CHINESE JOURNAL OF PHYSICS VOL. 13, NO.

z OCTOBER, 1975

Studies on a Spherical Probe Immersed in Plasma

D. C. AGARWAL*
J. K. Institute of Applied Physics and Technology
University of Allahabad, Allahabad-2, India

(Received 4 August 1975)

The paper deals with the studies on a spherical probe immersed in plasma.
The d. c. potential distribution and the space charge density have been computed
numerically for both the rigid and absorptive boundary conditions. The r.f.
characteristics of spherical antenna immersed in the plasma have also been
studied in detail.

1. INTRODUCTION

@xi is now well estabilished that when an electrode is immersed in a plasma,

it is enveloped by a plasma sheath which shields the .major. portion of the
plasma from the disturbing field caused by the electrode. If the electrode is
maintained negative, zero, or slightly positive with respect to plasma potential
the shielding effect of the plasma sheath continues to be effective. The sheath
is very thin-of the order of a Debye length and the major portion of the plasma
is undisturbed. In the region of the sheath the potential of the electrode in the
unperturbed case, increases monotonically from the negative value on the wall
to zero exponentially near the edge of the plasma sheath.
‘The importance of the study of the d. c. potential gap was pointed out by
Rabinowitz and Bernstein”) and then by several other investigators(2-4). All
these studies were concerned with the electrostatic probes and the starting
equation was the Boltzmann equation with the assumption that there is no
reflection
I of particle on the conductor surface. This boundary condition is based
on the assumption that the temperature of the conductor is low as compared to
that of plasma. However, for the study of the r. f. characteristics ‘of antennas
immersed in plasma the d. c. potential distribution is required to be simple as
much as possible and we have to calcu!ate it numerically. It is therefore purpose
of this paper to calculate this d. c. potential distribution and the space charge
density numerically and these values will be utilized to study the r. f. character-
istics of spherical antennas immersed in plasma.
* Lecturer, now on leave of absence at the Institute of Physikalische Weltraumforschung, Hidenhofstrasse
-8, 78, Freiburg, on a DAAD Fellowship.
( 1 ) I. N. Rabinowitz, and I. B. Bernstein, Phys. Fluids, 2 112, (1959).
.( 2 ) S. H. Lam, Phys. Fluids, 8 73, (1965).
( 3 ) S. H. Lam, Phys. Fluids 8 1002, (1965).
( 4 ) T. Aso, Radio Frequency Probe and its Applications to Ionospheric Probing, Ph. D. Thesis, Ionospheric
1. Research Laboratory, Kyoto University, Japan, 1974.
. . I
167 ’
 168 STUDIES ON A SPHERICAL PROBE IMMERSED IN PLASMA

2. STATEMENT OF EQUATIONS AND MODELS

The potential distribution for the antenna immersed in the plasma is shown
in Fig. 2.1. The basic equation governing the electric potential for such a
geometry may be represented as

Fig. 2.1. Ion sheath and d. c. potential distribution

along spherical antenna.
.

where & is the d. c. potential, e is the charge of electron, EO is the dielectric

constant in free space, N; and N, are the ion and electron density respectively.
For a spherical symmetry, the independent variable is simply r. Then we define
the normalizing parameters as

Then Eq (2.1) becomes

----=+w&?)
d2X
d2
(2.3)
In Eqs (2.2), k is the Boltzmann’s constant, T, is the electron temperature, 8
is the radius of the sphere, Ro is’ the Debye length, r is the distance of some
point from the center of the sphere, N, is the electron density at very far
position and x is the normalized d. c. potential distribution.
Eq (2.3) is to be supplemented by other equations governing the number
density distributions n; and %. For continnum plasmas these would be the
equations stating the conservation of mass and diffusion velocities. For a
collisionless plasma the number density distributions ni and ytB are to be
calculated by a detailed analysis of the trajectories 6f the particles. Under
the assumption of mono energetic ions, Bernstein and Rabinowitz”) give the
following expressions for ni.

