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NATL INST. OF STAND & TECH

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APR 2 7 1964

NBS

technical ^Tiote 91©. J2//

Volume 1

CONFERENCE ON NON-LINEAR

PROCESSES IN THE IONOSPHERE

DECEMBER 16-17, 1963

EDITORS

DONALD H. MENZEL AND ERNEST K. SMITH, IR.

Sponsored By

Voice of America

and

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
National Bureau of Star AUG 5 1965

150,748
NATIONAL BUREAU OF STANDARDS

technical ^ote 211, Volume 1

Issued April 17, 1964

CONFERENCE ON NONLINEAR PROCESSES

IN THE IONOSPHERE

DECEMBER 16-17, 1963

Sponsored by

Voice of America

and

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U. S. Government Printing Office Washington, D.C. 20402

Price 50$
PREFACE

This Conference was conceived by Professor Donald H. Menzel

about a year ago. In view of Dr. Ernest K. Smith's concern with the

effects of high-power transmissions in the ionosphere stemming from

his association with the Voice of America, I asked the two of them to

act as Co-chairmen for this Conference.

These gentlemen, together with an Advisory Committee made

up of Professor David Layzer of Harvard, George Jacobs of the Voice

of America and Dr. James R. Wait, Roger Gallet and myself from the

Central Radio Propagation Laboratory have been responsible for the

technical planning. Robert T. Frost was in charge of local arrangements and also served as Secretary/Treasurer for the Conference.

The non-technical editorial work was carried out under the

supervision of Mrs. Mildred F. Talbutt, ably assisted by Mrs. Dorothy

M. Burdick and Mrs. Anna M. von Kreisler. Dr. J. Robert Lebsack

of the Technical Information Office has contributed very usefully to

the planning of the publication phase.

Most of the papers presented at the meeting are published in

the six volumes of this Technical Note No. 211. Primary responsibility for their technical content must rest, of course, with the

individual authors and their organizations.

C. Gordon Little

Director, Central Radio

Propagation Laboratory

ii
Volume I

Table of Contents

Session 1. Collisional Radio -Wave Interactions, Part I

Chairman: O. E. H. Rydbeck

1.1 Some general remarks on non-linear processes.

Donald H. Menzel, Harvard College Observatory ... 1

1. 2 On some non-linear phenomena in the ionospheric

plasma.

P. Caldirola and O. De Barbieri, Istituto di Scienze

Fisiche dell ' Universita di Milano 27

Additional papers for Session 1 continued in Volume II.

Other Papers Presented at the Conference on Non-Linear Processes

in the Ionosphere*

Artificially stimulated VLF emissions. R. A. Helliwell, Stanford

University, Palo Alto, Calif.

Natural VLF emissions. Roger M. Gallet, CRPL-NBS.

Plasma resonances observed by Alouette satellite. Wynne Calvert,

CRPL-NBS.

Plasma instability in the whistler -mode caused by a gyrating electron

stream. T. F. Bell and O. Buneman, Stanford Electronics

Laboratory, Stanford University.

Ionospheric wave interaction using gyro-waves. Part II. The D region

R. A. Smith, The University of New England, Armidale,

N. S. W. , Australia

Copies of these particular papers are not available from the

Conference Organization; however, in some cases they may be

obtained directly from the authors at the addresses indicated.

in
Complete Contents

Session 1.

1.1 Some general remarks on non-linear processes,

Donald H. Menzel .Vol. L « . .

s ession 2.

2. 3

2. 5

s ession 3.

3. 1

3. 2

1. 2 On some non-linear phenomena in the ionospheric

plasma. P. Caldirola and O. De Barbieri. . . Vol. I. . . 27

1. 3 On some resonance phenomena near the gyrofrequency

obtained during the propagation of a radiowave in plasma

(ionosphere). M. Cutolo Vol. II 1

1. 4 Some non-linear phenomena in the ionosphere.

V. A. Bailey Vol. Ill 1 Remarks by Donald H. Menzel with reference to Bailey's

comments on solar electric fields. . . . Vol. Ill 61

1. 5 Ionospheric cross -modulation at the geomagnetic

equator. W. K. Klemperer Vol. Ill 63

Editor's Note Vol. Ill 83

VLF noise bands observed by the Alouette I satellite.

J. S. Belrose and R. E. Barrington . . Vol. Ill 85

The interaction of an antenna with a hot plasma and the

theory of resonance probes. J. A. Fejer. . . Vol. Ill . .101

Microscopic theory of ionospheric cross -modulation.

David Layzer and Donald H. Menzel . .Vol. IV 1 Ionospheric wave interaction using gyro -waves. Part 1. The night-time lower E -region.

R. A. Smith Vol. IV 39

3.3 Wave interaction at oblique incidence, near the gyro

frequency. F. H. Hibberd Vol. IV 45

3. 4 Non-linear propagation of electromagnetic waves in

magnetoplasmas

Mahendra S. Sodha and Carl J. Palumbo. . Vol. IV. . . 65

iv
3. 5 Collision effects in hydromagneto -ionic theory. Hari K.

\ Sen and Arne A. Wyller Vol. IV 73

3.6 A solution method for cold plasma. R. E. Hartle and

J. J. Gibbons Vol. IV 117

3. 7 Electromagnetic wave reflection from an oscillating

collision-free magneto -ionic medium.

O. E. H. Rydbeck Vol. V 1

Session 4.

4.1 Non-linear interaction coefficients for electrons in

nitrogen and air. A. V. Phelps Vol. V 49

4. 3 The excitation of optical radiation by high power density

radio beams. Lawrence R. Megill . . . Vol. V 69

4. 4 Alteration of the electron density of the lower ionosphere

with ground based transmitters.

Pietro P. Lombardini Vol. V 87

4. 7 Nonlinear effects in radiation generation through the

coupling of electron beams with diffraction gratings . Winfield W. Salisbury Vol. V Ill

4. 2 Use of radio transmitters to decrease D-region electron

density. Paul Molmud Vol. VI 1

4. 5 Non-linear ionospheric two-plasma processes.

Joseph E. Ro we Vol. vi 89

Concluding Remarks

Ernest K. Smith , Jr Vol. VI 113

D. T. Farley, Jr Vol. VI 114

THE GENERAL PROBLEM OF IONOSPHERIC NONLINEARITIES

Donald H. Menzel

I wish, first of all, to extend a welcome on behalf

of Dr. Ernest K. Smith and myself, co-chairmen of this

conference. My task this morning would have been much

lighter if only two distinguished Soviet scientists, Prof.

V. L. Ginsburg and Prof. A. V. Gurevich, had been able to

accept our invitation. Their joint paper*H)n "Nonlinear

Phenomena in a Plasma Located in an Alternating Electromagnetic Field" is a classic. It reviews and extends the

basic principles and applies them to an examination of

general non- linear problems.

Their paper is far too long and detailed for me to

review here. Perhaps I should just commend it to your

attention and then sit down. But such brevity is hardly

in the tradition of conference chairmen. Besides, I should

like to discuss some relatively unfamiliar non-linear processes that may occur in certain types of ionized plasmas.

Various contributors to this conference have asked

me to explain the qualifying adjective, "non-linear," which

defines the type of processes we are discussing here today.

Let me say, first of all, that "non-linear" does not signify

"irregular" or"non-uniform. " The mathematician, the physicist,

and the engineer each has his own definition of "non-linear."

At first sight these explanations may seem to be very different,

Harvard College Observatory, Cambridge, Mass.
but they are actually only alternate ways of expressing the

same basic facts.

. . dE

To the mathematician, quantities such as E, __

dt

d

2 E

,etc. , are linear in that they depend only on the first

dt 2 power of E. Differential equations involving these quantities

have solutions of the form:

E = f (t) .

And if two solutions are simultaneously present, the complete

solution is simply the sum of the two, or

E = f

t (t) + f2 (t) .

Mathematicians call this the principle of superposition.

Mathematical quantities involving the square, or

higher power, or cross-product of the foregoing, such as E

2

,

, dE % 2 dE

( —J , E — , etc. , are non-linear. Mathematically, we recogdt dt

nize the non-linearity because the complete solution is not

the sum of two elementary solutions. Cross-product terms

exist. The two disturbances react on one another. For example,

E

2 = f

2 (t) + 2f (t) f (t) + f

2 (t) ,

l 12 2

and so on.

Physically, E may represent an electric field that

varies with the time. Any number of electromagnetic fields

may exist simultaneously in any medium. As long as the disturbances are linear, we may represent their total effect as
the sum of the individual disturbances. But if the phenomenon

is non-linear, the total effect may depend on the cross-product

of the individual disturbances.

The engineer has a practical definition of non-linear

phenomena. As long as a simple filter can separate the different superposed periodic disturbances passing through a

circuit, they are linear. But when the two or more disturbances have interacted in such a way that they cannot be so

separated, then the phenomenon is non-linear. Under certain

circumstances where he wishes to induce artificial mixing,

the engineer will use a non-linear device such as a rectifier

or detector.

Many of the differential equations of mathematical

physics are linear in the first-order. They may contain,

however, certain non-linear terms that are negligible unless

the variable is exceptionally large. Consider the following

equation:

d

2 E , ,

, + w

2 E + BE 2 = , dT7

where 8 is small. When B is zero, this equation has the

solution*

E = A sin wt

But when we substitute this into the original equation, the

term in E becomes

BE 2 = BA2 sin 2 wt = | (1-cos 2ut)

2
In other words, the perturbations has caused the harmonic

of frequency 2oj to appear.

To obtain a general solution, we may employ the complex

Fourier series:

E - I ^1

iwnt .

Substituting this into the differential equation, we get

I An

u, 2 (n 2 -l) e lunt + ej I AmAm * e iw ( m+m, > t = 0-

Carry out the double summation by first summing over pairs

of m and m* such that

m + m' = n •

Then we can write

I [An

u 2(n2-l) +6 I AmAn_m] ei"nt = . n m

Hence we must have

A

n „2(n2-l) + 6 I Am

A

n.m

= . m

This infinite set of simultaneous equations is, of course,

nonlinear in the A 's. The solution contains higher harmonics, though they may have progressively smaller amplitudes.

The most familiar non-linear ionospheric process has

received the name Luxembourg Effect, because it first showed

up in connection with radio Luxembourg. The Luxembourg Effect

is a particular example of a general phenomenon that might
be more accurately termed "The Interaction of Acoustic and

Electromagnetic Waves in an Ionized Medium." The medium

may or may not contain a magnetic field.

In the Luxembourg Effect, a radio transmitter of high

power produces an intense electromagnetic field in the ionosphere. If an audio frequency modulates the carrier, the

electron velocity field responds to this modulation. In

effect, the temperature of the electron gas fluctuates by

an appreciable amount, according to the impressed audio

frequency. This temperature variation may also be described

as a pressure variation, a true acoustic wave. I shall refer

to the initial modulfeted wave and the acoustic disturbance

it produces in the medium as the "unwanted wave."

