= Bragg wavelength error propagation
{c}
{title2=$\delta\lambda/\lambda\simeq\delta d/d+\cot\theta\,\delta\theta$}
Differentiate <Bragg's law> at fixed correctly assigned <diffraction order>:
$$
n\lambda=2d\sin\theta,\qquad
n\,d\lambda=2\sin\theta\,dd+2d\cos\theta\,d\theta.
$$
Divide the second equality by the first. For small signed changes,
$$
\frac{\delta\lambda}{\lambda}\simeq
\frac{\delta d}{d}+\cot\theta\,\delta\theta.
$$
The <Bragg angle> error $\delta\theta$ must be in <radians>. An error quoted for detector deflection $2\theta$ must first be halved. The expression is a first-order <Taylor expansion>; an unknown <diffraction order> is a discrete assignment error, not an infinitesimal perturbation.
For error bounds, add magnitudes. For independent zero-mean random errors with <standard deviations> $\sigma_d,\sigma_\theta$, <variance> addition instead gives
$$
\left(\frac{\sigma_\lambda}{\lambda}\right)^2\simeq
\left(\frac{\sigma_d}{d}\right)^2+\cot^2\theta\,\sigma_\theta^2.
$$
Correlated errors require a <covariance> term. A shared calibration bias does not become independent merely because many line positions are measured.
With a chosen angular error $10$ <arcseconds>, $\delta\theta=4.84814\times10^{-5}\,\mathrm{rad}$, the angular contribution is $8.3972\times10^{-5}$ at $30^\circ$ and $1.7646\times10^{-5}$ at $70^\circ$. Larger <Bragg angles> reduce this contribution because $\cot\theta$ decreases. For a nearby <spectral doublet>, the inverse relation is $\Delta\theta\simeq\tan\theta\,\Delta\lambda/\lambda$: a fixed fractional separation spreads farther apart angularly at large <Bragg angle>.
This does not permit unlimited improvement. The geometry requires $n\lambda\leq2d$; a desired <wavelength> might not have an accessible large-angle reflection with the chosen <crystal>. Higher <diffraction orders> can reach larger <Bragg angles> but their strengths depend on the <crystal lattice structure factor>. Finite crystal response, slit widths and source size remain relevant. An error in $d$, including uncorrected <thermal expansion>, remains even when the angular term is small. Very near $90^\circ$, the leading angular derivative vanishes and the second-order <Taylor expansion> matters.
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