# Twisted electron impact single ionization coincidence cross-sections for
noble gas atoms

Nikita Dhankhar *  | Soham Banerjee *  | R. Choubisa *  |<EMAIL_ADDRESS>Department of Physics, Birla Institute of Technology and Science-Pilani, Pilani Campus, Pilani, Rajasthan, 333031, India

###### Abstract

We present the angular profiles of the triple differential cross-section
(TDCS) for the (e,2e) process on the noble gas atoms, namely He (1s), Ne (2s
and 2p), and Ar (3p) for the plane wave and the twisted electron impact. We
develop the theoretical formalism in the first-born approximation. The present
study compares the TDCS for different values of orbital angular momentum
number m and opening angles $\theta_{p}$ of the twisted electron beam with
that of the plane wave beam. In addition, we also investigate the TDCS for
macroscopic atomic targets to explore the influence of opening angle
$\theta_{p}$ of the twisted electron beam on the TDCS. Our results show that
the peaks in binary and recoil region shift from the momentum transfer
direction. The results also show that for larger opening angles the peaks for
p-type orbitals split into double-peak structures which are not observed in
the plane wave results for the given kinematics. The angular profiles for
averaged cross-section show the dependence of TDCS on the opening angles which
are significantly different from the plane wave TDCS.

##  I Introduction

The process of electron impact ionization (hereafter referred as an (e,2e)
process) of various atomic and molecular targets frequently finds an important
place in many fields. To name a few, electron impact ionization has practical
applications in plasma physics, atmospheric physics, radiation physics,
astronomy, and even biology [1, 2, 3, 4, 5]. A single electron is ejected from
the target in a coincident (e,2e) process when interacting with an incident
electron. The ejected and scattered electrons are detected with their momenta
fully resolved [6]. Experimental research in this field was first performed by
Erhardt et al. [7] and Amaldi et al. [8] independently. Since then, there has
been tremendous progress in the theoretical and experimental study of electron
impact ionization processes. The way we generally study an (e,2e) process is
through the triple differential cross-section (TDCS). The TDCS is proportional
to the probability of detecting the two outgoing electrons (the ejected
electron and the scattered electron) in coincidence with their momenta fully
resolved. Measuring and analyzing the TDCS of (e,2e) processes can provide
valuable insight into collision mechanisms and electronic structure of the
given target and serve as very strict tests on theoretical models of few-body
quantum dynamics.

After decades of progress in theoretical models aimed at describing (e,2e)
processes, we now have models that can fully explain the cross-sections of
simple targets such as H and He atoms [9, 10]. However, current theoretical
models still struggle to fully understand the dynamics for more complex
targets, such as molecules and multi-electron atoms. Among these complex
targets are inert gas atoms, such as Ne, Ar and Xe. Theoretical models for
atomic H and He that give good agreement with experimental results, such as
the BBK method [11], yield discrepancies for heavier atoms in the intermediate
and low energy regimes. Significant efforts have been made to develop
theoretical models that agree with experimental TDCS results of inert atoms.
Bell _et al._ [12] measured experimental TDCS results for Argon (3p) electrons
in co-planar geometry and compared them with theoretical results using
distorted wave Born approximation (DWBA). Brothers _et al._ [13] calculated
theoretical cross sections for Ne (2s), Ne (2p), Ar (2p), and Ar (3p) using
the factorized plane-wave Born approximation. The BBK and the DWBA$+R$-matrix
(close coupling) models were also used to describe TDCS of He (1s) and Ar (3p)
[14]. Kheifets _et al._ [15] used the DWBA-G (DWBA corrected by the Gamow
factor) model to include the post-collision interactions with the Gamow factor
for calculating TDCS of He (1s), Ne (2s), Ne (2p), and Ar (3p). Other
sophisticated models, like a convergent close-coupling, DWBA-G, and a hybrid
DWBA$+R$-matrix method, are also used for the calculation of the TDCS for Ne
and Xe ionization process [16]. Pflüger _et al._ [17] used the non-
perturbative B-spline R Matrix (BSR [18]) approach to calculate low incident
energy Ne (2p) TDCS. Ren _et al._ [19] used different models to study Ne (2p)
ionization at low energies, including second order DWBA-RM model, BSR model,
three-body distorted wave (3DW) model and distorted-wave Born approximation
with Ward-Macek (DWBA-WM) approximation. The TDCSs have also been calculated
for Ar (3p) and Ar (3s) atoms in the modified distorted wave formalism using
the second Born approximation [20]. Very recently, Gong _et al._ [21] used the
multi-center three distorted wave method(MCTDW) to calculate Ne(2p) TDCS at
low incident energies.

There have been several undertakings to improve the agreement between
experimental and theoretical data for the plane wave ionization of inert
atoms. However, it is still to be explored for the twisted beam impact
ionization of the same. Unlike plane waves, twisted electron beams carry a
well-defined orbital angular momentum (OAM) along the beam axis. This non-zero
OAM stems from the usual phase factor of $e^{im\phi}$. In addition to
quantized OAM, they also have a non-zero transverse linear momentum, and non-
uniform intensity in the transverse direction [22, 23]. Inspired by optical
vortex beams, many experimental groups have recently demonstrated the
production of electron vortex beams with several orders of phase singularity
and the corresponding angular momentum[24, 25]. Different experimental groups
have been successful in generating electron vortex with very high angular
momentum: up to $1000\hbar$ [26, 27]. These laboratory realizations of vortex
beams have stirred up a brand new area of research and boast promising
applications. There is a scope for applications in transmission electron
microscopy(TEM), fine probing of matter, surveying magnetic properties of
novel materials, investigating chiral properties of crystals, strong-field
ionization, transfer of angular momentum to nanoparticles, etc.[28, 29]. The
quantized OAM possessed by electron vortex beams acts as a new degree of
freedom in electron-target collision studies. Changing this new degree of
freedom may have interesting effects, and therefore, we must investigate the
impact of OAM on collision cross-sections.

