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HIGHER HARM ONICS G ENERATION IN W EAKLY IONIZED PLASMAS 5 4 5

strong interaction effects, such as reflected shocks and rarefaction waves.

Other experimental investigations, mainly quan­titative, are necessary, however, to prove that the observed reflected fronts are plane shock fronts and to determine all flow parameters. In addition, relaxation effects behind shock waves in plasmas can be studied.

Only then can the experimental results be reason­ably compared with the theory. But the steady state cannot be achieved, as the results show. Therefore the transient part of the theory of J o h s o n 2 must be taken as a basis. However, the assumption made in this theory that the current density can be ex­

pressed in terms of o • V • B is not valid. Further­more, ionization effects have to be taken into account. More experimental and theoretical in­vestigations are therefore required.

The author w ishes to thank Prof. R. W i e n e c k e for encouragement and for his continued interest in this work. He is indebted to his colleagues Mr. Z i m m e r m a n n

for his help with the measurements and Mr. S a r d e i for discussions concerning the theory. He is grateful to Mr. S c h m i d , Mr. S t e f f e s and Mr. L o e b e l for technical assistance.

This work has been undertaken as pa rt of the joint research programme of the In s titu t für Plasmaphysik and Euratom .

Higher Harmonics Generation in Weakly Ionized Plasmas in Time-Varying Crossed Fields

W o l f g a n g St il l e r * and Gü n t e r V o jt a

Arbeitsstelle für Statistische Physik der Deutschen Akademie der Wissenschaften zu Berlin705 Leipzig, DDR

(Z. Naturforsch. 24 a, 545— 555 [1969] ; received 21 November 1968)

The linearized Boltzmann equation of kinetic theory for the electrons of a weakly ionized plasma under the action of an alternating electric field E = cos co t {Ex , Ey , Ez} and a circularly polarized magnetic field = B {cos cug t, s i ncogt , 0} is solved. The direction-dependent p a r t /1 (r, v, t) of the distribution function contains especially higher harmonics with frequencies 2 a>ß , (O ± a>ß , co±2ci)g. These overtone frequencies are generated — unlike the higher harmonics obtained by Margenau and Hartman — by a cross drift mechanism and are present even if the isotropic part f° of the electron distribution function is assumed to be time-independent. The physical meaning of the various contributions to the distribution function f 1 is discussed in detail.

1. Introduction

The influence of crossed electric and magnetic external fields on plasmas has been studied for several reasons and with different methods. For example, it is interesting to learn at which electric field strengths the breakdown in a high-frequency low pressure gas discharge starts. Furthermore, it is important to study the plasma particle motion in crossed field geometries in order to get information about a possible particle acceleration. This is of in­terest in cosmic ray physics, electron and ion optics

* Based on a part of a thesis submitted to the Mathematisch- Naturwissenschaftliche Fakultät der Karl-Marx-Universität Leipzig in partial fulfillment of the requirements for the degree of Dr. rer. nat.

etc. Finally a large amount of effort has been de­voted to the examination of appropriate crossed field configurations for an effective heating and a stable confinement of high temperature plasmas. Often it is sufficient and less expensive to examine these problems in suitable model plasmas, e. g. using the weakly ionized regions of low pressure dis­charges 1.

Many investigations have been performed in which the crossed fields are assumed to be static. With respect to the applications mentioned it is also important to study the plasma behaviour if the ex-

1 F. K a r g e r , Z. Naturforsch. 22 a, 1890 [1967].

5 4 6 W. STILLER AND G. YOJTA

ternal electric field is time-dependent ( being mostly an alternating field in practice) and the magnetic field is constant2-4, or vice versa0 - '. Finally new phenomena are expected to appear if both the ex­ternal electric field and the magnetic field are time- varying and independently established 8.

An approximative theoretical treatment of such problems consists in using the single particle de­scription 9- 10. In a similar way the equations of mo­tion of magneto-hydrodynamics can be used6. Fi­nally it is possible to resort to the powerful methods of kinetic theory or statistical mechanics11-15. In this paper a weakly ionized plasma is investigated using Boltzmann’s kinetic equation. The plasma is assumed to be under the action of an alternating electric field, a circularly polarized magnetic field, and additional constant fields.

2. Basis Equations

Let us assume the heavy particles of a weakly ionized plasma to have a Maxwellian distribution function. For the electron distribution function the Lorentz ansatz is made

f ( r , v , t ) = /°(r, v, t) + ~ ' f1( r , v , t ) , \ v \ = v . (1)

The coupled system of differential equations for the calculation of the parts /° and f 1 is obtained from the Boltzmann equation 16:

where<5/° 1 m 3

<5/°, + ^ d i v r f 1 + - V " (v2[E + Ei] ’ f 1) = . at 3 3 tr av ot ,e-n

k T

(2 )

y fO I y

dt e — n v2 M 3v \ ' m dv

3, + ‘'g r « W ° + - (E + E,) -3 v (3)

+ — [ B x f 1] - - r i ' .m

[>’(f) : electron-neutral collision frequency, q/m: electronic charge-to-mass ratio, M: mass of a neutral particle, T : gas temperature, E : electric field, B : magnetic field.] The following field configuration is considered:

(a) homogeneous alternating electric field E (t) witha = a (f) = (q/m) E(t)

= (q/m) coso) t {Ex ,E y , E z} , (4)

