Complex approach of beam dynamic investigation in SC linac
Beam dynamic investigation is difficult for superconducting linac consisting from periodic sequences of independently phased accelerating cavities and focusing solenoids. The matrix calculation was preferably used for previous estimate of accelerating structure parameters. The matrix calculation doe...
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irk-123456789-1088012016-11-17T03:02:24Z Complex approach of beam dynamic investigation in SC linac Samoshin, A.V. Динамика пучков Beam dynamic investigation is difficult for superconducting linac consisting from periodic sequences of independently phased accelerating cavities and focusing solenoids. The matrix calculation was preferably used for previous estimate of accelerating structure parameters. The matrix calculation does not allow properly investigate the longitudinal motion. The smooth approximation can be used to investigate the nonlinear ion beam dynamics in such accelerating structure and to calculate the longitudinal and transverse acceptances. The potential function and equation of motion in the Hamiltonian form are devised by the smooth approximation. The advantages and disadvantages of each method will describe, the results of investigation will compare. Application package for ion beam dynamic analysis will create. A numerical simulation of beam dynamics in the full field will carry out for the different variants of the accelerator structure based on analytically obtained results. Анализ динамики ионного пучка в сверхпроводящем ускорителе, состоящем из последовательности независимо фазированных ускоряющих резонаторов и фокусирующих соленоидов, представляет сложную задачу. Для первоначальной оценки параметров ускоряющей структуры удобно использовать матричный метод расчета. Матричный метод не дает возможности корректно исследовать продольное движение. Для исследования нелинейной динамики в такой ускоряющей структуре и определения продольного и поперечного аксептанса может быть использовано гладкое приближение. В гладком приближении найдена потенциальная функция и уравнения движения в форме уравнения Гамильтона. Описываются достоинства и недостатки каждого метода, проводится сравнение результатов исследования. Создан пакет прикладных программ для анализа динамики пучка ионов. На основе полученных данных проведено численное моделирование динамики пучка в полном поле для различных вариантов ускорителя. Аналіз динаміки іонного пучка в надпровідному прискорювачі, що складається з послідовності незалежно фазованих прискорюючих резонаторів і фокусуючих соленоїдів, представляє складну задачу. Для початкової оцінки параметрів прискорюючої структури зручно використовувати матричний метод розрахунку. Матричний метод не дає можливості коректно досліджувати поздовжній рух. Для дослідження нелінійної динаміки в такій прискорюючій структурі та визначення поздовжнього і поперечного аксептанса може бути використано гладке наближення. У гладкому наближенні знайдено потенційну функцію і рівняння руху в формі рівняння Гамільтона. Описуються переваги і недоліки кожного методу, проводиться порівняння результатів дослідження. Створено пакет прикладних програм для аналізу динаміки пучка іонів. На основі отриманих даних проведено чисельне моделювання динаміки пучка в повному полі для різних варіантів прискорювача. 2012 Article Complex approach of beam dynamic investigation in SC linac / A.V. Samoshin // Вопросы атомной науки и техники. — 2012. — № 4. — С. 78-82. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 29.27.-A, 29.27.Bd http://dspace.nbuv.gov.ua/handle/123456789/108801 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Динамика пучков Динамика пучков Samoshin, A.V. Complex approach of beam dynamic investigation in SC linac Вопросы атомной науки и техники |
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Beam dynamic investigation is difficult for superconducting linac consisting from periodic sequences of independently phased accelerating cavities and focusing solenoids. The matrix calculation was preferably used for previous estimate of accelerating structure parameters. The matrix calculation does not allow properly investigate the longitudinal motion. The smooth approximation can be used to investigate the nonlinear ion beam dynamics in such accelerating structure and to calculate the longitudinal and transverse acceptances. The potential function and equation of motion in the Hamiltonian form are devised by the smooth approximation. The advantages and disadvantages of each method will describe, the results of investigation will compare. Application package for ion beam dynamic analysis will create. A numerical simulation of beam dynamics in the full field will carry out for the different variants of the accelerator structure based on analytically obtained results. |
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Samoshin, A.V. |