The various propagation constants that depend on temperature or pressure of the electron gas in the medium vary

with the frequency of the acoustic disturbance. If the density of the gas is high enough to make the medium dissipative,

the imaginary part of the complex index of refraction also

varies. A second radio wave, termed the "wanted wave,"

traversing the medium, will suffer from this variable absorption. Hence the wanted wave acquires some of the modulation of the unwanted wave. Since this superposed modulation

constitutes an interference with the wanted wave, it is

undesired or "unwanted." Thus the terminology.

The theory of this wave interaction was first given by

Martyn and Bailey-2

-v who pointed out the importance of magnetic

fields, especially for radio waves near the gyre frequency.
The formulas they derived, however, were over-simplified,

in that they linearized the equations describing the interaction. As a result, their formulas do not properly display

the phase lag of the acoustic disturbance relative to the

modulation by an amount dependent on the audio frequency.

Menzel and Layzer have developed a more detailed theory

which will be presented at this conference.

Experiments have shown that the Luxembourg variety

of wave interaction rarely leads to cross modulation in

excess of a few per cent. The amount is very sensitive to

a number of parameters. In the medium itself the significant

variables are: the electron density and temperature, the

frequency of dissipative electron collisions, and the magnetic field that fixes the electron gyrofrequency. The

phenomenon further depends on the power and polarization,

in addition to the radio and audio frequencies of the unwanted

wave. The frequency and polarization of the wanted wave are

also significant. The angle of incidence on the ionosphere

determines the degree of penetration. Since the unwanted

acoustic disturbance is localized, the greatest interaction

will occur when the wanted waves travel through a significant

portion of the disturbed volume.

Although the original theory and observations refer to

the interaction of two separate and independent waves, examinaa tion of the basic equation shows that/ single modulated wave

in the medium may interact with itself. As before, the
fluctuating field causes an acoustic disturbance in the

medium. This results in a variable attenuation coefficient,

which reacts on the original wave. Since the phase lag is

a function of the acoustic frequency, the degree of attenuation will similarly vary with the acoustic frequency. In

effect, some frequencies will experience demodulation whereas

others will be excessively modulated.

The foregoing phenomenon, termed "self interaction,"

indicates that any acoustic disturbance - of whatever origin -

can impress its modulation on a wave traversing the volume.

Rockets, missiles, explosions, collisions of solar ion

clouds with the upper atmosphere, etc., represent potential

sources of such acoustic waves. The question requiring study

is the magnitude of the interaction, not whether such interaction occurs. More specifically, of the various kinds of

interaction, are there any that could be studied experimentally?

Certain difficulties are obviously present, resulting

from fundamental differences in acoustic fields of radio

origin on one hand and of mechanical origin on the other.

For example, since the velocity of propagation of a radio

wave greatly exceeds that of sound, the acoustic field induced

by a modulated electromagnetic wave will be in phase over a

large volume of space. Hence the interaction with the wanted

wave will likewise be in phase over a large volume of space.

A mechanical disturbance, on the contrary, propagates

with a much smaller velocity. Hence a wanted wave, progressing
through the perturbed medium, will suffer high attenuation

in certain zones and low attenuation in ethers. The effect

resembles that of interference on optical frequencies. The

problem is complex and the total effect on the wanted wave

depends on the character and the extent of the basic disturbance.

The details of the interaction require precise analysis.

The limited volume of the mechanical disturbance may well be

offset by the intensity of the disturbance itself. In a region

where shock waves may exist, numerous non-linear effects may

occur, complicating the transport of electromagnetic waves

through the medium. The complex (dissipative) index of refraction will probably contain significant velocity-dependent

terms of non-linear character. The effect on a radio wave

transversing the medium may still be appreciable. Magnetohydrodynamic compression of the gas will significantly alter

the magnetic field, with attendant complication of the mathematical problem.

M. Cutolo has demonstrated experimentally the existence

of a phenomenon sufficiently different from the Luxembourg

Effect to deserve a special designation. He refers to it as

the "detection effect," since it depends on non-linear ionospheric properties for its occurrence. I prefer to call it

the "Cutolo Effect," after its discoverer. I shall describe

it briefly, since Cutolo plans to devote his paper to more

conventional non-linear problems.

Cutolo directs a beam of pulsed, modulated VHF waves

upon the ionosphere. Of significance is his use of the

8
gyrofrequency as the modulation frequency. In the Luxembourg terminology, this pulsed, modulated, VHF wave, further

modulated at audio frequencies to carry intelligence, represents

the "unwanted wave." Cutolo reports that a second HF wave,

the wanted wave, reflected from the ionospheric region traversed by the intense, focussed VHF signal, will have the audio

signal transferred to it. The amount of this cross modulation

is very sensitive to the exactness of match between the gyro

and modulating frequencies. Cutolo has found that the effect

occurs for both vertical and oblique incidence of the wanted

wave.

I have not seen, up to this time, any quantitative

physical explanation of the Cutolo effect. An unwanted wave

on the gyrofrequency would produce a large interaction of

the Luxembourg type. But a VHF wave, modulated at the gyrofrequency, is not equivalent to a pure electromagnetic wave

on the gyrofrequency. As is well known, such a wave can be

expressed as an unmodulated carrier and two unmodulated sidebands separated from the carrier by the gyrofrequency. To

account for the Cutolo effect we must suppose that some unknown ionospheric non-linearity so mixes the carrier and sidebands as to release the true gyrofrequency. It is difficult

to see, though, how such interactions occur because of the

very high frequencies employed. Cutolo used a carrier of

the order of 50 megacycles per second.
The following analysis will serve as an elementary

introduction to ionospheric theory for younger scientists.

The paper also considers various non-linear effects. An

electron of mass m and charge ~ e * f moving with vector

velocity y. in an electromagnetic field E and H and external acceleration F, obeys the well-known equation of motion:

dvx ei

m 4 -" = - £ <E v>x H + m.F . (1)

i dt 1- c "i- ~ i"

This equation applies along with Maxwell's equations to

define the physical state of the medium and its interaction

with the field.

The field F represents several varieties of mechanical

actions. In a macroscopic problem involving the entire atmosphere, F will include the gravitational acceleration g.

It also represents the collisional effects of nearby molecules,

which may act as a sort of resistance proportional to the

velocity and collision frequency v. As we shall note later

on, this collisional term may be anisotropic, because of

pressure gradients in the gas.

Maxwell's equations are:

k 3E 4ir curl H=VxH = + J , (2) c at c

curl E = VxE = --^S.,

c 9t

(3)

div H = V • h = , (4)

10
div E = V • E = 4?rp /< , (5) e

wherein J is the electric current, p the density of electric

charge, < the dielectric constant and y the magnetic permeability. Simultaneous solution of these equations leads to

V 2 e = ^

^-£ + V V • E , (6)

C2 3t2

v

2 H = JL!L ifi (7)

" c

2 at 2

When p is constant, the last term of (6) vanishes and we

e have the ordinary wave equations. For a plane wave, polarized

in the z x plane and traveling along the z axis, we have

E = i E e*

1<u(t " nZ/c) ; (8)

o

where w is the circular frequency and c/n the velocity of

light in the medium. Thus n is the index of refraction.

The quantity n may be complex. The unit vectors i, j, k

are the customary Cartesian set. This equation satisfies

( 6 ) when

n

2 = v k , (9)

For non-magnetic media, y = 1. Hence < = 1 for free space.

Part of our problem involves the determination of the effective k when the space contains electric currents and charges.

11
Return, now to equation (1) in the form:

itu -=A = -£< E--i.vxH-mi vvi . (10)

The first term on the right represents electric forces and

the second, magnetic forces. The third term, somewhat

schematically, represents the resistance that the charged

particle encounters as it moves through the medium. The

quantity v is the number of collisions per second experiencedby the moving ion. For convenience we shall drop the subscript i. The electrons, because of their great mobility,

are the major contributors to the current density J. Thus:

J = - N e y , (11)

where N is the number of electrons per cm3

. Under different

assumptions we shall calculate the variations of v and J with

the time. We shall then substitute the J, so calculated, into

Maxwell's equation (2).

Case I H = 0, v = 0, E = const: v = - e Et/m . (12)

Case II E = 0, v - 0, H = const (13)

Here take the derivative of (10) with respect to the time.

The resulting equation reduces as follows:

*? = " Z: tS ? S) = ( ~ )

2 (v x H) x H (14) dt mc dt mc - * -

Let y have two components yp parallel to and ys perpendicular

12
(German: senkrecht) to the field. The triple vector product

reduces to:

(v x H) x H = H2v

so that:

d

2 vs d

2 vD

and

Ys = Yos

eiULt

' Yp " Yop ? UL " eR/mc - < 16 >

The particle executes a spiral path with the Larmor frequency

w . The radius, a, and pitch, p, of the spiral are:

a = v /M , p = v /« . (17) OS l op L

Case III/ E=0,H=0,v= const; y = y e"

vt

, (18)

corresponding to simple damping.

Case IV, E = E e

ia)t

, H = 0, v = , (19)

y = (cE /iu m) e

Case V » At this point, let us consider a more general

problem for which we assume an electromagnetic wave traveling

parallel to the z-axis. Let Ev and E be the vector electric x y

fields and Hx and H

y

the variable magnetic fields associated

with the radiation. Let the magnetic field of the ionosphere

be of magnitude H, in the direction of the propagation.

13
The general equation of motion (10) breaks up into

equations for the three components: —dvx e e H

= - - E x v - wx dt m mc

dv

dt

e eH

Ev + — v_ - vvv m y mc x y(20)

dv

dt

= - vv » z

Multiply the second equation by i^ and add to the first, employing (16). The result is —

d e (v + iv ) = (iu> T - v) (v + iv ) (E + iE ) . (21) dt x y L x y' m x y'

Maxwell's field equations become:

3E

y

3 z

1 3H

C 3 t

3E

x

3 z

1 3H

I

C 3 t

3 z

K 3E 4irNeV

x c 3 t

(22)

3 Z

k 3E

c 3 t

4ttNeV

Multiply the second and fourth of these equations by i^ and

Ik
combine as before, to give: — (E + iE ) = - —(H + iH ) 3z x y c 3t x y

(23) —3 , . <i 3 4TriNe

(H + IH ) = (E + iE ) + (V + iv ) 3z x y c st x y c x y

Now introduce the complex quantities:

v + iv = V e

±iw(t - nz/c)

x y

E

x + iE

y = E e ±iu(t - nz/c > (24)

H + iH = H e

±ia) ( t -nz/c > x y

where V, E, and H are complex constants. With the aid of

(24) , (21) and (23) become:

[i(-wL

* u) + v] V = - e E/m (25)

inE = H (26)

itonH = -<uE * 4TriNeV. (27)

These equations contain 4 complex unknowns: E, H, V,

and n. Of these, n is the only one uniquely determined. H

and V are, as one would expect, merely proportional to E.

15
We may solve directly for n 2

:

n* - 1 -

4 ' Ne2 3L_ , (28)

mw o^w |iv

the complex index of refraction. Let

n = n

r I ik , (29)

where n r and n k are real. Then:

n 2 - k2 = 1 - iSiil " * "I. - , (30)

mu (wTul) 2 + v

2

o i, 47rN £

2 v 2n k = -. . (31)

r mw (oj^oj^) z +v z

The propagation factor becomes

e *iu)(t-nz/c) _ e

±iu)(t-n r z/c)-wkz/c » (32)

The last term of the exponent indicates an amplitude that

decreases exponentially with the distance. The effect results

from the collisions, which transfer a certain amount of energy

to the medium.

