Dynamics of the modulated electron beam in the inhomogeneous plasma barrier: one-dimentional simulation using PIC method
Evolution of the modulated electron beam moving through the plasma barrier was studied via computer simulation using PIC method. Inhomogeneous plasma barrier with parameters corresponding to experimental conditions [1-2] was studied. Dependencies of the maximal signal amplitude and coordinate of thi...
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| Cite this: | Dynamics of the modulated electron beam in the inhomogeneous plasma barrier: one-dimentional simulation using PIC method / I.O. Anisimov, M.J. Soloviova // Вопросы атомной науки и техники. — 2008. — № 4. — С. 209-213. — Бібліогр.: 12 назв. — англ. |
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Anisimov, I.O. Soloviova, M.J. 2017-01-04T20:42:51Z 2017-01-04T20:42:51Z 2008 Dynamics of the modulated electron beam in the inhomogeneous plasma barrier: one-dimentional simulation using PIC method / I.O. Anisimov, M.J. Soloviova // Вопросы атомной науки и техники. — 2008. — № 4. — С. 209-213. — Бібліогр.: 12 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/110592 PACS: 52.35Mw, 52.40Mj Evolution of the modulated electron beam moving through the plasma barrier was studied via computer simulation using PIC method. Inhomogeneous plasma barrier with parameters corresponding to experimental conditions [1-2] was studied. Dependencies of the maximal signal amplitude and coordinate of this maximum at the modulation frequency upon the initial beam modulation depth were obtained. Simulation results were compared with experimental results and outcomes of the previous simulations. Досліджено еволюцію модульованого електронного пучка в плазмовому бар’єрі шляхом комп’ютерного моделювання методом крупних частинок. Розглядається неоднорідний плазмовий бар’єр, що відповідає умовам лабораторного експерименту. Отримано залежності максимального значення амплітуди сигналу на частоті модуляції та положення даного максимуму від початкової глибини модуляції. Результати моделювання співставлено з експериментальними даними та результатами попередніх моделювань. Исследуется эволюция модулированного электронного пучка в плазменном барьере с помощью компьютерного моделирования методом крупных частиц. Рассматривается неоднородный плазменный барьер, соответствующий условиям лабораторного эксперимента. Получены зависимости максимальной амплитуды сигнала на частоте модуляции и положения данного максимума от начальной глубины модуляции. Результаты моделирования сопоставлены с экспериментальными данными и результатами предшествующих моделирований. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нелинейные процессы в плазменных средах Dynamics of the modulated electron beam in the inhomogeneous plasma barrier: one-dimentional simulation using PIC method Динаміка модульованого електронного пучка в неоднорідному плазмовому бар’єрі: одновимірне моделювання методом частинок Динамика модулированного электронного пучка в неоднородном плазменном барьере: одномерное моделирование методом частиц Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Dynamics of the modulated electron beam in the inhomogeneous plasma barrier: one-dimentional simulation using PIC method |
| spellingShingle |
Dynamics of the modulated electron beam in the inhomogeneous plasma barrier: one-dimentional simulation using PIC method Anisimov, I.O. Soloviova, M.J. Нелинейные процессы в плазменных средах |
| title_short |
Dynamics of the modulated electron beam in the inhomogeneous plasma barrier: one-dimentional simulation using PIC method |
| title_full |
Dynamics of the modulated electron beam in the inhomogeneous plasma barrier: one-dimentional simulation using PIC method |
| title_fullStr |
Dynamics of the modulated electron beam in the inhomogeneous plasma barrier: one-dimentional simulation using PIC method |
| title_full_unstemmed |
Dynamics of the modulated electron beam in the inhomogeneous plasma barrier: one-dimentional simulation using PIC method |
| title_sort |
dynamics of the modulated electron beam in the inhomogeneous plasma barrier: one-dimentional simulation using pic method |
| author |
Anisimov, I.O. Soloviova, M.J. |
| author_facet |
Anisimov, I.O. Soloviova, M.J. |
| topic |
Нелинейные процессы в плазменных средах |
| topic_facet |
Нелинейные процессы в плазменных средах |
| publishDate |
2008 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Динаміка модульованого електронного пучка в неоднорідному плазмовому бар’єрі: одновимірне моделювання методом частинок Динамика модулированного электронного пучка в неоднородном плазменном барьере: одномерное моделирование методом частиц |
| description |
Evolution of the modulated electron beam moving through the plasma barrier was studied via computer simulation using PIC method. Inhomogeneous plasma barrier with parameters corresponding to experimental conditions [1-2] was studied. Dependencies of the maximal signal amplitude and coordinate of this maximum at the modulation frequency upon the initial beam modulation depth were obtained. Simulation results were compared with experimental results and outcomes of the previous simulations.
