Mathematical optimization model for alternating-phase focusing (APF) linac
Mathematical model of beam dynamic optimization in an equivalent traveling wave is suggested. The problem of the APF linac parameters optimization is being discussed. Beam dynamics in the 14 MeV deuteron accelerator are considered.
Збережено в:
| Дата: | 2013 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2013
|
| Назва видання: | Вопросы атомной науки и техники |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/111927 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Mathematical optimization model for alternating-phase focusing (APF) linac / D.A. Ovsyannikov, V.V. Altsybeyev // Вопросы атомной науки и техники. — 2013. — № 4. — С. 93-96. — Бібліогр.: 19 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-111927 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1119272017-01-16T03:03:59Z Mathematical optimization model for alternating-phase focusing (APF) linac Ovsyannikov, D.A. Altsybeyev, V.V. Новые методы ускорения заряженных частиц Mathematical model of beam dynamic optimization in an equivalent traveling wave is suggested. The problem of the APF linac parameters optimization is being discussed. Beam dynamics in the 14 MeV deuteron accelerator are considered. Запропоновано математичну модель оптимізації на основі рівнянь динаміки часток в полі еквівалентної біжучої хвилі. Вирішена задача оптимізації параметрів прискорювача з ПФФ. Розглянуто динаміку пучка в прискорювачі дейтронів на 14 МеВ. Предложена математическая модель оптимизации на основе уравнений динамики частиц в поле эквивалентной бегущей волны. Решена задача оптимизации параметров ускорителя с ПФФ. Рассмотрена динамика пучка в ускорителе дейтронов на 14 МэВ. 2013 Article Mathematical optimization model for alternating-phase focusing (APF) linac / D.A. Ovsyannikov, V.V. Altsybeyev // Вопросы атомной науки и техники. — 2013. — № 4. — С. 93-96. — Бібліогр.: 19 назв. — англ. 1562-6016 PACS: 29.27.-a http://dspace.nbuv.gov.ua/handle/123456789/111927 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 |
| collection |
DSpace DC |
| language |
English |
| topic |
Новые методы ускорения заряженных частиц Новые методы ускорения заряженных частиц |
| spellingShingle |
Новые методы ускорения заряженных частиц Новые методы ускорения заряженных частиц Ovsyannikov, D.A. Altsybeyev, V.V. Mathematical optimization model for alternating-phase focusing (APF) linac Вопросы атомной науки и техники |
| description |
Mathematical model of beam dynamic optimization in an equivalent traveling wave is suggested. The problem of the APF linac parameters optimization is being discussed. Beam dynamics in the 14 MeV deuteron accelerator are considered. |
| format |
Article |
| author |
Ovsyannikov, D.A. Altsybeyev, V.V. |
| author_facet |
Ovsyannikov, D.A. Altsybeyev, V.V. |
| author_sort |
Ovsyannikov, D.A. |
| title |
Mathematical optimization model for alternating-phase focusing (APF) linac |
| title_short |
Mathematical optimization model for alternating-phase focusing (APF) linac |
| title_full |
Mathematical optimization model for alternating-phase focusing (APF) linac |
| title_fullStr |
Mathematical optimization model for alternating-phase focusing (APF) linac |
| title_full_unstemmed |
Mathematical optimization model for alternating-phase focusing (APF) linac |
| title_sort |
mathematical optimization model for alternating-phase focusing (apf) linac |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2013 |
| topic_facet |
Новые методы ускорения заряженных частиц |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/111927 |
| citation_txt |
Mathematical optimization model for alternating-phase focusing (APF) linac / D.A. Ovsyannikov, V.V. Altsybeyev // Вопросы атомной науки и техники. — 2013. — № 4. — С. 93-96. — Бібліогр.: 19 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT ovsyannikovda mathematicaloptimizationmodelforalternatingphasefocusingapflinac AT altsybeyevvv mathematicaloptimizationmodelforalternatingphasefocusingapflinac |
| first_indexed |
2025-07-08T02:54:30Z |
| last_indexed |
2025-07-08T02:54:30Z |
| _version_ |
1837045667181101056 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2013. №4(86) 93
MATHEMATICAL OPTIMIZATION MODEL FOR ALTERNATING-
PHASE FOCUSING (APF) LINAC
D.A. Ovsyannikov, V.V. Altsybeyev
Saint-Petersburg State University, Saint-Petersburg, Russia
E-mail: altsybeyev@gmail.com
Mathematical model of beam dynamic optimization in an equivalent traveling wave is suggested. The problem
of the APF linac parameters optimization is being discussed. Beam dynamics in the 14 MeV deuteron accelerator
are considered.
