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#  __
Spin effects in laser-assisted semirelativistic excitation of atomic hydrogen
by electronic impact

S. Taj1, B. Manaut1,2, M. El Idrissi1, Y. Attaourti2 and L. Oufni3
1 Université Sultan Moulay Slimane, FPBM, LIRST, BP : 523, 23000, Béni Mellal,
Morocco.
2 Université Cadi Ayyad, FSSM, LPHEA, BP : 2390, 40000, Marrakech, Morocco
3 Université Sultan Moulay Slimane, FSTBM, LPMM, BP : 523, 23000, Béni Mellal,
Morocco<EMAIL_ADDRESS>
###### Abstract

New insights into our understanding of the semirelativistic excitation of
atomic hydrogen by electronic impact have been made possible by combining the
use of polarized electron beams and intense laser field. The paper reviews
relativistic theoretical treatment in laser-assisted electron scattering with
particular emphasis upon spin effects. Different spin configurations for
inelastic electron-atom collisions is also discussed. The role of laser field
in such collision is of major importance and reveals new information on the
dynamics of the collision process. The examined modern theoretical
investigations of such relativistic laser-assisted collisions have shown that
the need for experimental data is of a paramount importance in order to asses
the accuracy of our calculations.
PACS number(s): 34.80.Dp, 12.20.Ds

##  1 Introduction

The spin is not only an indispensable ingredient in atomic physics but also
responsible for many phenomena observed in solid-state physics. In addition to
the uses of polarized electrons in studies of atomic physics, there have been
numerous studies of polarized electron scattering and polarized electron
emission from ferromagnetic solids over the past decade. In 1975, the purpose
of the Spin-Polarized Electron Source was to describe how this effect, which
had been discovered in spin-polarized photoemission experiments by Pierce et
al at the ETH-Zurich [1] , could be used to provide a compact spin-polarized
electron gun. Later on, an experimental work has been done to produce electron
beams in which the spin has a preferential orientation. They are called
polarized electron beams [2] in analogy to polarized light, in which the field
vectors have a preferred orientation. Extensive theoretical works have been
performed by introducing relativistic and spin effects in the collision
between incident particles and atoms [3-6].There are many reasons for the
interest in polarized electrons. One important reason is that in physical
experiments one endeavors to define as exactly as possible the initial and/or
final states of the systems being considered.

Since the 1960s when lasers became a worldwide-used laboratory equipment and
also large polarization effects in low-energy electron scattering were
ascertained, experimental and theoretical studies of laser-matter interaction
have witnessed continuous progress. By virtue of the increasing progress in
the availability of more powerful and tunable lasers , such processes are
nowadays being observed in laboratories [7-10]. Most experimental and
theoretical studies of laser-assisted electron-atom collisions were restricted
to the nonrelativistic regime and low-frequency fields, where it has been
already recognized that, as a general consequence of the infrared divergence
of QED, large numbers of photons can be exchanged between the field and the
projectile-target system. An extension of the first-Born nonrelativistic
treatment [11] to the relativistic domain was formally derived for unpolarized
electrons [12]. There have been also theoretical investigations of
relativistic scattering in multimode fields [13].

In the present paper we have to extended our previous results [14] to the case
of laser assisted inelastic excitation $1s$ to $2s$ of the atomic hydrogen by
polarized electrons. Therefore, we have begun with the most basic results of
our work using atomic units (a.u) in which one has ($\hbar=m_{e}=e=1$), where
$m_{e}$ is the electron mass at rest. We have used the metric tensor
$g^{\mu\nu}=diag(1,-1,-1,-1)$ and the Lorentz scalar product defined by
$(a.b)=a^{\mu}b_{\mu}$. The organization of this paper is as follows : the
presentation of the necessary formalism of this work in section 2, the result
and discussion in section 3 and at last a brief conclusion in section 4.

