ELECTRON CONDUCTIVITY MEASUREMENTS IN THE AFTERGLOW
OF A HELIUM DISCHADGE
by
LAWRENCE GOULD
B.S., Massachusetts Institute of Technology
(1950)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF
PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF
TECHNOLOGI
January, 1954
Signature redacted
Signature
Department/
Physics, January 11, 1954
Signature redacted
Certified
Signature redactedTheis
Accepted
by............
.*..............
Spevi
............
Chairman, Departmental Committee
on Graduate Students
ELECTRON CONDUCTIVITY MEASUREMENTS IN THE AFTERGLOW
OF A HELIUM DISCHARGE
by
LAWRENCE GOULD
Submitted to the Department of Physics on January 11, 1954
in partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
ABSTRACT
A microwave method previously reported for determining the
collision probability for momentum transfer of slow electrons has
been modified so that a variation in average electron energy from
0.012 to 3 electron volts may be obtained.
Measurements of the ratio
of the real part to the imaginary part of the electron conductivity
are performed in the afterglow of a pulsed helium discharge in a microwave resonant cavity.
The collision probability can then be inter-
preted from the conductivity measurements, provided that the experimental conditions are such that the electron energy distribution
function is known.
The electron conductivity measurements are obtained by
measuring the ratio of the microwave power transmitted through a
cavity to the power incident as a function of signal frequency in the
vicinity of cavity resonance.
The method described involves balancing
the transmitted and incident signals to zero at the cavity resonance
ii.
after they have passed through two separate receiving systems.
The
change in frequency from resonance and the corresponding change in
attenuation necessary to rebalance the two signals gives the required
data which will plot as a straight line whose slope yields the desired
information.
The average electron energy is varied by applying a microwave electric field in the afterglow, and, under appropriate assumptions, the average electron energy is determined theoretically from
this field.
Measurements from 0.012 to 0.068 electron volts are also
obtained by varying the gas temperature from 950K to 400 0K.
The
electron conductivity in the afterglow is studied as a function of
experimental parameters and the effects of pressure, electron energy,
impurities in the gas, ambipolar diffusion, non-uniform heating electric fields, and energy gradients are investigated.
The important
result of this investigation is that the value of the collision
probability for mementum transfer in helium is 18.3
2/0
cm2 /cm3
per mm Hg from 0 to 0.75 electron volts and increases slowly to a
peak value of 19.2 + 20/o at 2.2 electron volts.
Thesis Supervisor:
Title:
Sanborn C. Brown
Associate Professor of Physics
ACKNOWLEDGMENT
The author wishes to express his appreciation for the
continued guidance and encouragement by his thesis supervisor,
Professor S. C. Brown.
The assistance generously given by all
members of the Microwave Gas Discharge Group, in particular,
Mr. J. J. McCarthy and Mr. J. E. Coyle is gratefully acknowledged.
This investigation would not have been possible without the
generous financial assistance and excellent laboratory facilities
supplied by the Physics Department and the Research Laboratory of
Electronics.
l' I- a. umiin
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iii.
TABLE OF CONTENTS
Page
LIST OF FIGURES................................................ v
CHAPTER I
INTRODUCTION.......................................
1
CHAPTER II
ELECTRON CONDUCTIVITY IN THE AFTERGLOW....,......
7
II-1.
Electron Energy Distribution Function....... 9
11-2.
Electron Density Distribution........... .... 12
11-3.
Electron Conductivity Ratio for Constant
Collision Cross Section..............
CHAPTER III
22
THEORY OF ELECTRON CONDUCTIVITY MEASUREMENTS....... 29
III-1. Electron Conductivity by the Transmission
Method......
.......
.......
..........g.......
29
111-2. Electron Conductivity by the Resonance
Method..... ... ... ...................... 35
111-3. Comparison of Methods...................... 36
CHAPTER IV
APPARATUS AND PR.D..............
42
IV-l.
Measuring Mode.............................. 42
IV-2.
Power Measuring Section....................
46
IV-3.
Frequency Measuring Section................
52
IV-4.
Breakdown Mode.
53
IV-5.
Heating
IV-6.
Timing Apparatus for Transient Measurements. 55
IV-7.
Vacuum System and Gas Supply................
57
IV-8.
Microwave Cavities.................
60
d.................
54
iv.
Page
CHAPTER V
RESULTS................
..
.
............
65
V-1.
Effect of Impurities on Conductivity
V-2.
Thermal Conductivity Measurements.............. 71
V-3.
Heating Field Conductivity Measurements...... 75
V-4.
Density Decay Measurements..................... 84
V-5.
Proposals for Future Work..............
... 93
APPENDIX I
AVERAGE ELECTRON ENERGY DURING THE AFTERGLOW....... 95
APPENDIX II
SOLUTION OF SECOND ORDER DIFFERENTIAL EQUATION BY
RELAXATION METHOD.................................
BIBLIOGRAPH......................
BIOGRAPHICAL
.............................
SKETCH......................................
102
105
107
V.
LIST OF FIGURES
Page
Figure
2.1
Electron Density vs Position for Various Values of h...... 17
2.2
Electron Pressure vs Position for Various Values of h..... 18
2.3
Electron Density vs Position for Various Values of
Teo/Tg and h = 1 for Vacuum Cavity........................
2.4
20
Electron Density vs Position for Various Values of
Teo/Tg and h = 1 for Quartz Bottle........................ 21
/T
23
2.5
Electron Conductivity Ratio vs T
2.6
Electron Temperature vs Position.......................... 27
3.1
Equivalent Circuit for a Resonant Cavity.................. 30
3.2a
Percent Error in Conductivity Ratio vs Conductivity Ratio
.....................
for 10 db Overcoupled..................................... 38
3.2b
Percent Error in Conductivity Ratio vs Conductivity Ratio
for 10 db Undercoupled................................. 39
4.1
General Block Diagram of Experimental Equipment........... 43
4.2
Ratio of Transmitted to Incident Power as a Function of
4.3
Experimental Arrangement for Measuring Conductivity....... 47
4.4
Waveguide
4.5
Double Waveguide Directional Coupler...................... 50
4.6
Transition Unit from Mixer to I.F. Amplifier.............. 51
4.7
Timing Apparatus for Transient Operation................. 56
4.8
Block Diagram of Vacuum System and Gas Supply............
Mixer...........................................
48
58
vi.
Page
Figure
4.9
Cross Section of Copper Vacuum Cavity................... 61
4.10
Cross Section of Double Copper Cavity................... 64
5.1
Conductivity Ratio vs ?ressure........................... 67
5.2
Conductivity Ratio vs Time in Afterglow................. 68
1 Cr
as a Function of Cavity
Experimental Data of, o
5.3
Temperature.............................................. 72
5.4
5.5
Conductivity Ratio vs Time in Afterglow at T = 770K...... 74
1 ar
as a Function of Electron
Experimental Data of 7
Temperature at Center of Quartz Bottle................... 76
5.6
The Values of Il and 13 vs a............................. 79
5.7
Collision Cross Section as a Function of Electron
5.8
Conductivity Ratio as a Function of Electron Temperature
at Center of Vacuum Cavity............................... 83
5.9
Experimental Data of DaPo vs Pressure.................... 86
5.10
Electron Density vs Time in Afterglow.................... 87
5.11 Electron Density vs Time in Afterglow for Double Cavity.. 92
1.
CHAPTER I
INTRODUCTION
One of the important parameters in studying the interaction
between electron and gas molecules is the effective cross section for
elastic collisions between an electron and a single gas molecule which
depends only upon the atomic structure of the molecule and the kinetic
energy of the free electron.
This fundamental parameter, designated
by qc and expressed in units of area per molecule, can be determined
by measuring one of several related quantities.
For a particular
gas, when the gas density is given and the electron energy distribu.
tion function and the energy dependence of qc can be assumed, some of
the useful quantities which can be derived are the mean free path
the collision frequency (),
(i),
and the effective area for electron
collision with all the atoms in a unit volume at unit pressure
The last quantity, expressed in the units cm2 /cm3
normalized to 00C.
-l
per mm Hg or cm
per mm Hg has been designated by the symbol P c and
called the collision probability.
qC = 0.283 x 10-16 P
The relation between qc and Pc is
when c.g.s. system of units is used.
Monoenergetic beam methods have been utilized by Brode,1
Ramsauer,2,
Kollath,
and Normand5 for measuring the collision prob-
ability as a function of electron energy.
Basically, these methods
measure the decrease in electron beam current due to scattering in
the gas as the beam traverses through a given distance.
The
fractional decrease in beam current is directly related to P.,
2.
However, stray electric fields, contact potentials, and the inability
to obtain strictly monoenergetic beams at low energies are inherent
difficulties of this method which have prevented reliable measurements below 0.5 volts.
For example, the data published by Normand5
give curves showing practically identical shapes for energy dependence
of P
as those published by Ramsauer and Kollath4 but with a 0.4 volt
shift of the voltage scale.
Because of the uncertainty in low energy
measurements, the curves published by Brode
summarizing the accepted
results of various workers do not include data below an electron
energy of about 0.5 volts.
By considering the diffraction effects of the electron
6
wave interacting with the potential field of the atom, Allis and Morse
calculated theoretical curves for the energy dependence of Pc, when
values were assigned to the constants of the potential field in such
a manner as to produce the best fit with the experimuntal data known
at that time.
The shapes of their theoretical curves showed remarkable
good agreement with the experimental plots, thus confirming their
explanation of the "Ramsauer effect".
There is some uncertainty in
the theoretical curves as the electron energy approaches zero, because
both the magnitude of P
and the slope at the low energy limit depend
critically upon the values chosen for the potential constants in
obtaining the empirical fit at higher energies.
For example, the
tentative thermal value of the collision probability as indicated by
their theoretical curve for helium is 20 cm2 /cmA
per mm Hg and the
indicated slope of Pc versus energy is slightly negative.
.
3
Measurements of the diffusion of a swarm of electrons
through a gas under the influence of a constant electric field can
provide information about the collision cross sections of atoms and
molecules toward slow electrons.
performed by Townsend and Bailey,
Huxley and Zaazou.9
Measurements of this type have been
Wahlin,
and more recently, by
In this method, data of the average energy of
and the drift velocity of electrons in the d.c. field are used to
calculate the collision probability for as low a mean energy as 1/15
e.v.
Huxley and Zaazou carried out their calculations for the assump-
tion of a constant collision probability using both a Maxwell and a
Druyvesteyn electron energy distribution function.
However, in order
to obtain definite information about the collision probability, the
electron distribution function must be known.
The dependence of the
distribution function on the variation of the collision probability
with electron energy renders the results somewhat difficult to interpret, since the velocity variation of the collision probability is
itself the desired quantity.
However, if the collision probability
were known, the diffusion technique could be employed to obtain information about attachment and the loss of energy by rotational and
vibrational excitations in collisions between slow electrons and gas
molecules.
Interest in extending the energy range of measured collision
probability values to room temperature and below is enhanced by the
requirements of experiments in ionized plasmas where the collision
probability is an important parameter in the application of distribution
4.
theory and in determining experimental limits for certain techniques.
Margenau and Adler10 have used microwave measurements of
the high frequency conductivity of the positive column in a mercury
discharge to evaluate the mean free path for average electron
energies between about 0.6 and 1.1 volts.
These authors restricted
their discussion to a Maxwellian electron energy distribution function but treated both the cases of constant collision probability and
of constant collision frequency.
Phelps, Fundingsland, and Brown1 1
have described a microwave method for determining the probability of
collision by measuring the conductivity of a decaying plasma after
the electrons reach thermal equilibrium with the gas.
Both the micro-
wave method and the diffusion method measure the probability of collision for momentum transfer12 denoted by Pm.
Pm and Pc is
Fm
r
Pc
The relation between
2
f
10 ()(1
0
r
f f
0
- cos e)sin 9 de d(
1()sin 9 de d$
0
where 1(e)sin e dO d$ is the probability that, on collision, the
electron is scattered into the solid angle, sin 8 d8 do, about (.
The momentum transfer collision probability takes into account the
fact that the effectiveness of collisions in resisting current flow
increases as the scattering angle increases.
The electron beam experi-
ment determines a total collision probability, since electrons are
lost to the beam if they suffer any angular deflection greater than
5.
the angular aperture of the detector.
In general, Pm differs
appreciably from Pc only when there is a pronounced concentration of
scattering in either the background or forward directions.
is a constant, independent of 0, Pm and P
are equal.
If I (0)
For sufficiently
slow electrons, the scattering is independent of angle so, in such
cases, Pm need not be distinguished from Pe.
For most gases the
total collision probability is greater than the collision probability
for momentum transfer, except in helium, where for an electron energy
less than 10 e.v., Pm is less than Pc and for an electron energy
greater than 10 e.v., Pc is larger than Pm.
The values of Pm calcu-
lated from measurements of the distribution in angle of the scattered
electrons usually differ from the total collision probabilities by
a few percent at electron energies below 5 e.v.
The object of this thesis is to measure the collision probability for momentum transfer as a function of electron energy in
helium over as wide an energy range as possible using the microwave
method.
