# Study of Positron Impact Scattering from Methane and Silane Using an Analytically Obtained Static Potential with Correlation Polarization

^{*}

## Abstract

**:**

## 1. Introduction

_{4}) and silane (SiH

_{4}) which are simple symmetrical molecules. Interaction of positrons with these molecules casts light on the various processes taking places in the plasma and astronomical environments [9,10,11]. In fact, the scattering cross-section results from such molecules are the basic inputs for the characterization of their plasma.

_{4}and SiH

_{4}molecules, especially for the differential cross section (DCS). The only experimental DCS results for CH

_{4}are available from Przybyla et al. [12], who performed the relative DCS measurements with angular distribution ranging from 30° to 135° in the incident positron energy range 4–200 eV. On the theoretical end, a few old calculations from Jain [13], Jain and Thompson [14,15] and Jain and Gianturco [16] are available for positron-CH

_{4}scattering. These calculations are based on the optical model potential method using different polarization and correlation potentials. Out of these calculations, Jain and Gianturco [16] presented the detailed DCS and integrated cross section (ICS) results from 0.2 eV to 200 eV incident positron energy by employing a parameter-free model polarization potential within the density function description of the target correlation. Further, there are no specific experimental results available for ICS; however, a few theoretical calculations have been reported in the low energy region. Zecca et al. [17] calculated the ICS results in the energy range of 0.001 eV to 10 eV using the Schwinger multichannel method at both static and static plus polarization levels of approximation. Franz [18] performed the ICS calculations up to 10 eV by taking different forms of short-range correlation polarization potentials. In contrast, for the total cross section (TCS) results, several measurements and theoretical calculations have been reported in the past. Charlton et al. [19] measured the TCS for positron-CH

_{4}scattering in the incident positron energy range from 15 eV to 600 eV by using a slow positron beam time of flight (TOF) system. Further, with the minor modifications in the experimental set up, the same group of Charlton [20] reported TCS results in the low energy range from 2 eV to 20 eV. Floeder et al. [21] measured the TCS results in the incident positron energy from 5 eV to 400 eV by producing the low-energy positrons using a combination of the

^{22}Na positron emitter and tungsten-vane moderator. Dababneh et al. [22] used a beam transmission technique and carried out the measurements in the range of 1 eV to 500 eV positron energy. Sueoka and Mori [23] reported the positron impact total cross section in the energy range varying from 0.7 eV to 400 eV, using a retarding potential TOF method. Zecca et al. [17] reported the latest TCS measurements for positron-CH

_{4}scattering in the incident positron energy ranging from 0.1 eV to 50 eV. On the theoretical side, Jain [24] performed the TCS calculations for 2–600 eV positron energies by taking three different positronium formation energies and using a local spherically complex optical potential (SCOP). Further, Baluja and Jain [25] reported the TCS results up to 5000 eV. Raizada and Baluja [26] calculated TCS using the additivity rule in the energy range from 10 eV to 5000 eV. Recently, Singh et al. [27] employed a modified spherically complex optical potential (mSCOP) to calculate the TCS results for CH

_{4}in the incident positron energy range from the positronium formation threshold to 5000 eV.

_{4}scattering; only old theoretical calculations of Jain [28], Gianturco et al. [29], and Jain and Gianturco [16] are available in the literature. However, recently, Barbosa and Bettega [30] performed the Schwinger multichannel approach calculations to find the DCS and ICS results for positron scattering from SiH

_{4}in the incident positron impact energies up to 10 eV. Similar to CH

_{4}, there are no specific measurements for the ICS results. Further, only a few results have been reported for TCS. For example, Sueoka et al. [31] measured the TCS for positron-SiH

_{4}scattering in the 0.7–400 eV incident positron energy range, using a TOF set up with a retarding potential method (RP–TOF). Besides this experimental work, Baluja and Jain [25] calculated TCS up to 5 keV, using SCOP formalism. Further, Raizada and Baluja [32] reported the TCS results using the additivity rule in the energy range from 10 eV to 5000 eV. Sinha et al. [33] calculated the TCS for the 1 eV to 5 keV incident positron energy region with SCOP formalism.

