Coherent polarization radiation of relativistic electrons in crystals
A brief narration about the history of those heated arguments and discussions around the nature of so-called parametric X-radiation, which were concluded by the recognition of the discovery the phenomenon of coherent polarization bremsstrahlung of relativistic charged particles in crystals. Some imp...
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| Cite this: | Coherent polarization radiation of relativistic electrons in crystals / V.L.Morokhovskii// Вопросы атомной науки и техники. — 2014. — № 5. — С. 124-135. — Бібліогр.: 48 назв. — анг. |
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| citation_txt | Coherent polarization radiation of relativistic electrons in crystals / V.L.Morokhovskii// Вопросы атомной науки и техники. — 2014. — № 5. — С. 124-135. — Бібліогр.: 48 назв. — анг. |
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| description | A brief narration about the history of those heated arguments and discussions around the nature of so-called parametric X-radiation, which were concluded by the recognition of the discovery the phenomenon of coherent polarization bremsstrahlung of relativistic charged particles in crystals. Some important information and comments, which stay over of notice of specialists till now are reported.
Краткий рассказ об истории жарких споров и дискуссий вокруг природы так-называемого параметрического рентгеновского излучения, которые завершились признанием открытия явления когерентного поляризационного тормозного излучения релятивистских заряженных частиц в кристаллах. Изложены информация и комментарии, которые все еще находятся вне поля зрения специалистов.
Коротка розповiдь про гарячi суперечки та дискусiї вiдносно природи так-званого параметричного рентгенiвського випромiнювання, якi завершилися визнанням вiдкриття явища когерентного поляризацiйного гальмового випромiнювання релятивiстських заряджених часток в кристалах. Викладенi iнформацiя та коментарi, якi все ще залишаються поза увагою спецiалiстiв.
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COHERENT POLARIZATION RADIATION OF
RELATIVISTIC ELECTRONS IN CRYSTALS
V.L.Morokhovskii∗
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received July 15, 2014)
A brief narration about the history of those heated arguments and discussions around the nature of so-called paramet-
ric X-radiation, which were concluded by the recognition of the discovery the phenomenon of coherent polarization
bremsstrahlung of relativistic charged particles in crystals. Some important information and comments, which stay
over of notice of specialists till now are reported.
PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk
1. INTRODUCTION
In 1983-1985 the two pioneer experiments with the
goal to detect the coherent X-radiation of relativis-
tic particles in crystals were carried out in Kharkov
and Tomsk. For the first time the preliminary results
of these experiments simultaneously were reported in
Moscow at the Session of the Academy of Science of
the USSR in January 1986 [1]. The results of the
Tomsk experiment were reported by A.P.Potylitzin
[2], [3], and the results of the Kharkov experiment
were reported by me [4]. The discrepancy in the un-
derstanding and explanation the nature of observed
coherent radiation arose during discussions at that
Session.
The results of Tomsk experiment [2], [3] were in-
terpreted as: ”... the first experimental observation
of a new mechanism for X-ray emission by relativis-
tic charged particles in crystals: parametric Vavilov-
C̆erenkov X radiation”. They approved that: ”The
reason for the effect is that the periodic arrangement
of atoms (or nuclei) in a crystal causes the inequality
n > 1 to hold even in the X-ray range if the frequency
ω and the momentum of the photon are near the edge
of Brillouin zone” [3]. (It means, that n is the index
of refraction of X-rays in the crystal media.)
During previous ten years the series of the theoret-
ical works was done, where the so-called ”parametric
Vavilov-C̆erenkov X-radiation” has been predicted :
[5], [6], [7], [8], and the properties of such radiation
have been calculated. My colleagues expressed their
confidence about agreement between results of their
experiment and those theoretical predictions.
I assumed that the coherent polarization
bremsstrahlung had been observed in two our ex-
periments. Besides, I established, that in both
experiments the preliminary results were obtained:
they had vital experimental imperfections (in both
experiments γ-detectors were insufficiently shielded
from γ-radiation emitted by accelerator devices), and
therefore, we could not have the correct identification
of the radiation mechanism. To my mind, my col-
leagues were in a hurry to make conclusions in their
publications [2], [3]...
From that time the struggle for the priority in
the discovery of the super intensive sources in the
X-ray and γ-ray regions (based on supposed Vavilov-
C̆erenkov effect), which promised the financial sup-
port from military-industry complex1, began between
two groups, which belonged to the two different min-
istries: ”Education and Science” (Minsk and Tomsk)
and so-called ”Ministry of Middle Machine-Building”
(Moscow, Kharkov). I did not take part in such com-
petition.
I did not trust to the theoretical works, which had
been mentioned above, because their results were in
contradiction with Kharkov experiment. For exam-
ple:
1. the coherent maximum existed in the much
more wide region of angle deviation from the Bragg
angle, that was in contrast with theoretically pre-
dicted very narrow region in the series of the works
[5], [6], [7], [8];
2. the photon energy in the coherent maximum
can be changed by the crystal rotation in the wide re-
gion, photon energy depends on relativistic electron
energy, and its position on the energy scale is the
same as for coherent bremsstrahlung.
All these properties were in the contradiction
with theoretical predictions of [5], [6], [7], [8]. This
and other contradictions between experimental re-
sults and existed theoretical predictions, have forced
me to carry out my own theoretical work, where
the new model was developed, which I called as the
”...kinematical theory...” [10].
Studying the works of my predecessors and op-
ponents, I have discovered that they deny the law of
∗Corresponding author E-mail address: victor@kipt.kharkov.ua
1From March 1983 the word combination ”starlit wars” [9] became often used by the politicians...
124 ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2014, N5 (93).
Series: Nuclear Physics Investigations (63), p.124-135.
momentum-energy conservation not only using the
phrase: ”The phenomenon of radiation of electro-
magnetic waves by the charged particle, which moves
straightforward and uniformly in the space heteroge-
neous medium”. The denial of the fundamental law
is in the essence of these works (!).
The conflict of interests was the hindrance for
publication the experimental results obtained at
Kharkov 2GeV and 40MeV accelerators. Only
due to the support of academician Spartak Belyaev,
though with delay, our first preliminary experimental
results were published [4]. Our next more accurate
experimental results were forbidden for publication2.
Nevertheless, in June 8, 1989 we got the Author’s
certificate of the invention of ”The way (method)
of generation of the monochromatic directed X-ray
radiation” with priority from April 13 1987 [11],
which is based on the process of coherent polariza-
tion bremsstrahlung of relativistic charged particles
in crystals, and is described in [10] (Fig.1). It became
possible because of the different properties of radia-
tions, which were predicted within the framework of
models of parametric C̆erenkov radiation [5], [6], [7],
[8] and in [10]. For the first time in [10]3 it was shown
that expression of cross section, obtained within the
framework of classical electrodynamics and perturba-
tion (or kinematical) model agrees well with experi-
ments, which were carried out in Kharkov.
