Single Ionization of Methane Molecule by Directive Proton Impact: High Energies Domain

Abstract

The aim of this work is to give a simple and precise theoretical formalism to study the single ionization of small molecular targets by swift proton impact. The mathematical formalism given here for the sake to calculate the differential cross sections is based on the first-Born approximation using the Coulomb wave function. The incident and scattered continuum states of the proton are described by plane wave functions, and the ejected electron is described by a Coulomb wave function. The formalism under consideration is applied to study the single ionization of methane molecule. The comparison between our results and the experimental data showed good agreement.

1. Introduction

The study of ionization processes of atomic and molecular targets by impact of charged particles is actually considered as an important field of research, since the process is found in several domains of modern physics such as plasma physics, radiation physics, astrophysics, radiotherapy and planetary atmospheres [1, 2]. The interaction of charged particles with atomic and molecular targets leads to this kind of ionization reaction. In atomic and molecular collision theory, the analysis of the single ionization process constitutes an excellent tool to understand the structure of matter and the dynamics of atoms and molecules as well as the mechanism of the reaction. In addition, knowledge of total or differential cross sections in energy and angle plays an essential role to give substantial information of different ionization processes in many applications, for example, in astronomy [3, 4], medicine and biology [5, 6], irradiation of living matter [7, 8] and in cancer treatment [9]. Depending on the nature of the ionization process, we often need the corresponding mathematical formalisms to extract precise results based on accurate numerical calculations. However, the implementation of mathematical calculations is often complicated and the corresponding numerical codes prove to be very complex to realize. The majority of problems that appear during the mathematical calculations arise from the several interactions between the different particles. Regarding the numerical programing, in fact, it could be very complicated to write an optimized and accurate numerical code, as we often deal with complicated mathematical formulas and special functions in the complex space may be with contour integrals and complex singularities. The computation time is also a very important factor to obtain in-depth information. We frequently evade these problems by simplifying the mathematical formalism using theoretical and numerical approximations. So, the researcher has to choose the necessary approximations to make the calculations manageable, on the condition that the formalism gives good information on the nature of the process under study.

In the literature, we can find several theoretical and experimental studies of the ionization process by charged particles impact. These studies have been performed on atomic targets and developed to deal with molecular targets. In the experimental side, Langmuir and Jones (1928) [10], carried out the first work on simple ionization by electron impact of N₂ and H₂ molecules, quickly followed by those of Rudberg in 1930 [11]. Subsequently, no work would deal with this aspect of the research until the end of the sixties, where Ehrhardt et al. [12] and Amaldi et al. [13] have studied, respectively, the two electrons coincidence of the ionization reaction of helium and a thin film of carbon by electron impact. On the other hand, Bethe [14], then Massey and Mohr [15], have given series of works in the theoretical domain. They succeeded in establishing the theoretical bases to describe the ionization process using Born approximation to calculate cross sections. In 1932, Hughes and McMillen [15] measured total, simply and doubly differentials cross sections of the ionization of argon atom by electron impact [16]. In 1960, Peterkop [17] then Rudge and Seaton (1964, 1965) contributed by giving theoretical formulations of the single ionization problem, by proposing several approximations to study the collision problem as three-body systems. They were the first to establish the concept of the effective charge as a function of momentum vectors of the scattered and ejected electrons in the so called (e,2e) reaction. This would make it possible, not only to simplify the study of the ionization reaction but also to make it possible to describe the post-collision reaction between the outgoing particles [18].

Several works have been interested to the interaction of the CH₄ molecule with charged particles such as ions, electrons, positrons and protons [19, 20]. Most of these works are focused on the calculations of total and differential cross sections of the simple ionization of this molecule by electron impact. We can remark that, studies of the simple ionization of the molecule by proton impact are extremely rare. However, we can find the important works of: Senger [21] who measured the double differential cross section (DDCS) for proton projectile of energies from 0.25 to 2.0 MeV, Tachino et al. [22] that use the continuum distorted wave-eikonal initial state approximation (CDW-EIS) to calculate the DDCS for 2.0 MeV proton impact for different energies for the ejected electron. The contribution that we want to give here, is a study of the simple ionization of methane molecule by proton impact using the first-Born approximation 1 Coulomb wave model (FBA-CW) without and with Salin factor to take into account the transition to the continuum of the active electron. In this model, the incident and scattered proton are described by plane wave functions, whereas the ejected electron is described by a simple Coulomb wave function. This model has been used to study the single ionization of the water molecule by electron impact [23] where the model proves his power to give accurate results in the high energies domain with less difficulty in numerical computations. In this model, we use the description of Moccia for the molecular wave function where the ground state of methane molecule is described as an expansion over Slater type basis [24], centered on the carbon atom.

