#  __
Relativistic electron-impact ionization of hydrogen atom from its metastable
2S-state in the symmetric/asymmetric coplanar geometries

M Jakha1, S Mouslih2, S Taj1, and B Manaut1,
1 USMS, Faculté Polydisciplinaire, Équipe de Recherche en Physique Théorique
et Matériaux
(ERPTM), Béni Mellal, 23000, Morocco.
2 USMS, Faculté des Sciences et Techniques, Laboratoire de Physique des
Matériaux (LPM),
Béni Mellal, 23000, Morocco. Corresponding author, E-mail<EMAIL_ADDRESS>
###### Abstract

We analytically compute, in the first Born approximation for symmetric and
asymmetric coplanar geometries, the triple differential cross sections for
electron-impact ionization of hydrogen atom in the metastable 2S-state at both
low and high energies. The process is investigated by using the relativistic
Dirac-formalism and it is also shown that the nonrelativistic limit is
accurately reproduced when using low incident kinetic energies. At high
energies, relativistic and spin effects significantly affect the triple
differential cross sections. Our analytical approach which seems exact is
compared to some other results in the nonrelativistic regime for asymmetric
coplanar geometry. For this particular process and in the absence of any
experimental data and theoretical models at high energies, we are not in a
position to validate our model. We hope that the present study will provide
significant contribution to future experiments.

Keywords : Relativistic ionization ; Relativistic wave functions ; Analytic
calculations

##  1\. Introduction

In recent years, much attention has been paid to the experimental and
theoretical aspects of collision processes involving metastable atoms. This is
principally due to the fact that atoms in metastable states own some
properties such as long lifetimes, ability to transmit large amounts of
energy, low excitation and ionization potentials, resulting in very large
cross-sections. From the collision point of view, the ionization of metastable
atoms is very important to understand the mechanisms that occur in
astrophysical and fusion plasmas, in partially ionized systems, and also play
a major role in the gas discharge phenomenon. Apart from all of these,
metastable states of atoms are nowadays gaining increasing importance in many
areas of research, for example, cold atomic physics, in particular Bose-
Einstein condensation, nanolithography and also famously in laser physics [1].
Electron-impact ionization is the removal of one or more electrons from the
target resulting from the collision between it and an electron. We can
distinguish different types of ionization; single ionization, called (e, 2e)
process, which occurs when the resulting ion leaves the collision region with
a single positive charge, multiple ionization where several electrons in the
electronic cortege are ejected and the ion can have multiple positive charges.
In this work, however, we will only deal with the case of single ionization of
the hydrogen atom from its metastable 2S-state, when it is bombarded by an
electron of energy $E_{i}$ greater than the ionization potential. In the
collision zone, two electrons emerge with energies $E_{f}$ and $E_{B}$. Even
though these two electrons cannot be distinguished, it is convenient to call
the faster electron ”scattered electron” and the slower one ”ejected
electron”. Electron-impact ionization of atomic, ionic or molecular systems is
one of the important processes of collisional physics, in particular for the
study of the structure of matter. It also finds its application in various
fields such as astrophysics and plasma physics. Especially, the electron-
impact single ionization has proved to be a powerful tool for studying the
structure of atoms and their dynamics. Ionization of hydrogen atoms by
electron impact is the fundamental and simplest ionization process. The
hydrogen atom is an ideal target due to its analytically known wave functions,
although it is a particularly difficult target for experimentalists. At
present, there are many theoretical models to compute the cross sections of
hydrogen-atoms ionization in both the ground and metastable states at various
incident kinetic energies and under different kinematic conditions.
Unfortunately, ionization from metastable states has not been investigated to
the same extent, especially in the relativistic regime, as ionization from the
ground state; and this is mainly due to the lack of any experimental studies
on this type of ionization. The investigation of the ionization from
metastable states of hydrogen atoms by charged particles is now equally
interesting and experimental results will soon be available in this field. In
particular, the fully triple differential cross sections (TDCS) for the (e,
2e) process have been extensively studied for the ground state hydrogen atom
both theoretically [2, 3, 4, 5, 6, 7, 8, 9] and experimentally [10, 11, 12],
while for the ionization from the metastable state no such measurement of TDCS
is yet available in the literature, although the absolute total cross sections
have been measured much earlier [13, 14]. However, on the theoretical side,
quite a few calculations have been performed on the TDCS of the metastable
(2S) hydrogen atom using electron impact [15, 16, 17, 18, 19, 20, 21] and
significant differences were observed in the TDCS structures when compared to
the cross section of ground-state ionization. All these theoretical
calculations available in the literature to date have been done within the
framework of the asymmetric geometry and at low energies. To the best of our
knowledge, there is no study available to the ionization of the hydrogen atom
from its metastable 2S-state using relativistic formalism at high energies.
This work addresses, for the first time, a theoretical study and an analytical
calculation of the ionization of the hydrogen atom from its metastable
2S-state at high energies in both symmetric and asymmetric coplanar geometries
taking into account the effects of spin and relativity. In the asymmetric
coplanar geometry, we present a theoretical semirelativistic Coulomb Born
approximation (SRCBA) for the description of the ionization of hydrogen atom
by electron impact in the first Born approximation. In this approximation, the
incident and scattered electrons are described by Dirac plane relativistic
wave functions while the ejected electron is described by a Sommerfeld-Maue
semirelativistic Coulomb wave function and the hydrogen atom, in its
metastable state, is described by Darwin’s semirelativistic wave function. The
TDCS obtained in SRCBA will be compared with the corresponding one in the
nonrelativistic Coulomb Born approximation (NRCBA). In the symmetric coplanar
geometry, we present the relativistic formalism of the (e, 2e) reaction in the
relativistic plane wave Born approximation (RPWBA), where the incident,
scattered and ejected electrons are described by relativistic plane waves, and
the hydrogen atom in its metastable 2S-state is described by the relativistic
exact function, and it will be compared, in the nonrelativistic domain, with
the nonrelativistic plane wave Born approximation (NRPWBA). We confirm here
that our work is in fact an extension of a work published in $2005$ [2] by two
of our co-authors through which, for the first time, the SRCBA model was
applied to the ground state of the hydrogen atom and proved its validity
compared to experimental results and results of other theoretical approaches.
The same is true for the RPWBA model. A detailed account of the TDCS in the
ground state of the hydrogen atom was also presented, based on a relativistic
formalism, in the paper [3], and its validity was proven by its comparison
with the NRPWBA model in the non-relativistic domain due to the absence of
results at high energies. If these two models gave good results in the ground
state of the hydrogen atom, then the results obtained in the metastable
2S-state are also expected to be good and acceptable. The only difference
between our work and the previous ones [3, 2] is in the wave function that
describes the hydrogen atom in all computed models. This one difference is
capable enough to completely change the calculation and give rise to new
analytically complex integrals. We would like to point out here that our goal
in this research is not exactly to study ionization process in the non-
relativistic regime, but rather to study it at high energies by applying the
relativistic Dirac formalism. Thus, the NRCBA model that we have calculated in
this paper is only a way to compare the TDCS obtained in the SRCBA model.
There are many complex approaches that are applied when studying ionization
process in the non-relativistic domain, we mention here as example, the
R-matrix (RM) theory [22], the Convergent Close-Coupling (CCC) method [23],
and the distorted-wave Born approximation (DWBA) [24]. Since our goal is to
study this reaction at high energies, we decided to be satisfied only, in the
non-relativistic domain, with the application of the NRCBA model due to the
relative simplicity of its Coulomb wave function. We remind here that, in all
our calculations in the various models including the NRCBA model, we did not
take into account the residual ion H+, nor the exchange effects (since we work
at high energies in which the exchange effects are negligible) or the various
interactions that may occur between the two final electrons. We have found
that the relativistic and spin effects, become more and more important by
increasing the energy of the incident electron. All the appropriate numerical
tests to verify the validity of the analytical results we found were performed
with a very good degree of accuracy. The paper is constructed as follows. In
section 2, we deliver the different theoretical models in the asymmetric and
symmetric coplanar geometries and give, for each model, a detailed account of
the techniques which we have used to evaluate the TDCS. In section 3, we
discuss the numerical results we have obtained in each geometry. Finally,
section 4 is devoted to the conclusions. Atomic units $\hbar=m=e=1$ are used
throughout this work.

##  2\. Theoretical models

Let us consider a collision between a hydrogen atom in its metastable 2S-state
and an incident electron moving along the $z$-axis. As a result of this
collision, the hydrogen atom becomes ionized and the projectile electron
changes its four-momentum from $p_{i}$ to $p_{f}$. In the final state, two
electrons (scattered and ejected) emerge with four-momenta $p_{f}$ and
$p_{B}$. This reaction can be described, symbolically, as follows:

$\text{e}^{-}(p_{i})+\text{H(2S)}\longrightarrow\text{H}^{+}+\text{e}^{-}(p_{B})+\text{e}^{-}(p_{f}).$
(1)

All ionization reactions are studied within two geometric frameworks. The
first is called asymmetric geometry and the second is symmetric geometry, and
each of them may be coplanar or noncoplanar. In asymmetric geometries, a fast
electron of energy $E_{i}$ is incident on the target atom, and a fast
scattered electron is detected in coincidence with a slow ejected electron.
This kind of experiment was first performed by Ehrhardt et al. [25]. Symmetric
geometries, which are defined by the requirement that the two outgoing
electrons are detected with the same energies and equal scattering and
ejection angles (i.e. $E_{f}\simeq E_{B}$ and $\theta_{f}\simeq\theta_{B}$ ),
were introduced by Amaldi et al. [26]. In coplanar geometry, the three momenta
$\textbf{p}_{i}$, $\textbf{p}_{f}$ and $\textbf{p}_{B}$ are in the same plane,
whereas in noncoplanar geometry the momentum $\textbf{p}_{B}$ is out of the
($\textbf{p}_{i}$, $\textbf{p}_{f}$) reference plane.

