Observation of algebraic decay of positional order in a smectic liquid crystal

A smectic-$A$ liquid crystal in three dimensions has been predicted to exhibit algebraic decay of the layer correlations rather than true long-range order. As a consequence, the smectic Bragg peaks are expected to be power-law singularities of the form $q_{\parallel }^{-2+\eta }$ and $q_{\perp }^{-4+2\mathrm{}\eta }$, where $\parallel \left(\perp \right)$ is along (perpendicular to) the smectic density wave vector direction, rather than $\delta$ function peaks. Observation of these phenomena requires very high instrumental resolution together with a resolution function with wings which drop off much more rapidly than $q_{\parallel }^{-2\mathrm{}}\left(q_{\perp }^{-4\mathrm{}}\right)$. We show that these requirements may be met by using a three crystal x-ray spectrometer with multiple-reflection channel cut crystals as monochromator and analyzer. We find that the smectic-$A$ Bragg peaks observed in the liquid-crystal octyloxy-cyanobiphenyl are indeed consistent with the predicted power-law singularity form. Furthermore, the explicit values of $\eta$ required to describe the measured profiles are in accordance with calculations of $\eta$ using the harmonic approximation with empirically determined splay and layer compressibility elastic constants.