The second Born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal
The formulae for the cross section of bremsstrahlung by relativistic electrons and positrons taking into account the contribution of the second Born approximation are obtained. The dependence of the radiation cross section in the field of atomic plane on the sign of charge of the particle is conside...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2001 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Цитувати: | The second Born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal / N.F. Shul’ga, V.V. Syshchenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 131-134. — Бібліогр.: 9 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859755751229095936 |
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| author | Shul’ga, N.F. Syshchenko, V.V. |
| author_facet | Shul’ga, N.F. Syshchenko, V.V. |
| citation_txt | The second Born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal / N.F. Shul’ga, V.V. Syshchenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 131-134. — Бібліогр.: 9 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The formulae for the cross section of bremsstrahlung by relativistic electrons and positrons taking into account the contribution of the second Born approximation are obtained. The dependence of the radiation cross section in the field of atomic plane on the sign of charge of the particle is considered.
|
| first_indexed | 2025-12-02T01:01:03Z |
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| fulltext |
THE SECOND BORN APPROXIMATION IN THEORY OF
BREMSSTRAHLUNG OF RELATIVISTIC ELECTRONS AND
POSITRONS IN CRYSTAL
N.F. Shul’ga1, V.V. Syshchenko2
1National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: shulga@kipt.kharkov.ua
2Belgorod State University, Belgorod, Russian Federation
e-mail: syshch@bsu.edu.ru
The formulae for the cross section of bremsstrahlung by relativistic electrons and positrons taking into account
the contribution of the second Born approximation are obtained. The dependence of the radiation cross section in the
field of atomic plane on the sign of charge of the particle is considered.
PACS: 12.20.-m, 41.60.-m
In this paper we consider the second Born correction
to the process of bremsstrahlung of high energy
electrons and positrons in an external field. The account
of the second Born approximation leads to dependence
of the radiation cross section on the charge sign of
radiating particle. It is demonstrated that contribution of
the second Born approximation can be substantial for
the case of coherent interaction of radiating particle with
atoms of a crystal.
1. DIFFERENTIAL CROSS SECTION OF
THE RADIATION PROCESS
The cross section of the bremsstrahlung of electrons
and positrons in an external field is determined by the
relation [1]
kdpdMed 332
4
2
')'(
')2(4
ωεεδ
ω ε επ
σ −−= , (1)
where ),( pε and )','( p
ε are the energy and the
momentum of the initial and final particles, ω and k
are the frequency and the wave vector of the radiated
wave, )'( ωεεδ −− is the delta-function that determines
the energy conservation under radiation. According to
the rules of diagram technique [1] the squared matrix
element in (1) can be written with the account of the
contribution of the second Born approximation in the
form
∫ −−= 3
3
*
21
22
1
2
)2(
Re2
π
qdUUMMUUMM qgqgg ,
(2)
where gU is the Fourier component of the potential
energy of the electron (positron) in an external field,
),0( gg
=µ is the 4-momentum transferred to the
external field (it is assumed that the external field is
stationary), µµµµ kppg −−= ' , 1M and 2M are the
matrix elements which determine contributions of the
first and the second Born approximations (see Fig. 1):
u
egge
ebuM
gg
−−=
τε
γ
ε σ
γ
'2
ˆˆ
2
ˆˆ
ˆ' 00
1 , (3)
+
−+
−
−−+=
qqqg v
q
e
v
q
v
q
p
gmpeuM
σ
ε
γ
τ
ε
γ
σ
ε
γ
γ
σ
2
ˆ
1
ˆ
'
'2
'ˆ
1
2
ˆ
1
2
ˆˆˆ'
0
'
00
02
ue
p
gmp
v
q
gq
+++
+ ˆ
'2
ˆ'ˆ
'
'2
'ˆ
1
0
'
0
τ
γ
τ
ε
γ
, (4)
where µe is the photon polarization vector, v and v' are
the initial and final velocities of the electron,
µµµ qgq −=' . The values b, gσ and gτ in 1M and
2M are determined by the relations
gg
b
τσ
11 −= ,
p
g
gg 2
2
||
−=σ ,
'2
2
|| p
g
gngg
++= ⊥τ , (5)
where '/' ppn
= is the unit vector along the momentum
'p direction, and ⊥n are the components of this vector
orthogonal to the p .
