Appendix A: Ionization rates

  
A.1 Direct ionization (DI)

For the direct ionization (DI) cross sections we chose the fitting formula proposed by Arnaud & Rothenflug (1985) from the work of Younger (1981):

 

$$% \sigma_{\rm _{\rm DI}}(E) \sum_{j}\frac{1}{u_j I_j^{2}}\left[A_jU_j{+} B_j U_j^{2} {+}C_j \,\ln(u_j)+D_j\frac{\ln(u_j)}{u_j}\right] {\rm with}~~u_j=\frac{E}{I_j}~; ~~~U_j=1-\frac{1}{u_j}\cdot$$ (A.1)

The sum is performed over the subshells j of the ionizing ion. E is the incident electron energy and I_j is the collisional ionization potential for the level j considered.

The parameters A_j, B_j, C_j, D_j (in units of 10^-14cm²eV²) and I_j (in eV) are taken from the works of Arnaud & Raymond (1992) for iron, and of Arnaud & Rothenflug (1985) for the others elements. The parameters for elements not considered in these works are given in Mazzotta et al. (1998).

A.1.1 The Maxwellian electron distribution

For a Maxwellian electron distribution, Arnaud & Rothenflug (1985) obtained according to Eqs. (2), (8) and (A.1), the rate:

$$% C_{\rm _{\rm DI}}^{\rm M}(T)=\frac{6.692\times 10^{7}}{(kT)... ...-x_j}}{x_j} F^{\rm M}_{\rm _{\rm DI}}(x_j)~~~{\rm cm^3~s^{-1}}$$ (A.2)

where

$$% x_j =\frac{I_j}{kT}$$ (A.3)

 

$$F^{\rm M}_{\rm _{\rm DI}}(x_j)= A_j~[1-x_j f_{1}(x_j)]+B_j~[1+x_j -x_j(2+x_j)f_1(x_j)]+C_j~f_1(x_j)+D_j~x_j f_2(x_j)$$ (A.4)

where kT and I_j are in eV. The summation is performed over the subshells j of the ionizing ion. The mathematical functions, $f_1(x)={\rm e}^x \int_{1}^{\infty} \frac{{\rm e}^{-tx}}{t}~{\rm d}t$, and $f_2(x) ={\rm e}^x \int_{1}^{\infty} \frac{{\rm e}^{-tx}}{t}~\ln (t)~{\rm d}t$ can be computed from the analytical approximations given by Arnaud & Rothenflug (1985) in their Appendix B.

A.1.2 The Hybrid electron distribution

Similar to the Hybrid electron distribution, the direct ionization rate $C_{\rm _{\rm DI}}^{\rm H}(T,x_{\rm b},\alpha)$ is given by:

 

$$% C_{\rm _{\rm DI}}^{\rm H}(T,x_{\rm b},\alpha)=C(x_{\rm b},\alph... ...} \sum_{j}G^{\rm H}_{\rm _{\rm DI}}(x_j,x_{\rm b}, \alpha)~~~{\rm cm^3~s^{-1}}.$$ (A.5)

The function $G^{\rm H}_{\rm _{\rm DI}}(x_j,x_{\rm b}, \alpha)$ depends on the position of the power-law break energy as compared to the ionization potential:

• For $x_{\rm b}>x_j$:

  $G^{\rm H}_{\rm _{\rm DI}}(x_j,x_{\rm b}, \alpha)$ is the sum of the contribution of the truncated Maxwellian component and the power-law component:
  

  $$% G^{\rm H}_{\rm _{\rm DI}}(x_j,x_{\rm b},\alpha)=\frac{{\rm e}^{... ... b}) + {\rm e}^{-x_{\rm b}}\ F_{\rm _{\rm DI}}^{\rm PL}(u_{{\rm b}_{j}},\alpha)$$ (A.6)

