IEEE Transactions on Plasma Science, Vol. PS-6, No. 4, December 1978
AMBIPOLAR DIFFUSION IN A NONUNIORM PLASMA CONTAINING
DIFFERENT POSITIVE ION COMPONENTS*
B.M. WundererSektion Physik
der Universitat MunchenSchellingstra,Be 48000 Munchen 40
Received 1/5/78
Abstract-A plasma of low degree of ionization consisting of electrons and several positive ion components
cannot be described by a set of diffusion equations of the ions where the diffusion terms are given by co-
efficients of ambipolar diffusion and laplacians of the ion densities. This simple case is limited to proportionaldensity distributions of all particles; in the general case, the 'ambipolar" electric field couples the spatial
derivatives of the ions with those of the electrons. The conditions following from the proportionality assump-tion either restrict the usefulness of such a description to a few cases, the more important one being the funda-mental mode of diffusion or impose nonphysical correlations on the mobilities or coefficients of recombination
of the ion species.Computer calculations of one-dimensional axially symmetric density distributions show appreciable dif-
ferences in the wall currents and absolute densities between the general and simplified "proportional" model
especially for ions with low concentration. The interpretation of mass-spectroscopic effusion measurements
can be greatly affected by the choice of the description of the plasma.
1. INTRODUCTION
Several different positive ion components are normallypresent in plasma experiments carried out for the study of
ion-molecule reactions or recombination [ 1 - [5] . The descrip-tion of such plasmas by particle balance equations is treated
extensively in the paper of Oskam [6]. Oskam restricts the use
of the plasma model to situations where a so-called "propor-tionality condition" is valid: the normalized density distribu-
tions of the electrons (index e) and all ions (i = 1...N) are
assumed to be equal:
Vni =Vn (i = 1....N)n. n1 e
(1)
*Parts of this paper are a chapter of a thesis (Munich 1975)and were reported at the 2nd ESCAMPIG conference in Brati-
slava, CSSR 1976 as a contributed paper.
With this assumption balance equations suitable for analy-tical solution can be deduced. Eq. 1 holds, for example,when the spatial distribution of the plasma is the "funda-
mental mode of diffusion." Then every ion has a diffusionterm written as a product of a spatially uniform coefficientof ambipolar diffusion and the laplacian of its density:
D. ( : = D An.1 ai 1
(i = 1....N) (2)
In a numeric example Oskam showed already that in an
afterglow Eq. 1 need not be maintained throughout time
(although it was presumed as initial condition), if the ions
have different coefficients of ambipolar diffusion.
The objective of this paper is to derive particle balance
equations containing diffusion terms without the propor-
tionality restriction. It will be shown that in the generalcase the diffusion of the ions is coupled to the electron densitydistribution. A study of the conditions that are necessary
to hold for Eq. 1 shows that nearly every case of physicalinterest will violate this condition. The implications on theinterpretation of mass-spectroscopic measurements are dis-cussed by use of a numeric example.
2. PRESUMPTIONS AND FUNDAMENTAL EQUATIONS
We intend to neglect space-charge [7] -[9] and ion inertia[10] effects in the transport equations and therefore presumethe plasma Debye-length XD and the ion mean free pathsX; to be small compared to the diffusion length A of theplasma container. Then the plasma is quasi-neutral:
n = E n.e I
I
(3)
The conservation law of electric charge demands "ambi-polarity" for the particle currents:
V I (re - E r.) = 0 (4)e i1-
If the wall charge of the plasma container is stationary, Eq. 4has the boundary condition:
(Ie' r.) I- li Iwall -0 (5)
3. DERIVATION OF PARTICLE BALANCE EQUATIONS
The vectors reji and E have to be eliminated from the setof Eqs. 3, 6, 7, and 8. First, we derive an expression for theelectric field E:
U Vn +U pi Vn.e e eilieE =
n - I ;. ne i lie
(9)
The further assumption is made that all ions have the same
temperature U+. Since the ion mobilities are small comparedto that of the electrons, we expand Eq. 9 in powers of (Mui/Me)and retain only the zero-order term:
E = -uSe~- (+ o (;") )e n pe- ~en
(10)
For the derivation and applicability of this zero-order expan-sion the proportionality condition Eq. 1 is not necessary.
