Information-theoretical interpretation of level populations and charge distribution in beam-foil spectroscopy

The consequences of the assumption that, under prescribed experimental conditions, the ions emerge randomly excited from the foil are examined using the maximum entropy principle. For that purpose constraints are identified by examining the time-resolved intensity of the electromagnetic radiation emitted by the ions after the foil. This leads to the most likely level or multiplet population density matrix compatible with the experimental information. The main consequences of the foil excitation mechanisms are accounted for by incorporating the framework of atomic structure into our information-theoretical approach. In the case of transitions in individual Rydberg series it is shown that for large $n$ and a given $l$ the level population functions decrease as $n^{*-3\mathrm{}}$ and depend universally on the kinetic energy of the incoming beam. It is also shown that the identification of the radiation from the emerging ions in terms of one-electron states leads, according to the maximum entropy principle, to a unique factorized representation of the probability that an ion has a given configuration immediately after the foil. This representation is used for a derivation of the charge distribution which for heavy ions is shown to be approximately Gaussian or chi-squared in accordance with experiments. The passage of ${\mathrm{He}}^{+}$ ions through the foil is treated as an example of a consistency test based on the information-theoretical derivation of the charge distribution. Following Levine and co-workers a surprisal analysis of level population functions is suggested as a well-established alternative to detailed models of foil excitation which usually invoke perturbation theory and consequently often have an ill-defined range of validity.