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_^-2+ and q_^-4+2, where () is along (perpendicular to) the smectic density wave vector direction, rather than 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_^-2 (q_^-4). 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 required to describe the measured profiles are in accordance with calculations of using the harmonic approximation with empirically determined splay and layer compressibility elastic constants.