Scaling of collision strengths for highlyexcited states of ions of the H and Helike sequences^{⋆}
^{1} Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK
email: luis.fernandezmenchero@drake.edu
^{2} Present address: Department of Physics and Astronomy, Drake University, 2507 University Avenue Des Moines, IA 50311, USA
^{3} Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
Received: 10 March 2016
Accepted: 6 June 2016
Emission lines from highlyexcited states (n ≥ 5) of H and Helike ions have been detected in astrophysical sources and fusion plasmas. For such excited states, Rmatrix or distorted wave calculations for electronimpact excitation are very limited, due to the large size of the atomic basis set needed to describe them. Calculations for n ≥ 6 are also not generally available. We study the behaviour of the electronimpact excitation collision strengths and effective collision strengths for the most important transitions used to model electron collision dominated astrophysical plasmas, solar, for example. We investigate the dependence on the relevant parameters: the principal quantum number n or the nuclear charge Z. We also estimate the importance of coupling to highlyexcited states and the continuum by comparing the results of different sized calculations. We provide analytic formulae to calculate the electronimpact excitation collision strengths and effective collision strengths to highlyexcited states (n ≥ 8) of H and Helike ions. These extrapolated effective collision strengths can be used to interpret astrophysical and fusion plasma via collisionalradiative modelling.
Key words: atomic data / Sun: corona / techniques: spectroscopic
Tables of atomic data for Si xiii and S xv are only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/qcat?J/A+A/592/A135
© ESO, 2016
1. Introduction
Spectral emission lines of H and Helike ions have been used for diagnostics of both fusion and astrophysical plasmas for decades. Perhaps the most famous examples are the temperature and density diagnostics of the Helike ions in the Xrays, described by Gabriel & Jordan (1969). However, the status of the atomic data for these ions still requires improvement. As described below, atomic data for ions in these sequences are generally only available up to the principal quantum number n = 5. Atomic data for highlyexcited levels are needed for a variety of reasons. First, lines from these levels (up to n = 10) have been observed in laboratory plasma (see, e.g. the compilations in the NIST database Kramida et al. 2013) and recently also in Xray spectra of solar flares (see, e.g. Kepa et al. 2006). Second, transitions between highlyexcited levels should be included for any appropriate collisionalradiative modelling of these ions. Third, even if in most astrophysical spectra lines from these levels are not readily visible, they do contribute to the Xray pseudocontinuum, so they should be included in any spectral modelling.
Calculating atomic data for highlyexcited levels is not a trivial task and has various limitations, since it requires a significant increase in the size of the atomic basis set. In Sect. 2 we review previous calculations for the electronimpact excitation of He and Hlike ions, and present the results of test calculations made with larger basis sets. These calculations are performed to see how well the extrapolated data agree with the calculated ones for higher n. In Sect. 3 we study the behaviour of the electronimpact excitation collision strengths and effective collision strengths for several kinds of transitions. These are the most common transitions that decay producing the lines observed in astrophysics. We also estimate the importance of coupling to highlyexcited states and the continuum by comparing the results of differently sized distorted wave and Rmatrix calculations. We then provide analytic formulae to calculate electronimpact excitation collision strengths to highlyexcited states (n ≥ 8) of H and Helike ions. This is done by extrapolating the results obtained with the Rmatrix or distorted wave methods. Potentially, the method provides results up to n = ∞, although accuracy reduces as n increases. In Sect. 4 we compare the solar flare line intensities with those predicted by applying the extrapolation rules to the effective collision strengths. Finally, in Sect. 5 we summarise the main conclusions.
2. Atomic data
A number of calculations for the electronimpact excitation of ions of the H and Helike sequences can be found in the literature. Authors have used a number of different methodologies and different configuration interaction (CI) and close coupling (CC) basis sets.
Whiteford et al. (2001) calculated electronimpact excitation effective collision strengths for Helike ions. Whiteford et al. (2001) included in the CI/CC basis set all the singleelectron excitations up to principal quantum number n = 5 (49 finestructure levels) and used the radiationdamped intermediate coupling frame transformation ICFT Rmatrix method (Griffin et al. 1998). These data are the most rigorous and complete to date. They can be found in the UK APAP network database^{1}, as well as in OPENADAS^{2} and in the most recent version 8 (Del Zanna et al. 2015) of the CHIANTI database^{3}.
There are other studies in the literature. Aggarwal & Keenan (2005) calculated electronimpact effective collision strengths for Ar^{16 +} up to n = 5 using the Dirac Rmatrix method (Norrington & Grant 1987). Chen et al. (2006) calculated Dirac Rmatrix electronimpact excitation effective collision strengths for Ne^{8 +} up to n = 5. Kimura et al. (2000) performed Dirac Rmatrix calculations for the Helike ions S^{14 +}, Ca^{18 +} and Fe^{24 +} up to n = 4.
In the Hlike sequence, Ballance et al. (2003) performed a detailed study of hydrogenic ions from He^{+} to Ne^{9 +} plus Fe^{25 +}. Ballance et al. (2003) used a quite extensive basis set, including pseudostates. The basis set included all the spectroscopic terms up to n = 5 for all the ions except Ne^{9 +}. For Ne^{9 +} the basis set was extended up to n = 6. The pseudostate terms included in the calculations varied for each ion.
Even though the above calculations are quite extensive, they are still insufficient for the modelling of highlyexcited shells (n > 5), as noted in the introduction.
In the present work we have performed some test calculations with an extensive basis set up to n = 8. The calculations are focused on checking the validity of the extrapolation methods that we have developed, and discuss in the next section. As such, we do not include radiation damping of resonances. The target basis set includes all the possible l values for n = 1 − 6, and then up to 7g and 8f. We have performed both Rmatrix and distorted wave calculations with the same basis set. The Rmatrix suite of codes are described in Hummer et al. (1993) and Berrington et al. (1995). The calculation in the inner region was in LS coupling and included mass and Darwin relativistic energy corrections. The outer region calculation used the ICFT method (Griffin et al. 1998). The distorted wave calculations were carriedout using the autostructure program (Badnell 2011). The ICFT Rmatrix and distorted wave calculations were carried out with the same atomic structure to estimate the effects of the resonances and coupling in general.
To estimate the collision strengths for higher shells (n = 8 − 12) we performed a different distorted wave calculation. In this second calculation we used a configuration basis set consisting of 1s^{2} and 1snl for n = 8 − 12 and l up to 8h, 9h, 10g, 11f and 12d, that is, we neglect configurations with n = 2 − 7. Thus, although we automatically have CI between these more highlyexcited nshells, there is no mixing with lower ones, save for the ground. The atomic structure is oversimplified in order to get a description of the highlyexcited states which becomes increasingly demanding when retaining the full CI expansion. This calculation has a poorer atomic structure so it is expected that these results will not be of such high accuracy. It has been performed only to check if such an oversimplified atomic structure can give results for the effective collision strengths with an error which is acceptable for plasma modelling, and to compare that error with the one arising from the extrapolation of results obtained using the (smaller) fullCI expansion.
3. Extrapolation rules
The scattering calculations provide the collision strengths Ω as a function of the incident electron energy. The collision strengths are extended to high energies by interpolation using the appropriate highenergy limits in the Burgess & Tully (1992) scaled domain. The infinite energy limit points are calculated with autostructure. The temperaturedependent effective collisions strength Υ are calculated by convoluting these collision strengths with a Maxwellian electron velocity distribution.
The behaviour of the collision strengths Ω and effective collision strengths Υ for highlyexcited levels follows the semiempirical formula: (1)where n is the principal quantum number and A and α are parameters to be determined. Usually α is small and can be set to zero. The formula 1 is usually a good description for highlyexcited states, where the atom can be considered as a Rydberg one, meaning that for n at least two units more than the last atomic shell occupied by any inactive core electrons of the ion.
We used three models to determine the parameters of the formula 1:

Model 1:
leastsquares fit, including all the calculated values ofΩ and n.

Model 2:
two point extrapolation, calculate A and α from the values of Ω and n of the last two points calculated.

Model 3:
one point extrapolation, fix α = 0 and calculate A from the value of Ω and n of the last point calculated.
We have compared the results of these extrapolation Models with those obtained from explicit Rmatrix and distorted wave test calculations which we have performed. In the following sections we discuss the different cases.
3.1. Dipoleallowed transitions
These transitions are between states with opposite parity and with changes in total angular momenta ΔJ = 0,1 (and J = 0 → 0 forbidden) that is to say, electric dipole. The collision strength diverges logarithmically as the collision energy tends to infinity. (Burgess & Tully 1992). In general, the contribution from resonances will be small compared to the background, so distorted wave and Rmatrix methods produce similar results for the effective collision strengths.
Figure 1 shows the calculated collision strengths for the electric dipole transitions 1s^{2}^{1}S_{0} − 1snp ^{1}P_{1} of the moderatelycharged Si^{12 +} and the highlycharged Fe^{24 +} ions. We plot both the results of the Rmatrix and distorted wave methods. Both calculations were carried out with the same atomic structure, with a somewhat large CI/CC expansion, up to the atomic shell n = 8.
Fig. 1 Electronimpact excitation collision strengths for the electric dipole transition 1s^{2}^{1}S_{0} − 1snp ^{1}P_{1} of the ions Si^{12 +} and Fe^{24 +}. Curved lines: Rmatrix; ⋄: distorted wave calculation. 

