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AFCRL- 63-81

SAD

AV 61 (052) - 145

Ju 1962

9 4= TECHNICAL SUMM(ARY REPORT N.2

z 1 April 1960 - 31 December 1961

MICROWAVE INVESTIGATION OF THE DIELECTRIC WAVEGUIDE PROPAGATION BY

MAGNETO-IONIC DUCTS

lug. A. GILARDINI

SELENIA S.p.A.SVIA TIBURTINA Km 12,4

S*ROME - ITALY

The research reported in this document hasbeen sponsored by the Cambridge ResearchLaboratories, OAR through the 1kropean Offioe,Aerospace Research, United States Air Force.

mom

I

SUMMARY

The work performed under this Contract during the indicated

two years period is described. Theoretica] work includes : completion

of the basic theory for propagation along plasma columns in magnetic

fields, derivation of Brillouin diagrams, discussion of particularly

significant limits and preliminary analyses of the non-uniform plasma

case. Experimental work includes measurements of basic propagation

parameters : transmitted signal, wavelength and group velocity, and

of physical plasma parameters as the electron density.

TABLE OF CONTEN;TS

1. - Introduction ............... ................ pag. 2

2. - Comparison with beam waves ........ ............. pag. -

3. - Whistler propagation ......... ................ .. pat. 11

4. - Propagation in non-uniform colimns ..... ......... paC. 14

5. - Electron density measurements ..... ............ .. pa.,. 22

6. - Conclusions .... ................... .. ....... pag. 279

7. - References .. .......... ..................... ... . 30

33S1 d36

a i i m . . . • ,

INTRODUCTION

During the two years covered by the present report extensive

theoretical and experimental work has been performed on the propagation

of electromagnetic waves along plasma columns in longitudinal magnetic

fields.

In the theorj the most substantial contributions to the under-

standing of the propagation cLaracteristics are the discussion of the

dispersion equation solutions in the W6 e(L region, the evaluation of the

power ratios in the same region and the derivation of the Brillouin

diagrams. This analysis is reported in Technical Scientific Note N. 2.

A more complete discussion of the propagation characteristics of circular

ly symmetrical modes along an uniform plasma column of circular cross-

section will appear on the September issue of the Joarnal of Research,

edited by the Nationnl Bureau of Standards.

In the present report we describe other theoretical coroiier-

ations, which are of importance for understanding the ýro-agation theo-

retical and experimental results.

The Brillouin diagrams in the limits of a very thin hsnma

(d-_.o) and fora large cross-section plasma (d-p o) are discussed. 3y

means of a Doppler shift they become the corresponding diagrams for an

electron beam; it is interesting to rediscover in tLis woy the well-

known space-charge, cyclotron and syncronous waves.

As another special case, the Brillouin diaj-rams are discussed,

when the following typical ionospheric conditions for whistler propagatioa

are satisfied

Purpose of it is to find whether the delay versus frequency character-

istic of whistler atmosphericimay be explained also assuming propagation

raglio .2 di 36

"•SL S.p.A.

along well-defined plasma columns.

All the previous theoretical analyses assume an uniform density

plasma, an assumption which is not satisfied in the experiments. The ne-

cessity to explain the observed experimental data has led us to consider

also theoreticaly the effects of disuniformities.Results have been obtained

on simplified aspects of this problem; they are reported in this document.

A very long report would be necessary to describe fully all the

experiments and tentatives performed during these two years. Many

technical difficulties have been encountered, due to the RF high-power

involved in producing the discharge (gas purityand transfer of power

are the most important), to the difficulty of exciting selectively the

various propagation modes, to the necessary limitations in the dimensiona

of the experiment (which would have to be the scaled laboratory model

of an ionospheric duct) and finally to the complexity of some of the

me:asuring electronic equipment.

These experiments can be grouped in the followin~j cla•ses

a) Transmission measurements. Purpose of these measurements was to

collect more data over a wider range of experimental conditions, in

order to confirm the previous experim'ntal results and the general

correctness of their qualitative interpretation.

b) Propagation w:'velenght measurements. Previous data were insignificant,

due to the presence at the same time of various modes and of undesired

stray fields.

c) Electron density measurements. This is a basic quantity, which is

necessary to know in order to compare propagation experiments with

theory. Conventional methods had failed in aur geometrical and physical

conditions; new special methods had then to be used.

d) Group velocity measurements. From these data, knowing electron density

and the other physical parameters, we can derive experimentally

Brillouin diagrams. The same could be obtained in a more simple way

0 3 di 36

30- 3.-= S.p.A.

from the propagation wavelength measurements, but , as it is later

explained in this report, we can measure these wavelengths only for

the first mode, which is the less interesting one.

In the report detailed information is given on these experiments

and on their results. In all the experiments the plasma has been produced/

using a new RF generator capable of delivering up to 4 KW of power at

frequencies between 1.5 and 3 Mc.'fhen specific data of the operating

conditions will not be given, reference has to be made to the analogous

cases reported in Technical Summary Report N.1.

p

4 36fogioe d

SpLA.

COXPARISON WITH BEAU WAVYS

It is enlightening to relate our propagation modes to the

well-known space-charge, cyclotron and synchronous waves,which can be

excited in electron beams.

This relation is best shown on a /1, versus d/A diagram,

6, and E3 being the fix parameters for each curve. These curves can

be simply derived from the AI/ Xo versus d/ke plots, previously

obtained from the theoretical analysis. The usual beam waves are derived

in the microwave tube theory under the simplifying assumption of a phase

velocity much smaller than the light velocity, or <,/ K 1 1, which is

equivalent to our quasi-static approximation.

The result is that all the A •A, curves, near to the d/1

axis, which is the region where the quasi-static aprroximation holds, are

vertical and'given by the relation

where )/(")is the m - th root of the modified characteristic •quation:

The above vertical curves exist only when

- 6) i)e 3 ) , 1 f eO (an infinite set of curves exist4

b) e L 0 F, 2: 4 (an infinite set of curves exists)

c) "3 £ 0 , a&A/E,(only one curve exists m -

In an infinitely extended plasma d/x-tw. Then )/,, approaches

infinity and the second member of the characteristic equation becomes

equal to- 1.

5 ., 36

--- '•-•Sp.A.

Let us consider first the cases a) and b) above. The two

dielectric components e and E. have opposite signs and the equation

becomes s

('V' E3 14 (3)

Two cases exist, in which this equation is satisfiedas - aId Vv 0 asa where

is the m - th root of the Bessel function Jn.

