= Finite-band Cherenkov yield model
Choose an original transparent-band model, defined only for $400\leq\lambda\leq700\,\mathrm{nm}$:
$$
n(\lambda)=1.28+0.04\left(\frac{500\,\mathrm{nm}}\lambda\right)^2.
$$
This chosen <refractive index> is not a fit to water, glass or the historical liquids. It decreases from $1.3425$ to approximately $1.300408$, so the threshold $1/n$ increases across the band. For $\beta=0.75$, solving $n(\lambda_*)=1/\beta$ gives
$$
\lambda_*=500\sqrt{\frac{0.04}{1/0.75-1.28}}\,\mathrm{nm}
\simeq433.013\,\mathrm{nm}.
$$
Only $400\leq\lambda<\lambda_*$ emits. Faster particles with $\beta=0.78$ and $0.90$ emit throughout this chosen band. The <Cherenkov cone angle> and the <Frank-Tamm formula> are both evaluated with the same $n(\lambda)$.
\Image[/cherenkov-dispersive-threshold-angle-yield.png]
{title=Original dispersive <Cherenkov radiation> model: a wavelength-dependent threshold, emission <angle>, and produced <photon> spectrum}
{description=All curves are calculations for the chosen finite-band <refractive index>, with $z=1$ and $\alpha=1/137.036$. A missing angular segment means the threshold fails; the corresponding spectral yield is zero. The spectral axis is per centimetre of track and per nanometre of <vacuum> <wavelength>, so it is $10^{-11}$ times the density in <SI> units. No historical observations, measured material constants, <absorption (electromagnetic radiation)> or detector response are plotted.}
For a $1\,\mathrm{cm}$ track the chosen model predicts about $0.589$, $27.40$ and $142.85$ emitted <photons> for $\beta=0.75$, $0.78$ and $0.90$, respectively. An expected count below one is meaningful over repeated equivalent tracks; it is not a fractional <photon> observed on one track. The near-threshold example illustrates why extending a constant-index approximation across an entire optical band can give the wrong emitting interval.
Finite bandwidth is essential. The $1/\lambda^2$ factor cannot be extrapolated down to arbitrarily small <wavelength>: the chosen material model stops at its stated band. Real <optical dispersion> changes the threshold and direction, and appreciable <absorption (electromagnetic radiation)> requires a complex material response beyond the simple transparent-medium expression. A detected count must also include transport, collection geometry and the detector's wavelength-dependent response. Even a correct produced spectrum therefore does not, on its own, predict perceived colour or observed glow intensity.
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