Notes on coherent polarization X-ray radiation by relativistic electrons in a crystal
This paper deals with restrictions on the use of so-called kinematical theory of coherent polarization X-ray radiation by relativistic electrons in a single crystal target. Розглянуто обмеження застосування так названої кінематичної теорії когерентного поляризаційного рентгенівського випромінювання...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Cite this: | Notes on coherent polarization X-ray radiation by relativistic electrons in a crystal / V.L. Morokhovskii // Problems of atomic science and technology. — 2019. — № 3. — С. 94-99. — Бібліогр.: 30 назв. — англ. |
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| author | Morokhovskii, V.L. |
| author_facet | Morokhovskii, V.L. |
| citation_txt | Notes on coherent polarization X-ray radiation by relativistic electrons in a crystal / V.L. Morokhovskii // Problems of atomic science and technology. — 2019. — № 3. — С. 94-99. — Бібліогр.: 30 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | This paper deals with restrictions on the use of so-called kinematical theory of coherent polarization X-ray radiation by relativistic electrons in a single crystal target.
Розглянуто обмеження застосування так названої кінематичної теорії когерентного поляризаційного рентгенівського випромінювання релятивістських електронів в моно-кристалічній мішені.
Рассмотрено ограниченя применимости так называемой кинематической теории когерентного поляризационного рентгеновского излучения релятивистских электронов в моно-кристаллической мишени.
|
| first_indexed | 2025-11-24T05:08:37Z |
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NOTES ON COHERENT POLARIZATION X-RAY
RADIATION BY RELATIVISTIC ELECTRONS IN A
CRYSTAL
V.L.Morokhovskii∗
National Science Center ”Kharkiv Institute of Physics and Technology”, 61108, Kharkiv, Ukraine
(Received August 20, 2018)
This paper deals with restrictions on the use of so-called kinematical theory of coherent polarization X-ray radiation
by relativistic electrons in a single crystal target.
PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk
1. INTRODUCTION
It is common knowledge that the collision of relativis-
tic charged particles with atoms and nuclei results
in the emission of electromagnetic waves. The radi-
ation consists of two components. The first one is
the radiation emitted directly by the charged parti-
cle in the Coulomb field of the nucleus and its sur-
rounding atomic electrons. This part is well known as
the Bethe-Heitler bremsstrahlung. The second con-
tribution is caused by elastic scattering of the virtual
photons associated with the relativistic charged par-
ticle at the atomic electrons and nuclei. It has been
called polarization bremsstrahlung or simply polar-
ization radiation. Comparing both parts it is worth
to remark that while the part of bremsstrahlung,
described by Bethe-Heitler expressions, is emitted
mainly into the angle of order of 1/γ, with γ being the
relativistic Lorentz factor in the direction of the parti-
cle velocity. Polarization radiation appears, although
with varying intensity, at any direction [1, 2, 3].
The radiation pattern changes drastically if par-
ticle interacts not just with atom or an amorphous
medium, but with the constituents of a crystal.
In this case the charged particle passes with a
nearly constant velocity through the spatial periodi-
cally distributed atoms. Due to the electromagnetic
interaction, the charged particle and the atomic elec-
trons become sources of electromagnetic waves. And
due to the space periodicity of the crystal media and
the time periodicity of the particle-atomic collisions,
the constructive interference of these waves exists.
The constructive interference results in the ap-
pearance of coherent γ-radiation in the case of ra-
diation emitted by the charged particle, i.e. coher-
ent bremsstrahlung (CB) and coherent X-ray polar-
ization radiation (CPR)1 in the case of emission by
atomic electrons, respectively. CB is emitted mainly
into the angle of order of 1/γ in the direction of the
particle velocity. CPR emitted mainly into the an-
gle of order of 1/γ in different directions near the
Bragg angles, which are corresponded to the elastic
scattering of the virtual photons associated with the
relativistic charged particle at the different crystal
planes with low Miller indexes. Such properties of
CB and CPR give opportunity to separate these two
types of coherent radiations by the choice the angle
of photon observation θk ≫ 1/γ and angle between
electron momentum p⃗i and the reflected crystal plane
in the region θk/2− 2/γ ≤ ϕ ≤ θk/2 + 2/γ.
