Electron beam dynamics and channeling radiation simulation in crystal
2D code was designed for the simulation and simulation algorithms and methods are discussed in this report. The planar channeling is studying. Results of electron dynamics and radiation simulations are presented. The designed algorithm can be also used to separate channeled and dechanneled particl...
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irk-123456789-802452015-04-14T03:02:30Z Electron beam dynamics and channeling radiation simulation in crystal Bashmakov, Yu.A. Polozov, S.M. Динамика пучков 2D code was designed for the simulation and simulation algorithms and methods are discussed in this report. The planar channeling is studying. Results of electron dynamics and radiation simulations are presented. The designed algorithm can be also used to separate channeled and dechanneled particles. The dependence of the oscillation period versus electron oscillation amplitudes is simulated. It allows defining of the radiation spectrum. The fundamental and high order harmonics radiation probability is discussed. The probability of radiation spectrums is presented. Описываются алгоритмы и численные методы, использованные при разработке программы для двумерного численного моделирования. Рассматривается случай плоскостного каналирования и приведены результаты моделирования динамики электронов в кристалле и сопровождающего излучения. Выделяются частицы, захваченные в канал, и частицы, совершающие надбарьерное движение. Определяется зависимость периода колебаний от начальных условий. Эта зависимость существенна для определения спектральных свойств излучения. Рассмотрена возможность генерации основной и высших гармоник излучения. Приводятся спектры вероятности излучения. Описуються алгоритми і чисельні методи, що були використані при розробці програми для двомірного моделювання. Розглядається випадок площинного каналування і наведені результати моделювання динаміки електронів у кристалі і супровідного випромінювання. Виділяються частинки, захоплені в канал, і частинки, що роблять надбар’єрний рух. Визначається залежність періоду коливань від початкових умов. Ця залежність істотна для визначення спектральних властивостей випромінювання. Розглянуто можливість генерації основної і вищих гармонік випромінювання. Приводяться спектри ймовірності випромінювання. 2014 Article Electron beam dynamics and channeling radiation simulation in crystal / Yu.A. Bashmakov, S.M. Polozov // Вопросы атомной науки и техники. — 2014. — № 3. — С. 134-137. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 29.27.Bd http://dspace.nbuv.gov.ua/handle/123456789/80245 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Динамика пучков Динамика пучков |
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Динамика пучков Динамика пучков Bashmakov, Yu.A. Polozov, S.M. Electron beam dynamics and channeling radiation simulation in crystal Вопросы атомной науки и техники |
| description |
2D code was designed for the simulation and simulation algorithms and methods are discussed in this report. The
planar channeling is studying. Results of electron dynamics and radiation simulations are presented. The designed
algorithm can be also used to separate channeled and dechanneled particles. The dependence of the oscillation period
versus electron oscillation amplitudes is simulated. It allows defining of the radiation spectrum. The fundamental
and high order harmonics radiation probability is discussed. The probability of radiation spectrums is presented. |
| format |
Article |
| author |
Bashmakov, Yu.A. Polozov, S.M. |
| author_facet |
Bashmakov, Yu.A. Polozov, S.M. |
| author_sort |
Bashmakov, Yu.A. |
| title |
Electron beam dynamics and channeling radiation simulation in crystal |
| title_short |
Electron beam dynamics and channeling radiation simulation in crystal |
| title_full |
Electron beam dynamics and channeling radiation simulation in crystal |
| title_fullStr |
Electron beam dynamics and channeling radiation simulation in crystal |
| title_full_unstemmed |
Electron beam dynamics and channeling radiation simulation in crystal |
| title_sort |
electron beam dynamics and channeling radiation simulation in crystal |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2014 |
| topic_facet |
Динамика пучков |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/80245 |
| citation_txt |
Electron beam dynamics and channeling radiation simulation in crystal / Yu.A. Bashmakov, S.M. Polozov // Вопросы атомной науки и техники. — 2014. — № 3. — С. 134-137. — Бібліогр.: 10 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT bashmakovyua electronbeamdynamicsandchannelingradiationsimulationincrystal AT polozovsm electronbeamdynamicsandchannelingradiationsimulationincrystal |
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2025-07-06T04:12:49Z |
| last_indexed |
2025-07-06T04:12:49Z |
| _version_ |
1836869401098321920 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2014. №3(91) 134
ELECTRON BEAM DYNAMICS AND CHANNELING RADIATION
SIMULATION IN CRYSTAL
Yu.A. Bashmakov1,2, S.M. Polozov1
1National Research Nuclear University “Moscow Engineering Physics Institute”,
Moscow, Russia
E-mail: smpolozov@mephi.ru;
2P.N. Lebedev Physical Institute of RAS, Moscow, Russia
E-mail: bashm@x4u.lebedev.ru
2D code was designed for the simulation and simulation algorithms and methods are discussed in this report. The
planar channeling is studying. Results of electron dynamics and radiation simulations are presented. The designed
algorithm can be also used to separate channeled and dechanneled particles. The dependence of the oscillation peri-
od versus electron oscillation amplitudes is simulated. It allows defining of the radiation spectrum. The fundamental
and high order harmonics radiation probability is discussed. The probability of radiation spectrums is presented.
