Excitation of lf ion oscillations in the systems with relativistic electron beam
The instability of LF wave in ion flow during acceleration by space charge wave in collective ion accelerator based on high current REB, which is modulated in time and in space, is investigated.
Gespeichert in:
| Datum: | 2005 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2005
|
| Schriftenreihe: | Вопросы атомной науки и техники |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/79791 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Excitation of lf ion oscillations in the systems with relativistic electron beam / V.A. Balakirev, N.I. Onishchenko // Вопросы атомной науки и техники. — 2005. — № 2. — С. 161-163. — Бібліогр.: 4 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-79791 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-797912015-04-05T03:02:30Z Excitation of lf ion oscillations in the systems with relativistic electron beam Balakirev, V.A. Onishchenko, N.I. Plasma electronics The instability of LF wave in ion flow during acceleration by space charge wave in collective ion accelerator based on high current REB, which is modulated in time and in space, is investigated. Досліджена нестійкість НЧ-хвилі у потоці іонів при прискоренні їх хвилями густини заряду у колективному прискорювачі, що базується на сильнострумовому РЕП, який промодульований у часі та просторі. Исследована неустойчивость НЧ-волны в потоке ионов при ускорении их волнами плотности заряда в коллективном ионном ускорителе, базирующемся на сильноточном РЭП, промодулированном во времени и в пространстве. 2005 Article Excitation of lf ion oscillations in the systems with relativistic electron beam / V.A. Balakirev, N.I. Onishchenko // Вопросы атомной науки и техники. — 2005. — № 2. — С. 161-163. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 41.75.Lx, 41.85.Ja, 41.60.Bq http://dspace.nbuv.gov.ua/handle/123456789/79791 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 |
Plasma electronics Plasma electronics |
| spellingShingle |
Plasma electronics Plasma electronics Balakirev, V.A. Onishchenko, N.I. Excitation of lf ion oscillations in the systems with relativistic electron beam Вопросы атомной науки и техники |
| description |
The instability of LF wave in ion flow during acceleration by space charge wave in collective ion accelerator based on high current REB, which is modulated in time and in space, is investigated. |
| format |
Article |
| author |
Balakirev, V.A. Onishchenko, N.I. |
| author_facet |
Balakirev, V.A. Onishchenko, N.I. |
| author_sort |
Balakirev, V.A. |
| title |
Excitation of lf ion oscillations in the systems with relativistic electron beam |
| title_short |
Excitation of lf ion oscillations in the systems with relativistic electron beam |
| title_full |
Excitation of lf ion oscillations in the systems with relativistic electron beam |
| title_fullStr |
Excitation of lf ion oscillations in the systems with relativistic electron beam |
| title_full_unstemmed |
Excitation of lf ion oscillations in the systems with relativistic electron beam |
| title_sort |
excitation of lf ion oscillations in the systems with relativistic electron beam |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2005 |
| topic_facet |
Plasma electronics |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/79791 |
| citation_txt |
Excitation of lf ion oscillations in the systems with relativistic electron beam / V.A. Balakirev, N.I. Onishchenko // Вопросы атомной науки и техники. — 2005. — № 2. — С. 161-163. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT balakirevva excitationoflfionoscillationsinthesystemswithrelativisticelectronbeam AT onishchenkoni excitationoflfionoscillationsinthesystemswithrelativisticelectronbeam |
| first_indexed |
2025-07-06T03:46:15Z |
| last_indexed |
2025-07-06T03:46:15Z |
| _version_ |
1836867729366188032 |
| fulltext |
EXCITATION OF LF ION OSCILLATIONS IN THE SYSTEMS
WITH RELATIVISTIC ELECTRON BEAM
V.A. Balakirev, N.I. Onishchenko
NSC Kharkov Institute of Physics and Technology, Kharkov, Ukraine
The instability of LF wave in ion flow during acceleration by space charge wave in collective ion accelerator based
on high current REB, which is modulated in time and in space, is investigated.
PACS: 41.75.Lx, 41.85.Ja, 41.60.Bq
1. INTRODUCTION
At acceleration of an ion stream in the field of space
charge wave of high-current REB, for example, in the
model of collective ion accelerator [1], alongside with
forward motion ions make low-frequency (LF) transverse
oscillations with frequency ωi=2πne0 e 2/M 1/2 ,
where neo - density of REB, e - electron charge, M -
ion mass (ions are single-charge). It is obvious, that such
double-beam system is unstable concerning excitation of
LF oscillations. As a result of instability development LF
potential wave of electric field will be excited, which, on
the one hand, will destroy the dynamics of resonant
acceleration of ions, and on the another, can reduce in
ejection of ions on the walls of the drift chamber and,
accordingly, in current losses of the accelerated ions. It is
necessary to note, that the study of LF ion processes in
high-current REB of microsecond duration goes out far
off frames of collective ion accelerator physics. LF ion
processes may play also the important role, for example,
in powerful microwave plasma filled generators on the
basis of high-current REB (passotrons [2], vircators [3],
Cherenkov generators [4], etc.)
