Relativistic effects in a plasma at cyclotron resonance

Relativistic effects in a plasma at cyclotron resonance

Citation for published version (APA):

Schram, D. C. (1969). Relativistic effects in a plasma at cyclotron resonance. Utrecht University.

Document status and date: Published: 02/07/1969

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RELATIVISTIC EFFECTS

IN A PLASMA AT CYCLOTRON RESONANCE

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE WISKUNDE EN NATUUR-WETENSCHAPPEN AAN DE RIJKSUNIVERSITEIT TE UTRECHT, OP GEZAG VAN

DE RECTOR MAGNIFICUS, PROF. DR. J. LANJOUW,

VOLGENS BESLUIT VAN DE SEN AA T IN HET OPENBAAR TE VERDEDIGEN OP WOENSDAG 2 JULI 1969 DES NAMIDDAGS TE 2. 45 UUR.

DOOR

DANIEL CORN

ELIS SCHRAM

GEBOREN 1 AUGUSTUS 1940 TE RUSWUK

1969

DRUKKERU BRONDER-OFFSET N. V. ROTTERDAM

S T E L L I N G E N

1. Door zorgvuldige

me~ing

van de permittiviteit van plasma's bij

cyclotronresonantie is het mogelijk relativistische effecten vast

te stellen, terwijl de energie van de deeltjes slechts enkele

hon-derden elektronvolt bedraagt.

Deel III van dit proefschrift.

2. Het in deel V van dit proefschrift beschreven "trekken van een

ge-laden deeltje d66r cyclotronresonantie" kan, als dit relatief

snel gebeurt, worden opgevat als een quasi-elastische botsing. Dit

effect kan benut worden voor de verhitting van plasma's.

3. Bij de behandeling van de gemiddelde transmissiefactor voor

reso-nantiestraling houdt Zwicker ten onrechte geen rekening met de

invloed van de ruimtelijke verdeling van de dichtheid van de

stralende atomen.

Zwicker, H., hoofdstuk IV van Plasma Diagnostics,

editor W. Lochte-Holtgreven, North-Holland Publ. Co.,

Amsterdam

(1968).

4. Bij berekeningen aan het evenwicht van toroidale plasma's,

opge-sloten in helische magneetvelden, waarbij een reeksontwikkeling

met een ordening naar de kromming wordt toegepast, houdt Shafranov

geen rekening met een tweede-orde toename van de plasmastroom. In

hogere orde zal, bij geschikte opbouw van het primaire circuit, dit

effect het tot stand komen van het evenwicht van de kolom

vergemak-keli j ken.

Shafranov, V.D., Plasma Physics

(J.

Nucl. Energy, Part C)

5 (1963) 751.

5. Het verdient aanbeveling niet alleen voor verhitting, maar ook voor

opsluiting van plasma's met elektromagnetische velden, resonante

systemen met een lage kwaliteitsfactor te gebruiken of zelfs

lopen-de golven aan te wenlopen-den. Derhalve is het gebruik van trilholtes met

suprageleidende wanden niet aan te bevelen.

dienen gemeten te worden in een voor de verwachte frequenties

niet-resonant systeem. Met voordeel kan gebruik worden gemaakt

van een coaxiale structuur voor de bestudering van coaxiale

Penning-, magnetron- en invers magnetron ontladingen.

7. Als voor een plasma van lage

S

(kinetische druk klein ten

op-zichte van magnetische druk) behalve de totale stroom ook de

magnetisatiestroom in een magnetisch oppervlak ligt, zijn de

krachtlijnen geodetische krornmen in dat oppervlak.

8. De verliezen van een transmissielijn voor microgolven kunnen

eenvoudig gemeten worden door het achtereenvolgens plaatsen van

twee kortsluitingen op een afstand van een geheel aantal halve

golflengtes.

9. Een dun rooster met dikte den maaswijdte D, loodrecht geplaatst

op een magnetisch veld, heeft voor een bundel elektronen een

transparantie

d VJ. T " ' l -D

"§

waarbij v

.i.

en v;; de elektronen-snelheidscomponenten loodrecht op

en parallel aan het veld zijn. Hierbij wordt aangenomen dat de

gyratiestraal minstens zo groot is als de halve maaswijdte en

dat de spijlbreedte klein is ten opzichte van de maaswijdte.

10.

De zowel door Davydovskii als door Roberts

& Buchsbaum

voorge-stelde verklaring van de versnelling van geladen deeltjes in de

kosmische straling is weinig realistisch, zelfs indien men

reke-ning zou houden met inhomogeniteiten en elektrostatische velden.

Davydovskii, V.Ya., Sov. Phys. JETP

.!£

(1963) 629.

Roberts,

c.s.

en Buchsbaum, S.J., Phys. Rev. 135 (1964) A381.

11. Het fysisch rendement van kleine experimenten kan worden vergroot

door het gebruik van digitale

uitleesapparatuur~

Tevens verkleint

het de kan.s op psychokinese.

12. De hefboom van de Gis-klep van de moderne dwarsfluit dient

ver-lengd te worden.

D.C. Schram

2 juli 1969

This thesis concerns the interaction of plasmas with

electromagnetic waves of large amplitude in the presence of

a static magnetic field at cyclotron resonance (I-V). The

in-teraction is of a resonant type if the frequency of gyration

of the charged particles in the magnetic field, i~e. the

cy-clotron frequency, is eqial to the Doppler-shifted frequency of the wave. The particles will be accelerated to high

ener-gies if the cyclotron resonance conditi~n is sustained for a

sufficiently 16ng time (1-3). This is possible in the absence of desynchronizirig effects that cause the Doppler-shifted

fre-quency to differ from the cyclotron frequ~ncy. It is the

in-tention of this work to study the influence of the desynchro-nization on cyclotron resonance.

The study of this problem is of interest because in

the course of thermonuclear research the cyclotron resonance

inter~ction h~s been used to heat plasmas confined in inhomo-geneous magnetic fields (4-12). In some of these experiments

a small part of the electron population is accelerated to

very high energies, whereas the bulk is heated to a moderate

t~mperature only. The electron number d~nsity achie~ed is

low: the plasma frequency is usually of the order of the ap~

plied frequency. The resuits are s t i l l difficult to explain,

~ainly because a number of synchronizing and desynchronizing

eff~cts make~ the experiment~l situation very complex. A bet-ter understanding of the desyrichronization may therefore aid in the interpretation of these experiments.

A distinction must be made between inherently present

effects of a relativistic nature (1-3, 13-18, I-III) and

external disturbing effects, such as collisions (19),

inho-mogeneity. of the fields (20-25),and electrostatic fields (IV). Although the latter are also treated (IV-V), the main part of this work concerns the relativistic effects (I-III).