The objective of our work consists on studying the simple ionization of methane molecule by directive proton impact. Understanding this reaction is very important to study the different reaction in living mater and plasmas. This is because the molecule under consideration forms an important component in biological tissue and in certain planetary atmospheres, interstellar medium and even for medical considerations (see for example: [7, 25]).

2. Theoretical background

The simple ionization process of an atomic or molecular target by proton impact can be explained by the following Figure 1.

Figure 1.

The simple ionization process.

The Figure 1 above gives a simple representation for the simple ionization of atomic or molecular target by proton impact. This process can also be presented in the following equation:

$$p\left(E_i,{\mathit{k}}_i\right)+T\left(w_i\right)\to p\left(E_s,{\mathit{k}}_s\right)+T^{+}\left(w_f\right)+e^{-}\left(E_e,{\mathit{k}}_e\right)$$E1

where:

$T\left(w_i\right)$: the atomic or molecular target in the initial energy $w_i$.

$T^{+}\left(w_f\right)$: the atomic or molecular residual ion in the final energy $w_f$.

$p\left({\mathit{k}}_i\right)$: the incident proton with the momentum vector ${\mathit{k}}_i$ and energy $E_i$.

$p\left({\mathit{k}}_s\right)$: the scattered proton with the momentum vector ${\mathit{k}}_s$ and energy $E_s$.

$e^{-}\left({\mathit{k}}_e\right)$: the ejected electron with the momentum vector ${\mathit{k}}_e$ and energy $E_e$.

In a collision reaction, kinematic constraints are the conservation of the energy and the momentum:

$$E_i+w_i=E_s+E_e+w_f+E_r\text{and}{\mathbf{k}}_i={\mathbf{k}}_s+{\mathbf{k}}_e\mathbf{+}\mathit{Q}$$E2

Or

$${\mathrm{E}}_{\mathrm{i}}={\mathrm{E}}_{\mathrm{s}}+{\mathrm{E}}_{\mathrm{e}}+{\mathrm{E}}_{\mathrm{r}}+\mathrm{IP}\text{and}{\mathbf{k}}_i={\mathbf{k}}_s+{\mathbf{k}}_e\mathit{+}\mathit{Q}$$E3

Where $\mathrm{PI}=w_f-w_i$ represents the ionization energy of the active electron and ${\mathrm{E}}_{\mathrm{r}}$ and $\mathit{Q}$ are the recoil energy and recoil momentum of the ion. The first equation in (2) can be written as:

$$\frac{k_i^2}{2\mu }=\frac{k_s^2}{2\mu }+\frac{k_e^2}{2}+\frac{Q^2}{2M}+\mathrm{IP}$$E4

$\mu$ is the proton mass and M is the target mass. In the case of high incident energies, we generally have high scattering energies and the momentum $Q$ is very small compared to the momentums ${\mathrm{k}}_{\mathrm{s}}$ and ${\mathrm{k}}_{\mathrm{e}}$ and as the target is very heavy ($M\gg$) compared to the free particles (one proton is about 1837 times heavier than an electron) the recoil energy ${\mathrm{E}}_{\mathrm{r}}=Q^2/\left(2M\right)$ is often neglected. This is why in the literature we found Eq. (3) written in the form:

$${\mathrm{E}}_{\mathrm{i}}={\mathrm{E}}_{\mathrm{s}}+{\mathrm{E}}_{\mathrm{e}}+\mathrm{IP}\text{and}{\mathbf{k}}_i={\mathbf{k}}_s+{\mathbf{k}}_e\mathit{+}\mathit{Q}$$E5

