2.5D relativistic electromagnetic pic code for simulation of beam interaction with plasma in axial-symmetric geometry
2.5D relativistic electromagnetic PIC code for simulation of beam interaction with plasma in axial-symmetric geometry was developed. Accurate charge weighting scheme and difference schemes near the system axis were introduced. Simulation tests of electromagnetic wave interaction with inhomogeneous p...
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Tolochkevych, Yu.M. Litoshenko, T.E. Anisimov, I.O. 2011-02-25T11:20:25Z 2011-02-25T11:20:25Z 2010 2.5D relativistic electromagnetic pic code for simulation of beam interaction with plasma in axial-symmetric geometry / Yu.M. Tolochkevych, T.E. Litoshenko, I.O. Anisimov // Вопросы атомной науки и техники. — 2010. — № 4. — С. 47-50. — Бібліогр.: 7 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/17297 2.5D relativistic electromagnetic PIC code for simulation of beam interaction with plasma in axial-symmetric geometry was developed. Accurate charge weighting scheme and difference schemes near the system axis were introduced. Simulation tests of electromagnetic wave interaction with inhomogeneous plasma were carried out. 2.5D-релятивистский электромагнитный код методом частиц был разработан для моделирования взаимодействия пучка с плазмой в цилиндрической геометрии. Была применена точная схема взвешивания заряда в ячейках, а также модификация метода конечных разностей возле оси системы. Были проведены тестовые запуски для моделирования взаимодействия электромагнитной волны с неоднородной плазмой. 2.5D-релятивістський електромагнітний код методом частинок був розроблений для моделювання взаємодії пучка з плазмою в циліндричній геометрії. Було застосовано точну схему зважування заряду в комірках, а також модифікація методу скінченних різниць біля осі системи. Було проведено тестові запуски для моделювання взаємодії електромагнітної хвилі з неоднорідною плазмою. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Нерелятивистская электроника 2.5D relativistic electromagnetic pic code for simulation of beam interaction with plasma in axial-symmetric geometry 2.5D-релятивистский электромагнитный код методом частиц для моделирования взаимодействия пучка с плазмой в цилиндрической геометрии 2.5D-релятивістський електромагнітний код методом частинок для моделювання взаємодії пучка з плазмою в циліндричній геометрії Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
2.5D relativistic electromagnetic pic code for simulation of beam interaction with plasma in axial-symmetric geometry |
| spellingShingle |
2.5D relativistic electromagnetic pic code for simulation of beam interaction with plasma in axial-symmetric geometry Tolochkevych, Yu.M. Litoshenko, T.E. Anisimov, I.O. Нерелятивистская электроника |
| title_short |
2.5D relativistic electromagnetic pic code for simulation of beam interaction with plasma in axial-symmetric geometry |
| title_full |
2.5D relativistic electromagnetic pic code for simulation of beam interaction with plasma in axial-symmetric geometry |
| title_fullStr |
2.5D relativistic electromagnetic pic code for simulation of beam interaction with plasma in axial-symmetric geometry |
| title_full_unstemmed |
2.5D relativistic electromagnetic pic code for simulation of beam interaction with plasma in axial-symmetric geometry |
| title_sort |
2.5d relativistic electromagnetic pic code for simulation of beam interaction with plasma in axial-symmetric geometry |
| author |
Tolochkevych, Yu.M. Litoshenko, T.E. Anisimov, I.O. |
| author_facet |
Tolochkevych, Yu.M. Litoshenko, T.E. Anisimov, I.O. |
| topic |
Нерелятивистская электроника |
| topic_facet |
Нерелятивистская электроника |
| publishDate |
2010 |
| language |
English |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
2.5D-релятивистский электромагнитный код методом частиц для моделирования взаимодействия пучка с плазмой в цилиндрической геометрии 2.5D-релятивістський електромагнітний код методом частинок для моделювання взаємодії пучка з плазмою в циліндричній геометрії |
| description |
2.5D relativistic electromagnetic PIC code for simulation of beam interaction with plasma in axial-symmetric geometry was developed. Accurate charge weighting scheme and difference schemes near the system axis were introduced. Simulation tests of electromagnetic wave interaction with inhomogeneous plasma were carried out.
2.5D-релятивистский электромагнитный код методом частиц был разработан для моделирования взаимодействия пучка с плазмой в цилиндрической геометрии. Была применена точная схема взвешивания заряда в ячейках, а также модификация метода конечных разностей возле оси системы. Были проведены тестовые запуски для моделирования взаимодействия электромагнитной волны с неоднородной плазмой.
