Free Access
Issue
A&A
Volume 547, November 2012
Article Number A4
Number of page(s) 6
Section Atomic, molecular, and nuclear data
DOI https://doi.org/10.1051/0004-6361/201118418
Published online 19 October 2012

© ESO, 2012

1. Introduction

Generally, radiation damping of an excited atom leads to broadening of spectral lines. In dense plasmas, radiation damping stems from the interactions of an emitting atom or ion with perturbing plasma constituents. Among these interactions, the electron colliding with ions can lead to excitations and broadening of ionic spectral lines. The electron-impact broadening is a major contributor to radiative acceleration in stellar interiors and is of particular importance as a diagnostic tool in high-temperature plasmas. Not only is it important to precision spectroscopy, but also reveals dependence on major plasma characteristics, such as density and temperature. In the case of plasma diagnostics, the careful analysis of line profiles is a powerful technique for studying atomic and molecular interactions. The knowledge of electron-impact broadening is paramount for probing matter under extreme physical conditions such as in stellar atmospheres. In stellar interiors, electron-impact broadening contributes significantly to radiative acceleration, because of high electron density, and is imperative for evolutionary models, atomic abundances, and radiative opacity calculations.

To study electron-impact broadening, several theoretical methods and techniques have been used in the literature (Griem 1974; Dimitrijević & Konjević1980). However, there have been very few quantum mechanical calculations of electron-impact broadening and shift in spectral lines. Recently, the full quantum mechanical calculations for electron impact broadening of the 2s3s−2s3p transitions for the four Be-like ions from N IV to Ne VII have been performed by Elabidi et al. (2008). These calculations are based on the nonrelativistic quantum mechanics and intermediate coupling. In the context of quantum mechanics, electron-impact broadening and shift of spectral lines is studied by two different methods, i.e., calculating the scattering cross sections and generating the K-matrix. In the present work, we adopt the latter approach, defined by calculating K-matrix, which is similar to the method adopted by Seaton (1987).

The main aim of this paper is to perform quantum mechanical calculations of electron-impact broadening of the transition 2s3s ← 2s3p in Be-like ions, i.e., N IV, O V, F VI, and Ne VII. We also report new electron-impact broadening parameters for transitions in Be-like ions. We employ the impact approximation theory to obtain line broadening and shift parameters at various plasma conditions. In our numerical computations, Dirac Atomic R-matrix Codes (DARC) and General Purpose Relativistic Atomic Structure Package (GRASP0-10.10; Norrington 2009) are used to solve (N + 1)-electron colliding systems so that the scattering matrices (reactance matrices) are obtained. The paper is organized as follows. Section 2 gives details of the methods used to calculate electron-impact broadening parameters. It also outlines the impact approximation and scattering matrix theory. Section 3 explains our strategy for constructing the atomic structure of each ion and selection of configurations. The comparison between experimental and theoretical data, accuracy of our results, final selected data of spectral line widths, the content and organization of the delivered tables are described in Sect. 4. Our conclusions are given in Sect. 5.