Pioneering studies by Boxem et al. [30, 31, 28] inspired further research in
this direction. Serbo et al. [32] analyzed twisted electron scattering in the
relativistic regime and showed that the cross-section for bulk targets must be
integrated over all possible impact parameters. This integral eliminates any
dependence on the ‘m’ value or the amount of OAM associated with the twisted
beam. Matula et al. [33] studied the role of angular momentum in radiative
capture of twisted electrons by bare hydrogen-like ions. Using a theoretical
analysis of twisted Electron Energy-Loss Spectroscopy (EELS) from C60
fullerene, Schüler and Berakdar investigated the effects of OAM transfer on
plasmon generation [34]. Koshleva et al. calculated the cross-sections for
twisted electron Mott scattering by atomic targets in the fully relativistic
regime [35]. Using the first Born approximation, Maiorova et al. investigated
the elastic scattering of twisted electrons by $H_{2}$ molecule [36]. Harris
et al. [23] calculated the fully differential cross-section corresponding to
twisted electron wave ionization of hydrogen atom. The FDCS showed a shift in
recoil and binary peak positions due to the transverse momentum component.
Dhankhar _et al._ reported theoretical calculations of the five-fold
differential cross-section of double ionization of He atom by twisted waves in
$\theta$ variable mode and constant $\theta_{12}$-mode [37]. The study by
Mandal et al. (2020) showed the dependence of the Total Angular Momentum (TAM)
number (m) and opening angle ($\theta_{p}$) on the angular profile of the TDCS
and spin asymmetry for the relativistic electron impact ionization of the
heavy atomic targets [38]. Dhankhar _et al._ investigated the influence of the
OAM number ‘m’ and the opening angle $\theta_{p}$ single ionization cross-
sections of $H_{2}$ and $H_{2}O$ molecule by twisted electron waves
independently [39, 40].

In this communication, we extend our previous studies (for (e,2e) process on
$H_{2}$ and $H_{2}O$) of the differential cross-section by the twisted
electron beam on the inert gas atoms. We present the theoretical estimation
for the ionization of noble gas atoms, namely He, Ne (2s and 2p), and Ar (3p)
for the twisted electron beam. Our theoretical calculations are within the
first Born approximation framework for an incident plane wave electron beam
and an incident twisted electron beam. We describe the plane wave, Roothan-
Hartree-Fock wavefunction, Coulomb wave for the scattered electron, the bound
states of He, Ne, and Ar, and the ejected electron, respectively. We present
the theoretical formalism of our calculation of the TDCS in Sec.II. We present
our results of the TDCS for the different outer orbitals of the atoms for
different parameters of the twisted electron beam in Sec.III. Finally, we
conclude our paper in the Sec.IV. Atomic units are used throughout the paper
unless otherwise stated.

##  II Theoretical Formalism

In this study, we have calculated the TDCS for (e,2e) process on inert gas
atoms :

$e_{i}^{-}+X\rightarrow e_{s}^{-}+e_{e}^{-}+X^{+}.$ (1)

here, $e_{i}^{-}$, $e_{s}^{-}$, $e_{e}^{-}$, X and $X^{+}$ represents the
incident, scattered, the ejected electron, the target atom and the ionized
target respectively. The theory used here closely follows the approach used in
our previous studies [39, 40]. We have carried out the theoretical analysis in
the framework of the first Born approximation. We have also neglected the
exchange effects between the incident/scattered electron with the
bound/ejected electron here since the incident/scattered electron is faster
than the bound/ejected electron ($E_{s}>>E_{e}$).

The triple differential cross section for an (e,2e) process of an inert gas
can be written as,

$\frac{d^{3}\sigma}{d\Omega_{e}d\Omega_{s}dE_{e}}=(2\pi)^{4}\frac{k_{e}k_{s}}{k_{i}}|T^{pw}_{fi}(\mathbf{q})|^{2},$
(2)

here, $d\Omega_{e}$ and $d\Omega_{s}$ denote differential intervals of the
solid angles of the ejected and scattered electrons respectively. $dE_{e}$
denotes the energy interval of the ejected electron.
$\mathbf{k}_{i},\mathbf{k}_{s},$ and $\mathbf{k}_{e}$ are the momentum vectors
of the incident, scattered and ejected electrons respectively. The plane wave
scattering amplitude is given by,

$T^{pw}_{fi}(\mathbf{q})=\langle\phi_{f}|V|\phi_{i}\rangle,$ (3)

where $\mathbf{q}=\mathbf{k}_{i}-\mathbf{k}_{s}$ is the momentum transfer
vector, $\phi_{i}$ is the initial state wave function of the system, and
$\phi_{f}$ is the final state wave function. $V$ is the interaction potential
between the incident electron and the multi-electron target atom. In this
communication we use the frozen-core approximation, in which the problem of N
(total number of electrons in the target atom) electrons can be reduced to one
active electron problem. In this approximation, we assume that only one of the
target electrons (from the outer orbital of the target atom) participate in
the ionization process and is ejected in the final channel, while the other
electrons remain as frozen. The interaction potential under the approximation
is then defined as,

$V=\frac{-1}{r_{1}}+\frac{1}{r_{12}},$ (4)

In the above equation, the first term represents the Coulomb interaction of
the incident electron with the nucleus of the inert gas atom, with $r_{1}$ as
the distance from the nucleus to the incident electron. The second term is the
Coulomb interaction between the incident electron and the participating bound
electron, with $r_{12}$ as the separation between the two electrons.
For calculating the transition matrix element, we have used the following
assumptions:

  * •

The incident and the scattered electrons are both treated as plane waves.