PACS: 29.27.Bd
INTRODUCTION
The method of obtaining of the narrow-band X-rays
lies in utilizing the principle of so called channeling
radiation from crystals [1 - 3].
Channeling radiation is emitted by relativistic elec-
trons passing through single crystals along a direction of
high symmetry. The radiation is forward directed into a
narrow cone with an angle of emission Θ~γ-1.
There are two different types of channeling depend-
ent on the electron track – axial channeling and planar
channeling [4]. In the first case electron captured in the
channel is moving along the crystal axis and experience
the influence of the axially-symmetrical averaged cou-
lomb field of the crystal axis. In the planar channeling
the particle is forced by the fields of the atoms situated
on the crystalline plane.
The mechanism of channeling radiation can be de-
scribed in two principal ways: classical physics model
and quantum mechanics.
1. MAIN RELATIONS AND ANALYTICAL
STUDY
Electrical field formed between the crystallographic
planes forming the channel can be characterized with an
averaged potential ( )U x where x is transversal offset
from the channel central plane. As a rule ( )U x is
smooth, even and periodical function with period of 2d:
( ) ( )U x U x− = , ( 2 ) ( )U x dk U x+ = , where 2d is the
channel width, k − integer number. Potential value on
the border of the channel can be labeled as
0(| | )U x d U= = . Potential describing the electron chan-
neling phenomenon is often called “reversed parabola”
[3]. With fine accuracy it can be expressed as follows:
nN
n
n d
xaUxU ∑
=
=
0
0)( dx < , (1)
where U0 − potential of the field in crystal, 2d is the
distance between the crystal planes.
The schematic view of such potential is shown in
Fig. 1 for diamond crystal with U0=23.6 V and
d=0.67 Å.
Fig. 1. Schematic view of crystal potential [4]
In this field the particle with charge e, energy ε and
rest mass of m perform small transversal harmonic os-
cillations relative to channel central plane. If the magni-
tude of oscillations is much less then channel width
( mx d<< ) and the period of the oscillations can be ex-
pressed as:
0
0 0
2
2 , ,n
eUc
d ε
Ω = Ω Ω = (2)
here γ=ε/mc2=(1-β2)-1/2 is reduced particle energy, β=v/c,
v − particle velocity. It is obvious that the frequency of
transversal oscillations is reduced with energy gain as
~ γ-1/2 [4, 5].
Let us mention that the frequency of large transver-
sal oscillations ~mx d depends on their amplitude.
Therewith in particle motion Fourier harmonic expan-
sion appears higher harmonics of the fundamental fre-
quency and the particles motion becomes anharmonic.
This fact is significant for investigation of the radiation
spectral characteristics
Radiation frequency of k-harmonic propagating with
angle θ to the central plane of the channel in dipole
approximation for ultra-relativistic motion ( 1γ >> ) can
be written as:
2
2 2
2 , ( 1, 1)
1k
k γω θ γ
θ γ
Ω
= << >>
+
. (3)
Radiation frequency achieves its maximum at the ze-
ro angle: 22 γΩ=ω kk .
Based on mentioned equations maximum energy for
9 MeV electron channeling relative to (110) plane of the
diamond crystal estimation value of the radiated X-ray
mailto:smpolozov@mephi.ru
mailto:bashm@x4u.lebedev.ru
ISSN 1562-6016. ВАНТ. 2014. №3(91) 135
photons can be obtained. For this case maximum value
appears to be ћω=5.90 keV that qualitatively matches
with experimental results [3].
