= Franck-Hertz repeated-loss collection model
{c}
{title2=$E_n=E_0+e(V-V_c)-n\Delta$}
Define a signed offset $V_c$ so that the accelerating region adds <kinetic energy> $e(V-V_c)$. This convention concerns this gap; collector contact fields belong in the effective $V_r$. An <Electron> emitted with initial <kinetic energy> $E_0$ and undergoing $n$ identical excitation losses reaches the grid with
$$
E_n=E_0+e(V-V_c)-n\Delta.
$$
Each loss excites a fresh <chemical atom> initially in its <ground state>. These are repeated payments of the same excitation cost. The index $n$ counts events along an <Electron> trajectory, not the ordinal number of an atomic <excited state>. The outgoing <energy> also does not locate the last collision: after an early collision an <Electron> can regain enough <kinetic energy> to cross the barrier.
At fixed $V$, let $g(E_0)$ be the normalized initial-<energy> <probability density function> within the ensemble of <Electrons> reaching the grid, and $w_n(E_0,V)$ the <conditional probabilities> for $n$ losses within that ensemble. Then the ideal collection law is
$$
\frac{I(V,V_r)}{I_0}=\int_0^\infty g(E_0)\sum_{n=0}^\infty w_n(E_0,V)H\bigl(E_0+e(V-V_c)-n\Delta-eV_r\bigr)\,dE_0.
$$
Take $\sum_n w_n=1$ and exclude paths with insufficient <kinetic energy> for a loss at its actual collision position. The weights contain collision positions and rates, <mean free paths> and angular acceptance. This equation is a bookkeeping model, not a derivation of those weights. Multiplying all incident particles by one constant transmission is an additional approximation; real transmission and $I_0$ can vary with $V$.
One analytical extreme assumes $E_0=0$, one-dimensional motion and an immediate loss whenever the <kinetic energy> reaches $\Delta$. With $d=\Delta/e$, $x=V-V_c\geq0$ and $0<V_r<d$, the number of losses and remaining <kinetic energy> are
$$
n=\left\lfloor\frac{x}{d}\right\rfloor,\qquad \frac{E_n}{e}=r=x-d\left\lfloor\frac{x}{d}\right\rfloor,\qquad \frac{I}{I_0}=H(r-V_r).
$$
Here $\lfloor\cdot\rfloor$ is the <floor function>. Collection switches off for an interval of width $V_r$ after each reset, then recovers; the pattern repeats every $d$. For $d=4.9\,\mathrm V$, $V_c=1.2\,\mathrm V$ and $V_r=1.0\,\mathrm V$, an applied $V=11.4\,\mathrm V$ gives $n=2$, $r=0.4\,\mathrm V$ and rejection, whereas $V=12.2\,\mathrm V$ gives $n=2$, $r=1.2\,\mathrm V$ and collection. With $V_r=0$ the dips disappear in this idealization. With no collisions, there is only a single collection onset. Both limits expose the separate roles of loss and filtering.
To draw a smooth illustrative curve, extend this periodic rule mathematically to all $x$ and convolve it with a chosen <normal distribution> of voltage jitter, of <standard deviation> $\sigma$:
$$
C_\sigma(x)=\sum_{k\in\mathbb Z}\left[\Phi\!\left(\frac{x-kd-V_r}{\sigma}\right)-\Phi\!\left(\frac{x-(k+1)d}{\sigma}\right)\right].
$$
Here $\Phi$ is the <standard normal cumulative distribution function>. Each summand is the smoothed indicator of the accepted interval $kd+V_r<x<(k+1)d$. The result satisfies $0\leq C_\sigma\leq1$ and has period $d$. For the figure choose $\sigma=0.18\,\mathrm V$ and plot only well beyond the startup region. This is a chosen phenomenological curve, not a <Boltzmann equation> transport prediction or a fit to measured <electric current>. Its symmetric troughs center on $V_c+nd+V_r/2$, rather than the reset voltages $V_c+nd$. Finite excitation rates, several loss channels, <space charge>, initial angular and <energy> spreads, and real electrode geometry can change the trough shape and phase; a single convolution does not solve these effects.
\Image[/franck-hertz-repeated-loss-spacing-fit.png]
{title=Illustrative repeated-loss collection and an independent synthetic spacing fit}
{description=Left: the chosen Gaussian-smoothed threshold model, with $d=4.9\,\mathrm V$, $V_c=1.2\,\mathrm V$, $V_r=1.0\,\mathrm V$ and $\sigma=0.18\,\mathrm V$. Right: separately chosen synthetic trough positions, their <ordinary least squares> fit and residuals. The points are not extracted from the left curve or a historical trace.}
{source=franck-hertz-repeated-loss-spacing-fit.py}
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