Timely to remark, that in July 19, 1988 at the
Scientific Counsel of Kharkov Institute of Physics
and Technology academician Ja.B. Fainberg and pro-
fessor N.A. Khiznjak raised a claim for their rights
to the discovery [12]: ”The phenomenon of radia-
tion of electromagnetic waves by the charged parti-
cle, which moves straightforward and uniformly in the
space heterogeneous medium”, where they approved
that Tomsk and Kharkov experiments had confirmed
their theoretical predictions in [5]. Their suggestion
about discovery was not admitted by scientific com-
munity because it was erroneous.
Coat of Arms of the USSR
The Union of Soviet Socialist Republics
I
State Committee of Inventions and Discoveries
INVENTOR’S CERTIFICATE
N 1513528
Under the basis of authority, presented by the Government of the USSR,
In the Presence of the USSR State Committee of Science and Technology
(Statecominventions)
Statecominventions has issued this Certificate of authorship for the
Invention:
“The Way of Generation of Monochromatic
Directed X-rays”
Author (authors): D.I. Adeishvili, S.V. Blazhevich, G.L. Bochek,
V.I.Kulibaba, , G.L.Fursov, A.V.Shchagin.V.L.Morokhovskii
Claim N 4227556
Claimant:
The priority of the Invention from
April 13, 1987.
Registered in the Government Register of Inventions of the USSR
June 8, 1989.
The action of the Certificate is spread on the whole territory
Of the USSR
The Chairman of Committee Signature
The Head of the Department Signature
Fig.1.
2. QUANTUM-MECHANICAL THEORY
OF COHERENT X-RADIATION IN
CRYSTALS
2.1. Coherent X-radiation in Crystals as a Result
of Constructive Interference of Radiation Waves
from Crystal Atoms
Thus, for the first time the physical sense of the pro-
cess of the coherent X-radiation of relativistic charged
particles in crystals correctly was described as a co-
herent polarization bremsstrahlung within the frame-
work of classical electrodynamics in [10]. One can
find the qualitative description of such radiation pro-
cess in [10], (see paragraph: ”About the Nature of
X-Radiation of Relativistic Electrons in a Crystal”).
The nature of the considered type of radiation can
be clearly understood by the way of examination the
2My reviewers-opponents required to call this type of radiation certainly as: ”parametric Vavilov-C̆erenkov X-radiation”.
In the struggle with them the type of X-radiation was called as: ”parametric X-radiation” (abbreviation PXR), which does not
express the nature of radiation.
3In 1990 A.V. Shchagin, V.I. Pristupa and professor N.A.Khizhnyak (N.A.Khizhnyak was one of the pretenders to the dis-
covery of ”parametric Vavilov-C̆erenkov X radiation”) published the results from [10] in Phys. Lett.A 148 (1990) p.485 without
reference to the original work (!).
125
total bremsstrahlung of the fast charged particle in
the collision with crystal.
The quantum-mechanical model of the total co-
herent X-radiation, which arises as a result of col-
lision of relativistic charged particle and the crystal
target, will be briefly described in this section.
Coherent X-radiation results from constructive in-
terference of bremsstrahlung emitted as well by rel-
ativistic charged particle as by atomic electrons in
crystal.
1. Thus, for a better theoretical understanding we
will first consider the radiation caused by the interac-
tion of relativistic charged particle with isolated atom
only. Therefore let’s consider a system, composed of
three parts: the projectile (i.e. relativistic charged
particle), atom in rest, and the radiation field. The
Coulomb gauge should be used for potentials of in-
teraction.
Thus, one can represent the total Hamiltonian of
the given system in the following form:
Ĥtot = Ĥp + Ĥa + Ĥr + Ĥint , (1)
where Ĥp, Ĥa and Ĥr are the unperturbed Hamilto-
nians of the particle, of the atom and of the radiation
field, Ĥint is the Hamiltonian of interaction.
The unperturbed state functions for the particle
satisfy the stationary Dirac equation
Ĥpψp⃗,s(r⃗) =
{
−ih̄c ˆ⃗α∇⃗+mc2β̂
}
ψp⃗,s(r⃗) =
= E(p⃗)ψp⃗,s(r⃗) . (2)
Solutions of Dirac equation are:
ψp⃗,s(r⃗, t) = ψp⃗,s(r⃗)e
−iEt/h̄ =
=
√
E +mc2
2E
(
u⃗s
−ih̄cˆ⃗σ∇⃗
mc2+E u⃗s
)
ei(p⃗r⃗−Et)/h̄ , (3)
where ˆ⃗σ are Pauli matrices and u⃗s are vectors of elec-
tron polarization.
The unperturbed state functions φn(r⃗) for the
atom (i.e. atomic electrons in the field of the atomic
nucleus and electrons) satisfy the stationary one-
particle Schrödinger - Hartree equation:
Ĥaφn(r⃗) = ϵnφn(r⃗) , (4)
where:
Ĥa =
(−ih̄∇⃗)2
2m
+
Ze2
r
− e2
∑
n′(̸=n)
∫
|φn′(r⃗′)|2
|r⃗′ − r⃗| d3r′
.
Free radiation field should be possible to describe by
vector potential A⃗, which satisfies the wave equation and
gauge condition simultaneously:
△A⃗− 1
c2
∂2A⃗
∂t2
= 0 ,
∇⃗ · A⃗ = 0 . (5)
The general solution of the equation 5 is the superposition
of plane waves:
A⃗(r⃗, t) =
√
4πc2
V
∑
k⃗,λ
ε⃗k⃗,λ
(
qk⃗,λe
i(k⃗r⃗−ωt) + q∗
k⃗,λ
e−i(k⃗r⃗−ωt)
)
,
(6)
where ε⃗k⃗,λ is the unit vector of photon polarization, and
indexes λ = 1, 2 correspond to the two states of transverse
polarizations. Vectors ε⃗k⃗,λ satisfy conditions of orthogo-
nality:
ε⃗k⃗,1 · k⃗ = 0 , ε⃗k⃗,2 · k⃗ = 0 , ε⃗k⃗,1 · ε⃗k⃗,2 = 0 , (7)
which are consequence of the Coulomb gauge. For conve-
nience we choose the normalization multiplier in 6. Due
to the boundary condition of the periodicity inside the
cube volume V = L3, wave vectors k⃗ satisfy the equality
k⃗ =
2π
L
n⃗ , (8)
where ni = 0,±1,±2, ... The frequency of the wave is con-
nected with absolute value of the wave vector k⃗ by the law
of dispersion
ω(k⃗) = c|⃗k| . (9)
The interaction Hamiltonian Ĥint in 1 is given by:
Ĥint = Ĥp−n + Ĥp−a + Ĥp−r + Ĥa−r , (10)
where Ĥp−n describes the interaction between particle
and atomic nucleus, Ĥp−a describes the interaction be-
tween particle and atomic electrons, Ĥp−r describes the
interaction between particle and radiation field, Ĥa−r de-
scribes the interaction between atom and radiation field
(Ĥn−r is neglected because of infinitely large nuclear
mass). All these parts of interaction Hamiltonian should
be described as:
Ĥp−n =
Ze2
r
, (11)
Ĥp−a = −e2
∑
n′(̸=n)
∫
|φn′(r⃗′)|2
|r⃗′ − r⃗| d3r′ , (12)
Ĥp−r = −e ˆ⃗αA⃗(r⃗) , (13)
Ĥa−r = Ĥ
(1)
a−r + Ĥ
(2)
a−r =
= ih̄
ih̄e
mc
∑
n
A⃗(r⃗n)∇⃗n +
e2
2mc2
∑
n
A⃗2(r⃗n) .
