UK APAP R-Matrix Electron-Impact Excitation Cross-Sections for Modelling Laboratory and Astrophysical Plasma

Abstract

Systematic R-matrix calculations of electron-impact excitation for ions of astrophysical interest have been performed since 2007 for many iso-electronic sequences as part of the UK Atomic Process for Astrophysical Plasma (APAP) network. Rate coefficients for Maxwellian electron distributions have been provided and used extensively in the literature and many databases for astrophysics. Here, we provide averaged collision strengths to be used to model plasma where electrons are non-Maxwellian, which often occurs in laboratory and astrophysical plasma. We also provide many new Maxwellian-averaged collision strengths, which include important corrections to the published values. Recently, we made available the H- and He-like collision strengths. Here, we provide data for ions of the Li-, Be-, B-, C-, N-, O-, Ne-, Na-, and Mg-like sequences.

1. Introduction

Most analyses of astrophysical plasma emission rely on accurate atomic data and models. A major original contribution in terms of theory, codes, and production of a variety of atomic data for astrophysical applications began in the 1950s at University College London (UCL), led by a group of atomic physicists under the supervision of Prof. Mike Seaton, FRS. The group grew significantly and later joined forces with the team led by Prof. Phil Burke, FRS, at Queen’s University Belfast (QUB), leading to the development of a large suite of computational codes over several decades. Perhaps the most well-known ones are the Opacity Project (OP), aimed at calculating opacities (see [1]), and the Iron Project (IP) (see, e.g., [2]). The IP was primarily focused on computing radiative data and cross-sections for electron-impact excitation (EIE) of iron ions using close-coupling calculations with the R-matrix method. Later, the Iron Project codes were further improved and were used to calculate atomic data for other elements of astrophysical importance.

Nigel Badnell (NRB) provided many contributions to these codes (cf., [3]). He also developed, since the 1980s, a general-purpose non-relativistic atomic structure code, autostructure [4], which has been used worldwide to calculate photoionization cross-sections and radiative and dielectronic recombination rates and became the standard to calculate the wavefunctions and radiative data for the IP work.

During the early 2000s, NRB began contributing to a series of IP papers calculating EIE cross-sections, see e.g., [5,6], and formed the UK APAP collaboration, which originally included P.J. Storey, G. Del Zanna (GDZ), and H.E. Mason. With funding from PPARC/STFC for astrophysical applications, the UK APAP team produced a vast amount of EIE and radiative data. In addition to specific studies on complex iron and nickel ions (see the review in [7]), the majority of the work over the years focused on calculating data for the main isoelectronic sequences. An earlier review was given in [8].

NRB kept at Strathclyde1, the main repository of all the OP and IP codes and the main atomic data, which are still used in virtually all modelling codes for laboratory and astrophysical plasma worldwide. The EIE and associated radiative data were provided in a compact format for easy inclusion in ADAS2, adf04. As a set of EIE cross-sections for a single ion typically requires 5–30 GB of disk space, only Maxwellian-averaged rates over a range of temperatures have been provided.

Over the years, work by GDZ (e.g., [9,10]) on assessing the data produced by the UK APAP team and benchmarking against laboratory and astrophysical spectra for inclusion in the CHIANTI database3 identified a few errors in the data for some ions. These errors were corrected prior to inclusion in the CHIANTI database; however, the data on the UK APAP website were not updated. Below, we describe these issues and provide the corrected data.

However, the main aim of the present work is to provide bin-averaged cross-sections for all the data we were able to recover. These cross-sections are fundamentally important for analysing emissions from non-thermal plasma, but are generally not available in the literature or in any database. It is well known that, in most laboratory plasmas, the electron distribution is not Maxwellian, even in relatively stable plasmas such as those produced by Electron Beam Ion Traps. It is also well known that the rate coefficients for any distribution could, in principle, be recovered if the distribution can be approximated by a sum of Maxwellians. On the other hand, kinetic theory also shows that such an assumption is often not valid (cf. [11]), and in genera, more accurate results are obtained when original cross-sections are used.

Within solar physics, it is well known that during flares electrons are strongly non-thermal; see, e.g., the review by [12]. Recently, evidence has also been found that non-thermal electrons are present in quiescent solar active regions as well (cf. [13,14]). The possibility that non-thermal electrons are present even in planetary nebulae has been discussed in the literature (see, e.g., [15,16]), although no firm conclusions were reached. Clearly, however, there are astrophysical plasmas where electrons are non-thermal; the present dataset can then be used for accurate modelling. In the following sections, we briefly review the methods and the atomic data. We then draw the conclusions.

