Estimation of BBU threshold current in an inhomogeneous accelerating section
In this work we describe the procedure of estimation of regenerative beam blowup threshold current in an inhomogeneous accelerating section. The self-consistent problem of transverse motion of particles in the field of dipole mode excited by the beam is solved. The distribution of dipole field near...
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| Cite this: | Estimation of BBU threshold current in an inhomogeneous accelerating section / M.I. Ayzatsky, K.Yu. Kramarenko, V.V. Mytrochenko // Problems of atomic science and tecnology. — 2020. — № 3. — С. 86-91. — Бібліогр.: 12 назв. — англ. |
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| author | Ayzatsky, M.I. Kramarenko, K.Yu. Mytrochenko, V.V. |
| author_facet | Ayzatsky, M.I. Kramarenko, K.Yu. Mytrochenko, V.V. |
| citation_txt | Estimation of BBU threshold current in an inhomogeneous accelerating section / M.I. Ayzatsky, K.Yu. Kramarenko, V.V. Mytrochenko // Problems of atomic science and tecnology. — 2020. — № 3. — С. 86-91. — Бібліогр.: 12 назв. — англ. |
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| description | In this work we describe the procedure of estimation of regenerative beam blowup threshold current in an inhomogeneous accelerating section. The self-consistent problem of transverse motion of particles in the field of dipole mode excited by the beam is solved. The distribution of dipole field near the structure axis and the energy variation of beam particles were calculated before simulating the transverse dynamics. The threshold current for S-band industrial linac designed for radiation sterilization of medical products is estimated.
Описано процедуру оцінки порогового струму поперечної нестійкості пучка в неоднорідній прискорювальній секції. Вирішується самоузгоджена задача поперечного руху частинок у полі дипольної моди, яка збуджується пучком. Розподіл дипольного поля поблизу осі структури і зміна енергії частинок були розраховані до моделювання поперечної динаміки. Проведено оцінку порогового струму для промислового лінійного прискорювача S-діапазону, призначеного для радіаційної стерилізації медичної продукції.
Описана процедура оценки порогового тока поперечной неустойчивости пучка в неоднородной ускоряющей секции. Решается самосогласованная задача поперечного движения частиц в поле дипольной моды, которая возбуждается пучком. Распределение дипольного поля вблизи оси структуры и изменение энергии частиц были рассчитаны до моделирования поперечной динамики. Проведена оценка порогового тока для промышленного линейного ускорителя S-диапазона, предназначенного для радиационной стерилизации медицинской продукции.
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| first_indexed | 2025-12-01T15:02:42Z |
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ISSN 1562-6016. ВАНТ. 2020. №3(127) 86
ESTIMATION OF BBU THRESHOLD CURRENT
IN AN INHOMOGENEOUS ACCELERATING SECTION
M.I. Ayzatsky, K.Yu. Kramarenko, V.V. Mytrochenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine
E-mail: kramer@kipt.kharkov.ua
In this work we describe the procedure of estimation of regenerative beam blowup threshold current in an inho-
mogeneous accelerating section. The self-consistent problem of transverse motion of particles in the field of dipole
mode excited by the beam is solved. The distribution of dipole field near the structure axis and the energy variation
of beam particles were calculated before simulating the transverse dynamics. The threshold current for S-band in-
dustrial linac designed for radiation sterilization of medical products is estimated.
PACS: 29.20.Ej
INTRODUCTION
The phenomenon referred to as beam blowup (BBU)
was discovered due to observation of beam pulse short-
ening at the exit of accelerator at a pulse current above a
certain critical value. It was found that pulse shortening
is caused by beam destruction (particles are lost on
disks) with preceded asymmetric growth of beam trans-
verse dimensions. This BBU instability had been con-
vincingly identified as interaction between the beam and
the dipole deflecting mode, which is the next pass band
above the accelerating mode. A regenerative BBU (R-
BBU) is the BBU expected for a single-section linac.
Significant efforts, both experimental and theoretical,
were undertaken in order to study the detailed interac-
tion between the dipole mode and the beam and to de-
velop the methods of suppressing the R-BBU instability.