When E and E$ are both negative, I and I, functions replace

Jo and J1 and another poesibility exists. This is ,'j3- 1, a condition

which satisfies the characteristic equation~because the function ratio

Ij/Io equals unity when its argument becomes infinity.

In a stationary plasma, 6.0 when W , 4 5 W when w &it

and 64 3 = 1 when i = I )/L provided Cdj > W In -n

electron beam moving with velocity V& along the magnetic field axis

t

-4 (4)

(, (3 • (5)

Then e o when

di d 36

; p.A.

This is the well-known phase propagation charaoteristio of space charge

waves.

The 9,. oe condition is satisfied when t

(.,,_ o,-, ' o

This is the phase propagation characteristic of cyclotron waves.

The E1- 1 -condition implies t

Here too this case exists only if W tj For this reason and bectuse

such waves are of the surface type, they are not considered in the usual

microwave tube theory. They may however be of importance when a magnetic

beam focusing is not used and when proper excitation is provided.

Let us consider now the other limit case d/j-0 a •

The, JVý approaches zero and the KiKO ratio goes to infinity. One case,

which satisfies this limit, is 63 'O ( I being positive). Anothier

set of solutions is given by the equation

0(9

.Iis implies + *0 , beside the previous 1,+ -40 case.

0 u0_7... di 36

~7LSp.A.

In a stationary plasma, . - -. only when & - 0 and

e .0 when tj-

In an electron beam, E. = -a when

This is the phase propagation characteristic of synchronous waves.

II In an electron beam, 61--0 when:

z 2

P V& ) - t =(ii)

In the theory of transverse - field microwave tubes these waves are also

called cyclotron waves. In fact, we have usually t; a Ls t , so that the

phase characteristic at this limit coincides with the previously given

characteristic for these waves in an infinitely extended beam.

It is interesting to see how the above picture of beam waves is

modified, according to our analysis, when the complete theory is used

in -lace of the results of the quasi-static approximation.

In the infinitely thin column ( d -. o ) the complete theory

does not contribute any further solution, beyond those given by the

quasi-static analysis. In fact, the /X. versus d/A. curves

show clearly that, when d -+ o, A,/. 0 always, so that this case

implies always the validity of the quasi-static approximation.

8. d, 36

inA

The same )i//o •urves indicate instead more possibilities

for the infinitely extended plasma. In fact, if , I and eoae E Y

all the investigated circularly symmetrical modeeoexcept the first one)

approach, when 4 '+ to , the common ratio /a I/so . It is

worth of mention that this same ratio is attained also in the case of

a TEX plane uniform wave with a right hand polarization; however, fields

are different and in our case we obtain

E F 1___S. . . . .( 1 2 )

whereas in the TEM case one has t

0

When ~ 4and 63 -CEN the common ~/ .limit as d -P w is

I/o , for all modes except the first one ; in this limit A2 = -A, and

the total field then vanishes in the infinite plasma.

The propagation characteristics for a plasma column or for a

beam with an extremely large cross-section can be easily derived from

the above considerations (the first mode will be negleotedadue to its

low capacity of carrying microwave power inside).In order to have ,

we assume W 4 L46 . Starting from cj@ , 1% is a large negative number,

certainly less than • In this region, substituting for I and 3

" 9. d136

their values in the e formula, we obtain i

t "(14)

There will be a frequency at which this becomes equal to the value

similarly derived from the a( formula after substitution of the £ and

E 3 values

It

For &j values larger than this one, the versus 0 curve follows the

last formula up to w = Cj•

Waen ar(WL, as in most microwave tubes, the curve intersects

the W - W e horizontal line, given previously by the quasi-static

analysis; the diagram P versus W for very large d values shows the

continuity behaviour sketched in fig.1.

Wb

S~FIG 1

The corresponding diagram for a beam is obtained as usuallyby substitu-

ting the doppler shifted frequency (-Ptr, to signal frequency

10 36j m ,,, g, o. , i

WHISTLER PROPAGATION

The possibility of finding a region, on the kd versus Pdagram, where the propagation shows the typical f - I--.behaviour of

whistler propagation in an uniform ionosphere, has been reexamined on the

basis of our most recent results for uniform cylindrical plasme.,olumns.

We are assuming here the following conditions, which are satiefieý

along the ionospheric path of propagation of whistler atmospherics

< W4 a (16)

This means that E3 is negative, whereas 1 is positive; both, however,

are much larger than unity.

Recalling the shape of the previously discussed c( )f/A, versus

kd or P curves for constant E4 and ' 3 , we see that each of them

consistsbasically of two straight line parts. It is reasonable to expect

that over any significantly wide frequency range the shale of the re-

sulting kd versus ýk curves are correctly predicted by the elaboration

of one alone of the two sets of straight lines.

Let us consider first the constant *( part of the ('(versus kd)

curves, which is typical of the infinite plasma condition. We have q I

for the first solution ( &4 and Jej I), %(.or 0(ý for all the other

solutions. Then E4 is very large i

E0a - (17)

Then ft condition : 4 : - 3 • , for which the constant o value

I..1.1 . di 36

3il n1I

¶~3~9 p.A.

is es( , is attained when t

•. (18)

In our case ( £, and It],'.

c.J&L(19)

f t (20)

Correspondingly we have i

X &,/r first solution

p • other solutions,when w 4 ()/2- = •"1,(22)

.. r other solutionswhen C/a itJ (23)

The classical whistler propagation behaviour may thus be found

also in plasma columns, but only over the frequency region botweon the

cyclotron value and its half.Over this region whistlcr delay versus

frequency curves show a characteristic rising behaviour.

Let us consider now the set of straight lines diverging from

the origin in the O( versus k4 plane (quasi-static approximation

region).

For discussion simplicity it is' convenient to solve this problem

in the similar case of a plane geometry, where the basic equation of

IbIo12 .6

these lines is i

~ - (24)

Sbeing half the plasma thickness.

The solution is approximately t

P3 L F 3-S 'MWii + %I I3(25)

When m~

4 t!- ! (26)

"When m# # 0

the whistler characteristic, which is found in uniform plasmauis here

never attained.

It seems then possible to conclude that in general propagation

of circularly symmetrical fields along ionospheric plasma columns does

not explain whistler typical characteristics. In fact, we may justify

at most only the rising part of the whistler delay versus frequency

curve near cyclotron resonance. ,Moreover, our theoretical curve must

join, at frequencies lower than half the cyclotron value, a constant

delay instead of showing the typical behaviour(frequency drops as the

square of time delay).

rge. 13 di 36

LPA

PROPAGATION 3 NON-UNIORN COLDMNS

Various investigations on the chanjeo of the propagation oharac-

terietics due to a non - uniform electron density in the plasma column

have been performed for the purpose to explain some experimental results.

