Influence of normal and anomalous dopler effects on development of beam-plasma instability
The influences of normal and anomalous Dopler effects on development of a beam-plasma Cherenkov instability in the linear approximation is investigated. It is shown, that normal Dopler effect influences only on an absolute instability, leading to suppression of backward wave. The anomalous Dopler ef...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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nasplib_isofts_kiev_ua-123456789-788892025-02-09T13:38:20Z Influence of normal and anomalous dopler effects on development of beam-plasma instability Вплив нормального й аномального ефектів доплера на розвиток пучково-плазмової нестійкості Влияние нормального и аномального эффектов доплера на развитие пучково-плазменной неустойчивости Kuzelev, M.V. Rukhadze, A.A. Plasma electronics The influences of normal and anomalous Dopler effects on development of a beam-plasma Cherenkov instability in the linear approximation is investigated. It is shown, that normal Dopler effect influences only on an absolute instability, leading to suppression of backward wave. The anomalous Dopler effect influences not only on absolute, but also on convection instabilities and under the certain conditions it may lead to complete suppression of Cherenkov beamplasma instability. У лінійному наближенні досліджуються впливи нормального й аномального ефектів Доплера на розвиток пучково-плазмової нестійкості Черенкова в подовжньо обмежених системах. Показано, що нормальний ефект Доплера впливає лише на абсолютну нестійкість. Він призводить до непропускання зустрічної хвилі у визначеній області частот, зриваючи тим самим абсолютну нестійкість. Аномальний же 114 ефект впливає не тільки на абсолютну, але і на конвективну нестійкість і може у визначених умовах цілком задавити пучково-плазмову нестійкість Черенкова. В линейном приближении исследуются влияния нормального и аномального эффектов Доплера на развитие пучково-плазменной черенковской неустойчивости в продольно ограниченных системах. Показано, что нормальный эффект Доплера влияет лишь на абсолютную неустойчивость. Он приводит к непропусканию встречной волны в определенной области частот, срывая тем самым абсолютную неустойчивость. Аномальный же эффект влияет не только на абсолютную, но и на конвективную неустойчивость и может в определенных условиях полностью задавить черенковскую пучково-плазменную неустойчивость. 2005 Article Influence of normal and anomalous dopler effects on development of beam-plasma instability / M.V. Kuzelev, A.A. Rukhadze // Вопросы атомной науки и техники. — 2005. — № 1. — С. 110-113. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 52.35.-g https://nasplib.isofts.kiev.ua/handle/123456789/78889 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Plasma electronics Plasma electronics |
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Plasma electronics Plasma electronics Kuzelev, M.V. Rukhadze, A.A. Influence of normal and anomalous dopler effects on development of beam-plasma instability Вопросы атомной науки и техники |
| description |
The influences of normal and anomalous Dopler effects on development of a beam-plasma Cherenkov instability in the linear approximation is investigated. It is shown, that normal Dopler effect influences only on an absolute instability, leading to suppression of backward wave. The anomalous Dopler effect influences not only on absolute, but also on convection instabilities and under the certain conditions it may lead to complete suppression of Cherenkov beamplasma instability. |
| format |
Article |
| author |
Kuzelev, M.V. Rukhadze, A.A. |
| author_facet |
Kuzelev, M.V. Rukhadze, A.A. |
| author_sort |
Kuzelev, M.V. |
| title |
Influence of normal and anomalous dopler effects on development of beam-plasma instability |
| title_short |
Influence of normal and anomalous dopler effects on development of beam-plasma instability |
| title_full |
Influence of normal and anomalous dopler effects on development of beam-plasma instability |
| title_fullStr |
Influence of normal and anomalous dopler effects on development of beam-plasma instability |
| title_full_unstemmed |
Influence of normal and anomalous dopler effects on development of beam-plasma instability |
| title_sort |
influence of normal and anomalous dopler effects on development of beam-plasma instability |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2005 |
| topic_facet |
Plasma electronics |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/78889 |
| citation_txt |
Influence of normal and anomalous dopler effects on development of beam-plasma instability / M.V. Kuzelev, A.A. Rukhadze // Вопросы атомной науки и техники. — 2005. — № 1. — С. 110-113. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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| first_indexed |
2025-11-26T09:33:19Z |
| last_indexed |
2025-11-26T09:33:19Z |
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1849844940510593024 |
| fulltext |
PLASMA ELECTRONICS
INFLUENCES OF NORMAL AND ANOMALOUS DOPLER EFFECTS
ON DEVELOPMENT OF BEAM-PLASMA INSTABILITY
M.V. Kuzelev and A.A. Rukhadze 1
Physics Department of MSU, 119992, Leninskie Gory, Moscow, Russia;
1 Prokhorov’s General Physics Institute RAS, Moscow, Russia
The influences of normal and anomalous Dopler effects on development of a beam-plasma Cherenkov instability in the
linear approximation is investigated. It is shown, that normal Dopler effect influences only on an absolute instability,
leading to suppression of backward wave. The anomalous Dopler effect influences not only on absolute, but also on
convection instabilities and under the certain conditions it may lead to complete suppression of Cherenkov beam-
plasma instability.
