The Correction of Finite Thickness to the Vibrational Spectra of a Spherical Elastic Shell

概要

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The vibrational modes and corresponding eigenfrequencies in a spherical shell, such as C60, are obtained based on a continuum model. These modes are decomposed into the torsional and spheroidal modes (T-modes and S-modes), where S-modes are further decomposed into the upper and lower branches, a_{l} and b_{l}, respectively. The spectra dependence on shell thickness is taken into account to predict more accurate frequencies. The frequencies of T-modes decrease with shell thickness due to the purely transversal displacement. Moreover, the a_{l} S-mode bears a similar fashion to the T-mode, but the frequency of the b_{l} S-mode with a large l increases with shell thickness where the longitudinal displacement becomes predominant. The analytic expressions of corrections to vibrational frequencies are also derived up to third-order in the shell thickness. The differences between the present corrections and the previous results could be summarized as follows: (1) The leading correction to the frequency should be proportional to the value of the shell thickness-to-radius ratio instead of its square. (2) The sign of shift to the vibrational frequencies should depend on the vibrational modes, instead of positive shifts for all S-modes except the radial breathing mode. (3) The frequency of the radial breathing mode indeed gets modified by the inclusion of shell thickness, instead of no corrections.
2012-09-15