(14)
The Feynman diagrams that describe the radiation
process in the lowest approximation of the perturbation
theory are of the third order as displayed in Fig.2, where
generalized coordinates of system are: wave functions of
the free relativistic electrons |pi⟩ and |pf ⟩ in the initial
and final states, respectively; wave functions of atomic
electrons |ni⟩ and |nf ⟩ in the initial and final states, re-
spectively; and the four-vector of the free electromagnetic
field Aµ(r⃗, t). All coordinates are functions of the space
coordinates r⃗ and time t. The entire process consists of six
amplitudes (see Fig.2, couples of diagrams ”a”, ”b”, ”c”).
126
)c
)b
)a
*
a
k
A
ip
q
fp
0
e
0
e
||eZ
ipfp
*
a
k
A
q
0
e
||eZ
0
e
fn
*
a
k
A
ip
fp
e e
0
e
in
q
fn
*
a
k
A
ipfp
in
q
0
e
0
e
e
fn
0
e
0
e
*
a
k
A
ipfp
e
in
q
fn
ee
0
e
*
a
k
A
ipfp
in
q
´ ´
Fig.2. Feynman diagrams for electromagnetic radiation
from electrons interacting with atom
The first four diagrams describe the emission of photon
by the incident particle scattered at the nucleus (couple
of diagrams ”a”) and atomic electron (couple of diagrams
”b”). The last two diagrams (couple of diagrams ”c”)
describe the emission of photon by atomic electrons due
to the interaction with the incident particle field.
The differential cross section of the radiation process
can be then written in units of h̄ = c = 1 as
d6σ = 2π |Mat|2 δ(Ei − Ef − ω)
d3pfd
3k
(2π)6
, (15)
where Mat is the atomic matrix element and p⃗i, Ei and
p⃗f , Ef represent the initial and final momenta and ener-
gies of particle, respectively. Furthermore, k⃗ and ε⃗kλ are
the momentum and the polarization vector of the radiated
photon, and q⃗ = p⃗i − p⃗f − k⃗ is the recoil momentum. As-
suming, that the photon energies are much smaller than
the energy of the incident particle, i.e. ω ≪ Ei, Ef ,
but exceed the energies En,m for atomic transitions, i.e.
ω ≫ En,m, the atomic polarizability can be approximated
by the polarizability of the free electrons (it means, that
in the expression Ĥa−r in formula Eq.15 the contribution
from the Ĥ
(2)
a−r is much larger than from Ĥ
(1)
a−r). Such
problem was solved in: [13], [14], and in this case the
square of the total matrix element of the radiation pro-
cess for nonrelativistic atom has the form:
|Mat|2 =
∣∣∣MBS +MPR
∣∣∣2 =
=
32π3
ω
∣∣∣∣ε⃗kλ {
−ee20
m0
(Z − F (q⃗))
γq2
q⃗
ω − k⃗v⃗
+
+
e2e0
m
F (q⃗)
ω
v⃗ω − q⃗
(q⃗ + k⃗)2 − ω2
}∣∣∣∣∣
2
. (16)
Here v⃗ is the particle velocity, e0 and m0 are charge and
mass of the relativistic particle, while e and m are charge
and mass of the electron, F (q⃗) is the atomic formfac-
tor and Z is the atomic number. Furthermore, MBS
describes the bremsstrahlung of the relativistic charged
particle, i,e. couples of diagrams ”a” and ”b” in Fig.2,
and MPR results from polarization bremsstrahlung of the
atomic electrons, i.e. couple of diagrams ”c” in Fig.2.
One can see that in the expressions for amplitudes of ra-
diation 16: MBS ∼ m0
−1 and MPR ∼ m−1. It’s impor-
tant to accentuate, that the presence of masses m0 and
m in the denominators of the expressions for amplitudes
MBS and MPR gives evidence, that both amplitudes have
the same nature, i.e. both amplitudes correspond to the
bremsstrahlung processes.
2. Now we are interested in calculating the probability
of radiation of fast (v ∼ c) electrons (positrons) interact-
ing with many atoms arranged in a crystal (i.e. in a thin
crystal).
Let’s begin from estimations of the time charac-
teristics of such process. The transit time τ of rel-
ativistic particle across the atom is order of τ ∼
RBZ
−1/3c−1 ≈ 3 × 10−20s, where RB is a Bohr radius,
Z – atomic charge and c – velocity of light. Character-
istic vibration time for crystalline materials is order of
t ∼ ω−1 = h̄/(kΘ) ≈ 10−13s, where k – Boltzmann con-
stant, Θ – Debye temperature. One can see that τ ≪ t,
and the particle transit length is order of L ∼ 10−3cm
during the time t.
Let’s assume that crystal thickness Lc is so small, that
the inequality Lc ≪ L is satisfied.
The cross section measurements of radiation process
require the time T ≫ t. So one can see that a single rel-
ativistic particle interacts with static accidental configu-
ration of the crystal atoms, and different particles inter-
act with different accidental configurations. Therefore we
must calculate cross section of radiation of fast electron in
an accidental static crystal configuration, and after that
we must average cross section over all atomic configura-
tions, which depend on a crystal temperature.
So, following the procedure derived by Überall [15] in
his perturbation theory, we can apply the result of Eq.16
not only to the single atom but to the thin crystal as
well. For this purpose, in accordance with Huygens prin-
ciple, it’s necessary to take into account the interference
of amplitudes of radiation from all atoms of the crystal
target. This method was successfully applied calculating
the cross section of high-energy bremsstrahlung and elec-
tron pair production in the thin crystals by Überall [15]
and Diambrini [16].