(b) homogeneous time-independent electric field E 0 with

= ? ^ O y ) = ( (l / m ) {^Ox •> E « y , E ^ z \ (5)(c) homogeneous circularly polarized magnetic

field B K(f) withtoR = (q/m) B R = ojc { c o s o j b t, sin coB t, 0 } ,

ojc = ( q / m ) B . (6)

(d) electric field Ej1 induced by the time-varying magnetic field B li( t) with

a f = a f (r, t) = (q/m) E f (r, t)= coq cob\0 , 0 , x cos wB t + y sin coB t } . (7)

(e) homogeneous direction-constant time-indepen­dent magnetic field B0 with

wn = {0 , 0 , oj0} = (q/m) {0 , 0 , ß0} . (8 )

Making use of the abbreviations

w ( r , v , t ) = f l (r , v , t ) + i f l (r , v , t ),

■A-w = (q ltn ) (Ex + i Ey) = ctx + i A l , = ( q / m) (Efx + i Efy) = a f x -

2 W. P. A l l i s , in: Encyclopedia of Physics, S. F l ü g g e ed., Springer, Bedin —Gottingen —Heidelberg 1956, Vol. 21, p. 383.

3 A. W. G u r e v i c , Dokl. Akad. Nauk 104. 201 [1955].4 A . A i r o l d i - C r e s c e n t i n i , P. C a l d i r o l a , C. M a r o l i , and

E. S i n d o n i , Proc. 8th Intern. Conf. Phenomena Ionized Gases, Vienna, A u g . 1967, Contributed Papers, p. 168.

5 H. P f a u , G. V o j t a , and R. W i n k l e r , Kernenergie 7, 463 [1964].

6 E. L o r e n z , E. H a n t z s c h e , G. V o j t a , and R. W i n k l e r , Kernenergie 7. 469 [1964].

7 T. K i r s t e n and G. V o j t a , Kernenergie 7, 451 [1964].

Gw ~ V ( 3x + 4 dy ’■y, B?( t ) = - i c o c exp{icoBf } , (9)i cl \y , A qu- = o^)X -)- i

8 F. G. I n s i n g e r , H . J. H o p m a n , and J. K i s t e m a k e r . J. Nucl. Energy (Part C) 5,223 [1963],

9 E. K o c h , Beitr. Plasmaphys. 7, 479 [1967 ].10 G. V o j t a and J. W o n n , Beitr. Plasmaphys. 7, 501 [1967].11 G .V o j t a and T. K i r s t e n , Beitr. Plasmaphys. 5, 405 [1965].12 G. V o j t a and T. K i r s t e n , Proc. 7th Intern. Conf. Pheno­

mena Ionized Gases, Beograd, Aug. 1965, V o l . II, P- 80.13 W . S t i l l e r , Proc. 8th Intern. Conf. Phenomena Ionized

Gases, Vienna, Aug. 1967, Contributed papers, p. 270.14 W . S t i l l e r . Beitr. Plasmaphys. 7, 507 [1967].15 W . S t i l l e r and G. V o j t a , Physica 1969. to be published.16 V. L. G i n z b u r g and A. V. G u r e v i c , Fortschr. Phys. 8, 97

[I960].

HIGHER HARM ONICS GENERATION IN WEAKLY IONIZED PLASMAS 5 4 7

one can compose the first and second component equation of (3) to:

iR i 3/°31 ~~ i v) w — ~\Gw /° + [-Aqw + A w cos oj t -f- Aut ] — B-(t) f l j — Sw (T, v, t ) . (10)

In a similar fashion the z component of Eq. (3) can be written as

gf = - (c, /° + [A„s -M 2 cos <o / + /fg] - Re{ß? w) ) - S f (r , v , i ) (11)

where0

Gz = v "g“ , A02 = , Az = 3-z, A[z = cl\z . (12)

Let us consider now only the solutions m^w and Inll/z of the inhomogeneous differential equations of (10) and (11), respectively 15:

w ( f ° , f l ) = "‘M r, V, t ) = J j dr Su, ( r , v , t; /°, f \ ) exp[(v + i w0) r] J exp{ - (r -f i co0) *} , (13)

f l (f ° , w ) = inh/i ( r , v , t ) = | J'drS? ( r , v , r ; f ° , w ) exp|> r] j exp{ - v t } . (14)

Firstly it must be pointed out that these “solutions” are formal ones with respect to the unknown isotropic part /°. The difficulty that the quantity f l still contains w (and vice versa) can be overcome if one sub­stitutes w ( f ° , f l ) into the solution f l ( f ° , w ) . The resulting integrodifferential equation for f l ( r , v , t ; f ° ) has the form

3 fl

where

|y + v f \ + col Re j $dt f lexp( [v + i(co0 + a)B)] (t - i ) ) j =/(r, v, t; f°) (1 5 )

I ( r , v , t ; /°) = - G z f° - (A0z+ AZ cos <x>t + A iz)R\ 3/°av (16)

I - l 3 f° / 3 f ° \ / ]i J dt [Gwf° + Aow + A W cos (o t -g—j exp([v + i eo0] t ) j.exp { - [v + «(co0 + ojb) ] t } .

