Journal of Instrumentation
Coherent X-ray radiation excited by a beam of relativistic electrons in a
layered periodic structure
To cite this article: S. Blazhevich et al 2020 JINST 15 C05075
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2020 JINST 15 C05075
Published by IOP Publishing for Sissa Medialab
Received: January 13, 2020
Revised: March 7, 2020
Accepted: March 29, 2020
Published: May 29, 2020
XIII International Symposium on Radiation from Relativistic Electrons
in Periodic Structures — RREPS-19
September 16–20, 2019
Belgorod, Russian Federation
Coherent X-ray radiation excited by a beam of relativistic
electrons in a layered periodic structure
S. Blazhevich,
a
A. Noskov
a,b,1
and O. Shevchuk
a
a
Belgorod State University, Pobedy Str., 85, Belgorod 308015, Russia
b
Belgorod State Technological University named after V.G. Shukhov,
Kostyukova Str., 46, Belgorod 308012, Russia
E-mail: noskovbupk@mail.com
Abstract: The dynamic theory of coherent X-ray radiation, excited in a periodic layered medium
by a divergent beam of relativistic electrons in the Bragg scattering geometry, is developed. In
framework of the two-wave approximation of the dynamic theory of diffraction, expressions de-
scribing the spectral-angular and angular distributions of parametric X-ray radiation (PXR) and
diffracted transition radiation (DTR) are obtained taking into account the multiple scattering of the
relativistic electrons by the atoms of the target. Based on the expressions obtained, the possibilities
of the manifestation of the effects of dynamical diffraction in coherent X-rays were investigated. The
influence of the asymmetry of the electron Coulomb field reflection with respect to the target surface
(the reflecting layers situate nonparallel to target surface) on the spectral-angular characteristics of
the PXR and the DTR under conditions of the electron multiple scattering is estimated. It is shown
that at a fixed Bragg angle the width of the PXR spectrum increases when the angle of incidence of
the electron on the target decreases, which leads to an increase in the PXR angular density. In the
same conditions the width of the frequency domain of total external reflection and the amplitude of
the DTR spectrum also increase, which leads to a significant increase in the angular density of the
DTR. The obtained analytical expressions can be used to determine the optimal parameters of the
experiment on confirmation of the predicted dynamic effects.
Keywords: Models and simulations; Radiation calculations
1Corresponding author.
c
2020 IOP Publishing Ltd and Sissa Medialab
https://doi.org/10.1088/1748-0221/15/05/C05075
2020 JINST 15 C05075
Contents
1 Introduction 1
2 Geometry of the radiation process 2
3 Spectral-angular density 3
4 The influence of multiple scattering of relativistic electrons 4
5 Conclusions 8
1 Introduction
The coherent X-ray radiation excited by relativistic electrons in structured media is of current
interest for physicists and engineers from points of view both its fundamental research and its
different applications.
Parametric X-ray radiation (PXR) in a periodic layered medium arises due to diffraction of
pseudo-photons of the Coulomb field of a relativistic electron by the target layers, similar to how
PXR occurs in a single-crystal due to the diffraction by a system of parallel atomic planes [1, 2].
Diffracted transition radiation (DTR) is the result of diffraction of transition radiation photons
generated near the entrance surface of the target on the target layers by analogy with the DTR in a
single crystal [3, 4].
The dynamic theory of radiation of relativistic electrons in periodic layered media in the case
of symmetric reflection [5] well describes the experimental data presented in [6]. In the case
of symmetric reflection, the layers are parallel to the target surface. The process of coherent X-
ray radiation by a single relativistic electron in a periodic layered medium for the general case
of asymmetric reflection of the electron field relative to the target surface in the Laue scattering
geometry was considered in [7], and in the Bragg scattering geometry in [8]. In the work [7] it
was shown that the yield of the coherent radiation excited in layered target is considerably bigger
than in single-crystal one in the analogous conditions. This result opens the new perspectives for
application of such target in alternative source of X-ray radiation.
In [9], the dynamic theory of coherent X-ray radiation excited when a diverging beam of
relativistic electrons passes through a periodic layered target was developed. For the Bragg scattering
geometry, expressions are obtained that describe the spectral-angular characteristics of coherent X-
ray radiation.
In the present work, we have considered coherent X-ray radiation from a beam of the relativistic
electrons crossing the target in the form of a periodic layered medium in the Bragg scattering
geometry. The consideration has made for a general case of not parallel situation of the medium
layers in respect to the target surface. The main question under consideration in this work is how the
multiple scattering of the relativistic electrons affects the dynamical effects in the spectral-angular
distribution of the coherent radiation excited in the layered target. With this goal we investigate
– 1 –
2020 JINST 15 C05075
the effect of the relativistic electrons multiple scattering in the target by the medium atoms on the
spectral-angular characteristics of PXR and DTR. The significant growth of the coherent X-ray
radiation yield from the target with periodic layered structure in comparison with monocrystalline
one opens a perspective of application of such a target in intensive X-ray sources based on the
interaction of relativistic electron beams with periodic structures. The results of the investigation
of the radiation process features in such structures will be very useful in constructing the alternative
X-ray source based on interaction of relativistic electrons with periodic layered media.
