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REVIEW PAPER
Electron impact excitation of hydrogen atom with anomalous magneticmoment effects
M El Idrissi1, S Taj1, B Manaut1* and L Oufni2
1Laboratoire Interdisciplinaire de Recherche en Sciences et Techniques, Faculte Polydisciplinaire, Universite Sultan Moulay Slimane, Boite
Postale 523, 23000 Beni Mellal, Morocco
2Departement de Physique, Faculte des Sciences et Techniques, LPMM-ERM, Universite Sultan Moulay Slimane, BP: 523, 23000 Beni Mellal,
Morocco
Received: 11 June 2013 / Accepted: 09 September 2013 / Published online: 26 September 2013
Abstract: In framework of first Born approximation, Salamin wave states have been used to calculate the differential
cross section for laser assisted semirelativistic 1s–2s excitation of hydrogen atom by electron impact with anomalous
magnetic moment effects fully included. The obtained results have been compared with laser assisted relativistic differ-
ential cross section without such effects. It is found that the electron’s anomalous magnetic moment effects in laser assisted
1s–2s excitation is very important particularly with increase of intensity. Good agreement with laser assisted differential
cross section without anomalous magnetic moment effects is obtained when electron’s anomaly j ¼ 4ae� ¼ 2ðg� 2Þ is
taken to be zero.
Keywords: QED calculations; AMM effects; Relativistic scattering theory
PACS Nos.: 34.80.Dp; 12.20.Ds
1. Introduction
The study of relativistic atomic physics requires a thorough
knowledge of Quantum Electrodynamics (QED) formal-
ism. Precision tests of QED formalism require an appro-
priate inclusion of higher order effects and knowledge of
very precise input parameters. One of the basic input
parameters is anomalous magnetic moment (AMM) ae� ,
determined from fine-structure a and calculated straight-
forwardly by one-loop result ðae� ¼ a=ð2pÞ ’ 0:0011614Þ[1, 2]. Kinoshita et al. [3–7] have, after a many-year effort,
completed calculations including full five-loop effects. A
recent experimental value for ae� ¼ 0:001159652180ð73Þhas been obtained by Gabrielse [8]. It is now obvious that
whole apparatus and formalism of nonrelativistic quantum
collision theory [9] have to be revisited in order to extend
known nonrelativistic results to relativistic domain [10].
Many theoretical studies of laser-free and laser-assisted
electron-atom collision have been mainly carried out in
nonrelativistic regime [11–14]. Many other theoretical
investigations for 1s–2s processes have been confined
either to first Born approximation [15, 16] or to the use of a
numerical analysis [17]. Hydrogen has several purposes : it
has been used in collisions with charged ions from inter-
mediate to high energy region [18] and in a development of
palladium-based hydrogen thin film sensor using silicon
oxide substrate [19]. The main point in this study of laser
assisted electron atom scattering is to introduce electron’s
anomalous magnetic moment effects. The only method, by
which we can assess accuracy of various approximations, is
by comparing them with each other since there is no
experimental data in the relativistic regime. We proceed
with simplest approximations and progressive introduction
of more involved and in general, more accurate and
revealing approximation. Conceptual interest of the intro-
duction of anomalous magnetic moment of the electron is :
first, for affinity of theory; second, even if value of
anomaly of the electron is too small, it greatly affects
theory. In this paper, we use laser intensities below
threshold ionization *1016 W/cm2. It is worth noting that
beyond such intensity, electron acquires high quiver
momentum. Thus, single photon ionization channel could
be open. Moreover, for laser intensities of the order of*Corresponding author, E-mail: [email protected]
Indian J Phys (February 2014) 88(2):111–117
DOI 10.1007/s12648-013-0392-3
� 2013 IACS
*1016 W/cm2, kinetic energy transferred to atomic system
by field is of the order of rest mass of electron. Thus, laser-
atom interaction needs a relativistic treatment.
For pedagogical purposes, we begin by most basic
results of our work using atomic units (a.u) in which we
have �h ¼ me ¼ e ¼ 1, where me is electron mass at rest.
