Incoherent bremsstrahlung in flat and bent crystals
Incoherent bremsstrahlung by high-energy particles in crystal is due to the thermal spread of atoms in relation to their equilibrium positions in the lattice. The simulation procedure developed earlier for the incoherent radiation is applied to the case of the electrons and positrons motion in the s...
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| Zitieren: | Incoherent bremsstrahlung in flat and bent crystals / N.F. Shul'ga, V.V. Syshchenko, A.I. Tarnovsky // Вопросы атомной науки и техники. — 2012. — № 1. — С. 74-78. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859558418376818688 |
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| author | Shul'ga, N.F. Syshchenko, V.V. Tarnovsky, A.I. |
| author_facet | Shul'ga, N.F. Syshchenko, V.V. Tarnovsky, A.I. |
| citation_txt | Incoherent bremsstrahlung in flat and bent crystals / N.F. Shul'ga, V.V. Syshchenko, A.I. Tarnovsky // Вопросы атомной науки и техники. — 2012. — № 1. — С. 74-78. — Бібліогр.: 12 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | Incoherent bremsstrahlung by high-energy particles in crystal is due to the thermal spread of atoms in relation to their equilibrium positions in the lattice. The simulation procedure developed earlier for the incoherent radiation is applied to the case of the electrons and positrons motion in the sinusoidally bent crystal. The results of simulation are in agreement with the data of recent experiments carried out at the Mainz Microtron MAMI. The possibility of use of the sinusoidally bent crystals as undulators is discussed.
Некогерентное тормозное излучение частиц высокой энергии в кристалле обусловлено тепловым разбросом атомов относительно их равновесных положений в решетке. Развитая ранее процедура моделирования некогерентного излучения применена к случаю движения электронов и позитронов в синусоидально изогнутом кристалле. Результаты моделирования согласуются с данными недавних экспериментов на микротроне MAMI в Майнце. Обсуждается перспективность использования синусоидально изогнутых кристаллов в качестве ондуляторов.
Некогерентне гальмівне випромінювання частинок високої енергії у кристалі обумовлено тепловим розкидом атомів відносно їх рівноважних положень у решітці. Процедура моделювання некогерентного випромінювання, що розвита раніше, пристосовується до випадку руху електронів та позитронів у сiнусоiдально вигнутому кристалі. Результати моделювання знаходяться у відповідності до даних недавніх експериментів на мікротроні MAMI у Майнцi. Обговорюється перспективність використання сiнусоiдально вигнутих кристалів як ондуляторiв.
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Section B. QED Processes at High Energies
INCOHERENT BREMSSTRAHLUNG IN FLAT AND BENT
CRYSTALS
N.F. Shul’ga 1, V.V. Syshchenko 2∗, A.I. Tarnovsky 2
1National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
2Belgorod State University, 308015, Belgorod, Russian Federation
(Received September 26, 2011)
Incoherent bremsstrahlung by high-energy particles in crystal is due to the thermal spread of atoms in relation to
their equilibrium positions in the lattice. The simulation procedure developed earlier for the incoherent radiation is
applied to the case of the electrons and positrons motion in the sinusoidally bent crystal. The results of simulation
are in agreement with the data of recent experiments carried out at the Mainz Microtron MAMI. The possibility of
use of the sinusoidally bent crystals as undulators is discussed.
PACS: 02.70.-c, 41.60.-m, 61.85.+p
1. INTRODUCTION
The bremsstrahlung cross section for relativistic elec-
trons in a crystal is split into the sum of the coher-
ent part (due to the spatial periodicity of the atoms’
arrangement in the crystal) and the incoherent one
(due to the thermal motion of atoms in the crystal)
[1,2]. Although the spectrum of incoherent radiation
in crystal is similar to one in amorphous medium, the
incoherent radiation intensity could demonstrate sub-
stantial dependence on the crystal orientation due to
the electrons’ flux redistribution in the crystal (chan-
neling etc.). The simulation based on the semiclassi-
cal description of the radiation process [3–5] confirms
that viewpoint. The results of simulation are in a
good agreement with the corresponding early [6] and
recent [7] experimental data.
Here we present the results of simulation using the
improved procedure taking into account the crystal
deformations. The simulation was carried out under
the conditions of the recent experiment performed at
the Mainz Microtron MAMI [8] to explore the radi-
ation emission from periodically bent crystal. The
possibility of application of such crystals as undula-
tors is discussed during last years [8–10].
