PIC simulation of nonlinear regime wake field excitation in cylindrical resonator
The nonlinear mechanism of saturation of wake field amplitude, exited in the cylindrical resonator partially filled with dielectric by relativistic train of electron bunches numerically simulated by means of a specially elaborated 2.5- dimensional electromagnetic PIC-code.
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2006
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| Цитувати: | PIC simulation of nonlinear regime wake field excitation in cylindrical resonator / P.I. Markov, I.N. Onishchenko, A.F. Korzh, G.V. Sotnikov // Вопросы атомной науки и техники. — 2006. — № 5. — С. 199-202. — Бібліогр.: 2 назв. — англ. |
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irk-123456789-811812015-05-13T03:01:55Z PIC simulation of nonlinear regime wake field excitation in cylindrical resonator Markov, P.I. Onishchenko, I.N. Korzh, A.F. Sotnikov, G.V. Новые методы ускорения The nonlinear mechanism of saturation of wake field amplitude, exited in the cylindrical resonator partially filled with dielectric by relativistic train of electron bunches numerically simulated by means of a specially elaborated 2.5- dimensional electromagnetic PIC-code. С помощью специально разработанного нами 2.5-мерного электромагнитного PIC–кода численно промоделирован нелинейный механизм ограничения амплитуды кильватерного поля, возбуждаемого в цилиндрическом резонаторе, частично заполненном диэлектриком, релятивистской последовательностью электронных сгустков. Моделирование показало, что нелинейные захватные процессы, вызванные обратным влиянием поля большой амплитуды на заряженные сгустки, приводят к ограничению амплитуды электрического поля в системе. Получено время нарастания амплитуды поля, его амплитуда. Оценено количество электронных сгустков, которые следует инжектировать в резонатор для достижения максимума поля. За допомогою спеціально розробленого нами 2.5-мірного електромагнітного PIC-коду чисельно промодельовано нелінійний механізм обмеження амплітуди кільватерного поля, збуджуваного в циліндричному резонаторі, частково заповненому діелектриком, релятивістською послідовністю електронних згустків. Моделювання показало, що нелінійні захватні процеси, викликані зворотним впливом поля великої амплітуди на заряджені згустки, приводять до обмеження амплітуди електричного поля в системі. Отримано час наростання амплітуди поля, його амплітуду. Оцінено кількість електронних згустків, які слідує інжектувати у резонатор для досягнення максимуму поля. 2006 Article PIC simulation of nonlinear regime wake field excitation in cylindrical resonator / P.I. Markov, I.N. Onishchenko, A.F. Korzh, G.V. Sotnikov // Вопросы атомной науки и техники. — 2006. — № 5. — С. 199-202. — Бібліогр.: 2 назв. — англ. 1562-6016 PACS: 02.60.Cb, 07.05.Tp http://dspace.nbuv.gov.ua/handle/123456789/81181 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Новые методы ускорения Новые методы ускорения |
| spellingShingle |
Новые методы ускорения Новые методы ускорения Markov, P.I. Onishchenko, I.N. Korzh, A.F. Sotnikov, G.V. PIC simulation of nonlinear regime wake field excitation in cylindrical resonator Вопросы атомной науки и техники |
| description |
The nonlinear mechanism of saturation of wake field amplitude, exited in the cylindrical resonator partially filled
with dielectric by relativistic train of electron bunches numerically simulated by means of a specially elaborated 2.5-
dimensional electromagnetic PIC-code. |
| format |
Article |
| author |
Markov, P.I. Onishchenko, I.N. Korzh, A.F. Sotnikov, G.V. |
| author_facet |
Markov, P.I. Onishchenko, I.N. Korzh, A.F. Sotnikov, G.V. |
| author_sort |
Markov, P.I. |
| title |
PIC simulation of nonlinear regime wake field excitation in cylindrical resonator |
| title_short |
PIC simulation of nonlinear regime wake field excitation in cylindrical resonator |
| title_full |
PIC simulation of nonlinear regime wake field excitation in cylindrical resonator |
| title_fullStr |
PIC simulation of nonlinear regime wake field excitation in cylindrical resonator |
| title_full_unstemmed |
PIC simulation of nonlinear regime wake field excitation in cylindrical resonator |
| title_sort |
pic simulation of nonlinear regime wake field excitation in cylindrical resonator |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2006 |
| topic_facet |
Новые методы ускорения |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/81181 |
| citation_txt |
PIC simulation of nonlinear regime wake field excitation in cylindrical resonator / P.I. Markov, I.N. Onishchenko, A.F. Korzh, G.V. Sotnikov // Вопросы атомной науки и техники. — 2006. — № 5. — С. 199-202. — Бібліогр.: 2 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-06T05:35:32Z |
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| fulltext |
PIC SIMULATION OF NONLINEAR REGIME WAKE FIELD
EXCITATION IN CYLINDRICAL RESONATOR
P.I. Markov, I.N. Onishchenko, A.F. Korzh, G.V. Sotnikov
National Science Center “Kharkov Institute of Physics and Technology”
Kharkov, Ukraine
E-mail: pmarkov@kipt.kharkov.ua
The nonlinear mechanism of saturation of wake field amplitude, exited in the cylindrical resonator partially filled
with dielectric by relativistic train of electron bunches numerically simulated by means of a specially elaborated 2.5-
dimensional electromagnetic PIC-code.