2. Methods

Within the UK APAP scattering studies, autostructure has been used to calculate target wavefunctions. This was done using radial wavefunctions based on a scaled Thomas–Fermi–Dirac statistical model potential with a set of variational scaling parameters and a configuration–interaction (CI) expansion. The wavefunctions were used not only for the scattering calculations but also to provide, separately, a set of consistent radiative data. The EIE scattering calculations discussed here were large-scale, using the R-matrix codes combined with the intermediate coupling frame transformation (ICFT) method, as described by Griffin et al. [17]. The close-coupling (CC) expansion was typically the same as the CI expansion used in the structure calculation. The ICFT method first calculates the electron-impact excitation in pure $LS$-coupling and then transforms it into a relativistic coupling scheme via an algebraic transformation of the unphysical scattering or reactance matrices.

As a consequence, the ICFT method is much faster than the Breit–Pauli R-matrix (BPRM) method [18], the B-spline R-matrix (BSR) code (see, e.g., [19]), or the Dirac atomic R-matrix code DARC, originally developed by P. H. Norrington and I. P. Grant (see, e.g., [20]). Differences in the results obtained from these codes have been reported in the literature. However, comparisons of similar-size calculations for the same ion with the BSR, ICFT, and DARC, as, e.g., carried out by [21,22], found good agreement for the main transitions. Significant differences were found for weaker transitions and for those to the higher states. Such differences were mainly due to the structure description and correlation effects, rather than due to the different treatment of the relativistic effects in the three codes, as discussed, e.g., in [8].

The collisional excitation rate coefficient from a state i to a state j due to electron impact is

$$C_{ij}^{\mathrm{e}}={\int }_{v_0}^{\infty }v{\sigma }_{ij}\left(v\right)f\left(v\right)\mathrm{d}v\left[{\mathrm{cm}}^3{\mathrm{s}}^{-1}\right],$$

(1)

where ${\sigma }_{ij}$ is the cross section, v is the velocity of the electron, $f\left(v\right)$ its distribution function, and the limit of integration $v_0$ is the threshold velocity, i.e., the minimum velocity for the electron to be able to excite the atom from level i to j. It is common to introduce a dimensionless quantity, the collision strength ${\Omega }_{ij}\left(E\right)$ for electron excitation:

$${\sigma }_{ij}=\pi a_0^2{\Omega }_{ij}\left(E\right){\displaystyle \frac{I_{\mathrm{H}}}{g_{i}E}}$$

(2)

where $E={\displaystyle \frac{1}{2}}m{v}^2$ is the kinetic energy of the incident electron, $g_i$ is the statistical weight of the initial level, $I_{\mathrm{H}}$ is the ionization energy of hydrogen, and $a_0$ is the radius of the first Bohr orbit. With these definitions, the electron collisional excitation rate coefficient for a Maxwellian electron velocity distribution with an electron temperature $T_{\mathrm{e}}$ in Kelvin is

$$\begin{array}{cc}\hfill C_{ij}^{\mathrm{e}}& =a_0^2{\left({\displaystyle \frac{8\pi I_{\mathrm{H}}}{m}}\right)}^{1/2}{\left({\displaystyle \frac{I_{\mathrm{H}}}{k{T}_{\mathrm{e}}}}\right)}^{1/2}{\displaystyle \frac{{\mathsf{{\rm Y}}}_{ij}}{g_i}}exp\left(-{\displaystyle \frac{\Delta E_{ij}}{k{T}_{\mathrm{e}}}}\right)\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{8.63\times {10}^{-6}}{T_{\mathrm{e}}^{1/2}}}{\displaystyle \frac{{\mathsf{{\rm Y}}}_{i,j}\left(T_{\mathrm{e}}\right)}{g_i}}exp\left({\displaystyle \frac{-\Delta E_{i,j}}{k{T}_{\mathrm{e}}}}\right)\left[{\mathrm{cm}}^3{\mathrm{s}}^{-1}\right]\hfill \end{array}$$

(4)

where $\Delta E_{i,j}=E_j-E_i$ is the energy difference between the lower and upper states i and j, k is Boltzmann’s constant, and ${\mathsf{{\rm Y}}}_{i,j}\left(T_{\mathrm{e}}\right)$ is the dimensionless thermally averaged collision strength (or effective collision strength):

$${\mathsf{{\rm Y}}}_{i,j}\left(T_{\mathrm{e}}\right)={\int }_0^{\infty }{\Omega }_{i,j}exp\left(-{\displaystyle \frac{E_j}{k{T}_{\mathrm{e}}}}\right)d\left({\displaystyle \frac{E_j}{k{T}_{\mathrm{e}}}}\right),$$

(5)

where $E_j$ is the energy of the scattered electron relative to the final energy state of the ion.

