DETERMENATEON or PLASMA DENSITY: ;;A g; E.)
PROHLE AND OTHER PARAMETERS: ‘ (3-;
WITH AN ELECTROACOUSIIC‘PRQBE; g -'

Thesis for the Degree 0f Ph. D,
MICHIGAN STATE UNWERSETY
JACK G. 0U"

1974 '

Ike-'32. ' LIBRARY

Michigan State
Univers'ty

 

This is to certify that the

thesis entitled

DETERMINATION OF PLASMA DENSITY PROFILE AND
OTHER PARAIv'iETERS WITH AN ELECTROACOUSTIC PROBE

presented by

Jack G. Olin

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Electr. Engineering

 

 

144 /L(K 6/K

Major professor

 

 

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Plasma parameters

ABSTRACT

DETERMINATION OF PLASMA DENSITY PROFILE AND
OTHER PARAMETERS WITH AN ELECTROACOUSTIC PROBE

By

Jack C. Olin

The electron density profile and other plasma parameters of a
cylindrical warm-plasma column are studied through the excitation of
thermal resonances using an electroacoustic probe. The electromagnetic
field from the probe excites a series of thermal (Tonks-Dattner)
resonances as the current density is varied.

For each driving frequency, the dipole resonance and the first
three T-D resonances fire recorded. In this study, it is sufficient to
measure the relative magnitudes of the plasma densities at which these
resonances occur in order to determine the density profile and other
plasma parameters such as the temperature and the number density.

In the determination of the plasma density, the thermal
resonances are used to determine the unknown parameters appearing in the
solution of Poisson's Equation in the plasma column. The boundary
conditions for the thermal resonances in the plasma column are derived
and the total phase for the thermal resonances is determined using the
WKB approximation. The dipole resonance is used to determine the
average electron density in the plasma column. The analysis leads to

numerical values for the electron density profile parameters.

 

OThE

in

 

 

DETERMINATION OF PLASMA DENSITY PROFILE AND
OTHER PARAMETERS WITH AN ELECTROACOUSTIC PROBE

By

Jack C. Olin

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1974

To my family

Sigrid, Peter and Leslie

ii

 

The aul
guidance and G
Ccen.

He also
Sjvquist, J. A51

:raject.
The autI';

great effort pu

Jreen. Her hel

mvaluable. lie

and infinite par

ACKNOWLEDGMENTS

The author expresses his sincere appreciation and thanks for the
guidance and encouragement given him by his major professor, Dr. K. M.
Chen.

He also wishes to thank the committee members, Drs. P. D.
Nyquist, J. Asmussen, B. Ho, and G. Pollack for their interest in this
project.

The author furthermore wishes to express his gratitude for the
great effort put into the final typing of this thesis by Mrs. Roberta
Green. Her help in producing the final version of this thesis was
invaluable. He also thanks his wife, Sigrid, for the preliminary typing

and infinite patience.

iii

 

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LIST 0? FIGS.

2.1
2.2
2.3
M

2.5

 

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . .

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . .

INTRODUCTION 0 O O O O O O O O O I O O O O O O O O O O C .

BASIC THEORY OF TEMPERATURE RESONANCES IN PLASMA SHEATHS .

Introduction . . . . . . . . . . . . . . . . . . . .
General Theory . . . . . . . . . . . . . . . . . . . .
Determination of the Boundary Condition at the Wall .
Determination of the Total Phase for the Thermal
Resonances . . . . . . . . . . . . . . . . . . . . . .
Development of Relationships Between Dipole Resonance
Frequency and Plasma Frequency in a Cylindrical Plasma
Column . . . . . . . . . . . . . . . . . . . . . . . .

DETERMINATION OF ELECTRON DENSITY PROFILE IN CYLINDRICAL
PLASMA COLUMN BASED ON THERMAL RESONANCE DATA IN THE
SHEATH REGION 0 O O O O O I O O O O O I O O O O O O O O O 0

Introduction . . . . . . . . . . . . . . . . . . . . .
Experimental Procedure . . . . . . . . . . . . . . . .
Development of Functional Form for the Electron-
Density Profile . . . . . . . . . . . . . . . . .
Determination of Electron-Density Profile in a
Cylindrical Warm Plasma Column Based on a Parabolic
Approximation . . . . . . . . . . . . . . . . . . . .
Determination of the Electron-Density in a Warm
Plasma Cylinder Assuming Potential Distribution of

the Form (1 - 10(yr)) . . . . . . . . . . . . . . .

NUMERICAL RESULTS FOR THE ELECTRON DENSITY PROFILE IN A
CYLINDRICAL PLASMA COLUMN . . . . . . . . . . . . . . . . .

4.1
4.2

4.3

4.4

Introduction . . . . . . . . . . . . . . . . . . . . .
Numerical Results Based on Parabolic Electron-Density
Profile Approximation . . . . . . . . . . . . . . . .
Numerical Results Based on the Bessel Function
Approximation for the Static Electron—Density

Profile . . . . . . . . . . . . . . . . . . . . . . .
Graphical Presentation of Thermal Resonances Using

the WKB Approximation . . . . . . . . . . . . . . . .

iv

Page
iii

vi

19

31

41

41
43

46

56

66

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80

87

126

 

APPE‘CD IX A

APPENDIX B

REFERENCES

 

 

 

Page

APPENDIX A NUMERICAL COMPUTER READOUTS AND ADDITIONAL
COMMER GRAPHS O O O O C O O O O O O O O O O O O 13 3

APPENDIX B FORTRAN COMPUTER PROGRAMS WRITTEN
SPECIFICALLY FOR THE NUMERICAL ANALYSIS
IN THIS RESEARCH PROJECT . . . . . . . . . . . . 186

“WNCES O 0 O O O O O O O O O O O O O O O O O O O O O O O 203

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2.1.2

2.1.3

2.2.1

2.3.1

2.4.1

2.4.2

2.4.3

2.5.1

LIST OF FIGURES AND TABLES

Typical electron, ion and potential profiles in the
sheath region of a semiinfinite plasma in the
vicinity of a solid boundary. With the assumption
of ion drift towards the wall, the ion density is
not significantly changed in the sheath region . . .

Typical electron and ion density profiles in the
sheath region of a cylindrical plasma column.
Assuming ion drift towards the solid boundary, the
ion density does not significantly change in the
sheath region . . . . . . . . . . . . . . . . . . .

Typical sketches of the first three thermal resonances
(m - 1,2,3) occuring at a given frequency of the
incident EM field at three discharge current levels
producing density profiles ne}, n32, and n33. The
resonances occur when m2 a up at any current level

which corresponds to 2
mew

ne1(t1) - ne2(t2) = ne3(t3) =._E_%__ . . . . . . . .
e

Geometric arrangement used in the region where
thermal resonances occur. n1 represents a typical
waveform of a thermal resonance; t1 is the critical
point where w - mp. The one-dimensional approach
is justified in this region because t1 is typically
much smaller than the radius of the plasma column,

a . . . . . . . . . . O . I . I . . . I . . . . O . 0

Phase relation between electron density perturbation
n1 and associated electron velocity perturbation v1 .

The under-dense region in which thermal resonances
may occur if the phase conditions are satisfied and
an appropriate EM field illuminates the plasma
column . . . . . . . . . . . . . . . . . . . . . . .

Sketch of Airy function,
Ai(z) . %-cos(33/3 + s 2) dz . . . . . . . . . . .

Typical waveforms of the first three thermal
resonances. xpl, xpz, and xp3 are the critical
points at which kpl, kpz, and kp3 respectively

go to zero . . . . . . . . . . . . . . . . . . . . .

Geometric arrangement of cylindrical plasma column
contained in a cylindrical glass discharge tube of
wall thickness b. The inside radius is a while the
outside radius is c . . . . . . . . . . . . . . . . .

vi

Page

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20

21

29

32

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3.2.1

3.2.2

3.2.3

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3.1.1 A cylindrical plasma_column illuminated by TM field
as shown. not and E01 represent the transverse and
longitudinal components of electric field
respectively . . . . . . . . . . . . . . . . . . . . 42

3.2.1 Experimental arrangement for obtaining plasma
resonance data in a cylindrical plasma column.
An electroacoustic (E.A.) probe is used to excite
the dipole and thermal resonances in the plasma
column. The E.A. probe also picks up the
scattered field whose peaks indicate the presence
of resonances in the plasma . . . . . . . . . . . . . 44

3.2.2 Experimental results (data sets #1 and #2) for
the back scattered EM field from a cylindrical
plasma column as a function of discharge
current. f is the frequency of the incident
EM field. id, 11, 12, and 13 are the discharge
currents at which the dipole resonance and the
first three thermal resonances respectively
occur . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.3 Experimental results (data sets #3 and #4) for
the back scattered EM field from a cylindrical
plasma column as a function of discharge current.
f is the frequency of the incident EM field. id,
11, 12, and 13 are the discharge currents at which
the dipole resonance and the first three thermal
resonances respectively occur . . . . . . . . . . . . 48

3.2.4 Experimental results (data sets #5 and #6) for
the back scattered EM field from a cylindrical
plasma column as a function of discharge current.
f is the frequency of the incident EM field. id,
11, 12, and 13 are the discharge currents at which
the dipole resonance and the first three thermal'
resonances respectively occur . . . . . . . . . . . . 49

3.2.5 Experimental results (data sets #7 and #8) for
the back scattered EM field from a cylindrical
plasma column as a function of discharge current.
f is the frequency of the incident EM field. id,
11, 12, and 13 are the discharge currents at which
the dipole resonance and the first three thermal
resonances respectively occur . . . . . . . . . . . . 50

vii

 

 

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4.2.1 Normalized parabolic electron density profile as
a function of r/a, nel(r/a)/n01 = l - .83(r/a)2.
Also the normalized potential profile n1(r/a)/nw.
Based on data set #1 (f - 2.016 GHz, id - 270 ma,
11 - 185 ms, 12 - 150 ma, 13 - 125 ma) . . . . . . . 88

4.2.2 Normalized parabolic electron density profile as
a function of r/a, nel(r/a)/n01 - l - .82(r/a)2.
Also the normalized potential profile “1(r/a)/nw.
Based on data set #2 (f a 2.10 GHz, id = 290 ma,
11 - 190 ma, 12 = 150 ma, i3 3 120 ma) . . . . . . . 89

4.2.3 Normalized parabolic electron density profile as
a function of r/a, ne1(r/a)/nol 8 l — .83(r/a)2.
Also the normalized potential profile “1(r/a)/nw.
Based on data set #3 (f = 2.23 GHz, id 8 340 ma,
11 a 235 ma, 12 - 185 ma, 13 = 160 ma) . . . . . . . 90

4.2.4 Normalized parabolic electron density profile as
a function of r/a, ne1(r/a)/n01 a l — .8O(r/a)2.
Also the normalized potential profile “1(r/a)/Rw.
Based on data set #4 (f = 2.32 CH2, id = 355 ma,
11 - 245 ma, 12 - 200 ma, 13 - 175 ma) . . . . . . . 91

4.2.5 Normalized parabolic electron density profile as
a function of r/a, nel(r/a)/n01 = l - .86(r/a)2.
Also the normalized potential profile n1(r/a)/”w.
Based on data set #5 (f - 1.917 GHz, id - 270 ma,
11 - 180 ma, 12 . 135 ma, i3 . 110 ma) . . . . . . . 92

4.2.6 Normalized parabolic electron density profile as
a function of r/a, nel(r/a)/n01 = l - .83(r/a)2.
Also the normalized potential profile n1(r/a)/nw.
Based on data set #6 (f a 2.017 GHz, id = 285 ma,
11 - 19o ma, 12 - 150 ma, 13 - 120 ma) . . . . . . . 93

4.2.7 Normalized parabolic electron density profile as
a function of r/a, nel(r/a)/n01 . l - .84(r/a)2.
Also the normalized potential profile 01(r/a)/nw.
Based on data set #7 (f - 2.275 GHz, id = 290 ma,
11 - 195 ma, 12 - 150 ma, 13 - 120 ma) . . . . . . . 94

4.2.8 Normalized parabolic electron density profile as
a function of r/a, nel(r/a)/n01 - 1 - .85(r/a)2.
Also the normalized potential profile 01(r/a)/nw.
Based on data set #8 (f - 2.322 GHz, id - 320 ma,
11 = 210 ma, 12 B 160 ma, i3 = 135 ma) . . . . . . . 95

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4.3.1 Normalized Bessel series electron density profile
as a function of z/a,
nel(z/a)/n01 I exp(1 - Io(327(l - z/a))). Also the
normalized potential profile n1(z/a)/nw. Based
on data set #1 (f I 2.016 GHz, id I 270 ma,
11 I 185 ma, i2 I 150 ma, 13 I 125 ms) . . . . . . . 102

4.3.2 Normalized Bessel series electron density profile
as a function of z/a,
ne1(z/a)/n01 I exp(1 - 10(326(l - z/a))). Also the
normalized potential profile n1(z/a)/nw. Based
on data set #2 (f I 2.10 GHz, id I 290 ma,
11 I 190 ma, 12 I 150 ma, 13 I 120 ma) . . . . . . . 103

4.3.3 Normalized Bessel series electron density profile
as a function of z/a,
nel(z/a)/n01 I exp(l - Io(323(l - z/a))). Also the
normalized potential profile n1(z/a)/nw. Based on
data set #3 (f I 2.23 GHz, id I 340 ma, 11 I 235 ma,
12 I 185 ma, 13 I 160 ma) . . . . . . . . . . . . . . 104

4.3.4 Normalized Bessel series electron density profile
as a function z/a,
nel(z/a)/nol I exp(1 - 10(330(l - z/a))). Also the
normalized potential profile n1(z/a)/nw. Based on
data set #4 (f I 2.32 GHz, id I 355 ma, 11 I 245 ma,
12 I 200 ma, 13 I 175 ma) . . . . . . . . . . . . . . 105

4.3.5 Normalized Bessel series electron density profile
as a function of z/a,
ne1(z/a)/n01 I exp(l - 10(327(1 - z/a))). Also the
normalized potential profile n1(z/a)/nw. Based on
data set #5 (f I 1.917 GHz, id I 270 ma, 11 I 180 ma,
12 - 135 ma, 13 - 110 ms) . . . . . . . . . . . . . . 106

4.3.6 Normalized Bessel series electron density profile
as a function of z/a,
nel(z/a)/n01 I exp(l - 10(327(l - z/a))). Also the
normalized potential profile n1(z/a)/nw. Based on
data set #6 (f I 2.017 GHz, id I 285 ma, 11 I 190 ma,
12 I 150 ma, 13 I 120 ma) . . . . . . . . . . . . . . 107

4.3.7 Normalized Bessel series electron density profile
as a function of z/a,
ne1(z/a)/n01 I exp(l - 10(328(1 - z/a))). Also the
normalized potential profile n1(z/a)/nw. Based on
data set #7 (f I 2.275 GHz, id I 290 ma, 11 I 195 ma,
12 I 150 ma, 13 I 120 ma) . . . . . . . . . . . . . . 108

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4.3.8

4.3.9

4.3.10

4.3.11

4.3.12

4.3.13

4.3.14

Normalized Bessel series electron density profile

as a function of z/a,

ne1(z/a)/n01 I exp(l - 10(331(1 - z/a))). Also the
normalized potential profile n1(z/a)/nw. Based on
data set #8 (f I 2.322 GHz, id I 320 ma, 11 I 210 ma,
12 I 160 ma, 13 I 135 ma) . . . . . . . . . . . . . .

Normalized Bessel series electron density profiles

at resonances l and 2. Points t1 and t2 represent
the critical points in the plasma sheath at which
kpl and kpz respectively go to zero. Based on

data set #1 (f I 2. 016 GHz, id I 270 ma, 11 I 185 ms,
iz I 150 ma, 13 I 125 ma) . . . . . . . . . . . .

Normalized Bessel series electron density profiles
at resonances l and 2. Points t1 and t2 represent
the critical points in the plasma sheath at which
ROI and kpz respectively go to zero. Based on

data set #2 (f I 2.10 GHz, id I 290 ma, 11 I 190 ma,
iz I 150 ma, 13 I 120 ma) . . . . . . . . . . .

Normalized Bessel series electron density profiles
at resonances l and 2. Points t1 and t2 represent
the critical points in the plasma sheath at which
kpl and ksz respectively go to zero. Based on

data set 3 (f I 2.23 GHz, 1d I 340 ma, 11 I 235 ma,
12 I 185 ma, 13 I 160 ma) . . . . . . . . . . . . . .

Normalized Bessel series electron density profiles
at resonances l and 2. Points t1 and t2 represent
the critical points in the plasma sheath at which
land k2 respectively go to zero. Based on
data set #4 (f I 2.32 GHz, id I 355 ma, 11 I 245 ma,
12 I 200 ma, 13 I 175 ma) . . . . . . . . . . . . . .

Normalized Bessel series electron density profiles
at resonances l and 2. Points t1 and t2 represent
the critical points in the plasma sheath at which
kPl and k respectively go to zero. Based on

data set #5 (f I 1.917 CH2, id I 270 ma, 11 I 180 ma,
12 I 135 ma, 13 I 110 ma) . . . . . . . . . . . . . .

Normalized Bessel series electron density profiles

at resonances 1 and 2. Points t1 and t2 represent

the critical points in the plasma sheath at which
land k respectively go to zero. Based on

data set #S (f I 2.017 GHz, id I 285 ma, 11 I 190 ma,

12 3 150 ma, 13 3 120 ma) . . . . . . . . . . . . . .

Page

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112

113

114

115

 

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4.3.15

4.3.16

4.3.17

4.3.18

4.3.19

4.3.20

4.3.21

Normalized Bessel series electron density profiles
at resonances 1 and 2. Points t1 and t2 represent
the critical points in the plasma sheath at which
kpl and k respectively go to zero. Based on

data set 3; (f I 2.275 GHz, id I 290 ma, 11 I 195 ma,
12 I 150 ma, 13 I 120 ma) . . . . . . . . . . . . . .

Normalized Bessel series electron density profiles
at resonances l and 2. Points t1 and t2 represent
the critical points in the plasma sheath at which
kpl and k 2 respectively go to zero. Based on

data set 8 (f I 2.322 GHz, id I 320 ma, 11 I 210 ma,
12 I 160 ma, 13 I 135 ma) . . . . . . . . . . . . . .

Normalized Bessel series electron density profiles
at resonances 1 and 3. Points t1 and t3 represent
the critical points in the plasma sheath at which
kPl and k respectively go to zero. Based on

data set E? (f I 2.016 GHz, 1d I 270 ma, 11 I 185 ma,
12 I 150 ma, 13 I 125 ma) . . . . . . . . . . . . . .

Normalized Bessel series electron density profiles
at resonances l and 3. Points t1 and t3 represent
the critical points in the plasma sheath at which
kpl and k respectively go to zero. Based on

data set 53 (f I 2.10 GHz, id I 290 ma, 11 I 190 ma,
12 I 150 ma, 13 I 120 ma) . . . . . . . . . . . . . .

Normalized Bessel series electron density profiles

at resonances l and 3. Points t1 and t3 represent

the critical points in the plasma sheath at which
land k respectively go to zero. Based on

data set $3 (f I 2.23 GHz, id I 340 ma, 11 I 235 ma,

12 I 185 ma, 13 I 160 ma) . . . . . . . . . . . . . .

Normalized Bessel series electron density profiles

at resonances l and 3. Points t1 and t3 represent

the critical points in the plasma sheath at which
land k respectively go to zero. Based on

data set 32 (f I 2.32 GHz, id I 355 ma, 11 I 245 ma,

12 I 200 ma, 13 I 175 ma) . . . . . . . . . . . . . .

Normalized Bessel series electron density profiles

at resonances l and 3. Points t1 and t3 represent

the critical points in the plasma sheath at which
land kg respectively go to zero. Based on

data set 3 (f I 1.917 GHz, id I 270 ma, 11 I 180 ma,

12 I 135 ma, 13 I 110 ma) . . . . . . . . . . . . . .

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4.3.22

4.3.23

4.3.24

4.4.1

4.4.2

4.4.3

4.4.4

Table

3.2.1

Normalized Bessel series electron density profiles

at resonances 1 and 3. Points t1 and t3 represent
the critical points in the plasma sheath at which
kpl and k p3 respectively go to zero. Based on

data set #6 (f I 2. 017 GHz, id I 285 ma, 11 I 190 ma,
12 I 150 ma, 13 I 120 ma) . . . . . . . . . . . . . .

Normalized Bessel series electron density profiles
at resonances l and 3. Points t1 and t3 represent
the critical points in the plasma sheath at which
kpl and k respectively go to zero. Based on

data set #3 (f I 2.275 GHz, id I 290 ms, 11 I 195 ma,
12 I 150 ma, 13 I 120 ma) . . . . . . . . . . . . . .

Normalized Bessel series electron density profiles
at resonances l and 3. Points t1 and t3 represent
the critical points in the plasma sheath at which
kpl and kp3 respectively go to zero. Based on

data set #8 (f I 2. 322 GHz, id I 320 ma, 11 I 210 ma,
12 I 160 ma, 13 I 135 ma) . . . . . . . . . . . . . .

First thermal resonance for data set #1 based on a
WKB formulation using the parabolic electron density
Drafile O O O O O O O O O O O O O O O O O I O O O O 0

Second thermal resonance for data set #1 based on
a WKB formulation using the parabolic electron
den81ty prOfile O O O O I O O O O O O I O O O O O O 0

First thermal resonance for data set #1 based on
a WKB formulation using the Bessel series electron
density profile . . . . . . . . . . . . . . . . . . .

Second thermal resonance for data set #1 based on
a WKB formulation using the Bessel series electron
denSity prOfile O O O O O O I O O O O O O O O O O O 0

Experimental data set 1 through 8. Given are the
frequency of the incident EM field and the dis-
charge currents id, 11, 12 and 13 at which the
dipole resonance and the first three thermal
resonances respectively occur . . . . . . . . . . . .

xii

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sets
1.
f0:
YESC
res:

NUE£
aver

 

Table

4.2.1

4.2.2

4.2.3

4.2.4

4.2.5

4.3.1

4.3.2

4.3.3

4.3.4

Numerical results for the factor a for data sets

1 through 8. The columns identified by j I 2

and j I 3 represent numerical values for a obtained
from the use of combinations of resonances 1,2

(j I 2) and 1,3 (j I 3) respectively . . . . . . .

Numerical results for the ratio of peak to average
electron density n01/<nel(r)> and n02/<nez(r)>
for data sets 1 through 8 . . . . . . . . . . . . .

Numerical values for the ratio of critical radius
ta to the total radius a, rj/a, for data sets 1
rough 8 O I O O I O O O O I O O O C O O O O O 0

Numerical values for the ratio of critical dis-
tance zj measured from the wall for the jth
resonance to the total radius a as well as the
ratios 22/21 and 23/21 for the data sets 1

through 8 . . . . . . . . . . . . . . . . . . . . .

Numerical values of relative potential at the
wall, “w I eV(a)/kT and electron temperature

T for data sets 1 through 8. The columns
identified by j I 2 and j I 3 represent the
numerical values for "w and T based on the use

of combinations of resonances 1,2 (j I 2) and

1,3 (j I 3) respectively . . . . . . . . . . . . .

Numerical results for the factor Y for data

sets 1 through 8. The columns identified by

j I 2 and j I 3 represent the numerical values

for Y obtained from use of combinations of
resonances 1,2 (3 I 2) and 1,3 (3 I 3)

respectively . . . . . . . . . . . . . . . . . . .

Numerical results for the ratio of peak to
average electron density n01/<nel(r)> and
nozl<nel(r)> for data sets 1 through 8 . . . . . .

Numerical values for the ratio of the critical
distance 2 measured from the wall into the

plasma for the jth resonance to the total radius

a and also the ratios 22/21 and 23/21 . . . . . . .

Numerical values of the relative potential

"w I eV(a)/kT and the electron temperature T

for the data sets 1 through 8. The columns
identified by j I 2 and j I 3 represent the
numerical results based on the use of combina-
tions of resonances 1, 2 (j I 2) and 1, 3 (j I 3)
respectively . . . . . . . . . . . . . . . . . .

xiii

Page

81

82

83

84

85

96

97

98

99

Kntzvledfi"
in the walls
mlytical work
rezian has been
a;;-rcxinate the"
ing the static (1
bmfiaries has a
inhaledge of the
tyliadrical plat
:Clu-rns, a paraL
is frequently er

This rese-
“T-Sit? profile
iata for the di,
Mbe which illur
9f- . I
we bacxscatterer
31th “Perimentaj
Flam Column

at

refl‘mcy m

 

The theore

"

1“"

‘-
d,ti

onal exPt'esl

h

in.

.t

‘I
i
At

33a “01mm.

CHAPTER 1

INTRODUCTION

Knowledge of the static electron density profile of warm plasmas
in the so-called sheath region near solid boundaries is significant in
analytical work involving the plasma electron density. The sheath
region has been analyzed in plane geometries by researchers based on
approximate theoretical models.1 The more complex problems of determin-
ing the static electron density profile in warm plasmas with cylindrical
boundaries has also been treated theoretically by researchers.2-9 When
knowledge of the functional form of the electron density profile in a
cylindrical plasma column is needed for work involving such plasma

columns, a parabolic electron density profile of the form
ne(r) . no(1 - (ICE-)2)

is frequently employed using some typical value for the parameter 0.4’10

This research deals with the determination of the static electron
density profile in warm cylindrical plasma columns based on experimental
data for the dipole and thermal resonances induced by an electroacoustic
probe which illuminates the plasma column with an EM field and receives
the backseattered field. The experimental part of the research deals
with experimental determination of the discharge current levels in the
plasma column at which thermal resonances occur for a given excitation
frequency w.

The theoretical part of the research considers possible
functional expressions for the static electron density in warm cylin-

drical plasma columns based on a study of Poisson’s Equation in the

plasma column. The phase conditions for thermal resonances are studied

 

and the “13”"
I
exciting freq.“
aggroximtion i
sclutlon to 1’01
to the Poisson
The num
1

solution of sir.l
thermal resonanl
|

Graphs on the e
|

different anoro
|
aporopriate Bes-
I

well represent :
. I
is actual profi
Chapter 2
Sheath region of
Studied using m:
and the ratio of
column to the ex
Chapter 3
tr“

"I‘ndrical Plasr

h. "
rOSSlDlE solutior

of
all Parameters

Chapter 4
:19:

tron density

~ifi~e
““i‘mal forms

 

 

and the relationship between the average plasma frequency and the
exciting frequency is deve10ped. The commonly used parabolic profile
approximation is considered as an approximation to a Bessel function
solution to Poisson's Equation. Next, a Bessel function approximation
to the Poisson Equation is considered.

The numerical work done as part of this research deals with the
solution of simultaneous equations based on the phase condition for the
thermal resonances and the electron density profiles proposed above.
Graphs on the electron densities obtained on the basis of these
different approaches are presented and compared. It is found that an
appropriate Bessel function approximation of the profile density may
well represent a functional form considerably more representative of
the actual profile than the conventional parabolic profile.

Chapter 2 presents the basic theory of thermal resonances in the
sheath region of cylindrical plasma columns. Phase conditions are
studied using WKB approximations of the electron density perturbations
and the ratio of the average plasma frequency <mp(r)‘ in the plasma
column to the exciting frequency w is developed.

Chapter 3 deals with the formulation of Poisson's Equation in a
cylindrical plasma column and considers various functional forms as
possible solutions. Simultaneous equations are presented for each
assumed functional form whose numerical solution permits determination
of all parameters appearing in the pr0posed profile functions.

Chapter 4 presents the numerical results and shows graphs of the
edectron density profiles obtained. The profiles based on different

functional forms are compared.

BASIC 'l'li‘

2.1 Introductir
The

plasma hou-
well knom,
electron d.
associated
ShEath P110?
of the pro'.,
men: of the
Phenomenolo
Elec
DOSitlvely

DIOfile in

 

 

which dECre,
a tl’chal p
solid Wind,-
is shown ”he

The I
r . . to a r
model 353mg
an aooroXima
Typical Val“
Detential n

'd

Sh
”via to be

CHAPTER 2

BASIC THEORY OF TEMPERATURE RESONANCES IN PLASMA SHEATHS

2.1 Introduction

 

The occurrence of a plasma sheath in the vicinity of a
plasma boundary such as a solid wall, metallic or nonmetallic, is
well known. The plasma sheath represents a region of reduced
electron density due to the loss of electrons hitting the wall
associated with a negative potential region near the wall. The
sheath phenomenon is briefly discussed to establish the geometry
of the problem at hand. The well documented mathematical treat-
ment of the sheath problem is not presented here but a brief
phenomenological discussion appears in order.

Electrons hitting a nonmetallic wall mostly recombine with
positively charged ions. This leads to an electron density
profile in the vicinity of the wall, the so-called sheath region,
which decreases monotonically towards the wall. Figure 2.1.1 Shows
a typical plasma sheath for a semi-infinite plasma slab with a
solid boundary at x I O. The relative electron density ne(x)/no
is shown where n0 is the electron density as x approaches infinity.

The potential V(x) also goes monotonically from zero at
x I I to a negative wall potential. The commonly accepted sheath
model assumes an ion drift in the sheath region which results in
an approximately constant ion density also shown in Figure 2.1.1.
Typical values determined for the ratio of the relative wall

eV

potential "w I ETEI are in the neighborhood of 2. This value is

Shown to be independent of electron density profile parameters.

1.0

0.5

 

 

 

 

 

Fig.

