Evolution of electrons’ distribution function during their interactoin with plasma: numerical simulation
One-dimension numerical simulation of the beam-plasma system was carried out using the big-particles-in-cells method. Instantaneous velocity distribution of the beam electrons depending of the distance from injector was received. The distribution function was found to be oscillating and strongly irr...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2003
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| Цитувати: | Evolution of electrons’ distribution function during their interactoin with plasma: numerical simulation / I.O. Anisimov, S.M. Levitsky, D.V. Sasyuk, T.V. Siversky // Вопросы атомной науки и техники. — 2003. — № 4. — С. 78-80. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-1109902017-01-08T03:03:27Z Evolution of electrons’ distribution function during their interactoin with plasma: numerical simulation Anisimov, I.O. Levitsky, S.M. Sasyuk, D.V. Siversky, T.V. Нелинейные процессы One-dimension numerical simulation of the beam-plasma system was carried out using the big-particles-in-cells method. Instantaneous velocity distribution of the beam electrons depending of the distance from injector was received. The distribution function was found to be oscillating and strongly irregular. This result corresponds with the data of the previous calculations and laboratory experiments. After averaging over the sufficiently long time interval the distribution function becomes smoothed and similar to a plateau that was observed in the laboratory experiments. 2003 Article Evolution of electrons’ distribution function during their interactoin with plasma: numerical simulation / I.O. Anisimov, S.M. Levitsky, D.V. Sasyuk, T.V. Siversky // Вопросы атомной науки и техники. — 2003. — № 4. — С. 78-80. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: 52.35.Mw http://dspace.nbuv.gov.ua/handle/123456789/110990 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Нелинейные процессы Нелинейные процессы |
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Нелинейные процессы Нелинейные процессы Anisimov, I.O. Levitsky, S.M. Sasyuk, D.V. Siversky, T.V. Evolution of electrons’ distribution function during their interactoin with plasma: numerical simulation Вопросы атомной науки и техники |
| description |
One-dimension numerical simulation of the beam-plasma system was carried out using the big-particles-in-cells method. Instantaneous velocity distribution of the beam electrons depending of the distance from injector was received. The distribution function was found to be oscillating and strongly irregular. This result corresponds with the data of the previous calculations and laboratory experiments. After averaging over the sufficiently long time interval the distribution function becomes smoothed and similar to a plateau that was observed in the laboratory experiments. |
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Article |
| author |
Anisimov, I.O. Levitsky, S.M. Sasyuk, D.V. Siversky, T.V. |
| author_facet |
Anisimov, I.O. Levitsky, S.M. Sasyuk, D.V. Siversky, T.V. |
| author_sort |
Anisimov, I.O. |
| title |
Evolution of electrons’ distribution function during their interactoin with plasma: numerical simulation |
| title_short |
Evolution of electrons’ distribution function during their interactoin with plasma: numerical simulation |
| title_full |
Evolution of electrons’ distribution function during their interactoin with plasma: numerical simulation |
| title_fullStr |
Evolution of electrons’ distribution function during their interactoin with plasma: numerical simulation |
| title_full_unstemmed |
Evolution of electrons’ distribution function during their interactoin with plasma: numerical simulation |
| title_sort |
evolution of electrons’ distribution function during their interactoin with plasma: numerical simulation |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2003 |
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Нелинейные процессы |
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http://dspace.nbuv.gov.ua/handle/123456789/110990 |
| citation_txt |
Evolution of electrons’ distribution function during their interactoin with plasma: numerical simulation / I.O. Anisimov, S.M. Levitsky, D.V. Sasyuk, T.V. Siversky // Вопросы атомной науки и техники. — 2003. — № 4. — С. 78-80. — Бібліогр.: 15 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-08T01:28:39Z |
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| fulltext |
EVOLUTION OF ELECTRONS’ DISTRIBUTION FUNCTION DURING
THEIR INTERACTOIN WITH PLASMA: NUMERICAL SIMULATION
I.O. Anisimov, S.M. Levitsky, D.V. Sasyuk, T.V. Siversky
Taras Shevchenko National University of Kyiv, Radio Physics Faculty, Kyiv, Ukraine, ioa@u-
niv.kiev.ua
One-dimension numerical simulation of the beam-plasma system was carried out using the big-particles-in-cells
method. Instantaneous velocity distribution of the beam electrons depending of the distance from injector was re-
ceived. The distribution function was found to be oscillating and strongly irregular. This result corresponds with the
data of the previous calculations and laboratory experiments. After averaging over the sufficiently long time interval
the distribution function becomes smoothed and similar to a plateau that was observed in the laboratory experi-
ments.
PACS: 52.35.Mw
1. INTRODUCTION
The quasilinear theory created at the beginning of
the 1960-th [1-3] has foreseen that electron beam hav-
ing interacted with plasma acquire the electron’s energy
distribution function with the plateau due to the elec-
trons’ diffusion in the velocities’ space. Some experi-
mental investigations carried out in the same time con-
firmed fully this prediction [4-7]. But some disagree-
ment between the prerequisites of the quasilinear theory
and conditions of the experiments gave rise to the un-
certainty [8]. Later analytic calculation [9] and numeri-
cal simulation [10-12] showed that the instantaneous
distribution function should be highly non-monotonic.
Experimental measurements confirmed this conclusion
[13]. Consequently the problem of the agreement be-
tween the theory and experiment has not yet been
solved.