The technique described in this thesis enables the average
electron energy to be varied over a range of 0.012 to 3.3 electron
volts by a combination of thermal heating and cooling, and electrical
heating of the electrons.
An electronic field applied in the after-
glow of a pulsed helium discharge is used to increase the average
electron energy.
The electron conductivity in the afterglow was
studied as a function of experimental parameters and the effects of
pressure, electron energy, impurities in the gas, ambipolar diffusion,
non-uniform heating fields, and energy gradients were investigated.
6.
The condition that the collision frequency be less than the radian
frequency is maintained throughout the experiment, so that the electron energy distribution function is known independent of the variation of Pm with electron energy.
This enabled an expression for Pm
to be found from the experimental data.
In Chapter II the theoretical
expression for the electron conductivity in terms of the collision
probability is discussed taking into account the distribution of
electron velocity and the spatial distribution of the average energy
and the electron density.
The theory of the microwave measurements
of electron conductivity is discussed in Chapter III.
The apparatus
and measuring procedure is described in Chapter IV and the experimental results are summarized in Chapter V.
7.
CHAPTER II
ELECTRON CONDUCTIVITY IN THE AFTERGLOW
13
Margenau
has given a general theory for the behavior of
electrons in a gas under the action of a high frequency field when
only elastic collisions need be considered.
From his results the
complex electron conductivity, oc, may be written as
a
c'= a
2
1+-j
(,
me
=rr+Jai
o
/=
.rne2 2
= J/E = -*
(
m/o)
jvT 3df0
f 1 + (m))2
0
d
(2.1)
21
Here n is the electron density, e and m are the electronic charge and
mass, o is the radian frequency of the applied field, f0 is the first
term in the spherical harmonic expansion of the normalized electron
velocity distribution function for electrons of velocity, v, colliding
with neutral atoms.
The collision frequency for momentum transfer,
, is related to the probability of collision for momentum transfer,
Pm , by
m = Pmpov, where p0 is the pressure normalized to zero degrees
centigrade.
In a microwave cavity, the quantity which is measured4 is
the electron conductivity averaged with respect to the measuring electric field over the volume of the cavity and is given by the relation
fvacE dV
ac = r + Ja
E dV
V
(2.2)
8.
Since Vm is generally a complicated function of velocity, Eq. (2.1)
is difficult to manipulate mathematically.
The assumption that
2m( o over the velocity range covered by the distribution function,
f , simplifies the mathematics.
The ratio of the real part of the
conductivity to the imaginary part divided by the pressure, designated
by
,
is obtained by combining Eqs. (2.1) and (2.2) yielding
-
fn
~1 Cr
Pi
for the case
2
a2 .
V
r
00
J
1
Q0lCO
f f0d
v-Vd
EdV
(23)
J Dn3 dfodE
Cf
V
0
dv E~dV
When the electron distribution function is
independent of position in the cavity, the quantity e is independent
of the spatial variation of electron density and, hence, of averaging
with respect to the measuring field.
For this case, measurements of
as a function of energy can offer information about the velocity
dependence of Pm or f .
For the more complicated case where f0 is
a function of position in the cavity, the spatial variation of the
electron density and the measuring field must be known before any
information about Pm or f may be obtained.
Equation (2.3) and the
associated conductivity measurements are used in this experiment to
obtain the velocity dependence of P over as wide a range of velocity
as possible with the present microwave technique.
Therefore, the
experimental conditions must be arranged so that the electron energy
distribution function is known.
9.
II-1.
Electron Energy Distribution Function
In the afterglow of a pulsed discharge, the electrons whose
average energy is high during the discharge lose their energy through
elastic collisions with the gas atoms.
Eventually the electrons reach
energy equilibrium with the gas atoms and have a distribution function
which is Maxwellian with an electron temperature, Te, the same as the
gas temperature, Tg .
The time required for the electrons to reach
equilibrium with the gas atoms can be estimated from average electron
considerations.
The rate of this energy loss is equal to the product
of the fractional energy loss per collision, which is assumed to be
2m/M for monatomic gases, the average excess energy, u - u , and the
collision frequency,
m
d(-
For helium,
M
mp0
u
) .u4)
and Pm is approximately constant from five
electron volts energy to thermal energy.
From Eq. (2.4) one finds
that the time required for electrons in helium to cool within 20/0 of
thermal energies is 100/p0 microseconds.
In the range of one to ten
mm Hg pressure, conductivity measurements performed at least one millisecond after the discharge has ceased insures energy equilibrium between electrons and gas atoms.
By thermally heating and cooling the cavity, the electron
temperature may be varied over a range of 770K to 7000K and the distribution function will be Maxwellian over this temperature range.
10.
Application of an electric field in a plasma can also increase the
average electron energy.
In general, the energy distribution func-
tion and, hence, the average energy will depend upon a balance between the energy gained from the field and the energy lost due to
recoil with the gas and to energy transported to the walls by diffusion, conduction, and convection currents for the case where the
electron energy is a function of position.
Energy losses due to
inelastic collisions, attachment, and recombination will be neglected.
Margenau1 3 has shown the steady state distribution function for electrons in an atomic gas in the absence of inelastic collisions and large
2
m
diffusion losses to be Maxwellian under the assumption that 2
2
2
The equivalent electron temperature is given by
2
T
e
=T
g
+
(e)E(
(2.5)
6=2xk
where Eh is the applied heating field.
However, in a microwave cavity
the heating field is a function of position and deviations from Eq.
(2.5) due to energy gradients may become important.
The expression for the average energy when the energy is
a function of position is derived in Appendix I and is given by
T = T + a E+
a 0=
and
6mMok
b
=
.
where
. (nVT) + F
6mP
(2.6)
11.
An effective temperature, Te, is defined in terms of the average
electron energy by
i
= 2
2 kT e .
It is seen that the first two terms
which represent thermal energy and energy gained from the field are
identical with those in Eq. (2.5).
The term containing
V
. (nVTe)
represents the change in energy due to conduction and convection of
energy from regions of high energy to regions of low energy.
tion (2.6) is derived under the assumption that P
Equa-
is constant, which
The
is applicable to helium as a good approximation at low energies.
last term F, whose expression in terms of n and Te is given in
Appendix I, represents the energy lost due to the diffusion of electrons in a space charge field.
Calculations show that this energy
loss term is negligible compared to the other terms in Eq. (2.6) and,
hence, will be neglected.
The importance of the energy conduction
and convection term determines whether the distribution function changes
from that given by Ma genau.
This will be discussed more completely
later in the chapter.
In order to interpret Eq. (2.3), one must have knowledge
not only of the energy distribution function, but also the density
distribution.
The temperature spatial distribution given by Eq. (2.5)
will be used as a first approximation to determine the electron density
distribution when the average energy is a function of position.
It
will be assumed that the dominant electron density loss mechanism is
ambipolar diffusion.
12.
11-2.
Electron Density Distribution
The correct density distribution can be obtained from
solving the problem of ambipolar diffusion taking into account the
spatial variation of electron energy.
The equations15 governing the
diffusion of electrons in a space charge field, E,
are
=- VD n_ - pESn_
=
-.VD+n+ + p+Esn+
F
and F
(2.8)
](2.9)
Q.
=
where
(2.7)
are the electron and positive ion particle currents,
n- and n+ the electron and positive ion concentrations, D+ and p+ are
the positive ion diffusion coefficient and mobility, D_ and
_-are
the electron diffusion coefficient and mobility defined by
Dn
47Nrv2dv
=f
Jn~
-d -
4/
(2.10)
(2.11)
drv.
0
For the distribution function defined by Eq. (2.5), D_ and p_ are
In solving Eqs. (2.7), (2.9), and (2.9) the
usual assumptions of ambipolar diffusion are made, i.e.
F
+
functions of position.
13.
and n+
= n.
n
Eliminating E
-
from Eqs. (2.7) and (2.8) one finds
p._DV
D n
(2.12)
p+ +p
Since
+
- and D+ and
+ are essentially independent of position,
Eq. (2.12) becomes
-D [V n +
D-
.t
(2.13)
In order to evaluate Eas. (2.10) and (2.11), the velocity
variation of
m= cp 0
Vm
must be known. 'The simplest assumption is that
,valthough any power series in v may be manipulated
equally as well in the following treatment.
The ratio of V D~n/p_
in Eq. (2.13) is obtained from Eqs. (2.10) and (2.11) using the above
assumption for
m'
V D n
k
-- =
-
T -h/2n
h/
(2.14)
e
Combining Eqs. (2.9), (2.13), and (2.14) and using the relation
D+/ + = kT /e, one obtains
VT1-h/2n
-
- D+V.V
n +
Lt
/
TT-h/2
(2.15)
J
14.
It is assumed that the variation of density with time has the form
n = n e
t
(2.16)
and, hence, Eq. (2.15) becomes
T
T
(1 + T)
2n + (2 -
+ n
)Vn . 9
(
= 0.
)2
g
g
g
(2.17)
+
T
The problem consists of solving Eq. (2.17) for its characteristic
functions corresponding to the proper boundary conditions, proper
spatial variation of Te, and a given value of h.
In general, the
characteristic function corresponding to the lowest characteristic
value is the important physical solution.
solved for two different cavities.
Equation (2.17) will be
One is a copper vacuum cavity,
a rectangular parallelepiped in shape, in which the plasma fills the
entire cavity and the other is also a cavity of the same shape, but
the plasma is contained in a quartz cubic bottle concentric with the
cavity.
The dimensions of the cavities are the same as those described
in Section IV-8.
The applied electric field has a spatial configura-
tion corresponding to the fundamental mode of the cavity and, for a
rectangular cavity, is a function of only two coordinates.
The
boundary conditions for the vacuum cavity are that the electron density
vanish along the walls of dimensions A, B, and C and for the quartz
bottle the density vanish along the walls of dimension d.
15.
The solution of Eq. (2.17) will be obtained first for the
Expanding Eq. (2.17) one has
vacuum cavity.
2n j~
+
x2
2
+
2
2
R in + ?R
+2-h/2
1 + R ly 4y
Sz
nj
+ a2 R
+ 1 n
+ R1-h)(
R
2( C 2
z)
+
(2.18)
R = w = 1 + a cos2
TB
where
and ada
M
6nru k
2
(e 2
oh = 0.24 Eoh
Cos
at 3000 megacycles for helium.
Equation (2.18) may be separated by setting n = ncm(y,z)p(x).
The
solution for p(x) is
p(x) = cos
(2.19)
A
and the equation for m(y,z) is
g +2g
where
m + k
j
+)u
+
+
-)2
u
w = y/B,
2-h/2 =R
z/C,
+ g
(-h/2) (S&
2
C')
A = 7.16 cm,
(2.20)
g = (B/C)
2
%OR
k - g(2/2
1 + R' ju
1 + R aw
1+ R
=
+ Vm = 0
4)2
aU
B = 7.88 cm,
+
+
7-)B
A
C = 6.48 cm.
I-
16.
Since Eq. (2.20) is non-separable, there exist very few convenient
mathematical techniques for solving it.
The method of approach which
has been chosen consists of transforming Eq. (2.17) to a difference
equation which is solved using a relaxation technique.
is described in Appendix II.
The method
The results are best depicted by
plotting m(w,u) as a function of w for u
= 0.
Since the geometry
is roughly symmetrical around the z-axis of the cavity, m(w,u) has
approximately the same shape in any plane through the axis.
results are shown in Fig. 2.1.
The
The curves depict m(w,0) for various
values of h for Teo, the electron temperature at the center of the
cavity, equal to 4T .
The curves are compared with a cosine distribu-
tion which is the solution of Eq. (2.17) when the temperature distribution is independent of position.
For larger ratios of T,/T
, the
deviation from a cosine distribution increases.
The shape of the curves in Fig. 2.1 for the various values
of h is readily explainable.
The non-uniformity of the electron energy
gives rise to a larger electron diffusion coefficient at the center of
the cavity than near the walls.
The assumption of a quasi-equilibrium
state for ambipolar diffusion necessitates a change in the density
distribution to compensate for the rapid diffusion at the center and,
hence, a peak in the density distribution towards the walls is possible.
As the value of h decreases, the diffusion coefficient becomes a more
rapid function of energy, having an increasingly larger value at the
center, and a higher peaked distribution is necessary to balance this
effect.
Figure 2.2 shows a plot of the electron pressure, nT,,
1.8
1.6
h = -I
1.4
h=0
1.2
1.0
0.8
_
0.4
-
0.2
-
0.6
0
0
OSINE
0.1
0.2
0.3
0.4
0.5
w ( u= 0)
FIG. 2.1 - ELECTRON DENSITY VS. POSITION FOR VARIOUS VALUES OF h
-4
\.0
0-8 --
Tg 0 /T
4
0.4
H 2
C OSINE
0.2
0.
0
(.I
0.2
0.3
0.4
0.5
W (USO)
FIG. 22
PRESSURE VERSUS
ELECTRON
VALUES
OF H
POSITION
FOR
VARIOUS
I-i
19.
represented by m(wO)T
values of h and Teo /T
,
as a function of w for u = 0 for different
4.
