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A&A
Volume 631, November 2019
Article Number A42
Number of page(s) 9
Section Atomic, molecular, and nuclear data
DOI https://doi.org/10.1051/0004-6361/201935337
Published online 17 October 2019

© ESO 2019

1. Introduction

Carbon is the fourth most abundant element in nature after H, He, and O. It has a solar abundance of log[C/H] = −3.57 (Asplund et al. 2009) and thus contributes much to the emissivity of a plasma. Jointly with He, carbon contributes to the second peak maximum observed in the radiative cooling function of a plasma that evolves under collisional ionization equilibrium (see, e.g., Shapiro & Moore 1976; Boehringer & Hensler 1989; Schmutzler & Tscharnuter 1993; Sutherland & Dopita 1993; Gnat & Sternberg 2007; de Avillez et al. 2010). In addition, C I and C II ions (through fine-structure and metastable transitions) are the main contributors to radiative cooling at temperatures below 103 K. The other contributors are O I, Si I, Si II, S I, Fe I, and Fe II through fine-structure lines and O I, O II, Si I, Si II, Si I, S II, Fe I, and Fe II through metastable transitions (see, e.g., Dalgarno & McCray 1972; Wolfire et al. 1995, 2003). Furthermore, carbon plays an important role in the chemistry of molecular clouds (see, e.g., Larson 1981; Pavlovski et al. 2002; Glover 2007) as well as in shocks (see, e.g., Hollenbach & McKee 1989).

The cross sections of the electron-impact ionization of carbon ions have been extensively studied by means of theoretical calculations and experimental measurements. Since the seminal work of Bethe (1930), who derived the correct form of the cross section at high electron energies, much effort has been invested to calculate the cross sections using empirical and semi-empirical methods or more sophisticated quantum mechanical numerical calculations. Compilations comprising theoretical and experimental ionization cross sections for the carbon atom and ions, including the best fits to these, that have been popular in the Atomic Physics and Astrophysics communities, were published over the years by Lotz (1967, 1968), Bell et al. (1983; hereafter BL83), Arnaud & Rothenflug (1985, AR85), Lennon et al. (1988, L88), Suno & Kato (2006, SK06), Mattioli et al. (2007, M07), Dere (2007, D2007), to name only a few.

The recommended total (BL83; L88; SK06, and M07) and partial (AR85) cross sections are parametrized using the Younger (1981) formula:

σ ( E ) = 10 − 13 I Z , z 2 u [ ∑ i = 1 n max A i ( 1 − 1 u ) i + B ln ( u ) + C log u u ] cm 2 , $$ \begin{aligned} \sigma (E)=\frac{10^{-13}}{I^{2}_{Z,z} u}\left[\sum _{i=1}^{n_{\rm max}}A_{i}\left(1-\frac{1}{u}\right)^{i}+B\ln (u)+C\frac{\log u}{u}\right]\,\mathrm{cm}^{2} , \end{aligned} $$(1)

where u = E/IZ, z, IZ, z is the ionization potential (in eV) of the ion with atomic number Z and ionic state z, E is the incident electron kinetic energy (in eV); nmax = 2 in Younger (1981) and AR85 and 5 in BL83, SK06 and M07. The coefficient B (the Bethe constant) is determined from the photoionization cross section, σph, through

B = 4 I Z , z ∫ I Z , z ∞ 1 ϵ d f d ϵ d ϵ = I Z , z π α ∫ I Z , z ∞ σ ph ϵ d ϵ , $$ \begin{aligned} B=4I_{Z,z}\int _{I_{Z,z}}^{\infty } \frac{1}{\epsilon }\frac{\mathrm{d}f}{\mathrm{d}\epsilon }\mathrm{d}\epsilon =\frac{I_{Z,z}}{\pi \alpha } \int _{I_{Z,z}}^{\infty }\frac{\sigma _{\rm ph}}{\epsilon }\mathrm{d}\epsilon , \end{aligned} $$(2)

where df/dϵ is the optical differential oscillator strength. At high energies, the cross section tends to the Bethe limit (see, e.g., Younger 1981; Pradhan & Nahar 2011),

lim u → ∞ σ ( E ) = 1 u B ln ( u ) . $$ \begin{aligned} \lim _{u\rightarrow \infty } \sigma (E)=\frac{1}{u}B\ln (u). \end{aligned} $$(3)

When B is known, the coefficients Ai and C are determined directly from a least-squares fitting procedure. Hence, the fit to the cross section has the correct asymptote at high energies. In AR85, the ionization potential IZ, z and the coefficients Ai, B, and C refer to the subshell j of the initial ion. The total direct ionization cross section is then obtained by summing over all the subshells.

Bell et al. (1983) and Lennon et al. (1988) adopted the experimental data of Brook et al. (1978) for C I and extrapolated the data beyond 1 keV using a fitting equation to the Born approximation. The recommended curve for the C II ionization cross section follows the cross-beam measurements of Aitken et al. (1971) and at high energies the Coulomb-Born calculations of Moores (1972). For C III and C IV ionization, Bell and collaborators adopted the Coulomb-Born calculations of Jakubowicz & Moores (1981). In the case of C III ionization, a small contribution from inner-shell ionization was included. The cross section for ionization of C V was obtained by scaling the ionization cross section of B IV along the isosequence. For C VI the calculations by Younger (1980) have been adopted.