We also work with metric tensor glm = diag(1, - 1,
- 1, - 1) and Lorentz scalar product defined by
(a � b) = al bl.
2. Theory of inelastic collision in absence of AMM
effects
The second-order Dirac equation for electron in presence
of an external electromagnetic field is given by
p� 1
cA
� �2
�c2 � i
2cFlmr
lm
" #wðxÞ ¼ 0; ð1Þ
where rlm ¼ 12½cl; cm� are tensors related to Dirac matrices
cl, Flm = ql Am - qm Al is electromagnetic field tensor and
Al is four-vector potential. The plane wave normalized to
volume V and solution of second-order equation is known
as Volkov state [20]
wðxÞ ¼ 1þ k=A=
2cðk � pÞ
� �uðp; sÞffiffiffiffiffiffiffiffiffiffiffi
2VQ0
p
� exp �iðq � xÞ � i
Zkx
0
ðA � pÞcðk � pÞ du
24
35:
ð2Þ
In Eq. (2), u(p, s) represents a Dirac bispinor that satisfies
free Dirac equation and is normalized according to
uðp; sÞuðp; sÞ ¼ u�ðp; sÞc0uðp; sÞ ¼ 2c2. Quantity Q0 is
such as : Q0 = E/c - A2x/[2 (k�p) c3] and u ¼ ðk � xÞ ¼klxl is the phase. We now turn to calculation of transition
matrix element Sfi corresponding to process of laser
assisted electron-atomic hydrogen from initial state i to
final state f given by
Sfi ¼ �i
Zdthwqf
ðr1ÞUf ðr2ÞjVdjwqiðr1ÞUiðr2Þi:
¼ �i
Zdthwqf
ðr1Þc0wqiðr1ÞihUf ðr2ÞjVdjUiðr2Þi:
ð3Þ
The instantaneous interaction potential reads as
Vd ¼1
r12
� Z
r1
; ð4Þ
where r12 = |r1 - r2|;r1 are coordinates of incident and
scattered electron measured from center of hydrogen atom and
r2 are electron’s atomic coordinates measured from the same
origin. The functions Ui;f ðr2Þ are semirelativistic wave states
of hydrogen atom where the index i stands for initial state and
index f stands for final state [16]. We consider a circularly
polarized field and after some algebraic manipulation, the
spinorial part hwqfðr1Þc0wqi
ðr1Þi as given in Eq. (3) reads as
hwqfðr1Þc0wqi
ðr1Þi / uðpf ; sf Þ½C0 þ C1 cosðuÞþ C2 sinðuÞ�uðpi; siÞe�izðu�u0Þ; ð5Þ
with C0, C1 and C2 are such as
C0 ¼ c0 � 2xc
cðpiÞcðpf Þa2k=
C1 ¼ cðpiÞc0k=a=1 þ cðpf Þa=1k=c0
C2 ¼ cðpiÞc0k=a=2 þ cðpf Þa=2k=c0:
Now, we introduce well-known relations involving
ordinary Bessel functions as :
1
cosðuÞsinðuÞ
8<:
9=;e�iz sinðu�u0Þ ¼
Xþ1s¼�1
B0s
B1s
B2s
8<:
9=;e�isu; ð6Þ
with
B0s
B1s
B2s
8<:
9=; ¼
JsðzÞeisu0
Jsþ1ðzÞeiðsþ1Þu0 þ Js�1ðzÞeiðs�1Þu0
� �=2
Jsþ1ðzÞeiðsþ1Þu0 � Js�1ðzÞeiðs�1Þu0
� �=2i
8<:
9=;:ð7Þ
Proceeding along the lines of standard QED calculations,
spin unpolarized differential cross section (DCS) becomes
drdXf
¼Xþ1
s¼�1
drðsÞ
dXf
; ð8Þ
with
drðsÞ
dXf
¼jqf jjqij
1
ð4pc2Þ2
1
2
Xsisf
jMðsÞfi j2
!HinelðDsÞj j2
�����Qf¼QiþsxþE1s1=2�E2s1=2
: ð9Þ
Spinorial part 12
Psi
PsfjMðsÞfi j
2contains all information
about spin and laser-interaction effects, can be obtained
using REDUCE [21] and are explicitly given by
1
2
Xsi
Xsf
jMðsÞfi j2 ¼ 1
2Trfðp=f cþ c2ÞKðsÞðp=icþ c2ÞKðsÞg
¼ 2 J2s ðzÞA þ J2
sþ1ðzÞ þ J2s�1ðzÞ
� �B
��þ Jsþ1ðzÞJs�1ðzÞð ÞCþJsðzÞ Js�1ðzÞ þ Jsþ1ðzÞð ÞDgÞ; ð10Þ
with KðsÞ and KðsÞ
are given by
KðsÞ ¼ C0B0sðzÞ þ C1B1s þ C2B2s; KðsÞ ¼ c0KðsyÞc0:
ð11Þ
112 M El Idrissi et al.
Four coefficients A;B; C and D appearing in Eq. (10) are
such as
A ¼ c4 � ðqf � qiÞc2 þ 2Qf Qi �a2
2
ðk � qf Þðk � qiÞ
þ ðk � qiÞðk � qf Þ
� �
þ a2x2
c2ðk � qf Þðk � qiÞððqf � qiÞ � c2Þ þ ða2Þ2x2
c4ðk � qf Þðk � qiÞ
þ a2xc2ðQf � QiÞ
1
ðk � qiÞ� 1
ðk � qf Þ
� �ð12Þ
B ¼ � ða2Þ2x2
2c4ðk � qf Þðk � qiÞþ x2
2c2
ða1 � qf Þðk � qf Þ
ða1 � qiÞðk � qiÞ
�
þ ða2 � qf Þðk � qf Þ
ða2 � qiÞðk � qiÞ
Þ � a2
2þ a2
4
ðk � qf Þðk � qiÞ
þ ðk � qiÞðk � qf Þ
� �
� a2x2
2c2ðk � qf Þðk � qiÞðqf � qiÞ � c2� �
þ a2x2c2ðQf � QiÞ
1
ðk � qf Þ� 1
ðk � qiÞ
� �ð13Þ
C ¼ x2
c2ðk � qf Þðk � qiÞcosð2/0Þfða1 � qf Þða1 � qiÞ�
�ða2 � qf Þða2 � qiÞgþ sinð2/0Þfða1 � qf Þða2 � qiÞþða1 � qiÞða2 � qf Þg
�ð14Þ
D ¼ c
2ðA � qiÞ þ ðA � qf Þ� �
� c
2
ðk � qf Þðk � qiÞ
ðA � qiÞ þðk � qiÞðk � qf Þ
ðA � qf Þ� �
þ xc
QiðA � qf Þðk � qf Þ
þ Qf ðA � qiÞðk � qiÞ
� �ð15Þ
with A ¼ a1 cosðu0Þ þ a2 sinðu0Þ and more analytical
details can be found elsewhere [22]. We now turn to
function HinelðDsÞ of momentum transfer which is simply
proportional to Fourier transform of average (static)
potential felt by incident electron in field of hydrogen
atom. This quantity is obtained after long analytical
calculations [23] and it reduces to
HinelðDsÞ ¼ �4pffiffiffi
2p ½I1ðsÞ þ I2ðsÞ þ I3ðsÞ�; ð16Þ
with
I1ðsÞ ¼4
27c2
1
ðð3=2Þ2 þ jDsj2Þ
I2ðsÞ ¼2
27
1
c2� 4
� �3
ðð3=2Þ2 þ jDsj2Þ2
I3ðsÞ ¼8
91þ 1
8c2
� �jDsj2 � 27=4
ðð3=2Þ2 þ jDsj2Þ3;
ð17Þ
where jDsj ¼ jqf � qi � skj is momentum transfer in pre-
sence of laser field.