Fig. 1. Feynman diagrams corresponding to the first
and the second Born approximations in the process of
bremsstrahlung in an external field
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 131-134. 131
The matrix element of the radiation process
depends on the momentum transferred to the external
field g
in an explicit form. The cross section itself can
be also expressed directly through the transferred
momentum (and also through the angle ϑ between the
vectors k
and p ). Such presentation is especially
convenient in the range of small values of the
transferred momentum mg < <⊥ , because it is possible
to make an expansion in the matrix element by the
powers of ⊥g in this case. Transformation to the new
variables is described in [2,3]. The differential cross
section in this case takes the form:
gd
y
dyd
m
Med 3
22
2
4
4
1
'
)2( −
=
ω
ωδ
ε
ε
π
σ , (6)
where '2/2 ε εωδ m= . The variable y is connected to
ϑ by the relation
( ) ayfm +=2/ε ϑ , 11 ≤≤− y , (7)
where
−−= ⊥⊥
ε
δ
δ 2
4 2
||2
2 gg
m
ga ,
+−−= ⊥⊥
2
22
|| 2
1
m
gg
gf
δ
ε
δ
δ
.
From the fact that the value a in the radical in (7) must
be positive one can conclude that
εδ 2/2
|| ⊥+≥ gg . (8)
Note that Eq. (7) determines the possible values of
the radiation angle ϑ under given values of ||g and ⊥g
.
Eq. (4) can be simplified that makes the procedure
of summing over polarization of interacting particles
more easy than for original Eq. (4). Neglecting the
terms of the order of 22 / εm , we obtain after some
calculations the following expression for 2M :
uQ
qqeg
eQuM
qqg
+
+−= ⊥⊥
2
'
0
12 2
'
'2
ˆ
ˆ
'
'
σε σε
γ
τε ε
ω
,(9)
where 1Q is the spinor structure of 1M ,
gg
egge
beQ
τε
γ
ε σ
γ
'2
ˆˆ
2
ˆˆ
ˆ 00
1 −−= ,
and
qqqgqg
qeqegqqge
Q
στε ε
γγ
ττεσσε '
00
'
222 '4
ˆˆ'ˆ
'4
ˆˆ'ˆ
4
ˆˆˆ ⊥⊥⊥⊥ +−−= .
After summing over polarizations of final particles
and averaging over polarization of initial particle we
obtain with accuracy to terms of order of 22 / εm the
following equations for the values 2
1M and *
21MM
in (1):
−
+= ⊥
222
2
||
2
1 2
'
'
2|| bmg
g
M
ε
ε
ε
ε
, (10)
[
−−⋅−= ⊥⊥ bmppM
qq
MM
gqq
)2'(2
'
2||
2
' 22
1
'
*
21 τε
ω
σε σ
−
−+⋅+− ⊥ bgpg
ggg '
8
'
2
'
''
'
2
ε
ε
τε
ε
σε
εε
τε
ε
, (11)
Substituting these equations into (1), we obtain after
the integration over y and expansion on mg /⊥ the
following expression for the cross section of the
radiation with account of the second Born
approximation:
⊥⊥
⊥= dggdg
g
gd
m
ed ||2
||
2
22
2 '
2 ω
ωδ
ε
ε
π
σ +
2
gUF
×
−+
−−++
||
2
||||
3 1
'2
141
')2(
1
ggg
F δ
ε ε
ωδδ
ε
ω
επ
∫
++−
−
× −
⊥⊥⊥
qgqg UU
iqiqg
qqg
qdU
)0)(0(
)(
Re
||||||
3
,
(12)
where
−−+=
||||
2
12
'2
1
gg
F δδ
ε ε
ω
.