Arnaud & Rothenflug (1985) considered the ionization cross sections of the different subshells. The C VI cross-section (shell 1s) parameters were determined from the distorted wave-exchange approximation calculations of Younger (1981). For C V (shell 1s2) a fit to the measured cross section available at the Electron-Impact Ionization Data of Multicharged Ions database (EIIDMI; Crandall et al. 1979a)1 was carried out. The theoretical values of Younger (1981) for the direct ionization from shell 1s2 of the Li-like ion C IV were adopted, while for the 2s shell ionization, the Crandall et al. (1979b) data were kept. Excitation-autoiozation contributions to the C IV ion were taken into account; the derived formulae are presented in Appendix A of AR85. For C III (shells 1s2 and 2s2), AR85 adopted the calculations by Younger (1981) along the sequence. The fitting parameters for the C II and C I ionization of shell 2s2 were deduced by extrapolation from higher-z elements. The fit to the C II 2p shell ionization cross section is based on the measurements of Aitken et al. (1971), and those of C I were derived from the measurements by Brook et al. (1978). The parameter B was derived from the photoionization cross section of Reilman & Manson (1979).

Suno & Kato (2006) adopted for C VI and C V the cross sections calculated using the distorted-wave method with exchange (Pattard & Rost 1999) and the distorted-wave Born method (Fang et al. 1995), respectively. For C IV, the experimental data of Knopp et al. (2001) were selected. Because of excitation-autoionization, the cross section has two peaks. For C III, the experimental data of Woodruff et al. (1978) were chosen, and for C II and C I, the experimental data of Yamada et al. (1989) and Brook et al. (1978) were used, respectively.

Mattioli et al. (2007) adopted cross sections from theoretical and experimental data: the C VI and C V cross sections were taken from Arnaud & Rothenflug (1985). For C IV, the Crandall et al. (1979b) data were complemented with those available at the EIIDMI database. For C III, the data of Falk et al. (1983) for low metastable contributions and that of Loch et al. (2005) were adopted. Similarly to SK06, the cross sections of C II and C I were taken from Yamada et al. (1989) and Brook et al. (1978), respectively.

The Dere (2007) compilation combines experimental cross sections with those obtained with the flexible atomic code (FAC)2 (Gu 2002, 2008). For C VI, D2007 favored the parametric fit of Fontes et al. (1999) to their relativistic distorted -wave approximation cross-section calculations for ionization from the 1s shell; the C V cross section is determined with the FAC. For C IV and C III, direct ionization and excitation-autoionization cross sections calculated with the FAC were adopted; for C III, the 1s and 2s subshells were taken into account for the direct ionization cross sections, while for EA the 1s2l3 and 1s2l23l′ transitions were considered. The Yamada et al. (1989) and Brook et al. (1978) cross-section measurements were adopted for C II and C I, respectively. The total cross sections, available in version 8.0.7 of the CHIANTI atomic database3 (Del Zanna et al. 2015), are provided as spline nodes for a scaled energy U and cross section ΣZ, z given by

U = 1 − log f log ( u − 1 + f ) $$ \begin{aligned} U=1-\frac{\log f}{\log (u-1+f)} \end{aligned} $$(4)

and

Σ Z , z = u σ Z , z I Z , z log ( u ) + 1 , $$ \begin{aligned} \Sigma _{Z,z}=\frac{u \sigma _{Z,z}I_{Z,z}}{\log (u)+1}, \end{aligned} $$(5)

respectively. In these expressions f = 2 is an adjustable parameter, σZ, z is the unscaled cross section, and u, and IZ, z have the meanings related in previous paragraphs.

Further calculations using sophisticated quantum mechanical methods have been carried out in the last two decades by Bote et al. (2009), Abdel-Naby et al. (2013), and Wang et al. (2013) for C I, Ludlow et al. (2008), Ballance et al. (2011), Pindzola et al. (2012), and Lecointre et al. (2013) for C II, Fogle et al. (2008) for C III, Pindzola et al. (2012) for C IV, and Fontes et al. (1999) for C VI. Some of these authors also reported experimental measurements that were used to compare with the theoretical cross sections, for instance, Wang et al. (2013) for C I, Lecointre et al. (2013) for C II, and Fogle et al. (2008) for C III.

Empirical classical and semiclassical analytical methods have also been adopted over the years to calculate ionization cross sections of different ions. These include the binary-encounter-Bethe (BEB; Kim & Rudd 1994) model4 and its derivatives, for example, the relativistic BEB (Kim et al. 2000) and the modified relativistic BEB (Guerra et al. 2012, MRBEB) models. The advantage of these models is their simple analytical expressions and the small number of adopted parameters that depend on the binding energy, on the energy of the impacting electron, and on the shielding by inner electrons, for instance. The MRBEB model has been applied to calculate the K-, L- and M-shell ionization cross sections of several atoms, with Z varying between 6 and 83 (Guerra et al. 2012), and for several ionization stages of Ar, Fe, and Kr (Guerra et al. 2013), and U (Guerra et al. 2015). The model provides reliable direct ionization cross sections, and the relative differences to experimental data are smaller than 10% for the inner shells of neutral atoms and 20% for highly charged ions (Guerra et al. 2012, 2013). For a review of these models, see Llovet et al. (2014), for example.

We here use the MRBEB method to calculate the K- and L-shell cross sections of the carbon ions and convolve them with the Maxwell–Boltzmann electron distribution function in order to obtain the corresponding ionization rates. A further application is made to the evolution of an optically thin plasma in order to obtain the radiative losses due to electron-impact ionization, bremsstrahlung, and line emission. The structure of this paper is as follows. Section 2 describes the MRBEB model and the calculation of the K- and L-shell ionization cross sections of the carbon atom and ions. Section 3 describes the use of the calculated cross sections in determining the ionization structure and radiative losses of an optically thin plasma that evolves under collisional ionization equilibrium. Section 4 closes the paper with a discussion and final remarks.

