Structural Study of the Smectic-CTwist Grain Boundary Phase

We report the first three dimensional x-ray study of the recently discovered twist grain boundary smectic- $C$ phase $\left({\mathrm{TGB}}_C\right)$. Unlike the predictions of the original Renn-Lubensky model for TGB phases (and the experimental observations in the untilted ${\mathrm{TGB}}_A$ phase), the smectic layers in the ${\mathrm{TGB}}_C$ phase are shown to be tilted with respect to the helical TGB axis by an angle approximately equal to the smectic- $C$ director tilt angle. The reciprocal lattice consists of either two rings of equispaced Bragg spots or uniform rings confirming the existence of both commensurate and incommensurate TGB phases.