3 Calculations of the collisional ionization and recombination rates

Let us consider a collisional process of cross section $\sigma(E)$, varying with energy E of the incident electron. The corresponding rate coefficient (cm³s^-1), either for a Maxwellian distribution or a Hybrid distribution, f(x), is given by:

 

$$% {\rm Rate} \left(\frac{2kT}{m_{\rm e}}\right)^{\frac{1}{2}}\int_{x_{\rm th}}^{\infty}\ x^{\frac{1}{2}}\ \sigma(x kT)\ f(x)\ {\rm d}x$$ (8)

with $x_{\rm th}=E_{\rm th}/kT$. $E_{\rm th}$ corresponds to the threshold energy of the considered process (for $E<E_{\rm th}$, $\sigma(E)=0$). For the recombination processes, no threshold energy is involved and $x_{\rm th}=0$.

The rates for the Hybrid distribution depend on kT, $x_{\rm b}$ and $\alpha$and are noted $C_{\rm _{\rm I}}^{\rm H}(T,x_{\rm b},\alpha)$, $\alpha_{\rm _{\rm RR}}^{\rm H}(T,x_{\rm b},\alpha)$ and $\alpha_{\rm _{\rm DR}}^{\rm H}(T,x_{\rm b},\alpha)$ for the ionization, radiative and dielectronic recombination process respectively. The corresponding rates for the Maxwellian distribution which only depends on kT are $C_{\rm _{\rm I}}^{\rm M}(T)$, $\alpha_{\rm _{\rm RR}}^{\rm M}(T)$ and $\alpha_{\rm _{\rm DR}}^{\rm M}(T)$.

The ionization data are taken from Arnaud & Rothenflug (1985) and Arnaud & Raymond (1992), as adopted by Mazzotta et al. (1998) for the most abundant elements considered here. The recombination data are taken from the updated calculations of Mazzotta (1998). In the next sections we outline the general behavior of the rates with the electron distribution parameters, using mostly oxygen ions (but also iron) as illustration.

  
3.1 The electronic collisional ionization rates

The ionization cross sections present a threshold at the first ionization potential of the ionizing ion, $E_{\rm _{\rm I}}$. The cross sections always present a maximum, at $E_{\rm m}$, and decrease as $\ln(E)/E$ at very high energies (e.g., Tawara et al. 1985). The ionization rate is very sensitive to the proportion of electrons above the threshold and the modification of the ionization rate for the Hybrid distribution depends on how the high energy tail affects this proportion.

Parametric formulae for the ionization cross sections are available from the litterature and it is easy to derive the corresponding rates for the Hybrid distribution. This is detailed in Appendix A.

To understand the influence of the presence of a high energy power-law tail in the electron distribution, we computed the ratio $\beta_{\rm _{\rm I}}(T,x_{\rm b},\alpha)=C_{\rm _{\rm I}}^{\rm H}(T,x_{\rm b},\alpha)/C_{\rm _{\rm I}}^{\rm M}(T)$, of the ionization rate in a Hybrid distribution over that in a Maxwellian with the same temperature. This ratio is plotted in Figs. 3 to 7 for different ions and values of the parameters $x_{\rm b}$ and $\alpha$.

Figure 3: Variation of the ratio of the ionization rate in a Hybrid distribution over that in a Maxwellian with the same temperature, versus the break parameter $x_{\rm b}$ of the Hybrid distribution. The curves correspond to different values of the slope parameter $\alpha$ and are labeled accordingly. The ion considered is O+6, the temperature is fixed at 106 K.