This article presents results of the numerical simula-
tion of the electron beam interaction with plasma. Tem-
poral and spatial evolution of the instantaneous distribu-
tion function of the beam electrons was studied. Aver-
aging of this function over the sufficiently long time in-
terval gives the possibility to explain the experimental
results mentioned above.
2. PROGRAM DESCRIPTION
One-dimension numerical simulation of the beam-
plasma interaction was carried out using the big-parti-
cles-in-cells method (without external magnetic field).
The modified PDP1 package [14] was used. The pro-
gram is based on the 32-bit Windows operating system.
There are no implicit restrictions for the number of large
particles. This number is specified for every type of par-
ticles separately. The intermediate results of the simula-
tion can be preserved for the posterior treatment.
The developed diagnostic modules give the possibil-
ity to observe 3D plots of the spatial dependencies of
the instantaneous and averaged velocity distribution
functions for all types of particles (particularly for the
beam electrons and for the plasma electrons separately).
3. PARAMETERS SELECTION
The data mentioned below was obtained for such pa-
rameters of the beam-plasma system:
− density of the background plasma was ne=1014m-3,
corresponding Langmuir electron frequency was ω
p=5.65108s-1;
− initial beam velocity was v0=107m/s, corresponding
electrons’ energy was 275eV;
− current density of the beam was j=0.1A/m2, corre-
sponding ratio of the beam density to plasma densi-
ty was nb/ne=6⋅10-4.
Distance from the injector and the time were nor-
malized as x′=xωp/v0 and t′=ωp t/2π, respectively. The
time point t′ corresponds to the beginning of the beam
injection.
These parameters correspond to some real experi-
mental conditions and give the clearest picture of the ef-
fects observed in our numerical simulation.
4. PHASE PORTRAIT OF THE BEAM-
PLASMA SYSTEM
Fig.1 represents the phase portrait of the beam inter-
acting with plasma for t′=63.
Near the point of origin one can observe the linear
growth of the oscillations’ magnitude. Further linear
regime converts quickly into the regime of the wave
front tipping over. For larger distances from injector the
picture becomes more complicated. At the distance x′
>100 the electrons’ trajectories get mixed up and one
could not follow their course.
This result is similar to the numerous results of the
previous simulation (see, e.g., [15]).
Fig.1. Phase portrait of the beam electrons for t′=63
5. INSTANTANEOUS VELOCITY DIS-
TRIBUTION FUNCTION
Fig.2 demonstrates the three-dimensional plot of in-
stantaneous velocity distribution function of the beam
electrons for t′=63 from their injection in the plasma.
The datum lines correspond to the coordinate x′, nor-
malized velocity v′=2πv/v0 and number of electrons n
per the phase cell ∆x⋅∆v (in the arbitrary units). This di-
agram is a kind of 3D analog of the phase portrait
shown on fig.1. It shows the number of particles corre-
sponding to each cell of the phase space.
Fig.2. Instantaneous velocity distribution function of the
beam electrons depending on the distance from injector
for t′=63
The results presented on fig.2 are partially similar to
the data obtained in [8].
Observing on the display the time evolution of the
plot similar to fig.3 for the initial stage of the beam in-
jection one can see the motion of the forefront from the
injector. The highest peaks correspond to the velocities
less then v0 because the beam looses its energy during
the interaction with plasma.
The cross-sections of the plot shown on fig.2 (for
different x’) are presented on fig.3a-d.
a b
c d
Fig.3. Instantaneous velocity distribution function of the
beam electrons at different distances from injector (noted
at the plots) for t′=63
At a small distances from injector one can observe
the single peak that corresponds to the apart lines on the
phase portrait (fig.3a). After the wave front tripping
over the number of peaks is increased (see fig.3b-c). For
larger distances from injector separate peaks unite into
the continuos indented curve. The intensity of the spec-
trum is decreased with the distance from injector, and it
expands both to the sides of larger and smaller veloci-
ties.
The general picture of the velocity distribution evo-
lution is similar to the calculation for the stationary case
[9] and to the experimental results [13].
6. THE AVERAGED DISTRIBUTION
FUNCTION
Fig.4 shows the spatial dependence of the velocity
distribution function of the beam electrons averaged
over some tens of the Langmuir frequency periods. One
can see that the time averaging results to the smoothing
of the distribution function (compare fig.4 and fig.3).
Fig.4. The averaged velocity distribution function of
the beam electrons depending on the distance from
injector
The results presented on fig.4 are very much similar
to the experimental measurements [4-7]. At the large
distances from injector one can see some kind of
plateau. But this plateau is not formed due to the quasi-
linear beam relaxation in plasma. It appears due to the
time lag of the measuring elements that result to the
time averaging of the data.
7. CONCLUSION
Simulation of the beam-plasma interaction was car-
ried out using bib-particles-in-cells method. The spatial
and temporal evolution of the instantaneous velocity
distribution function of the beam electrons was studied.
The results obtained coincide to the results of the previ-
ous simulation and laboratory experiments.
Averaging of the instantaneous velocity distribution
over the interval of some tens of the Langmuir frequen-
cy periods results to the smoothing and formation of
some kind of plateau at the large distances from injec-
tor. This effect explains the plateau formation for
monokinetic electron beams that was observed in the
laboratory experiments [4-7].
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PACS: 52.35.Mw
Fig.1. Phase portrait of the beam electrons for t=63
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