The ionization process is one of the most significant reactions in atomic collisions. The study of multiple ionization processes of metastable atoms by electron impact is of considerable importance across various experimental fields, including astrophysics, plasma physics, radiation physics, and applied mathematics. A central challenge in this area lies in developing a general theoretical framework that can reliably predict ionization cross sections for a wide variety of atomic species over a practically relevant range of impact energies. Given the inherent complexity of the problem, a fully quantum mechanical treatment is currently feasible only for the simplest atoms, such as hydrogen and helium.

To gain deeper insights into the ionization mechanisms induced by electron impact, we employ hydrogen atoms as target systems. Ionization by fast particles was first planned quantum mechanically by Bethe [1] . For more than four decades, extensive research has focused on understanding the ionization process in hydrogen, particularly concerning both ground states [2] - [4] and metastable states [5] - [12] . The double differential cross-section (DDCS) for ground state hydrogen atoms has been comprehensively studied through (e, 2e) experiments, with both theoretical and experimental approaches making significant contributions to the field. Similarly, DDCS studies for hydrogenic metastable 3S-states. In this study, we aim to fill this gap by evaluating the DDCS for electron impact ionization of hydrogen atoms in the meta-stable 3S-state under various kinematic condition, our analysis is based on the multiple scattering theories developed by Das and Seal .

The results presented here aim to enhance the understanding of ionization processes in hydrogenic metastable states. Additionally, we compare those of established theories and results [13] - [15] . Lewis integral [16] is used in the present study for analytic calculation. The present new theoretical study on hydrogen 3S-state ionization by electrons offers an immense opportunity for further experimental study for ionization of hydrogen meta-stable 3S-state by electron.

The ionization cross section is the measure of the probability of the ionization process of an atom by electrons or molecule. Electrons impact ionization cross-section is estimated by taking the ratio of the number of ionization elements per unit time and per unit target to the incident electron flux. In this theory, we used the multiple scattering theories of Das and Seal [2] . The ionization of atomic hydrogen by electron in the most elaborate form is presently available in the following type

${\text{e}}^{-}+\text{H}\left(3\text{S}\right)\to {\text{H}}^{+}+2{\text{e}}^{-}$(1)

The symbol 3S represents the hydrogen meta-stable state and has been investigated in co-planar geometry through the analysis of Triple Differential Cross Sections (TDCS) obtained from (e, 2e) coincidence experiments. The direct T-matrix element for the ionization of hydrogen atoms by electron impact [2] can be expressed as

$T_{FI}=\langle {\Psi }_F^{(-)}\left({\bar{\gamma }}_a,{\bar{\gamma }}_b\right)|V_I\left({\bar{\gamma }}_a,{\bar{\gamma }}_b\right)|{\Phi }_I\left({\bar{\gamma }}_a,{\bar{\gamma }}_b\right)\rangle$ (2)

Here the perturbation potential $V_I\left({\bar{\gamma }}_a,{\bar{\gamma }}_b\right)$ is given by

$V_I\left({\bar{\gamma }}_a,{\bar{\gamma }}_b\right)=\frac{1}{{\gamma }_{ab}}-\frac{Z}{{\gamma }_b}$(3)

The nuclear charge of the hydrogen atom is ( $Z$) = 1, ${\bar{\gamma }}_a$ and ${\bar{\gamma }}_b$ are the distances of the two electrons from the nucleus and ${\gamma }_{ab}$ is the distance between the two electrons (see Figure 1 ).

The initial channel unperturbed wave function is

$\begin{array}{c}{\Phi }_I\left({\bar{\gamma }}_a,{\bar{\gamma }}_b\right)=\frac{{\text{e}}^{i\cdot {\bar{p}}_i\cdot {\bar{\gamma }}_b}}{{\left(2\pi \right)}^{\frac{3}{2}}}{\varphi }_{3S}\left({\bar{\gamma }}_a\right)\\ =\frac{{\text{e}}^{i\cdot {\bar{p}}_i\cdot {\bar{\gamma }}_b}}{{\left(2\pi \right)}^{\frac{3}{2}}}\cdot \frac{1}{81\sqrt{3\pi }}\left(27-18{\gamma }_a+2{\gamma }_a^2\right){\text{e}}^{-{\lambda }_a{\gamma }_a}\end{array}$(4)