##  2 Theory

The transition matrix element corresponding to the laser assisted inelastic
excitation of atomic hydrogen by electronic impact from the initial state $i$
to the final state $f$ is given by

$\displaystyle S_{fi}=-i\int
dt\;\langle\psi_{q_{f}}(\mathbf{r}_{1})\phi_{f}(\mathbf{r}_{2})|V_{d}|\psi_{q_{i}}(\mathbf{r}_{1})\phi_{i}(\mathbf{r}_{2})\rangle$
(1)

where $V_{d}=1/r_{12}-Z/r_{1}$ is the interaction potential, $\mathbf{r}_{1}$
are the coordinates of the incident and scattered electron, $\mathbf{r}_{2}$
the atomic electron coordinates, $r_{12}=|\mathbf{r}_{1}-\mathbf{r}_{2}|$ and
$r_{1}=|\mathbf{r}_{1}|$. Before we present the most interesting results of
our investigation regarding laser-assisted inelastic excitation of atomic
hydrogen by electronic impact, we sketch the principal steps of our
theoretical treatment. The solutions of the Dirac equation for an electron
with four-momentum $p^{\mu}$ inside an electromagnetic plane wave are well
known [15]. They read for the case of circular polarization of the field
propagating along the $Oz$ direction

$\displaystyle\psi_{q}(x)=\left(1+\frac{k\\!\\!\\!/A\\!\\!\\!/}{2c(kq)}\right)\frac{u(p,s)}{\sqrt{2VQ_{0}}}\exp\left[-i(qx)-i\int_{0}^{kx}\frac{(Ap)}{c(kq)}d\phi\right].$
(2)

where $u$ represents a free electron bispinor satisfying the Dirac equation
without field and which is normalized by
$\overline{u}u=u^{\dagger}\gamma^{0}u=2c^{2}$. Here the Feynman slash notation
is used, and $V$ is the normalization volume. The physical significance of
$q^{\mu}=(Q/c,\mathbf{q})$ is the averaged four-momentum (dressed momentum) of
the particle inside the laser field having a vector potential
$A^{\mu}=(0,a_{1}\cos(kx),a_{2}\sin(kx),0)$ with wave four-vector $k^{\mu}$ :
$q^{\mu}=p^{\mu}-k^{\mu}[A^{2}/2(kp)c^{2}]$.

In inelastic scattering, it is not only the state of the electron that is
changed, but also the state of the atom. Before starting the calculations, we
clarified the different configurations appearing with the orientations of the
electron’s spin polarizations. We have many possible scattering scenarios

1 | $e^{(\uparrow)}+H^{(\uparrow)}(1s)\longrightarrow e^{(\uparrow)}+H^{(\uparrow)}(2s)$
---|---
2 | $e^{(\downarrow)}+H^{(\uparrow)}(1s)\longrightarrow e^{(\uparrow)}+H^{(\uparrow)}(2s)$
3 | $e^{(\uparrow)}+H^{(\uparrow)}(1s)\longrightarrow e^{(\downarrow)}+H^{(\uparrow)}(2s)$
4 | $e^{(\uparrow)}+H^{(\downarrow)}(1s)\longrightarrow e^{(\downarrow)}+H^{(\downarrow)}(2s)$
6 | $e^{(\downarrow)}+H^{(\downarrow)}(1s)\longrightarrow e^{(\uparrow)}+H^{(\downarrow)}(2s)$
7 | $e^{(\downarrow)}+H^{(\downarrow)}(1s)\longrightarrow e^{(\downarrow)}+H^{(\downarrow)}(2s)$
⋮ | ⋮

Here, the up and down arrows indicate the direction of the electron’s and
atom’s spin polarization relative to some fixed axis. During the interaction,
the products of states with spin non flip are the same
$\varphi_{2s}^{{\dagger}(\uparrow})(\mathbf{r}_{2})\varphi_{1s}^{(\uparrow)}(\mathbf{r}_{2})=\varphi_{2s}^{{\dagger}(\downarrow)}(\mathbf{r}_{2})\varphi_{1s}^{(\downarrow)}(\mathbf{r}_{2})$
and the product of states with spin flip gives zero

