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2 Results and discussion

In order to determine the full line width at half maximum - W and the line shift - d of neutral calcium lines influenced by the Stark broadening mechanism, the semiclassical perturbation formalism has been used. This 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 1996). A review of the calculation procedure, with the discussion of updatings and validity criteria, is published, e.g., in Dimitrijevic (1996, 1997).

Atomic energy levels needed for calculations have been taken from Sugar & Musgrove (1995). The oscillator strengths have been calculated within the Coulomb approximation (Bates & Damgaard 1949, and the tables of Oertel & Shomo 1968). For higher levels, the method of Van Regemorter et al. (1979) has been used.

Electron-, proton-, and He II-impact broadening parameters for Zn I for perturber densities of 1013 cm -3 - 1019 cm-3 and temperatures from 2500 up to 50000 K, are presented in Table 1 (accessible only in electronic form). For perturber density of 1013 cm-3, only data for three multiplets are shown, since other data are linear with density for densities lower than 1014 cm-3. For perturber densities lower than 1013 cm-3, Stark broadening parameters for all tabulated multiplets are linear with perturber density. We also specify a parameter C (Dimitrijevic & Sahal-Bréchot 1984), which gives an estimate for the maximum perturber density for which the line may be treated as isolated when it is divided by the corresponding full width at half maximum. For each value given in Table 1, the collision volume (V) multiplied by the perturber density (N) is much less than one and the impact approximation is valid (Sahal-Bréchot 1969a,b). Values for NV > 0.5 are not given and values for $0.1 < NV \le 0.5$ are denoted by an asterisk. Stark broadening parameters for densities lower than tabulated are linear with perturber density. When the impact approximation is not valid, the ion broadening contribution may be estimated by using the quasistatic approach (Sahal-Bréchot 1991 or Griem 1974). In the region between where neither of these two approximations is valid, a unified type theory should be used. For example in Barnard et al. (1974), simple analytical formulas for such a case are given. The accuracy of the results obtained decreases when broadening by ion interactions becomes important.

There are three experimental studies with data of Stark widths and shifts of neutral zinc lines (Kusch & Oberschelp 1987; Fishman et al. 1979; Rathore et al. 1985). Theoretical data suitable for comparison with our results are published in Dimitrijevic & Konjevic (1983), Lakicevic (1983), and Rathore et al. (1985).

  
Table 2: Comparison between experimental and theoretical Stark full widths at half maximum. Experimental data:a-Kusch & Oberschelp (1967); b-Fishman et al. (1979) c-Rathore et al. (1985). Theoretical data: WDSB - present results; WDK - Dimitrijevic & Konjevic (1983)
\begin{table}\includegraphics[width=8.8cm]{1750t2.eps}\end{table}


  
Table 3: Comparison between experimental and theoretical Stark shifts. Experimental data: c-Rathore et al. (1985). Theoretical data: dDSB - present results; dDK - Dimitrijevic & Konjevic (1983)
\begin{table}\includegraphics[width=8.8cm]{1750t3.eps}\end{table}

In Tables 2 and 3, experimental (Kusch & Oberschelp 1987; Fishman et al. 1979; Rathore et al. 1985) Stark widths (Table 2) and shifts (Table 3) are compared with present results and with semiclassical Stark broadening parameters from Dimitrijevic & Konjevic (1983). In the experiments of Kusch & Oberschelp (1987) and Fishman et al. (1979), ion perturbers are protons, while in the experiment of Rathore et al. (1985) the carrier gas was neon. In order to make the adequate comparison, Stark broadening of neutral zinc by impacts with neon ions has been calculated and included in WDSB and dDSB (present calculation of Stark widths - WDSB and shifts - dDSB) values in Tables 2 and 3. For the 4p3P $^\circ - 5$d3D multiplet, the impact approximation is not valid for proton perturbers and the quasistatic ion broadening contribution is calculated according to Griem (1974). One can see in Table 2 that the agreement of all experiments with both calculations is very poor. The ratio of experimental widths of Kusch & Oberschelp (1967) and the theoretical ones vary from 0.25 up to 3.56. The experimental widths of Fishman et al. (1979) are two times larger than theoretical values from both approaches. The temperature trend of the experimental widths of Rathore et al. (1985) is in such disagreement with both theoretical approaches that the ratios of measured and calculated Stark widths vary e.g. for the 4722.16 Å line from 2.24 for $T = 13\,700$ K up to 0.77 for $T = 18\,100$ K for the present results, and from 2.02 up to 0.69 for the theoretical values of Dimitrijevic & Konjevic (1983). For the shift, ratios of experimental values of Rathore et al. (1985) and results of the present calculations vary from 1.24 to 0.46 for the same spectral line. The experimental results of Fishman et al. (1979) and Rathore et al. (1985) were not selected for critical compilations of reliable Stark broadening experimental data (Konjevic & Roberts 1976; Konjevic et al. 1984; Konjevic & Wiese 1990), while the results of Kusch & Oberschelp (1967) were selected with the attribution of the lowest accuracy (Konjevic & Roberts 1976). In the analysis of the Kusch & Oberschelp (1967) experiment, Konjevic & Roberts (1976) have found large variations of Stark widths within multiplets, and supposed that this may be caused by improper treatment of self-absorption. Moreover, Dimitrijevic & Konjevic (1983) have shown on the basis of the analysis of Stark width systematic trends within spectral series, that the experimental results of Kusch & Oberschelp (1967) are in disagreement with such trends. Lakicevic (1983) estimated on the basis of regularities and systematic trends Stark width and shift for the Zn I 4s2 1S -4p1P$^\circ$ transition for an electron temperature (T) of 20000 K and an electron density of 1017 cm-3. He obtained the value of 0.066 Å for the full width at half maximum, and 0.035 Å for the shift. We obtain the value of 0.039 Å for the width and 0.029 Å for the shift. On the basis of regularities and systematic trends as well, Stark widths and shifts for the Zn I 4p3P$^\circ$ -5s3S transition for electron temperatures of 10000 and 20000 K and an electron density of 1017 cm-3 have been estimated by Rathore et al. (1985). They obtained the value of 0.60 Å for the full width at half maximum, and 0.36 Å for the shift, for $T = 10\,000$ K, and our results are 0.371 Å for the width and 0.295 Å for the shift. For $T = 20\,000$ K, they obtained 0.61 Å for the width and 0.30 Å for the shift, and we 0.408 Å for the width and 0.336 Å for the shift. Particularly for the shift, this is in both cases an encouraging agreement of simple estimates with our semiclassical perturbation results.

