7
REVIEW PAPER Electron impact excitation of hydrogen atom with anomalous magnetic moment effects M El Idrissi 1 , S Taj 1 , B Manaut 1 * and L Oufni 2 1 Laboratoire Interdisciplinaire de Recherche en Sciences et Techniques, Faculte ´ Polydisciplinaire, Universite ´ Sultan Moulay Slimane, Boite Postale 523, 23000 Beni Mellal, Morocco 2 De ´partement 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 4a e 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) a e , determined from fine-structure a and calculated straight- forwardly by one-loop result ða e ¼ a=ð2pÞ’ 0:0011614Þ [1, 2]. Kinoshita et al. [37] have, after a many-year effort, completed calculations including full five-loop effects. A recent experimental value for a e ¼ 0:001159652180ð73Þ has 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 [1114]. 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 *10 16 W/cm 2 . 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 Ó 2013 IACS

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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

1,5

1,6

1,7

1,8

1,9

2,0

SRDCS with AMM SRDCS without AMM

DC

S (

a.u)

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