Circular polarization of the high energy electron bremsstrahlung with account of the second Born approximation
The estimation of contribution of the second Born approximation into circular polarization of bremsstrahlung from linearly polarized electrons is obtained. Account of this contribution could be important for correct interpretation of results of measurements of fine effects due to weak interactions.
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| Zitieren: | Circular polarization of the high energy electron bremsstrahlung with account of the second Born approximation / N.F. Shul’ga, V.V. Syshchenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 214-217. — Бібліогр.: 6 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1116012025-02-23T18:04:23Z Circular polarization of the high energy electron bremsstrahlung with account of the second Born approximation Цiркулярна поляризацiя гальмовного випромiнювання електронiв високоϊ енергiϊ з урахуванням другого борнiвського наближення Циркулярная поляризация тормозного излучения электронов высокой энергии с учетом второго борновского приближения Shul’ga, N.F. Syshchenko, V.V. QED processes in strong fields The estimation of contribution of the second Born approximation into circular polarization of bremsstrahlung from linearly polarized electrons is obtained. Account of this contribution could be important for correct interpretation of results of measurements of fine effects due to weak interactions. Отримано оцінку внеску другого борнiвського наближення до циркулярної поляризації гальмівного випромінювання лiнiйно-поляризованими електронами. Урахування цього внеску може бути важливим для коректної iнтерпретацiї результатів вимірювання тонких ефектів, що обумовлені слабкими взаємодіями. Получена оценка вклада второго борновского приближения в циркулярную поляризацию тормозного излучения линейно-поляризованными электронами. Учет этого вклада может быть важным для корректной интерпретации результатов измерения тонких эффектов, обусловленных слабыми взаимодействиями. 2007 Article Circular polarization of the high energy electron bremsstrahlung with account of the second Born approximation / N.F. Shul’ga, V.V. Syshchenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 214-217. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 12.20.Ds, 13.40.Ks, 41.60.-m https://nasplib.isofts.kiev.ua/handle/123456789/111601 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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English |
| topic |
QED processes in strong fields QED processes in strong fields |
| spellingShingle |
QED processes in strong fields QED processes in strong fields Shul’ga, N.F. Syshchenko, V.V. Circular polarization of the high energy electron bremsstrahlung with account of the second Born approximation Вопросы атомной науки и техники |
| description |
The estimation of contribution of the second Born approximation into circular polarization of bremsstrahlung from linearly polarized electrons is obtained. Account of this contribution could be important for correct interpretation of results of measurements of fine effects due to weak interactions. |
| format |
Article |
| author |
Shul’ga, N.F. Syshchenko, V.V. |
| author_facet |
Shul’ga, N.F. Syshchenko, V.V. |
| author_sort |
Shul’ga, N.F. |
| title |
Circular polarization of the high energy electron bremsstrahlung with account of the second Born approximation |
| title_short |
Circular polarization of the high energy electron bremsstrahlung with account of the second Born approximation |
| title_full |
Circular polarization of the high energy electron bremsstrahlung with account of the second Born approximation |
| title_fullStr |
Circular polarization of the high energy electron bremsstrahlung with account of the second Born approximation |
| title_full_unstemmed |
Circular polarization of the high energy electron bremsstrahlung with account of the second Born approximation |
| title_sort |
circular polarization of the high energy electron bremsstrahlung with account of the second born approximation |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2007 |
| topic_facet |
QED processes in strong fields |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111601 |
| citation_txt |
Circular polarization of the high energy electron bremsstrahlung with account of the second Born approximation / N.F. Shul’ga, V.V. Syshchenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 214-217. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-11-24T06:10:19Z |
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2025-11-24T06:10:19Z |
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| fulltext |
CIRCULAR POLARIZATION OF THE HIGH ENERGY ELECTRON
BREMSSTRAHLUNG WITH ACCOUNT OF THE SECOND
BORN APPROXIMATION
N.F. Shul’ga1, and V.V. Syshchenko2
1A.I. Akhiezer Institute of Theoretical Physics
National 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 estimation of contribution of the second Born approximation into circular polarization of bremsstrahlung
from linearly polarized electrons is obtained. Account of this contribution could be important for correct interpreta-
tion of results of measurements of fine effects due to weak interactions.
PACS: 12.20.Ds, 13.40.Ks, 41.60.-m
1. INTRODUCTION
It is well known [1-3] that the bremsstrahlung emit-
ted by polarized electrons could possess nonzero circu-
lar polarization. It is, particularly, an interesting fact
because the particles produced in weak processes turned
out to be polarized. Circular polarization of
bremsstrahlung was considered in many papers (see
article [2] and references therein) in the frameworks of
the first Born approximation of quantum electrodynam-
ics. However, it could be necessary to account for the
second Born approximation contribution for correct
interpretation of the results of measurements of fine
effects due to weak interactions. It is of interest also that
the second Born approximation leads to the dependence
of radiation characteristics (the cross section and polari-
zation) on the charge sign of the radiating particle.
