J. Plasma Fusion Res. SERIES, Vol. 2 (1999) 371-3'75

Nonlinear Dispersion Relation for Beam-Exited Plasma
OGAWA MaKoto, MORI Ichiro*, KAWAKAMI RETSUO ANd TOMINAGA KiKUO
Faculty of Engineering The University of Tokushima, Tokushima 770-8506, Japan(Received: 9 December 1998 / Accepted: 16 June 1999)

Abstract
Results of experimental study for generation of envelope soliton and the corresponding nonlinear
theory and also the analysis of dispersion relation in the beam-plasma system are discussed. We succeedto obtain the nonlinear dispersion relation with a Gaussian type Green's function. We also tried to solve

the nonlinear dispersion relation numerically. As an example, a solution of the dispersion relation
showing two curves in o>k plane is represented which correspond to the beam electron dispersion
diagram. Both curves are quite similar to the solution derived from the linear dispersion relation.

Keywords:
nonlinear, soliton, beam, coherent, dispersion

1. lntroduction

concerned with this emission and subsequent absorption

Hitherto, analysis of a dispersion relation has been
performed extensively because the dispersion relationshows an important character of the propagation ofwaves in a plasma. In the analysis, we usually use thelinearlization technique so that we can not discuss,for example, an amplitude dependent phenomenon(Kerr effect). Accordingly, we need something new
method to treat a nonlinear effect. We intend to solve

as Io which means collision frequency or self-energy
term.

2. The Nonlinear Theory
To introduce the curvature effect by the electricfield into electron's trajectory, we use the renormalization technique which is initially developed by theauthors: Karpman [], Dupree [2], Weinstock [3], Kono
and lchikawa [4]. However the method of these authors
can not be applied directly to our case, since we treat a
coherent interactions, i.e., k = ftq, between beam's
modulated wave number ft and the wave number k1 ofsurrounded plasma wave (plasmon's wave number) andthis problem by starting from a renormalization
technique and by using a Gaussian type Green'sfunction.

In this paper we consider a high frequency electricfield (elementary excitation) emitted by an electronis changed from linear straight line to curved line in thepresence of the electric field, i.e., when an electronemits a quantum (elementary excitation) its direction isfurther we require the finite frequency width atresonance, Imlo, where the life time of the elementary
excitation, r, is given as u = l/(2Relo), therefore we use
a Gaussian-type Green's function, which is utilized
initially by Horton, in ion-acoustic turbulence [5] andchanged to opposite side against the direction ofquantum by momentum conservation, while for thedefine a diffusion tensor D;; instead of scalar function,then the Green's function is described as:

beam by so called 'large signal theory', when the beam
injected into plasma. The trajectory of a beam electron

absorption of a quantum, the quantum strikes theelectron to bend its direction. We introduce a quantity*Corresponding author's e-mail: mori@ ee.tokushima-u'ac.jp371@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Ogawa M. el a/., Nonlinear Dispersion Relation for Beam-Exited plasmaGo(k. u.t : k, ui r\ =,{ffo,o,r,, ro. ur,}"z."rk. ntc"(k-k,,7t,k.u-Tki@-d)ikj(u-tJti. e.xp Il_'| 4kikjDij k,D\t(/,r).

SE(4r,- ,,). *1. (5)The eqs. (l)-(5) construct a closed set offundamental equation to analyse the nonlinear(k,o)

t22l ,, _ i *. 1u+ rtl rl_kikjDij

q = rftf p+ n r*,.,;

dispersion relation or the soliton emission. To discuss
the amplitude dependence, a coherent 2nd-order quantityao,r.

E (k, co) E (4 r, - a) 61,, 1,,6,.,, = 1E (k, a) 12 i s important. If we

Fourier transformation of the above Green's function

substitute a condition of coherence, k = kt, co = @r, into
the 2no-order quantity, eqs. (2) and (5), we find rhat anbecomes:

importance of the Green's function G"(k - k1-->0,a,k.a

G (k, o, ot; k', r/,s (-b)' .,Ama mt 'i(-(()+c)'.K- olr - iIa) is crucial, while in the Green's function, eq.
(l), the terms m i 0 are proportional to k2. then theya y = -2i t4'k'7T (k' U ) . D, (k, u)l- 3/2vanish when we consider the limit t -+ 0, therefore themost significant term is the ru = 0-th one.

t3nn-l/4

Gfn'&.u.at;k'.d,ri 'y=-.r

4nDij --l,lu_ dlr^-,o {2{iaCo + Ol . 6*.*, 6,,"t , (1)I r-#l"-rtl' exp
Iki(u- d)iki@- lhi
L. =.--k,k,. Di,(k,u)-4kikj . Dij(k,D)

u =l-\r/ {Dut2c =k . o-i2o,and Dii, which appears in the exponential function in eq.
(6) at resonance a- k.a - ilo = 0, then the following

function and m = 0 term is very important andKtn@) = K'z(r) = Jn/Q*). exp (-x). We denote rhatfl0)(k',a',a), 2o(k,a,ot), E (k,a) are the initial

relation is obtained:)=distribution function and the collision frequency, theelectric field respectively. Then we can add thefollowing fundamental equations (2), (3), (4) and (5).

