Deformation of the plasma concentration profile due to the modulated electron beam
Report is devoted to the numerical simulation of the electron beam with the longitudinal modulation moving along the concentration gradient of the planarly stratified plasma with the initially linear profile (one-dimensional model). Nonlinear modification of the plasma concentration profile due to t...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2002 |
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| Мова: | English |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2002
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| Цитувати: | Deformation of the plasma concentration profile due to the modulated electron beam / I.O. Anisimov, O.A. Borisov, O.I. Kelnyk, S.V. Soroka // Вопросы атомной науки и техники. — 2002. — № 5. — С. 107-109. — Бібліогр.: 8 назв. — англ. |
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Anisimov, I.O. Borisov, O.A. Kelnyk, O.I. Soroka, S.V. 2015-03-23T09:32:57Z 2015-03-23T09:32:57Z 2002 Deformation of the plasma concentration profile due to the modulated electron beam / I.O. Anisimov, O.A. Borisov, O.I. Kelnyk, S.V. Soroka // Вопросы атомной науки и техники. — 2002. — № 5. — С. 107-109. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 52.40.Mj https://nasplib.isofts.kiev.ua/handle/123456789/78921 Report is devoted to the numerical simulation of the electron beam with the longitudinal modulation moving along the concentration gradient of the planarly stratified plasma with the initially linear profile (one-dimensional model). Nonlinear modification of the plasma concentration profile due to the HF electric field excited by the beam is studied. The stationary case and initial problem were calculated. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Plasma electronics Deformation of the plasma concentration profile due to the modulated electron beam Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Deformation of the plasma concentration profile due to the modulated electron beam |
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Deformation of the plasma concentration profile due to the modulated electron beam Anisimov, I.O. Borisov, O.A. Kelnyk, O.I. Soroka, S.V. Plasma electronics |
| title_short |
Deformation of the plasma concentration profile due to the modulated electron beam |
| title_full |
Deformation of the plasma concentration profile due to the modulated electron beam |
| title_fullStr |
Deformation of the plasma concentration profile due to the modulated electron beam |
| title_full_unstemmed |
Deformation of the plasma concentration profile due to the modulated electron beam |
| title_sort |
deformation of the plasma concentration profile due to the modulated electron beam |
| author |
Anisimov, I.O. Borisov, O.A. Kelnyk, O.I. Soroka, S.V. |
| author_facet |
Anisimov, I.O. Borisov, O.A. Kelnyk, O.I. Soroka, S.V. |
| topic |
Plasma electronics |
| topic_facet |
Plasma electronics |
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2002 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Article |
| description |
Report is devoted to the numerical simulation of the electron beam with the longitudinal modulation moving along the concentration gradient of the planarly stratified plasma with the initially linear profile (one-dimensional model). Nonlinear modification of the plasma concentration profile due to the HF electric field excited by the beam is studied. The stationary case and initial problem were calculated.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/78921 |
| citation_txt |
Deformation of the plasma concentration profile due to the modulated electron beam / I.O. Anisimov, O.A. Borisov, O.I. Kelnyk, S.V. Soroka // Вопросы атомной науки и техники. — 2002. — № 5. — С. 107-109. — Бібліогр.: 8 назв. — англ. |
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| first_indexed |
2025-11-25T20:39:24Z |
| last_indexed |
2025-11-25T20:39:24Z |
| _version_ |
1850528944157097984 |
| fulltext |
DEFORMATION OF THE PLASMA CONCENTRATION PROFILE DUE TO
THE MODULATED ELECTRON BEAM
I.O.Anisimov1, O.A.Borisov, O.I.Kelnyk2, S.V.Soroka
Taras Shevchenko Kyiv National University, Radio Physics Faculty,
64, Volodymyrs’ka St., 01033, Kyiv, Ukraine, 1ioa@rpd.univ.kiev.ua, 2 oles@univ.kiev.ua
Report is devoted to the numerical simulation of the electron beam with the longitudinal modulation moving along the
concentration gradient of the planarly stratified plasma with the initially linear profile (one-dimensional model). Nonlin-
ear modification of the plasma concentration profile due to the HF electric field excited by the beam is studied. The sta-
tionary case and initial problem were calculated.
