Synchrotron X-ray study of the q-fold quasicrystalline symmetry of the smectic C twist grain boundary phase (TGBc)

We report the first X-ray scattering investigation of spatial variations of the q-fold quasicrystalline symmetry (so-called commensurability) of well-aligned TGB(c) samples. A spatial resolution of 20 x 40 mu m(2) was achieved using the ESRF microfocus beamline. The liquid crystal samples, contained between glass plates which were either parallel or in a wedge geometry, were scanned in order (i) to probe the mosaicity and (ii) to continuously change the balance between surface and volume effects. In the case of parallel plate cells, commensurability was observed everywhere throughout the sample, hence ruling out possible effects of mosaicity to explain the q-fold symmetry of the diffraction patterns previously reported when probed with a spatial resolution of 1 x 1 mm(2) in rotating anode experiments. In the case of wedge cells, the evolution of X-ray patterns with thickness suggested that commensurate lockin occurs for sufficiently thick samples with a width that is statistical.