Correlations in the thermal fluctuations of free-standing smectic-A films as measured by x-ray scattering

The displacement-displacement correlations in the thermal fluctuations of freely suspended smectic-A films of 3 to 34 layers have been determined using small angle specular and diffuse x-ray scattering. By choosing a symmetric resolution a simple separation of the resolution function into contributions parallel and perpendicular to the wave vector transfer is possible. This enables modeling of the scattered intensity without introducing an artificial separation of the specular and diffuse contribution. The data are interpreted using a continuous model to describe the displacement-displacement correlations, which is shown to be equivalent to the original discrete model of Holyst [Phys. Rev. A 44, 3692 (1991)], but computationally more efficient. Two characteristic in-plane lengths are introduced: R(1), above which the distance dependence of the correlation function follows a logarithmic law, and R(c), above which the layers throughout the film fluctuate in unison, i.e., conformally. Values for the smectic bend and compression elastic constants as well as the surface tension are obtained from the wavelength dependence of the correlations. The fluctuation profile depends only slightly on the film thickness and is nearly flat for the fluorinated compound investigated. All films investigated are conformal down to the smallest in-plane length scales measured. Furthermore, the collective long wavelength thermal fluctuations, which only depend on the diffuse scattering, can be separated from the local smectic disorder. The local contribution to the total fluctuation profile is found to be considerable.