Eigen-vibrations of isotropic multilayer spheres

General help

Using this javascript calculator, you can calculate the frequencies of the vibrations of a multilayer nanospheres assuming isotropic elasticity. These calculations are an extension of the model for the vibration of free spheres (see Lamb's modes of an isotropic sphere).

By using a large outer shell, it is possible to approximate the case of a matrix embedded multilayer sphere. For details, see the model introduced in the following paper which you may want to cite if you use this calculator: D. B. Murray & L. Saviot, Phys. Rev. B 69, 094305 (2004). This paper can also be downloaded for free from arXiv.

Input box help

One line for the core and one for each shell. At least one shell required so there should be at least 2 lines. The content of the lines is organized by column as follow:

1. ρ (g/cm³): mass density
2. v_L (m/s): longitudinal sound velocity
3. v_T (m/s): transverse sound velocity
4. R_ext (nm): external radius of the shell or of the core for the first line. This value should increase for every shell.
5. The rest of the line is ignored.

<ρu²>, the "mean square displacement", is calculated as: $\frac{{\int }_{r_{\mathrm{-}}<r<r_{\mathrm{+}}}\rho ⟨u|u⟩dV}{{\int }_{r<r_{\text{max}}}\rho ⟨u|u⟩dV}$. Note that it is weighted by the mass density so that the sum of <ρu²> for all the layers is 1.

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References

The implementation uses the linear algebra routines provided by the Numeric Javascript library.