= Finite-row electron diffraction model
{title2=$I(\phi)/I(0)=N^{-2}|\sum_{j=0}^{N-1}e^{ij\delta}|^2$}
Take $N$ equally spaced scattering sites at $x_j=jd$ on a line, a normally incident <plane wave>, equal site <scattering amplitudes>, and observation far enough away that outgoing directions are effectively parallel. Retain only one scattering event per amplitude, as in the <Born approximation>. For angle $\phi$ from the outward normal, write the adjacent-site <quantum phase> difference as $\delta=kd\sin\phi$, with $k=2\pi/\lambda$; reversing its overall sign leaves the intensity unchanged. Summing a <geometric series> gives
$$
\frac{I(\phi)}{I(0)}=\frac{1}{N^2}\left|\sum_{j=0}^{N-1}e^{ij\delta}\right|^2
=\left[\frac{\sin(N\delta/2)}{N\sin(\delta/2)}\right]^2,\qquad \delta=\frac{2\pi d}{\lambda}\sin\phi.
$$
At $\delta=2\pi m$ use the continuous limit $I/I(0)=1$, rather than treating $0/0$ as a missing peak. The integer $m$ is the <diffraction order>: $d\sin\phi=m\lambda$. This is a surface-row spacing, not the plane spacing in <Bragg's law>. The plotted intensity is normalized to its central value; it is not an integrated detection probability or an absolute collector <electric current>.
\Image[/electron-diffraction-row-geometry-intensity.png]
{title=Surface-row geometry and calculated <electron diffraction> intensity}
{description=Original idealized model: $N=12$, row spacing $d=0.250\,\mathrm{nm}$, equal site <scattering amplitudes>, normal incidence and field-free nonrelativistic <electron> <wavelengths> at $54$, $100$ and $400\,\mathrm V$. Straight arrows denote propagation directions, not sinusoidal particle paths. The first-order peak angles are about $41.9^\circ$, $29.4^\circ$ and $14.2^\circ$. Curves are calculated phase sums, not measured data or a fit to the <Davisson–Germer experiment>.}
{source=electron-diffraction-row-geometry-intensity.py}
At $100\,\mathrm V$, $\lambda\simeq0.12264\,\mathrm{nm}$ and the positive first-order angle is $\arcsin(\lambda/d)\simeq29.4^\circ$. At $400\,\mathrm V$, the <wavelength> halves and this angle becomes about $14.2^\circ$. Increasing $N$ narrows each peak through more selective <interference>, while increasing <momentum> brings a fixed-order peak nearer the normal. Unequal site amplitudes, multiple scattering, a three-dimensional <crystal lattice structure factor>, and finite detector acceptance can change this simple picture; it is an explanatory model with hypothetical inputs.
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