Let us consider some particular cases of (12). If the
condition eff
q ||< <δ is satisfied, where eff
q || are the
characteristic values of the longitudinal component of
the momentum q
in (12), we can neglect the
dependence of gU and qgU − on ||g in (12). After
integration over ||g we obtain that
2
22
)2('4
31)()(
πε ε
ωσ ⊥
⊥⊥ ⋅
+=
gd
gdwgd −
2
gU
+×
eff
eff
q
g
O
||
||
1 , (13)
where
ω
ω
ε
ε
π
d
m
gegdw 2
22 '
3
2)( ⊥
⊥ = . (14)
For εω < < Eq. (13) corresponds to the product of
the radiation probability ωddw / and the cross section
of elastic scattering of the particle in the external field
eldσ with account of contribution of the second Born
approximation,
×= ⊥
⊥ 2
2
4
)(
π
σ
gdgd el
−
−× ∫ ⊥⊥
'2
||
3
32
)0(
'
)2(
Re1
qqgg UU
iq
qqqdUU
πε .
For the Coulomb field of the nucleus with charge
|| eZ the last equation transforms to the form
132
−Ω=⊥ ϑπ
ϑε
πσ
2||
1)(
2
42
42 Ze
e
edeZgd el ,
where the scattering angle pg /⊥≈ϑ . The last result
coincides with the corresponding result of the paper [4]
obtained by different method. For arbitrary external
field the formula for eldσ was obtained in [5,6].
Note that radiation of electrons in Coulomb field
exceeds slightly the radiation of positrons. That is due to
the fact that electron attracted by the nucleus moves in
the region with larger gradient of the potential than the
positron.
So in the range of frequencies εω ~ the theorem
about factorization of the radiation cross section,
according to which
)()( ⊥⊥≈ gdgdwd elσσ , (15)
is justified with an accuracy to the correction which
determines the contribution of the second Born
approximation.
2. THE CROSS SECTION FOR RADIATION
OF RELATIVISTIC ELECTRONS AND
POSITRONS IN THE FIELD OF ATOMIC
PLANE IN A CRYSTAL
We can see that dependence of the radiation cross
section on the particle charge sign in the case of
radiation of high energy electrons and positrons in the
field of single atom is rather small. Different situation
arises for coherent interaction of relativistic particles
with atoms of crystal lattice. In this case, due to the
coherent effect the dependence of the radiation cross
section on the particle charge sign can be substantially
amplified in comparison with analogous dependence of
the radiation cross section in an amorphous medium.
The attention to this fact was paid in [7] during
consideration of contribution of the second Born
approximation into coherent radiation cross section of
relativistic electrons in the field of atomic plane of the
crystal. It was demonstrated that in considered case the
relative contribution of the second Born approximation
into coherent radiation cross section is determined by
the parameter
22
2
θε
α
a
RZe
p =
2
2
~
θ
θ c (16)
which represents by the order of value the ratio of the
squared critical angle of plane channeling [8] to the
squared angle of incidence θ of the beam to the atomic
plane (here || eZ is the charge of the nucleus of crystal
lattice atom, R is the screening radius of the atomic
potential, a is the average distance between atoms in the
crystal plane). In this case the Born expansion of the
radiation cross section is valid if 1< <pα . The
parameter pα rapidly increases with θ decrease. Under
1~pα the account of effects of channeling and above-
barrier motion of particles in respect to the crystal
atomic plane is necessary [3,8,9].
So consider the coherent radiation of electrons and
positrons in the field of continuous potential of one of
the atomic planes in a crystal under incidence of the
beam under small angle θ to this plane. The potential
energy of the particle in continuous potential of the
plane is determined by Eq. (8,9)
∫ ∑
=
−=
N
n
n
zy
rrudydz
LL
xU
1
)(1)(
, (17)
where )( nrru
− is the particle potential energy in field
of the single atom of crystal plane located in the point
nr
, yL and zL are the linear dimensions of the plane
and x is the coordinate, orthogonal to the atomic plane
of the crystal (summation in (17) is made over all atoms
of the crystal plane). Taking the atomic potential in the
form of the screened Coulomb potential, and
Rre
r
eZeru /||)( −= ,
we find the expression for the Fourier component of
(17):
g
zy
yzg u
aa
ggU 1)()()2( 2 δδπ= , (18)
where ya and za are the distances between atoms in
the plane along the axes y and z, and
22
||4
−+
=
Rg
eeZu g
π
.
