Mechanical vibrations of freely standing smectic-A films

The time-dependent layer displacement-layer displacement correlation functions are analyzed in a free standing thin smectic-A films with the help of the discrete layer model. The film motions are described using the Chebyshev polynomials of second kind, U_n(x). The dependencies of the eigenmodes on the wave number q_⊥ were analyzed. For small q_⊥ one mode is a low frequency acoustic wave and other modes are high frequency optical oscillations. The rather simple expression for the susceptibility matrix and for spectral densities of layer displacement correlation functions were obtained using the Chebyshev polynomials. The obtained results allow one to explain the experimental results on dynamic scattering and diffraction of X-rays in smectic-A films