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Solar UV and X-ray spectral diagnostics

Abstract

X-ray and ultraviolet (UV) observations of the outer solar atmosphere have been used for many decades to measure the fundamental parameters of the solar plasma. This review focuses on the optically thin emission from the solar atmosphere, mostly found at UV and X-ray (XUV) wavelengths, and discusses some of the diagnostic methods that have been used to measure electron densities, electron temperatures, differential emission measure (DEM), and relative chemical abundances. We mainly focus on methods and results obtained from high-resolution spectroscopy, rather than broad-band imaging. However, we note that the best results are often obtained by combining imaging and spectroscopic observations. We also mainly focus the review on measurements of electron densities and temperatures obtained from single ion diagnostics, to avoid issues related to the ionisation state of the plasma. We start the review with a short historical introduction on the main XUV high-resolution spectrometers, then review the basics of optically thin emission and the main processes that affect the formation of a spectral line. We mainly discuss plasma in equilibrium, but briefly mention non-equilibrium ionisation and non-thermal electron distributions. We also summarise the status of atomic data, which are an essential part of the diagnostic process. We then review the methods used to measure electron densities, electron temperatures, the DEM, and relative chemical abundances, and the results obtained for the lower solar atmosphere (within a fraction of the solar radii), for coronal holes, the quiet Sun, active regions and flares.

Introduction

The solar corona is the tenuous outer atmosphere of the Sun, revealed in its full glory during a total solar eclipse. The visible spectrum of the solar corona has two major components: the continuum (the K-corona) due to Thomson scattering of photospheric light by the free electrons in the corona; and weak absorption lines (corresponding to the Fraunhofer lines—the F-corona) superimposed on the continuum emission. The latter is due to scattering by interplanetary dust particles in the immediate vicinity of the Sun. From white light coronagraph observations, and using a model for the distribution of electrons in the corona (van de Hulst 1950), it is possible to estimate the electron number density, which has a value of the order of \(10^8\,\hbox {cm}^{-3}\) in the inner corona.

In addition, strong forbidden emission lines of highly-ionised atoms formed around 1–2 MK (e.g., the green and red coronal lines Fe XIV 5303 Å and Fe X 6374 Å) are also observed during eclipses. The forbidden lines allow measurements of electron densities and also of chemical abundances.

The solar corona is a very hot plasma (1 MK or more) that is mostly optically thin. The emission is due to highly-ionised atoms, which emit principally in the X-rays (5–50 Å), soft X-rays (50–150 Å), extreme ultra-violet (EUV, 150–900 Å) or far ultra-violet (UV, 900–2000 Å) region of the spectrum. Since radiation at these wavelengths cannot penetrate to the Earth’s surface, most of the observations and spectral diagnostics have been obtained from XUV (5–2000 Å) observations from space. These observations and associated spectroscopic diagnostics are the main focus for this review.

When imaged in the EUV at 1 MK, the solar corona shows a wide range of different structures that are magnetically linked to the underlying and cooler regions of the solar atmosphere, the chromosphere and the photosphere. Between the chromosphere and the corona there is a thin but highly complex region, the ‘transition region’ (TR, see e.g., Gabriel 1976), where the temperature dramatically increases. In addition we know, from both a theoretical and observational perspective, that there is a multitude of cooler loops at transition region temperatures that are not connected with the corona, as discussed e.g., by Feldman (1983), Antiochos and Noci (1986), Landi et al. (2000), Hansteen et al. (2014) and Sasso et al. (2015). The transition region emission is highly dynamic and very complex to interpret, with the likelihood of non-equilibrium and high-density effects that are normally not considered when studying the solar corona.

Much of the corona appears to have a diffuse nature (at modest spatial resolutions) and is referred to as ‘quiet Sun (QS)’. This quiet corona, which corresponds to a mixed-polarity magnetic field, is spattered with small bipolar regions which give rise to ‘bright points (BP)’ in the EUV and X-ray wavelength ranges. Then, in regions with enhanced magnetic field (which in the corresponding visible photosphere appear as sunspots), bright ‘active regions (AR)’ form, with a multitude of extended loop structures (see Fig. 1).

Fig. 1
figure 1

A TRACE composite image of the solar corona at 171 Å, formed at around 1 MK (courtesy of the TRACE consortium, NASA)

The other large-scale features of the solar outer atmosphere are the ‘coronal holes (CH)’, which appear as dark areas in EUV and soft X-ray images. At the photospheric level, they correspond to a prevalence of unipolar magnetic fields, corresponding to open magnetic field lines extending out into space. Inside polar coronal holes, large-scale ray-like extended features are usually observed, at various wavelengths (see e.g., DeForest et al. 2001). Due to their appearance these are named coronal hole plumes.

Remote-sensing XUV spectroscopy allows detailed measurements of plasma parameters such as electron temperatures and densities, the differential emission measure (DEM), the chemical abundances, Doppler and non-thermal motions, etc.

These topics have a vast literature associated with them. The present review aims to provide a synthetic up-to-date summary of some of the spectral diagnostics that have been used with data from recent missions or are currently routinely used, focusing on measurements of electron temperatures, electron number densities and chemical abundances in the lower solar corona. The diagnostic techniques used to study the plasma thermal emission measure (EM) distribution are only briefly described, as more emphasis is given to direct measurements of electron temperatures from ratios of lines from the same ion.

Although there are no current or planned instruments which will observe the X-rays with high-resolution spectrometers, we briefly discuss the rich set of diagnostics that in the past were available using satellite lines, i.e., lines formed by inner-shell excitation or dielectronic recombination.

We review the main processes underlying the formation of spectral line emission but we do not intend to replace in-depth presentations of basic material that can be found in books such as that one on solar UV and X-ray spectroscopy by Phillips et al. (2008) or specialised ones, such as the older but very good review on the transition region by Mariska (1992).

We provide a short review on atomic data, with emphasis on the most recent results. We also briefly describe some of the commonly-used atomic codes to calculate atomic data, and mention some of the issues related to uncertainties and line identifications, again not providing in-depth details on each of these topics, which were developed over more than 40 years. This review does not replace standard textbooks of atomic spectroscopy such as Condon and Shortley (1935), Grant (2006) and Landi Degl’Innocenti (2014), nor standard books and articles on atomic calculations in general.

The material presented here builds on and updates the useful review articles on atomic processes and spectroscopic diagnostics for the solar transition region and corona that have been previously written by Dere and Mason (1981), Gabriel and Mason (1982), Doschek (1985), Mason and Monsignori Fossi (1994), Del Zanna and Mason (2013) and Bradshaw and Raymond (2014).

We briefly discuss non-equilibrium effects such as time-dependent ionization and non-thermal distributions, two areas that have recently received more attention. Processes relating to hard X-ray emission from e.g., solar flares, as observed by e.g., RHESSI are not discussed in this review. Recent reviews on this topic have been provided by Benz (2017) and Krucker et al. (2008).

Plasma processes such as radiative transfer, relevant to the lower solar atmosphere (e.g., chromosphere), are not covered in this review. Such information can be found in standard textbooks such as Athay (1976) and Mihalas (1978). In the future, this Living review will be extended to cover the diagnostics of the outer solar corona, where densities become so low that photo-excitation and resonance excitation from the disk radiation need to be included in the modelling.

The solar XUV spectrum

We first briefly review some of the main spectrometers which have been used to observe the Sun from the X-rays to the UV. In this review, we do not discuss hard X-ray spectrometers, nor spacecrafts which carried spectrometers, but ultimately did not produce spectra. The emphasis is on high-resolution spectra. For most diagnostic applications, having an accurate radiometric calibration is a fundamental requirement, but particularly difficult to achieve, especially in the EUV and UV. Significant degradation typically occurs in space, due to various effects (see, e.g., the recent review of BenMoussa et al. 2013). We also mention some EUV imaging instruments which have been used extensively.

Historic perspective

The solar corona has been studied in detail since the early 1960s using data from a number of rocket flights. Some of them produced the best XUV spectra of the solar corona and transition region to date. An overview of these early days can be found in Doschek (1985), Mason and Monsignori Fossi (1994) and Wilhelm et al. (2004a).

Rocket flights

The X-rays are mostly dominated by L-shell (\(n=2,3,4 \rightarrow 2\)) emission from highly ionised atoms. Early (but excellent) X-ray spectra of the Sun were obtained by a large number of rocket flights, see for example Evans and Pounds (1968) and Davis et al. (1975), and the reviews of Neupert (1971b) and Walker Jr (1972).

The best X-ray spectrum of a quiescent active region was obtained with an instrument, built by the University of Leicester (UK), which consisted of Bragg crystal spectrometers with a collimator having a FOV (FWHM) of \(3{^{\prime }}\), and flown on a British Skylark sounding rocket on 1971 November 30 (Parkinson 1975). The instrument had an excellent spectral resolution, was radiometrically calibrated and the whole spectral region was observed simultaneously, unlike many other X-ray instruments which scanned the spectral regions.

The soft X-ray (50–170 Å) spectrum of the quiet and active Sun is rich in \(n=4 \rightarrow n=3\) transitions from highly ionised iron ions, from Fe vii to Fe xvi (see, e.g., Fawcett et al. 1968). Manson (1972) provided an excellent list of calibrated soft X-ray irradiances observed in quiet and active conditions in the 30–130 Å range by two rocket flights, on 1965 November 3 and 1967 August 8. The spectral resolution was moderate, about 0.23 Å (FWHM) for the quiet Sun, and 0.16 Å for the active Sun observation.

Behring et al. (1972) published a line list from a high resolution (0.06 Å) spectrum in the 60–385 Å region of a moderately active Sun. The instrument was built at the Goddard Space Flight Center (GSFC) and flown on an Aerobee 150 rocket flight on 1969 May 16. A similar line list for the EUV was produced by Behring et al. (1976). Behring et al. (1972, 1976) represent the best solar EUV line lists in terms of accuracy of wavelength measurements and spectral resolution. Unfortunately, these EUV spectra were not radiometrically calibrated.

Malinovsky and Heroux (1973) presented an integrated-Sun spectrum covering the 50–300 Å range with a medium resolution (0.25 Å), taken with a grazing-incidence spectrometer flown on a rocket on 1969 April 4. The photometric calibration of the EUV part of the spectrum was exceptionally good (about 10–20%), but the soft X-ray part was recently shown to be incorrect by a large factor (Del Zanna 2012a).

Acton et al. (1985) published a high-quality solar spectrum recorded on photographic film during the rocket flight on 1982 July 13, 2 min after the GOES X-ray peak emission of an M1-class flare. The instrument was an X-ray spectrometer/spectrograph telescope (XSST).

The spectrum was radiometrically calibrated, and it provided accurate line intensities from 10 Å up to about 77 Å. The spectral resolution was excellent, clearly resolving lines only 0.04 Å apart. Excellent agreement between predicted and measured line intensities has been found (see, e.g., the recent study of Del Zanna 2012a).

At longer wavelengths, the best EUV spectra have been obtained by the series of GSFC Solar Extreme Ultraviolet Rocket Telescope and Spectrograph (SERTS) flights. The one flown in 1989 (SERTS-89) (Thomas and Neupert 1994; hereafter TN94) observed the 235–450 Å range in first order. The SERTS-95 covered shorter wavelengths (Brosius et al. 1998). The SERTS-97 (Brosius et al. 2000) covered the 300–353 Å spectral region. Both SERTS-89 and SERTS-97 were radiometrically calibrated, although the calibration of the SERTS-89 spectra has been questioned (Young et al. 1998). Other SERTS and EUNIS (see, e.g., Wang et al. 2010, 2011) sounding rockets built at GSFC have served for the calibration of in-flight EUV spectrometers of several satellites, but have also returned many scientific results (Table 1).

Table 1 Some of the earlier space-borne spectroscopic instruments that observed the solar corona in the XUV

OSO

After the early rocket flights, the first series of small satellites were the Orbiting Solar Observatories (OSO). The first observations of \(n=3 \rightarrow 2\) lines in solar flares were made with the OSO-3 satellite in the 1.3–20 Å region (Neupert et al. 1967). OSO-5 produced spectra of solar flares in the 6–25 Å region (Neupert et al. 1973), and the first solar-flare spectra containing the \(n=2 \rightarrow 2\) L-shell iron emission, in the 66–171 Å range (Kastner et al. 1974a). OSO-6 also provided solar flare spectra (see the Doschek 1972; Doschek et al. 1973 line lists). OSO-7 produced EUV spectra in the 190–300 Å range of the coronal lines (Kastner et al. 1974b), later studied in detail by Kastner and Mason (1978).

OSO-8 (1975–1978) obtained the first high spatial (\(2{^{\prime \prime }}\)) and spectral observations of the chromosphere and the transition region with an UV spectrometer which operated in the 1200–2000 Å range (Bruner 1977). In its scanning mode, a line profile would typically be scanned across 1 Å with very high spectral resolution (0.02 Å) but low cadence (30–50 s). OSO-8 also carried a UV/visible polychromator which observed the H i Lyman \(\alpha ,\beta \), the Mg ii H, K, and Ca ii H, K lines. Results from these instruments are reviewed by Bonnet (1981). Unfortunately, the instruments suffered a drop in sensitivity very early on during the mission. The degradation was so dramatic that measures were put in place by Bonnet and others for a strict cleanliness program for the SOHO spacecraft. This cleanliness program was an overall success, as most instruments suffered little degradation, compared to other missions (see below).

OSO-8 also carried two X-ray spectrometers, with co-aligned graphite and PET crystals (see Parkinson et al. 1978, and references therein). The graphite had a large geometrical area (100 cm\(^2\)) but lower spectral resolution than the PET system. The spectrometers were uncollimated so viewed the whole Sun. The overlapping of spectra from active regions located in different parts of the solar surface was therefore a problem. The great advantages of the OSO-8 spectrometers over previous ones were the high sensitivity and the fact that the crystals were fixed, i.e., the entire wavelength ranges were observed with a high cadence, about 10 s. The graphite crystal spectrometer was well calibrated (10%) in the laboratory and in flight (Kestenbaum et al. 1976).

Skylab

More detailed studies of the solar corona from space started in May 1973, when Skylab, the first NASA space station, was launched. The Apollo Telescope Mount (ATM) on Skylab carried several solar instruments, observing from the UV to the X-rays between June 1973 and February 1974. Three successful series of Skylab workshops held in Boulder, Colorado, summarised the main results: coronal hole and high speed wind streams (Zirker 1977), solar flares (Sturrock 1980) and solar active regions (Orrall 1981).

The Harvard College Observatory (HCO) EUV spectrometer SO55 (Reeves et al. 1977a) on the ATM had a good spatial resolution in the 296–1340 Å range, but low spectral resolution (\(\simeq \) 1.6 Å or more, depending on the spectral range). In the standard grating position, the instrument scanned with a \(5{^{\prime \prime }}\times 5{^{\prime \prime }}\) slit covering six wavelengths typical of chromospheric to coronal temperatures: Ly\(\alpha \) (1216 Å, \(T \simeq 2 \times 10^4\) K); C II (1336 Å, \(T \simeq 3.5 \times 10^4\) K); C III (977 Å, \(T \simeq 7 \times 10^4\) K); O IV (554 Å, \(T \simeq 1.5 \times 10^5\) K); O VI (1032 Å, \(T \simeq 3 \times 10^5\) K) and Mg X (625 Å, \(T \simeq 1.1 \times 10^6\) K). With other grating positions, spectroheliograms of the lines Ne VII (465 Å, \(T \simeq 5 \times 10^5\) K) and Si XII (521 Å, \(T \simeq 1.8 \times 10^6\) K) were also recorded. The good spatial resolution of the HCO spectrometer enabled (Vernazza and Reeves 1978) to produce a list of line intensities for different solar regions, which was a standard reference for many years. The radiometric calibration of the instrument (about 35% uncertainty) is described by Reeves et al. (1977b). This instrument suffered severe in-flight degradation.

The Naval Research Laboratory (NRL) S082A slitless spectroheliograph on the ATM (Tousey et al. 1977) had a very good wavelength coverage (170–630 Å), with a spatial resolution reaching \(2''\) for small well defined features. The instrument obtained 1023 spectroheliograms of the whole Sun. However, the dispersion direction coincided with one spatial dimension (normally oriented E–W), so the images of the solar disk in the nearby spectral lines were overlapped (the instrument was fondly called the ‘overlappograph’). For this reason, most of scientific results have been obtained from spectra emitted by small well defined regions, such as compact flares, active region loop legs, the limb brightening, etc. For the smallest features, the instrument achieved an excellent spectral resolution of about 0.03 Å. As an example of the excellent quality of the instrument, a portion of a flare spectrum is shown in Fig. 2. A complete list of lines observed during flares was produced by Dere (1978b).

Fig. 2
figure 2

Image reproduced with permission from Feldman et al. (1988), copyright by OSA

A portion of the NRL S082A slitless spectrum of a solar flare

The NRL Skylab SO82B (Bartoe et al. 1977) had a \(2{^{\prime \prime }}\times 60{^{\prime \prime }}\) non-stigmatic slit and an excellent spectral resolution (0.04–0.08 Å) over the 970–3940 Å spectral range. Sandlin et al. (1977) and Sandlin and Tousey (1979) produced lists of coronal forbidden lines. Figure 3 shows a spectrum taken about \(30{^{\prime \prime }}\) off the solar limb, showing several coronal forbidden lines (Feldman et al. 1988). Line lists of chromospheric lines were provided by Doschek et al. (1977a) and Cohen (1981). Many excellent papers using the S082A and S082B instruments were produced by the groups at NRL and collaborators.

Fig. 3
figure 3

Image reproduced with permission from Feldman et al. (1988), copyright by OSA

A portion of the NRL S082B spectrum taken about \(30{^{\prime \prime }}\) off the solar limb, showing several coronal forbidden lines

P78-1

Significant improvements in terms of spectral resolution were achieved in the X-rays with the SOLEX and SOLFLEX crystal spectrometers aboard the US P78-1 satellite, launched in 1979 (for a description of the solar instruments on the P78-1 spacecraft see Doschek 1983).

The SOLEX crystals scanned the 3–25 Å spectral region with a resolution of \(10^{-3}\) Å at 8.2 Å and two multi-grid collimators of 20 or \(60{^{\prime \prime }}\). For more details, see Landecker et al. (1979). The SOLFLEX crystals observed the full Sun and covered four spectral bands in the 1.82–8.53 Å range (1.82–1.97; 2.98–3.07; 3.14–3.24; 8.26–8.53 Å) with a resolution varying from \(2.4 \times 10^{-4}\) Å at 1.9 Å to \(10^{-3}\) Å at 8.2 Å. The bands were chosen to observe the strong resonance lines from Fe xxv, Ca xx, and Ca xix with their associated satellite lines, as well as several other lines from high-temperature ions. The spectral ranges were scanned by rotating the crystals, with a typical cadence of 56 s.

These spectrometers allowed seminal discoveries related to solar flares. During the rise (impulsive) phase of solar flares, strong blue asymmetries in the resonance lines were observed, interpreted as upflows during the chromospheric evaporation. The crystals also enabled the observation of non-thermal broadening and the measurement of temperature and emission measure variations during flares. For a review of the SOLEX results see, e.g., McKenzie et al. (1980a, b, 1985) and Doschek et al. (1981b). For a review of SOLFLEX spectra and their interpretation see, e.g., Doschek et al. (1979, 1981a).

HRTS

Superb UV solar spectra were obtained by the series of High Resolution Telescope and Spectrometer (HRTS) instruments, developed at the Naval Research Laboratory (NRL) (Brueckner and Bartoe 1983). HRTS was flown eight times on sounding rockets between 1975 and 1992 which enabled many results. It was also flown on Spacelab 2, together with the CHASE instrument (see below) in 1985. HRTS covered the 1170–1710 Å spectral region with very high spectral (0.05 Å) and spatial (\(1{^{\prime \prime }}\)) resolution. HRTS had a stigmatic slit and also provided slit-jaw images. Each flight was unique, in that different slits, wavelength ranges or pointing were chosen. A good review of the flights can be found in http://wwwsolar.nrl.navy.mil/hrts_hist.html, written by K. Dere.

Sandlin et al. (1986) published a well-known list of HRTS observations of different regions on the Sun, with accurate wavelengths and line intensities, in the 1175–1710 Å spectral range. This list represents the most complete coverage in this wavelength range. A modification of the HRTS was flown on Spacelab 2 (Brueckner et al. 1986).

Brekke et al. (1991) published excellent HRTS spectra, obtained during the second rocket flight in February 1978. The spectrum was radiometrically calibrated by matching the quiet Sun intensities with those measured by the Skylab S082B calibration rocket flight CALROC. The long stigmatic slit of the HRTS instrument covered many solar regions. Figure 4 shows an averaged spectrum of an active region at the limb from HRTS.

Fig. 4
figure 4

Data from Brekke et al. (1991)

An averaged spectrum of an active region at the limb from HRTS, in two spectral regions, dominated by the important UV lines from Si iii, C ii, Si iv, O iv and S iv

CHASE

The Coronal Helium Abundance Spacelab Experiment (CHASE, see Breeveld et al. 1988) on the Spacelab 2 Mission (1985) was specifically designed to determine the helium abundance from the ratio of He ii 304 Å to Lyman-\(\alpha \) 1218 Å, on the disc and off the limb. A value of \(0.079\pm 0.011\) for the quiet corona was obtained by Gabriel et al. (1995). However, the instrument also recorded several spectral lines, used by Lang et al. (1990) to describe the temperature structure of the corona. One limiting aspect of CHASE was the lack of a specific radiometric calibration for the flight instrument.

SMM

The Solar Maximum Mission (SMM) was dedicated to the study of active regions and solar flares. SMM was launched in February 1980 but encountered some problems. It was repaired in-orbit by the NASA Space Shuttle in April 1984, and then resumed full operations until December 1989. It carried several instruments.

