Microwave resonant cavity measurements of radial electron density profile of positive column of gas discharg

Hsuan, H. C. S.

Vlachos, M. A. W.

English

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c 
U. of Iowa 68-30 
Profile Measurements of Plasma Columns 
Using Microwave Re sonant Cavit ie s* 
by 
Michael A. W. Vlachos'and 
Hulbert C. S. Hsuan 
The University of Iowa 
Iowa City, Iowa 52240 
May 1968 
* 
Research supported in part by the National Aeronautics 
and Space Administration under grant I- 
,vG,L-~ ~ - 0 ~ l - 0 0 ~  ( n , T c - a d  
NASA Trainee. + 
I 
r 
ABSTRACT 
The density profile of a posi t ive column of 
a gas discharge i s  measured by the comparison of two 
microwave cavi ty  measurements. The two cavity modes 
011 and m020. used are the TE 
INTRODUCTION 
The posi t ive column of a low pressure discharge has been 
diagnosed by various techniques. In par t icu lar ,  the r a d i a l  
densi ty  p ro f i l e  has been measured by Langmuir probe and 
interferometer techniques. No attempt, t o  our knowledge, has 
been made t o  make such measurements using resonant cavi t ies .  
The Langmuir probe method has been used successfully t o  measure 
the p ro f i l e  only f o r  large diameter tubes. 
techniques can be used f o r  small diameter tubes; they have been 
ca r r i ed  out by many authors and have been summarized i n  a book.’ 
However, it i s  known t h a t  interferometer measurements are  i n  general 
only 1% as sens i t ive  t o  plasma density changes a s  a re  resonant 
cavi ty  measurements (with the possible exception of a recent ly  
reported experiment using a Fabry-Perot resonate$ ). 
fore ,  desirable  t h a t  resonant cavi t ies  be used t o  make such 
measurements. 
Interferometer 
It is ,  there- 
Resonant cav i t i e s  have been used extensively t o  measure the 
density of plasma i n  the posit ive column. 
ments, the plasma was always assumed t o  be uniform, and so an 
e f f ec t ive  number density was actual ly  measured. But,  i t  i s  
noticed t h a t  d i f f e ren t  cav i t i e s  give d i f f e ren t  r e su l t s .  This i s  
In making these measure- 
4 
due t o  the d i f fe ren t  e l e c t r i c  f i e ld  configurations associated with 
the d i f fe ren t  cavi ty  modes, which i n t e r a c t  wi th  the inhomogeneity 
t o  give the observed difference i n  plasma frequencies. By using 
two or  more modes (one or  more cavi t ies) ,  t h i s  difference can be 
explored, and allows us t o  obtain the r ad ia l  density p ro f i l e .  
From various theories,  we can estimate the electron r ad ia l  
densi ty  p ro f i l e .  F i r s t ,  one can obtain the well-known €?esse1 
f'unction prof i le  n = n Jo (fi r/ro) based on the balance between 
the generation of e lectrons due to ionizat ion co l l i s ions  and the 
0 
l o s s  t o  the w a l l  due t o  amplipolar diff'usion. Second, one can 
estimate the p r o f i l e  of electrons i n  the bulk pa r t  of plssma 
through tha t  of the ions because of quasineutral i ty  conditions. 
It was shown i n  a recent calculation3 f o r  the posi t ive column tha t  
the ion density has a p ro f i l e  of the form 
n = n 0 (1 + Cr2/r2)-1+p,/2(ptv), where p and v are  co l l i s ion  and 
ionozation frequencies, respectively. Since p, i s  two orders of 
magnitude l e s s  than v f o r  the mercury plasma,* the density p r o f i l e  
becomes n = n . In e i t h e r  case, i f  we accept these 
formulasfrom anasymptotic point of view, we can wri te  
2 2 -1 
(1 + C r  / ro)  0 
2 2  n = n (1 - C r  /ro) 
0 
where ro i s  the tube radius and C i s  the "prof i le  parRmeter".' 
5 
EXPERIMENTAL THEORY 
For the chosen general form of the density prof i le ,  we 
