Coherent dynamic x-ray scattering has been used to study the thermally excited layer fluctuations in freely suspended smectic films of the compound 4O.8. Using 8-keV x rays and films with a thickness around 0.3 mm we resolve relaxation times down to a few ms. A combination of damped and oscillatory behavior is observed for the layer undulations, which can be attributed to inertial effects. These are due to the surface contribution to the free energy which cannot be disregarded for thin films.

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... CS numbers: 61.10. Kw, 42.25.Kb, If the dimensionality of a physical system is below the so-called marginal dimensionality, fluctuations can prevent the onset of true long-range order. Smectic-A (Sm-A) ordering, which consists of stacks of liquid layers, provides a nice example of such behavior. The reduced translational order leads to strong thermal fluctuations of the smectic layers, and the mean-square layer displacement ͗u 2 ͑r͒͘ diverges logarithmically with the sample size (Landau-Peierls instability) [1]. As a result the static density-density correlation function decays algebraically with distance ("quasi-long-range order"). These effects have been well documented experimentally. However, some of the fundamental questions regarding the dynamics of smectic systems have been addressed only recently. This concerns the behavior of the time-dependent layer displacement correlation function in the thermodynamic limit and the effects of finite sizes in freely suspended smectic films, including nonzero surface tension and viscous dissipation [2, 3] . This has been triggered off by new possibilities to study the dynamics of these fluctuations experimentally with coherent x rays. If coherent radiation is incident on a random medium, the scattered intensity shows a speckle pattern that reflects the instantaneous arrangement of the scattering centers. Movement of the scatterers causes a corresponding change in this pattern and thus contains information on the dynamics of the system. Photon correlation spectroscopy or dynamic light scattering (DLS) measures the time dependent intensity autocorrelation function of the speckle pattern. With visible light this is a well-established technique, while coherent dynamic x-ray scattering (DXS) has been developed only recently at third generation high-brilliance synchrotron sources [4] . In contrast to DLS, however, DXS can give access to dynamic processes on an atomic or molecular length scale. So far, the feasibility of DXS has been shown on various hard and soft condensed systems [5], with some emphasis on relatively slow dynamics (ms range) like those of colloidal particles in various liquids. Smectic liquid crystals can be suspended over an opening in a solid frame. Because of the controlled thickness (from two to over hundreds of layers) and the uniform ordering of the smectic layers parallel to the surfaces, these films provide perfect model systems [6] . From the dynamic point of view, the finite film thickness changes the continuous bulk response spectrum into discrete response modes that depend also on the surface parameters. DXS in the soft x-ray region has recently been applied to thick freely suspended smectic films [7] . Decay times typically in the range of tens of ms were reported to scale linearly with the film thickness (tens of mm). In this Letter we report DXS with hard x rays of thin smectic films. We observe an "oscillatory" behavior of the correlation function, which cannot be described by a simple exponential time dependence. This can be attributed to a complex behavior of the relaxation time originating from the inertial contribution to the dynamic response of thin films, in agreement with recent theory. Freely suspended smectic films of N-(4-n-butoxybenzilidene)-4-n-octylaniline, abbreviated as 4O.8, were spanned over a 10 3 23 mm 2 hole in a stainless steel frame and equilibrated as described earlier [8] . The films were mounted in a two-stage oven and evacuated in order to prevent possible sample degradation, to reduce air scattering, and to avoid acoustic low frequency dynamics. 4O.8 was obtained from Frinton Laboratories Inc., and has the following phase transitions (in ± C): K 33 Cr-B 48.5 Sm-A 63.5 N 78 I. DXS was performed at undulator beam line ID10A (Troika I) at the European Synchrotron Radiation Facility (ESRF, Grenoble) [4] . The sample was mounted vertically and illuminated with 7.98 keV radiation selected by a Si (111) monochromator. The longitudinal coherence 2316 0031-9007͞00͞85(11)͞2316(4)$15.00