Stark width regularities within magnesium spectral series

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1 Mon. Not. R. Astron. Soc. 415, (2011) doi: /j x Stark width regularities within magnesium spectral series Irinel Tapalaga, Ivan P. Dojčinović and Jagoš Purić University of Belgrade, Faculty of Physics, PO Box 44, Belgrade, Serbia Accepted 2011 March 15. Received 2011 March 1; in original form 2011 January 18 1 INTRODUCTION An addition to the theoretical and experimental investigation of Stark broadening of numerous spectral lines from different atoms and ions (Konjević & Wiese 1976a,b, 1990; Konjević, Dimitrijević & Wiese 1984a,b; Konjević et al. 2002) is the search for possible types of regularities of Stark parameters (Purić &Šćepanović 1999; Šćepanović & Purić 2003; Purić et al. 2008, and references therein). According to the latter authors approach to the problem of regularities, all relevant published papers can be divided into two groups. The first group of papers (see Purić et al and references therein) is devoted to the evaluation of different types of regularities, such as the dependences on atomic charge number, upper-level ionization potential, atomic polarizability, principal quantum number and rest core charge of the emitter (seen by an electron undergoing transition), based on the Stark width and shift theoretical formulae obtained in different approximations: semiclassical, semiempirical and adiabatic. In the second group of papers (Wiese & Konjević 1982, 1992), regularities were sought among experimental results and conclusions were drawn on the basis of analyses of the configuration of atomic energy levels and transition probabilities: namely, the expected regularities were studied and discussed by comparing existing experimental and theoretical data within similar spectra, without any attempt to find a particular functional relation between Stark parameters and the particular atomic structure parameter describing the atomic structure influence on such dependences appropriately. Many reviews (see e.g. Lanz & Artru 1987; Seaton 1987) indicate that experimentally determined Stark widths and also existing theoretical data cannot satisfy the need for all observed stellar lines and as used in opacity calculations relative radiation transfer modelling in different studies of astrophysical interest, although it is an active field of research. Therefore, it is of interest to exploit any irinelko@gmail.com ABSTRACT The dependences of electron- and proton-impact Stark width on the upper-level ionization potential within different series of neutral magnesium spectral lines have been evaluated and discussed. Dependences similar to those previously found for the electron-impact contribution were also obtained for the proton-impact contribution to the Stark broadening parameters. The emphasis is on the fine-structure influence on the studied Stark parameter dependences. The relations found can be used in both cases for the prediction of new Stark broadening data, thus avoiding much more complicated procedures. Key words: atomic data line: profiles opacity plasmas radiative transfer. possible theoretical approach that might provide simple relations, for example, from the systematic trends found in Stark broadening data in both simple and complex spectra. An approach based on the systematic trends found in Stark broadening parameters has been developed in a series of articles devoted to Stark parameter dependences on the upper-level ionization potential and rest core charge of the emitter (Purić et al and references therein). Such an approach differs from earlier Stark broadening trend analyses primarily in the choice of the variable conveying atomic structure information to Stark broadening parameter values (Purić, Miller & Lesage 1993). It is of interest to note that the work of several authors presented in Wiese & Konjević (1982, 1992) was based on the hydrogenic model, which uses integer principle quantum numbers instead of the upper-state ionization potential χ chosen here. Both variables take into account the density of states perturbing the emitting state. However, some advantages of the present