Excitation Of SurfaceWavesWith Different Azimuth Structure By Tubular Electron Beams
Yu.A. Akimov¤, V.P. Olefir¤ and N.A. Azarenkov¤
¤Karazin Kharkiv National University, 31 Kurchatov av., 61108 Kharkiv, Ukraine
Abstract. The excitation of eigen surface waves by tubular electron beams in cylindrical discharge devices is studied. The
system of nonlinear equations that describes the dynamics of the plasma-beam interaction is obtained. The influence of the
wave-field azimuth structure on the excitation efficiency and nonlinear stage of the instability is investigated both numerically
and analytically. It is derived analytical expressions for the saturation amplitude and excitation efficiency of the wave under
study. They are found to agree well with the results coming from the numerical modeling of the plasma-beam interaction
presented in this paper.
Keywords: Surface waves, electron beams, plasma-beam instability
PACS: 52.40.Mj, 52.80.-s, 29.27.Bd
Nowadays, gas discharges created and maintained by high-frequency surface waves are intensively studied both
theoretically and experimentally . The analysis of experimental data shows that spatial distribution of plasma
parameters essentially depends on a type of the wave maintaining a discharge and on its frequency. The wave-field
azimuth structure determines a wave energy absorbtion area and, as a result, a spatial distribution of the plasma density.
In plasma-beam discharge, the wave-field azimuth structure significantly influences also values of growth rates and
saturation amplitudes, determining the wave excitation efficiency . In what follows, we investigate the influence of
the azimuth structure of excited high-frequency potential surface waves on development of the plasma-beam instability.
FORMULATION OF THE PROBLEM
Let us consider a cylindrical plasma-beam discharge structure that consists of a metal waveguide of the radius b,
partially filled by a plasma with the radius a < b and the density averaged over the discharge cross-section, n0e. We
suppose that, in the plasma, there is a tubular non-relativistic electron beam with the internal radius R1b and external
radius R2b < a. The beam propagates with the initial velocity V0b along the axis of the discharge device. Its density,
n0b, is much less than the plasma density, n0e. In the area between the metal and plasma, a < r < b, a vacuum gap is.
Let us suppose also that the discharge is in an external axial steady magnetic field with the intensity H, which
meets the condition w2
pb, where wce = eH=(mec), wpe = (4pe2n0e=me)1=2, wpb = (4pe2n0b=me)1=2 are
the electron cyclotron and Langmuir frequencies of the plasma and beam. Here, e and me are the electron charge
and mass, accordingly. In that case, the magnetic field confines the beam particles motion in the transversal to the
discharge axis direction. Consequently, the beam motion can be treated as one-dimensional. At that, the influence of
such a magnetic field on the plasma electron motion is negligibly small.
A growth rate of the eigen surface wave field, in the considered discharge device, essentially depends on the wave
azimuth structure . At a given azimuth wavenumber, m, the maximum growth rate.
with Im and Km being the modified cylindrical Bessel function and McDonald function of the order m, accordingly.
The axial wavenumber of excited wave is determined by the plasma-beam resonance condition, k3 =w0(k3)=V0b.
In the used model, the influence of geometrical parameters of the beam on the growth rates and frequencies of
excited waves is described by the coefficient b , values of which significantly depend on the wave-field azimuth
structure. Influence of the azimuth wavenumber, m, on value of this coefficient is presented in fig. 1. One can see,
an increase of m results in a decrease of the parameter b in the whole range of axial wavenumbers, k3. The greatest
influence of m is observed in a range of small axial wavenumbers, k3. With an increase of k3 the influence of m on b
decreases. In the range of large values of k3a, k3a > 10, independently of the internal beam radius, value of b tends to
0.5, when the external beam radius is equal to the plasma column radius (fig. 1a), and goes to zero, when R2b=a < 1
(fig. 1b). Thus, both a decrease in the external radius and an increase of the internal beam radius result in a decrease
of the parameter b .
The numerical solution of the system (7) shows, the wave-field amplitude e oscillates, at nonlinear stage of the
plasma-beam instability, near a some steady-state value, esat , representing a wave-field saturation amplitude. This
value decreases with an azimuth wavenumber growth. It is caused by an essential difference in evolution of the wavefield
amplitude for the m = 0 and m 6= 0 modes. It is associated with the fact that the electric wave-field, in the center
of structure, is equal to zero for the modes m 6= 0, whereas it is finite for the mode m = 0. Because of it, internal part
of the beam weakly enough interacts with the waves m 6= 0. Moreover, this part extends with a wavenumber m growth.
Comparison of the normalized wave-frequency shift, (w ¡w0)=d0, obtained within the framework of the linear
theory (2), with that derived by numerical integration of the system (7) demonstrates that the former gives too low
estimation (by the factor 21=3). Use of this fact makes possible to build the approximation,
= ¡21=3[Fm(k3R2b)¡Fm(k3R1b)]1=3; (10)
which agrees well enough with the numerical results. Comparison of these data is shown in fig. 2, where the lines
and symbols correspond to the estimation (10) and numerical integration, accordingly. Their well agreement allows to
apply expression (10) to estimate the saturation amplitude,
esat = 22=3k3(R22
The latter expression is obtained from the momentum conservation law (9), at the supposition that, in the saturation
stage, all beam particles are trapped by the wave-field. Another assumption in (11) is a thermalization of the beam at
the saturation, when its average velocity becomes equal to the wave phase velocity. Thus, the existence of non-trapped
beam particles is not taken into account in (11). Therefore, expression (11) gives too high estimation of the saturation
amplitude, especially for the waves with m 6= 0.
The results of direct numerical integration and analytical expression (11) are shown in fig. 3 with the symbols and
lines, accordingly. It shows that, independently of the wave azimuth structure, the saturation amplitude grows with an
increase in the beam thickness, R2b¡R1b, or internal radius, R1b. The difference between the results of numerical and
analytical investigations depicted in fig. 3, at an increase of the beam thickness, is caused by several facts. First of all,
the phase velocity of excited wave at nonlinear stage slightly differs from that predicted by the estimation (10) (fig. 2).
Secondly, the existence of non-trapped particles in internal part of the beam, which, in fact, do not participate in the
plasma-beam interaction, is not taken into account in expression (11).
Mark also that the saturation amplitude dependence on the beam density, determined by the normalization (8), has
a usual form, y0 µ n2=3
The efficiency of the plasma-beam interaction, G, is determined by the ratio of surface wave energy at the saturation
stage to the initial beam energy. It can be reduced to
Therefore, the wave excitation efficiency grows with an increase in the beam density. According to the analysis of the
parameter b carried out earlier, the greatest excitation efficiency is reached for the m = 0 mode. For a given azimuth
wavenumber, value of G decreases, as the internal beam radius, R1b, increases or its external radius, R2b, decreases.
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