J. Plasma Fusion Res. SERIES, Vol.2 (1999\ 376-380

Anomalous Relaxation and Stopping Power forElectron Beam in a Beam-Plasma System
HARA Kazumasa, 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
Anomalous relaxation and stopping power, i.e., deformation of electron beam's distribution function
as a reaction of an emission of an elementary excitation (soliton) by the beam itself, in a beam-plasma
system are analyzed theoretically. The results are compared with that of our experiments. It is shown thatthe beam electron emits the soliton by coherent interaction, that is to say that modulated or bunched wave
numbe of the beam, ft, is equal to the wave number of surrounded plasma oscillation (emitted elementary

excitation or the soliton), ftl, and it is also shown that the beam slows down and its distribution function

is deformed completely when the beam collides to the steep negative front of the ion wave by the'Bremsstrahlung' in the deceleration field.

Keywords:
distribution function, anomalous relaxation, stopping power, renormalization, soliton

1. Introduction
A relaxation of the beam electron can be describedby the Fokker-Plank equation (FPE) for the beam

the fo. We neglect the first term under the assumptionthat induced phenomenon overtakes the spontaneousone. At the following section, we discuss the quantity

distribution /r:Du.

df"=-.Lts
^dt

i=l
1?-

2

2. The Nonlinear Theoryfi 6,f"1ti#

We start from the Vlasov and Poisson system andobtain the following set of self-consistent equations []:{D,ir"t, (1)D,, (k, v) =where the A,, D,, are the friction coefficient (vector) andthe diffusion coefficient (tensor) in the velocity space
respectively. The first term in right hand side in eq. (l)is linear term since the A, is related to the spontaneousemission or so called Cherenkov emission and does not+I#E(k,@)=-? z

include the fo or the intensity of the elemantary
excitation, while the second term is a nonlinear term
which is related to the induced emission and the Du isfunctions of intensity of the elementary excitation and*Corresponding(h)'author's e-mail: mori@ ee.tokushima-u.ac.jp
3',16+tE (k,, a,1t2

dk'*t a, Ia,'I Q/t)'@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research

HaraK. et al., Anomalous Relaxation and Stopping Power for Electron Beam in a Beam-plasma Systemf"(k,v, r>=quasilinear theory [2], if we exchange our correlational[ d,'f3'&',v', (f),2"(k, v, a)= (h),4(4)I + E (k,, a,1

length in velocity space, lu - v'1, with r,.
As for the initial distriburion function , f to)&', v', itt'),
we assume a delta function 6(u'- r,6), with initial beam
velocity, us and with spatial form factor f {0)(k'), rhen wecan obtain the beam distributionf (k, v, t):$ rc"r* -k,,v,k'v - a,)' E(-k,,-r, )' #),
47t

nb

(5)

Gc,ln(l -iletfi.l
Dij lv _ v'l.l tl
Ta,t,IIII(6)

ImI" l;" 2klu-rotfl'v -v"t.exp I1I\zc,l zl'1t - y) et ' t'exp(-ift'vor)'exp(-ReI"r), (9)where the /[0) (k' , v' , a'), Zo(k, v, a\, G"(k, v, a); k' , v' ,a') arc the Fourier transformation of the initial beam
distribution, the collision frequency between electrons
and the elementary excitations or self-energy term, andwhere the n6,ImI.o, ReIo are the beam electron density,

the Green's function respectively. In the original
Green's function |1, m = 0-mode is the most important

the width of frequency broadening, the collision
frequency between beam electron and a cloud ofelementary excitation, i.e., phonon cloud under theand we use only tbe m = 0-mode which is given in eq.

presence of ions when we consider the collision with ion

proportional to k2^, then they vanish when we consider
the coherent interaction, k = kr, in eqs. (2), (5) and takethe limit ft -+ 0 in the Go(k, v, a; k', v', a'). ^fhemeaning of the Green's function appears in the eq. (4)since the eq. (4) is transformation from the differential

waves. The Io is a function of frequency, and if beamelectron collide to plasmon cloud, its angular frequency,
ct, is given as a = {Dr"= k-vo (rrlo": plasma frequency),
then the elementary excitation (soliton) is not emitted
since sum of the kinetic energy and the plasmon energy
is conserved and fast energy exchange between them ispossible. However if beam electron collide to ion wave,(6). The terms m * 0 in the Green's function areequation, eq. (l), to integral equation, so that theGreen's function is a 'Kernel' or 'transition probability'
which connects the initial function state foQ = 0) to a
final statef,(r).
To solve the above set of equations we assumed firstly
that lE(kt, a)12 = lEl2 = const. in the expressions of

the total energy of electron and phonon cloud is notconserved and the reaction is progressive therefore theelementary excitation appears. In this case, Io =Function (yk.vd, where the y= "[^"1*,= (1/500).
Equation (9) includes an intensity of electric field lE(r)1'z.

