An experimental study of the absorptionof energy from a large amplitude

electromagnetic pulse by a collisionless plasma

Item Type text; Thesis-Reproduction (electronic)

Authors Hyde, Richard Montgomery

Publisher The University of Arizona.

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AN EXPERIMENTAL STUDY OF THE ABSORPTION OF ENERGY FROM A LARGE AMPLITUDE ELECTROMAGNETIC PULSE BY

A COLLISIONLESS PLASMA

byRichard Montgomery Hyde, Jr.

A Thesis Submitted to the Faculty of theDEPARTMENT OF ELECTRICAL ENGINEERING

In Partial Fulfillment of the Requirements For the Degree ofMASTER OF SCIENCE

In the Graduate CollegeTHE UNIVERSITY OF ARIZONA

1 9 7 8

STATEMENT BY AUTHOR

This thesis has been submitted in partial ful­fillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library■to be made available to borrowers under rules of the Library.

Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment' of source is made. Requests for permis­sion for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use.of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED:7 ^

/

APPROVAL BY THESIS DIRECTOR This thesis has been approved on the date shown below:

____________ . / A ? / ?R. N. CARLILE

Professor of Electrical Engineering- f .Date

This thesis is dedicated to the memory of the author's father, who impressed upon him the necessity of obtaining an education, and to the author’s son, for whom he obtained it. . Maranatha!

H i

ACKNOWLEDGMENTS

The work presented in this paper was the result of a three-year team effort. Contributions to this pro­ject have been made by Dr. Robert N. Carlile, Dr. William A. Seidler, Dr. Robert N. Piejak, Dr. Gary L. Jackson,Dr. Alexander Cavalli, William L. Cramer, and Gary L. Hagedon. The author's principal contribution to this effort was the diagnostics mentioned in this thesis.

The author would like to thank Dr. Robert N. Carlile and Dr. William A. Seidler for their help in collecting and assimilating the data in this paper.The author would also like to thank Dr. Robert Piejak for showing him how to operate the experimental system and Dr, Alexander Cavalli for his patience and under­standing.

This work has been made possible through a re­search contract with the United States Air Force Office of Scientific Research AFOSR-76-2949B.

iv

T A B L E O F C O N T E N T S

PageLIST OF I L L U S T R A T I O N S .................... .. viiLIST OF TABLES . .............. ..................... xABSTRACT . . . . . . . . . ................ .. . . . xi

CHAPTER1. INTRODUCTION ........... . . . . . . . . . 12. THE THEORY OF THE ABSORPTION OF ENERGY

FROM AN ELECTROMAGNETIC PULSE . . . . . . 33. THE EXPERIMENTAL S Y S T E M ........... 8

. 3.1. Vacuum System .................. 83.2. External Magnetic Field . . . . 103.3- The P l a s m a ........... 12

3.3.1. Plasma Generation . . . 143-3.2. Plasma Parameters . . . . 17

3.4. Electromagnetic Pulse Generator 243.5. Waveguide and Transitions . . . 283.6. System for the Measurement of

Electromagnetic PulseAbsorption.................... 28

4. THE LANGMUIR P R O B E S ............. 334.1. Theory of Langmuir Probes . . . 334.2. Design and Use of Langmuir Probe 374.3. Experimental Data Obtained from

Langmuir Probe . . ............ 405. EXPERIMENTAL RESULTS . . . . . . . . . . . 56

5.1. Experimental Data . ............ 565.2. Comparison of Theory with

Experimental Results ......... 626. C O N C L U S I O N S ............... 73

v

vi

PageAPPENDIX A: MAXWELL’S E Q U A T I O N S ............. .. . 77LIST OF REFERENCES . .............. . 90

TABLE OF CONTENTS— Continued

LIST OP ILLUSTRATIONS

Figure Page1. Schematic of experimental system showing

location of TEM waveguide . . . . . . . . 92. The experimental system at The University

of Arizona . .................... 113• External magnetic, field used to confine

the nitrogen plasma . 134. Three-inch, dispenser cathode with grids , 155 • Schematic of the electron beam produced

nitrogen plasma ......... 166. Microwave cavity used for number density

measurements . ......... 217. Equivalent circuit of microwave cavity . . 228. The charge-line pulse generator used to

generate the fast risetime, high ampli­tude pulses 26

9. TEM waveguide with transitions . . . . . . 2910. Fourier amplitude of the pulse showing

that the dominant frequency componentsare between 200 MHz and 2 GHz . . . . . . 31

11. Block diagram of measurement system . . . 3212. Voltage-current characteristic of a

Langmuir probe in a p l a s m a ........... , , 3413. Ion current versus probe potential, for

various ratios of probe radius to electron debye length ............. . . . . . . . . 38

14. Longitudinal Langmuir probe . . . . . . . 39

vii

yiii

Figure Page15. Block diagram of equipment for ion

saturation plots 4l16. Ion saturation plot for experimental run

number 22 . . . . . . . . . . . 4217. Radial profile of electron number density 4418. Longitudinal profile of electron number

density for 0.4 microns . . . . . . . . . 4919. Longitudinal profile of electron number

density for 6.0 m i c r o n s ............. 5020. Langmuir probe calibration curve for 0.4

microns ................ 5221. Langmuir probe calibration curve for 1.0

microns . . . . . . . . . . . . 5322. Langmuir probe calibration curve for 3.0

microns . . . . . . . . . . . . 5423. • Langmuir probe calibration curve for 6.0

microns . . . . ................. 5524. Photographs for experimental runs 31, 26,

21, 16, 11, and 6 at 0.4 microns ofpressure ............................ 58

25. Photographs for experimental runs 32, 27,22, 17) and 12 at 1.0 microns of pressure . 59

26. Photographs for experimental runs 333 28,23) 18, 13) and .8 at 3.0 microns ofpressure ...................... . . . . . . 60

27. Photographs for experimental runs 343 29,24, 19) 14, and 9 at 6.0 microns ofpres s u r e.............. .............. 6l

28. Comparison of experimental and digitizedwaveform which was input to waveguide . . 64

LIST OF ILLUSTRATIONS— Continued

ixLIST OF ILLUSTRATIONS— Continued

Figure Page ■29. Comparison of output waveforms for run 26. 6530. Comparison of output wave forms for run 31, • 6631. Comparison of output waveforms for run 14, • 6732. Comparison of output waveforms for run 19, 6833* Comparison of output waveforms for run 29? 6934. Comparison of output waveforms for run 34. 7035. Effect of including faraday rotation

(Ex ^ 0) on output waveform for run 34 ? , 7236. Comparison of linear collisionless propaga­

tion with nonlinear collisional propagationfor run 2 6 ................................. 74

37- Comparison of linear collisionless, propaga­tion with nonlinear collisional propagation for run 34 . . . . . . . . . . . . . . . . . 75

LIST OF TABLES

Table Page1. Experimental run numbers with correspond­

ing values of pressure and electron number density .................................... 57

x

ABSTRACT

A theory was proposed that indicated that the high amplitude electromagnetic pulse generated by a high altitude nuclear detonation would be absorbed by the ionosphere.. A system was constructed to experimentally prove this theory. The output of a fast risetime3 high amplitude pulse generator was the input to a one meter long, TEM waveguide that was completely contained in a stainless steel vacuum chamber. The waveguide was immersed in a magnetically confined, nitrogen plasma column produced by an electron beam from a hot dispenser cathode. A longitudinal Langmuir probe was designed and constructed to measure the electron.number density and the electron temperature. The input and output wave­forms of the waveguide were recorded and compared to the theory. The comparison showed that the theory qualitatively predicted the experimental results, but that collisions were not involved. Therefore, the plasma was classified as collisionless.

xi

CHAPTER 1

INTRODUCTION

A high altitude nuclear detonation generates a high amplitude electromagnetic pulse which can propagate through space and damage communication satellites (Carlile 1975, p. 1). A theory has been proposed (Seidler 1975) that indicated that the D-region of the ionosphere, which is a plasma, would effectively absorb the energy of such an electromagnetic pulse, thereby re­ducing the threat to the satellites. This thesis presents some experimental results which support that theory. A brief presentation of the theory is given in Chapter 2.