7.1.1

Typical electron,ion and potential
profiles in the sheath region of a
semiinfinite plasma in the vicinity

of a solid boundary. With the

assumption of ion drift towards the wall,
the ion density is not significantly

Changed in the sheath region.

Even thoug'
should new

The
conolex in '
Figure 2.1.l
assumed the
goal of thz'
the electrc
Vith other
this study
resonances ,
Section 3.2.
resonances :
drical plas.‘
the Sheath I
Numeric Va,
the Plasma i

frequfllcy h)

 

Show in Fig
a tl’plcal s‘r;
thermal r850
this ”pert
established.
333w
phenom‘enon,

sheaths is P

their

Even though it may vary somewhat in a cylindrical plasma sheath it
should nevertheless be approximately the same.

The electron density and potential distributions are more
complex in a cylindrical geometry as typically represented in
Figure 2.1.2. A parabolic electron density profile is frequently
assumed when cylindrical plasma columns are studied. The main
goal of this thesis is, in fact, the experimental determination of
the electron density profile assuming a parabolic profile, along
with other functional forms of the profile. The tool employed in
this study is an electroacoustic probe used to excite thermal
resonances in the plasma sheath region as discussed below in
section 3.2. Figure 2.1.3 shows typical sketches of thermal
resonances that may be excited in the sheath region of a cylin-
drical plasma column. The cylindrical column of warm plasma with
the sheath region as shown is illuminated by an incident electro-
magnetic wave of frequency w. The incident wave interacts with
the plasma in the sheath region near the wall where the plasma
frequency mp(r) is less than u to excite electroacoustic waves as
shown in Figure 2.1.3. Figure 2.1.3 is only intended to represent
a typical sketch of such resonances. The total phase of the mth
thermal resonance is assumed to be mn. In subsequent sections of
this report a more refined value for this total phase value is
established.

Based on this introductory discussion of the sheath
phenomenon, the basic theory of thermal resonances in plasma
sheaths is presented in this chapter. Boundary conditions for the

thermal resonances at the wall are examined. The phase condition

A norr
ele;

 

T:O

O .
Ventnh

Ԥ
Plasma

.17 ’1'1 e ]
>1 1

'0 \_’ 5""
I"

ll.,\

’1'
n

 

normalized static ion and
electron density

n,<r>/n10 l

 

fl-”

 

 

 

 

 

 

 

 

~_4>—r
r=O
Center of -—-’-1 .<__.
Wall
Plasma Column . Thicknes
of Glass
Tubing

Fig. 2.1.2 Typical electron and ion density
profiles in the sheath region of
a cylindrical plasma column.
Assuming ion drift towards the solid
boundary, the ion density does not

significantly change in the sheath
region.

 

 

."

  

Glass wall

 

 

 

 

 

 

)IZ

 

Fig. 2.1.3

 

 

 

Typical sketches of the first three thermal
resonances (m=1,2,3) occuring at a given
frequency of the incident EM field at
three discharge current levels producing
density profiles n , n , and n .
e1 82 283
The resonances occur when¢u2=wp' at any
current level which corresponds to
n (t mesons
e1 1>=ne (t2>=ne (t3): 2
9 5 e

’5
K

 

..
f-J

for the pt
I

the sheath
and finallr
the experi'
for the pr

frequency ‘

General Ii.
The

reirlon are

and

 

2.2

for the possible occurrence of electroacoustic thermal resonances in
the sheath region is examined using a WKB approximation technique,
and finally dipole resonances in the cylindrical plasma used in

the experiment are studied for the purpose of obtaining a value

for the proportionality constant Cp relating the average plasma
frequency <wp(r)> to the exciting frequency w by

<wp(r)2>
Cp 2 (__.______2 )

(a)

General Theory

 

The Maxwell and moment equations applicable to the plasma

region are

_ a ..
V X E - - SE'UOH (2.1)
and
VxlII-en 3+1- E (22)
(30 3t E:0 °

where E and K respectively represent the total electric field
intensity and total magnetic field intensity in the plasma; neo
represents the static electron density distribution in the plasma

which is non-uniform in the plasma sheath near a boundary; v
represents the mean ac electron velocity so that -eneév is the
leading term of the mean induced electron current. This formula-
tion is based on the assumption that the positive ion motion is
negligible in comparison to the electron motion. In the subsequent
analysis the total instantaneous electron density distribution
ne(x,t) will represent the dc component neo(;) plus the ac

perturbation term nl(x,t). All other quantities associated with

these two
field, anal
dc and per

In
moment ago

interest i

Since n (7.;
e

Since {'1 v .
e
term and

emutton (

From the v.

equation (.

The SeCond

gi‘len by

 

these two components of electron density such as the electric
field, and the velocity are also represented by a superposition of
do and perturbation terms.

In order to study perturbations in the plasma sheath, two
moment equations must be used. The first moment equation of
interest is the continuity equation

3n

3.;4- V - (“e-‘7) I O . (2-3)

Since ne(x,t) I neo(x) + n1(x,t), the continuity equation becomes

8n1(§,t)

at + V ° nev I O . (2-4)

1v, where nlv is a product of two perturbation

terms and therefore represents a negligible second order effect,

Since n v I ne v + n
e 0

equation (2.4) becomes

an

1 _.
Ia-E-I'i" V neov - O . (2.5)

From the vector identity
v-¢K=¢v-X+V¢-7\' (2.6)

equation (2.5) can be rewritten as follows

anl

at

+ neo V -‘$'+'3’- V ne0 O . (2.7)

The second moment equation based on the summation of momenta is

given by

——-—-+vv--EE -YkT
at m total mne

 

Vne . (2.8)
o

Here the

eradient

and

For the c

harmonic

In the pr

law

must be u
region of
of SPecif
of freed:

acouStic

the fallc

10

Here the density gradient Vne is associated with the pressure

gradient Vp. For an isothermal process,

p I nkT
and

VP I kTVn .
e

For the case of an ac perturbation, n1(§,t), due to an external

harmonic force, the total electron density is
ne(x,t) I neo(x) + n1(x,t) .

In the presence of ac perturbation at high frequency the adiabatic
law

p n Y I constant

must be used because the temperature is not equalized in the

region of high frequency electron perturbations. y is the ration
of specific heats and is given by (m + 2)/m where m is the degree
of freedom of the gas. For high frequency longitudinal electro-

acoustic plasma oscillations, m I 1, so that for these oscillations
Y I 3 . (2.9)
Separating equation (2.8) into its dc and ac components,

the following equations result. The dc equation is given by

e —- kT -
0 = - In. EdC " 1111',,_ Vneo(x) . (2.10)

The ac equ

Solution 0

i
‘dcm n

A one-dine

1
s and

the eleCtr

 

11

The ac equation is given by

 

+V;=-£-fi _3kT
“1

ac neom an(x,t). (2.11)

Solution of the dc equation for neo in terms of the potential

odc(§) in the plasma proceeds as follows:

 

Edc(x) I — vodc(x) (2.12)
' vadc(§) = n:Te Vneo(§) (2.13)
O

A one-dimensional component of equation (2.13) becomes

d kT d
‘5; ¢(x) I eneo dx neo(x) (2°14)

 

 

dfh¢(x) I 52- n1 dneo(x) + K

80

RT
¢(x) I-E— 1n neo(x) + K

e¢(x)
T? +

1n neo(x) I K'

¢(X)

kT

 

n x I K" e
e0( )

K, K', and K" are related arbitrary constants. Defining no to be

the electron density where ¢(x) I O, K" I no; therefore

which repr
which is 1:

In
necessary
here for r

Cc:

Ac

Sin
is eXClted
of the fOr

into the c.

 

and

In equatlo,

dropped f or

12

e¢(X)
kT *(2.15)

 

neo(x) I no e

which represents a Maxwellian dc electron density distribution

which is used in the subsequent plasma column analysis.

In order to analyze the ac behavior of the plasma it is

necessary to combine equations (2.5) and (2.11) which are repeated

here for reference:

Continuity Equation:

 

3n1(x,t) __
at + V ° neov I 0 (2.16)
Ac Momentum Transfer Equation:
3?] - e — 3kT —
at + vv - 5' ac - neom Vn1(x,t) (2°17)

Since the ac perturbation of the electron density n1(§,t)

is excited by a time harmonic incident EM wave with time dependence

of the form Re ejm , the system of equations may be transformed

into the complex phasor domain:

(2.18)

jwnl + V ° neov I 0-

and
3w? + v? a - 5’- E - 3” Vn (2.19)
m neom 1

 

In equations (2.18) and (2.19), the functional notation has been

dropped for simplicity with the understanding that

 

(3) I is t
only.
Ma )7

and

To Obta in

derived tai

Therefore

From EQUat;

 

Etuarion (

13

(1) n1 represents the phasor transform of n1(;,t) and is a function
of'x'only.
(2) 3 represents the phasor transform of 3(x,t) and is a function
of 3: only .
(3) E is the phasor transform of E(;,t) and is a function of x
only.
Maxwell's equations (2.1) and (2.2), for ac variations

only, become (after phasor transformation)

V x‘E I -jwu§fi' (2.20)
and

V x E-I -eneév + jmegE (2.21)

To obtain a solution for n1, a differential equation for n1 is

derived taking the divergence of equation (2.21), relating E'to V}

V 0 VxfiI—ev ° (neo;)+jw€ov °E

Therefore

 

v . (nee?) (2.22)

From equation (2.18)

V ° ne G’I -jwnl (2.23)
Equation (2.22) becomes
__ en1
V o E a -— --—-—€ (2.24)

 

In order

(2.19):

Combining:

From equa’.

and us ing

It (011mm

 

14

In order to eliminate E, the divergence is taken of equation

 

 

(2.19):
(jw-+V)Vov=-Ev-E-3kT Vzn
m ne m 1
O
Combining equations (2.25) and (2.24) yields
e2n
(jw+v)v-{7=+ 1-3kT V2n
meo neom 1

From equation (2.23)

v . neova -jwn1

and using vector identity equation (2.6),

V ° neov I neo V - G'+ 3" Vneo I —jwn1

It follows that

 

Substituting equation (2.29) into equation (2.26) yields

-— 2
-(jw + v)jwnl - +(jw + v)v Vneo ' e n1 _ 3kT
neo neo mso neom

 

 

 

V2n

(2.25)

(2.26)

(2.27)

(2.28)

(2.29)

(2.30)

After rearranging, a differential equation for n is obtained:

1

2 2
w - m - jwv
P

 

jw + v _
“1 + (35:) n1 ” (3kT/m) V
m

oVn
eo

(2.31)

 

 

 

{J
If as a ..

zero, equ.

 

2

where a;
inhomogonc
(jw/VZHV
force for
driving f9
satisfied;

(1) There
densir.
(2) there ,
densit:
The firs: ,
cylindrica
in the fad
metro“ v
by an 918C
radial dir
reptesEnts
lucident E
the radia1

dTiVing fc

15

If as a first approximation the collision frequency is set to

zero, equation (2.31) becomes (with V02 I 3kT/m)

2 2
w - w
V2n+——.—i—P——nI1-(2£v-°Vn (2.32)
V

e
l V 1 o
2 2
where Np is the plasma frequency (e ne)/(meco). This is an

inhomogeneous Helmholtz equation in n with a forcing function

1
(jw/V2)(; ° Vneo). This forcing function represents the driving

force for the perturbation in n Careful examination of this

1'

driving force shows that it is nonzero only if two conditions are

satisfied:

(1) There must exist a nonzero gradient of the static electron
density neo in the region of interest, and

(2) there must exist a component of 3 parallel to the electron
density gradient Vneo.

The first condition is satisfied in the sheath region of a

cylindrical plasma column where an electron density gradient exists

in the radial direction. The second condition is satisfied if an

electron velocity perturbation in the radial direction is set up

by an electric field component in the incident EM field in the

radial direction. Thus the velocity y'in the driving function

represents the coupling term between the radial component of the

incident EM field and the electron density perturbation n Here

1.
the radial component of the EM field represents physically the

driving force exciting the electron density perturbation.

cylinder
the vari;
uall (x =
character
compared

to treat

planar g.
rezion av

2 :
dnl u

~+§_

Equation

    

0f x in w.

the wall
2

°Scillato.

subjeCt t(

 

Of
“p(x),

 

In the So

wall is e

16

In the region of interest near the wall of the plasma
cylinder the geometry of interest is shown in Figure 2.2.1. Here
the variable x is introduced representing the distance from the
wall (x I 0) into the plasma normal to the wall. Since the
characteristic dimension of the sheath region is relatively small
compared to the radius of the plasma cylinder, it is justifiable
to treat the section of the sheath region shown in Figure 2.2.1 in
planar geometry. Thus equation (2.32) may be rewritten for that
region as a one-dimensional equation in x as

dzn wz - w 2(x) dne

+ P n = ((jw)/V2)(v
dx v2 1 x dx

 

 

 

°) (2.33)

The corresponding homogeneous equation is

dznl wz - w 2(x)
+ P n = o (2.34)

dx2 V2 l

 

Equation (2.34) has a natural oscillatory solution in the region
of x in which w2 is larger than m:(x). This is the region between
the wall (x I O) and the so-called critical point (x I xp) where

w I up. For values of x larger than xp, where m2 is less than
mp2, the solution represents an evanescent wave. The natural
oscillatory solution for n1 in the sheath region is of course
subject to boundary conditions at the wall and the functional form
of wp(x), where

2
2 e neo(x)
wp(x) I --————- (2.35)

me:
80

In the subsequent sections the boundary condition for n at the

1
wall is examined, followed by a study of the total phase require-

 

 

_!
’Igfllld

 

 

 

 

17

E-Glass wall

  

 

 

 

 

 

 

Q—GlQSS __1 ,r X
wall 0
thickness
l
l
l
l
“1
I >
I Q 13"]

2.2.1 Geometric arrangement used in the region
where thermal resonances occur. n
represents a typical waveform of 1
a thermal resonance; t is the critical
point where w=cu . The one—dimensional
approach is justified in this region
because t is typically much smaller than
the radius of the plasma column,a.

w

ment bet;

of nature

Determine
IL
on pnenOé
velocity
zero in 1
Standing -
(n90 indt

vwall . C

maXinum a

From equa

and “Sing

 

and IEtti

1" °“e-di.

dEDendenCe

WhEre A ah

L

2.3

18

ment between the wall and the critical point xp for the existence

of natural resonances.

Determination of the Boundary Condition at the Wall

The boundary conditions at the wall can only be established
on phenomenological grounds. It is reasonable to assume that the
velocity v associated with the electroacoustic wave motion goes to
zero in the immediate vicinity of the wall. For electroacoustic
standing wave perturbations in a uniform dc electron density
(neo independent of x) it can be shown that the boundary condition

- 0 corresponds to the boundary condition that n is a

vwall 1

maximum at the wall as follows:
From equation (2.18)

jwn + v - neaG'a o (2.36)

l

and using vector identity equation (2.6)

jwnl = - neo v . V + Vneo - V (2.37)

and letting Vneo 0 near the wall the following equation results

in one-dimensional form in x:

a - neo fL-v . (2.38)

jwn dx

1

Since we are assuming a standing wave in n and v, the functional

1

dependence of v on x is of the form
v(x) = A sin(kpx + 6) (2.39)

where A and 9 are the arbitrary magnitude and phase constants

respectiv

wall and

Substitut

Therefore

It is 1ND:

°f the or:

1
( 3;) st:

31 and v.

“1“) lead

 

(x a 0), n
at the Val
It
Figure 2.3
"all, It
Phase Cons

ESESEIEEflszg

2.4

19

respectively. For the assumed condition that v goes to zero at the

wall and letting x = O at the wall, equation (2.39) becomes:
v(x) = A sin(kpx) (2.40)

Substituting equation (2.40) into equation (2.38) yields

d
jwnl — - neo 52' A sin(kpx) (2.41)
Therefore
Ak
n1 = - neo (352) cos(kpx) . (2.42)

It is important to recall that n represents the phasor transform

1
of the original time harmonic function n1(x,t). The phase term

1
“32;

n1 and v. In addition, a spatial phase difference exists with

) shows that a n/Z radian time phase difference exists between

n1(x) leading v(x) by n/Z radians. This means that at the wall

(x I 0), n1 should have a maximum corresponding to the zero of v

at the wall. This phenomenon is shown graphically in Figure 2.3.1.
It should be understood that the sketches for v and n1 in

Figure 2.3.1 are only intended to show the relative phase at the

wall. It is clear that the actual thermal resonances have varying

phase constant and magnitude away from the wall which is not

represented here.

Determination of the Total Phase for the Thermal Resonances
Figure 2.4.1 shows the typical electron-density contour
expected in a cylindrical plasma column.
The propagation constant for electroacoustic waves in a

warm plasma, kp(x), is given by:

 

. [ill

7.?

l

I /...n;7c

fir...

 

Mn1(x,t1)
>k
D

 

.—
.—

O
NH
_ :1
mg
m
:1

5<

 

 

 

 

 

 

 

 

 

 

Fig. ?.3.1 Phase relation between electron density
perturbation n and associated electron
velocity perturbation v1.

 

 

 

 

 

 

 

2.4.1

21

 

 

 

 

  

Typical thermal
resonance in under
dense region

 

 

 

--—-—-—_——

I

' under—dense region
//L///////’ where kp(x) is real

'

 

 

 

O
x...
W

rig. 2.4.1 The under—dense region in which thermal
resonances may occur if the phase con-
ditions are satisfied and an appropriate
EN field illuminates the plasma column.

22

2
w (x)
kp(x) = ‘3— (1 - 1741/2 (2.43)

0 w

radian frequency of the electroacoustic wave

€
:3‘
0
H
m
8
ll

plasma frequency as a function of x

8
'U
A
>4
v
II

V = ’%%I- = thermal electron velocity

0 e

k = Boltzman constant

T = electron temperature
me = electron mass.

The prOpagation constant kp(x) is real only in regions in which

wp(X)2/w2

5,1. In Figure 2.4.1, kp(x) is real in the region

0 < x < xp, so that an electroacoustic wa”e can propagate between
x - 0 and x = xp. This permits electroacoustic standing waves of
a given frequency w to be excited in the sheath region between

x I 0 and x = xp as long as the total phase of the standing wave
satisfies the phase conditions to be derived. The boundary
condition at x a O was established in section 2.2. It is now
necessary to determine the total phase condition between x = 0 and
x - xp. ‘

The standard time-independent wave equation in one dimension
for electroacoustic waves, equation (2.34) is repeated here for
reference:

2

dn1

dx2

 

+ k 2(x) n = o (2.44)
n 1

where n
1
function

the WKB .

x-depend»

x
J[ kp(x I

where (k.

COUESPC'

directlc:
in ¢(x) :
by SUbst:

into the

 

23

where n1 represents the phasor transform of n1(x,t) and is a

function of x only. In order to establish the total phase of ml,
the WKB approximation is used; n1(x) is expressed in terms of an
x-dependent magnitude function ¢(x) and an x dependent phase term
er kp(x) dx as follows:11

x
11f k (x')dX'
p

n1(x) - ¢(x) e (2.45)

where the plus and minus signs in front of the phase term
correspond to waves propagating in the negative and positive x
directions respectively. It is now necessary to find an equation
in ¢(x) from which ¢(x) can be determined. This is accomplished
by substituting the assumed solution for n1(x), equation (2.45),

into the wave equation (2.44),

 

 

x x
dn1 12 ii] kp(x')dx' ti] kp(x')dx'
Fi— ' dx e + iikpbt) ¢e
2 x x _
d 2 11 k ' d ' ii k ' d '
“1.9.3. I v(x’xuie. / v“”‘(+aa»
d 2 2 dx ’ p
x dx
fx ( ) fx ( M
ii k x' dx' ~ ~ti k x' x'
d
+‘dfi e p (iikp(x)) + ¢e p
x
2 iif k (x')dx' dk (x)
- (iikp(x)) + ¢e P (:1 dx )
2
d n 2 dk (x)
21- (1%: 21k (309$ +¢(-k2(x) 1 1—P———))
dx dx p dx p dx

x
ii] k (x')dx'
* e p

A...
lkp(x) (
If, in t':
function
Compared
Hayes at
are cons;
of Electr
Therefme
Vicinity

deT1Vat1V

1“ W) i.

 

merefOre

24

Therefore equation (2.44) becomes

 

 

2 dk (x)
53.2 5151:- 2 _2____ 2 .
dx2 i Zikp(x) dx kp (x)¢ i i dx ¢ + kp (x)¢ 0 (2.46)

2 dk (x)
2;.2m.)g.2.1+..o
dx p x x

2 dk (x)
1 d ¢ d¢ 1 p
_. (2—+ o) = 0 (2.47)

ikp(x) dx2 dx kp(x) dx

If, in the region of interest, ¢(x) does not change rapidly as a
function of x, the first term in equation (2.47) is negligible
compared with the other terms. In the electroacoustic standing
waves at hand, the first two, or in some cases, three resonances
are considered, so that approximately one to three half-wavelengths
of electroacoustic standing wave are expected in the sheath region.
Therefore the variation of the peak magnitude of n1, ¢(x), in the

vicinity of the turning point is quite small and the second
2 .
derivative term, g—%-, may be neglected. The resulting equation
dx

in v(x) is given by

 

dk (x)
.2. £1.41 1 p ..
¢ dx + k (x) dx 0 (2°68)
Therefore
2d¢ dk (x) a 0

lntegrat:

Vhere K3

expressio-

“hm o

In

1maginary

X<x
p is

 

 

SlnCe °nly

positive t

25

Integration leads to the following solution:

d dk (x)
2 ¢ kp (x) w+ 1
1n(¢2) - - 1n(kp(x)) + ln(K2)

1n(¢2) = 1n( )

kp(X)

K3

JkP(X)

¢(X) = (2.49)

 

where K3 is an arbitrary integration constant. Thus, the

expression for n1(x) postulated in equation (2.45) takes the form:

x
‘ '
1 e ii I kp(x)dx

n1(x) - K-—-—-—— x (2.50)
Vkp(X) D
where kp(x) is real for x §_xp.
In the region where x is larger than xp,kp(x) 18
imaginary and may be written as ilkp(x)| so that n1(x) for
xp < x is most conveniently written as
X 0
1 ii] Ik (x')|dx
P
n1(x) = K-———-—- e x (2.51)

Since only an attenuated wave is expected in this region, the

positive term in the exponential is not applicable so that:

Thus the

Since the
r‘wresent

COUVenien

This is a

 

26

X
1 - f lkp(x')|dx'
(x) = K———————-— e (2.52)

Thus the expressions for n1(x) are summarized as follows:

 

x
K1 ii Jf k (x')dx'
——'_—_—__—" e X P for O<X<X
Vkp(x) p p
nl(x) =< )(2.53)
x
K2 8 - Jr Ikp(x')ldx' for x>xp
x
Jlkp(x)| P

 

 

Since the electroacoustic waves between x = 0 and x = x
represent standing waves, equation (2.53) for that region may be

conveniently written as

K1 p
n1(x) =-—-—-—-— sin(jrK k (x')dx' + 9) (2.54)
kp(x) p
x

where 6 represents an arbitrary phase constant. This expression
breaks down in the limit as x goes to xp where Kl/Jkp(x) becomes
unbounded. Therefore another formulation is required for the
2 l 2 2
vicinity of x - x : Since k (x) --——— (m - m (x)), where
p 1) v02 12

mp(x) is a slowly changing function of x, the expression for

kp2(x) can be linearized near x = xp as follows:

2 -a
kp (x) ;;72 (x - xp) (2.55)
o

This is a linear function with a value of zero at x a xp as

required 2

De

leads to:

"ansforns-

The wave e

Nov:

 

and

27

required and a slope equal to (--—2§).
V

0
Defining a new variable
2 = +035)“3 (x - x ) (2.56)
P
V
0
leads to:
k 2(z) - —<—°i§)2’3 z (2.57)
p v

0

Transformation of the original wave equation proceeds as follows:

The wave equation (2.34) from section 2.2 was

 

2
d n1 2
2 + k (x) n1 = 0
dx p
Now:
dn1 dn1 93.: _ £2_ a )1/3
dx dz dx dz V 2
o
and
dzn dn d“ ~ dn‘
.....!~. .. 5L (.__.1.) . £1... (4)93. .. .. £1... _l(_2_)1/3(_9_)1/3
2 dx dx dz dx dx dz dz 2 2
dx V0 V0

' 21 ( a2)
dz V

2/3

 

 

Thus the

x-x be
P

The soluti

function

Where N j
o

eQuation I

forz>0

and for z

28

Thus the wave equation in z applicable to the vicinity.of

x I x becomes:
P
d2n

dz2

 

— zn - 0 (2.58)

The solution to equation (2.58) is given in terms of the Airy

function as follows:

N 2
o s
n1(z) ;—- cos(3—-+ sz) ds (2.59)

o
where No is an arbitrary constant. For large values of [2],
equation (2.59) has the following asymptotic approximation:

for z > 0 which is equivalent to x > xp

No e -2/3z3/2

1/4

(2.60)
2/; z

n1(2) -

and for z < 0 which is equivalent to x < xp

N .
O

J; (_z)1/4

 

n1(z) - sin(-23- (-z)3/2 + n/4) (2.61)

See Figure 2.4.2 for a typical graph of the Airy function

in the vicinity of z - 0. Since equations (2.53) and equations
(2.60) and (2.61) should agree at some distance from x - xp, where
the linear approximation for kp2(x) still holds, the two solutions

may be compared. In the region x < xp, equation (2.53) gives

(in terms of the variable 2, using equation (2.54))

29

Ai(z)

0.6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—O.6 y. 7
a10 -a -6 -4 -2 q 2
I
I l
' L :X
0 XP

1
Ffigg. 2.4.2 Sketch of Airy function,1

Ai(z) = fi%r‘jr cos(s)/3 + s 2) dz
0

and fits

The ohas

Dhase to

Thus the

 

30

K 0
n1(z) I 1 .1 sin(J[ (-z')1/2 dz' + e) (2.62)
z

93
(Rn/v02) (ml/2

and after performing the integration in the phase term,

K1

21/3
(a/Vo )(-z)

 

 

3/2

 

 

n1(z) I sin(%-(—z) + e) (2.63)

1/2

The phase term in the argument of equation (2.63) agrees with the
phase term in equation (2.61) if
6 I n/4 (2.64)

Thus the WKB formulations for n1(x) in the two regions become

x
exp(- J{ kp(x') dx' for x > xp

1
Jkp(X) x
n1(x)- p 5 (2.65)

K2

Jkp(X)

Ff

 

x

p
sin(‘[- kpr') dx' + n/4) for 0 < x < xp

x

The significant result from this section needed in the subsequent
determination of the electron density profile from the thermal
resonance data is an expression for the total phase of these
thermal resonances between the wall and the critical point. This
phase expression is now obtainable as follows. From equation

(2.54) and (2.65) it is seen that at the wall where x I 0,

X

 

K p
n1(0) - ——1 sin( k (x') dx' + n/4) (2.66)
(n. (x) p
p 0
must represent a maximum of nl(x). This leads to the condition
that
xp
( kp(x') dx' + n/4) I (2m + 1) (n/2) (2.67)

O

where m

becomes

0!

Figure 2.
to be ex:
in Figure:
C”juncti
rePresent

T}
analytic;
of the 8:1
thermal r‘
thermal n
Plasma to;

density or

 

31

where m is a positive integer. Therefore the total phase integral

becomes

x
p

kp(x')dx' = <2m + 1)(n/2) - n/a

or

kp(x')dx' = (m + l/4)n (2.68)
0
Figure 2.4.3 shows typical wave forms of the thermal resonances
to be expected in the plasma sheath region. Only the phase shown
in Figure 2.4.3 for the various resonances is significant in
conjunction with this discussion; the magnitudes are merely
representative of typical waveforms.

The phase integral in equation (2.68) is used in the
analytical techniques developed in section 3 for the determination
of the electron density profiles in cylindrical plasma columns.

The WKB approximation developed in this section for the
thermal resonances is also used subsequently to graph examples of
thermal resonances with normalized magnitude for actual cylindrical
plasma columns based on the numerical results for the electron

density profile ne(r) presented in Chapter 4.

Development of Relationships between Dipole Resonance Frequency

 

and Plasma Frequency in a Cylindrical Plasma Column

 

In the determination of the electron density profile in a
cylindrical plasma column based on thermal resonance data, it is

necessary to know the relationship between the exciting EM wave

32

1n1(x)

first thermal resonance

 

 

 

 

 

 

 

U
l L '4~%>)t
I / I
I X I
I p1
U
A1126!) [second thermal. I
.resonance '
I \v I i) L_________>XX
x
p?
l
. I
#36:) 1 third thermal
' resonance

 

l j//f\l 5 1 ;.x

I ”I/ U XI ‘

 

 

 

 

p5 !
l
a (inside radius
._,. ...___ Glass of plasma. column)
wall
region

Fig. 2.4.3 Typical waveforms of the first three thermal

resonances. x , x , and x are the
P1 p2 p3

critical points at which k , k , and k
P1 P7 P3

respectively go to zero.

frequent

column.

where Cr
exact sc-
of the e
Such exa
Parabolic
Cvlindri‘
eqU8t1029

bEEVeen t

elECtron

 

XnaPDTt‘nr

an aPPfox
1‘ 18. to

average <

33

frequency w and the average plasma frequency <wp(r)> in the plasma

column.