A general method for taking into account longitudinal disuniformi

ties has been worked out by Prof. Cambi of the Rome University, on a

consultant basis for this Contract. His analysis is reported in Appendix A

More detailed analyses have been performed on the effects of

transverse disuniformities. ?or analytical simplicity we have always

considered a plane geometry, instead of our cylindrical one. In this case

we have assumed density to be dependent only on the coordinate of the

axis perpendicular to the plasma boundary.

As a simple case we have first investigated the propagation

conditions when the electron density drops linearly from a maximum at

the center to a finite value at the plasma boundary and when no magnetic

field is present. This same case has been investigated and similar resailts

reported at about the same time by Prof. Schumann. For this reason and

because of the scarce interest of this case for the interpretation of our

experiments, no further discussion will be presented here.

Details will instead be given of' a simple, but here more inte-

resting case.

As it is well-known, in the quasi-static approximation retar-

dation effects are neglected and the L.C. electric field is derived from

a scalar potential:

r -E (28)

W0.1 . Idi36

whereas a.o. magnetio fields are entirely neglected. We consider a slab

of plasma in free space; a coordinate axis system is chosen as shown

in figure 2. The constant magnetic field is directed along the poesitive

L. axis. Plasma electron density is a function of the transverse x ooordi

nate only, and so are e, Eand E3"

FIG. 2

Maxwell divergence equation provides then the following

differential equation for 0 :

t0- * ,~ ) ,~x(29)

In the plane geometry the modes, which correspond to the circularly

symmetrical modes of the cylindrical geometry, require 'O/ = 0 , and

then the differential equation for 0 becomes

4 4 E_ A + (30)

Let us assume now that typical whistler conditions (16) are

15 c, 36

satisfied, so that :

4 •(31)

3 -(32)

T heir ratio is thus density independent and is not function of position.

Eq. (30) becomes

- ej .~ e4. . (33)CD' ef x a t Ef Ox Ox

A simple solution is obtained if the density varies esponential-

ly

I 1f (34)

ie may call I/a the characteristic length of the density disuniformity.

Fq. (33) becomes

~J~L !Ix (35)

which has as a suitable solution

A Li[ IJ: e/ j3 (36)

16 36NOW ,. . di

For continuity E. (z - o) must vanish, and then )4/xrnO.Thi.leads to the following relation for I I

06A (37)

Outside the plasma a suitable solution for the potential in

* a PILo (38)

At the plasma boundary (x - b) Dx and E. must be continuous.

This leads to two relations between A and B, which are satisfied only

if the propagation constant P and the dielectric constant components

are related by the following dispersion relation

where I is evaluated at the plasma boundary

The second member of (39) can be rewritten as

1'3)P t (40)

S44

ýhe first of these two terms, as we shall show later, becomes negligible

for all modes except the lowest one, and eq. (39) has then the following

solutions s

S1 36

from which i

(I [(T4~' W i&~r*~ (42)

The first term of (40) is then :

4 F(11 I __ 4_6__.)_

which is obviously negligible in whistler conditions and provided k 4O.

Eq. (42) has to be compared with (27). The result is that in

the presence of an exponential density decay, the propagation oonstant

P given by the uniform plasma theory has to be multiplied by the factor

"• --- -- IT,, When the plasma thickness is a few times the

characteristic length of the disuniformity the effect may thus be large,

particularly for the lower ft modes.

18 36fgo di

PROPAGATION AND WAVELUGTH KEASURDIRITS

The experimental results reported in Technical Summary Report M.1

can be explained in a much more clear way with reference to the Brillouin

diagrams derived during this period and reported in Technical Scientific

Note n.2.

We recall that the transmitted signal is detected by an internal

probe and that propagation is observed only when Wd > &4.

Reference to figs. 3 and 4 indicates clearly that when UJ 4 cj

propagation may take place only t

a) In the first mode, when CJr> eg API. This is a surface wave and

as such carries very low microwave power along the central axial region

of the plasma column. For this reason this mode is not detected by the

probe.

b) In the upper branch modes, when YZC&4( r C In most cases these

waves are backward, and then are not properly launched by the helical

couplers. These waves may carry also forward power, but only at fre-

quencies very near to cyclotron resonance; near this resonance,

however, waves are very strongly attenuated by electron collisions.

All these facts may explain,why such waves have never clearly detected

in our experimental arrangement.

"When WC. Wj propagation takes place

a) when CJr .4 , over a limited range of densities and provided the plasma

diameter is sufficiently large

b) when (J always.

In Technical Summay Report n.1 we have given reasons to believe that

experimentally observed propagations are related to conditions b) above,

namely (•. L•,W!

.19 di 36

JI Ill A

One of the experiments, initiated during the first year of the

Contract and continued during the successive months of 1960, had the

purpose to measure wavelengths, when propagation was taking place along

the plasma column.These wavelengths are determined from the standing wave

patterns measured with a pick-up probe, which slides along the column

very closely to the outside surface of the plasma container.

To obtain significant data it was immediately apparent the necessity to

reduce the intensity of some undesired, stray fields, which were definite

ly found to be present outside the plasma.

S... The situation was improved by a better choice and arrangement of absorb-

ing materials around the tube. These materials reduce propagation

between input and output couplers, as well as spurious coaxial and

waveguide modes. However, it was found that residual propagation due

direct coupling and to dielectric modes along the container glass walls

could be effectively reduced only feeding the microwave signal directly

into the plasma, instead of injecting it through the glass walls by means

of an external helical coupler, as done in the previous experiments.

Many different types of inside couplers were tested for this purpose;

finally a plane spiral antenna of small dimensions was chosen for its

good coupling and mode selection properties.

Using thin couplers, properly placed absorbing materials and a

new accurate driving mechanism for the probe carriage, it was possible

to perform ainificant wavelength measurements. The result is that the

measured wavelengths are only slightly less than the corresponding free

apace wavelengths.

This is in agreement with our theoretical analysis, provided the

signal detected by the sliding external probe is propagating according

to the first circularly symmetrical mode. Such a possibility is highly

plausibile, because the first mode represents a surface wave; in this case

2C 36g di

fields just outside the plasma are strong and may mask easily higher

order mode fields.