Let us first consider O⁺⁶. Its ionization potential is $E_{\rm I}=739~{\rm eV}$ and the cross section is maximum at about $3~E_{\rm I}$. Its abundance, for a Maxwellian electron distribution, is maximum at $T^* \simeq 10^{6}$ K under ionization equilibrium (Arnaud & Rothenflug 1985). At this temperature, the threshold energy is well above the thermal energy ( $E_{\rm _{\rm I}}/kT\sim 8$) and only the very high energy tail of the Maxwellian contributes to $C_{\rm _{\rm I}}^{\rm M}(T)$, i.e. a small fraction of the electron distribution. This fraction is dramatically increased in the Hybrid distribution as soon at the break energy is not too far off from the threshold, $x_{\rm b}\sim 15$ for O⁺⁶(Fig. 3). The enhancement factor $\beta _{\rm _{\rm I}}(T,x_{\rm b},\alpha )$ naturally increases with decreasing break $x_{\rm b}$ and slope $\alpha$ parameters (Fig. 3), since the distribution median energy increases when these parameters are decreased (Fig. 2).

This behavior versus $x_{\rm b}$ and $\alpha$ is general at all temperatures as illustrated in Fig. 4, provided that the thermal energy is not too close to $E_{\rm m}$, i.e. that the majority of the contribution to the ionization rate is from electrons with energies corresponding to the increasing part of the ionization cross section. If this is no more the case, the ionization rate starts to decrease with increasing distribution median energy. Thus, for high enough values of the temperature (see the curve at T = 10⁸ K in Fig. 4), the factor $\beta _{\rm _{\rm I}}(T,x_{\rm b},\alpha )$ becomes less than unity and decreases with decreasing $x_{\rm b}$. The correction factor is small (around ${\sim} 10\%$) however in that case.

Figure 4: Same as Fig. 3 but the parameter $\alpha$ is now fixed to 2. The curves correspond to different values of the temperature.

More generally the enhancement factor $\beta _{\rm _{\rm I}}(T,x_{\rm b},\alpha )$ at fixed values of $x_{\rm b}$and $\alpha$, depends on the temperature (Fig. 4). It decreases with increasing temperature: the peak of the distribution is shifted to higher energy as the ratio $kT/E_{\rm _{\rm I}}$ increases and the enhancement due to the contribution of the hard energy tail decreases.

The qualitative behavior outlined above does not depend on the ion considered. We plotted in Figs. 5 and 6 the enhancement factor for the different ions of oxygen and a choice of iron ions at T^* (the temperature of maximum ionization fraction of the ion for a Maxwellian electron distribution under ionization equilibrium). $E_{\rm _{\rm I}}/kT^{*}$ is always greater than unity and the ionization rates are increased by the Hybrid distribution, the enhancement factor $\beta _{\rm _{\rm I}}(T,x_{\rm b},\alpha )$ increasing with decreasing $x_{\rm b}$. However this enhancement factor differs from ion to ion, it generally increases with increasing $E_{\rm _{\rm I}}/kT^{*}$ value (approximatively with an exponential dependence), as shown in Fig. 7. This is again due to the relative position of the peak of the distribution with respect to the threshold energy. Note that $E_{\rm _{\rm I}}/kT^{*}$ is generally smaller for more ionized ions (but this is not strictly true) so that low charge species are generally more affected by the Hybrid distribution.

In summary, the Hybrid rates are increased with respect to the Maxwellian rates except at very high temperature. The enhancement factor depends on the temperature, mostly via the factor $E_{\rm _{\rm I}}/kT$. It increases dramatically with decreasing temperature and is always important at T^*, where it can reach several orders of magnitude. The ionization balance is thus likely to be affected significantly, whereas the effect should be smaller in ionizing plasmas but important in recombining plasmas. For $x_{\rm b}$ typically lower than 10-20 (with this upper limit higher for lower temperature, see Fig. 4), the impact of the Hybrid rate increases with decreasing $x_{\rm b}$ and $\alpha$.

Figure 5: Same as Fig. 3 for the different ions of oxygen. Each curve is labeled by the charge of the ion considered. The slope parameter is fixed to $\alpha =2$ and the temperature for each ion is fixed at the value, T*, where the abundance of the ion is maximum for a Maxwellian electron distribution.

