= Frank-Tamm formula
{c}
{title2=$d^2N/(dx\,d\lambda)$}
{wiki=Frank–Tamm_formula}
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:
$$
\frac{d^2N}{dx\,d\lambda}
=\frac{2\pi\alpha z^2}{\lambda^2}
\left(1-\frac{1}{\beta^2n^2(\lambda)}\right),
\qquad \beta n(\lambda)>1.
$$
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 $dx\,d\lambda$ gives a dimensionless expected count. The explicit <SI> convention is
$$
\alpha=\frac{e^2}{4\pi\epsilon_0\hbar c},\qquad
\frac{d^2N}{dx\,d\omega}=\frac{\alpha z^2}{c}
\left(1-\frac{1}{\beta^2n^2(\omega)}\right).
$$
These conventions agree.<Passage of Particles Through Matter by Donald E. Groom and Spencer Klein>[§34.7.1, equations (34.43)–(34.44), p. 34: photon yields]{ref}
To check the convention, start from $E=\hbar\omega=hc/\lambda$ and $d^2N/(dx\,dE)=\alpha z^2\sin^2\theta/(\hbar c)$. Changing variables uses the positive density Jacobian $|dE/d\lambda|=hc/\lambda^2$; since $h/\hbar=2\pi$, 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 $\hbar\omega$ gives the corresponding energy loss density in this <SI> nonmagnetic convention:
$$
\frac{d^2W}{dx\,d\omega}
=\frac{z^2e^2\omega}{4\pi\epsilon_0c^2}
\left(1-\frac{1}{\beta^2n^2(\omega)}\right).
$$
For an original constant-index example, stipulate $n=1.50$, $\beta=0.90$, $z=1$ and $\alpha=1/137.036$. Integration from $400$ to $700\,\mathrm{nm}$ gives
$$
\frac NL=2\pi\alpha\left(1-\frac1{1.35^2}\right)
\left(\frac1{400\times10^{-9}\,\mathrm m}-\frac1{700\times10^{-9}\,\mathrm m}\right)
\simeq2.217\times10^4\,\mathrm m^{-1}.
$$
Thus a stipulated $1\,\mathrm{cm}$ track produces about $222$ <photons> in this band in the ideal model. This is an emitted count before <absorption (electromagnetic radiation)>, collection losses and detector response, not a brightness measurement or a measured water yield.
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