2.5D-релятивістський електромагнітний код методом частинок був розроблений для моделювання взаємодії пучка з плазмою в циліндричній геометрії. Було застосовано точну схему зважування заряду в комірках, а також модифікація методу скінченних різниць біля осі системи. Було проведено тестові запуски для моделювання взаємодії електромагнітної хвилі з неоднорідною плазмою.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/17297 |
| citation_txt |
2.5D relativistic electromagnetic pic code for simulation of beam interaction with plasma in axial-symmetric geometry / Yu.M. Tolochkevych, T.E. Litoshenko, I.O. Anisimov // Вопросы атомной науки и техники. — 2010. — № 4. — С. 47-50. — Бібліогр.: 7 назв. — англ. |
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| first_indexed |
2025-11-25T00:06:54Z |
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2025-11-25T00:06:54Z |
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| fulltext |
2.5D RELATIVISTIC ELECTROMAGNETIC PIC CODE FOR
SIMULATION OF BEAM INTERACTION WITH PLASMA
IN AXIAL-SYMMETRIC GEOMETRY
Yu.M. Tolochkevych, T.E. Litoshenko, I.O. Anisimov
Taras Shevchenko National University of Kiev, Radio Physics Faculty, Kiev, Ukraine
E-mail: yura.tolochkevych@gmail.com; taras.litoshenko@gmail.com; ioa@univ.kiev.ua
2.5D relativistic electromagnetic PIC code for simulation of beam interaction with plasma in axial-symmetric
geometry was developed. Accurate charge weighting scheme and difference schemes near the system axis were in-
troduced. Simulation tests of electromagnetic wave interaction with inhomogeneous plasma were carried out.
PACS: 52.65-y
1. INTRODUCTION
A lot of problems in plasma electronics (the nonlin-
ear stage of beam-plasma interaction, dynamics of elec-
tron bunches in wake fields excited by them in plasma,
the Bursian effect in injection of strong electron
bunches, etc.) can be solved only by means of computer
simulation.
The method of particles in cells is used intensively
numerical simulation of plasma [1]. This method is used
for the simulation of phenomena in space, ionosphere,
various discharges, plasma display cells etc.
Majority of existing programs for beam-plasma si-
mulation are electrostatic. But using such codes [2-3]
we cannot observe effects of electromagnetic waves
propagation or radiation in plasmas. The equations of
particles motion which are integrated are not relativistic,
but it is necessary to solve the relativistic equations of
motion in many cases.
The aim of this work is to present the relativistic
electromagnetic code for beam-plasma systems simula-
tion with axial symmetry using a method of particles in
cells with a longitudinal magnetic field.
2. THE DIFFERENCE SCHEMES
DESCRIPTION
The two-dimensional cell in cylindrical coordinates
(r-z) for the presented axial-symmetric code is showed
on Fig.1. Points are marked where each component of
field, potential, charge and current are calculated. Large
particle have a shape of the ring which can move along
z axis, vary its radius moving in radial direction and
rotate azimuthally. It is possible to simulate propagation
of electromagnetic waves of E type in two dimensional
systems without z component of magnetic field H ac-
cordingly. But in some cases the presence of this com-
ponent is important (e.g., dynamics of an electron bunch
injected along the magnetic field).
In the proposed code the space grid remains two-
dimensional, but large particles have three components
of velocity which results in three components of electric
and magnetic field. Therefore electromagnetic waves of
H type arise in the system.
For solution of Maxwell equations’ set the method
of finite differences in time domain (FDTD) was used
[4-5]. This method is based on Yee algorithm, which
allows finding both electrical and magnetic field in time
and space using first pair of the Maxwell equations (the
law of Ampere’s circuital law with Maxwell's correction
and the electromagnetic induction law).
Fig.1. Elementary cell for 2D model
Fig.2. Elementary 3D cell in cylindrical geometry
3D elementary cell in cylindrical geometry is repre-
sented on Fig.2. Each point where electric field compo-
nent is calculated is surrounded in a plane perpendicular
to its direction by four points, where magnetic field
components are calculated. Oppositely, each point
where the magnetic field component is calculated is
surrounded by four points where electric field compo-
nents are calculated.
