4 Method of calculation

The semiclassical perturbation formalism, as well as the corresponding computer code (Sahal-Bréchot 1969a,b), have been updated and optimized several times (Sahal-Bréchot 1974; Fleurier et al. 1977; Dimitrijevic & Sahal-Bréchot 1984; Dimitrijevic et al. 1991; Dimitrijevic & Sahal-Bréchot 1996b). The calculation procedure, with the discussion of updatings and validity criteria, has been briefly reviewed e.g. in Dimitrijevic & Sahal-Bréchot (1996a,b) so that only the basic details of calculations will be presented here. Stark full width (W) at the intensity half maximum (FWHM) and shift (d) of an isolated spectral line may be expressed as (Sahal-Bréchot 1996a,b; Fleurier et al. 1977):

$$N\int vf(v){\rm d}v (\sum_{i'\ne i}\sigma_{ii'}(v) +\sum_{f'\ne f}\sigma_{ff'}(v) + \sigma_{\rm el}) + W_{\rm R}$$ W =

$$N\int vf(v){\rm d}v\int_{R_3}^{R_D} 2\pi \rho {\rm d}\rho \sin 2\phi_{\rm p}$$ d = (1)

where N is the electron density, f(v) the Maxwellian velocity distribution function for electrons, $\rho$ denotes the impact parameter of the incoming electron, i and f denote the initial and final atomic energy levels, and i', f' their corresponding perturber levels, while $W_{\rm R}$ gives the contribution of the Feshbach resonances (Fleurier et al. 1977). The inelastic cross section $\sigma_{j,j'}(v)$ can be expressed by an integral over the impact parameter of the transition probability $P_{jj'}(\rho ,v)$ as

$$\sum_{j'\ne j}\sigma_{jj'}(v) = {1\over 2}\pi {R_1}^2 +\int_{R_1}^{R_{\rm D}}\sum_{j \ne j'}P_{jj'}(\rho ,v), j=i,f$$ (2)

and the elastic cross section is given by

$$\sigma_{\rm el} 2\pi R_2^2 + \int_{R_2}^{R_{\rm D}}8\pi \rho {\rm d}\rho\sin^2\delta$$ =

$$\delta (\phi_{\rm p}^2 + \phi_{\rm q}^2)^{1/2}.$$ = (3)

The phase shifts $\phi_{\rm p}$ and $\phi_{\rm q}$ due respectively to the polarization potential (r^-4) and to the quadrupolar potential (r^-3), are given in Sect. 3 of Chap. 2 in Sahal-Bréchot (1969a). $R_{\rm D}$ is the Debye radius. All the cut-offs R₁, R₂, R₃ are described in Sect. 1 of Chap. 3 in Sahal-Bréchot (1969b). For electrons, hyperbolic paths due to the attractive Coulomb force were used, while for perturbing ions the paths are different since the force is repulsive. The formulae for the ion-impact widths and shifts are analogous to Eqs. (1-3), without the resonance contribution to the width. The difference in calculation of the corresponding transition probabilities and phase shifts as functions of the impact parameter in Eqs. (2) and (3) is in the ion perturber trajectories which are influenced by the repulsive Coulomb force, instead of an attractive one as for electrons. Atomic energy levels have been taken from Moore (1985).

The contribution of ion impact widths and shifts to the total line widths and shifts can be neglected since our results for $T=20\,000$ K are two order of magnitude smaller.