For each atom of the crystal one obtains the radiation
amplitude Mate
−iq⃗R⃗
L⃗ , where R⃗L⃗ are the relative coordi-
nates of the crystal atoms. Thus, the cross section of
the process for the entire crystal as it is described by the
diagrams in Fig.2, differs from Eq.15 by the diffraction
factor ∣∣∣∣∣∣
∑
L⃗
exp(−iq⃗R⃗L⃗)
∣∣∣∣∣∣
2
. (17)
The crystal diffraction factor, corresponding to one crys-
tal cell and averaged over the thermal vibrations of the
crystal atoms, has the form:⟨∣∣∣∣∣∣
∑
L⃗
e−iq⃗R⃗
L⃗
∣∣∣∣∣∣
2⟩
u⃗T
= D(q⃗)S2(q⃗)e−q2u2
T +
+ ν
[
1− e−q2u2
T
]
, (18)
where ν represents the number of atoms in the elemen-
tary crystal cell, S(q⃗) is the structure factor of the crys-
127
tal, e−q2u2
T is Debye-Waller factor and u2
T is the mean
square of the thermal vibrational amplitude u⃗T of the
crystal atoms. Assuming the geometric shape of the crys-
tal to be a thin plate and there is one of the crystal lattice
basis vectors a⃗1 which forms a small angle with respect
to the normal to the crystal surface and the other basis
vectors a⃗2 and a⃗3. In this case the diffraction factor D(q⃗)
for an ideal crystal with N1 periods in the crystal direc-
tion a⃗1 and with large numbers of periods N2, N3 → ∞
in the perpendicular directions can be represented as:
D(q⃗) = lim
N2,N3→∞
3∏
j=1
sin2
(
Nj
2
q⃗a⃗j
)
Nj sin
2
(
Nj
2
q⃗a⃗j
)
=
=
4π2a1
V
∑
g⃗
δ(q⃗ − g⃗)⊥a⃗1
sin2
(
N1
2
q⃗a⃗1
)
Nj sin
2
(
N1
2
q⃗a⃗1
) ,
(19)
where V is the volume of the crystal elementary cell, g⃗
represents the reciprocal lattice vectors of the crystal, ar-
gument of the δ-function is the component of the vector
(q⃗ − g⃗)⊥a⃗1 , which is perpendicular to a⃗1.
From Eq.18 (and Eq.15) it becomes apparent that the
complete (BS and PR) bremsstrahlung in a crystal is di-
vided into two parts:
d6σcr = d6σcoh + d6σin . (20)
The first one is the coherent part. It is proportional to
e−q2u2
T and is correlated to the part of the crystal atoms
that contributes to the interference effect which results
in the production of coherent X-radiation. In this case
the crystal gets exactly a momentum q⃗ = g⃗, as it can be
seen from Eq.19 for N1 → ∞, and the recoil momentum
is transferred to this part of crystal atoms.
The second term, which is proportional to 1−e−q2u2
T ,
is the incoherent part. It appears due to the thermal vi-
brations of crystal atoms and corresponds to the part
of the atoms, that does not contribute to the coherent
radiation.
2.2. Coherent Cross Section in a Thin Crystal
For the calculation of the coherent cross section we as-
sume that the relativistic particle that causes the radia-
tion is not detected. An integration of Eq.15 over d3pf
after multiplication with the coherent term of the diffrac-
tion factor (see Eq.18) will yield the cross section de-
scribing the production of coherent radiation. The inte-
gration can be performed by introducing new variables4
p⃗′ = p⃗i − p⃗f , E′ = Ei − Ef and using p⃗′v⃗ = E′. Af-
ter the replacement of δ(q⃗ − g⃗)a⃗1δ(Ei − Ef − ω)d3pf by
δ(p⃗′− k⃗− g⃗)δ(E′−ω)(v⃗a⃗1/a1)
−1d2p′⊥a⃗1
dE′ and summing
over the two possible polarizations, denoted by λ = 1, 2,
the integration yields for N1 → ∞ (see Eq.19) the total
coherent spectral-angular cross section, normalized to the
volume of a crystal cell:(
d3σ
dωdΩ
)
coh
=
=
8πω
V (1− v⃗n⃗k)
∑
g⃗,λ
|MCR|2 S2(g⃗)e−g2u2
T ×
× δ
(
ω − g⃗v⃗
1− v⃗n⃗k
)
, (21)
where n⃗k = k⃗/k, and the sum includes all reciprocal lat-
tice vectors g⃗ of the crystal and the photon polarization
directions. Assuming for simplicity, that the charged par-
ticle is an electron, i.e. e0 = −|e| and m0 = m, the square
of the matrix element module in Eq.21 is of the form:
|MCR|2 =
∣∣∣MCBS +MCPR
∣∣∣2 =
=
e6
ω2m2
∣∣∣∣ε⃗kλ {
Z − F (g⃗)
γg2
g⃗
1− v⃗n⃗k
+
+F (g⃗)
v⃗ω − g⃗
(g⃗ + k⃗)2 − k2
}∣∣∣∣∣
2
. (22)
Like in Eq.16, the first part of the right hand side
of Eq.22 corresponds to the coherent bremsstrahlung of
the charged particle (CBS) and the second part corre-
sponds to the coherent polarization radiation (CPR). Due
to the condition q⃗ ≈ g⃗ for the momentum transferred the
crystal, the X-radiation becomes almost monochromatic.
This mechanism has been described in detail for coher-
ent bremsstrahlung in [15], [16]. The second part, i.e.
coherent polarization radiation in the sense of the coher-
ent component of the bremsstrahlung produced by atomic
electrons, exactly represents so-called PXR(B)5 [17].
If we discuss Eqs.21 and 22 it can be stated that both
CBS and CPR (or PXR(B)) contribute to coherent X-
radiation at the same photon energy, but they are emit-
ted with different angular distributions. In the case of
CBS all reciprocal lattice vectors contribute to the inten-
sity emitted forward into the angle cone with an opening
angle of ≈ 1/γ. On the other hand, CPR (or PXR(B)) is
emitted mainly into the cone of similar angular size but
which is aligned to the direction of v⃗ω − g⃗.
Thus, for the different reciprocal lattice vectors CPR
(or PXR(B)) can be obviously observed at different direc-
tions, 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 angles between two
reciprocal lattice vectors, the angular cones correspond-
ing to several reciprocal lattice vectors may partially over-
lap.6
4Such new variables represent momenta and energies of so-called pseudophotons of the field of relativistic charged particle.
5In [17] we introduced the abbreviation PXR(B), where the letter (B) is the first letter of the word ”Bremsstrahlung”, and
indicates the nature of radiation.
6Timely to remark, that in [10] in Fig.8 and Fig.9 the intersections of the reciprocal lattice plane with the ”pancake” of
recoil momenta, allowed by kinematics of the radiation process, are shown. And one can see, that Fig.8 corresponds to the case
that only one reciprocal lattice bend is inside the ”pancake” and Fig.8 corresponds to another case that the series of bends
is inside the ”pancake”. In the case of CBS [15], [16] the first case (Fig.8) is called as ”effect of point” and the second one
(Fig.9) is called as ”effect of row”. In the case of CPR we have the same kinematical pictures. However, essential difference in
angular distributions of radiation for CBS and CPR exists. For CBS all angular cones of partial radiations, which are caused
by contributions from the series of the reciprocal lattice bends, are coaxial. Therefore in the case of CBS we have the effective
interference of these partial radiations. This is the sense of the term of ”effect of row”. For CPR all angular cones of partial
radiations, which are caused by contributions from the series of the reciprocal lattice bends, have different directions. Therefore
the concept ”effect of row”, in contrast to CBS, can not be applied in the case of CPR.