method are that (1) χ-based trend analyses achieve better fits (compared with those obtained when the integer quantum number is used instead; Purić et al. 1987), (2) in χ values, the lowering of the ionization potential (Inglis & Teller 1939) can be taken into account, predicting merging with the continuum when the plasma environment causes a line s upper-state ionization potential to approach zero and (3) the Stark width (w) dependence on χ is theoretically expected (Purić et al. (1993, 2008). The proposed method is based on the fact that the Stark widths in angular frequency units exhibit a certain functional dependence on the upper-level ionization potential of the corresponding transition. In fact, it is the bounding energy of the electron on the upper level of the corresponding transition. In order to avoid misunderstanding, the positive value of this quantity (χ) is called the upperlevel ionization potential. Moreover, a systematic dependence has also been found on the rest core charge (Z c ) of the ionized emitter as seen by the electron undergoing transition. Simple relations based on such trends may be useful in astrophysics, when Stark broadening data for many lines are needed. Therefore, the Starkparameter dependences on χ and Z c were deduced from large-scale C 2011 The Authors

2 504 I. Tapalaga, I. P. Dojčinović and J. Purić Table 1. The appropriate parameters a and b for K and K temperature, together with the corresponding correlation factor R 2 for both the electronand proton-impact contributions to the Stark widths for all series studied (N e = m 3 ). Electron impact broadening Proton impact broadening Spectral (T = K) (T = K) (T = K) (T = K) series a (rad s 1 ) b R 2 a (rad s 1 ) b R 2 a (rad s 1 ) b R 2 a (rad s 1 ) b R 2 3d nf (3) 4.02E E E E d np (1) 2.62E E E E d np (3) 1.31E E E E p nd (1) 2.13E E E E p nd (3) 8.35E E E E p ns (1) 2.51E E E E p ns (3) 2.60E E E E s np (1) 2.50E E E E d nf (3) 8.66E E E E d np (1) 2.92E E E E d np (3) 1.61E E E E f nd (1) 3.36E E E E f nd (3) 1.05E E E E p nd (1) 2.90E E E E p nd (3) 1.11E E E E p ns (1) 2.73E E E E p ns (3) 4.05E E E E s np (1) 2.52E E E E s np (3) 1.07E E E E d nf (3) 1.31E N/A 1.29E N/A 8.30E N/A 1.43E N/A 5d np (1) 2.95E E E E d np (3) 2.00E E E E f nd (3) 1.17E E E E p nd (1) 3.72E N/A 6.76E N/A 9.31E N/A 9.93E N/A 5p nd (3) 1.28E E E E p ns (1) 3.19E E E E p ns (3) 8.49E E E E s np (1) 2.70E E E E s np (3) 1.11E N/A 9.85E N/A 3.60E N/A 5.24E N/A 6p nd (3) 1.47E N/A 1.32E N/A 3.32E N/A 4.73E N/A 6p ns (1) 3.54E N/A 7.19E N/A 5.39E N/A 6.40E N/A 6s np (1) 2.95E N/A 6.20E N/A 7.16E N/A 7.06E N/A 6s np (3) 1.09E E E E p nd (3) 2.25E N/A 1.94E N/A 4.49E N/A 6.47E N/A 7s np (1) 3.60E N/A 8.04E N/A 6.56E N/A 6.14E N/A semiclassical or semiempirical and modified semiempirical calculations (Griem 1974). The relations obtained were checked using existing experimental and theoretical data (Purić et al. 2008). Despite continuous acquisition of Stark broadening data (Purić et al and references therein), little is known about lines of heavy and multiply ionized emitters. This paucity of data particularly hampers opacity modelling of stellar atmospheres (Seaton 1987). Line widths assumed when analysing or synthesizing the majority of stellar spectra involve either unreliable approximations or outright ignoring of Stark broadening (even though it may exceed natural broadening several-fold). In some previous papers (Purić et al. 1991, 1993, 2008), it was shown that a physically based interpretation of the broadening regularities could provide Stark parameters with a useful combination of reliability and computational simplicity. This approach is focused on the strong connection between the Stark width and the energy required to ionize the emitter from the upper state of the transition. The accuracy of the data obtained using the procedure described is expected to be comparable with the accuracy of the data used in verification of Stark width dependences on the upper-level ionization potential. Therefore, it is very important for this method to have Figure 1. Typical Stark width temperature dependences of two Mg I lines: nm (singlet) and Mg I nm (triplet).