The collision strengths are typically calculated with a fine grid of incoming electron energy at lower energies and a coarser grid at higher energies, resulting in a total of typically several thousand bins. The target states, the incoming electron energies, and the collision strengths for all the transitions are written by the IP codes into an ‘OMEGA’ file. autostructure has also been used to calculate the high-energy limits for the collision strengths, following the methods described by [5,23]. These limits are typically added to the collision strengths OMEGA file. The FORTRAN code adasexj.f is then used to calculate the effective collision strengths $\mathsf{{\rm Y}}$ over a range of temperatures. The collision strengths are extended to high energies by interpolation using the high-energy limits in the [24] scaled domain before integration. These high-energy limits, the effective collision strengths $\mathsf{{\rm Y}}$, and the radiative data have been provided in the compact adf04 format.

The OMEGA file for each ion typically occupies 5–30 GB of disk space in a binary compressed form, which makes it difficult to publish. NRB developed and modified adasexj.f over the years to read the large collision strength files and compute their integral averages over a number of bins, making them suitable for distribution.

We provide, along with the attached data, the last public version (3.17, dated 22 October 2019) of adasexj.f that we used. The code produces an adf04_om file if the collision strengths are binned or an adf04 file with the effective collision strengths $\mathsf{{\rm Y}}$. After various tests, it was decided that when binning, the number of energies should be kept relatively small —101 bins—which, however, accurately reproduces the rates for thermal electrons. The same program can also be used to calculate the effective collision strengths $\mathsf{{\rm Y}}$ for a few non-thermal distributions and produce them in adf04 format. As input, adasexj.f can take either a full-resolution collision strength OMEGA file or the bin-averaged adf04_om file. Details can be found in the header of the file.

We have run adasexj.f on the full-resolution collision strength OMEGA files and cross-checked whether we obtained the same effective collision strengths $\mathsf{{\rm Y}}$ as those we published for a sample of ions. Occasionally, the code encounters some negative values for extremely weak transitions, in which case it sets them to zero. adasexj has been significantly modified by NRB over the years, and different versions can produce different effective collision strengths $\mathsf{{\rm Y}}$. The main changes over the years are related to how the collision strengths are extrapolated to zero energy and to the high-energy limits, which typically affect the low- and high-temperature values.

The adf04 file output of adasexj.f also contains Einstein’s spontaneous radiative transition probabilities, the A-values, for the main transitions, as well as the limit points, calculated by autostructure. By default, the structure run used for the scattering calculation only provides A-values for the dipole-allowed transitions. For the final production of the adf04 files, in addition, we typically run autostructure to calculate the A-values also for the weak forbidden transitions, up to at least a third-order multipole. These A-values were typically included in the published adf04 files but have not been included in the present datasets.

If users wish to add these A-values to the present adf04 files, they can use the FORTRAN program adf04mrgr.f. However, we recommend that users consult the literature to find more accurate A-values, particularly for ground configurations, as is commonly done for inclusion in the CHIANTI database. In fact, in most cases the A-values calculated by autostructure for forbidden lines are not very accurate. Sometimes the UK APAP published A-values have been calculated using experimental energies, and in those cases they are more accurate.

3. Atomic Data

3.1. H-like and He-like Ions

The earlier R-matrix calculations by [25], which included all the transitions among the 49 levels up to 1s 5l, were superseded by the larger calculations described by [26], where CI and CC expansions of all the configurations up to $n=6$ were included (36 states for the H-like ions and 71 for the He-like ions). The [26] calculations included radiation damping, which is an important effect for H- and He-like ions, as discussed, e.g., by [27,28]. The effective collision strengths $\mathsf{{\rm Y}}$, as well as the bin-averaged collision strengths, have been made available via ZENODO4.

As a side note, we point out that [29] showed that excitation rates to levels higher than $n=5$ can be estimated more accurately using extrapolation procedures rather than with direct calculations.