It is seen that the curves for nTe are
very much alike, indicating that despite the peaked density distribution, the electron pressure remains approximately the same.
In Fig.
2.3, curves of m(w,0) are shown as a function of w when h = 1, i.e.
P
constant, for a range of T /T
from 4 to 11.
The curves point
out the increase in the peaked density distribution as the electron
temperature increases.
From the results shown, it is obvious that
for any calculations involving the density averaged over the volume
of the cavity it is necessary to use the proper density distribution.
A similar calculation was performed for the cavity containing
a cubic quartz bottle of dimension d.
The equations are identical
except for the following changes in Eq. (2.20)
w
7ry/d,
= 1
u = rz/d,
g = 1
(-h/2)a
1 ++R2
+
A = 6.90 cm,
) + ( d
D
(2.21)
2
B =7.51 cm,
C = 6.28 cm.
The results are shown in Fig. 2.4 for h = 1 and TeT
= 11 and 81.
d =2.82 cm,
Since the non-uniformity of the heating field is small over the volume
of the quartz bottle, the deviation of the curves from a cosine is
negligible.
Hence, over this region of electron temperature, a cosine
distribution is sufficiently accurate for representing the density
distribution provided that ambipolar diffusion is the dominant loss
mechanism.
.8-
1.4
IA
1.1t-
-
-Te/T
LO
= |
-Tjo
TIt =6
o/Tj 4
T.6--
Os
0.4--
C OSIN E
-
O-Z
0
0.1
0.2
03
0.4
0.5
w (UzO)
FIG. 23
ELECTRON
OF
TO/TI
DENSITY
AND Hal
VERSUS
POSITION FOR VARIOUS
VALUES
FOR VACUUM CAVITY
0'
1.2 1--1.0
81
- -_Tr T_
0.8
0.6
0.4
COSINE
DISTRIBUTION-""
-
0.2
0
0.1
0
0.4
0.3
0.2
0.5
W (U=O)
FIG
2.4
ELECTRON
VALUES
DENSITY
OF T
/TW
VERSUS
AND
POSITION
FOR
Hul FOR QUARTZ
VARIOUS
BOT TLE
tNj
22.
II-3.
Electron Conductivity Ratio for Constant Collision Cross Section
When the electron distribution function is Maxwellian, the
conductivity ratio given by Eq. (2.3) becomes
T 1/2
) 12E2dV
S n(
V
)--
V(2.22)
nE dV
V
P 2kT
-9
-r
where
1/2
for the case when the collision cross section is constant.
The solu-
tion of Eq. (2.22) is considered when a heating field is present in
the copper vacuum cavity for three different cases.
If energy
gradients are negligible, two solutions of the equation are possible,
depending upon whether the density distribution is cosinesoidal or a
peaked distribution as predicted by the theory of ambipolar diffusion
in a non-uniform field.
The solutions are indicated in Case 1 and
Case 2, where Case 1 is for a cosine density distribution and Case 2
is for a peaked density distribution.
When energy gradients are
important, the solution of the equation is indicated in Case 3, using
a peaked density distribution.
Case 1
The density distribution is assumed to be n = nocos
0 A
and the electron temperature is defined by Te =
T
where f(y,z) = cos2
for 0 /0
kbl-
Cos2
l + af(yzI)
according to Eq. (2.5).
as a function of TeT
cos
The results
are shown in Fig. 2.5.
B
cos
C
3.5
3.0ENERGY DISTRIBUTION
-UNIFORM
2d5
N xNo CO S7TX COSmCOS
A
~
i1Z
C
2.0N a N, Co SX
m (YZ)
1.5-
I.0
04
a
6
10
1P
VERSUS
T ,VTv
14
TO/ Tb
F IG. 2.5
ELECTRON
CONDUCTIVITY
0
24.
The density distribution is taken to be the solution of
Case 2
Eq. (2.17) of the form n = n
jrm(y,z),
cos
which should be the
correct distribution if the theory of ambipolar diffusion in nonuniform fields is valid.
Also the electron temperature is
Te= T9l + af(y.z)] .
The results are shown in Fig. 2.5.
calculations for Cases 1 and 2 were done numerically.
The
The difference
in percent between the two cases is given by
percent difference in
(2.23)
= 5.1(T /T ) 1/20/0
so that at Teo = 10T , the difference in
is 160/0.
Hence, an
appreciable error cani be introduced into the interpretation of the
conductivity measurements if the proper density distribution is not
The curve for ? as a function of electron temperature, for
used.
the case when the electron temperature is uniform, i.e. independent
of position and, hence, averaging with respect to the density distribution and the measuring field configuration, is also shcwn in Fig. 2.5
for comparison with Cases 1 and 2.
It is necessary to calculate the conditions when energy
Case 3
gradients become important and have a measurable influence on the
The equation for g using Eqs. (2.6) and (2.22) is
values of ? .
o
T
C/2
B/2
I
.
+ af +
(n Y)
1/2
2
m(y,Z) cos
m(y,z)cos 2
0
0
dy dz
(2 .24)
C/2
S~B/2
B
dy dz
lpr""25.
where
b = 6
and
coo
E =
Cos2
mIn evaluating the above equation, it is assumed that (1) the electron
distribution function is still Maxwellian but with a perturbed temperature given by Eq. (2.6), (2) the peaked density distribution is
unaffected by the change in electron distribution function, and (3)
for a first order approximation the unperturbed expression for Te is
substituted into IV. (nVTe/T )
Under these conditions the equation for
term in Eq. (2.24).
0
e
becomes
C/2
(1 + af)l/2 (yz) + b 2
2 1/2 M(y-Z)
2p (1 + af)
0
*
2p
Cos 2
+
B/2
which is treated as a perturbation
(2.25)
dy dz
(1 + af)1
where
B/2
K
C/2
=
m(y,Z)cos2
0
dy dz.
0
The above equation must be solved numerically.
For T
/T greater
than 10, the solution of Eq. (2.25) is approximately
T
(g)
+
(2.26)
g
p0
26.
where p(T /Tg) is the solution of Eq. (2.22) for case 2. Below
2
T /T = 10, the correction factor, 3/p , decreases becoming zero at
T /T = 1. At a pressure of about 10 mm Hg, the influence of energy
eo g
gradients becomes important and the value of Q increases. The fact
that the presence of energy gradients produces an increase in V may
be contrary to what one would expect.
Ordinarily, for helium, an
increase in
is associated with an increase in electron average
energy, but
also depends upon the averaging with respect to the
density distribution and measuring electric field configuration.
Inasmuch as the effect of energy gradients is to conduct energy from
the center of the cavity to the walls, the net result is not only a
decrease of energy in the center but also a tendency towards a more
uniform energy distribution over the entire cavity.
Figure 2.6a shows
the electron temperature as a function of position when energy gradients
have no influence.
Figure 2.6b shows the electron temperature con-
figuration when energy gradients are important.
Although the electron
temperature at the center of the cavity decreases, the values of
are shifted towards the values of e for a uniform temperature
distribution.
Figure 2.5 depicts the effect clearly.
Similar calculations on the importance of energy gradients
have also been performed for the quartz bottle enclosed in a cavity.
For this case the evaluation of Eq. (2.24) can be done analytically,
yielding the following expression valid for a greater than 10
00.93al/2 Jl + 0.03
p0
(2.27)
A
(A)
I
__
0.8
0.8
0.6
0.6
0
0
I-
0.4
B)
0.4
I0.2
0.2
0
0
03
02
0.1
0.4
0.5
0
w
w (u0)
F IG. 2.6
ELECTRON
0.2
oj
TEMPERATURE
VERSUS
0.3
(U,0)
POSITION
0.4
0.5
28.
T
a = Te
g
.
where
At a pressure of 1 mm Hg or below, the perturbation term becomes
important.
It should be remembered that the condition of
is imposed throughout the discussion.
If
ao 0.03
2
2
m
, the relation
between the pressure and the electron temperature for a frequency of
3000 megacycles and 9 m
e 1/2
=
(2.28)
PO '-
Te
Therefore, in the vacuum cavity, 1500 K is approximately the maximum
temperature that can be measured before energy gradients have an
important influence and still maintain the condition of Eq. (2.28).
In the quartz bottle, 25,000 K is approximately the maximum temperature.
Accordingly, from the above discussion, it is evident that the
conductivity measurements should be taken in a quartz bottle so that
a wide range of electron energy can be obtained without introducing
complicated correction factors into the theory.
L
29.
CHAPTER III
THOEBI OF ELECTRON CONDUCTIVITY MEASUREMENTS
Methods of measuring the complex admittance and electron
density of a discharge are discussed in a series of articles by
Brown and Rose.16
17
Resonance curves obtained by means of standing
wave measurements give the data necessary for determining these
quantities.
A simpler and more precise technique for obtaining the
discharge characteristics will be considered here.
The method
consists of measuring the ratio of the power transmitted through a
microwave cavity to the power incident as a function of signal frequency in the vicinity of the cavity resonance. By proper choice of
variables, the data will plot as a straight line whose slope yields
the desired information.
This fact reduces the data necessary and
enhances the accuracy of the measurements. Both methods will be
discussed and a comparison between them will be made.
III-1.
Electron Conductivill by the Transmission Method
A microwave cavity containinq a discharge may be represented
by the equivalent circuit in Fig. 3.1.
A detailed discussion of the
general equivalence and the significance of the various quantities
in Fig. 3.1 are given in Part I of Brown and Rose. 16 The empty
cavity conductance and susceptance are presented by g and b,
and g.
represents the conductance of the line and of the input coupling.
Since the present technique involves measuring the transmitted power,
30.
4
KmtIJ~mm~
4
I
4
I
Q
FIG. 3.1
I
Equivalent Circuit for a Resonant Cavity
31.
If
the influence of the output loop and line has to be considered.
a very small output coupling is utilized, it may be represented by
g , which is the conductance of the output line reflected back into
the cavity.
The power transmitted through the cavity is represented
by the power absorbed in gn and is designated by Pt.
this analysis is to find an expression for P
The object of
as a function of the
equivalent circuit parameters and the incident power Pa*
If cavity methods apply, the discharge admittance is given
by1 9
E dV
V
(3.1)
gd + jbd
where
or
+
J/E is the complex conductivity of the ionized
plasma, J is the current density, and the integrals are to be taken
over the whole volume containing the measuring field E.
Since the
electrons are much more mobile than the ions, the conductivity at high
frequencies is due solely to their presence.
The imaginary part ja
of a gives rise to the discharge susceptance jbd and the shift in
cavity resonance.
ance gd'
The real part ar lowers the Q through the conduct-
In general, the quantities of interest are the electron
density obtained from a measurement of bd and the ratio of the real
part of the electron conductivity to the imaginary part, Cr
obtained from a measurement of gd/bd.
i'
The cavity method is applicable
only under the restriction that the absolute magnitude of the electron
32.
current be smaller than the displacement current, or that the
conductivity of the discharge Ji o<
w ((
The normalized impedance of the cavity and discharge terminating the line at PP' in Fig. 3.1 is
Z =
where g
Pa =
+
g
12)
g
+ gn + gd and bt = b
= g
(i -ir2) where 1
tion coefficient.
+ jbt
+ bd.
The power absorbed is
is the magnitude of the complex reflec-
The complex reflection coefficient is related
to the impedance by the expression
(3.3)
= (z-1)/(Z+l).
[(g
Pa
+b
g
(,
+ 1)gt + 11
+
The absorbed power is obtained by combining Eqs. (3.2) and (3.3).
+
+ b2(
1g
+ 1) 2*
Pa represents the power that is dissipated in g
and g
whereas the
power dissipated in gn represents the power transmitted through the
cavity.
The power transmitted is related to the power absorbed
through the expression
33.
21
2
gU
P
9
P
g
bti
g
-1
gj
+ bt)
(g
2 (
+
(3 5)
2
represents the fraction of the
where the factor
gt/(g2 + bt) + hg
8
and gn 9 represents that fraction of
absorbed power dissipated in g
the power dissipated in g which is dissipated in g .
Combining Eqs.
(3.4) and (3.5) yields
P
4g
P
'2
+ 1)gt + 1 2+ b2(
+ 1)22
(3.6)
where
b
(CO
-
) + bd '3,7)
co0 is the empty cavity resonant frequency and p is a dimensionless
quantity which depends upon the degree of coupling.
The cavity resonance frequency with a plasma present,
is defined by setting bt = 0 in Eq. (3.7) and is
denoted by o,
0
determined by
0o
bo
o
20
(3.8)
,
I
.
34
Since it is assumed that the presence of a discharge disturbs the
(o,
cavity characteristics only slightly, that is (o' - co)
near the resonant frequency 1o, Eq. (3.7) becomes
=.
(3.9)
bd
0
0)
Under these conditions, Eq. (3.6) takes the following form
P2 2(
P
If
=
2
+ 1)2
1+
+1)+1
2 .
2 [(l
(3.10)
+ 1)gt + 1]
is plotted as a function of x2, the square root of the reciprocal
of the slope of the line, represented by a is
i
~
-+
l)g + 1
(3.11)
p(--+ 1)
The difference between the value of a with and without a discharge is
a = ---
and also
a 9(3.13)
2(a
-
(3.12)
35.
