Cross-Sections for Projectile Ionization, Electron Capture, and System Breakdown of C⁵⁺ and Li²⁺ Ions with Atomic Hydrogen

Abstract

For many disciplines of science, all conceivable collisional cross-sections and reactions must be precisely known. Although recent decades have seen a trial of large-scale research to obtain such data, many essential atomic and molecular cross-section data are still missing, and the reliability of the existing cross-sections has to be validated. In this paper, we present projectile ionization, electron capture, and system breakdown cross-sections in carbon (C⁵⁺) ions and lithium (Li²⁺) ion collisions with atomic hydrogen based on the Monte Carlo models of classical and quasi-classical trajectories. According to our expectation, the QCTMC results show higher results in comparison to standard CTMC data, emphasizing the role of the Heisenberg correction constraint, especially in the low-energy regime. On the other hand, at high energy, the Heisenberg correction term has less influence as the projectile momentum increases. We present the total cross-sections of projectile ionization, electron capture, and system breakdown in C⁵⁺ ions and Li²⁺ ion collisions with atomic hydrogen in the impact energy range from 10 keV to 160 keV, which is of interest in astrophysical plasmas, atmospheric sciences, plasma laboratories, and fusion research.

1. Introduction

The divertor is the major part of a tokamak for plasma–surface interactions (PSI) since it is designed to manage the particle and energy exhaust from the core plasma as well as impurity screening [1,2]. Divertor plasma-facing components (PFCs) are installed to provide adequate in-vessel structural protection, sufficient heat exhaust capabilities, and compatibility with plasma purity criteria [1,3]. Controlling heat flow and material erosion at divertor target plates is a critical challenge in the design and operation of next-generation high-power steady-state fusion reactors. Emission spectroscopy at the plasma edge, particularly in the limiter and divertor regions, has provided evidence for the presence of molecules and neutral atoms, such as hydrogen atoms [4]. Ion recombination generates hydrogen atoms on the fusion reactor’s wall and other plasma-facing components (PFCs), which can then be re-emitted into the plasma. In addition to hydrogen atoms, contaminants, such as carbon, lithium, and oxygen, have been found at the plasma edge region. Carbon ions and lithium ions, for example, originated primarily in the limiter and divertor, where carbon composites were utilized in the first wall tiles [5], and lithium ions were employed as a potential solution to minimize heat flow to the divertor chamber. Furthermore, while the primary focus in fusion research is often on the behavior of deuterium and tritium, there can be some subtle isotope effects during inelastic processes, particularly when considering the interaction of these isotopes with other particles or materials in a fusion environment. One example where isotope effects may be relevant is in the study of fusion reactions involving different isotopic combinations of hydrogen and helium. Additionally, isotope effects may also be relevant in the study of fusion reactions involving hydrogen isotopes and heavier elements, such as lithium or boron, which are being explored as potential fuel sources for fusion reactors. In such cases, the behavior of the heavier isotopes and their interactions with hydrogen and hydrogen isotopes could exhibit isotope-dependent effects on the overall fusion process.

In this paper, we focus on the atomic cross-sections of collisions between neutral hydrogen atoms with carbon (C⁵⁺) ions and lithium (Li²⁺) ions in the limiter and divertor regions. These cross-sections are of significant importance in determining the reactor wall material and for modeling plasma diagnostic tools, such as beam emission spectroscopy devices (BES) [6,7]. This work was created to developing theoretical computation frameworks that can give the total cross-section for different inelastic processes accurately. In this paper, we studied the total cross-sections of the projectile ionization, the electron capture, and the system breakdown in a collision between carbon (C⁵⁺) ions and lithium (Li²⁺) ions with ground state hydrogen atoms, as described below (see Equations (1)–(4)).

$$A^{x+}+H\to A^{\mathrm{x}+1}+H+{\displaystyle e_A^{-}}\hspace{1em}(\mathrm{Projectile} \mathrm{ionization})$$

(1)

$$A^{x+}+H\to A^{x+1}+H^{-}\hspace{1em}(\mathrm{Target} \mathrm{electron} \mathrm{capture})$$

(2)

$$A^{x+}+H\to A^{(x+)-1}+H^{+}\hspace{1em}(\mathrm{Projectile} \mathrm{electron} \mathrm{capture})$$

(3)

$$A^{x+}+H\to A^{x+1}+H^{+}+{\displaystyle e_A^{-}}+{\displaystyle e_H^{-}}\hspace{1em}(\mathrm{System} \mathrm{breakdown})$$