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At low temperatures, the effects of the resonances is not significant. This is expected for strong dipole transitions, where the background is large in comparison. As we pointed out in FernándezMenchero et al. (2015), the Rmatrix calculations can not guarantee accuracy at very low temperatures, of the order of the energy of the first excited level. In fact, uncertainties associated with the position of the resonances can reach 100% for such low temperatures. However, for electron collision dominated plasmas, for example solar, the ions are mainly formed near the peak abundance temperature (vertical lines in the plots in Fig. 2), and at these temperatures the position of the resonances has a negligible effect on the effective collision strength.
Fig. 2 Electronimpact excitation effective collision strengths for the electric dipole transition 1s^{2}^{1}S_{0} − 1snp ^{1}P_{1} of the ions Si^{12 +} and Fe^{24 +}. Curved lines: Rmatrix; ⋄: distorted wave calculation basis n = 1 − 8; °: distorted wave calculation basis n = 8 − 12; vertical line: peak abundance temperature for electron collisional plasmas (Mazzotta et al. 1998). 

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Figure 2 shows the Maxwellintegrated effective collision strengths for the same transitions and ions. The difference in the Υ between both calculations for the transition from the ground level to 8p is around 15%.
Fig. 3 Electronimpact excitation effective collision strengths (× n^{3}) for the electric dipole transition 1s^{2}^{1}S_{0} − 1snp ^{1}P_{1} of the ions Si^{12 +} and Fe^{24 +} versus the principal quantum number n around peak abundance temperature. ×: Rmatrix results; ⋄: distorted wave results with basis set n = 1 − 8; °: distorted wave results with basis set n = 8 − 12; solid line: leastsquares fit using points n = 2 − 5; dashed line: extrapolation using the last two points; dotted line: extrapolation using the last point; see text. 

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Figure 3 shows the behaviour of the effective collision strengths Υ with respect to the principal quantum number n at the peak abundance temperature. We compare the extrapolations from n = 5 with the calculated values for the three models. For the lowercharged ion, Si^{12 +}, the disagreement between the Rmatrix and distorted wave results increases more rapidly for higher n, reaching 20% at n = 8. This is due to stronger coupling between the more highlyexcited states included in the closecoupling expansion. However, we note that the Rmatrix calculation cannot accurately describe transitions to the highest states included in the CI/CC expansion (FernándezMenchero et al. 2015). For Fe^{24 +}, the Rmatrix and distorted wave results agree better with each other, to 10% at n = 8, as coupling decreases with increasing charge.
Table 1 shows the different extrapolation parameters calculated with the three methods for Si^{12 +}, and choosing different reference points n_{0} for the extrapolation. The linear fit is performed taking into account all the points between n = 2 and n_{0}. The twopoint model takes the values of Υ(n) for n = n_{0} − 1 and n_{0} and calculates the parameters A and α through a twoequation and twounknown system (1). Finally, the onepoint model uses Υ(n_{0}) to calculate A and sets α = 0. Figure 4 displays the calculated analytic functions corresponding to each of the three models. The predicted extrapolation curves with the twopoint model vary considerably in terms of the reference point n_{0}, by more than 30%. They are influenced too much by the smaller values of n, which have yet to reach their asymptotic form. We see a similar variation in the twopoint model. On the other hand, the onepoint extrapolation gives more stable results. The predicted value of A changes by just 10% with the different choices of n_{0}.
With the linear fit it is necessary to include many points to obtain acceptable statistics and to reduce the associated error of kind β. In a linear leastsquares fit, an acceptable number of points is around twelve to get a β error under 20%. The computation cost of the linear fit is also larger. For such a small set of points, the linear fit is not an appropriate model. Thus, for strong dipole electric transitions we recommend use of the onepoint extrapolation.
The last calculated nshell (n = 8) is not a good reference point for the extrapolation due to the lack of convergence of the CI and CC expansions, compared to n ≤ 7. The parameters A and α calculated with the twopoint model are very similar using the second and the third last point as reference, and they are also with α closest to zero. These two curves for n_{0} = 6 and n_{0} = 7 shown in Fig. 4 are the best extrapolation models for this type of transition, if such data is available. If not, for smaller values of n_{0}, the onepoint extrapolation is the best model.
Fitting parameters for the extrapolation of the Υ at highn for the dipole electric transition of Si^{12 +}1s^{2}^{1}S_{0} − 1snp ^{1}P_{1} at a temperature of T = 3.4 × 10^{6} K, for different extrapolation reference points n_{0}.
Fig. 4 Extrapolation curves for the Υ × n^{3} displayed in Fig. 3, taking different extrapolation points n_{0}. ×: Rmatrix results; ⋄: distorted wave results. 