We now introduce Eq. 10 into the transport equations7 of the ions, which now contain only particle densities andtheir gradients:
A special case (and in one-dimensional problems the rigoroussolution) is "congruence" (Oskam [61), where no electriccurrent flows at all:
re = E r.e 1- 1-
(6)
This condition, although not necessary, is retained here forthe two-dimensional axially symmetric case. Without magneticfield and conducting walls, the deviations from Eq. 6 are
small.The particle currents themselves are given by transport
equations containing a drift and a diffusion term:
r e (n E + U Vn)(7)
F. = p. (ni E - U Vn
with mobilities Me,i and characteristic energies ("tempera-tures") Ue,i and the electric field E.
At last, we need the continuity equations for the singlecomponents including the influence of inelastic productionand loss processes:
-n+ V F. = (-8)i)col(Dt ~ 1 6t Coll
The processes contributing to the right hand sides are speci-fied below.
r. =-p. (n.U ne + U Vn.)1 1 1lene + I ( 11)
The proportionality condition reduces Eq. 11 to a formcontaining the usual (zero order in Mi/,ue) definition of theambipolar diffusion coefficient Dai:
r(P) = -V. (U +U )Vn. = -D .Vn.1 1 e + i al 1 (12)
The difference between these two expressions is the errorintroduced into the particle currents when the proportionalitycondition Eq. 1 is used inadequately. We define ("relative")ion concentrations:
C.: = ni (i = 1... N) Ec. = 11 n- .1Ie 1
(13)
in terms of which we can write this error:
r- r. (P) u1 1 iep.UeVc (14)
It vanishes only if the gradients of all ci are zero; which isequivalent to Eq. 1. It is usually assumed the ne is zero atthe wall of the plasma container. Then Eq. 12 can be usedlocally to calculate the wall currents.
To obtain the final balance equations of the ion densities,Eq. 11 has to be introduced into the continuity equations8. Here we only write down the diffusion term (V- Uj):
407
dent function.DCn ) : = -. = r n An -IVne2)1 i j ieC(i(n |yeI) +
e e
-t Ion
) + u An.lne +
(15)
This is reduced to Eq. 2 by the proportionality condition(Eq. 1), which is necessary and sufficient for the diffusionterms of the ions to be expressed by laplacians of their den-sities. This means that in the general case the electrostaticforces of the highly mobile electrons do not allow the ionsto be treated like normal gases with special (ambipolar)diffusion coefficients, but couples their diffusion in a rathercomplicated way. With only one ion component, Eq. 15is reduced to Eq. 2 by the neutrality condition (Eq. 3). Inmany experiments with different ion components [11] -
[13], the application of a "proportional" model for dataanalysis has not been proven experimentally. In some
cases [14],[15] different density profiles were measured forthe ions, so it is obviously impossible. In fact, diffusion terms
of the form of Eq. 15 were never used before, inspite of theirstraightforward derivation. Schmidt [16] and Smith [17]pointed out the many factors that influence the interpreta-tion of mass-spectroscopic experiments; a correct descriptionof the plasma is at least one important element for the correct
interpretation.
4. PRODUCTION AND LOSS PROCESSES
The production and loss processes contributing to the
particle balance of the single ions are described by five terms:
6St coll (tfon ( eact ( t )Recomb
The third and fourth terms account for the ion-molecule
reactions [1] of ions (j) leading to product ions (i) and forlosses of ions (i) respectively. In principle, we allow ail Torwiru
and backward reactions. In the most general case, the conver-
sion frequencies are given by sums over the products of rate
constants and densities of neutral gas constituents n1 (o):
V..: = EK(i + Q + j)n (o)31 0
(18)V.i. -+ nx (°)
By these reactions no positive charges are produced or lost.