PACS: 52.35.-g
The fundamental mechanisms of beam-plasma
instability, which is a basic for plasma relativistic
microelectronics [1], are the single particle and collective
Cherenkov effects, or in other words the Tomson and the
Ruman regimes of stimulated Cherenkov radiation. In the
limit of small beam density the resonance condition for
single particle Cherenkov instability looks as:
ω = Kz u, (1)
where ω -is a frequency, Кz- a longitudinal wave number,
u-beam velocity. The only single particle regime of
Cherenkov instability was realized in experiments [1].
Moreover, in experiments the external magnetic field is
usually strong and Larmour frequency 0
e
eB
mc
Ω = is
much higher than plasma frequency
24 p
p
e n
m
π
ω = ,
where Bо- is a strength of magnetic field, np-is a plasma
density. By this reason the most of theoretical
investigations were carried out under the assumption, that
Bo is infinite. At the same time, in the recent experiments
[2] it was shown, that the beam-plasma microwave
sources are efficiently working even when the Larmor
and Langmuir frequencies are of one order. Theoretically
it was predicted [3] and experimentally it was confirmed
[2] that the frequency spectrum of Cherenkov radiation in
a plasma waveguide practically does not depend on the
strength of magnetic field. But in the finite magnetic field
the new resonances and new mechanisms of beam-plasma
interaction arise. They are known as normal and
anomalous Dopler effects and take place when[4]:
e
zK uω
γ
Ω= ± (2)
110 Problems of Atomic Science and Technology. 2005. № 1. Series: Plasma Physics (10). P. 110-113
where γ=1 −u2
c2 −
1
2 - is the energy relativistic factor
of beam electrons. The resonances (1) and (2) are quite
different, but in spite of this it is possible of mutual
influence of Dopler and Cherenkov instabilities to each
other. It will be shown bellow, that normal Dopler effect
leads to forbidden of backward wave with Kz<0, exited by
the stimulated Cherenkov radiation (or when the
conditions (1) for Kz>0 and (2) for Kz<0 are satisfied
simultaneously). This may lead to suppression of
feedback coupling in a beam-plasma oscillator and even
to break its working. As far anomalous Dopler effect it
leads to increasing transverse of velocity of beam
electrons and decreasing of longitudinal velocity and
finally to violation of Cherenkov resonance condition (1)
and complete suppression of beam-plasma instability. It is
obvious, that the problem of influence of Dopler effects
on Cherenkov beam-plasma instability may be settled
only in the frame of general nonlinear theory.
Nevertheless in this report we restrict ourselves by
consideration this problem in linear approximation on the
basis of dispersion equation and qualitative analyzes of
nonlinear processes.
Let us now discuss the restrictions of linear
approximation. According to the conditions of real
experiments [2], we consider a cylindrical waveguide
with radius R, to be filled up by thin annular cylindrical
beam and plasma layers with:
Δb , Δ p << r br pR (3)
Here rp and rb –are the mean radiuses of layers, Δp and
Δb –their thicknesses. In fig.1 a principal scheme of beam-
plasma Cherenkov microwave source [2] is presented.
One of most important condition, which simplifies the
problem, is:
ω << c
Δp
(4)
In this limit it was predicted theoretically [3] and
confirmed in experiments [2], that the frequency of
excited waves does not depend on magnetic field.
Therefore the process of Cherenkov radiation may be
considered in infinite magnetic field. At the same time for
considering of Dopler effects the finite strength of
magnetic field must be taken into account. This
complicates the problem. But if the beam density is small,
nb
2n p
1
3 1
γ << 1 , (5)
111
Fig.1. A principal scheme of beam-plasma Cherenkov microwave source:1 - metallic waveguide with radius R; 2 –
plasma and beam layers with main radiuses rp and rb and thicknesses ∆ p and b∆ ; 3 – collector
-25 -20 -15 -10 -5 0 5 10 15 20 25
-15
-12,5
-10
-7,5
-5
-2,5
0
2,5
5
7,5
10
12,5
15
Fig.2.
then for simplification we can the use well known
perturbation theory.