At this point, note the basic philosophy behind these

elementary derivations of the propagation parameters. We have

solved the general equation of motion to determine the velocity

of the electron under the influence of external fields of force,

16
electromagnetic, magnetic, and collisional. Our first objective was the calculation of the current density, J. Then,

substituting this back into Maxwell's equation (2), we derived,

in effect the dielectric constant <, which by equation (9) is

equal to the square of the complex index of refraction. Equation (2) thus takes the successive forms:

1 3E 4 ir J

curl H = =- + —

- c 3t c

(33)

1 3E 4ttNev < 3E iuxE

c 3t c c 3t c

since v, E, and their derivations all have the same time

dependence, e

i(i)t

. The physical significance of < becomes more meaningful,

perhaps, if we follow the steps indicated by equation (33).

Here we may use the complex form of (2)

,

1 3E 4ttJ

curl H = + —

c 3t c

= (±

ia3E _4irN e V

j e±lw(t _ n2 / c) c c = * iw + 4*N£ 2 (34)

. iw 4ttN £

2 1 ±icj(t-nz/c)

at [

i — + -_—

' JE e

c mc i (-a). *u) +v

iwK *io)(t-nz /c) 3 c

17
where we have used (25) to express V in terms of E. Now,

solving for k we get

2 4TrNe 2 1 ,__.

< = n

z = 1 -

:— (35)

u) dipji iv

as before.

Now note some of the assumptions implicit in these

derivations of k. In the strictest sense we should have a

different velocity, v^ , for each electron. We would then

have evaluated the average v and J as follows:

J = - eNv = - e^Yi ~ ~ e[vf(v)dv, (36)

where f(v) is the velocity distribution function for the

electron gas.

The solution of this problem is far more difficult

than appears at first sight. The function, f = f(v), will

generally depend on time and position. The Boltzmann transport equation is basic for the determination of f.

dT

= ff + Y'?f+ S'?vf+S=0 ' (37)

In the above, a is the acceleration and vv the gradient in

velocity space. The quantity, S, is the collision integral,

which depends on the rate of change in the distribution function resulting from collisions. This term takes account of

the creation or destruction of electrons in a given range of

18
velocity. To calculate S we must know the physical details

of both elastic and inelastic collisions.

The first three terms of (37) are vector abbreviations

for the purely mathematical concept of the total derivative

of a function, f, that depends on the time, on the coordinates,

and the velocity components, in rectangular coordinates, we

may write —d 3f 3f dx 3f dy 3f dz

f (x,y,z,u,v,w,t) =^_+ + i + dt 3t 3x dt 3y dt 3Z dt

(38)

3f du 3f dv 3f du

+ + + +S = > 3u dt 3v dt 3o) dt

wherein u, v, and u are the three velocity components.

Let v* and v be the vector velocities of the electron

prior to and after the collision. Let v' and v be the initial

- 1 - l and final velocities of the colliding particle. Let f (v')dv'

and f(y)dv symbolically represent the number of electrons

within the ranges dv' and dv, and let F(v|)dvJ and Ffy^dVj

similarly represent the number of colliding particles in

ranges dv{ and dv 1#

The relative velocity of collision becomes:

u = v - yi and u = |y - yi|. (39)

Let <J(u,8) be the differential scattering cross-section describing the probability that a colliding electron will be

19
deflected through an angle 6, during collision. Define

the solid angle dfi = sineded<|>. Then we may express S as

follows:

S = (f dyjdftofuje) u [f(v)F(v 1 )- f(v')F(vJ)] (40)

The problem of calculating S and f depends on a precise

knowledge of the target cross section, a , as a function of

u and 6. We shall not consider it further here, except to

note that our arbitrary representation of the collisional

effects by the term mw in equation (10) was naive, to say

the least. It was particularly an oversimplification in

that we assumed v to be independent of v, in order to obtain

a linear equation.

V7e should have been much more realistic if we had

assumed v varied with the velocity. Certainly the collision

frequency will depend on the temperature of the electron gas.

This temperature, in turn, will vary with the impressed electric field. And if the field is modulated, the modulation

frequency will cause v to vary at that frequency. Thus

the absorption coefficient, k, equation (31) , should also

vary with the time. It is this variable absorption that

leads to cross-modulation, the Luxembourg effect, gyrointeraction and other non-linear responses, some of which

we shall be discussing during this conference.

But additional, still less obvious assumptions exist

20
in our over-simplified theoretical discussion. Maxwell's

equations apply, of course. We have ignored possible reactions

of the ionospheric plasma with the terrestrial magnetic field.

I refer particularly to phenomena lying in the realm of magnetohydrodynamics. We can at least infer the character of

some of these problems.

Rewrite equation (1) in the general form:

dv. e.v.xH

m. -^-i = e.E - 1 ~ 1 -~ + m.F , (41)

1 dt i ~ ci ~

wherein a positive e. refers to an ion and a negative e. to

an electron. We can take averages over a unit volume as

follows

p = Ln^

py = EmjVj^

(42)

J = ^c i v i

p = E e

wherein p is the mass per unit volume. These summations

imply, when relevant, integration over the velocity-distribution functions previously discussed.

The force-field F breaks up into two parts; external

fields such as gravity and internal fields which result from

colliding particles. One may write the latter force on a

given particle, i, as follows:

F int = J! F^ , (43)

21
where the summation is taken over all neighboring particles

j. This term cancels out only when the medium is uniform.

If electric currents are present, the term gives rise to

forces simulated in our earlier treatment by the collisional

term, m vv. This force can also be expressed, for the entire

medium, as a sort of electrical resistance, R, in the form

RJ. It also produces such major forces as pressure gradients.

Tangential forces, in the presence of sheer, lead to the

phenomenon of viscosity. In the presence of temperature

gradients it can even cause thermoelectric effect. For

example we may write

E = R (J - V T) , (44)

where 6 is a thermoelectric coefficient. In the strictest

sense both R and 9 will be tensors rather than scalars.

As we sum equation (41) over the individual particles

and employ Maxwell's equations, we obtain the general equation

of magnetohydrodynamic motion:

dv 1

P — = - Vp + pF + (7 X H) X H —1 , 1

(7 x E) x E + — 4tt "

4tt + — (7 x E) x E + — E 7T E (45)

1 3 - (E x H) . 4ttc at -

When H and E are constant or zero, the equation reduces to

that of ordinary hydrodynamics, less the second-order terms

related to viscosity. The terms represent the forces arising

22
from the electromagnetic fields. E x H is the Poynting vector,

which represents energy flowing out of the volume element.

Hence the partial time derivative of E x H represents the

forces caused by radiation pressure.

Now multiply (31) by e /m and sum ggain. The result

is

m dJ i —- ~ = E + I (vx H) - R(J - e V T) (46)

ne 2 dt c

"

where m is the mass of an electron and n the number of electrons per unit volume. In taking this sum we have allowed

for the fact that electrons, by virtue of their small mass,

are the main contributors to the electric currents. In passing

we may note that this equation has several obvious limiting

forms. It can reduce to Ohms law

E = R J , (47)

or to the electromotive force induced in a conductor moving

through a magnetic field

E = - (v x H)/c . (48)

Even equation (46) has its limitations. Acoustic waves

or hydrodynamic waves can directly produce changes in J, by

simple compression or expansion of the current-bearing regions.

Moreover, in a conducting medium, H itself may change under

magnetonydrodyn ami c forces. Such changes in H produce fluctuations in J. As we have just noted, variations in J directly

23
affect the dielectric constant. We must, therefore, set

up the equations governing the variations, not only of H

but also of E.

The following analysis is by no means complete. However, consider the steady-state situation, derived from

(46) ;

E + (v x H)/c - R J = (49)

Take the curl of this equation and reduce by means of the

Maxwell relationships. We get:

dH 3H

-=. = —a. + v • V H

dt 9t ' "

~

(50)

C 2R k 3 2 H

H. V V - H V • V + [V 2 H - -« *-] - - - - - 4 ir cr dtz

Now take the partial time derivative of (49) and

derive a similar equation for E. The result is:

3E 1 3 C2R k 3 2e -= + (v x H) = [ V

2 E - - w • E ] . (51)

3t C 3t - 4 tt c

2 3t2

These equations are also subject to the equation of

continuity,

dp 3p 3-=_. + v.yp = -pV.v,

or

3o (52)

-r + ?-<py) = o . 3 ^

2k
For regions where the conductivity is high, R -*

and we recover the ordinary equations of magnetohydrodynamics.

When the conductivity is low, divide through by R. The electromagnetic equations then reduce to the simple wave equations

for E and H. In other words, for media of low conductivity,

the hydrodynamic and electromagnetic fields are independent.

Otherwise the two fields are coupled. This means that, in

effect, we can combine the purely dynamic terms into a nonlinear wave equation, in which the dielectric constant is a function of the dynamical variations.

I shall not attempt to carry the analysis further.

My primary objective has been to show that non-linear effects abound in this general problem. I should point out

one additional fact. The electrical resistance R is closely

related to the damping factor k. Since k, in turn, depends

on both electrical and dynamical factors, the factor R introduces additional non-linearities. Detailed solutions

depend on assumptions concerning the basic physical conditions

in the medium. I have especially desired to emphasize the

importance of dynamical or magnetohydrodynamical effects on

radio propagation.

MvV. rL. r-Ginsberg k and A. V. Gurevich, Usp. Fiz. Nauk 70,

201-246, Feb. 1960 and 393-428, March 1960 and

english translation: Soviet Physics , Uspekhi, 3,

115-146 and 175-284, 1960.

Lf3j D. F. Martyn and V. A. Bailey, Phil Mag 18, 369, 1934.

V. A. Bailey, Nature , 139, 68, 838, 1937.

V. A. Bailey, Phil. Mag . 23, 774, 1937.

V. A. Bailey, Nuovo Cimento Suppl . £, 1430, 1956.

25

ON SOME NON-LINEAR PHENOMENA IN THE IONOSPHERIC PLASMA

P. Caldirola and 0. De Barbie ri

Istituto di Scienze Fisiche dell'Universita di Milano, Milano, Italy

In the first part of the paper we calculate and discuss

the distribution function of the electrons of a slightly

ionized plasma under the influence of an external magnetic

field and of an e.m. wave of the type E = E cos U)t. — —o

It is shown that, taking into account both elastic and

inelastic collisions between electrons and molecules, it is

possible to calculate explicitly the mean electronic energy due

only to the absorption of the extraordinary wave of the electric field and that it is maximum at the gyromagnetic resoej$)

nance (to = to = —^ ,The study of the effects of inelastic

H mc

collisions shows that in order to have an equal electronio

temperature, the intensity of the electric field must be, in

the case of inelastic collisions, about 5 times greater than

that for elastic collisions only.