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3.2. Bornallowed transitions
For nondipole Bornallowed transitions the collision strength tends to a constant value as the collision energy tends to infinity, given by the planewave Born (Burgess & Tully 1992). The is zero for doubleelectron jumps and spinchange transitions, in the absence of mixing. In the intermediate coupling scheme (IC), most transitions will be Bornallowed or dipoleallowed through configuration and/or spinorbit mixing.
Figure 5 shows the effective collision strengths for the onephoton optically forbidden transitions 1s^{2}^{1}S_{0} − 1sns ^{1}S_{0} for the ions Si^{12 +} and Fe^{24 +}. This kind of transition has a very weak background collision strength at all energies, so the enhancement due to resonances is large. This effect is largest at low temperatures; the Υ calculated using the distorted wave method are a considerable underestimate compared to those obtained with the Rmatrix method. The underestimation for the lowest temperatures (~10^{5} K) and lowest excited states can reach a factor of between two and ten. This effect is reduced progressively at higher temperatures and for more highly excited states. At the peak abundance temperature, the resonance enhancement is reasonably small, reduced to 10% for a moderatelycharged or a highlycharged ion, see Fig. 5.
Fig. 5 Electronimpact excitation effective collision strengths for the Born transition 1s^{2}^{1}S_{0} − 1sns ^{1}S_{0} of the ions Si^{12 +} and Fe^{24 +}. Curved line: Rmatrix; ⋄: distorted wave calculation basis n = 1 − 8; °: distorted wave calculation basis n = 8 − 12; vertical line: peak abundance temperature for electron collisional plasmas (Mazzotta et al. 1998). 

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Figure 6 shows the comparison between the Υ calculated with the Rmatrix method, with the distorted wave method using both basis sets, and for the three extrapolation models, all at the peak abundance temperature. For n ≥ 4, the Rmatrix and distorted wave results agree below 10%.
Fig. 6 Electronimpact excitation effective collision strengths (× n^{3}) for the Born transition 1s^{2}^{1}S_{0} − 1sns ^{1}S_{0} of the ions Si^{12 +} and Fe^{24 +} versus the principal quantum number n around peak abundance temperature. ×: Rmatrix results; ⋄: distorted wave results with basis set n = 1 − 8; °: distorted wave results with basis set n = 8 − 12; solid line: leastsquares fit using points n = 2 − 5; dashed line: extrapolation using the last two points; dotted line: extrapolation using the last point; see text. 

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To test the validity of the extrapolation rules for this type of transition we show again in Table 2 the calculated parameters for the three methods with different reference points n_{0} for Si^{12 +}. Fig. 7 shows the analytic curves.
Fitting parameters for the extrapolation of the Υ at highn for the dipole electric transition of Si^{12 +}1s^{2}^{1}S_{0} − 1sns ^{1}S_{0} at a temperature of T = 3.4 × 10^{6} K, for different extrapolation reference points n_{0}.
Fig. 7 Extrapolation curves for the Υ × n^{3} displayed in Fig. 6, taking different extrapolation points n_{0}. ×: Rmatrix results; ⋄: distorted wave results. 