The excitation of dipole backward waves by the con-
tinuous beam in homogeneous travelling wave acceler-
ating sections is considered in Refs. [1 - 5]. The
bunched beams and accelerating sections with losses are
considered in Refs. [6 - 8]. Using different approaches
the quantitative characteristics of R-BBU phenomenon
such as the growth rate or e-folding time (inverse of
growth rate) and the threshold (starting) current were
evaluated. The R-BBU instability in tapered accelerat-
ing sections (the dipole modes are trapped) also was
investigated, see Refs. [5, 9, 10]. The computed de-
pendence of dipole mode growth rate versus beam cur-
rent in SLAC injector is presented in Ref. [9]. The for-
mula for estimating the R-BBU starting current was
obtained by using the standing-wave analysis: the con-
dition for oscillation is that the power extracted from the
beam be equal to the power dissipated in the structure
walls, see Refs. [5, 10]. The estimated value of starting
current for 10 MeV 25 kW industrial electron linac is
presented in Ref. [11]. It was noted that although the
obtained value of starting current of 2 A is a rough
estimation, but a great margin is secured for 400 mA
operation.
In this work we describe the procedure of estimation
of regenerative beam blowup threshold current in an
inhomogeneous accelerating section. The self-consistent
problem of transverse motion of particles in the field of
dipole mode excited by the beam is solved. The distri-
bution of dipole field near the structure axis and the
energy variation of beam particles (energy varies due to
acceleration of particles) were calculated before simu-
lating the transverse dynamics.
1. PROBLEM FORMULATION
A one-sectional electron linac with beam energy up
to 10 MeV (average beam power is 20 kW) is under
development in «Accelerator» Science and Research
Establishment (see Ref. [12]). Linac is designed for ra-
diation sterilization of medical products. The operating
frequency opf is 2856 GHz. Travelling wave accelerat-
ing section with integrated buncher (phase shift per cell
at operating frequency is 2/3) is the inhomogeneous
(tapered) disk-loaded waveguide. To estimate the R-
BBU threshold current we study the possibility of expo-
nential growth of dipole mode amplitude:
, 0th e , (1)
where is the growth rate of dipole mode amplitude.
The value of the threshold current is obtained from the
condition 0 . If 0 , then the considered dipole
mode is not excited.
The frequencies and quality factors of the considered
dipole eigenmodes are listed in Table 2. In Figs. 1, 2 the
on-axis distributions of transverse components of elec-
tric and magnetic fields for the 1st and 5th eigenmodes
are shown.
Fig. 1. Ex on-axis distribution of the 1st and 5th dipole
modes (Im(Ex) 0)
The dependence of the energy of beam particles ver-
sus the longitudinal coordinate z at the initial part of the
accelerating section is shown in Fig. 3. This dependence
is obtained by simulating the longitudinal dynamics of
the beam particles in accelerating field.
ISSN 1562-6016. ВАНТ. 2020. №3(127) 87
Fig. 2. Hy on-axis distribution of the 1st and 5th dipole
modes (Re(Hy) 0)
Fig. 3. Dependence of beam energy versus
the longitudinal coordinate z at the initial part
of the accelerating section
The region of accelerating section of length L (see
Figs. 1-3) includes drift tube, input coupler and 12 cells
of integrated travelling wave buncher. Since the consid-
ered dipole eigenmodes are trapped (see Figs. 1, 2), fur-
ther we will consider the region of length L as a cavity
(resonant volume) in which dipole modes can be excited
by the beam.
The case of parametric resonance will not be consid-
ered because for each of the five modes the following
condition is fulfilled: 1.5res opf f .
We made several assumptions:
– the longitudinal component of the beam velocity is
much greater than the transverse one and beam current
density has only longitudinal component jz;
– the influence of dipole mode on longitudinal mo-
tion of particles is negligible and longitudinal dynamics
of the beam is assumed to be determined;
– the amplitude of dipole mode do not change during
the fly-time of one particle;
– at the entrance to the accelerating section the elec-
tron beam is continuous.
The interaction of beam particles with each of the
five dipole modes was considered.
2. EXCITATION OF THE DIPOLE MODE
BY THE BEAM AND EQUATIONS
OF PARTICLE TRANSVERSE MOTION
The electromagnetic field of dipole mode is
,
,
res
res
E A t E
H iB t H
(2)
where A t and B t are the time-dependent coeffi-
cients, resE
and resH
are the spatial distributions of
electric and magnetic field of dipole mode, respectively.
