Numerical simulation of the supercritical electron beam dynamics in magnetic field of finite size in the presence of plasma
The first results of development of a numerical electromagnetic 2.5-dimensional code «SOM 2.5» (3 dimensional by the velocities and 2 dimensional by the coordinates) for researching of a virtual cathode in the presence of plasma are submitted. Представлено перші результати створення чисельного елек...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2004
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| Cite this: | Numerical simulation of the supercritical electron beam dynamics in magnetic field of finite size in the presence of plasma / P.I. Markov, I.N. Onishchenko, G.V. Sotnikov // Вопросы атомной науки и техники. — 2004. — № 2. — С. 135-137. — Бібліогр.: 7 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859472430737653760 |
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| author | Markov, P.I. Onishchenko, I.N. Sotnikov, G.V. |
| author_facet | Markov, P.I. Onishchenko, I.N. Sotnikov, G.V. |
| citation_txt | Numerical simulation of the supercritical electron beam dynamics in magnetic field of finite size in the presence of plasma / P.I. Markov, I.N. Onishchenko, G.V. Sotnikov // Вопросы атомной науки и техники. — 2004. — № 2. — С. 135-137. — Бібліогр.: 7 назв. — англ. |
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| description | The first results of development of a numerical electromagnetic 2.5-dimensional code «SOM 2.5» (3 dimensional by the velocities and 2 dimensional by the coordinates) for researching of a virtual cathode in the presence of
plasma are submitted.
Представлено перші результати створення чисельного електромагнітного 2,5-мірного (3-х мірного по
швидкостях і 2-х мірного по координатах) коду для дослідження віртуального катоду в присутності плазми.
Представлены первые результаты создания численного электромагнитного 2,5-мерного (3-х мерного по
скоростям и 2-х мерного по координатам) кода для исследования виртуального катода в присутствии плазмы.
|
| first_indexed | 2025-11-24T10:51:27Z |
| format | Article |
| fulltext |
NUMERICAL SIMULATION OF THE SUPERCRITICAL ELECTRON
BEAM DYNAMICS IN MAGNETIC FIELD OF FINITE SIZE IN THE
PRESENCE OF PLASMA
P.I.Markov, I.N.Onishchenko, G.V.Sotnikov
NSC “Kharkov Institute of Physics and Technology” 61108 Akademicheskaya 1, Kharkov,
Ukraine, phone: (0572)356623, E-mail: sotnikov@kipt.kharkov.ua
The first results of development of a numerical electromagnetic 2.5-dimensional code «SOM 2.5» (3 dimension-
al by the velocities and 2 dimensional by the coordinates) for researching of a virtual cathode in the presence of
plasma are submitted.
PACS: 02.60.Cb, 07.05.Tp
1. INTRODUCTION
Basis of a method of collective acceleration of ions
is the slow wave of a spatial charge formed by high-cur-
rent electron beam as a result of its spatial and time
modulation. Methods of spatial [1, 2] and time [1, 3]
modulations are well-known. The use of beams with
current higher than limiting vacuum one opens the op-
portunity of time modulation by the virtual cathode field
in the presence of plasma [4, 5].
The virtual cathode is a strongly nonlinear formation
for complete description of which the numerical meth-
ods are used. In the present report the results of devel-
opment of the numerical electromagnetic code describ-
ing the self-consistent dynamics of a virtual cathode in
the cylindrical resonator are adduced.
2. THE NUMERICAL ALGORITHM
The theoretical analysis of dynamics of electron-ion
formation is based on a PIC method [6]. A statement of
the problem is the following.
The relativistic electronic beam having a ring section,
is injected in the cylindrical resonator. Beam thickness is
2 1r r∆ = − , where 1r and 2r are the inner and external
radii of E -beam. Beam current is bI . At the resonator
input (at 0z = ) injected beam is monoenergetic. The
transversal components of electron velocities are equal to
zero. In the drift space there is an external magnetic field
of intensity 0H that is directed along the longitudinal axis
z of the resonator.
The system is axially symmetrical. It allows being
restricted to the solution of set of Maxwell equations for
the E -type wave at numerical simulation of dynamics
of electromagnetic fields:
( )
( )
4 ;
4 ;
,
r r
z z
z r
E t c H z j
E t c r H r r j
H t c E r E z
ϕ
ϕ
ϕ
π
π
∂ ∂ = − ∂ ∂ −
й щ∂ ∂ = ∂ −л ы
∂ ∂ = ∂ ∂ − ∂ ∂
where rj , zj are the radial and azimuthal components
of macroparticles current density which are contained in
the resonator. They are calculated by means of the
mechanism of weighing of currents in nodes of a two-
dimensional spatial grid [6]. Thus it is necessary to
know a position and velocity of each macroparticle.
