Abstract

Keywords

Electron, Cross Sections, Ionization, Scattering

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1. Introduction

Bethe [1] was first introduced the theoretical non-relativistic studies for the atomic ionization problems. The triple differential cross-section (TDCS) in electron hydrogen atom ionization collision has become increasingly interesting over the last four to five decades both theoretically and experimentally for relativistic [2] - [9] as well as for non-relativistic energies [10] - [21] . Triple differential cross-section (TDCS), measured in (e, 2e) coincidence experiment investigated first by Ehrhardt et al. [22] and Amaldi et al. [23] . After that many researchers have been successfully investigated such experiments in a large extent for ionization process theoretically both in ground state [16] [24] - [32] and metastable state [33] - [43] of atomic hydrogen by electron and positron impact.

The first theoretical calculation of direct scattering amplitude of the TDCS for the coplanar asymmetric ionization of hydrogenic metastable 2S-state by electrons was calculated by Vučič et al. [20] in the coplanar asymmetric geometry. The present study of triple differential cross-section (TDCS) for ionization of metastable 3S state hydrogen atoms by electron exchange were never studied experimentally and theoretically. A few theoretical calculations for the TDCS of metastable 2S and 2P [45] - [50] state hydrogen atoms by electron exchange are observed. The theoretical results of Del et al. [21] , BBK model [34] , and the absolute data [15] for the ionization of hydrogen atoms by electrons from ground state have been considered in the present work for comparison.

The purpose of our present work is to calculate the triple differential cross sections (TDCS) for the electron impact ionization of hydrogen atoms in the metastable 3S state for coplanar asymmetric geometry with exchange effects. The present results give an interesting good qualitative fitness with the hydrogenic ground state ionization experimental data and some other hydrogenic ground state theoretical results as well as hydrogenic metastable 2S state and 2P state results. The new observation created a new dimension in this field of research.

2. Theory

We have considered here the direct and exchange amplitude of the T-matrix element. The T-matrix element for ionization of hydrogen atoms by electrons [17] can be written as,

$T_{fi}=\langle {\Psi }_f^{(-)}\left({\bar{r}}_1,{\bar{r}}_2\right)|V_i\left({\bar{r}}_1,{\bar{r}}_2\right)|{\Phi }_i\left({\bar{r}}_1,{\bar{r}}_2\right)\rangle$ (1)

Here the perturbation potential $V_i\left({\bar{r}}_1,{\bar{r}}_2\right)$ is given by

$V_i\left({\bar{r}}_1,{\bar{r}}_2\right)=\frac{1}{r_{12}}-\frac{Z}{r_2}$ (2)

For hydrogen atom nuclear charge (Z) = 1, $r_1$ and $r_2$ are the distance of the two electrons from the nucleus and $r_{12}$ is the distance between the two electrons. The initial channel unperturbed wave function is,

${\Phi }_i\left({\bar{r}}_1,{\bar{r}}_2\right)=\frac{{\text{e}}^{i\cdot {\bar{p}}_2\cdot {\bar{r}}_2}}{{\left(2\text{π}\right)}^{3/2}}{\phi }_{3S}\left({\bar{r}}_1\right)=\frac{{\text{e}}^{i\cdot {\bar{p}}_2\cdot {\bar{r}}_2}}{{\left(2\text{π}\right)}^{3/2}}\cdot \frac{1}{81\sqrt{3\text{π}}}\left(27-18r_1+2r_1^2\right){\text{e}}^{-{\lambda }_1{r}_1}$ (3)

where

${\phi }_{3S}\left({\bar{r}}_1\right)=\frac{1}{81\sqrt{3\text{π}}}\left(27-18r_1+2r_1^2\right){\text{e}}^{-{\lambda }_1{r}_1}$ (4)

and

${\lambda }_1=1/3$

Equation (4) is the hydrogenic 3S-state wave function, ${\bar{p}}_i$ is the incident electron momentum, ${\Psi }_f^{(-)}\left({\bar{r}}_1,{\bar{r}}_2\right)$ is the final three-particle scattering state wave function with the electrons being in the continuum with momenta ${\bar{p}}_1,{\bar{p}}_2$ . Co-ordinates of the two electrons taken to be ${\bar{r}}_1$ and ${\bar{r}}_2$ .