Figure 1: Schematic diagram of the coplanar geometry chosen for our
theoretical study. $\theta_{f}$ and $\theta_{B}$ are, respectively, the angles
of the scattered and ejected electrons with respect to the incident beam
direction $\textbf{p}_{i}$. For coplanar symmetric geometries,
$\theta_{f}\simeq\theta_{B}$ .

During this work, we will study the process (1) under two different
geometries. We will start first with asymmetric coplanar geometry and then
secondly with symmetric coplanar geometry. The detailed calculation of each
TDCS in each geometry will be presented.

###  2.1. Asymmetric coplanar geometry

We remember that in the case of the Ehrhardt coplanar asymmetric geometry, a
fast electron of kinetic energy $T_{i}$ is incident on the hydrogen target,
and a fast scattered electron of kinetic energy $T_{f}$ is detected in
coincidence with a slow ejected electron of kinetic energy $T_{B}$.
Additionally, the three momenta $\textbf{p}_{i}$, $\textbf{p}_{f}$, and
$\textbf{p}_{B}$ are in the same plane and the scattering angle $\theta_{f}$
of the scattered electron is fixed and small, while the angle $\theta_{B}$ of
the ejected electron is varied. In this geometry, we calculate step by step
the exact analytical expression of the semirelativistic spin-unpolarized TDCS
in the SRCBA approximation corresponding to the electron-impact ionization of
atomic hydrogen in its metastable 2S-state.

####  2.1..1 The S-matrix element

We begin with the first Born ionization S-matrix element for the process (1)
in the direct channel in which the exchange effects are neglected. It can be
written as [27]

$\begin{split}S_{fi}&=-\frac{i}{c}\int_{-\infty}^{+\infty}dx^{0}\langle\psi_{p_{f}}(x_{1})\phi_{f}(x_{2})|V_{d}|\psi_{p_{i}}(x_{1})\phi_{i}(x_{2})\rangle,\\\
&=-i\int_{-\infty}^{+\infty}dt\int
d\textbf{r}_{1}\psi_{p_{f}}^{{\dagger}}(t,\textbf{r}_{1})\psi_{p_{i}}(t,\textbf{r}_{1})\langle\phi_{f}(x_{2})|V_{d}|\phi_{i}(x_{2})\rangle.\end{split}$
(2)

Here, the potential $V_{d}=1/r_{12}-1/r_{1}$ presents the direct interaction
between the incident electron and the hydrogen atom, where
$r_{12}=|\textbf{r}_{1}-\textbf{r}_{2}|$ and $r_{1}=|\textbf{r}_{1}|$. The
nucleus of the target atom, which is assumed to be infinitely massive, is
chosen to be the origin of the coordinate system. The coordinates of the
incident and atomic electrons are labeled by $\textbf{r}_{1}$ and
$\textbf{r}_{2}$, respectively. $\psi_{p_{i}}$ and $\psi_{p_{f}}$ are the wave
functions describing, respectively, the incident and scattered electrons given
by a free Dirac solution normalized to the volume $V$

$\begin{split}\psi_{p_{i}}(x_{1})&=\frac{u(p_{i},s_{i})}{\sqrt{2E_{i}V}}e^{-ip_{i}.x_{1}},\\\
\psi_{p_{f}}(x_{1})&=\frac{u(p_{f},s_{f})}{\sqrt{2E_{f}V}}e^{-ip_{f}.x_{1}},\end{split}$
(3)

where $E_{i}$ and $E_{f}$ are, respectively, the total energies of the
incident and scattered electrons. $\phi_{i}(x_{2})=\phi_{i}(t,\textbf{r}_{2})$
is the semirelativistic Darwin wave function of atomic hydrogen in its
metastable 2S-state, which is accurate to the order $Z/c$ in the relativistic
corrections. It is given by

$\phi_{i}(t,\textbf{r}_{2})=\exp[-i\mathcal{E}_{b}(2S)t]\varphi^{(\pm)}_{2S}(\textbf{r}_{2}),$
(4)

where $\mathcal{E}_{b}(2S)$ is the binding energy of the metastable 2S-state
of atomic hydrogen given by

$\mathcal{E}_{b}(2S)=\frac{c^{2}}{\sqrt{2}}\sqrt{1+\sqrt{1-\alpha^{2}}}-c^{2},$
(5)

where $\alpha=1/c$ is the fine structure constant. For spin up,
$\varphi^{(+)}_{2S}(\textbf{r}_{2})$ is expressed by

$\varphi^{(+)}_{2S}(\textbf{r}_{2})=N_{D_{2}}\begin{pmatrix}2-r_{2}\\\ 0\\\
\frac{i(4-r_{2})}{4c}\cos(\theta)\\\
\frac{i(4-r_{2})}{4c}\sin(\theta)e^{i\phi}\end{pmatrix}\frac{1}{4\sqrt{2\pi}}e^{-r_{2}/2},$
(6)

where $\theta$ and $\phi$ are the spherical coordinates of $\textbf{r}_{2}$
and

$N_{D_{2}}=\frac{4c}{\sqrt{32c^{2}+10}}$ (7)

is the normalization constant. The wave function
$\phi_{f}(x_{2})=\phi_{f}(t,\textbf{r}_{2})$ in Eq. (2) is the Sommerfeld-Maue
wave function for continuum states [27], also accurate to the order $Z/c$ in
the relativistic corrections. We have

$\phi_{f}(t,\textbf{r}_{2})=e^{-iE_{B}t}\psi_{p_{B}}^{(-)}(\textbf{r}_{2}),$
(8)

where $\psi_{p_{B}}^{(-)}(\textbf{r}_{2})$ is given, in its final compact form
normalized to the volume V, by

$\begin{split}\psi_{p_{B}}^{(-)}(\textbf{r}_{2})&=e^{\pi\eta_{B}/2}\Gamma(1+i\eta_{B})e^{i\textbf{p}_{B}.\textbf{r}_{2}}\bigg{\\{}{}_{1}F_{1}(-i\eta_{B},1,-i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2}))+\frac{i}{2cp_{B}}(\alpha.\textbf{p}_{B}+p_{B}\alpha.\hat{\textbf{r}}_{2})\\\
&{}_{1}F_{1}(-i\eta_{B}+1,2,-i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2}))\bigg{\\}}\frac{u(p_{B},s_{B})}{\sqrt{2E_{B}V}}.\end{split}$
(9)

$\eta_{B}$ is the Sommerfeld parameter given by

$\eta_{B}=\frac{E_{B}}{c^{2}p_{B}},$ (10)

where $E_{B}$ is the total energy of the ejected electron and
$p_{B}=|\textbf{p}_{B}|$ is the norm of the ejected electron momentum. In Eq.
(9), the operator $[\alpha.\textbf{p}_{B}]$ acts on the free spinor
$u(p_{B},s_{B})$ and the operator $[\alpha.\hat{\textbf{r}}_{2}]$ acts on the
spinor part of the Darwin wave function.
The integral over the time coordinate in Eq. (2) can be separated yielding

$\int
dt\exp[i(E_{f}+E_{B}-E_{i}-\mathcal{E}_{b}(2S))t]=2\pi\delta(E_{f}+E_{B}-E_{i}-\mathcal{E}_{b}(2S)),$
(11)

while the integration over $d\textbf{r}_{1}$ can be performed by using the
well-known following Bethe integral

$\int
d\textbf{r}_{1}e^{i(\textbf{p}_{i}-\textbf{p}_{f}).\textbf{r}_{1}}\bigg{(}\frac{1}{r_{12}}-\frac{1}{r_{1}}\bigg{)}=\frac{4\pi}{\Delta^{2}}(e^{i\mathbf{\Delta}.\textbf{r}_{2}}-1),$
(12)

where the quantity $\mathbf{\Delta}=\textbf{p}_{i}-\textbf{p}_{f}$ is the
momentum transfer.
The direct S-matrix element in Eq. (2) becomes

$\begin{split}S_{fi}&=-i\int
d\textbf{r}_{2}\frac{\bar{u}(p_{f},s_{f})}{\sqrt{2E_{f}V}}\gamma^{0}\frac{u(p_{i},s_{i})}{\sqrt{2E_{i}V}}\bigg{\\{}{}_{1}F_{1}(i\eta_{B},1,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2}))-\frac{i}{2cp_{B}}(\alpha.\textbf{p}_{B}\\\
&+p_{B}\alpha.\hat{\textbf{r}}_{2}){}_{1}F_{1}(i\eta_{B}+1,2,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2}))\bigg{\\}}\frac{\bar{u}(p_{B},s_{B})\gamma^{0}}{\sqrt{2E_{B}V}}\varphi^{(+)}_{2S}(\textbf{r}_{2})e^{-i\textbf{p}_{B}.\textbf{r}_{2}}(e^{i\mathbf{\Delta}.\textbf{r}_{2}}-1)\\\
&\times\frac{8\pi^{2}}{\Delta^{2}}\delta(E_{f}+E_{B}-E_{i}-\mathcal{E}_{b}(2S))e^{\pi\eta_{B}/2}\Gamma(1-i\eta_{B}).\end{split}$
(13)

