Multi-beam accelerating module calculation
As has been pointed in [1], a multi-beam accelerating cavity with drift tubes is suggested to be used in a linac for the purpose of overcoming limitations on the mean current. In this paper the dispersion equation for such a construction was deduced and analyzed.
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| Дата: | 1999 |
|---|---|
| Автори: | , , , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
1999
|
| Назва видання: | Вопросы атомной науки и техники |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/81530 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Multi-beam accelerating module calculation / N.M. Gavrilov, S.V. Alexeyenko, D.A. Komarov, N.N. Netchaev, J.N. Struckov // Вопросы атомной науки и техники. — 1999. — № 4. — С. 52-53. — Бібліогр.: 6 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-81530 |
|---|---|
| record_format |
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| spelling |
irk-123456789-815302015-05-18T03:02:34Z Multi-beam accelerating module calculation Gavrilov, N.M. Alexeyenko, S.V. Komarov, D.A. Netchaev, N.N. Struckov, J.N. As has been pointed in [1], a multi-beam accelerating cavity with drift tubes is suggested to be used in a linac for the purpose of overcoming limitations on the mean current. In this paper the dispersion equation for such a construction was deduced and analyzed. 1999 Article Multi-beam accelerating module calculation / N.M. Gavrilov, S.V. Alexeyenko, D.A. Komarov, N.N. Netchaev, J.N. Struckov // Вопросы атомной науки и техники. — 1999. — № 4. — С. 52-53. — Бібліогр.: 6 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/81530 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 |
| description |
As has been pointed in [1], a multi-beam accelerating cavity with drift tubes is suggested to be used in a linac for the purpose of overcoming limitations on the mean current.
In this paper the dispersion equation for such a
construction was deduced and analyzed. |
| format |
Article |
| author |
Gavrilov, N.M. Alexeyenko, S.V. Komarov, D.A. Netchaev, N.N. Struckov, J.N. |
| spellingShingle |
Gavrilov, N.M. Alexeyenko, S.V. Komarov, D.A. Netchaev, N.N. Struckov, J.N. Multi-beam accelerating module calculation Вопросы атомной науки и техники |
| author_facet |
Gavrilov, N.M. Alexeyenko, S.V. Komarov, D.A. Netchaev, N.N. Struckov, J.N. |
| author_sort |
Gavrilov, N.M. |
| title |
Multi-beam accelerating module calculation |
| title_short |
Multi-beam accelerating module calculation |
| title_full |
Multi-beam accelerating module calculation |
| title_fullStr |
Multi-beam accelerating module calculation |
| title_full_unstemmed |
Multi-beam accelerating module calculation |
| title_sort |
multi-beam accelerating module calculation |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/81530 |
| citation_txt |
Multi-beam accelerating module calculation / N.M. Gavrilov, S.V. Alexeyenko, D.A. Komarov, N.N. Netchaev, J.N. Struckov // Вопросы атомной науки и техники. — 1999. — № 4. — С. 52-53. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT gavrilovnm multibeamacceleratingmodulecalculation AT alexeyenkosv multibeamacceleratingmodulecalculation AT komarovda multibeamacceleratingmodulecalculation AT netchaevnn multibeamacceleratingmodulecalculation AT struckovjn multibeamacceleratingmodulecalculation |
| first_indexed |
2025-07-06T06:32:47Z |
| last_indexed |
2025-07-06T06:32:47Z |
| _version_ |
1836878211529572352 |
| fulltext |
MULTI-BEAM ACCELERATING MODULE CALCULATION
N.M. Gavrilov, S.V. Alexeyenko, D.A. Komarov, N.N. Netchaev, J.N. Struckov
MIPhI, Moscow, Russia
INTRODUCTION
As has been pointed in [1], a multi-beam
accelerating cavity with drift tubes is suggested to be
used in a linac for the purpose of overcoming
limitations on the mean current.
The multi-beam accelerating cavity containing a
cylindrical screen 1 with end flanges 2 is shown in Fig.
1. Current conducting rings 3 are set inside the screen
perpendicularly to its axis. Drift tubes 4 with integral
constant magnet focussing quadrupoles are uniformly
distributed along the ring circumference. Each ring is
attached to the screen inner surface by means of two
diametrically placed supports 5 in such a way that
supports of neighbouring rings are mutually
perpendicular. At the operational mode the resonant
elements (ring 3 with supports 5) are exited so that the
oscillation phase is opposite.
In this paper the dispersion equation for such a
structure was deduced and analyzed.
DISPERSION EQUATION
The calculations have been carried out under the
following assumptions. The structure geometry is
approximated as shown in Fig. 2.
The drift tubes and the cavity body are assumed
to be ideally conducting. The structure length is
infinitely long. Only E and H modes can be exited in the
structure that follows from the module geometry. Let us
divide the structure into several regions. The complete
mathematical modeling of the electromagnetic wave
propagation requires some factors to be specifically
accounted for. Among them there are the boundary
conditions on the conductor surface (its geometrical
configuration), the conditions on the boundaries
dividing the regions, the two-dimensional periodicity.
Taking into account these requirements Helmholtz
equations were solved for each region with
corresponding boundary conditions. Thus, for the region
1 we have:
02 =+∆ I
z
I
z EkE , (1)
0),,( =zE I
z ϕη , (2)
),(),,( zfzRE I
z ϕϕ = , (3)
where the function f(ϕ,z) is given by
=−∈
−∈
−+∈
=
,...2,1,0},),({);,,(
};{);,,(
})1(;{,0
),(
νννϕ
ν θθν θϕϕ
θθνν θϕ
ϕ
TgTzzRE
zREzf
IV
z
т
II
z
т
(4)
It should be noted that E-modes are considered.
