Open M-shell Fe and Ni LTE opacity calculations with the code HULLAC-v9

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High Energy Density Physics journal homepage: www.elsevier.com/locate/hedp

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Open M-shell Fe and Ni LTE opacity calculations with the code HULLAC-v9

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D. Gilles a, *, M. Busquet b, M. Klapisch c, F. Gilleron d, J.-C. Pain d a

CEA/IRFU/SAp, F-91191 Gif-sur-Yvette, France Research Support Instruments, Lanham, MD 20706, USA c Berkeley Research Associates, Beltsville, MD 21042, USA d CEA, DAM, DIF, F-91297 Arpajon, France b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 25 April 2015 Accepted 26 April 2015 Available online xxx

It has been shown that a careful evaluation of the Dn ¼ 0 M-shell in Fe and Ni is necessary to obtain consistent stellar model predictions. At astrophysical conditions of Te ¼ 15.3 eV, ne ¼ 1013 1022cm3, the Dn ¼ 0 M-shell and absorption transitions among M to N and M to O shells arising from the ions Fe III through Fe XV, have been computed in the using detailed level accounting (DLA) with configuration interaction (CI), assuming LTE. The importance of extensive CI computations is shown by systematically increasing the size of the CI basis until convergence. Iron opacity results, including M-shell and M to upper shells transitions up to Dn ¼ 2, are presented and the influence of the number of atomic levels is discussed. These very large computations ee.g. ~7.3  105 levels or ~5700 relativistic configurations in Fe VIIe were performed using the flexibility and the efficiency of an updated version of the HULLAC-v9.5 code. Results for Ni are also reported. Some of our results are compared with those of other codes and databases, STA, SCO-RCG and OPCD3.3. © 2015 Elsevier B.V. All rights reserved.

Keywords: Opacity spectra Stellar plasma Nickel Iron HULLAC-v9 code SCO-RCG code

1. Introduction It is known for more than two decades that the iron and nickel intense Dn ¼ 0 M-shell and Dn ¼ 1 M to N shell transitions must be included in stellar opacity spectra to obtain consistent stellar model predictions [1e11]. The present studies are motivated by the improvement of the iron-group elements opacity which contributes to an opacity bump around T ¼ 15.3 eV. This feature is responsible for the pulsations induced by the Κ-mechanisms in Btype stars [12e14]. It is produced by intense peaks in the spectral opacity in the region of the maximum of the Rosseland weighting function. It has effects not only on the shape of the total opacity spectra but also on the Rosseland mean KR, a crucial ingredient in the LTE diffusion approximation used to model the transfer of radiation in stellar interiors. Such spectral opacity calculations present a challenge, not only due to the billions of transitions in the spectrum which have to be treated with detailed atomic structure codes, but also because these transitions are very sensitive to effects of Configuration Interaction (CI) [15].

* Corresponding author. Tel.: þ33 1 69 08 56 07; fax: þ33 1 69 08 65 77. E-mail address: [email protected] (D. Gilles).

Many statistical and detailed opacity codes have been developed and improved recently [16e23], see also reviews in Refs. [1,5,24e26], but only a few, namely OPAL [1] and OP [4], have produced tables for astrophysicists. Moreover, spectral monochromatic opacities dedicated to the calculation of radiative acceleration of stellar envelope are currently found only in OPCD3.3 opacities database. Recently, discrepancies between OP and OPAL tables have stimulated many discussions in the astrophysical community, because significant differences in the understanding of Slow Pulsating B and Cepheid stars were observed, depending of the choice of the opacity table. Moreover noticeable differences in the spectra between OP and other modern codes have been mentioned in recent works, where a few comparisons for iron and nickel were performed for temperatures between 15 and 40 eV and electronic densities around 3  1020cm3 [20,25,26]. The conditions of these comparisons were typical of experiments dedicated to improve our understanding of the iron and nickel opacities used in stellar envelope codes. As OP includes some CI, in the R-matrix formalism, one possible interpretation of the difference was the CI treatment. This was confirmed by HULLAC-v9 calculations on the n ¼ 3en ¼ 3 transitions [26] and more recently by ATOMIC calculations [16,17]. Other differences are still unexplained.

http://dx.doi.org/10.1016/j.hedp.2015.04.008 1574-1818/© 2015 Elsevier B.V. All rights reserved.

Please cite this article in press as: D. Gilles, et al., Open M-shell Fe and Ni LTE opacity calculations with the code HULLAC-v9, High Energy Density Physics (2015), http://dx.doi.org/10.1016/j.hedp.2015.04.008

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In a previous paper we used the code HULLAC-v9 [20e22], in which extensive CI treatment is possible, in order to investigate the influence of this effect on the opacity spectra. Results were presented for the n ¼ 3en ¼ 3 transitions for which all transitions can be calculated in a CI approach. Some effort to include part of the n ¼ 3en ¼ 4 in the spectra was also made, but only incomplete spectra were obtained due to the dramatic increase in the computation time and memory allocation. Note that although temperatures are similar, the densities found in the laboratory experiment [14,24] are about a factor of 1000 larger than those found in the stellar envelope. Consequently highly excited M-shell boundebound (beb) contributions, very sensitive to density, become more important and their inclusion in the calculation increases the difficulties. In the present study we will investigate the effect of a more complete CI treatment in less dense stellar envelope conditions using an improved version of HULLAC-v9 and we shall concentrate our illustrations to the temperature T ¼ 15.3 eV. The new capabilities afforded by the recent computer generation are also a critical point of such applications. Further, the substantial improvements in the algorithms and memory allocation in HULLAC-v9 are also central to the calculations presented here. In Appendix A we review the theoretical framework for CI with relativistic configurations (RCI) or Full CI, and introduce the notations of CIinNRC for CI in one Non-Relativistic Configuration, or RCM (relativistic configuration mixing) or “limited CI”. The way to implement these various CI treatments in HULLAC-v9 is described in Section 2. We then compare in Section 3, the n ¼ 3en ¼ 3 opacity calculations for Fe in different CI modes. Similar results and conclusions have been obtained for nickel and will be illustrated in the last section. In a CI treatment both the CI effects and the number of configurations play a role, but the difficulty increases with the value of the principal quantum number n of the upper shell of the transitions. Fortunately opacity and CI effects decrease with this upper shell n value. In this work, starting with Dn ¼ 0 M-shell contributions, we shall study the convergence of the ”full CI” opacity results by including more and more M to upper shell contributions in the calculation. As an example, we present in Fig. 1 Fe X boundebound (beb) calculations, at T ¼ 15.3 eV, for 3 increasing sets of levels, Dn ¼ 0 M-shell, Dn  1 and Dn  2. In this spectrum only one ion charge state is included. For illustration we add the bound-free (bef) for an electronic density ne of 3.2  1017 cm3. The overlay of the 3 calculations illustrates how the spectral beb opacity is formed and also how the inclusion of higher-n states in the Dn  2 set using a full CI treatment can modify the shape of the largest peak and shoulders of the first M-shell transitions, which is in the region of the maximum of the Rosseland weighting function. This is the goal of the present paper. The size of the calculations increases with the number of Mshell electrons to be treated. We shall first present, in Section 4, a complete study the for Fe XIII ion with 4 M-shell electrons. In this example it has been possible to explore the effect of adding more and more transitions in the full CI treatment, including also several electron jumps in the same transition. Such calculations take between a few minutes to a few hours on a single core computer, and present no difficulty. The same studies have been repeated for the different Fe XII to Fe VIII ions, with a dramatic increase in time and active memory needed in the calculation. Nevertheless it has been possible to evaluate the importance of (full) CI versus CI limited to one non-relativistic configuration treatment (RCM or CIinNRC mode in HULLAC-v9), on the first M-shell transitions for the different ions. This is illustrated in Section 5. In Section 6 we present a comparison between HULLAC-v9 with RCM and SCO-RCG iron opacity results for a typical stellar envelope condition,