At the University of Arizona Plasma Laboratory, an experimental system has been constructed (Carlile 1975, p • 1) which can create a weakly ionized nitrogen plasma having similar characteristics to the D-region of the ionosphere. Additionally, a high amplitude, fast risetime electromagnetic pulse generator has been con­structed (Carlile 1975, p. 18) to simulate the electro­magnetic pulse generated by a nuclear denotation in the atmosphere. The'experimental,system, including some diagnostic equipment, is described in Chapter 3-

1

2Chapter 4 is devoted to the Langmuir:.- probe which

was one of the more important items of diagnostic equip­ment. Two plasma parameters5 the electron number density and the electron temperature, were determined using this wire probei The theory, design, calibration, and use of the Langmuir: probe, is discussed in that chapter.

Chapter 5 contains the experimental results. It is there that the theory is modified to fit the limita­tions of the experimental system. Comparisons are made between the results predicted by the theory and the actual experimental results.

The conclusion is stated in Chapter 6 that the maximum pressure that was attainable in the system was too low for collisions to play a role in the experiment and therefore the results were collisionless.

THE THEORY OF THE ABSORPTION OF ENERGY FROM AN ELECTROMAGNETIC PULSE

The high amplitude electromagnetic pulse that is generated by a high altitude nuclear detonation can propagate into space and pose a potential threat to satellites. A theory has been developed (Seidler 1975) which predicts that an electromagnetic pulse will propa­gate nonlinearly through the D-region of the ionosphere and will cause rapid heating of the electrons in this region, causing considerable energy to be absorbed from the pulse. This theory uses the electron swarm model of the plasma:

'CHAPTER 2

(2.1)

(2.2)

g^r(nU) + -gV°H = —g-n6 E°v — nvuu (2.3)

3

4which are obtained by taking successive moments of the Boltzmann equation, and by making the assumption that the electron distribution function never deviates significantly from a Maxwellian (Carlile et al. 1978).In these equations, n, v, and U are the average electron number density, velocity, and energy in a small volume centered at r position and at time t; H is the heat flux; p is the isotropic electron pressure, assumed adiabatic; m and e are the mass and charge of the electron; E and B are the electric and magnetic fields at r and t; and , vm , and vu are the ionization, momentum transfer, and energy exchange collision fre­quencies, which are functions of the electron energy, U.

As has been suggested by Karzus and Latter (1965)5 the equations of the electron swarm model and the high frequency approximation of Maxwell's equations are transformed to retarded time x, where x = t - r/c and c is the velocity of light. . For plane waves in Cartesian coordinates, special variations of all quantities occur only in the z direction. There is a time independent electron number density nQ (z) and magnetic field Bq (z) in the z direction. Then, the x and y components of the wave equations for the electric and magnetic fields, transformed to retarded time, become

53Ex3z 2 0 (2.4)

+3z 2 noJy (2.5)

3Hx 1 . 3i“ " 2 Jy 0 (2.6)

9HZ + 0 (2.7)

where H = B(r,t)/u^ (yo is the permeability of free space) and j is the current density for electrons

j = -no (z)ev(r,t) (2.8)

Karzus and Latter (1965) point out that the transformation to retarded time requires that

3_ . 3_3t 3t

V ^ V - z(l/c)|- (2.9)

When this transformation is applied to the swarm equa­tions, Equation 2.3 and the x and y components of Equation 2.2 become

6

!y = e E*v - vu (.Usz)U (2.10)

(2 .11)

(2.12)

where = (e/m)B (z) is the electron cyclotron frequency. A complete derivation of Equations 2,ti to 2.8 and 2.10 to 2.12 Is contained in Appendix A.

model described above has been designed by Seidler (1977, p p . 8-15) which predicts the absorption*of energy from an electromagnetic pulse experiencing nonlinear propaga­tion in a plasma. The numerical technique employed is. a modification of the multistep predictor-corrector technique developed by Gear (1971) for solutions to stiff ordinary differential equations. The Equations 2.4 through 2.8 and 2.10 through 2.12 are adapted to this technique by dividing the transmission path along the z axis into 50 equal increments. The pulse input to this transmission path is at increment 1 and the output is at increment 50. Each increment is given a number and assigned a molecular number density of nitrogen, an ambient electron number density, and a non time-varying magnetic field.

A computer simulation using the electron swarm

This simulation will be utilized in Chapter 5 to compare the experimental results with the theory.It will also be used in Chapter 6 to show that3 while the nonlinear theory contains collision terms, the ex­periment was collisionless. This is the same as setting

and equal to zero in Equations 2.11 and 2.12, which causes Equations 2.4 through 2.8 and 2.11 and 2.12 to constitute a self-consistent linear set of equations.

7

THE EXPERIMENTAL SYSTEM

This chapter gives a description of the equip­ment used to conduct the absorption experiments.Included in this chapter is a discussion of the parame­ters of the plasma and a description of the electro­magnetic pulse generator.

3.1. Vacuum SystemThe vacuum chamber in which the experiments were

conducted was a stainless steel cylinder, consisting of two sections. The first section, which contained the cathode, was 46.5 inches in length with a n .8.125 inch inner diameter and a one-fourth inch thick wall. A cylindrical arm with radius of four inches and length of ten inches was located 29 inches from the cathode endplate and protruded from, one side of the section at an angle of 45° toward the cathode (see Figure 1).The second section was 28.5 inches in length and had an inner diameter of six inches with a 5/16 inch wall. The end plate of this section was the electromagnetic pulse input into the waveguide. A four-inch viewing port was

CHAPTER 3

8

TOP VIEW OF TEM WAVEGUIDE 301 STAINLESS 1/4" MESH .05"WIRE PLATE SEPERATION OF WAVEGUIDE IS 3.0 CM.

3 “ CATHODE

MAGNETSTO VACUUM PUMP

PLASMA

N GAS

INPUT

\ ^ r x n t x ^ \ f ^ x r i r x n i t x c i t r a n s it io n

EMP N OUTPUT

CONICAL TRANSITION

Figure 1. Schematic of experimental system showing location of TEM waveguide.

10located In this section, 16 inches from the end plate.