Thus difference schemes for Е and Н components of
the field take a form:
1
1/ 2, 1/ 2,
n n
r i k r i kE E+
+ += −
( )1/2 1/ 2 1/2
1/2, 1/2, 1/2 1/2, 1/2
0 1/2,
1 ;n n n
r i k i k i k
i k
tj H H
z ϕ ϕ ε ε
+ + +
+ + + + −
+
Δ⎡ ⎤− + −⎢ ⎥Δ⎣ ⎦
_______________________________________________________________
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2010. № 4.
Серия: Плазменная электроника и новые методы ускорения (7), с. 47-50. 47
mailto:yura.tolochkevych@gmail.com
mailto:taras.litoshenko@gmail.com
( )1 1/2 1/2 1/2
, , , , 1/2 , 1/2
1n n n n n
i k r i k i k r i k r i kE E j H H
zϕ ϕ
+ + +
+ −
⎡= − − −⎢ Δ⎣
48
+ +
( )1/2 1/2
1/2, 1/2,
0 ,
1 ;n n
z i k z i k
i k
tH H
r ε ε
+ +
+ −
Δ⎤+ − ⎥Δ ⎦
(1)
( )1 1/2 1/2 1/2
, 1/2 , 1/2 , 1/2 1/2, 1/2,
1
2
n n n n n
z i k z i k z i k i k i kE E j H H
ri ϕ ϕ
+ + +
+ + + + −
⎡= − − +⎢ Δ⎣
+ +
( )1/2 1/2
1/2, 1/2 1/2, 1/2
0 ,
1 ;n n
i k i k
i k
tH H
r ϕ ϕ ε ε
+ +
+ + − +
Δ⎤+ − ⎥Δ ⎦
0
;1/ 2 1/ 2
, 1/ 2 , 1/ 2 , 1 ,
tn n nH H E Er ri k i k i k i kz ϕ ϕμ
⎡ ⎤
⎢ ⎥⎣ ⎦
Δ+ −= + −+ + +Δ
n
1/ 2 1/ 2
1/ 2, 1/ 2 1/ 2, 1/ 2
n nH Hi k i kϕ ϕ
+ −=+ + + + − (2)
1
1/ 2, ,
n nE Er ri k i kz
⎡ ⎛ ⎞
⎢ ⎜ ⎟
⎝ ⎠⎣
− −+Δ
−
1 ;1, 1/ 2 , 1/ 2
0
tn nE Ez zi k i kr μ
⎤⎛
⎥⎜ ⎟
⎝ ⎠⎦
Δ− −+ + +Δ
⎞
1/ 2 1/ 2
, 1/ 2 1/ 2,
n nH Hz zi k i k
+ −= −+ +
1
1, ,2 ( 1/ 2)
n nE Ei k i kr i ϕ ϕ
⎡ ⎛ ⎞
⎢ ⎜ ⎟
⎝ ⎠⎣
+ ++Δ +
1 .1, ,
0
tn nE Ei k i kr ϕ ϕ μ
⎤⎛ ⎞
⎥⎜ ⎟
⎝ ⎠⎦
Δ+ −+Δ
3. FEATURES OF NEAR AXIS
CALCULATIONS
Solution of equations (1)-(2) has some specific fea-
tures in the cylindrical coordinate system connected
with the calculation of field’s components near the sys-
tem axis. One can see from Fig.2 that electromagnetic
field components Eφ, Ez and Hr are found on the axis,
and Er, Hφ and Hz components are found at the distance
∆r/2 from the axis.