Fig. 1. Fe X boundebound (beb) opacities, at T ¼ 15.3 eV, for 3 increasing set of transitions (Dn ¼ 0, Dn  1 and Dn  2). The Dn ¼ 0 bef and fef contributions at ne ¼ 3.2  1017 cm3 are added.

T ¼ 15.3 eV and ne ¼ 3.2  1017 cm3. Our calculations have been obtained using the energy and transition probabilities of a database with Dn  2 performed in the RCM approximation. SCO-RCG is a LTE hybrid opacity code, including also RCM treatment, that combines the statistical Super-Transition-Array approach and finestructure calculations and it is able to generate opacity spectra taking into account a large set of excited levels. All results will be compared to more complete CI calculations. In Section 7, we review comparisons between HULLAC-V9 (RCI and RCM for the n ¼ 3en ¼ 3 transitions), OPCD and the recently updated version of STA [18] of iron and nickel opacities, for a large range of electronic densities. The RCM mode is approximately implemented in STA [27e29], which includes within a statistical approach numerous transitions and excited levels in the calculation. The importance of CI and the influence of the number of excited levels are discussed. Some typical density domains, where strange behavior of OPCD spectral results has been found, will also be discussed. In the Appendix B, we comment on the energy and cross section generation data files in HULLAC-v9. The notion of user defined “group of configurations” is addressed and its interest for the case of large-scale calculations is illustrated. 2. CI options in HULLAC-v9 code For our LTE applications we have used a recently improved version of HULLAC-v9 [18,20e22] that consists of several modules. The first module ANGLAR reads the input data (name of the atom, a list of non-relativistic configurations and a list of processes) structured by keywords in free format. ANGLAR not only constructs the list of relativistic configurations, fine structure levels with their quantum numbers and the list of integrals necessary for the Hamiltonian and the requested processes, but also gives directives to the subsequent programs RELAC and PHOTION. Populations and spectral opacities are obtained in the final module MYCRM. The details of the theory and of the different modules of the code have been extensively described in many papers [19e22]. Some supplementary information is given in Appendix B. One feature of the input file relevant for the present work is the definition of groups of configurations (GroC). The configurations are introduced in a very compact notation reminiscent of superconfigurations. They can be separated by the keyword GROUP,

Please cite this article in press as: D. Gilles, et al., Open M-shell Fe and Ni LTE opacity calculations with the code HULLAC-v9, High Energy Density Physics (2015), http://dx.doi.org/10.1016/j.hedp.2015.04.008

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followed by a name. This defines the subsets of RC for which the (J, p) matrices of the DLA will be generated in ANGLAR, and further computed and diagonalized in RELAC. The above feature is used to limit RCI into smaller subsets of levels -giving "partial RCI". Another set of options limits the generation of the matrices themselves. The keyword CIinNRC generates coefficients of the Hamiltonian only between RC pertaining to the same NRC parent, whereas the default option CI lifts this restriction, and NOCI does not generate any CI matrix element. These options can differ from one GroC to another. Thus complete “full RCI” calculation, mentioned in appendix (A.4), means that a list of all possible configurations is given in the input file, all of them being included in one user defined GroC. This may give rise to large-scale calculations that are hardly tractable, as this implies the inclusion of all levels in one large matrix and allows for one or more "electron jumps". One way out of this difficulty is to reduce the number of configurations included in this single group and to check the convergence of the results by adding more and more configurations to the same group. Depending of the number of active electrons, here M-shell calculations, with n ¼ 1 and n ¼ 2 shells closed, the computation time, active memory (ram and swap) and storage may increase dramatically. We shall denote these calculations as “complete Full RCI” if results are converged or only “Full RCI” for a limited set of transitions (cf. results of Sections 4 and 5, for example). Complete Full RCI over all configurations forming the Dn ¼ 0 transitions is actually possible and is illustrated (Sections 3 and 7). Another way to limit the size of the matrices to be diagonalized is to split the set of configurations in two or more groups. In this case we shall speak of “limited RCI” or “partial RCI” calculations. Depending of the richness of the set, results can be slightly modified or not. An example is discussed in Appendix B. It is important to note that opacity spectra are sensitive to both the CI mode and the number of excited atomic levels selected [20,25,26]. 3. Including only Dn ¼ 0, n ¼ 3 transitions in Fe opacity As CI effects transfer oscillator strength from optically allowed transitions to forbidden transitions, it leads to numerous forbidden lines appearing in the spectra, including lines between levels differing by more than one electron jump. In Fig. 2 we compare RCI (“full CI”, solid line) to RCM (“CIinNRC”, dashed line) iron beb opacity calculations over the Dn ¼ 0 transitions, at T ¼ 15.3 eV and ne ¼ 1016 cm3 (mean charge ¼ 9.4), a condition relevant to stellar envelopes in astrophysics. All Dn ¼ 0, n ¼ 3 transitions of Fe III to Fe XV contributions have been included in the calculation. The only difference between the two calculations is the CI option used in HULLAC-v9 code, CI or CIinNRC. Also plotted is the corresponding bound-free (b-f) contribution (line with dots). We apply a Gaussian broadening (FWHM ¼ 0.1 eV) in order to smooth the spectra and make the difference clearly visible. In the RCI calculation the second, third and fourth bumps are due to double, triple and quadruple electronic jumps not allowed when only the CIinNRC mode is used. These spectral features are dominated by the b-f contributions at large energies (hn > 170 eV), see Fig. 2. Thus, CI plays a role in different parts of the n ¼ 3en ¼ 3 M-shell opacity spectrum. Globally the main CI effect results in a shift and an increase of the width of the n ¼ 3en ¼ 3 transition highest peak compared to the CIinNRC spectrum. However, although beb contributions to the opacity at hn > 100 eV are less intense and rapidly negligible compared to b-f contributions, these 2- and moreelectron jump transitions will interplay with M to upper shell contributions, Dn ¼ 1, 2, 3. The influence of CI on the b-f is negligible.