Here ${\Psi }_f^{(-)}\left({\bar{r}}_1,{\bar{r}}_2\right)$ is approximate wave function and is given by,

$\begin{array}{l}{\Psi }_f^{(-)}\left({\bar{r}}_1,{\bar{r}}_2\right)\\ =N\left({\bar{p}}_1,{\bar{p}}_2\right)\left[{\varphi }_{{\bar{p}}_1}^{(-)}\left({\bar{r}}_1\right){\text{e}}^{i\cdot {\bar{p}}_2\cdot {\bar{r}}_2}+{\varphi }_{{\bar{p}}_2}^{(-)}\left({\bar{r}}_2\right){\text{e}}^{i\cdot {\bar{p}}_1\cdot {\bar{r}}_1}+{\varphi }_{\bar{p}}^{(-)}\left(\bar{r}\right){\text{e}}^{i\bar{P}\cdot \bar{R}}-2{\text{e}}^{i{\bar{p}}_1\cdot {\bar{r}}_1+i{\bar{p}}_2\cdot {\bar{r}}_2}\right]/{\left(\text{2π}\right)}^3\end{array}$ (5)

where

$\bar{r}=\frac{{\bar{r}}_2-{\bar{r}}_1}{2},\bar{R}=\left({\bar{r}}_2+{\bar{r}}_1\right)/2,\bar{p}=\left({\bar{p}}_2-{\bar{p}}_1\right),\bar{P}=\left({\bar{p}}_2+{\bar{p}}_1\right)$

The normalization constant $N\left({\bar{p}}_1,{\bar{p}}_2\right)$ is given by

$\begin{array}{c}{\left|N\left({\bar{p}}_1,{\bar{p}}_2\right)\right|}^{-2}=|7-2\left[{\lambda }_1+{\lambda }_2+{\lambda }_3\right]-\left[\frac{2}{{\lambda }_1}+\frac{2}{{\lambda }_2}+\frac{2}{{\lambda }_3}\right]\\ +\left[\frac{{\lambda }_1}{{\lambda }_2}+\frac{{\lambda }_1}{{\lambda }_3}+\frac{{\lambda }_2}{{\lambda }_1}+\frac{{\lambda }_2}{{\lambda }_3}+\frac{{\lambda }_3}{{\lambda }_1}+\frac{{\lambda }_3}{{\lambda }_2}\right]|\end{array}$ (6)

here

${\lambda }_1={\text{e}}^{\frac{\text{π}{\alpha }_1}{2}}\Gamma \left(1-i{\alpha }_1\right),{\alpha }_1=\frac{1}{P_1}$

${\lambda }_2={\text{e}}^{\frac{\text{π}{\alpha }_2}{2}}\Gamma \left(1-i{\alpha }_2\right),{\alpha }_2=\frac{1}{P_2}$

${\lambda }_3={\text{e}}^{\frac{\text{π}\alpha }{2}}\Gamma \left(1-i\alpha \right),\alpha =-\frac{1}{P}$

The normalization constant $N\left({\bar{p}}_1,{\bar{p}}_2\right)$ is calculated numerically using Equation (6) and the approximated value of $N\left({\bar{p}}_1,{\bar{p}}_2\right)$ is 1.

${\varphi }_{\bar{q}}^{(-)}\left(\bar{r}\right)$ is the Coulomb wave function and is given by

${\varphi }_{\bar{q}}^{(-)}\left(\bar{r}\right)={\text{e}}^{\frac{\text{π}\alpha }{2}}\Gamma \left(1+i\alpha \right){\text{e}}^{i\bar{q}\cdot \bar{r}}{}_{1}F{}_1\left(-i\alpha ,1,-i\left[qr+\bar{q}\cdot \bar{r}\right]\right)$

For the electron impact ionization the parameters ${\alpha }_1$ , ${\alpha }_2$ and $\alpha$ are given below

${\alpha }_1=\frac{1}{p_1}$ for $\bar{q}={\bar{p}}_1$ , ${\alpha }_2=\frac{1}{p_2}$ for $\bar{q}={\bar{p}}_2$ and $\alpha =\frac{1}{p}$ for $\bar{q}=\bar{p}$

Equation (1) becomes,

$T_{fi}=N\left({\bar{p}}_1,{\bar{p}}_2\right)\left[T_B+T_{B^{\prime }}+T_i-2T_{PB}\right]$ (7)

where

$T_B=\langle {\Phi }_{{\bar{p}}_1}^{(-)}\left({\bar{r}}_1\right){\text{e}}^{i{\bar{p}}_2\cdot {\bar{r}}_2}|V_i|{\Phi }_i\left({\bar{r}}_1,{\bar{r}}_2\right)\rangle$ (8)