The opposite situation has to be found inside the plasma* where

the transmission detecting probe is placed. For this reason we believe

that the observed transmitted signal characteristics are in most oases

due to modes higher than the first, whereas the externally measured

wavelengths refer to first mode propagation.

During the two years covered by this Report further propagation

experiments have been performed. We shall mention here only two of them.

In one experiment, we have used our typical set-up but a different

frequency range, namely the 1300 Nc/s range. Basically, we have observed

the same phenomena as in the 5000 Mc/s range, the only difference being

an higher pumber of transmission peaks. This is in agreement with the

assumption that propagatioA is observed when &t, •

In a second experiment we have investigated propagation character-

istics using different tube diameters. We have found that the detected

signal power dependes on the tube diameter and that this signal practi-

cally disappears when the diameter is less than 4 cm. This experiment

was performed in pure Neon and in Neon contamined with Mercury, but no

significant difference was observed changing the gas. To explainwith

our physical picture, the observed behaviour we must suppose that the

electron density decreases with the tube diameter (for instance, this

may be due to larger diffusion losses); so that when d 4 4 cm the con-

dition Wt W.> is no longer satisfied.

21 i 36

ELECTRON DENSI7T MEASUMU=

In order to proceed from the previous qualitative considerations

into quantitative verifications of the theoretical analysis, it is neces-

sary to know the values of all basic physical parameters which charac-

terize the problem. Geometrical data, signal frequency and magnetic field

intensity are easily measured according to standard techniques. Diffioul-

ties arise, when we try to measure electron densities with conventional

methods; the failures of these tentatives have already been described

in Technical Summary Report N.1

A new approach was then chosen. The value of the average electron

density in the discharge tube is derived from measurements of the guided

wavelengths of microwave signals propagating along the plasma, when

metallic walls are placed around the tube in a waveguide arrangement.

Except for the presence of the external cylindrical metallic wall, con-

centricaly placed near to the discharge tube, all the other geometrical

and physical conditions are unchanged.

A.W. Trivelpiece and R.W. Gould 2), who first proposed this

method, compared it 4ith other microwave measuring techniques, as the

cavity and the scattering methods, and found that it provides results

in good agreement with those given by the other methods.

The measuring set-up consists of a cylindrical waveguide, 7.0 cm

I.D., with a longitudinal slot. A screw-drived carriage, traveling along

the guide, carries the probe. The cut-off frequency of the empty wave-

guide is at 2500 Mc/s; measurements have been taken at lower frequencies,

so that propagation modes derived from conventional waveguide modes do

not exist. At these frequencies and when the density becomes infinitely

large, the field configuration and the propagation characterietiec ap-

proach those of the classical T&' coaxial mode.

Measurementsare conveniently platted as Brillouin diagrams

2/. X3 versus k - 21/ Pomparing these curves with a set of

22 36

S-PA

theoretically evalusted diagrams for various plasma frequeneise the

experimental densities are determined. Theory necessarily assumes uniform

plasma, so that the described data must be regarded as average density

values.

The solution of the propagation problem in our real geometry

(plasma bounded by a glass tube, air and metallic waveguide) was first

carried out and solved with the aid of the so-called quasi-static approxi

mation and with the assumption of uniform plasma density. Whereas the

experimental conditions of other authors were such that the results of

the quasi - static approximation are sufficiently accurate, our compu-

tations have shown that this is not the case in our experiment, so that

we have faced the necessity of more complete computations based on the

rigorous characteristic equation of the plasma waveguile with metallic

walls.

Experimental P versus k data fall on a straight line, passing

through the origin; theoretically then, we must evaluate only the value

of the Brillouin diagram slope k/p. i/e( at the limit k - 0.

Assuming no glass walls around the plasma, the dispersion

equation is derived along the same lines used in the free space case

discussed in Technical Scientific Notesni1 and 2. For circularly symmetri

cal modes we obtain :

4 F/_

° Xf y;) J. X X N)

23 36fogho. d

lpAl

where

= .k, I., , (X,-9/ N~'.') * (44)

F" IF'<,<'. , ~~)1F ~ -___

"L L.(x.) .(.i) - (.)

b - metallic waveguide to plasma diameter ratio

XoXltx 2, - see Technical (Scientific) Note n. 1

No are interested in the low frequency range of each Brillouin

diagram, where the following approximations can be made

xo. • aL ,-4 ,--t-o

Under these conditions, the above defined parameters become

F•' ,,-. (&, >)"

F_ X £(e . 4).

('4) -(.4) A Vl(k4

It - V'4I- Ie.C•,-,)/(AI)

f +. ,

24 36No-di

When these expressions are substituted into (43), the dispersion

equation can be regarded as a relation between kpd and .4 (which is now

equal to the initial elope of the Brillouin diagram), provided we assume

that a- /'l1(JiS a constant parameter (kb a ZIN) The o( versus " d

curves can be found rather easily by graphical methods.

From these curves kpd - ((, 4. ) and the equation kpd =K1JAIr

a new set of curves kpd - Irb, KJL ) can be obtained.

Using these curvesfrom the measured 0( and kbd values the corre

sponding average plasma density is determined.

Following the above discussed procedure the X versus kPd curvei

for a constant a (fig. 5) and for a constant kbd (fig. 6) have been

computed, assuming for the diameter ratio b the experimental value 1.47.

The curves are those corresponding to the lowest x° solution of the dis-

persion equation. In fact we believe that our experimental data refer to

this solution, because the fields outside the plasma are in this cace

much larger than those of the higher order solutions.

Technically we had to face the problem of insuring a well defi-

ned standing wave pattern by using good reflective terminations at the

discharge ends. The best results were obtained with a mesh of tungsten

wires placed transversally inside the discharge tube together with an

external metallic iris.

Using this type of highly reflective terminations, measurements

have been performed using several frequencies in the 1 - 2 K.Mcis range.

To characterize discharge conditions the RF voltage applied to the plasma

between the electrodes was monitored. Typical results are as follows 2

RF discharge voltage 600 V peak, kbd - 5.86, 0( - 1.6 and then

nm3.4.10 1 1 ele.,/cm3 ; RF discharge voltage 1500 V peak, kbd - 5.86,

S- 1.3 and then nm 5.5.1011 elec./cm3 .

25 36

Sp.A.

UBOUP-V~LWCITY NEASUMOWETS

Wavelength measurements using a sliding antenna probe have pro-

vided information only on the first mode propagation. To obtain experi-

mentally Brillouin diagrams for the other modes a different approach must

be used. For this purpose, instead of measuring the phase c6nstant A

versus the signal frequency, we measure the derivative.&j / , that

is the group velocity.

