J. Plasma Fusion Res. SERIES, Vol. 4 (2001) 574-57'7

Effects of Atomic Processes on Sheath Formation
NISHINO Nobuhiro
HiroshimaUniversity,l-4-lKagamiyama,Higashi-Hiroshima'739-8527'Japan(Received:5December2000/Accepted:13September200l)

Abstract

using simplified MHDof atomic processes has
involving collision effects. The results showed that even a small rate

Effects of sheath tbrmation due to atomic processes were calculated

equationspresheath boundary to be smooth' Assuming
an effect on sheath depth and length, causes the sheath andcharge exchange processes increase ion
andionization
the Boltzmann distribution of electron density,
recombination processes prevent sheathhand,density, and thus stabilize sheath formation. on the other
does not exceed sound velocity
formation. When there is a large rate of charge exchange, the ion velocitywall).

at the wall (here the current density is assumed to be zero at thewas calculated by simple MHD
Consequently, in this paper, the condition of sheath formation
temperature gradient will be the next
equations. Therefore, the effects of pressure gradient and/or electron
rates'problem to solve in the near future due to the spatial profile of atomic reaction

Keywords:
atomic Drocess, sheath boundary, potential depth, ion velocity, twofluids model

sheath region and/or presheath region were not wide

1. Introduction

with respect to the typical length of the electron

Much research has been done with regard to sheathtemperature spatial profile, it would be important the effects of atomic processes on sheath formation assuming

formation at various conditions [1-10]' Recently K'U'
Riemann has investigated the charge exchange process
on sheath formation tll. X.P. Chen has also studied thecondition of sheath formation involving the charge
exchange process [2]. However, few considered thethat electron temperature remains constant'
In this paper, the effects of sheath formation due tothe ionization, recombination, and charge exchange
processes are calculated by simple MHD equations'

ionization and recombination processes in the above
studies [3]. Since, in general, atomic processes, except
charge exchange, indeed change local electron tem-2. Governing Equations
To simplify the calculation of the above-mentionedprocesses, one-dimensional two fluids MHD equations
were used with no consideration to the magnetic field'
The ion and neutral fluids are assumed to be cold in all
regions, with the ion having a small drift velocity to theperature and density, the assumption of the constantelectron temperature is no longer valid, especially whenneutral density is large. Therefore' it is believed that thetreatment of sheath formation by conventional methods
is not sufficient.

wall from the beginning. The electron density is

It is worthy to note, electron-electron collisionfrequency is larger than that of ions in most cases'

assumed to be Boltzmann disfibution' The conservation

Therefore, the Boltzmann disfibution regarding electron
density would be valid because of thermal equilibrium
dependent on the local electron temperature' If theequation required to be solved.

laws of ion particles and ion momentum' and Poisson

02001 by The Japan Society of Plasma

Corresponding author's e-mail : nishino@ hiroshima-u'ac'jp)t4

Science and Nuclear Fusion ResearchNishino N., Effects of Atomic processes on Sheath Formatron

dn. dnV.-+,dt dx -s,(t)choose the potential absolute value; then the initialpotential g is set to zero. The ionization and recom_Q)

bination rates represent the rate of increase of irIJ;byunit length, and the ionization and charge exchange rares
represent the rate of decrease of IJr2 by unit length.
If the atomic process is zero, these equations(3)

represent the collision-less sheath theory.

(4)

3. Calculational Results and Discussion(s)= rteo = 10re m-3, r?o = 0.1 nio, Te = 100 eV, g = 0, dtpldx

dn,m,V dn,m.W

dtdx^ d2o =-\nitod*,ne)en"= ilaexp(+$)
KI.