PACS: 02.60.Cb, 07.05.Tp
1. INTRODUCTION
The wakefield amplitude excited by a long train of
electron bunches in dielectric waveguide is restricted,
firstly, by attenuation due to low Q-factor of the resona-
tor and, secondly, by nonlinear wave-particle interaction
resulting in driver-beam trapping in the potential well of
the wake. In this presentation the second mechanism of
wakefield amplitude restriction is analyzed.
2. A STATEMENT OF PROBLEM
The investigation of a nonlinear stage of the slow
wave excitation in the resonator by charged particles
bunches (by a train of point electron bunches) was
firstly considered in the paper [1]. Maximum attainable
amplitude of an electric field was found to be not de-
pended on a beam current and the number of injected
bunches for obtaining amplitude saturation is inversely
depended on beam current.
At present along with analytical methods of research
there is a possibility of carrying out the detailed numeri-
cal simulation. In this presentation by means of a spe-
cially elaborated 2.5-dimensional electromagnetic PIC-
code the excitation of electromagnetic field by a train of
electron bunches in the cylindrical resonator partially
filled with dielectric was simulated.
The geometry of a calculated model is depicted in
the Fig.1. A statement of the problem is the following.
Fig.1. Geometry of a design model
The train of relativistic electron bunches is injected
in the drift chamber through the left-hand boundary of a
drift chamber. The bunch radii is . The average elec-
tron current is
br
bI . At the resonator input (at 0z = ) in-
jected bunch is monoenergetic. The transversal compo-
nents of electron velocities are equal to zero.
The system is axially symmetrical. It allows being
restricted to the solution of set of Maxwell equations
( )
( )
;
;
;
;
;
,
r
r
z
z
z r
r z
r
z
HE c j
t z
E c r H j
t r r
H E Ec
t r z
E H Hc j
t z r
EH c
t z
H c c E
t r r
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
π
π
π
∂∂
= − −
∂ ∂
∂ ∂
= −
∂ ∂
∂ ∂ ∂ ⎞⎛= −⎜ ⎟∂ ∂ ∂⎝ ⎠
∂ ∂ ∂ ⎞⎛= − −⎜ ⎟∂ ∂ ∂⎝ ⎠
∂∂
=
∂ ∂
∂ ∂
= −
∂ ∂
4
4
4
where , ,r zE E Eϕ and , ,r zH H Hϕ are the components
of electric and magnetic intensity in cylindrical coordi-
nate system and , ,r zj j jϕ are the components of cur-
rent density in the drift region, at numerical simulation
of dynamics of electromagnetic fields. They are calcu-
lated by means of the mechanism of «distribution» of
currents in nodes of a two-dimensional spatial grid
(Fig.2).
Fig.2. Spatial grid
Thus it is necessary to know a position and velocity
of each macroparticle. They are determined from the
solution of motion equations
( )
( ){ }
( )
;
;
,
r z
r z
z r
r E r r B z B c r
E r z B r B c r r
z E z r B r B c
ϕ
ϕ
ϕ
β δ ϕ ϕ
ϕ β ϕδ ϕ
β δ ϕ
⎧ ⎡ ⎤= − + − +⎣ ⎦⎪
⎪ ⎡ ⎤= − + − −⎨ ⎣ ⎦
⎪
⎡ ⎤⎪ = − + −⎣ ⎦⎩
2
2
___________________________________________________________
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2006. № 5.