In the second part of the paper v/e calculate, for the

wave E = E co§k>t, the components of the comnlex dielectric — —o y permittivity tenser, that are given by expressions dependent

in a rather complicated * way on E .It is shown that if E is J —

~o

sufficiently small they can be simplified and, using a suitable

perturbation technique, it is possible to calculate a non linear

dispersion relation. This relation contains explicitly E, and,

for E —.-0, becomes the Appleton-Hartree formula. Furthermore,

it shows that in the non—linear case, too, the electric field is

split up into two components whose propagation is never independent.

In the third part of the paper we calculate and study the

electronic distribution function and the complex dielectric

permittivity tensor for a wave of the type E=E [j+r)cos(at+[3)3costot

wi th to ~ to » a . H

27
Introduction .

The study of non-linear propagation of electromagnetic

waves in plasma has recently aroused the interest of many

scientists. In general^ the methods used in describing the

process of propagation of electromagnetic waves in plasma

are based on a procedure of linearization of the equations

that give the mathematical representation of the physical

problem. A very important physical characteristics of the

aforesaid non—linear effects rests on the fact that they can

be produced by relatively small electric fields. This can be

understood quite easily if we consider a slightly ionized

plasma in which the electron-electron and electron-ion collision frequency is much less than the electron-molecule collision frequency. In factj the propagation of an electromagnetic

wave in a plasma causes a relatively high increase of the

kinetic energy of the electrons. This is mainly due to two

facts: the first is the quite large value of the mean free

path \ of the electrons in the plasma (so that they can acquire considerable energy from the wave between two collisions), the second is that due to the smallness of the ratio

6 = -rr ^ 3 «4 *10 between the mass of the electron and the

mean mass of the molecules, the mean energy transferred in a

collision from the electrons to the neutral component of the

plasma is almost negligible. As a final result we see that

only the energy distribution of the electrons is altered

(with an increase of their temperature and mean energy). This

variation depends on the quantities E, H, w that characterize

the wave in the plasma. So that the parameters that characterize the plasma from the electromagnetic viewpoint (as the

dielectric permittivity e , the susceptibility X - v i "t he

28
conductivity a., and so on) will depend on these parameters

(E and oo mainly) and on the parameters that characterize the

plasma from the kinetic viewpoint (as the mean free path \ , the collision frequency v and so on). In this way^, in the relations that give the electric polarizability P or the conduction current density jj_ that is*

1

the electrical parameters e. n . y -, , <?• , will depend on E also

ik' ik' lk —

and the quantities e.. and o\ will no longer be proportional ik ik &

to E. Therefore_> the electrodynamic processes in the plasma

and the same propagation of electromagnetic waves will become

non-linear.

There are two types of approach to the theory of non-linear

effects in plasma: the first one is a generalization of the

mean free path method and the second one is the statistical

method based on Boltzmann equation.

The fundamental equation of the first method is Langevin '

equation —

dv e - + v v = —

dt - m

E + f~ X % (1)

e, m, v , v being respectively charge, aiass, collision frequency

and velocity of the electron; E is the electric field of the

radiowave

% the external magnetic field in which the plasma

is immersed. Knowing E eq. (1) allows us to calculate v, furthermore putting by definition ;

_j + = n e v

where n is the electronic density and j, is the total current

density, it is possible to calculate the conductivity a., and

the dielectric susceptibility X- hy means of the equation

1 K.

29
it

=

I

* 2

+ —

In this way are obtained the expressions of the dielectric

permittivity e. n (e.. = 4 k Tx-, + 6. n l) and of the conducti- * J ik ik LAik 1k-1 vitv a., that must, he substituted in the equations of the J lk

wave propagation . It is easy to see that these expressions

of e. n and of a., do not depend on E and so the propagation,

lk lk —

also of a very strong wave, is linear. To overcome this

difficulty []1 the electronic collision frequency v (that

enters into the expressions * of £., and of a.,) is considered

lk lk

dependent on the electronic temperature T e.g. in the following way*

v(T

e

) -v< V5 (2)

being T the molecular temperature and v the collision

frequency in the absence of an electric field (T = T) . In

order to calculate T a differential equation is established

equating dl /dt to the net power gained by the electron, in

the following way

Given the rather complicated structure of this equation, the

usual method to solve it consists in disregarding, at first , the dependence of v on T . In the final formulas obtained in

this way is introduced v , given by (2), and so it is possible

to calculate T as a function of the various parameters of

e * the plasma among which is the electric field. In this way £.,

and a., depend on E and the equations for the propagation

This dependence is valid for electron-molecule collisions,

see ref . £1 J

.

30
become non—linear . It is clear that this type of approach to

the theory of nor linear propagation presents defects and

contradictions that are typical of the mean free path method.

This method, worked out particularly by Townsend and Huxley,

is based on the calculation of the distance travelled by the

electron under the influence of the accelerating field during

the interval of time between two successive collisions. The

mean distance travelled by an ensemble of electrons per unit

time gives the diffusion velocity of the electrons through

the gas o The intensity of the field is always supposed to be

weak; the calculation is done by supposing that all the

electrons have the same velocity, then the mean value of the

results is calculated by means of an electronic distribution

function, previously unknown. A maxwellian distribution

function is employed to obtain results which do not contain

the above mentioned mean values. This method has the drawback

of being valid only for weak fields, which are not able to

deviate the electronic distribution function from its stationary form (maxwellian), without the possibility of exactly

defining the general validity and the order of magnitude of

the employed approximations. Furthermore, according to Huxley

^2 1 the method of the mean free paths can be employed whenever

we' suppose that among all the various electron-molecule types

of collision the binary ones are by far the most important

and that the motion of the particles can be divided into short

periods, in which the collisions occur, separated by comparatively long intervals during which the interactions with the

other particles can be neglected in comparison to the action

of the field.

It is evident that if the behaviour of a physical system can

be outlined in the above mentioned way, it can be analyzed by

31
means of Boltzmann ' s integrodifferential equation.

From a more general viewpoint, we do not see how it would

"be possibile to base the theory of non-linear propagation

on the equation (1) of dynamics, when we are faced with an

electron gas, with a statistical velocity distribution.

1 . - The Fundamental Equations of the Statistical Method.

We are interested in the lower E layer of the ionosphere,

i.e.j the zone "between 30 and 95 Km height.

The molecular temperature of this layer varies slowly from

205 °K (at 80 km) to 217 °K (at 95 km). Let us suppose that

the temperature gradient Is zero and let us take a mean molecular temperature equal to 210 °K. The molecular concentra14 / 3 tion is of about 10 mol/cm in this layer. The electron

density varies from 5-10 el/cm (at 80 km) to 5*10 el/cm

(at 95 km) and therefore it is much inferior to the molecular

one

The statistical state of the ionospheric plasma is described

by means of two distribution functions: one f(r, v, t) for

the electrons, the other F(R, V, t) for the molecules; these

are the solutions of the following system:

+ v .v f + Y'V f = fi + C

1 r - -v 11 12

-|2 + v-l£F + t -V = c

21

+ c 22

in which C represents the collision term for electron-electron

interaction , C the collision term for molecule-molecule

interactions and last C and C the collision terms for

electron-molecule interactions and where we have indicated

with jr and r_ the external accelerations acting on the electronic and molecular gases respectively. Taking into account

32
the characteristics of the ionospheric plasmaj the second of

the written equations can he simplified in the following

ways

C

22

"°

which, after integration, gives the following (maxwellian) distribution function for the molecular velocities,

3

*T f M ^ 2 I Jill

where N is the molecular density.

Moreover, as N is much greater than n (electronic density),

the first equation of the system can be simplified, neglecting C with respect to C. ; in this way we obtain s

of „ -p — e

- + v*V f + —

Dt — -r m

E+ ^-X% — c — • £vf " °12 &

C is given by the following integral operator that we will

indicate with J -[ f V

f (v 1 )F(V )-f (v)F(V) ga(#-g) dVdQ (5)

where g = v-V is the modulus of the relative velocity,

o($,g) is the differential cross-section, & the scattering

angle, v' and V 1 are the velocities of the electron and of

the molecule before the collision (after the collision they

become v and V respectively)

Taking into account the characteristics of the ionospheric

plasma it is possibile to simplify eq. (4) [_3 [J

• We have already said that 6 , the mean energy lost by an

electron in a collision is much less than 1 . In the case of

elastic collisions

33
6 = If - 3-4 • 10

in the case of inelastic collisions the energy lost by collision is noticeable, but because these collisions are much

less frequent than the first ones, the mean value of 6 is

always much less than 1. For this reason, the r.m.s. velocity

of an electron is much greater, also in strong fields, than

the mean directed velocity, so that the symmetric portion,

in v, of the electronic distribution function is much greater

than its antisymmetric portion. Moreover, as 6 is small, the

collision term (5) can be expressed in differential form; in

this way eq. (4) becomes the Boltzmann-Fokker-Planck equation.

For these reasons let us develop the electronic distribution

function is spherical harmonics in velocity space, putting;

f(r, v, t)= f

o (r, v, t)+ a

y

.-1 ( -' V ' t)+ X ^'^ < 6 ) v where a = -j—p. If we put this equation into (4) and integrate

all over dQ (differential of the solid angle in velocity

space) we obtain ; . . (7)

^+ j ir ^f-2

^ ¥*%} - f- — 2>mv v ; 2v2 1 vv

3 (1+ v 352 i)f

mv Ov

=0

Putting again eq. (6), multiplied by a into (4) and integrating again all over the solid angle dQ we have ;

Of Of

ot -^r m v — mc — —1 —1 (8)

where we have put: £l[] , £3] > C^l :

3^
I o J . 2 ov mv 0v ; o

J -Ui}"' _v£

i

In this way, starting from (4), with the approximation (6),

we have obtained (7) and (8). To establish these equations

we employed the properties of monodromy and orthogonality

of f and f expressed by (6) and we supposed that the

function x and the functions that can be obtained by it with

the application of the various operators of eq. (4) are

very small in respect to the function f and of the corresponding ones, obtained by applying to it the various

operators of (4)

It is easy to see that if the electronic density varies

smoothly and if the variation of the electronic current along

a mean free path \ is small with respect to the product of

the electronic density and the velocity, that is if :

dl

ox

«n <v>

then

So that the function X is small not only with respect to f

(as we supposed to derive eqs. (7) and (8)) but also with

respect to If J . It is clear that with the system composed

of (7) and (8) it is not possible to determine corrections

to f of the order of 6f . o o

35
2. - The Electronic Distribution Function for a Monochromatic

Wave . (*)

Let us suppose that the plasma is homogeneous, in this

case eqs. (7) and (8) become:

Of

o e at

+

, 2 3mv MM" 1+

kT _0_

mv Ov

f h=o (9)

Of Of

rZl + e-_o

E + -S.g6Xf 1 +vf. =0.

