Dynamics Of Beam Instability In A Bounded Volume Of Plasma: Numerical Experiment
V.P. Tarakanov1, E.G.Shustin2
1Institute of Thermal Physics of Extreme States of UIHT of RAS, Moscow
2 Institute of Radio Engineering and Electronics of RAS, Moscow
Keywords: low temperature plasma; electron beams; beam plasma discharge; ion flow; computer simulation.
PACS: 52.40.Mj; 52.65.Rr
Abstract. With the purpose of learning a nature of effect, detected in , of formation in beam plasma discharge of an ion
flow spreading on a normal from a discharge axis with energy, much exceeding a thermal energy of electrons, the numerical
simulation of dynamics of beam instability development in a limited volume of plasma (plasma resonator) at a low magnetic
field is carried out. Is shown, that in such system the accumulation of a field occurs because of a small group velocity of
plasma oscillations, excited by an electron beam. As the corollary, in area occupied by an electron beam, highly nonequilibrium
plasma with mean energy of electrons reaching hundreds of eV is formed. The electrons of this region create an
additional current of electrons from plasma to end plates, that results in increase of a potential of plasma in area occupied by
the beam. The acceleration of a flow of ions on a normal from an axis of a system is a consequence of presence of a potential
gradient between area occupied with the beam and peripheral area of plasma. Essential result of numerical experiment is
stochastic behavior of excited oscillations at temporal development, specific time of transition from regular oscillations to
stochastic ones being much smaller than it is required for development of slower types of instability in the system. It is
possible to guess, that the heating of plasma electrons observed in these numerical experiments using a model of collisionless
plasma, is determined by the mechanism of collisionless attenuation of stochastic oscillations.
The comparison of results of simulation to conclusions of the theory beam instability in a spatially bounded system, and also
with results of physical experiments is carried out.
Effect of beam instability at propagation of a beam in plasma and phenomenon, stipulated by this effect, of
beam plasma discharge (BPD) are known already more than 40 years and explicitly were researched under different
conditions and parameters of a system However perspective of BPD application in the technology of processing of a
surface of materials for micro- and nanoelectronics again has attracted attention to physics of interaction of a beam
with plasma at a low magnetic field. In particular, the problem on a nature of effect detected in  of formation in
BPD of an ionic flow spreading on a normal from a discharge axis with energy, much exceeding a thermal energy of
electrons has risen. With the purpose of refinement of the mechanism of acceleration of ions we have conducted
computer simulation of interaction in a plasma – beam system at parameters of model qualitatively appropriate to
conditions of experiments . In the given formulation of the task we aimed to reveal main features of interaction in
a longitudinally bounded system and their corollary, important for the final task: definition of a current balance and
energy relations for components of plasma in the system.
Below will be shown, that in conditions of these experiments:
1) At non-linear development of a beam instability in a bounded volume (plasma resonator) in area
occupied by the beam the strongly non-equilibrium plasma with mean energy of electrons reaching hundreds
electron-volts is builded up; at that distribution function of electrons of the beam and the plasma become physically
2) The electrons of this area create an additional current of electrons from plasma to end plates;
3) The increase of the electron current results in growth of a potential of plasma in area occupied by the
4) The potential gradient between area occupied by a beam and peripheral area of plasma determines
acceleration of a flow of ions on a normal to an axis of the system.
2. Numerical experiment
The simulation was conducted with usage of the “Karat” code . Mathematical model underlying the code, is
the Maxwell equations with different matter equations, including one in the form of kinetic equation solved by a method
of particles (PIC – method), and also ones in the form of different phenomenological models. The Maxwell equations
are solved by a plain finite-difference method on shifted grids having the second order of accuracy.
In the given work the two-dimensional version was used, in which all components of speed of particles were
taken into account.
The axisymmetrical task is considered. Countable area is a tube of 20 cm in length and 5 cm in radius. Its
surface is under a zero potential. From the left end face in a circle of 1 cm radius a beam with energy 2 keV and current
0.5 A is injected. Initially the cylinder is filled with plasma with density 1010 cm-3. All countable area is immersed in an
external permanent magnetic field 50 Gs.
Numerical and the physical parameters were selected so that the Debye screening distance was more than step
of a grid, and the number of macroscopic particles in a Debye orb was much more than unity.
On a surface of the cylinder and end faces limiting plasma, the condition of a total absorption of particles was
The main results of computer experiment are reduced to following.
The most fast process developing in the system, as well as follows from the theory of interaction in a beam –
plasma system, is the excitation of oscillations of electrons of a beam and plasma with frequency ω≈ωpe and
longitudinal wave number kz≈ω/V0, and appropriate generation of an electrical field. However already in time about
2-5 ttrans (ttrans is a transit time of the nonperturbed beam) obviously appears essential property of model: the researched
system represents a plasma resonator, the accumulation of energy in which results in constant change of conditions, in
which falls injected beam.. It is obvious visible from phase portraits of electrons in different instants after the injection
In time about ttrans=7,6 ns the typical bunching of a beam appropriate to development of instability on fixed
frequency is observed. It is important, that the wave is localized in the field of a beam: though because of that the
injected beam is limited on radius (rb0 < 2π/kz) and the magnetic field is low (rlarmor ~ rb), the electrons gain considerable
(transversal oscillation velocity, beam, being dilated, does not fall outside the limits ~ 2rb0. In the same limits the energy
of an electrical field of a wave is concentrated also.
At the subsequent stages of injection the new portions of a beam fall in a field generated at the previous stage,
therefore already through 10 ns the bunching occurs much faster, and at the end of an interaction region the phase
bunches practically get mixed up. In time ~100 periods of a principal frequency this is not only full chaotization of
motion of the beam that is observed, but also deceleration and even return of a part of the electrons: the electrons of the
beam become physically indistinguishable from electrons of plasma.