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Fig. 2. Iron beb HULLACev9 spectral dependent opacity at T ¼ 15.3 eV, ne ¼ 1016 cm3 (Gaussian broadening of FWHM ¼ 0.1 eV): full RCI (solid line) and RCM (dashed line). As the corresponding b-f contributions are very similar only one result is plotted (line with dots).

The beb and b-f contributions increase with density. Fig. 3 illustrates the influence of the electronic density on the full CI Dn ¼ 0 total opacity spectra for the same temperature of T ¼ 15.3 eV. When ne is increased by a factor 10 between successive results, from ne ¼ 3.2  1013 to 3.2  1024 cm3, the corresponding mean iron ionizations vary from ¼ 13 (3.2  1013) to ¼ 2 (3.2  1024 cm3). Each calculation includes all ion charge states from Fe III to Fe XV and all contributions (beb, bef, fef). Scattering is negligible in our applications. We can see that on the total opacity only the second 2-electron jump feature of the CI calculations remains visible, between 100 and 200 eV and only for densities smaller than 3  1017 cm3. This contribution disappears completely at high density when boundfree (bef) contributions become important. CI effects are concentrated for all densities on the two most intense peak contributions (hn < 100 eV). To summarize the CI effect plays an important role on the main peaks and the wings of the n ¼ 3en ¼ 3 M-shell opacity spectrum. The effect is not the same for the various transitions but the global result is a shift and an increase of the width, for all densities.

Fig. 3. Iron HULLACev9 Dn ¼ 0 full RCI spectral dependent opacity spectra at T ¼ 15.3 eV and for electronic densities ne varying by a factor of 10 between 3.2  1013 and 3.2  1024 cm3 (Gaussian broadening of FWHM ¼ 1 eV). Corresponding mean ion charge are respectively: 12.98, 12.12, 11.23, 10.32, 9.39, 8.46, 7.51, 6.5, 5.39, 4.16, 2.83, 2.15 and 2.02.

Please cite this article in press as: D. Gilles, et al., Open M-shell Fe and Ni LTE opacity calculations with the code HULLAC-v9, High Energy Density Physics (2015), http://dx.doi.org/10.1016/j.hedp.2015.04.008

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The less intense beb opacity contributions at hn > 100 eV are rapidly negligible, compared to bef contributions, for densities higher than 3.2  1018. Note that such large and complete calculations are tractable only for the n ¼ 3 to n ¼ 3 calculations or in the case of ions with a small number of M-shell electrons e.g., Fe XIII, and this is found true for more excited transitions. We have validated by these examples the possibility to neglect multiple electron jumps in much larger calculations. 4. Fe XIII: complete full RCI M-shell opacity calculations In all the calculations presented in this paper n ¼ 1 and n ¼ 2 shells are assumed to be closed. Thus, Fe XIII has 4 (¼26-12-10) open active M-shell electrons. The series of spectra listed below and plotted in Fig. 4(a)e(c) have been calculated for increasing sets of Dnmax transitions, with,

Dnmax ¼ all Dn  max from Dn0 ¼ 0 (n ¼ 3 / n ¼ 3) to Dn4. Here all the configurations are in one GroC and the CI option is used to obtain “full CI” opacity spectra for this set of levels. When max is increased, the highenergy range of the spectrum is enriched. The size of the matrices to be diagonalized is obviously increased. For each set of Dnmax levels, a number of boundebound transitions with multi-electron jump are allowed with full CI. However, very quickly the number of transitions becomes very large and the

calculation often becomes impossible in practice. The different opacity calculations correspond to:  Dn0 (n ¼ 3 / n ¼ 3) transitions. They include automatically all possible electron jumps (“3L” on the plot).  Dn1: Dn  1 transitions, by addition of the Dn ¼ 1 transitions. In this case, with 4 active electrons, we succeed to calculate up to 3, thus all, possible electron jumps (“34L-3j”).  Dn2: Dn  2 transitions. In this case we succeed to calculate only double electron jumps (“35L-2j”).  Dn3: Dn  3 transitions. Only one electron jump was tractable (denoted by “36L-1j”).  Dn4: Dn  4 transitions excluding 4f transitions (denoted by “37Lspd”). Also one electron jump. Complete sets with Dn2 or above require very large active memory (up to 200 Gb) or many computer cores. Calculation of the very numerous contributions, corresponding to more than one electron jump, becomes prohibitive. Fortunately their intensity decreases very quickly and it is possible to avoid the huge calculations. This is illustrated in Fig. 4(a) where the iron opacity has been plotted for the 3 first sets of Dnmax values and for corresponding calculations including more than one electron jumps. We have plotted the “best” calculation for each Dnmax set of transitions to illustrate the convergence with Dnmax of the Fe XIII beb opacity to the exact CI result. Moreover, bound-free (an example is given at ne ¼ 3.2  1013 cm3, thick black line with dots)

Fig. 4. (a). Convergence of the Fe XIII full RCI beb spectral opacity at T ¼ 15.3 eV; Dn1, (dashed), Dn2, (dot-dashed), Dn3, (thin solid), Dn4, (thick solid). Also plotted: an example of b-f contribution (ne ¼ 3.2  1013 cm3,, thick line with dots) and n ¼ 3 / n ¼ 3 contributions (3 dotted-dashed). We apply a Gaussian broadening of FWHM ¼ 1eV. (b): Fe XIII beb opacities at T ¼ 15.3 eV: Full RCI (Dn2, dot-dashed) and (Dn4, solid) versus RCM (CIinNRC, Dn2, dotted). We apply a Gaussian broadening of FWHM ¼ 1eV. (c): Zoom on Fig. 4(b). The Dn ¼ 0 calculation has been also plotted (dots). We apply a Gaussian broadening of FWHM ¼ 1eV.

