J. Plasma Fusion Res. SERIES, Vol.4 (2001) 395-398

Electron Injection in the Inward Shift configuration of LHD
for the Control of the Electric Field in plasma
SHISHKIN Alexander*, MoToJIMA Osamur, INAGAKI Shigerur and yAMAZAKI Kozor
Institute of Plasma Physics, National Science Center "Kharkov Institute of physics andTechnolo gy ", Academicheskaya str. l, Kharkov- I 08, 6 I I 0g, Ukraine
lNational InstituteFusion

forScience, Toki 509-5292, Japan(Received: 5 December 2000 / Accepted: 3 September 2001)

Abstract
High energy electron injection can be the way to control the electric field in plasma. Its effectiveness
depends on many factors and one of them is the type of magnetic configuration, particularly the particle
orbit properties, viscosity properties and others. Here the inward shift configuration ofthe Large Helical
Device is studied in the relation with the transfer of the electrons from helically trapped particles at thepoint of the injection into the passing particles in the center of magnetic configuration under the effect ofCoulomb scattering.

Keywords:
electron injection, particle orbits, Coulomb scattering, control of electric field in plasma, LHD

1. Introduction

Configuration (R*=3.9 m) and some advantages of theOutward Shifted Configuration of LHD were found.
Now the Inward Shift Configuration is under study.

One of the possiQle ways to affect the electric field

profile in plasma is to inject the high energy electrons orions [1,2]. In order the injected particles come to thecenter of the confinement volume it is possible to use
the particles with the transit orbits. If such particles startin the core of plasma they can transfer from the passing(or blocked particles) into helically trapped and escapefrom the confinement volume. These particles are

The viscosity in the Inward Shifted Configuration issmaller and hence it is favorable to introduce thebifurcation state in electric field 8,. In the Inward
Shifted Configuration the size of loss cone is reduced

compared to the Outward Shifted Configuration.
However this feature is not unfavorable to inject theparticle through the loss cone. In this paper it is shownthat the energy of injected particles can be reduced(from 100-150 keV in Outward Shifted Configuration to40 keV in the Inward Shifted Configuration) and theusually named as the particles that belong to the helical
Ioss cone. However it is possible to use the particles
with the transit orbits for the injection in plasma. Theparticles which belong to the loss cone can be injected
as the helically trapped outside the confinement volume
and then transfer into blocked (toroidally trapped) onesin the core of plasma. If these particles stay in the coreof plasma during long time they can drive the additional
fluxes which can lead to the bifurcation of the electricfield in plasma. This effect was studied for LHD in ourearlier paper [] on the base of the Outward Shifted*time of the particle staying in the center of theconfiguration is not smaller but larger (10 times) than in
the case of Outward Shifted Configuration. If the energyof injected electrons is reduced the pitch-angle
scattering can affect the particle penetration process. Itis found that the deposition patterns of the injected

Corresponding author's e-mail: shishkin@ vlink.kharkov.ua
395@2001 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchShishkin A. et aI., Electron Injection in the Inward Shift Configuration of LHD for the Control of the Electric Field in Plasma

particles are rather close to the center of the magnetic
configuration.

2. Particle Orbits

1\Each particle is traced by solving the guiding centerl'equations

I.
I:I

9^

g. y' e *rlE r^Bl *uc(?vi!vb w xyB),=dt-'ttB'"
82
2eB3[:a!t-'; )N-!

dl{=0.
ot

ff=0,(l)':.'where I,t/ is the kinetic energy of the particles, I{ and Va
are the parallel and the perpendicular velocities of theparticles, M and e arc the mass and charge of theparticles respectively, B is the magnetic field and E isthe electric field, p is the magnetic moment of thes-60f
330

390
R (cm)

Fig. 1 Electron trajectory in the vertical plane.

particles (Ir = MVzrl2B) and r is the particle guiding
center radius-vector.

The magnetic field is modeled with the scalart3l.

