= Gaussian spectral doublet resolution
{c}
{title2=$\Delta\theta>2\sigma_\theta$ for two maxima in the equal-Gaussian model}
A <spectral doublet> comprises two neighbouring <spectral lines>. Resolving it requires information about the line response, relative strengths and <noise>, not only precision in reading an angle. Choose two equal angular responses shaped as <Gaussian functions> with common <standard deviation> $\sigma$ and separation $\Delta$. If $x$ is angular displacement from their midpoint, their summed signal is
$$
f(x)=\exp\left(-\frac{(x-\Delta/2)^2}{2\sigma^2}\right)
+\exp\left(-\frac{(x+\Delta/2)^2}{2\sigma^2}\right).
$$
These are assumed response shapes, not a measured Siegbahn <frequency spectrum>. By symmetry $f'(0)=0$, and
$$
f''(0)=\frac{2e^{-\Delta^2/(8\sigma^2)}}{\sigma^2}
\left(\frac{\Delta^2}{4\sigma^2}-1\right).
$$
The midpoint changes from a maximum to a minimum at $\Delta=2\sigma$. For this particular equal-width, equal-height model, two distinct maxima occur when $\Delta>2\sigma$. Each isolated <Gaussian function> has <full width at half maximum> $2\sqrt{2\ln2}\,\sigma\simeq2.355\sigma$; the two-maxima threshold is therefore about $0.849$ times that width.
\Image[/bragg-angular-error-doublet-resolution.png]
{title=Angular error and the visibility of a chosen <spectral doublet>}
{id=image-bragg-angular-error-doublet-resolution}
{description=Original calculated models. Left: magnitude of the angular term in <Bragg wavelength error propagation>, alongside a separately chosen fractional spacing error of $20$ parts per million; the curves are not added. Right: explicitly chosen <interplanar spacing> $d=0.2000\,\mathrm{nm}$, first <diffraction order>, and lines at $0.2000$ and $0.2002\,\mathrm{nm}$. Equal <Gaussian function> responses use angular <standard deviations> $0.040^\circ$ and $0.008^\circ$; dotted curves show the narrower isolated components. None of these numbers is a historical crystal calibration or measured spectrum.}
In this chosen model, the two <Bragg angles> are $30.000000^\circ$ and $30.033085^\circ$. Their separation is $0.827$ times $\sigma$ for $\sigma=0.040^\circ$, giving one maximum, and $4.136$ times $\sigma$ for $\sigma=0.008^\circ$, giving two. Sharpening the response can reveal the doublet without changing either emitted <wavelength>.
The <Rayleigh criterion> uses a first-zero convention for specified <diffraction> patterns. A <Gaussian function> has no finite zero, so its two-maxima threshold is not the <Rayleigh criterion>. Neither convention is a universal inference limit: known response shapes and adequate <signal-to-noise ratio> can support estimating a separation before two visible maxima appear. Unequal strengths, background, finite sampling and uncertainty in the response can make that inference much harder. <Measurement resolution>, the precision of an isolated line's centre and absolute <wavelength> accuracy should therefore be reported separately.
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