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For this transition the twopoint model gives very similar results for extrapolation with n_{0} = 5,6,7. The twopoint model with n_{0} = 8 is slightly different, but the last point of the calculation should not be considered a good reference for the extrapolation. The error of the twopoint model extrapolation from n_{0} = 5,6,7 with respect to the calculated Rmatrix data is less than 10%. For Bornallowed transitions, we recommend a twopoint extrapolation model as the most accurate. The α parameters calculated with the twopoint model are also close to zero for n_{0} = 6,7, and so here the onepoint extrapolation also gives accurate results.
In general, the calculations with the oversimplified atomic structure give worse results than the extrapolated ones. We do not recommend such simplifications at all for estimation purposes.
3.3. Forbidden transitions
These transitions are characterised by zero limit points, electric dipole and planewave Born (Burgess & Tully 1992). They can arise for spinchanging and/or multipleelectron jump transitions. They are infrequent because of configuration and spinorbit mixing. For high impact energies, the collision strength decays with a power law Ω ~ E^{− γ}, with γ close to two. This rapid decay makes the resonance enhancement large, particularly at low temperatures.
Figure 8 shows the effective collision strengths for the pure spinchange transition 1s^{2}^{1}S_{0} − 1snp ^{3}P_{0} for the ions Si^{12 +} and Fe^{24 +}. As expected, at low temperatures there are differences up to a factor of two between distorted wave and Rmatrix due to the resonances.
Fig. 8 Electronimpact excitation effective collision strengths for the forbidden transition 1s^{2}^{1}S_{0} − 1snp ^{3}P_{0} of the ions Si^{12 +} and Fe^{24 +}. Curved lines: Rmatrix; ⋄: distorted wave calculation basis n = 1 − 8; °: distorted wave calculation basis n = 8 − 12; vertical line: peak abundance temperature for electron collisional plasmas (Mazzotta et al. 1998). 

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Figure 9 shows the comparison between the Υ calculated with the Rmatrix method, the distorted wave methods with both basis sets, and the three extrapolation models, at the peak abundance temperature. In this case, the models fit quite well the data for n = 6 − 8. For these transitions, the value of the parameter α is quite large, and the results of models type 2 and 3 differ significantly. The calculations with the simplified atomic structure again poorly reproduce the data.
Fig. 9 Electronimpact excitation effective collision strengths (× n^{3}) for the forbidden transition 1s^{2}^{1}S_{0} − 1snp ^{3}P_{0} of the ions Si^{12 +} and Fe^{24 +} versus the principal quantum number n around peak abundance temperature. ×: Rmatrix results; ⋄: distorted wave results with basis set n = 1 − 8; °: distorted wave results with basis set n = 8 − 12; solid line: leastsquares fit using points n = 2 − 5; dashed line: extrapolation using the last two points; dotted line: extrapolation using the last point; see text. 

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We show again in Table 3 the different extrapolation parameters calculated with the three methods and different reference point n_{0}. Figure 9 shows the extrapolation curves. Calculated values for the parameters A and α are quite similar for the different reference points n_{0} = 5,6,7 if the same model is used, one or twopoint. On the other hand, if we compare the results obtained with same n_{0} but different models, they are quite different. The onepoint model does not correctly reproduce the behaviour of the Rmatrix results for highn. For this type of transition we recommend a twopoint model.
Fitting parameters for the extrapolation of the Υ at highn for the dipole electric transition of Si^{12 +}1s^{2}^{1}S_{0} − 1snp ^{3}P_{0} at a temperature of T = 3.4 × 10^{6} K, for different extrapolation reference points n_{0}.
Fig. 10 Extrapolation curves for the Υ displayed in Fig. 9, taking different extrapolation points n_{0}. ×: Rmatrix results. 

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3.4. Lowcharged ions
The n^{3} behaviour of the collision strengths is fulfilled if we are in the limit of Rydberg atom, that is when the interaction of the core with the active electron can be considered a onebody Coulomb one. For lowercharged atoms the electronelectron interaction is of the same order as the nucleuselectron one. For such ions this Rydberg atom limit is reached at a higher value of the principal quantum number n. In principle, the above extrapolation rules should work in a lowercharged ion, but the reference point n_{0} should be high enough for it to be considered a Rydberg atom. This means that reference data for extrapolation are required up to n ≈ 8 − 10. This is a rather large Rmatrix calculation, because of the large boxsize. In addition, the coupling with the continuum increases as the charge decreases. So a good quality calculation for highn for a lowcharged ion must include pseudostates in the CI/CC expansions.
Figure 11 shows the effective collision strengths for dipole and Born allowed transitions of C^{4 +}. The background collision strength falls off as z^{2} while, initially, the resonance strength is independent of z, although at sufficiently high charge radiation damping usually starts to reduce the resonance contribution. So in C^{4 +} the resonance enhancement of the effective collision strengths at low temperatures is seen to be relatively smaller than for Si^{12 +} and Fe^{24 +}.
Fig. 11 Electronimpact excitation effective collision strengths for dipole allowed and Born transitions 1s^{2}^{1}S_{0} − 1sns ^{1}S_{0} of the ion C^{4 +}. Curved lines: Rmatrix; ⋄: distorted wave calculation basis n = 1 − 8; vertical line: peak abundance temperature for electron collisional plasmas (Mazzotta et al. 1998). 