The triple differential cross section (TDCS) that measures the probability for an incident proton with the energy $E_i$ and the momentum ${\mathit{k}}_i$ excites a target electron to the continuum with the energy $E_e$ and momentum ${\mathit{k}}_e,$ and the proton scatter with the energy $E_s$ and momentum ${\mathit{k}}_e$ is defined as:

$$\frac{{\mathrm{d}}^3\mathrm{\sigma }}{\mathrm{d}{\widehat{\mathbf{k}}}_{\mathrm{s}}\mathrm{d}{\widehat{\mathbf{k}}}_{\mathrm{e}}\mathrm{d}{\mathrm{E}}_{\mathrm{e}}}=\frac{k_e{k}_s}{k_i}\mathrm{S}\left(s\right){\left|⟨{\psi }_f|V|{\psi }_i⟩\right|}^2\mathrm{\delta }\left({\mathrm{E}}_{\mathrm{i}}-{\mathrm{E}}_{\mathrm{s}}-{\mathrm{E}}_{\mathrm{e}}+\mathrm{IP}\right)$$E6

Where $\mathrm{\delta }\left(\mathrm{x}\right)$ is the delta function and $\mathrm{S}\left(s\right)$ is Salin factor to take into account the transition to the continuum of the active electron:

$$\mathrm{S}\left(s\right)=\frac{s}{1-exp\left(-s\right)};s=\frac{2\mathit{\pi \mu }}{\left|{\mathbf{k}}_i-\mu {\mathbf{k}}_e\right|}$$E7

The TDCS is most sensitive test for the single ionization process, since it is a physical quantity that gives a complete description over the kinematic of the ionization process.

In the present work, the target under consideration is the CH₄ molecule ionized by proton impact:

$$p\left(E_i,{\mathit{k}}_i\right)+{\mathit{CH}}_4\left(w_i\right)\to p\left(E_s,{\mathit{k}}_s\right)+{\mathit{CH}}_4^{+}\left(w_f\right)+e^{-}\left(E_e,{\mathit{k}}_e\right)$$E8

In atomic unit we have:

$$k_i^2=2\mu E_i,k_s^2=2\mu E_s\mathit{\text{and}}{k}_e^2=2E_e$$E9

If we are interested to the ejected particle, the double differential cross section (DDCS) can be deduced from the TDCS by integration over the scattering solid angle ${\widehat{\mathit{k}}}_s$:

$$\frac{d^{\left(2\right)}\sigma }{d{\widehat{\mathit{k}}}_e{d\mathrm{E}}_e}=\int d{\widehat{\mathit{k}}}_s\frac{d^{\left(3\right)}\sigma }{d{\widehat{\mathit{k}}}_{s}d{\widehat{\mathit{k}}}_e{d\mathrm{E}}_e}$$E10

In Eq. (6), V represents the interaction potential energy:

$$V=\sum _{j=1}^N\frac{Z_j}{\left|{\mathit{r}}_0-{\mathit{R}}_j\right|}-\sum _{i=1}^n\frac{1}{\left|{\mathit{r}}_0-{\mathit{r}}_i\right|}$$E11

Where $Z_i$ is the charge of the nuclei j, ${\mathit{r}}_i$ is the position vector of the $i^{\mathit{th}}$ electron of the molecular target, ${\mathit{R}}_j$ is the position vector of the $j^{\mathit{th}}$ nuclei. The initial and final states can be written in the following forms:

$$\left|{\psi }_i⟩\right.=\left|{\varphi }_i\left({\mathit{k}}_i,{\mathit{r}}_0\right){\phi }_i\left({\mathit{r}}_1,{\mathit{r}}_2,\cdots ,{\mathit{r}}_n\right)\right.⟩$$E12

$$\left|{\psi }_f⟩\right.=\left|{\varphi }_s\left({\mathit{k}}_s,{\mathit{r}}_0\right){\phi }_f\left({\mathit{r}}_1,{\mathit{r}}_2,\cdots ,{\mathit{r}}_n\right)\right.⟩$$E13