Primarily, the m6tion of electrons in a right-handed

circularly polarized TEM-wave propagating along a uniform

static magnetic field is studied (I). One desynchronizing ef-fect is the variation of the cyclotron frequency due to the relativistic increase of the mass of the particle. A second effect, which may also be referred to as relativistic, is the Doppler shift caused by the variation of the axial velocity due to the Lorentz force exerted by the h.f. field. The two work in opposite directions and their relative importance de-pend~ on the index of refraction, n. In free space (n=l) the two effects cancel and, at cyclotron resonance, the energy increases indefinitely as i t would in a nonrelativistic

anal-ysis (1-2, 13, I). In a fast wa~e (n<l) t~e mass increase

dom-inates, whereas in a slow wave (n>l) the increase in axial ve-locity is more important (13, I). For fast as well as for slow waves the particle motion is periodically synchronized and de-synchronized; even at cyclotron resonance the energy oscillates

with a limited amplitude. The characteristic quantities of in~

terest here are the amplitude and the period of the

oscilla-tion. These have been calculated as functions of the field

parameter& under the assumptiori that the initial veiocities associated with the thermal motion can be neglected (I-II).

Using such a so-called "large-signal, cold-plasma" analysis the distribution function and the permittivity of a tenuous collisionless plasma are evaluated. The permittivity is expressed as a function of the h.f. electric field strength, of the field strength of the static magnetic field, and of the total index of refraction of the plasma in the structure (II). The expression is of a transcendental nature, because in gen-eral the total index of refraction is a function of the permit-tivity of the plasma.

A consistent solution is obtained for the interaction

in a free-space wave s~ructure, where the total index of

re-fraction is exclusively determined by the plasma itself (II). On the other hand, if the influence of the plasma on the in-dex of refraction can be neglected, the transcendental

char-acter of the expression is removed (III). This may be the

case for a tenuous plasma in a slow- or in a fast-wave struc-ture. A special case hereof is the long wavelength approxima-tion which has been used for the descripapproxima-tion of the interac-tion in a standing wave (26, III).

The predicted dependence of the dielectric behaviour of a plasma on the field parameters is confirmed by the re-sults of two experiments: one with standing and one with trav-elling waves (III). In both experiments the low-density, col-lisionless plasmas are produced by the h.f. fields themselves. The attenuation observed is found to be due to the escape of accelerated particles, and is not caused by collisions.

The simplified model in which only the relativistic desynchronization is taken into account, the "undisturbed case", may be invalidated by the disturbing effects mentioned above. By the comparison of two characterlstic times, the limit below which the results of the undisturbed case remain valid is estimated for various modes of disturbances (V), One of the characteristic times is associated with the un-disturbed motion and is equivalent to the energy oscillation time defined above, the other is associated with the distur-bance. If these characteristic times are equal, the desyn-chronizing effect of the disturbance cancels the relativistic effects for an extended period of time. Then a synchronous ac-celeration occurs (IV-V, 23). If the characteristic time of the disturbing effect is short compared to that of the undis-turbed motion, the external effect is defined as "strong". In that case the particle is pulled quickly through the res-onance and relativistic effects have no time to become of im-portance. Nonrelativistic calculations yield, at every pas-sage through the resonance, a net increase in energy for a population of particles which were initially distributed iso-tropically in velocity space (IV-V).

1. Davydovskii, V.Ya., Sov.Phys.-JETP 16 (1963) 629.

2. Kolomenskii, A.A., Lebedev, A.N., Sov.Phys.JETP 17 (1963)

179; Sov~Phys-Dokl. 7 (1963) 745.

-3. Schrader W.J., Thesis University of Utrecht 1967, Rijnhuizen Report 67-40 (1967).

4. Dandl, R.A., et al., Proc. IAEA Conf. on Plasma Physics and Contr. Nucl. Fusion Research, Novosibirsk 1968, Vienna 1969, Vol. II, 435.

5. Consoli, T., Proc. IAEA Conf. on Plasma Physics and Contr. Nucl. Fusion Research, Novosibirsk 1968, Vienna 1969, Vol. II, 361.

6. Ikegami, H., Proc. IAEA Conf. on Plasma Physics and Contr. Nucl. Fusion Research, Novosibirsk 1968, Vienna 1969, Vol. II, 423.

7. Budnikov, V.N., et al., Sov.Phys.-Tech.Phys. 12 (1967) 610. 8. Galktinov, B.V., et al., Sov.Phys.-Tech.Phys.l3 (1968) 838.

9. Hartman, C.W., Phys. Fluids 9 (1966) 821.

-10. Tuma, D.T., et al., Plasma Physics II (1969) II7.

II. Hooke, W.M. and Rothman, M.A., Nucl-.-Fusion 4 (I964) 33.

12. Rothman, M.A., et al., Princeton Univ~,Matt ~06 (1969).

13. Roberts,

c.s.

and Buchsbaum, S.J., Phys.Rev. 135 (I964) A381.

14. Hakkenberg, A.and Weenink, M.P.H., Physica 30-cf964) 2147,

Rijnhuizen Report 64-18 (1964). ~

15. Voronin, V.S. and Kolomenskii, A.A., Sov.Phys.-JETP 20

(1965) 1027.

-16. Kondratenko, A.N., Plasma Physics (J.Nucl.Energy Part C) 7 (1965) 89.

17. LutomTrski, R.F. and Sudan, R.N., Phys.Rev. I47 (1966) 156. 18. Gorbunov, L.M., Sov.Phys.-JETP 27 (I968)

IIO-.-I9. Hwa, R.C., Phys.Rev. IIO (I958)~07.

20. Seidl, M., Plasma Physics (J.Nucl.Energy Part C) 6 (I964) 597.

2I. Tuma, D.T. and Lichtenberg, A.J., Plasma Physics

9

(I967) 87.

22. Lichtenberg ,A.J., et al., Plasma Physics..!._!_ (1969) IOI. 23. Canobbio, E., Nucl. Fusion 9 (I969) 27.

24. Piliya, A.D. and Frenkel, V~Ya., Sov.Phys.Tech.Phys. 9

(1965) I356.

25. Grawe, H., Plasma Physics 11 (1969) I51.

26. Schram, D.C., Rijnhuizen Report 65-25 (1965).

CONTENTS

This thesis is a collection of five publications:

I. PARTICLE MOTION IN A TRAVELLING WAVE NEAR CYCLOTRON

RESONANCE, D.C. Schram.

Physica

].l_

(1967) pp. 617-631.

1. Introduction ..

2. The field configuration

3. The equations of motion

4. Solutions for space-charge-free electron plasmas

II. THE RELATIVISTIC INFLUENCE ON THE DIELECTRIC BEHAVIOUR

OF A PLASMA NEAR CYCLOTRON RESONANCE, D.C. Schram.

Physica

±Q

(1968) pp. 422-434.

I. Introduction

2. Method of determination of E

3. Electron motion

4. The distributi6n function

5. Numerical evaluation of E

III. THE PERMITTIVITY OF A PLASMA AT CYCLOTRON RESONANCE IN

LARGE AMPLITUDE E.M. FIELDS, D.C. Schram. Preprint Physica

l. Introduction

2. Standing wave experiment

3. Travelling wave experiment

IV. THE EFFECT OF AN ELECTROSTATIC FIELD ON PARTICLE

ACCELERA-TION AT CYCLOTRON RESONANCE, D.C. Schram, G.P. Beukema. Physica 42 (1969) pp. 247-290.