PACS: 29.27.-a
INTRODUCTION
At present time the design of accelerators with ac-
celerating field focusing has become important. For ex-
ample, feasibility of this focusing is possible in RFQ
and APF accelerators. Problems of design and optimiza-
tion of these structures were developed in [1 - 17].
Among these, we mention [15], which was first ob-
tained synchronous phase sequence for focusing period
in APF structure. RFQ structures are usually used as the
initial part of the high-energy accelerators. Also, they
are used independently for different application pur-
poses. However, their applying is limited to low ener-
gies (2…5 MeV). Therefore a combination of RFQ and
APF is considered to be a good solution for a high-
energy accelerator. A high acceleration rate and an ab-
sence of magnetic field focusing in APF structure makes
it more efficient and less expensive in comparison with
Alvarez linac. Therefore, the development problems of
these accelerators and improving the quality of beams
are still important and vital.
Various mathematical optimization models in RFQ
structures were proposed in [1, 9-12, 14, 16]. In particu-
lar, some models allow us to analyze the longitudinal
and transversal motions separately. Beam dynamic
optimization in an equivalent traveling wave is carried
out and the limitation of the defocusing factor is taken
into account. It is possible to obtain a guaranteed good
transversal dynamics. These ideas are considered in the
case of acceleration in the APF structures too.
In this paper we examine an approach of beam dy-
namics simulations in an equivalent traveling wave. To
estimate the beam quality we introduce the integral
functional. The numerical gradient method is suggested
to minimize this functional.
1. THE APF PRINCIPLE
The main parameter that determines the beam dynam-
ics in an APF accelerator is the synchronous phase se-
quence sϕ . The concept of APF is to obtain this se-
quence providing longitudinal and transversal motions
stability alternately so that in general motion is stable. For
example, we can obtain this sequence using the analysis
of stability diagrams for Hill equation. Next we consider
the possibility of optimizing the phase of synchronous
particle in order to improve the beam dynamics.
2. MATHEMATICAL MODEL OF BEAM
DYNAMICS
Denote by Li coordinate of the i -th acceletaring cell
and by Di coordinate of this center. Then accelerating
field approximation may be assumed as
.)(cos
)(
cos1)(=),(
1
t
LL
Dz
EtzE
ii
i
max
i
i ω
π
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
−
−
This standing wave approximation allows to accept
the following mathematical model of beam dynamics in
the equivalent traveling wave [18, 19].
)),((cos1= 2 τϕβα
τ
β
ss
s
d
d
−
(1)
,
)(12
= 2
3/22
p
d
d
s
s
β
βπ
τ
ψ −
(2)
)),)((cos))((cos(= ψτϕτϕαβ
τ
+− sssd
dp
(3)
,2= 21
11 S
d
dS
τ
(4)
,))((sin
)(1
= 2211
3/22
21 SS
d
dS
s
s
s ++
−
− ψτϕ
β
βαπ
τ
(5)
.))((sin
)(1
2= 21
3/22
22 S
d
dS
s
s
s ψτϕ
β
βαπ
τ
+
−
−
(6)
Where ,= sϕϕψ − ,= γγ −sp ),(τϕs ,/= λτ ct
)/(2= 2
0cmEq maxλα is the accelerating amplitude pa-
rameter, 222111 ,, SSS is the element of matrix
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
2221
2111=
SS
SS
G , that describe dynamics of initial
ellipse 0G in the radial plane ),,( ηρ where λρ /= r is
the radial position of particle, τρη dd /= . Step-wise
function )(τϕs defined as
[ ]
.)/2(=
2,2
],,[if,=)(
11
1
ππϕϕττ
ππϕ
τττϕϕ
+−−
−∈
∈
−−
−
iiii
i
iiit
Thus, )(τϕs is parameterized only by parameters )(i
sϕ .