| author_facet |
Samoshin, A.V. |
| author_sort |
Samoshin, A.V. |
| title |
Complex approach of beam dynamic investigation in SC linac |
| title_short |
Complex approach of beam dynamic investigation in SC linac |
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Complex approach of beam dynamic investigation in SC linac |
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Complex approach of beam dynamic investigation in SC linac |
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Complex approach of beam dynamic investigation in SC linac |
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complex approach of beam dynamic investigation in sc linac |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Динамика пучков |
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http://dspace.nbuv.gov.ua/handle/123456789/108801 |
| citation_txt |
Complex approach of beam dynamic investigation in SC linac / A.V. Samoshin // Вопросы атомной науки и техники. — 2012. — № 4. — С. 78-82. — Бібліогр.: 6 назв. — англ. |
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ISSN 1562-6016. ВАНТ. 2012. №4(80) 78
COMPLEX APPROACH OF BEAM DYNAMIC INVESTIGATION
IN SC LINAC
A.V. Samoshin
National Research Nuclear University – Moscow Engineering-Physics Institute,
Moscow, Russia
E-mail: AVSamoshin@mephi.ru
Beam dynamic investigation is difficult for superconducting linac consisting from periodic sequences of inde-
pendently phased accelerating cavities and focusing solenoids. The matrix calculation was preferably used for pre-
vious estimate of accelerating structure parameters. The matrix calculation does not allow properly investigate the
longitudinal motion. The smooth approximation can be used to investigate the nonlinear ion beam dynamics in such
accelerating structure and to calculate the longitudinal and transverse acceptances. The potential function and
equation of motion in the Hamiltonian form are devised by the smooth approximation. The advantages and
disadvantages of each method will describe, the results of investigation will compare. Application package for ion
beam dynamic analysis will create. A numerical simulation of beam dynamics in the full field will carry out for the
different variants of the accelerator structure based on analytically obtained results.
PACS: 29.27.-A, 29.27.Bd
INTRODUCTION
High-current accelerators have great perspectives for
problems of thermonuclear fusion, safe nuclear reactors,
transmutation of radioactive wastes and free electron
lasers. A large number of low energy particle accelera-
tors are applied in micro- and nanoelectronics, material
science, including the study of new construction materi-
als for nuclear industry, in medical physics, in particular
for cancer treatment by using of the accelerators of pro-
tons and light ions, in radiation technology. It is pro-
posed to use one universal accelerator, consisting of
independently phased cavities and solenoids sequence to
solve these problems.
An ion superconducting linac is usually based on the
superconducting (SC) independently phased cavities.
This linac consists of the niobium cavities which can
provide typically 1 MV of accelerating potential per
cavity. Such structures can be used for ion acceleration
with different charge-to-mass ratio in the low energy
region [1] and for proton linac in the high-energy region
(SNS, JHF, ESS project). It is desirable to have a con-
stant geometry of the accelerating cavity in order to
simplify manufacturing and to decrease the linac cost.
Such geometry leads to a non-synchronism but a stable
longitudinal particle motion can be provided by proper
RF cavities phasing. The beam can be both longitudi-
nally stable and accelerated in the whole system by con-
trol of the accelerating structure driven phase and the
distance between the cavities. In this paper two methods
of the beam dynamics investigation are compared for
low ion velocities and for the charge-to-mass ratio
Z/A = 1/66. This comparison can be demonstrated with
an example a post-accelerator of radioactive ion beams
(FRIB) linac, where beam velocity increases from
β = 0.01 to β = 0.06 [1].
Beam focusing can be provided with the help of SC
solenoid lenses, following each cavity and with the help
of special RF fields. As was shown early the beam fo-
cusing can be realized for the solenoid field near
B ~ 20 T. The value of magnetic field B can be reduced
by using of addition APF. The smooth approximation
has been applied to study the alternating phase focusing
(APF) in RIB linac. By adjusting the drive phase (φ1
and φ2) of the two cavities, we can achieve the accelera-
tion and the focusing by less magnitude of magnetic
field B [2]. Adding a focusing solenoid into focusing
period will also allow separate control of the transverse
and longitudinal beam dynamics. A schematic plot of
one period of the accelerator structure is shown in Fig.1.