_{4}and SiH

_{4}in a wide range of incident positron energies of 1–500 eV. We calculate the charge densities of these molecules by using the STO-6G Gaussian wavefunction obtained from GAUSSIAN 16 software with the Hartree–Fock method. Subsequently, the static potential is calculated analytically in a fixed molecular frame of reference. We construct the model potential consisting of the calculated static potential, adding to it the suitable polarization potential. This potential is further used in the Schrödinger equation to solve the scattering problem by adopting the method of partial wave phase shift analysis [37,38]. The scattering amplitude in terms of the phase shift is obtained. Thereafter, the differential and integrated cross-section results are calculated, utilizing the scattering amplitude in a conventional manner. Additionally, we also add the imaginary part of the potential, i.e., the absorption potential to the model potential, and calculate the TCS which takes into account all the inelastic scattering channels. In our present work, we do not explicitly consider the positronium formation contribution while calculating TCS. Here, our focus is to study positron-CH

_{4}/SiH

_{4}elastic scattering. In addition, the effect of the positronium formation is known to affect the magnitude and nature of TCS more only at low incident positron energies.

## 2. Theoretical Methodology

**r′**) at point

**r′**can be written in terms of the molecular wavefunction by taking its square modulus. Further, the molecular orbitals are expressed as a linear combination of the Gaussian atomic orbitals. As a result, the charge density p(

**r′**) can be written as [39]

_{kj}refers to the normalization constants and ${c}_{kj}$ denotes the coefficients of the un–normalized form of the Gaussian basis function ${\chi}_{kj}{r}^{\prime}$. Here, summations over i and k are, respectively, on the number of molecular orbitals and the total number of atomic orbitals, while j is over the Gaussian basis which is taken as 6 in the present case. We obtain the STO-6G Gaussian wavefunction and its related constants in Equation (2) using the GAUSSIAN 16 software with the Hartree–Fock method. The choice of taking the STO-6G basis set of the Gaussian wavefunctions provides quite accurate cross section results as demonstrated in our previous publications [34,35,36]. However, to check the accuracy of the wavefunction, we can compare our obtained ground state energies for CH

_{4}and SiH

_{4}molecules with other available reliable calculations. We find that the ground state energies for CH

_{4}and SiH

_{4}molecules in the present calculations are −40.11 and −290.21 Hartree, respectively. The corresponding values obtained from the DFT/B3LYP method with cc–pVTZ basis set [40] are −40.54 and −291.91 Hartree, and thus show close agreement with our results.

_{x}, m

_{y}and m

_{z}are non–negative integers and α

_{kj}is the exponents of the Gaussian basis function. The values of the exponents and coefficients of atomic and molecular wavefunctions viz. α

_{ij}, N

_{kj}, c

_{kj}and α

_{kj}are taken from our previous paper [39]. Now using the property of Gaussian multiplication, i.e., the product of the exponential parts of two orbitals with different centers, A and B can be written as an exponential of the same form at a center P on the line joining A and B,

_{p}is the position vector of P in the molecular frame and can be expressed as

^{2}2s

^{2}2p

^{2}), Si (1s

^{2}2s

^{2}2p

^{6}3s

^{2}3p

^{2})), the s-orbital of H atom with the p-orbital of C or Si atom, or two s-orbitals or two p-orbitals of C or Si atom. For CH

_{4}and SiH

_{4}molecules, the molecular frame is fixed with the C or Si atom at the origin and the four H atoms at the positions (a, a, a), (a, −a, −a), (−a, a, −a) and (−a, −a, a), where the values of ‘a’ are 0.6261 Å and 0.8544 Å for CH

_{4}and SiH

_{4}, respectively. We evaluate Equation (6) in a particular molecular frame of reference using the Euler angle transformation. The values of the calculated Euler angles are taken from our previous paper [39].

_{4}[41] and 32.24 a.u. for SiH

_{4}[41]. The expression for the short–range correlation potential is taken from Jain and Gianturco [16] as given below

_{4}[44] and 8.7 eV for SiH

_{4}[44]. ${\Delta}_{p}$ is the positronium formation energy given by ${\Delta}_{p}=I-6.8$ eV, where $I$ is the ionization energy of the target molecule. The ionization energies are 12.61 eV for CH

_{4}[45] and 11.7 eV for SiH

_{4}[45].