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The effective collision strengths for the last nshells included in the basis set show irregular behaviour. This is caused by the loss of quality for the description of the highest excited states. In lowcharged ions the inaccuracy in the description of the atomic structure is larger due to stronger coupling with more highlyexcited bound states and the continuum, which we neglect. For the present test calculations we did not include pseudostates in the description of C^{4 +} atomic structure or the closecoupling expansion. Thus, we do not recommend using this data in preference to Rmatrix with pseudostates data, rather it is a guide to extrapolating data in lowcharge ions. The uncertainty associated with an inaccurate atomic structure due to, for example, the lack of pseudostates in the case of a lowcharged ion, generates a much larger error than that associated with the use of an extrapolation formula.
Experimental and theoretical ratios (erg) for Si xiii.
Experimental and theoretical ratios (erg) for S xv.
3.5. Other sequences
As explained above, the behaviour of the collision strengths with respect to the quantum number tends to the form given by (1) when the active electron is highly enough excited, so the ion can be considered a Rydberg one. We need to consider transitions between Rydberg states with a difference in nvalues between the active electron and the highest coreelectron of at least two. For lowcharged ions, this difference in n may be necessary to be increased to up to four. In H and Helike sequences the ~n^{3} behaviour applies starting from n = 5, as shown before. For the Li and Belike sequences the starting shell is n = 6 and for Na and Klike the n = 7.
As the number of electrons increases, the size of the basis set required to obtain accurate results for the excited shells increases significantly. The complication of the core calculations increases and the application of the extrapolation rules becomes impossible. The present extrapolations provide good estimates only for the H and Helike sequences. For other sequences they may not be valid, and cannot be easily tested.
4. Comparisons with observations
In Tables 4 and 5 we show a comparison of the line ratios calculated with the extrapolation rules of the present work with the observed ones of Kepa et al. (2006) for the soft Xrays detected by RESIK coming from highlyexcited states of Si xiii and S xv in solar flares. The transitions involved in the line ratios are dipole allowed.
We have used as our starting point for the extrapolation calculations the n = 5Rmatrix data of Whiteford et al. (2001). We have used a twopoint extrapolation model and a reference point n_{0} = 5, and we have obtained extrapolated Υ from the ground state up to n = 10 the extrapolated values of energies, radiative parameters gf, and electronimpact excitation effective collision strengths Υ obtained with the extrapolation rules described here are available at the CDS. The modelling has been carried out by converting these data into CHIANTI Del Zanna et al. (2015) format and using the CHIANTI population solver to obtain the level populations.
Our theoretical line ratios are quite close (within 10% for most cases) with those estimated by Kepa et al. (2006). Regarding the observed values, Kepa et al. (2006) report a range. The lower values correspond to the peak and gradual phase of the flares. They are within the theoretical values in the 5 − 10 MK range. The higher values, instead, correspond to the early impulsive phase, and are in most cases outside the range of the theoretical values.
5. Discussion and conclusions
We carried out several new calculations, both Rmatix and distorted wave, for the electronimpact excitation of H and Helike ions. We have shown the dependence of the effective collision strengths Υ with respect to the principal quantum number n for several transition types of some benchmark ions of the Helike isoelectronic sequence. We tested three models to reproduce the behaviour of the Υ(n) and extrapolate them to more highlyexcited states.
In general, the extrapolation rules do not give such accurate values for the effective collision strengths as explicit calculations do; instead, they give an approximation that can be used for estimation purposes in modelling. Clearly, it is first necessary to have a good starting calculation before applying the extrapolations to the data. We note that Rmatrix results become increasingly uncertain for the highest energy states included in the CI and CC expansions of the target, due to their lack of convergence (FernándezMenchero et al. 2015). The description of the atomic structure, energy levels and radiative data, and the corresponding effective collision strengths, increasingly lose accuracy as we approach the last states included in the basis expansions. The same happens with the distorted wave calculations. Even when the distorted wave results are not affected by the coupling or resonances, which are included in the Rmatrix ones, the description of the atomic structure is its main limitation. In consequence, such calculations may not give more accurate results for highn than the extrapolation rules combined with accurate calculations for lower, but sufficiently excited, states.