The equations for A t and B t have the form
2
2
2
2
2
2
1 ,
.
res
res
res res
r
res
re
es
res res
s
dA
Q dt
Jd B dB B
Q dt Nd
d A dJA
N dtdt
t
(3)
Here res and Qres are the dipole mode resonant fre-
quency and quality factor, respectively, Nres is the dipole
mode norm * *
0 0res res res res res
V V
N E E dV H H dV
,
,z res z
V
J j E dV , zj is the beam current density and
,res zE is the longitudinal component of electric field.
We shall consider a particular polarization of dipole
eigenmode in the x-direction for which the longitudinal
component of electric field near the axis (in the vicinity
of electron stream) is of the form
, 0
0
, ,res z z
xE F z r
r
(4)
where 0,zF z r is the longitudinal field distribution at
0x r , 0y , and 0r is the characteristic transverse
displacement.
Expression for current density can be written as
2
,
b
z
res
z k k k
k
Ij
N
t x x t y y t z z t
(5)
where bI is the beam current, N is the number of parti-
cles on the dipole mode period.
It will be convenient to introduce the local time. The
dimensionless Lagrange time for kth particle is
0,
0
k res k res k c
dt t
, (6)
where 0,kt is the moment of time when kth particle en-
ters the cavity, z c , c is the velocity of light,
z L is dimensionless longitudinal coordinate and
res
c
L
c
. (7)
The solution of the system of Eqs. (3) is presented in
the following form:
1 2 1
1 2 1
( ) ( ) ( )
1 2 1
( ) ( ) ( )
1 2 1
2Re ,
2Re ,
i t i t i ta a a
i t i t i tb b b
A h t e h t e h t e
B h t e h t e h t e
(8)
where , , *
2 1
a b a bh h and
ISSN 1562-6016. ВАНТ. 2020. №3(127) 88
2
1,2
11
2 2
res
res
res res
i
Q Q
. (9)
Then from the system of Eqs. (3) we obtain the sys-
tem of equations for ( )
1
ah t and ( )
1
bh t :
1
1
( )
1
( )
1
,
2
.
2
a i t
res res
b i t
res
dh t e
dt i
dh t e J
dt i
N d
N
dJ
t
(10)
Consider the values of ( )
1
ah t and ( )
1
bh t at the
moments of time
p rest T p , (11)
where resT is the dipole mode period and p = 1, 2, 3, ... .
Then from (10) we obtain:
1
1
1
1
( ) ( )
1 1 1
( ) ( )
1 1 1
1 ,
2
1 .
2
p
p
p
p
t
i ta a
p p
res t
t
i tb b
p p
res t
res
h t h t dt e
i
h t h t dt e J
d
iN
dJ
N t
(12)
We introduce the following notation
2, ( , )
1
p res rest Qa b a b
p ph h t e . (13)
For complex amplitudes a
ph and b
ph the following
system of difference equations is obtained
1
1
( ) ( )
1
( ) ( )
1
1 ,
2
1 .
2
p
res res
p
p
res res
p
t
Q i ta a
p p
res t
t
Q i tb b
p p
res
res
t
h h e dt e
i
h h e dt
dJ
e J
iN
N dt
(14)
As res res resi t i t i t
res
dJ de e J i e J
dt dt
, then
1
1
1
( ) ( )
1
1
( ) ( )
1
1
2
1 ,
2
1 .
2
p
res res
p
res p res p
p
res res
p
t
Q i ta a
p p
t
i t i t
p p
res
t
Q i tb b
p p
res t
res
res
h h e dt e J
e J t e J t
i
h h e dt e J
iN
N
N
(15)
Only particles that leave the cavity to the moment in
time pt are taken into account when calculating values of
( )a
ph , ( )b
ph . Finally we obtain the following system of dif-
ference equations for complex amplitudes ( )a
ph and ( )b
ph :
, ,
1
1
, ,
1
( ) ( )
1
0
1
, ,
1 1
1 2
1
( ) ( )
1
01
1
,
.
res k
res k
Np
Q ia a
p p
N pNp
end end
k n k n
k k
N p N p
k n z n
k
N
Np
Q ib b
p
p
a
k n z n
k
N
p
p
h h e G e dx
x x
h h e
F
iG
xiG e dF
(16)
Here , 0k n kx x r is the normalized transverse dis-
placement of k-th particle,
,
end
k nx is the normalized trans-
verse displacement of k-th particle at the exit of the cav-
ity, , maxz n z zF F F and
2
maxb
sr
z
rees
G
N
LI F
f N
,
24
end en
a
re
d
b z
res s
zI F
f
G
N N
, (17)
where end
z is the longitudinal velocity of the particle
at the exit of the cavity, end
zF is zF at the exit of the
cavity.