In 1987 the author proposed ”The way (method)
of generation of the monochromatic directed X-ray
radiation”, based on the properties of the CPR phe-
nomenon, and got the ”Invention’s Certificate” [4].
The properties of CPR have been predicted in [6]
in the framework of the theoretical model, which is
called ”Kinematical theory” and has been at first con-
firmed by experiments at Kharkiv Linear Electron
Accelerators. This work was the physical ground
of the invention [4]. Thus in [4, 6] it was pro-
posed the smoothly-turnable highly-monochromatic
and highly-polarized X-ray source with negligibly
small contribution of the continuous background, and
adequate theoretical description of the CPR phe-
nomenon was made.
The CPR phenomenon then was investigated ex-
perimentally and theoretically in detail by interna-
tional collaboration in Darmstadt (Germany) at the
Superconductor Darmstadt Linear Electron Acceler-
ator S-DALINAC [7, 8, 9, 10, 11, 12, 13, 15]. In
[12] it was shown, that all properties of CPR, which
were measured in absolute units in Kharkiv, Darm-
stadt, Tomsk, Tokyo and the USA, were calculated
and found nearly the perfect agreement between the
∗Corresponding author E-mail address: victor@kipt.kharkov.ua
1The historical circumstances put to some different names of this phenomenon. In the literature are used the names:
”Parametric X-radiation (PXR)”; or ”Parametric X-radiation type B (PXR(B))”, where abbreviation (B) means the word –
”Bremsstrahlung”.
94 ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2019, N3(121).
Series: Nuclear Physics Investigations (71), p.94-99.
theory of perturbation and all referred experimental
results.
In particular, in [9] it was compared the intensity
of channeling radiation (CR) and intensity of CPR
from the same diamond target and at the same elec-
tron energy (3.0MeV ≤ Ee ≤ 9.0MeV ) and it was
shown, that the intensity of CPR is three order of
magnitude weaker than CR under the same condi-
tions.
In the publication [22] it was considered theoreti-
cally the case of CPR emitted by channeling electrons
with energies of several MeV, which during the radia-
tion process make the transition between two bound
states. Authors of this publication advocated: ”we
have an X-ray emission as a result of diffraction of
virtual CR”, and they called this radiation process
as ”diffracted channeling radiation (DCR)”. Further-
more the authors of [22] made the statement: ”It is
shown that, in comparison with PXR, the spectral
density of DCR is very large and the width of the an-
gular distribution is very narrow. The peak intensity
of DCR is about 10 times larger than that of PXR”.
The geometry of such radiation process is shown
in Fig.1.
3e
g
||g
r
iP
r
fP
r
2e
1e
Channeling
plane
Reflective
plane
Fig.1. Geometry of the experiment. The relativistic
electron with momentum p⃗i, which is parallel to the
crystal plane (e⃗2, e⃗3), penetrates into the crystal and
channeling in the plane (e⃗2, e⃗3). The virtual photon
emitted by the channeling electron and reflected by
the plane (e⃗1, e⃗3), which is perpendicular to the chan-
neling plane and creates the angle ̸ (p⃗i, e⃗3) ≈ θB
(θB is the Bragg angle), and the real photon γ
emitted under the near the mirror angle. Relativistic
electron leaves the crystal with momentum p⃗f . The
crystal got the recoil momentum, which is equal to
the reciprocal lattice vector g⃗∥
Intuition suggests that according to the physical
pattern of the CPR phenomenon, which is described
above, that the ideal dynamic of the incident rela-
tivistic electron, which gives the most intensive CPR,
is the uniform straight line motion. It is no doubt
that the character of charged particle dynamics has
an influence on the the coherent length in the coher-
ent radiation process. In [8] it’s shown, that the devi-
ations of the relativistic electron motion from the uni-
form straight line motion cause the value of the CPR
intensity to decrease. The phenomenon of channeling
corresponds to the particulary dynamic of relativis-
tic electron in crystal, when its motion perpendicular
to the channeling plane, which is characterized by
creation bound states in the averaged plane poten-
tial with discrete spectrum of the transverse energies.
Thus the statement made in [22] is not in agreement
with the habitual physical pattern of the CPR.