I. Introduction

2. Exact numerical treatment

3. The nonrelativistic approximation

V. THE ENERGY GAIN OF PARTICLES PULLED THROUGH

CYCLOTRON RESONANCE,

D.C. Schram, W. Strijland, L.Th.M. Ornstein.

Third European Conference on Controlled Fusion and Plasma Physics, Utrecht, 23-27 June 1969, Paper 71.

1967

PARTICLE MOTION IN A TRAVELLING WAVE NEAR

CYCLOTRON RESONANCE

by D. C. SCHRAM

FOM:-Instituut voor Plasma-Fysica Rijnhuizen, Jutphaas, r-rederland

Synopsis

The motion of a charged particle in a travelling e.m. wave propagating along a static magnetic field at cyclotron resonance is studied, taking into account the first-and second-order Doppler effects. As was derived in earlier treatments3) 4), the energy of the particle and the velocity component along the magnetic field are both periodic in time. However, few quantitative results in terms of attainable energy, oscillation time and longitudinal oscillation length wr-re available. With zero initial transverse velocity, v ..LO = 0, these quantities have been calculated numerically as functions of the field parameters; for refractive index n = 1, v ..LO =F 0 has been included. The results demonstrate the influence of the first- and second-order Doppler effects on the motion.

Also for v ..LO = 0, a Cerenkov-like condition for positive or negative energy exchange between the particle and the wave has been derived. The influence of a space charge field on the electron motion is indicated.

1. Introduction. The influence of the variation of the axial velocity and

of the mass of a particle (the first- and second-order Doppler effects) on the motion of a charged particle in a combination of a static magnetic field Bo

and a travelling transverse e.m. wave has bet:;n the subject of various in-vestigations!) 2) 3) 4). These effects have been studied in a single-particle approach under the assumption that the applied fields are spatially uniform, thus neglecting effects resulting from collisions and inhomogeneities of the applied fields. Though exact treatments of this problem exist 3) 4), it is difficult

to

obtain values for the characteristic quantities of the motion, as maximum attainable energy and specific time. The aim of this article is partly

to

solve this difficulty for a special case: zero initial perpendicular velocity. Further, it will be shown that under certain limitations the ef-fect of d.c. space charge fields on the motion can be incorporated in the

This work was performed as part of the research program of the association agreement of Euratom and the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek" (ZWO) and Euratom.

-618 D. C. SCHRAM

formalism. Finally, under some restrictions conditions for positive or negative energy exchange between the wave and the particle will be deduced. The energy of the particle and the longitudinal velocity are both periodic in time. So characteristic quantities of the motion are: minimum and maxi-mum energy, energy oscillation time t08 and the distance covered in this

time, the oscillation length

z

08 • They will be calculated as functions of the field

parameters.

In order to investigate the applicability of the model to an experimental situation the influences of collisions and inhomogeneities of the applied fields and of the electron density have to be estimated. If the ratio of the collision tinie tc to the energy oscillation tii-ne t08 is large, tclt08 ~ 1, the case

we study here, the influence of collisions may be neglected. If tcltos ~ I the energy gained by the particles will be dissipated by collisions before re-lativistic effects become imp9rtant. Then a finite-temperature small-signal non-relativistic theory is appropriate.

In the intermediate case tc ,...._, t08 both effects are important. Since a col..;

lision will generally disturb the phase relation between the electric field of the wave and the velocity of the particle, collisions may enhance the diffusion in velocity space. Higher temperatures than those following from a single particle approach are possible.

If the change of the amplitude of the electric field, the phase velocity, and the change of the field strength of the constant magnetic field are small over an oscillation length

z

08 , effects resulting from inhomogeneities of the

fields may be neglected. In the treatment below these quantities E, Vp and

Bo will be supposed to be uniform in space.

2. The field configuration. In the introduction it has been mentioned that we will assume a spatial homogeneity of the fields. This assumption will simplify the problem to a one-dimensional (z, t) problem. The fields to be acting on the particle are threefold:

-- a homogeneous constant magnetic field Bo, directed along the z coordinate; - a right-handed circularly polarized TEM wave propagating along this field (k

II

Bo). The fields Ej_, BJ_ are supposed to be independent of the transverse coordinates, x and y, while their amplitudes

E

and

B

do not chanl!e with - ----o- - z. - This nieans that neither amnlification nor attenuation of ---- - --- -- --- -- - -- .L

the wave is admitted;

- an electrostatic field, directed along the

z

coordinate and only dependent on this coordinate. This field is added to generalize the equations to actions of self-generated or applied d.c. space charges.

In formula:

Bx= Elvp·cos (wt - kz); Ex= E sin (wt - kz)}

_ _ e.m. wave

Bz =Bo Ez = Eo(z) = - ~ d<p(z) } static fields. (I)

3. The equations of motion. The relativistic form of the momentum equation is: dyv e - = - - ( E + v

x

B); dt m where y =

J

V•V I -c2 8 mc2 (2)

m is the rest mass of the particle and s is the total energy of the particle.

Insertion

of

the assumed fields yieids :

e

[A

E

J

(yi)' = - - E sin(wt - kz) - i -sin(wt - kz) -

Qy

m Vp

e [ A

E

J

(yy)' = - - -E cos(wt - kz)

+

i - cos(wt - kz) +!Ji

m Vp

eE[

'

J

eocp

(yi)' = - - - i sin( wt - kz)

-y

cos(wt - kz)

+ ..

-m Vp . m

oz

l

r

(3)

A dot signifies a total differentiation with respect to the time t and Q = = eB0/m is the cyclotron frequency related to the rest mass of the electron.

Assuming, that the amplitude E and the phase velocity Vp of the wave are constant over

z

the first two equations of (3) may be integrated directly. Since the problem is independent of the transverse coordinates x and y we are free in choosing xo and yo such that the integration constants are zero. Therefore they may be omitted without any loss of generality; in normq.lized form the equations thus become:

yi

- = g cos(wt - kz) - Qy/c

c

yy = g sin(wt - kz)

+

Qx/c

c

~

~(Y!_)\

= -

glf

__!_sin( wt - kz) - j_cos (wt- kz)Jl

+

tf>',

c dr\ c c c

where in definition

eE

g =

-mwc electric field parameter

eB0

b = Q/w = - - magnetic field parameter

mw

(4)

620 D. C. SCHRAM

T =wt reduced time

n = c/vp index of refraction

ft>'

=

d~z

(

~~

2

) electrostatic field parameter.

(5)

In order to elimjnate the quick (harmonic and cyclotron) time variation, we transform to velocity coordinates which rotate and travel with the e.m. field. The components of the velocity vector in this frame are related to Vx and Vy as follows:

= T where T =

(, vr; ') ('

Vx) (' cos (wt - kz)

Vn Vy ' -sin (wt - kz)

describes the orthonormal ve~tor transformation.

sin (wt - kz))'

cos (wt - kz)

We introduce normalized rotating momentum components:

{

x

Y .

