In high energy ion-atom collisions, ionization is one of the most momentous reactions. Electron impact ionization by charged particles is used in solving problems in isolated range of field like astrophysics, plasma physics, fusion technology, radiation physics, etc. The triple differential cross-sections have been attained in ejected electron energy and ejected angles in the electron hydrogen mechanism. The major challenge in the field of electron impact ionization is to develop a general theoretical framework, which will provide an accurate ionization cross sections for many atoms over a relevant impact energy range. Due to its perplexity, the fully quantum mechanical conduct of atomic ionization by electron is possible for the artless cases of hydrogen atom. In this work, atomic hydrogen is used as target in order to focus attention on the collision mechanism and to investigate the range of validity of various theoretical models.
Ionization by fast particle was first initiated quantum mechanically by Bethe . Electron atom ionization collision on different cross sections has become gradually interesting over the last four decades both theoretically and experimentally for non-relativistic  -  as well as relativistic  -  energies. Various theoretical models applying different kinematical conditions have been vastly used for experimental measurements. Ehrhardt et al.  have investigated this more successfully for different energies of the incident electrons from very low to high energies. The present new theoretical study on hydrogenic 3d state ionization by electron offers an immense opportunity for further experimental study for ionization of hydrogenic metastable 3d state by electron.
Ionization of the hydrogenenic atom by electron is a good image for perturbation theory because of the existence of empirical consequence. In this text, the electron-electron coincidence experiments called (e, 2e) experiments which offer a translucent idea of the kinematics of the collisions by delivering knowledge about the direction of the scattered and ejected electrons. The quantity measured in this kind of experiment is proportional to the TDCS, which represents the angular distribution of the ejected electron for selected incident and scattered electron momenta.
A multiple scattering wave function has been designed for two electrons moving in a coulomb field which include higher order and correlation effects. Lewis integral  has been used in the present study for analytic estimation. A multiple scattering principle  has been followed in the present study by 150 eV, 200 eV and 250 eV energies. The multiple scattering principle  plays a vital role in the study of hydrogenic ionization for ground state  and metastable states  and  - . So, the present results seem to be interesting. The existent novel theoretical results make a new dimension on ionization of hydrogenic metastable states. Current results are compared with previous related theories , , .
T-matrix element for ionization of hydrogen atoms by electrons pursuing Das and Seal  may be taken as
Here, and represent the coordinates of the atomic active electron and the incident electron, ( ,) and (,) represent the momenta and energies of the two electrons in the final state and (,) are the momentum and the energy of the incident electron.
Where the perturbation potential is given by
The nuclear charge of the hydrogen atom is Z = 1, and are the distance of the two electrons from the nucleus and is the distance between two electrons.
The initial channel unperturbed wave function is given in the following form
Here, is the hydrogenic 3d-state wave function and is approximate wave function is given by 
The normalization constant is calculated using Das and Seal  and Dhar and Nahar .
The Coulomb wave function is used from Das and Seal  and Dhar and Nahar .
The triple differential cross-sections for direct T-Matrix element is given by
Here the direct scattering amplitude is computed in .
The exchange scattering amplitude for hydrogen atom are
TDCS with exchange effects is given by
Right hand side of Equation (3) is computed numerically by exerting computer programming language Mat-Lab.
3. Results and Discussions
Ionization of hydrogenic metastable 3d state by electron impact is presented for different kinematic conditions. The existent new results are assimilated with the hydrogenic ground state theoretical results ,  and the absolute data . The ionization results of hydrogenic metastable 2S state , 3S state  and 3P state  are also covered here for comparison with our new theoretical study results. In this text, the ejected angle varies from 0˚ to 360˚ where scattering angles is fixed. The incident electron energy, 200 eV and 250 eV are taken here. In all diagram, and is indicated as recoil field while and is referred as binary field.
The obtained results are sketched corresponding to the different scattering angles Figure 1(a) for ejected electron energies as well as 15˚ Figure 1(b), 25˚ Figure 1(c) for considering the ejected angle from 30˚ to 100˚. Also the existent results are plotted for the scattering angle Figure 2(a), 7˚ Figure 2(b), 9˚ Figure 2(c), 11˚ Figure 2(d), 15˚ Figure 2(e), 20˚ Figure 2(f). Again, present results are plotted for incident electron energy and ejected electron energy for Figure 3(a), Figure 3(b), Figure 3(c) as well as sketched for incident electron energy and ejected electron energy for Figure 4(a), Figure 4(b), Figure 4(c).
In Figure 1(a) peak values of present first Born black dash curve, direct black
Figure 1. The TDCS impact as a function of the ejected electron angle for atomic hydrogen by incident electron energy with (a) and, (b) and, (c) and. Theory: Black dash curve, Black continuous curve, Red perfect curve illustrate present first Born, direct and exchange effect results respectively. Also Blue dash curve demonstrate Hydrogenic ground state Second Born results , Magenta dash dotted curve reveal Hydrogenic ground state BBK model  and Star indicated Hydrogenic ground state experiment  (multiplied by 0.00224).
continuous curve and exchange red perfect curve results show good qualitative agreement with those of the compared results , ,  in the recoil field but show somewhat vary in the binary field. This may be arised due to the change of the hydrogenic metastable states ionization. Current first Born, direct and exchange results are slightly shifted right from other compared results in the binary field. Present exchange results are conjoined with hydrogenic ground state experimental result  at about and where it is expected.
