INHOMOGENEOUS PLASMA PROFILE DEFORMATION DUE TO THE MODULATED ELECTRON BEAM: COMPUTER SIMULATION VIA PIC METHOD
I.O.Anisimov, O.I.Kelnyk, T.V.Siversky, S.V.Soroka
Taras Shevchenko National University of Kyiv, Radio Physics Faculty
Deformation of the plasma concentration profile due to the field of the incident electromagnetic
wave was often discussed (see, e.g., [1-3]). For the obliquely incident p-polarized waves of the
moderate power concentration profile deformation occurs mainly in the local plasma resonance region
(LPRR). Numerical solution of the initial-boundary problem results to the quasi-periodical
generation of the density cavities in LPRR [1-2]. Solution of the stationary problem results to formation
of the sharp concentration jumps . Relation between these solutions was not clarified .
Similar effects were predicted in  for the case of the modulated electron beam moving
through the inhomogeneous plasma. This problem was studied in  using numerical solution of
the non-linear equations’ set for the electric field and plasma concentration. In this case problem
was solved in the approximation of given beam current.
2. Problem definition and model description
Deformation of the inhomogeneous plasma profile due to the field of the modulated electron
beam was treated in this report. The numerical simulation of this problem was carried out using the
PIC method for the 1D case [7-8]. The initial plasma density profile was linear. The profile of the
current density with the initially harmonic modulation should not be distorted very much at the way
to LPRR. The characteristic length of the inhomogeneity L was chosen from this condition. All the
presented results were obtained for hydrogen plasma.
Typical profiles of the beam electrons’ density for its motion to dense plasma and to lowdensity
plasma are plotted on Fig. 1 a, b, respectively. Character of the processes in the beam (i.e.,
spreading or compression of the electron bunches) is defined by characteristics of plasma (namely,
subcriticality or supercriticality) where beam moved before it reached LPRR.
3. Initial stage of the beam-plasma interaction
Electric field increase in the LPRR took place at the initial stage after the beam injection
started (Fig. 2 a, b). It was accompanied by intensive oscillations of electron concentration. At the
same time Langmuir wave moving from LPRR against the plasma concentration gradient was excited
(see region AA′, Fig. 2 a) according to the prediction of the linear theory [9-11]. Deformation
of the ion concentration profile was observed in the LPRR later. Firstly the short-wave perturbation
(in the scale of the LPRR width) appeared, and then the cavity was formed in the same region. Deformation
of the ion density profile interrupted the Langmuir waves’ excitation.
Local maximum was often formed in the bottom of density cavity. Its position corresponded
to maximum of the HF electric field intensity (ponderomotive force caused by this field is directly
proportional to its intensity gradient). Density gradient of the cavity side from the dense plasma was
larger due to the similar character of spatial distribution of the electric field intensity.
Analytic estimations showed that plasma density profile deformation could be treated as
slow process relatively to electric field excitation in plasma with the given density profile. Consequently,
4. Late stage of interaction for plasma with hot electrons
In plasma with hot electrons (Те>>Ті) the ion-acoustic type perturbations propagated to both
sides from the cavity  (see Fig. 3 a).
The new peaks of electric field appeared on the local maximums of the subcritical plasma
resulting to new cavities formation. This process is demonstrated on Fig. 3 b where space-time distribution
of the electric field absolute value is presented. Line 1 corresponds to the boundary between
subcritical and supercritical plasma. Position of the electric field maximums coincide with
the region of plasma density in the range (0.93÷1.00)nс (black spots). Excitation of strong electric
fields on the local maximums of subcritical plasma (for nc-nmax<<nc) was also demonstrated by
computer simulation for the modeling profiles of plasma density.
Consequently, in the late stages the ion concentration profile in plasma with hot electrons
was strongly indented in the wide region around LPRR due to the short ion-acoustic pulses.
In the isothermal plasma (Те=Ті) due to the strong damping of ion-acoustic waves the cavity
was gradually transformed into the diffused jump of plasma density at the late stage. This jump
moved inside the dense plasma, its velocity was of order of ion thermal velocity. Parameters of concentration
jump correspond to analytic estimations based on the theory similar to .
Simulation demonstrated the similar character of plasma density profile deformation both
for the case of the field excitation by the modulated electron beam and by the uniform pumping
field. This conclusion is valid both for plasma with hot electrons and for isothermal plasma.
Consequently the relation between the stationary solution  and solution of initialboundary
problem [1-2] for inhomogeneous plasma excited by modulated electron beam or incident
p-polarized wave (or uniform pumping field) was clarified. The conditions and way of the stationary
solution formation was also found out.
Earlier it was proposed to use modulated electron beams as electromagnetic waves’ radiators
in the inhomogeneous plasmas (see, e.g., ). One of the most effective mechanisms such of radiation
is transition radiation from LPRR . But effects discussed above can restrain this radiation.
Beam injection duration and corresponding duration of radioemission should not exceed the time of
considerable deformation of the plasma density profile.
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