(4)

In the 1960s, the classical trajectory Monte Carlo calculations for the cross-section (CTMC) were developed [8]. This method of calculation has proven its effectiveness with high accuracy, especially in the medium energy region [7,8,9,10,11,12], where it is deemed a non-perturbative method and has the benefit of being able to compute more than one interaction channel at the same time [7,8,9,10,11,12]. In order to increase the validity of this method to cover a wider range of energy regimes, it was necessary to develop the calculation method by adding some quantum constraints, which played an important role in improving the accuracy of classical calculations in the low-energy regime [5,13,14]. These constraints impose on the energy of the system for a more stable atomic structure via applying a Kirschbaum and Wilets method [15,16], resulting in a quasiclassical trajectory Monte Carlo model (QCTMC). In this paper, we present the total cross-sections of projectile ionization, electron capture, and system breakdown in the collision of carbon (C⁵⁺) ions and lithium (Li²⁺) ions with atomic hydrogen in the range of energy from 10 keV to 160 keV, which is of interest in the field of fusion, astrophysical plasmas [17], and environmental sciences. The atomic unit is used throughout unless otherwise specified.

2. Theory

2.1. The CTMC Models

As is well-known, classical descriptions of collision processes work extremely well [5,7,18,19]. In this model, the H atom is presented by two particles representing the ionic core of H as well as one active electron; all particles can be described by their masses and charges [14,16]. Let P stands for the ionic core of the projectile, P_e stands for the electron of the projectile, T stands for the ionic core of the target, and T_e stand for the electron of the target [14]. Interactions between electrons are explicitly taken into account in our four-body calculations. At the time (t = −∞), we assume that the four-body collision system is made up of two isolated atoms: a projectile atom (P, P_e) marked as particles (1, 4) and a target atom (T, T_e) marked as particles (2, 3), as shown in Figure 1 [13,14]. At the beginning, both particles are in the ground state (nl = 1, 0) [13,14]. We used the Coulomb potential to describe all interactions [13,14]. Figure 1 depicts relative position vectors for four-body collision systems.

The initial electronic states can be determined by means of a microcanonical distribution. A microcanonical set represents the initial state of the target and projectile compelled by their binding energy in any given shell, which can be described as follows:

$${\rho }_{E_0}(\overrightarrow{A},\dot{\overrightarrow{A}})=K_1\delta (E_0-E)=\delta \left(E_0-{\displaystyle \frac{1}{2}}{\mu }_{T,Te,P,Pe}{\dot{\overrightarrow{A}}}^2-V(A)\right)$$

(5)

where K₁ serves as a normalization constant, while E₀ denotes the ionization energy [5,7,13,14]. According to Equation (5), the electronic coordinates are restricted within intervals, for which Equation (6) holds true, as shown below:

$${\displaystyle \frac{1}{2}}{\mu }_{Te}\dot{\overrightarrow{A}}=E_0-V(A)>0$$

(6)

The Hamilton equation is given as follows:

$$H_0=T+V_{coul},$$

(7)

where

$$T={\displaystyle \frac{{\displaystyle {\overrightarrow{P}}_p^2}}{2m_p}}+{\displaystyle \frac{{\displaystyle {\overrightarrow{P}}_{pe}^2}}{2m_{pe}}}+{\displaystyle \frac{{\displaystyle {\overrightarrow{P}}_T^2}}{2m_T}}+{\displaystyle \frac{{\displaystyle {\overrightarrow{P}}_{Te}^2}}{2m_{Te}}},$$

(8)

And

$$V_{coul}={\displaystyle \frac{Z_p{Z}_{Pe}}{\left|{\overrightarrow{r}}_p-{\overrightarrow{r}}_{Pe}\right|}}+{\displaystyle \frac{Z_P{Z}_T}{\left|{\overrightarrow{r}}_P-{\overrightarrow{r}}_T\right|}}+{\displaystyle \frac{Z_p{Z}_{Te}}{\left|{\overrightarrow{r}}_p-{\overrightarrow{r}}_{Te}\right|}}+{\displaystyle \frac{Z_{pe}{Z}_T}{\left|{\overrightarrow{r}}_{pe}-{\overrightarrow{r}}_T\right|}}+{\displaystyle \frac{Z_{Pe}{Z}_{Te}}{\left|{\overrightarrow{r}}_{pe}-{\overrightarrow{r}}_{Te}\right|}}+{\displaystyle \frac{Z_T{Z}_{Te}}{\left|{\overrightarrow{r}}_T-{\overrightarrow{r}}_{Te}\right|}}.$$