The low-charge-state low velocity beams require
stronger transverse focusing than one is used in existing
SC ion linac. Early investigation of beam dynamics
shows that for the initial normalized transverse emit-
tance εT = 0.1 π⋅mm⋅mrad and the longitudinal emit-
tance εV = 0.3π⋅keV/u⋅nsec the connection between the
longitudinal and transverse motion can be neglected if
maximum beam envelop Xm < 3…4 mm and inner ra-
dius of drift tubes a = 15 mm.
Beam dynamics in such systems cannot be studied
by means of analytical methods only. The initial setup
of the system consisting of different types supercon-
ducting resonators and focusing solenoids or quadru-
poles, can be performed using the transfer matrix calcu-
lation and the method of smooth approximation, and
then refined the beam dynamics simulation in polyhar-
monic field.
Fig.1. Layout of structure period
Code BEAMDULAC-SCL developed in the
laboratory DINUS allows comprehensive research ion
beam dynamics in a different structures that satisfy the
acceleration of many methods. To evaluate the
accelerator parameters implemented transfer matrix
calculation method. With a smooth averaging can
determine the stability region and to calculate the
dynamics of a single particle and beam. And for
completion, to verify selections, we perform the
calculation of beam dynamics in polyharmonic field.
ISSN 1562-6016. ВАНТ. 2012. №4(80) 79
1. MATRIX CALCULATIONS
The conditions of longitudinal and transverse beam
stabilities for the structure consisting from the periodic
sequence of the cavities and solenoids were studied
early using transfer matrix calculation [2]. The code
window with the plots of dependencies dimensionless
parameter α, phase advances per period (Floke parame-
ters), magnetic field which need for beam stable motion
with envelope value Xm = 3 mm is shown in the Fig.2.
Fig.2. Beam dynamic transfer matrix calculation
2. BEAM DYNAMICS IN SMOOTH
APPROXIMATION
The general axisymmetric equations of motion for
ion moving inside an accelerator can be written as
( )
( )( ) .2
γ222
22
ββ1,
d
dγ
d
d
,2
γ222
22
,
d
dγ
d
d
ϕ
ϕ
A
rmA
Ze
GtrrE
Am
eZ
t
r
t
A
zmA
ZetrzE
Am
eZ
t
z
t
∂
∂
−−=⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−=⎟
⎠
⎞
⎜
⎝
⎛
r
r
(1)
In every cavity the acceleration RF field of periodic
H-cavity can be represented as an expansion in spatial
harmonics
( ) ( )( ) ( )
( ) ( )( ) ( ) ⎪⎭
⎪
⎬
⎫
−=
−=
∑
∑
,ωcossinI
,ωcoscosI
10
00
tzzhrhEE
tzzhrhEE
innr
innz (2)
where E0 is the amplitude of RF field at the axis (E0 ≠ 0
if –Lr/2 < z–zi < Lr/2), hn = π/D + 2πn/D, n = 0, 1, 2, …,
Di = βGλ/2 is the cavity accelerating structure period
length, Lr is the cavity length, zi is the coordinate of the
i-th cavity center, I0, I1 are the modified Bessel func-
tions. Let we call particle which accelerating on axis
and does not have slow phase and transverse oscillation
term as reference. In our case the reference particle ve-
locity βc and the geometrical velocity βG are closely for
each class of the identical cavities. Retaining in (2) only
zeroth RF field harmonic we can use the traveling wave
system. In this system ωt can be replaced by
h0(z–zi) + φ0i, where φ0i is the RF phase when the refer-
ence particle traverses the cavity center. In equation (1)
the value Aφ is the azimuthal vector-potential of the
magnetic field in solenoid (B = rotA).
In SC linac design, it is very important to know the
bucket size since it relates to the longitudinal RF focus-
ing. But the linac longitudinal acceptance cannot be
obtained by matrix method because of the assumption
that the particles have small longitudinal oscillation
amplitude. In order to investigate the nonlinear ion
beam dynamics in such accelerated structure and to cal-
culate the longitudinal and transverse acceptances it can
be used smooth approximation [3,4]. In this paper,
equation of motion for ion beam in the Hamiltonian
form is derived in the smooth approximation for super-
conducting linac.
Let us consider the particle motion in the polyhar-
monic fields of the cavities and solenoids. The ion dy-
namics in such periodic structure is complicated. The
particles trajectories can be presented as a sum of the
slowly term and a fast oscillation term with a period L.