2 2
< > n C 2.69
mp (r) p w ( )

where Cp is a proportionality constant to be determined. An
exact solution for <wp(r)> as a function of m requires knowledge
of the electron density profile in the cylindrical plasma column.
Such exact analyses have been perfOrmed based on an assumed
parabolic electron density profile subdividing the plasma into
cylindrical sublayers and performing a numerical analysis on the
equations resulting from the boundary conditions at the walls and
between the strata.

Since it is the objective of this research to determine the
electron density profile in the plasma cylinder, it would be
inappropriate to presume any specific profile a priori. However,
an approximate value to Cp is sufficient for a profile analysis.
It is, therefore, appropriate to base the determination on a
uniform plasma with a uniform plasma density opuso that the
average <wp(r)> in the actual plasma cylinder corresponds to
wpuof the assumed uniform plasma.

It has been shown that a quasi-static approximation is
appropriate in many cases .13 The test for the validity of the
quasi-static approach in any specific case is based on an examina-
tion of Maxwell's Equations for the plasma region in the absence
of a uniform magnetic field. Maxwell's Equations in the plasma

region are:

Taking ti

results I

Letting I

“OH in t:

0f Labia.

ExpffisSi.

0f eqUat

In c0971a:
the 110130:
quasi‘st;

small.

Of the U.

34

v x B =- +jwuocpE (2.70)
V x E =—jn§ (2-71)
v . 13' - 0 (2.72)
v - F: = 0 (2.73)

Taking the curl of equation (2.70) and (2.71) and combining the

results leads to the homogeneous Helmholtz Equation
2 2-—
(V + cpuow )E I 0 (2.74)
2 2
Letting ke I w uocp, equation (2.74) becomes
(v2 + kez)E'- o (2.75)

Now in the quasi-static approach the system may be solved by use

of Laplace's Equation
2
v o = o . (2.76)

Expressing equation (2.76) in terms of EDby taking the gradient

of equation (2.76) leads to
V E I O (2.77)

In comparing equation (2.77) for the quasi-static approximation to
the homogeneous Helmholtz equation (2.75) it appears that the
quasi-static approximation is justified if ice2 is negligibly
small. Studying, for example, a one-dimensional application in x

of the two equations, equation (2.75) becomes

hm solut

Given the

can be deI

30 that

Thus the i

0“ the or:

mere fro!

35

d2 2

'——§ E(x) + k E(x) I 0 (2.78)
e

dx

The solution to equation (2.78) is

E(x) I K cos(kex) + K sin(kex) (2.79)

l 2

3E
Given the boundary conditions EO and (ax)o at x I 0, K1 and K2

can be determined as follows:

K a E (2.80)

(fig) I (Klke sin(kex) + K

8x 0 ke cos kex)o (2.81)

2
so that
BE 1
K2 (3x)o (Kg)

Thus the solution of equation (2.78) becomes

3E sin(kex)

E(X) '3 E0 COS(keX) + (5;)0 T (2.82)
On the other hand, the solution to

32 ‘ ‘

7 E(x) ‘3 0 (2.83)

8x
is

E(x) I le + K2 (2.84)
where from the boundary conditions Eo and (%§)o at x I 0:

K . E (2.85)

and

It follc

For vale
equation

(2.87),

Thus the

In terms

 

"here (1
ill 13 the

Penneabil:

36

and

3E
(-3—X)O = K1 (2.86)
It follows that
3E
E(x) E0 4' (SE-)0 x (2.87)

For values Ikexl2 << 1, the solution to the Helmholtz Equation,
equation (2.82), approaches the solution to Laplace's Equation

(2.87), because equation (2.87)

3E
E(x) = Eo + (3;)0 x

is in fact the first order Taylor series approximation of equation
(2.82)

sin k x

k
e

E(x) I Eo cos(kex) + (%§)o

Thus the condition for using a quasi-static approximation is:
2
Ikexl << 1 (2.88)
In terms of the cylindrical plasma column this means that
2 2
lepuow :1G | << 1 (2.89)

where dc represents the characteristic dimension of the system;
m is the incident EM wave frequency, no is the free space

permeability and

 

if the
system
order c
n, and
Thus It

that th

in Plan

in cyli

series

indicat

“hare

Since 1
excitlr

d
iDoleI

37

if the collision frequency v is assumed zero. In the experimental
system at hand, up is in the order of 20 x 109 rad/sec, w is in the
order of 10 x 109 rad/sec, dc may be taken as the radius a I .007
m, and so is the free space permittivity all taken in mks units.

” 2 1/2 2 2 -2
Thus Iso(l - —%—9 now dc I is in the order of l x 10 so

m
that the quasi-static approximation is justified in this analysis.

Consider the geometry of a cylindrical plasma column shown

in Figure 2.5.1. The solution of Laplace's Equation

V2¢ - o (2.90)

in cylindrical coordinates with z-independence can be expressed as

series solution

n in6

o = (Klnr + Kan‘“) e (2.91)

where n is an integer unequal zero. In regions 1 through 3 as

indicated in Figure 2.5.1, the solutions become:

61 I Anrn cos(n9) - (2.92)
n -n

62 Bar cos(n6) + Cnr ‘ cos(n9) (2.93)

¢3 ’ Dar-n COS(n9) + rn cos(n9) (2.94)

where an exciting field of the form rn cos(n6) is considered.
Since in the system at hand the free space wavelength of the
exciting EM wave is much larger than the radial dimension, the

dipolar contribution (n I l) is most significant so that the

 

 

‘.
'2‘
‘5' 2
. 1
I

38

     

Glass

l
Plasma

(3) (2) (1)

 

 

 

 

Glass wall
thickness b

Fig. 2.5.1 Geometric arrangement of cylindrical plasma
column contained in a cylindrical glass
discharge tube of wall thickness b. The
inside radius is a while the outside radius
is c.

Conti:
electI
solut:
becaus

resona

.‘D I

 

The Val
dipole

Rule. 1

T

tie detI

t0 zEro‘

39

problem can be simplified significantly by rewriting equations

(2.92) through (2.94) for n I l:

61 I Ar cos(0) (2.95)
62 I Br cos(9) + C-% cos(8) (2.96)
63 I D-% cos n(9) + r cos(8) (2.97)

Continuity of the potential o and the normal component of the
electric displacement at the two boundaries r I a and r I c permit

solution of the arbitrary constants. D is of primary interest

(1)

because it is maximum at the value-—EE at which the dipole

resonance OCCUI‘S .

The system of equations to be solved is:

 

 

 

 

 

 

I- I ~ I, - -
a -a -1/a 0 A 0
O c l/c -l/c B c
is
c -e 0 C I 0 2.98
p g a, ( )
eg to
0 Eg - "’2 —2‘ D so

The value of the arbitrary constant D must be maximum at the
dipole resonance. Since D can be expressed in terms of Cramer's
Rule, it is evident that its maximum value is obtained by setting
the determinant of the coefficient matrix in equation (2.98) equal

to zero,

 

Letting [qr

' c
£8 60 gr

Plasma , c

(l/a

EquitiOn (2
the um"leric

pemttivit

Since

40

 

 

a -a -l/a 0
0 c l/c ~1/c
I O (2.99)
e -e e /a2 0
P 8 8
0 e -e /c2 c /c2
8 8 0

Letting cgr represent the relative permittivity of the glass,

6 I s e and e
8 0 8:, pr

plasma, ep I soap , the expansion of equation (2.99) becomes
r

represent the relative permittivity of the

(U.2 + l/c2)€ (1 + e ) + (1/a2 — 1/c2)
gr pr

(epr - Egr ) I 0 . (2.100)
Equation (2.100 may be solved for Epr which in turn is used in
the numerical determination of mpz/w2 as follows. Given
numerical values for the radial dimensions and the relative
permittivity of the glass, egr I 5, a I .007 m, and c I .008 m:
c

J- -1.6
E

0

Since EB-I l - —%—-I 1 - C

This value for Cp I —§-is used in the subsequent numerical
m
analysis. The value for CI) agrees well with values obtained by Lee12

for similar discharge columns.

DETER
PMSF‘A COL

ll lntrogu

plasma

a frequ
the p13
sheath

and dis
Electrc
Hall ir
Center.
towards
from t}
ElEQtrc

rint}

finere n

is the

CHAPTER 3

DETERMINATION OF ELECTRON DENSITY PROFILE IN CYLINDRICAL
PLASMA COLUMN BASED ON THERMAL RESONANCE DATA IN THE SHEATH REGION

3.1 Introduction

 

When an electromagnetic wave is incident on a cylindrical
plasma as shown in Figure 3.1.1, a dipole resonance is excited at
a frequency w depending on the average plasma frequency mp(r) in
the plasma. Furthermore, thermal resonances may be excited in the
sheath region near the wall at certain combinations of frequency
and discharge current levels. These thermal resonancra represent
electroacoustic waves. The sheath rerion is the region near the
wall in which the electron density is reduced from its value at the
center. It is well known that the electron density decreases
towards the wall along with an increase in negative potential away
from the center:1 The prepngation constant associated with the
electroacoustic wave, kp(r), is a function of the radial distance

r in the plasma column and is given by:

2
w (r)
kp(r) a %_.(1 _._E_§__)1/2 . (3.1)

O (.0

Here mp(r) is the plasma frequency as a function of r defined as:

2
2 e ne(r)
mp (r) ' ‘FTET—I- (3.2)
e o
where ne(r) is the static electron density as a function of r, e

is the electron charge, me is the electron mass and so is the free

space permittivity; w is the frequency of the incident electro-

41

 

LTJI

 

42

 

  

 

Glass wall

A cylindrical plasma_column illuminated by
TM field as shown. B and E ‘represent

0t 01
the transverse and longitudinal components
of electric field respectively.

the w
frequ
say r
and k
reson
frequ
satis
shown
(m +

elect
aDPea
data

the e

it

3.2

43

magnetic field. Thermal resonances can exist, if in the so-called
sheath region near the wall, the electron density, and therefore
wp(r) is small enough to yield a real value for kp(r). Since in
fact ne(r) and therefore wp(r) increase monotonically away from
the wall as discussed in Chapter 2, there may exist for a given
frequency w of an incident EM wave a point in the plasma column,
say r I rp, at which w I wp(r), so that kp(r) is real for r > rp
and kp(r) is imaginary for r < rp. Under these conditions thermal
resonances may exist between r I rp and the wall where r I a for
frequencies m for which the total phase of such resonances
satisfies the total phase condition derived in Chapter 2. It was
shown there that the total phase for the mth resonance must be
(m'+ l/4)n. If an appropriate functional description of the
electron density profile can be formulated, the unknown parameters
appearing in such a formulation can be determined from pertinent
data regarding the thermal resonances. In the following section,
the experimental procedure is presented for collecting thermal
resonance data followed by a formulation of useful functional forms

of the electron density profile ne(r) and their analysis.

Experimental Procedure

 

The experimental arrangement for obtaining plasma resonance
data in a cylindrical plasma column is illustrated in Figure 3.2.1.
The experimental technique is based on the excitation of the dipole
resonance along with excitation of thermal resonances in the sheath
region in a bounded cylindrical plasma column in glass tubing by
use of an electroacoustic probe. The probe consists essentially of

an open-ended coaxial line fed by an RF generator through a direc-

, ..
'7'4

\_~l
h)

7:5.

n.
‘ n
" ‘.LI

3-!“

H...

 

.1

44

High Voltage
DC Power Supply

 

 

 

 

 

Current
Transformer

 

 

. , _——o
Sweep
Trans- E: 1 10 V

former _——o

 

pm

 

 

Plasma rad. a=7mm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Variac Glass
// thickness b=1mm ,
Filament
\5-f-ZPla§ma Column Supply
E.A. 7— Probe a
Anode
Glass mgr
thickness
b=1(mm) 63g?
Directional
Coupler CRO
-section ’
of plasma ggtector l L
cylinder of . 9 Q
inner radius ’- _1_ 1L
a I 7(mm) ’ ’

 

 

 

 

RF
Generator

 

 

 

Fig. 3.2.1 Experimental arrangement for obtaining plasma
resonance data in a cylindrical plasma column.
An electroacoustic (E.A.) probe is used to
excite the dipole and thermal resonances in the
plasma column. The E.A. probe also picks up
the scattered field whose peaks indicate the
presence of resonances in the plasma.

tiona
the p
glass
the c
of th
neces
longi
Refle
are d
tonne
densi
produ
catho
has a
The a
currg
“heme

the n

45

tional coupler. In order to excite electroacoustic resonances in
the plasma column, the open end of the probe is placed near the
glass wall containing the plasma column. The inner conductor of
the coaxial line is extended a small distance beyond the Open end
of the outer conductor so that the RF radiation contains the
necessary longitudinal component of‘E field to excite the desired
longitudinal electroacoustic resonances in the sheath region.
Reflections from the plasma cylinder are received by the probe and
are directionally coupled to an RF detector whose output is
connected to the vertical input of an oscillosc0pe. The electron
density in the plasma column is adjusted by a discharge current
produced by a high voltage source connected to the anode and
cathode of the plasma tube as shown in Figure 3.2.1. The current
has a low frequency (60 Hz) ac variation superposed on its dc level.
The ac component produces a variation in the plasma discharge
current and also produces the horizontal sweep on the oscilloscope.
Whenever the current level passes through a value which satisfies

the resonance condition

_ mp2(rm))1/2 d

r I (m + 1/4)n

at an excitation frequency w for the mth resonance, a peak is
observed in the reflected power level. In addition, the dipole
resonance is observed as the strongest resonance in the column.
The discharge current levels at the dipole resonance and the first
few thermal resonances are observed. In the subsequent numerical

analysis only the ratios of the discharge current levels are used.

shown
charg

freon

Devel

 

the e

poten

Where
assun
the v

inate

Since

DQCQS

plaSm

P0135

the S

3.3

46

Eight sets of data obtained in the experimentation are
shown in Figures 3.2.2 through 3.2.5. Table 3.2.1 shows the dis-
charge currents id, 11, 12, and i3 along with the excitation

frequency for each of the eight data sets.

DeveloPment of Functional Form for the Electron—Density_Profile

 

If a Maxwellian electron density distribution is assumed,
the electron density profile ne(r) is expressed in terms of the
potential profile, V(r), by equation (2.15) in section 2.2,

eV(r)

ne(r) I noe RT (3.3)

where no is the electron density at V(r) I 0. It is reasonable to
assume that in the plasma cylinder used in the experimentation,
the voltage at r I 0, V(O), is negligibly small and may be approx-

imated as zero,

V(O) I 0 (3.4)

Since the actual value of V(O) is not known, this approximation is

necessary to obtain a solution for the problem. Thus
no I ne(0) ‘ ‘ (3-5)

where no represents the electron density at the center of the
plasma column. The problem then is the formulation of a functional
form for V(r). This might best be arrived at by considering
Poisson's Equation in the region of interest and choosing a
functional relationship for V(r) which at least in form agrees with

the solution to Poisson's Equation. A complete solution of

47

Back scattered
Eh field
Data set #1 f=2.016 GHZ

 

Back scattered
. EM field
: Data set a? f:2.10 GHz

 
   

_I\
.I .l ‘ I

11
. i
12 d

Fig. 3.2.2 Experimental results (data sets #1 and 2) for
the back scattered Eh field from a cylindrical
plasma column as a function of discharge current.
f is the frequency of the incident EN field.
id, 11, i2, and 13 are the discharge currents

at which the dipole resonance and the first
three thermal resonances respectively occur.

Fig.

48

Back scattered
EN field
Data set 53 f:2.25 GHZ

 

i(ma)
Back scattered
EM field
Data set #4 f:?.5? GHz

i(ma)

 

3.2.3 Experimental results (data sets #3 and 4) for

the back scattered EM field from a cylindrical
plasma column as a function of discharge current.
f is the frequency of the incident EM field.

id, i1, 12, and 13 are the discharge currents

at which the dipole resonance and the first
three thermal resonances respectively occur.

 

A./.\

we)

49

+ Back scattered
Eh field
1 Data set as r=I.9I7 GHz

i(ma)

 

Back scattered
EM field
Data set #6 f=?.017 GHz

 

_> i(ma)

9 l 150‘ r 2 o 270' 550

5 l 1I .l
12 1d
Fig. 3.2.4 Experimental results (data sets #5 and 6) for
the back scattered EM field from a cylindrical
plasma column as a function of discharge current.
f is the frequency of the incident EM field.
id, 11, i2, and 13 are the discharge currents

at which the dipole resonance and the first
three thermal resonances respectively occur.

 

50

Back scattered
EM field
Data set #7 f=2.275 GHz

 

90‘ t 150 3910 270: 330 mm)
_ w ,e,
13 .i 11 v
12 1c1
1 Back scattered
EN field
Data set #8 f:2.322 GHz
i(ma)

 

Fig. 3.2.5 Experimental results (data sets #7 and 8) for
the back scattered ER field from a cylindrical
plasma column as a function of discharge current.
f is the ifrequency of the incident EM field.

1d, 11, 12, and iBare the discharge currents

at which the dipole resonance and the first
three thermal resonances respectively occur.

51

 

 

 

 

 

 

 

 

 

 

Data . - . . .

set # f (GHZ) ld (ma) 11 (ma) 12 (ma) 13 (ma)
1 2.016 270 188 150 125
2 2.100 290 190 150 120
3 2.230 340 235 185 160

4,.

4 2.320 355 245 700 175
5 1.917 270 190 135 110
6 2.017 285 190 150 120
7 2.275 290 195 150 120
8 2.322 320 210 160 155

 

 

 

 

 

 

Table 3.2.1 Experimental data set 1 through 8. Given
are the frequency of the inCident EN field
and the discharge currents id, i1, 1? and

13 at which the dipole resonance and the

first three thermal resonance respectively

occur.

 

Poi:
the

EXDE

Her

and

is
t 0;;

inc

In

50

52

Poisson's Equation in the plasma column is not possible because
the boundary condition for V(O) is not known and the available
experimental data are insufficient to determine it.

Poisson's Equation in cylindrical coordinates is given by:

gian+1gmi,_e_<gi

dr2 r dr to

(3.6)

slit).

Here 0(r) a eno(1 - e kT

) (3.7)

and T represents the electron temperature. This expression for
0(r) is based on the plasma sheath model in which the ion density
is nearly constant throughout the plasma region due to ion drift
towards the negative wall potential. Substituting equation (3.7)

into equation (3.6) yields:

2 en EELEL
d V(r) + 1 dV(r) = _ o (1 _ e kT

. )
dr2 Y dr 80

(3.8)

In the region away from the wall where eV(r) << kT, the following
approximation may be made:

en . .
____fi__ _ _ __g (l _ 1 _ eV(r)

-—-0
er r r to RT

(3.9)

so that we have the following approximation of Poisson's Equation:

2
e n

2
9.191.“). .1__V_§£_)._._....9. .-_-
er + r r cho V(r) O (3.10)

This is a Bessel Equation and the solution is in the form of a

zero order Bessel function with imaginary argument:

uhe
and

use

beco

Suff

53

V(r) = C IO(Klr) (3.11)

l
2
e n

is an arbitrary constant and K = ———g , containing no

where C 1 kTeo

1

and electron temperature as constants. If equation (3.11) were

used throughout the plasma column, C would represent the potential

1
V(O). As stated above, a value for V(O) is not available so that

equation (3.11) is merely used to show that a Bessel series is an
appronriate form for the potential V(r) near the center of the
plasma column. The approximations made in equation (3.10) do not
hold near the wall. The wall region is considered next.

In the sheath region near the wall, where the approximation

eV(r)
kT

<< 1 does not hold, the following alternate approximate
formulation may be used. Letting Vw be the wall potential,
V(a) a Vw, Poisson's Equation may be written as follows:

eV eV
2() 1d“ 9,, 913921-191?!
d V r V r _ o T T T
2 + .1?de ‘ - (l e e ) (3012)

5
dr 0

 

Defining a new variable v(r) = V(r) - Vw’ equation (3.12)

becomes: , .
2 ve eV(r)
l
111a+l§zitl=-_.°.(1-ek1‘ ekT ) (3.13)
drz r dr £0

 

Sufficiently close to the wall, v(r) is small enough to let

3313.2.

e kT é l + EE%32- . Therefore:

This eqna

On

4
-r'\.

“

Q-
Hf)

with K2
Electron

me solut

54

eV
2 '( k ) e“ w ‘( )
d v r) l.§3_f__= _ “.2. _ kT 9V C_
T4. r dl’ (1 e (1 + kT )) (3.14)
dr 0
This equation becomes:
eV eV
2 a k ) ezn 6—11 en -—11
d v(r) l.§X"£_._ o kT a _ ~_2 _ kT
drz + r dr .ETE; e v(r) co (1 e ) (3.15)

01'

dzvkr) + 1_dv(r) _

2 r dr

I _ _
K2v(r) — K3 (3.16)
dr

with K2 and K3 constants containing the wall potential, the
electron density at r - 0, no, and the electron temperature T.
The solution is again in the form of a zero order Bessel function

with imaginary argument in addition to a constant term:

v(r) - c 10(K2r) + K3/K2 (3.17)

2

The fact that the potential variation throughout the region is in
the form of Bessel function Io(x) and recalling that the only
available boundary condition for V(r) is based on the assumption
of zero potential at r - 0, a reasonable choice for curve fitting
the expected potential distribution is a Bessel function 10(2)
with a unity offset bringing it to zero at the origin as follows

(letting n(r) -

-E¥%£l for simplicity of notation and the argument

2 - yr):

n(r) ‘ 1 - 10(Yr) (3.18)

where Y
(3.18) 1
negative
lends it
profile

COPIESDC

n A
o and 1
the til,
(I
for n (I

e

DaraBOlj
driCal r
does inc

(3.19) a

LEtting

55

where y is an arbitrary constant to be determined. In equation
(3.18) it is anticipated on phenomenological grounds that V(r) is
negative for all 0 < r < a. The particular form of equation (3.18)
lends itself well to the determination of the electron density
profile from thermal resonance data as shown subsequently. The
corresponding electron—density distribution ne(r) is given by:

n (r) = n e(1 - 10(Yr)) ' (3 19)

e o ’ '

no and y must now be determined from numerical analysis based on
the thermal resonance data.

As an initial simplified approach, a parabolic approximation
for ne(r) is used in the next section. This is done because the
parabolic approximation for the electron density profile in cylin—
drical plasma columns has been used extensively in the past and it
does indeed represent an approximation of ne(r) given in equation
(3.19) as follows:

(1 - 10(Yr))

no e (3.20)

ne(r)

no(1 + 1 - 10(yr))

.. 1.: 2
1_2 2 . .
Letting (2) a a/a leads to the customarily used approximat1on:

necr) - no(1 - cog-)2) (3.22)

In the following section, a numerical solution technique is

deve10ped for no and a as well as the electron temperature T, the

relative
resonancc

R.

Determine
Plasma Cc

Ir
anpropria
The unkno

(l) n01 =

(3)

“04 E

(4 <
) ne(r

3.4

56

relative wall potential nw’ the turning points for the first m
resonances rm, and the ratio of peak to average electron density,

R.

Determination of Electron Density Profile in a leindrical‘Waqm

 

Plasma Column Based on a Parabolic Approximation

 

In order to solve for the pertinent parameters, an
apprOpriate system of simultaneous equations must be developed.
The unknown quantities are

(1) no - center peak electron density for the first thermal

1
resonance;

(2) no2 = center peak electron density for the second thermal

resonance;

(3) no a center peak electron density for the dipole resonance;

d
(4) <ne(r)d>av = average electron density for dipole resonance;

(5) r - value of r where kol(r) = 0 (critical turning point) for

1
first thermal resonance;
(6) r2 = value of r where ko2(r) = 0 (critical turning point) for
second thermal resonance;
r 2
(7) a constant in nem(r) “om(1 - o(;) ),
(8) T - electron temperature.

In order to solve for these eight unknown parameters, the

following eight independent simultaneous equations are necessary:

1 no
(1) J - ——1 (3.233)
n
d °d
i no
(2) 2 - ——?- (3.23b)

id nod

Equation
density

only rat

(3) <wp

Equation
resonancc
section 2

constant

(4) “pl(z
(5) mp1(r

EquatiOns

gOes to z

(6)

57

Equations (3.23a) and (3.23h) are valid because the electron
density is proportional to the plasma current level. In this study

only ratios of currents are needed.

2
2
<ne(r)d>av = Cp m (3.23c)

 

(3) (mp2(r)>av a
e 0

Equation (3.23c) is based on the relation between the dipole
resonance frequency and the average electron density discussed in
section 2.5 where a numerical value for the proportionality

constant Cp was found.
(4) mp1(r1) = w (3.23a)
(5) mp2(r2) . w (3.23e)

Equations (3.23d) and (3.23e) are based on the fact that kp(r)

goes to zero when mp(r) = w.

a

(6) kpl(r) dr = g-u (3.23f)
r1 .
a

(7) kp2(r) dr - g-n _ . (3.23g)
r2

Equations (3.23f) and (3.23g) represent the total phase spanned by
the first two thermal resonances respectively based on equation

(2.68) in section 2.4.

a
(8) <ned(r)>av - -l§- nod(l - a(§)2) an dr (3.23h)

Tia
O

In equat
average .
ship hol<

1:
equations

a numeri<

profile

the valuc
determine
experimen
electron
be develo
Computer.
It
square Of

OCCurs is

The plaSm.

58

In equation (3.23h), the peak electron density is related to the
average electron density for the dipole resonance; this relation-
ship holds equivalently for any thermal resonances.

In the following development, these eight simultaneous
equations are discussed in greater detail and are used to develop
a numerical solution for the desired parameters.

Using the parabolic approximation to the electron density

profile

ne(r) = no(1 - a(§)2), (3.24)

the values of no and a must be determined. These values can be
determined in terms of the thermal resonance data obtained in the
experimentation. To obtain the desired numerical solution for the
electron density profile, a system of simultaneous equations must
be developed which lends itself to a numerical solution on the
computer.

It was shown in section 2.5 that the average value of the
square of the plasma frequency, <wpd2(r)> when the dipole resonance

occurs is related to the resonance frequency w by the relation:

<wpd2> = Cp m2 ‘ ’ (3.25)

where Cp is a pr0portiona1ity constant determined in section 2.5.

The plasma frequency w is by definition given by

pd

 

<wpd2(r)> .. <ned(r)> (3.26)

e o
where <nod(r)> is the average electron density at the dipole

resonance. Letting nod represent the peak density at the center

of the cc

as follo:

Similarly

28:0 at r

F
rom basic

59

of the column at dipole resonance, <nod(r)> can be related to w

as follows:

2
(wpd (r)> meeo

 

 

 

 

<ned(r)> = 2
e
mesz
0 P
(ned(r)> = 2
e
2 a
9P meso w nod r.2
2 ='*—§ (l - 0(3) ) 2nrdr
e na
0
C com w <ne(r)>
._2_§_§__.= (1 - g) a ———————- (3.27)
n
e n od
°d
Therefore
2 (1 "%) 92 “0d
w = C e m (3.28)
p o e

The first thermal—resonance standing wave exists between the wall
(r - a) and the point rl in the plasma, at which the phase term

kp1(r) goes to zero:

k a 0 (3.29)

p1(‘1)

Similarly, for the second resonance, the phase term kp2(r) goes to

zero at r2:

kp2(r2) . o (3.30)

From basic theory, the phase term kp(r‘ for an electroacoustic wave

is given by

60

2

 

 

w (r)
k (r) = ge-(1 - —R—;——)1’2 (3.31)
P 0 wk
3kT
where Vo represents the thermal electron velocity ’;~— .
e
Therefore
2 r1 2
e “01(1 - a(;—) ) 2
m = m (3.32)
eEo
and
r
2 2 2
e n02(1 - at?) ) 2
m c = m (3.33)
e o
Combining equations (3.28) and (3.32) leads to
r n
2 1 od
(1 - a(—l) ) =-—— (1 - 9) - (——~) (3.34)
a C 2 no
p 1
and combining equations (3.28) and (3.33) leads to
r n
2 2 1 . 0d
(1 - a<—~) ) = E—-(1 - a/2) (———) (3.35)
a no
p 2
no id nod i _
Since -——-- -—-and --- -—-, where i , i and i are the currents
“02 11 1102 12 d 1 2

at which the dipole and first two thermal resonances occur,
equations (3.34) and (3.35) lead to the following expressions for

r and r :

1 2
1 1 1 1d 1/2
r1 ' 3‘; ' a" (a “ '2‘ 1'“) (3°36)
p l
1 1 1 1 1d 1/2
r2 ' 3‘2; ‘ a“ (I; ' ‘2‘ T) (337)

61

Since the total phase for the first two thermal resonances is

-%W and %m radius respectively, the following phase integrals

 

 

 

 

result:
2
a w w (r) 1/2
T (1 — J-T) dr = (5/4). (3.38)
r o w
l
and
2
a w m (r) 1/2
T (1 - 174 dr = (9/4)" (3.39)
r o w
2
Since for the first two thermal resonances:
2 r 2
2 e nolu - “(3" )
mp1 (r1) 2 m c ’
e o
and
2 r 2
2 e “02(1 - ((71...) )
mp2 (to) a m e ’
e 0
equation (3.38) and (3.39) become
2 2
a u) ‘3 “°1(1 ' “(”29 ) 1/2 5
"‘7“ (1 - 2 ) dr = (2')" (3.40)
r o w meeo
l
and
2 2
a w e “02‘1 ' “(1:3 ) 1/2 9
6"(1 ' 2 ) dr = (2)" (3.41)
r o m meeo
2
Combining equations (3.40) and (3.41), and expressing no1 and no2
no i
in terms of nod from equation (3.28), recalling that -—l>- ——-and

nod id

62

no2 i

-——-- —Z-, the following equation results:
“0d 1d
tl/a 11 C 1/2
9/1. (1-(——)<———L—)(1-a(—:;2)) d(—-)
id (1'50) a
l
(3.42)
- rZ/au - (~13) (——P————C ) (1 - (5)2 )dei) = 0
id (1 - .50) a a ) a
1

Equation (3.42) contains the three unknowns a, rl/a and rZ/a; rZ/a
can be expressed in terms of rl/a based on equations (3.36) and

(3.37) as follows:

r 2 1 1 1 1 1 d 1/2
“3’ “a“: (a‘a‘(;‘2) 1;)
1 1 1 1 1d 1/2
- (a -'E— (a - 2) if) (3-43)
p 1
Therefore equation (3.42) becomes:
r la
1 1 c
9/4 <1- (31) ($73—53) (1- a§2<)))1’2d(-§)
d .