-7A very low vacuum pressure of 7 x 10 milli­meters of mercury (Torr) was obtained in the system by means of a six-inch diffusion pump which had an un- baffled pumping speed of 1500 liters/second. A mechanical forepump with a displacement of 500 liters/minute was used in conjunction with the diffusion pump. The diffu­sion pump was connected to the vacuum chamber below the cathode through a six-inch "Tee" union, about 7.5 inches from the cathode end plate. A pneumatically operated gate valve and a liquid nitrogen cold trap were located between the chamber and the diffusion pump and can be seen at the left end of the system as shown in Figure 2.

A molecular sieve foreline trap was located in the roughing line of the mechanical pump. The trap pro­tected the vacuum system from back streaming oil from the mechanical pump. It also removed water vapor from the air which permitted lower roughing pressures.

3.2.. External Magnetic FieldThe plasma was magnetically confined in the

vacuum chamber by an external DC magnetic field. This magnetic field was coaxial with the longitudinal axis of the system and was created by seven coils, each with 800 turns of 11 AWG aluminum wire. An additional coil

Figure 2. The experimental system at The University of Arizona.

UNIVERSITY OF ARIZONAPHOTO CREDIT : GEO. KEY*

12of 900 turns of 11 AWG. aluminum wire was used as a mag­netic mirror to reflect particles at the end of the system and create a more uniform plasma column. This coil was tightly wound on the outside of the six-inch diameter section at the end plate. The seven other coils had an inner diameter of 14 inches, an outer diameter of 17 inches, and were 5.25 inches wide. These coils can be seen in Figure 2. The resistance of each of the large coils was 6.7 ohms, while the resistance of the mirror coil was 7.5 ohms. Figure 3 shows the strength of the magnetic field along the axis of the system. The effect of the mirror coil is readily apparent at the end of the system. Due to the support structure necessary for the vacuum chamber, there was little flexibility available for positioning the coils. Variable resistors were placed in series with each coil to control the current flow through each coil. This helped to compensate for the positioning of the coils . and thus created a magnetic field of reasonably uniform strength over the length of the waveguide..

3.3. The PlasmaIn this discussion of the plasma there are two

areas of particular interest: the generation of the plasma and the measurement of the plasma parameters.

3 320

<D 160

S 120

48 44 40 36 32 28 24 20 16 12 8 4 0 EndPlate

Distance along waveguide (inches)1—1Figure 3. External magnetic field used to confine the 00

nitrogen plasma.

143-3.1. Plasma Generation

The plasma used for the absorption experiments was a 1weakly ionized nitrogen gas. The ionization was accomplished by a beam of electrons from a hot, gridded , tungsten dispenser cathode with a three-inch diameter.The cathode had a flat surface that was impregnated with barium oxide and was 20 percent porous. The cathode surface was held normal to the axis of the system during operation. Typical operating temperature of the cathode was 1150°C which was obtained by passing AC current of 27 amps through the cathode filament. Figure 4 shows the

ientire cathode assembly which included three grids.The reason for using three grids was to obtain

a controllablereproducible uniform plasma. The first grid, which was closest to the cathode, accelerated, electrons emitted by the cathode. The primary electrons emerged from the grid system with energies of 50 to 200 electron volts. This accelerating voltage was controlled by the cathode voltage (V ;• see Figure 5). To produce a long plasma column of uniform electron number density, it was necessary that the .mean free path of the primary electrons be comparable with the length of the plasma column which was approximately one meter. Therefore, the magnitude of the cathode voltage had to be increased for higher gas pressures. The primary electrons ionized the

Figure 4. Three-inch dispenser cathode with grids.

UNIVERSITY OF ARIZONA PHOTO CREDIT : GEO. KEW

Cathode Grids Magnetic Plasma CollectorField

SecondaryElectronPrimary

Electron

Figure 5. Schematic of the electron beam produced nitrogen plasma.

. 17nitrogen gas molecules 3 producing secondary electrons with relatively low energies (about four electron volts).The primary and secondary electrons were constrained by the magnetic field to create a cylindrical plasma with a three-inch cross-section.

The first grid was operated positive with re­spect to the cathode and controlled the electron current emitted by the cathode, which controlled the electron number density of the plasma. This grid was made from a sheet of l/l6-inch thick molybdenum. Holes, l/l6-inch in diameter, were drilled through this plate at regular intervals in a circular area three inches in diameter.This grid was placed 1/8-inch from the cathode surface.The other two grids were made of a coarse tungsten mesh. The three grids were separated by 3/4-inch spacers. .During the absorption experiment, the last two grids were kept at ground potential.

3 .3.2. Plasma ParametersThere were four plasma parameters of interest

in the experiments. Only the pressure and electron number density will be discussed in this section. The strength of the external magnetic field was discussed in Section 3.2 and the electron temperature will be discussed in Section 4.3.

' 18Pressure was measured by two types of pressure

gauges. One type was an ionization gauge formerly manu­factured by National Research Corporation of Newton, Massachusetts and now available from Varian Vacuum Divisionin Palo Alto, California. . The other type of gauge was a-Schulz-Phelps gauge which was available from the Granville- Phillips Company of Boulder, Colorado. The controller for this gauge was also available from the Granville-Phillips Company. The different pressure ranges on these two types of gauges was the reason for their use with the experimental system. The effective range of the Schulz-Phelps gauge was from 1.0 Torr down to 1.0 x 10~^ Torr, while the

__ g Qionization gauge was effective from 5 x 10 to 1 x 10 Torr. The ionization gauge was utilized to monitor back­ground pressure during cathode heating prior to the in­letting of nitrogen gas. The Schulz-Phelps gauge was the source of pressure readings during the absorption experi­ment.

Typical background pressure in the system was 5 x 10"7 Torr, This was obtained after first filling the vacuum chamber with .dry electronic grade nitrogen gas and then pumping on the chamber with the diffusion pump over­night. The four specific pressures of experimental interest were 0.4, 1.0, 3.0, and 6.0 microns (1 micron - 10*" Torr).

Since the gas inlet was located at the end plate and the diffusion pump was located at the cathode end of the system, a pressure differential existed along the length of the vacuum chamber. Two■Schulz-Phelps gauges and two Granville-Phillips controllers were used to measure the magnitude of the pressure differential along the length of the waveguide. Initially the two gauges were placed side by side in the end plate to calibrate them to obtain identical readings over the range from 1 to 20 microns.Then one gauge was moved to a porthole located at the curve in the waveguide. The ratio of the pressure at the end of the waveguide to the pressure at the end plate was unity for pressures up to five microns. Above five microns, the ratio dropped slightly but was still 0.97^ at 20 microns. For the pressures of experimental interest, the pressure differential along the waveguide was negligible.

One of the most important plasma parameters was the electron number density, measured in electrons per cubic centimeter. The specific experimental values of interest were 1 x 10^, 3 x 10^, 1 x 10®, 3 x 10^, 1 x 10^ and 3 x 10®. Accurate measurement of electron number density was a very time-consuming process and was finally accomplished by use of a Langmuir probe which is discussed in the next chapter. Initially, however, number density was measured using a microwave cavity.