128
It should be pointed out, furthermore, that in
the two extreme cases Eqs.21 and 22 represent the
known, independent representations of CBS and CPR
(or PXR(B)). In the first case, for an observation angle
θ = ̸ (v⃗, n⃗k) ≈ 1/γ the first part of Eq.22 becomes dom-
inant, and thus Eqs.21 and 22 describe exactly CBS in
the low photon energy approximation analogous to [15],
[16]. On the other hand, if θ ≫ 1/γ, the contribution
of CBS becomes small and it can be neglected. In this
case Eqs.21 and 22 precisely represent the mathematical
description of CPR (or PXR(B)) which has been published
before in [10] and [17].
2.3. Influence of the Crystal Dielectric
Properties
Up to here we have neglected the influence of the dielectric
properties of the crystal, i.e. the radiation photons were
treated as traveling through vacuum. However, the influ-
ence of the crystal media can be taken into account by
changing the dispersion 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, which are
the same as for amorphous media. Thus, using this dis-
persion relation in Eq.22 we obtain the expression of the
square of the matrix element module:
|MCR|2 =
∣∣∣MCBS +MCPR
∣∣∣2 =
=
e6
ϵω2m2
∣∣∣∣ε⃗kλ {
Z − F (g⃗)
γg2
g⃗
1− v⃗n⃗k
+
+
F (g⃗)(v⃗
√
ϵω − g⃗)
ω2(1/(βγ)2 + 1− ϵ) + (
√
ϵω(n⃗k)⊥v⃗ + g⃗⊥v⃗)2
}∣∣∣∣2 .
(23)
Comparing Eqs.22 and 23, it becomes apparent that the
influence of the medium effects over the matrix 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 [10], [17]. In contrast to this, at high energies the
cross section is limited by χ and the shape of the angular
distribution becomes constant [18].
Timely to remark, that M.L.Ter-Mikhealyan in his
book [6] concluded that his so-called ”resonance radi-
ation” (or it is the same as CPR) is characterized by
the energy threshold (see formula (28.13) in [6]), and
”resonance radiation” exists only if electron energy is
larger, than the threshold energy. In our designations
M.L.Ter-Mikhaelyan’s condition is the same as the con-
dition 1/(βγ)2 ≪ |1 − ϵ| = |χ| in formula Eq.23. One
can see, that considered here type of coherent radiation
(i.e CPR) does not have a threshold. Such condition can
be called, for example, as an energy of saturation. In-
vestigations of the properties of the coherent polariza-
tion X-radiation, which were made in [10] and [17] cor-
respond to the case, when electron energies are less than
so-called ”threshold of resonance radiation”, i.e. when
1/(βγ)2 ≪ |1− ϵ| >> |χ| and we can assume, that in this
case the dielectric constant of the crystal media can be as-
sumed as ε0 = 1. Such conditions were briefly discussed
in [10] (see formula (34) in [10]).
V.G.Baryshevsky et al. in their work [20] made refer-
ence to the condition analogous to 1/(βγ)2 ≪ |1−ϵ| = |χ|
and pointed out that: ”... the PX intensity has no clearly
marked threshold character” and later they confirmed
that ”Vavilov-C̆erenkov condition is fulfilled only for”
1/(βγ)2 ≪ |1 − ϵ| >> |χ|. Such ”confirmations” respec-
tively to the Vavilov-C̆erenkov condition are erroneous.
Examining the coherent cross section Eq.21 one can
see, that the photon energy in the coherent maximum for
a specific reciprocal lattice vector g⃗ can be obtained by
turning the argument of δ-function to zero. So we have:
ω =
g⃗v⃗
1− v⃗n⃗k/c∗
, (24)
where c∗ = c/
√
ϵ.
Using δ-function, the cross section Eq.21 can be sim-
ply integrated over the photon energy and the differential
CPR yield per unit solid angle and per unit length can
be obtained:(
dNγ
LdΩ
)
g⃗
= σ(g⃗)
∑
λ
1
ω
∣∣∣∣∣ ε⃗kλ(ωv⃗ − g⃗)
(k⃗ + g⃗)2 − ω2 − ω2
p
∣∣∣∣∣
2
, (25)
where σ(g⃗) = (8πe6)/(V m2)S2(g⃗) exp(−g2ū2
T )F
2(g⃗), ωp
is the plasma frequency, and the relation ϵ = 1 − ω2
p/ω
2
has been used.
Simply to show, that using (k⃗+ g⃗)2−ω2 ≡ (ω/βγ)2+
(k⃗′
⊥v⃗)
2, where β = v/c and k⃗′
⊥v⃗ v⃗ = 0, this value becomes
identical to the one that describes by formula Eq.32 in
[18] (or Eq.37 in [19]) for the case ϵ = 1.
3. THE RESULTS OF THE FIRST KHARKOV
EXPERIMENT [21]
The next considerable work [21], in which the final results
of the first Kharkov experiment had been presented, was
published only in July 1993, after a prolonged struggle
with the opponents. In this paper it was shown, that
the cross section of the coherent polarization radiation of
relativistic charged particles (see formulae 21-23 with con-
tribution of only matrix element MCPR from the sum 22
or 23) can be simply obtained using Weizsäcker-Williams
method [22], and it is the same, as it was obtained within
the framework of the classical kinematical theory [10],
and these formulae are in good quantitative agreement
with results of the Kharkov pioneer experiments.
The first Kharkov experiment was performed on the linear
accelerator of electrons in Kharkov Physical Technical In-
stitute Academy of Science of Ukraine - LINAC−2GeV .
The single crystal Si target had the form of plane-parallel
plate, which was cut in the plane (110). Using 3-axes
goniometer and the method of crystal orientation [23],
crystal targets were placed on the trace of electron beam
so, that the crystal plane (1̄11) has been placed perpen-
dicular to the plane of radiation and it created the angle
ϕ with the beam direction. The flux of photons, which
were emitted under the angle θ > γ−1 respectively to
the electron beam direction, was measured. The photon
beam was formed with the help of the circular collimators
in the solid angle ∆Ω = π(∆θ)2, where the linear angle
of collimation was ∆θ ≪ θ. After passing through the
crystal target the electron beam and charged particles,
created by initial electron beam in the crystal target, were
turned by magnet into the beam absorber, where they
were absorbed. Spectra of photons were measured with
the help of semiconductor spectra-meter, placed at the
end of the photon channel inside the shield, which pro-
tected γ-spectra-meter from electromagnetic background
129
created by electron accelerator. The charge of the elec-
tron beam, which was passing through the crystal, was
measured by a thin monitor of secondary-emission, placed
behind the crystal target outside the photon channel, and
the beam charge was integrated. Spectra of radiation
were measured with electron beam current I ≤ 10−9 A.