The conditions of experiment [9] were very much
close to the conditions, which are necessary for ob-
servation so-called DCR, but it was not discovered.
Authors of [23] have performed the numerical cal-
culations of the spectral-angular distributions of the
DCR and at the end of their article they have formu-
lated the following question: ”Why in spite of sizable
amplitude of the DCR, it was not discovered up till
now?”
Let us make effort to answer this question.
2. CB+CPR IN THE FRAMEWORK OF
THE KINEMATICAL THEORY
Both differential cross sections of ordinary co-
herent bremsstrahlung (CB) and coherent polar-
ization radiation (CPR), obtained in the frame-
work of theory of perturbation and used in
[4, 5, 15, 19], contain the product of δ-functions:
δ(p⃗i − p⃗f − k⃗ − g⃗)δ(Ei − Ef − ω), which are the con-
sequence of the momentum-energy conservation laws
(here the system of units h̄ = c = 1 is used). It
means that: p⃗i and Ei are momentum and energy
of relativistic electron in initial state; p⃗f and Ef are
momentum and energy of the relativistic electron in
the final state; k⃗ and ω are momentum and energy of
the radiated photon; q⃗ ≡ p⃗i − p⃗f − k⃗ = g⃗ is the recoil
momentum received by crystal target; g⃗ is the recip-
rocal lattice vector. The kinematic of such processes
is shown in Fig.2.
eq
kq
j
ip
q
k
^k
^qfp
^fp
a
eq
kq
j
ip
q
k ^k
^q
fp ^fp
b
Fig.2. Kinematical sketches of the radiation process
of relativistic charged particle interacting with
target: a – radiation into the forward half-sphere; b
– radiation into the backward half-sphere
For the calculation of the spectral-angular distribu-
tion of the coherent x-ray radiation we assume, that
the relativistic particle that causes the radiation is
not detected. After integration of differential cross
section (see section 2 of [29]) over the final states
of relativistic electron we obtain the total coherent
spectral-angular cross section, normalized to the vol-
95
ume of a crystal cell, which looks like:(
d3σ
dωdΩ
)
coh
=
8πω
V (1− v⃗n⃗k)
∑
g⃗,λ
|MCR|2 ×
×S2(g⃗)e−g2u2(T )δ
(
ω − g⃗v⃗
1− v⃗n⃗k
)
, (1)
where n⃗k = k⃗/k, and the sum includes all reciprocal
lattice vectors g⃗ of the crystal and the photon polar-
ization directions ϵ⃗kλ. So as the charged particle is
an electron, i.e. its charge e0 = −|e| and its mass
m0 = m, the square of the matrix element module in
Eq.(1) is of the form:
|MCR|2 =
∣∣MBS +MPR
∣∣2 =
e6
ω2m2
×
×
∣∣∣∣∣ϵ⃗kλ
{
Z − F (g⃗)
γg2
g⃗
1− v⃗n⃗k
+ F (g⃗)
ωv⃗ − g⃗
(k⃗ + g⃗)2 − k2
}∣∣∣∣∣
2
.
(2)
The first part of the right hand side of Eq.(2)
corresponds to the coherent bremsstrahlung of the
charged particle (CB) and the second part corre-
sponds to the coherent polarization radiation (CPR)
emitted by crystal electrons. Both parts are propor-
tional tom−2, which is to say that both types of radia-
tion have the same bremsstrahlung nature [12, 27, 29].
Due to the condition q⃗ = g⃗ for the momentum
transferred to the crystal, the X-radiation becomes
almost monochromatic. This mechanism has been
described in detail for coherent bremsstrahlung in
[24, 26, 25]. The second part, i.e. coherent polariza-
tion radiation in the sense of the coherent component
of the bremsstrahlung produced by atomic electrons,
exactly represents CPR (and CPR identical with so-
called PXR(B) [6, 7]).