$\displaystyle\varphi_{2s}^{{\dagger}(\uparrow)}(\mathbf{r}_{2})\varphi_{1s}^{(\downarrow)}(\mathbf{r}_{2})$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{llll}2-r_{2},&0,&\frac{-i(4-r_{2})}{4r_{2}c}z,&\frac{(4-r_{2})}{4r_{2}c}(-y-ix)\\\
\end{array}\right)\left(\begin{array}[]{c}0\\\ 1\\\ \frac{i}{2cr_{2}}(x-iy)\\\
-\frac{i}{2cr_{2}}z\end{array}\right)\frac{1}{4\sqrt{2}\pi}e^{-2r_{2}}$
$\displaystyle=$
$\displaystyle\varphi_{2s}^{{\dagger}(\downarrow)}(\mathbf{r}_{2})\varphi_{1s}^{(\uparrow)}(\mathbf{r}_{2})$
$\displaystyle=$ $\displaystyle 0$ (9)

with $\varphi_{2s}(\mathbf{r}_{2})$ and $\varphi_{1s}(\mathbf{r}_{2})$ are the
wave functions of atomic hydrogen corresponding to $2s$ and $1s$ states
respectively. In this case, the probability that the bound electron changes
the orientation of its spin in the transition from the initial state $1s$ to
the final state $2s$ is zero. Thus, the number of realistic configurations
reduces many more. We review first the basics needed for the description of
spin polarization. Free electrons with four-momentum $p$ and spin $s$ are
described by the free spinors $u(p,s)$, the vector $s^{\mu}$ is defined by

$s^{\mu}=\frac{1}{c}\left(\left|\mathbf{p}\right|,\frac{E}{c}\widehat{\mathbf{p}}\right),$
(10)

(with $\widehat{\mathbf{p}}=\mathbf{p}/|\mathbf{p}|$ ) is a Lorentz vector in
a frame in which the particle moves with momentum $\mathbf{p}$. One easily
checks the normalization and the orthogonality conditions respectively

$s.s=s_{\mu}.s^{\mu}=-1\quad;\quad p.s=p_{\mu}.s^{\mu}=0.$ (11)

In practical calculations of quantum electrodynamic (QED) processes, we become
acquainted with a technique of calculation which allows the simple treatment
of complicated expressions especially the calculation of traces of products of
many $\gamma$ matrices. It is based on a projection procedure. The appropriate
operators which achieve this are called spin projection operators. In the
relativistic case, it is given by

$\widehat{\Sigma}(s)=\frac{1}{2}(1+\gamma_{5}s\\!\\!\\!/),$ (12)

with
$\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=-i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}$.
This operator has the following properties

$\widehat{\Sigma}(s)u(p,\pm s)=\pm u(p,s).$ (13)

One can also apply this formalism to helicity states where the spin points in
the direction of the momentum $\mathbf{p}$

$s_{\lambda}^{\prime}=\lambda\frac{\mathbf{p}}{\left|\mathbf{p}\right|}=\lambda\widehat{\mathbf{p}}\,\lambda=\pm
1.$ (14)

We can then define a four spin vector as

$s_{\lambda}^{\mu}=\frac{\lambda}{c}\left(\left|\mathbf{p}\right|,\frac{E}{c}\widehat{\mathbf{p}}\right).$
(15)

The starting point of our calculation is the laser-assisted DCS for atomic
hydrogen by an electron with well defined momentum $p_{i}$ and well defined
spin $s_{i}$. If the final spin $s_{f}$ is also measured, the polarized DCS
then reads as