Ot m Ov — mc — —1 —1 (v 10 )'

In order to solve this system of four equations we must define

two relaxation times. The first is the relaxation time x,,E

of the electric field that is defined as the time necessary

to change substantially the field (if E = E cos u>t then

t_ si to ) . The second is the relaxation time x of the

E r electronic energy; x is of the order of (6v)" as can be

seen by integrating (3) with E =0. Following Gurevich £5J] we can distinguish two cases : the first in which the electric

field varies slowly (x »x ) i the second in which it varies

quickly (x «x ) , Now we must consider the case in which the

field varies quickly because we study the phenomena related

with the propagation of a monochromatic wave :

E = E cos ojt

which pulsation is equal, or very near, to the gyropulsation

of the medium. In fact the value of the magnetic field of the

earth, that enters into (10) varies, in the zone of the

ionosphere in which we are interested, between 0.35 Gauss

(11)

(*) Some results of this section are already contained in the

paper "Sull' eccitazione dell' "airglow" per mezzo di radio

onde" of one of us (P.O.) published in Nuovo Cimento Suppl.

vol. XIX Serie X N° 2 (1961).

36
_7

and 0.45 Gauss so that, being to - Us , % ~ 1.4*10 sec , -1 -3 because 0) = e $&/mc ; moreover % «. (6 v) is about 4*1 0" sec,

— 5 r

6 being 6 =: 3.4 • 10~ and v « 8*10 coll/sec. From these

X E calculations v.e see that in our case ( —)«1. Therefore let

f r us develop f • and f into a power series of the parameter

x i E

(t~) putting : r

f

o = fo9

+ f

o1

+ f

o2

+ "*

we find that in the zero order approximation we can neglect

the variation of the distribution function due to collisions

(of the order of f/x ) in respect to the first term of eq.(9)

r „ Q foo

(because of the order of —- a f/x,_) so -= 0. Therefore :

f = f (v) 00 00

that is, in the zero order approximation, the symmetric part

of the distribution function is independent of time.

If we put into (9)

Of

-p oo £10 --a-rr

we see that u satisfies Langevm^s equation (1). This equation

can be solved for E given by (11) and gives the following

expression for the persistent part of f

10

Of Of

110= A

^ —~ C0SWt + B(w) —^ sinyt. (12)

The two vectors A(oj) and B(oj) are given by:

37
eE

aM= - -a

til C(w+u)H

) +v ]

[(u)-wH

) + v ]

5? X ^

2 2 2 ^o

+v(toH+(U> +v ) g- + "hK"^ + V ) "^E

w 2 , 2 2,

M

H

C03 ° W

+ v(o)

H

-3 a) +v ) — r -jf U) +v

< (13)

eE

b(u)-- ~r111 [(w^Sv^ [(u)-^) 2 + v 2

]

J&XE E

- 2uvir-u( vu - v ) r +

c

,2 2,2, *H oob2 °

+ w(wH-w +3v ) — —

W + V

where 5 is the angle between E and 56 . Given that f

Q

(v,t) ~ *

0Q

(v) + f

1 (v,t) and f

oo (v)>>f

01 (v ' t)

we have :

at 2 3v

2v

3/. kT3v„ 2e 2 „ _ vv (1+ )f - r—r v E«f ._ mv ov o 3tn6 10

2v

6__ _£#

2 av

from which :

f

01 (v,t)=K^-i

.2v J which is not limited for t—*• + oo . Stipulating then the

existence of the electronic distribution function we must

impose the condition J = 0, that is :

(^ kT a v vv(1 + — —-) f

2e

mv av oo 3m 6 10 E«f_ = .

38
This is the differential equation for f . Seemingly it is

oo o tf - - not self-consistent because the time is explicitly contained

in the expression E«f . "but noting that f must contain only — 10 7 oo J the r.m.s. value of the electric field, we obtain :

2e <E-f

3m 6 - 10 %

e E df

having put

cp(v) =

cos

2 2

OJ + V

so that ;

sin

+

E

a

a 2 A 3m 6

1 2 2 (u+u) ) +v

+

00

dv

1 (uj_wh

) 2+v2

(HO

f = C exp

oo - -/ mv dv

2 E2

k.T ; ~—°— cp (v)

3tn 6

(15)

where the constant C must be determined from the normalization

condition 5 /GD

4 7i v f (v) dv = 1

' 00

We see that the method suggested by Gurevich is particularly

clear and precise, because the distribution function in the

th

n approximation is obtained by imposing the condition of

boundedness for t

>+ Co of the subsequent approximation.

Keeping in mind that the magnetic field % produces a plasma

anisotropy breaking up the plane polarized electromagnetic

wave into two elliptically polarized waves, it is important

to get the electronic distribution function for an elliptically

polarized electric field E. We express E in terms of its

components along the three principal polarization axes t

E = E cos wt i e| + e 1 u+ E~ e~

X w

• — — I I o —lo —lo

39
Where E,, is the plane-polarized field parallel to $& and —Mo

E, and Et are two circularly-polarized fields in a plane

-JL -±o

perpendicular to ^o and rotating, respectively, in the same

sense or in the opposite sense of the electrons in the

magnetic field. For such an electric field the distribution

function (15) is still valid provided that E cp(v) is replaced

by:

E^cp(v)

2 E

+2

io 2 E

-2

(16)

2 2 2 2 2 2 w +

(w+w ) +v ( w- wtt) +v

n ri It is clear that at the gyromagnetic resonance (oj a oj ) the

rl energy transferred to the plasma is mainly due to the action

_ 2 of the E wave. Therefore let us consider in E cp(v) only the

lo o part due to this wave, in this case eq. (15) becomes:

f (v)=C exp -, oo oo * /' mv dv

kT +

2 -2

2 e E,1 o 3m 6

1

(,oj-oj„; + V \

S =

J.

=c

oo

1 + V

?e2"R~2

t ^ 2 1 o ,2

2^-2 ,2

e E, \ 1 o

2 2 36k^T

(17)

exp -< -mv

2kT

where we have put:

v(v) -5 (18)

with \ = (na N) being a the "radius" of the molecule, N

the molecular density and \ the mean free path of the electrons that we take as a constant. According to some authors

(see refs. £6j and f7j it would be better to choose v propor40
2 tional to v . We have chosen a linear dependence for the

greater simplicity of the final formulas obtained, this is

tantamount to substitutingD in them a suitable value X „_, eff 7 for the mean free path, that takes into account the dependence of X on v Putting:

2 -? 2 e Ei K *2

v

-l o ,. m A / v2

By means of (17) we can calculate the mean energy <e> absorbed by the electrons of the plasma under the action of the

component E. of the electric field, we get:

3 *

W ( 2 ' 4' I+

4

' Y + ">

W ( 2"4' 2 +

4 iY + lJ)

where W(k,m;z) is the Whittaker function \&\ of parameters

k, m and argument z. From (19) we have : <-t^W - ° (^.D

n

So that the energy absorption is maximal at the perfect resonance, w = co (See Pigs. N° 1j2). Prom the asymptotic

representation of the Whittaker functions_jit is possible to

se e that

lim <e(w) > = lim <e(uO >= ~ kT

n. c

U) —» od 00 _-* OD n

Prom which we see that if the frequency of the wave. is far

from the gyrofrequency, the electronic distribution function

tends to become maxwellian with a temperature equal to the

molecular one

At the perfect resonance eq. (17) becomes :

kl
f

(r) (v)=C (r)

00 00 exp if) J

kT+

mv dv

2-2 2

2 e E, X^

1 o

(20)

3m 6 v

If the electronic temperature is much greater than the molecular one, that is if

2 TM-2 ^2

, e E, X

| kT« —-M2 mo v

eq (20) becomes;

f*

(r)

(v) =C* (r)

00 00 exp* 1

m

" 8

6 ( 71T^) lo

2

V

4 k (21)

We have Druyvesteyn* s distribution function £9^ wherein

appears the r.m.s. value of the electric field as it should

be. With the aid of (21) we can calculate the fraction

P* (v ^v ) of electrons having a velocity exceeding a

(r) certain value v and the mean energy <e* > of the electrons

o

° These quantities are given by :

P* (r) (v^v ) =

' o

W (_ 1 1. 1 6 ( o_) 2

) m v

c

r(4)f#] (^^) 1A

V[8 eE. V lo

/ 2

3

mV

o 2

exP ^

+ IS

6 ( eEfl) lo

(22)

<E

*(r) >

r (4)

r (t)

TT" e E, X

3 6 lo

(r) Measuring <e* '> in eV, E~ in Volt/cm, X in cm and the lo 7 pressure p in mmHg we have

P <£ *

(r) >= 103.58 E, X= 6.73 lo

(23)

(23.1)

k2
We have found several expressions for the electronic distribution function in a slightly ionized gas under the action

of an alternating electric field and a constant magnetic field,

considering only elastic collision between electrons and gas

molecules. As a result of such calculations (see fig. 3 and

eq. (23)) it was found that for -^

s 1 (that is, for a field

intensity of some millivolt per cm and a pressure of some

10 mmHg) , it is possible to have electrons accelerated in

such a way that their mean energy is several eV so that they

can collide inelastically with the molecules.

Let us now study the effect of these collisions on the

electronic distribution function. In a number of researches

(Davydov £3} , Druyvesteyn F9~], Smit [jcT}), approximate

methods have been developed in order to take into account

this effect and recently a detailed study has been carried

out by Kovrizhnykh Fit]* The result achieved by him solves,in

principal the problem raised by us, even if the final formulas

are generally cumbersone to be numerically calculated. Due to

its laboriousness and to the uncertainty of the experimental

data required for its application, Kovrizhnykh' s method is

difficult to be employed for pratical purposes. Consequently,

we thought it advisable to adopt, for the evaluation of the

effect of inelastic collisions on the electronic velocity

distribution, a semiempirical procedure particularly useful

at least as far as our problem is concerned. Let us consider

a gas (like air or the ionosphere) consisting essentially of

poliatomic molecules. In this gas not only the levels of the

electronic configurations but also the rotation and oscillation levels whose energy is rather low (of the order of

(1Cf4 * 10~ 2

) eV for rotation levels and (0.1 * 0.5) eV for

oscillation levels) can be excited. Therefore inelastic

43
scattering with molecular excitation takes place also at low

temperature and can absorb a considerable amount of electron

energy. Keeping in mind the expression of the collision term

for elastic collisions

.(el) ^-72v2 ay2 c ,kT Of _ N v 6* <T ov

+ vf >

we write down in the same way the corresponding term for

inelastic collisions:

(inel)

1 j 2v d

2^/ v /kT Of „v v B(v) <T ^V + **

where R(v) is, corresponding to 6v, the electron energy loss

due to inelastic collisions. We can then take into account

this term, by substituting in the formulas for the energy

distribution of the electrons for elastic collision only, the

expression 6v with

R

tot

(v) =6v(v) + R(v)

which corresponds to the introduction of an "efficient" energy

loss for collisions given by :

R, ..(v)

6

(eff)

( v) = tot

(eff)/ N v

v '(v)

The energy loss 6 (v) as a function of the electron

velocity has been experimentally determined in many papers

(for instance | 12]] ). Prom the experimental data one can

deduce that, up to energies of about 2 eV, for air and the

ionosphere > 6 is nearly constant and in both cases it

corresponds to about 1.6*10 (to be compared with 6=3.4*10 '

for elastic collisions only) . Let us now note that in the

formulas for electronic energy distribution in the case of

kk
elastic collisions only, the parameter 6 appears only through

the ratio E /6 . This leads to the conclusion that, in order

to take into account the effect of inelastic collisions, it is

sufficient to jubstitute E /§ with E /6 in the final

formulas. Thus we can say that the effect of the inelastic

2 2 collisions is to reduce the value of E to a value E, \ 2 2 / (eff

)

(.red;

such that E, ,\/6 = E /6 . We have then

(red)' '

(red) c(eff)

For air and the ionosphere/ we shall obtain

(red) V 1i6 . 10

-3 5

The electronic mean energy is therefore reduced (see Fig. No. 4)

from its value (23.1) for

co = coH

2 TM-2 ,2

. e E, X

m 6 v

to

E,

<£ ** (r ) > = 1 < e *( r ) •>= 20.72 E, \= 1.345— (23.2)

-1 P

3 . - The Non Linear Dispersion Relation. ( *)

Taking (6) into account we have 1

£ - 1 ±t

= a -i-w

4-7C

/CD , E=f en/ v

J f dv

J

Putting the z axis along the direction of the external magnetic

field % we have :

{*) A. Airoldi of our Institute has contributed to the work

contained in this section. We take the occasion to thank

her for the aid given in the numerical computations.

