J. Plasma Fusion Res. SERIES, Vol.5 (2002) 346-350

Intermittent lon Trajectories and Focusing inSpherical Inertial Electromagnetic Confinement
BUDAEV Viatcheslav
Department of Energy Engineering and Science, Graduate School of Engineering,
Nagoya University, Nagoya 464-8603, Japan
Permanent address: Institute for Nuclear Fusion, RRC "Kurchatov Institute",

12i182, Kurchatov Sq. 1, Moscow, Russia(Received: 11 December 2001 / Accepted: l7 May 2O02)

Abstract
Performance of spherical magnetic electrostatic confinement fusion is analyzed. Magnetic field lines

in spherical cusps provide a chaotic scattering system that improves the ion focusing due to synergetic
effect of intermittent ion trajectories. Confined ions are bounced within electrostatic well reflecting frommagnetic barriers shaped by convex field lines. The concept offers a means of enhancing ion reactivity
provided by caustic formation that are resulted from an intersection of ion orbits within confinement
volume. Parameters of the experimental device are estimated.

Keywords:
magnetic inertial electrostatic confinement fusion, neutron source, spherical ion convergent flow, ion
focus

1. Introduction

schemes. Pseudo-spherical confinement volume ofmagnetic electrostatic confinement device was

Magnetic electrostatic plasma confinement (MEPC)
is an alternative plasma confinement scheme. In thisconcept the material grid using as cathode in inertial
electrostatic confinement (IEC), is replaced by virtual
cathode generated by confining electrons in multicusp
magnetic field. Potential advantages of MEPC are thatmany different fusion reactions (i.e., D-D, D-T, D-3He,

considered by Lavrentiev in [2], experimentally studied

in Poliwell [3], Penning traps [4]. In this paper thescheme of ion focus in spherical multicusp magnetic
electrostatic confinement [5] is analyzed.

2.Spherical Multieusp Magnetic
Electrostatic Plasma Confinement
The main idea described in this paper is totransform the plasma volume of the linear Magnetic

p-rtB, and p-7l.i; may occur from high-energy collisions
between the counter-streaming ions oscillating in theelectrostatic well formed by electrons. This devices can
produce high-energy neutrons, protons, alphas, etc. It
may be attractive source (106-1012 particles/s) suitable

Electrostatic Plasma Confinement device (see ref. [1])into pseudo-spherical volume by arrangement of ring
cusps on a sphere (Fig. I, Table l).
Electrons are produced by electron gun in point
cusps, by ionization of neutral gas and by secondary
emission from cathodes of linear cusps. Electrons arefor scattering analysis, neutron activation, isotope
production, medical therapy, oil well logging. Several
decades of small, cheap, pulsed experiments havedemonstrated the basic principles of MEPC [,2].
Further progress is possible by the use of improved
Corresponding author's e-mail: budaev @ ees.nagoya-u.acjp346@2002 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchBudaev V., Intermittent Ion Trajectories and Focusing in Spherical Inertial Electromagnetic Confinement

cusp gap reflect escaping electrons back into the plasma.
The cusp plugging is equivalent to magnetic shielding of

the grid wires of an electrostatic-inertial confinement
device. Described confinement concept offers a meansof enhancing ion reactivity provided by caustic
formation within a pseudo-spherical plasma. The dense
caustic regions is considered to be formed by bounced
ions reflected from convex magnetic barriers.
Magnetic fields. To describe magnetic fields created by
the current coils (Fig. l) we note an axial symmetry ofFig. 1 Schematic of the trap. The main idea is to trans-form a linear arrangement of the ring cusps to asherical one. The figure represents a cylindericalsymmetric configuration about z axis. The concept offers a means of enhancing ion reactivity
provided by caustic formation that are resulted
from an intersection of ion orbits within confine-the system. It means that magnetic surfaces will be created to confine the electrons. Only one component ofmagnetic vector potential describes the magnetic fieldstructure. In cylindrical coordinates (r, tp, z), magneticfield I(r, e) is created by the fields of N" coils,ment volume.
Table 1 Spherical cusps: experimental device. The

plasma parameters are estimated using approximate scalings of density, temperature [1],T. = f, = 0.05 0a, n n B', estimation of confine-Nc-s-H(r,z)= LH^,
n=lment time was made in text.