3. OPTIMIZATION PROBLEM
Let us consider the formalization of the beam
dynamics optimization problem. We may consider (1)-
(6) as equations that described program and disturbed
motions of charged particles [9].
)),(),(,(= tutxtf
dt
dx
)),(),(),(,(= tutytxtF
dt
dy
00 0(0) = , (0) = .x x y y M∈
ISSN 1562-6016. ВАНТ. 2013. №4(86) 94
Where x is a scalar and y is a n - dimensional
vector of phase coordinates, f is a scalar function and
F is a n - dimensional vector function. Control )(tu
is correspond to function )(τϕs . Accordingly, the con-
trol function will be parameterized as
[ ]
.)/2(=
2,2
],,[if,=)(
11
1
ππ
ππ
+−−
−∈
∈
−−
−
iiii
i
iii
uutt
u
tttutu
(7)
Such control is admissible. Let us consider the
following functional
.)()(=)( 21
,
TTT
uTM
xgdyyguI +∫ (8)
This functional is an integral estimate the beam
quality at the moment Tt = . Here the set uTM , is a
cross-section of the beam of trajectories at the moment
Tt = , i.e. uTT MTyy ,)(= ∈ , non-negative functions 1g
and 2g is characterization the dynamics of the dis-
turbed and program motions. We consider the problem
of functional (8) minimizing for the admissible controls.
To solve the problem of reducing the loss of parti-
cles and decrease the beam radius it is possible to take
),()(= 1122111 SFcFcg T +ψ
where
⎪⎩
⎪
⎨
⎧
−
⎪
⎩
⎪
⎨
⎧
−
∈
−+
.>if ,)(
,<if 0,
=
,> if ,)(
],,[ if ,0
,<if ,)(
=
11
2
11
11
2
2
2
2
21
1
2
1
1
SSSS
SS
F
F
TT
T
TT
ψψψψ
ψψϕ
ψψψψ
Where Scc , , , , 2121 ψψ are the non-negative
constants.
4. SOLUTION METHOD
The variation of functional (8) can be written as [9]
( ) ,=
,0
dtfdyfFI u
T
tu
T
u
T
utM
T
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
Δ+Δ+Δ− ∫∫ ψνμδ
where ,ψ ,μ ν are satisfying the following equations
.
,=
,=
ψψ
μνν
μμ
T
TT
T
x
f
dt
d
x
F
x
f
dt
d
y
F
dt
d
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−=
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
−
.)(=))(,(
,0=))(,(
,)(=))(,(
2
1
T
T
T
T
x
xgTxT
TxT
y
ygTxT
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
−
ψ
ν
μ
Having considered the parameterization (7) we can
obtain the functional gradient
( )
.]
)),(,(
)(
)),(),(,(
)([
])),(,()),(,(
)())),(),(,(
)),(),(,()(([
,2
1=
,
1
1
1
dtdy
u
utxtf
t
u
utytxtF
t
dyutxtfutxtf
tutytxtF
utytxtFt
uu
I
t
iT
iT
utM
it
it
tiiiiii
i
T
iiii
iiiii
T
it
Mi
∂
∂
+
+
∂
∂
−
−−×
×+−
−
∂
∂
∫∫
∫
+
−
−
ν
μ
ν
μ
π
(9)
By using the analytic representation (9) of the func-
tional gradient we can build gradient methods for opti-
mizing the accelerator parameters.
5. NUMERICAL SIMULATIONS
5.1. ACCELERATOR PARAMETERS
To illustrate the approach described earlier, we con-
sider the problem of design parameters for 433 MHz,
3.5 m long deuteron accelerator for 14 MeV output en-
ergy. Input energy is 3.5 MeV, amplitude of accelerat-
ing field is 110 kV/cm, and 6 focusing periods consist
of 115 accelerating cells. Provided NIIEFA input emit-
tances are present at Figs. 1, 2. Main parameters of
structure are present at Figs. 3, 4.