Power of radiation losses of electron due to channel-
ing radiation is defined by the equation:
4 2 2
2
2
3
e EP
m c
γ< >
= , (4)
here 2E< > is the mean square of electrical field along
the particle trajectory.
In the general the spectral and angle distribution of
the irradiated energy can be defined as the sum [5, 6]:
( )∑
∞
= σ
σπ
Ω
ϕθω
π
=
θω
ε
1
2
2
2
2
2
22 sin),,(
d
d
k k
kk K
c
e
d
a
. (5)
Here the vector ∫
Ωπ
Ω
π
Ω
=
/2
0
d)(
2
tet tik
k aa ,
[ ][ ]( ) ( )
δ++
ω
−β−ββ= − znynxn
c
it zyxexp1)( 2n-nna ,
( )[ ]Ω−β−ωΩ=θωσ − kn zzk 1),( 1 , ϕθ= cossinxn ,
ϕθ= sinsinyn , θ= cosxn , θ is the angle of normal n
and longitudinal axis z, φ is the angle of projection of n
to (x, y) plane, vector ),,( ϕθωka defines radiation char-
acteristics versus electron oscillations form and ampli-
tude.
In the dipole case for the ultra relativistic electron
we can to simplify the equation for spectral and angle
distribution and the irradiation spectrum can be defined
from (5) as:
[ ][ ]
2)1( β−
ββ
=
n
-nn
a k
k
, ∫
Ωπ
Ωβ
π
Ω
=β
/2
0
d
2
te tik
k
,
[ ]Ω−ωΩ=σ − kk '1 and we can to write equation of the
radiation energy spectrum
( )
( )
,d2211
)(
)(sin
8
d
d
22
1
2
22
42
u
uuuku
kuK
c
e
Ek
k
+−
−ξ
−ξπ
β×
×
Ω
γ
=
ξ
ε
∑
∞
+ξ=
(6)
here 22/ γΩω=ξ .
For the very large number of structure periods as it
is observed in crystal Eq. (6) can be rewritten as:
( )
d2218
d
d
2
2
2
1
2423
u
kkkc
Ke
Ek
k
ξ
+
ξ
−
ξ
β
Ω
γπ
=
ξ
ε ∑
∞
+ξ=
. (7)
The general view of electron phase trajectories in
ideal crystal with a harmonic potential are shown in
Fig. 2 [2].
Fig. 2. Analytically predicted electron phase space
trajectories in crystal with a harmonic potential [1]
2. CODE FOR NUMARICAL SIMULATION
The numerical simulation of electron oscillations in
a crystal was performed for planar channeling. The
equations of electron motion in a crystal in 2D Cauchy
form are:
d 0,
d
d ,
d
d ( ) ,
d
d .
d
z
x
x
t
z v
t
v U xe
t x
x v
t
γ
=
=
∂
= −
∂
=
(8)
Here γ=W/W0 – Lorenz-factor. The crystal potential is
presented in the generalized form Eq. 1, coefficients an
define the form of the crystal potential and are known
from numerous experimental data for a wide number of
the most useful crystals. It is proposed in the model that
the electron energy loss by radiation is much smaller
than beam energy and may be neglected.
New code for electron motion simulation in a crystal
was created using the BEAMDULAC code [7 - 9]. The
BEAMDULAC beam dynamics simulation code in-
cludes a basic core and a number of modules. It was
under development in DINUS research Laboratory at
MEPhI for the many years. The code has the modular
structure and numbers of routines to different task solve:
initial particles distribution (uniformly, Gauss, wa-
terbag), motion equation integration (4th order Runge-
Kutta method), beam emittance calculation, post pro-
cessing and etc. The code package has versions for own
space charge effects treatment both Coulomb part and
RF part (beam irradiation and beam loading).
The number of BEAMDULAC code modules (injec-
tion, integration, emittance calculation, post-processing)
were used to solve the channeling of electrons in crystal
problem.