The ionization rates for a Hybrid distribution are less dependent on the temperature than the Maxwellian rates, as illustrated in Figs. 8 and 9. This is a direct consequence of the temperature dependance of the enhancement factor: as this factor increases with decreasing temperature, the Hybrid ionization rate decreases less steeply with temperature than the Maxwellian rates. More precisely, as derived from the respective expression of the rates at low temperature (respectively Eqs. (A.2) and (A.10)), the Maxwellian rate falls off exponentially (as ${\rm e}^{-E_{\rm I}/kT}$) with decreasing temperature, whereas the Hybrid rate only decreases as a power-law. As expected, one also notes that the modification of the rates is more pronounced for lower value of $x_{\rm b}$ (compare the two figures corresponding to $x_{\rm b}=10$ and $x_{\rm b}=2.5$).

Figure 6: Same as Fig. 5 for the different ions of iron.

Figure 7: Maximum enhancement ratio of the ionization rate, $\beta_{\rm_{\rm I}}(T,x_{\rm b},\alpha)$, at T* versus $E_{\rm_{\rm I}}/kT^{*}$. T* is the temperature at which the abundance of the ion is maximum for a Maxwellian electron distribution and $E_{\rm_{\rm I}}$ is the first ionization potential of the ion. Each point corresponds to an ion of oxygen (open circle) or iron (filled circles). The points are labeled by the charge of the corresponding ion. The slope parameter is fixed to $\alpha =2$ and the break is $x_{\rm b}=\alpha+\frac{1}{2}$, corresponding to a maximum enhancement of the ionization rate.

  
3.2 The recombination rates

3.2.1 The radiative recombination rates

The radiative recombination rates are expected to be less affected by the Hybrid distribution, since the cross sections for recombination decrease with energy and no threshold exists. As the net effect of the high energy tail present in the hybrid distribution is to increase the median energy of the distribution (cf. Fig. 2), as compared to a Maxwellian, the radiative recombination rates are decreased.

To estimate the corresponding dumping factor, $\beta_{\rm _{\rm RR}}(x_{\rm b},\alpha)= \alpha_{\rm _{\rm RR}}^{\rm H}(T,x_{\rm b},\alpha)/\alpha_{\rm _{\rm RR}}^{\rm M}(T)$, we follow the method used by Owocki & Scudder (1983). We assume that the radiative recombination cross section varies as a power-law in energy:

$$% \sigma_{\rm _{\rm RR}}(E)\propto{E^{-a}}$$ (9)

which corresponds to a recombination rate (Eq. (8)), for a Maxwellian distribution (Eq. (4)), varying as:

$$% \alpha_{\rm _{\rm RR}}^{\rm M}(T)\propto{T^{\eta}}$$ (10)

with $\eta = a - \frac{1}{2}$.

The dumping factor computed for such a power-law cross section (Eqs. (8) with (9)) is:

$$% \beta_{\rm _{\rm RR}}(x_{\rm b},\alpha)= \frac{ \int_{0}^{\... ...}x} {\int_{0}^{\infty} x^{-\eta}\ f^{\rm M}(x)\ {\rm d}x}\cdot$$ (11)

Note that the dumping factor is independent of the temperature. It depends on the ion considered via the $\eta$ parameter.

Figure 8: Variation of the ionization rate with temperature. Each curve corresponds to an ion of oxygen and is labeled accordingly. Black lines: rates for a Hybrid electron distribution with $\alpha =2$ and $x_{\rm b}=10$. Black thick lines: rates for a Maxwellian distribution.

Replacing the Maxwellian and Hybrid distribution functions by their expression (respectively Eqs. (4) and (7)) we obtain:

$$% \beta_{\rm _{\rm RR}}(x_{\rm b},\alpha)= \frac{C(x_{\rm b},... ...c{3}{2}-\eta}~{\rm e}^{-x_{\rm b}}}{\alpha+\eta-1}\right]\cdot$$ (12)

This estimate of the dumping factor is only an approximation, since the radiative recombination has to be computed by summing over the various possible states of the recombined ions, taking into account the respective different cross sections. Furthermore, even if often the radiative recombination rate can be approximated by a power-law in a given temperature range, this does not mean that the underlying cross section is well approximated by a unique power-law.

Figure 9: Same as Fig. 8 for $\alpha =2$ and $x_{\rm b}=\alpha+\frac{1}{2}=2.5$.