A temporal structure of a HF field of an excited wave is shown in fig. 2. It is seen, that if near to a point of
injection the field on the initial stage has regular character appropriate to conception of the theory of hydrodynamic
beam instability (an amplitude-modulated wave on Langmuir frequency), in a further wave gets stochastic. Amplitude
of the field at stage, where it is regular, reaches 1000 V/cm, and then is reduced up to 200-300 V/cm.
It is possible to observe energy exchange of the beam with the field and with electrons of plasma in fig. 3. It is
seen, that up to 15 ns the shape of a cumulative distribution function of electrons on velocities (DF) qualitatively meets
to conception of the non-linear theory of hydrodynamic instability. 2 bunches are formed in the beam: in-phase
(delayed) and counter-phase (accelerated), more and more dispersing and extending in space of velocities in process of
the wave amplitude growth. Thus the distribution function of electrons of plasma practically does not vary. Further
chaotisation of oscillations results in intermixing bunches and formation of DF, almost monotonically falling up to
speeds ~ 1,5V0. Since t~30-40 ns, there is also heating of electrons of plasma by an intensive HF wave. As a result the
group of accelerated electrons of plasma (“a superthermal tail”) is created in the beginning, and then heating of all bulk
of plasma electrons occurs.
Fig.3. Velocity distribution function of electrons close to
back wall of the volume at different time moments: since
10 to 50 ns in 5 ns (curves 1-9) and since 50 to120 ns in 10
ns (curves 9-16). Dotted line – undisturbed beam velocity.
Returning to the analysis of an electrical field, generated in a system, (fig. 2), it is necessary to note, that HF
oscillations exist on a background of increasing quasi-steady potential: mean value of a potential on a time
interval>>1/ωpe reaches value ~150 V to the moment 120 ns. The growth of a potential is caused by increase of energy
of electrons of plasma and appropriate increase of a difference of flows of electrons and ions to interaction region
boundaries (fig. 4).
Fig. 4. Currents and mean energies of electrons (a) and ions (b) to different parts of plasma volume walls: 1 – to paraxial part
of back plate (R<2.5 cm); 2 – to peripheral part of back plate (R.2.5 cm); 3 – to side wall (5 cm<Z<20 cm).
As is already marked above, the generated fields are localized in area occupied in a beam. So the potential
gradient on radial boundary of this area accelerating ions from area of a beam to a lateral wall of a volume is created
that is obviously seen from a fig. 4.
Thus, as a result of development of instability at a low magnetic field it is possible to select two regions in a
volume of plasma with different parameters. Inside a beam the intensive stochastic oscillations in frequency band near
to Langmuir frequency are localized. In this area as a result of development of intensive instability the strongly nonequilibrium
plasma is formed that is characterized by rather high energy and non-isotropic electron component (the
mean energy reaches hundreds eV) and different directions of an drift of electrons and ions. On peripherals much more
feeble oscillations are excited which don’t influence essentially parameters of plasma in this area. The intensive going
away of electrons from paraxial area on end plates of the system boosts increase of a potential of this area of plasma
and, as a corollary, drift of nonmagnetized ions to periphery of this area. According to results of computer simulation,
the energy of thus accelerated ions can reach several tens eV.
In [3, 4] attention have been drawn first to the fact of a relaxation distance reduction for a monochromatic
beam in semi-bounded plasma. This is owing to effect of accumulation of oscillations in plasma near to a plane of
injection of a beam because of a smallness of their group velocity in comparison with speed of the beam. Value of a
maximum field is rated there: for conditions of our experiment (np=1010 cm-3, nb=108 cm-3, Wb=2*103 eV, Te=20 eV) in
absence of collisions of electrons with heavy particles E2/4π≈4nbWb~3÷5 kV/cm, that meets well enough to the data of
a fig. 2.
It is necessary to remind, that at stochastic oscillations one should account inverse of correlation interval as an
effective collision frequency . Both in numerical, and in physical experiment the effect of stochastization of excited
oscillations is obviously observed, thus width of a spectrum (i.e. the return interval of a correlation) reaches or even
exceeds 0,1ω0. According to , the accumulation of energy in this case is essentially reduced (compare to a fig. 2а, б).
Stochastization of excited oscillations is rather essential moment detected in numerical experiment. Let’s
remind, that in physical experiments the generation noise-like oscillations was observed regularly, including , and the
attempts were done to explain this effect by interaction of excited high-frequency waves with other types of oscillations.
From results shown above it is is clear, that stochastization comes already then, when any other instabilities in the
system do not appear yet.
As shown in , longitudinal restriction of the realistic the plasma – beam system results in feedback obviously
influential in dynamics of development of instability and time-space structure of generated fields. In this paper the
analogy of the given system to the generator of stochastic microwave oscillations executed as a travelling-wave tube
with the delayed feedback was offered.
It is necessary to note, that the stochastic character of excited oscillations appears for the broad class of the
determinate dynamic systems [7, 8]. The particular mechanism of stochastization in our case still should be researched.
The corollaries of described here development of process on the initial stage can apparently be observed and at
researches of a stationary system, including at beam plasma discharge. Let’s remind, at first, of measurements of
dissipation of a beam on angles and energies in BPD without a magnetic field , where were detected both
superthermal electrons and monotonically falling DF of a beam simultaneously with generation of a broadband
spectrum of RF radiation from plasma. Paper was already mentioned above also , containing the proofs of a
chaotization of generated oscillations due to feedback in a beam – plasma system. In  the formation of a peak in
distribution of HF fields in the beginning of a system with amplitude, sufficient for essential strengthening of ionization
in this area is shown. At last, experimentally observable acceleration of ions  on a normal
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