The nonlinear theory of interaction of ion streams with virtual cathode of the high-current relativistic electron beam
Nonlinear LF dynamics of interaction of the longitudinal ion stream with the electron virtual cathode formed by high-current relativistic electron beam which is injected in the cylindrical drift chamber is investigated. Both the case of coincided radii of electron and ion beams, and the case when...
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| Cite this: | The nonlinear theory of interaction of ion streams with virtual cathode of the high-current relativistic electron beam / V.A. Balakirev, I.N. Onishchenko, N.I. Onishchenko // Вопросы атомной науки и техники. — 2004. — № 2. — С. 21-23. — Бібліогр.: 3 назв. — англ. |
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Balakirev, V.A. Onishchenko, I.N. Onishchenko, N.I. 2015-03-31T08:41:22Z 2015-03-31T08:41:22Z 2004 The nonlinear theory of interaction of ion streams with virtual cathode of the high-current relativistic electron beam / V.A. Balakirev, I.N. Onishchenko, N.I. Onishchenko // Вопросы атомной науки и техники. — 2004. — № 2. — С. 21-23. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 52.40.Mj https://nasplib.isofts.kiev.ua/handle/123456789/79318 Nonlinear LF dynamics of interaction of the longitudinal ion stream with the electron virtual cathode formed by high-current relativistic electron beam which is injected in the cylindrical drift chamber is investigated. Both the case of coincided radii of electron and ion beams, and the case when the radius of ion beam exceeds the radius of electron beam are considered. Досліджена нелінійна НЧ динаміка взаємодії повздовжнього іонного потоку з електронним віртуальним катодом, утвореним потужнострумовим релятивістським електронним пучком, який інжектується в циліндричну камеру дрейфу. Розглянуто як випадок співпадаючих радіусів електронного та іонного пучків, так і випадок, коли радіус іонного пучка перевищує радіус електронного пучка. Исследована нелинейная НЧ динамика взаимодействия продольного ионного потока с электронным виртуальным катодом, образованным сильноточным релятивистским электронным пучком, который инжектируется в цилиндрическую камеру дрейфа. Рассмотрен как случай совпадающих радиусов электронного и ионного пучков, так и случай, когда радиус ионного пучка превышает радиус электронного пучка. Research was carried out under the support of STCU project №1569. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Сильноточные импульсные ускорители The nonlinear theory of interaction of ion streams with virtual cathode of the high-current relativistic electron beam Нелінійна теорія взаємодії іоних потоків з віртуальним катодом сильноточного релятивістського електроного пучка Нелинейная теория взаимодействия ионных потоков с виртуальным катодом сильноточного релятивистского электронного пучка Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
The nonlinear theory of interaction of ion streams with virtual cathode of the high-current relativistic electron beam |
| spellingShingle |
The nonlinear theory of interaction of ion streams with virtual cathode of the high-current relativistic electron beam Balakirev, V.A. Onishchenko, I.N. Onishchenko, N.I. Сильноточные импульсные ускорители |
| title_short |
The nonlinear theory of interaction of ion streams with virtual cathode of the high-current relativistic electron beam |
| title_full |
The nonlinear theory of interaction of ion streams with virtual cathode of the high-current relativistic electron beam |
| title_fullStr |
The nonlinear theory of interaction of ion streams with virtual cathode of the high-current relativistic electron beam |
| title_full_unstemmed |
The nonlinear theory of interaction of ion streams with virtual cathode of the high-current relativistic electron beam |
| title_sort |
nonlinear theory of interaction of ion streams with virtual cathode of the high-current relativistic electron beam |
| author |
Balakirev, V.A. Onishchenko, I.N. Onishchenko, N.I. |
| author_facet |
Balakirev, V.A. Onishchenko, I.N. Onishchenko, N.I. |
| topic |
Сильноточные импульсные ускорители |
| topic_facet |
Сильноточные импульсные ускорители |
| publishDate |
2004 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Нелінійна теорія взаємодії іоних потоків з віртуальним катодом сильноточного релятивістського електроного пучка Нелинейная теория взаимодействия ионных потоков с виртуальным катодом сильноточного релятивистского электронного пучка |
| description |
Nonlinear LF dynamics of interaction of the longitudinal ion stream with the electron virtual cathode formed by
high-current relativistic electron beam which is injected in the cylindrical drift chamber is investigated. Both the case of
coincided radii of electron and ion beams, and the case when the radius of ion beam exceeds the radius of electron beam
are considered.
Досліджена нелінійна НЧ динаміка взаємодії повздовжнього іонного потоку з електронним віртуальним
катодом, утвореним потужнострумовим релятивістським електронним пучком, який інжектується в
циліндричну камеру дрейфу. Розглянуто як випадок співпадаючих радіусів електронного та іонного пучків, так і
випадок, коли радіус іонного пучка перевищує радіус електронного пучка.
