J. Plasma Fusion Res. SERIES, Vol.4 (2001) 368-3'72

Steady State Formation of an Electron Plasma
in a Toroidal Geometry
NAKASHIMA Chihiro*, YOSHIDA Zensho, HIMURA Haruhiko,
MORIKAWA Junii. YAGI Keita and SAITOH Haruhiko
University o7 ,o,Oo, Totcyo, I I3-003J, Japan(Received: 5 December 2000 / Accepted: 27 August 2001)

Abstract
A novel system of trapping of energetic particles and/or multi-species nonneutral plasmas withdifferent charges has been developed. The system can trap energetic electrons emitted from an electron
gun, and produce a nonneutral plasma in a closed magnetic field (inside a separatrix). In contrast to
neutral plasmas, a nonneutral plasma has a deep potential well due to the charge nonneutrality, which
result in a formation of a strong self-electric field in the plasma. The self-electric field and an external
magnetic field produce a strong E x B drift flow. The peak value of negative potential formed by theelectron plasma is several hundred volts, thus the density of the electron plasmas is calculated to be -1013
m-3. Using a directional probe, both toroidal and poloidal flow measurements have been tried. This initial

data shows that the flow velocity is in the range between 106 and 107 m/s, being same order of E x Bdrift flow velocity.

Keywords:
nonneutral plasma, toroidal magnetic trap, electrostatic potential, radial self-electric field, E x B drift
flow. shear flow. toroidal eouilibrium. chaos in motion

1. lntroduction
There is an increasing interest in developing
toroidal trap of nonneutral plasmas. In contrast to thelost from the trap due to gradient and curvature drifts.
However, unlike neutral plasmas, nonneutral plasmas
can be confined in a toroidal geometry without the helpof an externally imposed rotational transform. This isdue to an E x B drift rotation of the particle generated
by the external toroidal magnetic field, the self-electricfield induced by the space charge and/or the external
applied electric field, which overcomes such drifts
responsible for particle loss. Although in the presence offirst order guiding center drifts the E x B drift surfaces
no longer coincide with the potential surfaces, but are
still closed (in the zero inertia limit), and thus theelectrons can be confined. In this case, the selfconsistently generated electric field with appropriate
charge density is necessary for the confinement.

linear traps the plasma is endless in toroidal direction,

therefore, no external electrostatic potentials arerequired for confinement. This property allows us toconfine energetic charged particles, such as antiparticles,
and/or multi-species nonneutral plasmas with different
charges simultaneously. Also, the toroidal nonneutral
plasma has been proposed as a high-B plasma sourcet 11.

Some experimental researches on confinement oftoroidal electron plasmas using pure toroidal magnetic
fields with/without the help of external electric fields
have been demonstrated [2]. In a single particle picture,
an electron in a toroidal magnetic field should simply be*C orresponding author's e -mail : nakasima@

plasma.q.t.u-tolcyo.ac.jp368@2001 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchNakashima C. et al., Steady State Formation of an Electron Plasma in a Toroidal Geometry

Therefore, the external electric field which can induce
the E x B drift of the particles before the formation ofnonneutral plasma should be applied in advance, and/orFig. 2), which is produced by the combination of aninternal ring conductor and vertical field coils. We also
apply a toroidal magnetic field to produce magnetic
shear. The magnetic field is of order 10-3 T and thecorresponding gyro-radius is of order l0-2 m. The
combination of these coils provides a great flexibility toproduce various magnetic field configurations such as
Dipole-like configuration with/without magnetic shear
and closed-field configurations with magnetic null (Xpoint). We use an X-point magnetic field configuration(shown in Fig. 2) in this experiment. The electrons canbe accelerated up to 2 keV and launched from theother methods which can inject electrons toward thetrapping region, for example, 'inductive charging
process' are needed to trap the electrons [2].

In our new method, in addition to a toroidal
magnetic field (8,), we can apply a poloidal magnetic
field (Bo), which can also produce magnetic shear. The

combination of Bo and 8, makes the orbits of electrons
closed. which enables the electrons to overcome thetoroidal drifts (gradient B drift and curvature drift). In a
toroidal magnetic trap, particles are required to begenerated outside the trapping region and propagate
across the magnetic fields to be trapped. In order tocause such a cross-field diffusion, an innovative method

electron gun installed on the mid-plane (Z = 0) 4.5 cm
inside the magnetic null point. The electrons are injected
45 degree against the mid-plane (and tangentially to theR-Z plane). The electron beam current is about 4 mA.