Досліджено еволюцію модульованого електронного пучка в плазмовому бар’єрі шляхом комп’ютерного моделювання методом крупних частинок. Розглядається неоднорідний плазмовий бар’єр, що відповідає умовам лабораторного експерименту. Отримано залежності максимального значення амплітуди сигналу на частоті модуляції та положення даного максимуму від початкової глибини модуляції. Результати моделювання співставлено з експериментальними даними та результатами попередніх моделювань.
Исследуется эволюция модулированного электронного пучка в плазменном барьере с помощью компьютерного моделирования методом крупных частиц. Рассматривается неоднородный плазменный барьер, соответствующий условиям лабораторного эксперимента. Получены зависимости максимальной амплитуды сигнала на частоте модуляции и положения данного максимума от начальной глубины модуляции. Результаты моделирования сопоставлены с экспериментальными данными и результатами предшествующих моделирований.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/110592 |
| citation_txt |
Dynamics of the modulated electron beam in the inhomogeneous plasma barrier: one-dimentional simulation using PIC method / I.O. Anisimov, M.J. Soloviova // Вопросы атомной науки и техники. — 2008. — № 4. — С. 209-213. — Бібліогр.: 12 назв. — англ. |
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| fulltext |
НЕЛИНЕЙНЫЕ ПРОЦЕССЫ В ПЛАЗМЕННЫХ СРЕДАХ
DYNAMICS OF THE MODULATED ELECTRON BEAM
IN THE INHOMOGENEOUS PLASMA BARRIER:
ONE-DIMENTIONAL SIMULATION USING PIC METHOD
I.O. Anisimov, M.J. Soloviova
Taras Shevchenko National University of Kyiv, Radio Physics Faculty, Kyiv, Ukraine
E-mail: ioa@univ.kiev.ua
Evolution of the modulated electron beam moving through the plasma barrier was studied via computer simula-
tion using PIC method. Inhomogeneous plasma barrier with parameters corresponding to experimental conditions
[1-2] was studied. Dependencies of the maximal signal amplitude and coordinate of this maximum at the modulation
frequency upon the initial beam modulation depth were obtained. Simulation results were compared with experi-
mental results and outcomes of the previous simulations.
PACS: 52.35Mw, 52.40Mj
1. INTRODUCTION
Problem of the modulated electron beam’s dynamics
in plasma is of interest in various branches of plasma
electronics such as electron beams’ using as emitters of
electromagnetic waves in ionosphere [3-4], transillumi-
nation of the plasma barriers for electromagnetic waves
using electron beams [1-2, 5], inhomogeneous plasma
diagnostics via transition radiation of electron beams
and electron bunches [6] etc.
___________________________________________________________
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2008. № 4.
Серия: Плазменная электроника и новые методы ускорения (6), с.209-213.
209
Evolution of the modulated electron beam in super-
critical plasma barrier was studied experimentally in
[2,7]. It was found out that signal at the modulation fre-
quency reached its maximum inside the barrier, and
magnitude of this maximum was directly proportional to
the initial beam modulation depth. These results were
explained in [8] by the concurrence between non-
resonant (signal) and resonant (noise) modes of the
beam-plasma system. But calculations presented in [8]
correspond to the initial problem, whereas results of
experiments [2, 7] correspond to the boundary problem.
Consequently it was impossible to compare results of
experiment and simulation.
Evolution of the linear space-charge waves (SCW)
of electron beam in the barrier with parabolic plasma
density profile was studied theoretically in [2]. Effect of
mutual transformation of SCW in the inhomogeneous
plasma was obtained.
In our previous works [9-10] evolution of the modu-
lated electron beam in plasma for the initial-boundary
problem was studied via computer simulation using PIC
method [11-12]. But homogeneous plasma barrier in [9-
10] doesn’t correspond to the experimental one that is
close to Gaussian shape [1-2, 7].
In this paper dynamics of the modulated electron
beam moving through the barrier with Gaussian plasma
density profile is studied via computer simulation using
PIC method. Initial-boundary problem is solved, and
results obtained are compared with results of experi-
ments and previous simulations.