19

20The technique of using a resonant cavity to

determine electron number density was used by Carlile (1963, P P • 51-58). He derived the following expression relating the perturbation frequency of the cavity with the plasma frequency (Carlile 19635 p. 53),

w—to ___ oCO 1 + V

(3.1)

In Equation 3.1, to is the perturbation frequency in the presence of a plasma, too is the cavity perturbation fre­quency in the absence of a plasma, to is the plasma . frequency, v is the collision frequency, and M is a con­stant that depends only on the unperturbed fields of the

2 2cavity. Provided v << to , as it was in this case, the value of M can be found experimentally. To do so, first a copper, cylindrical cavity was constructed which was 11 inches long and had an opening 3-5/16 inches in diameter (see Figure 6). It was a transmission coupled resonator with an equivalent circuit as shown in Figure 7. The center frequency was variable by means of a UHF frequency generator with a range of 800 to 2100 megahertz (MHz). The output of the cavity was fed into an oscil­loscope through a crystal detector to visually find the

Figure 6. Microwave cavity used for number density measurements.r \ j f—1

U m E H S l T Y OF AEIZ PEOIO CBEDII J GEO,-

22

Signal In

Signal Out

Figure 7. Equivalent circuit of microwave cavity.

resonant frequency of the cavity. In the absence of a plasma this frequency was found to be 1.233 gigahertz (GHz).

To experimentally find M, a rod of polystyrene with an outer diameter of 3.31 inches was placed in the cavity. It was found by Car'lile (19633 p . 5^0 that M is related to the perturbation frequency and the di­electric constant of the polystyrene by

23

M - fa )/fD£P 1 (3.2)

where f is the resonant frequency in the presence of Pthe polystyrene rod and f is the frequency in the absence of the rod, which was 1.233 GHz. The dielectric con­stant of the polystyrene was = 2.58 and M was experi­mentally found to be 0.1948. Using this value in Equation 3.1, it was possible to find the plasma fre­quency, (1) , for any given perturbation frequency. The plasma frequency is related to the number density by the formula (Chen 1974, p. 73)

,ne s1/2 ^me J (3.3)

where the units are in MKS.

2-4By substituting the value for M of 0,1948 into

Equation 3.1, the following relation is obtained

f = 2.2657/f(f-f0 ) (3.4)

Substituting Equation 3«4 into Equation 3,3 yields the relationship between number density in the cavity and the cavity resonant frequency in Hertz,

n = 6.36962 x 10“8 fCf-1.233 x 109) (3.5)

where n has the units of electrons per cubic centimeter.Equation 3.5 was then programmed into a computer

to obtain a listing of number densities for frequencies from 1.233 to 2.000 GHz with incremental steps of 0.1 MHz. The range of number densities that could be mea­sured with the cavity was from 7.85 x 108 to 9-767 x 10"^

„ The technique used for calibrating a Langmuir probe using the microwave cavity will be discussed in the next chapter.

3.4, Electromagnetic Pulse GeneratorIt is possible to simulate the electromagnetic

radiation released in a nuclear detonation with an electromagnetic pulse (Longmire 1978, p. 3). A pulse generator was designed and constructed by Hagedon (1975,

25p . 27) at The University of Arizona to generate a pulse with a 100 picosecqnd risetime and a variable falltime of 1 to 5 nanoseconds. This device was utilized to obtain the experimental results in this paper.

The pulse generator was a coaxial charged line utilizing a Gordos’ WR-126 mercury wetted reed switch (Gordos Corporation, Bloomfield, New Jersey) to obtain the fast risetimes with no switch bounce. The falltime was largely determined by the length of a brass slug (f) that was inside the coaxial cable (Hagedon 1975, p. 31).

tf = 2.2 RC

where

C =m ( ^ )1

and R = + Zq

From Figure 8 and the information from Hagedon (1975, p . 35)) the following values are obtained.

1

LARGE RESISTANCE ( » 5 0 r )

HIGHVOLTAGEINPUT

^ > > > > > > -> > > > > ^ ^

TYPE N CONNECTOR TO FIT HEREFigure 8, The charge-line pulse generator used to generate the

fast risetime, high amplitude pulses.

rv>o\

27Zq = 10.21 ohms

= 50 ohms Pq = 0.75 inches

= 0.53 inches

Hagedon (1975, p. 35) also found that for C = 8.816 picofarads, t equalled 1.062 inches. It was determined from the above information that to obtain a falltime of one nanosecond, £ must be 0.909 inches. It was decided to construct a switch having a falltime of about three nanoseconds. A slug of length 2.72 inches was computed to give a falltime of 2.992 nanoseconds. Such a slug was constructed and the falltime proved to be 3•0 nano­seconds .

The switching was accomplished by a solenoid aligned with the axis of the switch. . The solenoid was modulated by a multivibrator circuit with a variable oscillation frequency from 125 to 170 Hertz. The mercury reed switch was operated successfully with a maximum DC voltage of 25OO volts, negative. However, the switch would only function at that voltage with the repetition rate set at the minimum of 125 Her.tz. At the maximum voltage the risetime of the pulse was still less than 200 picoseconds (Hagedon 1975, p . 53)•

283.j5.. Waveguide and Transitions

A TEM waveguide was used to propagate the electromagnetic pulse through the plasma* This wave­guide consisted of two parallel plates of 0.25 inch mesh stainless steel screen seven centimeters wide and one meter long with a separation of three centimeters (see Figure 9). The characteristic impedance Of the waveguide was 109 ohms (Hagedon 1975, p . 34). The waveguide had a 45° bend at the output which fed into the 45° arm of the vacuum system (see Figure 1). At each end of the waveguide were 9•5-inch long conical tapers which provided a broadband transition to a twin- lead vacuum feed-through. The characteristic impedance of the waveguide in free space of 109 ohms was thus maintained inside the vacuum chamber (Hagedon 1975, p. 1). Outside the vacuum, broadband tapers designed by. Hagedon (1975, p * 34) provided a transition from 109 ohm twin-lead to 50 ohm coaxial cable. These transitions are the long, cylindrical objects in Figure 9 that are adjacent to the ends of the conical tapers.

3.6. System for the Measurement of Electromagnetic Pulse Absorption

The fast risetime of the electromagnetic 'pulsenecessitated using a sampling oscilloscope to observethe effects of the plasma on the pulse. The sampling

_Z

Figure 9• TEM waveguide with transitions.

035

oscilloscope used was a Tektronix type 66l with 4S2 and 5T1A plug-in units (Tektronix, Beaverton, Oregon).Since the maximum input to the 4.S2 sampling unit was one volt and the maximum trigger input to the 5.T1A timing unit was 250 millivolts, it was necessary to attenuate the pulse signal for use with the oscilloscope. Seidler.(1977/ P • 37) noted the high frequency components of the 'pulse (see Figure 10). Because of these high frequencies, special Weinschel broadband attenuators, rated from 0 to 12 GHz, were used (Weinschel Engineering, Los Angeles, California). In addition, a Microlab/FXR HM-30N sampling tee (Microlab/FXR, Livingston, New Jersey) was used to obtain a trigger for the 5T1A timing unit. A minimum of .40 nanoseconds of delay was required for the sampling oscilloscope. The delay lines that were constructed were one-half inch semiflexible, coaxial cable with a foam dielectric and were chosen to be 35 feet long (Hagedon 1975, p. 39).

The block diagram of the measurement system is shown in Figure 11.