However as we will see the correction factor is small, and we can reasonably assume that it allows a fair estimate of the true Hybrid radiative recombination rates. To minimize the errors, the Hybrid radiative recombination rate has to be calculated from the best estimate of the Maxwellian rates, multiplied by this approximation of the dumping factor:

$$% \alpha_{\rm _{\rm RR}}^{\rm H}(T,x_{\rm b},\alpha)= \beta_{... ...{\rm RR}}(x_{\rm b},\alpha)\ \alpha_{\rm _{\rm RR}}^{\rm M}(T)$$ (13)

where $\alpha_{\rm _{\rm RR}}^{\rm M}(T)$ is as given in Mazzotta et al. (1998).

The parameters $\eta$ for the various ions are taken from Aldrovandi & Péquignot (1973), when available. For other ions we used a mean value of $\eta=0.8$ corresponding to the mean value $<\eta>$ reported in Arnaud & Rothenflug (1985). The exact value has a negligible effect on the estimation of the radiative recombination rates.

Figure 10: Dumping factor for the radiative recombination rates, $\beta_{\rm_{\rm RR}}(x_{\rm b},\alpha)$, of oxygen ions versus the break parameter $x_{\rm b}$. The curves correspond to different values of the slope parameter $\alpha$ and are labeled accordingly. This dumping factor is independent of temperature.

The dumping factor is plotted in Fig. 10 for the various ions of oxygen. In that case a common $\eta$ value is used. The dumping factor decreases with decreasing values of $x_{\rm b}$ and $\alpha$, following the increase of the distribution median energy. The modification is however always modest, at most $15\%$ for $\alpha =2$. For iron, plotted in Fig. 11 for $\alpha =2$, the value of $\eta$ slightly changes with the considered ions, but this only yields negligible variations in the dumping factor.

3.2.2 The dielectronic recombination rates

The dielectronic recombination is a resonant process involving bound states at discrete energies E_i and the rates have to be computed by summing the contribution of many such bound states. According to Arnaud & Raymond (1992), and Mazzotta et al. (1998), the dielectronic recombination rates for a Maxwellian distribution can be fitted accurately by the formula:

$$% \alpha_{\rm _{\rm DR}}^{\rm M}(T)=T_{\rm eV}^{-3/2}\ \sum_{i}c_{i}~{\rm e}^{-x_{i}}~~~~~~~{\rm cm^{3}\ s^{-1}}$$ (14)

where $T_{\rm eV}$ is the temperature expressed in eV and x_i= E_i/kT. The numerical values for c_i and E_iare taken from Mazzotta et al. (1998). Only a few terms (typically 1 to 4) are introduced in this fitting formula. They roughly correspond to the dominant transitions for the temperature range considered.

Figure 11: Same as Fig. 10 for iron ions and $\alpha =2$. The slight variations among the species are due to the corresponding variations in the adopted $\eta$ values (see Eq. (12)).

Following again the method used by Owocki & Scudder (1983), we thus assume that the corresponding dielectronic recombination cross section can be approximated by:

 

$$% \sigma_{\rm _{\rm DR}}(E) \sum_{i} C_{i}~\delta(E-E_{i})~{\rm with}~ C_{i}=\frac{c_{i}\ (2\pi m_{\rm e})^{\frac{1}{2}}}{4~E_{i}}\cdot$$ (15)

The relation between C_i and c_i is obtained by comparing Eq. (14) with the equation obtained by integrating (Eq. (8)) the above cross section over a Maxwellian distribution (Eq. (4)). The dielectronic rates can then be computed from Eq. (8), with the cross section given by Eq. (15) and the distribution function given by Eq. (7):

 

$$% \alpha_{\rm _{\rm DR}}^{\rm H}(T,x_{\rm b},\alpha)=C(x_{\rm b},... ...>x_{\rm b}}c_{i}\left(\frac{x_{i}}{x_{\rm b}}\right)^{-\alpha-\frac{1}{2}}\cdot$$ (16)

Note that this estimate of $\alpha_{\rm _{\rm DR}}^{\rm H}(T,x_{\rm b},\alpha)$ is only an approximation, for the same reasons outlined above for the radiative recombination rates.