  
  
  where:
  

  $$% u_{{\rm b}_{j}} \frac{x_{\rm b}}{x_j}$$ = (A.7)

  $$F^{'}_{\rm _{\rm DI}}(x_j,x_{\rm b}) A_j \left[1-x_j f_{1}(x_{\rm b})\right]+B_j \left[1+\frac{x_j}{u_{{\rm b}_{j}}}-x_j (2+x_j) f_1(x_{\rm b})\right]$$ =

  $$+C_j \left[f_1(x_{\rm b}) + \ln\left(u_{{\rm b}_{j}}\right)\right... ...ft[x_j f_2(x_{\rm b})+\ln\left(u_{{\rm b}_{j}}\right) x_j f_1(x_{\rm b})\right]$$ (A.8)

  $$F_{\rm _{\rm DI}}^{\rm PL}(u_{{\rm b}_{j}},\alpha) A_{j}\left[ \frac{u_{{\rm b}_{j}}}{\alpha-1/2} - \frac{1}{\alpha+... .../2} {-} \frac{2}{\alpha+1/2} {+}\frac{u_{{\rm b}_{j}}^{-1}}{\alpha+3/2} \right]$$ =

  $$+\frac {C_j u_{{\rm b}_{j}}}{\alpha-1/2} \left[ \ln(u_{{\rm b}_{j... ...{D_j}{\alpha+1/2} \left[\ln(u_{{\rm b}_{j}}) + \frac{1}{\alpha+1/2}\right]\cdot$$ (A.9)

  
  

• For $x_{\rm b}<x_j$:

  Only the power-law component contributes of the electron distribution to the rate:
  

   

  $$% G^{\rm H}_{\rm _{\rm DI}}(x_j,x_{\rm b}, \alpha) {\rm e}^{-x_{\rm b}}\ \left(\frac{x_{\rm b}}{x_j}\right)^{\alpha+\frac{1}{2}}\ f_{\rm _{\rm DI}}(\alpha)$$ = (A.10)

  
  
  where
  

  $$% f_{\rm _{\rm DI}}(\alpha)=\frac{A_j}{\alpha^{2}-1/4} + \frac{2 ... ..._j}{\left(\alpha-1/2\right)^{2}} + \frac{D_j}{\left(\alpha+1/2\right)^{2}}\cdot$$ (A.11)

  
  

  
A.2 Excitation autoionization (EA)

For the excitation autoionization (EA) cross sections, we used the generalized formula proposed by Arnaud & Raymond (1992):

 

$$% \sigma_{\rm _{\rm EA}}(E) \frac{1}{u I_{\rm _{\rm EA}}}\left[A+B~U+C~U_{2}+D~U_{3}+E~\ln(u)\right]$$ =

$${\rm with}~~u=\frac{E}{I_{\rm _{\rm EA}}}~; ~~~~U_{n}=1-\frac{1}{u^n}$$ (A.12)

where $I_{\rm _{\rm EA}}$ is the excitation autoionization threshold and E is the incident electron energy.

The parameters A, B, C, D, E (in units of 10^-16cm²eV) and $I_{\rm _{\rm EA}}$ (in eV) are taken from the works of Arnaud & Rothenflug (1985) and Arnaud & Raymond (1992). The parameters for elements not considered in these works are given in Mazzotta et al. (1998).

A.2.1 The Maxwellian electron distribution

The excitation autoionization rate for a Maxwellian distribution is:

 

$$% C_{\rm _{\rm EA}}^{\rm M}(T) \frac{6.692 \times 10^{7}~{\rm e}^{-x_{\rm _{\rm EA}}}}{(kT)^{1/2}}~ F^{\rm M}_{\rm _{\rm EA}}(x_{\rm _{\rm EA}})~~~{\rm {cm}^3\,s^{-1}}$$ = (A.13)

where

 

$$% x_{\rm _{\rm EA}} \frac{I_{\rm _{\rm EA}}}{kT}$$ = (A.14)

$$F^{\rm M}_{\rm _{\rm EA}}(x_{\rm _{\rm EA}}) A+B[1-x_{\rm _{\rm EA}}f_1(x_{\rm _{\rm EA}})]+C[1-x_{\rm _{\rm EA}}+x_{\rm _{\rm EA}}^{2} f_1(x_{\rm _{\rm EA}})]$$ = (A.15)

$$+D\left[1-\frac{x_{\rm _{\rm EA}}}{2}+\frac{x_{\rm _{\rm EA}}^{2}... ..._{\rm _{\rm EA}}^3}{2} f_1(x_{\rm _{\rm EA}})\right]+ E f_1(x_{\rm _{\rm EA}}).$$

A.2.2 The Hybrid electron distribution

For the Hybrid electron distribution, the excitation autoionization rate $C_{\rm _{\rm EA}}^{\rm H}(T,x_{\rm b},\alpha)$ is given by:

