X-ray measurement of the twist grain boundary angle in the liquid crystal analog of the Abrikosov phase

We report the first experimental observation of the commensurate state of the recently discovered helical smectic-C phase. The projection of the reciprocal lattice on the plane perpendicular to the pitch axis consists of a ring of equispaced Bragg spots, as predicted by Renn and Lubensky for the ``twist grain boundary'' phase. On cooling, the number of spots q jumps from 16 to 18 and 18 to 20. A noticeable thermal hysteresis is observed on heating. The corresponding symmetries are quasicrystalline since the q-fold screw axes are not crystallographically allowed. These observations allow an accurate determination of all the parameters of the twisted lattice of screw dislocations.