  * •

The Roothaan-Hartree-Fock wave functions represent the participating bound
electron wave function of the target atom. These atomic wave functions are
given by a combination of Slater orbitals using Clementi-Roetti (CR)
coefficients [41], where Slater orbitals take the following general form [6],

$\chi_{nlm}(\mathbf{r})=N(n,\alpha)r^{n-1}e^{-\alpha r}Y_{lm}(\theta,\phi).$
(5)

Here n, l, m are the quantum numbers of the atomic orbital, $\alpha$ is the
orbital exponent, $Y_{lm}(\theta,\phi)$ are the complex spherical harmonics,
and

$N(n,\alpha)=\frac{(2\alpha)^{n+\frac{1}{2}}}{[(2n)!]^{\frac{1}{2}}}$ (6)

is the normalization constant.

  * •

The ejected electron is described by a Coulomb wave,
$\Psi_{k_{e}}^{-}(\mathbf{r}_{2})$. This way, the long-range effects of the
residual ion on the slow electron are taken into account.

We can then write the initial wave function as;

$\phi_{i}=\chi_{\mathbf{k}_{i}}(\mathbf{r}_{1})\Phi_{j}(\mathbf{r}_{2}),$ (7)

where $\chi_{\mathbf{k}_{i}}(\mathbf{r}_{1})$ is the normalized incident
electron plane wave, and $\Phi_{j}(\mathbf{r}_{2})$ is the $j^{th}$ Hartree-
Fock atomic orbital from CR coefficients of the target atom and is described
as;

$\Phi_{j}(\mathbf{r}_{2})=\sum_{i=1}^{N}c_{i}\chi_{n_{i}l_{i}m_{i}}(\mathbf{r}_{2}),$
(8)

where i is the $i^{th}$ basis function, $c_{i}$ is the expansion coefficient
of the atomic orbital for the $i^{th}$ basis function and
$\chi_{n_{i}l_{i}m_{i}}(\mathbf{r}_{2})$ is given by equation 5. Similarly,
the final state wave function can be written as

$\phi_{f}=\chi_{\mathbf{k}_{s}}(\mathbf{r}_{1})\psi_{\mathbf{k}_{e}}^{-}(\mathbf{r}_{2}),$
(9)

where $\chi_{\mathbf{k}_{s}}(\mathbf{r}_{1})$ is approximated by plane wave,
and $\psi_{\mathbf{k}_{e}}^{-}(\mathbf{r}_{2})$ is approximated by the Coulomb
wave, described as;

$\psi_{\mathbf{k}_{e}}^{-}(\mathbf{r}_{2})=\Gamma(1-i\eta)e^{-\frac{\pi\eta}{2}}\frac{e^{i\mathbf{k}_{e}.\mathbf{r}_{2}}}{(2\pi)^{3/2}}{}_{1}F_{1}(i\eta,1,-i(k_{e}r_{2}-\mathbf{k}_{e}.\mathbf{r}_{2})).$
(10)

Here, $\Gamma$ is the gamma function, $\eta=\frac{-Z}{k_{e}}$ is the
Sommerfeld parameter ( Z = Nuclear charge = 1 in our case because of the
frozen core approximation), and ${}_{1}F_{1}$ is the confluent hypergeometric
function of the first kind.
The transition matrix element can be simplified by analytically performing the
integral over incident electron coordinate $r_{1}$ [42] as follows;

$\int\frac{e^{i\mathbf{q.r}_{1}}}{|\mathbf{r}_{1}-\mathbf{r}_{2}|}d^{3}r_{1}=\frac{4\pi}{q^{2}}e^{i\mathbf{q.r}_{2}}.$
(11)

The transition matrix element is then given by the following expression

$T^{pw}_{fi}(\mathbf{q})=\frac{-2}{q^{2}}\langle\Psi^{-}_{\mathbf{k}_{e}}|e^{i\mathbf{q.r}_{2}}-1|\Phi_{j}(\mathbf{r}_{2})\rangle.$
(12)

The extension to twisted wave scattering in this formalism is straightforward.
We change the incident electron wave function from a plane wave to a twisted
wave propagating along z-axis. Similar to our previous studies, we have used
the Bessel beam [31] to represent the incident electron that carries quantized
angular momentum. In this case, the incident momentum vector $\mathbf{k}_{i}$
has a transverse component and is given by the following expression.