30

Ampl

itud

e (V

/m/H

z)

31

Fourier transform of input E-field

Frequency (Hz)

Figure 10. Fourier amplitude of the pulse showing that the dominant frequency components are between 200 MHz and 2 GHz. — Taken from Seidler (1977).

(Trigger Signal)

Delay Line Delay Line

30dBAttenuator

ImpedenceTransition

ImpedanceTransition

WaveguideSampler

60dBAttenuator

10 times Attenuator

PulseGenerator

PowerDivider

Type 66lSamplingOscilloscope

(input signal to Channel A) (Input signal to Channel B)

Figure 11. Block diagram of measurement system.

uoro

CHAPTER 4.

THE LANGMUIR PROBES

It was found, during experimental operation that it was difficult to measure electron number densities

Obelow 1 x 10° electrons per cubic centimeter using themicrowave cavity described in Section 3.3.2. To overcomethis limitation, a Langmuir probe was constructed thatwould accurately measure electron number densities down

6to 1 x 10 electrons per cubic centimeter. This chapter contains the theory and use of this probe, as well as some of the data obtained experimentally.

4.1. Theory of Langmuir Probes The Langmuir probe was an insulated wire probe

which was inserted into the plasma column. By varying a DC voltage applied to the probe and recording the current received, a plot with the shape of Figure 12 could be obtained (Mlllman and Seely 1951, p . 287).As long as the probe was at a negative potential with respect to the plasma, a sheath formed around the probe to isolate it from the plasma (Mlllman and Seely 1951, p. 289). Only electrons with sufficient energy to

33

34

Figure 12. VoItage-current characteristic of a Langmuir probe in a plasma.

35overcome the potential drop across the sheath could be collected by the probe. As the probe voltage was made more negative, the thickness of the sheath increased which retarded more electrons. At a sufficiently high enough negative voltage, the probe was collecting only the ion saturation current, which is region AB in Figure 12. As the,voltage on the probe was made more positive, the probe began to collect electrons. The electron current was the difference between the total collected current and the ion saturation current. When the probe voltage reached the floating potential, point F in Figure 12, the probe was collecting equal numbers of electrons and ions and the probe current was zero. As the positive probe voltage was increased, the electron current increased rapidly to values that were usually one or two orders of magnitude higher than the absolute value of the ion saturation current. At point C, the current broke off when the probe voltage was equal to the plasma potential. If the voltage was increased further, a point would be reached (point D in Figure 12) where the electrons collected by the probe would have sufficient energy to ionize the molecules in the sheath. This would neutralize the electron space charge, the probe current would rise rapidly, and the probe would become an auxiliary anode (Miliman and Seely 1951, p . 287)

36Two plasma parameters could be determined from

the plot of Figure 12. The electron temperature in electron volts (eV) could be calculated by plotting the points on the curve between point B and point C and di­viding the change in the probe voltage by the change of the natural logarithm of the electron current. If the plot of In I versus probe voltage was a straight line, then the plasma had a Maxwellian distribution (Miliman and Seely 1951, p . 288). The electron temperature for the experimental system ranged from 2 to 6 eV, A nominal value of 4 eV will be used in this paper (Seidler 1977, p. 51).

The electron number density could also be de­termined from the plot of Figure 12. An accurate measurement of the number density was essential to the experiment, so that comparison could be made between the experiment and the theory. Laframbiose (1966, p. 48) has shown that the number density varies linearly with the value of the ion saturation current. This assumed a weakly ionized, collisionless plasma having a Maxwellian distribution. It was also assumed that the plasma was at steady state, all particle velocities are much smaller than the speed of light, and no magnetic fields were present (Laframboise 1966, p. 4).

37Figure 13 (Laframboi.se 1966) is a normalized

plot of ion collection as a function of probe potential for various ratios of probe radius to electron debye length with the ratio of ion temperature (T+) to electron temperature (T-) approximately equal to zero.In the experimental system, T- = 4 eV and T+ = 0.025 eV, so the ratio was .00625, and may be approximated by zero. Figure 13 shows that the slope of the curves becomes a constant at sufficiently high values of probe potential. It was possible, therefore, to calibrate a Langmuir probe with a microwave cavity and then to extrapolate linearly from these known points with a high degree of certainty. This procedure is described in Section 4.3.

4.2. Design and Use of Langmuir Probe A Langmuir probe was constructed using a

molybdenum wire, .025 inch in diameter. This wire was connected to a ConHex right angle cable plug. A glass tube was used to insulate all but the two-inch long collecting portion of the wire probe (see Figure 14).The complete Langmuir probe was then mated to a hermeti­cally sealed connector which was vacuum sealed inside one end of a 3/8-inch outer diameter stainless steel tube. A RG-188/U coaxial cable was attached to the other side of the connector and was fed out through the open end of the stainless steel tube. The entire longitudinal

Figure 13. Ion current versus probe potential for various ratios of probe radius to electron debye length. — Taken from Laframboise (1966).

m

Figure 14. Longitudinal Langmuir probeU)VO

UNIVERSITY OF ARIZONA PEOTO CREDIT j GEO. KEW

40probe was 78.5 inches long from the wire probe to the end of the stainless steel tube. The longitudinal probe was then inserted in a feed-through in the endplate with the collecting portion of the probe centered on the three-inch cathode, perpendicular to the axis of the system, and parallel to the plates of the waveguide.By sliding the probe through the feed-through, it was possible to measure the electron number density from the end. plate, along the axis of the system, to the grids . of the cathode.

The equipment required to obtain experimental plots like.those shown in Figure 13 and to measure the ion saturation current was an X—Y plotter, a DC power supply variable from 0 to -200 volts, and an HP 425A DC microvolt-ammeter (Hewlett-Packard, Palo Alto, California). The block diagram of this equipment is shown in Figure 15. The ion saturation plots obtained using this setup compared well to Figure 13. Figure 16 is a representative example.

4.3. Experimental Data Obtained from Langmuir Probe

To calibrate the longitudinal Langmuir probe, the microwave cavity described in Section 3*3.2 was placed in the system between the cathode and the probe. The longitudinal probe was placed adjacent to and

4l

(output from meter)

Meter Probe

(input from Langmuir probe)

DC Supply Voltage

Figure 15. Block diagram of equipment for ion saturation plots .

Prob

e Cu

rren

t (m

illi

amps

)

-90 -120 Probe Potential (volts)

Figure 16. Ion saturation plot for experimental run number 22.-Crro

43centered at the opening of the cavity. The electron number density was changed by varying the cathode emis­sion current. The value of the number density was de­termined using the microwave cavity and the corresponding values of probe current were recorded for probe poten­tials of -100 and -200 volts. Six or seven values of number density were selected for each of the four pressures, 0.4, 1.0, 3.0, and 6.0 microns. The electron accelerating voltages used for each of these pressures was -60, -90, -150, and -200 volts, respectively. The higher voltages were necessary to obtain a reasonably uniform number density over the length of the waveguide at higher pressures, as was mentioned in Section 3.3.1.

The values of electron number density obtained using the microwave cavity were based - on the assumption of a uniform, cylindrical plasma column exactly 3.31 inches in diameter. To accurately calibrate the longi­tudinal probe, it was necessary to determine how much the actual plasma column deviated from the ideal. A short Langmuir probe was placed in. the system through a port in the side of the system about half way along the waveguide. The probe was moved radially ' across the vacuum chamber to measure the relative amplitude of the number density. The resulting radial profile is shown in Figure 17.