based on chaotic collisionless scattering near magnetic

We can produce the electron plasma steadily as long as
the electron gun is turned on in DC magnetic fields.
We have measured the dependence of the floatingnull (B = 0) has also been proposed in our system. In theneighborhood of the magnetic null point, theconservation of adiabatic invariants is broken. The
resultant increase in the degree of freedom causes chaosof the particle motion. Therefore, it would induce atransport of particles across magnetic fields. In fact,
numerical simulation shows that chaotic particles havelong orbit lengths [3,4].

potential (@r) of a trapped plasma on the injected
electron beam current (15"".) (Fig. 3), and radial floating
potential profiles @f(R, Z = 0) using high-impedance(500 MA) potential probes (Fig. a). The value of @yincreases monotonically until when the value of beam
current reaches at -1 mA. In this region, using Q rctar =Ib"u- X T.,up, (Qtotal = 8 n2 R q @i Q,orut is the total

2. Formation of a Electron Cloud in a
Toroidal Geometry
We have conducted the experiments on the ProtoRT device shown in Fig. l. The typical base pressure isabout 3 x l0-7 Torr. Particles are trapped by a static(DC) poloidal magnetic field with a separatrix (shown inB u coil

Ring conductor

|-..

/

Eo/NaElectron gun

-10

20

high impedance probeor DLP

30

40

5060

R (cm)

Fig. 1 Schematic diagram of experimental apparatus
Proto-RT; the typical magnetic surfaces (X-pointconfiguration) are shown in the chamber.

Fig.2 Numerically calculated orbit of electron; theparameters are the same as those of theexperiment.
369

Nakashima C. et aL, Steady State Formation of an Electron Plasma in a Toroidal Geometry

maximum value of @iis up to about -1000 V at R = 42cm when /b"u- = 3.7 mA. The relation between the
floating potential @y and space potential @o in thiselectron plasma is not clear, but recent experiments
indicate that the value of space potential @o is of order
10'z V [5]. Thus the inferred electron density is of order
10r3 m-3, which is approximately close to the Brillouindensity limit, while the electron beam density is -5 x100l0rr m-3.
Figure. 2 shows the numerically calculated electron

4 68

24 68

1o-4 Ib"u_ (A) 1o-3

beam orbit and magnetic surfaces using the present
experimental parameters. In Fig. 4, the electron emitted
at 4.5 cm inside the magnetic null is magnetized and has
a closed orbit of'trapped particle' inside the separatrix.
Finally, the orbit is terminated at the position of the gun
where the electron hits the source. In this case, we canestimate the trapping time of beam electrons (r"u1) from2

the numerical results. The value of ?"u1 is several
microseconds, which indicate that the trapping of theFig. 3 Floating potential V.S. Injected beam current

electron plasma is achieved because t,,up(-100 trrsec) >>2"4(-l psec).

3. Flow in a Toroidal Electron Plasma
To measure the primary direction of flow in;toroidal electron plasmas, a directional Langmuir probe(DLP) was applied in Proto-RT. Regarding the way todetermine the flow from the probe currents, many
theories and experiments of electrostatic probes havebeen performed on neutral plasmas for ion flow. Bothparallel and perpendicular velocities of the ion flow
have been estimated by a DLP or a Mach probe [6]. On
the other hand, for electron flow, no sufficiently precise

500

model has been available because of the faster thermalvelocity of electrons. However, in a nonneutral plasma,40

45

a self-electric field exists which drives strong

50

perpendicular E x B flow. And the faster perpendicularflow could be attained in the case where E is strong
enough and the strength ofB is relatively weak. In fact,R (cm)Fig. 4 Typical radial distribution of floating potential @r(R, Z = 0) with respect to the injected beam

recent experiments on Proto-RT showed distinct

cu rrent.

difference of the probe currents between the upstream
perpendicular- and the downstream perpendicular
direction [7].
The directional probe employed here consists offour cylindrical electrostatic probes. Each probe has thecharge of an electron plasma, R: is major radius of aplasma), the trapping time (z,.uo) of a beam electron isestimated as about 100 psec. Fig. 4 shows that theprofile @fbecome broad as the beam current increases.
In Fig. 4, profiles have approximately parabolic shapes
and some peaks, which imply that the plasma is consistof a bulk component with a spatially uniform density
and a beam component with a high density. The

diameter of 0.4 mm. All tips of the DLP areapproximately biased to plasma potential. In thisexperiment, the probe was inserted in the mid-plane (Z =0) of Proto-RT. And, from the obtained probe current,

flow velocities were calculated by the following
procedure.

