Understanding the time dependence of atomic level populations in evolving plasmas
Abstract
The time dependence of atomic level populations in evolving plasmas is studied using an eigenfunction expansion of the nonLTE rate equations. The work aims to develop understanding without the need for, and as an aid to, numerical solutions. The discussion is mostly limited to linear systems, especially those for optically thin plasmas, but the implicitly nonlinear case of nonLTE radiative transfer is briefly discussed. Eigenvalue spectra for typical atomic systems are examined using results compiled by Hearon. Diagonal dominance and sign symmetry of rate matrices show that just one eigenvalue is zero (corresponding to the equilibrium state), that the remaining eigenvalues have negative real parts, and that oscillations, if any, are necessarily damped. Gershgorin's theorems are used to show that many eigenvalues are determined by the radiative lifetimes of certain levels, because of diagonal dominance. With other properties, this demonstrates the existence of both “slow” and “fast” timescales, where the “slow” evolution is controlled by properties of metastable levels. It is shown that, when collisions are present, Rydberg states contribute only “fast” eigenvalues. This justifies use of the quasistatic approximation, in which atoms containing just metastable levels can suffice to determine the atomic evolution on timescales long compared with typical radiative lifetimes. Analytic solutions for two and threelevel atoms are used to examine the basis of earlier intuitive ideas, such as the “ionizing plasma” approximation. The power and limitations of Gershgorin's theorems are examined through examples taken from the solar atmosphere. The methods should help in the planning and interpretation of both experimental and numerical experiments in which atomic evolution is important. While the examples are astrophysical, the methods and results are applicable to plasmas in general.
 Publication:

Journal of Quantitative Spectroscopy and Radiative Transfer
 Pub Date:
 June 2005
 DOI:
 10.1016/j.jqsrt.2004.08.009
 Bibcode:
 2005JQSRT..92..479J