polarization dependence of the bremsstrahlung cross- section in...

JKAU: Sci., Vol. 23 No. 2, pp: 65-78 (2011 A.D./1432 A.H.)

DOI: 10.4197 / Sci. 23-2.5

65

Polarization Dependence of the Bremsstrahlung Cross-

Section in the Scattering of Electrons by Nuclei with

Electromagnetic Multipole Moments

S. A. Al-Khateeb

Department of Mathematics, Faculty of Science for Girls,

King Abdulaziz University, Jeddah, Saaudi Arabi

Abstract. The present work deals with one of the most important

problem in the quantum electrodynamics, 1.e the problem of

bremsstrahlung process due to the interaction between lepton

(electron) and the field of nuclei having electromagnetic multipole

distributions. The detaild calculations for the cross-sections and the

angular and energy distributions of the process are given in the first

Born approximation of the quantum perturbation theory in two cases:

With neglecting the polarizations of the photon and the electron, and

with studying these polarizations. Applying the obtained formulae to

some nuclei, the light nucleus ( )94Be and the heavier one ( )27

13Al , to

showing the effect of both circular photon polarization and

longitudinal electron polarization on the bremsstahlung cross-section,

and comparing the results obtained for the two nuclei graphically.

I-Introduction

The Study of the bremsstrahlung process was firstly done by Bethe and

Heitler[1]

in 1934, and many works about this process could have

appeared in the period (1934-1954)[2-4]

. In these works, the polarizations

of the photons of bremsstrahlung and of the electrons were not studied

because of the complicity of the calculations in this case.The circular

polarization of bremsstrahlung photons due to the collision of

longitudinally polarized electrons with nuclei, was firstly investigated by

Grodzins[5]

in 1959. At the same time, Olsen and Maximon[6]

studied the

circular polarization of the bremsstrahlung photons experimentally. Motz

66 S. A. Al-Khateeb

and Placious[7]

in 1960, studied the linear polarization of the

bremsstrahlung photons, and in 1962, Bisi and others[8]

made some

measurements about the dependence of the degree of circular polarization

of bremsstrahlung on the electric and magnetic field and on the atomic

number of the target nuclei. The intensity of bremsstrahlung due to the

interaction of electric and magnetic interactions between electrons and

target nucleus, had been calculated by Ginsberg and Pratt[9]

in 1964, and

in the next year, Mork and Olsen[10]

investigated the polarization effect

on bremsstrahlung, and showed that the study of polarization caused to

increase the cross-section of the process. The study of bremsstrahlung

due to the scattering of longitudinal polarized electrons by different

nuclei was done by Neumcke[11]

in 1966, and after many years, Behncke

and Nakel[12]

in1974 calculated the linear polarization of photons in the

scattering of electrons with carbon nucleus. In 1989, Caffo and others[13]

,

investigated the polarization of the scattered electrons and of the emitted

photons in the bremsstrahlung at high energies. The asymmetry of photon

polarization in the bremsstrahlung due to polarized electrons were

investigated in 1992 by Mergl and others[14]

and in 1996 by Haug[15]

. In

2002, Likhachev and others[16]

, calculated experimentally the degree of

linear and circular polarization of bremsstrahlung photons, and in 2005,

Lee and others[17]

investigated the production of bremsstrahlung in the

field of heavy atoms at intermediate range of energies. Haug[18]

and

Nukel[19]

, in 2006, both studied the photon polarization and the spin

asymmetry in the bremsstrahlung process in details. In 2009,

S.A.Alkhateeb[20]

, obtained the relation between the circular polarization

of photons and the longitudinal polarization of the scattered electrons in

the electromagnetic field of light nuclei.

II-Formulation of the Problem

The bremsstrahlung photons γ are produced in the process of the

interaction of electrons with the field of nuclei:

γ++→+−−

fieNNe (1)

Polarization Dependence of the Bremsstrahlung Cross-Section in the … 67

Feynman diagrams of this process are shown in Fig. a.