Sr= n"noKro*- n"n,K** - nf;nrK"*LS^= m,v n,no(ovlThe initial values of the calculation are set to be n16= O, Ui = 0.01, and deuterium plasmas. The normalized
ionization and charge exchange rates are the range of Ix l0-3 to I x l0-7 corresponding to the edge plasma

r*+(6)

parameters in tokamaks. The calculational results withIn above equations, ni, V and mi are the ion density,
velocity and mass respectively. n" and T. are electron

initial values could be set arbitrarily; however, if theatomic process is neglected, nothing will happen in thism rV ( n"n,K u + n2" n, K"*L)

atomic processes are shown in Fig.l (a), (b). Thesedensity and temperature respectively. S1 and S- are thecalculation. Fig.l (a), (b) show how the atomic process
forms the sheath, in particular if charge exchange rate ision particle source and momentum source respectively
due to the atomic processes. S1 includes the electron
impact ionization rate K1s1,I and all recombination rates

huge, even the ion velocity is smaller than soundvelocity, sheath formation occurs. Also, when recom_(i.e. radiative recombination rate Kp*, dielectronic
recombination rate Ks*, etc.). S- includes the charge
exchange rate (oV)gy and all recombination rates. E isthe electric field and @ is the potential. The neutral
density profile ne(-r) is needed to solve these above
equations. The profile of neutral density is assumed as
constant in this calculation.
Simplifying eq. (2)by subtracring eq. (l) and using
a steady state formula (i.e. dldt = 0), after normalization

bination processes exceed the ionization process, thesheath formation is unstable in this calculation (notshown). To understand this situation, the momentum
loss must be set to zero, and recombination rates mustnot be zero. In this case, the ion density decreases in thefirst step, and does not change, (i.e. electron density
becomes greater than that of the ions). At this point, thesheath is not formed any longer.

Due to the negative effects of recombination rateson sheath formation, we will investigate the ionization

process, we get:and charge exchange processes independently and morein-depth. The parameter scan ofionization (io) rates andcharge exchange (c.r) rates are shown in Fig.2-4. The
boundary of sheath and presheath is ambiguous in thisd(fi,u,) S,; S,: fi"^ fio Kion:ffix.iti K,u A)= , =,, dU,=-'-da 3.,
diS. = i" fio Kron+ n, n"\ o u,)"*d'o,,,2=-lni-ex77), q=*kTox(s)calculation due to the continual solution. We can thendefine the boundary as being the point where the valueof electron density is half that of the initial density thatmeet collision-less sheath theory. And we define the(9)wall where the current density is zero.
Figure 2 shows wall potential dependence on the io
and cx rates. When the io rate is greater than the cx rate,

where Si is the normalized particle source, and.i- is thenormalized momentum source. It is noteworthy thatrecombination rates are included in only

potential depth does not change. However, where the c-r

rate is greater than the io rate, the potential depth
increases drastically. On the other hand, ion velocity
does not change under these parameters. Figure 3 shows
the sheath length (i.e. from beginning to the wall) within
the parameters of io and cx rates. In these parameters^i,, whereas theionization rate is included in not only 51 but also.i-. U,
is the normalized ion speed by ion sound velocity, andii, i" and is fie the normalized ion, electron and neutral
density by initial ion density. We have freedom torange, sheath length increase with decrease of io and cx
575Nishino N., Effects of Atomic Processes on Sheath Formation

rates. Figure 4 shows the results of ion velocity at theboundary (half-n"6 point) within the parameters of io andcx rate. It is clear that ion velocity is close to soundvelocity where the io rate is greater than the cx rate.

This region looks collisionless.

Sheath formation, the conditionof ii > i.,

Pote n tio

Potentio I
2200

2_ -2:,-4'58o1003002oo400100500

150x(Debye-lenqth)x(DebYe-length)lon Velocil
0100150

X(Debye enqih)

1.2

1.2

1.016 08;v

3 0.4

P o.ao.2

4.2
0.oo.020ar000300

x(Debye length)

X(Debyelengih)

(b)

(a)

Fig. 1 Typical results of calculation including the atomic processes'
velocity;
Horizontal axis represents the length Ly Debye length; upper shows the potential; middle shows the ionprocesses
atomicotherthex
10-2,
and
1rate
=
lower shows ion and electron densities; (a). normalized ionization
to zero'are set to zero. (b). normalized charge exchange rate = 5, and the other atomic processes are set-3.0E-cPo)

c-4.0E-5.0E-9q)-tr
F^ >\:t -6.0E=c)
i:Joq)

FJ.!cic:f

1.OE-o6

1.oE-o7

11.0E-05.OE-07 l.oE_05
lo rateio rate

1.0E-03

1'oE-03cx rateFig. 3 Sheath length versus ionization (io) rate andcharge exchange (cx) rate.