Here

${\varphi }_{3S}\left({\bar{\gamma }}_a\right)=\frac{1}{81\sqrt{3\pi }}\left(27-18{\gamma }_a+2{\gamma }_a^2\right){\text{e}}^{-{\lambda }_a{\gamma }_a}$(5)

Here ${\lambda }_a=\frac{1}{3}$, ${\varphi }_{3S}\left({\bar{\gamma }}_a\right)$ is the hydrogen 3S-state wave function, and ${\Psi }_F^{(-)}\left({\bar{\gamma }}_a,{\bar{\gamma }}_b\right)$ is the final three-particle scattering state wave function [2] with the

electrons being in the continuum with momenta ${\bar{p}}_a$ and ${\bar{p}}_b$. And the coordinates of the two electrons are ${\bar{\gamma }}_a$ and ${\bar{\gamma }}_b$ respectively. Here the approximate wave function ${\Psi }_F^{(-)}\left({\bar{\gamma }}_a,{\bar{\gamma }}_b\right)$ is given by

$\begin{array}{c}{\Psi }_F^{(-)}\left({\bar{\gamma }}_a,{\bar{\gamma }}_b\right)=N\left({\bar{p}}_a,{\bar{p}}_b\right)[{\varphi }_{{\bar{p}}_a}^{(-)}\left({\bar{\gamma }}_a\right){\text{e}}^{i{\bar{p}}_b\cdot {\bar{\gamma }}_b}+{\varphi }_{{\bar{p}}_2}^{(-)}\left({\bar{\gamma }}_b\right){\text{e}}^{i{\bar{p}}_a\cdot {\bar{\gamma }}_a}\\ +{\varphi }_{\bar{p}}^{(-)}\left(\bar{\gamma }\right){\text{e}}^{i\bar{P}\cdot \bar{R}}-2{\text{e}}^{i{\bar{p}}_a\cdot {\bar{\gamma }}_a+i{\bar{p}}_b\cdot {\bar{\gamma }}_b}]/{\left(2\pi \right)}^3\end{array}$ (6)

Here $\bar{\gamma }=\frac{{\bar{\gamma }}_b-{\bar{\gamma }}_a}{2}$, $\bar{R}=\frac{{\bar{\gamma }}_b+{\bar{\gamma }}_a}{2}$, $\bar{p}=\left({\bar{p}}_b-{\bar{p}}_a\right)$, $\bar{P}=\left({\bar{p}}_b+{\bar{p}}_a\right)$

Here $N\left({\bar{p}}_a,{\bar{p}}_b\right)$ is the normalization constant, given by,

$\begin{array}{l}{\left|N\left({\bar{p}}_a,{\bar{p}}_b\right)\right|}^{-2}\\ =\left|7-2\left[{\lambda }_a+{\lambda }_b+{\lambda }_c\right]-\left[\frac{2}{{\lambda }_a}+\frac{2}{{\lambda }_b}+\frac{2}{{\lambda }_c}\right]+\left[\frac{{\lambda }_a}{{\lambda }_b}+\frac{{\lambda }_a}{{\lambda }_c}+\frac{{\lambda }_b}{{\lambda }_a}+\frac{{\lambda }_b}{{\lambda }_c}+\frac{{\lambda }_c}{{\lambda }_a}+\frac{{\lambda }_c}{{\lambda }_b}\right]\right|\end{array}$ (7)

Here

${\lambda }_a={\text{e}}^{\frac{\pi {\alpha }_a}{2}\Gamma \left(1-i{\alpha }_a\right)},{\alpha }_a=\frac{1}{p_a}$

${\lambda }_b={\text{e}}^{\frac{\pi {\alpha }_b}{2}\Gamma \left(1-i{\alpha }_b\right)},{\alpha }_b=\frac{1}{p_b}$

${\lambda }_c={\text{e}}^{\frac{\pi \alpha }{2}\Gamma \left(1-i\alpha \right)},\alpha =-\frac{1}{p}$