$\displaystyle\frac{d\sigma}{d\Omega_{f}}(\lambda_{i},\lambda_{f})=\sum_{n=-\infty}^{+\infty}\frac{d\sigma^{(n)}}{d\Omega_{f}}(\lambda_{i},\lambda_{f}),$
(16)

with

$\displaystyle\frac{d\sigma^{(n)}}{d\Omega_{f}}(\lambda_{i},\lambda_{f})=\left.\frac{|\mathbf{q}_{f}|}{|\mathbf{q}_{i}|}\frac{1}{(4\pi
c^{2})^{2}}\left|\overline{u}(p_{f},s_{f})\Gamma_{n}u(p_{i},s_{i})\right|^{2}\left|H_{inel}(\Delta_{s})\right|^{2}\right|_{Q_{f}=Q_{i}+n\omega+E_{1s1/2}-E_{2s1/2}}.$
(17)

The quantity $H_{inel}(\Delta_{s})$ which represents the integral part is
given by :

$\displaystyle H_{inel}(\Delta)=-\frac{4\pi}{\sqrt{2}}(I_{1}+I_{2}+I_{3})$
(18)

with $I_{1}$, $I_{2}$ and $I_{3}$ are as follow :

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{4}{27c^{2}}\int_{0}^{+\infty}dr_{1}\;r_{1}e^{-\frac{3}{2}r_{1}}j_{0}(\Delta
r_{1})=\frac{4}{27c^{2}}\frac{1}{((3/2)^{2}+\mathbf{\Delta}^{2})}$
$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\frac{6}{27}(\frac{1}{c^{2}}-4)\int_{0}^{+\infty}dr_{1}\;r_{1}^{2}e^{-\frac{3}{2}r_{1}}j_{0}(\Delta
r_{1})=\frac{2}{27}(\frac{1}{c^{2}}-4)\frac{3}{((3/2)^{2}+\mathbf{\Delta}^{2})^{2}}$
(19) $\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle-\frac{4}{9}(1+\frac{1}{8c^{2}})\int_{0}^{+\infty}dr_{1}\;r_{1}^{3}e^{-\frac{3}{2}r_{1}}j_{0}(\Delta
r_{1})=\frac{8}{9}(1+\frac{1}{8c^{2}})\frac{\mathbf{\Delta}^{2}-27/4}{((3/2)^{2}+\mathbf{\Delta}^{2})^{3}}.$

Using REDUCE [17], the spinorial part obtained after tedious calculations
reads as

$\displaystyle\left|\overline{u}(p_{f},s_{f})\Gamma_{n}u(p_{i},s_{i})\right|^{2}$
$\displaystyle=$
$\displaystyle\textbf{Tr}\\{\Gamma_{n}\frac{(1+\lambda_{i}\gamma_{5}s\\!\\!\\!/_{i})}{2}(cp\\!\\!\\!/_{i}+c^{2})\overline{\Gamma}_{n}\frac{(1+\lambda_{f}\gamma_{5}s\\!\\!\\!/_{f})}{2}(cp\\!\\!\\!/_{f}+c^{2})\\},$
(20) $\displaystyle=$
$\displaystyle\big{\\{}J_{n}^{2}(z)\mathcal{A}+\big{(}J^{2}_{n+1}(z)+J^{2}_{n-1}(z)\big{)}\mathcal{B}+\big{(}J_{n+1}(z)J_{n-1}(z)\big{)}\mathcal{C}$
$\displaystyle+J_{n}(z)\big{(}J_{n-1}(z)+J_{n+1}(z)\big{)}\mathcal{D}\big{\\}}.$

with $\overline{\Gamma}_{n}=\gamma^{0}\Gamma_{n}^{{\dagger}}\gamma^{0}$ and
$\Gamma_{n}$ is explicitly detailed in our previous work [18].
Before presenting our analytical results, we would like to emphasize that the
REDUCE code we have written for obtaining the four coefficients $\mathcal{A}$,
$\mathcal{B}$, $\mathcal{C}$ and $\mathcal{D}$ gave very long analytical
expressions which were difficult to incorporate in the corresponding latex
manuscript. Thus, we prefer to give below, for example, just the coefficient
$\mathcal{A}$ multiplying the Bessel function $J_{n}^{2}(z)$.