2 Geometry of the radiation process
Let us consider a beam of relativistic electrons with divergence ψ
0
intersecting a layered target
in a Bragg scattering geometry (figure 1). We consider the transverse size of the electron beam
incident on the target as very small and do not take into account its effect on the radiation angular
distribution. Periodic layered medium consists of alternating layers with thicknesses l
1
and l
2
with
dielectric susceptibilities, respectively χ
1
and χ
2
(T = l
1
+ l
2
is the period of the layered target). The
reflecting layers are located at a certain angle δ to the target surface (figure 1), which corresponds
to the case of asymmetric reflection of the radiation field (δ = 0 is a special case of symmetric
reflection). We consider that the process of the coherent radiation excited in the target by each
electron in the beam is independent. Therefore, the spectral-angular density of radiation, generated
by the electron beam can be obtained by averaging the expression for the spectral-angular density
of the radiation generated by a separate electron in the beam through all its possible trajectories in
the target. As well as the angular distribution of electron in the beam and radiation is very narrow
we introduce the vector variables ψ, θ and θ
0
in accordance with the definitions of the velocity of
the relativistic electron V and unit vectors: n — in the direction of the photon momentum emitted
near the direction of the electron velocity vector, and n
g
— in the direction of Bragg scattering,
using the small-angle approximation:
V =
1 −
1
2
γ
−2
−
1
2
ψ
2
e
1
+ ψ, e
1
ψ = 0, n =
1 −
1
2
θ
2
0
e
1
+ θ
0
,
e
1
θ
0
= 0, e
1
e
2
= cos 2θ
B
, n
g
=
1 −
1
2
θ
2
e
2
+ θ, e
2
θ = 0, (2.1)
where θ ≈
|
θ
|
is the radiation angle counted from the axis e
2
of the radiation detector, ψ ≈
|
ψ
|
is the deviation angle of the rectilinear trajectory of the electron from the beam axis e
1
, θ
0
≈
|
θ
0
|
is the angle between the direction n of the incident pseudo photon propagation and the axis e
1
,
γ = 1/
√
1 −V
2
≡ E
e
/m is the Lorentz factor of the relativistic electron. In accordance with the V
definition in (2.1), the
|
V
|
= V for any value of the angle ψ. The e
2
axis determines the direction
of specular reflection of photons incident on the target along the axis e
1
. In our notation the Bragg
angle θ
B
is the angle between the electron beam axis e
1
and the reflecting layers in the target
(see in figure 1). So the expression e
1
e
2
= cos 2θ
B
in (2.1) is the definition of θ
B
as well. The
corresponding photon energy is defined as ω
B
=
g
2 sin θ
B
. The introduced vectors are considered
as the sum of components parallel and perpendicular to the plane of the figure: θ = θ
| |
+ θ
⊥
,
θ
0
= θ
0| |
+ θ
0⊥
, ψ = ψ
| |
+ ψ
⊥
.
– 2 –
2020 JINST 15 C05075
Figure 1. Geometry of the radiation process. N is normal to the target boundary, e
1
is unit vector of the
electron beam axis, e
2
is unit vector of the Bragg scattering direction, near which PXR and DTR are observed,
L is the target thickness, ψ
0
is parameter of the electron beam divergence, n is vector in the direction of
diffracted pseudo photon of the relativistic electron coulomb field, n
g
is vector in the direction of diffracted
pseudo photon. θ
B
is the Bragg angle of the electron beam (angle of mirror reflection of the photon on the
beam axis e
1
from layers of the target to direction of axis e
2
).
3 Spectral-angular density
In the framework of the two-wave approximation of the dynamic theory of diffraction, the expres-
sions that describe the spectral-angular density of PXR and DTR in the case of a thin non-absorbing
target were obtained in [9]:
ω
d
2
N
(
s
)
PXR
dωdΩ
=
e
2
π
2
Ω
(
s
)
2
(Ω − χ
0
0
)
2
ξ
(
s
)
+
p
ξ
(
s
)
2
− ε
2
ξ
(
s
)
2
− ε + ε sin
2
b
(
s
)
√
ξ
(
s
)
2
−ε
ε
sin
2
b
(
s
)
2
ξ
(
s
)
+
√
ξ
(
s
)
2
−ε
ε
− σ
(
s
)
ξ
(
s
)
+
√
ξ
(
s
)
2
−ε
ε
− σ
(
s
)
2
,
(3.1a)
ω
d
2
N
(
s
)
DTR
dωdΩ
=
e
2
π
2
Ω
(
s
)
2
1
Ω
−
1
Ω − χ
0
0
2
ε
2
ξ
(s)
2
−
ξ
(s)
2
− ε
coth
2
b
(
s
)
√
ε−ξ
(s)
2
ε
. (3.1b)
– 3 –