Please cite this article in press as: D. Gilles, et al., Open M-shell Fe and Ni LTE opacity calculations with the code HULLAC-v9, High Energy Density Physics (2015), http://dx.doi.org/10.1016/j.hedp.2015.04.008

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and beb contributions are of the same order of magnitude in the energy range of these contributions. In any case, the changes in the shape of the intense first M-shell peaks by inclusion of higher Dnmax transitions, are very small, even with one electron jump. So we can reasonably assume converged full RCI for the last calculation, in the sense that inclusion of transitions to more excited levels than Dn4 will not modify the beb spectrum in the spectral range hn/Te < 20. We compared in Fig. 4(b), for the Fe XIII contribution, beb opacities at 15.3 eV in the modes full RCI (Dn_2, dot-dashed line) and (Dn_4, solid line) versus RCM (CIinNRC, Dn_2, dotted line). The difference between RCM and full RCI calculations, is much more important than very small deviations observed between successive Dnmax calculations (Fig. 4(a) and zoom in Fig. 4(c)). 5. RCI and RCM (CIinNRC) in Fe M-shell boundebound opacities We have compared RCI (solid line) to RCM (CIinNRC, dotted line)

Dn  1 M-shell beb opacity spectra for many Fe ions and many sets of transitions. Results for Fe VIII, Fe X and Fe XIII are reported in Fig. 5. From these comparisons we can conclude that all Fe ions beb opacities are sensitive to RCI treatment. It has been impossible to perform larger sets than (Dn  1) for ion Fe VIII (about 180 days on a one-core computer). Nevertheless we have seen in the preceding section that the shape of the intense peaks will not be modified by the inclusion of higher contributions in the energy range of interest. The CI treatment affects mostly the two main peak transitions (Dn ¼ 0) for ions with a large number of M-shell electrons (Fe VIeFe

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IX). For ions with a small number of M-shell electrons (as Fe XIII) the effect is distributed all along the different Dn contributions. In all cases, opacity spectra are much more sensitive to the CI treatment than to the neglecting of less intense transitions involving jumps of more than one electron (behavior similar to Fig. 4(a)). In Appendix B we show also that it is possible to gain in CPU time and memory allocation of the CI calculations using splitting into well-chosen sub groups of levels. Computation with partial CI are much less expensive and close to full CI but they differ noticeably from CIinNRC results. As we have obtained now exact reference calculations we will take benefit of the possibility to group configurations with respect to the electron jump, for example: 3 (s,p,d) / 4s; 3 (s,p,d) / 4d,…). We have tested using different ions that only a small change in the results were noticed, always much smaller than using RCM (CIinNRC) instead of RCI mode on the first M-shell transitions. This gain is crucial for ions such as Fe VIII for which the full CI calculation, for only Dn ¼ 0, 1 Mshell, can take more than 6 months (on one core of any computer). Comparatively Fe XIII, even for Dn  4 transitions (1 electron jump) never take more than 4 h computing time. 6. Dn ≤ 2Iron and nickel RCM (CIinNRC) HULLAC-v9 opacities We have computed with HULLAC-v9 RCM (CIinNRC) energy and spectra, including up to Dn ¼ 2 M-shell transitions, for Fe and Ni. Although the file sizes are large, e.g., up to 270 Gb for Fe VI to Fe VIII for the beb transitions, these calculations can be performed exactly and they include the most important contributions to the M-shell opacity. With these data we can compute spectra for all temperature and densities of interest. A few examples have been shown

Fig. 5. (a). HULLAC-v9 beb Fe VIII spectral opacity at T ¼ 15.3 eV: RCI (solid) and RCM (dotted). Charge state population ¼ 1. We apply a Gaussian broadening of FWHM ¼ 1eV. (b): same as Fig. 5(a) for Fe X. (c): same as Fig. 5(a) for Fe XIII.

Please cite this article in press as: D. Gilles, et al., Open M-shell Fe and Ni LTE opacity calculations with the code HULLAC-v9, High Energy Density Physics (2015), http://dx.doi.org/10.1016/j.hedp.2015.04.008

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above for 3 Fe ions. The comparison between HULLAC-v9 RCM and RCI results, calculated for the successive Dn transitions, is an indicator of the sensitivity of the different Dn transitions to CI treatment, in the continuation of earlier n ¼ 3 / n ¼ 3 comparisons. Moreover, it is interesting to compare HULLAC-v9 RCM to other code results where RCM approximation is also implemented, like SCO-RCG [23]. SCO-RCG is an LTE hybrid opacity code that combines the statistical Super-Transition-Array approach and finestructure calculations for intense and spectrally broad transition arrays, see Ref. [25] for details. Its authors provided us with the different opacity contributions (beb, bef, fef, scattering) together with the populations of the different ions relative to this calculation. Such detailed comparisons between codes give information on the atomic physics description and on the influence of the number of the excited levels included in opacity calculations. In Fig. 6(a) and (b) we show comparisons between HULLAC-v9 RCM (Dn  2 contributions) and SCO-RCG iron opacity result, for a typical astrophysical condition of interest (Fe, T ¼ 15.3 eV, ne ¼ 3.2  1017 cm3 and ¼ 8.45). The spectrum for these conditions shows many intense spectral lines. For clarity, we have chosen to overlay on SCO-RCG results a 300 points sliding average. HULLAC-v9 opacity has been calculated by adding each charge state contribution calculated separately, for ion Fe VIIIeFe XI with weights taken from the SCO-RCG charge state distribution. Values are displayed on Fig. 6(a) (total bb spectral opacity, with a Gaussian convolution of 0.1 eV). This choice is motivated by the desire to discriminate between the influence of the atomic physic and the influence of the charge state distribution called EOS by some authors. The agreement between SCO-RCG and HULLAC-v9 is quite good over the whole spectral range represented here, corresponding to the beb contributions. Electronic broadening is not included in the present HULLAC-v9 results and thus it is meaningless to compare lines in detail. Nevertheless, as CI effects decrease with increasing Dn, we see that it will be possible to use HULLAC-v9 RCM calculations for large Dn when full CI will become impossible and not necessary. We note that in HULLAC-v9 opacity calculations performed for different groups of levels with different CI options is possible and can be applied to include more and more transitions in future calculations. To show the good agreement also for the Dn ¼ 0 M-shell transitions we have plotted a SCO-RCG spectrum smoothed on 1630 points and compared them to HULLAC-v9 Dn ¼ 0 transition opacity applying a Gaussian convolution of 1 eV (Fig. 6(b)). The asymptotic part of HULLAC-v9 b-f points to the end of the spectral range of the M-shell transitions (around 300 eV).