It is assumed that there is the electric field in

potential (B =Y@s)

plasma and the electrical static potential is the functionof the magnetic surface. The electric potential is taken
as e@slW = 0.1. The more detail of the magnetic fieldmodel and electric field model can be found in [1,4].
The typical trajectory of the injected electron with@n=Bo(-, - *+ e,,.(fi)' sin(no - mrp) + e,,. r rino), {2)the moderate energy (W = 40 keV) is shown in Figure 1.
The starting position of the electron has the following

where Bs is the magnetic field at the circular axis of
torus, R and a1, are the major and minor radii of thesurface with the helical winding, m is the magnetic fieldcoordinates: ro = 54.58 cm, r)6 = 6.29 rad, th = 2.14 rad.
The electron starts outside the last closed magnetic
surface as the helically trapped particle, becomes theblocked particle and penetrates into the center of theconfinement volume and stays there until the re-trapping

number, r,8, e are the coordinates connected with thecircular axis of the torus. Then r is the radial variable, dand tp are the poloidal and toroidal angular variables
along the minor and major circumferences of the torus
respectively. The metric coefficients are the following:h, =1, ho= r, hE= R - rcosr). The index n = l, I - l, I =l, ..., where I is the helical winding pole number. For
this study I = 2, m = 10, R = 3.90 m, ah= 0.975 m, Bs =3 T are used. Numerical coefficients €n,. representing

occurs. The time of staying of such particle in the coreof plasma is almost 10 times larger than in the Outward
Shift Configuration. The staying time in the core region(in the state of blocked or passing particle) is evaluatedas ?retrap - ?det ap, where 7.",.u0 and f6",r"o are the times of

the re-trapping and de-trapping of the particles

the harmonics of the magnetic field arrp 82.rc = 0.'76, q.rc

respectively. The staying time of the injected electrons
with energy W = 40 keV is some less than I ms (Fig. 2).

-- 4.032, €r,ro = -0.056, €r,o = 0.05. The configuration

we study here is some different from the real Inward
Shifted Configuration of LHD (R- = 3.6 m) because of
the analytical approach for the description of themagnetic field. However some principal properties canbe found with the use of the simplified analytical model.
The most important characteristics of considered
configuration is good confinement of trapped particles.
It can be shown from the analysis of the loss cone losses

3. Effect of Goulomb Scattering
Injected particles collide with the bulk plasma
during the staying time and dispose in the some spaceinside the plasma.

3.1 Pitch-angle Scaftering Model
Collisions are taken into account in the following
396

Shishkin A. et al., Electron Injection in the Inward Shift Configuration of LHD for the Control of the Electric Field in plasma
0.3

cm; Under Coulomb scattering the moments of the test
particle change in accordance with the following rule[5,6]:

o{i=-o#vP *ffiu

0.1#=YZ Ln*2e*2 a@. v)-0.1n^ <D,(b.vl(4)where @1(-r) = iD(x) - x d@/dx, @(-x) is the error
integral. Parameter b* = ",[^*lTf : L is Coulomb-o.2

logarithm; ,*, **, Z- and n* are charge, mass,.0.3

temperature and density of the plasma particles; thefollowing dependence of the temperature 7" = l(OXl YIY(ap)l is supposed. Index (*) denotes the sort ofparticles in (2); it means summation.
Time evolution of I/11 and V1 (4) let us evaluate thecharacteristic times z1 and c1 which are roughly the timein which the test particle loses its ordered velocity in theinitial direction of motion (q) and the time in which thevelocity vector of the test particle is turned through anangle 900 (rr). These characteristic times are defined by>

the relations

0.1

0!ll -0.1

MV3

4re2l *U' n. <Dt(b. V)

(s)

-o.2
4!a-0.3

0200 400 600 800

M2 v3-8ne22 Le*2 n* <D(b. v)

(6)

1000

3.2 Deposition of the Iniected Particles
To study the effect of the scattering on theFig. 2 Time evolution of the electron velocity in thedeposition of particles here some different procedure isconsidered. The pitch velocity VxlV is changed troughthe equal time intervals. The size of 'Jumps" (A{1/V ) is
fixed. The sign of AI{,/V is chosen randomly. Adiabatic
invariant of the test particle remains constant between