From these diagrams we see that the momentum transfer to the

nucleus in the process is ifpkpq −+=

γ. The matrix element (s-matrix) for

the process could be written as the sum of two amplitudes corresponding

to the above diagrams (1) and (2) and has the form[3]:

2

4

1

4

2211

2 )(),()( xdxdxUxxxUeSifif

Γ= ∫ (2)

),()()()()(),(),(221241421121 γγ

γγ kxAxxGxAxAxxGkxAxxKfNNfK

−+−=Γ (3)

Where K

A is the electromagnetic potential of the emitted photon with

momentum γk ,

NA is the potential of the nuclear field, )(

21xxG

f− is the

Feynman propagator of the intermediate state between the two points 2x

and 1x .The potentials

KA and

NA are given in the form[5]:

)(2

4),( xkixki

Kee

VkxA

⋅−⋅

+=

ω

π

εμ

γ (4)

xqi

Ne

q

qdeZxA ⋅

∫−=

23

3

21

)2(4)(

π

π (5)

Where ),0( εεµ

�

= is the polarization vector of the photon, ω is the photon

energy, V is the normalization volume, Z is the atomic number of the

target nucleus, the potential )(xAN

given in (4) represent the coulomb

field of nucleus. The differential cross-section of the bremsstrahlung with

respect to photon energy is given by [4]:

γγ

ω

πω

σ

kefi

i

f

ddkppFqp

pmeZ

d

dΩΩ= ),,(

42

2

0

62

(6)

N

qf

pki

p −=−

γ

fp

ip

q

2x

1x

)1(

qi

pkf

p +=+γ

ip

q

2x

1x

γk

aFig .

γk

N

fp

)2(

68 S. A. Al-Khateeb

In(6): 0

m is the rest mass of electron, e is the electronic charge.

edΩ and

γkdΩ are the solid angles of the scattered electron and the

emmited photon respectively, ),,(γkppF

fi is a function depends on the

initial )(ip and final )(

fp momenta of electron and the photon

momentum )(γk . The final form of this function is given by [20]:

[{ ] [ ]⋅++−

−

=2222

0222

0

)()()()(4)()(

1

4

1γγγγ

γγ

kpkpEkpEkpmkpkpm

Fifiiff

fi

[ ] [ ]})()()(2))((2 2222

0

2

fifififippEEkpqkpmqkpkp −++−

γγγγ (7)

here:

[ ] [ ]ffffiiii

pEkppEkp θωθωγγ

cos)(,cos)( −=−= (8)

[ ]),cos(2cos2cos22222

fifiiifffippppppppq −−+++−= θωθωω (9)

φθθθθ cossinsincoscos),cos(fififi

pp += (10)

fiθθ , are the polar angles of

fipp , with respect to the photon direction,

),cos(fi

pp is the angle between ip and

fp , and φ is the angle between the

planes ),(γkp

iand ),(

γkp

f.

III-Analytical Results for the Problem

1- The basic Formula for the Differential Cross-Section

The differential cross-section of bremsstrahlung in the coulomb

field of the nucleus can be obtained from the above relations, and the

final form of this cross - section is given by:

γθθη

ω

σkefie

ddFqd

dΩΩ⎟⎟

⎠

⎞⎜⎜⎝

⎛= ),(

104

(11)

Which is known as Bethe - Hietler relation for the bremsstrahlung,

where:

( )i

f

p

pZ⎟⎠

⎞⎜⎝

⎛=

ωπ

αη

1

22

32

(12)


⋅ΔΔ

+−Δ

+−Δ

=

fi

i

f

ff

f

i

ii

fieqE

pqE

pF

2

22

2

22

22

2

22

0

2)4(

sin)4(

sin),(

ωθθθθ

−+ )sinsin( 2222

ffiipp θθ

)22(),(2

222 qEEf

fi

fi

fi−−

ΔΔ

θθ

(13)

ffffiiiipEpE θθ cos,cos −=Δ−=Δ (14)

φθθθθ cossinsin),(fififi

ppf = (15)

fiEE , and ω are the energies of initial and final electron and of the

emitted photon, where:

2

0

22

0

2

,, mpEmpEEEffiifi+=+=−=ω (16)

2- Study of the Electron and Photon Polarization and the Electro-

magnetic Structure of Target Nucleus

The studying of the interaction between the longitudinally

polarized electrons and the electromagnetic field of target nucleus

corresponding to the charge distribution (Ze) and the magnetic dipole

moment ( )I

µ of the nucleus, taking into consideration the circular

polarization of the bremsstrahlung photons, gives the following formula

for the differential cross-section of the process:

),,(),,(),,,(Ifi

m

essfi

e

essIfiessdZedZed μθθσθθσμθθσ

γγγ+= (17)

),(),(2

1),,(

10 fieefiefi

e

essdssdZed θθσθθσθθσ

γγ+= (18)

),(),(2

1),,(

10 fimefimIfi

m

essdssdd θθσθθσμθθσ

γγ+= (19)

Where BHe

dd σσ =0

is given by (11) and represent the cross-section due to

the charge distribution without studying the polarization of particles and

photons.