3 a set of data, calculated or measured, for the same plasma conditions, in particular electron temperature and density. This makes it possible to avoid the influence of density and temperature scaling, Debye screening effects and different ion contributions to the Stark widths used in the trend analysis. Otherwise, the existing data have to be normalized to a particular electron density and temperature and then to be used in the trend analysis. This is the case for the Stark width data of spectral lines originating from neutral magnesium studied here. They are taken from the Stark B data base calculated by a single group of authors (Dimitrijević&Sahal- Bréchot 1994a,b, 1996) separately for electron- and proton-impact contributions to the width of particular magnesium spectral lines. Therefore, here the results from the study of Stark parameter regularities are given in more detail within Mg I lines. The emphasis is Stark width regularities within magnesium 505 on the regularities of Mg I within different series and the influence of fine structure, such as different multiplicities (e.g. singlets and triplets), on the dependence of the Stark parameters on the upperlevel ionization potential. The electron- and proton-impact contributions to Stark broadening were studied simultaneously. The functional dependences of Stark broadening parameters obtained were compared with other theoretical (Griem 1974) and experimental (Konjević & Wiese 1976a; Lesage 2009) results. It was found that the agreement was within the theoretical uncertainties (±15 per cent) and the accuracy of the experimental results (less than 50 per cent as quoted by the authors Helbig & Kusch (1972)). Finally, the well-determined Stark width dependences on the upperlevel ionization potential are used to predict electron- and protonimpact contributions to the Stark widths of 12 Mg I spectral lines not Figure 2. The electron-impact contributions to Stark width dependences versus inverse upper-level ionization potential at K temperature for different Mg I spectral series with principal quantum number of the lower level equal to (a) n = 3, (b) n = 4, (c) n = 5, (d) n = 6,7. The numbers in brackets (1 or 3) indicate singlets or triplets, respectively. Values taken from (Griem 1974) are specified with 3G for triplets and 1G for singlets.

4 506 I. Tapalaga, I. P. Dojčinović and J. Purić calculated or measured so far, so as to demonstrate the method of prediction. 2 STARK WIDTH REGULARITIES 2.1 Theoretical background Theoretical relations for the line Stark width as a function of the upper-level ionization potential and the rest core charge of the emitter were evaluated by Purić et al. (1993) starting from equation (77) of Griem (1974). These relations were successfully fitted to a number of spectral-line Stark parameters as shown in a series of articles (see e.g. Purić et al. 1993; Purić & Šćepanović 1999; Purić etal. 2008) and were found to be of the form w = Z c 1 c a 1N e f (T e )χ b 1, (1) where w is the line width, χ is the corresponding upper-level ionization potential, a 1, b 1 and c 1 are coefficients independent of temperature, electron density and ionization potential for a particular transition and Z c is the rest core charge of the emitter, as seen by the electron undergoing transition (Z c = 1, 2, 3... for neutrals, singly charged ions,..., respectively). Equation (1) can be used (i) in the case of lines originating from the same type of transitions (e.g. resonances or off-resonances: Purić, Ćuk & Lakićević 1985); multiplets, supermultiplets, spectral series and transition within one stage of ionization (Z c = const) or within several stages of Figure 3. The electron-impact contributions to Stark width dependences versus inverse upper-level ionization potential at K temperature for different Mg I spectral series with principal quantum number of the lower level equal to (a) n = 3, (b) n = 4, (c) n = 5, (d) n = 6,7. The numbers in brackets (1 or 3) indicate singlets or triplets, respectively. The corresponding experimental values (Djeniže et al. 2004) are included.

5 ionization (Z c const) (Purić et al. 1991, 1993; Purić&Šćepanović 1999; Purić et al. 2008); (ii) within particular isoelectronic sequences (see e.g. Purićetal. 1988) and (iii) within a given isonuclear sequence (see e.g. Purić et al. 1988). Stark width regularities within magnesium 507 For the same plasma conditions and for exactly analogous transitions within different atomic spectra, corresponding a 1 and b 1 constants are the same. Consequently, one can determine empirically, from experiment or more sophisticated calculations, averaged values for a = a 1 N e f (T)andb = b 1. In the case of a neutral spectrum, such as Mg I spectra, Z c = 1 and consequently equation (1) can be written as w = aχ b. (2) In order to investigate different Stark parameter regularities, one requires an accurate set of theoretical and experimental data normalized to the particular electron density and temperature. The normalization to the same N e can be done by linear scaling due to the linear dependence of Stark widths on N e. However, the Stark width dependence on the electron temperature is different from line to line for all spectra. Therefore, the correction to the temperature dependence has to be done with great care for all data used Figure 4. The proton-impact contributions to Stark width dependences versus inverse upper-level ionization potential at K temperature for different Mg I spectral series with principal quantum number of the lower level equal to (a) n = 3, (b) n = 4, (c) n = 5, (d) n = 6,7. The numbers in brackets (1 or 3) indicate singlets or triplets, respectively.