7. Iron and nickel opacity calculations: comparisons with STA and OPCD results As shown before, intense and broad peak of the Dn ¼ 0 M-shell transitions, which provides a major contribution to the opacity for the stellar envelope conditions and temperatures (around 15 eV), is very sensitive to CI effect. Less intense Dn ¼ 1, 2, 3 transitions contribute only on the foot and the wings of the intense peaks. Therefore, we used the previous n ¼ 3 / n ¼ 3 spectra to perform extensive comparisons with OPCD database results [4]. OPCD results are the only monochromatic opacities available in the literature. They also include partial CI effect, in the context of the RMatrix formalism. We have described some noticeable differences with it in a previous work [24]. The numbers displayed on next figures correspond to the indices (element, temperature and density) of the OPCD3.3 data files. To have an idea of the complete spectra, with all excited transitions, we calculated a new set of opacities on the same grid points as before, with the recently updated version of the STA opacity code. In general STA spectral opacities appear to interpolate smoothly between the average-atom results and the DCA that underlies the unresolved transition array method. The CI effect is taken into account approximately in STA by using the RCM approximation. LTE spectra are built as the sum of the different charge state contributions weighted by the ionic abundances. We first compare the charge state distribution given by the different codes. We have found that HULLAC-v9, OPCD3.3 and STA results are very similar for all temperatures and densities below 1023 cm3, even for very low density. This has been previously shown in Fig. 4 and in Fig. 6 of Ref. [24]. For highest densities, the given by HULLAC-v9 and OPCD3.3 differ from those of STA and of average atom code results [38,39] as expected, because the HULLAC-v9 and OPCD3.3 codes do not include density effects. Values are displayed on the different figures in the next plot (Fig. 7). The OPCD3.3 grids cover a wide range of temperature and densities. We shall not present all the data, but we will focuss on T ¼ 15.3 eV results as before. In Fig. 7 (iron) and in next Fig. 8 (nickel) we plotted, for different electronic densities: OPCD (thin solid line), HULLAC-v9 results RCI (thick solid line) and RCM (dot-dashed line) and STA (dotted line). The goal of these comparisons was to determine for what conditions CI effect must be absolutely taken into account (difference between HULLAC RCI and RCM) and to compare HULLAC-v9 results with OPCD3.3 ones. As we are not discussing spectral line details, we have chosen to smooth OPCD results to clarify the comparisons. The long-dashed curve represents OPCD

Fig. 6. (a). SCO-RCG and SCO-RCG sliding average of spectral opacities compared to HULLAC-v9 RCM Dn  2 calculation at T ¼ 15.3 eV and ne ¼ 3.2  1017 cm3. (b): comparison between SCO-RCG (smoothed to 1 eV) and HULLAC-v9 Dn ¼ 0, with a Gaussian broadening (FWHM ¼ 1 eV).

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Fig. 7. Iron spectral opacities: HULLAC-v9 full RCI (thick solid) and RCM_(CIinNRC, dot-dashed) results compared to OPCD (thin solid), to a 300 points sliding average of OPCD (longdashed) and to STA (dotted) at T ¼ 15.3 eV and 4 densities: ne ¼ 3.2  1013 (a), 3.2  1016 (b), 3.2  1020 (c), 3.2  1022 cm3 (d). We use a Gaussian broadening (FWHM ¼ 1 eV).

smoothed results, a sliding average over 300 points, to be comparable to HULLAC-v9 results, illustrated with a convolution with a Gaussian broadening of 1eV. We recall that the sliding or moving average do not necessary cross the data as the median average. For the low density case of Fig. 7(a) (ne ¼ 3.2  1013 cm3, ¼ 2.9) CI is negligible. HULLAC v9 Dn ¼ 0 transitions are still in agreement with the STA peak and b-f results, with the small differences probably due to the difference in mean ionization. The deviation of

OPCD results is now totally unexplained as OPCD beb and b-f results are shifted to lower densities by a factor of 10. In Fig. 8, we plot similar comparisons for nickel at five densities: (a) ne ¼ 1013, (b) 3.2  1015, (c) 3.2  1017, (d) 3.2  1020, (e) 3.2  1022 cm3. We use linear and logarithmic grids to emphasize small and high spectral energy range of the opacities. HULLAC spectra are convolved with a Gaussian (FWHM ¼ 1 eV). The CI effect is noticeable for the first 3 densities typical of stellar envelopes (a, b, c). For the two highest densities the choice of the configuration interaction treatment has nearly no effect on the HULLAC-v9 spectra. The nickel ionizations are very similar to those of iron. Differences between RCM and RCI occur at densities smaller than for those in the iron case. This can be explained by the fact that, as the mean iron and nickel ionizations are similar for a given density and temperature, the number of active electrons for the most abundant ions is higher for the nickel case than for those in iron. As shown previously STA results show the importance of the remaining M-shell contributions and show good agreement with HULLAC-v9 RCM calculations in the energy range of the intense peaks of the Dn ¼ 0 M-shell transitions. For nickel, OPCD3.3 results show similar behavior but discrepancies with HULLAC-v9 are emphasized, as we can see comparing the thick solid and long-dashed curves. Although the intense Mshell peak can be observed in OPCD, at least below a few 1017 cm3 (Fig. 8(a)e(c)) we note an important shift towards highest energies. For the two highest densities OPCD3.3 nickel spectra appear to be multiplied by a vertical scaling factor, applied to the whole spectrum [20,26]. 8. Conclusion We have investigated the influence of Configuration Interaction effects in iron and nickel M and M to O shell LTE opacities in various

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Fig. 8. Nickel spectral opacities: HULLAC-v9 RCI (thick solid) and RCM_CIinNRC (dot-dashed) results compared to OPCD (thin solid), a sliding average over 300 points of OPCD (longdashed) and STA (dotted) at T ¼ 15.3 eV and 5 densities: ne ¼ 3.2  1013 (a), 3.2  1015 (b), 3.2  1017 (c), 3.2  1020 (d), 3.2  1022 cm3 (e). We use a Gaussian broadening (FWHM ¼ 1 eV).

