Maxwellian Rate Coefficients

The rate coefficient (\(k\)) for an electron-impact reaction can be expressed as,

\[k = \sqrt{\frac{2 e}{m_e}} \int_{0}^{\infty} E\;\sigma(E)f(E)\,dE,\]

where \(\sigma (E)\) is the cross-section for the reaction which is a function of \(E\) the electron energy, \(f(E)\) is the electron energy distribution function (EEDF) in \(eV^{-3/2}\), \(e\) and \(m_e\) are the charge and mass of the electron, respectively. The EEDF in \(eV^{-3/2}\) is also known as electron energy probability function.

Generalised form of EEDF

The generalized form of the EEDF (in \(eV^{-3/2})\) can be expressed as,

\[f(E) = c_1 T_{eff}^{-\frac{3}{2}} \; e^{-c_2 (E/T_{eff})^x},\]

where

\[\begin{split}c_1 = x \left(\dfrac{2}{3}\right)^{\dfrac{3}{2}} \dfrac{[\Gamma(5/2x)]^{(3/2)}}{[\Gamma(3/2x)]^{(5/2)}}; \\ c_2 = x \left(\dfrac{2}{3}\right)^{\dfrac{3}{2}} \left[\dfrac{\Gamma(5/2x)} {\Gamma(3/2x)}\right]^x.\end{split}\]

The \(x\) and \(T_{eff}\) are the two adjustable parameters; \(T_{eff}\) is the effective (or mean) electron temperature, and \(x\) [1] can be varied between 1 and 2. \(\Gamma(\xi)\) is the usual gamma function.

Maxwellian EEDF

When \(x = 1\), the generalised EEDF form presented in equation [Eq EEDF] reduces to Maxwellian distribution.

[1]

\(x=1\) reduces the generalised form to Maxwellian, and \(x=2\) to Druyvesteyn.