6 508 I. Tapalaga, I. P. Dojčinović and J. Purić in the particular case of the verification of certain types of abovementioned dependences and regularities. For instance, instead of the commonly adopted temperature dependence of T 1/2 for ion lines, one has to use, from line to line (Purić & Šćepanović 1999; Purić et al. 2008), the whole spectrum of functions given by f (T ) = A + BT C, (3) for a large temperature range (Griem 1974; 1997) given by 10 2 χ 0 kt e χ 0, (4) where A, B and C are coefficients independent of electron temperature and χ 0 is the ionization potential of a given emitter, the spectrum of which is used for plasma diagnostic purposes. It was found that the same type of functions can be used in the case of neutral spectral lines, and consequently in the case of Mg I spectral lines. 2.2 Results and discussion The main task of this paper is to investigate the relationship between Stark widths of spectral lines (full width at half-maximum, hereafter FWHM) and the upper-level ionization potential of the corresponding transition within spectral series of Mg I spectral lines which can be used for the prediction of Stark widths for missed Figure 5. The proton-impact contributions to Stark width dependences versus inverse upper-level ionization potential at K temperature for different Mg I spectral series with principal quantum number of the lower level equal to (a) n = 3, (b) n = 4, (c) n = 5, (d) n = 6,7. The numbers in brackets (1 or 3) indicate singlets or triplets, respectively.

7 spectral lines from these series. For this task, Stark width data from the Stark B data base were collected and matched with their corresponding energy levels taken from the NIST data base. A collection of 176 Mg I spectral lines with their corresponding Stark widths was used in further study of Stark width dependences on the upper-level ionization potential at different temperatures for lines originating from the following spectral series: 3d nf (3), 3d np (1), 3d np(3), 3p nd (1), 3p nd (3), 3p ns (1), 3p ns (3), 3s np (1), 4d nf (3), 4d np (1), 4d np (3), 4f nd (1), 4f nd (3), 4p nd (1), 4p nd (3), 4p ns (1), 4p ns (3), 4s np (1), 4s np (3), 5d nf (3), 5d np (1), 5d np (3), 5f nd (3), 5p nd (1), 5p nd (3), 5p ns (1), 5p ns (3), 5s np (1), 5s np (3), 6p nd (3), 6p ns (1), 6s np (1), 6s np (3), 7p nd(3)and7s np (1). Next to the series notation there is a number in parentheses, 1 or 3, standing for a singlet or triplet series, respectively. For all the studied series it was found that the relation given by equation (2) is appropriate for any particular temperature in cases of both electron- and proton-impact contributions to the Stark widths. During these analyses one has to use Stark width data in angular frequency units. The expected temperature dependences (equation 3) were verified for all 176 spectral lines used in the analysis. The dependences of the Stark width on the upper-level ionization potential were verified for all 36 spectral series studied here. In order to have the possibility of obtaining these data for any missed lines from the series, one has to use the obtained functional dependence expected according to equation (2) knowing only the upper-level ionization potential and to substitute it in the same equation. Using this procedure and the temperature dependence of the Stark widths given by equation (3), it is possible to obtain Stark broadening data by extrapolation or interpolation for any temperature of interest from the range defined by equation (4). An appropriate computer program has been designed in order to be able to obtain, in the first case, the Stark width dependence on the temperature of any particular spectral line originating from the abovementioned series of neutral magnesium. This gives an opportunity of obtaining the dependence of Stark width data on the upper-level ionization potential at any temperature, as given in Table 1. In this table the appropriate coefficients a and b are given for K and Stark width regularities within magnesium K temperature, together with the corresponding correlation factor R 2 for both electron- and proton-impact contributions to the Stark widths for all studied series. For some series, correlation factors R 2 are