domains of frequency and density, characteristic of stellar envelope conditions, with the code HULLAC-v9, both in full RCI (CI) and in RCM (CIinNRC) approximation, and we have compared our calculations with the OPCD, STA and SCO-RCG codes results. Our motivation was to improve the iron-group element M-shell opacities. These are well known to be sensitive to Configuration Interaction effects and to a precise atomic description of these transitions. Full RCI represents the case when every level can be mixed with any other level of same (J, p) until convergence. Such calculations are already tractable with the present computer generations. We focused our study around the temperature of the opacity bump, T ¼ 15.3 eV, where significant differences between spectral opacities have been mentioned in previous works. M-shell opacity results depend both on the CI treatment and on the possibility to include a complete atomic description of the excited levels of the ions. This very quickly gives rise to a huge number of atomic M-shell transitions. In this work, we have separately investigated the behavior of the intra-M-shell (Dn ¼ 0,

n ¼ 3) transitions, which are the most sensitive to CI and providing the major contribution to opacity. Then more excited M-shell (Dn  2, n ¼ 3) contributions were added. By diagonalization of the Hamiltonian on larger and larger basis of the Hilbert space of the N-electron wave-functions, convergence of the spectral opacity results in the region of the maximum of the Rosseland weighting function was obtained. We have thus demonstrated the possibility to evaluate the effect of adding more and more transitions in a full CI treatment, including also several electron jumps in the same transition. Full and complete CI results ewith only one large matrix per (J, p) for all levelse have been obtained for iron ions with few M-shell electrons, as illustrated for Fe XIII. We have repeated these calculations for ions with more and more electrons. Results were similar but less and less tractable, with dramatic increase in time and memory needed in the calculation. We have discussed and displayed the influence of the splitting of the group of levels in the Fe X case and demonstrated that it is possible to reduce the time spent in the calculation, with a

Please cite this article in press as: D. Gilles, et al., Open M-shell Fe and Ni LTE opacity calculations with the code HULLAC-v9, High Energy Density Physics (2015), http://dx.doi.org/10.1016/j.hedp.2015.04.008

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very small loss of precision, using well-chosen sub-groups for Dn > 0 transitions, which is also true for other iron ions. Therefore we can reasonably conclude that, in any case, changes in the shape of the intense first M-shell peaks by inclusion of higher Dn as Dn ¼ 3, 4 transitions, are very small, even in full CI treatment. These variations are always smaller than the large differences between full CI (but “incomplete” in the sense that they included all (but only) Dn  1 transitions) and RCM (CIinNRC) observed for all ions in the spectral range of the intense peaks. Good agreement between RCM HULLAC-v9, SCO-RCG and STA and the completeness of the spectral opacity has been illustrated for iron. We have created Dn  2 HULLAC-v9 RCM (CIinNRC) LTE databases of energies and cross sections for iron and nickel, which can be used to compute Mshell spectra for any temperature and density. Pointing out the importance of the Dn ¼ 0 contributions, we have also shown that it is possible to analyze and discuss the noticeable differences found with OPCD frequency-dependent opacity tables, for iron and nickel, using HULLAC-v9 intra M-shell calculations and STA complete spectral opacities. The influence of CI treatment on these major contributions to Rosseland and Planck means will be discussed in a future paper. Acknowledgments We thank T. Blenski and Q. Porcherot for many helpful discussions. We want also to acknowledge Patrick Hennebelle who let us use his computing facilities, operated under the contract FP7IDEAS-ERC which received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013 Grant Agreement No. 306483). Appendix A Perturbation theory for a multi-electron atom in a relativistic framework Since the same relevant atomic concepts may have different names and notations in the literature, e.g., between the Dirac or the €dinger equations, we define what we mean by, and how we Schro apply, perturbation theory within a relativistic framework. A.1. Relativistic Hamiltonian The theory of multi-electron atoms in a relativistic framework has been extensively developed and formulated by I.P. Grant [30,31]. The Hamiltonian can be written in the framework of perturbation theory, as:

H ¼ H0 þ H1 þ H2 In HULLAC, H0 is based on a central field model with a parametric potential U (r) [32e34],

H0 ¼

X

hD ðiÞ

i

(0.1)

2

hD ðiÞ ¼ ca$p þ bc þ Uðri Þ H1 ¼

X i

H2 ¼



 X Z 1  Uðri Þ þ ri r i;j ij

X ai $aj X  þ wðiÞ rij ij i

(0.2)

(0.3)

Here the perturbing operator H1 subtracts the parametric potential of H0, and replaces it with the electrostatic interaction between electrons. The first term of H2 is the Gaunt approximation to the Breit interaction and w includes electron self-energy, vacuum

9

polarization and other QED effects that are commonly called the "Lamb Shift". A.2. Relativistic configurations-jj coupling. The solutions of hD are 2-components spinors (see ref. [30], p 1626; Eq. (22.99)]). These one-electron solutions are characterized by the quantum numbers n, j and k. The latter is often used in textbooks, but it is more common to use l, which is the angular momentum of the large component. This choice has the advantage of easing the connection with the solutions of the non-relativistic Hamiltonian. The one electron wave-functions are usually noted as nlj or simply j. The multi-electron solutions of H0 are Slater determinants [15]. They are called relativistic configurations (RC), and are usually Q symbolized as nlji qi , for example 1sþ 2 2sþ 1 2p 1 , where qi is the i

occupation number of the relativistic shell nlj i. The degeneracy of the RC is lifted in first-order by the matrix elements of H1þH2 within an RC. It is then convenient to use a coupled representation of the wave functions because the different possibilities of coupling the nljq yield different coefficients of the Slater integrals. The first order energy description thus gives rise to "pure jej Terms", characterized by the total angular momentum J, parity p, and intermediate coupled angular momenta. In the case of many open shells, several possible intermediate coupled angular momenta may exist for the same RC and (J, p). Then the Hamiltonian can be diagonalized on that set, without departing from the "pure jej Term" description in one RC. The description of the atomic state beyond first order consists in an expansion in the complete basis of configuration states possessing the same (J, p). The configurations in this expansion are said (improperly) to "interact". This expansion can be achieved by building and diagonalizing the total Hamiltonian on some subset of the Hilbert space. The real interactions are between the electrons. These include electrostatic (H1), magneticemagnetic (H2), and mixed magnetic-electric. The latter, the spineorbit interaction, is implicit in the Dirac Hamiltonian, but it can be explained even with non-relativistic formalism [15,35]. These interactions have very different matrix elements between configurations and therefore will be relevant for different subsets of configurations. A.3. Relativistic configuration mixing and intermediate coupling. It was already noted by Condon and Shortley [36] that pure jej coupling is not adequate for most low ionization atomic systems. The simplest way to remedy this is to diagonalize the Hamiltonian on a restricted subset of the Hilbert space, namely, the set of the relativistic configurations that can be generated from the same Q parent non-relativistic configuration (NRC) nlk wk , i.e., the set of k