$\mathbf{k}_{i}=(k_{i}\sin{\theta_{p}}\cos{\phi_{p}})\hat{x}+(k_{i}\sin{\theta_{p}}\sin{\phi_{p}})\hat{y}+(k_{i}\cos{\theta_{p}})\hat{z}.$
(13)

Here, $\theta_{p}$ and $\phi_{p}$ are the polar and azimuthal angles of the
momentum vector respectively. $\mathbf{k}_{i}$ carves out the surface of a
cone when we change $\phi_{p}$. The longitudinal component of the momentum
$\mathbf{k}_{iz}$ is fixed, but the transverse momentum component
$\mathbf{k}_{i\bot}$ has a direction that depends on $\phi_{p}$. This means
that the direction of incident momentum is not well defined. However, the
magnitude of the transverse momentum is fixed, and is given by
$|\mathbf{k_{i\bot}}|=\varkappa=\sqrt{(k_{i})^{2}-(k_{iz})^{2}}$. The opening
angle of the cone can be calculated as
$\theta_{p}=\arctan(\frac{k_{i\bot}}{k_{iz}})$. The incident electron wave
function can be written as follows;

$\chi^{tw}_{\varkappa
m}(\mathbf{r_{1}})=\int^{\infty}_{0}\frac{dk_{i\bot}}{2\pi}\int^{2\pi}_{0}\frac{d\phi_{p}}{2\pi}a_{\varkappa
m}(k_{i\bot})e^{i\mathbf{k_{i}.r_{1}}}e^{-i\mathbf{k_{i}.b}}.$ (14)

This can be interpreted as a superposition of plane waves with amplitude
$a_{\varkappa m}(k_{i\bot})$,where this amplitude is given by;

$a_{\varkappa
m}(k_{i\bot})=(-i)^{m}e^{im\phi_{p}}\delta(|\mathbf{k}_{i\bot}|-\varkappa).$
(15)

The phase singularity of the $e^{im\phi_{p}}$ is responsible for the quantized
angular momentum content of the twisted wave, and the delta factor ensures
that the magnitude of the transverse momentum is well defined. Also, the
addition of the $e^{-i\mathbf{k}_{i}.\mathbf{b}}$ factor accounts for the
transverse orientation of the incident beam with respect to the target. Here,
$\mathbf{b}$ is the impact parameter vector measured with respect to the axis
of the incident wave. Unlike the plane wave, the Bessel beam has concentric
rings of high and low-intensity in the transverse direction [43]. Therefore,
we need to account for the orientation of the target with respect to the
incident beam.

Figure 1: TDCS as a function of ejected electron angle $\theta_{e}$ for the
plane wave (e,2e) process on He, Ne (2s and 2p shell) and Ar atoms in the co-
planar asymmetric geometry. The plane wave results of our theoretical model
are represented by black solid line and experimental results [15] by full
circles. The kinematics used here is : $E_{s}$ = 500eV, $E_{e}$ = 37eV and
74eV and $\theta_{s}$ = 6°. The arrows indicate the direction of plane wave
momentum transfer ($\theta_{\mathbf{q}}$) and recoil direction
($\theta_{-\mathbf{q}}$) for this and subsequent figures.

When we replace the incident plane wave by the twisted wave, we get the
following expression for twisted wave matrix element

$T^{tw}_{fi}(\varkappa,\mathbf{q,b})=(-i)^{m}\int^{2\pi}_{0}\frac{d\phi_{p}}{2\pi}e^{im\phi_{p}-i\mathbf{k.b}}T^{pw}_{fi}(\mathbf{q}).$
(16)

Therefore, we can evaluate the twisted wave matrix element $T^{tw}_{fi}$ in
terms of the plane wave matrix element $T^{pw}_{fi}$ [37]. The key difference
here is that the momentum transfer vector
$\mathbf{q}=\mathbf{k}_{i}-\mathbf{k}_{s}$ has to be calculated using the new
twisted wave momentum vector $\mathbf{k}_{i}$ from equation 13. We can then
write

$q^{2}=k^{2}_{i}+k^{2}_{s}-2k_{i}k_{s}\cos(\theta),$ (17)

where,

$\cos(\theta)=\cos(\theta_{p})\cos(\theta_{s})+\sin(\theta_{p})\sin(\theta_{s})\cos(\phi_{p}-\phi_{s}).$
(18)

$\theta_{s}$ and $\phi_{s}$ are the polar and azimuth angles of the scattered
electron momentum vector $\mathbf{k}_{s}$, which is described by a plane wave
in our treatment. Unlike the plane wave, the momentum transfer of a twisted
wave is not constant for a particular direction of $\mathbf{k}_{s}$, and
depends on the azimuthal angle of the incident wave vector $\mathbf{k}_{i}$.
This inherent uncertainty of momentum transfer direction for a twisted wave is
accounted for by taking an integral over the azimuthal angle $\phi_{p}$ in
equation 16.

We can now calculate the TDCS for twisted waves using equation 16 and equation
12 for the target fixed with respect to the incident beam. However, since
scattering experiments are usually characterized by a uniform distribution of
scattering centers ( inert atoms in our case) over a certain volume, we cannot
assume a fixed value of impact parameter $\mathbf{b}$. Instead, we need to
average over all possible values of $\mathbf{b}$ for meaningful results that
can be compared with experimental results. The details of averaging over all
possible orientations can be found in [32, 23], and average TDCS can be
written as ;

$(TDCS)_{av}=\frac{1}{2\pi\cos\theta_{p}}\int^{2\pi}_{0}d\phi_{p}\frac{d^{3}\sigma(\mathbf{q})}{d\Omega_{e}d\Omega_{s}dE_{e}}.$
(19)

Figure 2: Angular profile of the TDCS as a function of the ejection angle
$\theta_{e}$ for the twisted electron wave (e,2e) process on He atom in the
co-planar asymmetric geometry. The kinematics is $E_{s}$ = 500eV, $E_{e}$ =
37eV (left column) and 74eV (right column) and $\theta_{s}$ = 6°. The black,
orange, green, blue and red curves represent the TDCS for plane wave,
$\theta_{p}$ = 1°, 6°, 15° and 20°. Sub-figures (a) and (d) are for m = 1, (b)
and (e) for m = 2 and (c) and (f) are for m = 3.