Исследована нелинейная НЧ динамика взаимодействия продольного ионного потока с электронным
виртуальным катодом, образованным сильноточным релятивистским электронным пучком, который
инжектируется в цилиндрическую камеру дрейфа. Рассмотрен как случай совпадающих радиусов электронного
и ионного пучков, так и случай, когда радиус ионного пучка превышает радиус электронного пучка.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/79318 |
| citation_txt |
The nonlinear theory of interaction of ion streams with virtual cathode of the high-current relativistic electron beam / V.A. Balakirev, I.N. Onishchenko, N.I. Onishchenko // Вопросы атомной науки и техники. — 2004. — № 2. — С. 21-23. — Бібліогр.: 3 назв. — англ. |
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| first_indexed |
2025-11-24T11:37:29Z |
| last_indexed |
2025-11-24T11:37:29Z |
| _version_ |
1850845406173331456 |
| fulltext |
THE NONLINEAR THEORY OF INTERACTION OF ION STREAMS
WITH VIRTUAL CATHODE OF THE HIGH-CURRENT RELATIVISTIC
ELECTRON BEAM
V.A. Balakirev, I.N. Onishchenko, N.I. Onishchenko
NSC KIPT, Academic str. 1, Kharkov, 61108, Ukraine
Ph. 0572-356623, e-mail: onish@kipt.kharkov.ua
Nonlinear LF dynamics of interaction of the longitudinal ion stream with the electron virtual cathode formed by
high-current relativistic electron beam which is injected in the cylindrical drift chamber is investigated. Both the case of
coincided radii of electron and ion beams, and the case when the radius of ion beam exceeds the radius of electron beam
are considered.
PACS: 52.40.Mj
1. INTRODUCTION
Low-frequency (LF) interaction of ions with high-
current relativistic electron beams (HREB) of a large
duration (about µs and more) plays the important role in
beam dynamics and is explicitly exhibited in great
number of electron devices and facilities on HREB base.
Among them there are microwave generators with
microsecond duration, in particular, vircators [1],
pasotrons [2] and facilities for collective acceleration of
ions by quasi-static electric fields of a virtual cathode
(VC) [1], which is formed in the drift chamber when
HREB current exceeds the limiting vacuum current. The
purpose of the present paper is clearing up of the physical
pattern of electron VC evolution at its interaction with an
ion stream. Electron VC is a high-nonlinear multi-stream
formation therefore the examination of its behaviour at
interaction with ion streams is possible only with
engaging of numerical methods. The basic results of the
carried out examinations are presented in the given paper.
2. PHYSICAL MODEL.
BASIC EQUATIONS
The drift chamber represents a semi-infinite metal
pipe with the end face in the range of injection of high-
current relativistic electron beam. From the end face of
the drift chamber ( 0=z ) two tubular beams of charged
particles are injected: HREB and nonrelativistic ion beam.
The system is located in the homogeneous magnetic field,
directed along the axis of the system. Electrons of HREB
are magnetized, and the effect of magnetic field on ions
motion is neglected. Such situation, as a rule, is
implemented in the experimental conditions [3]. The
electron beam current exceeds the limiting vacuum
current so in the drift chamber electron VC is formed.
Injected ions will hit the area of electron VC and in part
or completely neutralize the field of VC space charge.
For theoretical description of nonlinear dynamics of
ion stream interaction with electron VC the following
approach is used. Let us define the indefinitely thin ring
macroscopic particles both for electrons, and for ions. The
laws of ring motion of macroscopic particles are
described by the time dependence of the longitudinal
coordinate ( )e,ie,Lie,i t,tzz 0= and the radius ( )iLii t,trr 0=
(for electrons only the longitudinal coordinate). The index
“ e ” corresponds to electrons of HREB, and “ i
”corresponds to ions, t is the current time, iet ,0 is the
injection time of corresponding particles. The electrical
potential of the system of electron and ion macroscopic
particles (Green function) is defined from the Laplace
equation with the corresponding right part. The boundary
conditions consist of vanishing of the electrical potential
on walls of the drift chamber. Summing (integration) over
all macroscopic particles, which are in volume of the drift
chamber in current time, gives the expression for the
electrical potential of the electron - ion system:
( )
( )
( )
( )
( )
∑
∫
−
−
∫
−
−
∞
=
−−
+−
−
−
−−
+−
−=Φ
1
0
000
0
000
1
02
n
t
Li
Li
i
Li
nii
t
Le
Le
eee
Le
n
nn
n
zz
a
n
zz
a
n
dt
a
rJtI
zz
a
n
zz
a
n
dttI
a
rJ
J
a
rJ
a
e
e
e
e
λ
λ
λ
λ
λ
λ
λλ
λ
(1)
The integration in (1) is executed over moments of
injection time of electrons and ions, nλ are the root of
cylindrical functions ( )xJ 0 , a is the radius of the drift
chamber, i,eI is the current of electron and ion beams.