$T_{B^{\prime }}=\langle {\Phi }_{p_2}^{(-)}\left({\bar{r}}_2\right){\text{e}}^{i{\bar{p}}_1\cdot {\bar{r}}_1}|V_i|{\Phi }_i\left({\bar{r}}_1,{\bar{r}}_2\right)\rangle$ (9)

$T_i=\langle {\Phi }_{\bar{p}}^{(-)}\left(\bar{r}\right){\text{e}}^{i\cdot \bar{P}\cdot \bar{R}}|V_i|{\Phi }_i\left({\bar{r}}_1,{\bar{r}}_2\right)\rangle$ (10)

$T_{PB}=\langle {\text{e}}^{i{\bar{p}}_1\cdot {\bar{r}}_1+i{\bar{p}}_2\cdot {\bar{r}}_2}|V_i|{\Phi }_i\left({\bar{r}}_1,{\bar{r}}_2\right)\rangle$ (11)

The direct scattering amplitude $f\left({\bar{p}}_1,{\bar{p}}_2\right)$ is the determined from

$f\left({\bar{p}}_1,{\bar{p}}_2\right)=-{\left(2\text{π}\right)}^2{T}_{fi}$ (12)

The exchange scattering amplitude is then approximated by

$g\left({\bar{p}}_1,{\bar{p}}_2\right)=f\left({\bar{p}}_2,{\bar{p}}_1\right)$ (13)

After analytical calculations using Lewis Integral [44] , the triple differential cross-sections (TDCS) with exchange effects is finally takes the following form,

$\frac{{\text{d}}^3\sigma }{\text{d}{\Omega }_1\text{d}{\Omega }_2\text{d}{E}_1}=\frac{p_1{p}_2}{p_i}\left[\frac{3}{4}{\left|f-g\right|}^2+\frac{1}{4}{\left|f+g\right|}^2\right]$ (14)

Here $E_1$ is the energy of the ejected electron. Hence, in our present study we have calculated the TDCS with exchange effects, given by the Equation (14) using computer programming language.

3. Results and Discussions

We have discussed here in this section the ionization of metastable 3S state hydrogen atoms by electrons with exchange effect. The triple differential cross-sections (TDCS) at $E_i=250\text{eV}$ incident energy with the ejected electron energy of 5 eV is calculated. The results of exchange effects are displayed by the Figure 1 ( ${\theta }_2=3˚$ ), Figure 2 ( ${\theta }_2=15˚$ ), Figure 3 ( ${\theta }_2=25˚$ ), Figure 4 ( ${\theta }_2=5˚$ ), Figure 5 ( ${\theta }_2=7˚$ ), Figure 6 ( ${\theta }_2=9˚$ ), Figure 7 ( ${\theta }_2=11˚$ ), Figure 8 ( ${\theta }_2=15˚$ ) and Figure 9 ( ${\theta }_2=20˚$ ). We consider here the incident electron energy is $E_i=250\text{eV}$ for some varied ejected angles ( ${\theta }_1$ ) and for a fixed scattering angle ( ${\theta }_2$ ). The ejection angle ${\theta }_1$ varies from 0˚ to 360˚. We consider here the recoil regions of the following figures from ${\theta }_1\left(0˚\text{-}150˚\right)$ and $\varphi =0˚$ on other hand the binary region is from ${\theta }_1\left(150˚\text{-}360˚\right)$ and $\varphi =180˚$ .

Electron-hydrogen ionization from ground state theoretical results of Dal et al. [21] , the BBK model of Brauner et al. [34] and the experimental results of Ehrhardt et al. [15] are presented here for comparisons. The earlier works of hydrogenic 2S-state [38] ionization results. The recent works on hydrogenic 2P-state [47] ionization results are also exhibited here for comparison with our present work. We have considered here the triple differential cross-sections (TDCS) with exchange effects for the ionization of metastable 3S-state hydrogen atoms by electrons for the incident electron energy of $E_i=250\text{eV}$ and ejected electron energies $E_1=5\text{eV}$ and 50 eV. The final continuum state of the present work shows a similar but shifted amplitude in the recoil lobe position as the earlier result of hydrogenic ground state second born approximation [21] .

In Table 1 we figure a comparison data for our present result of triple differential cross sections for ionization of hydrogenic 3S state by electron impact with exchange effects with the 2P state exchange results.