2. Theory and methods

2.1. Impact approximation theory

In dense plasmas, spectral line broadening arises from perturbations by the constituents of the plasma (both ions and electrons). Ions have large masses than electron and always move much slowly, so in evaluating line broadening parameters it is often a good approximation to treat them as being stationary. The effect of stationary ions is equal to an average ion microfield, introduced by Hooper (1966). This microfield perturbs the radiating ions by Stark effect and leads to spectral line broadening, known as Stark broadening. Similarly, the high-speed electrons also perturb the emitting ions through collisions, and cause the interruption of spontaneous emission. This interruption will broaden the widths of emission lines, and is known as electron-impact broadening. The radiative transition of spontaneous emission between upper (Xb) and lower state (Xa) of radiating ion is written as: (1)The impact approximation theory was first formulated by Baranger (1958a−c) and later reviewed by Griem (1974). The theory considers a single binary collision between the perturber and the perturbed ion. During the time of electron collision with a radiating ion, the interaction between perturbing ions is ignored, and at a time each electron interacts only with one perturbed ion. Within impact approximation and the isolated line approximation, the line profile is Lorentzian, i.e., (2)Where ω0 is the central line frequency, d is the shift, and 2w is the spectral line full width at half maximum. The width and shift of a spectral line are linearly proportional to electron density Ne and depend on the scattering matrix S for binary collisions. The final expression relating the shift and width of a spectral line could be obtained by averaging over Maxwell distribution of electron velocities. A simplified and compact expression has independently also been derived by Elabidi et al. 2004 in intermediate coupling scheme: (3)Where parameter α = 2.8674 × 10-23   eV   cm3. The units of electron density Ne is cm-3. Electron temperature Te and colliding electron energy ε are in Rydberg, and spectral line width w and shift d are in eV. The electron collision strength Ω(ε) in jj coupling is defined by Peach (1996) (if intermediate coupling is considered, Ω(ε) can be found in the works of Elabidi et al. 2004; or Elabidi et al. 2008), i.e., (4)Where two  { ... }  are the “6j” symbols. The upper and lower states of the transition are represented by the total angular momentum states |JiMi ⟩  and |JfMf ⟩  respectively, and is a scattering matrix element corresponding to an colliding electron scattered by the upper state Xb(5)Where ℓ,′ and j,j′ are the orbital and total angular momentum of colliding electrons before and after the collision with the target ion Xb respectively. During the colliding process, the total energy EI = Eb + ε = Ea + ε + ω, total parity Π, total spin angular momentum ST and total angular momentum  of (N + 1)-electron colliding system are conserved. The is the conjugate matrix of a scattering matrix S corresponding to colliding process of the lower state Xa(6)In Eq. (6), the total energy EF = Ea + ε, parity Π, total spin angular momentum ST and total angular momentum  of (N + 1)-electron colliding system are conserved. The above two scattering processes, described in Eqs. (5) and (6) will share the same total spin angular momentum ST and opposite parity Π when only electric dipole transition is considered. This information is useful for the later numerical calculations.

2.2. Scattering matrix

The calculation of scattering matrices SI and SF is necessary to obtain electron collision strength Ω(ε). These two matrices are closely related to collision processes by Eqs. (5) and (6), which are (N + 1)-electron colliding systems. Generally, a colliding system is constructed from N-electron target states Φi = |JiMi ⟩  (possibly including pseudo states) and single particle wavefunctions φij = |jm ⟩  of the continuum electron. A state of such colliding system is well defined by the wavefunction in Berrington (1977) and Norrington (2009) (7)Where ℑ is an antisymmetric operator to account for the electron exchange, j is the continuum basis function index for a particular κ value (κ = − − 1, if κ < 0, else κ = ), and θm is a capture state in which all of the (N + 1)-electrons are bound.

The (N + 1)-electron colliding system, defined by Eqs. (5) or (6), is solved by using DARC to obtain a reactance K-matrix which is real symmetric matrix. With K-matrix, scattering matrix S is defined as: (8)From Eq. (8), S is an unitary matrix, and its real part and imaginary part are written out as These two expressions are directly used in our numerical calculations.

3. Numerical calculations in detail

To obtain electron collision strength Ω(ε), we solved the colliding system which consists of an N-electron target ion and a colliding electron. The states of an N-electron target in DARC calculations are prepared first by a fully relativistic atomic structure code called GRASP0-10.10 as explained in the next section.

3.1. Atomic structure

In our calculations, the atomic structure data of four Be-like ions from N IV to Ne VII are obtained by using the multi-configuration Dirac-Fock (MCDF) atomic structure program GRASP0-10.10 developed by Dyall et al. (1989) and Parpia et al. (1996) included in DARC. The original GRASP0-10.10 have been partly improved by us according to our previous works (Duan et al. 2008; Bari et al. 2011) so that all the following atomic structure data can be calculated by extended average-level (EAL) mode. To calculate the atomic structure of the four Be-like ions, i.e., N IV, O V, F VI and Ne VII, we have selected the following electronic configurations: 1s22s2, 1s22s2p, 1s22s3s, 1s22s3p, 1s22s3d, 1s22s4s, 1s22s4p, 1s22s5s, and 1s22s5p. With the selection of these configurations, it is easy to obtain the electronic orbitals, the energy levels and states of target ions. These atomic data are incorporated as input in solving (N + 1)-electron colliding system by DARC.