The normalized particle velocity deviation with respect
to the reference particle velocity, Δβ, can be represented
as a sum of a slow motion term and a fast oscillation
term also.
Following Ref. [5] one can apply an averaging over
the fast oscillations and obtain the phase (ψ) and radial
(ρ = h0r) motion equations in smooth approximation:
( )
( ) ,
ρdξ
dρβγln
dξ
d
dξ
ρd
,
ψdξ
dψ
βγln
dξ
d3
dξ
ψd
2
2
2
2
∂
∂
−=⎥
⎦
⎤
⎢
⎣
⎡
+
∂
∂
−=⎥
⎦
⎤
⎢
⎣
⎡
+
eff
eff
U
U
(4)
ISSN 1562-6016. ВАНТ. 2012. №4(80) 80
where Ueff = U0 + U1 + U2 is effective potential function.
We use the following designations:
{ } ( ) ( ) ( ){ }0 1 2 0 1 24α ψ cosφ cosφ I ρ sin φ ψ sin φ ψ ,U ⎡ ⎤= − + − + + +⎣ ⎦
( )( ){
( ) ( ) ( )[ ]
( ) ( ) ( )[ ] }
( )( ){
( ) ( ) ( )[ ]
( ) ( ) ( )[ ] },ψφsinψφsinρI
ψφcosψφcosρI
ψφsinφsinφcosφcos2αχ
ψφsinψφsinρI
ψφcosψφcosρI
ψφsinφsinφcosφcos2αχ
2
21
2
1
2
21
2
0
2121
2
2
2
21
2
1
2
21
2
0
2121
2
11
+−++
++−++
+−−+
+++++
+++++
+++=U
(4)
( ) ( ) ( )[ ]
.
2
ρ~ρ~χ
ψφsinψφsinρρI~α
2
χ
2
2
2
2
2
4
211
3
2
L
L
B
L
L
B
L
L
BU
solsol
sol
+
++++−=
.
Here α = πeZUL/2Aλmc2βg
3γg
3 is the interaction pa-
rameter, B~ = (eZBL/2Amcβcγc)2 is the focusing parame-
ter.
In this expression for Ueff we take into account the
coherent oscillations of bunches and the effective poten-
tial function describe slowly oscillations in the reference
particle frame. Earlier, in [5] the effective potential
function was found in the frame where averaged veloc-
ity of the reference particle, 0β =c .
The analysis of the effective potential function
makes it possible to study the condition at which the
phase and radial stability of the beam is achieved and to
calculate the longitudinal acceptance.
( ) ...ρ
2
1ψ
2
10,0 2222 +Ω+Ω+= rzeffeff UU (5)
The code window with the frequencies of
longitudinal and transverse oscillations, the cross
section of the potential function and change the size of
the separatrix is shown in Fig.3.
Fig.3. The frequencies of longitudinal and transverse oscillations, potential function cross-section and separatrixes
for different approximation cases
3. NUMERICAL SIMULATION
OF ION BEAM DYNAMICS
3.1. SINGLE PARTICLE MOTION
For the analyses of longitudinal and transverse mo-
tions the beam dynamic was studied in averaged on fast
oscillations field. Field’s components can be obtained
from the effective potential function in smooth ap-
proximation. The solution can be obtained only by nu-
merical simulation because the field components are
nonlinear functions. The greatest interest represents
modeling of the dynamics of the full field. The results
of ion beam numerical simulations in polyharmonic
field with Z/A = 1/66 are shown in Fig.4.
ISSN 1562-6016. ВАНТ. 2012. №4(80) 81
Fig.4. Particle trajectories during acceleration
3.2. BEAM DYNAMICS
The initial parameters of numerical simulation for
this system are similar to the previous one. The initial
particle phase in the cavity are φ1 = –30° and φ2 = 20°,
and magnetic field of solenoid is B = 14 T. The beam
moves through 8 periods in this case. The numerical
simulation in polyharmonic field was performed to ver-
ify of the result obtained bellow. Geometrical velocity
βG varies in each cavity. The results of ion beam nu-
merical simulations in a polyharmonic field are close to
results received in smooth approximation. The beam
longitudinal volume increase is negligible and trans-
verse emittance slowly vying. The results are shown in
Fig.5 agree with previously calculation.