This velocity is determined from measurements of the trans-

mission time delay using free space propagation as reference. The signal

is detected by the same probe used for the transmission experiments, so

that we are certainly measuring the properties of those signals, which.

propagate through the main plasma body (second and higher order modes).

For measuring the delay the 5 k Mc/s signal is amplitude modu-

lated from a lower frequency sinusoidal signal (30 Mo/a) and the phase

variations of this modulating signal at the receiver are measured. To

achieve the derived accuracy, the modulating sigal is converted in the

receiver to a lower frequency (30 Kc/s ), so that the delay, which is

displayed on a CRT is larger than the propagation delay "s the frequency

ratio (103 in this case).

Group-velocity vg is given by the formula

CS= (45)

where A(iniuseC)is the variation of the displayed time delay passing

from free space to plasma guided propagation and 1 (in meters) is the

length of the changed path, which is set equal to the discharge tube

length.

26 36

The block diagram is shown in fig. 7.

The electrical oharacteristios of each main block are as follows t

a) !'ýrowave varactor modulator.- Its purpose is to amplitude modulate

at 30 Mc the microwave signal. The index of modulation is 1, up

to 100 mW of microwave power, and reduces to 0.5 at I W.

The levels of sideband harmonics are always at least 25 db lower than

the 30 Mc modulation sideband.

b) 30 Mc Amplif. and A.G.C. - This is a 120 db gain, 0,5 Mc band-widtb,

30 Mc center frequency amplifier; the output is held constant within

10 + 0,1 Volt over 90 db signal input va2.atf.ow.

3. 0 Mc quarts oscillator and O Mo + 30 Kc quartz L.O. - The two quartz

oscillators are built in the same thermostatically controlled containr,

so that the 30 Kc beating frequency can be held constant within 30/,

peak to peak frequency v~nriation over long time intervals.

The method and the equipment have been satisfactorily tested

measuring the propagation time delays in coaxial cables of known lengths

and in free space varying the path lengths. & maximum errors are of

the order of .1/usec, that is sufficient for our experiment.

The plane spiral antenna, placed inside the plasma, which was

chosen for the wavelength measurements, was no longer used, because of

its extremely chort life in the Iresence of strong discharges. A plane

iisk was used in some of the earlier tests, but it was also abandoned

because its design was found to be critical, the coupling characteristics

being different for each tube we have constructed.

Then a new design was chosen for both the transmitting and the

receiving antennas. It consists of a dielectric antenna, incorporated

into the end plates of the discharge tube (see fig.8). These new antennas

not only show more constant coupling characteristics, but, due to the

27 36S di

LpA

absence of metallio parts inside the discharge, allow to attain very

good vacuum, to maintain gas purity and t avoid the presence of sputtered

surfaces.

Using these antennas, however, the transmitted signals have

always been at low level. The detected power was found to increase with

the magnetic field up to the maximum attainable field (- 3000 Gauss). To

improve the signal to noise ratios at the receiver, a new larger power

supply for the solenoid was then built (50 V, 1000 A, so that fields

around 4500 Gauss are attained). Larger signals were detected at these

fields.

A large amount of data was taken using the curved and the

straight solenoids, discharge powers from a few hundred watts up to 4 IN,

neon gas at pressures in the millimeter range. Time delays from I to over

20!usec have been measured. Inconsistencies of these data, however, have

shown that operating conditiont are not uniquely determined by our

measured data : signal frequency, magnetic field intensity and RF voltage

across the discharge.

This implies that gas and glass surface conditions ire changing

..... with time and from one experiment to another. Consequently density and

its distribution are also changing. Ireviously discussed theories predict

that these variations may largely influence propagation characteristics.

This problem deserves then more work both theoretical and

experimental.

N6 28 d 36

CONCLUSIONS

The main results obtained during the two years period, covered

by the present report, have been discussed.

In the theory they are

1) The basic theory has been completed, including the C• c. region and

deriving Brillouin diagrams.

2) The relations with well-known aspects of the propagation in electron

beams have been discussed and understood; these results can be usefully

applied in microwave tube theory.

3) The differences between whistler characteristics for propagation in an

uniform ionosphere and in a well-defined ionospheric duct have been

reconsidered on more general grounds. These results are definitive for

the case of uniform plasma ducts, but it seems interesting to examine

from the same point of view the case of non-uniform ducts.

4) Preliminary theories on the propagation characteristics in non-uniform

plasma-columns have been worked out. In spite of the obvious analytical

difficulties of this problem, more work along these lines is highly

recommended.

£-xperiments have provided the following results z

1) Previous interpretation of transmission experiments has been confirmed

by further data, which cover a wiler range of experimental conditions.

2) Propagation wavelength measurements have been successfully performed

but they concern only first mode propagation.

3) A satisfactory method for electron density measurements has been found.

This method ia based on wavelength measurements in a circular waveguide

beyond out-offt which contains the plasma column concentrically around

the axis.

29 36NOO di

4) Group velocity measurements have been performed by measuring the

propagation time delays of the microwave signal in the plasma.

5) Operating conditions are not uniquely determined by the assumed funda-

mental parameters : signal frequency, magnetic intensity and RF voltage

across the discharge. There are reasons to believe that gas and glass

surface conditions are changing with time and from experiment to experi

ment. More work is required to insure known and repeatable conditions

in the future experiments.

These conclusions indicate also some of the basic work which next year

research program will have to accomplish. In particular we mention the

further development of non-uniform plasma theory and the quantitative

analysis of experiments performed under known and repeatable conditions.

REFERENCES

1) W.O. Schumann - Z. Angew. Phys. 12, 145 (1960)

2) A.4. Trivelpiece and R.N. Could - J. Appl. Fhys. 30, 1784 (1959)

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36

SINDEL&p. A

"PWPAGATION IN nIfM)OGSMOUS PILSA.S(By Prof. Cambi 1960)

1- Introduc tory

The presenoe of free charges in a plasma can notoriously be taken

into account by regarding the medium as possesuing a tensorial per-

mittivity as defined, in oartesian coordinates for instanoe, by

D.. a [a, E, +j,, E7 ]D,. 0 Z[-j, a& ., E +ti1

where 4E is the absolute dielectric constant of free space and

C, t. 6, are relative tensorial components defined by

W2 -• aj W (i( w

In these formulas, &jl is the square of the "plasma fre-

quency" related to the electron density n by

while W,, is the cyclotron frequency, related to the maSnetic

field by

44=eB/in.-

In an inhomogeneous plasma duct, the electron density

and/or the magnetic intensity are, generally speaking, functions

of the p*ints in t,.rms of the quantities

WP (2)

the components of the permittivity tensor becomei +b2 k ($)

Ifthe coordinates arc measured in torms of -- '

that is, if I & l- times the physical abscissa, and so on;

and if, further,t is written for the vector 49H if ,/to.

