Frequency of parametric X-ray radiation
The exact solution of the equation for the frequency of parametric X-ray radiation (PXR) of relativistic charged particles moving in a crystal is obtained and compared to the approximate solution. It is found that the exact solution is in good agreement with the approximate one and that the approxim...
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| Cite this: | Frequency of parametric X-ray radiation / A.V. Shchagin, G. Kube // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 85-87. — Бібліогр.: 8 назв. — англ. |
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Shchagin, A.V. Kube, G. 2023-12-11T11:54:06Z 2023-12-11T11:54:06Z 2023 Frequency of parametric X-ray radiation / A.V. Shchagin, G. Kube // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 85-87. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 41.60.-m, 61.80.Cb DOI: https://doi.org/10.46813/2023-146-085 https://nasplib.isofts.kiev.ua/handle/123456789/196180 The exact solution of the equation for the frequency of parametric X-ray radiation (PXR) of relativistic charged particles moving in a crystal is obtained and compared to the approximate solution. It is found that the exact solution is in good agreement with the approximate one and that the approximate PXR frequency solution is practically correct for a comparison to experimental data. Отримано точний розв’язок рівняння частоти параметричного рентгенівського випромінювання релятивістських заряджених частинок, що рухаються в кристалі, та порівняно з наближеним розв’язком. Встановлено, що точний розв’язок добре узгоджується з приблизним рішенням і що приблизний частотний розв’язок практично коректний для порівняння з експериментальними даними. A.V.S. is grateful to A.P. Potylitsyn for the discussion the work [6] and to V.A. Maisheev for the discussion the work [7] at the conference [8]. This project has received funding through the MSCA4Ukraine project, which is funded by the European Union. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Problems of Atomic Science and Technology Parametric radiation Frequency of parametric X-ray radiation Частота параметричного рентгенівського випромінювання Article published earlier |
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Frequency of parametric X-ray radiation |
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Frequency of parametric X-ray radiation Shchagin, A.V. Kube, G. Parametric radiation |
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Frequency of parametric X-ray radiation |
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Frequency of parametric X-ray radiation |
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Frequency of parametric X-ray radiation |
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Frequency of parametric X-ray radiation |
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frequency of parametric x-ray radiation |
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Shchagin, A.V. Kube, G. |
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Shchagin, A.V. Kube, G. |
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Parametric radiation |
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Parametric radiation |
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Problems of Atomic Science and Technology |
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Частота параметричного рентгенівського випромінювання |
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The exact solution of the equation for the frequency of parametric X-ray radiation (PXR) of relativistic charged particles moving in a crystal is obtained and compared to the approximate solution. It is found that the exact solution is in good agreement with the approximate one and that the approximate PXR frequency solution is practically correct for a comparison to experimental data.
Отримано точний розв’язок рівняння частоти параметричного рентгенівського випромінювання релятивістських заряджених частинок, що рухаються в кристалі, та порівняно з наближеним розв’язком. Встановлено, що точний розв’язок добре узгоджується з приблизним рішенням і що приблизний частотний розв’язок практично коректний для порівняння з експериментальними даними.
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1562-6016 |
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Frequency of parametric X-ray radiation / A.V. Shchagin, G. Kube // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 85-87. — Бібліогр.: 8 назв. — англ. |
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2025-11-25T23:31:25Z |
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ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 85
P A R A M E T R I C R A D I A T I O N
https://doi.org/10.46813/2023-146-085
FREQUENCY OF PARAMETRIC X-RAY RADIATION
A.V. Shchagin
1,2
, G. Kube
1
1
Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany;
2
National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine
E-mail: alexander.shchagin@desy.de
The exact solution of the equation for the frequency of parametric X-ray radiation (PXR) of relativistic charged
particles moving in a crystal is obtained and compared to the approximate solution. It is found that the exact solution
is in good agreement with the approximate one and that the approximate PXR frequency solution is practically cor-
rect for a comparison to experimental data.
PACS: 41.60.-m, 61.80.Cb
INTRODUCTION
Parametric Cherenkov radiation emitted by fast
charged particles moving in a layered dielectric medium
was first predicted in [1]. Later on, the equation for the
frequency of parametric X-ray radiation (PXR) emitted
by a relativistic charged particle moving in a crystal was
derived by Ter-Mikaelian [2]. The approximate solution
of the equation was confirmed in a number of experi-
mental researches of PXR properties, see e.g. [3 - 5].