}

Pr;= yvr;/c

=

y

7

cos(wt - kz)

+--;;-

sm(wt - kz) (6) (7a) = .Q { : sin(wt - kz) -

~

cos(wt'- kz)}

+

g; (7b) and

{

x .

y

}

~=~~=r--;;-~~-~+-;;-~~-~ (Ba) { x y . } = .Q ccos(wt -'·kz)

+

c sm(wt - kz) . (Bb)

By differentiating (7b) once a relation between Pr; and Pn may be obtained: 1 Pn = _{1 -} bf _{y -} '/ dpr;/dT. Z Vp (9) Differentiating (Bb) : dPn d { 1 dp r; } . . - d =-d

1

.

·;

. - d =-(1-b/y-z/vp)h+(l-z/vp)g. T T 1 - b y - Z Vp T (10) The combination of equations (4c) and (Ba) gives a relation between

Pz

=

= y

z/c

and pr;:

y

Vp

~(r.!_)-

yft>'

C dT C 1 dpr;

- - - ' - - - - ' - - - - = Pn = - . (11)

g 1 - b/y -

z/vp

dT

relation between y, p~ and z/c:

y2 - 1 - y2

~

·2 = y2 ( ·2

~

+

_!_

·2) =

p~

+

p2

=

p~

+

c2 c2 c2 '1

+

₍ 1

-

dp~)2

.

1 - b/y - z/vp dT (12)

The original set of equations with the three velocity components i,

y

and

z

as unknowns is thus transformed to a set of differential equations where x and

y

are replaced by slowly varying variables: an energy parameter y and a rotating velocity parameter

h·

In order to eliminate h out of this set of equations we multiply ( 10) by

1 dp~

Pn

= 1 _-

bf

_{y -} Z Vp

·1

- d T '

differentiate equation (12) with respect to time and combine this:

_!__~{y2

- 1 - y2

~}

= 1 - z/vp g

dp~

(13)

2 dT c2 , 1 - b/y - zfvp dT

in accordance with equation (11) it follows that:

dy Vp d . . ,

dT =-c- dT (y z/c) - (1 - z/vp) <P . (14)

This equation relates the change in axial velocity and the change in total energy; in other terms it relates the first and second order Doppler effects. This is somewhat more clear if we rewrite {14) in another form:

1 d(l/y) 1 d(l - z/cn) <P' 1 - z/vp

(1/y) dT (1 - z/cn) dT

+

y

1 - z/cn •

For the case <P' = 0, n = 1 the relative change of the effective cyclotron frequency fJ/y (the second-order Doppler effect) is equal to the relative change of the Doppler shifted frequency w(l - z/c). So if a particle starts at resonance, where the Doppler shifted frequency equals the cyclotron frequency

w - kzo = Q, it remains in resonance automatically. In that case its energy will grow until radiation effects become important. This result has been obtained previously by Davydovskiil) and Hakkenberg and Weenink2), and has been dealt with in some detail by Roberts and Buchsbaum3). However, it is clear that this so-called synchronous case will be disturbed if an applied or self-generated space charge field is present.

Combine the equations (13) and (14) and integrate:

g(p~

- ho)=

HY

2 - yg) -

t(r

2

~

_{-yg zg ) - b(y - yo)}

+

_{b(<P .,--<Po),}

cz c2

622 D. C. SCHRAM

(

·2 ) [ ( ·2 ·.2 )

g2 y2 - 1 - y2 !_ = i(y2 -

y~)

- i .y2 !_ -

y~ ~

-~ .· ~ .~

- b(y - yo)+ b(<J> - c/Jo) + gp1;0J + y2

[

d~

(y - <J>)

J,

(15)

as from the equations (11) and (14) it follows that gpn = y (d(y - <J>)/d-r). The equations (14) and (15) describe the particle motion if <J> is a known function of z.

In the remainder of this paper we will deal with plasmas in which no d.c. space charges occur. Nevertheless it appears that the influence of space charges may have interesting features and it v,rould be valuable to go through the rather complicated mathematics with a nonzero <J>'.

4. Solutions for space-charge.'-free electron plasmas. Under the restriction that no d.c. space charge field exists, <J>' = 0, solutions in terms of elliptic integrals are obtainable. Then the set of equations (14) and (15) reduces to:

dy 1 d ( z ) 1 ( z . zo ) - = - - - y - or y - yo = - y - --,. yo-d-r n d-r c n c c (16) and

(

·2) {

( ·2

·2)

. z . 2 Z 2 Zo g2 y2 ~ 1 _ y2 _ = .l(y2 _ Yo) _ .1 y2 _ - y - -c2 2 2 c2 o c2

}

2 (

dy

)2

--, b(y - yo)

+

gp1;0

+

Y2 d-r · (17) ' 2 1 (

z

2 2

z~)

g(pt; - P1;0) = i(Y2

- Yo) - 2 Y2~ -Yo~ - b(y - yo).

Elimination of z/c and introduction of p = 1 - zo/vp - b/yo, the initial resonance parameter (Doppler shifted frequency minus cyclotron frequency divided by applied frequency) results after some rearrangement in:

- {

d(y~:o)

2

r

= (1 - n2)2

(:a -

1)4

+

4p(l -n2{

:a -

1

)=

+

4-

A.(

'Y 1 \2

f

2 g2(1 - n2) ' gp1;0 ,, •• ~J

'

' •\Yo - )

lp -

y~ I

Yr

l' - ·r•-j

f

I /1 O\ ~10)

+

8

(___!_ -

1) {

gp~t;O

-

g:

(p

+

.!!__)} _

4

g

2

~~0

• Yo Yo Yo yo Yo

This equation is identical to equation (2.,-14) of Roberts and Buchs-baum3). Clearly the influence of the index of refraction n can be seen in equation (18). The axial velocity component and the rotating momentum component P1; may be expressed in y as follows:

z/vp = n2

+ {

(1 - n 2) - (p

+

b/yo)} :0 • (19)

g(pg - Pt;o) = ~(1 - n 2)(y - ro)2

+

_{pyo(r - yo). (20)}

The solution of equation {18) is a function of six parameters: - the electric field parameter

g;

- the resonance parameter p;

- the index of refraction n;

- the initial velocity components Vt;o

=

Vzo, v11o

=

Vyo, Vzo defining yo.

Such a large number of parameters prohibits a clear classification of the results; only incidental cases can be calculated numerically. It is, however, possible to restate the problem in a way where only four parameters occur. The cases n = 1 and .Zo = Vp have to be excluded and will be dealt with

separately.

4a. Case: n-=/= 1, zo-=/= Vp.