Substituting the Fourier component (18) into (12),
we obtain the following expression for the radiation
cross section
×= 22
32 '16
θω
ωδ
ε
επασ x
zy
dgd
maa
NZd
+
+
−−+× − 222
2
)(
112
'2
1
Rggg xxx θ
δ
θ
δ
ε ε
ω
×
+
+ − 22
12
|| Rgaa
Z
e
e
xzyε
α
+
−−+×
θ
δ
θ
δ
ε ε
ω
xx gg
12
'2
1
2
×
−+
−−+
θ
δ
ε ε
ω
θ
δ
θ
δ
ε
ω
xxx ggg
1
'2
141
'
2
+
× − 222 4
21
Rg
R
x
π
θ
. (19)
Here we have used the fact that in the case under
consideration xgg θ≈|| . The value xg here covers the
range θδ /≥xg . Under εω < < Eq. (19) transforms to
the corresponding result of the paper [7]. Note that in
the case of interaction of the particle with continuous
potential of the plane the radiation cross section cannot
be presented in the form (15) for any photon
frequencies. This is due to the fact that elastic scattering
on the continuous plane can take place only to some
fixed angles to the plane [5,6] because of energy and
momentum conservation laws in the process of elastic
scattering.
Eq. (19) demonstrates that for all frequencies the
cross section of radiation by positrons turns out larger
133
than the cross section of radiation by electrons, in
difference to the case of radiation in Coulomb field.
This result can be explained by the following way. The
sign of the effect is determined by competition of two
factors: (i) the electron is attracted to the plane and
moves in the region with larger gradient of the potential
than the positron, that leads to increase of radiation;
(ii) in distinct to the positron, it spends less time in the
region with large gradient of the potential, that leads to
decrease of radiaton. In Coulomb field the first factor
plays the determinative role, in the field of atomic plane
- the second one.
Eq. (19) demonstrates also that radiation spectrum
ωσω dd / posesses the maximum in the range of
frequencies satisfying the condition
θω
ωεε R
m
2~)(2
2
−
. (20)
With the particle energy growth the position of this
maximum moves to the region of high frequencies. For
θε /~ 2 Rm the maximum is located in the region of
frequencies for which the effect of recoil under radiation
is substantial. The parameter (16) that determines
dependence of the cross section on the particle charge
sign for θε /~ 2 Rm takes the form
θ
α 22
2
~
am
Ze
p .
So in the range of energies under consideration with
decrease of θ the dependence of coherent radiation
cross section on the charge sign of the particle becomes
substantial in the whole range of frequencies of radiated
photons.
3. COHERENT RADIATION ON A SET OF
ATOMIC PLANES IN THE SECOND BORN
APPROXIMATION
The cross section of bremsstrahlung on the crystal is
determined by relation [3]:
)( incohcoh ddNd σσσ += ,
where N is the whole number of atoms in crystal, cohdσ
is the coherent part of radiation cross section caused by
interference of radiation produced on different atoms
regularly arranged in the crystal, incohdσ is the
incoherent part caused by thermal spread of atom
positions in the crystal. For the case of interaction of the
particle with the set of parallel atomic planes in the
crystal, we can obtain the equation for cohdσ from (19)
by change of integration over xdg to the summation
over n
a
g
x
nx
π2)( = :
∑∫
≥
∞
→
θδθδ
π
//
...2...
xgx
x a
dg ,
where ax is the distance between atomic planes. The
cross section of coherent radiation of 1 GeV positrons
and electrons incident under the angle 4104 −⋅=θ
radians to the <011> plane of the Si crystal is shown on
the Fig. 2. We can see that the difference between
radiation cross sections for positrons and electrons in
the case illustrated is of order of 10%.
