IEPC-95-56 - 408 - I

On plasma flow instability in CDEA (for the experiment)E.B. Svirsky I

I. IntroductionSimultaneously with the studying of closed drift electron plasma accele- 3

rator by extended zone acceleration (CDEA) or low-density Hall accelerator

operation and with that of ion-magnetron plasma heating principle, as applied

to fusion problems,the problem of anomalous - great current transfer across

the magnetic field in these accelerators came into existence.By then the

plasma instability in the electric and magnetic fields, and the possibilities

of strong mixing due to emerging chaotic electric fields and,as result,an en-

hanced current transfer across the magnetic field (D.Bohm,1949 [93) were

known. It was noted dy the authors [2,5,63 at that times, but the theory mu-

tually - penetrating ion and electron fluxes C53 or that of a general hydro-

dynamical turbulence [63 were involved into an analysis. Same results of an

analysis plasma stability in transversely magnetic field (similar to CDEA),

in presence of ion beams in it,were known by that time (B.B.Kadomtsev,1958

C103).However, everybody was interested in the reason for their emergence in

connection with the experimental condition to be able to actively- interfere

-into the process of acceleration, impacting it. It has found one of the ne-

cessary conditions for the plasma flux stability in the crossed electric and

magnetic fields, dH/dx>O (where x is the coordinate of the flux propagation)

confirmed later in the number of studies made by the efforts of many researc-

hers in the USSR, in the experiment and the theory,and used in practice

[9,11-16,18,19,28;20,21, 23,24;25,27;29-313. Same theoretical studies on the

plasma flux stability in the crossed electric and magnetic fields were done

in parallel[13,14 ,17 ,253.CDEA in various modifications of its applications (technology, outer

space investigations etc.) manifests itself as a rather simple and versatile

instrument with the properties of a multi-purpose one. It serves as a plasma

generator (ionizer) and as an accelerator of this plasma simultaneously. When

a task for modifying such a versatile tool is posed, although looking simple

in design but complicated in the interaction of occurring phenomena, one ne-

eds the knowledge of those phenomena in detail to be able to compose a balan-

ce between the urgent ones and thous of secondary importance upon which one

can purposefully impact.According to the existing experimental data, one sho-

uld note the following features of the ionization zone:

1. As expected, the ionization in a number of axnpeting processes passesthrough the channeL A + e + E - A- + 2e, wher.. A,A are the atom and i ts

ion, e is the electron, E is its energy.The idea of an ionization length [333

was introduced for this purpose, and the accelerator operation efficiency un-

der working medium processing and was considered from this viewpoint. At the

same time, the estimation of mean free path for electron before ionization,after the energy consumption by it in the longitudinal electric field, about 3two orders of magnitudes exceeds the height of an accelerating channel , whe-

re electron can pass its acquired momentum to the wall, moving along the mag-netic field, not ionizing an atom.

2. The experimental data confirm the existence of an ionization zonecomparable with the accelerating channel length, according to both the con-rtact-free (optical) measurements C29,323, and the probe ones (20,233.

3. T'ese data indirectly confirm the experimental data on the char-

ge-mass-energy-analysis of a plasma flux [223. Moreover, in some cases

[22,323, ions have a chance to be ionized up to the second level and to the

third one. These data show that the mean free pass (range before ionizati-

on),based on the collisional ion ionization model up to the next level doesnot allow one to explain the phenomenon by the length of a range not ones for

electron but for ion as well.4. The experimental data C14,?03 confirm the presence of a zone with os-

cillation the intensity of witch c )ends on the magnitude and distribution of

SRRC "Kurchatov Institute",INF, Kurchatov Sq.l, Moscow, 123182, Russia.

II|

I -409 -

rragretIc -Feld along the channel, under which the level of those oscillationsis minimal and the magnitude of which is chosen by "a minimum in the opera-ting current under given voltage".

