3D intense beam dynamics simulation by using moments method
The program for 3D simulation of the intense charge particle beam dynamics on the base of the Multi-Component Ion Beam code is described. Fast analysis and study of the averaged beam characteristicsis performed by the moments method.
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| Zitieren: | 3D intense beam dynamics simulation by using moments method / N. Kazarinov, V. Aleksandrov, V. Shevtsov // Вопросы атомной науки и техники. — 2008. — № 5. — С. 140-142. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1115342017-01-11T03:03:41Z 3D intense beam dynamics simulation by using moments method Kazarinov, N. Aleksandrov, V. Shevtsov, V. Новые методы ускорения, сильноточные пучки The program for 3D simulation of the intense charge particle beam dynamics on the base of the Multi-Component Ion Beam code is described. Fast analysis and study of the averaged beam characteristicsis performed by the moments method. Дано опис програми 3D-чисельного моделювання інтенсивного пучку заряджених частинок, що створена на основі Multі-Component Іon Beam коду. Швидкий аналіз і дослідження середніх характеристик пучку проводиться методом моментів. Описывается программа 3D-численного моделирования интенсивного пучка заряженных частиц, созданная на основе Multi-Component Ion Beam кода. Быстрый анализ и исследование средних характеристик пучка проводится методом моментов 2008 Article 3D intense beam dynamics simulation by using moments method / N. Kazarinov, V. Aleksandrov, V. Shevtsov // Вопросы атомной науки и техники. — 2008. — № 5. — С. 140-142. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 29.17.+w http://dspace.nbuv.gov.ua/handle/123456789/111534 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Новые методы ускорения, сильноточные пучки Новые методы ускорения, сильноточные пучки |
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Новые методы ускорения, сильноточные пучки Новые методы ускорения, сильноточные пучки Kazarinov, N. Aleksandrov, V. Shevtsov, V. 3D intense beam dynamics simulation by using moments method Вопросы атомной науки и техники |
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The program for 3D simulation of the intense charge particle beam dynamics on the base of the Multi-Component Ion Beam code is described. Fast analysis and study of the averaged beam characteristicsis performed by the moments method. |
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Article |
| author |
Kazarinov, N. Aleksandrov, V. Shevtsov, V. |
| author_facet |
Kazarinov, N. Aleksandrov, V. Shevtsov, V. |
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Kazarinov, N. |
| title |
3D intense beam dynamics simulation by using moments method |
| title_short |
3D intense beam dynamics simulation by using moments method |
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3D intense beam dynamics simulation by using moments method |
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3D intense beam dynamics simulation by using moments method |
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3D intense beam dynamics simulation by using moments method |
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3d intense beam dynamics simulation by using moments method |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2008 |
| topic_facet |
Новые методы ускорения, сильноточные пучки |
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http://dspace.nbuv.gov.ua/handle/123456789/111534 |
| citation_txt |
3D intense beam dynamics simulation by using moments method / N. Kazarinov, V. Aleksandrov, V. Shevtsov // Вопросы атомной науки и техники. — 2008. — № 5. — С. 140-142. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT kazarinovn 3dintensebeamdynamicssimulationbyusingmomentsmethod AT aleksandrovv 3dintensebeamdynamicssimulationbyusingmomentsmethod AT shevtsovv 3dintensebeamdynamicssimulationbyusingmomentsmethod |
| first_indexed |
2025-07-08T02:18:04Z |
| last_indexed |
2025-07-08T02:18:04Z |
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1837043376022618112 |
| fulltext |
3D INTENSE BEAM DYNAMICS SIMULATION
BY USING MOMENTS METHOD
N. Kazarinov∗, V. Aleksandrov, V. Shevtsov
Joint Institute for Nuclear Research, 6, Joliot-Curie, 141980, Dubna, Russia
∗E-mail: nyk@lnr.jinr.ru
The program for 3D simulation of the intense charge particle beam dynamics on the base of the Multi-Compo-
nent Ion Beam code is described. Fast analysis and study of the averaged beam characteristicsis performed by the
moments method.
PACS: 29.17.+w
INTRODUCTION
Within the framework of the Multi-Component Ion
Beam code (MCIB04) [1] the program for 3D simula-
tion of the intense charge particle beam dynamics is cre-
ated.