2. COMPUTATION METHOD
Let us consider the high energy electron incidence on
the atomic string in the crystal. The two-dimensional
multiple scattering angle ϑ is equal to the sum of in-
dividual scattering angles on the atoms:
ϑ =
∑
n
ϑ(ρn), (1)
where ρn is the impact parameter of the collision with
the n-th atom of the crystal. The mean square of the
multiple scattering angle (averaged over the thermal
vibrations of atoms) can be expressed as the sum of
two blocks of terms:〈∣∣∣∣∣
∑
n
ϑ(ρn)
∣∣∣∣∣
2〉
=
∑
n,m
〈ϑ(ρn)〉 〈ϑ(ρm)〉
+
∑
n
{〈
ϑ(ρn)2
〉 − 〈ϑ(ρn)〉2
}
≡ 〈
ϑ2
〉
coh
+
〈
ϑ2
〉
incoh
. (2)
The first block describes the coherent scattering
which can be interpreted as a motion in the uniform
string potential [2,11]. The second one describes the
incoherent scattering of the electron on the thermal
vibrations of the lattice atoms.
The bremsstrahlung spectrum of the electrons
passing through the crystal also can be expressed
as the sum of the coherent and incoherent contribu-
tions [1, 2]. For the electrons of the energy ε ∼ 1
GeV the main contribution of the coherent effect is
made to the soft range of the spectrum (the photon
energy h̄ω is less or of the order of dozens of MeVs).
In the medium and hard ranges of the spectrum the
incoherent radiation is predominant.
The spectral density of the incoherent radiation
from the individual electron moving on the given tra-
jectory is described by the formula [4, 5](
dE
dω
)
incoh
=
2e2ε(ε − h̄ω)
3πm2c5
{
1 +
3
4
(h̄ω)2
ε(ε − h̄ω)
}
×
∑
n
{〈
ϑ(ρn)2
〉 − 〈ϑ(ρn)〉2
}
, (3)
∗Corresponding author E-mail address: syshch@yandex.ru
74 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 74-78.
where m and e are the electron’s mass and charge, c
is the velocity of light. It is convenient to compare
the incoherent radiation intensity of the uniform elec-
tron beam in the crystal to the corresponding inten-
sity in the amorphous medium (described by Bethe-
Heitler formula). The ratio of these two values is
equal to [4, 5]
Nγ =
1
2
πNR2 ln(mRc/h̄)
×
∫
d2ρ0
∑
n
{〈
ϑ(ρn)2
〉 − 〈ϑ(ρn)〉2
}
, (4)
where N is the total number of the electron’s colli-
sions with atoms under its motion through the crys-
tal, R is Thomas-Fermi radius of the atom, integra-
tion over d2ρ0 means the integration over all possible
points of incidence of the electron on the crystal in
the limits of one elementary cell.
The values of the function F (ρ) =
〈
ϑ(ρ)2
〉 −
〈ϑ(ρ)〉2 are determined by linear interpolation of the
values pre-calculated on the regular grid of impact
parameters [5]. The impact parameters ρn are found
using the electron’s trajectory obtained by numerical
integration of the equation of motion in the field of
the set of parallel uniform strings. The influence of
the incoherent scattering by the thermal vibrations of
atoms on the electron’s trajectory can be taken into
account by adding to each component of the elec-
tron’s transverse velocity the random value with the
dispersion c
√{
〈ϑ(ρ)2〉 − 〈ϑ(ρ)〉2
}
/2 after each col-
lision. For the further computational details see [4,5].
-1.5 -1 -0.5 0 0.5 1 1.5
0
0.5
1
1.5
2
2θ
c
θ, mrad
N
γ
c
-1.5 -1 -0.5 0 0.5 1 1.5
0
0.5
1
1.5
2 2θ
c
θ, mrad
N
γ
d
e
-
e
+
a
e
-
e
+
b
Fig. 1. (a) Typical trajectories of the electrons and positrons under planar channeling. Pluses mark the
positions of atomic strings (perpendicular to the figure plane) forming the atomic planes of the crystal. The
horizontal scale of the figure is highly compressed. (b) The same for above-barrier motion. (c) Simulated in-
coherent bremsstrahlung intensity (in ratio to the Bethe-Heitler intensity in amorphous medium, see Eq. (4))
from 1 GeV electrons vs incidence angle θ to (110) plane of 30 μm thick Si crystal [3]. Dotted line corre-
sponds to the trajectories simulated neglecting thermal vibrations of atoms. (d) The same for positrons
3. RESULTS OF SIMULATION
The origin of the orientation dependence of the in-
coherent radiation intensity is illustrated in Fig. 1.