(2.4a)
 D. C. AGARWAL
169
where ,8 =4T,/zTe the ratio of ion and electron temperature,
Z=I;/(nR2N~1/2(4kTi/n+kT~)/mi)
I;: being the total number of positive charges absorbed by the probe per set and
rni is the mass of ion (also called as normalized ion flux denity).
The expression for the electrons may also be represented as
7Z,=F (2.4b)

Substituting Eqs (2.4a) and (2.4b) into Eq (2.3) we get

(2.51

Now at a very far point from the sphere, ni=n, and so the right hand side of
Eq (2.5) becomes zero and we have

(2.61

Thus taking Eq (2.6) as the initial condition Es (2.5) can be solved nu-
merically. Here we have used Runge-Kutta method. First we assign some vaIues
to R and B as (5,7.5,10,20,50) and (O,O.Ol, O.l,l) respectively and then x and
x0 are calculated. (x, is the normalized antenna d. c. probe potential). Now,
putting ni - n, as a function of h at the conductor surface the new boundary
conditions may be represented as

hZ (12G.G)
ni-n,= (2.7)
0 (O<.Kzl)
h is some constant, and then the d. c. potential distribution becomes as

x= JE I+Z_-_z_
2 (2 z .zl 1
(l>‘ozo)
(2.81
I 0 (O>z=0)
Here & is the boundary between the sheath and plasma and may be represented
as
1
z= (2.9)
1+J$*+
Fig. 2.3, shows the tii-n, variation. (For calculating ni-n, first x is calculated.
Then nd is obtained and then ni- n, may be obtained). In Fig. 2.2 the variation
of h has been shown. In the calculations of Fig. 2.3, the parameters assumed
are R=lO, x,=5.9, h=0.39. Now the sheath thickness ds =R(l/Zo-1) and this
has been plotted in Fig. 2.5. When we do not give d. c. bias potential’to the
antenna, xa will be represented as floating potential and it can be determined
by calculating x,, versus x. But when R is very large floating potential x:,f is
determined by Es (2.10) given by LamC2)

_,. .., _, , .._

 STUDIES ON A SPHERICAL PROBE IMMERSED IN PLASMA
-
t 6
MO, x&=5.9

‘z=R/r
Fig. 2.2. Potential distribution 5,. ejection dekity ns,.iob density
ni, charge density ni--ne (all quantities are normalized).

,...

Fig. 2.3. Approximation of charge density tli-tie and potential

distribution x.

i2.10)

and there is little deviation when 2 is relatively small. Using Eq (2.10) we

obtain normalized floating potential which is shown in Fig. 2.4.

3. R. F. CHARACTERISTICS OF SPHERICAL PiiOBE IMMERSED

IN COMPRESSIBLE PLASMA

There are two methods for studying the r. f. characteristics of spherical

probe immersed in plasma, namely the hydrodynamical approach and the kinetiG
 D C. AGARWAI, 171

Normalized Antenna Potential

Fig. 2.4. Normalized antenna potential vs. normalized charge
density at antenna surface..

approach. The hydrodynamic approach has the advantage of simpbcity, of

course, not exact as that the kinetic approach. Here we have utilized this
hydrodynamical approach to derive the field equation for the study of the
antennas surrounded by sheath and immersed in plasma. Then it is solved
numerically with the parameters such as antenna ,radius and antenna d. c.
potential.
Using the linearized hydrodynamical equations the necessary equations may
be written as:
Electron Continuity Equation

~+v*[N.v]=0 (3.1)

The equation of Motion

VP + v (Xv, VI + -J$~%E = - Ym~,~ (3.2)

The poission’s equation

v.E= --eNc (3.3)

60

and the gas law as

P(+-)?= Constant in Time (3.4)

where E is the electric field, N, is the electron density, v is the mean of electron
velocity, p is the pressure, m is the mass of electron; e, is the charge of electron,

--_-a; __- .
I

172 STUDIES ON A SPHERICAL PROBE IMMERSED IN PLASMA

5-

4-

3-

2-

0.G l-

0. OL
0

Normalized Antenna Potential

Eig. 2.5. Thickness of ion sheath as a function of normalized antenna potential

~0 is the dielectric constant in free space, Y is the specific heat ratio and v is
the collision frequency. Now we take the following assumptions:
( i ) The field E depends on the r only that is the spherical symmetry has
been taken into account. Also the quasi-static nature of the approximation has
been utilized. In other words, it is assumed that there is no radiation of e. m.
wave, only the electron plasma wave being radiated.
(ii ) ion motion can be neglected, the frequency b&g very high.
(iii) The collision frequency of electrons is neglected.
(iv) The field is linear so that the tota field is composed of zeroth order
(d. c.) and the first order (perturbation component according to ~9”‘).
 D. C. AGARWAL 173