(9)

where T and V_coul, respectively, stand for the total kinetic energy and Coulomb potential term [5,7,13,14]. $\stackrel{⃑}{P}$, Z, $\stackrel{⃑}{r}$, and m stand for the momentum vector, charge, position vector, and mass of each particle [5,7,13,14]. Here, the equations of motion according to Hamiltonian mechanics are presented:

$${\dot{\overrightarrow{P}}}_p=-{\displaystyle \frac{\delta H_0}{\delta {\overrightarrow{r}}_P}}={\displaystyle \frac{Z_P{Z}_{Pe}}{{\left|{\overrightarrow{r}}_p-{\overrightarrow{r}}_{Pe}\right|}^3}}\left({\overrightarrow{r}}_P-{\overrightarrow{r}}_{Pe}\right)+{\displaystyle \frac{Z_P{Z}_T}{{\left|{\overrightarrow{r}}_P-{\overrightarrow{r}}_T\right|}^3}}\left({\overrightarrow{r}}_P-{\overrightarrow{r}}_T\right)+{\displaystyle \frac{Z_P{Z}_{Te}}{{\left|{\overrightarrow{r}}_P-{\overrightarrow{r}}_{Te}\right|}^3}}\left({\overrightarrow{r}}_P-{\overrightarrow{r}}_{Te}\right),$$

(10)

$${\dot{\overrightarrow{P}}}_{Pe}=-{\displaystyle \frac{\delta H_0}{\delta {\overrightarrow{r}}_{Pe}}}=-{\displaystyle \frac{Z_P{Z}_{Pe}}{{\left|{\overrightarrow{r}}_P-{\overrightarrow{r}}_{Pe}\right|}^3}}\left({\overrightarrow{r}}_P-{\overrightarrow{r}}_{Pe}\right)-{\displaystyle \frac{Z_T{Z}_{Pe}}{{\left|{\overrightarrow{r}}_T-{\overrightarrow{r}}_{Pe}\right|}^3}}\left({\overrightarrow{r}}_T-{\overrightarrow{r}}_{Pe}\right)-{\displaystyle \frac{Z_{Te}{Z}_{Pe}}{{\left|{\overrightarrow{r}}_{Te}-{\overrightarrow{r}}_{Pe}\right|}^3}}\left({\overrightarrow{r}}_{Te}-{\overrightarrow{r}}_{Pe}\right),$$

(11)

$${\dot{\overrightarrow{P}}}_T=-{\displaystyle \frac{\delta H_{H_0}}{\delta {\overrightarrow{r}}_T}}=-{\displaystyle \frac{Z_P{Z}_T}{{\left|{\overrightarrow{r}}_P-{\overrightarrow{r}}_T\right|}^3}}\left({\overrightarrow{r}}_P-{\overrightarrow{r}}_T\right)-{\displaystyle \frac{Z_{Te}{Z}_T}{{\left|Te-{\overrightarrow{r}}_T\right|}^3}}\left({\overrightarrow{r}}_{Te}-{\overrightarrow{r}}_T\right)+{\displaystyle \frac{Z_T{Z}_{Pe}}{{\left|{\overrightarrow{r}}_T-{\overrightarrow{r}}_{Pe}\right|}^3}}\left({\overrightarrow{r}}_T-{\overrightarrow{r}}_{Pe}\right),$$

(12)

$${\dot{\overrightarrow{P}}}_{Te}=-{\displaystyle \frac{\delta H_{H_0}}{\delta {\overrightarrow{r}}_{Te}}}=-{\displaystyle \frac{Z_P{Z}_{Te}}{{\left|{\overrightarrow{r}}_P-{\overrightarrow{r}}_{Te}\right|}^3}}\left({\overrightarrow{r}}_P-{\overrightarrow{r}}_{Te}\right)-{\displaystyle \frac{Z_{Te}{Z}_T}{{\left|Te-{\overrightarrow{r}}_T\right|}^3}}\left({\overrightarrow{r}}_{Te}-{\overrightarrow{r}}_T\right)-{\displaystyle \frac{Z_{Te}{Z}_{Pe}}{{\left|{\overrightarrow{r}}_{Te}-{\overrightarrow{r}}_{Pe}\right|}^3}}\left({\overrightarrow{r}}_{Te}-{\overrightarrow{r}}_{Pe}\right).$$