consider what cavi ty  modes should be used, and how they should be 
used. In  the  l i t e r a tu re ,  Brown' suggested using the TE and 
TMlll modes t o  obtain information about the r a d i a l  e lectron densi ty  
d is t r ibu t ion .  However, the three non-zero e l e c t r i c  f i e l d  com- 
ponents 
For our purpose the TMo20 mode i s  used. 
desirable  features  a s  the ~111 except t ha t  it has only one non- 
zero component. The f i e l d  components of both modes are given below: 
For the TMo20 mode: 
011 
of the TMllL mode, make calculat ions v i r t u a l l y  impossible. 
This mode has similar 
Er = 0 
HZ = 0 
6 
For the moll mode: 
UI E = -  r J1 (xo; r/ C ) s in  
Cp C 
EZ = 0 ..' 
H = O  
xO;2 
H~ - -   J~ (xo; r/rc) s i n  ( a z) Cp 
r 
C 
where r 
x '  are  zeros of the Bessel function. 
i s  the radius of the cavity, d i s  the cavi ty  height, and 
C 
01' x02 
We notice t h a t  these modes have two other  desirable  fea tures  
i n  addition t o  each having only one e l e c t r i c  f i e l d  component. 
shown i n  the diagram (Fig. l), the e l e c t r i c  f i e l d  in t ens i ty  of 
the TMo20 mode i s  approximately constant over the plasma region, 
while that  of the moll var ies  greatly,  being zero a t  the or igin,  
and growing as r . 
d i f f e ren t ly  w i t h  the plasma, whose density p r o f i l e  i s  a l so  shown 
i n  the same diagram. 
"weigh" the density p r o f i l e  d i f fe ren t ly  so a s  t o  determine the 
"prof i le  parameter" c given i n  equation (1). 
feature  i s  that i n  both cases the e l e c t r i c  f i e l d  component i s  
As 
2 It follows tha t  these two f i e l d s  i n t e r a c t  very 
It i s  t h i s  difference t h a t  allows us t o  
The second desirable  
7 
p a r a l l e l  t o  the plasma-glass boundary. Thus, there are no 
local ized e l e c t r o s t a t i c  modes generated a s  i n  the case of Tonks- 
Dattner r e ~ o n a n c e . ~  The strongest r e f l ec t ion  i s  from the plasma- 
g l a s s  interface,  and there i s  negligible re f lec t ion  from the 
densi ty  p r o f i l e  i t s e l f .  This i s  a very important point f o r  our 
experiment and w i l l  be made use of  i n  the appendix. 
But f o r  plasma tube currents < .1 amps, we can estimate 
t h a t  Er << E 
where Eo, Er, E .  are  the e l e c t r i c  f i e l d  components without the 
plasma column present,  t h a t  re f lec ted  from the plasma-glass 
and therefore sha l l  use the approximation Ei E Eo, 
0 
boundary, and t h a t  inside the plasma, respectively.  
The perturbation equation f o r  a cy l indr ica l  cavi ty  with 
a plasma tube symmetrically placed about i t s  center i s  given by: 
where f i s  the resonant frequency of the pa r t i cu la r  cavi ty  mode 
0 
A f  i s  the s h i f t  i n  resonant frequency 
Ilv i s  the volume occupied by the plasma i n  the cavi ty  
V i s  the volume of the cavity. 
From equation (l), one wri tes :  
2 2  f2  = f2  (1 - c r  /ro) 
P PO (4 )  
where f i s  the l o c a l  plasma frequency a t  r = 0. 
PO 
8 
After subst i tut ion,  the perturbation equation becomes: 
1 1 
f2 2fo Af 
PO 
- =  
2 
c Av 
dv 
- 7  
V 
( 5 )  
We now evaluate equation ( 5 )  for  the TM and TEOll modes, where 020 
A f  are superscripted w i t h  M o r  E, depending on whether they 
f O J  
a r e  of the TMo20 o r  moll mode, respectively: 
TMo20 Mode: 
1 
PO 
-
f2  
1 
2fm A f m  
0 
Jf (xo2 r/rc) r d r  
0 
c 2  r 
J~ (Xo2 r/rc) I' dr 
0 
We make the approximation 
where the higher oreer  terms contribute a negl igible  amount t o  the 
in t eg ra l .  Subst i tut ing the above, and evaluating the in tegra ls ,  
w e  obtain: 
9 
The various dimensions f o r  the experiment a re  l i s t e d  below: 
r = .54 cms 