1
,1 (3.44)
""* A(r/a)

- (1-(-1-2-)(——C-2-—-)(1-W)))1/2()-0

i 1 - 5a a

1

Solving equation (3.36) for a in terms of r1/a yields:

63

 

 

 

d
(1 - C 11>

a = r P 1 (3.45)
r_;92_ d
‘a 2i1C

Equations (3.44) and (3.45) represent two simultaneous equations in
two unknowns which may be solved numerically. After r1/a and a are

available, equation (3.40) can be solved for V0 which in turn gives
3kT

 

the electron temperature T from V0 = Er—-,
e
1 i C
= 52. _ .1 .__11_._ _ 3.2 1/2 .5
v0 5” (1 (id) (1 _ _50) (1 (a) )) d(a) (3.46)
rl/a
and:
mevo2
T 3k, (3.47)

The ratio of peak to average electron density nqkne(r)> is obtained

from equation (3.27) as

n
R=——o—--——= 1
<ne(r)> (1 - a/Z)

 

 

The equations deve10ped in this section for use in the computer
analysis are summarized here in the form in which they are

incorporated into the computer program for the numerical analysis.

rI/a 1 C
1 1/2
(1) 9/4 (1 - 13'1522372‘(1‘%(”2” d(:)
1
r (3.48a)
'2; A65) 12 C2 r2 1/2
' (l - 13.1 _ 0/2 (1 0*) )) d(a) ‘ O

64

 

 

 

d
1 " 11‘6“
(2) a a r 1 (3.48b)
( 1)2_ d
a 211C
(3) (/)-(l-}._(1__1_)f_d_)1/2
A r a a C a 2 i
P 2
a C 2 i . c
P l
n
._ l
(4) R ‘3 0 a: ( )
<ne(r)> 1 ‘3'
4w 1 11 C r 2 1/2 r
rl/a
and:
mevo2
(6) T ' 3k - (3.48e)

The experimental procedure also yields values for 13, the discharge
current level at which the third resonance occurs. These data are

not as reliable as those for i 11, and 1 because the third

2

thermal resonance is somewhat weak. It is nevertheless possible to

d!

check the results obtained from the numerical analysis of equations
(3.48) by performing a similar analysis based on the use of the
first and third resonance data. The corresponding equations differ
from equations (3.48) only in that the subscript (2) must be'

replaced by the subscript (3) as shown.

65

 

 

r /a
1 i C
1 p __ 1/2d
(1) 13/4 (1 - Ig‘ifjfj;fiz (1 - a( Z) 2)) d(:)
1
rr (3.498)
1 r
-———+ A(-—)
a a i C
- <1——3-——P———(1- a<~>2d£))1/2<)=o
i 1 - /2 a
J 1 d
1 _ .isL
11C
(2) a = Pi (3.491))
(r1)2_ d
a Zile
1 1 1 1 1d 1/2
(3) A(r/a) = (3" E*'(;" 59 "9
p 3
(l- l—(l— 5:511” (349)
- a ~ C o — 2 i ) ° C
p 1
n
1
<4) ———-—9—- = (——-—-—-)
<ne(r)> 1 -%
1 1 C
£9. 1 p 1/2d
(5) Vo ’ 5" (1 - {3'1 _ “/2 (1 a(£)2)) d(a) (3.49d)
rlla
and:
mevo2
(6) T - 3k (3.49e)

 

The numerical results obtained from the computer analysis of these
sets of simultaneous equations, (3.48) and (3.49), are presented

and discussed in Chapter 4.

66

3.5 Determination of the Electron Density in a Warm Plasma Cylinder

 

Assuming Potential Distribution of the Form (1 - 10(yr))

 

The assumption of the functional form:
:1(r) = (1 - 10(yr)) (3.18)

where n(r) 3 eV(r)/kT is based on the solutions of Poisson's
Equation in different regions of the cylinder in section 3.3. It
was seen there that this solution cannot represent an exact
solution for the potential distribution but it is of the correct
form especially in the sheath region where an offset Bessel
function was obtained as a solution. It furthermore satisfies the
approximate condition that V(o) and therefore n(o) - 0.

Although this approximation makes the necessary numerical
analysis somewhat complex, it is still sufficiently manageable to‘
be useful as a diagnostic technique which is the ultimate goal of
this thesis.

The known quantities from the experimental work with the
electroacoustic probe are:

m - the frequency of the incident radiation:

id - the current level at which the dipole resonance is observed;

11 I the current level at which the first thermal resonance is
observed;

12 - the current at which the second thermal resonance occurs.

The unknown quantities are:

(l) no1 - the peak electron density at the center of the plasma

column for the first thermal resonance:

67

(2) no2 = the peak electron density at the center of the plasma
column for the second thermal resonance;
(3) nod - the peak electron density at the dipole resonax e;
(4) <ne(r)d> - the average electron density at the dipole
resonance;

(5) r1 = the critical phase point (kpl(r1) = O) for the first
thermal resonance;

(6) r2 = the critical phase point (ko2(r2) = O) for the second
thermal resonance;

(7) y = the constant appearing in the Bessel function approximation
(1 - 10(yr)) for the potential profile;

(8) T = electron temperature.

Since eight unknowns appear in the analysis, eight independent

equations are needed; these equations are:

i no
(1) T1' =- TL (3.50.1)
d 0d
i no
(2) 33- - ‘ET’Z‘ (3.50b)
d 0d

Equations (3.50s) and (3.50b) are based on the fact that the peak
electron density in the plasma is proportional to the current
level. These equations also show that only the ratio of the

currents are used for the analysis.

(3) <w 2(r)> ‘ 82
p av meeo

2

 

<ne(r)d> s Cp w (3.50c)

Equation (3.50c) states that at the dipole resonance at a given

current level, and thus electron density level nod, the average

68

of the square of the plasma frequency is proportional to the
angular frequency'ufof the incident radiation. (The pr0portion-

ality constant Cp was found in section 2.5.)

(4) mp1(r1) = w (3.50d)
(5) mp2(r2) = w (3.50e)
Equations (3.50d) and (3.50e) relate the critical points r1 and r2
for the first and second thermal resonances respectively to the
2
incident radiation frequency w; here: w 2(r ) = _S__.oo
2 p1 1 meco 1
2 e

exp(l — 10(yr1)), and. wp(r2) - Egzz-noz exp(l - Io(yr2)).

a
(6) kp1(r) dr = (5/4)n (3.50f)

r1

a
(7) ko2(r) dr = (9/4)n (3.50g)

1'2

Equations (3.50f) and (3.50g) are based on the fact that the total
phase of the second thermal resonances span (5/4)n and (9/4)n

respectively based on equation (2.68) in section 2.4. Here:

wplz<r)
—-—————-—2)
0.)

m 1/2

and

kpz") ”'V‘ <1""“3r‘° -
O u)

69

l
a -—-——2- no
173

(8) <nod(r)>a exp(1 - Io(yr))2nr dr (3.50h)

v d

Equation (3.50h) relates the average electron density <nod(r)> to
the center peak electron density nod at the dipole resonance. The
ratio of peak to average electron density remains the same as the
current level is changed so that equation (3.50h) may be formulated
in terms of one of the thermal resonances. Equations (3.50) are
now used to develop a system of simultaneous equations suitable for
numerical analysis on the computer.

Since in this section the assumed functional relationship
for the relative potential distribution as a function of r,

n(r) - eZér) , is given by

 

n(r) 9 1 - Io(yr), (3.51)

the constant 1 appearing in the Bessel function is the primary
parameter of interest. The relative potential distribution appears

in the Maxwellian electron density distribution as follows

no(r) a no exp(1 - Io(yr)). . (3.52)

Here again no represents the electron density at the center of the
cylindrical plasma column where the potential V(o) is assumed zero
and therefore the relative potential n(0) is zero as a boundary
condition. Since Io(0) - 1, equation (3.52) shows that no(o) is
indeed no at the center of the column (r - O). The formulation of
no(r) in equation (3.52) introduces no as an additional parameter

that must be determined for any given electron density profile and

70

corresponding current level.

The relationship fundamental to this analysis is based on
the phenomenological argument, that the total phases of the
electroacoustic thermal resonances in the sheath region are
separated by n radians and that furthermore the fundamental thermal
resonance spans a total of one and one quarter n radians between

the wall and the critical turning point r where the propagation

1

constant goes to zero. This argument is based on equation (2.68)

in section 2.4. Now

lim P T
r + r1 ko(r) = 0 (3.53)
r > r1 L -

 

 

th
For the m resonance, the total phase can therefore be written as

follows:
a
kom(r) dr = (m + l/4)n (3.54)
r .
m
Since:
w 2(r)
:23— r)m -1/2
kpm(r) v0 (1 - -—;§———) .. (3.55)

Equation (3.54) becomes

a 2 (m + 1mmo

“‘7‘" d“ .1

 

(3.56)

From the definition of the plasma frequency wo(r)

71

e2n (r)

2 e
wpm (r) 3 "3i;§:*- (3.57)

and since from equation (3.52) repeated here for reference

no(r) = no exp(1 — Io(yr)), (3.58)

t
the total phase equation for the m h electroacoustic thermal
resonance becomes

a

 

2
e n exp(1 - I (yr))
(1 _ ( °m 2 0 )1/2 dr

wme
r 80

(m + 1/4)nV
= ° (3.59)

w

 

Here the electron density at the center, nom(0) = nom for the mth
thermal resonance, depends on the discharge current level
maintained in the plasma column; the current level resulting in
nom is im which is available from the experimental data. There
exists a direct proportionality between the current level 1m and
the electron density nom because the electron drift velocity may
be considered constant in a cylindrical plasma discharge column.
The relationship between the current 1m and the correspond~
ing dc electron density nom is established experimentally through
the dipole resonance frequency m which is related to the corre-

sponding plasma frequency wod(r) by

<mod2(r)> ‘ Cp wz (3.60)

72

Here Cp is a proportionality constant; wpd(r) is the plasma
frequency as a function of r at which a dipole resonance is
observed when the incident radiation frequency is w; <wpd2(r)>
represents the average of the square of the dipole resonance
plasma frequency. The relationship between w and (“Dd(r)> in
equation (3.60) was established in section 2.5, where a numerical

value for CD was obtained. Since

2 ezned(r)
it follows that
2m
w .850
<ned(r)> = Cp ~—-jr—— (3.62)
e

Similarly, because of the direct proportionality between the
current levels and the electron densities, equations for <ne1(r)>

and <nez(r)> can be written as follows

C mzmeco 11
<nel(r)> - (LT—a (3;) (3.63)
e

and

C wzmeco 12
<ne2<r>> - («IL—2*» (7) (3.64)
e d

th
and in general for the m resonance

C wzmeeo 1m
<nem<r>> = (43—2—9 (T) (3.65)

e d

73

In order to work with equation (3.59), it is necessary to obtain
an expression for nom; this can be accomplished in terms of
equation (3.65) by formulating <nem(r)> in terms of nom as

follows

a
l
<nem(r)> a ——§-J[. nom exp(1 - Io(yr))2wr dr (3.66)
na
0
Defining R to be the ratio of the peak electron density nom to the
average electron density <ne(r)>,

nom n32
(3.67)

R '<ne(r))- a
exp(1 - Io(yr))21r dr

0

 

nom can be expressed in terms of the frequency of the incident

radiation w and current ratios as follows:

C m meeo 1m
no = (-P-—-—--) (R) (-.-) (3.68)
m 82 1d

The phase integral in equation (3.59) furthermore contains rm and
V0 as unknown parameters. There exists no independent relationship
from.which rm and V0 can be determined but it is possible to

express rm in terms of r for example r2 in terms of r1. The

m-l’

condition leading to a functional relationship between rm and rm_1

is the following:

km(rm) I 0 (3.69)

where again rm is the critical turning point for the mth resonance.

74

 

Therefore:
2
mp (r )
w m m _
V- (1 "' 2 )— 09
o w
so that
2 2
wpm (rm) 3 m (3.70)
Since
2
2 e nom exp(1 - Io(yrm))
wpm (rm) 3 ——- m a —" , (3.71)
e o
it follows that
2
m e w
exp(1 - 1 (Yr )) - i3— (3 72)
o m 2 °
e n0
m
Defining
e nO
Am - 2 ’
w m c
e o
exp(1 - I (yr )> = —1- <3 73)
o m Am °

The value of Am can be determined numerically based on the value
of nom obtained through the solution of equations (3.65) through
(3.68). Since equation (3.73) contains both rm and the parameter
of final interest, 7, rm cannot be determined directly from

equation (3.73). However it is possible to determine rn in terms

of rm (n integer # m) by simultaneous solution of

(1) exp(1 - 10(an)) = 1/An (3.74)

75

and

(2) exp(1 - Io(yrm)) = l/Am (3.75)

Simultaneous solution of equations (3.74) and (3.75) leads to a
value for Ar defined by
m n

9

Ar = r - r (3.76)
m

In terms of rm and Arm n it is possible to write two simultaneous
’

phase integral equations in the form of equation (3.59) as follows:

 

 

 

a 1/2 (m + l/4)nV
(l - Am eXp(l - 10(yr))) dr a m (3.77)
r
m
and
a
(1 - An exp(1 - 10mm”2 dr
r +Ar
m m,n
(n + l/4)1rVo
. .. (3.78)
w
Forming the ratio of equations (3.77) and (3.78) yields
a
(l - Am exp(1 - 10(yr)))1/2 dr
r
m .. 11.1.14)...
a 1/2 (n + 1/4) (3.79)
(l - A exp(1 - I (yr))) dr
n o
r +Ar
m m

9

For any combination of m and n, m # n for which resonance data are
available, equation (3.79) still contains two unknown parameters,

rm and 1. If equation (3.79) is combined with equation (3.73),

76

repeated here for reference:

exp(1 - 10(Yrm)) = i— . ' (3.80)
m

equations (3.79) and (3.80) may be solved simultaneously for rm
and 7.
After obtaining values for rm and y, Vo can be calculated

from equation (3.77) as follows:

 

 

a
m 1/2
Vo (m + 1/4) (1 - Am exp(1 - 10(Yr)) dr (3.81)
r
m
Since:
3kT
V0 3 r , (3.82)
e
the electron temperature T can be calculated as:
Vozme .
T - 3k (3.83)

where k is Boltzmann's constant.

In the numerical analysis at hand,‘the first two electro-
acoustic thermal resonances are used so that m = 1 and n - 2.
The equations used in the subsequent computer analysis formulation,
written in terms of the first two thermal resonances, are

summarized here in the form used in the numerical analysis:

77

a
j[ (1 - A1 exv(1 - 10010))“2 dr

 

 

 

 

1’ _
1 g (1 + 1/4)
(1) a 1/2 (2 + 1/4) (3.84a)
(1 - A2 exp(1 - I (Yr))) dr
r1+Ar1 2 o
(2) exp(1 - 10(Yr1)) = %T- (3.84b)
1
(3) exp(1 - 10(Yr2)) .. 7t- (3.84s)
2
2
(3 n01
(9) A1 = 2 (3.84d)
w m C
e 0
e n02
(5) A2 = 2 (3.84e)
(l) m S
e O
2
C m meeo i1
(6) n01 = (~P—-2-—-—) (R) (1") - (3.84:)
e d
C m meeo i2 ‘
(7) n02 = (-L—2————> (R) (3") (3.849,)
e d
2
(8) R = "3 (3.84h)

 

a
J{ exp(1 - 10(Yr))2nr dr

78

 

 

a
(9) v0 = (1‘+“1/4)n (1 - A1 exp(1 - Io(yr)))1/2dr (3.841)
r
1
and
v0 2me
(10) T = 3k (3.84j)

A numerical analysis based on these equations is also performed
using a combination of the first and third resonance data. The
results from this analysis are used as a check on the results
obtained from the use of the first two resonances. In order to use
equations (3.84) for the first and third thermal resonance com-
bination, it is only necessary to replace the subscript (2) when-
ever it appears by the subscript (3). The corresponding set of
equations are:

a
(1 - A1 exp(1 - Io(yr)))1/2dr

(1) 1 - (1 + 1’41» (3.85a)

 

 

(l - A3 exp(1 - 10(Yr))) dr
r1+Ar1 3
(2) exp(1 - 10(le)) 9* [IT (3.85b)
1
(3) exp(1 - 10(yr3)) . %7- (3.85c)
3
2
e n01
(4) A1 ' -§—-—- (3.85d)
m m e .

eo

79

 

 

 

 

821103
(5) A3 = 2 (3.85e)
u) m E
e O
C m meeo i1
(6) n01 = (—P-—2———) (R) (T) (3.851?)
e d
C m meco i3
(7) no, = 62—7—9 (R) (T) (3.855;)
e d
n 2
(8) R .. a a (3.85h)
Jr. exp(1 - 10(Yr))2wr dr
0
a
(9) Vo = (1 +w1/4)n (l - A1 exp(1 - 10(Yr)))1/2dr (3.851)
1'
l
and:
V 2m
(10) T = 3k 6- (3.85j)

The numerical results obtained from the computer solution from
equations (3.84) and (3.85) are presented and discussed in the

following chapter.

4.1

4.2

CHAPTER 4

NUMERICAL RESULTS FOR THE ELECTRON DENSITY
PROFILE IN A CYLINDRICAL PLASMA COLUMN

Introduction

 

The simultaneous equations presented in section 3.4 and
section 3.5 are solved numerically using the data given in section
3.2. The solutions are presented in this chapter. The results

obtained for the different approaches are presented.

Numerical Results Based on Parabolic Electron Densityngrgfilg

 

.Approximation

 

The numerical results obtained in the simultaneous computer
solution of equations (3.48) and (3.49) are listed in Tables 4.2.1
through 4.2.5 for the eight sets of data analyzed. For ease of
identification, the data sets are identified throughout by two
numbers, 1, j; i = l to 8 represents the set number; j a 2
represents the use of the combination of the first and second
resonance (equations (3.48)) while j s 3 represents the use of the
combination of the first and third resonance (equations (3.49)).

The parameters listed in the Tables are:

(1) The factor a in the parabolic approximation

new) - neon - (“i—)2) .

(2) The calculated value of the ratio R - ne(r - 0)/<ne1(r)>.

(3) The critical points rm/a for the mth resonance.

(4) zm/a - (a - rm)/a.

8O

81

 

 

 

 

 

 

 

 

 

Data set # J j 2 3 3'3
1 .83 .83
2 .82 .85
3 .83 .83
4 .80 .86
5 .86 .86
6 .83 .83
7 84 .87
8 .85 85

 

 

 

 

 

Table 4.2.1 Numerical results for the factor¢£
for data sets 1 through 8. The columns
identified by 3:? and j=3 represent
numerical values for 4; obtained from
the use of combinations of resonances
1,2 (3:2) and 1,3 (jz3) respectively.

82

 

 

 

 

 

 

 

 

 

 

 

 

Data set # nO/zne1(r)> noé<n82(r)>
1 1.70 1.70
2 1.70 1.74
3 1.71 1.71
4 1.67 1.75
S 1.76 1.76
6 1.72 1.7?
7 1.73 1.77
8 1.74 1.74

 

 

Table 4.2.2 Numerical results for the ratio of peak
to average electron density nO A<ne (r)?

and n0 /<ne (r)>'for data sets11 th3ough 8.
2 2

83

 

 

 

 

 

 

 

 

 

 

 

 

 

Data set # r1/a r7/a rB/a
1 .88 .83 .77
2 .87 .80 .71
3 .88 .81 .77
4 .89 .84 .74
5 .86 .79 .71
6 .87 .81 .73
7 .87 .80 .72
8 .86 .79 .71

 

Table 4.2.3 Numerical values for the ratio of

critical radius r.

to the total radius

a, rj/a, for dataasets 1 through 8.

 

84

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Data set # z1/a 22/a 23/a 22/21 ZS/Z1
1 .12 .17 .23 1.44 1.95
2 .13 .19 .28 1.50 2.02
3 .12 .18 .23 1.51 1.90
4 .11 .16 .26 1.48 2.00
5 .14 .21 .29 1.53 2.07
6 .13 .19 .27 1.48 2.10
7 .13 .20 .28“ 1.53.. 2.02
8 .14 .21 .29 1.92 2.07

L_

 

 

Table 4.2.4

Numerical values for the ratio of
critical distance z- measured from the
wall for the jth regonance to the total
radius a as well as the ratios 2 /z1 and
23/z1 for the data sets 1 througg 8.

ti

85

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Data set # 3 1w? j':w j E 2 j E 3
1 -1.75 -1.75 14670 14670
2 -1.73 -1.90 19960 31630
3 -1.78 —1.78 18950 18950
4 -1.63 —1.96 11590 33070
5 -1.99 -1.99 29000 29000
6 -1.80 —1.80 20480 20480
7 -1.84 -2.00 27370 42820
8 -1-91 -1.91 39060 39060

 

 

Table 4.2.5

Numerical values of relative poten-

tial at the wall,'1w = eV(a)/kT

and electron temperature T for data

sets 1 through 8. The columns identi-
fied by j=2 and 3:3 represent the
numerical values for and T based

on the use of combinations of resonances
1,2 (j=2) and 1,3 (j=3) respectively.

86

(5) The ratios 22/21 where 21 - a - r1 and 22 - a — 22.
(6) The ratio 23/21, where 23 - a - r3.
(7) nw - ve/kT evaluated at the wall where Vw is the potential,

k is the Boltzman constant and T is the electron temperature.
(8) T, the calculated electron temperature.
The most significant parameter in the parabolic electron density
profile is the parameter a appearing in the functional formulation

of equation (2.44)

nee) = nolu - (Ag-)2)

The values for a obtained for any one data set using first the
combination of the first and second resonance and then the
combination of the first and third resonance are very close. Since
these two values for any one data set represent a mutual check, it
appears that the results obtained for a are correct. It must be
kept in mind, of course, that any calculations employing the third
resonance are only approximate, since the third resonances are
difficult to interpret from the oscillographs.

The ratio of peak electron density at the center of the
plasma column to the average static electrdn density in the column
for the discharge current level 11 was another of the parameters
obtained from the solution of the simultaneous equations (3.48) and
(3.49). Again this ratio is very close for data sets 1,2 and data
sets 1,3, indicating that the results are reliable. Good corre-
spondence for results using data sets 1,2 and 1,3 is also found
for the relative wall potential nw(nw - ve/kT). The temperature

T indicates some variation as seen in Table 4.2. The relative

4.3

87

variation is still insignificant considering how sensitive the
temperature is to variations in other plasma column parameters.
It should be recalled that the temperature is determined directly
from the phase integral.

The graphical results are shown in Figures 4.2.1 1‘hrough
4.2.8 for the parabolic electron density profiles and the relative
potential distributions for the eight data sets i,2 on a normalized
scale.

In conclusion, it is observed that some of the values
obtained in this analysis agree well with numerical values obtained
from approximate theoretical treatments or independent experimental
analyses. Theoretical analysis of a plasma sheath, for example,1
leads to a relative wall potential nw of approximately 2 which is
in agreement with the values obtained in this numerical analysis.
More significantly, the ratios of 22/21 obtained in this analysis
of approximately 1.5 agrees well with ratios of the distances from
the wall observed for the electric field perturbation for the
first and second thermal resonances in experimental work reported
earlier.14

The appendix contains complete computer readouts of all the

parameters for each data set.

Numerical Results Based on the Bessel Function Approximatigg;fpg_

 

the Static Electron Density Profile

 

The numerical results obtained in the simultaneous computer
analysis of equations (3.85) are listed in Tables 4.3.1 through
4.3.4 for the eight sets of data analyzed. For ease of identi-

fication, the data sets are identified throughout by two numbers

Fig.

-0.

-00

-00

-1
4.2.1

88

A.—
’

    
 

 

 
  

 

.
:
8 v:
.. b—
3 Plasma
t h ' Glass
: Thickness b
J _ r/a
: O 1 _ .
; Data Set #1.2
: Profile at 11
e. v r. 1 l f: 1 r. J .. 4 g
z 3 0.6 0.8 1 0 ,1 r/a
E n.(r/a)
. I-
2 : 7w
G. r—
; ..
J _
O ; u

 

Normalized parabolic electron density profile
as a function of r/a, . 2
n8 (r/a)/n = 1 - .83(r/a) .
1 - o1 .
Also the normalized potential profile
n1(r/a)/qw. Based on data set #1.(f=2:016 GHz,

id=270oma, 11:185 ma, 12:150 ma, i3=125 ma).'

Fig.

0.4

O
a)
.OIOOOIO..O..OOOOO‘

-00

-004

-008

I
_a

4.2.2

89

 

 
 

Glass
Thickness b

  

Data Set #2.2
Profile at i

=:r/a

2

.0.0.0.0.0000...0.0.0.0....000000.0.0.0....0.00.00.00.00.0.....0000000O
I

 

 

O
s
I

Normalized parabqlic electron density profile

as a function of r/a, ' 2
e101(r/a)/n = 1 - .82(r/a)

Also the normalized potential profile
q1(r/a)/qw. Based on data set #2 (f=2. 10 GHz,

id=290'ma, i1=190 ma, 12:150 ma, i3=120 ma).

 

Fig,

4.2.3

90

    
  
   

 

 

 

1.0 .9

0.8 3.. (rhfi

0.6 ; _
3 Plasma
: Glass

0.4 fl ~ Thickness b

0.2 a. - r/a
5 Data Set #3.2
; Profile at 11

0 { t’r/a

-o.2 {~

-004 ..

-006 .0

4L8 i.

-L0 3~

Normalized parabolic electron density profile
as a function of r/a, '
e(r/a)/no = 1 - .83(r/a) .

Also the normalized potential profile
n1(r/a)/qw. Based on data set #3 (f: 2. 23 GHz,

id=340 ma, i1z235 ma, 17:185 ma, 13: 160 ma).

_.__.____3

Fig,

91

‘
P

   

  

 

100 . ‘
0.8 {
0.6 J
; Plasma
E Class
0.4 a. - Thickness b
0.2 .. -- r/a
o O
2 Data Set #4.2
: Profile at 11
O {4 l $044‘Wi:& fivxiLijxl rr/a
-o,2 { _
-00'4 .. _
-006 fi‘p-
-008 .0“.

 

 

4.:2..4 Normalized parabolic electron density profile

as a function of r/a, . 2
e(r/a)/no = 1 - 80(r/a)

Also /the normalized potential profile

.q1( r/ a WA? Based on data set #4 .(f= 2. 32 GHz,

id =355.ma, 11:245 ma, 12:200 ma, 13:175 ma).

 

92

   

    

1.0 .“
0.8 .I
0'6 .0 '- Plasma
: Glass
0.4 J ._ Thickness b
o 2 .: _ r/a
5 Data Set #5,2 g
3 Profile at 11
O 9.
-o.2 {
-004 ..
-O.6 9'
-O.8 {
.1.0 ;

 

 

Fig' 4' 2?. 5 Normalized parabolic electron density profile
as a function of r/a,

e(r/a)/nO = .- 86(r/a)2
Also the normalized potential profile .
q1(r/a)fizw. Based on data set #5 (f: 1. 917 GHz,

id=270 ma, 11:180 ma, 12:135 ma, 13r110 ma).

Fig,

93

   

  

 

 

 

1.0.. 4‘
0.8 o.
0.6.:
: Plasma
3 Glass
0.4 fl - Thickness b
002 t — o r/a
: Data Set #6.2
S Profile at 11
0 € 24 _1.1 r—r/a
-002 .. P
-004 O. -
-o.6 .3—
4L8 {-
-,,o :L
4. 2 . 6

Normalized parabolic electron density profile
as a function of r/a,

“(r/a)/n = 1 - .83(r/a)2

Also the normalized potential profile .
n1(r/a)/qw. Based on data set #6 (f: 2. 017 GHz,

id=285 ma, 11:190 ma, 12:150 ma, 13:120 ma).

94

   
     

 

 

1.091-
0.8{
o.6{
: \l
3 Glass
0,4 1*. - \ Thickness b
o 2.: - o1 r/a
: Data Set #7.2 '
3 Profile at 11 f
O :1 r/a
-O.2€
“004.:
.006“:4
”008.0
-14);

Fig. 4w-2?.7 Normalized parabolic electron density profile

as a function of r/a, . 2
n6 (r/a)/n = 1 - .8fl(r/a) .
1‘ °1
Also the normalized potential profile _
n1(r/a)/qw. Based on data set #7.(f=2r275 GHz,.

id=290 ma, i1=195 ma, 12:150 ma, 13:120 ma).'