(13)

The Runge–Kutta method is typically utilized to numerically integrate equations of motion using an ensemble of approximately 5 × 10⁶ primary trajectories per energy [5,7,13,14]. Such an ensemble typically is required in order to ensure statistical uncertainties of less than 5% [5,7,13,14]. The total cross-section is given as follows:

$$\sigma ={\displaystyle \frac{2\pi b_{max}}{N}}{\sum }_j{b}_j,$$

(14)

where $b_j$ is the impact parameter corresponding to the trajectory associated with a specific process, such as projectile ionization, electron capture, or system breakdown. N is the total number of calculated trajectories. b_max is the maximum value for the impact parameter where the processes described can take place. The statistical uncertainty of the total cross-section can be calculated as follows:

$$∆\sigma =\sigma {\left[{\displaystyle \frac{N-N_P}{{NN}_P}}\right]}^{1/2}.$$

(15)

where $N_P$, is the number of trajectories that meet the requirements for a particular process.

2.2. The QCTMC Model

The QCTMC model was first introduced by Kirschbaum and Wilets [15,16] as an enhanced rendition of the conventional CTMC model, incorporating a quantum correction term that accounts for the Heisenberg uncertainty and Pauli principle [5,7,13,14]. In order to simulate the Heisenberg uncertainty and Pauli principle, a modified Hamilonian effective potential (V_H for Heisenberg and V_P for Pauli) is added to the pure Coulomb inter-particle potentials to represent a non-classical effect [15,16]. As a result, inter-particle interactions are enhanced. Thus,

$$H_{QCTMC}=H_0+V_H+V_P.$$

(16)

where H₀ is the standard Hamiltonian (see Equation (7)); the correction terms for H₀ include the following:

$$V_H={\displaystyle {\sum }_{i=1}^N}{\displaystyle \frac{1}{m{\displaystyle r_i^2}}}f\left({\overrightarrow{r}}_i,{\overrightarrow{p}}_i;{\xi }_H;{\alpha }_H\right),$$

(17)

And

$$V_p={\displaystyle {\sum }_{i=1}^N}{\displaystyle {\sum }_{j=i+1}^N}{\displaystyle \frac{2}{m{\displaystyle r_{ij}^2}}}f\left({\overrightarrow{r}}_{ij},{\overrightarrow{p}}_{ij};{\xi }_p;{\alpha }_p\right){\delta }_{s_i,s_j}.$$

(18)

where i, j are the electron’s index. Additionally, $r_{ij}=r_j-r_i$ and the relative momentum are determined as follows:

$${\overrightarrow{P}}_{ij}={\displaystyle \frac{m_i{\overrightarrow{p}}_j-m_j{\overrightarrow{p}}_i}{m_i+m_j}}.$$

(19)

If the spin of the ith and jth electrons is the same,${\delta }_{s_i,s_j}=1$; otherwise, if they have different spins, ${\delta }_{s_i,s_j}=0$. Finally, the selected constraining potential is as follows:

$$f\left({\overrightarrow{r}}_{\lambda \nu },{\overrightarrow{p}}_{\lambda \nu };\xi ,\alpha \right)={\displaystyle \frac{{\xi }^2}{4\alpha {\displaystyle r_{\mathsf{\lambda }\nu }^2}{\mu }_{\lambda \nu }}}exp\left\{\alpha \left[1-{\left({\displaystyle \frac{{\overrightarrow{r}}_{\lambda \nu }{\overrightarrow{p}}_{\lambda \nu }}{\xi }}\right)}^4\right]\right\}.$$

(20)

Due to the fact that a hydrogen atom is composed of a single electron and a single proton, the Heisenberg constraints are applied with a distinct scale parameter, a hardness parameter (${\alpha }_H$= 3.0), and a dimensionless constant (${\xi }_H$= 0.9258), which were applied to the four-body QCTMC model. As illustrated in a given Formula (21),

$$f\left({\overrightarrow{r}}_{T,Te},{\overrightarrow{P}}_{T,Te};{\epsilon }_H,{\alpha }_H\right)={\displaystyle \frac{{{\xi }_H}^2}{4{\alpha }_H{\displaystyle {\overrightarrow{r}}_{T,Te}^2}{\mu }_{T,Te}}}exp\left\{{\alpha }_H\left[1-{\left({\displaystyle \frac{{\overrightarrow{r}}_{T,Te}{\overrightarrow{P}}_{T,Te}}{{\xi }_H}}\right)}^4\right]\right\}.$$