By repeatedly differentiating Eq. (16) with respect to time and substituting terms from higher time derivatives by lower ones one can get a linear inhomogeneous third-order differential equation for f l ( r , v , t ; f°) :

fl + 3 v j] + (3 v2 + cOq + [cOq + coB] 2) fl + v(v2 -f ojg + [co0 + o>B] 2) fl = / + 2 v / + (v2 + [eo0 + coB] 2) / .(17)

With help of the Hurwitz criterion it can be easily proved that the solutions * fl (i = 1, 2 , 3) of the homo­geneous differential equation of (17) asymptotically disappear for t —> oo. Therefore the task consists in calculating the quantity mhfl (r , v, t; f°). Then one can find inhw(r, v, t; /°), too, by substituting inhfl (r, v, t; f°) into Eq. (13). The combination of the quantities inhtt; and inh/z with Eq. (2) then gives an equation for f° alone. The solution of this equation and the final calculation of the parts /° and f1 which are not necessary for the following considerations can be found in a further paper 17.

3. Calculation of the Distribution Function Part 'nhfl (r, v, t ; /°)

Now the isotropic part /° is assumed to be time-independent. This is tantamount to the fact that the relaxation time of the electronic energy is much greater than the momentum relaxation time.

17 W. S t i l l e r and G. V o j t a , Z. Naturforsdi. 24 a [1969], to be published.

5 4 8 W. STILLER AND G. VOJTA

In order to alleviate the solution procedure of Eq. (17) the term /(!*, v, t; /°) is transformed in an appro­priate manner. First one integrates over the time variable t . Only the solution (r, v, t; /°) for the stationary state is interesting; that means that terms with a factor exp[ — v t] appearing in I (r , v, t; /°) can be neglected. Equation (17) transforms into

f\ + 3 v j \ -f (3 v- + Wq + [co0 + coB] -) fl + v(v2 + (Op + [co0 + coB] -) flf 0 \ 0 | 0 fO

— I z 0 4* Gz f J 3-z cos o j t -0 - — ojq Re | i Aqw -g--- + Gwv + i co0

exp{ - i coB J3/° pcoc -3— Re ( r + i co0) c o s co t + co s in co t

}G

— oiq o jb (z cos coB t + y sin coB t) X'

(18))/°

(19)

( r + i o>0) 2 + co2

Now for inhfl (r, v, t; f°) the solution ansatz is made:j \ = inh/e {T,V, t ; /°) = ° f \ (T’ V) + ct (r, v) sin co t + c2( r ,v ) cos oj t

+ dx (r, v) sin o jb t + d2(r, v) cos coB t+ e1(r ,v ) sin{ (co + coB) 2} + e2( r ,v ) cos{(co + coB) t}+ gx(r, v) sin{(co-a>B) t} + g2( r ,v ) c o s { ( o j - o j b ) t ) .

Substituting this expression into (18) and equating the coefficients, the unknown quantities °fl , ct , et , d i , and gi ( i = l , 2) can be calculated containing in every case the isotropic part f ° ( r , v ) . Using the abbre­viations

;=1{Z„) = (Z 1 > - ®/i,(Z2) = (Z 2) = sin oj t + c2 cos co t ,(Z3) = (Z 3) = (<5j + Ax) sin coB t + (<52 + A2) cos coB t = d x sin coB t -f d2 cos coB t (Z4) = (Z 4) = e t sin{ ( o j + o jb ) t} + e2 cos{ ( o j + coB) t} ,(Z5) = (Z 5) = <jq sin{ ( o j — coB) f} + <79 cos{ (co — coB) t)

(20)

and XT f r r ri TT \ \ ( 3/° 3/° 3 /° 3 /° 3/° 3/°^ 0 — 5 U o y 5 ^ 02/ — n a 0 x + u 0 X 5 ~> r V 2 » . ’ u

3/" 3/" 3/« I a, , a„ , a, cos oj tx dv ’ v dv ’ * du ( 2 1 )

the solution inh/z ( r , v , t ; f°) can be given in terms of Table 1. The decomposition of the coefficient d1 (d2) into additive parts and At (d2 and A2) shall emphasize the influence of the induced electric field de­noted by Ax and Zl2 .

The physical meaning of the terms included in mhfl [r, v, t; /°) will be discussed in Section 5.

4. Calculation of the D istribution Function Part mhw { r , v , t ; f°)

The solution inh/* (r, v, t; f°) can be written for short in the forminh/l = O fl + e icjt + d e ioJBt + E e i(w + coB) t + g e i(o»-®B)f] } ( 2 2 )

with c = cl + i c 2 , d = d1 + i d 2 , £ = e1 + fe2 , g = g t + i g 2 .

Now (22) is substituted into (13) taking into consideration the meaning of according to Eq. (10) :

inhw(r , v, t; f°) = j — J dr 3/° + Gw f exp{ (v + i OJ0) r } \ exp{ - (v + i o j0 ) t }

Q TV I

A w J' dr cos oj t exp{ (v + i oj0) r}> exp{ — (v + i oj0) t} ( 2 3 )

+ j i ojq I’d r (% 1 +R e { - i [ c e i“' + (iei", r + £ e i(“+“»)' + g e i^ ~ w )T]} ) ( 6

For

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Cl - p ClCM 3 Cl

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3 Ö13

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iDh/l ( r , v , i ) OJ. U i * - .