95

     
  

 

 

 

1.0 o.
0.8 ;
0.6 .. ,_.
5 Plasma
E ‘\‘
. ” Thickness b
; Data Set #8.2 '
: Profile at: 11 P
0 .::-~1 minml “731 1 >r/a
-o,2 .5-
-o,'4 as...
4L6 ‘f—
toms {-
-100 :—

 

F’ ° '
lg- 4. 2 . 8 Normalized parabolic electron density profile

as a function of r/a,‘ 2
e1 01 .

Also the normalized potential profile .
q1(r/a)/hw, Based on data set #8.(f=2;322 GHz,

-id=320 ma, 11:210 ma, 12:160 ma, i3=135 ma).'

96

 

 

 

 

 

 

 

 

 

 

Data set # J _ 7 j 2 3
1 327 . 321
2 326 319
3 323 328
4 330 328
5 327 327
6 327 322
7 328 328
8 331 325

 

 

 

 

 

Table 4.3.1

Numerical results for the factorx
for data sets 1 through 8. The
columns identified by 3:2 and j=3
represent the numerical values for

X obtained from use of combinations
of resonances 1,2 (j=2) and 1,3 (jz3)
respectively.

97

 

 

 

 

 

 

 

 

 

 

 

 

Data set # no1/(nefr)> n02/<neér)>
1 1.99 1-94
2 1.98 1.93
3 1.96 1.99
4 2.01 1.99
5 1.99 1.99
6 1.99 1.95
7 1.99 1-99
8 2.02 1-97-

 

 

Talilea 4.3.? Numerical results for the ratio

of peak to average electron density
nO1/.<n.e (10> and no2ficn92(r)> for

data sets 1 through 8.

98

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2::a# z1/a z2/a zS/a zQ/z1 Z3/Z1

~—3 1 .14 .20 .23 1.49 2.02

2 .14 .21 .27 1.53 2.22

3 .11 .18 .26 1.60 1.94

4 .13 .20 .26 1.47 2.15

‘33 5 .14 .22 .28 1.59 2.08
6 .14 .21 .27 1.53 2.19

7 .14 .21 .28 1.53 2.06
h‘__8 .15 .24 .29 1.57 2.17

 

 

 

Table 4 ,33

Numerical values for the ratio of

the critical distance 2' measured from

the wall into the plasma for the jth

resonance to the total radius a and also

the ratios 22/z1 and 23/21'

“.3."

99

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Data Qw T T

set # j = j = j = 2 ' = 3
1 —1.8 —1.7 47380 30580
2 -1.8 —1.7 83690 74860
3 -1.8 -1.8 67470 57060
4 -1.9 -1.8 47950 77700
5 -1.8 -1.8 10350 10350
6 -1.8 -1.7 71630 66900
7 -1.8 —1.8 14400 14400
8 -1.9 -1.8 10200 10120

 

Table 4.3.4

Numerical values of the relative potential
eV(a)/kT and the electron tempera-

nw =

ture T for the data sets 1 through 8.

columns identified by j=2 and 3:3 represent
the numerical results based on the use of
combinations of resonances 1,2 (3:2) and

1,3 (3:3) respectively.

 

 

100

i,j; here i - 1 through 8 represents the set number; j - 2
represents the use of the combination of the first and second
resonance while 1 - 3 represents the use of the combination of the
first and third resonance.

The parameters listed in the Tables are:
(l) The calculated value of the ratio R 8 no(r a O)/<nel(r)>.
(2) The factor 7 in the Bessel series formulation in equation

(3.52)
ne(r) 8 n0 exp(1 - Io(yr)) .

(3) The ratios 22/21 and 23/21.

(4) The critical points zm/a for the mth resonance.

(5) 11V - ve/kT evaluated at the wall where Vw is the potential, k
is the Boltzmann constant and T is the electron temperature.

(6) T, the electron temperature.

The most important parameter in this analysis is y. The values

for 7 obtained for data sets i,2 and 1,3 compare well for the eight

sets analyzed and since sets 1,2 and 1,3 represent a mutual check

it appears that the functional form obtained is acceptable.

Good correspondence using data sets 1,1 and i,2 is also
obtained for the relative wall potential "w = ve/kT and to a
satisfactory extent for the electron temperature T. Since T is
very sensitive to other parameter variations, the difference
observed in some data sets between sets 1,2 and 1,3 is not very
significant.

The graphical results for the normalized electron density

profiles n31(z)/no1 (here nel(z) is the static electron density at

 

101

discharge current 11 where z - a - r) and the corresponding
relative potential n(r) - eV(r)/RT are shown for the eight data
sets in Figures 4.3.1 through 4.3.8. Subsequently, Figures 4.3.9
through 6.3.16 show simultaneous plots for the normalized electron
density profiles ne1(z)/n01 and nez(z)/nol for each data set.

These Figures also show the location of the critical turning points
zlla and 22/3 marked as t1 and t2. These must, of course, occur at
the same vertical magnitude on the graphs to be correct and indeed
good agreement with this requirement is observed indicating that
the numerical analysis is sufficiently accurate. Figures 4.3.17
through 4.3.24 show the corresponding simultaneous plots for
nel(z)/n01 and ne3(z)/n01. Again critical points zl/a and zala
marked at t1 and t3 closely satisfy the condition that the vertical

magnitudes are the same. It should be recalled from the theoret-

t and t occur at points at which

ical development that t1, 2 3

2 2
mp (tm) ' w ,
so that

2
nem(tm) . m meeo

 

m
which depends only on the excitation frequency m which is held
constant in any one data set.

In conclusion it is observed that the value of 11W agrees
with typical values predicted theoretically for plane plasma

sheaths which should not behave too differently near the wall in

 

102

¥
'—

  
 

_ Plasma

  
 

'Glass
Thickness b

00......‘000.0.00.00000000.

z/a
Data Set #1.2
Profile at i

O
N
.oooooooo.oooooooo.

-002

-0.4

0....OOOOO‘OOOOOOO0.0000000I

-O.6

-O.8

00.0....0000000
l

 

 

-100 :4

Fi5.4.3.1 Normalized Bessel series electron denSity profile
as a function of z/a,

ne1(z/a)/no1 = exp11 - 10(327(1-z/a))).

Also the normalized potential profile -
1(z/a)/ w‘ Based on data set #1 (f=2.016 GHz,

1d=270 ma, i1=185 ma, 12:150 ma,'i3=125 ma).

1.0

0.2

-004

-008

.OOOOOOIO‘.O...0.0.0.0....0.0IOOOO00‘...0.0.0.0...0000‘...0.000.000.0000...0......00000000.

L

-100

Fig. 4.3.2

103

‘
Ir

Plasma

   

Glass
Thickness b

Data Set #2.2
Profile at 11

1 b

 

 

Normalized Bessel series electron density profile

as a function of_z/a,

n (z/a)/n = eip(1 - I (326(1—z/a))).
e1 01 p 0 .

Also the normalized potential profile ' ‘
1(z/a)/,w. Based on data set #3 (f=2.10 GHz.

id=290 ma, 11:190 ma, 12:150 ma, 132120 ma).

104

1.0 *‘L

Plasma

  
 

Glass
Thickness b-

Data Set #3.2

Profile at i

A

1

00.000.00.00000‘00000000.0.0.0....00000000

--a 1...“ 1-4 a l A: 1L~AA1 >:2hi

w fi

002 0.4 0.6 o ‘ 1.0

000.0000...
0

-O.2

O

-004

.0...‘..’........

-0.6

OOOOOCOO‘

 

 

-100 . J—

Fig. 4.3.3 Normalized Bessel series electron density profile
as a function of z7a, - '
n (z/a)/n = exp(1 - I (3230-s/a))).
Also the normalized potential profile
1(z/a)/ w' Based on data set #3 Cf=2.23 GHz,

id=340 ma, 11:235 ma, 122185 ma, 13:160 ma).

 

105

L.
'-

  
 

 

 

1.0 g —
0 a i P
2 Plasma
0.6 Q. .— I
3 Glass
2 Thickness b
0.4 .: I—
0 2 5 Data Set #4.2
:4 Profile at 11 .
' _ 1.111--_l--1-l----11111 k
0 E 1.0 "Z/a
412 a
-004 ..
-006 t
-008 ;
-1.0 9.11-

 

Fig. 4. 3. 4 Normalized Bessel series electron density profile
as a function z/a

e(Z/as/nO = exp(1 ; I 0(330(1- z/a))).

Also the normalized potential profile
i(z/a)/ w‘ Based on data set #4 (f: 2. 32 GHz,

d=355 ma, 11:245 ma, 12:200 ma,1 3-..=175 ma).

106

0.8

0.6

0.4

  
  

Data Set #5.2
Profile at-i '

0.2

00......OOOOOOO0.00000IO0.0...0.00.00.00.00.

I I

O O

o o

.h N
oooooooo'oooooooo‘oooooooo‘o

I
o
o
m

I
O
O
a)

.0.....‘........*

 

 

-4. oJAL

Fig. 4.3.5 Normalized Bessel series electron density profile
as a function of z/a, '
“ ne1 (z/a)/nO = exp(1 - I o(327(1-z/a))).
01
Also the normalized potential profile . '
1(z/a)/w Based on data set #5 (f: 1. 917 Gfiz,

id=270 ma, oi1=180 ma, 12:135 ma, 13:110 ma).

Fig.'

-002

-004

-006

-008

-L0
4.3.6

t

.OOOOOOCO.....OCO0.0.0.0....000000.0.0....0......OOOO0.000I...0.0.0.0....OOIOOOO0.00......

107

 

 

A
ne1(2/a)
1
Plasma
..' Glass
Thickness b
———£——> z/a
Data Set #6.2
4 Profile at 11 '
J'z/a

 

 

/

Normalized Bessel series electron density profile
as a function ofz

ne1(Z/a)/n; = exp(1 - I O(327(1—2/a)))-
Also the normalized potential profile '
1(z/a)/w Based on data set #6 (f: 2. 017 GHz,

id=285 ma, 11:190 ma, 12:150 ma, 13:120 ma).

1.0

0.8

0.6

0.4

O
o
N

O

u
9
N

-004

I
C)
0\

-008

.00.0.0.0.0000...0.0.0.0....00000000.
L.

.00....C...00......OOOOOIOO.IOOOOC00.....00....OOOO...

108

Plasma

  
   

Glass
Thickness b

Data Set #7.2 .
Profile at i

A
I

 

 

 

1.

Normalized Bessel series electron density profile

as a function of zfia, .
ne (z/a)/no = exp(1 -'Io(328(1-z/a))).
1 - 1 -
Also the normalized potential profile .
1(z/a)/ w' Based on data set #7 (f=2.275 GHz,

ids290 ma, 11:195 ma, 12:150 ma, 13=i20 ma).

 

1.

O

O
(D

O
0‘

0
4s

0
o
N

-O.2

-004

-1..()
Fig. 4°308

109

1

Plasma

    

Glass
Thickness b

b Data Set #8.2

i Profile at 11

 

 

.OOOOCOO0.000....0.0.0.0....0000000......OOOO.COOOOCO0.0000.0.0.0000...0.00.00.00.00000000.

1.. .
Normalized Bessel series electron density profile
as a function of z/a,

ne (z/a)/nO =exp(1 - Ib(331(1-z/a))).

. . 1 . 1 1
Also the normalized potential profile _
1(z/a)/ w' Based on data set #8 Lf=2.322 GHz,

id=320 ma, 11:210 ma, 12:160 ma, 13s135 ma). 5

 

110

     

 

 

Lo .‘L

0.8 J _

(L6 ; _

O 4 3 2 Plasma

° 3 n01
I Glass
: Thickness b
2 , .

0.2 J' '
2. . I .
O1 '
E : Data Set #1.2
: 1 Profiles at 11 and 12

0 o 7“ {—A “1 “1 AA-ll-A :1 :z/a
‘ O , 012 0.4 0.6 0.8 1.0 -

t1‘2

Flg- 4u.3.9 Normalized Bessel series electron density
profiles at resonances 1 and 2. Points t1
and t2 represent the critical points in
the plasma sheath at which k and k
respectively to to zero. p1 p2
Based on data set #1.(f=2.016 GHz,

id =27o ma, i1=185 ma, 12:150 ma. 1 =1?5 ma)-

3

111

     

 

 

 

 

A

1.0 O b-
0.8 fl '
006 O. "'

; Plasma

5 Glass

, Thickness b
0.4 fl '

: o , . ~
0-2 g . Data Set #2.2 '

If 1 1 Profiles at 11 and 12
o J 411-- N114-111---11 A-” 1 ..H1 152/3
. o lcnr o 4 043 08 10

t1 t2

Fig. ‘1.i3.10 Normalized Bessel series electron density
profiles at resonances 1 and 2. Points t1
and t2 represent the critical points in
the plasma sheath at which~kp .and k

respectively go to zero. 1 p2
Based on data set #2.(f=2.10 GHz,
id=290 ma, i1=190 ma, 17:150 ma, i3=120 ma).

112

    

 

 

 

.1 .
.0 ., L
0.8 ‘2-
(L6 { —

. Plasma

: Glass
0.4 .: Thickness b

s ”/1
O 2 *3. Data Set #3.2

:4 Profi e

. 1 l l s at 11 and 12
O .0 1 - [v 1 ‘ - vi - A“ J A A 1.. l - Au ‘ :2/3

0 l 3 2 0.4 0.6 ()8 1 0
1:

F183 ‘4.-3.11 Normalized Bessel series electron density
profiles at resonances 1 and 2. Points t1
and t2 represent the critical points in
the plasma sheath at which‘k ‘ and k
respectively go to zero. p1 p2
Based on data set #3. (f=2.23 GHz,
id=340 ma, i1=235 ma, 17:185 ma, 13:160 ma).

Fig.

1.0

O
m

0
O
0‘

0.4

.0000...0......OOIOI.OOOO0......OOOOOOOO.IOOOOOI...

 

113

  
 

 

_ ne1(2/a)
n
01 .
_ Plasma
Glass.
"e (z/a) Thickness b
y:—
n03
--- -- O 1 z/a.
_ _ Data Set #4.2
: ; Profiles at 11 and i2
4 -1u11- u: 1u11+1¢$L :z/a
0 O 2 0.4 0.6 0.8 1.0

 

 

11. 3.12 Normalized Bessel series electron density

profiles at resonances 1 and 2. Points t1
and t2 represent the critical points in
the plasma sheath at which k and k
respectively go to zero. p1 p2
Based on data set #4. (f=2.32 GHz,
id=355 ma, 11:245 ma, 17:200 ma, 13:175 ma).

114

 

   
 

 

 

 

 

A
1-0. i ”
E n°1 .
0-6 t ” Plasma
5 n9 (z/a) Glass
: 2 Thickness b
E ‘ ' ’ o 1 ”La
0 2 ; 1 Data Set #5.2
° 1 * ' ‘ Profiles at i and i
.1 1 2.
.4 '
'1--11---1- 1-1 -u1-1..1 _
0 ’c) o o 4 0.6 ()s 1 o "z/3
t1 b2

Fig. 4.:3..13 Normalized Bessel series electron density
profiles at resonances 1 and 2. Points t
.and t2 represent the critical points in
the plasma sheath at which k and k
respectively go to zero. p1 p2
Based on data set #5. ( f=1.917 GHz,

id=270 ma, i1=180 ma, i2=135 ma, i3=110 ma).

1

. 1"

It.

 

1.0

Fig. 4.3.14

 

115

   
 

Data Set #6.?

Profiles at 11 and 12

>>z/a

 

 

 

0.6

0.8 1.0

Normalized Bessel series electron density

profiles at resonances 1 and 2.
and t2 represent the critical points in
plasma sheath at which k

respectively go to zero. 1

the

Points t1

‘and k
p-2

-Based on data set #6. (f=2.017 GHz,

162285 ma, 112190 ma, 1

=150 ma, 1 =120 ma).

2 3

1.0

0.6

0.4

’OOOOOOOOO’OOOIOOOO0.0COO...00.00.00.000‘000000000.

116

 

    

 

 

 

 

A .
- ne1(z/a)
§
n
°1
L Plasma
n82(z/a)
n Glass
01 Thickness b
_-I - - o 1 ”/3
. a
Data Set #7.2
I
4 1 , Profiles at 11 and 12
- llLl: A -u l - -- 11.1,1_ i..- n J 1::2/8
0 0.2 O 4 0.6 0 8 1 0
t1 t2

‘Fig. 4.3.15 Normalized Bessel series electron density

profile at resonances 1 and 2. Points t

- and t2 represent the critical points in

the plasma sheath at which k and k
respectively go to zero. ~p1. p2
Based on data set #7. (f=2,275 GHz, '

id=290 ma, 11:195 ma, i2=150 ma, 13:120 ma).

117

.1.

o
C
r"
.

 

   
   

 

 

 

 

0.8 fl " ne1(z/a)
5 n°1
; a
0.6 I Plasma
a ne2(z/af Glass
. Thickness b
0.4 t — no
. 1
3 _ - .
z 0 1 z/a
0.2 1: - Data Set #8.2 f
0‘
2. 1 1 Profiles at i1 and i2
° --_:1v-:1 -u11-1..111_1 _
0 f o o 2 0.4 0.6 ’6.8 "1.0 "z/a
t1 132

Fig. 4.3.16 Normalized Bessel series electron density
profiles a resonances 1 and 2. Points t1
and t represent the critical points in
the lasma sheath at which k 'and k
respectively go to zero. p1 p2
Based on data set #8.(fa2,3?7 GHz,
id=320 ma, 11:210 ma, i2=160 ma, 13:135 ma).

d
0
O

O
(D

O
0‘

O
o
h

'032

A

A

 

"O0.0...0......OCOO0.00COOOOOO'COOOOOOO0.0000COIIO.

Fig. 4.3.17

118

Plasma

Glass
Thickness b

1 : Data Set #1. 3
' Profiles at 11 and i3

  

e3(2fifl

 

28.
0 | 0.f 0.4 0.6 0.8 1.0 /

t t

1 3

Normalized Bessel series electron density
profiles at resonances 1 and 3. Points t
and t3 represent the critical points in the
plasma sheath at which k and k
respectively go to zero.p1 p3
Based on data set #1. (f- 2 016 GHz,

id=270 ma, 11:185 ma, 12:150 ma, i3=125 ma).

119

    
 

 

 

 

A
1.0 fl
0.3 .:
0 6 €
. Plasma
3 Glass
0 4 ; Thickness b
E z/a ,
0-2 1 Data Set #2.3 $1
I Profiles at i1 and 13
; 1-1 -1a4d.1a,1-- -1 .
O 0.4 0.6 Bo.s 1.0 "2/a

 

Fig. 4.3.18 Normalized Bessel series electron density
profiles at resonances 1 and 3. Points t
and t represent the critical points in the
plasmg sheath at which k and k
respectively go to zero. 1 p3
Based on data set #2. (f=2.10 GHz,
id=290 ma, i1=190 ma, i2=150 ma, i3=120 ma).

?ig.

 

120

  
 

 

 

 

Al
.0 9. " .
0.8 O: —- ne (z/a)
s
g .
0.6 1‘ -
3 Plasma
E n z a Glass
0.4 1g - e ( / ) Thickness b
: z/a
(L2 4 0 1
2 1 Data Set #3.3 1
: 1 Profiles at i and i
. . 1 3
0' .' --1 1-.--vl---v4-1--1411-1 i '
O 0.2 0.4 0.6 0.8 1.0 z/a
1:1 t3
4.3.19 Normalized Bessel series electron density

profiles at resonances 1 and 3. Points t
and t represent the critical points in the
plasma sheath at which k ‘ and k
respectively go to zero. 1 p3

Based on data set #3.(f=2.23 GHz,

id=340 ma, 11:235 ma, i2=185 ma, 13:160 ma).

 

 

121

    

 

 

 

1.0 o.“
0.8.“
0.6 ..’
0.4 4:
0.2 .{ 4 . Data Set #4.3
3‘ Profiles at 11 and i3
54 l
2 I
o .0 fLfil :1 :1. 47:1 :4 >Z/a
0 [0.2 0 4 0 6 ()8 10
t1 t3

Fig. 4.3.20 Normalized Bessel series electron density
profiles at resonances 1 and 3. Points t
and t represent the critical points in the
plasmg sheath at which k and k
respectively go to zero.p1 p3
Based on data set #4.(f=2.32 GHz,

id=355 ma, 11:245 ma, i2=2OO ma, i3=175 ma).

122

      

 

 

A
1A) a -
0.8 fl *
0.6 t ’ Plasma
; Glass
: Thickness b
0.4 fl
3 z/a
0.2 .1 Data Set 115.3
:4 Profiles at 11 and i3
:4
: 1 .
o l
0 .: “Ln- -~11---~.L-~e1~~~el >z/a
O 0.2 0.4 0.6 0.8 1.0
1 '%

 

.-

' Fig. 4.3.21 Normalized Bessel series electron density
profiles at resonances 1 and 3. Points t
and t represent the critical points in the
plasma sheath at which k and k
respectively go to zer0.p1‘ ‘ p3

Based on data set #5. ( f=1.917 GHz,

id=270 ma, i1=180 ma, 12:135 ma, 13:110 ma).

123

  
 

 

 

 

1.0 .. lil—
0.8 o: - ne (z/a)
. 1
o r—
! °1
;.c ‘
0'6 2 Plasma
5 Glass
; L ne 3(3/3) Thickness b
0.4 : T—
o
2 1
z _ _-__§
: 0 1 . z[?
0.2 't. , Data Set #6.3
I 1 Profiles at 11 and i3
04
: i
0 J QJHJA:1_11- “l :1fiil Pz/a
O 0 2 0.4 0 6 O 8 1.0
t1 t3

Fig. 4.3.22 Normalized Bessel series electron density
profiles at resonances 1 and 3} Points t
and t represent the critical points in the
p1asm2 sheath at which k and k
respectively go to zer0.p1 p3
Based on data set #6.(f=2.017 GHz,

id=285 ma, 11:190 ma, 12:150 ma, 13:120 ma).

124

     

 

 

 

 

1.0 .4—
043 { —
045 { —
: Plasma
: Glass
0.4 J Thickness b
: z/a
0-2 fl; Data Set #7.3
2 Profiles at i and i
.1 1 1 3
0 ; 1 -1 1,, 11.1-11-a11-111¥ 1-1.-- 1 ,.z/a
O O 2 0.4 O 6 0 8 1.0

Fig. 4.3.23 Normalized Bessel series electron density
profiles at resonances 1 and 3. Points t
and t represent the critical points in the
plasmg sheath at which k and k
respectively go to zer0.p1 p3
Based on data set #7. (f=2.275 GHz, '
id=290 ma, 11:195 ma, i?=1SO ma, i3=120 ma).

 

125

  
 

 

 

 

 

1.. .1.

008 .. ‘L n (z/a)
: e1
' n
3 01

0.6 o. _
i . Plasma
5 Glass

0,4 é __ n (z/a) Thickness b
E n
: -——- W
t o 1 ”hi.

0’2 ‘3‘ e ’ Data Set #8.3
: Profiles at i and i
04 . ' 1 3
I 1 '

0 5. “ 1‘ ‘Hl-A-Vls—thl-AL-th
0 0.2 0.4 0.6 018 1.0 z/a

t1 ‘5

Fig. 4.3.24

Normalized Bessel series electron density
profiles at resonances 1 and 3. Points t
and t3 represent the critical points in the
plasma sheath at which k and k
respectively go to zer0.p1 p3

Based on data set #8. (f=2.322 GHz,

id=320 ma, 11:210 ma, i7=160 ma, 13:135 ma).

4.4

126

the sheath region. More significantly, the ratio 22/21 agrees
well with observed values of approximately 1.5 from measurements

14

of the corresponding E field peaks in the thermal resonances.

Graphical Presentation of Thermal Resonances Using_the WKB

 

Approximation

 

Since the static electron profile analysis was based on the
phase integral in the underdense region, the WKB formulation for

the nth thermal resonance given in equation (2.65)

X
Ill

n1m(x) - E;:?;7 sin ( ‘kpm(x') dx' + n/4 )
x

should yield the correct form of the mth thermal resonance some
distance away from the critical point. Here x - 0 at the wall and
is positive into the plasma; kp(x) represents the phase constant
as a function of x. The mathematical formulation of the phase
integrals for the two profile formulations are, of course,
different. For the parabolic profile it is based on equations

(3.40) and (3.41) and is

x x 2 ‘
m m e nom

kp(x') dx' -

<|8
A
H
I

(1 - “£12111” dx (4.1)

For the Bessel function approximation the phase integral is based

on equation (3.77) and is

127

2
m m e n

e 0
Based on these phase integrals, the WKB form

X
m

1 ( kpmo.) dx + "/41 (4.3)

n (x) -
1m kpm(X)

x
is numerically evaluated and graphically presented in Figures 4.4.1
and 4.4.2 for the parabolic form and in Figures 4.4.3 and 4.4.4 for
the Bessel function formulation for data set #1. The Figures show
the first and second resonance. In the region near the critical
point where the WKB approximation fails, the expected section is
sketched in for completeness and does not represent a precise
solution. The interesting point is the phase of the perturbation
function n1m(x). The basic theory suggested that n1m(0) at the
wall (x . O) has a maximum so that a peak should be observed. In
fact. for the Bessel function formulation n1l(x) and n12(x) fall
slightly short of reaching a peak, while the parabolic approxima-
tion is slightly over the expected peak. It should be recalled
that the numerical analysis was based on the assumption that the
total phase for n1m(x) between x - 0 and xm is (m + l/4)n.

The deviation from the expected phase of n1m(0) at the wall
indicates a limitation in the accuracy of the numerical integration
techniques. Greater precision would not yield significant improve-
ment in the electron density profile in view of the approximate

nature of the available resonance data. It would, however, require

......0......OOOOO...O0....‘000....0.00.00.00.00000000.u

.oooooooo.ooooooo0.00.00.00.00.

 

128

O

Critical Point (2:21)

5
o
w
1
w
o
, +

l

1

Wall (220) 3 1
1

1

 

  

 

L ‘4‘. .l M 44:44.”); I #W-W+LA_¢+. #44

AAAMLAAA-IAAAH‘I
—~

 

(1 0,1 0,7 0.7» (1.1 0.9 0.4 0.1 0.8 0.9 1.0

z/z1————Ju—

Data Set #1
"irst Resonance
Parabolic Approximation

 

Fig. 4.4.1 First thermal resonance for data set #1 based

on a WKB formulation using the parabolic
electron density profile,

 

O

A

0.0.00.0000...0......I.0.00.00.00.00000000.

I
A

Q
O
O
A

 

Fig. 4.4.2

.OIOOOIOO‘OCOOOOO0.00....0......OOO0.00000000

129

O .
Critical Point (z-zp)-———\\\‘\|
' l
l
O
. l
O
9
\ \ |
\ I
\
\
I
l
I
I
5 0.6 0.7 0.8 0.9 1.0
z/z2

 

Data Set #1
Second Resonance
Parabolic Approximation

Second thermal resonance for data set #1 based
on a WKB formulation using the parabolic
electron density profile.

130

Critical Point (2:21)«n

    
    
  
 

 

-Wall (2:0)
/

 

 

 

 

Data Set #1
First Resonance
Bessel Approximation

00....00.000000...000000001 00......‘0000.0.0.00000000...0.00.0.00000000.

'0......0.00000000.

Fig. 4.4.3 First thermal resonance for data set #1 based
on a WKB formulation using the Bessel series
electron density profile.

0......0.0.0.0....00000000.

 

A

.0.....'........‘

 

/ Wall (2:0)
\

A

131

Critical Point (2:2?)——\\

Nix!

 

J’

 

AM »~—LM¢%1¢Avx!fv:=;-va£vv-V‘—v
0.6 0.7 0.8 0.9 1.0
z/z?—-—>
Data Set #1

Seoond-Resonance
Bessel Approximation

 

Fig. 4.4.4 Second thermal resonance for data set #1 based
on a WKB formulation using the Bessel series
electron density profile.

132

unreasonably long computer run times in view of the large number
of parameters determined simultaneously.

Figures 4.4.1 through 4.4.4 do show that, as expected, the
phase constant decreases and the magnitude of n1m(x) increases as

x goes from x - 0 to x - xm.