3. Theory of inelastic collision in presence of AMM
effects
We consider a numerical evaluation of laser-assisted
inelastic 1s–2s excitation of hydrogen atom involving
electron’s anomalous magnetic moment effects and present
a fully relativistic formula for differential cross section. In
weak-field approximation, laser-electron interaction is
taken into account exactly by using Salamin wave states
[24] for initial and final wave functions
WðxÞ ¼ ½1� ðak=A=þ bk=
þ dp=k=A=Þ� uðp; sÞffiffiffiffiffiffiffiffiffiffiffi2VQ0
p exp �iðq � xÞ � i
Zkx
0
ðA � pÞcðk � pÞ du
24
35;
ð18Þ
with
a ¼ jc
2� 1
c
� �=2ðk � pÞ; b ¼ jA2
4cðk � pÞ ; d ¼ j4ðk � pÞ :
ð19Þ
These wave functions represent the exact solution of
second-order Dirac equation for an electron with
anomalous magnetic moment in presence of an external
electromagnetic field
p� 1
cA
� �2
�c2 � i
2cFlmr
lm þ iae� p=� A=
cþ c
� �Flmr
lm
" #
wðxÞ ¼ 0; ð20Þ
with j ¼ 4ae� where j is electron’s anomaly. Term Flmrlm
stems from the fact that electrons have a spin-one-half, the
term multiplying ae� is due to their AMM.
We proceed now to evaluation of S-matrix element for
transition ði �! f Þ including electron’s anomalous mag-
netic moment
Sfi ¼ �i
ZdthWqf
ðr1ÞUf ðr2ÞjVdjWqiðr1ÞUiðr2Þi: ð21Þ
If we replace all wave functions and potential interaction
Vd in Sfi and using standard QED calculations, spin
unpolarized differential cross section reads
drðAMMÞ
dXf
¼Xþ1
s¼�1
jqf jjqij
1
ð4pc2Þ2
1
2
Xsisf
jMðAMMÞfi j2
!HinelðDsÞj j2
�����Qf¼QiþsxþE1s1=2�E2s1=2
: ð22Þ
Novelty in various stages of calculations is contained in
spinorial part
Electron impact excitation of hydrogen atom 113
1
2
Xsi
Xsf
jMðAMMÞfi j2 ¼ 1
2Tr ðp=f cþ c2ÞKðAMMÞ
s
n
ðp=icþ c2ÞKðAMMÞs
o; ð23Þ
with
KðAMMÞs ¼ C0B0sðzÞ þ C1B1sðzÞ þ C2B2sðzÞ þ C3B3sðzÞ
þ C4B4sðzÞ:ð24Þ
Five coefficients C0, C1, C2, C3 and C4, containing all
information about electron’s AMM effect and interaction
spin-laser, can be obtained using REDUCE [21] and are
explicitly given by
C0 ¼ 2ð2a=1k=p=ia1 � pf df dix� 2a=1k=p=f a1 � pidf dixh
þ2a=1k=c0a1 � pik � pf cdf di � 2a=1k=c0a1 � pf k � picdf di
� 2a=1k=a1 � piaf dixþ 2a=1k=a1 � pf aidf x
þ 2a=2k=p=ia2 � pf df dix� 2a=2k=p=f a2 � pidf dixþ 2a=2k=c0
� a2 � pik � pf cdf di � 2a=2k=c0a2 � pf k � picdf di
� 2a=2k=a2 � piaf dixþ 2a=2k=a2 � pf aidf x� 2k=p=iaf
� A2dixþ 2k=p=f p=iA2df dix� 2k=p=f c0k � piA
2cdf di
þ 2k=p=f aiA2df x� 2k=c0p=ik � pf A
2cdf di
þ 2k=c0k � piaf A2cdi � 2k=c0k � pf aiA
2cdf � k=c0bf c
� 2k=af aiA2xþ 2k=bf bix� c0k=bicþ c0cÞ�=ð2cÞ
ð25Þ
C1 ¼ ½2ð�a=1k=p=f c0cdf þ 2a=1k=p=f bidf x� 2a=1k=c0k � pf bicdf
�a=1k=c0af cþ 2a=1k=af bix þ 2k=a=1aibf x
�2k=p=ia=1bf dixþ 2k=c0a=1k � pibf cdi � c0k=a=1aic
�c0p=ik=a=1cdiÞ�=ð2cÞð26Þ
C2 ¼ ½2ð�a=2k=p=f c0cdf þ 2a=2k=p=f bidf x� 2a=2k=c0k � pf bicdf