(21)

Similar to the target atom, the correction term should be also added to the projectile atom as follows:

$$f\left({\overrightarrow{r}}_{P,Pe},{\overrightarrow{P}}_{P,Pe};{\epsilon }_H,{\alpha }_H\right)={\displaystyle \frac{{{\xi }_H}^2}{4{\alpha }_H{\displaystyle {\overrightarrow{r}}_{P,Pe}^2}{\mu }_{P,Pe}}}exp\left\{{\alpha }_H\left[1-{\left({\displaystyle \frac{{\overrightarrow{r}}_{P,Pe}{\overrightarrow{P}}_{P,Pe}}{{\xi }_H}}\right)}^4\right]\right\}.$$

(22)

As shown in Figure 1, equations of motion that incorporate Hamiltonian mechanics, as well as the correction terms for cross-section calculations, can be expressed as follows:

$$\begin{array}{c}{\dot{\overrightarrow{P}}}_p=-{\displaystyle \frac{\delta H_{QCTMC}}{\delta {\overrightarrow{r}}_P}}=\left[{\displaystyle \frac{Z_P{Z}_{Pe}}{{\left|{\overrightarrow{r}}_p-{\overrightarrow{r}}_{Pe}\right|}^3}}\left({\overrightarrow{r}}_P-{\overrightarrow{r}}_{Pe}\right)-\left(-{\displaystyle \frac{{{\xi }_H}^2}{2{\alpha }_H{\displaystyle {\overrightarrow{r}}_{P,Pe}^4}{\mu }_{P,Pe}}}-{\displaystyle \frac{{\left({\overrightarrow{P}}_{P,Pe}\right)}^4}{{\mu }_{P,Pe}{{\xi }_H}^2}}\right)exp\left\{{\alpha }_H\left[1-{\left({\displaystyle \frac{r_{P,Pe}{P}_{P,Pe}}{{\xi }_H}}\right)}^4\right]\right\}\right]+{\displaystyle \frac{Z_P{Z}_T}{{\left|{\overrightarrow{r}}_P-{\overrightarrow{r}}_T\right|}^3}}\left({\overrightarrow{r}}_P-{\overrightarrow{r}}_T\right)+\\ {\displaystyle \frac{Z_P{Z}_{Te}}{{\left|{\overrightarrow{r}}_P-{\overrightarrow{r}}_{Te}\right|}^3}}\left({\overrightarrow{r}}_P-{\overrightarrow{r}}_{Te}\right),\end{array}$$

(23)

$$\begin{array}{cc}\hfill {\dot{\overrightarrow{P}}}_{Pe}=-{\displaystyle \frac{\delta H_{QCTMC}}{\delta {\overrightarrow{r}}_{Pe}}}& =-\left[{\displaystyle \frac{Z_P{Z}_{Pe}}{{\left|{\overrightarrow{r}}_P-{\overrightarrow{r}}_{Pe}\right|}^3}}\left({\overrightarrow{r}}_P-{\overrightarrow{r}}_{Pe}\right)+\left(-{\displaystyle \frac{{{\xi }_H}^2}{2{\alpha }_H{\displaystyle {\overrightarrow{r}}_{P,Pe}^4}{\mu }_{P,Pe}}}-{\displaystyle \frac{{\left({\overrightarrow{P}}_{P,Pe}\right)}^4}{{\mu }_{P,Pe}{{\xi }_H}^2}}\right)exp\left\{{\alpha }_H\left[1-{\left({\displaystyle \frac{r_{P,Pe}{P}_{P,Pe}}{{\xi }_H}}\right)}^4\right]\right\}\right]\hfill \\ \hfill & -{\displaystyle \frac{Z_T{Z}_{Pe}}{{\left|{\overrightarrow{r}}_T-{\overrightarrow{r}}_{Pe}\right|}^3}}\left({\overrightarrow{r}}_T-{\overrightarrow{r}}_{Pe}\right)\hfill \\ \hfill & -\left[{\displaystyle \frac{Z_{Te}{Z}_{Pe}}{{\left|{\overrightarrow{r}}_{Te}-{\overrightarrow{r}}_{Pe}\right|}^3}}\left({\overrightarrow{r}}_{Te}-{\overrightarrow{r}}_{Pe}\right)-\left(-{\displaystyle \frac{{{\xi }_P}^2}{2{\alpha }_P{\displaystyle {\overrightarrow{r}}_{Te,Pe}^4}{\mu }_{Te,Pe}}}-{\displaystyle \frac{{\left({\overrightarrow{P}}_{Te,Pe}\right)}^4}{{\mu }_{Te,Pe}{{\xi }_H}^2}}\right)exp\left\{{\alpha }_p\left[1-{\left({\displaystyle \frac{r_{Te,Pe}{P}_{Te,Pe}}{{\xi }_P}}\right)}^4\right]\right\}\right],\hfill \end{array}$$

