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Time-resolved reflectivity changes for (a) 13 and 36 nm Pt(001) films on MgO substrate and (b) those for 20 and 27 nm Pd(111) films on Si substrates. (c) Their Fourier spectra including those from 5.6 and 14 nm Pt(111) films on Si substrates. The oscillation frequency appears to be inversely proportional to the film thickness.
Differences of interatomic distances of the slab structures after optimization from corresponding stress-free bulk materials. denotes the in-plane interatomic distance and is the interlayer distance. Their bulk values are obtained by the ab initio calculation as , 2.7846, and 2.7444 Å, and , 1.9340, and 2.2394 Å for Pt(111), Pt(001), and Pd(111), respectively.
Phonon-dispersion curves of 7-layer Pt(001) slab along  direction. Marked -point frequency is the longitudinal standing vibration, whose displacements are symmetric about the center of slab.
Measured and calculated phonon frequencies for ultrathin films and slab model. The colored broken lines represent the natural frequencies obtained by the one-dimensional lattice dynamics model (mass-spring model), whose spring constants are determined by the measurement of the thickest specimens. The black broken line is the resonant frequency of Pt(111) based on the continuum mechanics theory given by , where and represent sound velocity and average interplane distance, respectively.
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