�a=2k=c0af cþ 2a=2k=af bix þ 2k=a=2aibf x
�2k=p=ia=2bf dixþ 2k=c0a=2k � pibf cdi � c0k=a=2aic
�c0p=ik=a=2cdi�=ð2cÞð27Þ
C3 ¼ ½4a=1k=p=ia1 � pf df dix� a=1k=p=f a1 � pidf dix
þa=1k=c0a1 � pick � pf df di � a=1k=c0a1 � pf k � picdf di
�a=1k=a1 � piaf dixþ a=1k=a1 � pf aidf x
�a=2k=p=ia2 � pf df dixþ a=2k=p=f a2 � pidf dix
�a=2k=c0 a2 � pik � pf cdf di þ a=2k=c0a2 � pf k � picdf di
þa=2k=a2 � piaf dix� a=2k=a2 � pf aidf xÞ�=ð2cÞð28Þ
C4 ¼ ½4ða=1k=p=ia2 � pf df dix� a=1k=p=f a2 � pidf dix
þa=1k=c0a2 � pik � pf cdf di � a=1k=c0a2 � pf k � picdf di
�a=1k=a2 � piaf dixþ a=1k=a2 � pf aidf x
þa=2k=p=ia1 � pf df dix� a=2k=p=f a1 � pidf dix
þa=2k=c0 � a1 � pick � pf df di � a=2k=c0a1 � pf k � picdf di
�a=2k=a1 � piaf dixþ a=2k=a1 � pf aidf xÞ�=ð2cÞ:ð29Þ
More details are available in [25]. From Eq. (24),
five coefficients B0s(z), B1s(z), B2s(z), B3s(z) and B4s(z)
involving ordinary Bessel functions are given by
B0s
B1s
B2s
B3s
B4s
8>>>><>>>>:
9>>>>=>>>>;¼
JsðzÞeis/0
Jsþ1ðzÞeiðsþ1Þ/0 þ Js�1ðzÞeiðs�1Þ/0
� �=2
Jsþ1ðzÞeiðsþ1Þ/0 � Js�1ðzÞeiðs�1Þ/0
� �=2i
Jsþ2ðzÞeiðsþ2Þ/0 þ Js�2ðzÞeiðs�2Þ/0
� �=2
Jsþ2ðzÞeiðsþ2Þ/0 � Js�2ðzÞeiðs�2Þ/0
� �=2i
8>>>>><>>>>>:
9>>>>>=>>>>>;:
ð30Þ
The Integral part ðHinelðDsÞÞ remains intact since effect of
anomalous magnetic moment acts for present work just on
incident and scattered electrons.
4. Results and discussion
To test this new approach, we have used nonrelativistic
theoretical results. In limit between non-relativistic and
relativistic regimes, typically an incident electron kinetic
energy Ei = 100 a.u = 2700 eV, the effects of additional
spin terms and relativity are too small to be noticed. We
stress, however, that present approach can be directly
applied to fast electrons in intensive lasers (below
*1016 W/cm2) where the relativistic effects become
stronger and even dominant. The laser field is assumed to
be a monochromatic plane-wave classical field of circular
polarization, which is switched on and off adiabatically at
t �! �1 and t �! þ1, respectively. Its four-potential is
given by
AlðxÞ ¼ jajðel1 cosðuÞ þ e
l2 sinðuÞÞ: ð31Þ
In Eq. (31), |a| is amplitude, ejl (l = 0, 1, 2, 3, j = 1, 2)
with e1l and e2
l are polarization four-vectors and u ¼ klxl
is the phase, where kl = (x/c, k) is four-momentum of
laser photons verifying (ejl kl = 0), having a frequency x
and a three-momentum k = |k| (0, 0, 1) and xl = (ct;
x) are space-time coordinates. For following numerical
analysis of differential cross sections for a circularly
polarized electromagnetic plane wave field, we fix the
direction of laser field along Z-axis and take nuclear charge
as Z = 1. We use an angular frequency of x = 0.043
114 M El Idrissi et al.
a.u = 1.17 eV which corresponds to lasing transition of
Nd laser at a wavelength of 1064 nm.