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\frac{1}{(2(k.p_{f})^{2}(k.p_{i})^{2}c^{8})}\Big{[}2(k.p_{f})^{2}(k.p_{i})^{2}\lambda_{f}\lambda_{i}|\mathbf{p_{f}}|^{2}|\mathbf{p_{i}}|^{2}c^{8}\cos(\theta_{if})-2(k.p_{f})^{2}(k.p_{i})^{2}\lambda_{f}\lambda_{i}|\mathbf{p_{f}}|^{2}$
$\displaystyle\times
c^{6}\cos(\theta_{if})E_{i}^{2}+2(k.p_{f})^{2}(k.p_{i})^{2}\lambda_{f}\lambda_{i}|\mathbf{p_{f}}||\mathbf{p_{i}}|c^{10}-2(k.p_{f})^{2}(k.p_{i})^{2}\lambda_{f}\lambda_{i}|\mathbf{p_{i}}|^{2}c^{6}\cos(\theta_{if})E_{f}^{2}$
$\displaystyle+2(k.p_{f})^{2}(k.p_{i})^{2}\lambda_{f}\lambda_{i}c^{8}\cos(\theta_{if})E_{f}E_{i}+2(k.p_{f})^{2}(k.p_{i})^{2}\lambda_{f}\lambda_{i}c^{4}\cos(\theta_{if})E_{f}^{2}E_{i}^{2}+2(k.p_{f})^{2}$
$\displaystyle\times(k.p_{i})^{2}|\mathbf{p_{f}}||\mathbf{p_{i}}|c^{10}\cos(\theta_{if})+2(k.p_{f})^{2}(k.p_{i})^{2}c^{12}+2(k.p_{f})^{2}(k.p_{i})^{2}c^{8}E_{f}E_{i}+2(k.p_{f})^{2}(k.p_{i})$
$\displaystyle\times\lambda_{f}\lambda_{i}|a||\mathbf{p_{i}}|^{2}c^{4}\cos(\theta_{if})E_{f}\omega-2(k.p_{f})^{2}(k.p_{i})\lambda_{f}\lambda_{i}|a|c^{2}\cos(\theta_{if})E_{f}E_{i}^{2}\omega-2(k.p_{f})^{2}(k.p_{i})|a|c^{6}$
$\displaystyle\times
E_{i}\omega+2(k.p_{f})(k.p_{i})^{2}\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}|^{2}c^{4}\cos(\theta_{if})E_{i}\omega-2(k.p_{f})(k.p_{i})^{2}\lambda_{f}\lambda_{i}|a|c^{2}\cos(\theta_{if})E_{f}^{2}E_{i}\omega$
$\displaystyle-2(k.p_{f})(k.p_{i})^{2}|a|c^{6}E_{f}\omega-2(k.p_{f})(k.p_{i})(k.s_{f})\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}||\mathbf{p_{i}}|^{2}c^{6}\cos(\theta_{if})\omega+2(k.p_{f})(k.p_{i})$
$\displaystyle\times(k.s_{f})\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}|c^{4}\cos(\theta_{if})E_{i}^{2}\omega-2(k.p_{f})(k.p_{i})(k.s_{f})\lambda_{f}\lambda_{i}|a||\mathbf{p_{i}}|c^{8}\omega-2(k.p_{f})(k.p_{i})(k.s_{i})$
$\displaystyle\times\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}|^{2}|\mathbf{p_{i}}|c^{6}\cos(\theta_{if})\omega-2(k.p_{f})(k.p_{i})(k.s_{i})\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}|c^{8}\omega+2(k.p_{f})(k.p_{i})(k.s_{i})\lambda_{f}\lambda_{i}$
$\displaystyle\times|a||\mathbf{p_{i}}|c^{4}\cos(\theta_{if})E_{f}^{2}\omega-(k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}a^{2}|\mathbf{p_{f}}||\mathbf{p_{i}}|c^{2}\omega
a^{2}+k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}a^{2}\cos(\theta_{if})E_{f}E_{i}\omega^{2}$
$\displaystyle+2(k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}|^{2}|\mathbf{p_{i}}|^{2}c^{4}\cos(\theta_{if})\omega^{2}-2(k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}|^{2}c^{2}\cos(\theta_{if})E_{i}^{2}\omega^{2}$
$\displaystyle+2(k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}|a||\mathbf{p_{f}}||\mathbf{p_{i}}|c^{6}\omega^{2}-2(k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}|a||\mathbf{p_{i}}|^{2}c^{2}\cos(\theta_{if})E_{f}^{2}\omega^{2}-2(k.p_{f})(k.p_{i})$
$\displaystyle\times\lambda_{f}\lambda_{i}|a|c^{4}\cos(\theta_{if})E_{f}E_{i}\omega^{2}+2(k.p_{f})(k.p_{i})\lambda_{f}\lambda_{i}|a|\cos(\theta_{if})E_{f}^{2}E_{i}^{2}\omega^{2}+(k.p_{f})(k.p_{i})a^{2}c^{4}\omega^{2}$
$\displaystyle-2(k.p_{f})(k.p_{i})|a||\mathbf{p_{f}}||\mathbf{p_{i}}|c^{6}\cos(\theta_{if})\omega^{2}-2(k.p_{f})(k.p_{i})|a|c^{8}\omega^{2}+2(k.p_{f})(k.p_{i})|a|c^{4}E_{f}E_{i}\omega^{2}$
$\displaystyle+(k.p_{f})(k.s_{i})\lambda_{f}\lambda_{i}a^{2}|\mathbf{p_{f}}|c^{2}E_{i}\omega^{2}-(k.p_{f})(k.s_{i})\lambda_{f}\lambda_{i}a^{2}|\mathbf{p_{i}}|c^{2}\cos(\theta_{if})E_{f}\omega^{2}-(k.p_{i})(k.s_{f})\lambda_{f}\lambda_{i}$
$\displaystyle\times
a^{2}|\mathbf{p_{f}}|c^{2}\cos(\theta_{if})E_{i}\omega^{2}+(k.p_{i})(k.s_{f})\lambda_{f}\lambda_{i}a^{2}|\mathbf{p_{i}}|c^{2}E_{f}\omega^{2}+(k.s_{f})(k.s_{i})\lambda_{f}\lambda_{i}a^{2}|\mathbf{p_{f}}||\mathbf{p_{i}}|c^{4}\cos(\theta_{if})$
$\displaystyle\times\omega^{2}+(k.s_{f})(k.s_{i})\lambda_{f}\lambda_{i}a^{2}c^{6}\omega^{2}-(k.s_{f})(k.s_{i})\lambda_{f}\lambda_{i}a^{2}c^{2}E_{f}E_{i}\omega^{2}\Big{]}$