${\varphi }_i\left({\mathit{k}}_i,{\mathit{r}}_0\right)$ and ${\varphi }_s\left({\mathit{k}}_s,{\mathit{r}}_0\right)$ are the wave functions of the incident and scattered proton chosen here as plane wave functions $exp\left(i{\mathit{k}}_i·{\mathit{r}}_0\right)$ and $exp\left(i{\mathit{k}}_s·{\mathit{r}}_0\right)$. Now if we use the so-called frozen core approximation (FCA), we can reduce the initial and final states to the following forms:

$$\left|{\psi }_i⟩\right.=\left|{\varphi }_i\left({\mathit{k}}_i,{\mathit{r}}_0\right){\phi }_i\left({\mathit{r}}_1\right)\right.⟩$$E14

$$\left|{\psi }_f⟩\right.=\left|{\varphi }_s\left({\mathit{k}}_s,{\mathit{r}}_0\right){\phi }_f\left({\mathit{r}}_1\right)⟩\right.$$E15

where only the active electron is considered, and we can reduce the potential energy to

$$V=\frac{1}{r_0}-\frac{1}{\left|{\mathit{r}}_0-{\mathit{r}}_1\right|}$$E16

The molecular electrons are distributed among the five orbitals 1A₁, 2A₁, 1T_2x, 1T_2y and 1T_2z. If the reference origin is chosen on the carbon nuclei, each molecular orbital can be defined by linear combinations of Slater-type functions centered over the carbon atom [4]:

$${\phi }_i\left({\mathit{r}}_1\right)=\sum _{k=1}^{N_i}{a}_{\mathit{ik}}{\mathrm{\Phi }}_{n_{\mathit{ik}}{l}_{\mathit{ik}}{m}_{\mathit{ik}}}^{{\xi }_{\mathit{ik}}}\left({\mathit{r}}_1\right)$$E17

$a_{\mathit{ik}}$ is the contribution magnitude of the basis element ${\mathrm{\Phi }}_{n_{\mathit{ik}}{l}_{\mathit{ik}}{m}_{\mathit{ik}}}^{{\xi }_{\mathit{ik}}}\left({\mathit{r}}_1\right)$ given in the molecular frame, as follows:

$${\mathrm{\Phi }}_{n_{\mathit{ik}}{l}_{\mathit{ik}}{m}_{\mathit{ik}}}^{{\xi }_{\mathit{ik}}}\left({\mathit{r}}_1\right)=R_{n_{\mathit{ik}}}^{{\xi }_{\mathit{ik}}}\left(r_1\right)S_{l_{\mathit{ik}},m_{\mathit{ik}}}\left({\widehat{\mathit{r}}}_1\right)$$E18

$R_{n_{\mathit{ik}}}^{{\xi }_{\mathit{ik}}}\left(r_1\right)$ is the radial part chosen as a Slater function and $S_{l_{\mathit{ik}},m_{\mathit{ik}}}\left({\widehat{\mathit{r}}}_1\right)$ is the real spherical harmonic,

$$\left\{\begin{array}{c}{S}_{l_{\mathit{ik}},m_{\mathit{ik}}}\left({\widehat{\mathit{r}}}_1\right)={\left(\frac{m_{\mathit{ik}}}{2\left|m_{\mathit{ik}}\right|}\right)}^{\frac{-1}{2}}\left\{{\mathrm{Y}}_{l_{\mathit{ik}}-\left|m_{\mathit{ik}}\right|}\left({\widehat{\mathit{r}}}_1\right)+{\left(-1\right)}^{m_{\mathit{ik}}}\left(\frac{m_{\mathit{ik}}}{\left|m_{\mathit{ik}}\right|}\right){\mathrm{Y}}_{l_{\mathit{ik}}\left|m_{\mathit{ik}}\right|}\left({\widehat{\mathit{r}}}_1\right)\right\}\\ {\mathrm{S}}_{l_{\mathit{ik}}0}\left({\widehat{\mathit{r}}}_1\right)={\mathrm{Y}}_{l_{\mathit{ik}}0}\left({\widehat{\mathit{r}}}_1\right)\end{array}\right.$$E19