γωηθθσ

keefiedddF

qd ΩΩ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

141

1),( (20)

γωμηθθσ

kemIfimdddF

qad ΩΩ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

020

1)(4),( (21)

70 S. A. Al-Khateeb

γωμηθθσ

kemIfimdddF

qad ΩΩ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

121

1)(2),( (22)

Where I is the spin of target nucleon, epe

dd σσ =1

is the charged cross-

section with studying the polarization, m

d0

σ the magnetic cross-section

without studying polarization, and mpm

dd σσ =1

is the magnetic cross-

section with studying the polarization. The functionsme

FF01

, and m

F1

are

given by [20]:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Δ−

Δ++

ΔΔ−

−++Δ

+

+−+Δ

=

22

2

0

2

0

2

2

22

2

0

2

2

22

1

11),()coscos(

),(2

)coscos(sin

)coscos(sin

),(

if

fiffiiii

fi

fi

fffiiii

f

ff

iffiiii

i

ii

fie

fmpEpEf

mEpEpEp

mEpEpEp

F

θθωθωθωθθ

ωωθωθωθ

ωωθωθωθ

θθ

⎟⎟⎠

⎞⎜⎜⎝

⎛−

Δ+−

ΔΔ+

2

0

2222

22

011

)sinsin(m

Epp

mi

i

iiff

fiω

θθω

(23)

( ) ( )2

22

22

2

22

22

0

4

sin4

4

sin4),(

f

ff

i

i

ii

ffim

ppq

ppqF

Δ++

Δ+=

θθθθ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

Δ

Δ+

Δ

Δ+

ΔΔ+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

ΔΔ− 2

24

),(2 22

2

0

2

i

f

f

i

fi

fi

fi

fi qmEE

qf ωθθ (24)

( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+ΔΔ

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Δ+

Δ−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Δ+

Δ⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= 2

0

2

22

2

0

2

13

211112

4),( mEEEmE

q

pF

fii

fififi

ifim

ωθθ

( )⎟⎟

⎠

⎞

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Δ

Δ−

Δ

Δ−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

ΔΔ

−−

Δ

Δ−+

Δ

Δ+−

i

f

f

ii

fi

fi

f

fi

i

if EmEEppm

2

2222

0

2

2

2

2

2

0

ωωω

(25)

The quantity 1±=e

S represent the helicity of the initial electron.

1+=e

S for the right-longitudinal electron. 1−=e

S for the left-longitudinal

electron. The quantity 1±=γ

S represent the circular polarization of

bremsstrahlung photon. 1+=γ

S for the right circular photon polarization.

1−=γ

S for the left circular photon polarization.


3- The Angular Distribution of the Circular Polarized Bremsstrahlung

Photon

The integration of (17), (18) and (19) with respect to the solid angle

of the scattered electron e

dΩ gives the angular distribution of the circular

polarized bremsstrahlung photon into the form:

),(),(),,(Ii

m

essi

e

essIiessdZedZed μθσθσμθσ

γγγ+= (26)

)()(2

1)(

1 ieeioei

e

essdssdd θσθσθσ

γγ+= (25)

)()(2

1)(

1 imeiomi

m

essdssdd θσθσθσ

γγ+= (28)

Where:

γθηπθσ

kieioedd ΩΦ= )(4)(

0 (29)

γθηπθσ

kieiedd ΩΦ= )(2)(

11 (30)

γθμηπθσ

kimIiomdad ΩΦ= )()(4)(

0 (31)

γθμηπθσ

kimIimdad ΩΦ= )()(2)(

11 (32)

In the ultra relatives case ( )20

,, cmEEkfi>>

γ, the functions

mmee 1010,,, ΦΦΦΦ are given by:

⎩⎨⎧

Δ

Δ−+−+

Δ

−−

Δ

−−−

Δ=Φ

0

0

0

2

0

2

2

00

22

0

0

)2()22(127)1(2

8

1)( ε

γγ

β

γγγθ

amie

⎭⎬⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−−

−⎟⎟⎠

⎞⎜⎜⎝

⎛

Δ

+−+−+

β

ε

β

γ

γ

γ

γ

εγγγγ T

2

2

0

2 13

11

)22(2)2( (33)

