Channeling radiation beyond the continuum model: The phonon ‘‘Lamb shift’’ and higher-order corrections

The quantum theory for channeling radiation from mega-electron-volt electrons is extended to include higher-order corrections from phonon scattering. The formalism developed to describe radiation line profiles may be applied to any radiating system exposed to external perturbations. For channeling radiation, it is shown that at electron energies above a few mega-electron-volts, the dominant correction to line energies is of third order in the strength of the interaction giving rise to phonon scattering. Detailed analytical expressions as well as simple estimates are derived for both the second- and third-order thermal corrections. Their magnitude, relative to line energies, is typically of the order of ${10}^{\mathrm{-}3}$ or less. Corrections due to the periodic structure of axial and planar potentials are shown to be of the same order or smaller. Hence the present investigation results in a firm theoretical foundation for channeling-radiation spectroscopy which may then be used as a valuable probe for solid-state parameters, potentially in the study of vibrational anomalies in high-${\mathit{T}}_{\mathit{c}}$ superconductors. For electrons of a few mega-electron-volts, a shift due to photon refraction is important which previously has been included in an ad hocc way. The formalism contains this effect and the semiclassical formula for photon refraction in a free-electron gas is rederived within a fully-quantum-mechanical framework.