Fig. 2 Wall potential versus io rate and cx rate'

lo rate means normalized ionization rate, and cx
rate means normalized charge exchange rate.
Wall potential drastically changes in the region
where the cx rate is greater than the io rate.

Sheath will become large with io rate where thecx rate is greater than the io rate.

s76

Nishino N., Effects of Atomic processes on Sheath Formation

formation were investigated. The results of calculations
show that the ionization and charge exchange processes
permit the sheath formation, whereas on the other hand,g#'the recombination process inhibits the sheath formation.
The boundary of the sheath and presheath region is not9'E eclear in the existence of atomic processes. It is natural todefine the point where electron density is half of the.6-4

initial value as the boundary, which satisfies thecollisionJess theory.
r.E-07

Within the parameter range of ionization andcharge exchange rates of I x l0-3 to I x l0-7, it isr.E-05to rate; 1.E-03"c-03 cx rateconcluded that when the charge exchange rate is greaterthan ionization rate, the potential depth increases withthe charge exchange rate; whereas on the other hand, theFig. 4 lon velocity at half-n* point
lon velocity seems to be sound velocity where thewall potential and sheath structure seem to meet the re_quirement of the collision-less theory when theionization (io) rate is greater than the charge

exchange (cx) rate. lt looks collision-less.

ionization rate is greater than the charge exchange rate.
The effects of pressure gradient and/or electron
temperature gradient will be the next problem to solve

therefore, the increase of U1 and i or the increase of U1

in the near future due to the spatial profile of atomic

and the decrease of f" satisfies the stable sheath

reaction rates.

condition. In eq. (7) the io rate affects the iiUiincrement, but the cx rate does not. In eq. (g) both rates
relate the U,2 decrement. However, in eq. (9) i1 increase

References

tll K.-U. Riemann, Phys. plasmas 4,415g (lgg7).

means dgldr < 0, which overcomes the decrement due[2] X.P. Chen, Phys. Plasmas 5, 804 (199g).
[3] I. Senda, Phys. Plasmas 4, 1308 (199j).
t4l I.I. Bellis and M. Keidar, phys. plasmas 5, 1545

to the io rate, (Jiz will increase gradually. On the otherhand, cx rate decrease only U12, and does not change
irut. Therefore, i1 increase gradually, and then dqldx(1e98).

will be negative more gradually than that of io rate case.
Therefore, in the parameter range of both rates of I x[5] S. Takamura, M.Y. Ye, T. Kuwabara and N. Ohno,

10-3 tot6l G.C. Gas, B. Singha and J. Chutia, phys. plasmas

Phys. Plasmas 5,2151 (1993).

I x l0'7, the c-r rate mainly determine sheath

depth. In general, however, it is noted that this results

6,368s (1999).
I.
[7] Senda, Phys. Plasmas 2,6 (1995).

depend on the initial neutral density spatial profile, ion

drift velocity. Moreover, in this calculation, we neglect
electron temperature spatial profile and pressure term.
Even though, it is clear that atomic processes really doaffect sheath formation near the wall.

t8l S.K. Baishya, G.C. Das, J. Chutia and J. Sarma.
Phys. Plasmas 6,3678 (1999),

t9l I.I. Bellis, M. Keidar and S. Goldsmith, phys.
Plasmas 4,3461 (1997).

ll0l E. Ahedo, Phys. Plasmas 4,4419 (t997).

4. Conclusion
The effects of atomic processes on sheath577