#
Calculation of Energy and Angular Distributions of Electrons Produced in Intermediate-Energy p + H_{2} Collisions

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## Abstract

**:**

_{2}to intermediate projectile energies. The results for the doubly differential cross section at projectile energies from 48 to 200 keV are presented as a function of the energy and angle of emitted electrons. We consider a wide range of emission angles from 10 to ${160}^{\circ}$, and compare our results to experimental data, where available. Excellent agreement between the presented results and the experimental data was found, especially for emission angles less than ${130}^{\circ}$. For very large backward emission angles our calculations tended to slightly overestimate the experimental data when energetic electrons are ejected and the doubly differential cross section is very small. This discrepancy may be due to the large uncertainties in the experimental data in this region and the model target description. Overall, the present results show significant improvement upon currently available theoretical results and provide a consistently accurate description of this process across a wide range of incident energies.

## 1. Introduction

## 2. Two-Centre Wave-Packet Convergent Close-Coupling Method

#### 2.1. Close-Coupling Formalism

_{2}

^{+}ion by solving the Schrödinger equation. To this end, we apply an iterative Numerov approach. We take only those solutions with negative eigenvalues (energies) to construct the bound-state wave functions. To construct continuum pseudo-states, we subdivide the continuum of the electron into ${N}_{c}$ non-overlapping momentum intervals from ${k}_{\mathrm{min}}$ to ${k}_{\mathrm{max}}$. We then construct a wave packet ${\psi}_{n\ell m}^{\mathrm{WP}}\left(\mathit{r}\right)={\varphi}_{n\ell}^{\mathrm{WP}}\left(r\right){Y}_{\ell m}\left(\widehat{\mathit{r}}\right)/r$ for each interval, which represents the active electron having momentum anywhere within the boundaries of that interval. The wave-packet radial wave function is constructed by numerically integrating the continuum wave over the momentum interval $[{k}_{n},{k}_{n+1}]$ according to

#### 2.2. Scattering Amplitudes

#### 2.3. Differential Cross Sections

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Doubly differential cross section of ionisation for 48 keV collisions with $\begin{array}{c}\hfill {\mathrm{H}}_{2}\end{array}$ as a function of the ejected electron energy at various emission angles. Experimental data are by Gealy et al. [11]. Theoretical results: the WP-CCC approach. The present DI and ECC components are also shown.

**Figure 2.**Same as Figure 1 but for 67 keV.

**Figure 3.**Doubly differential cross section of ionisation for 75 keV collisions with $\begin{array}{c}\hfill {\mathrm{H}}_{2}\end{array}$ as a function of the ejected electron energy at various emission angles. Theoretical results: the WP-CCC approach. The present DI and ECC components are also shown.

**Figure 4.**Same as Figure 1 but for 95 keV.

**Figure 5.**Doubly differential cross section of ionisation for 100 keV proton collisions with $\begin{array}{c}\hfill {\mathrm{H}}_{2}\end{array}$ as a function of the ejected electron energy at various emission angles. Experimental data are by Kuyattand Jorgensen [14], and Ruddand Jorgensen [15]. Theoretical results: the WP-CCC approach and the first-order Born approximation by Ruddand Jorgensen [15]. The present DI and ECC components are also shown.

**Figure 6.**Doubly differential cross section of ionisation for 114 keV proton collisions with $\begin{array}{c}\hfill {\mathrm{H}}_{2}\end{array}$ as a function of the ejected electron energy at various emission angles. Experimental data are by Gealy et al. [11]. Theoretical results: the WP-CCC approach and the first-order Born approximation and continuum distorted wave–eikonal initial state molecular-orbital method by Galassi et al. [23]. The present DI and ECC components are also shown.

**Figure 7.**Doubly differential cross section of ionisation for 200 keV proton collisions with $\begin{array}{c}\hfill {\mathrm{H}}_{2}\end{array}$ as a function of the ejected electron energy at various emission angles. Experimental data are by Rudd et al. [16]. Theoretical results: the WP-CCC approach. The present DI and ECC components are also shown.

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**MDPI and ACS Style**

Plowman, C.T.; Spicer, K.H.; Kadyrov, A.S.
Calculation of Energy and Angular Distributions of Electrons Produced in Intermediate-Energy p + H_{2} Collisions. *Atoms* **2023**, *11*, 112.
https://doi.org/10.3390/atoms11080112

**AMA Style**

Plowman CT, Spicer KH, Kadyrov AS.
Calculation of Energy and Angular Distributions of Electrons Produced in Intermediate-Energy p + H_{2} Collisions. *Atoms*. 2023; 11(8):112.
https://doi.org/10.3390/atoms11080112

**Chicago/Turabian Style**

Plowman, Corey T., Kade H. Spicer, and Alisher S. Kadyrov.
2023. "Calculation of Energy and Angular Distributions of Electrons Produced in Intermediate-Energy p + H_{2} Collisions" *Atoms* 11, no. 8: 112.
https://doi.org/10.3390/atoms11080112