Semiclassical formula for the radiative mean lifetime of the excited state of the hydrogenlike atoms

Abstract

For hydrogenlike atoms, it is shown on the basis of Kramers's semiclassical formula for the oscillator strength that the mean lifetime of an excited state with a principal quantum number $n$ behaves with respect to $n$ as $\frac{{n}^{5}}{\mathrm{ln}n}$. In comparison with the exact calculations, it is shown that the $\frac{{n}^{5}}{\mathrm{ln}n}$ law provides a better and more accurate scaling law than the commonly used empirical ${n}^{4.5}$ scaling law. It is shown that the lifetime of an excited state $T(n,l)$, where $l$ is the angular momentum quantum number, increases for large $l$ as ${l}^{2}$. Previously, no law existed for the relationship between the lifetime and angular momentum. By writing $T(n,l)={c}_{\mathrm{nl}}{(\ensuremath{\mu}{Z}^{4})}^{\ensuremath{-}1}{n}^{3}{l}^{2}$, where $\ensuremath{\mu}$ and $Z$ are the atomic reduced mass and effective charge acting on the running electron, the coefficients ${c}_{\mathrm{nl}}$ for low and intermediate values of $n$ and $l$, and their asymptotic values for large $n$ and $l$, are tabulated. Within a factor of 2.35 the lifetime of any excited state, except the $l=0$ states, is given by the formula $0.847\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}{\ensuremath{\mu}}^{\ensuremath{-}1}{Z}^{\ensuremath{-}4}{n}^{3}{l}^{2}$ sec. Agreement is found with selected measured radiative lifetimes of excited helium and alkali-metal atoms. The spread in the experimental data is too large to allow deduction of a systematic deviation from the hydrogenic lifetimes.