The comparison between our semiclassical perturbation results and semiclassical results of Dimitrijevic & Konjevic (1983) is shown in Table 4. Differences between the present calculations and the semiclassical method described in Griem (1974) and used by Dimitrijevic & Konjevic (1983), have been discussed in detail in Dimitrijevic & Sahal-Bréchot (1995) and may be attributed to the theoretical differences and the differences in input data.

First of all, the lower cut-offs are different in both methods. In Dimitrijevic & Konjevic (1983), the same cut-off for widths and shifts as well as for both elastic and inelastic collisions has been used. The effect of the change of the set of values for the cut-offs R1, R2, and R3 has been studied and discussed in detail in Sahal-Bréchot (1969b): the final choice of the cut-offs was adopted for physical reasons (it allows for the unitarity of the S-matrix) and followed Seaton (1962).

Moreover, in the present method, with the help of the symmetrization procedure one takes into account the impact electron velocity change during an inelastic collision, which is not taken into account in Dimitrijevic & Konjevic (1983). The importance of the symmetrisation has been shown in Sahal-Bréchot (1969b). It has been demonstrated there that the symmetrisation improves considerably (factor two) semiclassical cross sections for small energies (close to the threshold), which are overestimated without symmetrisation. Also, for the difference of the semiclassical method (Griem 1974) used by Dimitrijevic & Konjevic (1983) we take into account explicitely the elastic collision contribution.


  
Table 4: Comparison between present Stark full widths at half maximum (WDSB) and shifts (dDSB) and the corresponding results (WDK, dDK) of Dimitrijevic & Konjevic (1983)
\begin{table}\includegraphics[width=8.8cm]{1750t4.eps}\end{table}

Another difference between methods used here and in Dimitrijevic & Konjevic (1983), is the Debye shielding effect. Griem (1974) gives the equation for the Debye shielding correction, which may be included when necessary, and here, the Debye cut-off is included in the calculations. Also, while in Dimitrijevic & Konjevic (1983) the ion-broadening contribution is only a correction within the quasistatic theory, the complete semiclassical perturbation calculation has been performed here for the ion-impact broadening, when the impact approximation is valid. The differences in the imput data are the more recent atomic energy level data (Sugar & Musgrove 1995).

Both methods have been compared with critically selected experimental data for 13 He I multiplets (Dimitrijevic & Sahal-Bréchot 1985) and it was found that the agreement between experimental data and both semiclassical methods is within the limits of 20 percent, which is the predicted accuracy of the semiclassical method (Griem 1974). One can see from Table 4 that for zinc differences are larger and increase with temperature, particularly for the shift. One must take into account that more recent and more complete energy levels have been used in our calculations.

The obtained Stark broadening data are of interest for a number of problems in astrophysics and plasma physics as e.g. abundance determinations, stellar spectra analysis and laboratory plasma diagnostics. Reliable experimental determinations of neutral zinc Stark broadening parameters will be of interest for checking and development of Stark broadening theory.

Acknowledgements
This work has been supported by the Ministry of Science and Technology of Serbia through the project "Astrometrical, Astrodynamical and Astrophysical Researches".


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