They are determined from the solution of motion equa-
tions which have been written down in cylindrical coor-
dinates:
2
2
2
0
2
02
2
02
1 ;
1 12 ;
1 ,
p p
r
p p p
p p
p p p p p
p p
p p p
z
d r d rq E
d t m d t
d d z d
r H H r
c d t d t d t
d d r d d r dq H
d t m c r d t d t r d t d t
d z d z d rq E H
d t m d t c d t
ϕ
δ
γ
ϕ ϕ
ϕ ϕ ϕ
δ
γ
δ
γ
й
= − +к
л
щц цж ж
ъ+ − +ч чз з
ъи иш ш ы
цж
= − −чзз ч
и ш
цж
= − + чз
и ш
where ( ) 1 22 21 pv cγ
−
= − ,
( ) ( ) ( )2 2 22
p p p p pv d r d t r d d t d z d tϕ= + + ,
( ) 2
r p z pE d r d t E d z d t cδ = + , q and m are the
charge and weight of the macroparticle.
The numerical solution of Maxwell equations for a E -
type wave and weighing of charges were carried out on
shifted spatial and time grids one from other. The spatial
grid for evaluation of these quantities is shown on Fig.1.
For a time discretization of motion equations the pre-
dictor-corrector method was used [7]. Values of the
macroparticles velocities are calculated in half-integer
time steps ( )1 2 1 2nt n τ+ = + , and coordinates ( ),p pz r -
in the integer time steps nt nτ= ( n is integer, τ is a
time step). Thus the values of the fields components,
contained in the motion equations , are calculated by the
linear interpolation from nodes of a grid. Owing to the se-
lected scheme, the solution of Maxwell equations is nec-
essary to carry out twice more often, than solution of mo-
tion equations. Function H ϕ is calculated in time steps
1 4nt ± , and zE and rE in time steps nt and 1 2nt + respec-
tively.
Boundary condition for fields consists in vanishing
of tangential components of an electromagnetic field on
walls of the drift chamber. At an initial time step the
value of electromagnetic fields components are equal to
zero; particles in the resonator are absent.
In the issue the operations flowchart on one time
step 1n nt tτ += − looks like this:
1. Finding the value of a magnetic field H ϕ at time
step 1 4nt +
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 2.
Series: Nuclear Physics Investigations (43), p.135-137. 135
mailto:sotnikov@kipt.kharkov.ua
( )1 4 1 4 , ,
ϕϕ ϕ
+ −=n n n n
H r zH F H E E .
2. Calculation the values of components of electric
field zE and rE at time step 1 2nt +
( )1 2 1 4 1 2, ,ϕ
+ + −=
r
n n n n
r E r rE F E H j ;
( )1 2 1 4 1 2, ,ϕ
+ + −=
z
n n n n
z E z zE F E H j .
3. Finding the value of a magnetic field H ϕ at time
step 3 4nt +
( )3 4 1 4 1 2 1 2, ,
ϕϕ ϕ
+ + + +=n n n n
H r zH F H E E .
Finding the value of a magnetic field 1 2nH ϕ
+ by av-
eraging.
4. Solving of motion equations by the predictor-correc-
tor method:
a. calculation a preliminary value of an angular ve-
locity of macroparticle 1 2nω + ;
b. the values of 1nr + , 1nz + , 1n
rv + , 1n
zv + ;
c. finding the values of the same quantities at time
step 1 2nt + by averaging;
d. calculation a final value of an angular velocity of
macroparticle 1 2nω + .
5. Injecting new macroparticles in the resonator and
with taking into account of newly injected particles
calculating values zj and rj in grid nodes for time
step 1 2nt + .
6. Calculation the values a components of an electric
field zE and rE at time step 1nt +
( )1 1 2 3 4 1 2, ,ϕ
+ + + +=
r
n n n n
r E r rE F E H j ;
( )1 1 2 3 4 1 2, ,ϕ
+ + + +=
z
n n n n
z E z zE F E H j .