Among the three extrapolation models considered, we recommend the second one. The Model 2 twopoint extrapolation provides results that are closer to the Rmatrix data, and reproduces the behaviour at highn. Nevertheless it is not a completely general rule, and each particular transition type should be analysed to check which is the most accurate extrapolation method for the ion. We do not recommend Model 1 the leastsquares fitting for several reasons. First, having just four points is not enough for a goodquality fitting. Second, the first data points n = 2,3 have not yet reached the required n^{3} behaviour. This occurs because they can not be considered Rydberg levels: they interact more strongly with the core and resonances play a larger role for excitation to these shells. Finally, the fitting method requires a considerably larger computational effort to obtain a result that may be worse than the simpler methods.
The Model 3 onepoint extrapolation gives, in general, a poorer estimate than the twopoint one, and the computational work is not substantially reduced. On the other hand, sometimes the parameters estimated with the onepoint rule are more stable with respect to the reference point n_{0} than the twopoint one, for example, the dipole transitions shown here.
Due to the inherent uncertainty in data for the most highlyexcited states, we do not recommend extrapolating the Υ from the last nshell, but suggest dropping form the extrapolation the last one or two nvalues. Also, the n^{3} behaviour applies only from a certain excited shell. The extrapolation has to start from a level when the atom can be considered as a Rydberg one. That is at least approximately two shells higher than the last occupied one (n = 3 for H and Helike sequences).
We also do not recommend the use of models with an oversimplified atomic structure so as to reach highn shells. The extrapolation from an accurate calculation to lowern provides in general a better estimate than a larger explicit calculation with a poorer atomic structure.
As an example application, we have compared lines ratios obtained with the presented extrapolationrule Model 2 with observations of Xrays by RESIK. We obtain good agreement with the observed Si xiii and S xv ratios during the peak phase of solar flares, but the values during the impulsive phase are still outside the theoretical range. It is expected that during the impulsive phase nonequilibrium effects are present. We investigated whether a nonMaxwellian distribution such as a κdistribution was able to increase the ratios, but did not find significant increases. We are currently investigating other possible causes.
For photoionised plasmas the ions exist at temperatures lower than the peak abundance in an electron collision
dominated plasma. At these lower temperatures, the Υ are affected more by resonance enhancement, and perhaps radiation damping thereof, and can depend greatly on the position of these resonances, which are determined by the atomic structure.
Resonance effects can cause the Υ to deviate from the n^{3} rule. The extrapolation rules should therefore only be applied starting from a higher excited shell, so these effects are minimised. This occurs even if the calculations are perfectly accurate at low temperatures. In these case, we suggest to start the extrapolation at least from four shells above the last occupied (n = 5 for H and Helike sequences).
We estimate that the accuracy of the extrapolation Model 2 is approximately 20% for all transition types of moderately and highlycharged ions, and approximately 50% for lowcharged ions. This estimate considers the core calculations to be perfect and the extrapolation carried out from a shell where the ion can be considered as Rydberg. The inaccuracies in the core calculations could lead to larger errors in some cases, especially for lowcharged ions and/or low temperatures.
Acknowledgments
This work was funded by STFC (UK) through the University of Strathclyde UK APAP network grant ST/J000892/1 and the University of Cambridge DAMTP astrophysics consolidated grant. Luis FernándezMenchero thanks the National Science Foundation (USA) for the grant PHY1520970.
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All Tables
Fitting parameters for the extrapolation of the Υ at highn for the dipole electric transition of Si^{12 +}1s^{2}^{1}S_{0} − 1snp ^{1}P_{1} at a temperature of T = 3.4 × 10^{6} K, for different extrapolation reference points n_{0}.
Fitting parameters for the extrapolation of the Υ at highn for the dipole electric transition of Si^{12 +}1s^{2}^{1}S_{0} − 1sns ^{1}S_{0} at a temperature of T = 3.4 × 10^{6} K, for different extrapolation reference points n_{0}.
Fitting parameters for the extrapolation of the Υ at highn for the dipole electric transition of Si^{12 +}1s^{2}^{1}S_{0} − 1snp ^{3}P_{0} at a temperature of T = 3.4 × 10^{6} K, for different extrapolation reference points n_{0}.
All Figures
Fig. 1 Electronimpact excitation collision strengths for the electric dipole transition 1s^{2}^{1}S_{0} − 1snp ^{1}P_{1} of the ions Si^{12 +} and Fe^{24 +}. Curved lines: Rmatrix; ⋄: distorted wave calculation. 