And ${\varphi }_{\bar{q}}^{(-)}\left(\bar{\gamma }\right)$ is the coulomb wave function, given by,

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

The general one-dimensional integral representation of confluent hyper geometric function is written by,

${}_1{}^{}F{}_1\left(a,c,z\right)=\frac{\Gamma \left(c\right)}{\left(a\right)\Gamma \left(c-a\right)}{\displaystyle {\int }_0^1\text{d}x}{x}^{\left(a-1\right)}{\left(1-x\right)}^{\left(c-a-1\right)}{\text{e}}^{\left(xz\right)}$

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

With ${\alpha }_a=\frac{1}{P_a}$ for $\bar{q}={\bar{p}}_a$, ${\alpha }_b=\frac{1}{p_b}$ for $\bar{q}={\bar{p}}_b$ and $\alpha =-\frac{1}{p}$ for $\bar{q}=\bar{p}$.

For the normalization constant $N\left({\bar{p}}_a,{\bar{p}}_b\right)$ of Equation (7) has been calculated numerically.

Now applying Equations (3), (4), (5) and (6) to the Equation (2), we get

$T_{FI}=N\left({\bar{p}}_a,{\bar{p}}_b\right)\left[T_B+T_{B^{\prime }}+T_I-2T_{PB}\right]$ (8)

where

$T_B=\langle {\Phi }_{{\bar{p}}_a}^{(-)}\left({\bar{\gamma }}_a\right){\text{e}}^{i{\bar{p}}_b\cdot {\bar{\gamma }}_b}|V_I|{\Phi }_I\left({\bar{\gamma }}_a,{\bar{\gamma }}_b\right)\rangle$ (9)

$T_{B^{\prime }}=\langle {\Phi }_{{\bar{p}}_b}^{(-)}\left({\bar{\gamma }}_b\right){\text{e}}^{i{\bar{p}}_a\cdot {\bar{\gamma }}_a}|V_I|{\Phi }_I\left({\bar{\gamma }}_a,{\bar{\gamma }}_b\right)\rangle$ (10)

$T_I=\langle {\Phi }_{\bar{p}}^{(-)}\left(\bar{\gamma }\right){\text{e}}^{i\cdot \bar{P}\cdot \bar{R}}|V_I|{\Phi }_I\left({\bar{\gamma }}_a,{\bar{\gamma }}_b\right)\rangle$ (11)

$T_{PB}=\langle {\text{e}}^{i{\bar{p}}_a\cdot {\bar{\gamma }}_a+i{\bar{p}}_b\cdot {\bar{\gamma }}_b}|V_I|{\Phi }_I\left({\bar{\gamma }}_a,{\bar{\gamma }}_b\right)\rangle$ (12)

For the first-born approximation, Equation (9) may be written as

because of the orthogonality of the initial and final target states. The above equations may be written by using Bethe integral [1] , as

$T_{B_1}=\frac{4{\text{e}}^{\left(\frac{\pi {\alpha }_a}{2}\right)}\Gamma \left(1-i{\alpha }_a\right)\left({\bar{t}}^2-{\bar{p}}_a\cdot \bar{t}-i{\alpha }_a{\bar{p}}_a\cdot \bar{t}\right){\text{e}}^{\left(i{\alpha }_a\ln \omega \right)}}{3\sqrt{6}{t}^2\left\{{\bar{t}}^2-{\left(i{\lambda }_a+{\bar{p}}_a\right)}^2\right\}{\left\{{\lambda }_a^2+{\left(\bar{t}-{\bar{p}}_a\right)}^2\right\}}^{2-i{\alpha }_a}}$ (20)