This S-matrix element contains two terms $S_{fi}^{(1)}$, $S_{fi}^{(2)}$. The
first one is given by

$\begin{split}S_{fi}^{(1)}&=-i\int
d\textbf{r}_{2}\frac{\bar{u}(p_{f},s_{f})}{\sqrt{2E_{f}V}}\gamma^{0}\frac{u(p_{i},s_{i})}{\sqrt{2E_{i}V}}\frac{\bar{u}(p_{B},s_{B})\gamma^{0}}{\sqrt{2E_{B}V}}\big{\\{}{}_{1}F_{1}(i\eta_{B},1,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2}))\big{\\}}\varphi^{(+)}_{2S}(\textbf{r}_{2})\\\
&\times
e^{-i\textbf{p}_{B}.\textbf{r}_{2}}(e^{i\mathbf{\Delta}.\textbf{r}_{2}}-1)\frac{8\pi^{2}}{\Delta^{2}}\delta(E_{f}+E_{B}-E_{i}-\mathcal{E}_{b}(2S))e^{\pi\eta_{B}/2}\Gamma(1-i\eta_{B}).\end{split}$
(14)

This first term can be reformulated in the following form

$\begin{split}S_{fi}^{(1)}&=-i[H_{1}(\textbf{q}=\mathbf{\Delta}-\textbf{p}_{B})-H_{1}(\textbf{q}=-\textbf{p}_{B})]\frac{\bar{u}(p_{f},s_{f})}{\sqrt{2E_{f}V}}\gamma^{0}\frac{u(p_{i},s_{i})}{\sqrt{2E_{i}V}}\frac{\bar{u}(p_{B},s_{B})\gamma^{0}}{\sqrt{2E_{B}V}}\\\
&\times\frac{8\pi^{2}}{\Delta^{2}}\delta(E_{f}+E_{B}-E_{i}-\mathcal{E}_{b}(2S))e^{\pi\eta_{B}/2}\Gamma(1-i\eta_{B}),\end{split}$
(15)

where $H_{1}(\textbf{q})$ is given by

$H_{1}(\textbf{q})=\int
d\textbf{r}_{2}e^{i\textbf{q}.\textbf{r}_{2}}{}_{1}F_{1}(i\eta_{B},1,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2}))\varphi^{(+)}_{2S}(\textbf{r}_{2}).$
(16)

According to the expression of $\varphi^{(+)}_{2S}(\textbf{r}_{2})$ given in
Eq. (6), $H_{1}(\textbf{q})$ can be written as

$H_{1}(\textbf{q})=\frac{N_{D_{2}}}{4\sqrt{2\pi}}(I_{1},0,I_{2},I_{3})^{T},$
(17)

and one has to evaluate

$I_{1}=\int
d\textbf{r}_{2}(2-r_{2})e^{-r_{2}/2}e^{i\textbf{q}.\textbf{r}_{2}}{}_{1}F_{1}(i\eta_{B},1,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2})).$
(18)

In this integral, we are confronted with the task of evaluating two types of
integrals, one of which is

$I^{\prime}_{1}=2\int
d\textbf{r}_{2}e^{-r_{2}/2}e^{i\textbf{q}.\textbf{r}_{2}}{}_{1}F_{1}(i\eta_{B},1,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2})),$
(19)

and the other one is

$I^{\prime\prime}_{1}=\int
d\textbf{r}_{2}r_{2}e^{-r_{2}/2}e^{i\textbf{q}.\textbf{r}_{2}}{}_{1}F_{1}(i\eta_{B},1,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2})).$
(20)

In order to evaluate the two integrals $I^{\prime}_{1}$ and
$I^{\prime\prime}_{1}$, we take recourse to the well-known integral [28]

$\begin{split}I(\lambda)&=\int
d\textbf{r}e^{i\textbf{q}.\textbf{r}}\frac{e^{-\lambda
r}}{r}{}_{1}F_{1}(i\eta_{B},1,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2})),\\\
&=\frac{4\pi}{q^{2}+\lambda^{2}}\exp\bigg{[}i\eta_{B}\ln\bigg{(}\frac{q^{2}+\lambda^{2}}{q^{2}+\lambda^{2}+2\textbf{q}.\textbf{p}_{B}-2i\lambda
p_{B}}\bigg{)}\bigg{]},\end{split}$ (21)

where $\lambda$ is a real variable. Looking at the expressions of the
integrals $I^{\prime}_{1}$ and $I^{\prime\prime}_{1}$ above, it becomes clear
that they are, respectively, the first and second derivatives of the integral
$I(\lambda)$. This yields

$I^{\prime}_{1}=2\bigg{(}-\frac{\partial
I(\lambda)}{\partial\lambda}\bigg{)}\bigg{|}_{\lambda=1/2}.$ (22)
$I^{\prime\prime}_{1}=\bigg{(}\frac{\partial^{2}I(\lambda)}{\partial\lambda^{2}}\bigg{)}\bigg{|}_{\lambda=1/2}.$
(23)

The other integrals $I_{2}$ and $I_{3}$ in (17) can be obtained by noting that

$\cos(\theta)e^{i\textbf{q}.\textbf{r}_{2}}=-\frac{i}{r_{2}}\frac{\partial}{\partial
q_{z}}e^{i\textbf{q}.\textbf{r}_{2}},$ (24)

and

$\sin(\theta)e^{i\phi}e^{i\textbf{q}.\textbf{r}_{2}}=-\frac{i}{r_{2}}\bigg{(}\frac{\partial}{\partial
q_{x}}+i\frac{\partial}{\partial
q_{y}}\bigg{)}e^{i\textbf{q}.\textbf{r}_{2}}.$ (25)

Thus we finally get

$\begin{split}I_{2}&=\frac{1}{c}\frac{\partial}{\partial
q_{z}}\bigg{[}I(\lambda)+\frac{1}{4}\frac{\partial
I(\lambda)}{\partial\lambda}\bigg{]}\bigg{|}_{\lambda=1/2},\\\
I_{3}&=\frac{1}{c}\bigg{[}\frac{\partial}{\partial
q_{x}}+i\frac{\partial}{\partial
q_{y}}\bigg{]}\bigg{[}I(\lambda)+\frac{1}{4}\frac{\partial
I(\lambda)}{\partial\lambda}\bigg{]}\bigg{|}_{\lambda=1/2}.\end{split}$ (26)

The second term in the S-matrix element given in Eq. (13) is

$S_{fi}^{(2)}=S_{fi}^{(2),1}+S_{fi}^{(2),2},$ (27)

with

$\begin{split}S_{fi}^{(2),1}&=-\int
d\textbf{r}_{2}\frac{\bar{u}(p_{f},s_{f})}{\sqrt{2E_{f}V}}\gamma^{0}\frac{u(p_{i},s_{i})}{\sqrt{2E_{i}V}}\frac{1}{2cp_{B}}\frac{\bar{u}(p_{B},s_{B})\gamma^{0}}{\sqrt{2E_{B}V}}\bigg{[}\gamma^{0}\frac{E_{B}}{c}-\not{p}_{B}\bigg{]}\varphi^{(+)}_{2S}(\textbf{r}_{2})\\\
&\times{}_{1}F_{1}(i\eta_{B}+1,2,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2}))e^{-i\textbf{p}_{B}.\textbf{r}_{2}}(e^{i\mathbf{\Delta}.\textbf{r}_{2}}-1)\\\
&\times\frac{8\pi^{2}}{\Delta^{2}}\delta(E_{f}+E_{B}-E_{i}-\mathcal{E}_{b}(2S))e^{\pi\eta_{B}/2}\Gamma(1-i\eta_{B}),\end{split}$
(28)

and

$\begin{split}S_{fi}^{(2),2}&=-\int
d\textbf{r}_{2}\frac{\bar{u}(p_{f},s_{f})}{\sqrt{2E_{f}V}}\gamma^{0}\frac{u(p_{i},s_{i})}{\sqrt{2E_{i}V}}\frac{1}{2c}\frac{\bar{u}(p_{B},s_{B})\gamma^{0}}{\sqrt{2E_{B}V}}\varphi^{{}^{\prime}(+)}_{2S}(\textbf{r}_{2}){}_{1}F_{1}(i\eta_{B}+1,2,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2}))\\\
&\times
e^{-i\textbf{p}_{B}.\textbf{r}_{2}}(e^{i\mathbf{\Delta}.\textbf{r}_{2}}-1)\frac{8\pi^{2}}{\Delta^{2}}\delta(E_{f}+E_{B}-E_{i}-\mathcal{E}_{b}(2S))e^{\pi\eta_{B}/2}\Gamma(1-i\eta_{B}).\end{split}$
(29)

The operator $[\alpha.\textbf{p}_{B}]$ in Eq. (28) is replaced by
$\big{[}\gamma^{0}\frac{E_{B}}{c}-\not{p}_{B}\big{]}$, and in Eq. (29)
$\varphi^{{}^{\prime}(+)}_{2S}(\textbf{r}_{2})$ is given by

