Phonon eigenfunctions of inhomogeneous lattices: Can you hear the shape of a cone?

TL;DR

This study investigates how the shape of a truncated cone influences phonon modes in a particle lattice, revealing that cone geometry affects vibrational properties and can be inferred from phonon spectra.

Contribution

The paper derives the phonon eigenfunctions of particles on a conical surface, linking cone shape to vibrational mode characteristics and localization phenomena.

Findings

A robust inhomogeneous triangular lattice forms under gravity.

Cone apex angle influences phonon energy crossings and localization.

Analytical and numerical methods agree on phonon localization boundaries.

Abstract

We study the phonon modes of interacting particles on the surface of a truncated cone resting on a plane subject to gravity, inspired by recent colloidal experiments. We derive the ground state configuration of the particles under gravitational pressure in the small cone angle limit, and find an inhomogeneous triangular lattice with spatially varying density but robust local order. The inhomogeneity has striking effects on the normal modes such that an important feature of the cone geometry, namely its apex angle, can be extracted from the lattice excitations. The shape of the cone leads to energy crossings at long wavelengths and frequency-dependent quasi-localization at short wavelengths. We analytically derive the localization domain boundaries of the phonons in the limit of small cone angle and check our results with numerical results for eigenfunctions.