The eigenvibrations and time-dependent layer displacement-layer displacement correlation functions are analyzed in a free-standing thin smectic-A films with the help of a discrete layer model. The film motions are described using the Chebyshev polynomials of second kind, U(n)(x). The eigenfrequencies problem is essentially simplified within the framework of this approach since the numerical solution of the high degree algebraic characteristic equation is replaced by the analytical solution of a rather simple trigonometrical equation. The dependences of eigenmodes on wave number q(perpendicular) were analyzed. For small q(perpendicular) one mode is a low frequency acoustic wave and other modes are high frequency optical oscillations. As the wave number q(perpendicular) increases all modes successively turn into relaxation when starting with the acoustic mode. The rather simple expression for susceptibility matrix and for spectral densities of layer displacement correlation functions were obtained using the Chebyshev polynomials. It was shown that the frequency dependences of spectral densities are sensitive to wave number q(perpendicular). For small q(perpendicular) the spectral densities of displacement-displacement correlation functions have a sharp peak and for large q(perpendicular) they turn into a contours of Lorentzian type.
The consequent description of the x-ray scattering from the free standing smectic-A films is suggested. Calculations are based on the discrete model for the film dynamics. The scattering intensity temporal autocorrelation function is obtained within the framework of this approach neglecting the multiple scattering and refraction effects. It is shown that the behavior of this function crucially depends on the film thickness. In particular, in thin films containing less than 10(3) layers the time dependence of has damping oscillation character. This behavior is determined by an acoustic mode that describes the film motion caused by the action of the surface tension. For thick films containing more than 10(4) layers the dynamics of the intensity temporal autocorrelation function is determined either by an acoustic mode or by a wide spectrum of modes depending on the x-ray geometry. In both the cases the autocorrelation function is a relaxation one. The results obtained are compared with the experiments on the coherent soft and hard x-ray scattering.
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