Eigenmodes and thermodynamics of a Coulomb chain in a harmonic potential

TL;DR

This paper analyzes the eigenmodes and thermodynamics of a one-dimensional Coulomb ion chain in a harmonic trap, deriving analytical expressions for mode spectra, thermodynamic functions, and stability criteria, applicable from large to small chains.

Contribution

It provides analytical calculations of eigenmodes, density of states, and thermodynamic properties of Coulomb chains in harmonic potentials, including stability conditions.

Findings

Eigenmodes differ from phonons due to non-uniform density.

Analytical spectra approximate small chains well.

Deviations from extensivity in thermodynamic quantities.

Abstract

The density of ions trapped in a harmonic potential in one dimension is not uniform. Consequently the eigenmodes are not phonons. We calculate the long wavelength modes in the continuum limit, and evaluate the density of states in the short wavelength limit for chains of $N\gg 1$ ions. Remarkably, the results that are found analytically in the thermodynamic limit provide a good estimate of the spectrum of excitations of small chains down to few tens of ions. The spectra are used to compute the thermodynamic functions of the chain. Deviations from extensivity of the thermodynamic quantities are found. An analytic expression for the critical transverse frequency determining the stability of a linear chain is derived.