5. Presence oscillation in a plasma by themselves does not speak about

its instability, since the oscillations can come from other regions, external

ones included.As the last resort, the energy consumption at its level spent on ioniza-

tion: Wi/Wo= rq Zi e UI/(U. I.) was assumed for CDEA,as a facility of defi-

nite complexity'(combination). Here, q is the flow rate of a working medium

per second, b is the working medium processing coefficient; Z is the ioniza-

tion level, e is the elementary charge; U, is the ionization potential;U, and

I. are the operating (working) voltage and current. The rest was explained bya low efficiency of the device.

These and other facts made us to reconsider the problems related with

the plasma flux instability in the crossed electric and magnetic fields.Therewas an opportunity enterprise to revise and specify the ideas about phenomena

proceeding in the accelerating channel, and was showed that in certain condi-

tions (slow alternating of the potential < 154-20 V/cm) the instability can to

appear because the pressing of the experimental facts C 14,20-23,30-32 etc.],i.e. the instability can to appear at the entrance and outlet of the accele-rator by the positive gradient of the magnetic field strength. Their analysishas shown that, indeed,the oscillating instability zone for a plasma flux canexist at a positive magnetic field gradient too, and that only one conditionfor the plasma flux stability in crossed electric and magnetic fields - outof a few ones - was found previously.In that case,such result is included in-to rather simple and descriptive model representation of the very "plasma ac-

celerator" C13,14a3. Let us propose a more versatile criterion of stabilityestimation for the accelerated plasma flux of the configuration under discus-sion with relation to the parameters which can be controlled by an eperimen-tal st.

2.Theoretical prerequisites to an analysis of experimental data.In the theoretical model [143 chosen for this purpose a 2D-dissipati-

on-free-cool plasma flux in the crossed electric and magnetic field has beenconsidered. The initial flux parameter do not depend on the transversely co-ordinate, and the lo- ale perturbations only (out of the whole range ofthem) are considered.Fc. am,the following conditions are assumed to be sa-

tisfied: Let us briefly uicuss these, Vo = const; dno/dx= const, (*)

chosen previously E14a,initial conditions for a physical model.Out of two

conditions (t), the first one is stronger. The condition v.=const means that

the plasma density,according to the stationary nature of the processes underconsideration, is constant along the discussed accelerating channel piece.In-deed, nooVo. = novo; no = n. v..o/v. = donst,(**)

where no.,v.o are the and density flux velocity at the beginning of the acce-lerating channel, respectively.

In order to make transition to local characteristics in the model "plas-

ma accelerator"which follow from the assumption (t),let us relate to an elec-tric field, E,and to the length, f, along which a change in the velocity, vo,occurs; then, multiplying this expression by v . ,one obtains the following ex-

pression J(2e/ME)dx = (2eEM)/l, and the criterion for the low scale nature of

perturbations is expressed by the smallness of the oscillation wave length, xrin co-mparison with the characteristic length, A, at which a change in the ve-

locity takes place.This conditions has the form: f /AX >-p-Mv=/(2eE >J )>1,(*t)Let us consider the first version of the condition, excluding the second

one as a version limiting the problem conditions, and make an analysis of in-terest for the experiment. The upper limitation on the problem under conside-ration is related with the made simplifications in the equations of motionfor an electron component, where we neglect the mass electron component, itmeans that the frequencies much lower than the electron-cyclotron and plasma

frequencies are under conside'-'- -n.

- 410 -

IThe set of equations for a plane representing such a model has the fol-

lowing form in the coordinates "t","x","y" (compare E14,253):

(vx)l+ Vo(v.).= -(eM)( I). (vy,))+ Vo(Vy)=-(e/M) x(), (A,B)

(ni)t+n,°oC(vH) +(vy)],+Vo(n),.= O; (ni)t+nioC(v.)+(vV),3+Vo(n),.= 0(C,D)

UM=-(c H) bt)i)v; U=(c/Ho)(()I; ( ).+(i)vy=0; n.=n*=n;n.=nio=nw (E,F,G)

The prim with subindex designate partial derivative.It includes (see,fig. 1):

a)initial parameters of the unperturbed flux,no.,n,,vo,Uo.@o,i.e.plasma

component densities, ion velocity and the electron drift velocity in the an-

perturbed electric and magnetic fields;electric potential;their initial equi-librium magnitudes are directed: (electric field E.) along the "x"-axis, (mag-netic field H. ) along the "z"-axis; (V) along the "x"-axis; (electron drift in

the mutually - perpendicular unperturbed electric and magnetic fields) along

the "y"-axis (Fig.1); c is velocity of light;b) and the perturbed magnitudes of plasma flux parameter n&,n.,v",vy,U,

