Polarization effects in synchrotron radiation in strong magnetic field
The electron self-polarization effect is found to be more essential in comparison with quasi-classical approximation. The results of ultra-quantum and quasi-classical calculations for the photon polarization coincide with each other. The rigid correlation between the photon linear polarization and t...
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| Zitieren: | Polarization effects in synchrotron radiation in strong magnetic field / P.I. Fomin, R.I. Kholodov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 154-156. — Бібліогр.: 9 назв. — англ. |
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Fomin, P.I. Kholodov, R.I. 2015-04-02T15:10:07Z 2015-04-02T15:10:07Z 2001 Polarization effects in synchrotron radiation in strong magnetic field / P.I. Fomin, R.I. Kholodov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 154-156. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 42.25-Ja; 41.60-Ap https://nasplib.isofts.kiev.ua/handle/123456789/79475 The electron self-polarization effect is found to be more essential in comparison with quasi-classical approximation. The results of ultra-quantum and quasi-classical calculations for the photon polarization coincide with each other. The rigid correlation between the photon linear polarization and the electron spin is found. The authors are grateful to V.E. Storizhko for his constant help in holding this work and also to S.P. Roshchupkin for useful discussions. The work was partly supported by the Grant for the young researchers of the National Academy of Sciences of Ukraine. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Electrodynamics of high energies in matter and strong fields Polarization effects in synchrotron radiation in strong magnetic field Поляризационные эффекты в синхротронном излучении в сильном магнитном поле Article published earlier |
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Polarization effects in synchrotron radiation in strong magnetic field |
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Polarization effects in synchrotron radiation in strong magnetic field Fomin, P.I. Kholodov, R.I. Electrodynamics of high energies in matter and strong fields |
| title_short |
Polarization effects in synchrotron radiation in strong magnetic field |
| title_full |
Polarization effects in synchrotron radiation in strong magnetic field |
| title_fullStr |
Polarization effects in synchrotron radiation in strong magnetic field |
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Polarization effects in synchrotron radiation in strong magnetic field |
| title_sort |
polarization effects in synchrotron radiation in strong magnetic field |
| author |
Fomin, P.I. Kholodov, R.I. |
| author_facet |
Fomin, P.I. Kholodov, R.I. |
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Electrodynamics of high energies in matter and strong fields |
| topic_facet |
Electrodynamics of high energies in matter and strong fields |
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2001 |
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English |
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Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Поляризационные эффекты в синхротронном излучении в сильном магнитном поле |
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The electron self-polarization effect is found to be more essential in comparison with quasi-classical approximation. The results of ultra-quantum and quasi-classical calculations for the photon polarization coincide with each other. The rigid correlation between the photon linear polarization and the electron spin is found.
|
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1562-6016 |
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https://nasplib.isofts.kiev.ua/handle/123456789/79475 |
| citation_txt |
Polarization effects in synchrotron radiation in strong magnetic field / P.I. Fomin, R.I. Kholodov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 154-156. — Бібліогр.: 9 назв. — англ. |
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2025-11-24T10:51:33Z |
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1850845410144288768 |
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POLARIZATION EFFECTS IN SYNCHROTRON
RADIATION IN STRONG MAGNETIC FIELD
P.I. Fomin1, R.I. Kholodov2
1N.N. Bogolyubov Institute for Theoretical Physics, NAS of Ukraine, Kiev, Ukraine
e-mail: pfomin@bitp.kiev.ua
2Institute of Applied Physics, NAS of Ukraine, Sumy, Ukraine
e-mail: fomin@ipfcentr.sumy.ua
The electron self-polarization effect is found to be more essential in comparison with quasi-classical
approximation. The results of ultra-quantum and quasi-classical calculations for the photon polarization coincide
with each other. The rigid correlation between the photon linear polarization and the electron spin is found.
PACS: 42.25-Ja; 41.60-Ap
1. INTRODUCTION
Study of the quantum-electrodynamic processes in
the presence of strong magnetic field close to the critical
value of about 1013 Gs is important for exploration of
astronomical objects such as pulsars. These sources of
radiation, it is well known, are connected to neutron
stars, where near to stars’ surface, the magnetic field has
value of such an order.