Серия: Плазменная электроника и новые методы ускорения (5), с.199-202. 199
mailto:pmarkov@kipt.kharkov.ua
where
( ) ( ) ( )
( )
, ;
;
; .
r z
z z z r r r
q v c m v r r z
r E r E z E c
B B H B B H
ϕ
β ϕ
δ ϕ
= − = + +
= + +
= + = +
2 22 2 2
2
0 0
1
200
2
The numerical solution of Maxwell equations and
«distribution» of charges were carried out on shifted one
from other spatial (Fig.2) and time (Fig.3) grids.
Fig.3. Time grid
For a time discretization of motion equations the
predictor-corrector method was used. Values of the
macroparticles velocities are calculated in half-integer
points of time (1 2 1 2nt n )τ+ = + , and coordinates
— in the integer points of time ( ,p pz r ) nt nτ= ( n is
integer, τ is a time step). Thus the values of the fields
components, contained in the motion equations, are cal-
culated by the linear interpolation from nodes of a grid.
Owing to the selected plan, the solution of Maxwell
equations is necessary to carry out twice more often,
than solution of motion equations. Magnetic field Hϕ is
calculated in points of time 1 4nt ± , and zE and in
points of time and
rE
nt 1 2nt + respectively.
Boundary conditions for fields consists in referenc-
ing to zero of tangential components of an electromag-
netic field on walls of the drift chamber. At an initial
point of time the value of electromagnetic fields com-
ponents are equal to zero; particles in the resonator are
absent.
3. THE NUMERICAL ALGORITHM
In the issue the operations flowchart on one time
step looks like this: 1nt tτ += − n
1. Finding the value of a magnetic field n 1 4
rH + ,
n 1 4
zH + , n 1 4Hϕ
+ at time step 1 4nt +
( )1 4 1 4 ,
r
n n
r H r
nH F H Eϕ
+ −= ;
( )1 4 1 4 ,
z
n n
z H z
nH F H Eϕ
+ −= ;
( )1 4 1 4 , ,n n n n
H r zH F H E E
ϕϕ ϕ
+ −= .
2. Calculation the values of components of electric bias-
ing vector n 1 2
rD + , n 1 2
zD + and n 1 2Dϕ
+ at time step
1 2nt +
( )1 2 1 4 1 2, ,
r
n n n
r D r rD F D H jϕ
+ += n− ;
( )1 2 1 4 1 2, ,
z
n n n
z D z zD F D H jϕ
+ += n− ;
( )1 2 1 4 1 4 1 2, , ,n n n n
D r zD F D H H j
ϕϕ ϕ
+ + += n
ϕ
− .
3. Calculation the values of components of electric field
n 1 2
rE + , n 1 2
zE + and n 1 2Eϕ
+ at time step 1 2nt +
,, ,
1 2, 1 21 2, 1 2 1 2, 1 2
1 2,1 2, 1 2,
1 2 1 2
1 2 1 2
1 2 1 2
,
,
.
j ij i j i
j ij i j i
j ij i j i
n n
rr r
n n
zz z
n n
E D
E D
E D ϕϕ ϕ
ε
ε
ε
+ ++ + + +
++ +
+ +
+ +
+ +
=
=
=
4. Finding the value of a magnetic field n 3 4
rH + ,
n 3 4
zH + , 3 4nHϕ
+ at time step 3 4nt +
( )3 4 1 4 1 2,
r
n n
r H rH F H Eϕ
+ += n+ ;
( )3 4 1 4 1 2,
z
n n
z H zH F H Eϕ
+ += n+ ;
( )3 4 1 4 1 2 1 2, ,n n n
H r zH F H E E
ϕϕ ϕ
+ + += n+ .