APPENDIX A

NUMERICAL COMPUTER READOUTS AND
ADDITIONAL COMPUTER GRAPHS

:33

flsiflflfilfli

134

NUMBER OF DATA SET 8 1

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINCRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATION
USING RESONANCES I AND 2

THE PHASE FOR RESONANCE I IS PI TIHES I.25
THE PHASE FOR RESONANCE 2 IS PI TIHES 2.25

THE SQUARE OF hPOVER H IS EQUAL TO 2.60

IO ' 0.2700E 00
II ' 0.I85OE 00
I2 ' O.I5OOE 00
H ' 0.12675 II
BETA’ATOR ' 0.IOOOE OI
RADIUS ' 0.7000E-OZ
ALFA ' 0.32595 00
RI 3 0.6160E'OZ
R2 ' 0.5767E‘02
ll . 0.3’000E'03
ZZ ‘ 0.IZI3E-OZ
NO DIPOLE 3 0.2233E I8
NO I RESONANCE ' 0.1530E I8
N0 2 RESONANCE ' 0.IZ§IE 18
22 TO ZI ' 0.I§~fiE OI
PEAK TC AVERAGE ' 0.17035 OI
V HALL ' 'O.ZZIOE OI
ETA'VH T0 KTTGO 3 ‘0.I748E OI
ELECTRON TEMP - 0.1567E 05

3.3383233

83"!Bfl‘.‘

83888

135

NUMBER OF DATA SET 8 I

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATION
USING RESONANCES I AND 3

THE PHASE FOR RESONANCE I IS PI TIMES 1.25
THE PHASE FOR RESONANCE 2 IS PI TIMES 3.25

THE SQUARE OF hPOVER H IS EQUAL TO 2.60

10 . 0.27005 00
11 - 0.18505 00
12‘ - 0.12505 00
u - 0.12.75 11
BETA-ATOR . 0.10005 01
RADIUS . 0.70005-02
ALFA - 0.02505 00
R1 . 0.61606-02
R2 . 0.53005-02
21 . 0.54005-03
zz - 0.10345-02
NO 019015 - 0.22335 10
N0 1 RESONANCE - 0.15305 18
NC 2 RESONANCE - 0.103~5 10
22 TC 21 . 0.19455 01
PEAK TD AVERAGE . 0.17035 01
v HALL - -0.22105 01
ETA-Vi To «7700 - -0.17405 01
515crao~ TEMP - 0.14.75 05

136

NUMBED OF DATA SET 8 2

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINORICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATICN
USING RESONANCES I AND 2

THE PHASE FOR RESONANCE I IS PI TINES 1.25
THE PHASE FOR RESONANCE 2 IS PI TIMES 2.25

THE SQUARE OF HPOVER H IS EQUAL TO 2.00_

ID ' 0.2000E 00
II ! 0.1QOOE 00
12 ' 0.1500E 00
U I 0.13I9E II
BETA-ATOR ‘ 0.IOOOE OI
RADIUS 8 0.10005-02
ALFA I 0.82I8E 00
RI ' 0.6090E-02
R2 l 0.5633E-02
lI - 0.9100E-03
12 I 0.1367E-02
NO DIPOLE ' 0.2412E 18
NO I RESONANCE ‘ 0.1560E 16
NO 2 RESONANCE ' 0.12686 18
12 TO II 3 0.1502E OI
PEAK TO AVERAGE 8 O.I697E OI
V HALL ' ~0.2965E OI
ETAtVH TO KTTOQ 3 ~O.I725E OI
ELECTRON TEMP 8 0.I990£ 05

137

NUMBER OF DATA SET 3 2

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATION
USING RESONANCES I AND 3

THE PHASE FOR RESONANCE I IS PI TIMES 1.25
THE PHASE FOR RESONANCE 2 IS PI TIMES 3.25

THE SQUARE OF HPOVER U IS EQUAL TO 2.60

332333333

3':

H
‘

83388

V HALL

-O.5I89E 01
‘0.1906E 01

IO 8 0.29006 00
II I 0.1900E 00
12 I 0.1200E 00
H 8 0.1319E 11
BETA'ATOR 8 0.I000E 01
RADIUS 8 0.7000E‘02
ALFA ' 0.8511E 00
R1 - 0.6020Er02
R2 . 0.5023E-02
21 ' 0.9600E-03
22 t 0.1977E°02
NO DIPOLE ' 0.247QE 18
NO I RESONANCE - 0.1621E 18
NO 2 RESONANCE 8 0.1024E 16
22 TO 21 ' 0.2018E 01
PEAK TO AVERAGE ' ’0.174IE 01

ETA-Yd TO KTTOQ
ELECTRON TEMP

0.3163E 05

I 838138 8

83888

138

NUflBER 05 DATA $51 . 3

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOIIC DENSITY PROFILE APPROXIMATICN
USING RESONANCES I AND 2

THE PHASE FOR RESONANCE 1 IS PI TIMES 1.25
THE PHASE FOR RESONANCE 2 15 PI TIMES 2.25

THE SQUARE OF HPOVER H IS EQUAL TO 2.00

ID 8 0.3100E GD
11 8 0.2350E 00
I2 8 0.1650E QC
H 8 0.1401E II
BETA8ATOR 8 0.I000E OI
RADIUS 8 0.7000E-02
ALFA 8 0.6317E 00
R1 8 0.6160E-02
R2 8 0.5735E'02
II 8 0.86OOE-O3
22 8 0.1265E-OZ
N0 DIPOLE 8 0.2745E 18
N0 1 RESONANCE 8 0.I697E 16
N0 2 RESONANCE 8 0.1493E 18
22 TO 21 8 0.1500E OI
PEAK TO AVERAGE 8 0.1712E 01
V HALL 8 '0.2909E 01
ETA8VH TO KTTOQ 8 -0.1782E OI
ELECTRON TEMP 8 0.1895E 05

NUMBER OF DATA SET

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATION
USING RESONANCES

THE PHASE FOR RESONANCE 1 IS PI TIMES 1.25
THE PHASE FOR RESONANCE 2 15 PI TIMES 3.25

THE SQUARE OF HPOVER H IS EQUAL TO 2.60

10
11
12

H
BETA8ATOR
RADIUS

ALFA

N0 DIPOLE

N0 1 RESONANCE
N0 2 RESONANCE
12 TO 21

PEAK TO AVERAGE
V HALL

ETA8VH TO KTTOQ
ELECTRON TEMP

139

AND 3

0.3400E OD
0.2350E DO
0.1bOOE 00
0.1401E II
0.1000E DI

0.6317E 00

0.5403E-02
0.8400E‘O3
0.2765E 18
0.1897E IB
0.1292E IB
0.1901E 01
D.I712E OI

0.1895E 05

NUMBER OF DATA SET

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATION

AND 2 '

USING RESONANCES

THE PHASE FOR RESONANCE 1 [S P! TIMES 1.25
YHE PHASE FOR RESONANCE 2 15 Pl IIMES 2.25

THE SQUARE OF hPOVER H 15 EQUAL TO 2.60

10

II

12

H
BETA8ATOR
RADIUS

ALFA

R1

R2

21

22

N0 DIPOLE

N0 1 RESONANCE
N0 2 RESONANCE
22 TO 21

PEAK TO AVERAGE
V HALL

ETA-VH TO KTTOQ
ELECTRON TEMP

140

0.3550E 00
0.2450E 00
0.2000E 00
0.1456E 11
0.1000E 01
0.7000E-02

0.6040E 00

0.623OE'02
0.5861E-02
0.77OOE-03
0.1139E-02
0.2904E 18
0.2006E 18
0.1636E 16
0.1679E 01
0.1672E 01
‘0.1627E OI
0.1159E 05

141

NUMBER OF DATA SET 8 6

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATION
USING RESONANCES I AND 3

THE PHASE FOR RESONANCE 1 IS PI TIMES 1.25
THE PHASE FOR RESONANCE 2 IS PI TIMES 3.25

THE SQUARE OF HPOVER H IS EQUAL TO 2.60

ID 8 0.3550E 00
II 8 0.2450E 00
12 8 0.1550E 00
H 8 0.1458E 11
BETA-ATOR 8 0.1000E OI
RADIUS 8 0.7000E-02
ALFA 8 0.6569E 00
R1 8 0.6090E*02
R2 8 0.5179E-02
21 8 0.9IOOE-O3
22 8 0.16215802
N0 DIPOLE 8 0.3OA3E 18
NO I RESONANCE 8 0.2100E 18
NO 2 RESONANCE 8 0.1329E 18
22 TO 21 8 0.2001E 01
PEAK TO AVERAGE 8 0.1753E 01
V HALL 8 80.55795 01
ETA8VH TO KTTOQ 8 -0.1959E 01
ELECTRON TEMP 8 0.3307E 05

142

NUMBER OF DATA SET 8 5

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATION
USING RESONANCES 1 AND 2

THE PHASE FOR RESONANCE 1 IS PI TIMES 1.25
THE PHASE FOR RESONANCE 2 IS PI TIMES 2.25

THE SQUARE OF hPOVER H IS EQUAL TO 2.60

10 8 0.2700E 00
II 8 0.1800E 00
12 8 0.1350E 00
H 8 0.1208E II
BETA8ATOR 8 0.1000E 01
RADIUS 8 0.7000E-02
ALFA 8 0.8641E 00
R1 8 0.6020E-02
R2 8 0.5496E-02
21 8 0.9800E-03
22 8 0.1504E-02
NO DIPOLE 8 0.2065E 18
NO I RESONANCE 8 0.1390E 18
N0 2 RESONANCE 8 0.1042E 18
22 TO 21 8 0.153hE 01
PEAK TO AVERAGE 8 0.1761E 01
V BALL 8 -0.4987E 01
ETA8VH TO KTTOQ 8 '0.1996E 01
ELECTRON TEMP 8 0.2900E 05

143

NUMBER OF DATA SET 8 5

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATIDN
USING RESONANCES I AND 3

THE PHASE FOR RESONANCE 1 IS PI TIMES 1.25
THE PHASE FOR RESONANCE 2 IS PI TIMES 3.25

THE SQUARE OF HPDVER H IS EQUAL TO 2.60

383888

I0 8 0.27006 00
11 8 0.1BOOE 00
I2 8 0.1100E 00
H . 0.120~E II
BETA-ATDR 8 0.10005 01
RADIUS 8 0.7000E’OZ
ALFA 8 0.8b51E 00
R1 . 0.60205802
R2 8 0.§9?4E-02
21 8 0.9800E-03
22 8 0.2026E-02
N0 DIPOLE 8 0.2065E 16
N0 1 RESONANCE 8 0.1390E 13
N0 2 RESONANCE 8 0.8496E 17
22 TO 21 8 0.2067E 01
PEAK TO AVERAGE 8 0.1761E 01
V HALL 8 -0.6987E 01
ETA-vu TD KTTOQ 8 -0.1996E 01
ELECTRON TEMP 8 0.2900E 05

~
d

vaaaslaz

144

NUMBER OF DATA SET 8 6

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATION
USING RESONANCES 1 AND 2

THE PHASE FOR RESONANCE 1 IS PI TIMES 1.25
THE PHASE FOR RESONANCE 2 IS PI TIMES 2.25

THE SQUARE OF EPOVER H IS EQUAL TO 2.60

ID 8 0.2850E 00
I1 8 0.1900E 00
I2 8 0.1500E 00
H 8 0.1267E 11
BETA8ATOR 8 0.1000E 01
RADIUS 8 0.70005-02
ALFA 8 0.8366E 00
R1 8 0.6090E-02
R2 8 0.5653E-OZ
ll . 0.9100E-03
22 8 0.1367E'02
NO DIPOLE 8 0.225DE 18
NO 1 RESONANCE 8 0.1500E 18
N0 2 RESONANCE 8 0.1164E 1B
22 TO 21 8 0.1480E 01
PEAK TO AVERAGE 8 0.1716E 01
V HALL 8 “0.317OE 01
ETA8VH TO KTTOQ 8 '0.1600E 01
ELECTRON TEMP 8 0.2048E 05

145

NUMBER OF DATA SET 8 6

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATION
USING RESONANCES 1 AND 3

THE PHASE FOR RESONANCE I IS PI TIMES 1.25
THE PHASE FOR RESONANCE 2 IS PI TIMES 3.25

THE SQUARE OF HPOVER H 15 EQUAL TO 2.60

10 8 0.2050E 00
11 8 0.19003 00
12 8 0.1200E 00
BETA8ATOR 8 0.10005 01
RADIUS 8 0.7000E“02
ALFA 8 0.83465 00
R1 8 0.6090E’02
R2 8 0.5037E‘02
21 8 0.9100E-03
22 8 0.1913E‘02
N0 DIPOLE 8 0.2250E 15
N0 1 RESONANCE 8 0.15005 13
N0 2 RESONANCE 8 0.9875E 17
22 TO 21 8 0.21025 01
PEAK TO AVERAGE 8 0.17165 01
V HALL 8 ‘0.317bE 01
ETA8VH TO KTTOQ 8 “0.18005 01
ELECTRON TEMP 8 0.20685 05

146

NUMBER OF DATA SET 8 7

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATION
USING RESONANCES 1 AND 2

THE PHASE FOR RESONANCE 1 IS PI TIMES 1.25
THE PHASE FOR RESONANCE 2 15 PI TIMES 2.25

THE SQUARE OF HPOVER H IS EQUAL TO 2.60

10 8 0.2900E 00
11 8 0.1950E 00
12 8 0.1500E 00
H . 0015296 11
BETA8ATOR 8 0.1000E 01
RADIUS 8 0.7000E-02
R1 8 0.6090E-02
R2 8 0.5007E'02
21 8 0.9100E‘03
22 8 0.13935802
N0 DIPOLE 8 0.2878E 18
N0 1 RESONANCE 8 0.1935E 18
N0 2 RESONANCE 8 0.1688E 18
22 TO 21 8 0.1531E 01
PEAK TO AVERAGE 8 0.1725E 01
V HALL 8 80.63335 01
ETA8VH TO KTTOQ 8 80.183BE 01
ELECTRON TEMP 8 0.27375 05

888833333323233333

288858:

NUMBER OF DATA SET

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATION

AND 3

USING RESONANCES

THE PHASE FOR RESONANCE 1 IS PI TIMES 1.25
THE PHASE FOR RESONANCE 2 15 P1 TIMES 3.25

THE SQUARE OF HPOVER H IS EQUAL TO 2.60

10

TI

12

H
BETA8ATOR
RADIUS

ALFA

R1

R2

21

22

N0 DIPOLE

N0 1 RESONANCE
N0 2 RESONANCE
22 TD 21

PEAK TO AVERAGE
V HALL

ETA8VH TO KTTOQ
ELECTRON TEMP

147

0.2900E 00
0.195OE 00
0.1200E 00
0.1429E 11
0.10005 01
0.7000E802

0.8704E 00

0.6020E-02
0.5018E-02
0.9600E-03
0.1982E-02
0.2953E 18
0.1986E 18
0.1222E 18
0.2022E 01
0.1771E 01

-0.2043E 01

0.4282E 05

2'386fl3I83383333333I

148

NUMBER OF DATA SET 8 8

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATICN
USING RESONANCES 1 AND 2

THE PHASE FOR RESONANCE 1 IS PI TIMES 1.25
THE PHASE FOR RESONANCE 2 IS PI TIMES 2.25

THE SQUARE OF EPOVER H IS EQUAL TO 2.60

10 8 0.3200E 00
11 8 0.2100E 00
12 8 0.1800E CO
H 8 0.1459E 11
BETA8ATOR 8 0.1000E 01
RADIUS 8 0.7000E-02
ALFA 8 0.8523E 00
R1 8 0.6020E-02
R2 8 0.5510E-02
21 8 0.9800E'D3
22 8 0.1490E-02
N0 DIPOLE 8 0.3030E 18
NO 1 RESONANCE 8 0.1988E 18
N0 2 RESONANCE 8 0.1515E 18
22 TO 21 8 0.1521E 01
PEAK TO AVERAGE 8 0.1743E 01
V HALL 8 80.6436E CI
ETA8VH TO KTTOQ 8 80.1913E 01
ELECTRON TEMP I 0.3900E 05

8882183833

888:

NUMBER OF DATA SET

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
PARABOLIC DENSITY PROFILE APPROXIMATICN

AND 3

USING RESONANCES

THE PHASE FOR RESONANCE I IS PI TIMES 1.25
THE PHASE FOR RESONANCE 2 IS PI TIMES 3.25

THE SQUARE OF HPOVER H IS EQUAL TO 2.80

10
II
12

H
BETA8ATOR
RADIUS

ALFA

R1

R2

21

22

N0 DIPOLE

N0 1 RESONANCE
N0 2 RESONANCE
22 TO 21

PEAK TO AVERAGE
V HALL

ETA8VH TO KTTOQ
ELECTRON TEMP

149

0.3200E‘00
0.2100E 00
0.13005 00
0.15596 11
0.1000E 01

0.8523E 00

0.6020E-02
0.4987E-02
0.9800E-03
0.2033E‘02
0.3030E 18
0.1988E 18
0.123IE 18
0.2076E 01
0.1743E 01
80.8436E 01
80.1913E 01
0.3906E 05

H
‘

IDIGESIII

150

NUMBER OF DATA SET 8 1

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED CN A
BESSEL FLNCTICN PRCFILE APPROXIMATICN

USING THERPAL RESONANCES 1

AND

2

TOTAL PHASE FCR FIRST RES IS PI TIPES 1.25
TOTAL PHASE FCR SEC RES IS PI TIMES 2.25

THE SQUARE OF hP OVER H

LOHER INT. LIMIT

INITIAL INCR. IN 21

NUMBER OF INTEGR. STEPS
VALUE OF HI

VALUE OF H2

DIPCLE CURRENT AT H1

CURRENT AT H1

CURRENT AT H2

AUMBER T. D. RESCNANCE 2ND H

CDEFF PEAK TC AVG EL DENS
GAMMA

ETA 8 VHALL TO KT OVER 0
GAMMA TIMES 21

GAMMA TIMES 22

22 TC 21

HPI

HP2

NO1

N02

A1 8 HPI CVER bl SQUARED
A2 8 HPZ CVER H2 SQUARED
21

22

VHALL

ELECTRON TEMPERATURE

2.60.

c.ccoce
c.1ccce,

C.1207E
C.1267E
C.Z7CCE
C.185CE
C.150CE

0.1988E
C.32taE
-0.18CaE
c.31COE
0.4619E
c.119ce
c.23$bE
c.21ZZE
0.1744E
6.14145
c.34595
0.26c«E
c.9485E
0.1413E
-c.73c3E
c.~7255

GI
-03
-02

C1

C5

151

NUMBER OF DATA SET . I

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
BESSEL FUNCTICN PRCFILE APPROXIMATION

USING THERMAL RESONANCES 1

AND

3

TOTAL PHASE FCR FIRST RES IS PI TIMES 1.25
TOTAL PHASE FOR SEC RES IS PI TIMES 3.25

THE SQUARE OF HP OVER H

LOHER INT. LIMIT

INITIAL INCR. IN 21

NUMBER OF INTEGR. STEPS
VALUE OF H1

VALUE OF H2

DIPOLE CURRENT AT H1

CURRENT AT H1

CURRENT AT H2

NUMBER T. D. RESONANCE 2ND H

COEFF PEAK TC AVG EL UENS
GAMMA

ETA - VHALL T0 KT CVER o
GAMMA TIMES 21

GAMMA IIMES 22

22 TC 21

HPI

HPZ

N01

N02

AI 8 HPI OVER H1 SOOARED
AZ 8 HPZ OVER H2 SQUARED
21

12

VHALL

ELECTRON TEMPERATURE

8 2.80

C.COOCE CD
C.IDDCE CG

0.1267E 11
0.1267E 11
C.270CE C0
C.1850E CC
0.13005 C0

0.1937E 01
0.3207E 03
-0.1717E 01
0.25COE 00
0.5050E CD
C.2020E CI
0.2392E 11
0.2005E 11
0.1797E 18
0.1263E 18
C.3563E 01
C.2504E 01
0.7795E-03
0.1575E‘02-
80.4523E CI
0.3058E 05

152

N

N¢M52« JF LATA SET 2

THIS IS AN ANALYSIS OF Tn: ELCCIRJM LENSITY
IN A CYLIMLRICAL PlASMA COLUMN bASFC LL A
sISSLL FUNCTILN PPCFILF APIROAIMAIICL

LSING THERI’AL RESONANCES I MIL) 2

TCTAL PHASE FER szcsl 4¢s 13 91 TIV:S 1.25
TLTAL PHASE FJR 3:: «E3 13 a; r was 4.26

II
N
o
3‘
0

THE SQUAJ: CF H9 OVER H

LOHER INT. LIVIT 8 C.CCOCE CC
INITIAL INCR. IN 21 8 0.1000E CO
NUMBER OF INTECR. STEPS 8 2C
VALLE OF HI = C.1319E 11
VALUE OF H2 = (3.131913 11
DIPOLE CURRENT AT H1 8 C.290CE CC
CURRENT AT HI = C.19CCE CC
CURRENT AT H2 8 C.15CCE CC
NUMBER T. O. RESCNANCE 2ND H = 2
CCEFF PEAK TC AVG EL OENS 8 0.1975E CI
GAMMA 8 C.3255E C3
ETA 8 VHALL TC KT EVER 0 = “0.1785E CI
GAMMA TIMES 21 8 C.31CCE CC
GAMMA TIMES 22 8 0.4743E 00
22 TO 21 = 0.1530E C1
HPI 8 C.2635E II
HPZ 8 C.2163E II
MCI 3 CQIOOZE 15
502 . 3 0.14705 18
AI 8 H91 EVER H1 SQUARED 8 0.39C7E C1
A2 8 HP? OVER 82 SQUAREO 3 C.2690E 01
21 8 0.9524[-o3
22 8 C.1457E-C2
VHALL = ‘C.1287E 02
ELECTRON TEMPERATURE 8

C.83¢9E C5

153

NUMBER OF DATA SET 8 2

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
BESSEL FUNCTICN PROFILE APPROXIMATICN

USING THERMAL RESONANCES 1

AND

3

TOTAL PHASE FOR FIRST RES IS PI TIMES 1.25
TOTAL PHASE FCR SEC RES IS PI TIMES 3.25

THE SQUARE OF HP OVER H

LOHER INT. LIMIT

INITIAL INCR. IN 21

NUMBER OF INTEGR. STEPS
VALUE OF H1

VALUE OF H2

DIPOLE CURRENT AT H1

CURRENT AT H1

CURRENT AT H2

NUMBER T. D. RESCNANCE 2ND H

COEFF PEAK TO AVG EL DENS
GAMMA

ETA 8 VHALL TO KT OVER Q
GAMMA TIMES 21

GAMMA TIMES 22

22 TC 21

HPI

HPZ

N01

N02

AI 8 HPI OVER H1 SGUARED
A2 8 HPZ OVER H2 SQUARED
21

22

VHALL

ELECTRON TEMPERATURE

2.60

C.CCCCE CD
C.1CCCE CO

c.13195 11
c.13195 11
c.2qcce cc
c.190ce cc
c.120ce co

0.192bE OI
0.3193E 03
-C.1097E C1
C.2680E CO
c.5950E CC
C.2220E C1
C.2«35E 11
0.19356 11
C.1862E 18
0.11756 16
0.39C7E 01
0.21526 01
c.33935—03
0.18635-02
-C.1095€ CZ
0.7486E CS

83888 I

Sta:

NUMBER OF DATA SET 8 3

154

THIS IS AN ANALYSIS OF THE ELECTRON CENSITY
IN A CYLINCRICAL PLASMA COLUMN BASED ON A
BESSEL FUNCTION PROFILE APPROXIMATION

USING THERPAL RESONANCES I

AND

2

TOTAL PHASE FOR FIRST RES IS PI TIMES 1.25
PI TIMES 2.25

TOTAL PHASE FOR SEC RES IS

THE SQUARE OF hP OVER H

LOHER INT. LIMIT

INITIAL INCR. IN 21

NUMBER OF INTEGR. STEPS
VALUE OF HI

VALUE OF NZ

DIPOLE CURRENT AT HI

CURRENT AT HI

CURRENT AT H2

NUMBER T. D. RESONANCE 2ND H

COEFF PEAK TO AVG EL SENS
GAMMA

ETA 8 VHALL TO KT OVER 0
GAMMA TIMES 2I

GAMMA TIMES 22

22 TO 21

HPI

HPZ

N01

N02

AI 8 HPI OVER HI SOUARED
AZ 8 HPZ OVER NZ SDUARED
21

22

VHALL

ELECTRON TEMPERATURE

2.60

C.CCOCE
C.ICOCE

C.IAOIE
C.IAOIE
c.3«oce
C.235CE
C.IB§CE

6.10566
0.323IE
-C.I751E
c.2ocoe
C.AI¢CE
C.leOE
C.2656E
c.2357E
C.2216E
0.17ASE
c.35cae
0.26296
c.80456
0.1287E
-c.10186
C.b7A?E

CO
CO
20
II
II
CC
CC
CO

2

CI
03

-C3
‘02
C2
C5

83‘38

3‘83

155

NUMBER OF CATA SET 8 3

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINCRICAL PLASMA COLUMN BASED ON A
BESSEL FUNCTION PROFILE APPROXIMATICN

USING THERMAL RESONANCES I AND

.3

TOTAL PHASE FOR FIRST RES IS PI TIMES [.25
TOTAL PHASE FOR SEC RES IS PI TIMES 3.25

THE SQUARE OF NP OVER N

LCHER INT. LIMIT

INITIAL INCR. IN ZI

NUMBER OF INTEGR. STEPS
VALUE OF NI

VALUE OF N2

DIPOLE CURRENT AT HI

CURRENT AT HI

CURRENT AT H2

NUMBER T. D. RESONANCE 2ND N

COEFF PEAK TO AVG EL DENS
GAMMA

ETA 8 VwALL TO KT OVER Q
GAMMA TIMES 21

GAMMA TIWES 22

22 TC 21

NPI

NPZ

NDI

NCZ

A1 8 UPI OVER NI SQUARED
A2 8 HPZ OVER NZ SQUARED
II

22

VHALL

ELECTRON TEMPERATURE

8 2.60

C.CGOCE
0.IOOOE

C.IAOIE
C.I¢OIE
C.3ACCE
C.235CE
C.IbOCE

c.1993E
C.3277E
-C.IBI7E
C.3ICDE
C.bOlAE
C.1940E
0.2617E
C.2159E
C.2151E
0.1‘65E
C.3§89E
C.2375E
c.9fi59E
0.1835E
-008931E
C.57C6E

CC
00
20
II
II
CC
CC
C0

-C3
-cz
CI
05

156

NUMBER OF DATA SET 8 6

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
BESSEL FUNCTION PROFILE APPROXIMATION

USING THERMAL RESONANCES 1

AND

2

TOTAL PHASE FOR FIRST RES IS PI TIMES 1.25
TOTAL PHASE FOR SEC RES IS PI TIMES 2.25

THE SQUARE OF NP OVER H

LONER INT. LIMIT

INITIAL INCR. IN 21

NUMBER OF INTEGR. STEPS
VALUE OF N1

VALUE OF H2

DIPOLE CURRENT AT N1

CURRENT AT H1

CURRENT AT H2

NUMBER T. D. RESONANCE 2ND H

COEFF PEAK TO AVG EL DENS
GAMMA

ETA 8 VHALL TO KT OVER Q
GAMMA TIMES 21

GAMMA TIMES 22

22 TC 21

UPI

HPZ

NCI

NCZ

AI 8 UPI OVER N1 SQUARED
A2 8 HPZ OVER NZ SQUARED
21

22

VHALL

ELECTRON TEMPERATURE

8 2.60

C.CCDCE CC
C.ICCCE CC

C.IASBE 11
C.1A58E 11
C.355CE CC
C.245CE CO
C.2COCE CD

C.2011E Cl
c.3299; 03
-C.18505 OI
c.31coe cc
0.a557£ CO
c.147CE Cl
C.2762E 11
0.2696E ll
C.2397E 18
0.195bE 16
6.35898 01
0.2930E 01
c.9396E-03
c.13516-c2
-c.7o«oe.c1
C.A7955 05

157

NUMBER OF DATA SET 8 A

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
BESSEL FUNCTION PROFILE APPROXIMATICN

USING THERMAL RESONANCES I AND 3

TOTAL PHASE FOR FIRST RES IS PI TIMES 1.25
TOTAL PHASE FOR SEC RES IS PI TIMES 3.25

THE SQUARE OF NP OVER H 8 2.60

LOHER INT. LIMIT 8 C.CCCCE

INITIAL INCR. IN ll 8 C.ICCCE

NUMBER OF INTEGR. STEPS 8

VALUE OF N1 8 C.IASBE

VALUE OF N2 8 C.IASBE

DIPOLE CURRENT AT H1 8 C.355CE

CURRENT AT N1 8 C.245CE

CURRENT AT N2 8 C.I55CE

NUMBER T. D. RESONANCE 2ND N 8

COEFF PEAK TO AVG EL DENS 8 C.IRRQE
GAMMA 8 0.328hE
ETA 8 VNALL TO KT OVER Q 8 80.1828E
GAMMA TIMES 21 8 C.2780E
GAMMA TIMES 22 8 C.5977E
22 TO ll 8 C.2150E
NPI 8 C.28C9E
HPZ 8 0.2234E
N01 8 C.2679E
N02 8 C.ISbBE
Al 8 UPI OVER N1 SQUARED 8 C.37IZE
AZ 8 HPZ OVER N2 SQUARED 8 C.2348E
ll 8 c.8665E
22 8 0.1820E
VHALL 8 '0.1223E
ELECTRON TEMPERATURE 8 C.727OE

CC
CC

II
11

~03
~02
C2
05

--

.._-

158

NUMBER OF DATA SET 8 5

THIS IS AN ANALYSIS OF THE ELECTRON

CENSITY

IN A CYLINDRICAL PLASMA COLUMN BASED ON A
BESSEL FUNCTION PROFILE APPROXIMATICN

USING THERVAL RESONANCES 1

AND

2

TOTAL PHASE FOR FIRST RES IS PI TIMES 1.25
TOTAL PHASE FOR SEC RES IS PI TIMES 2.25

THE SQUARE OF NP OVER U

LONER INT. LIMIT

INITIAL INCR. IN 21

NUMBER OF INTEGR. STEPS
VALUE OF N1

VALUE OF N2

DIPOLE CURRENT AT NI

CURRENT AT HI

CURRENT AT H2

NLMBER T. D. RESONANCE 2ND H

COEFF PEAK TO AVG EL DENS
GAMMA

ETA 8 VHALL TO KT OVER Q
GAMMA TIMES 21

GAMMA TIMES 22

22 TC 21

NPI

NPZ

N01

N02

A1 8 HPI OVER H1 SCUARED
A2 8 HPZ OVER N2 SQUARED
21

22

VNALL

ELECTRON TEMPERATURE

8 2.60

C.CCOCE
C.IOOCE

C.IZDAE
C.IZOAE
C.270CE
C.IBCCE
C.IBSCE

0.1986E
C.3268E
-C.1804E
C.31CDE
0.4929E
0.1590E
C.2262E
C.IQAIE
0.1579E
C.IIBAE
0.3467E
C.26COE
0.9485E
C.ISCBE
'C.16C9E
C.IDESE

CO
CO

II

01
'03
'62

02

C6

159

NUMBER OF CATA SET 8 5

THIS IS AN ANALYSIS or rue ELECTRON DENSITY
x~ A CYLINDRICAL PLASMA COLUMN BASED CA A
BESSEL FUNCTION PROFILE APPROXIMATICN

USING THERMAL RESONANCES 1

AND

3

TOTAL PHASE FOR FIRST RES 15 PI TIMES 1.25
TOTAL PHASE FOR SEC RES IS PI TIMES 3.25

THE SQUARE OF NP OVER H

LONER INT. LIMIT

INITIAL INCR. IN 21

NUMBER OF INTEGR. STEPS
VALUE OF d1

VALUE OF H2

DIPOLE CURRENT AT N1

CURRENT AT HI

CURRENT AT N2

NUMBER T. C. RESONANCE 2ND H

COEFF PEAK TO AVG EL DENS
GAMMA

ETA 8 VJALL TO KT OVER Q
GAMMA TIMES 21

GAMMA TIMES 22

22 TO 21

NPI

HPZ

N01

N02

AI 8 HPI OVER HI SQUARED
A2 8 HPZ OVER N2 SCUARED
21

22

VHALL

ELECTRON TEMPERATURE

8 2.60

C.CCCCE C0
C.ICCCE CO

0.120AE 11
C.IZOAE 11
C.270CE CC
C.IBOCE CC
C.IIOOE 00

C.IQflbE CI
c.32685 03
-O.18C46 OI
C.31COE C0
c.66A66 CC
0.20608 01
c.2242E II
C.ITSZE II
C.ISIRE 16
0.96AGE 17
6.34076 01
0.21196 01
0.1973E-02
-C.16C95'02
0.10356 0e

I‘. "

160

NUMBER OF CATA SET 8 6

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINORICAL PLASMA COLUMN BASED ON A

BESSEL FUNCTION PROFILE APPROXIMATICN
AND

USING THERMAL RESONANCES 1

TOTAL PHASE FOR FIRST RES IS PI TIMES 1.25
TOTAL PHASE FOR SEC RES IS PI TIMES 2.25

THE SQUARE OF NP OVER H

LCHER INT. LIMIT

INITIAL INCR. IN 11

NUMBER CF INTEGR. STEPS
VALUE OF HI

VALUE OF N2

DIPOLE CURRENT AT N1

CURRENT AT HI

CURRENT AT H2

NUMBER T. C. RESONANCE 2ND H

COEFF PEAK TC AVG EL DENS
GAMMA

ETA 8 VNALL TO KT OVER Q
GAMMA TIMES 21

GAMMA TIMES 22

22 TC 21

UPI

HPZ

NOI

NC2

A1 8 UPI OVER N1 SQUARED
A2 8 HPZ OVER N2 SQUARED
21

22

VHALL

ELECTRON TEMPERATURE

8 2.EC

C.CCCCE CD
0.1CO0E 00

20

C.IZETE 11
c.1267E 11
0.28506 00
C.IROCE C0
C.ISOCE 00

2

C.IQBbE C1
0.3268E C3

0.31COE CO
C.6763E C0
0.1530E C1
C.235°E II
C.2096E II
0.1768E 18
C.I380E 18
C.3AE7E CI
0.2737E 01
0.1451E8C2

80.1113E CZ

0.7163E c5

A)!"