Fig. 1. Input longitudinal emittance
Fig. 2. Input transversal emittance
Fig. 3. Synchronous particle velocity and lengh of cells
ISSN 1562-6016. ВАНТ. 2013. №4(86) 95
Fig. 4. Synchronous phase sequence
5.2. SIMULATIONS WITHOUT INTERACTION
OF CHARGED PARTICLES
The results of beam dynamics simulations without
interaction between charged particles are presented at
Figs. 5 - 9. There are no particle losses at this model.
Fig. 5. Beam envelopes for phase motion
Fig. 6. Beam envelopes for radial motion
Fig. 7. Beam envelopes for energy motion
Fig. 8. Output longitudinal emittance
Fig. 9. Output transversal emittance
5.3. SIMULATIONS WITH 14 mA BEAM
CURRENT
In this section the results of beam dynamics
simulations with 14 mA beam current are presented. It
should be noted that the optimization of the structure
was carried out without taking into account the
interaction of charged particles, but the results of this
simulations show good beam quality. Output emittances
are present at Figs. 10, 11. Particle losses are 8 %.
Fig. 10. Output longitudinal emittance
Fig. 11. Output transversal emittance
ISSN 1562-6016. ВАНТ. 2013. №4(86) 96
CONCLUSIONS
In this paper the problem of optimizing the parame-
ters of longitudinal and transverse motion of the beam
in an APF accelerator is examined. This problem is
formalized and the solution based on analyzing the
beam dynamics in an equivalent traveling wave is pro-
posed. This approach does not impose rigid restrictions
on the structure geometry, which later on allows to pro-
vide precision tuning of resonators to produce the de-
sired field distribution. Analytic representation of the
gradient of the quality functional allows us to provide
the numerical optimization by other motion parameters.
For instance it can be the width of the output energy and
phase spectrum, the radial divergence of the beam, the
effective emittance or the acceleration intensity.
ACKNOWLEDGEMENTS
This work was supported by St. Petersburg State
University, scientific project No. 9.38.673.2013. The
authors also thank professor Yu.A. Svistunov for the
deep interest he took in this work and for his active in-
volvement in its discussion.
REFERENCES
1. B.I. Bondarev, A.P. Durkin, A.D. Ovsyannikov.
New mathematical optimization models for RFQ
structures // Proceedings of the IEEE Particle
Accelerator Conference. 1999, v. 4, p. 2808-2810.
2. B. Bondarev, A. Durkin, Y. Ivanov, I. Shumakov,
S. Vinogradov, A. Ovsyannikov, D. Ovsyannikov.
The LIDOS.RFQ.Designer development // Particle
Accelerator Conference, 2001. PAC 2001.
Proceedings of the 2001, v. 4, p. 2947-2949.
3. Y. Iwata, T. Fujisawa, T. Furukawa, et al. Alternating-
phase-focused linac with interdigital H-mode struc-
ture for medical injector // Proceedings of the IEEE
Particle Accelerator Conference 2005, Volume
2005, Article number 1590666, p. 1084-1086.
4. R.A. Jameson. Design and Simulation of Practical
Alternating-Phase-Focused (APF) Linacs –
Synthesis and Extension in Tribute to Pioneering
Russian APF Research // Proceedings of RuPAC-
2012, Saint-Petersburg, Russia. 2012, p. 12-14.
5. V.V. Kapin, A.V. Nesterovich. Feasibility of
alternative phase focusing for a chain of short
independently-phased resonators // Proceedings of
RuPAC-2010, Protvino, Russia. 2010, p. 322-324.
6. V. Kapin, S. Yamada, Y. Iwata. Design of APhF-IH
Linac for a Compact Medical Accelerator // Na-
tional Institute of Radiological Sciences, Japan.
7. Y. Kondo, K. Hasegawa, T. Morishita, R.A. Jameson.
Beam dynamics design of a new radio frequency
quadrupole for beam-current upgrade of the Japan
proton accelerator research complex Linac //
Physical Review Special Topics-Accelerators and
Beams. August 2012, v. 15, issue 8, 23, Article
number 080101.