Thus from a measurement of P
/P
as a function of frequency for a
cavity with and without a discharge, it is possible to obtain the
ratio of discharge conductance to discharge susceptance and the
resonant frequency shift due to the presence of the discharge.
From
Eq. (3.1) one finds that the ratio of the real part of the electron
conductivity to the imaginary part,
%o/a,
averaged with respect to
the measuring field, is equal to the ratio of gd/bdo
111-2.
Electron Conductivity by the Resonance Method
Measurements of the standing wave ratio as a function of
frequency in the vicinity of the cavity resonance determines the
resonance curve.
The conductance g
is obtained from a measurement
of the db standing wave ratio on resonance, R0 , and the db standing
wave ratio far from resonance, R.O.
R = 20 log(
This is expressed as follows
)
undercoupled case
(3.14a)
)
overcoupled case
(3.14b)
+
1
g
R
= 20 log(
9
R.= 20 log g
A measurement of R
(3.14c)
.
with and without a plasma present and the
knowledge of R.0 yields the value of gd
From Eq. (3.8) we have
36.
2P(c
-o)
0
bd
The ratio of gd
b which is equal to
-bd
=
/
(ao - o )2p
-
(3.15)
is given by the expression
.
(3.16)
The coupling coefficient 0 is determined from the resonance curve
and is assumed to be independent of the discharge characteristics.
(1
It is given by the following relation
(3.17)
)t
Q g(l +
0
where Qu is the unloaded Q of the cavity.
111-3.
Comparison of Methods
Let us compare the accuracy of the two methods as a function
of experimental parameters.
a I
ad3.18)
(
--
is
r
C 2(oo
o
)
tion for
For the transmission method, the equa-
where ad is the value of a with a discharge and a. the value without
a discharge.
The fractional error in
a
is obtained from Eq. (3.18)
by the usual technique and by a simple transformation one has
37.
%o ad
ad
-1
r
a
1---
0i
ad
--
-"I ~
0
For the resonance wave method, the equation for
- r_
where g
(1
-
o'
/ci is
(3.20)
9
is the value of the total conductance with a discharge
present and g0 is the value without a discharge.
For the fractional
error in 0cr0., one obtains
A (ai
9t
a
9~
0 gt
g
r
g0
ai
9t
0
_
.
(3.21)
Equations (3.19) and (3.21) will be compared assuming the following
typical experimental conditions.
and an Rea = 40 db.
ratio is + 10/o,
The microwave cavity has a Qu = 5000
The accuracy in measuring a, P, and standing wave
10/o, and + 0.2 db respectively.
for
(co0 - o)
0
the transmission method is + 0.01 megacycle and for the resonance
curve method + 0.03 megacycle.
Figures 3.2 a and 3.2b show the percent
error for both methods as a function of rr/at and various values of
10 db OVERCOUPLED
14
5Mc
-
12 --
-
TRANSMISSION METHOD
RESONANCE CURVE METHOD
10
et
z
25Mc
8
bj b- ti15
5MC
50M c,
4
25Mc
-f
q
2
50MCT
C
I
0.05
I
0.10
I
I
0.15
0.20
I
0.25
Q30
1r
FIG. 3.2 a - PERCENT ERROR IN CONDUCTIVITY
FOR 10 4b OVERCOUPLED
RATIO VS CONDUCTIVITY
10 db UNDERCOUPLED
12
TRANSMISSION METHOD
-
at
10 k
--
--
RESONANCE CURVE METHOD
-
SOMc
2
8 F_-
25Mc
w-
F~ ~~.-- --
4
5 Mc
- ---
2
)
0'
50m
0.05
0.10
0.15
0.20
025
0.30
a..
FIG. 3.2b - PERCENT ERROR IN CONDUCTIVITY RATIO VS. CONDUCTIVITY
FOR 10 J6 UNDERCOUPLED
40.
frequency shift in megacycles (a' - o), i.e. electron density, for
0
0
the cases of 10 db overcoupled and 10 db undercoupled.
It is evident
that the transmission method is more accurate in the latter case
while the opposite is true for the overcoupled case.
For the values
considered, the transmission case is more accurate than the resonance
curve method by a factor of two under optimum conditions.
The
experimental data were taken in the region of the electron density
where the error in cy/cri is less than + 20/0.
The advantages of the transmission method are the following.
Plotting the experimental data as a straight line eliminates the
necessity of taking time consuming resonance curves and adds to the
accuracy and simplicity of the method.
Since the transmission method
is more accurate when the cavity is very much undercoupled, the
perturbation due to the coupling on the cavity characteristics will
be small.
To the contrary, the resonance curve method has its
greatest sensitivity when the cavity is overcoupled, and the coupling
perturbation may become important.
In order to eliminate the
possibility of disturbing the discharge with the measuring signal for
the case when the measuring signal is not used to maintain the dis-
charge, it is necessary to operate with a low level signal.
This can
be done conveniently since a null method is employed with the transmission method as described in Chapter IV and a lower signal to noise
ratio may be tolerated than with the resonance curve method.
The
problem of the effect of the slotted section and probe insertion on
the measurements is no longer encountered.
Unfortunately, the knowledge of a and P
is insufficient
for determining the value of the electric field, E, in the microwave
According to Rose and Brown,
cavity.
20
the electric field at
resonance is given by
E2
where P
ao
(3.22)
aou
is the power absorbed at resonance.
be computed for the cavity in question.
The quantity V
must
If measurements are made of
the reflected power at resonance, denoted by Pro, and off resonance,
denoted by Prea , the electric field can be determined.
the relation Pr
1
-
Combining
a with Eqs. (3.4), (3.11) and (3.17), the
electric field is given by
2
E
0
r
Pto I
a
P
ro
P
_
P
Pra
--.
I
(3.23)
Hence with this additional information concerning the reflected
power, it is possible to obtain all the desired cavity characteristics
using the transmission method.
42.
CHAPTER IV
APPARATUS AND PROCEDURE
The apparatus and procedure utilized for obtaining electron conductivity and electron density measurements in the afterglow of a discharge are described in this chapter.
The theory of
electron conductivity measurements was discussed in Chapter III.
The microwave cavity used in this experiment is a rectangular
parallelepiped in shape and is designed to resonate in its three
fundamental modes at wavelengths of 9.6, 10.0, and 10.6 cms.
The 9.6
cm mode is used to produce a pulse discharge in helium of variable
pulse length.
The 10.0 cm mode is used to increase the electron
average energy in the afterglow, and the 10.6 cm mode is used to
measure the characteristics of the plasma.
The apparatus and procedure
associated with each mode will be discussed separately and will be
referred to as the breakdown mode, heating mode, and measuring mode.
The general block diagram of the experimental microwave equipment is
shown in Fig. 4.1.
7/8" coaxial transmission line is used throughout
except where noted.
IV-l.
Measuring Mode
A continuous wave tunable magnetron (Raytheon, Qk59)
designated as number one in Fig. 4.1 supplies power to a coaxial line.
A power divider provides a continuously variable control over the
43.
M.L
PULSED MAMNETRON
d O.
ML.4
.L.
Dit
wAAVEMETER
PULSEDMMGNEERON
SECTION .C.ro~~.c.
O~tPOLEMEAER NG
TASIN
BR
BRDI
CTDGE
M
P.C.ND.LOOP
ED
DVIERRU ]
P.D
RON
PADRDVIE
FIG. 4.1 - GENERAL BLOCK DIAGRAM OF EXPERIMENTAL EQUIPMENT
ICTT
44.
fraction of power incident on the cavity, unused power being dissipated in a matched load.
A known fraction of power is coupled
from the line by a directional coupler to a power measuring thermistor
and associate measuring bridge.
In order to obtain microwave signals
of the order of microwatts incident on the cavity, an attenuation of
60 db exists between the cavity and the directional coupler.
This
attenuation is obtained by using a 20 db attenuator in conjunction
with a directional coupler as shown in the block diagram.
A direc-
tional coupler before the cavity samples a portion of the incident
power, which is transmitted to the power measuring section.
An out-
put loop on the cavity, adjusted for maximum power output for this
mode, transmits the output signal to the power measuring section.
In
all the measurements obtained, the value of the measuring electric
field is less than 0.1 volt/cm, so that the perturbation of the field
on the plasma characteristics is negligible.
The general measuring procedure consists of the following.
Samples of the incident and output signals are transmitted through
two separate superheterodyne receiving systems.
The two signals are
introduced to a push-pull input of an oscilloscope and their signal
levels adjusted for a null deflection.
The ratio of the incident
power to the transmitted power as a function of frequency in the
neighborhood of cavity resonance is shown in Fig. 4.2.
The frequency
of the signal is adjusted so that Pt is a maximum and o
may be
determined.
At this frequency, the input and output signals are
balanced for a null deflection on the oscilloscope by adjusting the
0
z
rn
L~ouanbaad jo uo1~cpufl
0
01
TRANSMITTED POWER
INCIDENT POWER
ia sv .ZvWd :tuoppqj- o;. p~q~l~. ttj o &v
3011
ODJJ
46.
gain of the two receiving systems.
the quantity J
is equal to one.
At this point, by definition,
When the attenuation in the
receiving system of the transmitted power is decreased, the frequency
must be changed to some value a)
There exists another frequency
true.
in order to have a null signal.
I for which the same conditions hold
The value of x is given byo
-
and
the decrease in attenuation between (P /P )
is obtained from
and (P t/P).
If the
attenuation is changed by 3 db, *6 will have the value of two.
By
changing the value of the attenuation and measuring the proper frequency changes, a complete set of data may be obtained.
The sections
for measuring input and output power and frequency will be considered
in the following.
A block diagram for this section of the equipment
is shown in Fig. 4.3.
IV-2.
Power Measuring Section
The microwave signals from the cavity are transmitted
through a double 10 cm band waveguide mixer unit in which the two
signals have separate mixers but utilize the same local oscillator.
The signals proceed through separate intermediate frequency amplifiers,
video amplifiers, and cathode followers and are connected to a balanced
input of a Techtronix Model 512 cathode ray oscilloscope.
The double waveguide mixer unit consists of two waveguide
mixers into which the local oscillator signal is inserted through a
double waveguide directional coupler.
the mixer is shown in Fig. 4.4.
A detailed cross section of
Each mixer contains two matched 1N23B
POWER MEASURING SECTION
-'
|AT TENUATOR
LOCAL,
OSCILLATOR
+
cr
I-.
II~iI.
ID
I-
F r|-
PAD
0
I.F
CA
MIXER
trj
--
TRANSITION CRO.
UNIT
--
L.O.
SWEEP
-RGE
INPUT
COUPLER
ci-
PAD
RECEIVER
ZJ
SIGNAL
GENERATOR
crINCIDENT: MAGIC
T'
SIGNAL i
FREQUENCY MEASURING SECTION
0
48.
*1
FIG. 4.4
Waveguids Mixer
49.
crystals which are matched to the guide by wedges as shown.
The two
crystals serve a two-fold purpose of increasing the mixer sensitivity
and enabling the mixer to be matched to the guide by means of symmetry.
The mixer was designed using the results of Hope
who had
constructed a balanced waveguide mixer of similar design.
The i.f.
signals are brought out from the mixer by a split rod whose sections
are insulated from one another.
The double waveguide directional
coupler is actually two directional couplers combined having only one
matched load and local oscillator input.
It has been designed22 to
have a coupling of 15 db and a directivity of 25 db.
double directional coupler is shown in Fig. 4.5.
A sketch of the
The crossed coupling
slots enable the low value of coupling to be obtained.
The slots are
so located that the power transmitted through them will be in phase
in one direction and out of phase in the other direction.
The above
characteristics are obtained by using slots 0.125" wide and 1.56"
long with the distance between the centers of the crossed slots
being 1.86".
A resistance card (USKON 9-112) is used for the matched
load.
The i.f. signals from the crystals are added together
through a transition unit from the mixer to the i.f. amplifier.
The
transition unit is shown in Fig. 4.6 and is designed so that the LC
network resonates at the i.f. amplifier frequency, which for this case
is 40 megacycles.
The d.c. crystal current is measured from the
terminals marked "to meter".
The i.f. amplifiers are type TAI-ISE
and have been modified so that the bandwidths are of the order of one
FIG. 4.5
-
/ \\\
-/VL
/
\_
%Q
Double Waveguide Directional Coupler
50.
H
0
L O-3O0
c
0.01
C+
0
TO
CRYSTAL
0.0
-1
0.01
47*~ TO METER
STR
TO IF STRIP
Z-,
0.01
C
L
11
~J1
H
0
52.
half megacycle.
Since transient measurements are being performed, 23
the local oscillator is frequency modulated by the oscilloscope
sweep, which is externally triggered.
The more rapidly the plasma
characteristics change with time in the afterglow of the pulsed
discharge, the shorter the sweep must be so that the plasma characteristics are essentially constant over the sweep time.