The influence of the medium where the radiation is
emitted, is neglected in the approximate solution, and
PXR is considered as emitted in vacuum due to the pe-
riodical interaction with crystallographic planes. How-
ever, sufficient influence of the medium on the radiation
frequency was found at the interaction of particles with
macroscopic periodical structures, as transition radiation
from a stack of thick foils [5, 6], and at undulator radia-
tion from an undulator filled by an amorphous medium
[7], and at undulator radiation from a volume reflection
undulator [8]. In the present paper, we consider an exact
analytical solution of the equation for the frequency of
PXR emitted in a crystal and compare it to the approxi-
mate solution and experimental data.
1. CALCULATIONS
The equation for the angular frequency of PXR,
excited by a relativistic charged particle in a crystal, was
derived by Ter-Mikaelian [2] and reads
2
1 cos
nV
l
V
c
, (1)
with V – the particle velocity, l – the distance between
crystallographic planes along the particle velocity vector
V , c – the speed of light,
1 – the observation
angle between particle velocity vector V and observa-
tion direction, 1 – the relativistic Lorentz factor of
the incident particles, n – the harmonic number (in
crystals n is associated with the crystal structure factor,
therefore PXR does not exists for every n ), and
2
1
p
(2)
the frequency dependent dielectric permittivity of the
crystal in the X-ray range for frequencies exceeding the
atomic frequencies in the crystal such that
p , (3)
p being the crystal plasma frequency. With 1
p
in
Eq. (2), the solution of Eq. (1) often can be used in the
approximate form
0
2
1 cos
nV
Vl
c
. (4)
The approximate solution 0 (4) can be considered
as “vacuum” solution because the dielectric permittivity
of the crystal (2) (where PXR is generated) is not taken
into account. Eq. (4) was confirmed experimentally in a
number of experimental investigations of PXR proper-
ties. In the following, taking Eq. (1) as start point it is
indicated how the exact solution differs from the ap-
proximate vacuum solution (4).
Eq. (1) together with the dielectric permittivity Eq.
(2) results in a quadratic equation that can be written in
the form
2 22
(1 cos ) cos 0
2
p
V Vn V
c l c
. (5)
The two solutions of Eq. (5) are
2 2
0 0
1,2
cos
2 2
2 1 cos
pV
V
c
c
. (6)
The second term under the root in Eq. (6) is much
smaller than the first one because of condition (3).
Therefore, one can write two separate solutions. The
low-frequency solution 1 (“-” sign in Eq. (6)) is
2
1
cos
4
pl
cn
, (7)
and the high-frequency solution 2 (“+” sign in
Eq. (6)) is
2
2 0 1
cos2
4
1 cos
plnV
Vl cn
c
. (8)
The low-frequency solution (7) does not correspond
to PXR because it does not satisfy the condition
p , see Eq. (3). The high-frequency solution
2
86 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146)
(8) differs from the vacuum solution (4) by the value of
the low-frequency solution
1 . The difference leads to
an increase in the PXR frequency (8) in the PXR reflec-
tion emitted in the backward hemisphere at
2
,
and to a decrease in the PXR frequency for radiation
emitted in the forward hemisphere at 1
2
. The
difference is absent if PXR is emitted at a right angle
with respect to the particle velocity vector, i.e.
2
.
The value of the difference 1 ( 1 is in the electron-
volt range in typical cases) between the exact (Eq. (8))
and the approximate vacuum solution (Eq. (4)) is much
smaller than the PXR quanta energy (typically 0 is
in the range exceeding a few kiloelectronvolt) calculated
by formula (4).
Let us consider an example for typical experimental
conditions described in [3, 4]. In those works, the PXR
spectral peak from the (111) crystallographic planes of a
Si single-crystal was observed in the vicinity of the cen-
ter of the PXR reflection at a fixed observation angle
305.9 mrad, and the angle of the crystal rotation
was around / 2 . PXR was generated by a 25.0 MeV
electron beam in a thin Si plate in Laue geometry. The
value l in (4) is
(111)
2
20.7
sin
2
l
g
Angstroms in the
center of the PXR reflection, where (111)g is the module
of the reciprocal lattice vector for the (111) Si crystallo-
graphic plane. The energy of the observed radiation [3,
4] and the calculated one by the approximate formula
(4) in the center of the PXR reflection spectral peak is
0 12.9 keV. The energy
1 calculated by formu-
la (7) is
1 0.8 eV for the plasma energy in Si crys-
tal 31.1p eV and n=1. The energy of the low-
frequency radiation (7) does not satisfy the condition (3)
and cannot be considered as a realistic solution. The
energy of the high-frequency radiation (8)
2 0 1 is less than
0 12.9 keV for
0.8 eV. The relative correction is
410 . The differ-
ence between the exact high-frequency solution
2
(8) and the approximate solution (4) amounts to only
0.8 eV, which is much less than the observed width of
the PXR spectral peak of 166 eV in [3, 4]. Thus, the
difference between the exact (8) and the approximate
(4) solution is negligibly small for PXR, and the approx-
imate solution (4) is practically correct.