Fig. 5
figure 5

Image reproduced with permission from Del Zanna and Mason (2014), copyright by ESO

The SMM/FCS AR spectrum of 1987 December 13

The SMM X-ray polychromator (XRP) Flat and Bent Crystal Spectrometers (FCS and BCS) instruments (Acton et al. 1980) produced excellent X-ray spectra of the solar corona, active regions and solar flares. The XRP/FCS had a collimator of about \(15{^{\prime \prime }}\times 14{^{\prime \prime }}\), so had a limited spatial resolution, although it could raster an area as large as \(7'\times 7'\) with \(5{^{\prime \prime }}\) steps. The FCS crystals could be rotated to provide the seven detectors access to the spectral range 1.40–22.43 Å. The FCS 5–25 Å spectral region is dominated by Fe xvii, O viii, Ne ix lines. A sample spectrum of a quiescent active region is shown in Fig. 5. The sensitivity of the FCS instrument decreased significantly early on in the mission, in particular at the longer wavelengths, so the O vii lines around 22 Å became very weak. The in-flight radiometric calibration was based on the assumption of carbon deposition on the front filter (see, e.g., Saba et al. 1999). The FCS data allowed measurements of the temperature distribution and electron density of solar active regions and flares together with measurements of relative coronal abundances of various elements. A complete list of lines observed with the SMM FCS during flares was published by Phillips et al. (1982). Another excellent SMM/FCS solar flare spectrum, this time with \(n=4 \rightarrow 2\) calibrated line intensities, was published by Fawcett et al. (1987). A significant limitation of these solar observations was that the spectral range of each detector was scanned, hence lines within the same channel were not observed simultaneously. This considerably complicated the analysis (cf. Landi and Phillips 2005), particularly for solar flares.

The XRP/BCS, with a collimator field of view of about \(6'\times 6'\) (the size of a large active region), was able to obtain spectra with eight position-sensitive proportional counters in the range 1.7–3.2 Å simultaneously, at a resolving power of about \(10^4\). The XRP/BCS observed X-ray line complexes of high temperature (in excess of 10 MK) coronal lines: Fe xxvi, Fe xxv, Ca xix, and S xv. This spectral range allowed a wide range of plasma diagnostics to be applied. The He-like ions allowed measurements of electron densities and temperatures (Gabriel and Jordan 1969). The satellite lines also allowed some important diagnostic measurements (see, e.g., Gabriel 1972a, b; Gabriel and Phillips 1979; Doschek 1985; Phillips et al. 2008).

The Ultraviolet Spectrometer and Polarimeter (UVSP) on board SMM was used to observe many features, including solar active regions and flares in the 1750–3600 Å range in first order of diffraction and 1150–1800 Å in the second (Woodgate et al. 1980). The spatial resolution was very good, about \(3{^{\prime \prime }}\), and the spectral resolution was excellent (0.04 Å in first order and 0.02 in second order). Various slits were available, from 1 to \(15{^{\prime \prime }}\) wide. As with previous UV instruments, the UVSP also suffered severe degradation in orbit (Miller et al. 1981) during its 10 months of operations.

Hinotori

Hinotori was a Japanese spacecraft in orbit between 1981 and 1982 which was used to observe high-energy X-ray emission produced by solar flares. The most important results were obtained with the Bragg spectrometers. The SOX2 scanning spectrometer had an excellent spectral resolution (0.15 mÅ) and produced excellent spectra of Fe xxvi and Fe xxv during large flares. Very high temperatures were measured. For a summary of the results from these spectrometers see Tanaka et al. (1982) and Tanaka (1986).

Yohkoh

Yohkoh (Japanese for sunbeam) was used to observe the Sun in X-ray emission from 1991 to 2001. The Bragg Crystal Spectrometer (BCS, see Culhane et al. 1991) produced similar measurements to the SMM/BCS of the X-ray H- and He-like line complexes. It used 4 bent crystals which observed the X-ray lines of highly ionized S, Ca, and Fe produced by flares and active regions in the 1.76–5.1 Å wavelength range (1.76–1.80: Fe xxvi; 1.83–1.89: Fe xxv; 3.16–3.19: Ca xix; 5.02–5.11: S xv). Many scientific results have been obtained, in particular those regarding the temperatures during flares (Phillips and Feldman 1995; Feldman et al. 1996).

One potential problem with instruments such as XRP/BCS and the Yohkoh BCS is that sources in different spatial locations produce superimposed spectra at different wavelengths. For example, extended sources produced a broadening of the lines, and complex spectra could arise in the case of multiple flares occurring at the same time within the field of view. This was not normally an issue for XRP/BCS, given its field of view (\(6\times 6\) arc minutes), but was occasionally more of a problem for Yohkoh/BCS which observed the full Sun.

SoHO

The Solar and Heliospheric Observatory (SoHO), a joint NASA and ESA mission which was launched in December 1995 to the L1 position, is still operational, although the attitude loss in 1998 caused significant degradation to some of its instruments, many of which have now been switched off. First results from SoHO were published in a special issue of Solar Physics in 1997 (volume 170). With 24 h monitoring, SoHO has produced a wealth of data and has changed our view of the Sun. SoHO carried a suite of several instruments, performing in-situ and remote-sensing observations. The radiometric calibration of various instruments on-board SoHO during the first few years of the mission was discussed during two workshops held at the International Space Science Institute (ISSI), in Bern. The results were summarised in the 2002 ISSI Scientific Report SR-002 (Pauluhn et al. 2002).

Here, we summarise the spectroscopic instruments used to study the solar corona. The Coronal Diagnostic Spectrometer (CDS), a UK-led instrument (Harrison et al. 1995) was routinely operated from 1996 to 2014. It comprised of a Wolter–Schwarzschild type II grazing incidence telescope, a scan mirror, a set of different slits (2, 4, \(90{^{\prime \prime }}\)), and two spectrometers, a normal incidence spectrometer (NIS) and a grazing incidence spectrometer (GIS). The wavelength range covered by the two detectors (150–800 Å) contains many emission lines emitted from the chromosphere, the transition region and the corona. The NIS had two wavelength bands, NIS-1, 308–381 Å and NIS-2, 513–633 Å. To construct monochromatic images (rasters), a scan mirror was moved across a solar region to project onto the detectors the image of a stigmatic slit (\(2''\) or \(4''\)). For the NIS instrument, the spectral resolution was about 0.3 Å before the SoHO loss of contact in 1998, then degraded to about 0.5 Å afterwards. The effective spatial resolution was about \(4{^{\prime \prime }}\).

The in-flight radiometric calibration of the CDS instrument was found to be very different from that which had been measured on the ground (actually better than expected, for the NIS see, e.g., Landi et al. 1997; Del Zanna et al. 2001a; Lang et al. 2007, while for the GIS see Del Zanna et al. 2001a).

The CDS team believed that the variation in the long-term radiometric calibration of the NIS instrument was mainly caused by the degradation of the microchannel plate detectors following the use of the wide slit. A standard correction was implemented in the calibration software. However, Del Zanna et al. (2010a) showed that this standard correction was quite different from expectation, and overestimated by large factors (2–3) for the stronger lines. The NIS instrument only degraded by about a factor of two in 13 years, which is quite remarkable. The new calibration by Del Zanna et al. (2010a) was confirmed by sounding rocket flights (e.g., EUNIS-2007, see Wang et al. 2011) and was adopted for the final calibration of the instrument.

The diagnostic potential of CDS was discussed by Mason et al. (1997). A NIS spectral line list for the quiet Sun can be found in Brooks et al. (1999), while more extended lists for different regions, with line identifications based on CHIANTI are given in Del Zanna (1999). Sample NIS spectra are given in Figs. 6 and 7.

Fig. 6
figure 6

Image adapted from Del Zanna (1999)

CDS NIS 1 averaged spectrum of the quiet Sun (off-limb)

Fig. 7
figure 7

Image adapted from Del Zanna (1999)

SoHO CDS NIS 2 averaged spectrum of the quiet Sun (on-disk)

The grazing incidence spectrometer used a grazing incidence spherical grating that disperses the incident light to four microchannel plate detectors placed along the Rowland Circle (GIS 1: 151–221 Å, GIS 2: 256–341 Å, GIS 3: 393–492 Å and GIS 4: 659–785 Å). The spectral resolution of the GIS detectors was about 0.5 Å. The GIS was astigmatic, focusing the image of the slit along the direction of dispersion but not perpendicular to it. The in-flight radiometric calibration of the GIS channels (only the pinhole \(2{^{\prime \prime }}\times 2{^{\prime \prime }}\) and \(4{^{\prime \prime }}\times 4{^{\prime \prime }}\) slits) is described in Del Zanna et al. (2001a) and Kuin and Del Zanna (2007). No significant degradation of the GIS sensitivity was found. The GIS suffered badly from ‘ghost lines’ due to the spiral nature of the detector. This made data analysis somewhat complicated. A full list of GIS spectral lines can be found in Del Zanna (1999).

Fig. 8
figure 8

An on-disc quiet Sun SUMER spectrum in the two spectral ranges which will be covered by the Solar Orbiter SPICE spectrometer

The Solar Ultraviolet Measurement of Emitted Radiation (SUMER) was a joint German and French-led instrument (Wilhelm et al. 1995). It was a high-resolution (\(1{^{\prime \prime }}\) spatial) spectrometer covering the wavelength range 450–1600 Å, dominated by lines emitted by the chromosphere and transition region. Detector A observed the 780–1610 Å range, while detector B covered the 660–1500 Å range. Second-order lines were superimposed on the first order spectra, however the second order sensitivity was such that only the strongest lines in the 450–600 Å range were observable.

SUMER had an excellent spectral resolution (\({\lambda \over d \lambda } = 19{,}000\)–40,000), and was able to measure Doppler motions (flows) with an accuracy better than 2 km/s, using photospheric lines as a reference. As in the case of the CDS, SUMER was able to scan solar regions to obtain monochromatic images in selected spectral lines. One main difference was that only lines within a wavelength range (typically 40 Å) could be recorded simultaneously by SUMER.

One disadvantage with SUMER was the amount of time it took to scan a spatial area. Some difficulties were encountered with the scanning mechanism, so it was used sparingly during the latter part of the mission. The SUMER radiometric calibration for the first few years of the mission was discussed at the ISSI workshops, see ISSI Scientific Report SR-002 (Pauluhn et al. 2002).

A spectral atlas of SUMER on-disk lines was published by Curdt et al. (2001). Figure 8 shows the spectrum in the two spectral ranges that will be observed with the Solar Orbiter SPICE instrument. Several strong transition region lines are present. A list of SUMER on-disk quiet Sun radiances in the 800–1250 Å range was published by Parenti et al. (2005), where radiances of a prominence were also provided. Within the SUMER spectral range, several coronal forbidden lines become visible off the solar limb, see e.g., the spectral atlases by Feldman et al. (1997) and Curdt et al. (2004). Some of the high-temperature forbidden lines become visible even on-disk in active regions and during flares (see, e.g., the lists of Feldman et al. 1998a, 2000). The last observations were carried out with SUMER in 2014, due to the significant degradation of the detectors.

The Ultraviolet Coronagraph Spectrometer (UVCS) was an instrument that was built and operated by a USA–Italy collaboration (Kohl et al. 1995). It observed the solar corona from its base out to 10 \(R_\odot \). Its heritage was the Spartan Ultraviolet Coronagraph Spectrometer, which flew several times between 1993 and 1998 (Kohl et al. 2006). Most of the UVCS scientific results are based on the measurements of the strong H I Lyman-\(\alpha \) and O vi (1032 and 1037 Å) lines, which are partly collisionally excited and partly resonantly scattered (Raymond et al. 1997). Several coronal lines such as Si xii (499 and 521 Å) and Fe xii (1242 Å) were also observed. UVCS produced measurements of chemical abundances, proton velocity distribution, proton outflow velocity, electron temperature, and ion outflow velocities and densities. Some of scientific results from UVCS have been reviewed by Kohl et al. (2006) and Antonucci et al. (2012). The instrument suffered a significant degradation (factor of ten) at first light, but continued to operate for a long time.

CORONAS

CORONAS-F, launched in 2001, provided XUV spectroscopy with the SPIRIT (Russian-led, see Zhitnik et al. 2005) and RESIK (REntgenovsky Spektrometr s Izognutymi Kristalami, Polish-led, see Sylwester et al. 2005) instruments, especially of flares during solar maximum. The SPIRIT spectroheliograph had two wavelength ranges, 176–207 and 280–330 Å, and a relatively high spectral resolution of about 0.1 Å. The instrument was slitless. The solar light was deflected at a grazing angle of about \(1.5^{\circ }\) by a grating, and then focused by a mirror coated with a multilayer with a high reflectivity in the EUV. This resulted in ‘overlappogram’ images of the spectral lines, highly compressed in the solar E–W direction, but with a good spatial resolution along the N–S direction. Most solar flares have a typical small spatial extension during the impulsive phase, so it was relatively straightforward to obtain flare spectra, only slightly contaminated by nearby emission at similar latitudes. The radiometric calibration was only approximate, obtained with the use of line ratios. SPIRIT observed several flares. For details and a line list see Shestov et al. (2014).

RESIK was a full-Sun spectrometer employing bent crystals to observe simultaneously four spectral bands within the range 3.4–6.1 Å, with a spectral resolution of about 0.05 Å. The instrumental fluorescence was a major limiting effect in the spectra, creating a complex background emission, which needed to be subtracted for continuum analysis, and to measure the signal from weaker lines.

The RESIK wavelength range (see Fig. 9) is of particular interest because line-to-continuum measurements can be used for absolute abundance determinations (see, e.g., Chifor et al. 2007) and because of the presence of dielectronic satellite lines (see, e.g., Dzifčáková et al. 2008). RESIK observed a large number of flares during 2001–2003.

Fig. 9
figure 9

Image adapted from Chifor et al. (2007)

RESIK spectrum of a flare. Different colours indicate the four channels

Some XUV spectroscopy was provided by CORONAS-Photon with the SPHINX (Polish-led) instrument (Sylwester et al. 2008), although the satellite, launched in 2009, unfortunately had a failure in 2010.

RHESSI

The NASA Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI, Lin et al. 2002), despite not being a purely high-resolution spectrometer, provided nevertheless very important X-ray spectral observations since 2002 at high energies, above 3 keV. The instrument achieved spatial and spectral resolutions significantly higher than those of earlier missions. Depending on the signal, it was possible to obtain imaging in selected energy bands at about \(2.5''\) resolution. The spectra have about 1 keV resolution, just allowing the Fe line complex to be resolved at 6.7 keV from the continuum emission.

Hinode

Hinode (Japanese for sunrise) is a Japanese mission developed and launched in September 2006 by ISAS/JAXA, collaborating with NAOJ as a domestic partner, NASA and STFC (UK) as international partners. Scientific operation of the Hinode mission is conducted by the Hinode science team organized at ISAS/JAXA. This team mainly consists of scientists from institutes in the partner countries. Support for the post-launch operation is provided by JAXA and NAOJ (Japan), STFC (UK), NASA, ESA, and NSC (Norway).

Initial results from the Hinode satellite have been published in special issues of the PASJ, Science and A&A journals in 2007 and 2008. Hinode (Kosugi et al. 2007) carried 3 instruments, the Solar Optical telescope (SOT, see Tsuneta et al. 2008), the X-ray imaging telescope (XRT, see Golub et al. 2007), and the Extreme-ultraviolet Imaging Spectrometer (EIS, see Culhane et al. 2007). We focus here on the latter instrument.

EIS has two wavelength bands, 170–211 and 246–292 Å (see Fig. 10), which include spectral lines formed over a wide range of temperatures, from chromospheric to flare temperatures (\(\log T ({\mathrm{MK}}) = 4.7{-}7.3\)). The instrument has an effective spatial resolution of about 3–\(4''\). The high spectral resolution (0.06 Å) allows velocity measurements of a few km/s. However with no chromospheric lines, these velocity measurements are difficult to calibrate. Rastering is normally obtained with the narrow slits (\(1''\) or \(2''\)).

The ground radiometric calibration (Lang et al. 2006) was revised by Del Zanna (2013b). A significant degradation e of about a factor of two within the first 2 years was found in the longer-wavelength channel. This degradation was confirmed by Warren et al. (2014).

Fig. 10
figure 10

Image adapted from Young et al. (2007a)

Hinode EIS spectrum of an active region. The dashed lines are the (scaled) effective areas of the two channels

Earlier spectroscopic diagnostic applications were described in Del Zanna and Mason (2005b) and Young et al. (2007a, b). A tabulation of spectral lines observed by Hinode/EIS was provided by Brown et al. (2008). A more complete list of coronal lines with their identification based on CHIANTI was provided by Del Zanna (2012b). A comprehensive list with identifications of cool (\(T \le 1\,\hbox {MK}\)) emission lines can be found in Del Zanna (2009a) and Landi and Young (2009). A discussion of the high-temperature flare lines and their blends can be found in Del Zanna (2008b) and Del Zanna et al. (2011b).

SDO

The NASA Solar Dynamics Observatory (SDO, see Pesnell et al. 2012) was launched in February 2010, carrying a suite of instruments. The Helioseismic and Magnetic Imager (HMI, see Schou et al. 2012), led from Stanford University in Stanford, California, measures the photospheric magnetic field. The Atmospheric Imaging Assembly (AIA, see Lemen et al. 2012), led from the Lockheed Martin Solar and Astrophysics Laboratory (LMSAL), provides continuous full-disk observations of the solar chromosphere and corona in seven extreme ultraviolet channels.

The only coronal spectrometer on SDO is the Extreme ultraviolet Variability Experiment (EVE) instrument (Woods et al. 2012). It provided solar EUV irradiance with an unprecedented wavelength range (1–1220 Å) and temporal resolution (10 s). The EVE spectra are from the Multiple EUV Grating Spectrographs (MEGS) and have about 1 Å spectral resolution. The MEGS A channel is a grazing incidence spectrograph for the 50–380 Å range. It ceased operations in May 2014. The MEGS B channel is a double pass normal incidence spectrograph for the 350–1050 Å range. A list of MEGS flare lines and their identifications is presented in Del Zanna and Woods (2013). Additionally, EVE has an EUV Spectrophotometer (ESP), a transmission grating and photodiode instrument similar to the SoHO Solar EUV Monitor (SEM). ESP has four first-order channels centered on 182, 257, 304, and 366 Å with approximately 40 Å spectral width, and a zero order channel covering the region 1–70 Å.

MEGS B suffered a significant degradation of its sensitivity from the beginning of the mission (factor of 10), while the degradation of MEGS A was more contained (see, e.g., BenMoussa et al. 2013). Various procedures such as the line ratio technique, previously applied to other instruments (as SoHO CDS, Del Zanna et al. 2001a), were used to obtain in-flight corrections. A recent evaluation of the EVE version 5 calibration showed relatively good agreement (to within 20%) with the SoHO CDS irradiance measurements for most lines (Del Zanna and Andretta 2015).

The combined spectral range of the two channels was observed with a sounding rocket in 2008 April 14 carrying a prototype EVE MEGS instrument (Woods et al. 2009; Chamberlin et al. 2009; Hock et al. 2010, 2012). The spectrum was radiometrically calibrated on the ground, and is shown in Fig. 11. On that day, and during the whole long deep solar minimum around 2008, the Sun was extremely quiet, so in principle the prototype EVE observation should represent the best EUV solar spectrum at minimum. However, significant discrepancies were found for several of the strongest lines when comparisons of the SoHO CDS and EVE observations were made (from May 2010, when the Sun started to be more active) as discussed in Del Zanna and Andretta (2015). The irradiances of the strongest lines appear to have been overestimated by 30–50%.

Fig. 11
figure 11

The EUV spectrum of the whole Sun, as measured by the prototype SDO/EVE instrument flown aboard a rocket in 2008 April 14

It is worth noting that the EVE spectra vary a lot in some spectral ranges, depending on the level of activity on the Sun. An example is shown in Fig. 12, where the quiet Sun spectrum is shown together with an X-class flare spectrum in the spectral ranges covered by the six SDO AIA EUV bands. Even the coarse EVE spectral resolution clearly shows that each AIA band has contributions from several spectral lines, and that different spectral lines become dominant under different solar conditions.

Fig. 12
figure 12

The EVE spectrum of the quiet Sun (above) and of an X-class flare (below, see Del Zanna and Woods 2013), in the spectral ranges covered by the six SDO/AIA EUV bands (their effective areas are shown, rescaled, with dashed lines)

IRIS

The Interface Region Imaging Spectrograph (IRIS, De Pontieu et al. 2014) was launched in July 2013, and has been producing excellent spectra and images of the solar atmosphere at very high temporal (2 s) and spatial (\(0.33{-}0.4{^{\prime \prime }}\)) resolution, but with a limited field of view. The high resolution has enabled many new scientific results. A special section of Science in October 2014 (volume 346) was dedicated to the first results from IRIS.

As with earlier UV instruments, IRIS suffered significant in-flight degradation during the first few years of the mission.

The IRIS Slit Jaw Imager (SJI) provides high-resolution images in four different passbands (C ii 1330 Å, Si iv 1440 Å, Mg ii k 2796 and Mg ii wing at 2830 Å). The IRIS spectrograph (SP) observes spectra in the 1332–1358, 1389–1407, and 2783–2834 Å spectral regions, where there are several emission lines formed in the photosphere, chromosphere, as well as in the transition region (Si iv, O iv, S iv). The highest temperature line observed by IRIS is the Fe xxi 1354.08 Å line, formed at high temperatures (12 MK) typical of flares (Young et al. 2015; Polito et al. 2015, see Fig. 13). This interesting flare line was previously observed with Skylab SO82B (see, e.g., Doschek et al. 1975) and SMM UVSP (see, e.g., Mason et al. 1986).

The use of a slit together with a slit-jaw image enables the precise location of the spectra to be established, and in addition high-cadence observations can be obtained. HRTS also had a slit-jaw camera but lacked the high-cadence which IRIS has, used photographic plates, and was generally flown on a rocket (with relatively short duration). Nonetheless, HRTS provided some interesting observations which can now be explored in more detail with IRIS.

Fig. 13
figure 13

Top: IRIS spectra in the Si iv spectral band (courtesy of V. Polito). The right panel shows the Si iv slit-jaw image and the location of the slit. The middle panel shows the detector image in the Si iv spectral window, and the left panel shows the averaged spectrum from the region indicated by the two parallel lines in the middle panel. Bottom: IRIS spectra in the Fe xxi band during a flare. The left image shows the C ii slit-jaw image and the location of the IRIS slit across two ribbons during a flare. The middle image shows a detector image with the Fe xxi 1354.08 Å line and photospheric lines from C i and O i. The rightmost panel shows the spectrum averaged along the slit in between the two lines shown in the middle panel

The formation of the XUV spectrum

Spectral line intensity in the optically thin case

In the majority of cases, the XUV emission from the solar corona and transition region is optically thin, i.e., all the radiation that we observe remotely has freely escaped the solar atmosphere. In this case, the observed intensity of a spectral line is directly related to its bound–bound emissivity. The radiance (or more simply intensity) \(I(\lambda _{ji})\) of a spectral line of wavelength \(\lambda _{ji}\) (frequency \(\nu _{ji} = c/\lambda _{ji}\)) is therefore:

$$\begin{aligned} {I(\lambda _{ji})} = {{{h \nu _{ji}\over {4\pi }}\;{\int \limits \,N_j(Z^{+r})\,A_{ji}\;ds}}} \quad (\mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}) \end{aligned}$$
(1)

where \(Z^{+r}\) indicates the element of atomic number Z which is r times ionized, ij are the lower and upper levels of the ion \(Z^{+r}, h\) is Planck’s constant, \(A_{ji}\)Footnote 1 is the transition probability for the spontaneous emission (Einstein’s A value), \(N_j(Z^{+r})\) is the number density (i.e., the number of particles per unit volume in \(\mathrm{cm}^{-3}\)) of the upper level j of the emitting ion, and s is the line-of-sight coordinate.