Here, $\frac{d^{3}\sigma(\mathbf{q})}{d\Omega_{e}d\Omega_{s}dE_{e}}$ is the
TDCS for a particular value of $\mathbf{q}$. Interestingly, averaging over
impact parameter $\mathbf{b}$ makes the TDCS independent of the OAM value
‘m’[32]. However, this does not mean that twisted wave TDCS is the same as
plane wave TDCS since the transverse momentum component of twisted waves plays
a direct role in the momentum transfer. The cross-section now depends on the
opening angle $\theta_{p}$ of the incident twisted electron beam.

##  III Results and discussions

Figure 3: Same as figure 2 except that the TDCSs are plotted for (e,2e) on Ne
(2s) atom.

We present the results of our calculations for the TDCS for noble gas atoms
viz., Helium (1s), Neon (2s and 2p orbitals), and Argon (3p) by a twisted
electron beam impact in this section. We benchmark our theoretical
calculations with the existing experimental data for the plane wave. We
present the results of the TDCS for twisted electron beam to study the effects
of different parameters of the beam on TDCS.

Figure 4: Same as figure 2 except that the TDCSs are plotted for (e,2e) on Ne
(2p) atom.

We compare our twisted electron beam TDCS with that of the plane wave TDCS for
different values of orbital angular momentum (OAM) number m and opening angle
$\theta_{p}$. For the given m, we compare the TDCS’s profiles for the
different values of $\theta_{p}$ , viz. 1°, 6°, 15° and 20°. The kinematics we
have used here is similar to Kheifets et al. [15] for the plane wave
ionization; scattered energy ($E_{s}$) = 500eV, ejected energy ($E_{e}$) =
37eV and 74eV and scattering angle ($\theta_{s}$) = 6° in the co-planar
asymmetric geometry.

Table 1: Scaling factors for the twisted electron TDCS of He (1s) atom.

m |  $E_{e}$ = 37eV |  $E_{e}$ = 74eV
---|---|---
1 |  TDCS ($\theta_{p}$ = 1°)$\times$ 150
TDCS ($\theta_{p}$ = 6°)$\times$ 10
TDCS ($\theta_{p}$ = 15°)$\times$ 20
TDCS ($\theta_{p}$ = 20°)$\times$ 30 |  TDCS ($\theta_{p}$ = 1°)$\times$ 150
TDCS ($\theta_{p}$ = 16°)$\times$ 10
TDCS ($\theta_{p}$ = 15°)$\times$ 20
TDCS ($\theta_{p}$ = 20°)$\times$ 30
2 |  TDCS ($\theta_{p}$ = 1°)$\times$ 7$\times 10^{3}$
TDCS ($\theta_{p}$ = 6°)$\times$ 30
TDCS ($\theta_{p}$ = 15°)$\times$ 50
TDCS ($\theta_{p}$ = 20°)$\times$ 80 |  TDCS ($\theta_{p}$ = 1°)$\times$ 1$\times 10^{4}$
TDCS ($\theta_{p}$ = 6°)$\times$ 50
TDCS ($\theta_{p}$ = 15°)$\times$ 40
TDCS ($\theta_{p}$ = 20°)$\times$ 20
3 |  TDCS ($\theta_{p}$ = 1°)$\times$ 5$\times 10^{5}$
TDCS ($\theta_{p}$ = 6°)$\times$ 50
TDCS ($\theta_{p}$ = 15°)$\times$ 100
TDCS ($\theta_{p}$ = 20°)$\times$ 150 |  TDCS ($\theta_{p}$ = 1°)$\times$ 1$\times 10^{6}$
TDCS ($\theta_{p}$ = 6°)$\times$ 150
TDCS ($\theta_{p}$ = 15°)$\times$ 70
TDCS ($\theta_{p}$ = 20°)$\times$ 40

###  III.1 Angular profiles of the TDCS for plane wave

Figure 1, shows the TDCS as a function of the ejected electron’s angle
($\theta_{e}$) for the plane wave electron impact ionization in the co-planar
asymmetric geometry for different noble gas atoms. The momentum transfer and
the recoil direction are represented by the arrows ($\theta_{\mathbf{q}}$ and
$\theta_{-\mathbf{q}}$) in figure 1 and subsequent figures. The left and right
columns show the TDCS for $E_{e}$ = 37eV and 74eV respectively in all the
figures. We benchmark the results of our theoretical calculations of the TDCS
for the plane wave with the experimental data [15] to validate our theoretical
model for the twisted electron impact ionization.

Figure 5: Same as figure 2 except that the TDCSs are plotted for (e,2e) on Ar
(3p) atom.

From figure 1, we observe that our theoretical model reproduces the
experimental results reasonably well in the binary region for all the cases.
The theory, however, underestimates the recoil peak for most of the cases.
This may be explained by the fact that our theoretical model uses a 1-Coulomb
wavefunction (1CW) for the ejected electron. The experimental results are on a
relative scale, so we have scaled them to compare the results of our
theoretical model in the binary peak region. The plane wave TDCS profile
represents a binary peak and a recoil peak along the momentum transfer
directions $\theta_{\mathbf{q}}$ and $\theta_{-\mathbf{q}}$ respectively for
both the kinematics. For the plane wave ionization, we also observe that the
magnitude of the TDCS decreases with an increase in the ejected electron
energy.