3.2. From K to S

With the atomic structure data of target ions, the (N + 1)-electron colliding systems of Eqs. (5) and (6) are solved by DARC, and their K matrices and quantum numbers of angular momentum are also obtained. Then K matrix is used to calculate the real and imaginary parts of scattering matrix S with Eqs. (9). The transformation from K to S is performed by a code developed by us.

4. Numerical results and discussion

In this paper, the electron-impact broadening widths of the transition arising from the 2s3s ← 2s3p configuration in Be-like ions N IV, O V, F VI and Ne VII are calculated. In these calculations, we have used the GRASP0-10.10 package included in DARC to provide the fine-structure wavefunctions, energy levels, wavelengths, and transition rates. We obtained the electron impact collision strengths by DARC. For those configurations, each target ion has 27 bound states. All of these target states are selected as target basis in our calculations. To generate and construct the (N + 1)-electron colliding systems defined in Eqs. (5) and (6), two parameters of the colliding electron are defined by specific boundary limits i.e., the quantum number of orbital angular momentum is  ≤ 10 and the total number of continuum basis function for a given κ is 14. The electron collision strength Ω(ε) defined in Eq. (4) is a function of colliding electron energy ε and is computed in an increasing energy sequence with an energy step of Δε = 0.02 Rydberg by the same procedure. To realize and confirm our numerical calculations, it is necessary for us to study the convergence of collision strength Ω(ε) with both the orbital angular momentum  and the colliding electron energy ε. In Fig. 1 the maximum of the colliding electron in Eq. (4) is limited by 10 or 15, and Ω(ε) are present as a function ε in transition in N IV ion. It can be seen that Ω(ε) converges with the orbital angular momentum  if  is large enough, and that it also decreases with the colliding electron energy ε. Thus the electron impact broadening in Eq. (3) is numerically calculated by the trapezoidal integration technique. The full width at half maximum (FWHM W = 2w) for transitions 2s3s ← 2s3p for various electron temperature Te and electron density Ne are given in Tables 17. The full quantum mechanical calculations of line widths w2 and its simplified calculations w1 by Elabidi et al. (2008) are also listed in Tables 17 for comparison purposes. These w2 and w1 were calculated in an intermediate coupling framework by including the fine-structure effects and relativistic corrections resulting from breakdown of the LS coupling approximation for the target ions. A comparison between these calculations and experimental results has proven to be a good agreement for the first time for highly charged ions. The line widths wc calculated by Ralchenko et al. (2001) with convergent close-coupling methods are also presented. These authors find (i) that calculated widths are generally smaller than both those measured and most of theoretical line widths from semiclassical calculations; and (ii) that the deviation between experiment and theory monotonically increases along the isoelectronic sequence, reaching a factor of 2 for Ne VII.

thumbnail Fig. 1

Real part of the collision strength as a function of the colliding electron energy for two values of the maximum angular momentum lmax. Squares: lmax = 10. Circles: lmax = 15.

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Table 1

Electron-impact widths W (FWHM) for the N IV transitions at various electron temperatures Te and electron densities Ne.

Table 2

Electron-impact widths W (FWHM) for 2s3s   1S1 ← 2s3p   1P1 transitions of N IV, where the line widths are listed as a function of electron temperature Te under electron density Ne = 1018   cm-3.

Table 3

Electron-impact widths W (FWHM) for the O V 2s3s   3S ← 2s3p   3P transitions.

Table 4

Electron-impact widths W (FWHM) for the O V 2s3s   1S ← 2s3p   1P transitions.

We performed calculations for each ion for the temperature values for which the experimental data are available. Principally, all the corresponding experimental results cited in the present work have been published by two groups. The first group (Blagojević et al. 1999, 2000) used a low-pressure pulsed arc and the second group (Wrubel et al. 1996, 1998) used a gas-liner pinch. The accuracy of the experimental line widths is generally given by Konjević et al. (2002) except for the few cases where other references have been mentioned. For 2s3s   1S ← 2s3p   1P transitions in F VI ions, we compared our results with w1 of Elabidi et al. (2008) as there are no experimental results available yet.