Fig.5. Beam dynamics in the polyharmonic field
ISSN 1562-6016. ВАНТ. 2012. №4(80) 82
4. NUMERICAL SIMULATION
IN POLYHARMONIC FIELD
The numerical simulation in polyharmonic field was
performed to verify of the result obtained bellow. The
simulation was spent for the same focusing periods and
the same initial parameters as in section 4 accordingly.
Geometrical velocity βG varies in each cavity too. The
results of ion beam numerical simulations in a polyhar-
monic field are close to results received in smooth ap-
proximation. The beam longitudinal volume increase
negligible and transverse emittance slowly vying.
CONCLUSIONS
The methods of the beam focusing in SC linac
analysis are compared for low ion velocities. By the
smooth approximation it was studied more detailed
nonlinear ion beam dynamics and founded the beam
stability area. It was done the recommendation for
choice of the reference particle phases and the value of
solenoid magnetic field B. It was shown that the smooth
approximation gives very good agreement with the
simulation in polyharmonic field. By the smooth ap-
proximation it is studied nonlinear ion beam dynamics
in linac with combined focusing.
REFERENCES
1. P.N. Ostroumov, et al. // Proc. of PAC’2001, p. 4080.
2. E.S. Masunov, D.A. Efimov, P.N. Ostroumov //
Proc. of EPAC’2004, p. 1405-1407.
3. E.S. Masunov, N.E. Vinogradov // Phys. Rev. ST-AB 4,
2001, 070101.
4. Ji Qiang, R.W. Garnett // Nucl. Instr. and Meth. A.
2003, v.496, p.33.
5. E.S. Masunov, A.V. Samoshin // Proc. of PAC’2007,
p.1568-1570.
6. E.S. Masunov, A.V. Samoshin // Proc. of RuPAC’2006,
p.162-164.
Статья поступила в редакцию 23.09.2011 г.
КОМПЛЕКСНЫЙ ПОДХОД К АНАЛИЗУ ДИНАМИКИ ПУЧКА В СВЕРХПРОВОДЯЩЕМ
ЛИНЕЙНОМ УСКОРИТЕЛЕ
А.В. Самошин
Анализ динамики ионного пучка в сверхпроводящем ускорителе, состоящем из последовательности не-
зависимо фазированных ускоряющих резонаторов и фокусирующих соленоидов, представляет сложную
задачу. Для первоначальной оценки параметров ускоряющей структуры удобно использовать матричный
метод расчета. Матричный метод не дает возможности корректно исследовать продольное движение. Для
исследования нелинейной динамики в такой ускоряющей структуре и определения продольного и попереч-
ного аксептанса может быть использовано гладкое приближение. В гладком приближении найдена потенци-
альная функция и уравнения движения в форме уравнения Гамильтона. Описываются достоинства и недос-
татки каждого метода, проводится сравнение результатов исследования. Создан пакет прикладных программ
для анализа динамики пучка ионов. На основе полученных данных проведено численное моделирование
динамики пучка в полном поле для различных вариантов ускорителя.
КОМПЛЕКСНИЙ ПІДХІД ДО АНАЛІЗУ ДИНАМІКИ ПУЧКА В НАДПРОВІДНОМУ ЛІНІЙНОМУ
ПРИСКОРЮВАЧІ
О.В. Самошин
Аналіз динаміки іонного пучка в надпровідному прискорювачі, що складається з послідовності незалеж-
но фазованих прискорюючих резонаторів і фокусуючих соленоїдів, представляє складну задачу. Для почат-
кової оцінки параметрів прискорюючої структури зручно використовувати матричний метод розрахунку.
Матричний метод не дає можливості коректно досліджувати поздовжній рух. Для дослідження нелінійної
динаміки в такій прискорюючій структурі та визначення поздовжнього і поперечного аксептанса може бути
використано гладке наближення. У гладкому наближенні знайдено потенційну функцію і рівняння руху в
формі рівняння Гамільтона. Описуються переваги і недоліки кожного методу, проводиться порівняння ре-
зультатів дослідження. Створено пакет прикладних програм для аналізу динаміки пучка іонів. На основі
отриманих даних проведено чисельне моделювання динаміки пучка в повному полі для різних варіантів
прискорювача.
|