$\begin{array}{c}{T}_{B_2}=\frac{16\sqrt{2}{\text{e}}^{\left(\frac{\pi {\alpha }_a}{2}\right)}\Gamma \left(1-i{\alpha }_a\right)\left({\bar{t}}^2-{\bar{p}}_a\cdot \bar{t}-i{\alpha }_a{\bar{p}}_a\cdot \bar{t}\right){\text{e}}^{\left(i{\alpha }_a\ln \omega \right)}}{3^{\frac{5}{2}}{t}^2\left\{{\bar{t}}^2-{\left(i{\lambda }_a+{\bar{p}}_a\right)}^2\right\}{\left\{{\lambda }_a^2+{\left(\bar{t}-{\bar{p}}_a\right)}^2\right\}}^2}\\ =\left[\frac{1}{{\lambda }_a}-\frac{4{\lambda }_a}{{\lambda }_a{}^2+{\left({\bar{p}}_a-\bar{t}\right)}^2}-\frac{2{\alpha }_a{\bar{p}}_a\left({\lambda }_a^2-p_a^2+t^2\right)}{{\left({\lambda }_a^2-p_a^2+t^2\right)}^2+4{\lambda }_a^2{p}_a^2}+\frac{4{\alpha }_a{\lambda }_a^2{p}_a}{{\left({\lambda }_a^2-p_a^2+t^2\right)}^2+4{\lambda }_a^2{p}_a^2}\right]\\ +i\left[\frac{2{\lambda }_a{\alpha }_a}{{\lambda }_a^2+{\left({\bar{p}}_a-\bar{t}\right)}^2}-\frac{4{\alpha }_a{\lambda }_a{p}_a^2}{{\left({\lambda }_a^2-p_a^2+t^2\right)}^2+4{\lambda }_a^2{p}_a^2}-\frac{2{\lambda }_a{\alpha }_a\left({\lambda }_a^2-p_a^2+t^2\right)}{{\left({\lambda }_a^2-p_a^2+t^2\right)}^2+4{\lambda }_a^2{p}_a^2}\right]\end{array}$(21)

$T_{B_3}=\frac{4{\text{e}}^{\left(\frac{\pi {\alpha }_a}{2}\right)}\Gamma \left(1-i{\alpha }_a\right)\left({\bar{t}}^2-{\bar{p}}_a\cdot \bar{t}-i{\alpha }_a{\bar{p}}_a\cdot \bar{t}\right){\text{e}}^{\left(i{\alpha }_a\ln \omega \right)}}{81\sqrt{2}\pi t^2\left\{{\bar{t}}^2-{\left(i{\lambda }_a+{\bar{p}}_a\right)}^2\right\}{\left\{{\lambda }_a^2+{\left(\bar{t}-{\bar{p}}_a\right)}^2\right\}}^{1+{\alpha }_a}}$ (22)

$T_{B_4}=0$ (23)

$T_{B_5}=0$ (24)

$T_{B_6}=-\frac{8{\text{e}}^{\left(\frac{\pi {\alpha }_a}{2}\right)}\Gamma \left(1-i{\alpha }_a\right)\left({\bar{t}}^2-{\bar{p}}_a\cdot \bar{t}-i{\alpha }_a{\bar{p}}_a\cdot \bar{t}\right){\text{e}}^{\left(i{\alpha }_a\ln \omega \right)}}{81\sqrt{2}\pi t^2\left\{{\bar{t}}^2-{\left(i{\lambda }_a+{\bar{p}}_a\right)}^2\right\}{\left\{{\lambda }_a^2+{\left(\bar{t}-{\bar{p}}_a\right)}^2\right\}}^{2-i{\alpha }_a}}$ (25)

Here $\omega =\frac{{\lambda }_a^2+{\left(\bar{t}-{\bar{p}}_a\right)}^2}{{\bar{t}}^2-{\left(i{\lambda }_a+{\bar{p}}_a\right)}^2}$ with $\bar{t}={\bar{p}}_i-{\bar{p}}_b$.