2. RESULTS AND DISCUSSION
We demonstrated a good agreement between the ex-
act (8) and the approximate vacuum solution (4) for
PXR frequencies produced by relativistic particles in a
crystal. Let us discuss the reason, why the exact and
approximate vacuum solutions for radiation frequencies
are practically the same for PXR in a crystal, and they
are sufficiently different at emission of radiation in for-
ward direction from macroscopic periodical structures,
as transition radiation from a stack of thick foils [5, 6],
undulator radiation from an undulator filled by amor-
phous medium [7], and undulator radiation from a vol-
ume reflection undulator [8].
The important thing is the period length l of the
structure. In the case of PXR the period is determined
by the distance between crystallographic planes, which
is usually in the Angstrom region. Therefore, the values
of the low-frequency solution (7) and the correction in
the high-frequency solution (8) are insignificant. In the
case of macroscopic structures [5 - 8], the period lengths
l can be in sub-mm or even larger range. Therefore, the
low-frequency solution and the related correction of the
high-frequency solution for macroscopic structures [5 -
8] can be sufficient to lead to significant changes of the
spectral distribution of the emitted radiation.
ACKNOWLEDGEMENTS
A.V.S. is grateful to A.P. Potylitsyn for the discus-
sion the work [6] and to V.A. Maisheev for the discus-
sion the work [7] at the conference [8]. This project has
received funding through the MSCA4Ukraine project,
which is funded by the European Union.
REFERENCES
1. Ya.B. Fainberg, N.A. Khizhnyak. Energy losses by a
charged particle passing through a laminar dielectric
// Sov. Phys. JETP. 1957, v. 5, p. 720; Translated
from Russian // Zh. Eksp. Teor. Fiz. 1957, v. 32,
p. 883.
2. M.L. Ter-Mikaelian. High-Energy Electromagnetic
Processes in Condensed Media, New York: Wiley-
Interscience, 1972; Translated from Russian:
Vliyanie Sredy na Elektromagnitnye Protsess(y pri
Vysokikh Energiyakh), Erevan: Izd. AN Arm. SSR,
1969 (in Russian).
3. A.V. Shchagin, V.I. Pristupa, N.A. Khizhnyak.
A fine structure of parametric X-ray radiation from
relativistic electrons in a crystal // Phys. Lett. A.
1990, v. 148, p. 485-488.
4. A.V. Shchagin, X.K. Maruyama. Accelerator-Based
Atomic Physics: Techniques and Applications / Eds.
S.M. Shafroth, J.C. Austin. New York: “AIP Press”,
1997, p. 279.
5. A.P. Potylitsyn. Electromagnetic Radiation of Elec-
trons in Periodic Structures, Springer Verlag Ber-
lin Heidelberg, 2011; Translated from Russian: Izlu-
chenie Elektronov v Periodicheskikh Strukturakh.
Tomsk: NTL, 2009 (in Russian).
6. V.N. Baier, V.M. Katkov. Transition radiation as a
source of quasi-monochromatic X-rays // Nucl. In-
struments and Methods. 2000, v. A 439, p. 189-198.
7. S. Bellucci, V.A. Maisheev. Radiation of relativistic
particles for quasiperiodic motion in a transparent
medium // J. Phys.: Condens. Matter. 2006, v. 18,
p. S2083-S2093.
8. A.V. Shchagin, G. Kube, S.A. Strokov. About fre-
quencies of radiation of relativistic particles in period-
ical crystalline structure. Oral paper at IX Internation-
al Conference Charged and Neutral Particles Channel-
ing Phenomena, June 4-9, 2023, Riccione, Italy // Sci-
entific Program and Abstract Book. 2023, p. 37.
Article received 30.06.2023
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 87
ЧАСТОТА ПАРАМЕТРИЧНОГО РЕНТГЕНІВСЬКОГО ВИПРОМІНЮВАННЯ
А.В. Щагін, Г. Кубе
Отримано точний розв'язок рівняння частоти параметричного рентгенівського випромінювання релятиві-
стських заряджених частинок, що рухаються в кристалі, та порівняно з наближеним розв'язком. Встановле-
но, що точний розв'язок добре узгоджується з приблизним рішенням і що приблизний частотний розв'язок
практично коректний для порівняння з експериментальними даними.
|