Hence, the production of all components j (* i) by reactionsof i must balance the total reaction loss of i, which defines a
"conversion loss frequency" vi-:
3 1
(i = 1....N) (19)
At last, recombination losses are included. Reactions and
recombination by triple collisions not discussed here expli-citly do not generate new problems.
The further discussion is restricted to constant coeffi-cients of conversion and recombination and to constant tem-
peratures. This excludes effects like cataphoresis or thermo-diffusion from the discussion, but is the normal simplificationin these types of models. If we summarize the set of equationsdescribing the plasma has the form
The first term describes production of electron-ion (number i)
pairs by plasma electrons (gi[nel ) and the second by particlesthat are not directly belonging to the plasma and inciuaea
in its particle balance (e.g. photons [18], fast electrons [19],excited or metastable neutrals [20]) . It is of no importancehere whether or not the spatial distribution of these particlesitself is coupled to the plasma (like metastables [21] ), in
any case the ionization can be described by a spatially depen-
v.n. - a.n.n11 l 1 e
(i = 1....N) (20)
with D(ni) given by Eq. 15.
5. CONDITIONS FOR PROPORTIONALITY
To justify the emphasis stressed on the importance of
the diffusion terms, we now search for necessary conditionsof proportionality that can be tested from the set of equa-tions (Eq. 20) describing an actual physical situation. The
violation of such conditions neccessitates the use of the dif-
fusion terms (Eq. 15).
408
= fi (r) (17)
The discussion is not possible in a general framework,but must distinguish several cases. Before starting the case
study, we assume proportionality. Then Eq. 20 can be written
as a set of linear equations for the ion concentrations c-
that now must be constants. The whole system is divided
by the electron density and the diffusion terms are abbre-viated:
L--D- (P)L.: ==I c.n
I P-
D -eai n
P.(21)
The set (Eq. 20) with these presumptions is written in matrix
form:(L1 -'v1 - a1ne) v1 2 ...... . v1N
V21 22 2ne) V23 . ..°2N
'VNl 00- N,N-1 ( N N Ne)
(D >: = ciDai (at> = c-Ciaa Jaii (24)
5.1 Steady-State-PlasmaFor the description of a steady-state-plasma, we can omit
the last two terms on the right-hand side. All off-diagonalelements of the matrix are constants, the only nonuniformterms being the Li and aine in the main diagonal. If we now
apply the gradient operator on these equations, we obtain:
cV (L. - n ) = -V n1 1 e n (i = 1...N)
(25)To describe an actual plasma, we must know the ionizationfunction on the right hand side. We discuss several cases
of the gi and fi related to special physical situations:
1. The positive (Schottky [221) column is described byionization proportional to the electron density. Then (g-=
Tine, fi = 0) the right hand sides in Eq. 25 vanish. It followb.
V(LicaIne) = 0 (i = 1...N) (26)
If we divide Eq. 26 by the appropriate Dai and subtract pairs
with different indices (i,j), we obtain:
+ (3log np d )
_
Ct
CN
(22)If we sum up all lines in this equation, we obtain a very simpleequation for the electron density, where the reactions do no
longer occur because of Eq. 19:
i C. (Da-n - acn) = _ gi+fi + Dlog np1 aifn 1e n ate 1 ~~~~e
(23)By this, average coefficients of diffusion and recombinationare defined:
/a.-1D .al
-D- )Vn = 0 for all pairs (i,j) (27)al
This means that for porportionality of the density distribution
in a Schottky column a correlation is necessary between thecoefficients of diffusion and recombination of all ions, since
only the round bracket can be zero anywhere.A special case is the normal column without any recom-
bination. Then Eq. 26 leads to the well-known eigenvalue
problem, the result of which is the fundamental mode pro-
file JO (1'r/R). This contradicts to the result of Popescu andMusa [231, who claim to have shown that a Schottky column
with reacting ions has a density profile deviating from the
fundamental mode. In fact, this deviation results from their
use of inappropriate current transport equations.IL In a plasma produced from a gas mixture by a beam of
light or of fast particles, we set the gi zero and define the f-:
fi (r):= y.f(r)
409
CII1 1
C2
l N
g I+f 1n.
