  We present a second-order perturbative analysis of the model describing a
two-level trapped ion interacting with a traveling laser field, in the
Lamb-Dicke regime. Unlike the customary approach, based on the interaction
picture and the rotating wave approximation, we reduce the original Hamiltonian
to a time-independent model, unitarily equivalent to a spin-boson model, which
features a form well suited to apply the perturbation theory. We determine the
first and second-order corrections of eigenstates and eigenvalues. By varying
the interaction parameters we identify four regimes characterized by different
eigenvalue distributions (possibly including level doublets) and suggest the
presence of an intermediate regime where the spectrum undergoes macroscopic
changes. In this regime, where the perturbation approach does not hold, our
reduced Hamiltonian is shown to reproduce the Jaynes-Cummings model.