(24)

$$\begin{array}{c}{\dot{\overrightarrow{P}}}_T=-{\displaystyle \frac{\delta H_{QCTMC}}{\delta {\overrightarrow{r}}_T}}=-{\displaystyle \frac{Z_P{Z}_T}{{\left|{\overrightarrow{r}}_P-{\overrightarrow{r}}_T\right|}^3}}\left({\overrightarrow{r}}_P-{\overrightarrow{r}}_T\right)-\left[{\displaystyle \frac{Z_{Te}{Z}_T}{{\left|Te-{\overrightarrow{r}}_T\right|}^3}}\left({\overrightarrow{r}}_{Te}-{\overrightarrow{r}}_T\right)+\left(-{\displaystyle \frac{{{\xi }_H}^2}{2{\alpha }_H{\displaystyle {\overrightarrow{r}}_{T,Te}^4}{\mu }_{T,Te}}}-{\displaystyle \frac{{{\overrightarrow{P}}_{T,Te}}^4}{{\mu }_{T,Te}{{\xi }_H}^2}}\right)exp\left\{{\alpha }_H\left[1-{\left({\displaystyle \frac{r_{T,Te}{P}_{T,Te}}{{\xi }_H}}\right)}^4\right]\right\}\right]+\\ {\displaystyle \frac{Z_T{Z}_{Pe}}{{\left|{\overrightarrow{r}}_T-{\overrightarrow{r}}_{Pe}\right|}^3}}\left({\overrightarrow{r}}_T-{\overrightarrow{r}}_{Pe}\right),\end{array}$$

(25)

$$\begin{array}{ll}{\dot{\overrightarrow{P}}}_{Te}=-{\displaystyle \frac{\delta H_{QCTMC}}{\delta {\overrightarrow{r}}_{Te}}}& =-{\displaystyle \frac{Z_P{Z}_{Te}}{{\left|{\overrightarrow{r}}_P-{\overrightarrow{r}}_{Te}\right|}^3}}\left({\overrightarrow{r}}_P-{\overrightarrow{r}}_{Te}\right)-\left[{\displaystyle \frac{Z_{Te}{Z}_T}{{\left|Te-{\overrightarrow{r}}_T\right|}^3}}\left({\overrightarrow{r}}_{Te}-{\overrightarrow{r}}_T\right)+\left(-{\displaystyle \frac{{{\xi }_H}^2}{2{\alpha }_H{\displaystyle {\overrightarrow{r}}_{T,Te}^4}{\mu }_{T,Te}}}-{\displaystyle \frac{{{\overrightarrow{P}}_{T,Te}}^4}{{\mu }_{T,Te}{{\xi }_H}^2}}\right)exp\left\{{\alpha }_H\left[1-{\left({\displaystyle \frac{r_{T,Te}{P}_{T,Te}}{{\xi }_H}}\right)}^4\right]\right\}\right]-\\ & \left[{\displaystyle \frac{Z_{Te}{Z}_{Pe}}{{\left|{\overrightarrow{r}}_{Te}-{\overrightarrow{r}}_{Pe}\right|}^3}}\left({\overrightarrow{r}}_{Te}-{\overrightarrow{r}}_{Pe}\right)-\left(-{\displaystyle \frac{{{\xi }_P}^2}{2{\alpha }_P{\displaystyle {\overrightarrow{r}}_{Te,Pe}^4}{\mu }_{Te,Pe}}}-{\displaystyle \frac{{\left({\overrightarrow{P}}_{Te,Pe}\right)}^4}{{\mu }_{Te,Pe}{{\xi }_H}^2}}\right)exp\left\{{\alpha }_p|\left[1-{\left({\displaystyle \frac{r_{Te,Pe}{P}_{Te,Pe}}{{\xi }_P}}\right)}^4\right]\right\}\right].\end{array}$$

(26)