The term

$$\begin{aligned} P_{ji} = h \nu _{ji}\;N_j(Z^{+r})\;A_{ji}. \end{aligned}$$
(2)

is the power (or emissivity) per unit volume (\(\mathrm{erg}\,\mathrm{cm}^{-3}\,\mathrm{s}^{-1}\)) emitted in the spectral line.

When the Sun is observed as a star, the flux (i.e., total irradiance in a line), \(F(\lambda _{ji})\), for an optically thin line of wavelength \(\lambda _{ij}\) is defined as:

$$\begin{aligned} F(\lambda _{ji}) = {1 \over 4 \pi d^2}~ \int _V P_{ji}~ dV \quad (\mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}) \end{aligned}$$
(3)

where d is the Sun/Earth distance.

There are many processes that affect the population of an upper level of an ion. Those processes occurring between the levels of an ion are, under normal conditions, much faster than those affecting the charged state of the ions, so the two groups of processes (described below) can usually be considered separately. For example, for C iv at \(N_{\mathrm{e}}=10^{10}\,\hbox {cm}^{-3}\) and \(T=10^5\,\hbox {K}\) we have Mariska (1992) for the allowed transition at 1548 Å the following time scales: collisional excitations and de-excitations: \(2 \times 10^{-3}\,\mathrm{s}\); spontaneous radiative decay: \(4 \times 10^{-9}\,\mathrm{s}\); radiative recombinations \(+\) dielectronic recombinations: \(88\,\mathrm{s}\); collisional ionization \(+\) autoionization: \(107\,\mathrm{s}\).

The population of a level is therefore normally calculated by separately calculating the excited level populations and the ion population. The radiance (intensity) of a spectral line (see Eq. 1) is therefore usually rewritten using the identity:

$$\begin{aligned} N_j(Z^{+r}) \equiv {N_j(Z^{+r})\over N(Z^{+r})}\,{N(Z^{+r}) \over N(Z)}\,{N(Z) \over N_{H}}\,{N_{H} \over N_{\mathrm{e}}}\,N_{\mathrm{e}} \end{aligned}$$
(4)

where the various terms are defined as follows:

  • \({N_j(Z^{+r})/N(Z^{+r})}\) is the population of level j relative to the total \(N(Z^{+r})\) number density of the ion \(Z^{+r}\). As we shall see below, the population of level j is calculated by solving the statistical equilibrium equations for the ion \(Z^{+r}\). It is a function of the electron temperature and density.

  • \({N(Z^{+r})/N(Z)}\) is the ion abundance, and is predominantly a function of temperature, but also has some electron density dependence.

  • \({N(Z)/N_{H}} \equiv Ab(Z)\) is the element abundance relative to hydrogen.

  • \({N_{H}/N_{\mathrm{e}}}\) is the hydrogen abundance relative to the electron number density. This ratio is usually in the range \(\sim 0.8{-}0.9\), since H and He, are fully ionized at coronal temperatures. If we neglect the contribution of the heavier elements to the electron density (\(\le \) 1%), and assume fully ionized H and He, the only parameter that changes this ratio is the relative abundance of He, which is variable. If we assume e.g., the Meyer (1985) abundances, \(N_{H}/N(\mathrm{He})=10\), then \({N_{H}/N_{\mathrm{e}}}=0.83\).

The intensity of a spectral line can then be written in the form:

$$\begin{aligned} I(\lambda _{ji})= \int _s G(N_{\mathrm{e}},T,\lambda _{ji})\,N_{\mathrm{e}}\,N_H\,\mathrm{d}s \end{aligned}$$
(5)

with the contribution function G given by:

$$\begin{aligned} G(N_{\mathrm{e}},T,\lambda _{ij}) = Ab(Z)~ A_{ji} ~{h \nu _{ij} \over {4\pi }} {N_j(Z^{+r})\over N_{\mathrm{e}} N(Z^{+r})} {N(Z^{+r}) \over N(Z)} \quad (\mathrm{erg~cm}^{3}~\mathrm{sr}^{-1} ~\mathrm{s}^{-1}) \end{aligned}$$
(6)

The contribution function contains all of the relevant atomic physics parameters and for most of the transitions has a narrow peak in temperature, and therefore effectively confines the emission to a limited temperature range. In the literature there are various definitions of contribution function, depending on which terms are included. Aside from constant terms, sometimes the elemental abundance and/or \(N_{H}/N_{\mathrm{e}}=0.8\) and/or the term \({1/4 \pi }\) are included in the definition.

Sometimes, as in the CHIANTI package, the contribution function \( C(N_{\mathrm{e}},T,\lambda _{ij})\) is calculated without the abundance factor:

$$\begin{aligned} G(N_{\mathrm{e}},T,\lambda _{ij}) = Ab(Z)~ C(N_{\mathrm{e}},T,\lambda _{ij}) \end{aligned}$$
(7)

Also note that sometimes (as in the CHIANTI emiss_calc program) the emissivity is defined as \({ Emiss} = h \nu _{ij}\,N_j(Z^{+r})/N(Z^{+r})\,A_{ji}\), i.e., with the fractional population of the upper level, in which case the emissivity is basically the contribution function without the elemental abundance Ab(Z) and the ion abundance \(N(Z^{+r})/N(Z)\), and without dividing for the electron density.

Collisional rates and Maxwellian distributions

The dominant mechanisms for the level populations in the low solar corona are collisional excitation and ionization of the ions by the free electrons. Considering excitation, the number of transitions in an ion from a state i to a state j due to electron collisions, per unit volume and time, is \(N_i \sigma _{ij} N_{\mathrm{e}} v f(v) \mathrm{d}v\), where \(N_i\) is the number density of the ion in the initial state, \(\sigma _{ij}\) is the cross section for the process, v is the velocity (in absolute value) of the electron, and f(v) the distribution function of the electrons.

In general, the collisional excitation rates are proportional to the total number of transitions integrated over the free electron distribution, the so-called rate coefficients

$$\begin{aligned} C^{\mathrm{e}}_{ij} = \int _{v_0}^\infty v\,\sigma _{ij} (v)\,f(v)\,\mathrm{d}v \quad ({\mathrm{cm}}^3~\mathrm{s}^{-1}), \end{aligned}$$
(8)

where 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:

$$\begin{aligned} {1 \over 2}\,m\,{v_0}^2 = E_j - E_i \end{aligned}$$
(9)

where m is the mass of the electron.

It is normally assumed that the electrons have enough time to thermalise, i.e., follow a Maxwell–Boltzmann (thermal) distribution (but see Sect. 6.2) in the lower solar corona. In this case, the probability \(f(v)\,\mathrm{d}v\) that the electron has a velocity (in the 3-D space) between v and \(v+\mathrm{d}v\) is:

$$\begin{aligned} f(v) = 4\,\pi \,v^2 \left( {m \over 2\,\pi \,kT_{\mathrm{e}}} \right) ^{3/2}\,{\mathrm{e}}^{-{m v^2/2\,kT_{\mathrm{e}}}}. \end{aligned}$$
(10)

where k indicates Boltzmann’s constant, and \(T_{\mathrm{e}}\) the electron temperature. On a side note, the most probable speed \(v_p\), i.e., the maximum value of the distribution is found by imposing that \( {\mathrm{d} f(v) \over \mathrm{d}v}=0\):

$$\begin{aligned} v_p = \left( {2\,k T_{\mathrm{e}} \over m}\right) ^{1/2}, \end{aligned}$$
(11)

while the average speed \(\langle v\rangle \) is:

$$\begin{aligned} \langle v\rangle = \int v\,f(v)\,\mathrm{d}v = \left( {8\,k T_{\mathrm{e}} \over \pi m}\right) ^{1/2} \end{aligned}$$
(12)

The collisional excitation rates can then be written using the Maxwell–Boltzmann distribution. As a function of the kinetic energy of the incident electron \(E = {1 \over 2}\,m\,{v}^2\), the rate coefficient can be written as:

$$\begin{aligned} C^{\mathrm{e}}_{ij}&= \left( {m \over 2 \pi \,k T_{\mathrm{e}}} \right) ^{3/2}\,4 \pi \,\int _{v_0}^\infty v^3\,\sigma _{ij} (v) \,{\mathrm{e}}^{-{m v^2/2\,kT_{\mathrm{e}}}}\,\mathrm{d}v \nonumber \\&= \left( {8 \over \pi \,m} \right) ^{1/2}\,(k T_{\mathrm{e}})^{-3/2} \int _{E_0}^\infty E\,\sigma _{ij} (E)\,{\mathrm{e}}^{-{E/kT_{\mathrm{e}}}}\,\mathrm{d}E \nonumber \\&= 5.287\times 10^{13}\,(k T_{\mathrm{e}})^{1/2} \int _{E_0}^\infty {E \over k T_{\mathrm{e}}}\,\sigma _{ij} (E)\,{\mathrm{e}}^{-{E/kT_{\mathrm{e}}}}\,\mathrm{d}\left( {E \over k T_{\mathrm{e}}}\right) , \end{aligned}$$
(13)

where \(E_0\) is the threshold energy of the electron, i.e., \(E_0 = E_j-E_i\), the energy difference between the ion states i and j.

In the case of collisional ionization of an atom or ion by a free electron, the expressions for the number of transitions are similar, as described below.

Excitation and de-excitation of ion levels

Inspection of Eq. (13) indicates a way to simplify the expression, by introducing a dimensionless quantity, the collision strength for electron excitation:

$$\begin{aligned} \sigma _{ij} = \pi a_0^2\,\varOmega _{ij}(E)\,{I_{\mathrm{H}} \over g_i E}, \end{aligned}$$
(14)

where \(g_i\) is the statistical weight of the initial level, \(I_{\mathrm{H}}\) is the ionization energy of hydrogen, and \(a_0\) the Bohr radius. The collision strength is a symmetrical quantity, such that \(\varOmega _{ij}(E)=\varOmega _{ji}(E^{\prime })\), where \(E^{\prime }=E-E_{ij}\) is the kinetic energy of the electron after the scattering.

The electron collisional excitation rate coefficient for a Maxwellian electron velocity distribution with a temperature \(T_{\mathrm{e}}\) (K) is then obtained by integrating:

$$\begin{aligned} C^{\mathrm{e}}_{ij}&= a_0^2 \left( {8 \pi I_{\mathrm{H}} \over m} \right) ^{1/2} \left( {I_{\mathrm{H}} \over k T_{\mathrm{e}}} \right) ^{1/2}{\varUpsilon _{ij} \over g_i} \exp \left( - {E_{ij} \over k T_{\mathrm{e}}} \right) \nonumber \\&={8.63\times 10^{-6} \over T_{\mathrm{e}}^{1/2}} {\varUpsilon _{i,j}(T_{\mathrm{e}}) \over g_i} \exp \left( {- \varDelta E_{i,j} \over kT_{\mathrm{e}}} \right) ~ \mathrm{cm}^3~\mathrm{s}^{-1}, \end{aligned}$$
(15)

where k is the Boltzmann constant and \(\varUpsilon _{i,j}\) is the thermally-averaged collision strength:

$$\begin{aligned} \varUpsilon _{i,j}(T_{\mathrm{e}}) = \int _0^\infty \varOmega _{i,j} \exp \left( -{E_{j} \over kT_{\mathrm{e}}}\right) d \left( {E_j \over kT_{\mathrm{e}}}\right) , \end{aligned}$$
(16)

where \(E_j\) is the energy of the scattered electron relative to the final energy state of the ion. Some details on electron-ion scattering calculations are provided in Sect. 5.

The electron de-excitation rates \(C_{j,i}^{\mathrm{d}}\) from the upper level j to the lower level i are obtained by applying the principle of detailed balance, assuming thermodynamic equilibrium, following Milne, who repeated Einstein’s reasoning on the radiative transition probabilities. In thermodynamic equilibrium, the processes of excitation and de-excitation must equal:

$$\begin{aligned} N_i N_{\mathrm{e}} C^{\mathrm{e}}_{i,j} = N_j N_{\mathrm{e}} C^{\mathrm{d}}_{j,i}, \end{aligned}$$
(17)

and the populations of the two levels are in Boltzmann equilibrium:

$$\begin{aligned} {N_i \over N_j} = {g_i \over g_j} \exp \left( {\varDelta E_{i,j} \over kT_{\mathrm{e}}} \right) , \end{aligned}$$
(18)

where \(g_i, g_j\) are the statistical weights of the two levels. So we obtain

$$\begin{aligned} C_{j,i}^{\mathrm{d}} = {g_i \over g_j} C_{i,j}^{\mathrm{e}} \exp \left( {\varDelta E_{i,j} \over kT_{\mathrm{e}}} \right) . \end{aligned}$$
(19)

This relation holds also outside of thermodynamic equilibrium, as long as the plasma is thermal. If the electron distribution is not Maxwellian, the definitions of the excitation and de-excitation rates are somewhat different, as described below in Sect. 6.2.

The ion level population and the metastable levels

The variation in \(N_j\), the population of level j of the ion \(Z^{+r}\), is calculated by solving the statistical equilibrium equations for the ion including all the important excitation and de-excitation mechanisms:

$$\begin{aligned} {{ dN}_j\over dt}&= \sum _{k>j} N_k A_{k,j} + \sum _{k>j} N_k N_{\mathrm{e}} C^{\mathrm{d}}_{k,j} + \sum _{i<j} N_i N_{\mathrm{e}} C^{\mathrm{e}}_{i,j} + \sum _{i<j} N_i B_{i,j}\,J_{\nu _{i,j}} \nonumber \\&\quad +\, \sum _{k>j} N_k B_{k,j}\,J_{\nu _{k,j}}- N_j \left( \sum _{i<j} A_{j,i} + N_{\mathrm{e}} \sum _{i<j} C^{\mathrm{d}}_{j,i} + N_{\mathrm{e}} \sum _{k>j} C^{\mathrm{e}}_{j,k}\right. \nonumber \\&\quad \left. +\, \sum _{i<j} B_{j,i}\,J_{\nu _{j,i}} + \sum _{k>j} B_{j,k}\,J_{\nu _{k,j}}\right) \end{aligned}$$
(20)

where the first five terms are processes which populate the level j: the first is decay from higher levels, the second is de-excitation from higher levels, the third is excitation from lower levels, and the other two are photo-excitation and de-excitation. The other five terms are the corresponding depopulating processes.

Note:

  1. *

    \(N_{\mathrm{e}}\) (\(\hbox {cm}^{-3}\)) is the electron number density.

  2. *

    \(C^{\mathrm{e}}, C^{\mathrm{d}}\) (\(\hbox {cm}^{-3}\,\hbox {s}^{-1}\)) are the electron collisional excitation and de-excitation rate coefficients defined above.

  3. *

    \(A_{j,i}\) (\(\hbox {s}^{-1}\)) are Einstein’s coefficients for spontaneous emission, also called transition probabilities or A-values.

  4. *

    \(B_{i,j}\) (\(i<j\)) are Einstein’s coefficients for absorption.

  5. *

    \(B_{k,j}\) (\(k>j\)) are Einstein’s coefficients for stimulated emission.

  6. *

    \(J_{\nu } = {1 \over 4 \pi } \int I_\nu ({\varvec{\Omega }})\,\mathrm{d}\varOmega ,\) i.e., is the average of the intensity of the radiation field over the solid angle.

Note that the terms associated with stimulated emission by radiation are normally negligible for the solar corona, while the terms associated with absorption of radiation are only important in the outer corona, where electron densities become sufficiently small. Also note that Einstein’s coefficient for stimulated emission is related to the A-value by:

$$\begin{aligned} B_{ul} = {c^2 \over 2\,h\,\nu _{ul}^3}\,A_{ul}, \end{aligned}$$
(21)

while Einstein’s coefficient for absorption is related to the other two coefficients by:

$$\begin{aligned} B_{lu} = {c^2 \over 2\,h\,\nu _{ul}^3}\,{g_u \over g_l}\,A_{ul} = {g_u \over g_l}\,B_{ul}, \end{aligned}$$
(22)

where we have indicated the lower level with l and the upper level with u for simplicity, and g indicates the statistical weight of the level. Typical \(A_{j,i}\) values for transitions that are dipole-allowed are of the order of \(10^{10}\,\hbox {s}^{-1}\), while those of forbidden transitions can be as low as \(100\,\hbox {s}^{-1}\).

For most solar (and astrophysical) applications, the time scales of the relevant processes are so short that the plasma is normally assumed to be in a steady state (\({{ dN}_j\over dt}=0\)). The set of Eq. (20) is then solved for a number of low lying levels, with the additional requirement that the total population of the levels equals the population of the ion: \(N(Z^r) =\sum _j N_j\).

In the simplified case of a two-level ion model (a ground state g and an excited state j) and neglecting other processes such as photoexcitation, we have:

$$\begin{aligned} N_{\mathrm{g}} N_{\mathrm{e}} C^e_{g,j} = N_j \left( N_{\mathrm{e}} C^e_{j,g} + A_{j,g}\right) \end{aligned}$$
(23)

so the relative population of the level j is

$$\begin{aligned} {N_j \over N_{\mathrm{g}}} = {N_{\mathrm{e}} C^e_{g,j} \over {N_{\mathrm{e}} C^e_{j,g} + A_{j,g}}} \end{aligned}$$
(24)

i.e., depends strongly on the relative values between the radiative rate \(A_{j,g}\) and the collisional de-excitation term \(N_{\mathrm{e}} C^e_{j,g}\). Levels that are connected to the ground state by a dipole-allowed transition have \(A_{j,i}\) values typically several orders of magnitude larger than the de-excitation term (at typical coronal densities, \(N_{\mathrm{e}}=10^8{-}10^{12}\,\hbox {cm}^{-3}\)), so their population is negligible, compared to the population of the ground state. When all the upper levels of the ion are of this kind, the statistical equilibrium equations are simplified, so that only direct excitations from the ground state need to be included. This is the so called coronal-model approximation, where only the electron collisional excitation from the ground state of an ion and the spontaneous radiative decay are competing.

However, ions often have so called metastable levels, m, which have a small radiative decay rate (e.g., corresponding to intersystem or forbidden transitions), so that collisional de-excitation starts to compete with radiative decay as a depopulating process at sufficiently high electron densities (\(A_{m,g}~\simeq ~N_{\mathrm{e}}~ C^e_{m,g}\)). In such cases, the population of the metastable levels becomes comparable to that of the ground state. Whenever ions have metastable levels, it is necessary to include all collisional excitation and de-excitation rates to/from the metastable levels when solving the statistical equilibrium equations. All the ions that have more than one level in the ground configuration have metastable levels (i.e., all the excited levels within the ground configuration), because the transitions within a configuration are forbidden, i.e., they have small radiative decay rates. The ion population is shifted from the ground level into the metastable(s) as the electron density of the plasma increases.

Proton excitation

Proton collisions become non-negligible when excitation energies are small, \(\varDelta E_{i,j} \ll kT_{\mathrm{e}}\). This occurs, for example, for transitions between fine structure levels, as in the Fe XIV transition in the ground configuration (\(3{s}^23{p}\,{}^2\hbox {P}_{1/2}\)\(^2\hbox {P}_{3/2}\)), as discussed in Seaton (1964a). Normally, only the fine structure levels within the ground configuration of an ion have a significant population, so only the proton collisions among such levels are important. The inclusion of proton excitation has some effects on the relative population of the levels. Proton collisional excitation and de-excitation are easily included as additional terms \(C^{\mathrm{p}}\) (\(\hbox {cm}^{3}\,\hbox {s}^{-1}\)) in the level balance equations.

Photoexcitation

Photoexcitation is an important process which needs to be included in the level balance equations when electron densities are sufficiently low. Photoexcitation is the process by which the excitation of an ion from a level i to a level j is caused by absorption of a photon. For this to occur, the photon has to have the same energy of the transition from i to j. From the statistical equilibrium equations (Eq. 20) and considering the relations between the Einstein coefficients, it is obvious that photoexcitation and de-excitation can easily be included as additional terms which modify the A-value. These terms are proportional to \(J_{\nu }\). It is common to assume that the intensity of the radiation field originating from the solar photosphere \(I_\nu \) does not vary with the solid angle (no limb brightening/darkening), in which case we have:

$$\begin{aligned} J_{\nu } = {\varDelta \,\varOmega \over 4 \pi }\,\overline{I_\nu } = W(r)\,\overline{I_\nu } \end{aligned}$$
(25)

where W(r) is the dilution factor of the radiation, i.e., the geometrical factor which accounts for the weakening of the radiation field at a distance r from the Sun, and \(\overline{I_\nu }\) is the averaged disk radiance at the frequency \(\nu \).

Assuming spherical symmetry (i.e., the solar photosphere a perfect sphere), and indicating with r the distance from Sun centre \(R_{\odot }\) the solar radius, we have:

$$\begin{aligned} W(r) = {1 \over 4 \pi } \int _0^{2 \pi } \int _0^{\theta _0} { sin}\;\theta \,d\theta \,d\phi = {1 \over 2}\,(1- { cos}\;\theta _0) = {1 \over 2} \left( 1- \left[ 1- \left( {R_{\odot } \over r}\right) ^2\right] ^{1/2} \right) \nonumber \\ \end{aligned}$$
(26)

where \(\theta _0\) is the angle sub-tending \(R_{\odot }\) at the distance r, i.e., \({ sin }\theta _0 = R_{\odot }/r \).

