ON PLASMA SHEATH PROBLEMS

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Submitted on 1 Jan 1979

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ON PLASMA SHEATH PROBLEMS

J.P. Lafon

To cite this version:

J.P. Lafon. ON PLASMA SHEATH PROBLEMS. Journal de Physique Colloques, 1979, 40 (C7),

pp.C7-843-C7-844. �10.1051/jphyscol:19797407�. �jpa-00219407�

JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6ment au n07, Tome 40, JuiZZet 1979, Fzge C7- 843

ON PLASMA SHEATH PROBLEMS

J.P. J. Lafon.

Observatoire de Meudon, Departement Recherches Spatiales (DESPA) C . N . R.S. L. A . 264, 92190 Meudon fiance.

The analysis of a large number of geo

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pnyslcal and astrophysical phenomena (such as electric charging of a body in a space plasma, alteration of the characteristics of wave emission by an antenna in a hot plasma, behaviour of a plasma probe

...

⁾ requires an accurate theory of the sheath surrounding a biased conductor in a collisionless plasma. One has to solve strongly self consistent, non linear and non local integro- differential equations : the sheath depends on the potential of the body which is self consistent with the sheath structure when it is not fixed;

the sheath governs the collected currents and so, sometimes, the emitted currents; all these cur- rents may conversely control the potential of the body (and so the sheath structure) when it is not imposed (cf fig. 1). Indeed, ignoring such pheno- mena may lead to erroneous analyses of what occurs.

The sheath structure and the collected currents have been investigated in a number of particular cases by various authors1 using dif- ferent (physical or mathematical) approximations depending on the cases considered. Roughly spea- king there are two classes of problems : those for which spherical or cylindrical symmetry can be assumed and the others. These problems can be solved using completely different formalisms. In particular problems of the second kind require the use of astill much heavier formalism. For problems of the first kind (cylindrical or sphe- rical bodies)

,

in non magnetized plasmas, elaborate, but also complicated theories have been developed by Bernstein and Rabinowitz 2

,

Moskalenko 3

,

and ~aframboise~. However surface effects are not taken into account without more or less crude approximations. On the other hand effects such as reflection or emission of parti- cles by the body may be important; besides there are few works concerning the case of magnetized plasmas and in this case the approximations used are rather crude 1

.

The equations governing the sheath struc- ture are a Vlasov equation for the distribution function F of the particles of each species s together with Poisson's equation which determines the electric potential in terms of the F

.

^{Even in}

spherical or cylindrical symmetry the solution of these equations is difficult because they are strongly non linear, non local and integro

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differential. Without crude approximations they can be solved only using numerical iteration in which the main problem is always the classifi

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cation of the particle orbits.

Now we have found a formalism in which the problem appears to be a particular case of a general problem which consists in finding .the intersection of an infinite number of domains depending on a continuous parameter in an n-di- mensional space as that of a finite (small) num

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ber of domains easily delimitablely5. We have proved a general theorem and we have derived from it a systematic method for solving the problem 6

.

The method is well fit for numerical computations and even reduces them to the minimum when they are necessary. In particular it is a powerful tool for solving the sheath problem completely for conductors with any surface effects as soon as the physics of these effects is known, even in some magnetized plasmas 9 .

Of course, strictly speaking, absolute self consistence requires complete radial (cylindrical or spherical) symmetry for exact solution. However, as a property of Poisson's equation which smoothes the space charge ir

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regularities, this condition is not so important as it seems at first glance and rather strong asymetries can be taken into account with very good accuracy8"' provided that they do not disturb too much the symmetry of the electric potential (which is the only thing truly necessary).

Using our new method we have investigated the behaviour of bodies reflecting some of the particles striking them (rather different effects are produced depending on the polarization of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797407

the b'~dy),~'~'and that of bodies emitting phyto

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electrons8w'''. We have also investigated the case of cylindrical bodies in some weakly magnetized plasmas9

.

Of course it is not possible to discugs all the results here : they are discussed in detail. elsewhere 1,s to 11

The figures shown hereafter illustrate some of the.characteristics of the sheath structure under various circumstances.