(homogeneous with 1), the adimensional Maxwell equations become

foglio 1 di 19

M"d. 8 • 210 ."S

SINDELp . A

C i- s -' " E., +ja E)by Zz

'a x 'a Y' a 31. -

As stated above, go, EIt, & are functions of the point through the

quantities b and k appearing in (3)s but it is quite obvious

that the analytical problem cannot be solved in the present ge=

nerality. Actuallythe problem is quite complicated even in the

most schematic asumptions: so that it is necessary to discuss

separately the most elementary laws of variability.

2- T,.M mde in a uniform plasma

ihe simplest conceivable case is that of a plasma where the elec-

tron density, or the magnetic induction, or both, are functions

of P qingle coordinate, say x. In this assumption, a TEM mode

independent of x and y is obviously possible, satisfying the

reduced system

'Z z "62

S{s)~

0 :•z 0 •,

P, and L. arc to be regarded as known functions of z. Elimina-

ting Kx and Ky, the system

j~ +,L (6)

dZt t E

is written at once.

foglio 2 di 19Me. 10 -210 * 295

SINDELiLp. A

In spite of the apparent simplicity, the ysteip in

ozoeedingly complicated, even for the simplest laws of varia-

bility of &I , and C.. In view of this, a rather extended

discussion is in order.

In a first instance, we consider the apparently trivial (but

still delicate enough) case where a, and &,are constant, that

is, the case of a TEX propagation in a uniform plasma. When

I, and 1. are constant, system (6) is solved by making E.

and E of the respective forms e, tipsI e J. •j provided

Sis a root of ( )' o.The possible exponents are, accordingly

From the general expression of E.,F x k , J 'e J k * " ) a & + Lk , + "e'4 ,

that of Ey is found at once to be

E7 jke, e6w" *,k, e _j ')ceýP& k epThe first Maxwell equations also give

g, . - a kit , + w k a t k. 3 e,•k 3 , lP, "'Ipk.*J P

If *t and P are taken positive (assuming that 9, > It, )

the terms with negative exponents represent a forward propagations

assuming that the field originates "from the left", and that

no discontinuities occur, these are the only components of in-

terest.

Of course, i:, t to excite,say at z-0, such a progrossive

field, the initial E and K should match the local characteristic

impedance; that is, they are not simultaneously arbitrary.

Taking as x- direction that of I at z-0 and denoting

by E the (constant) modulus, a progressive field is qbtained by

making k• •kj, o , ki , k4 s L: the propagating field is then

+ jF a C-, 0"' ;#a

joglio 3 di 19Med. - 210 .2908

SINDELLp. A

This can be regarded as resulting from the superposition of an

"alfa field"

EM' F. C-Of ore"

2E: .. i} e,;ig• /y rE °"

with a "beta field" similarly defined. The velocity of propa-

gation of the a-field is c/o( , that of the beta field o/.

In each of the two fields the vector K is perpendicular

to 1, but the Poynting vector is zeros in other words, the fields,

isolated, do not convey any power. Power propagates by virtue

of the superposition of the two fields. actually

-01 -.0 9

E

In the alfa field, the plane of polarization rotates ne-

gatively: actually, denoting by V the angle of E with x, we ha-

vsI

while in the beta field'f

When the two fields are superimposed the polarization is

fixed with time, but depends on as that is, the polarization

continuously rotates in the course of the propagation.

Actuallys. -( a, +. su". ( ,w-P)- ,2*±-,

1, w.. COs6.,f-p,/+CoJ(G.t.-o,, "V •.:.2

3- TFAI modes in a lona.tudinally variable plasma

The more general system (6) where A. and &.are generic functions

of s alone can be attacked in substantially the same way.

Looking for a function F of z such that EX . C,F and Ey a C2 F

foglio 4 di 19Me.d.18 -210 x295

SINDEL1. p. A

may satisfy the system, we find the conditions

-j• 6,eC, +( F,)C, -o0.

Non zero soia'don C and C2 exist if, and onlp ifS+ (f, -&,)1 F = 0 (7)

In spite of the apparent elementarity, the equation is

emeedingly complicated unless f, and tare constant (in which

case the classic solution is found). To make the problem practical,

we shall accept first the reasonable assumption that the varia-

tion of a and Lwith z be relatively slow (as referred to the

wavelength) so that in a substantial number ,of "wavelengths" the

variation may be regarded as linear. In this case the equation

becomes of the formS+ (A ÷S:) O o0)

and has the complicated solutions

PuTA + - (

where J is the Bessel fumction.

In the present case A + Bz is, alternatively, If, -

or V1,+a. ; so that, extending the value of the definitions

Mg~jiý:and Th'~+~to the present case where 01 and

ari functions of z, the general expression of, say, Ex is

wit b ej-

with bi- i, -i 2 , b2 2,it + is (The f•ormula, of course, is

only valid when &, and 12are linear functions of z). When F sati-

sfies (7) with the upper (minus) sign, the coefficients of Eyare + j times those of E ; the converse is true in the case of

the lower sign.

Accordingly

M21. 16.• 0.295

SINDELLp. A

It is seen that the knowledge of a closed solution of

equation (7) when a,* a& is linear is far from oonetiftking

an advantages in addition to being complioated, solutions of

form (9) are not physically evidents for instance, the way is

b• no means clear by which combinations of functions (9) may

tend to e tjlor e ,as is necessary, when w and P tend to

become constant (b1 and b2 to zero)

The remark suggests the opportunity of looking for a

more practical solution of the 2hysioal problem as reprosented

by the general equation

F U- ON)I

where g (z) - *.ts t a known function of a. The assumption

of a linear behavior of g is only an approximation which obviously

looses interest if it does not lead to a simpliftoation.

Writing F =•, as is natural to do, the equation transforms into

a Riccati equationa" -_o)

The fact that g is normally negative, and real, suggests

for ÷ a complex form a +j ; : this leads to

a .- 2j•; a. -I(%).