NUMBER OF

161

DATA SET 8 6

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINCRICAL PLASMA COLUMN BASED ON A
BESSEL FUNCTION PROFILE APPROXIMATION

USING THE

RMAL RESONANCES I

AND

3

TOTAL PFASE FOR FIRST RES IS PI TIMES 1.25
TOTAL PHASE FOR SEC RES IS PI TIMES 3.25

THE SQUAR

E OF NP OVER H

LOVER INT. LIMIT
INITIAL INCR. IN ZI
NUMBER OF INTEGR. STEPS
VALUE OF H1

VALUE OF NZ

DIPOLE CURRENT AT HI
CURRENT AT NI

CURRENT AT H2

NUMBER T.

O. RESONANCE 2ND H

COEFF PEAK TO AVG EL DENS

GAMMA

ETA 8 VHALL TO KT OVER Q
GAMMA TIMES 21
GAMMA TIMES 22

22 TC 21
UPI

NPZ

NOI

N02

AI 8 UPI
A2 8 HPZ
21

22

VHALL
ELECTRON

OVER NI SQJARED
OVER N2 SQUARED

TEMPERATURE

2.60

C.CCCCE
C.IDOCE

0.1267E
C.IZb7E
0.2850E
C.IROCE
0.1200E

C.ISABE
C.3221E
-0017376
CobebE
0.2190E
C.2359E
0.1875E
C.IT‘BE
C.IICQE
C.34t7E
C.2189E
c.0599E
0.1883E
-COICCIE
C.bb90E

C0
C0
2C
11
11
CC
CC
00

3

C1

~03
-02
C2
C5

NUMBER OF-DATA SET 8 7

162

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
BESSEL FUNCTION PROFILE APPROXIMATICN

USING THERMAL RESONANCES 1

AND

2

TOTAL PHASE FOR FIRST RES IS PI TIMES 1.25
TOTAL PHASE FOR SEC RES IS PI TIMES 2.25

THE SQUARE OF UP OVER H

LOVER INT. LIMIT

INITIAL INCR. IN 21

NUMBER OF INTEGR. STEPS
VALUE OF HI

VALUE OF H2

DIPOLE CURRENT AT H1

CURRENT AT HI

CURRENT AT H2

NUMBER T. D. RESONANCE 2ND U

COEFF PEAK TC AVG EL DENS
GAMMA

ETA I VHALL TO KT OVER 0
GAMMA TIMES 21

GAMMA TIMES 22

22 TC 21

UPI

HPZ

NOI

N02

AI 8 UPI OVER bl SOUARED
AZ 0 HPZ OVER H2 SQUARED
21

22

VHALL

ELECTRON TEMPERATURE

2.60

C.CCDCE
C.ICOCE

C.IQZQE
C.IAZQE
C.ZQOCE
C.IQSCE
C.ISCCE

0.19936
0.32776
-C.IEI?E
0.3ICOE
C.~743E
0.15306
C.2672E
C.234sE
c.22432
C.ITZSE
c.3497E
0.2090E
c.94596
0.1447E
“COZZSQE
C.IAACE

CO
CC

11

11
CO

-C3

-C2

C2
'C6

3833333333

883883383

NUMBER OF DATA SET ' t 7

163

THIS IS AN ANALYSIS OF THE ELECTRON CENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
BESSEL FUNCTION PROFILE APPROXIMATICN

USING THERMAL RESONANCES I

AND

3

TOTAL PHASE FCR FIRST RES IS PI TIMES 1.25
TOTAL PHASE FOR SEC RES IS PI TIMES 3.25

THE SQUARE OF UP OVER U

LOUER INT. LIMIT

INITIAL INCR. IN 21

NUMBER OF INTEGR. STEPS
VALUE OF UI

VALUE OF U2

DIPOLE CURRENT AT UI

CURRENT AT UI

CURRENT AT U2

NUMBER T. D. RESONANCE 2ND U

COEFF PEAK TC AVG EL DENS
GAMMA

ETA t VUALL TC KT OVER 0
GAMMA TIMES 21

GAMMA TIMES 22

22 TC 21 ‘

UPI

UPZ

NOI

NC2

AI 8 UPI OVER UI SOUARED
A2 ' dPZ OVER U2 SQUARED
2I

22

VHALL

ELECTRON TEMPERATURE

ill. I II M II I MIDI II a

3 2.60

C.CCOCE
C.ICCCE

C.IAZQE
C.IA29E
C.2GCCE
C.I95CE
C.IZOCE

c.1993E
C.3277E
“C.ISI7E
Co3ICCE
0.6386E
C.2060E
0.26725
CoZchE
0.2243E
C.I3EOE
0.3“97E
0.2I52E
COQ‘OSQE
C.IQAQE
-COZZS‘E
OoIEAOE

CO
CC

11
II

'C3
'C2
CZ
Ob

3‘88

NUMBER OF

164

CATA SET 8 8

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
BESSEL FUNCTION PROFILE APPROXIMATION

USING THERMAL RESONANCES I

TOTAL PHASE FOR FIRST RES

AND

2

IS PI TIMES 1.25

TOTAL PHASE FOR SEC RES IS PI TIMES 2.25

THE SQUARE CF LP EVER U

LOUER INT

. LIMIT

INITIAL INCR. IN ZI

NUMBER OF

INTEGR. STEPS

VALUE OF UI

VALUE OF U2

DIPOLE CURRENT AT U1
CURRENT AT uI
CURRENT AT U2

NUMBER T.

C. RESONANCE 2ND U

COEFF PEAK TC AVG EL DENS

GAMMA

ETA 8 VUALL TO KT OVER 0
GAMMA TIMES ZI
GAMMA TIMES 22

22 TO 21
UPI
UPZ
NOI
NC2

AI 8 UPI OVER UI SCUARED
A2 8 UPZ OVER U2 SQUARED

21

22

VUALL
ELECTRON

TEMPERATURE

2.60

C.CCOCE C0
C.IODCE CO

C.IASQE II
C.IASQE II
C.320CE CO
C.2ICCE CC
C.IbCCE C0

C.2C20E CI
C.33IOE 03
‘ColabbE OI
0.35COE 00
c.5495E CC
0.157CE OI
C.2695E II
C.2353E II
0.22825 18
c.1739E Ia
Co34I2E C1
0.2600E OI
C.IC57E'02
C.IbeE'OZ
C.IC2OE 06

fif‘?

.3333

III8332235I383

NUMBER OF DATA SET 8 8

165

THIS IS AN ANALYSIS OF THE ELECTRON DENSITY
IN A CYLINDRICAL PLASMA COLUMN BASED ON A
BESSEL FUNCTION PROFILE APPROXIMATICN

USING THERMAL RESONANCES I

AND

3

TOTAL PHASE FOR FIRST RES IS PI TIMES 1.25
TOTAL PHASE FOR SEC RES IS PI TIMES 3.25

THE SQUARE OF UP OVER U

LOUER INT. LIMIT

INITIAL INCR. IN 21

NUMBER OF INTEGR. STEPS
VALUE OF UI

VALUE OF U2

DIPOLE CURRENT AT U1

CURRENT AT U1

CURRENT AT U2

NUMBER T. D. RESONANCE 2ND U

COEFF PEAK TO AVG EL DENS
GAMMA

ETA 8 VaALL TO KT OVER Q
GAMMA TIMES 21

GAMMA TIMES 22

22 TO 21

UPI

UP2

NOI

N02

AI 8 UPI OVER U1 SQUARED
A2 8 UPZ OVER NZ SQUARED
21

22

VUALL

ELECTRON TEMPERATURE

8 2.60

C.DDOCE
C.IOOCE

0.1A59E
C.IASQE
C.320CE
C.210CE
C.IBOCE

0.1969E
0.3248E
-001775E
C.3060E
C.6bhOE
C.2170E
C.2695E
C.2121E
C.2282E
C.IAI3E
C.3AIZE
0.2112E
C.¢421E
0.2064E
-0015‘7E
C.IOIZE

CO
C0

11
11

01
-C3
-02

02

06

166

   

‘I

Thickness b

   

 

 

 

Data Set #1.}

 

'0

h _ _ h L

 

‘00000000

0 I
O
1

O

0.8

O

0.6

O00......00000.0...O0.0.00.00000000.00.0...............0......‘00000000.

A“ 2 2 4 6 8 0
O O O O C O

0 0 o O 0 o 0 4|
. . . . .

167

"r/a

   

(r/a)

 

Plasma‘
Thickness b

S
S
a
1
G

 

Data Set #2.}

 

_ _

P-

 

 

‘........

O...

.00.......................

0.8 8 F
0.6

0.4

0.2

O

1.

-O.2

-004

.1

...................O........................

-O.6

*—

-008

_

.........

168

   

ne1(r/a)
0

Plasma
Thickness b

Glass

      

 

Data Set #3.3

 

 

 

-100.-

L« _ p _ _ O b . p . .
.............................................*.................................0..0..OOOOO
4 6 8
O 8 6 4 2 6‘. . . .
. . . . . 0
1 0 0 O O 0 JV Aw n..u .

169

   

_

Plasma
Thickness b

Glass

l
‘I

K

0
Data Set #4.}

-

 
 

r/a

—

 

p b _

b

 

b

 

A“........‘........

0
.I

0.8

O

.................

.

0.6
0.4
0.2

........h.....................00...

2 4
O 0 0
_ .

O

-0.6
-008

.................

Q

-1 .0

17D

    

 

Glass
Thickness b

    

r/a

 

Data Set #5.}

 

 

Q
0
4|

.................

h
900.0000.

0.6

........ .................

0 0

—
“00000000

2
O
.

’h-

’004

.—

-006

‘-

-008

_
.........

0

.

1
_

171

4

L
J

o

—)-r/a

    

     

Plasma
Thickness b

Glass

  

 

Data Set #6.3

 

p B

 

...............................................................‘.................‘.........

O 8 6 4 2 2 4 6 8 0
1 o o o o o 4 4 4 4 ....

172

 

  
   

 

a
/
r
Ar
. 1.0.
=1
v
A
i
_
A
A
A
.0 L
8 _
a S A
m m .
3 5k 3 A
1 SC .
P 8.1 8 7 IA
1h / aw.
GT r A
t A
9
III" S A
a A
x a
K O D

 

..“......‘.......................................‘..........................‘..................

. . . . . . . .

173

   

Plasma

    

Glass
Thickness b

 

 

A _ _ . _ 4C, _ TC _ _ .
...................................0.....C......C..............‘.................‘........“
0 8 6 4 2 2 4 6 8 O

. . . . . . . . . .
1 0 O 0 0 0 Aw 0. 0. n.U 1..

a
/
2

Plasma
Glass
Thickness b

   

'1.3

         

   

Data Set

174
IAAA_IAAA..1

‘4‘- 1“!“

 

_

L—
1“. .................. ....................................................‘........‘........Ahl

0.8

1.

,& .4 .¢ .J A. ,o as nu
nv n. nu nu nv n. nw nu .u
_ . . . .

175

   
 

 

      

 

...D
a S a
m S /
S e z
a n
1 sk a
P «WC
Pi
1h 3 .0
GT / 2 .
2 us" 4.
1
\\ t .
e A
s s A
t 1
a
O D .
A
A
A
1
y
A
A
A
l
u
A
A
A
1
y
A
A
. .0 _ _
A. _ . A I _ L T _

 

 

‘.................’.....................................................‘..................
O 8 6 4 2 2 4 6 8 0
.. o o o o o 4 4 4 4 1.

176

Plasma

.OT Glass- I
‘l \ Thickness b

      

Data Set #3.}

 

z/a

 

 

.L p _ _ A w. A _ _ _
I 1 ‘ 4 I,
..................................O........................................................
nu q. .6 I. .4 a. 4. ,o 0
1 nv nu nu nu nu Aw Aw Aw A

177

.........

A.
O
R.
0.

Plasma

 

Thickness b

S
S
a
1
nu

0.6 . -
0.4 P

.3

Data Set #4

2
O

N
..........................

z/a

 

nv

TL.

4

_
o

-0.2 0
-0,4 0
O

-008
-1.0

-0.6

178

L

.0
0
1»

Plasma
z/a

‘ Glass
‘\ Thickness b

.3.

  

    

Data Set #5

§ 0
8 6
0 0

0.4

2
0

0
-002

-004

-006

-008

 

....... ........ .......................................................................H

-100

Plasma

.‘2 Glass
\ Thickness b

Data Set #6.}
AA_J
.0

     

179

 

..........................

A.
.

..........................

0.8
0.4 T. b
2

6
0

1.

O

0
0
-0.2

O

-004

.

-0.6

Q

8

O
.

 

 

 

180

a
/
2

Plasma
Thickness b

3
3
a
1
G

   
  

Data Set #7.3

 

 

nu TL _ _ _

4 . I.

............................................‘O....... .......O‘........

.h—

1.0.4.

. .0........

8 6 4 2 2 4 6 8 o
C . . . . C . . .
-

 

 

181

Plasma

  

Data Set #8.}

  
 

r'z/a

 

 

 

......... ‘........

........‘..........................

SUMMARY OF RES OF

NO ALFA

DATA SET NC I
0.153E 18 0.826E

DATA SET NC I
0.124E 16 0.8206

DATA SET NC I
0.103: 13 0.826E

DATA SET NC 2
0.1565 13 0.6225

DATA SET NC 2
0.125E IB 0.522E

DATA SET NC 2
0.102E 15 0.8516

DATA SET NC 3
0.1905 15 0.632E

DATA SET NC 3
0.149E Id 0.832E

DATA SET NC 3
0.129E 13 0.632E

DATA SET NC 6
0.200: Id 0.80RE

DATA SET NC 4
0.104E Ia 0.80~E

DATA SET N0 6
0.133E 18 0.859E

DATA SET NC 5
0.139E 18 0.8c6E

DATA SET NC 5
0.104E 18 0.6645

DATA SET N0 5
0.589E I7 0.86~E

182

PAR ANAALYSIS

U TEMP

RESONANCE N3 '1
00 0.127E 11 0.162E 05

RESONANCE NO 2
C0 0.127E 11 0.147E 05

RESONANCE NO 3
00 0.127E 11 0.147E 05

RESONANCE NO 1

00 0.132E 11 0.200E 05

RESONANCE NO 2
CO 0.132E II 0.200E 05

RESONANCE NO 3
CO 0.132E 11 0.3le 05

RESONANCE NO I
00 C.IhOE II 0.190E D5

RESONANCE NO 2’
00 0.1«CE 11 0.190E 05

RESONANCE N0 3
co 0.1«CE II 0.190E 05

RESONANCE NO I

00 0.1AOE II 0.116E 05
RESONANCE NO 2

00 0.1fibE 11 0.116E 05

RESONANCE NO 3
CD 0.1“6: 11 C.IICE 05

RESONANCE NO 1
00 0.120E 11 0.29CE 05

RESONANCE NO 2
00 0.120E II 0.290E 05

RESONANCE NO 3
CO 0.120E 11 0.290E 05

z CRIT.

o.aaoe-oa
c.1215—oz
0.163E-02
c.1375-oz
c.137e—oz
o.ane-oz
c.540e-03
c.1275-oz
C.Ioos-oé
0.770E-03

0.11AE-02

0.102E-02

C.980E 03
C.ISDE'DZ

0.203E-02

ETA

-00175E

-O.I75E

-0.175E

“0.173E

-0.173E

-001906

-00178E

-0.I78E

-00178E

-00163E

-0.IbBE

'O.I9OE

‘OOZOOE

-OOZOCE

-0.200E

01

CI

CI

01

C1

01

CI

CI

01

01

C1

CI

01

01

CI

 

 

 

 

U

C I w a fl . u no .-

DATA SET

C.IDCE

18

NO

DATA SET NC

0.1185

15

DATA SET N0

009.55

17

DATA SET NC

0.194E

18

DATA SET NC

0.1fi9E

15

DATA SET NC

0.122:

Id

DATA SET NC

0.1995

16

DATA SET

0.151E

18

N 0

DATA SET NC

0.123E

18

6
008326

b
0.839E

6
0.835E

7
0.841E
7
0.841E
7
0.873E
8
0.8525

8
0.852E

8
0.8523

CO

00

CD

00

CO

00

CO

CO

183 \

RESOIANCE NO I
0.127E 11 C.2CDE

RESONANCE NO 2
001275 11 002096

RESONANCE NO 3
0.127E 11 C.205E

RESONANCE NO 1
0.163E II 0.279E

RESONANCE NO 2
0.163E 11 0.2766

RESONANCE NO 3
0.193E II

RESONANCE NO I

RESONANCE NO 2
0.1n6E 11 0.3QIE

RESONANCE NO 3
0.1soE 11 0.391E

35

05

05

05

05

05

05

05

05

c.910£-03

C.I35E-02

001916-02

0.910E-03

C.I3QE‘02

0.1932-02

c.950E‘03

C.IRQE‘OZ

0.203E-02

‘0o15CE

-001806

-OOIBOE

-0016‘6

-0.18“E

-002046

-0.191E

-OQEQEE

-00191E

C1

C1

01

01

C1

CI

C1

CI

01

N

32

u

fl

N

SUHHARY OF RES OF

NO GAHHA
DATA SET NC 1
0.17~: 18 0.327E

DATA SET NC

1
0.161E 15 0.327

"I

DATA SET
0.126E 13

NC 1
0.321E

DATA SET
0.186E 18

NC 2
0.3205

DATA SET NC 2
0.147E 16 0.326E

DATA SET N0 2
0.110E 18 0.319E

DATA SET NC 3
0.222E 13 0.323E

DATA SET N0 3
0.17%E 15 0.323E

DATA SET NC 3
0.197E 18 0.328E

DATA SET NC 6
0.240E 18 0.330E

DATA SET NC 6
0.1906 18 0.3306

DATA SET NC A
0.157E 18 0.326E

DATA SET N0 5
0.153E 18 0.327E

DATA SET NC 5‘
0.116E 16 0.327E

DATA SET NC 5
0.965E 17 0.327E

184

BESSEL ANALYSIS

C3

C3

C3

C3

03

03

C3

03

03

C3

C3

03

03

03

« TEMP

NJ 1
00‘07'1C

RESONANCE
0.121E 11

RESONANCE NO 2
0.127E 11 0.97HE

RESONANCE NO 3
0.127E 11 0.3065

RESONANCE ND 1
0.1325 11 0.837E

RESONANCE NO 2
0.132E 11 0.837E

RESONANCE NO 3
0.1325 11 0.749E

RESONANCE NO 1
0.1~OE 11 0.675E

RESONANCE NO 2
0.1hCE 11 0.6756

RESONANCE NO 3
0.160E 11 0.310E

RESONANCE NO 1
0.1»6E 11 0.480E

RESONANCE N0 2
0.1aoE 11 0.~80&

RESONANCE NO 3
0.1»6E 11 0.777E

RESONANCE N0 1
0.120E 11 0.1C9E

RESONANCE NO 2
0.120E 11 0.104E

RESONANCE ND 3
0.120E 11 0.106E

05

05

05

05

05

05

O3

05

05

05

06

06

06

Z CRTT.
c.953e—ca
C.I«lE-CZ
0.157E-02
0.952E-03
c.106E-02
c.18bE-02

o
0.805E-03
C.129E-02
0.186E-02
C.940E°03
C.I38E‘02
0.182E-02
0.048E-O3

0.151E-OZ

c.197E-02

ETA

-C.18CE

-0018CE

-0.172E

-0.178E

-G.178E

“001706

’00E75E

-D.175E

-O.182E

-001855

“001855

-00183E

-001805

-O.180E

-OOISCE

C1

Cl

C1

01

C1

C1

01

Cl

01

Cl

Cl

C1

C1

01

C1

. . ‘ . - O U ~ .-

DATA SET N0
0.175E 18

DATA SET
0.136E 13

NC

DATA SET NC

0.11CE 15

DATA SET ND
0.226E 18
DATA SET NC
0.172: 18

DATA SET N0
0.138E 18

DATA SET NC
0.2235 13

DATA SET N0
0.1795 18
DATA SET N0
0.141E 13

6
0.327E

b
0.327:

b
3.322E

7
0.328E

7
0.326E

7
0.328E

6
0.331E

8
0.331E

6
0.325E

C3

C3

03

03

C3

C3

C3

C3

185

RESONANCE NO
0.127E 11
RESONANC: NC

0.1276 11

RESONANCE. NO
C.1Z7E 11
RESONANCE NO
001‘035 11

RESONANCE NO
0.143E 11

RESONANCE NO
0.143E 11

RESONANCE NO
0.166E 11

RESONANCE N0
0.146E 11

RESONANCE NO
0.1405 11

1
0.71oE

z
0.716E

3

300C;E

1
0.1945

2
0.14a6

3
0.164E

1
0.102E

2
0.1C2E

3
C.1C1E

05

’09

06

06

06

06

06

06

C.9§8E-O3
0.1k5E-02
C.1%SE-CZ
C.9~6Ef03
C.IASE-OZ
C.195£-02
0.106E-02
c.166E-02

C.206E-02

-D.1DCE

’00180:

”0.17wt

-O.182E

-00 182E

-00162E

‘0.187E

‘O.167E

-O.177E

Cl

01

C1

C1

C1

01

Cl

C1

C1

APPENDIX B

FORTRAN COMPUTER PROGRAMS WRITTEN SPECIFICALLY FOR
THE NUMERICAL ANALYSIS IN THIS RESEARCH PROJECT

187

ISYS T1HE'1C

ILOAC HATFTV

[OPT BOSCURCE

C..“‘THTS PROGRAM TS DESIGNED TO DETERMINE THE PARAHETERS

C....’OF A BESSEL SERTkS ELECTRON SENSTTY PROFTLE BASED ON

C..".THERFAL RESONANCE DATA OBTINEC thF AN ELECTROACCUSTTC PRCEE.
FUNCTION FIDTXT
1F1X‘.O11 10192

1 FIG-1.6X‘02I2.*‘2
GO TC 30

2 TFTX-STICDZOvZC

2C FTC'EXPTXTISORT(Z.*3.16159‘X)‘11.+1.18.IX1
GO TO 36

1C F1C'1.§X..ZIZ.“Z
0X“AI12¢“§‘2.“21
0X"bl12..‘b‘13..2.1.*21
0X“DI(Z..‘DOTA..3..Z.T..ZT
9X‘.1C/(Z..‘1D‘15..A.’3.’2.1.‘21
9X‘.12I(Zo..12’16..5..§.‘3o.201..21
9X“1A/12.‘.14.T7.‘6.‘5o‘§..3..2.1“21
30 RETURN
END
C GO ON
FUNCTTDN FTGZ1OTHTTCHDGZOA1DAZDHIIHZv
1H910HPZIGZDIFFT '
COFFCB ACAHHA
COPPER "TEST
YI'AGAHPA'GI
TFTTHTTCHT 13013315
13 D‘1o'AZ‘EXPT1o'F1C1Y1TT
TFTDT 18910323
23 F'SQRTTDT
GO TO 16
15 D'l.'A1‘EXP11.‘F101Y111
TFTDT 10913925
25 F'SORTTDT

OUJ‘UNF’

GO TC 16
18 F'Co
16 RETURA
END

FUNCTION FTNTTXvGAHHAvETAT
Yl'GAFHA..007'0AHHA‘X
F1NT'EXPTl.-F10|YIT1.1.007-X1
RETUPB
END

C STNOLE PRECTSTONPRODRAH
CO'HCN AGAHHA
COPFCA FTEST
DTFENSTCN AN2110C1
DIPENSTCA DTFFTZT
DTPEASTCN VTISOTIANTISDT
REA015001 NSET
READSSO.’ N00319020019110U1TpHZToNHARH
READTSo‘T AGAHHAOCOEFF
REA0150.1 PF10PHZDHPTHS
GZlT'C.
DGZlF'.01

9A9

G50

€30

73C
72

701

700
7C3

702

O\Ibcufia~

Odflml~m-

l

1

188

dRITEIprAQ) NSET

FORMATIIIoZXt'NUHBER OF DATA SET FOR 81C 0 '913'II1
HRITEIbuQSOI NHARNpPHlaPHZoHPTHS

FORHATIZXp'THIS IS AN ANALYSIS CF THE ELECTRON DENSITY'v/p

ZXD'IN A CYLINDRICAL PLASPA COLLFN BASED CN A'DID
ZXo'BESSEL FUNCTION PROFILE APPROXIHATION'DIO
ZXD'USINO THERFAL RESONANCES 1 AND '0139”:
ZXp'TOTAL PHASE FOR FIRST RES IS PI TINES '9FQ.20/o
Zfip'TOTAL PHASE FOR SEC RES IS PI TIHES 'vFfioZOIIO
ZXI.THE SQUARE OF UP OVER H ' '9F§.ZDIIII

HRITE1609301 021I'DGZ1F9NU0N19NZODIPI10N1IDHZIINHARH

FORMATIZXo'LOHER INT. LIHIT "9E15ofivlo
ZXI'INITIAL INCR. IN 21 "lE15voI0
ZXO'NUHBER OF INTEGR. STEPS "91159’0
ZXD.VALUE CF H1 "oE15oAv/u
ZXD'VALUE OF U2 “oE15.Ao/t
ZXO'OIPOLE CbRRENT AT HI 3'9E15.40I0
ZXI'CURRENT AT H1 "OE15.AOII
ZXO'CURRENT AT H2 "9E190Anlo
ZXu'NUHBER T. D. RESONANCE 2N0 N I'DIISOIT

AG'AGAHHA

DGZIFI.CI

GAPA'AGAPPA/a007

EH'QollE‘31

EPS'E.85E'12

0'1.EC2E’19

ANCI'?.‘N1‘.2‘EH‘EPSIO“Z‘U1IIOIPI1
’3..NPTHS

ANCI'ANCI‘CCEFF

ANCZ'ao‘HIFRZ‘EH‘EPSIQ.‘2.N2IIOIPII
I3.FHPTHS

ANOZ'ANCZ‘CCEFF

HPI'SCRTICFAZ‘ANOIIEHIEPSI

HPZ'SORTICF‘Z‘ANOZIENIEPSI

AIIHPIF‘ZIHIRRZ

A2'UP2“2/U2.‘2

Gll"CGZlF/2.