The molecular wave function given in Eq. (17) is defined in a molecular reference frame; we need then to transform it to the laboratory reference frame. This transformation is possible thanks to the relationship:

$$Y_{l_{\mathit{ik}},m_{\mathit{ik}}}\left({\widehat{\mathit{r}}}_1\right)=\sum _{\mu =-l_{\mathit{ik}}}^{l_{\mathit{ik}}}{D}_{{\mu }_{\mathit{ik}},m_{\mathit{ik}}}^{l_{\mathit{ik}}}\left(\alpha ,\beta ,\gamma \right)Y_{l_{\mathit{ik}},{\mu }_{\mathit{ik}}}\left({\widehat{\mathit{r}}}_1\right)$$E20

Where $D_{{\mu }_{\mathit{ik}},m_{\mathit{ik}}}^{l_{\mathit{ik}}}\left(\alpha ,\beta ,\gamma \right)$ is a rotation operator, $\alpha$, $\beta$ and $\gamma$ are the Euler angles. Because the target is randomly orientated, the measured cross section is an average over all possible orientations. This is why we need to average the theoretical cross section over all the Euler angles:

$$\frac{d^{\left(2\right)}\overline{\sigma }}{d{\widehat{\mathit{k}}}_e{d\mathrm{E}}_e}=\frac{1}{8\pi }\int \mathit{\text{dβsinβdα}}\mathit{d\gamma }\frac{d^{\left(2\right)}\sigma }{d{\widehat{\mathit{k}}}_e{d\mathrm{E}}_e}$$E21

In the present formalism, the ejected electron is described by a Coulomb wave function.

$${\phi }_f\left({\mathit{k}}_e,{\mathit{r}}_1\right)=exp\left(\frac{\pi }{k_e}\right)\mathrm{\Gamma }\left(1+\frac{i}{k_e}\right){}_1{\mathrm{F}}_1\left[\frac{-i}{k_e},1,-i\left({\mathit{k}}_e·{\mathit{r}}_1+k_e{r}_1\right)\right]exp\left(i{\mathit{k}}_e·{\mathit{r}}_1\right)$$E22

where the effective ionic charge is taken equal to 1.

3. Results and discussion

In Figures 2 and 3, we present the results concerning the angular distribution of the DDCS for the simple ionization of CH₄ molecule by proton impact. The incident energy is ${\mathit{E}}_{\mathit{i}}\mathbf{=}\mathbf{2}\mathbf{MeV}$, a very high energy in addition to the fact that proton mass is very large compared to the electron one, which justifies the choice of the first Born approximation as a good model in this case. Several results of the DDCS are extracted for ejection energies ${\mathit{E}}_{\mathit{e}}\mathbf{=}\mathbf{11.3}\mathbf{eV}$, 20 eV, 50 eV, 100 eV, 200 eV and 1000 eV. Furthermore, the incidence energy ${\mathit{E}}_{\mathit{i}}\mathbf{=}\mathbf{2}\mathbf{MeV}$ indicates that the time of the collision reaction is very short, we deal then with a very fast collision reaction which justifies that the FCA can be safely used [26]. In the goal to verify the formalism and the numerical code, our results extracted by the FBA-CW model are compared to the theoretical results of Tachino et al. [22] obtained by the CDW-EIS model and to the experimental data found in the paper of Senger [21] (in this paper Senger used the experimental data of Lynch and para [27]). In our numerical calculations of the DDCS, the description Moccia [19] for the molecular wave function is used for methane molecule. We recalled that in this description each molecular orbital is expanded over a Slater-type basis centered on the carbon atom under the assumption that this atom is the heaviest compared to hydrogen atoms.

Figure 2.

Angular distribution of the DDCS of the simple ionization of CH₄ molecule by proton impact of ${\mathbf{E}}_{\mathbf{i}}\mathbf{=}\mathbf{2}\mathbf{MeV}$ for different ejection energies ${\mathrm{E}}_{\mathrm{e}}$. Solid and dotted lines are, respectively, our results from the FBA-CW model with and without Salin factor. Dashed line: the results of Tachino et al. [22] given from the CDW-EIS model. Solid circles: the experimental data [21]. left in logarithmic scale and right in linear scale.