The monitor of secondary-emission was calibrated using
a standard Faraday cup. The calibration of γ-spectra-
meter was carried out with the help of standard γ-sources.
The scheme of measurements was described in Ref.[24]
in detail. Schematic view of the experimental setup on
the experimental area of the straight line beam exit of
the 2GeV Kharkov linear electron accelerator is shown
in Fig.3. The most important parameters of the scheme
of measurements on the accelerator LINAC − 2GeV are
represented in the table.
F
Q
1S
2S
2K
1K
1J
2J
3J
1M
2M
1G 2G 4K
8K
7K
6K
5K
3K
4M
3M
5M
Fig.3. Schematic view of the experimental setup on the experimental area of the straight line beam exit
of the 2GeV Kharkov linear electron accelerator. M1, M2 – magnets of beam parallel dislocation; M3 –
deflection magnet; M4, M5 – cleaning magnets; G1, G2-goniometers; K1,K2 – electron collimators; K3,K8
– photon collimators; J1, J2, J3 – electron beam current monitors (based on secondary-electron emission); F
– Faraday cup; Q – quantameter; S1-Ge(Li) – semiconductor gamma-spectrameter; S2 – NaJ total absorbing
gamma-spectrameter
Parameters of the scheme of measurements on the accelerator LINAC − 2GeV :
the energy of electrons in the maximum of the spectral distribution 1200MeV ;
the width of spectral distribution of the electron beam on the level
1/2 of maximum current 1%;
the middle divergence of electron beam 2 · 10−4 rad;
the duration of the current pulse 1...2µs;
the frequency of the current pulses 50 s−1;
the angle of photon registration θ 17.88mrad;
the accuracy of measurements of angles ϕ, ψ 5 · 10−5 rad;
the accuracy of measurements of angles α 10−3 rad;
the material and orientation of the crystal target Si(110);
the thickness of the crystal target 50µm;
the type of γ-spectra-meter DGDK − 50Al;
the energy resolution of γ-spectra-meter 104 eV ;
the efficiency of photon registration in the maximum of
the coherent radiation 10−2;
the frequency of counts of γ-spectra-meter 7 s−1;
the linear angle of photon channel acceptance ∆θ 8 · 10−4mrad
The spectra of electromagnetic radiation, which is
generated by relativistic electrons with different en-
ergies in the single crystal Si with orientation such,
that the crystal planes (1̄11) placed perpendicular to
the plane of radiation and under the angle ϕ respec-
tively to the electron beam direction, are measured.
One of them is shown in Fig.4. In Fig.4 we can ob-
serve one bright peak of CPR with energy 206 keV
from the planes (1̄11). ”Small” peak with energy
412 keV is stipulated by the generation of two pho-
tons with energies 206 keV during one pulse of the
accelerator current.
The energy dependence in the maximum
CPR from the angle ϕ between electron veloc-
ity direction and the crystalline plane (1̄11) is
shown in Fig.5 for electron energy 1200MeV .
130
0,0 0,2 0,4 0,6 0,8 1,0 1,2
0
200
400
600
800
N
um
be
r o
f p
ho
to
ns
Photon energy / mec2
Fig.4. The spectrum of coherent polarization radia-
tion generated by the electrons with energy 1.2GeV
in 50µm Si single crystal target under the angle
θ = 17.88mrad. The crystal plane (1̄11) has disposi-
tion perpendicular to the radiation plane and creates
the angle ϕ = 8.63mrad respectively to the electron
beam direction
6 7 8 9 10 11 12
0,25
0,30
0,35
0,40
0,45
0,50
0,55
0,60
P
ho
to
n
E
ne
rg
y
/ m
ec2
Angle mrad
Fig.5. The energy in the maxima of spectra of coher-
ent polarization radiation generated by electrons with
energy 1200MeV in 50µm Si single crystal target
under the angle θ = 0.3059 rad as a function of the
angle ϕ between the crystal plane (1̄11) and electron
beam direction
The solid line in Fig.5 is calculated using the for-
mula Eq.24, where ω = Eγ/mc
2. It is observed a
good agreement between experimental dependence
and the formula Eq.24.
For comparison experimental values of CPR yield
with the calculated yield using formula Eq.25, we ex-
cluded the continuous background from spectra (see
Fig.4) and then each spectrum was integrated over
the photon energy. So, the apparatus effect of the
line broadening, caused by the final spectra-meter
resolution, was excluded.
6 7 8 9 10 11
0
1
2
3
4
5
Angle mrad
dN
/
d
*L
-1
,
cm
-1
Fig.6. The total number of photons in coherent
maximum generated by the electrons with energy
1200MeV in 50µm Si single crystal target under
the angle θ = 0.3059 rad as a function of the an-
gle ϕ between crystal plane (1̄11) and electron beam
direction. Points-experiment; Dashed line-calculated
function using formula 21 integrated over photon en-
ergy (i.e. Eq.25); solid line-calculated function using
formula 21 integrated over photon energy with taking
into account the following: particle multiply scatter-
ing, photon absorption, crystal mosaicity, beam pa-
rameters, photon detector opening and efficiency by
Monte-Carlo manner
In Fig.6 it’s shown the dependence dNγ/dΩ/L (where
L = aN is the crystal thickness) from the angle ϕ
for electron energy 1200MeV . The experimental
values dNγ/dΩ/L are given by points. Nearly the
perfect agreement between experiment and the per-
turbative or kinematical theory, which is described
in the previous section (and which results for CPR
coincides with results obtained within the framework
or classical electrodynamics in [10]) was observed.
It means, that using the perturbative or kinemat-
ical theory, which was described in the previous sec-
tion and taking into account particle multiply scatter-
ing, photon absorption, crystal mosaicity, beam pa-
rameters, photon detector opening and efficiency, and
other experimental conditions by Monte-Carlo man-
ner [38] – is enough for exact calculation of proper-
ties of coherent polarization X-radiation of relativistic
particles in crystals.
131
3.1. Discussion of the Results of [21]
1. The agreement of experimental energy depen-
dence of CPR with dependence Eq.24 is the proof of
the radiation coherency. The kinematics of CPR (or
PXR(B)) is completely coincides with kinematics of
coherent bremsstrahlung (see formula Eqs.21, 22 and
[16]).
On the other hand, this dependence expresses
the law of pseudo-photon refraction from the atomic
planes of a crystal: k⃗ = k⃗0 + g⃗, where k⃗0v⃗ = ω,
k = ω/c∗, c∗ is the phase velocity of the free electro-
magnetic wave in a crystal. It is evident, that such
dependence can be used for experimental determina-
tion of the refraction coefficient in the direction of
radiation:
n =
c
c∗
=
(
c
v
− cg⊥ sinϕ
ω
)/
cos θ , (26)
so as at the right of (Eq.24) there are only values,
which were measured in the experiment. So, it can
be proved that n ≈ 1 − 3 · 10−6 for electron energy
of 1200MeV . Therefore the properties of observed
CPR are in the contradiction with radiation proper-
ties, predicted in [5]-[8], where n > 1 is the requesting
condition.
