Raman scattering of photons by hydrogenlike ions I: A semirelativistic and nondipole approach

Going beyond the usual nonrelativistic treatments of Raman scattering of light by a ground-state electron in hydrogenic ions, we propose in this work a semirelativistic and nondipole approach in the case of inelastic process under the ionization threshold. This scheme is based, on the one hand, on the discrete semirelativistic Coulomb Sturmian basis sets, and on the other hand, on the direct Foldy–Wouthuysen interaction Hamiltonian derived recently from the Dirac equation with electromagnetic potentials (Hinschberger and Hervieux 2012) and confirmed by us to seventh order in powers of $1∕m$. It includes the paramagnetic, the diamagnetic, the magnetic field–spin and the electric field–spin interactions. Moreover, the photon fields are expanded in electric and magnetic multipoles and nondipole effects are fully taken into account. On utilizing these ingredients, we build in detail the multipole expansion of the semirelativistic Raman scattering cross section when the photon energy is under the ionization threshold. For each type of the allowed multipole components, namely EE, MM and EM in the standard notations, suitable closed forms of total scattering cross sections are calculated for numerical computations. Such analytical expressions enable the evaluation of the contribution of each of these interaction operators and field multipoles in the scattering process.