Figure 1. Density effect on the 3d3/2 wavefunction in Al 12+. Wavefunction variation due to the plasma environment effect. The calculation is performed within the uniform free-electron gas model with density with Ne=1023 e/cm3 (a) or 1024 e/cm-3 (b). The solid black curves come from analytical computations, the red curves are from numerical integration using FAC [Belkhiri 2013].
This activity is aimed at studying the effects related to the presence of other ions and free electrons on the energies and wave-functions of an ion in a plasma. It is based on an ion-sphere type approach and concerns mostly moderate-density and high-temperature plasmas.
The simplest hypothesis consists in considering that the free electrons form a uniform-density gas which screens the nucleus-electron Coulomb interaction. The “plasma potential” is then analytically defined [Weisheit1996]. One of the interests of this approach is that it allows analytical calculations in the case of hydrogen-like ions, using time-independent perturbation theory [Belkhiri2013]. One may provide expressions for the perturbed energy at first and second orders, which allows one to define the validity of this perturbation approach. One may also obtain wave-functions perturbed up to first order as well as dipolar-electric matrix elements or radiative rates at this same order. A comparison of analytical expressions to numerical computations is presented on Figure 1.
A more involved description of the environment effect accounts for the polarization of the free electron density by the nucleus. A semiclassical method similar to the one use to describe the Thomas-Fermi atom — using a statistical expression for electron density, ionic sphere neutrality and Poisson equation — allows one to get implicitly the free-electron density and the “self-consistent” plasma potential. The obtained density accounts for the polarization of free electrons by the nucleus. This self-consistent potential is provided by a numerical integration. Nevertheless for hydrogen-like ions one may provide an approximate analytical expression valid for any value of nuclear charge and electron density, provided that the current ion-sphere hypothesis holds [Poirier2015].
These plasma potentials have been included in the Flexible Atomic Code [Belkhiri2013]. They allowed us to compute energy shifts, radiative and collisional rate corrections [Belkhiri2014, Belkhiri 2015]. An example of radiative rates including this effect in a helium-like ion is displayed on Figure 2.
Figure 2. Density effect on the radiative rates 1s4p 3PJ ->
1s4s 3S1 in Al11+.
The computation is done with FAC including a plasma potential, either of uniform electron gas type (UEGM), or Thomas-Fermi type (TF) [Belkhiri 2014]. The transition rate is plotted as a function of the spatially averaged free-electron density.