2.1. Influence of the Dielectric Properties of
the Crystal
Matrix elements 2 were calculated neglecting the in-
fluence of the dielectric properties of the crystal,
i.e. the radiation photons were treated as traveling
through vacuum. However, the influence of the crys-
tal can be taken into account by changing the dis-
persion relation for the radiated photon to k2 = εω2,
where ε = 1+χ is the dielectric constant with χ being
the electric susceptibility of the crystal. Thus, using
this dispersion relation in Eq.2, we obtain the ex-
pression of the square of the matrix element module:
|MCR|2 =
e6
εω2m2
∣∣∣∣⃗ϵkλ {Z − F (g⃗)
γg2
g⃗
1− v⃗n⃗k
+
+F (g⃗)
√
εωv⃗ − g⃗
ω2(1/(βγ)2 + 1− ε) + (
√
εω(n⃗k)⊥v⃗ + g⃗⊥v⃗)2
}∣∣∣∣2 .
(3)
Comparing Eqs.(2) and (3) it becomes apparent
that the influence of the medium effects the ma-
trix element mainly at high particle energies, i.e. if
1/(βγ)2 ≤ |1 − ε| = |χ|. For low energies, i.e. if
1/(βγ)2 ≫ |χ|, the influence of the medium can be
neglected and it turns out that the maximum of the
cross section increases proportional to ∼ γ2 [6, 7].
In contrast to this case, at high energies the cross
section is limited by χ and the shape of the angular
distribution becomes constant [12, 21, 27].2
3. ”EFFECT OF ROW”
If we discuss Eqs.(1) and (2), it can be stated
that both CBS and CPR contribute to coherent X-
radiation at the same photon energy ω − g⃗v⃗
1−v⃗n⃗k
, but
they are emitted with different angular distributions.
On the basis of laws of momentum-energy con-
servation one can determine the area of recoil mo-
menta q⃗, which are allowed by kinematics (see Fig.2)
and represent it (so-called ”Pancake”) inside the mo-
menta space together with the reciprocal lattice of
the crystal target. In Fig.3 is shown such region of
the momentum space, which confirms with geome-
try of experiment [9] shown in Fig.1 (the channeling
plane is the plane (001) of the diamond crystal, and
the main reflected plane is (22̄0)). One can see, that
in this case laws of momentum-energy conservation
allow for the contribution into the radiation process
the row of crystal planes, which correspond to the
reciprocal lattice vectors ..., g⃗22̄4, g⃗22̄0, g⃗22̄4̄, ....
In the case of CB [24, 26, 25] all these recipro-
cal lattice vectors contribute to the intensity emitted
forward into the angle cone with an opening angle of
≈ 1/γ. For CB all angular cones of partial radiations,
which are caused by contributions from the series of
the reciprocal lattice vectors, are coaxial. Therefore
in the case CB we have the effective summatiom of
these partial radiations. This is the sense of the term
of ”effect of row”.
On the other hand, CPR is emitted mainly into
the cone of similar angular size but which is aligned
to the direction of ωv⃗− g⃗. For CPR all angular cones
of partial radiations, which are caused by contribu-
tions from the series of the reciprocal lattice vectors,
have different directions and therefore the radiation
2Our formula 3 and M.L.Ter-Mikalian’s formula (28.160) in his book [14], which describes so-called ”resonance radiation”,
are similar in appearance. However they have vital differences. M.L.Ter-Mikalian has supposed, that the crystal dielectric
constant must be ϵ > 1. Therefore his formula is in the contradiction with laws of momentum-energy conservation. Because
of that M.L.Ter-Mikalian’s formula (28.160 of [14]) predicts unreal properties of coherent X-radiation, i.e. spectral-angular
distribution. ”Resonance radiation” is characterized by the energy threshold, which is absent for CPR, and so on. For the
repairing M.L.Ter-Mikalian’s formula (28.160) it is necessary to assume ϵ ≤ 1, to change the sign near the recoil momentum,
and to define more precisely the constant multiplier before this formula. In the several theoretical works on PXR made an exact
replica of M.L.Ter-Mikalian’s formula (28.160), i.e. used erroneous sign near the recoil momentum received by crystal in the
CPR process. Therefore formulae (3) in [16], (19) in [18] and (1.1) in [20] are in the contradiction with laws of momentum-energy
conservation and do not agree with experiment [17].
96
caused by the row of reciprocal lattice vectors overlap
weakly.