3. Result and Discussion

3.1. Carbon (C⁵⁺)–Hydrogen (H) Collision Cross-Sections

Most current fusion reactors employ carbon (C) and tungsten (W) as the primary PFCs. C-PFCs are primarily selected for their exceptional ability to withstand thermal shock and endure out-of-the-ordinary events without melting [1,2,20]. C-PFCs, on the contrary, experience significant erosion due to both physical sputtering and chemical erosion, even at low plasma edge temperatures. The presence of a small amount of carbon in the reactor area, such as a limiter and divertor, due to physical sputtering and chemical erosion, may enhance a variety of inelastic interactions with hydrogen atoms and other impurities, which may affect reactor operation efficiency [5]. In the previous paper [5], the ionization and the electron capture cross-sections of hydrogen targeted by the C⁵⁺ impact were discussed using both CTMC and QCTMC models. At low energies, the hydrogen target’s total ionization cross-section by the C⁵⁺ impact was observed to be elevated. This could be attributed to the prevalence of electron–electron interactions in the low-energy region. This claim was confirmed by a reduced version of the Monte Carlo code. Alongside this, in this section, we present the projectile ionization and the system breakdown cross-sections of a $C^{5+}+H$ collision; see Equations (27) and (28).

First, let us start with the projectile ionization channel when the projectile increased by one, and the hydrogen atom remained unchanged after the collision, see Equation (27),

$$C^{5+}+H\to C^{6+}+H+{\displaystyle e_C^{-}}$$

(27)

Figure 2 illustrates the projectile ionization cross-section as a function of impact energy within the energy range of 10 keV to 160 keV. These data are crucial for fusion research and have been calculated using both the CTMC and QCTMC models. The probability peak at 55 keV is also seen in Figure 2. The QCTMC findings show a slight elevation compared to the standard CTMC calculations in the low-energy regions below 100 keV, thereby affirming the significance of the Heisenberg correction term. On the other hand, in the high-energy range above 100 keV, when the projectile’s energy (velocity) was greater than that of an orbital electron (${v_p\gg v}_e$), the Heisenberg correction component was minimal (see Figure 2), which means that the CTMC and QCTMC calculations have roughly similar results, and this can be explained by two factors (1) The Heisenberg potential has less influence as the projectile momentum increases, and (2) the Heisenberg potential is inversely proportional to the square of the relative distance between colliders$(V_H\alpha {\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\displaystyle r_{ij}^2}$}\right.})$; see Equation (20). As a result, the $V_H(r,p,\alpha )$ potential contributes in the low-to-medium energy region and is negligible in the high-energy region. However, there were no previous theoretical or experimental data to compare this to.

Simultaneously, the breakdown channel of the $C^{5+}+H$ collision system was defined as follows:

$$C^{5+}+H\to C^{6+}+H^{+}+{\displaystyle e_C^{-}}+{\displaystyle e_H^{-}}$$

(28)

Figure 3 displays the system breakdown cross-sections of “full ionization” of both the target and the projectile using the CTMC and the QCTMC methods, yielding the final state of the four free particles as a function of the projectile impact. The peak maximum of the cross-section is about 30 keV, indicating that the system breakdown cross-sections have a high probability at low energy. Furthermore, in comparison to inelastic processes, such as projectile ionization, excitation, and electron capture, the $C^{5+}+H$ collision system has the lowest probability of system breakdown over a wide energy scale. Similarly, as noticed above, CTMC and QCTMC calculations produce the same results in the high-energy region above 100 keV, emphasizing how Heisenberg constraints can be negligible at the high-energy regime.

3.2. Lithium (Li²⁺)–Hydrogen (H) Collision Cross-Sections

The incredible effects of lithium on a fusion reactor’s plasma boundary were initially observed in a tokamak operating in the limiter mode without a divertor [21]. It was found that the energy confinement duration of the discharge in [21,22] a keV temperature plasma edge was significantly enhanced by the pulses that were preceded by lithium pellet injections [21,23]. In addition, lithium has become increasingly recognized as a potential solution to combat divertor heat flux issues within fusion reactors, as ionized lithium atoms form highly radiative plasma layers, which could significantly decrease heat flow into divertor surfaces [24,25,26,27,28]. As it happens, evaporated lithium in its ionic form (partially stripped ions) combined with hydrogen, carbon, and oxygen atoms increased the possibility for inelastic collisions between multiple charged ions and neutral atoms—stimulating inelastic collision research not only out of general scientific curiosity but also as essential tools in fusion-related research [5]. The interactions between ions and the H-target, for instance, have a vital role in determining radiation losses, beam penetration efficiency (ionization rate), and reducing the heat flux to the divertor surfaces [28] in tokamak plasmas. In this section, we present the projectile ionization, the target electron captures, the projectile electron captures, and the system breakdown cross-sections in a ${Li}^{2+}+H$ collision system; see Equations (29)–(32).

