The Triple Differential Cross Sections for Electron Impact Ionization of Metastable 3 s State Hydrogen Atoms with Exchange Effect

A final state wave function of multiple scattering theory developed by Das and seal is utilized in the present study to calculate the triple differential cross sections (TDCS) for the ionization of metastable 3S state hydrogen atoms at incident electron energy of 250 eV with the exchange effects in the asymmetric coplanar geometry for various kinematic conditions. Our present calculation results are compared with the available hydrogenic ground state experimental data and other existing theoretical results. A good qualitative agreement is shown with those of compared results of the present study specifically with hydrogenic ground state experimental data and metastable 2S and 2P state with exchange effect results. These new results offer an extensive scope for experimental verification in such ionization process.


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 ioniza-Open Journal of Microphysics tion 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.

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, Here the perturbation potential , i V r r is given by ( ) For hydrogen atom nuclear charge (Z) = 1, 1 r and 2 r are the distance of the two electrons from the nucleus and 12 r is the distance between the two elec- trons.The initial channel unperturbed wave function is, 27 18 2 e 81 3π 2π 2π where and 4) is the hydrogenic 3S-state wave function, i p is the incident electron momentum, ( ) ( ) where , N p p is given by here ( ) The normalization constant , N p p is calculated numerically using Equation (6) and the approximated value of ( ) , N p p is 1.
is the Coulomb wave function and is given by For the electron impact ionization the parameters 1 α , 2 α and α are given below where ( ) The direct scattering amplitude ( ) The exchange scattering amplitude is then approximated by ( ) ( ) After analytical calculations using Lewis Integral [44], the triple differential cross-sections (TDCS) with exchange effects is finally takes the following form, Here 1 E 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.

Results and Discussions
We have discussed here in this section the ionization of metastable 3S state hy-   .Theory: Full filled double curve reflects the present calculation of 3S state with exchange effect.Full single curve represent the first born 3S state result [48].Dash curve shows the second born 2P state result [47].Short dash curve focuses hydrogenic ground state second born result [21]; Dash-dotted curve shows hydrogenic ground state BBK model [34] and filled round shows hydrogenic ground state experiments [15]. .Theory: Full curve (m) reflects the present calculation of 3S state with exchange effect.Full single curve (red) represent the first born 3S state result [48].Dash curve shows the second born 2P state result [47].Short dash curve focuses hydrogenic ground state second born result [21]; Dash-dotted curve shows hydrogenic ground state BBK model [34], dash-dotted curve reflects the second born 2S-state result [38] and square shows hydrogenic ground state experiments [15]. .Theory: Full curve (m) reflects the present calculation of 3S state with exchange effect.Full single curve (red) represent the first born 3S state result [48].Dash curve shows the second born 2P state result [47].Short dash curve focuses hydrogenic ground state second born result [21]; Dash-dotted curve shows hydrogenic ground state BBK model [34], dash-dotted curve reflects the second born 2S-state result [38] and square shows hydrogenic ground state experiments [15]. .
Theory: Full curve reflects the present calculation.Dash curve represent the first born 3S-state result [48].Short dash curve shows the second born 2P-state result [47].Dash-dot curve exhibits the hydrogenic 2S-state result [38].Theory: Full curve reflects the present calculation.Dash curve represent the first born 3S-state result [48].Short dash curve shows the second born 2P-state result [47].Dash-dot curve exhibits the hydrogenic 2S-state result [38]. .
Theory: Full curve reflects the present calculation.Dash curve represent the first born 3Sstate result [48].Short dash curve shows the second born 2P-state result [47].Dash-dot curve exhibits the hydrogenic 2S-state result [38].
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 250 eV i E = and ejected electron energies 1 5 eV E = 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.Open Journal of Microphysics Table 1.Triple differential cross sections (TDCS) for ionization of atomic hydrogen atoms by electron impact with exchange effects are obtained by Equation ( 14) at metastable 3S-state.The incident energy is 250 eV, the scattering angle is 2 9 θ = n and the ejected electron energy is 1 5 eV E = . In Table In Figure 1 we consider the ejected electron energy 1 5 eV E = with a fixed scattering angle 2 3 θ =  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 250 eV i E = , scattered electron energy 1 50 eV E = and scattered angle 2 15 θ = .We also consider the ejected electron angle 1 θ 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 250 eV i E = , scattered electron energy ) the peak of our present study shows a smaller magnitude with hydrogenic ground state second born results [21].
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.

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 Open Journal of Microphysics wave function ( ) ( ) Figure 1 ( 2 3 θ = ), Figure 2 ( 2 15 θ = ), Figure 3 ( 2 25 θ = ), Figure 4 ( 2 5 θ = ),

Figure 1 .
Figure 1.Triple differential cross sections (TDCS) for ionization of atomic hydrogen by 250 eV electron impact for 2 3 θ =  vary against the ejected electron 1 θ relative to the incident electron direction.The ejected electron energy is 1 5 eV E =

Figure 2 .
Figure 2. Triple differential cross sections (TDCS) for ionization of atomic hydrogen by 250 eV electron impact for 2 15 θ =  vary against the ejected electron 1 θ relative to the incident electron direction.The ejected electron energy is 1 50 eV E =

Figure 3 .
Figure 3. Triple differential cross sections (TDCS) for ionization of atomic hydrogen by 250 eV electron impact for 2 25 θ =  vary against the ejected electron 1 θ relative to the incident electron direction.The ejected electron energy is 1 50 eV E =

Figure 4 .
Figure 4. Triple differential cross sections (TDCS) for ionization of atomic hydrogen by 250 eV electron impact with exchange effects for 2 5 θ =  vary against the ejected electron 1 θ relative to the incident electron direction.The ejected electron energy is 1 5 eV E =

Figure 5 .
Figure 5. Triple differential cross sections (TDCS) for ionization of atomic hydrogen by 250 eV electron impact with exchange effects for 2 7 θ =  vary against the ejected electron 1 θ relative to the incident electron direction.The ejected electron energy is 1 5 eV E =

Figure 6 .
Figure 6.Triple differential cross sections (TDCS) for ionization of atomic hydrogen by 250 eV electron impact with exchange effects for 2 9 θ =  vary against the ejected electron 1 θ relative to the incident electron direction.The ejected electron energy is 1 5 eV E =

Figure 7 .
Figure 7. Triple differential cross sections (TDCS) for ionization of atomic hydrogen by 250 eV electron impact with exchange effects for 2 11 θ =  vary against the ejected electron 1 θ relative to the incident electron direction.The ejected electron energy is 1 5 eV E =

Figure 8 .
Figure 8. Triple differential cross sections (TDCS) for ionization of atomic hydrogen by 250 eV electron impact with exchange effects for 2 15 θ =  vary against the ejected electron 1 θ relative to the incident electron direction.The ejected electron energy is 1 5 eV E =

Figure 9 .
Figure 9. Triple differential cross sections (TDCS) for ionization of atomic hydrogen by 250 eV electron impact with exchange effects for 2 20 θ =  vary against the ejected electron 1 θ relative to the incident electron direction.The ejected electron energy is 1 5 eV E =

Figure 4 and
Figure4and Figure5our 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].

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.
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.