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

Input

Mode:
ℓ=
number of modes:

Output

Status:

<ρu^{2}> for the

Calculated values (mode index, frequency (GHz),
<ρu^{2}> for
the core, <ρu^{2}> 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:

ρ (g/cm^{3}): mass density

v_{L} (m/s): longitudinal sound velocity

v_{T} (m/s): transverse sound velocity

R_{ext} (nm): external radius of the shell or of the
core for the first line. This value should increase for every
shell.

The rest of the line is ignored.

<ρu^{2}>, the "mean square displacement", is calculated as:
$\frac{{\int}_{{r}_{\mathrm{-}}<r<{r}_{\mathrm{+}}}\rho \u27e8u|u\u27e9dV}{{\int}_{r<{r}_{\text{max}}}\rho \u27e8u|u\u27e9dV}$.
Note that it is weighted by the mass density so that the sum of
<ρu^{2}> for all the layers is 1.

Shortcuts

References

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