The direct ionization rate and cross-section

The ionization rate due to the collisions with free electrons can be written as the integral of the velocity-weighted collisional cross-section over the Maxwell-Boltzmann distribution. Following Section 3.5.1 of Del Zanna and Mason [DZM18], this can be written as,

\[C^I = \sqrt{\frac{8}{\pi m_e}}(k_BT)^{-3/2}\int_I^{\infty}\mathrm{d}E\,E\sigma_I(E)\exp{\left(-\frac{E}{k_BT}\right)}\]

where \(E\) is the energy of the incident electron, \(I\) is the ionization energy of the initially bound electron, and \(\sigma_I\) is the ionization cross-section.

Making the substitution \(x=(E-I)/k_BT\), the above integral can be rewritten as,

\[\begin{split}\begin{aligned} C^I = \sqrt{\frac{8k_BT}{\pi m_e}}\exp{\left(-\frac{I}{k_BT}\right)}&\left(\int_0^{\infty}\mathrm{d}x\,x\sigma_{I}(k_BTx+I)e^{-x} \right. \\ &\left. + \frac{I}{k_BT}\int_0^{\infty}\mathrm{d}x\,\sigma_{I}(k_BTx+I)e^{-x}\right). \end{aligned}\end{split}\]

Each of these integrals is of the form such that they can be evaluated using Gauss-Laguerre quadrature,

\[\int_0^\infty\mathrm{d}x e^{-x}f(x) \approx \sum_{i=1}^n w_if(x_i),\]

where \(x_i\) is the \(i\)-th root of the Laguerre polynomial and \(w_i\) are weights. \(x_i\) and \(w_i\) can be computed using numpy.polynomial.laguerre.laggauss. Typically, using a degree of \(n=12\) is sufficient for this approximation.

Note

There is a typo in the expression for the ionization rate integral in Eq. 32 of Del Zanna and Mason [DZM18].

The direction ionization cross-section, \(\sigma_I\), is computed according to the method of Dere [Der07] which employs a scaling similar to that used by Burgess and Tully [BT92]. Rearranging Eq. 3 of Dere [Der07],

\[\sigma_I = \frac{\Sigma (\log{u} + 1)}{uI^2}\]

where \(u=E/I\) is the energy of the incident electron scaled by ionization potential and \(\Sigma\) is the scaled cross-section which is defined over,

\[U = 1 - \frac{\log{f}}{\log{u - 1 + f}}\]

where \(f\) is a fitting parameter. \(U,f,\Sigma\) are all stored in the CHIANTI database such that \(\sigma_I\) can be computed for a given \(E\). These scaled cross-section data are then interpolated to a given energy array.

The total rate is the summation of \(C^I\) over all electronic configurations of a given ion for which there is a defined cross-section. At a minimum, this includes the outer-shell electron though contributions from inner-shell electrons are also included for some ions. Sections 3.3 and 3.4 of Young [You25] provide more details on the calculation of the direct ionization cross-section and rate and how this is done in CHIANTI.