Fast analysis and study of the averaged beam char-
acteristics, such as root-mean-square (RMS) dimen-
sions, is performed by the moments method [2].
The main advantage of the moments method in com-
parison with macro particle one is fast calculation and
therefore applicability for transport line optimization.
The model describing the charge density of the
bunched beam is introduced. The external electromag-
netic fields are assumed to be linear. The approach of
effective linearization [2] of both longitudinal and
transversal beam self fields gives possibility to get the
closed system of the equations for second order mo-
ments.
The fitting procedure based on minimization of a
quadratic functional at any point of the beam line by us-
ing either gradient or simplex-method is available [3].
BEAM MODEL
Let consider the train of bunches (Fig.1), moving
with average velocity c0β with distance between its
center-of-mass 00λβλ = . Here 0λ is cyclotron RF field
wave length.
Fig.1.
The beam density may be defined as:
),()(),,( 0//0 yxctzNctzyx ⊥−=− ρβρβρ , (1)
where cZe
IN
0β
λ= – the number of particle at spatial
period λ , I – beam current, Ze – ion charge.
Longitudinal //ρ and transverse densities ⊥ρ are equal to:
∑
∞
− ∞=
−−=
n zz
nzz 2
2
// 2
)(exp
2
1)(
σ
λ
σπ
ρ (2.1)
−−=⊥ 2
2
2
2
22
exp
2
1),(
yxyx
yxyx
σσσπ σ
ρ (2.2)
According to formula (2.1) longitudinal density is
periodical function )()( //// λρρ += zz with a constant
number of particles at period λ :
∫
−
=
2/
2/
// )(
λ
λ
ρ Ndzz (3)
In the case σz ≳ λ this model describes the beam
with constant density and for σz << λ gives Gaussian
beam. The dependencies on z of the longitudinal beam
density for various values of ratio zσλ / are shown in
Fig.2.
-0.5 -0.25 0 0.25 0.5
z / λ
0
1
2
3
4
n
/ n
0
1
2
3
Fig.2. Longitudinal beam density
Curve 1 – zσλ / = 1; 2 – zσλ / = 4; 3 – zσλ / = 8
BEAM SELF FIELD
By using formulae (1, 2) the beam self field may be
represent in the following form [4]:
222
2
22
0//
22
2
2
2
2
0
20//
0
20//
,
22
1ln)(2
))(()(;
)4()(
)()(
)(2
)(
)()(
)(2
ayx
a
yx
a
bctzZeNE
sssR
s
y
s
xT
dsT
sRs
yctzZeNE
dsT
sRs
xctzZeNE
z
yx
yx
y
yxy
x
yxx
≤+
+−+−′≅
++=
+
+
+
=
+
−≅
+
−≅
⊥
∞
⊥
∞
∫
∫
βρ
σσ
σσ
ρ
σ
σσβρπ
ρ
σ
σσβρπ
Here )(2 22
yxa σσ += – RMS radius of the beam, b –
vacuum pipe radius and prime denotes derivative with
respect to z.
MOMENTS EQUATIONS
Let us define the second order moments M of the
beam distribution function f:
∫== dyfYY
N
YYM TT 1
, (5)
____________________________________________________________
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2008. № 5.
Series: Nuclear Physics Investigations (50), p.140-142.
z
140
λ λ
where superscript T denotes transpose vector or matrix,
),(),,(),,,,,( ////0
TTTTTT YYYVXctzyxyxY ⊥==−′′= δβ –
vector of phase space coordinates of the particle,
00 /)( βββδ −= – relative momentum spread. Integra-
tion in (4) is fulfilled over all phase space occupied by
bunch particles (at one spatial period), prime denotes
derivative with respect to longitudinal coordinate of the
bunch center-of-mass.
The equations for transverse second order moments
TYYM ⊥⊥⊥ = does not changed significantly in compari-
son with the case of non-bunched beam[2]:
TAMAMM ⊥⊥⊥ +=′ ;
+
=
extsext abb
E
A
0
(6)
Here M⊥, A are fourth order matrices, E is second order
unit matrix, aext and bext are 2×2 matrices defined by ex-
ternal fields. Second order matrix bs depends on RMS
dimensions and is defined by beam self fields:
+
= ⊥
y
x
yxA
s I
I
A
Zkb
σ
σ
σσβ /10
0/111
3
0
, (7)
where А – ion mass, IA = mc3/e – Alfven’s current.