In terms of the energy density per unit wavelength, \(U_\lambda \), the photoexcitation rate for a transition \(i\rightarrow j\) is:

$$\begin{aligned} P_{ij}=A_{ji}\,W(r)\,{g_j \over g_i}\,{\lambda ^5 \over 8\pi hc} U_\lambda \end{aligned}$$
(27)

where \(A_{ji}\) is the Einstein coefficient for spontaneous emission from j to \(i, g_j\) and \(g_i\) are the statistical weights of levels j and i, and W(r) is the radiation dilution factor.

Metastable levels affect the population of an ion, in particular those of the ground configuration. The transitions between these levels are normally in the visible and infrared parts of the spectrum, where almost all the photons are emitted by the solar photosphere. Therefore, the main contributions of photoexcitation to the level population is due to visible/infrared photospheric emission. A reasonable approximation for the photospheric radiation field at visible/infrared wavelengths is a black-body of temperature \(T_*\), for which the photoexcitation rate becomes:

$$\begin{aligned} P^{\mathrm{bb}}_{ij}=A_{ji} W(r) {g_j \over g_i} {1 \over \exp (E/kT_*) -1} \end{aligned}$$
(28)

The inclusion of photoexcitation can simply be carried out by replacing the \({A}_{ji}\) value in the statistical equilibrium equations with a generalized radiative decay rate (as coded in the CHIANTI atomic package), which in the black-body case is:

$$\begin{aligned} \mathcal{A}_{ij} = \left\{ \begin{array}{l@{\quad }l} W(r) A_{ji} {g_j\over g_i} {1 \over \exp ({\varDelta E}/kT_*) -1} &{} i<j \\ \\ A_{ji} \left[ 1 + W(r) {1 \over \exp ({\varDelta E}/kT_*) -1} \right] &{} i > j \end{array} \right. \end{aligned}$$
(29)

Clearly, the photospheric radiation field might depart significantly from black-body radiation, for example with absorption lines, so an observed solar spectrum should be used for more accurate calculations. Photoexcitation typically becomes a significant process at densities of about \(10^8\,\hbox {cm}^{-3}\) and below. It therefore becomes a non-negligible effect for off-limb observations of the corona above a fraction of the solar radius, where electron densities (hence electron excitations) decrease quasi-exponentially.

By affecting the populations of the levels of the ground configuration, photoexcitation has a direct effect on the intensities of the (visible and infrared) forbidden lines which are emitted by these levels. One typical example is for the Fe xiii infrared forbidden lines (see, e.g., Chevalier and Lambert 1969; Flower and Pineau des Forets 1973; Young et al. 2003, Fig. 14). These lines are used to measure electron densities (see, e.g., Fisher and Pope 1971), the orientation and strength of the magnetic field and small Doppler shifts in the solar corona. Such measurements are currently being made with the coronal multi-channel polarimeter (CoMP) instrument, now located at Mauna Loa (see, e.g., Tomczyk et al. 2007).

Fig. 14
figure 14

Ratios of the two infrared forbidden Fe xiii lines, as a function of density, without and with photoexcitation (assuming a blackbody spectrum seen at 1.1 \(R_\odot \)). CHIANTI version 8 was used. The wavelengths indicated are the wavelengths in air

By changing the population of the metastable levels of the ground configuration, photoexcitation also indirectly affects some EUV/UV spectral lines, in particular those connected with the ground configuration metastable levels.

Atomic processes affecting the ion charge state

Various processes can affect the ionisation state of an element. If we consider two ionization stages, we have the following processes, denoting for simplicity the population of the ion r-times ionised with \(N_{r}=N(Z^r)\):

  1. 1.

    radiative recombination, induced by the radiation field \(\sim N_{r+1} N_{\mathrm{e}} \alpha ^I_{r+1}\);

  2. 2.

    radiative recombination, spontaneous \(\sim N_{r+1} N_{\mathrm{e}} \alpha ^R_{r+1}\);

  3. 3.

    photoionisation, induced by the radiation field \(\sim N_{r} \alpha ^{\mathrm{PI}}_{r}\);

  4. 4.

    collisional ionization by direct impact of free electrons \(\sim N_{r} N_{\mathrm{e}} C^{\mathrm{I}}_{r}\);

  5. 5.

    dielectronic recombination \(\sim N_{r+1} N_{\mathrm{e}} \alpha ^{\mathrm{D}}_{r+1}\),

where \(C^{\mathrm{I}}_{r}\) (\(\mathrm{cm}^{3}\,\mathrm{s}^{-1}\)) are the rate coefficients for collisional ionization by electrons, and \(\alpha \) (\(\mathrm{cm}^{3}\,\mathrm{s}^{-1}\)) are the various recombination coefficients.

Collisional ionisation by electron impact and three-body recombination

The cross section for collisional ionization of an atom or ion from an initial state i to a final state j by a free electron, differential in the energy \(E_1\) of the incident and that of the ejected electron \(E_2\) can be expressed, as in the excitation process, in terms of a collision strength \(\varOmega \):

$$\begin{aligned} \sigma (E_1, E_2) = \frac{1}{k_1^2 g_i} \varOmega _{ij}, \end{aligned}$$
(30)

where \(g_i\) is the statistical weight of the initial state \(i, k_1\) is the kinetic momentum of the incident electron, while the collision strength \(\varOmega _{ij}\) can be calculated by replacing the bound orbital in the final state with the free orbital of the ejected electron and summing over its angular momentum (see, e.g., Gu 2008). The total ionization cross section is obtained by integrating over the energy \(E_2\) of the ejected electron:

$$\begin{aligned} \sigma _{ij}(E_1) = \int \nolimits _0^{\frac{E_1-I}{2}} \sigma _{ij}(E_1, E_2) \mathrm{d}E_2\,, \end{aligned}$$
(31)

where I is the ionization energy. Note that \(E_1 =I + E_s + E_2\), where \(E_s\) is the energy of the scattered electron. By energy conservation, indicating with \(E_i\) the energy of the i state of the ion, we have: \(E_j - E_i = E_1 - E_2 - E_s\). Also note that the energy of the incident electron must be above threshold: \(E_1 > E_j - E_i \), and for the ejected electron to exist it must have an energy \(E_2 > E_1 - (E_j - E_i)\).

The total number of ionisations is found by integrating over the distribution of the incident (free) electron. The total ionisation rate between two ions can be obtained by summing the rate coefficients for each initial state i over the final states j, and then over all the initial states, although the main contribution to the total is typically the ionisation between the two ground states of the ions.

Note that ionization by direct impact (DI) is the main process, although for some isoelectronic sequences additional non-negligible ionisation can occur via inner-shell excitation into a state above the ionization threshold which then auto-ionizes. This is referred as excitation–autoionization (EA). Goldberg et al. (1965) were among the first to point out the importance of this process, which was later confirmed with experiments (see, e.g., Crandall et al. 1979). The EA provides additional contributions at higher energies of the incident electron, and increases with charge. This contribution is calculated by multiplying the inner-shell excitation cross section with the branching ratios associated with the doubly-excited level.

The main DI process occurs via neighboring ionisation stages, however double ionisation processes can sometimes be non-negligible, as e.g., shown by Hahn et al. (2017). More details and references about these processes can be found below in the atomic data section.

The study by Bell et al. (1983) presented a review of calculated and measured cross sections between ground states of the main ions relevant for astrophysics. This was a landmark paper which formed a reference for a long time. A significant revision of the collisional ionization by direct impact was produced by Dere (2007), where most of the DI and EA cross sections between ground states were recalculated and compared to experimental data, whenever available. Urdampilleta et al. (2017) recently also provided a review of ionisation rates, but without providing new calculations.

Assuming a Maxwellian distribution, and indicating for simplicity with E the energy of the incident electron, we have, as in Eq. (13), that the rate coefficient for collisional ionization is:

$$\begin{aligned} C^{\mathrm{I}}&= \left( {8 \over \pi \,m} \right) ^{1/2}\,(k T)^{-3/2} \int _{I}^\infty E\,\sigma _{ij} (E)\,{\mathrm{e}}^{-{E/kT}} \mathrm{d}E \nonumber \\&= \left( {8 \over \pi \,m} \right) ^{1/2}\,(k T)^{1/2}\,{\mathrm{e}}^{-{I/kT}} \int _0^\infty (k T x + I)\,\sigma _{ij} (k T x + I)\,{\mathrm{e}}^{-x}\,\mathrm{d}x, \end{aligned}$$
(32)

where we have applied the substitution \(E = k T x + I\).

The function in the integrand is very steep. For typical temperatures where the ions are formed in the solar corona in equilibrium, the dominant values the cross section to the integral are those from threshold until the peak. The integral in Eq. (32) is of the type

$$\begin{aligned} \int _0^\infty f(x)\,{\mathrm{e}}^{-x}\,\mathrm{d}x, \end{aligned}$$
(33)

and is therefore often evaluated using a Gauss–Laguerre quadrature: \(\sum _i w_i f(x_i)\), where \(x_i\) is the root of a Laguerre polynomial, and \(w_i\) is the weight:

$$\begin{aligned} C^{\mathrm{I}} = \left( {8 \over \pi \,m}\right) ^{1/2}\,(k T)^{1/2}\,{\mathrm{e}}^{-{I/kT}} \sum _i w_i\,\left( x_i + {I\over k T} \right) \,\sigma _{ij} (k T x_i + I). \end{aligned}$$
(34)

The numerical factor \(\sqrt{8/(\pi \,m)} = 5.287 \times 10^{13}\) as before in the case of excitation. Also note that the analogous expression in the landmark paper by Bell et al. (1983) (their Eq. 8), is incorrect (surprisingly).

Three body recombination is the inverse process of collisional ionization. If we indicate with \(C^{\mathrm{I}}_{ij}\) the collisional rate coefficient for ionisation by direct impact of the ion \(Z^{+r}\) in its state i to the ion \(Z^{+r+1}\) in its state j, the rate coefficient for the three body recombination \(C^{\mathrm{3B}}_{ji}\) can be obtained by applying the principle of detailed balance:

$$\begin{aligned} N_{\mathrm{e}}\,N_i(Z^{+r})\,C^{\mathrm{I}}_{ij} = N_{\mathrm{e}}\,N_j(Z^{+r+1})\,C^{\mathrm{3B}}_{ji}, \end{aligned}$$
(35)

which leads to

$$\begin{aligned} C^{\mathrm{3B}}_{ji} = C^{\mathrm{I}}_{ij} = {g_i \over g_j} {N_{\mathrm{e}} \over 2} \left( {h^2 \over 2\,\pi \,m\,k T_{\mathrm{e}}} \right) ^{3/2}\,{\mathrm{e}}^{-{(E_i -E_f)/kT_{\mathrm{e}}}}\,\end{aligned}$$
(36)

in the case of a Maxwellian electron distribution. In the more general case of non-Maxwellian distributions, a simple relation between the rates does not hold. The detail balance applied to the two processes leads to the Fowler relation between the differential cross-sections and the three body recombination rate involves an integral over the energies of the electrons involved in the process.

Photoionisation and radiative recombination

Photoionisation is the process by which a photon of energy higher than the ionisation threshold is absorbed by an ion, leaving the ion in the next ionisation stage. For the solar corona, photoionisation is normally a negligible process. For this reason, it is not discussed in detail here.

We note, however, that photoionisation can become important in a number of cases, for example for cool prominence material in the corona, and for low-temperature lines formed in the chromosphere/transition-region, especially during flares, when the photoionising coronal radiation can become significant. This particularly affects lines from H I and He I, He II, as photoionisation is followed by recombination into excited levels, which can then affect the population of lower levels via cascading. For a discussion of this photoionisation–recombination process for He see e.g., Zirin (1975) and Andretta et al. (2003).

The photoionisation rate coefficient from a bound level i to the continuum c is:

$$\begin{aligned} \alpha ^{\mathrm{PI}}_{ic} = 4 \pi \,\int \limits _{\nu _0}^\infty {\sigma ^{\mathrm{(bf)}}_{ic}(\nu ) \over h\nu }\,J_\nu \,\mathrm{d}\nu , \end{aligned}$$
(37)

where \(\nu _0\) is the threshold frequency below which the bound–free cross section \(\sigma ^{\mathrm{(bf)}}_{ic}(\nu )\) is zero.

The photoionisation cross-sections increase roughly as the cube of the wavelength, until threshold. For H-like ions, the modified Kramers’ semi-classical expression is often used:

$$\begin{aligned} \sigma ^{\mathrm{(bf)}}_\nu = {64 \pi ^4 m e^{10} \over 3 \sqrt{3} c h^6 }\,{Z^4\,g^{\mathrm{(bf)}} \over n^5\,\nu ^3} = 2.815 \times 10^{-29} {Z^4\,g^{\mathrm{(bf)}} \over n^5\,\nu ^3} \quad \nu \ge \nu _0, \end{aligned}$$
(38)

where n is the principal quantum number of the level from which the ion of charge Z is ionised. \(g^{\mathrm{(bf)}}\) is the dimensionless bound–free Gaunt factor (which is close to 1.), introduced as a correction. Values of the bound–free Gaunt factor for H-like ions are tabulated by Karzas and Latter (1961).

Quantum–mechanical calculations of photoionisation cross-sections for ions in general are quite complex. There are two main approaches required to generate opacities: the R-matrix method (Berrington et al. 1987) and the perturbative, or “distorted wave” (DW) method (see, e.g., Badnell and Seaton 2003). The R-matrix method is accurate but computationally expensive. The perturbative approach is much faster but approximates resonances with symmetric line profiles and neglects their interaction with the direct background photoionization. There is generally good agreement between the DW cross sections and the background R-matrix photoionization cross sections.

The Opacity Project (OP, see Seaton et al. 1994) involved many researchers under the coordination of M.J. Seaton (UCL) for the calculations of the cross-sections with the R-matrix method. A significant improvement was the inclusion of inner-shell data calculated with the DW method, which formed the Updated OP data (UOP Badnell et al. 2005). Further updates are made available via the web site: http://opacity-cs.obspm.fr/opacity/index.html maintained by F. Delahaye at the Paris Meudon Observatory. On a side note, there is an extended recent literature where other groups used similar approaches but found discordant results. Work within several groups is ongoing, to try and resolve the various issues, as they are of fundamental importance to calculate the opacities for stellar interiors.

Photoionisation cross-sections for transitions from the ground level are often available in the literature as analytic fits to the slow-varying component of the cross-sections (see, e.g., Verner and Yakovlev 1995).

Radiative recombination is the inverse process of photoionisation, i.e., when a free electron recombines with the ion and a photon is emitted. The cross-sections for recombination from a level f to a level i are normally calculated from the photoionisation cross-sections using the principle of detailed balance in thermodynamic equilibrium, which leads to the Einstein–Milne relation:

$$\begin{aligned} \sigma ^{\mathrm{(RR)}}_{fi} (E) = {g_i \over g_f} {(h \nu )^2 \over 2 m E c^2} \sigma ^{\mathrm{(bf)}}_{if}(E). \end{aligned}$$
(39)

Radiative recombination rates for all ions of astrophysical interest have recently been obtained by Badnell (2006b) using the photoionisation cross-sections calculated with the DW method (Badnell and Seaton 2003) and the principle of detail balance (i.e., for Maxwellian electron distributions). Finally, we note that the process of radiative recombination, induced by the radiation field, should also be included, although it is normally a negligible process for the solar corona.

Dielectronic recombination

Dielectronic recombination occurs when a free electron is captured into an autoionization state of the recombining ion. The ion can then autoionize (releasing a free electron) or produce a radiative transition into a bound state of the recombined ion. The transition can only occur at specific wavelengths. The process of dielectronic recombination was shown by Burgess (1964a, 1965) and Seaton (1964b) to be very important for the solar corona.

By applying the principle of detailed balance, the rate for dielectronic capture should equal the spontaneous ejection of the captured electron, i.e., autoionization. If we indicate with \(C^{\mathrm{dc}}_{nj}\) the rate coefficient for the capture of the free electron by the ion \(Z^{r+1}\) in the state n into a doubly-excited state j of the recombined ion \(Z^{+r}\), we have

$$\begin{aligned} N_n(Z^{r+1}) N_{\mathrm{e}} C^{\mathrm{dc}}_{nj} = N_j(Z^{+r}) A^{\mathrm{auto}}_{jn} \end{aligned}$$
(40)

where \(A^{\mathrm{auto}}_{jn}\) is the autoionization probability for the transition from the doubly-excited state j to the state n.

By applying the Saha equation for thermodynamic equilibrium, we obtain

$$\begin{aligned} C^{\mathrm{dc}}_{nj} = {h^3 \over (2\pi m kT)^{3/2}} {g_j \over 2 g_n} A^{\mathrm{auto}}_{jn} \exp \left( - {E_j - E_n \over kT} \right) \end{aligned}$$
(41)

which is a general formula that also holds outside of thermodynamic equilibrium (as long as the electrons have a Maxwellian distribution), and relates the rate for dielectronic capture to the autoionisation rate (the two inverse processes). The dielectronic capture can then be followed by a radiative stabilization into a bound state of the recombined ion. For coronal plasmas, this normally occurs with a decay of the excited state within the recombining ion.

To calculate the dielectronic recombination coefficient \(C^{\mathrm{d}}_{p,u,l}\) of an ion in a state p that captures an electron to form a state u of the recombined ion, which then decays to a stable state l we have

$$\begin{aligned} C^{\mathrm{d}}_{p,l,u} (T) = C^{\mathrm{dc}}_{pu}(T) \left[ {A_{ul} \over \sum _k A_{uk} + \sum _q A^{\mathrm{a}}_{uq}} \right] \end{aligned}$$
(42)

where the sum over k is over all states in the recombined ion \(Z^{r}\) that are below u, and the sum over q is over all possible states of the ion \(Z^{r+1}\) and the free electron associated with the autoionization of the doubly-excited state u.

Burgess (1964a, 1965) showed that for coronal ions the main DR process is capture into high-lying levels (\(n\approx 10\)–100), and obtained a general formula for the DR rates that provides a good approximation at low densities and has been used extensively in the literature. More recent DR rates have been computed from the autoionization rate for the reverse process, as described in the atomic data section.

Dielectronic recombination for ions of several isoelectronic sequences have been calculated in a series of papers by Badnell and colleagues (see e.g., Badnell et al. 2003). More details on these atomic data can be found in Sect. 5.

Charge transfer

Charge transfer is also an efficient ionisation/recombination process, but only at very low temperatures, and is normally expected to be negligible in the solar corona. In principle, in the low transition region, charge transfer could be an important process for some ions. For example, Baliunas and Butler (1980) discuss the effect of charge transfer between H, He and low charge states of Si, finding that, for example, the Si iii ion population becomes broader in temperature, with a peak around 30,000 K instead of 50,000 K. Such an effect could be significant on a range of spectroscopic diagnostic applications. Such charge transfer effects were included in Arnaud and Rothenflug (1985) but not in subsequent tabulations of ion abundances. In principle, it would be possible to estimate these effects. For example, Yu et al. (1986) used Si iii line ratios to obtain a temperature of about 70,000 K in a fusion device. The temperature of low-Z ions in fusion devices is normally close to that of ionization equilibrium, although particle transport effects could shift the ion populations towards higher temperatures.

Charge state distributions

In the case of local thermodynamic equilibrium (LTE), if we write the detailed balance of processes 1., 2., 3., we obtain the Saha equation. At low densities, plasma becomes optically thin and most of radiation escape, therefore processes 1. and 3. are attenuated and the plasma is no longer in LTE. In this condition, the degree of ionization of an element is obtained by equating the total ionization and recombination rates that relate successive ionization stages:

$$\begin{aligned} {1 \over N_{\mathrm{e}}} {{ dN}_{r}\over dt} =N_{r-1} S_{r-1} - N_{r} (S_{r} +\alpha _{r}) + N_{r+1}\alpha _{r+1} \end{aligned}$$
(43)

for transitions of ion \(Z^r\) from and to higher and lower stages, obtaining a set of coupled equations with the additional condition \(N(Z)=\sum _r N_{r}\). Here, \(S_{r}\) and \(\alpha _{r}\) are the total ionization and recombination rates, i.e., those that include all the relevant processes.

Figure 15 shows as an example the total ionization and recombination rates for a few oxygen ions, as available with CHIANTI version 8 (black). Note that in addition the plots also show the ionization and recombination effective rates calculated at a density \(N_{\mathrm{e}}=10^{12}\,\hbox {cm}^{-3}\), as available in OPEN-ADAS (see next section).

Fig. 15
figure 15

Ionization and recombination (total) rates for a few oxygen ions, as a function of temperature. The first plot in the top left (O i) shows the total ionization rate for neutral oxygen. The second plot (O ii) shows the total ionization rate for O ii, and the total recombination rate from O ii to O i, and so on. The full curves are the ionization rates, while the dashed lines are the recombination rates. The black curves are the CHIANTI v.8 rates (zero density), while the blue ones are the effective rates obtained from OPEN-ADAS, and calculated at a density \(N_{\mathrm{e}}=10^{12}\,\hbox {cm}^{-3}\)

Whenever the time scales of the observed phenomena are less than those for ionization and recombination, we can assume that the population of ions lying in a given state is constant \(\left( {{ dN}_{r}\over { dt}} =0\right) \) and so the number of ions leaving this state per unit time must exactly balance the number arriving into that state. This is the so called collisional ionization equilibrium (CIE), which is normally assumed for the solar corona (for very low densities, as in the case of planetary nebulae, photoionisation becomes important and dominates the ion charge state distribution). In the case of two successive stages:

$$\begin{aligned} {N_{r+1}\over N_{r}} = {S_{r} \over \alpha _{r+1}}. \end{aligned}$$
(44)

Many ionization equilibrium calculations have been published, perhaps the most significant for the iron ions was that one from Burgess and Seaton (1964), where the newly discovered dielectronic recombination was included. Later ones include: Jordan (1969), Arnaud and Rothenflug (1985), Landini and Monsignori Fossi (1991), Arnaud and Raymond (1992) and Mazzotta et al. (1998). Other calculations which also included density effects are those from Summers (1972, 1974).

The improvements in the rates as recalculated in recent years have led to a revision of the ion populations in equilibrium, published by Dere et al. (2009) for CHIANTI version 6. In some cases, significant differences with previous ionisation tables were present. An example is given in Fig. 16. Clearly, such large differences affect any measurements that rely on the ion abundances, such as temperatures from DEM analyses, or relative elemental abundances. Bryans et al. (2009) ion populations are based on almost the same atomic data and rates as used in CHIANTI v.6, so are very similar.