*5
W r

0D zz i*of00

2 2 0V oj + v dv

a =o* = xx yy

2

3 JL 2

U /-CD

2 2 2 2

(u)+co ) + v (w-wH

) + v

Of

oo OV dv

^

xy yx 3 J 2

a =a =a =o = xz zx yz zy

GJ'+QJH

w-u».H

2 2 2 2

k(u*w ) + v (co-w ) +v ( H' H'

Of oo "ov

dv

£=£=£=£ = xz zx yz zy

(24)

£ ~ 1 U) ZZ £

4 71 f

Of

00

2 2 0v

w + v dv

£ -1 E -1 W rQO XX

= -XL

4 tx 4

2 r *— = -^r

3 — -<—

7i 37 2-w |/ N 2

0)-(jJ H

ajj) + NT (u»-i^) +

2

v 2

Of

00 .., >- —

dv Ov

2

£ £ U) -GO . . -£L =- -EE = -£ f v

3 _L

4tx In 3 J 2u

o (w-W ) +V (|j)+(i1l) +V ^

Of00 , > —t—dv Ov

where to is the plasma frequency, that is:

2 4 7i e n

0) = -

p m

The knowledge of the expressions ^ of £.. and o\, allows us to lk lk

study the problem of the propagation of electromagnetic waves

in a plasma. In order to solve this problem it is possible to

use in the linear approximation two methods that, in general,

are equivalent. The first consists in the integration of the

equations of the wave propagation, the second is related to

the calculation and the study of the dispersion relation. The

difference between these two methods lies in the fact that :

46
while the first onS (the integration of the equations of the

wave propagation) allows us to obtain quantitatively exact

results only in relatively few cases, the second, on the

contrary, gives a much more general description, also if it

is only qualitative, of the propagation. More precisely we

have in the case of a uniform plasma the two methods give

equal results, in the non-uniform case the knowledge of the

dispersion relation allows us, very often, to write down

approximate solutions of the equations of the propagation.

These solutions are in general the first term of a geometrical

optics series. We must stress the fact that if we base the

analysis of non linear propagation of a monochromatic wave

launched in a plasma on eqs (24) we are compelled to admit

that the deviations from linearity are not very strong. In

fact we must suppose that there is no harmonic generation so that the main effect of non linear propagation consists in

increasing the absorption coefficient and in modifying the

velocity of the wave and not in a substantial variation of

the shape of the monochromatic signal (caused by harmonic

generation). For these reasons it ought to be possible to

write down an approximate solution of the wave equation in

the following way :

E = E

q exp \± | Cot - riz)} (25)

where now : a =ft(E

o

)

it is clear that for E —°- this formula must become the

o Apple ton-Hartre e formla

We are now faced with the problem of calculating the

integrals (24). If we consider the collision frequency v a linear or a quadratic function of the velocity of the electron

kl
it is easy to see that, in the case $S^ , it is not

possible to give closed expressions for the components of

e and a . Furthermore, putting v= const., it is possible

to calculate all these integrals that are given by expressions

that do not contain E . On the other hand if we evaluate, in

—o 7 this case, the mean energy of an electron we easily arrive at

the following formula

<e >= * kT 1 +

e E

q

cp (v)

3m6kT

so that, also if we have a noticeable heating of the plasma,

the propagation is linear. All these difficulties can be

overcome if we consider only weak fields for which the electronic distribution function does not appreciably deviate

from its stationary form (maxwellian) . In this way it is possible to calculate this function, developing f in Taylor

series of E , stopping the expansion after the first two

terms and taking then v= const. If we indicate the new

distribution function obtained in this way with f(v,E ) we

have

Of

f(v,E ) = f (v,E =0) + (

' 00 ' o

' s 00'

2 2

OE E =0

O

E

that is:

f ^V-<2ffi>

2

1 +

2-2 , v 2 -, e E cp (v)v

o

Y

66(kT)

2 exp^ - mv

2k T

(26)

(26.1)

the mean energy of an electron calculated by means of this

function is given by :<£>= | kT ['*!] (27)

kd
where

a =

e

2

E

2

cp (v)

o 2m 6 kT

so that we define as a weak field a field satisfying the

following inequality (*) :

a < — >

(28)

(28.1)

By means of (26.1) it is possible to calculate all the integrals (24) and the following expressions are obtained:

1 . 2 v A w

ZZ 4 * P U2 + V

2

UT

1 A _£ c =a = -.— A -~r xx yy 4 7r 2 2 2 2 2

(u+toI ) + V (to- to ) +v

n rl

to.2 r

J&

w+w

xy yx 4 7i "2 V =a =a -a - xz zx yz zy

to zz A

.2 ..2

H

to -to.H

, v2 2 , N 2 2

e. =e =e =e = xz zx yz zy

to + v

2 r

to e =e =1-A -r^

xz yy 2to

w+w.H

to-Xi)

/ \2 2 , n2 2

_(to+toH

) +v (u-o^j) +v

to 2 -

e =e = A 7^ xy yx 2" to / \ 2 2 » k2 2

(to+ to ) +v (u-kW +v

where:

(24.1)

A = 1 + a (29)

(*) It must he said that this inequality can he too strong,

what is important is that the second term at the right

of (27) does not become much greater than the first.

^9
for E ~ 0, that is A = 1, these formulas become those of the

o '

' linear approximation.

Be fore calculating explicitly the dispersion relation it is

better to fine1, the expression of the components of £ and a in the usual reference system for the study of wave propagation. To this aim let us put the external magnetic field ^

in the plane y z of the new reference system at an angle

with z axis. It is easily seen that the matrix that allows

us to pass from the old reference system to the new one is

given by :

1 cos@ sin ©

-sin© cos@

If we indicate with e' the components of the complex dielecIjj- A trie permittivity tensor (defined by e* '==•£•,, -i — a., ) in

* J lk lk w lk

the new reference system and with eJ, the components of the

sale tensor in the old system, we have : m n

eik Yi

Yk ^mn

Introducing for simplicity's sake the notation

2

we have

u) Q OJ-lV a = A

m Q ( Q

2- or2

)

e xx " 1 "a Q

'xy = - e' - iaQUu cosq

yx n

£ xz

= ~ £ zx

= iaQwH

Sin0

e

' = 1 - a (Q - uTT sin ) yy H

a Mj sin© cos©

2

H

fc yz t zy

e zz

= 1 ~ a C0

2-»5 cos

2 ©) .

50
Let us consider the two Maxwell equations : vx h =±4| + 42 — — j c Ot c —

v xe ..111 - ' - c Ot

for the plane electromagnetic wave (25) propagating along

the z axis they give:

2 „ 2

30)

E

2

^ £>.E =

(30)

OZ c the two Maxwell equations and eq. (30) are equivalent if we suppose that there is not harmonic generation of the fundamental frequency oj Furthermore we have : oe w w on

+ i — ft, E = - i — z E

OZ c * — c

2

E oa2 — 2 - + — tX. E=-2

OZ C

OZ -

2

(J GO i - + -ttz

c

Ott „ w

2

2

c

otv

OZ

E-i —z—--E - c . tir- Oz

Tout if the electric field is sufficiently weak the propagation

will "be almost linear so that it ought to he possible to

disregard the right member of the last equations; this, on the other hand, agrees with the spirit of the perturbation

technique that we have adopted (*). So :

z c in this way eq. (30) becomes 1

(*) This approximation has also been adopted by Ginzburg &

Gurevich ref [l~j second part, see particularly eqs. (3.5)

an d (3.6).

51
[ XX

n/ E + £ 'E + e

» E =0

ox xy oy xz oz

e' E + yx ox

1 -tt Vy

E + e

' E = oy 7z oz

e« E + e

' E + £ 'E =0

zx ox zy oy zz oz

where, now, this is not a linear homogeneous system in the

three unknowns E , E , E as it happens in the linear

ox' oy' oz * case, hut it is a more complicated system owing to the pre2

sence of E in the expressions of e ' » However, in this case

o * ik ' too, it is easily shown that there exists a non zero solution for E , E , E provided that the following determinant

ox' oy' oz is equal to zero :

- • _

'XX

-yx

zx

tv -xy

yy

yz

xz

yz

zz

=

from which we have (compare with eq. (6.1) in ref. [j3]] ) J

ft. =1 - A X

1-iZ- Y sin 2 4 Y sin^G

2 [1-iZ-AX] V 4 [1-iZ-AXj

2 2 + Y cos

(31)

being : 2

X " 2

w. Y = _H

7 _ V Z ~ W

>r (compare with eq. (2.3) in ref. £14]] )

2

ft = 1 - A b

a -lr a - a c osG [B + VM+IT ]

(31.1)

being :

52
b =^Li ' H

r =

CO H

2 r sin

r

c

:: 2 cos » a = W

OJ H

B =

2 2 r a (a -A b + i r a

)

c _

/ 2 . , ^ v 2 <*. 2

(a - A b ] + r a

For A = 1 (linear case) eq, (31) becomes Apple ton-Hartree '

formula. In this case, as it is known [jl"], the results

obtained by the statistical theory and by the mean free path

theory must coincide.

The two signs that appear in eq. (31) show that in the non

linear case too, the field is split up into two components

whose propagation is never independent, given that under the

square root there appears the total value of the electric

field. We will call these two waves ordinary generalized

(O.G.) wave and extraordinary generalized (E.G.) wave,

identifying them with the corresponding ones of Appleton

(O.A. and E.A. respectively), in the limit E —*0 for every

propagation condition.

Let us put 2 „2

e E

w °

ff -

2

2m 6 kl w and A becomes

A=1+W

2rS cos g

1 + Z

2

sin J

1+Y)

2

+Z

2

(1-Y)

2

+Z

2

(32)

It is interesting to notice that the electronic density n , that is X, does not appear in (32) so that all the results

that are obtained for the Apple ton-Hartree formula when is

varied only n are valid also now if we subsitute X with A X.