At this stage, note that if $\lambda_{i}\lambda_{f}=1$ during the scattering
process, which physically means that there is no helicity flip occurring but
if $\lambda_{i}\lambda_{f}=-1$, this means that a helicity flip occurred. In
the absence of the laser field ($|a|=0)$, this coefficient reduces to

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\frac{1}{c^{4}}\Big{[}\lambda_{f}\lambda_{i}|\mathbf{p_{f}}|^{2}|\mathbf{p_{i}}|^{2}c^{4}\cos(\theta_{if})-\lambda_{f}\lambda_{i}|\mathbf{p_{f}}|^{2}c^{2}\cos(\theta_{if})E_{i}^{2}+\lambda_{f}\lambda_{i}|\mathbf{p_{f}}||\mathbf{p_{i}}|c^{6}-\lambda_{f}\lambda_{i}|\mathbf{p_{i}}|^{2}c^{2}\cos(\theta_{if})E_{f}^{2}$
$\displaystyle+\lambda_{f}\lambda_{i}c^{4}\cos(\theta_{if})E_{f}E_{i}+\lambda_{f}\lambda_{i}\cos(\theta_{if})E_{f}^{2}E_{i}^{2}+|\mathbf{p_{f}}||\mathbf{p_{i}}|c^{6}\cos(\theta_{if})+c^{8}+c^{4}E_{f}E_{i}\Big{]}$

We note that without laser field, equations (16) and (17) reduce to the well-
known polarized (first-Born) differential cross section of inelastic
excitation of atomic hydrogen by electronic impact.