Di1(k, v), and Zo(k, r, cr), then the E(k, t), f "(k, v, t) are

which can be obtained as:solved as initial value problems through inverse
transformation. We have a relation between D;; andl';.

E(k,,)=) (tr)(*) r*(4q nrv,- y) exp (y)Dij

Ct=tthla)t _.,2 -a-!

D,,trJ(t^r"t,')(10)

where l(q, x) is an incomplete Gamma function andvariables q, x are given as S = 1.3, x = allm2o/(2k)l', a1

lv -v'1"

where the y has a value of 0 -exP(-ift ror)sinh'(n'",1"1")(8).lnIplnl't111- "tl. The eq. (8) is=lll4CJlEl2tl,v=5.
At t = 0 E(k, t) has not value, however at the limit r -->anomalous collision frequency obtained by Tsytovich in377

HaraK. et al.. Anomalous Relaxation and Stopping Power for Electron Beam in a Beam-Plasma Systemi4+ 1 .L. -\A*.tt=-i:fm€o J (2ro3 k20, E(k, r) -+ 0 owing to sinh{(ImI"/2)t} andl(q, x -+-) -+ 0, while from the character ImLo - exp{-Relol}
and f(q, .x -i 0) -+ f(q) = 0.8975 = cohst., we get E(k'f --> -) -+ 0 then the elementary excitation has finite life\etk.k'v)JEez I' dk k= mtotime.

o(b.6tk.v-(D(k))
la*,e,(k,a)|,=,r*t

From the eq. (9), we can derive a stopping power
analytically when the beam passing through the plasma,
dWl0t or dWldx = (llv).dWldt, where the l4z is energy
per unit volume.

exceed the incoherent spontaneous effect. However, if

we consider a nonlinear theory by using A(v, 0, it issufficient to replace their &function with our Green's
function with finite width. The merit of our nonlinear
theory is 'no cut off in the /r-space even if at k = 0,
while in the above linear theories, there are cut off in thecoherent region such as ft.in1.u* with proper reason tonb

$c1nl'e -De,]1(r -v)e' {2c,1 El2(r - v) e't}i+#'(r2)where sr(ft, cr;) is a real part of permittivity. We haveneglect the term with respect to A(v, t) because coherent
interaction such an induced emission or absorption will

Y=*\[ +o"^"7=t?+(#,)'fr(ry)) erf k,avoid divergence.

l, fell'.r,^,f trtll

Figure I shows the results of a soliton obtained by['\*i-"'\*]ltheory at beam energy of 1920 eV and beam density n6- l0r4 m-3 by using the y:'l*Jm, = (1/500). The[, ,(u*lell')|2k |lvertical axis shows the electric field in unit volume of /rspace, E(k, t), i.e., if we represent E(r, r) with (Volt/2)
where i. is wave length, then the E(k, t) = F.ne, D.
The collision frequency or the self energy term, Io,Irtn \lo

is a function of time. The value of the Io becomesei rl,,."lgll"lt6oo'l,o'" 1(-Rerot'u l ll

a = | | I2cJ El'1t - y1 ,'t1.

(1 1)

The stopping power in the eq. (ll) is related toImIo which is a loss function as in the theory of
dielectric materials and the imaginary unit 'i' represents-:)

that an imaginary part of permittivity of electron gas iscontributed. Since the ImIo is width of frequency,
lml"l(2k) means phase velocity and we limit theinteraction in following velocities region: v s - lntl,"l (2k)-YUJ

( r ( /6 +lml"l(2k). The vs is initial beam velocity andwe assume also the inequality ImL"/(2k) << v6. In thestopping power, dWldx = (llvddWl0t depends on thequantity [2 {lmZ (2k) + {21 (3vil l. lm>"/(2ft) } 3 l. The-100"lfirst term is independent on rs, while the second termshows a character of (1/vZ). The character is the same as0.'15

0.20

0.25

0.30

Time fus)in the linear theory by Bethe [3]. In linear theoreticalresults of low velocity ion incidence into solid [4,5], theFig. 1 An example of the calculated wave packet(soliton) is shown. The following constants areused. lv- v'l = 10a (m/s), y= 0.72, y= {milm.lln =stopping power is proportional to incident velocity, 16.
These results are related to the quantity , A,(v, t), the firstterm in the Fokker-Planck eq. (1).

500.

378

HaraK. et al., Anomalous Relaxation and Stopping Power for Electron Beam in a Beam-Plasma System

suitable for soliton excitation, the soliton starts tosolve eq. (10) simultaneously. We choose the time thatthe beam's distribution function is proportional to 6(t -increase. The starting time is at 1.72 x 10-7s.