$\varphi^{{}^{\prime}(+)}_{2S}(\textbf{r}_{2})=[\alpha.\hat{\textbf{r}}_{2}]\varphi^{(+)}_{2S}(\textbf{r}_{2})=\frac{N_{D_{2}}}{4\sqrt{2\pi}}e^{-r_{2}/2}\begin{pmatrix}\frac{i(4-r_{2})}{4c}\\\
0\\\ (2-r_{2})\cos(\theta)\\\ (2-r_{2})\sin(\theta)e^{i\phi}\end{pmatrix}.$
(30)

$S_{fi}^{(2),1}$ can be recasted in the following form

$\begin{split}S_{fi}^{(2),1}&=-[H_{2}(\textbf{q}=\mathbf{\Delta}-\textbf{p}_{B})-H_{2}(\textbf{q}=-\textbf{p}_{B})]\frac{\bar{u}(p_{f},s_{f})}{\sqrt{2E_{f}V}}\gamma^{0}\frac{u(p_{i},s_{i})}{\sqrt{2E_{i}V}}\frac{1}{2cp_{B}}\frac{\bar{u}(p_{B},s_{B})\gamma^{0}}{\sqrt{2E_{B}V}}\\\
&\times\bigg{[}\gamma^{0}\frac{E_{B}}{c}-\not{p}_{B}\bigg{]}\frac{8\pi^{2}}{\Delta^{2}}\delta(E_{f}+E_{B}-E_{i}-\mathcal{E}_{b}(2S))e^{\pi\eta_{B}/2}\Gamma(1-i\eta_{B}),\end{split}$
(31)

where $H_{2}(\textbf{q})$ is the integral expressed by

$H_{2}(\textbf{q})=\int
d\textbf{r}_{2}e^{i\textbf{q}.\textbf{r}_{2}}{}_{1}F_{1}(i\eta_{B}+1,2,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2}))\varphi^{(+)}_{2S}(\textbf{r}_{2}).$
(32)

Replacing the Darwin function in Eq. (32) by its expression (6) leads to

$H_{2}(\textbf{q})=\frac{N_{D_{2}}}{4\sqrt{2\pi}}(J_{1},0,J_{2},J_{3})^{T},$
(33)

where

$\begin{split}J_{1}&=\int
d\textbf{r}_{2}e^{i\textbf{q}.\textbf{r}_{2}}e^{-r_{2}/2}(2-r_{2}){}_{1}F_{1}(i\eta_{B}+1,2,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2})),\\\
J_{2}&=\frac{i}{4c}\int
d\textbf{r}_{2}e^{i\textbf{q}.\textbf{r}_{2}}e^{-r_{2}/2}(4-r_{2})\cos(\theta){}_{1}F_{1}(i\eta_{B}+1,2,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2})),\\\
J_{3}&=\frac{i}{4c}\int
d\textbf{r}_{2}e^{i\textbf{q}.\textbf{r}_{2}}e^{-r_{2}/2}(4-r_{2})\sin(\theta)e^{i\phi}{}_{1}F_{1}(i\eta_{B}+1,2,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2})).\end{split}$
(34)

To evaluate this three integrals, we introduce a new integral that has been
calculated analytically by Attaourti et al in [2]

$\begin{split}J(\lambda)&=\int
d\textbf{r}e^{i\textbf{q}.\textbf{r}}\frac{e^{-\lambda
r}}{r}{}_{1}F_{1}(i\eta_{B}+1,2,i(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2})),\\\
&=\frac{4\pi}{(q^{2}+\lambda^{2})}{}_{2}F_{1}\bigg{(}i\eta_{B}+1,1,2,-2\frac{(\textbf{q}.\textbf{p}_{B}-i\lambda
p_{B})}{q^{2}+\lambda^{2}}\bigg{)},\end{split}$ (35)

where $\lambda$ is a real variable.
In the same way as before and after some manipulations, one gets

$\begin{split}J_{1}&=-2\bigg{(}\frac{\partial
J(\lambda)}{\partial\lambda}\bigg{)}-\frac{\partial^{2}J(\lambda)}{\partial\lambda^{2}}\bigg{|}_{\lambda=1/2},\\\
J_{2}&=\frac{1}{c}\frac{\partial}{\partial
q_{z}}\bigg{[}J(\lambda)+\frac{1}{4}\frac{\partial
J(\lambda)}{\partial\lambda}\bigg{]}\bigg{|}_{\lambda=1/2},\\\
J_{3}&=\frac{1}{c}\bigg{[}\frac{\partial}{\partial
q_{x}}+i\frac{\partial}{\partial
q_{y}}\bigg{]}\bigg{[}J(\lambda)+\frac{1}{4}\frac{\partial
J(\lambda)}{\partial\lambda}\bigg{]}\bigg{|}_{\lambda=1/2}.\end{split}$ (36)

For the term $S_{fi}^{(2),2}$, it can be written as

$\begin{split}S_{fi}^{(2),2}&=-[H_{3}(\textbf{q}=\mathbf{\Delta}-\textbf{p}_{B})-H_{3}(\textbf{q}=-\textbf{p}_{B})]\frac{\bar{u}(p_{f},s_{f})}{\sqrt{2E_{f}V}}\gamma^{0}\frac{u(p_{i},s_{i})}{\sqrt{2E_{i}V}}\frac{1}{2c}\frac{\bar{u}(p_{B},s_{B})\gamma^{0}}{\sqrt{2E_{B}V}}\\\
&\times\frac{8\pi^{2}}{\Delta^{2}}\delta(E_{f}+E_{B}-E_{i}-\mathcal{E}_{b}(2S))e^{\pi\eta_{B}/2}\Gamma(1-i\eta_{B}).\end{split}$
(37)

The quantity $H_{3}(\textbf{q})$ is given by

$H_{3}(\textbf{q})=\frac{N_{D_{2}}}{4\sqrt{2\pi}}(K_{1},0,K_{2},K_{3})^{T},$
(38)

where $K_{1}$, $K_{2}$ and $K_{3}$ are three integrals whose solutions are

$\begin{split}K_{1}&=-\frac{i}{c}\bigg{[}\frac{\partial
J(\lambda)}{\partial\lambda}+\frac{1}{4}\frac{\partial^{2}J(\lambda)}{\partial\lambda^{2}}\bigg{]}\bigg{|}_{\lambda=1/2},\\\
K_{2}&=-i\frac{\partial}{\partial q_{z}}\bigg{[}2J(\lambda)+\frac{\partial
J(\lambda)}{\partial\lambda}\bigg{]}\bigg{|}_{\lambda=1/2},\\\
K_{3}&=-i\bigg{[}\frac{\partial}{\partial q_{x}}+i\frac{\partial}{\partial
q_{y}}\bigg{]}\bigg{[}2J(\lambda)+\frac{\partial
J(\lambda)}{\partial\lambda}\bigg{]}\bigg{|}_{\lambda=1/2}.\end{split}$ (39)

####  2.1..2 Spin-unpolarized TDCS in the SRCBA

Using the standard procedures of QED [29], we obtain for the spin-unpolarized
TDCS

$\frac{d\bar{\sigma}^{(SRCBA)}}{dE_{B}d\Omega_{B}d\Omega_{f}}=\frac{1}{16\pi^{3}c^{6}}\frac{|\textbf{p}_{f}||\textbf{p}_{B}|}{|\textbf{p}_{i}|}\frac{e^{\pi\eta_{B}}}{\Delta^{4}}\big{|}\Gamma(1-i\eta_{B})\big{|}^{2}\big{|}\widehat{S}_{fi}^{(1)}+\widehat{S}_{fi}^{(2),1}+\widehat{S}_{fi}^{(2),2}\big{|}^{2}\bigg{|}_{E_{f}=E_{i}+\mathcal{E}_{b}(2S)-E_{B}},$
(40)

with

$\begin{split}\widehat{S}_{fi}^{(1)}&=\frac{1}{2}\sum_{s_{i},s_{f}}\sum_{s_{B}}\big{[}\bar{u}(p_{f},s_{f})\gamma^{0}u(p_{i},s_{i})\big{]}\big{[}\bar{u}(p_{B},s_{B})\gamma^{0}\big{]}\big{[}i(H_{1}(\textbf{q}=\mathbf{\Delta}-\textbf{p}_{B})-H_{1}(\textbf{q}=-\textbf{p}_{B}))\big{]},\end{split}$
(41)
$\begin{split}\widehat{S}_{fi}^{(2),1}&=\frac{1}{2}\sum_{s_{i},s_{f}}\sum_{s_{B}}\big{[}\bar{u}(p_{f},s_{f})\gamma^{0}u(p_{i},s_{i})\big{]}\big{[}\bar{u}(p_{B},s_{B})\gamma^{0}\big{]}\bigg{[}\gamma^{0}\frac{E_{B}}{c}-\not{p}_{B}\bigg{]}\frac{1}{2cp_{B}}\\\
&\times\big{[}H_{2}(\textbf{q}=\mathbf{\Delta}-\textbf{p}_{B})-H_{2}(\textbf{q}=-\textbf{p}_{B})\big{]},\end{split}$
(42)
$\begin{split}\widehat{S}_{fi}^{(2),2}&=\frac{1}{2}\sum_{s_{i},s_{f}}\sum_{s_{B}}\big{[}\bar{u}(p_{f},s_{f})\gamma^{0}u(p_{i},s_{i})\big{]}\big{[}\bar{u}(p_{B},s_{B})\gamma^{0}\big{]}\frac{1}{2c}\big{[}H_{3}(\textbf{q}=\mathbf{\Delta}-\textbf{p}_{B})-H_{3}(\textbf{q}=-\textbf{p}_{B})\big{]}.\end{split}$
(43)