2. MODEL DESCRIPTION, SIMULATION
METHOD AND PARAMETERS
Warm isotropic collisionless plasma with initial
Gaussian density profile is studied. Simulation is carried
out via particle-in-cell method using modified program
package PDP1 [11-12].
1D region between two electrodes is simulated. Inte-
relectrode space is filled with fully ionized hydrogen
plasma. Initial plasma density profile is obtained by the
approximation of experimental axial plasma density
profile [1-2, 7] by Gaussian function. So initial electron
and ion plasma density is set as
2
0
0( ) exp
2m
x xn x n n
⎡ ⎤−⎛ ⎞= + −⎢ ⎜ ⎟Δ⎝ ⎠⎢ ⎥⎣ ⎦
⎥ , (1)
where n0 is the plasma density for x→∞, n0 + nm is the
peak plasma density inside the barrier at x=x0, and Δ is
half-width of the plasma density profile. Simulation
parameters are presented in table.
Simulation parameters
n0 5.5·1010 cm-3
nm 2.04·1011 cm-3
x0 10 cm
Δ 3.87 cm
Simulation region length 20 cm
Plasma electrons' thermal velocity 6⋅107 cm/s
Plasma ions' thermal velocity 2,33⋅106 cm/s
Beam electrons velocity 2⋅109 cm/s
Electron beam modulation frequency 2.77 GHz
Electron beam modulation depth 0.01…0.3 with
the step 0.01
Simulation time step 10-13 s
Electron beam is injected into plasma barrier from
the left electrode. It moves to the right one. Electrodes
absorb both plasma and beam particles. Initially electron
beam is density-modulated:
0( ) (1 cos )t m tρ = ρ + ω , (2)
where m is the modulation depth.
Modulation frequency was selected in the range
ωp(n0)<ω<ωp(n0+nm), where ωp(n) is electron plasma
frequency corresponding to the plasma density n. So
two local plasma resonance regions are presented inside
the barrier at the modulation frequency.
The simulation was carried out during the time in-
terval of approximately 200 electron plasma periods or
5 ion plasma periods. During this time electron beam
reached the opposite electrode and quasi-stationary re-
gime was settled.
3. SIMULATION RESULTS
210
3.1. SPATIAL EVOLUTION OF SPECTRA OF
ELECTRON BEAM DENSITY AND ELECTRIC
FIELD STRENGTH FOR SMALL INITIAL
DEPTH OF THE BEAM MODULATION
All dependencies demonstrated in this section corre-
spond to the initial modulation depth m = 0.05.
Space-time distributions of the beam electrons' den-
sity and electric field strength were obtained from simu-
lation. Than temporal fast Furrier transformation was
carried out.
Fig.1 demonstrates the spatial evolution of spectra of
electron beam density (a) and electric field strength (b).
Arrows mark the modulation frequency. One can see
from Fig.1,a that only the modulation frequency is pre-
sented nearby the injector in the spectra of the electron
beam density. During the beam motion inside the barrier
in the region of plasma density increase the beam ex-
cites oscillations at the local plasma frequency accord-
ing to Cherenkov mechanism. As long as plasma den-
sity depends on coordinate, local plasma frequency
change up to coordinate too (in Fig.1,b it is noticeable
near the modulation frequency). For the given frequency
Cherenkov resonance conditions are not satisfied along
the beam trajectory after the vicinity of the first plasma
resonance point. So electric field in this region is de-
creased, but corresponding oscillations in the spectrum
of beam modes remain (see bottom of Fig.1,a). Conse-
quently oscillations with the wide band of frequencies
are presented in the beam density spectrum along with
the modulation frequency.
During the beam motion inside the barrier in the re-
gion of plasma density decrease the oscillations pre-
sented in the beam density spectrum excite the intensive
electric fields in corresponding regions of local plasma
resonance (top of Fig.1,b). At the same time the
modulation of the beam at corresponding frequencies is
increased (top of Fig.1,b).
The largest length of resonant beam-plasma interac-
tion is obtained near the maximum of plasma density,
and this length is decreased at the periphery. Therefore
maximal growth of the electric field of characteristic
oscillations is observed in the region of maximal plasma
density (Fig.1,b).
Signal at the modulation frequency is noticeable at
the whole simulation region both in spectra of electron
beam density perturbation and electric field strength
(Fig.1,a,b).