3. RESULTS OF THE NUMERICAL SIMU-
LATION
The mentioned algorithm has been implemented as a
complex of programs in C++ language. For numerical
calculations the following parameters of the experimen-
tal installation "Agat" [4, 5] has been chosen:
2,5R cm= , 1 1,5r cm= , 2 1,7r cm= , 15L cm= , energy
of beam electrons 280keV , intensity of external driv-
ing magnetic field 0 15H kOe= .
We carried out the calculations of the electron beam
dynamics at various values of input current of the beam.
In Fig.2 the results of simulation for beam current
0,5bI kA= , and in Fig.3 - for current 4,0bI kA= are
given.
For chosen sizes of the drift chamber and electron
beam the limiting vacuum current crI is equal to
3,76crI kA= .
The calculations which have been carried out by the
2.5-dimensional electromagnetic code "SOM 2.5" have
shown, that on taken times, which are equal approxi-
mately to one pass of the beam through the resonator,
when currents less than critical one the virtual cathode
is not forming. The motion of macroparticles along the
drift chamber is laminar, and the beam keeps the tubular
shape. During propagation along the resonator beam
thickness varies insignificantly.
At currents higher than critical one the forward front
of the beam moves as laminar one. The Coulomb field
of the forepart macroparticles decelerates the following
macroparticles and scatters them in a transverse direc-
tion. As a result the virtual cathode reflecting newly in-
jected electrons is forming. Eventually laminar motion
of beam particles is completely broken. Though for the
value of a magnetic intensity used in calculations the
beam remains tubular, its thickness varies essentially.
4. CONCLUSIONS
1. The numerical 2.5-dimensional electromagnetic
code "SOM 2.5", describing dynamics of a high-cur-
rent beam and fields in the cylindrical resonator,
based on a PIC method is created.
2. The numerical code adequately to physical insights
describes the process of creation of a virtual cathode
in the finite magnetic field. The limiting values of
currents received in 2.5-dimensional model, approxi-
mately coincide with calculated earlier in one-di-
mensional model.
3. Nevertheless, on large considered times at big values
of currents of injected beams the accumulation of
calculating errors takes place. In result there are re-
flected macroparticles even at b crI I< . Thus, there
is necessary to optimize the created numerical code
with the purpose of minimization of calculating er-
rors.
This work is supported by the STCU grant № 1569.
0,4t ns=
136
1,0t ns=
Fig.2. Results of simulation for the beam current 0,5bI kA=
0,4t ns=
1,0t ns=
Fig.3. Results of simulation for beam 4,0bI kA=
REFERENCES
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//VANT. Serija: Fizika Vysokih Energij I Atomnogo
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2. Yu.V.Tkach, Ya.B.Fainberg, E.A.Lemberg //Pisma
v ZhETF, 1978, v.28, No.9, p.580-584 (in Rusian).
3. A.N.Lebedev, K.N.Pazin //Atomnaya Energiya.
1976, v.41, No.4, p.244.
4. V.A.Balakirev, A.M.Gorban', I.I.Magda et. al. //
Fizika plazm., 1997, v.23, No 4 p.350–354 (in Ru-
sian).
5. P.I.Markov, I.N.Onishchenko, G.V.Sotnikov
//Problems of Atomic Science and Technology. Se-
ries: Plasma Physics. 2003, No 1(9). p.111-114.
6. Yu.A.Berezin, V.A.Vshyvkov. Metod chastits v di-
namike razregennoj plazmy, Novosibirsk: Nayka,
1980, p.98.
7. V.P.Il’in, Chislennye metody resheniyaa zadach
electrostatiki. Novosibirsk: Nauka, 1974.
ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ДИНАМИКИ СВЕРХКРИТИЧЕСКОГО ЭЛЕКТРОННОГО
ПУЧКА В МАГНИТНОМ ПОЛЕ КОНЕЧНОЙ ВЕЛИЧИНЫ В ПРИСУТСТВИИ ПЛАЗМЫ
П.И.Марков, И.Н.Онищенко, Г.В.Сотников
Представлены первые результаты создания численного электромагнитного 2,5-мерного (3-х мерного по
скоростям и 2-х мерного по координатам) кода для исследования виртуального катода в присутствии плазмы.
ЧИСЕЛЬНЕ МОДЕЛЮВАННЯ ДИНАМІКИ НАДКРИТИЧНОГО ЕЛЕКТРОННОГО ПУЧКА В
МАГНІТНОМУ ПОЛІ СКІНЧЕННОЇ ВЕЛИЧИНИ В ПРИСУТНОСТІ ПЛАЗМИ
П.І.Марков, І.М.Онищенко, Г.В.Сотників
Представлено перші результати створення чисельного електромагнітного 2,5-мірного (3-х мірного по
швидкостях і 2-х мірного по координатах) коду для дослідження віртуального катоду в присутності плазми.