$\begin{array}{c}{\bar{p}}_a\cdot \bar{t}={\bar{p}}_a\cdot \left({\bar{p}}_i-{\bar{p}}_b\right)\\ ={\bar{p}}_a\cdot {\bar{p}}_i-{\bar{p}}_a\cdot {\bar{p}}_b\\ =p_a{p}_i\cos {\theta }_a-p_a{p}_b\cos {\theta }_{ab}\end{array}$

Here $\cos {\theta }_{ab}=\cos {\theta }_a\cos {\theta }_b-\sin {\theta }_a\sin {\theta }_b$

For phase $\xi$ value we have

$\Gamma \left(1+i{\alpha }_a\right)={\text{e}}^{i\xi }\left|\Gamma \left(1+i{\alpha }_a\right)\right|$

$\Gamma \left(1-i{\alpha }_a\right)={\text{e}}^{-i\xi }\left|\Gamma \left(1-i{\alpha }_a\right)\right|$

$\begin{array}{c}\left|\Gamma \left(1-i{\alpha }_a\right)\right|=\sqrt{\Gamma \left(1-i{\alpha }_a\right)\Gamma \left(1+i{\alpha }_a\right)}\\ =\sqrt{\frac{i{\alpha }_a\pi }{\sin \left(i{\alpha }_a\pi \right)}}\\ =\sqrt{\frac{i{\alpha }_a\pi }{i\sinh \left(i{\alpha }_a\pi \right)}}\end{array}$

$\left|\Gamma \left(1-i{\alpha }_a\right)\right|=\sqrt{\frac{2\pi {\alpha }_a}{{\text{e}}^{\pi {\alpha }_a}-{\text{e}}^{-\pi {\alpha }_a}}}$

Taking the value of (20), (21), (22), (23), (24) and (25) in Equation (13) we find the final value of $T_B$ for First-Born result calculating in this work of Dhar et al. [8] . Following analytical evaluations using the Lewis integral [16] , the above expressions of Equation (13) has been calculated numerically.

Now, the Triple Differential Cross Section (TDCS) corresponding to the T-matrix element is given by:

$\frac{{\text{d}}^3\sigma }{\text{d}{E}_a\text{d}{\Omega }_a\text{d}{\Omega }_b}=\frac{p_a{p}_b}{p_i}{\left|T_{FI}\right|}^2$ (26)

For First-Born approximation we consider $T_{FI}=T_B$

Here the triple differential cross section (TDCS) is denoted by the symbol

$\frac{{\text{d}}^3\sigma }{\text{d}{E}_a\text{d}{\Omega }_a\text{d}{\Omega }_b}$ and $E_a$ is the energy of the ejected electron. Now the double

differential cross section (DDCS) is obtained by integrating [8] Equation (26) with respect to solid angle ${\Omega }_b$.

$\frac{{\text{d}}^2\sigma }{\text{d}{E}_a\text{d}{\Omega }_a}={\displaystyle \int \frac{{\text{d}}^3\sigma }{\text{d}{E}_a\text{d}{\Omega }_a\text{d}{\Omega }_b}\text{d}{\Omega }_b}$ (27)

Hence, the resulting expressions were numerically computed using a programming language MATLAB.

The ionization of hydrogen meta-stable 3S-state by electron impact is presented for different kinematic conditions. The existent new results are assimilated with the hydrogen ground state theoretical results [13] and the absolute data [15] . The Ionization results of hydrogen meta-stable 3d-state [12] and 2S-state result [14] are also covered here for comparison with our new theoretical study results. The emitted angle ${\theta }_a$ varies from 0˚ to 180˚ considered as a horizontal axis where DDCS is the vertical axis in all figures and the scattered angle ${\theta }_b$ varies from 0˚ to 100˚. The incident electron energy $E_I=250\text{eV}$ and 150 eV at taken here. In all diagram, ${\theta }_a$ (0˚ - 90˚) and $\varphi =0˚$ is indicated as recoil fields while ${\theta }_a$ (90˚ - 180˚) and $\varphi =180˚$ is referred as binary fields.

The DDCS for the ionization of meta-stable 3S-state hydrogen atoms by electrons at high incident energy of $E_I=250\text{eV}$ and 150 eV are presented (see Figures 2-7 ).

For an incident energy $E_I=250\text{eV}$ in Figure 2 , the present results exhibit two distinct peaks in the recoil and binary regions, consistent with the 2S-state results reported in [14] . The current first-born approximation results coincide with those for the ground state in [13] at angles ${\theta }_a=10˚$, ${\theta }_a=70˚$ and ${\theta }_a=177˚$. Moreover, the current result aligns well with the experimental ground state results from [15] in the ejected electron angular range of ${\theta }_a=70˚$ to ${\theta }_a=100˚$.