 

$$% C_{\rm _{\rm EA}}^{\rm H}(T,x_{\rm b},\alpha)=C(x_{\rm b},\alph... ..._{\rm _{\rm EA}}(x_{\rm _{\rm EA}},x_{\rm b}, \alpha)~~~~~~~~{\rm cm^3~s^{-1}}.$$ (A.16)

The function $G^{\rm H}_{\rm _{\rm EA}}(x_{\rm _{\rm EA}},x_{\rm b}, \alpha)$ depends on the position of the power-law break energy as compared to the ionization potential:

• For $x_{\rm b}>x_{\rm _{\rm EA}}$:

  $G^{\rm H}_{\rm _{\rm EA}}(x_{\rm _{\rm EA}},x_{\rm b}, \alpha)$ is the sum of the contribution of the truncated Maxwellian component and the power-law component:
  

  $$% G^{\rm H}_{\rm _{\rm EA}}(x_{\rm _{\rm EA}},x_{\rm b},\alpha)={... ...rm b}~{\rm e}^{-x_{\rm b}}~F^{\rm PL}_{\rm _{\rm EA}}(u_{\rm _{\rm EA}},\alpha)$$ (A.17)

  
  
  where:
  

  $$% u_{\rm _{\rm EA}} \frac{x_{\rm b}}{x_{\rm _{\rm EA}}}$$ = (A.18)

  $$F^{'}_{\rm _{\rm EA}}(x_{\rm _{\rm EA}},x_{\rm b}) A+B\left[1-x_{\rm _{\rm EA}}~f_1(x_{\rm b})\right]+C\left[1-\frac{x_{\rm _{\rm EA}}}{u_{\rm _{\rm EA}}}+x_{\rm _{\rm EA}}^{2}~f_1(x_{\rm b})\right]$$ = (A.19)

  $$+D\left[1{-}\frac{x_{\rm _{\rm EA}}}{2 u_{\rm _{\rm EA}}^{2}}{+} ... ...}{2}~f_1(x_{\rm b})\right]+E\left[ \ln(u_{\rm _{\rm EA}})+f_1(x_{\rm b})\right]$$

  $$F_{\rm _{\rm EA}}^{\rm PL}(u_{\rm _{\rm EA}},\alpha) A\left[\frac{1}{\alpha-1/2}\right]+B\left[\frac{1}{\alpha-1/2} - ... ...+C\left[\frac{1}{\alpha-1/2} - \frac{u_{\rm _{\rm EA}}^{-2}}{\alpha+3/2}\right]$$ = (A.20)

  $$+D\left[\frac{1}{\alpha-1/2} - \frac{u_{\rm _{\rm EA}}^{-3}}{\alp... ...\frac{1}{(\alpha-1/2)^2} +\frac{\ln(u_{\rm _{\rm EA}})}{\alpha-1/2}\right]\cdot$$

  
  

• For $x_{\rm b}< x_{\rm _{\rm EA}}$:

  Only the power-law component contributes of the electron distribution to the rate:
  

  $$% G^{\rm H}_{\rm _{\rm EA}}(x_{\rm _{\rm EA}},x_{\rm b},\alpha) x_{\rm b}\ {\rm e}^{-x_{\rm b}}\left(\frac{x_{\rm b}}{x_{\rm _{\rm EA}}} \right)^{\alpha-\frac{1}{2}}f_{\rm _{\rm EA}}(\alpha)$$ = (A.21)

  
  
  where
  

  $$% f_{\rm _{\rm EA}}(\alpha)=\frac{A}{\alpha-1/2}+\frac{B}{\alpha^... ...alpha+3/2)}+\frac{3~ D}{(\alpha-1/2)(\alpha+5/2)}+\frac{E}{(\alpha-1/2)^2}\cdot$$ (A.22)

  
  

A.3 Total ionization rates (DI + EA)

The total ionization rate $C_{\rm _{\rm I}}^{\rm H}(T,x_{\rm b},\alpha)$ is obtained by:

$$% C_{\rm _{\rm I}}^{\rm H}(T,x_{\rm b},\alpha) C_{\rm _{\rm DI}}^{\rm H}(T,x_{\rm b},\alpha)+C_{\rm _{\rm EA}}^{\rm H}(T,x_{\rm b},\alpha).$$ = (A.23)