Table 2: Scaling factors for the twisted electron TDCS of Ne (2s) atom.

m |  $E_{e}$ = 37eV |  $E_{e}$ = 74eV
---|---|---
1 |  TDCS ($\theta_{p}$ = 1°)$\times$ 150
TDCS ($\theta_{p}$ = 6°)$\times$ 10
TDCS ($\theta_{p}$ = 15°)$\times$ 20
TDCS ($\theta_{p}$ = 20°)$\times$ 30 |  TDCS ($\theta_{p}$ = 1°)$\times$ 150
TDCS ($\theta_{p}$ = 6°)$\times$ 10
TDCS ($\theta_{p}$ = 15°)$\times$ 20
TDCS ($\theta_{p}$ = 20°)$\times$ 30
2 |  TDCS ($\theta_{p}$ = 1°)$\times 10^{4}$
TDCS ($\theta_{p}$ = 6°)$\times$ 30
TDCS ($\theta_{p}$ = 15°)$\times$ 40
TDCS ($\theta_{p}$ = 20°)$\times$ 100 |  TDCS ($\theta_{p}$ = 1°)$\times 15\times 10^{3}$
TDCS ($\theta_{p}$ = 6°)$\times$ 80
TDCS ($\theta_{p}$ = 15°)$\times$ 50
TDCS ($\theta_{p}$ = 20°)$\times$ 50
3 |  TDCS ($\theta_{p}$ = 1°)$\times$ 1$\times 10^{6}$
TDCS ($\theta_{p}$ = 6°)$\times$ 150
TDCS ($\theta_{p}$ = 15°)$\times$ 200
TDCS ($\theta_{p}$ = 20°)$\times$ 300 |  TDCS ($\theta_{p}$ = 1°)$\times$ 5$\times 10^{6}$
TDCS ($\theta_{p}$ = 6°)$\times$ 500
TDCS ($\theta_{p}$ = 15°)$\times$ 150
TDCS ($\theta_{p}$ = 20°)$\times$ 150

Table 3: Scaling factors for the twisted electron TDCS of Ne (2p) atom.

m |  $E_{e}$ = 37eV |  $E_{e}$ = 74eV
---|---|---
1 |  TDCS ($\theta_{p}$ = 1°)$\times$ 120
TDCS ($\theta_{p}$ = 6°)$\times$ 8
TDCS ($\theta_{p}$ = 15°)$\times$ 20
TDCS ($\theta_{p}$ = 20°)$\times$ 30 |  TDCS ($\theta_{p}$ = 1°)$\times$ 150
TDCS ($\theta_{p}$ = 6°)$\times$ 8
TDCS ($\theta_{p}$ = 15°)$\times$ 20
TDCS ($\theta_{p}$ = 20°)$\times$ 30
2 |  TDCS ($\theta_{p}$ = 1°0$\times$ 600
TDCS ($\theta_{p}$ = 6°)$\times$ 15
TDCS ($\theta_{p}$ = 15°)$\times$ 60
TDCS ($\theta_{p}$ = 20°)$\times$ 150 |  TDCS ($\theta_{p}$ = 1°)$\times$ 1500
TDCS ($\theta_{p}$ = 6°)$\times$ 25
TDCS ($\theta_{p}$ = 15°)$\times$ 60
TDCS ($\theta_{p}$ = 20°)$\times$ 80
3 |  TDCS ($\theta_{p}$ = 1°)$\times$ 6$\times 10^{4}$
TDCS ($\theta_{p}$ = 6°)$\times$ 40
TDCS ($\theta_{p}$ = 15°)$\times$ 180
TDCS ($\theta_{p}$ = 20°)$\times$ 350 |  TDCS ($\theta_{p}$ = 1°)$\times$ 1$\times 10^{5}$
TDCS ($\theta_{p}$ = 6°)$\times$ 200
TDCS ($\theta_{p}$ = 15°)$\times$ 200
TDCS ($\theta_{p}$ = 20°)$\times$ 300

###  III.2 Angular profiles of the TDCS for the twisted electron wave

Table 4: Scaling factors for the twisted electron TDCS of Ar (3p) atom.

m |  $E_{e}$ = 37eV |  $E_{e}$ = 74eV
---|---|---
1 |  TDCS ($\theta_{p}$ = 1°)$\times$ 150
TDCS ($\theta_{p}$ = 6°)$\times$ 6
TDCS ($\theta_{p}$ = 15°)$\times$ 20
TDCS ($\theta_{p}$ = 20°)$\times$ 30 |  TDCS ($\theta_{p}$ = 1°)$\times$ 150
TDCS ($\theta_{p}$ = 6°)$\times$ 10
TDCS ($\theta_{p}$ = 15°)$\times$ 20
TDCS ($\theta_{p}$ = 20°)$\times$ 30
2 |  TDCS ($\theta_{p}$ = 1°)$\times$ 1200
TDCS ($\theta_{p}$ = 6°)$\times$ 20
TDCS ($\theta_{p}$ = 15°)$\times$ 80
TDCS ($\theta_{p}$ = 20°)$\times$ 200 |  TDCS ($\theta_{p}$ = 1°)$\times$ 1800
TDCS ($\theta_{p}$ = 6°)$\times$ 30
TDCS ($\theta_{p}$ = 15°)$\times$ 50
TDCS ($\theta_{p}$ = 20°)$\times$ 90
3 |  TDCS ($\theta_{p}$ = 1°)$\times$ 1$\times 10^{5}$
TDCS ($\theta_{p}$ = 6°)$\times$ 50
TDCS ($\theta_{p}$ = 15°)$\times$ 100
TDCS ($\theta_{p}$ = 20°)$\times$ 400 |  TDCS ($\theta_{p}$ = 1°)$\times$ 2$\times 10^{5}$
TDCS ($\theta_{p}$ = 6°)$\times$ 200
TDCS ($\theta_{p}$ = 15°)$\times$ 150
TDCS ($\theta_{p}$ = 20°)$\times$ 200

