Elasticity and the Landau-Peierls instability in the smectic twist-grain-boundary phase

The elasticity theory of the twist-grain-boundary phase in chiral smectic liquid crystals for the case of {α} (where 2{π}{α} is the twist-grain-boundary angle) irrational is developed. It implies that fluctuations destroy long-ranged translational order, leading to algebraic (rather than {δ}-function) singularities in the x-ray scattering near both a cylinder in reciprocal space and isolated Bragg peaks along its axis. These results, which could be experimentally tested through high-resolution x-ray scattering, also apply to ``nearly irrational'' {α}=J/s, with J and s integers and s{}1, out to length scales exponentially large in s.