= Lattice constant from two indexed reflections
Choose an original simulated model: a <primitive cubic lattice> with one identical point scatterer per <primitive unit cell>, edge $a=0.400\,\mathrm{nm}$, and incident <X-ray> <wavelength> $\lambda=0.200\,\mathrm{nm}$. These are stipulated lengths, not measurements of <sodium chloride>. Suppose orientation information has already assigned the two observed reflections the <Miller indices> $(100)$ and $(110)$. The detector deflection is $2\theta$, where $\theta$ is the <Bragg angle>.
Use the existing <interplanar spacing> formula and <Bragg's law>, with higher <diffraction orders> absorbed into the reflection indices:
$$
N=h^2+k^2+l^2,\qquad d_{hkl}=\frac{a}{\sqrt N},\qquad
\sin^2\theta_{hkl}=\frac{\lambda^2}{4a^2}N,\qquad
a=\frac{\lambda\sqrt N}{2\sin\theta_{hkl}}.
$$
The simulated $(100)$ reflection has $\sin\theta=1/4$, hence $2\theta=28.9550^\circ$. The $(110)$ reflection has $\sin\theta=\sqrt2/4$, hence $2\theta=41.4096^\circ$. Their squared <sines> are $0.0625$ and $0.125$, in ratio $1:2$. Substituting the unrounded angles independently gives
$$
a_{100}=\frac{0.200}{2(1/4)}=0.400\,\mathrm{nm},\qquad
a_{110}=\frac{0.200\sqrt2}{2(\sqrt2/4)}=0.400\,\mathrm{nm}.
$$
Agreement tests a proposed indexing and scale together; disagreement would prompt checks of the indexing, <wavelength>, angular zero, or the assumed <unit cell> geometry.
\Image[/x-ray-two-peak-lattice-inference.png]
{title=Two indexed reflections recover the same cubic repeat length}
{id=image-x-ray-two-peak-lattice-inference}
{description=Original simulation with $a=0.400\,\mathrm{nm}$ and $\lambda=0.200\,\mathrm{nm}$. Left: selected $(100)$ and $(110)$ positions on a detector-deflection axis; equal stick heights are only markers. Right: their squared <sines> lie on the line of slope $\lambda^2/(4a^2)=1/16$. These are selected reflections, not a complete scan, a powder trace, or measured <sodium chloride> data.}
Two peak positions alone do not establish the indexing. Assigning the same two angles to $(110)$ and $(200)$ would give $N=2,4$ and $a=0.400\sqrt2\simeq0.565685\,\mathrm{nm}$ instead. More reflections, orientation information and <systematic absences> help discriminate such assignments. An unobserved reflection can be inaccessible, weak or extinguished by the <atomic basis>; absence is not automatically evidence for a larger <interplanar spacing>.
Absolute scale also requires <measurement calibration> of the <wavelength>. To first order, <differentiation> of the inferred length gives
$$
\frac{\delta a}{a}=\frac{\delta\lambda}{\lambda}-\cot\theta\,\delta\theta,
$$
with angular changes in <radians>. A stipulated detector-zero error $\delta(2\theta)=+0.100^\circ$, with correct <wavelength>, biases the first inferred edge downward by approximately $0.338\%$ and the second by $0.231\%$. A common angular offset therefore need not cancel when two inferred lengths are compared. These are model sensitivity estimates, not historical instrument uncertainties.
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