2. Modified relativistic binary encounter Bethe model cross sections

2.1. The MRBEB model

The MRBEB model uses an analytical approach containing a single atomic parameter (the binding energy of the electron to be ionized) and takes into account the energy of the impacting electron and the shielding of the nuclear charge by the bound electrons of the target ion. Therefore, the number of electrons in the inner shells up to the subshell that is ionized acts as a screening of the nuclear potential as seen by the primary electron. In contrast to the MRBEB, the other binary encounter Bethe and relativistic binary encounter Bethe models require two input parameters (the binding energy and the kinetic energy of the bound electron).

The MRBEB cross section, denoted by σnlj,  LS, refers to the ionization of an nlj electron in an atom or ion in a given initial state LS. The cross section takes into account the relativistic interactions between the incident and target electrons during inner-shell ionization of heavy atoms or ions. The determination of the cross section requires knowledge of one single parameter: the binding energy of the target electron. The cross section is given by (Guerra et al. 2012)

σ n l j , LS = 4 π a o 2 α 4 N ( β t 2 + χ β b 2 ) 2 b ′ [ A ( β t , t , b ′ ) + B ( t , t ′ , b ′ ) ] cm 2 $$ \begin{aligned} \sigma _{nlj,\,\mathrm{LS}} = \frac{4\pi a_{o}^{2}\alpha ^{4}N}{\left(\beta _{t}^{2}+\chi \beta _{b}^{2}\right)2 b^{\prime }} \left[A(\beta _{t}, t, b^{\prime })+B(t, t^{\prime },b^{\prime })\right]\,\mathrm{cm}^{2} \end{aligned} $$(6)

with

A ( β t , t , b ′ ) = 0.5 [ ln ( β t 2 1 − β t 2 ) − β t 2 − ln ( 2 b ′ ) ] ( 1 − 1 t 2 ) $$ \begin{aligned} A(\beta _{t},t,b^{\prime })= 0.5\left[\ln \left(\frac{\beta _{t}^{2}}{1-\beta _{t}^{2}}\right)-\beta _{t}^{2}-\ln (2b^{\prime })\right]\left(1-\frac{1}{t^{2}}\right) \end{aligned} $$(7)

and

B ( t , t ′ , b ′ ) = 1 − 1 t − ln t 1 + t 1 + 2 t ′ ( 1 + t ′ / 2 ) 2 + b ′ 2 ( 1 + t ′ / 2 ) 2 t − 1 2 $$ \begin{aligned} B(t, t^{\prime },b^{\prime })=1 -\frac{1}{t}-\frac{\ln t}{1+t}\frac{1+2t^{\prime }}{(1+t^{\prime }/2)^{2}}+ \frac{b^{\prime \, 2}}{(1+t^{\prime }/2)^{2}}\frac{t-1}{2} \end{aligned} $$(8)

with

t = E B , β t 2 = 1 − 1 ( 1 + t ′ ) 2 , b ′ = B m e c 2 , t ′ = E m e c 2 , β b 2 = 1 − 1 ( 1 + b ′ ) 2 · $$ \begin{aligned}&t =\frac{E}{B}, ~\beta _{t}^{2}=1-\frac{1}{(1+t^{\prime })^{2}},~ b^{\prime }=\frac{B}{m_{\rm e} c^{2}}, \\&t^{\prime } =\frac{E}{m_{\rm e} c^{2}},~ \beta _{b}^{2}=1-\frac{1}{(1+b^{\prime })^{2}}\cdot \end{aligned} $$

In these equations, E denotes the kinetic energy of the impacting electron, B is the binding energy of the target electron, and c stands for the speed of light, while me is the electron mass; ao and α are the Bohr radius and the fine-structure constant, respectively. The scaling 1 β 2 + χ β b 2 $ \displaystyle \frac{1}{\beta^{2}+\chi \beta_{b}^{2}} $ includes the effects of the shielding of the nucleus by the target-bound electron through the parameter χ, which is therefore related to the shielding coefficient Cnlj(Z) and the binding energy B of the target electron through the relation χ = 2 Ry Cnlj(Z)/B (Ry is the Rydberg energy). The shielding of the nucleus is described by

C nlj ( Z ) = 0.3 Z e f f , n l j 2 2 n 2 + 0.7 Z eff , n ′ l ′ j ′ 2 2 n ′ 2 , $$ \begin{aligned} C_{nlj}(Z)=0.3 \frac{Z^{2}_{eff,\,nlj}}{2n^{2}}+0.7 \frac{Z^{2}_{{\mathrm{eff},\,n^{\prime }l^{\prime }j^{\prime }}}}{2n^{{\prime }2}} , \end{aligned} $$(9)

where n′l′j′ stands for the next subshell after subshell nlj, ordered in energy, and Zeff is the screening effect (Guerra et al. 2017).

2.2. Cross-section calculations

Using the MRBEB model, we calculated the K- and L-shell ionization cross sections for carbon and its ions. The binding energies of each target electron were calculated using the multi-configuration Dirac-Fock (MCDF) theoretical framework with the multi configuration Dirac-Fock and general matrix element (MCDFGME) code (Indelicato & Desclaux 1990), which evaluates level energies, including correlation and first- and second-order quantum electrodynamics corrections. Table 1 displays the binding energies obtained with the MCDF method without electronic correlation beyond the intermediate coupling using the Rodrigues et al. (2004) ground-state configurations for the different ions. First-order retardation terms of the Breit operator and the Uelhing contribution to the vacuum polarization terms were included self-consistently. The Wichmann-Kroll and Kallen-Sabry contributions, as well as higher-order Breit retardation terms and other QED effects, such as self-energy, were included as perturbations.