Fig.2 presents the spatial evolution of spectra of
electron beam density at the modulation frequency (a),
electric field strength at the modulation frequency (b)
and at the doubled modulation frequency (c).
Similarly to the previous simulation [9-10], inside
the barrier spectral amplitude of the electron beam den-
sity at the modulation frequency reaches the maximum
value (Fig.2,a). Spatial evolution of the electric field
strength spectrum at the modulation frequency (fig.2,b)
doesn’t demonstrate a maximum. Dependence of the
field amplitude at the modulation frequency on coordi-
nate contains a lot of fluctuations with considerable am-
plitudes.
a
b
Fig.1. Spatial evolution of spectra of electron beam
density (a) and electric field strength (b).
Arrows mark the signal modulation frequency
The largest values of the field at the modulation fre-
quency are reached in the regions of local plasma reso-
nance at the edges of the barrier. High level of fluctua-
tions is caused by presence of the wide-band oscillations
in the electron beam spectrum. These fluctuations arise
according to the mechanism of polarization beam-
plasma instability (part of them are excited according to
Cherenkov mechanism in local plasma resonance region
as it was mentioned above). These fluctuations form the
homogeneous gray background in the left side of Fig.1,b
bounded by the frequency of local plasma resonance.
But at the doubled modulation frequency (Fig.2,c)
there are no oscillations at the local plasma frequency
as well as strong fluctuations. That’s why spatial de-
pendence of the signal at this frequency has the clear
maximum, and it‘s position coincides with the position
of the maximum in Fig.2,a. Notice that only a weak
local peak is presented at this point in Fig.2,b.
a
b
c
Fig.2. Spatial evolution of spectra of electron beam
density at the modulation frequency (a), electric field
strength at the modulation frequency (b) and at the
doubled modulation frequency (c)
The position of the signal maximum at the modula-
tion frequency (Fig.2,a,c) corresponds to x ≈ 11 cm. It is
situated in the region where amplitude of the electric
field strength at resonant frequencies grows considerably
(Fig.1,b). So concurrence between resonant modes and
signal at the modulation frequency takes place. As a re-
sult restriction of the signal amplitude occurs as it was
described in [8]. But the shape of plasma barrier density
profile determines the position where this effect takes
place for the parameters of our simulation. Notice that
for homogeneous plasma density in the barrier [8] this
coordinate is determined by the start of non-linear stage
of the beam-plasma instability for the resonant mode.
3.2. SPATIAL EVOLUTION OF SPECTRA OF
ELECTRON BEAM DENSITY AND ELECTRIC
FIELD STRENGTH FOR LARGE INITIAL
DEPTH OF THE BEAM MODULATION
All dependencies discussed in this section correspond
to the initial modulation depth m=0.28. They were ob-
tained in the same way as in the previous section.
Spatial evolution of spectra of electron beam density
and electric field strength doesn’t differ drastically from
the case of small initial depth of the beam modulation.
а
b
c
Fig.3. Spatial evolution of spectra of electron beam
density at the modulation frequency (a), electric field
strength at the modulation frequency (b) and at the
doubled modulation frequency (c)
Just as in previous case maximal growth of the field of
the characteristic oscillations is observed in the range of
maximal plasma density – in the middle of the barrier.
In contrast to the case of small initial beam modula-
tion depth, position of the maximal amplitude of the
signal at the modulation frequency doesn’t coincide
with the region of the maximal growth of the field of
characteristic oscillations (Fig.3,a). From comparison
Fig.2,a,c and Fig.3,a,c one can conclude that in this case
distance from injector to maximum of the signal ampli-
tude at the modulation frequency is much smaller than
in the case of small initial beam modulation depth. So
maximum of the signal at the modulation frequency is
reached due to the non-linear saturation of its instability.
Furthermore in Fig.3,a,c oscillations of the signal ampli-
tude at the modulation frequency are observed. These
oscillations are not observed in the case of small initial
beam modulation depth (compare with Fig.2,a,с). But
similar oscillations take place for strong beams in the
homogeneous supercritical plasma [10].
Spatial evolution of spectra of electric field strength
at the modulation frequency (Fig.3,b) doesn’t demon-
strate any local maximums (contrary to Fig.2,b). This
effect can be explained both by considerable removal of
this maximum to injector and by its closeness to the first
region of the local plasma resonance.