Оmitt the details of calculations we write here the
dispersion equation for cable (symmetric) wave, excited
by electron beam in a plasma waveguide:
( ) ( )
2 2 1 2 2 3
2 2
2 2
2
2
p b p b
p dp ch
e z
z
G G
K uK u
ω ω γ ω ω γ
ω
ωω
γ
− −
− Ω = +
Ω −− −
(6)
The first term in the right side of this equation
describes Dopler effects and the second term-Cherenkov
effect. In (6)
24 b
b
e n
m
πω = is the Langmuir
frequency of beam electrons, the quantities
2
0 02 2 20 0 0
0 02 2
0 0 0 0
( ) ( )1, ( )
( ) ( )
p
p p p p p
p p p
K x rx K x Rr I x r
K K I x r I x R
ω
й щ
Ω = = ∆ −к ъ
к ъл ы
(7)
determine the frequency Ω p and transverse wave number
Kp of exited cable wave,
2
2 2
0 2
p
zx K
c
Ω
= − ,
2 2
2 21 0 0 0
0 02 2 2 2
0 0 0 0
( ) ( ),
( ) ( )
b b b b b b
dp ch
p p p p p p p p
r I x r r I x rG x Q G x
r K I x r r K I x r
∆ ∆= =
∆ ∆
, (8)
where
2
2(1 )p
z
uQ
c K u
Ω
= − .
The results of numerical solutions of (6)-(7), which
will be discussed in the next section, are presented in
Fig.2-4. Here we well give the growth rates of
Cherenkov and anomalous Dopler instabilities and their
analysis. From (6) it follows:
1
2 2 3
3
1
2 2 2
,
1 3 1 ,
2 2
.
2
p
ch p b
e
dp p b
p e
Gi for Cherenkov
Gi for anomalous Dopler
ω ω δω δω
ω ω
γ
δω
ω ω
→ + = Ω +
+ − Ω =
− Ω Ω
(9)
For normal Dopler effect there is no instability in the
beam-plasma system. As we can see from Fig.2-4
suppression of backward cable wave with Kz<0 takes
place. This means that when
2
e
p γ
ΩΩ = (10)
the backward cable wave can’t propagate. This cuts off
feedback coupling in a beam-plasma oscillator and breaks
its work. At the same time normal Dopler effect does not
influence the amplifiers work.
112
As for anomalous Dopler effect, when the second
condition (2) is satisfied, the beam-plasma system
becomes unstable with growth rate (8). It leads to
increasing of transverse velocity of beam electrons and
decreasing of longitudinal one. As a result the resonance
condition (1) may violate and Cherenkov beam-plasma
instability stops. In this sense anomalous Dopler effect
may turned out to be catastrophic for both beam-plasma
oscillator and amplifier. Of course, it is possible only if
the external magnetic field is weak and growth rate of
Cherenkov instability is less, then Dopler instability.
Let us now demonstrate the above statements by
discussing the numerical solutions of equation (6)
presented in Fig.2-4. The invariable parameters of the
system taken from the experiments [2] are: radius of
waveguide R=2cm., main radius of beam rb=1сm, Δb = Δp
=0,1 сm, Langmuir frequency of beam electrons
ωb=2∙10+10 с-1 and velocity u=2∙6∙1010 сm/с (γ=2). The
cyclotron Ω e and plasma ωp frequencies and main radius
of plasma rp, were varying. In the figures the
dependencies ω(Kz) in units 1010с-1 and wave numbers Кz
in сm-1 are presented.
In Fig.2 the case of strong magnetic field is shown: Ω
e=10∙1010с-1 аnd ωb=6∙1010с-1. We see 2 regions of
instability for Kz≥ 0, which are marked by vertical lines
«а», «b», «с» and Кz=0 (picture is asymmetric relative to
co-ordinates): between the Кz=0 and «а» the instability is
113
-25 -20 -15 -10 -5 0 5 10 15 20 25
-15
-12,5
-10
-7,5
-5
-2,5
0
2,5
5
7,5
10
12,5
15
Fig.3
-25 -20 -15 -10 -5 0 5 10 15 20 25
-15
-12,5
-10
-7,5
-5
-2,5
0
2,5
5
7,5
10
12,5
15
Fig.4.
stipulated by Cherenkov effect, whereas between «b» and
«с»-by anomalous Dopler effect.
In Fig.3 the opposite case of relatively weak magnetic
field is shown: Ω e =6∙1010с-1, and ωp=10∙1010c-1.. Here
again we have 2 regions of instability, but now they are
wider, that corresponds to expressions (9). In Fig.3 as in
Fig.2 the Cherenkov and Dopler instability regions are
separate, but now their growth rates become of one order.