3.2. Li-like Ions

Ref. [30] used the radiation- and Auger-damped ICFT R-matrix approach to calculate EIE data for all Li-like ions from Be⁺ to Kr³³⁺. The targets included 204 close-coupling (CC) levels, with valence electrons up to $n=5$ and core-electron excitations up to $n=4$.

During the assessment for CHIANTI v.8 [9], small inconsistencies were identified for a few ions between the highest temperature effective collision strengths $\mathsf{{\rm Y}}$ and their high-temperature limits. These discrepancies were caused by an unintended repetition of the last few collision strength values. The data were corrected by GL. Some of the corrected ions were included by GDZ in CHIANTI v.8. Here, we provide the bin-averaged collision strengths for the outer shell calculations. The entire dataset was processed with the corrected OMEGA files. We found that the differences between the corrected effective collision strengths $\mathsf{{\rm Y}}$ and those reprocessed for Fe are within 1–2 percent; see Figure 1 (left). Therefore, we provide the older but corrected values, together with the bin-averaged collision strengths for the 33 ions in this sequence.

3.3. Be-like Ions

Following the earlier work on Mg ix [31] and Fe xxiii [6], the ICFT R-matrix method was used by [32] to calculate EIE rates for all the ions between ${\mathrm{B}}^{+}$ and ${\mathrm{Zn}}^{26+}$ in the sequence. CI and CC expansions of atomic states up to $nl=7\mathrm{d}$ were included, for a total of 238 fine-structure levels.

The entire dataset was processed, and a few tests indicate close agreement (within a few percent) with the published effective collision strengths $\mathsf{{\rm Y}}$, as shown in Figure 1 (right), so we only provide the bin-averaged collision strengths for the ions in this sequence.

3.4. B-like Ions

The EIE rates for the boron-like ions from C⁺ to Kr³¹⁺ were calculated by [33] using the ICFT R-matrix method. A total of 204 close-coupling levels were included in the target, following the Fe xx ion model of [34].

During the assessment for CHIANTI v.8 [9], errors in the published data were found: the 2s 2p^{2 2}S_1/2 and ²P_1/2 levels (No. 8 and 9) were inverted by mistake (as the experimental and theoretical energies had different orderings); hence, the collision strengths and A-values for transitions to/from these levels were incorrect. The data for these transitions were recalculated by GL, and a few ions were included in CHIANTI by GDZ.

However, the checks indicated significant discrepancies in the effective collision strengths, as displayed in Figure 2 (left) for B-like iron. Therefore, the entire dataset of 31 ions was reprocessed with the corrected OMEGA files, and the thermally averaged collision strengths were made available, along with the bin-averaged ones.

3.5. C-like Ions

Following the previous study of C-like Fe xxi by [35], R-matrix ICFT calculations of C-like ions from N ii to Kr xxxi (i.e., N⁺ to Kr³⁰⁺) were reported in [36]. A total of 590 fine-structure levels (282 terms) were included in the configuration–interaction target expansion and close-coupling collision expansion for all the ions. These levels arise from twenty-seven configurations $2l^{3}n{l}^{\prime }$ with $n=2-4$, $l=0-1$, and $l^{\prime }=0-3$, plus the three configurations $2s^{2}2p5l$ with $l=0-2$. A supplementary package can be found at Zenodo5. This package includes the inputs for the autostructure and R-matrix ICFT calculations, as well as the effective collision strengths $\mathsf{{\rm Y}}$.

The entire dataset of 29 ions was processed, and differences with the published data were found. We found that the published data were processed with an older version of the code: adasexj_2.11.f. As Figure 2 (right) shows, the differences are only of a few percent for transitions within the $n=3$ states but become significant for transitions to higher levels. We have therefore recalculated the effective collision strengths $\mathsf{{\rm Y}}$ for all ions and provide these along with the bin-averaged collision strengths.

3.6. N-like Ions

Following the earlier calculations for Fe xx by [37], which included all the main levels up to $n=4$, ref. [38] provided a larger ICFT calculation involving CI and CC expansions of 27 configurations containing up to three promotions from the ground configuration $2s^{2}2p^3$, up to $n=5$, giving rise to 725 fine-structure states. All the N-like ions from N⁺ to Zn²³⁺ were calculated. A supplementary package available at Zenodo6 includes the effective collision strengths $\mathsf{{\rm Y}}$ in adf04 format.

The entire dataset was processed, and a few tests indicate exact agreement with the published values, so we only provide the bin-averaged collision strengths for all the ions in this sequence.