,IgN+fN
nI e.I
(28)
f gives the spatial distribution of ionizing particles and 'Y, thecoefficient of the ionization that produces component iThe comparison of two equations (25) leads to:
(Yi ZLL- v f- = i( V' n
DcD. Dc ) n (D D eai 1 aj 3 e al aj
(29)We can assume that f is not proportional ne (this would becase 1). Then both gradients can neither vanish nor be pro-
portional simultaneously and everywhere, so the round brac-kets must disappear in both sides to fulfill Eq. 29. Besidethe correlation between the ai and Dai concluded from Eq. 27,a second condition results from Eq. 29 that gives the c; direct-ly. Because the c; are normalized, we can write:
i D (zID i )(30)ai ai
It follows from Eq. 30 that all c; are directly proportional tothe rates of production Yi divided by the Dai* This means thatthe plasma particle balance is not affected by the reactions:The plasma must behave as if no reactions occurred at all.Mathematically, this leads to a further restriction imposed
on the vi'j. Generally, there are (N2- N) coefficients l-ij free tobe chosen. It follows from Eq. 30 that N-1 of them are nolonger free.
A further consequence of Eq. 30 is the fact that ions
produced by reactions only (secondary or tertiary ions)cannot be dealt with in the framework of a "proportional"model, because their ci had to be zero.
Ill. The most general case of arbitrary production func-tions gi and fi leads to:
1 g__f 1 gj j=D c.
vn D cc. n D
ail e ajj e ai
D )Ve (31)a3
This condition cannot be tested without the knowledgeof the solution ne, [cII of Eq. 22, but it seems very impro-bable that it is fulfilled in the general case. The safe approachis to use the balance equations without the proportionalityrestriction. By the results the application of the proportionalmodel could possibly be justified a posteriori, which is useless.
5.2 Afterglow PlasmaFor the treatment of proportionality in an afterglow,
we omit the steady-state production rates g; and f; in Eq. 22and retain instead the time dependent terms. Application ofthe gradient operator leads to:
V (L. - a.n - atogne) = 0 (i = 1....N)
(32)By comparison by pairs, we can eliminate the time-dependentterm:
(Dai - D ) V ( An ) = (a -- a.) Vna1 aj3 ne 1i j e (33)
Equation 33 is fulfilled for arbitrary ne if all ai and Dai areequal and independent of the ion species. In this case, the ioncomposition does not affect the plasma decay at all, whateverthe reactions are.
A second case must be discussed: Let us assume that neis separable in the space and time coordinates:
n (r,t) = a (r) * b (t) (34)
Then the last terms in Eq. 32 vanish and it remains:
V (Da -- a. ab) = 0ai a 1 (35)
If we resolve Eq. 35 for b, we see that it is no longer time
dependent. Therefore we must exclude recombination com-pletely. Thus, ne must be given by the fundamental modeof diffusion.
5.3 ConclusionsThe physical situations where the application of a propor-
tional model can be justified a priori are restricted to eithercorrelations between coefficients that are normally uncor-related or the fundamental mode of the plasma density distri-bution (including the steady-state Schottky column). There-fore, in all real cases, diffusion terms (Eq. 15) have to beused in Eq. 20.
6. MODEL CALCULATIONS FOR ANELECTRON-BEAM-PLASMA WITH TWO
ION COMPON ENTS AND RECOMBI NATION
6.1 Physical Foundations and Solutions of the Model Equa-tions
A simple model is established for a long cylindricalplasma produced by an electron beam. In order to save com-
puter time, the one-dimensional approach is used. The reactionscheme given in Figure 1 indicates the electrons and ions 1are formed by ionization, whereas component 2 is producedby reactions of 1; both components have different mobilitiesand coefficients of recombination. The particle balance
410
equations have the form:
ianl = D(n1) + f1 (r) -vn1n 1a e
ain2 = D (n2) + vcn a2n2ne
T: = * az u:ct D 12 at
vstwe obtain a normalized system:
AgN + f (9) - G(t-D)N1 - SNe(Ne-Ni (1-T)) = 0
ne = n + n2 (36)
As before, we shall specialize this set either for the steadystate (a/at = 0) or the afterglow (f1 = 0) case. This reactionscheme applies for example to a noble gas plasma containingatomic and molecular ions (provided that the influence ofmetastables can be neglected) [211.