370

Nakashima c. et al., Steady State Formation of an Electron plasma in a Toroidal Geometry

As the probe is almost at plasma potential theconductor). For the toroidal flow, the magnitude of Bo isabout -3 x l0-3 T at R = 43 cm and the strength of thecurrents collected to the probe are mainly carried by freeelectrons. Thus, as first approximation, we estimate theradial electric field E is estimated to be -3 x 103 V/mfrom the value of the V@p (@p -102 V). Thus, the E x Bflow velocity in toroidal direction is expected to be 106m/s, which is comparable to the value measured by thedirectional probe. Also, the E x B flow velocity inpoloidal direction is of order 106 m/s because B, is oforder 10-3 T.

flow velocity from the difference of the values between
the upstream- and the downstream currents. Assuming

6V - il / e n" S, where dI is the difference of DLp
currents, n" is the electron density and S is the area of
the DLP tip, then the radial distributions of vertical (v.)
and toroidal (v,) flow are obtained as shown in Fig. 5.
On the mid-plane (Z = 0) radial E x I flow should benegligible due to the vertical symmetry of a toroidal

Using the relation of Q - v,(v,)/R,or, the radial
distribution of angular velocity can roughly be obtained
as shown in Fig. 6. (R.o, is a distance between a centerof the rotation and the probe position.) As recognizedfrom the profiles, the velocity fields seem to have strong
shear in both toroidal and poloidal directions, although

plasma, then vertical flow (v.) assumed to correspond topoloidal (azimuthal) E x B flow. Both toroidal andpoloidal flow velocities are calculated to be of order106-7 m/s. It should be note that in this calculation theelectron density is assumed to be -1013 m-3 which seems
to be adequate in the Proto-RT experiments [7].

more accurate measurements and analyses are required
to conclude it confidently.

For the radial profile of @y in Fig. 4 the measured

toroidal flow direction at Z = 0 corresponds to the-V<Dr, Bo direction, which implies that rhe trapped

4. Summary
We have successfully produced a toroidal electron
plasma steadily by trapping electrons emitted from anelectron gun. As a result, The negative potential @;iselectron plasma exists outside the ring conductor (likethe electron orbit plotted in Fig. 2). On the other hand,

the measured poloidal flow direction does notcorrespond to the -V@y x B, direction exactly. This

deeply formed in the Proto-RT device. The value of @i

is approximately monotonous increase function of theinjected beam current when the beam current is small(less than - I mA). In this region, the trapping time ofbeam electron is estimated to be -100 psec. The

discrepancy seems to be caused by the large parallel
particle flux (the beam electron velocity is about 107 m/s
and the density is -5 x 1013 m-3) from the electron gun

in a strong poloidal field region (near the ringIrOaX

fi7=>lootl

B040

4244

39 40 41 42 43 44 45 46

46

R (cm)R (cm)

Fig. 6 Typical radial distribution of angular velocity O =Fig. 5 Typical radial distribution of flow in a toroidal

VooJR-, in a toroidal electron plasma (X-pointconfiguration).

electron plasma (X-point configuration).
371

Nakashima C. et al., Steady State Formation of an Electron Plasma in a Toroidal Geometry

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tll Z. Yoshida et al., in Proceedings of lTth IAEA

plasma density is -1013 m-3 which is close to theBrillouin density limit.
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drives strong perpendicular -E x B flow. Using the DLP,
toroidal and poloidal flow measurements have beentried. An initial data shows that the flow velocity is inFusion Energy Conference, IAEA-CN-69/ICP/l0(R), Yokohama, Japan October 1998.
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the range between 106 and 107 m/s, which seems to beconsistent with E x B drift flow velocity. From theroughly calculated radial distribution of angular velocity
the velocity fields seem to have strong shear in both
toroidal and poloidal directions. Although the data show
the existence of equilibrium flow with shear, furthertsl H. Himura et al., 'Electrically Nonneutralized
Plasmas for Hydrodynamic Confinement' inProceedings of 1Sth IAEA Fusion Energy
Conference, (to be published), So lento, Italy

studies are required to conclude those confidently.

October 2000.

Acknowledgments
The authors are grateful to Professor T.M. O'Neil,
UCSD nonneutral plasma group, and Professor Yuichi
Ogawa for their suggestions and discussions. This work

t61

K. Nagaoka et al., J. Phys. Soc. Japan. 70, 131(2001). H. Van Goubergen et al., Plasma Phys.
Control. Fusion 4l,Ll7 (1999). H. Amemiyaet al.,Contrib. Plasma Phys. 39, 515 (1999). R.
Hatakeyama et aI.,Pbys. Plasmas 4,2947 (1997).

was supported by Grabt-in Aid for Scientific Researchfrom the Japanese Ministry of Education, Science andCulture No. 09308011. The work of CN was supported

by Research Fellowships of the Japan Society for theH. Himura, C. Nakashima, H. Saitoh and Z.
Yoshida, 'Probing of flowing electron plasmas'

Promotion of Science (JSPS) for Young Scientists.

Phys. Plasmas (to be publishe[1.

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