Figure 1: The different TDCSs (Unpolarized DCS, Spin polarized DCS with
($\lambda_{i}=\lambda_{f}=0$)) scaled in $10^{-9}$ as a function of the angle
$\theta_{f}$ . The relativistic parameter is $\gamma=1.0053$, the electrical
field strength is $\mathcal{E}=0.05\;a.u$. The geometric parameters are
$\theta_{i}=45^{\circ}$, $\phi_{i}=0^{\circ}$, $\phi_{f}=45^{\circ}$ and the
number of photons exchanged are $n=\pm 100$.

##  3 Results and discussions

The spin-polarized differential cross sections $d\sigma^{(n)}/d\Omega_{f}$ are
computed for the laser-assisted semirelativistic excitation of atomic hydrogen
by electronic impact, where $n$ denotes the number of photons absorbed or
emitted. With a view to qualitative comparison with our previous work [14],
kinematics and the geometry parameters are chosen in accordance with those
used in [14]. The direction of the laser field is chosen parallel to $Oz$
axis. The corresponding spin unpolarized differential cross section
$d\overline{\sigma}^{(n)}/d\Omega_{f}$ results are also presented for
comparison.

Figure 2: The different TDCSs (Unpolarized DCS, Spin flip DCS and Spin non
flip DCS) scaled in $10^{-9}$ as a function of the angle $\theta_{f}$ . The
relativistic parameter is $\gamma=1.0053$, the electrical field strength is
$\mathcal{E}=0.05\;a.u$. The geometric parameters are $\theta_{i}=45^{\circ}$,
$\phi_{i}=0^{\circ}$, $\phi_{f}=45^{\circ}$ and the number of photons
exchanged are $n=\pm 100$.

In view of equation (20) if the mathematical condition
($\lambda_{i}=\lambda_{f}=0$) is used, the spinorial part takes the following
form

$\displaystyle\left|\overline{u}(p_{f},s_{f})\Gamma_{n}u(p_{i},s_{i})\right|^{2}=\frac{1}{4}\textbf{Tr}\\{\Gamma_{n}(cp\\!\\!\\!/_{i}+c^{2})\overline{\Gamma}_{n}(cp\\!\\!\\!/_{f}+c^{2})\\}$
(21)

In comparison with the unpolarized spinorial part

$\displaystyle\frac{1}{2}\sum_{s_{i}s_{f}}\left|\overline{u}(p_{f},s_{f})\Gamma_{n}u(p_{i},s_{i})\right|^{2}=\textbf{Tr}\\{\Gamma_{n}(cp\\!\\!\\!/_{i}+c^{2})\overline{\Gamma}_{n}(cp\\!\\!\\!/_{f}+c^{2})\\}$
(22)

it may be noted from equations (21) and (22) that the spin unpolarized DCS is
equal to two times to the spin polarized DCS but only under the condition
($\lambda_{i}=\lambda_{f}=0$)

$\displaystyle\frac{d\overline{\sigma}^{(n)}}{d\Omega_{f}}=2\times\frac{d\sigma^{(n)}}{d\Omega_{f}}(\lambda_{i}=0,\lambda_{f}=0),$
(23)

this equation represents our first consistency check. The numerical results
are displayed in figure 1, where we have plotted the two DCSs (2 times spin
polarized DCS with ($\lambda_{i}=\lambda_{f}=0$) and spin unpolarized DCS ) by
varying the angle $\theta_{f}$ of the scattered electron. As it is seen in
this figure, we have indistinguishable curves. The most important result
(second consistency check) is the sum of polarized DCS (spin flip) and
polarized DCS (spin non flip) always gives the spin unpolarized DCS as this is
shown in figure 2.