For the solution of Eq. 36 in the steady-state form,we can use the balance equations for ions 1 and electrons.The latter is obtained by adding together the two equations(Eq. 36) after dividing them by the pi and taking into accountthe neutrality condition (Eq. 3). The advantage resulting fromthis procedure is that the diffusion term of the electronsis now a laplacian: An + f 1 - v n ( 1 1 )
al al a2
A,N1 + ULN1-(e
VSNJ) +
Ne
V?Ne VgN1 + (U+1) [f(g) - GN1-TSN N = °
(41)
Since, for comparison, we intend to show the differenceseffected by the "proportional" diffusion operator, we writedown also the second equation in its proportional form(the electron equation is not chanqed):
A N1 +f() -GN1 - TSNeN, = 0 (41b)
ne (n1
+ (n -n ).2 )=al e 1iD
U+U [U (n(Ane I1)e + e e
Vn *Vnlne
+An I D nl _ Dalnn =e
al al alWe now normalize the radial coordinate to the tube radiusR and extract from f1 the normalized radial profile:
ri-: = R;', (38)
= : Af () ; 2Trf (')fds = 1Dal
0
The systems (Eq. 41) respectively (Eq. 41b) are solved withhomogenuous boundary conditions on a computer. A totalstep iteration scheme with under-relaxation accounting forthe nonlinear recombination terms is used; with ten pivotpoints and a convergence limit of 10-5 about 100 iterationsare necessary to obtain a solution.
For calculation of the time dependence in the afterglow,we use the steady-state solutions as initial conditions. Thebeam f1 is switched off at time zero, and a sudden thermali-zation to gas temperature of the electrons is assumed. Con-sequently, we have in the afterglow (index [a] ):
U = 1; D (a) =2_ . Uat 1 +(39)
By dividing the whole system by A, multiplying it with R2and with the abbreviations [ 1 9]:
2
G. =Dal
R2S:
D
a2
D: = Dl= ill
Da2 1'2
N : -n= . Ie, 1 A
(42)
Therefore, the parameters G and S must be modified. Besides,a U -2 dependence is assumed for the coefficients of recombi-nation:
(a) U+ G;S (a) = u+ 1/2SG G= ~~U (43)
The time scale is normalized to the diffusion relaxationtime of ion species 1: T: = t Dl t * Dal 2
A2 R
411
=W eU+
2A9: = R A
(40): = RV
with Jo(1o) = 0 , Bo = 2.405 (44)
With these assumptions the normalized afterglow equationsare written as: aN1 - 1 1 1+N (A9Na -
mI, B2 ( 2LLN 1+_ 1 Ne
IN2 + G NiN11-Ne Ne
TS (a) N Ne)
iUN v1 F (a)- A2NG (1--0
2
1- e e- 1-T)) I + D UT
Recombination Wall Recombination
(45)The electron equation which seems rather complicated has
nevertheless the most useful form for a computer solution,
since the square bracket contains no new derivatives and the
second term in given by the first equation.
The system (Eq. 47) is solved by a linear extrapolation
technique with 100 steps for the unit interval of T. In paral-
lel, also here the proportional system is solved with the ion
equation;
UNT - 1[2 N1 - G N -TS N1NJ
6.2 Parameters (4 5b)The parameters chosen apply to a negative glow in hydro-
gen with a water contamination at a pressure of about 10 Paand currents of some mA. Ion component 1 can be identifiedwith H3, which is formed from the primary ions H2 sofast (mean life of H2+ less 1 ps) that it can be assumed to be2 ~~~~~+the primary ion itself; component 2 by H30 that is theproduct of an ion-molecule reaction between H3+ and H20with a rate constant of (4-5) x 10 9 cm3 s-5 [24],[25].The mobilities of the ions are due to Miller et al. [26] andFleming et al. [27] respectively, the coefficients of recombina-tion are taken from Leu et al. [18],[19] . The (rounded)reduced parameters D and T are therefore .9 and .2, thereduced electron temperature U is assumed to be 5. Thereduced conversion frequency G (which is proportional to
the H20 partial pressure) is varied between 0.1 and 100
(according to H20 concentration in H2 of about 10 -6 to
10 -3). The coefficient of recombination S (proportional to
Figure 1. Reaction scheme for an electron-beam plasmawith ion conversion and recombination.