U,~,4, i.e. plasma component densities, and longitudinal transversely ion and

electron velocity components,as well as the electric field potential, respec-tively,for which the plasma flux stability condition is determined in a line-

ar approximation.Then,using the designation:6 = n*/n.;k..k as the wave vec-

tor () components along the corresponding coordinate axes and L-"-(Ho/n)

d/dx(n0 /Ho ); ka= kx+ ky; U.= -cE./H0; wp2=41h oe 2 /M. where M is the mass

of ion. Let us make simplification and, substituting the values for the per-turbed quantities the amplitude values of which are multiplied by

exp{iwt+ik.x+ikyv), we get the dispersion relationship(i.e. a linear analysis

stability of the system equation):S a

2 + (2k.vo + S. L k 2 /kv) a + k. 2v.o + k 2 44. L. U. - 0 , (1)

Let us analyze the relationship. It gives an expression for a frequency in the

spectrum of perturbations at the dependence similar to that given inE14a3:w = -0,5 [2 kM v. + L. ,. k2/kv +

+ -L (2k.v.+ L. a k 2/ky)z - 4(v.2 kI + LkOiUJJo)J , (2)

The condition4 vokx/ kv + L M. k2 /kv, ; >0, (3)

is the stability criterion. Or, using a = . / k,. it can be written in a more

compact form F(v.,Uo,a)= v.- Uo+ 0,25 L. wl.(l + J.) > 0 (3')

The parameter d with the dependence Uo= U.(v.) in a given expression isa generatrix of the family of straight lines (symbol of equality).Let us dis-

cuss the problems related with unstable plasma flux regimes in the accellera-

tor.Let this regime be determined by the parameters (U,,v.) (See fig.2).Any

regime determined by a point in an external region "B" respective to an enve-

lop is stable for same values of the parameter"C'satisfying the condition (3)

(non-shadowed semispace over each straight line) and unstable respective to

others contradicting this condition. The features of that motion in more de-

tail are determined by tangents to the envelop passing through the chosen po-

int of the regime (U.,v.) that is found from the condition (10)(See Fig.2).

This condition determines the tangent slopes. These angles are two in number:

dr ,z - 2(+± -vo0 /(W. L ) 2+U.o/0,/L- 0,25 - v.o/./L. ) (4)The angles between the tangents

U + LU = a ( v - vo) ; U + U. = ca( v - Vo), ( # )the slope of which is between "d1 " and "az"(and covers the region "A"), de-termine the spectrum of unstable oscillation. An increment for the unstable

flux regime (characterized by = -U 0 /( win LN);n = v./(wimLH );o = kM/ ky).is

equal -= k . Lm - ay/ - - 0,25(1 + d") ; / > > =d, (5)and the spectrum of unstable oscillation is determined by the relationship

w = + I y- ( 4= m /-1 + ac 2 + 0,5 -1 + =. ) Lm wSn k (6)

In this case the wave is rising in time,on the one hand, (the sign "minus" in

front of the first radical), on the other, it mr- that the medium is an ab-

- L411-

sorbina one for the electror gnetic oscillations in this frequency-wave ran-

ge, which do not propagate fa- from the place of their emergence, that is

confirmed by the refraction index equal ck/4 and produced from relationship

(2),being a complex one for all frequencies not satisfying the condition (3).

Vertical angeles complimenting thous which are resting upon the stability

parabola(region"A") determine the generation spectrum of the possible oscil-

lations provided by the plasma flux instability and having an increment de-

pending on the propagation orientation of the emerging perturbations(See Fig.

3)(azimuthal dispersion).It gives the values of "W' representing the spectrum

of these oscillations, the mine L-value which possesses of extremal proper-

ties included.Let us determine this direction:F' (vo,UL.,a=-v.-,5L*"n=1F" (v.,U.,d )= - 0,5 L.Wiw> d.=-2 v./L.Wi, (7)

p.,s, = arctq(-.<,t)=arctq (-2 fo),

Since there is no distinct answer to the determination of "min-max"

under real changes in LH, let us substitute this extremely value into the

inequality (10) and, in combination with the other parameters, obtain the ge-

neral instability criterion written in dimensional quantities:

f H d O ( ) e

v= + 00,. L. - 0,25 ( 0,. L-) 2 = H--- + 2E , - (x)3 - <0 0

d H M

where ,. is the potential and operating potential. H is the magnetic field.

