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:The simulated $(100)$ reflection has $\sin \theta =1/4$, hence $2\theta =28.9550^{\circ }$. The $(110)$ reflection has $\sin \theta =\sqrt{2}/4$, hence $2\theta =41.4096^{\circ }$. Their squared sines are $0.0625$ and $0.125$, in ratio $1:2$. Substituting the unrounded angles independently givesAgreement tests a proposed indexing and scale together; disagreement would prompt checks of the indexing, wavelength, angular zero, or the assumed unit cell geometry.

Two peak positions alone do not establish the indexing. Assigning the same two angles to $(110)$ and $(200)$ would give $N=2,4$ and $a=0.400\sqrt{2}\simeq 0.565685\mathrm{nm}$ 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 giveswith 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.