>o(X I ) (X 2)

0 TJU n> + 4 t o _

+ " » ) c o s tü ( + <U ( ^ + (u2_ (02) s i n w 3/0 O,0{ ( 0, 2 - 4 - r * ) c o s « * - 2 V 0, s in ft) /} 3 /»[ ”2 + » 2 - 4 ] 2 + 4 v 2 (Ug 3 „ + ---------- «* 3 „

(X3>co,G

[r2 + C02 _ f02 j2 + 4 );2 w 2

(X 4)„ (v2 + 0 J2 + + Wb) ( - V sin coB t + (co0 + ojb) cos coB t) U0z

(X 5)

+ __________ c1( r , v ) - 0JCl 2[ v2 + co2 - (co0 + C0ß) 2]2 + 4 v 2(to0 + COB) 2

(X 6)

_i c2(r, v) -coc/2 o 2 + CO2 - ((O0 + coB) 2] 2 + 4 v2 (co0 + coB) 2

+ ___________ J *1 (**, v) • coc/2O 2 + coB - ( c o0 + ojb) 2]2 + 4 ^ 2 ( c o0 + coB) 2

(X 7) d2(r,v) -coc/2

(^2[co0 + wB + w] + [ (oj0 + (oB) 2 - co2] [co0 + w B - co] ) sin{ (co + coB) t j

v ( (f00 + 0)b) 2 + V2 + CO2 — 2 (co0 + OJß) co)cos{ (co -f- COB ) z} + (v [co0 + coB-co] + [(oj0 + ojb) 2-c o 2] [co0 + coB + co]) sin{(co-coB) t}

~ v ( (O)o + cob) 2 + v2 + co2 + 2 (co0 + coB) co)cos{ (co — eoB) t}- v ( (co0 + coB) 2 + v2 + co2 - 2 co (co0 + cOß) )sin{ (co + coß) z}

+ ( ^ K + Wß + W] + [(w 0 + cob) 2-c o 2] [co0 + coB-co])cos{(co + coB) t] + }' ( ( 0)o + wß)2 + v~ + co2 + 2 co(co0 -f coß) )sin{ (co — coB) z}

+ (^ O o + Wß-co] + [(co0 + cob) 2-c o 2] [co0 + cO ß-co])cos{(co-coB) Z>

[ - v(4 co| + 4 co0 ojb + r2 + cog) + (r2 + <o{j) (co0 + 2 coB) sin{2 coB t ] + r ( v 2 + coß) cos{2 coB z}]

______________ “ 2 V ? “J WC 2O2 + c o l - (co0 + COß)2] 2 + 4 r2(ft>0 + COß)2 *-+ w o(4 cob + 4 co0 coB + v2 + cog) _ v (v 2 + cü(5) sin{2 coß z} + (v2 + col) (co0 + 2 coB) cos{2 coBz}]

+ ej(r, 17) -coc/2[j’2 + (oj + COß) 2 - (C0q + COß) 2] 2 + 4 V2 (OJ0 + COß) 2

(X 8)

+

e2(r, V) - co.c/2 [?'“+ (co + coB) 2- (o j0 + ojB) 2] 2-(-4r2(co0 + coB) 2

f / i ( r , v ) ;ojc/2 [)'2 + (co coB) 2 (c.o0 + coß) 2] 2 + 4 r 2(co0 + coB) 2

(X 9)

g»(r , v) -coc/ 2[ v 2 -f ( oj — COß) 2 — (co0 + COß) 2] 2 + 4 V2 (cOq + COß) 2

( o 2 + 2 coB] [co0 - oj] + [ojq - co2] [co + co0 + 2 ojb] ) Sin co t - r (p2 + (co0 + coB) 2+ (oj + 0Jß)2 + 2 (co0 + COß) (co + COß)) cos co f + P2 [co + co0 + 2 ojb] + (co - co0) ( (oj + COß)2 - (" 0 + °>b) 2) sin{ (co + 2 ojß) /} + v (v2 + (co - co0) 2) cos{ (co + 2 coB) «}

( - „ [ „ 2+ (co -j- ojß)2 (oj0 + cob) 2] + 2 v(co0 + cob) [co + oj0 + 2coB])s in o jf + ( - r 2(co-co0)+ [ (co0 + OJß) 2 - (oj + COß) 2] [co + co0 + 2 COß] ) COS OJ t + ( - V[v2 + (co + OJß)2 _ (co0 + COß) 2]

+ 2 V(ötf + COß) (co - OJ0) ) sin{ (co + 2 coB) «} + (2 v2(co0 + coB) + [v2 + (oj + coB) 2 — (C00 + COß) 2] (co — CO0) ) COs{ (co + 2 COß) f}

(2 V2 (oj0 + COß) + (co - OJ0) [r2 + (co - coB) 2 - (co0 + coB) 2] ) sin co t + v (v2 + (co - coB) 2- (oj0 + ojß)2 + 2 (oj0 + ojß) [oj0 + 2 ojj{ - co]) cos co f + (2 r2 (oj0 + coB) - (co + co0) [v2 + (co - COn)- (<w + COß)2] ) sin{ (co - 2 coB) /} - v (v2 + (co - coB) 2- (C00 + C/Jß) “ + 2 (cO0 + COß) (co + C00) ) COs{ (co — 2 COß) /}