When the electron is incident to the atomic plane of
the crystal under angle θ less than some critical an-
gle θc, it can be captured by the attractive potential
of the plane. The finite motion in that potential is
called as the planar channeling (see, e.g., [2, 11]).
75
Under planar channeling the electrons collide
with atoms under small impact parameters more fre-
quently than in amorphous medium, that leads to
the increase of the incoherent bremsstrahlung inten-
sity; for above-barrier motion the situation is oppo-
site. The account of the incoherent scattering of the
particles on the thermal vibrations of the atoms leads
to dechanneling and, hence, to the smoothing of the
describer orientation dependence (compare solid and
dotted lines in Fig. 1, (c) and 1, (d)).
In the present article the simulation was carried
out under the conditions of the experiment [8], where
the radiation from ε = 855 MeV electrons under their
incidence onto the silicon crystal with sinusoidally
bent (110) planes had been studied. The yield of
photons with the energy h̄ω = ε/2 (for which the
incoherent mechanism of bremsstrahlung is predom-
inant) had been registered.
-60 -40 -20 0 20 40
0
1
2
3
4
φ, mrad
N
γ
(1
1
0
)
-- (5
5
1
)
-- (3
3
1
)
-- (1
1
1
)
(0
0
1
)
(1
1
1
)
(3
3
1
)
Electrons in flat crystal
-60 -40 -20 0 20 40
0
0.5
1
1.5
2
φ, mrad
N
γ
(1
1
0
)
-- (5
5
1
)
-- (3
3
1
)
-- (1
1
1
)
(0
0
1
)
(1
1
1
)
(3
3
1
)
Electrons in bent crystal
Fig. 2. The incoherent bremsstrahlung intensity (in ratio to Bethe-Heitler intensity in amorphous medium)
from 855 MeV electrons in flat (upper plot) and sinusoidally bent (lower plot) silicon crystals under scanning
of the goniometric angle φ like in the experiment [8]
The results of simulation (Fig. 2) demonstrate the
qualitative agreement with the experimental data [8].
We can see characteristic structures similar to one
in Fig. 1, (c), generated by different crystallographic
planes with the common [11̄0] axis.
The decrease of the radiation intensity in com-
parison to the reference flat crystal permits to esti-
mate the reduction of the dechanneling length due to
the crystal bending. We can see that the bending of
the crystallographic planes increases the dechannel-
ing rate so highly that the scattering on the thermal
vibrations of atoms already have no substantial influ-
ence on the incoherent radiation intensity (compare
solid and dashed curves on the lower panel of Fig. 2).
The main cause of the rapid dechanneling in the
bent crystal lies in the arising of the centrifugal ad-
dition to the planar potential [11]:
Ueff = U(x) − ε
x
Rb
under |x| � Rb (5)
(where Rb is the bending radius) and, as a conse-
quence, to the elimination of the potential barriers
between which the channeling could take the place.
In our case the height of the potential barrier in the
flat crystal is about Umax ≈ 23.5 eV, and the centrifu-
gal correction on the half of the interplanar distance
ΔUc(Δx = dp/2) ≈ 13.2 eV for the maximal bend-
ing of the sinusoid, that corresponds to the curve 2
76
in Fig. 3. So, only a small part of incident particles
could be captured into the channeling regime.
Fig. 3. Potential energy of the electron in the
planar potentials of the flat (curve 1) and bent
(curve 2) crystals
4. CONCLUSIONS
The results of simulation demonstrate that the pla-
nar channeling effect leads to the orientation depen-
dence of the incoherent bremsstrahlung intensity not
only on flat, but also in bent crystals. However, in
the bent crystal the electrons rapidly leave the planar
channels. That leads to doubt about the efficiency of
sinusoidally bent crystals as undulators. As we can
see from Fig. 4, the use of the positron beam instead
of the electron one could not seriously improve the
situation.
Qualitatively similar results could be expected for
the nuclear reactions yield in bent crystals [12].