Table 1.

Binding energies (eV) calculated with the MDFGME code for the ground-state configurations of C I–C VI.

Figure 1 displays the inner shell (dashed black lines) and total (solid black lines) cross sections of the carbon ions calculated with the MRBEB model. In addition, the figure also shows the total cross sections of Arnaud & Rothenflug (1985, blue lines), Dere (2007, red lines), and those calculated with the GIPPER code5 (Fontes et al. 2015, green lines). In general, the total MRBEB, GIPPER, and D2007 cross sections have a similar distribution for C II, C III, and C IV ions. Small deviations occur during the approach to the peak maximum (C II), in the peak maximum (C III ans C IV), and after the peak maximum (C III and C IV). The total MRBEB cross sections for C II–C VI ions dominate the others for T >  108 K because of the relativistic effects, which are taken into account in the calculations.

thumbnail Fig. 1.

K- and L-shell (dashed black lines) and total (solid black lines) ionization cross sections of the carbon atom and ions calculated with the MRBEB model and total cross sections in AR85 (blue lines) and D2007 (red lines) calculated with the GIPPER code (green lines).

The MRBEB and GIPPER cross sections have small deviations for C I, C II, C III, and C IV near and at the peak maximum. The largest deviations between the MRBEB and GIPPER cross sections occur for C V and C VI ions starting at the approach to the peak maximum. The GIPPER cross sections overlap those discussed in AR85. The cross sections in D2007 have the highest values for C V and C VI ions at peak maximum and at high energies up to 108 K when the MRBEB cross section takes over. Clearly, a simple analytical method depending on a single atomic parameter gives similar results to those provided by the more sophisticated GIPPER and FAC methods.

2.3. Fits to the cross sections

The MRBEB cross sections were fit (for C I see Fig. 2) with the functional (Bote et al. 2009)

σ ( E ) = 4 π a o 2 U − 1 U 2 ( a 1 + a 2 U + a 3 1 + U + a 4 ( 1 + U ) 3 + a 5 ( 1 + U ) 5 ) 2 $$ \begin{aligned} \sigma (E)=4\pi a_{o}^{2} \frac{U-1}{U^{2}}\left(a_{1}+a_{2}U+\frac{a_{3}}{1+U}+\frac{a_{4}}{(1+U)^{3}}+\frac{a_{5}}{(1+U)^{5}}\right)^{2} \end{aligned} $$(10)

thumbnail Fig. 2.

Fit (dotted lines) to the electron-impact ionization cross sections (solid lines) of C I using Eqs. (10) and (11).

for U = E/Enlj ≤ 16, while for higher energies a variation of their Eq. (5) was used,

σ ( E ) = U U + 1.513 4 π a o 2 β 2 [ b 1 χ + b 2 χ P + b 3 + b 4 ( 1 − β 2 ) 1 / 4 + b 5 1 P ] , $$ \begin{aligned} \sigma (E)=\frac{U}{U+1.513}\frac{4\pi a_{o}^{2}}{\beta ^{2}}\left[ b_{1} \chi +b_{2}\frac{\chi }{P}+b_{3}+b_{4}\left(1-\beta ^{2}\right)^{1/4}+b_{5}\frac{1}{P}\right], \end{aligned} $$(11)

with χ = 2lnP − β2, and where β = v / c = E ( E + 2 m e c 2 ) / ( E + m e c 2 ) $ \beta=\mathit{v}/c=\sqrt{E(E+2m_{\mathrm{e}}c^{2})}/(E+m_{\mathrm{e}}c^{2}) $ and P = p / ( m e c ) = E ( E + 2 m e c 2 ) / m e c 2 $ P=p/(m_{\mathrm{e}}c)=\sqrt{E(E+2m_{\mathrm{e}}c^{2})}/m_{\mathrm{e}}c^{2} $ are the velocity and momentum of the impacting electron, respectively, a0 is the Bohr radius and the parameters a1 − a5 and b1 − b5 (displayed in Tables 2 and 3) are characteristic of each element and electron shell. These parameters were calculated using a least-squares fit. The two right columns in the tables represent the maximum relative differences below and above 0, respectively. The absolute relative difference is the largest of the absolute of these two maxima. The relative difference is given by

Δ σ = ( 1.0 − σ fit σ ) 100 % . $$ \begin{aligned} \Delta \sigma = \left(1.0-\frac{\sigma _{\rm fit}}{\sigma }\right)100\%. \end{aligned} $$(12)

Table 2.

a1 − a5 fit coefficients for Eq. (10).

Table 3.

b1 − b5 fit coefficients for Eq. (11).

The maximum relative differences above and below zero are labeled Δσ− and Δσ+, respectively. The maximum absolute relative differences occur for U = E/Enlj ≤ 16, that is, with the fitting of Eq. (10). The maximum error is found to be 1.315% and occurs for the cross section associated with shell 2p1/2 of C I. This is followed for absolute relative differences of 0.915% and 0.81% for the fit of the EII cross sections of the 2s shell of C I and C III. For the fits associated with energies of U = E/Enlj >  16, the maximum absolute relative difference is lower than 0.07%.