001
101
333
331
331
333
224
220
224
113
111
004 224
113
111
111 111
113 333
331
331
000
004 224
-- -
--
-
-
- - - --
- -
-
--
--
- --
- -
-
-
-
-
-
-
F
a
p2
a
22p
g
fd sin/
a
22p
kq
f
dg
110
ip k
a
22p
101
Fig.3. Two projections of crossing the ”pancake” of
allowed recoil momenta by kinematics and reciprocal
lattice of Diamond crystal. Vectors p⃗i and k⃗ are
in the plane with Miller indexes (001). ”Pancake”
is perpendicular to p⃗i and placed on the distance
δ ≈ ω/(2EiEf ) + ω(1 − cos θk) from the origin
of coordinate system. The thickness of ”pancake”
is ∼ δ. The grey vertical strip in the horizontal
projection shows the crossing of ”pancake” and the
main plane of the reciprocal lattice [6]
Thus, for the different reciprocal lattice vectors
CPR can be obviously observed at different directions
and several individual maxima of coherent radiation
appear. For high particle energies these maxima are
well separated from each other. In contrast, for low
energies, i.e. if 1/γ becomes comparable to the an-
gles between two reciprocal lattice vectors, the angu-
lar cones corresponding to several reciprocal lattice
vectors may partially overlap (Fig.4).
It should be pointed out, furthermore, that in
the two extreme cases Eqs.(1) and (2) represent
the known, independent representations of CB and
CPR. In the first case, for an observation angle
θ = ̸ (v⃗, n⃗k) ≈ 1/γ the first part of Eq.(2) becomes
dominant, and thus Eqs.(1) and (2) describe exactly
CB in the low photon energy approximation analo-
gous to [24, 25, 26]. On the other hand, if θ ≫ 1/γ
the contribution of CB becomes small and can be ne-
glected. In this case Eqs.(1) and (2) precisely repre-
sent the mathematical description of CPR which has
been published before in [6, 7].
ip
r
k
r
ip
r
k
r
a
b
Fig.4. Schematic sketch of radiation contributions
from crystal planes with different orientations and
with condition g⃗iv⃗ = g⃗j v⃗, where i, j - denote the
indexes of planes, which are described by their
reciprocal lattice vectors g⃗i; v⃗ is the charged particle
velocity: a – for CB; b – for CPR
If we selected the radiation angle θk ≫ 1/γ, then
we eliminated not only CB background, but the con-
tribution of CPR stipulated by the reciprocal lattice
vectors, which are not in the plane of radiation. In
this case ”effect of row” for CPR is absent of fact,
and we have ”effect of point”. So in the case, which
is described in Fig.3 we may take into account only
dominant contribution of vector g⃗22̄0.
4. CPR GENERATED BY CHANNELING
RELATIVISTIC ELECTRONS
The EM field inside the crystal matter can be written
in two-wave approximation in the following form [27]:
A⃗ω(r⃗) =
4∑
s=1
Cs
[
ϵ⃗
(s)
1 exp
(
ik⃗1r⃗
)
+
+αsϵ⃗
(s)
2 exp
(
ik⃗2r⃗
)
+ c.c.
]
, (4)
where k⃗2 = k⃗1 + g⃗, k = ω/c∗ , αs denotes the am-
plitudes of the reflected waves, and Cs are constants.
s = (1, 2, 3, 4) because of two photon polarizations
and two possible refraction factors.
Now let us mean, that the arbitrary reciprocal
lattice vector is g⃗ = g⃗∥ + e⃗1 · gh, where g⃗∥ is in the
channeling plane, and gh is the projection g⃗ on the
unit vector e⃗1, and suppose that initial electron mo-
mentum p⃗(i) is parallel to the channeling plane (see
Fig.1). The coordinate wave functions of the channel-
ing electron ([28, 30, 5]) in the initial and final states
may be written in the form of Bloch functions:
ϕ(m)(r⃗) =
∑
h
A
(m)
h e
i(p⃗
(m)
∥ +e⃗1·gh)r⃗ . (5)
Passing over the constant, matrix element MCR can
be written in the form:
MCR ∼
∫
d3rϕ∗
f (r⃗)
ˆ⃗p · A⃗∗
ω(r⃗)ϕi(r⃗) . (6)
Substituting A⃗∗
ω(r⃗) in the form of (4) and ϕi,(f)(r⃗)
in the form of Bloch functions, we obtain sum of
two matrix elements, which origin from two terms
with k⃗1 and k⃗2 in the sum of (4). These ma-
trix elements differ from each other by δ-functions:
δ(p⃗
(i)
∥ − p⃗
(f)
∥ − k⃗1,(2)∥). The first matrix element repre-
sents the ordinary channeling radiation. The second
97
one describes the coherent polarization radiation of
the channeling electron.