Finding the value of a magnetic field n 1 2
rH + ,
n 1 2
zH + , 1 2nHϕ
+ at time step 1 2nt + by averaging
( )
( )
( )
, , ,
1 2, 1 2 1 2, 1 2 1 2, 1 2
1 2, 1 2, 1 2,
1 2 1 4 3 4
1 2 1 4 3 4
1 2 1 4 3 4
2,
2,
2.
j i j i j i
j i j i j i
j i j i j i
n n n
r r r
n n n
z z z
n n n
H H H
H H H
H H Hϕ ϕ ϕ
+ + + + + +
+ + +
+ + +
+ + +
+ + +
= +
= +
= +
5. Solving of motion equations by the predictor-
corrector 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 + .
6. Calculating values of a current density zj , rj and
ϕj in grid nodes for time step 1 2nt + .
7. Injecting new macroparticles in the drift chamber.
8. Computing the values of a charge density of ρ for
time step 1nt + .
9. Calculation the values a components of electric bias-
ing vector n 1
rD + , n 1
zD + and n 1Dϕ
+ at time step 1nt +
( )1 1 2 3 4, ,
r
n n n n
r D r rD F D H jϕ
+ + += 1 2+ ;
( )1 1 2 3 4, ,
z
n n n n
z D z zD F D H jϕ
+ + += 1 2+ ;
( )1 1 2 3 4 3 4, , ,n n n n n
D r zD F D H H j
ϕϕ ϕ ϕ
+ + + += 1 2+ .
10. Calculation the values of components of electric
field n 1
rE + , n 1
zE + and n 1Eϕ
+ at time step + 1nt
, , ,
1 2, 1 2 1 2, 1 21 2, 1 2
1 2, 1 2, 1 2,
1 1
1 21
1 1
,
,
.
j i j i j i
j i j ij i
j i j i j i
n n
r r r
nn
z zz
n n
E D
E D
E Dϕ ϕ ϕ
ε
ε
ε
+ + + ++ +
+ + +
+ +
++
+ +
=
=
=
11. Carrying-out the correction of an electric field
and rE for time step 1nt + according to the Boris
scheme
zE
D δ
∗
= −∇ ΦE E ,
where
∗
E and E are initial and corrected electric field
values, δ Φ is correction to electric field potential, D∇
is a difference analog of the Hamilton operator.
For preventing occurrence of electromagnetic field
noise with grid period, on each ten-thousand step of
evaluations the nine point fields averaging was fulfilled.
3. RESULTS OF NUMERICAL SIMULATION
The algorithm described above has been imple-
mented as a complex of C ++ programs. The main pa-
rameters were chosen close to existing ones in the in-
stallation "Almaz-2", namely, the radius of the drift
chamber is 4.3 cm, permittivity is equal 2.1, radius of
the channel for beam in dielectric is 1.05 cm, length of
the chamber is 55.3 cm, electron bunch radius is 0.5 cm,
electron energy is 5 MeV, bunch duration is 0.078 ns,
pulse repetition period is 0.37 ns. For reduction of cal-
culating time and numerical errors the average injected
current was chosen 10 A, i.e. 20 times greater than ex-
perimental value.
Results of simulation showed that during the first
85.5 ns, i.e. when 230 bunches were injected practically
linear growth of electric field amplitude up to saturation
value of 95 kV/cm is observed (see Fig.4). We assume
that the expected number of clots at current of 0.5 A is
about 4600.
Fig.4. Time dependence of intensity of an electric field
on an axis at a right end face of the resonator
The electric field spectrum corresponding to Fig.4 is
depicted on Fig.5. It’s visible, that the basic maximum
on an electric field oscillation spectrum has frequency
2.7 GHz. More high-frequency maximums with ampli-
tude approximately 10 times smaller are spread up to
frequency 16.2 GHz.
Fig.5. The electric field spectrum
Linear growth of field amplitude results in square-
law change of field energy that it is possible to see on
the energy diagram (Fig.6). We shall notice a good per-
formance of the energy conservation law in our simula-
tion.
Fig.6. The energy diagram: W is the electromagnetic
field energy, P is the particles energy losses
After achieving of maximum the electric field ampli-
tude oscillates slowly with frequency of about 11 MHz.
Self-consistent influence of excited field on bunches
motion dynamics leads to essential spreading of longi-
tudinal and transversal velocities of electrons on the
phase plane (see Fid. 7, where spreading of longitudinal
velocities is depicted).