Open with DEXTER  
In the text 
Fig. 2 Electronimpact excitation effective collision strengths for the electric dipole transition 1s^{2}^{1}S_{0} − 1snp ^{1}P_{1} of the ions Si^{12 +} and Fe^{24 +}. Curved lines: Rmatrix; ⋄: distorted wave calculation basis n = 1 − 8; °: distorted wave calculation basis n = 8 − 12; vertical line: peak abundance temperature for electron collisional plasmas (Mazzotta et al. 1998). 

Open with DEXTER  
In the text 
Fig. 3 Electronimpact excitation effective collision strengths (× n^{3}) for the electric dipole transition 1s^{2}^{1}S_{0} − 1snp ^{1}P_{1} of the ions Si^{12 +} and Fe^{24 +} versus the principal quantum number n around peak abundance temperature. ×: Rmatrix results; ⋄: distorted wave results with basis set n = 1 − 8; °: distorted wave results with basis set n = 8 − 12; solid line: leastsquares fit using points n = 2 − 5; dashed line: extrapolation using the last two points; dotted line: extrapolation using the last point; see text. 

Open with DEXTER  
In the text 
Fig. 4 Extrapolation curves for the Υ × n^{3} displayed in Fig. 3, taking different extrapolation points n_{0}. ×: Rmatrix results; ⋄: distorted wave results. 

Open with DEXTER  
In the text 
Fig. 5 Electronimpact excitation effective collision strengths for the Born transition 1s^{2}^{1}S_{0} − 1sns ^{1}S_{0} of the ions Si^{12 +} and Fe^{24 +}. Curved line: Rmatrix; ⋄: distorted wave calculation basis n = 1 − 8; °: distorted wave calculation basis n = 8 − 12; vertical line: peak abundance temperature for electron collisional plasmas (Mazzotta et al. 1998). 

Open with DEXTER  
In the text 
Fig. 6 Electronimpact excitation effective collision strengths (× n^{3}) for the Born transition 1s^{2}^{1}S_{0} − 1sns ^{1}S_{0} of the ions Si^{12 +} and Fe^{24 +} versus the principal quantum number n around peak abundance temperature. ×: Rmatrix results; ⋄: distorted wave results with basis set n = 1 − 8; °: distorted wave results with basis set n = 8 − 12; solid line: leastsquares fit using points n = 2 − 5; dashed line: extrapolation using the last two points; dotted line: extrapolation using the last point; see text. 

Open with DEXTER  
In the text 
Fig. 7 Extrapolation curves for the Υ × n^{3} displayed in Fig. 6, taking different extrapolation points n_{0}. ×: Rmatrix results; ⋄: distorted wave results. 

Open with DEXTER  
In the text 
Fig. 8 Electronimpact excitation effective collision strengths for the forbidden transition 1s^{2}^{1}S_{0} − 1snp ^{3}P_{0} of the ions Si^{12 +} and Fe^{24 +}. Curved lines: Rmatrix; ⋄: distorted wave calculation basis n = 1 − 8; °: distorted wave calculation basis n = 8 − 12; vertical line: peak abundance temperature for electron collisional plasmas (Mazzotta et al. 1998). 

Open with DEXTER  
In the text 
Fig. 9 Electronimpact excitation effective collision strengths (× n^{3}) for the forbidden transition 1s^{2}^{1}S_{0} − 1snp ^{3}P_{0} of the ions Si^{12 +} and Fe^{24 +} versus the principal quantum number n around peak abundance temperature. ×: Rmatrix results; ⋄: distorted wave results with basis set n = 1 − 8; °: distorted wave results with basis set n = 8 − 12; solid line: leastsquares fit using points n = 2 − 5; dashed line: extrapolation using the last two points; dotted line: extrapolation using the last point; see text. 

Open with DEXTER  
In the text 
Fig. 10 Extrapolation curves for the Υ displayed in Fig. 9, taking different extrapolation points n_{0}. ×: Rmatrix results. 

Open with DEXTER  
In the text 
Fig. 11 Electronimpact excitation effective collision strengths for dipole allowed and Born transitions 1s^{2}^{1}S_{0} − 1sns ^{1}S_{0} of the ion C^{4 +}. Curved lines: Rmatrix; ⋄: distorted wave calculation basis n = 1 − 8; vertical line: peak abundance temperature for electron collisional plasmas (Mazzotta et al. 1998). 

Open with DEXTER  
In the text 