  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.