Now the zeroth order component of electron density (HO) can be written as
(see appendix)
ed
no=N,exp ~
( kT, ) (3.5)

where N- is the plasma density at a far region, T, is the electron temperature,

k is the Boltzmann’s constant and $0 is the d. c. potential distribution.
Now from Eqs (3.1) and (3.3) we get the first order terms as
r2 (-jo~0& + elzovl) =const. in space (3.6)

Now expressing the first order field at the surface of sphere as J% we can have
-j@eo&~[-jc&&i + e%&l]r- R (3.7)

Now the relationship between the r. f. electron current -e%ul and the r.f.
electric field El can be expressed as

C ;;
-enovl = joco El- __ Eis
1
(3.3)

From Eqs (3.1) to (3.4) and Eq (3.8) after some algebraic simplifications the
r. f. electric field EL can be expressed as (see appendix)

+02(1+)E,= 02+ (3.9)

where op is the electron plasma angular frequency at a far region ( = /‘e2N,/eom).

Now we normalize + and R by the Debye length of far region (AD =-v’/~okT,/e~N~).
The normalized d. c. potential x will also be used. All these three quantities
can be expressed as _’
;=?-/n D I

e40
x= --
kT,
Then Eq (3.9) can be expressed as

In the plasma region, t is nearly equal to zero and so Es (3.11) becomes (Plasma
region k%, where >c, is the boundary between sheath and plasma region)

(3.13)

^ i _.f -.._ . ~. .
 174 STUDIES ON A SPHERICAL PROBE IMMERSED IN PLASMA

whose solution may be represented as

(3.13)

where
2
K=+&-1 (3.14)

K being the normalized propagation constant of electron plasma wave (normalized

by the inverse of debye length). A is the integral constant and cp=l- c$/c.P is
the effective relative dielectric constant of the plasma. Now the specific ratio
may be represented as a complex function of frequency and we, use Ravikovich’sC2’
method for deciding this function. The dispersion’ function for e. p. wave from
kinetic theory can be represented as
1
__-!z!_~
2 Km, ( v’ ~KOP >
_* +K2+1=0 (3.15)

where Z(C) is the plasma dispersion function. While Eq (3.14) is obtained by

the hydrodynamic theory and in principle it should be in agreement with Eq
(3.15), so by comparing Eq (3.15) and (3.14) we can get Y (the specific heat
ratio). Fig. 3.1 shows the dispersion relation obtained from Eq (3.15) and Bohm
and Gross dispersion relation. Fig. 3.2 shows the specific heat ratio r as decided
by the above method. Essentially Y must be decided from fi which is the first
order electron velocity distribution function. But it is difficult to obtain fi and
so we use the above described method.
Now, in the sheath region (r,(k) it is impossible to calculate r. f. electric
field analytically and so we solve the equation by the Runge Kutta Gill method.
Let the solution of Es (3.11) may be expressed as

(3.16)

“P

Fig. 3.1. Dispersion curve of Landau and Bohm and Gross (Solid
Line: Landau, dash line: Bohm and Gross).

_ .
 D. C. AGARWAL- 175

UP
Fig. 3.2. Effective specific heat ratio Y.

Here FI(r) is a general solution of Eq (3.11) when the r. h. s. of Eq (3.11) is

equat: to zero and EI,F2(r) is a particular solution of Eq (3.11). A is the integral
constant. Now since E1,F2(r) must be equal to the second term of Eq (3.13) at
the boundary between sheath and plasma (r=?o) so ,Eq (3.17) must be derived

Similarly equating the first term of Eq (3.13) and that of Es (3.16) we can get
I (3.17)

Es (3.18) as

Now, we can obtain Fl(q), -F?+(G) by setting the initial condition in Eq (3.18) and
dsing the Runge-Kutta-Gill method. To calculate El, it is required to decide
the integral constant A and since it is arbitrary so the boundary condition at
the conductor surface must be decided. The boundary conditions which we use
here, may be (i) rigid boundary or (ii) the absorptive boundary.
Case 1: Rigid Boundary Condition: This was proposed by Fejer”’ and it means
that the r. f. electron current must be reflected back elastically by the sharp d. c.
potential wall (Eq 3.19)
&R; rtovi=o (3.19)
( 5 ) J. A., Fejer, Radio Science, 68D 1171, (1964).
176 STUDIES ON A SPHERICAL PROBE IMMERSED IN PLASMA

