J. Plasma Fusion Res. SERIES, Yol.2 0999\ 350-351

lon-Beam Focusing by Plasma BGK Low-Frequency
oscillations in the Nonhomogeneous Magnetic Field
EGoRov Alexej M., KISELEV valentin A., MAsLov vasyl I.* and oNISHCHENKo Ivan N.
NSC Kharkov Institute of Physics &Technology, Kharkov 310108, ILkraine(Received: 9 December 1998 / Accepted: 6 Mav 1999)

Abstract
The formation of the radial electric field which is sufficient for ion beam focusing in the currentcarrying plasma in the vicinity of a magnetic barrier is investigated. It is shown that the instability
development in the current-carrying plasma, plasma electron trapping by wave field and their propagation
along the magnetic coil axis lead to the formation of a radial electric field.

Keywords:
ion beam focusing, electric field ofpolarization, low-frequency instability

1. Introduction
It is well-known that in the plasma being in theexternal electric field in the vicinity of the magnetic
barrier a formation of polarization electric fields canfor the electron distribution function the followine
expression

take place. These fields are sufficiently strong to enable

fr"(nthe ion beam focusing. Also, in the current-carrying
plasma the excitation of a wave field (instability) was
observed (see, for example, []).
Let us show that this instability can lead to theformation of the focusing electric field in the vicinity ofthe magnetic barrier (for instance a coil).

*r,^ -u,). r"(o- v,n*v))rz.

lal<v.

(1)"(- + %n- % ). lr l> y*
I.\u

2. Development of the lon-Acoustic
Instability in the Current-Carrying Plasma

1*1

u=7'l (v -Vo)d€.

The ion-acoustic instability is developed when theelectron drift velocity Vo is less than the electron thermal{velocity V,r,. At the linear stage the wave amplitude

The nonlinear stage begins with the small amplitude @"uincreases with the growth rate y= (7tm"l8m)tt2kvd[l].

= (y/a)2(m,/m)(TJe) << (7"/e). The amplitude saturation proceeds through the mechanism of plasma ion
trapping which takes place when the amplitude value
reaches h = (Vpn - V7)2m1/e, here V21; = Tilmi. Thisvalue corresponds to the trapping of a major part ofplasma electrons by the nonlinear BGK wave field. The

At the nonlinear stage the plasma electrons are trappedby a wave field. The nonlinear stage of the wave
evolution begins when the frequency of resonant
electron oscillation in the wave field Oi. = k(eQu/m,)r/2 =kVr,/2 exceeds the growth rate 7. Taking into accountthat the trapped particles are phase mixed we can obtain*

Corre sponding author' s e-mail : maslov @ rocke tmail.com

350@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Researchin the Nonhomogeneous Magnetic
Esorov A.M. et al., Ion-BeamFocusing by Plasma BGK Low-Frequency Oscillations

polarization of plasma flow appears [2]. The electrons'in the crossed radial electric field E and the longitudinalmagnetic field, move in the azimuth direction andoscillate along the radius. To describe the electron

trapped electrons are moving with a mean velocity equal
to the wave phase velocitY Vo6.
As the magnetic coil is a barrier for electrons thelatters are decelerated in the longitudinal direction
depending on the invariant values Vfi + V|= const,Vll
H = const. Hete V11, V1 are the transversal andlongitudinal electron velocities, respectively, ,FI is thedynamics we use the equations

i, +[Vu a".- (o?r /2 +V] / r)l = 0,Ve+V,(Vs lr-0.")=0

magnetic field strength of the coil. Consequently, tomaintain the mean energy of trapped electrons, the wave
loses its energy by decreasing its amplitude. However,there is the mechanism of amplitude sustaining at an
initial level, namely, the amplitude is pumped by theextemal electric field. The trapped electrons transfer theenergy from Es to the wave by oscillating in the wave== 4ne26nlm.,6n = n" - ni. From these equations it follows that the electron oscillatory velocity is equal to V,= (elm{Z")E,ocos(O"t), {2"= (a2u- 2a7)tn and the electron oscillation radius is R.. = eE,lm,(raf," - Za'zr)Therefore and because of the fact that the radius of acylinder filled with electrons is less than the radius of acylinder filled with ions, R"" ( 11, and taking into account the balance of forces, we derive the inequality for

Let us determine the growth rate of the instability
development and the energy transfer to the wave. The

expressions for the electron and ion density
perturbations can be obtained from the Vlasov equation-- (ttt2k)(et m")'Qr Eo d1/3 | v=v,,- i(e I m.)nQu d,,,fi I u=u,,-e' E; Qu k' TJ + eQyl 7",

radial plasma polarization 6n < Hzol 8nm"c2. When 6n <1,25.10'0 cm-3 this inequality provides the radial electricfield 4 = -errH2l4m.c2 as great as l0 kV/cm.

Q)

We have performed the computer simulation of ion
beam focusing in the field of a radial polarization ofelectrons relatively to ions in the plasma. In this case theelectrons trapped by the low-frequency wave field are
moving to the coil region. The simulation shows that the6 n|lno=eQrlmtV|nThe substitution of (2) into the Poisson equation

size of a focused spot is not strongly influenced by thephase of the ion beam injection if the length of the coilgives the expression for the growth rate of instability 71|,^\v2LT*=T*Te=lnl6J\l/2\metmi)(Vd lVr)kVE, - eEoVo /27"

(4)(where " . " is the time derivative) similar to [3] for theelectron azimuth motion and electron radial oscillations
with velocitie s V 6 r 0 and V, -- r, respectively . Hete a2,field and 86.

6 nf I no -e2 i\k Eol kT:

Field

is much longer than the wavelength.

(3)

3. Conclusion
The results of this investigation show that if thewave with trapped electrons runs into the magnetic
barrier, the polarization electric field can be formed.
This field is sufficient for the focusing of the high-Here yis the linear growth rate and 76 is the growth rateof wave amplitude pumping due to the extemal electricfield E6, Vr. is the group velocity of the wave.
The fields in the plasma outside the coil, Eo, andinside the coil, 81, are connected by means of theenergy ion beams.

plasma conductivities Er= [161160. Now, we use theexpression oylor= (v"sla.")2(nylno). For v"r << a)ce theplasma conductivity in the coil region o1 is much less

References

t1l A.B. Mikhajlovsky, Theory of plasma instabilities

than the plasma conductivity outside the coil 01 (( do.
Hence, all external potential is concentrated in the coilregion. Inasmuch as, the electrons are magnetized, the(Consultants Bureau, New York, 1974) Yol.2.

t2l K.D. Sinelnikov et a/., Nuclear Physics 1, ll0(re59).

t3l R.C. Davidson, Theory of nonneutral plasmas

trapped electrons moving with the wave into the coilinterior are directed to the coil axis in conformity withthe magnetic field structure. In this way the radial(W.A. Benjamin, Massachusetts 1974).

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