H01

00 1C I'Foz

021-52100021F

DOZ'GZIIIC.

522‘021

022'0229052

YIOAGAHNA'GII

YZ'AGAHPA‘GZZ

DClT'Al‘ElPII-‘FIOI'III-AZ‘EXPI[o‘FICIYZII

IFIDGZT170097029701

IFICGZ'GZIIQDoI 70207020703

022-022-002

DOI'DGZIIC.

GO TC 701

CONTINUE

IUITCFI'I

UL'CZZ

CALL INTIGZIIvNUnULnAINT'GZIOHIOHZDHPIDHPZDAIDAZDIHITCH'GZDIFFI

AINTI'AINT

IUITCFil

UL'GZI

1C

4C
9C

ICC

711
721

720

7A1

74C

879

1

1
2

1

macaw-

189

CALL INTICZ1InNUoULoAINTuCZIohI0U20HPIOHPZOA19AZIIHITCFIGICIFFI
AIATZ'AINT
DIFFIII'AINTI'Hl/HZ‘AINTZ‘
PHZIPHI
IFIGZI'A.) E0n609¢1
HRITEIOOTSQI CAHA'COEFF
FURNATIZXO'FOR CAHA ' '9E15.inv
ZXI'ANO COEFF ' '0E15oAn/o
ZXv'THE DIFFERENCE DIVERGES FOR ALL POSITIVE GZl'vI/I
IFIDIEFIII’DIFFIZII AODZO'ZC
ERR-021
H'Z
DIFFIII‘DIFFIZI
GO TO 30
IFIDCZIF-.01I 100990990
521'621‘DC21F
DGllE-DGZIFIICo
GO TO 30
CONTIBUE
CAT"CZI‘.OCOC1
DGAT'GZI
CAT'GATOCGAT
Y3'CAT-GZI
21'FIC1731'I1c‘ALOOI1.IAII)
IFIZII 71Co711'71&
IFIDGAT'021I9.I 711,711'713
GAI'GAT‘DCAT
DGAT'DGATI10.
GO TC 71C
CONTINUE
IEIABSICAT-AGAHHAI-ABSIGATISC.11 720:72Cv721
AGANFA'IAGAFHAOGATIIZo
GO TC 73C
CONTINUE
BCCNST'1038E‘23
YY'ACAHPA’CZI
ETA'1.'FICIAGANHA1
BZ'AZ
81‘A1
lZTOlI'GlZ/GZI
TE’HI“2‘EH‘AINTZ‘*2/GAHA**ZI3.14159‘FZ/3olBCCNSI
VHALL'ETA‘BCONST‘TEIQ
211‘C21/CAHA
ZZZ'GZZICAHA
CALL INTEIETADGAHAvSI
COEFFT'.0C7“2/2.IS
IFIABSICCEEF'COEEFTI‘ABSICOEEFIZC.II 74C'7ACD741
COEFF'ICOEFF‘COEFFTI’Z.
00 TC 72
CONTINUE
HRITEIboe791 COEFETOGAHADETA'5219022022TC210HP19HP20ANC1IAFC20
A19A20211'222'VHALL0TE
FORMATIZXv'COEFF PEAK TO AVG EL DENS
2X9'GANHA
ZXD'ETA I VHALL TO KT EVER 0
ZXI'GAHHA TIHES 21
ZXt'OAHNA TIHES 22
ZXD'ZZ TO 21

.OE1SQ‘DIO
'0E15.A0I9
.DE150‘AID
'pElfivolo
'uE15.AoIn
'0E15.§0ID

190

6 2X9.NP1 ' .0E15ofip’9
7 ZXv'hPZ ' 'DE15ofioIo
8 ZXD'NOI ' 'OEI5ofioI0
1 ZXO.NOZ ' 'pE15.§oI0
2 ZXO'AI ' UPI OVER HI SCLARED ' 'DE1saAoIn
3 2X0'A2 ' HPZ OVER H2 SQUAREO ' '9E15ofitlo
6 2X1'21 ' '0E15.4ulp
5 2x.'zz U '.El$.4.l.
b ZXO'VNALL ‘ ' '0E19000/9
7 ZXD'ELECTRON TEMPERATURE I '0E1504'I'
8 III I

OZ'.CC7IZ§.

2"02

DO 60C 1.1026

2.2952

R1'AGAHHA'GAHA.Z

VIII'I.’FIOIR11

ANZIII'ANOZ‘EXPIVIIII
ECC ANIII'ANCI’EXPIVIIII

CALL PLOTAIVOANDZbI

CALL PLCTZIANOAN29261
777 STOP

END

SUBRCLTINE INTIXIDN'XFDSOGZ10IIDNZIUP1'NP2'A1'A2'INITCH'OICTFFI

DIFENSICN X131

COHHCN ACANNA

COPPER PTEST

NINI2‘291

XN'N

DX'IXF-XIIIIXN-1oI

NCCUNT'C

XIII'AI'Zo’DX

XIZI'fil'DX

XI31'XI

S'C.

DO 10 I'3vN32

XIII'XIIIOZo‘OX

XIZI-XI2192o‘DX

XI31'XI3102.‘0X .

DSIFIGZIDINITCHoXIlItALvAZobIDHZINPI'HP29GZDIFF1

1+Ao‘FICZIDIIITCH0XI210A10A29HIOHZDNPIvHPZOGZDIFFI

ZOFIGZIOINITCHOXI310A1vAZoHIDUZvNPIDHPZvCIDIEFI
1C S'S’Dfil3o‘DS
AC RETURN

END

SUDRCLTINE INTEIETAvOAHHAvSI

DIPENSICN X131

N'ZO

N'N/2.2+1

XN'N

x1'00

XF.O7CE-2

DX'IXF-XIIIIXN-1.1

NCCUNT'C

XIII'XI'Zo‘DX

X121'71‘Ol

XI3I'XI

$.00

191

DO 1C I'31N'2
XIII-X11102.*DX
XIZI'XI2I‘Zo‘OX
XIBI'AI3IAZo‘DX
DS'FIATIXII1:5AFHA0ETA1’A.‘FIAT(X1219
1 CAHPA'ETAI ‘FINTIXI319GAHHAuETA1
1C S‘S‘DXI3o‘DS
AC RETURN
ENC

192

00.. Pic JACK OLIN BSSR

ILCAD HAIFIV

ICPT NOSOURCE
cooOtotHIS PROGRAR IS DESIGNED IO DEIERPINE IPE PARAHEIERS
CRRR‘RDF A PARABOLIC tLECtRCNDENSlIY FRCFILE BASED ON THERMAL
COOOOoRESONANCE DATA OBTAINED HITH AN ELLECIROACOUSTIC PROBE.

13
23

15

18
16

17

94.9

SEC

97C

HA1

(”NOW-bulwa-

\IOUNI-

FUNCTICN EINTIRTAI
COPHCA FT
COFHCN AICDAI10AIZ'RITA
COPHON ALFAnaETA
COPHCD A
IEINTT 15:15:13
0'10'F1ll‘10.30’110‘oS‘ALFA7.110-‘LFA.RTA“Z”BET‘..2
IFIDI 18018v23
FINT-SORTIDI
GO TC 16
D'1.'A12IAID‘3.II1o“.5‘ALFA1‘11.‘ALFA‘RTA“21/OETA.‘2
IFIDI 18918025
FINT'SORTIDI
GO TC 16
FINT'Co
RETURB
ENC
N PRCGRAP
DIPENSION DIFFIZI
DIPEDSICN AN110019ETAR1100)
COPPER HT
COPPER AICDAI1DA129R1TA
COFHCB ALFAnBETA
CONHCF A
CONTIBUE
READISA‘I NSET
READISI‘I AID-AI1'AIZDH9RADIUS'AHARH
READISO‘I DRITADRITAI'OETA
REAC‘SO.7 RHIDPHZDHPTHS
HRITEIODSAQI NSET
FORMATIIIIZXp'NUNBER OF DATA SET ‘ .013'IIT
HRITEIODQSOI NHARHQPP10PHZ'HPTHS
FORMATIZXI'THIS IS AN ANALYSIS CF THE ELECTRCN DENSITY '0’,

ZXA'IN A CYLINDRICAL PLASPA COLUFN BASED CN A '0’:
ZXv'PARABOLIC DENSITY PRCFILE APPROXIVATICN '0],
ZXp'USINC RESONANCES 1 AND .913!!!
’9
ZXI'THE PHASE FOR RESONANCE 1 IS PI TIHES .DF‘QZDID
219'THE PHASE FOR RESCDANCE 2 IS PI TIHES 'DFA.ZOID
I9
2A9'THE SQUARE OF hPCVER 5 IS ECLAL TC 'DFA.2!/’/1
URITE1609701 AID-AI10A129H98ETA'RADIUS
FORMATIZXO'ID ' °1E15ofivlv
ZXD.11 ' '0E15o40l9
ZXD'TZ ' '0EI§.A'ID
ZXD'N ' '9E15oADIp
ZXv'DETA'ATOR U '9E15.§9I0
ZXp'RADIUS ' 'DE15.§'III7
A'BETA‘RACIUS
R1TA'R1TAI
fit!

00 1C IIPpZ

193

1 R1TAIR1TA§DR1TA
ALFA'110-P10’HPTHS,A11‘BETA.‘ZII1R1TA‘.2'A10’60’A11‘OET‘..27
ADIII-IALFA'I1.IALFA'.5IIHPTNSPAICIAIZ’BETAPRZ
ADZI1clALFA’11.IALFA'oSIIUPTHSPAID/A11’DETA.’2
IEIADII 19192

2 IFIADZI 19193

DRTAISORTIAD1I’SORTIAOZI

HTI'I

UL'DRTA‘RITA

CALL 1NT1UL9A1NTI

AINTIIAINT

NTII

ULIRITA

CALL INTILL9AINT1

AINTZIAINT

1C DIFFIIIIAINTI’AINTZ‘PHZIPH1
IFIRITA'2.I 5C951951

h)

51 URITEIboGEOI
98C F0RHATIZX9'DIFFERENCE DIVERGES'9/11
GO TC 52
5C IFIDIFFIIT‘DIFFIZ)! AC92092€
20 ERR-RITA
DIFFIIIIDIFFIZI
HIZ
GO TO 3C
40 IFIABSIRITA-ERR1-.011 10099099C
9C RlTA-RITA-DRITA
DRITA-DRITAIlD.
GO TC 30

ICC R1IR1TA‘A011.IBETA-1.IIA
RZIRIODRTAIA+API1.I8ETA'1.1
21IA-Rl
22IA-RZ
ANCDIbPTHS/I1.‘.5‘ALFA1.8.85E-12‘9.115"![1.602E‘19‘.2
1 ‘H‘IZ
ANOlIANOC/AID‘AII
ANCZIANCO/AID‘AIZ
ETEHP'9.11E-31/3./1.386‘23‘H“2l3.16159.92
1 *(AINTZ‘A1“2
HRITEIb99851 ALFA
C85 FORMATIZX9'ALFA I '9E15.A9//1
COEFF '10/110’ALFAIZOI
ZZTCZIIZZIZI
ETA-ALOGIlo-ALFAI
VHALLIETA'I.386'23IETEHP/1.6C2E'19
HRITE1699601 919R29219229ANOD9ABC19AN02922T0219CCEFF9VIALL9

1 ETA9ETEHP

960 FORNAT12X9'R1 I '9E15oA9/9
1 2X9‘R2 I '9E15.69I9
2 2X9'21 I '9E15009I9
3 2X9'22 I '9E15gfi9l9
A 2X9'N0 DIPOLE I '9E15oQ9/9
5 2X9'N0 1 RESONANCE I '9E15.§9I9
7 2X9'N0 Z RESONANCE I '9E15ofi9’9
8 2X9'22 TO 21 I '9E15ofi9/9
9 2X9'PEAK TO AVERAGE I '9E15.A9/9
9 2X9'V hALL I '9E15.§9/9
9 2X9'ETAIVH TO KTTOQ I .DE150‘9’9

pppp—ppppppppppp

80C

\R

1c
£0

194

an'ELECTRGN TEHP - '9E15.4:III)
DZ-RADILSIZS.
ZI-OZ
DO BCC 1'1926
Z-ZOCZ
ANIID-ANOIIIl.-ALFA‘(1IRADIUSDIIZI
EYARIII-ALOGIl.-ALFA‘(l/RADIUS)II2)
COhTIhUE
CALL PLCTAIETARoANprI
GO TC 17
SICP
END ,
SUBRCLTINE lNTIXI-S)
DIPENSICN XI31
COPHON HT
COPHCB A109A119A129R1TA
COPHCA ALFApBETA
CUPPLB A
‘F.10
NISO
NsN/20241
XN-N
DXIIXF-XIIIIXN-l.)
NCCUNT-C
xxt»-xl-2.90x
xt2)-)l-Dx
X(31-¥1
SIC.
DO 10 1I39N92
XIII-XIIIAZ.IDX
XIZ)-X(21*Z.IDX
XIBD-XI3102.IDX
DS-FIBT(X(11)+6.IF1NT(X(21)AFIKT(X(31)
s-Soox13.005
RETURN
ENC

u-‘o-‘___“-_~—“~F~F”~—-—~FFPp—pp

tttt

195

FOZGRF JACK OLIN BSSR

ILCAD HATFIV

C
C
C

90

91
S)
96

99
ICC

1"-

THIS SUBROUTINE PLOTS THC VARIABLES ON THE SAFE PLCT
hITH THE ZERO AXIS AS TEE CENTER IFAX VALUES ARE
CALCLLATED AUTOMATICALLY FOR V - ZNAX I YNAX
SUERCLTIKE FLOTZIYoloNI
OIPEBSICN CCLIIOZI:YIIOOI.Z(ICCI
INTEGER STAR900198LANK9COL9PLLS
STARI'. '

STARI 1547716624
DOTI'. '

DOTI 1262501952
BLANKI' '

BLANRI lC7795257b
PLUSI'A '

PLUSI 1312633600
XAXAII'X ‘

XXXXXI -4152196A8

YHAXIC.CC

ZHAXI(.CC

DO 95 KII9N
XIABSIYIKII-ABSIT"AXI
1‘11193995993
YNAAIYIKI

CONTIDUE
VHAXIAOSIVNAXI

DO ICC L‘I9N
OIABSIZILII‘ABSIZHAXI
IFIOI 1CC91CO999
ZHAX'ZILI

CONTINUE
ZHAXIAOSIZMAXI
WRITE1692COI THAXoZHAX
IFIZFAX‘YFAXI 70971971
YHAXIIHAX

ZHAXITHAX
FORHATIIII918X9'PXNAX I'9EIQ.095X9'*YHAX I'9E1A.6943X9'X’91CX9'Y'I
WRITE1694COI

FURNATI'I'I

URITE169ZI
FORH‘IO'....O.........0.....‘........0‘...................

00.0.0.0..0000OOI00......O0.0.0...0.0.0.0.;000000..’

DO 3 1'19101

COLIII I BLANK
COLISIIIDCT

IIIA

DO A III:A
JI50.'IVIIIIYHAXOI.)+I.5
KISD.‘IZIIIIZHAX+1.I+I.5
COLIJI I STAR
COLIKIIPLLS
HRITEIboSIICOL(IJIoIJI191C119Y111921II
F0RHATIIX9ICIA19IPZqulI
COLIJIIBLANK
COLTKIIBLANK

IFII’III 25C93009300
COLISIIIXXXXX

300

250

990

196

c COLIJIIPLANK
a COLIKIIELANK

IFII-III 25C930093OO
COLIACIIXXXXX
IIIIIAS

GO TC A

COLIAEIIDCT

CONTINUE
HRITEIonGOI
FORMATIIIII/IIII/I
RETURN

END

00‘.
[LOAD
C
C
C

90

9?
s:
96

99
ICC

1t)

32
35

42
45

197

EOZORF JACK OLIN BSSR
HATEIV
THIS SUBROUTINE PLOTS THC VARIABLES ON THE SAME PLCT
NIT" THE ZERO AXIS AS TEE CENTER ’HAX VALUES ARE

CALCLLATED AUTOMATICALLY FOR Y ' ZHAX I YHAX '
SUEROLTINE ELOTZIY929NI

DIFENSICN CCL110219Y110019211COI

INTEGER STAR9OOT98LANK9CDL9PLUS

STARI'I '

STARI 19Q7714024

DOT... .

DOTI 1262301952

BLANKI' '

BLANKI 1C77S52576

PLUSI'9 '

PLUSI 1312033¢00

XIXX’I'X '

XXXXXI ‘515219658

YHAx-C.CC

ZHAXIC.CC

DO 95 KIloN

XIABSIYIKII'AOSIT"AXI

IFIXI 95995993

YNAXIYIKI

CONTINUE

YHAXIADSIYNAXI

DO ICC LI19N

QIAOSIIILIIIADSIZHAXI

IFIQI 1CC9ICD999

ZHAXIIILI

CONTINUE

2HAXIADSIZNAXI

HRITEIO9ZCOI YNAX92HAX

IEIZFAX‘YPAXI 70971971

YHAXIZNAX

IHAXIYHAX

EORNATI/II918X9‘IANAX I'9E1§.O95X9'*YNAX I'9E14ob943X9'X'9ICX9'Y'1
URITE1694COI

FORHATI'I'I

NR1TE169ZI
Fonn’TO....COOOOOC....00....‘C...0.........................
1 9 '
1......00000.....0....‘0....0.0.‘.....O...’.........‘.’
DO 3 1'19101

COLIII I BLANK

COLISIIIDCT

IIIA

DO A 1I19N

JISO.PIYIIIIYNAX*I.I*I.5
KI50.IIZIIIIZHAx+I.I+I.5

COLIJI I STAR

COLIKIIPLLS
HRITE169511COLIIJI9IJI191C179YIII921II
FORMAT11X9101A191P2E9.11

COLIJIIDLANK

COL1NIIDLANK

IEII‘III 25C93009300

COLISIIIXXXXX

198

IIIIIIS

GO TC '0
COLISIIICCT
CONTINUE
HRITEIO-977)
FURNATI/IIII/IIIII
RETURN

END

199

IIII HKB JACK OLIN 855R

ISYS TIME-1C

ILCAD HATFTV

IOPT NOSCURCE

(cooooTHlS PRCGRAH PLCTS THERMAL RESCNALCES BASEC LN A hKB
CIIIIOAPFRCXIHATICN AWAY FROM THE CRITICAL POINT FOR A GIVEN
C‘III‘BESScL SERIES ELECTRON DENSITY PRCFILE

1C

23

16
lb

(T‘U'IJ‘LLJNP-

FUNCTICN FICIXI

IFIK‘.OII 19197

FIC=1.IX“2/2.‘.2

2U TC 5C

IFEX’511C92C'21)

FICIEXPIII/SCITIZ.‘3.IQISQIXI°II.*I./3./XI

GO TC 3C

EICIlo‘X.‘2/2.“2
OXIIAIIZ.ҤI2.II21
IXIIbIIZ."b’I3-‘2.I*‘21
OXIIBIIE...8.14.‘3..2.IIIZI
RXIRICIIZ.“ICII5.I4.‘3.‘2.I.‘21
IXIIIZI12o“12‘16-‘5o‘§o‘3o.2.I.*ZI
IXRRIA/12.I.1A‘(7.‘b.‘5.‘4.‘3.‘2.1“21

RETURN

ENC

FUNCTION FIX)

CONPLN ANCDOAMNA9A9H9TEHP9EN9EPS'CI8CONST

YI A‘OAHHA'GANHA‘X

DII.‘1.IH’IZICIIZIEN/EPS‘ANO‘EXPIIo-FIOIYII

IFICI 18918923

FISORIICI‘H/SORT(3.‘BCONST*TEVP/EFI

GO TO 16

FIC.

RETURN

END

C"“‘NAIN PRCGRA"

C96

997

998

I

NOUIl-ri-I

DINENSICN BETAPIICOI9ANIIIOCI

CUHHCN ANCIGANMA9A9H9TEHP9EH9EPS9C9OCONST

READISO‘I NSET9NRES9ANO9GAHHA9H9TEPP92P

READISD‘I A9N9H

RE‘0159.7 TEST

URITE169QQ61 NSEToNRES

FORMATIII92X9'SET NUHBER IS '9139/9
2X9'RES NUMBER IS .9139/II

HRITEIO99§7T ANU9OAHHA9A9U9TEPP92P9N3ET

FORPATIZXI'NO I '9E15o‘9/9
2X9'OAH"A I '9E15.A9I9
2X9'RADIUS I '9E15.A9/9
2X9'RADIAN FREQUENCY I '9E15oA9/9
2X9'TENPERATURE I '9El5oA9/9
2X9'2 CRITICAL I '9E15.‘9/9
ZXO'NUHBER OF DATA SET I '9139/9
III

NRITE1699987

FORMATIZX9'DISTANCE FRO" HALL'93X9'PERTLROEO ELECTR CENSITY'vllI
EHI9.11E'31

EPSID.85E'12

OII.bCZE‘19

BCCNSTIIoBSE‘23

KNNIN

\'

1C
AC

200

JZIZNIIXNN'IoI

l"DZ

00 1C 1319N

ZIZIDZ

CALL INTIZI7NIS9MI

Y-A‘GAHFA-GAHNA*2
D'Io‘I.INRIZIOIIZIEH/EPSIANC’EXPI1.‘FIOIYII
IEIDT IC9§09§1

BETAPIIIICo

GO TO SC
DETAPTIIISORTIDTIN/SORTI3.‘OCCNSTRTENPIENI
IEIBETAPIII‘TEST160960961

ANIIITIC.

GO TC 62
ANIIIII1./SCRTIBETAPIIII‘SINISoIIISQIAo‘SI
URITE1699991 29AN1117

CONTINUE

CALL PLOT‘IAN19AN19NI

STCP

END

SUORCLTINE TNTIXI9XF9S9NI

DIVENSICN X137

COFNCN ANC9GAHHA9A9H9TENP9EP9EPS9C98CONST
N‘ZC

NIN/ZRZII

XNIN

DXIIrF-XIT/IXN-1.1

NCCUNTIC

XTITIXI‘Zo'DX

XIZIIXI‘CX

XI3IIFI

SIC.

DO 10 1'39N92

XIITIATITIZ.‘DX

XIZTIAIZT‘ZoRDX

XISIIXI3TIZoRDX
DSIFIXIIIIII.RFIXIZII*FIXI3II
SISICA/3.RDS

RETURN

END

201

.... UKBPAW JACK UL‘N BSSQ
ISYS TIHE'IC
[LOAD HATFIV
ICPT NOSULPCE
C..’.‘THIS FRCGRAp PLOTS THERHAL RESCBANCES EISEU CN A “KB
C.‘...APPRCXIFATTCN AHAY FRO" THE CRTTTCAL POINT FCR A GTVEN
COOtOOPAPABCLlC ELECTRCN DENSITY FFCFILE
FUNCTION FTXT
COPPER ANC.ALFA.A:H.I£HP.EHpEPS'OpBCCNST
Z'x
O‘lo'l.IU‘*2‘C"(/tH/EPS*ANC“1o‘ALFfi‘(l.“Z/A)“Z)
IF(D’ 18913023

(3 F=SC91(C)‘u/SCHI(3.‘BCCNSTtTth/EP)
GU 1C 10

18 Pic.

1t RETLRN
END

C.‘..‘HAIN PRCGRAH
DIVERSICN BETAP(IOO’9ANI(IOCT
CUHHCR ANCoALFADAvHDTEHPOEH'EPSOODBCUNST
pE‘O‘5'.) NSETDNRESOANUOALF‘phnTEFPrZN
READ‘59‘, AoNpH.
REflOC5o.) TEST
WRITETODQQb’ NSEToNRES

996 FURNAT‘IIDZXD'SET NUHBER IS .0139]!

l 2X0'RES NUHBEQ IS '913'19 ~
2 va'PARABULIC APPROXIHATXCN 0F PRUFILE'DIIT
HRXTEC60997’ ANUDALFADA'H'TEHPOZHvNSET

997 FDRHAT(ZXO'NU ' .0515o‘plp
1 ZXO'ILFA 3 'vEl5afivIv
2 ZXp'RACIUS 3 '9515.Qp/D
3 ZXp'RADIAN FREQUENCY ' '9515.49/9
6 ZXo'TEHPtRATURE. ' 'pEl5-‘nln
5 ZXa'l CRTTICAL * 'iElS.‘v/0
t ZXD'hUHBER OF DATA SET I '013010
7 [IT

HRITE‘vaQS)
998 FURNAT‘ZXo'OISTANCE FRUH UALL'93X0'PERTURBEO ELECTR DENSITY'I”.
EHtG.llE-31
EPS'E.35£‘12
O‘loéCZE‘lq
BCUNST'lo3BE'23
XNN'N
OlllFITxhh-loi
1"02
00 1C T'lvN
231*C1
CALL TNT‘ZDZH051H,
D'lo‘loIU“2‘Q..Z/EHIEPS‘ANC‘(lo‘ALFA‘Tlo‘l/A)“ZT
IF(0, 4C940'Ql

fiC BETAP‘IT'Co
GU TC 5C
41 BETAPTII'SQRT(D)‘HISQRT(3g‘BCUNST‘TEHP/EH)
50 IF‘BETAP‘IT-TEST’60'60:6I
50 TC 62
Cl ANI‘IT'lo/SORTTBETAPTI”‘SINK3.14159/4.0S)

t2 HRITElboGGQT ZoANltl)

<59
10

\fl

1C
6C

202

FUR"IT(2’OE[SobprOEl5ob’
CONTINUE

CALL FLUT4(‘Nl-AN10N!
5TCP

tNL

SUfiiiLTINE TNTTKI'XFn59N)
JIFENSILN 1(3)

CUFHGh LAD.ALFA.A.upTEHP-E”-EF3prQC£NST
N'ZC

NIH/2.2‘l

XN'N

OX'TXF-Xl’/(XN'1.’
NCCUNT'C

XTIT'7I'2o.0X

X'ZT'}I'CI

X(3)*)I

5'0.

00 1C ['3thZ
X(l)*’(l)02.‘DX
X‘2)'X(Z)9Z.‘DX
X(3)'X(3l‘2.‘DX
DS'F!)(l!T+§.*F(X(2))*F(X(3)i
S'S‘CXI3.‘US

PETURB

END

REFERENCES

6

1O

11

1?

REFERENCES

B. S. Tanenbaum, "Plasma Physics", McGraw - Hill 00.,
N. Y., 1967. '

A. Dattner, "Plasma Resonance," Ericsson Tech.,
Vol. 3, No. 2, pp. 309-350, 1957.

"Resonances Oscillations in a Hot Nonuniform Plasma",
The Physics of Fluids, Vol. 7, No. 9, pp. 1489-1500,
Sep. 1964.

P. a. Vandenplas, "Electron Waves and Resonances in
Rounded Plasma", Interscience Publishers, N. Y., 1968.

F. w. Crawford and G. S. Kino, "The Mechanism of Tonks-
Dattner Resonances of a Discharge Column", Proc. of

the Sixth International Conference on Ionization
Phenomena in Gases, Paris, July 1963.

B. Ho and K. M. Chen, "Electroacoustic Resonance in
Plasma Layer Surrounding a Metallic Cylinder",
Proc. of IEEE, Vol. 56, No. 9, pp. 1600-1602, Sept. 1968.

P. D. Golden and E. J. Yadlowsky, "Plasma Resonance and
Standing Longitudinal Electron Waves", The Physics of
Fluids, Vol. 14, No. 9, pp. 1990-1996, Sept. 1971.

A. W. Baird III, "Determination of Electron Density
Profiles from Tonks—Dattner Resonance Data in Plasmas",
J. of Appl. Phys., Vol. 42, No. 13, pp. 5358-5361.

Apr. 1971.

J. V. Parker, "Collisionless Plasma Sheath in
Cylindrical Geometry", The Physics of Fluids, Vol. 6,
Research Notes, pp. 1657-1658, Nov. 1963.

F. W. Crawford, G. S. Kino, and A. B. Cannara, "Dipole
Resonances of a Plasma in a Magnetic Field", J. of
Appl. Phys., Vol. 34, No. 11, pp. 3168-3175, Nov. 1963.

J. L. Powell, and B. Crasemann, "Quantum Mechanics", Addison Weslev
Publ. Co., Inc., Reading, Mass., 1961.

C. Y. Lee, "Electromagnetic Scattering from a Plasma-
Coated Cylinder", Ph.D. Thesis, M.S.U., 1971.

204

13

14

205

P. H. Vandenplas, "Oscillations de plasmas finis,
inhomogenes et de temperature non nulls", Ph. D.
Thesis, Universite de Bruxelles, April 1961.

K. J. Parblakar and Brian C. Gregory, "Direct
Measurement of Dipolar Radial Electric Fields in a
resonating Plasma Column", The Physics of Fluids,
Vol. 14, No. 9, pp. 1984-1989, September 1971-