Fig. 16
figure 16

Ion populations of some important coronal ions in equilibrium in the low-density limit. Results from the tabulations of CHIANTI v.6, v.7.1 and Mazzotta et al. (1998) are shown

Finally, a few remarks about the formation temperature of a spectral line. This can be quite different from the temperature \(T_{\mathrm{max}}\), corresponding to the peak charge state of an ion, a common assumption in the past. This is often found especially in transition region lines, where the temperature gradient of the solar atmosphere is very steep. As we shall see below, there are indications that several structures in the solar corona are nearly isothermal, with temperatures typically very different from \(T_{\mathrm{max}}\). This issue is particularly relevant when lines from different ions are used for diagnostic applications such as measuring relative elemental abundances (see Sect. 14).

As a first approximation, an effective temperature \(T_{\mathrm{eff}}\) (cf. Eq. 93), where most of the line is formed, can be defined when a continuous distribution of temperatures in the plasma volume is present, as described below in Sect. 7.

Diffusion processes within the transition region, where strong flows and strong temperature gradients exist can also affect the temperature of formation of a line, see e.g., Tworkowski (1976).

Density-dependent effects on the ion balances

Most ionization equilibrium calculations are in the so-called low-density limit (the coronal approximation), where all the population in an ion is assumed to be in the ground state and all the rates are calculated at low densities.

However, as shown by Burgess and Summers (1969), the dielectronic recombination rates decrease significantly at high densities. This is caused by the fact that the intermediate excited states below the ionization limit can easily be re-ionized via electron impact, since this ionization increases linearly with the electron density. When this re-ionization occurs, recombination does not happen. The effect is particularly strong for lower charge states, i.e., transition-region ions, since at typical TR densities the dielectronic recombination becomes suppressed significantly. As a consequence, these ions become populated at progressively lower temperatures. To estimate this suppression, collisional-radiative (CR) modelling was carried out by Summers and Burgess, as described in Summers (1972, 1974), and further refined in following studies.

Recently, Nikolić et al. (2013) suggested an empirical formula to reproduce the suppression factors as calculated by Summers (1974) as a function of the ion charge, isoelectronic sequence, electron density, and temperature. The principle idea was that one would apply these suppression factors to the most recent dielectronic recombination results from the DR project (Badnell et al. 2003), to assess for the importance of this effect for a particular application. If the effect is found to be important, appropriate CR modelling should be carried out. We note that such suppression should only be applied to the total DR rates. One problem with the approach is that the recombination rates calculated by Summers (1974) were actually effective rates, i.e., included many other density effects in addition to the DR suppression, such as the three body recombination, and the changes related to the presence of metastable levels. Another problem is that the Nikolić et al. (2013) formulae were trying to reproduce the Summers (1974) tables which neglected secondary autoionization, with the result that the suppression would be over-estimated, if those factors were applied to the DR project rates, as they included secondary autoionization. Indeed for the boron, carbon, aluminium and silicon sequences Summers (1974) produced tables with secondary autoionization which show less suppression with density. A revision of the Nikolić et al. (2013) formulae is underway.

The effect of including the metastable levels in the ion balance calculations for transition region ions can be as important as the suppression of dielectronic recombination. An approximate calculation for C iv was carried out by Vernazza and Raymond (1979) using a rough estimate of the DR suppression based on Summers (1974) (assuming that it would be the same for all fine-structure levels), and adding collisional ionisation from the metastable levels in C iii. The two effects appeared equally important.

One way to estimate these effects is the generalised collisional-radiative (GCR) modelling (McWhirter and Summers 1984; Summers et al. 2006) which has been implemented within the Atomic Data and Analysis Structure (ADAS). Once the level-resolved ion balance equations are solved, effective ionisation and recombination rates can be obtained. These rates are currently available via OPEN-ADAS.Footnote 2

Figure 15 shows as an example the total effective ionization and recombination rates (blue curves) for a few oxygen ions, calculated at a density \(N_{\mathrm{e}}=10^{12}\,\hbox {cm}^{-3}\), as available in OPEN-ADAS (1996 version), compared to the CHIANTI v.8 rates (black curves, zero density). The effect of the suppression in the DR rates is clearly present in the ADAS effective recombination rates.

The effect of these processes on diagnostic ratios and predicted radiances has been discussed in the literature (cf. Vernazza and Raymond 1979; Del Zanna et al. 2002; Doyle et al. 2005). Generally, the predicted radiances increase by factors of 2–3, thus reducing the discrepancies with observation for the so-called anomalous ions, i.e., those that have predicted radiances typically a factor of 5–10 lower than observed, as discussed below in Sect. 7.3.

The density effect on the ion charge state distribution are particularly important for the IRIS O iv and S iv lines, as shown by Polito et al. (2016a). Figure 17 shows as an example the fractional O iv abundance in equilibrium at different electron densities (blue) as calculated with the OPEN-ADAS rates and the low-density value as in CHIANTI version 8.

Fig. 17
figure 17

Fractional O iv abundances in equilibrium at different electron densities (blue, from OPEN-ADAS), and the low-density values as in CHIANTI version 8 (black)

Optical depth effects

In most cases, spectral lines in the XUV originating from the solar corona and transition region are optically thin, i.e., the emitted photons freely escape. There are however several cases, especially for strong transition-region lines, where lines could be optically thick. One question naturally arises: how can we check if a spectral line is optically thin? There are various ways. One obvious effect of optical depth effects is a flattening, widening (or even self-reversal of the core) of the line profile. Estimates of the optical depth of lines from their width and their centre-to-limb variation have been obtained by e.g., Roussel-Dupre et al. (1979) and Doschek and Feldman (2004). However, a shape that is for example non-Gaussian does not necessarily mean that a line is not optically thin. In fact, a non-Gaussian shape can be produced by a superposition of Doppler motions. We further discuss such effects together with other broadening mechanisms below in Sect. 13.

Table 2 Some UV line ratios that can be used to investigate optical depth effects

A direct way to estimate if lines are optically thin is to consider the ratios of lines that originate from a common upper level (a branching ratio). In the optically thin case, the ratios are equal to the ratios of the A-values (see, e.g., Jordan 1967), which are normally known to a good accuracy. If opacity effects are present, the ratio would be different. Table 2 lists a few commonly used ratios from C II, Si II, Si III, and C III (see, e.g., Doyle and McWhirter 1980; Keenan and Kingston 1986; Brooks et al. 2000; Del Zanna et al. 2002).

Another direct method applies to the doublets of the Li-like ions. In the optically thin case, the ratios of the two lines of the doublet should be equal to the ratio of the oscillator strengths, i.e., two. Due to opacity effects, the intensity of the brightest component, the \(^2\hbox {S}_{1/2}{-}{}^2\hbox {P}_{3/2}\) typically decreases (relative to the weaker one), because it has a larger oscillator strength.

From the observed departure of a ratio from the optically thin case, the optical depths of the lines and the path lengths of the emitting layer can be obtained. The process is not straightforward though. Various approximations and approaches exist in the literature, involving the probability that a photon emitted from a layer of certain optical depth will escape along the line of sight. For details, see, e.g., Holstein (1947), Jordan (1967), Irons (1979), Doyle and McWhirter (1980), Kastner and Kastner (1990) and Brooks et al. (2000) and references therein.

We now introduce a simplified discussion of optical depth. The one-dimensional radiative transfer equation for the specific intensity of the radiation field that propagates, at frequency \(\nu \) along the direction \({\varvec{\Omega }}\) inside a plasma is:

$$\begin{aligned} {\mathrm{d} \over \mathrm{d}s} I_\nu ({\varvec{\Omega }}) = -k_\nu ^{(\mathrm{a})}\,I_\nu ({\varvec{\Omega }})\,+k_\nu ^{(\mathrm{s})} I_\nu ({\varvec{\Omega }})\,+\epsilon _\nu , \end{aligned}$$
(45)

where s is the spatial coordinate measured along the direction of propagation, and the three quantities \(k_\nu ^{(\mathrm{a})}, k_\nu ^{(\mathrm{s})}\), and \(\epsilon _\nu \), are the absorption coefficient, the coefficient of stimulated emission, and the emission coefficient, respectively. The equation is often written in this form:

$$\begin{aligned} {\mathrm{d} \over \mathrm{d}s} I_\nu ({\varvec{\Omega }}) = -k_\nu \,[I_\nu ({\varvec{\Omega }}) - S_\nu ], \end{aligned}$$
(46)

where the source function \(S_\nu = \epsilon _\nu /k_\nu \) and \(k_\nu = k_\nu ^{(\mathrm{a})} - k_\nu ^{(\mathrm{s})}\).

The specific optical depth

$$\begin{aligned} \mathrm{d} \tau _\nu = - k_\nu \,\mathrm{d}s \end{aligned}$$
(47)

is defined in the direction opposite to that of the propagation of the radiation, which reflects the point of view of an observer receiving the radiation. The optical thickness at frequency \(\nu \) of a plasma slab of geometrical thickness D is defined as:

$$\begin{aligned} \tau _\nu = \int _0^D k_\nu (s)\,\mathrm{d}s. \end{aligned}$$
(48)

A plasma is optically thick when \(\tau _\nu \gg 1\), i.e., when the photon has a probability practically equal to unity to be absorbed within the slab.

Considering a simple two-level atom with lower and upper level populations \(N_l, N_u\), it is possible to show that these monochromatic coefficients are related to Einstein’s coefficients:

$$\begin{aligned} k_\nu ^{(\mathrm{a})}= & {} {h\,\nu \over 4 \pi }\,N_l\,B_{lu}\,\phi (\nu - \nu _0); \quad k_\nu ^{(\mathrm{s})} = - {h\,\nu \over 4 \pi } \,N_u B_{ul}\,\chi (\nu - \nu _0);\quad \nonumber \\ \epsilon _\nu= & {} {h\,\nu \over 4 \pi }\,N_u\,A_{ul}\,\psi (\nu - \nu _0) \end{aligned}$$
(49)

where \( \phi (\nu - \nu _0), \chi (\nu - \nu _0), \psi (\nu - \nu _0)\) are the area-normalised line profiles for the extinction, induced emission, and spontaneous emission, respectively. \(\nu _0\) indicates the line centre frequency.

When the incident (exciting) and emitted photons are not correlated, we have complete redistribution and the three local line profiles are equal, in which case the source function \(S_\nu \) becomes frequency independent:

$$\begin{aligned} S_{\nu _0} = {N_u\,A_{ul} \over N_l\,B_{lu} - N_u B_{ul}}, \end{aligned}$$
(50)

and the bound–bound opacity of a line can be expressed in terms of the A-value as:

$$\begin{aligned} \tau _\nu = \int _0^D {c^2 \over 8\,\pi \,\nu ^2}\,{g_u \over g_l} \,N_l\,A_{ul}\,\left\{ 1- {N_u g_l \over N_l g_u} \right\} \,\phi (\nu - \nu _0) \mathrm{d}s, \end{aligned}$$
(51)

which could also be written in terms of the oscillator strength \(f_{lu}\) of the transition, using the relation:

$$\begin{aligned} g_u\,A_{ul} = {8 \pi ^2\,e^2\,\nu _{ul}^2 \over m\,c^3}\,g_l \,f_{lu}. \end{aligned}$$
(52)

The line centre optical thickness (or opacity) is an useful quantity to assess if a spectral line is optically thin:

$$\begin{aligned} \tau _{\nu _0} = \int k_{\nu _0} N_l \mathrm{d}s, \end{aligned}$$
(53)

where \(k_{\nu _0}\) is the absorption coefficient at line center frequency \(\nu _0\) which can be written, neglecting the induced emission and assuming a Doppler line profile:

$$\begin{aligned} k_{\nu _0} = {h\,\nu _0 \over 4 \pi }\,B_{lu} {1 \over \pi ^{1/2} \,\varDelta \nu _D}, \end{aligned}$$
(54)

where \( \varDelta \nu _D\) is the Doppler width of the line, in frequency (for more details on Doppler widths see Sect. 13).

Considering the relations between Einstein’s coefficients and the oscillator strength, and the expression for the population of the lower level (cf. previous notation):

$$\begin{aligned} N_l = {N_l \over N(Z^{+r})}\,{N(Z^{+r}) \over N(Z)}\,Ab(X)\,\frac{N_{\mathrm{H}}}{N_{\mathrm{e}}}\,N_{\mathrm{e}}, \end{aligned}$$

the opacity at line centre can be written as:

$$\begin{aligned} \tau _{0} = {\pi \,e^2\,\over m\,c} {1 \over \pi ^{1/2}\,\varDelta \nu _D} f_{lu}\,D = {\pi ^{1/2}\,e^2\,\over m\,c\,\varDelta \nu _D} f_{lu}\,D\,{N_l \over N(Z^{+r})}\,{N(Z^{+r}) \over N(Z)}\,Ab(X)\,\frac{N_{\mathrm{H}}}{N_{\mathrm{e}}}\,N_{\mathrm{e}}, \nonumber \\ \end{aligned}$$
(55)

where \(N_{\mathrm{e}} \) is the average electron density of the plasma emitting the line, and D is the path length along the line of sight through the source. This is a very useful definition commonly used, and easy to estimate, once the various factors are known. If \(\tau _{0}\) is of the order of one or less, opacity effects can be neglected. Even if \(\tau _{0}\) is much larger, it does not mean that photons do not escape. For example, values of \(\tau _{0} \simeq 10^4\) are needed for no photons to escape (Athay 1971).

With the above definition, we can show that even the strongest lines in the transition region are not affected by opacity in typical quiet Sun regions. Following Mariska (1992), we consider one of the strongest lines, the C iv 1548 Å resonance transition, which has \(f=0.2\). Assuming typical quiet Sun values of \(N_{\mathrm{e}}=10^{10}\,\hbox {cm}^{-3}\) and \(T=10^5\,\hbox {K}\) and observed values of the line width we have \(\tau _{0} \simeq 10^{-8}\,D\) as an order of magnitude (this value can differ depending on the chosen element and ion abundances). Therefore, a slab thickness larger than \(10^{8}\,\hbox {cm}\) would be needed before opacity effects come into play. This is more than the typical size of the region where C iv is emitted. Indeed opacity effects are typically seen in lower temperature lines formed in the chromosphere or occasionally in active regions.

Another effect that often decreases the intensity of a spectral line is absorption by cool plasma that is along the line of sight. The main absorption is by neutral hydrogen (with an edge at 912 Å), neutral helium (with an edge at 504 Å) and ionised helium (with an edge at 228 Å). Such effects were noticed long ago from Skylab observations (see e.g., Orrall and Schmahl 1976). They are particularly evident when surges or filaments are present.

From the observed absorption at different wavelengths and some assumptions about the underlying emission, it is possible to estimate the column density of the hydrogen and helium absorbing plasma. Kucera et al. (1998) used SOHO CDS while Del Zanna et al. (2004b) combined SOHO CDS with SUMER. Others have used the absorption seen in EUV images, such as SDO AIA (see, e.g., Williams et al. 2013).

Continuum radiation

Free–free emission is produced when an electron interacts with a charged particle Z and looses its kinetic energy \(E= m v^2/2\) releasing a photon of energy \(h\nu \): \( Z + e(E) \Longrightarrow Z + e(E') + h\nu \). The process is called bremsstrahlung (‘braking radiation’). The emission is a continuum. The calculation of the free–free emission in the classical way can be found in textbooks, and is based on the radiation emitted by a charged particle in the Coulomb field of the ion, with an impact parameter b. Classically, the minimum impact parameter is set so the kinetic energy of the free electron is greater than the binding energy: \(E > {e^2 Z \over b}\), otherwise we would have recombination. With a quantum–mechanical treatment, a correction factor needs to be introduced. This need to take into account that \(m v^2/2 \ge h\nu \), otherwise a photon could not be created. The correction is the free–free Gaunt factor \(g_{\mathrm{ff}}(\nu v)\) which is close to unity and has a weak dependence on the frequency \(\nu \) and the electron velocity/energy. The energy emitted per unit time, volume and frequency is

$$\begin{aligned} \frac{dW}{dt dV d\nu }= \frac{16 \pi }{3^{3/2}} \frac{Z^{2}e^{6}}{m^{2} v c^{3}} N_{\mathrm{e}} N_{\mathrm{i}} g_{\mathrm{ff}} \end{aligned}$$
(56)

where \(N_{\mathrm{e}} N_{\mathrm{i}}\) are the electron and ion number densities. We note that slight different definitions are found in the literature, depending on how the Gaunt factor is defined. We follow Rybicki and Lightman (1979).

For a Maxwellian velocity distribution of electrons the process is called thermal bremsstrahlung. The calculation of the emitted energy per unit of time and volume requires an integration over the electron velocity distribution. The integral with the Gaunt factor produces the mean Gaunt factor \(\langle g_{\mathrm{ff}}\rangle \) (see, e.g., Karzas and Latter 1961). We obtain:

$$\begin{aligned}&\displaystyle \frac{dW}{dt dV d\nu } = \frac{32 \pi e^6}{3 m c^3} \Big (\frac{2 \pi }{3 k m}\Big )^{1/2}~\frac{Z^{2} N_{\mathrm{e}} N_{\mathrm{i}}}{T_{\mathrm{e}}^{1/2}}~{\mathrm{e}}^{-{h\nu /kT_{\mathrm{e}}}}~\langle g_{\mathrm{ff}}\rangle \end{aligned}$$
(57)
$$\begin{aligned}&\displaystyle \frac{dW}{dt dV d\nu } = 6.8\times 10^{-38} \frac{Z^{2} N_{\mathrm{e}} N_{\mathrm{i}}}{T_{\mathrm{e}}^{1/2}}~{\mathrm{e}}^{-{h\nu /kT_{\mathrm{e}}}}~\langle g_{\mathrm{ff}}\rangle \,\,\, \mathrm{erg\,cm}^{3}\,\mathrm{s}^{-1}\,\mathrm{Hz}^{-1} \end{aligned}$$
(58)

where k is Boltzmann constant and h is the Planck constant.

Free–free emission is the main radiative loss mechanism for low density plasmas at \(T>10^7\,\hbox {K}\).

Free–bound emission is produced when a free electron of energy E is captured by an ion (\(Z^{r+1}\)) into a bound state of \(Z^{r}\):

$$\begin{aligned} Z^{r+1} + e(E) \Rightarrow Z_{n}^{r} + h \nu \end{aligned}$$

a photon of energy \(h \nu = E + I_{n}\) is emitted and \(I_n\) is the ionization energy of the bound state n. For a Maxwellian electron velocity distribution, the continuum emission is characterized by discontinuities at the ionization thresholds.

The free–bound continuum emissivity produced from recombination onto an ion of charge Z can be written as

$$\begin{aligned} P_{\mathrm{fb},\lambda }= & {} 3.0992 \times 10^{-52} N_{\mathrm{e}} N_{Z+1} {E_\lambda ^5 \over T^{3/2}} \sum _i {\omega _i \over \omega _0} \sigma ^{\mathrm{bf}}_i \exp \left( - {E_\lambda - I_i \over kT} \right) \quad \nonumber \\&\times (\mathrm{erg}~\mathrm{cm}^{-3}~\mathrm{s}^{-1}~\mathrm{~\AA }^{-1}) \end{aligned}$$
(59)

where \( N_{\mathrm{e}}\) and \(N_{Z+1}\) are the number densities of electrons and recombining ions, respectively, in units of \(\hbox {cm}^{-3}; E_\lambda \) is the energy in \(\hbox {cm}^{-1}\) of the emitted radiation; T is the plasma temperature in K; \(\omega _i\) is the statistical weight of the level i in the recombined ion; \(\omega _0\) is the statistical weight of the ground level of the recombining ion; \(\sigma ^{\mathrm{bf}}_i\) is the photoionization cross-section from the level i in the recombined ion to the ground level of the recombining ion, in units of Mb (\(=10^{-18}\,\hbox {cm}^2\)); \(I_i\) is the ionization energy in units of \(\hbox {cm}^{-1}\) from the level i in the recombined ion, and the sum is over all levels i below the recombined ion’s ionization limit.

Finally, there is another process which produces continuum radiation, the so-called Two-photon continuum. It is caused by two-photon decay processes in H-like and He-like ions. Compared to the free–free and the bound–free, this continuum is nearly negligible, except at low temperatures \(\hbox {T}\le 3 \times 10^4\,\hbox {K}\). However, it is important in the population modelling of such ions. The transition from the metastable \(1s\,2s\, ^1\hbox {S}_0\) state of Helium-like ions to their ground state \(1{s}^2\,{}^1\hbox {S}_0\) is strictly forbidden, hence the two-photon process becomes an important depopulating process. The same occurs in H-like ions, where the transition from the metastable \(2s\,{}^2\hbox {S}_{1/2}\) state to the ground state \(1s\,{}^2\hbox {S}_{1/2}\) is also strictly forbidden. Calculations of the rates for these processes have been carried out by several authors, see e.g., Drake (1986) for He-like and Parpia and Johnson (1982) for the H-like ions.

The available atomic data (including relativistic effects) for the continuum were assessed for CHIANTI version 3 by Young et al. (2003). Figure 18 shows the continuum calculated with CHIANTI version 8 at a temperature of 10 MK, from 5 to 200 Å. The black curves are calculated with the photospheric abundances recommended by Asplund et al. (2009), while the blue ones with the ‘coronal’ abundances of Feldman (1992a). Note that at the EUV wavelengths, most of the continuum is free–free radiation. Also note the significant variation with elemental abundances in the X-rays. This means that when chemical abundances are estimated from line-to-continuum measurements (see below in Sect. 14), a self-consistent approach needs to be adopted.

Fig. 18
figure 18

The continuum calculated with CHIANTI version 8 at a temperature of 10 MK, from 5 to 200 Å. The black curves are calculated with the photospheric abundances recommended by Asplund et al. (2009), while the red ones with the ‘coronal’ abundances of Feldman (1992a)

The satellite lines

Satellite lines were discovered by Edlén and Tyrén (1939) in laboratory vacuum spark spectra of the He-like carbon. They were called satellites because they were close to the resonance line, mostly at longer wavelengths, although some were also present at shorter wavelengths. They were correctly interpreted as \(1{s}^2\,nl\)\(1s\,2p\,nl\) transitions, satellites of the He-like resonance transition \(1{s}^2{-}1{s}\,2p\) (the so-called parent line). They also observed the \(1{s}^2\,nl\)\(1s\,3p\,nl\) transitions, satellites of the He-like \(1{s}^2\)\(1s\,3p\) transition, and the \(1s\,nl{-}2{p}\,nl\) satellites of the H-like 1s–2p resonance line.