r = 8.12 crns 
0 
C 
xo2 = 5.52 
Subst i tut ing 
1 
P O  
- =  
f2  
TEoll Mode: 
1 
P O  
- =  
f 2  
- 
these i n  the perturbations equation ( 8 ) ,  one obtains 
1 
2fE hfE 
0 
0 I 
10 
Once again we make the reasonable approximation 
and carry out the integration 
where r = .54 as before 
r = 5.4 cms 
0 
C 
Substituting the above in (12), we obtain 
11 
We have evaluated the two perturbation equations for the  
TMo20 and TEoll modes. 
included i n  these equations v i a  the parameter c. 
The plasma r a d i a l  inhomogeneity has been 
By carrying out 
2 perturbat ion experiments such t h a t  f 
we can eliminate it, and hence have an expression of c i n  terms 
of the measured quant i t ies  <, fE, A?, A?. 
i s  the same for both modes, 
PO 
0 
12 
EXPERIMENT AND RESULTS 
Two cy l ind r i ca l  cav i t ies ,  exci ted i n  the TMop0 and !l!Eoll 
modes, respectively,  were placed i n  turn over a chosen region of 
length 8 cms of the posi t ive column. 
by a 2-4 GHz o sc i l l a to r ,  and the tube current read with a 
ca l ib ra t ed  m e t e r .  
The cav i t i e s  were exci ted 
After the posi t ive column had warmed up and 
reached a steady s t a t e ,  measurements of Af against  plasma tube 
cur ren ts  f o r  current < .12 amps were made. The data  i s  shown - 
be low : 
. i o  + .005 
. la  + . O O ~  .Ob 7.6 - + .15 - 
- 4.4 + .1 - .06 
1 .12 11.3 + .2 - .27 + .010 I - 
The three points  are  plot ted on the same graph using 
the same current ax is  but different  u n i t s  of frequency s h i f t  so 
t h a t  both p l o t s  may appear on the graph. 
13 
As was already s ta ted,  it i s  reasonable t o  assume t h a t  f o r  
Hence, we obtain 2 a given current  each cavi ty  sees the same f 
an expression f o r  C i n  terms of the known parameters. 
PO 
We choosevalues of 118, A? f o r  currents  .070, .075, .080 amps, 
f ind  C fo r  each, and average t o  obtain a C corresponding t o  .75 amps. 
The values a re  tabulated below: 
Average 
.07 .127 5.5 .0230 - 45 
.075 .140 6.0 .0231 .435 
-075 -- -- -- .44 
.08 . i52  6.5 .0230 .440 
We can evaluate the '$ difference which, including the r ad ia l  
inhomogeneity i n  our perturbation equation, i s  produced i n  evaluat-  
We use the value of C f o r  0.015 amps, 2 instead of f 2 
ing (fP),,. PO' 
namely c = .44. 
For the mode: 
1 3.68 - 1.a: 5 difference = lOC$ (1 - 
.$ . 68 
For the TEoll mode: 
1 -113 - .0/:25 c .113 $ difference = lo@ (1 - 
= 3@ 
CONCLUSION 
The r a d i a l  e lectron density d i s t r ibu t ion  of the pos i t ive  
column was measured using resonant cavi t ies .  For a current of 
0.075 amp, the ' p ro f i l e  parameter' c characterizing t h i s  d is t r i -  
bution was found t o  be 0.44 + .l. The e r r o r s  arose from two 
sources, namely, the measurement of the tube radius (which was 
made w i t h  a t r ave l l i ng  microscope a f t e r  the tube had reached a 
steady s t a t e ) ,  and the approximation E = E i n  the perturbation 
equation. It should be emphasized tha t  some assumption had t o  
be made a s  t o  the form of the d is t r ibu t ion  function. The general 
method can obviously be applied t o  other choices of the d i s t r i -  
bution forms such as trapezoid.1 Furthermore, one could pick a 
more complicated form of the d is t r ibu t ion  f'unction, characterized 
by say, two ' p ro f i l e  parameters', and by using more than two cavi ty  
modes, estimate both of them. 
- 
0 
16 
APPENDIX 
In  equation ( 3 ) ,  we have made the approximation tha t  the 
f i e l d  inside the plasma tube remains approximately unchanged, 
i . e . ,  E - E o .  I n  t h i s  appendix, the e r r o r  due t o  such an 
approximation i s  calculated.  I n  the calculat ion,  we sha l l  choose 