In Eq. (40), $|\textbf{p}_{i}|$ and $|\textbf{p}_{f}|$ are, respectively, the
norms of the initial and final electron momenta. All the calculations in Eq.
(40) can be done analytically and only five terms out of nine are nonzero, the
diagonal terms $\big{|}\widehat{S}_{fi}^{(1)}\big{|}^{2}$,
$\big{|}\widehat{S}_{fi}^{(2),1}\big{|}^{2}$,
$\big{|}\widehat{S}_{fi}^{(2),2}\big{|}^{2}$, and
$\widehat{S}_{fi}^{(1){\dagger}}\widehat{S}_{fi}^{(2),1}$, as well as
$\widehat{S}_{fi}^{(2),1{\dagger}}\widehat{S}_{fi}^{(1)}$. In Eqs. (41)-(43),
the different sums over spin states give the following results:

$\frac{1}{2}\sum_{s_{i},s_{f}}\big{|}\bar{u}(p_{f},s_{f})\gamma^{0}u(p_{i},s_{i})\big{|}^{2}=2c^{2}\bigg{(}\frac{2E_{i}E_{f}}{c^{2}}-(p_{i}.p_{f})+c^{2}\bigg{)},$
(44)
$\sum_{s_{B}}\bigg{|}\bar{u}(p_{B},s_{B})\gamma^{0}\bigg{[}\gamma^{0}\frac{E_{B}}{c}-\not{p}_{B}\bigg{]}\bigg{|}^{2}=4E_{B}\bigg{(}\frac{E_{B}^{2}}{c^{2}}-c^{2}\bigg{)},$
(45) $\sum_{s_{B}}\big{|}\bar{u}(p_{B},s_{B})\gamma^{0}\big{|}^{2}=4E_{B},$
(46) $\frac{1}{2}\sum_{s_{t}}(...)=1(...),$ (47)

where $(p_{i}.p_{f})$ in Eq. (44) is the scalar product of initial and final
four-momentum, and $\sum_{s_{t}}(...)/2$ denotes the averaged sum over the
spin states of the target atomic hydrogen.
We have to compare the TDCS in Eq. (40) with the corresponding one in the Non-
Relativistic Coulomb Born Approximation (NRCBA), where the incident and
scattered electrons are described by non-relativistic plane waves:

$\psi_{p_{i,f}}(\textbf{r}_{1})=(2\pi)^{-3/2}e^{i\textbf{p}_{i,f}.\textbf{r}_{1}},$
(48)

whereas the ejected electron is described by a Coulomb wave function:

$\psi_{c,p_{B}}(\textbf{r}_{2})=(2\pi)^{-3/2}e^{i\textbf{p}_{B}.\textbf{r}_{2}}e^{\pi/2p_{B}}\Gamma\Big{(}1+\frac{i}{p_{B}}\Big{)}{}_{1}F_{1}\Big{(}-\frac{i}{p_{B}},1,-(p_{B}r_{2}+\textbf{p}_{B}.\textbf{r}_{2})\Big{)},$
(49)

and the hydrogen atomic in its metastable 2S-state is described by the non-
relativistic (NR) wave function [30]

$\psi_{2S}^{NR}(\textbf{r}_{2})=\frac{1}{4\sqrt{2\pi}}(2-r_{2})e^{-r_{2}/2}.$
(50)

Thus, the TDCS in the NRCBA is given by:

$\frac{d\bar{\sigma}^{(NRCBA)}}{dE_{B}d\Omega_{B}d\Omega_{f}}=\frac{p_{f}p_{B}}{p_{i}}\big{|}f_{ion}^{CBA}\big{|}^{2},$
(51)

where $f_{ion}^{CBA}$ is the first Coulomb-Born amplitude corresponding to the
ionization of metastable 2S-state hydrogen atom by electron impact which is
given by:

$\begin{split}f_{ion}^{CBA}&=-\frac{2}{\Delta^{2}}\frac{e^{\pi/2p_{B}}}{4(2\pi)^{2}}\Gamma\bigg{(}1-\frac{i}{p_{B}}\bigg{)}\bigg{[}-2\frac{\partial
I(\textbf{q}=\mathbf{\Delta}-\textbf{p}_{B})}{\partial\lambda}-\frac{\partial^{2}I(\textbf{q}=\mathbf{\Delta}-\textbf{p}_{B}}{\partial\lambda^{2}}\bigg{]}\bigg{|}_{\lambda=1/2},\end{split}$
(52)

where the integral $I(\textbf{q})$ is the same as that given previously in Eq.
(21), but here $\eta_{B}=1/p_{B}$.

###  2.2. Symmetric coplanar geometry

The symmetric coplanar geometry can be considered as a particular case of
asymmetric coplanar geometry. Let us first remind that the symmetric geometry,
also called binary geometry, is defined by the requirement that the kinetic
energies of the scattered and ejected electrons are nearly the same, and the
scattered and ejected electron angles with respect to the incident beam
direction are equal to each other. In this section, we present the
relativistic formalism of the $(e,2e)$ reaction in the Relativistic Plane Wave
Born Approximation (RPWBA), where the incident, scattered, and ejected
electrons are described by relativistic plane waves, and the hydrogen atom in
its metastable 2S-state is described by the relativistic exact function given
by:

$\phi_{i}(t,\textbf{r}_{2})=\exp[-i\mathcal{E}_{b}(2S)t]\varphi^{(\pm),2S}_{Exact}(\textbf{r}_{2}),$
(53)

where $\mathcal{E}_{b}(2S)$ is the binding energy of the metastable 2S-state
of atomic hydrogen given in (5). For spin up,
$\varphi^{(+),2S}_{Exact}(\textbf{r}_{2})$ is expressed by:

$\begin{split}\varphi^{(+),2S}_{Exact}(\textbf{r}_{2})&=\frac{1}{2\sqrt{4\pi}}\frac{(2Z)^{\gamma_{H}+1/2}}{a_{2S}^{\gamma_{H}+1}}\sqrt{\frac{2\gamma_{H}+1}{(a_{2S}+1)\Gamma(2\gamma_{H}+1)}}r_{2}^{\gamma_{H}-1}e^{-Zr_{2}/a_{2S}}\\\
&\times\begin{pmatrix}ig_{2S_{1/2}}(r_{2})\\\ 0\\\
f_{2S_{1/2}}(r_{2})\cos(\theta)\\\
f_{2S_{1/2}}(r_{2})\sin(\theta)e^{i\phi}\end{pmatrix},\end{split}$ (54)

where $\theta$ and $\phi$ are the spherical coordinates of $\textbf{r}_{2}$.
The two quantities $g_{2S_{1/2}}(r_{2})$ and $f_{2S_{1/2}}(r_{2})$ are such
as:

$\begin{split}g_{2S_{1/2}}(r_{2})&=\sqrt{1+\frac{Z\alpha}{\sqrt{2(1-\gamma_{H})}}}\bigg{[}\bigg{(}1-\frac{2Zr_{2}}{a_{2S}(2\gamma_{H}+1)}\bigg{)}\big{(}a_{2S}+1\big{)}-1\bigg{]},\\\
f_{2S_{1/2}}(r_{2})&=\sqrt{1-\frac{Z\alpha}{\sqrt{2(1-\gamma_{H})}}}\bigg{[}\bigg{(}1-\frac{2Zr_{2}}{a_{2S}(2\gamma_{H}+1)}\bigg{)}\big{(}a_{2S}+1\big{)}+1\bigg{]},\end{split}$
(55)

where $Z$ is the atomic number, and the two parameters $\gamma_{H}$ and
$a_{2S}$ are given by:

$\begin{split}\gamma_{H}&=\sqrt{1-Z^{2}\alpha^{2}},\\\
a_{2S}&=\sqrt{2(\gamma_{H}+1)},\end{split}$ (56)

with $\alpha=1/c$ is the fine structure constant.
Substituting all these expressions into the first Born S-matrix element (2)
and after some manipulations, one gets

$\begin{split}\frac{d\bar{\sigma}^{(RPWBA)}}{dE_{B}d\Omega_{B}d\Omega_{f}}&=\frac{1}{2}\frac{p_{f}p_{B}}{c^{6}p_{i}\Delta^{4}}\bigg{(}\frac{1}{2}\sum_{s_{i},s_{f}}\big{|}\bar{u}(p_{f},s_{f})\gamma^{0}u(p_{i},s_{i})\big{|}^{2}\bigg{)}\sum_{s_{B}}\big{|}\bar{u}(p_{B},s_{B})\gamma^{0}\big{|}^{2}\\\
&\times\big{|}\Phi_{2,1/2,1/2}(\textbf{q}=\mathbf{\Delta}-\textbf{p}_{B})-\Phi_{2,1/2,1/2}(\textbf{q}=-\textbf{p}_{B})\big{|}^{2}.\end{split}$
(57)