not available (N/A) because there were only two points in that series so R 2 would have a trivial value. However, the parameters of linear best-fitting a and b are listed because based on available series we assume that all series have a high value of R 2,sotwo points are enough to predict the behaviour of other lines that belong to those particular series. During these analyses the influence of fine structure on these dependences (singlets or triplets) was observed and found to be of the form presented in Fig. 1 as far as temperature dependence is concerned. This kind of temperature dependence is typical for most spectral lines studied here. Namely, it was found that in the case of lines from singlet and triplet series the Stark width temperature dependences are increased and decreased functions, respectively. This is not the case only with the np n s triplet spectral series, because the Stark width dependences on the temperature are increased functions, as was found for singlet series. From this figure one can also conclude that these dependences are far from being weak functions of temperature, as has usually been accepted in the literature so far. Having found these temperature dependences, one can find the corresponding dependences of the Stark widths on the upper-level ionization potential for any spectral line within the spectral series studied here for every temperature, as is shown in Figs 2 5. In Figs 2 and 3, electron-impact contributions to the Stark width dependences versus the inverse value of the upper-level ionization potential are given for all investigated spectral series at two different electron temperatures, and K, respectively. For example, it was found that for seven spectral lines of Mg I belonging to the 3s np singlet series, the corresponding correlation factors in both cases (electron- and proton-impact contributions) are equal to unity. The same behaviour is observed for all other series studied. It is worth noting that the electron-impact contributions to the Stark widths of the lines originating from the np n s triplet spectral series follow the same Stark width dependence of the upper-level ionization potential as obtained for the Mg I singlet series. Figure 6. Best-fitting Stark widths for singlet Mg I series, electron and proton impacts, at (a) and (b) K. The corresponding experimental values (Helbig & Kusch 1972) are included.

8 510 I. Tapalaga, I. P. Dojčinović and J. Purić Values taken from Griem (1974) are also included in Fig. 2(a) and (b) for the sake of comparison. The agreement is within the uncertainties of these two theories (±20 per cent). The corresponding experimental values (Djeniže, Bukvić & Srećković 2004) are included in Fig. 3(a). The agreement of these experimental results with theory is within the experimental accuracy (±20 per cent). Similarly, Figs 4 and 5 present graphically the dependences of the proton-impact contribution on the upper-level ionization potential at temperatures of and K, respectively. It is worth noting that these dependences are almost the same in both cases (electron impact and proton impact) for every singlet spectral series (with correlation factor R ), as is shown in Fig. 6. In this figure the results for 16 singlet spectral series are presented, together with the existing experimental values at a temperature of K (Helbig & Kusch 1972) for the sake of comparison. These experimental results were obtained with an estimated error of more than ±50 per cent. As an example, the function w = χ 2.24, (5) where χ has to be taken in ev in order to get w in angular frequency units, can be used at K temperature for all singlet series treated together. When the spectral series studied (singlets and triplets) were treated separately, in almost all cases the corresponding correlation factors R 2 were better then 0.99, as shown in Table 1. As far as Figure 7. Best-fitting Stark widths for triplet Mg I series, for (a) and (c) electron and (b) and (d) proton impacts, at K temperature: (a) and (c) 11 series: np n d(n = 3,4,5,6,7), ns n p(n = 4,5,6,7) and nf n d(n = 4,5); (b) and (d) Mg I triplet series nd n p(n = 3,4,5), nd n f(n = 3,4,5) and np n s(n = 3,4,5) with the best fit for 11 series (thick solid line).