PACS: 52.40.Mj
1. INTRODUCTION
The problem of the Langmuir waves excitation in
non-uniform plasma due to the modulated electron beam
is studied during a long time [1–3] because this phe-
nomenon is typical for plasma electronics. But only the
asymptotic solutions for stationary problem of Langmuir
waves excitation were obtained previously. However, for
the large magnitudes of the beam current the electric field
excited in the local plasma resonance region (LPRR) can
modify the plasma concentration profile [5-8].
2. MODEL DESCRIPTION AND BASIC EQUA-
TIONS
Isotropic warm (Te≠0) weakly inhomogeneous
planarly stratified plasma is considered (density depends
on z only). Near the plasma resonance region the depen-
dence of plasma density upon z is linear:
np(z)=n0(1+z/L), (1)
where L is the characteristic length of the plasma non-uni-
formity, n0=np(0) is the plasma density in the LPRR where
the modulation frequency ω of the electron beam coin-
cides with the local Langmuir frequency. Modulated elec-
tron beam moves along z-axis. Its alternative current den-
sity can be represented as
)v()exp(),( om ztzitijtzj ±Θ−= κω , (2)
where vo is the beam velocity, Θ(x) is the step function
that describes the beam front motion, κ=ω/vo is the wave
number.
The system under consideration is described by a
set of Maxwell equations, continuity equation and lin-
earized equation of plasma electrons’ motion:
−−−=
=+
=−+
.v3
;0
;0414
2 umnn
z
mEen
t
umn
z
un
t
n
c
uen
t
E
c
j
c
oTeoo
o
o
νδ
∂
∂
∂
∂
∂
∂
∂
∂ δ
π
∂
∂π
(3)
Here no and δn are the time averaged plasma
density and its deviation, u and ν are the plasma electrons’
instantaneous velocity and collision frequency, respective-
ly.
Excluding u and δn from (3) gives
( )
,
v
v34
v3
2
2
2
22
2
2
−+−=
=+−+
z
j
t
jj
t
E
z
EEz
t
E
o
Te
Tep
∂
∂
∂
∂νπ
∂
∂ν
∂
∂ω
∂
∂
(4)
where ωp
2(z)=4πn0(z)e2/m is the local Langmuir plasma
frequency. Inhomogeneous wave equation (4) describes
the excitation of Langmuir waves in the warm inhomoge-
neous plasma by the modulated electron beam. After in-
troducing dimensionless variables
mj
E
π
ωε
4
= ,
mj
jj =~ , tωτ = ,
o
z
v
ω
ς = , 2
2
v
3~
o
TevT = ,
ω
νν =~ ,
o
poL
v
ωλ = , ( ) 2)( ωως pn = ,
equation (4) takes the form
,~~)~1(
~
~~)( 2
2
2
2
+±−=
=+−+
jTj
Tn
ν
∂ τ
∂
∂ τ
∂ εν
∂ ς
ε∂ες
∂ τ
ε∂
(5)
where upper and lower signs correspond to the positive
and negative beam velocity respectively. The deformation
of the plasma concentration profile can be described by
the inhomogeneous equation for ion-acoustic waves:
[ ],)(~ 2
02
2
2
2
2
2
ες
∂ ς
∂Λ
∂ ς
δ∂
∂ τ
δ∂ nnnM =− (6)
When transient processes caused by the electron
beam front are finished and only oscillations on the modu-
lation frequency remain than equations set (5-6) can be re-
duced to the single equation
Problems of Atomic Science and Technology. 2002. № 5. Series: Plasma Physics (8). P. 107-109 107
mailto:2oles@univ.kiev.ua
).exp()~)~1((
)]1)((~1[~ 2
02
2
ςν
εεΛςν
ς
ε
iTi
ni
d
dT
+±=
=−−−+
(7)
In the next section we examine this stationary
regime.