In Figure 3 , at an ejection energy $E_a=10\text{eV}$, the present outcome intersects with the ground state result [13] at ${\theta }_a=10˚$ and with the 3d-state results [12] at both ${\theta }_a=10˚$ and ${\theta }_a=180˚$. Furthermore, it crosses the ground state experimental values [15] three times, approximately at ${\theta }_a=80˚$, ${\theta }_a=90˚$, ${\theta }_a=170˚$, indicating a good overall assessment. Additionally, while the present results and the 2S-state result [14] exhibit a similar shape throughout the entire angular region, they differ in magnitude.

In Figure 4 , $E_a=20\text{eV}$, the current result shows, the current result show similar nature specially in the recoil region with ground state experimental result , 3d-state result [12] and 2S-state result [14] but in the binary region it is not agree well with experimental result due to different state. It meets with ground state theoretical result [13] at ejected angles 30˚, 50˚, and 150˚.

For incident energy $E_I=150\text{eV}$, as shown in Figure 5 , the present results, along with the 3d-state result [12] and the 2S-state result [14] , exhibit similar shapes in both the recoil and binary regions, though with differing magnitudes. The current first-born approximation results coincide with the ground state theoretical result [13] at angles ${\theta }_a=10˚$ and ${\theta }_a=180˚$. Furthermore, the present results align well with the experimental ground state data from [15] in the ejected electron angular range of ${\theta }_a=70˚$ to ${\theta }_a=100˚$.

Let us consider the case of Figure 6 , the current Double Differential Cross Section (DDCS) result and the 3d-state results [12] exhibit a similar shape in the recoil and binary regions but differ in magnitude from the 2S-state results [14] . Notably, the present DDCS results share a similar pattern, especially in the recoil region, with both the ground state results and the corresponding experimental results . However, in the binary region, there is a lack of agreement with those results due to the different states involved.

At last, we consider ejection energy as $E_a=50\text{eV}$, in Figure 7 , The present calculation and the 3d-state meta-stable result from [12] show a similar shape to the ground state theoretical result [13] in the recoil region, while differing notably in the binary region. The present DDCS magnitudes are higher than those of the ground state experiment [15] and 2S-state result [14] in both regions.

Here is a table (see Table 1 ) of comparison results for ionization of hydrogen 3S-state for incident energy $E_I=250\text{eV}$ is presented.

Finally, meta stable 3S-state is an excited state of an atom or other system with a longer lifetime than the other excited states. However, it has a shorter lifetime than the stable ground state. The peak structure of the present results shows good qualitative agreement with compared results in the recoil region but shows somewhat disagreement in the binary region. This may have happened due to the change of the hydrogen meta-stable states by electrons. It is remarked that the peak structure for both in recoil and binary region, the First-Born results are very close to the meta-stable 2S-state with different magnitudes for all scattering angles. But in the binary region, the opposite peak patterns of the meta-stable 2S-state [10] are much sharper than the corresponding ground state experimental result [15] and 3S-state results. However, the limitation of the theory is that at low energy, the theory of Das and Seal [2] does not give significant results.

This study presents the calculation of Double Differential Cross Sections (DDCS) for the ionization of meta-stable 3S-state hydrogen atoms impacted by electron at incident energies of 150 eV and 250 eV. These new theoretical results significantly contribute to the ongoing exploration of meta-stable state ionization. Currently, no experimental data is available to validate these DDCS results for hydrogen 3S-state. The findings are expected to stimulate further theoretical and experimental efforts to study ionization cross-sections of various meta-stable states in hydrogen due to electron and positron impact. Such studies are vital for advancing our understanding of atomic scattering phenomena. Investigations under different kinematic conditions and for other atomic species would also be of considerable interest.

The computational work was conducted in the Simulation Lab, Department of Mathematics, Chittagong University of Engineering and Technology, Chittagong, 4349, Bangladesh.