211
212
3.3. INFLUENCE OF THE INITIAL BEAM
MODULATION DEPTH ON THE MAXIMAL
SIGNAL AMPLITUDE AT THE MODULATION
FREQUENCY AND POSITION OF THIS
MAXIMUM
Fig.4 presents dependencies of maximal signal am-
plitude (a) and coordinate of this maximum (b) at the
modulation frequency upon the initial beam modulation
depth. Two characteristic regions of the initial beam
modulation depth can be marked out from Fig.4.
а
b
Fig.4. The dependencies of maximal signal amplitude
(a) and coordinate of this maximum (b) at the modula-
tion frequency on the initial beam modulation depth
For initial modulation depths m≤0.1 the signal
maximal amplitude is reached due to the concurrence of
the signal at the modulation frequency with the reso-
nant modes. Electric field of the resonant modes traps
beam electrons, and as a result modulation at the signal
frequency is suppressed. These processes occur just
after electron beam passing through the plasma density
maximum. That’s why maximal signal position remains
constant in this range of initial modulation depths
(Fig.4,b).
For initial modulation depths m≥0.1 formation of
the signal amplitude maximum at the modulation fre-
quency is caused by the non-linear saturation of insta-
bility. As a result maximum amplitude of the signal
becomes approximately constant (Fig.4,a), and its posi-
tion gradually moves to the injector (Fig.4,b).
CONCLUSIONS
Evolution of the modulated electron beam moving
through the inhomogeneous plasma barrier was studied
via computer simulation for 1D model using PIC
method.
1. Spectrum of characteristic oscillations of the
beam-plasma system competitive with the signal at the
modulation frequency varies in space and depends on
the barrier shape. The amplitude of these oscillations
rises steeply in the region of plasma density decrease
along the beam trajectory. Upper harmonics of charac-
teristic oscillations of the beam-plasma system are pre-
sented in this region.
2. Concurrence between resonant modes and signal
at the modulation frequency takes place for small initial
modulation depths (m≤0.1). This effect moves to
restriction of the signal amplitude at the modulation
frequency, as it was observed earlier in case of
homogeneous barriers [9-10]. But now the shape of
plasma barrier density profile determines the position,
where maximal signal amplitude at the modulation
frequency is reached. 3. For large initial modulation depths (m≥0.1) the
signal amplitude maximum at the modulation frequency
is reached as a result of beam-plasma instability satura-
tion at this frequency. Oscillations of the signal ampli-
tude at the modulation frequency are observed. In gen-
eral the beam dynamics doesn’t differ from the case of
homogeneous barriers [9-10].
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modulated electron beam in plasma for different
modes of beam-plasma turbulence // Ukr. Fiz.
Zhurn. 2008, v.53, №4, p.382-388. Статья поступила в редакцию 08.05.2008 г.
ДИНАМИКА МОДУЛИРОВАННОГО ЭЛЕКТРОННОГО ПУЧКА В НЕОДНОРОДНОМ
ПЛАЗМЕННОМ БАРЬЕРЕ: ОДНОМЕРНОЕ МОДЕЛИРОВАНИЕ МЕТОДОМ ЧАСТИЦ
И.А. Анисимов, М.И. Соловьёва
Исследуется эволюция модулированного электронного пучка в плазменном барьере с помощью
компьютерного моделирования методом крупных частиц. Рассматривается неоднородный плазменный
барьер, соответствующий условиям лабораторного эксперимента. Получены зависимости максимальной
амплитуды сигнала на частоте модуляции и положения данного максимума от начальной глубины
модуляции. Результаты моделирования сопоставлены с экспериментальными данными и результатами
предшествующих моделирований.
ДИНАМІКА МОДУЛЬОВАНОГО ЕЛЕКТРОННОГО ПУЧКА В НЕОДНОРІДНОМУ
ПЛАЗМОВОМУ БАР’ЄРІ: ОДНОВИМІРНЕ МОДЕЛЮВАННЯ МЕТОДОМ ЧАСТИНОК
І.О. Анісімов, М.І. Соловйова
Досліджено еволюцію модульованого електронного пучка в плазмовому бар’єрі шляхом комп’ютерного
моделювання методом крупних частинок. Розглядається неоднорідний плазмовий бар’єр, що відповідає
умовам лабораторного експерименту. Отримано залежності максимального значення амплітуди сигналу на
частоті модуляції та положення даного максимуму від початкової глибини модуляції. Результати
моделювання співставлено з експериментальними даними та результатами попередніх моделювань.
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Simulation parameters
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