Wednesday, 25 December 2013

Atomic partition functions in IDL

Many books on radiative transfer and spectroscopy introduce the partition function ($U$, PF) as a statistical weight or probability that an atom is in the ionization stage $r$ at the temperature $T$. It is defined as the sum over all excitation states ($i$) of their statistical weights ($g_i$) multiplied by an
exponential factor (cf. Rutten, 2003, p.30, Eq.289):
$$U_r \equiv \sum_i g_i \mathrm{e}^{-\frac{\chi_{r, i}}{kT}},$$ where $\chi_{r, i}$ is the excitation potential of the state $i$ in the ionization stage $r$, $k$ is the Boltzmann constant) and $T$ is the temperature.

The trouble with PF is that there is an infinite number of the energy levels. No matter how small their contribution becomes (the exponential factor is decreasing with the excitation potential), PF computed after this definition goes to infinity. Farther, that means that the probability of an atom to occupy any finite state would be zero. Since this is clearly unphysical, thus there is a need to define a reliable procedure to cut off the partition function. There are many approached described in literature. However, they are usually rather non-trivial and, therefore, computing partition function even for a limited range of temperature and for a specific atomic element can be a very tedious job. The results of these computations are often published as the coefficients of the best polynomial fits for a range of temperature values. The IDL procedure that you can find below is written to load these tables from several different sources (see below) and to interpolate them for given a temperature, an atomic specie and an ionization stage.