Quantum-electrodynamic processes with magnetic
field occur in heavy ion collision too. The magnetic
field produced by colliding nuclei in the region between
them at the moment of the closest approach has order of
magnitude about 1012 Gs in the case, when that region
has the size of Compton wavelength of electron (see
Fig. 1). The electric fields of nuclei mutually
compensate one another in that region.
Fig. 1. Magnetic field of colliding nuclei and e+e– pair
in that field
We consider, that the series of quasi-equidistant
narrow peaks in the electron-positron distribution of
total energy, observed in heavy ions collision at GSI,
Darmstadt [1,2], is a result of movement of an electron-
positron pair in such magnetic field in that region (see
Fig. 2). Narrow lines are the resonant pair production on
the Landau levels. The resonant pair production by two
equivalent photons is the second order quantum-
electodynamic process. But in resonant conditions, as is
known, this process breaks up to two independent first
order processes.
Therefore it is meaningful to study the most
elementary processes in detail, with magnetic field,
which has order of magnitude about 1012 Gs.
Fig. 2. Narrow peaks in heavy ions collision
experiments at GSI, Darmstadt [1, 2] are the resonant
pair production on the Landau levels:
E E En e e
= + =+ − 2 22 2 2m neH Lz+ + π /
2. ELECTRON SELF
POLARIZATION EFFECT
Now we pass to consideration of the synchrotron
radiation process. Certainly this process has been well
investigated. Its general relativistic theory for a long
time had been constructed [3,4]. But only
ultrarelativistic quasi-classical approximation was
studied in detail [5,6]. Other approximations were
studied too [7,8], but there were some questions.
The quantum effects in the process have dual
character. The quantum, that is the single photon, is
radiated. And movement of the electron is also
quantized. The dispersion law of the electron has the
form (=1, c=1):
ε n zm n h p m= ⋅ + +2 2 21 2 / , (1)
h H H= / 0 , H m e0
2= / .
The movement is characterized by the principle
quantum number n that is number of the Landau level. H
is magnetic field in units of critical one. The case of the
big value of Landau level is the case of quasi-classical
electron behavior.
But we examine a case of strong field, when the
individual energy levels are experimentally
154 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 154-156.
distinguishable. Thus the number of final states of
electron is equal to one. This approximation is called
ultra-quantum approximation. However, the magnetic
field that is 1012 Gs in units critical field is a good small
parameter of the problem. Due to the presence of the
small parameter all calculations can be made
analytically and it is possible to receive in Furry picture
the simple expressions for probabilities of the process.
Formulas (2-5) describe the probabilities of the
process with transition of electron on the next level:
)1()1(
2
1 22 unmh
du
dW +⋅−⋅α=
++
, (2)
)1(
2
1 22 unmh
du
dW +⋅⋅α=
−−
, (3)
)1(
4
1 23 umh
du
dW +⋅α=
−+
, (4)
dW
du
mh n n u u u
− +
= − + − +1
64
1 1 11 55 2 4 6α ( )( ) .
(5)
The signs «+» and «-» designate the direction of initial
and final electron spins along and against field
respectively, a is the fine structure constant, u is the
cosine of the angle q between directions of the magnetic
field and the photon motion.
The formulas (2) and (3). It is so called spin-up state
and spin-down state of electron.
The formulas (4) and (5) describe probabilities of
spin reorientation. These probabilities are not equal,
what is the essence of the electron self-polarization
effect, which had been found by Sokolov and Ternov in
quasi-classical approximation.
You can see, the probabilities have different powers
of magnetic field [8,9]. It is influence of quantum
character of the particles’ movement. For example: In
field about 1012 Gs one probability is three order of
magnitude greater then another. The dependence on an
angle of photon radiation in these formulas (4), (5) is
also different, what is shown in Fig. 3.
Fig. 3. Probability of spin-flip process versus both
angle of photon radiation u = cosθ and electron
energy level n
So, the self-polarization effect is more essential in
comparison with quasi-classical approximation.
3. PHOTON POLARIZATION
The following question is the photon polarization.
The polarization vector has the form:
e eλ
ν
λ= ( , )0
, e A A eiB
λ = = ⋅1 0(cos , sin , ) ,
e e A Aλ λ π= == → +2 1 2( / ) (6)
in the case, when the photon is radiated along the axis z
(along magnetic field). If the photon’s wave vector k is
directed arbitrarily it is necessary to perform the
corresponding rotation of polarization vector. A and B
are the parameters, which determine the type of
polarization. The Stokes parameters are more known.