(2 v (oj0 + ojß) [co-o>0- 2 ojb] - v[v2 + (co - coB) 2 _ (co0 + ojb) 2] ) sin oj t + (2 2 (co0 + coB)+ [v + (co - COß) 2 - (OJ0 + OJß) 2] [co - CO0 - 2 OJß] cos OJ t+ (2 r(co0 + COß) (co + co0) + P[r2+ (co-coB) 2 - (co0 + coB) 2]) sin{ (co - 2 coB) t }+ (2 ^(cop + coß) - (co + co0) [v2+ (co-coB) 2_ (co0 + ojb) 2] cos{ ( c o - 2 coB) /}

rahlo 2. Solution

■/'•/ uoijnjog g 9iqi?x

{1 (3(0 Z — m ) }SOD ((3(0 + °(0) (°(0 + 0))<t Z + [3 (3(0 + 0(o) ~ z ( ^ m ~ m ) + Zd ] d ) +{7 (aro z ~ (o)}uts ( [ s (aro + °ro) — s (ao? — co) + zd] (°ro + ro) 4- (3 (o4°o? )s,t g —) +

7 co s o d ((3(o + ° ( o ) ( ° ( o — co)d z — [ s ( H(o + ° c o ) ~ z ( Sco ~ m ) + s '1 ] '* ) +

7 co uis ([«co z — °(o — (o] [z (aro 4 °o?) — z (»co — co) + zd ] + (3(o + °(o)zd z )

{ 7 (Ho? z ~ ( o ) } s o d ( [ s ( a co — ° ( o ) — z (»co — co) 4 zd] ( ° ( o 4 co) — (»co + °co)zd z ) +{? (a(o g - o ; > ) } u i s ( [ s ( Ho;> 4 ° ( o ) - z ( » m — c o ) + z< t ] < i+ [ m — » co £ + ° w ] ( » m + <)c o ) d z ) +

7 co s o d ( [ s ( a « ) 4 ° r o ) — z (»co — co) + zd] [co — »CO z + ° ( o ] ( a o? + °co)zd z — ) +

7 co uis ( s ( a w + ° r o ) — z (»co — 0 ? ) 4 zd ]d + [co — »co z + °ro] ( a (o + °m)<i z ) _

{ l ( a (o £ 4 o > ) } s o d ( ( 3 ( o 4 ° o ? ) (°co — co)d z — [ s ( a w + ° r o ) — z (»oy + co) 4 zd]d) 4

{ 7 ( a ro £ + ( o ) }uis ( [ _ ( « 0 ? + ° r o ) — z ( a ro + co) 4 zd] ( ° r o — co) + ( a o? 4 ° ( o ) zd z ) +

7 co s o d ( [ « o ? £ 4 ° ( o 4 ( o ] ( a w + ° ( o ) z + z ( a w + ° c o ) — z ( a ro + c o ) + zd ) <1 +

7 co uis ( [s («0? + °(o) — z (ao? 4 co) + s<t] [aw z + °m + co] + (H(o + 0(o) zd Z ~ ) _

{? (ao? z + m ) } SOD ( [ s (ao> + °w) — z (aro + m) + zd] (°m — co) — (aro + °co)zd z — ) +{? (ao? z + m ) } u!s ( + (o) (a(o + °w ) Z ~ z (U(o + m ) + z (9fo + °ro) + z‘l ) d +

j 0 0 s o d ( [ j ( f l fo + o } ) — s ( a 09 + ° r a ) + z< t\ ( t f o ) + ° o ? ) + [ s ( a r a + ° a ? ) — s ( a w + o ? ) + z< i] ( a o? + w ) — ) +

jco uts ((«to + to) (aro + °ro)<t z + g (aw + co) + s (aw + °ra) + zd )d

; (3(0 + °co)zd f + z [- (affl + °ffl) - g ( a r o - ro) + zd]

Z/:yo>■ (a ‘Jl) 7' 6

(6A)

. (a09 + ° m ) zd |7 + s [s (a o? + °ro) - s (a r o - 0>) + zd]

+

Z/°co. (a U ) lß

z ('d(0 + °ro) zd V + s iz (9ffl + °ffl) ~ z ( n m + m ) + zd l Z/°co. (^

( 8A)s (a ro + ° w ) zd f7 + s [- (a <o I °(o) - g (a w + ffl) + zd]

Zpco. (a ‘j ) t3

+

[ { / a(0 ^}S0D (Oo? + ~ d ) d + {7 a(0 2}UIS (3(0 £ + °0?) (g(0 + g,t) + (0CO + - d + »CO °(0 f + »CO f ) d ]

[{7 3(19 £}sOO (aCO Z + °(0) (%<0 + zd ) + { 1 s } uls (zm + Zd )d + (Zm + Zd + 9ro °m ^ t ) ° ro]

{1 (3(0 — co) }sOD ( ( 3(0 + °ro)o? Z + Zc0 + zd + s (gf0 + °(o))<t +[7 (3(0 + co) }sOD ( (3(0 + 0(0) CO Z — Zm 4 zd 4 z (tIf0 + 0o9) )d +

{; (3(o — (o )}u is ( [ ( o 4 a ( o 4 ° w ] [ j W - j ( 3( o 4 °o))] 4 [co — »co 4 ° w ] s<t) —{ 7 (3(0 4 co) }uis ( [co — 3(0 4 Oco] [s(0 — z (3(0 4 °(o) ] 4 (co 4 3(0 4 °(o) -<()

{ 7 (3(o — co) j-soD ( [co — 3(0 4 °co] [zco — z (3(o 4 °co) ] 4 [ro 4 »co 4 °(o]zd) 4 | ; (3(0 4 o?) }sod ([(o — 3o? 4 oTO] [ zco — z (3(o 4 °ro) ] 4 [co 4 a(o 4 °w ]s^) — {7 (»co — co) }uis