Define:

1 - n2

- - - - (y /yo - 1)

==

(J, energy or mass variable

1 - zo/vp g2(1 - n2)

2(

=

G, electric field parameter

y0 1 - zo/vp)2 b -p

-1

-

1 = ( /

==

{3, resonance parameter yo(l - Zo Vp) p

+

b yo) _ gpgo(I - n2 )

n _

gPno(l - n2) Ilr;o = ₂ , 110 = ₂ •

ro(l - zo/vp)2 _{ro(l - zo/vp)}2

(21)

Note that both the new electric field and tile new resonance parameter embody the Doppler shift at t = 0. Then equation (18) becomes:

2{(J

+

1 -. n2 } d(J' = J-()4

+

4(3()3 - 4()2((32 - G

+

Ilgo)

+

1 - zo/vp dT

+

8(J{Ilr;o{3

+

G}

+

4II;0 • (22) We will suppose here that Ilr;o

=

Il11o

=

0, so v .L (t

=

0) = 0. The differential

equation does not simplify very much, but the number of parameters is again reduced by two. This is anappropriaterestriction for a low temperature, low density plasma in a strong wave. The specific velocities that the electrons attain after acceleration by the fields will be much larger than the thermal velocities. The rare collisions will not change this situation, if the lifetime of the electrons is not too large. They will occur most likely near the energy minimum, since there the. cross section is largest. Moreover, in general the

624 D. C. SCHRAM

electrons spend more time near the energy minimum than near the maxi-mum 6).

Under this restriction v J..O

=

0 the equations become:

( J - n2 ) 0 ₄

o

+

_do T' = (I - n2) T =

f

I - zo/vp

.J

p 4(0) 0 Ii 7 _ kz =

J

4(o

+

I) do ; . .JP4(0) 0

where P4(o) = - o{o3 - 4f3o2

+

4((32 - G) o - BG}.

z

,

(J-n2)(o+I) --:;;; = 1 -

{o

+

1 - n2 } · J - Zo/Vp p~ I i - = - b-02

-f3o}.

g

G

(23) (24) (25) (26)

The general solution of equation (23) is a sum of elliptic integrals. The energy oscillates as a function of time and of kz. The energy maxima and minima may be found byputting (do/dT)omax,min = O; omax and omin are roots of the fourth-degree polynomial P4(0). One of these roots is

o

= 0, or y =yo, so if v J..0 = 0 the initial energy of the particle is either an energy maximum or

a minimum, depending on the sign of the other root. The polynomial P4(0)

may have two or four real roots. In the first case

o

oscillates between

o

= 0 and the other root, in the latter case between

o

= 0 and the nearest other root.

The sign of the roots (see appendix) depends on the sign of G. If G is positive (fast wave, I - n2

>

0) the roots are nonnegative and

I - zo/Vp I - io/Vp

y /yo - I =

o

where

2

I - n2 I - n

cannot be negative, so y /yo - 1

>

0. Tl1is· means that for v j_O

=

0 the

parti-cle always gains energy in a fast waye. For negative G (slow wave, I - n 2

<

<

0) the situation is different and both energy gain or loss may occur depending on the ratio

z

0/vp. In this case the roots

o

are nonpositive, so two cases may be distinguished:

I - zo/vp

>

O(zo

<

vp), again the initial energy is minimum;

I - io/vp

<

O(zo

>

vp), here the initial energy is maximum and the particle will lose energy. (See fig. 1).

1i

I

liextr. ··--·---G >O Cn<1)

(

0 'tos .,...----.. 't ii

_ ,

I

0 liextr. G<O (n>1)

Fig. 1. Sketch of energy behaviour of a particle with zero initial transverse velocity.

This Cerenkov-like condition for energy loss of the particle is valid under the restrictions made: v .LO = 0 and no d.c. space charges. If v .LO ;:/= 0 such a

condition is difficult to obtain.

The exact location of the roots depends only on G and {J, involving the electric and magnetic field strengths, the index of refraction n and the initial axial velocity .Zo. The critical value of {3, where the number of roots changes from two to four and which is ne?-rest to the nonrelativistic resonance value {3 = 0, f3c1, indicates the shift and the broadening of the resonance (see figure 2). An approximate value may be given by f3c1:::::::

,...., 3~ G/4 (appendix). As can be seen in the figures 2a to 2d the resonance shifts to positive values of {3 for positive G (fast wave) and to negative values of {3 for negative G (slow wave). In the first case the effect of the relativistic mass change, the second-order Doppler effect, is predomi-nant, while for slow wave interactions the first-order Doppler effect over-balances the second-order Doppler effect. In a free space interaction both effects cancel each other as we will see in the case n = 1. In the figures 2a to 2d, oextr is given as function of {3 with G as parameter. With help of these figures and the definitions (21) Yextr may be evaluated. The axial and perpendicular velocity at the time that

a

is extreme may be found by the equations (25) and (26).

626 ~ G

1

k G

I

-02 -001 G=O .. ·G=10.5 1:" D. C. SCHRAM G=-10~ 102 '"::::::::::··· 101 0 +0.2 +0.4 +0.6 -0.6 -0.4 -0.2 - P 0 +02 - l l . G=10"10

G:O ·:· G=-10-io :· G=O f".::G=109 € G=10"8 .··: G=1C>7 ,·:.::::. (·" 0 +001 +0.02 +0.03 - l l 107 -0.02 -0.01 Fig. 2. Oextr as function of {J, with Gas parameter;

2a and 2c: G > O; 2b and 2d: G < 0.

0 +0.01

- l l

G = 0 is the case in which both Doppler effects are neglected.

the elliptic type; they may be transformed into standard elliptic integrals. In order to obtain half the oscillation time and length the integration has to be performed between

o

=

0 and

o

=

oextr• so

~extr Tos =

211

2Io 1 - n2

+

_{1 -}

zo/vp

4

J

odo

+

- n2 -JP4(0) 0 (27) 0

(-r ~ kz)os = 2(li +lo). (28)

I ₁ and I 0 may be expressed in complete elliptic integrals of the first and third kind 7). They have been calculated and are shown in fig. 3 as a function of

f3

and with G as parameter. It appears, that for values of

f3

which are not too close to

f3c,

I 1 ~I 0. The oscillation time can therefore be approximated

104 Io 10

I

lo

110'

G=10-2 101 _-Ii. +11

t

!

1 lo

r

•!, 101 10-1 '---0.0.._1_--J.o___:_•_o.o.._1__,,_--l_---'"'"' - P

Fig_ 3. lo and Ii as function of fJ with Gas parameter;

3a and 3c; G > 0; 3b and 3d: G < 0.

+001

-~

G = O is the case in which both Doppler effects are neglected.

by:

2Io

Tos

~----1 - zo/vp

If zo/Vp is not too small, kzos ~ Tos zo/Vp, while for Zo = 0 there holds: 2n2fi

kzos

=

1 _ n2 .

The exact values for Tos and kz08 may be calculated with help of the

equations (27), (28) and (21), and the figures 3a to 3d. .

4b. Case z0 = Vp. It has already been noticed that for slow wave

interaction (n

>

I) the sign of the energy change depends on the ratio

z0/vp. If z0

>

Vp the electron will lose energy (the extreme value of y is

smaller than the initial value); in the opposite case zo

<

Vp, the electron will gain energy.