Knowing the electrical potential of the system of electron
and ion beams, it is easy to find the longitudinal and
radial component of self-consistent electric field which
will be determined by the position of all electrons and
ions injected in the drift chamber from the moment 0=t
up to the current t . Further, having substituted
expressions for electric field in motion equations for
electrons and ions, the set of self-consistent equations
may be obtained.
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 2.
Series: Nuclear Physics Investigations (43), p.21-23. 21
mailto:onish@kipt.kharkov.ua
3. THE RESULTS OF THE NUMERICAL
ANALYSIS
Numerical simulation of ion relaxation of electron
VC has been carried out for the case of electron and ion
beams with equal initial radiuses, as well as for the case
when the initial ion radius exceeds the HREB radius.
Calculations were carried out for the following
parameters of the system: HREB current is 5.6 кА, energy
of electrons is 280 keV, ion beam currents are 370 А and
1.1 кА, energy of ions is 25 keV, radius of the electron
beam is 1.6 сm, radius of the drift chamber is 2.5 cm. The
model ratio of ion mass to electron mass is equal to 40.
The ratios of particles number per unit of length of ion
beams to HREB
e
i
N
N are equal 1.5 and 4.5. We will dwell
on the most simple case when HREB and ion stream have
conterminous initial radii at
e
i
N
N =4.5. The numerical
analysis has shown, that in this case the ion virtual anode
(VA) is formed as well as the electron VC in the drift
chamber. Ion VA is located close to the end face of the
drift chamber (on the distance, approximately, 1.5 mm),
therefore it simulates well the plasma emitter of ions with
unlimited emissive capacity. The ion current is limited by
a space charge (the plasma anode). On the initial stage of
the process the electron VC starts to move deep into the
drift chamber under action of the electric field of ion
space charge (see fig.1).
-1 0 1 2 3 4 5 6 7 8 9 10
-1,0
-0,8
-0,6
-0,4
-0,2
0,0
0,2
0,4
0,6
0,8
1,0
400
320
240
160
80
0
80
160
240
320
400
32
1
32
1
p
ot
en
tia
l,
kV
v/
c
z, cm
Fig.1. Phase portrait of electron beam (1–t=1.6 ns;
2–t=2 ns; 3–t=2.4 ns). Electric potential (lower curves
1-3 correspond to above-mentioned moments of time.
Ie=5.6 kA, Re=R0i=1.6 cm, Ni /Ne=4.5
Here behind the electron VC the wave perturbation of
the potential is formed with a length, approximately, 8cm,
containing three spatial periods, which propagates deep
into the drift chamber with the phase velocity essentially
exceeding the velocity of the electron VC and close to the
velocity of the accelerated ion stream. The energy of the
accelerated ions in the field of the electron VC space
charge is about the initial energy of electrons – 280 keV.
As ions accumulate the wave perturbation of potential
damps, the electron VC moves slowly deep into drift
chamber and, eventually, disappears. As a result of the
complex process of electron VC relaxation the double-
flow electron-ion stream is formed in the system.
0 2 4 6 8 10
0,0
0,5
1,0
1,5
2,0
2,5
r,
cm
z,cm
Fig.2 Longitudinal and radial coordinates of ions at the
moment of time. t=2.6 ns, Ie=5.6 kA, Re=1.6 cm,
R0i=2.3 cm, Ni /Ne=1.5
-1 0 1 2 3 4 5 6 7
-0,10
-0,05
0,00
0,05
0,10
0,15
0,20
0,25
v z/c
z,cm
Fig.3 Phase portrait of ions at the moment of time
t=1.6 ns, Ie=5.6 kA, Re=1.6 cm, R0i=2.3 cm, Ni /Ne=1.5
In the case, when the initial radius of ion beam (
cmri 3.2= ) is more than HREB radius, the non-stationary
ion VA is formed at
e
i
N
N =1.5. The position of the ion
VA oscillates with time both in longitudinal, and in radial
directions. Accordingly the angle of ion beam injection
into the drift chamber makes oscillations. Radial motion
of ions in the field of the electron VC space charge leads
to their focusing on the axis of the system (fig.2). In the
focus the positive potential of ions essentially increases
and second ion VA is appeared. At the same time the part
of ions on the leading edge of ion beam gains additional
acceleration in the field of ion VA space charge during its
formation (fig.3).