A and therefore to the current of the fast electrons [19] )
spans one order of magnitude between 100 and 1000.
The shape of the electron beam is given by a bell curve:
f() oa exp (-( /YB) ) (46)
with beam radius PB = 0.35 and m = 2.2 [19]. The rather
contracted beam deviates clearly from the fundamental mode
profile, which is essential for the effect of the diffusionterms to be demonstrated. Although calculations with otherbeam profiles were carried out, the results are not presentedhere for reasons of the size of this paper.
6.3 Results for the Steady-State PlasmaThe results are first presented for one example with a
special set of parameters (S = 100 and G = 0.178) and thensummarized for the whole range of S and G.
In Figure 2 the normalized profiles of the ionizing beamf and of the densities ne and n1 (not distinguishable) and
n2 for the case of the correct (c) and the "proportional"model (p) are shown. The most striking feature is the greatdifference between the n2-profiles. By plotting the concentra-
tion of ion 2 relative to the electrons c2 versus the radius(Figure 3), we see a much steeper increase towards the wall
for the correct than for the "proportional" model. The factthat the "proportional" model gives an increase at all isa contradiction to the assumption under which it was built
up. The wall current of ion 2 is about 30% higher for the
412
1
Figure 2. Density profiles normalized to the axis for the beamof ionizing eletrons (f1), electrons and ions 1 (Ne) and thecorrect (c) and "proportional" (p, dashed) solution for ions2 (N2).
correct than for the proportional model. These observationsare summarized for the variation of G and S in Figures 4 and5. The densities ne, n1 and n2 in the tube axis (p = 0) arenormalized to ne(O)IG = 0.1 and the currents r1, r2, andrat the tube wall (p = 1) are normalized to re(1)IG = 0.1.Their logarithms are plotted against the logarithm of G;S is the parameter for the two sets of curves. For ion 2 thereduced wall current is higher than the axis density, bothcurves run parallel up to G of about 10 and then converge.The wall current of ion 1 decreases much steeper than itsdensity in the axis. This can be understood from the con-version losses of these ions on their way to the tube wall.The proportional model underestimates the differences in bothcases; giving too high the axis density and too low the wallcurrent for ion 2 and the reverse for ion 1. With increasingrecombination (Figure 5) the differences of the densitiesbecome smaller but those of the currents increase. The de-crease of the electron densities with increasing G followsfrom the higher recombination rate (by a factor of 5) of ions2 and is more pronounced for the high S.
The consequence of this behavior for the interpretationof mass-spectroscopic measurements is the following: If weassume that the wall currents directly reflect the behaviorof the bulk plasma, we would misestimate the densities byfactors ranging from up to more than 10 depending on theactual situation. But even the proportional model still giveslarge errors. If we define Qi as the ratio of axis density to wall
I' ----
0 0.5 1pFigure 3. Radial distribution of the concentration c2 of ions2 relative to the electrons for the two models.
-current for ion i:
n.
liv_,Oi I (47 )
we can conclude from wall currents to axis densities by amodel calculation. Now, we can introduce the relative errorAQG caused by the proportional model
I (P)Q =-Qi- - 1i Qi (48)
This error is plotted versus G in Figure 6 for the two sets ofFigures 4 and 5. We note that AQ is positive for ion 2 andnegative for ion 1, and can amount to 50%. Its absolutevalue is small for the dominating ion. This is natural since,to a first approximation, we could regard the plasma as con-taining only this one species. When the densities of bothcomponents are of the same order, the associated AQ arealso nearly equal in magnitude but of opposite sign. Largedeviations always occur for the minority ions, since theircontribution to the ambipolar electric field is negligible.With increasing recombination the AQ becomes smaller,
413
ll
1
Figure 4. Axis densities and wall currents for electrons and theion species versus reduced conversion frequency G for thecorrect and the proportional model (dashed). Recombinationparameter S is 100.