2 The zeroes of fl are located at the points s

AX + iZ = 1 AX + iZ = 1 + Y

53
and one of the values of ft is infinite for ;

AX = (1-iZ) (l-iZ)^-Y 2

(1-iZ) Y

2

cos

2

In this way we see that for A = 1 we obtain the well known

results of the linear theory []l3j • Furthermore the two

2 values of ft are infinite for :

Y = + (-1 + iZ)

Z = + i

if sinO ^

if cos-5 ^

71 these poles are also branch points for ft if = — . It is easily seen from (15) and (26.1) that in these points

we have f (v) = 1, f(v,E ) = od , for these reasons all the

oo ' ' o ' integrals (24) diverge. Moreover A = oo and in this case the

perturbation technique that we have used is no more valid.

Given that the dispersion relation (31) depends on w in a

much more complicated way than Appleton-Hartree ' s formula,

we have studied it using an electronic computer. By means

of (31) we have calculated the refraction index \x and the

absorption index x using the relation s

u.2 / • n2

a = ( u - i x) = iN

from which

H =

X,

i\f7T?'<

i

/m2 + N

2 - l

In the numerical calculations we have taken as a constant : 0) the molecular temperature T, T = 200 °K

(2) the external magnetic field % , = 0.42 Gauss for which

w

R

= 7.388-106 puls/sec

5h
(3) the electronic density n, n = 10 el/cm

(A) the mean energy lost by an electron in a collision,

6 = 1 .6*10~ 3

(5) the direction of the propagation, we have put 0=0

(longitudinal propagation) in this case we have, for

symmetry, § = ~ ,

We have varied ;

("l) The amplitude E of the electric field

(2) Its pulsation around the chosen value of oj n

(3) The electronic collision frequency v , The figures show the results that we have obtained.

Prom the figures N° 5, 6. 7 it is possible to see that

the absorption index is in the non linear case greater than

that calculated by means of Apple ton-Hartree ' s formula,

moreover^ it increases with increasing electric field. For a -4 / field equal to, or le.ss than, 10 Volt/cm we do not have

appreciable deviations from the linearity. For low values of

the collision frequency (Fig.. N° 5, v= 2.5*10 coll/sec) the

absorption index of the 0.C-. wave increases very much for

oj ~ to j with increasing v this increase disappears (Figs.

N° 6 } 7) and for equal EQ and — the ratio between the

H

values of x for the O.G wave (and E 5 G, wave) and the values

of x for the 0,A, wave (and E.A. wave) tends to 1 with

increasing v . As it is seen from Figs. N° 8, 9,. 10 the behaviour of

the refraction index \i is similar to that illustrated for

X when E , w, v are varied.

The fact that the propagation tends to become linear

with increasing v is easily understandable. In fact if E

q and

go remain fixed the power transferred from the wave to the

electrons is unaltered, while with increasing v also the

55
power dissipated by the electrons increases due to the collisions, in this way the electronic temperature tends to become

equal to the molecular one

Prom the Figs. N° 8, 9 it can be seen that with increasing

non linearity, also the wave number of the O.G. wave increases.

This fact generalizes a preceding result £15]] according to

which with increasing non linearity the wave number of an

electromagnetic wave passing through an isotropic plasma is

also increased.

The value of a given by (28) varies along the curves

traced in the figures, for every curve the maximum value of

this quantity is indicated that, in general, is achieved by

a for a) ^ ug, It is seen that for some propagation cases the

condition (28.1) is not fulfilled. For this reason it is

interesting to see if we obtain different results taking into

account other terms in the development of f in series of

2 ^ oo E . To this aim we have calculated the third term of the

expansion that gives for A the value

2

A = 1 + a + %- (33)

6

and the fourth term for which

a

2 a

3

A=1 + a + -£--— (34)

If we consider the method used to calculate (31) we see immedjLately that we obtain the new dispersion relations taking

for A the values given by (33) and (34). The Figs. N° 11, 12,

13, 14 show the results obtained in this way. For electric

fields equal to, or less than, 10~ Volt/cm the results

calculated by means of the values of A given by (29), (33),

(34) are practically equal. For stronger electric fields, of

the order of 10 " Volt/cm, or for low collision frequencies

56
(Figs. N° 11, 13) the results are quite different. With

increasing v (Pigs.. N° 12, 14) these differences disappear,

given that the non linearity of the medium is decreased for

the reasons explained before.

The problem that arises when we wg^t to treat strong

fields and low values of the collision frequency does not

Consist in choosing a particular form for A hut in showing

that, in this case, we can still use the dispersion relation.

In fact, as it can be seen from Fig. N° 8 it is possible to

find some propagation conditions for which there is a strong

difference between the phase velocities of the O.G. and E.G.

waves. This causes, of course, a strong deformation of the

monochromatic wave launched in the plasma. In this case we

have harmonic generation and eqs (25), (30) loose their

meaning. From this analysis we infer that it is not possible

to treat, using a dispersion relation, the problem of non

linear propagation of a rather strong electromagnetic wave

in a plasma with a rather low value of the electronic collision frequency. We think that this problem could be solved

in a satisfactory way by trying to integrate directly the

system constituted by the equations of Boltzmann and of

Maxwell. In fact, it is clear, that the electronic distribution function (15) from which we started in order to obtain

eq.. (31) is valid only if the wave remains monochromatic.

4. - The Electronic Distribution Function for an Amplitude

Modulated Electric Field. (*)

The system of equations that we must integrate is given

by eqs (9) and (10) that is ; Of

e. 3 I 2- - + ~ r— H v E«f

at ' 2 av 1 . ^ av

3mv L I 2v

3/- kT v vv ( 1+ — T")f * mv av

= (9)

(*) All the results contained in this section will appear, in

a work by one of us (O.D.B.), in a more complete and elaborate form*

57
Of Cf

—T- + I

° E + — 5SXf,+vf =0 (10)

c)t m ov - mc - -1 -1

where now

'E = E 1 + T) cos ( a t+ p ) cos tot (35)

with [if ~ u) »a 7 0^T)<1, 0^P<2ti.

We begin "by defining three relaxation times; the first is the

quick * relaxation time of the electric field t , it is

E.q.

determined by the frequency of the carrier and is of the order

-1 -1

of u) , that is t_ ~ w The second is the slow relaxation

E.q.

time of the electric field ?„ , it is determined "by the

iJj .s . frequency of the modulating signal, and is of the order of

-I -

a. , that is L ~ a . The third is the relaxation time

E.s.

of the electronic energy x that, as we know, is of the order

of (6 v)" 1

. We shall suppose that

Wu *Vq. <<T

r« T E.s. < 36> ±

J E

If we remember the calculations of secT/e// 2 we see that (36) can be satisfied for values of a till to some kc/sec. From

the fact that "V, »t derives that now the electronic disE.s. r tribution function must depend explicitly on time by means

of the function cos( at+p ). In this way we, have the first

term of (9) of the order 0: the monochromatic wave (*

)

1 1 term of (9) of the order of f /- ^ - •£ . Let us put, as for

a T E. S .

Of —1 — OV

*) It is easy to see that this method of integration is tantamount to developing the functions f and f -j in a double

series of powers of the parameters

( ^

q

) « 1 and ( TE,q<

) « 1 . We shall calculate the first

r E.s.

terms of these developments.

58
We have

at

u o

a

2

f

"U at ov — etav

"but u (which gives the current) must vary in time at the rate

of the frequency uuof the carrier, so au/at ~ u> u-, in this

way, disregarding terms of the order of ct/w, we have:

^1 3U at

at at av

and eq, (10) hecomes the usual Langevin's equation :

a u —r + v u = —

at — m E + ^X% — c

We already know (see section 2 ) the solution of this equation

for a monochromatic wave, taking into account that (35) can

"be written as;

E=E - tjcos j(co + a)t+p}= +E cosarfc+E -ncos {(w-a)t-pj

and remembering eq. (13) we have the persistent part of f

given by;

V A(co)+ —t) 4A(o)+a)+A(o)-a) Kcos (at+p)

+ -T] -|B(.(jJ+a)-B(w-a) I sin (at+P)

+

-r 3f

o av —

cosoot+

(37)

B(co)+ -ir) jB(w+a)+B(w- a) l C os (at+p) +

1

° f

o + |n lA((i)-a)-A(&H-a) [sin (at+p) -~ sinwt

In the calculation of E»f we disregard termr of the order

of ^ = -

i^-L that iSj we put : E.s.

59
E •A(w-a) = E 'A(w) = E «A(co+a)

-< E -.B(w-a) = E -B(u) = E •B(w+a)

and we define a function S (t) "by :

I

2

S(t) = 1+T) cos(at+p)

(38)

(39)

in this way eq (9) becomes :

Of

Ot 3mv

6

2 v

2 Ov

2 2

HE »A(cjJ)cos wt+E *B(w)cosoJt sin'Wt

"O — —o —

1 Of -i

r 3/„ kT v _ - —— i vv (1+ — -~) f

2 Ov |_ mv Ov o =

On account of (36) we see that the first term of this equation

(of the order of f /

) can be disregarded with respect to

E , S . the second term (of the order of f A_ ) and to the third

o' E.q.

term (of the order of f /t ) so that taking the mean, over the

period of the carrier, of the resulting equation we obtain : —6 r 3 L

~ T~ vv -41 + kT

2 -.2

e E

v

+ fe + 72— *

(v > E(t))

2v - I 3m 6 v

ov

=

a first integration gives

e 2 E

2 vv

3 f + vv

2

(kT+

of.

3m 6

} ?(v) E(t)) -t~ = A(t) Ov

where A(t) is an unknown function of time, it can be easily

seen that, unless it is not identically zero, f diverges

strongly for v = 0. so that A (t) * and

f

o (v,t) = C(t) exp J -f

mv dv

2 -2

e E

kT+ jsr?*TM*TM

(ho)

&
The study of this distribution function is made easy "by the

fact that it is identical with (15) provided that we substi2 2

tute E with E S(t). In particular we have the mean energy

<£> depending on time, <e (t) > , (* )

, and taking as "before

only the wave E^ it is given by (19) where now;

e 2

eT 2

X

2

y= —=4-^— s (io

3 6k¥

furthermore we have, as before :

( 9<£ <*» ) =0

n

Taking into account (38), eo (37) for f can be simplified

in the following way

n Of

^1

= 1 + T}C0s( at+p) A(w) coswt+B(w) sin'cot v (37-1)

furthermore putting ;

4 71 2t = — en

00 > DP

v f . dv = a • E + —

— 1 = — t

it is possible to evaluate the conductivity and the dielectric permittivity of the plasma. It is easy to see that

disregarding terms of the order of j =

T

* ^* the components

-t-j • s • of these tensors are given by eqs (24) provided that we

2 2 substitute E with E E (t)

o o We can still use approximation (26), so that the components

of the complex dielectric permittivity tensor can be written

as

(*0 ^k

(o),a,t) = ej^U) + A ik (w,a,t)

(*) See Fig. N° 15.

61
being

e E cp(v)

A <u,«,t) = e. k

(o.) 2m6

°

k!t

S(t)

where e.' (u) is the expression that we obtain in the linear

lk

theory for the carrier. By eq (41 ) we see that e' is varied,

around its steady value e! ( u>) , by the quantity A., (o),a,t).