= (a), ?

I

Io- tll ^/t",=-'Q){ D,,

As the eq. (5) is a operator, eigenvalue must be derived
for Xo, i.e., we multiply Green's function from the rightside to the eq. (5) and after integration, we divide theGreen's function again. It can be obtained also D;; by+ ffF w,.,,'1'

C-4Qc-kr,u,k.D- atrt li,d,a;,E(k,(//!=I

We define that '1,' equals the square root of the ratio: Io

where the function K:._rz(x) is the modified BesselD,i &,1))\\/-i\-o + O l 6r.16,r. (6)substitution of eq. (6) into eq. (2) then the resultant
relation becomes:Q)c,ln6,*.oy l'2 l"- ,tl'D,,1k,a1=L' -'-. (r- y). st.

(8)-12 4kr fo,Jlou ltL

where the y (in eqs. (7), (8)) has a value of 0-l and theG"(k,u, a;k',D',of). ff1k', tl,ofS, (3)C1 is denored as C, = lll(2n)l(e"lm)2{kikjlk2l.lf we)enf

f"(k,o,a4=I ou U(k,a,a;k',ff)1H,t,a;,define a new quantity: Cf = {ll(2n)l(eolm)2Ikikjlk2l(1 - y)er, then the D4 in eq. (6) can be replaced withtl,olSlE(k,k.a\12.

(4)372

OgawaM. et al., Nonlinear Dispersion Relation for Beam-Exited Plasma

G &.1). D'. k'.

relation becomes as following:rl.i\ - -)- lfil
4n=e,{k.p.tEi.2o)= l +?

c,.lEl2

I lr- rtl'''
lp+ic ) 6*,160.a. €)trt-'l l\nu . (o,\''t / rr r" \.2 .

' exp(- L+'y'c'"where p = -iro and lElz = lE(k,p)12.

t (E )i rp*i,t \" )r(0"'19:1")-p ?

#,F" I,utzK t)\The electric field can be calculated from eq. (3), theinitial distribution function, flf,o) (k',a',p), (p' = -ial), isa a/'',{ft)given as:f'Qr', ri p) = fio)1k' 1 n u6 1tl - uoy*,.(:=\""^01-4\.
'\zftkT"J '\ 27tkT"l(lo)l.+1".^o/?11"\=0,'\2kr"\27Tkr"))where the /{0)(k'), nr, n, are the no-dimensional beam
forming factor, the beam electron density and theplasma density respectively. In actual calculation, weputflor(/r) = o(l) = 1. Thus the electric field, E(k,p),n =9r, , =2, p =-iri', c =k'u+iLo,

can calculate as:

a1 =--E{k.p'1=12

ftf,tr 1o.1,, --.
{ c'' lnlrJ|,Jia"c,'=leY @\o -y)er' 2r1m"ttt2[rtsn\inolu-ool'.
'exp J lr-".1"-'\/ ,p+rc_1-

tEr I vc; trt-+Z -J I^" I^u 4ftc,

in^'-'eXDrl'-tYl'
4r c:lEl'1,(r2)I\'" l''.^o /.dq\\zftkr"i '\2ttkT"J\*,I

(13)

JI
The equations (l l), (12) include the Io and it isimportant especially when a resonance condition, ro =ft.u, is satisfied, however an eigenvalue of the operator
eq. (5) includes also lEl2,a,k,u so that the situation isl

complicated, so, we will discuss the most probable valueof them from our experimental data.

\l/l

3. The Experimental Data
In experiment we use mirror magnetic field of 80(Gauss) in center and mirror ratio of 1.4. A stainlessvessel for plasma region is 16 cm in diameter and 42(1 1)(cm) in length. The region is fed with argon gas froml0-4 to 7 x l0-a Torr. A beam of 8 (mm) in diameter,
1.92 keV-18 (mA), is injected into plasma. Any plasma
density of l0r4-5 x l0r5 (m-3) in center is controllable.
A solitary wave appears as burst with time width
150-500 (nsec). The carrier frequency is =400 (MHz).
The solitary waves are constantly generated as anEquation (11) is a nonlinear integral equation forelectric tield, E(k,p), the first term of which shows acontribution of beam with initial speed u6 and it behave
actively by generating mainly the electric field, while

the second term represent the contribution ofmaxwellian plasma. The both term interact each otherthrough the electric field. After some calculation withintermittent burst in suspended time of every 4=8

respect to the first term, a relative permittivity,e,(k,p,lEl2,2o), is obtained. Therefore dispersiongrsec). The carrier frequency is ar = 2.505 x lOe (rad/s)

while the accelerating voltage is given by 1.92 (kV), so373

Ogawa M. el a/., Nonlinear Dispersion Relation for Beam-Exited Plasmathat the beam velocity uo= 2.59 x 107 (m/s) then fromthe resonance condition, (D = k1.t) o, we can get the valueof k11 as ftrr = 103.3 (rad/m) = 100 (rad/m), while by ourreport [6], the ltl is given as: lkl = l20n (rad/m).