For a given
sweep time, a minimum bandwidth of the receiver system is necessary
so that the signal is not distorted as it traverses the receiving
system.
For the case of a receiver bandwidth of about one half
megacycle, the minimum sweep time allowable is approximately two
microseconds which is sufficient for all the measurements performed
in this experiment.
Another important criterion is that the amplifiers
be adjusted so that their bandwidths and i.f. frequencies are essentially identical.
This is necessary in order to balance the signals
to zero over the bandwidth of the receiver.
A precision waveguide
attenuator, calibrated24 to a precision of + 0.05 db, is inserted
in the section of guide containing the transmitted power and is used
IV-3.
6
.
to produce the desired change in
Frequena Measuring Section
The frequency is measured by means of a standard cavity
wavemeter using a superheterodyne receiver on the output to denote
the wavemeter resonance.
Rather than measuring the signal frequencies
directly, it is more accurate and convenient to measure the difference
between the signal frequency and the resonance frequency.
A sample
53.
of the incident signal is transmitted through a magic tee into the
wavemeter.
The side arms of the magic tee contain crystals which are
matched to the tee by double stub tuners.
The crystals are connected
to a General Radio signal generator, operating in the megacycle
region.
The signal generator, coupled with the magic tee, is utilized
to amplitude modulate 2 5 the measuring signal before it arrives at the
wavemeter.
With no amplitude modulation, the wavemeter is adjusted
for the frequency at which the transmitted power is a maximum, which
corresponds to co
0
in Fig. 4.2.
When the frequency is at a value x),
the signal to the wavemeter is amplitude modulated so that the first
side band is transmitted through the wavemeter.
quency gives the difference co
- ca.
The modulating fre-
In a similar manner,
is obtained and their sum gives the appropriate value of x.
I- 0
If the
wavemeter is adjusted exactly to the cavity resonance, the two readings
will be identical.
If they are not similar, their difference divided
by two yields the deviation of the wavemeter reading from the true
resonance.
Thus an accurate determination of the resonance frequency
is possible within + 10 k.c.
IV-1.4.
Breakdown Mode
A high power tunable pulse magnetron (Raytheon 2J54)
designated as number four in Fig. 4.1 supplying 100 kw peak power is
used to produce a two microsecond discharge in the cavity.
A model 12
modulator unit supplies the high voltage pulse to the magnetron.
A
power divider in the line varies the power incident on the cavity.
54.0
A low power tunable magnetron (QK61), supplying 100 watts peak power
is introduced through a waveguide directional coupler of the type
discussed in Section IV-2 and is designated as number three.
The
magnetron is pulsed for a duration varying between 0.1 millisecond
and 6 milliseconds.
A well regulated pulsed voltage supply is used
to modulate the magnetron output.
The stability of the pulse is
measured by the stability of the output of a wavemeter in the line.
A directional coupler before the cavity samples power from both
magnetrons so that the pulse forms can be viewed on a transient
superheterodyne receiver.
The waveguide directional coupler provides
a certain degree of isolation between the two magnetrons so that the
operation of one will not affect the operation of the other.
After the high power magnetron produces a discharge in the
cavity, the low power magnetron is applied and the frequency and power
incident on the cavity is adjusted so that the ionization in the
cavity increases slightly.
The low power magnetron helps in stabilizing
the discharge so that accurate measurements in the afterglow can be
performed.
The combination of both magnetrons facilitates controlling
the electron density during the discharge to any value desired over
a range of 10
to 10 0 electrons per c.c.
IV-5. -Heating Mode
A continuous wave tunable magnetron (Raytheon, QK60)
designated as number two in Fig. 4.1 supplies power to a coaxial line
system similar to that for the measuring mode.
The cavity characteristics
55.
are measured by the resonance method and, hence, a slotted section
in the line is used to take standing wave ratio measurements.
An
output loop connected to a matched load is adjusted so that it only
couples out the power in the heating mode.
By changing the coupling
between the cavity and the matched load, an additional loss is
reflected back into the cavity so that Qu for this mode, measured at
the input terminals of the cavity decreases.
A decrease in Qu from
5000 to 200 is easily obtained by this method.
In the afterglow of a discharge, the electric field in the
cavity, for a constant incident power, will be a function of electron
density.
If an electric field is applied to the cavity, its magnitude
has a maximum value for a particular density corresponding to the
frequency of the field and decreases for larger and smaller densities.
It is desirable that the electric field remain constant for a sufficiently long period of time so that the electron may reach equilibrium with the electric field.
For the applied fields used in this
experiment, the time necessary for equilibrium is of the order of one
tenths of milliseconds.
QU i of the order of several hundreds allow
the electric field to remain constant for the order of milliseconds
thus insuring equilibrium with the field.
,IV-6. Timing Apparatus for Transient Measurements
In order to perform afterglow measurements, it is necessary
that the apparatus have the proper timing mechanism.
Figure 4.7 shows
a block diagram of the apparatus giving the proper triggers and pulses
56.
SYNC. SIGNAL
POWtA MEASRING SECTION
TIMM00
TRHHIER
1102
GENERATOR
TRIGGE"
NO I
ELAT L ONE
TRANSIENT NECEIVER
MARKER SENERATOR
OSCILLOSCOPE
TRISGER AMPLIFIER
PULSE
UNIT
MA
E
E
--
NO 4
O
3
FIG. 4.7 - TIMING APPA RATUS FOR TRANSIENT OPERATION
57.
for transient operation.
A timing generator, synchronized with a
120, 60, 30, or 20 c.p.s. signal, supplies two triggers designated
by numbers one and two.
Trigger no. 1 is transmitted through a
variable delay line, which can delay the trigger up to a time of
50 m.s. for the 20 c.p.s. repetition rate.
The delayed trigger is
used to trigger the sweep of the oscilloscope in the power measuring
section of the measuring mode and to trigger the oscilloscope of the
transient receiver for the heating mode.
The delayed trigger is also
placed on the vertical plates of a Dumont Type 304 oscilloscope.
The
undelayed trigger no. 1 triggers a marker generator whose markers are
superimposed upon the delayed trigger on the oscilloscope screen.
Thus the time at which the transient receivers are operative is
determined by setting the delayed trigger on the appropriate marker.
In addition trigger no. 1 is transmitted through a trigger amplifier
which provides the proper shape and magnitude of trigger for the
model 12 modulator of the high power pulsed magnetron no. 4.
Trigger
no. 2 initiates a pulse forming unit which provides a variable pulse
up to 6 milliseconds in length to the regulated power supply of the
low power magnetron no. 3.
Trigger no. 2 can be delayed with respect
to trigger no. 1 so that the magnetron no. 3 will be in operation
after the high power magnetron has produced a discharge in the cavity.
IV-7,
Vacuum System and Gas Supply
A block diagram of the vacuum system is shown in Fig. 4.8.
58.
M.cLEO
SAME
OIL DIFFUSIOW
METAL VALVE
LITER
EETALALALLE
TR
LLIDQ.UIEA LVAL
ALSAK
METALVALV
METAL
VALVE
FIG. 4.8 - VACUUM SYSTEM AND GAS SUPPLY
59.
A standard forepump and three stage oil diffusion pump are used in
conjunction with metal valves
wherever necessary.
The helium
pressure is measured by a McLeod gauge which is calibrated to an
accuracy of + 10/o over a range of 0.01 to 20 mm Hg.
A liter pyrex
flask is placed near the cavity in order to minimize the fluctuations
in the gas pressure.
For studies in helium, an activated charcoal
trap cooled by liquid nitrogen is used.
In order to obtain as pure
helium as possible, it is necessary to produce the helium and store
it directly on the vacuum system.
The method for producing the helium is as follows.
A
"fritted" glass filter with one end having a break-off seal is placed
in a liquid helium Dewar flask and is connected to the main vacuum
system by a metal valve.
A liter pyrex bottle connected to the cavity
side of the system by another valve, serves as a reservoir for the
helium.
After the system is baked, a vacuum of about 10
mm Hg is
obtained when the system is isolated from the pumps, liquid helium is
placed in the Dewar flask.
When the "fritted" glass filter is completely
immersed in the liquid helium, the break-off seal is broken and helium
evaporates into the system.
After filling the system with helium,
the valve between the glass filter and the system is closed and the
glass filter must be sealed off from the system before the liquid
helium evaporates below the glass filter.
This is necessary because
the valve does not produce a complete cut off and there would be a
danger of air diffusing into the system.
Hence, a forepump is connected
to this part of the vacuum system in order to maintain a vacuum of
60.
10-
mm Hg in this region.
The purity of the helium is such that
consistent measurements can be obtained and appears to be independent
of the presence of the activated charcoal trap.
IV-8.
Microwave Cavities
Three types of cavities were used during the course of the
experiment.
All the cavities were vacuum tight and capable of being
outgassed at 430 C.
This necessitated using high temperature solder
whenever necessary.
Each cavity will be described separately.
A copper vacuum cavity was constructed with glass bubble
inserts so that the coupling loops used for coupling power in or out
of the cavity might be varied externally.
cavity is shown in Fig. 4.9.
A cross section of the
The cavity is rectangular parallelepiped
in shape and has inside dimensions of 2.82" x 2.55" x 3.10".
0.F.H.C.
copper was used throughout the .construction. The main base of cavity
(1) was milled out of a solid copper bar in order to minimize the
number of joints to be made vacuum tight.
brazed to the base in two operations.
The end plates (2) were
The first operation used gold-
copper solder (melting point at 960 0 C) and the second operation used
gold-nickel solder (m.p. at 950 C).
The pumping lead consisted of
20 mil kovar tubing (3) which was brazed into a copper tubing (4).
After the brazing of the cavity was completed, glass tubing was sealed
to the kovar which connected the cavity to the vacuum system.
The
pumping lead was brazed to the main base using BT solder (m.p. at
780 0C).
The coupling assembly is the following.
A kovar sleeve (5)
61.
TO VACUUM SYSTEM
GLASS
7
COUPLING LOOP HOLDER
(COPPER)
T4
COPPER
GLASS
FOR
COUPLING
LOOPS
ARC WELD/
COPPER-GOLD
SOLDER
FIG. 4.9 - CROSS -SECTION
NICKEL- GOLD
SOLDER
OF COPPER VACUUM CAVITY
62.
was brazed to the cavity with BT solder.
After this operation, a
kovar fitting (6) with a hemispherical glass bubble sealed at one
end was inserted into the kovar sleeve so that the other end made a
tight fit with the sleeve.
joint was vacuum tight.
without a single failure.
This end was arc-welded so that the
This operation has been highly successful
A coupling loop holder (7) was formed
as shown so that the coupling loops could be screwed into place.
The holder was held firmly to the cavity with screws and could be
removed when the cavity was baked.
were used on the cavity.
Four such coupling assemblies
The use of three solders reflected the
inexperience of the writer at the time of fabrication.
A similar
construction, using only one type of solder, which was built at a
later date will be discussed shortly.
Another type of cavity used was of a simpler construction.
It consisted of a quartz bottle which was attached to the vacuum
system, enclosed in a microwave cavity of the same shape as the
previous one.
a side.
The quartz bottle was cubical in shape and 1.18" on
The wall thickness was 1 millimeter.
dimensions were 2.47" x 2.71" x 2.96".
The cavity inside
It was constructed of brass
plates, soft-soldered together except for one side which was held
in place by screws.
This is necessary so that the quartz bottle
may be introduced into the cavity.
The bottle was positioned so that
its center coincided with the center of the cavity.
The cavity was
silver plated on the inside and finished in a high polish so that a
high Q was obtained.
63.
The third cavity consisted of a double O.F.H.C. copper
vacuum cavity in which the two cavities were separated by a fine
copper mesh.
A cross section of the cavity is shown in Fig. 4.10.
The coupling assemblies are not shown on the sketch.
The copper
mesh decouples the two cavities so that the electromagnetic interaction between them is negligible.
The inside dimensions of the two
cavities are the same as the vacuum cavity discussed in the first
part of the section.
The main base (1) was fabricated from a single
plate of copper 1/2" thick which had deep V-shaped grooves so that
the plate could be bent into the shape of a box.
the bottom of Fig. 4.10.
This is shown in
In the process of bending the plate, the
copper mesh was slid into place and the joining corners were held
together with copper screws.
brazing.
Thus only one of the four sides required
The end plates (2) were fitted into place in addition to
the pumping lead and the fittings for the coupling mechanism.
whole assembly was held together with stainless steel jigs.
The
Gold-
copper solder wire and gaskets were used throughout the assembly
whenever necessary.
In this way, the entire cavity could be success-
fully brazed with gold-copper solder in one operation so that the
cavity was vacuum tight.
After the brazing processi the glass bubble
assemblies were placed into position and arc-welded, thus completing
the cavity.
64.
/
PUMPING LEAD
2
//
COPPER MESH
I
HOLE FOR
COUPLING
MECHANISM
CROSS-SECTION OF CAVITY
GROOVE FOR MESH
HOLES FOR COUPLING MECHANISMS
SKETCH OF PLATE BEFORE BENDING
FIG. 4.10 - CROSS-SECTION FOR DOUBLE CAVITY
65.