The different sums over spin states $s_{i}$, $s_{f}$ and $s_{B}$ are given
before in Eqs. (44-46). The functions $\Phi_{2,1/2,1/2}(\textbf{q})$ are the
Fourier transforms of the relativistic atomic hydrogen wave functions

$\Phi_{n=2,j=1/2,m=1/2}(\textbf{q})=(2\pi)^{-3/2}\int
d\textbf{r}_{2}e^{i\textbf{q}.\textbf{r}_{2}}\varphi^{(+),2S}_{Exact}(\textbf{r}_{2}),$
(58)

and $\mathbf{\Delta}=\textbf{p}_{i}-\textbf{p}_{f}$ is the momentum transfer.
Replacing the exact function $\varphi^{(+),2S}_{Exact}(\textbf{r}_{2})$ by its
expression (54) yields

$\begin{split}\Phi_{n=2,j=1/2,m=1/2}(\textbf{q})&=(2\pi)^{-3/2}\frac{1}{2\sqrt{4\pi}}\frac{(2Z)^{\gamma_{H}+1/2}}{a_{2S}^{\gamma_{H}+1}}\sqrt{\frac{2\gamma_{H}+1}{(a_{2S}+1)\Gamma(2\gamma_{H}+1)}}\\\
&\times\begin{pmatrix}\int
d\textbf{r}_{2}e^{i\textbf{q}.\textbf{r}_{2}}r_{2}^{\gamma_{H}-1}e^{-Zr_{2}/a_{2S}}ig_{2S_{1/2}}(r_{2})\\\
0\\\ \int
d\textbf{r}_{2}e^{i\textbf{q}.\textbf{r}_{2}}r_{2}^{\gamma_{H}-1}e^{-Zr_{2}/a_{2S}}f_{2S_{1/2}}(r_{2})\cos(\theta)\\\
\int
d\textbf{r}_{2}e^{i\textbf{q}.\textbf{r}_{2}}r_{2}^{\gamma_{H}-1}e^{-Zr_{2}/a_{2S}}f_{2S_{1/2}}(r_{2})\sin(\theta)e^{i\phi}\end{pmatrix}.\end{split}$
(59)

The expression of the TDCS in the Semi-Relativistic Plan Wave Born
Approximation (SRPWBA) remains similar to that given in the RPWBA (57), except
the expression of the Fourier transform which changes since the wave function
describing the hydrogen atom in the SRPWBA is replaced by the Darwin wave
function that we have previously expressed in Eq. (6). This TDCS in Eq. (57)
is to be compared with the corresponding one in the Non-Relativistic Plane
Wave Born Approximation (NRPWBA), where the incident, scattered, and ejected
electrons are described by non-relativistic plane waves:

$\begin{split}\frac{d\bar{\sigma}^{(NRPWBA)}}{dE_{B}d\Omega_{B}d\Omega_{f}}&=\frac{2^{10}}{\pi^{2}\Delta^{4}}\frac{p_{f}p_{B}}{p_{i}}\bigg{[}\frac{4\textbf{q}^{2}-1}{(1+4\textbf{q}^{2})^{3}}-\frac{4\textbf{q}_{0}^{2}-1}{(1+4\textbf{q}^{2}_{0})^{3}}\bigg{]}^{2},\end{split}$
(60)

where $\textbf{q}=\mathbf{\Delta}-\textbf{p}_{B}$ and
$\textbf{q}_{0}=-\textbf{p}_{B}$.

##  3\. Results and discussion

In this paper, we develop an exact relativistic model, in the first Born
approximation, to study the ionization of the metastable 2S-state hydrogen
atom by electron impact at high energies in the asymmetric and symmetric
coplanar geometries. The required derivatives of hypergeometric functions and
all integrals resulting from the Fourier transforms of the relativistic and
semirelativistic atomic hydrogen wave functions are computed in closed
analytic forms using the programming language MATHEMATICA, which is also used
to plot the various figures of the present work. In this section, we will
present all the numerical results obtained in both asymmetric and symmetric
geometries; during that, we will follow the same arrangement that we adopted
in the previous section. We will start first with the results obtained in the
case of asymmetric geometry and then symmetric geometry. All the TDCSs are
given in atomic units.

###  3.1. Asymmetric coplanar geometry

We will begin our discussion, in this case, by comparing our results with
those obtained by Hafid et al. [15] in the nonrelativistic domain. Hafid’s
results were obtained using the well-known approximation BBK model of Brauner
et al. [17], and when Hafid presented his results, he also compared with those
obtained by Coulomb wave function and second born calculations of Vucic et al.
[16] with respect to the incoming electron kinetic energy of $250$ eV and the
ejected electron kinetic energy of $5$ eV. In the following figures (Fig. (2),
Fig. (3) and Fig. (4)), which contain the comparison with other theoretical
calculations, the angular choice is as follows: $p_{i}$ is along the $z$-axis
and $\theta_{i}=0^{\circ},\phi_{i}=0^{\circ}$. For the scattered electron, we
choose $\phi_{f}=0^{\circ}$ and $\theta_{f}$ is fixed in Figs. (2) and (3),
respectively, to the values $\theta_{f}=3^{\circ}$ and $\theta_{f}=5^{\circ}$,
while in Fig. (4) $\theta_{f}$ varies from $-12^{\circ}$ to $12^{\circ}$. For
the ejected electron, we choose $\phi_{B}=0^{\circ}$ and $\theta_{B}$ varies
from $0^{\circ}$ to $360^{\circ}$ in Figs. (2) and (3) and it is fixed to the
value $\theta_{B}=20^{\circ}$ in Fig. (4).

Figure 2: The TDCS of the (e, 2e) ionization of hydrogen 2S in terms of the
ejection angle $\theta_{B}$. The incident and the ejected electron kinetic
energies are $250$ eV and $5$ eV respectively and the scattering angle
$\theta_{f}=3^{\circ}$. The solid red line gives our results (NRCBA) given in
Eq.(51), the solid black line those of Hafid et al. [15] and the dashed line
results obtained by Coulomb wave function.

We compare, in Fig (2), our results in the NRCBA (Eq.(51)) with those of Hafid
et al. and those obtained by Coulomb wave function (where a Coulomb wave is
used for the ejected electron and plane waves for the incident and scattered
electrons: this model is called 1CW, one coulomb wave) for the incident
kinetic energy of $250$ eV, ejection kinetic energy value of $5$ eV and the
scattering angle of $\theta_{f}=3^{\circ}$. Our results in the NRCBA model
were obtained, as we have seen in the theoretical calculations in the previous
section, by using a Coulomb wave function to describe only the ejected
electron, whereas the fast incident and scattered electrons are described by
non-relativistic plane wave functions, thus neglecting the Coulomb interaction
of the fast scattered electron with the system. Thus, our NRCBA model is the
same as the well-known 1CW model. Therefore, the results obtained in both
models should be compatible with each other. As we can see in Fig. 2, the two
curves representing these results (red and dashed curves respectively) have
good agreement and convergence. They are very close, both in the shape of the
curve and the location of the peaks, as well as in the order of magnitude.
These two results obtained using the Coulomb wave functions remain different
in magnitude, as well as in the height of the binary peak from the result
obtained by Hafid et al. using BBK approximation.

Figure 3: The TDCS of the (e, 2e) ionization of hydrogen 2S in terms of the
ejection angle $\theta_{B}$. The incident and the ejected electron kinetic
energies are $250$ eV and $5$ eV respectively and the scattering angle
$\theta_{f}=5^{\circ}$. The solid red line gives our results (NRCBA) given in
(51), the solid black line those of Hafid et al. [15], the dotted line those
of Vucic et al. [16] and the dashed line results obtained using Coulomb wave
function.

Figure (3) represents similar parameterization as in Fig. (2), but with the
scattering angle $\theta_{f}=5^{\circ}$. We have also included here the second
Born results of Vucic et al. [16]. Again, it is clearly seen from this figure,
that our model NRCBA still gives the same results compared to the model 1CW
with an apparent difference between them and the results obtained from other
calculations. In the recoil region, all results remain close in form and
magnitude. It is interesting to see that the results of Hafid et al. reveal a
peak which is present also in the second Born calculation of Vucic et al. and
absent in the calculation used in the first Born Coulomb approximation (the
ejected electron is described by a Coulomb wave and the incident and scattered
electrons by plane waves). Comparing Figs. (2) and (3), we note that the
magnitude of the two peaks decreases with increasing the scattering angle
$\theta_{f}$.