In this section, we present the results of our calculations for TDCS with the
twisted electron beam for $E_{s}$ = 500eV, $E_{e}$ = 37eV and 74eV and
$\theta_{s}$ = 6° in the co-planar asymmetric geometrical mode for He, Ne (2s
and 2p sub-shell) and Ar (3p). Here, we study the effect of twisted electron
beam on the TDCS by varying OAM number m as well as the opening angle
$\theta_{p}$. Figures 2-5 represent the TDCS for an incident twisted electron
beam for the noble gas atoms for different values of m and $\theta_{p}$. In
figures 2-6, the black, orange, green, blue and red curves represents the TDCS
for plane wave (pw), $\theta_{p}$ = 1°, 6°, 15° and 20° respectively. In the
figures 2-5, frames (a),(d); (b),(e) and (c),(f) depict the TDCS for m = 1, 2
and 3 respectively for the ejected electron energies 37eV and 74eV.

We have used different scaling factors to compare the TDCS for the twisted
electron beam with the plane wave TDCS as shown in tables 1-4. For all the
cases considered here, the magnitude of TDCS is the lowest for the smallest
opening angle i.e. $\theta_{p}$ = 1° (see scaling factors for $\theta_{p}$ =
1° in tables 1-4). The angular profiles for the TDCS at $\theta_{p}$ = 1°and
for m = 1 and 3, predict peaks in the binary and recoil regions for He and Ne
(2s), and a peak in the binary region only for Ne(2p) and Ar (3p) (see orange
curves for m = 1 and 3 in figures 2-5). The peaks, however, are shifted from
the momentum transfer direction (see orange curves in figures 2-5 for m = 1
and 3 around the arrows). At the same opening angle and m = 2, we observe the
peaks along the momentum transfer directions for He, Ne(2p) and Ar (see orange
curves in figures 2, 4, 5 for m = 2 around the arrows). However, for Ne (2s)
at $\theta_{p}$ = 1° and m = 2, we observe peaks in the forward and backward
regions (see orange curves in figure 3 (b) and (e)).

When the opening angle is same as the scattering angle i.e., $\theta_{p}$ =
$\theta_{s}$, we observe peaks in the forward and backward direction for most
of the cases (see green curves in the figures 2-5 around $\theta_{e}$ = 0°(or
360°) and 180°). However, for He and Ne (2s), we observe peaks in the forward
and perpendicular directions for m = 1 only (see green curves in the figures 2
(a) and (d) and 3 (a) and (d) around $\theta_{e}$ = 0°(or 360°) and 90°). For
all the cases, a minimum is observed around the plane wave momentum transfer
direction. Similar observation has also been accounted in our previous
communication for the water molecule [40] and by Harris et al. for the
hydrogen atom [23].

Figure 6: (TDCS)av as a function of the ejected electron angle $\theta_{e}$
for the plane wave and twisted electron beam for different opening angles (as
shown in the frames) respectively. The kinematics used here is same as used in
figure 2-5. We have scaled up the magnitude of the (TDCS)av by a factor of 5
for Ne (2p) and Ar (3p) for all the values of $\theta_{p}$.

On further increasing the opening angle of the incident beam to 15°, some
interesting results are observed. The magnitude of the TDCS decreases as
$\theta_{p}$ is increased from 6°. For the s-orbitals (He and Ne (2s)), the
angular profile of the TDCS at $\theta_{p}$ = 15° represents a two peak
structure around the perpendicular directions with the dominant peak close to
$\theta_{e}$ = 270° and a shallow peak around $\theta_{e}$ = 90° (see blue
curves in figures 2 and 3), except for $E_{e}$ = 74eV and m = 1. For $E_{e}$ =
74eV and m = 1, the dominant peak is around 90° (see blue curve in figure 2(d)
and 3(d) around $\theta_{e}$ = 90° and 270°). For Ne (2p) and Ar (3p) at
$\theta_{p}$ = 15°, the peak around $\theta_{e}$ = 90° splits and we observe a
double-peak structure which is a characteristic of the p-dominant atomic
orbitals. The double-peak structure in the TDCS is observed only for m = 1 for
$E_{e}$ = 37eV and for all values of m in the TDCS for $E_{e}$ = 74eV for Ne
(2p) and Ar (3p) (see blue curves in figures 4 and 5).

For an opening angle $\theta_{p}$ = 20°, the angular profiles of the
ionization cross-sections is somewhat similar to those observed for
$\theta_{p}$ = 15°. We observe peaks approximately around $\theta_{e}$ = 90°
and 270° as observed earlier for He and Ne (2s) (see red curves in figures 2
and 3). For Ne (2p) and Ar (3p), at lower energy of the ejected electron i.e.
$E_{e}$ = 37eV, we observe a shallow peak and a double-peak structure (similar
to the one observed at $\theta_{p}$ = 15° for the same atoms) around
$\theta_{e}$ = 90° and 270° respectively for most of the cases. However, for
$E_{e}$ = 74eV and m = 1, we observe a dominant peak and a shallow double-peak
structure around $\theta_{e}$ = 270° and 90° respectively. For the same
ejected electron energy and m = 2 and 3, we observe a dominant peak and a
shallow peak in the perpendicular directions (see red curves in figures 4-5
around $\theta_{e}$ = 90° and 270°). Due to the additional transverse
momentum, we observe peak splitting for the twisted electron TDCS at higher
opening angles only, i.e., for $\theta_{p}$ = 15° and 20° for the Ne (2p) and
Ar (3p). The present calculations of the TDCS for twisted electron wave on the
noble gas atoms clearly show the dependence of TDCS on both m and
$\theta_{p}$.

