O.V.Samchuk, O.I.Kelnyk
Taras Shevchenko National University of Kyiv, Radio Physics Faculty
60 Volodymyrs’ka St., 01033, Kyiv, Ukraine
Whistler waves’ radiation is one of the results obtained from the experiments when the electron
beams were injected into the ionosphere plasma [1]. Transition radiation of the modulated
electron beams can be one of the mechanisms of this radioemission. This kind of radiation was
studied theoretically in the couple of works [2,3] but mostly for the sharp plasma boundary model.
But, one can see that the efficiency of the whistler transition radiation can be increased for the case
of weakly inhomogeneous plasma, where the Cherenkov resonance conditions are satisfied in some
region. Our work [4] was devoted to that case, but for the unbounded electron stream model that
isn’t applicable for the ionosphere experiments’ conditions. This work contains the calculation of
the whistlers’ radioemission for the more realistic model of cylindrical modulated electron beam.
The considered model presupposes the radiation exciting by the cylindrical modulated electron
beam with given current that moves in the weakly inhomogeneous anisotropic plasma along the
constant magnetic field direction and concentration gradient.Inhomogeneous plasma.

Here W(ξ) is Wronsky determinant, Wp(ξ) – determinant obtaining from Wronskyan via replacing
its р-th column with {0,0,0,γjm}. γ is some coefficient that varies for different field components.
The sum (6) contains one item corresponding to the whistlers propagating in the beam motion
direction. Cherenkov resonance conditions can be satisfied only for this item. So the expressions for
the radiated field have a shape of double integrals upon ξ (inner integral in the sum (6)) and upon ρ
(Fourier-Bessel integral). Both of these integrals can be calculated using the stationary phase
method. Note that the stationary phase condition coincides with the condition of Cherenkov resonance
The radiation patterns for the obtained radioemission have been build as the angular dependencies
of Pointing vector for the typical active ionosphere experiments’ conditions (beam modulation
frequency is considered 2⋅106 rad/s, beam velocity – 3⋅108 cm/s). The radiation pattern characteristics
is studied for different values of parameters (Fig.2). One can see that the radiation patterns
for all cases have many maximums and are similar to the corresponding spatial spectra of the beam
electromagnetic field. It can be explained according to the fact that the radiation flows from the
large region with spatial sizes much greater than the radioemission wavelength.
At first, let’s examine the radiation patterns’ dependence on the beam radius. For large radiuses
those much greater than the radiation wavelength, large number of these wavelength confine
oneself to the beam transversal size. As a result, the number of maxima in the radiation pattern becomes
even greater, as it is shown on Fig.2a-b. Note that the radioemission flows into the narrow
angular band for these large transversal sizes of electron beam in full accordance with the fact that this radiation flows under one given angle for the transversally unbounded electron beam. For the
substantially smaller beam radiuses (15-200 cm, beam width is smaller than radiation wavelength),
the radiation pattern shape is practically independent on this radius value.

For the intermediate case (r0=3000 cm, about several wavelength) one can see that the beam
radius increase leads to the appearing of additional periodicity of the radiation pattern maxima
groups (see Fig.2c-d). This periodicity corresponds to the number of wavelength those confine oneself
to the beam transversal size.
According to the whistler dispersion equation, the wavelength increases with increasing of
electron cyclotron frequency. Contrary, the frequency of the radiation pattern maxima groups increases
with cyclotron frequency increasing (see Fig.3a-b). This fact take place because the length
of the Cherenkov resonance region becomes larger and shifts to the direction of larger ξ with the
increase of electron cyclotron frequency. Such a dependence exists for both large (Fig.3a-b) and
small (Fig.3c-d) beam radiuses (relatively to the radiation wavelength). Also, one can obtain similar
dependencies upon the concentration gradient g.
Fig.3 also shows that the radiation patterns’ maxima groups condense with disappearing of
small oscillations. For the large beam radiuses, maxima width is practically independent on cyclotron
frequency and angle. For smaller radiuses, this width decreases with cyclotron frequency increase
as well as with the increase of angle. One can see the dependence mentioned above on the
radiation patterns (Fig.3c-d) for smaller angular band (see. Fig.3e-f).
Estimations of the whistlers’ total radiated power from the Cherenkov resonance region were
also carried out. It was found out that this radioemission is much more effective than the transition
radiation on the corresponding sharp plasma border. But, for the conditions of the real beam-plasma
experiments, the calculated radiation efficiency is found too large (about 10%). So the given current
approach is not applicable for the calculation of total radiated power. This calculation requires the
self-consistent approach.

r0=7500m; c – g=120000, Ωc=3, r0=0.15m; d – g=120000, Ωc=5, r0=0.15m; e – g=120000, Ωc=3,
r0=0.15m; f – g=120000, Ωc=5, r0=0.15m
1. J. Lavergnat, T. Lehner. // IEEE Trans.-1984.-Vol. AP-32.-#2. P.p.177-181.
2. І.О.Anisimov, О.P.Vygovs’ka, О.І.Kelnyk. // Ukr.phys. zhurn. 2001. Vol.46. #5, P.p.557-566.(in
3. І.О.Anisimov, О.І.Kelnyk.// Visnyk Kiev. Univ. Series: Phys.&Math. 1998. #4. P.p. 238-242.(in
4. I.O.Anisimov, O.I.Kelnyk. // Proc. of the I Int. Young Scientists Conf. on App. Phys. Taras
Shevchenko National University of Kyiv, Faculty of Radiophysics. Kyiv, 2001. P.p.17-18.
5. E.Kamke. Ordinary Differential Equations Reference Book. Moscow, Nauka, 1976. (in Russian)

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