Fig.7. Phase plane for macroelectrons
Despite of essential spread of particles on velocities,
their relativistic factor remains high. Therefore in con-
figuration space the spread of particles is observed only
on cross coordinate, while shift of particles in a longitu-
dinal direction is absent, that it is possible to view in
Fig.8.
201
202
Fig.8. Configuration space for macroelectrons
CONCLUSIONS
The executed numerical simulation has shown, that
nonlinear trapping processes caused by back influence
of field of big amplitude on charged bunches, result in
restriction of amplitude of an electric field in system.
Time of field amplitude growth, and its value are re-
ceived. The number of electron bunches which should
be injected in the resonator for achievement of a maxi-
mum of a field is estimated.
This study was partly supported by CRDF Grant #
UP2-2569-KH-04.
REFERENCES
1. V.I. Kurilko, J. Ullschmied // Nuclear Fusion. 1969.
№9, p.129-135.
2. V.A. Balakirev, I.N. Onishchenko, D.Yu. Sidorenko,
G.V. Sotnikov // Pisma v JTF. 2003, v.29, №14, p.39-
45 (in Russian).
МОДЕЛИРОВАНИЕ НЕЛИНЕЙНОГО РЕЖИМА ВОЗБУЖДЕНИЯ КИЛЬВАТЕРНОГО ПОЛЯ В
ЦИЛИНДРИЧЕСКОМ РЕЗОНАТОРЕ С ПОМОЩЬЮ PIC-КОДА
П.И. Марков, И.Н. Онищенко, А.Ф. Корж, Г.В. Сотников
С помощью специально разработанного нами 2.5-мерного электромагнитного PIC–кода численно промо-
делирован нелинейный механизм ограничения амплитуды кильватерного поля, возбуждаемого в цилиндри-
ческом резонаторе, частично заполненном диэлектриком, релятивистской последовательностью электрон-
ных сгустков. Моделирование показало, что нелинейные захватные процессы, вызванные обратным влияни-
ем поля большой амплитуды на заряженные сгустки, приводят к ограничению амплитуды электрического
поля в системе. Получено время нарастания амплитуды поля, его амплитуда. Оценено количество электрон-
ных сгустков, которые следует инжектировать в резонатор для достижения максимума поля.
МОДЕЛЮВАННЯ НЕЛІНІЙНОГО РЕЖИМУ ЗБУДЖЕННЯ КІЛЬВАТЕРНОГО ПОЛЯ В
ЦИЛІНДРИЧНОМУ РЕЗОНАТОРІ ЗА ДОПОМОГОЮ PІC-КОДУ
П.І. Марков, І.М. Онищенко, О.Ф. Корж, Г.В. Сотніков
За допомогою спеціально розробленого нами 2.5-мірного електромагнітного PIC-коду чисельно промо-
дельовано нелінійний механізм обмеження амплітуди кільватерного поля, збуджуваного в циліндричному
резонаторі, частково заповненому діелектриком, релятивістською послідовністю електронних згустків. Мо-
делювання показало, що нелінійні захватні процеси, викликані зворотним впливом поля великої амплітуди
на заряджені згустки, приводять до обмеження амплітуди електричного поля в системі. Отримано час нарос-
тання амплітуди поля, його амплітуду. Оцінено кількість електронних згустків, які слідує інжектувати у ре-
зонатор для досягнення максимуму поля.
PIC SIMULATION OF NONLINEAR REGIME WAKE FIELD EXCITATION IN CYLINDRICAL RESONATOR
1. INTRODUCTION
2. A STATEMENT OF PROBLEM
3. THE NUMERICAL ALGORITHM
3. RESULTS OF NUMERICAL SIMULATION
CONCLUSIONS
МОДЕЛИРОВАНИЕ НЕЛИНЕЙНОГО РЕЖИМА ВОЗБУЖДЕНИЯ КИЛЬВАТЕРНОГО ПОЛЯ В ЦИЛИНДРИЧЕСКОМ РЕЗОНАТОРЕ С ПОМОЩЬЮ PIC-КОДА
МОДЕЛЮВАННЯ НЕЛІНІЙНОГО РЕЖИМУ ЗБУДЖЕННЯ КІЛЬВАТЕРНОГО ПОЛЯ В ЦИЛІНДРИЧНОМУ РЕЗОНАТОРІ ЗА ДОПОМОГОЮ PІC-КОДУ
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