In

the intermediate case zo = Vp, it can be established that neither gain nor loss -will occur.

628 D. C. SCHRAM

The equation (18) becomes with p = 1 - zo/vp - b/yo = - b/yo and

again v J.O

=

0:

- { 2y : :

r

=

(y - yo)2 (I - n2)2 { (y - yo)2 - 4 1

~

n2 (y - yo)+

( b2 g2 )} ,

+

4 (1 - n2)2 - I - n2 = P4(y).

Only two real roots of .f4(y) exist: Yi, ₂ = yo. Since these roots are the

extrema of y there is no other solution than y - yo for all times.

Thus the equations ( 19) and (20) become for this case:

z/vp = n2

+

_{(1 - n2) yo/y}

₌

₁

and

gp~ =

f(I -

n2_{)(y -yo) 2}

+

_{pyo(y -yo)}

₌

_0.

The energy of a particle starting with an initial velocity io = vp, v J.O = 0 remains unchanged! In physical terms it means that the force -eE J. is

exactly counterbalanced by the magnetic force

ez

x

BJ.

if

z

= Vp.

4c. Case n = 1. If the index of refraction n equals the free space value

n = 1 (vp = c), analytic results for T(y) are obtainable. The equations involved are similar to those following from a quasistatic nonrelativistic analysis; so it appears that the influence of the relativistic mass increase and the variation of the axial velocity cancel each other. For this special value of n a resonance occurs for p

=

1 - b/y0 - io/c

=

0 (Doppler shifted frequency equal to cyclotron frequency), the case called synchronous by Roberts and Buchsbaum3). The energy does not exhibit a maximum, but increases indefinitely.

The equations (18) to (20) simplify for n

=

1 to:

-{y :: }

2

= (y -yo) 2 (pyo)2

+

_{2(y -yo) yo {gppgo - g}2_{(1 - io/c)}-g2}

_{P;o ·}

z/c

=

1 - (1 - io/c)

~.

y g(h - P~o) = pyo(y - yo).

Out of resonance: p = 1 -

z

0

/c -

b/y0

=F

0, these equations may be

rewritten into:

0

{ do

}2

J

(o

+

1) do

- (o+ I ) - = o2--2N10 -N~; so: p = I

dpr

v

-02+2N1o+N~ 0 and 0

I

Nsdo · p(T-kz) = .

.j

-02+2N1o+N~ . 0

Here is:

c5 = y/yo - I,· N1 = -gpp;;o + g2(I - zo/c) . N2 = gzp;o. N - 1 - .

I

2 2 ' 2 _{4 2 '} 3 - Zo C.

YoP YoP

After integration it follows:

p-r =

-[..J -

c52_{+2N1c5 +}

N~

_{- (1 + N 1) arc sin ---,=} 1c5=N=i

=J

0

vNi

+ N;

o [ c5 -N

1 Jo

p(-r -kz) = N3 arc sin

,J

.

Ni+N;

o

The energy oscillates as a function of T and kz; the maximum and minimum

can be found by solving the quadratic equation c52 + 2N 1c5 - N~ = O:

c5max = +N1 + .JNi +

N~;

c5min = +N1 - .JNi

+N~.

The oscillation time pros= (1 + N1) 2n, while pkZos = (I + N1 - N3) 2n. If v l.O

=

0 the solutions are:

2g2_{(1 - zo/c)}

c5max

=

2 2'· ' c5min

=

0;

YoP

p-ros={l

+

g2(1 -_{2 2}

z

0/c) } 2:n, . pkzos = { I - ( 1 - _{2 2} g2 ) (1 - zo/c) 2n. . }

YoP YoP

If p = 0 (resonance) the equation (18) simplifies to:

1 { d(y/yo)2 }2 b 2 .,

4

d(gT/yo)

=

2

Yo

(y/yo - 1) + Prio!Yo· So

0

f

(c5 + 1) dc5 [ 1 2 2

gT/yo

=

,J

bf c5 2

I

2

=

6(b/ )2 .J(2M/yo + Prio1Yo)3 +

2 Yo

+

Prio Yo yo ,

·o

+

2b/yo ₂(b/yo)2

-P;o!Y~

'\/ /2M/ Yo

+

p2

rio Yo

I

2Jo

0•

This equation may be inverted into the form c5 = c5(T) with help of cubic equation solutions:

{2M/yo +

P~o/r5}t

=A+ B

where:

A, B

=

~C

+,Jez+

D3

and where

C = 3 _b_. Pno -

P~o

+ 3 (}!__)2 2:_ and D =

!!!_ -

p;o

3 . 2 •

Yo Yo Yo yo yo yo Yo

For large times an approximate solution can be obtained:

' 9 g27'2b

(yfyo - I)"' s - - 3 - ·

630 D. C. SCHRAM

Since the energy does not oscillate no oscillation amplitude and time can be defined. However,it is possible that the energy decreases to a minimum:

(y/yo - I)min = -

~;

0

before it starts to increase. The tim€ necessary to reach this minimum is:

gT(min) = -1 Pno {bfy _

P;o}

.. yo (b/yo)2 _{Yo 3y5}

and is only positive for a limited range of values for

P

11o.

Acknowledgements. The author is indebted to ProfessorC.M. Braams and dr. W. ]. Schrader for reading the manuscript. He gratefully ac-knowledges the numerical work performed by mr. G. Bos of the Technical University Delft, and the assistance ofmr. E .. Oord.

This work was performed as part of the research program of the as-sociation agreement of Euratom and the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "N ederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (ZWO) and Euratom.

APPENDIX

We will investigate the number and location' of the roots of the polynomial

P₄(o)

=

-o{o3 - 4{3o2

+

_{4((32 - G) & - 8G}}

₌

_-oP

3(o), as function of the

parameter {3 for a fixed value of G. This polynomial P 4(o) has been introduced

in the equations (23) and (24). One of the roots of P4(0) is

o

= 0, so we have to solve Ps(o) = 0. The third degree polynomial P3(o) may have either one

real and two complex roots or three real roots. For a fixed value of the electric field parameter G this depends on the resonance parameter (3. The

roots can be found by using Cardan's method S):

A 1

+

A 2 . A 1 - A 2 1 3 -,. 01 =

tfJ

+

A

1

+

A

2, 02, a =

tfJ

+

2

±

i 2 Y where Ai,2 =

~c

±.Jn, c

= - 287(33

+

tfJG

+

4G

and D =

c2 _

(tf32

+

tG)3.

Evidentiallytherootso2, 3arecomplexifA1 as wellasA2 arerealandA2 *A1; in other terms if D

>

0.

In

the opposite case D

<

0

o

2, 3 are real.

In

the

intermediate case D

=

0, A1

=

A2, there are three real roots of which at

= f3c· In order to find these values for {3 the following equation has to be

solved:

D = {3~

+

{3~ - 2{3~G - 9f3cG - 2

47G

+

c2 = 0.

This biquadratic equation has been solved according to the method of EulerS). It appears that only two real roots exist if

c2

d = -₂₇ (G

+

ll)3

>

0 so if G

>

- l l .

