The comprehensive details of modified JK semiempirical formulation to evaluate differential and integral ionization cross sections of molecules due to electron impact are discussed in our previous articles [26–28]. In brief, (e, 2e) processes for the production of SE of energy (𝜖) by an incident electron energy (E), the differential ionization cross-section per energy is given by

$${\text{Q}}_{\text{i}}\left(\text{E},\in \right)=\frac{{4{\pi }\text{a}}_{0}^{2}\text{R}}{\text{E}}\left[\left(1-\frac{\text{ϵ}}{\left(\text{E}-{\text{I}}_{\text{i}}\right)}\right)\frac{\text{R}}{\text{W}}\frac{{\text{d}\text{f}}_{\text{i}}\left(\text{W},0\right)}{\text{d}\text{W}}\text{ln}\left[1+{\text{C}}_{\text{i}}\left(\text{E}-{\text{I}}_{\text{i}}\right)\right]+\frac{\text{R}}{\text{E}} {\text{S}}_{\text{i}}\frac{(\text{E}-{\text{I}}_{\text{i}})}{\left({\text{ϵ}}^{3}+{\text{ϵ}}_{0\text{i}}^{3}\right)}\left(\text{ϵ}-\frac{{\text{ϵ}}^{2}}{\left(\text{E}-\text{ϵ}\right)}+\frac{{\text{ϵ}}^{3}}{{(\text{E}-\text{ϵ})}^{2}}\right)\right]$$

1

Where, i symbolize the formation of ith type cations and the first and second parts deal with the Bethe and Möller cross-sections. The symbols a0, R, 𝜖0i, Ii, \(\frac{{df}_{i}\left(W,0\right)}{dW}\), Ci and Si represents the first Bohr radius, Rydberg’s constant, energy parameter, ionization threshold, oscillator strengths, collisional parameter and the number of electrons participating in collisional processes respectively [26].

The averaged SE energy <𝜖> are evaluated with the help of the energy-dependent cross sections Qi (E, 𝜖) i.e. an important parameter to explore plasma physics related calculations [19]. The averaged SE energy <𝜖> is defined as

$$<\in > =\frac{{\int }_{0}^{\left(E-{I}_{i}\right)}W{Q}_{i}(E,\in )d\in }{{\int }_{0}^{\left(E-{I}_{i}\right)}{Q}_{i}\left(E,\in \right)d\in }$$

2

Where, energy loss (W) is the sum of ionization threshold (Ii) and the SE energy (𝜖), Mathematically, W = Ii + 𝜖.

The TICS is given by

$${\text{Q}}_{\text{i}}^{\text{T}}\left(\text{E}\right)={\int }_{0}^{(\text{E}-{\text{I}}_{\text{i}})}{ \text{Q}}_{\text{i}}\left(\text{E},\in \right)dϵ$$

3

In the present model, the ionization threshold and the dipole oscillator strengths \(\frac{{df}_{i}\left(W,0\right)}{dW}\) are the major input parameter which has one to one mapping with the photoionization cross-sections Qiph(in mega barn) as [29]

$${Q}_{i}^{ph}\left(Mb\right)=109.75 \frac{{df}_{i}\left(W,0\right)}{dW}\left({eV}^{-1}\right)$$

4

For individual atoms C, H and Si, the photoionization cross sections (PICS) are derived from experimental measurement [30]. The PICS for C and Si atoms are calculated from the Hartee-Slater central field calculations by Railman et al. [31] along with the work of Sakamoto et al. [32] and Berkowitz et al. [33] for H atom, from ionization threshold to 50 keV. For W > 50 eV, the PICS are extrapolated using Thomas-Reike-Kuhn (TRK) model with ⁓0.1% accuracy [34].

The collisional parameter Ci is a very sensitive parameter, that depends on the type of the molecular ions species, energy and the energy parameters 𝜖0i [17, 18, 26, 27]. The ionization threshold Ii or appearance energies [35], Ci and 𝜖oi are tabulated in Table 1.

Table 1

Table for ionization potential Ii, Collision parameter Ci, and energy parameter 𝜖0i for CH3 and SiH3 radicals.

Molecules | Ii [35] | Ci | 𝜖0i |

CH3 | 9.84 | 0.034 | 60 |

SiH3 | 8.14 | 0.075 | 35 |

In view of various applications in plasma processes, we have calculated the ionizations rate coefficients by using present calculated TICS which is based on Maxwell distribution law [36, 37] given as

$${\text{R}}_{\text{i}}=\underset{-{\infty }}{\overset{+{\infty }}{\int }}4{\pi }{\left(\frac{1}{2{\pi }\text{m}{K}_{B}\text{T}}\right)}^{\frac{3}{2}}{m}_{e} exp\left(\frac{-\text{E}}{{K}_{B}\text{T} }\right){\text{Q}}_{\text{i}}\left(\text{E}\right) \text{E}\text{d}\text{E}$$

5

The factor Ri is the function of energy and temperature. Where, \({K}_{B}\), me and T are Boltzmann coefficient, mass of electron and temperature, respectively.