Finally there is a wide field of possible applications of the theorem and of the method, in particular for investigating all plasma sheath problems which can be stated under a simi-

1,12 ,lar form

.

REFERENCES

1

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LAFON J.-P. J.

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Etude d'une classe de syst8- mes ob6issant B un ensemble continu de condi- tions dans un ensemble 'continu dlbtats. T h h e de Doctorat d'Etat Qs Sciences. Universitk de Paris VII

.

1976. Rpt Observatoire de Mkudon DESPA159bis.

2

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^{BERNSTEIN I. B. aab}

RABINOWITZ

I G*;

-

~ h y s . Fluids, 1959,

2,

112-121.

3

-

MOSKALENKO A. M.

- ^xvth

Int. Astron. Cong., Warsaw, Poland, 1959, 7/12 September 1964.

4

-

LAFRAMBOISE J.

-

Theory of cylindrical and spherical Langmuir probes in a collisionless Maxweilian plasma at rest. Rpt nOIOO.of the

Institute for aerospace studies of the University of Toronto, Canada (UTIASIOO), 1966.

5

-

LAFON J.-P. J.

-

Journal of Math. Phys.,.

18,

1977, 1178-1187.

6

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LAFON J.-P. J.

-

Plasma Phys.,

17,

1975, 731-740 and 1175.

7

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LAFON J.-P. J.

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Plasma Phys.,

11,

1975, 741-756.

8

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LAFON J.-P. J.

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Radio Science,

11,

1976, 4 8 3 - 4 9 3 .

9

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^{LAFON J .-P. J.}

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Journal of Plasma Phys., 10, 1973, 383-396.

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10- LAFON J.-P. J.

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To appear.

11- LAFON J.-P. J.

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To appear.

Fig 3

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Spherxcal body emitting photoelecrrons zn a p l a m a . Electran density profiles i a planes thxau%gh the center of the sphere and a t a qegularly increasing angle ( 18' Istap) with the direction af the ligrh.(TeleccrOns

= Tions = T, rsdibs of the body = Debye length) Fig. 4

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Spherical body emitting phoraelecrronr i n a plaonp. 1 3 0

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electron density curves; T -

ions - =elecCronr = T Tph,,toelectrons ST; Radius o f the body = p (in Debye lenarhs)

Dlsiribution function of =he parricles chat leave a mersllic body in a plasma .

g, denotes rhe o ~ s r r ~ b u t i o n function a € the emrrted parizcles ( p h o t o e n ~ s r ~ o n , secondary emxssron, t h e m o e ~ i s s ~ o n

...

⁾

ygd denotes the dircriburion funcrlon of the diffusely reflected particles ( I = depends on that of the particles rcri!hng the body)

ll denotes the c o e f f i c ~ e n r of rpecular reflecrion

L~-.,,i.r.i;i~n~.~~) cd1~ctc.1 iq rlru ,LIOI~L.

L I ' . n l i d i \ cl,liffr<l h LIE ~ l i v L c

The brackets are equal to 1 or 0 depending on the energy a?,$* the mopyRntm &the particles.

-.

Effect oC parrlal reflectran ( r e f l e c r ~ m coefficient 1) of rono of the parclclea strrking a c y l ~ n d r i c a l body. T i o n s =

T.leC~rons =

=.

p, Yp, denote respecrively the renperarure o€ the parcicler:

cha radius of the body (1" Debye lenpcps), the poceniral of rhe body ( r n kT

13.

nd the charge of a particle; index a for parrxclcr such chat .q V C 0, b for rhore for vhrch q V > 0.

= P b P

The current collected is O M L (Orbital Motion L ~ m i r e d ) i.e.

independent of rhe sheath structure in the regLon9 i n d ~ c a r e d i n the folloulng table :

B + C + D

parrlcles b everywhere

lIa = 0 va ^{= o}

1, = 0 lib = 0

C o n L r a c t ~ o n of a c y l i n d r ~ c a l sheath by an axial magnerlc freld.

Curves i , i v , ~ i . ~ i i correspond respectively t o rhe valuer 0.,0.1.

0 . 6 , 2 . o f the ratio (Debye lengrh/lannor radius); (TLons = T e l e c t r o n S ; rzdlus of the body s Debye lenechJ