Thus we can to say that it receipt the condition of the stability for every

point of a channel. One can see that, with the accuracy up to the third term

(the value of which, is negligible for a main regimes of accelerator operati-

ons),the necessary and sufficient stability condition is the following: a ne-

gativ sign of comparison between the potential difference passed by the flux

and a half of the potential drop derivative under logarithmic magnetic field

change, i.e. the value and sign of a derivative logarithmic potential drop

under logarithmic magnetic field energy change d(lnE 4[-(x)3)

d(lnE H=3)

is the decisive factor.The corresponding value of a maximal increment will be

equa Y- = LH , k C - 9 -0,25 9)

were - role of the criterion (8) is evidently seen.

The xuced expression for the phase and group velocities has the form:

IV= V,= W,. Lm+ + O2 + d) ,] (10)

the dependence of which (spectrum) on the parameter d is given in Fig.4-S,un-

der unstable accelerator operation regime, they have the form:

v 7r,,= , L.4 , - - - 0,25(1 + ) - - + )I

= 1/k ( + i y - wo ), (11)

The phase and group velocities for the extremely increment are equal zero,

i.e. oscillation exist as standing waves.

One should note that the dependence of frequency (Fig.5) on the directi-

on of propagation ( c ), under unstable oscillations

Re (= G4o = -w.L.kE-/jn d / (1 + J) - 0,5 1 + )] (12)

has a maximum for waves propagating in the direction of the main plasma beam

acceleration the dependence of which on jn (Fig.5),

Z R. A - /p. + _ 1 /27 -N . + + 127 (13)

is about equal (2fn.) 1 ^/ for large values of pn and the whole spectrum is li-

mited the equation, equalizing the expression (21) to zero:

ci.- = -no+a.nZ- 1 = I o ° o( 1- -- )

*~~~r 'n~ ' " ^ ' no

- 412 - 34-12

IThere is no such a limitation for the inverse direction on the wave propaga-

tion. Experime n t a 1 r e s s d

The plasma accelerator P-130 C35 where the plasma was proxced and its

Xeions were accelerated, was use4 for the studies. The ion beam was extrac-

ted th-io e an cele in the limiter, imm in diameter,located at the distan-

Sthrouh an orifice inthelrator (ig.6) then the registration took place.

ce of 2 fr the accelerator (Fi _ 22 As a result of measurement

The experiment in detail is represented in tZJ. As a res-lt of ent

with the mass-spectrometer, M-7303, the Xe-ion were detected, up to

Xe-(Fig.7), under discharge voltage 220 V See, expere proc2ed. It ture-

rating characterti of the accelerated flux Fig8) were pr ced. It tur-

red out that the particle energy spread was expanded up to the .rgY t3J

(Fig.9). The rulti-h, nature of the spectrum has allowed one to estimate

the dominant places of the proction for ions of various multiplicity res-

petive to the main potential relief, since the mass-spectrmeter has not

descerned the charge exchanged ions and those which have rt exchanged the

charge under egtation. The results of such a recalculation are given in

charge From the gieastratint it follows that the dominant place production,

respective to the potential distribution in the accelerator, counting off

from the anode is, at U.

= 2 20 V.

a). for Xe Z 0.55 U ; 6) w the accelerated Xe , registered after the.-2 s) f t

chage exchange as Xe, it follows that they should be born as Xe closer to

0.14 U., i.e. they do not belong to the main group., at

a). Xe--- graph: for the accelerated X , registered as the Xe 44 , at

the places closer to the anode than xb t 0.62 U., i.e. they can be related

to the main group (a), born as Xe, at the place xob

0.55 U. and ionized up to

Xe-at the point xb62 U.; r).X -graph for the Xe-- registered as Xe-

closer to the ande than x .65 U - they are covered by the se d group

4. Discuss ion

Let us analyze the oppr- nity to implement the produced conclusion,

1. Let the applie .t0ential difference, U. = 20 V, total Ocurrent thro

ugh the acceleratoIi maximal magnetic field strength,HO=10 e; coimperrt

sation current, I 1 A, working gas expenditure, qa 0.3 0u ^/S..

