= Franck-Hertz retarding collector transfer
{c}
{title2=$T(E)=H(E-eV_r)$}
For <Electrons> directed normally toward an ideal planar collector, a retarding <electric potential difference> $V_r>0$ imposes the barrier $eV_r$. Write its transfer as $T(E)=H(E-eV_r)$, where $H$ is the <Heaviside step function> and $e$ the positive <elementary charge>. Thus a perfectly sharp loss need not produce a sharp drop in <electric current>: the drop also depends on where the outgoing <energy> lies relative to the barrier. For an oblique incident <Electron> at angle $\theta$ to the normal, the same planar model instead tests $E\cos^2\theta>eV_r$; angular scattering can change collection even at unchanged total <kinetic energy>.
\Image[/franck-hertz-collector-threshold.png]
{title=<Franck-Hertz energy-loss model>: separate acceleration, loss and collection}
{description=Original functional schematic, not a scale reconstruction of a historical tube. The lower panel applies an ideal normally incident retarder to the explicitly chosen one-collision example. Equal vertical strokes mark outgoing <energies>; their labeled weights give <probabilities>.}
{source=franck-hertz-collector-threshold.py}
For example, choose a monoenergetic incoming beam with $u=5.6\,\mathrm{eV}$, $p=0.75$ and $\Delta=4.9\,\mathrm{eV}$. After one encounter, $25\%$ have $5.6\,\mathrm{eV}$ and $75\%$ have $0.7\,\mathrm{eV}$. A $1.0\,\mathrm V$ retarder collects only $0.25I_0$, whereas a $0.5\,\mathrm V$ retarder collects $I_0$; here $I_0$ is the incident <electric current> with fixed geometrical acceptance and no background. The mean outgoing <kinetic energy> is $1.925\,\mathrm{eV}$, independently checked by $5.6-0.75(4.9)$. These are chosen inputs and calculated fractions, not historical measurements.
More generally, let $f(E;V)$ be the normalized outgoing <probability density function> at a fixed accelerating <electric potential difference> $V$, for the normally incident idealization. A retarding scan gives
$$
\frac{I(V,V_r)}{I_0}=\int_{eV_r}^{\infty}f(E;V)\,dE,\qquad
f(eV_r;V)=-\frac{1}{eI_0}\frac{\partial I}{\partial V_r}.
$$
The second equation follows by differentiating the lower integration limit, with <Dirac delta functions> understood for sharp beams. It explains why an integrated collector readout can reveal an <energy> distribution. With angles present it probes the distribution of normal <kinetic energy>; finite instrumental resolution smooths it, and numerical <differentiation> amplifies noise. Keep the incoming beam fixed during this scan. A scan of $V$ changes $f$ itself and cannot be differentiated by the same inversion formula.
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