8. V.V. Kushin. On improving the phase-alternating
focusing in linear accelerators // Nuclear Energy.
1970, v. 29, № 2, p. 123-124.
9. A.D. Ovsyannikov. Control of program and
disturbed motions // St.Petersburg, Vestnik SPbGU,
2006, v. 10 (4), p. 111-124.
10. A.D. Ovsyannikov, D.A. Ovsyannikov, M.Yu. Balabanov,
S.-L. Chung. On the beam dynamics optimization
problem // International Journal of Modern Physics A.
20 February 2009, v. 24, Issue 5, p. 941-951.
11. A.D. Ovsyannikov, D.A. Ovsyannikov, S.-L. Chung.
Optimization of a radial matching section //
International Journal of Modern Physics A.
20 February 2009, v. 24, issue 5, p. 952-958.
12. A.D. Ovsyannikov, D.A. Ovsyannikov, A.P. Durkin,
S.-L. Chang. Optimization of Matching Section of an
Accelerator with a Spatially Uniform Quadrupole
Focusing // Technical Physics. The Russian Journal
off Applied Physics. 2009, v. 54, № 11, p. 1663-1666.
13. D.A. Ovsyannikov. Modeling and Optimization of
Charged Particle Beam Dynamics. Leningrad State
University, Leningrad, 1990, 312 p.
14. D.A. Ovsyannikov, A.D. Ovsyannikov,
M.F. Vorogushin, Yu.A. Svistunov, A.P. Durkin.
Beam dynamics optimization: Models, methods and
applications // Nuclear Instruments and Methods in
Physics Research, Section A: Accelerators,
Spectrometers, Detectors and Associated
Equipment. 1 March 2006, v. 558, Issue 1, p. 11-19.
15. D.A. Ovsyannikov, V.G. Papkovich. On the design
of structures with accelerating field focusing //
Problems of Atomic Science and Technology.
Section: Linear accelerators. Kharkov. 1977,
Issue 2(3), p. 66-68.
16. D.A. Ovsyannikov, A.D. Ovsyannikov, I.V. Antropov,
V.A. Kozynchenko. BDO-RFQ code and
optimization models // Physics and Control,
Proceedings 2005 International Conference. 2005,
p. 282-288.
17. Yu.A. Svistunov, Yu.V. Zuev, A.D. Ovsyannikov,
D.A. Ovsyannikov. Compact deuteron accelerator
design for 1 MeV neutron source // Vestnik SPbGU.
2011, v. 10(1), p. 49-59.
18. I.M. Kapchinsky. Theory of linear resonance
accelerator. Moscow: «Energoizdat», 1982, 310 p.
19. A.D. Ovsyannikov, A.Y. Shirokolobov. Mathematical
model of beam dynamic optimization in traveling
wave // Proceedings of RuPAC-2012, Saint-
Petersburg, Russia. 2012, p. 355-357.
Article received 24.05.2013.
МАТЕМАТИЧЕСКАЯ МОДЕЛЬ ОПТИМИЗАЦИИ ПАРАМЕТРОВ УСКОРИТЕЛЯ С ПЕРЕМЕННО-ФАЗОВОЙ
ФОКУСИРОВКОЙ
Д.А. Овсянников, В.В. Алцыбеев
Предложена математическая модель оптимизации на основе уравнений динамики частиц в поле эквивалентной
бегущей волны. Решена задача оптимизации параметров ускорителя с ПФФ. Рассмотрена динамика пучка в ускорителе
дейтронов на 14 МэВ.
МАТЕМАТИЧНА МОДЕЛЬ ОПТИМІЗАЦІЇ ПАРАМЕТРІВ ПРИСКОРЮВАЧА З ПЕРЕМІННО-ФАЗОВИМ
ФОКУСУВАННЯМ
Д.О. Овсянников, В.В. Алцибєєв
Запропоновано математичну модель оптимізації на основі рівнянь динаміки часток в полі еквівалентної біжучої хвилі. Ви-
рішена задача оптимізації параметрів прискорювача з ПФФ. Розглянуто динаміку пучка в прискорювачі дейтронів на 14 МеВ.
|