Now, if the electric field at the antenna surface may be represented as Elo, then

EiS =E10 (3.20)

and then the integral constant may be decided by

A= E10-ElO$R) (3.21)
E;(R)
where PI(&), Ij;(E) are the values at the antenna surface and are being de-
termined by the numerical method, and then the electric field is represented as

El(r) = (3.22)

Case 2: Absorptive Boundary Condition:

This boundary condition was proposed by Balmain@’ and is based on the
assumption that all the electrons must be absorbed at the conductor surface
and (ii) the perturbation of electron is in the form of Maxwellian distributio?
In such a case the condition is represented as Eq (2.3)

ra2; n@1=-
J 2k T, n1
___
xm
(3.23)

while n1 is obtained from Eq (3.3) and is given by

nl=_L.2-
e AD [
+ -$ W] (3.24)

Now, we define the electric field at the antenna surface as EIO and then from
Eqs (3.7) and (3.24), El,
-
may be represented as Eq (3.25)

El,=Elo+jOP 2 +

Eq (3.25) may be represented as

0 J 7r c (3.25)

&,=&o(l+B) (3.26)
Substituting Eqs (3.26) and (3.16) into Eq (3.25) with the setting r=R we get

Since PI (.@ and Fz(R) can be obtained numerically so we can decide A and B
from Es (3.27). In this case the electric field is represented as

El(r) = (3.28)
I - --
APO(T) +Elo&t-~)Fa(r), rsro
(6) K. G., Bal_main, Radio Science, 2, 1 (1966).
 D. C. AGARWAi 177
3.1. The Field From Antenna: Figs. 3.3, to 3.5 show the examples of electric
field when we supply the unit electric field into antenna surface and we use the
two types of boundary condition. Fig. 3.3. shows the real imaginary and absolute
values of the electric field. From this figure it is evident that for the case of
rigid boundary condition there is no static electric field component in the
imaginary part of the electric field whereas such a static electric field exists by
a little amount in the case of absorptive boundary condition. However, for the
real part of the electric field, static electric field exists by some amount in both
the rigid and absorptive boundary conditions. For absorptive values of the
electric field it is seen that there exists some interference pattern which is
composed of the static electric field component and the wave field component.
3.2. The Impedance of Spherical Antenna Immersed in Compressible Plasma:
The input impedance 2 of the spherical antenna may be described as

- IEI
---- Ret?
- - - - ImE

\
\ _# N’ Distqc e i n Debye (ength i;

Rigid model
c

Absorp*ive m o d e l

Fig. 3.3. Real, Imaginary and absolute values of the electric field.
 ‘1 78 STUDIES ON A SPHERICAL PROBE IMMERSED IN PLASMA

’ OS3
In Xa=
I'--\ 0
I -I '.
MO, +$,=I6
--- Absorptive
- Rigid mode!.
modet

LJ ,, JU
..
-0.1 -

Length 7
-0.2
t
Fig. 3.4. Imaginary part of electric field.

Absorptive mode{
- Rigid mode(

0.0

-0.05

-0.1 t
Fig. 3.5. Imaginary part of the electric field.

(3.29)

where $lS is the r. f. potential at the antenna surface. Under the quasistatic
approach $lS can be represented as

&.=J;&(W (3.30)

In Eq (X29), Jls is the summation of displacement current density and the

electron current density and can be given as
Jls = jocoE1- ertovl
= jwcoE1, (3.31)
Now, we define the normalized impedance given by

-- L
 D. C. AGARWAL 179
ZN=RN+jxN
=~~EE,,Ro~Z (3.32)

(3.33)

Taking the parameters R=5, 7.5, 10, 20, 50 and x,=0, 1, 2, 3, 4, 5, 7 and 8, ZN
has been calculated by Es (3.33) and is plotted in Fig. 3.6 for both the case of
rigid and absorptive boundary condition. It is seen from this figure that when
o>op the resistance curves for both the rigid and absorptive boundary coincide
if X. is greater than 5. The reactance curves follow the same pattern. However,
there exists a difference that in the absorptive surface there exists some resistance
whereas it does not exist for the rigid surface. Also it is seen that only when
xa is high (>5) there exists much variation in the resistance for the rigid
boundary case whereas for the absorptive boundary case such a variation exists
for all the values of 1~~. Therefore in all our later results we have presented
the case of absorptive boundary. A similar calculation was done for R=5, 7.5,
10, 20 and 50 keeping x,=0, 2, 4, 6 and 8. It is found that in the radiation
region (o>op) resistance is owed to the radiation of e. p. wave and decreases
as xa increases. However, it is not true for R=50. But in the cutoff region