Figure 4: The TDCS of the (e, 2e) ionization of hydrogen 2S in terms of the
scattering angle $\theta_{f}$. The incident and the ejected electron kinetic
energies are $250$ eV and $5$ eV respectively and the ejection angle
$\theta_{B}=20^{\circ}$. The solid red line gives our results (NRCBA) given in
(51) and the solid black line those of Hafid et al. [15]

We study in Fig. (4), for the same kinetic energy, the variation of the NRCBA
in terms of the scattering angle $\theta_{f}$ for the ejection angle
$\theta_{B}=20^{\circ}$. The comparison with the results of Hafid et al. is
also included. This actually gives a sharp peak, higher than the other peaks
in the previous figures. We observe also that the scattered electron, which is
relatively faster than the ejected one, goes out with small angles. The
discrepancies among the results obtained using the Coulomb wave functions and
other theoretical results obtained from BBK and second Born approximations
shown in Figs. (2) and (3) are expected due to the difference of the
approaches used in each theoretical study. The final state wave function in
the BBK results of Hafid et al. is improved by including the effects of all
long range Coulomb interactions and the repulsion between the two final
electrons, leading to results comparable to the second Born approximation
(Vucic et al.). It has been proved, for the ionization of the hydrogen (1S)
atom by electron impact, that it is the BBK model which gives the excellent
agreement with experiment at impact energies greater than $150$ eV [17]. From
a theoretical point of view, the difficulty resides in the description of the
three-body final state (residual ion, scattered and ejected electrons) in
Coulomb interaction. The question of the correlations of the different wave
functions involved in its description remains, nowadays, an unanswered
question. Many attempts have been made to determine the TDCS by neglecting the
interaction of the fast scattered electron with the system or by using a
product of two Coulomb wave functions for the scattered and ejected electrons.
Only an experimental benchmark study could judge the validity or not of a
theoretical approach, and therefore our only hope is in future experiments
that will give us the opportunity to validate our results at different
energies.

Figure 5: The two TDCSs as a function of the ejection angle $\theta_{B}$. The
incident and the ejected electron kinetic energies are $250$ eV and $5$ eV
respectively and the scattering angle $\theta_{f}=3^{\circ}$. The other angles
are chosen as follows: $\theta_{i}=0^{\circ},\phi_{i}=0^{\circ}$,
$\phi_{f}=0^{\circ}$ and $\phi_{B}=180^{\circ}$.

Figure (5) depicts the TDCS in the SRCBA and the corresponding one in the
NRCBA for the scattering angle $\theta_{f}=3^{\circ}$. The incident electron
kinetic energy is $T_{i}=250$ eV and the ejected electron kinetic energy is
$T_{B}=5$ eV. We see, as in the case of the ground state [2], that the two
curves are identical and have two peaks, one in the interval between
$-180^{\circ}$ and $0^{\circ}$ (recoil peak) due to projectile-nucleus
interaction and the other in the range between $0^{\circ}$ and $180^{\circ}$
(binary peak) due to the electron-electron interaction. The fact that the
SRCBA gives the same results compared to NRCBA at low energies can be
considered as a criterion for checking its consistency and its validity.
However, even in the nonrelativistic regime, small effects are presented; due
to the semirelativistic treatment of the wave functions that we have used in
the SRCBA, and these can only be related to the spin effect.

###  3.2. Symmetric coplanar geometry

In symmetric geometry, as we mentioned previously, the kinetic energies of
both scattered and ejected electrons are required to be approximately equal.
The TDCS, in all models studied in the previous section, depends explicitly on
the kinetic energy values of the scattered and ejected electrons, in addition
to the different spherical coordinates related to each electron. Therefore,
care must be taken when choosing the values of these kinetic energies, so that
the above-mentioned geometry condition is fulfilled. We remind the reader here
of the relation that allows us to obtain these values without violating the
requirement of symmetric geometry. Using the kinetic energy conservation
$T_{f}=T_{i}+\varepsilon_{2S}-T_{B}$, we find that, according to the condition
$T_{f}=T_{B}$, $T_{B}=(T_{i}+\varepsilon_{2S})/2$, where
$\varepsilon_{2S}=-3.4$ eV $=-0.125$ a.u. is the nonrelativistic binding
energy of atomic hydrogen in its metastable 2S-state. Thus, every kinetic
energy of the incoming electron corresponds to a kinetic energy of the
scattered electron determined from that relation so that the condition of
symmetric geometry always remains true. For the symmetric coplanar geometry,
we choose the following angular situation where $p_{i}$ is along the $z$-axis
($\theta_{i}=0^{\circ},\phi_{i}=0^{\circ}$). For the scattered electron, we
choose ($\theta_{f}=45^{\circ},\phi_{f}=0^{\circ}$) and for the ejected
electron we choose $\phi_{B}=180^{\circ}$ and the angle $\theta_{B}$ varies
differently from a figure to another. First of all, we will try to clarify the
limit between the relativistic and non-relativistic domains in the case of the
ionization of the hydrogen atom from its metastable 2S-state. Because,
compared to the results of the ground state, we found that there is a
significant difference between the two non-relativistic limit values. If the
hydrogen atom is ionized from its ground 1S-state, the non-relativistic limit
value is defined by the relativistic parameter
($\gamma=[1-(\beta/c)^{2}]^{-1/2}$) value of $1.0053$ which corresponds to an
incident electron kinetic energy of $2700$ eV [2]. We recall here that, in
atomic units, the kinetic energy is related to $\gamma$ parameter by the
following relation: $T_{i}=c^{2}(\gamma-1)$. It means that when the value of
the relativistic parameter $\gamma$ is greater than $1.0053$, a difference
between the relativistic and non-relativistic kinetic energies will appear. In
the case of the ionization of the hydrogen atom from its metastable 2S-state,
we found that the non-relativistic limit changed and increased slightly from
$2700$ eV until it reached the value of $4250$ eV.

Figure 6: The Three TDCSs of the (e, 2e) ionization of hydrogen 2S as a
function of the ejection angle $\theta_{B}$. The incident and the ejected
electron kinetic energies are $4250$ eV and $2123.3$ eV respectively and the
scattering angle $\theta_{f}=45^{\circ}$.

(a)

(b)

Figure 7: Same as in Fig. (6) but for the incident and the ejected electron
kinetic energies of (a) $10000$ eV and $4998.3$ eV and (b) $20000$ eV and
$9998.3$ eV respectively.

In Fig. (6), it can be seen that there is no difference at all between the
TDCSs (RPWBA, SRPWBA and NRPWBA) in the non-relativistic limit, since all the
curves of the three TDCSs are almost equal and identical. This figure
represents the first check of our models in particular in the non-relativistic
limit ($T_{i}=4250$ eV, $T_{f}=T_{B}=2123.3$ eV). But, we note that when we
pass this limit by raising the kinetic energy of the incoming electron to $10$
keV and $20$ keV, the non-relativistic TDCS begins to differ from the other
two TDCSs that remain equal as depicted in Fig. (7). Thus, the agreement
between the relativistic and nonrelativistic models is good from the
nonrelativistic limit and below ($T_{i}\leq 4250$ eV), but the disagreement
increases at high energies. It appears from Fig. (7) that at the relativistic
domain, the effects of the spin terms and the relativity begin to be
noticeable and that the non-relativistic formalism is no longer valid. We
notice from Fig. (6) that there is a parfait symmetry around the value
$\theta_{B}=45^{\circ}$ and the three TDCSs are all peaked in the vicinity of
the same value. We also note from Fig. (7) that the binary peak position in
the relativistic domain begins with a shift towards smaller values than
$45^{\circ}$. Comparing Figs. (6) and (7), it is clearly seen that the
magnitude of the binary peak decreases with increasing the kinetic energy of
the incident electron, which is the usual behavior in charged particle-impact
ionization of an atom. By the way, these two relativistic and
semirelativivstic TDCSs (RPWBA and SRPWBA) remain the same and equal,
regardless of the kinetic energy value of the incoming electron. For example,
we give in Fig. (8) a representation of the RPWBA and SRPWBA at high incident
kinetic energy of $511002$ eV. It appears to us through Fig. (8) that the two
TDCSs (RPWBA and SRPWBA), despite the different wave functions used to
describe the hydrogen atom in each of them, give the same results even at high
energies. This fact was proven and applied in more than one place when
studying the excitation or ionization of the hydrogen atom where it is
sufficient to use only the Darwin wave function, instead of the exact
analytical wave function, as a semirelativistic state to represent the atomic
hydrogen, and it was found that this gives nearly the same results as the
exact description only when the condition $Z\alpha\ll 1$ is fulfilled. This is
precisely the reason why, when studying theoretically asymmetric geometry in
the previous section, we were satisfied with only the treatment of the SRCBA
model without the corresponding one in the Relativistic Coulomb Born
Approximation (RCBA), so there is no need to complicate the calculation more
as long as both give the same results.

Figure 8: The two TDCSs in the relativistic regime as a function of the
ejection angle $\theta_{B}$. The incident and the ejected electron kinetic
energies are $511002$ eV and $255499.3$ eV respectively and the scattering
angle $\theta_{f}=45^{\circ}$. Figure 9: The TDCS, in RPWBA, as a function of
the angle $\theta_{B}$ of the ejected electron and the incident electron
kinetic energy $T_{i}$ varying from $1000$ eV to $5000$ eV. We have used the
condition $40^{\circ}\leq\theta_{f}=\theta_{B}\leq 50^{\circ}$.

For the sake of illustration, in a similar way to the 2D-plot, the contour
plot in Fig. (9) exhibits more information on the variation and the shape of
the TDCS in the RPWBA versus both incident electron kinetic energy and angle
$\theta_{B}$ in the binary coplanar geometry. For the variation with respect
to $\theta_{B}$, we observe that the TDCS decreases at small and large angles.
We see also that the TDCS presents a maximum only at the particular point of
$\theta_{B}=\theta_{f}=45^{\circ}$, and its magnitude at this particular point
decreases as the electron kinetic energy increases.

Figure 10: The TDCS, in RPWBA, as a function of the angle $\phi_{B}$ of the
ejected electron and the incident electron kinetic energy $T_{i}$ varying from
$1000$ eV to $5000$ eV for $\theta_{B}=\theta_{f}=45^{\circ}$.