3.7. O-like Ions

Following an earlier calculation for Fe xix by [39], where the target included 342 close-coupling levels up to $n=4$, R-matrix ICFT calculations for all the O-like ions from Ne²⁺ to Zn²²⁺ were reported in [40]. The CI and CC expansions included a total of 630 fine-structure levels arising from 27 configurations containing up to three promotions from the ground configuration $2s^{2}2p^4$, up to $n=5$. A supplementary package available at Zenodo7 includes the input files of the R-matrix calculations and the effective collision strengths $\mathsf{{\rm Y}}$ in adf04 format.

The entire dataset was processed, and a few tests indicate exact agreement with the published values, so we only provide the bin-averaged collision strengths for all the ions in this sequence.

3.8. Ne-like Ions

Ref. [41] calculated with the ICFT R-matrix method, the EIE effective collision strengths $\mathsf{{\rm Y}}$ for all the ions from Na⁺ to Kr²⁶⁺. The target included 209 levels, up to outer-shell promotions to $n=7$.

As shown by [42], the long-standing discrepancies between predicted and observed intensities for the important Fe xvii X-ray lines in astrophysical observations were finally resolved when considering solar flares and either the [41] or the Breit–Pauli R-matrix calculations by [43].

As in the case of the Li-like ions mentioned above, some of the collision strengths were affected by the error in the high-temperature values. The data were corrected by GL for Mg iii, P vi, Mn xvi, Si v, S vii, Ar ix, and Ni xix. The corrected data were included in CHIANTI v.8 [9]. Fe xvii and Kr xxvii were also affected and were fixed. As shown in Figure 3 (left), the issue occurred at temperatures much higher than those at which the ions are typically formed.

Figure 3 (right) shows the differences between the present effective collision strengths $\mathsf{{\rm Y}}$ and those that were corrected by the high-temperature error. Such differences are clearly large. We also found that the data not affected by the high-T problem were processed with the older adasexj_2.11.f version, resulting in significant differences, as shown in Figure 4 (left).

The entire dataset was therefore reprocessed, and both collision strengths and thermally averaged ones for all 26 ions are provided.

3.9. Na-like Ions

Ref. [44] calculated with the ICFT R-matrix method the EIE effective collision strengths $\mathsf{{\rm Y}}$ for all the Na-like ions from Mg⁺ to Kr²⁵⁺. The close-coupling expansion included configurations up to $n=6$. Inner-shell excitation data with the ICFT R-matrix method, with both Auger and radiation damping included, were also produced by [45].

We are providing here the bin-averaged collision strengths for the outer shell calculations. We found relatively small differences between the present and the published effective collision strengths $\mathsf{{\rm Y}}$, as shown in Figure 4 (right). Therefore, we only provide the bin-averaged collision strengths for the 25 ions in this sequence.

3.10. Mg-like Ions

Ref. [46] described the ICFT R-matrix calculations for all the Mg-like ions from ${\mathrm{Al}}^{+}$ to ${\mathrm{Zn}}^{18+}$. The target included a total of 283 fine-structure levels in both the CI target and CC collision expansions, from the configurations $1{\mathrm{s}}^{2}2{\mathrm{s}}^2{\mathrm{p}}^{6}3\{\mathrm{s},\mathrm{p},\mathrm{d}\}nl$ with $n=4,5$, and $l=0-4$.

The entire dataset was processed, and a few tests revealed differences compared with the previously published effective collision strengths $\mathsf{{\rm Y}}$, as shown in Figure 5 (left). As a result, we have recalculated all the thermally averaged collision strengths and provide them, along with the bin-averaged values, for all 24 ions in this sequence.

Finally, in Figure 5 (right), we show that the differences between the effective collision strengths $\mathsf{{\rm Y}}$ obtained from the full OMEGA file and those derived from the bin-averaged collision strengths are of the order of one to two percent at the temperatures of interest.

4. Conclusions

The UK APAP calculations described here are only part of the work carried out over the years under the supervision of Nigel Badnell. These calculations have significantly improved upon previous work for all isoelectronic sequences, where, in most cases, only a limited number of R-matrix calculations for a few ions were available, often supplemented by interpolated data or data of lower quality.

Much of the data described here has already been incorporated into various databases and modelling codes used to study laboratory and astrophysical plasma. The present work corrects several errors and provides effective collision strengths $\mathsf{{\rm Y}}$ and a comprehensive set of bin-averaged collision strengths to be used for modelling non-thermal plasma. Where possible, we will continue to process more data and make them available.