Smectic Liquid Crystals as Model Systems of Low-dimensional Order 

In this paper we discuss the absence of true long-range order (Landau-Peierls instability) in smectic liquid crystals, which is a direct consequence of the strong thermal fluctuations associated with ordering in less than three dimensions. While the resultant X-ray lineshape has been confirmed already quite some time ago, recently some new aspects have been discovered. In the first place, thanks to modern synchrotron sources, the spectral dependence of the fluctuations in freely suspended smectic films has been determined down to molecular dimensions. While at long wavelengths top and bottom of a film fluctuate in unison, at shorter wavelengths a cross-over to independent fluctuations could be observed. Secondly, in smectic elastomers the stability of the layered structure against fluctuations is dramatically enhanced by the crosslinked architecture. As a result the usual Landau-Peierls instability is no longer observed. Hence we have found the first evidence for existence of long range translational order in soft '1-D' layered systems embedded in 3-D space.