3. Application to an optically thin plasma

We discussed different sets of cross sections for the ionization of the carbon atom and ions and now compare their effects on the ionization structure and on the cooling of an optically thin plasma characterized by a Maxwell–Boltzmann (MB) electron distribution function. The determination of the ionic state of the plasma as well as its emissivity is important for theoretical models as well as observations of absorption features in the diffuse medium where Li-like ions (C IV, N V, and O VI) are important to distinguish between different ionization mechanisms, such as turbulent mixing layers (Slavin et al. 1993), shock ionization (Dopita & Sutherland 1996), conduction interfaces (Borkowski et al. 1990), and radiative cooling (Edgar & Chevalier 1986). These processes in turn are used to study high-velocity clouds (see, e.g., Indebetouw & Shull 2004) or gas in the Local Bubble (de Avillez & Breitschwerdt 2009). In addition, the emission caused by C V and C VI is important to understand the spectra associated with the processes described above, as well as with that of ionizing (Masai 1984) or recombining (de Avillez & Breitschwerdt 2012) plasmas or in the soft X-ray spectra that are observed in supernova remnants. Spectral fitting codes for X-ray emitting space plasmas are often used to determine the temperature and ionization state of a plasma. Orbiting X-ray observatories such as Chandra and XMM-Newton have chip-based X-ray detectors that can barely resolve individual lines for diffuse low surface brightness plasmas like the hot interstellar medium. Therefore it is important to quantify as well as possible the contribution of various ions such as carbon to the total spectrum.

For each set of cross sections we followed the evolution of a gas parcel cooling under collisional ionization equilibrium conditions from an initial temperature of 109 K where it is completely ionized. The plasma was composed of hydrogen and carbon with solar abundances (Asplund et al. 2009) and a hydrogen particle density, nH, of 1 cm−3. These calculations are referred to as the MRBEB, GIPPER, and D2007 models.

The processes we took into account are electron-impact ionization, radiative recombination including cascades into the ground state, dielectronic recombination, bremsstrahlung, and line emission. No charge-exchange reactions were considered. Hence, and because the evolution is in collisional ionization equilibrium, the ionization structure of each element (composed of atoms and ions) can be treated independently of the other elements. Thus, the evolution of the carbon atom and ions is the same whether the plasma is composed only of carbon or of any set of elements including carbon.

3.1. Thermal model

The ionization structure and emission properties of the gas parcel were followed using the thermal model described in de Avillez et al. (2018; for further details, see de Avillez in prep.). We therefore present a summary of this model in this subsection.

The density nZ, z (cm−3) of an ion with atomic number Z and charge state z (z = 0,  …, Z) is determined from the populations of the neighboring charge states z − 1, z, and z + 1 through recombination (αZ, z, which includes radiative and dielectronic recombination) and collisional ionization (SZ, z) rates from state z to z − 1 and z + 1, respectively. Hence, nZ, z is given by the system of equations

S Z , z − 1 n Z , z − 1 n e − ( S Z , z + α Z , z ) n Z , z n e + α Z , z + 1 n Z , z + 1 n e = 0 , $$ \begin{aligned} S_{Z,z-1}n_{Z,z-1}n_{\rm e}-(S_{Z,z}+ \alpha _{Z,z})n_{Z,z}n_{\rm e}+\alpha _{Z,z+1}n_{Z,z+1}n_{\rm e}=0, \end{aligned} $$(13)

where ne is the electron density (cm−3), which is obtained from

n e = ∑ z = 1 Z z n Z , z . $$ \begin{aligned} n_{\rm e}=\sum _{z=1}^{Z}z\,n_{Z,z}. \end{aligned} $$(14)

By multiplying both sides of Eq. (13) by 1/nZ (nZ is the number density (cm−3) of the species of atomic number Z) and factorizing ne, the system of equations simplifies to

S Z , z − 1 x Z , z − 1 − ( S Z , z + α Z , z ) x Z , z + α Z , z + 1 x Z , z + 1 = 0 , $$ \begin{aligned} S_{Z,z-1}x_{Z,z-1}-(S_{Z,z}+ \alpha _{Z,z})x_{Z,z}+\alpha _{Z,z+1}x_{Z,z+1}=0 , \end{aligned} $$(15)

where xZ, z = nZ, z/nZ is the ion fraction (which varies between 0 and 1). This system of equations may be cast into the matrix form

AX = 0 , $$ \begin{aligned} {AX}=0, \end{aligned} $$(16)

where X is a vector comprising all ion fractions xZ, z and A is a tridiagonal matrix with elements SZ, z − 1, −(SZ, z + αZ, z), and αZ, z + 1 at each row populating the diagonal band. The solution of this system of equations is straightforward using any Gauss-elimination method with or without pivoting. The final solution is then transformed into the ion density through

n Z , z = x Z , z n Z = x Z , z A ( Z ) n H , $$ \begin{aligned} n_{Z,z}=x_{Z,z}\,n_{Z}=x_{Z,z}\,A(Z)n_{\rm H}, \end{aligned} $$(17)

where A(Z)=nZ/nH is the abundance of the species and nH is the hydrogen number density (cm−3).