64 ' 64

This is the case for all fast wave interactions and for most slow wave interaction problems;

\G\

=

:r

involves an extremely high value for the e.m. electric field strength. The two real roots

/3ci,

2 may be calculated as

follows: f3c1, 2 = -

-./Pi

± (

.Jh

+ -./

p3) where

Pi

= a1

+

a2

+

c, a1

+

a2 . a1 - a2 r

P2,

3

=

-2

+

c

±

i 2 v 3 and a1,2

=

~217G3

+

is6c2 - s21'.2G

±

-./d;

c =

~

+

1~.

If

\G\

~ Io-2 an approximate solution is 5):

. , 3 / - 3 /

-f3c1 1:::::; "2V 2G

+ "\/

4G2; f3c2 1:::::; -1.

The orientation of the real roots of P 4(b) may be found in figure 2 for some

values of G.

f3

= 0 is the nonrelativistic resonance value. Clearly f3c1, the

absolute smallest critical value of

f3

is a measure of the resonance shift and the resonance broadening caused by the first-and· second-order Doppler effects.

Received 31-8-67

REFERENCES

1) Davydovskii, V. Ya., Soviet Phys. JETP 16 (1963) 629.

2) Hakkenberg, A. and Weenink, M. P. H., Rijnhuizen Report 64-18.

3) Roberts, C. S. and Buchsbaum, S. J., Phys. Rev. 135 (1964) A381. 4) Lutomirski, R. F. and Sudan, R. N., Phys. Rev. 147 (1966) p. 156.

5) Schrader, W. J., C. R. Coll. Interaction Champs H. F.-Piasmas, Saciay (1964) p. 74. 6) Schram, D. C., Rijnhuizen Report 65-25.

7) Grobner, W. and Hofreiter, N., Integraltafel I, II Wien, Springer Verlag (1961).

Physica 40 (1968) 422-434 ©North-Holland Publishing Co., Amsterdam

THE RELATIVISTIC INFLUENCE ON THEDIELECTRIC BERA VI OUR OF A PLASMA NEAR CYCLOTRON RESONANCE

D. C. SCHRAM

Association Euratom-FOM, FOM-Instituut voor Plasma-Fysica, Rijnhuizen, jutphaas, Nederland

Received 18 March 1968

Synopsis

The interaction between an electron plasma and a transverse e.m. wave propagating along a stationary magnetic field exhibits a resonance if the Doppler shifted frequency equals the electron cyclotron frequency. The interaction will become nonlinear if the electric field of the wave is so large, that the variation of the axial velocity, first-order Doppler-effect, and the relativistic mass change, second-order Doppler-effect, have to be taken into account. These effects are of comparable importance and will determine the permittivity of an electron plasma near cyclotron resonance in case collisions and inhomogeneities of the fields and of the density can be neglected. Here, a method is

presented by v.rich the permittivity can be calculated for given values of the v1ave

amplitude, of the density, of the magnetic field strength, and of the applied frequency.

1 . Introduction

The permittivity of a plasma, e, in a stationary homogeneous magnetic field, B0, along which a right-handed circularly polarized TEM wave

propagates (k// Bo) exhibits an infinite resonance if collisions, inhomoge-neities of the fields, and relativistic effects are neglected:

co;(co '--' kvzo) ·

Br ..L

=

1 - (I)

co2_{(co - kvzo - .Q)}

The resonance occurs when the electron cyclotron frequency .Q is equal to the Doppler shifted frequency co - kvzo, where co is the applied frequency, k is the propagation constant, and Vzo is a drift velocity along k. The square of the angular plasma frequency co; is equal to co; = Ne2/mso, where N is the electron number density. The resonance is damped if either of the perturbing effects mentioned above is not negligible. Here, we will restrict ourselves to the relativistic effects, i.e. the variation of the mass and of the axial velocity of the electron. So, we assume a collisionless, infinite, homo-geneous plasma in spatially uniform fields.

The permittivity, e, describes the reaction of the plasma to the wave and

can be calculated from the results of the single particle motion studies. In

an earlier publication 1) the motion of an electron with zero initial transverse energy has been investigated in a similar field configuration. It has been shown 1-3) that the energy of the electron oscillates in time and also along the z-coordinate. The characteristic quantities of the motion (the energy, the oscillation time, t08 , and the oscillation length, Zos) are calculated as

functions of the amplitude,

E,

and of the frequency of the wave, w, of the magnetic field strength, Bo, of the drift velocity, Vzo, and of the index of

refraction, n =

.J

er J_. The choice of n as a parameter leads to a mathematically

simple but possibly inconsistent formulation of the form:

, 2

n = n(E, w, Bo, Vzo, wP, n). (2)

It will prove to be difficult, if not impossil;>le, to ·separate the parameter n at the right-hand side of this equation. Therefore, a consistency check must be made: can the calculate9. value of n be made identical to the originally assumed value? It is found, that at least certain assumed values of n lead to an inconsistency, e.g.: n = I. In this publication, an approximate so-lution of the transcendental equati?n (2) has been obtained.

2. Method of determination of£

Two restrictions have to be imposed on .the character of £ in connection

with the choice of the e.m. field in which the permittivity £ of the plasma ·

will be studied. The first restriction is, that the permittivity is independent of time, so the electric field of the wave must satisfy (\7 · E = 0):

£ o2E

\f2E=-·--.

c2

ot

2

The field B and the permittivity may then be calculated from:

£

oE

I

oE

\7

x

B = - · -= -

- +

µoj.

c2

ot

c2

ot

(3)

(4) As a solution of the wave equation (3) we choose a right-handed circularly polarized wave propagating along Bo - Boiz (see fig. I):

E = E{ix sin( wt - kz) - iy cos( wt - kz)}

=

-Ein, B = E {ixcos(wt- kz)

+

iysin(wt- kz)}

=

E

i~;

Vn _¥ Vn _¥ (5) so oE - w -=wEi~ and k = - .

ot

Vp

424 D. C. SCHRAM

ls. lz

..

8(1

j'"

ts ..

§o

Fig. 1. The field configuration.

In assuming the field, we have made the second restriction: E: is a diagonal

tensor of the form

( er.L E:

=

~

0

0)

0 . Sr// (6)

Since we deal with a TEM wave, the tensor element er// is not of interest. The permittivity may be regarded as a scalar with magnitude er _u which we will

call briefly s. From eq. (4) it follows that the current density is:

j = (t: - 1) ·wsoEh.

Thus,

jn,

the component which is in phase with E is zero if the permittivity E: is assumed to be a diagonal tensor. This is in agreement with the assump-tion of constant amplitude of the wave. It still must be shown that the

restrictions imposed upon € are consistent v1ith the response of the plasma

to the wave. With respect to the first restriction it can be remarked that the resulting permittivity is nonlinear, but will be independent of time, since the field of the wave is constant in 'magnitude. Furthermore, it will be shown in section 4 that indeed

jn

=

0. We find:

u

s - 1 = - - A o

wsoE

In the rotating frame (i~, ·in) the velocity components are:

Ve= Vx cos(wt - kz) +Vy sin(wt - kz)

=

V·ie, Vn

=

-Vx sin( wt - kz)

+

Vy cos( wt - kz)

=

V•i71.