Fig. 2. The cross section of coherent radiation of
1 GeV positrons (solid line) and electrons (dashed line)
incident under the angle 4104 −⋅=θ radians to the
<011> plane of the Si crystal. The dotted line shows the
Bethe-Heitler cross section
This work is supported in part by Russian
Foundation for Basic Research (Project № 00-02-
16337).
REFERENCES
1. A.I. Akhiezer, V.B. Berestetskii. Quantum
Electrodynamics. New York: “Interscience Publ.”,
1965, 868 p.
2. M.L. Ter-Mikaelian. High-Energy Electro-
dynamic Processes in Condensed Matter. New
York: “Wiley Interscience”, 1972, 457 p.
3. A.I. Akhiezer, N.F. Shul’ga. High-Energy
Electrodynamics in Matter. Amsterdam: “Gordon
and Breach”, 1996, 388 p.
4. W.A. McKinley, G. Feshbach. The Coulomb
Scattering of Relativistic Electrons by Nuclei //
Phys. Rev. 1948, v. 74, p. 1759-1763.
5. V.V. Syshchenko, N.F. Shul’ga. Elastic scat-
tering of high energy charged particles in an
external field in the second Born approximation //
Ukr. Fiz. Zh. 1995, v. 40, №1-2, p. 15-21.
6. A.I. Akhiezer, N.F. Shul'ga, V.I. Truten',
A.A. Grinenko, V.V. Syshchenko. Dynamics of
high-energy charged particles in straight and bent
crystals // Physics-Uspekhi. 1995, v. 38, №10,
p. 1119-1145.
7. A.I. Akhiezer, P.I. Fomin, N.F. Shul’ga.
Coherent bremsstrahlung of electrons and positrons
of ultrahigh energy in crystals // JETP Lett. 1971,
v. 13, №12, p. 506-508.
8. J. Lindhard. Influence of Crystal Lattice on
Motion of Energetic Charged Particles // K. Dan.
Vidensk. Selsk. Mat.-Fyz. Medd. 1965, v. 34, №14.
9. D.S. Gemmell. Channeling and related effects
in the motion of charged particles through crystal //
Rev. of Mod. Phys. 1974, v. 46, №1, p. 129-228.
134
THE SECOND BORN APPROXIMATION IN THEORY OF BREMSSTRAHLUNG OF RELATIVISTIC ELECTRONS AND POSITRONS IN CRYSTAL
N.F. Shul’ga1, V.V. Syshchenko2
1National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: shulga@kipt.kharkov.ua
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-79412 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-02T01:01:03Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Shul’ga, N.F. Syshchenko, V.V. 2015-04-01T18:55:49Z 2015-04-01T18:55:49Z 2001 The second Born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal / N.F. Shul’ga, V.V. Syshchenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 131-134. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 12.20.-m, 41.60.-m https://nasplib.isofts.kiev.ua/handle/123456789/79412 The formulae for the cross section of bremsstrahlung by relativistic electrons and positrons taking into account the contribution of the second Born approximation are obtained. The dependence of the radiation cross section in the field of atomic plane on the sign of charge of the particle is considered. This work is supported in part by Russian Foundation for Basic Research (Project № 00-02- 16337). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Electrodynamics of high energies in matter and strong fields The second Born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal Второе борновское приближение в теории тормозного излучения релятивистских электронов и позитронов в кристалле Article published earlier |
| spellingShingle | The second Born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal Shul’ga, N.F. Syshchenko, V.V. Electrodynamics of high energies in matter and strong fields |
| title | The second Born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal |
| title_alt | Второе борновское приближение в теории тормозного излучения релятивистских электронов и позитронов в кристалле |
| title_full | The second Born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal |
| title_fullStr | The second Born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal |
| title_full_unstemmed | The second Born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal |
| title_short | The second Born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal |
| title_sort | second born approximation in theory of bremsstrahlung of relativistic electrons and positrons in crystal |
| topic | Electrodynamics of high energies in matter and strong fields |
| topic_facet | Electrodynamics of high energies in matter and strong fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79412 |
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