Satellite lines have later been observed in solar spectra, and soon it was recognised that they have many important and unique diagnostic applications.

Fig. 19
figure 19

A sketch of a few of the main satellite lines of the He-like resonance line w

The satellite lines of the He-like ions involve the presence of doubly-excited states of the Li-like ions, which can then auto-ionize or produce the satellite line (see Fig. 19). The doubly excited states exist because of the interaction with at least one continuum, in this case, the continuum of the ground state of the He-like ions. The formation of the doubly-excited state follows the usual selection rules in that the autoionising state and the final state plus free electron have to have the same parity and total angular momentum J (and L and S in case of LS coupling).

In principle the doubly excited states can be formed by inner-shell electron impact ionisation of the Be-like ion or by two-electron impact excitation, but these processes are normally negligible, compared to the inner-shell excitation of one electron in the Li-like ion (which can subsequently auto-ionize or produce the satellite line), or dielectronic capture, the inverse process of autoionisation. Both processes need to be calculated. The relative importance of the two then depends on the relative values of the relevant rates, as well as the status of the ion (i.e., if it is out of equilibrium, ionising or recombining).

Dielectronic capture occurs when a free electron is captured into an autoionization state of the recombining Li-like ion. The ion can then autoionize (releasing a free electron) or produce a satellite line, i.e., a transition into a bound state of the recombined ion. The transitions occur at specific wavelengths, close to the wavelength of the He-like resonance line. As we have mentioned in Sect. 3.5, Burgess with Seaton developed the theory of dielectronic recombination, however those calculations were focused on the states where the spectator electron is in a highly excited state with n typically around 100, as these give the highest contribution to the total dielectronic recombination of an ion.

On the other hand, for spectral diagnostic purposes, the transitions that are important and are observed are those arising from lower autoionising levels.

Early theories and computations for these satellite lines were initially carried out by Gabriel and colleagues in the late 1960s and early 1970s, and later by a number of researchers, in Europe, Russia and USA. Notable papers, where the main diagnostics and notation were developed for the He-like lines are Gabriel and Paget (1972) and Gabriel (1972a). Several excellent laboratory observations of the satellite lines exist, taken by Doschek, Feldman, Presnyakov, Boiko, Aglitskii and several other colleagues (see, e.g., Aglitskii et al. 1974; Feldman et al. 1974; Boiko et al. 1977, 1978a, b). There are several excellent in-depth reviews on the satellite lines, such as Dubau and Volonte (1980), Doschek (1985, 1990), Mewe (1988) and Phillips et al. (2008).

Below, we summarise the main processes and highlight some of the main diagnostic applications for the solar plasma. Many more diagnostic applications are available for other types of plasma.

Inner-shell of Li-like ions

We consider first the inner-shell excitation. The simultaneous excitation of two electrons has a low probability and therefore the \(1s\,2{p}^2\) population comes mainly from excitation of the excited state \(1{s}^2\,2p\). On the other hand, the \(1s\,2s\, 2p\) can be excited from either \(1{s}^2\,2s\) or \(1{s}^2\,2p\), although the first one usually dominates. Following Gabriel (1972a), the intensity of the satellite line from level s to the final level f produced by inner-shell excitation is (in photon units):

$$\begin{aligned} I^{\mathrm{inner}}_{{ sf}} = \beta \,N_{\mathrm{Li}{\text {-}}\mathrm{like}}\,N_{\mathrm{e}}\,C^e_{{ is}} \,{A_{{ sf}} \over \left( \sum _k A^{\mathrm{auto}}_{sk} + \sum _{f<s} A_{{ sf}} \right) } \sim \beta N_{\mathrm{Li}{\text {-}}\mathrm{like}}\,N_{\mathrm{e}}\,C^e_{{ is}}\,{A_{{ sf}} \over A^{{ tot}}_s}, \end{aligned}$$
(60)

where \( N_{\mathrm{Li}{\text {-}}\mathrm{like}}\) is the abundance of the Li-like ion; \( C^e_{{ is}}\) is the electron impact excitation rate coefficient by inner-shell; \(A_{{ sf}}\) the transition probability by spontaneous emission from level s to the final level \(f; \sum _k A^{\mathrm{auto}}_{{ sk}}\) is the total decay rate via autoionisation from level s to into all available continua k in the He-like ion; and \(\sum _{f<s} A_{{ sf}} \) is the total decay rate by spontaneous radiative transitions to all possible final states f. For He-like ions, the main autoionisation is to the ground configuration of the recombining ion (which is a single level), so the \(\sum _k A^{\mathrm{auto}}_{sk}\) is normally replaced by \(A^{\mathrm{auto}}_s\), the total autoionisation rate into the He-like ground-level continua. The ratio of A values \(A_{{ sf}}/A_s^{{ tot}}\) on the right of the equation is basically a branching ratio. This is normally small, meaning that the majority of inner-shell excitations decay by autoionisation, rather than by emission of the satellite line.

\(\beta \,N_{\mathrm{Li}{\text {-}}\mathrm{like}}\) is the population of the lower level which populates the level s by collisional excitation. The value of \(\beta \) is obtained by solving the level population for the Li-like ion. At coronal densities, the population of the Li-like ions is all in the ground state, so \(\beta =1\) for the \(1s\,2s\, 2p\), and \(\beta =0\) for the \(1s\,2p^2\). At increasing densities, \(\beta \) varies but can easily be calculated.

The ratio of the satellite line produced by inner-shell excitation to the resonance line w in the He-like ion is an excellent diagnostic for measuring the relative population of the Li-like versus the He-like ion, i.e., to assess if departures from ionisation equilibrium are present. In fact, since most of the population of the He-like ion \( N_{\mathrm{He}{\text {-}}\mathrm{like}}\) is in the ground state g, the intensity of the resonance line w can be approximated with

$$\begin{aligned} I_w = N_u A_{ug} \simeq N_{\mathrm{He}{\text {-}}\mathrm{like}} N_{\mathrm{e}} C^e_{{ gu}} \end{aligned}$$
(61)

where \(C^e_{{ gu}}\) is the excitation rate from the ground state. Therefore, the ratio of the satellite line with the resonance is approximately

$$\begin{aligned} {I^{\mathrm{inner}}_{{ sf}} \over I_w} = {\beta N_{\mathrm{Li}{\text {-}}\mathrm{like}} \over N_{\mathrm{He}{\text {-}}\mathrm{like}}} {C^e_{{ is}} \over C^e_{{ gu}}} {A_{{ sf}} \over A^{\mathrm{tot}}_s} \end{aligned}$$
(62)

which is independent of electron density. We recall that the electron collisional excitation rate \(C^e_{ij} \sim T_{\mathrm{e}}^{-1/2} {\varUpsilon _{i,j}(T_{\mathrm{e}}) \over \omega _i} \exp \left( {- \varDelta E_{i,j} \over kT_{\mathrm{e}}} \right) \), so the ratio does not depend much on electron temperature, because the excitation energies \(\varDelta E_{i,j}\) of the two transitions are very similar, so also the exponential factors are very similar. In summary, the ratio of the satellite line with the resonance line depends directly on the relative population of the Li-like versus the He-like ion, aside from some factors which depend on the atomic data. In the original study by Gabriel (1972a), the actual collision strengths were approximated with effective oscillator strengths.

The above ratio is usually small, because the branching ratio \(A_{{ sf}}/A^{\mathrm{tot}}_s\) is small, and because the \(N_{\mathrm{Li}{\text {-}}\mathrm{like}}\) abundance is normally lower than \(N_{\mathrm{He}{\text {-}}\mathrm{like}}\). Finally, we note that a correction factor to the resonance line intensity needs to be included. This factor should take into account the increase in the line intensity due to \(n>3\) satellite lines formed by dielectronic recombination (see below). Once the ratio of the satellite line with the resonance is measured, the relative population of the Li-like versus the He-like ion can be obtained, and compared to the values predicted by assuming ionisation equilibrium. In this way, one can assess if the plasma is ionising or recombining.

He-like satellite lines formed by dielectronic capture

As mentioned earlier, the dielectronic capture of a free electron colliding with the He-like ion can produce an autoionising state s of the Li-like ion, which can then produce a satellite line. The intensity of the satellite line, decay to the final level f is

$$\begin{aligned} I^{\mathrm{dr}}_{{ sf}} = N_s A_{{ sf}} \end{aligned}$$
(63)

where as before \(A_{{ sf}}\) the transition probability by spontaneous emission from level s to the final level f, and \(N_s\) is the population of the autoionising state, which is determined by the balance between the dielectronic capture (with rate \(C^{\mathrm{dc}}\)), autoionisation and radiative decay to all possible lower levels:

$$\begin{aligned} N_{\mathrm{He}{\text {-}}\mathrm{like}} N_{\mathrm{e}} C^{\mathrm{dc}} = N_s \left( \sum _k A^{\mathrm{auto}}_{sk} + \sum _{f<s} A_{{ sf}} \right) \end{aligned}$$
(64)

where we have seen that the sum of autoionising rates in this case reduces to a single term \(A^{\mathrm{auto}}_s\), the total autoionisation rate into the He-like ground-level continua. The intensity of the satellite line can therefore be written as

$$\begin{aligned} I^{\mathrm{dr}}_{{ sf}} = N_{\mathrm{He}{\text {-}}\mathrm{like}} N_{\mathrm{e}} C^{\mathrm{dc}} {A_{{ sf}} \over A^{\mathrm{auto}}_s + \sum _{f<s} A_{{ sf}}}. \end{aligned}$$
(65)

We have seen in Sect. 3.5 that, by applying the Saha equation for thermodynamic equilibrium we obtain a direct relation between the rates of the two inverse processes of dielectronic capture and autoionisation (Eq. 41), which in our case is

$$\begin{aligned} C^{\mathrm{dc}} = {h^3 \over (2\pi \,m\,kT_{\mathrm{e}})^{3/2}} {\omega _s \over \omega _1} \exp \left( - {E_s - E_1 \over k T_{\mathrm{e}}} \right) A^{\mathrm{auto}}_{s} \end{aligned}$$
(66)

where \(\omega _s, \omega _1\) are the statistical weights of the level s and the ground state of the He-like ion, and \(E_s - E_1\) is the energy difference between the two states.

The intensity of the satellite line formed by dielectronic recombination can therefore be written as

$$\begin{aligned} I^{\mathrm{dr}}_{{ sf}} = 3.3\times 10^{-24} N_{\mathrm{He}{\text {-}}\mathrm{like}} N_{\mathrm{e}} {I_H \over (kT_{\mathrm{e}})^{3/2}} {\omega _s \over \omega _1} \exp \left( - {E_s - E_1 \over k T_{\mathrm{e}}} \right) {A_{{ sf}} A^{\mathrm{auto}}_s \over A^{\mathrm{auto}}_s + \sum _{f<s} A_{{ sf}}}\nonumber \\ \end{aligned}$$
(67)

where \(I_H\) is the ionisation energy of neutral hydrogen (13.6 eV).

The ratio of the intensities of the satellite line formed by dielectronic recombination with the parent line is therefore

$$\begin{aligned} {I^{\mathrm{dr}}_{{ sf}} \over I_w} \simeq 3.3\times 10^{-24} {I_H \over (kT_{\mathrm{e}})^{3/2} C^e_{{ gu}}} {\omega _s \over \omega _1} \exp \left( - {E_s - E_1 \over kT_{\mathrm{e}}} \right) {A_{{ sf}} A^{\mathrm{auto}}_s \over A^{\mathrm{auto}}_s + \sum _{f<s} A_{{ sf}}}\qquad \end{aligned}$$
(68)

i.e., is independent of the population of the ions and the electron density. We recall the electron collisional excitation rate \(C^e_{{ gu}}\) depends on the electron temperature, so in effect the above ratio depends only on \(T_{\mathrm{e}}\), aside from the transition probabilities. In other words, the ratio is an excellent temperature diagnostic. It depends on \(T_{\mathrm{e}}\) as

$$\begin{aligned} {I^{\mathrm{dr}}_{{ sf}} \over I_w} \sim T_{\mathrm{e}}^{-1/2} \exp \left( {E_0 - E_s + E_1 \over kT_{\mathrm{e}}} \right) \end{aligned}$$
(69)

where \(E_0\) is the energy of the resonance transition. In normal coronal conditions, \(E_0 - E_s + E_1 < kT_{\mathrm{e}}\) so the exponential factor is a slowly varying function of \(T_{\mathrm{e}}\), so effectively the intensities of the satellite lines increase, with respect to the resonance line, when the electron temperature is lower. Indeed, observations show that during the gradual phase of solar flares, when the temperature decreases, the satellite lines become prominent.

When considering ions along the isoelectronic sequence, since \(A^{\mathrm{auto}}_s \) does not depend much on Z, the main dependence with Z comes from the transition probability \(A_{{ sf}}\), which scales as \(Z^4\). Therefore, while the intensities of the satellite lines of say oxygen are weak, they become significant in heavier elements like iron. Indeed the best measurements of the satellite lines are from iron. The literature usually follows the notation from Gabriel (1972a), which is noted here in Table 3.

Table 3 List of the main He-like satellite lines, following the notation of Gabriel (1972a)

Satellite lines of other sequences

The process of formation of satellite lines is a general one, and satellite lines are present also for ions of other sequences. We refer to the above-cited reviews for more details, but would like to mention two important sets of satellites. The most important ones are those of H-like ions.

H-like satellite lines have some similarities and differences with the He-like ones. The main difference is that the only process of formation is dielectronic capture. The main satellites have, as in the He-like case, the \(n=2\) parent line, the \(\hbox {L-}\alpha \) doublet, the decay of the \(2p\,^2\hbox {P}_{3/2,1/2}\) to the ground state (see Fig. 20).

The \(2p\,{}^2\hbox {P}_{3/2,1/2}\) and \(2s\,{}^2\hbox {S}_{1/2}\) are mainly collisionally excited from the ground state. The 2s \(^2\hbox {S}_{1/2}\) decays mainly via a two-photon emission at low densities (even for solar flare conditions), and collisional transfer between the \(2s\,{}^2\hbox {S}_{1/2}\) and \(2p\,{}^2\hbox {P}\) is negligible. For a list of the main transitions and description of the calculations see, e.g., Boiko et al. (1977, 1978b).

Fig. 20
figure 20

A sketch of the process of formation of satellite lines of the H-like \(n=2\) resonance line

Satellite lines in ions of other sequences are also important at some wavelengths. Most notably, the region between 1.85 and 1.9 Å (cf. Fig. 22) is reach in satellite lines from Fe xx, Fe xxi, Fe xxii, and Fe xxiii, as discussed e.g., in Doschek et al. (1981a), where a detailed list of transitions is provided. The most notable transition is the Fe xxiii inner-shell \(\beta \) line, which can be used, in conjunction with the He-like Fe xxv, to assess for departures from ionisation equilibrium (Gabriel 1972b).

Measuring departures from Maxwellian distributions

The fact that the satellite lines can only be excited by free electrons having specific energies means that, if one could measure the intensities of several satellites, information on the energy distribution of the free electrons could be obtained. A method to assess if non-Maxwellian distributions exist during solar flares was suggested by Gabriel and Phillips (1979). It relies on the measurement of the j and \(n=3\) complex of satellites in the He-like Fe, of which the transition d13 is the main one.

Fig. 21
figure 21

A Maxwellian distribution of electrons at 15 MK, with the energies of the satellite lines j and d13, and the threshold energy to excite the resonance line w

The j and \(n=3\) complex are produced by electrons having two specific energies, shown in Fig. 21 with two vertical lines, superimposed on a Maxwellian distribution of a plasma with 15 MK, a temperature typical of moderately-large solar flares.

On the other hand, any electron with energy above the resonance w threshold can excite the line. Those energies are indicated with the grey area under the Maxwellian curve. If the electron distribution is Maxwellian, then the temperatures obtained from the observed j / w and d13 / w and the above prescriptions should be consistent, assuming that all the atomic data are accurate. On the other hand, if a tail of high-energy electrons is present (as it is known to be present for large flares from hard X-ray bremsstrahlung observations), then the resonance line would be greatly enhanced, as all the electrons above the threshold would increase the collisional population of the upper level of the resonance line. The temperatures obtained (with the Maxwellian assumption) from the observed j / w and d13 / w should therefore be different.

The method does not allow one to infer the complete electron distribution, but just to confirm if the distribution is Maxwellian or not. Several measurements of satellite lines that can only be excited at progressively higher energies would be needed to describe the electron distribution.

The d13 line complex has been resolved in laboratory plasmas, and the effects of non-thermal electrons on the spectra have been detected (see, e.g., Bartiromo et al. 1983, 1985; Lee et al. 1985). Satellite lines have also been observed in solar spectra (see next section), and some applications of the present diagnostics have been published. They are summarised later in Sect. 6, where we discuss non-equilibrium effects.

Observations

Among the earliest observations, there are those reported by Neupert and Swartz (1970) and Neupert (1971a) obtained with the OSO-5, where the satellite lines from the \(1s\,2s\,2p\) configuration in the He-like Si, S, Ar, and Fe ions were observed. Walker and Rugge (1971) observed several satellites of the H- and He-like Mg, Al, Si and S ions, form the OV1-17 satellite. Acton et al. (1972) observed the satellites of the He-like O and Ne using a rocket, while Parkinson (1971, 1972) and Parkinson et al. (1978) provided excellent spectra of He-like satellites of Ne, Mg and Si obtained with Skylark rockets and with OSO-8 observations of active regions and flares. Parkinson et al. (1978) also reported observations of the H-like Si xiv with OSO-8.

Doschek et al. (1971a, b) presented an analysis of the observations of the He-like Ca and Fe obtained with the NRL Bragg crystal spectrometer on-board OSO-6. Further earlier observations were those of the He-like Fe obtained by the Intercosmos-4 Satellite and the ‘Vertical-2’ Rocket (Grineva et al. 1973), and of the H-like lines by the same rocket and Intercosmos-4 and Intercosmos-7 (Aglitskii et al. 1978).

Later, several observations were obtained with the excellent NRL SOLFLEX instrument (cf. Doschek et al. 1980, 1981a; Feldman et al. 1980) and then from the crystals on-board the Hinotori, Yohkoh and SMM satellites.

Fig. 22
figure 22

An SMM BCS spectrum during the peak phase of a flare, containing the He-like lines from iron and many satellites, some of which are labelled

Fig. 23
figure 23

An SMM BCS spectrum during the peak phase of a flare, containing the Ca He-like lines and associated satellites

Fig. 24
figure 24

Figure from Phillips et al. (2004)

A Yohkoh BCS spectrum of the He-like lines from S and associated satellites. The dashed line is a theoretical spectrum

Figure 22 shows as an example a spectrum obtained by SMM BCS, the He-like lines from iron, while Fig. 23 shows the corresponding spectrum of the He-like Ca lines. In addition, Fig. 24 shows a Yohkoh BCS spectrum of the sulphur He-like lines and associated satellites during a flare.

Atomic calculations and data

This review is primarily concerned with diagnostics of electron densities and temperatures from spectral lines of the same ion. In this case, the primary atomic data are the cross-sections for electron-ion collisions and the radiative data (A-values), with their uncertainties. We therefore mainly review the latest work on these atomic data, and where references and data can be obtained, with emphasis on the CHIANTI database. We also introduce to the unfamiliar reader a few basic aspects about the relevant atomic calculations for such data. Finally, we also briefly review ionization and recombination data, line identification and benchmarking studies.

The CHIANTI atomic package

The CHIANTI package consists of a critically evaluated set of atomic data for a large number of ions of astrophysical interest. It also includes a quantity of ancillary data and a suite of Interactive Data Language (IDL) programs to calculate optically thin synthetic spectra (including continuum emission) and to perform general spectral analysis and plasma diagnostics, from the X-rays to the infrared region of the spectrum.

The CHIANTI database started in 1995 as a collaboration between K. P. Dere (NRL, USA), H. E. Mason (University of Cambridge UK), and B. Monsignori Fossi (Arcetri Observatory, Italy). Sadly, the latter died prematurely in 1996. M. Landini (University of Florence, Italy) then worked with the CHIANTI team for several years. Currently, the project involves K. P. Dere and P. R. Young (George Mason University, USA), E. Landi (University of Michigan, USA), and G. Del Zanna and H. E. Mason (University of Cambridge, UK).

The database relies almost entirely on the availability of published atomic data. Each version release is accompanied by a publication that discusses the new atomic data and any other new features of CHIANTI. References to the original calculations, line identifications, wavelength measurements and benchmark work are provided in the published papers and throughout the database. It is very important to acknowledge the hard work of the atomic data providers.

The CHIANTI package is freely available at the web page: http://www.chiantidatabase.org/ and within SolarSoft, a programming and data analysis environment written in IDL for the solar physics community. There is also a Python interface to some of the CHIANTI routines (CHIANTIPy: http://chiantipy.sourceforge.net/).

The first version of the CHIANTI database was released in 1996 and is described in Dere et al. (1997). v.2 is described in Landi et al. (1999). In v.3 (Dere et al. 2001) the database was extended to wavelengths shorter than 50 Å, while v.4 (Young et al. 2003) included proton excitation data, photoexcitation, and new continuum routines. v.5 (Landi et al. 2006) included new datasets, while v.6 (Dere et al. 2009) included a complete database of ionization and recombination coefficients. v.7 (Landi et al. 2012) improved ions important for the EUV and the UV, while v.7.1 (Landi et al. 2013) included new DW atomic data for the soft X-ray lines and new identifications from Del Zanna (2012a). V.8 (Del Zanna et al. 2015b) included new atomic data for several isoelectronic sequences and for a few coronal ions from iron and nickel.

As with any atomic data package, CHIANTI has been developed to suit some specific applications in solar physics and astrophysics, although with time it has become more generally used and is often used as a reference atomic database rather than a modelling code. CHIANTI basic data are in fact included in many other atomic databases and modelling packages, for example:

The main assumptions within the CHIANTI codes are that the plasma is optically thin and in thermal equilibrium. The plasma ionization is dominated by collisions (i.e., no photo-ionization is included). Tabulations of ion populations in equilibrium are provided, in the low-density limit.

Line emissivities are reliable only within given temperature and density ranges. The maximum densities for an ion model depend on which excitations/de-excitations are included. Normally, all those from the ground configuration and the main metastable levels are included, although for most ions the complete set of excitation rates has been included in v.8 (Del Zanna et al. 2015b). This allows the use of the CHIANTI data for higher densities. The electron collision strengths were, until v.8, interpolated over a fixed temperature grid in the scaled domain as formulated by Burgess and Tully (1992). However, this occasionally reduced the accuracy at lower temperatures, typical of photoionised plasma. The rates for the new ions in v.8 were normally released as published, without interpolation.