as well as magnetic vector potential,NcCusp anode radius, mAr{r, z) =2 Ar,{r, Z- z,) ,Cusp magnetic induciton, Tfor coils centered at z, vectot potential of a coil centered at r = 0, z = 0, defines as [7],Volume magnetic induction, T
Cusp anode gap width, cm
anode voltage, Qo, kY

Ar(r,z)=Hr/9

Electron injection energy, keVElectron injection current, A

k2=[rt('I-r\- 5l K&) - na<t|:

Pulse duration, ms

Volume plasma density no, m-3
Central plasma density, m-3

where a - coil radius, ./-cunent in the coil, K and E elliptic integrals. Field lines are defined by equations, g= const, rA6= gsrt1.

aA, a(a.)
o, H,(r.7'1=i-!
H,(r,z)=.dz'drEnergy confinement time ?,, msec

confined in a trap by magnetic fields. Negative space

Field lines topology for spherical system of 5 ring

charge of these contained electrons forms ancusps and point cusps at the axes is shown in Fig. 2. It

electrostatic well for ion confinement, Ions are born inthe edge of the device by the ionization and fall in theelectrostatic well. Estimation of space charge needed fornegative potential well [6] is of dn- = 3Qpl2neRz, where
0o - potential in the center, R-plasma radius. For theexperimental trap (Table l) is needed only a slight
imbalance of electron charge of 6n - 106-107 1/cm3.
Effects of collisions should be taken into account onlyin the edge and small central region with dense plasma.
In main central part of the confinement volume the ion
motion is collisionless. Electrostatic plugging is used toreduce the loss rate of plasma flowing along magnetic
field lines out of the cusp gaps. Cathodes outside theillustrates the minimum-B configuration that offers agood magnetohydrodynamical stability. Radial
distribution of beta (a ratio of plasma pressure tomagnetic pressure) is defined by this minimum-B
configuration. Plasma is a diamagnetic matter andvacuum magnetic fields will be modified by theconfined hot plasma. For the effect considered in thispaper, it is important only general topology, it is aconvex nature, of the magnetic field lines. There are nostrong diamagnetic currents and this topology is notexpected to be destroyed by confined plasma. The finitebeta effect is expected to be important for the strengthof magnetic field inside the confinement volume rather347

Budaev V., Intermittent Ion Trajectories and Focusing in Spherical Inertial Electromagnetic Confinementof ion motion.
Time scales. There are several time scales of the process. (l) The ion-electron momentum-transfer collision
time. (2) The ion-ion collision time. (3) The ion bounce
time z6 : time that an ion takes to complete a closed orbit in the spherical electrostatic well. 16 - 4alu6, Lto =(Zrpslm)ttz, (p0 - the ion well depth. For typical operating
conditions of experimental device (Table 1), the timescales hierarchy estimated (using formula from [1]): zu- 10-6 s, Ti1 - O.l s, Ti" - 100_1200 s. Hence, ?;1 ( ?ie suggesting that the ions and electron physics are decoupledin the ion-ion collision time, accordingly ion-electron
collisional interactions can be neglected in the central
plasma confinement region.