RC with the same nlk wk but with all possible ji qi combinations with P wk ¼ qi . Then, the real interaction taken into account is the i ½nlji 2nlk 

spineorbit mentioned above. The result of this operation is a set of energy levels that can be put in one-to-one correspondence with "intermediate coupling" LSJ levels of the non-relativistic Hamiltonian with spineorbit interaction. In this work, we call this relativistic configuration mixing (RCM) in order to distinguish it from the classical "configuration interaction" (CI) originating from (H1), described below. It will also be called CIinNRC (Configuration Interaction in one Non Relativistic configuration). Note that RCM does not shift the average energy of the parent NRC. It only modifies the intensity ratio and position of some lines of the transition array. A.4. Configuration interaction. It is well known [36,37] that when applying perturbation theory, one finds non-diagonal matrix elements of H between NRC's,

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because the individual quantum numbers n and l are not conserved when applying the electrostatic interactions H1. As a consequence, the selection rule jDlj ¼ 1 is no longer valid, and it is also possible that the number of "jumping electrons" in a transition is larger than 1. Before the availability of powerful computers, almost all the computations were non relativistic, and the term CI was unambiguously referring to the above effect. For clarity, when using a RC expansion, we will call the same effect full CI or Relativistic Configuration Interaction (RCI), in order to distinguish it from RCM mentioned above. The latter RCM is the result of a different real interaction than H1. According to Ref. [15] H2 should not be included in this process. In this work, we consider higher orders of perturbation theory by diagonalizing the Hamiltonian on larger and larger bases of the Hilbert space of the N-electron wave-functions. In other words, in “partial RCI” and RCM (CIinNRC) the matrix of the Hamiltonian for all levels is split in smaller matrices, corresponding to several RC (J, p) subsets or to equivalent NRC subsets. Appendix B. HULLAC-v9 code B.1. HULLAC-v9 is an integrated code for calculating atomic structure and cross sections for collisional and radiative atomic processes. The code consists in a set of 8 programs based on a consistent model that uses the same set of wave-functions for all the different processes with the same level of accuracy. For LTE applications only four modules have to be operated consecutively (ANGLAR, RELAC, PHOTION and MYCRM). HULLAC-v9 is open to contributors on a basis of collaboration: contact [email protected]. It has been widely described in many papers [18e22]. Special efforts have been made to insure the simplicity of use. Data file input are built using the user interface for Master program data definitions, check of consistency [19e21]. The atomic structure part of the HULLAC package includes the graphical angular momentum coupling code ANGLAR which generates fine structure levels in a jj coupling scheme for a set of user specified electron configurations. It deals with the Racah algebra calculations of the angular coefficients of the matrix elements for the Hamiltonian and for collisional and radiative cross sections. This corresponds to the “L” mode or detailed Level description mode and allows taking into account CI effects inside user defined groups of configurations or GroC (see illustration at the end of the Appendix). RELAC is based on the relativistic parametric potential method and uses the results of ANGLAR to calculate radiative transition probabilities. Two output files are generated (energy “enr.crm” and beb transitions rates in “ga.crm”). PHOTION uses the corresponding angular results of ANGLAR together with the continuum and bound orbital's to calculate photo-ionization level-to-level transitions and rates and generates a third output file “rr.crm” containing the bound-free contributions. As CI effects are mainly concentrated on beb contributions we have not, in this study, systematically calculated the photo-ionization contributions (rr.crm files turns huge). We have performed our calculations on one core of a 8 processors (80 cores) CPU Intel Xeon E7 8870 of 1 Tb of Ram memory. We want to point out that these large files for the different ions are stored after the calculations. Using the data files (3 for LTE per charge state, 6 for NLTE) MYCRM module calculates the ionic populations for user given thermodynamic conditions and computes the frequency dependent opacity (or emissivity) spectra using the energy and transitions rate from output files of RELAC and PHOTION. Time spent in calculating the spectra varies between several minutes to a few hours depending on the number of energy points. So once the data

have been calculated the production of the spectra are not time consuming. M-shell calculations for Fe VI-XI or Ni VIII-XIII ions can lead to very large ga.crm and rr.crm files (up to 1 Tb each). This means huge files and computation times spent from RELAC and PHOTION for each ion state and, at the end, the necessity to read once again and store simultaneously all the files. Thus the calculation of an opacity spectrum (approximately 4 or 6 ions around mean ionization state) becomes barely tractable, even in CIinNRC mode. However, in LTE applications, it is always possible to calculate separately each ion contribution and to add them at the end to obtain the total spectrum, using LTE populations obtained from other sources or from Saha equation. Even if our work is motivated by the production of complete spectra and of their corresponding Rosseland or Planck means, such results are out of the scope of the present paper focused on large CI effects upon the important M-shell group of transitions. Influence of spectral line profiles will be discussed elsewhere. Thus, for clarity, we use in the illustrations a convolution of the spectra by a Gaussian profile (generally with an FWHM of 0.1 or 1 eV) to smooth the spectra. Values of the energy grid and of the FWHM used for the calculations are given in the captions for each illustration. B.2. Defining partial RCI in HULLAC-v9 In previous papers we have described and illustrated the different CI modes implemented in HULLAC-v9. The diagonalization of eigenvectors for all levels of same J (in one charge state) is performed in each GroC. Exact full RCI treatment between a set of fine structure levels requires the inclusion of all levels in the same GroC. To lighten the burden, we may choose to group configurations in several GroC (limited RCI mode), as has been illustrated in the example of the Fe X, n ¼ 3 / n ¼ 3, 4 transitions, with the n ¼ 1, 2 shells closed. Note that RCM is obtained by using one GroC per NRC, which can be done automaticaly using the keyword CIinNRC. Fe X has Na ¼ 7 “active” open M-shell electrons (“3L^7” in the data), defined by Na ¼ Z e Q e 10, where Z is the atomic number, Q the charge state (here 9 for Fe X ion) and 10 is the number of electrons in closed shells. The beb full RCI opacity results are compared in Fig. 9 to different calculations with different number of CI sub-groups. The choice is not unique so we have chosen to associate a group to a given n (n ¼ 3 or n ¼ 4) and to a given l (s, p, d or f) value for n ¼ 4 transitions. In any case for intra M-shell (n ¼ 3 to n ¼ 3) transitions,

Fig. 9. Fe X HULLAC-v9 RCI beb spectral opacities for the Dn  1, n ¼ 3 transitions: one group or full RCI (black solid), 2 (dotted) and 5 (dashed) groups or "partial" RCI. Charge state population ¼ 1. We use a Gaussian broadening (FWHM ¼ 0.1 eV) and Dhn ¼ 103 eV.