Electrons and protons have Maxwellian distribution functions. Indeed CHIANTI data include Maxwellian-averaged electron and proton collision strengths. However, it is possible to study the effects of particle distributions that are linear combinations of Maxwellians of different temperatures. An implementation of the CHIANTI programs that calculates line emissivities assuming \(\kappa \) distributions has recently been made available (Dzifčáková et al. 2015), based on v.7.1.

Finally, various other databases of atomic data exist, e.g., ADAS (http://open.adas.ac.uk, primarily for fusion work) and ATOMDB (http://www.atomdb.org, primarily for high energies).

Atomic structure calculations

We refer the reader to standard textbooks of atomic spectroscopy such as Condon and Shortley (1935), Grant (2006) and Landi Degl’Innocenti (2014), the various articles cited below, and the Springer Handbook of Atomic and Molecular Physics (Drake 2006), in particular Chapter 21 on atomic structure and multi-configuration Hartree–Fock theories by Charlotte Froese Fischer, and Chapter 22 by Ian Grant on relativistic atomic structure. For recent reviews of multiconfiguration methods for complex atoms see Bieroń et al. (2015) and Froese Fischer et al. (2016).

The atomic structure of an ion is obtained by solving the time-independent Schrödinger equation:

$$\begin{aligned} H\varPsi _i = E_i \varPsi _i \end{aligned}$$
(70)

where \(\varPsi _i\) are the wavefunctions of the system, \(E_i\) are the eigenvalues, and H is the Hamiltonian.

For lighter elements of astrophysical importance, the approach that is often adopted is to describe the system with a non-relativistic Hamiltonian, and then add relativistic corrections using nonrelativistic wavefunctions and the Breit–Pauli approximation. For heavier elements, a fully relativistic approach is usually needed. This involves solving the Dirac equation and taking into account the Breit interaction (see, e.g., Grant 2006).

The non-relativistic Hamiltonian describing an ion with N electrons and a central nucleus having charge number Z can be written in the form

$$\begin{aligned} {H} = \sum _{i=1}^N \left( {p_i^2 \over 2\,m} - {Z e^2 \over r_i} \right) + \sum ^{N-1}_{i < j} {e^2 \over r_{ij}}, \end{aligned}$$
(71)

where \(\mathbf {r}_i\) is the position vector of the ith electron (relative to the nucleus), \(\mathbf {p}_i\) is its momentum, and \(r_{ij}\) is the distance between the i and j electrons: \( r_{ij} = r_{ji} = \left| \mathbf {r}_i - \mathbf {r}_j \right| \). The terms in the Hamiltonian are the total kinetic and potential energy of the electrons in the field of the nucleus, and the repulsive Coulomb energy between the electrons. This equation neglects the spin of the electron.

It can be shown that this Hamiltonian commutes with the total angular momentum L and the total spin S of the electrons. Additionally, it commutes with the parity operator. This allows the usual description of the wavefunction in terms of the associated good quantum numbers LSp.

For complex highly-ionised atoms, it is common to assume that the electron repulsions are small perturbations relative to the much stronger nuclear central potential, and to simplify the equation by assuming that each electron experiences a central potential, caused by the electrostatic interaction with the nucleus, screened by the other electrons. In this way, the zero-order Hamiltonian is first solved:

$$\begin{aligned} {H}_0 = \sum _{i=1}^N \left( {p_i^2 \over 2\,m} + V_{\mathrm{c}}(r_i) \right) \end{aligned}$$
(72)

where \(V_{\mathrm{c}}(r_i)\) is the potential energy of each electron in the central field. The first-order Hamiltonian

$$\begin{aligned} {H}_1 = \sum _{i=1}^N \left( -V_{\mathrm{c}}(r_i) -{Z e^2 \over r_i} \right) + \sum _{i<j} {e^2 \over r_{ij}}. \end{aligned}$$
(73)

is then considered as a perturbation. There are several approximate numerical methods to find a solution for the central potential and to evaluate \(V_{\mathrm{c}}(r)\). For example, the Thomas–Fermi statistical method and the Hartree–Fock self-consistent method using the variational principle.

When configuration interaction (CI) is considered, one atomic state is described by a linear combination of eigenfunctions of different configurations (but with the same parity, as \(H_1\) commutes with the parity operator). Configuration interaction becomes increasingly important for excited states whenever the difference in energy between the states becomes small. The Hamiltonian \({H}_1\) commutes with the operators \(L^2\) and \(S^2\), so the total L and S values are still good quantum numbers representing the state.

There are several relativistic corrections (see, e.g., Bethe and Salpeter 1957), some of which arise directly from the solution of the Dirac equation. They are normally grouped into two classes, one-body and two-body operators (see, e.g., Badnell 1997). The non-fine-structure operators commute with the operators associated with the total angular momentum L and the total spin S of the electrons (plus their azimuthal components) so they do not break the LS coupling and are often included as an additional term in the Hamiltonian \(H_0\). On the other hand, the fine-structure operators only commute with the operators associated with the total angular momentum J and its azimuthal component. The nuclear spin–orbit interaction, i.e., the interaction between the orbital angular momentum in the field of the nucleus and the intrinsic spin of the electron, is the operator causing the main splitting of the J-resolved levels. Normally, atomic structure calculations have to be carried out in intermediate coupling, where the atomic states, characterized by the quantum numbers J and M are expressed as linear combinations of the form (in Dirac notation)

$$\begin{aligned} |\alpha J M \rangle = \sum _{{ LS}} \mathcal{C}_{{ LS}}\,|\alpha L S J M \rangle , \end{aligned}$$
(74)

where the sum is extended to all the L and S values that are compatible with the configuration \(\alpha \) and with the value of J. The \(\mathcal{C}_{{ LS}}\) are the coefficients of the expansion, obtained by diagonalising the Hamiltonian of the perturbations.

Once the Schrödinger equation is solved and the eigenfunctions and eigenvalues are calculated, it is relatively straightforward to calculate the radiative data such as line strengths and A values for the different types of transitions.

The first requirement in any structure calculation is a good representation of the target, i.e., accurate wavefunctions for the target ion. Such representation requires the inclusion of configuration interaction (CI) with a large set of configurations (see, e.g., Layzer 1954).

Often, significant discrepancies between the experimental energies and those calculated ab-initio have been found. Various semi-empirical corrections based on the observed energies have therefore often been applied to the calculations. This improves the oscillator strengths. One example is the term energy correction (TEC) to the Hamiltonian matrix, introduced by Zeippen et al. (1977) and Nussbaumer and Storey (1978) within the superstructure program to improve the description of the spin–orbit mixing between two levels, which requires their initial term separation to be accurate. Semi-empirical corrections are often applied within the other atomic structure programs, see e.g., B. C. Fawcett work using the Cowan’s Hartree–Fock code (see the Fawcett 1990 review).

Traditionally, there have been two sets of atomic structure calculations that have been carried out in the literature. The first one is to provide the wavefunctions to be used for the scattering calculation. For such calculations, the emphasis is to include in the CC expansion all the levels that are deemed important (e.g., by cascading or by adding resonances in the excitation cross sections) for the spectroscopically-relevant levels.

The second set of calculations normally include a large set of CI and are focused on obtaining the most accurate energies and A-values for the lower levels in an ion, especially for the ground configuration and in general for the metastable levels. In fact, such sets of A-values establish the population of the lower levels, so it is important to obtain accurate values.

Codes

Over the years, many codes have been developed to calculate the atomic structure and associated radiative data. Among the most widely used earlier codes are Cowan’s Hartree–Fock (Cowan 1967, 1981), the superstructure program (Eissner et al. 1974), the CIV3 program (Hibbert 1975), and the HULLAC code (Bar-Shalom et al. 1988).

Autostructure (Badnell 2011), originally a development of the superstructure program, has now become a multi-purpose program for the calculations of a wide range of atomic processes. The code and relative information are available at http://www.apap-network.org/codes.html. Ian Grant’s general-purpose relativistic atomic structure program (GRASP) (Grant et al. 1980; Dyall et al. 1989), based on multiconfiguration Dirac–Hartree–Fock theory, was significantly modified and extended (GRASP2K, see Jönsson et al. 2007, 2013c). Among other features, the new version implements a bi-orthogonal transformation method that allows initial and final states in a transition array to be optimized separately. This often leads to an improvement in the accuracy of the rates. The multi-configuration Hartree–Fock (MCHF) (Froese Fischer 1991) code has also been improved and modified over the years. The most up-to-date version is called ATSP2K. As in GRASP2K, to calculate transition probabilities the orbitals of the initial and final state do not need to be orthogonal. The international collaboration on computational atomic structure (COMPAS) has a useful description of various codes (especially GRASP2K and ATSP2K) and their applications. The codes can be downloaded from their website: ddwap.mah.se/tsjoek/compas/software.php. The Flexible Atomic Code (FAC) (Gu 2004) is available at GitHub: https://github.com/fnevgeny/fac/.

Uncertainties in atomic structure calculations

The accuracy of a particular atomic structure calculation depends on several factors. One important factor is the representation which is used for the target wavefunctions. The target must take into account configuration interaction and allow for intermediate coupling for the higher stages of ionization (cf. Mason 1975a).

It is customary to assess the accuracy of the target wavefunctions by comparing the level energies with the experimental ones. This is a good zero-order approach, although there are cases when, despite relatively good agreement between theoretical and experimental energies, significant problems are still present.

In the literature, the studies which focus on the atomic structure of an atom normally perform a series of calculations, where the size of the CI (that is the number of configurations) is increased until some convergence is achieved for the (lower) levels of spectroscopic importance. It is also common to study the effects of which electrons are allowed to be promoted/excited. One recent example is given by Gustafsson et al. (2017a), where the MCDHF method, as implemented in the GRASP2K program, was used to study correlation effects for Mg-like iron.

Fig. 25
figure 25

A comparison between experimental and ab-initio energies for the lowest 27 levels (\(3{s}^2\,3{p}^2, 3{s}\,3{p}^3, 3{s}^2\,3{p}\,3{d}\)) in Si-like Fe xiii with different target wavefunctions (see text)

As an example of how different calculated energies can be, we consider here the ab-initio energies for the lowest 27 levels (\(3{s}^2\,3{p}^2, 3{s}\,3{p}^3, 3{s}^2\,3{p}\,3{d}\)) as calculated by different authors with different codes in recent years for Si-like iron (Fe xiii). These are plotted in Fig. 25, relative to the experimental energies (all known) as available in CHIANTI v.8. Gupta and Tayal (1998) used the CIV3 computer code and considered for the CI expansions one- and two-electron excitations. Aggarwal and Keenan (2005) used the GRASP code to calculate the ab-initio energies for the lowest 97 levels. Del Zanna and Storey (2012) used the autostructure code and the Thomas–Fermi–Amaldi central potential, with scaling parameters, including in the CI expansion all the configurations up to \(n=4\), giving rise to 2186 fine-structure levels. Vilkas and Ishikawa developed a multiconfiguration Dirac–Fock–Breit self-consistent field plus state-specific multireference Møller–Plesset (MR–MP) perturbation procedure to obtain very accurate level energies for open-shell systems. They applied this procedure to Si-like ions as described in Vilkas and Ishikawa (2004). Recently, Jönsson et al. (2016) applied the GRASP2K code to carry out large-scale calculations for Si-like ions, including correlations up to \(n=8\). It is clear that the GRASP2K and MR–MP calculations are outperforming the other codes, achieving spectroscopic accuracy, i.e., deviations from the experimental energies of a few hundred Kaysers. However we should note that the other calculations were mostly concerned in defining the wavefunctions for the scattering calculations. For this relatively simple ion, the autostructure energies are close to those from CIV3, while larger deviations are clear for the GRASP calculations. In turn, such deviations often affect the oscillator strengths to the levels, and as a consequence the cross-sections for electron excitations, as pointed out by Del Zanna (2011a) for this particular ion, and also as discussed below.

Another way to assess the accuracy of the target wavefunctions is to compare oscillator strengths (gf values) for the dipole-allowed transitions as computed with different sets of configurations or parameters. Similarly, A-values for the forbidden transitions are usually compared. Typically, the gf values for transitions to lower levels do not vary significantly, while it is always the transitions to the highest levels in the calculations that are more uncertain. They are usually also the weakest. Figure 26 shows as an example the gf values for transitions from the ground configuration of Fe xiv as calculated with a set of configurations giving rise to 136 fine-structure levels (Del Zanna et al. 2015a), and a much larger CI (2985 levels) as calculated by Liang et al. (2010). There are some clear discrepancies for transitions to the higher levels, despite the fact that the same code (autostructure) and approximations were used.

Fig. 26
figure 26

Image adapted from Del Zanna et al. (2015a)

Oscillator strengths (gf values) for all transitions from the ground configuration of Fe xiv as calculated with a smaller CI (136 levels) versus those calculated with a much larger CI (2985 levels) by Liang et al. (2010). The dashed lines indicate \(\pm \,20\%\)

Generally, larger CI calculations lead to more accurate oscillator strengths. However, for complex ions (e.g., the coronal iron ions) a large calculation does not necessarily provide accurate values. For example, the spin–orbit mixing among fine-structure levels of the same J and parity is very sensitive to the difference between energy level values, and small variations can lead to large differences in oscillator strengths. This can even occur for some of the strongest transitions to low-lying levels, as shown, e.g., for Fe xi (Del Zanna et al. 2010b) and Fe viii (Del Zanna and Badnell 2014).

Any difference in the gf values is directly reflected in differences in the A values, which directly affect the line intensity. One way to assess the accuracy in the oscillator strengths (or A values) is to compare the values as computed in the Babushkin (length) and the Coulomb (velocity) gauges. A recent example is from the Fe xiv GRASP2K calculations by Ekman et al. (2018), where the accuracy was estimated from

$$\begin{aligned} { dT} = \frac{|A_{B}-A_{C}|}{\max (A_{B},A_{C})}, \end{aligned}$$
(75)

where \(A_{B}\) and \(A_{C}\) are the transition rates obtained in the Babushkin and Coulomb gauge, respectively (Fig. 27).

Fig. 27
figure 27

Image reproduced with permission from Ekman et al. (2018), copyright by Elsevier

Scatter plots of accuracy estimates \({ dT}\) versus A-value for E1 and E2 transitions in Fe xiv

One other way to assess the accuracy of the A-values is to compare the results of different calculations, where e.g., the size of the CI is different, but the same methods and codes are used. Figure 28 shows as an example the percentage difference in the A values of Fe xiii transitions within the energetically lowest 27 levels, as obtained by Storey and Zeippen (2010) and Del Zanna and Storey (2012). In both cases, the AUTOSTRUCTURE code was used. In Storey and Zeippen (2010), 114 fine-structure levels within the \(n=3\) complex were included in the CI expansion. In Del Zanna and Storey (2012), 2985 \(n=3,4\) levels were included. Clearly, very good agreement within 5–10% is present for the strongest transitions. A similar agreement is found with the A-values calculated by Young (2004) with SUPERSTRUCTURE.

Fig. 28
figure 28

Percentage difference in the A values of Fe xiii transitions within the energetically lowest 27 levels, as obtained with two different calculations

Measurements of level lifetimes

Laboratory measurements of radiative lifetimes of atomic levels have been very important in order to assess the accuracy of the theoretical radiative rates. Even with large computations, it is often found that theoretical values disagree significantly with the observed ones, especially for forbidden and intersystem transitions.

Many results have been obtained using beam-foil spectroscopy. Indeed there is a large body of literature on beam-foil measurements of level lifetimes. For recent reviews see, e.g., Träbert (2005, 2008). Beam-foil spectroscopy is based on a production of beams of fast ions that target a thin foil, where they are excited by the collisions with the electrons that are present in the foil (at the solid density). As a result, the ions reach a status where high-lying levels (with large values of the quantum numbers nlJ) are multiply excited (i.e., several electrons are excited). The experimental setup is such that the ions travel through the thin foil and continue to travel in a low-density environment. During this travel, the various excited levels decay to the ground state in a complex way. The allowed transitions quickly depopulate most of the levels, while the metastable levels last much longer. In the mean time, the high-lying levels tend to decay via \(\varDelta n=1\) transitions, and maintain the populations of the low-lying levels for a while via cascading. The beam-foil technique consists in measuring the transitions at varying distance from the foil. The decay curves provide measurements of the lifetimes of the levels, while the spectra at larger distances have been extremely useful to identify the transitions from metastable levels.

From the lifetimes of the levels, direct measurements of the A-values are sometimes possible. There is an extensive literature on theoretical and experimental lifetime measurements for important metastable levels, mostly in the ground configurations of abundant ions. Such levels typically have large quantum numbers J. Figure 29 shows as an example the results for the A-value measurements for the famous green coronal line in Fe xiv, obtained directly from the lifetime of the first excited level within the ground configuration of this ion. Theoretical values are also plotted in the figure. We can see good agreement between most calculations and experimental data, although a scatter in the values is present. Note that Brenner et al. (2007) used the Heidelberg EBIT to obtain a lifetime of 16.73 ms (equivalent to an A-value of \(59.8\,\hbox {s}^{-1}\)) with a very small uncertainty, which would indicate disagreement with theory (blue line, taking into account the experimental transition energy and the anomalous magnetic moment of the electron). However, this small uncertainty has been questioned (Träbert 2014b).

Fig. 29
figure 29

Image reproduced with permission from Träbert (2014a)

A comparison of theoretical and experimental A-value measurements for the green coronal line in Fe xiv. The red line indicates the value obtained from ab-initio calculations, adjusted to the experimental transition energy. The blue line indicates the value obtained by additionally taking into account the anomalous magnetic moment of the electron

Quite often, however, levels decay via multiple branching pathways, so in the literature it is common to compare level lifetimes as calculated by theory (taking into account the branching ratios) and by experiment. This offers an excellent way to benchmark/validate the atomic structure calculations. For general reviews of how atomic structure calculations can be assessed by, e.g., lifetime measurements see, e.g., Träbert (2010, 2014a). Träbert (2005) reviewed lifetime measurements for the important iron ions. Table 4 shows as an example the lifetime measurements and predictions for Fe xii, as described in Träbert et al. (2008).

Table 4 Fe xii level lifetimes in the millisecond range and known or predicted principal decay channels (M1, E2, or M2)

The ion beam energy determines the average charge state reached. Therefore, by tuning the energy, it is possible to obtain spectra where particular charge state ions are enhanced. Beam-foil spectroscopy has therefore been fundamental for the identification of spectral lines, in particular those arising from levels with long lifetimes. However, the spectra always contain a mixture of charge states. A sample spectrum is shown in Fig. 30.

Many other methods and devices have been used to measure atomic lifetimes, mostly with heavy-ion storage rings and electron beam ion trap (EBIT) devices. EBIT devices are extremely useful for assessing solar spectra because the plasma density is normally in the range \(10^{11}{-}10^{13}\,\hbox {cm}^{-3}\), i.e., not far from typical densities found in active regions and flares. The beam energy can be adjusted so only a range of charge states from an element (all those below the threshold for ionization of the highest charge) are observed. This aids the identification of the spectral lines. Figure 30 (below) shows a spectrum obtained with an electron beam ion trap (EBIT), as an example.

For a review of measured lifetimes for iron ions using heavy-ion storage rings see Träbert et al. (2003), while Träbert (2002) also discusses EBIT measurements.

Magnetically-induced transitions

We note here a relatively new and potentially interesting diagnostic to measure solar magnetic fields using XUV lines. A few cases have been discovered where transitions which are strictly forbidden are actually observed in laboratory plasma, being induced by very strong magnetic fields. They are called magnetically-induced transitions (MIT). For sufficiently strong magnetic fields and atomic states which are nearly degenerate and have the same magnetic quantum number and parity, a strong external field could break the atomic symmetry (by mixing the atomic states), and induce new magnetically-induced transitions.

Fig. 30
figure 30

Image reproduced with permission from Träbert (2017), copyright by CSP

A beam-foil spectrum of iron ions (top), compared to an EBIT spectrum

Beiersdorfer et al. (2003) reported an observation of an X-ray MIT transition at 47 Å in Ne-like Ar by an EBIT with strong magnetic fields of the order of 30 kG. Li et al. (2013) reported calculations of the strength of MIT along the Ne-like sequence. The same transition in Ne-like iron (Fe xvii) originates from the metastable \(2{s}^{2}2{p}^{5}\,{}^3\hbox {P}_{0}\) level (hereafter metastable \(J=0\) level), which normally decays with a well-known M1 forbidden transition (at 1153.16 Å) to the lower \(^1\hbox {P}_{1}\) level, see Fig. 31 (top). The decay to the \(J=0\) ground state is strictly forbidden, however the MIT line at 16.804 Å was recently measured by Beiersdorfer et al. (2016) using the Livermore EBIT in the presence of a strong magnetic field, allowing them to provide an estimate of the lifetime of the metastable \(J=0\) level in Fe xvii. This MIT transition is very close to one of the strong X-ray lines from this ion, at 16.776 Å, usually labelled with (3F). Figure 31 (top) also shows two other main X-ray lines (see Del Zanna and Ishikawa 2009 for a discussion on line identifications and wavelength measurements for this ion).

Another example of a MIT occurs in Cl-like iron. The Fe x spectrum was discussed in detail in Del Zanna et al. (2004a). This ion gives rise to famous red coronal line, within the \(3{s}^2\,3{p}^5\,{}^2\hbox {P}_{3/2, 1/2}\) ground configuration states, and several important forbidden transitions in the EUV and visible which ultimately decay into the \(3{p}^{4}\,3{d}\,{}^4\hbox {D}_{5/2, 7/2}\) levels (see Fig. 31, bottom). The \(3{p}^{4}\,3{d}\,{}^4\hbox {D}_{7/2}\) level is a metastable, and can only decay to the ground state via a M2 transition. This line is the fourth strongest line in the EUV spectrum of Fe x. The \(^4\hbox {D}_{5/2}\) level decays via an E1 transition to the ground state. This transition is relatively strong, but much weaker than the M2. Smitt (1977) was the first to tentatively assign an energy (of \(388{,}708\,\hbox {cm}^{-1}\)) to both \(3{p}^{4}\,3{d}\,{}^4\hbox {D}_{5/2, 7/2}\) levels, which were then confirmed in Del Zanna et al. (2004a). The Hinode EIS measurements however showed a discrepancy of about a factor of 2 between predicted and observed intensities of these lines (Del Zanna 2012b), and it was only with the latest large-scale scattering calculation (Del Zanna et al. 2012a) that agreement has been found.