9 Stark width regularities within magnesium 511 Table 2. The calculated values for the electron-impact contribution to the Stark widths (FWHM) w (nm) of 12 Mg I spectral lines at T e = and K normalized to an electron density of N e = m 3 are given. Ion λ(å) Transition Terms w (nm) T = K w (nm) T = K Mg I s 10p 1 S 1 P o Mg I s 11p 1 S 1 P o Mg I s 12p 1 S 1 P o Mg I p 10d 1 P o 1 D Mg I p 11d 1 P o 1 D Mg I p 12d 1 P o 1 D Mg I p 10d 3 P o 3 D Mg I p 11d 3 P o 3 D Mg I p 12d 3 P o 3 D Mg I p 10s 3 P o 3 S Mg I p 11s 3 P o 3 S Mg I p 12s 3 P o 3 S the Stark width dependences on the upper-level ionization potential are concerned, regarding electron- and proton-impact contributions one can conclude that all singlets can be treated together in both cases. However, in the case of lines originating from 20 triplet series, one has to use different equations for every particular series in both cases (if higher precision is needed, e.g. R 2 > 0.99) or to divide them into four groups (if smaller precision is acceptable). The first group consists of 11 series np n d(n = 3, 4, 5, 6, 7), ns n p (n = 4, 5, 6, 7) and nf n d(n = 4, 5) (Fig. 7a and c); the second of three series nd n p(n = 3, 4, 5); the third of three series nd n f (n = 3, 4, 5); the fourth of three series np n s(n = 3, 4, 5) (Fig. 7b and d). In all these cases the corresponding correlation factors R 2 for electron-impact contributions are , , and , respectively. The same conclusion can be drawn for the proton-impact contribution. In general, the dependences of the obtained Stark width on the upper-level ionization potential can be used for the prediction of Stark width data for lines not investigated so far, as given in Table 2. In this table the predicted Stark width values for 12 spectral lines missed so far from the studied series are given. Based on the above-described analysis it is possible to predict Stark widths at any temperature, but the results in this paper are given only for T = and K, as presented in Table 2. It is expected that the predicted data are of the same accuracy as the data used in the verification of Stark width theoretical dependences on the upper-level ionization potential. The existing experimental (Helbig & Kusch 1972; Djeniže et al. 2004) and theoretical data (Griem 1974) are in good agreement (within the quoted experimental and theoretical uncertainty) with the obtained Stark width regularities based on the theoretical data used in this analysis (Dimitrijević & Sahal-Bréchot 1994a,b, 1996). 3 CONCLUSION Searching for different types of regularities and systematic trends that can simplify complicated theoretical calculations, especially those used in astrophysics, is of great interest. Therefore the aim of this paper was to establish as precisely as possible the Stark parameter dependence on the upper-level ionization potential for 36 Mg I spectral series, so as to demonstrate the capabilities of the method described. This work successfully proves the existence of strong functional Stark width dependences on the upper-level ionization potential for lines originating from the same series. These dependences were obtained and found to be in the form given by equation (2). It was found that temperature dependence is very important for studying Stark parameter regularities and therefore we have used theoretical values obtained by different authors for 176 Mg I spectral lines originating from 36 different series in order to determine the Stark parameter temperature dependence through introduced coefficients A, B and C for these lines, used in systematic trend analysis for temperature-data normalization. Normalization at an electron density was assumed to be a linear function for non-hydrogenic emitters. On the basis of the above results, one can draw the following conclusions. First, equation (2) can be successfully used for the prediction of Stark widths using the coefficients a and b. Secondly, the influence of fine structure (singlet and triplet) on Stark width dependences on the upper-level ionization potential is very important: namely, the singlet series can be treated all together, in contrast to the triplet ones. Thirdly, in general the best precision can be obtained using equation (2) for any particular series separately. The theoretical dependence obtained has been compared with the experimentally and theoretically determined Stark widths published so far. The agreement was found to be within the theoretical uncertainty and experimental error. After analysing the relationship between the Stark width of spectral lines and the upper-level ionization potential, the same conclusions can be drawn for the behaviour of both electron- and protonimpact contributions. An almost linear (log log scale) connection between FWHM and upper-level ionization potential can be established according to equation (2), not only within a particular series of spectral lines but also among all singlet series studied here. As far as the triplet series is concerned, the same conclusion stands for any particular series studied, but not for all series treated together. Namely, series have to be treated separately if better precision is necessary, or divided into four groups (np n d, ns n pandnf n d together; nd n p, nd n fandnp n s) with less precision. Finally, the dependences of the obtained Stark widths on upperlevel ionization potential can be used for prediction of Stark width values, as is done for 12 magnesium spectral lines belonging to the same series that are studied here but have not been measured or theoretically calculated so far. It is expected that the accuracy of the data obtained using the procedure described is comparable with the accuracy of the data used in the verification of Stark width dependences on upper-level ionization potential, taking into

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