In the present article outcomes of the theory of LF
instability of ion stream propagated along high-current
REB are presented.
2. THE MAIN PART
The theory of LF transverse ion instability is
developed within the frames of the following model. In
the flat drift chamber, consisting of two parallel ideally
conducting planes, high-current REB and ion stream are
propagated. The REB is homogeneous and completely
fills in the drift chamber. The thickness of an ion stream is
arbitrary with respect to the distance between walls of the
drift chamber (the size of the drift chamber). The ion
stream is symmetric concerning the plane of symmetry of
the drift chamber. The system is located in exterior
magnetic field, directional along the propagation of
particle flows. The effect of magnetic field on ion
movement is neglected.
On the initial stage of interaction of electronic and ion
beams instability is developed and wave amplitude grows
in time to exponential law. At the nonlinear stage phase
displacement of ions in the field of wave reduces in
stabilization of instability. For waves with antisymmetric
distribution of a potential in the transverse direction the
requirement of synchronism between ions and LF wave is
possible only for odd harmonics of the frequency of
transverse ion oscillations
ω=kvi02s1 ωi s=0,1 ,2 . .. ,
where ω− LF wave frequency, kn - its longitudinal
wave number, v0i - longitudinal ion velocity. At s=0
synchronism with first harmonic occurs.
The initial set of equations contains the equation of
excitation for amplitude of LF wave
dCn
dτ
=σ 1
2π∫0
2π
J s a e
−iϑ dϑ 0 , (1)
σ= 1 for fast charge density waves and σ= -1 for
slow waves, and also equation of ion motion in
Lagrangian variables
dϑ
dτ
=μu−[ i
2
C n
s
a
J s
' a e iϑк . с . ] , (2)
a da
dτ
=−1
2
C n sJ s a e
iϑк .с . , (3)
du
dτ
=−1
2
C n J sa e iϑк . с . , (4)
ϑ− phase coordinate of ion, u− the dimensionless
longitudinal velocity, a− the dimensionless amplitude
of transverse ion oscillations,
μ=ω i
k
n2
k
xn2 ωi
2ω
pi 2
b
x i
1
k
xn2 χ ,
where for cherenkov branches χ=
ωb
γ
03/2
k n
k
xn3
and for
cyclotron ones χ=
ω
b2
ωHe γ0
k
xn2
k
xn2k
n2
2 ,
ωb , ωHe - Langmuir and cyclotron frequencies of REB,
γ 0 - the relativistic factor of REB.
In the set of equations (1) - (4) the dimensionless
variables are used.
Cn=ϕ n /ϕ *, τ=t / t *,
u=v iz−v i0 /v∗¿
¿
,
where
Problems of Atomic Science and Technology. Series: Plasma Physics (11). 2005. № 2. P. 161-163 161
ϕ∗¿ M
e
ω pi 2ωi
x i
b
χ
k
xn2
,
t∗¿ 1
ω pi ωi
2
b
x i
1
k
xn2 χ ,
v∗¿
ωi k n
k
xn2
ω pi= 4πe2 ni
M
,
x i− half-thickness of ion stratum, k xn− transverse
wave number.
The set of equations (1) - (4) has integrals
a2
2
−su=Const , (5)
∣Cn∣
2σ
s
1
2π∫0
2π
a2 dϑ 0=Const . (6)
From the first integral it follows, that reduction of the
energy of transverse motion is accompanied by
deceleration of ions and vice-versa. The second integral
reflects the law of energy conservation in the system. In
the case of fast charge density waves (Cherenkov and
cyclotron waves, σ=1 ), which energy is positive,
reduction of the energy of transverse motion reduces in
increase of wave energy. For a slow charge density wave
σ=−1 , which energy is negative, the increase of wave
energy is accompanied by increase of the energy of
transverse motion of ion oscillators.
The right member of the equation for phase coordinate
of ions (2) contains two items, responsible for two
mechanisms of ion grouping in the field of LF wave. The
item proportional to parameter μ takes into account ion
grouping, stipulated by longitudinal motion, and second
item – phase grouping, connected with transverse motion
of ions in the field of LF wave. The indicated set of
nonlinear equations has been solved by numerical
methods for the basic resonance s=1 and the lowest
transverse harmonic n=1 with transverse wave number
k x1=π /b . Initial value of LF wave amplitude is
chosen equal to ∣C1∣=10−2 , number of ions on a
period of LF wave is 300.
Let's analyze outcomes of numerical accounts for the
case μ=0 . Initial value of a τ=0 =a i=π /2 that
corresponds to initial thickness of ion stratum x i=b /2 .
For value a i=π /2 the slow charge density wave which
energy is negative is unstable.
On fig.1 the dependence of the amplitude module of
excited slow charge density wave on time is presented.
Wave amplitude on initial stage of instability
exponentially rises with time, attains maximum value, and
then makes damped oscillations. As it was already noted,
growth of wave amplitude with the negative energy is
accompanied by increase of transverse energy of ion
oscillators a2 or increase of amplitude of transverse
oscillations of ions.
0 50 100 150 200
0,0
0,5
1,0
1,5
2,0
2,5
C
n, a
.u
.
