Frank-Tamm formula ($d^{2}N/(dxd\lambda )$)

For constant particle speed and electric charge $q=ze$, in a transparent, linear, nonmagnetic medium with isotropy, the Frank-Tamm formula gives the produced photon density:Set it to zero below threshold. Here $e$ is the positive elementary charge, $x$ is track length, $\lambda =2\pi c/\omega$ is vacuum wavelength, and $\alpha$ is the fine-structure constant. With SI lengths in metres, the density has units ${\mathrm{m}}^{-2}$; multiplying by $dxd\lambda$ gives a dimensionless expected count. The explicit SI convention isThese conventions agree.^[ref]

To check the convention, start from $E=\hslash \omega =hc/\lambda$ and $d^{2}N/(dxdE)=\alpha z^2{\sin }^2\theta /(\hslash c)$. Changing variables uses the positive density Jacobian $|dE/d\lambda |=hc/{\lambda }^2$; since $h/\hslash =2\pi$, the coefficient becomes $2\pi \alpha z^2$. No factor of the material refractive index belongs in this Jacobian: $\lambda$ is not the in-medium wavelength. Multiplication by $\hslash \omega$ gives the corresponding energy loss density in this SI nonmagnetic convention:

For an original constant-index example, stipulate $n=1.50$, $\beta =0.90$, $z=1$ and $\alpha =1/137.036$. Integration from $400$ to $700\mathrm{nm}$ givesThus a stipulated $1\mathrm{cm}$ track produces about $222$ photons in this band in the ideal model. This is an emitted count before absorption, collection losses and detector response, not a brightness measurement or a measured water yield.

Choose an original transparent-band model, defined only for $400\le \lambda \le 700\mathrm{nm}$: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 $n({\lambda }_{\ast })=1/\beta$ givesOnly $400\le \lambda <{\lambda }_{\ast }$ 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 $n(\lambda )$.

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 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.