The consistent solution of the undamped cyclotronwave propagation problem

54 (197 1) 325-328

Physica

0 North-Holland

LETTER

THE

CONSISTENT

Co.

TO THE EDITOR

SOLUTION

WAVE

Publishing

OF THE UNDAMPED

PROPAGATION

CYCLOTRON-

PROBLEM

D. C. SCHRAM Association

Euratom-FOM, FOM-Instituut VOOYPlasma-Fysica, Rijnhuizen, Jutphaas, Nederland Received 26 March 197 1

In a recent publicationi) the nonlinear dielectric behaviour of a cold plasma drifting along a static magnetic field (Be) in a large amplitude E.M. wave (I?, w, k) near and at cyclotron resonance has been treated. The permittivity, E, of the plasma has been calculated from the reactive current; this current has been found by the summation over velocity space of the appropriate velocity component of the individual particles. To that end, the characteristic quantities, which define the individual motion of the particles such as the amplitude and the period of the energy oscillation have been expressed as functions of 8, Be and, close to the resonance, of the index of refraction, n. As a result, also the expressions for the reactive currents are functions of the permittivity, and close to the resonance: n = J&r = rc(g, /I, d, 92)

(1)

(cf. eqs. (2) and (34) of ref. I), w h ere the symbols have the same meaning as in ref. 1 : eE g =

the EM. B=

g

-

yomwc(l

Vzo/Yp) -

field-strength

-

the magnetic-field 2 &A!!Lwp YO02

the electron-density

-

%0/z+) ’

parameter t,

eBo yom@(l

yo(l

-

%o/~p)

1,

parameter, n,e2 ms0o2y0

’

parameter.

t For convenience a slightly different E.M. field parameter g instead of g has been used: the parameter g” contains a small correction factor due to the zero-order drift velocity along the static magnetic field. As in ref. 1 the zero-order transverse velocity is supposed to be zero, so y,~- m(O)/m = l/(1 - v~~/c~)+. 325

D. C. SCHRAM

326

I

0.6

/-----:

t-

0.8i

I

I

I

1.0

1.0

0 -

PI Pot

Fig. 1. The dielectric function jz as a function of the normalized magnetic-field meter B/f%.

para-

In ref. 1 an elaborate numerical procedure is described for the solution of eq. (1) in terms of vz = n(p) for a given set of values of g”and of B (cf. fig. 8 of ref. 1). Once this has been done, also the characteristic quantities of the motion can be calculated as functions of p. Though, in principle, the solution of eq. (1) can be obtained, one is forced to solve the problem again for any new set of values of the E.M. field strength and density, i.e. the parameters g’ and FL To avoid this cumbersome numerical procedure, an alternative, approximate, procedure is presented here which results in generalized expressions for all quantities. It has been established, that under the condition that 5’ > the quantity cz [( 1 - &)/a] C is a function of b/flat only (cf. fig. 1) :

[(I- &)/al c = iz(B/Pac),

(2)

where

c= /3;tcis the This is values of pressions

(Ep)*,

and

/& = 0.73 c

(3)

width of the nonlinear resonance. the consistent solution of E - 1 as a function of /l for any set of g and Z Now it is possible to substitute this solution in the exfor the energy oscillation (cf. eqs. (18)-(23) of ref. 1). This enables

UNDAMPED

CYCLOTRON

WAVE

PROPAGATION

I

I

0

1.0

327

I

1.0

PfPoc

-

Fig. 2. The normalized energy amplitude as a function of the magnetic-field parameter /?I&.

‘1‘Jo t

lo'

16

16’

I

-1.0

Fig. 3. Oscillation-time

0

I

1

1.0

P~POC functions jo and jl as functions of magnetic-field

B/BBC.

parameter

UNDAMPED

328

CYCLOTRON

WAVE

PROPAGATION

to express the energy-amplitude oscillation time and length in new functions of A, jo and ji which depend only on p/BBC. In terms of these functions which are shown in figs. 2 and 3 we find as a complete solution of us

the problem the following : the energy amplitude :

Y&r - YOw the oscillation

A(B/Bac) yo(l - %0/7+) g2 ; C

time :

30 lid,, = TlJ* = c the oscillation

(4

1 1 -

v&p

I il

C"

jo

B

1 .

’

(5)

length :

kzos = This work was performed under the association agreement and FOM with financial support from ZWO and Euratom.

REFERENCE 1) S&ram,

D. C., Physica 40 (1968) 422.

of Euratom