The maximum energy of accelerated ions at the
leading edge of the ion beam mounts to 400 keV. In the
radial direction HREB and accumulated ions form the
radial potential well in which ions make transversal
oscillations. The phase portrait of ions ( r,vr ) and the
shape of the potential well are presented in fig.4. As the
angle of injection of passed ion current into the drift
chamber oscillates, the position of the ion focus located
22
behind the electron VC, oscillates in the longitudinal
direction. The ion focus approaching to the area of the
electron VC increases the potential there and the electron
VC disappears, that, in turn, leads to ion focus moving
deep into the drift chamber. Thus the electron VC is
recovered. There were observed four such relaxation
oscillations leading to LF modulation of HREB. After that
accumulation of ions in the area of electron VC leads to
VC disappearing. The frequency of HREB modulation for
the chosen model ratio of ion and electron masses is
equal, approximately, 800 МHz. In recalculation on
nitrogen ions this frequency makes 32 MHz.
0,0 0,5 1,0 1,5 2,0 2,5
-0,4
-0,3
-0,2
-0,1
0,0
0,1
0,2
-160
-120
-80
-40
0
40
80
po
te
nt
ia
l,
K
v
V
r/c
r, cm
Fig.4. Phase portrait of ions rvr , and potential
distribution in the point z=0.9 cm in the cross-section of
the drift chamber at the moment of time t=11.6 ns
Ie=5.6 кА, =er 1.6 cm, =ir 2.3 сm, Ni /Ne=1.5
4. CONCLUSIONS
Thus, LF electron VC relaxation under the action of
the ion stream was investigated. The self-consistent
system of the nonlinear integral differential equations,
which describes the above-mentioned process, was
obtained and examined by numerical methods. It is
shown, that in the case of conterminous radii of electron
and ion beams, the ion stream leads to excitation of
coherent wave structure behind the area of the electron
VC. Accumulation of ions in the system eventually leads
to the damping of this structure and forming of laminar
electron-ion flow. If the radius of ion stream exceeds the
HREB radius the pattern of nonlinear interaction of the
ion stream with the electron VC becomes essentially
richer. In the area of the ion stream injection the non-
stationary ion virtual anode is formed, the position of
which oscillates both in the longitudinal, and in the radial
directions. The radial focusing of ions in the area of the
electron VC space charge leads to formation of ion VA in
the area of focusing. The position of ion focus oscillates
with time in the longitudinal direction. At the same time,
the electron VC periodically disappears and appears
again, that is the use of deep LF modulation of HREB
density.
Research was carried out under the support of STCU
project №1569.
REFERENCES
1. A.E.Dubinov, I.J.Kornilova, V.D.Selemir Collective
acceleration of ions in systems with a virtual cathode
// UFN, 2002, v.172, №11, p.1225-1245.
2. Yu. P.Bliokh and G.S.Nusinovich, Fellow, IEEE
Temporal Evolution of Electron Beam Transport in
PASOTRON Microwave Sources // IEEE
Transactions on Plasma Science, v.29, No.6,
December 2001, p.951-959.
3. V.A.Balakirev, A.M.Gorban, I.I.Magda,
V.E.Novikov, I.N.Onishchenko, S.S.Pushkarev,
Collective acceleration of ion-modulated high-current
RAP // Plasma physics, 1997, v.23, No.4, p.350-354.
НЕЛИНЕЙНАЯ ТЕОРИЯ ВЗАИМОДЕЙСТВИЯ ИОННЫХ ПОТОКОВ С ВИРТУАЛЬНЫМ КАТОДОМ
СИЛЬНОТОЧНОГО РЕЛЯТИВИСТСКОГО ЭЛЕКТРОННОГО ПУЧКА
В.А. Балакирев, И.Н. Онищенко, Н.И. Онищенко
Исследована нелинейная НЧ динамика взаимодействия продольного ионного потока с электронным
виртуальным катодом, образованным сильноточным релятивистским электронным пучком, который
инжектируется в цилиндрическую камеру дрейфа. Рассмотрен как случай совпадающих радиусов электронного
и ионного пучков, так и случай, когда радиус ионного пучка превышает радиус электронного пучка.
НЕЛІНІЙНА ТЕОРІЯ ВЗАЄМОДІЇ ІОНИХ ПОТОКІВ З ВІРТУАЛЬНИМ КАТОДОМ
СИЛЬНОТОЧНОГО РЕЛЯТИВІСТСЬКОГО ЕЛЕКТРОНОГО ПУЧКА
В.А. Балакірєв, І.М. Онищенко, М.І. Онищенко
Досліджена нелінійна НЧ динаміка взаємодії повздовжнього іонного потоку з електронним віртуальним
катодом, утвореним потужнострумовим релятивістським електронним пучком, який інжектується в
циліндричну камеру дрейфу. Розглянуто як випадок співпадаючих радіусів електронного та іонного пучків, так і
випадок, коли радіус іонного пучка перевищує радіус електронного пучка.
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 2.
Series: Nuclear Physics Investigations (43), p.21-23. 23
BASIC EQUATIONS
REFERENCES
|