Non-Local Theory of Spectra of “Modified” Ion Cyclotron Waves in Plasma with Large Ion Orbits in Crossed Fields
Yu. N. Yeliseyev
National science centre “Kharkov Institute of Physics and Technology”
Akademicheskaya St.,1, 61108 Kharkov, Ukraine, tel. +38-057-335-61-02
Abstract. The non-local problem on the stability of the nonneutral cylinder plasma, consisting of magnetized electrons
and unmagnetized collisionless ions, is considered on the ground of kinetic treatment of ions, having been born at rest.
For the plasma with “hot” electrons the dispersion equation is solved numerically for various azimuth wave numbers. The
solutions are presented in a graphic form. It is shown that for the lowest radial modes the instability of «negative mass»-
type is possible having the frequency of the order of ion radial oscillation one and the maximal growth rate up to 0.5 of it.
Keywords: Non-Neutral Plasma, Crossed Fields, Unmagnetized Ions, Large Ion Larmor Radii, Ion Cyclotron Instability.
PACS: 52.20.Dq; 52.25.Dg; 52.27.Jt; 52.35.-g; 52.35.Fp; 52.35.Qz; 41.20.Cv.
The plasmas in crossed fields consisting of magnetized electrons and unmagnetized collisionless ions are widely
used in various devices. These are plasma lenses  and plasma ion sources based on Penning cell . Such plasmas
are formed in the channel of electron and ion beams (secondary plasmas) [2-5]. Though such plasmas are investigated
already more than 50 years, the interpretation of the observed ion oscillation spectra in the whole range of field
strengths, of the electron and ion densities poses an actual not solved until now problem yet. The reason of such a
state of affairs consists in complexity of the kinetic description of unmagnetized ions. The peculiarity of ions in
these plasmas is that they have been born at rest under the ionization of the residual gas. Being trapped in the potential
well of the radial electric field the ions perform radial oscillation motion along strongly extended, nearly
straight-line trajectories with the amplitude comparable to the radius of the plasma cylinder. The frequency of these
oscillations is called the “modified” cyclotron frequency of a particle Ω . The equilibrium distribution function of
such ions was not known for a long time. It was often taken in the form of a “rigid rotator” , however it did not
follow from anywhere. The transversal wave length of the oscillations is of the same order of magnitude as the radius
of the plasma cylinder. Under these circumstances the non-local consideration of plasma stability is required.
The observed spectra of oscillations were interpreted in 60-th, 70-th on the ground of a local hydrodynamics description
of ions [4-6, 8]. However this description is inapplicable for the unmagnetized ions and gives spectra of
oscillations badly agreed with observable ones .
In [9-12] equilibrium distribution function of the ions having been born at rest in volume of the plasma cylinder
in crossed fields has been determined. The non-local problem on the stability of plasma in a strong radial electric
field is solved on the ground of kinetic treatment of unmagnetized ions. The linearized Vlasov kinetic equation for
ions and the Poisson equation are solved jointly. The solution of Vlasov equation in variables “cylindrical coordinates
– momentum” ( , , , , , r z r v v z v ϕ ϕ ), found in , is used. Because of its complex form the stability of low
frequency oscillations ( ω << Ω) and of even azimuth wave numbers m was investigated only. Such oscillations
were observed in Penning discharge at low pressure .
In the present report the non-local problem on the stability of such a plasma is considered using the solution of
linearized Vlasov equation in the independent variables R,θ (cylindrical coordinates of Larmor centre of a particle
in crossed fields), ρ,ϑ (its coordinates on Larmor circle), , z z v , obtained in . In these variables it is possible
to obtain analytically the dispersion equation valid within the total range of the ion “modified” cyclotron frequency
Ω, for arbitrary strength of magnetic B and radial electric ( < 0 r E ) fields (i.e. for magnetized and unmagnetized
ions), azimuth wave number m and longitudinal wave vector z k . The oscillations spectra of plasma consisting of
unmagnetized ions and “hot” electrons are found numerically from obtained dispersion equation.
Plasma Model and Solution of Stability Problem
It is supposed, that the electric potential in plasma Ψ(r) obeys a square-law function of radius r
(Ψ(r) = Ψ(a)(r2 / a2 ) ). The plasma cylinder is bounded with a metal casing of radius a.
The electrons are magnetized being distributed uniform in radius with density ne . In crossed fields they rotate
rigidly around the axis of the plasma cylinder with the frequency of – / 2 ( ) /( 2 ) e r ω = cE Br = cΨ a a B having
the Maxwellian distribution in this rotating frame of reference. In present report the electrons are assumed to be
“hot” ( – )/( ) 1 e zTe ω mω k v
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