Figure (10) shows that when the incident electron kinetic energy increases
provided that $\theta_{B}=\theta_{f}=45^{\circ}$, the peak of the TDCS
decreases and remains nearly around $\phi_{B}=180^{\circ}$. From Fig. (10), we
see that as the energy increases, the probability to observe the ejected
electron in the direction $\phi_{B}=180^{\circ}$ diminishes progressively.
Figure (10) also shows that for all figures in the symmetric geometry, in
which we choose $\phi_{B}$ to be constant, it must be equal to $180^{\circ}$
since the pick is clearly located at the same value.

Figure 11: The TDCS, in RPWBA, as a function of the angle $\theta_{B}$ of the
ejected electron and the angle $\theta_{f}$ of the scattered electron for
$T_{i}=5000$ eV and $T_{B}=2498.3$ eV.

Figure (11) represents the variations of TDCS in the RPWBA in terms of the
scattered and ejected electron angles at the energies $T_{i}=5000$ eV and
$T_{B}=2498.3$ eV. The purpose of including this figure is to show how
important the condition on both angles to be verified in the symmetric
coplanar geometry. Through this figure, it becomes clear to us that the TDCS
represents a maximum value at $\theta_{f}=\theta_{B}=45^{\circ}$ and begins to
decrease directly in the areas where this condition is broken. From here it
becomes evident that we must always respect this requirement and take it into
account when working within the symmetric coplanar geometry.

(a)

(b)

(c)

Figure 12: The two TDCSs in symmetric coplanar geometry as a function of the
ejection angle $\theta_{B}$ for the scattering angle $\theta_{f}=45^{\circ}$.
The incident and the ejected electron kinetic energies are (a) $T_{i}=10000$
eV and $T_{B}=4998.3$ eV, (b) $T_{i}=15000$ eV and $T_{B}=7498.3$ eV and (c)
$T_{i}=20000$ eV and $T_{B}=9998.3$ eV.

In Fig. (12), we plot the two TDCSs (SRCBA and NRCBA) with respect to the
symmetric coplanar geometry at relativistic energies. In the relativistic
regime, by increasing the value of the incident kinetic energy ($10$ keV, $15$
keV, $20$ keV), we notice the shift of the maximum of the TDCS in the SRCBA
towards smaller values than $\theta_{B}=45^{\circ}$, as well as the fact that
the SRCBA is always lower than the NRCBA.

(a)

(b)

Figure 13: The four TDCSs in symmetric coplanar geometry as a function of the
ejection angle $\theta_{B}$ for the scattering angle $\theta_{f}=45^{\circ}$.
The incident and the ejected electron kinetic energies are (a) $T_{i}=250$ eV
and $T_{B}=123.3$ eV and (b) $T_{i}=3250$ eV and $T_{B}=1623.3$ eV.

Recall that in the non-relativistic domain, we have already compared our
results for the two asymmetric and symmetric coplanar geometries. In Fig.
(13), we plot the RPWBA and the NRPWBA with the two TDCSs (SRCBA and NRCBA).
We see that the NRCBA and the SRCBA are the same and they give a lower TDCS
due to the fact that the ejected electron still feels Coulomb effect of the
residual ion as much as its kinetic energy ($T_{B}=123.3$ eV in this case) is
insufficient to cross the Coulomb barrier imposed by the residual ion. By
increasing the kinetic energy of the incident and ejected electrons
simultaneously (always checking the symmetric coplanar geometry), we find that
the ejected electron begins slowly to escape from the Coulomb effect until it
completely crosses it at kinetic energy $T_{B}=1623.3$ eV. In Fig. (13b),
there is a very good agreement between the four models and they produce the
same results as the use of the Coulomb wave function is no longer necessary.
For the sake of comparison with the experimental results available in the
literature for the total cross section, we attempted to calculate our total
cross section in the RPWBA by performing the numerical integration of the TDCS
(57) over all outgoing scattering angles and energies. After that, the total
cross section depends only on $E_{i}$, the incident electron kinetic energy.

$\begin{split}\bar{\sigma}_{tot}^{RPWBA}&=4\pi^{2}\int_{0}^{E_{B}^{max}/2}\frac{d\bar{\sigma}^{(RPWBA)}}{dE_{B}d\Omega_{B}d\Omega_{f}}dE_{B}\int_{0}^{\pi}\sin(\theta_{B})d\theta_{B}\int_{0}^{\pi}\sin(\theta_{f})d\theta_{f},\end{split}$
(61)

where $E_{B}^{max}$ is the maximum value of the ejected electron kinetic
energy. It is chosen, according to the kinetic energy conservation, to be
$E_{B}^{max}=E_{i}+\mathcal{E}_{b}(2S)$ where $\mathcal{E}_{b}(2S)$ is the
binding energy of the metastable 2S-state of atomic hydrogen given in (5). The
division by $2$ in the maximum limit of integral over $E_{B}$ is inspired by
the author Prasad [31]. In Table 1, we compare our results (61) with those
obtained theoretically by Mukherjee et al. [32] using a rigorous distorted
wave method in which the effects of both the initial and final channel
distortions are taken into account. Table 2 summarizes the comparison of our
results with the experimental ones measured by Defrance et al. using a crossed
beam method where an electron beam intersects a beam of atomic hydrogen at
$90^{\circ}$ [14]. The notable differences that appear between our results and
Mukherjee’s results or the experimental ones may be mainly due to two main
factors, the first of which is that the experimental results are obtained by
colliding a beam of electrons with a beam of hydrogen atoms, and thus
extracting the total cross section data in terms of the center of mass energy,
while we have assumed the hydrogen atom to be stable and bombarded it with an
electron. This means that we are inputting the kinetic energy of the incoming
electron. This difference in the input parameters will inevitably lead to
different final results. The second is that the maximum limit of the ejected
electron kinetic energy $E_{B}^{max}$ cannot be determined in an agreed and
controlled method. Each one and the method he thinks is logical and follows to
define this maximum value. This difference, along with the difference in the
theoretical approach applied to study the ionization process, may constitute
an obstacle to comparing the theoretical results with each other.
Nevertheless, all results will be acceptable as long as they are approximately
equal, at least, in order of magnitude about $10^{-16}\text{cm}^{2}$.

Table 1: Comparison of our results (61) with the theoretical ones of Mukherjee et al. [32] for total cross section $\text{e}^{-}-\text{H(2S)}$ ionization. Incident electron energy [eV] | Total cross section $[10^{-16}\text{cm}^{2}]$
---|---
Theoretical results | Our results
5.1 | 4.665 | 2.43
7.65 | 8.416 | 7.13
10.2 | 10.03 | 9.48
13.6 | 9.424 | 10.36
17 | 8.672 | 10.19
20.4 | 7.968 | 9.67
30.6 | 6.608 | 7.93
40.8 | 5.360 | 6.56
68 | 3.792 | 4.32
Table 2: Comparison of our results (61) with the experimental ones of Defrance et al. [14] for total cross section $\text{e}^{-}-\text{H(2S)}$ ionization. Center of mass energy [eV] | $\sigma_{tot}^{2S}$ $[10^{-16}\text{cm}^{2}]$ |  Incident electron energy [eV] |  $\bar{\sigma}_{tot}^{RPWBA}$ $[10^{-16}\text{cm}^{2}]$
---|---|---|---
Experimental results | Our results
6.3 | 5.94 | 6.3 | 4.31
8.3 | 8.75 | 8.3 | 6.98
10.3 | 10.5 | 10.3 | 8.39
12.3 | 7.67 | 12.3 | 8.99
14.3 | 8.06 | 14.3 | 9.13
18.3 | 7.56 | 18.3 | 8.81
23.3 | 6.22 | 23.3 | 8.06
25.3 | 6.92 | 25.3 | 7.75
31.8 | 6.63 | 31.8 | 6.82

##  4\. Conclusion

In this work, we have calculated the triple differential cross sections (TDCS)
for the ionization of hydrogen atom by electron impact in the metastable
2S-state for asymmetric and symmetric coplanar geometries. In the asymmetric
coplanar geometry, we have compared our nonrelativistic results with those of
other theories and found that the present model is very close to that obtained
by Coulomb wave function at the scattering angles $\theta_{f}=3^{\circ}$ and
$\theta_{f}=5^{\circ}$. In the symmetric coplanar geometry, a new
nonrelativistic limit value is determined theoretically to be $4250$ eV, which
is very different from that known for the ground state ($2700$ eV) [2].
Relativistic triple differential cross section have been evaluated within the
relativistic model (RPWBA) in the first Born approximation. The consistency of
this theoretical model is checked by taking the nonrelativistic limit.
Semirelativistic TDCS in the SRPWBA gives nearly the same results, regardless
of the kinetic energy of the incoming electron, as the RPWBA if the condition
$Z\alpha\ll 1$ is satisfied. It is shown that the nonrelativistic formalism is
no longer valid, in both geometries, for incident kinetic energies higher than
$10$ keV, due to the spin and relativistic effects which begin to appear at
high energies. Comparing our results for the two asymmetric and symmetric
coplanar geometries, we found that the use of the Coulomb wave function to
describe the ejected electron is no longer necessary as long as its kinetic
energy $T_{B}\geq 1623.3$ eV. The validation of this work requires an
experimental study. We hope that our results should serve as a motivation to
perform such collisions experiments in the future.

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