Figure 3: The different TDCSs (Unpolarized DCS, Spin flip DCS and Spin non
flip DCS) scaled in $10^{-13}$ as a function of the relativistic parameter
$\gamma$, the geometric parameters are $\theta_{i}=45^{\circ}$,
$\phi_{i}=0^{\circ}$, $\phi_{f}=45^{\circ}$ and $\theta_{f}=45^{\circ}$. The
electrical field strength is $\mathcal{E}=0.5\;a.u$ and the number of photons
exchanged are $n=\pm 50$.

Figure 3 shows the different spin polarized and unpolarized DCSs versus the
relativistic parameter $\gamma$. It is apparent from this figure that the
kinetic energy of the incident electron has an important effect in the spin
orientation. For much of the high energy range, the spin polarized DCS (spin
non flip) is approximately equal to the spin unpolarized DCS and the spin
polarized DCS (spin flip) is equal to zero. The physical meaning of this
result is that : in high energy, the probability that the incident electron
changes its spin is zero.

Figure 4: The different TDCSs (Unpolarized DCS, Spin flip DCS and Spin non
flip DCS) scaled in $10^{-22}$ as a function of the relativistic parameter
$\gamma$, the geometric parameters are $\theta_{i}=45^{\circ}$,
$\phi_{i}=0^{\circ}$, $\phi_{f}=45^{\circ}$ and $\theta_{f}=45^{\circ}$. The
electrical field strength is $\mathcal{E}=0.5\;a.u$ and the number of photons
exchanged are $n=\pm 50$.

In order to clarify the situation in which we have seemingly overlapping
curves for the three approaches in figure 3, we give in figure 4 the three
approaches (spin polarized (spin flip and spin non flip) and the spin
polarized DCS). As it is noticed, the spin polarized DCS (spin non flip) and
the spin unpolarized DCS overlap but the spin polarized DCS (spin flip)
converges to zero.

Figure 5: The behavior of the degree of polarization $P$ as a function of the
angle $\theta_{f}$ varying from $-180^{\circ}$ to $180^{\circ}$ and the
relativistic parameter $\gamma$ scaled in $10^{-2}$ in absence of the laser
field.

Our main interest is the polarization degree of the electron after the event.
This quantity is defined as

$\displaystyle P=\frac{\frac{d\sigma^{(n)}}{d\Omega_{f}}(\textit{spin non
flip})-\frac{d\sigma^{(n)}}{d\Omega_{f}}(\textit{spin
flip})}{\frac{d\sigma^{(n)}}{d\Omega_{f}}(\textit{spin non
flip})+\frac{d\sigma^{(n)}}{d\Omega_{f}}(\textit{spin flip})},$ (24)

figure 5 shows the polarization degree, which is related by equation (24) to
the spin polarized differential cross section ratio. A tree dimensional plot
of this quantity is given versus the angle $\theta_{f}$ and the relativistic
parameter $\gamma$. The first observation that can be made concerns the shape
of the polarization degree that is strongly changed with the relativistic
parameter $\gamma$. An interesting behavior emerges with increasing $\gamma$,
particularly for $\theta_{f}=0^{\circ}$, where the polarization degree
performs a plateau-like behavior. This emphasizes the fact that the spin
polarized DCS is very sensitive to the variation of the relativistic parameter
$\gamma$ and this fact has to remain true for the case in the presence of the
laser field. This is consistent with the results shown in figure 3 and 4.

##  4 Conclusion

We have studied the laser assisted inelastic excitation of atomic hydrogen by
polarized electrons. We unraveled the influence of the orientation of the
spin-polarization of the incoming and scattered electrons relative to the
orientation of the spin-polarization of the bound electron. These spin effects
depend strongly on the energies of the incoming electron. It is observed that
in the transition from $1s$ state to $2s$ state, the electron’s probability
for changing its spin is zero.

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