{iEFigure I shows a envelope soliton obtained by thezoexperiment. The probes are used as antennae. The signal
pass through coaxial cable with 50-O impedance and thecable is directly connected to the HP-54542A 2GHzoscilloscope. The total length of probe (antenna) is 5!
faE(cm), however only =1 (cm) of the tip will besubmerged into plasma. The terminal voltage at theoscilloscope is =0.1 (V), so that the electric field is
expected as =0.1 (V/cm).

Time (ts)

4. Soliton from the Theory
To get a soliton we limit the interacting velocity

Fig.1 A solitary wave detected by 2-GHz oscilloscope isshown. Since the carrier frequency is 400 MHz,
then the phase is crowded, however. the phase isconfirmed to be continuous.

region,(*)(0(0..(*)",

then the main contribution comes from the first term ofceq. (l l), and we get E(k,t) by inverse transformation.

E(k.t1=3(lbor- r /., \-€ttrl lf Jrl'tb4n'r,'uoq)!

))E.

\r-tcllt,,t"'- l(*).

exp(- ik Dot),'"n,(n,", Im I" l"\,k ll,

at=--J-4C,lEl.t

E

(9,)

0.2

(14)

Fig.2 An example of the calculated wave packet(soliton) is shown. The following constants areused. lu - u'l = 10a (m/s), y = 0.508, 7= {m"lmil1/2(1/500).

, Q=1.3, y=5,

where the f(4,x) is an incomplete Gamma function. At
the limit t - O, E(k,t) -+ 0 owing to sinh [(ImL"l2)t)
and l(q,x -+ -) --> 0, while from the character ImIo *exp{-Relot} we get E(k,t --> -) -+ 0 then theelementary excitation has finite life time.

-becomes usually a function of carrier frequency, ft.us,

in the electron-electron collision, then the soliton does
not appear because the k.ao, oscillates so fast, however

if the response is retarded by ions corresponding to thefrequency {*JA' (ft.u0), then the soliton, having
a time width (l/roi), appears.

Figure 2 shows the calculated E(k,t) when thebeam electrons collide to ion wave. The following
constants are used: correlative length in velocity
space lu - v'l = lOa (m/s), the slowing down ratio{*J "40.4

Time (psec)

5. Dispersion Relation
As shown in the nonlinear dispersion relation eq.

(12) with eq. ( 13), the relative permittivity= (1/500), no dimensional constant y = 0.508.

e"(k,rtt,lEl2,2) is a function of a Io while the EoWe find that the collision frequency, 2" (k,a,t),374

Ogawa M. et al., Nonlinear Dispersion Relation for Beam-Exited Plasma

depends on k, a and lEl2, so we must solve themsimultaneously. If we slice a manifold e"(k,a,lEl2.2o)with the surface lEl = constant and y = constant, then thesituation is some what simpler and we can get a crosssection of zero-plane which will be expected to showe, {Arbitrnry Unit)
Ix !Ot?lD.U?.5

the dispersion relation under the condition lEl =constant. Matters are too complicated, so, as an example

0.0{xldl-2.$

we neglect maxwellian term in eq. (12) and leave only abeam term and examine to solve hereafter by numerical

10method.

-J.U

2mIffiFigure 3 represents the case lBl = 0.02 and y - 0.6.
Figure 4 shows two curves (I, il) in o>k plane. It seems
that the curve (tr) is a slow space charge wave of beamelectron and the other curve (I) is a fast space charge
wave. Both curve are quite similar to those derived fromthe linear dispersion relation.

e.r {radls)& (rad/m) -100-200

Fig.3 The relative permittivity e"lk,a,lEl2,2"l of electron
gas is shown, where the following constants areused. lEl = 0.01, lu - u'l = 104 (m/s) and y = 0.72.

References

Ixld]

lll I.M. Al'tshul and V.I. Karpman, Sov. Phys. JETP(r) (r)22,36r (1966).

12[2] T.H. Dupree, Phys. Fluids 9,1773 (1966).
[3] J. Weinstock, Phys. Fluids 12, 1045 (1969).
[4] M. Kono and Y.H. Ichikawa; Prog. Theor. Phys.

849,7s4 (1973).

[5] C.W. Horton, Longtime Prediction in Dinamicsq(John Wiley, 1983) p.3l l.

Eo[6] I. Mori, T. Morimoto, R. Kawakami and K.

3Tominaga, to be published in J.Plasma Fusion Res.
2 (1999\.

-200-100

0100200e (rad/m)

Fig.4 The cross section cutting by zero plane in themanifold of fig. 3, i.e., e,k,al = 0 is represented.
1There are two curves. where the curve (tr) will bea slow space charge wave of beam electron andthe curve (I) seems to be a fast space charge
WAVE.

375