2. The half-width of a magnetic field rise, x .5cm;halfbwidth of mag-

retic field decay, X2M = 1 cm are quantities, to a corsiderable degree, on

the choice of a compensator design and on the umber of magnetic field coils

3. The nature of a working gas,X (atomic weight is M Xe =131.3 a.m.u.)

half-width of a magnetic field rise is oxiat half-width of a magnetic fiel

decay is in of a sngle magnetic field coil.

The opportunity to iplement the produced solutions,first of all, is re

ferred to the scale of the oscillation under consideration, it is determine

by the assigrlents of a scale for magnetic field (and for the electric ore)

strengthe anges in the system: k rag/ 1 3 ( k > 30 ), that gives an es

timate for the prosu scales undr corsideratia according to (9),(14)(16)5

.... t-. 30 L.. A, k ga e 0,a th og -

For presetting soma magnetic field strength dependenc on the longitudi

nal coordinate, let us represent this function in the form (see, Fig. 11) (4

H = H. / t 1 + ( x / xIua)tiv to the magnetic field ma

where the origin of coordinateis chosen resth anet

xii equal H., xM.2 is the absolute value of a coordinate, where H =

and the indices designate the magnetic field either in the first zone or

the second oie. These data are quite sufficient to make same estimates of in-

terest. Preliminary e that, in connection with a stationary nature of the

process and with a continuity of the current lines in the circuit, 1

I - -41 -

j= corst= E 6 = E/y - E/H , (4.2)

the electric field strength is proportional to the magnetic field strength

(Fig.11).Such consideration are confirmed by the experiment (See, measurement

in [34). Proceeding from this, the electric potential distribution along the

channel calculated and shown in the same Figure. The positions of anode, ac-

celerator output and magnetic field maximum are also shown there. A quantity

proportional to the expression for the stability criterion respective to the

magnetic field maximum position and to the anode is shown in Fig.12. One can

see here that there is an instability zone occupying 5/B of the channel from

the anode to the magnetic field maximum the zone, were the derivative dH / dx

>0, not mentioning the fact that the second zone (at the accelerating chan-

nel output), were dH / dx < 0, completely under unstable conditions.

However, it is important that the instability region in the first zone

is separated from the anode and from the external instability region, being

an "island". The "slit" along the border with the external instability zone

is narrow, being present always but losing it sense, i.e it is short-circui-3 ted, when its dimension to be comparable with the long wavelength perturbati-

on, especially with the dimensions of the Larmor radius. These gaps can be

found with the criterion (8). In spite of the fact that the produced equation

is a transcendent one the position of a border can be determined with a suf-

ficient accuracy, expanding (8) in to series in the vicinity to the maximum.

The position of the maximum depends on the magnetic field strength distribu-

tion half-width only, being located at the depth x.= 0.58 xmu (4.2)

respective to the magnetic field maxium, and the stability boundaries are

about at the distances (Fig.12): 1 1 x xW. 2 = . 0(4.3).,2=-- + 0.1 + /2 -(- + - )

I 3 x xM/ 3 IxNu

deep in the channel respective to the magnetic field maximum. In this case,in the same Figure an increment in relative units is shown so that the ordi-

nate is proportional to the increment value at various pieces (Fig. 12). The

instability zone should make an effect on the resistive properties of the ac-

celerator ( a rise in oscillation in the passing electron flux occurs in this

zone due to oscillations in the electric potential). It means that it will be

manifested upon transient characteristics by a reduction in the current until

an equilibrium is installed due to a rise in the number of current ca -ers

(ionization). This time of a current rise should coincide with the t. ne-

cessary for a flux coming from the anode side and having velocity v,. in-

tersection of the instable zone in order to pass the distance equal to a ran-

ge before ionization, equal ),: 't = )i/vo. A stable level of ionization is

set during this time in a flux passing through the instability zone (it can

be compared with the experimental results in [303), where they have been mea-

sured).The instability "islands" in the plasma flux are not only the oscilla-

tors in the channel, absorbing the energy from discharge but the localized

pieces,where heavy particles acquire additional momenta, not related with the

main accelerating field (their spread occurs)(See, 233):dv. = Av, ( xjon.z,Z).It would be hard to explain a double-hunched or multi-hunched nature of

energy distribution for outgoing Xe'-, Xe--ions by any other statistical

processes (See, experimental paper C223) except of competing ionization ones:1) at the specific place in the instability zone (respective to the main

potential distribution) those electrons or h.f. electromagnetic field oscila-

tion possessing of resonance properties (having the corresponding energy qu-

antum) respective to ionization of Xe up to the level Xe', X**, Xe' are used

by the passing flux of neutrals and exited atoms. In this case, one should

note that the dependence of plasma oscillation energy (amplitude) distributi-

on on the longitudinal coordinate should exist in the zone, since the elect-

rons and ions passage through the zone takes place and oscillations - risingin time in their medium by crossing the unstable zone - are realized into the