( ( 3(0 4 °oi)o) z 4 zm 4 zd 4 z (Uo:> + °f0) ) d + { 1 (üffl + ro) } UIS ( (ao? + °m ) m Z ~ zm + zd + z (yro + °m ) ) d_

( SA)

g (3 (o 4 ° m ) z d f + z [z (3(o 4 °fo) - 3(0 4 s-(] Z/° m. (a ‘A ) zp

(L A>z (3(0 4 °o?) t' 4 j [j (3(o 4 °o?) - % m + zd]

H-

Z/°(*>. ( a ‘d ) xp

z (3(0 4 ° ( 0 ) f 4 z [z (3(0 4 °ro) - zco 4 zd]Z/°co. ( a ‘j,)z3

(9 k)

z (3(0 4 °C0 ) zd f 4 s [z (3(0 4 °(o) - zco 4 zd]

Z°ß (7 3(0 SOD d 4 7 3(0 UIS (3(0 4 °(o) )

Z/°co. (a ‘j ) To

(s (3(0 4 °ro) 4 0(0 4 zd ) d

+

+

0(0

(tA)

0(o zd f + z [ ° c o - zco + zd] a Q x—, - xX3

( 8 A )

Oo> zd V + z\-zm ~ zm + zd ]

(zx) (I a )

o/0 1 co u i s ( O c o - z co + z d ) CO+ 7 CO s o o (Oco + z m + z d ) d 0/ g {? m u is m a z — 1 co sod (g/t — §(o — z m ) } ° J(0(7 ‘ a M )

5 5 2 W. STILLER AND G. VOJTA

The final form ofinhw i r , v, t; /°) (r , v , t ; f°) + i inh/J (r,v,t; /«) ( 2 4 )

can be got by calculating the time integrals; see Tables 2 and 3.Knowing the solution vector ‘^ f 1 (V, v, t; /°) one can discuss the physical meaning of the several terms

of inhf1.

5. Physical Meaning of the Solution

For the purpose of simplifying the discussion some arrangements are made, permitting a more systematic treatment.— Firstly each term shall be characterized with re­

spect to its kind of time dependence (abbrevia­tion K: “kind”).

— The presence of each term originates in a physi­cal mechanism, which shall be given next (abbre­viation 0 : “origin”).

— Each term of the solution mhf1 has its influence on the current density distribution j according to the relation

OO

j ( r , t ) = q $ d v v f ( r , v , t ) = § q j d v v s f l { r , v , t )(25)

(abbreviation C: “consequence”).If necessary the statements K, 0 , and C will be completed by annotations (abbreviation A ).

It should be pointed out that conclusions from the solution f1(/°) to thet temporal behaviour of the current density only can be drawn if the isotropic part /° is not time-dependent.

Explanation of the terms of mh/J (compare Table 1). <Z1)K : This term is not-time-dependent.0 : The term is caused by the constant electric field

and the spatial gradient of the distribution function in direction of the z axis. (Z 1) is de­pendent on the parameters co0 , oj.q , coB of the magnetic fields.

C: The result is a time-constant drift current in direction of the z axis.

(Z 2)K: These parts of the solution oscillate with the

fundamental frequency o) of the alternating electric field.

0 : They are caused by the z component of the alternating electric field. (Z2) is influenced by the parameters « 0 , ojq , coB of the magnetic fields.

C: This results in an alternating current (in z direction) oscillating with the fundamental fre­quency co of the electric a.c. field.

A: In the limit oj—> 0, az -+a<)z the term (Z2) reduces to the term (Z 1} if only homogeneous plasmas are under consideration.

(Z 3)K: These parts oscillate with the fundamental ro­

tation frequency coB of the circularly polarized magnetic field.

0 : 1) Terms sin coB t and d2 cos coB t . The simultaneous presence of the driving forces Uqx and U,Qy [compare Eq. (21)] and the ro­tating magnetic field originates an E X B drift (cross drift) 2 in direction of the z axis. If the quantities Uox and Uoy disappear or the quan­tity ojq is equal to zero (that is tantamount to the fact, that one of the both reasons necessary to the development of a cross drift mechanism disappears) the terms sin coB t and (52 cos eoB t are zero, too.2) Terms At sin coB t and A2 cos coB t. The time- dependent circularly polarized magnetic field produces an alternating electric field according to Faraday’s induction law. This a.c. field is the origin for a particle drift. A disappearing rotating field (a)c ==0) involves the disappear­ance of the induced electric field, i. e.

= A2 = 0 .

C: The result is an alternating current oscillating with the fundamental frequency ojb in z direc­tion. This current is a cross drift current in case 1), a simple drift current in case 2).

(Z 4) [ (Z 5)]K: These parts oscillate with the sum [the differ­

ence] of the fundamental frequency o j and the fundamental rotation frequency o jb .

0: The simultaneous presence of the “driving forces” Ux and U,y [compare Eq. (21)] and the rotating magnetic field originates a cross drift in the direction of the z axis. The dis­appearance of one of these causes (or both of

HIGHER HARMONICS G ENERATION IN W EAKLY IONIZED PLASM AS 5 5 3

them) makes the terms (Z4) and (Z 5) van­ishing.

C: There will appear a time-periodic cross drift term in z direction with the frequency co + coB[co-coB].

It is now necessary to discuss the parts of inh/ i , (the terms of mhf l can be discussed in a similary way).