Prob

e Cu

rren

t (M

icro

ampe

res)

44

Radial Distance (inches)

Figure 17. Radial profile of electron number density.

45To obtain accurate calibration plots, it was

necessary to find the ratio between the actual number density seen by the microwave cavity to the number density seen.by the two-inch Langmuir probe. The first step was to normalize the radial profile of Figure 17. The curve was drawn on 10 cm x 10 cm graph paper with the vertical scale in microamperes of probe current and the horizontal scale in inches measured radially. The ordinate of this curve is f(r) with the amplitude of the function normalized to 1.0 by dividing by 14.3 micro­amperes . The curve is related to number density by

n(r) = no . C.l)

where nQ is the maximum value of number density of the radial profile. The assumed value of number density measured by the cavity could now be related to the actual value of number density inside the microwave cavity.It was assumed that the radial profile inside the cavity was the same as that of Figure 17. The number of elec­trons in the cavity was then.

N = /n(v )dV

46

N - /n(r)dzrd(f>dr (4.2)

Equation 4.2 is in cylindrical coordinates. Integrating with respect to dz and d*# yields

N = L/n(r) 27rrdr (4.3)

where L is the length of the cavity. Substituting Equation 4.1 into 4.3 gives

2irLnN = g ■ /f(r)rdr (4.4)

By definition, the number density measured by the cavity is equal to the average number density in the cavity times the volume of the cavity.

N = n tt a^L (4.5)

In the above equation, a is the cavity radius which was 1.655 inches.

Equating 4.4 and 4.5 and solving for n yields,

2nnav = (1.655)(14.3) /f(r)rdr (4.6)

Using Figure 17, f(r)rdr was determined graphi­47

cally by letting the three-inch point on the horizontal axis to be. equal to r = 0 and using dr = 0.25 inches .The total value of the integral was then 27.06 square inches. Substituting this value into Equation 4.6 gives

Therefore the average value of number density measured by the microwave cavity was 1.38 times the peak value of number density in the radial profile.

The average number density measured by the Langmuir probe was

where £ is the probe collection length, two inches.

(4.7)

1i "?np = j f n (r ) dr

-l2

(4.8)

Substituting Equation 4.1 into Equation 4.8gives

_ 1 no f 1 f(r)drnp - 2 1473

(4.9)-1

48Again the value of the integral was determined graphi­cally from Figure 17 and was found to be 27.45 inches. Equation 4.9 now becomes

n = 0.956 nQ (4.10)

The number density measured by the probe could now be related to the number density measured by the microwave cavity from Equations 4.7 and 4.10.

nav0.9561.38 0.695

So the number density measured by the probe was 0.695 times the number density measured by the microwave cavity.

A longitudinal number density profile was obtained for each of the experimental runs using the longitudinal probe described in the previous section. It was found for the three lower pressures that the number density varied by less than ten percent over the length of the waveguide (see Figure 18)3 but for six microns the number density at the end of the waveguide was twice the value at the end plate (see Figure 19).

The radial and longitudinal profiles were used to properly adjust the measurements taken with the

49

Distance (inches)

Figure 18. Longitudinal profile of electron number density for 0.4 microns. — Taken from Seidler (1977).

50

Distance (inches)

Figure 19. Longitudinal profile of electron number density for 6.0 microns. — Taken from Seidler (1977).

51Langmuir probe and the microwave cavity. Plots of number density versus probe current were made for each of the four pressures and are shown In Figures 20, 21, 22, and 23. These plots were used to find the correct amount of probe current for any given value of electron number density.

(ele

ctro

ns/c

m52

-100 Vprobe voltage

-200 Vprobe voltage

- - - - - - G 0

0.01Probe Current (milliamps)

Figure 20. Langmuir probe calibration curve for0-4 microns.

(ele

ctro

ns/c

m53

-100 Vprobe voltage

-200 Vprobe voltage

0.01Probe Current (milliamps)

Figure 21. Langmuir probe calibration curvefor 1.0 microns.

(ele

ctro

ns/c

m

-100 V •probe voltage

-200 Vprobe voltage

0.01Probe Current (milliamps)

Figure 22. Langmuir probe calibration curve for 3-0 microns.

(ele

ctro

ns/c

m55

-100 Vprobe voltage

-200 Vprobe voltage —

0.01Probe Current (milliamps)

Figure 23. Langmuir probe calibration curvefor 6.0 microns.

CHAPTER 5

EXPERIMENTAL RESULTS

5.1. Experimental Data Using the experimental system described in

Section 3.6, 24 runs were made at six different values of electron number density and four different pressures.In Table 1, the run numbers are shown with their corre-

__ ospending values of pressure in microns (1 micron = 10

Torr) and electron number density in electrons per cubic centimeter (cm-^). The magnetic field was not changed during the 24 runs and is shown in Figure 3• The pulse generator described in Section 3.4 was set at a poten­tial of -1420 volts. For each experimental run, the input and output pulses were displayed on the sampling oscilloscope and a photograph was taken. These photo­graphs are shown in Figures 24, 25, 26, and 27 for pressures of 0.4, 1.0, 3.0, and 6.0 microns, respectively. The photographs in each pressure range are shown in decreasing value of number density. In all cases the input pulse remained the same and is shown as the top trace in the photographs. The horizontal scale is one

56

Table 1. Experimental run numbers with corresponding values of pressure and electron number density.

Pressure (microns)

Electron Number Density (cm ~>)1 x 109 3 x 10y .1 x 10ti 3 x 107 1 x 107 3 x 106

0.4 31 26 21 16 11 61.0 32 27 22 17 12 73.0 33 28 ' 23 18 13 86.0 34 29 24 19 14 9

VJ1

58

bSHIHI BSiiHBSSSIB^ ■ e e i s s

m s s i K aFigure 24. Photographs for experimental runs 31, 26,

21, 1 6 , 11, and 6 at 0.4 microns of pressure.

UNIVERSITY OF ARIZONA££020 CREDIT % GEO. KEW

Figure 25. Photographs for experimental runs 32, 27, 22, 17, and 12 at 1.0 microns of pressure.

UNIVERSITY OF WIZOKA PHOTO CREDIT ; GEO. EE6

BaBBBSBgjg

■ B a a■ B B S !

laeBBSsal a s a a a

Figure 26. Photographs for experimental runs 33, 28,2 3 , 1 8 , 13, and 8 at 3.0 microns of pressure.

UNIVERSITY OF ARIZONAFBOIO CREDIT i GEO. KE»

6l

BBSSSIh

BSBHeEHiaiSB SFigure 27. Photographs for experimental runs 3 , 29,

24, 19, 14, and 9 at 6.0 microns of pressure.

#3% "039 : $10330 0$0HdVNOZIHY 30 HISHSAINn

62nanosecond per division and the vertical scale is 200 millivolts per division. Figures 243 25, 26 and 27 show that the pulse amplitude decreases with increasing number density and pressure. How well the theory presented in Chapter 2 could predict the experimental results, is the subject of the next section.