Please cite this article in press as: D. Gilles, et al., Open M-shell Fe and Ni LTE opacity calculations with the code HULLAC-v9, High Energy Density Physics (2015), http://dx.doi.org/10.1016/j.hedp.2015.04.008

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Table 1 Dn ¼ 0 and Dn  1 HULLAC-v9 RCI and RCM calculations and Dn  2 HULLAC-v9 RCM calculations for ion Fe X: number of levels, size of the large arrays, number of relativistic configurations, time spent in the calculations and typical size of the file storage. Fe X Number of levels Size of large arrays Number of relativistic configurations Time spent in Anglar Time spent in Relac Time spent in Photion GA file size (Gb) RR File size (Gb)

Dn ¼ 0 RCI 1 group

Dn ¼ 0 RCM 1g/NRC

Dn ¼ 0,1 RCI 1 group

Dn ¼ 0,1 RCI 2

Dn ¼ 0,1 RCI 5

groups

groups

7255 2.3 M 295 36 min 4 min 32 min 0.2 2.3

7255 0.5 M 295

117 427 560 M 1870

117 427 494 M 1870

1 min 40 s 7 min 0.1 0.6

36 min 40 days Not done 54 Not done

30 min 30 days Not done 50 Not done

M-shell configurations should be kept in one GroC as full CI among them is always important. To summarize.  5 RCI groups (dashed curve, n ¼ 3 / n ¼ 3, n ¼ 3 / 4s, n ¼ 3 / 4p, n ¼ 3 / 4d, n ¼ 3 / 4f),  2 RCI groups (dotted curve, n ¼ 3 / 3, n ¼ 3 / 4s, p, d, f,  1 RCI group (solid curve, n ¼ 3 > 3, 4) We can see that most important effects are visible at highest energies (but more excited levels and b-f contributions have to be taken into account in this spectral range). 5 CI sub-groups seems to be a good compromise for such a large and otherwise intractable calculation, e.g., for Fe VI to Fe VIII. In Table 1 we compare Dn ¼ 0 and Dn  1 HULLAC-v9 RCI and RCM calculations and Dn  2 HULLAC-v9 RCM calculations for ion Fe X. We can see that including the Dn ¼ 1 transitions in full CI increases the time spent in RELAC by a factor 14400! The RCM RELAC calculations never take more than 6 h, but even these calculations can need active memory of 100 Gb (resident and swap). Much more active memory is necessary for ions Fe VIeFe VIII. For example: 5730 relativistic configurations (7.3 105 levels) for the Fe VII Dn  2 HULLAC-v9 RCM ANGLAR and RELAC calculations. References [1] F.J. Rogers, C.A. Iglesias, Science 263 (1994) 50; (a) C.A. Iglesias, F.J. Rogers, B.G. Wilson, Ap. J. 360 (1990) 221; (b) C.A. Iglesias, F.J. Rogers, Ap. J. Suppl. Ser. 79 (1992) 507; (c) C.A. Iglesias, F.J. Rogers, Ap. J. 443 (1995) 469. [2] A.N. Cox, S.M. Morgan, F.J. Rogers, et al., Ap. J. 393 (1992) 292. [3] N.R. Simon, Ap. J. Lett. 260 (1982). [4] M.J. Seaton, R.M. Badnell, Mon. Not. R. Astron. Soc. 354 (2004) 457; (a) M.J. Seaton, et al., Mon. Not. R. Astron. Soc. 354 (2004) 457. http://cdsweb. u-strasbg.fr/topbase/. [5] J.-L. Zeng, F.-T. Jin, J.-M. Yuan, Front. Phys. China 4 (2006) 468; (a) F. Jin, J. Zeng, T. Yuan, et al., Ap. J. J. 693 (2009) 597. [6] L.B. Da Silva, B.J. MacGowan, D.R. Kania, et al., Phys. Rev. Lett. 69 (1992) 438. [7] P.T. Springer, D.J. Fields, B.G. Wilson, et al., Phys. Rev. Lett. 69 (1992) 3735; (a) P.T. Springer, K.L. Wong, C.A. Iglesias, et al., J. Quant. Spectrosc. Radiat. Transf. 58 (1997) 927. [8] J.-L. Zeng, F.-T. Jin, G. Zhao, et al., Chin. Phys. Lett. 20 (6) (2003) 862. [9] G. Winhart, K. Eidmann, C.A. Iglesias, et al., J. Quant. Spectrosc. Radiat. Transf. 54 (1995) 437. [10] C. Chenais-Popovics, H. Merdji, T. Missalla, et al., Ap. J. Suppl. 127 (2000) 275. [11] G. Loisel, P. Arnault, S. Bastiani-Ceccotti, et al., High. Energy Density Phys. 5 (2009) 173;  Paris XI- Paris Sud-UFR Orsay, (2011); (b) (a) G. Loisel, Ph D Thesis, Universite ze, T. Blenski, et al., Experiment Proposal (Private F. Thais, S. Turck-Chie Communication), 2010. [12] S. Salmon, S. Montalban, J. Morel, et al., Mon. Not. R. Astron. Soc. 422 (2012) 3460. [13] A.A. Pamyatnykh, Acta Astron 49 (1999) 119; (a) P. Walczak, J. Daszynska-Daszkiewicz, A.A. Pamyatnykh, T. Zdravkov, Mon. Not. R. Astron. Soc. 432 (2013) 822; (b) J. Daszynska-Daszkiewicz, W. Szewczuk, P. Walczak, Mon. Not. R. Astron. Soc. 431 (2013) 3396.

Dn ¼ 0,1 RCM 1g/NRC

Dn ¼ 0,1, 2 RCM 1g/NRC

117 427 154 M 1870

117 427 37 M 1870

227 598 74 M 3445

30 min 2 days Not done 41 Not done

92 min 175 min 1120 min 16 43

255 min 361 min 860 min 40 85

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Please cite this article in press as: D. Gilles, et al., Open M-shell Fe and Ni LTE opacity calculations with the code HULLAC-v9, High Energy Density Physics (2015), http://dx.doi.org/10.1016/j.hedp.2015.04.008

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