###  III.3 Angular profiles for the (TDCS)av for macroscopic atomic targets

In figure 6, we present the results of our calculations for the TDCS averaged
over the impact parameter b, (TDCS)av, as a function of the ejected electron
angle $\theta_{e}$ for He, Ne (2s and 2p shells) and Ar (3p) noble gases in
the co-planar asymmetric geometry. As shown by the equation 19, the (TDCS)av
depends only on the opening angle $\theta_{p}$ of the incident twisted
electron beam. We keep the same kinematics as used in the earlier figures. We
compare the results of the averaged cross-section ((TDCS)av) with the plane
wave TDCS. We present the angular profiles for $\theta_{p}$ = 1°, 6°, 15° and
20°. For a better comparison of the (TDCS)av with the plane wave TDCS, we have
scaled the (TDCS)av for $\theta_{p}$ = 15° and 20° by a factor of 5. We have
not used any scaling factors for $\theta_{p}$ = 1° and 6°. From figure 6, we
observe that for the smallest opening angle, i.e. $\theta_{p}$ = 1°, the
angular profile of the (TDCS)av is similar to that of the plane wave TDCS for
all the atoms (see orange curves in the figure 6). For $\theta_{p}$ =
$\theta_{s}$ = 6°, we observe peaks in the forward and backward direction with
a small peak around the plane wave momentum transfer direction for He only
(see green curves around the arrows in figure 6 (a) and (e)). For higher
opening angles, like 15° and 20°, we observe that the angular profiles
represent a two peak structure with peaks around the perpendicular directions
($\theta_{e}$ = 90° and 270°). For both the opening angles, the dominant peak
is approximately around $\theta_{e}$ = 270° and the shallow peak is located
close to $\theta_{e}$ = 90° except for He, wherein for $E_{e}$ 74eV and
$\theta_{p}$ = 15° the dominant and shallow peak is located around
$\theta_{e}$ = 90° and 270° which is in contrast to all the other cases for
$\theta_{p}$ = 15°. We also observe that the magnitude of (TDCS)av decreases
with an opening angle. It can be seen from equations 2 and 12 that the plane
wave TDCS is inversely proportional to the quad of momentum transfer
magnitude. Thus, in computing the (TDCS)av, the plane wave TDCS from azimuthal
angles with smaller momentum transfer magnitude will dominate the average,
thus resulting in a shift of binary peak to the recoil direction for higher
$\theta_{p}$ (for more details see [44]).

##  IV Conclusion

In this paper, we have presented the theoretical study of the triple
differential cross-sections (TDCS) for an (e,2e) process on different noble
gas atoms (He, Ne (2s and 2p), and Ar (3p)) by the twisted electron beam. We
studied the angular distributions of the TDCS for the co-planar asymmetric
geometry in the first Born approximation for both the plane and twisted
electron beam. We have studied the effect of the OAM number, m and the opening
angle $\theta_{p}$ of the incident twisted electron beam on the TDCS. We have
benchmarked our theoretical results with the experimental data for the plane
wave electron beam.
For the (e,2e) ionization of the noble gas atoms by twisted electron impact,
the angular profile of the TDCS is altered from the plane wave case. We also
observe that the magnitude of the TDCS for twisted electron wave impact is
smaller. For the plane waves, the angular profile of the TDCS peaks around
$\theta_{\mathbf{q}}$ (binary peak) and $\theta_{-\mathbf{q}}$ (recoil peak).
However, for the twisted electron beam, due to the additional transverse
momentum and the phase (OAM (m)) dependence of the incident twisted electron
beam, the angular profiles of the TDCS for twisted electron case the peaks are
shift from the momentum transfer directions for a smaller opening angle. A
minimum occurs around the plane wave momentum transfer direction when
$\theta_{p}$ = $\theta_{s}$. For higher opening angles, like 15° and 20°, we
observe peaks in the perpendicular directions. At these opening angles, we
also observe that the peaks split into double-peak structures for p-type
orbitals (Ne (2p) and Ar (3p)). In addition, we discuss the (TDCS)av (averaged
over the impact parameter b) as a function of the opening angle $\theta_{p}$
of the twisted electron beam. For a macroscopic target, the angular profiles
of (TDCS)av significantly depend on the opening angle ($\theta_{p}$) of the
twisted electron beam. At smaller opening angle, like $\theta_{p}$ = 1°, the
(TDCS)av is similar to the plane wave TDCS. However, for higher opening angles
i.e. 15° and 20° the peaks shift to perpendicular directions. The results of
(TDCS)av show the dependence of momentum transfer and thus the averaged cross-
section on the opening angle of the incident twisted electron beam.

Our present communication attempts to investigate the (e,2e) process on the
noble gas atoms to unravel the effects of the twisted electron’s different
parameters on the angular profile of the TDCS. We have used the 1CW wave
function in our theoretical model to study the TDCS. The present study can
also be extended for other complex theoretical models, like DWBA, CCC, 2C,
etc.. We are working on this aspect.

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