= Franck-Hertz spacing fit and contact offset
{c}
{title2=$V_n=b+nd,\quad\Delta=ed$}
Fit the positions of successive, consistently defined troughs by <linear regression>, $V_n=b+nd+\epsilon_n$. A constant intercept $b$ absorbs the <contact potential difference>, collection phase and initial <energy> effects; $d$ estimates the repeated loss expressed as a voltage. Subtracting successive positions cancels $b$. A line fit uses every position and leaves residuals that can reveal curvature or a misidentified trough. It still assumes that the phase is constant over the fitted range and that all troughs belong to the same loss channel.
For an original numerical example, choose $n=1,2,3,4,5$ and synthetic positions $V_n=(6.62,11.46,16.48,21.30,26.14)\,\mathrm V$. Their successive spacings are $(4.84,5.02,4.82,4.84)\,\mathrm V$. The <ordinary least squares> formulas give
$$
\widehat d=\frac{\sum_n(n-\bar n)(V_n-\bar V)}{\sum_n(n-\bar n)^2}=4.888\,\mathrm V,\qquad
\widehat b=\bar V-\widehat d\,\bar n=1.736\,\mathrm V.
$$
The fitted residuals are $(-0.004,-0.052,0.080,0.012,-0.036)\,\mathrm V$, with zero sum and zero index-weighted sum. Their <residual sum of squares> is $0.01056\,\mathrm V^2$. If independent, equal-<variance>, zero-mean errors describe these positions, the residual variance estimate is $s^2=0.01056/(5-2)=0.00352\,\mathrm V^2$ and the slope <standard error> is $s/\sqrt{10}=0.01876\,\mathrm V$. Thus the inferred model loss is $4.888\,\mathrm{eV}$ with conditional <standard error> $0.01876\,\mathrm{eV}$. These errors summarize only the chosen position scatter; a voltage-scale error or a drifting collection phase is not included. Adjacent spacings share a position error and are therefore correlated even when position errors are independent.
The offset has a sharper <identifiability> limitation. Even if a constant-phase model were exact, an initial beam <kinetic energy> $E_0$ shifts the troughs by $-E_0/e$. For the symmetric illustrative curve, $b=V_c-E_0/e+V_r/2$. The transformation $V_c\mapsto V_c+\alpha$, $E_0\mapsto E_0+e\alpha$ leaves every trough position unchanged. Measuring $b$ alone therefore cannot determine the <contact potential difference>. Relabeling $n\mapsto n+m$ also changes $b$ by $-md$ without changing $d$. An independent offset calibration and a justified collection model are needed before turning a first trough into an excitation threshold. Varying $V_r$ tests this model's predicted half-voltage shift of the trough centers; it is a diagnostic prediction of this chosen curve, not a universal correction for experimental minima.
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