For the reality of the l.fthand Uide, b +2U must vanish:

this gives = and

The further position a = In u yields

44--. C •

The physical problem is thus redused to that of assuming

a convenient form of u which, whert substituted into (11), may

yield a reasonable approximation to g(s). This can be done in

many different ways. For example, a function u -e , that is,

42 , oould approximate g(s) by the expression

foglio 6 di 19

MeW. 1. 4 210.29

SINDELLp. A

.I.the parameters of which can be so chosen as to reasonablY

"1matohthe actual g(s). For a match near the orlgin, for instanoe,

the conditions

are written.

The corresponding i is aN"214 s as the components of

the field have been assumed to be of the form +t the

"*einstantaneous wavelength" is 2x/i.(2ry/h) a

From the initial value 2X/h the wavelength increases if is

positives in this case the amplitude also increases.

The complete expression is

that is, neglecting a multiplicative constant

F *xp (yyz - h C'For given g(O) and g'(O), is the real root of the

equation

4 . t - . N '(0) 0.o

As h is real by assumptionj is negative or positive according

as g'(O) is positive or negative.

According to (3), the two physical expressions for the

function g are

,k- 4 k+1i , k-t

where k -4i/4 is proportional to the static magnetic induction

and *l, OlZn/m1t, Wt is proportional to the electron den-

sity. If the variation of g (referred to the wavelength) is slow,

Sis small and h 2 is close to g(o), that is, to ealor P .

"Given the four constants as defined by the

foglio 7 di 19

Mod. 18 - 210 * 295

SINDEL

L.p. A

initial values of I,, ,and their derivatives we thus have

ba

x aC, e. (, z .- p e'") +j e, cw(, (.jh. X 2 v" ) -

if the constants ]', • are relatc4d to t,- an•,od kz , to

L,+ E.2 "

The Maxwell equations -ive.e,( ý, . , () X-£ ( C -z L

•, • je , ( &Z*•.'") C p (•. - -• e•

- ie,(j,-2 e,,"' ) , •. (•:, "

It is easily recojnized that 1 hl and h2 are chosen

positive, the progressive fields are those involving + J inthe exponentials: assuminr that the entering field at t o 0

is matched to the local impedance the conditions for the pro-

&-ressive field where I~x(O) - E, Ey(O) - 0 are

foglio 8 di 19

Med. 21 • 210 x 295

SINDELL p. ASpACI lik/1'.O lhal ,

es alk/l e. h~ie 40r

C, 2,! no.

Accordingly, and writing for brevity

PC = GWp ; = expe a

the field is

EX -81 l CXF1 4...L. ewp'

2

-Ey, E g/. if, ex"p,' T, Lvp).lt,•72 2 , h

-Z 8 1 2

In every expression, the first term represents the "alfa

field", the other the "beta field".As usual, the Poynting vector

pertaining to e:ch separzte field vanishes identically, while4 -•. • J= -• "3

2. 4 woo•,iEp . 2 etc.

The discu'xion of Lhe field: rotation, attenuation, etc.

is carried out without difficulty: but the consideration of a

more ooncrute c se is of greater interest.

4- Propagation in a longitudinally variable laminar plasma duct

An a next case, in order of increasing complication, we

can consider Lhat of a plasma istribution that, in a cartesian

space, may show a variation ir the z-direotion alone, as above,

but is confined in a finite width in the x-di..eotion being unli-

mited, however, along the y-axis. If we look for fields whioh

are independent of y, the Kaxwell equations become

foglio 9 di 19

Mod. 16 2- 0 x 29 3

SINDELS. p. A

'6 -n~jc"b UJIz y* "z"aK E, .U E,.A E:

-9

that is, eliminating K,

V5Y + "a' E Y

V'E2 'b& Ea-A -6 -A z t16 : 3

It is recalled that 9, LA ta are functions of z alone.

The question is spontaneous whether a solution can exist

of he form E - AF where the vector A is function of x and the

scalar F of z only. Writing for brevity F/F -yas before, and

F/F' - -• . .. •i - i the system becomes

.A,(. • +j,,t , A , = ÷11

-AY (2 -A)y ,A,. - 0

ItAX 0 -As + 13 A

faglio 10 di 19

Mod. Is • 210 X 295

SINDELLt. A

(Accents lenote differentiation in respect to x, dots in re-

spect to a).

System (13), where the ooefficients are functions of z,

should admit of solutions A , Ay, Az, functions of x only. The

requirement is very -rostkictive, and for given 1E, at L I cannot

be satisfied by a single function +. This means that the exist-

ence of a solution of type tF is an exceptional event, if even

possible.

The investigation whether of not such a plane propagation

can take place is r,.latively simple. If the first two equations

(13) aro written

AX + A ,, A'

we find, upon differentiation in respect to z:

S .A + A; 0

j. A,.I. AY .0d •X

this requires, first, that

foglio 'I di 19MAd. Sl •210 x 295

SINDELL p. A

Furthermore, the ratio

~(S -gjCOSJIS/dx

must be independent of ma that is, as &, and &.depend on u only,

must be a constant. This means

- e, 1.e 3 (14)

Replacing in the first condition, we find

) AL e, e.. •ii•,o

and

Ax = ,•

The equations are necessary consequences of the assumption

of a form IF(-) for the vector I (plane waves).

Replacing in the first two (13), we have

A" -eIAY -0.

The second relation is perfectly compatible with the ins

dependence of A from sj the first equation requires that they

ratio

D'/' (15)

be a constant. iy is connected to 1, and Eby (14))

Ioglio 12 di 19M"d. 1S - 210 x 295

SINDELLp. A

From A' J- D A. we find, replaoing in the third (13)

-jD (Ag' + S3 Ay)-4

that is

.)D (e, .+ 3)A, j' e,A; CA . Ay,4•,

or finally

D (Ct.+ 9j) + CC 0 o. (16)

(14),(15) and (16) represent the necessary conditions for the

existence of plane wavess 4 can readily be eliminated, writing

from (15)

(e. -( Oas) +e1 ,0. +,,+

that is, replacing into(14),

-(Ha.s+ ) Hi,,s (17)

and,replacing into(16)

D (et * t... + ee, I•,.a +. K] .o.(,8)

The conditions should be satisfied by suitable values

of the constants C1 C2 and D (to which H and K are obviously

connected). As 1j is expressible in terms of E, and 1,, the

two equations, with given constants, completely determine the

plasma configuration. it is possible that the conditions be

less reatrictive thqn they appear at first, owing to the pre.

sence of three arbitrary constants which could possibly match

well enough the actual situation.