4.1. Nonrelativistic regime
We have computed laser-assisted DCS for electron-impact
excitation 1s–2s of atomic hydrogen at intermediate elec-
tron energies for fixed angles hi = 90�, /i = /f = 45�.
Before starting a discussion, we should mention some
abbreviations: (SRDCS) stands for semirelativistic differ-
ential cross section and (NRDCS) stands for nonrelativistic
differential cross section. Figure 1 contains results of the
following DCSs (SRDCS with AMM, SRDCS without
AMM, SRDCS without laser and NRDCS without laser)
where relativistic parameter is c = 1.0053. We have
computed these results by setting electron’s anomaly ae�
and intensity of laser field e equal to zero in our general
computer program. One main feature of present DCS
result, which illustrates the first check of consistence of our
new formalism, is overlapping curves of four approaches.
This consistency shown by the overlap of four approaches
should be verified in nonrelativistic regime when electric
field strength and anomaly are taken to be zero. Figure 2
reveals dependence of SRDCS with and without AMM
effects on laser field strength. Ratio of summed SRDCS
increases steadily by about 1.6 order of magnitude, when
electric field strength increases from 0 a.u to 0.25 a.u. This
figure shows clearly that electron’s AMM effects is
strongly dependent on electric field strength e. Figure 3
depicts difference between two approaches when variation
depends on angle hf. Similarly to electric field strength
dependence, for value e ¼ 0:25 a:u ratio SRDCS (with
AMM)/SRDCS (without AMM) is approximatively 1.47 in
vicinity of hf = 90�. Figure 4 shows measurements taken
by Williams et al. [26] limited to electrons with incident
kinetic energy Ei = 100 eV. Generally, agreement between
80 85 90 95 100
0,0
0,2
0,4
0,6
0,8
1,0
1,2 DCS without AMM DCS with AMM Nonrelativistic DCS Semirelativistic DCS
DC
S (
a.u)
Angle θf (degree)
Fig. 1 Comparison between SRDCS with AMM (ae� = 0 and
e ¼ 0:0 a:u), SRDCS without AMM (e ¼ 0:0 a:u), SRDCS without
laser and NRDCS for a relativistic parameter c = 1.0053. Geometric
parameters hi = 90�, /i = /f = 45� and 80� B hf B 100�
1,0
1,1
1,2
1,3
1,4
1,5
1,6
1,7
SRDCS without AMM SRDCS with AMM
DC
S (
a.u)
Electric field strength ε (a.u)
0,00 0,05 0,10 0,15 0,20 0,25
Fig. 2 DCSs (with and without AMM) as a function of electric field
strength e for a relativistic parameter c = 1.0053 and geometric
parameters hi = hf = 90�, /i = /f = 45�. The corresponding num-
ber of photons exchanged is ±100
80 85 90 95 100
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
1,8 SRDCS without AMM SRDCS with AMM
DC
S (
a.u)
Angle θf (degree)
Fig. 3 DCSs (with and without AMM) as a function of the angle hf
for electric field strength e ¼ 0:25 a:u, relativistic parameter
c = 1.0053 and geometric parameters hi = 90�, /i = /f = 45� and
80� B hf B 100�. The corresponding number of photons exchanged
is ±100
Electron impact excitation of hydrogen atom 115
two models (NRDCS and SRDCS) is good and, in some
cases, both of them are clearly beyond a perfect agreement
with data. Hence, agreement between NRDCS and SRDCS
results can lead to conclusion that approximations of
Darwin wave function are justified in nonrelativistic
regime. In fact, difference between theoretical models and
experimental data are significant for small and high angles.
To minimize this difference, it is necessary at lower
energies, to add Higher-order corrections and Coulomb
effects in order to obtain satisfactory agreement between
theory and experiment which is not the subject of this
study.