Some modifications were made in the code to ana-
lyze results of the beam dynamics simulation of elec-
trons in crystal. First we should define the oscillation
period. It can be calculated as the distances between of
two zeros of trajectories ><− xx~ . Second the phase
trajectories ( r
•
β ) of each particle were expanded to the
Fourier series. The oscillations period of electron should
be defined for it and Fast Fourier Transform (FFT) algo-
rithm applying to define the oscillation harmonics. Fol-
lowing the averaged electron’s energy loses to radiation
are simulated as [5]:
,
3
2
0
2
0 ><
γ
>=δ< ch
cr
rad E
W
lrW (9)
where >< chE is the averaged field acting to the elec-
tron, lcr is the crystal length, W0=mc2,
r0=e2/4πε0m2c4=1.82·10-15 m. The number of photons
radiated by each electron can be calculated and the total
number of generated photons can be defined to the each
frequency.
ISSN 1562-6016. ВАНТ. 2014. №3(91) 136
3. EXAMPLES OF SIMULATION
The test simulation was made for diamond crystal
having crystal potential U0=23.6 V and coefficients
a1=4.0, a3= -8.0, d=0.67 Å. The crystal length was cho-
sen as 55 µm because of the multiple electrons scattering
will influence sufficiently to results for thicker crystals.
Fig. 3. Trajectories of channeled electrons in (z, x)
plane (a) and phase trajectories in (βr, r)
phase plane (b) are shown
The results of simulation are presented in Fig. 3. The
trajectories in (z, r) plane (a) and phase trajectories in
(βr, r) phase plane (b) are shown. The beam energy is
21 MeV, waterbag initial distribution was used for sim-
ulation. Trajectories in Fig. 3,a are detailed because of
the number of oscillations is a thousands for 55 µm
length crystal. The values of length and transverse coor-
dinate are normalized: β=v/c, ρ=x/d. The phase motion
stability region and the separatrix are clear seen. Note
that the electrons phase trajectories in the crystal are
similar to the analytically predicted (see Fig. 2 and [1]).
Both the fundamental and the third spectrum harmonics
can be seen from the figure.
The maximal angle of electron’s trajectory and crys-
tal plane θm can be easily defined with initial distribu-
tion conditions. An electron can be dechanneled in case
when this angle will be bigger than critical angle θcr. As
the first example the maximal injection angle is restrict-
ed by θm 2.5·10-3 and the part of dechanneled electrons
is about 7%.The dechanneling is more intensive in case
when injection angle increased up to 7.5·10-3. The part
of dechanneled particles grows up to 30% this case (Fig.
4). Trajectories of unchanneled electrons can be easily
seen in Fig. 4,a and they have the large angles with
crystal planes and unstable phase trajectories in (βr, r)
phase plane are placed top and bottom of separatix. A
dechanneled particle cross a number of channels. The
oscillation period of such particles is much smaller than
of channeled one. The multiple scattering of electrons
moving in crystal close to the separatrix can brought to
the particle dechanneling.
Fig. 4. Channeling of electrons in case of large value
of injection angle θ, dechanneled particles can be seen
Fig. 5. Typical energy distribution of radiated photons
The total energy distribution of radiated photons nph
is shown in Fig. 5. The distribution is calculated with
W=21 MeV, injection angle is equal to 2.5·10-3, number
of electrons in the bunch is equal to ne=6·1011 with cor-
responds to 1 mA of pulse beam current and 10 µs pulse
length. Such bunch parameters are typical for electron
linacs. The dechanneled particles are excludes from
spectrum calculation after particles dynamics simulation
by the simple algorithm. Following the probability of
one electron radiation is calculating using known oscil-
lation frequencies distribution and Eq. (4). At last the
probability density distribution is multiplies on the par-
ticles number in the bunch.
The simulation shows that the frequency density dis-
tribution maximums are shifted right with electrons en-
ergy increase which corresponds to the theory.
If we can to define the phase trajectories of chan-
neled electrons the period of oscillations can be calcu-
lated for all particles. The connection of electrons oscil-
lation period T and amplitude xmax in crystal is shown in
Fig. 6.
a
b
b
a
ISSN 1562-6016. ВАНТ. 2014. №3(91) 137
Fig. 6. Dependence of electrons oscillation period (T/d)
of oscillation amplitude (xmax/d)
CONCLUSIONS
The oscillations of electrons in crystal are discussed
in the effective potential approach. The general analyti-
cal description was presented and main predictions are
made. Following the especially designed code to simu-
late electrons dynamics in crystal was discussed. Results
of simulation are presented and they are very close to
analytically proposed.
