Calculations at Series Limits in One-electron Systems

The behavior of atoms in high quantum levels is generally well understood— in principle. For sufficiently large values of the principle quantum number, discrete levels that would exist for an isolated atom, do not. They are said to be dissolved, or quenched by perturbations from surrounding atoms and ions. In practice, the calculation of relevant atomic properties can become quite involved, especially if one strives for accuracy. Detailed references may be found in the papers by Stehlé and Jacquemot, Hummer and Mihalas, or the book by Griem. We draw attention here to a general principle that should be of practical value in the calculation of opacity near the series limits. We show that as soon as the Lyman or Balmer lines overlap and the individual profiles wash out, the net absorption cross section rapidly approaches the value at the series limit. It has been known for more than seventy years that the line and continuous opacities merge smoothly. We are unaware of a discussion of the rapidity of this convergence, which has a practical value for the calculation of opacity. Independent of the difficult question of the relative contributions of line and continuous opacity, we can get a good estimate of their sum. The accuracies of estimates for two series are discussed below. We offer the following thoughts as a computational aid. They may also be of heuristic value.