The Validity of Long Wavelength Approximation in the Evaluation of Two-Photon Decay Rate

This paper investigates the validity of the long wavelength approximation in the calculation of two-photon decay of $2s_{1/2}$ level in hydrogen-like ions with nuclear charge $Z=1-100$ based on time-dependent second-order perturbation theory and angular momentum algebra. While the relativistic structure effects on the two-photon decay rates are highlighted in the literature, the role of slowing effects in the photon electric dipole operators are not discussed extensively. The rate is computed by the sum-over-states method, with bound-bound and bound-free electric dipole matrix elements obtained in the Babushkin and Coulomb gauges, which satisfy the Lorenz gauge condition, as well as their non-relativistic limits in the long-wavelength approximation (Length and Velocity forms, respectively). The present results explicitly show how this approximation breaks gauge invariance by overestimating the Babushkin values by ∼24%${\left(\alpha Z\right)}^2$ while underestimating the Coulomb rates by ∼$31\%{\left(\alpha Z\right)}^2$. Using analytical eigenfunctions of the Dirac equation, we found that the contributions of the negative continuum states to the rate scale are ∼$0.0134{\left(\alpha Z\right)}^4$ in the Babushkin gauge and ∼$1.46{\left(\alpha Z\right)}^4$ in the Coulomb gauge, making the latter gauge more susceptible to errors when attempting to achieve basis completeness in multiphoton calculations. The present results are useful in assessing the complexity requirements of radiative transition rates for atomic systems of interest.