Let us discuss one possible channeling radiation ap-
plication. Nowadays angiography has become one of the
most commonly used medical procedures. However the
X-ray tubes are mostly used in angiography imaging
systems. The problem that encounters in using X-ray
tubes is low monochromaticity due to bremsstrahlung
while angiography imaging requires quasimono-
chromatic energy spectrum for better image quality and
lower dose rate obtained by the patient. There are sever-
al methods of eliminating undesirable spectrum parts of
the radiation: synchrotron or undulator radiation, Comp-
ton scattering, K-capture and radiations in aligned crys-
tals (such as channeling radiation, coherent bremsstrah-
lung, parametric X-ray radiation, etc.).
The new angiography facility was proposed in [10].
The channeling radiation and polycapillary optics to
prevent the irradiation of the patient by bremsstrahlung
from crystal is proposed to utilize in such new facility.
REFERENCES
1. Yu.A. Bashmakov, E.G. Bessonov // Rad. Eff. 1982,
v. 66, p. 85-94.
2. J. Freudenberger, H. Genz, L. Groening, et al. // NIM
B 119. 1996, p. 123-130.
3. S.B. Dabagov, N.K. Zhevago // Rivesta del nuovo
cimento. 2008, v. 31, № 9, p. 491-529.
4. J. Lindhard // Kgl. Dan. Vid. Selsk. Mat.-Fys. Medd.
1965, v. 34, №14.
5. D.F. Alferov, Yu.A. Bashmakov, P.A. Cherenkov //
Sov. Phys. Usp.. 1989, v. 32, p. 200; Usp. Fiz. Nauk.
1989, v. 157, p. 389 (in Russian).
6. D.F. Alferov, Yu.A. Bashmakov, E.G. Bessonov:
Preprint LEBEDEV-72-23. 1972.
7. E.S. Masunov, S.M. Polozov // NIM A. 2006, v. 558,
p. 184.
8. E.S. Masunov, S.M. Polozov // Problems of Atomic
Science and Technology. Series “Nuclear Physics
Investigations”. 2008, № 5, p. 136.
9. E.S. Masunov, S.M. Polozov // Phys. Rev. ST AB.
2008, v. 11, p. 074201.
10. Yu.A. Bashmakov, T.V. Bondarenko, S.M. Polozov,
G.B. Sharkov // Proc. of. RuPAC’2012. 2012, p. 406.
Article received 05.09.2013
ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ДИНАМИКИ И ИЗЛУЧЕНИЯ ЭЛЕКТРОННОГО ПУЧКА
ПРИ КАНАЛИРОВАНИИ В КРИСТАЛЛЕ
Ю.А. Башмаков, С.М. Полозов
Описываются алгоритмы и численные методы, использованные при разработке программы для двумер-
ного численного моделирования. Рассматривается случай плоскостного каналирования и приведены резуль-
таты моделирования динамики электронов в кристалле и сопровождающего излучения. Выделяются части-
цы, захваченные в канал, и частицы, совершающие надбарьерное движение. Определяется зависимость пе-
риода колебаний от начальных условий. Эта зависимость существенна для определения спектральных
свойств излучения. Рассмотрена возможность генерации основной и высших гармоник излучения. Приво-
дятся спектры вероятности излучения.
ЧИСЕЛЬНЕ МОДЕЛЮВАННЯ ДИНАМІКИ І ВИПРОМІНЮВАННЯ ЕЛЕКТРОННОГО ПУЧКА
ПРИ КАНАЛУВАННІ В КРИСТАЛІ
Ю.А. Башмаков, С.М. Полозов
Описуються алгоритми і чисельні методи, що були використані при розробці програми для двомірного
моделювання. Розглядається випадок площинного каналування і наведені результати моделювання динаміки
електронів у кристалі і супровідного випромінювання. Виділяються частинки, захоплені в канал, і частинки,
що роблять надбар’єрний рух. Визначається залежність періоду коливань від початкових умов. Ця залеж-
ність істотна для визначення спектральних властивостей випромінювання. Розглянуто можливість генерації
основної і вищих гармонік випромінювання. Приводяться спектри ймовірності випромінювання.
introduction
1. main relations and analytical study
2. code for numarical simulation
3. examples of simulation
conclusions
references
численное моделирование динамики и излучения электронного пучка при каналировании в кристалле
|