5.2. Comparison of Theory with Experimental Results

The nonlinear theory discussed in Chapter 2 does not include the effects produced by the metal plates of the TEM waveguide. According to the theory, as the pulse propagates in the z direction, the electric field will slowly rotate with distance in the XY plane (Carlile et al. 1978). This phenomenon is commonly called faraday rotation. Therefore, in the theory, the electric field will'.'develop an x component and the time- varying magnetic field will develop a y component.

The plates of the waveguide prevented faraday rotation by constraining the electric field to be in the y direction and the time-varying magnetic field to be in the x direction. The computer simulation was modified to account for the waveguide effects by setting the x component of the electric field ( E ) and the y component of the magnetic field (H ) equal to zero at the output of each of the 50 incremental distances. In this way.

the computer code prevented faraday rotation just as the TEM-waveguide did in the experiment.

The input pulse to the waveguide is shown in Figure 28 as well as the digitized waveform that was used by Seidler (1977, p . 36) as the input for the com­puter simulation. To make a comparison between the theory and the experiment, six representative runs were selected. They were runs 26, 31, 14, 19, 29, and 34, and are in Figures 29, 30, 31, 32, 33, and 34, respec­tively. The waveforms for 6.0 microns. Figures 31, 32, 33 and 34, are particularly interesting. In Figure 31, the output pulse closely follows the input pulse shown in Figure 28. However, as the number density increases, the output pulse first decreases in amplitude (Figure 32) and then begins to ring (Figures 33 and 34). The ringing suggests that the plasma may be excluding bands of frequencies from the pulse (Carlile et al. 1978).

It should be observed in Figures 29 through 34 that the theory predicts the experimental runs, quali­tatively. The theory waveforms have the same shape as the experimental waveforms, but they have different full-widths-half-maxima. This difference appears to be due to the band pass of the experimental system, since the pulse is broadened by its propagation through a low electron density plasma (Seidler 1977, P- 60).

63

64

ExperimentalDigitized

3 080

1.0 ‘

Time (nanoseconds)Figure 28. Comparison of experimental and digitized

waveform which was input to waveguide.— Taken from Seidler (1977).

pressure N2 -OA/jl

THEORY

EXPERIMENT

T I M E ( n s )

Figure 29. Comparison of output waveforms for run 26.— Taken from Seidler (1977).

OUT

PUT

(vol

t/cm

)

pressure N2 s OAfj.

THEORY

EXPERIMENT

T I M E ( n s )

Figure 30. Comparison of output waveforms for run 31 — Taken from Seidler (1977).

Outp

ut v

olta

ge67

ExperimentPrediction

Time (nanoseconds)

Figure 31- Comparison of output waveforms for run 14. — Taken from Seidler (1977).

OU

TPU

T (v

olt/

cm)

68

p r e s s u r e N g = 6 . 0 yx

THEORY

EXPERIM ENT

T I M E ( n s )

Figure 32. Comparison of output waveforms for run 19.— Taken from Seidler (1977).

OUT

PUT

(vol

t/cm

)

69

pressure Ng = Q.OfJ. A ^THEORY

EXPERIMENT

T I M E ( n s )

Figure 33. Comparison of output waveforms for run 29.— Taken from Seidler (1977).

OUT

PUT

(vol

t/cm

)

pressure N 2 z 6.0yu.

-THEORY

EXPERIMENT

T I M E ( n s )

Figure 3^. Comparison of output waveforms for run 34— Taken from Seidler (1977).

Figure 35 uses the data from run 3^ to show the difference in the theoretical predictions for the case where faraday rotation is not allowed (E = 0) and the case where faraday rotation is allowed (E ^0). The experimental data of Figure 34 is observed to correspond more closely to the E^ =

71

0 theory.

OUT

PUT

(vol

t/cm

)

pressure N2 = 6.0/x

E. # 0

0 . 5 1 .0 1.5 2 . 0 2 . 5 3 . 0T I M E ( n s )

Figure 35. Effect of including faraday rotation (Ex / 0) on output waveform for run — Taken from Seidler (1977).

CHAPTER 6

CONCLUSIONS

The experimental results presented in Chapter 5 have been shown to agree.qualitatively with, the theory of nonlinear propagation discussed in Chapter 2. The experimental evidence definitely supports the belief that the high amplitude electromagnetic pulse generated by a high altitude nuclear detonation would be absorbed by the plasma of the ionosphere.

A comparison of the experimental results with the theory was made to determine if the pulse propagation was linear or nonlinear. If the experiment was linear, then the collision terms, vm and in Equations 2.10 through 2.12, could be set equal to zero with little change in the output waveform. This comparison is shown in Figures 36 and 37 using the data of Figures 29 and 34. It is apparent from the figures that the experiment was conducted in a collisionless plasma. For collisions to play a role in the experimental results, the pressure would have to be increased by about an order of magnitude

There were two factors at the University of Arizona Plasma Laboratory that prevented obtaining higher

73

74

-120 Collisional— Collisionless

Time (nanoseconds)Figure 36. Comparison of linear collisionless propaga­

tion with nonlinear collisional propagation for run 26. — Taken from Seidler (1977).

80

60

40

20

0

20

40

60

80

75T T

CollisionalCollisionless

_______ !________ 1________ I________ I________ 1_________0 0.5 1.0 1.5 2.0 2". 5 3.0

Time (nanoseconds)37- Comparison of linear collisionless propaga­

tion with nonlinear collisional propagation for run 34. — Taken from Seidler (1977).

operating pressures than 6.0 microns. First5 the liquid nitrogen cold trap that isolates the vacuum chamber from the diffusion pump was not designed to operate at pressures higher than ten microns. Secondly, the hot dispenser cathode did not emit properly at higher pressures. Both of these problems could be solved if a baffle were placed between the cathode and the wave-r guide such that a pressure of 100 microns could be maintained on the waveguide side while a pressure of less than 1.0 micron existed in the area of the cathode and cold trap. Such a baffle has been designed by Dr. R. N. Carlile at The University of Arizona. If successful, the baffle could generate operating pressures high enough to prove the collisional aspects of the absorption theory.

76

APPENDIX A

MAXWELL»S EQUATIONS

A uniform plasma filling all space with a uni- __

form magnetic field, Bq = z Bq , Is assumed. The plasma is excited by an infinite plane source parallel to the y axis. Let y be normal to the source plane. Let (w,y,u) be a new coordinate system (Cartesian) where wy plane is parallel to the source plane. In the source plane, it is required that•

M M = m = m = o3w 3y 3w 3y

so E and H are the same at every point in the source plane. Thus,

E = E(u) and H = H(u) .