3. LINEAR STATIONARY EXCITATION OF
LANGMUIR WAVES
For δnp=0, t→∞ equation (5) describes the linear
stationary excitation of Langmuir waves by the alternative
current (2). Outside LPRR the solution can be presented
as a superposition of the field of current (2) and Langmuir
wave excited by this current (fig. 1). Conversion of the
modulated electron beam field into Langmuir waves is
most efficient in the vicinity of Cherenkov resonance
point where the Langmuir wave phase velocity is equal to
the electron beam velocity. Predominantly the accompa-
nying wave is excited. This phenomenon determines the
dependence of the LPRR field magnitude and magnitude
of the Langmuir wave that propagates to subcritical plas-
ma on the direction of the electron beam velocity (see fig.
1a,b). The dependence on the beam velocity direction
vanishes when ~ ~ν > > T .
a
b
Fig. 1. Spatial distribution of electric field in the warm
plasma caused by the modulated electron beam for
~ .T = 0 01 , ωL/vo=10: a – ~ν = 0 ; b – ~ .ν = 0 02
4. TRANSIENT PROCESSES CAUSED BY
THE FOREFRONT OF ELECTRON BEAM
The forefront of electron beam excites the Lang-
muir oscillations in plasma at the local electron plasma
frequencies. For the small collisions’ frequency ( ~ ~ν < < T )
they leak to the subcritical plasma (fig. 2). The motion of
the forefront of the Langmuir wave with the modulation
frequency that is excited in the resonance region can be
observed for this case. As a result the stationary distribu-
tion of electric field is formed (see fig.1).
Fig.2. Electric field excited by electron beam moving into
plasma for ωL/vo=10, ~ .T = 0 01 , ~ .ν = 0 001
5. NONLINEAR DEFORMATION OF THE
CONCENTRATION PROFILE
It was already noticed that for the large magni-
tudes of the beam current the electric field excited in the
LPRR could modify the plasma concentration profile.
a
b
Fig. 3. Electric field (a) and nonlinear deformation of
the concentration profile near the LPRR (b); dashed
line – initial profile, solid line – disturbed profile;
005.0~ =T , Λ=3⋅10–7, ωL/vo=50
The concentration disturbance in the stationary
case can be presented in a form:
( ) 2
0~ εςΛδ nn −= . (8)
Fig.3 shows the disturbance of the plasma con-
centration profile in the LPRR in the stationary case that
is obtained from the numerical solution of equation (7).
Deformation of the plasma concentration profile
strongly depends on the direction of the electron beam ve-
locity. It is more significant for the beams moving into
plasma because the electric field in the LPRR is stronger
for this case (see fig.1).
a
b
Fig.4. Spatial and temporal dependence of the plasma
density variation for positive (a) and negative (b) beam
velocity direction; Λ=3⋅10–4, М=6⋅105, L=10, ν=0.02,
T=0.01
6. EVOLUTION OF THE PLASMA DENSITY
PROFILE DEFORMATION
The nonlinear deformation caused by the elec-
tron beam’s eigen field and Langmuir waves is evolving
in time. Fig. 4 shows the spatial and temporal dependence
of the concentration disturbance. That result was obtained
by the numerical integration of the equations’ set (5-6)
For the case of the positive beam velocity plasma
concentration profile deformation is mostly localized near
the LPRR and the local concentration minimum is formed.
If the beam velocity is negative, main concentration mini-
mum is much less relatively to the previous case, and also
this minimum is shifted toward the subcritical plasma.
REFERENCES
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1447 (1973) (in Russian).
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636-646 (1975) (in Russian).
3. K.P.Shamrai. J. Plasma Phys., 31, #2, 301-311
(1984).
4. I.O.Anisimov, O.A.Borisov. Physica Scripta, 62, #5,
375-380 (2000).
5. V.B.Gildenburg, G.M.Fraiman. ZhETF, 69, #5(11),
1601-1606 (1975) (in Russian).
6. G.J.Morales, Y.S.Lee. Phys. Fluids (1977), 20, #2,
1135-1147.
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#9, 1336-1342 (1989) (in Russian).
I.O.Anisimov1, O.A.Borisov, O.I.Kelnyk2, S.V.Soroka
Taras Shevchenko Kyiv National University, Radio Physics Faculty,
|