1 Introduction

Hydrogen far infrared, submillimetric and radiofrequency lines have been the object of a large number of works during the last three decades. The importance attributed to such studies lies in the fact that they may provide valuable diagnostics for the thermal structure of the solar atmosphere (Hoang Binh 1982; Brault & Noyes 1983; Boreiko & Clark 1986; Hoang Binh et al. 1987; Carlsson & Rutten 1992), and also of some stellar atmospheres (see Waters & Marlborough 1992); they may also help in the understanding of the proton-electron recombination process in the H II regions. In their paper published in 1987, Hoang Binh et al. have given theoretical data for the calculation of hydrogen solar line broadening in the far infrared and submillimetric spectra $\left( n\sim 5-20\right)$, within the impact approximation for ions. We aim to study the linewidths and lineshapes of the same lines for a wide range of densities, scaling from low to mean densities - for which the impact approximation breaks down for ions, but is still valid for electrons. However, the densities must be chosen below the Inglis-Teller limit, such as to avoid or reduce the line dissolution (or overlap) under the broadening process.

In fact, according to Hoang Binh (1982), most of the solar infrared hydrogen lines are emitted in the chromo-sphere at heights between 750 and 1600 km above the photosphere, where the electron density $N_{\rm e}$ and ion density $N_{\rm p}$vary between 6 10¹⁰ and 10¹¹ cm^-3, and where the temperature is of the order of 6 10³ K (Vernazza et al. 1981). In such conditions, the impact approximation is valid or almost valid for ions, and - of course for electrons - for the $n_{\alpha}$ lines and most of the $n_{\beta}$ lines $\left( 5\leq n\leq 20\right)$. For the lines emitted in the neighbourhood of the phosphere - $N_{\rm e}\sim N_{\rm p}\sim 10^{14}$ cm^-3 and $T\sim 6500$ K - the impact approximation is not valid for ions, even for n=5, but is still valid for electrons.

In a previous paper (Motapon et al. 1997), we used a quantum formalism to calculate the linewidths and lineshapes of the lower infrared hydrogen lines broadened by electrons, ions and both electrons and ions. This formulation, using the no-quenching approximation, has been found to be not suitable for the description of electron broadening of lines emitted from transitions between atomic levels above n=5, since in such cases, electron broadening is mainly due to inelastic collisions. On the other hand, it is expected to apply conveniently to these lines, broadened by impact with ions since this last process is dominated by elastic ion-atom collisions. However, quantum calculations are long for ions and may be costly.

There is a need for simple and efficient approaches for the determination of linewidths. Even the semi-classical calculations may be doubtful if an adequate choice of the cut-offs is not made in the integration over the impact parameters. In this work, a collision rate approach is used to describe electron impact widths. Electronic linewidths are determined from the collision rates via the electron-atom excitation cross sections. In a few cases, the traditional semi-classical theory is used to calculate ion and electron linewidths for comparison with the collision rate method, and for quantitative comparison of the electron and ion contributions to the total impact linewidths.

In the second section of this paper, the semi-classical method is briefly described. The linewidth calculation is presented, and the limits of the approach are discussed. The third section deals with the collision rate methods. The comparison of the collision rate approach and the semi-classical method in the fourth section is followed by the conclusion. Values of collision rates are provided in Tables 4 to 6, for further calculations of electron impact widths and approximate profiles.