Let us now launch in the plasma, besides the wave (35), also

another wave of the type :

E = E cos( v t+ A) — Y ""O y

m

Indicating withcp(w,v) the function given by eq (14) and

with cp (y,v) the same function with u> replaced by y a*id where

$ is substituted with the angle between E and $& , it is

-"O y

easily seen that the electronic distribution function f (v,t)

is either:

f (v,t)=C(t)exp j. -i

mv dv

(^3a)

^m6 E^9(ui.,v)S(t) + 2Ej9 (Y,v)cos

2

(Yt+A)

. ° °Y

or

f

o (v,t)=C(t)exp J

-J"

mv dv

o , _ e kT+ -r~r 3m6 rE^9(w,v)E(t)+E 2

cp( Y ,v)

(^3b)

Eq (43 .a) is valid if \»x > (43 .b) is valid if x„ «t

t „ being the relaxation time of the electric field (42) which

-

is: "L ~ y .Furthermore it must be noticed that in the calculation of eqs. (43 a,b) we have neglected terms proportional

to the product of the two waves (35) and (42); it is possible

62
to make this approximation if there is no phase correlation

between the two fields, let us suppose now that the electric

field (42) is small and that y is ?aT from the gyromagnetic

frequency so that we can neglect in eqs (43) the part due

to this wave, in this way f (v,t) is given "by eq (40). The

equation for the propagation of the wave (42) is

VXVXE + -i- E

Y

.±=l + 4| -%Sa = M c

2 =

at

2

o

2 at

and given that the wave (35) contributes to j .,. the pro-

•^cond' * pagation conditions of the wave (42) will depend on those of

wave (35). We can still use the approximation (26); an easy

calculation shows that in this case eq. (44) becomes :

2

VXVXE + -1 |'y)--=| + *f a( Y )'-^ + 5=

Y c at c

where e(y) and g(y) a^€ the expressions which we obtain in

the linear theory for the wave (42) and where £J» is given by :

e E^ cp (w, v) -< aE

1 + ~2mTM S(t) >y Tt

c It is clear that the wave (42) does not remain monochromatic

and that these considerations constitute the starting point

for a microscopic theory of cross-modulation.

/tW jLaosaayj oj; The. Idu'lltto a* Sde^^e u^iciie eWo ILviVercxtA.

63
REFERENCES

[l] V.L. Ginzburg, A.V. Gurevich - Soviet Physics. USPEKHI

3, 115, 175 (1960).

[2] L.G.H. Huxley - Phyl.Mag. 2£, 210, 442 (1937); 29, 313(1940)

Proc.Phys.Soc. 64B, 844 (1951)

see also:

L.G.H. Huxley, R.W. Crompton - The Motion of Slow Electrons

in Gases - in "Atomic and Molecular Processe"

ed. by D.R. Bates - Academic Press (1962).

[3j B.J. Davydov - Zurn .Eksp .Teor .Piz . (U.S.S.R.) 6, 463(1936);

7, 1069 (1937).

£43 S. Cnapman, T.G. Cowling - "The Mathematical Theory of

Non-Uniform Gases" Cambridge U.P. (i960).

CO A * v * Gurevich- Sov.Phys .JETP 3, 895 (1957).

[6] A.V. Phelps - Journ.of Applied Phys . 31, 1723 (1960)

[j] H.K.Sen, A. A. Wyller - Journ.of Geophysical Res. 65 3931

(1960).

[]8] E.T. Whittaker, G.N. Watson - Modern Analysis - Cambridge

U.P. (1958).

[9] M.J. TJruyvesteyn, P.M. Penning - Rev. Mod. Phys . J2, 87(1940).

[10] J. A. Smit - Physica, £, 543 (1936).

£11] L.M. Kovrizhnikh - Sov.Phys. JETP j_0, 347 (1960)

&
£12] E.W. Crompton, L.G. Huxley, D.J. Sutton - Proc .Roy .Soc

218A , 507 (1953).

| 133 K.G. Budden - "Radio V/aves in the Ionosphere", Cambridge

U.P. (1961),

f"l4l Re Gallet - Whistlers in "Geophysics, the Earth's Environment" ed. by De Witt, Hieblot, Lebeau

Gordon & Breach (1963).

[J5] M. Epstein - The Phys . of Fluids 5, 492 (1962)

65
List of Figures

Pig. N° 1 - Values of < e > calculated "by means of eq. (19) in

-5 6 whichs 6= 3.4*10 , oj = 7.388' 10 puls/sec corresponding to

<$>= 0,42 Gauss, X- 10 cm corresponding to a pressure

p = 6.7- 10""* mmHg, T = 210 °K. It is possible to see clearly

that the energy absorption maximal for w = w . Moreover for

n

00 weak electric fields and for increasing jjj— all the curves

H

tend to the same orizontal asymptote that is equal to the

value of the energy of the electrons in thermal equilibrium

with the molecules.

The various curves correspond to the following values of the

E| component of the electric field:

Curve N° 1 E7 = 10~ Volt/cm

Curve N° 2 E, = 7.5- 10~ 4 Volt/cm

Curve N° 3 E

l = 5 '10 Volt/cm

Curve N° 4 ET = 2.5 -10~ 4 Volt/cm

Curve N° 5 ET = 10~ 4 Volt/cm

-° -5

Curve N° 6 E, = 5*10 Volt/cm

Curve N° 7 E 1 = 10 J Volt/cm

±0

Fig. N° 2 - Values of < £ > calculated by means of eq. (19)

with the same values of the parameters as for Fig. N° 1.

For this figure we can make the same observations as those

that we made for Fig. N° 1. The various curves correspond to

different values of -jt- namely ; Curve N° 1 oj = w^

Curve N° 2 oj = 0,7 Ug, w= 1 .3 wH

Curve N° 3 w = 0.5 oj'h? w = 1 . 5 wH

Curve N ° 4 ui = 2 w H

Curve N° 5 cu = 3 w

H

Curve N° 6 w = 4 oJtt Curva N° 7 w = 5 Uu

66
Fig. N° 3

(t) Shape of the function P* v

" (v >v ) given by

eq. (22) for 6 = 3.4*10~ 5

, Ej = 10~ 3 Volt/cm, p = 10~ 3 mmHg

(that is for Ei /p = 1 corresponding, according to eq. (23.1),

to a mean electronic energy equal to 6,73 eV) , \ = 66 cm. From

this figure it is possible to see that in this case we are

compelled to take into account also of inelastic collisions.

Fig. N° 4 - Behaviour of <e>(for u = Ujj) as a function of E,

Curve N° 1 s values of < e> calculated by means of eq. (19)

_5

in which 6= 3.4*10

values of < e > calculated by means of eq.(23.l)

-5

in which 6= 3.4*10

values of < e> calculated by means of eq. (19)

in which 6= 1.6° 10

values of <e> calculated by means of eq. (23.2)

in which 6= 1 .6*10~ 3

Curve N° 2

Curve N° 3

Curve N° (

Fig. N° 5 - Values of X for v = 2.5*10 coll/sec

Curves N° 1 (O.G. and E.G.) E = 10~ 3 Volt/cm a 1max

= 2.49

Curves N° 2 (O.G. and E.G.) E =7.5*10 4

" o Curves N° 3 (O.G. and E.G.) E = 5*10~ 4

" Curves N° 4 (O.G. and E.G.) E =2.5*10~ 4

" o

c Fig. N° 6 - Values of x for v= 5*10 coll/sec

Symbols same as in Fig. N° 5

a_ = 1.40

2 max

a, = 0.62

3 max

a A = 0.15

4 max

a, = 0.67 1max a„ = 0.38

2 max

a, = 0.17

3 max

Fig. N° 7 - Values of X for v= 7.5*10 coll/sec

Symbols same as in Fig. N° 5 a, = 0.32 1max

a =0.18

2 max

a^ = 0.08

3 max

67
Fig. N° 8 - Values of \i for v= 2.5*10 coll/sec

Symbols same as in Pig. N° 5

a, = 2.49 a_ = 1.40 a. = 0.62 a„ = 0.15 1max 2 max 3 max 4max

c Fig. N° 9 - Values of u for v= 5*10 coll/sec

Symbols same as in Fig. N° 5 a, = 0.62 a = O.38 a^ = 0.17 1max 2max 3max

c. Fig. N° 10 - Values of u for v= 7.5-10 coll/sec

Symbols same as in Fig. N° 5

a, = 0.32 a = 0.18 a. = u.08

1max 2max 3 max

c Fig. N° 11 - Values of X for v= 2.5*10 coll/sec

1(0. G. and E.G.) a with A given by (29) E

q = 10~3 Volt/cm

1(0. G. and E.G.) a

2 with A given by (33) E

q = 10

-3 Volt/cm

1(0. G. and E.G.) a

3 with A given by (34) E^= 10~ 3 Volt/cm

3(0. G. and E.G.) a with A given by (29) E

o =5'10~4 Volt/cm

O A. 3(0. G. and E.G.)a with A given by (33) E

q

=5'10 * Volt/cm

Fig. N° 12 - Values of x for v= 5*10 coll/sec

Symbols same as in Fig. N° 11.

Fig. N° 13 - Values of u for- v= 2.5*10 ooll/sec

Symbols same ae in Fig. N° 11

Fig. N° 14 - Values of u for v= 5*10 coll/sec

Symbols same as in Fig. N° 11

Fig. N° 15 - Behaviour of <e(t)>for ET = 5*10" Volt/cm,

other constants same as in Fig. N° 1, for various values

Of T) .

68
Pig. 8° 1

69
Fig. IT 2

70

Ei in Volt/cm
*(r)

P (v>V

c> )

0.75-

0.50 -

0.25-

e in eV

Fig. N° 3

71
Fig. N° 4

10 7-5 io"4

2.5 5 7-5 10

3 2.5

- ; i Ei in Volt/cm

72
Fig. N° 5

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

JaL

OJtt

73
Fig. N° 6

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1 . S

U)H

7^
Pig. N° 7

0.5 0.6 0.7 0.8 0.9 1.1 1.2 1.3 1.4 1-3

03

OJ.H

75
Fig. N

c

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

76
.2

.1

2.0

•9

.8

• 7

.6

• 5

• 4

.3

.2

.1

1.0

• 9

.8

• 7

.6

• 5

• 4

.3

.2

0.1

••1;^0".-Gi l

Fig. N° 9

0.5 0.6 0.7 0.I 0.9 1 1.1 1.2 1.3 1.4 1.5

0)H

77
.2

.1

2.0

.9 i

: j~ErQ j | 2-eLg. !

Fig. N° 10

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

OJ H

78
19

8

7

6

5

4

3

10

9

8

76

5

4

10

-2

Fig. N° 11

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1-5

GO

OJ.H

79
Fig. N° 12

0.5 0.6 0.7 0.

80
Fig. N° 13

0.5 0.6 0.7 0.

81
Fig. N° 14

82
<c(t)>

in eV

1.25

0.75

0.50 -

0.25

"0= 0.99

T)= 0.9

T)= 0.8

T)- 0.6

T)- 0.4

T)= 0.2

n «

Fig. W° 15

OOOOOOOOOOOO flL.fi W N N ft 83