The further decreasing of magnetic field leads to overlap
the instability regions. It is obvious, that in this case only
in the frame of nonlinear theory problem may be solved.
At the same time one can easily suppress the
anomalous Dopler instability by separation of beam and
plasma layers. In the cases, which are presented in Fig.2
and 3 the layers were very close, whereas in the case
presented in the Fig.4 they were separated- the mean
radius of plasma was rp =1,2 сm,. or the clearance
between the layers was 1mm.We see very essential
changes: the instability regions become very narrow,
especially for Dopler instability. This means that by
separation of beam and plasma layers one can
successfully suppress anomalous Dopler instability.
Finally let us discuss very shortly the influence of
normal Dopler effect on the beam-plasma instability. As it
was noticed above and as it is seen in Fig.2-4 the normal
Dopler effect leads to suppression of backward wave with
Kz<0. In Fig.2-4 the frequency range of normal Dopler
effect is marked by arrow. We see that this region is very
narrow near the resonance frequency (10). Nevertheless
this phenomenon may suppress the undesirable modes in
a beam-plasma amplifier.
From the above analysis one can make the following
conclusions:
1. Normal and anomalous Dopler effects can
essentially influence the character of development of
Cherenkov beam-plasma instability, and thus the work of
Cherenkov plasma sources of the microwave radiation
(generators and amplifiers) only in conditions of moderate
magnetic fields when Larmor frequency electrons is
comparable with plasma frequency.
2. Normal Dopler effect can suppress the backward
cable plasma wave excited by a beam, at performance of a
condition (10) and by that to break generation, having
suppressed a feedback in the microwave generator.
However, action of normal Dopler effect is shown in very
narrow range of frequencies of generation near frequency
(10). On the forward wave the normal effect of influence
does not render and consequently it does not influence
work of plasma Cherenkov amplifier.
3. Anomalous Dopler effect is one of dangerous
instabilities of beam-plasma system and consequently its
influence on Cherenkov instability can appear more
dramatic. The anomalous Dopler effect leads to increasing
of transverse velocity of beam electrons and consequently
to full failure of Cherenkov beam-plasma instability. In
this sense anomalous Dopler effect can affect essentially
work of plasma microwave sources like generators and
amplifiers.
REFERENCES
1. M.V. Kuzelev, A.A. Rukhadze, P.S. Strelkov. Plasma
Relativistic Microwave Electronics. M.: Publ. House
MSTU, 2002, p. 544.
2. P.S. Strelkov, D.K. Ulyanov // Plasma Phys. Reports
(26). 2000, №4, p.329
3. I.N. Kartashov,M.V. Kuzelev, A.A. Rukhadze
//Plasma Phys. Reports (30). 2004, №1,30, р.60
4. M.V. Kuzelev, A.A. Rukhadze. Basic of Plasma Free
Electron Lasers. Edition Frontieres, France, 1995,
p.246.
ВЛИЯНИЕ НОРМАЛЬНОГО И АНОМАЛЬНОГО ЭФФЕКТОВ ДОПЛЕРА НА
РАЗВИТИЕ ПУЧКОВО-ПЛАЗМЕННОЙ НЕУСТОЙЧИВОСТИ
М.В. Кузелев, А.А. Рухадзе
В линейном приближении исследуются влияния нормального и аномального эффектов Доплера на
развитие пучково-плазменной черенковской неустойчивости в продольно ограниченных системах. Показано,
что нормальный эффект Доплера влияет лишь на абсолютную неустойчивость. Он приводит к непропусканию
встречной волны в определенной области частот, срывая тем самым абсолютную неустойчивость. Аномальный
же эффект влияет не только на абсолютную, но и на конвективную неустойчивость и может в определенных
условиях полностью задавить черенковскую пучково-плазменную неустойчивость.
ВПЛИВ НОРМАЛЬНОГО Й АНОМАЛЬНОГО ЕФЕКТІВ ДОПЛЕРА НА РОЗВИТОК
ПУЧКОВО-ПЛАЗМОВОЇ НЕСТІЙКОСТІ
М.В. Кузельов, A.А. Рухадзе
У лінійному наближенні досліджуються впливи нормального й аномального ефектів Доплера на
розвиток пучково-плазмової нестійкості Черенкова в подовжньо обмежених системах. Показано, що
нормальний ефект Доплера впливає лише на абсолютну нестійкість. Він призводить до непропускання
зустрічної хвилі у визначеній області частот, зриваючи тим самим абсолютну нестійкість. Аномальний же
114
ефект впливає не тільки на абсолютну, але і на конвективну нестійкість і може у визначених умовах цілком
задавити пучково-плазмову нестійкість Черенкова.
115
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