⎩⎨⎧

Δ−

Δ−−+

Δ

−−

Δ

−+

Δ=Φ L

amie

0

0

0

2

2

00

22

0

1

)1(

)2()2(1614

8

1)(

γ

γγ

β

γγγθ

⎭⎬⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−−

β

ε

β

γ

γ

γ0

2

2

13

1

(34)

Timε

βγ

γγγε

γ

γ

β

γγθ

3

0

22

0

2

0

2

0

)1(2

)1(

)1(42

)1(

2

3)(

−

Δ+−+

−−

Δ

Δ−−+

−=Φ

0

0

0

2

)1(4

)2()22(ε

γ

γγ

Δ−

Δ++−+ (35)

Timε

βγ

γγγε

γ

γ

β

γγγθ

3

0

0

2

0

1

)1(

)1(

)1(2

)1(1

2)(

−

Δ+−+

−−

⎩⎨⎧

Δ

Δ−−+−=Φ

⎭⎬⎫

Δ−

Δ+−+

0

0

0

)1(2

)2()2(ε

γ

γ (36)

72 S. A. Al-Khateeb

where:

i

i

icm

Ea

Eθ

εγ

γ

cos1,,02

0

−=Δ== (37)

γ

γγγβ

)1(2ln2,2)1(

0

2−

=Δ+−=a

L (38)

[ ] γεγβ

γβεγε ln2,

1

1ln,)1(2ln2

0−=

−+

+−

=−=T

a (39)

The second term in (27) and (28) represents the effect of the spin

correlation between the electron and photon spins on the charge and

magnetic distributions of the target nucleus respectively.

4- Calculation of the Degree of Circular Polarization of the

Bremsstrahlung Photons

The degree of circular polarization in the case of the

bremsstrahlung emitted from the collision of electron with nuclei having

charge distribution and magnetic dipole moment, is give by:

[ ] [ ][ ] [ ]

11

11

),,(),,(

),,(),,(),,(

−==

−==

+

−

=

γγ

γγ

γγ

γγ

μθσμθσ

μθσμθσμθ

sIiess

sIiess

sIiess

sIiess

Ii

ZedZed

ZedZed

ZeP (40)

Using the relations (26) - (36) we get the following relation for (P):

⎥⎦

⎤⎢⎣

⎡

+

+=

)()(1

)()(1)(),,(

0

1

iI

iI

ieeIi

Ra

RapsZeP

θμ

θμθμθ (41)

Where: e

m

i

e

m

i

e

e

ieRRp

1

1

1

0

0

0

0

1 )(,)(,)(Φ

Φ=

Φ

Φ=

Φ

Φ= θθθ

)(ie

p θ represents the degree of charged circular polarization of

bremsstrahlung produced in the collision of longitudinally polarized

electrons )1,1( −=es with the field of nuclei having zero magnetic dipole

moment. If we take the averge on the spin state of electron )1( ±=es and

the summation over the photon spin )1( ±=γs , we have the relation:

[ ]0000

)(1)()(),,( RadddZedIeimieIi

μσθσθσμθσ +=+= (42)

Which represents the angular distribution of the bremsstrahlung

produced by un-polarized electron in the field of a nucleus with spin


I.The second term in (42) gives the magnetic correction to Bethe-Heitler

relation for bremsstrahlung. This correction is more effective for the light

nuclei, which has a large magnetic moment at large angles of the photon

emission.

5- Numerical Application to some Nuclei (Results and Discussion)

We shall now consider some applications of the obtained formulas

to particular nuclei. The electric and magnetic scattering cross-sections

for the two nuclei ( )27

13Al and ( )9

4Be at different values of energy

)5000,3000,1000( Mev=ε and angles )60,30,10( 000

=θ , and for the polarized

and unpolarized cases are considered, and we have the following results:

1- the polarized electric cross-sections for the two nuclei are

decreased with increasing both energies and scattering angles, such that

edσ is more effective at energy Mev1000=ε and angle 0

10=θ [Fig.1+3],

for ex:

