhancement — can climb back across those orders of magnitude. A single slice cannot answer that; only the full integral, with realistic spectral and angular distributions, can. You can vary all five factors yourself in the parameter explorer and watch the exponent move.

Why one slice is not the whole story

The representative-slice estimate deliberately discards the two features that do the real work in a genuine evaluation. First, the true flux spans many orders of magnitude in energy, so the integral over $E$ accumulates contributions from a vast number of slices, not one. Second, the coupling $\sigma(E)$ may be sharply peaked or coherently enhanced at particular energies, in which case the effective product is set by a narrow band that a single average badly misrepresents. A faithful number therefore requires the full $\iint \cdots \, dE\,d\theta$ with measured or modelled distributions — which is precisely the calculation the framework exists to organise, and which remains open to independent validation.