% A&O
40 -
0.'
Figure 5. The same as in Figure 4 for S = 1000.
0.1 1 10 100 G
Figure 6. Relative error of the ratio of axis density to wallcurrent for the two ion species 1 and 2 with parameter S.
Figure 7. Electron density profiles in the afterglow with timeas parameter.
414
l
because the influence of diffusion is less pronounced.
6.4 Results for Afterglow PlasmaIn Figures 7 and 8 the time development of the electron
0.1 - - n -
Figure 8. Time development of the radial distribi
C2 for the two models (dashed = [P]).
density profiles and of concentration c2 are shown for one
set of parameters (G = 1, S = 100). Figure 9 gives the decaycurves of the logarithms of densities and currents. Up to re-
duced times of about 3, the density and current of ion 2
show a very different behavior. The former decaying very
slowly after a short steep decrease caused by the enhancementof recombination (see Eq. 43), the decay of the latter cannotbe distinguished from a straight line indicating an exponentialdecay. The time constant of this curve, however, is greaterthan that of a pure n2-plasma in the fundamental mode
(by a factor of 1.35 in our example). This is due to the pro-
duction of ions 2 from reactions of 1 that continues duringthe afterglow. From Figure 8 one sees that the nonproportion-ality error is decreasing absolutely and relatively in time,also Figure 9 shows that in the late afterglow the currents
and densities run parallel. All calculations, if carried out fora time interval long enough, end in density profiles given bythe radial fundamental mode, which can be seen from Figure7. This proves the consistency of the thoretical considerationsof Section 5.2. Every decaying plasma had to end up in thisdensity distribution, if no other effects (development ofspace charges [10] and transition to free electron diffusion
Figure 9. Decay of densities N1, N2, and Ne and wall currents
ri and r2 in the afterglow: semilogarithmic plot versus
reduced time t.
1 p [7] ) would gain importance in the real physical situation.Finally, the time history of the density/current ratio
uttion of error is shown in Figure 10. Again, the tendency of the plasma
%AQ
40 -
0
-40
-60'
S=100
AQ2
a1
S=1000
1 2 r
Figure 10. Time history of the error of the axis density/wallcurrent ratio with parameter G for different S.
to become more "proportional" can be seen from this figure,with errors less than 5% after two time constants (some100 ps) in all cases considered here. The cause for the error
of ion 1 decaying much slower for G = 10 then that of ion 2
415
0-In (n/no)(r/ro)
- 1
2 -X 2.5
may be due to the fact that ion 1 rapidly becomes the mino-
rity ion, because it is lost by diffusion, recombination, and
conversion.
7. CONCLUSION
45
678
From a straightforward derivation of the particle balance
equations for a plasma with several different ion components,it has been shown that in the general case the density distribu-
tions of the ions cannot be proportional. Some of the special
cases where proportionality is possible have been discussedin detail.
A numeric comparison of the correct and the "propor-
tional" diffusion models for a simple reaction scheme gives
appreciable differences in the ratio of axis density to wall
current, the key variable for the interpretation of mass-
sprectroscopic data. Although these results cannot be genera-
lized, they illustrate the importance of describing a multi-
ion-plasma by a "non-proportional" system of particle balance
equations.
ACKNOWLEDGMENT
The author wishes to express thanks to Prof. W. Roll-wagen and Prof. A Heisen, who suggested this work, to Prof.G. Pfirsch for valuable discussions, the Leibniz-Rechenzentrumfor computer time, and to the Deutsche Forschungsgemein-schaft for financial support.
REFERENCES
1 E.W. McDaniel et al., Ion-Molecule Reactions, New
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1972.3 G.E. Veatch and H.J. Oskam, Phys. Rev. A 1, 1498
(1970).
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