For . jiven configuration the problem only imposes two

conditions upon the three constants, one of which could possibly

be left free for a best pos.ible matcht if the configuration

varies rather slowly with z, a solution by steps is then conceivable.

loglio 13 d 19

Mad. 18 • 210.295

SINDELLSpA

- 5-Suggested method for the solution of more meneral oarteaian oases

Cases of wider generality than those that have been cons

sidered hitherto cannot be trateby methods of comparable aim-

plioity. The description of a computational method that may pre-

sumably be valid in a larger variety of cases is thus in order.

We assume, as before, that s, , o, ,o be functions of z only,

but their generality is not restriteed at the moment.

The solutions of Lhe Maxvell equations are dupposed to be

expanded ih series of orthogonal functions of x, having functions

of z as coefficients. If the width of the channel is indicated

by 2w, cos x and sin k can constitute such a set of

orthogonal functions. A simple inspection of equations (12) shows

that E x and Ey, as functions of x, have the same parity, while Ez

is opposite: the same is true of the components of K. Accordingly,

"even" harmonic components can be written as

"(22)E a a os -0-wx E - e 00s-~-WA E - esi=x' E~x = x co y = y co~ z = e Wsn

"odd" components as

E - a sin -r E a e sin±!-!-- E. -- e cos-1=xX X w y y w z W

Similar expression are valid for Kx, K , Kz in terms of harmonic

components k , k y, ke. (These components are actually functions

of the order ns the corresponding index, however, is omitted for

simplicity.) With the notations accepted in (22), the harignic

components appear to be connected by the ordinary system:

W I

.iy r iC 04 (23)j

The equations are valid both the even and for the odd components.

foglio '4 di 19

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SINDELLp. A

Even in the simplest assumptions as to the form of the

funotionse 1,,€s of a, the resolution of (23) ie oxeoeed1ly

complicated. If attention is given to the fact that one system

(23) exists for any positive value of n, the neoessity of soe

standard method of solution, at least approximated, is obvious.

Regarding the variation of the parameters as a relatively

small perturbation in respect to the constancy, we write 0- -

. ,. '...9-0+.t51, etc., where, of course, T, Ta, 2

Assuming that •J, .. ' k• satisfy the static system.

4X nr( F-

'.V W + A, Cy (24)

M Tr J . kZj-'r

We look for solutions of system (23) written in the form

x + f • I kz + z, where it is supposed that products of

small terms of the form. of f times z , are negligible in the

whole useful runge cx" z in comparison to the p•'idriFal ter:.ia of

4forn e times S. It is readily founA that the corrective terms f

and 1 have to satisfy the system

•r~

If the functions - , k, solutions of (23), have been

so constructed as to satisfy the boundary conditions at too, the

correcting functions f and £ can be subject to the simple condition

of having zero initial values.

foglio 15 di 19

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Application of the method first requires solution of

system (24) where the ooefficients are constant. (Of course, in

the case n - 0, the field is TM). The constant-coefficient systom

obviously ---its of solutions of the formhA

y 1L ai.. constant v C

provided j has Auch a value as to make zero the determinant of

the algebraic system of the"constants".

Before entering in further detail, we remark that the

equation for r is biquadr.itic and that for every value of r a

solution of the form

is available, with' constant ax, . . b (one of them arbitrary)

System (25) is subs~antially of the same form as (24), but

is complete instead of homogereous, the "forcing terms" buing of

'he form

that is, of the form

constant x Z e

The analytical problem is thus solved as soon ab one

particular solution of the complete system is known. This can

be done with ý1t andqrd rules as soon as "a fundamont 1 system"

of solutions of the homogeneous system is known. In the present

case, the "fundam-n al system" is merely cons 4 Iuted by the set

of four sextuples of functions corresponding to the four

different values of 5.In a somewhat more detailed discussion, we may observe,

first, that the unkno~n functions are actually four rather

than six, since two, out 0f equaions (24), are in fixite terms.

"Eliminating e and kz, (24) is replaced by the :;ystom of four equations:

foglio di

Met. 11 - 210a 295

SINDEL•pA

Y j~ e y(z ,)

ky k. - j' (A ) j,

where • is written for brovity in the place of A11T/*'. Similarly,

system (25) becomes

I4 S= 2-

~~~ý a~(>,.i IF- - Z3, e

I ) Y+.ft.fx 1

The solutions of (26) are of the 'form

(i-1,2,3,4,)wherel: is a root of the biquadratic equation ob-

tained by making zero the determinant of the algebraic system

obtained by replacing the last expression in (26), and BI, C|, D,

arc the corresponding solutions. Writing for brevity

F, . o F .2e2e,, F.%=j5.4

the general solution of the complete system (27) is given by

el r~l [f/A,1 o, l,] + el"EfhA ý4.1a + - e fJai$/,a 'i a*g +

+8e41fA /Az4i+

I = as above with coefficients C1 , C2 , C 3 , 04.

fir as above with coeffioients D1, D2e D3 , D4 19#ogliA di 19

SINDELLp.A

4 is the determinant of the "fundamental system" namely

X g B.s4 lz J) eaz

A as (it

4z 4& X • 4 2

As (,... -4 are solutions of a biquadratio equation, the sum

,+r,4r3 +r* is sere: so that A, reduces to the determinant

of ,he constants. Similarly,

4F 4

4 4 c J) 12

while A,, 4j, £14 are 6imilar in form but have fF,,F3, F4 respectively

at the second, third, fourth row.

Functions F have obviously the form of . sum of termsJ. z

ez where 5 is a known function of z. The determinants Ai

can thus be split in sums of four determinants the first

of which has the first row proportional to ea , the 3econd to

e a•z and so on. Out of these determinants the first is pro-

portional to e + ' )21 the second to e etc. In any

case, the solutions (29) are explicit and can be directly evalu-

ated, for any given set of functions "•t / 5 / 1 '.

Of course, the solution is valid in the limit of the ap-

proximation consisting in regarding the pertubations as first

order quantities. The functions ; + f.,K + 9 which by virtuex x z

of the presence of the arbitrary constants R,9 5 ...,R4 have just

the same forms (29) are solutions of a system of form (23) where,

however, ,, ,,• ar• not the diven functions of z but differ from

them b:! terms of the order of .he square of he perturbations.

joglio i19

Mod. 1. • 210 x 29S

SINDELSp.A

g

The prooedure oan premzmably be iterated, thus inoreasing

the.complioation, but not the difficulty of "he solution.

loglio 19 di 19

MA. 1 - 210 x 29S

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