0Power balance isszas. Ion thermalization: collisional
degradation of the beam-like ion distribution function isa crucial issue in the assessment of the physical feasibility of all IEC schemes, because it may preclude adequate ion concentration at the spherical center. Spherical magnetic electrostatic multicusp considered in thispaper, gives a chance to improve the system performance by dense caustic formation in the central region.
Synchrotron radiation losses in magnetic IEC have been
estimated to be negligible [6]. An important power sinkmay be from bremsstrahlung radiation losses, that canbe minimized by utilizing low-Z fuels (D-D, D-T) [4].
For the ring cusps confinement time is assumed toscale according the classical diffusion estimated by
Pastukhov [8], raiir = 2r"i aVlSaptP" Ul.The nonradiative cross-field electron energy loss time lll, Ilrtl= 7/T.on7 + llrdfir, where T.on6 is the time for heat loss byconduction -1.5 r611, t6ss is the characteristic time forthe diffusion across the magnetic field. For sphericalcusps (Table 1), raur = 0.144 r.i aRlp6po = 6.6 x l0-3
sec, it is above 50 7o higher then in a device of the same
scale of linear set of ring cusps (like Jupiter-2M).
The theoretical study in [4] illustrated efficientzFig. 2 Magnetic field lines (constant r,4r) in the sphericalcusps system. The plot represents a cylindricalsymmetric configuration about z axis. Axis aremeasured in units of the radius of the sphericallyconfined plasma. Convex magnetic line provide astochasticity system for ions bounced within.

not for the magnetic lines topology defined only by thetopology of coils. The number of ring cusps can beoptimized by detailed simulation of the system.
Radial scales. (l) The boundary (collisional plasma)layer thickness is estimated by magnetic flux conservation, / = a(BoR"/Br), a - anode gap half-width, Bomagnetic induction in cusp gaps, Ro-cusp anode radius.
(2) R" is a radius of central dense region (collisionalplasma) assuming a focusing in pure electrostatic wellof potential 0o of spherical symmetry. Initial ion energyof the order of ionization energy Erc - 5 eY, R" = (En/2eQ)ttzRo t1l. (3) The central collisionless plasma re-gion R" < r < A. (4) A radius R,, when the iongyroradius P = u]t/a becomes larger than E x B drift
pass over the gyration time -1lro, p > DExBlrrl., 1)ExB =operating regimes of spherical magnetic IEC (Penningtrap) with high ratio of fusion power to ion input powercE x BlB2. Assuming Q?) = Qo (l - 12/R2), E = 2Qo QlR2), maximal angular velocity can be estimated, u1(r) (2eQo Q2lR2)/*r)t''. Assuming a multipole magnetic
field dependence, B(r) - B,oQlR) , m = 7-lO, whereboundary magnetic field 86 - (2m"T"o)tt2l€pe6, ntg, T96,

(C > 100) identifying beam-like solution for the ion
distribution. Even the pessimistic estimation [9] offusion breakeven leaves open the development of driven
neutron sources in a pure electrostatic well predicting pvalue (fusion power over ion input power) of Q < 0.21

p"5, electron mass, electron temperature at the edge andfor a 50 kV square electrostatic well. More optimistic
estimation [0] is considered a scenario of the processof ion-ion collisions providing Q - 1.3 for the samesystem. It is needed detailed analysis of the powerelectron gyroradius in the edge, respectively. The radius
is estimated by R- - R(p.,o/R (eQolTd)ttz)r/^. For the experimental device (Table l), A = 10-20 mm, R" - 3-5mm., (rz = 7-10, T.b - lO eY, p.t - 410-4 m), R. - 0.8
R. It means that for R" ( r ( 0.8 R, we need not apply aguiding center approximation for numerical simulation

balance for the spherical MEPC to evaluate a reactivity

in dense regions.
348

B:daev V., Intermittent Ion Trajectories and Focusing in Spherical Inertial Electromagnetic Confinement

3. lon Focusing and Caustic Formation in
Spherical Magnetic Inertial Gonfinement
Magnetic field lines in spherical cusps provide achaotic scattering system that improves the ion focusing
due to synergetic effect of intermittent ion trajectories.

o.2

Confined ions are bounced within electrostatic well

4.2

reflecting from magnetic barriers shaped by convex fieldlines. The corrcept offers a means of enhancing ion
reactivity provided by caustic formation that are resulted
from an intersection of ion orbits within confinement
volume. Internrittency of the ion trajectories results in

4.200.2

0.4

x/R

-5104

05104

x/R

Fig. 3 Single ion test trajectory in the quasisphericalelectromagnetic multicusp system with a plasma
radius of R = 0.3 m, cusp magnetic field I = 1 T,
electrostatic well depth of 1 kV of the parabolicform. 2D cross-section. Intermittency and mixing

the singular areas (referred as caustics with denseplasma). The .on motion in this system refers to thesystem known lrom nonlinear dynamics as a star-shaped
billiard, it belongs to the stochastic K-systems and many
examples of caustic formation in a stochastic scatteringsystems are kn,rwn [ll,l2].

indicates a formation of the focusing regions(caustics) near the center.