_{4}and SiH

_{4}molecule can be described in a non-relativistic manner using the model potential (Equations (9) and (10)) in the Schrödinger equation (in atomic units),

## 3. Results and Discussion

#### 3.1. Positron-CH_{4} Scattering

#### 3.1.1. Differential Cross Sections

_{4}scattering in a wide range of incident positron energies from 1 eV to 500 eV. These DCS results are compared with the only available measurement of Przybyla et al. [12] and theoretical calculation of Jain and Gianturco [16]. The experimental cross-section results of Przybyla et al. [12] are relative measurements; therefore, we normalized them with our calculated DCS at 90° angle to obtain an overall best agreement with the experiment.

_{4}scattering.

#### 3.1.2. Elastic Integrated and Total Cross Sections

_{4}scattering in the incident positron energy range from 1 eV to 500 eV. In Figure 5a, we show our elastic ICS results in the entire energy range, and we compare these with the theoretical calculations of Jain and Gianturco [16], Zecca et al. [17] and Franz [18] available in the range of up to 10 eV only. All the calculations show deviations up to 10 eV, which may be either due to the difference in the methods of calculations or the utilized charge densities obtained from different wavefunctions. The cross sections reported in [16] have smaller values in comparison to our results, except at 10 eV. The calculated ICS results of Zecca et al. [17] also have lower cross sections than the present results but the nature of the curve is very similar. However, the ICS results of Franz [18] are higher than all the cross sections reported.

_{4}scattering are shown in Figure 5b, and these are compared with the measurements of Charlton et al. [19,20], Floeder et al. [21], Dababneh et al. [22], Sueoka and Mori [23] and Zecca et al. [17], and the theoretical calculations of Baluja and Jain [25] and Singh et al. [27]. Below the positronium formation energy (i.e., 5.81 eV), the experimental results have different values and are not in agreement among themselves. In this region, our TCS results are in good agreement with the measurements of Dababneh et al. [22], and Sueoka and Mori [23]. The measurements of Charlton [20] are lower than the present TCS results, whereas those of Zecca et al. [17] are somewhat higher. In fact, Zecca et al. [17] measured the cross sections with the angular resolution of $\sim $4° with an estimated energy-dependent angular distribution varying from ∼17° at 1 eV to 5.4° at 10 eV positron energy. Thus the experimental results of Zecca et al. [17] are more accurate in comparison with the previously available other measurements, where the angular resolutions were relatively smaller. Above the positronium formation energy, our TCS results are in very good agreement with all the measurements [17,19,20,21,22,23] except in the energy region from 10 eV to 30 eV, where these are slightly higher. On comparison with the theoretical calculations of Singh et al. [27], we see that our TCS results are in overall good agreement with only small differences, whereas the TCS results of Baluja and Jain [25] have lower cross sections below 30 eV, but match with our results as the energy increases. This may be due to the different absorption potentials taken in these calculations [25] as compared to ours.

_{4}scattering in a wide range of incident positron energy from 1 eV to 500 eV.

#### 3.2. Positron-SiH_{4} Scattering

#### 3.2.1. Differential Cross Sections

_{4}scattering from 1 eV to 4 eV incident positron energies and their comparison with the theoretical calculations of Jain and Gianturco [16] and Barbosa and Bettega [30]. The DCS curves from all the three calculations show similar behavior as well as positions of the minimum, almost at the same angle. As far as the magnitude of the cross sections is concerned, the calculations of Barbosa and Bettega [30] are higher, while those of Jain and Gianturco [16] are lower than the present DCS results.

_{4}at intermediate and higher energies for comparison. Since our results for CH

_{4}at higher energies, as presented above, compare with the available experimental results in a favorable manner, it would be worthwhile to also present similar results for SiH

_{4}so that these are available for future comparison purposes. We show our DCS results for SiH

_{4}in the intermediate (10–50 eV) and high (100–500 eV) energy regions in Figure 8a,b. We observe the following general features here, i.e., the minimum position of the DCS curves shift toward the forward angles as the energy increases, and at higher energies the DCS curves fall off smoothly with respect to the scattering angles. Although the reason for this shift of the minimum is not very clear to us, this effect may be attributed to the behavior of the polarization potential [16] as pointed out from the similar positron-molecule scattering calculations.