It should be noted that for 1p the fourth term on
the right-hand side of the 1st equation in Eqs. (16) is
absent and ( , )
0 0a bh .
To determine the dependence of transverse dis-
placement ,k nx on for each particle, the system of
equations for dipole mode amplitudes (16) should be
completed by the equations of transverse motion. As
relativistic factor depends only on the longitudinal
component of the beam velocity, the system of equa-
tions for transverse motion of k-th particle has the fol-
lowing form:
( ) ( )
1, ,
( ) ( )
1 0
, ,
Re
Re
,
,
k
k
ia re
p xk x k x
ib im
p y
k n k x
h e Edq qd K
d d
h e Z H
K
dx q
K
d
(18)
where z c and 21 1 are the functions of
(see Fig. 3), ( )re
xE and ( )im
yH are the on-axis distribu-
tion of electric and magnetic fields, respectively, Z0 is
the vacuum resistance;1 ( 1)N p k Np and
, 0
,
,0
k x
k x
z
q
, (19)
0
2
0
2
e
L e
K
m c
, 0
0 0
L
K
r
. (20)
Here ,0z is the longitudinal velocity of the particles
at the entrance of the cavity, 0 ,0z c ,
2
0 01 1 , me and e are the electron mass and
charge, consequently.
To solve the equations of transverse motion, the fol-
lowing initial conditions are used: , 0 0k xq ,
1, 0 0k nx , 1, 0 1nx .
Since the complex amplitudes of dipole mode ( )
1
a
ph
and ( )
1
b
ph do not change during the fly-time of the parti-
cle and k is not a function of ,k nx , ,k nq (see Eq. (6)),
the system of Eqs. (18) is linear.
A program for numerical solution of systems of
Eqs. (16) and (18) has been written. It was tested on
dipole eigenmodes of pillbox cavity for which the ana-
lytical expression of R-BBU threshold current can be
obtained.
ISSN 1562-6016. ВАНТ. 2020. №3(127) 89
3. PILLBOX CAVITY. TEST CALCULATION
Consider pillbox cavity of radius b and length d.
Beam particles are passing at the axis of the cavity with
constant longitudinal velocity 0 . The electromagnetic
field distribution of x-polarized TM1mp-mode near the
axis is
0
cos sin ,
2 2
cos ,
2
z x
res
y
kk xE e A t k z e A t k z
k
H e B t k z
Z ck
(21)
where mk b , k p d , 0, 1, 2 ...p ,
2 2
res c k k is the resonant frequency, m is the
m-th root of the Bessel function 1J .
Applying the Fourier transform to the left and right
parts of equations in system (3), we obtain the following
system of equations for amplitudes of Fourier harmon-
ics A and B :
2 2
2 2
,
,
res res
res
res res
res
res
res res
i JB B B
Q N
i i JA A A
Q N
(22)
where
2
0 0, 2 2
04
p res
res m
m
b
N db J
c
,
0,
2, 0,
1, 0p
p
p
and
cos
2 2
t
z
V
ij dVkJ x k z e dt
, (23)
1
2
x x d
, (24)
0 0 ,
t
b
z l
I
j dt t t t z
S
. (25)
It follows from the system of Eqs. (22) that
resA i B .
To obtain x from Eq. (24) the following equation
of transverse motion of the particle should be solved
with zero initial conditions:
22
2 2
0 0
0
2
sin
2
cos ,
2
i
e
i res
kd ed x
A e p
kd m
B e p
c k
(26)
where 0 , 0 0t is the particle injection
phase at frequency , 0d is the particle fly-
angle at frequency ; z d is dimensionless longi-
tudinal coordinate.
After some transformations we obtain
1
0
s
2
,co
res
res
res
res
ib
J
N B
d
I k d
e x p
BN
(27)
where 0ix x e
.