CHAPTER V
RESULTS
The theory of electron conductivity measurements and the
apparatus and procedure associated with the measurements have been
described in the previous chapters.
The results and interpretation
of the measurements performed in the three cavities, described in
Section IV-8, will be presented in this chapter.
The important
result obtained is the value of the collision probability for
momentum transfer over a range of electron energy from 0 to 3
electron volts.
The value of the collision probability in helium is
18.3 + 20/o from 0 to 0.75 e.v. and increases slowly to a peak value
of 19.2 + 20/o at 2.2 e.v.
V-1.
Effect of Impurities on Conductivity Measurements
During the initial stages of the experiment, measurements
of
Po
C-i, as a function of pressure were taken at room temperature
and at pressures below 15 mm Hg to verify that ( was independent of
pressure and to check the performance of the measuring apparatus.
Under these conditions,
E
should be independent of electron density
distribution and measuring field configuration, and according to
Eq. (2.3), independent of gas pressure.
The helium samples used were
flasks of reagent grade gas from Air Reduction Company.
of these measurements showed that ?
The results
was not only a function of
66.
pressure but also a function of time in the afterglow.
tion of
The varia-
for two different gas samples is shown in Fig. 5.1.
typical curve for
A
as a function of time is shown in Fig. 5.2.
The values of
e in Fig.
in Fig. 5.2.
Inasmuch as one of the objectives of the experiment
5.1 are averages of data similar to those
was to obtain as accurate measurements of e as possible, it was
necessary to eliminate or control the cause of the above effects.
The experimental data seemed to indicate that the anomalies in the
data were produced by impurities present in the helium.
The im-
purities in the gas sample according to Air Reduction specifications
are N2 , 02, A, and carbon bearing gases, present at least one part
in 105.
In order to verify this hypothesis, helium was produced by
the method described in Section IV-7.
The impurity content27in
7
the helium was probably less than 1 part in 10 and will be referred
to as "pure" helium in the discussions to follow.
The results of
the measurements in "pure" helium are shown in Figs. 5.1 and 5.2.
is seen that
It
eis independent of pressure and time in the afterglow
as predicted, thus verifying that impurities produced the discrepancy
in the measurements.
The mechanism by which the impurities can effect higher
values of
is not definitely established.
Since the pressure of
impurities is much smaller than the pressure of helium, the value of
Pm is determined by the helium atoms.
(2.3), a change in
Therefore, according to Eq.
when impurities are present can only be produced
by a change in the electron velocity distribution function.
The
GAS
SAMPLE
(I)
*1GAS
0.0170
SAMPLE
(2)
0.0160F4
c.-
PURE"
00150
5
I
0
z
0
0
HELIUM
x
0.0140
0
2
4
PRESSURE
FIG.
5.1
CONDUCTIVITY
l0
8
6
(MM
RATIO
12
HG)
VERSUS
PRESSURE
-ZJ
QOl8I-
N
MM HG
( P, a 7.5
)
HELIUM
-BOTTLE
0.017-
a
Q1060
F
4
0.015-
e
t,
I-
,
"PURE"
/
0
II
II
HELIUM
( P, a 7.0
MM HG)
0
()Q
I0
0
U
0.014F-
0
I
0
I
I
I
I
I
4
I
I
II
8
I
I
II
12
I
I
TIME
FIG. 5.2
CONDUCTIVITY
16
II
.!
20
I
24
I
I
28
I
I
32
I
w
(MILLISECONDS)
RATIO VERSUS
TIME
IN
AFTERGLOW
(11)
69.
introduction of fast electrons into the plasma or the removal of
more slow electrons than fast ones can produce higher values of
average electron energy, and hence higher values of
Pm constant,
energy.
e
, since for
is proportional to the square root of the average
Helium metastables produced during the discharge can ionize
those impurities whose ionization potential is less than the energy
of the metastable level.
Imprisoned resonance radiation of helium,
which has a wavelength of 5300 A, can also ionize impurities.
The
energy of the electrons produced by both of the reactions is of the
order of 5 to 10 electron volts.
The balance between the total elec-
tron energy introduced into the afterglow by these processes and
the total energy lost by elastic collisions with the gas will determine the importance of the reactions.
However, a quantitative analy-
sis to check which phenomenon is important would require a knowledge
of the type and number of impurities, the number of metastables or
the intensity of the imprisoned radiation, and the rates of the
reactions.
ment.
Unfortunately, these factors were not known in the experi-
Moreover, if the ionization of impurities either in the after-
glow or during the discharge is important, the concentration of the
impurity ion may be of the same order of magnitude as the electron
density.
Under such conditions, recombination of electrons and
impurity ions can be an important electron loss mechanism.
Indeed,
measurements of electron density as a function of time in bottle
helium at pressures below 15 mm Hg yielded curves which indicated a
recombination phenomenon at high electron densities and a diffusion
70.
phenomenon at low electron densities.
Above 15 mm Hg, the density
decay curves gave straight lines on reciprocal density versus time
plots, yielding recombination coefficients of the order of 10
to
107 cc/ion-sec, which are the same order of magnitude as the
recombination coefficients of N2 and 02 measured by Biondi.2 8
29
Theory and experiment
28 indicate that the recombination coefficient
varies inversely as the electron energy.
Under this condition, a
greater number of slow electrons will recombine than fast ones, thus
shifting the average electron energy to a higher value.
This shift
will be counteracted by elastic collisions which tend to maintain
the electron distribution function Maxwellian in equilibrium with
the gas atoms.
The experimental data were not sufficient to give a
consistent-indication of which mechanism might be the dominant one.
Indeed, all three mechanisms might be equally important considering
the heterogeneous mixture of impurities present in the helium gas
samples.
Careful measurements under controlled conditions would be
necessary to separate the various processes.
The rate of change of
and electron density with time and their dependency on pressure for
helium with known amounts of different impurities can give insight
to the processes producing this effect.
Regardless of the mechanisms
involved, it is still important to eliminate the effect whenever
possible so that false values of e
are not measured.
71.
V-2,
Thermal Conductivity Measurements
Measurements of
as a function of gas temperature and
pressure were obtained in the copper vacuum cavity.
Provisions were
made for cooling the cavity at dry ice (1950K) and liquid air (770K)
temperatures and for heating the cavity from room temperature to
40000.
During the measurements all the 7/8" coaxial lines entering
the cavity were filled with one atmosphere of dry nitrogen to
prevent the condensation of water vapor in the coaxial linesat the
low temperatures and the oxidation of the lines at the high
temperatures.
The results are shown in Fig. 5.3.
Included in the
figure is a sketch of the cavity showing the relative positions of
the coupling lines for the measuring mode and the pulsed breakdown
The crosses represent the experimental points and the solid
line represents the theoretical curve for
equal to 18.3 cm2/cm 3 per mm Hg.
case of Pm constant,
when Pm is constant and
According to Eq. (2.3), for the
is given by
m r(3)
w ri(5/2)
2kT
m
(
5
.
1
)
mode.
It is seen that the experimental points agree with the above equation over the range of 770K to 4000 K.
Above 1000K,
the values of
4
obtained are higher than the theoretical curve.
It
is believed that
impurities liberated from the walls of the cavity at the higher
temperatures produced the higher values of C.
At a given temperature
-
x
0.026-
x
x
x
-x
0.024-
0.0220.020-
x xxx
0.018-
EXPERIMENTAL POINTS
THEORETICAL CURVE
FOR PM 18.3
x
x
0.016
TO VACUUM
SYSTEM
0.014-
0.012
-10
MEASUF ING
INPU T
10.5 Cw
MEASURING
OUTPUT
0.010
0.008
COPPER VACL UM
CAVITY
PULSED
BREAKDOWN
9.5 CM
0.004
0002
0
SI
too
t
200
I
I
300
i
400
i
i
500
t
6UU
rwU
awO
TEMPERATURE OF CAVITY (OK)
FIG. 5.3 - EXPERIMENTAL DATA OF
Er- AS A FUNCTION OF CAVITY TEMPERATURE
Po 0 -i
-
73.*
the value of g could be lowered by outgassing the cavity for several
days at a temperature of 4300C.
In addition, the plasma character-
istics drifted slowly during the measuring time, indicating that
impurities were constantly being introduced into the gas.
At 7000K
the plasma characteristics drifted so rapidly that accurate measurements were difficult to obtain.
The direction of the drift was always
such as to decrease the electron density due to a more rapid electron
decay and a lower initial density for a constant amount of microwave
power absorbed in the cavity.
Both facts indicate an increasing
amount of impurities in the discharge.
The data for P as a function of time in the afterglow at
770K are shown in Fig. 5.4. for pressures of 6.7 and 12.8 mm Hg
normalized to 000.
A large variation of p with time is observed which
has an asymptotic value of 0.0765 for
pressure.
at 770K.
, which is independent of
It is this value which is taken to be the correct one for
At liquid air temperature, the cavity can behave as a trap
for any impurities that may be liberated from the walls during the
pulsed discharge and for any impurities in the gas.
tion of ( with time is to be expected.
Thus the varia-
It should be noted that the
value of i at the lower pressure is higher than the value of
e at
the higher pressure, contrary to the results shown in Fig. 5.1 where
increases with pressure.
However, the data in Fig. 5.1 were obtained
under the condition that the ratio of the impurity pressure to the
helium pressure is constant.
Hence, the ratio can vary from one gas
sample to the other, whereas when the cavity behaved as a trap, this
e
0.011
0
0
.O,01-
: 6.7 MM HG
F
*A P.
=12. 8 MM HG
T =77 OK
!a- 0.0090
0.008-0
U
0.007
T
0
I
I
II
4
Ii
II
8
IE
II
I2
iI
TIME
FIG. 5.4
CONDUCTIVITY
II
II
II
20
16
(MILLISECONDS)
Ii
II
24
i
I
28
RATIO VERSUS TIME IN AFTERGLOW AT T=77*K
32
75.
condition need not have been the case.
Therefore, in order to
interpret the electron conductivity measurements properly, it is
important that care be taken to determine whether any impurities
present have any appreciable effect.
values of P
V-3
If this is not done, erroneous
as a function of temperature can be obtained.
Heating Field Conductivity Measurements
Measurements of e as a function of the heating electric
field were obtained for the case of the quartz bottle enclosed in
the cavity.
The electric field was measured according to the usual
microwave techniques.20
Transient operation was used to measure the
Q of the heating mode at the same time in the afterglow as when the
electron conductivity measurements were taken.
Even though the Q of
the heating mode was of the order of 200 or so, a change in Q of 10
to 200/o was obtained for electron densities of the order of 108
electrons/cc at a gas pressure of 5 mm Hg.
results for e
The averaged experimental
as a function of the electron temperature and the
electric field at the center of the quartz bottle are shown in Fig.
5.5.
The scatter in data is + 20/o.
The relation between the elec-
tron temperature and the electric field is given by Eq. (2.5) which,
2
-C
for helium, is Teo = Tg (1 + 0.24 EOh) where the field is in volts/cm.
The subscript zero represents quantities evaluated at the center of
the cavity.
Included in Fig. 5.5 is a sketch of the cavity and bottle
showing the relative positions of the coupling lines for the measurirg,
heating, and breakdown modes.
L
0.12
EXPERIMENTAL CURVE
-THEORETICAL CURVE FOR Pm -18.3
x x x xx POWER SERIES APPROXIMATION
-
-
0.10
-
--
0. 08
-'MEASURING
TO VACUUM
OUTPUT
HEATING FIELD
INPUT
0 .06
MAT CHED
LO AD
HEATING FIELD
OUTPUT
0.04
-MEASURING
INPUT 10.5 CM
PULSED
BREAKDOWN
9.5 CM
0.02
I
0
QUARTZ
BOTTLE
I '
zuuu
I
I
FIG. 5.5 -EXPERIMENTAL
I
I
I
I
6000
I
DATA OF
I
I
10,000
--
ar
-
I
I
I I I I I I I i
II I II I II
i
I
i
I
14,000
18,000
22,000
26,000
Te o (*K)
AS A FUNCTION OF ELECTRON TEMPERATURE AT
CENTER OF QUARTZ BOTTLE
-.21
a)
S
77.
The relation between (
and Pm depends upon the spatial
distribution of the electron density and the electric field, since
the velocity distribution function depends upon position.
The
electric field configuration is assumed to be that of the fundamental
mode.
When a dielectric is inserted into a microwave cavity, the
change in the resonant frequency of the cavity will be a first order
effect, while the change in electric field will be a second order
effect. 1 9
Since the quartz bottle introduces only a 30/o shift in
the resonant frequency, the perturbation of the electric field is
negligible.
In addition, the electron density distribution is assumed
to be cosinesoidal, characteristic of ambipolar diffusion.
Density
decay curves with zero electric field yield straight lines on a semilog plot of density versus time, indicating ambipolar diffusion as
According to the theory of ambipolar
the dominant loss mechanism.
diffusion in non-uniform fields, outlined in Chapter II, the deviation of the density distribution from a cosine distribution in the
quartz bottle is negligible over the range of electric fields used
in the experiment.