1.72 x l}t). The beam is relaxed out within 100ns after

Figure 2 shows the beam distribution function,/(ft,v, l), in the vertical axis which means an electron

the starting.

number in the cubic, ,1,3, composed of the wave length,

Analytical results of dWlOt are shown in fig. 3.
Mechanism of the stopping power is the following: at

2. As the eq.(9) is the function ot lE(k, r)12, we must

least, 2nd-order nonlinear quantity in the eqs. (2) and (5)ois fE(/c1, @)12.G,(k - kr, v, k.v - il.o- ar1) which meansthat two electrons having (a;, /c) and (ar, k), inreract
trough exchanging the phonon under the presence ofion. The coupled electrons may behave as Bose particle
in which the force is gravitation. The Io includes theabove quantity and is accompanied with a factor (eolmo)2, moreover, E(k, cl) is proportional to (eoleo) andthe Io includes the factor lE(kr raor)|2, so that if charge

TLby the 4th order of accumulated charge even in the 2ndorder nonlinear while the soliton occurs by 3'd-order|.72 x 1O -ts-t
1.g5 x 1o

2.00 x 10-t?

fr.

accumulation occurs, then the stopping power increases

nonlinear. For the actual calculation of the Io, one mustobtain a eigen value since eq. (5) is a operator.

3. Experiment

A apparatus consists of a beam gun and a

2.597 2.598 2.599 2,600 2.601

cylindrical container in which plasma is generated by anelectron beam. The container is made of stainless steel,which is 45cm in length and 16cm in diameter, and isfilled with Ar gas of 10-4 - 10-3Torr. A mirror magnetic
field with 8OGauss at a central part and mirror ratio of
1.4 is used. The electron beam with 1920V and lSmA
injected into plasma in the density of 108 - 10ecm-3.
Figure 4 shows the solitons and the signal of I
MHz component of the ion wave detected with 300kHz
band width observed in our experiments. The solitons
are indicated by arrows. which are detected by a mixer
with passive device only, while the ion wave is obtainedby the frequency analyzer, HP-8554L-8552B system.
The signal of the ion wave delays from that of thesoliton, the reason is the following: The soliton has large
intensity, then the passive device can be used so that
there is no delay, however the signal of ion wave passthrough the frequency analyzer by amplification andsome delay is occurred but the main delay comes from'the uncertainty' and also a sampling rule that the band
width Aal = 300kHz corresponds to time uncertainty(time delay) A/ with the relation A,aLt > O.5 (cycle) sothat delay time is to be 1.7ps.
Figure 5 shows a oscilloscope trace of the solitons
and also the ion wave in rather high plasma density. Thesoliton is not hyperbolic-secant function, in contrast to
the lower density case, but has many fine splits,Vo ( x 1o'm/s)
Fig.2 Relaxation of a beam is shown. At t = 1.72 x10-?(s), where the soliton starts to increase as infig. 1, the beam function 6(t - 1.72 x 10r) also
stafts up.

40j(brb0.15

0.20

0.25

0.30

Time (ys)Fig.3 The electronic stopping power dWldt for beamelectron is shown as a function of time.
379

HaraK. et al., Anomalous Relaxation and Stopping Power for Electron Beam in a Beam-Plasma Systemoshowing that the heavy load, such as the nonlinear
Landau damping and the ion wave turbulence, makedeform the shape. The ion wave appears as a triangleowith steepening negative front. The appearance of)oE-c)soliton coincide actually with the ion wave. This meansthat the beam electron emits the soliton when the beamcollide to the steep negative front of the ion wave by the35
gco.

'6NIc

Bremsstrahlung in the deceleration field.

c).cE,lta=,o:=oReferencesc[]oa.ETominaga, to be published in J. Plasma Fusion Res.
2 (1999).
t2l V.N. Tsytovitch, Nonlinear Effects in Plasmasq)

oc=050(Trans. by M. Hamberger, New York-London,o100

Plenum Press, 1970).
t3l H.A. Bethe, Ann. Phys. (Leipzig) 5,325 (1930).
t4l O.B. Firsov, Zh. Eksp. Teor. Fiz. 36, 1517 (1959)[Sov. Phys. JETP 9, 1076 (1959)].

Time (ps)Fig. 4 The soliton and the signal of lMHz-component ofthe ion wave in the experiment is shown. The
soliton is indicated by arrow while the ion wave is
detected by 300kHz-band width so that the signal
is delayed.

[5] J. Lindhard, M. Scharff and H.E. Schigtt, Kgl.
Danske. Videnskab. Sel-skab. Mat-fvs. Meddele.,
33, No.14 (1963).

0.

0.01

oo.=
CLE-0.01

lrl-I--

Tll-010

L Mori, T. Morimoto, R. Kawakami and K.

15Time (ps)Fig. 5 The coupling between the soliton and the ionwave is shown. The coupling appears when theplasma density which is proportional to the gaspressure is higher in such the pressure of 4 x 10-a
Torr. The ripple of horizontal line shows the ionwave.

380