Fig. 31
figure 31

Grotrian diagrams (not to scale) of the lowest levels in Fe xvii and Fe x, indicating the levels involved in the magnetically-induced transitions

The MIT in Fe x was discussed in detail by Li et al. (2015). In the presence of a strong magnetic field, the two \(3{p}^{4}\,3{d}\,{}^4\hbox {D}_{5/2, 7/2}\) states will mix, which will produce a new MIT (E1) from the \(^4\hbox {D}_{7/2}\) level to the ground state \(3{s}^2\,3{p}^5\,{}^2\hbox {P}_{3/2}\), see Fig. 31 (bottom). Li et al. (2015) carried out increasingly large atomic structure calculations, which were used to estimate the splitting of the \(3{p}^{4}\,3{d}\,{}^{4}\hbox {D}_{5/2}\) and \(3{p}^{4}\,3{d}\,{}^{4}\hbox {D}_{7/2}\) levels, which turns out to be a major uncertainty in the diagnostic. The best calculations provided a splitting of \(20\,\hbox {cm}^{-1}\). The authors did not take into account the review by Del Zanna et al. (2004a) of previous experimental data, where a much smaller energy difference of \(5\,\hbox {cm}^{-1}\) was suggested. The assessment was partly based on previous work, which included Skylab observations. Judge et al. (2016) carried out a similar analysis of the same Skylab observations to obtain \(3.6 \pm 2.7 \hbox { cm}^{-1}\). A similar energy difference was estimated from recent EBIT measurements (Li et al. 2016), although a more accurate measurement would be useful. It remains to be seen if magnetic fields can be measured with Hinode EIS, given the various uncertainties: aside from the uncertainty in the energy splitting, and the difficulty in obtaining accurate atomic data for these levels, the transitions from the \(^4\hbox {D}\) are significantly affected by variations in the electron temperature and density. They are also sensitive to non-Maxwellian electron distributions (Dudík et al. 2014b). Finally, a significant uncertainty is associated with the Hinode EIS calibration.

Finally, we point out that in principle the MIT are a general tool available for other states/sequences. For example, Grumer et al. (2013) presented calculations for the \(2s\,2p\,{}^3\hbox {P}_{0} \rightarrow 2{s}^2\,{}^1\hbox {S}_{0}\) MIT transition in Be-like ions. while Grumer et al. (2014) and Johnson (2011) also discuss the effects that are present in ions/isotopes with non-zero nuclear spin, which cause hyperfine structure and hyperfine-induced E1 transitions.

Atomic structure data

Excellent sets of A-values have been obtained with semi-empirical corrections and the superstructure and CIV3 programs. The MCHF have also been very accurate for the ground configurations, and there are several studies by C. Froese Fischer and collaborators. An on-line database of MCHF calculations is available at http://nlte.nist.gov/MCHF/. Very accurate ab initio multi-reference Møller–Plesset calculations have been carried out for a limited set of ions and configurations by Y. Ishikawa and collaborators. For example, see Ishikawa and Vilkas (2001) for the Si-like ions, and Ishikawa and Vilkas (2008) for S-like ions.

The new development of the GRASP program, GRASP2K (Jönsson et al. 2007), allows large-scale calculations to be performed even for the most complex ions. Such calculations reach spectroscopic accuracy, in the sense that theoretical wavelengths in the EUV can be within a fraction of an Å of the observed values. Previous ab-initio calculations typically had an accuracy of a few Å, which is often not sufficient to enable the line identification process (see below). A recent review of GRASP2K calculations can be found in Jönsson et al. (2017). As a measure of the accuracy, one could look at the differences between the length and velocity forms of the oscillator strengths, or at the differences in the A-values as calculated by different codes. Typical uncertainties of the order of a few percent are now achievable for the main transitions.

The literature on radiative data on each single ion is too extensive to be provided here. We refer the reader to the on-line references (http://www.chiantidatabase.org/chianti_direct_data.html) in the CHIANTI database, where details about the radiative data for each ion are provided. We also note that the NIST database at http://physics.nist.gov/PhysRefData/ASD/lines_form.html provides an extensive and very useful bibliography of all the published radiative data for each ion. NIST also provides a series of assessment reports of theoretical and experimental data.

Here, we just mention the most recent GRASP2K calculations not just because they are very accurate, but also because they cover entire isoelectronic sequences, and references to previous studies on each ion in the sequence can be found.

Jönsson et al. (2011) carried out GRASP2K calculations for states of the \(2s^{2}\,2p^{2},\,2s\,2p^{3}\), and \(2p^{4}\) configurations in C-like ions between F iv and Ni xxiii. These calculations have been supplemented with those from Ekman et al. (2014) for ions in the C-like sequence, from Ar xiii to Zn xxv, by including levels up to \(n=4\).

Rynkun et al. (2012) carried out calculations for transitions among the lower 15 levels (\(2{s}^{2}\,2{p}, 2{s}\,2{p}^2, 2{p}^3\)) in ions of the B-like sequence, from N iii to Zn xxvi. Rynkun et al. (2013) calculated radiative data for transition originating from the \(2{s}^{2}\,2{p}^{4}, 2{s}\,2{p}^{5}\), and \(2{p}^{6}\) configurations in all O-like ions between F ii and Kr xxix. These calculations have recently been extended to include many \(n=3\) levels, for ions from Cr xvii to Zn xxiii by Wang et al. (2017). Rynkun et al. (2014) calculated transition rates for states of the \(2{s}^{2}\,2{p}^{3}, 2{s}\,2{p}^{4}\), and \(2{p}^{5}\) configurations in all N-like ions between F III and Kr xxx. These calculations have recently been extended to include levels up to \(n=4\), for ions from Ar XII to Zn XXIV, by Wang et al. (2016b). Jönsson et al. (2013a) carried out GRASP2K calculations for the \(2{s}^{2}\,2{p}^{5}\) and \(2{s}\,2{p}^{6}\) configurations in F-like ions, from Si vi to W lxvi. The calculations on F-like ions has recently been extended by Si et al. (2016) to include higher levels up to \(n=4\) for ions from Cr to Zn. Jönsson et al. (2013b) calculated radiative data for levels up to \(n=4\) for the B-like Si x and all the B-like ions between Ti XVIII and Cu XXV. Jönsson et al. (2014) carried out GRASP2K calculations for the \(2{p}^{6}\) and \(2{p}^{5}\,3{l}\) configurations in Ne-like ions, between Mg III and Kr XXVII. These calculations have recently been extended to include levels up to \(n=6\), for all Ne-like ions between Cr xv and Kr xxvii by Wang et al. (2016a).

Wang et al. (2015) calculated radiative data for ions along the Be-like sequence.

Jönsson et al. (2016) calculated radiative data for the \(3{s}^2\,3{p}^2, 3{s}\,3{p}^3\) and \(3{s}^2\,3p\,3d\) configurations of all the Si-like ions from Ti ix to Ge xix, plus Sr xxv, Zr xxvii, Mo xxix. Gustafsson et al. (2017b) performed GRASP2K calculations for 3l 3\(l'\), 3l 4\(l'\), and \(3s\,5l\) states in Mg-like ions from Ca ix to As xxii, and Kr xxv. Ekman et al. (2018) performed GRASP2K calculations for ions in the Al-like sequence, from Ti X through Kr XXIV, plus Xe XLII, and W LXII. The radiative data are for a large set of 30 configurations: \(3s^2\{3l,4l,5l\}, 3p^2\{3d,4l\}, 3s\{3p^2,3d^2\}, 3s\{3p3d,3p4l,3p5s,3d4l'\}, 3p\,3d^2, 3p^3\) and \(3d^3\) with \(l=0,1,\ldots ,n-1\) and \(l'=0,1,2\).

Electron-ion scattering calculations

As the electron-ion collisions are the dominant populating process for ions in the low corona, significant effort has been devoted to calculate electron-ion scattering collisions over the past decades. The collision strengths usually have a slowly varying part and spikes, that are due to resonances (dielectronic capture). The theory of electron-ion collisions has received significant contributions by M. J. Seaton (University College London), P. Burke (Queens University of Belfast), and A. Burgess (University of Cambridge), among many other members of a large community of atomic physicists. Standard textbooks and reviews on electron-ion collisions are Mott and Massey (1949), Burgess et al. (1970), Seaton (1976), Henry (1981) and Burke (2011).

Using Rydberg units, the nonrelativistic Hamiltonian describing an ion with N electrons and a central nucleus having charge number Z and a free electron (i.e., the \((N+1)\)-electron system) is

$$\begin{aligned} {H (Z,N+1)} = - \sum _{i=1}^{N+1} \left( \nabla _i^2 + {2Z \over r_i} \right) + \sum ^{N}_{i < j} {2 \over r_{ij}}. \end{aligned}$$
(76)

We search for solutions of the time-independent Schrödinger equation for the \((N+1)\)-electron system

$$\begin{aligned} H(Z,N+1) \varPsi _c = E \varPsi _c \end{aligned}$$
(77)

where E is the total energy of the \((N+1)\)-electron system, and \(\varPsi _c\) is the wavefunction of the \((N+1)\)-electron system. The \(\varPsi _c\) are usually expanded in terms of products of wavefunctions of the N-electron target and those of the free electron.

Within the so called close-coupling (CC) approximation, the scattering electron sees individual target electrons, and a set of integro-differential equations need to be solved. An efficient way to solve the scattering problem is to use the R-matrix method, described in Burke et al. (1971), Burke and Robb (1976) and Burke (2011). Such method was further developed over many years by the Iron Project, an international collaboration of atomic physicists, which was set up to calculate electron excitation rates for all the iron ions. A significant contribution to the development of the Iron Project programs was made by Berrington, see e.g., Berrington et al. (1987).

The analysis of the scattering interaction

$$\begin{aligned} \langle \varPsi _i | H(Z,N+1) - E | \varPsi _{i'} \rangle \end{aligned}$$
(78)

involves the calculation of the elements of the so-called reactance matrix, \( K_{ii'}\). The transmission matrix \(T_{ii'}\) is related to the reactance matrix:

$$\begin{aligned} T = \frac{-2\mathrm{i} K}{1-\mathrm{i} K}, \end{aligned}$$
(79)

and the resulting scattering matrix \( \mathcal S\)

$$\begin{aligned} \mathcal{S} = 1-T = {1+ \mathrm{i} K \over 1- \mathrm{i} K} \end{aligned}$$
(80)

is unitary. The dimensionless collision strength (\(\varOmega _{if}\)) for any transition \(i-f\) from an initial i to a final state f is then obtained from the transmission matrix

$$\begin{aligned} \varOmega _{if} \propto |T_{if}|^2. \end{aligned}$$
(81)

The collision strengths are usually calculated over a finite range of the energy of the incoming electron. Within the Iron Project codes, the collision strengths are extended to high energies by interpolation using the appropriate high-energy limits in the Burgess and Tully (1992) scaled domain. The high-energy limits are calculated following Burgess et al. (1997) and Chidichimo et al. (2003).

DW UCL codes

In the Distorted Wave (DW) approximation, the coupling between different target states is assumed to be negligible and the system of coupled integro-differential equations is significantly reduced. A general method which includes the effects of exchange between the free and a bound electron, was developed by Eissner and Seaton (1972) at UCL. The method requires that the free-electron wavefunction \(\theta _i\) are orthogonal to those of the one-electron bound orbitals with the same angular momentum. This is achieved by adopting a general expression for the total wavefunction of the \(N+1\) system of the form

$$\begin{aligned} \varPsi =\mathcal{A} \sum _i \chi _i(\mathbf {x}_1,\ldots , \mathbf {x}_{N}) \theta _i(\mathbf {x}_{N+1}) + \sum _j \phi _j c_j, \end{aligned}$$
(82)

where \(\phi _j \) is a function of bound state type for the whole system. The functions \(\chi _i\) and \(\phi _j\) are constructed from one-electron orbitals, and \(\mathcal{A}\) is an operator that guarantees antisymmetry. The functions \(\phi _j\) are called correlation functions, which are mainly introduced to obtain orthogonality. Note that the above general expression with the correlation functions was adopted by the UCL DW code, mainly developed by Eissner and Seaton (1972) and Eissner (1998). This code has been widely used for a very long time and has produced a very large amount of atomic data.

The first accurate scattering calculations for the coronal iron ions were carried out with the UCL DW codes (see, e.g., Mason 1975a). These were prompted by ground-based eclipse observations of the forbidden lines. These were also needed for the early EUV Skylab and OSO-7 observations in the 1970s. There was a series of papers from Bhatia and Mason. Such calculations were further revised and improved by several groups. These calculations were generally carried out in intermediate coupling that is transforming the LS coupling calculation using term coupling coefficients. This approximation was found adequate for most coronal ions.

Although the effects of resonance enhancement can be included in DW calculations, usually it is not. DW calculations typically agree with the background collision strength values at lower energies, lower than the maximum threshold. At energies higher than the maximum threshold, DW and R-matrix calculations should agree, if the same wavefunctions are used. For a comparison between DW and R-matrix calculations see, e.g., Burgess et al. (1991). Excellent agreement between the collision strengths calculated with the UCL DW and the R-matrix codes was found, to within a few percent. Figure 32 shows as an example the partial collision strengths calculated at 10 Ry for the Mg vii \(2{s}^2\,2{p}^2\,{}^1\hbox {S}-2{s}~2{p}^3\,{}^1\hbox {P}\) transition.

Fig. 32
figure 32

Reproduced with permission from Burgess et al. (1991)

A comparison of partial collision strengths calculated with the DW and R-matrix codes, for the Mg vii \(2{s}^2\,2{p}^2\,{}^1\hbox {S}-2{s}\,2{p}^3\,{}^1\hbox {P}\) transition at 10 Ry

Other codes: excitation by electron impact

Several other codes have been developed over the years to calculate cross-sections for excitation by electron impact, and have produced a large amount of atomic data. For example, D. Sampson and H. L. Zhang developed several codes based on the Coulomb–Born-exchange method (cf. Sampson et al. 1979). Widely used DW codes are the Flexible Atomic Code (FAC) (Gu 2003, 2008), and HULLAC (Bar-Shalom et al. 1988). Recently, the autostructure DW code (Badnell 2011) has also been implemented. FAC and autostructure are publicly available, as the UCL DW.

We note that different assumptions are made in the various DW codes. For example, the autostructure DW code does not impose the orthogonality condition and the second term in the expansion is not present, but it does calculate all the appropriate exchange overlaps. In the DW approximation, the scattering equations are uncoupled and not all the elements of the reactance matrix \( K_{ii'}\) need to be calculated. Another important difference is whether the DW method is unitarized. Some of the DW codes, such as the FAC and HULLAC, use the relation

$$\begin{aligned} T = \frac{-2\mathrm{i}K}{(1-\mathrm{i} K)} \times \frac{(1+\mathrm{i} K)}{(1+\mathrm{i} K)} = \frac{-2\mathrm{i} K + 2K^2}{1 + K^2} \approx -2\mathrm{i} K, \end{aligned}$$
(83)

in which case the DW method is called non-unitarized. As pointed out by Fernández-Menchero et al. (2015a), the non-unitarized DW method can lead to large errors (factors of 10) for a few weak transitions, where the coupling in the scattering equations becomes important. The original and default version of the autostructure DW was also non-unitarized, but a unitarized option, which provides good agreement with the fully close-coupling calculations, has recently been implemented (Badnell et al. 2016).

The R-matrix codes, used within the Iron Project for the scattering calculations, are described in Hummer et al. (1993), Berrington et al. (1995), Burgess (1974) and Badnell and Griffin (2001), and have been applied to the calculations of many ions. The main repository for the codes is the UK APAP web page maintained by N. R. Badnell at http://www.apap-network.org/codes.html.

Many of the calculations for the astrophysically important (not heavy) ions are carried out in LS-coupling, and relativistic corrections are applied later. The R-matrix calculations are carried out in an inner and outer region. The R-matrix calculations in the inner region normally include the mass and Darwin relativistic energy corrections. Many of the outer region calculations are carried out with the intermediate-coupling frame transformation method (ICFT), described by Griffin et al. (1998). This method is computationally much faster than the Breit–Pauli R-matrix method (BPRM).

A fully relativistic Dirac R-matrix code, called DARC, was developed by P. H. Norrington and I. P. Grant (see, e.g., Norrington and Grant 1981), and is also available at the UK APAP web page. Several comparisons between the results obtained by the various codes have been carried out, and are normally satisfactory. For example, Badnell and Ballance (2014) compared the R-matrix results of ICFT, BPRM and DARC on Fe iii. However, discrepancies for weaker transitions have also been reported (see, e.g., the discussion in Badnell et al. 2016).

A different approach, the B-spline R-matrix (BSR) method was developed by other authors (see, e.g., Zatsarinny 2006; Zatsarinny and Bartschat 2013, for details). The method uses term-dependent non-orthogonal orbital sets for the description of the target states. The wave functions for different states are then optimized independently, which result in a much better target representation. This method is computationally demanding and has mostly been used for the calculations of cross sections for neutrals and low charge states, where an accurate representation of the target states is typically difficult (see, e.g., Wang et al. 2013). In a recent work, a detailed comparison of similar-size calculations for the same ion with the BSR, ICFT, and DARC was carried out by Fernández-Menchero et al. (2017). As in the previous comparisons, good agreement was found among the lower and stronger transitions. Significant differences were however found for the weaker transitions and for transitions to the higher states. The 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.

Other approaches and codes exist. For example, the Dirac R-matrix with pseudo-states method (DRMPS), described in Badnell (2008). For H-like systems, the convergent close-coupling (CCC) method of Bray and Stelbovics (1992) and the relativistic convergent close-coupling (RCCC) method (Bostock 2011) provide extremely accurate cross sections.

Uncertainties in electron-ion excitation rates and in the level population

In this section, we provide some examples on various effects which can reduce the accuracy of the collision strengths and the derived rates. Clearly, if the atomic structure is not accurate, this can have a direct effect on the rates. For example, the high-energy limits of the collision strengths for dipole-allowed transitions are directly related to the gf values (Burgess et al. 1997). Therefore, any differences in gf values obtained with different structure calculations are directly reflected as differences in the collision strengths. The comparisons of the gf values in Fig. 26 were used by Del Zanna et al. (2015a) to explain the discrepancies in the rates for Fe xiv calculated by Aggarwal and Keenan (2014).

The calculations of dipole-allowed collision strengths for simpler ions obtained with different targets are normally in agreement, within 10–20%, while results for weaker transitions can differ by significant factors. However, even strong transitions in complex ions can sometimes be difficult to calculate accurately. For example, as we have previously mentioned, transitions to levels with a strong spin–orbit interaction are very sensitive to the target wavefunctions. Figure 33 shows as an example a comparison between rates for Fe xi lines, as calculated in two different ways. The largest differences are for a few of the brightest transitions, and are related to decays from three \(J=1\) levels which have a strong spin–orbit interaction. Indeed, as shown in Del Zanna et al. (2010b), large differences in the gf values for these transitions are present.

Semi-empirical corrections are normally not applied to the scattering calculations. However, they can provide significant improvements to the rates (see, e.g., Fawcett and Mason 1989; Del Zanna and Badnell 2014).

Fig. 33
figure 33

Image adapted from Del Zanna et al. (2010b)

Effective collision strengths for the strongest Fe xi transitions as calculated by Aggarwal and Keenan (2003) and Del Zanna et al. (2010b). The dashed lines indicate \(\pm \,20\%\). The largest differences for the strongest transitions are related to decays from \(J=1\) levels (e.g., 4–37, 1–39, 3–39, 4–39, 2–41, 3–41, 4–41 indicated in the figure)

Fig. 34
figure 34

Adapted from Del Zanna et al. (2012a)

Above: collision strengths, averaged over 0.1 Ryd in the resonance region. The data points are displayed in histogram mode. Boxes indicate the DW values. Below: thermally-averaged collision strengths, with other calculations. The plots on the left are for the Fe x forbidden red coronal line within the ground configuration, those on the right for the Fe x forbidden 257.26 Å transition

Fig. 35
figure 35

Adapted from Del Zanna et al. (2012a)

Above: collision strengths, averaged over 0.1 Ryd in the resonance region. The data points are displayed in histogram mode. Boxes indicate the DW values. Below: thermally-averaged collision strengths, with other calculations. The plots on the left are for the Fe x allowed 345.7 Å transition, those on the right for the allowed 1–30 174.5 Å transition, the strongest in the EUV

Another factor in the accuracy of a calculation is the type of scattering approximation used. DW calculations are known to considerably underestimate collision strengths for some types of transitions. The resonances at lower energies typically affect the rates at lower temperatures, where the DW and R-matrix calculations can provide very different values for the rates. The contribution of the resonances varies with the type of transition. For dipole-allowed transitions, resonance effects can be very small. On the other hand, they are usually significant for forbidden and intersystem lines, because the background collision strength values are much lower. Figure 34 shows as an example the collision strengths and rates for forbidden transitions in Fe x, one within the ground configuration, and one not. In contrast, Fig. 35 shows collision strengths and rates for two allowed transitions in Fe x. We can see that resonances can occasionally have a significant contribution to the collision strengths even for strong dipole-allowed transitions.

We note that resonances can be included within DW calculations with a perturbative treatment (DW \(+\) resonances), as e.g., in H. L. Zhang and D. Sampson codes and FAC (although they are not normally included). We also note that generally the resulting collision strengths, even with the resonances included, are lower than those calculated with the close-coupling R-matrix calculations. An example is shown in Fig. 36, where the collision strengths for a He-like forbidden transition as calculated by Zhang and Sampson (1987) is compared to the result of an R-matrix calculation by Whiteford et al. (2001) (see further details in Badnell et al. 2016). This issue has been studied by Badnell et al. (1993), where significant differences between the DW \(+\) resonances treatment and close-coupling R-matrix calculations for Mg-like ions were found. Recently, Fernández-Menchero et al. (2016b) compared two similar large-scale calculations for Fe xxi, an R-matrix one and an earlier DW \(+\) resonances calculation of Landi and Gu (2006). Again, the comparison showed a systematic underestimate of the cross sections by the perturbative results.

Fig. 36
figure 36

The rate for the forbidden line in Si xiii as calculated with the R-matrix codes by Whiteford et al. (