S dependence on the coordinate because the increment and inverse time of flight

- 414 - I

Ithrough the instability "island" are comparable.

2) the presence of two correlating group of Xe'and Xe "different ion

energies corresponds to the conclusion 1) about the correspondence of two

places with the same oscillation increment level in a plasma flux(See,Fig.12)

respective to the potential relief, to the ionization processes at two pla-

3) letter on, by this reason,the particles with different charges obtain

the same portion of additional momenta (or energies, +Ze) from the side of

low frequency potential oscillations which throw them to about the same placein the energy spectrum after their passage from the accelerating channel, in

accordance with their charge Ze (then, they pass the same sequence of accele-

rations and decelerations in the localized field of oscillations);The measurement made in C14,14a,203, confirmed by cur measurements, corn-

firmed the presence of a zone with enhanced mean square root amplitude of po-tential oscillation even under conditions which are determined as stable ones

( L> 0 ), on the basis of accelerating channel up to a "mouth" ( and up to

the magnetic field strength maximum).There are demonstration to do of influence the size of concentrating

magnetic field strength on the longitudinal size "island" and the amplitude

of increment one in the kw*.L. units at the graphics of Fig. 13.

5. Con c lus i o n s.In conclusions it is necessary to emphasize once more that:1.In a plasma flux under crossed electric and magnetic fields some uns-

table oscillations possessing of a maximal increment depending on the orien-

tation of perturbations and affecting, in a radical way, the general picture

of oscillations are singled out. That has given an opportunity, using the

expression for them, to produce other necessary plasma flux stability condi-tions, along with the known one (dH /dx > O) in a boundary the problem to

stay:2.Importance of a role for the logarithm potential difference derivative

with respect to the logarithm magnetic field energy (not the spatial size).3.The sufficiency criterion includes an absolute value of the potential

difference under which the ion passes up to the place of observation(or the

energy acqu I-1 by ion up to the place of observation).4. So -idences are represented for comparison of the ionization zone

with the regi cn of unstable oscillations in the passing fluxes (the main cur-

rent) of electrons and ions in to main plasma flow, as a zone with standingwave in the laboratory coordinate system. They exist in the general spectrumof an oscillatory instability, their waves are propagating in both directions

- up and down along the flux. A correlation is possible in their medium only.5. Another approach to the problem of oscillation suppression is propo-

sed, if we contirue to consider COEA to be the combine instrument. Namely, it

is proposed to determine the level, size and the position of the oscillatoryzone sufficient for the instrument operation under optimal conditions respec-tive to the task necessary to be solved and to use the criteria produced abo-

ve as the stability ones (8).The projections of oscillations possessing of a maximal increment upon

the longitudinal and transversely coordinate axes allow one to consider

them,taking their activity into account (if one wishes), as "time-of--lightoscillation", or an "ionization instability",since the electric power is ta-

ken for ionization from the reservoir of unstable oscillations, closely rela-

ted with the power supply source via an electron component. The significanceof a given approach is not worth of overestimating in the problem of dissipa-tive processes: they play the role of an estimate, since they are outside thelimits of the model under consideration..

In conclusion I express my gratitude to Prof. A.I.Morozov who stimulatedthe emergence of this paper after discussion about experimental results [223

and expressed his desire to have the result of these studies in more detailed

- 415--

Irepresentation.

I acknowledges Mr. A.Veselovzorovfor formulation o-f question, participa-tion in the experiment, at some stages, and the constant participation in thediscussion about the experimental results of him, as well as the members ofthe team for the useful discussion about the produced data.

The discussion about the produced results with G.N.Tilinin turned out tobe also useful, his remarks concerning the representation of the results im-

portant for experimenter were taken into account.

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