To assess the quality of our calculations, we performed systematic comparisons with earlier experimental and theoretical data. Our results in Tables 17 agree well with both the experimental results and the previous theoretical calculations. Our results for the triplet transitions of N IV, for both the singlet and triplet transitions of O V, and singlet transitions of Ne VII are found to be in excellent agreement with the corresponding experimental results. However, the line widths for triplet transitions of Ne VII revealed a significant difference between our calculations and the corresponding measurements (see Table 6). Our calculated width of the transition line at vacuum wavelength 1992.06 Å is about 0.308 Å, while the corresponding measurement is about 0.45 Å and the difference amounts to 32%. We summarize our results by examining them in the following section by presenting line widths W = 2w (FWHM) of singlet and triplet transitions 2s3s ← 2s3p in four Be-like ions from N IV to Ne VII:

  1. Line widths of N IV. In Tables 1and 2, we present our results forthe line widths of singlet and triplet transitions2s3s ← 2s3p of N IV ion for various electron temperatures and densities along with the experimental results (Blagojević et al. 1999, 2000; Wrubel et al. 1996, 1998). The previous theoretical calculations wc of Ralchenko et al. 2001), the full quantum mechanical calculations w2 and its simplified calculations w1 of Elabidi et al. (2008) are also listed in Tables 1 and 2.

    Table 5

    Electron-impact widths W (FWHM) for the F VI 2s3s   1S ← 2s3p   1P transitions listed as a function of electron temperature Te under electron density Ne = 1018   cm-3.

    The experimental measurements wm give singlet and triplet transitions 2s3s ← 2s3p line widths of N IV ions at different electron temperatures. The measurements by Milosavljević & Djenize (1998) were carried out for electron temperature Te = 5.4 × 104 K, while Blagojević et al. (1999) considered some lower temperatures, and Wrubel et al. (1998) measured at higher temperatures. From Table 1, our results are in good agreement with the experimental values. In detail, most of the other spectral line widths are in good agreement with wm within 5%. The difference for the majority of the spectral lines amounts to 10% and the most significant difference reaches to 18%. Additionally, there are a few cases where the difference between our results and existing experimental values is large. For example, in the transition under plasma condition Te = 9.63 × 104 K and Ne = 8.1 × 1017   cm-3, the difference in the line width reaches to 18%. Similarly, the transitions and in Table 1 have two and four exceptional cases, respectively, which our results show as large deviations from experimental values. For singlet transitions listed in Table 2, it can be seen that the discrepancies between W and two experimental results wm cited by Bochum Group (Wrubel et al. 1996, 1998) are within 33%. Our results are less than experimental wm in contrast to w1. When compared with wm, our results are better than w2, but worse than w1. Overall, our results W for triplet transitions are always better than those of singlet transitions.

  2. Line widths of O V. In Tables 3 and 4, we compare our theoretical calculations of line widths for the singlet and triplet transitions 2s3s ← 2s3p of O V ion for various electron temperatures and densities. We also note that the measured widths are determined with an error of 23%: the error of the temperature determination is 18−27% and 15% for the density determination (Elabidi et al. 2008).

    Table 6

    Electron-impact widths W (FWHM) for the Ne VII 2s3s   3S ← 2s3p   3P transitions under electron density of Ne = 3 × 1018   cm-3.

    Table 7

    Electron-impact widths W (FWHM) for the Ne VII 2s3s   1S ← 2s3p   1P transitions under electron density of Ne = 3.5 × 1018   cm-3 and electron temperature Te = 2.205 × 105   K.

    In triplet transitions in O V, our results are consistent with w1 and agree with each other, but w2 are closer to the experimental values when compared to our calculations. However, in singlet transitions, w1 is the best in two first lines of the table, and our results are the best in two last lines, while w2 is the worst. Most of our line width results are within an error of 15%, and the maximum error is about 25% in comparison to the experimental results.

