Eigen-vibrations of isotropic multilayer spheres

Disclaimer: use at your own risk. Help available at the bottom of this page.


Mode: ℓ=
number of modes:



<ρu2> for the

Calculated values (mode index, frequency (GHz), <ρu2> for the core, <ρu2> for the first shell, ...

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. vL (m/s): longitudinal sound velocity
  3. vT (m/s): transverse sound velocity
  4. Rext (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: r - < r < r + ρ u | u V r < r max ρ u | u V . Note that it is weighted by the mass density so that the sum of <ρu2> for all the layers is 1.



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