#### 3.2.2. Elastic Integrated and Total Cross Sections

_{4}scattering at the incident positron energies from 1 eV to 500 eV. There are no experimental results available for ICS; therefore, we compare in Figure 9a the present elastic ICS results with the theoretical results of Jain and Gianturco [16] and Barbosa and Bettega [30] that are available up to 10 eV. We see a perfect agreement of our results with the theoretical calculations of Barbosa and Bettega [30]. Further, all the theoretical ICS curves show a similar nature, but the results of Jain and Gianturco [16] have lower magnitude of the cross section than the other two calculations. This difference can also be traced back to the DCS results comparisons as shown in Figure 6 and Figure 7, where the DCS results of [16] also show a lower cross section, especially in the forward direction. We report our ICS calculations up to 500 eV incident positron energies for the sake of future comparisons, in case further studies are performed theoretically or experimentally in this energy range.

_{4}scattering, we show the comparison of our calculated DCS, ICS, and TCS results, wherever possible. We find reasonably good agreement with the previously reported measurements and calculations.

## 4. Conclusions

_{4}and SiH

_{4}molecules in a wide range of incident energies from 1 eV to 500 eV. We obtained the static potential for CH

_{4}and SiH

_{4}molecules analytically, utilizing the Gaussian wavefunctions. A model potential consisting of static and polarization potential was constructed to study the scattering problem. Further, using the model potential, the Schrödinger equation was solved with the partial wave analysis method to obtain the scattering amplitude and phase shifts with which the DCS and ICS results were calculated. Further, the absorption potential was added to the model potential to include the inelastic part in the scattering process. Thereafter, the TCS results, that have both the inelastic and elastic components included, were obtained for positron-CH

_{4}/SiH

_{4}scattering. Where possible, we compared our present cross section results with the available experimental and other theoretical results and found quite good agreement. More experimental and theoretical studies are needed for the positron-CH

_{4}/SiH

_{4}scattering in future to verify the present cross section results, especially for SiH

_{4}. Thus, we successfully applied our analytical static potential approach to describe the positron scattering elastic scattering from CH

_{4}and SiH

_{4}molecules in the present work. We feel that our present method can be further extended and applied to more complex molecules.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**(

**a**) Elastic integrated and (

**b**) total cross section for positron-CH

_{4}scattering as a function of incident positron energy. In (

**a**); solid line, present ICS; dashed line, Jain and Gianturco [16]; dashed dot line, Zecca et al. [17]; dashed double dot line, Franz [18]; in (

**b**); solid line, present TCS, dashed dot line, Baluja and Jain [25]; dashed line, Singh et al. [27]; open star, Zecca et al. [17]; solid sphere Charlton [19]; solid up triangle Charlton [20]; open solid sphere, Floeder et al. [21]; open diamond, Dababneh et al. [22]; solid square, Sueoka and Mori [23].

**Figure 8.**Differential cross sections for positron-SiH

_{4}elastic scattering at incident positron energies (

**a**) 10–50 eV and (

**b**) 100–500 eV.

**Figure 9.**(

**a**) Elastic integrated and (

**b**) total cross section for positron-SiH

_{4}scattering as a function of incident positron energy. In (

**a**); solid line, present ICS; dashed line, Jain and Gianturco [16]; dashed dot line, Barbosa and Bettega [30]; in (

**b**); solid line, present TCS; solid sphere, Sueoka et al. [31]; dashed line, Baluja and Jain [25]; dashed dot line, Raizada and Baluja [32]; dashed double dot, Sinha et al. [33].

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**MDPI and ACS Style**

Mahato, D.; Sharma, L.; Srivastava, R.
Study of Positron Impact Scattering from Methane and Silane Using an Analytically Obtained Static Potential with Correlation Polarization. *Atoms* **2021**, *9*, 113.
https://doi.org/10.3390/atoms9040113

**AMA Style**

Mahato D, Sharma L, Srivastava R.
Study of Positron Impact Scattering from Methane and Silane Using an Analytically Obtained Static Potential with Correlation Polarization. *Atoms*. 2021; 9(4):113.
https://doi.org/10.3390/atoms9040113

**Chicago/Turabian Style**

Mahato, Dibyendu, Lalita Sharma, and Rajesh Srivastava.
2021. "Study of Positron Impact Scattering from Methane and Silane Using an Analytically Obtained Static Potential with Correlation Polarization" *Atoms* 9, no. 4: 113.
https://doi.org/10.3390/atoms9040113