- - 
2 2 
P 
a f ixed current of 0.075 amps and wri te  f = ( fp)ave. .  This 
choice i s  good f o r  the case of no generation of l oca l  modes, 
which i s  t rue  f o r  our choice o f  Cavity modes. 
The exact form f o r  equation ( 3 )  i s  
+ . Eo dv 2 
N (fp)ave. Av 
- 2f2 I E o l e  dv 
0 
V I  I 
where E i s  the f i e l d  i n  the plasma 
Eo i s  the unperturbed f ie ld .  
The general scheme i s  t o  write E i n  terms of E and t o  
0 
compare with the case where E i s  s e t  equal t o  Eo. The calcula- 
t i o n  i s  car r ied  out fo r  the TM mode with the remark tha t  the 020 
r e s u l t s  f o r  the moll mode are  similar. 
The incident,  ref lected,  and transmitted f i e l d s  a re  given, 
respectively,  as:  
Einc. a Jo (kr )  Hint. kJl(kr) 
Eref .  a R J o  ( k r )  Href .  cT kJl(kr)  
Etrans.  a TJo ( k ' r )  Htrans.  OC k'J1 ( k ' r )  
( B )  
where 
Matching solut ions a t  the boundary r = r 
magnetic components, we obtain 
fo r  the e l e c t r i c  and 
0 
.*. E = TE 
0 
18 
We have here imp l i c i t l y  assumed tha t  because the g l a s s  tube 
i t s e l f  i s  thin-walled, there i s  negligible re f lec t ion  due t o  it. 
Our perturbation equation becomes: 
2 
(fp)ave. 2 Av f m o d v  
Af - =  
2fE E2 dv 
0 
f O  
V 
Thus, the $ e r r o r  i n  Af on making the approximation E = E 
numerator of the perturbation equation is :  
i n  the 
0 
(T - 1) - lo& where T i s  given ir? equation ( D ) .  
For a current of .075 amps, about which our measurements were made 
as the mean between f 2  and i t s  value near and i f  we take (fp)ave. 2 
PO 
the plasma boundary, we obtain from equation ( 9 )  
But, f o r  the TMoz0 mode, 
-1 k = .66 cm 
r = .54 cm 
0 
and we f i n d  tha t  the % e r r o r  i s :  
- 1 ) - 10046 z 3$l . 2 ( i - rg  
which i s  s m a l l ,  a s  desired. 
ACKNOWLEDGEMENTS 
The author wishes t o  express h i s  appreciation t o  
D r .  Peter  Noerdlinger f o r  h i s  helpful suggestions; t o  D r .  Karl 
Ionngren f o r  reading the manuscript; and t o  D r .  J. A. Van Allen 
f o r  h i s  support of t h i s  research under h i s  National Aeronautics 
and Space Administration grant NsG 2.33-62. 
20 
REFERENCES 
hl. A. Heald and C. B. Wharton, Plasma Diagnostics with Micro- 
waves. New York: John Wiley and Sons, 1965. 
R. J. Chaffin and J. B. Bayer, IEEE Trans. on Microwave Theory 2 
and Techniques, M I T - ~ ~ ,  3’7, 1968. 
’A. Cavaliere, F. Engelmann, and A. Sestero, Phys. Fluids, - 11, 158, 
1968. 
F. W. Crawford and G. S. Kino, Proc. IRE, 49, 17(6-i, 1961. 4 -
’K. A. Groif and J. H. de Leeuw, J. Appl. Phys., - j8, 4466, 1967. 
, 
‘S. C. Brown, Proc. Second U. N. International Conference on the 
Peaceful Uses of Atomic Energy, Vol. 32, 294, 19%. 
’L. Tonks, Phys. Rev., 37, 1458, 1931. 
A. Dattner, Phys. Rev., - 10, 205, 1963. 

668-569 
.25 
.20 
.I5 
. I  
.05 
0 
A 
9: I t  O 
45 9 
- 
- 8  
- 7  
- - 6  
- 5  
- - 4  
- 3  
- - 2  
- I  
4 - 
// 
/ 
-----  I I I I I I 
-06 .07 -08 -09 .IO .I I 
Fig. 2 .  Frequency shift of the TM 020 and TE 011 cavity 
modes as a function of current. 
U. o f I o w a 6 8 - 3  
Profile Measurements of Plasma Columns Using 
Microwave Resonant Cavities --MICHAEL A. W. 
VLACHOS and HULBERT C. S. HSUAN 
p. 9, equation 8: Close curly bracket a t  end of equation. 
p. 10, equation 12: 
As above. 
p. 12, bottom: L a s t  sentence should be followed by the 
notation (Figure 2). 
p. 13, table: Third number under "C" should read .45. 
p. 19, Acknowledgement: 
The acknowledgement should read: 
authors wish t o  express the i r  appreciation 
t o  . . ." 
'(The ' 


Profile measurements of plasma columns using microwave resonant cavities

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