X-ray scattering in smectic-Aliquid crystals

The two-dimensional ($2D$) crystal and the $3D$ smectic-$A$ liquid crystal are examples of Landau-Peierls systems, where long-wavelength fluctuations wash out long-range order of the order parameter. However, the decay of the order-parameter correlation functions is of the power-law type, rather than exponential. The result is that in x-ray scattering characteristic power-law singularities appear instead of the usual Bragg peaks. The details of these singularities, which have recently been observed experimentally in a smectic-$A$ liquid crystal by Als-Nielsen et al., are analyzed for the smectic-$A$ case. A new scaling property of the structure factor is derived as a function of the momentum-transfer components parallel ($\kappa$) and perpendicular ($K_{\perp }$) to the smectic layers. Very close to the Bragg points, finite-size effects become important, including a new and unusual effect when $K_{\perp }$ is proportional to the inverse square root of the finite thickness of the specimen. The crossover in the Bragg peaks due to order-restoring effects of an external magnetic field is presented.