2. The model of CPR was developed within the
framework of perturbation theory, which had been
used for development of: the model of resonance ra-
diation [6]; X-ray transition radiation(XTR) [7]; and
quasi-C̆erenkov radiation, which is far from Bragg di-
rections [25]. However CPR can not be identified as:
a resonance radiation; XTR; and quasi-C̆erenkov ra-
diation, which is far from Bragg directions, because
CPR is generated on the crystal planes, for which
g⃗ · n⃗ = 0 (where n⃗ = a⃗1/a1, see formula Eq.19),
but for the other mentioned types of radiation it
is necessary to be g⃗ · n⃗ = g and g ̸= 0. CPR is
accompanied with recoil momentum received by a
crystal with transverse component, which is much
larger than longitudinal component, and it is not
typical for C̆erenkov type of radiation. The existing
of large transverse recoil momentum leads to the re-
sult, that the effective dielectric constant, which was
introduced in Ref.[25], is the following ϵeff = ϵ0 (in
the X-ray region of photon energy for the Si crystal
ϵ0 ≈ 1 − 5 · 10−6). The application of the perturba-
tion theory in the CPR model is possible due to the
condition of weak bond of incoming and scattered
electromagnetic waves [10], which is also correct for
the thick crystals. On the contrary, the models of:
a resonance radiation, XTR [7], and quasi-C̆erenkov
radiation, which is far from Bragg directions [25]
suppose the strong bond between incoming and scat-
tered waves, and in this case the scattered wave can
be small only for a thin crystal.
3. The spectral-angular distribution of radiation
[10] (see Fig.6.) is proportional to the flux of virtual
photons of the field of relativistic electron through
the crystal atomic plane. Thus, the mechanism of
radiation can be interpreted as a coherent scattering
of virtual photons, associated with the relativistic
charge, on the crystal electrons. The flux of vir-
tual photons in the direction v⃗ is equal zero and
has the maximum on the cone with the generatrix,
which is directed under the angle 0.5γ−1 respectively
to v⃗. The refraction of the virtual photons from
atomic planes according to the law k⃗ = k⃗0+ g⃗, where
k⃗0v⃗ = ω, k = ω/c, is not mirror, so as v ̸= c. The
minimum of the orientational dependence Fig.6. is
in the angle ϕ3 = θ/2 − 1/(2θγ2), which is in accor-
dance with our theoretical model [10]. The difference
of the values in maxima of orientational dependence
(see Fig.6., dashed line) is stipulated by different
directions of generatrix of cone, where the flux of
virtual photons (associated with relativistic charge)
has its maximum. They are directed under the dif-
ferent angles ϕ respectively to the atomic plane, and
the coefficient of reflection depends on the angle ϕ.
Taking into account the multiple scattering of rela-
tivistic electrons in the crystal target, the divergence
of electron beam and the angle of the acceptance
of the photon channel, leads to the reduction the
”height” of maxima and the ”depth” between them
(Fig.6, solid line). The agreement between theoreti-
cal and experimental orientational dependence gives
a possibility to conclude, that formulae, which are
presented in the theoretical part of the present paper,
give the adequate description of the spectral-angular
distribution of CPR.
4. The spectral-angular density of CPR, is pro-
portional to α3, and it is nearly the same order as
the spectral-angular density of channeling radiation,
which is proportional to α2. It depends on small pop-
ulations of the bound states of the fast electrons in
crystal potential. CPR differs from channeling radia-
tion by the possibility of changing the photon energy
by changing the crystal orientation in the wide region;
CPR is much more monochrome ∆Eγ/Eγ ≈ 7 · 10−3;
and its continuous background is two-order less than
for channeling radiation. CPR has weak dependence
from such parameters of electron beam as the width
of its spectrum and angular divergence. Such prop-
erties of CPR make it possible to create the coherent
sources of monochromatic directed X-rays, suitable
in operation, which have continuous changing of the
photon energy from units of keV to the units of
MeV [11], and they can be applied for calibration
of γ-spectra-meters, selective raising the resonance
systems, and others.
At that time we had two theoretical models,
which described quite different physical processes,
and which were used for the explanation the prop-
erties of the so-called PXR. The first of them is
predicted in [5], [6], [7] [8] relates to the Vavilov-
Čerenkov type of radiation, and the second one re-
lates to the coherent polarization bremsstrahlung
[10], [21].
132
4. THE RESULTS OF THE
INTERNATIONAL COLLABORATION
The situation changed drastically at the beginning
of 1993 when due to professor N. Shul’ga, who is an
active propagandist of the achievements of Kharkov
physicists7, and because of professor A.Richter who
showed his interest to my work [10] (Professor
A.Richter is the possessor of the encyclopedic knowl-
edge and, besides, learned Russian). A.Richter initi-
ated the translation of my paper [10] into English and
placed it in the site of Darmstadt Kernphysik Insti-
tute (Germany), where he was a director. A.Richter
invited me to the Darmstadt Kernphysik Institute
for experimental investigation of PXR.8 Due to the
great talent for organization of professor A.Richter
and doctor H.Genz, the powerful international scien-
tific group was created, and the program of experi-
ments at the Darmstadt superconducting accelerator
S-DALINAC was proposed [27].
Using high-energy electron beam from
S-DALINAC, the most considerable experiments
were carried out and properties of coherent polar-
ization radiation were investigated in detail.
Let’s only briefly enumerate the main results, ob-
tained within the framework of this international col-
laboration. Spectra of the CPR as functions from:
electron beam parameters [17]; crystal parameters
and crystal orientation and direction of radiation re-
spectively to the electron beam [17]; shape and line-
width of coherent maxima [28]; absolute intensity and
spectral density [29]; polarization properties of the
CPR [30], [31], [32], [33]; interference between the
CBS and the CPR [34], [35] were measured with high
accuracy and they were compared with theoretical
models.
Within the framework of this collaboration the
following dissertations were defended: [36], [37], [38],
[39], [40].
5. CONCLUSIONS
In the latest decade of the past century physicists
intensively discussed the nature of co-called ”Para-
metric X-radiation of relativistic charged particles in
crystals”. It was a unique situation, that the sev-
eral experimental groups from the USSR (Tomsk,
Yerevan, Kharkov) in their experimental works ”con-
firmed” incorrect predictions of the erroneous theo-
ries. At that time the physical phenomenon was de-
scribed correctly only in one publication [10]. The
Darmstadt group (wide international collaboration)
carried out the most considerable and exact exper-
iments and step by step disproved wrong confirma-
tions.
Besides, professor Hideo Nitta (Tokyo Gakugei
University) made an important contribution into the-
oretical description of CPR (PXR(B)) in his works
[18], [41], [42].