In Figure 1(b), current first Born result show flat peak magnitude and direct results show minimum peak magnitude than the hydrogenic ground state experimental results , hydrogenic ground state BBK model  and present exchange result. In the recoil field, present exchange results show uniquely closer with low magnitude to the hydrogenic ground state experimental result  but in the binary field, present direct and exchange outcomes are moved forward direction from the compared results , .
In Figure 1(c), Existence first Born result is assembled with hydrogenic ground state experimental result , BBK result  and current direct result. The peak pattern of present direct and exchange results provide similar but shifted onward direction from hydrogenic ground state second Born results . Peak magnitude of existence exchange results which is upper than other compared results ,  in both recoil and binary field.
In Figure 2(a), in the recoil field, about, present first Born results make a petty peak and on the other side, current direct configuration make a short lobe with 3S-state results  and current exchange result. In the binary field, the magnitude of obtained exchange outcome is decreased from other compared results , , .
In Figure 2(b), present first Born curve express identical behavior with 3P-state curve  and direct curve depict similar conduct with 3S-state curve  in the recoil field. The present exchange result is reversely jointed with 3P—state results  at higher ejected angle about.
In Figure 2(c), in the recoil field at approximate, existent first Born structure create same peak pattern with 2S-state structure  whereas present direct curve conjoined contrarily with 2S-state  and existent exchange curve make a nice lobe. New exchange result form minimum peak magnitude than 2S-state result  in the binary field.
In Figure 2(d), current attained first Born configuration conjoined with 2S-state  and 3S-state  configurations in the recoil field near about. Present direct including exchange results make flat form where 2S-state  result create a lower dip at about. Present direct and exchange structure give a dull peak where 2S-state  structure shows clear peak at ejected angle about 288˚.
In Figure 2(e), existent first Born curve provides almost similar behavior with 3S-state , 3P-state , present direct and exchange curves but shows a gross difference with the results of 2S-state  in recoil field at about. The existent first Born diagram give same peak with 3P-state , present direct and exchange diagram but 2S-state  diagram reveal small lobe at approximate 252˚.
In Figure 2(f), present first born configuration meets contrarily with 3S-state  configuration while present direct and present exchange configuration meets reversely with 3P-state  configuration in the recoil field at about . Present exchange diagram display opposite form with 3P-state  diagram in both recoil and binary field.
Figure 2. The TDCS impact as a function of the ejected electron angle for atomic hydrogen by incident electron energy with ejected electron energy and (a), (b) (c), (d), (e), (f). Theory: Black dash curve, Black continuous curve, Red perfect curve exhibit present first Born, direct and exchange effect results respectively. Also Green dash dotted curve display hydrogenic 3P-state result , Magenta dotted curve expose hydrogenic 3S-state result  and Blue dash dotted curve demonstrate hydrogenic 2S-state result .
To understand these structures, a table (please see Table 1) which is given for assimilation of hydrogenic ionization of 3d state with 2S-state, 3S-state, 3P-state results.
In Figure 3(a), It is observed that current first Born configuration coincides with present direct and exchange configurations at. Present direct curve exhibit similar nature with exchange curve in the recoil field but shows a little bit different nature in the binary field.
Table 1. Electron impact hydrogenic ionization of 2S-state, 3S-state direct, 3P-state exchange and 3d state results are discriminated for, and.
Figure 3. The TDCS impact as a function of the ejected electron angle for atomic hydrogen by incident electron energy and ejected electron energy with (a) (b) (c) Theory: Black dash curve, black continuous curve and red perfect curve expose present first Born, direct & exchange results respectively.
In Figure 3(b), first Born TDCS curve create a lower dip at about and whereas present direct and exchange results make flat. Direct curve with high magnitude display almost similar pattern with present exchange curve in both recoil and binary field.
In Figure 3(c), The first Born diagram is overlapped with direct and exchange diagrams many times at distinct ejected angle both in recoil and binary field. The first Born and exchange results create two lower dip in the binary field at approximate and.
Here a table (please see Table 2) which is presented for assimilation of first Born, direct and exchange results of hydrogenic 3d-state ionization.
In Figure 4(a), current attained exchange configuration conjoined with present direct configurations whereas first Born configurations with high
Table 2. First Born, direct and exchange results of H(3d) are distinguished for, and.
Figure 4. The TDCS impact as a function of the ejected electron angle for atomic hydrogen by incident electron energy and ejected electron energy with (a) (b) (c) Theory: Black dash curve, black continuous curve and red perfect curve expose present first Born, direct & exchange results respectively.
magnitude display almost similar pattern with present exchange configurations in recoil field. Exchange results make flat with Present direct curve but first Born curve create a dull lobe at about in the binary field.
In Figure 4(b), The existent exchange diagram give same peak with existent direct diagram in the recoil field but first Born diagram reveal gross difference with direct and exchange diagram both in recoil and binary field. At about, existent exchange and direct diagram create short lobe but first Born result make a long lobe.
In Figure 4(c), The first Born diagram is overlapped with direct and exchange diagrams many times at distinct ejected angle both in recoil and binary field. The first Born and exchange results create two lower dip in the binary field at approximate and.
The present calculation exposes additional possible structure of the cross-section curves for intermediate momentum transfer in the hydrogenic 3d-state ionization at 150 eV, 200 eV and 250 eV impact energy. In the present estimation, the correlated three particle final state wave function of Das and Seal  has been followed. New theoretical computational observations for ionization of hydrogenic 3d state by electrons may be generalized for application to the other atom as well as ions and which may play a vital role to provide much interesting and potential results in this field of research.
The computational works are executed in the Simulation Lab of Department of Mathematics, Chittagong University of Engineering and Technology, Chittagong-4349, Bangladesh.
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