The connection of these parameters is defined by
formula:
ξ 1 2= ⋅sin cosA B , ξ 2 2= ⋅sin sinA B ,
ξ 3 2= cos A . (7)
The probabilities of polarized radiation have the form
shown in formulas (2-4), where it is necessary to make a
replacement of (1 + u2) by the following expression:
( ) ( ( )( ) )1 1 1
2
1 12 2
3 2+ → − − + +u u uξ ξ , (8)
if µ µ= ' ; and
( ) ( ( )( ) )1 1 1
2
1 12 2
3 2+ → − − − +u u uξ ξ , (9)
if µ µ= − ' .
You can see, that the probabilities don’t depend on
parameter ξ1. It is an obvious result, as this parameter
characterizes polarization on the angle + 45 and -45
degrees relatively to the plane (k, H), and these are
symmetrical situations. The degree of polarization is
equal to ratio of the difference to the sum of
probabilities with the opposite polarizations:
β
ξ ξλ λ
λ λ
=
−
+
=
− −
+
= =
= =
dW dW
dW dW
u u
u
1 2
1 2
2
2
3
2
2 1
1
( )
. (10)
The degree values as function of parameters ξ2 and ξ
3 lie in some plane, as shown in Fig. 4.
The type of polarization is defined by ξ2 and ξ3, at
which modulo degree of polarization | β | has the
maximal value. For example: In the case of forward
radiation (along field) (u=1, ξ2=1, ξ3=0) we have the
right circular polarization. In the case sideways radiation
(u=0, ξ2=0, ξ3=-1) we have the linear polarization, when
the vector of electric field of the photon is perpendicular
to the plane (k, H). So we have received the result
known in classical consideration of the problem. Ultra-
quantum and quasi-classical results coincide with each
other. The reason is in following: the type of photon
polarization does not depend on the Landau level
numbers of initial and final electrons. From every
energy level the radiation has the similar polarization,
which depends only on a direction of photon motion.
155
Fig. 4. Degree of photon polarization versus Stokes
parameters, a) u = 1, b) u = 1/3, c) u = 0
However, the rigid correlation between the photon
linear polarization and the electron spin is found. You
can see in formulas (8) and (9), that in a spin-flip
process (9) the sign of linear polarization (the sign of ξ3)
is opposite in comparison with a process without spin
reorientation (8).
4. CONCLUSION
In conclusion the idea to measure polarization of the
X-ray pulsar radiation is proposed. There are two types
of the directional radiation pattern of pulsar (see Fig. 5):
pencil-shaped, when radiation is along the field
direction and knife-shaped, when radiation is
perpendicular to the field. In a radiation spectrum of an
X-ray pulsar the cyclotron lines were found (see Fig. 6).
They are hyrolines, predicted by Gnedin. The cyclotron
radiation is principle in vicinity of these lines. We
propose to select one of lines, to measure its
polarization and to define the Stokes parameters. Then it
is not difficult to calculate the angle between directions
of magnetic field and radiation. It will allow confirming
the type of the directional radiation pattern of pulsars
(pencil or knife), and may allow proposing a new
radiation scheme.
Fig. 5. Directional radiation pattern of X-ray
pulsar (cross-section), a) pencil-shaped, b) knife-
shaped
Fig. 6. Spectrum of X-ray pulsar,Hercules X-1
ACKNOWLEDGMENT
The authors are grateful to V.E. Storizhko for his
constant help in holding this work and also to
S.P. Roshchupkin for useful discussions.
The work was partly supported by the Grant for the
young researchers of the National Academy of Sciences
of Ukraine.
REFERENCES
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Investigations of correlated e+e– emission in heavy-
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2. R. Bar, A. Balanda, J. Baumann et al. Experiments
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emission in magnetic field // Zh. Eksp. Teor. Fiz.
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4. A.A. Sokolov, I.M. Ternov. Relativistic electron.
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8. I.G. Mitrofanov, A.S. Pozanenko. Generation of
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156
1. INTRODUCTION
POLARIZATION EFFECT
3. PHOTON POLARIZATION
4. CONCLUSION
acknowledgment
REFERENCES
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