The bunching factor k⊥ is connected with changing
of the transverse beam self fields due to changing of the
longitudinal density:
== ⊥
−
⊥ ∫ 2
0
22/
2/
2
2
02
// )(
z
zF
z
zdzzk
λ
λ
ρλ (8)
Here 2z is current longitudinal RMS dimension of
the bunch:
∫
−
=
2/
2/
//
22 )(
λ
λ
ρ dzzzz (9)
and 3/2
0 λ=z its value for non-bunched beam. The
plot of function )(xF⊥ is shown in Fig.3,a.
0 0.2 0.4 0.6 0.8 1
x
0.92
0.94
0.96
0.98
1
F ⊥
a
0 0.2 0.4 0.6 0.8 1
x
0
0.2
0.4
0.6
0.8
1
F /
/
b
Fig.3.
As may be seen from Fig.3,a function )(xF⊥ is ap-
proximately equal to unity with difference does not
greater than 6%. In the program this function is repre-
sented as the sixth order polynomial.
The equations for the longitudinal second order mo-
ments M// has the following form:
==
2
2
//////
δδ
δ
z
zzYYM T
(10.1)
δzz 22 =
′
(10.2)
( ) zzE
cAm
Zez 22
0
2
β
δδ +=
′
(10.3)
δ
β
δ z
z
zE
cAm
Ze z
222
0
2 =
′
(10.4)
Computation of average zzE in accordance with formu-
lae (4, 5) results in:
+
+
=
4
1
)(2
ln1
223
0
//22
0 yxA
z
b
I
I
A
ZkzE
cAm
Ze
σσββ
(11)
The bunching factor of the longitudinal motion k// is
defined by formula:
=−= ⊥
−
∫ 2
0
2
//
2/
2/
2
//
2
//// )]2/()([
z
zFkdzzk
λ
λ
λρρλ (12)
The plot of function )(// xF is shown in Fig.3,b. In
the case x ∼ 1 function F// is close to zero because of the
longitudinal electric field of non-bunched beam is equal
to zero. For the well bunched beam(x << 1) due to small
longitudinal density at point z = λ/2 formulae (11) and
(12) become identical and function F// is close to unity.
In the program function )(// xF is approximated by the
fifth order polynomial for all values of x.
MCIB04 CODE MODIFICATION
The 3D moments equations were introduced into ex-
isting program library code MCIB04 [1]. The interface
of the program is shown in Fig.4.
Fig.4. Interface of the program
Before launching of the program the files containing
the beam-line lattice, initial beam parameters and (op-
tionally) the longitudinal magnetic field distribution
have to be created.
During working of the program the changes of the
second order moments along the beam-line are comput-
ed. The plots of the longitudinal magnetic field distribu-
tion (green line in Fig.4) and RMS dimensions of the
beam (red – x and blue – y) are given at monitor. The
special windows are intended for values of the beam
RMS dimensions at the exit of the channel (RMSX,
RMSY) and initial parameters – RMS dimensions(X,Y),
emittances (Xemit, Yemit), mass-to-charge ratio (A/Z),
kinetic energy (Energy) and beam current (Current).
The fitting procedure based on minimization by us-
ing either gradient or simplex-method of a quadratic
functional computed for every second order moments at
any point of the beam line is available [3].
____________________________________________________________
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2008. № 5.
Series: Nuclear Physics Investigations (50), p.140-142.141
The dependencies on distance along the channel of
the beam envelopes, emittances, momentum spread and
other parameters are writing to the file and processing
by the graphing program package.
BUNCHING SYSTEM COMPUTATION
The simulation of the bunching system of the
DC350 cyclotron axial injection beam-line [5] was ful-
filled by using created 3D version of MCIB04 code.
The bunching system consists of linear and sinu-
soidal bunchers. The linear buncher is placed at 275 cm
and sinusoidal – at 80 cm from median plane of the cy-
clotron. In the simulation all bunchers were replaced by
infinitesimal width gap with variable voltage.