4.2. Relativistic regime
At larger impact energies and larger laser intensities (below
0.25 a.u), two approaches of SRDCS (with and without
AMM) demonstrate qualitatively a different behavior
whose magnitude increases very rapidly. We present in
Fig. 5, two SRDCSs with and without AMM. It is observed
that, compared to the case shown in nonrelativistic regime,
magnitude of differential cross section with AMM effects
has increased roughly by 2.5 orders with dependence on
collision geometry remaining basically the same. One of
the most visible differences between Figs. 3 and 5 is that
the peak around hf = 90� has increased by more than an
order of magnitude and becomes less narrower with
increase of the energy. In order to illustrate strong depen-
dence of SRDCS with AMM effect on electric field
strength, we give in Fig. 6 SRDCSs versus electric field
strength (e). It shows on one hand that in case of relativistic
regime, it is well known that electron’s AMM effect is
large particularly in high intensities. This is not only
10-4
10-3
10-2
10-1
100
101
Experimental data SRDCS (Darwin) NRDCS
Log 10
(DC
S)
(a.u
)
Scattering angle (degree)0 10 20 30 40 50 60
Fig. 4 Variation of differential 1s–2s cross section of e- -
H scattering at 100 eV. The dots are observed values of Williams
[26], solid line represents semi-relativistic approximation and dashed
line corresponds to nonrelativistic DCS
87 88 89 90 91 92 93
0
1
2
3
4
5 SRDCS without AMM SRDCS with AMM
DC
S (
a.u)
Angle θf (degree)
Fig. 5 DCSs (with and without AMM) as a function of the angle hf
for electric field strength e ¼ 0:25 a:u, relativistic parameter c = 2.0
and geometric parameters hi = 90�, /i = /f = 45� and
87� B hf B 93�. The corresponding number of photons exchanged
is ±100
3,8
4,0
4,2
4,4
4,6
4,8
5,0
5,2
5,4
5,6
5,8
6,0
SRDCS without AMM SRDCS with AMM
DC
S (
a.u)
Electric field strength ε (a.u)0,00 0,05 0,10 0,15 0,20 0,25
Fig. 6 DCSs (with and without AMM) as a function of electric field
strength e for a relativistic parameter c = 2.0 and geometric
parameters hi = hf = 90�, /i = /f = 45�. The corresponding num-
ber of photons exchanged is ±100
116 M El Idrissi et al.
explained in this case but also in the case of nonrelativistic
regime (see Fig. 2). On the other hand, SRDCS with
electron’s AMM effects depends on initial-kinetic energy
and also on laser field intensity. The electron’s AMM effect
acts on both electron’s states and ratio of SRDCS with
AMM to SRDCS without AMM. AMM effect changes
with variation of laser intensity. Comparing Figs. 2 and 6,
it is observed that increase of differential cross section in
relativistic regime becomes very important for high laser
intensities. Figure 7 shows SRDCSs dependence on fre-
quency in that with low frequency, a significant difference
appears whereas with high frequency insignificant differ-
ence is observed.
5. Conclusions
In order to calculate differential cross section for semirel-
ativistic 1s–2s collision and when the condition (Za� 1) is
fulfilled, we have used simple Darwin wave functions
which are expected to lose validity as a relativistic states
with very large atomic number Z. We have also focused on
the calculation of inelastic (1s–2s) SRDCS of hydrogen
atom by electronic impact with electron’s anomalous
magnetic moment effects fully included. Different analyt-
ical details for calculation of SRDCS with electron’s AMM
effects are presented. It is found that electron’s AMM
effects increases with laser field intensity e as well as with
low frequency x. We have shown that addition of elec-
tron’s AMM effects considerably enhances analytical
model and gives more accurate results.
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0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,201,4
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1,6
1,7
1,8
1,9
2,0
SRDCS with AMM SRDCS without AMM
DC
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Laser angular frequency ω (a.u)
Fig. 7 DCSs (with and without AMM) as a function of x for an
electric field strength of e ¼ 0:2 a:u, a relativistic parameter c = 1.2
and geometric parameters hi = hf = 90�, /i = /f = 45�. The corre-
sponding number of photons exchanged is ±100
Electron impact excitation of hydrogen atom 117