(7)

The current density

h

may be calculated if the velocity distribution function

f is known:

h

= -e

f

fve dv; similarly j₇₁ = -e

f

_{fv 71} dv. The permittivity iS:

s = 1 -

_e_A-f

fve dv.

wsoE (8)

The distribution function

f

will be evaluated from the results of the study of the motion of a single particle.

3. Electron motion

In an earlier publication!) an extensive treatment has been given of the motion of a single particle and the variation of its energy in the e.m. wave and the stationary magnetic field. A short survey of the method of calcu-lation and of the results will be given here and will serve as an introduction to the evaluation of the permittivity. The fields Bo = Boiz and the e.m.

field (eq. 3) are inserted into the equation of motion:

d~tv)

= -

~

(E

+

v

x

B),

where

y =

----,;====--J

1 - v·v/c2 mc2

(9)

is the total energy of the particle, <ff, normalized to mc2; m is the rest mass of the electron. It has been found that the motion of the particle can be de- · scribed by the equation:

2 (-(;-

+

I ) d(; = J-c5{(;3 - 4,8(;2

+

4(,82 - G) (; - 8G}. ( 10) I - n2 _{I - Vzo/vp d-r} Here 1 _ n2 I . - ) (; = 1 / ( _!_ - 1 is an energy function; 1 - Vzo Vp \yo g2(I _ n2)

G = ₂ is a parameter of the oscillatory electric field,

Yo(! - Vzo/Vp)2 _.

with g

=

eE/mwc; (11)

b

,8

= - I is a parameter of the static magnetic field,

yo(! - Vzo/vp)

with b = eBo/mw = !2/w;

-r = wt the reduced time and

n = c/vp the refractive index, where Vp is the phase velocity and c

is the velocity of light in free space; yo and Vzo are the initial values of y and Vz respectively*.

In deriving eq. (10) it has been assumed, that the initial velocity components

Vxo == v;o and Vyo = v110 are both equal to zero: v J.o

==

0. Accordingly, vve

assume that the thermal velocity of the particle is negligible compared to the velocity attained by the acceleration in the fields. No assumption about

*

The special cases Vp = c and Vp = Vzo have to be excluded and will be dealt with

426 D. C. SCHRAM

the magnitude of a drift velocity, Vzo, along Bo has been made. Therefore we consider a cold plasma drifting along the magnetic field Bo in a strong e.m. wave: we deal with a large-signal cold-plasma analysis. The velocity com-ponents are related to the parameters as follows:

l Vg 02 - 2130

-2Gro(o l - Vzo/Vp

+

1)'

g c l - n2 Vri

do

g c 0o(l - Vzo/vp)

dr

Vzo , (1 - n2_)(o

+

₁₎ -;;- = 1

~

{o

+

l - n2 } · 1 - Vzo/vp . '

The integral formulation of eq. (10) is:

r

20 2 }

6

+

do

T

=I

dr =

f

tl -

n2 J;4;)Vzo/Vp '

0

whereas from eq. (14) it follows that:

d

7 _ kz =

f

2(1

+

o) do ,

J

JP4(0)

0

where the fourth-degree polynomial P 4(0) is:

P 4(o) = -o{oa - 4{302

+

4(f32 - G) o - BG}. (12) (13) (14) (i5) (16) (17)

From these equations it follows, that the energy and the axial and perpen-dicular velocity components are oscillating in time, and thus periodic in kz. The maximum and minimum values for the energy variable

o

are roots of

P 4(

o)

= 0. One of the extrema is

o

= 0, the other will be referred to as oextr-If P4(0) has not only two but four real roots, then oextr is the root with the smallest absolute value out of the three nonzero roots. It has been found that oextr

>

0 if G

>

0 and oextr

<

0 if G

<

0. From expressions (1 l) it follows that for a fast wave (s

<

!) as well as for a slow wave (s

>

!) in case

Vzo

<

Vp, the initial energy is minimum. If, however, Vzo

>

Vp, which is only

possible in a slow wave, the initial energy is maximum and the particle will lose some of its inital (axial) energy. The change of the energy of the particle is sketched as functions of r and of kz in figs. 2a and 2b. The curve described in (vg, vri, Vz) space is shown in fig. 6a and will be discussed in section 4.

---kz Fig. 2 io'~~~~~~~~~~~~~~ 6~xtr.

t

/NR •' _, -0.5 0 Fig. 3 ... _.,

:

(, '" ,·: ....

'

---

,,... ... / ' / : / :; :1 ' ... . ·· .... "· ·· .. 0.5 , 1.5 2 -~/~c

Fig. 2. Sketch of the energy behaviour, a) as a function of time, b) as a function of z. Fig. 3. Energy function o~xtr ({J/{Jc); the dotted curve labelled NRrepresents the

non-relativistic equivalent. .

The amplitude of the energy oscillation, oextr• can be written approximately as:

(18) where o:xtr is a function of {3/f3c as shown in fig. 3. The critical value /3c of the magnetic field parameter {J, is a measure of the resonance broadening. In a good approximation:

(19)

A reduced oscillation time Tos and oscillation length kz08 (indicated in

figs. 2a and 2b) can be found by integrating eqs. (i5) and (i6) from

o

=

0

'[

:i~~/~·

;(]

,,'°l

·~

1

t,:,r--==~~\j

~ -M 0 M 1 ~

-~/~c

Fig. 4. Oscillation time functions io({J/{Jc) and i1({J/flc); the dotted curve is the non-relativistic (NR) equivalent of io.

428 D. C. SCHRAM

too = bextr and by multiplying the result by a factor 2:

2b 2

0 Oextr

+

-i

f

I - n2 I - Vzo/vp {

Ji

Tos =

J

= 2 .JP4(0) do =2 I - n2

+

_{I -} lo _{Vzo/Vp '} } (20)

0 0

Tos - kzos =

2fi

+

21 O·

The integrals l 1 and lo can be approximated:

lo'"" io(fJ/fJc) ·

/G/

4 ;

Ji

'""ii({J/fJc) ·

/~/

·

The functions i0 and i1 are shown in fig. 4.

The following integral will be used for the calculation of s:

'Tos Oextr (21) (22) (23)

f

~ ~dT=

₂

J (02 - 2f3o) do _ 212(f3, G) ( 24)

g c Gyo(I - Vzo/vp) .JP4(0) Gyo(I - Vzo/vp)

0 0

The quantity l 2 can likewise be split into two parts, one of which depends

on {J/{Jc whereas the other is a power of G:

(25) In fig. 5 i2 is shown as a function of {J/{Jc. In all approximate separations

:l

!'

I

i2

_/NR

J

: 1 .· 0 -1 -2

·3f

t

::t

I

1

-1 -0.5 0 0.5 1 1.5 2 -~/~c