)10,1000(107982.6 031

==×=−

θεσ Mevatdep

,

)60,5000(1072937.4 036

==×=−

θεσ Mevatdep

2- the polarized magnetic cross-sections for the two nuclei are also

decreased with increasing both energies and scattering angles, such that

edσ is more effective at energy Mev1000=ε and angle 0

10=θ [Fig(2) +

(4)], for ex:

)10,1000(107122.5 019

==×=−

θεσ Mevatdmp

,

)60,5000(1005474.1 022

==×=−

θεσ Mevatdmp

10 20 30 40 50 60

Angle Degrees q

- 1´ 10 - 23

0

1´ 10 - 23

2´ 10- 23

3´ 10- 23

4´ 10- 23

ssor

Cnoit

ceSHds

peL-g

=05.

Polarization Bremsstrahlung Radiation- AL2 7

1 3

He=0001

,0003

,0005

ve

ML

10 20 30 40 50 60

Angle Degrees q

- 1´ 10 - 24

0

1´ 10 - 24

2´ 10 - 24

3´ 10 - 24

4´ 10 - 24

ssor

Cnoit

ceSHds

peL

Polarization Bremsstrahlung Radiation- Be94

He=0001

,0003

,0005

ve

ML

Fig.3 Fig.1

74 S. A. Al-Khateeb

3- comparing the polarized ep

dσ and unpolarized e

dσ electric cross-

sections for the two nuclei ( )27

13Al and ( )9

4Be [Fig(5) + (6)], we see that:

i) the values of cross-sections for ( )27

13Al are larger than the value for

( )94

Be .

ii) the value of polarized ep

dσ electric cross-section for the two

nuclei is larger than the value for the unpolarized e

dσ cross-section, for

ex:

)10,1000(107982.6 031

==×=−

θεσ Mevatde

,

)10,1000(1075822.4 023

==×=−

θεσ Mevatdep

10 20 30 40 50 60

Angle Degrees q

- 1´ 10 - 19

0

1´ 10 - 19

2´ 10- 19

3´ 10- 19

4´ 10- 19

5´ 10- 19

ssor

Cnoit

ceSHds

pmL-g

=05.

Polarization Bremsstrahlung Radiation - AL2 7

1 3

He=0001

,0003

,0005

ve

ML

10 20 30 40 50 60

Angle Degrees q

- 2´ 10 - 20

0

2´ 10 - 20

4´ 10 - 20

6´ 10 - 20

8´ 10 - 20

1´ 10 - 19

ssor

Cnoit

ceSHds

pmL-g

=05.

Polarization Bremsstrahlung Radiation - Be94

He=0001

,0003

,0005

ve

ML

10 20 30 40 50 60

Angle Degrees q

0

2 ´ 10 -31

4 ´ 10 -31

6 ´ 10 -31

ssor

Cnoit

ceSH-

LA

72

31

,-e

B94L

Bremsstrahlung Radiation ds e

He=0001

ve

ML

10 20 30 40 50 60

Angle Degrees q

0

5 ´ 10 -24

1 ´ 10 -23

1.5 ´ 10 -23

ssor

Cnoit

ceSH-

LA

72

31

,-e

B94L

Polarization Bremsstrahlung Radiation ds ep

He=0001

ve

ML

Fig.4 Fig.2

Fig.6 Fig.5


4- comparing the polarized mp

dσ and un polarized m

dσ magnetic

cross-sections for the two nuclei ( )27

13Al and ( )9

4Be [Fig(7) + (8)],we see

that:

i) the value of cross-sections for ( )27

13Al are larger than the value for

( )94

Be .

ii) the value of polarized mp

dσ magnetic cross-section for the two

nuclei is larger than the value for the un polarized m

dσ cross-section, for

ex:

)10,1000(1073899.4 023

==×=−

θεσ Mevatdm

,

)10,1000(107122.5 019

==×=−

θεσ Mevatdmp

5- comparing the total electric cross-section epe

dddE σσ += and the

total magnetic cross-section mpm

dddM σσ += for the two nuclei [Fig(9) +

(10)],we see that:

The values of E

dσ and M

dσ for the ( )27

13Al nucleus are larger than

that for the ( )94

Be nucleus, and have the largest value at 01000 , 10 .Mevε θ= =

10 20 30 40 50 60

Angle Degrees q

0

1 ´ 10 -23

2 ´ 10 -23

3 ´ 10 -23

4 ´ 10 -23

ssor

Cnoit

ceS

-L

A7

23

1,-

eB

94

Bremsstrahlung Radiation dsm

e=

0001

ve

M

10 20 30 40 50 60

Angle Degrees q

- 1´ 10- 19

0

1´ 10- 19

2´ 10- 19

3´ 10- 19

4´ 10- 19

5´ 10- 19

ssor

Cnoit

ceS

-L

A7

23

1,-

eB

94

Polarization Bremsstrahlung Radiation dsmp

e=

0001

ve

M

Fig.8 Fig.7

76 S. A. Al-Khateeb

6- comparing between the electric and magnetic parts of the cross-

section for the two nuclei [Fig (11) + (12)], where

mpmepedddMdddE σσσσ +=+= , . We see that:

i) the magnetic part is more effective (i.e has larger values) in the

polarized and un polarized cross for the two nuclei.

ii) in the polarized case, the values of cross-sections (electric or

magnetic) are larger than the unpolarized case, which means that the

study of polarization in the bremsstrahlung process gives us best result,

and make the process more evident than that if we neglect the

polarization effect, a result in consistent with the experimental results.

10 20 30 40 50 60

Angle Degrees q

- 1´ 10 - 23

0

1´ 10 - 23

2´ 10 - 23

3´ 10 - 23

4´ 10 - 23

ssor

Cnoit

ceS

EdH-

LA

72

31

,-e

B94L

Polarization Bremsstrahlung Radiation

He=0001

ve

ML

10 20 30 40 50 60

Angle Degrees q

- 1´ 10 - 19

0

1´ 10 - 19

2´ 10 - 19

3´ 10 - 19

4´ 10 - 19

5´ 10 - 19

ssor

Cnoit

ceS

MdH-

LA

72

31

,-

eB

94L

Polarization Bremsstrahlung Radiation

He=0001

ve

ML

10 20 30 40 50 60

Angle Degrees q

0

1 ´ 10 -31

2 ´ 10 -31

3 ´ 10 -31

4 ´ 10 -31

5 ´ 10 -31

6 ´ 10 -31

ssor

Cnoit

ceS

EdH-

LA

72

31

,-e

B94L

Bremsstrahlung Radiation

He=0001

ve

ML

10 20 30 40 50 60

Angle Degrees q

0

1 ´ 10-23

2 ´ 10-23

3 ´ 10-23

4 ´ 10-23

ssor

Cnoit

ceS

MdH-

LA

72

31

,-

eB

94L

Bremsstrahlung Radiation

He=0001

ve

ML

Fig.11 Fig.12

Fig.9Fig.10


Acknowledgements

The author would Like to thank the Deanship of Scientific

Research at King Abdulaziz University to support this research (no. 21-

h008/429) and encourage researchers to continue the march of scientific

research.

References

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[4] Bethe, H. and Maximon, L. (1954) Phys. Rev. 93: 768.

[5] Grodzins, L. (1959) Progr. Nucl. Phys. 7: 163.

[6] Olsen, H. and Maximon, L. (1959) Phys. Rev. 114: 887.

[7] Motz, J. and Placious, R. (1960) Nuovo. Cimento. 15: 572.

[8] Bisi, A. et al. (1962) Nucl. Phys. 36: 320.

[9] Ginsberg, E. and Pratt, R. (1964) Phys. Rev. 134: 773.

[10] Mork, K. and Olsen, H. (1965) Phys. Rev. B140: 1661.

[11] Neumcke, B. (1966) Phys. Lett. 23: 382.

[12] Behncke, H. and Nakel, W. (1974) Phys. Lett. A47: 149.

[13] Caffo, M. et al. (1989) Nucl. Phys. B327: 93.

[14] Mergl, E. et al. (1992) Phys. Rev. Lett. 69: 901.

[15] Haug, E. (1996) Z. Phys. D37: 9.

[16] Likhachev, V. et al. (202) Nucl. Inst. Meth. A495: 139.

[17] Lee, R. et al. (2005) J. Exp. Theor. Phys. 100: 1.

[18] Haug, E. (2006) J. Rad. Phys. and Chem. 75: 1330.

[19] Nakel, W. (2006) J. Rad. Phys. and Chem. 75: 1164.

[20] Alkhateeb, S.A. (2009) Quantum Electrodynamical treatment of leptonic pair production

processes, Ph.D. Thesis, K.S.A.

78 S. A. Al-Khateeb

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