2. BREMSSTRAHLUNG MATRIX ELEMENT
AND CROSS SECTION
The cross section of bremsstrahlung in an external
field is determined by formula [3]
kdpdMed if
332
4
2
')'(
')2(4
ωεεδ
ωεεπ
σ −−= , (1)
where e is the electron charge, (ε, p) and (ε', p') are the
energy and momentum of the initial and final particles,
(ω, k) are the frequency and wave vector of the photon
emitted, δ(ε - ε' - ω) is the delta-function expressing the
energy conservation under radiation, and Mf i is the ma-
trix element of the radiation process. Since the main
contribution to the bremsstrahlung cross section is made
by small values of the momentum g = p - p' - k trans-
ferred to the external field (the last is assumed to be
stationary and potential), g << m, it is convenient to
express the matrix element as a function of the trans-
ferred momentum. This permits to make an expansion
in the matrix element by the powers of small parameter
g / m.
According to the rules of diagram techniques [3] the
squared absolute value of the matrix element in (1)
could be written in the form
∫ −+
=
3
3
**21
22
1
2
)2(
Re2
||
π
qdUUMMU
UMM
qgqg
gif
, (2)
where Ug is Fourier component of the particle potential
energy in the external field, M1 and M2 are the matrix
elements that determine contributions of the first and the
second Born approximations (see Figure).
Feynman diagrams corresponding to the first and
the second Born approximations in the description of
bremsstrahlung in the external field
Discriminating in the propagator in M1 the depend-
ence on the longitudinal and transverse components of
the transferred momentum g in an explicit form we
could write
uQuM 11 '= , (3)
where
gg
eggebeQ
τε
γ
εσ
γ
'2
ˆˆ
2
ˆˆˆ 00
1 −−= ,
gµ = (0, g) = pµ - p'µ - kµ is the transferred 4-
momentum, pµ , p'µ , kµ are the 4-momenta of the initial
and final electrons and photon, eµ is the photon’s polari-
zation vector, , γµ
µγpp =ˆ µ are Dirac matrices, v and v'
are the velocities of the initial and final electrons. The
values b, σg and τg are determined by the formulae
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (1), p. 214-217 214
−=
gg
gb
τσ
11
|| ,
p
gg 2
2
||
g
−=σ ,
'2
2
|| p
gg
ggn ++= ⊥τ ,
where is the unit vector along the momen-
tum p' direction, and n
|'|/' ppn =
⊥ is the component of this vector
orthogonal to p.
Neglecting the terms of order m 2 / ε 2 and m 2 / ε' 2
the formula for M2 could be derived to the form [4,5]
uQegeQuM
qqg
+
+−= ⊥⊥
2
'
0
12 2
'
'2
ˆˆˆ
'
'
σεσε
γ
τε
ω qq ,(4)
where q 'µ = gµ - qµ and
qqqgqg
qeqegqqgeQ
στεε
γγ
ττεσσε '
00
'
222 '4
ˆˆ'ˆ
'4
ˆˆ'ˆ
4
ˆˆˆ ⊥⊥⊥⊥ +−−= .
The cross section itself also could be expressed
through the transferred momentum (and the angle ϑ
between the vectors k and p). Transformation to new
variables is described in [6]. After that the differential
cross section gets the form
gd
y
dyd
m
Med if
3
22
2
4
4
1
'
)2( −
=
ω
ωδ
ε
ε
π
σ , (5)
where . The variable y is connected to ϑ
by the relation
'2/2 εεωδ m=
( ) ayfm +=2/εϑ , , (6) 11 ≤≤− y
where
−−= ⊥⊥
ε
δ
δ 2
4 2
||2
2 gg
m
ga ,
+−−= ⊥⊥
2
22
|| 2
1
m
gggf δ
ε
δ
δ
,
||g and g ⊥ are the components of g that parallel and
orthogonal to the momentum p of the projectile particle.
Eq. (6) determines the possible values of the radiation
angle under given values of and and gϑ ||g ⊥. One
could conclude from the condition of positiveness of the
value a under radical in (6) that
εδ 2/2
|| ⊥+≥ gg .
The formulae that describe the bremsstrahlung cross
section averaged over polarizations of initial electron
and summed over polarizations of final particles were
obtained in [4, 5].
3. ACCOUNT OF POLARIZATION EFFECTS
Remember how to take into account the polarization
of interacting particles. Matrix element of a process
with single electron in initial and final states has the
form uQuM if '= . The squared absolute value of such
matrix element is equal to (see, e.g., [3])
{ }QuuQuuMMM ififif ''Sp*2
== ,
where
00 γγ += QQ .
