A new undulator producing an X-ray beam of high brilliance has been used at the European Synchrotron Radiation Facility (ESRF, Beamline-9, Troika) for investigating the structure of two-dimensional crystals of 1-tetradecanol, 1-dodecanol, 1-decanol and 1-nonanol. These fatty alcohols with relatively short chains crystallize in a hexagonal structure akin to that of alkanes in the Rotator II phase. The high flux obtained under grazing incidence conditions has permitted a detailed investigation to be made of the in-plane (Q(x)) and out-of-plane (Q(z)) diffraction from the monolayers. When approaching the 2D melting temperature the diffracted intensity decreases, and this pre-melting effect, which is more pronounced for the shorter chains, is attributed to a critical increase of the Debye-Waller factor. In addition, the Bragg rod profiles along Q(z) show that, in the hexagonal structure, the chain molecules do not remain perfectly perpendicular to the water surface.