From Spheres to Ellipsoids: The Story of the Density of States*

Packings of frictionless ellipsoids have not only captured the imagination of the public, but also bring up a number of fundamental issues regarding the properties of jammed media. For instance, the average contact number $Z$ of such packings at jamming varies continuously between the spherical isostatic value $Z_{\rm iso}=6$ and the value $Z_{\rm iso}=10$ for ellipsoids of revolution if the ellipticity $\epsilon-1$ is turned on. Here we study the vibrational spectra of soft ellipsoids both as a function of density and $\epsilon$. Our spectra show a two-band structure. For small aspect ratios there is first a rotational band, then a gap and then a second band of translational character. As we increase the aspect ratio, the gap closes and the remaining band has a mixed character. We discuss various surprising features of the spectrum in detail and show how the changes in the gap are related with the change of $Z$ with $\epsilon$.