  3. Line widths of F VI. In Table 5, we list our results W as a function of electron temperature under an electron density Ne = 1018   cm-3. There are no experimental results available for F VI  line widths. Only the quantum mechanical calculations w1 of Elabidi et al. (2008) are listed in Table 5 for comparison purposes. Our calculations show large differences from w1 results. Regarding this discrepancy between the quantum mechanical calculation w1 and our theoretical calculations, the reader may wonder about the cause of such a difference. At present, we cannot understand this discrepancy. The reasons of this discrepancy need to be investigated both experimentally and theoretically in future.

  4. Line widths of Ne VII. In Table 6, we report our results for the triplet transitions of Ne VII along with the experimental line widths measured by the Bochum group (Wrubel et al. 1996, 1998) at electron density Ne = 3 × 1018   cm-3 and electron temperature Te = 2.38 × 105 K. A comparison between our results and the measurements yields a maximum difference of 32%. Our results are in good agreement with the theoretical calculations w1. In Table 7, we present the line widths for singlet transition of Ne VII ion at electron density Ne = 3.5 × 1018   cm-3 and electron temperature Te = 2.205 × 105 K. A comparison between W and wm (Wrubel et al. 1998) shows that the difference is within 22%. It indicates that our results W for line widths of Ne VII are in good agreement with wm and are more accurate than w1 but less accurate than w2.

5. Conclusions

In the present work, a new set of electron-impact broadening of the singlet and triplet transition 2s3s ← 2s3p in four Be-like ions from N IV to Ne VII have been obtained by solving the (N + 1)-electron colliding systems of Be-like ions with DARC on the basis of relativistic quantum mechanical approach. Spectral line widths of triplet transitions of N IV ion are in best agreement with experimental values. A maximum discrepancy of 32% appears between W and wm for triplet transitions of Ne VII ion at electron density Ne = 3 × 1018   cm-3 and electron temperature Te = 2.38 × 105 K, which can be considered as tolerable agreement with respect to the accuracy of the experiment. Overall, one can conclude that our results for all defined temperatures agree with the experimental values within 10%, 20%, and 30% for N IV, O V, and Ne VII respectively. For singlet transitions in F VI ion (see Table 5), our results are quite different from w1, which needs clarification in future studies. Our calculations further support the argument of good agreement between quantum mechanical calculations and experiments for line widths of  △ n = 0 transitions in highly charged ions in accordance with the conclusion of Elabidi et al. (2008). Our present calculations will be used for plasma diagnostics.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11275029) and the Foundation for Development of Science and Technology of Chinese Academy of Engineering Physics (Grant Nos. 2011A0102007 and 2009A0102006).

References

All Tables

Table 1

Electron-impact widths W (FWHM) for the N IV transitions at various electron temperatures Te and electron densities Ne.

Table 2

Electron-impact widths W (FWHM) for 2s3s   1S1 ← 2s3p   1P1 transitions of N IV, where the line widths are listed as a function of electron temperature Te under electron density Ne = 1018   cm-3.

Table 3

Electron-impact widths W (FWHM) for the O V 2s3s   3S ← 2s3p   3P transitions.

Table 4

Electron-impact widths W (FWHM) for the O V 2s3s   1S ← 2s3p   1P transitions.

Table 5

Electron-impact widths W (FWHM) for the F VI 2s3s   1S ← 2s3p   1P transitions listed as a function of electron temperature Te under electron density Ne = 1018   cm-3.

Table 6

Electron-impact widths W (FWHM) for the Ne VII 2s3s   3S ← 2s3p   3P transitions under electron density of Ne = 3 × 1018   cm-3.

Table 7

Electron-impact widths W (FWHM) for the Ne VII 2s3s   1S ← 2s3p   1P transitions under electron density of Ne = 3.5 × 1018   cm-3 and electron temperature Te = 2.205 × 105   K.

All Figures

thumbnail Fig. 1

Real part of the collision strength as a function of the colliding electron energy for two values of the maximum angular momentum lmax. Squares: lmax = 10. Circles: lmax = 15.

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