The ionization rates, SZ, z, are determined by convolving σ(E)v with the MB electron distribution function,

f ( E ) d E = 2 E 1 / 2 π 1 / 2 ( k B T ) 3 / 2 e − E / k B T d E , $$ \begin{aligned} f(E)\mathrm{d}E=\frac{2E^{1/2}}{\pi ^{1/2}(k_{\rm B}T)^{3/2}} \mathrm{e}^{-E/k_{\rm B}T}\mathrm{d}E, \end{aligned} $$(18)

and are given by

⟨ σ v ⟩ = ( 2 m e ) 1 / 2 ∫ Φ Z , z + ∞ σ ( E ) E 1 / 2 f ( E ) d E cm 3 s − 1 , $$ \begin{aligned} \langle \sigma { v}\rangle =\left(\frac{2}{m_{\rm e}}\right)^{1/2}\int _{\Phi _{Z,z}}^{+\infty } \sigma (E) E^{1/2} f(E) \mathrm{d}E\quad \,\mathrm{cm}^{3}\mathrm{s}^{-1}, \end{aligned} $$(19)

where me is the electron mass (g), Φz, z is the ionization threshold (eV), and σ(E) is the electron-impact ionization cross section (cm2).

Radiative and dielectronic recombination rates used in these calculations are based on calculations with the AUTOSTRUCTURE code6 (Badnell 2011) and are taken from Badnell (2006a) for H II and C II through C VII recombining to H I and C I through C VI, respectively. The radiative recombination rates have the functional

α Z , z RR = A [ ( T T 0 ) 1 / 2 ( 1 + ( T T 0 ) 1 / 2 ) 1 − b ( 1 + ( T T 1 ) 1 / 2 ) 1 + b ] − 1 , $$ \begin{aligned} \alpha _{Z,z}^\mathrm{RR}=A\left[\left(\frac{T}{T_{0}}\right)^{1/2}\left(1+\left(\frac{T}{T_{0}}\right)^{1/2}\right)^{1-b} \left(1+\left(\frac{T}{T_{1}}\right)^{1/2}\right)^{1+b}\right]^{-1}, \end{aligned} $$(20)

where A (cm3 s−1), T0, 1 (K), and b (dimensionless) are fit coefficients. The latter is replaced by b + Cexp(T2/T) (C is dimensionless and T2 is given in K) for low-ionization stages. The dielectronic recombination rates are given by the Burgess (1965) general formula

α DR MB = 1 ( k B T ) 3 / 2 ∑ j c j e − E j / ( k B T ) cm 3 s − 1 , $$ \begin{aligned} \alpha _{\rm DR}^\mathrm{MB}=\frac{1}{(k_{\rm B}T)^{3/2}}\sum _{j}c_{j} \mathrm{e}^{-E_{j}/(k_{\rm B}T)} \quad \mathrm{cm}^{3} \mathrm{s}^{-1} , \end{aligned} $$(21)

whose coefficients are taken from Badnell (2006b) for H II and C VI, Bautista & Badnell (2007) for C V, Colgan et al. (2004, 2003) for C IV and C III, respectively, and Altun et al. (2004) for C II.

The cooling due to electron-impact ionization ( Λ Z , z EII $ \Lambda^{\mathrm{EII}}_{Z,z} $), bremsstrahlung ( Λ Z , z FF $ \Lambda^{\mathrm{FF}}_{Z,z} $), and line (permitted, forbidden, and semi-forbidden) emission ( Λ Z , z LE $ \Lambda^{\mathrm{LE}}_{Z,z} $) shown in Fig. 4 are given by

Λ Z , z EII = n e n H A ( Z ) x Z , z S Z , z Φ Z , z erg cm − 3 s − 1 , $$ \begin{aligned} \Lambda ^\mathrm{EII}_{Z,z}=n_{\rm e}n_{\rm H}A(Z)x_{Z,z}S_{Z,z} \Phi _{Z,z}\quad \mathrm{erg~cm}^{-3}\,\mathrm{s}^{-1}, \end{aligned} $$(22)

with SZ, z denoting the ionization rate (cm3 s−1) and ΦZ, z is the ionization threshold (erg) of the ionizing ion,

Λ Z , z FF = C z 2 n e n H A ( Z ) x Z , z T 1 / 2 ⟨ g ff ( γ 2 ) ⟩ erg cm − 3 s − 1 , $$ \begin{aligned} \Lambda ^\mathrm{FF}_{Z,z}=C\, z^{2} n_{\rm e}n_{\rm H}A(Z)x_{Z,z}T^{1/2} \langle { g}_{_{\rm ff}}(\gamma ^{2})\rangle \quad \mathrm{erg~cm}^{-3}\, \mathrm{s}^{-1}, \end{aligned} $$(23)

where C = 1.4256 × 10−27 erg cm3 s−1 K−1/2, T is the temperature (K), γ2 = z2Ry/kBT (Ry is the Rydberg energy) is the normalized temperature, and ⟨gff(γ2)⟩ is the total free-free Gaunt factor (see the details of its calculation in, e.g., de Avillez & Breitschwerdt 2015), and

Λ Z , z LE = n e n H A ( Z ) x Z , z ∑ i , j , i < j n Z , z , j A ji E ij erg cm − 3 s − 1 , $$ \begin{aligned} \Lambda ^\mathrm{LE}_{Z,z}=n_{\rm e}n_{\rm H}A(Z)x_{Z,z}\sum _{i,j,i<j}n_{Z,z,j}A_{ji}E_{ij} \quad \mathrm{erg~ cm}^{-3}\,\mathrm{s}^{-1}, \end{aligned} $$(24)

where nZ, z, j is the population density of level j of the ion, Aji is the spontaneous decay rate from level j to level i (i <  j), and Eij is the excitation energy between the two levels.