To understand the effect of the hybrid distribution, let us assume that only a single energy $E_{\rm _{\rm DR}}$ is dominant, corresponding to a simple Dirac cross section at this energy. In that case, from Eq. (8), the ratio of the dielectronic recombination rate in a Hybrid distribution over that in a Maxwellian with the same temperature, $\beta_{\rm _{\rm DR}}(T, x_{\rm b},\alpha)= \alpha_{\rm _{\rm DR}}^{\rm H}(T,x_{\rm b},\alpha)/\alpha_{\rm _{\rm DR}}^{\rm M}(T)$, is simply the ratio of the Hybrid to the Maxwellian function at the resonance energy. Its expression depends on the position of the resonance energy with respect to the energy break. In reduced energy coordinates, we obtain from Eqs. (4) and (7):

$$% \beta_{\rm _{\rm DR}}(T, x_{\rm b},\alpha) C(x_{\rm b},\alpha) ~~~~~~~~~{\rm for}~\frac{E_{\rm _{\rm DR}}}{kT} \leq x_{\rm b}$$ =

$${\beta_{\rm _{\rm DR}}(T, x_{\rm b},\alpha)} C(x_{\rm b},\alpha)\ {\rm e}^{\left(\frac{E_{\rm _{\rm DR}}}{kT}-... ...ht)}\left(\frac{E_{\rm _{\rm DR}}}{x_{\rm b}kT}\right)^{-(\frac{1}{2} +\alpha)}$$ =

$$~~~~~~~~~~~~~~~~~~~~~{\rm for}~\frac{E_{\rm _{\rm DR}}}{kT} \geq x_{\rm b}.$$ (17)

Figure 12: Variation of the ratio of the DR rate in a Hybrid distribution over that in a Maxwellian with the same temperature, versus the break parameter $x_{\rm b}$ of the Hybrid distribution. The curves correspond to different values of the slope parameter $\alpha$ and are labeled accordingly. The ion considered is O+6, the temperature is fixed at 106 K.

For $(x_{\rm b}~kT) > E_{\rm _{\rm DR}}$ (i.e. at high temperature or high value of $x_{\rm b}$), the resonance lies in the Maxwellian part of the distribution. $\beta_{\rm _{\rm DR}}(T, x_{\rm b},\alpha)$ is independent of the temperature and the dielectronic recombination rates are decreased, following the variation of the normalisation factor, $C(x_{\rm b},\alpha)$, i.e. the decrease is modest (see Fig. 2).

For $(x_{\rm b}~kT) < E_{\rm _{\rm DR}}$ the resonance lies in the power-law part of the distribution. The increase of the dielectronic recombination rate can be dramatic, increasing with decreasing $x_{\rm b}$ and $\alpha$.

These effects of the Hybrid distribution on the dielectronic recombination rates are illustrated in Figs. 12 to 15, where we plotted the factor $\beta_{\rm _{\rm DR}}(T, x_{\rm b},\alpha)$ for various ions and values of the parameters. The factors are computed exactly from Eqs. (14) and (16).

In Fig. 12 we consider O⁺⁶ at the temperature of its maximum ionization fraction, $T^*=10^{6}~{\rm K}$. For this ion only one term is included in the rate estimate, with $E_{\rm _{\rm DR}}= 529~{\rm eV}$, and $E_{\rm _{\rm DR}}/kT=6.1$ at the temperature considered. We plotted the variation of $\beta_{\rm _{\rm DR}}(T, x_{\rm b},\alpha)$ with $x_{\rm b}$ for $\alpha =3$, $\alpha =2$ and $\alpha =1.5$. For $x_{\rm b}> 6.1$ the "resonance'' energy $E_{\rm _{\rm DR}}$ lies in the Maxwellian part of the distribution and the dielectronic recombination rate is decreased as compared to a Maxwellian, but by less than $10\%$, following the variation of the normalisation factor $C(x_{\rm b},\alpha)$. For smaller values of $x_{\rm b}$, the rate is increased significantly, up to a factor of 5 for $\alpha =1.5$.