τ, a.u.
a
Fig. 1. The dependence of wave amplitude on time
On fig.2 phase portraits a ,ϑ in various moments
of time are presented. On a phase portrait a ,ϑ the
line a=π corresponds to the wall of the drift chamber.
During numerical calculation ions which hitted on walls
of the drift chamber were output from the system and did
not contribute to amplitude of LF wave. At the stage of
exponential growth of amplitude on the indicated phase
plane the figure such as an ellipse is formed.
In the moment of time τ=31 (fig. 2a)
corresponding to maximum value of amplitude, all ions
have increased the transverse energy in comparison with
initial value. Thus, approximately, 30 % of ions will be
rejected on the walls of the drift chamber. In the point of
minimum of amplitude τ=38 transverse energy of ion
oscillators according to integral (6) is minimal (fig. 2b).
As a whole on a phase plane a ,ϑ two bunches are
formed, which rotate in opposite directions. At major
times (fig. 2c) τ=114 , bunches have complex
multiflow structure. Partial phase mixing of ions reduces
in reduction of phase oscillations of slow REB charge
density wave amplitude.
Above the dynamics of LF slow charge density wave
excitation with negative energy by ion oscillator stream
was investigated. Let us consider the case a i=3π /4 .
For such initial value of half-thickness of ion stratum the
fast REB charge density wave is unstable, which energy
is positive ( σ0 in integral (5)).
At a nonlinear stage of instability the amplitude makes
deep in high-scale regular phase oscillations. Maximum
and especially average value of amplitude are much
lower, than in the case of slow charge density wave.
Though times τ m , during which amplitudes of fast and
slow density charge waves attain the first maxima are
close ( τ m=31 for slow and τ m=30 - for fast charge
density waves) the loss of ions on the walls of the drift
chamber decreased up to 15 %.
162
1 2 3 4 5
2,0
2,2
2,4
2,6
2,8
3,0
3,2
a,
a
.u
.
θ, a.u.
a
0 1 2 3 4 5 6
0,5
1,0
1,5
2,0
2,5
3,0
a,
a
.u
.
b
θ, a.u.
0 1 2 3 4 5 6
0,5
1,0
1,5
2,0
2,5
3,0
a,
a
.u
.
θ, a.u.
c
Fig. 2. Phase portraits of ions a ,ϑ
at instants τ = 31, 38, 114, correspondingly for a, b, c
3. CONCLUSION
Thus, LF instability of ion oscillator stream
concerning excitation of LF REB charge density waves is
investigated. The self-consistent nonlinear set of
equations is obtained, which describes nonlinear
dynamics of ion oscillator instability, formed by ions that
make transverse oscillations in a static electric field of
REB. It is shown, that for slow and for fast charge density
waves the patterns of instability development qualitativly
differ. The slow charge density wave has negative energy.
Due to this, growth of wave amplitude is accompanied by
increase of transverse energy of ion oscillators and,
accordingly, amplitudes of transverse LF oscillations of
ions. The longitudinal velocity of ions also will increase.
In the case of fast charge density wave instability
develops under the traditional scenario. Increase of wave
amplitude accompanies reduction of transverse energy of
ion oscillators and their longitudinal velocity.
REFERENCES
1. V.A.Balakirev, A.M.Gorban, I.I.Magda,
V.E.Novikov, I.N.Onishchenko, S.S.Pushkarev,
Collective acceleration of ions modulated by high-
current REB // Plasma Physics. 1997, v23, N4,
p.350-354.
2. Yu Bliokh, G. Nusinovich, J. Felsteiner, V.
Granatstein // Physical review E. 2002, v.66,
(056503).
3. A. Sabkin, A. Dubinov, V. Zhdanov, et al // Plasma
Phys. Reports. v.23, №4, p.316-322.
163
4. V. Balakirev, N. Karbushev, A. Ostrovskij, Yu.Tkach
Theory of amplifiers and generators based on
relativistic beams. Kiev: “Naukova dumka”, 1993.
ВОЗБУЖДЕНИЕ НИЗКОЧАСТОТНЫХ ИОННЫХ КОЛЕБАНИЙ В СИСТЕМАХ
С РЕЛЯТИВИСТСКИМ ЭЛЕКТРОННЫМ ПУЧКОМ
В.А. Балакирев, H.И. Онищенко
Исследована неустойчивость НЧ-волны в потоке ионов при ускорении их волнами плотности заряда в
коллективном ионном ускорителе, базирующемся на сильноточном РЭП, промодулированном во времени и в
пространстве.
ЗБУДЖЕННЯ НИЗЬКОЧАСТОТНИХ ІОННИХ КОЛИВАНЬ У СИСТЕМАХ
З РЕЛЯТИВИСТСЬКИМ ЕЛЕКТРОННИМ ПУЧКОМ
В.А. Балакірєв, М.І. Онищенко
Досліджена нестійкість НЧ-хвилі у потоці іонів при прискоренні їх хвилями густини заряду у колективному
прискорювачі, що базується на сильнострумовому РЕП, який промодульований у часі та просторі.
164
|