The following characteristic equation for is ob-
tained from the system of Eqs. (22):
2 2res
res
res
i
Q
. (28)
We seek the solution of characteristic Eq. (28) in the
form:
res . (29)
Then using Eqs. (27) – (29) we obtain (in linear ap-
proximation):
4 4 3
0 0
2 2
0, 0 0
Im
.
2
res
re
b re
ms
s
p m
I Z k
f
Q
c
J E
(30)
Here 2
0 0eE m c e is the beam energy in volts,
0 0 c and
( ) ( )1 cos 1 1 sin
2
p p
c s
p p
f
k k
, (31)
where
0
resd
,
2
( )
22
2
01
1
pp
s
p
k
, (32)
2 2
0
2 2 4
( 0)
32
21
1
p p pp
c
p
k
, p
p
. (33)
If 0f the instability of the considered dipole
mode is absent at any current.
We obtain the expression for threshold current from
the Eq. (30) under the condition 0 :
2 2 4
0, 0 0
4 4
0 02
p m m re
t
s
s
h
reQ
E
I
Z cf
J
k
. (34)
Fig. 4. Dependence of vs current for TM110 (b=4 cm,
d=20 cm) and TM111 (b=4 cm, d=17.8 cm) modes
in pillbox cavity at beam energy of 1 MeV;
solid line – Eq. (30), circ. – systems of Eqs. (16), (18)
The dependencies of versus current for TM110 and
TM111 modes in pillbox cavity at electron beam energy
of 1 MeV are shown in Fig. 4. These dependencies are
obtained from numerical solutions of Eq. (30) and sys-
tems of Eqs. (16), (18), where
ISSN 1562-6016. ВАНТ. 2020. №3(127) 90
1ln lnb b
p p
res
h h
T
. (35)
Threshold current values are given in Table 1.
Table 1
R-BBU threshold current for dipole modes in pillbox
cavity (b=4 cm) at beam energy of 1 MeV
Threshold current, A
TM110-mode TM111-mode
d, cm
Eq.(34) Eqs.(16), (18) Eq.(34) Eqs.(16), (18)
17.8 6.04 5.92
20.0 2.06 2.07
As one can see from Fig. 4 and Table 1 different
procedures give R-BBU quantitative characteristics that
are in good agreement.
4. SIMULATION
According to the problem formulation we have been
performed the numerical calculations of the systems of
Eqs. (16), (18) for the first five dipole modes which are
trapped in the initial part of the accelerating section (see
Figs. 1, 2). The dependencies of versus current are
shown in Fig. 5. From Fig. 5 it follows that for the first
mode 1 0 at thI I . For all other modes 2 5 0 at
any beam current. Thus only the first dipole mode can
be excited by the beam in the accelerating section (beam
energy variation is shown in Fig. 3). The R-BBU
threshold current value is shown in Table 2.
Table 2
R-BBU threshold current for dipole modes
in the accelerating section
d. m. fres, GHz Qres Ith, A
1 3.785729 9561.5 6.3
2 3.821361 9841.1
3 3.871112 10113
4 3.932035 10507
5 3.991609 11049
The estimated value of threshold current is 6 A.
Since the value of operating current is 400 mA there is a
small possibility to obtain the unstable beam dynamics.
Fig. 5. Dependence of vs current for dipole modes
in the cavity
CONCLUSIONS
The procedure of estimation of R-BBU threshold
current in tapered accelerating sections is presented. The
estimated value of threshold current for electron linac
which is under development in «Accelerator» Science
and Research Establishment is 6.3 A.
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Article received 19.02.2020
ISSN 1562-6016. ВАНТ. 2020. №3(127) 91
ОЦЕНКА ПОРОГОВОГО ТОКА ПОПЕРЕЧНОЙ НЕУСТОЙЧИВОСТИ ПУЧКА
В НЕОДНОРОДНОЙ УСКОРЯЮЩЕЙ СЕКЦИИ
Н.И. Айзацкий, Е.Ю. Крамаренко, В.В. Митроченко
Описана процедура оценки порогового тока поперечной неустойчивости пучка в неоднородной уско-
ряющей секции. Решается самосогласованная задача поперечного движения частиц в поле дипольной моды,
которая возбуждается пучком. Распределение дипольного поля вблизи оси структуры и изменение энергии
частиц были рассчитаны до моделирования поперечной динамики. Проведена оценка порогового тока для
промышленного линейного ускорителя S-диапазона, предназначенного для радиационной стерилизации
медицинской продукции.
