Phonon eigenfunctions of inhomogeneous lattices: Can you hear the shape of a cone?
arXiv:2107.11431 · doi:10.1103/PhysRevE.104.065005
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.
arXiv comment20 pages, 12 figures
References in corpus (10)
- Continuum limit of the vibrational properties of amorphous solids
- Low-frequency vibrational modes of stable glasses
- Trapping of Topological-Structural Defects in Coulomb Crystals
- Sonic Landau-level lasing and synthetic gauge fields in mechanical metamaterials
- Two-Dimensional Clusters of Colloidal Spheres: Ground States, Excited States, and Structural Rearrangements
- Topological defects in the crystalline state of one-component plasmas of non-uniform density
- Defects in Conformal Crystals: Discrete vs. Continuous Disclination Models
- Statistical Mechanics of Low Angle Grain Boundaries in Two Dimensions
- Transitions between phyllotactic lattice states in curved geometries
- Statistical Mechanics of Dislocation Pileups in Two Dimensions