Figure 3.

The same as in Figure 2.

In Figures 2 and 3, the DDCS results are presented in logarithmic scales (left side) and linear scales (right side) in order to have a clear view of the difference between the theoretical models. In these figures, our results are also presented with and without Salin factor (see Eq. (8) of (6)) to demonstrate the effect of this factor on the cross section, or in other terms to clearly see the effect of the transfer to the continuum of the active electron. Generally speaking, from the comparison between our results and the experimental data, we can notice a good agreement between them for the ejection energies from 100 eV to 1000 eV, and an acceptable agreement for energies 20 eV and 50 eV. We also clearly observe that our results are close to the experimental data in the peaks compared to theoretical results of Tachino et al. [22] obtained from the CDW-EIS model.

Concerning the results where the ejection energy $E_i=11.3\mathrm{eV}$, we can clearly see that our formalism and that of Tachino et al. [22] cannot correctly describe the collision reaction for this scale of ejection energy or for lower energies. We believe that when the electron leaves the molecule with a low velocity, it will undergo different interactions, such as the distortion of the wave function due to the interaction with the residual ion and the interaction with the proton. We also think that there are many transitions of the active electron from the intermediate state before to be ejected, which requires calculations of the transition amplitude with another model that takes in to account this kind of transitions or use higher orders Born approximation.

Contrary to the numerical computation of the TDCS, the computation time of the DDCS is very important; this is why in the work of Tachino et al. [22] where the CDW-EIS model is used, the molecular wave function was truncated by excluding the contribution of Slater type orbitals of principle number n ≥ 7. In the work of Tachino et al. [22], as in ours, the wave function of the CH₄ molecule is described using the Slater type basis given by Moccia [19]. However, in our formalism, all the elements of this basis are used to compute the DDCS. For the purpose to study the truncation effect of this molecular basis on the computation accuracy, we compare in Figure 4 our theoretical results without and with truncation by excluding the contribution of Slater type orbitals of principle number n ≥ 7.

Figure 4.

The DDCS of single ionization of CH₄ by proton impact of ${\mathrm{E}}_{\mathrm{i}}=2\mathrm{MeV}$ for different ejection energies ${\mathrm{E}}_{\mathrm{e}}$. Solid and dotted lines are, respectively, our results from the FBA-CW model with and without truncation of the original basis of Moccia [19] by excluding the contribution of slater type orbitals of principle number n ≥ 7.

Tachino et al. [22] considered that in the original description of Moccia [19], the contribution of Slater type molecular orbitals of principal numbers n ≥ 7 is negligible compared to the contribution of those of $\mathrm{n}<7$. From the results given in Figure 4, we can clearly see that there is just a very little difference between the curves with and without truncation. This ensures that the difference between our results and those of the Tachino et al. [22] is not due to the truncation of the original description of the molecular wave function given by Moccia [19]. We can say then that our formalism of the FBA-CW model clearly gives precise results for high ejection energies compared to the CDW-EIS model used by Tachino et al. [22].

4. Conclusion

Single ionization of methane molecule by 2 MeV proton impact was considered in the present paper. The results given here are extracted using our formalism of the FBA-CW model. These theoretical data are compared to the theoretical results of Tachino et al. [22] obtained from the CDW-EIS model. All these findings are compared to the experimental data given by Senger [21]. We generally found that our results are in well agreement with the experiment and better than those of Tachino et al. [22] for high ejection energies.

This proves once more that the simple formalism that we have developed in the frame of the FBA-CW model can provide very accurate results, under certain geometric and energetic conditions. However, as any other approximated model certain corrections are needed to improve the accuracy of the formalism in high incident energies domain. For example, we can take into consideration the distortion of the ejected electron wave function. We may also take into account the capture phenomenon.

Acknowledgments

The work of Pr A K Ferouani was supported by DGRSDT, Algerian Ministry of Higher Education and Research, under Project PRFU-code A11N01EP130220230001.