The level populations were calculated as described in de Avillez et al. (2018) by assuming that there is an equilibrium between excitation by electron impact and de-excitation by electron impact and spontaneous decay. Hence, the population of level j is obtained from the equation

∑ m < j C mj e n e n Z , z , m + ∑ n > j ( A nj + C nj d n e ) n Z , z , n − n Z , z , j [ ∑ j < n C jn e n e + ∑ j > m ( A jm + C jm d n e ) ] = 0 , $$ \begin{aligned}&\sum _{m<j}C_{mj}^\mathrm{e}n_{_{\rm e}}n_{Z,z,m}+\sum _{n>j}\left(A_{nj}+C_{nj}^\mathrm{d}n_{\rm e}\right)n_{Z,z,n} \nonumber \\&\quad - n_{Z,z,j}\left[\sum _{j<n}C_{jn}^\mathrm{e}n_{\rm e}+\sum _{j>m}\left(A_{jm}+C_{jm}^\mathrm{d}n_{\rm e}\right)\right] = 0, \end{aligned} $$(25)

coupled to the equation of mass conservation

∑ j n Z , z , j = n Z , z . $$ \begin{aligned} \sum _{j}n_{Z,z,j}=n_{Z,z}. \end{aligned} $$(26)

In these equations C mj e $ C^{\mathrm{e}}_{mj} $ and C jn e $ C^{\mathrm{e}}_{jn} $ denote the excitation rates (cm3 s−1) from levels m to j (m <  j) and j to n (j <  n), respectively, while C nj d $ C^{\mathrm{d}}_{nj} $ and C jm d $ C^{\mathrm{d}}_{jm} $ denote the de-excitation rates (cm3 s−1) from levels n to j and j to m, respectively; Anj and Ajm are the Einstein spontaneous decay coefficients (s−1) from levels n to j and j to m, respectively; and nZ, z, m and nZ, z, n are the population densities (cm−3) of levels m and n, respectively. The excitation and de-excitation rates are given by

C ij e = 8.629 × 10 − 6 T − 1 / 2 ω i − 1 e − y Υ ij ( T ) $$ \begin{aligned} C^\mathrm{e}_{ij}=8.629\times 10^{-6}\,T^{-1/2}\,\omega ^{-1}_{i}\, \mathrm{e}^{-{ y}}\, \Upsilon _{ij}(T) \end{aligned} $$(27)

and

where y = Eij/kBT, U = Ee/Eij is the reduced electron energy (Ee is the energy of the impacting electron), ωi and ωj are the statistical weights of levels i and j, respectively, and T is the temperature (K). The forms of the effective collision strength, Υij(T), and of relate to the collision strengths and are given by

Υ ij ( T ) = y e y ∫ 1 + ∞ Ω ij ( U ) e − y U d U $$ \begin{aligned} \Upsilon _{ij}(T) = { y} \mathrm{e}^{{ y}}\int _{1}^{+\infty }\Omega _{ij}(U)\, \mathrm{e}^{-{ y}U}\mathrm{d}U \end{aligned} $$(29)

The wavelengths, coefficients of spontaneous transitions, and the effective collision strengths were taken from version 8.0.7 of the CHIANTI atomic database.

3.2. Calculations and methods

The evolution of a gas parcel is calculated as follows: (i) at each temperature calculate first the ionization and recombination rates and solve the system of equations (15) to obtain the ion fractions, (ii) determine the ions and the electron densities from Eqs. (17) and (14), respectively, and (iii) obtain the emissivity due to the different processes using Eqs. (22)–(24).

The numerical methods used in these calculations are the same as in de Avillez et al. (2018), that is, (i) numerical integrations in semi-finite intervals, such as the ionization rates, were calculated with a precision of 10−15 using the double-exponential transformation method of Takahasi & Mori (1974), Mori & Sugihara (2001), and (ii) the system of equations (15) is solved using a Gauss-elimination method with scaled partial pivoting (Cheney & Kincaid 2008) and a tolerance of 10−15.

3.3. Results

The carbon-ion fraction variations with temperature calculated with the MRBEB, GIPPER, and D2007 ionization cross sections are displayed in Fig. 3. Although there are some differences between the location of the ion profiles, the ionic structure shows similar properties in the three models: the same profiles for all the ions that are characterized with the dominance of C VII above 106 K, the dominance of C I below 104 K, and the classic C V plateau resulting from the K-shell ionization potential. The differences between the three models are reflected in the small deviation to the right for the MRBEB model with regard to the others, while the GIPPER model shows profiles that in some cases are to the left of the D2007 profiles and in other cases to the right, but always to the left of the MRBEB profiles. The reason is that the threshold ionization energies in the MRBEB model are higher than those of the GIPPER and D2007 models, which leads to a delayed ionization of the carbon ions in comparison to the other cases.

thumbnail Fig. 3.

Carbon ionic fraction variation with temperature evolving under collisional ionization equilibrium calculated with the MRBEB, GIPPER, and D2007 cross sections.

Figure 4 displays the cooling (erg cm3 s−1), normalized to nenH, due to electron-impact ionization (EII), bremsstrahlung (FF) using the total Gaunt factors calculated in de Avillez & Breitschwerdt (2017), for instance, and line (allowed, forbidden, and semi-forbidden) emission (LE) calculated with the three models (MRBEB, GIPPER, and D2007). Emissivities associated with different processes in different ranges in temperature overlap. For instance, above 105 K, the bremsstrahlung in the three models overlaps, except around 106. A complete overlap is visible above 2 × 106 K in electron-impact ionization and bremsstrahlung. In the latter the overlap extends to 105 K, except around 106 K. Deviations among the three models are seen in electron-impact ionization below 106 K and in line emission above 105 K, where the second peak maximum noticeably increases. For lower temperatures the line emission seems to be the same for all the models. The excess in emissivity in the MRBEB model is similar to that seen in the ionization structure (i.e., the deviation to the right). Although there are deviations in the models, they predict in general the same behavior and profile for the emissivities due to the different processes.

thumbnail Fig. 4.