If an electron is in a mixed (partially polarized)
state, the products of bispinor amplitudes are to be re-
placed by the corresponding density matrices:
ρ→uu , ''' ρ→uu .
The density matrix for polarized electron is described by
Eq. (29.13) from [3]:
( )[ smp ˆ1ˆ
2
1
5γρ += ]− , (7)
where
+
⋅
+
⋅
=
)(
,
εµ mmm
s ζppζζp , , 0=µµ ps
ζ is the double average value of the spin vector in the
electron rest frame (in pure state 1=ζ , in mixed one
1<ζ ). It is easy to see that
ε
ps ⋅
=0s , and that in ul-
trarelativistic case pvζs
m
⋅
≈ .
For nonpolarized electron ( mp += ˆ
2
1
ρ ) . Substitu-
tion of this formula is equivalent to averaging over the
electron’s polarizations. Since we need to calculate the
cross section of the process with arbitrary polarization
of final electron, let us put ( )mp += 'ˆ
2
1'ρ and multiply
the result by the factor 2 that would be equivalent to
summation over final electron polarizations.
The polarization of final photon is present in the
value Q as 4-vector e , and in *
µ Q as eµ . So, in the
squared absolute value of the matrix element we get the
tensor . For description of the case of arbitrary
partially polarized state this tensor ought to be replaced
by the density matrix [3]:
νµee*
( ))1()2()2()1(1*
22
1
νµνµνµνµ
ξ eeeegee ++−→
( ))1()2()2()1(2
2 νµνµ
ξ eeeei −+ (8)
( ))2()2()1()1(3
2 νµνµ
ξ eeee −+ .
Here ξ 1, ξ 2, ξ 3 are the Stocks parameters, and
are the polarization 4-vectors.
)1(
µe )2(
µe
Note that here ξ is the polarization discriminated by
the detector. Polarization of the final photon as well could
be easily found if we know the squared absolute value of
the matrix element as a function of parameters ξ :
ξβ ⋅+=α
2
ifM . (9)
Then the polarization of the final photon will be de-
scribed by Stocks parameters [3]
α
βξ =)( f . (10)
Since the circular polarization degree is of our interest,
we need to calculate only the values α and β 2 , that
have their origin from the substitution of the first and
the third terms of the photon density matrix, respec-
tively.
215
Substituting the expression for the matrix element in
the form (2) with account of Eqs. (3), (4) for M1 and M2
and (7), (8) for density matrices into Eq. (5) for the ra-
diation cross section, we obtain (after simple but rather
awkward calculations) for the values α and β 2 , deter-
mined in (9), integrated over the variables y and ||g 1:
=
−
∫∫
−
∞
α
δ
1
1
2||
1 y
dydg
δ
π
2
2 ⊥g
+ 2
2
'23
2
gU
εε
ω
)(
)2(
11
3 ⊥+ gIU g
πε
+++
'23
2
'2'23
2 22
εε
ω
ε
ω
εε
ω ;
2
1
1 2
||
1
β
δ
∫∫
−
∞
− y
dydg
δ
π
2
2 ⊥=
g ( ) ( ) 2
3
1
'2
'
gU
−
+
⋅
ε
ω
εε
εεωvζ
)(2
)2(
1
2
1
3 ⊥+ gIU g
πε
( )
+−−
+
×
'2
1
3
2
'2
1
3
1
'2
'
ε
ω
ε
ω
ε
ω
ε
ω
εε
εεω ,
where for the screened Coulomb potential
Rre
r
eeZrU /||)( −= , 22
||4
−+
=
R
eeZ
g g
πU
as the potential of atom in which the electron radiates,
⊥
⊥ =
g
eZgI
2
422)4()( π
π
+
−
−−
+
−× −
⊥
−
⊥
−
⊥
⊥
22
22
22 4
4arcsin
24
/4
Rg
Rg
Rg
Rg π . (11)
This result is justified with the accuracy up to the terms
of order of m 2 / ε 2, m 2 / ε' 2 and m 2 / ωε .
Under integration of the expressions obtained over
d 2g⊥ there arises logarithmic divergence under large g⊥
in the first Born approximation, and linear divergence in
the second one. That divergence is connected to the ap-
proximation g⊥ << m used above. Hence for estimation
of the contribution of the second Born approximation
into the radiation polarization one needs to integrate
over g⊥ from 0 to m introducing the cut-off on the upper
limit.
Substituting the result of such integration into (10)
we get the following formula for the circular polariza-
tion degree:
( )
'
3
2'
'6
5'
3
1
22
)(
2
εεεε
ε
ε
ωεεω
ξ
−+
+−+
⋅=
F
f vζ , (12)
where we have denoted
1 Since in a field of an individual atom the characteristic value
of appreciably exceeds ,
-dependence of U can be neglected [5,6].