First, let us commence by discussing the projectile ionization channel, wherein the projectile charge is incremented by one while the hydrogen target remains unchanged after the collision; see Equation (29),

$${Li}^{2+}+H\to {Li}^{3+}+H+{\displaystyle e_{Li}^{-}}$$

(29)

Figure 4 illustrates the projectile ionization cross-section in a ${Li}^{2+}+H$ collision as a function of impact energy using CTMC and QCTMC calculation methods. The projectile ionization cross-sections were fitted by Gaussian curves. The peak maximum appeared at 30 keV, and this can be explained using the basic kinematic picture. During the collision, at low impact energy, the projectile overlapped with the target atom and remained close to the target for an extended period of time, increasing the possibility of the ionization process owing to the increased probability of interactions among ionic cores, electrons, and ionic cores-electrons (slow-collision). Furthermore, the QCTMC results, as expected, were greater than the regular CTMC results; this observation clearly illustrates the impact of the quantum correction term in the low-energy regime. For the high-energy regime above 100 keV, both the CTMC and QCTMC models provide identical results due to the fact that the Heisenberg correction term influence decreases as the projectile momentum increases. Our model was validated against both experimental and theoretical calculation approaches [5,13,14] despite the absence of any previous data for comparison with the present calculation.

Simultaneously, we considered the target electron capture cross-sections; see Equation (30),

$${Li}^{2+}+H\to {Li}^{3+}+H^{-}$$

(30)

Figure 5 depicts the target electron capture cross-section in a ${Li}^{2+}+H$ collision as a function of impact energy using both CTMC and QCTMC calculation methods. The probability peak at 60 keV is also seen in Figure 5. However, according to our expectation, the QCTMC results are slightly higher than typical CTMC calculations, emphasizing the role of the Heisenberg correction term in the low-energy regime. In the high-energy region, the CTMC and QCTMC results are roughly the same.

Similarly, as in the case of the target electron capture, we also calculated the projectile electron capture cross-section, as described in Equation (31),

$${Li}^{2+}+H\to {Li}^{+}+H^{+}$$

(31)

Figure 6 depicts the projectile electron capture cross-section as a function of the impact energy between 0.1 keV and 400 keV using the CTMC and QCTMC methods. The probability peak appeared at 160 keV. At low-to-intermediate energy regions, the QCTMC results were higher than the CTMC data, indicating the relevance of quantum features in enhancing the cross-section results, particularly for the low-energy regime. On the other hand, in the high-energy area, both QCTMC and CTMC produced the same results, demonstrating that the Heisenberg correction term can be insignificant in this energy zone, as mentioned earlier. Furthermore, as we noticed, the probability of the electron capture process for the projectile in a ${Li}^{2+}+H$ collision is dominant, and this can be justified by considering the nuclear charge effect, as when the nuclear charge increases, the capability to attract high-velocity electrons increases.

Finally, the breakdown “full ionization” channel of the ${Li}^{2+}+H$ collision system was defined as follows:

$${Li}^{2+}+H\to {Li}^{3+}+H^{+}+{\displaystyle e_{Li}^{-}}+{\displaystyle e_H^{-}}$$

(32)

Figure 7 shows the complete break of the ${Li}^{2+}+H$ collision system as a function of impact energy using both QCTMC and CTMC calculation models. The peak maximum of the cross-section was about 10 keV. As we expected, QCTMC improved the cross-section results at a low-energy regime.

4. Conclusions

We presented the total cross-sections in C⁵⁺ + H and Li²⁺ + H collisions using both four-body CTMC and four-body QCTMC calculation methods. Our calculations were performed for the impact energy ranging from 10 to 160 keV, where the cross-sections were predicted to be relevant to astronomical plasmas, atmospheric sciences, plasma laboratories, and fusion research interests. We found that the CTMC calculations underestimated the recent QCTMC results slightly, demonstrating the role of quantum features on the cross-section calculations in the low-energy zone. Nonetheless, the CTMC results showed an identical result to the new model (QCTMC) in the energy high-energy region. In conclusion, we found that the cross-section results at low-energy regimes were enhanced by the Heisenberg correction term, while its impact diminished with the increase in the projectile momentum.