Comparison of the cooling (erg cm3 s−1), normalized to nenH, due to electron-impact ionization (EII), bremsstrahlung (FF), and line emission (LE)) in a gas parcel evolving under collisional ionization equilibrium and calculated with MRBEB, GIPPER, and D2007 electron-impact ionization cross sections.

4. Discussion and final remarks

We applied the modified relativistic binary encounter Bethe model to calculate the K- and L-shell ionization cross sections of the carbon atom and ions and compare their variation with energy of the impacting electron with those calculated using the GIPPER code and those published by Dere (2007), which includes cross sections calculated with the flexible atomic code and the relativistic distorted wave approximation of Fontes et al. (1999), and experimental data.

In general, the three sets of cross sections have a similar profile for C II, C III, and C IV ions, and small deviations occur during the approach to the peak (C II), in the peak (C III and C IV), and after the peak (C III and C IV) maximum. The total MRBEB cross sections for C II–C VI ions dominate the others for T >  108 K because of the relativistic effects. The MRBEB and GIPPER cross sections have small deviations for C I, C II, C III, and C IV near and at the peak maximum, while the largest deviations occur in the approach to the peak maximum for C V and C VI. The D2007 cross sections have the highest values for C V and C VI ions at peak maximum and at high energies up to 108 K when the MRBEB cross sections take over. Although the FAC and GIPPER code use the distorted-wave method for the electron-impact ionization, the D2007 and GIPPER cross sections still show differences that can stem from the different numerical methods and the adopted atomic model.

We further explored the effects of the three sets of cross sections on the ion fractions of the carbon ions and on the cooling due to electron-impact ionization, bremsstrahlung, and line emission by an optically thin plasma that evolves under collisional ionization equilibrium and cooling from a temperature of 109 K. The three calculations, using the MRBEB, GIPPER, and D2007 cross sections, show deviations in the ion fractions of the same ion that decrease with increase in ionization degree: the strongest deviations occur in the lowest ionization states (C I–C III), and the smallest deviations in the highest ionization states. These differences in the ion fractions propagate to the emissivities. At high temperatures the emissivities are similar in the three calculations, while in other temperature regimes noticeable differences are observed, for instance, in the second peak maximum of the line emission around 106 K. The agreement between the emissivities calculated with the three sets of cross sections is nevertheless good overall.

The results show that a simple analytical model that only depends on one atomic parameter (the electron binding energy) is capable of providing electron-impact ionization cross sections similar to those calculated with the more sophisticated quantum mechanical methods in GIPPER and FAC. This shows that it is possible to build a database of cross sections associated with the atoms and ions of the ten most abundant elements in nature with the MRBEB method that can be used by any spectral emission code.

4

BEB combines the Mott cross section with the high-incident energy behavior of the Bethe cross section.

5

GIPPER provides electron-impact ionization, photoionization, and autoionization data using the distorted-wave approach.

Acknowledgments

This research was supported by the projects “Enabling Green E-science for the SKA Research Infrastructure (ENGAGE SKA)” (M. A.; reference POCI-01-0145-FEDER-022217, funded by COMPETE 2020 and FCT) and “Ultra-high-accuracy X-ray spectroscopy of transition metal oxides and rare earths” (M. G.; reference PTDC/FIS-AQM/31969/2017, and SFRH/BPD/92455/2013 funded by FCT), and by the project UID/FIS/04559/2013 (LIBPhys). Partial support for M. A. and D. B. was provided by the Deutsche Forschungsgemeinschaft, DFG project ISM-SPP 1573. The calculations were carried out at the ISM – Xeon Phi cluster of the Computational Astrophysics Group, University of Évora, acquired under project “Hybrid computing using accelerators & coprocessors – modelling nature with a novell approach” (M. A.), InAlentejo program, CCDRA, Portugal.

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All Tables

Table 1.

Binding energies (eV) calculated with the MDFGME code for the ground-state configurations of C I–C VI.

In the text
Table 2.

a1 − a5 fit coefficients for Eq. (10).

In the text
Table 3.

b1 − b5 fit coefficients for Eq. (11).

In the text

All Figures

thumbnail Fig. 1.

K- and L-shell (dashed black lines) and total (solid black lines) ionization cross sections of the carbon atom and ions calculated with the MRBEB model and total cross sections in AR85 (blue lines) and D2007 (red lines) calculated with the GIPPER code (green lines).
In the text
thumbnail Fig. 2.

Fit (dotted lines) to the electron-impact ionization cross sections (solid lines) of C I using Eqs. (10) and (11).
In the text
thumbnail Fig. 3.

Carbon ionic fraction variation with temperature evolving under collisional ionization equilibrium calculated with the MRBEB, GIPPER, and D2007 cross sections.
In the text
thumbnail Fig. 4.

Comparison of the cooling (erg cm3 s−1), normalized to nenH, due to electron-impact ionization (EII), bremsstrahlung (FF), and line emission (LE)) in a gas parcel evolving under collisional ionization equilibrium and calculated with MRBEB, GIPPER, and D2007 electron-impact ionization cross sections.
In the text

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