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

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Abstract
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.