Explanation of the terms of inh/* (compare Table 2 ).(X I)K: This part is time-independent.0: The term is produced by the driving force

X I) is influenced by the magnetic field B 0 perpendicular to Uox [compare factor (v2 + co02) -1].

C: The arising drift current in direction of the x axis is time-independent.

(X 2)K: The term (X 2) is not time-dependent.0: The simultaneous presence of the force U

and the constant magnetic field B 0 perpendicu­lar to U$y yields a cross drift in x direction.

C: The cross drift current appearing in x direc­tion is constant with respect to time.

(X 3)K: These parts oscillate with the fundamental fre­

quency oj of the alternating electric field.0: This term is caused by the x component E x of

the alternating electric field. (X 3) is influenc­ed by the constant magnetic field being perpen­dicular to E x [compare the factor:

( [„2 + co2 _ W(2] 2 + 4 v 2 - 1

in (X 3 )] .C: This results in a drift current oscillating with

the fundamental frequency co in x direction. (X 4)K: These parts oscillate with the frequency oj of

the alternating electric field.0: The simultaneous presence of Uy and B 0 being

perpendicular to \Jy gives rise to a cross drift in x direction.

C: The arising cross drift current oscillates with a frequency co in a: direction.

(X 5)K: These parts oscillate with the fundamental fre­

quency coB of the rotating magnetic field.

0 : The simultaneous presence of U qz and the cir­cularly polarized magnetic field being perpen­dicular to Uqz leads to a cross drift in x direc­tion.

C: The alternating cross drift current oscillates with the frequency coB in x direction.

(X 6)K: These terms involve oscillations with frequen­

cies oj ± coB .0 : The simultaneous presence of the driving force

Uz and the circularly polarized magnetic field gives rise to a cross drift in direction of the x axis.

C: The arising alternating cross drift currents in x direction have frequencies oj + oj-q .

(X 7)K: These parts are either time-independent or pe­

riodical in time with the frequency 2 coB .0 : The existence of these terms has two origins:

01: The forces in (Z3) characterized by d1 and d2 yield in connection with the rotating magnetic field a cross drift in x direction.

02: The simultaneous presence of the induced elec­tric field E;R (given by / l1 and A2 in (Z 3)) and the circularly polarized magnetic field ro­tating perpendicular to EiR give rise to a cross drift in direction of the x axis, too.

C: This will lead to cross drift currents along the x axis which are either time-independent or os­cillating with the frequency 2 coB .

Expla- F irst influence of the Second influence of the nations rotating magnetic rotating magnetic

to <(X 7} field field

0 1 : The driving forces \1qx and Uoy cause in con­nection w ith the ro­tating magnetic field a cross drift in z direction.

0 2 : The time-dependentrotating magnetic field induces an electric field in z direction.

The force which drives the electrons in z direc­tion according to the first influence of the ro ­tating magnetic field causes moreover in con­nection with the circu­larly polarized magne­tic field a cross drift in x direction.The induced electric field and the polarized magnetic field rotating perpendicular to it give rise to a cross dirft in x direction.

Table 4.

5 5 4 HIGHER HARMONICS GENERATION IN WEAKLY IONIZED PLASMAS

A: In this connection it should be emphasized that the rotating field takes part in the development of a cross drift mechanism in a twofold man­ner:

The physical situation described in Table 4 can be proved in some mathematical respect by investi­gating the terms of (X 7):

All the frequency functions (i = 1 . . . 4) are homogeneous functions of first degree with respect to oj.q . Therefore the quantities dx and d.2 are pro­portional to (compare Table 1; the additional bearing of oj . in "Ba11 and is not importanthere and can be ignored). Another mathematical hint for the twofold influence of the rotating field is the fact that the products from sin coB t and cos coB t — which must be calculated according to Eq. (23) — yield constant parts and terms with2 eoB in agreement with (X 7).(X 8) [<X9>]K: These parts are periodical in time with fre­

quencies oj or o) + 2 coB [co or oj — 2 coB] .0: This case is completely analogous to that of

(X 7). This can be easily seen if one replaces the forces Uqx and U$y in (X 7) by Ux and Uy . The forces in {Z 4) and {Z 5} charac­terized by , e.2 and , g2 yield in connection with the rotating magnetic field a cross drift in x direction.

C: The mechanism causes cross drift currents os­cillating with frequencies co or oj + 2 coB [ oj or oj — 2 coB] . The proof that the circularly po­larized magnetic field is engaged in the cross drift processes in a twofold manner can be given similar to (X 7}.

Overlooking the terms of mhfl it can be stated that the terms from (X I ) to (X 4) are caused by

the constant fields (constant electric field, density - and temperature gradients), the alternating electric field, and by the cross drift mechanism produced by these constant fields and the constant magnetic field. On the other hand the terms from (X 5) and (X 9 ) exclusively give rise to cross drift processes in which the rotating magnetic field plays an im­portant role ( (X 5) — (X 9) disappear if the circu­larly polarized magnetic field is not present).

In Table 5 all terms of f1 are listed with respect to their time behaviour.

6. Conclusion

In the last section it was shown that higher cur­rent harmonics are produced by a cross drift me­chanism if a weakly ionized plasma is under the action of an alternating electric field and a circu­larly polarized magnetic field. It is worth noting that the higher harmonics disappear when one re­places the time-dependent rotating magnetic field by a constant magnetic field by putting the rotation fre­quency coB = 0 in the solution mhf1 (r, v, t ) . Further, it should be mentioned that higher harmonic terms with a frequency of 2 coB are present in inhf1 if only a constant electric field and a rotating magnetic field are taken into account. All these facts show that the time-dependent circularly polarized magnetic field (in connection with its induced electric field) plays the decisive role for the generation of “higher mag­netic cross harmonics”.