”Recently, a new coherent radiation process
from crystals irradiated with relativistic electrons
was discovered [3], [10], [17], [43], [44], which is
called parametric X-ray radiation (PXR)”. This
statement was made by professor H. Nitta in the origin of
his work ”Theoretical notes on parametric X-radiation”
[41] in 1996.
H. Nitta pointed to the five pioneer experimental
works, where two of them [10], [17] were initiated and
carried out by the author of this paper.
About 30 years have passed since that time when the
first experiments had been started. My former colleagues
from the deceased USSR, who were my opponents for-
merly, wrote the series reviews on PXR [45], [46] [47],
[48]9, where they passed into silence the discussions on
the nature of the PXR and about their participation in
those discussions.
Here I use my right to comment some important in-
formation, which stays over of notice of specialists till
now.
Our work within the framework of the wide international
collaboration was supported by BMFT under Contract
N. 06DA641I, DFG Contract N. 436UKR113-19 and a
Max-Plank-Forschungspreis of Germany. Besides, our
work was supported by the Ukrainian Academy of Sci-
ence, The State Committee of Science and Technology of
Ukraine (Projects ”Semiclassical” and ”Quanta”), and by
Grants N.UA3000 and N.UA3200 from the International
Science Foundation.
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ÊÎÃÅÐÅÍÒÍÎÅ ÏÎËßÐÈÇÀÖÈÎÍÍÎÅ ÈÇËÓ×ÅÍÈÅ ÐÅËßÒÈÂÈÑÒÑÊÈÕ
ÝËÅÊÒÐÎÍÎÂ Â ÊÐÈÑÒÀËËÅ
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Êðàòêèé ðàññêàç îá èñòîðèè æàðêèõ ñïîðîâ è äèñêóññèé âîêðóã ïðèðîäû òàê-íàçûâàåìîãî ïàðàìåòðè-
÷åñêîãî ðåíòãåíîâñêîãî èçëó÷åíèÿ, êîòîðûå çàâåðøèëèñü ïðèçíàíèåì îòêðûòèÿ ÿâëåíèÿ êîãåðåíòíîãî
ïîëÿðèçàöèîííîãî òîðìîçíîãî èçëó÷åíèÿ ðåëÿòèâèñòñêèõ çàðÿæåííûõ ÷àñòèö â êðèñòàëëàõ. Èçëîæå-
íû èíôîðìàöèÿ è êîììåíòàðèè, êîòîðûå âñå åùå íàõîäÿòñÿ âíå ïîëÿ çðåíèÿ ñïåöèàëèñòîâ.
ÊÎÃÅÐÅÍÒÍÅ ÏÎËßÐÈÇÀÖIÉÍÅ ÂÈÏÐÎÌIÍÞÂÀÍÍß ÐÅËßÒÈÂIÑÒÑÜÊÈÕ
ÅËÅÊÒÐÎÍIÂ Â ÊÐÈÑÒÀËI
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Êîðîòêà ðîçïîâiäü ïðî ãàðÿ÷i ñóïåðå÷êè òà äèñêóñi¨ âiäíîñíî ïðèðîäè òàê-çâàíîãî ïàðàìåòðè÷íîãî
ðåíòãåíiâñüêîãî âèïðîìiíþâàííÿ, ÿêi çàâåðøèëèñÿ âèçíàííÿì âiäêðèòòÿ ÿâèùà êîãåðåíòíîãî ïîëÿðè-
çàöiéíîãî ãàëüìîâîãî âèïðîìiíþâàííÿ ðåëÿòèâiñòñüêèõ çàðÿäæåíèõ ÷àñòîê â êðèñòàëàõ. Âèêëàäåíi
iíôîðìàöiÿ òà êîìåíòàði, ÿêi âñå ùå çàëèøàþòüñÿ ïîçà óâàãîþ ñïåöiàëiñòiâ.
135
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| id | nasplib_isofts_kiev_ua-123456789-80496 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:14:50Z |
| publishDate | 2014 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Morokhovskii, V.L. 2015-04-18T15:00:08Z 2015-04-18T15:00:08Z 2014 Coherent polarization radiation of relativistic electrons in crystals / V.L.Morokhovskii// Вопросы атомной науки и техники. — 2014. — № 5. — С. 124-135. — Бібліогр.: 48 назв. — анг. 1562-6016 PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk https://nasplib.isofts.kiev.ua/handle/123456789/80496 A brief narration about the history of those heated arguments and discussions around the nature of so-called parametric X-radiation, which were concluded by the recognition of the discovery the phenomenon of coherent polarization bremsstrahlung of relativistic charged particles in crystals. Some important information and comments, which stay over of notice of specialists till now are reported. Краткий рассказ об истории жарких споров и дискуссий вокруг природы так-называемого параметрического рентгеновского излучения, которые завершились признанием открытия явления когерентного поляризационного тормозного излучения релятивистских заряженных частиц в кристаллах. Изложены информация и комментарии, которые все еще находятся вне поля зрения специалистов. Коротка розповiдь про гарячi суперечки та дискусiї вiдносно природи так-званого параметричного рентгенiвського випромiнювання, якi завершилися визнанням вiдкриття явища когерентного поляризацiйного гальмового випромiнювання релятивiстських заряджених часток в кристалах. Викладенi iнформацiя та коментарi, якi все ще залишаються поза увагою спецiалiстiв. Our work within the framework of the wide international collaboration was supported by BMFT under Contract
 N: 06DA641I, DFG Contract N: 436UKR113-19 and a Max-Plank-Forschungspreis of Germany. Besides, our
 work was supported by the Ukrainian Academy of Science, The State Committee of Science and Technology of
 Ukraine (Projects "Semiclassical" and "Quanta"), and by Grants N: UA3000 and N: UA3200 from the International Science Foundation. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Электродинамика Coherent polarization radiation of relativistic electrons in crystals Когерентное поляризационное излучение релятивистских электронов в кристалле Когерентне поляризацiйне випромiнювання релятивiстських електронiв в кристалi Article published earlier |
| spellingShingle | Coherent polarization radiation of relativistic electrons in crystals Morokhovskii, V.L. Электродинамика |
| title | Coherent polarization radiation of relativistic electrons in crystals |
| title_alt | Когерентное поляризационное излучение релятивистских электронов в кристалле Когерентне поляризацiйне випромiнювання релятивiстських електронiв в кристалi |
| title_full | Coherent polarization radiation of relativistic electrons in crystals |
| title_fullStr | Coherent polarization radiation of relativistic electrons in crystals |
| title_full_unstemmed | Coherent polarization radiation of relativistic electrons in crystals |
| title_short | Coherent polarization radiation of relativistic electrons in crystals |
| title_sort | coherent polarization radiation of relativistic electrons in crystals |
| topic | Электродинамика |
| topic_facet | Электродинамика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80496 |
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