Here Laplacians are expressed in a cylindrical
coordinate system, its center being at the module axis.
The solution of Eq. (1) is represented as a series of own
functions. The required dispersion equation is obtained
by means of using expressions of magnetic field
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. № 4.
Серия: Ядерно-физические исследования (35), с. 52-54.
52
strength components and determination of series
coefficients Aik.
For the purpose of analysis it seems to be
convenient to consider the harmonic (0,0) for which the
dispersion equation can be written as
( ) ( ) ( )
+
++′
ΘΘ ∫ ∫ ∫ ∫
Θ
Θ−Θ
Θ
Θ
L L
Tr G
zRudzdzaRuzaRK
LT 0
00
0
0000 ,,,,,,11 ϕϕϕϕξ
( ) ( ) ( ) ( ) +
++′
Θ
+ ∫ ∫
Θ
Θ
dzdzRzRudzdzaRuzaRK
L
G
L
mlml
T
ϕϕνϕϕϕϕξ ,,,,,,,,1
0000
0
00
( ) ( ) ( )
+Γ
+Γ+
Θ
+ ∫ ∫
Θ
zRdzdzaRzaRK
g
T
L
,,,,,,1
00
0
0000 ϕϕϕϕξ
( ) ( ) ( ) ( ) =
Γ
+Γ+
Θ
+ ∫ ∫
Θ
dzdzRzRdzdzaRzaRK
g
T
L
l ϕϕνϕϕϕϕξ ,,,,,,,,1
0000
0
0000
( ) ( ) ( )
+′
+′+′
ΘΘ
= ∫ ∫ ∫ ∫
Θ
Θ
Θ
Θ
L L
Tr G
zRudzdzaRuzaRK
LT 0
00
0
0000 ,,,,,,11 ϕϕϕϕξ
( ) ( ) ( ) ( ) +′
′
+′+′
Θ
+ ∫ ∫
Θ
Θ
dzdzRzRudzdzaRuzaRK
L
G
L
T
ϕϕνϕϕϕϕξ ,,,,,,,,1
0000
0
0000
( ) ( ) ( )
+Γ ′
+Γ ′+
Θ
+ ∫ ∫
Θ
zRdzdzaRzaRK
g
T
L
,,,,,,1
00
0
0000 ϕϕϕϕξ
( ) ( ) ( ) ( )
′
Γ ′
+Γ ′+′
Θ
+ ∫ ∫
Θ
dzdzRzRdzdzaRzaRK
g
T
L
ϕϕνϕϕϕϕξ ,,,,,,,,1
0000
0
0000 (5)
where the functions ξ, u, Г, v – are taken to be Bessel
and Neumann functions, such as (6).
if
ij
ij
ij
ij exI
cI
cK
xK −
− )(
)(
)(
)( , (6)
That equation was solved graphically (Fig 3,
Table 1) for λ=10 cm. For any module geometry chosen
in advance (Q, Qт, L, g) such a graph makes it possible
to determine the required diameter of drift tubes and to
analyze the dynamics of its variation with changing β:
2
0
2 2
−
=
β λ
πk
consta
, (7)
where the const is defined by graph (Fig. 3).
Table 1.
νa Lgт θθ
1 2 3
0,5 5,513 1,508 0,811
1,0 4,742 2,112 0,813
1,5 6,803 1,834 0,605
2,0 9,610 2,122 0,523
Fig. 3. The parameters are: 1 – R/a=1.25, a/η=1, b/a=4:
2 — R/a=1.5, a/η=1.25, b/a=5: 3 — R/a=2.0, a/η=1.5,
b/a=6.
It is of interest to note that in the limiting case
when 1→тQQ Eq. (5) can be written as
1
)}()]([)()]([{ )]([
)}()]([)()]([{ )]([
00000
11111
2 =
+−++
+−++
RKaRKRIaRIaRK
RIaRIRKaRKaRK
ννννν
νννννξ (8)
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. № 4.
Серия: Ядерно-физические исследования (35), с. 52-54.
52
Eq. (8) is similar to that obtained by P. Alliot
for such a structure [2].
CONCLUSION
The correlation of geometrical parameters of
the structure was demonstrated in this work. In addition,
recommendations are given for the determination of the
drift tube diameter for cases of given geometry and
dynamics. The limiting case 1→тQQ was
investigated. The obtained results could be used in the
development of accelerators.
REFERENCE
1. Gavrilov N.M., Minaev S.A., Shalnov A.V. High
current accelerator of ions. Application for patent
№92-007144/21-052566 18.11.92.
2. Tikhonov A.N., Samarsky A.S. Equations of
mathematical physics. Nauka, Moscow, 1972 (in
Russian).
3. Milovanov O.S., Sobenin N.P. Microwave
engineering. Atomizdat, Moscow, 1981 (in
Russian).
4. Elliot R.S. Azimuthal surface waves on circular
cylinders.-Journal of applied physics/ April 1955,
vol.26, N.4.
5. Silin R.A., Sazonov V.P. Slow-wave systems.
Sovetskoye radio, Moscow, 1966 (in Russian).
6. Sivukhin N.A. Electrodynamics of periodical
structures. Sovetskoye radio, Moscow, 1987 (in
Russian).
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. № 4.
Серия: Ядерно-физические исследования (35), с. 52-54.
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