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 2.
Series: Nuclear Physics Investigations (43), p.135-137. 137
numerical simulation OF THE SUPERCRITICAL ELECTRON BEAM DYNAMICS IN MAGNETIC FIELD OF finite SIZE IN the PRESENCE OF PLASMA
P.I.Markov, I.N.Onishchenko, G.V.Sotnikov
1. INTRODUCTION
2. THE NUMERICAL ALGORITHM
3. Results of the numerical simulation
4. conclusionS
REFERENCES
ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ДИНАМИКИ СВЕРХКРИТИЧЕСКОГО ЭЛЕКТРОННОГО ПУЧКА В МАГНИТНОМ ПОЛЕ КОНЕЧНОЙ ВЕЛИЧИНЫ В ПРИСУТСТВИИ ПЛАЗМЫ
ЧИСЕЛЬНЕ МОДЕЛЮВАННЯ ДИНАМІКИ НАДКРИТИЧНОГО ЕЛЕКТРОННОГО ПУЧКА В МАГНІТНОМУ ПОЛІ СКІНЧЕННОЇ ВЕЛИЧИНИ В ПРИСУТНОСТІ ПЛАЗМИ
П.І.Марков, І.М.Онищенко, Г.В.Сотників
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| id | nasplib_isofts_kiev_ua-123456789-79368 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-24T10:51:27Z |
| publishDate | 2004 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Markov, P.I. Onishchenko, I.N. Sotnikov, G.V. 2015-03-31T15:14:09Z 2015-03-31T15:14:09Z 2004 Numerical simulation of the supercritical electron beam dynamics in magnetic field of finite size in the presence of plasma / P.I. Markov, I.N. Onishchenko, G.V. Sotnikov // Вопросы атомной науки и техники. — 2004. — № 2. — С. 135-137. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 02.60.Cb, 07.05.Tp https://nasplib.isofts.kiev.ua/handle/123456789/79368 The first results of development of a numerical electromagnetic 2.5-dimensional code «SOM 2.5» (3 dimensional by the velocities and 2 dimensional by the coordinates) for researching of a virtual cathode in the presence of plasma are submitted. Представлено перші результати створення чисельного електромагнітного 2,5-мірного (3-х мірного по швидкостях і 2-х мірного по координатах) коду для дослідження віртуального катоду в присутності плазми. Представлены первые результаты создания численного электромагнитного 2,5-мерного (3-х мерного по скоростям и 2-х мерного по координатам) кода для исследования виртуального катода в присутствии плазмы. This work is supported by the STCU grant № 1569. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Динамика пучков Numerical simulation of the supercritical electron beam dynamics in magnetic field of finite size in the presence of plasma Чисельне моделювання динаміки надкритичного електронного пучка в магнітному полі скінченної величини в присутності плазми Численное моделирование динамики сверхкритического электронного пучка в магнитном поле конечной величины в присутствии плазмы Article published earlier |
| spellingShingle | Numerical simulation of the supercritical electron beam dynamics in magnetic field of finite size in the presence of plasma Markov, P.I. Onishchenko, I.N. Sotnikov, G.V. Динамика пучков |
| title | Numerical simulation of the supercritical electron beam dynamics in magnetic field of finite size in the presence of plasma |
| title_alt | Чисельне моделювання динаміки надкритичного електронного пучка в магнітному полі скінченної величини в присутності плазми Численное моделирование динамики сверхкритического электронного пучка в магнитном поле конечной величины в присутствии плазмы |
| title_full | Numerical simulation of the supercritical electron beam dynamics in magnetic field of finite size in the presence of plasma |
| title_fullStr | Numerical simulation of the supercritical electron beam dynamics in magnetic field of finite size in the presence of plasma |
| title_full_unstemmed | Numerical simulation of the supercritical electron beam dynamics in magnetic field of finite size in the presence of plasma |
| title_short | Numerical simulation of the supercritical electron beam dynamics in magnetic field of finite size in the presence of plasma |
| title_sort | numerical simulation of the supercritical electron beam dynamics in magnetic field of finite size in the presence of plasma |
| topic | Динамика пучков |
| topic_facet | Динамика пучков |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79368 |
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