K= 5

I 1 I I

w
I I l\I ,I
2.0
i,6
0.0
..
0.5
w
1.0 1.5 20

UP "P
Fig. 3.6, Variation of normalized resistance as a function of O/O+. . ,’ 1
180 STUDIES ON A SPHERICAL PROBE IMMERSED IN PLASMA

.!
R=S
4

8 1
9
‘;
‘;1
c2 c
P
2
-z
E -1
2
-2

-3

-4

-5
0.5 1.0

w
-
UP
Fig. 3.7. Variation of normalized reactance as function of W/W).

the resistance is owed to the loss of small signal conductance and this also
decreases as x0 increases. The condition is not true at o=O, o=og. It is also
seen that in the cut-off region, reactance shows the sheath resonance which is
affected by xa and this is remakable when R is small. However, in the region
of radiation, almost there is no effect of sheath on the reactance and it is nearly
equal to that of cold plasma model. This phenomena seems to be remarkable
when R is high.

APPENDIX
Deduction of eqs (3.5) and (3.9):
Electron continuous Equation

++v- (Nev) =0 (11

Equation of Electron Motion

-$(N”v) +-$vp+v(N,~, V) +$~E=o (2)

Poission’s Equation
 D. C. AGARWAL 181

vE=--eNa (3)
EO
Gas Equation

p(+J=const. (4)

Now, from Eq (l), we get

v++v(N& V) -N&M) =o (5)

Also, from Ea (2) we can have

(6)

Now, from Ess (5) and (6) we get

(7)

Now we represent V, N,, P and E as the summation of zeroth and first order
. as follows :
v=vo+v1

N,=No+nl
(81
P=Po -l-PI
E=Eo+Ei
First we consider the zeroth order terms: Putting the left hand side equation
to zero, we get
VP0 + eNoE0 = 0 (9)
Then putting po=NokT, we get from Es (2)
kT,vNo-eNovch=O (10)

whose solution may be represented as

No=N, exp -$- (111

( >
Now we consider the first order terms. Then from Eqs (l), (3), (4) and (7)
we can get
84%
ar+~’ (Nod =0 (12)

a= -+-El (13)

p,=rkT,nl (14)

-__..,I
182 STUDIES ON A SPHERICAL PROBE IMMERSED IN PLASMA

(15)

Substituting Eqs (13), (14) into (15) we get

Now using the assumption that the direction of the field vector is radial and
that it depends on r only, Es (16) can be written as (after t;me factor taken
as ei”‘) . .
rk Te
m
l -$--$---$-(p”l,-5 % $ -$(r2EI)-~e2NOEI=j~~.Novl~
l

cow j A,
(17)

Alsp Eqs (12) and (13) can be represented as

i
d r2 -jw*E1+rtovl
dr II( e )I =O .
(18)I
_
That is
r ,. ., .
i r2 ( - jwcoE1 -t e&u,) = Const. in Space ..
(19

N OW on the assumption that at the antenna surface i.e. at r=R the Es (19)
may be represented as -jaoEls. Then we can get

00)
>
I& 3(20) is satisfied all around the region r>R and so the wave equation (17)
becomes

SUMMARY

This paper has been concerned with the studies on a spherical probe immersed
in plasma. The d. c. potential distribution and the space charge density have
been computed numerically for both the rigid and absorptive boundary conditions.
The r. f. characteristics of spherical antenna immersed in the plasma have also
been studied in detail.

ACKNOWLEDGEMENTS

This work was started by the author when he was at the Department of
Electrical Engineering, Tohoku University, Sendai, Japan on a visiting research
fellowship during March 1973-74. He wishes to thank Prof. S. Adachi for sug-
gesting the problem then. He also wishes to thank Prof. Dr. X. Rawer for
providing research facilities to him in his Institute now a days.
. . i

,‘. .