If a power series in velocity is assumed for Pm, the
collision probability, of the form
Pm
= b 1 + b2v + b3V2 + b v3 + ...
(5.2)
Eq. (2.3) becomes
=
SA
I5
s=1
(5.3)
r
78.
where
2hT
Am
(2 2
'r/2
s/2
f T )E1
2dV
( )
2 s/2
r(
ol'(5/2)
S=
f
s/2
2
ir/2
(
+ a cos 0.393u cos 0.449w)
I--
0938
5
fnE 2dV
dV
3
(cos w cos u cos o.393u dw du)
V
a'= 0.241 E
and
T = 300 0K.
when
The expression I can be calculated analytically when s is an even
number and must be calculated numerically when s is an odd number.
I is a function only of the electric field, Eoh.
The expressions
for I2 and I are
I2
1 + 0.86a;
The expressions for I
I = 1 + 1.73a + 0.754a2.
and I3 are shown in Fig. 5.6.
(5.4)
When Pm is
equal to 18.3 and constant, i.e. s equal to one, the plot of f as
a function of Teo , and hence a, is shown in Fig. 5.5 as the dotted
curve.
It is seen that the experimental and the theoretical curves
agree up to a temperature of 4000 0 K.
At low electron energies, both
the thermal and the heating field measurements yield the same value
for Pm.
This indicates that the higher values of
e
obtained in the
thermal measurements above 400 0K are not the true values, but are probably
L
8
6
t-44
0
(0-2
2
0
10
'0405 20
40
30
50
60
70
tsu
EA
FIG.
5.6
THE
VALUES
OF
1,
AND 12
VERSUS
0
-13
F
80.
due to impurities.
For a more accurate determination of the velocity
dependence of Pm, the experimental curve for e
terms of a series in 18 .
can be expressed in
The coefficients, A, are determined from
the experimental curve in Fig. 5.5.
The series approximation for
using four terms is found to be the expression
=
1.47 x 10-2I
The values of b
- 9.4l x 10-12 + 8.66 x l0-513 - 9.76 x 10-I4.
(5.5)
can be obtained from the values of A , according to
Eq. (5.3), giving the following expression for Pm
Pm = 18.5 - 7.38 x 10~ 9 v + 3.96 x 10-16 2 - 2.4
x
10-24v
(5.6)
where v is in cm/sec.
A plot of the momentum transfer cross section as a func-
tion of electron velocity in square root of volts is shown as the
solid curve in Fig. 5.7.
This plot is compared with the total col-
lision cross section data of Normand5 using the d.c. method.
Below
a velocity of 1.5 square root of volts, Normand's data have an
oscillatory behavior.
Since the microwave method cannot distinguish
such a behavior, the curve shown in Fig. 5.6 is an average of Normand's
data.
In addition, the momentum transfer cross section for the d.c.
method is derived by using Normand's data and the angular distribution data of Ramsauer and Kollath.3 0
It is seen that there is good
agreement between the microwave method and the d.c. method.
1-- 04
20
I
2
2
2
15V-
U
MOMENTUM TRANSFER CROSS SECTION
MICROWAVE METHOD
U
w
0
0
1o0k
TOTAL COLLISION GROSS SECTION
DC METHOD (NORMAND)
MOMENTUM TRANSFER CROSS SECTION
DC METHOD (NORMAND, RAMSAUER B
KOLLATH)
0
0
U
z
0
5k
0
0
I
0.5
I
1.0
I
1.5
2.0
I
I
2.5
3.0
IVOLTS
FIG. 5.7 - COLLISION CROSS SECTION AS A FUNCTION OF ELECTRON VELOCITY
82.
Measurements of P as a function of heating field were
also obtained in the copper vacuum cavity.
Fig. 5.8.
points.
The results are shown in
The experimental data are represented by the crosses and
The dotted curves are theoretical results for e
corresponding
to Case 3 in Section 11-3 in which the effects of energy gradients
are included.
It is seen that the predicted increase in e due to
the importance of energy gradients is evident from the data at
5 mm Hg.
At T
= 3300 0 K, the difference between the theoretical
curves for p = 5mm and 10 mm is 120/o.
The experimental data verify
that the density distribution predicted from the theory of ambipolar
diffusion in non-uniform fields is the proper one.
In general,
interpretation of measurements in the vacuum cavity would be rather
difficult for gases in which P
is not known.
A knowledge of PM is
necessary in order to calculate the proper density distribution for
the evaluation of ? .
A method of successive approximations can be
used in which a cosine distribution is assumed for the density and
a first order approximation is obtained for Pm from the measurements
of Q .
This velocity variation for Pm is used to calculate a second
order approximation for the density distribution.
This process is
continued until self consistent expressions for P
and density distribu-
tion are obtained.
This procedure is tedious and cumbersome and not
nearly as direct as are the corresponding measurements in the quartz
bottle.
00
0.035
0 00-
@
-0-
0.030F-00
0
e
--
.0-
oe-
o
Ale oo
0.025
4
0
5.0 MMHG
0
=
()Pe
=10 MM HG
0.020
THEORETICAL
CURVES
0.051I
0
t'00
Ta
FIG. 5.8
RATIO
CONDUCTIVITY
AT CENTER
2100
1500
900
OF
I
2700
I
3300
(* K
)
0
-
.0 -0
AS A
FUNCTION
VACUUM CAITY
OF ELECTRON TEMPERATURE
S4*
V-4. -Density Deay Measurements
During the course of measuring () as a function of time
in the afterglow, the data of the shift in the resonant frequency of
the cavity, and hence, the electron density as a function of time
are also obtained.
Density decay curves were taken in tank helium,
The
bottle helium, and "pure" helium, in the copper vacuum cavity.
decay curves in tank helium and bottle helium below 15 mm Hg gave
straight lines on a semilog plot of density versus time only over a
portion of the decay curve at late times.
At earlier times, the
decay curves were characteristic of recombination
higher modes. 31
or diffusion of
At pressures above 15 mm Hg, it was found that
plots of 1/n versus t were linear, which indicated that a recombination process was taking place.
The values of the recombination
coefficients obtained are of the order of 5 x 10~7 to 10-
cc/ion-sec,
where the values varied from one gas sample to the other and appeared
to be a function of the electron density during the discharge.
An
adequate explanation of these characteristics is not known, mainly
because of the lack of knowledge of the nature and amount of
impurities present.
The values of the ambipolar diffusion coefficients
times pressure, Dap0, for tank helium and bottle helium, obtained at
+
2
-
the late times are shown in Fig. 5.8. The Da o for He2 is 780 cm
0
2
12
mm Hg/sec12 and Dapo for He+ is 490 cm -mm Hg/sec as shown on the
graph.
The discrepancy between the measured values of Dapo and the
value of D p for He
above 3 mm Hg is believed due to attachment of
electrons to impurities.
k
The presence of 02 of the order of one
.
85
part in 104 or 10-5 can give rise to the measured discrepancies.
The amount of impurities is not necessarily the same for the data
shown for the tank and bottle helium over the pressure range and,
hence, only general trends can be predicted from the data.
Measurements of DaPo as a function of pressure for "pure"
helium are also shown in Fig. 5.9.
were used.
Two different breakdown conditions
The crosses represent data taken with short breakdown
pulses (1 is) while the points represent data taken with long breakdown pulses (several milliseconds).
The data for the short break-
down pulses agree with the measurements of Phelps. 3 2 At high pressures,
He2 is the dominant ion and at low pressures, He + is the dominant ion.
The discrepancy between the Da po for long and short pulses can best
be seen from a comparison of the density decay curves for the two
pulses.
Typical decay curves for long and short pulses are shown in
Fig. 5.10 in which the frequency shift in megacycles is plotted as
a function of time on a semilog plot.
The characteristic difference
between the two curves is the large production of electrons due to
metastable-metastable collisions for the long pulse discharge.
The
time constant for the loss of electrons through ambipolar diffusion
TD
1 a-
A2
A2 p
(5.6)
where Da is the ambipolar diffusion coefficient of He2 since the
.
pressure is greater than 2 mm Hg so that the dominant ion is He2
/\ is the characteristic diffusion length of the container.
L
The time
is
tooj
TANK
HELIUM
-
soo
It00
BOTTLE
N
C%4
U
HELIUM
8001
.. "PURE"
400-
I
0
I
2
I
I
4
I
PRESSURE
FIG. 5.9
EXPERIMENTAL
I
I
I
DATA
X SHORT
PULSE
* LONG
PULSE
I
(MM
OF
I
12
10
8
6
HELIUM
I
I
14
HG)
Dap
VERSUS
PRESSURE
0)
S
-
40
3020
+-SHORT
PULSE
(P,
8.I
M M HG)
lo76
LONG
PULSE
(P, 282 MM HG)
54S3-
0
8
16
TIME
FIG. 5.10
ELECTRON
32
24
40
48
56
(MILLISECONDS)
DENSITY
VERSUS
TIME IN
AFTERGLOW
Co
pr
88
constant for the production of electrons through metastablemetastable collisions is
2
D(M
2Cp
2
2
30)
+ 0.2p
(5.7)
where Dm is the metastable diffusion coefficient of the triplet
metastable atom since the pressure is sufficiently high so that the
triplet metastable atom is dominant over the singlet metastable atom.
cm 2/sec. 33 The
The triplet metastable diffusion coefficient is
P
2
term Cp represents the destruction of metastables by three body
-3.
2
collisions with two neutral atoms and is equal to 0.42p sec . The
factor of two in the time constant arises from the fact that it takes
two metastables to produce one electron.
For the short pulse decay curves, where the metastable
concentration is small and, hence, the electron production by metastable-metastable collisions is small, the decay of electrons is
dominantly a loss by diffusion and the time constant for this case
is given by Eq. (5.6).
The data in Fig. 5.9, represented by the
crosses, verify this conclusion, yielding a value of DaPo independent
of pressure (for pressures greater than 2 mm Hg) concurring with the
measurements of Phelps.
For the long pulse decay curves, where the
metastable concentration is large and, hence, the electron production
by metastable-metastable collisions is appreciable, the decay of electrons is a balance between the rate of loss by diffusion and the rate
of production by metastable collisions.
At early times in the
O9
afterglow, the rate of production is larger than the rate of loss of
electrons, producing the increase in electron concentration.
At
later times, the rate of loss becomes larger than the rate of production of electrons, but* if the decay time of diffusion is of the
same order as the decay time of production, the ratio of the rate of
loss to the rate of production remains approximately constant over
the measuring time.
The equation for the rate of change of electron
density with time is
-
Dn
where a
=
a
2
at
-i+2M
2
(5.8)
YD +i
is the ionization frequency per unit metastable concentration.
The time variation of M, the metastable concentration is
-t/T
M = M e(5.9)
At late times in the afterglow, Dan//\2 is greater than a M2so
that the electron density is assumed to have the form
n
n e~
(5.10)
.
Substituting Eqs. (5.9) and (5.10) into (5.8), one finds
a M2
1l
-T D
aiM e
T
D
m
D(5.11)
no
r
90.
If T is of the order of TD within 100/o or so, then for example at
pO = 5 mm Hg, 2/T
2
2
= 100 and l/T = 90 using /N = 1.7 cm.
The
time factor in Eq. (5.11) has the form e-lot which will decrease to
l/e of its value in 100 milliseconds.
Therefore, in Fig. 5.10 over
the time interval of 25 ms to 50 me, the decay time 1/T remains
roughly constant for the long pulse decay curve.
The effect of the
metastables will be to decrease the values of 1/T and accordingly
cause lower values of Da p
to be measured.
The true value of DaPo
can be obtained only by waiting a sufficiently long time so that the
rate of production of electrons is negligible compared to the rate
of loss by diffusion.
The data for the Dapo measured using long
breakdown pulses are shown in Fig. 5.9 and are represented by dots.
It is seen that these values of Dapo are lower than the Da p
He+ and cover a range from 600 to 750 cm 2 /sec-mm Hg.
2
for
The lower
values of DaPo corresponded to the presence of a larger value of
metastable concentration.
In order to present additional proof that the lower values
of Da
0
which were measured can be attributed to the presence of
metastables, the following experiment was attempted.
If the meta-
stable concentration could be decreased by properly irradiating the
plasma, then the Dapo should increase.
The triplet metastable atom
is difficult to de-excite by radiation since radiative transitions
from the triplet to the singlet state are forbidden, but it is
possible if the radiation can ionize the triplet metastable atom.
A double cavity as described in Section IV-8 was constructed.
If
91.
cavity one is operated in a steady state discharge, while cavity two
is operated in a pulsed discharge, the wire mesh separating the two
cavities allows all the radiation from cavity one to traverse into
cavity two.
Under these circumstances, the steady state discharge
radiation has the possibility of not only quenching but also producing metastables in the afterglow of cavity two.
The experiment
was performed to see which of the two effects is important.
At a pressure range of 5 to 10 mm Hg, it was found that
the radiation from cavity one increased the electron density in the
afterglow of cavity two and also decreased the value of Da o.
For
example at 8.0 mm Hg, the Dapo changed from 710 to 605 cm2/sec-mm Hg
while the electron density increased by 200/o.