This work is supported in part by internal grant
of Belgorod State University and the federal program
“Academic and Teaching Staff of Innovation Rus-
sia”, the government contract 16.740.11.0147 dated
02.09.2010.
-60 -40 -20 0 20 40
0
1
2
3
4
φ, mrad
N
γ
(1
1
0
)
-- (5
5
1
)
-- (3
3
1
)
-- (1
1
1
)
(0
0
1
)
(1
1
1
)
(3
3
1
)
Positrons in flat crystal
-60 -40 -20 0 20 40
0
0.5
1
1.5
2
φ, mrad
N
γ
(1
1
0
)
-- (5
5
1
)
-- (3
3
1
)
-- (1
1
1
)
(0
0
1
)
(1
1
1
)
(3
3
1
)
Positrons in bent crystal
Fig. 4. The same as in Fig. 2 for positrons
77
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12. M.V. Bondarenco. Nuclear interactions and mul-
tiple coulomb scattering at volume reflection //
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|
| id | nasplib_isofts_kiev_ua-123456789-106986 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-26T14:12:46Z |
| publishDate | 2012 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Shul'ga, N.F. Syshchenko, V.V. Tarnovsky, A.I. 2016-10-10T15:54:44Z 2016-10-10T15:54:44Z 2012 Incoherent bremsstrahlung in flat and bent crystals / N.F. Shul'ga, V.V. Syshchenko, A.I. Tarnovsky // Вопросы атомной науки и техники. — 2012. — № 1. — С. 74-78. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 02.70.-c, 41.60.-m, 61.85.+p https://nasplib.isofts.kiev.ua/handle/123456789/106986 Incoherent bremsstrahlung by high-energy particles in crystal is due to the thermal spread of atoms in relation to their equilibrium positions in the lattice. The simulation procedure developed earlier for the incoherent radiation is applied to the case of the electrons and positrons motion in the sinusoidally bent crystal. The results of simulation are in agreement with the data of recent experiments carried out at the Mainz Microtron MAMI. The possibility of use of the sinusoidally bent crystals as undulators is discussed. Некогерентное тормозное излучение частиц высокой энергии в кристалле обусловлено тепловым разбросом атомов относительно их равновесных положений в решетке. Развитая ранее процедура моделирования некогерентного излучения применена к случаю движения электронов и позитронов в синусоидально изогнутом кристалле. Результаты моделирования согласуются с данными недавних экспериментов на микротроне MAMI в Майнце. Обсуждается перспективность использования синусоидально изогнутых кристаллов в качестве ондуляторов. Некогерентне гальмівне випромінювання частинок високої енергії у кристалі обумовлено тепловим розкидом атомів відносно їх рівноважних положень у решітці. Процедура моделювання некогерентного випромінювання, що розвита раніше, пристосовується до випадку руху електронів та позитронів у сiнусоiдально вигнутому кристалі. Результати моделювання знаходяться у відповідності до даних недавніх експериментів на мікротроні MAMI у Майнцi. Обговорюється перспективність використання сiнусоiдально вигнутих кристалів як ондуляторiв. This work is supported in part by internal grant of Belgorod State University and the federal program “Academic and Teaching Staff of Innovation Russia”, the government contract 16.740.11.0147 dated 02.09.2010. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Section B. QED Processes at High Energies Incoherent bremsstrahlung in flat and bent crystals Некогерентное тормозное излучение в прямых и изогнутых кристаллах Некогерентне гальмівне випромiнювання у прямих та вигнутих кристалах Article published earlier |
| spellingShingle | Incoherent bremsstrahlung in flat and bent crystals Shul'ga, N.F. Syshchenko, V.V. Tarnovsky, A.I. Section B. QED Processes at High Energies |
| title | Incoherent bremsstrahlung in flat and bent crystals |
| title_alt | Некогерентное тормозное излучение в прямых и изогнутых кристаллах Некогерентне гальмівне випромiнювання у прямих та вигнутих кристалах |
| title_full | Incoherent bremsstrahlung in flat and bent crystals |
| title_fullStr | Incoherent bremsstrahlung in flat and bent crystals |
| title_full_unstemmed | Incoherent bremsstrahlung in flat and bent crystals |
| title_short | Incoherent bremsstrahlung in flat and bent crystals |
| title_sort | incoherent bremsstrahlung in flat and bent crystals |
| topic | Section B. QED Processes at High Energies |
| topic_facet | Section B. QED Processes at High Energies |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106986 |
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