If we suppose wave functions ϕi,(f)(r⃗) to be plane
waves, we obtain result, which exactly coincides with
(3) (see [27]).
Insertion of the EM field component A⃗
(s)
ω (r⃗) from
Eq.(4) and channeling electron wave functions in the
form of (5) into Eq.(6) gives:
M
(s)
CR ∼ αsϵ⃗
(s)
2
∑
h,h′
A
(m)
h (p⃗
(i)
∥ − k⃗1 − g⃗∥ − e⃗1gh)A
(m′)
h′
×δ(p⃗
(i)
∥ − p⃗
(f)
∥ − k⃗∥ − g⃗∥)δ(gh − p
(f)
⊥ − k
(1)
⊥ − gh′) , (7)
where p
(f)
⊥ = p⃗f · e⃗1 and k
(1)
⊥ = k⃗1 · e⃗1. After inte-
gration |M (s)
CR|2 over d3pf we obtain the multitude of
reflections in different crystallographic planes, which
do not contribute into selected direction of photon
registration. So effective summation of different re-
flections is absent and the situation occurs analogous
to depicted above in the subsection ”EFFECT OF
ROW”.
5. CONCLUSIONS
It is safe to repeat the following quotation from
the paper [12]: ”So it’s shown, that simple and clear
so-called kinematical theory [6, 13] is enough for ex-
act calculation of all properties of the present type of
radiation”.
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| id | nasplib_isofts_kiev_ua-123456789-195148 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-24T05:08:37Z |
| publishDate | 2019 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Morokhovskii, V.L. 2023-12-03T14:11:04Z 2023-12-03T14:11:04Z 2019 Notes on coherent polarization X-ray radiation by relativistic electrons in a crystal / V.L. Morokhovskii // Problems of atomic science and technology. — 2019. — № 3. — С. 94-99. — Бібліогр.: 30 назв. — англ. 1562-6016 PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk https://nasplib.isofts.kiev.ua/handle/123456789/195148 This paper deals with restrictions on the use of so-called kinematical theory of coherent polarization X-ray radiation by relativistic electrons in a single crystal target. Розглянуто обмеження застосування так названої кінематичної теорії когерентного поляризаційного рентгенівського випромінювання релятивістських електронів в моно-кристалічній мішені. Рассмотрено ограниченя применимости так называемой кинематической теории когерентного поляризационного рентгеновского излучения релятивистских электронов в моно-кристаллической мишени. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Electrodynamics Notes on coherent polarization X-ray radiation by relativistic electrons in a crystal Нотатки про когерентне поляризаційне рентгенівське випромінювання релятивістських електронів у кристалі Заметки о когерентном поляризационном рентгеновском излучении релятивистских электронов в кристалле Article published earlier |
| spellingShingle | Notes on coherent polarization X-ray radiation by relativistic electrons in a crystal Morokhovskii, V.L. Electrodynamics |
| title | Notes on coherent polarization X-ray radiation by relativistic electrons in a crystal |
| title_alt | Нотатки про когерентне поляризаційне рентгенівське випромінювання релятивістських електронів у кристалі Заметки о когерентном поляризационном рентгеновском излучении релятивистских электронов в кристалле |
| title_full | Notes on coherent polarization X-ray radiation by relativistic electrons in a crystal |
| title_fullStr | Notes on coherent polarization X-ray radiation by relativistic electrons in a crystal |
| title_full_unstemmed | Notes on coherent polarization X-ray radiation by relativistic electrons in a crystal |
| title_short | Notes on coherent polarization X-ray radiation by relativistic electrons in a crystal |
| title_sort | notes on coherent polarization x-ray radiation by relativistic electrons in a crystal |
| topic | Electrodynamics |
| topic_facet | Electrodynamics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/195148 |
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