ОЦІНКА ПОРОГОВОГО СТРУМУ ПОПЕРЕЧНОЇ НЕСТІЙКОСТІ ПУЧКА
В НЕОДНОРІДНІЙ ПРИСКОРЮВАЛЬНІЙ СЕКЦІЇ
М.І. Айзацький, К.Ю. Крамаренко, В.В. Митроченко
Описано процедуру оцінки порогового струму поперечної нестійкості пучка в неоднорідній прискорюва-
льній секції. Вирішується самоузгоджена задача поперечного руху частинок у полі дипольної моди, яка збу-
джується пучком. Розподіл дипольного поля поблизу осі структури і зміна енергії частинок були розрахова-
ні до моделювання поперечної динаміки. Проведено оцінку порогового струму для промислового лінійного
прискорювача S-діапазону, призначеного для радіаційної стерилізації медичної продукції.
|
| id | nasplib_isofts_kiev_ua-123456789-194532 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-01T15:02:42Z |
| publishDate | 2020 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Ayzatsky, M.I. Kramarenko, K.Yu. Mytrochenko, V.V. 2023-11-27T12:14:56Z 2023-11-27T12:14:56Z 2020 Estimation of BBU threshold current in an inhomogeneous accelerating section / M.I. Ayzatsky, K.Yu. Kramarenko, V.V. Mytrochenko // Problems of atomic science and tecnology. — 2020. — № 3. — С. 86-91. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 29.20.Ej https://nasplib.isofts.kiev.ua/handle/123456789/194532 In this work we describe the procedure of estimation of regenerative beam blowup threshold current in an inhomogeneous accelerating section. The self-consistent problem of transverse motion of particles in the field of dipole mode excited by the beam is solved. The distribution of dipole field near the structure axis and the energy variation of beam particles were calculated before simulating the transverse dynamics. The threshold current for S-band industrial linac designed for radiation sterilization of medical products is estimated. Описано процедуру оцінки порогового струму поперечної нестійкості пучка в неоднорідній прискорювальній секції. Вирішується самоузгоджена задача поперечного руху частинок у полі дипольної моди, яка збуджується пучком. Розподіл дипольного поля поблизу осі структури і зміна енергії частинок були розраховані до моделювання поперечної динаміки. Проведено оцінку порогового струму для промислового лінійного прискорювача S-діапазону, призначеного для радіаційної стерилізації медичної продукції. Описана процедура оценки порогового тока поперечной неустойчивости пучка в неоднородной ускоряющей секции. Решается самосогласованная задача поперечного движения частиц в поле дипольной моды, которая возбуждается пучком. Распределение дипольного поля вблизи оси структуры и изменение энергии частиц были рассчитаны до моделирования поперечной динамики. Проведена оценка порогового тока для промышленного линейного ускорителя S-диапазона, предназначенного для радиационной стерилизации медицинской продукции. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Beam dynamics Estimation of BBU threshold current in an inhomogeneous accelerating section Оцінка порогового струму поперечної нестійкості пучка в неоднорідній прискорювальній секції Оценка порогового тока поперечной неустойчивости пучка в неоднородной ускоряющей секции Article published earlier |
| spellingShingle | Estimation of BBU threshold current in an inhomogeneous accelerating section Ayzatsky, M.I. Kramarenko, K.Yu. Mytrochenko, V.V. Beam dynamics |
| title | Estimation of BBU threshold current in an inhomogeneous accelerating section |
| title_alt | Оцінка порогового струму поперечної нестійкості пучка в неоднорідній прискорювальній секції Оценка порогового тока поперечной неустойчивости пучка в неоднородной ускоряющей секции |
| title_full | Estimation of BBU threshold current in an inhomogeneous accelerating section |
| title_fullStr | Estimation of BBU threshold current in an inhomogeneous accelerating section |
| title_full_unstemmed | Estimation of BBU threshold current in an inhomogeneous accelerating section |
| title_short | Estimation of BBU threshold current in an inhomogeneous accelerating section |
| title_sort | estimation of bbu threshold current in an inhomogeneous accelerating section |
| topic | Beam dynamics |
| topic_facet | Beam dynamics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/194532 |
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