The initial parameters of the beam are contained in
Table.
48Ca beam initial parameters
Injected beam 48Ca6+
Mass, A 48
Charge, Z 2…8
Injected current, µA 0…190
Ca beam current, µA 0…700
He beam current, µA 200
48Ca6+ kinetic energy, keV/u 3.1375
Diametr, mm 8
Emittance, π mm×mrad 142
The initial conditions for the moments were defined at
the entrance of the linear buncher and were found by
macro-particle simulation. Charge state distributions for
ion beam and its self fields were taken into account in
this simulation.
The beam focusing is provided by two solenoids. The
longitudinal magnetic field of the cyclotron is consid-
ered also.
225 235 245 255 265 275
distance, cm
0
10
20
30
40
En
ve
lo
pe
s,
m
m
H
V
A
Fig.5. Apertutre (A), horizontal (H) and vertical (V) 48-
Ca6+ beam envelopes near inflector
The matching condition at the entrance of the spiral
inflector corresponds to the steady state of the beam
(without envelopes oscillation) in the uniform magnetic
field with magnitude to be equal to the field in the cy-
clotron center. The amplitude of the voltage at linear
buncher was found to provide the equality k⊥ = 2 at the
entrance of sinusoidal buncher.
The beam envelopes near spiral inflector of the cy-
clotron are shown in Fig.5.
Let define the bunching efficiency as ratio of the
number of particles within RF phase interval 015≤∆ ϕ
to non-bunched beam one. This quantity shows the pos-
sible increasing of the number of particle captured into
acceleration in the cyclotron due to the bunching sys-
tem. The dependence of the bunching efficiency on the
48Ca6+ beam current is shown in Fig.6.
0 40 80 120 160 200
I, µ A
1
3
5
7
9
Bu
nc
hi
ng
ef
fic
ie
nc
y
Fig.6. Bunching efficiency versus beam current
REFERENCES
1. V. Aleksandrov, N. Kazarinov, V. Shevtsov. Multi-
Component Ion Beam code–MCIB04 // Proc. XIX
Russian Particle Accelerator Conference (RuPAC-
2004). Dubna, Russia, 2004, p.201.
2. N.Yu. Kazarinov, E.A. Perelstein, V.F. Shevtsov.
Moments method in charged particle beams dyna-
mics // Particle Accelerators. 1980, v.10, p.33-48.
3. N.Yu. Kazarinov, V.F. Shevtsov. Optimization of
transportation channel parameters for beam with
big spatial charge // JINR communication. P9-2002-
148, Dubna, 2002.
4. A.W. Chao. Physics of Collective Beam Instabili-
ties in High Energy Accelerators // Wiley Series in
Beam Physics and Accelerator Technology. ISBN
0-471-55184-8, New-York, 1993.
5. G. Gulbekyan, et al. Axial injection channel of the
DC-350 cyclotron // 18-th International Confer-
ence on Cyclotron and their Application.
30 September – 5 October, Giardini Naxos, Italy,
2007.
Статья поступила в редакцию 07.09.2007 г.
3D-МОДЕЛИРОВАНИЕ ДИНАМИКИ СИЛЬНОТОЧНОГО ПУЧКА С ИСПОЛЬЗОВАНИЕМ МЕТОДА
МОМЕНТОВ
Н. Казаринов, В. Александров, В. Шевцов
Описывается программа 3D-численного моделирования интенсивного пучка заряженных частиц, созданная на осно-
ве Multi-Component Ion Beam кода. Быстрый анализ и исследование средних характеристик пучка проводится методом
моментов
3D-МОДЕЛЮВАННЯ ДИНАМІКИ ПОТУЖНОСТРУМОВОГО ПУЧКА З ВИКОРИСТАННЯМ МЕТОДУ
МОМЕНТІВ
Н. Казарінов, В. Олександров, В. Шевцов
Дано опис програми 3D-чисельного моделювання інтенсивного пучку заряджених частинок, що створена на основі
Multі-Component Іon Beam коду. Швидкий аналіз і дослідження середніх характеристик пучку проводиться методом
моментів.
142
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