On the axis Hr=0, but the set (1)-(2) cannot be used
for Ez and Eφ calculation near the axis. Therefore, the
first Maxwell equation in the integral form should be
used in order to obtain the field near the axis:
dS
t
DIHdl
l
S∫ ∫∫ ∂
∂
+= . (3)
Calculating integral for the cell allocated on the sys-
tem axis (see Fig. 3,a) and taking into account that its
radius is equal to ∆r/2, it is possible to obtain Ez:
1
0, 1/2 0, 1/2
n n
z k z kE E+
+ += − (4)
1/ 2 1/ 2
0, 1/ 2 0 1/ 2, 1/ 2
0 0 1/ 2,
4 .n n
z k k
k
tj H
r ϕ ε ε
+ +
+ + +
+
Δ⎡− −⎢ Δ⎣
⎤
⎥⎦
Eφ is calculated similarly (see Fig. 3,b):
1 1/2 1/2
0, 0, 0, 0, 1/2 0, 1/2
1n n n n n
k r k k r k r kE E j H H
zϕ ϕ
+ + + +
+ −
⎡ 1/2⎡ ⎤= − − −⎢ +⎣ ⎦Δ⎣
1/2
0, 1/2
0 ,
2 .n
z k
i k
tH
r ε ε
+
+
Δ⎤+ ⎥Δ ⎦
(5)
On the system axis jφ=0 аnd Hr=0, equation (5) can
be rewritten in the form:
ki
n
kz
n
kr
n
k
tH
r
EE
,0
2/1
2/1,0,0
1
,0
2
εεϕ
Δ
⎥⎦
⎤
⎢⎣
⎡
Δ
−= +
+
+ . (6)
а
b
Fig.3. Calculation of electric field components Ez (а)
and Eφ (b) near the axis of the system
4. SPECIFICITY OF WEIGHTING
PROCEDURE
Procedure of current density weighting also has the
specific features in cylindrical geometry. The current
density in each node
St
qj
ΔΔ
Δ
= , (7)
should be defined so that the continuity equation is ful-
filled.
Weighting of the first order was used in the pro-
gram, where large particles’ cross-section has the square
shape in the plane (r, z). Therefore, they can distribute
the charge to four cells simultaneously. In order to find
the current density caused by each large particle it is
necessary to trace how the particle passes through each
edge of the elementary cell (Fig. 4) during each time
step of simulation [6].
Maximum distance that particle can pass during one
time step should not exceed the cell size (so called Cou-
rant condition):
max xv tΔ ≤ Δ . (8)
Let's consider the possible variants of particle cur-
rent distribution to the neighboring nodes. To reduce the
quantity of the possible variants we use Courant condi-
tion in more hard form relatively to (8): limit maximum
distance the particle passes per time step with to 2/xΔ .
During its move per time step the particle can give
the contribution to 4 sides (Fig.4,a), to 7 sides (Fig.4,b),
or to 10 sides (Fig.4,c).
Fig. 4,a
b
c
Fig.4. Weighting of the current density in the system
It is also necessary to consider that large particle
charge density is a function of its radius, in contrast to
be constant in the rectangular geometry.
5. TEST SIMULATIONS OF
ELECTROMAGNETIC WAVE INTERACTING
WITH INHOMOGENEOUS PLASMA
A simulation volume has a form of cylindrical reso-
nator with radius of 0,4 m and length of 12,8 m. A cy-
lindrical waveguide mode E01 with frequency
f = 400 MHz is excited by the rod with alternating cur-
rent. It is located on the axis of the resonator and has a
length of 0,37 m. Electromagnetic absorbing layers is
imposed on the both sides of resonator.
The system is partly filled with inhomogeneous
plasma with linear density profile. The plasma consists
of electrons and ions of hydrogen with near zero tem-
perature.
The spatial distribution of z component of electric
field for the case of wave interaction with subcritical
inhomogeneous plasma is showed on Fig.6. The plasma
density changes from np(z = 4,5 m) = 0,2⋅1015 m -3 to
np(z = 12,8 m) = 0,6⋅1015 m-3. One can see that in this
case the wave remains periodical but changes its wave-
length while propagating through the plasma. This hap-
pens because subcritical plasma has lower dielectric
permeability than vacuum that leads to the decrease of
phase velocity of the wave.
The distribution of z component of electric field for
the case of interaction with subcritical inhomogeneous
plasma with the reflection point on the density profile is
showed on Fig.5.
Fig.5. Spatial distribution of z component of electric field for the case of electromagnetic wave interaction
with subcritical plasma
Fig.6. Spatial distribution of z component of electric field for the case of reflection of electromagnetic wave
from the inhomogeneous plasma
Fig.7. Spatial distribution of intensity of electromagnetic field for the case of reflection of electromagnetic
wave from the inhomogeneous plasma
The plasma density changes from
np(z = 4,5 m) = 0,6⋅1015 m -3,
to np(z = 12,8 m) = 1,5⋅1015 m-3.
One can see that now the electromagnetic wave re-
flects from the inhomogeneous plasma. There are also
low intensity oscillations of electric field beyond the
reflection point which could be explained by generation
of second harmonic.