Time (ps)Outward Shifted Configuration (a) and Inward
Shifted Configuration (b).

way.

the collisions. The value LV,/V is taken as 0.2 that
Test particle is scattered by the particles of the bulk

corresponds to the helical magnetic well value. The time

plasma of the density n- = n-(O)tl - YIY(ap)lk.
Parameters z.(0) and ft can be varied; the function Y
describes the cross-section of the magnetic surfaces, it isinterval between the jumps is near 1.45 x l0-5 s. It ismuch smaller compared the ra that is near 0.1 s under
the density n* - l0t3 cm-3. One example of the particle
trajectory with the consequences of the kicks is shownin Figure 3. Its trajectory is some different from theinitial trajectory (Fig.l), the particle becomes thepassing earlier than the particle without collisions.
The deposition of the passing particles can be seenon Figure 4. These are the footprints of the passing

r \'*r/R \'t - tr\n cos{no-mqv-(- \an)'-\anmG)) Tntn'^1fi)
Y(ao) is the function that describes the cross-section ofthe magnetic surface which surrounds the plasma; ao1 isthe plasma radius, in the case considered bere aor= 4J
397

Shishkin A. et aL, Electron Injection in the Inward Shift Configuration of LHD for the Control of the Electric Field in Plasma

OU'lr

9n-:UN

N'Starting-!,,.-...'

Point-60 1
320-bu340 360 380 400 420

320 340 360 380 400 420

440

440

R (cm)R (cm)

Fig. 4 Deposition of the passing particles in one verticalcross section.

Fig. 3 Typical trajectory of the particle transformingfrom the helically trapped into passing due topitch angle scattering.

In such way the idea to control of the electric field with
the injection of high energy particles can be checked.

particle trajectories in one cross-section of the torus on
the background of the same cross-section of the4.5 Calculating of the radial electric fieldmagnetic surface.

generated by injected electrons is under the performance

4. Conclusionsnow.

4.1 In the LHD configuration with the inward shiftof the magnetic axis all effects of the electric field

References

control in plasma can take place as it was shown before

t1l O.Motojima, A.A. Shishkin, Sh. Inagaki and K.Y.

for the configuration with the outward shift of theWatanabe, Nuclear Fusion 40, 833 (2000).

I2l Sh. Inagaki, O. Motojima and A.A. Shishkin, J.

magnetic axis shift.

4.2 The energy of injected electrons can be

Plasma Fusion Res. SERIES 3, 485 (2000).

reduced to the moderate values (40 keV) instead thet3l O. Motojima, K.Y. Watanabe, A. Sagara, K.

high energy (100-150 keV). Such particles have thetransit orbits and can penetrate in the core of plasma.
The total power of the injection can be smaller 2.5 - 4times in comparison with the case as it was considered

Yamazaki, A.A.Shishkin et al., Proceedngs of the
6th Workshop on WENDELSTEIN 7-X and Helias

Reactors, Schloss Ringberg, Bavaria, 20-24
November 1995, Editors F. Herrnegger and F. Rau,
Max-Planck-Institut fur Plasmaphysik, Report IPP

earlier.

4.3 The deposition of the injected particles is21331,20 January 1996, pages 227-236.

obtained when the collisions are taken into account. The
location of the deposition of injected particles is shifted
relatively the position of the particles without collisions

t4l O. Motojima and A.A. Shishkin, Plasma Phys.

but not so strong. The penetration length of injectedparticles with collisions becomes shorter than one
without collisions but not so strong.

Ionized Plasma, Reviews of Plasma Physics (edit.
by Acad. M.L. Leontovich) vol.4, Consultants
Bureau, New York, 1996,p.93-241.
t6l A.A. Shishkin, O. Motojima and E.I. Polunovskij,Tech. Phys. Lett.25,725 (1999).

Control. Fusion 41,227 (1999).

t5l D.V. Sivukhin, Coulomb Collisions in a Fully

4.4 For the study here we select electrons because
in the experimental way it is more simple to arrange theelectron guns instead of ion gun and the high energy
particles can not be effected strongly by the collisions.
398