~⊥g
||
δ>>−1R δ~||g
g g
∫
∫
⊥⊥⊥
⊥⊥⊥⊥
⋅
⋅
= m
g
m
g
dgggU
dgggUgI
F
0
22
0
2
3
)(
)2(
1
ε
π
. (13)
It is easy to see that in the first Born approximation
the formula (12) manifests agreement with well known
result [2].
Substituting I (g ⊥) in the form (11) into (13), we ob-
tain the following estimate for the value of F:
)ln(
/
2||
~
mR
mZ
e
eF ε
α
π
− , (14)
where α ≈ 1/137 is the fine structure constant. For ex-
ample, the silicon atom has Z = 14, ln(mR) ≈ 4, and for
electrons of energy 100 MeV
||/102~ 4 eeF −⋅− .
Account of the second Born approximation leads to
the dependence of the polarization on the radiating parti-
cle charge sign. It is easy to see from (12), (14) that the
degree of circular polarization of radiation from positrons
exceeds the same from electrons by the relative value
'
3
1
'6
5
)ln(
/
εε
ε
ε
εω
απ
+
+
mR
mZ .
For soft photons (ω << ε ) that relative difference is of
order of Zα mω / ε 2.
4. CONCLUSION
We see that the contribution of the second Born ap-
proximation into circular polarization of the photons
emitted by polarized electrons of high energy on an
amorphous target is rather small. However, such small
correction could be important for correct interpretation
of the results of measurement of fine effects due to
weak interactions.
It should be mentioned also that the relative contri-
bution of the second Born approximation could substan-
tially grow in the case of radiation of the electrons in
oriented crystal due to coherent effects [6], like it takes
the place for the bremsstrahlung cross section summed
over polarizations of particles participating in the proc-
ess [4,5].
This work is accomplished in the content of the Pro-
gram “Advancement of the scientific potential of high
education” by Russian Ministry of Education and Sci-
ence (project RNP.2.1.1.1.3263), Russian Foundation
for Basic Research (project 05-02-16512) and the inter-
nal grant of Belgorod State University
REFERENCES
1. Ia.B. Zel’dovich. //Doklady Akad. Nauk SSSR. 1952,
v. 83, p. 63 (in Russian).
2. H. Olsen, L.C. Maximon. Photon and electron po-
larization in high-energy bremsstrahlung and pair
production with screening //Phys. Rev. 1959, v. 114,
p. 887-904.
216
3. A.I. Akhiezer, V.B. Berestetskii. Quantum Electro-
dynamics. New York: “Interscience”, 1965.
4. N.F. Shul’ga, V.V. Syshchenko. On the coherent
radiation of relativistic electrons and positrons in
crystal in the range of high energies of gamma-
quanta //Nucl. Instr. and Methods B. 2002, v. 193,
p. 192-197.
5. N.F. Shul’ga, V.V. Syshchenko. The second Born ap-
proximation in theory of bremsstrahlung of relativistic
electrons and positrons in crystal //Problems of Atomic
Science and Technology. 2001, N6(1), p. 131-134.
6. A.I. Akhiezer, N.F. Shul’ga. High-Energy Electro-
dynamics in Matter. Amsterdam: “Gordon and
Breach”, 1996, 388 p.
ЦИРКУЛЯРНАЯ ПОЛЯРИЗАЦИЯ ТОРМОЗНОГО ИЗЛУЧЕНИЯ
ЭЛЕКТРОНОВ ВЫСОКОЙ ЭНЕРГИИ С УЧЕТОМ ВТОРОГО БОРНОВСКОГО ПРИБЛИЖЕНИЯ
Н.Ф. Шульга, В.В. Сыщенко
Получена оценка вклада второго борновского приближения в циркулярную поляризацию тормозного из-
лучения линейно-поляризованными электронами. Учет этого вклада может быть важным для корректной
интерпретации результатов измерения тонких эффектов, обусловленных слабыми взаимодействиями.
ЦIРКУЛЯРНА ПОЛЯРИЗАЦIЯ ГАЛЬМОВНОГО ВИПРОМIНЮВАННЯ
ЕЛЕКТРОНIВ ВИСОКОΪ ЕНЕРГIΪ З УРАХУВАННЯМ ДРУГОГО БОРНIВСЬКОГО НАБЛИЖЕННЯ
М.Ф. Шульга, В.В. Сищенко
Отримано оцінку внеску другого борнiвського наближення до циркулярної поляризації гальмівного ви-
промінювання лiнiйно-поляризованими електронами. Урахування цього внеску може бути важливим для
коректної iнтерпретацiї результатів вимірювання тонких ефектів, що обумовлені слабкими взаємодіями.
217
|