Finally we mention the important condition of time-independence of the isotropic part /° used in calculating the solution vector P(r, v, t; /°). In 1948 M a r g e n a u and H a r t m a n n 18 found that higher

f 1 Time-Independent

Terms CO COß

Time-Dependent Terms

2 COB CO + COB CO — COB CO + 2 COB CO — 2 COB

11 <X1><X2><X7>

<X3><X4><X8><X9>

<X5> <X7> <X 6> <X 6> <X8> <X9>

n <Y 1><Y2><Y7>

<Y3><Y4><Y8><Y9>

<Y 5> <Y7> < Y 6 > <Y 6> <Y8> <Y9>

n <Z1> <Z2> <Z3> <Z 4> <Z 5>

Table 5.

18 H . M a r g e n a u and L. M . H a r t m a n . Phys. Rev. 73, 309 [1948].

GENERALIZED ELECTRON CYCLOTRON RESONANCE 5 5 5

current harmonics appear in plasmas subjected to an a.c. field. These authors made the following an- satz for the isotropic part /°:

/ ( v, t) = /o (v) + (t;) exp (2 i co t)+ vx\ fI (v) exp(i co t) + fl O) exp(3 i co<)].

In recent time similar series expansions hav been applied in order to calculate the amplitudes of higher harmonics 19-22 even if a constant magnetic field is taken into account 23, 24. The essential result of all these papers following the treatment of Margenau and Hartmann can be derived as follows:

Suppose the isotropic part /° to be time-depen­dent. By expanding f° into a Fourier series it is pos­sible to obtain an infinite number of current har­monics in general. If in addition a constant mag-

19 P. R o s e n , Phys. Fluids 4, 341 [1961].20 J. K r e n z , Phys. Fluids 8, 1871 [1965].21 T in g -W ei T a n g , Phys. Fluids 9 , 415 [1966].

netic field is taken into account this will modify the amplitudes of the higher harmonics without chang­ing the time-behaviour, however, which is exclusi­vely determined by the alternating electric field it­self.

On the contrary the field configuration investigat­ed in this paper (a.c. and d.c. field, rotating mag­netic field, induced electric field) gives rise to a finite number of “higher magnetic cross harmonics” in the plasma under the condition that /° is time- independent. In another paper17 it is planned to give an explicit calculation of the isotropic part /° for the field configuration here discussed making use of the additional assumption that certain rela­tions of homogeneity 14,15 are valid.

22 T. M o r r o n e , Phys. Fluids 10, 1507 [1967],23 K. C h i y o d a , J. Phys. Soc. Japan 20, 290 [1965].24 M. S . S o d h a , J. Phys. S o c . Japan 21, 2674 [1966].

Generalized Electron Cyclotron Resonance in Weakly Ionized Time-Varying Magnetoplasmas

W o l f g a n g St il l e r * and Gü n t e r V o jt a

Arbeitsstelle für Statistische Physik der Deutschen Akademie der Wissenschaften zu Berlin,705 Leipzig, DDR

(Z. Naturforsch. 24 a, 555— 559 [1969] ; received 21 Decem ber 1968)

The electron distribution function is calculated explicitly for a weakly ionized plasma under the action of an alternating electric field E = {0 , 0 , Eoz cos co t} and a circularly polarized magnetic field B R = ß c{cos cob t , sin cob J, 0} rotating perpendicular to the a .c . field. Furthermore, a con­stant magnetic field B 0 = {0, 0, ß 0} is taken into account. The isotropic part /° of the electron dis­tribution function which contains, in special cases, well-known standard distributions (distributions of D r u y v e n s t e y n , D a v y d o v , M a r g e n a u , A l l is , F a i n , G u r e v ic ) shows a resonance behaviour if the frequencies co, a>c = iq /™ ) B c , <w„ = (q / m ) B 0 , and cob satisfy the relation

c o = V c o c2 + ( co0 + cob) 2.

This can be understood as a generalized cyclotron resonance phenomenon.

A general treatment of weakly ionized time-vary- ing magnetoplasmas by means of the kinetic theory (Boltzmann equation) was given in the paper 1. The application to a special field configuration consisting of an alternating electric field and a circularly po­larized magnetic field was described in a following paper 2. Assuming the isotropic part f° of the elec­tron distribution function to be time-independent one can determine the direction-dependent part f1 in terms of the unknown part /°. This leads to higher

harmonic terms in the solution f1! /0} which can be explained by a cross drift mechanism. It is worth noting that an explanation of these terms does not require the explicit knowledge of the quantity /°.

In this paper we are interested in a precise de­termination of /° in order to calculate f1 (and also / as a whole) in its explicit form. This will enable us to investigate a special cyclotron resonance phe­nomenon appearing when the characteristic frequen­cies of the external fields satisfy a simple relation.

* Based on a part of a thesis submitted to the Mathematisch- 1 W. S t i l l e r and G. V o j t a , Physica (to appear 1969). Naturwissenschaftliche Fakultät der Karl-Marx-Universität 2 W. S t i l l e r and G. V o j t a . Z. Naturforsch. 24 a, 545 [1969]. Leipzig in partial fulfillment of the requirements for the degree of Dr. rer. nat.