Gaussian approximations to equilibrium charge-state distributions with the maximum entropy method

Abstract

Charge-state distributions for ions in beam-foil experiments are discussed in detail and analysed using the maximum entropy method (MEM). The symmetric distributions can be well described by one Gaussian in the whole charge-state interval. It is found that asymmetric distributions, exhibiting the shell effect can be treated using two disjoint Gaussians, in two separate intervals. The relative weights of these Gaussians are interpreted by using Fermi Dirac statistics. Physically this corresponds to varying occupancy of some of the shells in the ions. The fact that no more than two moments, in one interval or two, is sufficient is of significant importance in the analysis of charge-state distributions in general. An interpolation procedure to obtain the charge-state distribution for an energy is developed and tested. The procedure, which is made possible by the Gaussian character, uses MEM and experimental information. It is only possible to predict distributions in energy regions where experimentally measured values for other energies exist. The method is tested on asymmetric distributions of copper ions in the energy range 0.902-1.505 MeV u-1 and symmetric distributions for calcium, titanium, yttrium, rhodium, holmium and bismuth ions for energies from 0.191 to 3.184 MeV u-1.