For plane waves, ■

77

78In what follows 3 let u = z so

Retarded Time Transformation

g(z,t) = f(z,T)T = t - Z/C

where c equals the velocity of light.

is , !£9Z dZ

, 9f 9t + 3? 3i

9f9z

Ay At Ay

11 = - 1 9z c

9z t1 9f c 9t

Vt z i d_ c 9t

i£31 i£9t

A t = 0

Also

Vt X E79

= -y o3t

V X H - I + e0||

V ° E ■= p/e

V - H = 0

Transformed equations

z 35 x Etn 9Et 9^

z 5 x a / = - % sT

- 3 - * 1 3Ht - 3E.z 3 i x H T - z c X 'fi---- 3t + eo'TT

z 3 i • E t - z • c — = p T / e o

z f c " H t - z • i 0

3Hz = 1 3Hz 3z c 3t

Hz = 0 since there is no source for Hz

80Wave equation for H

s x cS x ? H]z i z x j + e o z i z x l l

A(z x z x H> - z k x z x ! #; x j + £o || + z | x ||]

k z x J - V o . 7 7

+ z x I k^ k x H - z x H? - J]Since . z x: z x H = z Hz - H

H f z H z - R] +1 k[k(x Hx + ? Hy>]k z x J = -k 77 ; H, - I k(y j* -x 3y)

81x component:

32H x a , _ 1 3 3Hx ^" : 2 - + o' 3f(2 J z ~ ~ Jy)

y component:

- f e i x = - & & ( 2 + v

z component:

H = 0 z

Wave Equation for E x component:

92Ex , 1 3 rn^ + =

3Exno V

y component:

i!ii + i |?[2 li£ + '1o:;jy:i 0

82z 'component:

o

]_t ^where n = [■ ■]!= 377 ohms

o

Averages and Definitions

L e t v t - [ ^ ] V 2

f = fx fy fz

_ rn(.r,t)-,1/3 1 (WX -fx - C no ] - , 2

fy - [ ^ 3 1/3 7 ^ - e -

f - rn(r,t)1l/3 1fz " L n J V tTv~ e(wz " V

w = position vector in velocity spacex wx + y wy + z wz

83nQ = average election number density.V = v(r,t) = average velocity.It is assumed that the form of the distribution function is a displaced Maxwellian,Averages:

nQ/f d^ w = n(r,t)

n/w q f d°w = n(r3t)q v(r,t)

= j(r,t) (coul/sec/m )

n"^n^f^ m w 2f d^w = ^ k T + ^ m v2

= j U(r3t ) (joule)

H(r,t) = heat flux

H(r,t) = n /w(5- m w 2)f d^ w

= ^n(r,t)v(r,t)U(r,t)

+ n(r,t)v(r,t)k T (joule/sec/m2)

It is assumed that while v may respond rapidly to ex­ternal forces on the plasma, k T = remains constant since is determined by electron-electron collisions which is a slow process.

84H = n /w-^-mw^ f d ^ w o d

1 2 8 Hz = n m w w f d w o 2 z

2 2 2 2Since w = w + w + w , Hz can be determined by x y zevaluating the x, y s and z components of the integral,

'*z ”x2 f d3 w = I Vz(v r + vx2>o

/wz wy 2 t w = s- vz(5LX + vy 2)

f w = §— v„(3 + v^2)n z m zo

H z = I ™ n v z[¥ + vx2 + V T + vy 2 + ^ + ^

Hz = n vzC|- k T + |- m v2]

Since U = k T.} o

n vz[2 Uo + I m v +

H = - n v U + n v U z 2 z z o

85Moments of the Boltzmann Equation1. First Moment

V n v = n Vj

2. Second Moment

n m. + v - V v] = - n e [E + v x B] - Vp

- n m vM » v

3. Third Moment

■|+r(n U) + & V » H = -^ n e E « v - n v U d U j j U

' Transformation of Moment Equations to Retarded Time1. First Moment

3n , 3_ 9t 9z (n v z ) 1 9_

c 9t (n vz) = n v I

2. Second Moment

9 —v v z v

86v ° v " 1 9

Z c 9? v

_ ' 9v Iz 9v. z 9z c 9t

9v , 9v vz 9v 9t z 9z c 9t

- ^[E + v x B(r,t) + v x BQ(r)]

k ( Y n k T ) + s i I j r ( r n k T) - =Vm

Y = = ratio of specific heatscv •

y = (2 + N)/N (adiabatic compression)

N = degrees of freedom

v

Y = 5/3 for N = 3

For Bo (r) = Bq z

x component:

9vx9t + 3 ^ v 9v z x

c 9t — E + ^ v B m x m z y

87em vy Bo vX

y component:

3v 3v 3v 3v_ X + - J L v z3t 3z vz c 3t Bx

+ £m Bo Vy

z component:

- E m z

- 5 ( v x B y - v y B x >Y k t 3n nm 3z

, y k T 1 3n+ "T5i c W Tz

3. Third Moment

3(n U) , 2 3Hz 2 1 3Hz3?---- + 3 3i 3 o St-

n v u u

88Summary of Transformed Equations

(1)32E 9E'X

3i~ + % J’x ] = 0

(2) 32Ey _L i a_3%__+ — -2— r p v2 c 3t L 3z + V y ]

3EZ ^(3) z T * 7 ^ 3 z

(4) l!!x + M x3z2 ^

1 3_[2 ! 5 lc 3t 3z jy ]

C5) 8 H y _ M i3z2 3z - ? I?C2 E 1 + Jx]

(6) Hz = 0

(7) (n v w) - % -|t (n ) - n.v.c 3t

(8) !Z* + ! Z i v . i ! ! i3t 3 z z c 3t

- - E + - v B - - v B m x m z y m y o XVx

89(9) f Z z + f Z z V _ I z3t 3z z c 3t

* I Ey " I Vz Bx + m Vx Bo " VMy Vy

|i*» ■, ■ ’, » » > - Ii H S

+ T K.-— — - v vmn c 3t Mz z

(11) & ( n u) + | | M - | i |f£

- ne E • v - n vu U

(12) j = - n e v

LIST OF REFERENCES

Carlile, R. N. Experimental Investigation of Microwave. Propagation in a Plasma Waveguide. Internal Technical Memorandum No. TM-30, Electronics Research Laboratory, University of California, Berkeley, California,August 1963.

Carlile, R. N. An Experimental Study of the Nonlinear • Propagation of an Electromagnetic Pulse' through' the Ionosphere. AFWL-6TR-75-H5, Air Force Weapons Laboratory, December, 1975.

Carlile, R. N ., ¥. L. Cramer, R. M. Hyde, Jr., and W. A. Seidler. Absorption of Energy from a Large Ampli­tude Electromagnetic Pulse by a Collisionless Plasma. Unpublished manuscript, University of Arizona, Tucson, 1978.

Chen, F. F . Introduction to Plasma Physics. New York: Plenum Press, 1974. ' ~

Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equations. Englewood Cliffs, N.J.: Prentice-Hall,. Inc., 1971.

Hagedon, G. L. Instrumentation for an Electron Beam Plasma System. Unpublished thesis. University of Arizona, Tucson, 1975. '

Karzus, W. S . and R. Latter. Detection of the Electro­magnetic Radiation from Nuclear Explosions in Space. Physics Review, Vol. 137, No. 5B (March 8, 1965), pp. B1369-1378.

Laframboise, J. G. Theory of Spherical and Cylindrical Langmuir Probes in a Collisionless, Maxwellian Plasma at Rest. Report No. 100, Institute for Aerospace Studies, University of Toronto, June 1966.

Longmire, C. L. On the Electromagnetic Pulse Produced by Nuclear Explosions. IEEE Transactions on Antennas and Propagation, Vol. AP-26, No. I- (January 1978), pp. 3-13.

90

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McGraw-Hill Company, 1951.Seidler, W. A. Pulsed Power Heating of the D-Reglon.

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