The distribution of the intensity of the electric field
is showed on the Fig.7. One can see that the standing
wave is formed in plasma by the incident and the re-
flected wave. The reflection point is defined by the
transversal wave number k⊥. It’s well known from the
theory [7] that in the case of non-zero angle θ between
the wave number and the density gradient the reflection
49
50
density is
2
2 01
2
2 /k R
k
πν
θ
−
= =
(9)
sinref cr crn n n ,
2. I.O. Anisimiv, T.E. Litoshenko. Dynamics of a
modulated electron beam in homogeneous plasma:
2d simulation // Ukrainian Journal of Physics. 2008,
v.54, №4, p.388-392.
where ncr = 1,9⋅1015 m-3 is critical density, R – radius of
the resonator, ν01 = 2.4 – root of the Bessel’s function.
For parameters of simulation nref = 0,97⋅1015 m-3. This
value is close to the observed value of
nref, exp = 0,8⋅1015 m-3.
CONCLUSIONS
The 2.5D electromagnetic relativistic PIC code for
simulation of beam- and wave-plasma interaction in
cylindrical geometry is presented in the paper. The ac-
curate charge weighting scheme is introduced which
allows the charge conservation law to be automatically
fulfilled. The correctness of the program is confirmed
by the comparison of simulation results with theoretical
predictions.
REFERENCES
1. C.K. Birdsall, A.B. Langdon. Plasma Physics via
Computer Simulation. McGraw-Hill book company,
1985.
3. I.O. Anisimov, Yu.M. Tolochkevych. Dynamics of
1D electron bunch with the initially rectangular den-
sity profile injected into homogeneous plasma //
Ukrainian Journal of Physics. 2009, v.54, №5,
p.454-460.
4. A. Taflove, S.C. Hagness. Computational Electrody-
namics: The Finite Difference Time Domain Me-
thod, Second edition. Artech House, 2000.
5. P. Russer, U. Siart. Finite Domain Methods in Elec-
trodynamics // Springer Proceeding in Physcis.
2008.
6. J. Villasenor, O. Buneman. Rigorous charge conser-
vation for local electromagnetic field solvers //
Computer Physics Communications. 1992, v.69,
p.306-316.
7. V.L. Ginzburg. Propagation of electromagnetic
waves in plasmas. 2nd edition. Moscow: «Nauka»,
1967.
Статья поступила в редакцию 02.06.2010 г.
2.5D-РЕЛЯТИВИСТСКИЙ ЭЛЕКТРОМАГНИТНЫЙ КОД МЕТОДОМ ЧАСТИЦ
ДЛЯ МОДЕЛИРОВАНИЯ ВЗАИМОДЕЙСТВИЯ ПУЧКА С ПЛАЗМОЙ В ЦИЛИНДРИЧЕСКОЙ
ГЕОМЕТРИИ
Ю.М. Толочкевич, Т.Е. Литошенко, И.А. Анисимов
2.5D-релятивистский электромагнитный код методом частиц был разработан для моделирования взаимо-
действия пучка с плазмой в цилиндрической геометрии. Была применена точная схема взвешивания заряда в
ячейках, а также модификация метода конечных разностей возле оси системы. Были проведены тестовые
запуски для моделирования взаимодействия электромагнитной волны с неоднородной плазмой.
2.5D-РЕЛЯТИВІСТСЬКИЙ ЕЛЕКТРОМАГНІТНИЙ КОД МЕТОДОМ ЧАСТИНОК
ДЛЯ МОДЕЛЮВАННЯ ВЗАЄМОДІЇ ПУЧКА З ПЛАЗМОЮ В ЦИЛІНДРИЧНІЙ ГЕОМЕТРІЇ
Ю.М. Толочкевич, Т.Є. Літошенко, І.О. Анісімов
2.5D-релятивістський електромагнітний код методом частинок був розроблений для моделювання взає-
модії пучка з плазмою в циліндричній геометрії. Було застосовано точну схему зважування заряду в комір-
ках, а також модифікація методу скінченних різниць біля осі системи. Було проведено тестові запуски для
моделювання взаємодії електромагнітної хвилі з неоднорідною плазмою.
http://www.amazon.com/exec/obidos/search-handle-url/ref=ntt_athr_dp_sr_1?%5Fencoding=UTF8&search-type=ss&index=books&field-author=C.K.%20Birdsall
http://www.amazon.com/exec/obidos/search-handle-url/ref=ntt_athr_dp_sr_2?%5Fencoding=UTF8&search-type=ss&index=books&field-author=A.B%20Langdon
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