= Basis intensities and phase ambiguity
Keep the cubic translations above, but attach an <atomic basis> with A at <fractional crystal coordinates> $(0,0,0)$ and B at $(u,0,0)$. Stipulate real, angle-independent relative <scattering amplitudes> $f_A=1$ and $f_B=0.6$. Neglect <wave absorption>, repeated <crystal scattering> and thermal motion. This is a point-scatterer model under the <Born approximation>, not a claim about the scattering strengths of named elements. Its <crystal basis structure factor> is
$$
F_{hkl}(u)=1+0.6e^{2\pi ihu},\qquad
J_{hkl}(u)=|F_{hkl}(u)|^2=1.36+1.2\cos(2\pi hu).
$$
Here $J$ is a dimensionless squared motif amplitude. Comparing it with measured integrated <wave intensities> requires the experimental geometry and detector response as well. The algebra follows directly by multiplying $F$ by its <complex conjugate>; the cross term records <wave interference> between the two sites.
With $u=1/2$, reflections with even $h$ have $J=2.56$, whereas odd $h$ gives $J=0.16$, a ratio of $16:1$. With $u=1/4$, the first four $(h00)$ values instead are $1.36,0.16,1.36,2.56$. Their geometrically possible positions are unchanged, although their strengths change. For equal stipulated amplitudes $f_A=f_B=1$ and $u=1/2$, the odd-$h$ amplitude vanishes exactly: an example of a <systematic absence>. Different motif weights can thus expose a repeat that equal weights conceal.
\Image[/x-ray-basis-intensities-and-phase-ambiguity.png]
{title=The motif changes intensities, while inversion can preserve them}
{id=image-x-ray-basis-intensities-and-phase-ambiguity}
{description=Original point-scatterer simulations with $f_A=1$ and $f_B=0.6$. Top: the same $(h00)$ reflection labels have different squared motif amplitudes for $u=1/2$ and $u=1/4$. Bottom left: moving B from $u=1/4$ to $u=3/4$ inverts its offset modulo a translation. Bottom right: the two $(100)$ <crystal basis structure factors> are <complex conjugates>, $1\pm0.6i$, with the same squared modulus $1.36$. No detector profiles or real-material measurements are plotted.}
The same formula also proves an ambiguity: $u=1/4$ and $u=3/4$ give identical $J_{hkl}$ for every <integer> $h,k,l$, because $\cos(2\pi h(1-u))=\cos(2\pi hu)$. Yet $F_{100}$ changes from $1+0.6i$ to $1-0.6i$, reversing its <wave phase>. This particular ambiguity is inversion of the motif; it does not prove that these two simple motifs are distinct chemical substances. The conclusion is conditional on the stipulated real <scattering amplitudes>.
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