In Figure 1 we consider the ejected electron energy $E_1=5\text{eV}$ with a fixed scattering angle ${\theta }_2=3˚$ and the incident electron energy is 250 eV. It is exciting to observe that our present results show a good qualitative agreement with the compared results of present first born result [48] , the hydrogenic ground state result of BBK model [34] , the second born approximation [21] , the experimental data [15] and the second born experiment of 2P-state [47] exchange effects results. Our present result shows a fall in the recoil region and two prominent peaks in binary region.

We consider for Figure 2 the ejected electron energy $E_i=250\text{eV}$ , scattered electron energy $E_1=50\text{eV}$ and scattered angle ${\theta }_2=15˚$ . We also consider the ejected electron angle ${\theta }_1$ from 30˚ to 100˚. In this figure our present results gives small magnitude compared with the previous theoretical results [21] [38] [47] [48] and shows a good qualitative similarity with the experimental results [15] [34] .

In a similar way Figure 3 shows us the 3S metastable state with exchange effects for ejected electron energy $E_i=250\text{eV}$ , scattered electron energy $E_1=50\text{eV}$ and scattered angle ${\theta }_2=25˚$ . As we increase our scattering angle ( ${\theta }_2$ ) the peak of our present study shows a smaller magnitude with hydrogenic ground state second born results [21] .

From Figures 4-9 we compared the present exchange effects results with previous theoretical results like 2P and 2S metastable states [38] [47] . We also include here the present first Born result [48] . The scattering angle varies from ${\theta }_2=5˚$ (Figure 4), ${\theta }_2=7˚$ (Figure 5), ${\theta }_2=9˚$ (Figure 6), ${\theta }_2=11˚$ (Figure 7), ${\theta }_2=15˚$ (Figure 8), ${\theta }_2=20˚$ (Figure 9) in the fixed incident energy $E_i=250\text{eV}$ , ejected energy $E_1=5\text{eV}$ and the ejected electron angle is vary from ( ${\theta }_1=0˚$ to 360˚).

Figure 4 and Figure 5 our present result shows a good agreement of our present result with 2S-state metastable exchange results [38] . The figure displayed a deep lobed peak structure which shows a good qualitative similarity with the compared results [38] [47] [48] .

When we increase our scattering angle in Figure 6 our present result and the present first Born result remains same and give a good qualitative agreement with the compared 2P state exchange result [47] .

In Figure 7 our present TDCS exchange curve shows a very interesting result. It exhibits two falls in recoil region whereas the compared 2S state exchange result [38] there is only one fall in the recoil region. The result shows a bit different from 2P state exchange result [47] .

In the Figure 8 and Figure 9 the magnitude of the present results smaller than the present first Born results [48] . The characteristic features of the cross section curves of the present calculation shows a good improvement comparing with the previous results as the scattering angles are increasing.

The present result of 3S-state exchange effects gives a good qualitative improvement comparing with the previous 2S and 2P state exchange effect results. The present result also show a similar conduct with the hydrogenic ground state result [21] in the binary region. Moreover there needs more experimental works in this field for further investigation.

Finally, the scattering mechanism for the ionization of metastable 3S state with exchange effects for 250 eV incident electron energy is presented here in this study. The scattered electrons are described by a plane wave in the first Born term of Equation (5) whereas the ejected electrons are defined by a Coulomb wave. In the second term of the Equation (5) the scattered electrons are defined by the Coulomb wave while the ejected electrons are defined by the plane wave. The projectile electron interaction appeared in the third term shows almost similar behavior in the final channel wave function. The fourth term represents two plane waves for both ejected and scattered particles. The above results gives us a strong view of peaks both in recoil region and binary region. We can conclude that the present peak values gives us a good agreement with our compared experimental results as well as the theoretical results. In Our present study the measurements of peak values gives us the encouragement for further research in this field of interest. Moreover it needs more experimental works in this field for further investigation.

4. Conclusion

Our present calculation on the triple differential cross sections for ionization of atomic hydrogen by electron impact with exchange effects in metastable 3S-state exposes a thinkable additional structure of the cross-section curves for small momentum transfer in the ionization of the hydrogen atoms. The final state wave function ${\psi }_f^{(-)}\left({\bar{r}}_1,{\bar{r}}_2\right)$ of Das and Seal gives a good qualitative result with the hydrogenic ground state experiment as well as with the BBK model of ground state hydrogen atoms. For good qualitative agreement, the present study are very encouraging for the future experiments which may play a vital role to give interesting and significant results in this field of research.

Acknowledgements

The computational works have been performed in the Simulation Lab of the Department of Mathematics, Chittagong University of Engineering and Technology Chittagong-4349, Bangladesh.

Conflicts of Interest

The authors declare no conflicts of interest.

References