If the number of
metastables produced is greater than the number of metastables
quenched, the metastable concentration in cavity two will increase.
The increase in metastable concentration produces an increase in the
electron density and decreases the apparent Dap0 measured according
to the discussion in the previous paragraphs.
At higher pressures,
the opposite effect occurs, for the electron density in cavity two
decreases when a steady state discharge is in cavity one.
Typical
decay curves at pressures of 21 and 47 mm Hg are shown in Fig. 5.11.
Both the decay curves with and without radiation applied are shown.
At these pressures, the number of metastables quenched is much larger
than the number of metastables produced in cavity two, so that the
decrease in metastable concentration produces a decrease in electron
density.
The experiment is useful from a qualitative point of view,
-
10
MM HG
8 -21
NO
ADIATION
76-
P, = 21 MM HG
PLUS RADIATION
W400
P, =47 MM HG
NO RADIATION
3-
z
z
P. P47 MM HG PLUS RADIATION
4
9
12
16
TIME
FIG. 5.11
ELECTRON
DENSITY
20
24
28
30
(MILLISECONDS)
VERSUS TIME IN AFTERGLOW FOR DOUBLE
CAVITY
34
93.
although nothing quantitative can be obtained from the measurements.
V-5.
Proposals for Future Work
The results reported in this thesis show that the velocity
variation of the collision probability for momentum transfer can be
obtained over a range of 0 to 3 e.v. electron energy.
The method
used is applicable not only for helium but for any gas as long as
the experimental conditions are such that the electron energy can be
related to the applied electric field.
This means that effects'of
impurities, energy conduction through gradients, changes in electron
density through non-uniform heating, and for molecular gases, energy
losses through vibrational and rotational excitations must all be
negligible.
For molecular gases, where vibrational and rotational
excitations are important, a measurement of the ratio m/M can be
obtained.
From the thermal measurements of the electron conductivity
information on the collision probability can be obtained.
Measure-
ments of the electron conductivity as a function of electric field,
coupled with a knowledge of the collision probability can determine
the ratio, m/M.
The deviation of m/M from the actual ratio of masses
gives a measure of the importance of inelastic collisions at low
electron energies.
The mechanism by which impurities increase the average
energy in a helium afterglow should be investigated.
By introducing
controlled amounts of impurities, while measuring the rate of change
of electron density and average energy with time, an insight into
94.
the important mechanisms involved may be obtained.
Different gases,
such as N2 , 02, A, whose ionization potential is less than the energy
of the helium metastable state, should be utilized.
In such an
experiment, the electron conductivity ratio would be used as a
detector of average electron energy.
In fact, in any gas where the
collision probability is known, the electron conductivity ratio can
be used as a measure of the average energy.
r
95.
APPENDIX I
AVERAGE ELECTRON ENERGY DURING THE AFTERGLOW
An expression for the average energy in the afterglow will
be derived.
According to the standard technique,
f is expanded
into spherical harmonics and into a Fourier series in time, i.e.
.
v
v
f= f+
0
V
f0
f0 + f e
~t0
ej0
and
0~~ 1+ i
+ ...
f=
f
+
e
+
(1)
From the Boltzmann Transport Equation, a set of four equations is
0l
, f and fl, assuming the following mechanisms:
obtained for f0 ,
a)
diffusion of electrons in a space charge field, E0
b)
application of a small high frequency field, Eh, which
is a function of position
c)
electrons undergo elastic collisions only
d)
the collision frequency is less than the radian frequency,
2
9 m <
2
Calculations show that f is negligible in our region of investiga0
tion but becomes important when the oscillation amplitude limit is
approached, i.e.
/u)=A
.
The three basic equations for the dis-
tribution function under the assumptions that f
is negligible,
96.
jo)f are
+
0
0f+
0
1
+ t. 3 -f-- =0
0) f
(JO)
+
(j +
where
G = G
2 [eo
Lv
l
S 3
C')G
(2)
1 edv
0 E0
-
6E=
Eh
1 = 00
(
e (real
+ m
M
6f+
6
Eq. (2) is the density equation, Eq. (3)
=
(4)
+ V (m
0 M1
m M
+ kT Z
M 43v
'
(
fF , and
is the d.c. current equa-
tion, and Eq. (4) is the a.c. current equation.
The only quantity
varying with time will be the electron density, so that in Eq. (2)
C fV,?t may be replaced by (On/at)(fg/n). The equation for the
-
average electron energy u
the condition that P
= my2 /2 is obtained from Eq. (2) under
is constant, which is a good approximation for
helium over the energy range of interest.
Equation (2) is multiplied
by 4rv3dv and integrated from 0 to O.
%In
-
ad
G dv
v + V.H -
0
where
H
J
0
7rv4fdv.
0
I
r
917.
Integrating by parts the last term in Eq. (5), we find
@0
a; +
.H
Gdv = 0.
+
(6)
0
The terms in the above equation will be evaluated separately.
Using
m= v/1 where f is the mean free path and independent
Eq. (4) and
becomes
Gd
of velocity, the term
0
fee
JGdv
0
0
V
df0
E
02
E
C
of2vFe21
)
+
\
Mvf0t
dv
;602
kT
0
0
4rv dv
2
4r2d
+()T
)+
=f
Gd
2n [I-v2
A M 1
22 m- a -.
where
(j
+ (
E)
0
**
Eof0
I
+I
*
1
(7)
47rv 2 dv.
The term I represents the energy lost to the space charge field,
E0
and will be treated as a perturbation term in Eq. (7). It will now
2
2
2
2
) and,
(ME)/42
kT +
be shown that I is small compared to
hence, can be neglected.
I will be evaluated using Eq. (3) and
assuming f6 to be Maxwellian with a temperature, Te.
I
=
-
5
2
iiE 02Vf0+
0
e 2dfO
)d
d(vE 0)dv
(8)
F
98.
df0
Substituting d
ef, we have
= -
d(v2)e
M
MI2
~ 6 ;
Assuming E
eE
Vn
n+kTi
e
e~
(9)
eE 2
kT en
j*
E
is of the form
D
0
T n
D )
(10)
.
a
K
D
kT
Substituting E0 into Eq. (9) and using-
-
Vn 2
n
-
I=
we find
,
Ke
(11)
^100D
,
D
Since D
1
D
and I becomes
MI2kT Da V 2
n
6m D
(12)
2
00
2n
f 0LGdv = M [
2 MF
-V
-2
3kT + (
M)
2
)
To a first approximation '2kT e
2
kT + (1)
m
2
4Cd
,
so that Eq. (7) is
2
2)(
40i
D Vn
1 - MQ2
D- n 2)
9m -A(-)
. (13)
11
r
99.
For an approximate evaluation of the perturbation term for helium,
0 0/400p2, D^ 100 D
andassume
Vn/n = tan
M/m ~,ooo,
22
2
1/40
a
/\
considering a one dimensional case and
1 cm.
~A
A
The perturbation
term becomes approximately
)Vl2
2 x 10- 2
n
2
D
2 x
A
Over a pressure range of 1 to 10 mm Hg and up to distance x ^ /
,
M
9m
the perturbation term is negligible compared to one and, hence, will
be neglected in the remainder of the derivation.
The next term in Eq. (6) which is to be evaluated is H,
which is defined in Eq. (5).
,
tion for f
IWe have, assuming a Maxwellian distribu-
0
(fg+o +-.0 f
dv
H
R
eE
00
0
eE
0
k
(15)
0)2
2
3
.
kTe''
Next we evaluate the product v Fwhere
particle flow defined by r
=
J
f0dv.
F
is the d.c.
The result is
0
2eE
Usin
1 + )m. ((16)
th=
Using the relation
v =
v
,
r n
multiply Eq. (16) by 3wr/S and
100.
subtract from Eq. (15), yielding
-.
3T -
eE
jo
v f-
2 n) -
+)(v
(
(17)
+V) (Vn).
(
H -
(
e
The terms containing E
H -
v
cancel and the remaining expression is
~v
vViij
v-
(18)
vn + ;n V - : Vvn.
8 (V
Cancelling terms in the above equation, we find the following
expression for H
H =
V r -
vn
v.
(19)
Combining Eqs. (13) and (19) into Eq. (6) using the continuity equation i n/gt = -'7.
, we have
9L
v
+
u=kT+( )
nv(v)-
. (v)
+31.
(20)
In order to evaluate the last two terms in the bracket, assume
F=
u
-VDan and from Eq. (16) D_ =
kT + ()
2 ME
2
R
. (D n
,
we have for the above equation
+(
-)v
2 Dn
+
3VDan .7i
(21)
101.
Using the same assumptions and limits as in the evaluation of the
perturbation term in Eq. (13), it can be shown that in the above
equation the last two terms in the bracket are small compared to
the first term for the case where the energy spatial distribution
is determined by the heating field distribution.
The last two
terms in the bracket are proportional to the term F introduced in
Eq. (2.6).
Therefore, the expression for the average energy including
the effects of energy gradients can be written as
kT + (e)
u
.
+
n
u)
(22)
46
or in terms of an equivalent temperature
T = T + a F + ;
. (nVT.)
where
a
0
M
6m= 2 k
and
b=
M
6mP2ii
.
0
(23)
102.
APPENDIX II
SOLUTION OF SECOND ORDER DIFFERENTIAL EQUATION BY RELAXATION METHOD
A solution of the following differential equation is
desired35
+ g
+ j-
+ k 0-- +9=
0
(1)
The characteristic
where m, g, j, k, and 9 are functions of w and u.
.
value, P, for the equation is contained in the expression for R
Equation (1) may be transformed to the following difference equation
by approximating the first and second derivatives at a point w,u in
terms of the values of the function at a distance h from w,u by the
following relations
9 m
-m(w
+
h~u) - 2m(wu) + m(w
h2
w2
-
h,u)
(2)
M- mw + hu) - m(w - hu)
2h
m(w + h,u)
+
l + h(Wu
m(w,u + h)
+ m(w,u - h)
+ m(w - h,u) [
-
(w,
)
With these transformations, Eq. (1) becomes
g + 1 k(w,u)
g
-
k(w,u) + 2 (w,u)m(w,u)
= 0.
(3)
103.
The problem consists of calculating the proper function m and value p
that will satisfy Eq. (3) over the region of interest.
The boundary
conditions are that m vanish along the border of a rectangle with
sides B and C.
Symmetry considerations can narrow the region in
which Eq. (3) must be solved.
The following technique is used.
The region of interest
is divided into a network of points separated by a distance h.
As
a first approximation to the function, it is assumed that m(w,u) =
cos irw cos ru.
Values of m(w,u) are assigned to each part in the
net and an arbitrary value for A is chosen.
The residual, R0 , at
any point is calculated in the following manner.
If Eq. (3) is
written as
m(w + h,u)a(w,u) + m(w - h,u)b(w,u) + m(w,u + h)c(w,u)
+ m(w,u - h)d(w,u) + m(w,u)e(w,u) = 0
(4)
then the residual at a point wu is defined as
R0 (w,u) = m(w + h,u)a(w,u) + m(w - h,u)b(w,u) + m(w,u + h)c(w,u)
+ m(w,u - h)d(w,u) - e(w,u)m(w,u).
(5)
The object of the calculation is to reduce all the residuals to
zero simultaneously for this occurs when the proper eigenfunction
and eigenvalue are used.
If, for example, the residual at
(wu) is
104.
made to vanish by decreasing m(wu) by the amount F%(w,u)/e(wu),
a change in the neighboring residuals must be made.
shown that the general rule is as follows.
It can be
When the function m
at a given point is decreased or increased by a quantity & , the
residual of a neighboring point must be decreased or increased by
an amount ).I where 'fL is the coefficient of the given point in the
difference equation for the neighboring point and is evaluated at
the neighboring point.
After relaxing the residuals somewhat, a new value of p
is obtained by averaging the values of P calculated from Eq. (5)
for every point.
This new value of 0 is used to calculate new
residuals which have to be relaxed.
This process is continued
until 0 and m(w,u) are essentially independent of the relaxation
method and hence the residuals are small.
It is advisable to
begin with a net containing a few points and increase the number
of points until an asymptotic value for 0 and m(wu) is reached.
105.
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3.
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106.
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107.
BIOGRAPHICAL SKETCH
The author was born on November 28, 1930 in Boston,
Massachusetts.
He attended the W. L. Garrison Grammar School and
in 1947 graduated from Boston Public Latin School.
He then
attended M.I.T. and received a B.S. in Physics in 1950.
He continued
in the M.I.T. Graduate School and was appointed a teaching fellow
for the 1950 fall term.
He was appointed a research assistant and
maintained the position from 1951 to 1953.
Sigma Xi at M.I.T.
He was elected to
He is coauthor of a paper entitled "Methods of
Measuring the Properties of Ionized Gases at High Frequencies. IV."
published in the Journal of Applied Physics, Vol. 24, August, 1953.
He presented a paper entitled "A Microwave Measurement of the Velocity
Dependence of the Collision Cross Section of Slow Electrons in Helium"
to the 1953 Conference on Gaseous Electronics.