The energy levels have been taken from Martin (1973). For the g-levels which are missing, we have calculated the energies by using an asymptotic expression of the quantum defect and the polarisation potential (Deutsch 1969).
Calculations of oscillator strengths which enter the expressions of the semi-classical cross-sections and the A parameters have been obtained with the Bates & Damgaard approximation. Owing to the high levels involved, this approximation is sufficient. Moreover these transitions and the corresponding high levels are not included in the recent and sophisticated data of TOPbase (Cunto et al. 1993). Thus for the calculations of the oscillator strengths we have updated a code given by Dimitrijevic who used the tables by Oertel & Shomo (1968) and the Van Regemorter et al. (1979) formula adapted for high levels. For the transitions between the very high levels nf-n'g (n, n' > 7), we have used the Coulomb approximation, the corresponding quantum defects being negligible.
C1 is the impact validity criterion.
gives the ratio of the contribution of inelastic collisions versus elastic ones for the width.
C3 is the criterion for the "one state'' approximation validity criterion.
C4 is the ratio of the contribution of strong collisions to the total impact width and thus gives a validity criterion for the perturbation theory approximation.
C5 is the criterion for the isolated line approximation, defined by (Dimitrijevic & Sahal Bréchot 1984; Baranger 1958b).
where W is the total width,
is the width due to electron collisions, that due proton collisions and that due to collisions with He+. We neglect collisions with He++ in this criterion and we assume that protons and He+ ions are in proportion of 90% and 10% respectively.
is the energy distance between the upper level and the nearest perturbing level.
Our calculations show that for all studied lines, when the impact approximation is not valid (C1>1), the elastic collisions are generally dominant (C2<1). Consequently in most cases, the quasistatic effect is quadratic. The opposite case (C2>1) only occurs for high densities and for low temperatures, for the two following lines: 8651 Å for collisions with He++ and 10028 Å for collisions with protons, He+ and He++. For these two cases of minor importance, the quasistatic treatment involves the dipolar interaction and is not treated in the present paper.
According to Dimitrijevic & Sahal-Bréchot (1984), if C5< 1, energy levels broadened by collisions do not overlap. Then the line is isolated. If C5> 1 we have in principle to treat the problem of overlapping lines. We show that, for all the studied lines of the present paper, the problem does not appear. The isolated line criterion is always satisfied for the lines: 10138 Å, 8779 Å, 8651 Å, and 8584 Å. In fact, the criterion is not satisfied for the line 10028 Å at densities equal or higher than 1012 cm-3. For the 8736 Å line, the criterion is not satisfied for densities equal or higher than 1011 cm-3. However, if we look at the ratio of the contribution of the upper level to that of lower level to the impact width (the C3 criterion), we conclude that the "one state'' approximation is always valid, because the lower level contribution is always negligible (). Therefore the off-diagonal elements (Baranger 1958b) which enter the expression of the line profiles of overlapping lines in the "two state case'' can be neglected. Consequently it is possible to continue to use the isolated line approximation in the "one state case'' for the present studied lines which should be a priori not isolated.
In addition, we have checked the validity condition of the perturbation theory. The validity criterion is given by C4, which represents the relative contribution of strong collisions to the total impact width. It is well satisfied (C4 does not exceed a few 20%) for collisions with electrons. For collisions with ions, the strong collision contribution is more important (C4 is of the order of 65%) and the perturbation approximation is not so good.
Finally, it can be noticed that the ionic shifts of 8651 Å and 8736 Å lines present some deviations from linearity at 1013 cm-3 due to the Debye cut-off.
Considering the validity conditions discussed above, our data can then be used for astrophysical purposes as follows:
1. If the impact approximation criterion () is satisfied, the resulting profiles are lorentzian, and the total widths and shifts are given by:
2. If the impact approximation criterion is satisfied for electronic collisions and not satisfied for the collisions with ions, then the quasistatic approximation can be used:
2a. If C2> 1, the quasistatic interaction is dipolar. This case is not treated in this paper.
2b. If the quasistatic interaction is quadratic (C2< 1), which is the predominant case, the resulting widths and shifts are given by:
However, it must be noticed that an intermediate region between the impact and quasistatic approximations is expected for collisions with ions, for which neither of the two limiting approximations is valid. For such conditions the static model employed can only give an estimate for the line shape.
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