Cornputer simulation. In principle, to model the systemaccurately, the general form of the Boltzmann transport
equation woultl have to be solved. However, this prob-effects only the radial position of caustics rather not theformation itself. It reflects a property of magnetic
topology being an example of stochastic scatteringsystem. A variation of the confinement characteristics,
such as magnetic field strength and the potential profile
shape, do not effect on the process of the focusing

lem is difficult to solve. Ion-electron collisional interaction can be neElected and the problem can also be modeled by considt:ring the two-dimensional Fokker-Plancktransport equatLon of the ion species alone. The ions andelectron physir;s are decoupled in the ion-ion collisiontime and belovr only single ion three-dimension trajec-

regions formation: the caustics formation was observedwith the cusp magnetic field strength of 0.5 Tesla andthe potential profile of the form QQ) = Qr(l - (rlR)'), d= 3,5,7. The fusion reactivity is expected to be increasedin the dense caustic regions leading to an enhancement

tory is tested in vacuum magnetic and electrostatic
fields. Potential profile formed by electrostatic plugging
of cusps was studied theoretically and experimentally(see ref. []). Flere is considered an idealized version of
spherical symnletry when the perturbation of the ion self
space charge on the potential well profile is negligible
and the well is parabolic one. In this case, it is assumed
an electrostatic potential in the form of @(r) = QoQ - (r/of performance.
Heuristic comment. There are several issues of caustic
formation in a real spherical cusps. One of them is driving of a plasma to a Maxwellian distribution in the edge.
In a pure IEC system it leads to the spread of the focus
in the geometric center of sphere due to angular momentum change in the edge collisional plasma. In sphericalmulticusp system the performance enhancement due toR)2), R-plasma radius. The three dimension motion
equations,bounced ion orbits topology and caustic formation is notm,dildt.=-uiQ+6xE

expected to depend on collisions in edge plasma. According to the theory of singularities and caustics [1 1],
the singularity formation depends on the curvature ofreflecting boundary. A stochasticity parameter [12] estimated for the magnetic barriers curvature, is over unity
suggesting that magnetic barriers are strong stochastic
region for the ion motion within. If the system is characterized as stochastic scattering system, mixing trajectoryis expected to produce the dense caustic-like channels in
the confinement volume. Other issue is a potential profile forming by cusps. In many experiments performedin MEPC devices, a deep negative potential well has
been sustained for many milliseconds [1]. From thedrldt=uwere numerically solved using subroutine ode45 (fromtoolbox of MI.TLAB-6 package) based on an explicit
Runge-Kutta (,1,5) formula, the Dormand-Prince pair. It
is a one-step solver in a computing for nonstiff problems.

An example of the bounced ion orbit illustrates anintermittency and mixing and indicates a formation of
the focusing regions (caustics) near the center R" < r <0.1fi (Fig. 31,. The topology of this focusing isdetermined by a set of cusps (3D symmetry of systemand multipole). The strength of current in the coils
349

Budaev V., Intermittent Ion Trajectories and Focusing in Spherical Inertial Electromagnetic Confinement

theory of caustics [11] it is known that the topology ofReferences

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4. Conclusion
Performance of spherical magnetic electrostatic
confinement fusion is analyzed. Magnetic field lines inspherical cusps provide a chaotic scattering system that
improves the ion focusing due to the synergetic effect ofintermittent ion trajectories. Confined ions are bounced
within electrostatic well reflecting from magnetic
barriers shaped by convex field lines. The concept offersa means of enhancing ion reactivity provided by caustic
formation that are resulted from an intersection of ion

Termoyaderny Sintes 1,36 (1999).
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This work was partially supported by Japan Society
for the Promotion of Science (JSPS).

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