We consider other temperatures, fixing $\alpha$ to $\alpha =2$, in Fig. 13. Since we only consider the parameter range $x_{\rm b}>\alpha+1/2$, there is a threshold temperature, $kT > E_{\rm _{\rm DR}}/(\alpha+1/2)$, above which the resonance always falls in the Maxwellian part. The dielectronic recombination rate is decreased via the factor $C(x_{\rm b},\alpha)$. This factor slightly decreases with decreasing $x_{\rm b}$ (cf. Fig. 2). At lower temperature, the resonance energy can fall above the break, provided that $x_{\rm b}$ is small enough ( $x_{\rm b}<E_{\rm _{\rm DR}}/kT$). This occurs at smaller $x_{\rm b}$ for higher temperature and the enhancement at a given $x_{\rm b}$ increases with decreasing temperature.

Figure 13: Same as Fig. 12 but the parameter $\alpha$ is now fixed to 2. The curves correspond to different values of the temperature and are labelled accordingly.

Figure 14: Same as Fig. 12 for the different ions of oxygen. Each curve is labeled by the charge of the ion considered. The slope parameter is fixed to $\alpha =2$ and the temperature for each ion is fixed at the value, T*, where the abundance of the ion is maximum for a Maxwellian electron distribution.

Figure 15: Same as Fig. 14 for iron.

We display in Figs. 14 and 15 the variation of the factor $\beta_{\rm _{\rm DR}}(T, x_{\rm b},\alpha)$ with $x_{\rm b}$ (for $\alpha =2$), for the different ions of oxygen and iron, at the temperature of maximum ionization fraction for a Maxwellian distribution under ionization equilibrium. For most of the ions this temperature is above the threshold temperature, $kT = E_{\rm _{\rm DR}}/(\alpha+1/2)$, for all the resonances and the dielectronic rate is decreased. For the ions for which this is not the case (O⁺¹, O⁺⁶ and from Fe⁺¹ to Fe⁺⁵), the dielectronic rate can be increased significantly (by a factor between 2 to 5) provided $x_{\rm b}$ is small enough (typically $x_{\rm b}=2.5-5$). The increase starts as soon as $x_{\rm b}<E_{\rm _{\rm DR}}/kT^*$ for the oxygen ions. The behavior of $\beta_{\rm _{\rm DR}}(T, x_{\rm b},\alpha)$ is more complex for the iron ions (two breaks in the variation of $\beta_{\rm _{\rm DR}}(T, x_{\rm b},\alpha)$), due to the presence of more than one dominant resonance energy (more than one term), taken into account in the computation of the dielectronic rate.

Figure 16: Variation of the total recombination rate with temperature. Each curve corresponds to an ion and is labeled accordingly. Black lines: rates for a Hybrid electron distribution with $\alpha =2$ and $x_{\rm b}=\alpha+\frac{1}{2}=2.5$. Black thick lines: rates for Maxwellian distribution.

In conclusion, the effect of the hybrid distribution on the dielectronic rate depends on the position of the resonance energy as compared to the power-law energy break. It can only be increased if $kT < E_{\rm _{\rm DR}}/(\alpha+1/2)$. At high temperature, the dielectronic recombination rate is slightly decreased.

3.2.3 The total recombination rates

At $x_{\rm b}=10$, the total rates are basically unchanged by the Hybrid distribution. For $x_{\rm b}=2.5=\alpha+1/2$ (Fig. 16), the total rates are more significantly changed. The radiative recombination rate increases with decreasing temperature and it usually dominates the total recombination rate in the low temperature range. As the dielectronic rate is increased by the Hybrid distribution only at low temperature, there are very few ions for which the total recombination rate can be actually increased. This only occurs in a small temperature range, in the rising part of the dielectronic rate. One also notes the expected slight decrease of the radiative recombination rates (when it is dominant at low temperature) and of the dielectronic rate at high temperature.